A new equivariant cohomology ring

Abstract

In this paper, motivated by Chen–Ruan’s stringy orbifold theory on almost complex orbifolds, we construct a new equivariant cohomology ring ${\mathscr {H}}^*_G(X)$ for an equivariant almost complex pair (X, G), where X is a compact connected almost complex manifold, G is a connected compact Lie group which acts on X and preserves the almost complex structure.

Appendix A: Existence of equivariant volume form

In this appendix we show the existence of the equivariant volume form that we used in the definition of equivariant product $\star _m$. Let G be a connected compact Lie group and T be its maximal torus. Let $\mathfrak g$ and $\mathfrak t$ be their Lie algebras. G acts on G by conjugation and on $\mathfrak g$ by the adjoint representation.

Proposition 5

For every $h\in T$, there exists an $\alpha \in \mathfrak t$ such that $C(h)=C(\alpha )$.

Corollary 3

For every $h\in T$, G / C(h) is a Kähler manifold. Moreover, it has a G-equivariant volume form ${\varOmega }^G_h$ with equivariant integration 1.

Proof

By Proposition 5, $G/C(h)=G/C(\alpha )$ for some $\alpha \in \mathfrak t$. Since $G/C(\alpha )$ is a (co)-adjoint orbit, it is Kähler and the G-action on it is Hamiltonian. Denote its Kähler form by $\omega _\alpha$ and by $\omega _\alpha + \mu _\alpha$ the equivariant extension of $\omega _\alpha$, where $\mu _\alpha$ is the moment map for the Hamiltonian G-action on $G/C(\alpha )$. Then $(\omega _\alpha +\mu _\alpha )^d$ is an equivariant volume form, where d is the complex dimension of $G/C(\alpha )$. Suppose the equivariant integration of $(\omega _\alpha +\mu _\alpha )^d$ is $\theta \in {\mathbb {R}}$. Then $(\omega _\alpha +\mu _\alpha )^d/\theta$ has equivariant integration 1. $\square$

Proof of Proposition 5

Since T is abelian, its adjoint action on $\mathfrak g$ induces a splitting

$$\begin{aligned} \mathfrak g = \mathfrak t \oplus {\mathbb {C}}_1\oplus \cdots \oplus {\mathbb {C}}_k \end{aligned}$$

and the action is given by $1\oplus \phi _1\oplus \cdots \oplus \phi _k$. Let $T_i=\{t\in T\mid \phi _i(t)=1\}$, and for any $I\subseteq \{1,\ldots ,k\}$, set

$$\begin{aligned} T_I=\bigcap _{i\in I}T_i \end{aligned}$$

and $T_\varnothing =T$. Then T is stratified by

$$\begin{aligned} T'_I=T_I{\setminus } \bigcup \limits _{J\supseteq I} T_J. \end{aligned}$$

Similarly, let $\mathfrak t_I$ be the Lie algebra of $T_I$ and $\mathfrak t'_I$ forms a stratification of $\mathfrak t$.

Suppose that $h\in T'_I$, then choose an $\alpha \in \mathfrak t'_I$. It is sufficient to show that $\mathfrak c(h)=\mathfrak c(\alpha )$ which implies that $C(h)=C(\alpha )$, since both C(h) and $C(\alpha )$ are connected subgroup of G (cf. [20, Corollary 2, §3.1]). In fact,

$$\begin{aligned} \mathfrak c(h)=\{\xi \mid h\xi h^{-1}=\xi \}, \end{aligned}$$

since $h\in T'_I$, it fixes $\mathfrak t$ and all ${\mathbb {C}}_i$ for $i\in I$. Hence $\mathfrak c(h)=\mathfrak t\oplus \bigoplus _{i\in I}{\mathbb {C}}_i$. Similarly, $\mathfrak c(\alpha )$ is the same space.

Theorem 4

Suppose that $h_1, h_2\in T$ and ${\mathbf {h}}=(h_1,h_2)$. The fibration $p_1:G/C(h_1, h_2)\rightarrow G/C(h_1)$ admits a fiber-wise G-equivariant volume form $\text {vol}({\mathbf {h}},h_1)$, such that $p_{1,*}(\text {vol}({\mathbf {h}},h_1))=1$.

Proof

The fiber of $p_1:G/C(h_1,h_2)\rightarrow G/C(h_1)$ is $C(h_1)/C(h_1,h_2)$. Set $K=C(h_1)$. Since $h_1$ and $h_2$ commutes, $h_2\in K$. Hence $C(h_1,h_2)=C_K(h_2)$ and $C(h_1)/C(h_1,h_2)=K/C_K(h_2)$. By Corollary 3, there exists a K-equivariant volume form ${\varOmega }^K_{h_2}$ on $K/C_K(h_2)$. Note that

$$\begin{aligned} G/C(h_1,h_2)=K/C_K(h_2)\times _K G. \end{aligned}$$

For each point $[h]\in G/C(h_1)$, write the fiber over [h] by $F_{[h]}$. Then by setting $\text {vol}({\mathbf {h}},h_1)|_{F_{[h]}}=h^*{\varOmega }^K_{h_2}$ for each point $[h]\in G/C(h_1)$ we get this volume form $\text {vol}({\mathbf {h}},h_1)$. We can also get this volume form $\text {vol}({\mathbf {h}},h_1)$ via the canonical isomorphism

$$\begin{aligned} H^*_G(G/C(h_1,h_2))=H^*_G(K/C_K(h_2)\times _K G)\cong H^*_K(K/C_K(h_2)). \end{aligned}$$

Since the equivariant integration of ${\varOmega }^K_{h_2}$ over $K/C_K(h_2)$ is 1, $p_{1,*}(\text {vol}({\mathbf {h}},h_1))=1$. The proof is complete. $\square$

By induction on the length of ${\mathbf {h}}=(h_1,\ldots ,h_m)\in G_m^{\textsf {f} }$, this theorem could be generalized to $G/C({\mathbf {h}})\rightarrow G/C({\mathbf {h}}_{i_1,\ldots , i_k})$ and $G/C({\mathbf {h}})\rightarrow G/C({\mathbf {h}}_{\infty })$. Therefore all kinds of fibrations used in the definition of the equivariant product $\star _m$ have fiber-wise G-equivariant volume forms with equivariant push-forwards being 1. For example for $p_1:G/C({\mathbf {h}})\rightarrow G/C(h_1)$ with ${\mathbf {h}}=(h_1,h_2,h_3)$, we could decompose $p_1$ into

$$\begin{aligned} G/C({\mathbf {h}})\xrightarrow {p_{1,2}} G/C(h_1,h_2)\xrightarrow {p_1} G/C(h_1). \end{aligned}$$

By similar proof as Theorem 4 we get the volume form $\text {vol}({\mathbf {h}},(h_1,h_2))$ for $p_{1,2}:G/C({\mathbf {h}})\rightarrow G/C(h_1,h_2)$. The fiber-wise G-equivariant volume form for $p_1:G/C({\mathbf {h}})\rightarrow G/C(h_1)$ is

$$\begin{aligned} \text {vol}({\mathbf {h}},h_1)=\text {vol}({\mathbf {h}},(h_1,h_2))\wedge p_{1,2}^*\text {vol}((h_1,h_2),h_1). \end{aligned}$$

Moreover,

$$\begin{aligned} p_{1,*}(\text {vol}({\mathbf {h}},h_1))=p_{1,*}\circ p_{1,2,*}(\text {vol}({\mathbf {h}},(h_1,h_2))\wedge p_{1,2}^*\text {vol}((h_1,h_2),h_1))=1. \end{aligned}$$