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\begin{document}

\title{The Knizhnik-Zamolodchikov equation 
}
 \author{Speaker:  Anatoly Preygel
 \\ Typist:  Emily Peters
 }
 \date{\today}
 \thanks{Available online at \texttt{http://math.mit.edu/$\sim$eep/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}

%%% Start typing here!

\section{Motivation:}

Why do we expect a differential equation to be useful in this formalism Wasserman's using?

\begin{recall}
$\lambda$ is a tableau or ``signature''; it gives rise to $V_\lambda$.  If $\lambda$ admissible for level $\ell$, we get $H_\lambda$, an irrep of $\tilde{LG}_\ell \rtimes S^1_{rot^n}$.  This has $H_\lambda(0)=V_\lambda$.
\end{recall}

Heuristically, what are primary fields?  $\operatorname{Hom}_G(V_{\lambda}\otimes W, V_{\mu} )$.  Under favorable circumstances, get a map to $\operatorname{Hom}_{\tilde{LG}\rtimes S^1_{rot}}(H_\lambda \tensor C^\infty(S^1,W), H_\mu)$.  Say this takes $\phi$ to $\varphi$.

Think of $\varphi$ as $W^u \tensor \operatorname{Hom}^{unbd}(H_\lambda,H_\mu)$-value distribution.

Fourier modes:  $\varphi(n)=``\int_{S^1} \varphi(\zeta) \zeta^{-n} \frac{d \zeta}{2 \pi \zeta}'' = \varphi(\zeta^{-n})$.

This is not so bad; for $w \in W$, $\varphi(n)(w)LH_\lambda(k) \rightarrow H_\mu(k-n)$

Some notation:
$\lambda_1, \lambda_2, \mu$ are tableau; $W_1$, $W_2$ are $G$-representations.  Set $\mathcal{U}=\operatorname{Hom}_G(V_{\lambda_1}\tensor W_1 \tensor W_2 , V_{\lambda_2})$.
\begin{align*}
 \phi^1:  V_{\lambda_1}\tensor W_1&  \rightarrow V_\mu \\
 \phi^2:  V_{\mu}\tensor W_2&  \rightarrow V_{\lambda_2} \\
 \phi^2 \circ \phi^1 \in \mathcal{U}
\end{align*}

replace $\phi^i$ by $\varphi^i$; consider $\varphi^2 \circ \varphi^1$.  We take $(f,g)$ to $\varphi^2(f) \circ \varphi^1(g)$, for $f \in C^\infty(S^1,W_2)$, $g \in C^\infty(S^1,W_1)$, still ignoring issues about unbounded operators.

$\mathcal{U} \simeq \DirectSum_{\mu'} \Hom_G(V_{\mu'} \tensor W_1, V_{\lambda_2}) \tensor \Hom_G(V_{\lambda_1} \tensor W_2, V_{\mu'})$

With superscripts on  $\varphi$ being charges, subscripts being targets and sources.

$\phi^2 \circ \phi^1=\Sigma _{\mu'} (?) \phi^{w_1}_{\lambda_2 \mu'} \tensor \phi^{w_2}_{\mu' \lambda_1}$
gets sent to same think, with $\varphi$s.

Play with power series:  $f=\Sigma f_n z^n$, $g=\Sigma g_n w^n$.

$\varphi: f \mapsto \varphi(f)$.

\includegraphics[scale=.3]{Wednesday9-45amPicture1.jpg}

$\underline{\varphi}$ defined to the the image under composition of the diagram above.

Four-point function:   $F_\mu = \underline{\varphi^2(f) \circ \varphi^1(g)}$.  By playing with power series, $F_\mu$ is a function of $z/w$ with corefficients of the form $f_n g_{-n}$ times some number.  ie, $F_\mu(\zeta)=\Sigma_{n \geq 0} \underline{\varphi^2(n) \circ \varphi^1(-n)} \zeta^n$.

\begin{lemma}
$\underline{\varphi^2(f) \varphi^1(g)} \in \mathcal{U}$ and $\underline{\varphi^2(f) \varphi^1(g)} = \int_{S^1-\{1\}} (\tilde{f} \star g)(\zeta) F_{\mu}(\zeta) \frac{d \zeta}{2 \pi \zeta}$, where $\tilde{f}$ is $f$ `backwards' around the circle.  This is assuming $f$ and $g$ have disjoint support, and $F_\mu$ extends continuously to $S^1-\{1\}$.
\end{lemma}

It's easy to check

$(\tilde(f) \star g)(\zeta)=(\tilde(g) \star f)(\zeta^{-1})$;

This means we have relations between holomorphic functions $F_\mu(\zeta)$ and $G_\mu(\zeta^{-1})$;

\includegraphics[scale=.5]{Wednesday9-45amPicture2.jpg}

And, relations on these  things that are called fourpoint functions ($F_\mu$) give relations between primary fields.

Question:  what's a four point function?  why's it called that?  What's it do?

Answer:  So, given $f,g \in C^\infty(S^1,\C)$ and $w_1, w_2 \in W_1, W_2$, we have $ \underline{\varphi^2(w_2 f) \circ \varphi^1(w_1g)}$.

underline here means to restrict to vectors $v_\lambda$ in $H_\lambda(0)$; do this projection by taking inner products

$\langle  \underline{\varphi^2(w_2 f) \circ \varphi^1(w_1g)} v_{\lambda_1}, v_{\lambda_2}
,  \rangle$ -- four inputs, hence `four point function.'


Motivation: What do differential equations have to do with anything?

{\color{blue} Answer:  transport coefficients?}

\section{Generalities}

Wanted ``nice'' OCE setisfied by $F_\mu$ and $G_\mu(\zeta^{-1})$.  Wasserman's ``Basic ODE:''
$$ \frac{df}{dz} = (\frac{P}{z} + \frac{Q}{1-z})f
$$
$f$ is $\mathcal{U}$-valued funtion, $P, Q \in \operatorname{End}(\mathcal{U})$.

This equations is linear, first order, with regular singularities.

\begin{example}
$\mathcal{U}=\C$.
$$\frac{df}{f}=\frac{dz}{z}(P+Q\frac{z}{1-z})
$$
\
here $\frac{df}{f}=d \log f$, $\frac{dz}{z}=d \log z$, and $P+Q\frac{z}{1-z}$ is holomorphic near 0.

This gives us $f=\zeta^P$ -- a function homomorphic near 0 and unique up to scalars.
\end{example}

\begin{note}
If $P$ has eigenvalues not differing by integers, then solutions near 0 look like
$f=\Sigma a_i \tilde{f}(\zeta) \xi_i \zeta$

Canonical basis of $U$ corresponding to eigenvalues of $P$, and corresponding to solutions.
\end{note}

General fact:  Such a ``nice'' differential equation of $\C P^1$ with regular singularities at $0$, $1$, $\infty$ correspond to local systems on $\C P^1-\{ 0, 1, \infty \}$ (with variables $\mathcal{U}$).  These correspond to maps $\pi_1(\C P^1-\{ 0, 1, \infty \}) \rightarrow GL(\mathcal{U})$.

\section{Motivation}  

Why do we get a differential equation?  

Suppose we're given $n+1$ points in $\C P^1$ and coordinates $D \rightarrow \C P^1$; this gives us a vectors space non-canonical $\mathcal{U}$.

A change of coordinates gives us an isomorphism of vectors spaces.

This sort of data gives us a variable on $M_{0,n+1}$ -- the moduli space of $(n+1)$ distinct ordered points in $\C P^1$.

A change of coordinates not necessarily preserving $0$ also gives an isomorphism of vector spaces.

Think of this giving us a parallel transport corresponding to some connection on this vector bundle.

So, ``nice'' differential equations means something coming from a flat connection.

$n=3$ (4 points).  $M_{0,4} \simeq \C P^1 - \{ 0,1,\infty\}$.

The KZ equation in Wasserman is this construction on $M_{0,4}$.

Other KZ equations in the literature:  consider the configuration space of $n$ points in $\C$, $\C^n - \Delta$.  Take $\cup \{ \infty\}$ to get $M_{0,n+1}$.  

The setup for KZ:  $G=SU(N)$; $\g$; $X_k$ is orthonormal basis for $\g$.  

The {\em Casimir} $\Omega$ (not to be confused with vacuum!) is $\Sigma X_k \tensor X_k$.

\begin{lemma} $\Omega$ is in the center of $\mathcal{U}(\g)$ (universal enveloping algebra).
\end{lemma}

$\Delta_\lambda$ is the scalar by which $\Omega$ acts of $V_\lambda$.

Let $\mathcal{U}=\Hom(V_{\lambda_1} \tensor W_1 \tensor W_2 \otimes \cdot \tensor W_{n-1},V_{\lambda_2})$

$\g$ acts on each vector space inside there.

$\Omega_{i,j}= - \Sigma \pi_i(X_k) \tensor \pi_j(X_k) \in \operatorname{End}(\mathcal{U})$; the Casimir element acting separately on the $i,j$ components.

\begin{theorem}
$F_\mu$ satisfies
$$ (N+\ell) \frac{d F_\mu}{dz } = (\frac{\Omega_{23}-(\Delta_\mu-\Delta_{w_1} - \Delta_{w_2})/2}{z} + \frac{\Omega_{12}}{z-1})F
$$
\end{theorem}

\begin{theorem}
$f_\mu=\zeta^{(\Delta_\mu-\Delta_{w_1} - \Delta_{w_2})/(2(N+\ell))} F_{\mu}(\zeta)$ satisfies
$(N+\ell)\frac{df_\mu}{dz} = (\frac{\Omega_{23}}{\zeta} + \frac{\Omega{12}}{\zeta-1})f_\mu$
\end{theorem}


\end{document}