Mirror symmetry for Nahm branes

The Dirac-Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We extend this construction to the case of arbitrary rank $n$ and degree $0$, studying the associated connection and curvature. We then generalize to the case of rank $n > 1$ 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 spaces of rank $1$ Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle 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. Our 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 then use the Fourier-Mukai and Fourier-Mukai-Nahm transforms to describe the corresponding dual branes restricted to the smooth locus of the Hitchin fibration. The dual branes are checked to be $(BAA)$-branes supported on a complex Lagrangian multisection of the Hitchin fibration.Comment: 34 pages. Exposition greatly improve

O'Grady spaces and symplectic resolution of moduli spaces of Higgs bundles

We describe here a degeneration of the symplectic desingularization of the moduli spaces of topologically trivial $GL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$-Higgs bundles over a hyperelliptic curve, into O'Grady's ten and six dimensional exceptional examples of irreducible holomorphic symplectic manifolds. Most of this note is a survey of work on these degenerations by Donagi-Ein-Lazarsfeld, de Cataldo-Maulik-Shen and Felissetti-Mauri, although in certain cases we provide details that were missing the previous articles

Higgs bundles over elliptic curves

In this paper we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study the generic and non-generic fibres