Quantization commutes with reduction for coisotropic A-branes

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 $B_{\mathrm{red}}$ on $X//G$ from $B$, recovering the usual construction when $B$ is Lagrangian. For a canonical coisotropic A-brane $B_{\mathrm{cc}}$ on a holomorphic Hamiltonian $G_{\mathbb{C}}$-manifold $X$, there is a fibration of $(B_{\mathrm{cc}}{)}_{\mathrm{red}}$ over $X//G_{\mathbb{C}}$. We also show that `intersections of A-branes commute with reduction'. When $X=T^{\ast }M$ for $M$ being compact Kähler with a Hamiltonian $G$-action, Guillemin-Sternberg `quantization commutes with reduction' theorem can be interpreted as ${\mathrm{Hom}}_{X//G}(B_{\mathrm{red}},(B_{\mathrm{cc}}{)}_{\mathrm{red}})\cong {\mathrm{Hom}}_X(B,B_{\mathrm{cc}}{)}^G$ with $B=M$.

27 pages