Berkeley String-Math Seminar

Organized by Mina Aganagic, Dima Galakhov, Vivek Shende and Andrey Smirnov

Mondays, 2:00-3:00, 402 Le Conte Hall.

Schedule of talks for Spring 2017:

Jan 11 Daniel Halpern-Leistner The non-abelian localization theorem and the Verlinde formula for Higgs bundles
Jan 16 Eric Zaslow Framing Duality
Jan 23 No Seminar
Jan 30 Harold Williams Cluster Theory of the Coherent Satake Category
Feb 6 Eugene Gorsky Khovanov-Rozansky homology and Hilbert schemes
Feb 13 Nikolai Reshetikhin Limit shapes and fluctuations in dimer models
Feb 20 No Seminar No Seminar
Feb 27
Peter Koroteev Elliptic algebras and instantons in large-N limit [NOTES]
Mar 6 No Seminar No Seminar
Mar 13 Alexander Braverman Some recent advances in the mathematical construction of Coulomb branches of 3 and 4-dimensional gauge theories
Mar 20 Hiraku Nakajima Geometry of bow varieties [NOTES]
Mar 27 No Seminar
Apr 3 Aaron Lauda
A new look at quantum knot invariants CANCELED!
Apr 10 Constantin Teleman
Dualities in Topological Field Theory
Apr 17 Clay Cordova
BPS Particles, Superconformal Indices, and Chiral Algebras
Apr 24 Dan Xie
Wild Hitchin system and Argyres-Douglas theory
May 1 Erik Carlsson
Geometry behind the shuffle conjecture
May 8 Shamil Shakirov
Integrability of higher genus refined Chern-Simons theory

A note to the speakers: 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.

The seminar archive can be found: here


Jan 11
The Verlinde formula is a celebrated explicit computation of the dimension of the space of sections of certain positive line bundles over the moduli space of semistable vector bundles over an algebraic curve. I will describe a recent generalization of this formula in which the moduli of vector bundles is replaced by the moduli of semistable Higgs bundles, a moduli space of great interest in geometric representation theory and mathematical physics. A key part of the proof is a new ``virtual non-abelian localization formula" in K-theory, which has broader applications in enumerative geometry. The localization formula is an application of the nascent theory of Theta-stratifications, and it serves as a new source of applications of derived algebraic geometry to more classical questions.

Jan 16
In this talk, I will explain and generalize the following curious observation. First, two things: 1. The quiver Q with one node and g arrows has invariants which are conjecturally dimensions of parts of the middle dimensional cohomology of the twisted character variety of a genus-g surface. 2. There is a famous special-Lagrangian submanifold of six-space defined by Harvey and Lawson, and used by Aganagic, Klemm and Vafa in the context of string theory. The open Gromov-Witten invariants of this submanifold depend on an integer-valued "framing" at infinity. Their generating function (superpotential) can be written as a power series in dilogarithms with integer coefficients, the BPS numbers. If we choose framing g, then the BPS numbers are the quiver invariants of Q. Note that g does *not* represent any kind of genus in the physics story!? In general, the invariants of *all* symmetric quivers with n nodes are controlled by a single moduli space associated to a genus-n surface. Go figure. (This is based on joint work with Linhui Shen and David Treumann.)​

Jan 30
We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric avatars of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory). This is joint work with Sabin Cautis.

Feb 6
Khovanov and Rozansky introduced a knot homology theory generalizing the HOMFLY polynomial. I will describe a conjectural relation between the Khovanov-Rozansky homology and the homology of sheaves on the flag Hilbert scheme of points on the plane. The talk is based on the joint work with Andrei Negut and Jacob Rasmussen.

Feb 27
I will discuss some mathematical aspects of instanton counting in two different physical theories- one with gauge group of rank N, the other of small fixed rank. It will be shown that instanton sectors of both theories are equivalent in the N to infinity limit.

Mar 13
I will start the talk by reviewing our recent work with M.Finkelberg and H.Nakajima on the mathematical construction of Coulomb branches of 3-dimensional N=4 super-symmetric gauge theories as affine complex (possibly singular) symplectic algebraic varieties admitting a canonical quantization (no physics background will be assumed). I will then proceed to the discussion of the generalization of this construction to Coulomb branches of 4-dimensional gauge theories compactified on a circle. I will present a set of conjectures about the resulting varieties and give some examples

Mar 20
Bow varieties were introduced by Cherkis as analog of ADHM type description of instanton moduli spaces on the Taub-NUT space ( $\mathbb C^2$ with a hyper-Kaehler metric, not an Euclidean one.) We study these varieties from more algebro-geometric point of view, and introduce their `multiplicative' analog. Applications are their identifications with Coulomb branches of 3d and 4d gauge theories respectively. The first part is joint work with Y.Takayama, while the second is with D.Yamakawa.

Apr 3
The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison's novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (skew Howe) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the existence of an (a,q)-super polynomial conjectured by physicists (joint with Garoufalidis and Lê), and leads to a new elementary approach to Khovanov homology and its sl(n) analogs (joint with Queffelec and Rose).

Apr 10
Kramers-Wannier duality is a symmetry relating the high-and low-temperature phases of the 2-dimensional lattice Ising model. Electric-Magnetic duality is a 3-dimensional duality between abelian (flat) gauge theories for Pontryagin dual abelian groups. Both dualities generalize to higher-dimensional manifolds. We describe the relation between them using the notion of relative field theory. The order and disorder operators of the Ising model are endpoints of Wilson and t’Hooft electro-magnetic loops, respectively. There is a higher-dimensional generalization to finite homotopy types. This is based on joint work with Dan Freed.

Apr 17
I will describe several recently posed conjectures that relate BPS states in four-dimensional N=2 quantum field theories to representation theory of non-unitary chiral algebras. These conjectures construct wall-crossing invariant generating functions of refined BPS indices which surprisingly are equal to characters of chiral algebras. These characters frequently enjoy nice modular properties. I will also discuss extensions to BPS states in the presence of defects which further enrich the correspondence.

May 1
The original "shuffle conjecture" of Haglund, Haiman, Loehr, Ulyanov, and Remmel predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A, in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the ideas that went into my recent proof of this conjecture with Anton Mellit, namely the construction of a new algebra which has many elements in common with DAHA's, and which has been expected to have a geometric construction. I will then explain a new result with Eugene Gorsky and Mellit, in which we construct this algebra using the torus-equivariant K theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals in C[x,y] of finite codimension.

May 8
TIt is known that Chern-Simons topological quantum field theory admits a one-parameter deformation -- refinement -- in the genus 1 sector. I will tell about a genus 2 generalization of this fact. Just like in the torus case, the crucial role is played by an interesting quantum-mechanical integrable system, which is a genus 2 generalization of Ruijsenaars-Schneider-Macdonald system. The knowledge of this quantum mechanics allows us to prove mathematically the corresponding mapping class group action, which is a genus 2 generalization of SL(2,Z)