13th Seminar on Conformal Field Theory January 17th, 2020 Technical University Darmstadt ************************************************************************ ------------------------------------------------------------------------ Abstracts of the talks ------------------------------------------------------------------------ ************************************************************************ ************************************************************************ Drazen Adamovic: On logarithmic and Whittaker modules for affine vertex operator algebras and beyond ************************************************************************ 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 vertex operator algebras L_k(sl(2)) 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 algebra techniques. We present explicit realizations of Whittaker modules for some vertex algebras. We will discuss our recent efforts to generalize these realizations to higher rank cases. ************************************************************************ Fabian Kertels: BPS algebras for heterotic theories on tori ************************************************************************ Bogomol'nyi-Prasad-Sommerfield states are one-particle states saturating the lower bound on the mass present in N = 2 supersymmetric field theories. In 1996, Harvey and Moore introduced a multiplication on these BPS states and claimed to obtain a Borcherds-Kac-Moody algebra. I will report on my findings in making their still mysterious construction rigorous in the case of the CFT of a heterotic string compactified on a torus, and in exploring how close the resulting BPS algebras actually are to Borcherds-Kac-Moody algebras. ************************************************************************ Giovanna Carnovale: Jordan classes in Lie algebras and algebraic groups ************************************************************************ Reductive algebraic groups and Lie algebras can be stratified by means of irreducible, locally closed, smooth unions of orbits of elements that have similar Jordan decomposition, called Jordan classes. They were introduced for the solution of representation theoretic problems: in the Lie algebra context they appeared in the work of Borho and Kraft on sheets and the module structure of rings of regular functions on adjoint orbits. In the algebraic group context they made their first appearance in the work of Lusztig on the generalised Springer correspondence. After illustrating differences and similarities between the group and the Lie algebra situation, I will show how locally the two stratifications can be related and how to deduce geometric properties of the group stratification from properties of the Lie algebra one. The talk is based on joint work with Filippo Ambrosio and Francesco Esposito.