Seminar on Conformal Field Theory December 13th, 2013 Technische Universität Darmstadt ************************************************************************ ------------------------------------------------------------------------ Abstracts of the talks ------------------------------------------------------------------------ ************************************************************************ ************************************************************************ Fiebig: Recent development around Lusztig's conjecture ************************************************************************ Lusztig stated in 1981 a formula for the simple rational characters of reductive algebraic groups defined over a field of positive characteristic p. In the case of GL(n) it was expected that the formula holds for p>n. In 1990 it was shown in a combined effort of several authors that the conjecture is true if the characteristic is big enough. In 2007 a first upper bound on the exceptions was found, which, however, is an enormously huge number. Very recently, Geordie Williamson found a series of counterexamples with p>n, which came as a big surprise for representation theorists. I will report on these events and maybe speculate about what we can expect to be a plausible answer now. ************************************************************************ Gray: Classifying superconformal field theories via chiral rings ************************************************************************ I will introduce some of the main mathematical ideas behind N=2 superconformal field theory, using the N=2 minimal models as a concrete example. To illustrate these ideas we will focus on the so-called chiral rings, which arise from basic Lie algebra representation theory plus ideas from quantum field theory. Then I'll describe how chiral rings appear in Cecotti and Vafa's famous ADE classification of a special subclass of N=2 minimal models (those with space-time supersymmetry), and we'll briefly reacquaint ourselves with some other prominent appearances the ADE pattern in maths and physics. Finally I'll present the chiral rings of the entire class of N=2 minimal models (when the extra assumption of space-time supersymmetry is dropped) and report on ongoing work with K. Wendland concerning a geometric classification of these theories. ************************************************************************ Yamauchi: Transposition automorphisms of vertex operator algebras ************************************************************************ A mysterious relation is known to exist between finite simple groups and vertex operator algebras. The study of centralizers of involutions played a central role in the classification of finite simple groups. In this talk I will explain Miyamoto's construction of involutions of a vertex operator algebra based fusion algebras. Then I will exhibit several examples of finite simple groups acting on vertex operator algebras and corresponding results of the Conway-Miyamoto type.