Electric-Magnetic Duality And The Geometric Langlands Program

By A. Kapustin, E. Witten. 2006

1,119 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/718fd8fcc35118b1784df5ca4040b2ff7978ca0d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Sergei Gukov
2. E. Witten
3. Tudor Dimofte
4. D. Gaiotto
5. C. Vafa
6. A. Kapustin
7. J. Teschner
8. N. Nekrasov

In Nuclear and High Energy Physics

1. Sergei Gukov
2. E. Witten
3. D. Gaiotto
4. Marcos Mariño
5. Tudor Dimofte
6. C. Vafa
7. N. Seiberg
8. Yuji Tachikawa

In Statistical and Nonlinear Physics

1. Marcos Mariño
2. E. Witten
3. D. Gaiotto
4. Sergei Gukov
5. C. Vafa
6. Tudor Dimofte
7. N. Seiberg
8. Dan Xie

Citations in the corpus

• 2d index and surface operators (2013, background): “(3)In fact, these equations belong to a one-parameter family of BPS equations introduced in [43].”
• On D = 5 super Yang-Mills theory and (2, 0) theory (2010, background): “One might be able to test this idea by reducing to D = 4 and making contact with the discussion of S-duality for BCFG groups in [16].”
• Constructible sheaves and the Fukaya category (2006, methodology): “For example, according to Kapustin-Witten [16], given a generic eigen-brane for the ’t Hooft operators, there is a corresponding regular, holonomic Hecke eigen-D-module.”
• Homological algebra of knots and BPS states (2011, methodology): “duli space of solutions on R2 with a single defect operator in the adjoint representation of the gauge group G = SU(2) is the weighted projective space WCP2 (1,1,2) (= the space of Hecke modifications [49], see also [50]). In this approach, the colored homology Hsl(2), ( ) is given by the L2 cohomolo”
• p-Adic mathematical physics: the first 30 years (2017, background): “ representations. Therefore it can be viewed as part of the Langlands program in number theory and the representation theory [287] (for a recent consideration of the geometrical Langlands program see [292]). Let Gbe the Galois group of a Galois extension of Qand αa representation of Gin Cn. There is”
• Topology of character varieties of Abelian groups (2013, background): “We remark that these character varieties are also main players in mirror symmetry and the geometric Langlands programme (see, for example [19, 25, 29]).”
• N$$ \mathcal{N} $$ = 1 supersymmetric indices and the four-dimensional A-model (2017, background): “ion [47], which is naively a dierent object than the partial twist considered here. More relatedly, partial topological twists of N= 2 theories along a Riemann surface have been considered, e.g., in [48, 49, 50]. Surface operator expectation values play an important role in our story, since they are”
• Fivebranes and 4-manifolds (2013, background): “GL twist) was first considered by Marcus [107] and related to the geometric Langlands program in [40].”
• Chern-Simons theory and S-duality (2011, methodology): “Following [40, 41], we interpret this duality as mirror symmetry between moduli spaces Y =MH(G,C) and Ỹ =MH(G,C).”
• The finiteness conjecture for skein modules (2019, background): “(6) Vafa and Witten [VW94] have studied another topological twist (first described by Yamron in [Yam88]) of the N = 4 supersymmetric 4d Yang–Mills theory.”