Shift operators and connections on equivariant symplectic cohomology

TL;DR

This paper develops shift operators in equivariant symplectic cohomology, generalizing known structures from algebraic geometry, and proves their properties including commutation with a flat connection, with applications to toric manifolds.

Contribution

It introduces and constructs shift operators on equivariant symplectic cohomology, extending the concept from algebraic geometry, and proves the flatness of the associated connection.

Findings

Constructed shift operators on equivariant symplectic cohomology.

Proved the shift operators commute with a flat, multivariate Seidel $q$-connection.

Computed algebraic structures for toric manifolds.

Abstract

We construct shift operators on equivariant symplectic cohomology which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus $T$, we assign to a cocharacter of $T$ an endomorphism of $(S^1 \times T)$-equivariant Floer cohomology based on the equivariant Floer Seidel map. We prove the shift operator commutes with a connection. This connection is a multivariate version of Seidel's $q$-connection on $S^1$-equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology. We prove that the connection is flat, which was conjectured by Seidel. As an application, we compute these algebraic structures for toric manifolds.