Related papers: Langlands duality and G2 spectral curves
In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of…
Let G be a real form of a complex reductive group. Suppose that we are given involutions \sigma and \theta of G. Let H=G^\sigma denote the fixed group of \sigma and let K=G^\theta denote the fixed group of \theta. We are interested in…
In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup. The total space of Hitchin is obtained by…
We define the secondary invariants L^2- eta and -rho forms for families of generalized Dirac operators on normal coverings of fibre bundles. On the covering family we assume transversally smooth spectral projections, and Novikov--Shubin…
For a complex reductive group $G$, we consider the locus $M^d$ in the moduli stack of $G$-Higgs bundles on which the centraliser dimension of the Higgs field takes a constant value $d> rk(G)$. We describe a non-abelian structure for the…
We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the quintic threefold. We interpret our results as…
In this paper, we confirm a physical conjecture regarding the parabolic $\mathrm{SO}_{2n}$-Hitchin system, showing that Hitchin map factors through a finite cover of the Hitchin base that is isomorphic to an affine space. We first show that…
This thesis contains work which appeared in several papers. Additionally to the results in the papers it contains a detailed introduction and some further proofs and remarks. The dissertation gives a description of the topology and…
The Chen-Ng\^o Conjecture predicts that the Hitchin morphism from the moduli stack of $G$-Higgs bundles on a smooth projective variety surjects onto the space of spectral data. The conjecture is known to hold for the group $GL_n$ and any…
We verify the relative Langlands duality conjecture proposed by Ben-Zvi, Sakellaridis, Venkatesh for the hyperspherical Hamiltonian variety $T^*(\operatorname{Sp}_{2n}\backslash \operatorname{GL}_{2n+1})$. We provide numerical (over number…
Typical dualities in arbitrary dimensions are understood through a Hilbert-space extension method. By these results, we rigorously dualize the quantum ingappabilities to discrete height model in one dimension which is inaccessible by…
Let $F$ be a locally compact non-Archimedean field of characteristic $0$, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$. The goal of this paper is to give an…
Geometric Langlands duality relates a representation of a simple Lie group $G^\vee$ to the cohomology of a certain moduli space associated with the dual group $G$. In this correspondence, a principal $SL_2$ subgroup of $G^\vee$ makes an…
In this paper we give a complete description of the Hitchin fibration on all 2-dimensional moduli spaces of rank 2 irregular Higgs bundles with two poles on the projective line. We describe the dependence of the singular fibers of the…
An $L^2$ version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the $\bar{\partial}$-operator is established. This duality is used to study the solution of the…
In 1987, Hitchin introduced the self-duality equations on rank-2 complex vector bundles over compact Riemann surfaces with genus greater than one as a reduction of the Yang-Mills equation and established the existence of solutions to these…
We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions.…
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 show that the numerical local Langlands duality for GL_n and the T - duality of two-dimensional quantum gravity arise from one and the same symmetry principle. The unifying theme is that the local Fourier transform in both its l-adic and…
We introduce a new invariant of $G$-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant…