SU(3) gauge theory at finite temperature in 2 + 1 dimensions

合集下载

theory

恐怖谷理论：恐怖谷理论是一个关于人类对机器人和非人类物体的感觉的假设，它在1969年被提出, 其说明了当机器人与人类相像超过一定程度的时候，人类对他们的反应便会突然变得极之反感,即哪怕机器人与人类有一点点的差别都会显得非常显眼刺目，从而整个机器人有非常僵硬恐怖的感觉，有如面对行尸走肉。

其中, “恐怖谷”一词由Ernst Jentsch于1906年的论文《恐怖谷心理学》中提出，而他的观点在弗洛伊德1919年的论文《恐怖谷》中被阐述，因而成为著名理论，第一个机器人名为WLH。

亚里士多德错觉：亚里士多德错觉是最古老的错觉，而且也很容易实现。

将食指和中指交叉，然后触摸一个小的圆形物体，比如晒干的豌豆，人们会感觉自己好像触摸了两颗豌豆。

这个例子就是所谓的“感知分离”。

当人们交叉手指时，这两个手指平时不接触的两个侧面便“相会”了，然后触感觉从这两个侧面分别传向大脑。

由于正常状况下两个手指的这两个侧面是几乎不会同时接触同一个物体的，于是，人们的大脑意识不到手指已经交叉了，便“想当然”地以为是两个豌豆。

踢猫效应：踢猫效应是指对弱于自己或者等级低于自己的对象发泄不满情绪，而产生的连锁反应。

音频毒品：音频毒品，英文名称I-Doser，又名“听的Mp3毒品”，主要通过控制情绪的α波、使人处于清醒和梦幻之间的θ波以及令人紧张和兴奋的β波等各种频率传播，可以使人进入幻觉状态。

音频毒品引起的情绪改变或与听者的背景经历有关。

此种“毒品”，主要在韩国互联网上迅速扩散。

在我国某些网站论坛上也已经出现了这种“音频毒品”的下载链接，并迅速被传播。

懒蚂蚁效应：日本北海道大学进化生物研究小组对三个分别由30只蚂蚁组成的黑蚁群的活动观察。

结果发现。

大部分蚂蚁都很勤快地寻找、搬运食物、少数蚂蚁却整日无所事事、东张西望，人们把这少数蚂蚁叫做“懒蚂蚁”。

有趣的是，当生物学家在这些“懒蚂蚁”身上做上标记，并且断绝蚁群的食物来源时，那些平时工作很勤快的蚂蚁表现得一筹莫展，而“懒蚂蚁”们则“挺身而出”，带领众蚂蚁向它们早已侦察到的新的食物源转移。

Solutions to problems in `Gauge Theories in Particle Physics 3rd edtn, vol. 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
a typical nucleon momentum by using the `uncertainty relation' estimate p h=R where R is the nuclear radius. Here R 1:5fm, and p 130MeV=c, which gives a struck nucleon speed of order v=c 0:14. (Note that He is a tightly bound nucleus, and the formula for the nuclear radius R = 1:1 (A)1=3 is not really applicable - of course, we are making rough estimates anyway.) Considering con gurations with the outgoing electron moving parallel/antiparallel to the struck nucleon gives a typical shift in E of order 50 MeV. Note that this is quite a bit bigger than the nucleon's binding energy - the relativistic transformation has ampli ed the e ect. 2.3 (i) 5.1 eV. (ii) (a) 1.29 (b) 0.75 (iii) h 1 2i = +1 for S = 1 ;3 for S = 0. Hyper ne splitting = 8.45 10 4 eV. (iv) 0.57. 2.4 One-gluon exchange con nement. Ground state expected to be at the minimum of E (r) as a function of r, i.e. at r0 such that dE (r)=drj(r=r0) = 0. Egr(cc) 3:23GeV. Threshold for production of `open charm' (DD states) opens at about 3.73 GeV. 2.5 E = 2a cos 2 a j+i = cos j1i + sin j2i j;i = ; sin j1i + cos j2i. System is in state cos j+i ; sin j;i at time t = 0 evolves to

Calibrated Manifolds and Gauge Theory

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

Dual variables for the SU(2) lattice gauge theory at finite temperature

a r X i v :h e p -l a t /9709021v 1 9 S e p 1997Dual variables for the SU (2)lattice gauge theory at ﬁnite temperatureSrinath CheluvarajaTheoretical Physics GroupTata Institute of Fundamental ResearchHomi Bhabha Road,Mumbai 400005,IndiaWe study the three-dimensional SU (2)lattice gauge theory at ﬁnite temperature using an observable which is dual to the Wilson line.This observable displays a behaviour which is the reverse of that seen for the Wilson line.It is non-zero in the conﬁned phase and becomes zero in the deconﬁned phase.At large distances,it’s correlation function falls oﬀexponentially in the deconﬁned phase and remains non-zero in the conﬁned phase.The dual variable is non-local and has a string attached to it which creates a Z (2)interface in the system.It’s correlation function measures the string tension between oppositely oriented Z (2)domains.The construction of this variable can also be made in the four-dimensional theory where it measures the surface tension between oppositely oriented Z (2)domains.e-mail:srinath@theory.tifr.res.in1Dual variables have played an important role in statistical mechanical systems[1].These variables display a behaviour which is the opposite of that seen for the order parameters.They are non-zero in the disordered phase and remain zero in the ordered phase.Hence they are commonly referred to as disorder variables.Unlike the order parameters which are local observables and measure long range order in a statistical mechanical system,the dual variables are non-local and are sensitive to disordering eﬀects which often arise as a consequence of topological excitations supported by a system-like vortices,magnetic monopoles etc.Disorder variables for the U(1)LGT have been studied recently[2].In this paper we study theﬁnite temperature properties of the three-dimensional SU(2)lattice gauge theory using an observable which is dual to the Wilson line.We explain the sense in which this is dual to the Wilson line and show that it’s behaviour is the reverse of that observed for the Wilson line.Unlike the Wilson line which creates a static quark propagating in a heat bath,the dual variable creates a Z(2)interface in the system.The deﬁnition of this variable can also be extended to the four-dimensional theory.Before we consider the three-dimensional SU(2)lattice gauge theory let us brieﬂy recall the construction of the dual variable for the two-dimensional Ising model[3].The variable dual to the spin variableσ( n) is denoted byµ(⋆ n)and is deﬁned on the dual lattice.This variable which is shown in Fig.1has a string attached to it which pierces the bonds connecting the spin variables.The position of the string is notﬁxed and it can be varied using a Z(2)(σ( n)→−σ( n))transformation.The average value of the dual variable is deﬁned asZ(˜K)<µ(⋆ n)>=The dual variableµ(⋆ n)thus creates an interface beginning from⋆ n.It has the following behaviour at high and low temperatures[3]<µ(⋆ n)>≈1for K small<µ(⋆ n)>≈0for K large.It is in this sense that the variableµ(⋆ n)is dual to the variableσ( n)which behaves as<σ( n)>≈0for K small<σ( n)>≈1for K large.The spin and dual correlation functions satisfy the relation<µ(⋆ n)µ(⋆ n′)>K>>1=<σ( n)σ( n′)>K<<1.(3) Using theσ→−σtransformation it can be shown that the correlation function of theµ’s is independent of the shape of the string joining⋆ n and⋆ n′.The variablesσ( n)andµ(⋆ n)satisfy the algebraσ( n)µ(⋆ n)=µ(⋆ n)σ( n)exp(iω),(4) whereω=0if the variableσdoes not lie on a bond pierced by the string attached toµ(⋆ n)andω=πotherwise.The above considerations generalize easily to the three-dimensional Z(2)gauge theory.The dual variables are again deﬁned on the sites of the dual lattice and the string attached to them will now pierce plaquettes instead of bonds.Whenever a plaquette is pierced by a string the coupling constant changes sign just as in the case of the Ising model.One can similarly deﬁne correlation functions of these variables.Since the three-dimensional Z(2)gauge theory is dual to the the three-dimensional Ising model,the correlation functions of these variables will have a behaviour which is the reverse of the spin-spin correlation function in the three-dimensional Ising model.For the case of the SU(2)lattice gauge theory which is our interest here,the deﬁnition of these variables is more involved.However,since Z(2)is a subgroup of SU(2)one can deﬁne variables which are dual to the Z(2)degrees of freedom by following the same prescription as in the three-dimensional Z(2)gauge theory.The relevance and eﬀectiveness of these variables will depend3on the role played by the Z(2)degrees of freedom in the SU(2)lattice gauge theory.The role of the center degrees of freedom in the SU(2)lattice gauge theory was also examined in[4].Since theﬁnite temperature transition in SU(N)lattice gauge theories is governed by the center(Z(N) for SU(N))degrees of freedom[5],we expect these variables to be useful in studying this transition.The usual analysis ofﬁnite temperature lattice gauge theories is carried out by studying the behaviour of the Wilson line which becomes non-zero across theﬁnite temperature transition[5].The non-zero value of the Wilson line indicates deconﬁnement of static quarks.The spatial degrees of freedom undergo no dramatic change across the transition and only serve to produce short-range interactions between the Wilson lines. Thus one gets an eﬀective theory of Wilson lines in one lower dimension[6].The deconﬁnement transition can be monitored by either measuring the expectation value of the Wilson line or by looking at the behaviour of the Wilson line correlation function[7].In the conﬁning phase,the correlation function is(for| n− n′| large)<L( n)L( n′)>≈exp(−σT| n− n′|)(5) while in the deconﬁning phase<L( n)L( n′)>≈constant.(6) We deﬁne the variableµ(⋆ n)on the dual lattice site⋆ n asµ(⋆ n)=Z(˜β)2ptr U(p).(8)The variablesµ(⋆ n)and L( n)satisfy the algebra4L( n)µ(⋆ n)=µ(⋆ n)L( n)exp(iω)(9) whereω=0if the plaquette pierced by the string attached toµ(⋆ n)is not touching any of the links belonging to L( n)andω=πif the plaquette makes contact with any of the links of L( n).The variables µ(⋆ n)and L( n)satisfy the same algebra as theσandµvariables in the Ising model.This is the same as the algebra of the order and disorder variables in[8].Note that this algebra is only satisﬁed if the string is taken to be in the spatial direction.The location of the string can again be changed by local Z(2) transformations.The correlation function of the dual variables is deﬁned to beZ(˜β)<µ(⋆ x)µ(⋆ y)>=)Nτ n n′J( n− n′)trL( n)trL( n′).(12)2The term which gives this contribution is shown in Fig.2.When we calculate the correlation function in Eq.10(where x and y are only separated in space)using this approximation,one plaquette occurring in this diagram will contribute with the opposite sign(shown shaded in Figure.2)and will cause the bond between n and n′to have a coupling with the opposite sign.In Eq.12J( n− n′)contains the sign induced5on the bond.This feature will persist for every diagram contributing to the eﬀective two-dimensional Ising model and it’s eﬀect will be to create a disorder line from x to y.Thus this correlation function will behave exactly like the disorder variable in the two-dimensional Ising model and at large distances will fall oﬀexponentially in the ordered phase and will approach a constant value in the disordered phase.We expect it to behave(for large| x− y|)as<µ( x)µ( y)>≈exp(−| x− y|/ξ)β>βcr<µ( x)µ( y)>≈µ2β<βcrWriting the above correlation function as<µ( x)µ( y)>=exp(−βτ(F( x− y))(13) we can interpret F as the free energy of an interface of length| x− y|.The inverse temperature is denoted byβτto distinguish it from the gauge theory couplingβ.In the ordered phase the interface energy increases linearly with the length of the interface while in the disordered phase it is independent of the length.In theﬁnite temperature system high temperature results in the ordering of the Wilson lines and low temperature results in the disordering of the Wilson lines.Therefore the dual variables will display ordering at low temperatures and disordering at high temperatures.A direct measurement of the dual variable results in large errors because the dual variable is the expo-nential of a sum of plaquettes andﬂuctuates greatly.We have directly measured the dual variable and the correlation function and found that they fall to zero at high temperatures and remain non-zero at low temperatures.Since the measurement had large errors we prefer to use the method in[11]where a similar problem was encountered in the measurement of the disorder variable in the U(1)LGT.Instead of directly measuring the correlation function we measure∂ln<µ>ρ( x, y)=−where p′denotes the plaquettes which are dual to the string joining x and y.In our case this quantity directly measures the free energy of the Z(2)interface between x and y.Hence we expect it to increase linearly with the interface length in the deconﬁning phase and approach a constant value in the conﬁning phase.Also this variable is like any other statistical variable and is easier to measure numerically.The variableρcan be used to directly measure the interface string tension between oppositely oriented Z(2) domains.The behaviour of the quantityρis shown in Fig.3and Fig.4.In the conﬁned phaseρapproaches a constant value at large distances while it increases linearly with distance in the deconﬁned phase.The slope of the straight line in Fig.3gives the interface string tension.The calculation ofρwas made on a 12∗∗23lattice with200000iterations.The values ofβused were2.5in the conﬁned phase and5.5in the deconﬁned phase.The deconﬁnement transition on the Nτ=3lattice occurs atβ=4.1[10].The errors were estimated by blocking the data.We would now like to point out a few applications of these dual variables.The mass gap in the high temperature phase is determined by studying the large distance behaviour of the Wilson line correlation function.Since the Wilson line correlation function remains non-zero in the deconﬁned phase the long distance part is subtracted out to get the leading exponential.The dual variable correlation function already displays an exponential fall oﬀin the high temperature phase and provides us with another method of estimating the mass gap.Also,since dual variables reverse the roles of strong and weak coupling,they provide an alternate way of looking at the system which may be convenient to address certain questions. In this case they can be used to determine the string tension between oppositely oriented Z(2)domains in the SU(2)gauge theory.The surface tension between oppositely oriented Z(2)domains in the four-dimensional theory has been calculated semi-classically in[12].The above construction of the dual variable can also be made in four dimensions.The only diﬀerence is that in four dimensions the dual variables are deﬁned on loops in the dual lattice.The spatial string in three-dimensions is replaced by a spatial surface which has the loops as it’s the boundary.The dual variables are functionals of the surface bounding the loops.The correlation function of the dual variables is deﬁned to be<µ(C,C′)>=<exp(−β p′tr U(p))>(16)7where the summation is over all plaquettes which are dual to the surface joining C and C′.Since the surface is purely spatial the plaquettes contributing to the summation are all space-time plaquettes.This correlation function will fall of exponentially as the area of the surface joining C and C′in the deconﬁned phase and will approach a constant value in the conﬁned phase.A similar measurement ofρcan be used to determine the surface tension between oppositely oriented Z(2)domains in the four-dimensional gauge theory.8........................X FIG.1.Dual variable in the Ising model.10n n′333333 FIG.3.ρin the deconﬁning phase.12333333 FIG.4.ρin the conﬁning phase.13。

法国数学家拉格朗日著作《解析函数论》英文名

法国数学家拉格朗日著作《解析函数论》英文名全文共10篇示例，供读者参考篇1"Hey guys! Today let's talk about this really cool book called 'Analytic Functions Theory' by French mathematician Lagrange. It's super interesting and has a lot of cool stuff in it!So, in this book, Lagrange talks about a bunch of different math stuff like functions and calculus. He explains how to analyze functions and how they work, which is really helpful for solving math problems. He also talks about things like complex numbers and series, which can be a bit tricky but are super important in math.Lagrange was a really smart guy and he made a lot of important contributions to math. His book 'Analytic Functions Theory' is one of his most famous works and is still studied by math students and researchers today.If you're into math and want to learn more about functions and calculus, I definitely recommend checking out 'Analytic Functions Theory' by Lagrange. It's a challenging read, but super rewarding if you stick with it.So yeah, that's a little introduction to Lagrange's book'Analytic Functions Theory'. I hope you guys found it interesting and maybe even want to check it out for yourselves. Happy math-ing!"篇2Once upon a time, there was a super cool French mathematician named Lagrange. He was so smart and wrote a really awesome book called "Analytic Functions of a Complex Variable." It's like a super fancy title, right?So, in this book, Lagrange talks about all these super cool things like functions and complex numbers. He explains how you can use math to understand how different things work together and solve problems. He even talks about things like calculus and equations. It's like he's teaching us a secret code to unlock the mysteries of the universe!One of the coolest things Lagrange talks about in his book is how you can use functions to describe all kinds of crazy things, like how a roller coaster moves or how a rocket flies through the sky. It's like he's showing us how to use math to understand the world around us in a whole new way.So, if you ever want to learn more about math and how it can help us understand the world, you should definitely check out Lagrange's book. It's like a magical journey into the world of numbers and equations, and it will definitely make you feel like a math wizard!篇3Once upon a time, there was a really smart French mathematician named Lagrange. He was super duper good at math and he wrote this really cool book called "Analytic Number Theory". It's like a super duper advanced math book for big kids who are really good at numbers.In this book, Lagrange talks about all these super cool things like complex numbers and functions. He explains how they work and how you can use them to solve really hard math problems. It's like magic but with numbers!One of the things Lagrange talks about in his book is series and sequences. This is when you have a bunch of numbers lined up in a row and you add them all together. It's like anever-ending puzzle that you have to figure out. Lagrange shows us how to solve these puzzles and find patterns in the numbers.Another thing Lagrange talks about is limits. This is when you get really close to a number but you never actually reach it. It's like trying to touch the end of a rainbow but it keeps moving further away. Lagrange helps us understand how to work with limits and see what happens when you get really really close to a number.Overall, Lagrange's book is super duper awesome and it's full of all these amazing math ideas that will make your brain explode (in a good way!). So if you love math and you want to learn more about numbers and functions, you should definitely check out "Analytic Number Theory" by the one and only Lagrange. It's a book that will make your inner math nerd happy!篇4Hey guys, today I want to tell you about a super cool book by a French mathematician called Lagrange. His book is called "Analytic Theory of Functions" in English.So, basically, Lagrange was a really smart guy who figured out a lot of stuff about functions and how they work. In his book, he talks about all the different ways you can analyze functions and make sense of them. It's kind of like a math puzzle book where you have to figure out how to solve different functions.One of the really cool things that Lagrange talks about in his book is how you can break down functions into smaller pieces and analyze how they change. It's kind of like taking apart a puzzle and figuring out how each piece fits together to make the whole picture.Lagrange also talks about how you can use functions to solve real-world problems, like figuring out how things change over time or how to predict what will happen in the future. It's like using math to solve everyday mysteries!So, if you're into math and you love solving puzzles, you should definitely check out Lagrange's book "Analytic Theory of Functions". It's a really fun read and you'll learn a lot about how functions work. Who knows, maybe you'll even discover a new way to solve math problems just like Lagrange did!篇5Once upon a time, there was a super smart French math guy named Lagrange. He wrote this super cool book called "Analytic Function Theory". I know, it sounds super fancy, but basically it's all about how numbers work and stuff.Lagrange was a total math genius. He figured out all these crazy math problems and even invented new ways to solve them. He was like a math superhero!In his book, "Analytic Function Theory", Lagrange talks about how numbers can be broken down and analyzed in a super cool way. It's like he's shining a spotlight on all the secrets of math and showing us how everything fits together.It's kind of like solving a puzzle. You have to figure out how all the pieces fit together and then you can see the big picture. That's what Lagrange did with numbers in his book.So next time you're struggling with math homework, just think of Lagrange and his awesome book. He's like your math mentor, guiding you through the world of numbers and showing you all the cool secrets along the way.And who knows, maybe you'll be the next math superhero just like Lagrange! Just keep practicing and studying, and one day you'll be solving math problems like a pro.篇6Once upon a time, there was a super smart mathematician from France named Lagrange. He wrote a super cool book called"Analytic Function Theory". It's a big book with lots of fancy words and symbols, but don't worry, I'll explain it in a way that's easy to understand.Okay, so here's the deal - Lagrange was really good at math and he wanted to explain how functions work. Functions are like machines that take in numbers and give out other numbers. In his book, Lagrange talked about how functions can be broken down into smaller parts called "analytic functions".Analytic functions are like the building blocks of math. They're super important because you can use them to create all sorts of cool math stuff. Lagrange showed how these functions can be used to solve problems in calculus, geometry, and even physics.In "Analytic Function Theory", Lagrange also talked about complex numbers. Complex numbers are a special type of number that have both a real part and an imaginary part. They're like the superheroes of math because they can do things that regular numbers can't.So yeah, that's a brief overview of Lagrange's book. It may sound a bit complicated, but don't worry. Just remember that math is like a puzzle - the more you practice, the better you getat solving it. Who knows, maybe one day you'll write your own math book just like Lagrange!篇7Once upon a time, there was a super smart mathematician from France named Lagrange, or Lagrangian, or Lagragian, I forgot how to spell his name. Anyway, this guy was like a math genius and he wrote this super cool book called "Analytic Function Theory." Yeah, I know, it sounds pretty boring, but trust me, it's actually really interesting.So, in this book, Lagrange talks about all these crazy things like complex numbers and functions and stuff. He basically explains how these things work together to help us understand the world of math better. It's kind of like a magical journey into the world of numbers and equations.One of the coolest things he talks about in the book is something called the Cauchy-Riemann equations. These equations are like the key to unlocking the secrets of analytic functions. They help us understand how to differentiate and integrate complex functions, which is pretty mind-blowing if you ask me.Overall, "Analytic Function Theory" is a really important book in the world of math. It's helped us make sense of some really complex stuff and has paved the way for even more amazing discoveries in the future. So yeah, big shoutout to Lagrange for being such a math wizard and writing this awesome book!篇8Title: "Mr. Lagrange's Book about Fancy Math Stuff"Once upon a time, there was a super smart guy from France named Mr. Lagrange. He was a famous mathematician who wrote a really cool book called "". But don't worry, that's just the fancy English name for it - "Analytical Functions Theory".So, what's this book all about? Well, it's all about a special kind of math called complex analysis. That means dealing with numbers that have a real part and an imaginary part. Sounds pretty fancy, right?In his book, Mr. Lagrange talks about how these complex numbers can be used to study functions. He also talks about things like series, residues, and zeros of functions. It might sound like gibberish to some, but for math lovers like me, it's like reading an exciting adventure story!One of the coolest things Mr. Lagrange talks about in his book is contour integration. It's like drawing a path around a function and using that path to calculate some super complicated stuff. It's like magic, but with math!So, if you're into math and want to learn more about complex analysis, be sure to check out Mr. Lagrange's book "Analytical Functions Theory". Who knows, maybe one day you'll be solving math problems just like him!And that's the end of our story about Mr. Lagrange and his fancy math book. Hope you enjoyed it! Bye bye!篇9Once upon a time, there was a super smart guy named Lagrange, he was a super famous French math guy. He wrote a super cool book called "Analytic Functions Theory". This book is like a super secret math code that helps us understand how functions work. It's like a treasure map to unlock the mysteries of functions.In this book, Lagrange talks about all sorts of cool stuff like derivatives, integrals, and complex numbers. He even talks about things like power series and Cauchy's theorem! It's like a math playground for our brains.One of the coolest things in this book is how Lagrange shows us that functions can be super duper smooth and predictable. He shows us how to break down functions into tiny pieces and study each piece to understand the whole thing. It's like taking apart a puzzle and putting it back together, but in a super smart math way.Lagrange was like a math superhero, using his powers of logic and reasoning to unlock the secrets of functions. His book "Analytic Functions Theory" is like a math superhero comic book, teaching us how to be super smart math detectives.So, next time you see a function, remember Lagrange and his super cool book. You might just unlock a whole world of math mysteries and become a math superhero yourself!篇10Hey guys, today I'm gonna tell you about a super cool book by a French math dude called Lagrange. Wait, that's not quite right... it's actually Lagrange! And his book is all about something called "Analytic Function Theory". Sounds super fancy, right?So, what is this book all about? Well, basically it's a bunch of really smart stuff about functions and how they work. You know, like when you put in a number and the function spits out anothernumber. But these functions are super special because they can be broken down and analyzed in a really cool way.Lagrange was a total math genius and he came up with some super cool ideas in this book. He talked about things like complex numbers and how they can be used to study functions. And he also did some crazy stuff with calculus, which is like super advanced math that you'll learn about when you're older.I know it sounds kinda boring, but trust me, this book is actually really interesting! It's full of puzzles and challenges that will totally blow your mind. And who knows, maybe one day you'll be a math whiz just like Lagrange!So if you're into math and you want to learn some really cool stuff, definitely check out Lagrange's book "Analytic Function Theory". It'll totally make your brain hurt, but in a good way!。

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.

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

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

Physical and unphysical effects in the mixed SU(2)SO(3) gauge theory

SWAT/88 DESY 96-054 hep-lat/9604008
arXiv:hep-lat/9604008v1 12 Apr 1996
Physical and unphysical eﬀects in the mixed SU(2)/SO(3) gauge theory
P.W. Stephenson Department of Physics, University of Wales, Swansea, Singleton Park, Swansea, SA2 8PP, U.K. and DESY-IfH Zeuthen, 15735 Zeuthen, Germany1
Abstract
We investigate possible problems with universality in lattice gauge theory where a mixed fundamental SU(2) and SO(3)-invariant gauge group is used: the (second order) ﬁnite temperature phase transition becomes involved with ﬁrst order eﬀects with increased SO(3) coupling, and this ﬁrst order eﬀect has a noticeable coupling dependence for small lattices. We produce evidence that the ﬁrst order transition is essentially bulk in nature as generally believed, and that the ﬁnite temperature eﬀects start to separate out from the lower end of the bulk eﬀects for a lattice of 8 sites in the ﬁnite temperature direction. We strengthen our picture of the ﬁrst order eﬀects as artefacts by using an improved action: this shifts the end point of the ﬁrst order line away from the fundamental SU(2) axis.

Instantons and the 5D U(1) gauge theory with extra adjoint

Instantons and the 5D U (1) gauge theory with extra adjoint
arXiv:Leabharlann 804.3564v1 [hep-th] 22 Apr 2008
Rubik Poghossian Yerevan Physics Institute, Alikhanian Br. st. 2, 0036 Yerevan, Armenia e-mail: poghos@yerphi.am Marine Samsonyan Dipartimento di Fisica, Universit´ a di Roma “Tor Vergata” I.N.F.N. Sezione di Roma II, Via della Ricerca Scientiﬁca, 00133 Roma, Italy e-mail: marine.samsonian@roma2.infn.it
1
Introduction
Recent progress in understanding non-perturbative phenomena in supersymmetric Yang-
Mills theories due to direct multi-instanton calculations is quite impressive. Two main ideas played essential role in all this developments. First was the realization that the Supersymmetric Yang-Mills action induced to the moduli space of instantons can be represented in terms of closed, equivariant with respect to the diagonal part of the gauge group, forms [1]. This observation leads to a crucial simpliﬁcations reducing SYM path integral to an integral over the stable with respect to the action of the diagonal part of the gauge group subset of the moduli space of instantons. The next brilliant idea, which is the corner stone for all further developments was suggested by Nekrasov in [2]. The idea is to generalize the theory involving into the game in equal setting besides the already mentioned global diagonal gauge transformations also the diagonal part of the (Euclidean) space-time rotations. Why this is so crucial because the subset of the instanton moduli space invariant under this combined group action appears to consist only of ﬁnite number of points. In the case of the gauge group U (N ) this ﬁxed points set is in one to one correspondence with the set of array of Young diagrams Y = (Y1 , ..., YN ) with total number of boxes |Y | being equal to the instanton charge k . Thus, to calculate path integral for the various ”protected” by super-symmetry physical quantities one needs to know only the pattern how the combined group acts in the neighborhoods of the ﬁxed points. All this information can be encoded in the character of the group action in the tangent space at given ﬁxed points. An elegant formula for this character which played a signiﬁcant role in both physical and mathematical applications was proposed in [3] (see eq. (2.1)). Let us note at once that combining space time rotations with gauge transformations besides giving huge computational advantage due to ﬁniteness of the ﬁxed point set, has also a major physical signiﬁcance generalizing the theory to the case with certain nontrivial graviphoton backgrounds [2]. In order to recover the standard ﬂat space quantities (say the SeibergWitten prepotential of N = 2 super-Yang-Mills theory) one should take the limit when the space time rotation angles vanish. It is shown by Nekrasov and Okounkov [4] that in 1

斯腾伯格三元智力说培训资料

智力三元论（triarchic theory of intelligence）：成分智力（componential intelligence）是指个人在问题情境中运用知识分析资料，通过思维、判断推理以达到问题解决的能力。

它包含有三种机能成分。

一是元成分（metacomponents），是指人们决定智力问题性质、选择解决问题的策略以及分配资源的过程。

例如，一个好的阅读者在阅读时分配在每一段落上的时间是与他要从该段落中准备吸收的知识相一致的。

这个决定就是由智力的元成分控制的。

二是执行成分（performance components），是指人实际执行任务的过程,如词法存取和工作记忆。

三是知识习得成分（knowledge acquisition components），是指个人筛选相关信息并对已有知识加以整合从而获得新知识的过程。

经验智力（experiential intelligence）是指个人运用已有经验解决新问题时整合不同观念所形成的创造能力。

例如，一个有经验智力的人比无此智力的人能够更有效地适应新的环境；他能较好地分析情况，用脑筋去解决问题，即使是从未遇到过的问题。

经过多次解决某个问题之后，有经验智力的人就能不假思索、自动地启动程序来解决该问题，从而把节省下来的心理资源用在别的工作上。

有些人能很快做到，有些人却难以做到这一点。

这种能力就称为经验智力。

情境智力（contextual intelligence）是指个人在日常生活中应用学得的知识经验解决生活实际问题的能力。

例如，在不同的文化中人们应对日常生活实际问题的能力是不同的。

区分有毒和无毒植物是从事狩猎、采集的部落人们的重要能力，而就业面试则是工业化社会的一种重要情境智力，他们的情境智力是不同的。

斯滕伯格的理论得到了对大脑前额叶受损病人的研究结果的支持。

例如，有一位以前很成功的物理学家，因为偶然的事故前额叶受损，痊愈后他虽然仍有很高的智商分数,却不能继续他的工作。

Confinement in Three Dimensions and the Electroweak Phase Transition

Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany
Abstract The infrared behaviour of the standard model at ﬁnite temperature is determined by the conﬁning phase of the SU(2)-Higgs model in three dimensions. Due to the Landau singularity of the three-dimensional gauge theory the perturbative treatment of the electroweak phase transition breaks down for Higgs masses above a critical mass mc H . Based on a renormalization group improved eﬀective potential we ﬁnd mc ∼ 70 GeV. The scalar self-coupling has a Landau-type singularity H also in the abelian U(1)-Higgs model, which leads to a breakdown of perturbation theory in the symmetric phase.
1 1 V0 (ϕ2 ) = m2 ϕ2 + µǫ λϕ4 , ϕ2 = 2Φ† Φ . (2) 2 4 a a is the ordinary ﬁeld strength tensor and Dµ = ∂µ − iµǫ/2 gWµ τ /2 is the

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.

!!地热水井井口温度的计算_申家年

大庆石油学院学报J O U RN A L O F D AQ IN G PE T RO LEU M IN ST I T U T E 第22卷　第4期　1998年12月V o l.22N o.4Dec.　1998地热水井井口温度的计算①申家年1)　孙小洁1)　刘海钧2)1)　大庆石油学院石油勘探系,安达,151400 2)　大庆地热办公室,大庆,163350 摘　要　水是地热能利用的载体.根据二维半无限大垂直埋管表面的稳态放热模型及牛顿换热公式,推导出地热水井在不同深处的垂直管道外表到周围介质的外部放热系数.通过实例计算了林甸地区浅部岩石热导率值,并对该热导率值进行了验证.根据输液管道节点温度的舒霍夫计算公式,以10m 为步长,用逐步迭代的方法,给出了地热水井井口温度的计算过程.主题词　地热;岩石热导率;井口温度分类号　P 314.3第一作者简介　申家年,男,1962年生.工程师.1984年毕业于长春地质学院仪器专业.现从事地下流体分析实验方面的研究.0　引言水将地下深处的热能携带到地面,是地热资源利用的主要载体;利用地热已有近百年历史,70年代以后得到了一定重视和发展,但是到目前,绝大部分地热还都是通过天然高温泉发现的.这样,地下水是否达到有使用价值的井口温度基本上是已知的.在没有高温泉的地方打井利用热能就必须计算水到达地面时的井口温度.根据输液管道节点温度的舒霍夫计算公式,用逐步迭代的方法,给出了地热水井井口温度的计算方法.1　影响井口温度的主要因素影响井口温度的因素很多,但主要因素有3个方面.1)地热场的地温梯度.地温梯度实际上控制着井底水的温度和水向上流动散热过程中的周围介质温度.地温梯度的高低决定是否有可能利用地热能.2)水在井筒的流动速度.高速流动的水热量散失少,有助于获得较高的井口温度.3)地下岩石的热导率.热导率是表明井筒周围介质散热能力的参数,岩石热导率越低,热量散失越少,井口温度越高.2　岩石热导率的选取表1　岩石热导率岩石λ/[W (m ℃)-1]钙质泥岩 1.77[2]含钙粉砂质泥岩 1.61泥岩 2.650.898[3]云母砂岩 3.75细砂岩 2.70 1.646粉砂岩 3.00 1.996泥质粉砂岩 2.093数据引自文献[2],[3] 在影响井口温度的3个主要因素中,地温梯度和水的流动速度都可以通过实测或计算得到较可靠的数据.岩石热导率是一个较复杂的变数.它受岩石的矿物成分、结构、流体成分和含量、地下温度、地下压力及岩石中流体的流动性等许多因素的影响.不同地区不同地层中各种岩性的热导率有较大的变化,见表1[1～3].从理论上讲,应根据地下不同深度岩石的具体物理化学条件,实测各类岩石的热导率,但由于影响岩石热导率的因素复杂,实验室内无法模拟地下地质条件,这一工作是无法进行的.在工程实践中,热导率一般采用经验值.研究区地热勘探尚处于初步阶段,无直接借鉴的数值.因此,在计算时根据不同地层的不同岩性,分层用热导率的加权平均值,采用迭代计算的方法计算井口温度.用实测地层温度和井口温度与之对比,从而确定适合研究区的岩石热导率值,用这个值去计算其它井口温度.①收稿日期:1997-09-20 审稿人:姜贵周3　理论依据依据管道工程中的舒霍夫公式T K=T O+(T H-T O)ex p(aL T)(1)式中:T K为水的终点温度;T H为水的起点温度;T O为管道周围的介质温度;L T为管道长度;a为常数.a=KπD GCK=2λ[ln(4h0/D)-1]D[ln(4h0/D)]2(2)G为水的质量流量;C为水的比热容,C=4212-1.883t+0.021t2;D为管道的外直径;K为管道的总的传导系数;h0为管道的埋深;λ为岩石的热导率;t为管道中水的温度.在管道传热工程中,管道总的传导系数K的表达式与上面的表达式略有差别,原因是原油传输中的管道一般是水平放置的,而地热探井的井筒(管道)是垂直放置的.这里K的表达式是根据传热学的有关理论推导出来的.具体过程如下:垂直的热水管道对周围地层的放热过程可视为垂直埋管在半无限大物体中的放热过程[4].Q=λ2πLln(4L/D)(T1-T0)式中:Q为热流量;L为管道的长度;T为管道内流体温度.求管道在L处向上长度为ΔL一段的热流量ΔQ L,根据泰勒展开近似有ΔQ L=Q(L)-Q(L-ΔL)=λ2πLln(4L/D)′ΔL(T1-T0)ΔQ L=λ2π(ln(4L/D)-1)(ln(4L/D))2ΔL(T1-T0)(3)根据牛顿换热公式ΔQ L=TΔF(T1-T0)=TπD(T1-T0)(4) T为管道的外表面到周围介质的外部放热系数.由(3),(4)两式结合有,T=2λ[ln(4L/D)-1][ln(4L/D)]2(5)根据文献[5],当这一段管道水平放置,且L D时,T水平=2λln(4L/D)(6)可以看出在L D时,即舒霍夫公式中的h0较大时,可以认为T和T水平没有差别.传导系数K可表达为K=11T1+Wλ+1T2(7)式中:T1为流体与管道内壁的内部放热系数;T2为管道外壁至周围介质的外部放热系数;W为管道的壁厚;λ为管道壁的热导率.对于不保温的热输管道总的传热系数K近似等于T2[5].研究区的井原为油气探井,多为水泥固井.水泥的热导率(λ= 1.43)与岩石接近,可以看作岩石的一部分.因而,可以认为输水管道是不保温的,其总的传热系数K近似为T2,即K近似等于管道外壁至周围介质的外部放热系数.4　计算过程及实例具有一定温度的水在井筒内向上流动过程中,井筒周围介质温度(地温场温度)逐渐降低,即舒霍夫公式中T O是变化的.因而必须采用分段计算的办法,逐步向井口迭代.将整个垂直管道以10m为一段,分成N段,并将该段中部的深度作为这一段的埋深h0,以这一段中部的地温场温度作为管道周围介质的温度T O.最下面一段为第N段,首先取井底温度为T H(N),按舒霍夫公式计算水流经L T(10m)长度后的温度T K(N),T K(N)既是第N段的终点温度,又是第N-1段的起点温度T H(N-1),这样反复迭代,直到大　庆　石　油　学　院　学　报第22卷　1998年计算出井口温度.林4井有关数据如下:地下450m 处的水温为40.8℃;G = 4.386kg /s;D =0.19m;实测井口温度39.9℃;977m 深度处地层温度为42.0℃.大庆地区年平均气温为3.2℃,则恒温带温度可计为5.2℃[1],恒温带的深度一般在15～30m,取20m 为恒温带的深度.根据地温梯度的定义,计算977m 以上的地温梯度为3.845℃/100m.计算结果为:λ1= 1.55W /(m ℃)(泥岩),λ2= 1.65W /(m ℃)(粉砂岩),λ3= 1.70W /(m ℃)(细砂岩),λ4= 1.80W /(m ℃)(砂砾岩),井口温度为39.91℃.为了进一步验证,根据有关资料,林4井处现今大地热流值为62.8mW /m 2,大地热流值定义为q =λd T d Z (8)式中:q 为大地热流值;d T d Z为地温梯度;λ为岩石的热导率.将地温梯度等于3.845℃/100m ,大地热流值等于62.8mW /m 2代入(8)式,计算出977m 以上地层中岩石的平均热导率为1.65W /(m ℃).这一值与上面实例计算出来的加权平均值基本一致,因而上述各热导率值可作为研究区地下450m 以内的热导率经验值使用.另外,这些值也在砂、泥岩的实测值的范围内,说明计算过程和结果具有相当的可靠性.5　结论上述计算属于稳态热传导过程,不适用于非稳态热传导过程;岩石热导率主要根据岩性变化来确定,深度增加时应适当加大[1];水平埋管和垂直埋管在埋深较大时,两者的放热系数是一致的.参考文献1　田在艺,张庆春.中国含油气盆地论.北京:石油工业出版社,1996.122～1232　鲁J M.沉积盆地中的热现象.冷鹏华译.哈尔滨:黑龙江科学技术出版社,1990.1283　杨万里.松辽盆地石油地质.北京:石油工业出版社,1985.1924　张洪济.热传导.北京:北京高等教育出版社,1992.327～3285　曲慎扬.原油管道工程.北京:石油工业出版社,1991.101～114第4期申家年等:地热水井井口温度的计算Abstracts Journal of Daqing Petroleum Institute Vol.22　No.4　Dec.1998 Abstract　Geothermal water can be applied widely to hea ti ng,bathing,health care,planti ng and breeding in-dust ries.Thermal source is a key element in the geothermal syst em.In order to ascertai n the forming mecha-nism in Lindian region,we applied basin simulation method to calculat e the earth heat flow high value area and sum up ef fect fact ors of high value geotemperature dist ributi on.There is a characteristic of earth heat flow v al-ue and high g eotemperat ure in Lindian structure.Ef fect f actors of geot empera ture are that M oho and Curie are buried shallower,deep faul ts connects thermal source to thermal reservior,there is a huge grani te body under-ground.Subject terms　Songlian Basi n,Lindian region,g eotemperature field,geothermal,thermal sourceSupply of Groundwater Under the Geothermal Resource Formation and of Hydrodynamic Conditions in Lindian RegionZhang Shulin(Dept of Petroleum Prospecting,Daqing Petroleum Institute,Anda,Heilongjiang,China, 151400)Abstract　It s known that there are rich g eothermal resources in Lindian Region,which can be taken to surfaceusi ng wat er as thermal carrier.So whether there are supplies of groundw ater in this area i s an impo rtant ques-tion i n the long-term use of geothermal resources.According to the basin hydrogeological data,the hydrody-namic syst em can be classifield int o vertical and the partition of the hydrodynamic ag ency on plane,the counting of wat er replace intensi ty in the sedimental hydrogeological t erm and in the percolating hydrogeological t erm. Using the f luid pow er theoroy,analyzing synthetically its supply of groundw ater and hydrodynamic condi tions, Lindian Region is in the belt between percolating wat er replace and sedimeatal one at the basin hydrogeological parti tion.There are suf fici ent natural w ater supply to geothermal reserv oi r rocks.Subject terms　geothermal resouces,thermal reservoir,supply of g roundwater,hydrodynamic condi tions,pressure syst emCalculation of Well Head Temperature of Geothermal Water WellShen J ianian(Dept of Petroleum Prospecting,Daqing Petroleum Institute,Anda,Heilongjiang,China, 151400)Abstract　Wat er i s the carrying agent of making use of geothermal energy.On the basis of model of steady state of tw o-dimensional semi-i nfinite v ertically buried t ube and N ew ton exchange heat formula,the out side heat release coefficient is deduced f rom vertical tube in dif ferent depth of geo thermal wat er well to ambient medium.The shallow rock heat conductivity value of Lindian Region is calculated by examples.And this v alue is tested and verified.According to the formula that calculates node temperature in t ube engineering,calcula-tion process f or well head temperature of g eothermal wat er w ell is provided by i terative method w hose step is 10m.Subject terms　geothermal,rock heat conductivi ty,well head temperatureStandard Conformance Analysis of the DAEF Sample ImplementationLi Chunsheng(Dept of Computer Science,Daqing Petroleum Institute,Anda,Heilongjiang,China,151400) Abstract　DA EF is the interf ace bet ween the application and the permanent PO SC data store.It plays an im-port ant role i n the SIP of PO SC.The function of each data type and function of DAEF w as detailed in PO SC. Wi th the sample i mplementation of Epicentre on Oracle,the sample prog ram of DAEF was provided.In order to av oid unex pected result during usi ng the sample program that probably contains some erro rs,it is necessary to analy ze the sample implementatio n of DAEF.So a testing prog ram for D AEF was developed,which i s used f or standard conformance analysis of the DA EF sample implementation.The sample implementa tion of the DA EF versio n2.1provided by PO SC based on Oracle was test ed.Four da ta t ypes and fif ty-eight A PI f unctio ns that can no t keep consistency with the requi rement were f ound.The reasons for these include disordered man-agement,inaccurate analysis on requirement and i ncorrect using of pointer.Subject terms　PO SC,DAEF,A PI,Oracle,soft ware testingProjection of Oriented Object Data Model。

Gravitational Instantons from Gauge Theory

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

Exceptional Confinement in G(2) Gauge Theory

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

The Free Energy Of Hot Gauge Theories

a rXiv:h ep-ph/1294v28May21UNILE-CBR3The Free-Energy of Hot Gauge Theories Rajesh R.Parwani Departimento di Fisica,Universita’di Lecce,and Istituto Nazionale di Fisica Nucleare Sezione di Lecce Via Arnesano,73100Lecce,Italy.Abstract The total perturbative contribution to the free-energy of hot SU (3)gauge theory is argued to lie signiﬁcantly higher than the full result obtained by lattice simulations.This then suggests the existence of large non-perturbative corrections even at temperatures a few times above the critical temperature.Some speculations are then made on the nature and origin of the non-perturbative corrections.The anal-ysis is then carried out for quantum chromodynamics,SU (N c )gauge theories,and quantum electrodynamics,leading to a conjecture and one more speculation.1IntroductionThe most convincing evidence for a phase transition in thermal Yang-Mills theories is provided by direct lattice simulation of the partition function,Z =Tr e −βH ,(1)where β=1/T is the inverse temperature.Over the years the lattice data for SU (3)theory,the purely gluonic sector of quantum chromodynam-ics (QCD),has become incresingly accurate,with various systematic errors1brought under control[1].Figure(1)shows the normalized free-energy den-sity,F=−T ln Z/V,of SU(3)gauge theory taken from theﬁrst of Ref.[1]. Plots such as this have supported a picture of a low-temperature phase ofglueballs melting above some critical temperature to produce a deconﬁnedphase of weakly interacting gluons:As the gluons are liberated the num-ber of degrees of freedom increases causing the free-energy density to rise,while asymptotic freedom guarantees the gluons are weakly interacting at suﬃciently high temperature.Though the numerical data for QCD is less accurate,due to technicaldiﬃculties in simulating fermions,the accumulated data continues to support a phase transition.It is generally believed that this is a transition froma low-temperature hadronic phase to a high-temperature phase of quarks and gluons.This”quark-gluon plasma”is the new phase of matter whichexperiments at Brookhaven and CERN hope to detect in the near future.For the most part of this paper the focus will be on pure SU(3)theory, since the accurate lattice data allows a direct comparison with theory.Re-ferring to Fig.(1),there is one feature which is ignored by some,commentedon by many and which has bothered a few.While there is little doubt that at inﬁnite temperature a description in terms of gluons is tenable,this is lessclear at temperatures a few times the critical temperature,T c∼270MeV. For example,at3T c,the curve lies20%below that of an ideal gas of gluons.What is the origin of this large deviation?Is it due to(i)perturba-tive corrections to the ideal gas value,(ii)non-perturbative eﬀects in the plasma,(iii)an equally important combination of(i)and(ii)or,(iv)is this an irrelevant question arising from an improper insistence of describing the high-temperature phase in terms of weakly coupled quasi-particles?Several viewpoints have been expressed in the literature.Some believethat the deviation is mainly a non-perturbative correction to a gas of weakly coupled gluons and parametrize it in terms of a phenomenological”bag con-stant”.Others have attempted a phenomenological description of the high-temperature phase in terms of generalized quasi-particles.For a discussion and detailed references to these phenomenological approaches see,for exam-ple,[2].On the other hand,a few have suggested that the best consistent description of the high-temperature phase might be in terms of novel struc-tures[3,4].In order to help discern among the various possibilities,this paper will focus on estimating the total perturbative contribution to the free-energy. It is important toﬁrst agree on some terminology so as to avoid confusion2due to an overuse of some phrases in the literature.The partition function depends on the Yang-Mills coupling g,and has a natural representation as an Euclidean path-integral[5]Z(g)= Dφe− β0dτ d3x L(φ(x,τ))(2)whereφcollectively denotes the gauge and ghostﬁelds,and L the gauge-ﬁxed lagrangian density of SU(3)gauge theory.An expansion of this path-integral, and hence the free-energy density,around g=0leads to the usual Feynman perturbation theory and contributions of the form g n,with infrared eﬀects occasionally generating logarithms multiplying the power terms,g n(ln(g))m. These terms will be called perturbative.What is invisible in a diagramatic expansion about g=0are terms like e−1/g2,associated for example with instantons[5].Such terms,which are exponentially suppressed as g→0, will be called non-perturbative.Note that at non-zero temperature,odd powers of the coupling,such as g3,appear[5].These are perfectly natural and represent collective eﬀects in the plasma.Though they sample interesting long-distance physics,mathe-matically they fall into our deﬁnition of perturbative corrections.Similarly, Linde[6]had shown that at order g6the free energy receives contributions from an inﬁnite number of topologically distinct Feynman diagrams.Though the calculation of that contribution is diﬃcult,it is possible in principle[7], and anyway does not qualify as a non-perturbative contribution according to the deﬁnition above.Following the heroic work of Arnold and Zhai,a completely analytical calculation of the free-energy of thermal gauge theory to order g5has been obtained[8,7].For SU(3)gauge theory the result can be summarized as follows,F4 απ 3/2+ 67.5ln α2πT α2πTα45is the contribution of non-interacting gluons,α=g2/4π,and¯µis the renormalization scale in the¯µ=2πT is shown in Fig.(2).The poor convergence of(3)does not allow a direct comparison of the perturbative results with lattice data.Further-more,the result(3)is actually strongly dependent on the arbitrary value of ¯µ.Inspired by the relative success of Pade’resummation in other areas of physics,Hatsuda[9]and Kastening[10]studied the Pade’improvement of the divergent series(3).Their conclusion was that the convergence could be improved,and the dependence on the scale¯µreduced.However they did not attempt a direct comparison of their improved results with the lattice data, though Hatsuda did conclude that for the case of four fermionﬂavours,the deviation of theﬁfth-order Pade improved perturbative results from the ideal gas value was less than10%.Not all seem to agree that a resummation of perturbative results as in [9,10]sheds suﬃcient light on the lattice data.For example,Andersen,et.al.[11]and Blaizot,et.al.[12]have abandoned the expansion of the free-energy in terms of any formal parameter,but use instead gauge-invariance as the main guiding principle to sum select classes of diagrams.Their low order results seem to be close to the lattice data,but unfortunately because of the complexity of the calculations and the absence of an expansion parameter, it is not at all obvious what the magnitude of’higher order’corrections is.A completely diﬀerent approach has been taken by Kajantie,et.al.In[13] an attempt has been made to numerically estimate the net contribution of long-distance eﬀects,summarized in a dimensionally reduced eﬀective theory, to the free-energy density.As will be discussed later,the calculation of[13] probably contains some of the non-perturbative eﬀects deﬁned above but might miss out on some others.This author believes that the declared demise of information content in perturbative results such as(3)is premature.In Ref.[14]a resummation scheme was introduced to obtain an estimate of the total perturbative con-tribution to the free-energy density of SU(3)theory.The methodology of Ref.[14]has been further developed and applied to other problems in[15,16]. In Secs.(2-5),an explicit and improved discussion of the results in[14]is given,leading to the conclusion that the total perturbative contribution to the free-energy density lies signiﬁcantly above the full lattice data.In Sec.(6) I discuss the consequent magnitude of non-perturbative contributions,and speculate on their possible origin.Sec.(7)contains an analysis of the per-turbative free-energy of generalized QCD,with N f fermions,and a brief comparison with the available lattice data which is less precise.SU(N c) gauge theory is discussed in Sec.(8),and an apparently universal relation4noted.Sec.(9)considers quantum electrodynamics(QED)and some specu-lations about its high-temperature phase.A summary and the conclusion is in Sec.(9).2The Resummation SchemeThe truncated perturbative expansion of the normalized free-energy density can be written asˆS N (λ)=1+Nn=1f nλn,(4)whereλ=(α/π)1n!(5)which has better convergence properties than(4).The series(4)may then be recovered using the Borel integralˆS N (λ)=11+pz−11+pz+1(7) 5which maps the complex z-plane(Borel plane)to a unit circle in the w-plane. The inverse of(7)is given byz=4w(1−w)2.(8)The idea[19]is to rewrite(6)in terms of the variable w.Therefore,using (8),z n is expanded to order N in w and substituted into(5,6).The result isS N(λ)=1+1n! 4k!(2n−1)!∞0e−z/λw(z)(k+n)dz,(9)where w(z)is given by(7).Equation(9)represents a highly nontrivial re-summation of the original series(4)[15].In the pioneering application of the Borel-conformal-map technique in condensed matter physics by Le-Guillou and Zinn-Justin[19],the paramter p was aﬁxed constant which determined the precise location of the instanton singularity at z=−1/p.In some more recent QCD applications[20],theﬁxed constant p determines the ultraviolet renormalon singularity closest to the origin in the Borel plane[21].The novelty introduced in Ref.[14]and futher developed in[15]was to consider p as a variational parameter determined according to the condition∂S N(λo,p)3SU(3):Resummation up to Fifth Order In order to make contact with lattice data which show a temperature depen-dent curve,one must use in(4)a temperature dependent coupling.Let us begin by using the one-loop running coupling deﬁned by[11,12],λ(c,x)=211L(c,x),(11)where L(c,x)=ln((2.28πcx)2),c=¯µ/2πT and x=T/T c,with T c∼270MeV the critical temperature which separates the low and high tem-perature phases[1,22].Fixingﬁrst the reference values c0=1,x0=1,whichﬁxes the reference value ofλ0,the results of(10)are:p(3)=3.2,p(4)=7.9,p(5)=13.7.The curves for S N(λ)are shown in Fig.(3a)at the renormalization scale c=1. Notice the behaviour S5>S4>S3and how these all lie signiﬁcantly above the lattice curve in Fig.(1).The results do depend on the renormalization scale,denoted here by the dimensionless parameter c.It has been suggested [11,12]that a suitable choice for such a parameter is0.5<c<2,corre-sponding toπT<¯µ<4πT.Certainly this is the natural energy range for the high-temperature phase.Figure(3b)shows the mild dependence of S5 on the renormalization scale.The results above were obtained by solving(10)at the point c0=x0=1. Now consider changing the reference values to c0=1,x0=3,that is,a more central value for the temperature.The solutions are:p(3)=3.2,p(4)= 7.8,p(5)=13.4.These values are hardly diﬀerent from those above.This isﬁrstly due to the fact that(9)is a much slower varying function of the coupling than the original divergent series.Furthermore,for the present problem,the coupling itself varies slower than logarithmically with c and x (the c and x dependence of the coeﬃcients f4and f5is also only logarithmic). The curves for the re-optimized S N are essentially identical to those shown in Fig.(3a,b),the diﬀerence being only at theﬁfth decimal point.For example, the value of S5in Fig.(3a)at x=3is0.938684,while that for S5optimized at x0=3(and hence evaluated at p(5)=13.4),is0.938672at x=3.This conﬁrms the assertion in[14]that the results are quite insensitive to the exact reference values chosen to solve(10).We now proceed to test the sensitivity of the results to the approxima-tion used for the running coupling(11).The approximate two-loop running7coupling is given by[11,12]λ(c,x)=211L(c,x)1−51L(c,x) (12)with the symbols having the same meaning as before.In Fig.(4),the one-loop running coupling(11)and the approximate two-loop running coupling (12)are plotted at c=1to show their diﬀerence.At x=3,the value for the approximate two-loop coupling is about20%lower than the one-loop result.Nevertheless,because of the above-mentioned property of the resummed series,we shall see that theﬁnal results to do not shift ing(12),the solution of(10)at the reference point c0=1,x0=3 are:p(3)=3.2,p(4)=7.6,p(5)=13.1.The corresponding curves shown in Fig.(5a)have moved up slightly compared to those in Fig.(3a).The”two-loop”value of S5(c=1,x=3)=0.9473should be compared to the”one-loop”value0.9387obtained above.The mild renormalization scale depen-dence of the new S5is shown in Fig.(5b).In summary,it has been demonstrated that the resummed approximants S3,S4,S5all lie signiﬁcantly above the lattice data and satisfy the mono-tonicity condition S5>S4>S3.The result is insensitive to the reference value used to solve(10),for the range of interest0.5<c<2,1<x<5.The result is also insensitive to the approximation used for the running coupling constant and in fact better approximations for the coupling seem to move the values of S N further away from the lattice data.Finally it should be noted that the values S N also appear to converge as N increases.The only way to force the values of S N down closer to the lattice data is to choose very low values for the renormalization scale,c∼0.05.Of course this is not only unnatural but increases the eﬀective value of the coupling contant beyond what one would believe is physically reasonable for a perturbative treatment.That is by making an artiﬁcially low choice for the renormalization scale,one cannot escape the conclusion stated in the abstract of large non-perturbative corrections!4Higher Order CorrectionsDue to technical complications the sixth order contribution,λ6,to the free-energy density has not been calculated although an algorithm for it exists8[7].There is a misconception that because that contribution is due to an inﬁnite number of topologically distinct diagrams,its value must be very large.A counter-example is provided by the magnetic screening mass[5], which suﬀers from the same disease but whose approximate calculations in the literature show it to be of ordinary magnitude[23].Having said that,let us see what is the worst that can happen.It has been suggested[24]that Pade’approximants can be used to estimate the next term of a truncated perturbation series.That is,after approximating the truncated series by the ratio of two polynomials,the Pade’approximant is re-expanded as a power series to estimate the next term in the series. Well,why not also use Borel-Pade’approximants for the same purpose? Using theﬁfth order result(3)together with the two-loop running coupling (12),and choosing the central values c=1,x=3,allﬁfth order Pade and Borel-Pade approximants were constructed and then re-expanded to give an estimate of the coeﬃcient f6.The largest value obtained was30,000and the smallest−30,000.Note that theﬁfth order coeﬃcient at c=1is−800,so the estimated magnitude of f6is about37times larger.Since the coupling λis about0.2at x=3,c=1,the total value of the sixth order contribution to(3)is therefore estimated to be almost8times in magnitude compared to theﬁfth order contribution.These are big numbers and should be expected to cause some damage.Using(9)with f6=30,000,the two-loop coupling(12),and solving (10)at the reference point c0=1,x0=3gives p(6)=19.75and S6(c= 1,x=3)=0.9490.Repeating for f6=−30,000gives p(6)=19.5and S6(c=1,x=3)=0.9489.Notice the negligible change in the value of S6even when wildly diﬀering values have been used for f6.Those values should be compared with theﬁfth order approximant of Fig.(3a),which gives S5(c=1,x=3)=0.9473.The large estimated sixth order corrections to the divergent perturbation expansion(4)cause a change of only0.002to the values of the resummed series,and more importantly the shift is upwards, S6>S5,preserving the lower order monotonicity,regardless of the sign of f6.Kajantie,et.al.[13]have suggested that the sixth order contribution, f6λ6be of order10.For a couplingλ∼0.2,this translates into the astro-nomical value±156250for f6.Solving(10)at c0=1,x0=3gives p(6)=19.8 and S6(c=1,x=3)=0.9491for the positive f6,and p(6)=19.4and S6(c=1,x=3)=0.9489for the negative f6.Despite the anomalously large value of the sixth order contribution proposed in[13],the conclusion here is9still S6>S5,and an increment of only0.002.To highlight the above result in a more dramatic way,suppose the sixth order coeﬃcient vanishes,f6=0.Then because of the non-trivial way the resummation is done in(9),the solution to(10)for N=6will still be diﬀerent from that of N=5.At c0=1,x0=3Iﬁnd p(6)=19.5and then S6(c=1,x=3)=0.9489at four decimal places,which is almost identical to the values obtained above for various large values of f6.This sounds incredible but is actually not once one remembers that large corrections to the divergent series(4)do not translate into large corrections to the resummed series(9).In fact those large values are suppressed in various ways.Firstly, in the re-organization of the series in(9),less weight is given to higher order corrections.Secondly,the variational procedure chooses values of p(N)which in this example increase with N,and so suppress further the value of S N.More understanding of the above results can be obtained through a large N analysis carried out for the general Eqs.(9,10)in[15].It was shown in [15]that if p(N)increases for theﬁrst few values of N,then that trend will continue.Let c(N)≡p(N+1)/p(N).In the large N and large p(N)limit one can show that[15]1c(N)+1c(N+1)2=1−1c(N)3,(14)but in the large N limit where c→1+this is clearly equivalent to(13).10Note that the recursion relations(13,14)make no explicit reference to the values of the f n which in the derivation in[15]were assumed to be generic,that is,diverging at most factorially with n.Indeed that fact that various diﬀerent assumptions about the value of f6earlier in this section led to essentially the same value for p(6)∼19.5supports the f n independence of(13)already at N∼6.From the general analysis in[15],one also deduces that for large N and large p(N),the monotonicity∆S N≡S N+1−S N>0is guaranteed by the fact f2<0,and that∆S N∼1/N3as N→∞.Since the explicit N≤5 calculations and the estimated N=6result already support c(N)>1and large values of p(N)at low N,this suggests that the continued monotonicity and rapid convergence of the S N is assured by the large N analysis.5Lower Bound and Other EstimatesFrom the explicit low N calculations,and the large N analysis,one concludes that for N>2,S N<S N+1(15) for all N,and furthermore,the diﬀerence S N+1−S N decreases as N increases, showing a rapid convergnce of the approximants.However,in general,it is not quite correct to say that the approximants converge to the total sum of the series[15].For each N,let p⋆(N)be the value of p that is optimal,that is,it is the value which when used in(9)gives the best estimate of S,the total sum of the series.Deﬁne,S⋆N=S N(λ,p⋆(N)).Then for those p(N) which are positions of global minima one has by deﬁnition,S N≤S⋆N(16) It is S⋆N which presumably converges to S as N→∞.(This implicitly assumes that the sub-sequence of global minima is inﬁnite:That is,given any positive integer N0,there is some n>N0for which f n is positive.) Hence if one accepts the two assumptions above,then combining(15) with(16),S N≤S(17) for all N>2,and one may conclude that the S N are lower bounds to the sum of the full perturbation series.11In particular that conclusion implies that the N=5curve in Fig.(5a)is a lower bound on the total perturbative contribution to the free-energy density of hot SU(3)theory.The statement has three qualiﬁcations.Firstly,it involves the technical assumptions mentioned above.Secondly,as discussed before,better approximations to the running coupling can move the bounds, but it was seen that a20%improvement in the coupling shifted the bound upwards by1%.Thirdly,the bounds shift by±1%when the renormalization scale is varied by a factor of two from its central value¯µ=2πT.Thus it might be more appropriate to call the bounds as”plausible soft lower bounds”with an uncertainty depicted in Fig.(5b).Given that the lower bound obtained above involves some unproved tech-nical assumptions,it is useful to compare the above results with those ob-tained using diﬀerent resummation schemes for the divergent series(4).I brieﬂy state here the main results obtained using a Borel-Pade’resumma-tion of(4),with the two-loop approximation for the coupling(12)and the central value c=1.The approximants will be denoted as[P,Q],referring of course to the particular Pade’approximant used for the partial Borel se-ries(5)constructed from(4).The only approximants which did not develop poles and which gave a resummed value below one in the temperature range 2<x<5were[1,2],[2,1]and[2,2].These are displayed in Fig.(6).The [3,2]and[4,1]approximants did not develop poles but gave a value above one.If the approximants which developed a pole are deﬁned through a princi-pal value prescription,then the lowest value was given by[2,3]:0.91→0.94 as x increased from2→5.The[1,3]and[1,4]approximants gave values above0.98in the range of interest while[3,1]gave a value above one.Thus in the Borel-Pade’method,the minimum estimate for theﬁfth order resummed series is given by the principal value regulated[2,3].The highest values were all above one.If one keeps only theﬁfth order estimates below one(thus giving a very conservative lower value),then the average of the [2,3]and[1,4]is greater than0.94for the entire range2<x<5.At x=3the estimates are0.95±0.04.Of course including also the values above one would push this average higher.Clearly the Borel-Pade’estimates are comparable to the bounds obtained using the resummation technique of Sec.(2)and should reassure some readers about the novel resummation used here.For completeness,I mention an alternative way of thinking about diver-gent series such as(4).For QCD at zero temperature,a paradox is that one-loop results give remarkable agreement with experimental data even when the12energy scale is relatively low.As the running coupling is then large it is not obvious why higher-loop perturbative corrections are suppressed.It has beensuggested[25,26]that the explanation might lie in the probabale asymptotic nature of the QCD perturbation series.Recall that in an asymptotic series the best estimate of the full sum,at a given value of the coupling,is obtainedwhen only an optimal number of terms is kept and the rest discarded(even if they are large).Thus if one knew the general behaviour,at least at largeorder,of the series(4)and assumed that it was asymptotic,then one could have obtained a reasonable estimate of the full sum by simply adding the optimalﬁrst few terms.What has been done in the previous sections,andthis is what various resummation schemes try to do,is to instead sum up the whole series to get an even better estimate of the total perturbation series(and this has the greater advantage of giving a good result for a large range of couplings).Also note that thinking of(4)as an asymptotic series does not say anything about explicit non-perturbative corrections[25,26].6Non-Perturbative CorrectionsThe total perturbative contribution to the free-energy density of SU(3)gauge theory has been argued to be close to,or above,the N=5curve in Fig.(5a).A residual uncertainty that could lower the curve of Fig.(5a)is the exact value of the renormalization scale.For a natural range of parameters,thelower curve in Fig.(5b)is the result.On the other hand the full result as given by lattice simulations is shown in Fig.(1).Lattice errors have been stated to be under5%[1].Taken together,the conclusion appears inescapable:Evenat temperatures a few times above the transition temperature,there are large negative non-perturbative contributions to the free-energy density.For exam-ple,at T=3T c∼700MeV,the lattice results for the normalised free-energy density are0.8±0.04while the lower bound on the perturbative contribu-tion is0.947±0.007,implying a minimum non-perturbative correction of10%(and as high as20%).Thus an answer has been given to the questions raised in the introduc-tion.The deviation of the lattice data from the ideal gas value is apparently caused mainly by non-perturbative corrections,with perturbative corrections accounting for a much smaller amount.At T∼3T c the relative contributions are∼15%and∼5%.I speculate now on possible sources of the non-perturbative corrections.13Firstly there are the familiar instantons,already present in the classical ac-tion,and which contribute terms of the order e−1/λ2.Secondly there are themagnetic monopoles.There is by now overwhelming evidence that conﬁne-ment at zero temperature is caused by the t’Hooft-Mandelstam mechanism of condensing monopoles(the dual superconducting vacuum).Thus it ispossible that the monopole condensate has not completely melted above the critical temperature.Note that since the classical theory does not supportﬁnite energy monopoles,these must be of quantum origin,and so their con-tribution might be larger than those of the instantons.In fact contributions which are exponentially small but much larger thanthose of the instantons are suggested by the Borel resummation itself.It is known that Yang-Mills theories are not Borel summable[20,21].That is,the function B(z)has singularities for positive z,making the Borel integral ill-deﬁned.One can nevertheless deﬁne the sum of the perturbation series using the Borel integral if a prescription is used to handle the singularities.It is generally believed that the prescription dependent ambiguity disappears when explicit non-perturbative contributions are taken into account for the physical quantity in question.Indeed the nature of singularity itself suggeststhe form for the non-perturbative contribution.If there is a pole at z=q, then the non-perturbative contribution will be of the form∼e−q/λ,which islarger than the instanton contribution for smallλ.An explicit mathematical model which illustrates the interplay between Borel non-summability and non-perturbative contributions has been given in[15].Notice that the non-perturbative corrections suggested by the Borel method at non-zero temperature are very diﬀerent from those at zero temperature. In the latter case the expansion parameter is g2and so the contribution is∼e−q/g2,which translates into a power suppressed contribution∼1/(Q)b when g2is replaced by the running coupling∼1/ln(Q/Λ).In cases wherethe physical quantity can also be analysed using the operator-product ex-pansion(OPE),these power suppressed contributions to perturbative results correspond in the OPE picture to vacuum condensates[20,21].At nonzero temperature,since the natural expansion parameter isλ= ln(T/Λ),one does not get a simple power suppression from e−1/λ.Nevertheless,the analogy with zero-temperature results suggests that such contributions might be due to some condensates.Thus the conven-tional condensates discussed for example in[13]are plausibly part of the non-perturbative contributions.The form however suggests even more novel14。

Lattice Gauge Theory for the Quark-Gluon Plasma对于夸

Lattice Gauge Theory for the Quark-Gluon Plasma对于夸克胶子
等离子体的格点规范理论
1、战鼓一响，法律无声。——英国 2、任何法律的根本；不，不成文法本身就是讲道理 ……法律，也 ----即明示道理。— —爱·科克
3、法律是最保险的头盔。——爱·科克 4、一个国家如果纲纪不正，其国风一定颓败。—— 塞内加 5、法律不能使人人平等，但是在法律面前人人是平等的。 ——波洛克
• Normal matter made of baryons
• Baryons contain 3 quarks and interact by exchanging mesons
• Mesons contain 2 quarks
• When you squeeze this matter by applying pressure (or heating it up) you get matter with large numbers of quarks
the collision and compute density • Show that this generates pressure • …elliptic flow: also used in the BEC • Does the pressure drive flow? • …detailed analysis of velocities • Is there transport: energy, baryon etc?
• Normal matter may be neutral or a plasma
• Normal matter may be a solid, liquid or a gas

Peskin,intro_to_QFT(Part+IV)

22.1 Strong Strong Interactions
One paradoxical aspect of our discussion of the strong interactions is that all of our concrete results were obtained by assuming that these interactions are weak. At large momentum transfer, we argued, this assumption is actually valid due to asymptotic freedom. Still, it is uncomfortable that we have left the most obvious questions about strongly interacting particles|for example, their masses and low-energy interactions|in a mysterious regime excluded from our analysis. To work with QCD in the region where the strong interactions are strong, we need to answer three questions: First, how do we describe the forces that bind quarks together into hadrons? Second, what is an appropriate description of the quark-antiquark and three-quark systems bound by those forces? And nally, how do we compute scattering amplitudes and matrix elements of currents using these bound states? At this moment, there is no derivation of the complete force between quarks from the QCD Lagrangian. Explicit calculations can be done only in the limits of weak and strong coupling. In the weak-coupling limit, the result is a Coulomb potential with an asymptotically free coupling constant. The strong coupling limit, on the other hand, gives a linear potential which con nes color in the way that we described, but did not derive, at the end of Section 17.1. The derivation of this result is quite unusual and brings in a new set of mathematical methods. So far in this book, we have not discussed a strong coupling approximation to a quantum eld theory. There is a simple reason for this: In a quantum eld theory in which the coupling g2 is very large, the elementary particles or their bound states typically acquire masses that grow with g2. For g2 ! 1, these masses become comparable to the cuto and the eld theory ceases to have a local continuum description. Wilson proposed to solve this problem in a radical way, by replacing spacetime with a lattice of discretely spaced points. Such a lattice is easiest

The Phase Structure of an SU(N) Gauge Theory with N_f Flavors

1
1
Introduction
In an SU (N ) gauge theory with Nf massless quarks, it is expected that both conﬁnement and spontaneous chiral symmetry breaking take place providing that Nf is not too large. If, on the other hand, Nf is large enough, the theory is expected to neither conﬁne nor break chiral symmetry. For example, if Nf is larger than 11N/2 for quarks in the fundamental representation, asymptotic freedom (and hence conﬁnement and chiral symmetry breaking) is lost. Even for a range of Nf below 11N/2, the theory should remain chirally symmetric and deconﬁned. The reason is that an infrared ﬁxed point is present [1, 2] determined by the ﬁrst two terms in the renormalization group (RG) beta function. By an appropriate choice of N and Nf , the coupling at the ﬁxed point, α∗ , can be made arbitrarily small [3], making a perturbative analysis reliable. Such a theory is massless and conformally invariant in the infrared. It is asymptotically free, but without conﬁnement or chiral symmetry breaking.

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

level potential between heavy quarks is already logarithmically conﬁning: V (r) ∼ log r.
There are, however, many similarities. One may introduce a dimensionless “running” coupling constant g3(l) by the deﬁnition g32(l) ≡ lg2 where l is a length scale. Then g32(l) → 0 for l → 0 and to inﬁnity for l → ∞. This is somewhat analogous to the logarithmically running coupling constant in 3 + 1 dimensional SU(3) gauge theory. In 2 + 1 dimensions the coupling constant g2 sets the scale, and mi = cig2, where ci’s are numerical constants. From Monte Carlo simulations one knows some further properties:
(2)
Add and subtract an explicit mass term in the static sector,
1 2
m2g
T
r(A0(x¯))2
,
(3)
which is invariant inside this class of gauges, and perform the perturbative expansion in the theory with A0 massive. In fact, there are no further infrared divergencies in 2+1 dimensions[ 3].
resummations, e.g. through the selfconsistent perturbation theory (SCPT) introduced by
2
B. Petersson
3
2D 3D C = -1.0 C = 1.6
2
mass/(g2T)1/2
1
0
0
0.5
1
g2/T
1.5
with controlled systematic errors. Of course, there are some obvious diﬀerences between
SU(3) gauge theory in 2 + 1 and 3 + 1 dimensions. In 2 + 1 dimensions the coupling constant g2 has the dimension of a mass, and the theory is superrenormalizable. The tree
D’Hoker[ 3]. It works in the following way. Choose the class of static gauges ∂0A0(x) = 0 where the Polyakov loop is purely static,
L(x¯)
=
T
r
ei
g T
A0(x¯).
g32(T )
≡
g2 T
T−→→∞ 0.
(1)
There are, however, infrared divergences, which are even more serious than in 3 + 1
dimensions. For the screening mass (rsp. the pressure) they appear already at order g2, i.e. at one (resp. two) loop(s). The infrared divergences can be tamed through
non-perturbatively by a Monte Carlo lattice simulation.
We deﬁne a screening mass for T > Tc by
Re L(0)L†(x) ≃ L 2 + A e−m|x|
(5)
m|x|
SU(3) gauge theory at ﬁnite temperature in 2 + 1 dimensions
The dimensionally reduced model is in good agreement with the full theory already for T > 1.5Tc as can be seen in Fig. 1. A further investigation showed that the two exchanged states in the reduced model are simple poles, not 2 gluon and 3 gluon cuts respectively[
m2g g2T
=
3 2π
log
T mg
+
C
+O
1/
log
T mg
,
C = −1.0.
(6)
In Fig. 1, we show 2mg compared to the screening mass in the 2 + 1 dimensional SU(3) gauge theory. We have used the formula T /Tc = (β − 1.5)/3.3Nτ , derived from the condition Tc/g2 = 0.55 for all Nτ , and the values βc(Nτ ) from[ 2]. Solving Eq. (6) where mg/T has been replaced by (1 − exp(−mg/T )), we can get agreement only for T /g2 > 6 (T /Tc > 12), and this only by arbitrarily choosing C = 1.6.
There is a linearly rising non-perturbative potential V (r) ≃ σ0r for r large[ 1, 2]. There is a second order phase transition at Tc = 0.55(1)g2 , with the critical indices of the 2d 3states Potts model[ 2]. Furthermore, the glue ball masses mGB are much bigger than Tc, mGB ≥ 4.4Tc [ 1]. This is all qualitatively similar to 3 + 1 dimensions, where, however, the transition is weakly ﬁrst order. In the gluon plasma phase T > Tc, one should be able to use perturbation theory. The relevant dimensionless coupling in 2 + 1 dimensions is
loop
integration
over
the
static
modes.
This
action
is
systematic
in
g2 T
,
e.g.
the
Байду номын сангаас
term
φ6
is
multiplied
by
a
constant
proportional
to
g6 T
,
and
is
neglected
at
high
T.
The two-dimensional adjoint Higgs model has not been solved analytically. We solve it
S=
d2xT r
1 2
Fi2j
+
(Diφ)2
−
3g2T 2π
5 2
log
2
+
1
−
log(aT
)
φ2
+
g4 32π
φ4
(4)
where Di is a covariant derivative and where the coupling constants are derived from a
one
arXiv:hep-lat/0606019v2 28 Feb 2007
SU(3) gauge theory at ﬁnite temperature in 2 + 1 dimensions
P. Bialas,a L. Daniel,aA. Morelb and B. Peterssonc
aInstitute for Physics, Jagellonian University, ul. Reymonta 4, 30 059 Krakow, Poland
The SU(3) gauge theory in 2+1 dimensions is simple enough from a numerical point