Tag - Vertex operator algebras

Ching Hung Lam: A lattice theoretical interpretation of generalized deep holes

6th June, 2022

We will give a lattice-theoretical interpretation of generalized deep holes of the Leech lattice VOA VΛ. We show that a generalized deep hole defines a 'true' automorphism invariant deep hole of the Leech lattice. We will also discuss a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge 24 having non-abelian V1 and the set of equivalence classes of pairs (𝜏,β̃ ) satisfying certain conditions, where 𝜏∈Co0 and β̃ is a 𝜏-invariant deep hole of squared length 2. It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge 24. Finally, we will discuss an observation of G. Höhn, which relates the weight one Lie algebra of holomorphic VOAs of central charge 24 to certain codewords associated with the glue codes of Niemeier lattices.

Sergei Gukov: Surprising VOA Structures from Quantum Topology

6th June, 2022

In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a 3-manifold and a choice of Spin-C structure, the q-series invariant turns out to be a character of a (logarithmic) vertex algebra that depends on the 3-manifold.

Angela Gibney: Vector bundles on the moduli space of curves from representations of VOAs

6th May, 2022

Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This gives rise to a functor taking n-tuples of A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g. If V is strongly rational (in which case A(V) is finite and semi-simple), much is known about these sheaves, including that they are coherent and satisfy a factorization property. Factorization ultimately allows one to show the sheaves are vector bundles with Chern classes in the tautological ring. In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality. As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type. As an application, we show that if V is strongly finite, then sheaves of coinvariants and conformal blocks are coherent. This is a preliminary description of new and ongoing joint work with Krashen and Damiolini, extending work with Damiolini and Tarasca.

Iana Anguelova: Microlocality and Chiral Algebras

16th October, 2021

In one of my last conversations with Ben Cox, we discussed our mutual desire to work together on the axiomatic approach to multilocal and quantum chiral algebras. We both had worked already on issues related to multilocality; situations where the fields/vertex operators in question have Operator Product Expansions (OPEs) with more than the one singularity at 'z=w'. In particular, we worked together on the theory of N-point local chiral algebras, i.e., algebras that are 'complete' with respect to OPEs, and have singularities at roots of unity. But we were planning to work on the outstanding case where the OPEs have singularities at infinite multiplicative lattices. Such is the example of the Frenkel-Jing quantum vertex operators. In this talk I will discuss some problems arising in the axiomatic approach to multilocal chiral algebras, both N-point local, and quantum.

Dražen Adamović: On logarithmic and Whittaker modules for affine vertex algebras

6th August, 2020

Simple affine vertex algebras at admissible levels are semi-simple in the category O, but beyond the category O they contain interesting categories of representations with many new research challenges. We will first present our explicit lattice realizations of simple affine VOA Lk(𝔰𝔩2) at arbitrary admissible level k, and their modules in certain categories. Then we discuss the existence and explicit realization of logarithmic modules which appear as extensions of weight modules. The next natural task is to include Whittaker modules in the representation category. Although Whittaker modules are constructed using standard Lie-theoretic constructions, we will show that in order to understand the structure of affine Whittaker modules, one needs to apply vertex-algebraic techniques. We present explicit realization of Whittaker modules for some vertex algebras. We will discuss our recent efforts to generalize this realization in higher-rank cases.