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Register a new userMirror symmetry with branes by equivariant Verlinde formulae
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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.
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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 physicists 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.
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 as branes in gauged Landau-Ginzburg models with neutral matter. We then study correlation functions in the topological A-twisted version of the GNLSM, and identify their values with open Hamiltonian Gromov-Witten invariants. Supersymmetry breaking can occur in the A-twisted GNLSM due to nonperturbative open symplectic vortices, and we canonically BRST quantize the mirror theory to analyze this phenomenon.
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 forms which are all related to the Weierstrass elliptic function. We also conjecture that the generating function of virtual cobordism classes of these moduli spaces depends only on $chi(mathcal{O}_S)$ and $K_S^2$ via two universal functions, one of which is determined by the cobordism classes of Hilbert schemes of points on $K3$. We present generalizations of these conjectures, e.g. to arbitrary surfaces with $p_g>0$ and $b_1=0$. We use a result of J. Shen to express the virtual cobordism class in terms of descendent Donaldson invariants. In a prequel we used T. Mochizukis formula, universality, and toric calculations to compute such Donaldson invariants in the setting of virtual $chi_y$-genera. Similar techniques allow us to verify our new conjectures in many cases.
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 ingredients are the Bialynicki-Birula theory of ${mathbb C}^*$-actions on semiprojective varieties, ${mathbb C}^*$ characters of indices of ${mathbb C}^*$-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier-Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles.
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 equivariant K-theory is invariant upon interchanging its Kahler and equivariant parameters as well as inverting the weight of the $mathbb{C}^times_hbar$-action. First, we find a two-parameter family $X_{k,l}$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb$^k[mathbb{C}^2]$ is obtained via direct limit $ltoinfty$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted $hbar$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$N$ sheaves on $mathbb{P}^2$ with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.
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