Seminar on Conformal Field Theory
December 13th, 2013
Technische Universität Darmstadt
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Abstracts of the talks
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Fiebig: Recent development around Lusztig's conjecture
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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.
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Gray: Classifying superconformal field theories via chiral rings
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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.
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Yamauchi: Transposition automorphisms of vertex operator algebras
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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.