Local geometric Langlands correspondence and affine Kac-Moody algebras

Frenkel, Edward; Gaitsgory, Dennis

doi:10.1007/978-0-8176-4532-8_3

Cited by 96 publications

(426 citation statements)

References 29 publications

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“…This generalizes some known relations between Whittaker and tilting/projective modules in the classic BGG category O [81][82][83]. For abelian theories, the conjecture is proven in [84].…”

Section: Jhep10(2016)108supporting

confidence: 65%

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Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

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Abstract:We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs-and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality -an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

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“…This generalizes some known relations between Whittaker and tilting/projective modules in the classic BGG category O [81][82][83]. For abelian theories, the conjecture is proven in [84].…”

Section: Jhep10(2016)108supporting

confidence: 65%

“…In particular, all projective and tilting modules in O C arise this way. In geometric representation theory, it is known that a non-degenerate Whittaker module over a semisimple Lie algebra can be averaged or degenerated to give the "big" projective module in the BGG category O [81][82][83]. Our analysis of Neumann b.c.…”

Section: Jhep10(2016)108mentioning

confidence: 99%

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

View full text Add to dashboard Cite

show abstract

“…Recall that this construction gives aĝ-module W with an embedding V → W (this is the vacuum module associated to a chiral algebra constructed, e.g., in [FG2,Section Remark 4.2. This theorem was proved originally in [BD1,5.4.8].…”

Section: We Have the Following Essential Computationmentioning

confidence: 99%

A geometric proof of the Feigin-Frenkel theorem

Raskin¹

2012

Represent. Theory

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Abstract. We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands duality through the geometric Satake theorem.

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“…We expect that it is possible to apply somewhat similar ideas to the ramified case. A second major gap is that we do not shed light on the utility of two-dimensional conformal field theory for the geometric Langlands program [5,9,[27][28][29][30] . The last of these references applies conformal field theory to the ramified case.…”

Section: Introductionmentioning

confidence: 99%

Electric-magnetic duality and the geometric Langlands program

Kapustin

Witten

2007

Communications in Number Theory and Physics

835

2,044

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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 Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

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Local geometric Langlands correspondence and affine Kac-Moody algebras

Cited by 96 publications

References 29 publications

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

A geometric proof of the Feigin-Frenkel theorem

Electric-magnetic duality and the geometric Langlands program

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