Lorentz gauge and Gribov ambiguity in the compact lattice U(1) theory

合集下载

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.。

Gribov Copies in Lattice QCD

a rXiv:h e p-la t/93610v115J un1993Gribov Copies in Lattice QCD Paolo Marenzoni Dipartimento di Fisica Universit`a di Parma,Viale delle Scienze 43100Parma (Italy )Pietro Rossi CRS4Via Nazario Sauro,1009123Cagliari (Italy )Abstract We have performed an exhaustive search for Gribov copies on the lattice and their possible dependence from ﬁnite temperature eﬀects.We show that,for each value of lattice size,Gribov copies are dense in conﬁguration space at low temperature but their density tend to lower when the temperature increases.We have investigated lattice sizes running from 163×8to 164.Gribov[1]has shown that Landau’s gauge deﬁned by the equation∂·A=0(1) does notﬁx the gauge uniquely,that is given a gauge orbit A,satisfying the Landau condition we can alwaysﬁnd a gauge transformation g,such that A g also satisﬁes the equation(1).The gaugeﬁxing deﬁnition can be sharpened further by following Zwanziger[2]idea of requiring that,besides the Landau condition,all the eigenvalues of the Fadeev-Popov operator be positive.The domain deﬁned by the condition∂·A=0,and|∂·D|>0,is called theﬁrst Gribov horizon.Zwanziger was able to show that the Gribov horizon is a convex region and every gauge orbit crosses it at least once.From the point of view of weighting correctly conﬁguration in the functional integral it would be im-portant to show that every orbit goes through theﬁrst Gribov horizon only once.This statement has to be given meaning within measure theory in function space,that is all orbit contributing to the functional integral with non zero measure must cross theﬁrst Gribov horizon only once.As of today this statement can’t be proven in the continuum theory.By looking at the lattice formulation of the theory we can gain some insight.As many author have already shown,on the lattice we do have conﬁguration with Gribov copies as well as conﬁguration for which no Gribov’s copy has been found. Since,every conﬁguration partakes to the functional integral for every value of the coupling constant we have to discern whether they do with or without a measurable contribution.As we all know,the lattice theory presents two distinct phases in the physical temperature,and an interesting question is whether the measure on conﬁgurations admitting Gribov copies is temperature dependent and whether is aﬀected by the temperature transition.Secondly and more inter-estingly,we would like to know the measure in orbit space of conﬁgurations admitting Gribov copies.Beside these theoretical aspects of the quantumﬁeld theory,there is a pragmatic reason behind this investigation.One of the most eﬀective ways to increase the overlap between physical states and operators used to excite these states from the vacuum,is given by the smearing procedure[3].This consist of constructing non local operators in a way that the physical extent of these operators is constant and does not go to zero with the lattice spacing,thus maintaining a good overlap between the operator and the desired phys-ical state.This procedure,though,is not gauge invariant,since any attempt to construct smeared gauge-invariant operators is very costly and not very eﬀective.Therefore weﬁx the gauge,typically Landau or Coulomb,and we build our non local wave functions.This procedure would be questionable if we were to show thatﬁxing to a given gauge does not uniquely determine the point in orbit space.We have studied the problem of Gribov copies in the four dimensional SU(3)theory with the standard Wilson action.We are in agreement with previous results(see[4]and[5])where it’s found that the value ofβaﬀects the density of Gribov copies.In eﬀect weﬁnd that,while at lowβthese copies have an high density,at higher values they become much more rareﬁed,even if their density never goes to zero.The conjecture that Gribov copies would disappear at the critical temperature,does not seem to be true,and this qualitative behavior seems to be volume independent.In chapter one we brieﬂy review Zwanziger’s formulation of the Gribov problem and discuss the lattice implementation of the same,in chapter two we describe the numerical implementation of the gaugeﬁxing algorithm and ﬁnally in chapter three we discuss our results.1Landau’s Gauge Condition on the Lattice Let us consider the functionalF c[g]= d4xT rA gµ(x)A g†µ(x),(2)whereA gµ=g−1Aµg+g−1∂µg,(3) andg(x)=e−w(x),(4) is an SU(3)matrix and w is antihermitian and traceless.If we expand F c in w(x)we have:F c[g]=F c[1]+ d4xT r[∂·A(x)w(x)]− d4xT r w†(x)∂µDµw(x) +◦(w3),(5)whereDµw(x)=∂µw(x)+[Aµ(x),w(x)],(6) is the covariant derivative and∂·D,is the Faddeev-Popov operator.Zwanziger’s prescription to gaugeﬁx consists in choosing as representative on the orbit theﬁeld A such that F c attains a local minimum.It follows from()that Zwanziger’s condition is equivalent to∂·A=0and∂·D positive deﬁnite.This variational formulation of the gaugeﬁxing condition can be easily generalized to the lattice.If we look at the functional:F l[g]=Re x,µT rU gµ(x) ,(7)where(x)=g(x)Uµ(x)g†(x+µ),(8)U gµand againg(x)=e−w(x).(9) in the limit of a→0(where a is the lattice spacing)we haveF l[g]=F c[g]+o(A4).(10)Expanding in w(x)we can obtain the lattice deﬁnition of∂·A,that is∂·A a l=T r T a µ Uµ(x)−U†µ(x−µ) (11)and the stationary condition will be∂·A l=0.The lattice expression for the Faddeev-Popov operator is quite cumber-some,so we will not write it here but it is quite easy to obtain.2The Gauge Fixing Algorithm and Search for CopiesOur algorithm to gaugeﬁx an SU(3)conﬁguration is implemented on massive parallel machines,with a checkerboard updating of the gauge links.We startby observing that the functional F l[g]can be rewritten as a sum over even sites or odd sites only,for instance:F l[g]=Re x EV EN,µT r U gµ(x)+U g†µ(x+µ) ,(12)and a similar expression for odd sites.We can now apply,alternatively,gauge transformations which are non trivial on even sites and the identity on odd sites and viceversa as we switch from the deﬁnition for F l as a sum over even or odd sites.In this framework, each site contributes to the functional with an independent term and the functional dependence of each term from the gauge transformation is given by:Re g(x) µT r Uµ(x)+U†µ(x+µ) .(13) To implement our gaugeﬁxing prescription we only need to produce an algorithm that at each step increases the value of the local sum alternatively on each checkerboard,since this implies that the functional F l will increase monotonically.Since F l is also bounded this will guarantee the convergence of the algorithm.There are many such algorithms and we have implemented a variety of them.We perform such a gauge ransformation g(x)ﬁrst on a checkerboard and after on the other one.The iterations has been carried out until|∂·A l|2<10−14.(14) We,then,apply to the gaugeﬁxed conﬁguration U,a”large”random gauge transformation,then we repeat the gaugeﬁxing procedure generating a new conﬁguration U′.We are interested in whether U′is the same or not as U.In fact,since F l does not distinguish between conﬁguration that diﬀer for a global gauge transformation,we are only interested whether U and U′diﬀer for a global or a local gauge transformation.The simplest observable that can answer this question is1∆(U,U′)=As an insurance against round-oﬀerrors we have kept track of the gauge transformation needed toﬁx the gauge and,at the end,applied its inverse to the gaugeﬁxed conﬁguration.We have always been able to return to the original conﬁguration within an accuracy of10−13−10−14.3ResultsWe have generated conﬁgurations on lattices163×N t,with N t=8,10,14and 16,for a large set ofβvalues.The conﬁgurations have been generated with the Local Hybrid Monte Carlo algorithm and were separated by250−500 trajectories.For all theβconsidered we checked,for a set of observables, that the autocorrelation has never been larger than15.For each conﬁgu-ration we haveﬁxed to the lattice Landau gauge,then,starting from the original conﬁguration we have performed a random gauge transformation. Afterﬁxing this newly obtained conﬁguration we have compared the two gaugeﬁxed ones and determined whether we were in presence of a copy.The number of conﬁguration tried and the number of copies is summarized in table1.From the results reported on this table we could observe that,on all the lattices we have simulated,if at the lower values ofβthe percentage of Gribov copies over attempts is next to the100%,at higherβ’s it decreases to 10−20%.This happens smoothly and we do not observe any sharp transition around the critical temperature.References[1]V.N.Gribov,Nucl.Phys.B139(1978)1.[2]D.Zwanziger,Nucl.Phys.B209(1982)336.[3]P.Bacilieri et al.Nucl.Phys.B317(1989)509.[4]E.Marinari,C.Parrinello,R.Ricci,Nucl.Phys.B362(1991)487.[5]M.L.Paciello et al.Phys.Lett.B276(1992)163.Lattice N o of Conf 163×88888 163×10888 163×148888 163×168888。

验压仪测量仪说明书

Calibration Technology Starts HereDigital Test Gauge takes the concept of an analog Test Gauge,and brings it to a new level.The NEW BetaGauge P.I.PRO combines the accuracy of digital technology with the simplicity of an analog gauge and achieves performance,ease-of-use,and a feature set unmatched in the pressure measurement world.The BetaGauge PI PRO builds on the proven performance of the BetaGauge PI with a newpurpose designed Zinc-Aluminum Alloy High Pressure Cast housing.The new housing improves theenvironmental integrity of the gauge in a lighter and smaller yet more rugged form factor.Setup of the BetaGauge P.I.PRO is fast and straightforward,through a menu-driven display,that is simple enough to allow the gauge to be used anywhere in the world.Ease-of-use,performance,and features set the BetaGauge P.I.PRO Digital Pressure Test Gauge apart,just as you’ve come to expect from Martel Electronics!The BetaGauge P.I.PRO first and foremost,provides 0.05%of full scale accuracy of pressure in any one of seventeen (17)ranges.Readings may be displayed in any one of 18standard engineering units,or in 1custom unit you define,toeliminate tedious conversion calculations.The 5-1/2digit,0.65"high digital display and companion percent-of-range bar graph provide an easy to read display,even from a distance.A number of features provide additional ease of use.The reading sample rate can be user adjusted to match the type of measurement required.A power saving mode maximizes battery life to 2,000hours.Other featuresinclude MIN/MAX function,Auto Power Off and backlight control.Continued on backPrecision Digital Pressure Test Gauge The BetaGauge PI PROSingle-handed process validationGeneral Features.Accuracy of ±0.05%of F.S.(±0.04%of Reading for Reference Class Verse).Temperature compensated accuracy over 0to 50°C .17standard pressure ranges available .Data logging.Displays in 18standard or 1custom engineering unit .Displays ambient temperature in °C or °F.Large,back-lit,5-1/2digit display with 0.65"high digits and 20segment bar graph.Adjustable TARE zeros large system offsets.User-conﬁgurable sample rate maximizes measurement performance and battery life.User-conﬁgurable damping smooths readings from pulsating/plant air sources.Auto Shut-off for extended battery life .Low battery indicator.CSA intrinsically safe,Class 1,Div.2Groups A,B,C &D;ATEX approved;CE approved.Available with optional 24V external power input.Rubber boot standard on all stand-alone bottom mount gaugesFor more information:-Call us toll free at 800-821-0023or 603-434-1433-Fax us at 603-434-1653-Email us at ********************and ask for a BetaGauge PI Data Sheet -Or go to our website, and click on BetaGauge PI PRO -Online product comparisonBetaGauge PI PRO(Shown with optional pump)PROPump Kits shown with optionalOrder#BetaGauge-PI-xxx10”H20*/20mBAR,xxx=010;1psi*/70mBAR,xxx=001;15.000psi/1BAR,xxx=015;30.000psi/2BAR,xxx=030;100.00psi/7BAR,xxx =100;300.00psi/20BAR,xxx=300;500.00psi/30BAR,xxx=500;1,000.00psi/70BAR,xxx=01K;2,000.0psi/140BAR,xxx=02K;3,000.0psi/200BAR,xxx=03K;5,000.0psi/300BAR,xxx=05K;10,000psi/700BAR,xxx=10K;15,000psi/1035BAR,xxx=15K(Q42010);-15to+15psi*,xxx=015C;-15to+30psi*,xxx=030C-xPort Position Bottom(standard);x=__(leave blank);Back(option);x=RM -KitxKit(includes BetaGauge PI,case,rubber boot,pressure calibration handpump,hose andﬁttings)With MECP500pneumatic test pump(500psi and below);Kit5With MECP10K hydraulic test pump(above500psi);Kit3-24OPTIONAL:External24VDC Power InputATTENTION:All Rear Mount and24VDC External Input gauge configurationsare supplied in the previous BetaGauge PI housing.Only the standard bottommount configured gauges are supplied in the new BetaGauge PI PRO housing.BetaGauge P.I.PRO may beallows pressure reading dataoff-line analysis.(Requiresto a ZERO function thatsystem pressures can be nulledwhen combined with thedisplay,allows the BetaGaugeand volume measurements,easilytank,directly.Damping can be setsources,such as plant air.Anavailable for applications where theincorporated into a process line.Athe BetaGauge P.I.from damageavailable in a self-contained calibrationassociatedﬁttings.P.I.PRO gauges,the BetaGauge PIRGauge and combines it with theReference Class gauges providereading accuracy,especiallydigital test gauges arepsig,100psig,50010,000psig.Setup ofstraightforward,through aintuitive functions,thatanywhere in thedisplays.Pressure readingsengineering units,or in user-conversion calculations.high digital display andis temperature-compensatedtemperature sensor measurement can beBetaGaugePIR PROReferenceClassDistributed by:。

Theories with global gauge anomalies on the lattice

a r X i v :h e p -l a t /0010003v 1 3 O c t 20001Theories with global gauge anomalies on the latticeP.Mitra a∗aSaha Institute of Nuclear Physics,Block AF,Bidhannagar,Calcutta 700064,INDIAA global anomaly in a chiral gauge theory manifests itself in diﬀerent ways in the continuum and on the lattice.In the continuum case,functional integration of the fermion determinant over the whole space of gauge ﬁelds yields zero.In the case of the lattice,it is not even possible to deﬁne a fermion measure over the whole space of gauge conﬁgurations.However,this is not necessary,and as in the continuum,a reduced functional integral is suﬃcient for the existence of the theory.Presented at Lattice 2000,Bangalorehep-lat/00100031.IntroductionAnomalies are of two diﬀerent types.Local or divergence anomalies have been known since 1969[1]:classically conserved symmetry currents cease to be conserved after quantization if there are anomalies of this kind.For example,the the-oryL =¯ψ(i∂/−eA /)ψ−116π2F µνF µν=0.(2)If an anomalous current is associated with a gauged symmetry ,it leads to an apparent problem in quantization because the equations of motion of the gauge ﬁelds require the current to be con-served.A treatment of such a theory just like a usual gauge theory shows an inconsistency.This problem can be sorted out by paying proper at-tention to phase space constraints,as suggested by [2].The anomaly itself can be made to vanish in a sense by going to the constrained subman-ifold of classical phase space.However,theories with anomalous gauge currents are to be distin-guished from theories with nonanomalous gauge currents.If a theory is nonanomalous ,it pos-sesses gauge freedom,and is describable in any22.Functional integral in the continuum The full partition function of the gauge theory with fermions may be written asZ= D AZ[A],(3) Z[A]≡e−S eff= D¯ψDψe−S(ψ,¯ψ,A)(4)An anomaly-free theory has Z[A]gauge invariant. If there is a gauge anomaly,Z[A]varies under gauge transformations of A:Z[A g]=e iα(A,g−1)Z[A],(5) whereαmay be regarded as an integral represen-tation of the anomaly.It obeys some consistency conditions(mod2π):α(A,g−12g−11)=α(A,g−11)+α(A g1,g−12)α(A,g−1)=−α(A g,g).(6) The case becomes one of a global anomaly ifαis independent of A,and vanishes for g connected to the identity but not for some g which cannot be continuously connected to the identity.A-independence implies an abelian representation satisfyingα(g2g1)=α(g1)+α(g2).(7) In the SU(2)case,the two components of the gauge group manifest themselves in two possible values of the phase:e iα=±1.In an anomaly-free theory,the partition func-tion factorizes into the volume of the gauge group and the gauge-ﬁxed partition function:Z= D AZ[A]= D AZ[A] D gδ(f(A g))∆f(A)= D g D AZ[A g−1]δ(f(A))∆f(A) = D g D AZ[A]δ(f(A))∆f(A)=( D g)Z f(8)This is the standard Faddeev-Popov argument. Here,δ(f)represents a gauge-ﬁxing operation and∆f is the corresponding Faddeev-Popov de-terminant.This decoupling of gauge degrees of freedom does not occur if a local anomaly is present.For a global anomaly however,the partition function factorizes again:Z= D ge−iα(g) D AZ[A]δ(f(A))∆f(A)(9)As the phase factors form a representation of the gauge group,D ge−iα(g)= D(gh)e−iα(gh)=e−iα(h) D ge−iα(g)(10)where h stands for aﬁxed gauge transformation. If h is not connected to the identity,e−iα(h)=1, and consequently D ge−iα(g)=0,which in turn means that Z=0.Does this mean that the the-ory cannot be deﬁned?Let us look at expectation values of gauge invariant operators.D AZ[A]OD ge−iα(g) D AZ[A]δ(f(A))∆f(A)(11) The expression on the right is of the form0D AZ[A]δ(f(A))∆f(A).(12)The right hand side is precisely what one gets in the canonical approach to quantization where gauge degrees of freedom are removed byﬁxing the gauge at the classical level and only physi-cal degrees of freedom enter the functional inte-gral.The Faddeev-Popov determinant arises in the canonical approach as the determinant of the matrix of Poisson brackets of what may be called the”second class constraints”,i.e.,the Gauss law3operator and the gaugeﬁxing condition f,which is of course introduced by hand and not really a constraint of the theory.There are both ordinary ﬁelds and conjugate momenta,but the latter are easily integrated over.The point is that the full functional integral is not needed in the canonical approach and there is no harm if it vanishes!A trace is left behind by the global anomaly. One may imagine a classiﬁcation of the gauge-ﬁxing functions f where f,f′are said to belong to the same class if there exists a gauge transfor-mation connected to the identity to go from a con-ﬁguration with f=0to one with f′=0.Then Z f=Z f′.More generally,when such a transfor-mation is not connected to the identity,Z f=e−iα(g0)Z f′,(13) where g0is determined by f,f′.These factors e−iα(g0)occurring in partition functions cancel out in expectation values of gauge invariant oper-ators,so that Green functions of gauge invariant operators are fully gauge independent[4]. There is an assumption in all this:that there is a possibility ofﬁxing the gauge.A general the-orem[6]asserts that gauges cannot beﬁxed in a smooth way.For the construction of functional integrals,however,it is suﬃcient to have piece-wise smooth gauges.It should also be remem-bered that these questions arise even for theories without disconnected gauge groups and are not speciﬁc to the context of global anomalies.ttice formulationOn going to the lattice,one starts to use group-valued variables associated with links instead of A deﬁned at points of the continuum.The topol-ogy also changes:the gauge group becomes con-nected on the lattice:it becomes possible to go to any gauge transformation from the identity in a continuous manner.Thus there are no large gauge transformations any more.Does it mean that there is no global anomaly on the lattice? The issue is complicated because chiral symme-try is not straightforward here.Chiral symmetry on the lattice has begun to make more sense in the last few years thanks to the Ginsparg-Wilson relation imposed on D,the euclidean lattice Dirac operator:γ5D+Dγ5=aDγ5D,(14) where a is the lattice spacing.An analogue ofγ5 appears from the above relation:γ5D=−DΓ5,Γ5≡γ5(1−aD).(15) It satisﬁes(Γ5)2=1,(Γ5)†=Γ5,(16) and can be used to deﬁne left-handed projection: P−ψ≡12[1+γ5]=¯ψ.(17) In this way of deﬁning chiral projections,P−,but not P+,depends on the gaugeﬁeld conﬁguration. Nontriviality of chirality on the lattice stems from this P−.A fermion measure is deﬁned by specifying a basis of lattice Diracﬁelds v j(x)satisfyingP−v j=v j,(v j,v k)=δjk.(18) One has to integrate over Grassmann-valued ex-pansion coeﬃcients inψ(x)= j a j v j(x).(19)Expansion coeﬃcients also come from the expan-sion of¯ψin terms of¯v j satisfying¯v j P+=¯v j, but these are as usual,i.e.,do not involve gauge ﬁelds.Questions of locality and integrability arise be-cause of the gaugeﬁeld dependence in P−.Ab-sence of a local anomaly appears to be suﬃcient to ensure locality[7].Global anomalies are man-ifested as a lack of integrability.Consider,following[5],a closed path in the SU(2)gauge conﬁguration space,with the param-eter t running from0to1.Deﬁnef(t)=det[1−P++P+D(t)Q t D(0)†],(20) with D(t)the Dirac operator corresponding to gaugeﬁelds at parameter value t,and Q t the uni-tary transport operator for P−(t)deﬁned by∂t Q t=[∂t P−(t),P−(t)]Q t,Q0=1.(21)4Then f(t)is real,positive and satisﬁesf(1)=T f(0).(22)HereT=det[1−P−(0)+P−(0)Q1]=±1(23)depending on the topology of the considered path in the gauge conﬁguration space.f changes sign an even or odd number of times along path de-pending on T and while det D(t)is related to f2, det D(t)det D(0)†=f2(t),(24)the chiral fermion determinant det Dχ(t)behaves like f:det Dχ(t)det Dχ(0)†=f(t)W(t)−1, (Dχ)ij≡a4 x¯v i(x)Dv j(x).(25) Here W(t)is a phase factor arising from the gauge ﬁeld dependence of v j.It is a lattice artifact and may be taken to reduce to unity near the contin-uum limit.Then det Dχchanges sign,i.e.,fails to return to its starting value after transportation along a closed path if the path hasT=−1.(26)Such paths have been shown to exist in the SU(2) theory.A part of such a path lies along a gauge orbit,and a part is non-gauge.2Thus det Dχis multivalued,implying that the fermion measure is not well deﬁned,and hence the functional integral does not make sense.This is roughly similar to the continuum.The Dirac operator is gauge-invariant and its determinant and f can change only on non-gauge portions of the closed path.So the problem of sign change of f occurs once again in non-gauge paths connect-ing gauge-related conﬁgurations.However,in the continuum,the sign change occurs between con-ﬁgurations which can be connected only by a non-gauge path.On the lattice,the sign change oc-curs when conﬁgurations are connected by a non-gauge path,though a connection is also possible。

Hofstede's Cultural Dimensions

What are Hofstede's five Cultural Dimensions?Read the About . . . section on the right side of this page. Then review the definitions of each Hofstede Dimension listed below. Following that, you can select the country or countries you're interested in from the list in the left margin of this page.On each country page you will find the unique Hofstede graphs depicting the Dimension scores and other demographics for that country and culture - plus an explanation of how they uniquely apply to that country.* Description for each of Hofstede's Dimensions listed belowPower Distance Index (PDI) that is the extent to which the less powerful members of organizations and institutions (like the family) accept and expect that power is distributed unequally. This represents inequality (more versus less), but defined from below, not from above. It suggests that a society's level of inequality is endorsed by the followers as much as by the leaders. Power and inequality, of course, are extremely fundamental facts of any society and anybody with some international experience will be aware that 'all societies are unequal, but some are more unequal than others'.Individualism (IDV) on the one side versus its opposite, collectivism, that is the degree to which individuals are inte-grated into groups. On the individualist side we find societies in which the ties between individuals are loose: everyone is expected to look after him/herself and his/her immediate family. On the collectivist side, we find societies in which people from birth onwards are integrated into strong, cohesive in-groups, often extended families (with uncles, aunts and grandparents) which continue protecting them in exchange for unquestioning loyalty. The word 'collectivism' in this sense has no politicalmeaning: it refers to the group, not to the state. Again, the issue addressed by this dimension is an extremely fundamental one, regarding all societies in theworld.Masculinity (MAS) versus its opposite, femininity, refers to the distribution of roles between the genders which is another fundamental issue for any society to which a range of solutions are found. The IBM studies revealed that (a) women's values differ less among societies than men's values; (b) men's values from one country to another contain a dimension from very assertive and competitive and maximally different from women's values on the one side, to modest and caring and similar to women's values on the other. Theassertive pole has been called 'masculine' and the modest, caring pole'feminine'. The women in feminine countries have the same modest, caring values as the men; in the masculine countries they are somewhat assertive and competitive, but not as much as the men, so that these countries show a gap between men's values and women's values.Uncertainty Avoidance Index (UAI) deals with a society's tolerance for uncertainty and ambiguity; it ultimately refers to man's search for Truth. It indicates to what extent a culture programs its members to feel eitheruncomfortable or comfortable in unstructured situations. Unstructured situations are novel, unknown, surprising, different from usual. Uncertainty avoiding cultures try to minimize the possibility of such situations by strict laws and rules, safety and security measures, and on the philosophical and religious level by a belief in absolute Truth; 'there can only be one Truth and we have it'. People in uncertainty avoiding countries are also more emotional, and motivated by inner nervous energy. The opposite type, uncertainty accepting cultures, are more tolerant of opinions different from what they are used to; they try to have as few rules as possible, and on the philosophical and religious level they are relativist and allow many currents to flow side by side. People within these cultures are more phlegmatic and contemplative, and not expected by their environment to express emotions.Long-Term Orientation (LTO) versus short-term orientation: this fifth dimension was found in a study among students in 23 countries around the world, using a questionnaire designed by Chinese scholars It can be said to deal with Virtue regardless of Truth. Values associated with Long Term Orientation are thrift and perseverance; values associated with Short Term Orientation are respect for tradition, fulfilling social obligations, and protecting one's 'face'. Both the positively and the negatively rated values of this dimension are found in the teachings of Confucius, the most influential Chinese philosopher who lived around 500 B.C.; however, the dimension also applies to countries without aConfucian heritage.Websites directly related to Geert HofstedeGeert Hofstede's personal WebsiteItim - An international consulting organization utilizing Prof. Hofstede's concepts.Itimfocus - An international consulting organization in the field of culture & change management that uses scans and tools based on scientific research undertaken by Professor Hofstede.Gert Jan Hofstede's personal WebsiteGeert Hofstede™ is a trademark of Geert Hofstede BV, Velp, the Netherlands What are the practical applications for Geert Hofstede's research on cultural differences?Geert HofstedeFrom Wikipedia, the free encyclopediaJump to: navigation, searcherard Hendrik Hofstede (born 3 October 1928, Haarlem) is an influential Dutch psychologist, who studied the interactions between national cultures and organizational cultures. He is also an author of seve ng Culture's Consequences[1] and Cultures and Organizations,Software of the Mind, co-authored by his son Gert JanHofstede.[2] Hofstede's study demonstrated that there arenational and regional cultural groupings that affect thebehaviour of societies and organizations, and that arevery persistent across time.Contents[hide]• 1 Hofstede's Framework for Assessing Culture• 2 Criticism• 3 Bibliographyo 3.1 Articles• 4 See also• 5 Notes• 6 External links[edit] Hofstede's Framework for Assessing Cultures found five dimensions of culture in his study of n ational work related values:•Low vs. high power distance - This dimension measures how much the less powerful members of institutions and organizations expect and accept that power isdistributed unequally. In cultures with low power distance (e.g. Ireland, Austria, Australia, Denmark, New Zealand), people expect and accept power relations that are moreconsultative or democratic. People relate to one another more as equals regardless of formal positions. Subordinates are more comfortable with and demand the right tocontribute to and critique the decisions of those in power. In cultures with high power distance (e.g. Malaysia), the less powerful accept power relations that are autocratic orpaternalistic. Subordinates acknowledge the power of others based on their formal,hierarchical positions. Thus, Low vs. High Power Distance does not measure or attempt to measure a culture's objective, "real" power distribution, but rather the way people perceive power differences.•Individualism vs. collectivism - This dimension measures how much members of the culture define themselves apart from their group memberships. In individualist cultures, people are expected to develop and display their individual personalities and to choose their own affiliations. In collectivist cultures, people are defined and act mostly as amember of a long-term group, such as the family, a religious group, an age cohort, a town, or a profession, among others.•Masculinity vs. femininity - This dimension measures the value placed on traditionally male or female values (as understood in most Western cultures). In so-called 'masculine' cultures, people value competitiveness, assertiveness, ambition, and the accumulation of wealth and material possessions. In so-called 'feminine' cultures, people value relationships and quality of life. This dimension is often renamed by users ofHofstede's work, e.g. to Quantity of Life vs. Quality of Life. Another reading of the same dimension holds that in 'M' cultures, the differences between gender roles are moredramatic and less fluid than in 'F' cultures•Low vs. high uncertainty avoidance - This dimension measures how much members of a society attempt to cope with anxiety by minimizing uncertainty. In cultures with high uncertainty avoidance, people prefer explicit rules (e.g. about religion and food) and formally structured activities, and employees tend to remain longer with their present employer. In cultures with low uncertainty avoidance, people prefer implicit or flexible rules or guidelines and informal activities. Employees tend to change employers morefrequently.Michael Harris Bond and his collaborators subsequently found a fifth dimension which was initially called Confucian dynamism. Hofstede later incorporated this into his framework as:•Long vs. short term orientation - This dimension describes a society's "time horizon," or the importance attached to the future versus the past and present. In long term oriented societies, people value actions and attitudes that affect the future:persistence/perseverance, thrift, and shame. In short term oriented societies, people value actions and attitudes that are affected by the past or the present: normative statements, immediate stability, protecting one's own face, respect for tradition, and reciprocation of greetings, favors, and gifts.ndencies and not characteristics of individuals. A Japanese person for example canhave a very low 'uncertainty avoidance' compared to a Filipino even though their'national' cultures point strongly in a different direction. Consequently, a country'sscores should not be interpreted as deterministic.[edit] CriticismHofstede's conceptualization of culture as static and essential has attracted somecriticism. In a recent article in the Academy of Management's flagship journal, TheAcademy of Management Review, Galit Ailon deconstructs Hofstede's book Culture'sConsequences by mirroring it against its own assumptions and logic[3]. Ailon findsseveral inconsistencies at the level of both theory and methodology and cautionsagainst an uncritical reading of Hofstede's cultural dimensions.Hofstede's work has not just also been criticized because he seems to identify cultureswith nations based on the supposition that within each nation there is a uniformnational culture. Other types of cultures are acknowledged to exist but allowed little,if any, influence.[4][edit] Bibliography日本The Buddhist-Shinto societies also have an additional Dimension, that of LongTerm Orientation (LTO). Geert Hofstede added this Dimension after the originalstudy, and it was applied to twenty-three of the fifty original countries in hisstudy. The Buddhist/Shinto Countries of Taiwan and Japan have LTO as themost closely correlating Dimension.* * *Power Distance Index (PDI) that is the extent to which the less powerfulmembers of organizations and institutions (like the family) accept and expectthat power is distributed unequally. This represents inequality (more versusless), but defined from below, not from above. It suggests that a society's levelof inequality is endorsed by the followers as much as by the leaders. Power and inequality, of course, are extremely fundamental facts of any society and anybody with some international experience will be aware that 'all societies are unequal, but some are more unequal than others'.Individualism (IDV) on the one side versus its opposite, collectivism, that is the degree to which individuals are inte-grated into groups. On the individualist side we find societies in which the ties between individuals are loose: everyone is expected to look after him/herself and his/her immediate family. On the collectivist side, we find societies in which people from birth onwards are integrated into strong, cohesive in-groups, often extended families (with uncles, aunts and grandparents) which continue protecting them in exchange for unquestioning loyalty. The word 'collectivism' in this sense has no political meaning: it refers to the group, not to the state. Again, the issue addressed by this dimension is an extremely fundamental one, regarding all societies in the world.Masculinity (MAS) versus its opposite, femininity, refers to the distribution of roles between the genders which is another fundamental issue for any society to which a range of solutions are found. The IBM studies revealed that (a) women's values differ less among societies than men's values; (b) men's values from one country to another contain a dimension from very assertive and competitive and maximally different from women's values on the one side, to modest and caring and similar to women's values on the other. The assertive pole has been called 'masculine' and the modest, caring pole 'feminine'. The women in feminine countries have the same modest, caring values as the men; in the masculine countries they are somewhat assertive and competitive, but not as much as the men, so that these countries show a gap between men's values and women's values.Uncertainty Avoidance Index (UAI) deals with a society's tolerance for uncertainty and ambiguity; it ultimately refers to man's search for Truth. It indicates to what extent a culture programs its members to feel either uncomfortable or comfortable in unstructured situations. Unstructured situations are novel, unknown, surprising, different from usual. Uncertainty avoiding cultures try to minimize the possibility of such situations by strict laws and rules, safety and security measures, and on the philosophical and religious level by a belief in absolute Truth; 'there can only be one Truth and we have it'. People in uncertainty avoiding countries are also more emotional, and motivated by inner nervous energy. The opposite type, uncertainty accepting cultures, are more tolerant of opinions different from what they are used to; they try to have as few rules as possible, and on the philosophical and religious level they are relativist and allow many currents to flow side by side. Peoplewithin these cultures are more phlegmatic and contemplative, and not expected by their environment to express emotions.Long-Term Orientation (LTO) versus short-term orientation: this fifth dimension was found in a study among students in 23 countries around the world, using a questionnaire designed by Chinese scholars It can be said to deal with Virtue regardless of Truth. Values associated with Long Term Orientation are thrift and perseverance; values associated with Short Term Orientation are respect for tradition, fulfilling social obligations, and protecting one's 'face'. Both the positively and the negatively rated values of this dimension are found in the teachings of Confucius, the most influential Chinese philosopher who lived around 500 B.C.; however, the dimension also applies to countries without a Confucian heritage.文化维度文化维度是荷兰国际文化合作研究所所长霍夫斯塔德(Geert Hofstede)及其同事在对文化因素进行定量研究时采用的概念。

D=9 supergravity and p-brane solitons

a r X i v :h e p -t h /9703063v 1 8 M a r 1997CTP TAMU-15/97hep-th/9703063D =9supergravity and p -brane solitonsN.Khviengia and Z.KhviengiaCenter for Theoretical Physics,Texas A&M University,College Station,TX77843-4242AbstractWe construct the N =2,D =9supergravity theory up to the quartic fermionic terms and derive the supersymmetry transforma-tion rules for the ﬁelds modulo cubic fermions.We consider a class of p -brane solutions of this theory,the stainless p -branes,which cannot be isotropically oxidized into higher dimensions.The new stainless elementary membrane and elementary particle solutions are found.It is explicitly veriﬁed that these solutions preserve half of the supersym-metry.1IntroductionThe recent progress in the understanding of M-theory revived the interest in the supergravity theories in diverse dimensions.Much work has been done in constructing and classifying the p-brane solutions to these theories. However,the extended supergravity in nine dimensions has not yet been fully constructed and investigated.Purely bosonic N=2,D=9action in the context of type-II S-and T-duality symmetries has been discussed in[1,2], but the full action of the theory and the supersymmetry transformation rules has not appeared in the literature.The goal of this paper is toﬁll this gap and present an explicit construction of the N=2,D=9supergravity up to the quartic fermionic terms and provide an exhaustive classiﬁcation of stainless p-brane solutions of this theory.With few exceptions,the lower dimensional supergravity theories can be obtained by dimensional reduction of the eleven dimensional Cremmer-Julia-Scherk(CJS)supergravity[3].Some examples of such exceptions in D=10 are provided by type IIB and massive type IIA supergravity theories[4,5]. Dimensional reduction of CJS action,apart from massless supergravities, can also give massive theories in D≤8[6,7].As one descends through the dimensions to obtain lower dimensional supergravities,a plethora of isotropic p-brane solutions arises[8,9,10].Solutions of the dimensionally reduced theory are also solution of the higher-dimensional theory,however,in higher dimension these solutions may or may not exhibit isotropicity.The solutions which cannot be isotropically lifted to the higher dimension and,therefore, cannot be viewed as p-brane solutions in the higher dimension,are called stainless solutions[11].The extended N=2supergravity in nine dimensions can be truncated to N=1supergravity whose stainless solutions consist of an elementary particle and a solitonic5-brane[11].It turns out that,whilst the elementary particle remains stainless in N=2theory,the solitonic5-brane becomes rusty and can be obtained from the type IIA D=10solitonic 6-brane.We explain this phenomenon by employing a two-scalar5-brane solution of N=2supergravity.The paper is organised as follows.In sections2,using the ordinary Scherk-Schwarz dimensional reduction procedure[6],we obtain the bosonic Lagrangian of N=2,D=9supergravity theory by dimensionally reducing the eleven-dimensional CJS Lagrangian from eleven directly to nine dimen-sions.It is of interest to note that in D=9,unlike D=8case[12], nontrivial group manifolds do not arise.In section3,we perform an analo-gous dimensional reduction for fermions and obtain the fermionic part of the nine-dimensional supergravity up to quartic fermions.The supersymmetry transformation rules for the bosonic and fermionicﬁelds,modulo trilinear fermions,are derived in secion4.In section5,stainless solutions to the ob-tained N=2,D=9supergravity are analysed.New stainless elementaryparticle and elementary membrane solutions are found,and it is shown that they preserve half of the supersymmetry.A stainless solitonic6-brane is also discussed.It is noted that if one is to include type IIB chiral supergrav-ity into consideration,the solitonic6-brane and the elementary membrane solutions become rusty and can be treated as descendants of the type IIB solitonic7-brane and self-dual3-brane in D=10.2Bosonic SectorThe bosonic part of the N=1,D=11supergravity Lagrangian is given by [3]L=ˆe48ˆFˆµˆνˆρˆσˆFˆµˆνˆρˆσ+2κ(ˆeˆµˆrˆeˆνˆs−ˆeˆµˆsˆeˆνˆr)∂ˆµˆeˆtˆν.(5)2We shall now perform the ordinary Scherk-Schwarz dimensional reduction of CJS Lagrangian(1)directly to D=9dimensions.We begin by dimen-sionally reducing the Einstein-Hilbert term in(1).It is convenientﬁrst to consider reduction from an arbitrary D+d to D dimensions,and then apply obtained formulas to our case where D+d=11and D=9.The Lorentz invariance of the supergravity theory in D+d dimensions, enables one to cast the vielbein into the triangular formˆeˆrˆµ= ˆe rµˆA iµ0ˆe iα .(6)Hereµ,r=0,1,···,D−1;α,i=1,···,d,the hatted indices belong to (D+d)-dimensional space,andˆA iµare the vector gaugeﬁelds that give rise to2-indexﬁeld strengths in the dimensionally reduced theory.Then the inverse veilbein is given byˆeˆµˆr= ˆeµr−ˆAαr0ˆeαi ,(7) where internal indices are raised and lowered by the metricˆgαβ=ˆe iαˆe jβδij.(8) The components of the spin-connectionˆωˆrˆsˆt are given by[6]ˆωrsj=1ˆeαiˆeµr∂µˆe jα−(i↔j),2ˆωjrs=−ˆωrsj,(9)ˆωirj=−1ˆgαβˆFµναˆFβµν(11)4+1d L iα,(13)ˆAαµ=2κAαµ,where L iαis a unimodular matrix detL iα=1,andγis a free parameter that determines an exponential prefactor of the Einstein-Hilbert term in a lower dimensional theory.As a result of rescaling,the vielbeins are brought to the following form ˆeˆrˆµ= δγe rµ2κδ1d L iα ,ˆeˆµˆr= δ−γeµr−2κδ−γAαr0δ−1Under the rescaling of the type(13),the metric in D dimensions rescales asˆgµν=δ2γgµν,and a Ricci scalar changes asR→δ−2γ R−2γ(D−1)gµν∇µ∇νlnδ−γ2(D−1)(D−2)gµν∇µlnδ∇νlnδ .(15)Using(12)and(15),we obtain the expression for the dimensional reduction of the Einstein-Hilbert action from D+d to D dimensions(for d>1)that generalises the result of ref.[6]for the arbitrary value of parameterγ d D+d xˆeˆR= d D xeδγ(D−2)+1 R−κ2δ−2γ+2gµν∂µgαβ∂νgαβ ,(16)4where1β≡γ2(D−1)(D−2)+2γ(D−1)+1−d F rsj,ˆωrij=δ−γQ rij,ˆωjrs=−ˆωrsj,(18)ˆωijr=δ−γ(2P rij+1Lαi∂r Lαj+(i↔j),Qµij=12invariant under the general coordinate transformation[6].Upon compact-iﬁcation,this symmetry becomes D=9general coordinate transformation and a set of the[U(1)]2reparametrization transformations.Denoting a pa-rameter in D=11byξˆµ,one can show that under theξα-reparametrization transformation,the gaugeﬁelds deﬁned in(20)transform noncovariantly in-volving derivatives of the parameterξα.Nevertheless,the supersymmetry transformation rules and the dimensional reduction procedure is somewhat simpliﬁed by this choice[8,11].In order to obtain results in terms of the covariant gaugeﬁelds,used in[6],one has to change conventions by identi-fying the nine-dimensional gaugeﬁelds as B rst=δ3γˆA rst,B rsi=δ2γˆA rsi and B rij=δγˆA rij.This amounts to the following redeﬁnitions of theﬁelds:Bµνρ=Aµνρ−6κA i[µAνρ]i+12κ2A i[µA jνAρ]ij,Bµνi=Aµνi−4κA j[µAνi]j,(21)Bµij=Aµij.Dimensionally reducing theﬁeld strength,weﬁndˆF=δ−4γ F rstu−2kF rsti A i u+4k2F rsij A i t A j u ≡δ−4γF′rstu, rstuˆF=δ−3γ−12F′rsti,rstiˆF=δ−2γ−1F rsij,(22) rsijˆF=0,rijkˆF=0.ijklAbove rules reﬂect the fact that theﬁeld strengths in D=9do not transform covariantly under the U(1)transformations arising from eleven-dimensional general coordinate transformation.This U(1)reparametrization invariance should not be confused with another U(1)symmetry which is a gauge symmetry of the antisymmetricﬁeld in D=9.In D=11,the U(1)gauge transformation is given by[6]δΛˆAˆµˆνˆρ=3∂[ˆµΛˆνˆρ],(23) which upon reduction gives the following U(1)gauge transformationsin D=9:δΛAµνρ=3∂[µΛνρ],δΛAµνα=2∂[µΛν]α,(24)δΛAµαβ=∂µΛαβ.Notice that since we are performing the ordinary dimensional reduction, none of theﬁelds and parameters in D=9depend on extra compactiﬁcation coordinates,in other words,all∂αderivatives are identically zero.Turning to the kinetic term for the antisymmetric tensorﬁeld in(1)and performing the straightforward reduction of theﬁeld strengths using(22), we obtain−e(12)4εµ1···µ9εαβ 3Fµ1µ2µ3µ4Fµ5µ6µ7µ8Aµ9αβ+24Fµ1µ2µ3µ4Fµ5µ6µ7αAµ8µ9β−16Fµ1µ2µ3αFµ4µ5µ6βAµ7µ8µ9+12Fµ1µ2µ3µ4Fµ5µ6αβAµ7µ8µ9 ,(26) where the totally antisymmetric tensor in nine dimensions is deﬁned as εµ1···µ9εαβ=εµ1···µ9αβ.In order to have the canonical normalization of a scalarﬁeld kinetic term in(16),we introduce a dilaton in D dimensionsδ=e βκφ(27) and chooseγ=−1D−2=−174κ2R−1√2(∂µφ)2−148e47κφF′µνρσF′µνρσ−1√8e−87κφFµναβFµναβ+L F F A ,(28) where L F F A is given by(26).The bosonic action(28)is invariant under the Abelian U(1)gauge trans-formations(24).We shall consider the supersymmetry and Lorentz invari-ance of the full N=2,D=9ation in section4where the supersymmetry transformation rules for theﬁelds will be derived.Using the general expression for dimensional reduction of the Einstein-Hilbert term(16),one can rewrite the action(28)in the p-brane metric[8] which appears naturally in the p-braneσ-models and is related to the canon-ical gravitational matric in D dimensions as gµν(p−brane)=e a/(p+1)gµν, with[11]a2=∆−2(p+1)˜dwhere ˜d=D −p −3and ∆in maximal supergravity theories is equal to 4.Then for the p -brane metric,the parameter γis deﬁned by the equation:2κ(p +1)(7γ+1)(−2/β)1/2=−(D −2)a.(30)In the following,we use the canonical value of γ,which in D =9is −1/7.3Fermionic sectorWe shall now compactify the fermionic part of the elven-dimensional super-gravity Lagrangian which reads[3]LF =L (1)F +L (2)F +quartic fermions,(31)L (1)F =ˆe 98ˆ¯ψˆr ˆΓˆr ˆs ˆt ˆu ˆv ˆw ˆψˆs +12ˆ¯ψˆt ˆΓˆu ˆv ˆψˆwˆF ˆt ˆu ˆv ˆw ,(33)where the covariant derivative is given byˆD ˆs ˆψˆt=∂ˆs ˆψˆt +13√73√77Γr Γiχie 17κφ,ψi −→χi e17κφ.(39)In the kinetic term for the fermions,for example,the exponential in(39) cancels against the corresponding factors coming from the determinantˆe and the covariant derivativeˆDˆs,and the shift of the fermionicﬁeld ensures that the Lagrangian is diagonalised.In the arbitrary dimension D,one has to make the following redeﬁnitionsψr−→ ψr−12(γ(D−1)+1),(40)ψi−→χiδ−12¯ψµΓµνρDνψρ+e7ΓiΓj+δij Dµχj+κe√2¯ψρΓ[ρΓµνΓσ]Γjψσ+¯ψρΓµνΓρ 149¯χiΓµν −23Γkδij+59Γjδik χk ,(43)L(2)F=κe√7¯ψλΓµνρσΓλΓiχi+¯χiΓµνρσ 1124e−17κφF′µνρi −¯ψλΓ[λΓµνρΓσ]Γiψσ+2¯ψλΓµνρΓλ δij−27¯χk GammaµνρΓk δij−816e−47κφFµνij ¯ψλΓ[λΓµνΓσ]Γijψσ−2449ΓkΓi+δki χj ,(44)where the covariant derivetives Dµψν=e sµe tνD sψt and Dµχi=e sµD sχi are deﬁned asD sψt=∂sψt+14Q sijΓijψt,(45)D sχj=∂sχj+14Q sikΓikχj+Q sj kχk.(46) In deriving(43)and(44),we have used the followingﬂipping property of Majorana spinors in nine dimensions¯ψΓr1···r nη=(−1)n¯ηΓr n···r1ψ.(47)4Supersymmety transformationsIn this section,we obtain the supersymmetry transfomation laws in nine dimensions.In order to preserve the triangular form of the veilbeinˆe rα=0, one has to consider combined Lorentz and supersymmetry transformation laws.As we shall see,the requirement of the oﬀ-diagonal part of the veilbein be zero,imposes an additional constrain on the Lorentz group parameters. This,in its turn,aﬀects the supersymetry transformation laws of theﬁelds.Combining the supersymmetry and the Lorentz tranformation laws in eleven dimensions,we have[3]δˆeˆµˆr=−¯ηΓˆrˆψˆµ+Λˆrˆsˆeˆsˆµ,(48)δˆAˆµˆνˆρ=−3144 ˆΓˆνˆρˆσˆγˆµ+8ˆΓˆνˆρˆσδˆγˆµ ˆFˆνˆρˆσˆγη+13√¯εΓjχjδrs+Λ′rs,(52)3√7where symmetrization is performed with the unit strength andΛ′sr is the redeﬁned local SO(1,8)Lorentz transformation parameterΛ′sr=Λsr−¯εΓ[sψr]+1¯εe−37κφ Γiψµ+Γµ(δij+172κ−¯εΓ(iχj)A jµ+Λ′i j A jµ,(54)κδij(δφ)−¯εΓ(iχj)+Λ′ij,(55) LαjδLαi=−√3where the redeﬁned SO(2)Lorentz parameter isΛ′ij=Λij−¯εΓ[iχj].(56) Tracing(55)withδij,oneﬁnds the transformation law forδφ,which upon substitution into(52),puts the veilbein transformation law into canonical form.Suppressing theΛ′rs andΛ′ij transformations,we obtain the supersym-metry transformation rules for the bosonicﬁelds:δe rµ=−¯εΓrψµ,(57)δφ=−3¯εΓiχi,(58) 7κLαjδLαi=−¯ε Γ(iχj)−1¯εe−37kφ Γiψµ+Γµ(δij+12κ¯εe47κφΓαβψµ−¯εe47κφ 8√2¯εe17κφ ΓαΓ[µ−2κe37κφΓαγAγ[µ ψν]2−1√7ΓαΓβ χβ−4κ¯εe47κφ 6√¯εe−27κφΓ[µνψρ]+3√2√√√ΓαΓγ)+24√7κe37κφF iνρ Γµνρ+12Γνδρµ Γiε28−1√7Γνρσλµ+48e−17κφF′νρσj Γνρσµ+1542e−47κφFνρij 3Γνρµ+36Γνδρµ Γijε+cubics,7×24δχi=−1√7e27κφF′µνρσΓµνρσΓiε+1√144e−47κφFµνjiΓµνΓjε+cubics,(64)6whereDµε=∂µε+14QµijΓijε(65) 5Stainless p-brane solutionsMost supergravity theories in D<11dimensions can be obtained by dimen-sionally reducing D=11supergravity theory.Therefore,solutions of the dimensionally reduced theory are also solutions of the higher-dimensional theory.However,in higher dimension these solutions may or may not ex-hibit isotropicity.The solutions which cannot be isotropically lifted to the higher dimension and,therefore,cannot be viewed as p-brane solutions in the higher dimensions,are called stainless solutions[11].In other words,stain-less p-branes are genuinely new solutions of the supergravity theory in the given dimension and should not be treated on the same footing as solutions which are descendants of the higher dimensional p-branes.Weﬁrst brieﬂy review some of the main results on p-brane solutions [8,10,11]and then apply general results to constructing and classifying stainless p-branes in N=2,D=9theory.The p-brane solutions in general involve the metric tensor g MN,a dilatonφand an n-index antisymmetric .The Lagrangian for theseﬁelds takes the formtensor F M1M2···M ne−1L=R−12n!e−aφF2n,(66) where a is a constant given by(29)[8,11].In D=11,the absence of a dilaton implies that∆=4.The value of∆is preserved under the dimen-sional reduction procedure and,hence,all antisymmetric tensors in maximal supergravity theories have∆=4.However,if an antisymmetric tensor used in a particular p-brane solution is formed from a linear combination of the originalﬁeld strengths,then it will have∆<4.An example of this is a solitonic5-brane in N=1,D=10supergravity considered below.We shall be looking for isotropic p-brane solutions for which the metric ansatz is given by[8,11]ds2=e2A dxµdxνηµν+e2B dy m dy m,(67) where xµ(µ=0,···,d−1)are the coordinates of the(d−1)-brane world volume,and y m are the coordinates of the(9−d)-dimensional transverse space.The functions A and B,also the dilatonφ,depend only on r=√This ansatz for the metric preserves an SO(1,d−1)×SO(9−d)subgroup of the original SO(1,8)Lorentz group.For the elementary p-brane solutions,the ansatz for theﬁeld strenght isgiven by[8,11]F mµ1···µn−1=εµ1···µn−1∂m e C,(68)whereεµ1···µn−1≡gµ1ν1···εν1···withε012···=1,and C is a function of r only. The dimension of the brane world volume is d=n−1.For the solitonic p-brane solution,the ansatz for the antisymmetric tensor is given by[8,11]F m1···m n =λεm1···m n py p∆(D−2)ln 1+k˜d A,φ=7ǫa2˜d√√r˜d −1.(72)The solutions(70)-(72)are valid for an n-indexﬁld strength with n>1. When n=1,i.e.˜d=0,there only exists a solitonic solution described by (70)with kr−˜d→klogr and˜d→0.Stainlessness of a p-brane solution crucially depends on a degree of the antisymmtric tensor involved in a solution,and the value of constant a oc-curring in the exponential prefactor.There are two diﬀerent situations when a stainless p-brane solution may arise in a given dimension.In theﬁrst sce-nario,no(D+1)-dimensional theory contains the necessaryﬁeld strength for a brane solution.In particular,if in D dimensions the solution is elemen-tary,the(D+1)-dimensional theory must have aﬁeld strength of degree one higher than that in D dimensional theory.If it is a solitonic solution,the (D+1)-dimensional theory must contain aﬁeld strength of the same degree as in D dimensional theory.In the second case,the requiredﬁeld strengthexists in the (D +1)-dimensional theory,but a p -brane is stainless only if the constant ˆa of a corresponding antisymmtric tensor in (D +1)dimensions is not related to the constant a of the D -dimensional theory as [11]ˆa 2=a 2−2˜d24E 2e −2κϕ12e −2κϕe −2κϕ.(75)The internal metric (75)is not diagonal,therefore,the terms in the La-grangian containing g αβwill not be diagonal as well.To diagonalize theLagrangian,the following redeﬁnitions have to be made:F 2MN +E F 1MN →F (2)MN,F 1MN→F (1)MN ,F ′2MNP +EF ′1MNP →F ′(2)MNP ,F ′1MNP →F ′(1)MNP .(76)Here and throughout this subsection M,N,P =0,···,8denote the curvednine-dimensional world volume indices,whilst R,S,T =0,···,8denote the ﬂat indices.Then the Lagrangian (28)can be written ase −1L =R −12(∂ϕ)2−1√4e 37φ−ϕ(F (2)2)2−18e −47φ(F 2)2−1√12e −17φ−ϕ(F ′(2)3)2−1√whereF2n≡F M1M2···M n F M1M2···M ndenotes the square of an n-indexﬁeld strength,H M=∂M E and the parameter κis set to1/2.If in(77)one retains only oneﬁeld strength and a corresponding dilaton, which for F(i)2and F′(i)is a linear combination ofφandϕ,one arrives at the Lagrangian of the form(66).Then general results described above can be applied to constructing single p-brane solutions in the given supergravity theory.It should be noted that for the purposes ofﬁnding a purely elementary or a purely solitonic p-brane solution,the F F A-term in the Lagrangian and the Chern-Simons modiﬁcations of theﬁeld strengths can be disregarded due to the fact that the constraints implied by these terms are automatically satisﬁed in D=9.However,in general,for certain p-brane solutions,the L F F A term and the Chern-Simons modiﬁcations to theﬁeld strengths give rise to nontrivial equations in some dimensions[10].Good examples illustrating this point are dyonic p-branes in D=4and D=6dimensions.Since our principal interest lies in the stainless p-brane solutions,we have ﬁrst to determine which of the p-branes of N=2,D=9supergravity are stainless.For this,one recalls that the N=2,D=10supergravity contains a2-indexﬁeld strength,a3-indexﬁeld strength and a4-indexﬁeld stregthing the criteria for stainlessness with theˆa2values1,14of a p-brane and the equation(73),weﬁnd that the stainless solutions of N=2,D=9supergravity theory are an elementary particle,an elementary membrane and a solitonic6-brane.Applying(70),we obtain the metrics for these solutionsparticle:ds2= 1+k r6 2/7dy m dy m,(78) membrane:ds2= 1+k r4 3/7dy m dy m,(79) 6−brane:ds2=dxµdxνηµν+ 1+klogr 3/7dy m dy m.(80)It is of interest to note that a solitonic5-brane,which is stainless as a solution to N=1,D=9supergravity theory,does not remain stainless in N= 2,D=9supergravity.This seeming paradox can be resolved if we consider details of the truncation of N=2to N=1supergravity,which contains a dilaton,a2-indexﬁeld strength and a3-indexﬁeld strength.One cannot consistently truncate out either two2-index antisymmetric tensors or a scalar ﬁeld.Nonetheless,it is possible to make a consistent truncation if weﬁrst rotate the scalarﬁelds:ϕ= 8φ1− 8φ2,φ= 8φ1+ 8φ2(81)and then setφ2=F4=H1=F(1)3=F(1)2=0which now is consistent with the equations of motion.Deﬁning˜F2≡√2F(2)2,we get the Lagrangian for the bosonic sector of N=1,D=9supergravity[11,14]:L=eR−112ee−√7φ1(F(2)3)2−12r −1/7 1+k2r 6/7 1+k273e−A−B+C−27φ∂m Cγmγ9ǫijΓjε,δψ0=114e−B+C−27φ∂m CγmǫijΓijε,(85)δψm=∂mε+128e−A+C−27φ∂m Cγmnγ9ǫijΓijε+3√2Aε,γ9ε0=ε0,ǫijΓijε0=ε0,(86)whereε0is a constant spinor.Thus the elementary particle solution preserves half of the supersymmetry.To varify that the elementary membrane solution also preserves half of the supersymmetry,we make a3+9split of the gamma matrices:Γµ=γµ⊗γ7,Γm=1l⊗γm,(87) whereγ7=γ0γ1...γ5in the transverse space andγ1γ2γ3=1l on the world volume.The transformation rules for the fermions becomeδχi=−√6e−B∂mφγ7⊗γmΓiε−1√7e−B−3A+C+17φγµ⊗γ7γmε,(88)δψm=∂mε+128e−B−3A+C+17φ∂n C1l⊗γmnε+2√Scherk-Schwarz dimensional reduction procedure which gives an advantage of constructing the lower dimensional theory in one step,as opposed to the standard Kaluza-Klein step by step dimensional reduction,and also enables one to consider compactiﬁcation on a non-trivial group manifolds[12].We explored the stainless p-brane solutions to the obtained N=2supergravity in D=9.Having derived the supersymmetry transfornation laws for theﬁelds, we were in the position to examine the sypersymmetry of the found stainless p-branes.Discussing the relation of the N=2solutions to the N=1 stainless solutions,it was observed that the stainless solutions of truncated theory may or may not remain stainless in the extended supergravity.The notion of stainlessness was discussed in the case when,along with D=11 supergravity,type IIB supergravity was taken into consideration. AcknowledgementsWe are grateful to E.Sezgin,C.N.Pope,H.Lu and M.J.Duﬀfor useful discussions.References[1]E.Bergshoeﬀ,C.Hull and T.Ortin,Duality in the type-II superstringeﬀective action,QMW-PH-95-2,hep-th/9504081.[2]A.Das and S.Roy,On M-theory and the symmetries of type II stringeﬀective actions,Nucl.Phys.B482(1996)119,hep-th/9605073. [3]E.Cremmer,B.Julia and J.Scherk,Supergravity theory in eleven di-mensions,Phys.Lett.B76(1978)409.[4]L.Romans,Massive N=2a supergravity in ten dimensions,Phys.Lett.169B(1986)374.[5]E.Bergshoeﬀ,M.De Roo,M.B.Green,G.Papadopoulos andP.K.Townsend,Duality of type II7-branes and8-branes,UG-15-95, hep-th/9601150.[6]J.Scherk and J.H.Schwarz,How to get masses from extra dimensions,Nucl.Phys.B153(1979)61.[7]P.M.Cowdall,H.Lu, C.N.Pope,K.S.Stelle and P.K.Townsend,Domain walls in massive supergravities,CTP-TAMU-26-96A,hep-th/9608173.[8]M.J.Duﬀ,R.Khuri,J.X.Lu,Phys.Rept.String solitons259(1995)213.[9]N.Khviengia,Z.Khviengia,H.Lu and C.N.Pope,Intersecting M-branesand bound states,Phys.Lett.B388(1996)21,hep-th/9605077. [10]H.Lu and C.N.Pope,p-brane solitons in maximal supergravities,Nucl.Phys.B465(1996)127,hep-th/9512012.[11]H.Lu,C.N.Pope,E.Sezgin and K.S.Stelle,Stainless super p-branes,Nucl.Phys.B456,(1995)669,hep-th/9508042.[12]A.Salam and E.Sezgin,d=8supergravity,Nucl.Phys.B258(1985)284.[13]H.Lu and C.N.Pope,Multi-scalar p-brane solitons,CTP-TAMU-52/95,hep-th/9512153.[14]S.J.Gates,H.Nishino and E.Sezgin,Class.Quantum Grav.3(1986)21.。

浅析《西游记》俄译本中民间俗语的翻译

Modern Linguistics 现代语言学, 2023, 11(9), 4188-4193 Published Online September 2023 in Hans. https:///journal/ml https:///10.12677/ml.2023.119562浅析《西游记》俄译本中民间俗语的翻译夏东旭新疆大学外国语学院，新疆乌鲁木齐收稿日期：2023年8月15日；录用日期：2023年9月14日；发布日期：2023年9月28日摘要科米萨罗夫认为在翻译过程中进行适当的语用适应性调整，将原文的语用潜力最大限度表达出来，使译文为读者理解接受。

《西游记》是我国四大名著之一，前苏联汉学家罗高寿的俄译本在俄语世界广受欢迎，影响深远。

本文以科米萨罗夫的语用适应原则为指导，对俄译本中的民间俗语的翻译所采用的语用适应策略如，保障译文读者对于原作的对等理解；将原作的情感效果传达给译文读者，促进其对原文理解正确；要针对具体情境下的具体接受者；实现“翻译外的最高任务”进行了研究，并且分析了其中的替换法，概括化，具体化等翻译方法。

关键词《西游记》，俗语，翻译，语用适应Analyzing the Translation of Folk Sayings in the Russian Translation of Journey to the WestDongxu XiaForeign Languages Institute, Xinjiang University, Urumqi Xinjiang Received: Aug. 15th , 2023; accepted: Sep. 14th , 2023; published: Sep. 28th , 2023AbstractKomissarov believes that appropriate linguistic adaptation in the translation process maximizes the linguistic potential of the original text and makes the translation understandable and accepta-ble to the readers. Journey to the West is one of the four great masterpieces of China, and the Rus-sian translation by the Soviet sinologist А. P. Rogachev is popular and influential in the Ru s-sian-speaking world. This paper, guided by Komissarov's principle of pragmatic adaptation, ex-夏东旭amines the pragmatic adaptation strategies used in the translation of folklore in Russian transla-tion, such as guaranteeing the reciprocal understanding of the original work by the translated readers; conveying the emotional effect of the original work to the translated readers to promote their correct understanding of the original text; targeting the specific receiver in the specific con-text; and realizing the “supreme task of extra-translational translation”, and also analyzes the methods of substitution, generalization, and concretization in the translation.KeywordsJourney to the West, Colloquialisms, Translation, Pragmatic AdaptationThis work is licensed under the Creative Commons Attribution International License (CC BY 4.0)./licenses/by/4.0/1. 引言近代国学专家陈代湘在《国学概论》中表示，《西游记》是我国第一部浪漫主义章回体长篇神魔小说。

The Light Front Gauge Propagator The Status Quo

a rXiv:h ep-th/048135v118A ug24THE LIGHT FRONT GAUGE PROPAGATOR:THE STATUS QUO A.T.Suzuki a and J.H.O.Sales b Abstract.At the classical level,the inverse diﬀerential operator for the qua-dratic term in the gauge ﬁeld Lagrangian density ﬁxed in the light front through the multiplier (n ·A )2yields the standard two term propagator with single unphysical pole of the type (k ·n )−1.Upon canonical quantization on the light-front,there emerges a third term of the form (k 2n µn ν)(k ·n )−2.This third term in the propagator has traditionally been dropped on the grounds that is exactly cancelled by the “instantaneous”term in the interaction Hamil-tonian in the light-front.Our aim in this work is not to discuss which of the propagators is the correct one,but rather to present at the classical level,the gauge ﬁxing conditions that can lead to the three-term propagator.It is revealed that this can only be acomplished via two coupled gauge ﬁxing condi-tions,namely n ·A =0=∂·A .This means that the propagator thus obtained is doubly transversal. 1.INTRODUCTION As early as 1970with J.B.Kogut and D.E.Soper [1]and a little later with E.Tomboulis in 1973[2],light-front gauge propagator for Abelian and Non-Abelian gauge ﬁelds derived via canonical quantization was known to have a (third)term proportional to (k 2n µn ν)(k ·n )−2.According to the latter,“The third term rep-resents an instantaneous “Coulomb”-type interaction.”Moreover,he (see also [3])showed then that “We will now show that all graphs representing the Coulomb term ...precisely cancel the contributions from the last term of the propagator ...so that we are left with an eﬀective interaction Hamiltonian density ...i.e,the two usual vertices,and a propagator...”where the so-called eﬀective propagator is the traditional two-term light-front propagator:(1.1)G µνab (k )=−δabk ·nMore recently,P.P.Srivastava and S.J.Brodsky [4]rederived the three-term “doubly transverse gauge propagator”(1.2)G µνab (k )=−i δabk ·n +k 2n µn ν2 A.T.SUZUKI A AND J.H.O.SALES Bfrom which the canonical quantization is performed via correspondence principle between Poisson brackets and Dirac commutators for theﬁeld operators.Their derivation clearly shows the conspicuous instantaneous interaction terms(the so-called tree-level seagull diagram terms)present in the interaction Hamiltonian in the light-front.Their explicit calculations for the electron-muon scattering in the Abelian QED theory in the light-front as well as the one-loopβ-function for the non-Abelian Yang-Millsﬁelds,with gluon vacuum polarization tensor,three-point vertex functions and gluon self-energy corrections from the quark loop,show us the subtle cancellations that come to play a crucial role into the game of light-front renormalization program with instantaneous interaction terms in the Hamiltonian and the third term of the gluon propagator.On the other hand,if one uses the classical approach of inverting the diﬀerential operator sandwiched between the quadratic term in the Lagrangian density plus the gaugeﬁxing term of the form(n·A)2in order to obtain the gaugeﬁeld propagator, the result is straightforwardly given by(1.1).There is no way-classically-to arrive at(1.2)with only the gaugeﬁxing Lagrangian of the form(n·A)2.This means that,as it stands,there is an anomaly between the classical and the quantum propagator.2.CLASSICALLY DEDUCIBLE THREE-TERM L.F.PROPAGATORSince at the classical level we just look for the inverse operator sandwiched between the quadratic term in the Lagrangian density plus the gaugeﬁxing term, in order to get a three-term propagator we need to incorporate not only the usual n·A=0condition into the gaugeﬁxing part,but couple it to the Lorentz condition ∂·A=0.The reason why we need the latter condition becomes clear when one understands that the Lorentzian condition coupled to the former gauge condition is nothing more than the constraint equation for the unphysicalﬁeld component A−, which is not a dynamical variable in the light-front formalism.Note that this does not remove too many degrees of freedom from the gaugeﬁelds as one would naively think,but that both of those two are in fact necessary to completelyﬁx the gauge in the light-front with no residual gauge freedom left.In a recent work,we have shown how this can be accomplished[5]via considering one Lagrange multiplier of the form(n·A)(∂·A)/α,whereαis the single gaugeﬁxing parameter.In this work we show that the one gaugeﬁxing term above referred to can be generalized to a two term general gaugeﬁxing term of the form(n·A)2/α+(∂·A)2/β, where nowαandβare two independent gaugeﬁxing parameters,yielding the same result,namely,the three-term,“doubly transverse propagator”(1.2)..The Lagrangian density for the vector gaugeﬁeld(for simplicity we consider an Abelian case)is given by(2.1)L=−12β(∂µAµ)2−12Aµ ∂2gµν−∂µ∂ν Aν3 andL GF=−12αnµAµnνAν(2.3)=12αAµnµnνAν(2.4) so that(2.5)L=1β∂µ∂ν−1(k·n)(β−1)D which inserted into(2.12)yields(2.16)D=−(k·n)(β−1)k·nC4 A.T.SUZUKI A AND J.H.O.SALES Band(2.18)C=−(β−1)(k·n)k2(2.19)B=0(2.20)C=D=1(k·n)2(2.22)Therefore,the relevant propagator in the light-front gauge is:(2.23)Gµν(k)=−1k·n+nµnν5References[1]J.B.Kogut and D.E.Soper,Phys.Rev.D1(1970)2901.[2]E.Tomboulis,Phys.Rev.D8(1973)2736.[3]J.M.Cornwall,Phys.Rev.D10(1974)500.[4]P.P.Srivastava and S.J.Brodsky,Phys.Rev.D64(2001)045006.[5]A.T.Suzuki and J.H.O.Sales,Nucl.Phys.A725(2003)139.[6]A.T.Suzuki and A.G.M.Schmidt,Prog.Theor.Phys.103(2000)1011.[7]A.T.Suzuki and J.H.O.Sales,hep-th/0303129(2003)a,b Instituto de F´ısica Te´o rica-UNESP,01405-900,S˜a o Paulo,Brazil.。

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.。

Gaugegravity duality

a rXiv:g r-qc/6237v 318Apr261Gauge/gravity duality GARY T.HOROWITZ University of California at Santa Barbara JOSEPH POLCHINSKI KITP,University of California at Santa Barbara Abstract We review the emergence of gravity from gauge theory in the context of AdS/CFT duality.We discuss the evidence for the duality,its lessons for gravitational physics,generalizations,and open questions.1.1Introduction Assertion:Hidden within every non-Abelian gauge theory,even within the weak and strong nuclear interactions,is a theory of quantum gravity.This is one implication of AdS/CFT duality.It was discovered by a circuitous route,involving in particular the relation between black branes and D-branes in string theory.It is an interesting exercise,how-ever,to ﬁrst try to ﬁnd a path from gauge theory to gravity as directly as possible.Thus let us imagine that we know a bit about gauge theory and a bit about gravity but nothing about string theory,and ask,how are we to make sense of the assertion?One possibility that comes to mind is that the spin-two graviton mightarise as a composite of two spin-one gauge bosons.This interesting idea would seem to be rigorously excluded by a no-go theorem of Weinberg &Witten (1980).The Weinberg-Witten theorem appears to assume noth-ing more than the existence of a Lorentz-covariant energy momentum tensor,which indeed holds in gauge theory.The theorem does forbid a wide range of possibilities,but (as with several other beautiful and pow-erful no-go theorems)it has at least one hidden assumption that seems so trivial as to escape notice,but which later developments show to be unnecessary.The crucial assumption here is that the graviton moves in the same spacetime as the gauge bosons of which it is made!12Gary T.Horowitz and Joseph PolchinskiThe clue to relax this assumption comes from the holographic prin-ciple(’t Hooft,1993,and Susskind,1995),which suggests that a grav-itational theory should be related to a non-gravitational theory in one fewer dimension.In other words,we mustﬁnd within the gauge theory not just the graviton,but aﬁfth dimension as well:the physics must be local with respect to some additional hidden parameter.Several hints suggest that the role of thisﬁfth dimension is played by the energy scale of the gauge theory.For example,the renormalization group equation is local with respect to energy:it is a nonlinear evolution equation for the coupling constants as measured at a given energy scale.†In order to make this precise,it is useful to go to certain limits in which theﬁve-dimensional picture becomes manifest;we will later return to the more general case.Thus we consider four-dimensional gauge theories with the following additional properties:•Large N c.While the holographic principle implies a certain equiva-lence between four-andﬁve-dimensional theories,it is also true that in many senses a higher dimensional theory has more degrees of free-dom;for example,the one-particle states are labeled by an additional momentum parameter.Thus,in order toﬁnd aﬁfth dimension of macroscopic size,we need to consider gauge theories with many de-grees of freedom.A natural limit of this kind was identiﬁed by’t Hooft(1974):if we consider SU(N c)gauge theories,then there is a smooth limit in which N c is taken large with the combination g2YM N c heldﬁxed.•Strong coupling.Classical Yang-Mills theory is certainly not the same as classical general relativity.If gravity is to emerge from gauge theory, we should expect that it will be in the limit where the gaugeﬁelds are strongly quantum mechanical,and the gravitational degrees of freedom arise as eﬀective classicalﬁelds.Thus we must consider the theory with large t Hooft parameter g2YM N c.•Supersymmetry.This is a more technical assumption,but it is a natural corollary to the previous one.Quantumﬁeld theories at strong coupling are prone to severe instabilities;for example,particle-antiparticle pairs can appear spontaneously,and their negative poten-tial energy would exceed their positive rest and kinetic energies.Thus, QED withﬁne structure constant much greater than1does not exist, even as an eﬀective theory,because it immediately runs into an in-stability in the ultraviolet(known as the Landau pole).The Thirring †This locality was emphasized to us by Shenker,who credits it to Wilson.Gauge/gravity duality3 model provides a simple solvable illustration of the problem:it existsonly below a certain critical coupling(Coleman,1975).Supersym-metric theories however have a natural stability property,because theHamiltonian is the square of a Hermitean supercharge and so boundedbelow.Thus it is not surprising that most examples ofﬁeld theorieswith interesting strong coupling behavior(i.e.dualities)are super-symmetric.We will therefore start by assuming supersymmetry,butafter understanding this case we can work back to the nonsupersym-metric case.We begin with the most supersymmetric possibility,N=4SU(N c)gauge theory,meaning that there are four copies of the minimal D=4supersymmetry algebra.The assumption of N=4supersymmetry hasa useful bonus in that the beta function vanishes,the coupling does notrun.Most gauge theories have running couplings,so that the strongcoupling required by the previous argument persists only in a very nar-row range of energies,becoming weak on one side and blowing up onthe other.In the N=4gauge theory the coupling remains strong andconstant over an arbitrarily large range,and so we can have a largeﬁfthdimension.The vanishing beta function implies that the classical conformal in-variance of the Yang-Mills theory survives quantization:it is a conformalﬁeld theory(CFT).In particular,the theory is invariant under rigid scaletransformations xµ→λxµforµ=0,1,2,3.Since we are associating theﬁfth coordinate r with energy scale,it must tranform inversely tothe length scale,r→r/λ.The most general metric invariant under thisscale invariance and the ordinary Poincar´e symmetries isds2=r2r2dr2(1.1)for some constantsℓandℓ′;by a multiplicative redeﬁnition of r we can setℓ′=ℓ.Thus our attempt to make sense of the assertion at the beginning has led us(with liberal use of hindsight)to the following conjecture:D=4,N=4,SU(N c)gauge theory is equivalent to a grav-itational theory inﬁve-dimensional anti-de Sitter(AdS)space.Indeed, this appears to be true.In the next section we will make this statement more precise,and discuss the evidence for it.In theﬁnal section we will discuss various lessons for quantum gravity,generalizations,and open questions.4Gary T.Horowitz and Joseph Polchinski1.2AdS/CFT dualityLet us deﬁne more fully the two sides of the duality.†The gauge theory can be written in a compact way by starting with the D=10Lagrangian density for an SU(N c)gaugeﬁeld and a16component Majorana-Weyl spinor,both in the adjoint(N c×N c matrix)representation:L=1Gauge/gravity duality5 completely crazy comes from comparing the symmetries.The D=4, N=4,SU(N c)super-Yang-Mills theory has an SO(4,2)symmetrycoming from conformal invariance and an SO(6)symmetry coming from rotation of the scalars.This agrees with the geometric symmetries of AdS5×S5.On both sides there are also32supersymmetries.Again on the gravitational side these are geometric,arising as Killing spinors on the AdS5×S5spacetime.On the gauge theory side they include the 16‘ordinary’supersymmetries of the N=4algebra,and16additional supersymmetries as required by the conformal algebra.The precise(though still not fully complete)statement is that the IIB supergravity theory in a space whose geometry is asymptotically AdS5×S5is dual to the D=4,N=4,SU(N c)gauge theory.The metric(1.1) describes only a Poincar´e patch of AdS spacetime,and the gauge theory lives on R4.It is generally more natural to consider the fully extended global AdS space,in which case the dual gauge theory lives on S3×R. In each case the gauge theory lives on the conformal boundary of the gravitational spacetime(r→∞in the Poincar´e coordinates),which will give us a natural dictionary for the observables.The initial checks of this duality concerned perturbations of AdS5×S5. It was shown that all linearized supergravity states have corresponding states in the gauge theory(Witten,1998a).In particular,the global time translation in the bulk is identiﬁed with time translation in the ﬁeld theory,and the energies of states in theﬁeld theory and string theory agree.For perturbations of AdS5×S5,one can reconstruct the background spacetime from the gauge theory as follows.Fields on S5 can be decomposed into spherical harmonics,which can be described as symmetric traceless tensors on R6:T i···j X i···X j.Restricted to the unit sphere one gets a basis of functions.Recall that the gauge theory has six scalars and the SO(6)symmetry of rotating theϕi.So the operators T i···jϕi···ϕj give information about position on S5.Four of the remaining directions are explicitly present in the gauge theory,and the radial direction corresponds to the energy scale in the gauge theory. In the gauge theory the expectation values of local operators(gauge invariant products of the N=4ﬁelds and their covariant derivatives) provide one natural set of observables.It is convenient to work with the generating functional for these expectation values by shifting the Lagrangian densityL(x)→L(x)+ I J I(x)O I(x),(1.3) where O I is some basis of local operators and J I(x)are arbitrary func-6Gary T.Horowitz and Joseph Polchinskitions.Since we are taking products of operators at a point,we are perturbing the theory in the ultraviolet,which according to the energy-radius relation maps to the AdS boundary.Thus the duality dictionary relates the gauge theory generating functional to a gravitational theory in which the boundary conditions at inﬁnity are perturbed in a speciﬁed way(Gubser et al.,1998,and Witten,1998a).As a further check on the duality,all three-point interactions were shown to agree(Lee et al., 1998).The linearized supergravity excitations map to gauge invariant states of the gauge bosons,scalars,and fermions,but in fact only to a small subset of these;in particular,all the supergravity states live in special small multiplets of the superconformal symmetry algebra.Thus the dual to the gauge theory contains much more than supergravity.The identity of the additional degrees of freedom becomes particularly clear if one looks at highly boosted states,those having large angular momentum on S5and/or AdS5(Berenstein et al.,2002,and Gubser et al.,2002).The ﬁelds of the gauge theory then organize naturally into one-dimensional structures,coming from the Yang-Mills large-N c trace:they correspond to the excited states of strings.In some cases,one can even construct a two dimensional sigma model directly from the gauge theory and show that it agrees(at large boost)with the sigma model describing strings moving in AdS5×S5(Kruczenski,2004).Thus,by trying to make sense of the assertion at the beginning,we are forced to‘discover’string theory.We can now state the duality in its full form(Maldacena,1998a):Four-dimensional N=4supersymmetric SU(N c)gauge theory is equivalent to IIB string theory with AdS5×S5boundary conditions. The need for strings(though not the presence of gravity!)was already anticipated by’t Hooft(1974),based on the planar structure of the large-N c Yang-Mills perturbation theory;the AdS/CFT duality puts this into a precise form.It alsoﬁts with the existence of another important set of gauge theory observables,the one-dimensional Wilson loops.The Wilson loop can be thought of as creating a string at the AdS5boundary, whose world-sheet then extends into the interior(Maldacena,1998b,and Rey&Yee,2001).See also Polyakov(1987,1999)for other perspectives on gauge/string duality and the role of theﬁfth dimension.We now drop the pretense of not knowing string theory,and outline the original argument for the duality in Maldacena(1998a).He con-sidered a stack of N c parallel D3-branes on top of each other.EachGauge/gravity duality7 D3-brane couples to gravity with a strength proportional to the dimen-sionless string coupling g s,so the distortion of the metric by the branesis proportional to g s N c.When g s N c≪1the spacetime is nearlyﬂat and there are two types of string excitations.There are open strings on thebrane whose low energy modes are described by a U(N c)gauge theory.There are also closed strings away from the brane.When g s N c≫1,the backreaction is important and the metric describes an extremal black 3-brane.This is a generalization of a black hole appropriate for a three dimensional extended object.It is extremal with respect to the charge carried by the3-branes,which sources theﬁve form F5.Near the hori-zon,the spacetime becomes a product of S5and AdS5.(This is directly analogous to the fact that near the horizon of an extremal Reissner-Nordstrom black hole,the spacetime is AdS2×S2.)String states near the horizon are strongly redshifted and have very low energy as seen asymptotically.In a certain low energy limit,one can decouple these strings from the strings in the asymptoticallyﬂat region.At weak cou-pling,g s N c≪1,this same limit decouples the excitations of the3-branes from the closed strings.Thus the low energy decoupled physics is de-scribed by the gauge theory at small g s and by the AdS5×S5closed string theory at large g s,and the simplest conjecture is that these are the same theory as seen at diﬀerent values of the coupling.†This con-jecture resolved a puzzle,the fact that very diﬀerent gauge theory and gravity calculations were found to give the same answers for a variety of string-brane interactions.In the context of string theory we can relate the parameters on thetwo sides of the duality.In the gauge theory we have g2YM and N c.Theknown D3-brane Lagrangian determines the relation of couplings,g2YM=4πg s.Further,each D3-brane is a source for theﬁve-formﬁeld strength,so on the string side N c is determined by S5F5;this integratedﬂux is quantized by a generalization of Dirac’s argument for quantization of the ﬂux S2F2of a magnetic monopole.The supergravityﬁeld equations give a relation between thisﬂux and the radii of curvature of the AdS5 and S5spaces,both being given byℓ=(4πg s N c)1/4ℓs.(1.4) Hereℓs is the fundamental length scale of string theory,related to the string tensionµbyµ−1=2πℓ2s.Notice that the spacetime radii are large in string units(and so the curvature is small)precisely when the’t Hooft †The U(1)factor in U(N c)=SU(N c)×U(1)also decouples:it is Abelian and does not feel the strong gauge interactions.8Gary T.Horowitz and Joseph Polchinskicoupling4πg s N c=g2YM N c is large,in keeping with the heuristic argu-ments that we made in the introduction.It is also instructive to express the AdS radius entirely in gravitational variables.The ten-dimensional gravitational coupling is G∼g2sℓ8s,up to a numerical constant.Thusℓ∼N1/4c G1/8,G∼ℓ8ℓ2+1−r20ℓ2+1−r20Gauge/gravity duality9 isS BH=Ag2sℓ8s∼T3Hℓ114S YM(Gubser et al.,1996).The numerical disagreement is not surprising,as the Yang-Mills calculation is for an ideal gas,and at large g s the Yang-Mills degrees of freedom are interacting.Thus one expects a relation of the form S BH=f(g s N c)S YM,ideal,where f(0)=1;the above calculation implies that f(∞)=34,but theﬁrst correction has been calculated both at weakand strong coupling and is consistent with f(g s N c)interpolating in a rather smooth way.Hawking&Page(1983)showed that for thermal AdS boundary con-ditions there is a phase transition:below a transition temperature of order1/ℓthe dominant conﬁguration is not the black hole but a gas of particles in AdS space.The low temperature geometry has no horizon and so its entropy comes only from the ordinary statistical mechanics of the gas.The same transition occurs in the gauge theory(Witten, 1998b).The N=4gauge theory on S3has an analog of a conﬁnement transition.At low temperature one has a thermal ensemble of gauge-invariant degrees of freedom,whose entropy therefore scales as N0c,and at high temperature one has the N2c behavior found above—the same scalings as on the gravitational side.There is another test one can perform with the gauge theory atﬁ-nite temperature.At long wavelengths,one can use a hydrodynamic approximation and think of this as aﬂuid(for a recent overview see Kovtun et al.,2003).It is then natural to ask:What is the speed of sound waves?Conformal invariance implies that the stress energy ten-sor is traceless,so p=ρ/3which implies that v=1/√10Gary T.Horowitz and Joseph Polchinskiseem to be diﬃcult since the bulk does not seem to have any preferred speed other than the speed of light.But recent work has shown that the answer is yes.The AdS/CFT duality also gives an interesting perspective on the black hole membrane paradigm(Thorne et al.,1986).The black hole horizon is known to have many of the properties of a dissipative sys-tem.On the dual side it is a dissipative system,the hot gauge theory. One can thus compute such hydrodynamic quantities such as the shear viscosity.These are hard to check since they are diﬃcult to calculate directly in the strongly coupled thermal gauge theory,but,rather re-markably,the numerical agreement with the observed properties of the real quark-gluon plasma at RHIC is better than for conventionalﬁeld theory calculations(for a discussion see Blau,2005).There is also aﬁeld theory interpretation of black hole quasinormal modes(Horowitz&Hubeny2000).A perturbation of the black hole decays with a characteristic time set by the imaginary part of the low-est quasinormal mode.This should correspond to the timescale for the gauge theory to return to thermal equilibrium.One can show that the quasinormal mode frequencies are poles in the retarded Green’s func-tion of a certain operator in the gauge theory.The particular operator depends on the type ofﬁeld used to perturb the black hole(Kovtun& Starinets,2005).Finally,consider the formation and evaporation of a small black hole in a spacetime which is asymptotically AdS5×S5.By the AdS/CFT correspondence,this process is described by ordinary unitary evolution in the gauge theory.So black hole evaporation does not violate quan-tum mechanics:information is preserved.This also provides an indirect argument against the existence of a‘bounce’at the black hole singular-ity,because the resulting disconnected universe would presumably carry away information.1.3.2Background independence and emergenceThe AdS/CFT system is entirely embedded in the framework of quan-tum mechanics.On the gauge theory side we have an explicit Hamil-tonian,and states which we can think of as gauge invariant functionals of theﬁelds.Thus the gravitational theory on the other side is quan-tum mechanical as well.In particular the metricﬂuctuates freely except at the AdS boundary.One is not restricted to perturbations about a particular background.Gauge/gravity duality11 This is clearly illustrated by a rich set of examples which provide a detailed map between a class of nontrivial asymptotically AdS5×S5supergravity solutions and a class of states in the gauge theory(Lin et al.,2004).These states and geometries both preserve half of the supersymmetry of AdS5×S5itself.On theﬁeld theory side,one restricts toﬁelds that are independent of S3and hence reduce to N c×N c matrices. In fact,all the states are created by a single complex matrix,so can be described by a one-matrix model.This theory can be quantized exactly in terms of free fermions,and the states can be labeled by a arbitrary closed curve(the Fermi surface)on a plane.On the gravity side,one considers solutions to ten dimensional supergravity involving just the metric and self-dualﬁve form F5.Theﬁeld equations are simply dF5=0 andR MN=F MP QRS F N P QRS(1.9) There exists a large class of stationary solutions to(1.9),which have an SO(4)×SO(4)symmetry and can be obtained by solving a linear equation.These solutions are nonsingular,have no event horizons,but can have complicated topology.They are also labeled by arbitrary closed curves on a plane.This provides a precise way to map states in theﬁeld theory into bulk geometries.Only for some“semi-classical”states is the curvature below the Planck scale everywhere,but the matrix/free fermion description readily describes all the states,of all topologies, within a single Hilbert space.Thus the gauge theory gives a representation of quantum gravity that is background independent almost everywhere—-that is,everywhere ex-cept the boundary.Conventional string perturbation theory constructs string amplitudes as an asymptotic expansion around a given spacetime geometry;here we have an exact quantum mechanical construction for which the conventional expansion generates the asymptotics.All lo-cal phenomena of quantum gravity,such as formation and evaporation of black holes,the interaction of quanta with Planckian energies,and even transitions that change topology,are described by the gauge the-ory.However,the boundary conditions do have the important limitation that most cosmological situations,and most compactiﬁcations of string theory,cannot be described;we will return to these points later.To summarize,AdS/CFT duality is an example of emergent gravity, emergent spacetime,and emergent general coordinate invariance.But it is also an example of emergent strings!We should note that the terms‘gauge/gravity duality’and‘gauge/string duality’are often used,12Gary T.Horowitz and Joseph Polchinskiboth to reﬂect these emergent properties and also the fact that(as weare about the see)the duality generalizes to gravitational theories withcertain other boundary conditions,and toﬁeld theories that are notconformally invariant.Let us expand somewhat on the emergence of general coordinate in-variance.The AdS/CFT duality is a close analog to the phenomenonof emergent gauge symmetry(e.g.D’Adda et al.,1978,and Baskaran&Anderson,1988).For example,in some condensed matter systems inwhich the starting point has only electrons with short-ranged interac-tions,there are phases where the electron separates into a new fermionand boson,e(x)=b(x)f†(x).(1.10) However,the newﬁelds are redundant:there is a gauge transformationb(x)→e iλ(x)b(x),f(x)→e iλ(x)f(x),which leaves the physical elec-tronﬁeld invariant.This new gauge invariance is clearly emergent:it iscompletely invisible in terms of the electronﬁeld appearing in the orig-inal description of the theory.†Similarly,the gauge theory variables ofAdS/CFT are trivially invariant under the bulk diﬀeomorphisms,whichare entirely invisible in the gauge theory(the gauge theoryﬁelds dotransform under the asymptotic symmetries of AdS5×S5,but these are ADM symmetries,not gauge redundancies).Of course we can alwaysin general relativity introduce a set of gauge-invariant observables bysetting up eﬀectively a system of rods and clocks,so to this extent thenotion of emergence is imprecise,but it carries the connotation that thedynamics can be expressed in a simple way in terms of the invariantvariables,as the case in AdS/CFT.‡1.3.3GeneralizationsThus far we have considered only the most well-studied example ofgauge/gravity duality:D=4,N=4,Yang-Mills⇔string theorywith AdS5×S5boundary conditions.Let us now ask how much more general this phenomenon is(again,for details see the review by Aharony et al.,2000).†This‘statistical’gauge invariance is not to be confused with the ordinary electro-magnetic gauge invariance,which does act on the electron.‡Note that on the gauge theory side there is still the ordinary Yang-Mills gauge redundancy,which is more tractable than general coordinate invariance(it does not act on spacetime).In fact in most examples of duality there are gauge symmetries on both sides and these are unrelated to each other:the duality pertains only to the physical quantities.Gauge/gravity duality13 First,we imagine perturbing the theory we have already studied, adding additional terms(such as masses for some of theﬁelds)to the gauge theory action.This is just a special case of the modiﬁcation(1.3), such that the functions J I(x)=g I are independent of position.Thus we already have the dictionary,that the the dual theory is given by IIB string theory in a spacetime with some perturbation of the AdS5×S5 boundary conditions.In general,the perturbation of the gauge theory will break conformal invariance,so that the physics depends on energy scale.In quantum ﬁeld theory there is a standard procedure for integrating out high en-ergy degrees of freedom and obtaining an eﬀective theory at low energy. This is known as renormalization group(RG)ﬂow.If one starts with a conformalﬁeld theory at high energy,the RGﬂow is trivial.The low energy theory looks the same as the high energy theory.This is because there is no intrinsic scale.But if we perturb the theory,the RGﬂow is nontrivial and we obtain a diﬀerent theory at low energies.There are two broad possibilities:either some degrees of freedom remain massless and we approach a new conformal theory at low energy,or allﬁelds become massive and the low energy limit is trivial.Since the energy scale corresponds to the radius,this RGﬂow in the boundaryﬁeld theory should correspond to radial dependence in the bulk.Let us expand a bit on the relation between radial coordinate and energy(we will make this argument in Poincar´e coordinates,since the perturbed gauge theories are usually studied on R4).The AdS geom-etry(1.1)is warped:in Poincare coordinates,the fourﬂat dimensions experience a gravitational redshift that depends onﬁfth coordinate,just as in Randall-Sundrum compactiﬁcation.Consequently the conserved Killing momentum pµ(Noether momentum in the gauge theory)is re-lated to the local inertial momentum˜pµbyrpµ=14Gary T.Horowitz and Joseph Polchinskioﬀin such a way that the warp factor(which is r/ℓin AdS spacetime) has a lower bound.The former clearly corresponds to a new conformal theory,while the latter would imply a mass gap,by the argument fol-lowing eq.(1.11).In the various examples,oneﬁnds that the nature of the solution correctly reﬂects the low energy physics as expected from gauge theory arguments;there is also more detailed numerical agree-ment(Freedman et al.,1999).So the classical Einstein equation knows a lot about RGﬂows in quantumﬁeld theory.A notable example is the case where one gives mass to all the scalars and fermions,leaving only the gaugeﬁelds massless in the Lagrangian. One then expects the gauge theory toﬂow to strong coupling and pro-duce a mass gap,and this is what is found in the supergravity solution. Further,the gauge theory should conﬁne,and indeed in the deformed geometry a conﬁning area law is found for the Wilson loop(but still a perimeter law for the’t Hooft loop,again as expected).In other ex-amples one alsoﬁnds chiral symmetry breaking,as expected in strongly coupled gauge theories(Klebanov&Strassler,2000).As a second generalization,rather than a deformation of the geometry we can make a big change,replacing S5with any other Einstein space; the simplest examples would be S5identiﬁed by some discrete subgroup of its SO(6)symmetry.The product of the Einstein space with AdS5 still solves theﬁeld equations(at least classically),so there should be a conformally invariant dual.These duals are known in a very large class of examples;characteristically they are quiver gauge theories,a product of SU(N1)×...×SU(N k)with matterﬁelds transforming as adjoints and bifundamentals(one can also get orthogonal and symplectic factors). As a third generalization,we can start with D p-branes for other val-ues of p,or combinations of branes of diﬀerent dimensions.These lead to other examples of gauge-gravity duality forﬁeld theories in various dimensions,many of which are nonconformal.The case p=0is the BFSS matrix model,although the focus in that case is on a diﬀerent set of observables,the scattering amplitudes for the D0-branes themselves.A particularly interesting system is D1-branes plus D5-branes,leading to the near-horizon geometry AdS3×S3×T4.This case has at least one advantage over AdS5×S5.The entropy of large black holes can now be reproduced exactly,including the numerical coeﬃcient.This is related to the fact that a black hole in AdS3is a BTZ black hole which is locally AdS3everywhere.Thus when one extrapolates to small coupling,one does not modify the geometry with higher curvature corrections.We have discussed modiﬁcations of the gauge theory’s Hamiltonian,。

Quiver Gauge Theory and Noncommutative Vortices

for obtaining classical solutions it suﬃces to solve the DUY equations rather than the full second-order ﬁeld equations (but it is by no means necessary). As a special case, on M4 (n=2) the 3 DUY equations reduce to the famous self-duality equations F = ∗F which yield instantons and monopoles.
§2. K¨ ahler times coset space G/H
G , with M2n being To be concrete, let us consider U(k ) Yang-Mills theory on M2n × H a real 2n-dimensional K¨ ahler manifold with K¨ ahler form ω and metric g . For cosets, we
and has since been extended supersymmetrically5) and embedded into superstring theory.6) In the present talk, for Lie groups G of rank one and rank two, we shall apply this scheme to perform a G-equivariant reduction of Yang-Mills theory over G/H to a quiver gauge theory on M,7)–10) formulate its BPS equations and show how to construct a certain class of solutions, which admit a D-brane interpretation. These solutions, however, only exist when the system is subjected to a noncommutative deformation. Therefore, about half-way into the talk we specialize to M = Cn and apply a Moyal deformation. Most material presented here has appeared in Refs. 11)–13), some is work in progress.

Inflation During and After the Zero Lower Bound

Inﬂation During and After the Zero Lower Bound
S. Bora˘ gan Aruoba University ofபைடு நூலகம்Maryland ∗
Frank Schorfheide University of Pennsylvania, NBER August 7, 2015
∗
This Version: August 7, 2015
2
been at the ZLB since 1999 except for two brief stints, inﬂation has been negative, long-run inﬂation expectations demonstrate signiﬁcant ﬂuctuations, the and the real interest rate has remained positive. During the 2009-2015 ZLB episode in the U.S., on the other hand, except for two quarters early on, inﬂation has been positive but real rates have been consistently negative. Inﬂation and real rates in the Euro Area have behaved qualitatively similar to the U.S. Another crucial diﬀerence between the U.S. and Europe on the one hand, and Japan on the other is the remarkable stability of long-run inﬂation expectations in the former two economies despite fairly large swings in actual inﬂation. Using a ﬂexible time-series model with a good inﬂation forecasting record, we extract a low frequency trend-inﬂation component, which remains positive in the U.S. and the Euro Area throughout the sample, but has been negative in Japan since the late 1990s. Looking into the future, the time series model predicts a substantial probability of deﬂation for Japan over the next ﬁve years, while for the U.S. and Europe these probabilities are no more than 20%. Second, we turn to one of the workhorse models for monetary policy analysis to understand the diﬀerences among the three economies and to study a possible exit from the ZLB: a textbook-style New Keynesian dynamic stochastic general equilibrium (DSGE) model with ZLB constraint. It is well known that the ZLB generates multiple equilibria: the model predicts a set of diﬀerent economic outcomes conditional on the same set of fundamentals.1 Of course, in reality only one of these outcomes is observed. Thus, it is common to augment a model that lacks a unique equilibrium with a probabilistic selection mechanism, which is often called a sunspot shock. This sunspot shock is a placeholder for a more complete theory of how ﬁrms and households coordinate their beliefs and actions.

Supersymmetric Gauge Theories and Gravitational Instantons

a rXiv:h ep-th/987237v13J ul1998CALT-68-2190SUPERSYMMETRIC GAUGE THEORIES AND GRA VITATIONAL INSTANTONS SERGEY A.CHERKIS California Institute of Technology,Pasadena CA 91125,USA E-mail:cherkis@ Various string theory realizations of three-dimensional gauge theories relate them to gravitational instantons 1,Nahm equations 2and monopoles 3.We use this correspondence to model self-dual gravitational instantons of D k -type as moduli spaces of singular monopoles,ﬁnd their twistor spaces and metrics.This work provides yet another example of how string theory unites seem-ingly distant physical problems.(See references for detailed results.)The central object considered here is supersymmetric gauge theories in three di-mensions.In particular,we shall be interested in their vacuum structure.The other three problems that turn out to be closely related to these gauge theories are:•Nonabelian monopoles of Prasad and Sommerﬁeld,which are solutions of the Bogomolny equation ∗F =D Φ(where F is the ﬁeld-strength of a nonabelian connection A =A 1dx 1+A 2dx 2+A 3dx 3and Φis a nonabelian Higgs ﬁeld).•An integrable system of equations named after Nahm dT i2εijk [T j ,T k ],(1)for T i (s )∈u (n ).These generalize Euler equations for a rotating top.•Solutions of the Euclideanized vacuum Einstein equation called self-dual gravitational instantons ,which are four-dimensional manifolds with self-dual curvature tensorR αβγδ=1The latter provide compactiﬁcations of string theory and supergravity that preserve supersymmetry and are of importance in euclidean quantum gravity.The compact examples are delivered by a four-torus and K3.The noncom-pact ones are classiﬁed according to their asymptotic behavior and topology.Asymptotically Locally Euclidean (ALE)gravitational instantons asymptoti-cally approach R 4/Γ(Γis a ﬁnite subgroup of SU (2)).These were classiﬁed by Kronheimer into two inﬁnite (A k and D k )series and three exceptional (E 6,E 7and E 8)cases according to the intersection matrix of their two-cycles.Asymp-totically Locally Flat (ALF)spaces approach the R 3×S 1 /Γmetric.(To be more precise S 1is Hopf ﬁbered over the two-sphere at inﬁnity of R 3.)Sending the radius of the asymptotic S 1to inﬁnity we recover an ALE space of some type,which will determine the type of the initial ALF space.For example,the A k ALF is a (k+1)-centered multi-Taub-NUT space.Here we shall seek to describe the D k ALF space.M theory on an A k ALF space is known to describe (k+1)D6-branes of type IIA string theory.Probing this background with a D2-brane we obtain an N =4U (1)gauge theory with (k+1)electron in the D2-brane worldvolume.As the D2-brane corresponds to an M2-brane in M theory,a vacuum of the above gauge theory corresponds to a position of the M2-brane on the A k ALF space we started with.Thus the moduli space of this gauge theory is the A k ALF.Next,considering M theory on a D k ALF one recovers 4k D6-branes parallel to an orientifold O 6−.On a D2-brane probe this time we ﬁnd an N =4SU(2)gauge theory with k matter multiplets.Its moduli space is the D k ALF.So far we have related gauge theories and gravitational instantons .D3NS5NS5pp p k D3Figure 1:The brane conﬁguration corresponding to U (2)gauge theory with k matter mul-tiplets on the internal D3-branes.There is another way of realizing these gauge theories.Consider the Chalmers-Hanany-Witten conﬁguration in type IIB string theory (Figure 1).2In the extreme infrared limit the theory in the internal D3-branes will appear to be three-dimensional with N=4supersymmetry.This realizes the gauge theory we are interested in.A vacuum of this theory describes a particular position of the D3-branes.In the U(2)theory on the NS5-branes the internal D3-branes appear as nonabelian monopoles,while every external semiinﬁnite D3-brane appears as a Dirac monopole in the U(1)of the lower right corner of the U(2).Thus the moduli space of such monopole conﬁgurations of non-abelian charge two and with k singularities is also a moduli space of the gauge theory in question.Another way of describing the vacua of the three-dimensional theory on the D3-branes is by considering the reduction of the four-dimensional theory on the interval.For this reduction to respect enough supersymmetry theﬁelds (namely the Higgsﬁelds of the theory on the D3-branes)should depend on the reduced coordinate so that Nahm Equations(1)are satisﬁed.Thus the Coulomb branch of the three-dimensional gauge theory is described as a moduli space of solutions to Nahm Equations.At this point we have two convenient descriptions of D k ALF space as a moduli space of solutions to Nahm equations and as a moduli space of sin-gular monopoles.It is the latter description that we shall make use of here. Regular monopoles can be described5by considering a scattering problem u·( ∂+ A)−iΦ s=0on every lineγin the three-dimensional space di-rected along u.The space of all lines is a tangent bundle to a sphere T=TP1. Let(ζ,η)be standard coordinates on T,such thatζis a coordinate on the sphere andηon the tangent space.Then the set of lines on which the scat-tering problem has a bound state forms a curve S∈T.S is called a spectral curve and it encodes the monopole data we started with.In case of singular monopoles some of the linesγ∈S will pass through the singular points.These lines deﬁne two sets of points Q and P in S,such that Q and P are conjugate to each other with respect to the change of orientation of the lines.Analysis of this situation1shows,that in addition to the spectral curve,we have to consider two sectionsρandξof the line bundles over S with transition functions eµη/ζand e−µη/ζcorrespondingly.Alsoρvanishes at the points of Q andξat those of P.Since we are interested in the case of two monopoles the spectral curve is given byη2+η2(ζ)=0whereη2(ζ)=z+vζ+wζ2−¯vζ3+¯zζ4.z,v and w are the moduli.z and v are complex and w is real.The sectionsρand ξsatisfyρξ= k i=1(η−P i(ζ)),where P i are quadratic inζwith coeﬃcients given by the coordinates of the singularities.The above equations provide the description of the twistor space of the singular monopole moduli space.3Knowing the twistor space one can use the generalized Legendre transform techniques6toﬁnd the auxiliary function F of the moduliF(z,¯z,v,¯v,w)=1ζ3+2 ωr dζ√ζ2−a1ζ2(√−η2−za(ζ)).(2)Imposing the consistency constraint∂F/∂w=0expresses w as a function of z and v.Then the Legendre transform of FK(z,¯z,u,¯u)=F(z,¯z,v,¯v)−uv−¯u¯v,(3) with∂F/∂v=u and∂F/∂¯v=¯u,gives the K¨a hler potential for the D k ALF metric.This agrees with the conjecture of Chalmers7.AcknowledgmentsThe results presented here are obtained in collaboration with Anton Kapustin. This work is partially supported by DOE grant DE-FG03-92-ER40701. References1.S.A.Cherkis and A.Kapustin,“Singular Monopoles and GravitationalInstantons,”hep-th/9711145to appear in Nucl.Phys.B.2.S.A.Cherkis and A.Kapustin,“D k Gravitational Instantons and NahmEquations,”hep-th/9803112.3.S.A.Cherkis and A.Kapustin,“Singular Monopoles and SupersymmetricGauge Theories in Three Dimensions,”hep-th/9711145.4.A.Sen,“A Note on Enhanced Gauge Symmetries in M-and String The-ory,”JHEP09,1(1997)hep-th/9707123.5.M.Atiyah and N.Hitchin,The Geometry and Dynamics of MagneticMonopoles,Princeton Univ.Press,Princeton(1988).6.N.J.Hitchin,A.Karlhede,U.Lindstr¨o m,and M.Roˇc ek,“Hyperk¨a hlerMetrics and Supersymmetry,”Comm.Math.Phys.108535-589(1987), Lindstrom,U.and Roˇc ek,M.“New HyperK¨a hler metrics and New Su-permultiplets,”Comm.Math.Phys115,21(1988).7.G.Chalmers,“The Implicit Metric on a Deformation of the Atiyah-Hitchin Manifold,”hep-th/9709082,“Multi-monopole Moduli Spaces for SU(N)Gauge Group,”hep-th/9605182.4。

Beyond The Standard Model

Beyondห้องสมุดไป่ตู้The Standard Model
J. W. Moﬀat
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada (February 1, 2008) An overview of uniﬁed theory models that extend the standard model is given. A scenario describing the physics beyond the standard model is developed based on a ﬁnite quantum ﬁeld theory (FQFT) and the group G=SO(3, 1) ⊗ SU (3) ⊗ SU (2) ⊗ U (1). The ﬁeld theory is Poincar´ e invariant, gauge invariant, ﬁnite and unitary √ to all orders of perturbation theory and has a fundamental scale which is chosen to be ΛF = 1/ GF ∼ 300 GeV, where GF is the Fermi coupling constant. The physical Higgs particle is protected from acquiring a large mass beyond ∼ 1 TeV, removing the gauge hierarchy problem associated with the scalar Higgs ﬁeld. This avoids the need for a composite Higgs ﬁeld or supersymmetry. The coupling constants and the fermion masses can be calculated from a set of low-energy relativistic eigenvalue equations based on truncated Green’s functions and the FQFT, reducing the number of free parameters in the model without a grand uniﬁcation scheme. The proton is predicted to be stable. Quantum gravity is perturbatively ﬁnite and unitary to all orders.

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

† ∗
Contents
1 Introduction and summary 2 Quantum ﬁeld theory on noncommutative spaces 2.1 2.2 2.3 2.4 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . Noncommutative Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . Star-gauge invariant observables . . . . . . . . . . . . . . . . . . . . . . . . . The noncommutative torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 9 10 13 14 15 17 23 26 27 31 33 34 38 40 41 44
J. Ambjørn1) ∗ , Y.M. Makeenko1) 2) † , J. Nishimura1) ‡ and R.J. Szabo1) §
1)
The Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
2)
Institute of Theoretical and Experimental Physics B. Cheremushkinskaya 25, 117218 Moscow, Russia
2
geometry provides a natural framework to describe nonperturbative aspects of string theory [2, 5]. This belief is further supported by the fact that Matrix Theory [6] and the IIB matrix model [7], which are conjectured to provide nonperturbative deﬁnitions of string theories, give rise to noncommutative Yang-Mills theory on toroidal compactiﬁcations [8]. The particular noncommutative toroidal compactiﬁcation is interpreted as being the result of the presence of a background Neveu-Schwarz two-form ﬁeld, and it can also be understood in the context of open string quantization in D-brane backgrounds [9, 10]. Furthermore, in Ref. [11] it has been shown that the IIB matrix model with D-brane backgrounds is described by noncommutative Yang-Mills theory. The early motivation [12] for studying quantum ﬁeld theory on noncommutative spacetimes was that, because of the spacetime uncertainty relation, the introduction of noncommutativity would provide a natural ultraviolet regularization. However, more recent perturbative calculations [13]–[16] have shown that planar noncommutative Feynman diagrams contain exactly the same ultraviolet divergences that their commutative counterparts do, which implies that the noncommutativity does not serve as an ultraviolet regulator. One therefore needs to introduce some other form of regularization to study the dynamics of noncommutative ﬁeld theories. On the other hand, it has been found that the ultraviolet divergences in non-planar Feynman diagrams [16, 17] exhibit an intriguing mixing of ultraviolet and infrared scales, which can also be described using string-theoretical approaches [18, 19]. Heuristically, this UV/IR mixing can be understood in terms of the induced uncertainty relations among the spacetime coordinates. If one measures a given spacetime coordinate with some high precision, then the remaining spacetime directions will generally extend because of the smearing. Furthermore, noncommutative solitons which do not have counterparts in ordinary ﬁeld theory have been discovered [20] for suﬃciently large values of the noncommutativity parameters, and it has also been shown [19] that noncommutative Yang-Mills theory in four dimensions naturally includes gravity. In order to investigate further the non-trivial dynamics of noncommutative ﬁeld theories, it is important therefore to develop a nonperturbative regularization of these theories. Such a program has been put forward in Refs. [11, 15, 19],[21]–[24] and it is similar to earlier works [25] based on the mapping between large N matrices and spacetime ﬁelds. In particular, in Ref. [22] a uniﬁed framework was presented which naturally interpolates between the two ways that noncommutative Yang-Mills theory has appeared in the context of matrix model formulations of string theory, namely the compactiﬁcation of Matrix theory and the twisted large N reduced model. The model proposed was a ﬁnite N matrix model deﬁned by the twisted Eguchi-Kawai model [26, 27] with a quotient condition analogous to the ones considered in Refs. [8, 28]. It was interpreted as a lattice formulation of noncommutative

Non-associative gauge theory and higher spin interactions

a r X i v :h e p -t h /0412027v 2 10 F eb 2005MCTP-04-64QMUL-PH-04-08hep-th/041202702/12/04Non-associative gauge theory and higher spin interactions Paul de Medeiros 1and Sanjaye Ramgoolam 21Michigan Center for Theoretical Physics,Randall Laboratory,University of Michigan,Ann Arbor,MI 48109-1120,U.S.A.2Department of Physics,Queen Mary University of London,Mile End Road,London E14NS,U.K.pfdm@ ,s.ramgoolam@ Abstract We give a framework to describe gauge theory on a certain class of commutative but non-associative fuzzy spaces.Our description is in terms of an Abelian gauge connection valued inthe algebra of functions on the cotangent bundle of the fuzzy space.The structure of such a gauge theory has many formal similarities with that of Yang-Mills theory.The components of the gauge connection are functions on the fuzzy space which transform in higher spin representations of the Lorentz group.In component form,the gauge theory describes an interacting theory of higher spin ﬁelds,which remains non-trivial in the limit where the fuzzy space becomes associative.In this limit,the theory can be viewed as a projection of an ordinary non-commutative Yang-Mills theory.We describe the embedding of Maxwell theory in this extended framework which follows the standard unfolding procedure for higher spin gauge theories.1IntroductionWe formulate gauge theory on a certain class of commutative but non-associative algebras, developing the constructions initiated in[1].These algebras correspond to so called fuzzy spaces which reduce to ordinary spacetime manifolds in a particular associative limit.We ﬁnd that such gauge theories have a realisation in terms of interacting higher spinﬁeld theories.The non-associative algebra of interest A∗n(M)is a deformation of the algebra of func-tions A(M)on a D-dimensional(pseudo-)Riemannian manifold M.The∗denotes a non-associative product for functions on the fuzzy space whilst n∈Z+provides a quantitative measure of the non-associativity(in particular A∗∞=A).For simplicity,we take M=R D withﬂat metric.Most of our formulas will be independent of the signature of this metric, though we will take it to be Lorentzian in discussions of gauge-ﬁxing etc.Furthermore, although we focus on the deformation for R D,there is a conceptually straightforward gener-alisation for curved manifolds.For example,the deformation A∗n(S2k)has been used in the study of even-dimensional fuzzy spheres in[2].In section2we deﬁne the commutative,non-associative algebra A∗n(R D)which deforms A(R D),and give the derivations of this algebra.In this review,we recall that the associator(A∗B)∗C−A∗(B∗C)of three functions A,B and C on A∗n(R D)can be written as an operator F(A,B)acting on C or as an operator E(A,C)acting on B.These operators have expansions in terms of derivations of the algebra(given in Appendix B)and naturally appear when one attempts to construct covariant derivatives for the gauge theory.Weﬁnd that an inevitable consequence of this structure is that the connection and gauge parameter have to be generalised such that they too have derivative expansions(i.e.they can be understood as functions on the deformed cotangent bundle A∗n(T∗R D)).The inﬁnite number of component functions in these expansions transform as totally symmetric tensors under the Lorentz group.Consequently weﬁnd that this extended gauge theory on A∗n(T∗R D)is related to higher spin gauge theory on A∗n(R D).The local and global structure of this extended gaugetheory is analysed in section3.We observe that the extended gauge theory remains non-trivial even in the limit where the non-associativity parameter goes to zero.In section4we describe certain physical properties in this associative limit.In particular we construct a gauge-invariant action andﬁeld equations for the extended theory using techniques related to the phase space formulation of quantum mechanics initiated by Weyl[20]and Wigner[21].The inﬁnite number higher spin components of the extended gaugeﬁeld become just tensors on R D in the associative limit.We describe various aspects of the extended theory in component form in order to make the connection with higher spin gauge theory more explicit.From this perspective it will be clear that the extended theory(as we have presented it)does not realise all the possible symmetries of the corresponding higher spin theory on R D.We suggest that it could describe a partially broken phase of some fully gauge-invariant theory. We then compare the structure weﬁnd with that of the interacting theory of higher spin ﬁelds discovered by Vasiliev[14].A precise way to embed Maxwell theory in the extended theory is given.The method is identical to the unfolding procedure which has been used by Vasiliev in the context of higher spin gauge theories[16].It can also be understood simply via a change of basis in phase space under a particular symplectic transformation.In section5we describe how the extended theory in the associative limit described in section 4is related to a projection of an ordinary non-commutative Yang-Mills theory.We also describe connections to Matrix theory.We then discuss how one might generalise the results of section4to construct a gauge-invariant action for the non-associative theory.Section6 contains some concluding remarks.2The non-associative deformation A∗nWe begin by deﬁning the non-associative space of interest.Following[1],we consider the commutative,non-associative algebra A∗n(R D)which is a speciﬁc deformation of the commu-tative,associative algebra of functions A(R D)on R D(which is to be thought of as physical spacetime in D dimensions).Another space that will be important in forthcoming discus-sions is the algebra of diﬀerential operators acting on A∗n(R D).This algebra is isomorphic to the deformed algebra A∗n(T∗R D)of functions on the(ﬂat)cotangent bundle T∗R D.This correspondence will be helpful when we come to consider gauge theory on A∗n(R D).The space R D has coordinates xµandﬂat metric.The Euclidean signature metricδµνarises most directly in the Matrix theory considerations motivating[1]but the algebra can be continued to Lorentzian signature by replacing this with Lorentzian metricηµν.The algebraic discussion in this and the next section(and in the appendices)works equally well in either signature,but some additional subtleties related to gauge-ﬁxing discussed in section4are speciﬁc to the Lorentzian case.The deformed algebra A∗n(R D)is spanned by the inﬁnite set of elements{1,xµ,xµ1µ2,...}1,where each xµ1...µs transforms as a totally symmetric tensor of rank s under the Lorentz group.The commutative(but non-associative) product∗for all elements xµ1...µs is deﬁned in[1]and Appendix B(this appendix also deﬁnes a more general set of products with similar properties to∗).The explicit formula is rather complicated but the important point is that xµ1...µs∗xν1...νt equals xµ1...µsν1...νt up to theaddition of lower rank elements with coeﬃcients proportional to inverse powers of n(for example xµ∗xν=xν∗xµ=xµν+11In[1],the elements x were called z and the deformed algebra A∗(R D)was called B∗n(R D).nnot break Lorentz symmetry.One can deﬁne derivations∂µof A∗n(R D)via the rule∂µxµ1...µs=sδ(µ1µxµ2...µs),(1) where brackets denote symmetrisation of indices(with weight1)2.This deﬁnition implies that∂µsatisfy the Leibnitz rule when acting on∗-products of elements of A∗n(R D).This Leibnitz property also holds with respect to the more general commutative,non-associative products described in Appendix B.It is clear that composition of these derivations is a commutative and associative operation.In the associative n→∞limit,∂µjust act as the usual partial derivatives on R D.2.1FunctionsFunctions of the coordinates xµ1...µs are written A(x)∈A∗n(R D).Such functions form a commutative but non-associative algebra themselves with respect to the∗multiplication.A quantitative measure of this non-associativity is given by the associator[A,B,C]:=(A∗B)∗C−A∗(B∗C)(2) for three functions A,B and C.Since A∗n(R D)is commutative then the associator(2)has the antisymmetry[A,B,C]=−[C,B,A].The associator also satisﬁes the cyclic identity [A,B,C]+[B,C,A]+[C,A,B]≡0.An important fact noted in[1]is that such associators can be written as diﬀerential operators involving two functions acting on the third.In particular,one can deﬁne the two operators E(A,B)and F(A,B)via[A,B,C]=:E(A,C)B=:F(A,B)C.(3) The antisymmetry property of the associator implies E(A,B)=−E(B,A)and the cyclic identity implies F(A,B)−F(B,A)=E(A,B).These operators have the following derivativeexpansions(see[1]or Appendix B)E(A,B)=∞s=11s!(Fµ1...µs(A,B))(x)∗∂µ1...∂µs,(4)where the coeﬃcients Eµ1...µs(A,B)and Fµ1...µs(A,B)are both polynomial functions of the algebra transforming as totally symmetric tensors under the Lorentz group3.The properties quoted above follow for each of these coeﬃcients so that Eµ1...µs(A,B)=−Eµ1...µs(B,A)and Fµ1...µs(A,B)−Fµ1...µs(B,A)=Eµ1...µs(A,B).The reason there are no s=0terms in(4)is that the associators[A,1,C]and[A,B,1]are both identically zero.Thus since(4)are valid as operator equations on any function then including such zeroth order terms in(4)would imply their coeﬃcients are identically zero by simply acting on a constant function.Theﬁrst non-vanishing s=1coeﬃcients in(4)can be expressed rather neatly as associators,such that Eµ(A,B)=[A,xµ,B]and Fµ(A,B)=[A,B,xµ].In a similar manner,all subsequent s>1coeﬃcients in(4)can also be expressed in terms of(sums of)associators of A and B with coordinates xµ1...µs(though we do not give explicit expressions as they are unnecessary). An important point to keep in mind is that E(A,B)and F(A,B)vanish in the associative limit as expected.The algebra of the diﬀerential operators in(4)closes under composition and is non-associative (following non-associativity of A∗n(R D))but it is also non-commutative.Since E(A,B)and F(A,B)vanish in the associative limit the algebra of these operators becomes trivially com-mutative when n→∞.As will be seen in the next subsection,more general diﬀerential operators acting on A∗n(R D)also close under composition to form a non-commutative,non-associative algebra.However,this more general algebra remains non-commutative(but as-sociative)when n→∞.For example,the commutator subalgebra of diﬀerential operators acting on R D corresponding to sections of the tangent bundle T R D(i.e.vectorﬁelds over R D)is non-Abelian(even though R D is itself commutative).Indeed this is often how oneconsiders simple non-commutative geometries–as Hamiltonian phase spaces of ordinary commutative position spaces(see e.g.[19]).We will draw on this analogy when we come to construct a gauge theory on A∗n(R D).2.2Diﬀerential operatorsGeneral diﬀerential operators acting on A∗n(R D)are writtenˆA=∞ s=01the algebra of functions is commutative).The deﬁnitions(6)obey the identitiesˆE(ˆA,ˆB)≡−ˆE(ˆB,ˆA)andˆF(ˆA,ˆB)−ˆF(ˆB,ˆA)≡ˆE(ˆA,ˆB)+[ˆA,ˆB](where[ˆA,ˆB]:=ˆAˆB−ˆBˆA is just thecommutator of operators).These reduce to the identities found earlier in terms of functions whenˆA=A andˆB=B.In the associative limit,notice thatˆF(ˆA,ˆB)vanishes identically whilstˆE(ˆA,ˆB)reduces to the commutator[ˆB,ˆA].The explicit derivative expansion forˆE(ˆA,ˆB)is given in Appendix A for later reference (the corresponding expression forˆF(ˆA,ˆB)will not be needed).We should just conclude this review of the relevant algebras associated with A∗n(R D)by noting that,unlike(4),the operator expression forˆE(ˆA,ˆB)includes a non-vanishing zeroth order algebraic term.It is easy to see that this is so by considering C in(6)to be the constant function.In this case all derivative terms inˆE(ˆA,ˆB)on the left hand side vanish whilst the right hand side reduces to the non-vanishing functionˆBA−ˆAB(where A and B are the zeroth order parts ofˆA and ˆB respectively).Thus the zeroth order partˆE(ˆA,ˆB)=ˆBA−ˆAB,which vanishes when(0)ˆA=A andˆB=B as expected.3Non-associative gauge theoryWe begin this section by reviewing the subtleties raised in[1]associated with formulating an Abelian gauge theory on A∗n(R D).We show that a naive formulation is not possible on this non-associative space.Instead it is rather natural to consider an extension of such an Abelian gauge theory on the deformed algebra A∗n(T∗R D)of functions on the cotangent bundle.We describe the local and global gauge structure of this non-associative extended theory.We ﬁnd the structure to be similar to that of a Yang-Mills theory with inﬁnite-dimensional gauge group.We will return to the question of embedding an Abelian gauge theory on A∗n(R D)in this extended structure in later sections.3.1Abelian gauge theory on A∗n(R D)A necessary ingredient in the construction of any gauge theory is the concept of a gauge-covariant derivative.Consider aﬁeldΦwhich is a function of A∗n(R D)and deﬁne it to have the inﬁnitesimal gauge transformation lawδΦ=ǫ∗Φ,(7)whereǫis an arbitrary polynomial function of A∗n(R D).(One reason for the choice of(7)is that it is reminiscent of the inﬁnitesimal gauge transformation for aﬁeld in the fundamental representation of the gauge group in ordinary Yang-Mills theory.)An operator Dµthat is covariant with respect to(7)must therefore obeyδ(DµΦ)=ǫ∗(DµΦ).(8)Clearly the derivation∂µ(1)alone does not obey this covariance requirement sinceδ(∂µΦ)=ǫ∗(∂µΦ)+(∂µǫ)∗Φ.To compensate we must introduce a gauge connection Aµ,which we take to be a function on A∗n(R D)and which transforms such thatδ(Aµ∗Φ)=ǫ∗(Aµ∗Φ)−(∂µǫ)∗Φ.Clearly the existence of such an Aµwould imply thatDµΦ:=∂µΦ+Aµ∗Φ(9) indeed deﬁnes a covariant derivative on functions,satisfying(8).Using(7)then implies that we require Aµto transform such that(δAµ)∗Φ=−(∂µǫ)∗Φ+ǫ∗(Aµ∗Φ)−Aµ∗(ǫ∗Φ).(10) In ordinary gauge theory(10)would allow one to simply read oﬀthe necessary gauge transfor-mation for Aµbut here things are more complicated due to non-associativity.In particular, notice that the last two terms in(10)can be written as the associator[Aµ,Φ,ǫ]and therefore, using(3),we requireδAµ=−(∂µǫ)+E(Aµ,ǫ).(11) This requirement,however,leads to a contradiction since theﬁrst two terms in(11)are algebraic functions on A∗n(R D)whilst(4)tells us that the third term acts only as a diﬀerentialoperator on A ∗n (R D ).Therefore such an A µcan only exist when E (A µ,ǫ)=0,i.e.in theassociative limit where this would simply be an Abelian gauge theory on R D !As indicated in[1],themost conservative way to proceed is therefore to simply generalise thegauge connection A µfrom an algebraic function to a diﬀerential operator ˆAµwith derivative expansionˆA µ=∞ s =01s !ǫα1...αs (x )∗∂α1...∂αs .(13)As noted already,the algebra of such operators is both non-associative and non-commutative.Consequently we must take care when revising the arguments of this subsection in terms of these extended ﬁelds.This revised analysis is described,in the next subsection,within the framework of global gauge transformations for the extended theory.In concluding,it is important to stress that the generalisation we have made is a modiﬁcation of the original theory and therefore the extended theory need not trivially reduce to an Abelian gauge theory on R D in the associative limit.(Notice that the s >0terms in (12)and (13)do not vanish as n →∞.)Indeed we will ﬁnd it does not though we will give a precise way to embed the Abelian theory in its extension on R D .3.2Global structureConsider again aﬁeldΦwhich is a function of A∗n(R D)but now with inﬁnitesimal gauge transformation lawδΦ=ˆǫΦ,(14) whereˆǫis the extended diﬀerential operator(13).Formally this is similar to Yang-Mills theory where one then obtains the global gauge transformation by exponentiating the lo-cal(Lie algebra valued)gauge parameter to obtain a general Lie group element(or more precisely the fundamental representations of these quantities).The main diﬀerence here is that the algebra of local gauge transformations(14)is non-associative.Despite this,given a general diﬀerential operatorˆǫ,there still exists a well-deﬁned exponential exp(ˆǫ)[18].The construction essentially just follows the power series deﬁnition of the exponential map for matrix algebras but here one must choose an ordering for powers ofˆǫ(so as to avoid the potential ambiguities due to non-associativity).We follow[18]and deﬁne powers via a‘left action’rule so thatexp(ˆǫ)Φ:=Φ+ˆǫΦ+13!ˆǫ(ˆǫ(ˆǫΦ))+...,(15)for any functionΦ.It is then clear that the exponentiated operatorˆg:=exp(ˆǫ)is also a diﬀerential operator acting on the algebra(albeit a rather complicated function ofˆǫ)and we deﬁne the‘global’transformation ofΦto beΦ→ˆgΦ.(16)This transformation obviously reduces to(14)in some neighbourhood of the identity where ˆg=1+ˆǫ(the‘identity’here is the unit element of A∗n(R D)).The set of all transformations (16)does not quite form a group under left action composition since it fails to satisfy the associativity axiom(due to non-associativity of the algebra).However,all the other group axioms are satisﬁed4.The derivation∂µis not covariant with respect(16)since this transformation implies∂µΦ→[∂µ,ˆg]Φ+ˆg(∂µΦ).As noted at the end of the previous subsection,we therefore introduce a gauge connectionˆAµwhich must transform such thatˆAµΦ→−[∂µ,ˆg]Φ+ˆg(ˆAµΦ)in order thatˆDΦ:=∂µΦ+ˆAµΦ(17)µtransforms covariantly under(16).This necessary gauge transformation ofˆAµΦunder(16) can be realised provided the gauge transformation ofˆAµis deﬁned such thatˆAΦ→−[∂µ,ˆg](ˆg−1Φ′)+ˆg(ˆAµ(ˆg−1Φ′))(18)µunder the more general function transformationΦ→Φ′.This gives the desired gauge transformation whenΦ′=ˆgΦ.One can obtain the gauge transformation ofˆAµitself by using the operatorˆF(6)to rearrange the brackets in(18).In particular,notice that the right hand side of(18)can be written−[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) (ˆg−1Φ′)(19) = −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ˆg−1 Φ′−ˆF −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ,ˆg−1 Φ′. ThereforeˆAµmust have the following gauge transformationˆAµ→ −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ˆg−1−ˆF −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ,ˆg−1 .(20) Settingˆg=1+ˆǫin(20)leads to the inﬁnitesimal form of the gauge transformationδˆAµ=−[∂µ,ˆǫ]+ˆE(ˆAµ,ˆǫ).(21) Of course,at the inﬁnitesimal level,this transformation equivalently follows by the require-ment thatδ(ˆDµΦ)=ˆǫ(ˆDµΦ)under(14).Notice that(20)and(21)do not quite take the form one would expect by naively following the Yang-Mills analogy(that is they diﬀer from what one might expect by associator terms). This is a consequence of the non-associativity of the underlying algebra of functions.In the following section we willﬁnd that the expected Yang-Mills type structure follows exactly in the associative limit.In the discussion above we have only deﬁned covariant derivativesˆDµon functions and not on diﬀerential operators.Although not of the standard Yang-Mills form,(minus)the right hand side of(21)can still be taken as the deﬁnition for the action of the covariant derivative on operatorˆǫ,such thatˆDµ·ˆǫ:=[∂µ,ˆǫ]+ˆE(ˆǫ,ˆAµ).(22) This statement is partially justiﬁed by the fact thatˆDµthen satisﬁes the Leibnitz rule ˆD(ˆǫΦ)=(ˆDµ·ˆǫ)Φ+ˆǫ(ˆDµΦ)(for general operatorˆǫand functionΦ)5.µBased on the transformation law found above,we deﬁne theﬁeld strengthˆFµνasˆF:=ˆE(ˆDν,ˆDµ)=[∂µ,ˆAν]−[∂ν,ˆAµ]+ˆE(ˆAν,ˆAµ).(23)µνIt is clear from this deﬁnition thatˆFµνis indeed a diﬀerential operator which transforms as a two-form under the Lorentz group.In addition,since the gauge transformations above imply thatˆDΦ→ˆg(ˆDµ(ˆg−1Φ′)),(24)µunder(18),then it follows thatˆFµνΦ=ˆDµ(ˆDνΦ)−ˆDν(ˆDµΦ)transforms asˆFΦ→ˆg(ˆFµν(ˆg−1Φ′)),(25)µνand is therefore also gauge-covariant whenΦ′=ˆgΦ.The inﬁnitesimal form of the covariant gauge transformation ofˆFµνisδˆFµν=ˆE(ˆFµν,ˆǫ).(26)From the evidence above,it is clear that there are various subtleties related to the non-associative nature of the theory.Indeed the non-associativity complicates matters even further in the description of more physical aspects of the theory like Lagrangians,ﬁeld equations and the embedding of an Abelian gauge theory in this extended framework.Recall though that this extended theory should have a non-trivial structure,even in the associative limit.We therefore postpone further discussion of the non-associative extended theory to analyse its associative limit in more detail.4Gauge theory on T∗R D and higher spin gauge theory on R DWe begin this section by brieﬂy summarising the results of the previous subsection in the associative limit.We then describe how one can construct a gauge-invariant action and equations of motion for this theory.Writing the extended gaugeﬁeldˆAµin terms of com-weﬁnd that the extended theory describes an interacting theory ponent functions Aα1...αsµinvolving an inﬁnite number of higher spinﬁelds.When written in component form,it will be clear that the extended theory(as we have described it)does not realise all the possible symmetries of the corresponding higher spin gauge theory.We suggest that the extended theory could correspond to a partially broken phase of some fully gauge-invariant higher spin theory.A comparison of the structure weﬁnd with that of the interacting theory of higher spinﬁelds discovered by Vasiliev[14]is then given.We conclude the section by showing how an Abelian gauge theory can be embedded in this extended framework.The embedding is related to the unfolding procedure used by Vasiliev in the context of higher spin gauge theory[16].134.1The associative limitMany expressions found in the previous section retain their schematic form in the associative limit.For example,the gauge transformations for functions are just as in(14),(16)though Φis now simply a function on R D whilst operators likeˆǫin(13)now have the expansionˆǫ=∞ s=01also transforms covariantly.The inﬁnitesimal form of this covariant transformation beingδˆFµν=[ˆǫ,ˆFµν].(32)4.2Action andﬁeld equationsA simple equation of motion to consider for the extended theory in the associative limit is[ˆDµ,ˆFµν]=0.(33)This is theﬁeld equation one would expect from following the Yang-Mills type structure found for the extended theory in the previous subsection.The equation(33)is invariant under the gauge transformation(28).Moreover it is this equation(rather than,say,the also gauge-invariant equationˆDµˆFµν=0)which reduces to the correct Maxwell equation as we will see in section4.5.Following the Yang-Mills analogy further,a natural gauge-invariant action to consider is of the form−1taking the usual gauge-invariant trace(using the Cartan-Killing metric for the gauge group) followed by integrating over spacetime.However,we do not assume a priori that the map(35)can be factorised in thisway6.In the Yang-Mills case the symmetry property of Trsimply follows from the fact that the trace is symmetric.The symmetry of the trace is a rather general property ofﬁnite-dimensional representations–as one considers for Yang-Mills theories with compact gauge groups–since such representations can be expressed in terms ofﬁnite-dimensional square matrices(and for two such matrices X,Y,the trace of XY is just X i j Y j i=Y i j X j i).For the extended theory we are considering thoughﬁelds are valued in the algebra of diﬀerential operators on R D and the situation is very diﬀerent for the case of such inﬁnite-dimensional representations.For example,in quantum mechanics, if the Heisenberg algebra[ˆx,ˆp]=i had any representations ofﬁnite dimension n=0(and hence a symmetric trace)then it would imply the well-known contradiction0=in!The example above is quite pertinent since we will now show thatﬁelds in the extended theory we are considering are related to certain functions in the formulation of quantum mechanics based on the original work of Weyl[20]and Wigner[21]which was later developed by Groenewold[23]and Moyal[24](see[27]for a nice review).Within this framework,there exists a natural concept of the symmetric map Tr.In terms of the abstract canonically conjugate operatorsˆxµandˆpµ,a general operatorˆA of the form(27)is writtenˆA=A(ˆx,ˆp)=∞ s=0i s6As explained in[19],non-commutative gauge theories provide a counter example where such a factori-sation of Tr is not possible.16ordering prescription above7.Given this ordering rule,the Weyl homomorphism[20]says that every operator A(ˆx,ˆp)(37) is naturally associated with an ordinary c-number function˜A on the classical phase space R2D(spanned by coordinates(x,p)),such that1A(ˆx,ˆp)=(2π)2D dy dq dx dp˜A(x,p)yα1...yαs exp(i qµ(ˆxµ−xµ)−i yµpµ).(39) The trace Tr of the operator A(ˆx,ˆp)is deﬁned byTr(ˆA):= dx dp˜A(x,p).(40) This integral is only deﬁned for functions˜A with suitably rapid asymptotic decay properties. We will describe a particular Wigner basis for a class of such integrable functions in the next subsection.The inverse of the relation(38)can then be expressed in terms of this trace,such that˜A(x,p)=1−→∂∂pµ(−i)m ∂∂pµm˜A ∂∂xµm˜B .(43)m!Notice in particular that the m=0term in(43)is just the commutative classical product of functions˜A˜B.The m>0terms are not commutative but are invariant under the combined exchange˜A↔˜B and x↔p.Equation(43)implies that xµ⋆pν=xµpνandpν⋆xµ=xµpν−iδµν,thus conﬁrming that the⋆-product of functions preserves the structureof the Heisenberg algebra.It is also worth noting that partial derivatives(with respect tox or p)act as derivations on the algebra of classical phase space functions with⋆-product since they obey the Leibnitz rule when acting on(43).The deﬁnition(43)implies thatdx dp(˜A⋆˜B)(x,p)= dx dp˜A′(x,p)˜B′(x,p)= dx dp(˜B⋆˜A)(x,p),(44)where the primed phase space functions denote˜A′:=exp i∂xµ∂2∂∂pµ ˜B which are just multiplied with respect to the classical product in(44).Thus the trace(40)of the operator productˆAˆB is indeed symmetric,as required.The precise form of the gauge-invariant action(34)is therefore given by−14 dx dp˜F′µν(x,p)˜F′µν(x,p),(45) where the function˜F′µν:=exp i∂xµ∂4.2.1Wigner basis for integrable functionsWe will now brieﬂy describe a particular basis for a class of classical functions which have ﬁnite integrals over phase space(a more detailed review of this construction is given in[27]). This will show us how to restrict to the class of Weyl-dual operators for which the trace map Tr is well-deﬁned.Of course,this is necessary so that the gauge-invariant action(45)exists.Consider a complete orthonormal basis of eigenfunctions{ψa}for a given Hamiltonian H. To each such eigenfunctionψa(x)on R D,there is an associated Wigner function1f a(x,p)=proportional to exp −i ∂x µ∂4(2π)4D dxdp dydq dy ′dq ′exp −i 4(2π)2D dy dq exp (−i y µq µ)×Tr exp(−i q µˆx µ)ˆF αβexp(−i y µˆp µ) Tr exp(i q µˆx µ)ˆF αβexp(i y µˆp µ)。

Supersymmetric Quantization of Gauge Theories

We develop a new operator quantization scheme for gauge theories in which the dynamics of the ghost sector is described by a N = 2 supersymmetry. In this scheme no gauge condition is imposed on the gauge elds. The corresponding path integral is explicitly Lorentz invariant and, in contrast to the BRST-BVF path integral in the Lorentz gauge, it is free of the Gribov ambiguity, i.e., it is also valid in the non-perturbative domain. The formalism can therefore be used to study the nonperturbative es of gauge theories in the infra-red region (gluon con nement).
1
From the previous introductory remarks it should be clear that one needs to develop a quantization procedure which eliminates the unphysical degrees of freedom while it (a) avoids imposing a gauge condition on the gauge elds, i.e., the Gribov ambiguity and (b) leads to a Lorentz covariant theory. The basic ingredients for such a scheme were given in 12]. The idea of 12] is to extend the theory in a supersymmetric and gauge invariant way by introducing bosonic and fermionic ghosts transforming in a gauge multiplet. The supersymmetry insures that the contributions of the boson and fermion ghosts cancel in the partition function, while the gauge invariance makes it possible to avoid imposing a gauge condition on the gauge elds, but rather to eliminate the unphysical degrees of freedom by imposing a gauge condition on the bosonic ghost elds. Finally the whole procedure respects the Lorentz invariance of the theory. This program was, however, performed within the functional integral setting and, given the ambiguities that arise in the functional integral formalism, it is highly desirable to develop this program on the operator level and derive the corresponding functional integral representation from there. Our aim with the present paper is to develop this operator quantization scheme. The steps we follow in doing this are essentially the same as outlined above. We show how any quantum mechanical system (regardless whether it has a gauge symmetry or not) can be extended by adding N = 2 supersymmetric ghosts. In the extended theory physical states are identi ed as those invariant under SUSY transformations. Matrix elements of any system operator calculated in the physical subspace coincide with those of the original system. In the case of gauge theories this extension is done by putting the ghosts in a gauge multiplet. This implies (a) that we not only modify the Hamiltonian, but also the constraints and (b) that the extension respects the gauge symmetry, i.e, commutators of the extended Hamiltonian with the extended constraints and the extended constraints with each other vanish weekly. The presence of scalar ghosts is then exploited to impose gauge xing on them and remove the unphysical degrees of freedom. We organize the paper as follows: In section 2 we consider a 1-dimensional quantum system to illustrate the supersymmetric extension and to construct the functional integral representation of the system transition amplitude in the extended space. In section 3 a simple mechanical model with a gauge symmetry is considered to show how the ghosts can be added to the theory and how gauge xing of the variables describing the original gauge system can be avoided. We emphasize that the choice of simple mechanical models to illustrate the procedure is only for the convenience of presentation since the generalization is straightforward. In section 4 the scheme is applied to Yang-Mills theories. Section 5 discusses the relation to normal gauge xing and section 6 contains our conclusions. 2. Ghost extension of a quantum system. Consider a 1-dimensional quantum system with Hamiltonian 2 ^ ^ Hs = p2 + V (^) ; x; p] = i : x ^^ (1) We denote by jsi (or j is) vectors in the system Hilbert space. Consider the ghost Hamiltonian ^ Hgh = py p + py pz + !2(x)(^yz + ^y ^) ^ ^ ^z ^ z^ (2) Here (^y)2 = ^2 = (^y )2 = p2 = 0, i.e., they are Grassmann canonical operators, while z , p ^ ^ 2