Vertex Algebras and Poisson Geometry

20 Feb 2022, 19:00 → 25 Feb 2022, 12:00 Europe/Zurich

Anton Alexeev (Universite de Geneve (CH)), Leonid Rybnikov (NRU HSE Moscow)

The purpose of this conference is to bring together experts in Poisson Geometry,
Geometric Representation Theory, and Conformal Field Theory to discuss the
recent progress and various interactions of these highly active areas of research.

Monday 21 February

09:30 → 10:30 Involutions on symplectic singularities - Hiraku Nakajima

Involutions on symplectic singularities
Okounkov said, "Symplectic singularities are the Lie algebras of the 21st century". Then involutions on symplectic singularities are the symmetric spaces of the 21st century. I will study the case of certain (Q)uiver varieties of type A, which are also intersections of (N)ilpotent cone of GL and slices, affine (G)rassmannian slices of GL, and also (C)oulomb branches of quiver gauge theories of type A. Either description, except (C), gives natural involutions whose fixed point sets are the same type of varieties (Q),(N),(G) associated with classical groups. Since fixed point sets are different, they are different involutions. I will explain how to understand all involutions in quiver varieties (myself, Yiqiang Li) and also identify fixed point sets with Coulomb branches of certain variants of quiver gauge theories (on-going projects with other people).

11:00 → 12:00 Diagrams, fission spaces and global Lie theory - Philip Boalch

I'll recall how to construct algebraic Poisson varieties by gluing pieces of surface with wild boundary conditions (extending the q-Hamiltonian framework), and then move on to discuss the link to quiver varieties and how this may be generalised, leading to a theory of "diagrams" for the wild character varieties (i.e. the wild nonabelian Hodge moduli spaces in their Betti algebraic structure). Much of this is motivated by quite straightforward questions about classifying rational Lax representations of finite dimensional integrable systems.
Some references:
--The first examples of fission spaces were really in Birkhoff's 1913 paper; they were shown to be q-Hamiltonian in arXiv:math/0203161, but they weren't given this name until Ann. Inst. Fourier 59, 7 (2009), and the story was then extended to the general case in arXiv:1111.6228, arXiv:1512.08091 (joint with Yamakawa)
--The theory of diagrams is in arXiv:1907.11149 (with Yamakawa), and has been extended by Doucot arXiv:2107.02516.

16:30 → 17:00 Coffee break

17:00 → 18:00 Parabolic restriction for Coulomb branch algebras and categorical g-actions - Joel Kamnitzer

Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This
variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians. Motivated by the geometric Satake correspondence, we define a categorical g-action on modules for these truncated shifted Yangians. Our main tool is the study of how the Coulomb branch algebra changes when we pass from G, V to L, U, where L is a Levi in G and U is the invariants for a coweight whose centralizer is L. I will discuss the relation of our work to the geometric Satake conjecture of Braverman-Finkelberg-Nakajima.

18:15 → 19:15 Harish-Chandra modules over quantizations of nilpotent orbits - Ivan Losev

Let O be a nilpotent orbit in a semisimple Lie algebra over the complex numbers. Then it makes sense to talk about filtered quantizations of O, these are certain associative algebras that necessarily come with a preferred homomorphism from the universal enveloping algebra. Assume that the codimension of the boundary of O is at least 4, this is the case for all birationally rigid orbits (but six in the exceptional type), for example. In my talk I will explain a geometric classification of faithful irreducible Harish-Chandra modules over quantizations of O, concentrating on the case of canonical quantizations -- this gives rise to modules that could be called unipotent. The talk is based on a joint paper with Shilin Yu (in preparation).

Tuesday 22 February

09:30 → 10:30 Coulomb branches of noncotangent type- Alexander Braverman

We propose a construction of the Coulomb branch (as an affine singular symplectic variety) of a 3d N=4 gauge theory corresponding to a choice of a connected reductive group G and a symplectic
finite-dimensional representation M of G, satisfying certain anomaly cancellation condition. This extends previous work of Braverman, Finkelberg and Nakajima (which dealt with the case when M was the cotangent bundle of another representation).

We shall discuss the relation of our construction with conjectures of Ben-Zvi, Sakellaridis and Venkatesh.

11:00 → 12:00 Twisted Fock module of toroidal algebra via DAHA and vertex operators - Mikhail Bershtein

We construct the twisted Fock module of quantum toroidal gl_1 algebra with a slope n′/n using vertex operators of quantum affine gln. The proof is based on the q-wedge construction of an integrable level-one Uq(glˆn)-module and the representation theory of double affine Hecke algebra. The results are consistent with Gorsky-Neguţ conjecture (Kononov-Smirnov theorem) on stable envelopes for Hilbert schemes of points in the plane and can be viewed as a manifestation of (gl1,gln)-duality.
Based on joint work with R. Gonin

16:30 → 17:00 Coffee break

17:00 → 18:00 Deligne categories and deformed double current algebras - Pavel Etingof

Deligne categories are tensor categories, introduced by P. Deligne, which provide a formal way to interpolate representation-theoretic structures attached to classical groups and supergroups (such S_n, GL(n),Sp(2n),O(n),GL(n|m),OSp(n|2m),etc.) to complex values of the integer "rank parameter" n. I will first review them and then explain how to use them to construct and study deformed double current algebras which have recently become increasingly popular in the mathematics and physics literature. This is joint work with D. Kalinov and E. Rains.

18:15 → 19:15 Stokes phenomena, Poisson-Lie groups and quantum groups - Valerio Toledano-Laredo

Let G be a complex reductive group, G its dual Poisson-Lie group, and g the Lie algebra of G. G-valued Stokes phenomena were exploited by P. Boalch to linearise the Poisson structure on G. I will explain how Ug-valued Stokes phenomena can be used to give a purely transcendental construction of the quantum group Uhg. I will also show that the semiclassical limit of this construction recovers Boalch’s. The latter result is joint work with Xiaomeng Xu.

Wednesday 23 February

09:30 → 10:30 Derived and deformable 3d TQFT - Tudor Dimofte

VOA's often appear on holomorphic boundary conditions of 3d topological QFT's. A classic example involves WZW models on the boundary of Chern-Simons theories. In such a setup, algebraic structures in the boundary VOA (e.g. categories of modules and conformal blocks) can be used to reconstruct structures of the bulk 3d TQFT. I will discuss a modern generalization of this boundary-bulk correspondence, involving logarithmic VOA's and topological twists of 3d supersymmetric gauge theories. Two subtle and beautiful new features that arise are the presence of dg/derived structure, and deformations over certain moduli spaces of connections. I will also discuss a key example, introduced in recent work with T. Creutzig, N. Garner, and N. Geer, relating Feigin-Tipunin algebras, a supersymmetric generalization of Chern-Simons theory, and the derived category of representations of a quantum group at a root of unity.

11:00 → 12:00 Restricted dynamical quantum groups - Giovanni Felder

I will review the theory of the Yang-Baxter equation and its Interaction-Round-A-Face version as they appeared in statistical mechanics. I will then concentrate on the case of (restricted) solid-on-solid models and explain the related representation theory of dynamical quantum groups on groupoid-graded vector spaces. The talk is based on joint work with Muze Ren.

Thursday 24 February

09:30 → 10:30 Nilpotent orbits arising from admissible affine vertex algebras - Anne Moreau

In this talk, I will give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals. I will also connect these varieties with the cohomology of the small quantum groups associated with an l-th root of unity.

This is a joint work with Tomoyuki Arakawa and Jethro van Ekeren.

11:00 → 12:00 Weight representations of affine Kac-Moody algebras and small quantum groups - Tomoyuki Arakawa

Motivated by the 4d/2d duality discovered by Beem et al, I will talk about the category of weight representations of affine Kac-Moody algebras (that does not necessarily belong to the category O) and its connection with the small quantum groups.
This is a joint work with Kazuya Kawasetsu and Thomas Creutzig.

16:30 → 17:00 Coffee break

17:00 → 18:00 Vertex algebras from gauge theories- Davide Gaiotto

I will discuss some constructions of vertex algebras from supersymmetric gauge theories

18:15 → 19:15 A Rogers-Ramanujan-Slater type identity related to the Ising model - Reimundo Heluani

We prove three new q-series identities of the Rogers-Ramanujan-Slater
type. We find a PBW basis for the Ising model as a consequence of one of these
identities. If time permits it will be shown that the singular support of the
Ising model is a hyper-surface (in the differential sense) on the arc space of
it's associated scheme. This is joint work with G. E. Andrews and J. van Ekeren
and is available online at [4]https://arxiv.org/abs/2005.10769

Friday 25 February

09:30 → 10:30 On the WKB approximation of a system of ordinary differential equations - Xiaomeng Xu

This talk concerns the global aspects of the WKB analysis of meromorphic linear systems of ordinary differential equations with Poncar\'{e} rank 1. It proposes a connection between the WKB approximation of the monodromy data and the integral periods over the associated spectral curves, which generalizes the asymptotic and jump properties of Voros symbols in the exact WKB analysis of Schrödinger equations. It is based on a joint work with Anton Alekseev and Yan Zhou.

11:00 → 12:00 Fermionic formulas and knot invariants - Boris Feigin