The focus of this one day workshop is to discuss several recent topics on vertex algerbas and related topics.
Date and location
Part I: February 29th, 2024
Part II: March 12th, 2024
CAB 572, University of Alberta, Edmonton, Canada
Schedule (Part I)
14:00~ 14:50 Shashank Kanade (University of Denver)
Title: Torus knots and W algebras
Abstract: I'll explain the connection of colored invariants of torus knots (colored with finite-dimensional irreducible modules of a simply-laced finite-dimensional Lie algebra g) with characters of the corresponding principal W algebras. This connection rests on a conjecture about asymptotic weight multiplicities in g modules.
15:00~ 15:25 Tea/coffee break in CAB 680
15:30~ 16:20 Xuanzhong Dai (RIMS, Tokyo)
Title: On recent progress of chiral de Rham complex and modular forms
Abstract: In 1994, D. Zagier, Y. Manin, and W. Eholzer speculated that the Rankin-Cohen brackets of modular forms should be related to vertex operator algebras. However it looks very difficult to connect the axioms of operations of modular forms and vertex algebras. Nevertheless, efforts have been made to construct vertex algebras associated with modular curves. One such endeavor involves sheaf constructions known as the chiral de Rham complex, whose cohomology is linked to the Witten genus. Within these sheaves lies a purely even subsheaf termed chiral differential operators, interpreted as the chiralization of differential operators, which locally manifest as certain copies of beta-gamma systems. In this talk, we will introduce the CDO/CDR on the upper half plane, as modular forms appears naturally on invariant global sections.
16:30~ 17:20 Jinwei Yang (Shanghai Jiao Tong University)
Title: How modular data uniquely determines a ribbon fusion category of type A and its applications.
Abstract: It is believed that modular data, i.e., the set of S and T matrices, uniquely determine a modular tensor category. We prove that this conjecture is true for the ribbon fusion category arising from simple affine VOA at admissible levels. As a consequence, we prove a few Kazhdan-Lusztig type correspondences.
Schedule (Part II)
10:00~ 10:50 David Ridout (University of Melbourne)
Title: Modularity and meromorphicity
Abstract: I will review some old work that explains why the Verlinde formula fails for certain logarithmic VOA-module categories unless one is careful about convergence.
11:00~ 11:50 Chelsea Walton (Rice University)
Title: H-module algebras
Abstract: By Harshit’s request, I will chat about various older constructions of H-module algebras, where H is a Hopf algebra. A need for examples (of VOAs) seems to be a theme of the workshop; see below. Perhaps these old results could prompt some new ideas…
12:00~ 14:00 Lunch
14:00~ 14:50 Terry Gannon (University of Alberta)
Title: Jacobi characters and exotic vertex operator algebras
Abstract: A frustrating aspect of our current understanding of CFT and VOAs is a lack of examples. More precisely, there are very few if any examples of VOAs which are independent of classical math like Lie algebras or lattices or finite groups. Is this because that is all the VOAs there are? Or are there whole worlds of new families of VOAs, and we are just too dumb (too classical) to find them? If we look at some shadows cast by VOAs (e.g. their tensor categories), we find several hints that such exotic VOAs should exist. The best way I know of for probing their existence are the Jacobi characters of the VOA (which are multi-variable generalizations of the familiar characters of VOAs). These store rich information on the VOA, including a full description of a subVOA. Their theory has evolved to a point where they are now quite effective at eliminating several candidates for exotic VOAs, and severely constraining the hypothetical structure of the others. In my talk I'll sketch this story, and apply it to the most interesting exotic candidate: the so-called Haagerup CFT, which has been pursued by both mathematicians and physicists for well over a decade.
Organizer: Thomas Creutzig, Shigenori Nakatsuka, Harshit Yadav