Branes in the moduli space of framed instantons

TL;DR

This paper explores the geometric structures called branes within hyperkähler manifolds, specifically in Nakajima quiver varieties and moduli spaces of sheaves, revealing new instances of these submanifolds through involutions.

Contribution

It characterizes branes in Nakajima quiver varieties and demonstrates their existence in moduli spaces of sheaves via involution actions, extending the understanding of branes in these settings.

Findings

Branes are described as fixed loci of involutions in quiver varieties.

Existence of branes in moduli spaces of sheaves is established.

Involutions act on sheaves, leading to new geometric structures.

Abstract

In the physicist's language, a brane in a hyperkahler manifold is a submanifold which is either complex or lagrangian with respect to three Kahler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.