Commun. Math. Phys. 193, 47 – 67 (1998) Communications in Mathematical Physics c Springer-Verlag 1998
Symplectic Structures on Gauge Theory?
Nai-Chung Conan Leung
School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455, USA
Received: 27 February 1996 / Accepted: 7 July 1997
[ ] Abstract: We study certain natural differential forms ∗ and their equivariant ex- tensions on the space of connections. These forms are defined usingG the family local index theorem. When the base manifold is symplectic, they define a family of sym- plectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations).
Contents 1 Introduction ...... 47 2 Higher Index of the Universal Family ...... 49 2.1 Dirac operators ...... 50 2.2 ∂-operator ...... 52 3 Moment Map and Stable Bundles ...... 53 4 Space of Generalized Connections ...... 58 5 Higer Level Chern–Simons Forms ...... 60 [ ] 6 Equivariant Extensions of ∗ ...... 63
1. Introduction
In this paper, we study natural differential forms [2k] on the space of connections on a vector bundle E over X, A
i [2k] FA (A)(B1,B2,...,B2 )= Tr[e 2π B1B2 B2 ] Aˆ(X). k ··· k sym ZX They are invariant closed differential forms on . They are introduced as the family local indexG for universal family of operators. A
? This research is supported by NSF grant number: DMS-9114456 48 N-C. C. Leung
We use them to define higher Chern-Simons forms of E:
[ ] ch(E; A0,...,A )=∗ L(A0,...,A ), l l ◦ l l and discuss their properties. When l = 0 and 1, they are just the ordinary Chern character and Chern-Simons forms of E. The usual properties of Chern-Simons forms are now generalized to
(1) ch(E) ∂ = d ch(E), A ◦ ◦ s (2) ch(E; s α)=( 1)| |ch(E; α). · − When we restrict our attention to the case when X is a symplectic manifold (or [2] a Kahler manifold) with integral symplectic form ω,[ω]=c1(L). We study on (E Lk) for large k. Asymptotically, they define the symplectic form on (E Lk). A ⊗ A ⊗ Using a connection DL on L, we can identify all these ’s with (E) and get a one parameter family of symplectic forms on (E), A A k A
i FA +kωIE k(DA)(B1,B2)= Tr [e 2π B1B2]symAˆ(X). ZX Their -moment maps are G i FA+kωIE (2n) 8k(A)=[e2π Aˆ(X)] . When we let the parameter k go to infinity, we will obtain the standard symplectic form on (E): A n 1 ω − (A)(B1,B2)= Tr B1B2 . ∧ (n 1)! ZX − Notice that is an affine constant form in the sense that it is independent of the point A . The moment map equation associated to is ∈A ωn 1 ωn F − = µ I , A ∧ (n 1)! E n! E − which is equivalent to the Hermitian Einstein equation (a semi-linear system of partial differential equations)
FA = µEIE.
Nevertheless, the moment map equation^ for 8k is a fully non-linear system of equa- tions (of Monge-Ampere type) which are closely related to stability of E as will be explained in Sect. 4. Then we explain how to remove the assumption of integrality of ω in Sect. 5. More- over, all ’s will be canonically embedded inside a larger space ˜[E], the space of generalizedA connections. Generalized connections satisfy a weaker constraint:A
1 1 Aj = gij− Aigij + gij− dgij + dθij . However, they share many properties of an ordinary connection. The first Chern class defines a surjective affine homomorphism ˜ 2 c1 : [E] H (X, C), A → Symplectic Structures on Gauge Theory 49
and [2] gives a relative symplectic form on it. We define the notion of extended moduli space and polarization which links to our previous picture on stability. In Sect 6, we will generalize the moment map construction. We study equivariant ex- [2k] [2i,2j] 2i j tensions of all and their properties. They are ( ,Symm(Lie )∗)G given by ∈ A G [2i,2j] (DA)(B1,...,B2i)(φ1,...,φj)
i FA = Tr [e 2π B1 B2 φ1 φ ] Aˆ(X) ··· i ··· j sym ZX In the last section, we will briefly explain how these higher [2i,2j] are related to family stability. We need to look at all connections on the universal bundle which induces the canonical relative universal connection Drel. Let D + S be such a connection, then its curvature is given by:
2,0 F (x, A)=FA at x, 1,1 F (x, A)(v, B)=B(v)+DA,v(S(A)(B)) at x, 2 F0,2 = d S + S . A [ ] By twisted transgression of ∗ (defined using D+S), we get Map(M, )-invariant [ ] G closed forms on Map(M, ), namely, M∗ . We will explain how these forms are related to stability for a family ofA bundles over X parametrized by M when both X and M are symplectic manifolds.
2. Higher Index of the Universal Family
In this section we consider the universal connection and the universal curvature. They [ ] induce canonical differential forms ∗ on the space of connections which are important for our later discussions. These forms are closely tied with the local index theorem for the universal family of certain operators. We shall discuss the Dirac operator and the ∂ operator in detail. Let X be a compact smooth manifold of dimension 2n and E be a rank r unitary vector bundle over X. We denote the space of connections on E by and the space of automorphisms of E by , which is also called the group of Gauge transformationsA on E. The tangent space ofG at a point A is canonically isomorphic to the space of one forms on X with End(AE) valued, that∈A is, T = 1(X, End(E)). AA In particular, is an affine space. (Later in this paper, we shall construct a natural affine extensionA of by an one dimensional affine space.) is an infinite dimensional Lie Group with LieA algebra being the space of sections of End(G E) over X, that is, Lie = 0(X, End E). G The center of can be identified as the space of non-vanishing functions on X and hence the centerG of its Lie algebra can be identified as the space of all functions on X, C(X), or the space of zero forms on X, 0(X). By integrating over X, we can identify the dual of Lie as the space of 2n forms on X with End(E) valued in the sense of distributions G 2n (Lie )∗ = (X, End(E)) . G dist 50 N-C. C. Leung
It has a dense subspace 2n(X, End(E)) consisting of smooth 2n forms which are enough in most purposes. The above identification follows from the fact that
0 2n (X, End E) (X, End E)dist C , ⊗ −→ φ α Tr φα ⊗ −→ Z X is a perfect pairing. By putting all connections together, one can form a universal connection as follow: Let E = π1∗E be the universal bundle, where π1 : X Xis the projection to the first factor. Then we can patch up all connections on×A−→E to form a partial connection on E and since the bundle E is trivial along , we get a (universal) connection D on E. A To be more precise, let us describe D in terms of horizontal subspaces. A connection A on E is given by a U(r) equivariant subbundle H of TE over E such that TE is isomorphic to TvE H. Here TvE is the vertical tangent bundle of E with respect to E X .Now,lete Ebe a point on the fiber of E over (x, A) X . Then TeE is canonically−→ isomorphicL∈ to T E T and we can choose H ∈T ×Aas a horizontal e AA AA bundle of T E (where H is the horizontal subbundle of TEdefined by A). This horizontal subspace defines our universal connectionL D. L The curvature F of D will be called the universal curvature, it is a End(E) valued two form on X . With respect to the natural decomposition of two forms on X : ×A ×A 2(X )=2(X) 1(X) 1( ) 2( ), ×A A A M O M we decompose F into corresponding three components,
2,0 1,1 0,2 F = F + F + F . They are given explicitly by
2,0 F (x, A)=FA at x, F1,1 (x, A)(v, B)=B(v)atx, F0,2 =0. 1 where (x, A) X ,v TxXand B (X, End(E)). Now, we want∈ to×A use ∈to parametrize∈ certain families of first order elliptic operators on X by coupling differentA connections with a fixed differential operator. The following three situations are of most interest to us.
2.1. Dirac operators. Let X be a spin manifold of even dimension m =2n. For any Riemannian metric g on X, we get a Dirac operator D on X. By coupling D with a connection on E, A , we get a twisted Dirac operator DA on X. Varying these connections, we then∈A get a family of Dirac operators parametrized by . A The index of DA (Index DA = dim KerDA dim CokerDA) can be computed by the Atiyah-Singer Index theorem: −
Index DA = ch(E)Aˆ(X), ZX which can be expressed in terms of curvature of E via the Chern-Weil theory: Symplectic Structures on Gauge Theory 51
i FA Index DA = Tr e 2π Aˆ(X). ZX The virtual vector space (the formal difference of two vector spaces) KerD A − CokerDA varies continuously with respect to A and forms a virtual vector bundle over (despite the fact that KerD or CokerD might jump in dimensions separately). A A A This is called the index bundle over , Ind (D). We could formally regarded Ind (D)as an element in the K group of , KA( ), even though is of infinite dimension. This A A A can be interpreted as Ind (D) giving an element in K(Y ) for any compact subfamily Y . Although Ind (D) is only a virtual bundle, its determinant is an honest complex ⊂A line bundle over , called the determinant line bundle for Dirac operators, Det (D). By the familyA index theorem of Atiyah and Singer, we have (for any compact sub- family Y ) ⊂A i ch(Ind (D)) = Tr e 2π FAˆ(X) ZX as cohomology classes. Bismut and Freed [B+F] strengthens this result to the level of differential forms and give the local family index theorem. They introduce superconnection At on the infinite dimensional bundle π E and they proved the following equality on the level of differential forms: ∗ i 2π F ˆ lim ch(At)= Tr e A(X). t 0 & ZX We decompose this differential form into a sum according to their different degrees [0] + [2] + , where [2k] 2k( ). ··· ∈ A Proposition 1.
i i [2k] 2k FA (A)(B1,B2,...,B2 )=( ) Tr[e 2π B1B2 B2 ] Aˆ(X), k 2π ··· k sym ZX where A and B T = 1(X, End (E)). ∈A i ∈ AA The notation [ ] denotes the graded symmetric product of those elements inside. ··· sym Proof of Proposition.
[2k] (A)(B1,B2,...,B2k)
i 2,0 1,1 2 (F +F )(x,A) = Tr [e π ](B1,B2,...,B2k)Aˆ(X) ZX 1,1 2k i F i 2k (F ) = Tr [e 2π A ( ) (B ,B ,...,B )] Aˆ(X) 2π (2k)! 1 2 2k sym ZX i i 2k FA =( ) Tr [e 2π B1B2 B2 ] Aˆ(X). 2π ··· k sym ZX 2,0 For the second equality, we used the fact that F has no effect on the Bi’s and the last equality holds since F1,1 is the tautological element which can be described as below. Under the following relations: 52 N-C. C. Leung
1(X) 1( ) End(E) ⊗ A ⊗ = Hom(T ,1(X, End (E))) ∼ A Hom(T ,1(X, End (E))) ⊃ AA = Hom(1(X, End (E)), 1(X, End (E))),
1 F1,1 comes from the identity endomorphism of (X, End (E)). Hence we have proved the proposition.
Corollary 1. [2k] is a invariant closed form on and such that [2k] =0if k>n. G A Proof of Corollary. Both [2k] = 0 for k>nand the invariance of [2k] with respect to action are clear from the above proposition. We will use the Bianchi identity and theG simple fact that the antisymmetrization of a symmetric expression is always zero to show the closedness of [2k]. Consider
2k [2k] i [2k] d (A)(B0,...,B2 )= ( 1) B (A)(B0,...,Bˆ ,...,B2 ) k − i i k i=0 X i+j [2k] + ( 1) (A)([B ,B ],B0,...,Bˆ ,...,Bˆ ,...,B2 ). − i j i j k i i 2n n k 1 =( ) (n k) Tr[F − − (D B )B0 Bˆ B2 ] . 2π − A A i ·· i ·· k sym ZX because all B ’s are independent of the connection A . Hence, i ∈A 2k i [2k] ( 1) B (A)(B0,...,Bˆ ,...,B2 ) − i i k =0 Xi i 2n n k 1 =( ) (n k) d Tr[F − − B0 B2 ] 2π − A ··· k sym ZX Here, we have used the Bianchi identity DAFA = 0 and the fact that d Tr = Tr DA. Now the vanishing of the above expression follows from the Stokes’s theorem. Hence we have showed that [2k] is closed. An alternative way to prove the closedness is by observing that these forms come from the pushed forward of certain closed differential forms on X . ×A 2.2. ∂-operator. The second case is when X is a compact complex manifold. Then (complexified) differential forms on X can be decomposed into holomorphic and anti- holomorphic components: Symplectic Structures on Gauge Theory 53 l i,j (X) C = (X). ⊗ + = iXj l As an End(E)-valued two form on X, the curvature form of a Hermitian connection 2,0 1,1 0,2 can be decomposed accordingly: FA = FA + FA + FA . One should not confuse this 0,2 decomposition of FA with the previous decomposition of F.IfFA = 0, then the (0, 1) 0, part of DA defines a differential complex on ∗(X, E). Let us denote p DA by ∂A, 1 0 , 1 ◦ where p : (X, E) C ( X, E) is the projection to the anti-holomorphic ⊗ −→ 0,2 2 part. ∂A gives a complex because FA = ∂A. hol 0,2 hol Let = DA FA =0 , then parametrizes a family of elliptic com- plexes overA X.Now,{ ∈A|is the space} of connectionsA on E, not necessary Hermitian. However, in order forA it to parametrize elliptic complexes, we need to choose a com- patible Hermitian metric on E (which in fact exists and is unique up to unitary gauge transformations) and a Hermitian metric on the base manifold X. Such a metric induces a Todd form on X representing the Todd class of X. As in the case of Dirac operators, we get a sequence of differential forms on hol: A i [0] [2] [4] Tr e 2π FTd = + + + . X ··· ZX Here the only difference is the replacement of Aˆ(X) by the Todd form on X, TdX. Notice that the odd class and Aˆ class are closely related to each other: Td = ec1Aˆ. If X is Spin Kahler manifold then this equality holds on the level of differential forms. 1 − 2 In this case, the ∂ operator on X is equal to the Dirac operator on X twisted by KX , where the canonical line bundle KX is determined by the (almost) complex structure on X and its square root is given by the chosen spin structure on X. In this identification, Kahlerian is important, which ensures that the complex structure and the metric structure on X are compatible to each other. Later on, we shall explain that asymptotic behaviors of [2] are closely related to stability in Mumford’s Geometric Invariant theory [M+F]. The importance of higher [2k] will be briefly discussed in the last section. 3. Moment Map and Stable Bundles In this section, we will explain the relationship between the form [2] we found earlier and stability properties of the bundle E. This relation is originally discovered in the author’s thesis [Le1] and explained in [Le2]. Here, we shall give a global picture in this section and the next one. Then we shall also discuss higher [2k] later in this paper. Now, we suppose X is a symplectic manifold with symplectic form ω. We also assume ω is integral which means [ω] represents an integer cohomology class in X, [ω] H2(X, Z). The general case will be treated in the next session. H2(X, Z) classifies ∈ complex line bundles over X up to isomorphisms, that is, [ω]=c1(L) for some L over X. In fact, we can find a Hermitian connection DL on L such that the last equality holds i on the level of forms, that is ω = 2π FL, where FL is the curvature form of a certain connection DL on L. We are interested in the behaviors of [2] on the space of connections on E Lk for large k. In order to distinguish connections on different bundles, we write (E Lk) A N N 54 N-C. C. Leung k for the space of connections on E L . However, by tensoring with DL, we can identify different (E Lk), A N(E) ( E L k ) , N A −→ A ⊗ D D D k . A 7→ A ⊗ L i i Under this identification, the curvature 2π FA of DA will be sent to 2π FA + kωIE. [2] We shall denote the corresponding 0 = by k. We therefore have i FA +kωIE k(DA)(B1,B2)= Tr [e 2π B1B2]symAˆ(X) ZX in the spin case (and similarly formula for other cases by replacing the Aˆ by suitable characteristic classes). Notice that, at a general point DA ,k may be degenerated. However, it will become non-degenerate if k is chosen large∈A enough and therefore, in a suitable sense, k is asymptotically symplectic form on . (For the rest of this paper, a symplectic form may possobly degenerate but this deficiancyA can usually be rescued by asymptotic non-degeneracy.) Moment maps. For any k, k is always a an invariant form and a moment map exists. That is, can be extended to a equivariantG closed form on . k G A Lemma 1. The moment map for the action on with symplectic form k is given by 8 : 2n(X, End(E)), G A k A−→ i FA+kωIE (2n) 8k(A)=[e2π Aˆ(X)] . Proof of Lemma. First, we need to show that 8 is a equivariant map, but this is clear k G from the definition of 8k. Next, we want to show that k + 8k is a equivariant form. That is, for any φ Lie = 0(X, End(E)), G ∈ G ιφk = d(8k(φ)) as an ordinary one form on ,or, A (A)(D φ, B)=d( Tr 8 (A) φ)(B) k A k · ZX for any A and B T = 1(X, End(E)), ∈A ∈ AA d( Tr 8k (DA)φ)(B) ZX d i 2 2 (FA +tDAB+t B )+kωIE = =0 Tr e 2π φAˆ(X) dt|t ZX i i F +kωI = Tr[e 2π A E D B] φAˆ(X) 2π A sym ZX i i F +kωI = Tr[e 2π A E D Bφ] Aˆ(X) 2π A sym ZX i i F +kωI = Tr[e 2π A E D φB] Aˆ(X) 2π A sym ZX = k(DA)(DAφ, B) Symplectic Structures on Gauge Theory 55 Hence, k + 8k is a equivariant closed form and 8k is the moment map for the gauge group action on Gwith the symplectic structure given by . A k Remark . For each k, we obtain a equivariant closed form on , namely + 8 . G A k k It depends only on the choice of a connection DL on L. Neverthless, the equivariant 2 cohomology class it represents, [k + 8k] H ( ), is independent of such a choice. ∈ G A 2 Proposition 2. [k + 8k] H ( ) is independent of the choice of the Hermitian ∈ G A connection DL on L. Proof of Proposition. If DL and DL0 are two connections on L, so they differ by a one form on X, say α. They induce k + 8k and k0 + 8k0 on . Their difference is a equivariant exact form on which can be written down explicitlyA (like the Chern-SimonsG construction). More precisely,A (0 + 80 ) (k + 8k)=dG2k,D ,D , k k − L L0 where 1 i F +k(ω+tdα)I 2 (A)(B)=k dt α Tr e 2π A E BAˆ(X). k,DL,DL0 Z0 ZX Hence, the cohomology classes they represented are the same. Large k limit of k + 8k. We now consider k and 8k for large k. If we expand k in powers of k,wehave n 1 n 1 ω − n 2 (A)(B1,B2)=k − Tr B1B2 + O(k − ). k · ∧ (n 1)! ZX − The leading order term of it defines a (everywhere non-degenerate) symplectic form on , moreover, it is a constant form on in the sense that there is no dependence of A inA its expression. The moment map definedA by this constant symplectic form is ωn 1 8(A)=F − . A∧(n 1)! − If we expand 8k for large k ,wehave n nω n 1 n 2 8 (A)=k I +k − 8(A)+O(k − ). k n! E The first term is a constant term and can be absorbed in the definition of 8k (since the moment map is uniquely determined only up to addition of any constant central element) and therefore, the leading (non-trivial) term for 8k becomes 8 for large k. We now look at the symplectic quotient of by Hamiltonian actions with respect A G 1 to different k. First choose a coadjoint orbit (Lie )∗, the inverse image 8− ( ) O⊂ G 1 O of under the moment map 8 is invariant. Then one can show that 8− ( )/ inherits a naturalO symplectic form (on theG smooth points) from the original symplecticO G space. We can now apply this procedure to any of these k or since they are all invariant symplectic forms on . G A 1 The simplest choice of the coadjoint orbit would be 0, however, 8− (0) may not be non-empty in general. We would choose those coadjoint orbits which consist of a single 56 N-C. C. Leung point only. The set of all these coadjoint orbits can be identified with the space of 2n forms on X (in the sense of distributions). Hermitian Einstein metrics. We first consider 8 which depends on A is a (semi-)linear 2n 1 fashion. Let ζ (X), then D 8− (ζ) if and only if ∈ A ∈ ωn 1 F − = ζI . A ∧ (n 1)! E − 1 By taking the trace and integrating it over X, we get a necessary condition for 8− (ζ) being non-empty which is ζ = µE, ZX n 1 1 c1(L) − where µ = the Euler-Lagrange equation implies that FA is a parallel endomorphism and therefore their eigenspaces are all holomorphic subbundles of E. When E is irreducible, E itself V is the only non-trivial subbundle. So, the endomorphism FA has to be constant which is fixed by the topology of the bundle. From another point of view, the Yang-Millsequation onVE (that is the Euler-Lagrange 2 ωn hol equation of the functional F )isD∗F = 0. When X is Kahler and A , X | A| n! A A ∈A we have D∗ =[DA, ]. So, A R V 0=D∗F =D ( F ) (D F )=D ( F ) A A A − A A A A ^ ^ ^ by the Bianchi identity. Hence, the two equations are equivalent. Stability. Existence of solutions to FA = µIE on E can be rephased in algebraic geometric languages. By [N+S],[Do] and [U+Y], there exists a Hermitian Einstein metric on E if and only if E is a MumfordV stable bundle. In such a case, the metric is also unique up to scaling by a global constant. (When E is not irreducible, then existence of the Hermitian Einstein metric on E is equivalent to E being a direct sum of irreducible Mumford stable bundles over X with the sum slope, we called such a bundle E a Mumford poly-stable bundle.) Mumford stability is only the “linearization” of stability in studying moduli spaces of vector bundles using Mumford’s Geometric Invariant theory. In geometric invariant theory of vector bundles, Gieseker [Gi] found the correct notion of stability and stated them in geometric terms. In the thesis, the author found that stability is in fact equivalent to existence of solutions to moment map equations for 8k with k large (and control of the curvature of the solutions as k goes to infinity). To solve that (fully-nonlinear) equation, the author used a very involved singular perturbation method which identified the obstruction for perturbations is precisely the unstability [Le1, Le2]. The basic idea is when k goes to infinity, we expected the family of solutions will blow up (provided it exists). Different directions will blow up according to different rates. However, this information can be captured from algebraic geometry. Then we try to perturb the singular solution (whose existence can be proved by using a theorem of Uhlenbeck and Yau) to finite k; there will be numerous obstructions. We can identify all these obstructions. In fact, obstructions vanish precisely when the bundle is Gieseker (poly-)stable. Instead of trying to solve the equations, we shall discuss the geometry underlying these k’s, 8k ’s and their higher degree cousins. ωn First we want to set 8k equal to a constant multiple of n! IE. The constant has to be 1 χ(X, E Lk), where χ is the index of the corresponding operator given by rk(E) ⊗ the Atiyah-Singer index theorem. Therefore, the equations defined by the moment map would become n i F +kωI (2n) 1 k ω [e 2π A E Td(X)] = χ(X, E L ) I . rk(E) ⊗ n! E These equations describe the stability properties of the hlomorphic bundles. Unlike the Hermitian Einstein equations, this system of equations is fully nonlinear unless n =1. Instead of ∂¯ operator, we can use the Dirac operator and replace Tdform by Aˆ forms. 58 N-C. C. Leung 4. Space of Generalized Connections In this section, we shall explain a setting which is more suitable for studying the symplec- tic quotient of (or hol)by with respect to [2]. All the proofs here are elementary and sometimesA we onlyA sketchG the reasonings. From the last section, we know that it is natural to put (E Lk)’s together for large k. Stability depends only on the ‘direction’ L and one shouldA not⊗ distinguish E among E Lk’s. We are going to explain how to achieve these in a natural manner by allowing k⊗to be any complex number. Moreover, in this approach, we do not need to assume that the symplectic form ω is an integral form. To do this, we need to define generalized connections on E. Review of connections. Let us first recall the definition of connections in Cˇ ech languages. Let be a good cover of X. That is, for any U ,U ,...,U ,U U U U i j k ∈U i,j,...,k ≡ i ∩ j ∩ Uk is a contractible set. Then the bundle E is determined by a collection of ···∩ 1 gluing functions: g : U GL(r) which satisfies g = g− and g g g =1. { ij ij → } ij ji ij jk ki Two collections g and g0 are equivalent if there exists hi : Ui GL(r) satisfying 1 { → } gij0 = hi− gij hj. A connection D is a collection of matrix-valued one form on , A 1(U ,gl(r)) . A U { i ∈ i } On Uij , they satisfy 1 1 Aj = gij− Aigij + gij− dgij . Now, a generalized connection DA is essentially the same thing except that we relax the last equality to 1 1 Aj = gij− Aigij + gij− dgij + dθij for some collection θij : Uij C which satisfies θij = θji and d(θij +θjk+θki)=0. The definition of gauge{ equivalent→ } for two generalized− connections are the same as 1 1 ordinary connections: Ai0 = gi− Aigi + gi− dgi for some automorphism gi : Ui GL(r) . { → } 1 Let F = dA + A A , then one can prove that we still have F = g− F g . That i i i ∧ i j ij i ij is F 2(X, End (E)). One can also verify that the notion of generalized connections is well-defined∈ and independent of the choice of good cover or the gluing functions g of E. U { ij } Generalized connections. 1 Definition 1. A collection DA = Ai (Ui,gl(r)) is called a generalized connec- tion if { ∈ } 1 1 Aj = gij− Aigij + gij− dgij + dθij for some collection θ as before. { ij } The curvature F 2(X, End (E)) of D is defined to be A ∈ A F = dA + A A i i i ∧ i on U . ˜[ ] denotes the space of all generalized connections on E. i A E We collect some basic facts about ˜[ ] in the following lemma. A E Lemma 2. (1) (E) ˜[ ]. A ⊂ A E (2) ˜[E] ˜[E L] for any line bundle L. A ≡ A ⊗ Symplectic Structures on Gauge Theory 59 (3) ˜[ ] has a natural affine structure such that (E) is an affine subspace of it of A E A codimension b2(X). k In a sense, ˜[ ] should be regarded as (E L ) where the L’s forms a base A E L,k A ⊗ of H2(X, Z)/T or and k C. Previously, in order to identify (E) and (E L), ∈ S A A ⊗ we have to pick a connection on L. But now, ˜[E] and ˜[E L] are always canonically isomorphic to each other. So it is natural to considerA equivalentA ⊗ classes of vector bundles which are isomorphic to each other modulo tensoring with line bundles. We denote the equivalent class of E by [E]. That is [E L]=[E] for any line bundle L. Notice that End([E]) is a well-defined⊗ bundle on X and the group of all gauge transformations of [E] is also a well-defined notion for the equivalent class.G i Let c1([E],DA)= 2πTr FA be the first Chern form of any generalized connection DA ˜[E]. It is always a closed two form on X. However, its cohomology class is not determined∈ A by [E]. In fact, it induces a surjective affine homomorphism ˜ 2 c1 : [E] H (X, C) A → and acts on ˜[ ] preserving the fiber of c1. G A E Now, [2] can be naturally extended to a invariant relative closed two form ≡ G on ˜[ ]. A E Relative forms. Let us first define the relative notion which will also be used in later sections. Suppose that X P π M is a fiber bundle over M. We have a canonical exact sequence of vector bundles→ → over P : 0 T P TP π ∗ TM 0 , −→ vert −→ −→ −→ where TvertP denote the vertical tangent bundle (or the relative tangent bundle). In fact, we can regard this exact sequence as the definition of TvertP. Recall that a k-form is a section of k(TP) over P . k DefinitionV 2. A relative k-form is a section of (TvP ) over P . rel A relative connection DA on a bundle E over P is a first order relative differential operator V rel D : 0(P, E) 0 ( P, T ∗ P E), A −→ vert ⊗ that is, rel rel DA (fs)(v)=v(f)s+f(DA s)(v) for any f ∞(P),s 0(P, E) and v TvertP. ∈C ∈rel rel rel∈ 2 The curvature of DA is FA =(DA ) . An ordinary k-form (resp. connection of E)onP induces a relative k-form (resp. relative connection of E). Notice that the curvature of a relative connection is a relative two form on P with End(E)-valued. If we are given a splitting of TP as a direct sum of TvertP and π∗TM then a relative form can be extended to a form on P by assigning zero to π∗TM. For example, when P has a Riemannian metric or P is a product of X and M, then TP T P π∗TM. ≡ vert ⊕ Extended moduli space. Now, is a invariant relative closed two form on ˜[E]. (The theory of deRham model of equivariantG cohomology can be extended to theA relative 60 N-C. C. Leung case.) Then admits a -equivariant closed extension + 8. Just as before, for any 0 G D ˜[ ] and φ (X, End([E])), 8 is given by A ∈ A E ∈ i FA 8(DA)(φ)= Tr e2π φAˆ(X). ZX 1 For any non-vanishing top form µ on X, 8− (C µIE) would be preserved by . · G ˜ 1 Definition 3. µ := 8− (C µIE)/ is called the extended moduli space. M · G ˜ 2 Now, c1 descends to give a map c1 : µ H (X, C). To study the stability problem, we haveM to→ introduce polarization to specify a partic- ular direction in ˜[ ]. A E Definition 4. A polarization is a line field in ˜[ ] which is invariant under affine P A E translations. is called regular if it is transverse to the fiber of c1. Two regular polar- P 2 izations are equivalent to each other if they induce the same line field in H (X, C). It is easy to see that given a line bundle L and a connection DL on it. They induce a regular polarization in ˜[E]. Different choices of connections always give equivalent polarizations. A Remark . The word polarizatioon we used here comes from algebraic geometry which means a choice of an ample line bundle up to numerical equivalency. This should not be confused with the notion of polarizations used in geometric quantization. We fix a i line bundle L on X and choose a connection DL on L. Let ω = 2π FL be its curvature. ωn Then the top form µ = n! is non-vanishing if and only if ω is a symplectic form. We are interested in the ends of µ. It being non-empty near the + ends is related to stability properties of E as explainedM in the last section. However,∞ now the picture we have here is more geometric. All the symplectic quotients (E Lk)// ’s (k Z) are embedded inside a bigger continuous family which isA the symplectic⊗ G quotient∈ of the space of generalized connections. Moreover, “ µ c1=+ ” should be regarded as the moduli space of Hermitian Einstein connections onME. Loosely| ∞ speaking, asymptotically [2] near c1 =+ , defines a Poisson structure on ˜[ ] and restricting to each fiber on ∞ A E c1, it is a symplectic form. 5. Higer Level Chern–Simons Forms In this section, we shall show that these deteremine Chern character and Chern– Simons forms of E. t the sma time, we define∗ higher level generalzytions of ern–Simons forms. In order to simplify the diskussion below, we shall assume that the characteristic form of X is one. The author would like to thank the referee who pointed ot that this section is closely related to earlier work of Gefand and Wang. In fact, the author found out later that this is also related to certain previous work of BOtt. Inside , there is a canonical affine foliation such that any two connections D and A A DA0 are in the dame leaf if and only if DA0 = DA + αIE for some one form α on X. 1 1 Interm of an (integrable) subbundle of A, it is given by (X) End(E) = TA . [ ] ⊂ A We restrict ∗ to a differential form along the foliation (see relative forms in Sect. 3). [ ] We shall still denote this relative differential form an ( )by∗ . We are going to see that [ ] A ∗ is equivalent to the operation of taking the Chern character form of any connection. Symplectic Structures on Gauge Theory 61 [2k] For exaple, ω 0 , 2kT ∗ now gives a homomorphism: ⊂ A A V 2k [2k] 2k , γ( , (T calA)) C A× → A −→ ^ By abuse of languages, we also denote this homomorphism by [2k]. The we have i [2k] FA (DA,γ)= Tr e 2π γ. Zx [2k] 2n 2k If we regard as a map − (X)dist, the the image of DA is just given th A→ by the n k Chern character form of DA. Therefore, taking the Chern character form − [ ] of any connection is equivalent to the restriction of the differential form ∗ to the above foliation of . To recoverA secondary (or higher level generalized) characteristic forms on E,we [ ] look at the chain complex of , S ( ) and ∗ defines a chain homomorphism A ∗ A [ ] [ ] ∗ : ( ) ∗ (X). S A → Recall that an element in l( ) (a singular chain) is a finite linear combination of singular l-simplexes σ’s on SwithA complex coefficient. Where a singular l-simplex is A a smooth map from the standard l-simplex 1l to , σ : 1l . There is a boundary 2A →A homomorphism ∂ : ( ) l 1( ) with ∂ = 0 and its homology is called the Sl A →S− A singular homology of ,H∗ ( ). For completeness, let us recall the notations: A sin A l 1 = t P suml t =1 . L j j | j=j j =0 Xj th where P0 is the origin of R∞ andPi in the i standard basis vector in R∞. To define ∂, it is sufficient to know its effect on a singular l-simplex σ : 1 .Wehave: l →A l ∂σ = ( 1)iσ ∂i, − ◦ l =0 Xi i th where ∂l : δl i 1l is te i face map given by − → i=1 l i 0 ∂l j =0tjPj = + tj 1Pj. − ! =0 = +1 X Xj jXi Now we will imitate the previous constructions to obtain characteristic forms for any singular chain in , A [ ] ω ∗ : ∗(X) . l Sl → dist We no longer nedd to assume that X is even dimensional. It is enough for us to construct it for a single singular l-simplex σ : 1 ansd extend it by linearity. l →A For any such σ, we get a connection on π∗E X 1 by pulling back the Aσ 1 → × l universal connection A over E X A. We denote its curvature by Fσ. → × 62 N-C. C. Leung [ ] Definition 5. We define ∗ : (A) ∗(X) by l Sl → dist 1 [ ] 2 Fα l∗ (σ)(γ)= Tr e γ. X 1 Z × L [ ] Lemma 3. ∗ commute with the (co-)boundary operators of both complexes. That is, ∗ [ ] [ ] d l∗ = l∗ 1 ∂. ◦ − ◦ [ ] This lemma follows from the Stokes’s theorem. Next, we shall use l∗ define secondary characteristic forms (Chern–Simons forms) and their higher level generalizations. We first define a map L which is similar to the homotopy operator K : : ( ) 0 ( ) defined by one construction. (since [K, ∂]=1or 1, this was usedS ∗ toA prove→S that∗ theA singular homology of an affine space is trivial.) Now,− l+1 L : ( ). l A→SL A Y Roughly speaking, Ll is given by the complex polyhedron spanned by l + 1 points in . Notice that both K and L use strongly the affine structure on . Let us write A A σ = Ll(DA0 ,...,DAl), then L is defined by as l l σ tjPj = tjDAj . =0 ! j=o Xi X Here, l t D is a well-defined element in because of l t =1. j=o j Aj A j=0 j We can extend Ll by linearity to the free Abelian group l generated by elements l+1P AP in , that is, A l+1 Q Ll : := Z l( ) A " A# →S A Y On , there is also a boundary operator ∂ : L l 1 defined by A∗ A A →A− l i ∂ (A0,...,A:l)= ( 1) (A0,...,Aˆi,...,Al) A − =0 Xi on any generator (A0,...,Al)in L. The proof of the following lemma is straightfor- ward. A Lemma 4. (1) ∂2 ==, (2)L ∂ = ∂A L. ◦ A ◦ Now we can define a higher Chern–Simons form as follow: Definition 6. The higher Chern–Simons is the linear homomorphism [ ] ch(E)=∗ L : ∗(X)dist ∗ ◦ ∗ A∗ → such that on any generator (A0,...,A )in , it is given by l Al [ ] ch(E; A0,...,A:l)=L∗circLL(A0,...,Al). Symplectic Structures on Gauge Theory 63 In fact, ch(E; A0,...,Al) i always a smooth diffrential form on X. Noticce that ch(E; A=) is just the ordinary Chern character form of A0 and ch(E; A0,A1)isthe ordinary Chern–Simons form of the pair A0 and A1. Proposition 3. ch(E) ∂ = d ch(E). ◦ A ◦ This proposition follows from the previous two lemmas. When l = 0, this proposition implies that the ordinary Chern character form is always a closed differential form. When L = 1, the proposition says that ch/E; A1) ch(E; A0)=d?,ch(E;,A0,A1) − , which is the most important property of Chern–Simons forms as the image under d of a canonical diffferential form (by the proposition for l =2: ch(E; A1,A2) ch(E; A0,A2)+ch(E; A0,A1)=dch(E;A0,A1,A2). − Using Stokes’ theorem we have ch(E; A1,A2) ch(E; A0,A2)+ ch(E; A0,A1)=0, − ZX ZX ZX which is useful when we use X ch(E; A1,A2) as an action functional. For instance, the Chern–Simons theory over a three manifold uses such a functional and has important impacts on knot theory and threeR dimensional topology in recent years. These ch(E)’s have another symmetry property. The symmetry group of l+1 elements +1, acts on and we have the following proposition. Sl Al Proposition 4. For any s S +1 and α , we have ∈ l ∈Al s ch(E; s α)=( 1)| |ch(E; α). · − The proof of this proposition is left to the readers. When we look at the case for l =1,weget ch(E; A0,A1)= ch(E; A1,A0), − which is a well-known property for Chern–Simons forms. [ ] Hence, we have finished constructing higher Chern–Simons forms of E using ω ∗ ’s. [ ] 6. Equivariant Extensions of ∗ In the last section, we analyze [2](= ) and its -moment map 8. We showed that how they are related to the Hermitian Einstein metricG and stability when X is a symplectic [ ] manifold. In this section, we first generalize the moment map construction to all ∗ ’s [ , ] and study their equivariant extensions ∗ ∗ . Let us first review some basic materials about equivariant cohomology. (See, for example [A+B] [M+Q] for details.) We shall decribe the Cartan model here. When M admits a G (compact connected Lie Group) action, then the equivariant cohomology of M is defined as H∗ (M)=H∗(EG M), G ×G where EG is the Universal space for G. It can be computed as the cohomology of the G following differential complex on (∗(M) Symm∗(Lie G)∗) . Choose any base φ ’s ⊗ i 64 N-C. C. Leung i of Lie G and let φ ’s be its dual base. For any α ψ ∗(M) Symm∗(Lie G)∗,we define the differential D to be ⊗ ∈ ⊗ D(α ψ)=dα ψ ι α (φ ψ). ⊗ ⊗ − φi ⊗ i i X 2 It is not difficult to check that D vanishes on G invariant elements in ∗(M) ⊗ Symm∗(Lie G)∗. Therefore, it defines a differential complex on (∗(M) G ⊗ Symm∗(Lie G)∗) whose cohomology can be proved to be HG∗ (M). Although, in our situation, the group is not a compact finite dimensional Lie Group, we shall still consider the same complexG and call it the equivariant complex. Elements that are D-closed (-exact) are called equivariant closed (exact) forms. [ ] [2k] Equivariant extensions of ∗ . Now, we want to extend each as a equivariant closed form on for any k. That is, we should have [2i,2j], whereG A i+j=k [2i,2j] 2i j ( ,Symm(LieP )∗)G, ∈ A G such that [2k,0] = [2k] and for any φ Lie = 0(X, End (E)), we have ∈ G [2i,2j] [2i 2,2j+2] ιφ = d( − (φ)). Theorem 1. [2i,2j] (DA)(B1,...,B2i)(φ1,...,φj) i FA = Tr [e2π B1 B2 φ1 φ ] Aˆ(X), ··· i ··· j sym ZX where D ,B T =1(X, End (E)), φ Lie = 0(X, End (E)). A ∈A ∈ AA ∈ G Proof of Theorem. Before proving the theorem, let us first explain some notations. [2i,2j] isa2i-form on valued in the jth symmetric power of the dual of Lie algebra of and A [2i,2j]G it is also -invariant. At a point DA and 2i tangent vector B’s at DA, gives G j ∈A an element in Symm (Lie )∗. Evaluating it on the j element of Lie , then it gives us a number which we claimedG to be given by the above formula. G First, invariancy of [2i,2j] is clear from the formula. [2k,0] = [2k] can also be seen directly.G It remains to verify that j [2i,2j] ˆ [2i 2,2j+2] ιφl (φ0,...,φl,...,φj)=d( − (φ0,...,φj)), =0 Xl 0 where each φl Lie = (X, End (E)). Now, we take any B1,...,B2i 1 TA = 1(X, End (E))∈ and extendG it to vector fields over all of by parallel transport.− ∈ ThenA A [Bi,Bj] is always a zero vector field on . (Here, [ , ] means the bracket of two vector fields on the space .) Therefore, A A [2i 2,2j+2] d( − (φ0,...,φj))(B1,...,B2i 1) − 2i 1 − l+1 [2i 2,2j+2] = ( 1) Bl( − (φ0,...,φj)(B1,...,Bˆl,...,B2i 1)) − − =1 Xl 2i 1 − i l+1 FA = ( 1) Tr[e2π (DABl)B1 Bˆl B2i 1φ0 φj]symAˆ(X) − ·· ·· − ·· =1 X Xl Z Symplectic Structures on Gauge Theory 65 On the other hand, [2i,2j] ˆ ιφ (φ0,...,φl,...,φj)(B1,...,B2i 1) l − i 2π FA ˆ = Tr[e (DAφl)B1 B2i 1φ0 φl φj]symAˆ(X) ·· − ·· ·· ZX Hence, j [2i,2j] ˆ [2i 2,2j+2] ιφ (φ0,...,φl,...,φj) d( − (φ0,...,φj)) (B1,...,B2i 1) l − − =0 Xl i FA = Tr[e2π DA(B1 B2i 1φ0 φj)]symAˆ(X) ·· − ·· ZX i FA = dTr[e2π B1 B2i 1φ0 φj]symAˆ(X) ·· − ·· ZX =0 by the Stokes’s theorem. So we have proved the theorem. [2] In particular, when we look at = 0, we recover the moment map 80 as [0,2] [ ] . The extension of the total ∗ as equivariant closed form on would be [2i,2j] G [ ] A i+j n since the highest degree form in ∗ is of degree 2n. We can relax the restriction≤ i + j n and obtain an equivariant closed form of arbitrart large total degree, P ≤ [ , ] we shall denote it by ∗ ∗ . [ , ] [ ] ∗ ∗ as equivariant pushforwards. From previous section, we know that ∗ occurred as the Chern character of the virtual bundle by the family local theorem. Recall that [ ] i ∗ = Tr e2π FAˆ(X). ZX [ ] and weextended ∗ as a equivariant closed form on . Now, we are going to G i A extend the results in section two. So we want to extend Tre2π FAˆ(X)toa equivariant [ , ] G closed form on X such that we get ∗ ∗ by integrating it over X. Let us first consider×A an example where we want to find an element in 2n(X × , (Lie )∗)G such that it gives us φ by integrating it over X. This is clear that this is i A G FA (2n) given by the 2n-form Tr[e2π φ] (x) at a point (x, DA) X when evaluating 0 ∈ [ ×A, ] it on φ Lie = (X, End (E)). To generalize it to other ∗ ∗ ’s, we introduce 1 ∈ G i +1 and such that Tr e2π F Aˆ(X) would be our equivariant form on X . G ×A 0 Definition 7. Let 1 (X ,End(E))(Lie )∗ be given by ∈ ×A G 1(x, DA)(φ)=φx, where (x, D ) X and φ Lie = 0(X, End (E)). A ∈ ×A ∈ G By evaluating 1 on φ at (x, DA), we should arrive at an element in (EndE)(x, A)= EndEx, and 1 is defined such that this element is just given by φ at x. Therefore, 1 is an tautological element and independent of the variable DA . Put it another way, 1 is the pullback from the identity element in ∈A 0 0 0 (X, End (E))(Lie )∗ Map( (X, End (E)), (X, End (E))). G ≡ 66 N-C. C. Leung i 2π F+1 ˆ Now we consider TrE e A(X). In the exponential sums, we multiple forms on X by exterior product and elements in (Lie )∗ by symmetric product. Therefore, we×A have G i 2π F+1 ˆ Tr e A(X) ∗(X ,End(E))(Symm∗(Lie )∗). E ∈ ×A G i +1 Theorem 2. Tr e2π F Aˆ(X)is a equivariant closed form on X . E G ×A Proof of Theorem. To prove the theorem, we can forget the Aˆ(X) term. It is rather straight i 2π F+1 forward to check that TrE e is equivariant. To prove that it is equivariantly closed, 0 G we choose any φ Lie = (X, End (E)). Then, by Bianchi identity DF =0,we have ∈ G i i 2 F+1 2 F+1 d(Tr e π (φ)) = Tr [e π D(1φ)]sym, where D is the universal connection on E. On the other hand, we have i i 2 F+1 2 F+1 ιφTr e π = Tr [e π ιφF]sym. i +1 Therefore, in order to prove that Tr e2π F is equivariantly closed, it is enough to check that ιφF = D(1φ). At a point (x, D ) X ,φinduces the vector field (0,D φ) T( )(X ) A ∈ ×A A ∈ x,DA ×A = T X T = T X 1(X, End (E)). If we choose any other vector (v, B) x × DAA x × ∈ T( )(X ), then x,DA ×A ιφF(x, DA)(v, B) = F(x, DA)((v, B),(0,DAφ)) 2,0 1,1 =(F +F )(x, DA)((v, B),(0,DAφ)) = FA(x)(v,0)+(DAφ)(x, v) =(DAφ)(x, v). For D(1φ), we have D(1φ)(x, A)(v, B) =(DA,xφ)(x) =(DAφ)(x, v). i 2π F+1 Therefore, we have ιφF = D(1φ) and TrE e is a equivariantly closed form on . G A [ , ] Next, we shall show that the pushforward of this form gives ∗ ∗ on which i [ ] A 2π F ˆ generalizes X TrE e A(X)=∗. Theorem 3.R i 2π F+1 ˆ [ , ] TrE e A(X)=∗∗. ZX Symplectic Structures on Gauge Theory 67 i Proof of Theorem. For simplicity, we shall omit some factors of the power of 2π in the i +1 2i j calculation below. We consider the component of Tr e2π F in ( ,Symm X E A (Lie )∗). We evaluate it on 2i tangent vectors B’s on and j elements φ’s in Lie at a pointG A , R A G ∈A i 2π F+1 ˆ TrE e A(X) (DA)(B1,...,B2i)(φ1,...,φj) ZX i 2,0 1,1 (2 (F +F )+1)(x,DA) = Tr e π (B1,...,B2i)(φ1,...,φj)Aˆ(X) ZX 1,1 2k 2j i F (F ) 1 (x, DA) = Tr [e2π A (B ,...,B ) (φ ,...,φ )] Aˆ(X) (2k)! 1 2i (2j)! 1 j sym ZX i FA = Tr [e2π B1 B2 φ1 φ ] Aˆ(X) ·· i ·· j sym ZX [2i,2j] = (DA)(B1,...,B2i)(φ1,...,φj). Hence the theorem. Acknowledgement. This work is done during my visit to the Courant Institute of Mathematical Science. Courant created an excellent research environment for me and my work. 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