Organized by Mina Aganagic, Semeon Artamonov, Miroslav Rapcak, and Vivek Shende

Mondays 2:00-3:00 PM at 402 Le Conte Hall

*Special Day/Time.

This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

In this talk we discuss Koszul duality from a physics perspective, and emphasize its role in coupling quantum field theories to topological line defects. Using this physical translation of Koszul duality as inspiration, we propose a physical definition of Koszul duality for vertex algebras. The appearances of vertex algebras (physically: holomorphic conformal field theories) in physics are legion; one particularly interesting context in which they appear is (a simplified version of) the three dimensional Anti-de Sitter (AdS)/ two-dimensional conformal field theory (CFT) holographic correspondence. One may be tempted to propose that algebras of operators in AdS and in CFT are Koszul dual to one another in this sense. We will find instead, by studying a popular physical example of AdS(3)/CFT(2), that a deformation of this version of Koszul duality is required to relate the two. This talk is based on work in collaboration with K. Costello.

We will discuss the definition of Coulomb branches in N=4 SUSY field theories in 3 dimensions, adapting the Braverman-Finkelberg-Nakajima construction to the case of non-polarized representation. Their computation can be done by an Abelianization theorem. Time permitting, I will discuss the associated TQFT.

It is well-known that the GIT quotient depends on a choice of an equivariant ample line bundle. Various different quotients are related by birational transformations, and their B-models (D^bCoh) are related by semi-orthogonal decompositions, or derived equivalences. If we apply mirror symmetry, it is natural to ask how the A-models of the mirror of various quotients are related. We give a description in the case of toric variety, where the A-side is described using constructible sheaves and Lagrangian skeleton. This is a work in progress.

Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers the Yangian of a quiver Q as defined by Maulik-Okounkov. However, for general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem.

One can define a K-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to K-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.