Equivariant Lagrangian Floer homology via cotangent bundles of EGN

Internet publication

We provide a construction of equivariant Lagrangian Floer homology HFG(L0,L1), for a compact Lie group G acting on a symplectic manifold M in a Hamiltonian fashion, and a pair of G-Lagrangian submanifolds L0,L1⊂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 in the auxilliary choices involved in their construction, and are H∗(BG)-bimodules. In the case when L0=L1, we show that their chain complex CFG(L0,L1) is homotopy equivalent to the equivariant Morse complex of L0. Furthermore, if zero is a regular value of the moment map μ and if G acts freely on μ−1(0), we construct two "Kirwan morphisms" from CFG(L0,L1) to CF(L0/G,L1/G) (respectively from CF(L0/G,L1/G) to CFG(L0,L1)). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat SU(2)-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.

Authors: Cazassus, G

• Publication date: 2022-02-21
• Language: English