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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.
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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.
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 study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchins integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence in support of SYZ. Moreover, they throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.
We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing a tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach.
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.
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