Organized by Mina Aganagic, Ivan Danilenko, 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.

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

Given a polarized hyperplane arrangement, Braden-Licata-Proudfoot-Webster defined two convolution algebras: A & B. We show that both of them can be realized as Fukaya categories of hypertoric varieties. This proves a conjecture of Braden-Licata-Phan-Proudfoot-Webster in 2009 and gives a geometric interpretation of the Koszul duality between A & B. The proof relies on the construction of non-commutative vector fields on Fukaya categories (Abouzaid-Smith), the surgery quasi-isomorphism for singular Legendrians (Asplund-Ekholm), and the cobar interpretation of the Chekanov-Eliashberg DGA (Ekholm-Lekili). Time permitted, I'll also talk about the homological mirror symmetry for hypertoric varieties and the relation to knot Floer homology. This is based on joint works in progress with Sukjoo Lee, Siyang Liu and Cheuk Yu Mak.

In 2013, Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees constructed a correspondence between four dimensional N=2 SCFTs, a certain kind of supersymmetric quantum field theory, and vertex algebras. When applied to the theories of class S, one obtains a rich family of vertex algebras which furnish novel representations of critical level, affine Kac-Moody algebras. Moreover, these vertex algebras satisfy an intricate set of gluing relations, arising from the geometric origins of class S.

I will review a construction of this family, due to Arakawa. Then I will explain an extension of this work to the setting of twisted class S, where modules of two different affine Kac-Moody algebras are sewn together. This talk is based on joint work with Christopher Beem.

By now it is known that many interesting phenomena in geometry and representation theory can be understood as aspects of mirror symmetry of 3d N=4 SUSY QFTs. Such a QFT is associated to a hyper-Kähler manifold X equipped with a hyper-Hamiltonian action of a compact Lie group G and admits two topological twists. The first twist, which is known as the 3d B-model or Rozansky-Witten theory, is a TQFT of algebro-geometric flavor and has been studied extensively by Kapustin, Rozansky and Saulina. The second twist, which is known as the 3d A-model or 3d Seiberg-Witten theory, is a more mysterious TQFT of symplecto-topological flavor. The 2-categories of boundary conditions for these two TQFTs are expected to provide two distinct categorifications of category O for the hyperkahler quotient X///G and 3d mirror symmetry is expected to induce the the Koszul duality between categories O for mirror symplectic resolutions.

In this talk I will explain recently completed work with Ben Gammage realizing these expectations for abelian gauge theories. The key inputs are the theory of perverse schobers, which gives us a combinatorial description of the 3d A-model, and the compatibility of 2-categorical 3d mirror symmetry with the 3-categorical geometric Langlands program. The latter was formalized in my prior work with Gammage and Mazel-Gee. If time permits I will also discuss what is known about quiver gauge theories and the 3-Lie algebra action on 2-category O coming from Nakajima’s correspondences. These lift the well known 2-Lie algebra actions due to Webster and Kamnitzer-Webster-Weekes-Yacobi.

We study the quantum connection in positive characteristic for conical symplectic resolutions. We conjecture the equivalence of the p-curvature of such connections with (equivariant generalizations of) quantum Steenrod operations of Fukaya and Wilkins. The conjecture is verified in a wide range of examples, including the Springer resolution, thereby providing a geometric interpretation of the p-curvature and a full computation of quantum Steenrod operations. The key ingredients are a new compatibility relation between the quantum Steenrod operations and the shift operators, and structural results for the mod p quantum connection recently obtained by Etingof--Varchenko.

Two-dimensional chiral/holomorphic quantum field theories serve as an arena where exact algebraic techniques offer a great deal of control over the spectrum of the theory. The underlying algebraic structure, known as a vertex algebra or vertex operator algebra, has seen countless applications ranging from superstring theory and pure mathematics to statistical and condensed matter physics. In this talk I will describe recent work with B. Williams developing an analogous algebraic structure for three-dimensional QFTs that are partially topological and partially holomorphic, leading to the notion of a raviolo vertex algebra.

Fixed point Floer cohomology of a symplectic automorphism categorifies the Lefschetz trace formula. On the categorical side, twisted Hochschild homology of an automorphism on an A infinity category can be understood as a categorical trace. I will explain how the twisted open-closed maps are used to relate these two invariants in the setting of Fukaya categories of Landau-Ginzburg models. Applications to symplectic topology and mirror symmetry will be discussed, especially for Lefschetz fibrations. This is joint work with Paul Seidel.

In its simplest incarnation, the geometric Langlands program was defined by Beilinson and Drinfeld in the 90’s as relating, on one side, a flat connection on a Riemann surface, and on the other side, a more sophisticated structure known as a D-module. A recent generalization of the correspondence, due to Aganagic-Frenkel-Okounkov, establishes an isomorphism between q-deformed versions of conformal blocks on the Riemann surface, for a W-algebra on one side, and a Langlands dual affine Lie algebra on the other side.

In this talk, we will elucidate the meaning of tame ramification in this correspondence. The crucial new ingredient will be a definition of q-primary vertex operators on the W-algebra side, which we argue to be determined entirely from the representation theory of the dual quantum affine algebra, much like the celebrated relation between W-algebra generators and the q-characters of Frenkel-Reshetikhin. As an application, we propose a construction of fundamental representations of quantum algebras via the supersymmetric Higgs mechanism in gauge theory with 8 supercharges, on an Omega-background.

Like many physical objects, under small perturbations, black holes possess the property of vibrating at discrete characteristic frequencies (known as QNM frequencies). They are complex and depend on the type of black hole and the boundary conditions imposed on the perturbation.

As we anticipate “hearing” these frequencies through new experiments more distinctly, it becomes essential to understand their mathematical structure better or, in other words, learn the”language” of black holes. Of course, there is no single ”language” for many different regimes where the perturbations arestudied. In this talk, we will present one language that covers a wide range of regimes for four-dimensional Schwarzschild black holes in three different backgrounds (flat, dS, and AdS).

We discuss joint work with Tom Gannon, showing that the algebra \(D(SL_n/U)\) of differential operators on the base affine space of \(SL_n\) is the quantized Coulomb branch of a certain 3d \(\mathcal{N} = 4\) quiver gauge theory. In the semiclassical limit this confirms a conjecture of Dancer-Hanany-Kirwan on the universal hyperkähler implosion of \(SL_n\). In fact, we prove a generalization interpreting an arbitrary unipotent reduction of \(T^* SL_n\) as a Coulomb branch. These results also provide a new interpretation of the Gelfand-Graev Weyl group symmetry of \(D(SL_n/U)\).

Mirror symmetry for 3d N=4 SUSY QFTs has recently received much attention in geometry and representation theory. Theories within this class give rise to interesting moduli spaces of vacua, whose most relevant components are called the Higgs and Coulomb branches. Nakajima initiated the mathematical study of Higgs branches in the 90s; since then, their geometry has been pivotal in diverse areas of enumerative geometry and geometric representation theory. On the other hand, a mathematically precise definition of the Coulomb branch has only recently been proposed, and its study has started.

Physically, 3d mirror symmetry is understood as a duality for pairs of theories whose Higgs and Coulomb branches are interchanged. Mathematically, it descends to a number of statements relating invariants attached to the dual sides. One of its key predictions is the identification of dual pairs of elliptic stable envelopes, which are certain topological classes intimately related to elliptic quantum groups. In this talk, I will first explain how to use brane diagrams and Cherkis bow varieties to describe both branches of a type A gauge theory. Then, I will discuss the main ideas behind the proof of mirror symmetry of sable envelopes for bow varieties (joint with Richard Rimanyi). A crucial step of our proof involves the process of ”resolving” large charge branes into multiple smaller ones. This phenomenon turns out to be the geometric counterpart of the algebraic fusion for R-matrices (and its dual). Time permitting, I will also hint at recent work in progress to extend this result to affine type A (with Rimanyi) and prove its enumerative counterpart, known as mirror symmetry of vertex functions (with Dinkins and Rimanyi).

We link several constructions of vertex operator algebras (VOA) in supersymmetric QFT. One is the SCFT/VOA correspondence of Beem et al, identifying VOA inside the protected sector of a 4d N=2 SCFT. Using the Omega-background approach to the SCFT/VOA, and compactifying the 4d theory on the infinite cigar geometry, we relate this to the construction of boundary VOAs in topologically twisted 3d N=4 theories due to Costello-Gaiotto. When the cigar is of finite size, the 3d theory gets further reduced on the interval, and we end up relating this to chiral algebras in 2d N=(0,2) theories as well. This allows to explain, among other things, some earlier observations on TQFTs originating from 4d SCFTs of Argyres-Douglas type. Another interesting connection is to the recently discovered rank-0 3d N=4 SCFTs.

My talk is based on the joint project with Lev Rozansky. I will explain how we rigorously construct 3D TQFT, which is a KRS theory with targets Hilbert scheme of points on a plane. Defects in the last theory encode a knot in the three-space and we rediscover the HOMFLY-PT homology of a link as a part of the TQFT. The construction is very flexible and allows us to construct the annular \(\mathfrak{gl}_n\)-homology as well as a categofified quantum super-group \(\mathfrak{gl}(m|n)\) link invariant. If time permits, I will explain how one can use Elliptic Hall Algebra (that acts on coherent sheaves on the Hilbert scheme) to compute knot homology a large class of knots.