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\begin{document}

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\section*{Verlinde ring}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $G$ a suitable [[Lie group]], the \emph{Verlinde ring} is the collection of [[isomorphism classes]] of [[positive energy representations]] of the corresponding [[loop group]], equipped with the ``fusion'' [[tensor product]].

The Verlinde ring is also understood as being the ring of [[equivariant cohomology|equivariant]] [[twisted K-theory]] classes on $G$ (\hyperlink{FHT}{FHT}) and, essentially equivalently, of [[Chan-Paton gauge fields]] over [[D-branes]] in the [[WZW model]] (see there for further references).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[fusion ring]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Due to

\begin{itemize}%
\item [[Erik Verlinde]], \emph{Fusion rules and modular transformations in 2D conformal field theory}, Nuclear Physics B \textbf{300} (3): 360--376, (1988) \href{https://doi.org/10.1016/0550-3213(88}{doi}90603-7)

\end{itemize}
\begin{quote}%
We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.


\end{quote}
See also

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Alessandro Valentino]], \emph{$(3,2,1)$-TQFTs and Verlinde algebras} (\href{http://mathoverflow.net/q/74593/381}{MO question}, \href{http://mathoverflow.net/a/263829/381}{MO answer})


\item [[Dan Freed]], [[Mike Hopkins]], [[Constantin Teleman]], \emph{[[Loop Groups and Twisted K-Theory]]}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Verlinde_algebra}{Verlinde algebra}}



\end{itemize}
[[!redirects Verlinde rings]]

[[!redirects Verlinde algebra]] [[!redirects Verlinde algebras]]



\end{document}