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Register a new userMirror symmetry for Nahm branes
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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.
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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.
We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-Hubsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism $s$ of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-Hubsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.
Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.
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.
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Eulers $Gamma$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.
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