Abstracts for Fall 2021

Thomas Creutzig, U. Alberta, The category O of affine osp(1|2n) at admissible level.

osp(1|2n) behaves in many respects similar to finite dimensional simple Lie algebras. The same is expected to be true for its affine vertex algebra and we will see that this is indeed true for the category O at admissible level. I will explain how to construct the universal affine vertex superalgebra of osp(1|2n) by translating the equivariant CDO of sp(2n). This construction gives valuable information about the simple affine vertex superalgebra at admissible level, in particular we will be able to understand that the category O at admissible level is a braided fusion supercategory. 

Gurbir Dhillon, Yale University, The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras.

Fix an affine Lie algebra ̂gκ with associated principal affine W-algebra Wκ. A basic conjecture of Frenkel–Kac–Wakimoto asserts that Drinfeld–Sokolov reduction sends admissible ̂gκ-modules to zero or cohomological shifts of minimal series Wκ-modules. In recent work, we proved this conjecture and a natural generalization to the spectrally flowed Drinfeld–Sokolov reduction functors and to a larger family of ̂gκ-modules. This extends the previous results of Arakawa and Arakawa--Creutzig--Feigin. In the talk, we review the history and statement of the conjecture, discuss the form the answer takes, and highlight a few ingredients of its proof which may be of use elsewhere. 

Matthew Rupert, Utah State University, Uprolling Unrolled Quantum Groups. 

I will discuss joint work with Thomas Creutzig where we construct families of commutative (super) algebra objects in the category of weight modules for unrolled restricted quantum groups of a simple Lie algebra at roots of unity. We study their categories of local modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Based on motivation from the rank one examples, we expect that these categories should be equivalent to module categories for vertex operator algebras, and we present conjectures for the structure of module categories for the higher rank Triplet and Bp vertex operator algebras. 

Veronika Pedić, University of Zagreb, Representation theory and fusion rules for Weyl vertex algebras and beyond.

Weyl vertex algebra is an interesting example of a non-rational and non C2-cofinite vertex algebra. We describe fusion rules in the category of Weyl vertex algebra weight modules and explicitly construct the intertwining operators appearing in these equations. We describe applications of our methods to other VOAs, in particular the M(p) singlet. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules of the affine Lie superalgebra gl(1|1). This part of the talk is based on joint work with D. Adamović.

In the second part we present results of a joint project with D. Addabbo, K. Barron, K. Batistelli, F. Orosz Hunziker and G. Yamskulna. Among other things we calculate the first Zhu algebra of the Weyl vertex algebra. 

Juan Villareal, North Carolina State University, Logarithmic vertex algebras.

In this talk, I want to explain a generalization of vertex algebras called logarithmic vertex algebras, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields. In this work, we develop a framework that allows many results about vertex algebras to be extended to logarithmic vertex algebras. Finally, I will mention one example which is motivated by physics,  this example exhibits some unexpected new features that are peculiar to the logarithmic case. This is joint work with Bojko Bakalov.

Kathrin Bringmann, University of Cologne, Modularity of class number generating function.

I my talk I will speak about various results related to the modularity of the class number generating function and some applications. 

Reimundo Heluani, IMPA, Borcherds identity in logarithmic coordinates.

The exponential change of coordinates z = exp(t) induces an automorphism on every conformal vertex algebra. Vertex operators in these new coordinates play an essential role in Zhu's proof of modularity of conformal blocks. In this talk we'll take a look at a version of Borcherds formula for these operators. Unlike the usual formula involving Laurent expansions of rational functions, this formula uses Fourier expansion and explicit domains of convergence.

Qing Wang, Xiamen University, Trigonometric Lie algebras, affine Lie algebras, and vertex algebras. 

We present natural connections among trigonometric Lie algebras, affine Lie algebras, and vertex algebras. More specifically, we prove that restricted modules for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebra. Furthermore, we prove that every quasi-finite unitary highest weight irreducible module of type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi module for the simple affine vertex algebra. This is a joint work with Haisheng Li and Shaobin Tan.

Niklas Garner, University of Washington, Non-semisimple 3d TQFTs for the Feigen-Tipunin algebras and quantum groups.

 I will describe a class of physical 3d QFTs that conjecturally serve as non-semisimple, derived generalizations of Chern-Simons theory with compact gauge group SU(n). These 3d QFTs admit two different boundary conditions furnishing VOAs, one of which being a Feigen-Tipunin algebra, and we conjecture a novel logarithmic level-rank-like duality that relates them. Modules for the Feigen-Tipunin algebra are expected to be related to modules for the quantum group via a logarithmic Kazhdan-Lusztig-like correspondence, thereby connecting our physical QFT to mathematical TQFTs built from modules of the quantum group. Our proposed physical QFT offers a new perspective on these VOAs and mathematical TQFTs and allows for the use of techniques in supersymmetric QFT to analyze their properties. This is based on joint work with T. Creutzig, T. Dimofte, and N. Geer. 

Haisheng Li, Rutgers University, Deforming vertex algebras by module and comodule actions of vertex bialgebras.

Previously, we studied a notion of vertex bialgebra and a notion of module vertex algebra for a vertex bialgebra, and  gave a smash product construction of nonlocal vertex algebras. Here, we introduce a notion of right comodule vertex algebra  for a vertex bialgebra. Among the main results,  we give a construction of quantum vertex algebras from vertex algebras with a right comodule vertex algebra structure and a compatible  (left) module vertex algebra structure  for a vertex bialgebra. As an application, we obtain a family of deformations of the lattice vertex algebras.  This is based on a joint work with Naihuan Jing, Fei Kong, and Shaobin Tan.