# Branes on $G$-manifolds

## TL;DR

This paper defines G-equivariance for branes on Calabi-Yau manifolds with group actions, linking brane charges to equivariant cohomology and analyzing string spaces as G-representations, especially on flag manifolds.

## Contribution

It introduces a new notion of G-equivariance for branes and relates brane charges to equivariant cohomology, providing formulas for string space dimensions on flag manifolds.

## Key findings

- Branes on G-manifolds can be assigned equivariant charges in cohomology.
- String spaces between equivariant branes form G-representations.
- Explicit dimension formulas are derived for string spaces on flag manifolds.

## Abstract

Let $X$ be Calabi-Yau manifold acted by a group $G$. We give a definition of $G$-equivariance for branes on $X$, and assign to each equivariant brane an element of the equivariant cohomology of $X$ that can be considered as a charge of the brane. We prove that the spaces of strings stretching between equivariant branes support representations of $G$. This fact allows us to give formulas for the dimension of some of such spaces, when $X$ is a flag manifold of $G$.