ANR GRAW

Presentation

Affine W-algebras are vertex algebras that extend the Virasoro algebra. They were introduced by Zamolodchikov in physics and developed further in mathematics by Arakawa, Feigin, Frenkel, Kac, Wakimoto, etc. They are significant for the representation theory of affine Lie algebras or affine quantum groups. Recently new constructions of affine W-algebras in quantum field theories have garnered the attention of many mathematicians. The project GRAW aims to strengthen collaborations, in particular between mathematicians and physicists, and create new ones between reserachers with complementary skills around the representation theory of affine W-algebras. This workshop is supported by the project ANR-24-CE40-3389 (GRAW)

LOCALIZATION

University Paris-Cité, Building Sophie Germain, Amphithéatre TURING, Ground level. The plan to access the university is here and here

SPEAKERS

D. Butson
T. Creutzig
J. Grimminger
T. Kimura
V. Krylov
L. Topley

TALKS

D. Butson: Inverse Hamiltonian reduction for affine W-algebras in type A
T. Creutzig: W-algebras via 2-parameter families of vertex algebras
J. Grimminger: 3d Coulomb branches
T. Kimura: QQ-character and geometric representation theory
V. Krylov: Quantum Hikita conjecture and traces for quantized Coulomb branches via enumerative geometry
L. Topley: Symplectic singularities, twisted Yangians and presentations of finite W-algebras

SCHEDULE

WEDNESDAY 24
9h00: T.Creutzig
10h00: COFFEE BREAK
11h00: L.Topley
12h00: LUNCH TIME
14h00: T.Creutzig
15h00: COFFEE BREAK
THURSDAY 25
9h30: T.Kimura
10h30: COFFEE BREAK
11h00: V.Krylov
12h00: LUNCH TIME
14h00: T.Creutzig
15h00: COFFEE BREAK
15h30: T.Kimura
FRIDAY 26
9h30: D.Butson
10h30: COFFEE BREAK
11h00: T.Kimura
12h00: LUNCH TIME

ABSTRACTS

D. Butson: I will explain a recent proof, joint with Sujay Nair, of inverse Hamiltonian reduction for affine W-algebras in type A at generic level, a certain embedding of the affine W-algebra corresponding to an arbitrary nilpotent into that corresponding to a larger nilpotent with respect to the closure order on orbits, tensored with an auxiliary algebra of so-called free fields. I will also remark on the relation of this result to generalized coproducts on (not necessarily dominantly) shifted variants of the affine Yangian in type A, and the analogy between inverse Hamiltonian reduction and parabolic induction maps between affine W-algebras.
T. Creutzig: A 2-parameter family of vertex algebras is a vertex algebra over rational functions in two variables. They become powerful tools if one can show that they satisfy certain universal properties, in particular in that case they can be used to classify vertex algebras of a certain type. In these lectures I would like to introduce strong generating types of vertex algebras, give a status report on known 2-parameter vertex algebras and then if time permits I could explain how they can be used to deduce dualities of W-algebras and these dualities in turn give us a lot of information about representation theory.
J. Grimminger: 3d Coulomb branches are mathematical objects (often symplectic singularities) motivated by physical systems. I will give a physicists perspective geared towards mathematicians, introducing the Coulomb branch and the context within it arises. I will then talk about 3d mirror symmetry and symplectic duality, as well as the stratification of the Coulomb branch into symplectic leaves. If time permits I will address deformations of the Coulomb branch, and the Namikawa Weyl group.
T. Kimura: The qq-character, introduced by Nekrasov in the framework of supersymmetric gauge theory, can be viewed as a further deformation of Frenkel-Reshetikhin's q-character for finite-dimensional modules of quantum affine algebras. It plays a central role in constructing the generating currents of deformed W-algebras, serving as a q-analogue of the Sugawara construction. Moreover, as for the q-character, the qq-character admits a geometric realization via the equivariant K-theory of quiver varieties. In this talk, I will address the following three aspects of the qq-character 1. Algebraic construction and its relation to deformed W-algebras 2. Geometric realization and equivariant integral formulas 3. Irreducibility properties of the qq-character.
V. Krylov: Graded traces on algebras generalize the notion of characters. They are closely related to q-characters introduced by Frenkel and Reshetikhin. Kamnitzer, McBreen, and Proudfoot introduced a particular D-module, known as the D-module of graded traces, whose solutions are these graded traces. The quantum Hikita conjecture predicts an isomorphism between the D-module of graded traces for the quantized Coulomb branch and certain specialization of the quantum D-module of the resolved Higgs branch. This is an incarnation of a 3D mirror symmetry, linking the algebra and geometry of dual spaces. In this talk, I will formulate a general conjecture relating graded traces of Verma modules for quantized Coulomb branches to certain specializations of so-called vertex functions coming from the quasimap counts to Higgs branches. I will explain how this conjecture can be proved for ADE quivers, and how the quantum Hikita conjecture then follows as a corollary. If time permits, I will also discuss applications to classifying graded traces and will raise a question of possible extensions to the setting of vertex algebras. Our proof uses recent results of Kamnitzer, Kalmykov, Leroux-Lapierre, Pinet and Weekes. Based on joint work with Hunter Dinkins and Ivan Karpov.
L. Topley: In 2006 Brundan and Kleshchev famously introduced the (dominantly) shifted Yangian for gl_n, and its truncations, to give presentations of finite W-algebras in type A, associated to any nilpotent element. In 2009 Brown showed that a certain truncation of a twisted Yangian is isomorphic to a finite W-algebra of type B, C or D, associated to a rectangular nilpotent element. Both of these works constructed an explicit homomorphisms from a Yangian to a W-algebra. I will describe a recent joint work with Kang Lu, Yung-Ning Peng, Lukas Tappeiner and Weiqiang Wang. We have defined a shifted analogue of the twisted Yangian, and used this to provide presentations of almost all finite W-algebras for classical Lie algebras, associated to even nilpotent elements. One of the interesting features of our work is that we did not write down an explicit homomorphism from a Yangian to a W-algebra; instead we relied upon the uniqueness of Losev's quantization of the universal Poisson deformation of nilpotent Slodowy varieties, which are an interwsting family of conic symplectic singularities.