Equivariant Lagrangian Floer homology via cotangent bundles of E G N $EG_N$

Abstract

We provide a construction of equivariant Lagrangian Floer homology $H{F}_G(L_0,L_1)$, for a compact Lie group $G$ acting on a symplectic manifold $M$ in a Hamiltonian fashion, and a pair of $G$-Lagrangian submanifolds $L_0,L_1\subset M$. We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of $EG$. Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are $H^{\ast }\left(BG\right)$-bimodules. In the case when $L_0=L_1$, we show that their chain complex $C{F}_G(L_0,L_1)$ is homotopy equivalent to the equivariant Morse complex of $L_0$. Furthermore, if zero is a regular value of the moment map $\mu$ and if $G$ acts freely on ${\mu }^{-1}\left(0\right)$, we construct two ‘Kirwan morphisms’ from $C{F}_G(L_0,L_1)$ to $CF(L_0/G,L_1/G)$ (respectively, from $CF(L_0/G,L_1/G)$ to $C{F}_G(L_0,L_1)$). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat $SU\left(2\right)$-connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.