ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1, Issue 2, 1995

INTERSECTION PAIRINGS IN MODULI SPACES OF HOLOMORPHIC BUNDLES ON A RIEMANN SURFACE

LISA C. JEFFREY AND FRANCES C. KIRWAN

Abstract. We outline a proof of formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n, d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) and fixed determinant on a Riemann surface of genus g ≥ 2.

1. Introduction

The moduli space M(n, d) of semistable rank n degree d holomorphic vector bundles with fixed determinant on a compact Riemann surface Σ is a smooth Kähler manifold when n and d are coprime. This space had long been studied by algebraic geometers (see for instance Narasimhan and Seshadri 1965 [22]), but a new viewpoint on it was revealed by the seminal 1982 paper [1] of Atiyah and Bott on the Yang-Mills equations on Riemann surfaces. In this paper a set of generators for the (rational) cohomology ring of M(n, d) was given and formulas for the Poincaré polynomial were proved. Given the specification of a set of generators, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring.

Little progress was made on the problem of determining these intersection pairings until 1991, when Donaldson [7] and Thaddeus [24] gave formulas for the intersection pairings in H^{∗}(M(2, 1)) (in terms of Bernoulli numbers). Then, using physical methods, Witten [26] found formulas for the evaluation of any product of generators on the fundamental class of M(n, d) for arbitrary rank n. These generalized his (rigorously proved) formulas [25] for the symplectic volume of M(n, d): for instance, the symplectic volume of M(2, 1) is given by

vol(M(2, 1)) = 2\frac{(−1)^{g−1}}{(2π^{2})^{g−1}} (1 − \frac{1}{2^{2g−3}} )ζ(2g − 2)

where g is the genus of the Riemann surface and ζ is the Riemann zeta function.

In this article we outline a mathematically rigorous proof of Witten’s result. Full details, in particular for the case of rank n at least three, will appear later [18]. The key idea is to use a description of M(n, d) as a symplectic quotient and to apply the nonabelian localization theorem (Witten [26]; Jeffrey-Kirwan [17]), which is a generalization of the Duistermaat-Heckman theorem [8]. A different approach to the nonabelian localization principle has been given recently by Guillemin-Kalkman

\begin{matrix}c \\ ◯ \end{matrix}1995 American Mathematical Society

Received by the editors June 28, 1995.

1991 Mathematics Subject Classification. 58F05, 14F05, 53C05.

Key words and phrases. Moduli spaces, symplectic geometry, intersection pairings.

This material is based on work supported by the National Science Foundation under Grant.

No. DMS-9306029.

[11] and independently by Martin [21]; their ideas are crucial in the technicalities of our proof since they can be adapted to certain noncompact situations where the approach of [17] is not valid.

This paper is organized as follows. In Section 2 we describe the generators for the cohomology ring H^{∗}(M(n, d)). In Section 3 we outline tools from the Cartan model of equivariant cohomology and some different versions of localization. In Section 4 we recall properties of the extended moduli space, a finite dimensional symplectic space equipped with a Hamiltonian action of SU(n) for which the symplectic quotient is M(n, d). Finally Section 5 gives an outline of the proof of Witten’s formulas.

2. Generators for the cohomology ring

Let us assume throughout, in order to avoid exceptional cases, that the Riemann surface Σ has genus g ≥ 2.

In [1], Atiyah and Bott gave a set of generators for H^{∗}(M(n, d)); their procedure may be described as follows^{1}. We may form a normalized universal rank n vector bundle

𝕌 → Σ × M(n, d)

(see [1], p. 582). Then there is the following description [1] of a set of generators of H^{∗}(M(n, d)):

f_{r} = ([Σ], c_{r}(𝕌)),  (1)
b_{r}^{j} = (α_{j}, c_{r}(𝕌)),  (2)
a_{r} = (1, c_{r}(𝕌)).  (3)

Here, [Σ] ∈ H_{2}(Σ) and α_{j} ∈ H_{1}(Σ) (j = 1, …, 2g) form standard bases of H_{2}(Σ, ℤ) and H_{1}(Σ, ℤ). The bracket represents the slant product H^{N}(Σ × M(n, d)) ⊗ H_{j}(Σ) → H^{N−j}(M(n, d)). More generally if K = SU(n) and Q is an invariant polynomial of degree s on its Lie algebra k = su(n) then there is an associated element of H^{∗}(BSU(n)) and hence an associated element of H^{∗}(Σ × M(n, d)) which is a characteristic class Q(𝕌) of the universal bundle 𝕌. Hence the slant product gives rise to classes

Q([Σ]) ∈ H^{2s−2}(M(n, d)),  (4)
Q(α_{j}) ∈ H^{2s−1}(M(n, d)),  (5)
Q(1) ∈ H^{2s}(M(n, d)).  (6)

In particular, letting τ_{r} ∈ S^{r}(k^{∗})^{K} denote the invariant polynomial associated to the r-th Chern class, we recover our generators of H^{∗}(M(n, d)):

f_{r} = τ_{r}([Σ]),  (7)
b_{r}^{j} = τ_{r}(α_{j}),
a_{r} = τ_{r}(1).

1In this paper, all cohomology groups are assumed to be with complex coefficients.

A special role is played by the invariant polynomial τ_{2} = \frac{1}{2}⟨·, ·⟩ on k given by an invariant inner product proportional to the Killing form. We normalize the inner product as follows for K = SU(n):

⟨X, X⟩ = −Trace(X^{2})/(4π^{2}).

The associated class f_{2} is the cohomology class of the symplectic form on M(n, d).

In Sections 4 and 5 of [26], Witten obtained formulas for generating functionals from which one may extract all intersection pairings

∏_{r=2}^{n} a_{r}^{m_{r}}f_{r}^{n_{r}} ∏_{k_{r}=1}^{2g}(b_{r}^{k_{r}})^{p_{r,kr}} [M(n, d)].

Let us begin with pairings of the form

∏_{r=2}^{n} a_{r}^{m_{r}} exp f_{2}[M(n, d)].  (8)

For m_{r} sufficiently small to ensure convergence of the sum, Witten obtains ([26], (4.74))

∏_{r=2}^{n} a_{r}^{m_{r}} exp f_{2}[M(n, d)] = Γ(∑_{λ} \frac{c^{−λ} ∏_{r=2}^{n} τ_{r}(2πiλ)^{m_{r}}}{D^{2g−2}(2πiλ)} ),  (9)

where Γ is a universal constant depending on n, d and g, and the Weyl odd polynomial D on t^{∗} is defined by

D(X) = ∏_{γ>0} γ(X)

where γ runs over the positive roots. The sum over λ in (9) runs over those elements of the weight lattice Λ^{w} that are in the interior of the fundamental Weyl chamber.^{2} The element

c = e^{2πid/n}diag(1, …, 1)  (10)

is a generator of the centre Z(K), so since λ ∈ t^{∗} is in Hom(T, U(1)), we may evaluate λ on c as in (9): c^{λ} is defined as exp λ(c̃) where c̃ is any element of the Lie algebra of T such that exp c̃ = c.

There are similar formulas ([26], (5.15) and (5.18)) for pairings involving the f_{r} for r > 2 and the b_{r}^{j} as well as f_{2} and the a_{r}.

For concreteness it is worth examining the special case n = 2, d = 1. Here the dominant weights λ are just the positive integers. The relevant generators of H^{∗}(M(2, 1)) are

f_{2} ∈ H^{2}(M(2, 1))  (11)

(which is the cohomology class of the symplectic form on M(2, 1)) and

a_{2} ∈ H^{4}(M(2, 1)) :  (12)

2The weight lattice Λ^{w} ⊂ t is the dual lattice of the integer lattice Λ^{I} = Ker(exp) in t under the inner product ⟨·, ·⟩.

these arise from the invariant polynomial τ_{2} = ⟨·, ·⟩ by a_{2} = τ_{2}(1), f_{2} = τ_{2}([Σ]) (see (7)). We find then that the formula (9) reduces for m ≤ g − 2 to^{3} ([26], (4.44))

a_{2}^{j} exp(f_{2})[M(2, 1)] = \frac{2^{2g}(−1)^{g−1−j}}{2(8π^{2})^{g−1}} (∑_{n=1}^{∞}(−1)^{n+1} \frac{π^{2j}}{n^{2g−2−2j}} ).  (13)

Thus one obtains the formulas found in [24] for the intersection pairings a_{2}^{m}f_{2}^{n}[M(2, 1)]; these intersection pairings are given by Bernoulli numbers, or equivalently are given in terms of the Riemann zeta function ζ(s) = ∑_{n≥1} 1/n^{s}.

3. Equivariant cohomology and localization

The methods we shall use involve the application of the nonabelian localization theorem [17, 26]. In [17] we considered a compact symplectic manifold M equipped with the Hamiltonian action of a compact group K. We expressed the cohomology ring H^{∗}(M_{red}) of the reduced space or symplectic quotient M_{red} = μ^{−1}(0)/K of M by K in terms of the equivariant cohomology of M. (This was under the assumption that 0 is a regular value of μ, which implies that M_{red} is a symplectic orbifold.) There is a surjective ring homomorphism Φ from the equivariant cohomology H_{K}^{∗}(M) of M to the cohomology H^{∗}(M_{red}) of the reduced space. In terms of this map, we derived a formula for the evaluation η_{0}[M_{red}] of a cohomology class η_{0} ∈ H^{∗}(M_{red}) on the fundamental class of M_{red}. This formula involves data that enter the Duistermaat-Heckman formula, and its generalization the abelian localization formula [2, 4, 5] for the action of a maximal torus T of K on M: that is, the components F of the fixed point set M^{T} of T on M, and the equivariant Euler classes e_{F} of the normal bundles to F in M. In the following paragraphs we sketch this construction.

The K-equivariant cohomology of M is the cohomology of a certain chain complex Ω_{K}^{∗}(M), which can be expressed as

Ω_{K}^{∗}(M) = (S(k^{∗}) ⊗ Ω^{∗}(M))^{K}  (14)

(where Ω^{∗}(M) denotes the space of differential forms on M, and S(k^{∗}) denotes the ring of polynomial functions on the Lie algebra k of K). An element f ∈ Ω_{K}^{∗}(M) may be thought of as a K-equivariant polynomial function from k to Ω^{∗}(M), or alternatively as a family of differential forms on M parametrized by X ∈ k. The differential D on the complex Ω_{K}^{∗}(M) is then defined by

(Dα)(X) = d(α(X)) − ι_{X^{#}}(α(X))  (15)

where X^{#} is the vector field on M generated by the action of X.

In terms of this notation, the map Ω_{K}^{∗}(M) → Ω_{K}^{∗}(pt) = S(k^{∗})^{K} given by integration over M passes to H_{K}^{∗}(M). Thus for any D-closed element η ∈ Ω_{K}^{∗}(M) representing a cohomology class [η], there is a corresponding element ∫_{M} η ∈ Ω_{K}^{∗}(pt) which depends only on [η]. The same is true for any D-closed element η = ∑_{j} η_{j} which is a formal series of elements η_{j} in Ω_{K}^{j}(M) without polynomial dependence on X: we shall in particular consider terms of the form η(X)e^{i(ω+μ(X))} (where η ∈ Ω_{K}^{∗}(M)), since the element

ω̄(X) = ω + μ(X) ∈ Ω_{K}^{2}(M)

satisfies Dω̄ = 0.

3Here, we have identified a_{2} with Witten’s class Θ and f_{2} with Witten’s class ω.

If X lies in t, the Lie algebra of a chosen maximal torus T of K, then there is a formula for ∫_{M} η(X) (the abelian localization formula [1, 3, 4, 5]) which depends only on the fixed point set of T in M. It tells us that

∫_{M} η(X) = ∑_{F ∈F} ∫ \frac{i_{F}^{∗} η(X)}{e_{F} (X)}  (16)

where F indexes the components F of the fixed point set of T in M, the inclusion of F in M is denoted by i_{F} and e_{F} ∈ H_{T}^{∗} (M) is the equivariant Euler class of the normal bundle to F in M. In particular, applying (16) with η replaced by the formal sum ηe^{iω̄} we have

h^{η}(X) =^{def} ∫_{M} η(X)e^{iω̄(X)} = ∑_{F ∈F} h_{F}^{η} (X),  (17)

where

h_{F}^{η} (X) = e^{iμ(F)(X)} ∫_{F} \frac{i_{F}^{∗} η(X)e^{iω}}{e_{F} (X)} .  (18)

Note that the moment map μ takes a constant value μ(F) ∈ t^{∗} on each F ∈ F, and that the integral in (18) is a rational function of X.

We assume that η_{0} ∈ H^{∗}(M_{red}) comes via the surjective homomorphism Φ from a class in H_{K}^{∗}(M) which is represented as an equivariant differential form by η ∈ Ω_{K}^{∗}(M), and denote the symplectic form on M_{red} by ω_{0}. Let us recall the main result (the residue formula, Theorem 8.1) of [17]:

Theorem 3.1. [17] Let η ∈ H_{K}^{∗}(M) induce η_{0} ∈ H^{∗}(M_{red}). Then we have

η_{0}e^{iω_{0}}[M_{red}] = n_{0}C^{K}Res(D^{2}(X) ∑_{F ∈F} h_{F}^{η} (X)[dX]),  (19)

where n_{0} is the order of the subgroup of K that acts trivially on^{4} M, and the constant C^{K} is defined by

C^{K} = \frac{(−1)^{n_{+}}}{|W | vol (T)}.  (20)

We have introduced s = dim K and l = dim T; here n_{+} = (s − l)/2 is the number of positive roots^{5}. Also, F denotes the set of components of the fixed point set of T, and if F is one of these components then the meromorphic function h_{F}^{η} on t ⊗ ℂ is defined by (18) and the polynomial D : t → ℝ is defined by D(X) = ∏_{γ>0} γ(X), where γ runs over the positive roots of K.

4The subgroup of K which acts trivially on M is the same as the subgroup, N say, of K which acts trivially on μ^{−1}(0), and is a finite central subgroup of K. For since 0 is a regular value of μ it follows that N is a finite normal subgroup of K, and because K is connected N is therefore contained in the center of K. Thus the coadjoint action of N on k^{∗} is trivial, so N acts trivially on the normal bundle to μ^{−1}(0) in M, and as M is connected this means N acts trivially on M.

5Here, the roots of K are the nonzero weights of its complexified adjoint action. We fix the convention that weights β ∈ t^{∗} satisfy β ∈ Hom(Λ^{I}, ℤ) rather than β ∈ Hom(Λ^{I}, 2πℤ) (where Λ^{I} = Ker(exp : t → T) is the integer lattice). This definition of roots differs by a factor of 2π from the definition used in [17]: there, the roots γ satisfy γ(Λ^{I}) ⊂ 2πℤ. This is why the constant C^{K} differs from that of [17] by a factor of (2π)^{s−l}.

The above formula was called a residue formula in [17] because the quantity Res (whose general definition was given in Section 8 of [17]) can be expressed as a multivariable residue (or alternatively in terms of iterated 1-variable residues).

Here we shall make particular use of the case where K has rank 1, for which the results are as follows.

Corollary 3.2. In the situation of Theorem 3.1, let K = U(1). Then

η_{0}e^{iω_{0}}[M_{red}] = n_{0}Res_{X=0}( ∑_{F ∈F_{+}} h_{F}^{η} (X)).

Here, the meromorphic function h_{F}^{η} on ℂ was defined by (18), and Res_{X=0} denotes the coefficient of 1/X, where X ∈ ℝ has been identified with 2πiX ∈ k. The set F_{+} is defined by F_{+} = {F ∈ F : μ_{T} (F) > 0}. The integer n_{0} is as in Theorem 3.1.

Corollary 3.3. (cf. [17], Corollary 8.2) In the situation of Theorem 3.1, let K = SU(2). Then

η_{0}e^{iω_{0}}[M_{red}] = −\frac{n_{0}}{2} Res_{X=0}((2X)^{2} ∑_{F ∈F_{+}} h_{F}^{η} (X)).

Here, Res_{X=0}, h_{F}^{η} and F_{+} are as in Corollary 3.2, and X ∈ ℝ has been identified with diag(2πi, −2πi)X ∈ t. The integer n_{0} is as in Theorem 3.1.

Remark: Notice that we shall not only be considering the reduced space M_{red} = μ^{−1}(0)/K with respect to the action of the nonabelian group K, but also μ^{−1}(0)/T and M_{red}^{T}(t) = μ_{T}^{−1}(t)/T for regular values t of the T -moment map μ_{T} which is the composition of μ with restriction from k^{∗} to t^{∗}. We shall use the same notation η_{0} for the image of η under the surjective homomorphism Φ for whichever of the spaces μ^{−1}(0)/K, μ^{−1}(0)/T or μ_{T}^{−1}(0)/T we are working with, and the notation η_{t} if we are working with μ_{T}^{−1}(t)/T. It should be clear from the context which version of the map Φ is being used.

We shall need the following very recent additional results, which are expressed in the above notation:

Proposition 3.4. (Reduction to the abelian case) (S. Martin [21]) If T is a maximal torus of K, then we have that

∫_{μ_{K}^{−1}(0)/K}(ηe^{iω̄})_{0} = \frac{1}{|W |} ∫_{μ_{K}^{−1}(0)/T} (Dηe^{iω̄})_{0} = \frac{(−1)^{n_{+}}}{|W |} ∫_{μ_{T}^{−1}(t)/T} (D^{2}ηe^{iω̄})_{t}

for any regular value t of μ_{T} sufficiently close to 0.

Proposition 3.5. (Dependence of symplectic quotients on parameters) (Guillemin-Kalkman [11] ; S. Martin [21]) Consider a symplectic manifold M acted on in a Hamiltonian fashion by T = U(1) and let μ denote the moment map for this action. Let t_{0} < t_{1} be two regular values of μ. Then we may subtract t_{i} from the moment map to get a modified symplectic quotient M_{red}^{T}(t_{i}) = μ^{−1}(t_{i})/T for i = 0, 1 with maps

Φ : H_{T}^{∗} (M) → H^{∗}(M_{red}^{T}(t_{i}))

sending η to (η)_{t_{i}}. We then have

∫_{M_{red}^{T}(t_{0})}(ηe^{iω̄})_{t_{0}} −∫_{M_{red}^{T}(t_{1})}(ηe^{iω̄})_{t_{1}} = n_{0} ∑_{F ∈F:t_{0}<μ(F)<t_{1}} Res_{X=0}e^{iμ(F)(X)} ∫_{F} \frac{η(X)e^{iω}}{e_{F} (X)} .

Here, X ∈ ℂ has been identified with 2πiX ∈ t ⊗ ℂ.

Corollary 3.6. If t_{1} ∈ t is close enough to t_{0} ∈ t that there is a path between t_{0} and t_{1} in μ(M) consisting entirely of regular values of μ, then

∫_{M_{red}^{T}(t_{0})}(ηe^{iω̄})_{t_{0}} = ∫_{M_{red}^{T}(t_{1})}(ηe^{iω̄})_{t_{1}}.

Remark: In fact both the equality

∫_{μ_{K}^{−1}(0)/K}(ηe^{iω̄})_{0} = \frac{(−1)^{n_{+}}}{|W |} ∫_{μ_{T}^{−1}(t)/T} (D^{2}ηe^{iω̄})_{t}

of Proposition 3.4 and the residue formula of Proposition 3.5 can be deduced easily from the residue formula of [17] when M is a compact symplectic manifold. However the proofs of Propositions 3.4 and 3.5 can be adapted to apply in certain circumstances when M is not compact and the residue formula of [17] is not valid. This will be crucial later.

4. Extended moduli spaces

To make progress we invoke a description of a symplectic space M(c) equipped with a Hamiltonian action of K = SU(n) such that the symplectic quotient of M(c) at 0 is M(n, d), and explain our general strategy for obtaining (9) and its generalizations. The moduli space M(n, d) was described by Atiyah and Bott [1] as the symplectic reduction of an infinite dimensional symplectic vector space A with respect to the action of an infinite dimensional group G (the gauge group).^{6} We however shall exhibit M(n, d) as the symplectic quotient of a finite dimensional symplectic space M(c) by the Hamiltonian action of a finite dimensional group K. In this case the group K is SU(n). One characterization of the space M(c) is that it is the symplectic reduction of the infinite dimensional affine space A by the action of the based gauge group G_{0} (which is the kernel of the evaluation map G → K at a prescribed basepoint: see [14]). Now if a compact group G containing a closed normal subgroup H acts in a Hamiltonian fashion on a symplectic manifold Y, then one may “reduce in stages”: the space Y//H = μ_{H}^{−1}(0)/H has a residual Hamiltonian action of the quotient group G/H with moment map μ_{G/H} : Y//H → (𝔤/𝔥)^{∗}, and μ_{G}^{−1}(0)/G is naturally identified as a symplectic manifold with μ_{G/H}^{−1}(0)/(G/H). Similarly M(c) has a Hamiltonian action of G/G_{0} ≅ K, and the symplectic reduction with respect to this action is identified with the symplectic reduction of A with respect to the full gauge group G.

A more concrete (and entirely finite dimensional) characterization of M(c) is given in [14]. The space is defined by

M(c) = (ε_{r} × e_{c})^{−1}(△) ⊂ Hom (𝔽, K) × k.  (21)

Here, we identify Hom(𝔽, K) with K^{2g} through a choice of generators {x_{1}, …, x_{2g}} for the free group 𝔽 on 2g generators; then ε_{r} : Hom (𝔽, K) → K is the evaluation map on the relator r = ∏_{j=1}^{g}[x_{2j−1}, x_{2j}]

ε_{r}(h_{1}, …, h_{2g}) = ∏_{j=1}^{g}[h_{2j−1}, h_{2j}].  (22)

6To obtain his generating functionals, Witten formally applied his version of nonabelian localization to the infinite dimensional space A.

The map e_{c} : k → K is defined by

e_{c}(Y) = c exp(Y),  (23)

where the generator c of the centre of K was defined at (10) above. The diagonal in K × K is denoted △. The space M(c) then has canonical projection maps pr_{1}, pr_{2} which make the following diagram commute:

M(c)  ⟶^{pr_{2}}  k
pr_{1} ↓  ↓ e_{c}
Hom(𝔽, K) ⟶^{ε_{r}} K

(24)

In other words, M(c) is the fibre product of Hom(𝔽, K) and k under the maps ε_{r} and e_{c}. The action of K on M(c) is given by the adjoint actions on K and k. The space M(c) has the following properties (see [14] and [15]):

Proposition 4.1. (a) The space M(c) is smooth near all (h, Λ) ∈ Hom(𝔽, K) × k for which the linear space z(h) ∩ ker(d exp)_{Λ} ≠ {0}. Here, z(h) is the Lie algebra of the stabilizer Z(h) of h.

(b) There is a K-invariant 2-form ω on Hom(𝔽, K) × k whose restriction to M(c) is closed and which defines a nondegenerate bilinear form on the Zariski tangent space to M(c) at every (h, Λ) in an open dense set in M(c) which includes the subset of M(c) where Λ = 0. (Thus the form ω gives rise to a symplectic structure on an open dense subset of M(c).)

(c) With respect to the symplectic structure given by the 2-form ω, a moment map μ : M(c) → k^{∗} for the action of K on M(c) is given by σpr_{2}, where pr_{2} : M(c) → k is the projection map to k (composed with the canonical isomorphism k → k^{∗} given by the invariant inner product on k) and σ = −2.

(d) The space M(c) is smooth in a neighbourhood of μ^{−1}(0).

(e) The symplectic quotient M_{red} = μ^{−1}(0)/K can be naturally identified with Y_{c}/K = M(n, d).

Using our description of M(c) as a fibre product, it is easy to identify the components F of the fixed point set of the action of T. We examine the fixed point sets of the action of T on Hom(𝔽, K) and k and we find

M(c)^{T}  ⟶^{pr_{2}}  t
pr_{1} ↓  ↓ e_{c}
Hom(𝔽, T) ⟶^{ε_{r}} 1 ∈ T

(25)

(Notice that ε_{r} sends Hom(𝔽, T) to 1 because T is abelian.) Thus

M(c)^{T} = Hom(𝔽, T) × e_{c}^{−1}(1) = T^{2g} × {δ − c̃: δ ∈ Λ^{I} ⊂ t}  (26)

where c̃ is a fixed element of t for which exp c̃ = c. (Here, Λ^{I} denotes the integer lattice Ker(exp) ⊂ t.) If we ignore the singularities of M(c), this description also enables us to identify the equivariant Euler class e_{F_{δ}} of the normal bundle of each component T^{2g} × (δ − c̃) in M(c)^{T} (indexed by δ ∈ Λ^{I}). This should be simply the equivariant Euler class of the normal bundle to T^{2g} in K^{2g}, implying that e_{F_{δ}} is in fact independent of δ and is given by

e_{F_{δ}}(X) = (−1)^{n_{+}g}D(X)^{2g}.  (27)

The symplectic volume of the component F_{δ} is independent of δ (indeed these components are all identified symplectically with T^{2g}): we denote the volume of F_{δ} by ∫_{F} vol_{ω}.

The constant value taken by the moment map μ_{T} on the component F = F_{δ} is given by σ(δ − c̃) where σ = −2 as in Proposition 4.1(c).

We shall need also the following property ([16]):

Proposition 4.2. The generating classes a_{r}, b_{r}^{j} and f_{r} (r = 2, …, n, j = 1, …, 2g) extend to classes ã_{r}(X), b̃_{r}^{j}(X) and f̃_{r}(X) ∈ H_{K}^{∗}(M(c)).

Indeed, the equivariant differential form ã_{r}(X) ∈ Ω_{K}^{∗}(M(c)) whose restriction represents the cohomology class a_{r} ∈ H^{∗}(M(n, d)) is simply the invariant polynomial τ_{r} ∈ S^{r}(k)^{K} ≅ H_{K}^{∗}(pt) which is associated to the rth Chern class.

Finally we shall need to work with the symplectic subspace Mt(c) = μ_{K}^{−1}(t) of M(c), which is no longer acted on by K but is acted on by T. The space Mt(c) has an important periodicity property:

Lemma 4.3. Suppose Λ_{0} ∈ Ker(exp) ⊂ t. Then there is a homeomorphism s_{Λ_{0}} : Mt(c) → Mt(c) defined by

s_{Λ_{0}} : (h, Λ) ↦ (h, Λ + Λ_{0}).

5. Proof of the residue formula

Naïve application of the residue formula (Theorem 3.1) to the space M(c), ignoring the fact that it is noncompact and has singularities, would thus yield

∏_{r=2}^{n} a_{r}^{m_{r}} exp(f_{2})[M(n, d)] =

n_{0}C^{K}Res(D^{2}(X)(∫_{F} vol_{ω}) ∑_{δ∈Λ^{I}} \frac{∏_{r=2}^{n} τ_{r}(X)^{m_{r}}e^{i(−c̃+δ)(X)}}{(−1)^{n_{+}g}D^{2g}(X)} ). (28)

The main problem with (28) (related to the noncompactness of M(c), which permits the fixed point set M(c)^{T} to consist of infinitely many components F_{δ}) is that the sum over δ does not converge for X ∈ t. In this section we shall sketch a sequence of results that enable us nonetheless to conclude that (28) is true if interpreted appropriately. We shall concentrate mainly on the case when n is 2 but the argument can be generalized to higher n; details will be given in a subsequent paper. (See Theorem 5.10 and the remarks following it.)

Lemma 5.1. For generic ξ ∈ t,

M(c exp(ξ)) =^{def} {(h_{1}, …, h_{2g}, Λ) ∈ K^{2g} × k :

∏_{j=1}^{g} h_{2j−1}h_{2j}h_{2j−1}^{−1}h_{2j}^{−1} = c exp(ξ) exp(Λ)}

is a smooth manifold.

Proof: We see that M(c) = F^{−1}(c) where F : K^{2g} × k → K is defined by

F (h_{1}, …, h_{2g}, Λ) = ∏_{j=1}^{g} h_{2j−1}h_{2j}h_{2j−1}^{−1}h_{2j}^{−1} exp(−Λ).

By Sard’s theorem, for generic ξ ∈ k, c exp(ξ) is a regular value of F (although c itself cannot be a regular value of F since M(c) is not smooth, see [15],[16]). Moreover, since F is K-equivariant, we may assume without loss of generality that ξ ∈ t. □ Remark: Note that when exp(ξ) = 1 we recover the definition of M(c).

Lemma 5.2. Define Mt(c exp(ξ))) = μ^{−1}(t)∩M(c exp(ξ)), where μ : K^{2g} ×k → k is defined by

μ(h_{1}, …, h_{2g}, Λ) = σΛ

(see Proposition 4.1(c)) and μ_{T} is the projection of μ onto t. Then there is a T -equivariant homeomorphism s_{ξ} : Mt(c exp(ξ)) → Mt(c) which sends μ_{T}^{−1}(0) ∩ Mt(c exp(ξ)) to μ_{T}^{−1}(σξ) ∩ Mt(c).

Proof: One may define the homeomorphism as follows:

s_{ξ} : (h_{1}, …, h_{2g}, Λ) → (h_{1}, …, h_{2g}, Λ + ξ). □

Lemma 5.3. We have

∫_{M_{red}} Φ(ηe^{iω̄}) = \frac{1}{|W |} ∫_{N} s_{ξ}^{∗}Φ(Dηe^{iω̄})

where

N = μ_{T}^{−1}(0) ∩ Mt(c exp(ξ))/T

and |W | = n! is the order of the Weyl group of K = SU(n).

Proof: We first identify ∫_{M_{red}} Φ(ηe^{iω̄}) with

\frac{1}{|W |} ∫_{μ^{−1}(0)/T} Φ(Dηe^{iω̄}) = \frac{1}{|W |} ∫_{μ_{T} |_{Mt(c)}^{−1}(0)/T} Φ(Dηe^{iω̄}),

via Proposition 3.4, whose proof can be made to work in this situation, even though M(c) is noncompact and singular, because μ^{−1}(0) is compact and M(c) is nonsingular in a neighbourhood of μ^{−1}(0). Then we observe that for sufficiently small ξ ∈ t,

∫_{μ_{T} |_{Mt(c)}^{−1}(0)/T} Φ(Dηe^{iω̄}) = ∫_{μ_{T} |_{Mt(c)}^{−1}(σξ)/T} Φ(Dηe^{iω̄})

(by Corollary 3.6). Finally we use the homeomorphism s_{ξ} of Lemma 5.2 to establish an identification between μ_{T} |_{Mt(c)}^{−1}(σξ)/T and N = μ_{T} |_{Mt(c exp(ξ))}^{−1}(0)/T.□

Let us now examine the behaviour of the images in H_{T}^{∗} (Mt(c)) of the generating classes ã_{r}(X), b̃_{r}^{j}(X), f̃_{r}(X) ∈ H_{K}^{∗}(M(c)) under pullback under the homeomorphisms s_{Λ_{0}} : Mt(c) → Mt(c) defined at Lemma 4.3. By abuse of language, we shall refer to these images also as ã_{r}(X), b̃_{r}^{j}(X) and f̃_{r}(X). It follows from [16] that the classes ã_{r}(X) are the images in H_{K}^{∗}(M(c)) of the polynomials τ_{r} ∈ H_{K}^{∗} = S(k^{∗})^{K}. Moreover ([16], (8.18)) the classes b̃_{r}^{j}(X) ∈ H_{K}^{∗}(M(c)) are of the form b̃_{r}^{j}(X) = pr_{1}^{∗}(b̃_{r}^{j}(X))_{1} where (b̃_{r}^{j}(X))_{1} ∈ H_{K}^{∗}(K^{2g}) and pr_{1} : M(c) → K^{2g} is the projection in (24). It follows that s_{Λ_{0}}^{∗}b̃_{r}^{j}(X) = b̃_{r}^{j}(X) and s_{Λ_{0}}^{∗} ã_{r}(X) = ã_{r}(X).

Furthermore we see from (8.30) of [16] that f̃_{2}(X) is of the form

f̃_{2}(X) = pr_{1}^{∗}f_{2}^{1} + ⟨μ, X⟩  (29)

where f_{2}^{1} ∈ H_{K}^{∗}(K^{2g}) and μ : M(c) → k is the moment map (which is σ times the restriction to M(c) of the projection K^{2g} × k → k: see Proposition 4.1(c)). It follows from the definition (29) of f̃_{2} that for any Λ_{0} in the integer lattice of t (the kernel of the exponential map),

s_{Λ_{0}}^{∗} f̃_{2}(X) = f̃_{2}(X) + σ⟨Λ_{0}, X⟩.  (30)

Note also that there is a commutative diagram of homeomorphisms

Mt(c exp(ξ)) ⟶^{s_{ξ}} Mt(c)
s_{Λ_{0}} ↓  ↓ s_{Λ_{0}}
Mt(c exp(ξ)) ⟶^{s_{ξ}} Mt(c)

(31)

For simplicity, we restrict from now on to K = SU(2), and let Λ_{0} = 2πimdiag(1, −1) (for m ∈ ℤ) and identify X ∈ ℝ with 2πiXdiag(1, −1) ∈ t.

Lemma 5.4. Suppose η is a polynomial in the ã_{r}(X). Then for any positive integer m we have that

∫_{N} s_{ξ}^{∗}Φ(ηe^{iω̄}e^{imσX}) = ∫_{Mt(c exp(ξ))∩μ_{T}^{−1}(σm)/T} s_{ξ}^{∗}Φ(ηe^{iω̄})

= ∫_{N} s_{ξ}^{∗}Φ(ηe^{iω̄}) + 2 sgn(σ) ∑_{F ∈F ,0<|μ_{T} (F)|<|σ|m} Res_{X=0}(∫_{F} \frac{η(X)e^{iω̄(X)}γ(X)}{e_{F} (X)}).

Here, F is the set of connected components of the fixed point set of the action of T on M(c) and if F ∈ F then e_{F} denotes the equivariant Euler class of the normal bundle of s_{ξ}^{−1}(F) in M(c exp ξ). Also γ(X) = 2X denotes the positive root of SU(2) and σ = −2 is the normalization constant resulting from the normalization of the symplectic form (see Proposition 4.1 (c)). The overall factor of 2 multiplying the sum over F results from the fact that the element −1 ∈ U(1) acts trivially on M(c).

Proof: We see that

∫_{Mt(c exp(ξ))∩μ_{T}^{−1}(σm)/T} s_{ξ}^{∗}Φ(ηe^{iω̄}) = ∫_{N} s_{ξ}^{∗}Φ(s_{m}^{∗}ηe^{iω̄})

= ∫_{N} s_{ξ}^{∗}Φ(ηe^{iω̄}e^{imσX})

by (30). By (26) the set of components of the fixed point set of the action of T on M(c exp ξ) is {s_{ξ}^{−1}(F) : F ∈ F}. Using the fact that the root −γ(X) = −2X of SU(2) represents the Poincaré dual of Mt(c exp ξ) in M(c exp ξ), Guillemin and Kalkman’s proof of Proposition 3.5 can be applied to compactly supported equivariant forms in a neighbourhood of Mt(c exp(ξ)) ∩ μ_{T}^{−1}[σm, 0] in the smooth manifold M(c exp ξ) to show that

−sgn(σ)(∫_{Mt(c exp(ξ))∩μ_{T}^{−1}(σm)/T} s_{ξ}^{∗}Φ(ηe^{iω̄}) − ∫_{N} s_{ξ}^{∗}Φ(ηe^{iω̄}))

= −2 ∑_{F ∈F ,0<|μ_{T} (F)|<|σ|m} Res_{X=0}(∫_{F} \frac{η(X)e^{iω̄(X)}γ(X)}{e_{F} (X)}).□

Corollary 5.5. For all j ≥ 0,

−sgn(σ) ∫_{N} s_{ξ}^{∗} Φ(X^{j}(e^{imσX} − 1)ηe^{iω̄}) = \frac{(−1)^{g−1}}{2^{2g−2}} ∑_{k=1}^{m} Res_{X=0} \frac{e^{iσ(k−1/2)X} ∫_{F} ηe^{iω}}{X^{2g−1−j}} .

Proof: This follows from Lemma 5.4 using the fact that e_{F} (X) = (−1)^{g}γ(X)^{2g} where γ(X) = 2X is the positive root of SU(2).□

We shall encode this information by defining a formal Laurent series L_{η} in an indeterminate x as follows:

L_{η}(x) = ∑_{j≥0} \frac{∫_{N} s_{ξ}^{∗}Φ(ηX^{j}e^{iω̄})}{x^{j+1}}.  (32)

We appeal to the following elementary observation:

Lemma 5.6. Suppose f(x) = ∑_{j∈ℤ} f_{j}x^{j} is a formal Laurent series in x. Then

∑_{j≥0} \frac{Res_{X=0}(X^{j}f(X))}{x^{j+1}} = P(f(x)),

where P denotes the principal part.

We have also

Proposition 5.7. For any formal power series g(x) = ∑_{j≥0} g_{j}x^{j} in x,

P(L_{g(X)η}(x)) = P(g(x)L_{η}(x)).

Proof: This follows because for any k ≥ 0,

L_{X^{k}η}(x) = ∑_{j≥0} \frac{∫_{N} s_{ξ}^{∗}Φ(ηX^{k+j})e^{iω̄}}{x^{k+j+1}} x^{k}.

Proposition 5.8.

−sgn(σ)P((e^{iσx} − 1)L_{η}) = \frac{(−1)^{g−1}}{2^{2g−2}} P \frac{(∫_{T^{2g}} ηe^{iω})e^{iσx/2}}{x^{2g−1}} .

Proof:

P((e^{iσx} − 1)L_{η}(x)) = P(L_{(e^{iσX}−1)η}(x)) by Proposition 5.7

= P(∑_{j≥0} x^{−j−1} ∫_{N} s_{ξ}^{∗}Φ((X^{j}ηe^{iω̄})(e^{iσX} − 1)) by definition of L_{η}

so that

−sgn(σ)P((e^{iσx}−1)L_{η}(x)) = \frac{(−1)^{g−1}}{2^{2g−2}} P ∑_{j≥0} x^{−j−1}Res_{X=0}(\frac{(∫_{T^{2g}} η(X)e^{iω})e^{iσX/2}}{X^{2g−1−j}} )

(by Corollary 5.5 when m = 1 )

= \frac{(−1)^{g−1}}{2^{2g−2}} P \frac{(∫_{T^{2g}} η(x)e^{iω})e^{iσx/2}}{x^{2g−1}} by Lemma 5.6. □

Corollary 5.9.

−sgn(σ)L_{η}(x) = \frac{(−1)^{g−1}}{2^{2g−2}} \frac{∫_{T^{2g}} ηe^{iω}e^{iσx/2}}{(e^{iσx} − 1)x^{2g−1}} + h(x)

where h is a formal Laurent series with at most a simple pole at 0.

Proof: This follows immediately when we divide through the result of Proposition 5.8 by (e^{iσx} − 1). □

Unwinding the definition of L_{η} we have from Corollary 5.9 (provided that k ≥ 1) that

∫_{N} s_{ξ}^{∗}Φ(X^{k}ηe^{iω̄}) = Res_{x=0}(x^{k}L_{η}(x))

= \frac{(−1)^{g}}{2^{2g−2}} Res_{x=0} \frac{1}{2ix^{2g−1−k} sin(|σ|x/2)}(∫_{T^{2g}} η(x)e^{iω}).  (33)

It follows finally that

Theorem 5.10.

∫_{M(2,1)} a_{2}^{j}e^{if_{2}} = \frac{(−1)^{g}}{2^{2g−1}i} ∫_{T^{2g}} e^{iω}Res_{X=0}(\frac{1}{X^{2g−2−2j} sin |σ|X/2}),

where the normalization is such that the class a_{2} = Φ(X^{2}).

Proof: This is true since

∫_{M(2,1)} a_{2}^{j}e^{if_{2}} = ∫_{M(2,1)} Φ(X^{2j}e^{iω̄})

(by the remarks in the paragraph above (29), and noting that τ_{2}(X) = X^{2})

= ∫_{N} s_{ξ}^{∗}Φ(X^{2j+1}e^{iω̄}) (by Lemma 5.3).

(Notice that D(X) = γ(X) = 2X in our chosen normalization.) We then combine this fact with (33). □

A short calculation using the results of [16] (particularly the description of the symplectic form using the formulas (7.11)-(7.12) for the fundamental class) shows that

∫_{T^{2}} ω = −2.

This gives vol_{ω}(T^{2g}) = (−1)^{g}2^{g}. So, recalling from Proposition 4.1(c) that σ = −2, we find finally

∫_{M(2,1)} a_{2}^{j}e^{if_{2}} = \frac{i^{g−1}}{2^{g−1}} Res_{X=0}(\frac{1}{X^{2g−2−2j} sin X} ).  (34)

Thus we have recovered Witten’s formula (see (37) below): this result was first obtained by Donaldson [7] and Thaddeus [24].

Theorem 5.11. The intersection numbers of the classes a_{2} and f_{2} on M(2, 1) are given by

∫_{M(2,1)} a_{2}^{j}e^{f_{2}} = \frac{(−1)^{g−1−j}}{2^{g−1}} Res_{X=0}(\frac{1}{X^{2g−2−2j} sin X} ).  (35)

Remark: Notice that formally we can express the result of Theorem 5.10 as

∫_{M(2,1)} a_{2}^{j}e^{if_{2}} = (−1)^{g}Res_{X=0}(∑_{k≥0} \frac{(2X)^{2}X^{2j}e^{(2k+1)iσX/2} ∫_{T^{2g}} e^{iω}}{(2X)^{2g}} ),

which is what we would expect from applying the residue formula of [17] formally to M(c). However this sum does not converge in a neighbourhood of 0 and the sum of the residues at 0 does not converge.

Witten’s result (using (4.29) and (4.44) of [26]) may be expressed in the following form for j ≤ g − 2:

∫_{M(2,1)} Θ^{j}e^{ω} = \frac{(−1)^{g−1−j}}{2^{g−2}π^{2(g−1−j)}} ∑_{n=1}^{∞} \frac{(−1)^{n+1}}{n^{2(g−1−j)}}  (36)

(taking into account orientations and the fact that M(2, 1) is a 2^{2g}-fold cover of the moduli space Witten studies). To show this agrees with our result (35), we use the following fact which can be established by an elementary contour integral argument:

Lemma 5.12. For any positive integer m, we have

∑_{n>0} \frac{(−1)^{n+1}}{n^{2m}} = πiRes_{X=0} \frac{e^{iπX}}{X^{2m}(e^{2iπX} − 1)} = \frac{π^{2m}}{2} Res_{Y =0} \frac{1}{Y^{2m} sin Y} .

Using Lemma 5.12, (36) becomes

∫_{M(2,1)} Θ^{j}e^{ω} = \frac{(−1)^{g−1−j}}{2^{g−1}} Res_{Y =0} \frac{1}{Y^{2(g−1−j)}(sin Y)},  (37)

in agreement with (35).

The sum in (36) may be expressed in terms of zeta functions through the following fact which follows from an easy computation:

Lemma 5.13. For any positive integer m, we have

∑_{n>0} \frac{(−1)^{n+1}}{n^{2m}} = (1 − \frac{1}{2^{2m−1}} )ζ(2m).

Remark: The proof of Theorem 5.10 given above can be generalized to give formulas for intersection pairings involving the b_{2}^{j} as well as a_{2} and f_{2}. These were already treated in Thaddeus’ work [24]. Remark: Lemma 5.12 establishes directly that for rank 2 and degree 1 the formula (13) obtained by Witten (expressing intersection numbers in terms of a sum over irreducible representations of SU(2)) may be rewritten as the residue that appears in our proof. We can generalize this argument to cover n higher than 2, getting formulas involving iterated 1-variable residues similar to the residue in Theorem 5.10. Details will appear in a later paper [18]. For general coprime n and d the work of Szenes [23] establishes the equivalence of the sum (9) obtained by Witten and the residue (28) which appears in our proof. Szenes’ proof is a multidimensional generalization of the contour integral argument that establishes Lemma 5.12.

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Lisa C. Jeffrey, Mathematics Department, Princeton University, Princeton, NJ 08544, USA

E-mail address: jeffrey@math.princeton.edu

Frances C. Kirwan, Balliol College, Oxford OX1 3BJ, UK

E-mail address: fkirwan@vax.ox.ac.uk