13th Seminar on Conformal Field Theory
January 17th, 2020
Technical University Darmstadt
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Abstracts of the talks
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Drazen Adamovic: On logarithmic and Whittaker modules for affine
vertex operator algebras and beyond
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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.
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Fabian Kertels: BPS algebras for heterotic theories on tori
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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.
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Giovanna Carnovale: Jordan classes in Lie algebras and algebraic groups
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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.