Quantization commutes with reduction for coisotropic A-branes

TL;DR

This paper explores how quantization and reduction processes for coisotropic A-branes on Hamiltonian G-manifolds are compatible, extending classical results and defining new notions of invariance and reduction.

Contribution

It introduces a notion of G-invariance for coisotropic A-branes, constructs reduced branes via a Marsden-Weinstein-Meyer type process, and interprets quantization commutes with reduction in this context.

Findings

Construction of reduced coisotropic A-branes from G-invariant ones.

Establishment of a fibration of reduced branes over the quotient space.

Interpretation of Guillemin-Sternberg quantization commutes with reduction theorem in terms of Hom spaces.

Abstract

On a Hamiltonian $G$-manifold $X$, we define the notion of $G$-invariance of coisotropic A-branes $B$. Under neat assumptions, we give a Marsden-Weinstein-Meyer type construction of a coisotropic A-brane $B_{\operatorname{red}}$ on $X // G$ from $B$, recovering the usual construction when $B$ is Lagrangian. For a canonical coisotropic A-brane $B_{\operatorname{cc}}$ on a holomorphic Hamiltonian $G_\mathbb{C}$-manifold $X$, there is a fibration of $(B_{\operatorname{cc}})_{\operatorname{red}}$ over $X // G_\mathbb{C}$. We also show that `intersections of A-branes commute with reduction'. When $X = T^*M$ for $M$ being compact K\"ahler with a Hamiltonian $G$-action, Guillemin-Sternberg `quantization commutes with reduction' theorem can be interpreted as $\operatorname{Hom}_{X // G}(B_{\operatorname{red}}, (B_{\operatorname{cc}})_{\operatorname{red}}) \cong \operatorname{Hom}_X(B, B_{\operatorname{cc}})^G$ with $B = M$.