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

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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。

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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