Gauge Theory, Ramification, And The Geometric Langlands Program
By Sergei Gukov, E. Witten. 2006
570 citations recorded by the source corpus.
In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves. We describe the relevant surface operators in N=4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters. Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification. The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space.
Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/3aaf2e55ffd243618046f9d0e3654f208613fb6b
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Citations in the corpus
- Wall-crossing in coupled 2d-4d systems (2011, background): “Deeper developments along this direction have come with the inclusion of surface defects [6, 7, 8].”
- Wild quiver gauge theories (2011, methodology): “Then, by compactifying on the Riemann surface with a suitable twist, we get a sigma model whose target space is the Hitchin moduli space MH [31, 32, 33, 34, 35].”
- The virtue of defects in 4D gauge theories and 2D CFTs (2010, methodology): “A parallel reasoning to the Liouville case identifies the simplest M2 brane defects in the AN−1 theory with (N, 1) degenerate vertex operators in Toda theory (N represents the fundamental representation) and also with the simplest surface operators in SU(N), defined by [45], which break the gauge sy”
- 2d index and surface operators (2013, methodology): “Following [29, 30] we would like to describe the surface operator as a coupled 2d-4d system.”
- Generalized global symmetries (2014, background): “J H E P 0 2 ( 2 0 1 5 ) 1 7 2 surface operators [33, 34] associated to elements of Z(G).”
- Reading between the lines of four-dimensional gauge theories (2013, background): “7 The case of g = su(2) was discussed in [28,7].”
- Vortex Counting and Lagrangian 3-Manifolds (2010, background): “In other words, they are halfBPS surface operators of the four-dimensional N =2 gauge theory [10].”
- Coupling a QFT to a TQFT and duality (2014, background): “Examples of such surfaces and the way they can end on lines were considered in [14,15].”
- Affine SL(2) Conformal Blocks from 4d Gauge Theories (2010, methodology): “Surface operators for N = 4 super Yang-Mills theory were studied by Gukov and Witten [11].”
- General Argyres-Douglas theory (2012, methodology): “The identification of the mass with the coefficient of the simple pole is the following: the two form on the Seiberg-Witten geometry should depend linearly on the mass [27, 28] while the Hitchin moduli space in complex structure I do depend linearly on the coefficient of the simple pole [29].”
- Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories (2012, background): “This surface operator is closely related to those studied by Gukov and Witten [31, 32].”
- Defects and quantum Seiberg-Witten geometry (2014, background): “They can be described either as Gukov-Witten monodromy defects [3, 4] or by coupling to a class of 3d N = 4 supersymmetric gauge theories.”
- Analytic Continuation Of Chern-Simons Theory (2010, background): “In effect, the flow equations have a codimension two singularity in four dimensions that can be modeled by the same surface operators studied in [55].”
- Iterated destabilizing modifications for vector bundles with connection (2008, background): “This could help in the ramified case [21]—when there are no semistable bundles, it would seem logical to consider the lowest stratum P0 parametrizing variations of Hodge structure.”