Geometrically describes Higgs bundle moduli spaces for orthogonal groups.
problem Understanding moduli spaces of Higgs bundles for orthogonal groups.
method Cayley and Langlands type correspondences for real orthogonal and symplectic Higgs bundles.
result Complete abelianization of real slices for all quasi-split real forms.
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 of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years …
Paper describes integrable structure of Hitchin moduli spaces.
problem Integrable structure of Hitchin moduli spaces.
method Explicit parameterizations and Separation of Variables method.
result Clear analogy with Drinfeld's geometric Langlands correspondence.
Established a correspondence between Higgs bundles and Bogomolny equations.
problem Geometric Langlands program and S-duality in N=4 super Yang-Mills theory.
method Kobayashi-Hitchin correspondence between Higgs bundles and solutions of the extended Bogomolny equation.
result Proved the conjecture by Witten linking Higgs bundles and Bogomolny equations.
Revisits Vafa-Witten theory, deriving new invariants and homologies.
problem Understanding Vafa-Witten equations and their invariants.
method Physical derivation and mathematical analysis of Vafa-Witten equations.
result Novel invariants and Floer homologies derived from Vafa-Witten equations.
Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.
problem Realizing geometric Langlands correspondences for Higgs bundles and connections.
method Using transversal Higgs bundles and holomorphic connections, induced divisors and parameters.
result Evidence suggests generic realization of Dolbeault geometric Langlands correspondence.
A theory linking invariants, Floer homologies, and Higgs bundles.
problem Understanding complex geometric structures and their invariants.
method Categorification and geometric Langlands correspondence.
result Established a new geometric Langlands correspondence.
String theory connects lattice models, links, and geometric Langlands.
problem Connecting lattice models, links, and geometric Langlands.
method T-duality and worldvolume theories in string theory.
result Unified understanding of various mathematical concepts.
Reconstructs Shelstad's character identity using index theory.
problem Classifying group representations via the Langlands program.
method Geometric proof using index theory of elliptic operators in K-theory. result Evidence that index theory can be used in representation classification.
Study Higgs bundles and their Langlands duals on surfaces.
problem Matching components of Higgs bundle moduli spaces under Langlands duality.
method Defined antiholomorphic involutions on Higgs bundle moduli spaces and analyzed fixed point loci.
result Components of fixed point loci match under Langlands duality for specific groups.
New quantum integrals discovered for a spin chain model.
problem Exploring quantum integrals for a spin chain model.
method Using surface defects and observables in 4D N=2 super-QCD. result First construction of quantum integrals and their joint eigenvectors.
5D gauge theories are dual to 3D and 2D models via Floer homologies.
problem Exploring dualities in 5D gauge theories and their 3D and 2D counterparts.
method Using Landau-Ginzburg models and Floer homologies, the paper establishes dualities between different gauge theories and their associated homologies.
result Dual A∞-categories of Floer homologies are derived, proving mirror symmetry and Langlands duality. Researchers construct a probabilistic model for a WZW theory on hyperbolic space and link it to Liouville theory.
problem Rigorous probabilistic construction of WZW models on curved spaces.
method Path integral approach on closed Riemann surfaces twisted by gauge fields.
result Correspondence between correlation functions of H3-WZW and Liouville CFT. We give a geometric characterisation of the topological invariants associated to SO(m,m+1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)-Higg…
New Langlands duality conjectures for 3-manifold skein modules.
problem Understanding Langlands duality in 3-manifold skein modules.
method Combining representation theory of double affine Hecke algebras and 1-form symmetry structure.
result Recent special cases confirmed, with detailed proofs.
This article addresses the question of whether Langlands duality for complex reductive Lie groups may be implemented by T-dualization. We prove that for reductive groups whose simple factors are of Dynkin type A, D, or E, the answer is yes.
S-dual of Hamiltonian spaces connects to Langlands duality.
problem Hamiltonian spaces and their S-duals.
method Definition and properties of S-dual of Hamiltonian spaces.
result S-dual of Hamiltonian spaces is related to Langlands duality.
The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of t…
The paper explores gauge theory invariants and their duals via topological-holomorphic twist.
problem Understanding gauge theory invariants and their duals in 4d and 2d.
method Topological-holomorphic twist of N=4 supersymmetric gauge theory.
result Derived novel topological and holomorphic invariants and their Langlands duals.
We construct and study a closed, two-dimensional, quasi-topological (0,2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, pu…
Novel gauge-theoretic Floer homologies defined from 5d N=2 theory, linking 4, 3, and 2-manifolds.
problem Defining and linking novel Floer homologies for different manifold dimensions.
method Physics of a topologically-twisted 5d N=2 gauge theory, Vafa-Witten, Hitchin, and BF configurations.
result Derived novel gauge-theoretic and symplectic Floer homologies, and Atiyah-Floer correspondences.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS q-series for 3-manifolds with line defects. method Proving homomorphism from skein module to space of q-series, conjecturing holomorphic modularity. result Holomorphic quantum modularity of q-series suggests new approach to Langlands duality. Defines and parametrizes sl(2)-type singular fibres in symplectic and odd orthogonal Hitchin systems.
problem Characterizing and understanding singular fibres in Hitchin systems.
method Stratification by semi-abelian spectral data, study of irreducible components, global description of degenerations.
result Extension of Langlands duality to sl(2)-type Hitchin fibres. The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.
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…
We show that the resolvent of the Laplacian on SL(3,R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…
Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.
problem Equidistribution of geodesics and holonomies in Anosov homogeneous spaces.
method Analyzes maximal flat cylinders and their holonomies for Anosov subgroups.
result Joint equidistribution of maximal flat cylinders and holonomies as circumference tends to infinity.
This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.
problem Mathematical definition of the algebra of BPS states.
method Construction of cohomological Hall algebras for 3-Calabi-Yau categories.
result Construction of cohomological Hall algebras and proof of Joyce's conjecture.
Study skein modules via gauge theory, finding non-TQFT dimensions.
problem Understanding skein modules of 3-manifolds using gauge theory.
method Embed skein modules into 4d N=4 super-Yang-Mills theories, using N=1 supersymmetry. result Find non-standard dimensions of skein modules, differing from TQFT predictions.
New probabilistic method constructs Kähler-Einstein metrics and suggests zero-free properties of zeta functions.
problem Existence and explicit formulas for Kähler-Einstein metrics on Fano varieties.
method Probabilistic construction involving canonical random point processes.
result Zero-free properties of Archimedean zeta functions and their relation to Langlands program.
The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program…
The paper explores de Rham theory for singular spaces and stacks.
problem Identifying de Rham theory for singular differentiable spaces.
method Identifying two potential answers and studying them, including the exterior algebra of the cotangent complex and de Rham stacks.
result There exists a version of the de Rham theorem for singular differentiable spaces with almost no restrictions.
The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.
problem Integrating Rota-Baxter operators into Lie group structures and geometries.
method Introducing Rota-Baxter operators on Lie groups, Lie algebroids, and groupoids.
result Geometrization of Rota-Baxter Lie algebras and groups.
We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the const…
Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semis…
Geometrically interprets and computes intersection pairings for higher laminations.
problem Understanding and computing intersection pairings for higher laminations.
method Realization of higher laminations as points in the affine building, geometric interpretation of pairings, use of combinatorial results.
result Intersection pairings can be computed as the length of minimal weighted networks in the building.
In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, a…
We study twisted N=2 superconformal gauge theory on a product of two Riemann surfaces Sigma and C. The twisted theory is topological along C and holomorphic along Sigma and does not depend on the gauge coupling or theta-angle. Upon Kaluza-Klein reduction along Sigma, it becomes equivalent to a topological B-model on C …
We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold.…
The paper decomposes spectral functions on marked tori strata.
problem Decomposing square-integrable functions on strata of differentials.
method Spectral decomposition and analysis of differential operators.
result The continuous spectrum of the foliated Laplacian is larger than Siegel-Veech transforms.
Defines rho numbers for metrics with positive scalar curvature.
problem Computing index classes for manifolds with boundary and cuspidal parabolics.
method Uses orbital integrals, delocalized eta invariants, and index theorems.
result Defines higher rho numbers for metrics with positive scalar curvature.
The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.
problem Decomposing the cohomology of smooth symmetric stacks into invariant parts.
method Using cohomological Hall induction and intersection cohomology of moduli spaces.
result Establishes the BPS decomposition theorem for various symplectic stacks.
Constructs cohomology decompositions for symmetric stacks.
problem Cohomology of symmetric stacks.
method Constructs decompositions of cohomology, Borel--Moore homology, and vanishing cycle cohomology.
result Defines BPS cohomology and proves its equivalence to intersection cohomology for smooth stacks.
This paper connects symplectic and Kähler manifolds via brane quantization.
problem Quantizing Kähler manifolds using brane techniques.
method Using physical proposals and geometric quantization, the authors relate A-model morphism spaces to quantizations of symplectic and Kähler manifolds.
result Chan-Leung-Li's work provides a mathematical realization of the action of A-branes on B-branes, linking deformation quantizations of symplectic and Kähler manifolds.
Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …
Generalizes uniformization to algebraic correspondences.
problem Uniformizing non-homeomorphic genus zero orbifolds.
method Constructs algebraic correspondences to simultaneously uniformize orbifolds.
result Realizes Teichmüller space of a punctured sphere in correspondences.
Study of holomorphic correspondences combining entire maps and Fuchsian groups.
problem Understanding dynamics of entire maps and their interactions with Fuchsian groups.
method Systematic study of (∞:∞) holomorphic correspondences arising from conformal combinations of transcendental entire maps and Fuchsian groups. result The resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function with a simple pole.
Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.
problem None explicitly stated, focuses on introduction.
method Pedagogical introduction of concepts linking algebraic structures to geometry.
result Explains the non-abelian Hodge correspondence and its connections.