Summary and Brief Background

The Volume Measure for Flat Connections as Limit of the Yang–Mills Measure Ambar N. Sengupta Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA e-mail: [email protected] Abstract We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang– Mills measure defined in terms of the heat kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space. 2000MSC: 81T13 1 Introduction Summary and Brief Background We work with a closed, oriented surface Σ of genus g ≥ 2, and a compact, connected, semisimple Lie group G equipped with a biinvariant metric. The space A of all connections on a principal G– bundle over Σ has a natural symplectic structure which is preserved by the pullback action ω 7→ φ∗ ω of the group G of bundle automorphisms φ. The moment map turns out to be J : ω 7→ Ωω , where Ωω denotes the curvature of any connection ω. In this setting, the Marsden-Weinstein procedure can be carried out rigorously [19] and produces a symplectic structure Ω on the smooth strata of J −1 (0)/G. Since J −1 (0) is the set of connections with zero curvature, J −1 (0)/G 1 is the moduli space of flat connections. This space, along with the symplectic structure Ω on it, is of interest from many different points of view (as attested to by the collection [14]). To be precise, J −1 (0)/G is not, in general, a smooth manifold but there is a subset M0g (arising from points of J −1 (0) of “minimal” isotropy) which is a manifold and Ω is a symplectic structure on M0g . In this paper we • give a rigorous proof of Witten’s formula ([24, formula (4.72)]) volΩ (M0g ) = |Z(G)| vol(G)2g−2 1 α (dim α)2g−2 P (1) for the symplectic volume of the moduli space M0g of flat connections, for a compact, semisimple gauge group G, over a closed oriented surface of genus g ≥ 2 (terminology, notation, and hypotheses are explained in detail later in this introduction; note also that M0g is actually a subset of the full moduli space of flat connections), and • prove Forman’s theorem [8, Theorem 1] that Wilson loop expectations in the quantum Yang-Mills theory converges to the corresponding symplectic integrals. We will keep things as self-contained as reasonably possible and no knowledge of the moduli space of flat connections is actually necessary to understand the technical content of this paper. Indeed we shall work with a standard realization of M0g as a finite-dimensional manifold. Our proof has two main ingredients: (i) a determinant identity (Proposition 1), (ii) careful accounting of certain dense subsets of full measure in the moduli space M0g where nice properties hold. Witten [24] determined the symplectic volume of the moduli space of flat connections in several different ways. One way involves the limit of the partition function of the quantum Yang-Mills theory over the surface. It is this approach, involving the heat kernel on the structure (gauge) group, that we follow here. Forman used this approach and Witten’s volume formula to prove the convergence of the Wilson loop expectations. Liu [15, 16] used Forman’s approach along with other ideas to study the symplectic volume and related integrals. We refer to the collection [23], and the bibliography therein, for other works concerning the symplectics of the moduli space of flat connections. 2 In the present paper we restrict our attention to the moduli space of flat connections without distinguishing between bundles of different topological type. The methods used here should extend to bundles of specified topology and also to the case of surfaces with boundary but this is not carried out here. The limiting result we prove can be reformulated to give the limit of the discrete Yang-Mills measure for cell-complexes but we do not describe how this is done and deal only with the case where the surface of genus g is obtained by appropriate pasting of 1–cells on the boundary of a single 2–cell. (The method is described in the proof of [18, Lemma 8.5].) We use, in several places, the existence of appropriate dense subsets. We give either proofs or exact references to proofs, when we state or use such density results. It is widespread practice in the literature on this subject to state or use without clear justification results concerning certain subsets of the moduli space of flat connections which are assumed to be dense and, implicitly, of full measure. But much of the technical difficulty in proving the volume formula lies in taking proper account of such subsets (which need also to be of full measure) and so we have strived to be careful about this issue. (I am thankful to the anonymous referee for stressing the necessity of having sets of full measure.) Statement of Results We work with a compact, connected, semisimple Lie group G, whose Lie algebra LG is equipped with an Ad–invariant inner-product. The heat kernel on G is a function Qt (x), for t > 0 and x ∈ G, satisfyt (x) ing the heat equation ∂Q∂t = 21 ∆G QRt (x), where ∆G is the Laplacian on G, and the initial condition limt↓0 G f (x)Qt (x) dx = f (e) for every continuous function f on G, where e is the identity in G and dx is R Haar measure on G of unit total mass G dx = 1. For any integer g ≥ 1, let Kg : G2g → G be the product commutator map given by −1 −1 −1 Kg : G2g → G : (a1 , b1 , ..., ag , bg ) 7→ b−1 g ag bg ag ...b1 a1 b1 a1 (2) The subset Kg−1 (e), where e is the identity in G, of G2g will be of special interest to us. The group G acts by conjugation on G2g . If A ⊂ G2g is preserved by this action, denote by A0 the set of all points on A where the isotropy is Z(G), the center of G. The quotient Mg = Kg−1 (e)/G 3 (3) is identifiable in a standard way with the moduli space of flat G– connections over a closed, connected, oriented two–dimensional manifold of genus g, but we shall not need any detail of this. The subset M0g = Kg−1 (e)0 /G (4) (when non-empty) has a manifold structure and on M0g there is a natural symplectic form Ω. Let volΩ be the volume form corresponding to this symplectic structure; i.e. volΩ = Our main result is: 1 d/2 , d! Ω where d = dim M0g . Theorem 1 Suppose g ≥ 2. Let f be a continuous G–conjugationinvariant function on G2g , and f˜ the function induced on M0g = Kg−1 (e)0 /G. Then vol(G)2−2g f (x)Qt (Kg (x)) dx = lim t↓0 G2g |Z(G)| Z Z M0g f˜ dvolΩ (5) where the integration on the left is with respect to unit mass Haar measure, the integration on the right is with respect to the symplectic volume measure, |Z(G)| is the number of elements in the center Z(G) of G, and vol(G) is the volume of G with respect to the Riemannian structure on G given by the Ad–invariant metric on LG. The integral on the left in (5) arises from integration with respect to the Yang–Mills measure in the euclidean quantum field theory of the Yang–Mills field on a compact oriented surface of genus g. We shall not need this; for more details see [17] or the review [23]. Setting f = 1 leads, after some computation (detailed in (42) below) to Witten’s formula ([24, formula (4.72)]) for the symplectic volume of the moduli space M0g : volΩ (M0g ) = |Z(G)| vol(G)2g−2 X α 1 (dim α)2g−2 (6) where α runs over all irreducible representations of G. In essence, equation (5) for f = 1 is one of the approaches used by Witten in [24] to determine the symplectic volume of moduli space. For general f , Theorem 1 was proven by Forman in [8] using Witten’s volume formula (in fact, this is also what we shall do, but we shall also prove the volume formula (6)). For G = SU (2) and G = SO(3), the result was proven in [21]. 4 What we shall prove in this paper is actually the limit formula : f (x) dvol(x), t↓0 |dKg (x)∗ | (7) for any continuous function f on G2g , where the linear map dKg (x)∗ : LG → (LG)2g is the adjoint of the derivative (LG)2g → LG : H 7→ Kg (x)−1 Kg′ (x)(xH), and dvol is Riemannian volume measure on the submanifold Kg−1 (e)0 ⊂ G2g . The known result (34) then implies (5). The main difficulty in proving (7) lies in taking proper care of the critical points of Kg and it is to this technical issue that most of the work in this paper is devoted. Now we give a quick definition of the symplectic structure Ω. It will be useful to think of G2r as a subset of G4r via the map lim Z G2g 1−2g f (x)Qt (Kg (x)) dx = vol(G) Z Kg−1 (e)0 Φ : G2r → G4r −1 −1 −1 : (a1 , b1 , ..., ar , br ) 7→ (a1 , b1 , a−1 1 , b1 , ..., ar , br , ar , br ) For any 1 ≤ i ≤ 4r, and x ∈ G4r , we write fi = Ad(xi−1 ...x1 ) : LG → LG with f1 being the identity map. Next let Ω̃ be the 2–form on G4r specified by Ω̃x (xH, xH ′ ) = 1 X −1 −1 ǫij hfi−1 Hi , fj−1 Hj′ iLG 2 1≤i,j≤4r ′ ) ∈ (LG)4r , and ǫ equals 1 where H = (H1 , ..., H4r ), H ′ = (H1′ , .., H4r ij for i < j, equals −1 if i > j, and is 0 if i = j. Finally, define Ω = Φ∗ Ω̃ a 2–form on G2r (8) The quotient space M0g = Kg−1 (e)0 /G, if non-empty, has a unique smooth manifold structure for which the quotient map q : Kg−1 (e)0 → Kg−1 (e)0 /G is a submersion. The restriction of Ω to Kg−1 (e)0 drops down to a 2–form Ω on M0g = Kg−1 (e)0 /G: q ∗ Ω = Ω|Kg−1 (e)0 (9) It was shown in [11, 12] (with more details in [19]) that Ω is a symplectic form on M0g , and, as proved in [19], is induced Marsden-Weinsteinstyle from the Atiyah-Bott symplectic structure [1] on the space of all connections. 5 Other Remarks We take this opportunity to correct in this paper Corollary 3.2 and Lemma 4.4(ii) of [22]. The correct forms involve sets of full measure and we have stated the correct result here as Proposition 7. It is this form, using sets of full measure, which is useful both for the results of [22] and for our results here. I am very grateful to an anonymous referee for pointing out this error which was present in an earlier version of this paper. It should be noted that what we compute is the volume of M0g and not of the full moduli space Mg . The latter is not, in general, a smooth manifold but is believed to be the union of symplectic manifolds, called symplectic strata, of different dimensions, these manifolds corresponding to the different isotropy groups for the action of G on Kg−1 (e). Volumes of all the strata have been calculated for G = SU (2) and G = SO(3) in [21]. 2 Summary of technical tools In this section we collect together some results, proven elsewhere, which we will need. 2.1 A determinant identity Let V and W be finite dimensional real inner-product spaces, and A : V → W a linear map. If A : V → W (6= 0) is a linear isomorphism onto its image A(V ), then by the determinant of A we shall mean det A = the determinant of a matrix of A relative (10) to orthonormal bases in V and A(V ). If ker(A) 6= {0}, or if V = {0}, then we define det(A) = 0. Thus det A is determined up to a sign ambiguity, and | det A| is independent of the choice of bases. Let A : V → W and B : W → Z be linear maps between finite dimensional inner-product spaces. If A is an isomorphism onto W or if B is an isometry (in which case | det B| = 1 unless W = {0}) then | det(BA)| = | det(B)|| det(A)| Consideration of matrices shows that   det A|(ker A)⊥ = det (A∗ |Im A) 6 (11) The following is a slightly sharpened form of Proposition 2.1 of [22]. (It is this sharper statement which was used in [22].) Proposition 1 Let X, Y (6= {0}) be finite dimensional vector spaces equipped with inner-products, and let V be a subspace of X, and Z a subspace of Y . Let L1 : X → Z and L2 : X → Y be linear maps such that L1 |V ⊥ = 0 and L2 |V = 0 (12) Let L = L1 + L2 , (13) and N = ker(L). Then : (i) there exists a I : N ⊕ N⊥ → V ⊕ V ⊥ unitary isomorphism and a linear isomorphism J : Z ⊕ Y → Z ⊕ Y, with | det J| = 1, such that   J (L1 |V ) ⊕ (L2 |V ⊥ ) I = (L1 |N ) ⊕ (L|N ⊥ ) (14) (ii) The maps L1 |V : V → Z and L2 |V ⊥ : V ⊥ → Y are both surjective if and only if L1 |N : N → Z and L|N ⊥ : N ⊥ → Y are both surjective. (iii) The following equality of determinants holds : | det L∗1 || det L∗2 | = | det(L1 |N )∗ || det L∗ | (15) Here L∗1 : Z → X, L∗2 : Y → X, (L1 |N )∗ : Z → N and L∗ : Y → X. Since the statement is slightly sharper than the one in [22] (where this sharper form is used) we include the full proof, though it is almost identical to that given in [22]. Proof. (i) Let   I : N ⊕ N ⊥ → V ⊕ V ⊥ : (a, b) 7→ (a + b)V , (a + b)V ⊥ , 7 wherein the subscripts signify orthogonal projections onto the corresponding subspaces. Since N ⊕ N ⊥ ≃ X ≃ V ⊕ V ⊥ isometrically, by means of (x, y) 7→ x + y, I corresponds to the identity map on X and is thus a unitary isomorphism. Let Ll : Y → N ⊥ ⊂ X, be a linear left-inverse for the injective map L|N ⊥ ; thus Ll L(b) = b for every b ∈ N ⊥ . Next define J = J2 J1 : Z ⊕ Y → Z ⊕ Y, where J1 : Z ⊕ Y → Z ⊕ Y : (a, b) 7→ J1 (a, b) = (a, a + b) J2 : Z ⊕ Y → Z ⊕ Y : (a, b) 7→ J2 (a, b) = (a − L1 Ll b, b). It is clear that both J1 and J2 are injective. Moreover, they are also surjective, because for any (z, y) ∈ Z ⊕ Y , J1 (z, y − z) = (z, y) and J2 (z +L1 Ll y, y) = (z, y); note that z +L1 Ll y ∈ Z because L1 (X) ⊂ Z. So J1 and J2 are isomorphisms and hence so is J. By considering matrix representations for J1 and J2 , we have | det J1 | = | det J2 | = 1, and so | det J| = | det J2 | | det J1 | = 1 (16) For any (a, b) ∈ N ⊕ N ⊥ , we have :    J (L1 |V ) ⊕ (L2 |V ⊥ ) I(a, b) = J L1 (a + b)V , L2 (a + b)V ⊥   = J L1 (a + b), L2 (a + b)   = J2 L1 (a + b), L(a + b)   = J2 L1 (a + b), L(b) = =     L1 (a + b) − L1 Ll L(b), L(b)  L1 (a), L(b) . This proves equation (14), and part (i). (ii) follows directly from (i). (iii) Since L1 |V ⊥ = 0 it follows that L∗1 (Z) ⊂ V . Similarly, ∗ L2 (Y ) ⊂ V ⊥ and L∗ (Y ) ⊂ N ⊥ . So, with appropriately restricted codomains (for instance we are taking L∗1 : Z → V instead of L∗1 : Z → X) : 8 (L1 |V )∗ = L∗1 , (L2 |V ⊥ )∗ = L∗2 , and (L|N ⊥ )∗ = L∗ In view of this, we may take adjoints in equation (14) to obtain : I ∗ (L∗1 ⊕ L∗2 )J ∗ = (L1 |N )∗ ⊕ L∗ , as maps Z ⊕ Y → N ⊕ N ⊥ wherein again some of the operators are taken with restricted codomains. Taking determinants (which, by our definition, is not affected by restriction of codomains), and using the determinant of products given in (11), and the fact that | det J| equals 1, we obtain the determinant formula (15). QED We will use the preceding proposition in a specific context. Let G be a compact, connected, semisimple Lie group with Lie algebra LG equipped with an Ad–invariant metric. Let g1 and g2 be positive integers, and g = g1 + g2 . We have the product commutator maps Kgi : G2gi → G and Kg : G2g → G specified through (2). Let xi ∈ G2gi and x = (x1 , x2 ). Define C1 : G2g → G : (x1 , x2 ) 7→ Kg1 (x1 ), C2 : G2g → G : (x1 , x2 ) 7→ Kg2 (x2 ) Then Kg (x) = C2 (x)C1 (x) and we have the derivative maps Kg (x)−1 dKg (x) : Tx G2g → LG, Ci (x)−1 dCi (x) : Tx G2g → LG which are related by Kg (x)−1 dKg (x) = C1 (x)−1 dC1 (x) + Ad(C1 (x)−1 )C2 (x)−1 dC2 (x) We will apply Proposition 1 with X = (LG)2g ≃ (LG)2g1 ⊕ (LG)2g2 , V = (LG)2g1 ⊕ 0 and L1 = C1 (x)−1 dC1 (x), L2 = Ad(C1 (x)−1 )C2 (x)−1 dC2 (x) Specializing Proposition 1 to this situation gives us the following result: Proposition 2 Let x = (x1 , x2 ) ∈ G2g1 × G2g2 . Then Kgi is submersive at xi , for both i = 1 and i = 2, if and only if Kg is submersive at x and C1 |Kg−1 (e) : Kg−1 (e) → G is submersive at x. Furthermore, | det dKg (x)∗ | det [dC1 (x)| ker dKg (x)]∗ = | det dC1 (x)∗ | | det dC2 (x)∗ | = | det dKg1 (x1 )∗ | | det dKg2 (x2 )∗ | (17) 9 2.2 A disintegration formula The following disintegration formula, proved in Proposition 3.1 of [22] will be useful. (The formula (19) is proved for vastly more general K by Federer [7]). Proposition 3 Let K : M → N be a smooth mapping between Riemannian manifolds. Let NK = K (M \ CK ), where CK is the set of points where K is not submersive, i.e. the rank of dK is less than dim N . Assume that CK 6= M . Suppose φ is a continuous function of compact support on M . Let vol denote Riemannian volume measure. (For example, on the submanifold K −1 (h) \ CK ⊂ M , for h ∈ NK , which is given the metric induced from M . If dim K −1 (h) = 0, the Riemannian volume is understood to be counting measure.) If φ vanishes in a neighborhood of CK , then h 7→ Z K −1 (h)\CK φ dvol is continuous on NK (18) and Z M φ dvol = Z NK "Z K −1 (h)\CK # φ dvol | det(dK)∗ | dvol(h) (19) In our application, every open subset U of M can be expressed as the union of a sequence of open subsets Un with compact closure, and there is a sequence of continuous functions 0 ≤ φ1 ≤ φ2 ≤ ... ≤ φn ↑ 1U , where φn is 0 outside Un . Then, for f any continuous non– negative function on M , using φn f in place of f in (19), and letting n → ∞, monotone convergence shows that (19) holds for f 1U in place of φ, if U is any open subset of M \ CK . In particular, (19) holds for 1U in place of φ and hence, if vol(M ) < ∞, also for 1U −V for any open sets U, V ⊂ M \ CK . 2.3 Some dense sets of full measure We shall describe some useful subsets which are dense and of full measure in appropriate sets. The group G acts by conjugation on G2r : G × G2r → G2r : (h, x) 7→ hxh−1 = (hx1 h−1 , ..., hx2r h−1 ) where x = (x1 , ..., x2r ). 10 (20) Semisimplicity of the compact group G (i.e. that the center G is finite) is important in the following. We equip G with an Ad–invariant metric. Proposition 4 Let G be a compact, connected, semisimple Lie group and T a maximal torus in G, acting on G by conjugation. Then the set of points in G where the isotropy is Z(G) is an open set of full measure. By “full measure” we mean a measurable set whose complement has measure zero. In particular, an open set of full measure is automatically dense since the measures under consideration assign positive measure to non-empty open sets. Proof. Under the adjoint action of the compact abelian group T , the Lie algebra LG splits up as a direct sum of LT and two-dimensional spaces R1 , ..., Rk on each of which T acts by ‘rotations.’ The compact Lie group G, equipped with the Ad-invariant metric on the Lie algebra LG, is a complete Riemannian manifold. We shall use a result concerning the exponential map for such manifolds. For each unit vector u ∈ LG let δ(u) be the infimum of all real numbers r > 0 such that the distance of exp(ru) from the identity e is r. Now let B be the subset of LG consisting of 0 and all v 6= 0 such that |v| < δ(v/|v|), and let W = exp(B). Then it is known (see, for instance, Chavel [5, Theorem 3.2 and Proposition 3.1]) that B is open, W is an open set of full measure in G, and B → W : v 7→ exp(v) is a diffeomorphism onto W (21) For any t ∈ T , the conjugation map G → G : x 7→ txt−1 is an isometric isomorphism and so the function δ is invariant under the adjoint action of T on LG. Therefore, Ad(t)B = B for all t ∈ B. Let W 0 = exp(W ′ ) (22) where W ′ is the subset of B consisting of all points of the form v = vLT + v1 + · · · + vk , with vLT ∈ LT and each vi ∈ Ri being non-zero: W ′ = {vLT + v1 + · · · + vk | vLT ∈ LT , each vi ∈ Ri and vi 6= 0} (23) Suppose t ∈ T commutes with x ∈ W 0 . We know that x = exp(v) for a unique v ∈ W ′ . Moreover, since exp is injective on B and Ad(t)v ∈ B, the relation txt−1 = x implies that Ad(t)v = v. Since 11 Ad(t) preserves each subspace Ri , whose direct sum along with LT is LG, it follows that Ad(t)vi = vi for each i ∈ {1, ..., k}. Since T acts on the two-dimensional spaces Ri by rotations and fixes the nonzero vector vi , this means that in fact Ad(t) is actually the identity on each Ri . Therefore, Ad(t) is the identity on all of LG and so t ∈ Z(G). Thus the T –isotropy at each point of W 0 is Z(G). Now W ′ is clearly an open subset of full measure in B, and so, since exp is a diffeomorphism on B, it follows that W 0 is a subset of full measure in W . Since W is of full measure in G, we conclude that W 0 is of full measure in G. By a general result of transformation group theory (Bredon [3, Theorem 4.3.1 and Corollary 6.2.5], Kawakubo [10, Theorem 4.27], Bourbaki [2, IX.96,No.4,Thèoréme 2]) for compact Lie groups acting on connected manifolds, the set of points of minimal isotropy is (dense and) open in the whole space. QED We apply this to show that the conjugation action of G on Gr has minimal isotropy Z(G) on a set of full measure: Proposition 5 Let G be a compact, connected, semisimple Lie group, and k any integer ≥ 2. For the conjugation action of G on Gk , the subset on which the isotropy group is Z(G) is a dense open set of full measure in G. Proof. As noted earlier, the set of points of minimal isotropy (for a compact Lie group acting on a connected manifold) is open, being a consequence of a general result on transformation groups (Bourbaki [2, IX.96,No.4,Thèoréme 2]). So we focus on the measure theoretic issue. Since Gk = G2 × Gk−2 , it will suffice to prove the result for k = 2. Let U be the subset of G2 consisting of all points where the isotropy group of the conjugation action of G is Z(G). The subset G0 of G which consists of points which generate maximal tori is of full measure in G (see, for example, Bröcker and tom Dieck [4, Theorem IV.2.11(ii)]). If x ∈ G0 then the preceding lemma implies that for almost every y ∈ G the isotropy group at (x, y) is Z(G) (any element which commutes with x lies in the maximal torus generated by x; see, for example, [4, Theorem IV.2.3(i)]). So, by Fubini’s theorem, (G0 × G) ∩ U is of full measure in G2 . So U is of full measure in G2 . QED The preceding result has the following consequence: 12 Proposition 6 For any integer r ≥ 1 and compact, connected semisimple group G, the critical points of the mapping Kr : G2r → G form a set of measure 0 in G2r . Proof. There is a remarkable relationship, stated below in (32), between the the derivative dKr and the istropy of the conjugation action of G on G2r . The relation (32) implies that at any critical point x of Kr the isotropy group {g ∈ G : gxg −1 = x} has a non-trivial Lie algebra, and so, in particular, the isotropy group is not equal to Z(G). The preceding proposition then implies that the set of critical points of Kr is contained in a set of measure 0. QED Next we show that almost every point on almost every level set Kr−1 (h) is a point of isotropy Z(G). For this we use the important fact that the product commutator map Kr : G2r → G is surjective. This is proven in [20, Proposition 4.2.4] and uses semisimplicity of G (as I learnt later, this result also appears in Bourbaki [2, Lie IX.33 Corollaire to Proposition 10]). Proposition 7 For any integer r ≥ 1, let Ur0 be the subset of G2r where the isotropy of the conjugation action of G is Z(G). Then for almost every h ∈ G the set Kr−1 (h) ∩ Ur0 is of full measure in Kr−1 (h). Proof. Let Ur be the set of all non-critical points of Kr . Then Ur0 ⊂ Ur (24) because of the striking relation (32) between the behavior of dKr and the isotropy of the conjugation action. The mapping Kr |Ur : Ur → G is an open mapping. We have the co-area/disintegration formula giving the volume of any open set A ⊂ Ur : vol(A) = Z Kr (Ur ) "Z Kr−1 (h)∩A # dvol dvol | det(dKr )∗ | (25) where vol always denotes Riemannian volume arising, in our situation, from any choice of Ad-invariant metric on G. Since vol(G2r ) < ∞, the formula (25) holds when A is the difference of open sets. Taking A to be the set Ur − Ur0 of measure 0, it follows that for almost every h ∈ Kr (Ur ) the set Kr−1 (h)∩Ur0 is of full measure in Kr−1 (h)∩Ur . Now Kr (Ur ) contains all regular values of Kr : here we use the surjectivity of Kr which assures that every regular value of Kr is in fact a value of Kr . Moreover, by Sard’s theorem, the set of all regular values of Kr is 13 a set of full measure in G, and, furthermore, note that Kr−1 (h) ⊂ Ur for any regular value h of Kr . Thus almost every point h ∈ G satisfies the condition that Kr−1 (h)∩Ur0 is of full measure in Kr−1 (h)∩Ur = Kr−1 (h). QED Using the notation from the preceding result we also have: Proposition 8 For g1 , g2 ≥ 1 and g = g1 + g2 , let Kg−1 (e)0,0 = (Ug01 × Ug02 ) ∩ Kg−1 (e) (26) and let Ci : G2g1 × G2g2 → G : (x1 , x2 ) 7→ Kgi (xi ), for i ∈ {1, 2}. Then Kg−1 (e)0,0 is not empty and the set def U12 = C1 (Kg−1 (e)0,0 ) = C2 (Kg−1 (e)0,0 ) = Kg1 (Ug01 ) ∩ Kg2 (Ug02 ) (27) is a dense open subset of full measure in G. Proof. Let Di be the set of all regular values of Kgi . If h ∈ Di is in the complement of Kgi (Ug0i ) then Kg−1 (h) ∩ Ug0i = ∅, while, by surjectivity i −1 of Kgi , the level set Kgi (h) is a non-empty closed submanifold of G2gi and so has positive volume. So by the preceding result, the set of all such elements h has measure 0. Thus Kgi (Ug0i ) ∩ Di is of full measure in Di . By Sard’s theorem, Di is a set of full measure in G, and so Kgi (Ug0i ) has full measure in G. Since Kgi is submersive on Ug0i it follows that the image Kgi (Ug0i ) is an open subset of G. So the sets Kgi (Ug0i ), for ∈ {1, 2}, are open sets of full measure on G and hence so is their intersection U = Kg1 (Ug01 ) ∩ Kg2 (Ug02 ) The relation Kr (br , ar , ..., b1 , a1 ) = Kr (a1 , b1 , ..., ar , br )−1 (28) shows that Kr (Ur0 ) = Kr (Ur0 )−1 and so U = U −1 Let h ∈ U . Then there is, for i = 1, 2, an xi ∈ Ug0i with Kg1 (x1 ) = h and Kg2 (x2 ) = h−1 . Then x = (x1 , x2 ) is a point in Kg−1 (e)0,0 whose 14 image under C1 is h and whose image under C2 is h−1 . This, together with the inversion property (28) implies Ci (Kg−1 (e)0,0 ) ⊃ U for i = 1, 2. Conversely, suppose h ∈ C1 (Kg−1 (e)0,0 ). This means that there is a point (x1 , x2 ) ∈ Kg−1 (e)0,0 with C1 (x1 , x2 ) = h. Since Kg (x1 , x2 ) = C2 (x2 )C1 (x1 ), it follows that C2 (x2 ) = h−1 . The condition (x1 , x2 ) ∈ Kg−1 (e)0,0 says also that xi ∈ Ug0i , for i = 1, 2, and so h ∈ Kg1 (Ug01 ) and h−1 ∈ Kg2 (Ug02 ). The inversion property (28) then implies that h ∈ U . The argument works if we start with h ∈ C2 (Kg−2 (e)0,0 ). QED 2.4 Facts about Ω and Ω The compact, semisimple group G acts by conjugation on G2g . Let Ug0 be the set of all points where the isotropy is Z(G). Clearly, this is carried into itself by the conjugation action. Moreover, Ug0 is a dense open subset of full measure in G2g , as we have shown. Let Kg−1 (e)0 = Ug0 ∩ Kg−1 (e), the set of points on Kg−1 (e) where the isotropy group of the conjugation action of G is Z(G), and assume that it is non-empty (Proposition 8 implies that this is so when g ≥ 2). Let Kg−1 (e)0 be the set of points x in Kg−1 (e) where Kg is submersive, i.e. dKg (x) : Tx G2g → TKg (x) G is surjective. It is a consequence of Theorem 2(v) below that Kg−1 (e)0 is a subset of Kg−1 (e)0 . Then Kg−1 (e)0 , being a level set of a smooth submersion Kg |Ug0 : 0 Ug → G, is a smooth submanifold of G2g . The quotient M0g = Kg−1 (e)0 /G being a quotient of a smooth manifold by a compact Lie group, having the same isotropy subgroup Z(G) everywhere, is a smooth manifold (sections 16.14.1 and 16.10.3 in [6]). The conjugation action of the group G on G2r , gives for any x = (x1 , ..., x2r ) ∈ G2r the orbit map γx : G → G2r : h 7→ hxh−1 = (hx1 h−1 , ..., hx2r h−1 ) (29) The derivative at x of the product commutator map Kg : G2g → G is, technically, a map Tx G2g → TKg (x) G, but by means of appropriate left translations to the identity we shall sometimes view it as a map 15 (LG)2g → LG and sometimes as (LG)2g → TKg (x) G. Its adjoint, relative to the given Ad–invariant metric on LG, is then a linear map dKg (x)∗ : LG → (LG)2g (30) Recall from (8) the 2–form Ω on G2g . We summarize some facts about Ω, γ, and Kg : Theorem 2 Let g ≥ 1 and assume that Kg−1 (e)0 is not empty. Then: (i) there is a unique smooth manifold structure on M0g = Kg−1 (e)0 /G such that the quotient map q : Kg−1 (e)0 → Kg−1 (e)0 /G is a submersion (ii) there is a unique smooth 2–form Ω on Kg−1 (e)0 /G such that q ∗ (Ω) = Ω|Kg−1 (e)0 (iii) the 2–form Ω is closed and non–degenerate, i.e. it is symplectic on M0g (Proposition IV.E in [11] and [12, Proposition 3.3]) (iv) Ω satisfies the “moment map” formula Ωx (xY, γx′ H) = hY, dKg (x)∗ Hi(LG)2g (31) for all x ∈ Kg−1 (e), H ∈ LG and Y ∈ (LG)2g ([11, Proposition IV.G]) (v) for any x = (x1 , ..., x2g ) ∈ G2g , the kernel of γx′ : LG → (LG)2g equals the kernel of dKg (x)∗ : LG → (LG)2g : ker γx′ = ker dKg (x)∗ = {H ∈ LG : Ad(x1 )H = · · · = Ad(x2g )H = H} (32) ([11, Proposition IV.C] and also in [9]) (vi) If x ∈ Kg−1 (e)0 then |Pfaff(Ωq(x) )| = | det γx′ | | det dKg (x)∗ | (33) where the Pfaffian is, as usual, the square root of the determinant of the matrix of Ωq(x) relative to an orthonormal basis ([12, Proposition 3.3]) (vii) If f is a measurable function on Kg−1 (e)0 , invariant under the conjugation action of G, and f˜ the induced function on M0g = Kg−1 (e)0 /G then Z M0g f˜ dvolΩ = 1 vol(G/Z(G)) 16 Z Kg−1 (e)0 f dvol, | det dKg∗ | (34) whenever either side is defined, where volΩ is symplectic volume for the symplectic structure Ω, while vol by itself always denotes Riemannian volume. (Essentially [12, Proposition 3.5] or by part (vi) and [22, Lemma 3.4].) 2.5 An application We shall “prefabricate” a result that will go into the proof of Theorem 1. Let g1 , g2 be positive integers and g = g1 + g2 . Let Kg−1 (e)0,0 the subset of Kg−1 (e) consisting of all points (x1 , x2 ) ∈ G2g1 × G2g2 such that the isotropy of the G–conjugation action on Ggi is Z(G) at xi , for i = 1, 2. We have the maps Ci : G2g → G specified by C1 (x1 , x2 ) = Kg1 (x1 ), C2 (x1 , x2 ) = Kg2 (x2 ) Recall from Proposition 8 (equation (27)) that def U12 = C1 (Kg−1 (e)0,0 ) = C2 (Kg−1 (e)0,0 ) = Kg1 (Ug01 ) ∩ Kg2 (Ug02 ) is an open subset of full measure in G. Let Di be the set of all regular values of Kgi . By Sard’s theorem, Di is a subset of full measure in G. The maps Kgi being surjective, Di is contained in the image of Kgi . (The set Di is also open in G.) The inversion relation Kr (br , ar , ..., b1 , a1 ) = Kr (a1 , b1 , ..., ar , br )−1 (35) implies that Di = Di−1 . Therefore, def D = D1 ∩ D2−1 (36) is also a subset of full measure in G. Proposition 9 The following disintegration formula holds: Z Kg−1 (e)0,0 dvol = | det dKg∗ | vol(G) Z "Z D Kg−1 (h) 1 (37) dvol(x1 ) | det dKg1 (x1 )∗ | # "Z Kg−1 (h−1 ) 2 # dvol(x2 ) dh | det dKg2 (x2 )∗ | where dh is the unit-mass Haar measure on G and vol(G) is the volume of G with respect to the given Ad-invariant metric on the Lie algebra of G. 17 Proof. Let Ur0 be the subset of G2r consisting of all points where the isotropy of the conjugation action of G is Z(G). Then Ur0 is a non-empty (in fact, dense) open subset of G2r (this is a special case of a general theorem on group actions: Bredon [3, Theorem 4.3.1 and Corollary 6.2.5], Kawakubo [10, Theorem 4.27], Bourbaki [2, IX.96, No.4, Thèoréme 2]). By Theorem 2(v), the map Kg : G2g → G is a submersion at every point in Ug01 × Ug02 , and so, Kg−1 (e)0,0 , being a level set of a submersion, is a smooth submanifold of G2g . From Proposition 2 it follows that C1 |Kg−1 (e)0,0 is submersive at every point. Therefore, by the disintegration formula in Proposition 3, we have Z Kg−1 (e)0,0 dvol = | det dKg∗ | vol(G) Z U12 (38) "Z C1−1 (h)∩Kg−1 (e)0,0 # dvol dh | det dKg∗ | | det(dC1 | ker dKg )∗ | Next we use the determinant identity from Proposition 2 to obtain: Z Kg−1 (e)0,0 dvol = | det dKg∗ | vol(G) Z U12 "Z (39) C1−1 (h)∩Kg−1 (e)0,0 # dvol(x1 , x2 ) dh | det dKg1 (x1 )∗ | | det dKg2 (x2 )∗ | Now the identity map (h−1 )0 : (x1 , x2 ) 7→ (x1 , x2 ) (h)0 × Kg−1 C1−1 (h) ∩ Kg−1 (e)0,0 → Kg−1 2 1 is an isometry (the metric on the left is inherited from that on G2g ). So we have Z Kg−1 (e)0,0 dvol = | det dKg∗ | vol(G) Z U12 "Z Kg−1 (h)0 1 (40) dvol(x1 ) | det dKg1 (x1 )∗ | # "Z Kg−1 (h−1 )0 2 Since both U12 and D are subsets of full measure in G, the integration R · · · dh above can be replaced by · U12 D · · dh. Finally, by Proposition 0 is of full measure in K −1 (c) for almost every c, and 7, the set Kg−1 (c) gi i so we obtain the desired formula (37). QED R 18 # dvol(x2 ) dh | det dKg2 (x2 )∗ | 2.6 A heat-kernel integral and its limit If X1 , ...., Xd is an orthonormal basis of the Lie algebra of G, and α P an irreducible representation of G then di=1 α∗ (Xi )2 is of the form −Cα I, where Cα is a scalar (‘Casimir’ ) and I is the identity operator on the representation space of α. The heat kernel Qt has a standard character expansion: Qt (x) = X (dim α)e−Cα t/2 χα (x), α where χα is the character of the representation α. The following is a very useful formula: Z −1 −1 −1 Qt (hb−1 g ag bg ag . . . b1 a1 b1 a1 ) da1 . . . dbg = X e−Cα t/2 χα (h) , (dim α)2g−1 (41) where the sum is over all inequivalent irreducible representations α of G. This can be verified using : (i) the identity (see Ex 4.17.3 in Bröcker and tom Dieck [4]) G2g α Z G χα (aba−1 c) da = (dim α)−1 χα (b)χα (c), (ii) repeated application of standard convolution properties of characters, and (iii) commuting integral and a series sum. Integral and sum can be commuted because X e−Cα t/2 (dim α) α Z |χα (· · ·)| d · · · ≤ X e−Cα t/2 (dim α)2 = Qt (e) < ∞ α Formula (41) is due to Witten ([24, equation (2.51)]) who determined it in his exact evaluation of the partition function of two– dimensional quantum Yang–Mills theory (the heat-kernel was not used explicitly in [24]). P It is known (Knapp[13, Lemma 10.3]) that α (dim1 α)k < ∞ for k ≥ 2. So, for g ≥ 2, using dominated convergence in (41) gives lim t↓0 Z G2g −1 −1 −1 Qt (hb−1 g ag bg ag . . . b1 a1 b1 a1 ) da1 . . . dbg = X α χα (h) , (dim α)2g−1 (42) Proposition 10 The limit formula (42) continues to hold, with the P limit limt↓0 and the sum α being both in the L2 (G, dh)–sense. 19 Proof. Let k = 2g − 1, and dα = dim α. Then X e−Cα t α dkα χα − X 1 α dkα χα 2 L2 (G) X (e−tCα − 1)2 = d2k α α which,for t > 0, is bounded, term by term, by the convergent series 1 α d2k . QED P α 3 Evaluation of Limits With notation and assumptions as before, let Kg−1 (h)0 def = the set of all non-critical points of Kg : G2g → G which lie on (43) Kg−1 (h) for any h ∈ G. A point x ∈ G2g is a non-critical point of Kg if and only if the isotropy group at x of the conjugation action of G on G2g is discrete, an observation immediate from Theorem 2 (v). Therefore, in particular, Kg−1 (e)0 ⊂ Kg−1 (e)0 (44) If g ≥ 2 then, by Proposition 8 (also Proposition III B of [11]), Kg−1 (e)0 is not empty and hence also Kg−1 (e)0 6= ∅. As a consequence of the disintegration formula, we have the following result (mentioned in [11, section IV]): Lemma 1 Suppose g is an integer ≥ 2. Let f be a continuous function on G2g which is 0 in a neighborhood of the critical points of Kg . Then lim t↓0 Z G2g f (x)Qt (Kg (x)) dx = vol(G)1−2g Z Kg−1 (e)0 f dvol | det dKg∗ | (45) Proof Let C be the set of all critical points of Kg . Then the complement G2g \ C is open and the image Kg (G2g \ C) is an open subset of G of full measure (by Sard’s theorem, since it contains all regular values of the surjective map Kg ) and hence is also dense in G. By Proposition 3 we have the disintegration Z G2g f (x)Qt (Kg (x)) dx = vol(G)−2g Z 20 Kg (G2g \C) F (h)Qt (h) dvol(h),(46) where def F (h) = Z Kg−1 (h)0 f dvol | det dKg∗ | (47) is a continuous function of h ∈ Kg (G2g \ C). The identity e belongs to Kg (G2g \ C) since Kg−1 (e)0 6= ∅. Moreover, F (h) is 0 when h is outside the compact set Kg (support(f )) ⊂ Kg (G2g \ C). So F extends to a continuous function on G, 0 outside Kg (G2g \ C). So, remembering that the Riemannian volume on G is vol(G) times the normalized Haar mass dh, Z G2g f (x)Qt (Kg (x)) dx = vol(G)1−2g Z G F (h)Qt (h) dh (48) and, by the initial–condition property of the heat kernel Qt , this approaches the limit 1−2g vol(G) 1−2g F (e) = vol(G) f dvol | det dKg∗ | Z Kg−1 (e)0 as t ↓ 0. QED Things are much easier when we deal with a regular value of Kg : Lemma 2 Let r be any integer ≥ 1, f a continuous function on G2r , and c a regular value of Kr : G2r → G. Then lim t↓0 Z G2r f (x)Qt (Kr (x)c−1 ) dx = vol(G)1−2r Z Kr−1 (c) f dvol | det dKr∗ | (49) Proof. The argument is essentially the same as in the preceding lemma, but we no longer have to worry about critical points of Kr since there are none on Kr−1 (c). Let U and V be neighborhoods of c, with V ⊂ U , and U consisting only of regular values of Kr . Let φ be a continuous function on G, with 0 ≤ φ ≤ 1 everywhere, equal to 1 on V and equal to 0 outside U . Let ψ = 1 − φ. Then f = (φ ◦ Kr )f + (ψ ◦ Kr )f , and Z G2r   f (x)ψ(Kr (x))Qt Kr (x)c−1 dx ≤ |f |sup sup Qt (yc−1 ) y∈G\V → 0, as t ↓ 0 by a uniform-limit property of the heat kernel Qt as t ↓ 0. 21 On the other hand, the integrand in Z G2r f (x)φ(Kr (x))Qt (Kr (x)c−1 ) dx is 0 near the critical points of Kr . Note also that φ(Kr (x)) = 1 when x ∈ Kr−1 (c), and Kr−1 (c) contains no critical point of Kr . So, by Proposition 3 and the argument used in Lemma 1, as t ↓ 0, this integral approaches the limit vol(G)1−2r Z Kr−1 (c) f dvol | det dKr∗ | Combining all these observations, we obtain the desired result. QED The preceding result is essentially present in Forman [8]. 4 Proof of the main result Let g be a positive integer. Recall that Kg−1 (e) ⊂ G2g . The set of points on Kg−1 (e) where dKg (x) : Tx G2g → TKg (x) G is surjective is denoted Kg−1 (e)0 . The set of points on Kg−1 (e) where the isotropy group of the G–conjugation action is Z(G) is denoted Kg−1 (e)0 . Now suppose g1 and g2 are positive integers with g = g1 + g2 . We denote by Kg−1 (e)0,0 the subset of Kg−1 (e) consisting of all points (x1 , x2 ) ∈ G2g1 × G2g2 such that the isotropy of the G–conjugation action on Ggi is Z(G) at xi , for i = 1, 2. Thus (c)0 (c−1 )0 × Kg−1 Kg−1 (e)0,0 = ∪c∈G Kg−1 2 1 (50) The subset Ug0i of G2gi where the isotropy group is Z(G) is (dense and) open in G2gi , as proven in Proposition 5. So Kg−1 (e)0,0 = (Ug01 × Ug02 ) ∩ Kg−1 (e) = (Ug01 × Ug02 ) ∩ Kg−1 (e)0 is an open subset of Kg−1 (e)0 . Theorem 3 For any integer g ≥ 2, and integers g1 , g2 ≥ 1 with g = g1 + g2 , Z Kg−1 (e)0 dvol | det dKg∗ | dvol | det dKg∗ | Z dvol = −1 0,0 | det dKg∗ | Kg (e) = Z Kg−1 (e)0 = vol(G)2g−2 lim t↓0 22 Z G2g (51) (52) Qt (Kg (x)) dx (53) Proof. If f is a continuous function on G2g , with 0 ≤ f ≤ 1, which is 0 in a neighborhood of the critical points of Kg then vol(G)1−2g Z Kg−1 (e)0 f dvol | det dKg∗ | Z = lim t↓0 G2g Z ≤ lim t↓0 G2g f (x)Qt (Kg (x)) dx(54) Qt (Kg (x)) dx The right side was noted in (42) to be finite. Taking appropriate f , with f = 1 at distances beyond 1/n from the critical points of Kg , and then letting n → ∞ we have, by dominated convergence, 1−2g vol(G) Z Kg−1 (e)0 dvol ≤ lim t↓0 | det dKg∗ | Z G2g Qt (Kg (x)) dx (55) Next, observing that Kg (x1 , x2 ) = Kg2 (x2 )Kg1 (x1 ) for x1 ∈ Gg1 and x2 ∈ Gg2 , and using the convolution property of the heat kernel Z G Qt (ac)Qs (c−1 b) dc = Qt+s (ab) = Qt+s (ba) we have Z Z −1 G2g1 G Qt (Kg1 (x1 )c ) dx1 Z G2g2  Qt (cKg2 (x2 )) dx2 dc = Z G2g (56) Q2t (Kg (x)) dx Then lim Z t→0 G2g lim Qt (Kg (x)) dx = Z Z t→0 G = Z  G G2g1 lim Z Qt (Kg1 (x1 )c−1 ) dx1 t→0 G2g1  ··· lim Z t→0 G2g2 Z g2 G  ···  Qt (cKg2 (x2 )) dx2 dc dc (57) because of the L2 (G, dc)–convergence of the limits limt→0 noted in Proposition 10. Let Di be the set of all regular values of Kgi : G2gi → G, and def D = D1 ∩ D2 , 23 (58) which, as we have already noted in the context of (36), is a dense open subset of full measure in G. R R Since D is of full measure in G, we can replace G · · · dc by D · · · dc on the right side in (57). Then using the limit value computed in Lemma 2 we have lim Z t→0 G2g Qt (Kg (x)) dx = 2−2g vol(G) Z "Z D Kg−1 (c) 1 (59) dvol | det dKg∗1 | Z Kg−1 (c−1 ) 2 # dvol dc | det dKg∗2 | Now inserting our “prefabricated” piece Proposition 9, we see that the R integral D [· · ·] dc on the right side in (59) is equal to [vol(G)]−1 Z Kg−1 (e)0,0 dvol | det dKg∗ | Combining this with (55), we write 1−2g vol(G) Z Kg−1 (e)0 dvol | det dKg∗ | ≤ lim Z t→0 G2g = Z Qt (Kg (x)) dx Kg−1 (e)0,0 (60) dvol vol(G)1−2g | det dKg∗ | Since Kg−1 (e)0,0 ⊂ Kg−1 (e)0 , it follows that the inequalities in (60) are equalities. QED Since the middle integral in (60) is finite so are the others. As consequence, we have Corollary 1 For any integer g ≥ 2, the sets Kg−1 (e)0,0 and Kg−1 (e)0 open, dense subsets of full measure in Kg−1 (e)0 . Now we are ready for Proposition 11 For any integer g ≥ 2 and any continuous function f on G2g , lim t↓0 Z G2g f (x)Qt (Kg (x)) dx = vol(G)1−2g 24 Z Kg−1 (e)0 f dvol |dKg∗ | (61) Proof. We have proved this (in Lemma 1) when f is zero near the critical points of Kg . We have also proved this for f = 1 in Theorem 3. Now by Proposition 6, the set Ug of non-critical points of Kg is of full measure in G2g , and so Z G2g f (x)Qt (Kg (x)) dx = Z Ug f (x)Qt (Kg (x)) dx Since Kg−1 (e)0 is a subset of Ug , the task reduces to proving a limiting result for integrals over Ug , given that the limiting formula holds for continuous functions of compact support as well as for the constant function 1. The proof is finished by using Lemma 3 below (take X to be Ug , which is an open subset of G2g ). QED Lemma 3 Let µt , for t ≥ 0, be finite Borel measures on a locally compact Hausdorff space X such that limt↓0 µt (X) = µ0 (X) and lim t↓0 Z X f dµt = Z X f dµ0 for every continuous function f of compact support in X. Assume that X is the union of a countable collection of compact sets. Then lim t↓0 Z X f dµt = Z X f dµ0 for every bounded continuous function f on X. Proof. Let ǫ > 0. Since X is the union of a countable number of compact sets, and µ0 (X) < ∞, there is a compact set K ⊂ X for which µ0 (K c ) < ǫ By local compactness there is an open set U ⊃ K with compact closure U , and, by Urysohn’s lemma, there is a continuous function Φ with 1K ≤ Φ ≤ 1U First we demonstrate that lim supt↓0 µt (U ) is < ǫ. For s > 0 we have c µs (U ) = µs (X) − µs (U ) ≤ µs (X) − 25 Z X Φ dµs and so, for any t > 0, Z c sup µs (U ) ≤ sup µs (X) − inf 0<s≤t 0<s≤t X 0<s≤t Φ dµs which implies c lim sup µt (U ) ≤ lim sup µt (X) − lim inf t↓0 t↓0 t↓0 = µ0 (X) − Z X Z X Φ dµt Φ dµ0 < µ0 (K c ) < ǫ Now choose an open set V ⊃ U with compact closure V , and a continuous function ψ with i.e. 1V c ≤ ψ ≤ 1U c 1U ≤ 1 − ψ ≤ 1V , (62) Let f be a continuous function on X and write it as f = ψf + (1 − ψ)f Since (1 − ψ)f is continuous and of compact support, lim t↓0 Z X R Now we must bound | (1 − ψ)f dµt = X Z ψf dµt − X R X Z X (1 − ψ)f dµ0 ψf dµ0 . To this end, we have c f ψ dµt | ≤ |f |sup µt (U ) for all t ≥ 0. Combining all this, we have lim sup | t↓0 Z X f dµt − Z X h c c i f dµ0 | ≤ lim sup |f |sup µt (U ) + |f |sup µ0 (U ) t↓0 ≤ 2|f |sup ǫ and since ǫ > 0 is arbitrary, this is all we needed. QED Finally, we can turn to Proof of Theorem 1 Let f be a continuous function on G2g , invariant under the conjugation action of G, and f˜ the function induced on 26 M0g = Kg−1 (e)/G. Then lim t↓0 Z G2g f (x)Qt (Kg (x)) dx 1−2g = vol(G) Z Kg−1 (e)0 1−2g vol(G) = vol(G) |Z(G)| Z f dvol |dKg∗ | (by equation (61)) f˜ dvolΩ (by Theorem 2(vii)) M0g which is what we had set out to prove. QED Acknowledgements. I am thankful to Jeff Mitchell for comments and the reference [5]. Parts of this paper were written while I was visiting the Indian Statistical Institute, Kolkata, and the University of Bonn, and I thank these institutions and my hosts K.B. Sinha and S. Albeverio, respectively, at these institutions. I also thank the Alexander von Humboldt Foundation for support during my visits in Germany and the US National Science Foundation for grants DMS 9800955 and DMS 0201683. I am most grateful to an anonymous referee for a very careful examination of this paper, for pointing out a serious gap in an earlier version of this paper and for several useful comments. References [1] M. Atiyah and R. Bott, The Yang-Mills Equations over Riemann Surfaces, Phil. Trans. R. Soc. Lond. A 308, 523-615 (1982) [2] N. Bourbaki: Groupes et Algebrès de Lie, Chapitre 9. Masson, 1982. [3] G. Bredon: Introduction to Compact Transformation Groups. Academic Press (1972). [4] T. Bröcker and T. tom Dieck: Representations of Compact Lie Groups. Springer-Verlag, 1985. [5] I. Chavel: Riemannian geometry: A modern introduction. Cambridge University Press, 1993. [6] J. Dieudonne: Treatise on Analysis, Vols III and IV, transl. I.G. Macdonald, Academic Press, New York and London, 1972. [7] H. Federer: Geometric Measure Theory, Springer-Verlag, New York, 1969. 27 [8] R. Forman: Small volume limits of 2-d Yang-Mills, Commun. Math. Phys. 151, 39-52 (1993) [9] W. Goldman, The Symplectic Nature of Fundamental Groups of Surfaces, Adv. Math. 54, 200-225 (1984) [10] K. Kawakubo: The Theory of Transformation Groups. Oxford University Press, 1991. [11] C. King and A. Sengupta, An Explicit Description of the Symplectic Structure of Moduli Spaces of Flat Connections, J. Math. Phys. (Special Issue on Topology and Physics) 35, 5338-5353 (1994) [12] C. King and A. Sengupta: The Semiclassical Limit of the Two Dimensional Quantum Yang-Mills Model, Journal of Mathematical Physics (Special Issue on Topology and Physics) 35, 5354-5361 (1994) [13] A.W. Knapp : Representation theory of Semisimple Groups: An Overview Based on Examples. Princeton University Press. [14] Deformation Quantization of Singular Symplectic Quotients, (2001) edited by N. P. Landsman, M. Pflaum and M. Schlichenmaier. Progress in Mathematics, vol. 198. Publisher: Birkhäuser. [15] K. F. Liu: Heat Kernel and Moduli Space, Letters 3, 743-762 (1996) Math. Research [16] K. F. Liu: Heat Kernel and Moduli Spaces II, Math. Research Letters 4, 569-588 (1997) [17] A. Sengupta: Gauge Theory on Compact Surfaces, Memoirs of the Amer. Math. Soc. 126 number 600 (1997) [18] A. Sengupta, Yang-Mills on Surfaces with Boundary :Quantum Theory and Symplectic Limit, Commun. Math. Phys., 183 (1997), 661-705. [19] A. Sengupta: The Moduli Space of Yang-Mills Connections over a Compact Surface, Reviews in Mathematical Physics 9, 77-121 (1997) [20] A. Sengupta: A Yang-Mills Inequality for Compact Surface, Infinite Dimensional Analysis, Quantum Probability, and Related Topics 1, 1-16 (1998) [21] A. Sengupta: The Moduli Space of Flat SU(2) and SO(3) Connections over Surfaces, J. Geom. Phys. 28, 209-254 (1998) 28 [22] A. Sengupta: Sewing Symplectic Volumes for Flat Connections over Compact Surfaces J. Geom. Phys. 32, 269-292 (2000) [23] A. Sengupta: The Yang-Mills Measure and Symplectic Structure on Spaces of Connections, in [14] [24] E. Witten: On Quantum Gauge Theories in Two Dimensions, Commun. Math. Phys. 141, 153-209 (1991) [25] E. Witten: Two Dimensional Quantum Gauge Theory revisited, J. Geom. Phys. 9, 303-368 (1992) 29