Abstracts for Spring 2023

Florencia Orosz Hunziker, University of Denver, C-graded vertex algebras and Weyl vertex algebras under conformal flow.

We prove that a class of  C-graded vertex algebras satisfying suitable conditions are rational, i.e., have semisimple representation theory, with a one-dimensional level zero Zhu algebra. We apply our result to the rank one Weyl vertex algebra and show that its representation theory changes significantly under conformal flow. This talk is based on joint work with K. Barron, K. Batistelli, V. Pedic, and G. Yamskulna.

Thomas Creutzig, University of Alberta, Quantum groups and VOAs

There are two natural sources of braided tensor categories: categories of modules of VOAs and quantum groups. Jointly with Matt Rupert and Simon Lentner, we are developing a theory on relating the two and in this talk I will explain that under suitable conditions a VOA category is a relative Drinfeld center, while this is always true for quantum groups. This implies nice correspondences as e.g. the one between singlet algebra and unrolled small quantum group of sl(2) at root of unity.

Maryam Khaqan, Stockholm University, Vertex operators for imaginary gl2-subalgebras in the Monster Lie algebra. 

The Monster Lie algebra is a quotient of the physical space of the vertex algebra tensor product of a specific rank-2 lattice vertex algebra with the Moonshine module of Frenkel, Lepowsky, and Meurman. 

In this talk, I will describe elements in the tensor product vertex algebra that project onto generators of gl2-subalgebras corresponding to each imaginary simple root of the Monster Lie algebra. Furthermore, for a fixed imaginary simple root, I will illustrate how the action of the Monster simple group on the Moonshine module induces an orbit of each gl2-subalgebra of the Monster Lie algebra constructed in this way. We conjecture that this Monster action is non-trivial. 

This talk is based on joint work with Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, and Scott H. Murray.

Shigenori Nakatsuka, University of Alberta, On Feigin-Tipunin type extension of W-algebras. 

The triplet algebra is an extension of the (p,1)-model of Virasoro algebra, which is a famous example of C2-cofinite but irrational VOA. Feigin and Tipunin gave a construction and generalization of this algebra to simply-laced principal W-algebras by using VOA bundles over flag varieties. In this talk, we'll generalize their construction for all the W-algebras and then explain their properties for $sl(2)$. The talk is based on my on-going joint work with Thomas Creutzig and Shoma Sugimoto. 

Ole Warnaar, The University of Queensland, Cylindric partitions and character formulas for W algebras.

Cylindric partitions are an affine analogue of plane partitions. In this talk I will explain the role cylindric partitions have played in recent years in the computation of characters of the vertex operator algebra W_r(p,p'). 

Pedram Hekmati, University of Auckland, Distributional characters on toric contact manifolds. 

Toric contact manifolds have a nice combinatorial description in terms of their moment cones. This paves the way for an explicit computation of their various invariants, such as the fundamental group, cohomology ring and contact homology. In this talk, I will discuss the transverse Dirac operator on these manifolds. It features notably in certain supersymmetric gauge theories and its T-equivariant index character determines a distribution on the torus T, for which we derive a simple and explicit formula. 

Daniele Valeri, Sapienza University of Rome, Deformations of W-algebras and differential-difference equations.

In this talk I will review some results about q-deformations of W-algebras and their relations with differential-difference equations.

Anna Lachowska, EPFL Lausanne, The small quantum group and related geometry.

Let u_q(g) denote the small quantum group associated to a semisimple complex Lie algebra g and a root of unity q. I will describe some recent results on the structure of the center of u_q(g) and discuss its relation  to the geometry of the Springer resolution and the affine Springer fibers. 

A lower bound on the dimension of the center of u_q(g) suggests a connection with the representation theory of the rational Cherednik algebra.  

Simon Lentner, University of Hamburg, A theory of logarithmic Kazhdan-Lusztig correspondences. 

We want to understand braided tensor categories U that have a commutative algebra and a known braided tensor category C of local modules. Our first result is that U is a relative Drinfeld center of the category B of twisted modules, our second result is that in good cases we can understand B in terms of a Hopf algebra in C and we develop methods to determine this Hopf algebra. Hence we can determine U explicitly. In particular we can prove that if U is equivalent as an abelian category to representations of a quantum group, or certain generalizations, and if it has a commutative algebra as above, then it is equivalent already as braided tensor category. The main application we have in mind are the categories of representations of certain vertex alebras, which are defined as kernel of screenings in a free field realization. In this case the latter gives by construction a commutative algebra with known C and conjecturally the Hopf algebra above should be the Nichols algebra of screenings. With our results above we can prove in some cases braided category equivalences to certain quantum groups, which are instances of logarithmic Kazhdan Lusztig conjectures.

Ivana Vukorepa, University of Zagreb, Tensor category KL_k(sl_{2n}) via minimal affine W-algebras at the non-admissible level k=-\frac{2n+1}{2}.

Representation theory of simple affine vertex algebra $L_k(g)$, for arbitrary simple Lie algebra g and general complex level k, is a very important direction in the theory of vertex algebras. Some of the best understood cases are non-negative integer levels k \in \mathbb{Z}_{\geq 0} and so-called admissible levels. In the present talk we consider special non-admissible levels for g =sl_m. We prove thatKL_k(sl_m) is a semi-simple, rigid braided tensor category for all even m greater or equal than 4, and k=-\frac{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 \hookrightarrow sl_{m+1} at level k=-\frac{m+1}{2}. For this we inductively identify minimal affine W-algebra W_{k-1}(\mathfrak{sl}_{m+2},\theta) as simple current extension of L_k(sl_m) \otimes H \otimes 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. Adamovic, T. Creutzig and O. Perse.

Masoumah Al-Ali, Saudi Electronic University, Orbifolds of Gaiotto-Rapčák Y-algebras.

Gaiotto and Rapčák introduced an important family of vertex algebras called Y_ {N_1,N_2, N_3}[\psi]-algebras where N_1, N_2, N_3 are nonnegative integers and \psi is a complex parameter. These vertex algebras arise as a simple one-parameter quotients of the universal two-parameters W_\infity-algebra and serve as building blocks for many interesting vertex algebras. The universal two parameters W_\infity-algebra has a full automorphism group Z_2 and these algebras inherit this action. We shall study the structure of their orbifolds. Regardless of the extremal cases, we show that these orbifolds are generated by a single field of weight four, and we give strong finite generating set.  

Jehanne Dousse, University of Geneva, Characters of level 1 C_n^{(1)}-modules, integer partitions, and the Capparelli-Meurman-Primc-Primc conjecture.

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.