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. Fill the following form to join.

*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.

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

Braverman, Finkelberg, and Nakajima define the K-theoretic Coulomb branch of a 3d N=4 SUSY gauge theory as the affine variety M_{G,N} arising as the equivariant K-theory of certain moduli space R_{G,N}, labelled by the complex reductive group G and its complex representation N. It was conjectured by Gaiotto, that (quantized) K-theoretic Coulomb branches bear the structure of (quantum) cluster varieties. I will outline a proof of this conjecture for quiver gauge theories, and show how the cluster structure allows to count the BPS states (aka DT-invariants) of the theory. Time permitting, I will also show how the above cluster structure relates to positive and Gelfand-Tsetlin representations of quantum groups, and higher rank Fenchel-Nielsen coordinates on moduli spaces of PGL_n local systems. This talk is based on joint works with Gus Schrader.

I will explain a physically motivated construction of Ricci-flat K3 metrics via a hyper-Kahler quotient, which yields the first examples of explicit Ricci-flat metrics on compact non-toroidal Calabi-Yau manifolds. I will also relate it - both physically and mathematically - to a second such construction, which is as yet not completely explicit: the missing data is the BPS index of a little string theory on T^2. Via string dualities, we can reformulate this data in terms of both open string reduced Gromov-Witten theory of K3 and generalized Donaldson-Thomas theory of an auxiliary non-compact threefold. I will show that these invariants may be extracted from the metrics produced by the first approach.

The monodromy of quantum difference equations is closely related to elliptic stable envelopes invented by M.Aganagic and A.Okounkov. In the talk I will explain how to extract these equations from the monodromy using the geometry of the variety X and of its symplectic dual Y. In particular, I will discuss how to extend the action of representation-theoretic objects on K(X), such as quantum groups, quantum Weyl groups, R-matrices, etc, to their action on K(Y). As an application, we will consider the example of the Hilbert scheme of points in the complex plane, where these results allow us to prove the conjectures of E.Gorsky and A.Negut about the infinitesimal change of the stable basis. Based on joint work with A.Smirnov.

This talk is concerned with non-abelian mirrors. We define (non)abelian gauged linear sigma models (GLSMs) by providing some data first, and then we study T-dual theories of GLSMs. The construction of nonabelian mirrors is related to abelian mirrors construction found by Hori and Vafa in 2000. Thus, we will review abelian mirrors first and then give a framework for nonabelian mirrors. Two concrete examples will be discussed in the talk, the projective space and Grassmannian. Finally, I will briefly summarize some applications of nonabelian mirrors such as 2d Hori-Seiberg duals in mirrors, mirrors of pure gauge theories.

The 4D/2D correspondence recently discovered by Beem et al. constructs representation theoretical objects, such as representations of affine Kac-Moody algebras, as invariants of 4 dimensional superconformal field theories with N = 2 supersymmetry. Furthermore, it is expected that there is a remarkable duality between the representation theoretical objects constructed in this way and the geometric invariants called Higgs branch of the original 4 dimensional theory. In this talk, I will discuss about this duality from a mathematical perspective.

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle.

While the BPS state counting problem in general toric Calabi-Yau manifolds was solved a decade ago, it has long remained an unsolved problem to identify the underlying BPS state algebra. Recently, we solved this problem by introducing a new infinite-dimensional algebra, the BPS quiver Yangian. We constructed representations of the algebra in terms of crystal melting, and derived the representations physically from supersymmetric quantum mechanics. This talk is based on two papers arXiv:2003.08909 and arXiv:2008.07006, in collaboration with Wei Li and Dmitry Galakhov.

It is a general expectation that the refined topological string theory of local Calabi-Yau 3-folds produces quasimodular forms which are solutions of some refined holomorphic anomaly equation. I will present a mathematical derivation that it is indeed the case for the refined topological string on local P^2 in the Nekrasov-Shatashvili limit. The rather indirect proof exploits mathematical incarnations of various string dualities. Along the way, I will present a new way to compute the D4-D2-D0 BPS spectrum of local P^2 in the large volume limit. Work partly joint with Honglu Fan, Shuai Guo, and Longting Wu.

We analyze symmetries corresponding to separated topological sectors of 3d N= 4gauge theories with Higgs vacua, compactified on a circle. The symmetries are encoded in Schwinger-Dyson identities satisfied by correlation functions of a certain gauge-invariant operator, the "vortex character." Such a character observable is realized as the vortex partition function of the 3d gauge theory, in the presence of a 1/2-BPS line defect. The character enjoys a double refinement, interpreted as a deformation of the usual characters of finite-dimensional representations of quantum affine algebras. We derive and interpret the Schwinger-Dyson identities from various physical perspectives: in the context of the 3d gauge theory itself, in a 1d gauged quantum mechanics, in 2d q-Todatheory, and in 6d little string theory. We establish the dictionary between all approaches.

In the second part of my talk I would start by giving a reasonably brief summary of the first part, followed by a discussion of the known or expected relations to some other lines of research, which could include (a selection of) the following points:

- Relation to geometry of hypermultiplet moduli spaces
- Relation to the spectrum of BPS-states, geometry of space of stability conditions
- Relation classical-quantum
- Relation with Theta-functions on intermediate Jacobian fibration
- Interplay between 2d-4d wall-crossing and free fermion picture.

Starting with some context and motivation from a mathematical and physical perspective, I will discuss recent work with Raphael Rouquier on a higher tensor product operation for 2-representations of Khovanov's categorification of U(gl(1|1)^+), examples of such 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensor-product-based gluing formula for these 2-representations expanding on work of Douglas-Manolescu.

Stacks of M2-branes in Omega background lead to a class of Coulomb branch algebras due to Kodera and Nakajima. On the other hand, stacks of M5-branes in Omega background lead to a class of vertex operator algebras due to a version of the AGT correspondence. I will provide a new interpretation of key objects in the theory of vertex operator algebras, such as Miura operators and degenerate fields, as describing various junctions of M2 and M5 branes in M-theory. The new perspective leads to a rather close interplay between the relevant class of Coulomb branch algebras and vertex operator algebras with many striking consequences.