8 Equivariant Cohomology References: Audin §5, Berline-Getzler-Vergne §7 . Definition 8.1 Let G be a compact Lie group. The universal bundle EG is a con- tractible space on which G acts freely. Definition 8.2 The classifying space BG is BG = EG/G. Proposition 8.3 H∗(BG)= S(g∗)G = S(t∗)W (polynomials on g invariant under the adjoint action, or polynomials on t invariant under the Weyl group action) Here, the degree in H∗(BG) is twice the degree as a polynomial on g. (Chern-Weil: evaluate polynomials on curvature) Example 8.4 S1 acts freely on all S2n+1, and these have homology only in dimen- sions 0 and 2n + 1. The universal space EU(1) is ∞ 2 S = {(z1, z2,...) ∈ C ⊗ Z : only finitely many nonzero terms, |zj| = 1} Xj = S1 ∪ S3 ∪ . 2n−1 2n+1 ∞ where S → S via (z1,...,zn) 7→ (z1,...,zn, 0). The space S is in fact contractible, so it is EU(1). Lemma 8.5 BU(1) = EU(1)/U(1) = CP ∞ Proposition 8.6 H∗(BU(1)) = C[x] where x has degree 2. Recall C[x] H∗(CP n)= . < xn+1 = 0 > 8.1 Homotopy quotients Suppose M is a manifold acted on by a compact Lie group (not necessarily freely). We want to find a substitute for the cohomology of M/G, which is in general not a manifold. Definition 8.7 ∗ ∗ HG(M)= H (MG) where we define the homotopy quotient MG =(M × EG)/G. 35 Proposition 8.8 If G acts freely on M then M/G is a smooth manifold and ∗ ∗ HG(M)= H (M/G) ∗ More generally, HG(M) is a module over the ring ∗ ∗ ∗ HG = HG(pt) = H (BG). 8.2 The Cartan model ∗ De Rham cohomology version of H (MG): Definition 8.9 ∗ ∗ ∗ G ΩG(M)=(Ω (M) ⊗ S(g )) where we have defined S(g∗)= {f : g → R : fis a polynomial} Here, S(g∗) is acted on by G through the coadjoint action of G on g∗. Example 8.10 T abelian: ∗ ∗ T ∗ ΩT (M) = Ω (M) ⊗ S(t ) since all polynomials on t are automatically invariant, because the action of T on t is trivial. Lemma 8.11 ∗ S(t )= C[x1,...,xℓ] where ℓ = dim(T ) Proposition 8.12 S(g∗)G = S(t∗)W where the Weyl group W acts on t ∗ Let X be a vector field on M. An element f ∈ ΩG(M) may be thought of as a G- equivariant map f : g → Ω∗(M), where the dependence of f(X) ∈ Ω∗(M) on X ∈ g ∗ is polynomial. The grading on ΩG(M) is defined by deg(f)= n + 2p if X 7→ f(X) is p-linear in X and f(X) ∈ Ωn(M). We may define a differential ∗ ∗ D : ΩG(M) → ΩG(M) by (Df)(X)= d(f(X)) − ιX# f(X) where X# is the vector field on M generated by the action of X ∈ g and ι denotes ∗ the interior product. Then D ◦ D = 0 and D increases the degree in ΩG(M) by 1. 36 ∗ Theorem 8.13 (Cartan [13]) HG(M) is naturally isomorphic to the cohomology ∗ ∗ H (ΩG(M),D) of this complex. ∗ ∗ ∗ ∗ G In particular, since D = 0 on ΩG, we see that HG = HG(pt)= S(g ) . If (M,ω) is a symplectic manifold equipped with the Hamiltonian action of a compact group G, with moment map Φ, we define 2 ω¯(X)= ω + ΦX ∈ ΩG(M). Lemma 8.14 Dω¯ = 0 Proof: (Dω¯)(X)= dω − iX# ω + dΦX But iX# ω = dΦX by definition of Hamiltonian group actions. 2 So Dω¯ = 0 and we can define [¯ω] ∈ HG(M). Definition 8.15 A principal circle bundle over a point p equipped with the action of a torus T is P = S1 equipped with a weight β ∈ Hom(T, U(1)). Thus T acts on S1 by t ∈ T : z ∈ S1 7→ β(t)z. Denote this bundle by Pβ. Example 8.16 T = U(1), β(t)= tm for m ∈ Z Denote this bundle by Pm. 8.3 Characteristic classes of bundles over BU(1) and BT Example 8.17 EU(1) ×U(1) Pm is the homotopy quotient of Pm. Explicitly this means {(z,w) ∈ EU(1) ×U(1) Pm}/ ∼ where (z,w) ∼ (zu−1, umw). Every point (z,w) is equivalent to a point (z′, 1) by choosing u = w−1/m so (z,w) ∼ (zw1/m, 1). 37 Since there are m solutions to u = w−1/m, any two of which differ by multiplication by a power of e2πi/m, we see that EU(1) ×U(1) Pm = EU(1)/Zm where 2πir/m Zm = {e , r = 0,...,m − 1}. Connection form θm on EU(1) ×U(1) Pm: Note that a connection form θ on EU(1) satisfies π−1(b) θ = 1. We also expect θm to satisfy R θm = 1. −1 Zπm (b) But since each fibre of EU(1) may be written as {eiφ,φ ∈ [0, 2π]} and the fibre of πm iφ 1 EU(1)/Z → BU(1) corresponds to {e ,φ ∈ [0, 2π/m]} , we have −1 θ = so m πm (b) m we need R Lemma 8.18 θm = mθ in terms of our earlier θ. It follows that the first Chern class c1(Pm) of the principal circle bundle Pm (which is represented in Chern-Weil theory by the curvature of dθm) satisfies Lemma 8.19 c1(Pm)(X)= mX ∗ where c = c1(EU(1) → BU(1)) is the generator of H (BU(1)). This is a special case of Lemma 8.20 Let T → P → M be a principal T -bundle with connection 1 θ =(θ1,...,θℓ) ∈ Ω (P ) ⊗ t. 1 Let β ∈ Hom(T, U(1)). Form the associated principal circle bundle P ×T S := {(p,s) ∈ P × S1}/ ∼ where (p,s) ∼ (pt,β(t−1)s) for t ∈ T . Write Lie(β) : t → R. Lie(β)= {(b1, . , bℓ)} 1 (bj ∈ Z). Then a connection form on P ×T S is ℓ bjθj = Lie(β)(θ). Xj=1 38 2 ∗ Lemma 8.21 If (x1,...,xℓ) ∈ H (BT ) are the generators of H (BT ) for a torus T of rank ℓ, then the first Chern class of the associated principal circle bundle 1 ET ×T (S )β specified by the weight β is ℓ 1 c1(ET ×T (S )β)= bjxj. Xj=1 8.4 Characteristic classes in terms of the Cartan model Definition 8.22 A G-equivariant vector bundle over a G-manifold M is a vector bundle V → M with an action of G on the total space covering the action of G on M. An equivariant principal circle bundle P → M is a principal circle bundle with the action of G on the total space P covering the action of M. Lemma 8.23 Suppose P → M is a principal circle bundle with connection θ ∈ 1 Ω (P ). Its first Chern class is represented in de Rham cohomology by c1(P ) = [dθ]. Lemma 8.24 If P →π M is a G-equivariant principal U(1)-bundle then its equivari- ant first Chern class is represented in the Cartan model by G c1 (P ) = [Dθ] = [dθ − iX# θ] where X# is the vector field on P generated by X ∈ g. ∗ Proof: For a collection of sections sα : Uα ⊂ M → P , sαDθ is closed but not ∗ exact in ΩG(M). In particular, if M is a point and P = Pβ = U(1) equipped with β ∈ Hom(T, U(1)), then c1(Pβ) = [Dθ]= −(Lie(β))(X)= −iX# θ (since dθ =0 on M). Lemma 8.25 If P → M is a principal U(1) bundle with T action, and the T action on M is trivial (but the T action on the total space of P is not trivial), then on each fibre π−1(m) ∼= S1, the T action is given by a weight β ∈ Hom(T, U(1)). Then T c1 (P ) = [Dθ] = [dθ − β(X)]. Remark 8.26 Atiyah-Bott p. 9 have a different convention on characteristic num- bers. One obtains their convention from ours by replacing X by −X. Our convention is consistent with Berline-Getzler-Vergne §7.1. The situation of the preceding Lemma arises in the following context: Lemma 8.27 Let M be equipped with a T action, and let F be a component of M T . ∗ For α ∈ HT (M) and iF : F → M the inclusion map, ∗ ∗ ∗ ∗ ∗ R iF α ∈ HT (F )= H (F ) ⊗ HT (pt) = H (F ) ⊗ [x1,...,xℓ]. 39 8.5 Equivariant first Chern class of a prequantum line bundle Definition 8.28 Let (M,ω) be a symplectic manifold with a Hamiltonian action of a group G. A prequantum bundle with connection is a principal circle bundle P → M ∗ for which c1(P ) = [ω], equipped with a connection θ for which dθ = π ω. Lemma 8.29 If we impose the condition that LX# θ = 0, then diX# θ = −iX# dθ = −iX# ω = −dΦX . It is thus natural to also impose the condition iX# θ = −ΦX . Thus the specification of a moment map for the group action is equivalent to specifying a lift of the action of T from M to the total space P . Lemma 8.30 If F ⊂ M is a component of the fixed point set, then ∗ 2 iF ω¯(X) ∈ ΩT (F ) = ω|F + ΦX (F ). 1 Proof: For any f ∈ F , P |f is a copy of S on which T acts using a weight βF ∈ Hom(T, U(1)).