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

Weekly on **Mondays 1:10 PM** (Pacific Time). Talks will be given in Zoom using the following link: http://berkeley.zoom.us/j/93328405860?pwd=Um1GbHBCSUJMdUlWWnd0ZVMxQmwwdz09

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*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.

Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016

Knizhnik-Zamolodchikov (KZ) equation was derived in late 1980s from Ward identities of two dimensional conformal field theory with a current algebra symmetry. It is obeyed by the WZNW conformal blocks but remarkably admits analytic continuation in spins, magnetic quantum numbers and the level. In this talk I will review the work of recent years in which this extension meets the relevant physics. I will derive equations obeyed by the expectation values of the surface defect(s) in four dimensional supersymmetric gauge theory, and demonstrate its isomorphism to the KZ equation for 4- (joint work with O.Tsymbaliuk) and 5-point (joint work with S.Jeong and Norton Lee) conformal blocks in genus zero, with infinite dimensional representations associated to the vertices. We'll find a few surprising twists not seen in the rational conformal field theory. If time permits I will discuss the critical level limit and the results for the eigenfunctions of quantum Gaudin and XXX models (joint work with Norton Lee).

I will discuss my work on the \Omega-deformation of brane intersections in M-theory and its applications to vertex algebras.

Two-dimensional integrable field theories are characterised by the existence of infinitely many integrals of motion. Recently, two unifying frameworks for describing such theories have emerged, based on four-dimensional Chern-Simons theory in the presence of surface defects and on Gaudin models associated with affine Kac-Moody algebras. I will explain how these formalisms can be used to construct infinite families of two-dimensional integrable field theories. The latter can all naturally be formulated as so-called E-models, a framework for describing Poisson-Lie T-duality in sigma-models. The talk will be based on the joint work [arXiv:2008.01829] with M. Benini and A. Schenkel and [2011.13809] with S. Lacroix.

We present a construction of topological quantum gravity, which connects three previously unrelated areas: (1) Topological quantum field theories of the cohomological type, as developed originally by Witten; (2) the mathematical theory of the Ricci flow on Riemannian manifolds in arbitrary spacetime dimension, developed originally by Hamilton and later by Perelman in his proof of the Poincare conjecture; and (3) nonrelativistic quantum gravity of the Lifshitz type. This connection should be useful both for physics and for mathematics: It puts the mathematical literature on the Ricci flow into a new perspective using the methods of path integrals and quantum field theory, and sheds new light on puzzles of quantum gravity (spacetime topology change, short-distance completeness, etc) in a controlled setting in which many powerful theorems have been proven by the mathematicians since Perelman.

A Looijenga pair is a pair (X,D) with X a smooth complex projective surface and D a singular anticanonical divisor in X. I will describe a series of correspondences relating five different classes of string-theory motivated invariants specified by the geometry of (X,D):

- the log Gromov--Witten theory of (X,D),
- the Gromov--Witten theory of X twisted by the sum of the dual line bundles to the irreducible components of D,
- the open Gromov--Witten theory of special Lagrangians in a toric Calabi--Yau 3-fold determined by (X,D)
- the Donaldson--Thomas theory of a symmetric quiver specified by (X,D), and
- a class of BPS invariants considered in different contexts by Klemm--Pandharipande, Ionel--Parker, and Labastida--Marino--Ooguri--Vafa.

We first explain how counting holomorphic curves ending on a Lagrangian in a Calabi-Yau 3-fold by the values of their boundaries in the skein module of a Lagrangian leads to a definition of open Gromov-Witten invariants. We study basic non-compact Lagrangians where the count can be carried out directly by looking at curves at infinity. The calculation leads to a local description of 1-parameter families of holomorphic curves near basic disks. We use this description to explain how basic holomorphic disks give the geometry underlying quiver descriptions of generating functions for colored HOMFLY polynomials, as well as other related topological string partition functions. The talk reports on several joint projects with Gruen, Gukov, Kucharski, Longhi, Shende, Sulkowski, Stosic, and Park.

After reviewing homological mirror symmetry for toric varieties (with a quick new proof), I will explain how to glue together large volume mirrors to large complex structure toric degenerations. Homological mirror symmetry for these mirrors then follows from known local-to-global principles.

For any complex reductive group G one can associate a geometric object called the Affine Grassmannian. The transveral slices are interesting affine Poisson varieties appearing in this geometry. In this talk I'll focus on the equivariant quantum cohomology of these spaces. In particular, I'll show how the quantum differential equation in types ADE can be identified with the trigonometric Knizhnik-Zamolodchikov equation.

I will review Braverman-Finkelberg's geometric Satake correspondence conjecture for affine Lie algebras via instanton moduli spaces on C^2/(Z/\ell) and their refinement by the use of Coulomb branches of affine quiver gauge theories. Most of the statements were proved in type A, viewing Coulomb branch as quiver varieties and use the level-rank duality. Then I would like to spend most of my time explaining a new approach to one remaining statement. (This is a joint work with Dinakar Muthiah.) It is a description of the intersection cohomology as a graded vector space, which is given in terms of Brylinski-Kostant filtration in the usual geometric Satake. The new approach is regarded as an affine Lie algebra version of a work by Ginzburg-Riche. It gives a coset VOA module structure on the equivariant intersection cohomology groups, as conjectured by Belavin-Feigin, Nishioka-Tachikawa, as a higher level AGT. It is also regarded as a Coulomb branch type construction of the cotangent bundle of an affine flag variety and its quantization.

Deligne categories are tensor categories, introduced by P. Deligne, which provide a formal way to interpolate representation-theoretic structures attached to classical groups and supergroups (such S_n, GL(n),Sp(2n),O(n),GL(n|m),OSp(n|2m),etc.) to complex values of the integer "rank parameter" n. I will first review them and then explain how to use them to construct and study deformed double current algebras which have recently become increasingly popular in the mathematics and physics literature. This is joint work with D. Kalinov and E. Rains.

I will discuss constructions of supersymmetric interfaces in quantum field theories with eight supercharges, both within one theory and between different theories, that provide physical realizations of certain geometric constructions in the context of quiver varieties (such as stable envelopes and actions of Yangians/quantum loop/quantum elliptic algebras). The basic building blocks are supersymmetric Janus interfaces that realize stable envelopes introduced by Maulik and Okounkov. 3d theories give rise to their elliptic versions as in Aganagic-Okounkov, and the K-theoritic and cohomological versions follow by the dimensional reduction to 2d and 1d. Based on the work in progress with N.Nekrasov.

I will discuss supergroup gauge theory with emphasis on the following aspects: instanton counting, Seiberg-Witten geometry, brane construction, realization in topological string and intersecting defects, Bethe/Gauge correspondence, and quantum algebraic structure. Based on joint works with H.-Y. Chen, N. Lee, F. Nieri, V. Pestun, and Y. Sugimoto (mostly summarized in https://arxiv.org/abs/2012.11711).