## 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.