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 …
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48 results for “geometric Langlands correspondence”
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.
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.
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.
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.
Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to…
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.
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…
We establish a Kobayashi-Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. This correspondence was conjectured by Witten and plays an important role in the physical description of the the geometric Langlands program in ter…
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…
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…
Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding -packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic…
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 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 -categories of Floer homologies are derived, proving mirror symmetry and Langlands duality.
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…
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 -WZW and Liouville CFT.
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.
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.
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.
Given a compact Riemann surface and a complex reductive Lie group equipped with real structures, we define antiholomorphic involutions on the moduli space of -Higgs bundles over . We investigate how the various components of the fixed point locus match up, as one passes from to its Langlands dual $^LG…
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 super-Yang-Mills theories, using supersymmetry.
result Find non-standard dimensions of skein modules, differing from TQFT predictions.
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 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.
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 -series for 3-manifolds with line defects.
method Proving homomorphism from skein module to space of -series, conjecturing holomorphic modularity.
result Holomorphic quantum modularity of -series suggests new approach to Langlands duality.
Defines and parametrizes -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 -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.
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.
One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups and . These pairings are a generalization of the intersection pairing between measured laminatio…
We show that the resolvent of the Laplacian on SL(3,)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of . The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…
Joint equidistribution of maximal flat cylinders and holonomies for Anosov homogeneous spacesmath.DS
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.
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 …
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.
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
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.
Quilted Floer Cohomologymath.SG
We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…
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.
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…
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.
A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter space-time are obtained in order to illustrate this correspondence.
Combines Gaussian process and Geometric Harmonics for better uncertainty estimation.
problem Uncertainty estimation in kernel-based methods.
method Combines Gaussian process and Geometric Harmonics.
result Alternative interpretations of uncertainty and accelerated Bayesian Optimization.
Geometric approach finds correspondences between different conditions.
problem Integrating multiple biological datasets.
method Fibered latent space with pull-back metric, diffeomorphism flows.
result Minimizing energy functional yields diffeomorphism flows.
The paper explains geometric correspondences for homothetic navigation.
problem Understanding geodesic and Jacobi field correspondences in homothetic navigation.
method Providing conceptual explanations and shortcuts to formulas.
result Directly seeing local correspondence between isoparametric functions or hypersurfaces.
Geometries and dual field theories linked by AdS/CFT.
problem Understanding the AdS/CFT correspondence.
method Geometric extremization principles informed by physical considerations.
result Key role of Sasaki-Einstein and GK geometry.
Geometrically describes the linear and quadratic forms for rational links.
problem Predicting generating functions for colored HOMFLY-PT polynomials of rational links.
method Direct geometric description of linear and quadratic forms in terms of configuration spaces.
result Direct geometric description of forms for rational links.
High-dimensional ConvNets detect patterns in 32+ dimensions for geometric registration.
problem Detecting geometric patterns in high-dimensional spaces.
method High-dimensional convolutional networks applied to geometric registration problems.
result High-dimensional ConvNets outperform global pooling approaches in 3D registration and image correspondence.