Mathematical Physics

The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory.  I will explain how string theory unifies these phenomena.  This is based on my work with Kevin Costello [arXiv:1810.01970].

Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic ``canonical form" whose residues reflect
the boundary structure of the space.  Familiar examples include
polytopes and the positive parts of toric varieties.  A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.

I will talk about the definition and examples of positive geometries,
and discuss what is known about the geometry and combinatorics of the
amplituhedron.  The talk will be based on various joint works with
Arkani-Hamed, Bai, Galashin, and Karp.

One can associate to any finite graph Q the skew-symmetic Kac-Moody Lie algebra g_Q. While this algebra is always infinite, unless Q is a Dynkin diagram of type ADE, g_Q shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to Q gives a geometric representations of g_Q. Encouraged by this story, one could hope to define the Yangian of g_Q, for general Q, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties. In fact there are two approaches to doing this: firstly via the stable envelope construction of Maulik and Okounkov, secondly via the preprojective Hall algebra of Schiffmann and Vasserot. Via another Hall algebra construction, due to Kontsevich and Soibelman, and work of myself and Meinhardt on BPS Lie algebras, these approaches turn out to be more or less the same. In showing this we show that the correct Lie algebra of endomorphisms associated to Q is cohomologically graded, with zeroeth piece the old Kac-Moody Lie algebra - the best way to build a (Borcherds) Lie algebra out of Q is directly from geometric representation theory.

Work of Bezrukavnikov on local geometric Langlands correspondence and works of Gorsky, Neguţ, Rasmussen and Oblomkov, Rozansky on knot homology and matrix factorizations suggest that there should be a categorical version of a certain natural homomorphism from the affine Hecke algebra to the finite Hecke algebra in type A, sending basis lattice elements on the affine side to Jucys-Murphy elements on the finite side. I will try to explain some of the structures involved and will talk about recent progress towards a construction of such a categorification in the setting of Hecke categories.

I will sketch why self-dual versions of the moduli of G-Higgs bundles are expected to arise physically from the study of 4d theories of class S. I will then describe an extension of the Langlands duality results of Hausel-Thaddeus (G=SL(n)) and Donagi-Pantev (arbitrary reductive G) that yields self-dual moduli spaces as a corollary.

The Alday-Gaiotto-Tachikawa correspondence connects gauge theory on a fourfold with conformal field theory. We are interested in a certain algebro-geometric incarnation of this framework, where the fourfold is an algebraic surface and instantons/differential geometry are replaced with sheaves/algebraic geometry. In this talk, we will present a certain approach to AGT that yields partial results for quite general surfaces, and ask questions about what still needs to be done to state and prove the full correspondence in the language of algebraic geometry.