Abstracts

- Dražen Adamović: "Some new constructions of vertex algebras and their modules in LCFT"

We shall first review some basic constructions in the representation theory of C_2 cofinite VOAs in LCFT. They carry the structure of certain extensions of non-rational Virasoro vertex algebras. Next we shall discuss our recent realizations of affine vertex algebras V_k(sl(2)), and show how one can obtain realization of their logarithmic modules. We will also present a properties of new logarithmic VOAs, called the R^(p) algebras, present their different realizations and prove certain interesting conjectures related to them (this part is based on recent joint work with T. Creutzig, N. Genra and J. Yang). Some applications in the representation theory of affine vertex algebras will be also discussed.

- Sebastiano Carpi: "Unitary W_3 algebras beyond rationality"

I will explain a recent result on the unitarity of simple vertex operator algebras associated to the W_3 algebra with central charge greater than or equal to 2.

(Based on a joint work with Yoh Tanimoto and Mihály Weiner).

- Ilaria Flandoli: "Algebras of screenings and Nichols algebras"

We introduce in a vertex algebra setting non-local screening operators associated to non-integral lattices. The algebra generated by these screenings is, under certain conditions, a Nichols algebra with diagonal braiding. We look at examples of such screening algebras: for positive definite lattices we obtain the positive part of the associated quantum group, for negative definite lattices extensions thereof.

- Azat Gainutdinov: "On deformation of tensor categories"

Tensor categories give us a very convenient language both in representation theory and in mathematical physics. For example, many algebraic aspects of two-dimensional conformal field theories can be formulated in the language of tensor categories. I am interested in the problem of deformation of such categories and will talk about new results in this direction. The Hochschild type complexes called «Davydov-Yetter» classify infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the comonad cohomology of the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of Ocneanu rigidity which says «no deformations for semisimple categories». Second, it allows to use standard methods from comonad cohomology theory to compute Davydov-Yetter cohomology for a family of non-semisimple finite-dimensional Hopf algebras generalizing Sweedler's four dimensional Hopf algebra. This is a joint work with J. Haferkamp and Ch. Schweigert.

- André Henriques: "Liouville-type CFTs in algebraic quantum field theory"

Liouville-type CFTs are distinguished by a number of features, such as the non-existence of a vacuum vector, and the non-surjectiveness of the state-field correspondence.

I'll explain how to define (not compute!) the state-space of an arbitrary Liouville type CFT, and present a conjecture due to J. Teschner. I will define the notion of field, and explain how to construct fields that don't come from states.

- Ana Kontrec: "Classification of irreducible modules for Bershadsky-Polyakov algebra at certain levels"

This is joint work with Drazen Adamovic. We study the representation theory of the Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$. In particular, Zhu algebra of $\mathcal W_k$ is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector.

We classify all modules in the category $\mathcal{O}$ for the Bershadsky-Polyakov algebra $\mathcal W_k$ for $k=-5/3, -9/4, -1,0$. In the case $k=0$ we show that the Zhu algebra $A(\mathcal W_k)$ has 2--dimensional indecomposable modules.

- Paolo Papi: "Conformal embeddings in basic classical Lie superalgebras"

We will discuss the conformal embeddings between basic classical Lie superalgebras with emphasis on the case of the embedding of the even part in the whole Lie superalgebra. We classify such embeddings and in many relevant cases, we compute the decomposition of the affine vertex algebra of the ambient Lie superalgebra as a module for the affine vertex algebra of the embedded subalgebra. The main tool is a fusion rules argument that proves to be very powerful. Joint work with D. Adamović, V. Kac, P. Moseneder Frajria, O. Perse.

- Nils Scheithauer: "Eisenstein series, dimension formulas and orbifolds"

Modular forms for the Weil representation impose restrictions on the characters of orbifolds. For a particular Eisenstein series this gives a formula for the dimension of the weight-1 subspace of an orbifold. The formula implies an upper bound which we use to define generalised deep holes. Up to the uniqueness of the moonshine module every strongly rational, holomorphic vertex operator algebra of central charge 24 can be obtained by orbifolding the vertex operator algebra of the Leech lattice with a generalised deep hole. This gives the first uniform construction of the vertex operator algebras on Schellekens' list.