Equivariant branes

TL;DR

This paper introduces the concept of equivariant branes on Calabi-Yau manifolds with group actions, extending the theory of equivariant sheaves, and analyzes the representation structure of string spaces between such branes.

Contribution

It defines equivariant branes and their charges, and demonstrates that string spaces between them carry natural group representations, with explicit decompositions in special cases.

Findings

String spaces support group representations.

Decomposition into invariant subspaces is possible.

Explicit decompositions for toric varieties and flag manifolds.

Abstract

Given a Calabi-Yau manifold $X$ acted by a group $G$ and considering the $B$-branes on $X$ as objects in the derived category of coherent sheaves, we give a definition of equivariant branes, which generalizes the concept of equivariant sheaves. We also propose a definition of equivariant charge of an equivariant brane. The spaces of strings joining the branes ${\mathcal F}$ and ${\mathcal G}$, are the groups $Ext^i({\mathcal F},\,{\mathcal G})$. We prove that the spaces of strings between two $G$-equivariant branes support representations of $G$. Thus, these spaces can be decomposed in direct sum of invariant spaces for the $G$-action. We show some particular decompositions, when $X$ is a toric variety and when $X$ is a flag manifold of a semisimple Lie group.