skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus

Abstract

After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction-restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure)

Authors:
;  [1];  [2]
  1. Centre de Physique Theorique, UMR 6207 du CNRS et des Universites Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var, affilie a la FRUMAM (FR 2291), Luminy Case 907, F-13288 Marseille Cedex 9 (France)
  2. (CBPF), Rua Dr. Xavier Sigaud, 150 Rio de Janeiro (Brazil)
Publication Date:
OSTI Identifier:
20929689
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 4; Other Information: DOI: 10.1063/1.2714000; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGEBRA; CONFORMAL INVARIANCE; QUANTUM FIELD THEORY; SL GROUPS

Citation Formats

Coquereaux, R., Schieber, G., and Centro Brasileiro de Pesquisas Fisicas. Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus. United States: N. p., 2007. Web. doi:10.1063/1.2714000.
Coquereaux, R., Schieber, G., & Centro Brasileiro de Pesquisas Fisicas. Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus. United States. doi:10.1063/1.2714000.
Coquereaux, R., Schieber, G., and Centro Brasileiro de Pesquisas Fisicas. Sun . "Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus". United States. doi:10.1063/1.2714000.
@article{osti_20929689,
title = {Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus},
author = {Coquereaux, R. and Schieber, G. and Centro Brasileiro de Pesquisas Fisicas},
abstractNote = {After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction-restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure)},
doi = {10.1063/1.2714000},
journal = {Journal of Mathematical Physics},
number = 4,
volume = 48,
place = {United States},
year = {Sun Apr 15 00:00:00 EDT 2007},
month = {Sun Apr 15 00:00:00 EDT 2007}
}
  • Using the fact that the fusion algebra of a rational conformal field theory is specified in terms of integers that are related to modular transformation properties, the authors completely classify 2-field chiral RCFT's. The authors show that the only possibilities for the non-trivial fusion rule are {phi} {times} {phi} = 1 or {phi} {times} {phi} = 1 + {phi}. We reduce the 3-field classification to a set of algebraic equations and solve them in a few cases.
  • We discuss chiral zero-rest-mass field equations on six-dimensional space-time from a twistorial point of view. Specifically, we present a detailed cohomological analysis, develop both Penrose and Penrose-Ward transforms, and analyse the corresponding contour integral formulae. We also give twistor space action principles. We then dimensionally reduce the twistor space of six-dimensional space-time to obtain twistor formulations of various theories in lower dimensions. Besides well-known twistor spaces, we also find a novel twistor space amongst these reductions, which turns out to be suitable for a twistorial description of self-dual strings. For these reduced twistor spaces, we explain the Penrose and Penrose-Wardmore » transforms as well as contour integral formulae.« less
  • We present a solution of the problem of a free massless scalar field on the half line interacting through a sinusoidal potential on the boundary. For a critical value of the period, this system is a conformal field theory with a nontrivial and explicitly calculable [ital S] matrix for scattering from the boundary. It describes the critical behavior of a number of condensed matter systems, including dissipative quantum mechanics and of barriers in quantum wires.''
  • A method to obtain the boundary states and the crosscap states explicitly in various conformal field theories, is presented. This makes it possible to construct and analyse open string theories in several closed string backgrounds. The authors discuss the construction of such theories in the case of the backgrounds corresponding to the conformal field theories with SU(2) current algebra symmetry.
  • The coupling of conformal field theories to 2-d gravity may be studied in the conformal gauge. As an application, the results of Knizhnik, Polyakov and Zamolodchikov for the scaling dimensions of conformal fields are derived in a simple way. Their conjecture for the susceptibility exponent ..gamma.. of strings is proven and extended to arbitrary genus surfaces. The result agrees with exact results from random lattice models.