Lie monoid actions in Floer theory

Establish that for nice symplectic manifolds M and M′ with symplectic actions of compact Lie monoids G×H and G′×H′, invariant Lagrangians L0, L1⊂M, and a (G×H)-invariant Lagrangian correspondence Λ⊂M−×M′ well-composable on the left, the following hold: (i) CF(L0) and CF(L1) carry u-bimodule structures; (ii) CF(L0, L1) is a quadrimodule over the square built from CM(G), CF(L0), CF(L1), and CM(H) as specified; (iii) the subcategories of the Fukaya co-categories F(M) and F(M′) consisting of invariant Lagrangians upgrade to u-bimodule d-categories over (CM(G), CM(H)) and (CM(G′), CM(H′)); and (iv) the A∞ functor Φ_Λ: F(M)→F(M′) upgrades to a (φ*, ψ*)-equivariant functor of u-bimodule d-categories.

Background

Motivated by the Morse-theoretic framework of f-bialgebras, the authors propose Floer-theoretic analogues capturing how Lie monoid actions interact with Fukaya-type structures. The proposed structures rely on moduli of biforests and their compactifications acting as parameter spaces for operations on Floer chains.

The conjecture specifies that, in the presence of suitable symplectic group actions and invariant Lagrangians, Floer complexes should admit u-bimodule and quadrimodule structures compatible with the monoid actions, and Lagrangian correspondences should induce equivariant A∞-functors at the chain level.