---

Conformal Field Theory / Théorie des champs conformes
(Org: Terry Gannon and/et Mark Walton)

---

C. CUMMINS, Concordia University, Montreal, Quebec
Congruence subgroups

In recent years modular invariance has been recognised as playing an important role in many areas of mathematics and physics. In this talk I shall discuss the computation and classification of congruence subgroups of groups commensurable with the modular group. Part of the aim of this project is to find a complete list of low genus subgroups and in particular a listing of all Hauptmoduls ( which arise, for example, in the study of moonshine).

MARC DE MONTIGNY, Alberta

BENJAMIN DOYON, Department of Physics, Rutgers University, Piscataway, New Jersey  08854, USA
Twisted modules for vertex operator algebras and Bernoulli polynomials

In this work, joint with J. Lepowsky and A. Milas, we describe how, using general principles of the theory of vertex operator algebras and their twisted modules, one can obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call "modified weak associativity."

T. GANNON, University of Alberta, Edmonton
The classification of fusion rings, I

In this introductory talk I will define from first principles the concepts of fusion rings and modular data, which play fundamental roles in e.g. conformal field theory. I will run over the basic examples and their elementary properties. Very little is known in the literature regarding their classification, and this is a glaring open problem. I'll review what is known, and describe the methods those researchers used. Then I'll introduce the new methods which we (my collaborator Mark Jackson and I) believe will permit considerable further progress. In his talk ("The classification of fusion rings, II"), Jackson will describe the results we have obtained thus far.

J. GEGENBERG, New Brunswick

MARK JACKSON, University of Alberta
The classification of fusion rings, II

Although many examples of fusion rings and modular data do exist(e.g. those arising from Kac-Moody algebras and finite groups) a classification, except in dimensions up to 3, has yet to completed. In an earlier talk ("The classification of fusion rings, I"), my collaborator Terry Gannon sketched our new approach to attacking this fundamental problem. In this talk I will describe our results. In particular, I will illustrate our approach with a proof that classifies 3 dimensional fusion rings which is considerably shorter and more transparent than any appearing in the literature. Early results for dimensions 4 and 5 will also be presented.

J. LEPOWSKY, Rutgers University, Piscataway, New Jersey  08854-8019, USA
Intertwining operators for vertex operator algebras and recursions for q-series

Vertex operator constructions of affine Lie algebras and "Z-algebras" have been used to prove, interpret or conjecture many interesting combinatorial partition identities, including the classical Rogers-Ramanujan identities. More recently, Feigin and Stoyanovsky have interpreted certain such combinatorial identities in terms of what they call the "principal subspaces" of certain modules. One of the classical methods for studying such identities is the "Rogers-Ramanujan recursion" for certain formal power series, and generalizations of this recursion discovered by Rogers and Selberg. In the present work, joint with S. Capparelli and A. Milas, we introduce such recursions into the theory of vertex operator algebras by recovering them by means of the theory of intertwining operators for modules for vertex operator algebras.

R. MANN, University of Waterloo, Waterloo, Ontario  N2L 3G1
From soup to NUTS

Studies of black hole pair production in the cosmic soup of the early universe provided the first suggestive hints that de Sitter spacetime had maximal entropy. Insights from Holographic and the (A)dS/CFT correspondence later resulted in a formalization of this notion into the N-bound conjecture, which states that that any spacetime with positive cosmological constant L = 3l^-2 has observable entropy S £ S_dS=pl². From this followed the notion that dS spacetime also had maximal mass/energy, expressed in the form of a conjecture that any asymptotically dS spacetime with mass greater than pure dS has a cosmological singularity.

Asymptotically dS spacetimes with NUT charge, however, provide counterexamples to these conjectures under certain circumstances. Such spacetimes can have a mass greater than dS spacetime whilst respecting the N-bound on entropy, and can also violate the N-bound in certain circumstances. I shall discuss the basic geometric structure of NUT-charged dS spacetimes, illustrating by example how such violations occur, and discuss the implications for holography.

P. MATHIEU, Laval

J. RASMUSSEN, CRM, Universite de Montreal, Montreal, Quebec
On the su(2)_-1/2 WZW model

The bosonic bg ghost system is equivalent to an su(2)_-1/2 WZW model, and is found to have an infinite number of representations with increasingly negative dimensions. This contradicts the traditional belief (based on the Kac-Wakimoto characters) that the four admissible representations are the only ones present. To facilitate computations, a faithful free-field representation is available. It may be extended by extra zero modes, allowing lifts of the model. These new models contain so-called relaxed representations. Indecomposable representations are also encountered, reflecting that the associated lifted model is a logarithmic CFT. Modular invariance will be addressed briefly.

GORDON SEMENOFF, UBC

M. WALTON, University of Lethbridge, Lethbridge, Alberta  T1K 3M4
Affine fusion

The fusion of the Wess-Zumino-Novikov-Witten conformal field theories is associated with their chiral algebras, the nontwisted affine Kac-Moody algebras. Properties of this affine fusion will be reviewed. Results and conjectures from the mathematical physics literature will be emphasized.