Quantization commutes with reduction again: the quantum GIT conjecture I

TL;DR

This paper proves a version of the 'quantization commutes with reduction' principle for symplectic manifolds with Hamiltonian group actions, relating gauged A-models to reduced models and establishing equalities in quantum cohomology.

Contribution

It establishes a new quantum GIT conjecture relating gauged A-models to reduced models, extending the classical quantization commutes with reduction theorem.

Findings

Equality of quantum cohomology spaces for gauged and reduced models

Extension of the quantization commutes with reduction to the symplectic and quantum setting

Outline of proof and future directions for non-monotone cases

Abstract

For a compact monotone symplectic manifold $X$ with Hamiltonian action of a compact Lie group $G$ and smooth symplectic reduction, we relate its gauged $2$-dimensional $A$-model to the $A$-model of $X/\!/G$. This (long conjectured) result is parallel to the ($B$-model!) \emph{quantization commutes with reduction} theorem of Guillemin and Sternberg in quantum mechanics. Here, we spell out some of the precise statements, and outline the proof of equality for the spaces of states (quantum cohomology). We also indicate the way to some related results in the non-monotone case. Additional Floer theory details will be included in a follow-up paper.