The goal of this mini-conference is to explore the newly found connections between various dualities in physics (such as the S-duality in 4D supersymmetric gauge theory and Mirror Symmetry of 2D sigma models) and in mathematics (geometric Langlands Program). Among the topics considered will be the work of Gaiotto and Witten on duality of boundary conditions in 4D super-Yang-Mills theory, as well as the wall-crossing formulas studied in mathematics by Joyce, Bridgeland, Toledano Laredo, Kontsevich, and Soibelman and in physics by Gaiotto, Neitzke, Moore, and others.

Davide Gaiotto, S-duality and boundary conditions (two lectures)

N=4 Super Yang Mills theory has a large set of interesting half BPS, conformally invariant boundary conditions. S-duality acts in surprising ways on that set, mapping even elementary boundary conditions, as Neumann or Dirichlet, to rather intricate ones. We will give detailed examples of S-dual pairs of boundary conditions for U(1), SU(2) gauge groups, and introduce a general recipe for any gauge group. Besides the physical interest, the subject has various mathematical applications, including symplectic duality (aka 3d mirror symmetry) and geometric Langlands, and can be formulated as an interesting mathematical problem in its own sake.

Valerio Toledano Laredo, Stability conditions and Stokes factors

I will explain how Joyce's wall-crossing formulae for invariants counting semistable objects in an abelian category A may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of A.
This allows one in particular to interpret his holomorphic generating functions as defining an isomonodromic family of such connections parametrised by the space of stability conditions of A.
This is joint work with T. Bridgeland (arXiv:0801.3974).

Ori Ganor, From S-Duality to Chern-Simons via Minimal Strings

There are two special values of the coupling constant for which there exist noncentral elements of SL(2,Z) that map N=4 Super Yang-Mills theory with gauge group U(n) to itself. At these values, the field theory can be compactified on a circle with duality-twisted boundary conditions. The low-energy limit of this model directly probes the S-duality operator. Augmented by an R-symmetry twist, and with additional restrictions on the rank n, this low-energy limit appears to be a nontrivial topological field theory. Upon further compactification on a torus, the Hilbert space of the low-energy theory can be mapped, using U-duality, to the finite dimensional space of minimal string states on a three-dimensional manifold that is a torus fibre-bundle over a circle. Using the string theory realization, I'll compare the low-energy theory with Chern-Simons theory. Also, compactification on a Riemann surface of higher genus suggests a relation between the dimension of the Hilbert space of certain Chern-Simons theories on the Riemann surface and the supertrace of the action induced by mirror symmetry on the appropriate cohomology of the appropriate Hitchin space.

Joerg Teschner, Quantum geometric Langlands vs. nonperturbative CFT dualities

Interesting nonperturbative CFT dualities like the FZZ duality between the 2d black hole and Sine-Liouville can be understood as a combination of two remarkable phenomena: The first is the modular (self-)duality of Liouville or conformal Toda theory, the second is the relation between Liouville theory and the sl_2-WZNW model. The latter can be seen as a natural "quantum" analog of the geometric Langlands duality. It may be reformulated as a correspondence between quantum Teichm\"uller theory and the sl_2-WZNW model.

Jaume Gomis , 't Hooft Loop Correlators and S-duality

We make a proposal for the computation of correlators of 't Hooft loop operators, which were originally introduced as magnetic order parameters of phases in gauge theories. We show that the correlators of 't Hooft loop operators in N=4 Super Yang Mills theory are in complete agreement with the S-duality conjecture, thus exhibiting the action of Electric-Magnetic Duality at the level of correlation functions.

Masahito Yamazaki, Crystal Melting and non-commutative Donaldson-Thomas theory

Motivated by recent mathematical developements in non-commutative Donaldson-Thomas theory, we construct a new statistical mechanical model of crystal melting to count BPS branes wrapping an arbitrary toric Calabi-Yau 3-fold. We also discuss the wall crossing phenomena, which are crucial for the proper understanding of the relation between the crystal melting and the topological string theory. (Joint work with H. Ooguri, arXiv:0811.2801 and arXiv:0902.3996.)

## Mini-conference on Dualities in Physics and Mathematics (March 16-20)

## All talks last 1 hour, and will be held in the Main Seminar Room.

DateTimePlace*SpeakerTitleComments, Links,...Edward FrenkelIntroduction and overviewDavide GaiottoS-duality and boundary conditions, IYan SoibelmanWall-crossing formulas for motivic Donaldson-Thomas invariantsSergei GukovBranes, Duality, and QuantizationDavide GaiottoS-duality and boundary conditions, IIGreg MooreWall-crossing, field theory, and 5-branesAndy NeitzkeWall-crossing in moduli spaces of Higgs bundlesDaniel JafferisDualities in three dimensional gauge theoriesValerio Toledano LaredoStability conditions and Stokes factorsAnton KapustinQuantum Geometric Langlands and Gauge TheoryOri GanorFrom S-Duality to Chern-Simons via Minimal StringsJoerg TeschnerQuantum geometric Langlands vs. nonperturbative CFT dualitiesMaxim ZabzineChiral de Rham complex and generalized geometryJaume Gomis't Hooft Loop Correlators and S-dualityMasahito YamazakiCrystal Melting and non-commutative Donaldson-Thomas theory## AUDIO/VIDEO/NOTES FOR ALL MINI-CONFERENCE LECTURES

## MINI-CONFERENCE INFORMATION

The goal of this mini-conference is to explore the newly found connections between various dualities in physics (such as the S-duality in 4D supersymmetric gauge theory and Mirror Symmetry of 2D sigma models) and in mathematics (geometric Langlands Program). Among the topics considered will be the work of Gaiotto and Witten on duality of boundary conditions in 4D super-Yang-Mills theory, as well as the wall-crossing formulas studied in mathematics by Joyce, Bridgeland, Toledano Laredo, Kontsevich, and Soibelman and in physics by Gaiotto, Neitzke, Moore, and others.

## MINI-CONFERENCE PARTICIPANTS

## ABSTRACTS OF LECTURES

Davide Gaiotto,S-duality and boundary conditions(two lectures)N=4 Super Yang Mills theory has a large set of interesting half BPS, conformally invariant boundary conditions. S-duality acts in surprising ways on that set, mapping even elementary boundary conditions, as Neumann or Dirichlet, to rather intricate ones. We will give detailed examples of S-dual pairs of boundary conditions for U(1), SU(2) gauge groups, and introduce a general recipe for any gauge group. Besides the physical interest, the subject has various mathematical applications, including symplectic duality (aka 3d mirror symmetry) and geometric Langlands, and can be formulated as an interesting mathematical problem in its own sake.

Valerio Toledano Laredo,Stability conditions and Stokes factorsI will explain how Joyce's wall-crossing formulae for invariants counting semistable objects in an abelian category A may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of A.

This allows one in particular to interpret his holomorphic generating functions as defining an isomonodromic family of such connections parametrised by the space of stability conditions of A.

This is joint work with T. Bridgeland (arXiv:0801.3974).

Ori Ganor,From S-Duality to Chern-Simons via Minimal StringsThere are two special values of the coupling constant for which there exist noncentral elements of SL(2,Z) that map N=4 Super Yang-Mills theory with gauge group U(n) to itself. At these values, the field theory can be compactified on a circle with duality-twisted boundary conditions. The low-energy limit of this model directly probes the S-duality operator. Augmented by an R-symmetry twist, and with additional restrictions on the rank n, this low-energy limit appears to be a nontrivial topological field theory. Upon further compactification on a torus, the Hilbert space of the low-energy theory can be mapped, using U-duality, to the finite dimensional space of minimal string states on a three-dimensional manifold that is a torus fibre-bundle over a circle. Using the string theory realization, I'll compare the low-energy theory with Chern-Simons theory. Also, compactification on a Riemann surface of higher genus suggests a relation between the dimension of the Hilbert space of certain Chern-Simons theories on the Riemann surface and the supertrace of the action induced by mirror symmetry on the appropriate cohomology of the appropriate Hitchin space.

Joerg Teschner,Quantum geometric Langlands vs. nonperturbative CFT dualitiesInteresting nonperturbative CFT dualities like the FZZ duality between the 2d black hole and Sine-Liouville can be understood as a combination of two remarkable phenomena: The first is the modular (self-)duality of Liouville or conformal Toda theory, the second is the relation between Liouville theory and the sl_2-WZNW model. The latter can be seen as a natural "quantum" analog of the geometric Langlands duality. It may be reformulated as a correspondence between quantum Teichm\"uller theory and the sl_2-WZNW model.

Jaume Gomis,'t Hooft Loop Correlators and S-dualityWe make a proposal for the computation of correlators of 't Hooft loop operators, which were originally introduced as magnetic order parameters of phases in gauge theories. We show that the correlators of 't Hooft loop operators in N=4 Super Yang Mills theory are in complete agreement with the S-duality conjecture, thus exhibiting the action of Electric-Magnetic Duality at the level of correlation functions.

Masahito Yamazaki,Crystal Melting and non-commutative Donaldson-Thomas theoryMotivated by recent mathematical developements in non-commutative Donaldson-Thomas theory, we construct a new statistical mechanical model of crystal melting to count BPS branes wrapping an arbitrary toric Calabi-Yau 3-fold. We also discuss the wall crossing phenomena, which are crucial for the proper understanding of the relation between the crystal melting and the topological string theory. (Joint work with H. Ooguri, arXiv:0811.2801 and arXiv:0902.3996.)