Related papers: Geometric Endoscopy and Mirror Symmetry
Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold $C$. But understanding these statements is extremely difficult without picking…
The first part of this paper is a survey of mathematical results on mirror symmetry phenomena between Hitchin systems for Langlands dual groups. The second part introduces and discusses multiplicity algebras of the Hitchin system on…
We prove the Strominger--Yau--Zaslow and topological mirror symmetries for parabolic Hitchin systems of types B and C. In contrast to type A, a geometric reinterpretation of Springer duality is necessary. Furthermore, unlike Hitchin's…
In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\textrm{GL}_n$-local systems. Certain hypergeometric local systems admit a…
We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m+1). We then show that duality for G2 is implemented by an involution on the base space which takes one…
We propose a geometric interpretation of the theory of elliptic endoscopy, due to Langlands and Kottwitz, in terms of the Hitchin fibration. As applications, we prove a global analog of a purity conjecture, due to Goresky, Kottwitz and…
Gross and Siebert identified a class of singular Lagrangian torus fibrations which arise when smoothing toroidal degenerations, and which come in pairs that are related by mirror symmetry. We identify an immersed Lagrangian in each of these…
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…
We study the geometry of singular $\text{SL}_n$-Hitchin fibres over the elliptic locus. We show that orbifold singularities appear in the $\text{PGL}_n$-moduli space $M^{ell}(\text{PGL}_n)$ exactly when the $\text{SL}_n$ side…
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…
The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional K\"ahler tori and stable…
We investigate the local geometry of a pair of independent contact structures on 3-manifolds under maps that independently preserve each contact structure. We discover that such maps are homotheties on the contact 1-forms and we discover…
We prove a homological mirror symmetry equivalence between an $A$-brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a $B$-brane category for a mirror LG model, understood as a category of matrix…
We study aspects of Heterotic/F-theory duality for compactifications with Abelian discrete gauge symmetries. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with n-sections, associated with the…
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…
We prove three new results about the global Springer action defined in \cite{GSI}. The first one determines the support of the perverse cohomology sheaves of the parabolic Hitchin complex, which serves as a technical tool for the next…
We discuss geometrical aspects of different dualities in the integrable systems of the Hitchin type and its various generalizations. It is shown that T duality known in the string theory context is related to the separation of variables…
Recently, mirror symmetry is derived as T-duality applied to gauge systems that flow to non-linear sigma models. We present some of its applications to study quantum geometry involving D-branes. In particular, we show that one can employ…
We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential…
The geometric Langlands correspondence was described some years ago in terms of $S$-duality of $\N=4$ super Yang-Mills theory. Some additional matters relevant to this story are described here. The main goal is to explain directly why an…