Mini-Conference on Vertex Algebras

University of Denver, October 10-11, 2016

All talks will be held in Nelson Hall, Room 119

• 8:30 - 9:20 am, Oct. 10: Geoffrey Mason (UC Santa Cruz)
  
  
  • Title: Pierce bundles of vertex rings
    
  • Abstract: Pierce sheaves and their corresponding etale bundles are special kinds of geometric structures over a Boolean base space that have proved useful for studying certain classes of associative rings. Pierce introduced and used them to study commutative von Neumann regular rings (vNr). We show that this theory extends to the category of vertex rings. For example, there is an equivalence of categories that says that every vertex ring is the space of global sections of a certain type of etale bundle over a Boolean base space. We define and characterize von Neumann regular vertex rings, which correspond to etale bundles of simple vertex rings. This generalizes Pierce's original theorem, which is the case when the vertex ring is a commutative vNr and the bundle is a bundle of fields.
    
    
  
  

• 9:30 - 10:20 am, Oct. 10: John Duncan (Emory)
  
  
  • Title: Genus Zero Groups and Umbral Moonshine
    
  • Abstract: Recent joint work with Miranda Cheng reveals a natural association of mock modular forms to (certain) genus zero groups of isometries of the hyperbolic plane. The mock modular forms arising are distinguished in that their Fourier coefficients are all rational integers, and essentially all of the classical mock theta functions of Ramanujan are included. In this talk we describe some of the implications of this for umbral moonshine.
    
    
  
  

• 11:00 - 11:50 am, Oct. 10: Bob Griess (Michigan)
  
  
  • Title: About group-invariant integral forms in VOAs
    
  • Abstract: Review of work on integral forms in VOAs, especially ones which are invariant under finite groups. Constructions and characterizations. Realizations of Chevalley and Steinberg groups as automorphism groups of naturally associated VAs in positive characteristic. A variety of examples including a few related to the Monster.
    
    
  
  

• 3:00 - 3:50 pm, Oct. 10: Robert McRae (Vanderbilt)
  
  
  • Title: Tensor categories for vertex operator algebra extensions: Theory
    
  • Abstract: This talk will be an expanded version of my AMS Special Session talk on Saturday. I will discuss Kirillov and Ostrik's procedure for obtaining a monoidal category of representations of an algebra in a braided tensor category and explain how it applies to vertex operator algebra extensions in the case that a suitable module category for the smaller vertex operator algebra admits vertex tensor category structure. As an example, I will illustrate how this procedure gives a natural monoidal category structure on twisted modules for a vertex operator algebra in certain cases. This is joint work with Thomas Creutzig and Shashank Kanade.
    
    
  
  

• 4:00 - 4:50 pm, Oct. 10: Shashank Kanade (Alberta)
  
  
  • Title: Tensor categories for vertex operator algebra extensions: Continued
    
  • Abstract: In this talk, I will present further properties of the induction functor introduced in Kirillov and Ostrik's highly influential work. Kirillov and Ostrik work in semi-semisimple categories, and our aim is to work without this assumption as much as possible. Time permitting, I will present some ideas about a Verlinde formula for super-algebras. This is a joint work with Thomas Creutzig and Robert McRae.
    
    
  
  

• 8:30 - 9:20 am, Oct. 11: Ching Hung Lam (Academia Sinica)
  
  
  • Title: Construction of a holomorphic VOA of central charge 24 whose weight one Lie algebra has the type F_{4,6}A_{2,2}
    
  • Abstract: I will discuss a construction of a holomorphic VOA of central charge 24 whose weight one Lie algebra has the type F_{4,6}A_{2,2} using a Z_2-orbifold construction from a holomorphic VOA V with V_1 ≅ A_{8,3}A_{2,1}^2.
    
    
  
  

• 9:30 - 10:20 am, Oct. 11: Chongying Dong (UC Santa Cruz)
  
  
  • Title: Congruence property in orbifold conformal field theory
    
  • Abstract: The congruence property is established for rational orbifold theory. That is, the kernel of the representation of the modular group on the orbifold conformal blocks of any rational, C_2-cofinite vertex operator algebra with a finite automorphism group G is a congruence subgroup. In particular, the q-characters of any irreducible g-twisted module for g in G and V^G are modular functions on the same congruence subgroup.
    
    
  
  

• 11:00 - 11:50 am, Oct. 11: Bojko Bakalov (NC State)
  
  
  • Title: Logarithmic Vertex Algebras
    
  • Abstract: In logarithmic conformal field theory the operator product expansion (OPE) of quantum fields involves logarithms. I will introduce a notion of a logarithmic vertex algebra, which provides a rigorous algebraic formalism for studying such OPEs. I will derive a Borcherds identity for logarithmic vertex algebras, and present the examples of free bosons and symplectic fermions.
    
    
  
  

• 3:00 - 3:50 pm, Oct. 11: Tomoyuki Arakawa (RIMS, Kyoto and MIT)
  
  
  • Title: Chiral algebras of class S and symplectic varieties
    
  • Abstract: Recently, Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees constructed a map from four dimensional N=2 superconformal field theories to vertex algebras in such a way that the character of the vertex algebra gives the Schur limit of the superconformal index of the four dimensional theory, which predicts the existence of a large number of interesting vertex algebras. This conjecture was in part reformulated by Y. Tachikawa as the existence of 2d TQFT whose targets are vertex algebras. In my talk I will prove the Tachikawa conjecture, and reduce another conjecture on symplectic varieties by Moore and Tachikawa to a statement on vertex algebras.
    
    
  
  

• 4:00 - 4:50 pm, Oct. 11: Antun Milas (Albany)
  
  
  • Title: Unrolled quantum groups and vertex algebras
    
  • Abstract: We first introduce unrolled quantum groups and discuss a conjectural correspondence between them and certain vertex subalgebras of the Heisenberg vertex algebra. Then we focus on the simplest unrolled quantum group, U_q^H(sl_2), q = e^{πi/p}, and the corresponding vertex algebra, called the (1,p)-singlet algebra. We show how to use deformable families of modules to efficiently compute tangle invariants colored with projective U_q^H(sl_2)-modules. We also discuss logarithmic open Hopf link invariants in connection to asymptotic (or quantum) dimensions of the singlet vertex algebra. In the last part, we present evidence for higher rank generalization of these results. This talk is based on a joint project with T. Creutzig and, in part, also jointly with M. Rupert.