Categories in Algebra, Geometry and Mathematical Physics
Conference in honour of Ross Street's sixtieth birthday
July 11-16 2005, Macquarie University, Sydney
Tue, 12 July: 15:20 - 16:00
Conformal field theory and Frobenius algebras in modular tensor categories
Fuchs, Jürgen (Karlstads Universitet, Sweden)
A class of Frobenius algebras in modular tensor categories provides the right tools for proving the existence of two-dimensional rational conformal quantum field theories. For an algebra $A$ in this class, the construction used in the proof provides in particular a dictionary between algebraic structures and physical concepts, part of which looks as follows:\\[1ex]
\newcommand{\PBS}[1]{\let\temp=\\#1\let\\=\temp}
\begin{tabular}{>{\PBS\raggedright\hspace{0pt}}m{13em}@{\quad}p{1em}@{\quad}>{\PBS\raggedright\hspace{0pt}}p{10.5em}}
category of left $A$-modules &= &boundary conditions \\[.5ex]
category of $A$-bimodules &= &defect lines \\[.5ex]
bimodule morphisms between
certain induced bimodules &= &bulk fields \\[.5ex]
Picard group of the
bimodule category &= &internal symmetries
\end{tabular}\\[1ex]
Morita-equivalent algebras yield the same conformal field theory; the Brauer group of the modular tensor category describes the subset of modular invariant partition functions that arise from an automorphism of the fusion rules.