Equivariant Lagrangian Floer homology via cotangent bundles of EG_N

We provide a construction of equivariant Lagrangian Floer homology (Formula presented.), for a compact Lie group (Formula presented.) acting on a symplectic manifold (Formula presented.) in a Hamiltonian fashion, and a pair of (Formula presented.) -Lagrangian submanifolds (Formula presented.). We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of (Formula presented.). 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 (Formula presented.) -bimodules. In the case when (Formula presented.), we show that their chain complex (Formula presented.) is homotopy equivalent to the equivariant Morse complex of (Formula presented.). Furthermore, if zero is a regular value of the moment map (Formula presented.) and if (Formula presented.) acts freely on (Formula presented.), we construct two ‘Kirwan morphisms’ from (Formula presented.) to (Formula presented.) (respectively, from (Formula presented.) to (Formula presented.)). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat (Formula presented.) -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.