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\begin{document}

\title{Primary fields, boundedness of smeared primary fields
}
 \author{Speaker:  Arturo Prat-Waldron
 \\ Typist:%%%
 }
 \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!

 Goal:  $H_\lambda, H_\mu \in$ IPER; we want to study their Connes fusion, relatedly their intertwiners.  These are an honest rep of $\tilde{LG\rtimes\Trot}$.  $I \subset S^1$.
 
 We want to study $\Hom_{L_I G}^{bndd}(H_\lambda, H_mu)$ and get explicit elements.
 
 \begin{example}
 $G=SU(N)$, $V=\C^N$ the vector representation.  $H=L^2(S^1,V)$ and $Cliff(H) \acton \cF_V$
 
 $LG \rtimes \Trot \rightarrow U_{res}(H) \rightarrow PU(\cF_V)$
 
 We have $H_\lambda, H_\mu \subset \cF_V^{\tensor \ell}$ and $P_\lambda$, $P_\mu$ projections onto these subspaces.
 
 Given $f \in L^2(S^1,V)$, send it to creation $a(f) \in B(\cF_V^{\tensor \ell})$.
 
 we have an equivariance condition, $\pi(g) a(f) \pi(g)^*=a(gf)$, with $g \in \tilde{LG \rtimes \Trot}$ and $\pi: \tilde{LG \rtimes \Trot}\rightarrow U(\cF_V^{\tensor \ell})$.

 
 Define $\phi_{\lambda,\mu}(f)=P_\mu a(f) P_\lambda^* \in \Hom^{bdd}(H_\lambda, H_\mu)$.
 
  If $g \in \tilde{L_I G}$ and $supp(f) \subset I^c$ then $gf = g$; So
 $\pi(g)\phi(f)\pi(g)^*=\phi(f)$.  This implies $\phi(f) \in \Hom_{\tilde{L_I G}}(H_\lambda, H_\mu)$; we also have boundedness, $\norm{\phi(f)} \leq \norm{f}_{L^2}$.
 
 This gives us a procedure for construction bndd intertiwners; start with a function, construct an operator; as long as it's equivariant, we get an element of the intertwining space.
 
  \end{example}

 
 \begin{defn}
 fields take functions on $S^1$ and produce an element of the intertwiner $\Hom(H_\lambda, H_\mu)$.  {\em Primary} means they are $LG \rtimes \Trot$ equivariant.
 \end{defn}
 
\begin{defn}
$V$ is a $G$-module, $H_\lambda$, $H_\mu$ are IPERs of level $\ell$.  Define $V^{fin}:=V[z,z^{-1}]$ which is acted on by 
$L^{poly} \g_\C \rtimes i \R d$.  A {\em primary field of charge $V$ and level $\ell$} is a linear map
$$ \phi:  V[z,z^{-1}] \tensor H_\lambda^{fin} \rightarrow H_\mu^{fin}
$$
which is $L^{pol}\g_\C \rtimes i \R d$ equivariant.  

\end{defn}

$H^{fin}_\lambda = \DirectSum_{N\geq 0}^{alg} H_\lambda(n)$ are modes of the primary field $\phi$.  $\phi(v,n):=\phi(v\tensor^n):  H_\lambda^{fin} \rightarrow H_\mu^{fin}$.  The $L^{pol}\g_\C \rtimes i \R d$ equivariance is equivalent to

(1) $[X[n],\phi(v,m)] = \phi(Xv,m+n)$
(2) $[d,\phi(v,n)] = -n \phi(v,n)$

For $X \in \g_\C$, $X(n)=X \tensor z^n \in L^{pol} \g_\C$.

(2) implies $\phi(v,n)$ lowers energy by $n$:
$\phi(v,n):  H_\lambda(k) \rightarrow H_\mu(k-n)$.
(check:   $d(\phi(v,n)\xi) = \phi(v,n) d\xi - n \phi(v,n)\xi=(k-n)\phi(v,n)\xi$.)

In particular, $\phi(v,0):H_\lambda(0) \rightarrow H_\mu(0)$.

$\phi_0: V \tensor V_\lambda \rightarrow V_\mu$ in the initial term of $\phi$.

Equation (1) implies equivariance w.r.t.~$\g$, which impies $\phi_0 \in \Hom_G(V \tensor V_\lambda, V_\mu)$.

\begin{lemma}
The map $$\Hom_{L^{pol}\g_\C \rtimes i \R d}(V^{fin} \otimes H_\lambda^{fin}, H_\mu^{fin}) \rightarrow \Hom_G(V \tensor V_\lambda, V_\mu)$$
coming from $\phi \mapsto \phi_0$ is injective.
\end{lemma}

\begin{proof}
$H_\lambda^{fin}$ is generated by the $H_\lambda(0)$ operators as a $L^{pol}(\g_\C)$-module.
\end{proof}

Denote the image of $\Hom_{L^{pol}\g_\C \rtimes i \R d}(V^{fin} \otimes H_\lambda^{fin}, H_\mu^{fin})$ in $\Hom_G(V \tensor V_\lambda, V_\mu)$ by $\Hom_G^\ell(V \tensor V_\lambda, V_\mu)$.

\begin{prop}
$\Hom_G^\ell(V \tensor V_\lambda, V_\mu)=\Hom_G(V_\lambda \tensor V_\mu,V_\nu)$ if $ \lambda, \mu,\nu$ are admissible at level $\ell$ and at least one is {\em minimal} $G$-module:  means highest weight is dominant.  {\color{blue} (audience doesn't think that's what this means.  Miniscule?  something about exterior algebra?)}

\end{prop}

\section{$G=SU(N)$.  Vector primary fields of $SU(N)$: }

\begin{defn}
vector=charge of primary field is the vector rep $V=V_\Box=\C^N$.
\end{defn}

If $f,g$ are signatures of admissible $G$-modules of level $\ell$, we want to study $\Hom_{SU(N)}(V \otimes V_f, V_g)$.

Consider $V_\Box \tensor V_f = \DirectSum_{g>f} V_g$ where $g$ can be obtained from $f$ by adding one box.

Let $W=V \tensor \C^\ell$; $V$ injects into $W$.  $\Lambda W=(\Lambda V)^{\tensor \ell}$ and 
\begin{align*}
S:W \tensor (\Lambda W) & \rightarrow \Lambda W \\
w \tensor x & \mapsto w \wedge x
\end{align*}
$V_f, V_g \subset (\Lambda V)^\ell$.

\begin{lemma}
Let $T \in \Hom_{SU(N)}(_\Box \tensor V_f, V_g)$ and $T \neq 0$.  WE can find $SU(N)$-equivariant projections
$P_\Box: W \rightarrow V_\Box$, $P_f: W \rightarrow V_f$, $P_g: W \rightarrow V_g$, such that
$T=P_g S(P_\Box^* \tensor P_f^*)$.
\end{lemma}

\begin{proof}
consider a signature $f=(f_1 \geq f_2 \geq \cdots \geq f_N)$ which is admissible, ie $f_1 - f_N \leq \ell$.  Let $(e_1, \ldots e_N )$ be a basis of $V_\Box$.

We let $e_f = e_1^{\tensor(f_1 - f_2)} \tensor (e_1 \wedge e_2)^{\tensor(f_2 - f_3)} \cdots (e_1 \wedge \cdots \wedge e_N)^{\tensor(\ell-f_1+f_N)}$.

for $e_g$ we have a similar formula; the only way we can get a non-zero intertwiner is when $g$ has one more box than $f$.  ie, $g_i = f_i$ if $i \neq k$ and $g_k=f_k+1$.  We get from $e_f$ to $e_g$ by adding $\pm e_k$ in the $(f_1-f_k)$ copy of $\Lambda V \subset (\Lambda V)^{\tensor \ell}$.

\begin{example}
(On camera)
\end{example}

Now take $P_\Box:W = V_\Box \tensor \C^\ell = \DirectSum_{\ell} V_\Box \rightarrow V_\Box$, projection onto the $(f_1-f_k)$ copy of $V_\Box$.  

We have $SU(N) e_f \hookrightarrow V_f \subset \Lambda W$, similarly for $G$, and $P_f$ or $P_g$ going the other direction:  $\Lambda W \rightarrow V_f$.  

$S(P_\Box^* \tensor):V_\Box \tensor (\Lambda W) \rightarrow \Lambda W$ (recall $\Lambda(W)=(\Lambda V)^{\tensor \ell}$) which implies esterior multiplication by an element of $V_\Box$ in the $f_1 - f_k$ copy of $\Lambda V \subset (\Lambda V)^{\tensor \ell}$.

\end{proof}

\begin{theorem}
Any $SU(N)$-intertwiner $\phi_0: V_\Box \tensor H_\lambda(0) \rightarrow H_\mu(0)$ is the initial term of a unique vector primary field.  All vector primary fields arise as ``compressions of fermions'' so satisfy the $L^2$ bound $\norm{\phi(f)}\leq c \norm{f}_2$ for all $f \in V[z,z^{-1}]$.
the map extends continuously to $L^2(S^1,V_\Box)$ and satisfies the global equivariance relation $\phi_\mu(G) \phi(f) \pi_\lambda(g)^*=\phi(gf)$ for all $g \in \tilde{(LG \rtimes \Trot)}_\ell$
\end{theorem}

\begin{proof}
$\Lambda W = \cF_W(0)$, $W \subset L^2(S^1, W)$ as constant functions --  $V_\lambda, V_\mu \subset W$, $H_\lambda, H_\mu \subset \cF_W$ giving projections $P_\lambda$, $P_\mu$.

Define 
$\phi_{\lambda, \mu}(v,n):=P_\mu a(v \tensor z^n) P_\lambda^*$.
\end{proof}

Punchline:  if I have a smeared primary field, I evaluate it on a complementary interval to get one of these intertwiners.


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