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

TL;DR

The paper introduces a new $S^1$-localisation method called localisation by pseudocycles, enabling the lifting of localisation procedures to moduli spaces of holomorphic curves, and applies it to relate symplectic classes and Gromov-Witten invariants, confirming Seidel's conjecture.

Contribution

It develops a novel $S^1$-localisation technique for moduli spaces of holomorphic curves and applies it to establish new relations in symplectic geometry.

Findings

Established localisation by pseudocycles (LbP) for semifree $S^1$-actions.

Lifted $S^1$-localisation from parameter spaces to moduli spaces.

Proved a conjecture relating equivariant symplectic classes and Gromov-Witten invariants.

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.