A representation theoretic approach to the WZW Verlinde formula

37 pages, LaTeX2e. wrong statement in subsection 4.2 corrected and
  rest of the paper adapted

Abstract:
  By exploring the description of chiral blocks in terms of co-invariants, a
derivation of the Verlinde formula for WZW models is obtained which is entirely
based on the representation theory of affine Lie algebras. In contrast to
existing proofs of the Verlinde formula, this approach works universally for
all untwisted affine Lie algebras. As a by-product we obtain a homological
interpretation of the Verlinde multiplicities as Euler characteristics of
complexes built from invariant tensors of finite-dimensional simple Lie
algebras. Our results can also be used to compute certain traces of
automorphisms on the spaces of chiral blocks. Our argument is not rigorous; in
its present form this paper will therefore not be submitted for publication.

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\begin{document}

\begin{flushright}  {~} \\[-15 mm]  {\sf hep-th/9707069} \\[1mm]
{\sf May 1998 (revised)} \\[1 mm] {\sf (original version: CERN-TH/97-152)}
\end{flushright}

\begin{center} \vskip 13mm
{\Large\bf A REPRESENTATION THEORETIC APPROACH}\\[2.8mm]
{\Large\bf TO THE WZW VERLINDE FORMULA }\\[16mm]
{\large J\"urgen Fuchs} \\[3mm]
DESY\\[.6mm] Notkestra\ss e 85, \ D -- 22603~~Hamburg\\[11mm]
{\large Christoph Schweigert} \\[3mm] CERN \\[.6mm] CH -- 1211~~Gen\`eve 23
\end{center}
\vskip 20mm

\begin{quote}{\bf Abstract}\\
By exploring the description of chiral blocks in terms of co-invariants,
a derivation of the Verlinde formula for WZW models is obtained which
is entirely based on the representation theory of affine Lie algebras.
In contrast to existing proofs of the Verlinde formula, this approach
works universally for all untwisted affine Lie algebras. As a by-product
we obtain a homological interpretation of the Verlinde multiplicities
as Euler characteristics of complexes built from invariant tensors of
finite-dimensional simple Lie algebras.\\
Our results can also be used to compute certain traces of automorphisms
on the spaces of chiral blocks.
\end{quote}
\newpage

\Sect{Introduction and summary}I

Spaces of chiral blocks are finite-dimensional\ vector spaces that arise naturally
in the study of moduli spaces of flat connections over complex curves;
therefore they emerge in various contexts in physics and mathematics.
These vector spaces form a vector bundle over the moduli space of
smooth projective complex curves with marked points.
The aim of this paper is to further elucidate the
structure of such spaces by using the rep\-re\-sen\-ta\-ti\-on\ theory of affine Lie algebra s.

In physics, spaces of chiral block s appear in the following guises.
In three-dimensional topological Chern$\mbox{-\hspace{-.66 mm}-}$ Simons gauge theories
with space-time equal to the product of $\dl R$\ (describing time) and a
complex curve, they arise as the spaces of physical states
that are obtained when quantizing the theory in the temporal gauge.
In two-di\-men\-si\-o\-nal\ conformal field theory\ the chiral blocks are the basic constituents
of correlation functions, which are the quantities of prime interest in
any quantum field theory.
More precisely, correlation functions on closed orientable Riemann surfaces are
obtained as bilinear combinations of chiral block s, while correlators on surfaces
that have boundaries and\,/\,or are unorientable can be expressed in terms
of linear combinations of chiral block s. Correlators satisfying boundary conditions
that correspond to the presence of so-called $D$-branes are expressible in
terms of chiral block s, or close relatives thereof, as well.

In this paper we consider the spaces of chiral block s which are associated to
WZW (Wess\hy Zu\-mi\-no\hy Wit\-ten) conformal field theories. A WZW theory is specified by
the choice of a finite-dimensional\ simple Lie algebra\ \mbox{$\bar{\mathfrak  g}$}\ and a positive integer {\rm k}. In this
case of our interest each of the marked points of the curve is labelled with
an integrable weight at level ${\rm k}$ of the untwisted affine Lie algebra\ \mbox{$\mathfrak  g$}\ that is
associated to \mbox{$\bar{\mathfrak  g}$}\ via the loop construction. The chiral block s of WZW theories\
are also of interest in algebraic geometry.
Namely, the space of WZW chiral block s can be interpreted as the space of holomorphic
sections in the {\rm k}-th tensor power of a line bundle over the
moduli space of flat \mbox{$\bar{\mathfrak  g}$}-connections over the curve. Chiral WZW blocks can
therefore be regarded as non-abelian generalizations of theta functions. In this
paper we use the fact that the space of chiral block s can be described in terms of
co-in\-va\-ri\-ant s of certain integrable modules over \mbox{$\mathfrak  g$}\ (this characterization of
chiral block s will be reviewed in \ssref d). Based on this description we can
apply tools from the rep\-re\-sen\-ta\-ti\-on\ theory of the affine Lie algebra\ \mbox{$\mathfrak  g$}\ to study the
structure of chiral block s.  As a crucial ingredient we will introduce a suitable
central extension of the so-called block algebra\ $\mbox{$\bar{\mathfrak  g}$}({\cal C})$, which by definition
(for details see subsection \ref{gf}) consists of the \mbox{$\bar{\mathfrak  g}$}-valued algebraic
functions on the (punctured) curve $\cal C$.

The most fundamental information about a sheaf of chiral block s is its rank; there
exist closed expressions for this quantity, which are commonly referred to as
{\em Verlinde formul\ae\/} \cite{verl2,sorg}. Owing to factorization theorems
\cite{tsuy,falt,beau,tele2}, the problem of computing these numbers can be
reduced to the case of a curve of genus zero. Accordingly it will be assumed
throughout this paper that the genus is zero.
The main result of this paper is a purely rep\-re\-sen\-ta\-ti\-on-theoretic argument
for deriving the Verlinde formula for chiral blocks at genus zero.
In contrast to existing algebraic proofs of the Verlinde formula (which we will
briefly list in \sref S) we work as long as possible in the framework
of infinite-dimensional\ Lie algebra s. This enables us to obtain the Verlinde formula in a
uniform manner for {\em all\/} choices of the underlying finite-dimensional\ simple Lie algebra\
\mbox{$\bar{\mathfrak  g}$}, including the cases of $\mbox{$\bar{\mathfrak  g}$}=F_4,\,E_6,\,E_7,\,E_8$ for which no rigorous
algebraic proof had been known so far. Another advantage of our approach
is that we can derive the formula for an arbitrary number of marked points
without invoking (genus-preserving) factorization rules; this can be
interpreted as an independent check of these factorization rules at fixed
genus zero. As a further by-product we obtain a description of the spaces of
chiral block s in terms of a complex of invariant tensors of \mbox{$\bar{\mathfrak  g}$}\ with vanishing
Euler characteristic.

The structure of our approach is as follows. We first determine
the spaces of (genus zero) two-point blocks by making use of the explicit form
of the two-point block algebra; this is done in \sref T. In \sref A
the problem for an arbitrary number of marked points is reduced to the
case of two marked points and to the calculation of certain branching rules.
We then derive an integral formula for the latter and use it to find an
integral formula for the dimension of the space of chiral block s.
When doing so, we have to make an assumption about
the existence of a suitable completion of the modules. Finally,
in \sref F, we employ a generalized Poisson resummation rule to cast the
integral formula into the usual form of the Verlinde formula, i.e.\ as a finite
sum over elements of the matrix $S$ that describes the modular transformation
properties of the characters (which are the one-point blocks on the torus).

We conclude the paper in \sref S with some remarks which set our approach into
the context of related work. In particular we present a complex of co-in\-va\-ri\-ant s
that characterizes the chiral block s, which has the property that the vanishing of
its Euler characteristic is equivalent to the Verlinde formula.
We also comment briefly on a possible extension to non-unitary theories.
In \sref C the necessary rep\-re\-sen\-ta\-ti\-on-theoretic background is summarized.
Some technical aspects have been relegated to appendices.

\Sect{Chiral blocks as co-in\-va\-ri\-ant s}C

\subsection{The algebra s \mbox{$\mathfrak  g$}\ and \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}}\label{gf}

A basic ingredient in the definition of chiral block s is
an untwisted affine Lie algebra\ \mbox{$\mathfrak  g$}. (In conformal field theory, the semidirect sum of \mbox{$\mathfrak  g$}\ with the Virasoro algebra\
plays the r\^ole of the chiral symmetry algebra\ of WZW theories.) For the
purposes of this paper we regard \mbox{$\mathfrak  g$}\ as the centrally extended loop algebra
  \begin{equation}  {\mathfrak  g}:= {\bar{\mathfrak  g}}\otimes{\dl C}\twobrac t \oplus {\dl C} K  \,,  \labl g
where ${\dl C}\twobrac t$ denotes the ring of Laurent series in some
indeterminate $t$ and \mbox{$\bar{\mathfrak  g}$}\ is a finite-dimensional\ simple Lie algebra,
which is isomorphic to, and will be identified with, the horizontal (i.e.
zero mode) subalgebra of \mbox{$\mathfrak  g$}. The Lie bracket relations of \mbox{$\mathfrak  g$}\ read $[{\bar x}\raisebox{.07em}{$\scriptstyle\otimes$} f,
\bar y\raisebox{.07em}{$\scriptstyle\otimes$} g]\,{=}\,[{\bar x},{\bar y}]\raisebox{.07em}{$\scriptstyle\otimes$} fg + \kappa({\bar x},{\bar y})\,{\rm Res}_0({\rm d}f\,g)\,K$
for ${\bar x},{\bar y}\,{\in}\,\mbox{$\bar{\mathfrak  g}$}$ and $f,g\,{\in}\,{\dl C}\twobrac t$ (here $\kappa$ denotes
the Killing form of \mbox{$\bar{\mathfrak  g}$}) and $[K,{\bar x}\raisebox{.07em}{$\scriptstyle\otimes$} f]\eq0$,
i.e.\ $K\,{\in}\,{\mathfrak  g}$ is a central element. We also introduce the subalgebra s
  \begin{equation}  {\gM_{}^+}:={\bar{\mathfrak  g}}\otimes t\,{\dl C}[[t]] \,, \qquad
  {\gM_{}^-}:={\bar{\mathfrak  g}}\otimes t^{-1}{\dl C}[t^{-1}]    \labl{gpm}
of \mbox{$\mathfrak  g$}\ (${\dl C}[t]$ and ${\dl C}[[t]]$ denote polynomials and
arbitrary power series in $t$, respectively);
then as a vector space \mbox{$\mathfrak  g$}\ can be decomposed as
  \begin{equation}  {\mathfrak  g} \,=\, {\gM_{}^-} \oplus{\bar{\mathfrak  g}} \oplus {\dl C} K \oplus {\gM_{}^+} \,. \labl{gM}
The subalgebra s ${\mathfrak  g}^\pm$ must not be confused with the maximal nilpotent
subalgebras ${\mathfrak  g}_\pm$ that appear in the triangular decomposition
\begin{equation}  {\mathfrak  g} \,=\, {\mathfrak  g}_-\oplus\mbox{$\gM^{}_\circ$}\oplus{\mathfrak  g}_+  \Labl1t
of \mbox{$\mathfrak  g$}\ into the Cartan subalgebra\ \mbox{$\gM^{}_\circ$}\ and the
nilpotent subalgebra s ${\mathfrak  g}_\pm$ that correspond to the positive and negative
\mbox{$\mathfrak  g$}-roots, respectively; one has ${\mathfrak  g}_\pm\cong {\mathfrak  g}^\pm\,{\oplus}\,{\bar{\mathfrak  g}}_\pm$.

Concerning our characterization \erf g of \mbox{$\mathfrak  g$}, two remarks are in order.
First, apart from defining the grading that corresponds to the power of $t$,
in the present context the outer derivation $D=-L_0$ of the affine
algebra will not play any particular r\^ole.
Accordingly we did not include $D$ in the definition \erf g, even though it
is e.g.\ needed in order for \mbox{$\mathfrak  g$}\ to possess a non-degenerate invariant
bilinear form and roots of finite multiplicity \cite{KAc3}.
Second, strictly speaking it is the subalgebra\ $\oo{\gM}{1.6}{10.3}={\bar{\mathfrak  g}}\otimes
{\dl C}[t,t^{-1}] \oplus {\dl C} K$ of \mbox{$\mathfrak  g$}\ that is generated
when one allows only for Laurent polynomials rather than arbitrary Laurent
series which should be referred to as the affine Lie algebra. However,
as will be explained in the following subsection, we will only deal with
\mbox{$\mathfrak  g$}-rep\-re\-sen\-ta\-ti\-on s for which
every vector of the associated module (rep\-re\-sen\-ta\-ti\-on\ space) is annihilated by
all but finitely many generators of the subalgebra\ $\oo{\gM}{1.6}{10.3}{}_{}^+$ of $\oo{\gM}{1.6}{10.3}$.
As a consequence, any such rep\-re\-sen\-ta\-ti\-on\ of $\oo{\gM}{1.6}{10.3}$ can be naturally promoted
to a rep\-re\-sen\-ta\-ti\-on\ of the larger algebra\ \mbox{$\mathfrak  g$}. For the present purposes
the distinction between \mbox{$\mathfrak  g$}\ and $\oo{\gM}{1.6}{10.3}$ is therefore immaterial.
Note that unlike in the case of $\oo{\gM}{1.6}{10.3}$, the subalgebra s
${\gM_{}^+}$ and ${\gM_{}^-}$ of \mbox{$\mathfrak  g$}\ are not isomorphic;
in particular, ${\gM_{}^-}$ is a subalgebra\ of $\oo{\gM}{1.6}{10.3}$, while ${\gM_{}^+}$ is not.

The physical states of a WZW conformal field theory\ can be completely described in terms of the
rep\-re\-sen\-ta\-ti\-on\ theory of the affine algebra\ \mbox{$\mathfrak  g$}; e.g.\
the WZW primary fields $\Phi\,{\equiv}\,\Phi_\Lambda$ correspond to the highest weight vector s
of integrable irreducible highest weight module s \mbox{${\cal H}_\Lambda$}\ of \mbox{$\mathfrak  g$}.
However, the description of chiral blocks also involves
another infinite-dimensional\ Lie algebra, which we will call the {\emblock algebra\/} and
denote by \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}. The block algebra\ is the Lie algebra\ of \mbox{$\bar{\mathfrak  g}$}-valued algebraic functions on the
(punctured) curve. More precisely, it is defined as follows.
To any open subset $U$ of the Riemann sphere
\mbox{${\dl P}^1$}\ one associates the ring ${\cal F}(U)$ of algebraic functions on
$U$ and the vector space ${\gB(U)}:={\bar{\mathfrak  g}}\,{\otimes}\,{\cal F}(U)$; here and
below all tensor products are taken over the complex numbers, unless
stated otherwise. The open subset of interest to us
  \begin{equation}  {{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}} := \mbox{${\dl P}^1$} \setminus \{ p_1,p_2,...\,, p_{m} \}  \,,  \end{equation}
where $\{p_i\,{\mid}\, i\,{=}\,1,2,...\,,\mn\}$ is a finite set of pairwise distinct
non-singular points on \mbox{${\dl P}^1$}. The points
$p_i$ correspond to the positions of the WZW primary fields $\Phi_{\Lambda_i}$
whose correlation function is obtained from the chiral blocks we are interested
in; they are called the {\em insertion points\/} or the {\em parabolic
points\/}, and ${{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}}$ is referred to as a punctured Riemann sphere.
The corresponding vector space
  \begin{equation}  \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}:= {\bar{\mathfrak  g}}\otimes{\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})  \Labl25
becomes a Lie algebra\ when endowed with the natural bracket
  \begin{equation}  [{\bar x}\ot f,\bar y\ot g]:=[{\bar x},{\bar y}]\,\ot\, fg
  \qquad{\rm for}\;\ {\bar x},{\bar y}\,{\in}\,{\bar{\mathfrak  g}} \;\ {\rm and}\;\ f,g\,{\in}\,{\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}}) \,.
  \Labl xy
A basis ${\cal B}$ of the block algebra\ \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}\ is given by
  \begin{equation}  {\cal B} = \bar{\cal B} \times \mbox{\large(} \{z^0\} \cup \bigcup_{i=1}^m
  \{ \zz i^n\,{\mid}\, n\,{\in}\,\mbox{$\zet_{<0}$} \} \mbox{\large)} \,,  \end{equation}
where $\bar{\cal B}\,{=}\,\{{\bar x}^a\,{\mid}\, a\eq1,2,...\,,{\rm dim}\,{\bar{\mathfrak  g}}\}$ is a basis of
the Lie algebra\ \mbox{$\bar{\mathfrak  g}$}, $z$ is the global coordinate of ${\dl C}\,{\subset}\,\mbox{${\dl P}^1$}$, and
$z_i$ are the values of $z$ at the insertion points $p_i$.

As a side remark we mention that the block algebra \erf{25} admits a natural
central extension, which we will employ in \sref T. A similar remark applies
to the block algebras that arise when \mbox{${\dl P}^1$}\ is replaced by some
Riemann surface of higher genus. The corresponding centrally extended
Lie algebra s are known as higher genus affine Lie algebra s or as generalized
Krichever$\mbox{-\hspace{-.66 mm}-}$ Novikov algebras of affine type \cite{scHl1,shei3}.

\subsection{\mbox{$\mathfrak  g$}-modules}

For the applications we have in mind, we will need to consider modules
(rep\-re\-sen\-ta\-ti\-on\ spaces) over the block algebra which come from modules over the affine
Lie algebra\ \mbox{$\mathfrak  g$}. Most of the \mbox{$\mathfrak  g$}-modules that we are interested in here
share the specific properties that the action of
the Cartan subalgebra\ $\mbox{$\gM^{}_\circ$}$ can be diagonalized in such a way that the resulting weight
spaces are finite-dimensional\ when the full Cartan subalgebra\ (i.e.\ including the derivation $D$)
is considered, and such that each weight of the module can be obtained from
a finite set $\{\mu_\ell\}$ of weights by subtraction of a finite number of
positive \mbox{$\mathfrak  g$}-roots. The collection of all such \mbox{$\mathfrak  g$}-modules forms the objects of a
category, called the {\em category\/} ${\cal O}$ (see e.g.\ Chap.\,9.1 of
\cite{KAc3}); this category is closed under forming finite direct sums or
tensor products, submodules and quotients. Every module $V$ in ${\cal O}$
is in particular {\em restricted\/}, i.e.\ each element $v\,{\in}\, V$ is annihilated
by the step operators for all but a finite number of positive \mbox{$\mathfrak  g$}-roots;
moreover, the subalgebra\ ${\gM_{}^+}$ of \mbox{$\mathfrak  g$}\ acts locally nilpotently.

Among the modules in ${\cal O}$ there are in particular the highest weight module s $V$
for which by definition
there exists a highest weight vector \mbox{$v^{}_\Lambda$}\ which is annihilated by \mbox{$\gM^{}_+$}, i.e.\
$\mbox{$\mathfrak  g$}_\alpha\mbox{$v^{}_\Lambda$}\eq0$ for all positive \mbox{$\mathfrak  g$}-roots $\alpha$, which is an eigenvector
of \mbox{$\gM^{}_\circ$}, i.e.\ $h\mbox{$v^{}_\Lambda$}\,{=}\,\Lambda(h)\mbox{$v^{}_\Lambda$}$ for all $h\,{\in}\,\mbox{$\gM^{}_\circ$}$, and for which the
action of \mbox{${\mathfrak  g}^{}_-$}\ yields the whole module, ${\sf U}(\mbox{${\mathfrak  g}^{}_-$})\mbox{$v^{}_\Lambda$}= V$.
In particular, for highest weight module s the set $\{\mu_\ell\}$ of distinguished weights
contains only a single element, the highest weight\ $\Lambda$. Every \mbox{$\mathfrak  g$}-module with highest weight\
$\Lambda$ can be obtained as a suitable quotient from the {\emVerma module\/}
${\cal V}_\Lambda\,{:=}\, {\sf U}({\mathfrak  g}){\otimes_{{\sf U}({\mathfrak  g}^{}_+\oplus{\mathfrak  g}^{}_\circ)}}\mbox{$v^{}_\Lambda$}$,
which as a $\mbox{$\mathfrak  g$}_-$-module is isomorphic to the free module
${\sf U}({\mathfrak  g}_-){\otimes_\complex^{}}\mbox{$v^{}_\Lambda$}$.

The space of physical states of a (chiral) WZW theory\ is the direct sum of
integrable irreducible highest weight module s \mbox{${\cal H}_\Lambda$}\ of \mbox{$\mathfrak  g$}\ which all have one and the same eigenvalue of
the central generator $K\,{\in}\,{\mathfrak  g}$, i.e.\ the same level. Throughout the paper we
therefore keep some fixed value {\rm k}\ of the level. For integrability of
the module \mbox{${\cal H}_\Lambda$}, the highest weight\ $\Lambda$ must be dominant integral, which implies
that {\rm k}\ must be a non-negative integer, and the horizontal projection
${\bar\Lambda}$ of $\Lambda$ must lie in the set
  \begin{equation}  {{\rm P}_{\!\kV}} := \{ {\bar\Lambda}\,{\in}\,\mbox{$\bL_{\rm w}$} \,{\mid}\, ({\bar\Lambda},\bar\alpha^{(i)}_{})\,{\ge}\,0\;
  {\rm for\;all}\;i\eq1,2,...\,,{\rm rank}\,\mbox{$\bar{\mathfrak  g}$},\;({\bar\Lambda},{\bar\theta})\leq{\rm k} \}
  \,, \Labl pk
i.e.\ belong to the integral weights in the closure
of the dominant Weyl alcove at level {\rm k}\ (here \mbox{$\bL_{\rm w}$}\ denotes the weight lattice
of the horizontal subalgebra\ \mbox{$\bar{\mathfrak  g}$}, i.e.\ the lattice in \mbox{$\gB^\star_\circ$}\ that is spanned
by the fundamental \mbox{$\bar{\mathfrak  g}$}-weights, and $\bar\alpha^{(i)}$ are the simple roots
and ${\bar\theta}$ the highest root of \mbox{$\bar{\mathfrak  g}$}).
Note that ${{\rm P}_{\!\kV}}$ is a finite set (e.g.\ at level 0 there is
only a single integrable module, the trivial one-dimen\-sional\ module ${\cal H}_0$
with highest weight\ $\Lambda\eq0$).

In conformal field theory\ terms, the WZW chiral blocks are the chiral constituents of the
$m$-point correlation functions
  \begin{equation}  \mbox{\sf B}_{\{\Lambda_i\},\{p_i\}} = \langle\,
  \Phi_{\Lambda_1}(p_1)\,\Phi_{\Lambda_2}(p_2)\cdots \Phi_{\Lambda_m}(p_m)
  \,\rangle  \end{equation}
for primary WZW fields. These fields are associated with integrable irreducible highest weight module s
of \mbox{$\mathfrak  g$}. As a consequence, when studying chiral blocks, one has to
deal with a collection of irreducible highest weight module s \mbox{${\cal H}^{}_{\Lambda_i}$}\ ($i\,{=}\,1,2,...\,,\mn$) of \mbox{$\mathfrak  g$}\ which are at
level {\rm k}\ and satisfy ${\bar\Lambda}_i\,{\in}\,{{\rm P}_{\!\kV}}$,
and analyze the tensor product space (over $\dl C$)
  \begin{equation}  \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em:=\mbox{$\cal H$}_{\Lambda_1} \otimes \mbox{$\cal H$}_{\Lambda_2}
  \otimes\cdots\otimes\mbox{$\cal H$}_{\Lambda_m}  \,.  \labl{hvl}
This tensor product is in a natural way a module over the ${m}$-fold direct sum
${\mathfrak  g}^m\equiv{\mathfrak  g}\oplus{\mathfrak  g}\oplus\cdots\oplus{\mathfrak  g}$.

In addition to irreducible highest weight module s, we will occasionally also have to deal with
other objects in the category ${\cal O}$, namely with the
so-called {\em generalized\/} or {\em parabolic Verma modules\/}
  \begin{equation}  {\cal P}_\Lambda:= {\sf U}({\mathfrak  g}) \otimes^{}_{{\sf U}({{\liefont p}_{}^+})} \mbox{$\bar{\cal H}_{{\bar\Lambda}}$}  \,. \Labl VV
Here $\mbox{$\bar{\cal H}_{{\bar\Lambda}}$}$ denotes the irreducible \mbox{$\bar{\mathfrak  g}$}-module with highest weight\
${\bar\Lambda}$, while ${{\liefont p}_{}^+}$ is an arbitrary parabolic subalgebra of \mbox{$\mathfrak  g$}.
More specifically, we consider the case where
  \begin{equation}  {{\liefont p}_{}^+}={\bar{\mathfrak  g}} \oplus {\dl C} K \oplus {\gM_{}^+} \,.  \labl{pM}
Then ${\mathfrak  g}\,{=}\,{{\liefont p}_{}^+}\,{\oplus}\,{\gM_{}^-}$ with ${\gM_{}^-}\,{=}\,{\bar{\mathfrak  g}}\otimes t^{-1}{\dl C}[t^{-1}]$
as defined in \erf{gpm}, and for any integrable
\mbox{$\mathfrak  g$}-weight $\Lambda$ the Poin\-ca\-r\'e\hy Birk\-hoff\hy $\!$Witt theorem\ implies a natural isomorphism
  \begin{equation}  {\cal P}_\Lambda \equiv {\sf U}({\mathfrak  g}) \otimes^{}_{{\sf U}({{\liefont p}_{}^+})} \mbox{$\bar{\cal H}_{{\bar\Lambda}}$}
  \,\cong\, {\sf U}({\gM_{}^-}) \otimes \mbox{$\bar{\cal H}_{{\bar\Lambda}}$}  \labl{vuu}
of ${\gM_{}^-}$-modules. In particular, ${\cal P}_\Lambda$ is free as a
${\gM_{}^-}$-module.

Finally another class of modules will play a r\^ole, which are not restricted
and hence in particular not in the category ${\cal O}$, but still integrable.
These \mbox{$\mathfrak  g$}-modules, called {\em evaluation modules\/}, are finite-dimensional\ and of level
zero; we will encounter them in subsection \ref{sB}.

For later reference we also present the characters of some of the \mbox{$\mathfrak  g$}-modules of
our interest. The character of an integrable irreducible highest weight module\ \mbox{${\cal H}_\Lambda$}\ is given by the Weyl\hy Kac character formula
  \begin{equation}  \chii_\Lambda^{} = {\cal X} \cdot \sum_{w\in W} \epsilon(w)\,{\rm e}^{w(\Lambda+\rho)} \,,  \Labl wk
where the summation is over the Weyl group $W$ of \mbox{$\mathfrak  g$}, $\rho$ is the Weyl vector
of \mbox{$\mathfrak  g$}\ and
  \begin{equation}  {\cal X}:= \mbox{\large(} \sum_{w\in W} \epsilon(w)\, {\rm e}^{w(\rho)}_{} \mbox{\large)}^{-1}_{}
  = {\rm e}^{-\rho} \prod_{\alpha>0}(1-{\rm e}^{-\alpha})^{-{\rm mult}\,\alpha}_{}
  \,.  \labl{uvc}
Both ${\cal X}$ and the second factor in \Erf wk are totally antisymmetric under
the Weyl group $W$, so that $\chii_\Lambda^{}$ is $W$-invariant.
The character of the Verma module with highest weight\ $\Lambda$ is
${\rm e}^{\Lambda+\rho}{\cal X}$. Accordingly we will refer to the quantity \erf{uvc}
as the `universal Verma character' of \mbox{$\mathfrak  g$}.

\subsection{The tensor product ${\cal H}_{\vec\Lambda}^{\scriptscriptstyle(m)}$\ as a \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}-module}\label{HVL}

We would now like to endow the vector space \erf{hvl} with the structure
of a \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}-module. To this end we have to choose a local holomorphic
coordinate $\zeta_i$ around each insertion point $p_i$
such that $\zeta_i(p_i)\eq0$; for instance, in terms of the global coordinate
$z$ of ${\dl C}\,{\subset}\,\mbox{${\dl P}^1$}$, we can take $\zeta_i=z\,{-}\, z_i$ when $p_i\,{\not=}\,\infty$,
while for $p_i\,{=}\,\infty$ we can take $\zeta_i=z^{-1}$.

For any ${\bar x}\ot f$ with ${\bar x}\,{\in}\,{\bar{\mathfrak  g}}$ and $f\,{\in}\,{\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$ we expand $f$ in
these local coordinates
so as to obtain Laurent series $f_{p_i}\,{=}\,\fpi(\zeta_i)$. By linear extension
this defines, for each $i\iN\{1,2,...\,,{m}\}$, a ring homomorphism from ${\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$
to ${\dl C}\twobrac{\zeta_i}$, and hence by identifying the indeterminate $t$ of
the loop construction with the local coordinate $\zeta_i$, the local realizations
  \begin{equation}  x_i:={\bar x}\,\ot\, f_{p_i}  \labl{xi}
can be regarded as elements of the loop algebra
  \begin{equation}  \mbox{$\bar{\mathfrak  g}_{\rm loop}$} = {\bar{\mathfrak  g}}\otimes{\dl C}\twobrac t = {\gM_{}^-} \,{\oplus}\, {\bar{\mathfrak  g}} \,{\oplus}\, {\gM_{}^+}
  \,,  \end{equation}
and, as such, as elements of the affine Lie algebra\ \mbox{$\mathfrak  g$}\ \erf{gM}. Doing so, along with
any vector $v_i\,{\in}\,\mbox{${\cal H}^{}_{\Lambda_i}$}$ also $R_{\Lambda_i}(x_i) v_i$, where $R_{\Lambda_i}$ is
the \mbox{$\mathfrak  g$}-rep\-re\-sen\-ta\-ti\-on\ associated to the irreducible \mbox{$\mathfrak  g$}-module $\mbox{${\cal H}^{}_{\Lambda_i}$}$, is a vector in
\mbox{${\cal H}^{}_{\Lambda_i}$}. Moreover, even though the block algebra\ \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}\ is not centrally extended,
we can obtain a rep\-re\-sen\-ta\-ti\-on\ \mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em\ of \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}\ on \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em, namely by defining
the action of ${\bar x}\ot f\,{\in}\,\mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}$
on the element $v_1\ot v_2 \ot\cdots\ot v_m\,{\in}\,\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em$ by
  \begin{equation}   \mbox{\large(} \mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em({\bar x}\ot f)\mbox{\large)} (v_1\ot v_2\ot\cdots\ot v_m)
  := \sum_{i=1}^m v_1\ot v_2 \ot\cdots\ot\, R_{\Lambda_i}(x_i) v_i\,\ot
  \cdots\ot v_m \,.  \labl{rxf}
To verify that this yields a \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}-rep\-re\-sen\-ta\-ti\-on\ we compute
  \begin{equation}  \hsp{-2}\begin{array}{l}
  \mbox{\large(} \mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em({\bar x}\ot f)\mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em({\bar y}\ot g)\mbox{\large)} (v_1\ot v_2\ot\cdots\ot v_m)-
  \mbox{\large(} \mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em({\bar y}\ot g)\mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em({\bar x}\ot f)\mbox{\large)} (v_1\ot v_2\ot\cdots\ot v_m)
  \\{}\\[-.7em] \hsp{12}
  = \dsum_{i=1}^m v_1\ot v_2 \ot\cdots\,\ot\,
    [R_{\Lambda_i}(x_i),R_{\Lambda_i}(y_i)] v_i\,\ot\, \cdots\ot v_m
  \\{}\\[-.7em] \hsp{12}
  = \dsum_{i=1}^m v_1\ot v_2 \raisebox{.07em}{$\scriptstyle\otimes$}\cdots\,\ot\, R_{\Lambda_i}([{\bar x},{\bar y}]\raisebox{.07em}{$\scriptstyle\otimes$} f_{p_i}g_{p_i})
    v_i\,\ot\, \cdots\ot v_m \\[.2em] \hsp{13.6}
    + {\rm k}\,\kappa({\bar x},{\bar y})\, \mbox{\large(} \dsum_{i=1}^m {\rm Res}_{p_i}({\rm d}f\,g) \mbox{\large)}
    \, v_1\ot v_2 \ot\cdots\ot v_m \,, \end{array} \labl{xfyg}
where in the first equality we observed that terms acting on different tensor
factors of \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em\ cancel and in the second equality we inserted
the bracket relations of the affine Lie algebra\ \mbox{$\mathfrak  g$}.
Now the terms in \erf{xfyg} that involve the level {\rm k}\ cancel as a
consequence of the residue formula, while the other terms add up to
$\mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em([{\bar x}\ot f, {\bar y}\ot g])(v_1\ot v_2 \ot\cdots\ot v_m)$, where the
Lie bracket is the one of \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}\ as defined in \Erf xy. Hence as promised,
for any choice $\vec\zeta\,{\equiv}\,(\zeta_1,\zeta_2,...\,, \zeta_m)$ of local
coordinates at the parabolic points we have a rep\-re\-sen\-ta\-ti\-on\ $\mbox{$R_{\vec\Lambda}\!\!\!\raisebox{.5em=\mbox{$R_{\vec\Lambda;\vec\zeta}\hsp{-1.05}\raisebox{.55em$ of the block
algebra \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}.

Note that the cancellation of the terms coming from the central extension
only works if all modules \mbox{${\cal H}^{}_{\Lambda_i}$}\ have the same level. Under the same condition
the result does not only hold for irreducible highest weight module s, but analogously also for any module
on which the central element $K$ acts as a multiple of the identity,
and hence in particular for Verma module s and their quotients and for direct sums
of such modules.

\subsection{Co-in\-va\-ri\-ant s}

Before we can introduce chiral blocks, we need one more ingredient, the notion
of a co-in\-va\-ri\-ant. For any Lie algebra\ ${\mathfrak  h}$ we denote by ${\sf U}({\mathfrak  h})$ its universal enveloping
algebra and by
  \begin{equation}  \U^+({\mathfrak  h}):={\mathfrak  h}\,{\sf U}({\mathfrak  h})  \end{equation}
the augmentation ideal of ${\sf U}({\mathfrak  h})$. Then for any ${\mathfrak  h}$-module $V$, the quotient
module
  \begin{equation}  \coi V{\mathfrak  h} := V \,/\, \U^+({\mathfrak  h})\,V   \labl,
is known as the {\em space of co-in\-va\-ri\-ant s\/} of $V$ with respect to  ${\mathfrak  h}$.
(Strictly speaking, in place of \erf, one should write
  \begin{equation}  \coi V{\mathfrak  h} := V \,/\, R(\U^+({\mathfrak  h}))\,V  \,,  \end{equation}
where $R$ denotes the rep\-re\-sen\-ta\-ti\-on\ by which ${\mathfrak  h}$ acts on the module $V$. Different
actions of ${\mathfrak  h}$ on one and the same underlying vector space will of course
give rise to different spaces of co-in\-va\-ri\-ant s.)

Let us list briefly a few basic facts about co-in\-va\-ri\-ant s (for further properties
of co-in\-va\-ri\-ant s see Appendix \ref{acoin}).\\
$\bullet$\,~The vector space $\coi V{\mathfrak  h}$ can be characterized as the largest
quotient module of $V$ on which ${\mathfrak  h}$ acts trivially. \\
$\bullet$\,~The space of co-in\-va\-ri\-ant s of the tensor product $V \,{\otimes}\, W$ of two
${\mathfrak  h}$-modules $V$ and $W$ equals their tensor product over ${\sf U}({\mathfrak  h})$:
  \begin{equation}  \coi{V\,{\otimes}\, W}{\mathfrak  h}\,{\equiv}\,\coi{V\,{\otimes_\complex^{}}\,W}{\mathfrak  h} = V\,{\otimes^{}
  _{{\sf U}({\mathfrak  h})}}\,W \,.  \end{equation}
Indeed, both spaces are by definition equal to the quotient of
$V \,{\otimes}\, W$ by the subspace that is spanned by the vectors $(xv)\ot w+
v\ot(xw)$ with $v\,{\in}\, V$, $w\,{\in}\, W$ and $x\,{\in}\,{\mathfrak  h}$.\,%
 \futnote{In the definition of the tensor product $\otimes^{}_{{\sf U}({\mathfrak  h})}$ we
have to include the canonical anti-involution of ${\sf U}({\mathfrak  h})$ that is
defined by ${\bf1}\mapsto{\bf1}$ and $x\mapsto-x$ for all $x\,{\in}\,{\mathfrak  h}$, so as
to obtain the structure of a right ${\sf U}({\mathfrak  h})$-module on $V$.}
\\
$\bullet$\,~The concept of co-in\-va\-ri\-ant s generalizes the notion of
invariant tensors to the case of non-fully reducible modules. More precisely,
when ${\mathfrak  h}$ is a finite-dimensional\ semi-simple Lie algebra\ and $V$ is fully reducible, then
$\coi V{\mathfrak  h}$ is isomorphic to the space of invariant tensors of $V$. Thus
the dimension of $\coi V{\mathfrak  h}$ is given by the number of singlets
contained in $V$; in particular, one has the formula
  \begin{equation}  {\rm dim}\, \mbox{\large(} \coi{\mbox{$\bar{\cal H}_{{\bar\Lambda}}$} \,{\otimes}\, \mbox{$\bar{\cal H}_{{\bar\Lambda}'}$}}{\mathfrak  h}\mbox{\large)}
  = \delta^{}_{{\bar\Lambda}',{\bar\Lambda}^+_{\phantom|}}  \Labl0f
for the ${\mathfrak  h}$-co-in\-va\-ri\-ant s of the tensor product of two finite-dimensional\ highest weight module s over ${\mathfrak  h}$;
a distinguished representative for the corresponding non-trivial co-in\-va\-ri\-ant\ is
given by $v_{\bar\Lambda}\raisebox{.07em}{$\scriptstyle\otimes$} v_{-(\bar\Lambda')^+}$, where $v_\Lambda$ is the
highest weight vector of $\mbox{$\bar{\cal H}_{{\bar\Lambda}}$}$ and $v_{-(\bar\Lambda')^+}$ the lowest
weight vector of $\mbox{$\bar{\cal H}_{{\bar\Lambda}'}$}$.

\subsection{The definition of chiral blocks}\label{ss.d}

We are now in a position to give a precise definition of the chiral blocks
\cite{tsuy,falt,beau,beLa,fesv2}. We have seen that any choice $\vec\zeta\,
{\equiv}\,(\zeta_1,\zeta_2,...\,, \zeta_m)$ of local coordinates at the parabolic
points leads to a representation \mbox{$R_{\vec\Lambda;\vec\zeta}\hsp{-1.05}\raisebox{.55em\ of the block algebra $\mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}$ on \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em.
We start by introducing the spaces
  \begin{equation}  \mbox{$B$}_{\vec\zeta} = \coi\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em{{R^{\scriptscriptstyle(m)}_{\vec\Lambda;\vec\zeta}}({\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})})} \equiv \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em \,/\,
  \mbox{$R_{\vec\Lambda;\vec\zeta}\hsp{-1.05}\raisebox{.55em(\U^+(\mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}))\, \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em \,  \Labl B'
of co-in\-va\-ri\-ant s. Now these spaces of course depend on the choice of local
coordinates $\vec\zeta$.
On the other hand, as already mentioned the chiral blocks play the r\^ole of
the building blocks for correlation functions in conformal field theory; it is a fundamental
physical requirement that those correlation functions
should depend covariantly on the choice of coordinates. Here we will impose the
natural stronger requirement that even the chiral blocks transform covariantly
under a change of the local coordinates at the parabolic points.

To get rid of the coordinate dependence we make use of the group
  \begin{equation} U := \{u\,{\in}\,{\dl C}[[z]] \,|\, u(0)\eq0,\, \Frac{{\rm d}u}{{\rm d}z}(0)\nE0
  \} \end{equation}
of local coordinate changes. The group $U^m\equiv U\,{\times}\, U\,{\times}\,\cdots\,{\times}\, U$
acts transitively on the set of all collections $\vec\zeta$ of local
coordinates by sending each local coordinate $\zeta_i$ to the local
coordinate $\zeta_i\,{\circ}\, u_i$ with $u_i\,{\in}\, U$. Moreover, for any $\vec u\,{\in}\, U^m$
one can find a map $\Gamma_{\!\vec u}{:}\ \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em\,{\to}\,\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em$ with the property that
  \begin{equation}  \Gamma_{\!\vec u}\, \mbox{$R_{\vec\Lambda;\vec\zeta}\hsp{-1.05}\raisebox{.55em(x) = \mbox{$R_{\vec\Lambda;\vec\zeta\circ\vec u}\hsp{-1.85}\raisebox{.55em(x)\, \Gamma_{\!\vec u} \end{equation}
for all $x\,{\in}\,\mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}$, and this map is unique up to a scalar.
Now via the local realizations \Erf xi (obtained by identifying the local
coordinates at the parabolic points with the indeterminate of the loop
construction for the relevant summand of ${\mathfrak  g}^m\equiv{\mathfrak  g}\oplus{\mathfrak  g}\oplus\cdots
\oplus{\mathfrak  g}$) one associates to any choice $\vec\zeta$ of local coordinates a
subalgebra\ $\gM_{\vec\zeta}\!\!\raisebox{.5em}{$\scriptscriptstyle(m)$}$ of ${\mathfrak  g}^m$
which is isomorphic to the block algebra\ \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}. The map $\Gamma_{\!\vec u}$
has the property that it restricts to an isomorphism
  \begin{equation}  \Gamma_{\!\vec u}:\quad \gM_{\vec\zeta}\!\!\raisebox{.5em}{$\scriptscriptstyle(m)$} \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em\stackrel\cong\to \gM_{\vec\zeta\circ\vec u}\hsp{-1.1}\raisebox{.5em}{$\scriptscriptstyle(m)$\hsp{.3}} \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em  \end{equation}
and therefore induces a map on spaces of co-in\-va\-ri\-ant s taken with respect
to the different actions of the block algebra on the space \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em\
that are associated to the different choices of local coordinates.

We now interpret the chiral blocks as the {\em equivalence classes\/} of co-in\-va\-ri\-ant s
under the action of the group $U^m$, and denote the blocks by
  \begin{equation}  \mbox{$B$} = \coi\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em{\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})} \equiv \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em \,/\, \U^+(\mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$})\, \mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em \,,  \labl B
where we think of $\mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}$ as the abstract block algebra\ ${\bar{\mathfrak  g}}\,{\otimes}\,{\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$
without specifying its embedding into  $\mbox{$\mathfrak  g$}^m$.
For the purposes of this paper, we regard this prescription as the definition
of the spaces $\mbox{$B$} \equiv \mbox{$B$}^{}_{\!\{\Lambda_i\}, \{p_i\}}$  of chiral blocks.

In conformal field theory\ terminology, taking \mbox{$\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$}-co-in\-va\-ri\-ant s corresponds to the procedure of
imposing the Ward identities of the current algebra\ on the correlation functions.
Let us also remark that in certain contexts it is actually more natural to
consider the dual of the space $\coi\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em{\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})}$, i.e.\ the space of
${\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})}$-invariants in the (algebraic) dual of $\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em$. As already mentioned, in
algebraic geometry and in quantum Chern$\mbox{-\hspace{-.66 mm}-}$ Simons theory, chiral block s describe
holomorphic sections in line bundles over moduli spaces; the relation to
invariants is via infinite-dimensional\ Borel$\mbox{-\hspace{-.66 mm}-}$ Weil$\mbox{-\hspace{-.66 mm}-}$ Bott theory in which the full
algebraic dual of a highest weight module appears. Here we are only interested
in the dimensions, which are finite and hence are the same for invariants and
co-in\-va\-ri\-ant s.

Manifestly, the definition \erf B does not depend on the choice of local
coordinates. Moreover, any two spaces of chiral blocks with the same number of
insertion points and the same highest weights are isomorphic.
(The relevant isomorphism is canonical when there is a conformal mapping
of \mbox{${\dl P}^1$}\ that maps the two sets of insertion points bijectively on each other.)
Later on we will, however, often work with specific representatives, i.e.\
prescribe a specific choice of coordinates.  In this paper, we will focus on a
genus zero curve for which a (quasi-)\,global holomorphic
coordinate exists. Any choice of such a global coordinate gives a natural
set of local coordinates at the parabolic points.

\Sect{Two-point blocks}T

The Verlinde formula provides a closed expression for the dimensions ${\rm dim}\,\mbox{$B$}$
of spaces of chiral blocks. To enter the calculation of such dimensions,
let us first investigate the special situation where the number of insertion
points is $m\eq2$. Recall that
by definition the chiral blocks are independent of the choice of the
global holomorphic coordinate $z$. For definiteness, we will work with specific
representatives of the blocks, namely by choosing the global coordinate in
such a way that the two insertion points are at $z\eq0$ and at $z\,{=}\,\infty$.
Then the block algebra is nothing but the (polynomial) loop algebra
  \begin{equation}  \mbox{$\bar{\mathfrak  g}$}(\mbox{${\dl P}^1$}{\setminus}\{0{,}\infty\}) = {\bar{\mathfrak  g}}\otimes{\dl C}[z{,}z^{-1}]
  =: {\mathfrak  z} \,, \Labl gt
where the indeterminate of the loop construction is given by the global
coordinate $z$ on ${\dl C}\,{=}\,\mbox{${\dl P}^1$}{\setminus}\{\infty\}$.
Now as we have seen in Subsection \ref{HVL}, the tensor product
$\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em\equiv\HL1{\otimes}\,\HL2$ is a $\mathfrak  z$-module. In order to describe
the $\mathfrak  z$-co-in\-va\-ri\-ant s of this module it is, however, most desirable that not only
the tensor product, but both factors $\HL1$ and $\HL2$
can be regarded as modules over the block algebra\ individually.
As is clear from the calculation \erf{xfyg}, this can definitely not be
achieved with the block algebra\ as defined so far. Therefore in addition to $\mathfrak  z$\
we introduce a central extension of $\mathfrak  z$\ by a one-dimensional center
${\dl C}\hat K$, so as to obtain a centrally extended loop algebra
${\hat\gt}={\mathfrak  z}\oplus{\dl C}\hat K$. The bracket relations of ${\hat\gt}$ read
  \begin{equation}  [{\bar x}\raisebox{.07em}{$\scriptstyle\otimes$} f,\bar y\raisebox{.07em}{$\scriptstyle\otimes$} g]=[{\bar x},{\bar y}]\,\ot\, fg + \kappa({\bar x},{\bar y})\,
  {\rm Res}^{}_0({\rm d}f\,g) \,\hat K \,.  \end{equation}

By identifying the local coordinates at $0$ and $\infty$, respectively, with the
indeterminate of the loop construction, we obtain two different embeddings
$\imath_0$ and $\imath_\infty$ of ${\mathfrak  z}$ as a vector space into \mbox{$\mathfrak  g$}. We can
extend these maps to two isomorphisms between  ${\hat\gt}$ and \mbox{$\mathfrak  g$}\ as Lie algebra s,
which read explicitly
  \begin{equation}  \imath_0({\bar x}\ot z^n):= \xb\ot t^n\,, \hsp{2.9} \imath_0(\hat K):= K  \hsp{1.6}  \labl8
(${\bar x}\,{\in}\,\mbox{$\bar{\mathfrak  g}$}$, $n\,{\in}\,{\dl Z}$) and
  \begin{equation}  \imath_\infty({\bar x}\ot z^n):= \xb\ot t^{-n}\,, \qquad \imath_\infty(\hat K):= -K \,,  \labl9
respectively. (Note that one should carefully distinguish between the block algebra\ as an
abstract Lie algebra\ and the embedding of the block algebra\ into $\mbox{$\mathfrak  g$}^m$ via local coordinates.
For instance, in the two-point situation considered here, the block algebra\
$\mathfrak  z$\ is isomorphic to the algebra\ ${\bar{\mathfrak  g}}\,{\otimes}\,{\dl C}(t)$ of \mbox{$\bar{\mathfrak  g}$}-valued
Laurent {\em polynomials\/}
that is contained in $\oo{\gM}{1.6}{10.3}$. But for a generic choice of the coordinate $z$,
$\mathfrak  z$\ is embedded into \mbox{$\mathfrak  g$}\ as a subalgebra\ that is isomorphic but not identical
to ${\bar{\mathfrak  g}}\,{\otimes}\, {\dl C}(t)$ and in particular involves arbitrary Laurent series,
as is generically needed in order to be able to define inverses and hence to
have a {\em group\/} of local coordinate changes. This is the main motivation
why in this paper we regard \mbox{$\mathfrak  g$}\ rather than its subalgebra\ $\oo{\gM}{1.6}{10.3}$ as the affine
Lie algebra.)\,%
 \futnote{As a side remark, we mention that we can extend the automorphism
$\omega:=\imath_0\circ(\imath_\infty)^{-1} =\imath_\infty\circ(\imath_0)^{-1}$, which acts as
$\omega(\xb\ot t^n)\,{=}\,\xb\ot t^{-n}$ for ${\bar x}\,{\in}\,\mbox{$\bar{\mathfrak  g}$}$ and $n\,{\in}\,{\dl Z}\,$ and as
$\,\omega(K)\,{=}\,{-}K$, to an automorphism that includes the outer derivation on
the centrally extended loop algebra, namely via $\,\omega(D)\,{:=}\,{-}D$.}

Using the embeddings \erf8 and \erf9, each of the two irreducible highest
weight modules $\HL1$ and $\HL2$ of \mbox{$\mathfrak  g$}\ can be
separately endowed with the structure of a ${\hat\gt}$-module. With the help of
the isomorphism $\imath_0$ we can transport the triangular decomposition \Erf1t
of \mbox{$\mathfrak  g$}\ to a triangular decomposition of ${\hat\gt}$. (Alternatively we could
choose to use $\imath_\infty$ for this purpose; this would result in a different
triangular decomposition.) It then makes sense to talk about highest and lowest
weight modules over the centrally extended block algebra\ ${\hat\gt}$.
Let us analyse the structure of the vector spaces $\HL1$ and $\HL2$
in this spirit. One easily sees that also as a ${\hat\gt}$-module, $\HL1$ is an
irreducible highest weight module of level {\rm k}\ with highest weight $\Lambda_1$,
while as a ${\hat\gt}$-module $\HL2$ is a {\em lowest\/} weight module, with lowest
weight $-\Lambda_2^+$ and at level $-{\rm k}$.\,%
 \futnote{Thus the horizontal part of the weight is $-{\bar\Lambda}_2^+$, which is
the lowest weight of the finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-module with highest weight\ ${\bar\Lambda}_2$ or, in
other words, $-{\bar\Lambda}_2^+\,{=}\,\bar w_{{\rm max}}({\bar\Lambda}_2)$ with $\bar w_{{\rm max}}$ the longest
element of the Weyl group of \mbox{$\bar{\mathfrak  g}$}.
Also, even though $\HL2$ as a ${\hat\gt}$-module is a lowest weight module
and hence in particular not in the category ${\cal O}$, the distinction
between the algebra s \mbox{$\mathfrak  g$}\ and $\oo{\gM}{1.6}{10.3}$ is again immaterial, because the module is
integrable and restricted.}
We are interested in the tensor product of these two ${\hat\gt}$-modules, and in the
co-in\-va\-ri\-ant s of the tensor product with respect to the  action of ${\mathfrak  z}$. Now we make the simple
but crucial observation that
the tensor product $\HL1{\otimes}\,\HL2$ has level $\,{\rm k}+(-{\rm k})\eq0\,$ as a
${\hat\gt}$-module, or in other words, that the ${\hat\gt}$-module $\HL1{\otimes}\,\HL2$
factorizes to a $\mathfrak  z$-module. As a consequence, the $\mathfrak  z$-co-in\-va\-ri\-ant s of the tensor
product {\em coincide\/} with its ${\hat\gt}$-co-in\-va\-ri\-ant s.

We can conclude that we are left with the task of finding the ${\hat\gt}$-co-in\-va\-ri\-ant s of
$\HL1{\otimes}\,\HL2$. Now it is not difficult at all to determine these co-in\-va\-ri\-ant s
explicitly. Namely, every vector in $\HL1{\otimes}\,\HL2$ is a finite sum of
vectors $v_1\raisebox{.07em}{$\scriptstyle\otimes$} v_2$ with $v_1\,{\in}\,\HL1$ and $v_2\,{\in}\,\HL2$, and as a consequence
without loss of generality we can take such vectors $v_1\raisebox{.07em}{$\scriptstyle\otimes$} v_2$ as
representatives for the ${\hat\gt}$-co-in\-va\-ri\-ant s. Now since $\HL1$ is a highest weight module\ over ${\hat\gt}$,
we can write $v_1\,{=}\, x_-v_+$, where $v_+$ is a highest weight vector of $\HL1$
and $x_-\,{\in}\,{\sf U}({\hat\gt}_-)$ with ${\hat\gt}_- \equiv \imath_0^{-1}(\mbox{$\mathfrak  g$}_-)$.
Moreover, as representatives of ${\hat\gt}$-co-in\-va\-ri\-ant s, we have the equivalence
  \begin{equation} v_1\,\ot\, v_2 = x_- v_+ \,\ot\, v_2 \;\sim\; v_+ \,\ot\, y_- v_2 \,,  \end{equation}
where $y_-\,{=}\, (-1)^n\,x_n x_{n-1}\cdots x_2 x_1$ when $x_-\,{=}\, x_1 x_2\cdots x_n$
with $x_i\,{\in}\,{\hat\gt}_-$.
Without loss of generality we can therefore assume that $v_1\,{=}\, v_+$, i.e.\
restrict our attention to representative vectors of the form $v_+ \raisebox{.07em}{$\scriptstyle\otimes$} v_2$.
Next we write $v_2\,{=}\, y_+ v_-$, where $y_+\,{\in}\,{\sf U}({\hat\gt}_+)$ and $v_-$ is the lowest
weight vector
of $\HL2$, regarded as a ${\hat\gt}$-module. Since any part of $y_+$ that is in the
augmentation ideal of ${\sf U}({\hat\gt}_+)$ annihilates $v_+$, we can use the same
argument as before to conclude that the only possible representative for a
${\hat\gt}$-co-in\-va\-ri\-ant\ is $v_+\raisebox{.07em}{$\scriptstyle\otimes$} v_-$. Finally we impose invariance under the Cartan subalgebra\
of ${\hat\gt}$, which amounts to the requirement that $v_+\raisebox{.07em}{$\scriptstyle\otimes$} v_-$ has weight zero,
i.e.\ that
  \begin{equation}  \Lambda_1+(-\Lambda_2^+)=0 \,.  \end{equation}
Thus we conclude that the space of co-in\-va\-ri\-ant s is zero-dimensional unless
$\Lambda_1\,{=}\,\Lambda_2^+$, in which case it is one-dimen\-sional\ and has $v_+\ot v_-$
as a (distinguished) representative. This result (and its derivation, too) is in
complete analogy with the formula \Erf0f for co-in\-va\-ri\-ant s of finite-dimensional\ semi-simple
Lie algebra s.

In short, we have shown that
  \begin{equation}  {\rm dim}\,\mbox{$B$}_{\{\Lambda_1,\Lambda_2\}} = \delta^{}_{\Lambda_1^{},\Lambda_2^+}
  \,.  \Labl2d
Of course, this simple result can also be obtained by various other means.
The reason why we presented this particular derivation is that it sets the
stage for a similar analysis that can be performed for any arbitrary value ${m}$
of insertion points.

\Sect{An integral formula for ${\rm dim}\, B$}A

\subsection{The space \mbox{$B$}\ as a two-point co-in\-va\-ri\-ant}\label{sBH}

Let us now turn to the $m$-point situation, where $m>2$.
Our aim is to describe the space
  \begin{equation}  \mbox{$B$}= \coi\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em{\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})} \equiv \coi{\, \bigotimes_{i=1}^{m} \mbox{${\cal H}^{}_{\Lambda_i}$}}{\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})}
  \labl{BH}
of chiral blocks. Because of the independence of \mbox{$B$}\ of the choice of
coordinates, we can assume that the first and ${m}$th insertion points $p_1$
and $p_{m}$ are at $z_1\eq0$ and $z_{m}\,{=}\,\infty$, respectively.
It can then be shown that there is a natural isomorphism
  \begin{equation}  \mbox{$B$} \;\cong\; \coi{\HL1 \,{\otimes}\, {\overline{\cal H}^{\scriptscriptstyle({m}-2)}} \,{\otimes}\, \HL{m}}{\mathfrak  z}  \labl{biso}
of vector spaces, where $\mathfrak  z$\ is the block algebra\ \Erf gt corresponding to only
two insertion points at zero and infinity,
and where ${\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ stands for the tensor product
  \begin{equation} {\overline{\cal H}^{\scriptscriptstyle({m}-2)}} := \bigotimes_{i=2}^{m-1} \mbox{$\bar{\cal H}^{}_{{\bar\Lambda}_i}$}   \end{equation}
of finite-dimensional\ irreducible \mbox{$\bar{\mathfrak  g}$}-modules.
Here \mbox{$\bar{\cal H}^{}_{{\bar\Lambda}_i}$}\ denotes the irreducible highest weight module\ of \mbox{$\bar{\mathfrak  g}$}\ whose highest weight\ ${\bar\Lambda}_i$ is the horizontal
projection of the highest weight\ $\Lambda_i$ of the \mbox{$\mathfrak  g$}-module \mbox{${\cal H}^{}_{\Lambda_i}$}. Furthermore,
$\mathfrak  z$\ is defined to act on ${\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ by evaluation in the obvious manner,
i.e.\ ${\bar x}\raisebox{.07em}{$\scriptstyle\otimes$} f\,{\in}\,{\mathfrak  z}$ with ${\bar x}\,{\in}\,{\bar{\mathfrak  g}}$ acts as
$\sum_{i=2}^{m-1} f_{p_i}(0)\cdot {\bf1}\,\ot\, \cdots \,\ot\, {\bf1}\,\ot\,
  \bar R_{{\bar\Lambda}_i}({\bar x})\,\ot\,{\bf1} \,\ot\, \cdots \,\ot\, {\bf1} $,
where $\bar R_{{\bar\Lambda}_i}$ denotes the \mbox{$\bar{\mathfrak  g}$}-rep\-re\-sen\-ta\-ti\-on\ carried by \mbox{$\bar{\cal H}^{}_{{\bar\Lambda}_i}$}\
and $f_{p_i}(0)\,{=}\, f(z_i)$ is the value of $f\,{\in}\,{\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$ at the
insertion point.\,%
 \futnote{Note that the result \erf{biso} implies in particular that co-in\-va\-ri\-ant s
taken with respect to different actions of the block algebra, corresponding to
different sets of local coordinates, are isomorphic. Namely, after replacing
the relevant affine module by the corresponding finite-dimensional\ module the only data
entering the description of the block are the values of the functions
$f\,{\in}\,{\cal F}({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})$ at the insertion point; manifestly these values do not
depend on the choice of local coordinates.}

The proof of the isomorphism \erf{biso} between the space of
chiral blocks and the space $\coi{\HL1\,{\otimes}\,{\overline{\cal H}^{\scriptscriptstyle({m}-2)}}\,{\otimes}\,\HL{m}^{}}{\mathfrak  z}$
of $\mathfrak  z$-co-in\-va\-ri\-ant s is not difficult, but a bit lengthy
(compare also proposition 2.3.\ of \cite{beau}).
Therefore we present here only an outline of the proof and defer most details
to Appendix \ref{aproof}.

Rather than proving the isomorphism \erf{biso} directly, we start from a
somewhat more general setting. We consider two finite sets
$P:=\{p_1,p_2,...\,, p_n \}$ and $Q:=\{q_1,q_2,...\,, q_l\}$ of pairwise
distinct points of $\mbox{${\dl P}^1$}$, where we assume that the set $Q$ is not empty,
while $P$ may be empty. To each point we
associate an integrable highest weight of \mbox{$\mathfrak  g$}, and introduce the tensor product
  \begin{equation}  {\tilde {\cal H}} := {\tilde {\cal H}_{\scriptscriptstyle(1)}} \otimes {\tilde {\cal H}_{\scriptscriptstyle(2)}}  \qquad{\rm with}\qquad
  {\tilde {\cal H}_{\scriptscriptstyle(1)}} := \bigotimes_{i=1}^n \mbox{$\bar{\cal H}$}_{{\bar\Lambda}_i} \, ,  \quad
  {\tilde {\cal H}_{\scriptscriptstyle(2)}} := \bigotimes_{j=1}^l \mbox{$\cal H$}_{\Lambda'_j}  \Labl ht
of irreducible modules of the horizontal subalgebra \mbox{$\bar{\mathfrak  g}$}\ for the points in $P$
and of irreducible modules of the affine Lie algebra\ \mbox{$\mathfrak  g$}\ for the points in $Q$, respectively.
Finally, we fix an additional insertion point on $\mbox{${\dl P}^1$}\,{\setminus}(P\cup Q)$,
with an associated integrable highest \mbox{$\mathfrak  g$}-weight $\Lambda$. Without loss of
generality we can assume that this insertion point is $z=\infty$.

Moreover, in addition to \mbox{${\cal H}_\Lambda$}\ we also consider the corresponding parabolic Verma
module
${\cal P}_{\!\Lambda}={\sf U}(\mbox{$\mathfrak  g$})\,{\otimes_{{\sf U}({{\liefont p}_{}^+})}}\,\mbox{$\bar{\cal H}_{{\bar\Lambda}}$} \,{\cong}\, {\sf U}({\gM_{}^-})\,{\otimes}\,\mbox{$\bar{\cal H}_{{\bar\Lambda}}$}$
as introduced in formula \erf{vuu}. As we will show in Appendix \ref{aproof}
by employing special properties of co-in\-va\-ri\-ant s of free modules and the
behavior of exact sequences under the operation of taking co-in\-va\-ri\-ant s,
we have the isomorphisms
  \begin{equation}  \coi{{\tilde {\cal H}}\otimes\mbox{$\bar{\cal H}$}_{\bar\Lambda}}{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus Q)} \;\cong\;
  {\lfloor \coi{{\tilde {\cal H}}\otimes {\cal P}_{\Lambda}}{\gM_{}^-} \rfloor}_{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus Q)}
  \;\cong\; \coi{{\tilde {\cal H}}\otimes{\cal P}_{\Lambda}}{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus (Q\cup\{\infty\}))}
  \labl{ihV}
and
  \begin{equation}  \coi{{\tilde {\cal H}}\,{\otimes}\,{\cal P}_{\!\Lambda}}{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus(Q\cup\{\infty\}))} \;\cong\;
  \coi{{\tilde {\cal H}}\,{\otimes}\,\mbox{${\cal H}_\Lambda$}}{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus(Q\cup\{\infty\}))} \,.  \labl{hV}
Combining these results, we learn that there is an isomorphism
  \begin{equation}  \coi{{\tilde {\cal H}}\,{\otimes}\,\mbox{${\cal H}_\Lambda$}}{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus(Q\cup\{\infty\}))} \;\cong\;
  \coi{{\tilde {\cal H}}\otimes\mbox{$\bar{\cal H}$}_{{\bar\Lambda}}}{{\bar{\mathfrak  g}}({{\dl P}^1_{}}\setminus Q)} \,.  \labl{47}
Employing this isomorphism we replace successively all but the first and last\,%
 \futnote{If we so wished, we could also replace one of the remaining \mbox{$\mathfrak  g$}-modules
$\HL1$ and $\HL{m}$ by the corresponding \mbox{$\bar{\mathfrak  g}$}-module, thereby obtaining co-in\-va\-ri\-ant s
with respect to the  one-point block algebra\ ${\bar{\mathfrak  g}}(\mbox{${\dl P}^1$}{\setminus}\{\infty\})\,{=}\,{\bar{\mathfrak  g}}({\dl C})\,{=}\,{\bar{\mathfrak  g}}
\,{\otimes}\,{\dl C}[[z]]\,{\cong}\,{\bar{\mathfrak  g}}\,{\oplus}\,{\gM_{}^+}$. In contrast, according to the
discussion in Appendix \ref{aproof}, it is not possible to apply this
manipulation also to {\em both\/} of the remaining \mbox{$\mathfrak  g$}-modules.}
affine irreducible module s \mbox{${\cal H}^{}_{\Lambda_i}$}\ that appear in the space \erf{BH} of co-in\-va\-ri\-ant s by
the corresponding irreducible module s \mbox{$\bar{\cal H}^{}_{{\bar\Lambda}_i}$}\ of the horizontal subalgebra \mbox{$\bar{\mathfrak  g}$}.\,%
This way one finally arrives at the isomorphism \erf{biso}.

\subsection{Branching rules}\label{sB}

Now we realize that
in view of the isomorphism \erf{biso} we are actually in a situation very
similar to the one investigated in \sref T. The extension to the general case
amounts to take properly into account the additional finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-modules.
We will argue that this can be achieved by means of suitable branching rules
for completed tensor products involving finite-dimensional\ ${\hat\gt}$-modules.
First we note that all elements of ${\mathfrak  z}$ are represented
on the tensor product ${\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ of finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-modules by evaluation at the
respective insertion points $p_2,p_3,...\,, p_{m-1}$, so that
no central term can arise in their Lie brackets. Therefore the action of
$\mathfrak  z$\ on ${\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ can be extended to an action of ${\hat\gt}$ with $\hat K$ represented by
zero. We call the thus obtained finite-dimensional\ level-zero ${\hat\gt}$-module ${\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$,
as well as any other ${\hat\gt}$-module that is obtained in an analogous manner
from a finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-module, an {\em evaluation module\/}.
Evaluation modules are not restricted, hence not in the category ${\cal O}$,
but the step operators of ${\hat\gt}$ still act locally nilpotently.

As a consequence, the tensor product $\HL1\,{\otimes}\,{\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ of ${\hat\gt}$-modules
isn't an object in the category ${\cal O}$ either; but
it is easily checked that this ${\hat\gt}$-module is still integrable (this fact is
also used in the proof of \Erf hV in Appendix \ref{aproof}),
and that the central element $\hat K$ acts as multiplication by {\rm k}.
Not surprisingly, it turns out to be rather difficult to compute the co-in\-va\-ri\-ant s
of the module $\HL1{\otimes}\,{\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ directly.
In order to determine the co-in\-va\-ri\-ant s, we would therefore like to replace
$\HL1{\otimes}\,{\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$ by a direct sum of irreducible highest weight module s of ${\hat\gt}$.
To this end we work with a completion \mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\ of $\HL1{\otimes}\,{\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$.
More specifically, we assume that there exists a suitable a completion such
that \mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\ becomes reducible and can be written as a direct sum
  \begin{equation}  \mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\ \,\cong\, {\cal J} \oplus \bigoplus_\ell\HM\ell \,,  \labl{deco}
where each of the summands $\HM\ell$ is an integrable highest weight
${\hat\gt}$-module, while ${\cal J}$ is a direct sum of
${\hat\gt}$-modules which are irreducible, but are not weight modules.
(Note that for the uncompleted module such a decomposition
typically does not exist.)

Now the tensor product $(\bigoplus_\ell\HM\ell)\otimes\HL{m}
\subseteq\mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\,{\otimes}\,\HL{m}$ is at level ${\rm k}+(-{\rm k})=0$\, as a ${\hat\gt}$-module,
and hence by repeating the arguments of \sref T one finds that its $\mathfrak  z$-co-in\-va\-ri\-ant s
are the same as its ${\hat\gt}$-co-in\-va\-ri\-ant s. The ${\hat\gt}$-co-in\-va\-ri\-ant s in turn can be
computed in exactly the same manner as in the two-point situation of
\sref T. More precisely, among the integrable highest weight module s in the decomposition \erf{deco}
only those modules contribute to the ${\hat\gt}$-co-in\-va\-ri\-ant s whose highest weight is
$\Lambda_{m}^+$, and each of these yields precisely one independent co-in\-va\-ri\-ant.
In addition, the same analysis indicates
that the submodule ${\cal J}$ of \mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\ does not contribute any co-in\-va\-ri\-ant s, and also
that the ${\hat\gt}$-co-in\-va\-ri\-ant s of $(\mbox{$\HL1\!{\otimes}\barB$})\,{\otimes}\,\HL{m}$ are the same as those of
$\mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\,{\otimes}\,\HL{m}$.

Unfortunately, so far we cannot describe a topology on the module
${\cal H}_{\Lambda_1}\otimes\bar{\cal H}^{(m-2)}$ such that in the
completion with respect to  that topology the relation \erf{deco} holds. The existence
of such a topology should therefore be regarded as a conjecture.
{}In the sequel we will assume that this conjecture is valid; then
the computation of the dimension ${\rm dim}\,\mbox{$B$}$ of the space \Erf BH of co-in\-va\-ri\-ant s
merely amounts to determining the integrable highest weight module s in the decomposition \erf{deco} of
$\HL1{\otimes}\,{\overline{\cal H}^{\scriptscriptstyle({m}-2)}}$. Rewriting the branching rule \erf{deco} as
  \begin{equation}  \HL1\otimes{\overline{\cal H}^{\scriptscriptstyle({m}-2)}} \,\cong\, {\cal J} \oplus
  \bigoplus_{\mu\in{{\rm P}_{\!\kV}}} \NN\mu\,\mbox{$\cal H$}_\mu \,, \labl-
our arguments amount to the formula
  \begin{equation}  {\rm dim}\,\mbox{$B$}_{\!\{\Lambda_i\},\{p_i\}} = \NN{\Lambda_\mn^+} \,.  \labl+

Notice that on the module ${\cal H}_{\Lambda_1}\,{\otimes}\,\bar{\cal H}^{(m-2)}$
(as on any other evaluation module of the affine Lie algebra, except for the
trivial module) we do not have an action of the derivation in the Cartan subalgebra\
of the affine Lie algebra\ \mbox{$\mathfrak  g$}. However, a posteriori, we can define such an
action on a submodule of the completion of
${\cal H}_{\Lambda_1}\,{\otimes}\,\bar{\cal H}^{(m-2)}$, namely on the direct sum of the
integrable highest weight module s $\mbox{$\cal H$}_\mu$ over the centrally extended block algebra\ ${\hat\gt}$. We will
assume from now on that this has been done and that the eigenvalues of the
derivation on the highest weight vector s are chosen in the standard way such that
the characters furnish a module of \mbox{SL$(2{,}{\dl Z})$}.
This, together with the assumption that the module ${\cal J}$ is not a weight
module and hence does not contribute to characters, opens the possibility to
compute the branching coefficients $\NN\mu$ by manipulating characters, and
indeed this will be achieved in the next subsections. Notice that the
character of $\bigoplus_{\mu\in{{\rm P}_{\!\kV}}} \NN\mu\,\mbox{$\cal H$}_\mu\subseteq\mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}$ as
introduced in formula \erf- is given by the product of the character $\chil1$
of $\HL1$ and the \mbox{$\bar{\mathfrak  g}$}-characters $\chibl2$ to $\chibl{{m}-1}$,
   \begin{equation}  {\raisebox{.15em}{$\chi$}}^{}_{\oplus^{}_\mu\NN\mu{\cal H}_\mu} = \chil1(h)\cdot \chibl2\chibl3{\cdots}\,\chibl{{m}-1}(\bar h)
   \,,  \Labl0c
where it is understood
that the eigenvalues of the derivation are chosen as explained above.

\subsection{A projection formula}

Our task is now to find the integrable highest weight module s in
the decomposition \erf{deco}. In this subsection
we present a projection formula which can be used to that effect. We start by
introducing a certain linear functional ${Plenum}$ on the space of class functions
$\varphi$ on $G$, where $G$ is the connected, simply connected
compact real Lie group whose Lie algebra\ is the compact real form
of the horizontal subalgebra \mbox{$\bar{\mathfrak  g}$}\ of \mbox{$\mathfrak  g$}. We denote by $T$ the maximal
torus of $G$ that corresponds to the chosen Cartan subalgebra\ \mbox{$\gB^{}_\circ$}\ of \mbox{$\bar{\mathfrak  g}$}. Thus the
elements $t$ of $T$ are group elements of the form
  \begin{equation}  t(\bar h) = \mbox{\sc Exp}(2\pi{\rm i}\bar h)  \quad{\rm with}\;\ \bar h\,{\in}\,(\mbox{$\gB^{}_\circ$})_{\dl R}^{} \,,
  \end{equation}
and the normalized Haar measure $\dt$ on $T$ is the flat measure
$\int_T\!\dt\;\,{=}\,\prod_{i=1}^{{\rm rank}\,{\bar{\mathfrak  g}}}(\int_0^1\!{\rm d}\bar h^i)$.

We then define, for each integrable weight $\Lambda$ of \mbox{$\mathfrak  g$}, the linear
functional {Plenum}\ by
  \begin{equation}  {Plenum}[\varphi] := \int_T\!\dt\; \mbox{\large(} {\cal X}^{-1} {\rm e}^{-(\Lambda+\rho)}\mbox{\large)}(2\pi{\rm i} h)
  \cdot \varphi(\bar h)  \,.  \end{equation}
Here ${\cal X}$ is the `universal Verma character' that appears in the Weyl\hy Kac character formula\
\Erf wk. Also, concerning the relation between elements of \mbox{$\gM^{}_\circ$}\ and of \mbox{$\gB^{}_\circ$}\
and between elements of their dual spaces we adhere to the following
conventions. The elements $\bar h\,{\in}\,\mbox{$\gB^{}_\circ$}$ and $h\,{\in}\,\mbox{$\gM^{}_\circ$}$ are related by
  \begin{equation}  h = \bar h - \tau L_0 + \varpi K  \end{equation}
with $\tau,\varpi\,{\in}\,{\dl C}$, and the analogous decomposition of
\mbox{$\mathfrak  g$}-weights reads
  \begin{equation}  \mu = {\bar\mu}+\mu^0\Lambda_{(0)}+\nu\delta =: {\bar\mu} + \mu^\natural \,,  \end{equation}
where $\Lambda_{(0)}$ is the zeroth fundamental \mbox{$\mathfrak  g$}-weight and $\delta$ the null root of
\mbox{$\mathfrak  g$}. Thus the \mbox{$\mathfrak  g$}-weight $\mu\,{\in}\,\mbox{$\gM^\star_\circ$}$ acts on $h\,{\in}\,\mbox{$\gM^{}_\circ$}$ as
  \begin{equation}  \mu(h)={\bar\mu}(\bar h)+\mu^\natural({-\tau L_0{+}\varpi K})={\bar\mu}(\bar h)+\varpi\mu^0+\nu\tau  \,.  \end{equation}
Notice that strictly speaking, $\varphi$ has to be a functional on the full
Cartan subalgebra\ of the affine Lie algebra\ \mbox{$\mathfrak  g$}, including the derivation; we will
not always explicitly display this dependence in the sequel.

Let us now consider the special case that $\varphi$ is the character of
a Verma module, $\varphi\equiv\varphi(\bar h{;}\tau{,}\varpi)
={\cal X}_{\Lambda'}(2\pi{\rm i} h)$; we get immediately
  \begin{equation} \begin{array}{ll}  {Plenum}[{\cal X}_{\Lambda'}] \!\!
  &=\displaystyle\int_T\!\dt\; \mbox{\large(} {\cal X}^{-1} {\rm e}^{-(\Lambda+\rho)}\mbox{\large)}(2\pi{\rm i} h) \cdot
    \mbox{\large(} {\cal X} {\rm e}^{\Lambda'+\rho}\mbox{\large)}(2\pi{\rm i} h)  \\{}\\[-.8em]
  &=\displaystyle\int_T\!\dt\; {\rm e}^{\Lambda'-\Lambda}(2\pi{\rm i} h)  \\{}\\[-.8em]
  &={\rm e}^{2\pi{\rm i}(\Lambda'-\Lambda)^\natural({-\tau L_0{+}\varpi K})}_{}\,\delta_{\bar\Lambda,\bar\Lambda'}
    \equiv {\rm e}_{}^{2\pi{\rm i}\varpi((\Lambda')^0-\Lambda^0)} {\rm e}_{}^{2\pi{\rm i}\tau
    (\nu_{\Lambda'}-\nu_\Lambda)}\, \delta^{}_{\bar\Lambda,\bar\Lambda'}
  \,.  \end{array} \labl{vere}
As a consequence, the operator ${Plenum}$ picks the $\Lambda$-isotypic
component in the resolution of any module. In particular, we can analyze
the irreducible characters, i.e.\ take
$\varphi\,{=}\,{\raisebox{.15em}{$\chi$}}_\Lambda(2\pi{\rm i} h)$. By the Weyl\hy Kac character formula\ we know that
${\raisebox{.15em}{$\chi$}}_\Lambda = \sum_{w\in W} \epsilon(w)\,{\cal X}_{w(\Lambda+\rho)-\rho}$,
and hence \erf{vere} tell us that
  \begin{equation} \begin{array}{ll} {Plenum}[{\raisebox{.15em}{$\chi$}}_{\Lambda'}] \!\!
  &=\displaystyle\sum_{w\in W} \epsilon(w)\, {Plenum}[{\cal X}_{w(\Lambda'+\rho)-\rho}]  \\{}\\[-.8em]
  &=\displaystyle\sum_{w\in W} \epsilon(w)\,
    {\rm e}^{2\pi{\rm i}[w(\Lambda'+\rho)-(\Lambda+\rho)]^\natural({-\tau L_0{+}\varpi K})}_{}\,
    \delta^{}_{{\hat w}({\bar\Lambda}'+\bar\rho),{\bar\Lambda}+\bar\rho} \,.  \end{array}  \Labl0h
Here ${\hat w}$ denotes the induced affine action of $w\,{\in}\, W$ on the weight space
\mbox{$\gB^\star_\circ$}\ of the horizontal subalgebra, i.e.\
the action of $w$ on the horizontal part ${\bar\mu}$ of a \mbox{$\mathfrak  g$}-weight $\mu\,{\in}\,\mbox{$\gM^\star_\circ$}$
yields the horizontal part of the $w$-transformed \mbox{$\mathfrak  g$}-weight:
  \begin{equation}  {\hat w}({\bar\mu})\,{=}\,\overline{w\mbox{{\footnotesize(}$\mu${\footnotesize)}}} \,.  \Labl wh
In the following, for notational simplicity we suppress the hat-symbol and
just write $w(\bar\mu)$ for the horizontal projection $\overline{w\mbox{{\footnotesize(}$\mu${\footnotesize)}}}$ of $w(\mu)$.

Next we restrict to the case of our interest, where both $\Lambda$ and
$\Lambda'$ are integrable weights, and
use the fact that the Weyl group $W$ of \mbox{$\mathfrak  g$}\ acts freely on the interior
of the Weyl alcoves. This property of $W$ implies that
in fact only a single Weyl group element, namely
$w\,{=}\,\mbox{\sl id}$, can give a non-zero contribution to the sum; as a consequence\ \Erf0h reduces to
  \begin{equation}  \mbox{${\rm P}_{\!\Lambda'}$}[\chii_\Lambda^{}] = {\rm e}^{2\pi{\rm i}(\Lambda-\Lambda')^\natural({-\tau L_0{+}\varpi K})}_{}\cdot
  \delta^{}_{{\bar\Lambda},{\bar\Lambda}'}  \,.  \end{equation}
Moreover, we are only interested in weights $\Lambda$ at some fixed value {\rm k}\
of the level. Accordingly we now further
restrict our attention to functionals $\mbox{${\rm P}_{\!\Lambda'}$}$
for which $\Lambda'$ is at level {\rm k}\ as well; then ${\bar\Lambda}\,{=}\,{\bar\Lambda}'$
implies that also $\Lambda^0\,{=}\,\Lambda'{}^0$.
Finally we require that the $\delta$-component of the weights $\Lambda$
and $\Lambda'$ is already specified by the horizontal part of the weights
(e.g.\ that it is given by $-\Delta_\Lambda\,{=}\,{-}({\bar\Lambda},{\bar\Lambda}\pl2\bar\rho)
/4({\rm k}\,{+}\,{\rm h})$, the natural value in conformal field theory; owing to the fact that
for all $\nu\,{\in}\,{\dl C}$, $\mbox{${\cal H}_\Lambda$}$ and $\mbox{$\cal H$}_{\Lambda+\nu\delta}$
are isomorphic as \mbox{$\mathfrak  g$}-modules,
this does not result in any loss of generality). Then the equality of
${\bar\Lambda}$ and ${\bar\Lambda}'$ also enforces equality of the $\delta$-components,
so it follows that in fact we not only have ${\bar\Lambda}\,{=}\,{\bar\Lambda}'$, but even
$\Lambda\,{=}\,\Lambda'$.

Summarizing, we have proven that the projection formula
  \begin{equation}  \mbox{${\rm P}_{\!\Lambda'}$}[\chii_\Lambda^{}] \equiv
  \int_T\!\dt\; \mbox{\large(} {\cal X}^{-1} {\rm e}^{-(\Lambda'+\rho)}\,\chii_\Lambda^{}\mbox{\large)} (2\pi{\rm i} h)
  =\delta_{\Lambda,\Lambda'}^{}  \labl1
holds for all integrable \mbox{$\mathfrak  g$}-weights $\Lambda$ and $\Lambda'$
at fixed level whose $\delta$-components depend only on the horizontal part.
More generally, acting with the operator ${Plenum}$ on the character of a direct
sum of integrable highest weight module s provides us with the multiplicity of the module \mbox{${\cal H}_\Lambda$}\
in that sum.\,%
 \futnote{Incidentally, the identity \erf1 can also be used to derive an
integral formula for branching functions for embeddings of affine Lie algebra s.
Such expressions for branching functions have also been obtained in
\cite{haso,hwrh2}.}

Note that when we interpret the parameter $\tau$ as a complex variable (rather
than some formal indeterminate), in order for the character $\chii_\Lambda^{}(2\pi{\rm i} h)$ to
be a convergent power series in $\exp(2\pi{\rm i}\tau)$, it must be required that
$\tau$ has positive imaginary part. However, the result \erf1 tells us
e.g.\ that the expression
  \begin{equation}  \mbox{${\rm P}_{\!\Lambda'}$}[\chii_\Lambda^{}]
  = \displaystyle \sum_{w\in W} \epsilon(w)\,{\rm e}^{2\pi{\rm i}[w(\Lambda+\rho)-(\Lambda'+\rho)]^\natural({-\tau L_0{+}\varpi K})}
  _{} \int_T\!\dt\; {\rm e}^{2\pi{\rm i} (w({\bar\Lambda}'+\bar\rho)-({\bar\Lambda}+\bar\rho))(\bar h)}_{} \end{equation}
is independent of $\tau$, so that in the present context
this restriction is in fact rather irrelevant.
Similarly, in various formul\ae\ that we will deal with below a priori the
imaginary part of $\tau$ must be kept strictly positive in order that the
affine characters, respectively\ the sums over the infinite Weyl group $W$, converge,
and it is not guaranteed that the limit $\tau\,{\to}\,0$ of the whole expression
exists.

\subsection{The integral formula}

We now employ the formula \erf1 to count the number of integrable highest weight module s
with highest weight\ $\Lambda_{m}^+$ in the completed tensor product \mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}.
{}From our assumption \Erf0c about the characters we deduce
with the help of the projection formula \erf1 that the branching coefficients
$\NN\mu$ appearing introduced in equation \erf- satisfy
  \begin{equation}  \NN{\Lambda_\mn^+} = \int_T\!\dt\; \mbox{\large(}{\cal X}^{-1}{\rm e}^{-\Lambda_{m}^+-\rho}\mbox{\large)}(2\pi{\rm i} h) \cdot
  \chil1(2\pi{\rm i} h)\, \chibl2\chibl3{\cdots}\,\chibl{{m}-1}(2\pi{\rm i}\hb)  \,.  \end{equation}
We now manipulate the right hand side\ of this identity as follows.
We first insert the Weyl\hy Kac character formula\ \Erf wk for $\chil1$; next we introduce a dummy
summation $|\overline W|^{-1}\sum_{{\bar w}\in\overline W}$ and substitute $w\,{\mapsto}\,{\bar w}^{-1}w$, where we
consider the Weyl group ${\overline W}$ of \mbox{$\bar{\mathfrak  g}$}\ as canonically embedded in the Weyl
group $W$ of \mbox{$\mathfrak  g$}; and finally we set $\bar h\,{=}\,{\bar w}^{-1}(\bar h')$ and use
invariance of the measure $\dt$ under the transformation from $\bar h$ to $\bar h'$.
This way we arrive at
  \begin{eqnarray}  & \NN{\Lambda_\mn^+} \!\!&
  = \displaystyle \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}(2\pi{\rm i}\hb) \sumw w \epsilon(w)\,{\rm e}^{w(\Lambda_{m-1}+\rho)
  -(\Lambda_{m}^++\rho)}(2\pi{\rm i} h)
  \nonumber\\{}\nonumber\\[-.3em]
  &&= \displaystyle\Frac1{|\overline W|}\, \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}(2\pi{\rm i}\hb) \sum_{w\in W} \sum_{{\bar w}\in\overline W}
  \epsilon(w{\bar w})\, {\rm e}^{{\bar w}^{-1}w(\Lambda_1+\rho)-\Lambda_{m}^+-\rho}(2\pi{\rm i} h)
  \nonumber\\{}\nonumber\\[-.6em]
  &&= \displaystyle\Frac1{|\overline W|}\, \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}(2\pi{\rm i}\hb)
  \nonumber\\{}\nonumber\\[-.98em]  &&\hsp{4.5}
  \cdot \displaystyle\sum_{w\in W} \epsilon(w)\, {\rm e}^{w(\Lambda_1+\rho)}(2\pi{\rm i} h) \cdot\sum_{{\bar w}\in\overline W}
  \epsilon({\bar w})\, {\rm e}^{-{\bar w}(\Lambda_{m}^++\rho)}(2\pi{\rm i} h)
  \,. \end{eqnarray}
Next we use the Weyl character formula for the characters $\bar\chii_{\bar\Lambda}^{}$ of finite-dimensional\
\mbox{$\bar{\mathfrak  g}$}-modules \mbox{$\bar{\cal H}_{{\bar\Lambda}}$}\,%
 \futnote{In agreement with the remarks made above the final result, as
displayed in formul\ae\ \Erf eN and \erf V below, does not depend
on $\tau$ and $\varpi$ at all. Accordingly, for ease of notation we suppress
the superficial dependence on these variables from now on.}
to obtain
  \begin{equation} \begin{array}{ll}  \NN{\Lambda_\mn^+} \!\!
  &= \displaystyle\Frac1{|\overline W|}\, \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}(2\pi{\rm i}\hb)\,{\bar{\raisebox{.15em}{$\chi$}}}_{{\bar\Lambda}^+_{m}}(-2\pi{\rm i}\hb)
  \\{}\\[-.98em]&\hsp{11}
  \cdot \bar\uvc^{-1}(-2\pi{\rm i}\hb)\,\displaystyle\sum_{w\in W} \epsilon(w)\,{\rm e}^{w(\Lambda_1+\rho)}(2\pi{\rm i}\hb)
  \,, \end{array} \Labl*1
where
  \begin{equation}  \bar\uvc = \mbox{\large(}\! \sum_{{\bar w}\in\overline W} \epsilon({\bar w})\, {\rm e}^{{\bar w}(\bar\rho)}_{} \mbox{\large)}^{-1}_{}
  = {\rm e}^{-\bar\rho} \prod_{{\bar\alpha}>0}(1-{\rm e}^{-{\bar\alpha}})^{-1}_{}  \end{equation}
is the universal Verma character of \mbox{$\bar{\mathfrak  g}$}, the horizontal analogue of ${\cal X}$
\erf{uvc}.

In a final step we rewrite \Erf*1 in the form
  \begin{eqnarray}  && \phantom x\hsp{-2.5}
  \NN{\Lambda_\mn^+} = \displaystyle\Frac1{|\overline W|} \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}\chibl{m}(2\pi{\rm i}\hb)
  \cdot \bar\uvc^{-1}(-2\pi{\rm i}\hb)\! \sum_{w\in W}\! \epsilon(w)\, {\rm e}^{w(\Lambda_1+\rho)}(2\pi{\rm i}\hb)
  \,.\nonumber\\[-.6em] {}  \label*  \end{eqnarray}
Here we employed the identity
  \begin{equation}  \bar{\raisebox{.15em}{$\chi$}}_{{\bar\Lambda}^+_{}}^{}(\bar h) = \bar\chii_{\bar\Lambda}^{}(-\bar h) \end{equation}
for characters of irreducible highest weight module s of \mbox{$\bar{\mathfrak  g}$}. This relation follows by making the
substitution ${\bar w}\,{\mapsto}\,{\bar w}\bar w_{{\rm max}}$, with $\bar w_{{\rm max}}$ the longest element of the
Weyl group of \mbox{$\bar{\mathfrak  g}$}, in the Weyl character formula, and by using the fact that this element acts as
$\bar w_{{\rm max}}({\bar\mu})\,{=}\,{-}{\bar\mu}^+$ on \mbox{$\bar{\mathfrak  g}$}-weights:
  \begin{equation}  \bar{\raisebox{.15em}{$\chi$}}_{{\bar\Lambda}^+_{}}^{}(\bar h)
  = \frac {\sum_{{\bar w}\in\overline W} \epsilon({\bar w}) {\rm e}^{{\bar w}({\bar\Lambda}^++\bar\rho)(\bar h)}}
  {\sum_{{\bar w}\in\overline W} \epsilon({\bar w}) {\rm e}^{{\bar w}(\bar\rho)(\bar h)}}
  = \frac {\epsilon(\bar w_{{\rm max}}) \sum_{{\bar w}\in\overline W} \epsilon({\bar w}) {\rm e}^{-{\bar w}({\bar\Lambda}+\bar\rho)(\bar h)}}
  {\epsilon(\bar w_{{\rm max}}) \sum_{{\bar w}\in\overline W} \epsilon({\bar w}) {\rm e}^{-{\bar w}(\bar\rho)(\bar h)}} = \bar\chii_{\bar\Lambda}^{}(-\bar h) \,.  \end{equation}

\Sect{From the integral formula to the modular $S$-matrix}F

\subsection{${\rm dim}\,\mbox{$B$}$ in terms of weight multiplicities}\label{dbf}

Our aim is now to deduce the Verlinde formula from the integral formula \erf*.
This can be achieved by using information about the structure of the Weyl
group $W$ of \mbox{$\mathfrak  g$}\ or, more precisely, about the relation between $W$ and the
Weyl group ${\overline W}$ of the horizontal subalgebra\ \mbox{$\bar{\mathfrak  g}$}\ which is canonically embedded as a
subgroup into $W$. There are essentially two different
possibilities to implement this relationship. The first amounts to
working with a special set $\fc W{.7}{.78}{.2}$ of representatives of the coset of
$W$ by its subgroup ${\overline W}$. By definition, $\fc W{.7}{.78}{.2}$ is
that subset of $W$ which consists of those representatives of the elements of
$W/{\overline W}$ that have minimal length. One knows
(see e.g.\ \cite{iwah2} and also \cite[remark\,8.1.]{gale}) that
every element of $W/{\overline W}$ has a unique representative with this
property, that for any integrable \mbox{$\mathfrak  g$}-weight $\Lambda$ and any $w\,{\in}\, W$ the
\mbox{$\bar{\mathfrak  g}$}-weight $w({\bar\Lambda}\,{+}\,\bar\rho)\,{-}\,\bar\rho$ is dominant integral if and only if
$w\,{\in}\,\fc W{.7}{.78}{.2}$, and that each element $w\,{\in}\, W$ can be uniquely represented in
the form $w\,{=}\,\bar w \fc w{.5}{.48}{.1}$ with ${\bar w}\,{\in}\,{\overline W}$ and $\fc w{.5}{.48}{.1}\,{\in}\,\fc W{.7}{.78}{.2}$.
Together with the Weyl character formula, which absorbs the summation over ${\overline W}$,
it then follows that our result \Erf*1 can be rewritten as
  \begin{equation} \begin{array}{ll}  {\rm dim}\,\mbox{$B$} \!\!
  &= \displaystyle\Frac1{|\overline W|}\, \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}\, \bar\uvc^{-1}(2\pi{\rm i}\hb)
  \sum_{\fc w{.37}{.32}{.1}\in\fc W{.55}{.52}{.2}}\epsilon({\bar w}{\fc w{.5}{.48}{.1}})\,{\bar{\raisebox{.15em}{$\chi$}}}_{{\fc w{.37}{.32}{.1}}({\bar\Lambda}_1+\bar\rho)-\bar\rho}(2\pi{\rm i}\hb)
  \\{}\\[-1.41em]&\hsp{20} \cdot\bar\uvc^{-1}(-2\pi{\rm i}\hb)\,{\bar{\raisebox{.15em}{$\chi$}}}_{{\bar\Lambda}^+_{m}}(-2\pi{\rm i}\hb)
  \\{}\\[-.7em]
  &= \displaystyle\Frac1{|\overline W|}\, \int_T\!\dt\; J(2\pi{\rm i}\hb)\, \chibl2\chibl3{\cdots}\,\chibl{{m}-1}{\bar{\raisebox{.15em}{$\chi$}}}_{{\bar\Lambda}_m}(2\pi{\rm i}\hb)
  \sum_{\fc w{.37}{.32}{.1}\in\fc W{.55}{.52}{.2}} \epsilon({\fc w{.5}{.48}{.1}})\, {\bar{\raisebox{.15em}{$\chi$}}}_{{\fc w{.37}{.32}{.1}}({\bar\Lambda}_1+\bar\rho)-\bar\rho}(2\pi{\rm i}\hb)
  \,. \end{array} \Labl0d
Here we introduced the function
  \begin{equation}  J(2\pi{\rm i}\hb) := \bar\uvc^{-1}(2\pi{\rm i}\hb)\, \bar\uvc^{-1}(-2\pi{\rm i}\hb) = \prod_{{\bar\alpha}\in
  \bar\Delta_+} \mbox{\Large$|$} {\rm e}^{\pi{\rm i}{\bar\alpha}(\bar h)}-{\rm e}^{-\pi{\rm i}{\bar\alpha}(\bar h)}
  \mbox{\Large$|$}^2 \,,  \labl{Jh}
where the product extends over the set $\bar\Delta_+$ of positive \mbox{$\bar{\mathfrak  g}$}-roots.
This function can be recognized as the Jacobian factor that, together with
$|{\overline W}|^{-1}$, appears in Weyl's integration formula that relates
the integral of a class function
over the whole group $G$ to an integral over its maximal torus $T$.
Thus we can rewrite \Erf0d as an integral over $G$:
  \begin{equation}  {\rm dim}\,\mbox{$B$} = \sum_{\fc w{.37}{.32}{.1}\in\fc W{.55}{.52}{.2}}\epsilon({\fc w{.5}{.48}{.1}})\, \int_G\!\dg\,
  {\bar{\raisebox{.15em}{$\chi$}}}_{{\fc w{.37}{.32}{.1}}({\bar\Lambda}_1+\bar\rho)-\bar\rho}\, \chibl2\chibl3{\cdots}\,\chibl{{m}-1}{\bar{\raisebox{.15em}{$\chi$}}}_{{\bar\Lambda}_m}  \,, \end{equation}
where ${\rm d}\mu_G$ is the normalized Haar measure of $G$. From the orthogonality
property of $G$-characters with respect to  ${\rm d}\mu_G$ it then follows immediately that
  \begin{equation}  {\rm dim}\,\mbox{$B$} = \sum_{\fc w{.37}{.32}{.1}\in\fc W{.55}{.52}{.2}} \epsilon({\fc w{.5}{.48}{.1}})\,
  \bar N_{\!{\fc w{.37}{.32}{.1}}({\bar\Lambda}_1+\bar\rho)-\bar\rho,{\bar\Lambda}_2...{\bar\Lambda}_{m}}  \,,
  \Labl eN
where $\bar N_{\!{\bar\mu}_1{\bar\mu}_2...{\bar\mu}_l}$ denotes the number of singlets in
the tensor product $\mbox{$\bar{\cal H}$}_{{\bar\mu}_1}\,{\otimes}\,\mbox{$\bar{\cal H}$}_{{\bar\mu}_2}{\otimes}\cdots{\otimes}\,
\mbox{$\bar{\cal H}$}_{{\bar\mu}_l}$.

The right hand side\ of expression \erf{eN} actually constitutes a finite sum,
even though ${\fc W{.7}{.78}{.2}}$ is an infinite set. To see this, we
observe that the \mbox{$\bar{\mathfrak  g}$}-module $\mbox{$\bar{\cal H}$}_{{\bar\Lambda}_2}{\otimes}\cdots{\otimes}\,
\mbox{$\bar{\cal H}$}_{{\bar\Lambda}_{m}}$ can be decomposed into a direct sum of finitely many
irreducible \mbox{$\bar{\mathfrak  g}$}-modules. According to the identity \erf{0f},
the module $\mbox{$\bar{\cal H}$}_{{\fc w{.37}{.32}{.1}}({\bar\Lambda}_1+\bar\rho)-\bar\rho}$
has to appear in this decomposition in order to give a non-zero contribution to
$\bar N_{\!{\fc w{.37}{.32}{.1}}({\bar\Lambda}_1+\bar\rho)-\bar\rho,{\bar\Lambda}_2...{\bar\Lambda}_{m}}$. Hence
only finitely many ${\fc w{.5}{.48}{.1}}$ can contribute and we obtain a manifestly integral
expression for ${\rm dim}\,\mbox{$B$}$ in terms of multiplicities of invariant tensors of
finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-modules. (In the special case of $m\eq3$ this description of the
WZW fusion rules is known as the Kac$\mbox{-\hspace{-.66 mm}-}$ $\!$Walton formula \cite{KAc3,walt3}.)
As we will see in Subsection \ref{seul}, the right hand side\ of \erf{eN} can also be
interpreted as the Euler characteristic of a certain complex of \mbox{$\bar{\mathfrak  g}$}-co-in\-va\-ri\-ant s.

If we so wish, we can proceed by expressing the number
$\bar N_{\!{\bar\mu}_1{\bar\mu}_2...{\bar\mu}_l}$ through weight multiplicities of finite-dimensional\
\mbox{$\bar{\mathfrak  g}$}-modules. To this end one has to implement the so-called Racah$\mbox{-\hspace{-.66 mm}-}$ Spei\-ser
formula for the number of
invariant tensors in a tensor product of finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-modules. We find
  \begin{equation}  \begin{array}{ll}  {\rm dim}\,\mbox{$B$} \!\! &= \displaystyle\sum_{\fc w{.37}{.32}{.1}\in\fc W{.55}{.52}{.2}}\epsilon({\fc w{.5}{.48}{.1}})\sum_{{\bar w}\in\overline W}\epsilon({\bar w})\,
  {\rm mult}_{{\bar\Lambda}_2...{\bar\Lambda}_{{m}-1}}({\bar w}{\fc w{.5}{.48}{.1}}({\bar\Lambda}_1+\bar\rho)\,{-}\,\bar\rho
  \,{-}\,{\bar\Lambda}_{m}) \\{}\\[-.8em]
  & \equiv \displaystyle\sum_{w\in W}\epsilon(w)\, {\rm mult}_{{\bar\Lambda}_2...{\bar\Lambda}_{{m}-1}}
  (w({\bar\Lambda}_1+\bar\rho)\,{-}\,\bar\rho\,{-}\,{\bar\Lambda}_{m})  \,,  \end{array} \end{equation}
where ${\rm mult}_{{\bar\Lambda}_2...{\bar\Lambda}_{{m}-1}}({\bar\mu})$ denotes the
multiplicity of the weight ${\bar\mu}$ in the tensor product module
$\bar{\cal H}_{{\bar\Lambda}_1}{\otimes}\,\bar{\cal H}_{{\bar\Lambda}_2}{\otimes}\cdots{\otimes}\,
\bar{\cal H}_{{\bar\Lambda}_{{m}-1}}$.

\subsection{Resummation}

Alternatively, we can also express $\,{\rm dim}\,\mbox{$B$}\,$ completely in terms of elements
of the modular matrix $S$ that describes the transformation properties
of the characters of the \mbox{$\mathfrak  g$}-modules \mbox{${\cal H}_\Lambda$}\ under the modular transformation
$\tau\mapsto-1/\tau$. To this end, we recall that the affine Weyl group $W$
can be regarded as the semi-direct product of ${\overline W}$ and of translations by
elements of the coroot lattice $\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}$ of \mbox{$\bar{\mathfrak  g}$}. We can therefore write the
elements of the affine Weyl group $W$ in \erf* as pairs $w=({\bar w};{\bar\beta})$
of elements ${\bar w}\,{\in}\,{\overline W}$ of the Weyl group of \mbox{$\bar{\mathfrak  g}$}\ and of coroot lattice
elements ${\bar\beta}\,{\in}\,\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}$. Then the induced action \Erf wh of $w\,{\in}\, W$ on the
weight space of \mbox{$\bar{\mathfrak  g}$}\ is given by
  \begin{equation}  w(\bar\mu)={\bar w}(\bar\mu)+{\rm k}{\bar\beta} \,.  \end{equation}
Also taking care of the shift in the effective level that results from the
shift of the weights by the Weyl vector $\rho$ (which has level {\rm h}) in the
Weyl group action, we can then rewrite \erf* in the form
  \begin{equation} \begin{array}{ll}  \NN{\Lambda_\mn^+} \!\!
  &= \displaystyle\Frac1{|\overline W|}\, \int_T\!\dt\; \chibl2\chibl3{\cdots}\,\chibl{{m}-1}\chibl{m}(2\pi{\rm i}\hb)
  \\{}\\[-.8em]&\hsp{3.8}
  \cdot \bar\uvc^{-1}(-2\pi{\rm i}\hb) \displaystyle\sum_{{\bar w}\in\overline W}\epsilon({\bar w})\, {\rm e}^{{\bar w}({\bar\Lambda}_1+\bar\rho)}
  (2\pi{\rm i}\hb) \sum_{{\bar\beta}\in\bL^{\!\vee_{}}_{{}^{}}} {\rm e}^{({\rm k}+{\rm h}){\bar\beta}}(2\pi{\rm i}\hb) \,. \end{array} \Labl dj

Relation \erf{dj} expresses the dimension of the space of
chiral block s as the integral of a certain function over a torus, namely the maximal
torus of the Lie group\ $G$. We will now show that one can replace this
integration by a finite summation over suitable \mbox{$\bar{\mathfrak  g}$}-weights.
This is achieved with the help of (a slightly generalized version of)
Poisson resummation which states that
for any function $f{:}\; (\mbox{$\gB^\star_\circ$})_{\dl R}\cong{\dl R}^{{\rm rank}\,{\bar{\mathfrak  g}}}\to{\dl C}\,$
that is periodic with respect to  \mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}\ and for any natural number $l$, the formula
  \begin{equation}  \int_T\!\dt\;\! \sum_{{\bar\beta}\in\bL^{\!\vee_{}}_{{}^{}}}f(\bar h)\, {\rm e}^{{\bar\beta}}(2\pi{\rm i} l\bar h)
  = |l\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}/\mbox{$\bL_{\rm w}$}|\!\! \sum_{{\bar\lambda}\in l^{-1}\bL_{\rm w}/\bL^{\!\vee_{}}_{{}^{}}}\!\! f({\bar\lambda}) \labl{lem}
holds. Here as before $\dt$ is the normalized Haar measure on $T$ and
$\bar h\,{\in}\,\mbox{$\gB^{}_\circ$}$ is related to $t\iNT$ by $t\,{=}\,\mbox{\sc Exp}(2\pi{\rm i}\hb)$. Also, the
integrand of the left hand side\ is single-valued on $T$ because $f$ is by assumption
\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}-periodic, and because for different elements
$\bar h_1,\,\bar h_2\,{\in}\,\mbox{$\gB^\star_\circ$}$
which satisfy $\mbox{\sc Exp}(2\pi{\rm i}\hb_1)\,{=}\,t\,{=}\,\mbox{\sc Exp}(2\pi{\rm i}\hb_2)$
the values of ${\bar\beta}(\bar h_1)$ and ${\bar\beta}(\bar h_2)$ for ${\bar\beta}\,{\in}\,\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}$
differ by an integer, the exponential function is \mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}-periodic as well.

To derive relation \erf{lem}, we introduce the Fourier transform $\tilde f$
of $f$. The Fourier components are labelled by the lattice dual to \mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}, i.e.\
by the weight lattice \mbox{$\bL_{\rm w}$}:\,%
 \futnote{We identify \mbox{$\gB^{}_\circ$}\ and the weight space \mbox{$\gB^\star_\circ$}\
via the non-degenerate invariant bilinear form.}
$f(\bar h)\,{=}\,\sum_{{\bar\lambda}\in\bL_{\rm w}} {\rm e}^{2\pi{\rm i}{\bar\lambda}(\bar h)} {\tilde f}_{\bar\lambda}$.
Then the left hand side\ of \erf{lem} can be written as
  \begin{equation}  \int_T\!\dt\;\! \sum_{{\bar\beta}\in\bL^{\!\vee_{}}_{{}^{}}}\sum_{{\bar\lambda}\in\bL_{\rm w}}
  {\tilde f}_{\bar\lambda} \, {\rm e}^{2\pi{\rm i} ({\bar\lambda}+l {\bar\beta})(\bar h)}
  = \sum_{{\bar\lambda}\in\bL_{\rm w}} {\tilde f}_{\bar\lambda} \sum_{{\bar\beta}\in\bL^{\!\vee_{}}_{{}^{}}}
  \int_T\!\dt\; {\rm e}^{2\pi{\rm i}({\bar\lambda}+l{\bar\beta})(\bar h)}
  = \sum_{{\bar\lambda}\in l\bL^{\!\vee_{}}_{{}^{}}} {\tilde f}_{\bar\lambda} \,.  \labl{ls}
Here the last equality holds because the integral over the torus is non-zero
only when the integrand is constant, which happens only when ${\bar\lambda}=
-l{\bar\beta}$, i.e.\ when ${\bar\lambda}$ lies in $l$ times the coroot lattice.
On the other hand, for the right hand side\ of \erf{lem} we compute
  \begin{equation}  \sum_{{\bar\lambda}\in l^{-1}\bL_{\rm w}/\bL^{\!\vee_{}}_{{}^{}}}\!\! f({\bar\lambda}) =
  \sum_{{\bar\mu}\in\bL_{\rm w}} {\tilde f}_{\bar\mu} \!\!\sum_{{\bar\lambda}\in l^{-1}\bL_{\rm w}/\bL^{\!\vee_{}}_{{}^{}}}\!\!
  {\rm e}^{2\pi{\rm i}({\bar\lambda},{\bar\mu})} = \mbox{\Large$|$} \Frac{\bL_{\rm w}}
  {l^{\phantom|}_{\phantom|}\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}{}\!} \mbox{\Large$|$}\!\sum_{{\bar\mu}\in l\bL^{\!\vee_{}}_{{}^{}}}
  {\tilde f}_{\bar\mu}\,, \labl{rs}
where we use the fact that the sum over ${\bar\lambda}$ only gives
contributions if the summand is constant and hence equal to one, which
happens precisely if the inner product $({\bar\lambda},{\bar\mu})$ is an integer.
The resummation formula \erf{lem} now follows by equating
relations \erf{rs} and \erf{ls}.

Applying now the result \erf{lem} to formula \erf{dj} for ${\rm dim}\,\mbox{$B$}$, we get
  \begin{eqnarray}
  & \hsp{-.5} {\rm dim}\,\mbox{$B$} \!\!&= \hllw \!\displaystyle\sum_{{\bar\lambda}\in{({\rm k}+\gV)}^{-1}\bL_{\rm w}/\bL^{\!\vee_{}}_{{}^{}}}\!\!\!\!
  \bar\uvc^{-1}(-{\bar\lambda})\,
  \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_2}({\bar\lambda})\, \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_3}({\bar\lambda})\cdots
  \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_{m}}({\bar\lambda}) \!\sum_{{\bar w}\in{\overline W}} \!\epsilon({\bar w}) \,
  {\rm e}^{2\pi{\rm i} {\bar w}({\bar\Lambda}_1+\bar\rho,{\bar\lambda})}  \nonumber\\[.3em]
  &&= \hllw \!\displaystyle\sum_{{\bar\lambda}\in{({\rm k}+\gV)}^{-1}\bL_{\rm w}/\bL^{\!\vee_{}}_{{}^{}}}\!\! J({\bar\lambda})\,
  \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_1}({\bar\lambda})\, \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_2}({\bar\lambda})\cdots
  \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_{m-1}}({\bar\lambda})\, \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_{m}}({\bar\lambda})
  \,, \label{dp}\end{eqnarray}
where in the second line we used once more the Weyl character formula.

\subsection{Characters and the modular matrix $S$}

The final step in establishing the Verlinde formula
is now to express the result \erf{dp} in terms of the entries of the modular
matrix $S$ that governs \cite{KAc3} the transformation of the affine characters
${\raisebox{.15em}{$\chi$}}_\Lambda$ ($\Lambda\,{\in}\,{{\rm P}_{\!\kV}}$) under the modular transformation
$\tau\mapsto-1/\tau$. This is possible because the generalized quantum
dimensions, i.e.\ ratios of $S$-matrix elements, coincide with the characters
of the horizontal subalgebra, evaluated at suitable arguments.

We first rewrite the summation in expression \erf{dp}, which extends over
the \mbox{$\bar{\mathfrak  g}$}-weights
${\bar\lambda}\,{\in}\,({({\rm k}+\gV)}^{-1}\mbox{$\bL_{\rm w}$})/\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$} \equiv \mbox{$\bL_{\rm w}$}/({({\rm k}+\gV)}\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$})$, in such a way that only
weights in the dominant Weyl alcove appear. The function $J$ \Erf Jh is the
square of a function that is odd under the Weyl group ${\overline W}$ and hence
vanishes when $\bar h$ is on the boundary of some Weyl
chamber. Owing to the ${\overline W}$-invariance of the characters and of $J$,
it follows that each Weyl chamber gives the same contribution to the sum.
Restricting the summation to those weights that belong to the dominant Weyl
alcove therefore just amounts to cancelling the prefactor $1/|{\overline W}|$. In the
summation we are then left with the weights of the form
  \begin{equation}  {\bar\lambda} = \frac{{\bar\Lambda}+\bar\rho}{{\rm k}+\gV} \,,  \labl4
where the weights ${\bar\Lambda}+\bar\rho$ are the integral weights in the interior of
the dominant Weyl alcove at level ${\rm k}+{\rm h}$. The weights ${\bar\Lambda}$ are thus
the integral weights in the dominant Weyl alcove at level {\rm k}, i.e.\ precisely
the elements of the set ${{\rm P}_{\!\kV}}$ \erf{pk}. Formula \erf{dp} can therefore
be rewritten as
  \begin{equation}  {\rm dim}\,\mbox{$B$}= \hllw\sumk{\bar\Lambda} J(\Frac{\Lambdab+\bar\rho}{\kV+\gV})\,\raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_1}\!(\Frac{\Lambdab+\bar\rho}{\kV+\gV})\,\raisebox{.15em}{$\bar\chi$}_
  {{\bar\Lambda}_2}\!(\Frac{\Lambdab+\bar\rho}{\kV+\gV}) \cdots \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_{m}}\!(\Frac{\Lambdab+\bar\rho}{\kV+\gV}) \,. \labl{dq}

A closed expression for the modular matrix $S$ is provided by the Kac\hy Peterson formula\
  \begin{equation}  \begin{array}{l}  S_{\Lambda,\Lambda'}^{}=(-{\rm i})^{({\rm dim}\,{\bar{\mathfrak  g}}-{\rm rank}\,{\bar{\mathfrak  g}})/2_{}}\,
  |\bL_{\rm w}/\bL^{\!\vee_{}}_{{}^{}}|_{}^{-1/2}({\rm k}+{\rm h}_{})^{-{\rm rank}\,{\bar{\mathfrak  g}}/2} \\{}\\[-.66em] \hsp{9}
  \cdot \displaystyle\sum_{{\bar w} \in{\overline W}} \epsilon({\bar w})\,
  \exp\mbox{\large[} -\Frac{2\pi{\rm i}}{{\rm k}+{\rm h}}\,({\bar w}({\bar\Lambda}+\bar\rho)
  ,{\bar\Lambda}'+\bar\rho) \mbox{\large]} \,.  \end{array}  \labl{kpf}
By comparison with the Weyl character formula, it follows that the characters
$\raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_i}$ evaluated at the weights \erf4 satisfy
  \begin{equation}  \raisebox{.15em}{$\bar\chi$}_{{\bar\Lambda}_i}(\Frac{\Lambdab+\bar\rho}{\kV+\gV}) =
  \mbox{\LARGE(} \frac{S_{\Lambda_i,\Lambda}}{S_{{{\rm k}\Lambda_{(0)}},\Lambda}} \mbox{\LARGE)}^* \,. \labl i
Similarly, together with the denominator identity one finds that
  \begin{equation} \begin{array}{ll}
  S_{{{\rm k}\Lambda_{(0)}},\Lambda} \!\!&= {|\mbox{$\bL_{\rm w}$}/{({\rm k}+\gV)}\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}|}^{-1/2}_{}\displaystyle\prod_
  {{\bar\alpha}>0} 2 \sin \Frac{(\bar\Lambda+\bar\rho,{\bar\alpha})\pi}{{{\rm k}+\gV}}  \\[.8em]
  &= {\rm i}^{-|\bar\Delta_+|}\, {|{({\rm k}+\gV)}\mbox{$\bL^{\!{\scriptscriptstyle\vee}_{}}_{\phantom|}$}/\mbox{$\bL_{\rm w}$}|}^{1/2}_{} \displaystyle\prod
  _{{\bar\alpha}>0} \mbox{\Large(} \!\exp\mbox{\large[} {\rm i}\pi\mbox{\large(}\Frac{(\alphab,{\bar\Lambda}+\bar\rho)}{{\rm k}+{\rm h}}\mbox{\large)} \mbox{\large]} -\exp\mbox{\large[} -{\rm i}\pi\mbox{\large(}\Frac{(\alphab,{\bar\Lambda}+\bar\rho)}{{\rm k}+{\rm h}}\mbox{\large)}
  \mbox{\large]} \mbox{\Large)}  \,, \end{array}\end{equation}
which in turn implies
  \begin{equation}  |S_{{{\rm k}\Lambda_{(0)}},\Lambda}|^2 =  \mbox{\large$\left|\frac{({\rm k}+\GV)\bL^{\scriptscriptstyle\vee} J(\Frac{\Lambdab+\bar\rho}{\kV+\gV})  \,.  \labl0

We now insert the identities \erf i and \erf0 into formula \Erf dq for
${\rm dim}\,\mbox{$B$}$. We then finally obtain (using also the fact that $S_{{{\rm k}\Lambda_{(0)}},\Lambda}$
and ${\rm dim}\,\mbox{$B$}$ are real)
  \begin{equation} {\rm dim}\, \mbox{$B$}^{}_{\{\Lambda_i\},\{p_i\}} =
  \sumk{\bar\Lambda} (S_{{{\rm k}\Lambda_{(0)}},\Lambda})^2 \, \prod_{i=1}^{m}
  \frac{S_{\Lambda_i,\Lambda}}{S_{{{\rm k}\Lambda_{(0)}},\Lambda}} \,. \labl V
This is our desired result, the Verlinde formula that expresses the
dimension ${\rm dim}\,\mbox{$B$}$ of spaces of genus zero chiral WZW blocks through the
entries of the matrix $S$ \erf{kpf}. Note that the right hand side\ of \erf V is
typically a complicated combination of complex numbers (actually all entries
of $S$ lie in a cyclotomic extension of the rationals), i.e.\ unlike the
alternative expression on the right hand side\ of \Erf eN, it is not manifestly integral.

\Sect{Homological aspects of the Verlinde formula}S

In this final section we add a few comments on general aspects of the Verlinde
formula and present an interpretation of the Verlinde multiplicities ${\rm dim}\,\mbox{$B$}$
as expressed by formula \erf{eN} in terms of Euler characteristics of certain
complexes of co-in\-va\-ri\-ant s.

\subsection{The Verlinde formula}

In the literature, derivations of the Verlinde formula have been described
at various levels of rigor, ranging from more heuristic considerations
to mathematically complete proofs. It seems fair to say that typically
the more rigorous these proofs are, the smaller the number of situations is to
which they apply. A conformal field theory\ deduction which does not use any specific properties
of the chiral symmetry algebra of the theory, and hence applies to arbitrary
rational conformal field theories, was given in \cite{verl2,mose,bryz};
it involves certain formal manipulations with chiral blocks and hence may
be regarded as somewhat heuristic.
All other proofs known to us involve (implicitly or explicitly) the block
algebra ${\gB(U)}$,  and thereby the rep\-re\-sen\-ta\-ti\-on\ theory of affine Lie algebra s; as a consequence\ they
work exclusively for WZW conformal field theories.
(The two kinds of approaches to the Verlinde formula also differ drastically in
another aspect: those applicable to arbitrary conformal field theories\ only aim at
establishing a relation between the dimensions of spaces of chiral blocks and
the modular transformation properties of the characters, i.e.\ the zero point
blocks at
genus one, but they do not predict any concrete expressions for these modular
transformations. In contrast, the proofs that only work in the WZW setting do
provide such expressions, in particular the explicit form of the modular
transformation matrix $S$; roughly speaking, they combine the information
encoded in the Verlinde formula for general conformal field theories\ with the Kac\hy Peterson formula\ \erf{kpf}
for $S$.) Among these approaches which apply to the WZW case, there is one
\cite{dawe,blth} that is based on the path-integral
quantization of Chern$\mbox{-\hspace{-.66 mm}-}$ Si\-mons\  gauge theories and
is accordingly affected by the usual difficulties in setting
up a rigorous quantization procedure, e.g.\ concerning the proper mathematical
setting for path integrals. A related approach \cite{witt27,axdw} combines
holomorphic quantization of Chern$\mbox{-\hspace{-.66 mm}-}$ Si\-mons\  theories and surgery manipulations on
three-manifolds. There is also an approach \cite{fink} which combines the
isomorphism \cite{kalu3+4,kalu5+6} between the tensor categories of certain
modules over affine Lie algebra s \mbox{$\mathfrak  g$}\ at negative level and categories of modules
over quantum groups ${\sf U}_q({\mathfrak  g})$ with $q$ a root of unity with an isomorphism
between categories of \mbox{$\mathfrak  g$}-modules at positive and negative levels to deduce the
Verlinde formula from the rep\-re\-sen\-ta\-ti\-on\ theory of quantum groups.

Another possibility is to exploit the fact that the spaces of chiral blocks are
isomorphic to the spaces of holomorphic sections in line bundles
over certain projective varieties, namely over moduli spaces of semi-stable
principal bundles with structure group $G$, where $G$
is the real, compact, connected and simply connected Lie group with Lie algebra\ \mbox{$\bar{\mathfrak  g}$}.
(In the mathematics literature, sometimes this isomorphism between chiral
blocks and geometric objects, rather than an expression for their common
dimension, is
meant when the term `Verlinde conjecture' is used.) One can attempt
to analyze these spaces with various methods of traditional finite-dimensional\ algebraic
geometry; this way several proofs have been established for the case of WZW theories\
based on ${\bar{\mathfrak  g}}=\mbox{$\liematrixfont{sl}(2)$}$, where the vector bundles have rank two and their moduli
space can be described rather explicitly (see e.g.\ \cite{besz,thad2,jewe}, as
well as \cite{sorg}\,%
 \futnote{Other sources for references are \cite{beau2} and \cite{SCho2},
and also the WWW pages\\ {\tt http:/$\!$/www.ictp.trieste.it/\~{}mblau/ver.html}
\,and\, {\tt http:/$\!$/www.desy.de/\~{}jfuchs/Vfcb.html}\,.}
for a more detailed exposition and for more references). Finally, using
the methods of algebraic geometry, proofs that work for $\mbox{$\liematrixfont{sl}(N)$}$ with
arbitrary $N$ have been obtained in \cite{falt,beLa,beau}; these arguments start
from the description of the moduli space of flat connections over a curve
in a double coset form. The proof of \cite{falt} which uses torsion-free
sheaves even applies to all simple Lie algebra s \mbox{$\bar{\mathfrak  g}$}, except for
${\bar{\mathfrak  g}}=F_4$, $E_6$, $E_7$
or $E_8$. While these proofs are formulated in a purely algebraic setting, a
derivation where topological tools play an essential r\^ole was given
in \cite{tele}; there the modules are completed to Hilbert spaces
and the Verlinde formula is obtained from certain vanishing theorems for
complexes of these Hilbert spaces (compare subsection \ref{seul} below).

The argument that we presented in sections \ref{s.T}\,--\,\ref{s.F}
makes extensive use of the rep\-re\-sen\-ta\-ti\-on\ theory of affine Lie algebra s and is
in most aspects rather different from all those mentioned above. In
particular, all our arguments work simultaneously for WZW theories\ based on
arbitrary simple Lie algebra s \mbox{$\bar{\mathfrak  g}$};\,%
 \futnote{This applies likewise to the vanishing theorem in \cite{tele}.}
at no step in the derivation is there any need to distinguish between
different cases that have to be treated separately. Furthermore,
when combined with the results of \cite{fusS6} and \cite{fusS4}, respectively, our
methods should provide a possibility to characterize the spaces of chiral blocks\ and to prove the
Verlinde formula not only for WZW theories, but also for two other classes of
conformal field theories. The first of these classes is given by the so-called integer spin simple
current extensions of WZW theories; these correspond to such
WZW theories\ which in a Lagrangian setting are associated to
non-simply connected group manifolds, while for ordinary WZW theories\ one always
deals with the simply connected covering group of the simple Lie algebra\ \mbox{$\bar{\mathfrak  g}$}.
The second class of theories consists of all coset conformal field theories, including in
particular those coset
theories in which so-called field identification fixed points are present.
 %
As a final application we mention that the present representation theoretic
approach, combined with the results obtained in \cite{fusS3,furs},
should be helpful in verifying the conjectures made in \cite{fuSc8} concerning
the values of certain traces on the spaces of chiral block s.
(These traces have interesting applications to the construction of chiral block s for
integer spin simple current extensions of WZW theories, to the classification
of boundary conditions for conformal field theories\ \cite{fuSc5}, and in algebraic geometry
to the derivation of a Verlinde formula for non-simply connected groups.)

\subsection{Complexes of co-in\-va\-ri\-ant s}

We will now present a  homological interpretation of formula \erf{eN} which
expresses the Verlinde
multiplicities ${\rm dim}\,\mbox{$B$}$ as an alternating sum of non-negative integers. To this
end we use the BGG resolution of an irreducible highest weight module of
\mbox{$\mathfrak  g$}\ in terms of parabolic Verma modules \erf{vuu} which are free
$\mbox{$\mathfrak  g$}^-$-modules. We introduce the (finite) direct sums
  \begin{equation}  {\cal P}_{\!\Lambda}^{(j)}:= \sumwlj {\cal P}^{}_{\!\fc w{.37}{.32}{.1}(\Lambda+\rho)-\rho}  \end{equation}
of generalized Verma modules.
Here $\ell(w)$ denotes the length of the Weyl group element $w$.
Note that the modules ${\cal P}_{\!\Lambda}^{(j)}$ are still free ${\gM_{}^-}$-modules, and that
for every integrable \mbox{$\mathfrak  g$}-weight $\Lambda$ and every $\fc w{.5}{.48}{.1}\,{\in}\,\fc W{.7}{.78}{.2}$ the \mbox{$\bar{\mathfrak  g}$}-weight
$w({\bar\Lambda}\,{+}\,\bar\rho)\,{-}\,\bar\rho$ is dominant integral. In particular we have
  \begin{equation}  {\cal P}_{\!\Lambda}^{(j)} \,\cong\, {\sf U}({\gM_{}^-}) \otimes \mbox{\large(}\!\! \sumwlj
  \! \mbox{$\bar{\cal H}$}_{\fc w{.37}{.32}{.1}({\bar\Lambda}+\bar\rho)-\bar\rho} \mbox{\large)}
  = {\sf U}({\gM_{}^-}) \otimes \mbox{$\bar{\cal H}$}^{(j)}_\Lambda   \end{equation}
as ${\gM_{}^-}$-modules, where $\mbox{$\bar{\cal H}$}^{(j)}_\Lambda$ is the finite-dimensional\ \mbox{$\bar{\mathfrak  g}$}-module
  \begin{equation} \mbox{$\bar{\cal H}$}^{(j)}_\Lambda:= \sumwlj \mbox{$\bar{\cal H}$}_{\fc w{.37}{.32}{.1}({\bar\Lambda}+\bar\rho)-\bar\rho} \, . \end{equation}

The {\em parabolic BGG resolution\/} of an irreducible highest weight module\ \mbox{${\cal H}_\Lambda$}\ over \mbox{$\mathfrak  g$}\ with integrable
highest weight\ $\Lambda$ states \cite{gale} that there exists a semi-infinite exact
sequence of \mbox{$\mathfrak  g$}-modules with \mbox{$\mathfrak  g$}-module homomorphisms of the form
  \begin{equation}  \ldots \to {\cal P}_{\!\Lambda}^{(j)} \to {\cal P}_{\!\Lambda}^{(j-1)} \to \ldots \to {\cal P}_{\!\Lambda}^{(0)}
  \equiv {\cal P}_{\!\Lambda}^{} \to \mbox{${\cal H}_\Lambda$} \to 0 \,. \labl{133}
Note that this complex is governed by the same set ${\fc W{.7}{.78}{.2}}$ of representatives
of the coset $W/{\overline W}$ that we encountered in Subsection \ref{dbf}.
When we tensor the sequence \erf{133} with the finite-dimensional\ $\mbox{$\bar{\mathfrak  g}$}$-module
${\overline{\cal H}^{\scriptscriptstyle({m}-2)}}\equiv\bigotimes_{i=1^{}}^{{m-1}_{}}\mbox{$\bar{\cal H}^{}_{{\bar\Lambda}_i}$}$, we obtain a
semi-infinite exact sequence
  \begin{equation} \begin{array}{l}  \ldots \to {\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}^{(j)}_{\Lambda_m} \to
  {\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}^{(j-1)}_{\Lambda_m} \to \ldots \\[.7em] \hsp8 \ldots \to
  {\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}^{(1)}_{\Lambda_m} \to
  {\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}_{\Lambda_m} \to {\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\cal H$}_{{\Lambda_m}} \to 0
  \,. \end{array}  \labl{tgse}
We now observe that (by using the isomorphism \erf{47} to replace also the
first affine module in the description \erf{biso} of the chiral block s by a finite-dimensional\
\mbox{$\bar{\mathfrak  g}$}-module) the space \mbox{$B$}\ of chiral block s is isomorphic to the co-in\-va\-ri\-ant s of
${\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\cal H$}_{{\Lambda_m}}$ with respect to the  block algebra\ ${\tilde\gM}$ for a single insertion
point. (${\tilde\gM}$ is the algebra of \mbox{$\bar{\mathfrak  g}$}-valued algebraic functions that are
allowed to have a pole of finite order at a single distinguished point
and is hence isomorphic to $\mbox{$\bar{\mathfrak  g}$}\oplus\mbox{$\mathfrak  g$}^-$.)
We would like to combine this information with the exact sequence \erf{tgse}.
To this end, we first observe that the spaces in the sequence \erf{tgse}
(except for the last two) carry the structure of ${\tilde\gM}$-modules which
are free $\mbox{$\mathfrak  g$}^-$-modules.
On the other hand, taking co-in\-va\-ri\-ant s of an exact sequence does in general not
produce another exact sequence, but only a complex. However, one can show (see
Appendix \ref{arie}) that this procedure still constitutes a right-exact
functor, so that the exact sequence \erf{tgse} provides us, by taking co-in\-va\-ri\-ant s
with respect to  ${\tilde\gM}$, with a semi-infinite complex
  \begin{equation} \begin{array}{l}  \hsp{-1.1}
  \ldots \to \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}^{(j)}_{\Lambda_m}}{\tilde\gM} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}^{(j-1)}_{\Lambda_m}}{\tilde\gM} \to \ldots \\[.7em]
  \hsp{5.5} \ldots\to \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}^{(1)}_{\Lambda_m}}{\tilde\gM} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}_{\Lambda_m}}{\tilde\gM} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\cal H$}_{\Lambda_m}}{\tilde\gM}  \equiv \mbox{$B$} \to 0    \end{array} \labl{res0}
of spaces of ${\tilde\gM}$-co-in\-va\-ri\-ant s that is still exact at the two right-most
positions.

Next we note that using the two fundamental facts \erf{hhh} and \erf{17} about
co-in\-va\-ri\-ant s of free modules, one can derive the isomorphism
  \begin{equation} \begin{array}{ll} \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, {\cal P}_{\!\mu}}{\tilde\gM} \!\!&
  \;\cong\;  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\otimes (\mbox{$\U(\gminus)$}\,{\otimes}\,\mbox{$\bar{\cal H}$}_{\bar\mu})}{\tilde\gM} \\{}\\[-.7em]&
  \;\cong\;  \coi{\mbox{$\U(\gminus)$} \otimes ({\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\bar{\cal H}$}_{\bar\mu})}{{\gM_{}^-}\oplus{\bar{\mathfrak  g}}}
  \;\cong\;  \coi{ {\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\bar{\cal H}$}_{\bar\mu}}{\bar{\mathfrak  g}}  \end{array} \end{equation}
of vector spaces. Hereby we can simplify the various spaces that occur in the
complex \erf{res0}, so that it can be written as a complex that, except for
$B\,{=}\,\coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\cal H$}_{\Lambda_m}}{\tilde\gM}$, involves only \mbox{$\bar{\mathfrak  g}$}-co-in\-va\-ri\-ant s:
  \begin{equation} \hsp{-5.6} \begin{array}{l}  \ldots \to \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\bar{\cal H}$}^{(j)}_{\Lambda_m}}{\bar{\mathfrak  g}} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\bar{\cal H}$}^{(j-1)}_{\Lambda_m}}{\bar{\mathfrak  g}} \to \ldots \\[.7em]
  \hsp{7.8} \ldots\to \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\bar{\cal H}$}^{(1)}_{\Lambda_m}}{\bar{\mathfrak  g}} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\, \mbox{$\bar{\cal H}$}_{\Lambda_m}}{\bar{\mathfrak  g}} \to \mbox{$B$} \to 0  \,.   \end{array} \labl{erf}

Thus we have finally arrived at a complex of \mbox{$\bar{\mathfrak  g}$}-co-in\-va\-ri\-ant s which is governed by
the subset $\fc W{.7}{.78}{.2}$ of the Weyl group $W$ of \mbox{$\mathfrak  g$}. Moreover, by the
same arguments as in the previous section, we deduce from formula \Erf0f
for \mbox{$\bar{\mathfrak  g}$}-co-in\-va\-ri\-ant s that this semi-infinite complex is not really infinite, but
actually constitutes a finite complex of \mbox{$\bar{\mathfrak  g}$}-co-in\-va\-ri\-ant s, i.e.\ we even have
  \begin{equation} \begin{array}{l}  0 \to \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\,\overline{\cal H}^{({j_{\rm max}})}_{{\bar\Lambda}_m}}{\bar{\mathfrak  g}} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\,\overline{\cal H}^{({j_{\rm max}}-1)}_{{\bar\Lambda}_m}}{\bar{\mathfrak  g}} \to \ldots  \\[.7em] \hsp9
  \ldots \to \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\,\overline{\cal H}^{(1)}_{{\bar\Lambda}_m}}{\bar{\mathfrak  g}} \to
  \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}\,{\otimes}\,\mbox{$\bar{\cal H}$}_{{\bar\Lambda}_m}}{\bar{\mathfrak  g}} \to \mbox{$B$} \to 0   \end{array} \labl{fin}
for some non-negative integer ${j_{\rm max}}$.

\subsection{Verlinde multiplicities and Euler characteristics}\label{seul}

Owing to the finiteness of the complex \erf{fin}, we can express
the dimension of the space \mbox{$B$}\ of chiral blocks as
  \begin{equation}  {\rm dim}\,\mbox{$B$} = \mbox{\sc x} + \sum_{j=0}^{j_{\rm max}} (-1)^j \, {\rm dim}\, \mbox{\large(} \coi{{\overline{\cal H}^{\scriptscriptstyle({m}-1)}}{\otimes}
  \overline{\cal H}^{(j)}_{{\bar\Lambda}_m}}{\bar{\mathfrak  g}} \mbox{\large)}  \end{equation}
through the Euler characteristic \mbox{\sc x}\ of the complex \erf{fin} and an alternating
sum of dimensions of spaces of \mbox{$\bar{\mathfrak  g}$}-co-in\-va\-ri\-ant s. The latter coincide with the
number of singlets in the respective \mbox{$\bar{\mathfrak  g}$}-modules. Together with the identity
$(-1)^{\ell(w)}\,{=}\,\epsilon(w)$ it then follows that the result \Erf eN for the
Verlinde multiplicities can be rephrased as the statement that the Euler
characteristic \mbox{\sc x}\ of the complex \erf{fin} vanishes.

Actually, we conjecture that not only $\mbox{\sc x}=0$, but that
the whole homology of the complex is zero, i.e.\ that
\erf{fin} is in fact an {\em exact\/} sequence. Note that
it is immediate that the sequence
is indeed exact as long as the number of insertion points is $m\leq2$.
In these cases the only non-vanishing spaces in the complex \erf{erf} are those
involving the modules ${\cal P}_{\Lambda_m}$ or $\mbox{$\cal H$}_{\Lambda_m}$ (for $m\eq2$ this is
easily seen by considerations similar to those that led to formula \Erf2d;
the case $m\eq1$ can be regarded as a special case of the $m\eq2$ situation
where the second insertion point carries the weight $\Lambda\,{=}\,{{\rm k}\Lambda_{(0)}}$).
Exactness then already follows from the fact (compare Appendix \ref{arie}) that
taking co-in\-va\-ri\-ant s constitutes a right-exact functor.
Another situation where exactness can be established directly (though in a
somewhat lengthy manner, by using the explicit form of the BGG maps and of the
Clebsch$\mbox{-\hspace{-.66 mm}-}$ Gordan decomposition) is $m\eq3$ for ${\bar{\mathfrak  g}}\,{=}\, A_1$.

This conjecture about the vanishing of the homology of \erf{fin} has also
been made, from a different perspective, in \cite{kuma5}.
We do not know of any direct Lie algebraic proof of our conjecture. However, a
similar result has been obtained in \cite{tele}, based on a completion of the
highest weight modules \mbox{${\cal H}_\Lambda$}\ to topological Hilbert spaces. The vanishing
theorem of \cite{tele} refers to the cohomology of a complex that involves
dual space of the space $\coi\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em{\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})}$ of co-in\-va\-ri\-ant s considered here, i.e.\
the space of ${\gB({{\dl P}^1_{\!\!\scriptscriptstyle(\mn)}})}$-invariants in the algebraic dual $(\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em)^\star_{}$
of $\mbox{${\cal H}_{\vec\Lambda}\!\!\!\raisebox{.5em$. The vanishing statement for the homology should be related, of
course, to the corresponding statement for the cohomology in \cite{tele}.
 %
Conversely, a proof of our conjectures concerning the structure of the completed
tensor product module \mbox{$(\HL1\!{\otimes}\barB)^{\overline{\phantom w}}$}\ in subsection \ref{sB}, and hence of the
vanishing of the Euler number of the complex \erf{fin}, might constitute a first
step towards a more direct derivation of the vanishing theorem of \cite{tele}.
In this context it could also be interesting to analyze
the lowest weight socle of the algebraic dual \mbox{$(\HL1\!{\otimes}\barB)_{\phantom|}\dual$}\ of \mbox{$\HL1\!{\otimes}\barB$}\
(compare appendix \ref{sscl}).

We regard the results of \cite{tele} also as an indication that unitarity
should be a crucial input for the vanishing of the whole homology.
Indeed, this fits nicely with the observation \cite{mawa2}
that when one inserts the modular matrix $S$ for so-called admissible
\cite{kawa2} \mbox{$\mathfrak  g$}-modules (which appear in WZW theories\ at fractional level and are
not unitarizable), one still obtains integers, which however
can now also be negative,\,%
 \futnote{The proper definition of $S$ in these cases is, however, problematic;
compare e.g.\ section 5 of \cite{dolm4}.}
and as already noted in \cite{fema3},
this suggests that in this situation the Verlinde multiplicities may still
possess a homological interpretation.
More specifically, note that when one deletes the space $B$ in the complex
\erf{fin} one obtains another complex whose Euler characteristic does not
vanish any more but is given by ${\rm dim}\,\mbox{$B$}$; in the non-unitary case we would
expect the presence of a similar complex whose Euler characteristic is still
given by the Verlinde multiplicity, but whose
homology is no longer concentrated at the last position.

The interpretation of such a complex and its relation to the true fusion
coefficients (which are dimensions and therefore are manifestly non-negative,
and which can for example be deduced by explicitly
solving the constraints implied by null vector decoupling, in particular the
Knizh\-nik$\mbox{-\hspace{-.66 mm}-}$ Za\-mo\-lod\-chi\-kov equation \cite{fema3,awya,pery3,fugp6})
remains to be clarified. In this context the observation \cite{mawa2} that
in many cases the Verlinde
multiplicities are controlled by the integral parts of the weights, and hence
by fusion rules of unitary theories, is particularly interesting.
In accordance with general experience with Borel$\mbox{-\hspace{-.66 mm}-}$ Weil$\mbox{-\hspace{-.66 mm}-}$ Bott  theory, it might
find its explanation by the property that in the non-unitary case the
homology is still concentrated
at a single position of the complex, albeit not at the last one.

We finally remark that the traces of the action of certain outer automorphisms
on the spaces of chiral block s \cite{fuSc8} are (possibly negative) integers.
This suggests that they might find a natural interpretation as (possibly
twisted) Euler characteristics as well. Such an interpretation would not
only explain the rather surprising integrality properties of these traces,
but may also be helpful for a proof of the conjectures in \cite{fuSc8}.

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