Related papers: Mirror symmetry, Langlands duality, and the Hitchi…
In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect…
We generalize the classical theory of Higgs bundles, spectral curves and Norm maps to Deligne-Mumford curves. As an application, under some mild conditions, we prove the Strominger-Yau-Zaslow duality for the moduli spaces of Higgs bundles…
By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in…
The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a…
In this note we present pairs of hyperkaehler orbifolds which satisfy two different versions of mirror symmetry. On the one hand, we show that their Hodge numbers (or more precisely, stringy E-polynomials) are equal. On the other hand, we…
We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for…
We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is Mirror…
This paper provides an introduction to non-abelian Hodge theory and moduli spaces of Higgs bundles on compact Riemann surfaces. We develop the moduli theory of vector bundles and Higgs bundles, establish the main correspondences of…
We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies…
The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners.…
We extend Hitchin's results on "The self-duality equations on a Riemann surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove existence results for orbifold solutions of the Yang-Mills-Higgs equations and construct…
This thesis is dedicated to the study of certain loci of the Higgs bundle moduli space on a compact Riemann surface. Motivated by mirror symmetry, we give a detailed description of the fibres of the $G$-Hitchin fibration containing…
The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg…
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…
The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are…
Exploiting the description of rings of differential operators as Azumaya algebras on cotangent bundles, we show that the moduli stack of flat connections on a curve (allowed to acquire orbifold points) defined over an algebraically closed…
The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories…
Mirror symmetry relates type IIB string theory on a Calabi-Yau 3-fold to type IIA on the mirror CY manifold, whose complex structure and Kaehler moduli spaces are exchanged. We show that the mirror map is a particular case of a more general…
It is argued that every Calabi-Yau manifold $X$ with a mirror $Y$ admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space $Y$. The mirror…
In this paper, we study the moduli space of Higgs pairs, which can be considered as a generalization of holomorphic pairs. Higgs pairs are an example of quiver bundles. We introduce the notion of $\tau$-stability of Higgs pairs for…