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We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchins proposal.
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 curvatu
We describe supersymmetric A-branes and B-branes in open N=(2,2) dynamically gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold target spaces. For a subset of toric manifolds, these equivariant branes have a mirror description
We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces $S$ of general type. We express our conjecture in terms of the Igusa cusp form $chi_{10}$ and Borcherds type lifts of three quasi-Jacobi fo
We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ing
We prove that the Hilbert scheme of $k$ points on $mathbb{C}^2$ (Hilb$^k[mathbb{C}^2]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivarian