WZW action in odd dimensional gauge theories

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On the Gauge Aspects of Gravity

a rXiv:g r-qc/96213v18Feb1996On the Gauge Aspects of Gravity †Frank Gronwald and Friedrich W.Hehl Institute for Theoretical Physics,University of Cologne D-50923K¨o ln,Germany E-mail:fg@thp.uni-koeln.de,hehl@thp.uni-koeln.de ABSTRACT We give a short outline,in Sec.2,of the historical development of the gauge idea as applied to internal (U (1),SU (2),...)and external (R 4,SO (1,3),...)symmetries and stress the fundamental importance of the corresponding con-served currents.In Sec.3,experimental results with neutron interferometers in the gravitational ﬁeld of the earth,as interpreted by means of the equivalence principle,can be predicted by means of the Dirac equation in an accelerated and rotating reference ing the Dirac equation in such a non-inertial frame,we describe how in a gauge-theoretical approach (see Table 1)the Einstein-Cartan theory,residing in a Riemann-Cartan spacetime encompassing torsion and curvature,arises as the simplest gravitational theory.This is set in con-trast to the Einsteinian approach yielding general relativity in a Riemannian spacetime.In Secs.4and 5we consider the conserved energy-momentum cur-rent of matter and gauge the associated translation subgroup.The Einsteinian teleparallelism theory which emerges is shown to be equivalent,for spinless mat-ter and for electromagnetism,to general relativity.Having successfully gauged the translations,it is straightforward to gauge the four-dimensional aﬃne group R 4⊃×GL (4,R )or its Poincar´e subgroup R 4⊃×SO (1,3).We brieﬂy report on these results in Sec.6(metric-aﬃne geometry)and in Sec.7(metric-aﬃne ﬁeld equations (111,112,113)).Finally,in Sec.8,we collect some models,cur-rently under discussion,which bring life into the metric-aﬃne gauge framework developed.Contents1.Introduction2.Remarks on the history of the gauge idea2.1.General relativity and Weyl’s U(1)-gauge theory2.2.Yang-Mills and the structure of a gauge theory2.3.Gravity and the Utiyama-Sciama-Kibble approach2.4.E.Cartan’s analysis of general relativity and its consequences3.Einstein’s and the gauge approach to gravity3.1.Neutron matter waves in the gravitationalﬁeld3.2.Accelerated and rotating reference frame3.3.Dirac matter waves in a non-inertial frame of reference3.4.‘Deriving’a theory of gravity:Einstein’s method as opposed to thegauge procedure4.Conserved momentum current,the heuristics of the translation gauge4.1.Motivation4.2.Active and passive translations4.3.Heuristic scheme of translational gauging5.Theory of the translation gauge:From Einsteinian teleparallelism to GR5.1.Translation gauge potentialgrangian5.3.Transition to GR6.Gauging of the aﬃne group R4⊃×GL(4,R)7.Field equations of metric-aﬃne gauge theory(MAG)8.Model building:Einstein-Cartan theory and beyond8.1.Einstein-Cartan theory EC8.2.Poincar´e gauge theory PG,the quadratic version8.3.Coupling to a scalarﬁeld8.4.Metric-aﬃne gauge theory MAG9.Acknowledgments10.ReferencesFrom a letter of A.Einstein to F.Klein of1917March4(translation)70:“...Newton’s theory...represents the gravitationalﬁeld in a seeminglycomplete way by means of the potentialΦ.This description proves to bewanting;the functions gµνtake its place.But I do not doubt that the daywill come when that description,too,will have to yield to another one,for reasons which at present we do not yet surmise.I believe that thisprocess of deepening the theory has no limits...”1.Introduction•What can we learn if we look at gravity and,more speciﬁcally,at general relativity theory(GR)from the point of view of classical gaugeﬁeld theory?This is the question underlying our present considerations.The answer•leads to a better understanding of the interrelationship between the metric and aﬃne properties of spacetime and of the group structure related to gravity.Furthermore,it •suggests certain classicalﬁeld-theoretical generalizations of Einstein’s theory,such as Einstein–Cartan theory,Einsteinian teleparallelism theory,Poincar´e gauge theory, Metric-Aﬃne Gravity,that is,it leads to a deepening of the insight won by GR.We recently published a fairly technical review article on our results29.These lectures can be regarded as a down-to-earth introduction into that subject.We refrain from citing too many articles since we gave an overview a of the existing literature in ref.(29).2.Remarks on the history of the gauge idea2.1.General relativity and Weyl’s U(1)-gauge theorySoon after Einstein in1915/16had proposed his gravitational theory,namely general relativity(GR),Weyl extended it in1918in order to include–besides grav-itation–electromagnetism in a uniﬁed way.Weyl’s theoretical concept was that of recalibration or gauge invariance of length.In Weyl’s opinion,the integrability of length in GR is a remnant of an era dominated by action-at-a-distance theories which should be abandoned.In other words,if in GR we displace a meter stick from one point of spacetime to another one,it keeps its length,i.e.,it can be used as a standardof length throughout spacetime;an analogous argument is valid for a clock.In con-trast,Weyl’s uniﬁed theory of gravitation and electromagnetism of1918is set up in such a way that the uniﬁed Lagrangian is invariant under recalibration or re-gauging.For that purpose,Weyl extended the geometry of spacetime from the(pseudo-) Riemannian geometry with its Levi-Civita connectionΓ{}αβto a Weyl space with an additional(Weyl)covectorﬁeld Q=Qαϑα,whereϑαdenotes theﬁeld of coframes of the underlying four-dimensional diﬀerentiable manifold.The Weyl connection one-form reads1ΓWαβ=Γ{}αβ+ψ,D ψA)mat L (DJ=0A theorem local gauge symmetry coupling A Noether’s J <dJ=0of Lagrangian(d ψ),L mat ψgauge potentialsymmetry rigid ConservedJA(connection)current Fig.1.The structure of a gauge theory `a la Yang-Mills is depicted in this diagram,which is adapted from Mills 53.Let us quote some of his statements on gauge theories:‘The gauge principle,which might also be described as a principle of local symmetry ,is a statement about the invariance properties of physical laws.It requires that every continuous symmetry be a local symmetry ...’‘The idea at the core of gauge theory...is the local symmetry principle:Every continuous symmetry of nature is a local symmetry.’The history of gauge theory has been traced back to its beginnings by O’Raifeartaigh 69,who also gave a compact review of its formalism 68.the electromagnetic potential is an appendage to the Dirac ﬁeld and not related to length recalibration as Weyl originally thought.2.2.Yang-Mills and the structure of a gauge theoryYang and Mills,in 1954,generalized the Abelian U (1)-gauge invariance to non-Abelian SU (2)-gauge invariance,taking the (approximately)conserved isotopic spin current as their starting point,and,in 1956,Utiyama set up a formalism for the gauging of any semi-simple Lie group,including the Lorentz group SO (1,3).The latter group he considered as essential in GR.We will come back to this topic below.In any case,the gauge principle historically originated from GR as a concept for removing as many action-at-a-distance concept as possible –as long as the group under consideration is linked to a conserved current.This existence of a conserved current of some matter ﬁeld Ψis absolutely vital for the setting-up of a gauge theory.In Fig.1we sketched the structure underlying a gauge theory:A rigid symmetry ofa Lagrangian induces,via Noether’s theorem,a conserved current J ,dJ =0.It can happen,however,as it did in the electromagnetic and the SU (2)-case,that a conserved current is discovered ﬁrst and then the symmetry deduced by a kind of a reciprocal Noether theorem (which is not strictly valid).Generalizing from the gauge approach to the Dirac-Maxwell theory,we continue with the following gauge procedure:Extending the rigid symmetry to a soft symmetry amounts to turn the constant group parameters εof the symmetry transformation on the ﬁelds Ψto functions of spacetime,ε→ε(x ).This aﬀects the transformation behavior of the matter La-grangian which usually contains derivatives d Ψof the ﬁeld Ψ:The soft symmetry transformations on d Ψgenerate terms containing derivatives dε(x )of the spacetime-dependent group parameters which spoil the former rigid invariance.In order to coun-terbalance these terms,one is forced to introduce a compensating ﬁeld A =A i a τa dx i (a =Lie-algebra index,τa =generators of the symmetry group)–nowadays called gauge potential –into the theory.The one-form A turns out to have the mathematical mean-ing of a Lie-algebra valued connection .It acts on the components of the ﬁelds Ψwith respect to some reference frame,indicating that it can be properly represented as the connection of a frame bundle which is associated to the symmetry group.Thereby it is possible to replace in the matter Lagrangian the exterior derivative of the matter ﬁeld by a gauge-covariant exterior derivative,d −→A D :=d +A ,L mat (Ψ,d Ψ)−→L mat (Ψ,A D Ψ).(4)This is called minimal coupling of the matter ﬁeld to the new gauge interaction.The connection A is made to a true dynamical variable by adding a corresponding kinematic term V to the minimally coupled matter Lagrangian.This supplementary term has to be gauge invariant such that the gauge invariance of the action is kept.Gauge invariance of V is obtained by constructing it from the ﬁeld strength F =A DA ,V =V (F ).Hence the gauge Lagrangian V ,as in Maxwell’s theory,is assumed to depend only on F =dA ,not,however,on its derivatives dF,d ∗d F,...Therefore the ﬁeld equation will be of second order in the gauge potential A .In order to make it quasilinear,that is,linear in the second derivatives of A ,the gauge Lagrangian must depend on F no more than quadratically.Accordingly,with the general ansatz V =F ∧H ,where the ﬁeld momentum or “excitation”H is implicitly deﬁned by H =−∂V /∂F ,the H has to be linear in F under those circumstances.By construction,the gauge potential in the Lagrangians couples to the conserved current one started with –and the original conservation law,in case of a non-Abelian symmetry,gets modiﬁed and is only gauge covariantly conserved,dJ =0−→A DJ =0,J =∂L mat /∂A.(5)The physical reason for this modiﬁcation is that the gauge potential itself contributes a piece to the current,that is,the gauge ﬁeld (in the non-Abelian case)is charged.For instance,the Yang-Mills gauge potential B a carries isotopic spin,since the SU(2)-group is non-Abelian,whereas the electromagnetic potential,being U(1)-valued and Abelian,is electrically uncharged.2.3.Gravity and the Utiyama-Sciama-Kibble approachLet us come back to Utiyama(1956).He gauged the Lorentz group SO(1,3), inter ing some ad hoc assumptions,like the postulate of the symmetry of the connection,he was able to recover GR.This procedure is not completely satisfactory, as is also obvious from the fact that the conserved current,linked to the Lorentz group,is the angular momentum current.And this current alone cannot represent the source of gravity.Accordingly,it was soon pointed out by Sciama and Kibble (1961)that it is really the Poincar´e group R4⊃×SO(1,3),the semi-direct product of the translation and the Lorentz group,which underlies gravity.They found a slight generalization of GR,the so-called Einstein-Cartan theory(EC),which relates–in a Einsteinian manner–the mass-energy of matter to the curvature and–in a novel way –the material spin to the torsion of spacetime.In contrast to the Weyl connection (1),the spacetime in EC is still metric compatible,erned by a Riemann-Cartan b (RC)geometry.Torsion is admitted according to1ΓRCαβ=Γ{}αβ−b The terminology is not quite uniform.Borzeskowski and Treder9,in their critical evaluation of diﬀerent gravitational variational principles,call such a geometry a Weyl-Cartan gemetry.secondary importance in some sense that some particularΓﬁeld can be deduced from a Riemannian metric...”In this vein,we introduce a linear connectionΓαβ=Γiαβdx i,(7) with values in the Lie-algebra of the linear group GL(4,R).These64components Γiαβ(x)of the‘displacement’ﬁeld enable us,as pointed out in the quotation by Einstein,to get rid of the rigid spacetime structure of special relativity(SR).In order to be able to recover SR in some limit,the primary structure of a con-nection of spacetime has to be enriched by the secondary structure of a metricg=gαβϑα⊗ϑβ,(8) with its10componentﬁelds gαβ(x).At least at the present stage of our knowledge, this additional postulate of the existence of a metric seems to lead to the only prac-ticable way to set up a theory of gravity.In some future time one may be able to ‘deduce’the metric from the connection and some extremal property of the action function–and some people have tried to develop such type of models,but without success so far.2.4.E.Cartan’s analysis of general relativity and its consequencesBesides the gauge theoretical line of development which,with respect to gravity, culminated in the Sciame-Kibble approach,there was a second line dominated by E.Cartan’s(1923)geometrical analysis of GR.The concept of a linear connection as an independent and primary structure of spacetime,see(7),developed gradually around1920from the work of Hessenberg,Levi-Civita,Weyl,Schouten,Eddington, and others.In its full generality it can be found in Cartan’s work.In particular, he introduced the notion of a so-called torsion–in holonomic coordinates this is the antisymmetric and therefore tensorial part of the components of the connection–and discussed Weyl’s uniﬁedﬁeld theory from a geometrical point of view.For this purpose,let us tentatively callgαβ,ϑα,Γαβ (9)the potentials in a gauge approach to gravity andQαβ,Tα,Rαβ (10)the correspondingﬁeld ter,in Sec.6,inter alia,we will see why this choice of language is appropriate.Here we deﬁnednonmetricity Qαβ:=−ΓD gαβ,(11) torsion Tα:=ΓDϑα=dϑα+Γβα∧ϑβ,(12)curvature Rαβ:=′′ΓDΓαβ′′=dΓαβ−Γαγ∧Γγβ.(13)Then symbolically we haveQαβ,Tα,Rαβ ∼ΓD gαβ,ϑα,Γαβ .(14)By means of theﬁeld strengths it is straightforward of how to classify the space-time manifolds of the diﬀerent theories discussed so far:GR(1915):Qαβ=0,Tα=0,Rαβ=0.(15)Weyl(1918):Qγγ=0,Tα=0,Rαβ=0.(16)EC(1923/61):Qαβ=0,Tα=0,Rαβ=0.(17) Note that Weyl’s theory of1918requires only a nonvanishing trace of the nonmetric-ity,the Weyl covector Q:=Qγγ/4.For later use we amend this table with the Einsteinian teleparallelism(GR||),which was discussed between Einstein and Car-tan in considerable detail(see Debever12)and with metric-aﬃne gravity29(MAG), which presupposes the existence of a connection and a(symmetric)metric that are completely independent from each other(as long as theﬁeld equations are not solved): GR||(1928):Qαβ=0,Tα=0,Rαβ=0.(18)MAG(1976):Qαβ=0,Tα=0,Rαβ=0.(19) Both theories,GR||and MAG,were originally devised as uniﬁedﬁeld theories with no sources on the right hand sides of theirﬁeld equations.Today,however,we understand them10,29as gauge type theories with well-deﬁned sources.Cartan gave a beautiful geometrical interpretation of the notions of torsion and curvature.Consider a vector at some point of a manifold,that is equipped with a connection,and displace it around an inﬁnitesimal(closed)loop by means of the connection such that the(ﬂat)tangent space,where the vector‘lives’in,rolls without gliding around the loop.At the end of the journey29the loop,mapped into the tangent space,has a small closure failure,i.e.a translational misﬁt.Moreover,in the case of vanishing nonmetricity Qαβ=0,the vector underwent a small rotation or–if no metric exists–a small linear transformation.The torsion of the underlying manifold is a measure for the emerging translation and the curvature for the rotation(or linear transformation):translation−→torsion Tα(20) rotation(lin.transf.)−→curvature Rαβ.(21) Hence,if your friend tells you that he discovered that torsion is closely related to electromagnetism or to some other nongravitationalﬁeld–and there are many such ‘friends’around,as we can tell you as referees–then you say:‘No,torsion is related to translations,as had been already found by Cartan in1923.’And translations–weFig.2.The neutron interferometer of the COW-experiment11,18:A neutron beam is split into two beams which travel in diﬀerent gravitational potentials.Eventually the two beams are reunited and their relative phase shift is measured.hope that we don’t tell you a secret–are,via Noether’s theorem,related to energy-momentum c,i.e.to the source of gravity,and to nothing else.We will come back to this discussion in Sec.4.For the rest of these lectures,unless stated otherwise,we will choose the frame eα,and hence also the coframeϑβ,to be orthonormal,that is,g(eα,eβ)∗=oαβ:=diag(−+++).(22) Then,in a Riemann-Cartan space,we have the convenient antisymmetriesΓRCαβ∗=−ΓRCβαand R RCαβ∗=−R RCβα.(23) 3.Einstein’s and the gauge approach to gravity3.1.Neutron matter waves in the gravitationalﬁeldTwenty years ago a new epoch began in gravity:C olella-O verhauser-W erner measured by interferometric methods a phase shift of the wave function of a neutron caused by the gravitationalﬁeld of the earth,see Fig.2.The eﬀect could be predicted by studying the Schr¨o dinger equation of the neutron wave function in an external Newtonian potential–and this had been veriﬁed by experiment.In this sense noth-ing really earth-shaking happened.However,for theﬁrst time a gravitational eﬀect had been measured the numerical value of which depends on the Planck constant¯h. Quantum mechanics was indispensable in deriving this phase shiftm2gθgrav=gpath 1path 2zx~ 2 cm~ 6 cmA Fig.3.COW experiment schematically.the neutron beam itself is bent into a parabolic path with 4×10−7cm loss in altitude.This yields,however,no signiﬁcant inﬂuence on the phase.In the COW experiment,the single-crystal interferometer is at rest with respect to the laboratory,whereas the neutrons are subject to the gravitational potential.In order to compare this with the eﬀect of acceleration relative to the laboratory frame,B onse and W roblewski 8let the interferometer oscillate horizontally by driving it via a pair of standard loudspeaker magnets.Thus these experiments of BW and COW test the eﬀect of local acceleration and local gravity on matter waves and prove its equivalence up to an accuracy of about 4%.3.2.Accelerated and rotating reference frameIn order to be able to describe the interferometer in an accelerated frame,we ﬁrst have to construct a non-inertial frame of reference.If we consider only mass points ,then a non-inertial frame in the Minkowski space of SR is represented by a curvilinear coordinate system,as recognized by Einstein 13.Einstein even uses the names ‘curvilinear co-ordinate system’and ‘non-inertial system’interchangeably.According to the standard gauge model of electro-weak and strong interactions,a neutron is not a fundamental particle,but consists of one up and two down quarks which are kept together via the virtual exchange of gluons,the vector bosons of quantum chromodynamics,in a permanent ‘conﬁnement phase’.For studying the properties of the neutron in a non-inertial frame and in low-energy gravity,we may disregard its extension of about 0.7fm ,its form factors,etc.In fact,for our purpose,it is suﬃcient to treat it as a Dirac particle which carries spin 1/2but is structureless otherwise .Table 1.Einstein’s approach to GR as compared to the gauge approach:Used are a mass point m or a Dirac matter ﬁeld Ψ(referred to a local frame),respectively.IF means inertial frame,NIF non-inertial frame.The table refers to special relativity up to the second boldface horizontal line.Below,gravity will be switched on.Note that for the Dirac spinor already the force-free motion in an inertial frame does depend on the mass parameter m .gauge approach (→COW)elementary object in SRDirac spinor Ψ(x )Cartesian coord.system x ids 2∗=o ij dx i dx jforce-freemotion in IF (iγi ∂i −m )Ψ∗=0arbitrary curvilinear coord.system x i′force-free motion in NIF iγαe i α(∂i +Γi )−m Ψ=0Γi :=1non-inertial objects ϑα,Γαβ=−Γβα16+24˜R(∂{},{})=020global IF e i α,Γi αβ ∗=(δαi ,0)switch on gravity T =0,R =0Riemann −Cartang ij |P ∗=o ij , i jk |P ∗=0ﬁeld equations 2tr (˜Ric )∼mass GR2tr (Ric )∼massT or +2tr (T or )∼spinECA Dirac particle has to be described by means of a four-component Dirac spinor. And this spinor is a half-integer representation of the(covering group SL(2,C)of the)Lorentz group SO(1,3).Therefore at any one point of spacetime we need an orthonormal reference frame in order to be able to describe the spinor.Thus,as soon as matterﬁelds are to be represented in spacetime,the notion of a reference system has to be generalized from Einstein’s curvilinear coordinate frame∂i to an arbitrary, in general anholonomic,orthonormal frame eα,with eα·eβ=oαβ.It is possible,of course,to introduce in the Riemannian spacetime of GR arbi-trary orthonormal frames,too.However,in the heuristic process of setting up the fundamental structure of GR,Einstein and his followers(for a recent example,see the excellent text of d’Inverno36,Secs.9and10)restricted themselves to the discussion of mass points and holonomic(natural)frames.Matter waves and arbitrary frames are taboo in this discussion.In Table1,in the middle column,we displayed the Ein-steinian method.Conventionally,after the Riemannian spacetime has been found and the dust settled,then electrons and neutron and what not,and their corresponding wave equations,are allowed to enter the scene.But before,they are ignored.This goes so far that the well-documented experiments of COW(1975)and BL(1983)–in contrast to the folkloric Galileo experiments from the leaning tower–seemingly are not even mentioned in d’Inverno36(1992).Prugoveˇc ki79,one of the lecturers here in Erice at our school,in his discussion of the classical equivalence principle,recognizes the decisive importance of orthonormal frames(see his page52).However,in the end,within his‘quantum general relativity’framework,the good old Levi-Civita connection is singled out again(see his page 125).This is perhaps not surprising,since he considers only zero spin states in this context.We hope that you are convinced by now that we should introduce arbitrary or-thonormal frames in SR in order to represent non-inertial reference systems for mat-ter waves–and that this is important for the setting up of a gravitational gauge theory2,42.The introduction of accelerated observers and thus of non-inertial frames is somewhat standard,even if during the Erice school one of the lecturers argued that those frames are inadmissible.Take the text of Misner,Thorne,and Wheeler57.In their Sec.6,you willﬁnd an appropriate discussion.Together with Ni30and in our Honnef lectures27we tailored it for our needs.Suppose in SR a non-inertial observer locally measures,by means of the instru-ments available to him,a three-acceleration a and a three-angular velocityω.If the laboratory coordinates of the observer are denoted by x x as the correspond-ing three-radius vector,then the non-inertial frame can be written in the succinct form30,27eˆ0=1x/c2 ∂c×B∂A.(25)Here ‘naked’capital Latin letters,A,...=ˆ1,ˆ2,ˆ3,denote spatial anholonomic com-ponents.For completeness we also display the coframe,that is,the one-form basis,which one ﬁnds by inverting the frame (25):ϑˆ0= 1+a ·c 2 dx 0,ϑA =dx c ×A dx A +N 0.(26)In the (3+1)-decomposition of spacetime,N and Ni βαdx0ˆ0A =−Γc 2,Γ0BA =ǫABCωC i α,with e α=e i ,into an anholonomic one,then we ﬁnd the totallyanholonomic connection coeﬃcients as follows:Γˆ0ˆ0A =−Γˆ0A ˆ0=a A x /c 2 ,Γˆ0AB =−Γˆ0BA =ǫABC ωC x /c 2 .(28)These connection coeﬃcients (28)will enter the Dirac equation referred to a non-inertial frame.In order to assure ourselves that we didn’t make mistakes in computing the ‘non-inertial’connection (27,28)by hand,we used for checking its correctness the EXCALC package on exterior diﬀerential forms of the computer algebra system REDUCE,see Puntigam et al.80and the literature given there.3.3.Dirac matter waves in a non-inertial frame of referenceThe phase shift (24)can be derived from the Schr¨o dinger equation with a Hamilton operator for a point particle in an external Newton potential.For setting up a grav-itational theory,however,one better starts more generally in the special relativistic domain.Thus we have to begin with the Dirac equation in an external gravitational ﬁeld or,if we expect the equivalence principle to be valid,with the Dirac equation in an accelerated and rotating,that is,in a non-inertial frame of reference.Take the Minkowski spacetime of SR.Specify Cartesian coordinates.Then the ﬁeld equation for a massive fermion of spin1/2is represented by the Dirac equationi¯hγi∂iψ∗=mcψ,(29) where the Dirac matricesγi fulﬁll the relationγiγj+γjγi=2o ij.(30) For the conventions and the representation of theγ’s,we essentially follow Bjorken-Drell7.Now we straightforwardly transform this equation from an inertial to an accel-erated and rotating frame.By analogy with the equation of motion in an arbitrary frame as well as from gauge theory,we can infer the result of this transformation:In the non-inertial frame,the partial derivative in the Dirac equation is simply replaced by the covariant derivativei∂i⇒Dα:=∂α+i previously;we drop the bar for convenience).The anholonomic Dirac matrices are deﬁned byγα:=e iαγi⇒γαγβ+γβγα=2oαβ.(32) The six matricesσβγare the inﬁnitesimal generators of the Lorentz group and fulﬁll the commutation relation[γα,σβγ]=2i(oαβγγ−oαγγβ).(33) For Dirac spinors,the Lorentz generators can be represented byσβγ:=(i/2)(γβγγ−γγγβ),(34) furthermore,α:=γˆ0γwithγ={γΞ}.(35) Then,the Dirac equation,formulated in the orthonormal frame of the accelerated and rotating observer,readsi¯hγαDαψ=mcψ.(36) Although there appears now a‘minimal coupling’to the connection,which is caused by the change of frame,there is no new physical concept involved in this equation. Only for the measuring devices in the non-inertial frame we have to assume hypotheses similar to the clock hypothesis.This proviso can always be met by a suitable con-struction and selection of the devices.Since we are still in SR,torsion and curvatureof spacetime both remain zero.Thus(36)is just a reformulation of the‘Cartesian’Dirac equation(29).The rewriting in terms of the covariant derivative provides us with a rather ele-gant way of explicitly calculating the Dirac equation in the non-inertial frame of an accelerated,rotating observer:Using the anholonomic connection components of(28) as well asα=−i{σˆ0Ξ},weﬁnd for the covariant derivative:Dˆ0=12c2a·α−ii∂2¯hσ=x×p+1∂t=Hψwith H=βmc2+O+E.(39)After substituting the covariant derivatives,the operators O and E,which are odd and even with respect toβ,read,respectively30:O:=cα·p+12m p2−β2m p·a·x4mc2σ·a×p+O(1Table2.Inertial eﬀects for a massive fermion of spin1/2in non-relativistic approximation.Redshift(Bonse-Wroblewski→COW)Sagnac type eﬀect(Heer-Werner et al.)Spin-rotation eﬀect(Mashhoon)Redshift eﬀect of kinetic energyNew inertial spin-orbit couplingd These considerations can be generalized to a Riemannian spacetime,see Huang34and the literature quoted there.。

Space-Time Symmetries, T-Duality and Gauge Theory

2
Gauge transformations of the spacetime
1 Aµ = − Fµν X ν . 2
A U (1) gauge ﬁeld Aµ with the constant ﬁeld strength Fµν can be written as (1)
Let det Fµν = 0, therefore X µ = −2Gµν Aν , spacetime coordinates. Under the gauge transformation ˜µ = Aµ + ∂µ Λ , Aµ → A the coordinate X µ has the following transformation ˜ µ = X µ − 2Gµν ∂ν Λ . Xµ → X 2 (4) (3) (2)
µ ν X(µ r) = A ν X .
where Jµν is the operator Jµν = Xµ ∂ν − Xν ∂µ , the square length X µ Xµ remains invariant.
˜ µ as a linear combination of {X µ }. For a special gauge function Λ, it is possible to write X (8)
Invariance of the action (16) under the gauge transformations (9) and (17), leads to the equations AT g ˜A = g , ˜=φ. φ
F M + M T F = 2F , can be interpreted he spacetime with the

Absence of the Effects of Vortices in the Gauge Glass

a r X i v :c o n d -m a t /0207310v 1 [c o n d -m a t .d i s -n n ] 12 J u l 2002Absence of the Eﬀects of Vortices in the Gauge Glass Toshiyuki Hamasaki and Hidetoshi Nishimori Department of Physics,Tokyo Institute of Technology,Oh-okayama,Meguro-ku,Tokyo 152-8551,Japan 1Introduction The gauge glass,which is an extension of the XY model to the random spin system,attracts both theoretical and practical interest.This model provides good descriptions of some physical situations,for example,the XY magnet with random Dzyaloshinskii-Moriya interactions [1]and a granular supercon-ductivity and Josephson-junction arrays with positional disorder [2][3].Espe-cially,the two-dimensional gauge glass has been studied actively because of the interesting relation between the Kosterlitz-Thouless (KT)transition and the eﬀect of disorder.This relation was ﬁrst discussed by Rubinstein,Shraimanand Nelson [1].They have shown that for small amount of randomness,as the temperature is decreased,there appears ﬁrst the paramagnetic phase,then a KT-like phase,and ﬁnally again a paramagnetic phase.Thus a reentrant transition appears in the gauge glass.After their work,Natterman,Scheidl,Korshunov and Li [4]corrected their description of the reentrant transition to the one without reentrance.These latter authors pointed out an overesti-mation of vortex pair density in the previous work.Furthermore,Sheidl [5]found a new ordered phase in which single vortex excitations occur in thereentrant phase,and Maucourt and Grempel[6]suggested from Monte Carlo simulations that there is no indication of a low-temperature reentrant phase. Today,the two-dimensional gauge glass at low temperatures is generally be-lieved to have no reentrance[7],except for some researchers[8].Hence,the present major interest in the two-dimensional gauge glass is the structure of the KT phase,namely,the boundary between the freezing phase of the vortex-pair excitations and the non-freezing phase.The gauge glass has also been investigated in theﬁeld of the spin glass theory because the gauge glass is an extension of the Ising spin glass with continuous spin variables[9].Particularly,the method of gauge transformation,which was developed in the study of the spin glass,is a powerful technique for deriving analytical results for gauge glass.Ozeki and Nishimori[10]found the exact solution of the internal energy of the gauge glass under a special condition using the method of gauge transformation.This special condition relevant to the exact solution corresponds to a line in the phase diagram called the Nishi-mori line.Ozeki and Nishimori also showed that the phase boundary between the ordered and the disordered phases runs parallel to the temperature axis. Although they could obtain the exact solution on the Nishimori line,there exists a mysterious property associated with the line:For example,the exact solution of the internal energy on the Nishimori line has no singularity as a function of temperature although the the line runs across the phase boundary. From these facts in mind,we aim to clarify what occurs under the special condition of this line.In this paper,we calculate some gauge invariant quantities on the Nishimori line.First,we deﬁne the gauge glass and show that the exact solution of the internal energy can be calculated under the special condition in Section 2.Next,we introduce the gauge invariant correlation functions and calculate them using the spin wave approximation,gauge transformation and the Villain model in Section3.The gauge invariant distribution functions are introduced and calculated in Section4.From the results of Sections3and4,we discuss the physical properties of the Nishimori line in Section5.2General properties2.1Gauge glassThe gauge glass is deﬁned by the HamiltonianH=−J i,j cos(θi−θj−A ij),(1)where A ij is the quenched random phase shift and the coupling constant J is positive.The sum runs over all nearest-neighbor pairs on a lattice.Note that we do not specify the spatial dimensionality explicitly here.We assume that A ij is independently distributed at each bond as follows,P(A ij)=12πe K p cos A ij2πH e K i,j cos(θi−θj−A ij)2πe K i,j cos(θi−θj−A ij),(4)where the ranges of integration over A ij andθi are from0to2π.Here we apply the gauge transformation to eqn(4).Since the gauge transformation is just a change of running variables,the value of eqn(4)is independent of the choice of{φi}and therefore we may integrate eqn(4)over all gauge variables {φi},each from0to2π,and divide each integration ofφi by2π.Using the special condition K=K p,we obtain the exact internal energy asFig.1.The schematic phase diagram and the Nishimori line(dashed).The spin glass phase exists in three dimensions or over.E=−JN B I1(K p)Fig.2.Schematic diagram of the correlation function C p(r)and chirality.Leftﬁgure shows a path from site0to site r.The rightﬁgure shows two states with diﬀerent chiralities.3Gauge invariant correlation functions3.1DeﬁnitionsIn order to investigate the physical signiﬁcance of the result in the previous section,we introduce the gauge invariant correlation functions asC p= exp i(θ0−θr−r k=1A k−1,k) (6) C le= cos(θ0−θ1−A01)cos(θr−θr+1−A r,r+1) (7) C ch= sin(θ0−θ1−A01)sin(θr−θr+1−A r,r+1) .(8)Equation(6)is the gauge-invariant correlation function which measures the correlation along a path from site0to site r taking the phase twist of A ij into consideration.In Figure2,we sketch a schematic picture of a path of corre-lation function C p.Equation(7)means the local energy correlation function. The local energy correlation function is also gauge invariant.Equation(8)rep-resents the chirality correlation function.Chirality introduced by Villain[12] is an Ising-like degree of freedom.The chirality distinguishes the two states which are transformed to each other using the mirror transformation(Figure 2).The chirality correlation function is also gauge invariant.3.2Spin wave approximationLet us now consider the Hamiltonian introduced in eqn(1)in two dimensions. At suﬃciently low temperatures,the eﬀect of thermalﬂuctuations is small,so that the spins tend to align to each other with the twist A ij taken into account. Hence,we are able to expand the cosine term near(θi−θj−A ij)=0, H≃J2 1K G0r−σr2exp 1K G(−)0r−σ +1K p−1 2exp 1K G(−)0r−σ −1K p−1N k1−cos k·rN k(1−cos k1)(1±cos k·r)Now,applying the special condition K=K p,we can obtain these correlation functions as follows,C p=e−K−1p r/2,C le=e K−1p,C ch=0.(16) Surprisingly,the chirality correlation function vanishes under the condition K=K p.The chirality degrees of freedom behave perfectly independently from place to place.3.3Gauge transformationFrom comparison of the results of the spin wave approximation with the exact results,we can investigate the diﬀerence between the model with periodic-ity and the model without periodicity.In this section,we calculate the exact solutions of C p,C le and C ch for models with periodicity using gauge trans-formation under the special condition K=K p.After the derivations of the exact solutions of correlation functions,we study the asymptotic forms of the exact solutions at low temperatures and compare them with the solutions in the previous section using the spin wave approximation.The correlation functions which we introduced in Section3.1are invariant un-der the gauge transformation,and therefore the same calculation method as for the internal energy(5)in Section2can be applied to the calculation of the correlation functions.As a consequence,we obtain the following expression under the special condition K=K p,C p= I1(K)I0(K) 2,C ch=0.(17)When K=K p,the chirality correlation vanishes exactly and this solution is consistent with the result of spin wave approximation(16).At low tem-perature,the asymptotic behaviors of C p and C le in eqn(17)are written as follows,C p∼e−K−1p r/2,C le∼e−K−1p.(18) Equation(18)is also consistent with the solution of spin wave approxima-tion expressed in eqn(16).This comparison suggests that under the special condition K=K p,the system forms a characteristic structure which is not inﬂuenced by the vortices.The analysis of the Villain model in the following reinforces this picture.3.4Villain modelThe Hamiltonian of the spin wave approximation (9)neglects periodicity in-cluded in the cosine term (1).If we need to describe the system at low tem-peratures more accurately,it should be necessary to consider periodicity in the Hamiltonian.Accordingly,we consider the Villain model which adds the periodicity to the spin wave HamiltonianZ V =2π 0 i d θi 2 i,j (θi −θj −A ij −2m ij π)2.(19)where m ij is an integer between −∞and ∞.The random phase shift (10)also neglects the periodicity of modulo 2π.There-fore we also add 2m ij πinto the probability distribution of A ij (10).The con-ﬁgurational average is described as[···]=2π 0 i,j d A ij2πK −1p (···)∞ {m ij =−∞}e −K p2πe i(θ0−θr − k A k −1k )e −K p2(θi −θj −A ij −2m ij π)2=∞ −∞d A ij 2πK −1p e −K pSubstituting eqn(22)into eqn(21),we obtain the gauge-invariant correlation function asC p=e−K−1p r/2,(23) where we have used the relation K=K p.The result of this calculation agrees with the one of the spin wave approximation(16)and the asymptotic form of the exact solution(18)under the condition K=K p.The local energy correlation function and the chirality correlation function are derived by the same calculation as in the gauge invariant correlation.These results are written by simple formulas asC le=e−K−1p,C ch=0.(24) Those two correlation functions also agree with the results of the spin wave approximation(16)and asymptotic exact solution(18)under the condition K=K p.We may conclude from these results as follows.The spin wave approximation neglects periodicity,and the results of the spin wave approxi-mation do not include the eﬀects of vortices.The results for the Villain model in contrast include the eﬀects of vortices.Accordingly,the agreement of the results of spin wave approximation with those of the Villain model suggests the absence of the eﬀects of vortices on the correlation functions we calculated under the condition K=K p.Vortices are perfectly irrelevant to the gauge invariant correlation functions.4Distribution functionsIrrelevance of vortices in the gauge invariant correlation functions on the Nishi-mori line can be veriﬁed by using a distribution function with gauge invari-ance.This gauge invariant distribution function measures the distribution of the random phase twist between site i and j.The deﬁnition isP(x)= δ(x−(θi−θj−A ij)) .(25)Using the spin wave approximation and the Villain model,we obtain the comparable results as in the gauge invariant correlation functions when K= K p.The results using the spin wave approximation and the Villain model are, respectively,P sw(x)= 2πexp −K pK p2x2 ,(27)the latter being the exact solution.These two expressions are completely iden-tical.The distribution function of two bonds can also be calculated by the spin wave approximation and the Villain model.Weﬁrst show the deﬁnition of the two-bond distribution functionP(x,y)= δ(x−(θk−θl−A kl))·δ(y−(θm−θn−A mn)) .(28)Equation(28)is calculated by using the spin wave approximation and the Villain model for K=K p as,again the latter being the exact solution,P sw(x,y)=K p2(x2+y2) (29)P Villain(x,y)=K p2(x2+y2) .(30)Similarly to eqns(26)and(27),these results agree with each other.We there-fore see immediately that vortices have no eﬀects on the distribution function (28).It also follows that two-bond variables behave independently because eqns(29)/(30)are just products of eqns(26)/(27).It is instructive to consider another two-bond extension,P′(x,y)= δ(x−(θk−θl−A kl)) · δ(y−(θm−θn−A mn)) .(31)Although eqn(31)is gauge invariant,we can not calculate the exact solution even if we apply gauge transformation to eqn(31).Nevertheless,we can obtain the result by the spin wave approximation asP′sw (x,y)=K p1−A2exp −K pof the thermal averages of single-bond distribution function at two diﬀerent locations as in eqn(31),these are correlated:P′(x,y)=P(x)P(y).Physical signiﬁcance of this counter-intuitive results needs further clariﬁcation.5Summary and discussionsIn this paper,we have investigated the properties of the gauge glass on the Nishimori line through the gauge invariant quantities.Firstly we calculated the gauge invariant correlation functions using the spin wave approximation, gauge transformation and the Villain model.From these calculations,we found that the results of the spin wave approximation are identical to the results by the method taking periodicity into account.Thus,we expect that the peri-odicity or vortices has no inﬂuence on the system(gauge invariant quantities to be precise)under the condition K=K p.The eﬀect of vortices can be ne-glected on the Nishimori line.Next,we calculated the gauge invariant distribution functions using the spin wave approximation and the Villain model.We found the same results as for the gauge invariant correlation functions.Moreover,we showed that the dis-tribution function(28)is independently distributed at each bond.An variant of eqn(28)was also calculated by using the spin wave approximation.We may conclude from these results as follows.In the two-dimensional gauge glass,the eﬀect of vortices plays an important role because the phase transi-tion of the model is described by vortex-pair unbinding.Accordingly,the spin wave approximation which neglects the eﬀect of vortices is considered in gen-eral not to be able to describe the singularity related to the phase transition. However,our analysis based on the gauge invariant quantities predicts that under the condition K=K p,the singularity related to the phase transition vanishes for gauge invariant correlation functions even if we consider the eﬀect of vortices in the system.Thus,the renormalization group arguments[7][13], which analyze the eﬀects of vortices asymptotically,should be reconsidered seriously under the present perspective.Further investigation of the property of the Nishimori line is necessary.References[1]M.Rubinstein,B.Shraiman and D.R.Nelson,Phys.Rev.B27,1800(1983).[2] E.Granato and J.M.Kosterlitz,Phys.Rev.B33,6553(1986).[3]W.Y.Shih,C.Ebner and D.Stroud,Phys.Rev.B30,134(1984).[4]T.Nattermann,S.Scheidl,S.E.Korshunov and M.S.Li,J.Phys.(France)5,565(1995).[5]S.Sheidl,Phys.Rev.B55,457(1997).[6]J.Maucourt and D.R.Grempel,Phys.Rev.B56,2572(1997).[7] D.Carpentier and P.Doussal,Nucl.Phys.B588,565(1993).[8] C.Mudry and X-G.Wen,Nucl.Phys.B549,613(1999).[9]H.Nishimori,Prog.Theor.Phys.66,1169(1981).[10]Y.Ozeki and H.Nishimori,J.Phys.A26,3399(1993).[11]H.Nishimori,“Statistical Physics of Spin Glasses and Information Processing:An Introduction”Oxford University Press,(2001).[12]J.Villain,J.Phys.C10,4793(1977).[13]L-H.Tang,Phys.Rev.B54,3350-3366(1996).。

One-loop effective potential of N=1 supersymmetric theory and decoupling effects

a r X i v :h e p -t h /9906141v 2 19 O c t 1999UPR-0849-T hep-th/9906141One-loop eﬀective potential of N=1supersymmetrictheory and decoupling eﬀectsI.L.Buchbinder ∗,M.Cvetiˇc +and A.Yu.Petrov ∗∗Department of Theoretical Physics,Tomsk State Pedagogical University634041Tomsk,Russia+Department of Physics and Astronomy,University of PennsylvaniaPhiladelphia,PA 19104–6396,USAAbstractWe study the decoupling eﬀects in N =1(global)supersymmetric theories with chiral superﬁelds at the one-loop level.Examples of gauge neutral chiral superﬁelds with minimal (renormalizable)as well as non-minimal (non-renormalizable)couplings are considered,and decoupling in gauge theories with U (1)gauge superﬁelds that cou-ple to heavy chiral matter is studied.We calculate the one-loop corrected eﬀective Lagrangians that involve light ﬁelds and heavy ﬁelds with mass of order M .Elimina-tion of heavy ﬁelds by equations of motion leads to decoupling eﬀects with terms that grow logarithmically with M .These corrections renormalize light ﬁelds and couplings in the theory (in accordance with the “decoupling theorem”).When the ﬁeld theory is an eﬀective theory of the underlying fundamental theory,like superstring theory,where the couplings are calculable,such decoupling eﬀects modify the low energy predictions for the eﬀective couplings of light ﬁelds.In particular,for the class of string vacua with an “anomalous”U (1),the vacuum restabilization triggers decoupling eﬀects,which can signiﬁcantly modify the low energy predictions for couplings of the surviving light ﬁelds.We also demonstrate that quantum corrections to the chiral potential depend-ing on massive background superﬁelds and corresponding to supergraphs with internal massless lines and external massive lines can also arise at the two-loop level.Contents1Introduction22Eﬀective action in the model of interacting light and heavy superﬁelds42.1General structure of the eﬀective action ......................52.2The eﬀective equations of motion .........................82.3Calculation of the one-loop k¨a hlerian eﬀective potential .............93One-loop eﬀective action for minimal models133.1Calculation of the eﬀective action (13)3.1.1The one-loop k¨a hlerian eﬀective potential (13)3.1.2Corrections to the chiral potential (14)3.2The eﬀective action for light superﬁelds (18)3.2.1Contribution of the self-interaction of the light superﬁeld (20)3.2.2Absence of the self-interaction of the light superﬁeld (23)4One-loop eﬀective action for non-minimal models244.1The model with heavy quantum superﬁelds and external light superﬁelds (25)4.1.1Calculation of the eﬀective action (25)4.1.2Solving the eﬀective equations of motion (27)4.2The model with light and heavy quantum superﬁelds and light and heavyexternal superﬁelds (29)4.2.1Calculation of the eﬀective action (29)4.2.2Solution of the eﬀective equations of motion (32)5Quantum corrections to the eﬀective action in gauge theories335.1Gauge invariant model of massive chiral superﬁelds (33)5.2One-loop k¨a hlerian potential in supersymmetric gauge theory (35)5.3Chiral potential corrections (40)5.4Strength depending contributions in the eﬀective action (41)6Summary45 Appendix A47 Appendix B49 Appendix C52 1IntroductionThis paper is devoted to the calculation of the one-loop eﬀective action for several models of the global N=1supersymmetric theory with chiral superﬁelds and a subsequent study of some of their phenomenologically interesting aspects.In particular,we investigate in detail the decoupling eﬀects due to the couplings of heavy and light chiral superﬁelds in the theory and subsequent implications for the low energy eﬀective action of light superﬁelds.In principle the decoupling eﬀects of heavyﬁelds inﬁeld theory are well understood. According to the decoupling theorem[1,2](for additional references see,e.g.,[3])in theﬁeld theory of interacting light(with masses m)and heavyﬁelds(with masses M)the heavyﬁelds decouple;the eﬀective Lagrangian of the lightﬁelds can be written in terms of the original classical Lagrangian of lightﬁelds with loop eﬀects of heavyﬁelds absorbed into redeﬁnitionsof new lightﬁelds,masses and couplings,and the only new terms in this eﬀective Lagrangianare non-renormalizable,proportional to inverse powers of M(both at tree-and loop-levels).In aﬁeld theory as an eﬀective description of phenomena at certain energies,the rescaling of theﬁelds and couplings due to heavyﬁelds does not aﬀect the structure of the couplings,since those are free parameters whose values are determined by experiments.On the other hand if theﬁeld theory is describing an eﬀective theory of an underlying fundamental theory,like superstring theory,where the couplings at the string scale are calculable,the decouplingeﬀects of the heavyﬁeld can be important and can signiﬁcantly aﬀect the low energy pre-dictions for the couplings of lightﬁelds at low energies.Therefore the quantitative study ofdecoupling eﬀects at the loop-level in eﬀective supersymmetric theories is important;it should improve our understanding of such eﬀects for the eﬀective Lagrangians from superstring the-ory and provide us with calculable corrections for the low energy predictions of the theory.We also note that as the decoupling theorem is based onﬁnite renormalization ofﬁelds and parameters as all parameters in the eﬀective theory(ﬁelds,masses,couplings)are determinedfrom the corresponding string theory and hence cannot be renormalized.Therefore we willuse consistence with the decoupling theorem only to check the results.Eﬀective theories of N=1supersymmetric four-dimensional perturbative string vacuacan be obtained by employing techniques of two-dimensional conformalﬁeld theory[4].In particular the k¨a hlerian and the chiral(super-)potential can be calculated explicitly at thetree level.While the chiral potential terms calculated at the string tree-level are protectedfrom higher genus corrections(for a representative work on the subject see,e.g.,[5,6],and references therein),the k¨a hlerian potential is not.Such higher genus corrections to thek¨a hlerian potential could be signiﬁcant;however,their structure has not been studied verymuch.In this paper we shall not address these issues and assume that the string theory calculation provides us with a(reliable,calculable)form of the eﬀective theory at M string,which would in turn serve as a starting point of our study.One of the compelling motivations for a detailed study of decoupling eﬀects is the phe-nomenon of vacuum restabilization[7]for a class of four-dimensional(quasi-realistic)heteroticsuperstring vacua with an“anomalous”U(1).(On the open Type I string side these eﬀects are closely related to the blowing-up procedure of Type I orientifolds and were recently stud-ied in[8].)For such string vacua of perturbative heterotic string theory,the Fayet-Iliopoulos (FI)D-term is generated at genus-one[9],thus triggering certainﬁelds to acquire vacuumexpectation values(VEV’s)of order M String∼g gauge M P lanck∼5×1017GeV along D-and F-ﬂat directions of the eﬀective N=1supersymmetric theory.(Here g gauge is the gauge coupling and M P lanck the Planck scale.)Due to these large string-scale VEV’s a numberof additionalﬁelds obtain large string-scale masses.Some of them in turn couple through(renormalizable)interactions to the remaining lightﬁelds,and thus through decoupling ef-fects aﬀect the eﬀective theory of lightﬁelds at low energies.(For the study of the eﬀective Lagrangians and their phenomenological implications for a class of such four-dimensional string vacua see,e.g.,[10]–[12]and references therein.)The tree level decoupling eﬀects within N=1supersymmetric theories,were studiedwithin an eﬀective string theory in[13].In a related work[14]it was shown that the lead-ing order corrections of order1important next order eﬀects in the eﬀective chiral potential[15].In addition,in[15]the nonrenormalizable modiﬁcations of the k¨a hlerian potential(as was also pointed out in[16]) were systematically studied.These tree level decoupling eﬀects(as triggered by,e.g.,vac-uum restabilization for a class of string vacua)lead to new nonrenormalizable interactions which are competitive with the nonrenormalizable terms that are calculated directly in the superstring theory.In this paper we consider one-loop decoupling eﬀects in N=1supersymmetric theory. We study both the eﬀects on chiral(gauge neutral)superﬁelds and on the eﬀects of gauge superﬁelds.(In another context see[17].)It turns out that an essential modiﬁcation of low energy predictions takes place not only for chiral superﬁelds[18]but also for gauge superﬁelds. As stated earlier such eﬀective Lagrangians arise naturally due to the vacuum restabilization for a class of supersymmetric string vacua and trigger couplings between heavyﬁelds with mass scale M∼1017GeV and the light(massless)ﬁelds[19].(Note however,that we do not include supergravity eﬀects which could also be signiﬁcant.)As a result weﬁnd that the one-loop eﬀective action after a redeﬁnition ofﬁelds,masses and couplings coincides with the one-loop eﬀective action of the corresponding theory where heavy superﬁelds are completely absent,in accordance with the decoupling theorem.How-ever,since the masses and the couplings of theﬁelds are calculable in string theory(at the mass scale M string),the decoupling eﬀects add additional corrections to the eﬀective action of the light superﬁelds.These corrections grow logarithmically with M(mass of the heavy superﬁelds)and modify the eﬀective couplings in an essential way,which for a class of string vacua under consideration can be signiﬁcant.Another interesting result presented in this paper pertains to the chiral eﬀective potential. When the chiral potential depends on massive superﬁelds,quantum corrections due to these ﬁelds appear earlier than in the case when one considers lightﬁelds only.This paper is organized as follows.In Section2the general structure of the eﬀective action studied is given and the general approach to addressing the decoupling eﬀects is presented. Section3is devoted to the study of the eﬀective action for the“minimal”model with one heavy and one light(gauge neutral)chiral superﬁeld.In Section4the leading order decou-pling corrections to the eﬀective action for non-minimal models(with more general couplings) are considered.Section5is devoted to the investigation of the one-loop decoupling eﬀects in N=1supersymmetric theory with U(1)gauge vector superﬁelds and chiral superﬁelds charged under U(1).A summary and discussion of the obtained results are given in Section 6.In Appendix A details of the calculation of the one-loop k¨a hlerian eﬀective potential for the minimal model are presented,in Appendix B the calculation of the one-loop k¨a hlerian eﬀective action via diagram technique for the minimal model is described,and in Appendix C details of the calculation for the eﬀective action of non-minimal models are given.2Eﬀective action in the model of interacting light and heavy superﬁelds2.1General structure of the eﬀective actionN=1supersymmetric actions with chiral supermultiplets arise as a subsector of an eﬀective theory of N=1supersymmetric string vacua.Such calculations are carried out for per-turbative string vacua primarily by employing conformalﬁeld theory techniques.(Though less powerful techniques,e.g.,sigma-model approach,in which the integration over massivestring modes is carried out in the the background of the ten-dimensional manifold with thestructure M4×K where M4is a four-dimensional Minkowski space and K is a suitable six-dimensional compact(Calabi-Yau)manifold,can also be employed.)The resulting eﬀectivetheories contain as an ingredient N=1chiral superﬁeldsΦi with actionS[Φ,¯Φ]= d8zK(¯Φi,Φi)+( d6zW(Φi)+h.c.)(2.1) HereΦi=Φi(z),z A≡(x a,θα,¯θ˙α);a=0,1,2,3;α=1,2,˙α=˙1,˙2,d8z=d4xd2θd2¯θ. Real function K(¯Φi,Φi)is called the k¨a hlerian potential,the holomorphic function W(Φi)is called the chiral potential[20].Expression(2.1)represents the most general action of gauge neutral chiral superﬁelds which does not contain higher derivatives at a component level[20]. We refer to this action as the chiral superﬁeld model of a general form.In a special case K(¯Φi,Φi)=Φ¯Φ,W(Φi)∼Φ3we obtain the well-known Wess-Zumino model.For W(Φi)=0 the present theory represents itself as a N=1supersymmetric four-dimensional sigma-model (see,e.g.,[6]).The action(2.1),which originates from superstring theory,can be treated as a classical eﬀective action of the fundamental theory,suitable for description of phenomena at energies much less than the Planck scale.Such models of chiral superﬁelds are widely used for the study of possible phenomenological implications of superstring theories(see,e.g.,recent papers[8,10,11,12,18]and references therein).One of the most important aspects of the study of these models pertains to the investigation of the decoupling eﬀects,which is the main subject of the present paper.The starting point in the study of the decoupling eﬀects is the model with the classicalaction(2.1)and,for the sake of simplicity,two chiral superﬁelds:a light one,φ,and a heavyone,Φ,i.e.Φi={Φ,φ}.The aim is to to calculate the low-energy eﬀective action in the one-loop approximation and to compute the one-loop corrected eﬀective action of light superﬁeld, only.We refer to the model in which the k¨a hlerian potential is of the canonical(minimal)form:K(Φ,¯Φ,φ,¯φ)=Φ¯Φ+φ¯φ(2.2) as the minimal model,and the model in whichK(Φ,¯Φ,φ,¯φ)=Φ¯Φ+φ¯φ+˜K(Φ,¯Φ,φ,¯φ)(2.3) with˜K=0–as the non-minimal one(in analogy with[10]).We assume that the function ˜K(Φ,¯Φ,φ,¯φ)can be expanded into power series in superﬁeldsΦ,¯Φ,φ,¯φwhere the leading order term is at least of the third order in the chiral superﬁelds(and thus proportional to at least one inverse power of M)˜K(Φ,¯Φ,φ,¯φ)=φ¯Φ2M...(2.4)The chiral potential M is taken to be of the form:W=MM+...(2.6) with M as a massive parameter.Hence the possible vertices of interaction of superﬁelds havethe formφΦ2,Φφ2,Φ¯φ2M...The eﬀective actionΓ[Φ,¯Φ,φ,¯φ]is deﬁned as the Legendre transform from the generating functional of connected Green functions[21]W[J,¯J]:exp(i¯h(S[Ψ,¯Ψ,ϕ,¯ϕ]++( d6z(JΨ+jϕ)+h.c.)))(2.7)Γ[Φ,¯Φ,φ,¯φ]=W[J,¯J]−( d6z(JΦ+jφ)+h.c.)Γ[Φ,¯Φ,φ,¯φ]can be calculated using the loop-expansion method.This method employs the splitting of all the chiral superﬁelds into a sum of the background superﬁeldsΦ,φand the quantum onesΦq,φq,using the ruleΦ→Φ+√¯hφqAs a result the action(2.1)after such changes can be written asS q= d8zK(Φ+√¯h¯Φq,φ+√¯h¯φq)++[ d6zW(Φ+√¯hφq)+h.c.](2.9) and the eﬀective action takes the form:exp(i¯hS[Φ+√¯h¯Φq,φ+√¯h¯φq]−−√δΦ(z)Φq(z)+δΓ(for details see[20,21]).The eﬀective action(2.10)can be cast in the formΓ[Φ,¯Φ,φ,¯φ]= S[Φ,¯Φ,φ,¯φ]+˜Γ[Φ,¯Φ,φ,¯φ].Here˜Γ[Φ,¯Φ,φ,¯φ]is a quantum correction in eﬀective action which can be expanded into power series in¯h as˜Γ[Φ,¯Φ,φ,¯φ]=∞ n=1¯h nΓ(n)[Φ,¯Φ,φ,¯φ](2.11) The one-loop quantum correctionΓ(1)to the eﬀective action is deﬁned through the fol-lowing expression[21]:e iΓ(1)= DΦq D¯Φq Dφq D¯φq exp(iS(2)q)(2.12) Here S(2)q corresponds to the part of S q(2.9)which is quadratic in quantum superﬁelds.It is of the formS(2)q= d8z(KΦ¯ΦΦq¯Φq+Kφ¯Φφq¯Φq+KΦ¯φΦq¯φq+Kφ¯φφq¯φq)++[ d6zWΦΦΦ2q+WφΦΦqφq+Wφφφ2q+h.c.](2.13) As a result we arrive at the one-loop eﬀective action of the formΓ[Φ,¯Φ,φ,¯φ]=S+¯hΓ(1)= d8zK(Φ,¯Φ,φ,¯φ)+[ d6zW(Φ,φ)+h.c.]++¯h( d8zK(1)(Φ,¯Φ,φ,¯φ)+( d6zW(1)(Φ,φ)+h.c.))(2.14)Here we suppose that the one-loop correction in the eﬀective actionΓ(1)can be represented in the formΓ(1)= d8zK(1)(Φ,¯Φ,φ,¯φ)+( d6zW(1)(Φ,φ)+h.c.)+...(2.15) Dots denote terms that depend on the supercovariant derivatives of the chiral superﬁelds.The loop corrected eﬀective action has the following structureΓ[Φ,¯Φ,φ,¯φ]= d8zL eff(Φ,D AΦ,D A D BΦ,¯Φ,D A¯Φ,D A D B¯Φ,φ,D Aφ,D A D Bφ,¯φ,D A¯φ,D A D B¯φ)+( d6zL(c)eff(Φ,φ)+h.c.)+...(2.16)Here D A are supercovariant derivatives,D A=(∂a,Dα,¯D˙α).L eff is the eﬀective super-Lagrangian that we write in the formL eff=K eff(Φ,¯Φ,φ,¯φ)+...K=K(Φ,¯Φ,φ,¯φ)+∞n=1¯h n K(n)(2.17)and L(c)is the eﬀective chiral LagrangianL(c)=W eff(Φ,φ)+...(2.18)K eff is the k¨a hlerian eﬀective potential that depends only on the chiral superﬁeldsΦ,¯Φ,φ,¯φbut not on their(covariant)derivatives.W eff is the chiral eﬀective potential that depends on on(holomorphic)chiral superﬁelds{Φ,φ},only.Dots denote the terms that depend on the the space-time derivatives of chiral superﬁelds only.Furthermore,one can prove that the one-loop correction to the chiral potential is zero(for the N=1supersymmetric theory which does not include gauge superﬁelds).However,higher corrections can exist(cf.[22]–[24]),i.e.W eff(Φ,φ)=W(Φ,¯φ)+∞n=2¯h n W(n)(Φ,φ)(2.19)Here K(n)and W(n)are loop corrections to the k¨a hlerian and chiral potential,respectively.Since in this paper we concentrate on the one-loop corrected eﬀective action only,we are mainly interested in the correction to the k¨a hlerian potential which is the leading term in the one-loop corrected low-energy eﬀective action.(At low energies(E≪M)higher derivative terms are suppressed.)Our ultimate goal is to obtain the eﬀective action for light superﬁelds,only.For that purpose one must eliminate heavy superﬁelds from the one-loop eﬀective actionΓ[Φ,¯Φ,φ,¯φ] (2.14)by means of the eﬀective equations of motion.These equations can be solved by an iterative method up to a certain order in the inverse mass M of heavy superﬁeld.Substituting a solution of these equations into the eﬀective action(2.14)we then obtain the one-loop corrected eﬀective action of light superﬁelds only.In the following subsection we shall describe the procedure in detail.2.2The eﬀective equations of motionThe eﬀective equations of motion for heavy superﬁelds in the model with the eﬀective action (2.14,2.15)are of the formδΓ4¯D2(∂K∂Φ)+∂Wδ¯Φ=0:−1∂¯Φ+∂K(1)∂¯Φ=0(2.20)The eﬀective equations of motion for light superﬁelds have an analogous form.We consider the case when the interactions with the gauge superﬁelds are absent(see however Section5) and W(1)=0(which is absent at one-loop level(cf.,discussion above)).The equations(2.20)can be solved via an iterative method,described below.We can represent the heavy superﬁeldΦin the formΦ=Φ0+Φ1+...+Φn+...(2.21) whereΦ0is zeroth-order approximation,Φ1isﬁrst-order one,etc..We assume that|D2Φ|≪M¯Φsince the superﬁeldΦis heavy,and thus the assumption is valid.The zeroth-order approximationΦ0can be found from the condition∂WAfter a substitution of the expansion(2.21)into equations(2.20)we arrive at the following equation for the(n+1)-th-order solution for¯Φn+1(∂¯W∂¯Φ)|Φ=Φ0+...+Φn=(2.22)=¯D2∂Φ|Φ=Φ0+...+Φn−∂K∂Φ|Φ=Φ0+...+Φn−∂K(1)2Φ2+˜W(see eqs.(2.5-2.6),eq.(2.23))can be rewritten in the formM¯Φn+1+(∂¯˜W∂¯Φ)|Φ=Φ0+...+Φn=(2.23)=¯D2∂Φ|Φ=Φ0+...+Φn−∂K∂Φ|Φ=Φ0+...+Φn−∂K(1)which represents itself as a column q =uv.The action for q reads asS 0q =−14¯D 2q −ψand ¯χ[q ]=14D 2q −¯ψ)δ(14¯D2−1214D 2q −¯ψ)δ(12T r log ∆(2.32)Here T r is a functional supertrace,andS [q ]=116(K ψ¯ψ−1){D 2,¯D2}−14¯W′′D 2(2.34)The terms proportional to the supercovariant derivatives of K ψ¯ψ,W ′′and ¯W′′are omitted since the one-loop k¨a hlerian eﬀective potential by deﬁnition does not depend on the deriva-tives of superﬁelds.In order to determine T r log ∆we use the Schwinger representationT r log ∆=trds∂sΩ=Ω˜∆(2.35)Here ˜∆is a matrix operator of the form ˜∆=−14W ′′¯D2−116A (s )D 2¯D2+18B α(s )D α¯D2+14C (s )D 2+1i˙A=F +AF 2−CW ′′(2.38)1i˙C=−¯W ′′−A ¯W ′′2+CF 2and an analogous system of equations for ˜A,˜B,˜C ,which can be obtained from this one by changing W ′′into ¯W ′′and vice versa.Here F =1−K ψ¯ψ.Since the initial condition for ΩisΩ|s =0=1the initial conditions for A,˜A,B,˜B,C,˜C )are A |s =0=˜A |s =0=C |s =0=˜C |s =0=0.The solution for B α,˜B ˙αevidently has the form B α=˜B ˙α=0.The manifest form of thematrices A,˜AC,˜C ,necessary for exact calculations,is of the form A =A 11A 12A 21A 22;C =C 11C 12C 21C 22(2.39)Here index 1denotes the sector of heavy superﬁeld Φand 2the sector of light superﬁeld φ.Now let us solve the system for matrices A,C .The solution for ˜A,˜C can be easily obtained in an analogous way since the system with B α=˜B ˙α=0is invariant under the change A →˜A,C →˜C.Let us study the solution for A,C which should be chosen in the formA =A i +A 0(2.40)C =C i +C 0Here A i ,C i is a partial solution of the inhomogeneous system,and A 0,C 0is a general solution of the homogeneous system.It is straightforward to see that A i =−12−1,C i =0.And A 0,C 0should satisfy the system of equations1i˙C0=A 0¯G 2+C 0F 2A 0,C 0should be chosen to be of the form A 0=a 0exp(iωs ),C 0=c 0exp(iωs )where a 0,c 0,ωare some functions of the background superﬁelds and the d’Alembertian operator,but are independent of s .As a result we arrive at the equations for a 0,c 0:a 0(ω1−F 2)+c 0W ′′=0(2.42)c 0(ω1−F 2)+a 0¯W′′2=0This system of equations has a non-trivial solution atdetω12−F 2W′′¯W ′′2ω12−F 2=0(2.43)In principle,parameters ωcan be found from this equation.Their exact form is determined by the structure of the matrix W ′′and F .It turns out that for the speciﬁc cases studied in detail the subsequent sections (minimal (Section 3)and non-minimal (Section 4)cases)these parameters are diﬀerent.As a result the ﬁnal solution can be cast in the form:A =ka 0k exp(iωk s )−12∞dsi∂16π2(is )2.A ,and ˜Aare functions of 2.Hence in order to calculate the one-loop k¨a hlerian eﬀective potential it is necessary to ﬁnd A and ˜Aand to expand them into a power series in 2.In this section we addressed in detail the techniques needed to calculate the one-loop corrected eﬀective action,to eliminate the heavyﬁelds and to obtain the eﬀective action of lightﬁelds only.In the subsequent sections these techniques will be applied to obtain the explicit form of the one-loop corrected actions for speciﬁc models.In Section5we shall also include interactions with the U(1)vector superﬁelds and modify the procedure accordingly. 3One-loop eﬀective action for minimal modelsIn this section we study decoupling eﬀects for the model with minimal k¨a hlerian potential (2.2)K=Φ¯Φ+φ¯φ.Theﬁrst part consists of calculating the one-loop correction to the k¨a hlerian eﬀective potential.In the second part we solve the eﬀective equations of motion for the heavy superﬁelds.As a result we arrive at the eﬀective action of the light superﬁelds.3.1Calculation of the eﬀective action3.1.1The one-loop k¨a hlerian eﬀective potentialHere we are going to calculate the one-loop contribution to k¨a hlerian eﬀective potential by means of the eﬀective equations of motion.We study the minimal model with the chiral potential in the formW=13!gφ3(3.1)with the corresponding functions in W′′(see eq.(2.25))Wφφ=λΦ+gφWφΦ=λφWΦΦ=M(3.2) The total classical action with the chiral potential(3.1)is of the formS= d8z(φ¯φ+Φ¯Φ)+[ d6z(13!gφ3)+h.c.](3.3)Note that the chiral potential used is of the“minimal”form:it involves the renormalizable terms only and the renormalizable coupling between the light and heavy superﬁelds is linear in the heavyﬁelds,which yields a dominant contribution in the study of the decoupling eﬀects.These types of couplings are typical for a class of eﬀective string models after the vacuum restabilization was taken into account,and thus this minimal model provides a prototype example for the study of decoupling eﬀects in N=1supersymmetric theories. (The results for this model and the physics consequences were presented in[18].For the sake of completeness we present here the intermediate steps in the derivation.)In order toﬁnd the one-loop k¨a hlerian eﬀective potential we should determine the operator Ω(s)that satisﬁes the equation(2.35).For the case of this minimal model this equation leads to the following system of equations for matrices A,C:1i˙C=−¯W′′−A¯W′′2with the analogous equations for˜A,˜C.Initial conditions are A|s=0=˜A|s=0=C|s=0=˜C|s=0=0.Calculations described in Appendix A show that the one-loop contribution to k¨a hlerian eﬀective potential is of the form:K(1)=−1(|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2)|2)××log(|λΦ+gφ|2+2λ2φ¯φ+M2+ µ2+ +(|λΦ+gφ|2+2λ2φ¯φ+M2− (|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2|2)2MΦ2+Φφ2)+h.c.)−1(|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2)|2)××log(|λΦ+gφ|2+2λ2φ¯φ+M2+ µ2+ +(|λΦ+gφ|2+2λ2φ¯φ+M2− (|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2|2)corrections to the chiral eﬀective potential one must set¯Φ=¯φ=0.Possible vertices contributing to one-loop eﬀective potential should be quadratic in quantum superﬁelds[21]. They have the formK¯φ¯φ¯φ2,Kφφφ2,(Kφ¯φ−1)φ¯φ,12WΦφΦφ,K¯Φ¯Φ¯Φ2,KΦΦΦ2,(KΦ¯Φ−1)Φ¯Φ,142)g(Φ)(3.6)Namely,after a transformation to an integral over the chiral superspace by the ruled8zF(Φ,¯Φ)= d6z(−¯D24(2−m2))g(Φ)(3.8) A transformation to the form of an integral over the chiral superspace leads tod6zf(Φ)(2(2π)4f(Φ)(p2decreases a number of D,¯D-factors by4and the corresponding scaling dimension by2.Each propagator of a massless superﬁeld gives no contribution(scaling dimension0)since it has the form(cf.[20])G(z1,z2)=−D21¯D2216δ12=0.Hence a contribution of such a diagram is equal to zero,and a one-loop contribution to the chiral eﬀective potential is absent:W(1)(Φ)=0.We note that this situation is analogous to the general model of one chiral superﬁeld studied in[27].However,higher order(loop)corrections to the chiral eﬀective potential can arise not only for diagrams with external massless lines but also for those with heavy external lines, in spite of the fact that it was commonly believed that quantum corrections to the chiral eﬀective potential for massive superﬁelds are absent.For example,consider the supergraph|¯D 2||¯D 2¯D2D 2D 2D 2D 2−−−−Here a double line denotes the external superﬁeld Φ,and a single line corresponds to thepropagator <φ¯φ>of the massless superﬁeld φ.A contribution of such a supergraph is of the form I =d 4p 1d 4p 2(2π)8d 4θ1d 4θ2d 4θ3d 4θ4d 4θ5(gk 2l 2(k+p 1)2(l +p 2)2(l +k )2(l +k +p 1+p 2)2××δ13¯D 2316δ14δ42D 21¯D 253!)2λ3d 4p 1d 4p 2(2π)8d 2θΦ(−p 1,θ)Φ(−p 2,θ)Φ(p 1+p 2,θ)××k 2p 21+l 2p 22+2(kl )(p 1p 2)3!)2λ3d 2θd 4p 1d 4p 2(2π)8k 2p 21+l 2p 22+2(kl )(p 1p 2)3!)2λ3d 2θd 4x 1d 4x 2d 4x 3d 4p 1d 4p 2。

Calibrated Manifolds and Gauge Theory

a rX iv:mat h /42368v9[mat h.GT]19A ug27CALIBRATED MANIFOLDS AND GAUGE THEORY SELMAN AKBULUT AND SEMA SALUR Abstract.By a theorem of Mclean,the deformation space of an associative submanifold Y of an integrable G 2-manifold (M,ϕ)can be identiﬁed with the kernel of a Dirac operator D /:Ω0(ν)→Ω0(ν)on the normal bundle νof Y .Here,we generalize this to the non-integrable case,and also show that the defor-mation space becomes smooth after perturbing it by natural parameters,which corresponds to moving Y through ‘pseudo-associative’submanifolds.Inﬁnitesi-mally,this corresponds to twisting the Dirac operator D /→D /A with connections A of ν.Furthermore,the normal bundles of the associative submanifolds with Spin c structure have natural complex structures,which helps us to relate their deformations to Seiberg-Witten type equations.If we consider G 2manifolds with 2-plane ﬁelds (M,ϕ,Λ)(they always exist)we can split the tangent space T M as a direct sum of an associative 3-plane bun-dle and a complex 4-plane bundle.This allows us to deﬁne (almost)Λ-associative submanifolds of M ,whose deformation equations,when perturbed,reduce to Seiberg-Witten equations,hence we can assign local invariants to these ing this we can assign an invariant to (M,ϕ,Λ).These Seiberg-Witten equations on the submanifolds are restrictions of global equations on M .We also discuss similar results for the Cayley submanifolds of a Spin (7)manifold.0.Introduction We ﬁrst study deformations of associative submanifolds Y 3of a G 2manifold (M 7,ϕ),where ϕ∈Ω3(M )is the G 2structure.We prove a generalized version of the McLean’s theorem where integrability condition of the underlying G 2structure is not necessary.This deformation space might be singular,but by perturbing it with some natural parameters it can be made smooth.This amounts to deforming Y through the associatives in (M,ϕ)with varying ϕ,or alternatively deforming Y throughthe pseudo-associative submanifolds (Y ’s whose tangent planes become associative after rotating by a generic element of the gauge group of T M ).Inﬁnitesimally,these perturbed deformations correspond to the kernel of the twisted Dirac operator D /A :Ω0(ν)→Ω0(ν),twisted by some connection A in ν(Y ).2SELMAN AKBULUT AND SEMA SALURThe associative submanifolds with Spin c structures in(M,ϕ)are useful objects to study,because their normal bundles have natural complex structures.Also we can view(M,ϕ)as an analog of a symplectic manifold,and view a non-vanishing 2-planeﬁeldΛon M as an analog of a complex structure tamingϕ.Note that 2-planeﬁelds are stronger versions of Spin c structures on M7,and they always exist by[T].The data(M7,ϕ,Λ)determines an interesting splitting of the tangent bundle T M=E⊕V,where E is the bundle of associative3-planes,and V is the complementary4-plane bundle with a complex structure,which is a spinor bundle of E.Then the integral submanifolds Y3of E,which we callΛ-associative submanifolds,can be viewed as analogues of J-holomorphic curves;because their normal bundles come with an almost complex structure.Even if they may not always exist,their perturbed versions,i.e.almostΛ-associative submanifolds,always do. AlmostΛ-associative submanifolds are the transverse sections of the bundle V→M. We can deform such Y by using the connections in the determinant line bundle of ν(Y)and get a smooth deformation space,which is described by the twisted Dirac equation.Then by constraining this new variable with another natural equation we arrive to Seiberg-Witten type equations for Y.So we can assign an integer to Y, which is invariant under small isotopies through almostΛ-assocative submanifolds. In fact it turns out that(M7,ϕ,Λ)gives aﬁner splitting T M=¯E⊕ξ,where¯E is a6-plane bundle with a complex structure,andξis a real line bundle.In a way this structure of(M,ϕ)mimics the structure of(Calabi-Yau)×S1manifolds,and by‘rotating’ξinside of T M we get a new insight for so-called“Mirror manifolds”which is investigated in[AS1].There is a similar process for the deformations of Cayley submanifolds X4⊂N8of a Spin(7)manifold(N8,Ψ),which we discuss at the end.So in a wayΛ-associative(or Cayley)manifolds in a G2(or Spin(7))manifold,behave much like higher dimensional analogue of holomorphic curves in a Calabi-Yau manifold.We would like to thank MSRI,IAS,Princeton and Harvard Universities for pro-viding a stimulating environment where this paper is written,and we thank R.Kirby and G.Tian for continuous encouragement.Theﬁrst named author thanks to R. Bryant and C.Taubes for stimulating discussions and useful suggestions.31.PreliminariesHere weﬁrst review basic properties of the manifolds with special holonomy(most material can be found in[B2],[B3],[H],[HL]),and then proceed to prove some new results.Recall that the set of octonions O=H⊕l H=R8is an8-dimensional division algebra generated by<1,i,j,k,l,li,lj,lk>.On the set of the imaginary octonions im O=R7we have the cross product operation×:R7×R7→R7,deﬁned by u×v=im(¯v.u).The exceptional Lie group G2can be deﬁned as the linear automorphisms of im O preserving this cross product operation,G2=Aut(R7,×). There is also another useful description in terms of the orthogonal3-frames in R7: (1)G2={(u1,u2,u3)∈(im O)3|<u i,u j>=δij,<u1×u2,u3>=0} Alternatively,G2can be deﬁned as the subgroup of the linear group GL(7,R) whichﬁxes a particular3-formϕ0∈Ω3(R7).Denote e ijk=dx i∧dx j∧dx k∈Ω3(R7), thenG2={A∈GL(7,R)|A∗ϕ0=ϕ0}(2)ϕ0=e123+e145+e167+e246−e257−e347−e356Deﬁnition1.A smooth7-manifold M7has a G2structure if its tangent frame bundle reduces to a G2bundle.Equivalently,M7has a G2structure if there is a 3-formϕ∈Ω3(M)such that at each x∈M the pair(T x(M),ϕ(x))is isomorphic to(T0(R7),ϕ0).Here are some useful properties,discussed more fully in[B2]:Any G2structure ϕon M7gives an orientationµ∈Ω7(M)on M,and thisµdetermines a metric g= , on M,and a cross product structure×on its tangent bundle of M as follows:Let i v denote the interior product with a vector v then(3) u,v =[i u(ϕ)∧i v(ϕ)∧ϕ]/6µ(4)ϕ(u,v,w)= u×v,wTo emphasize the dependency onϕsometimes g is denoted by gϕ.In particular,the 14-dimensional Lie group G2imbeds into SO(7)subgroup of GL(7,R).Note that because of the way we deﬁned G2=Gϕ02,this imbedding is determined byϕ0. Since GL(7,R)acts onΛ3(R7)with stabilizer G2,its orbitΛ3+(R7)is open for dimension reasons,so the choice ofϕ0in the above deﬁnition is generic(in fact it has two orbits containing±ϕ0).G2has many copies Gϕ2inside GL(7,R),which are all conjugate to each other,since G2has only one7dimensional representation.Hence the space of G2structures on M7are identiﬁed with the sections of the bundle: (5)RP7≃GL(7,R)/G2→Λ3+(M)−→M4SELMAN AKBULUT AND SEMA SALURwhich are called the positive3-forms,these are the set of3-formsΩ3+(M)that can be identiﬁed pointwise byϕ0.Each Gϕ2imbeds into a conjugate of one standard copy SO(7)⊂GL(7,R).The space of G2structuresϕon M,which induce the same metric on M,that is allϕ’s for which the corresponding Gϕ2lies in the standard SO(7),are the sections of the bundle(whoseﬁber is the orbit ofϕ0under SO(7)): (6)RP7=SO(7)/G2→˜Λ3+(M)−→Mwhich we will denote by˜Ω3+(M).The set of smooth7-manifolds with G2-structures coincides with the set of7-manifolds with spin structure,though this correspondence is not1−1.This is because Spin(7)acts on S7with stabilizer G2inducing the ﬁbrationsG2→Spin(7)→S7→BG2→BSpin(7)and so there is no obstruction to lifting maps M7→BSpin(7)to BG2,and there are many liftings.Cotangent frame bundle P∗(M)→M of a manifold with G2 structure(M,ϕ)can be expressed as P∗(M)=∪x∈M P∗x(M),where eachﬁber is: P∗x(M)={u∈Hom(T x(M),R7)|u∗(ϕ0)=ϕ(x)}Throughout this paper we will denote the cotangent frame bundle by P∗(M)→M and its adapted frame bundle by P(M).They can be G2or SO(7)frame bundles; to emphasize it sometimes we will specify them by the notations P SO(7)(M)or P G2(M).Also we will denote the sections of a bundleξ→Y byΩ0(Y,ξ)or simply byΩ0(ξ),and the bundle valued p-forms byΩp(ξ)=Ω0(Λp T∗Y⊗ξ),and the sphere bundle ofξby S(ξ).There is a notion of a G2structureϕon M7being integrable, which corresponds toϕbeing an harmonic form:Deﬁnition2.A manifold with G2structure(M,ϕ)is called a G2manifold if the holonomy group of the Levi-Civita connection(of the metric gϕ)lies inside of G2. Equivalently(M,ϕ)is a G2manifold ifϕis parallel with respect to the metric gϕi.e.∇gϕ(ϕ)=0;this condition is equivalent to dϕ=0=d(∗gϕϕ).In short one can deﬁne a G2manifold to be any Riemannian manifold(M7,g) whose holonomy group is contained in G2,thenϕand the cross product×come as a consequence.It turns out that the conditionϕbeing harmonic is equivalent to the condition that at each point x0∈M there is a chart(U,x0)→(R7,0)on which ϕequals toϕ0up to second order term,i.e.on the image of U(7)ϕ(x)=ϕ0+O(|x|2)Remark1.For example if(X6,ω,Ω)is a complex3-dimensional Calabi-Yau man-ifold with K¨a hler formω,and a nowhere vanishing holomorphic3-formΩ,then X×S1has holonomy group SU(3)⊂G2,hence is a G2manifold.In this case (8)ϕ=ReΩ+ω∧dt.5 Deﬁnition 3.Let(M,ϕ)be a manifold with a G2structure.A4-dimensional submanifold X⊂M is called an co-associative ifϕ|X=0.A3-dimensional submanifold Y⊂M is called an associative ifϕ|Y≡vol(Y);this condition is equivalent toχ|Y≡0,whereχ∈Ω3(M,T M)is the tangent bundle valued3-form deﬁned by the identity:(9) χ(u,v,w),z =∗ϕ(u,v,w,z)The equivalence of these conditions follows from the‘associator equality’of[HL] (10)ϕ(u,v,w)2+|χ(u,v,w)|2/4=|u∧v∧w|2In general,if{e1,e2,..,e7}is any orthonormal coframe on(M,ϕ),then the expres-sion(2)forϕhold on a chart.By calculation∗ϕ,and using(9)we can calculate the expression ofχ(note the error in the the second term of6th line of the corresponding formula(5.4)of[M]):(11)∗ϕ=e4567+e2367+e2345+e1357−e1346−e1256−e1247χ=(e256+e247+e346−e357)e1+(−e156−e147−e345−e367)e2+(e245+e267−e146+e157)e3+(−e567+e127+e136−e235)e4+(e126+e467−e137+e234)e5+(−e457−e125−e134−e237)e6+(e135−e124+e456+e236)e7Alsoχcan be expressed in terms of cross product operation(c.f.[H],[HL],[K]): (12)χ(u,v,w)=−u×(v×w)− u,v w+ u,w vWhen dϕ=0,the associative submanifolds are volume minimizing submanifolds of M(calibrated byϕ).Even in the general case of a manifold with a G2structure (M,ϕ),the formχimposes an interesting structure near associative submanifolds: Notice(9)implies that,χmaps every oriented3-plane in T x(M)to the orthogonal subspace T x(M)⊥,so if we choose local coordinates(x1,...,x7)for M7we get (13)χ= aαJ dx J⊗∂6SELMAN AKBULUT AND SEMA SALURFrom(9)it is easy to calculate aαijk=∗ϕijks g sα,where g−1=(g ij)is the inverse of the metric g=(g ij),and of course the metric g can be expressed in terms ofϕ.By evaluatingχon the orientation form of Y we get a normal vectorﬁeld so: Lemma1.To any3-dimensional submanifold Y3⊂(M,ϕ),χassociates a normal vectorﬁeld,which vanishes when Y is associative.Henceχdeﬁnes an interestingﬂow on3dimensional submanifolds of(M,ϕ),ﬁxing associative submanifolds.On the associative submanifolds with a Spin c structure,χrotates their normal bundles and imposes a complex structure on them: Lemma2.To any associative manifold Y3⊂(M,ϕ)with a non-vanishing oriented 2-planeﬁeld,χdeﬁnes an almost complex structure on its normal bundleν(Y) (notice that in particular any coassociative submanifold X⊂M has an almost complex structure if its normal bundle has a non-vanishing section).Proof.Let L⊂R7be an associative3-plane,that isϕ|L=vol(L).Then to every pair of orthonormal vectors{u,v}⊂L,the formχdeﬁnes a complex structure on the orthogonal4-plane L⊥,as follows:Deﬁne j:L⊥→L⊥by(15)j(X)=χ(u,v,X)This is well deﬁned i.e.j(X)∈L⊥,because when w∈L we have:<χ(u,v,X),w>=∗ϕ(u,v,X,w)=−∗ϕ(u,v,w,X)=<χ(u,v,w),X>=0Also j2(X)=j(χ(u,v,X))=χ(u,v,χ(u,v,X))=−X.We can check the last equality by taking an orthonormal basis{X j}⊂L⊥and calculating<χ(u,v,χ(u,v,X i)),X j>=∗ϕ(u,v,χ(u,v,X i),X j)=−∗ϕ(u,v,X j,χ(u,v,X i))=−<χ(u,v,X j),χ(u,v,X i)>=−δijThe last equality holds since the map j is orthogonal,and the orthogonality can be seen by polarizing the associator equality(10),and by noticingϕ(u,v,X i)=0. Observe that the map j only depends on the oriented2-plane l=<u,v>generated by{u,v}.So the result follows.In fact,for any unit vectorﬁeldξon an associative Y(i.e.a Spin c structure) deﬁnes a complex structure Jξ:ν(Y)→ν(Y)by Jξ(z)=z×ξ,and the complex structure deﬁned in Lemma2corresponds to J u×v,because from(12):χ(u,v,z)=χ(z,u,v)=−z×(u×v)− z,u v+ z,v u=J v×u(z).Also recall that the complex structures on any SO(4)bundle such asν→Y are given by the unit sections of the associated SO(3)bundleλ+(ν)→Y,which is induced by the left reductions SO(4)=(SU(2)×SU(2))/Z2→SU(2)/Z2=SO(3).7 Deﬁnition4.A Riemannian8-manifold(N8,g)is called a Spin(7)manifold if the holonomy group of its Levi-Civita connection lies in Spin(7)⊂GL(8,R). Equivalently a Spin(7)manifold(N,Ψ)is a Riemannian8-manifold with a triple cross product×on its tangent bundle,and a harmonic4-formΨ∈Ω4(N)withΨ(u,v,w,z)=g(u×v×w,z)It is easily checked that if(M,ϕ)is a G2manifold,then(M×S1,Ψ)is a Spin(7) manifold whereΨ=ϕ∧dt−∗ϕ.Deﬁnition 5.A4-dimensional submanifold X of a Spin(7)manifold(N,Ψ)is called Cayley ifΨ|X≡vol(X).This is equivalent toτ|X≡0whereτ∈Ω4(N,E) is a certain vector-bundle valued4-form deﬁned by the“four-fold cross product”of the imaginary octonionsτ(v1,v2,v3,v4)=v1×v2×v3×v4(see[M],[HL]).2.Grassmann BundlesLet G(3,7)be the Grassmann manifold of oriented3-planes in R7.Let M7be an oriented smooth7-manifold,and let˜M→M be the bundle oriented3-planes in T M,which is deﬁned by the identiﬁcation[p,L]=[pg,g−1L]∈˜M:(16)˜M=P SO(7)(M)×SO(7)G(3,7)→M.This is just the bundle˜M=P SO(7)(M)/SO(3)×SO(4)→P SO(7)(M)/SO(7)=M. Letξ→G(3,7)be the universal R3bundle,andν=ξ⊥→G(3,7)be the dual R4 bundle.Therefore,Hom(ξ,ν)=ξ∗⊗ν−→G(3,7)is the tangent bundle T G(3,7).ξ,νextendﬁberwise to give bundlesΞ→˜M,V→˜M respectively,and letΞ∗be the dual ofΞ.Notice that Hom(Ξ,V)=Ξ∗⊗V→˜M is the bundle of vertical vectors T v(˜M)of T(˜M)→M,i.e.the tangents to theﬁbers ofπ:˜M→M,hence (17)T˜M∼=T v(˜M)⊕π∗T M=(Ξ∗⊗V)⊕Ξ⊕V.That is,T˜M is the vector bundle associated to principal SO(3)×SO(4)bundle P SO(7)→˜M by the obvious representation of SO(3)×SO(4)to(R3)∗⊗R4+R3+R4. The identiﬁcation(17)is deﬁned up to gauge automorphisms of bundlesΞand V. Note that the bundle V=Ξ⊥depends on the metric,and hence it depends onϕwhen metric is induced from a G2structure(M,ϕ).To emphasize this fact we can denote it by Vϕ→˜M.But when we are considering G2structures coming from G2 subgroups of aﬁxed copy of SO(7)⊂GL(7,R),they induce the same metric and so this distinction is not necessary.8SELMAN AKBULUT AND SEMA SALURLet P(V)→˜M be the SO(4)frame bundle of the vector bundle V,identify R4 with the quaternions H,and identify SU(2)with the unit quaternions Sp(1)=S3. Recall that SO(4)is the equivalence classes of pairs[q,λ]of unit quaternionsSO(4)=(SU(2)×SU(2))/Z2Hence V→˜M is the associated vector bundle to P(V)via the SO(4)representation (18)x→qxλ−1There is a pair of R3=im(H)bundles over˜M corresponding to the left and right SO(3)reductions of SO(4),which are given by the SO(3)representations(19)λ+(V):x→qx q−1λ−(V):y→λyλ−1The map x⊗y→xy gives actionsλ+(V)⊗V→V and V⊗λ−(V)→V;by combining we can think of them as one conjugation action(20)(λ+(V)⊗λ−(V))⊗V→VIf the SO(4)bundle P(V)→˜M lifts to a Spin(4)=SU(2)×SU(2)bundle (locally it does),we get two additional bundles over˜M(21)S:y→qy E:y→yλ−1They identify V as a tensor product of two quaternionic line bundles V=S⊗H E.In particular,λ+(V)=ad(S)andλ−(V)=ad(E),i.e.they are the SO(3)reductions of the SU(2)bundles S and E.Also there is a multiplication map S⊗E→V.Recall the identiﬁcations:Λ2(V)=Λ2+(V)⊕Λ2−(V)=λ−(V)⊕λ+(V)=λ(V)=gl(V)=ad(V).2.1.Associative Grassmann Bundles.Now consider the Grassmannian of associative3-planes Gϕ(3,7)in R7,con-sisting of elements L∈G(3,7)with the propertyϕ0|L=vol(L)(or equivalently χ0|L=0).G2acts on Gϕ(3,7)transitively with the stabilizer SO(4),so it gives the identiﬁcation Gϕ(3,7)=G2/SO(4).If we identify the imaginary octonions by R7=Im(O)∼=im(H)⊕H,then the action of the subgroup SO(4)⊂G2on R7is (22) ρ(A)00Awhereρ:SO(4)=(SU(2)×SU(2))/Z2→SO(3)is the projection of theﬁrst factor ([HL]),that is for[q,λ]∈SO(4)the action is given by(x,y)→(qxq−1,qyλ−1).So the action of SO(4)on the3-plane L=im(H)is determined by its action on L⊥. Now let M7be a G2manifold.Similar to the construction before,we can construct the bundle of associative Grassmannians over M(which is a submanifold of˜M):(23)˜Mϕ=P G2(M)×G2Gϕ(3,7)→M9which is just the quotient bundle˜Mϕ=P G2(M)/SO(4)−→P G2(M)/G2=M.Asin the previous section,the restriction of the universal bundlesξ,ν=ξ⊥→Gϕ(3,7) induce3and4plane bundlesΞ→˜Mϕand V→˜Mϕ(by restricting from˜M).Also (24)T˜Mϕ∼=T v(˜Mϕ)⊕Ξ⊕VFrom(22)we see that in the associative case,we have an important identiﬁcation:Ξ=λ+(V)(as bundles over˜Mϕ),and the dual of the actionλ+(V)⊗V→V givesa Cliﬀord multiplication:(25)Ξ∗⊗V→VIn fact this is just the map induced from the cross product operation[AS2].Recall that T v(˜M)=Ξ∗⊗V→˜M is the subbundle of vertical vectors of T(˜M)→M. The total space E(νϕ)of the normal bundle of the imbedding˜Mϕ⊂˜M should be thought of an open tubular neighborhood of˜Mϕin˜M,and it has a nice description: Lemma3.([M])Normal bundleνϕof˜Mϕ⊂˜M is isomorphic to V,and the bundle of vertical vectors T v(˜Mϕ)is the kernel of the Cliﬀord multiplication c:Ξ∗⊗V→V. We have T v(˜M)|˜Mϕ=T v(˜Mϕ)⊕νϕ,and the following exact sequence over˜MϕT v(˜Mϕ)→Ξ∗⊗V|˜Mϕc−→V|˜Mϕ→0Hence the quotient bundle,T v(˜M)/T v(˜Mϕ)is isomorphic to V.Proof.This is because the Lie algebra inclusion g2⊂so(7)is given byaβ−βtρ(a)where a∈so(4)is y→qy−yλ,andρ(a)∈so(3)is x→qx−xq.So the tangent space inclusion of G2/SO(4)⊂SO(7)/SO(4)×SO(3)is given by the matrix β∈(im H)∗⊗H.Therefore,if we writeβas column vectors of three queternions β=(β1,β2,β3)=i∗⊗β1+j∗⊗β2+k∗⊗β3,thenβ1i+β2j+β3k=0([M],[Mc]). The reader can consult Lemma5of[AS2]for a more self contained proof of this fact,where the Cliﬀord multiplication is identiﬁed with the cross product operation.3.Associative SubmanifoldsAny imbedding of a3-manifold f:Y3֒→M7induces an imbedding˜f:Y֒→˜M:(26)˜M⊃˜Mϕ˜fր↓Y f−→M10SELMAN AKBULUT AND SEMA SALURand the pull-backs˜f∗Ξ=T(Y)and˜f∗V=ν(Y)give the tangent and normal bundles of Y.Furthermore,if f is an imbedding of an associative submanifold into a G2manifold(M,ϕ),then the image of˜f lands in˜Mϕ.We will denote this canonical lifting of any3-manifold Y⊂M by˜Y⊂˜M.Also since we have the dependency V=Vϕ,we can denoteν(Y)=ν(Y)ϕ=νϕwhen needed.˜Mϕcan be thought of as a universal space parameterizing associative submani-folds of M.In particular,if˜f:Y֒→˜Mϕis the lifting of an associative submanifold, by pulling back we see that the principal SO(4)bundle P(V)→˜Mϕinduces an SO(4)-bundle P(Y)→Y,and gives the following vector bundles via the represen-tations:(27)ν(Y):y→qyλ−1 T(Y):x→qx q−1where[q,λ]∈SO(4),ν=ν(Y)and T(Y)=λ+(ν).Also we can identify T∗Y with T Y by the induced metric.From above we have the action T∗Y⊗ν→νinducing actionsΛ∗(T∗Y)⊗ν→ν.Let L=Λ3(Ξ)→˜M be the determinant(real)line bundle.Recall that the deﬁnition(9)implies thatχmaps every oriented3-plane in T x(M)to its comple-mentary subspace,soχgives a bundle map L→V over˜M,which is a section of L∗⊗V→˜M.SinceΞis oriented L is trivial,soχactually gives a section(28)χ=χϕ∈Ω0(˜M,V)Clearly˜Mϕ⊂˜M is the codimension4submanifold which is the zeros of thissection.Associative submanifolds Y⊂M are characterized by the conditionχ|˜Y =0,where˜Y⊂˜M is the canonical lifting of Y.Similarlyϕdeﬁnes a mapϕ:˜M→R.3.1.Pseudo-associative submanifolds.Here we generalize associative submanifolds to a moreﬂexible class of submani-folds.To do this weﬁrst generalize the notion of imbedded submanifolds.Deﬁnition 6.A Grassmann-framed3-manifold in(M,ϕ)is a triple(Y3,f,F), where f:Y֒→M is an imbedding,F:Y→˜M,such that the following commute(29)˜M Fր↓Y f−→MWe call(Y,f,F)a pseudo-associative submanifold if in addition Image(F)⊂˜Mϕ. So a pseudo-associative submanifold(Y,f,F)with F=˜f is associative.11 Remark2.The bundle˜M→M always admits a section,in fact the subbundle ˜Mϕ→M has a section.This is because by[T]every orientable7-manifold admits a non-vanishing linearly independent2-frameﬁeldΛ={v1,v2}1.By Grahm-Schmidt process with metric gϕ,we can assume thatΛis orthonormal.The cross product assignsΛto an orthonormal3-frameﬁeld{v1,v2,v1×ϕv2}on M,then3-plane gen-erated by{v1,v2,v1×ϕv2}:=<v1,v2,v1×ϕv2>gives a section ofλϕ:M→˜Mϕ.LetFigure1.Z(M)and Zϕ(M)denote the set of Grassmann-framed and the pseudo-associative submanifolds,respectively,and let Aϕ(M)be the set of associative submanifolds. We have inclusions Aϕ(M)֒→Zϕ(M)֒→Z(M),where theﬁrst map is given by (Y,f)→(Y,f,˜f).So there is an inclusion Im(Y,M)֒→Z(M),where Im(Y,M)is the space of imbeddings.This inclusion can be thought of the canonical sections ofa bundle(30)Z(Y)π−→Im(Y,M)withﬁbersπ−1(f)=Ω0(Y,f∗˜M).We also have the subbundle Zϕ(Y)π−→Im(Y,M) withﬁbersπ−1(f)=Ω0(Y,f∗˜Mϕ).So Z(Y)is the set of triples(Y,f,F)(in short just set of F’s),where F:Y→˜M is a lifting of the imbedding f:Y֒→M. Also Zϕ(Y)⊂Z(Y)is a smooth submanifold,since˜Mϕ⊂˜M is smooth.There is the canonical sectionΦ:Im(Y,M)→Z(Y)given byΦ(f)=˜f.Therefore,Φ−1Zϕ(Y):=Imϕ(Y,M)is the set of associative imbeddings Y⊂M.Also,any 2-frameﬁeldΛas above gives to a sectionΦΛ(f)=λϕ◦f.To make these deﬁnitions parameter free we also have to divide Im(Y,M)by the diﬀeomorphism group of Y.12SELMAN AKBULUT AND SEMA SALURThere are also the vertical tangent bundles of Z(Y)and Zϕ(Y)T v Z(Y)π−→Z(Y)∪∪T v Zϕ(Y)π|−→Zϕ(Y)withﬁbersπ−1(F)=Ω0(Y,F∗(Ξ∗⊗V)).By Lemma3theﬁbers of T v(Zϕ)can be identiﬁed with the kernel of the map induced by the Cliﬀord multiplication (31)c:Ω0(Y,F∗(Ξ∗⊗V))→Ω0(Y,F∗(V))One of the nice properties of a pseudo-associative submanifold(Y,f,F)is that there is a Cliﬀord multiplication action(by pull back)(32)F∗(Ξ∗)⊗F∗(V)→F∗(V)If F is close to˜f,by parallel translating theﬁbers over F(x)and˜f(x)along geodesics in˜M we get canonical identiﬁcations:(33)F∗(Ξ)∼=T Y F∗(V)∼=νfinducing Cliﬀord multiplication between the tangent and the normal bundles.So if ∀x∈Y the distance between F(x)and˜f(x)is less then the injectivity radius j(˜M), there is a Cliﬀord multiplication between the tangent and normal bundles of Y.3.2.Dirac operator.The normal bundleν=ν(Y)of any orientable3-manifold Y in a G2manifold (M,ϕ)has a Spin(4)structure(e.g.[B2]).Hence we have SU(2)bundles S and E over Y such thatν=S⊗H E(18),with SO(3)reductions adS=λ+(ν),and adE=λ−(ν)which is also the bundle of endomorphisms End(E).If Y is associative, then the bundle ad(S)becomes isomorphic to T Y,i.e.S becomes the spinor bundle of Y,soν(Y)becomes a twisted spinor bundle.The Levi-Civita connection of the G2metric of(M,ϕ)induces connections on the associated bundles V andΞon˜M.In particular,it induces connections on the tangent and normal bundles of any submanifold Y3⊂M.We will call these connections the background connections.Let A0be the induced connection on the normal bundleν=S⊗E.From the Lie algebra decomposition so(4)=so(3)⊕so(3),we can write A0=B0⊕A0,where B0and A0are connections on S and E, respectively.Let A(E)and A(S)be the set of connections on the bundles E and S.Hence A∈A(E),B∈A(S)are in the form A=A0+a,B=B0+b,where a∈Ω1(Y,ad E) and b∈Ω1(Y,ad S).SoΩ1(Y,λ±(ν))parametrizes connections on S and E,and the connections onνare in the form A=B⊕A.To emphasize the dependency on b and a we sometimes denote A=A(b,a),and A0=A(0,0)=A0.13 Now,let Y3⊂M be any smooth manifold.We can ex press the covariant derivative∇A:Ω0(Y,ν)→Ω1(Y,ν)onνby∇A= e i⊗∇e i,where{e i}and{e i} are orthonormal tangent and cotangent frameﬁelds of Y,respectively.Furthermore, if Y is an associative submanifold,we can use the Cliﬀord multiplication of(25)(i.e. the cross product)to form the twisted Dirac operator D/A:Ω0(Y,ν)→Ω0(Y,ν) (34)D/A= e i.∇e iThe sections lying in the kernel of this operator are usually called harmonic spinors twisted by(E,A).Elements of the kernel of D/Aare called the harmonic spinors twisted by E,or just the twisted harmonic spinors.4.DeformationsIn[M],McLean showed that the space of associative submanifolds of a G2mani-fold(M,ϕ),in a neighborhood of aﬁxed associative submanifold Y,can be identiﬁed with the harmonic spinors on Y twisted by E.Since the cokernel of the Dirac op-erator can vary,the dimension of its kernel is not determined(it has zero index since Y is odd dimensional).We will remedy this problem by deforming Y in a larger class of submanifolds.To motivate our aproach we willﬁrst sketch a proof of McLean’s theorem(adapting the explanation in[B3]).Let Y⊂M be an associative submanifold,Y will determine a lifting˜Y⊂˜Mϕ.Let us recall that the G2structure ϕgives a metric connection on M,hence it gives a connection A0and a covariant diﬀerentiation in the normal bundleν(Y)=ν∇A:Ω0(Y,ν)→Ω1(Y,ν)=Ω0(Y,T∗Y⊗ν)Recall that we identiﬁed T∗y(Y)⊗νy(Y)by the tangent space of the Grassmannian of3-planes T G(3,7)in T y(M).So the covariant derivative lifts normal vectorﬁeldsv of Y⊂M to vertical vectorﬁelds˜v in T(˜M)|˜Y .We want the normal vectorﬁelds v of Y to move Y in the class of associative submanifolds of M,i.e.we want the liftings˜Y v of the nearby copies Y v of Y(pushed oﬀby the vectorﬁeld v) to lie in˜Mϕ⊂˜M upstairs,i.e.we want the component of˜v in the direction of the normal bundle˜Mϕ⊂˜M to vanish.By Lemma3,this means∇A(v)should be in the kernel of the Cliﬀord multiplication c=cϕ:Ω0(T∗(Y)⊗ν)→Ω0(ν),i.e.D/A0(v)=c(∇A(v))=0,where D/Ais the Dirac operator induced by thebackground connection A0,i.e.the composition(35)Ω0(Y,ν)∇A0−→Ω0(Y,T∗Y⊗ν)c→Ω0(Y,ν)The condition D/A(v)=0impliesϕmust be integrable at Y,i.e.the so(7)-metric connection∇Aon Y coincides with G2-connection(c.f.[B2]).Now we give a general version of the McLean’s theorem,without integrability assumption onϕ:Recall from(Section3.1)thatΦ−1Zϕ(Y)is the set of associative14SELMAN AKBULUT AND SEMA SALURsubmanifolds Y ⊂M ,where Φ:Im (Y,M )→Z (Y )is the canonical section (Gauss map)given by Φ(f )=˜f.Therefore,if f :Y ֒→M is the above inclusion,then Φ(f )∈Z ϕ.So this moduli space is smooth if Φwas transversal to Z ϕ(Y ).MM ~~G (3,7)Figure 2.Theorem 4.Let (M 7,ϕ)be a manifold with a G 2structure,and Y 3⊂M be an associative submanifold.Then the tangent space of associative submanifolds of M at Y can be identiﬁed with the kernel of a Dirac operator D /A :Ω0(Y,ν)→Ω0(Y,ν),where A =A 0+a ,and A 0is the connection on νinduced by the metric g ϕ,and a ∈Ω1(Y,ad (ν)).In the case ϕis integrable a =0.In particular,the space of associative submanifolds of M is smooth at Y if the cokernel of D /A is zero.Proof.Let f :Y ֒→M denote the imbedding.We consider unparameterized deformations of Y in Im (Y,M )along its normal directions.Fix a trivialization T Y ∼=im (H ),by (17)we have an identiﬁcation ˜f ∗(T v ˜M )∼=T Y ∗⊗ν+T Y +ν.We ﬁrst claim Π◦d Φ(v )=∇A (v ),where d Φis the induced map on the tangent space and Πis the vertical projection.Ω0(Y,ν)=T f Im (Y,M )d Φ−→T ˜f Z (Y )=Ω0(Y,˜f ∗(T v ˜M ))Π→Ω0(Y,T ∗Y ⊗ν)↓exp↓exp Im (Y,M )Φ−→Z (Y )。

Superradiant Instability of Five-Dimensional Rotating Charged AdS Black Holes

which must be accompanied by their spontaneous emission as well [2]. The spontaneous superradiance arises due to quantum instability of vacuum in the Kerr metric, leading to a pair production of particles. When leaving the ergosphere these particles carry positive energy and angular momentum from the black hole to inﬁnity, whereas inside the ergosphere they form negative energy and angular momentum ﬂows into the black hole [8]. The phenomenon of superradiance, after all, has a deep conceptual signiﬁcance for understanding the stability properties of the black holes. As early as 1971 Zel’dovich [2] noted that placing a reﬂecting mirror (a resonator) around a rotating black hole would result in re-ampliﬁcation of superradiant modes and eventually the system would develop instability. The eﬀect of the instability was later studied in [9] and the system is now known as a “black hole bomb.” This study has also created the motivation to answer general questions on the stability of rotating black holes against small external perturbations. Using analytical and numerical methods, it has been shown that the Kerr black holes are stable to massless scalar, electromagnetic, and gravitational perturbations [10]. However, the situation turned out to be diﬀerent for perturbing massive bosonic ﬁelds. As is known, classical particles of energy E and mass m, obeying the condition E < m, perform a ﬁnite motion in the gravitational potential of the black hole. From quantum-mechanical point of view there exists a certain probability for tunneling such particles through the potential barrier into the black hole. In consequence of this, the bound states of the particles inside the potential well must become quasistationary or quasinormal (see, for instance [11] and references therein). Similarly, for ﬁelds with mass µ, the wave of frequency ω < µ can be thought of as a “bound particle” and therefore must undergo repetitive reﬂections between the potential well and the horizon. In the regime of superradiance, this will cause exponential growth of the number of particles in the quasinormal states, developing the instability [12, 13, 14, 15]. Thus, for massive bosonic ﬁelds the potential barrier of the black hole plays the role of a mirror in the heuristic model of the black hole bomb. There are also alternative models where a reﬂecting mirror leading to the instability arises due to an extra dimension which, from Kaluza-Klein point of view, acts as a massive term (see for instance, Ref. [16]). In recent years, the question of the stability of black holes to external perturbations has been the subject of extensive studies in four and higher-dimensional spacetimes with cosmological constant. In particular, analytical and numerical works have revealed the perturbative stability of nonrotating black holes in de Sitter or anti-de Sitter (dS/AdS) 3

AdS_3, Black Holes and Higher Derivative Corrections

a r X i v :0705.0735v 2 [h e p -t h ] 19 M a y 2007arXiv:0705.0735AdS 3,Black Holes and Higher Derivative Corrections Justin R.David,Bindusar Sahoo and Ashoke Sen Harish-Chandra Research Institute Chhatnag Road,Jhusi,Allahabad 211019,INDIA E-mail:justin,bindusar,sen@mri.ernet.in Abstract Using AdS/CFT correspondence and the Euclidean action formalism for black hole entropy Kraus and Larsen have argued that the entropy of a BTZ black hole in three dimensional supergravity with (0,4)supersymmetry does not receive any correction from higher derivative terms in the action.We argue that as a consequence of AdS/CFT correspondence the action of a three dimensional supergravity with (0,4)supersymmetry cannot receive any higher derivative correction except for those which can be removed byﬁeld redeﬁnition.The non-renormalization of the entropy then follows as a consequence of this and the invariance of Wald’s formula under a ﬁeld redeﬁnition.1BTZ solution describes a black hole in three dimensional theory of gravity with nega-tive cosmological constant[1]and often appears as a factor in the near horizon geometry of higher dimensional black holes in string theory[2].Furthermore the entropy of a BTZ black hole has a remarkable similarity to the Cardy formula for the degeneracy of states in the two dimensional conformalﬁeld theory[3].For these reasons computation of the entropy of BTZ black holes has been an important problem,both in three dimensional theories of gravity and also in string theory.Initial studies involved computing Bekenstein-Hawking formula for BTZ black hole entropy in two derivative theories of ter this was generalized to higher derivative theories of gravity[4,5,6,7,8,9,10],where the lagrangian density contains arbitrary powers of Riemann tensor and its covariant derivatives as well as gravitational Chern-Simons terms[11],both in the Euclidean action formalism[12]and in Wald’s formalism[13,14,15,16].While the above mentioned formalism tells us how to calculate the entropy of a BTZ black hole for a given action with arbitrary higher derivative terms,it does not tell us what these higher derivative terms are.It was however argued by Kraus and Larsen[5,6]using AdS/CFT correspondence that if the three dimensional theory under consideration has at least(0,4)supersymmetry then the entropy of a BTZ black hole of given mass and angular momentum is determined completely in terms of the coeﬃcients of the gravitational and gauge Chern-Simons terms in the action and hence does not receive any higher derivative corrections.This result is somewhat surprising from the point of view of the bulk theory, since for a given three dimensional theory of gravity the entropy does have non-trivial dependence on all the higher derivative terms.Thus one could wonder how the dependence of the entropy on these higher derivative terms disappears by imposing the requirement of(0,4)supersymmetry.In this note we shall propose a simple explanation for this fact:(0,4)supersymmetry prevents the addition of any higher derivative terms in the supergravity action(except those which can be removed byﬁeld redeﬁnition and hence give an equivalent theory).Our argument is based on the following observation.In AdS/CFT correspondence the bound-ary operators dual to theﬁelds in the supergravity multiplet are just the superconformal currents associated with the(0,4)supersymmetry algebra.The correlation functions of these operators in the boundary theory are determined completely in terms of the central charges c L,c R of the left-moving Virasoro algebra and the right-moving super-Virasoro algebra.Of these c R is related to the central charge k R of the right-moving SU(2)currents2which form the R-symmetry currents of the super-Virasoro algebra and hence to the coef-ﬁcient of the Chern-Simons term of the associated SU(2)gaugeﬁelds in the bulk theory. On the other hand c L−c R is determined in terms of the coeﬃcient of the gravitational Chern-Simons term in the bulk theory.Thus the knowledge of the gauge and gravitational Chern-Simons terms in the bulk theory determines all the correlation functions of(0,4) superconformal currents in the boundary theory.Since by AdS/CFT correspondence[17] these correlation functions in the boundary theory determine completely the boundary S-matrix of the supergravityﬁelds[18,19],we conclude that the coeﬃcients of the gauge and gravitational Chern-Simons terms in the bulk theory determine completely the boundary S-matrix elements in this theory.Now the boundary S-matrix elements are the only perturbative observables of the bulk theory.Thus we expect that two diﬀerent theories with the same boundary S-matrix must be equivalent.(We shall elaborate on this later.)Combining this with the observation made in the last paragraph we see that two diﬀerent gravity theories,both with(0,4)supersymmetry and the same coeﬃcients of the gauge and gravitational Chern-Simons terms,must be equivalent.Put another way,once we have constructed a classical supergravity theory with(0,4)supersymmetry and given coeﬃcients of the Chern-Simons terms,there cannot be any higher derivative corrections to the action involvingﬁelds in the gravity supermultiplet except for those which can be removed byﬁeld redeﬁnition.1 The non-renormalization of the entropy of the BTZ black hole then follows trivially from this fact.The complete theory in the bulk of course can have other matter multiplets whose action will receive higher derivative corrections.However since restriction to the ﬁelds in the gravity supermultiplet provides a consistent truncation of the theory,2and since the BTZ black hole is embedded in this subsector,its entropy will not be aﬀected by these additional higher derivative terms.Our arguments will imply in particular that theﬁve dimensional supergravity action constructed in[21],after dimensional reduction on a sphere,must be equivalent to the three dimensional supergravity action given in eq.(2)below with the precise relationship between the various coeﬃcients as given in eq.(4).This in turn would explain why the analysis of the black hole entropy given in[22,23](see also[24])agrees with the expected result.We should caution the reader however that theﬁeld redeﬁnition needed to arrive at the action given in(2)may not be invertible on allﬁeld conﬁgurations.For example if we take a Chern-Simons action and add to it the usual kinetic term for a gaugeﬁeld then the kinetic term can be removed formally by aﬁeld redeﬁnition.However the theory with the kinetic term has an extra pole in the gaugeﬁeld propagator corresponding to a massive photon which is absent in the pure Chern-Simons theory.This happens because theﬁeld redeﬁnition that takes us from the theory with the gauge kinetic term to pure Chern-Simons theory is not invertible on the plane wave solution describing the propagating massive photon.This however does not aﬀect our argument as long as theﬁeld redeﬁnition is invertible on slowly varyingﬁeld conﬁguration around the AdS3background.In this context we note that suchﬁeld redeﬁnitions are carried out routinely in string theory,e.g. in converting a term in the gravitational action quadratic in the Riemann tensor to the Gauss-Bonnet combination.The former theory typically has extra poles in the graviton propagator which are absent in the latter theory.For completeness we shall now describe this unique(0,4)supergravity action and compute the entropy of a BTZ black hole from this action.The action was constructed in [25,26]by regarding the supergravity as a gauge theory based on SU(1,1)×SU(1,1|2) algebra.3IfΓL andΓR denote the(super-)connections in the SU(1,1)and SU(1,1|2) algebras respectively,then the action is taken to be a Chern-Simons action[30]of the form:S=−a L d3x T r(ΓL∧dΓL+2ΓR∧ΓR∧ΓR ,(1)3where a L and a R are constants.Note that the usual metric degrees of freedom are encodedin the connectionsΓL andΓR[31].Thus there is no obvious way to add SU(1,1)×SU(1,1|2)invariant higher derivative terms in the action involving theﬁeld strengths associated with the connectionsΓL andΓR.From this viewpoint also it is natural that the supergravity action does not receive any higher derivative corrections.The bosonicﬁelds of this theory include the metric G MN and an SU(2)gaugeﬁeld A M(0≤M≤2),represented as a2×2anti-hermitian matrix valued vectorﬁeld.After expressing the action in the component notation and eliminating auxiliaryﬁelds using their equations of motion we arrive at the actionS= d3x √4πǫMNP T r A M∂N A P+2m =12(a L−a R),(3) k R=4πa R=4π 12 ΓR MS∂N ΓS P R+1−det G L(3)0+KΩ3( Γ) ,(6)where L(3)0denotes an arbitrary scalar constructed out of the metric,the Riemann tensor and covariant derivatives of the Riemann tensor.A general BTZ black hole in the three dimensional theory is described by the metric:G MN dx M dx N=−(ρ2−ρ2+)(ρ2−ρ2−)(ρ2−ρ2+)(ρ2−ρ2−)dρ2+ρ2dy−ρ+ρ−where l,ρ+andρ−are parameters labelling the solution.Of these the parametersρ±can be removed locally by a coordinate transformation,so that any scalar combination of the Riemann tensor and metric computed for this metric is a function of the parameter l only.We deﬁneh(l)=L(3)0,(8) evaluated in the background(7),andg(l)=πl3c L q L c R q R2(M−J),q R=1πg(l0).(14)3.The parametersρ±are related to M and J via the relationsM±J=2π(C∓K)4l3(−6l−2+2m2),l0=1m(16)andc L=24π 1m−K =24πa R,(17) where in(17)we have used(3).Using(4)we getc R=6k R,c L=48πK+6k R.(18) Eqs.(11),(12),(18)give the desired expression for the entropy of a BTZ black hole in terms of the coeﬃcients of the gauge and gravitational Chern-Simons terms.By our previous argument addition of higher derivative terms do not change this result as long as they respect(0,4)supersymmetry.Since the crux of our argument has been the relationship between non-renormalization of the boundary S-matrix and the non-renormalization of the classical action,we shall now elaborate on this by examining how this works for the gauge sector of the theory.In this case the Chern-Simons theory has equation of motion F MN=0whereF MN≡∂[M A N]+[A M,A N](19) is the gaugeﬁeld strength.Any additional gauge invariant term in the action will involve the gaugeﬁeld strength and hence will vanish when F MN=0.A standard argument then shows that such terms can be removed from the action using aﬁeld redeﬁnition.We shall now see how the vanishing of the additional terms in the action for F=0 is related to the non-renormalization of the boundary S-matrix.For this weﬁrst review the computation of the boundary S-matrix from pure Chern-Simons theory.We begin by writing the Euclidean AdS3metric in the Poincare patchl2ds2=The gauge ﬁeld action in the Euclidean space takes the form 5S gauge =−i k R3A M A N A P +k RδA (0)a 1z ( z 1)···δA (0)a n z ( z n )e −I [A (0)z ]A (0)z ( z )=0,(24)where ¯J a ( z )are the SU (2)currents of the CFT at the boundary and the A a M are deﬁned throughA M =12(d ΦΦ−Φd Φ)+ (26)whereΦ( z ,x 0)= d 2wK ( z ,x 0; w )B (0)z ( w),(27)K ( z ,x 0; w )=1(x 0)2+|z −w |2 ,(28)and B(0)z is chosen such that(26)satisﬁes the boundary condition(23).Eq.(28)giveslimx0→0∂z K( z,x0; w)=δ(2)( z− w),(29)lim x0→0∂¯z K( z,x0; w)=−1(¯z−¯w)2.(30)Using eqs.(26)-(29)weﬁnd that A z(x0=0, z)is equal to B(0)z( z)toﬁrst order in an expansion in a power series in B(0)z.Thus to this order(23)is satisﬁed for B(0)z=A(0)z. The higher order contributions to B(0)z can be obtained by iteratively solving eq.(23)with the ansatz for A M given in(26)-(28).The result isB(0)z ( z)=A(0)z( z)+1(¯z−¯w) A(0)z( w)A(0)z( z)−A(0)z( z)A(0)z( w) + (31)where···denote higher order terms.We can now substitute the solution given in(26)-(31) into(22)to evaluate the on-shell action I[A(0)z].Evaluation of the boundary contribution is straightforward.In evaluating the contribution from the Chern-Simons term weﬁrst use the equation of motion to express it asik R3ǫMNP∂t T r U−1t∂M U t U−1t∂N U t U−1t∂P U t =ǫMNP∂M U−1t∂t U t U−1t∂N U t U−1t∂P U t (33) and that U−1t∂t U t=Φ,we can express(32)as a pure boundary term−k R4π2 d2zd2w(¯z−¯w)−2A(0)a z( z)A(0)a z( w)−k R2δa1a2(¯z1−¯z2)−2,(36)9¯J a1(¯z1)¯J a2(¯z2)¯J a3(¯z3) =−ik R[5]P.Kraus and rsen,JHEP0509,034(2005)[arXiv:hep-th/0506176].[6]P.Kraus and rsen,JHEP0601,022(2006)[arXiv:hep-th/0508218].[7]S.N.Solodukhin,Phys.Rev.D74,024015(2006)[arXiv:hep-th/0509148].[8]B.Sahoo and A.Sen,JHEP0607,008(2006)[arXiv:hep-th/0601228].[9]P.Kraus,arXiv:hep-th/0609074.[10]Y.Tachikawa,Class.Quant.Grav.24,737(2007)[arXiv:hep-th/0611141].[11]S.Deser,R.Jackiw and S.Templeton,Annals Phys.140,372(1982)[Erratum-ibid.185,406.1988APNYA,281,409(1988APNYA,281,409-449.2000)];Phys.Rev.Lett.48,975(1982).[12]J.M.Maldacena and A.Strominger,JHEP9812,005(1998)[arXiv:hep-th/9804085].[13]R.M.Wald,Phys.Rev.D48,3427(1993)[arXiv:gr-qc/9307038].[14]T.Jacobson,G.Kang and R.C.Myers,Phys.Rev.D49,6587(1994)[arXiv:gr-qc/9312023].[15]V.Iyer and R.M.Wald,Phys.Rev.D50,846(1994)[arXiv:gr-qc/9403028].[16]T.Jacobson,G.Kang and R.C.Myers,arXiv:gr-qc/9502009.[17]J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998)[Int.J.Theor.Phys.38,1113(1999)][arXiv:hep-th/9711200].[18]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,Phys.Lett.B428,105(1998)[arXiv:hep-th/9802109].[19]E.Witten,Adv.Theor.Math.Phys.2,253(1998)[arXiv:hep-th/9802150].[20]A.Dabholkar, A.Sen and S.P.Trivedi,JHEP0701,096(2007)[arXiv:hep-th/0611143].[21]K.Hanaki,K.Ohashi and Y.Tachikawa,Prog.Theor.Phys.117,533(2007)[arXiv:hep-th/0611329].11[22]A.Castro,J.L.Davis,P.Kraus and rsen,arXiv:hep-th/0702072.[23]A.Castro,J.L.Davis,P.Kraus and rsen,arXiv:hep-th/0703087.[24]M.Alishahiha,arXiv:hep-th/0703099.[25]M.Nishimura and Y.Tanii,Int.J.Mod.Phys.A14,3731(1999)[arXiv:hep-th/9904010].[26]J.R.David,Mod.Phys.Lett.A14,1143(1999)[arXiv:hep-th/9904068].[27]A.Achucarro and P.K.Townsend,Phys.Lett.B180,89(1986).[28]A.Achucarro and P.K.Townsend,Phys.Lett.B229,383(1989).[29]A.Giacomini,R.Troncoso and S.Willison,arXiv:hep-th/0610077.[30]E.Witten,Commun.Math.Phys.121,351(1989).[31]E.Witten,Nucl.Phys.B311,46(1988).[32]P.Kraus and rsen,JHEP0701(2007)002[arXiv:hep-th/0607138].[33]S.Elitzur,G.W.Moore,A.Schwimmer and N.Seiberg,Nucl.Phys.B326,108(1989).12。

Gauge-fixing approach to lattice chiral gauge theories

a r X i v :h e p -l a t /9804015v 1 11 A p r 1998Gauge-Fixing Approach to Lattice ChiralGauge TheoriesWolfgang Bock 1,Maarten F.L.Golterman 2,Yigal Shamir 31Institute of Physics,Humboldt University,Invalidenstr.110,10115Berlin,Germany 2Department of Physics,Washington University St.Louis,MO 63130,USA 3School of Physics and Astronomy,Tel-Aviv University Ramat Aviv,69978Israel Abstract:We review the status of our recent work on the gauge-ﬁxing approach to lattice chiral gauge theories.New numerical results in the reduced version of a model with a U(1)gauge symmetry are presented which strongly indicate that the factorization of the correlation functions of the left-handed neutral and right-handed charged fermion ﬁelds,which we established before in perturbation theory,holds also nonperturbatively.1IntroductionThe nonperturbative formulation of chiral gauge theories on the lattice is a long-standing and,to date,still unsolved problem.The local chiral gauge invariance on the lattice is broken for non-zero values of the lattice spacing,even in models with an anomaly-free spectrum.The failure of many proposals in the past was connected to the fact that the strongly ﬂuctuating longitudinal gauge degrees of freedom alter the fermion spectrum leading to vector-like instead of chiral gauge theories.Already several years ago the Rome group proposed to use perturbation theory in the contin-uum as guideline and to transcribe the gauge-ﬁxed continuum lagrangian of a chiral gauge theory to the lattice [1].The hope is that,for a smooth gauge-ﬁxing condition (like e.g.the Lorentz gauge),preferably smooth gauge conﬁgurations are selected from a gauge orbit,whereas rough gauge ﬁeld conﬁgurations are suppressed by a small Boltzmann weight.The gauge-ﬁxed model in the continuum is invariant under BRST symmetry.This symmetry is broken on the lattice,but the hope is that it canbe restored by adding all relevant and marginal counterterms which are allowed bythe exact symmetries of the lattice theory.Concrete lattice implementations of thenonlinear gauge µ{∂µAµ+A2µ}=0and of the Lorentz gauge µ∂µAµ=0were ﬁrst given in refs.[2]and[3].As aﬁrst step we have studied a model with U(1)gauge symmetry.The im-portant advantage of the abelian case is that the ghost sector drops out from the path integral and no Fadeev–Popov term needs to be included in the action.As a second simpliﬁcation we included only a gauge-boson mass counterterm which is the only dimension-two counterterm and ignored all dimension-four counterterms.We notice already here that the lattice action can be formulated such that a fermion-mass counterterm(which is the only dimension-three counterterm)needs not to be added to the action.The coeﬃcient of the gauge-boson mass counterterm has to be tuned such that the photons are massless in the continuum limit(CL).This value of the coeﬃcient corresponds,for suﬃciently small values of the gauge coupling,to a continuous phase transition between a“ferromagnetic”(FM)and a novel,so-called ferromagnetic“directional”(FMD)phase.The CL has to be taken from the FM side of the phase transition where the photon mass is larger than zero and the expectation value of the vectorﬁeld vanishes.The FMD phase is characterized by a non-vanishing condensate of the vectorﬁeld and a broken hypercubic rotation invariance.Motivated by previous investigations of lattice chiral gauge theories weﬁrst re-stricted the gaugeﬁelds to the trivial orbit,where only the dynamics of the longi-tudinal gauge degrees of freedom is taken into account[4].We shall refer to this model in the following as the“reduced model.”In perturbation theory and by a high-statistics numerical simulation we could show in this reduced model that1.the U(1)L,global⊗U(1)R,global is restored on the FM-FMD phase transition whichis a central prerequisite for the construction of a chiral gauge theory on the lattice,and2.the fermion spectrum in the CL contains only the desired left-handed chargedfermion and a right-handed neutral“spectator”fermion.Theﬁrst statement applies also to the strongly coupled symmetric phase of the Smit-Swift model in which the unwanted species doublers were shown to decouple.The fermion spectrum in this phase however contains only a neutral Dirac fermion which decouples completely from the gaugeﬁelds when they are turned on back again. Later it was argued that a lattice chiral gauge theory can indeed not be deﬁned within a symmetric phase or on its boundaries[5].The outline of the rest of the paper is as follows:In Sect.2.we introduce the fully gauged U(1)model and its reduced version.In Sect.3we review our previous results for the phase diagram and the fermion spectrum in the reduced model.Our new numerical results which further substantiate the above statement about the fermion spectrum are presented in Sect.4.In the last section,we brieﬂy summarize our results and give a brief outlook to future projects.2The ModelThe fully gauged U(1)lattice model is deﬁned by the following action S V =12ξg 2 x,y,z 2xy (U )2yz (U )− x B 2x (V (U )) −κ µx(U µx +U †µx )+ x,y 24 µ(V µx −ˆµ+V µx )2,V µx =Im U µx .(2)The action in eq.(1)includes the following terms(from the left to the right):the usual plaquette term (∝1/g 2),the Lorentz gauge-ﬁxing term (∝1/2ξg 2),the gauge-boson mass counterterm (∝κ),the “naive”kinetic term for the fermions and the Wilson term (∝r )which we use to remove species doublers.U µx =exp(igA µx )is the lattice link variable,U µνx the plaquette variable,g is the gauge coupling,ξis the gauge-ﬁxing parameter,r is the Wilson parameter and P L ,R =12ξd 4x (∂µA µ)2,and has an absolute minimum at U µx =1,validating weak coupling perturbation theory [2,3].We also notice that the fermionic part of (1)is invariant under the shift symmetry ψR →ψR +ǫR ,ψR +ψD ψe −S V (U ;ψL ,ψR )= D φD U D2xµ ψx +µγµψx −r ((ψx (φ†x P L +φx P R )ψx ) ,(4)where V rµx=Im(φ†xφx+µ).The reason why this reduced model is of interest is that itshould lead in the CL to a theory of free chiral fermions in the correct representationof the gauge group.This is a necessary condition for the construction of chiral gauge theory with unbroken symmetry.The failure of many previous proposals of chiralgauge theories,like e.g.of the Smit-Swift model,was connected to the fact that the fermion spectrum is altered by the stronglyﬂuctuating gauge degrees[4].In thefollowing sections we shall reexamine this important question in our model.3Phase Diagram and Fermion Spectrum of the Reduced ModelLet’sﬁrst consider the phase diagram of the pure bosonic part of the action(4) which only includes the gauge-ﬁxing term and the gauge-boson mass counterterm.The(κ, κ)-phase diagram of this higher derivative scalarﬁeld theory contains at large κa FM phase atκ>κc,where V rµx =0and φ >0,and a FMD phase atκ<κc with V rµx =0[2].Both phases are separated by a continuous phase transition atκc( κ)[7].To one-loop order weﬁndκc=0.02993+O(1/ κ)[7].As explained in Sect.1,in the fully gauged modelκhas to be tuned from the FM side towards this phase transition in order to obtain massless photons.In the reduced model we have computed the order parameter φ in the FM phase,both in perturbation theory in1/ κand also numerically,andﬁnd that it vanishes in the limitκցκFM−FMD [7,8].This phenomenon is associated with the1/(p2)2propagator for theφ-ﬁeld ﬂuctuations.The vanishing of φ implies that the U(1)L,global⊗U(1)R,global symmetry which is broken to its diagonal subgroup in the FM phase,is restored on the FM-FMD phase transition line,an essential prerequisite for the construction of a chiral gauge theory with unbroken gauge symmetry.We now introduce the four fermion operatorsψn R=ψR,ψn L=φ†ψL,ψc L=ψL and ψc R=φψR.Theﬁelds with the superscripts c(charged)and n(neutral)transform nontrivially under the U(1)L,global and U(1)R,global subgroups respectively.We have calculated the neutral and charged fermion propagators both to one-loop order in perturbation theory in1/ κ[8],and also numerically[9].We could show that1.) the unwanted species doublers decouple in all cases and2.)only theψn R-andψc L-propagators exhibit in the CL isolated poles at p=(0,0,0,0),which correspond to the desired massless fermion states.We found that non-analytic terms occur in the ψn L-andψc R-propagators and that there are no poles in these channels in the CL.If theψn R-andψc L fermions are the only free fermions that exist in the CL of the reduced model,we would expect that theψn L-andψc R-correlation functions in coordinate space factorize for suﬃciently large separations|x−y|in the following manner ψn L,xψc L,y φ†xφy , ψc R,xψn R,y φxφ†y .(5) We were able to show in one-loop perturbation in1/ κthat the two relations in eq.(5)hold both in the FM phase and also in the CL,i.e.forκցκFM−FMD[8].It is important to conﬁrm these relations also nonperturbatively.Figure1:The ratios R R(a)and R L(b)as function of t at( κ,κ)=(0.2,0.3)(r= 1).The lattice size is6324.The data points are connected by solid lines and the horizontal dotted lines are to guide the eyes.4Numerical ResultsTo this end we have performed a quenched simulation at the point( κ,κ)=(0.2,0.3) in the FM phase.We set r=1and determined theﬁve correlation functions ψn L,xψn R,y , ψc L,xψc R,y and φ†xφy on an cylindrical lattice of size L3T with L=6,T=24.For the fermionﬁelds we used antiperiodic(peri-odic)boundary conditions in the temporal(spatial)directions,whereas for the scalar ﬁeld we used periodic boundary conditions in all directions.To compute the four fermionic correlation functions numerically we have to em-ploy point sources and sinks which implies that a very high statistics is required to obtain a satisfactory signal to noise ratio.We sety(x,t)=x1ˆ1+x2ˆ2+x3ˆ3+mod(x4+t,T)ˆ4,(6) with t=1,...,T−1.For a given scalarﬁeld conﬁguration we have randomly picked a source point on each time slice and averaged over the resulting T correlation functions.For the computation of the fermionic correlation functions we used in total 1300scalar conﬁgurations which were generated with a5-hit Metropolis algorithm and,in order to reduce autocorrelation eﬀects,were separated by2000successive Metropolis sweeps.In the case of the bosonic correlation function we summed,for a given conﬁgura-tion,over all lattice sites x, (1/(L3T) xφ†xφy(x,t) ,where y(x,t)is given again by eq.(6),and measured the bosonic correlation function on each of the1300×2000 conﬁgurations.To check if relation(5)holds also nonperturbatively we have computed the tworatiosR L=ψn L,xψc L,xψc R,yψn R,y φ†xφy,(7)which should approach a constant at suﬃciently large separations t.The two ratios are displayed inﬁg.1as a function of t.The two graphs clearly show that both ratios start toﬂatten oﬀat t≈5and are,within errors,indeed constant at larger separations.The fact that R R=1(for larger t)is consistent with shift symmetry. 5ConclusionThe quenched results presented in the last section suggest that the factorization of theψn L-andψc R-correlation functions(cf.eq.(5))which we established before in 1-loop perturbation theory(cf.ref.[8])remain valid also nonperturbatively.As future direction of research we plan to study the U(1)model with full dy-namical gaugeﬁelds.This requires the fermion representation to be anomaly free. We furthermore want to extend the gauge-ﬁxing approach to the nonabelian gauge theories.This is a non-trivial issue,because it is not known whether the BRST for-mulation of gauge theories can be deﬁned consistently beyond perturbation theory. Acknowledgements:WB is supported by the Deutsche Forschungsgemeinschaft under grant Wo389/3-2,MG by the US Department of Energy as an Outstanding Junior Investigator,and YS by the US-Israel Binational Science Foundation,and the Israel Academy of Science.References[1]A.Borelli,L.Maiani,G.-C.Rossi,R.Sisto,M.Testa,Phys.Lett.B221(1989)360;Nucl.Phys.B333(1990)335.[2]Y.Shamir,Phys.Rev.D57(1998)132.[3]M.Golterman,Y.Shamir,Phys.Lett.B399(1997)148.[4]Y.Shamir,Nucl.Phys.B(Proc.Suppl.)47(1996)212.[5]Y.Shamir,Phys.Rev.Lett.71(1993)2691.[6]M.Golterman,D.Petcher,Phys.Lett.B225(1989)159.[7]W.Bock,M.Golterman,Y.Shamir,hep-lat/9708019.[8]W.Bock,M.Golterman,Y.Shamir,hep-lat/9801018.[9]W.Bock,M.Golterman,Y.Shamir,hep-lat/9709154.。

Abelian projection and studies of gauge-variant quantities in lattice QCD without gauge fix

a rXiv:h e p-la t/961129v127Nov1996Abelian projection and studies of gauge-variant quantities in the lattice QCD without gauge ﬁxing Sergei V.SHABANOV 1Institute for Theoretical Physics,Free University of Berlin,Arnimallee 14,WE 2,D-14195,Berlin,Germany Abstract We suggest a new (dynamical)Abelian projection of the lattice QCD.It contains no gauge condition imposed on gauge ﬁelds so that Gribov copying is avoided.Conﬁgurations of gauge ﬁelds that turn into monopoles in the Abelian projection can be classiﬁed in a gauge invariant way.In the continuum limit,the theory respects the Lorentz invariance.A similar dynamical reduction of the gauge symmetry is proposed for studies of gauge-variant correlators (like a gluon propagator)in the lattice QCD.Though the procedure is harder for numerical simulations,it is free of gauge-ﬁxing artifacts,like the Gribov horizon and copies.1.One of the important features of the QCD conﬁnement is the existence of a stable chromoelectrical ﬁeld tube connecting two color sources (quark and antiquark).Numerical studies of the gluon ﬁeld energy density between two color sources leave no doubt that such a tube exists.However,a mechanism which could explain its stability is still unknown.It is believed that some speciﬁc conﬁgurations (or excitations)of gauge ﬁelds are re-sponsible for the QCD conﬁnement,meaning that they give a main contributions to the QCD string tension.Numerical simulations of the lattice QCD shows that Abelian (com-mutative)conﬁgurations of gauge potentials completely determine the string tension in the full non-Abelian gauge theory [1].This phenomenon is known as the Abelian dominance.Therefore one way of constructing eﬀective dynamics of the conﬁgurations relevant to the QCD conﬁnement is the Abelian projection [2]when the full non-Abelian gauge group SU(3)is restricted to its maximal Abelian subgroup (the Cartan subgroup)U(1)×U(1)by a gauge ﬁxing.Though dynamics of the above gauge ﬁeld conﬁguration cannot be gauge dependent,a right choice of a guage condition may simplify its description.There is a good reason,supported by numerical simulations [3],[4],to believe that the sought conﬁgurations turn into magnetic monopoles in the eﬀective Abelian theory,and the conﬁnement can be due to the dual mechanism [5]:The Coulomb ﬁeld of electric charges is squized into a tube,provided monopole-antimonopole pair form a condensate like the Cooper pairs in superconductor.It is important to realize that the existence of monopoles in the eﬀective Abelian theory is essentially due to the gauge ﬁxing,in fact,monopoles are singularities of thegaugeﬁxing.Note that monopoles cannot exist as stable excitations in pure gauge the-ory with simply connected group like SU(3).Since the homotopy groups of SU(3)and of U(1)×U(1)are diﬀerent(the one of SU(3)is trivial),a gauge condition restricting SU(3)to U(1)×U(1)should have singularities which can be identiﬁed as monopoles[2].A dynamical question is to verify whether all conﬁgurations of non-Abelian gaugeﬁelds relevant to the conﬁnement(in the aforementioned sense)are”mapped”on monopoles of the Abelian theory(the monopole domimance[4]).It appears that monopole dynam-ics may depend on the projection recipe[6].There are indications that some Abelian projections exhibit topological singularities other than magnetic monopoles[7].Though the lattice QCD is,up to now,the only relible tool for studying monopole dynamics,the true theory must be continuous and respect the Lorentz invariance.In this regard,Abelian projections based on Lotentz invariant gauge conditions play a dis-tinguished role.For example,the gauge can be chosen as follows D HµA offµ=0where D Hµ=∂µ+igA Hµ,A Hµare Cartan(diagonal)components of guage potentials Aµ,while A offµare its non-Cartan(oﬀ-diagonal)components.This gauge restricts the gauge sym-metry to the maximal Abelian(Cartan)subrgoup and is manifestly Lorentz invariant. The lattice version of the corresponding Abelian theory is known as the maximal Abelian projection.The above homotopy arguments can be implemented to this gauge to show that it has topological singularities and Gribov’s copying[9](in the continuum theory, zero boundary conditions at inﬁnity have to be imposed[10]).The Gribov copying makes additional diﬃculties for describing monopole dynamics(even in the lattice gluodynamics [11]).In this letter,a new(dynamical)Abelian projection is proposed.It involves no gauge condition to be imposed on gaugeﬁelds.The eﬀective Abelian theory appears to be non-local,though it can be made local at the price of having some additional(ghost)ﬁelds.All conﬁgurations of gaugeﬁelds that turn into magnetic monopoles in the eﬀective Abelian theory are classiﬁed in a gauge invariant way.The eﬀective Abelian theory fully respects the Lorentz symmetry and the Gribov problem is avoided.Another important aspect of the QCD conﬁnement is the absence of propagating color charges,meaning that a nonperturbative propagator of colored particles,gluons or quarks, has no usual poles in the momentum space.It has been argued that such a behavior of a gluon propagator in the Coulomb gauge could be due to an inﬂuence of the so called Gribov horizon on long-waveﬂuctuations of gaugeﬁelds[9],[12].The result obviously depends on the gauge chosen,which makes it not very reliable.The situation looks more controversial if one recalls that a similar qualitative behavior of the gluon propagator has been found in the study of Schwinger-Dyson equations[13]. In this approach,the Gribov ambiguities have not been accounted for.So,the speciﬁc pole structure of the gluon propagator occurred through a strong self-interaction of gauge ﬁelds.In this letter,we would also like to propose a method for how to study gauge-variant quantities,like a gluon propagator,in the lattice QCD,avoiding any explicit gaugeﬁxing. The method is,hence,free of all the aforementioned gaugeﬁxing artifacts.It gives a hope that dynamical contributions(self-interaction of gaugeﬁelds)to the pole structure of the gluon propagator can be separated from the kinematical(gauge-ﬁxing)ones.2.To single out monopoles in non-Abelian gauge theory,oneﬁxes partially a gauge so that the gauge-ﬁxed theory possesses an Abelian gauge group being a maximal Abelian subgroup of the initial gauge group.The lattice formulation of the Abelian projection has been given in[8].The idea is to choose a function R(n)of link variables Uµ(n),n runs over lattice sites, such thatR(n)→g(n)R(n)g−1(n)(1) under gauge transformations of the link variablesUµ(n)→g(n)Uµ(n)g−1(n+ˆµ),(2) where g(n)∈G,G is a compact gauge group,andˆµis a unit vector in theµ-direction.A gauge is chosen so that R becomes an element of the Cartan subalgebra H,a maximal Abelian subalgebra of a Lie algebra X of the group G.In a matrix representation,the gauge condition means that oﬀ-diagonal elements of R are set to be zero.Clearly,the gaugeﬁxing is not complete.A maximal Abelian subgroup G H of G remains as a gauge group because the adjoint action(1)of G H leaves elements R∈H untouched.A conﬁguration Uµ(n)contains monopoles if the corresponding matrix R(n)has two coinciding eigenvalues.So,by construction,dynamics of monopoles appears to be gauge-dependent,or projection-dependent.It varies from gauge to gauge,from one choice of R to another[6].Yet,the monopole singularities are not the only ones in some Abelian projections[7].In addition,Abelian projections may suﬀer oﬀthe Gribov ambiguities [11].To restrict the full gauge symmetry to its maximal Abelian part and,at the same time, to avoid imposing a gauge condition on link variables,we shall use a procedure similar to the one discussed in[14]in the framework of continuumﬁeld theory.A naive continuum limit of our procedure poses some diﬃculties.To resolve them,a corresponding operator formalism has to be developed.It has been done in[15]for a suﬃciently large class of gauge theories.Consider a complex Grassmannﬁeldψ(n)(a fermion ghost)that realizes the adjoint representation of the gauge group:ψ(n)→g(n)ψ(n)g−1(n),(3)ψ∗(n)→g(n)ψ∗(n)g−1(n).(4) Let the fermion ghost be coupled to gaugeﬁelds according to the actionS f= n,µtrDµψ∗(n)Dµψ(n),(5)where Dµψ(n)=ψ(n+ˆµ)−U−1µ(n)ψ(n)Uµ(n)is the lattice covariant derivative in the adjoint representation.We assume thatψ(n)=ψi(n)λi,whereλi is a matrix represen-tation of a basis in X normalized as trλiλj=δij,andψi(n)are complex Grassmann variables.The partition function of the fermion ghostﬁeld readsZ f(β)= n(dψ∗(n)dψ(n))e−βS f=detβD TµDµ,(6)where the integration over Grassmann variables is understood,and D Tµdenotesa trans-position of Dµwith respect to a scalar product induced by n,µtr in(5).Note that the action(5)can be written in the form S f= ψ∗D TµDµψ.Consider a pair of real Lie-algebra-valued scalarﬁeldsϕ(n)andφ(n)(boson ghosts) with an actionS b=1(2π)dim G e−βS b=(detβD TµDµ)−1.(10) We have the identityZ b(β)Z f(β)=1.(11) By making use of this identity,the partition function of gaugeﬁelds can be transformed to the formZ Y M(β)=v−L G µ,n dUµ(n)e−βS W Z b(β)Z f(β)=(12)=v−L G D UµDψ∗DψDϕDφe−β(S W+S b+S f),(13)where S W is the Wilson action of gaugeﬁelds,v G a volume of the group manifold G,L a number of lattice sites,and D denotes a product of correspondingﬁeld diﬀerentials over lattice sites.The eﬀective actionS eff=S W+S b+S f(14) is invariant under gauge transformations(2)–(4)and(8),(9).The factor v−L G is included to cancel the gauge group volume factorizing upon the integration overﬁeld conﬁgurations in(13).Now we may take the advantage of having scalarﬁelds in the adjoint representation and restrict the gauge symmetry to the Cartan subgroup without imposing gauge conditions on the link variables.We make a change of the integration variables in(13)φ(n)=˜g(n)h(n)˜g(n)−1,(15) where˜g(n)belongs to the coset space G/G H,dim G/G H=dim G−dim G H,and h(n)∈H.Other newﬁelds denoted˜Uµ(n),˜ϕand˜ψ∗,˜ψare deﬁned as the corresponding gauge transformations of the initialﬁelds with g(n)=˜g−1(n).No restriction on their values is imposed.Relation(15)determines a one-to-one correspondence between old and new variables if and only if˜g(n)∈G/G H and h(n)∈K+,where K+is the Weyl chamber in H.An element h of the Cartan subalgebra H belongs to the Weyl chamber K+⊂H if for any simple rootω,(h,ω)>0;(,)stands for an invariant scalar product in X.In a matrix representation of X,it is proportional to tr(see[16],pp.187-190).With the help of the adjoint transformation,any element of a Lie algebra can be brought to the Cartan subalgebra.Since the Cartan subalgebra is invariant under the adjoint action of the Cartan subgroup,˜g(n)must be restricted to the coset G/G H.There are discrete transformations in G/G H which form the Weyl group W[16].Any element of W is a composition of reﬂections in hyperplanes orthogonal to simple roots in H.Its action maps H onto H itself.The Weyl group is a maximal isomorphism group of H[16].Therefore, a one-to-one correspondence in(15)is achieved if h(n)∈H/W≡K+.Due to the gauge invariance of both the measure and exponential in(13),the integral over group variables˜g(n)is factorized and yields a numerical vector that,being divided by v L G,results in(2π)−Lr,r=dim H=rank G.This factor is nothing but a volume of the Cartan gauge group G H.The integration over h(n)inquires a nontrivial measure,and the integration domain must be restricted to the Weyl chamber K+.So,in(13)we havev−1G dφ(n)=(2π)−r K+dh(n)µ(n).(16) The measure has the form[17]µ(n)= α>0(h(n),α)2,(17)whereαranges all positive roots of the Lie algebra X.The Cartan subalgebra is isomor-phic to an r-dimensional Euclidean space.The invariant scalar product can be thought as an ordinary vector scalar product in it.Relative orientations and norms of the Lie algebra roots are determined by the Cartan matrix[16].The integration measure for the otherﬁelds remains unchanged.For example,G=SU(2),then r=1,µ=h2(n)where h(n)is a real number because H SU(2)is isomorphic to a real axis.The Weyl chamber is formed by positive h(n).The su(3)algebra has two simple rootsω1,2(r=2).Their relative orientation is determined by the Cartan matrix,(ω1,ω2)=−1/2,|ω1,2|=1.The Weyl chamber is a sector on a plane(being isomorphic to H SU(3))with the angleπ/3.The algebra has three positive rootsω1,2andω1+ω2.So,the measure(17)is a polynom of the sixth order.Its explicit form is given by(28).Theﬁeld h(n)is invariant under Abelian gauge transformationsg H(n)h(n)g−1H(n)=h(n),g H(n)∈G H.(18) Therefore,after integrating out the coset variables˜g(n)in accordance with(16),we represent the partition function of Yang-Mills theory as a partition function of the eﬀective Abelian gauge theoryZ Y M(β)=(2π)−Lr D˜Uµe−βS W F(˜U),(19)whereF(˜U)=(detβD TµDµ)1/2 K+ n(dh(n)µ(n))e−βS H,(20)S H=1/2 n,µtr h(n+ˆµ)−˜U−1µ(n)h(n)˜Uµ(n) 2.(21)To obtain(19),we have done the integral over both the Grassmann variables and the boson ghostﬁeld˜ϕ(n),which yields(detβD TµDµ)1/2.The function F(˜U)is invariant only with respect to Abelian gauge transformations,˜Uµ(n)→g H(n)˜Uµ(n)g−1H(n+ˆµ).It provides a dynamical reduction of the full gauge group to its maximal Abelian subgroup.Since no explicit gauge condition is imposed on the link variables˜Uµ(n),the theory do not have usual gaugeﬁxing deceases,like the Gribov copies or horizon.We shall call the Abelian projection thus constructed a dynamical Abelian projection.3.Making a coset decomposition of the link variables[8]˜Uµ(n)=U Hµ(n)U chµ(n),(22) where U Hµ(n)=exp u Hµ(n),u Hµ(n)∈H and U chµ(n)=exp u chµ(n),u chµ(n)∈X⊖H,we conclude that lattice Yang-Mills theory is equivalent to an Abelian gauge theory with the actionS A=S W−β−1ln F.(23) The link variables U chµ(n)play the role of chargedﬁelds,while U Hµ(n)represents”electro-magnetic”ﬁelds.In the naive continuum limit,U Hµbecome Abelian potentialsU Hµ(n)→exp n+ˆµndxµA Hµ,A Hµ∈H.(24)Note that theﬁeld h(n)carries no Abelian charge and does not interact with U Hµas easily seen from(22)and(21)because(U Hµ)−1(n)h(n)U Hµ(n)=h(n).Bearing in mind results on simulations of the Polyakov loop dynamics on the lattice, one should expect that the Coulombﬁeld of charges in the eﬀective Abelian theory is squeezed into stable tubes connecting opposite charges.A mechanism of the squeezing has to be found from a study of dynamics generated by(23).First,one should verify if the dual mechanism can occur in the eﬀective Abelian theory.In our approach,conﬁgurations U Hµ(n)containing monopoles can exist.Kinematical arguments for this conjecture are rather simple.Let G be SU(N).In a matrix represen-tation,the change of variables(15)becomes singular at lattice sites where theﬁeldφ(n) has two coinciding eigenvalues.This condition implies three independent conditions on components ofφ(n)which can be thought as equations for the singular sites.At each moment of lattice time,these three equations determine a set of spatial lattice vertices (locations of monopoles).Therefore on a four-dimensional lattice,the singular sites form world-lines which are identiﬁed with world-lines of monopoles[2].The new link variables˜Uµ(n)=˜g(n)Uµ(n)˜g−1(n+ˆµ)(25)inquires monopole singularities via˜g(n).Their density can be determined along the lines given in[8].So,monopole dynamics is the dynamics of conﬁgurationsφ(n)with two equal eigenval-ues in the full theory(13).If such conﬁgurations are dynamically preferable,then one can expect that in the dynamical Abelian projection,eﬀective monopoles and antimonopoles form a condensate.All monopole-creating conﬁgurations of the scalarﬁeldφ(n)can easily be classiﬁed in a gauge invariant way.First of all we observe that the change of variables(15)is singular if its Jacobian vanishes nµ(n)=0.(26) We have to classify all conﬁgurationsφ(n)which lead toµ(n)=0.The polynom(17)is invariant with respect to the Weyl group.According to a theorem of Chevalley[16],any polynom in H invariant with respect to W is a polynom of basis(elementary)invariant polynoms tr h l(n)with l=l1,l2,...,l r being the orders of independent Casimir operators of G[16].Therefore,µ(n)=P(tr h l1(n),tr h l2(n),...,tr h l r(n))==P(trφl1(n),trφl2(n),...,trφl r(n))=0.(27) Solutions of this algebraic equation determine all conﬁgurationsφ(n)which will create monopoles in the dynamical Abelian projection(19).For G=SU(3),we have r=2,l1= 2,l2=3and[18]1µsu(3)(n)=As follows from(21)and(22),the Abelianﬁeld U Hµ(n)and the Cartanﬁeld h(n) are decoupled because[U Hµ(n),h(n)]=0.So,in the full theory,we deﬁne Abelian link variables by the relation[Uφµ(n),φ(n)]=0.(29) The coset decomposition assumes the formUµ(n)=Uφµ(n)U chµ(n).(30) One can regard it as a deﬁnition of chargedﬁelds U chµ(n)for given Uµ(n)andφ(n).Consider a vector potential corresponding to Uφµ(n)as determined by(24).It has theformAφµ(n)=rα=1Bαµ(n)eφα(n),(31)where Bαµ(n)are real numbers,and Lie algebra elements eφα(n)form a basis in the Cartan subalgebra constructed in the following wayeφα=λi trλiφlα−1.(32) It is not hard to be convinced that[18][eφα,eφβ]=0.(33) Since for any group G one of the numbers lαis equal to2,one of the elements(32) coincides withφitself.The elements(32)are linearly independent in X becausedet Pαβ≡det tr eφαeφβ=const·P.(34) So,a generic elementφof X has a stationary group Gφ⊂G with respect to the adjoint action of G in X,gφφg−1φ=φ,gφ∈Gφ.This stationary group is isomorphic to the Cartan subgroup G H.All linear combinations of the elements(32)form a Lie algebra of Gφ∼G H.In fact,the basis(32)can be constructed without an explicit matrix representation ofλi.We recall that for each compact simple group G and its Lie algebra X,there existr=rank G=dim H symmetrical irreducible tensors of ranks lα,d i1,i2,...,i lα,invariant withrespect to the adjoint action of G in X.Clearly,(eφα)i=d ij1...j lα−1φj1···φjlα−1.Now it is easy to see that the Abelian potentials Bφµ(n)are singular at lattice sites whereφ(n)satisﬁes(27).Indeed,from(31)we getBαµ(n)=Pαβ(n)tr eφβ(n)Aφµ(n),(35) where PαβPβγ=δαγ.The determinant of the matrix Pαβ(n)vanishes at the sites where µ(n)=P(n)=0.At these sites,the inverse matrix Pαβ(n)does not exist,and theﬁelds Bαµ(n)are singular.For unitary groups SU(N),lα=2,3,...,N,the singular sites form lines in the four-dimensional lattice[2],[8].These lines are world-lines of monopoles.5.The above procedure of avoiding explicit gaugeﬁxing can be implemented to re-move the gauge arbitrariness completely and,therefore to study gauge-variant correlators,like the gluon propagator,or some other quantities requiring gaugeﬁxing on the lattice [20].The advantage of dynamical gaugeﬁxing is that it is free of all usual gaugeﬁxing dynamical artifacts,Gribov’s ambiguities and horizon[14].It is also Lorentz covariant.Recent numerical studies of the gluon propagator in the Coulomb gauge[19]show that it can beﬁt to a continuum formula proposed by Gribov[9].The same predictions were also obtained in the study of Schwinger-Dyson equations where no eﬀects of the Gribov horizon have been accounted for[13].The numerical result does not exclude also a simple massive boson propagator for gluons[19].So,the problem requires a further investigation.Gaugeﬁxing singularities(the Gribov horizon)occur when one parametrizes the topo-logically nontrivial gauge orbit space by Cartesian coordinates.So,these singularities are pure kinematical and depend on the parametrization(or gauge)choice.They may,how-ever,have a dynamical evidence in a gauge-ﬁxed theory[21].For example,a mass scale determining a nonperturbative pole structure of the gluon propagator in the infrared region(gluon conﬁnement)arises from the Gribov horizon[9],[12]if the Lorentz(or Coulomb)gauge is used.From the other hand,no physical quantity can depend on a gauge chosen.There is no gauge-invariant interpretation(or it has not been found yet) of the above mass scale.That is what makes the gluon conﬁnement model based on the Gribov horizon looking unsatisfactory.Here we suggest a complete dynamical reduction of the gauge symmetry in lattice QCD,which involves no gauge condition imposed on gaugeﬁelds and,hence,is free of the corresponding kinematical artifacts.For the sake of simplicity,we discussﬁrst the gauge group SU(2).Consider two auxiliary(ghost)complexﬁeldsψandφ,Grassmann and boson ones,respectively.Let they realize the fundamental representation of SU(2),i.e.they are isotopic spinors.The identity(11)assumes the formZ b(β)Z f(β)= Dφ+DφDψ+Dψe−β(S b+S f)=1,(36) where S f= n(∇µψ)+∇µψand S b=1/2 n(∇µφ)+∇µφ,and the lattice covariant deriva-tive in the fundamental representation is deﬁned by∇µφ(n)=φ(n+ˆµ)−U−1µ(n)φ(n). Inserting the identity(36)into the integral representation of the Yang-Mills partition func-tion(12),we obtain an eﬀective gauge invariant action.The ghostﬁelds are transformed asφ(n)→g(n)φ(n)andψ(n)→g(n)ψ(n).In the integral(13),we go over to new variables to integrate out the gauge group volumedφ+(n)dφ(n)=v su(2)∞dρ(n)ρ3(n),(37)whereφ(n)=˜g(n)χρ(n),χ+=(10),ρ(n)is a real scalarﬁeld,and˜g(n)is a generic element of SU(2).A new fermion ghostﬁeld and link variables˜Uµare related to the old ones via a gauge transformation with g(n)=˜g−1(n).Since the eﬀective action is gaugeinvariant,the integral over˜g(n)yields the gauge group volume v Lsu(2).We end up withthe eﬀective theoryZ Y M(β)= D˜Uµe−βS W F(˜U),(38)F(˜U)=(detβ∇+µ∇µ)1/2∞0 ndρ(n)ρ3(n) e−βS(ρ),(39)S(ρ)=1/2 n,µ ρ(n+ˆµ)−χ+˜U−1µ(n)χρ(n) 2.(40)The function(39)is not gauge invariant and provides the dynamical reduction of the SU(2) gauge symmetry.A formal continuum theory corresponding to(38)has been proposed and discussed in[14].Expectation values of a gauge-variant quantity G(U)are determined byG(U) ≡ F(U)G(U) W= D Uµe−βS W F(U)G(U).(41)For example,for the gluon two-point correlator one sets G(U)=Aµ(n)Aµ′(n′)where the gluon vector potential on the lattice reads2iaAµ(n)=Uµ(n)−U+µ(n)−1Though the integration domain is restricted in the sliced path integral(20),this re-striction will disappear in the continuum limit because of contributions of trajectories reﬂected from the boundary∂K+[17],[18].It is rather typical for gauge theories that a scalar product for physical states involves an integration over a domain with boundaries which is embedded into an appropriate Euclidean space.The domain can even be com-pact as,for example,in two-dimensional QCD[22].In the path integral formulation,this feature of the operator formalism is accounted for by appropriate boundary conditions for the transition amplitude(or the transfer matrix)rather than by restricting the integration domain in the corresponding path integral[22],[23].In turn,the boundary conditions are to be found from the operator formulation of quantum gauge theory[18],[22],[23].So,a study of the continuum limit requires an operator formulation of the dynamical reduction of a gauge symmetry,which has been done in[15].The dynamical Abelian projection can be fulﬁlled in the continuum operator formal-ism.The whole discussion of monopole-like singular excitations given in sections3and4 can be extended to the continuum theory.So,it determines Lorentz covariant dynamics of monopoles free of gaugeﬁxing artifacts.To study monopole dynamics in the continuum Abelian gauge theory,one has to introduce monopole-carrying gaugeﬁelds[24].AcknowledgementI express my gratitude to F.Scholtz for valuable discussions on dynamical gauge ﬁxing,to A.Billoire,A.Morel and V.K.Mitrjushkin for providing useful insights about lattice simulations,and D.Zwanziger and M.Schaden for a fruitful discussion on the Gribov problem.I would like to thank J.Zinn-Justin for useful comments on a dynamical evidence of conﬁguration space topology in quantumﬁeld theory.I am very grateful to H.Kleinert for a stimulating discussion on monopole dynamics.References[1]T.Suzuki and I.Yotsuyanagi,Phys.Rev.D42(1990)4257.[2]G.’t Hooft,Nucl.Phys.B190[FS3](1981)455.[3]H.Shiba,T.Suzuki,Phys.Lett.B333(1994)461.[4]J.D.Stack,S.D.Neiman and R.J.Wensley,Phys.Rev.D50(1994)3399.[5]S.Mandelstam,Phys.Rep.23(1976)245;’t Hooft,in:High Energy Physics,ed.M.Zichichi(Editrice Compositori,Bologna, 1976).[6]see,for example,L.Del Debbio,A.Di Giacomo,G.Pafutti and P.Pier,Phys.Lett.B 355(1995)255.[7]M.N.Chernodub,M.I.Polikarpov and V.I.Veselov,Phys.Lett.B342(1995)303.[8]A.S.Kronfeld,G.Schierholz and U.-J.Wiese,Nucl.Phys.B293(1987)461.[9]V.N.Gribov,Nucl.Phys.B139(1978)1.[10]I.M.Singer,Commun.Math.Phys.60(1978)7.[11]S.Hioki,S.Kitahara,Y.Matsubara,O.Miyamura,S.Ohno and T.Suzuki, Phys.Lett.B,271(1991)201.[12]D.Zwanziger,Nucl.Phys.B378(1992)525.[13]M.Stingl,Phys.Rev.D34(1986)3863.[14]F.G.Scholtz and G.B.Tupper,Phys.Rev.D48(1993)1792.[15]F.G.Scholtz and S.V.Shabanov,Supersymmetric quantization of gauge theories, FU-Berlin preprint,FUB-HEP/95-12,1995.[16]S.Helgason,Diﬀerential Geometry,Lie Groups,and Symmetric Spaces(Academic Press,NY,1978).[17]L.V.Prokhorov and S.V.Shabanov,Phys.Lett.B216(1989)341;pekhi34(1991)108.[18]S.V.Shabanov,Theor.Math.Phys.78(1989)411.[19]C.Bernard,C.Parrinello and A.Soni,Phys.Rev.D49(1994)1585.[20]Ph.de Forcrand and K.-F.Liu,Nucl.Phys.B(Proc.Suppl.)30(1993)521.[21]V.G.Bornyakov,V.K.Mitrjushkin,M.M¨u ller-Preussker and F.Pahl,Phys.Lett.B 317(1993)596.[22]S.V.Shabanov,Phys.Lett.B318(1993)323.[23]S.V.Shabanov,Phys.Lett.B255(1991)398;Mod.Phys.Lett.A6(1991)909.[24]H.Kleinert,Phys.Lett.B293(1992)168.。

Higher dimensional geometries related to fuzzy odd-dimensional spheres

a r X i v :h e p -t h /0207111v 2 5 A u g 2002Brown Het-1309Higher dimensional geometriesrelated toFuzzy odd-dimensional spheresSanjaye Ramgoolam Department of Physics Brown University Providence,RI 02912ramgosk@We study SO (m )covariant Matrix realizations ofmi =1X 2i =1for even m as can-didate fuzzy odd spheres following hep-th/0101001.As for the fuzzy four sphere,these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset.The fuzzy S 2k −1is related to a higher dimensional cosetSO (2k )1.INTRODUCTIONFuzzy spheres provide many interesting solutions to Matrix Brane actions[1,2,3,4,5] They have also been suggested to play a role in the context of a spacetime explanation of the stringy exclusion principle[6,7,8,9].A class of odd-dimensional fuzzy spheres was deﬁned in[10]and studied in more detail in[11].The detailed SO(m)decomposition of the Matrix algebras related to the fuzzy S m was given in[11].For m>2the Matrix algebras contain more representations than is necessary to describe functions on the sphere,and a projection is needed to get the desired traceless symmetric representations.The geometry of general even dimensional fuzzy spheres was studied in[12]and it was found that the Matrix algebras related to the fuzzy sphere of2k dimensions approaches,in a limit of large matrices,the algebra of functions on a higher dimensional space SO(2k+1)/U(k),the coset of SO(2k+1)by the right action of U(k).This lead to the statement,for k≥2,thatﬂuctuations around the fuzzy sphere solution can be described equivalently in two ways.On the one hand there is an abelian theory on the higher dimensional coset.On the other hand there is a non-abelian theory on the sphere.Atﬁnite n this non-abelian theory on the fuzzy sphere has to be formulated on a commutative but non-associative algebra[11].The structure constants of the associative algebra for the higher dimensional coset have been described more explicitly in[13].The abelian theory on the higher dimensional coset,in the case k=2has been further studied in[14].A connection between the fuzzy four-sphere and the4D quantum Hall eﬀect[15]was found in[16]and higher even fuzzy spheres were used to explore6D and8D generalizations of the quantum Hall eﬀect.The extra degrees of freedom of the fuzzy4-sphere have been discussed in the context of uncertainty relations in[17].In this paper we explore the extent to which the picture of physics on even dimensional fuzzy spheres developed in[12]can be extended to fuzzy odd spheres.Weﬁnd that,as in the even case,there is an underlying higher dimensional geometry related to the Matrix algebra.For the case of the sphere S2k−1with symmetry SO(2k),the relevant geometry is SO(2k)is,however,diﬀerent from the role played by theU(k−1)×U(1)SO(2k+1)/U(k)in the case of even dimensional spheres.To explain this,we need torecall some details about the construction of even and odd fuzzy spheres.In all cases one starts with matrices X i,where i transforms in the vector of SO(m),andmi=1X2i=1(1.1)The X i are in End(R n),i.e Matrices which are transformations of the vector space R n, which is a representation of SO(m).The X i in(1.1)are related by a rescaling to the X i used in the bulk of this paper.R n is a vector space whose dimension N depends on an integer n.The precise dependence of N on n can be found in[11].The crucial diﬀerence between even and odd spheres is that,in the case of even spheres R n is an irreducible representation of SO(2k+1),whereas in the case of odd spheres R n is a direct sum of irreducible representations R+n and R−n of SO(2k),that is,R n=R+n⊕R−n.The matrices X i are maps from R+n to R−n and vice versa from R−n to R+n.For even fuzzy spheres,End(R n)becomes commutative in the large n limit and approaches the algebra of functions on a classical space SO(2k+1)/U(k).In the odd sphere case,both End(R+n)and End(R−n)become commutative in the large n limit and approach the algebra of functions on SO(2k)hasU(1)×U(1)a simpler description as S2×S2when we use the isomorphism SO(4)≡SU(2)×SU(2). Section3gives two ways to prove that End(R±n)approaches the algebra of functions on S2×S2in the large N limit.One uses the techniques of Kaluza-Klein reduction[18]or equivalently harmonic analysis on homogeneous spaces,as described for example in[19]. Another uses the analysis of the stabilizer group of a particular solution to the algebraic relations satisﬁed by the generators of the Matrix algebra.Section8proves,by similar methods,that the higher dimensional geometry related to the fuzzyﬁve-sphere is SO(6)further.In par-U(k−1)×U(1)ticular with describe some bundle structures they admit.For the even sphere case,the bundle structure of SO(2k+1)/U(k)was used to map abelianﬁeld theory on the fuzzycoset to non-abelianﬁeld theory on the fuzzy sphere base[12].In the odd sphere case, the relevant higher dimensional geometry is not in general a bundle over the sphere,so the same strategy as[12]cannot be used to obtain aﬁeld theory on the sphere.However some bundle structures on the coset do exist and we describe them.An interesting feature is that base andﬁbre are both hermitian symmetric spaces,which come up in N=2 sigma models for example[20].In section9.3we give a qualitative comparison of some basic counting of degrees of freedom which explains why it appears diﬁcult to obtain an ordinary non-abelian theory on the fuzzy odd sphere.We observe some intriguing similar-ities to discussions of the entropy of M5-branes.Section9also describes some avenues for future research.2.Review of fuzzy odd spheresTo get a fuzzy three sphere[10]we look for matrices which satisfy the equation i X2i=c,where c is a constant.The matrices are constructed by starting withV=V+⊕V−.The vector space V+is the two-dimensional spinor representation of SO(4) of positive chirality.The space V−is the two-dimensional representation of negative chi-rality.The projector P+acting on V projects onto V+,and P−projects onto V−.In terms of the SO(4)=SU(2)×SU(2)isomorphism,these have spins(2j L,2j R)=(1,0)and (2j L,2j R)=(0,1)respectively.For every odd integer n,the space Sym(V⊗n)(which is the symmetrized n-fold tensor product space)contains a subspace where there are(n+1)factors of negative chirality.This subspace is an2irreducible representation of SO(4)labelled by(2j L,2j R)=((n+1)),which we will2call R+n.The projection operator in End(Sym(V⊗n))which projects onto this irrep is called P.In[10]it was called P R+,but here we are using notation which makes the n R+ndependence explicit.Sym(V⊗n)also contains a subspace where there are(n+1)factors of positive chirality.This subspace is an irreducible2representation of SO(4)labelled by(2j L,2j R)=((n−1)),which we will call R−n.2.The space R n is deﬁned to be a direct sum The projector for this subspace is called PR−nR n=R+n⊕R−n(2.1) The projector for this space is a sumP R n=P R+n+P R−n(2.2)The Matrices X i are linear transformations of R n ,i.e they are in End (R n ).X i =P R nrρr (Γi )P R n(2.3)More precisely,X i map R +n to R −n and R −n to R +n .This can be expressed by saying it isa sum of matrices in Hom (R +n ,R −n )and in Hom (R +n ,R −n )X i =P R +n X i P R −n +P R −n X i P R +n(2.4)In other words,if we arrange the vectors of R +along the upper rows of a column vector and those of R −along the lower rows,the matrices X i are non-zero in the oﬀ-diagonal blocks.Let us prove that i X 2i commutes with generators of SO (4).[ iX 2i ,P R ntρt (Γk Γl )P R n ]=i[X i ,P R ntρt (Γk Γl )P R n ]X i+X i [X i ,P R ntρt (Γk Γl )P R n ]=iP R nr[ρr (Γi ),t ρt (Γk Γl )]P R nsρs (Γi )P R n+P R nrρr (Γi )P R n [sρs (Γi ),tρt (Γk Γl )]P R n=P R n rρr ([Γi ,Γk Γl ])P R nsρs (Γi )P R n+P R nrρr (Γi )P R nsρs ([Γi ,Γk Γl ])P R n=P R nr ρr (Γl δik −Γk δil )P R nsρs (Γi )P R n+P R nrρr (Γi )P R ns(ρs (Γl )δik −ρs (Γk )δil )P R n=0(2.5)Since X 2i commutes,we know it is a constant in each irrep.X 2i =a +P R +n+a −P R −n (2.6)Now consider the action on X 2i on R +.We ﬁndX 2i P R +n=P R +nrρr (Γi P −)P R −n sρs (Γi P +)P R +n =a +P R +n (2.7)and likewise :X 2i P R +n=P R −nrρr (Γi P +)P R +nsρs (Γi P −)P R −n =a −P R −n(2.8)Now consider the operation of exchanging P +with P −,which leaves X 2i as deﬁned in (2.3)invariant.Nowpeform this operation on the equation (2.7)to ﬁndX 2i P R −n=P R −nrρr (Γi P +)P R +nsρs (Γi P −)P R −n=a +P R −n (2.9)Comparing with (2.8)we ﬁnd that a +=a −.Thus X 2i is proportional to the identity.The radius is easily calculated as follows :X 2i P R +n=rρr (Γi Γi P +)+r =sρr (Γi P −)ρs (Γi P +)(2.10)In the second term,the index s can take n +12values.Also we havei (Γi ⊗Γi )P −⊗P +=2(P +⊗P −).The ﬁrst term gives(n +1)2(4+(n −1))=(n +1)(n +3)2,n −32,n +32(n +2k −1)(2.12)3.Fuzzy three-sphere and S 2×S 2The Matrix Algebras related to fuzzy S m ,for m >2contain more representationsthan required to give the algebra of functions on a sphere [11].In the even sphere case,the higher dimensional manifolds related to the Matrix algbras have been identiﬁed [12].We recall some relevant facts here.In the case of the even fuzzy sphere S 2k it was justSO(2k+1)/U(k).Several diﬀerent proofs of this result were given.Theﬁrst proof follows immediately once we know[11]that the Matrix algebra contains every representation of SO(2k+1)with unit multiplicity.We then used the result of[21]that the algebra of functions on SO(2k+1)/U(k)has precisely this property.In section3.1we will follow this line of argument,i.e we will compare the SO(4)decomposition of the functions on S2×S2to the representations found in End(R±n)at large n andﬁnd agreement.Insection3.2we will develop an analogous argument for Hom(R+n,R−n)or Hom(R−n,R+n), showing that the space of sections of a bundle on S2×S2agrees with the SO(4)content of these oﬀ-diagonal matrices.A second line of argument to relate fuzzy even spheres to the appropriate coset will be brieﬂy described in section3.3and the analog will be developed for the fuzzy three-sphere.3.1.Spectrum of End(R±n)and Functions on S2×S2Rotations L12and L34form a U(1)×U(1)subgroup of SO(4).In a self dual represen-tation associated with an SO(4)Young diagram of row lengths(r1,r2)the eigenvalues of these generators on the highest weight vector are r1,r2.In an anti-self dual representation associated with the same Young diagram the eigenvalues on the highest weight vector are (r1,−r2).The combination L12+L34generates a U(1)subgroup of the left SU(2)in the SU(2)×SU(2)description of SO(4).The SU L(2)weight is,therefore,described by 2J L=L12+L34.The combination L12−L34generates a U(1)subgroup of the right SU(2). The SU R(2)highest weight is,therefore,described by2J R=L12−L34.For self-dual reps, we have then2J L=r1+r2and2J R=r1−r2,where r2is positive.For antiself-dual reps, 2J L=r1−r2and2J R=r1+r2for positive r2.To get the spectrum of representations of SU(2)×SU(2)present among the harmonics of(SU(2)×SU(2))SO(4)decompositon of End(R+n)or End(R−n)at large n[11].We conclude that large n SO(4)decomposition of End(R+n)or End(R+n)has the same representations as the space of functions on S2×S2.Since the structure constants of the algebra are determined bythe Clebsch-Gordan coeﬃcients,once we have established that the Matrix algebra has the same repesentations as F un(S2×S2)we know that it is the same algebra.3.2.Spectrum of Hom(R+n,R−n)and sections on S2×S2In the large n limit of Hom(R+n,R−n)(or Hom(R−n,R+n))we have self-dual reps associated with Young Diagrams labelled by non-negative row lengths(r1,r2)(which have highest weights(r1,r2)),and obeying the condition r1−r2odd.We also have antiself-dual reps.associated with Young Diagrams labelled by non-negative row lengths (r1,r2)(which have highest weights(r1,−r2)),and obeying r1−r2odd.These results are found in[11].This approaches a space of sections on SO(4)matrices Xµand Xµνin one particular state,the n-fold tensor product of the v0(see appendix for notation).We will do something similar for the fuzzy3-sphere.Identify a set of generators for the complete Matrix algebra,and take expectation values in the large n limit,and analyze the stabilizer group if the expecttaion values.A complete set of generators is:X+i=PR−n rρr(Γi P+)P R+nX−i=PR+n rρr(Γi P−)P R−nX+ij=PR+n rρr(12[Γi,Γj]P−)PR+nX−ij=PR−n rρr(12[Γi,Γj]P+)PR−n(3.1)The coordinates of the sphere X i are related to the above by X i=X+i+X−i.It is also useful to deﬁneX ij=X+ij+X−ijY ij=Y+ij+Y−ijY i=X+i−X−i˜Xij=X+ij−X−ij˜Y ij =Y+ij−Y−ij(3.2)Take a state|s>in R+n containing(n+1)2copies of a†1v0.Weﬁnd that<s|i X+122<s|i Y+122<s|iX+342<s|−iY+342(3.3)Other expectations values are zero.The algebraic equations obeyed by X+ij and Y+ij have a solution with the above numerical values.These values are stabilized by the rotations L12and L34which generate a U(1)×U(1) subgroup of SO(4).The SO(4)acts transitively on the solutions,so the set of solutions is SO(4)2i=j X+ij X+ij=−(n+1)(n+5)i=j X−ij X−ij=−(n+1)(n+5)i=j Y+ij Y+ij=−(n−1)(n+3)i=j Y−ij Y−ij=−(n−1)(n+3)(4.1)There are self duality or antiself-duality relationsX+ij=12ǫijkl X−klY+ij=−12ǫijkl Y−kl(4.2)Incidentally,the(anti)self-duality equations show that there are only3independent components for X+ij and3for Y+ij.We are using conventionsΓ5=−Γ1Γ2Γ3Γ4with Γ5v0=v0.Together with the equations in(4.1)this shows explicitly the origin of the S2×S2in the upper block.Similarly there is the S2×S2in the lower block deﬁned bythe X−ij and Y−ij variables.Consider multiplying X ±i with X ±j (X +iX −j −X +j X −i )+ǫijkl2ǫijkl X −kl +2(X −ij+ǫijkl2(X +k X −l −X +l X −k )=−n +12X −kl )(4.3)By converting all +to −and at the same time converting ǫijkl to −ǫijkl ,we can write the folllowing equation :(X −i X +j −X −j X +i )−ǫijkl2ǫijkl X +kl +2(X +ij −ǫijkl2(X −k X +l −X −l X +k )=n +12X +kl )(4.4)From these we can write down(X −iX +j −X −j X +i )=n 4ǫijkl (−X +kl +Y +kl )+2X +ij (X +i X −j −X +j X −i )=−n4ǫijkl (X −kl −Y −kl )+2X −ij(4.5)By adding the two equations above we get :[X i ,X j ]=−n4ǫijkl (X +kl −X −kl )−(Y +kl −Y −kl ) +2X ij=n 4ǫijkl (˜Xkl −˜Y kl )+2X ij (4.6)By taking the diﬀerence of the two eqs.in (4.5)we ﬁnd{X +i ,X −j }−{X +j ,X −i }=−n4ǫijkl (X kl −Y kl )−2(˜Xij )(4.7)This can be expressed as a commutator of X i and Y i .{X i ,Y j }=n4ǫijkl (X kl −Y kl )+2(˜Xij )(4.8)We of course have relations likeX +i X +j =0X −iX −j =0(4.9)which follow fromP R+n P R−n=P R−n P R+n=0(4.10) Now we consider multiplying X±ij and Y±ij with X±kX+klX−i−X−i Y−kl=δil X−k−δik X−lX−klX+i−X+i Y+kl=δil X+k−δik X+lY−klX+i−X+i X+kl=δil X−k−δik X−lY+ kl X−i−X−i X−kl=δil X+k−δik X+l(4.11)Deﬁning C jkli≡δilδkj−δikδjl we can rewrite the above as:X+klX−i−X−i Y−kl=C j kli X−jX−klX+i−X+i Y+kl=C j kli X+jY−klX+i−X+i X+kl=C j kli X+jY+ kl X−i−X−i X−kl=C j kli X−j(4.12)Combining these with the facts:X+kl X+i=X+i X−kl=0X−kl X−i=X+i X−kl=0(4.13)which follow from(4.10)these imply that X±kl +Y±klacts by commutators as generators ofSO(4)rotations.[X+kl +Y−kl,X−i]=C jkliX−j[X−kl +Y+kl,X+i]=C jkliX+j[X+kl +Y−kl,X+i]=C jkliX+j[X−kl +Y+kl,X−i]=C jkliX−j(4.14)The diﬀerences X±kl−Y±kl act by anticommutators as a rotation combined with change of X+i to X−i:{X+kl−Y−kl,X−i}=C j kli X−j{X−kl−Y+kl,X−i}=−C j kli X−j{X−kl−Y+kl,X+i}=C j kli X+j{X+kl−Y−kl,X+i}=−C j kli X+j(4.15)Recalling from(3.2)that X i=X+i+X−i and Y i=X+i−X−i allows us to write some interesting anti-commutator actions:{X+kl−Y−kl,X i}=−C j kli Y j{X−kl−Y+kl,X i}=C j kli Y j{X+kl−Y−kl,Y i}=−C j kli X j{X−kl−Y+kl,Y i}=C j kli X j(4.16)It follows that:[X kl+Y kl,X i]=2C jkliX j[X kl+Y kl,Y i]=2C jkliY j{X kl−Y kl,X i}=0{X kl−Y kl,Y i}=0(4.17)Recalling from(3.2)that˜X kl=X+kl−X−kl and˜Y kl=Y+kl−Y−kl we have{˜X kl+˜Y kl,X i}=−2C j kli Y j{˜X kl+˜Y kl,Y i}=−2C j kli X j[˜X kl−˜Y kl,X i]=0[˜X kl−˜Y kl,Y i]=0(4.18)So we have interesting anticommutator actions which are SO(4)rotations combined with an X−Yﬂip.Finally we need to consider multiplication X ij X kl and X ij Y kl Some of these are fa-miliar from the fuzzy S4case:[X ij,X kl]=δjk X il+δil X jk−δjl X ik−δil X jk[Y ij,Y kl]=δjk Y il+δil Y jk−δjl Y ik−δil Y jk(4.19)Further relations are[X ij,Y kl]=0X+(ij Y+kl)=−δjk(X−i X+l+X−lX+i)−δil(X−j X+k+X−kX+j)+δjl(X−i X+k +X−kX+i)+δik(X−j X+l+X−lX+j)+(n+1)(n+3)2δjl P R+n(4.20)By taking +to −we haveX −(ij Y −kl )=−δjk (X +i X −l +X +l X −i )−δil (X +j X −k +X +k X −j )+δjl (X +i X −k +X +k X −i )+δik (X +j X −l +X +l X −j )X −ij Y −il +X −il Y −ij =2(X +j X −l +X +l X −j )−(n +1)(n +3)2(−δik δjl +δil δjk )(4.22)A special case of (4.22)is X (ij Y il )=2{X j ,X l }−(n +1)(n +3)2δjl (4.23)It is worth pointing out that the equations of this section correct some equations in the appendix of [10]and also the remarks in section 7.3of [11].5.Algebra of fuzzy 3-sphere at large NIt is useful to deﬁne,as in the case of fuzzy even spheres,rescaled variables.A surprising feature will be that,after the standard rescalings,some non-commutativity wil remain at large n in the fuzzy odd cases.We can deﬁne variablesA ±ij =iX ±ijn C ±i =√n(5.1)and the following linear combinations are useful.A ij =A +ij +A −ij˜A ij =A +ij−A −ij B ij =B +ij +B −ij˜B ij =B +ij −B −ijC i =(C +i +C −i )˜C i =i (C +i−C −i )(5.2)The sphere-like relations are of the form:i=j(A±ij)2=P R±ni=j(B±ij)2=P R±ni=j(A ij)2=1i=j(B ij)2=1i C2i=1i˜C2i=1(5.3)We have the following vanishing commutators and anticomutators:[A,A]=0[B,B]=0[A,C]=[A,˜C]=0[˜A+˜B,C]=[˜A+˜B,˜C]=0{˜A−˜B,C}={˜A−˜B,˜C}=0(5.4)There are also some interesting commutators of C,˜C which survive in the large N limit(C−i C+j−C−j C+i)=−i2ǫijkl(A−kl+B−kl)(5.5)[C i ,C j ]=−i2ǫijkl (A kl +B kl )A (ij B kl )=−{˜A ij ,˜B kl }=δjk 2{C j ,C k }−δjl2{C j ,C l }+12{˜Ci ,˜C l }+δil 2{˜Ci ,˜C k }−δik 3(δik δjl −δil δjk )A (ijB il )=−{C j ,C l }+δjlfuzzy three-sphere do not allow it to be a solution of such a Matrix model.It is a very interesting problem to try toﬁnd any physical Matrix model where the fuzzy three-sphere (or the fuzzyﬁve sphere discussed later in this paper)is a solution.It would be even more fascinating toﬁnd these general fuzzy spheres as supersymmetry-perserving solution of any physical Matrix model.A physical Matrix model would be one that comes up as an action for branes(parallel or intersecting)or as a dual to some M-theory background. In the light of the diﬃculty of constructing a transverseﬁve-brane from the BFSS matrix model,ﬁnding such a fuzzyﬁve sphere as a physical Matrix model solution(or proving it is not possible)would give important information about the relation between M-theory and large N matrix systems.We will not solve this problem in this paper but we will exhibit some toy Matrix models which do have the fuzzy three sphere as a solution,and we will observe some qualititative similarities between these toy Matrix models and some others that have appeared in recent literature.Given the relations we wrote down in(4.6)and(4.7)it is easy to obtain some Matrix Actions which admit as solutions the matrices of the fuzzy three-sphere.Consider a Matrix action with a symmetry group which contains SO(4))and considerﬁelds transforming in the vector of SO(4),labelledΦi as in the BFSS[23]or IKKT[24][25]matrix models. Suppose,unlike these models there is an additional Matrix variable transforming in the antisymmetric of SO(4)which we will denote P ij.Suppose the action isT R [Φi,Φj]−P ij 2(6.1) The variation of this action with respect toΦi or P ij will be proportional to[Φi,Φj]−P ijFrom(4.6)it follows that if we setΦi=X iP ij=n4ǫijkl(˜X kl−˜Y kl)+2X ij(6.2)where X i,X kl,Y kl,˜X kl,˜Y kl are deﬁned in(3.1)(3.2),we will have a solution to the action (6.1).Another way to construct an action which is solved by the matrices of section3is to start with variablesΦi and˜Φi and antisymmetric Matrix variables Q ij which have an action T R({Φi,˜Φj}−Q ij)2(6.3)In this case if we useΦi=X i˜Φi=Y iQ ij=n4ǫijkl(X kl−Y kl)+2(˜X ij)(6.4)It is interesting that in recent literature Matrix actions have been considered which involve matrices transforming in the vector of the SO group as well as other antisymmetric tensors[26][27][28][29].It would be interesting to see if the fuzzy odd sphere solutions of model actions of the form(6.1)could also be found for actions involved in the above papers. Perhaps relating terms of the form in(6.1)to terms in these actions could be useful.The role of P ij could also be played by compositeﬁelds such as¯ψΓijψ.7.Fluctuatingﬁelds around solution7.1.Physics on S2×S2Useful information about the physics of a solution deﬁned by the X i matrices is obtained by consideringﬂuctuations around such a solution.In the even sphere case the action for theﬂuctuations is a U(1)theory on SO(2k+1)/U(k)with SO(2k+1)symmetry. The existence of a hidden higher dimensional coset in the odd sphere allows a somewhat analogous result.Imagine weﬁnd a solution to Matrix theory which uses these matricesΦi=X iΦa=0(7.1)TheΦi transform as a vector of the SO(4)symmetry group of the fuzzy three sphere,and we call them parallel scalars.TheΦa are invariant under that symmetry group and are called transverse scalars.Consider theﬂuctuationsΦi=X i+φ+i(A+,B+)+φ−i(A−,B−)+φ+ij(A+,B+)C−i+φ−ij(A−,B−)C+jΦa=φ+a(A+,B+)+φ−a(A−,B−)+φ+ai(A+,B+)C−i+φ−ai(A−,B−)C+i(7.2)Eachφis aﬁeld living on S2×S2.Notice that the transverse scalars contain compo-nents which transform as vectors under the parallel SO(4).The parallel scalars give risetoﬁeldsφ±ij which is a two-index tensor which can be reduced to symmetric traceless part, antisymmetric part and a scalar.By expanding a Matrix action which is solved by the fuzzy three-sphere Matrices,around the solution,we would get aﬁeld theory,in analogy to the analogous results for fuzzy even spheres[30][12][14].We outline some features of theﬁeld theory on S2×S2that results from theseﬂuc-tuations.The kinetic term ofΦa isdt T R(∂tΦa∂tΦa)(7.3)Using the expansion forΦa in(7.2),we will get,from the diagonal terms,dtdA+dB+ ∂tφ+a(A+,B+) 2+ dtdA−dB− ∂tφ−a(A−,B−) 2= dtdAdB ∂tφ+a(A,B) 2+ ∂tφ−a(A,B) 2(7.4) dA+dB+=dA−dB−is an SO(4)invariant measure for S2×S2.These diagonal terms coming from the large n limit of End(R+n)and End(R−n).From squaring the oﬀdiagonal terms,φ+ai(A+,B+)C−i+φ−ai(A−,B−)C+i we obtain terms includingdt T R R+n(∂tφ+ai(A+,B+)C−i∂tφ−ai(A−,B−)C+j)(7.5)After rescalings of(5.1)the relations in(4.12)allow us to show that polynomials in A−,B−can be pulled to the left of C−i at the cost of converting the pair(A−,B−)to(B+,A+) (not the switch).There are extra terms coming from the RHS of(4.12)which can be ignored in the leading large n limit.The terms in(7.5)can then be written asdt T R R+n(∂tφ+ai(A+,B+)∂tφ−aj(B+,A+)C−i C+j)(7.6)It is useful to write this as a product of parts symmetric in(ij)and a part antisymmetric in these indices.The symmetric part is simpliﬁed using(4.20)to replace(C−i C+j+C−j C+i)=12ǫijkl(A+kl+B+kl)≡H ij(7.8)We have deﬁned in(7.7)and(7.8)a symmetric tensor and an antisymmetric tensor living on S2×S2.After converting the trace to an integral using SO(4)invariance,we can writethe kinetic terms in(7.5)asdt dA dB∂tφ+ai(A,B)∂tφ−aj(B,A)(G ij+H ij)(7.9)Analogous to the term in(7.5)there is a trace over R−n which leads by similar steps as above to dt dA dB∂tφ−ai(A,B)∂tφ+aj(B,A)(G ij−H ij)(7.10) Although H contains i the action is hermitian since hermitian conjugation convertsφ+to φ−.The above arguments illustrate the use of the algebraic relations of section4in deriving the action forﬂuctuations.It appears that there is some U(2)symmetry which mixes the R+n and the R−n but which is broken to U(1)×U(1).We leave it to the future to elucidate all the symmetries of such an action.A further interesting problem is to relate this theory on S2×S2to someﬁeld theory on S3.In the fuzzy even sphere case there was a U(1) theory on the higher dimensional geometry and a non-abelian theory on the sphere itself [12].We would expect a generalization of that correspondence to yield someﬁeld theory on the sphere S3.The counting of degrees of freedom in section9shows that it cannot be a standard non-abelian theory.The discussion in section9will also show that the analogous ﬁeld theory on the coset SO(6).This works in the case of k=2,U(k−1)×U(1)and gives the right counting of degrees of freedom,i.e N2scales like n10while quadratic expressions like X2i scale like n2.This means we should expect a10dimensional space which is indeed the dimension of the above coset for k=3.Here we will develop an argument similar to section3.3based on the expectation values of generators of the matrix algebra evaluated in a state in R n.We pick a state of the form v0⊗···⊗v0⊗a†1v0···a†1v0, where there are(n+1)copies of a†1v0.We sum over diﬀerent ways2of embedding the copies v0and the a†1v0in V⊗n in order to make sure we have a state in Sym(V⊗n).Considering normalized quantities analogous to(5.1),we have in the large n limit<s|A+12|s>=1<s|A+34|s>=<s|A+56|s>=1<s|C21|s>=1/2<s|C22|s>=1/2<s|C1C2+C2C1|s>=0<s|C1C3+C3C1|s>=0...<s|B+12|s>=−1<s|B+34|s>=<s|B+56|s>=1(8.1)Other A ij and B ij have zero expectation value.The above variables are deﬁned in the same way as for the fuzzy3-sphere.These expectation values in(8.1)are only preseved by U(1)×U(2).This means that the complete set of variables describing the matrix algebra End(R+n)in the large n limit have a solution which is stabilized by the subgroup U(1)×U(2).The full set of solutions,obtained by action of SO(6)on the irrep R+n,is acted on transitively by the SO(6).Hence the set of solutions is the coset SO(6)/U(2)×U(1). The U(1)is generated by rotations in the12direction.In the3546block there is a U(2) subgroup which preserves the expectation values.Consider<X21>=<X22>=1.Theseare matrices of the form P Q−Q P (8.2)where P is real antisymetric and Q is real symmtric.8.2.Group Theory Proof:Harmonics of SO(6)/U(2)×U(1)from Matrix realizationIn the next two sections,we develop the arguments analogous to sections3.1and3.2, now for the case of the fuzzyﬁve-sphere.We recall from[11]the list of representations of SO(6)which appear in End(R+n)at large n.There are self-dual representations associated with highest weights λ=(p1+p2+p3,p1+p2,p1),and antiself-dual reps with highest weights λ=(p1+p2+p3,p1+p2,−p1),where p1,p2,p3are positive.The associated Young diagrams have row lengths r=(p1+p2+p3,p1+p2,p1)Unlike the case of the fuzzy three sphere,we now have representations with multiplicity more than2.The multiplicity is。

Phase Diagram of SO(3) Lattice Gauge Theory at Finite Temperature

a r X i v :h e p -l a t /9708016v 2 23 A u g 19971Phase Diagram of SO(3)Lattice Gauge Theory at Finite TemperatureTIFR/TH/97-43Saumen Datta a and Rajiv V.Gavai a ∗aTheory Group,Tata Institute of Fundamental Research,Homi Bhabha Road,Mumbai 400005,IndiaThe phase diagram of SO (3)lattice gauge theory at ﬁnite temperature is investigated by Monte Carlo techniques with a view i)to understand the relationship between the deconﬁnement phase transitions in the SU (2)and SO (3)lattice gauge theories and ii)to resolve the current ambiguity of the nature of the high temperature phases of the latter.Phases with positive and negative adjoint Polyakov loop,L a ,are shown to have the same physics.A ﬁrst order deconﬁning phase transition is found for N t =4.1.IntroductionSince the continuum limit of a lattice gauge theory is governed by its 2-loop β-function,one expects the physics of conﬁnement and deconﬁne-ment for pure SU (2)gauge theory to be identi-cal to that of pure SO (3)gauge theory.On the other hand,SO (3)does not have the Z (2)center symmetry whose spontaneous breakdown in the case of the SU (2)theory indicates its deconﬁne-ment transition.This makes the investigation of the phase diagram of the SO (3)gauge theory es-pecially interesting and important.It has been argued[1]that the deconﬁnement transition for the SO (3)lattice gauge theory may show up as a cross over which sharpens in the continuum limit to give an Ising-like second order phase transition.Another reason for investigating the ﬁnite tem-perature transition in SO (3)gauge theory is that it is supposed[2]to have a bulk phase transition and may thus provide a test case for studying the interplay between these diﬀerent types of phase transitions.Recently,simulations of the Bhanot-Creutz action for SU(2)gauge theory[2],S =pβf (1−13Tr a U p ),(1)at ﬁnite temperature revealed[3]that the known deconﬁnement transition point in usual Wilson action becomes a line in the βf -βa plane and joins3Tr U p ),(2)where U p denotes the directed product of the link variables,U µ(x )∈SO (3),around an elementary plaquette p.The action (1)for βf =0also cor-responds to an SO (3)gauge theory which was found in [2]to have a ﬁrst order bulk transition at βa ∼2.5.A third action we used is the Halliday-Schwimmer action [5]S =βv p(1−12ables,the partition function in this case also con-tains a summation over all possible values of{σp}. It too shows[5]aﬁrst order bulk phase transition atβv∼4.5.The chief advantage of this action is that both the link variables Uµandσp can be updated using heat-bath algorithms.We studied the adjoint plaquette P,deﬁned as the average of13 phase for the sameβa.The mass gap,obtainedfrom the connected parts of the correlator aboveor from their zero momentum projected versions,was similar for both the positive and negative L astates corresponding to bothβa=2.6and3.5,asexpected for states with same physics.It is,how-ever,considerably diﬀerent forβa=2.3.4.Order and Nature of the TransitionIn simulations on43×4,63×4and83×4lat-tices with the actions(1)and(3),long metastable states were observed on all lattices near the tran-sition region,signaling a possibleﬁrst order tran-sition. L a was seen to tunnel between all the three states,two of which correspond to the same value of the action.Runs on smaller lattices show more tunnellings and largerﬂuctuations in the positive L a-phase.The estimated transition points for43×4,63×4and83×4lattices are βvc=4.43±0.02,4.45±0.01and4.45±0.01 respectively.Fig.2displays distributions of L a from the runs made at the critical couplings but from dif-ferent starts.We performed about100K-400K heat-bath sweeps depending on the size of the lat-tice.While the frequent tunnelling smoothens the peak structure for the43×4lattice considerably, a clear three-peak structure is seen for both the 63×4and the83×4lattices.The stability of these peaks under changes in spatial volume sug-gests the phase transition to be ofﬁrst order.The estimates of the discontinuities in the plaquette, L a +and L a −are0.0575±0.0030,0.87±0.04 and0.28±0.04respectively.It is also interesting to note that i)the peak for the conﬁned phase is almost precisely at zero and ii)normalising by the maximum allowed L a in each phase,the discon-tinuties for both the positive and negative phases are equal,being0.29±0.01and0.28±0.04re-spectively.We also studied the theory on83×2,44,64and 84lattices.On all these lattices,only one tran-sition point was found,where both the plaquette and L a show a discontinuity.A clear shift inβc was found in going from N t=2to N t=4but no perceptible change inβc was found in going from N t=4to6and8for both actions(1)and(3).1234567-0.6-0.4-0.200.20.40.60.81 1.2 1.4 1.6 N(L)aL aN = 8sN = 6sN = 4sFigure2.The distribution of L a on N3s×4lattices at their critical couplings.This is in sharp contrast to the SU(2)case,and is also unexpected for a deconﬁnement transition.5.SummaryOur simulations with a variety of actions showed the negative L a -state to be present for all of them.However,using a‘magneticﬁeld’term to polarise,we found a unique L a state depending on the sign of theﬁeld.The correla-tion function measurements in both the phases of positive and negative L a indicated that the two states are physically identical high temper-ature deconﬁned phases of SO(3)gauge theory. Although a shift inβc was observed in changing N t from to2to4,no further shift was seen for N t =6and8which is characteristic of a bulk phase transition.REFERENCES1. A.V.Smilga,Ann.Phys.234(1994)1.2.G.Bhanot and M.Creutz,Phys.Rev.D24(1981)3212.3.R.V.Gavai,M.Grady and M.Mathur,Nucl.Phys.B423(1994)123;R.V.Gavai and M.Mathur,Phys.Rev.D56(1997)32.44.S.Cheluvaraja and H.S.Sharatchandra,hep-lat/9611001.5.I.G.Halliday and A.Schwimmer,Phys.Lett.B101(1981)327.。

Gravitational Instantons from Gauge Theory

NC instantons, is related to the size of a minimal two-
sphere known as a “Bolt” in the gravitational instantons.
Substituting the ansatz (11) into Eq.(12), we get
A general strategy was suggested in [6] to solve the self-duality equation (9). For example, let us consider the anti-self-dual (ASD) case. Take a general ansatz for the ASD Fµν as follows
Consider a NC spacetime deﬁned by
[yµ, yν ]⋆ = iθµν
(1)
with a constant 4 × 4 matrix θµν . Gauge theories can
be constructed on this NC spacetime. For example, the action for NC U (1) gauge theory in ﬂat Euclidean R4 is
The equation (13) was solved in [6] for the single instanton case. It was found there that the eﬀective metric (6) for the single U (1) instanton is related to the Eguchi-Hanson (EH) metric [8], the simplest ALE space, and that the family of the EH space is parameterized by the instanton number. In this paper we will show that the connection between NC U (1) instantons and hyperKa¨hler geometries is more general. More precisely, we will see that the NC self-duality equation (4) is mapped

The superconducting phase transition and gauge dependence

|x−y|→∞
2 where the zeroes denote bare quantities, F0 is a short for µν µν 0 0 F0 F0 and Dµ = ∂µ + ie0 Aµ . The Sgf is the gauge ﬁxing part and is given by
The superconducting phase transition and gauge dependence
Claude de Calan and Flavio S. Nogueira
Centre de Physique Th´ eorique, Ecole Polytechnique, F-91128 Palaiseau Cedex, FRANCE (Received February 1, 2008)
Sgf =
dd x
1 2 (∂µ A0 µ) . 2 a0
(4)
lim
< G(x, y) >,
(1)
where the operator G(x, y) is given by a smeared string: G(x, y) = φx exp −ie
µ,z
Aµz hAµx are the scalar and gauge ﬁelds, respectively, deﬁned in a lattice. The gauge group is noncompact. The ﬁeld hµz = ∆µ Vz−x − ∆µ Vz−y , with ∆µ being a lattice derivative and V is the kernel of ¯ −1 , where ∆ ¯ is the lattice Laplacian. The operator (−∆) G so deﬁned is gauge invariant and in the Landau gauge µ,z Aµz hµz vanishes [2]. By using the so deﬁned order parameter, Kennedy and King have shown that there is true long range order for d ≤ 4 (G∞ = 0) only if the Landau gauge is ﬁxed. Note that this result was proved for a lattice GL model only and is not a trivial matter to extend the analysis of Ref. [2] to the continuum. The aim of this report is to provide an analysis of the gauge dependence directly in the continuum. Also, we will deﬁne a local, gauge invariant OP. To this end, we will employ the Ward-Takahashi (WT) identities. The study that will be undertaken here was initiated recently by one of us [4]. Let us consider the following bare action for the GL model: S= 1 2 M2 0 0 F0 + (Dµ φ0 )† (Dµ φ0 ) + 0 A0 A0 4 2 µ µ u0 2 (3) |φ0 |4 + Sgf , + m2 0 |φ0 | + 2 dd x 1

Gravity Dual to Pure Confining Gauge Theory

a rXiv:0712.4269v2[he p-th]3Ja n28Preprint typeset in JHEP style -HYPER VERSION Girma Hailu ∗Newman Laboratory for Elementary Particle Physics Cornell University Ithaca,NY 14853Abstract:We ﬁnd a dual gravity theory to pure conﬁning N =1supersymmetric SU (N )gauge theory in four dimensions which has the correct gauge coupling running in addition to reproducing the appropriate pattern of chiral symmetry breaking.It is constructed in type IIB string theory on R 1,3×R 1×S 2×S 3background with N number of electric D5and 2N number of magnetic D7-branes ﬁlling four dimensional spacetime and wrapping respectively two and four cycles.Introduction.—The theory of quantum chromodynamics(QCD)of the strong nuclear interactions becomes highly nonperturbative and hard at low energies.The gauge/gravity duality[1,2,3]relates a gauge theory in strongly coupled nonperturba-tive region to a gravity theory in weakly coupled perturbative region and,therefore, provides the possibility for a calculable classical gravity description to low energy QCD.Theﬁrst example of gauge/gravity duality in[1]involves conformalﬁeld the-ory with N=4supersymmetry.A gravity dual to pure N=1supersymmetric SU(N)gauge theory is highly desirable for several reasons.First,N=1super-symmetric SU(N)gauge theory exhibits phenomena such as conﬁnement and chiral supersymmetry breaking and could serve as a laboratory to gain new insight into QCD.Second,there is a possibility that N=1supersymmetry may be part of nature at energies accessible in the coming generation of experiments at the Large Hadron Collider(LHC)and a gravity description is useful for calculating physical quantities such as glueball mass spectra in the N=1gauge theory.Third,if a suit-able supersymmetry breaking scheme which removes the gaugino in the pure N=1 theory is found,it could be used to study the real world QCD itself.Fourth,the supergravity background has nonsingular geometry due to nonperturbative quan-tum eﬀects which is useful for studying early universe cosmological scenarios and possibilities that a universe like ours could reside on the background.Indeed,there has been extensive eﬀort towardsﬁnding a gravity dual to pure N= 1supersymmetric gauge theory,[4,5,6]most notably laid the foundational work.The work in[5]produced a gravity dual to N=1supersymmetric SU(N+M)×SU(N) gauge theory with N number of D3and M number of D5-branes on AdS5×T1,1 conifold background involving novel cascading renormalization groupﬂow towards pure N=1supersymmetric SU(M)gauge theory in the infrared and deformation of the conifold,but the gravity theory has constant dilaton and does not reproduce the gauge coupling running of a pure conﬁning gauge theory.The work in[6]produced supergravity solutions with N number of NS5or D5-branes involving running dilaton and appropriate pattern of chiral symmetry breaking,but it does not reproduce the gauge coupling running of pure N=1supersymmetric SU(N)gauge theory in four dimensions.In this note,weﬁnd a dual gravity theory to pure N=1supersymmetric SU(N) gauge theory in four dimensions which has the correct gauge coupling running and which reproduces the appropriate pattern of chiral symmetry breaking robustly.The important new ingredients which facilitate our construction are the gauge/gravity duality mapping with running dilaton and running axion obtained recently in[7] with magnetic D7and Dirac8-branes playing crucial role and the set of equations obtained in[8]which allows studying systematically type IIBﬂows with N=1 supersymmetry.Our starting point is the running of the Yang-Mills coupling in the gauge theory and its mapping to the running of the dilaton in the gravity theory.Weﬁnd asupergravity dual in type IIB string theory on R1,3×R1×S2×S3background with N number of D5and2N number of D7-branesﬁlling four dimensional(4-d) spacetime and wrapping respectively2and4-cycles.The gauge theory is engineered by wrapping N electrically charged D5-branes on non-zero S2cycle with S3of zero-size in the base at the tip which leads to blown-down S2and R-R3-form F3ﬂux through blown-up S3after the familiar geometric transition which deforms the tip as in[5].See[9]for a conifold transition on a setting in the topological A-model with S2and S3interchanging roles.The running of the gauge coupling leads to a running dilaton on the gravity side which is related to R-R F1ﬂux such that the background follows the equations for the class ofﬂows with imaginary self-dual3-formﬂux in[8]and the supergravity solutions are read oﬀfrom[7]with appropriate changes of variables.The runnings of the dilaton and the axion are due to magnetic coupling of the axion to D7-branes and Dirac8-branes which emanate from the D7-branes.Demanding that the correct renormalization groupﬂow of the gauge theory living on the N electrically charged D5-branes be reproduced leads to2N magnetic D7-branesﬁlling4-d spacetime and wrapping4-cycles at the ultraviolet edge of the background.The background with the F1and the F3ﬂuxes induces3-from NS-NS H3ﬂux and also5-form R-R F5ﬂux which can be viewed as coming from the wrapped D5-branes,which are fractional D3-branes,via backreaction NS-NS2-form potential and the F3ﬂux.The axion potential C0in the axion-dilaton coupling coeﬃcient is related to the Yang-Mills angle and preserves only a Z2N discrete symmetry in the ultraviolet which matches with the anomaly-free R-symmetry in the gauge theory. The supergravity solutions preserve only a Z2symmetry in the infrared and the breaking of the Z2N symmetry down to Z2gives N discrete vacua,reproducing the same pattern of symmetry breaking by gaugino condensation in the gauge theory.Gauge theory.—Consider N=1supersymmetric pure SU(N)gauge theory.The classical theory has global U(1)R-symmetry which is anomalous in the quantum theory.The anomaly-free quantum theory has a reduced Z2N discrete symmetry. Gaugino condensation breaks the Z2N symmetry down to Z2giving N number of discrete vacua.The low energy infrared dynamic of this theory at the scaleΛis described by the Veneziano-Yankielowicz superpotential[10],W VY=NS−NS log(S32π2Tr WαWα,(1)where S is the glueball superﬁeld deﬁned in terms of the gauge chiral superﬁeld Wαcontaining the gauge and the gauginoﬁelds in the N=1vector multiplet.Ex-tremizing W VY with S gives the vacuum expectation value of the glueball superﬁeld corresponding to the N vacua,S =Λ3e2πik/N,k=1,2,···,N.(2) Let us deﬁne T=8π2/g2,where g is the Yang-Mills coupling constant in the gauge theory.The quantum loop corrections to the running of the gauge coupling areexhausted at one loop and with the exactβfunction we havedTdτ(e−Φ)=g s N2e1,G2=A e x+g2˜ǫ2,G3=ex−g2e2−A e x−g2dτ,G6=e−6p−xF3=−12α′Ne−Φ(h1+bh2),F1=−Ne−Φc+g s N4α′g s N(1−τcothτ)cschτ,h1=h2coshτ,K=−12πe6p+x2π˜ǫ3.(14)Deﬁning v=e6p+2x,u=e2x and h=e−4A,the equations for v,u and h arev′+(2cothτ+3g s Ndτln(uv−g s N4g s eΦKh2πeΦh=0,(17)withΦgiven by(11)and K given in(13).A consistent set of solutions requires the boundary conditions v(0)=0,u(0)=0,and h(0)=h0.Equation(15)is easily solved for v,(16)is then solved for u/h and the result is used for h/u in(17)to write integral solution for h which is then used with the expression for u/h toﬁnd u.The values of u and v increase towards the ultraviolet asτincreases.The value of h also starts increasing asτincreases fromτ=0because of the sign of the third term in (17)which comes from the running of the dilaton in the asymptotically-free gauge theory here and is opposite to that in[7]where h decreases.D7and Dirac8-branes.—As it is shown in[7],the runnings of the axion and the dilaton are due to magnetic D7-branesﬁlling4-d spacetime and wrapping the 4-cycle1ω4=sinθ1sinθ2dθ1∧dφ1∧dθ2∧dφ2=ψ,˜Q7= 04πdC0=2N.(20)2πTherefore,we have2N number of D7-branes.Bottom of background.—Now we explore the geometry at the bottom of the background.Let us expand the variables in the metric given by(12)and solving(15)-(17)to leading order inτnearτ=0,e g=τ+O(τ3),a=−1+O(τ3),A=τ+O(τ3),B=1+O(τ2),h=h0+O(τ),v=τ+O(τ2),u=e2Φc h0τ2+O(τ3),(21) where h0=h(0).Therefore,nearτ=0,we have e x+g=u1/2e g=eΦc h1/20τ2+O(τ3), e x−g=u1/2/e g=eΦc h1/20+O(τ),e−6p−x=u1/2/v=eΦc h1/20+O(τ).The1-forms in the metric given by(5)nearτ=0have the forms G1∼(e2Φc h0)1/4τe1,G2∼(e2Φc h0)1/4˜ǫ2,G3∼(e2Φc h0)1/4˜ǫ1,G4∼−(e2Φc h0)1/4τǫ2,G5∼(e2Φc h0)1/4dτ,and G6∼(e2Φc h0)1/4˜ǫ3.The metric nearτ=0is thenηµνdxµdxν+eΦc h1/20 dτ2+˜ǫ21+˜ǫ22+˜ǫ23+τ2(e21+ǫ22) ,(22) ds2∼h−1/2where the˜ǫi are given by(7)with a∼−1.Therefore,the geometry atτ=0is R1,3×S3with the radius of S3a function of the magnitude of the vacuum expectation value of the glueball superﬁeld S atτ=0which is simply the’t Hooft coupling[14] eΦc g s N and conﬁnement via gaugino condensation is the source for the deformationof the tip as established in[5].The boundary value h0in solving(15)-(17)is then related to the radius of S3atτ=0and,therefore,a function of the’t Hooft coupling.Large N and large’t Hooft coupling.—Let us check the conditions on the param-eters in the theory for the supergravity description to be good.We need a large value of N so that only planar Feynman graphs survive and a large’t Hooft coupling.The ’t Hooft coupling is related to the magnitude of the vacuum expectation value of the glueball superﬁeld and the size of the holes in’t Hooft’s ribbon graphs have large size and can deﬁne Riemann surface for a string worldsheet for large’t Hooft cou-pling ifﬁlled by D-brane disks[15].On the other hand,for small’t Hooft coupling, the gauge theory has a good perturbative description.Let us see the implication of the large’t Hooft coupling constraint in our case,particulary now that we have a dilaton whose magnitude decreases asτincreases.The running of the dilaton is given by(11).In order for eΦg s N>>1,we need eΦc g s N(1−τg s e−Φ+C0=i2πψ,(23)where we have used(20)for C0and the subscript inτad is for axion-dilaton in order to avoid confusion in notation with the radial variableτ.Note that the second term in(23)corresponds to a Yang-Mills angle ofΘ=−Nψand clearly shows thatψ→ψ+c,whereψ=ψ+4π,is anomalous U(1)symmetry andΘis left invariant underψ→ψ+4π2N n,where1≤n≤2N.This discretesymmetry inτad corresponds to the symmetry in the locations onψ,where dψis related to G6as given in(6)and(7),at which the D7-branes in the ultraviolet edge wrap theω4cycle.Both the F1ﬂux and the F3ﬂux in the ultraviolet contain dψand are single-valued.Theﬂuxes in the supergravity solutions have non-vanishing components which explicitly contain sinψand cosψin the infrared as we see in the expressions(8)and(9)together with(12)and(13)and preserve only a Z2symmetrycorresponding toψ→ψ+2π.Therefore,the Z2N symmetry is broken down to Z2 by the solutions in the gravity theory and gives the same N number of discrete vacua in the infrared as in the gauge theory.Conclusions.—Theﬁnal picture we have is that the electric N=1supersym-metric SU(N)gauge theory lives on the electrically charged N number of D5-branes ﬁlling4-d spacetime and wrapping S2with vanishing S3at the infrared end before geometric transition.The supergravity solutions involve the background withﬂuxes after the transition with S2blown-down and S3ofﬁnite size at the tip which gives the familiar gravitational description to conﬁnement via gaugino condensation in the gauge theory.The2N number of magnetic D7-branesﬁll up4-d spacetime and wrap 4-cycles at the ultraviolet edge with invisible Dirac8-branesﬁlling4-d spacetime and emanating from the D7-branes and the F1ﬂux through G6which is related to the running of the dilaton.We also have the backreaction NS-NS H3ﬂux and the F5ﬂux eﬀectively coming from the wrapped D5fractional D3-branes.The quan-tum theory has Z2N discrete symmetry in the ultraviolet and arises robustly from our solution for the axion potential.The Z2N symmetry is broken down to Z2by the supergravity solutions in the infrared giving N vacua as in the gauge theory.It is satisfying that the C0potential which comes from the F1ﬂux which is a crucial component of our construction obtained using the equations for type IIBﬂows with N=1supersymmetry we obtained in[8]and the gauge/gravity duality mapping with running dilaton and running axion we obtained in[7]has provided a consistent picture.Most importantly,the renormalization groupﬂow of the gauge theory is reproduced in the gravity theory.What can we say about the gauge theory which lives on the2N magnetic D7-branes?Because the number of D7-branes is the same as the order in the Z2N discrete symmetry in the ultraviolet,it is convenient to wrap each one of the D7-branes over a4-cycle at each one pointψ=4πReferences[1]J.Maldacena,Adv.Theor.Math.Phys.2(1998)231–252,[hep-th/9711200].[2]S.S.Gubser,I.R.Klebanov,and A.M.Polyakov,Phys.Lett.B428(1998)105–114,[hep-th/9802109].[3]E.Witten,Adv.Theor.Math.Phys.2(1998)253–291,[hep-th/9802150].[4]J.Polchinski and M.J.Strassler,hep-th/0003136.[5]I.R.Klebanov and M.J.Strassler,JHEP08(2000)052,[hep-th/0007191].[6]J.M.Maldacena and C.Nunez,Phys.Rev.Lett.86(2001)588–591,[hep-th/0008001].[7]G.Hailu,arXiv:0711.1298[hep-th].[8]G.Hailu,JHEP10(2007)082,[arXiv:0709.3813[hep-th]].[9]R.Gopakumar and C.Vafa,Adv.Theor.Math.Phys.3(1999)1415–1443,[hep-th/9811131].[10]G.Veneziano and S.Yankielowicz,Phys.Lett.B113(1982)231.[11]G.Hailu and S.H.H.Tye,JHEP08(2007)009,[hep-th/0611353].[12] A.Butti,M.Grana,R.Minasian,M.Petrini,and A.Zaﬀaroni,JHEP03(2005)069,[hep-th/0412187].[13]G.Papadopoulos and A.A.Tseytlin,Class.Quant.Grav.18(2001)1333–1354,[hep-th/0012034].[14]G.’t Hooft,Nucl.Phys.B72(1974)461.[15]H.Ooguri and C.Vafa,Nucl.Phys.B641(2002)3–34,[hep-th/0205297].。

Exceptional Confinement in G(2) Gauge Theory

The triviality of the center has profound consequences for the way in which conﬁnement is realized. In particular, a static “quark” in the fundamental {7} representation of G(2) can be screened by three G(2) “gluons” in the adjoint {14} representation. As a result, in G(2) Yang-Mills theory the color ﬂux string connecting two static G(2) “quarks” can break due to the creation of dynamical gluons. This phenomenon is reminiscent of full QCD (with an SU (3) color gauge group) in which the string connecting a static quark and anti-quark can break due to the pair creation of light dynamical quarks. Indeed, 6 of the 14 G(2) gluons transform as {3} and {3} under the SU (3) subgroup of G(2) and thus qualitatively behave like dynamical quarks and anti-quarks. In particular, they explicitly break the Z Z(3) center symmetry of the SU (3) subgroup down to the trivial center of G(2). The remaining 14 − 6 = 8 G(2) “gluons” transform as {8} under the SU (3) subgroup and hence resemble the ordinary gluons familiar from QCD. It should be pointed out that — despite the broken string — just like full QCD, G(2) Yang-Mills theory is still expected to conﬁne color. In particular, one does not expect colored states of single G(2) “gluons” in the physical spectrum. The triviality of the center of G(2) Yang-Mills theory also aﬀects the physics at high temperatures. In particular, there is not necessarily a deconﬁnement phase transition, and we expect merely a crossover between a low-temperature “glueball” regime and a high-temperature G(2) “gluon” plasma. Due to the triviality of the center, unlike e.g. for SU (Nc ) Yang-Mills theory, there is no qualitative diﬀerence between the low- and the high-temperature regimes because the Polyakov loop is no longer a good order parameter. It is often being asked which degrees of freedom are responsible for conﬁnement. Popular candidates are dense instantons, merons, Abelian monopoles and center vortices. Center vortices (and ’t Hooft twist sectors) are absent in G(2) gauge theories, while the other topological objects potentially exist, although their identiﬁcation is a very subtle issue that often involves unsatisfactory gauge ﬁxing procedures. At strong coupling G(2) lattice gauge theories still conﬁne without a center. Hence, center vortices should not be necessary to explain the absence of colored states in the physical spectrum [13]. Still, the center plays an important role for the ﬁnite temperature deconﬁnement phase transition in SU (Nc ) Yang-Mills theory, and center vortices may well be relevant for this physics. If G(2) Yang-Mills theory indeed has no ﬁnite temperature deconﬁnement phase transition, one might argue that this is due to the absence of center vortices and twist sectors. Assuming that they can be properly deﬁned, Abelian monopoles are potentially present in G(2) gauge theory, and might be responsible for the absence of colored states. On the other hand, if — despite of the existence of Abelian monopoles — a deconﬁnement phase transition does not exist in G(2) Yang-Mills theory, monopoles might not be responsible for the physics of deconﬁnement. In any case, quantifying these issues in a concrete way is a very diﬃcult task. The exceptional conﬁnement in G(2) gauge theory can be smoothly connected with the usual SU (3) conﬁnement by exploiting the Higgs mechanism. When a

Low--dimensional sisters of Seiberg-Witten effective theory

b ˙ ¯ ¯ aD ¯ bα D aα Dα W = D W , α ˙
(5)Βιβλιοθήκη where a, b = (nothing, tilde) are the global SU (2) indices. The superﬁeld W can be naturally expressed in the framework of harmonic superspace approach (see the monography [4] and also recent [5] ), but do not themselves depend on harmonics in the chosen basis. This theory has (inﬁnitely) many diﬀerent classical vacua. Supersymmetric vacuum has zero energy. At the classical level, it has zero potential energy. Note ¯ φ] = 0, which now that the potential commutator term in (2) vanishes when [φ, implies that φ belongs to the Cartan subalgebra of the corresponding Lie algebra.
α ˙ ¯θ ¯2 = 1. In ¯2 = θα d2 θ θ2 = d2 θ Our convention is close to that of Ref. [2], θ2 = θα θα θ ˙θ , ˙ ˙ the following we will also use (σ µ )αβ σ µ )βα = {1, −τ }βα . But our Minkowski metric ˙ = {1, τ }αβ ˙ , (¯ ηµν = diag(1, −1, −1, −1) diﬀers in sign from Wess and Bagger’s conventions and we include the extra factor 2 in the deﬁnition of V . 3

The QCD vacuum

a r X i v :h e p -l a t /9709066v 1 18 S e p 19971INLO-PUB-7/97The QCD vacuum ∗Pierre van BaalaaInstituut-Lorentz for Theoretical Physics,University of Leiden,PO Box 9506,NL-2300RA Leiden,The NetherlandsWe review issues involved in understanding the vacuum,long-distance and low-energy structure of non-Abelian gauge theories and QCD.The emphasis will be on the role played by instantons.1.INTRODUCTIONThe term “QCD vacuum”is frequently abused.Only in the case of the Hamiltonian formulation is it clear what we mean by the vacuum:it is the wave functional associated with the lowest energy state.Observables create excitations on top of this vacuum.Knowing the vacuum is knowing all:We should know better.Strictly speaking the vacuum is empty.Nev-ertheless its wave functional can be highly non-trivial,deviating considerably from that of a non-interacting Fock space,based on a quadratic the-ory.Even in the later case the result of probing the vacuum by boundaries is non-trivial as we know from Casimir.The probe is essential:one needs to disturb the vacuum to study its prop-erties.Somewhat perversely the vacuum may be seen as a relativistic aether.It promises to mag-ically resolve our problems,from conﬁnement to the cosmological constant.For the latter super-symmetry is often called for to remove the other-wise required ﬁne-tuning.It merely hides the rel-ativistic aether,even giving it further structure.Remarkably it seems to have enough structure to give a non-trivial example of the dual supercon-ductor at work [1].Most will indeed put their bet on the dual su-perconductor picture for the QCD vacuum [2],and this has motivated the hunt for magnetic monopoles using lattice techniques,long before supersymmetric duality stole the show [1].The deﬁnitions rely on choosing an abelian projec-tion [3]and the evidence is based on the no-tion of abelian dominance [4],establishing the dual Meissner eﬀect [5],or the construction of22.V ACUUM DEMOCRACYThe model we wish to describe here starts from the physics in a small volume,where asymptotic freedom guarantees that perturbative results are valid.The assumption is made,that at least for low-energy observables,integrating out the high-energy degrees of freedom is well-deﬁned pertur-batively and all the non-perturbative dynamics is due to a few low-lying modes.This is most easily deﬁned in a Hamiltonian setting,since we are in-terested in situations where the non-perturbative eﬀects are no longer described by semiclassical methods.plete gaugeﬁxingDue to the action of the gauge group on the vectorﬁelds,aﬁnite dimensional slice through the physical conﬁguration space(gauge inequivalent ﬁelds)is bounded.One way to demonstrate this is by using the complete Coulomb gaugeﬁxing, achieved by minimising the L2norm of the gauge ﬁeld along the gauge orbit.At small energies,ﬁelds are suﬃciently smooth for this to be well deﬁned and it can be shown that the space under consideration has a boundary,deﬁned by points where the norm is degenerate.These are by deﬁ-nition gauge equivalent such that the wave func-tionals are equal,possibly up to a phase factor in case the gauge transformation is homotopically non-trivial.The space thus obtained is called a fundamental domain.For a review see ref.[10].2.2.Non-perturbative dynamicsGiven a particular compact three dimensional manifold M on which the gauge theory is deﬁned, scaling with a factor L allows one to go to larger volumes.It is most convenient to formulate the Hamiltonian in scale invariantﬁeldsˆA=LA.Di-viding energies by L recovers the L dependence in the classical case,but the need of an ultraviolet cutoﬀand the resulting scale anomaly introduces a running coupling constant g(L),which in the low-energy eﬀective theory is the only remnant of the breaking of scale invariance.When the volume is very small,the eﬀective coupling is very small and the wave functional is highly localised,staying away from the bound-aries of the fundamental domain.We may com-pare with quantum mechanics on the circle,seen as an interval with identiﬁcations at its bound-ary.At which points we choose these boundaries is just a matter of(technical)convenience.The fact that the circle has non-trivial homotopy,al-lows one to introduce aθparameter(playing the role of a Bloch momentum).Expressed inˆA,the shape of the fundamental domain and the nature of the boundary condi-tions,is independent of L.Due to the rise of the running coupling constant with increasing L the wave functional spreads out over the fundamental domain and will start to feel the boundary iden-tiﬁcations.This is the origin of non-perturbative dynamics in the low-energy sector of the theory. Quite remarkably,in all known examples(for the torus and sphere geometries),the sphalerons lie exactly at the boundary of the fundamental domain,with the sphaleron mapped into the anti-sphaleron by a homotopically non-trivial gauge transformation.The sphaleron is the saddle point at the top of the barrier reached along the tun-nelling path associated with the largest instanton, its size limited by theﬁnite volume.For increasing volumes the wave functionalﬁrst starts to feel the boundary identiﬁcations at these sphalerons,“biting its own tail”.When the en-ergy of the state under consideration becomes of the order of the energy of this sphaleron,one can no longer use the semiclassical approximation to describe the transition over the barrier and it is only at this moment that the shift in energy be-comes appreciable and causes sizeable deviations from the perturbative result.This is in particular true for the groundstate energy.Excited states feel these boundary identiﬁcations at somewhat smaller volumes,but nodes in their wave func-tional near the sphaleron can reduce or postpone the inﬂuence of boundary identiﬁcations.This has been observed clearly for SU(2)on a sphere[11].The scalar and tensor glueball mass is reduced considerably due to the boundary identi-ﬁcations,whereas the oddball remains unaﬀected (seeﬁg.1).These non-perturbative eﬀects re-move an unphysical near-degeneracy in perturba-tion theory(with the pseudoscalar even slightly lower than the scalar glueball mass).The dom-inating conﬁgurations involved are associated to3instantonﬁelds,in a situation where semiclassical techniques are inappropriate for computing the magnitude of the eﬀect.When boundary identi-ﬁcations matter,the path integral receives large contributions from conﬁgurations that have non-zero topological charge,and in whose background the fermions have a chiral zero mode,its conse-quences to be discussed later.fFigure1.The low-lying glueball spectrum on a sphere of radius R as a function of f=g2(R)/2π2 atθ=0.Approximately,f=0.28corresponds to a circumference of1.3fm.From ref.[11].At some point technical control is lost,since so far only the appropriate boundary conditions near the sphalerons can be implemented.As soon as the wave functional starts to become apprecia-ble near the rest of the boundary too,this is no longer suﬃcient.This method has in particular been very suc-cessful to determine the low-lying spectrum on the torus in intermediate volumes,where for SU(2)agreement with the lattice Monte Carlo re-sults has been achieved within the2%statistical errors[10,12].In this case the non-perturbative sector of the theory was dominated by the en-ergy of electricﬂux(torelon mass),which van-ishes to all orders in perturbation theory.The leading semiclassical result is exp(−S0/g(L)),due to tunnelling through a quantum induced barrier of height E s=3.21/L and action S0=12.5.Al-ready beyond0.1fm this approximation breaks down.Oneﬁnds,accidentally in these small vol-umes,the energy to be nearly linear in L.The eﬀective Hamiltonian in the zero-momen-tum gaugeﬁelds,derived by L¨u scher[13],and later augmented by boundary identiﬁcations to include the non-perturbative eﬀects[12],breaks down at the point where boundary identiﬁca-tions in the non-zero momentum directions as-sociated with instantons become relevant.The sphaleron has an energy72.605/(Lg2(L))and was constructed numerically[14].Its eﬀect be-comes noticeable beyond volumes of approxi-mately(0.75fm)3.For SU(3)this was veriﬁed directly in a lattice Monte Carlo calculation of theﬁnite volume topological susceptibility[15]. The results for the sphere have shown that also these eﬀects can in principle be included reliably, but the lack of an analytic instanton solution on T3×I R has prevented us from doing so in practise.2.3.Domain formationThe shape of the fundamental domain depends on the geometry.Assuming that g(L)keeps on growing with increasing L,causing the wave func-tional to feel more and more of the boundary,one would naturally predict that the inﬁnite volume limit depends on the geometry.This is clearly unacceptable,but can be avoided if the ground state obtained by adiabatically increasing L is not stable.Thus we conjecture that the vacuum is unstable against domain formation.This is the minimal scenario to make sure that at large vol-umes,the spectrum is independent of its geome-try.Domains would naturally explain why a non-perturbative physical length scale is generated in QCD,beyond which the coupling constant will stop running.However,we have no guess for the order parameter,let alone an eﬀective theory de-scribing excitations at distances beyond these do-mains.Postulating their existence,nevertheless a number of interesting conclusions can be drawn. The best geometry to study domain formation is that of a box since it is space-ﬁlling.We can exactlyﬁll a larger box by smaller ones.This is not true for most other geometries.In small to intermediate volumes the vacuum energy density is a decreasing function[12]of L,but in analogy to the double well problem one may expect that at stronger coupling the vacuum energy density rises again with a minimum at some value L0, assumed to be0.75fm.For L suﬃciently larger than L0it thus becomes energetically favourable to split the volume in domains of size L30.Since the ratio of the string tension to the4scalar glueball mass squared shows no structure around(0.75fm)3,we may assume that both have reached their large volume value within a domain. The nature of theirﬁnite size corrections is suﬃ-ciently diﬀerent to expect these not to cancel ac-cidentally.The colour electric string arises from the fact thatﬂux that enters the box has to leave it in the opposite direction.Flux conservation with these building blocks automatically leads to a string picture,with a string tension as com-puted within a single domain and a transverse size of the string equal to the average size of a domain,0.75fm.The tensor glueball in an in-termediate volume is heavily split between the doublet(E+)and triplet(T+2)representations of the cubic group,with resp.0.9and1.7times the scalar glueball mass.This implies that the tensor glueball is at least as large as the average size of a domain.Rotational invariance in a domain-like vacuum comes about by averaging over all orien-tations of the domains.This is expected to lead to a mass which is the multiplicity weighted average of the doublet and triplet,yielding a mass of1.4 times the scalar glueball mass.Domain formation in this picture is driven by the largeﬁeld dynam-ics associated with sphalerons.Which state gets aﬀected most depends in an intricate way on the behaviour of the wave functionals(cmp.ﬁg.1). In the four dimensional euclidean context,O(4) invariance makes us assume that domain forma-tion extends in all four directions.As is implied by averaging over orientations,domains will not neatly stack.There will be dislocations which most naturally are gauge dislocations.A point-like gauge dislocation in four dimensions is an in-stanton,lines give rise to monopoles and surfaces to vortices.In the latter two cases most natu-rally of the Z N type.We estimate the density of these objects to be one per average domain size. We thus predict an instanton density of3.2fm−4, with an average size of1/3fm.For monopoles we predict a density of2.4fm−3.If an eﬀective colour scalarﬁeld will play the role of a Higgsﬁeld,abelian projected monopoles will appear.It can be shown[16]that a monopole (or rather dyon)loop,with its U(1)phase rotat-ing Q times along the loop(generating an elec-tricﬁeld),gives rise to a topological charge Q.In abelian projection it has been found that an instanton always contains a dyon loop[17].We thus argue this result to be more general,leading to further ties between monopoles and instantons.2.4.Regularisation andθIt is useful to point out that the non-trivial ho-motopy of the physical conﬁguration space,like non-contractable loops associated to the instan-tons(π1(A/G)=π3(G)=Z Z),is typically de-stroyed by the regularisation of the theory.This is best illustrated by the example of quantum me-chanics on the circle.Suppose we replace it by an annulus.As long as the annulus does notﬁll the hole,or we force the wave function to vanish in the middle,theta is a well-deﬁned parameter associated to a multivalued wave function.We may imagine the behaviour for small instantons in gauge theories to be similar to that at the cen-ter in the above model.Indeed,the gauge in-variant geodesic length of the tunnelling path for instantons on M×I R,given byℓ= ∞−∞dt4) 2)), with the instanton size deﬁned byρ≡(1+b2)−15ter by mixing the electricﬁeld withθtimes themagneticﬁeld,E→E−θB/(2π)2.In these ap-proaches theta is simply a parameter added to the theory.Whether or not one will retrieve the ex-pected periodic behaviour in the continuum limit becomes a dynamical question.It should be pointed out that in particular forSU(N)gauge theories in a box(in sectors with non-trivial magneticﬂux)there is room to arguefor a2πN,as opposed to a2π,periodicity fortheθdependence.However,the spectrum is pe-riodic with a period2π,and the apparent dis-crepancy is resolved by observing that there is a non-trivial spectralﬂow[19].This may lead tophase transitions at some value(s)ofθ,relatedto the oblique conﬁnement mechanism[3].Sim-ilarly for supersymmetric gauge theories this in-terpretation,supported by the recent discoveryof domain walls between diﬀerent vacua[20],re-moves the need for semiclassical objects with acharge1/N.Such solutions do exist for the torus, but the fractional charge is related to magneticﬂux and the interpretation is necessarily as statedabove!The“wrong”periodicity in theta has long been used to argue against the relevance of in-stantons,but in the more recent literature this isnow phrased more cautiously[21,22].3.INSTANTONSInstantons are euclidean solutions responsiblefor the axial anomaly,breaking the U A(1)sub-group of the U(N f)×U(N f)chiral symmetry for N fﬂavours of massless fermions[23],as dictatedby the anomaly, f∂µ¯Ψfγµγ5Ψf(x)=2N f q(x). The breaking of U A(1)manifests itself in thesemiclassical computations through the presenceof fermion zero modes,with their number and chiralityﬁxed by the topological charge,through the Atiyah-Singer index theorem[24].Integra-tion over the fermion zero modes leads to the so-called’t Hooft vertex or eﬀective interaction[23]. The integration over the scale parameter of the instanton ensemble is infrared dominated and a non-perturbative computation is desirable.In addition it is believed that the instantons are responsible for chiral symmetry breaking,where a chiral condensate is formed,which breaks the axial gauge group U A(N f)completely.This spon-taneous breaking is dynamical and it is less well established that instantons are fully responsible.It is the basis of the instanton liquid model as developed by Shuryak over the years.For a com-prehensive recent review see ref.[9].The details of the instanton ensemble play an important role. Only a liquid-like phase,as opposed to the di-lute or crystalline phases,will give rise to a chiral condensate.The model also makes a prediction for the average size and the topological suscep-tibility.In particular the latter quantity should be well-deﬁned beyond a semiclassical approxima-tion.For large sizes the instanton distribution is exponentially cut-oﬀand instantons do not give rise to an area law for the Wilson loop.Whenlarge instantons are more weakly suppressed the situation may diﬀer[25],but a semiclassical anal-ysis in this case should not be trusted. Remarkably the topological susceptibility in pure gauge theories can be related to theη′mass through the so-called Witten-Veneziano relation,f2π(m2η+m2η′−2m2K)/2N f= d4x<T(q(x)q(0))>R ≡χt,leading to the predictionχt∼(180MeV)4. This is based on the fact that the U A(1)symme-try is restored in the planar limit[26,27],withχt of order1/N2.From the requirement that in the presence of massless quarksχt(and the thetadependence)disappears,the pure gauge suscep-tibility can be related to the quark-loop contri-butions in the pseudoscalar channel.Pole dom-inance requires the lightest pseudoscalar meson to have a mass squared of order1/N.Relating the residue to the pion decay constant gives the desired result[27].The index R indicates the ne-cessity of equal-time regularisation[26].A deriva-tion on the lattice using Wilson and staggered fermions was obtained in ref.[28],making use of Ward-Takahashi identities.Finally,also the coarse grained partition function of the instan-ton liquid model[9]allows one to directly deter-mine the’t Hooft eﬀective Lagrangian[23],from which the Witten-Veneziano formula can be read oﬀ[29].This formula is almost treated as the holy grail of instanton physics.It is important to realise that some approximations are involved, although it is gratifying there are three indepen-dent ways to obtain it[26,27,28,29].63.1.Field theoretic methodA direct computation ofχt= d4x<q(x)q(0)> on the lattice requires a choice of discretisation for the charge density.A particularly simple one is[30]q L(x)=− T r(Uµν(x)˜Uµν(x))/16π2, where Uµν(x)=17 cutoﬀbreaks the scale and rotational invariance,we would expect that the action is no longer con-stant on the continuum moduli space.Indeed,fora smooth instanton the Wilson action behaves asS W(ˆρ≡ρ/a)=8π2(1−ˆρ−2/5+O(ˆρ−4))and causesthe instanton to shrink,until it becomes of thesize of the cutoﬀand falls through the lattice.Cooling willﬁrst remove high-frequency modesand one is left with a slow motion along the mod-uli space,giving rise to a plateau in the coolinghistory,used to identify the topological charge.One will miss instantons smaller than someﬁxedvalueˆρc.Assuming asymptotic freedom,one eas-ily shows that the error vanishes in the contin-uum limit.Note that by construction,the coolingmethod never will associate charge to a disloca-tion with an action smaller than96π2/11N,theentropic bound,which would spoil scaling[36].For extracting the size distribution,cooling andunder-relaxed(or slow)cooling[37]is problematicas the size clearly will depend on where alongthe plateau one analyses the data[38].The sizedistribution can be made to scale properly onlyat the expense of carefully adjusts the number ofcooling steps[39]when going to diﬀerentβ.One can avoid loosing instantons under coolingby modifying the action such that the scaling vio-lations change sign[35],for example by adding a2×2plaquette to the Wilson action.This so-calledover-improved action has the property that in-stantons grow under cooling,until stopped by theﬁnite volume.Consequently it would still muti-late the size distribution.This can be avoided byimproving the action so as to minimise the scalingviolations[40].A particularly eﬃcient choice isthe so-calledﬁve-loop improved(5Li)lattice ac-tion:S5Li= m,n c m,n x,µνTr(1−P mµ,nν(x)),where P mµ,nν(x)is the m×n plaquette andc1,1=65720,c1,2=−890andc3,3=18bining results forβ=2.4(averaged over the diﬀer-ent lattice types and boundary conditions after20 cooling sweeps)and forβ=2.6(after50cooling sweeps).The solid curve is aﬁt to the formula P(ρ)∝ρ7/3exp(−(ρ/w)p),with w=0.47(9)fm and p=3(1),which atsmall sizes coincides withthe semiclassical result[23].The peak of this dis-tribution occurs atρ=0.43(5)fm.Under pro-longed cooling,up to300sweeps,I-A instantonannihilations and in particularﬁnite size eﬀectsin the charge one sector do aﬀect the distribu-tion somewhat,but not the average size,whichtherefore seems to be quite a robust result.Figure3.SU(2)instanton size distribution forβ=2.4(squares)and2.6(crosses)in a volume1.44fm across at lattice spacings a=0.12and0.06fm.The dotted and dashed lines representthe cutoﬀat aˆρc for both lattices.From ref.[40].It would be advantageous if one could come upwith a deﬁnition for the size that is related to aphysical quantity,since now the notion is basedon the semiclassical picture.This is neverthe-less appropriate for the comparison with the in-stanton liquid.The relatively large value of theaverage size as compared to that of1/3fm pre-dicted by the instanton liquid[9]is a point ofworry,typically leading to stronger interactionsthat may lead to a crystal(without chiral sym-metry breaking),rather than a liquid.Neverthe-less,in ref.[40]it has been tested that the pseu-doparticles are homogeneously distributed with adensity of2−3fm−4and occupying nearly halfthe volume.This is the case when only close I-Apairs have annihilated and therefore depends onthe amount of improved cooling.It does,how-ever,show that the pseudoparticles are relativelydense(more so than assumed in the instanton liq-uid[9]).The value of3.2fm−4for the density,de-rived earlier in the context of the domain pictureis quite realistic in the light of these results.3.3.SmoothingAnother method to study instantons on thelattice is based on the classicalﬁxed point ac-tions[41],deﬁned through the saddle point equa-tion S F P(V)=min{U}(S F P(U)+κT(U,V)).It isobtained as the weak coupling limit of the block-ing transformation with a positive deﬁnite ker-nelκT(U,V),which maps a lattice{U}to{V},coarser by a factor two.Reconstructing theﬁnefrom the coarse lattice is called inverse blocking.It can be shown to map a solution of the latticeequations of motion to a solution on theﬁner lat-tice with the same action.Iterating the inverseblocking,the lattice can be made arbitrarilyﬁne,thereby proving the absence of scaling violationsto any power in the lattice spacing[41].Thisclassically perfect action still looses instantons be-low a critical size,which is typically smaller thana lattice spacing.For solutions this most likelyhappens at the point where the continuum inter-polation of the latticeﬁeld is ambiguous,causingthe integer geometric charge[42]to jump.Forrough conﬁgurations that are not solutions,in-verse blocking typically reduces the action by afactor32and makes it more smooth.Theﬁxedpoint topological charge is deﬁned as the limit-ing charge after repeated inverse blockings.Thisguarantees no charge will be associated to disloca-tions(of any action below the instanton action).The classicalﬁxed point action,although op-timised to be short range,still has an inﬁnitenumber of terms andﬁnding a suitable trunca-tion is a practical problem.From examples ofparametrisations,the success of reducing scalingviolations in quantities like the heavy quark po-tential(tested by restoring rotational invariance)is evident,for recent reviews see ref.[45].In prac-tise only a limited number of inverse blockings isfeasible and theﬁxed point topological charge hasto rely on a rapid convergence.The closer one isable to construct theﬁxed point action the bet-9ter this convergence is expected to be.For two di-mensional non-linear sigma models suﬃcient con-trol was achieved to demonstrate that more than one inverse blockingdid not appreciably change the topological charge [43].In four dimensional gauge theories,both ﬁnd-ing a manageable parametrisation and doing re-peated inverse blockings is a major eﬀort.It goes without saying that if no good approximation to the ﬁxed point action is used,one cannot rely on its powerful theoretical properties.Studies of in-stantons for SU (2)gauge theories were performed in ref.[44].A 48term approximation to the ﬁxed point action was used to verify the theoretical properties.The geometric charge was measured after one inverse blocking and it was shown that for Q =1the instanton action was to within a few percent from the continuum action (slightly above it due to ﬁnite size eﬀects),whereas for Q =0the action was always lower.Subsequently a simpliﬁed eight parameter form was used on which the instanton action was somewhat poorly reproduced,but such that the Q =1boundary stayed above the entropic bound for the action.A value of χt =(235(10)MeV)4was quoted on a 84lattice with physical volumes of up to 1.6fm,taking full advantage of the fact that ﬁxed point actions can be simulated at rather large lat-tice spacings.However,<Q 2>measured on the coarse lattice was up to a factor 4larger than on the ﬁne lattice (for the two dimensional study much closer agreement was seen [43]).Further inverse blocking to check stability of the charge measurement was not performed.The same eight parameter action was used in ref.[46],but they did their simulations on the ﬁne lattice and performed an operation called smoothing:ﬁrst blocking and then inverse block-ing.They changed the proportionality factor κin the blocking kernel,requiring that the saddle point condition is satisﬁed for the blocked lat-tice.Due to the change of κthe properties of the ﬁxed point action that inspired these authors can unfortunately no longer be called upon as a justiﬁcation.This smoothing satisﬁes the prop-erties of cooling (the action always decreases and stays ﬁxed for a solution)and should probably by judged as such.(See for further comments be-low.)They consider their study exploratory and concentrate on ﬁnite temperature near and be-yond the deconﬁnement transition.In ref.[47]the number of terms to parametrise the ﬁxed point action was extended to four powers of resp.the plaquette,a six-link and an eight-link Wilson loop.The latter was required to improve on the properties for the classical solutions.They achieved ˆρc =0.94,still considerably smaller than for S 5Li ,and reproduced the continuum instan-ton action to a few percent for ρ>ρc .To increase the quality of the ﬁt a constant was added,which should vanish in the continuum limit (as it drops out of the saddle point equation).Possible rami-ﬁcations of this at ﬁnite coupling are not yet suf-ﬁciently understood.After one inverse blocking insuﬃcient smooth-ing is achieved to extract the pseudoparticle po-sitions and sizes and further inverse blocking was considered computationally too expensive.Like in ref.[46],they also introduced a smoothing cy-cle,but now by blocking the ﬁne lattice back to the coarse one.Such a cycle would not change the action when the blocking is indeed to the same coarse lattice.However,there are 24diﬀerent coarse sublattices associated to a ﬁne one and in ref.[47]the smoothing cycle involved blocking to the coarse lattice shifted along the diagonal over one lattice spacing on the ﬁne lattice.Unlike in ref.[46],the smoothing cycle will be repeated.Figure 4.Example of the smoothing after 1and 9cycles,shown on the ﬁne lattice.From ref.[47].Although the ﬁxed point nature of the action guarantees it is close to a perfect classical action it needs to be demonstrated that it preserves the topological charge at suﬃciently large scales.For cooling this is argued from the local nature of the updates,not aﬀecting the long distance be-haviour.Improved cooling is in this sense less。

The odd-dimensional Goldberg Conjecture

Theorem 1.2. Let (M, g, ξ) be a compact Sasakian–Einstein manifold of dimension 2n + 1. Then any contact metric structure (ξ′, g) on (M, g) is Sasakian. Moreover, if ξ′ is diﬀerent from ±ξ, then the following two cases occur:
Theorem 1.1. [5] Any compact Einstein K–contact manifold (M, g, ξ) is Sasakian.
Their proof goes roughly as follows. First, an Einstein K–contact manifold has prescribed (positive) Einstein constant. If the K–contact structure is quasi–regular (i.e. the orbits of the Reeb vector ﬁeld ξ are closed), then the quotient of M by the ﬂow of ξ is an almost Ka¨hler orbifold [9] which is Einstein with positive scalar curvature by the O’Neill formulas. One then applies Sekigawa’s proof to obtain that the almost Ka¨hler structure is integrable, which in turn means that the K–contact structure is Sasakian. If the K–contact structure is not quasi– regular, the space of orbits of ξ is not an orbifold (and may not be even a tractable topological space). To overcome this diﬃculty, the authors of [5] provide a beautiful argument showing that the Reeb vector ﬁeld ξ can be approximated (in a suitable sense) by a sequence of quasi– regular Reeb vector ﬁelds ξi which deﬁne K–contact structures on a sequence of (no longer Einstein) metrics gi approaching g. Then for the sequence of orbifolds thus obtained, one can use “approximative” Sekigawa formulas and eventually show that the K–contact structure is integrable.