$S^1$-localisation by pseudocycles, lifts to $S^1$-localisation of moduli spaces, and application to invariants of $S^1$-equivariant symplectic cohomology

Abstract

We demonstrate a way to apply $S^1$-localisation to moduli spaces of holomorphic curves. We first prove a reinterpretation of Atyiah-Bott $S^1$-localisation, called {\it localisation by pseudocycles} (LbP), for a smooth semifree $S^1$-action on a manifold. We demonstrate that, for certain moduli spaces of holomorphic curves parametrised by some stratum of the homotopy quotient of a manifold, we may ``lift" the LbP procedure from the parameter space to the moduli space. As an application we deduce relations between equivariant symplectic classes and Gromov-Witten invariants, thus proving a conjecture of Seidel.

arXiv comment42 pages, 3 figures. v2: most of the paper rewritten for quality of life improvements (no changes in mathematics behind proofs, but corrections to errors, restructured, improved notation, removed irrelevant information, added details to crucial arguments, etc). Changed title and abstract. Please do let me know of any errors that are spotted, or any abstruse sections of writing