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\begin{document}

\title{Quantum dimension
}
 \author{Speaker: Josiah Thornton
 \\ Typist: Brent Pym
 }
 \date{\today}
 \thanks{Available online at \texttt{http://math.mit.edu/CFTworkshop}.  Please email \texttt{eep@math.mit.edu} with corrections and improvements!}

\maketitle

\begin{abstract}
Notes from the ``Conformal Field Theory and Operator Algebras workshop," August 2010, Oregon.
\end{abstract}

Outline for the talk

\begin{enumerate}
\item An example
\item Semi-simple ribbon categories
\item Continuation of example
\item Definition of quantum dimension
\item Properties thereof
\item Computations for IPERs of $LSU(2)$
\end{enumerate}

\section{Basic example}

Let $V = \C^2$ and choose a basis $\{e_2,e_2\}$ with dual basis $\{\epsilon_1,\epsilon_2\}$ for $V^*$.  We have an evaluation map
$$
e : V \otimes V^* \to \C
$$
defined by eating a vector with a linear functional.  There is also an embedding
$$
i : \C \to V \otimes V^*
$$
defined by
$$
i(1) = e_1 \otimes \epsilon_2 + e_2 \otimes \epsilon_2.
$$
The composition
$$
e \circ i : \C \to V \otimes V^* \to \C
$$
yields
$$
e(e_1 \otimes \epsilon_1 + e_1\otimes \epsilon_2) = 2.
$$

\section{Semi-simple ribbon categories}
Let $\cC$ be a $\C$-linear abelian category equipped with the following structure:
\begin{itemize}
\item $\otimes : \cC \times \cC \to \cC$, a bilinear functor
\item $\alpha_{U,V,W} : (U\otimes V) \otimes W \isomto U \otimes (V \otimes W)$, a functorial isomorphism for all $U,V,W \in \cC$
\item $I \in \cC$ such that $End(I) = \C$, together with functorial isomorphisms $$\lambda_V : I \otimes V \isomto V$$ and $$\rho_V : V \otimes I \to V$$ for all $V \in \cC$
\end{itemize}
Such a structure is called a \emph{monoidal structure}.

\begin{example}$Vec(\C)$, together with the tensor product $\otimes$.\hfill$\Box$\end{example}
\begin{example}
$Rep(SU(N))$ with tensor product $\cH_1 \otimes \cH_2$ of the underlying Hilbert spaces $\cH_1,\cH_2 \in Rep(SU(N))$ and the action $$g(x\otimes y) = g(x)\otimes g(y)$$ for $g \in SU(N)$ and $x\in \cH_1$, $y \in \cH_2$. \hfill$\Box$.
\end{example}

\begin{definition}
A \emph{braiding} is given by isomorphisms
$$
\sigma_{VW} : V \otimes W \isomto W \otimes V,
$$
which are functorial in $V,W \in \cC$, and satisfy some commutative diagrams.  A category with a monoidal structure and a braiding is called a \emph{braided monoidal category}.
\end{definition}
\begin{example}
For the categories $Vec(\C)$ and $Rep(SU(N))$, the isomorphisms
$$
\tau_{VW} : V \times W \to W \times V
$$
given by
$$
\tau(v,w) = (w,v)
$$
give a braiding.\hfill$\Box$
\end{example}

\emph{Rigidity} is given by homomorphisms
$$
e_V : V^* \otimes V \to I
$$
and
$$
i_V : I \to V \otimes V^*,
$$
where $V$ is the right dual of $V$.  These maps must make the following diagrams commute:

\includegraphics[scale=.5]{Thursday4pmPicture1.jpg}


\begin{example}
For $Vec(\C)$, we have $V^* = Hom(V,\C)$ and the maps $e_V$ and $i_V$ as before.\hfill$\Box$
\end{example}
\begin{example}
For $Rep(G)$, the representation on the dual of $\cH$ is given by continuous linear functionals $\cH^* = C(\cH,\C)$ with the representation $$(gf)(v) = f(g^{-1}v)$$ for $f \in \cH^*$, $g \in G$ and $v \in \cH$. \hfill$\Box$
\end{example}

Suppose that we have a map $f : U \to V$.  We can construct a dual map $f*$ by the composition
$$
V^* \overset{\id \otimes i_U}{\longrightarrow} V^* \otimes U \otimes U^* \overset{\id \otimes f \otimes \id}{\longrightarrow}  V^* \otimes V \otimes U^* \overset{e_V \otimes \id}{\longrightarrow} U^*
$$
\begin{definition}
A \emph{ribbon category} is a rigid braided tensor category with a functorial isomorphism $\delta_V : V \isomto V^{**}$ such that
\begin{align*}
\delta_{V\otimes W} &= \delta_V \otimes \delta_W \\
\delta_I &= \id_I \\
\delta_{V^*} &= \left(\delta_V^*\right)^{-1}
\end{align*}

\end{definition}

\section{Quantum dimension}
Let $\cC$ be a semi-simple ribbon category.

\begin{definition}
Let $V \in \cC$ and $f \in End(V)$.  Define the \emph{trace of $f$} to be the composition
$$
I \to V \otimes V^* \overset{f \otimes \id}\longrightarrow V\otimes V^* \to V^{**} \otimes V^* \overset{e_{V^*}}{\longrightarrow} I
$$
\end{definition}

One important property of the trace is
$$
tr(f\otimes g) = tr(f)tr(g),
$$
a fact which needs the ribbon structure.

\begin{definition}
The \emph{quantum dimension of $V \in \cC$} is defined to by $tr(\id_V)$.
\end{definition}

If there are finitely many simple objects, the quantum dimension is a real number.  For $LSU(N)$, the quantum dimension is, in fact, positive.

\section{Calculations}

Let
$$
N^h_{fg} = \dim\left(Hom(V_h,V_f\otimes V_g)\right)
$$

\begin{theorem}[Wassermann]
The Connes fusion satisfies
$$
\cH_f \boxtimes \cH_g = \bigoplus N^h_{fg} \mathrm{sgn}(\sigma_h) \cH_{h'}
$$
where 
$$
h' = \sigma(h+\delta) - \delta
$$
is a permutation.
\end{theorem}
\begin{corollary}
For $G = SU(2)$, we have
$$
\cH_l \boxtimes \cH_l = \cH_0
$$
\end{corollary}

\begin{theorem}
For the permutation signature $f$, we have
$$
\cH_\Box \boxtimes \cH_f = \bigoplus_{g = f + \Box} \cH_g
$$
\end{theorem}

\begin{corollary}
For $G = SU(2)$ and $1 \le \lambda \le l-1$,
$$
\cH \boxtimes \cH_{\lambda} = \cH_{\lambda-1} \oplus \cH_{\lambda+1}
$$
\end{corollary}

\section{IPERs of $LSU(2)$}
Fix a level $l \in \Z$, consider
$$
\cH_0 ,\dotsc,\cH_l,
$$
and define
$$
d_i = \dim \cH_i
$$
and
$$
d_l =1.
$$

\begin{proposition}
For $0 \le i \le l$, we have
$$
d_i = d_{l-i},
$$
so that
$$
d_1 d_i = d_{i-1} + d_{i+1}
$$
and
$$
d_1d_{l-i} = d_{l-i-1}+ d_{l-i+1},
$$
and hence
$$
d_{i-1} = d_{l-i+1}
$$
for $1 \le i \le l-1$.
\end{proposition}

We can use this proposition to calculate the quantum dimensions, as in the following picture:

\includegraphics[scale=.5]{Thursday4pmPicture2.jpg}

\end{document}

