Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structures (Q1983575)

The authors define equivariant Langrangian Floer cohomology for an equivariant almost complex structure that is not necessarily regular by constructing equivariant semi-global Kuranishi structures similar to those used in [the authors, ``Semi-global Kuranishi charts and the definition of contact homology'', Preprint, \url{arXiv:1512.00580}]. Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) that is either closed or has contact type boundary, and assume that \(L_0\) and \(L_1\) are oriented Lagrangian submanifolds that intersect transversally. Let \(G\) be a group that acts symplectically on \((M,\omega)\) such that \(g(L_0) = L_0\) and \(g(L_1) = L_1\) for all \(g \in G\), and assume that \(G\) fixes the orientations on \(L_0\) and \(L_1\). Let \(J\) be a \(G\)-invariant and \(\omega\)-compatible almost complex structure on \(M\). The authors make several standard assumptions such as (S) The maps \(\pi_2(M) \stackrel{\int \omega}{\rightarrow} \mathbb{R}\) and \(\pi_2(M,L_i) \stackrel{\int \omega}{\rightarrow} \mathbb{R}\) for \(i=0,1\) have image 0. (O) The pair \((L_0,L_1)\) is equipped with a relative spin structure which is preserved by \(G\). (J) an assumption concerning the compatibility of the almost complex structure \(J\) with the contact form and Reeb vector field on a collar neighborhood of the boundary when \(\partial M \neq \emptyset\). The chain groups in the Lagrangian Floer cochain complex \(CF^\bullet(L_0,L_1)\) are free modules over a Novikov ring \(R\) generated by points in the intersection \(L_0 \cap L_1\), and the boundary operator is defined by counting \(J\)-holomorphic strips between intersection points \(p,q \in L_0 \cap L_1\). This count is given by the number of elements in a moduli space \(\mathcal{M}_J(p,q;A)\), where \(A\) represents a homotopy class of continuous strips connecting \(p\) and \(q\). In general, a \(G\)-invariant almost complex structure \(J\) won't be regular, and hence certain transversality conditions commonly used to prove that \(\mathcal{M}_J(p,q;A)\) is a manifold won't hold. In the words of the authors, ``The main contribution of this paper is to obtain a \(G\)-equivariant cochain complex \(CF^\bullet(L_0,L_1)\) when \(J\) is not regular by constructing an equivariant version of a semi-global Kuranishi structure, initially developed in [\textit{E. Bao} and \textit{K. Honda}, ``Semi-global Kuranishi charts and the definition of contact homology'', Preprint, \url{arXiv:1512.00580}] for contact homology''. The bulk of the paper consists of detailed constructions of equivariant semi-global Kuranishi structures used to define boundary maps, chain maps, and chain homotopies for equivariant Lagrangian Floer cohomology. For instance, to define the boundary operator the authors replace the moduli space \(\mathcal{M}_J(p,q;A)\), which might not be a manifold, with a space \(\mathcal{Z}(\mathscr{K}(p,q;A), \mathfrak{G})\) defined by the Kuranishi structure which is a manifold by Lemma 2.7.3 of the paper. The constructions of the required Kuranishi structures imply the following main theorem: \textbf{Theorem 1.0.3} (equivariant Lagrangian Floer cohomology) Suppose \(G\) acts on \((M,\omega)\) symplectically and, for each \(i=0,1\), \(L_i\) is oriented, \(g_i(L_i) = L_i\) for each \(g \in G\), and \(G\) fixes the orientation of \(L_i\). If (S) and (O) hold, then there exists an \(R\)-module \(HF_G^\bullet(L_0, L_1)\) which is an invariant of \((L_0,L_1)\) under \(G\)-equivariant Hamiltonian isotopy. Moreover, when there exists a regular \(G\)-invariant \(\omega\)-compatible almost complex structure on \(M\) satisfying (J), the usual definition of equivariant Lagrangian Floer cohomology can be made and agrees with \(HF_G^\bullet(L_0, L_1)\).