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.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0001/CNTP-2007-0001-0001-a001.pdf

Electric-Magnetic Duality And The Geometric Langlands Program

We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in 0807.4723. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

Wall-crossing, Hitchin Systems, and the WKB Approximation

We find the gravity dual of a marginal deformation of = 4 super Yang Mills, and discuss some of its properties. This deformation is intimately connected with an SL(2,) symmetry of the gravity theory. The SL(2,) transformation enables us to find the solutions in a simple way. These field theory deformations, sometimes called β deformations, can be viewed as arising from a star product. Our method works for any theory that has a gravity dual with a U(1) × U(1) global symmetry which is realized geometrically. These include the field theories that live on D3 branes at the conifold or other toric singularities, as well as their cascading versions.

Deforming field theories with U(1) × U(1) global symmetry and their gravity duals

These lecture notes are based on the master class given at the Center for the Topology and Quantization of Moduli Spaces, University of Aarhus, August 2007. I provide an introduction to the recent work on the Montonen-Olive duality of N = 4 super-Yang-Mills theory and the Geometric Langlands Program.

http://www.ctqm.au.dk/research/MCS/LectureNotesKapustin.pdf

Lectures on Electric-Magnetic Duality and the Geometric Langlands Program

1. Opening remarks 2. A heavy ion phenomenology primer 3. Results from lattice QCD at nonzero temperature 4. Introducing the gauge/string duality 5. A duality toolbox 6. Bulk properties of strongly coupled plasma 7. From hydrodynamics for far-from-equilibrium dynamics 8. Probing strongly coupled plasma 9. Quarkonium mesons in strongly coupled plasma 10. Concluding remarks and outlook Appendixes References Index.

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Gauge/String Duality, Hot QCD and Heavy Ion Collisions

Deformations of N = 4 SYM and integrable spin chain models

We propose a generalization of the Bagger-Lambert-Gustavsson action as a candidate for the description of an arbitrary number of M2-branes. The action is formulated in terms of N=2 superfields in three dimensions and corresponds to an extension of the usual superfield formulation of Chern-Simons matter theories. Demanding gauge invariance of the resulting theory does not imply the total antisymmetry of the underlying 3-Lie algebra structure constants. We relax this condition and propose a class of examples for these generalized 3-Lie algebras. We also discuss various associated ordinary Lie algebras.

Multiple M2-branes and generalized 3-Lie algebras

Singular monopoles and supersymmetric gauge theories in three dimensions

We construct the supergravity solution for fully localized D2/D6 intersection. The near horizon limit of this solution is the supergravity dual of supersymmetric Yang-Mills theory in 2+1 dimensions with flavor. We use this solution to formulate mirror symmetry of 2+1 dimensional gauge theories in the language of AdS/CFT correspondence. We also construct the supergravity dual of a non-commutative gauge theory with fundamental matter.

Supergravity Solution of Intersecting Branes and AdS/CFT with Flavor

We study Bogomolny equations on R^2×S^1. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using the Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperkahler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of k periodic monopoles provides the exact solution of N=2 super Yang–Mills theory with gauge group SU(k) compactified on a circle of arbitrary radius.

Sergey A. Cherkis

Papers

Deformations of N = 4 SYM and integrable spin chain models

Multiple M2-branes and generalized 3-Lie algebras

Singular monopoles and supersymmetric gauge theories in three dimensions

Supergravity Solution of Intersecting Branes and AdS/CFT with Flavor

Nahm transform for periodic monopoles and N = 2 super Yang-Mills theory