Organized by Mina Aganagic, Ivan Danilenko, Andrei Okounkov, and Peng Zhou

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

Meetings are in-person, at **402** Physics South

We have a lunch on the 4th floor of Physics South before the seminar. If you are interested in join us for the lunch or dinner, please sign-up at here.

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For those joining us remotely, we have a Zoom link: https://berkeley.zoom.us/j/99373923587?pwd=SUhXamdtbHhJMUJERlJ4NVJHL1Jtdz09

*Special Day/Time/Location.

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 2022 Spring 2022 Fall 2021, Spring 2021, Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016

I will report on a joint work in progress with Pablo Boixeda Alvarez, Michael McBreen and Zhiwei Yun where categories of microlocal sheaves on some affine Springer fibers are described in terms of the Langlands dual group. In particular, in the slope 1 case we recover the regular block in the category of (graded) modules over the small quantum groups. Assuming a general formalism connecting microlocal sheaves to Fukaya categories, this yields new examples of homological mirror symmetry, including the mirror dual to \(T^*(G/B)\).

In work to appear with Ballin-Creutzig-Dimofte, we constructed vertex operator algebras associated to A and B twists of 3d N=4 abelian gauge theories. These are boundary VOAs supported on holomorphic boundary conditions of Costello-Gaiotto. For the B twist, the vertex algebra \(V_B\) is a simple current extension of an affine Lie superalgebra, and using the work of Creutzig-Kanade-McRae, we can study its representation theory using this simple current extension. An analogous extension procedure for quantum groups was developed by Creutzig-Rupert. I will explain how to apply their strategy to \(U_q^H(\mathfrak{sl}(2))\), the unrolled restricted quantum group at 4-th root of unity, and obtain a quantum supergroup whose category of representations is equivalent to that of \(V_B\). This is joint work in progress with T. Creutzig and T. Dimofte.

In a landmark work, Frances Kirwan described the relation between the cohomology of a GIT quotient of a smooth projective variety X and the equivariant cohomology of X by what is known as the ‘subtraction method’: this relies on the equivariant perfection of the destabilizing stratification of X. Later work geared toward describing the cohomology ring structure relied on Poincare duality and explicit integration formula, but did not produce explicit answers. It has long been suspected that a better answer applies to quantum cohomology, and a precise conjecture to that effect was proposed by the speaker some time ago. I will describe the background to the conjecture and outline the recent proof, which is joint work with Dan Pomerleano.

Recent advances around Fukaya categories can be used to (mathematically rigorously) produce sheaves on \(\mathrm{Bun}_G\) from smooth fibers of Hitchin fibrations. The resulting sheaves are presumably Hecke eigensheaves; I’ll explain why I don’t know how to prove this, and discuss various related questions.

In my talk I will first review the interpretation of the counts of solutions of Kapustin-Witten equations on a 3-manifold times a line as Stokes coefficients associated to the perturbative expansion of Chern-Simons theory. These counts can be naturally combined into q-series with integral coefficents, labelled by an ordered pair of flat connections. I will then present an explicit algorithm/formula for computing them for a large class of closed 3-manifolds. If time permits I will also make some comments about categorification of these counts and relation to Fukaya-Seidel category of connections on the 3-manifold.

Link homology is the space of states of BPS particles in a special 5d QFT. The BPS particles can be presented as D2 branes attached to Chern-Simons-related D4 branes and link-related NS5 branes. We assume that D2 branes form a stack split into domains by D4 and NS5 interfaces. The vibrations of the stack are described by a 3d topological B-model with target spaces being Nakajima quiver varieties. We use the associated 2-categories in order to describe the categorical braid group actions and find the link homology. The talk is based on my joint work with A.Oblomkov, R.Rimanyi and T.Dimofte.

Top down (i.e., stringy) constructions of quantum field theories provide a general template for constructing and studying a wide variety of different strongly coupled systems which are otherwise difficult to study using “textbook” methods based on perturbation theory of a Lagrangian field theory. In this talk we use the geometry of extra dimensions in string theory to study generalized symmetries of such systems, including their action on local operators, extended operators, and the interplay between these structures. Using this perspective, we also extract the corresponding topological operators associated with these symmetry generators. This provides a method to read off further properties such as non-trivial fusion rules for various categorical symmetries.

Kontsevich and Soibelman suggested a correspondence between Donaldson-Thomas invariants of Calabi-Yau 3-folds and holomorphic curves in complex integrable systems. After reviewing this general expectation, I will present a concrete example related to mirror symmetry for the local projective plane (partly joint work with Descombes, Le Floch, Pioline), along with applications in enumerative geometry (partly joint work with Fan, Guo, Wu). I will end by an “explanation” of the general correspondence based on the physics of 4d N=2 quantum field theories and holomorphic Floer theory.

Stable envelopes are correspondences useful for constructing geometric action of quantum groups and solutions to quantum Knizhnik-Zamolodchikov (qKZ) equations. I will review basic aspects of this and explain the construction in a novel class of examples consisting of certain vortex (also known as quasimap) moduli spaces. The main technical result is that K-theoretic curve counts in these varieties are controlled by qKZ equations with vertex operators of Verma module type (in this talk, I will focus on the sl(2) case). Applications of the construction include the ramified version of quantum q-Langlands correspondence of Aganagic-Frenkel-Okounkov, and a proposal for categorification of the Gukov-Manolescu invariant of knot complements. This is work in progress with Mina Aganagic.

Deeper structures behind BPS counting on toric Calabi-Yau 3-folds have recently been realized mathematically in terms of the quantum loop group associated to a certain quiver drawn on a torus, which is endowed with an action on the BPS vector space via crystal melting. In this talk, we identify the annihilator of the aforementioned action, thus leading to the definition of a reduced quantum loop group associated to a toric Calabi-Yau 3-fold satisfying a certain consistency condition we call “shrubbiness”.

Computing the \(L_2\) cohomology of moduli spaces of monopoles and instantons is a challenging problem. It is significant in physics having an interpretations as counting the BPS states in quantum gauge theories, as well as in mathematics, manifesting itself in the geometric Langlands correspondence for complex surfaces.

We propose a rather unconventional compactification of these moduli spaces in terms as exploded geometry. The corresponding De Rham theory was developed by Brett Parker and, as we expect, should produce the desired cohomology spaces. In this talk I shall describe how exploded geometry naturally emerges in the study of monopole walls (doubly periodic monopoles). Then I shall describe the exploded twistor description of monopoles in \(\mathbb{R}^3\), which automatically incorporates their asymptotic form and provides the desired compactification of their moduli space.

Quasimaps to GIT quotients were defined by Ciocan-Fontanine--Kim--Maulik. Following their ideas, we define quasimaps to moduli spaces of sheaves. We show that such quasimaps are intrinsically governed by sheaves, state some basic properties of their moduli spaces and discuss a few examples.

In the second talk, using quasimaps, we establish a very general wall-crossing formula relating Donaldson-Thomas theory of Surface x Curve and Gromov-Witten theory of moduli spaces of sheaves on Surface. The wall-crossing is governed by Vertex (also known as I-function). We then discuss the old and the new instances of this wall-crossing.

I will report an on-going joint project with Hanany and Finkelberg. We identify Coulomb branches of orthosymplectic quiver gauge theories with orthosymplectic bow varieties. Then we use this identification to realize closures of nilpotent orbits for SO, and more as Coulomb branches.

In the third talk, we discuss how quasimaps can be used to translate S-duality of Vafa-Witten invariants of Curve x Curve’ to a duality of quasimap invariants of moduli spaces of Higgs bundles on Curve. This provides an enumerative realization of Kapustin-Witten’s dimensional reduction in the case of SL and PGL.

Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of length d sheaves is p(d), the number of plane partitions of d. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about categorical and K-theoretic refinements of DT invariants and of related enumerative invariants, such as Pandharipande-Thomas (PT) and BPS invariants. In the first talk, I will focus on the explicit case of C^3. In particular, we show that the K-theoretic DT invariant for d points on C^3 also equals p(d). A central construction here is that of quasi-BPS categories, which is the categorical version of BPS invariants. In the second talk, I will talk about a categorical version of the DT/ PT (Pandharipande-Thomas) correspondence for local surfaces. Time permitting, I will also discuss the construction of quasi-BPS categories for quivers with potential and for K3 surfaces. The talks are based on joint work with Yukinobu Toda.

I will give an update on the dictionary between the representations of quantum toroidal algebras and branes of Type IIB string theory. I will argue that brane crossings correspond to R-matrices, of which the "degenerate resolved conifold" is one example. A more interesting example is given by the "Hanany-Witten R-matrix", for which the brane creation effect and Hanany-Witten rules become algebraic identities.

In a few cases, for a quasiprojective suface S it is known that there are vertex algebras which provide formulas for enumerative invariants of moduli spaces of sheaves on S. I will explain a number of examples where this is known or expected to hold and focus on extending this structure to the case of elliptic surfaces. I will describe a close relationship between the geometry of moduli spaces of sheaves on elliptic surfaces and the construction of vertex representations of toroidal lie algebras and their supersymmetric and logarithmic extensions.