An intertwining relation for equivariant Seidel maps

TL;DR

This paper extends Seidel maps to $S^1$-equivariant settings, reveals their non-commutativity with quantum products, and introduces an intertwining relation to describe this failure, with explicit computations on classical examples.

Contribution

It introduces an equivariant Seidel map framework, proves an intertwining relation for its non-commutativity, and demonstrates applications through explicit computations.

Findings

Equivariant Seidel map does not commute with the quantum product.

An intertwining relation describes the failure of commutativity.

Explicit calculations on complex plane and projective space.

Abstract

The Seidel maps are two maps associated to a Hamiltonian circle action on a convex symplectic manifold, one on Floer cohomology and one on quantum cohomology. We extend their definitions to $S^1$-equivariant Floer cohomology and $S^1$-equivariant quantum cohomology based on a construction of Maulik and Okounkov. The $S^1$-action used to construct $S^1$-equivariant Floer cohomology changes after applying the equivariant Seidel map (a similar phenomenon occurs for $S^1$-equivariant quantum cohomology). We show the equivariant Seidel map on $S^1$-equivariant quantum cohomology does not commute with the $S^1$-equivariant quantum product, unlike the standard Seidel map. We prove an intertwining relation which completely describes the failure of this commutativity as a weighted version of the equivariant Seidel map. We will explore how this intertwining relationship may be interpreted using connections in an upcoming paper. We compute the equivariant Seidel map for rotation actions on the complex plane and on complex projective space, and for the action which rotates the fibres of the tautological line bundle over projective space. Through these examples, we demonstrate how equivariant Seidel maps may be used to compute the $S^1$-equivariant quantum product and $S^1$-equivariant symplectic cohomology.