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70
ElectricMagnetic duality and the geometric Langlands program
, 2006
"... The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super YangMills theory in four dimensions. The key ingredients are electricmagnetic duality of gauge theory, mirror symmetry of sigmamodels, branes, Wilson and ’t H ..."
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Cited by 294 (26 self)
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The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super YangMills theory in four dimensions. The key ingredients are electricmagnetic duality of gauge theory, mirror symmetry of sigmamodels, branes, Wilson and ’t Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke
Noncommutative Counterparts of the Springer Resolution
, 2006
"... Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular represe ..."
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Cited by 45 (3 self)
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Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebrogeometric problems, such as the derived equivalence conjecture and description of T. Bridgeland’s space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a tstructure on the derived category of the resolution. The intriguing fact that the same tstructure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.
ToledanoLaredo V., Gaudin model with irregular singularities
"... Abstract. We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac–Moody algebra at the critical level, extending the construction of higher ..."
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Cited by 24 (3 self)
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Abstract. We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac–Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from [FFR] to the case of nonhighest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P 1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finitedimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.
Geometric Endoscopy And Mirror Symmetry
, 2007
"... Abstract. The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T–duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the ..."
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Cited by 21 (2 self)
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Abstract. The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T–duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the Bmodel side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of Abranes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.C. Ngô, a fact which
Quantum Curves and DModules
, 2008
"... In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an Ibrane configuration, which consists of D4 and D6branes intersecting along a holomorphic curve in a complex surface, together with a Bfield. Mathematically, it is described ..."
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Cited by 14 (0 self)
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In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an Ibrane configuration, which consists of D4 and D6branes intersecting along a holomorphic curve in a complex surface, together with a Bfield. Mathematically, it is described by a holonomic Dmodule. Here we focus on spectral curves, which play a prominant role in the theory of (quantum) integrable hierarchies. We show how to associate a quantum state to the Ibrane system, and subsequently how to compute quantum invariants. As a first example, this yields an insightful formulation of (double scaled as well as general Hermitian) matrix models. Secondly, our formalism elegantly reconstructs the complete dual NekrasovOkounkov partition function from a quantum SeibergWitten curve.
A brief review of abelian categorifications
"... This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various repr ..."
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Cited by 12 (3 self)
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This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra sln. The review is intended to give nonexperts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles. 1
Quantization of soliton systems and Langlands duality
 LANGLANDS CORRESPONDENCE FOR LOOP GROUPS, CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 103
, 2007
"... We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac–Moody algebras. Our experience with the Gaudin models associated to finitedimensional simple Lie algebras su ..."
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Cited by 11 (2 self)
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We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac–Moody algebras. Our experience with the Gaudin models associated to finitedimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra bg should be encoded by affine opers associated to the Langlands dual affine algebra L bg. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding affine opers may be expressed as Schrödinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this and other recently found examples of the correspondence between quantum integrals of motion and differential operators may be viewed as special cases of the Langlands duality.