# Mathematical Physics

This series consists of talks in the area of Mathematical Physics.

## Seminar Series Events/Videos

Feb 25 2019 - 2:00pm
Room #: 400
Speaker(s):
Scientific Areas:

## Categorical Bernstein Operators and the Boson-Fermion correspondence.

Thursday Jan 17, 2019

Bernstein operators are vertex operators that create and annihilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category.

Collection/Series:
Scientific Areas:

## REPRESENTATIONS OF THE ELLIPTIC QUANTUM GROUP AND RELATED GEOMETRY

Monday Jan 14, 2019
Speaker(s):

Collection/Series:
Scientific Areas:

## Categories of line operators in 3d N=4 gauge theories

Monday Oct 29, 2018
Speaker(s):

A 3d N=4 gauge theory admits two topological twists, which we'll simply call A and B. The two twists are exchanged by 3d mirror symmetry. It is known that local operators in the A (resp. B) twist include the Coulomb-branch (resp. Higgs-branch) chiral rings.  In this talk I will discuss the *line* operators preserved by the two twists, which in each case should have the structure of a braided tensor category.

Collection/Series:
Scientific Areas:

## Unification of integrability in supersymmetric gauge theories

Monday Oct 22, 2018
Speaker(s):

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].

Collection/Series:
Scientific Areas:

## Positive geometries and the amplituhedron

Monday Oct 22, 2018
Speaker(s):

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.

Collection/Series:
Scientific Areas:

## Yangians from Nakajima quiver varieties

Monday Oct 15, 2018
Speaker(s):

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.

Collection/Series:
Scientific Areas:

## Towards a categorification of a projection from the affine to the finite Hecke algebra in type A

Monday Sep 24, 2018

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.

Collection/Series:
Scientific Areas:

## Langlands duality and self-duality for Hitchin systems

Monday Sep 24, 2018
Speaker(s):

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.

Collection/Series:
Scientific Areas:

## Toward AGT for general algebraic surfaces

Thursday Sep 20, 2018
Speaker(s):

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.

Collection/Series:
Scientific Areas:

## Localization theory for W-algebras

Monday Sep 10, 2018
Speaker(s):

The localization theorem, which has played a central role in representation theory since its discovery in the 1980s, identifies a regular block of Category O for a semisimple Lie algebra with certain D-modules on its flag variety. In this talk we will explain work in progress which produces a similar picture for the Virasoro algebra and more generally for affine W-algebras.

Collection/Series:
Scientific Areas:

## LECTURES ON-DEMAND

### Jocelyn Bell Burnell: University of Oxford

Speaker: Jocelyn Bell Burnell