ar X iv : m at h / 98 04 13 5 v 1 [ m at h . D G ] 2 9 A pr 1 99 8 DEGENERATE CHERN - WEIL THEORY AND EQUIVARIANT COHOMOLOGY

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. As an application, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions on compact manifolds with boundary. One of the formulas is then used to yield a very simple proof of a localization formula due to Jeffrey-Kirwan [10] in the case of G = SU(2) or SO(3). Throughout this paper, G will be a compact connected Lie group, with g as its Lie algebra. Assume that G acts freely on a smooth manifold P . Then the quotient map P → P/G = M gives P a structure of principal G-bundle. The celebrated Chern-Weil theory gives us a homomorphism cw : S(g) → H(M), (1) called the Chern-Weil homomorphism. Here S(g) is the algebra of polynomials on g which is invariant under the adjoint representation of G on g. The Chern-Weil construction uses a connection 1-form ω ∈ (Ω(P ) × g) and its curvature 2-form Ω = dω + 1 2 [ω, ω]. The equation dΩ = [Ω, ω] can be used to show that for any invariant polynomial F ∈ S(g), F (Ω) is the pullback of a closed form on M . This defines the homomorphism (1). Furthermore, for two connections ω and ω with curvatures Ω and Ω respectively, there is a canonically defined differential form T(ω0,ω1)F on M , called the transgression form, such that dT(ω0,ω1)F = F (Ω )− F (Ω). Therefore, the Chern-Weil homomorphism is independent of the choice of ω. We call this Chern’s formulation. Cartan [5] presented Weil’s formulation, which we review in §1. Through Weil’s formulation, Cartan (§5 in [6]) discovered that the Chern-Weil homomorphism can be factored as S(g) φ → H G(P ) r → H(M), whereH G(P ) is the equivariant cohomology of P , and φ is the homomorphism which gives H G(P ) the structure of an H (BG) ∼= S(g)-module. The homomorphism r can be obtained by a similar Chern-Weil construction. In this paper, we shall generalize the above picture to the case that the G-action on a smooth manifold W is only locally free on a dense open set W 0 ⊂ W . Using a connection ω on W , and a cutoff function f , we shall construct homomorphisms cw f : S(g ) → H G(W ), 1991 Mathematics Subject Classification: Primary 55N91, 57R20, 57S15, 58F05. The authors are supported in part by NSF 1 2 HUAI-DONG CAO & JIAN ZHOU and r f : H ∗ G(W ) → H G(W ), such that cw f = r G f ◦ φ. We shall also show that these homomorphisms are independent of the choices of connection ω and the cut-off function f . The main results are stated in Theorem 2.1-2.5. We call these results the degenerate Chern-Weil theory. We remark that our approach corresponds to Chern’s formulation. It is interesting to find a Weil’s formulation. The constructions in this paper are motivated by our earlier work [4], where a localization formula for circle action due to Kalkman [12] is used to obtain wall crossing formulas in Seiberg-Witten theory due to Li-Liu [15] and Okonek-Teleman [18]. As applications of our degenerate Chern-Weil theory, we prove two nonabelian localization formulas (Theorem 3.1 and Theorem 3.2) of Kalkman type for G = SU(2) and SO(3). Theorem 3.1 is very useful in the study of various wall crossing phenomenon. In a forthcoming paper, we shall apply Theorem 3.1 to study wall crossing phenomenon in symplectic reduction. On the other hand, though moduli spaces in Donaldson theory are in general noncompact and our results do not yet readily apply to the study of wall crossing phenomenon of Donaldson invariants, we believe suitable modifications should yield some results in this direction. We shall leave this issue for future investigations. As an application of Theorem 3.2, we shall give a very simple proof of the nonabelian localization formula of Jeffrey-Kirwan [10] in the case of Hamiltonian SU(2) or SO(3)-actions. The rest of the paper is organized as follows. In §1 we review the equivariant cohomology and fix some notations. The degenerate Chern-Weil theory is presented in §2. In §3 we prove two nonabelian localization formulas of Kalkman type. The application of the second formula to symplectic reduction is given in §4. Acknowledgements. We would like to thank Professor Blaine Lawson for his interest in our work. The research in this paper is carried out during the second author’s visit at Texas A&M University. He thanks the Department of Mathematics and the Geometry, Analysis and Topology group for hospitality and financial support. He also finds the lecture notes from Professor Blaine Lawson’s courses on Chern-Weil theory [14] extremely useful. 1. Preliminaries on equivariant cohomology We will use two differential geometric models, the Weil model and the Cartan model, for equivariant cohomology. We refer the reader to Atiyah-Bott [1], Cartan [5, 6], Kalkman [11], Lawson [14] and Mathai-Quillen [17] and the references therein for more details. The Weil algebra [5] is the Hopf algebra W (g) = Λ(g)⊗ S(g), where elements in Λ(g) have degree 1, and elements in S(g) have degree 2. Let {ξa} be a basis of g, such that [ξa, ξb] = f c abξc, where f c ab’s are the structure constants. Let {θa} be a dual basis in Λ(g), and {Θa} a dual basis in S(g). Define the Weil differential dw : W (g) → W (g) by DEGENERATE CHERN-WEIL THEORY AND EQUIVARIANT COHOMOLOGY 3 setting dwθ a = − 2 f bcθ θ +Θ, dwΘ a = −f bcθΘ and extending it as a derivation of degree 1. There are also contractions ia and Lie derivatives La on W (g) defined by iaθ b = δ a, Laθ b = −f b acθ, iaΘ b = 0, LaΘ b = −f b acΘ. Notice that G acts on W (g) by extending the co-adjoint representation. Its linearization can be identified with La’s. It is easy to verify the homotopy formula La = dwia + iadw. Let X be a compact smooth G-manifold. Denote by (Ω(X), d) the de Rham complex of X . The G-action on X induces a homomorphism from the Lie algebra g to the Lie algebra of vector fields on X . Denote by ιa and La the contraction and the Lie derivative by the vector field corresponding to ξa ∈ g respectively. Let (W (g)⊗ Ω(X))basic be the subalgebra of W (g)⊗Ω(X) consisting of elements fixed by the G-action, and annihilated by ia⊗1+1⊗ ιa. This subalgebra is invariant under dw ⊗1+1⊗d, the corresponding complex is called the Weil model. The cohomology of Weil model is called the equivariant cohomology and denoted by H G(X). Cartan model is given by the complex (ΩG(X), DG), where ΩG(X) = (S(g ) ⊗ Ω(X)), andDG = 1⊗d−Θ⊗ιa, called the Cartan differential. When there is only one Lie group involved, we will use D for DG. Since D is a G-invariant operator on S(g) ⊗ Ω(X), it then maps ΩG(X) to itself. Furthermore, since Θ ⊗ La acts as zero on S(g), we have D = −Θ ⊗ La = −Θ(La ⊗ 1 + 1⊗ La). Therefore, D = 0 on ΩG(X) = (S(g ∗)⊗ Ω(X)). It is possible to identify H G(X) with H (ΩG(X), D) through an isomorphism Ψ : W (g)⊗ Ω(X) → W (g)⊗ Ω(X) defined by Ψ = ∏