Connections on equivariant Hamiltonian Floer cohomology (Q1989597)

In this article, the author constructs several endomorphisms of the \(S^1\)-equivariant Hamiltonian Floer cohomology of a symplectic manifold \((M,\omega)\). The \(S^1\)-equivariant Floer cohomology is a module over the ring \(\Lambda[[u]]\), the power series ring in the variable \(u\) over the Novikov ring \(\Lambda\) over a variable \(q\). The variable \(q\) corresponds to evaluating the symplectic class on a trajectory. The variable \(u\) can be thought of as corresponding to the generator of \(H^*(BS^1)\). By formally differentiating the counts of trajectories with respect to the variables \(q\) or \(u\), the author constructs two endomorphisms, \(\Gamma_q\) and \(\Gamma_u\), of the \(S^1\)-equivariant Hamiltonian Floer cohomology. The map \(\Gamma_q\) satisfies \[ \Gamma_q(f x)=f \Gamma_q(x)+u (\partial_q f) x \] for \(f\in \Lambda[[u]]\) and \(x\) in the equivariant Floer cohomology. The connection \(\Gamma_q\) reduces to the quantum cap product with the symplectic class on the non-equivariant version of Hamiltonian Floer homology. When the symplectic manifold is monotone, the \(S^1\)-equivariant Floer cohomology can be defined without the Novikov parameter \(q\), and there is a map \(\Gamma_u\) which satisfies \[ \Gamma_u(f x)=f \Gamma_u(x)+2 u^2 (\partial_u f) x, \] for \(f\in \mathbb{Z}[[u]]\). Furthermore, the map \(\Gamma_u\) reduces to the quantum cap product with the Chern class of the symplectic manifold on non-equivariant Hamiltonian Floer cohomology. The construction is inspired by the Gauss-Manin connection on algebraic De Rham cohomology [\textit{N. M. Katz} and \textit{T. Oda}, J. Math. Kyoto Univ. 8, 199--213 (1968; Zbl 0165.54802)], as well as a connection on periodic cyclic homology of a family of \(dg\)-algebras, due to \textit{E. Getzler} [in: Quantum deformations of algebras and their representations. Papers presented at the BSF-Gelbart Institute workshop held at Bar-Ilan University, Ramat-Gan and the Weizmann Institute, Rehovot, Israel, Dec. 29-Jan. 3, 1992. Ramat-Gan: Bar-Ilan University; Providence, RI: American Mathematical Society. 65--78 (1993; Zbl 0844.18007)].