Supersymmetric field theories and geometric Langlands: The other side of the coin

Balasubramanian, A. R.; Teschner, J.

doi:10.48550/arxiv.1702.06499

Cited by 5 publications

(14 citation statements)

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“…which may be recognizable as the WKB condition for associating a quantum state to the Lagrangian submanifold F . 19 As remarked in footnote 18, S and likewise S −1 is canonically a Spin c structure on F , not a line bundle. Likewise,…”

Section: Quantum States and The Hitchin Fibrationmentioning

confidence: 97%

“…The brane B F is of type (B, A, A). For such a brane, with support F and CP bundle S, the leading σ-model approximation to Hom(B F , B cc ), where B cc has CP bundle L, is the ∂ cohomology of F with values in 19…”

Section: Quantum States and The Hitchin Fibrationmentioning

confidence: 99%

“…The left hand side of (8. 19) is a half-density in x and a density in µ. The right hand side involves a half-density in y = x µ .…”

Section: Hecke Operators As Central Vertex Operatorsmentioning

confidence: 99%

“…As usual, quantization means that a suitable class of smooth functions -in general neither holomorphic nor antiholomorphic -become operators on a Hilbert space. See also [13][14][15][16][17][18][19][20] for prior work on the gauge theory interpretation of the spectrum of quantum Hitchin Hamiltonians.…”

Section: Introductionmentioning

confidence: 99%

“…(A. 19) describes two complex Lagrangian submanifolds L and L r that are defining the boundary condition at t = 0 and t = 1.…”

mentioning

confidence: 99%

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Gauge Theory and the Analytic Form of the Geometric Langlands Program

Gaiotto¹,

Witten²

2021

Preprint

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We present a gauge-theoretic interpretation of the "analytic" version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction -such as electric-magnetic duality between Wilson and 't Hooft line operators in four-dimensional gauge theory -are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.

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Section: Quantum States and The Hitchin Fibrationmentioning

confidence: 97%

Section: Quantum States and The Hitchin Fibrationmentioning

confidence: 99%

“…The left hand side of (8. 19) is a half-density in x and a density in µ. The right hand side involves a half-density in y = x µ .…”

Section: Hecke Operators As Central Vertex Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…(A. 19) describes two complex Lagrangian submanifolds L and L r that are defining the boundary condition at t = 0 and t = 1.…”

mentioning

confidence: 99%

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Gauge Theory and the Analytic Form of the Geometric Langlands Program

Gaiotto¹,

Witten²

2021

Preprint

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Teichmüller TQFT vs. Chern-Simons theory

Mikhaylov¹

2018

J. High Energ. Phys.

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Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by Liouville conformal blocks. It is related but not identical to PSL(2, R) ChernSimons theory. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of Virasoro conformal blocks. We propose that this theory can be defined by an analytically-continued Chern-Simons path-integral with an unusual integration cycle. On hyperbolic three-manifolds, this cycle is singled out by the requirement of invertible vielbein. Mathematically, our proposal translates a known conjecture by Andersen and Kashaev into a conjecture about the Kapustin-Witten equations. We further explain that Teichmüller TQFT is dual to complex SL(2, C) Chern-Simons theory at integer level k = 1, clarifying some puzzles previously encountered in the 3d-3d correspondence literature. We also present a new simple derivation of complex Chern-Simons theories from the 6d (2,0) theory on a lens space with a transversely-holomorphic foliation.

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di-Langlands correspondence and extended observables

Jeong,

Lee,

Nekrasov

2024

J. High Energ. Phys.

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We explore the difference Langlands correspondence using the four dimensional $$ \mathcal{N} $$ N = 2 super-QCD. Surface defects and surface observables play the crucial role. As an application, we give the first construction of the full set of quantum integrals, i.e. commuting differential operators, such that the partition function of the so-called regular monodromy surface defect is their joint eigenvectors in an evaluation module over the Yangian Y$$ \left(\mathfrak{gl}(2)\right) $$ gl 2 , making it the wavefunction of a N-site $$ \mathfrak{gl}(2) $$ gl 2 spin chain with bi-infinite spin modules. We construct the Q- and $$ \overset{\sim }{\textbf{Q}} $$ Q ~ -surface observables which are believed to be the Q-operators on the bi-infinite module over the Yangian Y$$ \left(\mathfrak{gl}(2)\right) $$ gl 2 , and compute their eigenvalues, the Q-functions, as vevs of the surface observables.

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Supersymmetric field theories and geometric Langlands: The other side of the coin

Cited by 5 publications

References 0 publications

Gauge Theory and the Analytic Form of the Geometric Langlands Program

Gauge Theory and the Analytic Form of the Geometric Langlands Program

Teichmüller TQFT vs. Chern-Simons theory

di-Langlands correspondence and extended observables

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