A survey of equivariant operations on quantum cohomology for symplectic manifolds

TL;DR

This survey reviews various ideas and results on equivariant operations in quantum cohomology and Floer theory for symplectic manifolds, highlighting general notions, examples, and connections to Floer invariants.

Contribution

It compiles and discusses different approaches and results on equivariant quantum operations, including finite and circle group actions, providing a comprehensive overview.

Findings

Discussion of equivariant quantum operations for finite groups

Connection between equivariant operations and Floer invariants

Outline of $S^1$-equivariant operations and mod-$p$ pseudocycles

Abstract

In this survey paper, we will collate various different ideas and thoughts regarding equivariant operations on quantum cohomology (and some in more general Floer theory) for a symplectic manifold. We will discuss a general notion of equivariant quantum operations associated to finite groups, in addition to their properties, examples, and calculations. We will provide a brief connection to Floer theoretic invariants. We then provide abridged descriptions (as per the author's understanding) of work by other authors in the field, along with their major results. Finally we discuss the first step to compact groups, specifically $S^1$-equivariant operations. Contained within this survey are also a sketch of the idea of mod-$p$ pseudocycles, and an in-depth appendix detailing the author's understanding of when one can define these equivariant operations in an additive way.