Mirror symmetry for Nahm branes

TL;DR

This paper generalizes the Nahm transform to higher rank Higgs bundles, constructing hyperholomorphic bundles called Nahm branes, and analyzes their behavior under Fourier--Mukai transform, linking geometric and physical perspectives.

Contribution

It extends the Dirac--Higgs bundle construction to higher rank and degree zero cases, defining Nahm branes and studying their properties under Fourier--Mukai transform.

Findings

Construction of hyperholomorphic bundles (Nahm branes) over moduli spaces of Higgs bundles.

Demonstration that Nahm branes transform to Lagrangian multisections under Fourier--Mukai.

Connection of these geometric objects to BAA- and BBB-branes in physics.

Abstract

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.