Brane involutions on irreducible holomorphic symplectic manifolds

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 respec...

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Bibliographic Details
Published in:arXiv.org 2017-09
Main Authors: Franco, Emilio, Jardim, Marcos, Menet, Grégoire
Format: Article
Language:English
Subjects:
Manifolds (mathematics)
Mathematical analysis
Physicists
Sheaves
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1606.09040