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\begin{document}

\title{Lecture 2: Modular Functor}
\author{Alexander A. Voronov}
%\address{Department of Mathematics\\ M.I.T.,
%2-246\\ 77 Massachusetts Ave.\\ Cambridge, MA 02139-4307}
%\curraddr{}
%\email{voronov@math.mit.edu}
%\urladdr{http://www-math.mit.edu/~voronov/}
%\thanks{Research supported in part by an AMS Centennial Fellowship.}

%\subjclass{Primary 14H10; Secondary 32G15, 55P62}
\date{September 17, 1997}

\maketitle

\section{Modular Functor}

In physical examples, especially when one deals with the structure of
chiral, holomorphic CFT, it is more natural to consider a more general
theory, where a Riemann surface $\Sigma$ with holomorphic holes is
assigned a finite-dimensional space $V_\Sigma$ of operators rather
than a unique operator $|\Sigma\rangle$. This situation is formalized by the
notion of a modular functor, which starts with a finite set $I$ of
indices, with an involution $\alpha \mapsto \bar \alpha$ for $\alpha
\in I$ defined and a unit element $\mathbf{1} \in I$, such that $\bar
\mathbf{1} = \mathbf{1}$, fixed. The physical meaning of this set $I$
is the space of quantized momenta of the string. A \emph{modular
functor} assigns a Riemann surface $\Sigma$ with $n$ holomorphic
holes labeled not only by the numbers 1, \dots, n, but also by a
vector $\alpha = (\alpha_1, \dots, \alpha_n)$ of elements of the index
set $I$, a vector space $V_{\Sigma, \alpha}$ depending holomorphically
on $\Sigma$:
\[
(\Sigma,\alpha) \mapsto V_{\Sigma,\alpha}, \qquad \dim
V_{\Sigma,\alpha} < \infty.
\]
The holomorphic dependence is understood as follows. Every Riemann
surface gives rise to a vector space holomorphically depending on the
class of a Riemann surface means that the moduli space of Riemann
surfaces with holes (or just the base $S$ any holomorphic family $X
\to S$) is provided with a holomorphic vector bundle $\mathcal V$
whose fiber over the Riemann surface $\Sigma$ is $V_\Sigma$.

This assignment must satisfy the following axioms.

\begin{enumerate}

\item \textbf{Disjoint union}:
$V_{\Sigma_1 \coprod \Sigma_2,\alpha_1 \coprod \alpha_2} = V_{\Sigma_1,\alpha_1} \otimes V_{\Sigma_2,\alpha_2}$.

\item \textbf{Sewing}:
If $\Sigma$ is a Riemann surface with $n+2$ holes, and $\widehat \Sigma$ is the result of sewing the $n+1$st hole to the $n+2$nd hole, and $\alpha$ is an indexing of the remaining $n$ holes on $\widehat \Sigma$, then
\begin{equation}
\label{sewing}
V_{\widehat \Sigma,\alpha} = \bigoplus_{\beta \in I} V_{\Sigma,(\alpha, \beta, \bar \beta)}.
\end{equation}

\item \textbf{Normalization 1}:
If $D$ is the unit disk, understood as the Riemann sphere with one hole around infinity: $D = S^2 \setminus \{\abs{z} > 1\}$, then
\[
V_{D,\alpha} = \begin{cases}
\nc & \text{if $\alpha = \mathbf{1}$},\\
0 & \text{otherwise}.
\end{cases}
\]

\item \textbf{Normalization 2}:
If $\Sigma$ is an annulus, then 
\[
\dim V_{\Sigma, (\alpha_1, \alpha_2)} = 
\begin{cases}
1 & \text{if $\alpha_2 = \bar \alpha_1$},\\
0 & \text{otherwise}.
\end{cases}
\]

\end{enumerate}

Here is a number of consequences of the above axioms, which make the
story more similar to the set of axioms of a CFT.

\begin{xca}
If $\Sigma_1$ and $\Sigma_2$ are sewn end-to-end, then there exists a
composition map
\[
V_{\Sigma_1,(\alpha,\beta)} \otimes V_{\Sigma_2,(\bar \beta, \gamma)}
\to V_{\Sigma_1 \cup \Sigma_2,(\alpha, \gamma)}.
\]
\end{xca}

\begin{xca}
If $\Sigma$ is a torus, then $V_\Sigma = \langle I \rangle_{\nc}$, the
linear span of the index set $I$.
\end{xca}

\begin{thm}[Physics folklore and G. Segal]
If $X \to S$ is a holomorphic family of surfaces, there exists a
canonical flat projective connection in the holomorphic vector bundle
$\mathcal V$ defined by the modular functor.
\end{thm}

\begin{rem}
Here a \emph{projective connection} means an isomorphism $p_*:
V_{X_s,\alpha} \to V_{X_s',\alpha}$ , defined for any path $p$
connecting points $s$ and $s' \in S$ up to a nonzero scalar factor. A
projective connection is \emph{flat}, if $p_*$ does not change if the
path $p$ is deformed smoothly leaving its ends fixed.
\end{rem}

\begin{proof}
The proof is based on the following exercise on differential geometry.
\begin{xca}
\begin{enumerate}

\item
If a Lie algebra $\gtg$ acts locally transitively on a manifold $M$,
\emph{i.e}., there is a morphism $\gtg \to \Vect(M)$, such that the
evaluation map $\gtg \to T_m M$ is surjective for any point $m \in M$,
and the action of $\gtg$ on $M$ lifts to an action on a vector bundle
$V$ over $M$, then there exists a natural flat connection on $V$

\item
Same, assuming the action of $\gtg$ on $V$ is projective: show that
there is a flat projective connection on $V$.

\end{enumerate}
\end{xca}

Assume for simplicity that the Riemann surfaces in question have only
one holomorphic hole.
\begin{lm}
The tangent space to the space of Riemann surfaces $X$ with one
holomorphic hole is naturally isomorphic to $\Vect_\nc
(S^1)/\Vect(X)$, where $\Vect_\nc(S^1)$ is the space of smooth vector
fields on the circle $S^1$ (in fact, those which come from smooth
vector fields on an annulus containing $S^1$), and $\Vect(X)$ is the
space of holomorphic vector fields on the complement of the disk in
$X$ identical on the boundary of the hole.
\end{lm}

\begin{proof}[Proof of Lemma]
The classical theory of Beltrami differentials suggests that an
infinitesimal deformation of the complex structure on a Riemann
surface preserving the holomorphic hole is governed by the class of a
smooth $(-1,1)$-form $\mu d\bar z /dz$, $z$ being a local holomorphic
coordinate on the Riemman surface, $\mu = 0$ on the hole, modulo the
$(-1,1)$-forms of the type $\bar \partial \eta d\bar z /dz$, where
$\eta/dz$ is a smooth vector field on the surface vanishing on the
hole, an infinitesimal diffeomorphism. Computing this \coh\ group in
\v{C}ech \coh\ using the covering by two open sets, a neighborhood of the
hole and a neighborhood of the complement, we see that the tangent
space is given by $\Vect(S^1)/\Vect(X)$.
\end{proof}
\end{proof}

\begin{cor}
\label{indep}
Given a modular functor, the projective space
$\mathbb{P}(V_{\Sigma,\alpha})$ is naturally associated to a smooth
surface $\Sigma$ with a labeling $\alpha$ independently of the choice
of a complex structure.
\end{cor}

\begin{proof}
The fact is that the Teichm\"uller space, the space of conformal
classes of complex structures on a Riemann surface (before taking the
quotient by the diffeomorphism group of the Riemann surface) is
contractible. Therefore, lifting the projective connection to the
Teichm\"uller space, we get an identification of the fibers of the
corresponding projective bundle.
\end{proof}

\section{The Affine Example}

Before building up on the axioms of the theory, let us show that it
exists in nature. Here is an example showing up in the
Wess-Zumino-Witten model. Let $\gtg$ be a simple finite-dimensional
Lie algebra with the inner product normalized so that the long root
$\theta$ has the inner square $(\theta, \theta) = 2$. Fix a nonnegative
integer $k$, which is going to be the \emph{level} of an irreducible
highest-weight representation of the affine Kac-Moody algebra
$\widehat{L\gtg}$, a central extension of the current algebra $L\gtg =
\gtg \otimes \nc((t))$. Take the set of all irreducible integrable
highest-weight representations of $\widehat{L\gtg}$ of level $k$ as
the label set $I$. These representations correspond bijectively to the
set of irreducible finite-dimensional representations $L(\lambda)$ of
$\gtg$ with the highest weight $\lambda$ satisfying the condition
$(\theta, \lambda) \le k$, which implies that the set $I$ is finite,
%$I = \{1, \dots, N\}$,
see \cite{kac,ps}. Define the involution on $I$ which takes a
representation $L(\lambda)$ of $\gtg$ to its Hermitian dual
$\overline{L(\lambda)}^*$ and the unit element $1 \in I$ corresponds
to the trivial representation $L(0)$ of $\gtg$.

Given a Riemann surface $\Sigma$ with $n$ holomorphic holes, consider
the group $G(\Sigma)$ of meromorphic $G$-valued functions
on $\Sigma$ with the only singularities at the centers of the
holes. For any labeling $\alpha$ of the holes on $\Sigma$ by elements
$\widehat{L}_1$, \dots, $\widehat{L}_n$ of $I$, the group
$G(\Sigma)$ acts on the tensor product $\widehat{L}_1 \otimes \dots
\otimes \widehat{L}_n$ via the Laurent expansions at the poles. This
action is \emph{a priori} projective, because each $\widehat{L}_i$ is
a representation of level $k$. However, the Residue Theorem gives a
canonical splitting of the restriction of the central extension of
$\bigoplus_{i=1}^n L\gtg$ to $G(\Sigma)$. Thus, $\widehat{L}_1
\otimes \dots
\otimes \widehat{L}_n$ becomes a representation of $G(\Sigma)$, and
we define the value of the modular functor at the pair $(\Sigma,
\alpha)$ as the space of invariants
\[
V_{\Sigma,\alpha} = (\widehat{L}_1 \otimes \dots \otimes
\widehat{L}_n)^{G(\Sigma)}.
\]


\begin{thm}
This correspondence defines a modular functor.
\end{thm}

\begin{proof}
Here we will follow Segal's paper \cite{segal}.  The loop group
version of the Peter-Weyl Theorem asserts that
\[
\Hol_k (\widehat{LG}) = \overline{\bigoplus_{i \in I}
\overline{\widehat{L}_i}^* \otimes \widehat{L}_i},
\]
where $\Hol_k (\widehat{LG})$ denotes the holomorphic functions $f:
\widehat{LG} \to \nc$ on the central extension of the loop group such
that $f(u \gamma) = u^k f(\gamma)$ for $u \in \nc^*$ and $\gamma \in
\widehat{LG}$, $I$ the set of irreducible highest-weight
representations of $\widehat{LG}$ of level $k$, the bar over the
right-hand side denotes completion in an appropriate topology of the
function space, and $\bar V^*$ for a representation $V$ means its
Hermitian dual.

Let us prove the Sewing Axiom \eqref{sewing}. Suppose that a Riemann
surface $\Sigma$ is getting sewn along its $n+1$st hole $D_1$ and
$n+2$nd hole $D_2$ to form a surface $\hat \Sigma$ with $n$
holes. Note that the group $G(\Sigma)$ acts transitively on $LG$ by
$g\gamma = g_1 \gamma g_2^{_1}$, where $g_1$ is the restriction of $g$
to $D_1$ and $g_2$ to $D_2$. The isotropy group of 
\end{proof}

\section{Verlinde's Algebra}

Suppose a modular functor is given. For any Riemann surface $\Sigma$
with $n$ holes and a collection $\alpha_1, \dots, \alpha_n$ of labels
thereof, define positive integers
\begin{equation}
\label{correlators}
\langle \alpha_1, \dots, \alpha_n | \Sigma \rangle = \dim V_{\Sigma, \alpha}.
\end{equation}
This dimension is independent of the complex structure on $\Sigma$ and
the choice of holes, because of Corollary~\ref{indep}. Consider the
linear span $V$ of the label set $I$. This space carries a symmetric
bilinear inner product $(\alpha, \beta) = \langle \alpha, \beta |
\Sigma\rangle$, where $\Sigma$ is the Riemann sphere with two holes, a
distinguished element 1, and a multiplication
\[
\alpha \cdot \beta = \sum_{\gamma \in I} 
\langle  \alpha,  \beta, \bar \gamma |S^2 \rangle \gamma,
\]
\begin{xca}
where $S^2$ is the Riemann sphere with three holes. Deduce from the
axioms of a modular functor that this defines on $V$ the structure of
a commutative associative algebra with an inner product satisfying the
invariance condition $( \alpha \beta, \gamma) = (\alpha, \beta
\gamma)$. This algebra is called \emph{Verlinde's algebra}. In fact,
this is a \emph{Frobenius algebra}, and the reason why it comes about
is that the correspondence
\begin{align*}
\left\{\text{\parbox{1.2in}{\center{The moduli space of Riemann surfaces with
 $n$ holomorphic holes}}}\right\} & \to \Hom(V^n, \nc),\\
\makebox[1.2in]{$\Sigma$} & \mapsto \langle \alpha_1, \dots, \alpha_n \rangle,
\end{align*}
defines a \emph{topological quantum field theory}. This will be the
topic of the next section.
\end{xca}

In fact, the structure of Verlinde's algebra determines all the
\emph{correlators} \eqref{correlators}, because any Riemann surface
may be cut into Riemann spheres with one, two, and three holes. Thus
Verlinde's algebra encodes the complete list of dimensions for the
modular functor $V_{\Sigma, \alpha}$.


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