Course Abstracts and References

David Ben-Zvi - Theory X and Geometric Representation Theory

References:

• E. Witten - Geometric Langlands from Six Dimensions http://arxiv.org/abs/0905.2720
• J. Lurie - On the Classification of Topological Field Theories http://arxiv.org/abs/0905.0465
• D. Freed - The Cobordism Hypothesis http://www.ams.org/journals/bull/2013-50-01/S0273-0979-2012-01393-9/
• A. Kapustin - Topological Field Theory, Higher Categories, and Their Applications http://arxiv.org/abs/1004.2307

Kevin Costello - Factorization Algebras Associated to the (2,0) Theory

Abstract: Kevin will discuss theorems of Beem, Rastelli and van Rees, showing that W-algebras appear as protected sectors of the operators in a 6d (2,0) superconformal field theory. Hide abstract

Show references

• C. Beem, L. Rastello, B. van Rees - W Symmetry in Six Dimensions http://arxiv.org/abs/1404.1079
• C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, B. van Rees - Infinite Chiral Symmetry in Four Dimensions http://arxiv.org/abs/1312.5344

Opers

• http://arxiv.org/abs/math/0501398
• http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf

Finite and Affine W-algebras:

• http://front.math.ucdavis.edu/0105.5225
• http://arxiv.org/abs/0707.3108
• References listed at http://ncatlab.org/nlab/show/Drinfel%27d-Sokolov+reduction

Tudor Dimofte - 3d Gauge Theories, Symplectic Duality and Knot Homology

I will discuss a circle of ideas related to boundary conditions and interfaces in 3d gauge theories with N=4 supersymmetry. Such gauge theories flow at low energy (or long distance) to sigma-models, whose target spaces contain at least two noncompact hyperkahler components, the so-called Higgs and Coulomb branches. Boundary conditions in the gauge theory descend to geometric objects on the Higgs and Coulomb branches, which can be described as modules over a deformation quantization of the ring of functions on the the respective branches. Three-dimensional mirror symmetry maps categories of boundary conditions associated to the Higgs branch to those associated to the Coulomb branch, and vice versa. Part of my aim is to explain how these categories are related to generalizations of BGG Category O, recently studied by Braden, Licata, Proudfoot, and Webster (BLPW), and how 3d mirror symmetry induces the "symplectic duality" of PLPW.

I will begin by reviewing some of the basics of 3d N=4 theories and their moduli spaces, and then discuss two important classes of examples:

1. For abelian gauge theories, the moduli spaces are hypertoric manifolds, and boundary conditions are related to "hypertoric category O." Physics predicts a new, explicit form of the symplectic duality map relating categories on the Higgs and Coulomb branches of a given theory.
2. For quiver gauge theories, with non-abelian gauge groups, the same basic symplectic duality map can be worked out. More interestingly, in this case, the Coulomb branch is a moduli space of singular monopoles. The 3d gauge theory suggests a new, explicit description of the Coulomb branch as a complex manifold. Quantization of the algebra of functions produces a Yangian (related to work of Kamnitzer-Webster-Weekes-Yacobi), and boundary conditions are related to quantizations of moduli spaces of vortices, on which the Yangian acts.

Finally, I will discuss the reduction of 3d N=4 theories to two-dimensional Landau-Ginzburg models. The 2d models provide a new, simple way to study many features of boundary conditions in 3d, and their morphisms. One important application is a new description of braiding actions on "3d" categories, which relates mathematical definitions of knot homology (using these categories) to a physical proposal of Gaiotto and Witten (using LG models). Hide Abstract

Show references

In physics:

1. M. Strassler - An Unorthodox Introduction to Supersymmetric Gauge Theory http://arxiv.org/abs/hep-th/0309149 (basics of supersymmetric QFT and renormalization group flow)
2. K. Intriligator and N. Seiberg - Mirror Symmetry in Three Dimensional Gauge Theories http://arxiv.org/abs/hep-th/9607207
3. J. de Boer, K. Hori, H. Ooguri, and Y. Oz - Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes http://arxiv.org/abs/hep-th/9611063 (original discussions of mirror symmetry in three dimensions)
4. D. Gaiotto and P. Koroteev - On Three Dimensional Quiver Gauge Theories and Integrability http://arxiv.org/abs/1304.0779 (a nice review of mirror symmetry for quiver gauge theories, and reduction to two dimensions)
5. D. Gaiotto and E. Witten - Knot Invariants from Four-Dimensional Gauge Theory http://arxiv.org/abs/1106.4789 (using 2d Landau-Ginzburg models to obtain quantum braiding actions)

In mathematics:

1. P. Kroheimer - The Construction of ALE spaces as Hyper-Kähler Quotients, J. Diff Geom 29 (3) 1989
2. H. Nakajima - Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (2) 1994 (the constructions that inspired 3d mirror symmetry in physics)
3. T. Braden, A. Licata, N. Proudfoot, and B. Webster - Gale duality and Koszul duality http://arxiv.org/abs/0806.3256
4. T. Braden, A. Licata, N. Proudfoot, and B. Webster - Hypertoric category O http://arxiv.org/abs/101.2001 (categories of D-modules related to boundary conditions in 3d abelian theories, and their symplectic duality)
5. T. Braden, A. Licata, N. Proudfoot, and B. Webster - Quantizations of Conical Symplectic Resolutions II: Category O and Symplectic Duality http://arxiv.org/abs/1407.0964 (categories related to boundary conditions in more general 3d theories)
6. J. Kamnitzer, B. Webster, A. Weekes, O. Yacobi - Yangians and Quantizations of Slices in the Affine Grassmannian http://arxiv.org/abs/1209.0349 (the mathematics related to a quantization of the Coulomb branch in quiver gauge theories)

Andrew Neitzke - Hitchin Systems in Supersymmetric Field Theory

References:

• E. Witten - Geometric Langlands from Six Dimensions http://arxiv.org/abs/0905.2720
• D. Gaiotto, G. Moore, A. Neitzke - Wall-crossing, Hitchin Systems, and the WKB Approximation http://arxiv.org/abs/0907.3987
• A. Neitzke - Hitchin systems in N=2 field theory https://www.ma.utexas.edu/users/neitzke/expos/hitchin-systems.pdf
• A. Neitzke - Cluster-like Coordinates in Supersymmetric Field Theory http://www.pnas.org/content/111/27/9717.abstract
• G. Moore - PITP lectures on BPS states and Wall-Crossing http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf
• D. Gaiotto, G. Moore, A. Neitzke - Spectral Networks http://arxiv.org/abs/1204.4828