Organized by Mina Aganagic, Ivan Danilenko, Peter Koroteev, and Miroslav Rapcak

Weekly on **Mondays 2:10 PM** (Pacific Time)

**Most talks starting Oct 11 will be given in Zoom** using the following link: http://berkeley.zoom.us/j/93328405860?pwd=Um1GbHBCSUJMdUlWWnd0ZVMxQmwwdz09

For in-person meetings: at 402 Physics South

*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.

Spring 2021, Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016

Khovanov showed, more than 20 years ago, that Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. In these lectures, I will describe two solutions to this problem, which originate in string theory, and which are related by a version of (homological) mirror symmetry. The first approach was recently proven by Ben Webster to be equivalent to his purely algebraic approach to categorification using KRLW algebras. The second approach is completely new and much simpler. The resulting theory describes a precise way in which Chern-Simons (or quantum group) knot invariants derive from a more fundamental theory. It makes many predictions both for algebraic and symplectic geometry, two areas of mathematics connected by mirror symmetry, and for knot theory.

I will describe this story in the course of several lectures, pointing out open problems along the way. The lectures are based on:

- M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves,’' arXiv:2004.14518
- M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part II: Lagrangians,’' arXiv:2105.06039

While the subject brings together substantial amount of mathematics and physics, the end result is accessible to graduate students. Correspondingly, the lectures should get more accessible as we go along.

We introduce the notions of (G,q)-opers and Miura (G,q)-opers, where G is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q)-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. Additionally we associate to a (G,q)-oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as generalized q-Wronskians. We show that the QQ-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.

Based on recent paper with Frenkel, Sage and Zeitlin

In the ``Euclidean'' approach to quantum gravity, it sometimes seems useful to include complex saddle points. But what class of complex spacetime metrics is physically sensible? That will be the topic of this lecture.

Given a 3d N=4 supersymmetric quantum field theory, there is an associated Coulomb branch, which is an important reflection of the A-twist of this theory. In the case of gauge theories, this Coulomb branch has a description due to Braverman-Finkelberg-Nakajima; I'll discuss how we can generalize this geometric description in order to construct non-commutative resolutions of Coulomb branches (giving a more physical meaning to a previously known construction of Kaledin), and its connection to wall-crossing functors arising from varying Kahler moduli (and thus to the construction of knot homology).

Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants.

By the MNOP conjecture these rank 1 “abelian” invariants are determined by the GW invariants of X. Along the way we also express rank r DT invariants in terms of invariants counting D4-D2-D0 branes: rank 0 sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore.

Over the past decade we have witnessed the emergence of a plethora of correspondences between QFTs in various dimensions arising from higher dimensional SCFTs. In this talk I will overview another strategy (well-known to experts) to obtain correspondences building upon geometric engineering techniques. Several new applications and examples will be presented, involving supersymmetric theories in different dimensions. In particular, we will include some results about the higher Donaldson-Thomas theory for Calabi-Yau three-folds, generalizations of level/rank dualities, and an algebra organizing G(2) instantons.

The Reshetikhin-Turaev construction for the quantum group U_q(gl(1|1)) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. Tangle Floer homology is a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. In earlier work with Ellis and Vertesi, we show that tangle Floer homology categorifies a Reshetikhin-Turaev invariant arising naturally in the representation theory of U_q(gl(1|1)); we further construct bimodules \E and \F corresponding to E, F in U_q(gl(1|1)) that satisfy appropriate categorified relations. After a brief summary of this earlier work, I will discuss how the horizontal trace of the \E and \F actions on tangle Floer homology gives a gl(1|1) action on annular link Floer homology that has an interpretation as a count of certain holomorphic curves. This is based on joint work in progress with Andy Manion and Mike Wong.