More On Gauge Theory And Geometric Langlands

  title={More On Gauge Theory And Geometric Langlands},
  author={Edward Witten},
  journal={arXiv: High Energy Physics - Theory},
  • E. Witten
  • Published 13 June 2015
  • Mathematics, Physics
  • arXiv: High Energy Physics - Theory
The geometric Langlands correspondence was described some years ago in terms of $S$-duality of $\N=4$ super Yang-Mills theory. Some additional matters relevant to this story are described here. The main goal is to explain directly why an $A$-brane of a certain simple kind can be an eigenbrane for the action of 't Hooft operators. To set the stage, we review some facts about Higgs bundles and the Hitchin fibration. We consider only the simplest examples, in which many technical questions can be… Expand

Figures from this paper

Quantum $q$-Langlands Correspondence
We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affineExpand
Supersymmetric field theories and geometric Langlands: The other side of the coin
This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin andExpand
Hecke modifications of Higgs bundles and the extended Bogomolny equation
Abstract We establish a Kobayashi–Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. ThisExpand
Torsion line bundles and branes on the Hitchin system
We study the fixed loci for the action of tensorisation by a line bundle of order $n$ on the moduli space of Higgs bundles for the Langlands self-dual group $\mathrm{GL}(n,\mathbb{C})$. We equipExpand
S-duality of boundary conditions and the Geometric Langlands program
  • D. Gaiotto
  • Physics, Mathematics
  • Proceedings of Symposia in Pure Mathematics
  • 2018
Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in aExpand
Green-Schwarz superstring as subsector of Yang-Mills theory
We consider Yang-Mills theory with N=2 super translation group in ten auxiliary dimensions as the structure group. The gauge theory is defined on a direct product manifold \Sigma_2\times H^2, whereExpand
Quantum Hall Effect and Langlands Program
Recent advances in the Langlands program shed light on a vast area of modern mathematics from an unconventional viewpoint, including number theory, gauge theory, representation, knot theory and etc.Expand
Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence
Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltonians is proposed as a natural quantisation condition. Separation of Variables can be used to relate the classification ofExpand
Mirror symmetry for Nahm branes
Using the Dirac--Higgs bundle and the morphism given by tensorization, we consider a new class of virtual hyperholomorphic bundles over the moduli space of M of degree 0 semistable Higgs bundles.Expand
The reverse Yang–Mills–Higgs flow in a neighbourhood of a critical point
The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the $C^\infty$ topology to a critical point. The construction uses only the complexExpand


Lectures on the Langlands program and conformal field theory
These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to theExpand
Electric-Magnetic Duality And The Geometric Langlands Program
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 areExpand
Geometric Langlands From Six Dimensions
Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gaugeExpand
Gauge Theory, Ramification, And The Geometric Langlands Program
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 orExpand
Some comments on Chern-Simons gauge theory
Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describeExpand
Mirror Symmetry, Hitchin's Equations, And Langlands Duality
Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold $C$. But understanding theseExpand
Geometric Langlands duality and the equations of Nahm and Bogomolny
  • E. Witten
  • Mathematics, Physics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2010
Geometric Langlands duality relates a representation of a simple Lie group Gv to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2Expand
Reducing S duality to T duality.
Yang-Mills S-duality makes predictions for all correlators of this effective conformal field theory that Montonen-Olive duality between electric and magnetic states reduces to the standard two-dimensionalDuality between momentum and winding states. Expand
The Yang-Mills equations over Riemann surfaces
  • M. Atiyah, R. Bott
  • Mathematics
  • Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1983
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gaugeExpand
Gauge theory and wild ramification
The gauge theory approach to the geometric Langlands program is extended to the case of wild ramification. The new ingredients that are required, relative to the tamely ramified case, areExpand