Organized by Mina Aganagic, Sujay Nair, 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 want to be added to the seminar **mailing list**, use this link https://groups.google.com/g/berkeley-string-math

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 2023 Spring 2023 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

We revisit supersymmetric localization with monopole operators in 3D N=4 gauge theories, arriving at a new point of view on their quantized Coulomb branch algebras. Our construction agrees with and recovers the existing mathematical definition. We apply the machinery to define a nonabelian analog of shift operators in enumerative theory of quasimaps, and illustrate the construction in detail for curve counts in GL(n)/B.

The talk will consist of two parts. In the first, longer, part I will review the basics of quantum toroidal algebras and their representation theory focusing on the simplest example of type gl(1). I will also discuss the identification between the representations of the algebra and branes of Type IIB string theory, how brane interactions manifest themselves in this setup and lead to integrability.

In the second part, if time permits, I will introduce new brane configurations inspired by algebraic considerations which I call spiralling branes. I will compute the corresponding partition functions and show that they coincide with the K-theoretic vertex function of Nekrasov and Okounkov and the magnificent four partition function recently introduced by Nekrasov, elucidating the algebraic meaning of both of them.

There exists a well-known similarity between the Kloosterman sum in number theory and the Bessel differential equation. This connection was explained by B. Dwork in 70s by discovering the Frobenius structures in the p-adic theory of the Bessel differential equation. In my talk I will speculate that this connection extends to the equivariant quantum differential equations for a wide class of varieties, which includes the Nakajima quiver varieties. As the main result, I will give an explicit conjectural description of the Frobenius structure for the equivariant quantum connection. The traces of the Frobenius structures are natural finite field analogs of the mirror integral representations of the J-function in quantum cohomology. As an example, I will discuss how, in the case of Grassmannians Gr(k,n), these considerations bring us to the B-model side of Gr(k,n) discovered earlier by Marsh and Rietsch.

I will explain how to associate a spectral network to a Demazure weave via Floer theory. Specifically, the talk will present how spectral networks, as introduced by Gaiotto-Moore-Neitzke, arise when computing J-holomorphic strips associated to augmentations of Lagrangian fillings of Legendrian links. In particular, this provides a Floer-theoretical description of Stokes lines in higher rank and with arbitrary irregular Stokes data, not necessarily associated to an algebraic braid. The talk will contain motivation and examples, including the Berk-Nevins-Roberts collision lines in higher rank WKB problems from the symplectic viewpoint as well as connections to cluster algebras and wall-crossing.

The count of microstates for supersymmetric black holes is typically obtained from a supersymmetric index in weakly-coupled string theory. I will explain how to find the saddles of the gravitational path integral corresponding to this index in a general theory of supergravity in asymptotically flat space. Such saddles exhibit a new attractor mechanism which explains the agreement between the string theory index and the macroscopic entropy. Time permitting, I will also describe how the phenomenon of wall crossing, in which the index changes discontinuously, appears at the level of the gravitational path integral.

There is a hope that some of the technology developed for counting curves may be also applicable to certain number-theoretic counting problems. While the exact contours of this hope remain a bit vague at the moment, I will focus on a handful of actual mathematical statements proven in our ongoing joint work with David Kazhdan.

I'll recall some basics about Slodowy slices, generalized slices in the affine Grassmannian, and quantizations thereof called W-algebras and Yangians, respectively, as well as their analogues for affine Lie algebras which are naturally described using the theory of vertex algebras. Then I'll explain a construction of vertex algebras associated to divisors in toric Calabi-Yau threefolds, which include affine W-algebras in type A for arbitrary nilpotents, and outline a dictionary between the geometry of the threefolds and the representation theory of these algebras. I'll also explain the physical interpretation of these results, as an example of twisted holography for M5 branes in the omega background.

The “Algebra of the Infrared” refers to a collection of homotopical algebra structures (discovered by Gaiotto-Moore-Witten) that one associates to a massive two-dimensional N=(2,2) quantum field theory (subject to certain constraints). This provides a powerful framework for working out the category of boundary conditions of such QFTs. Specializing to the example of massive Landau-Ginzburg models, one is lead to a novel “web-like” construction of the Fukaya-Seidel category. In this talk, after reviewing these developments, I will discuss work-in-progress with Greg Moore which seeks to generalize the web framework to N=(2,2) theories with non-trivial twisted masses. In the Landau-Ginzburg context this amounts to studying LG models defined by a holomorphic one-form dW which is closed but not necessarily exact. Among the results we will announce are a Koszul duality theorem for boundary algebras in such theories, and a categorification of the wall-crossing formula for the CP^1 model with twisted masses.

In the first part of the talk, I will recall Kazhdan-Lusztig's geometric realization of the affine Hecke algebra H_q as well as Bezrukavnikov's categorification of the statement. One of the fundamental tools of the theory is the so-called asymptotic affine Hecke algebra introduced by Lusztig (it can be thought of as a "limit'' of H_q as "q goes to 0''). I will explain the geometric realization of this algebra (joint work with Bezrukavnikov and Karpov) and mention applications. In the second part, we will discuss one very concrete example of this story. Understanding of this example will lead us to explicit character formulas for all irreducible modules (with integral highest weights) in categories O for certain Vertex algebras coming from the 4D/2D correspondence. The main “geometric” object of the second part of the talk will be very simple: the equivariant K-theory of the resolution of (arbitrary type) Kleinian singularity. The second part of the talk is based on joint works with Bezrukavnikov, Kac, and Suzuki.

To each compact, orientable surface S whose connected components have non-empty boundary, we define a dg category whose objects are neatly embedded 1-manifolds in S. In the case when S is a disk, this recovers the so-called "Bar-Natan category": the natural setting for Khovanov's celebrated categorification of the Jones polynomial. We expect that our dg categories form part of the 2-dimensional layer of a categorified version of the Turaev--Viro TQFT associated to the quantum group U_q(sl(2)). In particular, the (twisted) Hom-pairing on our dg categories recovers the canonical bilinear form on the state spaces of the Turaev--Viro theory. (This is joint work with M. Hogancamp and P. Wedrich.)

Given a Riemann surface C and a central charge c, one can define the notion of Virasoro conformal block. Virasoro conformal blocks capture universal features of conformal field theory on C. I will describe a new scheme for constructing Virasoro conformal blocks at central charge c=1, by relating them to simpler “abelian” objects, namely conformal blocks for the Heisenberg algebra, on a branched double cover of C. The key new ingredient is a spectral network on the surface C. Some particularly important Virasoro conformal blocks at c=1 are also known as “tau functions”, and I will explain what abelianization tells us about them. This is joint work in progress with Qianyu Hao, inspired by work of Coman-Longhi-Pomoni-Teschner, Iwaki, Marino and others.

Many interesting varieties can be realized as the Coulomb branch of a 3d N=4 gauge theory, and this realization can give us some very interesting information. One of the most familiar varieties that appears this way is the cotangent bundle of the Grassmannian of k-planes in C^n. I'll explain this realization as a special case of the more general theory of bow varieties and discuss how this allows us to construct tilting generators and non-commutative resolutions on T^*Gr(k,n).

In this talk I will start with introducing a new presentation of deformed double current algebra of type \(gl_k\), denoted by \(A^{(k)}\), which is motivated from the study of M2 branes in the twisted M-theory. Then I will explain how to find an algebra embedding from \(A^{(k)}\) to the mode algebra of \(W^{(k)}_\infty\), which is a matrix-extended generalization of \(W_{1+\infty}\) algebra. The embedding is motivated from the M2 ending on M5 configuration. \(A^{(k)}\) can be "doubled" to get an algebra \(Y^{(k)}\). When \(k\) is not equal to 2, we can show that \(Y^{(k)}\) is isomorphic to affine Yangian of type \(A_{k-1}\). Conjecturally, such isomorphism also holds for \(k=2\). I will also discuss various coproducts of these algebras along the way, which are motivated from the fusions of branes.