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 Fall 2016:

Sep 12	Mina Aganagic	Two mathematical applications of little string theory
Sep 19	Vivek Shende	Categories and moduli spaces associated to singular Lagrangians
Sep 26	Andrei Negut	q-deformed W-algebras, Ext operators and the AGT-W relations
Oct 3	Dmitry Galakhov	The two-dimensional Landau-Ginzburg approach to link homology
Oct 10	No Seminar
Oct 17	Lotte Hollands	Opers, spectral networks and the T[3] theory
Oct 24	Michael Viscardi	Equivariant quantum cohomology and the geometric Satake equivalence
Oct 31	Pavel Putrov	Resurgence in Chern-Simons theory
Nov 7	Nikita Nekrasov	How I learned to stop worrying and to love both instantons and anti-instantons
Nov 14	Andrey Smirnov	Quantum K-theory and Nekrasov-Shatashvili conjecture
Nov 21	No Seminar
Nov 28	No Seminar
Dec 5	Tudor Dimofte (2:30-4) and Alexander Braverman (4-5)	TBA

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


Abstracts

Sep 12
I will describe two mathematical applications of little string theory. The first leads to a variant of AGT correspondence that relates q-deformed W algebra conformal blocks to K-theoretic instanton counting. This correspondence can be proven for any simply laced Lie algebra. The second leads to a variant of quantum Langlands correspondence which relates q-deformed conformal blocks of an affine Lie algebra and a W algebra, associated to a Langlands dual pair of Lie algebras. The proof of the correspondence for simply laced Lie algebras involves, in a crucial way, the recently discovered elliptic stable envelopes. This is based on joint works with Nathan  Haouzi and with Edward Frenkel and Andrei Okounkov.

Sep 19
I describe certain categories which arise from the consideration of singular Lagrangian geometries in symplectic manifolds, in 4 and 6 dimensions. These are (mathematically) the Fukaya category of a neighborhood of the singular space; presumably in physics, some category of branes. I will explain how moduli spaces usually associated with irregular singularities, cluster varieties, knot homology, and various other subjects arise in this manner, and then pose some questions regarding the connection to brane-tiling, higher genus effects, and quantization.

Sep 26
I will present a construction of the q-deformed W-algebra of gl_r and its Verma module, that does not use the free field realization or screening charges. The upshot is that our method allows us to directly compute the commutation relations between the Carlsson-Okounkov Ext operator on the moduli space of rank r sheaves and the defining currents of the q-deformed W-algebra. This implies a geometric representation theory interpretation of the AGT-W relations.

Oct 3
We describe rules for computing a homology theory of knots and links in R^3. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of SU(2) magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.

Oct 17
In this seminar I will explain how certain partition functions of four-dimensional quantum field theories, such as the non-Lagrangian T[3] theory, have a geometric interpretation as generating functions of so-called opers. This will reveal close ties to spectral networks and the exact WKB method. (This is joint work with Andy Neitzke.)

Oct 24
Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a semisimple group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These slices arise as Coulomb branches of 3D N=4 SUSY framed quiver gauge theories, so that our result is an analogue of (part of) the work of Maulik and Okounkov in the Higgs branch setting.

Oct31
In my talk I will consider resurgence properties of Chern-Simons theory on compact 3-manifolds. I will also describe what role resurgence plays in the problem of categorification of Chern-Simons theory, that is the problem of generalizing Khovanov homology of knots to compact 3-manifolds.

Nov7
In quantizing classical mechanical systems to get (non-perturbative in hbar corrections to) the eigenvalues of the Hamiltonian one often sums over the classical trajectories as in localisation formulas, but also take into account the contributions of the so-called "instanton-antiinstanton gas". The latter is an ill-defined set of approximate solutions of equations of motion. The talk will attempt to alleviate some of the frustrations of this 40+ yrs old approach by making use of honest solutions of equations of motion of complexified classical mechanical system. The examples will include algebraic integrable systems, from the abstract Hitchin systems to the well-studied anharmonic oscillator. If time permits, I will explain the origin of these ideas in the Bethe/gauge correspondence of Nekrasov-Shatashvili.

Nov14
In my talk I will discuss the construction of quantum K-theory using the moduli spaces of quasimaps. This construction works well for Nakajima quiver varieties and I will illustrate it on the simplest example: the cotangent bundles over grassmannians.