190 research outputs found
Hopf algebras and finite tensor categories in conformal field theory
In conformal field theory the understanding of correlation functions can be
divided into two distinct conceptual levels: The analytic properties of the
correlators endow the representation categories of the underlying chiral
symmetry algebras with additional structure, which in suitable cases is the one
of a finite tensor category. The problem of specifying the correlators can then
be encoded in algebraic structure internal to those categories. After reviewing
results for conformal field theories for which these representation categories
are semisimple, we explain what is known about representation categories of
chiral symmetry algebras that are not semisimple. We focus on generalizations
of the Verlinde formula, for which certain finite-dimensional complex Hopf
algebras are used as a tool, and on the structural importance of the presence
of a Hopf algebra internal to finite tensor categories.Comment: 46 pages, several figures. v2: missing text added after (4.5),
references added, and a few minor changes. v3: typos corrected, bibliography
update
From non-semisimple Hopf algebras to correlation functions for logarithmic CFT
We use factorizable finite tensor categories, and specifically the
representation categories of factorizable ribbon Hopf algebras H, as a
laboratory for exploring bulk correlation functions in local logarithmic
conformal field theories. For any ribbon Hopf algebra automorphism omega of H
we present a candidate for the space of bulk fields and endow it with a natural
structure of a commutative symmetric Frobenius algebra. We derive an expression
for the corresponding bulk partition functions as bilinear combinations of
irreducible characters; as a crucial ingredient this involves the Cartan matrix
of the category. We also show how for any candidate bulk state space of the
type we consider, correlation functions of bulk fields for closed oriented
world sheets of any genus can be constructed that are invariant under the
natural action of the relevant mapping class group.Comment: 41 pages, several figures. version 2: typos corrected, bibliography
updated, introduction extended, a few minor clarifications adde
Eisenstein series and quantum groups
We sketch a proof of a conjecture of [FFKM] that relates the geometric
Eisenstein series sheaf with semi-infinite cohomology of the small quantum
group with coefficients in the tilting module for the big quantum group
Quantum group symmetry of integrable models on the half-line
This contribution to the Proceedings of the Workshop on Integrable Theories,
Solitons and Duality in Sao Paulo in July 2002 summarizes results from the
papers hep-th/0112023 and math.QA/0208043. We derive the non-local conserved
charges in the sine-Gordon model and affine Toda field theories on the
half-line. They generate new kinds of symmetry algebras that are coideals of
the usual quantum groups. We show how intertwiners of tensor product
representations of these algebras lead to solutions of the reflection equation.
We describe how this method for finding solutions to the reflection equation
parallels the previously known method of using intertwiners of quantum groups
to find solutions to the Yang-Baxter equation.Comment: Contribution to the Proceedings of the Workshop on Integrable
Theories, Solitons and Duality in Sao Paulo in July 2002, 11 pages, JHEP3
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