Brane involutions on irreducible holomorphic symplectic manifolds

TL;DR

This paper explores brane involutions on irreducible holomorphic symplectic manifolds, their construction from K3 and abelian surfaces, and their behavior under mirror symmetry, with concrete examples on K3 surfaces and related manifolds.

Contribution

It introduces a method to construct brane involutions on moduli spaces from surfaces and analyzes their transformation under mirror symmetry, including explicit examples.

Findings

Brane involutions can be constructed from surfaces to their moduli spaces.

The behavior of brane involutions under mirror symmetry is characterized.

Examples are provided for K3 surfaces and K3^{[2]}-type manifolds.

Abstract

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists' language, i.e. a submanifold which is either complex or lagrangian submanifold with respect to each of the three K\"ahler structures of the associated hyperk\"ahler structure. Starting from a brane involution on a K3 or abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier--Mukai transform. Later, we recall the lattice-theoretical approach to Mirror Symmetry. We provide two ways of obtaining a brane involution on the mirror and we study the behaviour of the brane involutions under both mirror transformations, giving examples in the case of a K3 surface and $K3^{[2]}$-type manifolds.