Mini-workshop on vertex algebras and applications

Speakers

• Jehanne Dousse (10:00-10:50) - Characters of level 1 C_n^{(1)}-modules, integer partitions, and the Capparelli-Meurman-Primc-Primc conjecture
• Sven Moller (11:10-12:00) - Sheaves of vertex operator algebras and 4d superconformal field theories
• Coffee break 12:00-12:30
• Ivana Vukorepa (12:30-13:20) - Tensor category KL_k(sl(2n)) via minimal affine W--algebras at the non-admissible level k=−(2n+1)/2
• Free discussion 13:30-14:30

Abstracts

• Jehanne Dousse

Title: Characters of level 1 C_n^{(1)}-modules, integer partitions, and the Capparelli-Meurman-Primc-Primc conjecture

Abstract: A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. Since Lepowsky, Milne, and Wilson's seminal work in the 1980's, several connections have been established between integer partitions and characters of standard modules of affine Lie algebras. Among these, an approach initiated by Primc in the 1990’s and developed by the speaker and Konan in the past few years consists in studying crystal bases of the modules to obtain character formulas which can be expressed in terms of generalised partitions.
In this talk, we show how our method applies to level 1 standard modules of C_n^{(1)} to give several expressions for their characters as generating functions for generalised partitions. Doing this, we prove a recent conjecture of Capparelli-Meurman-Primc-Primc on characters of level k standard modules of C_n^{(1)} in the particular case of level 1.
This is joint work with Isaac Konan.

• Sven Moller

Title: Sheaves of vertex operator algebras and 4d superconformal field theories

Abstract: Recently, vertex operator algebras (VOAs) were shown to describe certain (e.g. geometric) aspects of 4d superconformal field theories (SCFTs) in physics with striking implications for quantum gravity and the holographic principle.
One important task is hence to construct the VOAs in the image of this 4d/2d-duality and understand associated data such as the associated symplectic variety due to Arakawa.
In this talk I will describe a novel construction of a class of VOAs (that describe N=4 supersymmetric Yang–Mills theories) as global sections of sheaves of VOAs over Hilbert schemes.
This is joint work in progress with Tomoyuki Arakawa and Toshiro Kuwabara.

• Ivana Vukorepa

Title: Tensor category KL_k(sl(2n)) via minimal affine W--algebras at the non-admissible level k=−(2n+1)/2

Abstract: In the first part of the talk we study the representation theory of nonadmissible simple affine vertex algebra L_{-5/2}(sl(4)). This case is of particular interest since it appears in conformal embeddings of affine vertex algebras. We determine the fusion rules between irreducible modules in the category of ordinary modules KL_{-5/2} and prove that KL_{-5/2} is a semi-simple, rigid braided tensor category.
In the second part of the talk we prove that KL_k(sl(m)) is a semi-simple, rigid braided tensor category for all even m ≥ 4, and k = −(m+1)/2 . Moreover, all modules in KL_k(sl(m)) are simple-currents and they appear in the decomposition of conformal embeddings gl_m → sl_{m+1} at level k = −(m+1)/2. For this we inductively identify minimal affine W–algebra W_{k−1}(sl(m+2), θ) as simple current extension of L_k(sl(m)) ⊗ H ⊗M, where H is the rank one Heisenberg vertex algebra, and M the singlet vertex algebra for c = −2.
This is joint work with D. Adamović, T. Creutzig and O. Perše.