A spin decomposition of the Verlinde formulas for type A modular categories
Christian Blanchet
A modular category is a braided category with some additional algebraic features. % (duality, twist, a finite set of simple objects %satisfying a domination property and a non-degeneracy axiom). The interest of this concept is that it provides a Topological Quantum Field Theory in dimension $3$. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with $SU(N)$. Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.
spin_verlinde.tex
%latex
%version 10 02 2005


\documentclass[12pt]{amsart}
\usepackage[latin1]{inputenc}
\usepackage{amssymb,pstricks}
\usepackage{graphicx}
\setlength\unitlength{1mm}


%\nocom
%\PSTricksOff
\newtheorem{thm}{Theorem}[section] 
\newtheorem{pro}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}

\theoremstyle{definition}
\newtheorem{dfn}{Definition}[section]
\newtheorem{exa}{Example}[section]

\theoremstyle{remark}
\newtheorem*{rem}{Remark}
\newtheorem*{rems}{Remarks}
%\newcommand{\be}{\begin{equation}}
%\newcommand{\ee}{\end{equation}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\V}{\mathbf{V}\hspace{-2.5pt}}

\newcommand{\ra}{\rangle}\newcommand{\la}{\langle}


\newcommand{\fk}{\mathbf{k}}
\newcommand{\esp}{\vspace{5pt}}
\newcommand{\one}{{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.7,0.65)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }\!{}}

\newcommand{\oneun}{\one_1}

\begin{document}
\title[Spin Verlinde formulas]{A spin decomposition
of the Verlinde formulas for type A modular categories}

\author{Christian Blanchet}
\address{L.M.A.M., Universit\'e de Bretagne-Sud,
BP 573, F-56017 Vannes, France  }
\email{Christian.Blanchet@univ-ubs.fr}
\date{February 2005}
\subjclass[2000]{57R56}
\keywords{Verlinde formula, modular category, modular functor,
TQFT, 3-manifold, quantum invariant, spin structure}

\begin{abstract}
 A modular category is a braided category with some additional
algebraic features.
% (duality, twist, a finite set of simple objects
%satisfying a domination property and a non-degeneracy axiom).
The interest of this concept is that it provides
a Topological Quantum Field Theory in dimension $3$.
 The Verlinde formulas associated with a modular category are the dimensions
 of the TQFT modules.
  We  discuss reductions and
  refinements of these formulas for   modular categories related with $SU(N)$.
 Our main result is a splitting of the Verlinde formula,
corresponding to a brick decomposition of the TQFT modules
whose summands are indexed by spin structures modulo
an even integer.
 We introduce here the notion of a spin modular category, and give 
the proof of the decomposition theorem in this general context.  
\end{abstract}

\maketitle\setcounter{section}{-1}
\section{Introduction}
Given a simple, simply connected complex Lie group $G$,
 the Verlinde formula \cite{Ver}
    is a combinatorial function $\mathcal{V}_G: 
(K,g)\mapsto \mathcal{V}_G(K,g)$
associated
    with $G$ (here the integers $K$ and $g$ are respectively
the level and the genus). In conformal field theory this formula
gives the dimension of the so called { conformal blocks}.
 Its combinatorics was intensively studied 
since this formula  has a deep interpretation 
as the rank of a space of { generalized theta functions}
(sections of some bundle over the moduli space of $G$-bundles
over a Riemann surface) \cite{BL,Beau,Fal,So}.
See \cite{BiL,BiL2}, for a  development using
methods of sympleptic geometry. 


We will consider here  a purely topological approach
to Verlinde formulas related with  $SU(N)$.
 The genus $g$ Verlinde formula associated with a modular category 
\cite{Tu} is
 the dimension
 of the TQFT-module of a genus $g$ surface; the general formula is
given in \cite[IV,12.1.2]{Tu}.
 Various constructions of modular categories are known, either
from quantum groups \cite{AP,BK,Sa} or from skein theory \cite{TW2,Bhec,BB}.  
The geometric Verlinde formula for the group $SU(N)$
at level $K$ is recovered from the so called $SU(N,K)$
modular category. This modular category can be 
 obtained either
from the quantum group $U_qsl(N)$ when $q=s^2$ is a primitive
$(N+K)$-th root of unity or from Homfly skein theory. Its simple objects correspond to
the weights in the fundamental alcove.
 
 One may also consider a modular category with less simple objects.
This was done for $gcd(N,K)=1$ by restricting to representations whose heighest weight is in the root lattice, and was called the  projective or $PSU(N)$ theory \cite{KT,Yo,MW,Le,LT,Sa2}.
 Using an appropriate choice of the framing parameter in Homfly skein theory,
we have obtained in \cite{Bhec} a variant which is defined
for all $N,K$. We are not aware of a quantum group approach
to these reduced modular categories for $\gcd(N,K)>1$. Nevertheless we find it convenient to call them $PU(N,K)$ modular categories. In our construction
the simple object corresponding to the deformation of the determinant of
the vector representation of $sl(N)$ may be non trivial; we think
that a version of the quantum group $U_q(gl(N))$ could be used here. 

As well known, the Verlinde formula  for the $SU(N,K)$ modular category
coincides with the formula in conformal fields theory for the group $SU(N)$;
 $$d_{N,K}(g)=\mathcal{V}_{SU(N)}(K,g).$$
 We show that for the $PU(N,K)$ modular category 
 the Verlinde formula is
 $$\tilde{d}_{N,K}(g)=\frac{d_{N,K}(g)}{{N'}^g},$$
 where $N'=\frac{N}{\gcd(N,K)}$.
 These integral numbers satisfy the level-rank duality relation
 $$\tilde{d}_{N,K}(g)=\tilde{d}_{K,N}(g)\ ,$$
 which is an integral version
 of a reciprocity formula in \cite{rec} (see also \cite{KT}).

 Our main contribution here is to show that under certain condition the TQFT modules decompose in blocks indexed by spin type structures (respectively $1$-dimensional
 cohomology classes) on the surface, and compute the corresponding refined
 Verlinde formulas. 
 
An important part of this paper is devoted to the spin decomposition theorem.
 The proof  is given in the general case of a 
modulo $d$ spin  modular category; this notion, developed in Section
\ref{spin-g}, is new and appears in the  $\mathbb{Z}/d$ graded cases which are  
not weakly non-degenerate in \cite{LT}. 
As a motivation, we give below the combinatorial counterpart
of  this theorem for the A series, in the special case where the rank
is even and 
divides the level (Theorem \ref{spin_A}).
 We consider the action of $\mathbb{Z}/N$ on the set
$$\Gamma_{N,K}=\{\lambda=(\lambda_1,\dots,\lambda_{N}),\
K\geq\lambda_1\geq\dots\geq \lambda_{N-1}\geq\lambda_N=0\}\ ,$$ 
given for the generator 
 of the cyclic group
$\mathbb{Z}/N$ by
$$(\lambda_1,\dots,\lambda_{N-1},0)\longmapsto
(K,\lambda_1,\dots,\lambda_{N-1})-(\lambda_{N-1},\dots,\lambda_{N-1})\ .$$ 
 We denote by $\sharp\, \mathrm{orb}(\lambda)$
 the cardinality of the orbit of $\lambda$,
and by $Stab(\lambda)$ the stabilizer subgroup. 
 For $\mathfrak{a},\mathfrak{b}\in\mathbb{Z}/N$, the numbers 
$\epsilon_\lambda(\mathfrak{a},\mathfrak{b}) \in
\{0,1,-\frac{1}{2},\frac{1}{2}\}$ are defined  as follows.

\esp\noindent If $\sharp\, \mathrm{orb}(\lambda)$ is even, then
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
1\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$  are zero modulo $|Stab(\lambda)|$,}\\
0 \ \text{ else.}\end{array}\right.$$
If $\sharp\, \mathrm{orb}(\lambda)$ is odd, then
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
\frac{1}{2}(-1)^{\frac{2\mathfrak{a}}{|Stab(\lambda)|}\frac{2\mathfrak{b}}{|Stab(\lambda)|}}
\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ 
are  zero modulo $\frac{|Stab(\lambda)|}{2}$,}\\
0\ \text{ else.}\\
\end{array}\right.$$

\begin{thm} Suppose that
$N$ is even, and that $K/N$ is an odd integer.\\
a) For $(a,b)\in(\mathbb{Z}/N)^g\times (\mathbb{Z}/N)^g$, the formula 
\begin{eqnarray*} d_{N,K}^{(a,b)}(g)&=&
\left((N+K)^{N-1}N\right)^{g-1}
  \sum\limits_{\lambda\in\Gamma_{N,K}}\ \prod\limits_{\nu=1}^g
\frac{\epsilon_\lambda(a_\nu,b_\nu)}{(\sharp\, \mathrm{orb}(\lambda))^2}\\
& &\times\prod\limits_{1\leq i<j\leq N}
  \left({2\sin{(\lambda_i-i-\lambda_j+j)\frac{\pi}{N+K}}}
\right)^{2-2g} 
\ .\end{eqnarray*}
defines a natural number $d_{N,K}^{(a,b)}(g)$,\\
b) There exists a splitting of the  $SU(N)$ Verlinde formula at level $K$
$$ \mathcal{V}_{SU(N)}(K,g)=\sum\limits_{(a,b)\in (\mathbb{Z}/N)^g\times (\mathbb{Z}/N)^g}
d_{N,K}^{(a,b)}(g)\ .$$
\end{thm}


For $N=2$, the spin TQFT producing the above decomposition 
was studied in \cite{BM}, and a nice 
algebrico-geometric interpretation  was obtained by 
Andersen and Masbaum \cite{AM}.
We quote that for $N>2$, the involved spin structures are not the usual ones.
 These structures  have coefficients modulo an even integer; they
can be understood as something intermediate between usual spin structures
(with modulo $2$ coefficients) and complex spin structures.
 The convenient formalism for the TQFT involving these structures
should be a slightly extended version of Homotopy Quantum Field Theory
as developed by Turaev \cite{Tu3,Tu4}.

The paper is organized as follows. In Section 1 we study spin structures
modulo an even integer. In Section 2 we define our spin modular categories.
 In Section 3 we establish the spin decomposition of the TQFT in a general
context. In Section 4 we consider Verlinde formulas for modular
categories of the A series. 
In Section 5 we establish similar decomposition theorems based
on $1$-dimensional cohomology classes. In Section 6 we give computer results for small
values of $N$ and $K$. 
\section{Spin structures modulo an even integer}
Let $d$ be an even integer. We recall 
 here the topological definition for modulo $d$  spin  structures
that was given in \cite{varso,Bhec}.

There exists, up to homotopy, a
unique non-trivial map
$g$ from the classifying space $BSO$ to the Eilenberg-MacLane space $K(\mathbb{Z}/d,2)$.
 Define the fibration
$$\pi_d: BSpin(\mathbb{Z}/d)\rightarrow BSO$$
to be the pull-back, using $g$, of the path
fibration over $K(\mathbb{Z}/d,2)$.
 The space $BSpin(\mathbb{Z}/d)$ is a classifying space
for the non-trivial central extension
of the Lie group $SO$ by $\mathbb{Z}/d$,
which we denote by $Spin(\mathbb{Z}/d)$. For $d=2$, this group
$Spin(\mathbb{Z}/2)=Spin$ is the universal cover of $SO$, and for general
$d$, we have
$$ Spin(\mathbb{Z}/d)=\frac{Spin\times \mathbb{Z}/d}{(-1,d/2)}\ .$$


Now we can use the fibration $\pi_d$ to define structures.  Let \mbox{$E_{Spin(\mathbb{Z}/d)}=\pi_d^*(E_{SO})$}
be the pull-back of the canonical vector bundle over $BSO$.

\begin{dfn} A   modulo $d$ spin structure 
(or $Spin(\Z/d)$  structure) on a manifold $M$
is an homotopy class
of fiber maps from the stable tangent bundle $\tau_M$ to
$E_{Spin(\mathbb{Z}/d)}$.
\end{dfn}
If non-empty the set of these structures, denoted by
$Spin(M;\mathbb{Z}/d)$, is affinely isomorphic to
$H^1(M;\mathbb{Z}/d)$, by obstruction theory.
 Moreover the obstruction for existence is a class
 $w_2(M;\mathbb{Z}/d)\in H^2(M;\mathbb{Z}/d)$, which is the image of
 the Stiefel-Whitney class $w_2(M)$ under the homomorphism
 induced by the inclusion of coefficients
 $\mathbb{Z}/2\hookrightarrow \mathbb{Z}/d$.
 The Stiefel-Whitney class $w_2(M)$ is zero for every compact oriented
manifold whose dimension is lower or equal to $3$, hence
 spin  structures modulo $d$ exist on these manifolds.

The various descriptions of  usual spin structures \cite{Milnor} apply
to modulo $d$ spin structures.
 The above definition defines, up to equivalence, a $Spin(\mathbb{Z}/d)$ principal
bundle over the stable oriented framed bundle $PTM$ (with fiber $SO$)
whose restriction to the fiber is equivalent to
the cover map $Spin(\mathbb{Z}/d)\rightarrow SO$.
 The cover of $PTM$ defined by the modulo $d$ spin structure
is classified by a cohomology class $\sigma\in H^1(PTM,\mathbb{Z}/d)$
whose restriction to the fiber is non-trivial. The above correspondence
is   one to one;
this gives an
alternative definition, and we will identify
$Spin(M;\mathbb{Z}/d)$ with the corresponding affine sub-space of
$H^1(PTM,\mathbb{Z}/d)$.
\begin{dfn}[Alternative definition of modulo $d$ spin structures]
A   modulo $d$ spin structure 
  on an oriented manifold $M$
is  a cohomology class $\sigma\in H^1(PTM,\mathbb{Z}/d)$
whose restriction to the fiber is non-trivial.
\end{dfn}
 Observe that a spin structure can be evaluated on
a framed $1$-cycle in the manifold.

Let us consider an oriented surface $\Sigma$.
 An immersed curve has a preferred framing defined by using the tangent
vector. If a closed embedded curve $\gamma$ bounds a disc, then the evaluation
of a modulo $d$ spin structure on the corresponding framed
$1$-cycle $\tilde{\gamma}$ is $\frac{d}{2}$.
 Following \cite{A,J}, we get the theorem below which gives a convenient
description of modulo $d$ spin structures on the oriented surface $\Sigma$.
\begin{thm}
a) Let $\gamma$ denotes an embedded closed curve with $\sharp\gamma$ components.
 The assignement $\gamma\mapsto\sigma(\tilde{\gamma})+
(\sharp\gamma)\frac{d}{2}$
extends to a well defined map 
$q_\sigma: H_1(\Sigma,\mathbb{Z}/d)\rightarrow \mathbb{Z}/d$.\\
b) The map  $\sigma\mapsto q_\sigma$ defines
a canonical bijection between $Spin(\Sigma,\mathbb{Z}/d)$
and the set of maps
$q:H_1(\Sigma,\mathbb{Z}/d)\rightarrow \mathbb{Z}/d$
 such that for all $x$, $y$ one has
\begin{equation} \label{int}q(x+y)=q(x)+q(y)+\frac{d}{2}x.y\ .\end{equation}
Here $.$ denotes the intersection form
on $H_1(\Sigma,\mathbb{Z}/d)$.
\end{thm}
\begin{proof}
 The formula $\sigma(\tilde{\gamma})+(\sharp\gamma)\frac{d}{2}$ 
is unchanged if we add or remove to the embedded curve $\gamma$
a trivial component.
 Let us denote by $\gamma$ (resp. $\gamma'$) the left handed (resp. right handed) curve
 in the band move represented in Figure \ref{bandmove}.
We have that $\sharp\gamma'-\sharp\gamma=\pm 1$. By considering the Gauss map,
we see that the  cycle $\tilde \gamma' -\tilde \gamma$ is homologous
in $PT\Sigma$
to $\tilde u$ where $u$ bounds a disc  on the surface.
 We get that
 the formula is also
unchanged under this band move.
 We deduce that homologous curves in $\Sigma$ give the same result; hence we have that
$q_\sigma$ is well defined on $H_1(\Sigma,\mathbb{Z})$.
% Using that $q_\sigma([\gamma])$ is additive for disjoint unions,
%we show that $q_\sigma$ is well defined on $H_1(\Sigma,\mathbb{Z}/d)$.
 \begin{figure}
\centerline{
\begin{pspicture}[.6](0,-0.5)(1,1)
\pscurve{->}(0,0)(.5,.3)(1,0)
\pscurve{<-}(0,1)(.5,.7)(1,1)
\end{pspicture}\hspace{18pt}
$\ \longleftrightarrow\ $
\begin{pspicture}[.6](0,-0.5)(2,1)
\pscurve{->}(1,0)(1.3,.5)(1,1)
%\pscurve{->}(.6,.5)(1,.9)(1.4,.5)(1,.1)(.6,.5)
%\pscurve{->}(.63,.42)(.615,.47)(.61,.50)
\pscurve{<-}(2,0)(1.7,.5)(2,1)
\end{pspicture}}
\caption{\label{bandmove}Band move}
\end{figure}
 
Let $\gamma$ be  a generic immersed curve. 
 Smoothing a crossing changes $\sharp\gamma$
by $\pm 1$ and does not change the $1$-cycle $\tilde \gamma$. Hence one has $q_\sigma([\gamma])=
\sigma(\tilde{\gamma})+(\sharp\gamma+I(\gamma))\frac{d}{2}$,
where $I(\gamma)$ is the number of 
double points. 
 It follows that for any $x,y\in H_1(\Sigma,\mathbb{Z})$, 
Property  (\ref{int}) holds.
 We deduce that $q_\sigma$ is well defined on $H_1(\Sigma,\mathbb{Z}/d)$.
Bijectivity is established by using that
 the map  $q_\sigma$ commutes with the action of $H^1(\Sigma,\mathbb{Z}/d)$.
\end{proof}


 Let $M=\mathbf{S}^3(L)$ be obtained by surgery on the framed link
$L$ in the $3$-sphere.
 We want to give a combinatorial description
for modulo $d$ spin structures on $M$.  Recall that $M$ is the boundary of a
$4$-manifold $W_L$ called the trace of the surgery.
To each $\sigma\in Spin(M;{\mathbb{Z}/d})$ is associated
a relative obstruction $w_2(\sigma;{\mathbb{Z}/d})$
in $H^2(W_L,M;\mathbb{Z}/d)$. 
 The group $H^2(W_L,M;\mathbb{Z}/d)$ is a free $\mathbb{Z}/d$ module
of rank $m=\sharp L$.
 Taking the coordinates of the relative obstruction
 in the preferred basis (the basis which is Poincar� dual to the cores of the handles), 
we get a map $\psi_L: Spin(M;{\mathbb{Z}/d})
\rightarrow\left(\mathbb{Z}/d\right)^m$.

The following theorem is proved in \cite{Bhec}.{Here  $B_L=(b_{ij})$ is
the linking matrix.}
\begin{thm} The map $\psi_L: Spin(M;{\mathbb{Z}/d})
\rightarrow\left(\mathbb{Z}/d\right)^m$ is injective, and its image
is the set of those $(c_1,\dots,c_m)$ which are
solutions of the following $\mathbb{Z}/d$-characteristic
equation
$$B_L\left(\begin{array}{c} c_1\\ \vdots\\ c_m\end{array}\right)=
{\frac{d}{2}}\left(\begin{array}{c} b_{11}\\ \vdots\\
 b_{mm}\end{array}\right)\ \ ({\rm mod}
\ d)\ .$$
\end{thm}


\section{Spin modular categories}
 A ribbon category is a category equipped with  tensor product,
braiding, twist and duality satisfying compatibility conditions
\cite{Tu}.
 If we are given a ribbon category $\mathcal{C}$, then we can define
an invariant of links whose components are colored with  objects of
$\mathcal{C}$. This invariant extends to a representation of the
$\mathcal{C}$-colored tangle category and more generally
to a representation of the category of $\mathcal{C}$-colored ribbon graphs
\cite[I.2.5]{Tu}.
 In a ribbon category there is a notion of trace of morphisms
and dimension of objects. 
 The trace of a morphism $f$ is denoted by $\langle f\rangle$.

\centerline{\raisebox{12mm}{$\langle f\rangle=$} \begin{picture}(10,28)
\put(6,12 ){$f$}
\put(1,2){\includegraphics[height=24mm]{tr.eps}}
\end{picture}}
 The dimension
of an object $V$ is the trace of the identity morphism $\one_V$; we will use the notation
$\langle V\rangle$ as well as  $\langle \one_V\rangle$.
 We often say {\em quantum} trace and dimension
to distinguish from the usual trace and dimension in vector spaces. 

Let $\fk$ be a field. A ribbon category  is said to be
$\fk$-additive if the Hom sets are 
$\fk$-vector spaces, composition and tensor product are
bilinear, and $End(\text{trivial object})=\fk$.


We first recall the definition of a modular category
\cite{Tu,PS}. 
A {\em modular category} 
over $\fk$ 
is a $\fk$-additive ribbon category in which there
exists a finite family $\Gamma$ of simple objects $\lambda$
(here simple means that $u\mapsto u \one\!_{\lambda}$
from $\fk=End(\mathrm{trivial\ object})$ to $End(\lambda)$ is an isomorphism)
satisfying the  axioms below. 
\begin{itemize}
%\item (Normalization axiom) The trivial object 
%is in $\Gamma$.
%\item (Duality axiom) For any object $\lambda\in \Gamma$,
%the dual  $\lambda^\ast$ is isomorphic to an object in
%$\Gamma$.
\item (Domination axiom) For any object $V$ in the category
there exists a finite decomposition
$\one\!_V=\sum_i f_i\one\!_{\lambda_i}g_i$,
with $\lambda_i\in\Gamma$ for every $i$.
\item (Non-degeneracy axiom) The following matrix is invertible.
$$S=(S_{\lambda\mu})_{\lambda,\mu\in \Gamma}$$
where $S_{\lambda\mu}\in \fk$ is the endomorphism
of the trivial object
associated with the $(\lambda,\mu)$-colored,
$0$-framed Hopf link  with linking $+1$. 
\end{itemize}
It follows that $\Gamma$ is a representative
set of isomorphism classes of simple objects; note that the trivial
object $\Theta$ is simple, so that we may suppose that $\Theta$
is in $\Gamma$.
 If we replace the non-degeneracy axiom
by the non-singularity condidition below
then  we have the definition of a 
{\em pre-modular category} (
a morphism $f\in Hom(V,W)$ is called negligible if
for any $g\in Hom(W,V)$ we have $\la fg\ra=0$): 
\begin{itemize}
\item  (Non-singularity) The category has no non-trivial negligible morphism.
\end{itemize}
 
 A general modularization procedure for pre-modular categories, 
and a criterion for existence are developed by Brugui\`eres 
\cite{Bru}, and by M\"uger in the context of $*$-categories
\cite{Mu}.
 Note that after quotienting by negligible we get the
non-singularity condition. This property gives that the 
pairing 
$$ \begin{array}{ccc}
Hom(V,W)\otimes  Hom(W,V)&\mapsto &\fk\\
f\otimes g&\mapsto&\la fg\ra
\end{array}$$
is non singular. We can deduce that there exists no non-trivial morphism
between non-isomorphic simple
objects.



  One may ask further that the category has direct sums.
 In fact direct sums may be added in  a formal way,
and a pre-modular category with direct sums is abelian. This latter
fact was pointed out to us by  Brugui�res.


 In a modular category $\mathcal{C}$, with representative set of simple objects
$\Gamma$, the {\em Kirby color} $\Omega=\sum_{\lambda\in\Gamma}\langle \lambda \rangle \lambda$
 is used to define an invariant of  closed
oriented manifolds  with colored graph.
 If $M=S^3(L)$ is obtained by surgery on the framed link
$L$ in the sphere and contains a colored graph $K$, then a formula for this invariant is
$$\tau_{\mathcal{C}}(M,K)=
\frac{{\langle L(\Omega,
\dots,\Omega),K\rangle}}{\langle U_1(\Omega)\rangle^{b_+}
\langle U_{-1}(\Omega)\rangle^{b_-}}\ .
$$
Here $b_+$ (resp. $b_-$) is the number of
positive (resp. negative) eigenvalues of the linking
matrix $B_L$, and $U_{\pm1}$ denotes the unknot with framing $\pm1$.

Modular $G$-categories, with $G$ a group have been introduced
by Turaev in \cite{Tu4}; details in the case of an abelian group $G$,
 and examples derived
from quantum groups are given in \cite{LT}.

Let $G$ be an abelian  group. A $G$ {\em grading} of a $\fk$-additive
 monoidal category $\mathcal{C}$
is a family of full sub-categories $\mathcal{C}_j,\ j\in G$,
such that \begin{itemize}
\item[(i)] for any pair of objects $V\in Obj(\mathcal{C}_j)$,
$V'\in Obj(\mathcal{C}_{j'})$, one has  $V\otimes V'\in Obj(\mathcal{C}_{j+j'})$;\\
\item[(ii)] if for some pair of objects $V\in Obj(\mathcal{C}_j)$,
$V'\in Obj(\mathcal{C}_{j'})$ one has   $Hom_{\mathcal{C}}(V,V')\neq\{0\}$,
then $ j=j'$;\\
\item[(iii)] each object of $\mathcal{C}$ is either in  $\cup_j Ob(\mathcal{C}_j)$,
or  a direct sum of objects in $\cup_j Obj(\mathcal{C}_j)$.
\end{itemize}

Axiom (iii) asks that every object splits as a direct sum of homogeneous
objects. Axiom (i) asks that tensor product is homogeneous, and axiom (ii) that
any non-zero morphism with source or target
an homogeneous object is homogeneous; this implies that the dual of an homogeneous object 
has opposite grading.

Let $\mathcal{C}$ be a modular category. We denote
by $\mathcal{U}(\mathcal{C})$ the abelian group of isomorphism
classes of invertible objects in $\mathcal{C}$
(the law is tensor product).
 If $U$ is a subgroup of ${\mathcal{U}}(\mathcal{C})$ and 
$G=\hat U$ is
the group of characters $\chi:U\rightarrow \fk^*$, then  the category
$\mathcal{C}$ is  $G$ graded. A simple object $\lambda$
is an object in $\mathcal{C}_\chi$
 if and only if for every $J\in U$
equality 
in Figure \ref{grading} holds.


 
\begin{figure}
\centerline{
%\raisebox{20mm}{$\displaystyle \forall J\in U\ \ ,\ \ $}
\begin{picture}(60,40)
\put(0,0){\includegraphics[height=40mm]{grad.eps}}
\put(9,5){$\lambda$}
\put(21,18){$J$}
\put(35,20){$=\ \ \ \chi(J)$}
\put(57,5){$\lambda$}
\end{picture}
}
\caption{\label{grading}}
\end{figure}


A {\em modular $G$-category} \cite{LT}
over $\fk$ 
is a $G$ graded  $\fk$-additive ribbon category 
$(\mathcal{C};\mathcal{C}_j,\ j\in G)$
 in which there
exists  finite families $\Gamma_j\subset Ob(\mathcal{C}_j)$, $j\in G$,
 of simple objects $\lambda$
satisfying the  axioms below.
\begin{itemize}
%\item (Normalization axiom) The trivial object 
%is in $\Gamma_0$.
%\item (Duality axiom) For any object $\lambda\in \Gamma_j$,
%the dual  $\lambda^\ast$ is isomorphic to an object in
%$\Gamma_{-j}$.
\item (Domination axiom) For any object $V$ in $\mathcal{C}_j$
there exists a finite decomposition
$\one\!_V=\sum_i f_i\one\!_{\lambda_i}g_i$,
with $\lambda_i\in\Gamma_j$ for every $i$.
\item  (Non-degeneracy axiom) The following matrix is invertible.
$$S=(S_{\lambda\mu})_{\lambda,\mu\in \Gamma_0}$$
where $S_{\lambda\mu}\in \fk$ is the endomorphism
of the trivial object
associated with the $(\lambda,\mu)$-colored,
$0$-framed Hopf link  with linking $+1$. 
\end{itemize}



 It is shown in \cite{Tu4} that a modular $G$-category
with $G$ an abelian group
gives  invariants of $3$-manifolds
equipped with a $1$-dimensional cohomology class.
 A modular $G$-category may not be a modular category, even
in the case where $G$ is a finite abelian group
(see \cite[Section 1.6]{LT}).
 We point out that a modular category with a $G$ grading
is not necessarily a modular $G$-category.
 The reason is that the $S$-matrix restricted to zero graded
objects may be non-invertible. 
 In addition, the  zero graded
subcategory may be non-modularizable, so that there is no hope to get
a modular $G$-category by using some modularization procedure.
 The latter fact implies that the modular $G$-category
is not weakly non-degenerate \cite{LT};
it is verified for the class of modular categories
we consider below. These categories  have a $\Z/d$
grading with $d$ even and  give invariants of $3$-manifolds
equipped with modulo $d$ spin structures; the relevant version 
of Homotopy Quantum Field Theories as considered by Turaev,
 should be understood in relation with
\cite[Remark 7.4.6]{Tu4}. 
 
For a simple object $\lambda$ the twist coefficient $\theta_\lambda$
is defined by the figure \ref{twist}.
 In the quantum group context, this coefficient is given by the action of the 
so called quantum Casimir.
\begin{figure}
$$\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{-}(.1,0)(.15,.45)(.45,.75)(.7,.5)(.4,.2)(.20,.38)
\pscurve{->}(.14,.62)(.1,.9)(.1,1)
 \put(-.2,0){$\lambda$}
\end{pspicture}
\ =\ \theta_\lambda\ 
\begin{pspicture}[.5](0,-.5)(1,1)
\psline{->}(.1,0)(.1,1)
 \put(.2,0){$\lambda$}
\end{pspicture}$$
\caption{\label{twist}}
\end{figure}
\label{spin-g}
\begin{dfn}
Let $d$ be an even integer (resp. an integer). 
A  modular category is modulo $d$ spin\footnote{Invariants associated with a (mod. 2) spin modular category
are considered in \cite{Sa3}[Theorem 2b]. Our definition here is more general.} (resp. modulo $d$ cohomological) 
if it is equipped
with
 an invertible object $\varrho$ whose order is $d$ and whose twist coefficient
is $\theta_\varrho=-1$ (resp. $\theta_\varrho=1$).
\end{dfn}

In the following we will mainly discuss the spin case; the cohomological case
will be considered in section \ref{coho_}.
  
Let $(\mathcal{C},\varrho)$ be a modulo $d$ spin  modular category,
with $\Gamma$ as a representative set of simple objects. 
 The object $\varrho^d$ is isomorphic to the trivial, hence we have
$\la \varrho^{d}\ra=1$. The dual objects $\varrho$ and 
$\varrho^{d-1}$ have the same quantum dimension.
We deduce that $\mathfrak{d}=\la \varrho\ra=\pm 1$.
 Note that invertible objects are simple, hence the braiding for
$\varrho^2$ is identity up to a scalar.
 By closing we get this scalar and  establish the  following identity.\\[10pt]
\begin{equation}\label{cross_rho}
\begin{pspicture}[.5](0,-0.5)(1,1)
\psline{->}(1,0)(0,1)
\psline{-}(0,0)(.35,.35)
\psline{->}(.65,.65)(1,1)
 \put(0,.3){$\varrho$} \put(.8,.3){$\varrho$}
\end{pspicture}
\ =\ -\ 
\mathfrak{d}\ \ 
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.3,.5)(0,1)
\pscurve{->}(1,0)(.7,.5)(1,1)
 \put(0,.4){$\varrho$} \put(.8,.4){$\varrho$}
\end{pspicture}
\end{equation}

The next identity is obtained in a similar way.\\[10pt]

\begin{equation}\label{band_rho}
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.3,.5)(0,1)
\pscurve{<-}(1,0)(.7,.5)(1,1)
 \put(0,.4){$\varrho$} \put(.8,.4){$\varrho$}
\end{pspicture}
\ =\ \mathfrak{d}\ \ \ 
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.5,.3)(1,0)
\pscurve{<-}(0,1)(.5,.7)(1,1)
 \put(.3,0){$\varrho$} \put(.4,.9){$\varrho$}
\end{pspicture}
\end{equation}
 
 It is convenient to fix a primitive $d$-th root of unity $\zeta$,
and to identify the group of characters 
$\chi :\{\varrho^j,j\in\mathbb{Z}/d\} \rightarrow \fk^*$ with $\Z/d$.
 Then the category $\mathcal{C}$ is $\Z/d$ graded.
 A simple object $V$ has degree equal to $j$ if and only if the equality
in figure \ref{varrho} holds.
\begin{figure} 
\centerline{
        \begin{pspicture}(0,-.5)(5,2)
        %\psgrid
        %\psframe(1.5,1.5)(2.5,2)
        \put(2.1,0){$\varrho$}\put(3.0,0){$V$}
        \psline[linearc=.3](2,0)(2,1.05)
        %\psline[linearc=.3](2,2)(2,2.6)
        \psline[linearc=.3]{->}(2,1.35)(2,1.7)
        \pscurve(2.8,0)(2.7,.5)(2.2,.7)
        \pscurve{->}(1.9,.8)(1.7,1)(2.75,1.4)(2.8,1.7)
\put(4,.7){ $=\ \ \zeta^{j}$}
\end{pspicture}
        \begin{pspicture}(1,-0.5)(5,2)
        %\psgrid
        %\psframe(1.5,1.5)(2.5,2)
        \put(2.1,0){$\varrho$}\put(3,0){$V$}
        %\psline[linearc=.3](2,1)(2,1.5)
        \psline[linearc=.3]{->}(2,0)(2,1.7)
        \psline{->}(2.8,0)(2.8,1.7)
        \end{pspicture}}
\caption{\label{varrho}}
\end{figure}

 The Kirby color  decomposes according to
this grading.
$$\Omega=\sum_{\lambda\in \Gamma}
\la \lambda \ra \lambda\ =\sum_{j\in \Z/d}\ \Omega_j$$
 Here the  notation $\la \lambda \ra$ is the quantum dimension of $\lambda$.

The proof of the  theorem below
is the same as in the ungraded case (see e.g. \cite{BB}). 
 The statement holds for
any  $G$ graded pre-modular category with $G$ an abelian group
\cite[Prop. 1.4]{LT}.

\begin{thm}(Graded sliding property)
Suppose that $V_j$ is an object in $\mathcal{C}_j$, then the equality
in Figure \ref{graded_sliding} holds for any $j'\in\mathbb{Z}/d$.
 Here the framed knot labeled with $\Omega_{j'}$ may be knotted or 
linked with the other component labeled $V_{j}$; this fact is represented by the dashed part in the figure.
\begin{figure} 
\centerline{\includegraphics[height=50mm]{slid.eps}}
\caption{\label{graded_sliding}Graded sliding property.}
\end{figure}
\end{thm}


%\section{Spin refined $3$-manifold invariants}
\label{spinref}
The following theorem is proved from the graded sliding property
as was done in \cite[Theorem 4.2]{Bhec}.
 We suppose 
that $\mathcal{C}$ is a modulo $d$ spin  modular
category, and that 
$$\Omega=\sum_{\lambda\in \Gamma}
\la \lambda \ra \lambda\ =\sum_{j\in \Z/d}\ \Omega_j$$
is the graded decomposition of the Kirby color;
note \cite[lemma 4.5]{Bhec} that $\langle U_{\pm 1}(\Omega)\rangle=
\langle U_{\pm 1}(\Omega_{d/2})\rangle$.
\begin{thm}Let $\mathcal{C}$ be a modulo $d$ spin modular category,
and 
\mbox{$\Omega=\sum_{j\in \Z/d}\ \Omega_j$} be the graded decomposition of the Kirby
element.
Provided $c=(c_1,\dots,c_m)$ satisfies the modulo $d$
characteristic condition,
the formula
$$\tau_{\mathcal{C}}^{\mathrm{spin}}(M,\sigma)=
\frac{{\langle L(\Omega_{c_1},
\dots,\Omega_{c_m})\rangle}}{\langle U_1(\Omega)\rangle^{b_+}
\langle U_{-1}(\Omega)\rangle^{b_-}}
$$
defines an invariant of the surgered manifold $M={\bf S}^3(L)$
equipped with the modulo $d$ spin  structure
$\sigma=\psi_L^{-1}(c_1,\dots,c_m)$
.\\
Moreover,
$$\forall M\ \ \tau_{\mathcal{C}}(M)=\sum_{\sigma\in Spin(M;\mathbb{Z}/d)}
\tau_{\mathcal{C}}^{\mathrm{spin}}(M,\sigma)$$
\end{thm}
\section{The spin decomposition of the Verlinde formula}\label{decomp}
If we are given a modular category $\mathcal{C}$ then we get
a TQFT. In brief we have a functor $\V_\mathcal{C}$
from a cobordism category in dimension $3$
to vector spaces. If $\mathcal{C}$ is a 
modulo $d$ spin modular category, then we will construct here a  decomposition
of the TQFT modules $\V_\mathcal{C}(\Sigma_g)$
of a genus $g$ surface
and compute the ranks of the summands.

The TQFT gives a normalized invariant for a closed $3$-manifold $M$
equipped with $p_1$-structure or $2$-framing $\alpha$ and colored graph $K$. 
We extend the scalar field $\fk$ if necessary, and fix $\kappa$ such that
$\kappa^6=\frac{\la U_1(\Omega)\ra }{\la U_{-1}(\Omega)\ra}$.
 Let $\mathcal{D}=\kappa^{-3}\la U_1(\Omega)\ra=
\kappa^{3}\la U_{-1}(\Omega)\ra$; note that $\mathcal{D}^2=\la \Omega \ra$.

The normalized invariant of a connected closed $3$-manifold
\mbox{$M=(M,\alpha,K)$} is then \cite{BHMV}
\begin{equation} \label{normalized} Z_\mathcal{C}(M,\alpha,K)=\mathcal{D}^{-1-b_1(M)} \kappa^{\sigma(\alpha)}
\tau_{\mathcal{C}}(M,K) \ .\end{equation}
Here $b_1(M)$ is the first Betti number, and $\sigma(\alpha)$
is the sigma invariant:
$\sigma(\alpha)=3signature(W_L)-\langle p_1(W_L,\alpha),[W_L]\rangle$,
where $W_L$ is the trace of the surgery and $p_1(W_L,\alpha)\in H^4(W_L,S^3(L))$ is the relative
obstruction to extending $\alpha$.



Let $\Sigma$ be an oriented surface with
structure (a marking \cite{Tu} or a $p_1$-structure
\cite{BHMV}). We  use the object $\varrho$
to define  a group action on $\V_\mathcal{C}(\Sigma)$ as follows.
 To an embedded oriented curve $\gamma$ in $\Sigma$ we  associate
the TQFT operator
$$\phi_\gamma :\V_\mathcal{C}(\Sigma)\rightarrow \V_\mathcal{C}(\Sigma)$$
corresponding to a trivial cobordism $[0,1]\times \Sigma$ equipped with
a colored link $\gamma_{\frac{1}{2}}(\varrho)$.
 Here $\gamma_{\frac{1}{2}}$ is the link  ${\frac{1}{2}}\times\gamma$
 equipped with the framing given by the orientation and the
normal vector parallel to $\Sigma$.
The components of this link are colored with $\varrho$.

 The spectral projector of $\phi_\gamma$ corresponding to the
eigenvalue $\zeta^\nu$ is 
equal to $\frac{1}{d}\sum_{j=0}^{d-1} \zeta^{-\nu j}\phi_\gamma^j$.
 This projector is represented by a trivial cobordism
with colored link $\gamma(\pi_\nu)$ where
the color $\pi_\nu$ is defined by
$$\pi_\nu=\frac{1}{d}\sum_{j=0}^{d-1} \zeta^{-\nu j}\varrho^j\ .$$
 Using the definition of the grading, we get the following
lemma.

\begin{lem}\label{proj}
Let $V$ be an object in $\mathcal{C}_j$; denote by
 $\delta_{\nu j}$   the Kronecker symbol. One has the equality
in Figure \ref{proj_fig}.
\end{lem}

\begin{figure}
\centerline{
\begin{picture}(70,45)
\put(8,7){$V$}
\put(16,20){$\pi_\nu$}
\put(35,20){$=\ \ \delta_{\nu j}$}
\put(57,7){$V$}
\put(0,5){\includegraphics[height=40mm]{grad.eps}}
\end{picture}}
\caption{\label{proj_fig}}
\end{figure}
We denote by $PT\Sigma$ the principal $SO$-bundle
of oriented orthonormal frames in the stabilized tangent
bundle to $\Sigma$ (we could stabilize only once).

We denote by $\tilde\gamma$ the lift in $PT\Sigma$, using the unit
tangent vector,
of the embedded  curve $\gamma$.
\begin{pro}
There exists a well defined action
of the group $H_1(PT\Sigma,\mathbb{Z}/d)$ on $\V_\mathcal{C}(\Sigma)$, 
which maps $x=[\tilde\gamma]$ to the operator
$\psi_x=(-\mathfrak{d})^{\sharp\gamma}\phi_\gamma$.
\end{pro}
\begin{proof}
The $\mathbb{Z}/d$ module $H_1(PT\Sigma,\mathbb{Z}/d)$
is generated by the $1$-cycles $\tilde \gamma$ associated with embedded curves $\gamma$.
 A trivial circle represents the generator on the fiber;
 this generator has order $2$. A disjoint union represents the sum.
 All the other 
 relations are given by the modified band move
 in Figure \ref{mbandmove}.
 \begin{figure}
\centerline{
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.5,.3)(1,0)
\pscurve{<-}(0,1)(.5,.7)(1,1)
\end{pspicture}
$\ =\ $
\begin{pspicture}[.5](0,-0.5)(2,1)
\pscurve{->}(0,0)(.3,.5)(0,1)
%\pscurve{->}(.6,.5)(1,.9)(1.4,.5)(1,.1)(.6,.5)
\pscircle(1,.5){.4}
\pscurve{->}(.63,.42)(.615,.47)(.61,.50)
\pscurve{<-}(2,0)(1.7,.5)(2,1)
\end{pspicture}}
\caption{\label{mbandmove}Modified band move}
\end{figure}
By relation (\ref{band_rho}) the modified band move doesn't change the number of components
modulo $2$, hence by relation (\ref{cross_rho}) $\psi_x$ is well defined
by the formula $\psi_x=(-\mathfrak{d})^{\sharp\gamma}\phi_\gamma$.
Here $\gamma$ is an embedded curve such that
the lift $\tilde\gamma$ represents $x$.
 A crossing resolution changes by $\pm 1$ the number of components,
 hence in the above formula we can use an immersed
 curve as well. If $\gamma,\gamma'$ represent
$x$ and $x'$, then we can isotope $\gamma'$ so that
$\gamma\cup \gamma'$ is an immersed
curve.  This shows that for all $x,x'$, one has
 $\psi_x\psi_{x'}=\psi_{x+x'}$.
 \end{proof}
As a consequence, we have a decomposition of $\V_\mathcal{C}(\Sigma)$ indexed by
the group $H^1(PT\Sigma,\mathbb{Z}/d)$ identified with the characters
on $H_1(PT\Sigma,\mathbb{Z}/d)$.
 Recall that we have chosen a primitive $d$-th root of unity
denoted by $\zeta$. A vector $v$ belongs to the component
indexed by $\sigma$ if and only if
for every $x\in H_1(PT\Sigma,\mathbb{Z}/d)$ one has
$\psi_x v=\zeta^{\sigma(x)}v$ .
 Since the generator of the fiber acts by $-1$, only the classes
whose restriction to the fiber is non-trivial,
i.e. $Spin(\Z/d)$ structures,
will correspond to non-trivial summands.

If $\sigma$ is a modulo $d$ spin structure  on the genus $g$ oriented surface
$\Sigma_g$, we denote by ${\V}_\mathcal{C}(\Sigma_g,\sigma)$
the corresponding summand and by $ d_\mathcal{C}(g,\sigma)$ its dimension.
$${\V}_\mathcal{C}(\Sigma_g,\sigma)=\{v\in {\V}_\mathcal{C}(\Sigma_g),
\forall x\in H_1(PT\Sigma,\mathbb{Z}/d)\  \psi_x v=\zeta^{\sigma(x)}v\}$$
\begin{thm}\label{spin}
a) There exists a splitting of the TQFT module
$$ \V_\mathcal{C}(\Sigma_g)=
\oplus_{\sigma\in Spin(\Sigma_g,\mathbb{Z}/d)}\ 
\V_\mathcal{C}(\Sigma_g,\sigma)\ .$$
b) Suppose that the scalar field $\fk$ has characteristic zero,
 then the refined Verlinde formula is the following
\begin{eqnarray*} d_\mathcal{C}(\Sigma_g,\sigma)&=&
\langle \Omega\rangle^{g-1}
  \sum\limits_{\lambda\in\Gamma}\ 
  \la\lambda\ra^{2-2g} \times\prod\limits_{\nu=1}^g
\frac{\epsilon_\lambda(a_\nu(\sigma),b_\nu(\sigma))}
{(\sharp\, \mathrm{orb}(\lambda))^2}
\ .\end{eqnarray*}
\end{thm}
Here $(a(\sigma),b(\sigma))\in (\mathbb{Z}/d)^g\times (\mathbb{Z}/d)^g$
is given by the values of $q_\sigma$ on a sympleptic basis.

\esp\noindent If $\sharp\mathrm{orb}(\lambda)$ is even, then
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
1\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ are zero modulo $|Stab(\lambda)|$,}\\
0 \ \text{ else.}\end{array}\right.$$
If $\sharp\mathrm{orb}(\lambda)$ is odd, then
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
\frac{1}{2}(-1)^{\frac{2\mathfrak{a}}{|Stab(\lambda)|}\frac{2\mathfrak{b}}{|Stab(\lambda)|}}
\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ 
are  zero modulo $\frac{|Stab(\lambda)|}{2}$,}\\
0\ \text{ else.}\\
\end{array}\right.$$
\begin{rem}
Any element in $Stab(\lambda)$ has quantum dimension equal to one. In the case
where $\mathfrak{d}=-1$, the group $Stab(\lambda)$ is generated by an even power
 of $\varrho$, and $\sharp \mathrm{orb}(\lambda)$
is even.  We do not know examples with $\mathfrak{d}=-1$.
\end{rem}\begin{rem}
 If the scalar field $\fk$ has  characteristic $p>0$, then
statement b) computes the dimension mod. $p$.
\end{rem}
\begin{proof}
The formula in a) follows from the decomposition of the vector space
$\V_{\mathcal{C}}(\Sigma_g)$ described above.
 Moreover the dimension $d_\mathcal{C}(\Sigma_g,\sigma)$
 of a summand is the trace
of the corresponding projector. This projector can be represented
by a cobordism $ [0,1]\times\Sigma_g$ in which we have inserted a convenient
skein element. 
 By a standard TQFT argument we get
$$d_\mathcal{C}(\Sigma_g,\sigma)
=Z_\mathcal{C}(S^1\times \Sigma_g,\text{skein element})\ .$$
The $3$-manifold $S^1\times \Sigma_g$ is obtained by surgery on the borromean
link with $2g+1$ components represented in Figure \ref{cborr} \cite[Th. 14.12]{Li}.

 In this presentation, a meridian around the bigger 
component corresponds to $S^1\times pt$, and the $2g$ meridians around the
other components correspond to a system of $2g$ fundamental curves in $\Sigma_g$; these curves are framed by using the meridian disc.
 The skein element which arises here is represented by these
$2g$ curves, decorated with some $\pi_\nu$.
 
If $\mathfrak{d}=1$, then $\nu$ is the value of
the quadratic form $q_\sigma$ on the curve, and
if $\mathfrak{d}=-1$, then $\nu$ is the value of $\sigma$
on the $1$-cycle represented by the curve.

By using (\ref{normalized}) and lemma (\ref{proj}) we get
\begin{equation}d_\mathcal{C}(\Sigma_g,\sigma)=\mathcal{D}^{-(2g+2)}\sum_\lambda \langle \lambda \rangle
B_\lambda
=\langle \Omega\rangle^{-1-g}
\sum_\lambda \langle \lambda \rangle
B_\lambda\ ,\end{equation}
where $B_\lambda$ is the invariant of the colored borromean link in figure
\ref{cborr}. Here $(a_1,b_1),\dots,(a_g,b_g)$ are given by the values
of the quadratic form $q_\sigma$ on the corresponding curves
if $\mathfrak{d}=1$, and are equal to the value of $\sigma$
on the $1$-cycle represented by the curve
if $\mathfrak{d}=-1$.

\begin{figure}
\centerline{
\begin{picture}(40,65)
\put(0,5){\includegraphics[height=60mm]{bor.eps}}
\put(1,10){$\lambda$}
\put(37,54){$\Omega_{a_1}$}\put(38,18){$\Omega_{a_g}$}
\put(32,60){$\Omega_{b_1}$}\put(33,25){$\Omega_{b_g}$}
\end{picture}
}
\caption{\label{cborr} Colored borromean link}
\end{figure}
Recall that the cyclic group generated by the
object $\varrho$, identified with $\mathbb{Z}/d$, acts on the set $\Gamma$ of
(representatives of) isomorphism classes of simple objects.
 If $j$ is in the stabilizer subgroup of $\lambda$,
then we choose a basis for the $1$-dimensional vector spaces
 $Hom_\mathcal{C}(\varrho^j,\lambda^*\otimes\lambda)$
and the dual basis for $Hom_\mathcal{C}(\lambda^*\otimes\lambda,\varrho^j)$.
 We denote these bases by the trivalent vertices in Figure \ref{tri}.

\begin{figure}\centerline{
\begin{picture}(30,30)
\put(3,3){\includegraphics[height=24mm]{tri1.eps}}
\put(1,20){$\lambda$}
\put(22,20){$\lambda$}
\put(14,8){$\varrho^j$}
\end{picture}
\hspace{2cm} 
\begin{picture}(30,30)
\put(3,3){\includegraphics[height=24mm]{tri2.eps}}
\put(1,7){$\lambda$}
\put(24,7){$\lambda$}
\put(16,18){$\varrho^j$}
\end{picture}
}
\caption{\label{tri}Trivalent vertices.}
\end{figure}

\begin{figure}
\centerline{
\begin{picture}(30,50)
\put(3,3){\includegraphics[height=40mm]{rel1.eps}}
\put(0,25){$\lambda$}
\put(7,6){$\lambda$}\put(7,37){$\lambda$}
\put(15,25){$\varrho^j$}
\end{picture}
\raisebox{20mm}{$\displaystyle \ =\frac{1}{<\lambda>}\ $}
\begin{picture}(30,50)
\put(5,3){\includegraphics[height=40mm]{rel3.eps}}
\put(8,27){$\lambda$}
\end{picture}}


\vspace{10pt}

\centerline{
\begin{picture}(30,50)
\put(3,3){\includegraphics[height=40mm]{rel2.eps}}
\put(1,27){$\lambda$}
\put(7,6){$\varrho^j$}\put(7,37){$\varrho^j$}
\put(20,27){$\lambda$}
\end{picture}
\raisebox{20mm}{$\displaystyle \ =\ \ $}\begin{picture}(30,50)
\put(5,3){\includegraphics[height=40mm]{rel3.eps}}
\put(8,27){$\varrho^j$}
\end{picture}}
\caption{\label{rel23}Relations for trivalent vertices.}

\end{figure}
We then have the  relations in Figure \ref{rel23};
recall that in the case $\mathfrak{d}=-1$, $j$ must be even. 

The proposition \ref{keypro} below is the key point in the computation.
 By using this proposition, we get
\begin{equation}\label{blambda}B_\lambda=\langle \lambda\rangle
\prod\limits_{\nu=1}^g
\sum_{j,j'\in Stab(\lambda)} \zeta^{ja_\nu+j'b_\nu}\frac{(-1)^{jj'}}
{\langle \lambda\rangle^2}
\frac{\langle \Omega\rangle^2}{d^2}
\end{equation}
Let $l=\sharp\mathrm{orb}(\lambda)$ and $l'=\frac{d}{l}=|Stab(\lambda)|$. The stabilizer subgroup
is then $Stab(\lambda)=\{ ls, \ 0\leq s <l'\}$.
 
If $l$ is even then $\sum_{j,j'\in Stab(\lambda)} \zeta^{ja_\nu+j'b_\nu}$
 is zero unless $\zeta^{la_\nu}=\zeta^{lb_\nu}=1$, and we get
$$\sum_{j,j'\in Stab(\lambda)} \zeta^{j a_\nu+j'b_\nu}(-1)^{jj'}=
\left\{
\begin{array}{ll}
|Stab(\lambda)|^2\ &\text{if $a_\nu\equiv b_\nu\equiv 0$ mod. $|Stab(\lambda)|$,}\\
0& \text{else.}
\end{array}
\right.$$


If $l$ is odd, then we decompose the sum 
$\sum_{j,j'\in Stab(\lambda)} \zeta^{ja_\nu+j'b_\nu}(-1)^{jj'}$ according to the parity of the indices.
 The sum  is zero if $\zeta^{2la_\nu}\neq 1$ or $\zeta^{2lb_\nu}\neq1$.
It remains four cases to consider according to  $\zeta^{la_\nu}=\pm 1$, $\zeta^{lb_\nu}=\pm 1$. 

Case $a_\nu\equiv b_\nu\equiv 0$ mod. $|Stab(\lambda)|$.
$$\begin{array}{ll}
\displaystyle\sum_{j,j'\in Stab(\lambda)} \zeta^{j a_\nu+j'b_\nu}(-1)^{jj'}&=\displaystyle\sum_{j,j'\text{ even}}
+\sum_{j\text{ even ,}j' \text{ odd}}+\sum_{j'\text{ even ,}j \text{ odd}}-\sum_{j,j' \text{ odd}}\\
&=\displaystyle\frac{{l'}^2}{4}+\frac{{l'}^2}{4}+\frac{{l'}^2}{4}-\frac{{l'}^2}{4}\\
&=\displaystyle\frac{{l'}^2}{2}\ .\end{array}$$
Case $a_\nu\equiv b_\nu\equiv  \frac{|Stab(\lambda)|}{2}$ mod. $|Stab(\lambda)|$.
$$\begin{array}{ll}
\displaystyle\sum_{j,j'\in Stab(\lambda)} \zeta^{j a_\nu+j'b_\nu}(-1)^{jj'}&=\displaystyle\sum_{j,j'\text{ even}}
+\sum_{j\text{ even ,}j' \text{ odd}}+\sum_{j'\text{ even ,}j \text{ odd}}-\sum_{j,j' \text{ odd}}\\
&=\displaystyle\frac{{l'}^2}{4}-\frac{{l'}^2}{4}-\frac{{l'}^2}{4}-\frac{{l'}^2}{4}\\
&=-\displaystyle\frac{{l'}^2}{2}\ .\end{array}$$
Case $a_\nu\equiv 0$ , $b_\nu\equiv  \frac{|Stab(\lambda)|}{2}$ mod. $|Stab(\lambda)|$.
$$\begin{array}{ll}
\displaystyle\sum_{j,j'\in Stab(\lambda)} \zeta^{j a_\nu+j'b_\nu}(-1)^{jj'}&=\displaystyle\sum_{j,j'\text{ even}}
+\sum_{j\text{ even ,}j' \text{ odd}}+\sum_{j'\text{ even ,}j \text{ odd}}-\sum_{j,j' \text{ odd}}\\
&=\displaystyle\frac{{l'}^2}{4}+\frac{{l'}^2}{4}-\frac{{l'}^2}{4}+\frac{{l'}^2}{4}\\
&=\displaystyle\frac{{l'}^2}{2}\ .\end{array}$$
Case $b_\nu\equiv 0$ , $a_\nu\equiv  \frac{|Stab(\lambda)|}{2}$ mod. $|Stab(\lambda)|$.
$$\begin{array}{ll}
\displaystyle\sum_{j,j'\in Stab(\lambda)} \zeta^{j a_\nu+j'b_\nu}(-1)^{jj'}&=\displaystyle\sum_{j,j'\text{ even}}
+\sum_{j\text{ even ,}j' \text{ odd}}+\sum_{j'\text{ even ,}j \text{ odd}}-\sum_{j,j' \text{ odd}}\\
&=\displaystyle\frac{{l'}^2}{4}-\frac{{l'}^2}{4}+\frac{{l'}^2}{4}+\frac{{l'}^2}{4}\\
&=\displaystyle\frac{{l'}^2}{2}\ .\end{array}$$

In all cases we get the formula below.
\begin{equation}\label{sum}
\sum_{j,j'\in Stab(\lambda)} \zeta^{j a_\nu+j'b_\nu}(-1)^{jj'}=
\epsilon_\lambda(a_\nu,b_\nu)\ |Stab(\lambda)|^2\ .
\end{equation}
 
In the case where $\mathfrak{d}=-1$, then  $l$ is even, $l'=|Stab(\lambda)|$ divides $\frac{d}{2}$ and we have
$\epsilon_\lambda(a_\nu,b_\nu)=\epsilon_\lambda(a_\nu+\frac{d}{2},b_\nu+\frac{d}{2})$.
 So that we may define $(a_1,b_1),\dots,(a_g,b_g)$  by the values
of the quadratic form $q_\sigma$ as well.

We will now establish statement b). 
$$\begin{array}{ll}d_\mathcal{C}(\Sigma_g,\sigma)&\displaystyle
=\langle \Omega\rangle^{-1-g}
\sum_\lambda \langle \lambda \rangle^{2-2g}
\prod_{\nu=1}^g \epsilon_\lambda(a_\nu,b_\nu)\ |Stab(\lambda)|^2\ \frac{\langle \Omega\rangle^2}{d^2}
\\&\displaystyle
=\langle \Omega\rangle^{g-1}
\sum_\lambda \langle \lambda \rangle^{2-2g}
\prod_{\nu=1}^g \frac{\epsilon_\lambda(a_\nu,b_\nu)}{\sharp \mathrm{orb}(\lambda)^2}\ .
\end{array}$$
\end{proof}
\begin{lem}\label{exists}
 For any $i\in\mathbb{Z}/d$ the subcategory $\mathcal{C}_i$ contains at least
one simple object, and for any simple object  $\lambda_i$ in $\mathcal{C}_i$, 
one has 
$$\langle \lambda_i\rangle \Omega_{i+j}=\lambda_i\otimes \Omega_j\ .$$
\end{lem}
In a modular category the dimension of a simple object is non-zero,
hence we have that  for any $i$
\begin{equation}\langle \Omega_i\rangle =\langle \Omega_0\rangle=
\frac{1}{d}\langle \Omega\rangle\end{equation}
\begin{proof}
Let $\nu$ be a generator for the subgroup
of $\mathbb{Z}/d$ formed with all $i$ such that $\mathcal{C}_i$
contains at least one non-trivial object.
 Suppose that $\nu$ has order $d'$, then $\varrho^{d'}$
is a simple object whose contribution in the $S$ matrix is the
same as that of the trivial. This object is isomorphic to the trivial,
and we deduce that $d'=d$.
 This proves the first part of the lemma. The second part follows from
the graded sliding property (see \cite[Section 1.3]{LT}).
\end{proof}
\begin{lem}\label{vanish}
 Let $\lambda$ be a simple object in $\mathcal{C}$, for any $i$ in $\mathbb{Z}/d$   
 the following morphism 
is non-zero   if and only if $\lambda$ is isomorphic to
$\varrho^j$ for some $j$.

\centerline{\includegraphics[height=25mm]{circbcd.eps}}
\end{lem}
\begin{proof}
If $\lambda$ is equal to $\varrho^j$ then the morphism
is equal to $\frac{1}{d}\langle\Omega\rangle \zeta^{ij}\one\!_\lambda$,
and so is not zero.
 
 Suppose now that for some simple object $\lambda$
the above morphism is not zero.
 By using Lemma \ref{exists}, we obtain a scalar $t_\lambda$ such that

\vspace{5pt}
\centerline{\includegraphics[height=25mm]{circbcd.eps}
\raisebox{11mm}{$=
t_\lambda^i$}\includegraphics[height=25mm]{circbcd0.eps}\ .}
\vspace{5pt}

Note that $t_\lambda^d=1$, hence there exists $j$ such that
 $t_\lambda=\zeta^j$. By the graded sliding property
we deduce that the contribution of $\lambda$ in the $S$ matrix is the same
as that of $\varrho^j$, and we get the required isomorphism.
\end{proof}
\begin{lem}
\label{rfusion}
 Let $\lambda$ be a simple object in $\mathcal{C}$, then one has the relation
in Figure \ref{rfus}.
\begin{figure}
\centerline{
\begin{picture}(30,50)
\put(3,5){\includegraphics[height=40mm]{fus1.eps}}
\put(4,27){$\Omega_a$}
\put(11,40){$\lambda$}
\put(28,40){$\lambda$}
\end{picture}
\raisebox{22mm}{$\ \ \ =\ \ \sum_{j\in Stab(\lambda)}
\zeta^{aj}\ \frac{\langle \Omega \rangle}{d}\ $}
\begin{picture}(30,50)
\put(4,5){\includegraphics[height=40mm]{fus2.eps}}
\put(14,27){$\varrho^j$}
\put(3,40){$\lambda$}
\put(21,40){$\lambda$}
\put(3,7){$\lambda$}
\put(20,7){$\lambda$}
\end{picture}
}
\caption{\label{rfus}}
\end{figure}
\end{lem}
\begin{proof}
We first use the domination axiom. The decomposition of the identity
of $\lambda^*\otimes\lambda$ is given by a so called {\em fusion formula}
(see e.g. \cite[Section 1.2]{BB}). Note that in this formula the multiplicity
of an invertible objet is one if it belongs to the stabilizer subgroup
of $\lambda$ and zero else.
 We then apply Lemma \ref{vanish}.
 The result follows.
\end{proof}
\begin{lem}\label{tetra} For $i,j$ in $Stab(\lambda)$, one has
the relation in Figure \ref{tet}.
\end{lem}
\begin{figure}
\centerline{
\begin{picture}(25,50)
\put(10,5){\includegraphics[height=40mm]{rel4.eps}}
\put(8.5,15){$\lambda$}
\put(19,34){$\varrho^i$}
\put(18,12){$\varrho^j$}
\end{picture}
\raisebox{22mm}{$
 \ =\ {(-1)^{ij}}\ $
} 
\begin{picture}(18,50)
\put(3,5){\includegraphics[height=40mm]{rel5.eps}}
\put(1,15){$\lambda$}
\put(11,34){$\varrho^i$}
\put(10,13){$\varrho^j$}
\end{picture}
\raisebox{22mm}{$
\  =\ \frac{(-1)^{ij}}{\langle \lambda\rangle^2}\ $
} 
\begin{picture}(20,50)
\put(3,5){\includegraphics[height=35mm]{rel3.eps}}
\put(7,25){$\lambda$}
\end{picture}
}
\caption{\label{tet}}
\end{figure}
\begin{proof}
The first equality uses the defining property of a 
modulo $d$ spin modular category.
 The second one comes from the definition of the trivalent vertices.
\end{proof}
\begin{pro}\label{keypro}
The  formula in Figure \ref{key} holds.
\begin{figure}
\centerline{\begin{picture}(30,50)
\put(3,5){\includegraphics[height=40mm]{key1.eps}}
\put(3,20){$\lambda$}
\put(16,34){$\Omega_b$}
\put(23,27){$\Omega_a$}
\end{picture}
\raisebox{22mm}{$\ =\ \sum_{j,j'\in Stab(\lambda)}\ \
\zeta^{aj+bj'}\ \frac{(-1)^{jj'}}
{\langle \lambda\rangle^2}
\ \frac{\langle \Omega \rangle^2}{d^2}
\ $}
\begin{picture}(30,50)
\put(3,5){\includegraphics[height=35mm]{rel3.eps}}
\put(7,20){$\lambda$}
\end{picture}
}
\caption{\label{key}}
\end{figure}
\end{pro}
\begin{proof}
By using Lemma \ref{rfusion} twice (firstly for the component colored by $\Omega_b$), we get
the formula in Figure \ref{keycalc}.
\begin{figure}
\centerline{
\raisebox{22mm}{$lhs\  =\ \sum_{j,j'\in Stab(\lambda)}
\zeta^{aj+bj'}\ \frac{\langle \Omega \rangle^2}{d^2}\ $}
\begin{picture}(50,50)
\put(3,5){\includegraphics[height=40mm]{keyeps.eps}}
\put(24,35){$\varrho^{j'}$}
\put(23,22){$\varrho^{j}$}
\put(17,10){$\lambda$}
\end{picture}
}
\caption{\label{keycalc}}
\end{figure}
After an isotopy, we  apply Lemma \ref{tetra}. The result follows.
\end{proof}




\section{Verlinde formulas for type A modular categories}
\subsection{The $SU(N,K)$ modular category}
We first consider the so called $SU(N,K)$ modular category.
 The construction can be done either from the representation theory
of the quantum group $U_qsl(N)$ at a convenient root of unity
\cite{AP,TW,BK} or from skein theory \cite{Yo,Bhec}.
 In the following we will use Young diagrams to denote the corresponding simple object.
 Here a Young diagram (or partition)
$\lambda$ is a finite non-increasing sequence
of non-negative integers.
A cell for this partition is a pair $c=(i,j)$
with $1\leq j\leq \lambda_i$.
 We denote by $\lambda^{\vee}$ the tranpose of $\lambda$;
$(i,j)$ is a cell in  $\lambda^{\vee}$ if and only if $(j,i)$
is a cell in  $\lambda$.
 The content and hook-length for a cell $c=(i,j)$
are defined respectively by
$$cn(c)=j-i\ , \ hl(c)=\lambda_i+\lambda^{\vee}_j-i-j+1\ .$$
The size of $\lambda$ is $|\lambda|=\sum_i \lambda_i$.

The following theorem is proved in \cite{Bhec}. The result can also be obtained
from \cite[Th. 3.3.20]{BK} ($A_{N-1}$ case).
\begin{thm} \label{modcatH}
Let $N,K\geq 2$. Suppose that $a$ is a $2N(N+K)$-th root
of unity in the scalar field and $s=a^{-N}$.

There exists a modular category
 $\mathcal{C}^{SU(N,K)}$ whose set of distinguished  simple objects
 is 
 $$\Gamma_{N,K}=\{\lambda=(\lambda_1,\dots,\lambda_{N}),\
K\geq\lambda_1\geq\dots\geq \lambda_{N-1}\geq\lambda_N=0\}\ .$$

The quantum dimension and framing coefficient
of a simple object $\lambda\in \Gamma_{N,K}$ are given by the following formulas
(here $[n]=\frac{s^n-s^{-n}}{s-s^{-1}}$ denotes the quantum integer).

$$\langle \lambda \rangle=
\prod_{\mathrm{cells}}\frac{[N+cn(c)]}{[hl(c)]}$$

\centerline{\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{-}(.1,0)(.15,.45)(.45,.75)(.7,.5)(.4,.2)(.20,.38)
\pscurve{->}(.14,.62)(.1,.9)(.1,1)
 \put(-.2,0){$\lambda$}
\end{pspicture}
$\ =\ a^{|\lambda|^2}s^{N|\lambda|+2\sum_{\mathrm{cells}}cn(c)}\ \ $
\begin{pspicture}[.5](0,-.5)(1,1)
\psline{->}(.1,0)(.1,1)
 \put(.2,0){$\lambda$}
\end{pspicture}
}
\end{thm}
\begin{rems}1. In the quantum group approach, a Young diagram in $\Gamma_{N,K}$
gives a highest weight module, which is irreducible and has non zero quantum dimension. 
 The quantum dimension follows from Weyl's character formula and computation with symmetric functions in \cite[section I.3]{Macdo}. The value of the twist is obtained by the action of Drinfeld quantum Casimir.\\
2. In the skein theoretic approach the Young diagram gives a minimal idempotent in Hecke algebra (the deformation of the Young symmetrizer
in the symmetric group algebra); this idempotent becomes a simple object
in the so called Karoubi completion of the Hecke category.
\end{rems}
We denote by $\V_{N,K}(\Sigma_g)$
the TQFT vector space, associated with a genus $g$ surface
$\Sigma_g$,
 for the modular category $\mathcal{C}^{SU(N,K)}$, and by
$d_{N,K}(g)$
 its rank. We give below the well known computation for this formula.

\begin{thm}
The rank $d_{N,K}(g)$ is
equal to the Verlinde number for the group
$SU(N)$ at level $K$.
$$\begin{array}{l}
d_{N,K}(g)=\mathcal{V}_{SU(N)}(K,g)=\hfill \\
  \left((N+K)^{(N-1)}N\right)^{g-1}
  \sum\limits_{\lambda\in\Gamma_{N,K}}\ \prod\limits_{1\leq i<j\leq N}
  \left({2\sin{(\lambda_i-i-\lambda_j+j)\frac{\pi}{N+K}}}
\right)^{2-2g} 
  \end{array}$$
\end{thm}
\begin{proof}
We can use Turaev's formula \cite[Corollary 12.1.2]{Tu}.
Note that this formula computes the TQFT-invariant of the manifold
$S^1\times \Sigma_g$.
$$d_{N,K}(g)=\left(\sum_{\lambda\in\Gamma_{N,K}}\langle \lambda\rangle^2\right)
^{g-1}\sum\limits_{\lambda\in\Gamma_{N,K}}
\langle \lambda \rangle^{2-2g}$$
The computation  is achieved with the lemma below.
Statement a) is a standard fact on symmetric functions; statement b)
is contained e.g. in  the proof of lemma 2.8
in \cite{Bhec}.
 Note that the result does not depend on the choice
of the root of unity with required order; it is also unchanged if $s$ is replaced by $\tilde s=-s$.
 The formula agrees with the Verlinde
number $\mathcal{V}_{SU(N)}(K,g)$ \cite{Beau,So}.
\end{proof}
 \begin{lem}
a) $$\langle \lambda\rangle^2=
\frac{a_{\rho+\lambda}\overline a_{\rho+\lambda}}
{a_{\rho}\overline a_{\rho}}$$
with $\rho=(N-1,N-2,\dots,0)$ and, for
$l=(l_1,\dots,l_N)$
$$a_l=det\left(s^{2(i-1)l_j}\right)
_{1\leq i,j\leq N}$$
b) $$ \sum_{\lambda\in\Gamma_{N,K}}\langle \lambda\rangle^2
=\frac{N(N+K)^{N-1}}{a_{\rho}\overline a_{\rho}}.$$
\end{lem}
\subsection{Spin decomposition of the Verlinde formula for ${SU(N,K)}$ modular category.}
In the category $\mathcal{C}^{SU(N,K)}$, the object $(K)$ (a $K$ cells Young diagram with only 
one row) is an invertible object whose
order is $N$. It is a generator of the group of invertible
objects, and has quantum dimension $1$. Its framing coefficient is
equal to 
$$\theta_K=a^{K^2}s^{NK+K(K-1)}=(-a^{N+K})^K\ .$$ 
If $N=jl$, and $(-a^{N+K})^{Kj^2}=-1$, then the category 
$\mathcal{C}^{SU(N,K)}$ equipped with the invertible object $\varrho=(K)^{\otimes j}$ is a 
modulo $l$ spin modular category.

Recall that $a$ is a $2N(N+K)$-th root of unity. Let $d=gcd(N,K)$, $N=dN'$,
$K=dK'$. A convenient integer $j$ exists if and only if
\begin{verse}
either $d$ is even, $N'$ is odd and the exponent of $2$ in  $K'$ is even 
($K'=2^{2n}(2m+1)$),\\
or $d$ is odd and the exponent of $2$ in $N$ is an even positive number.
\end{verse}
We emphasize the simplest case in the theorem below.
\begin{thm}
If $N$ is even and $K'=\frac{K}{N}$ is an odd integer, then the category
$\mathcal{C}^{SU(N,K)}$ equipped with the invertible object $\varrho=(K)$
 is a modulo $N$ spin  modular category.
\end{thm}
 The following theorem is an application of \ref{spin}.
\begin{thm}\label{spin_A}
Suppose that  $N$ is even and $K'=\frac{K}{N}$ is an odd integer.
a) There exists a splitting of the Verlinde formula
$$d_{N,K}(g)=\sum\limits_{\sigma\in Spin(\Sigma_g,\mathbb{Z}/d)}
d_{N,K}(g,\sigma)\ .$$
b) The refined Verlinde formula is the following
\begin{eqnarray*} d_{N,K}(g,\sigma)&=&
\left((N+K)^{N-1}N\right)^{g-1}
  \sum\limits_{\lambda\in{\Gamma}_{N,K}}\ \prod\limits_{\nu=1}^g
\frac{\epsilon_\lambda(a_\nu(\sigma),b_\nu(\sigma))}
{(\sharp\, \mathrm{orb}(\lambda))^2}\\
& &\times\prod\limits_{1\leq i<j\leq N}
  \left({2\sin{(\lambda_i-i-\lambda_j+j)\frac{\pi}{N+K}}}
\right)^{2-2g} 
\ .\end{eqnarray*}
\end{thm}
Here $(a(\sigma),b(\sigma))\in \mathbb{Z}/N)^g\times (\mathbb{Z}/N)^g$
are given by the values of $q_\sigma$ on a sympleptic basis.
 We consider the action of $\mathbb{Z}/N$ on the set
$$\Gamma_{N,K}=\{\lambda=(\lambda_1,\dots,\lambda_{N}),\
K\geq\lambda_1\geq\dots\geq \lambda_{N-1}\geq\lambda_N=0\}\ ,$$ 
given for the generator 
 of the cyclic group
$\mathbb{Z}/N$ by
$$(\lambda_1,\dots,\lambda_{N-1},0)\longmapsto
(K,\lambda_1,\dots,\lambda_{N-1})-(\lambda_{N-1},\dots,\lambda_{N-1})\ .$$ 
 We denote by $\sharp\, \mathrm{orb}(\lambda)$
 the cardinality of the orbit of $\lambda$,
and by $Stab(\lambda)$ the stabilizer subgroup. 
 The numbers $\epsilon_\lambda(\mathfrak{a},\mathfrak{b}) \in
\{0,1,-\frac{1}{2},\frac{1}{2}\}$ are defined  as follows.

\esp\noindent If $\sharp\mathrm{orb}(\lambda)$ is even, then
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
1\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ are zero modulo $|Stab(\lambda)|$,}\\
0 \ \text{ else.}\end{array}\right.$$
If $\sharp\mathrm{orb}(\lambda)$ is odd, then
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
\frac{1}{2}(-1)^{\frac{2\mathfrak{a}}{|Stab(\lambda)|}\frac{2\mathfrak{b}}{|Stab(\lambda)|}}
\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ 
are  zero modulo $\frac{|Stab(\lambda)|}{2}$,}\\
0\ \text{ else.}\\
\end{array}\right.$$
\begin{rem}
In the general case, one can use the reduction formula \cite[Theorem 3.6]{Bhec}
in order to establish a tensor product decomposition of the $SU(N,K)$
TQFT functor $\V_{N,K}$
$$\V_{N,K}=\V^{U(1)}_{N'}\otimes \widetilde{\V}_{N,K}\  ,$$ 
where $\widetilde \V_{N,K}$ is the TQFT functor
associated with the modular category
$\mathcal{C}^{PU(N,K)}$ discussed below, and $\V^{U(1)}_{N'}$
(known as a $U(1)$ theory)
is associated with a modular category based on linking numbers. The latter
involves a root of unity $\eta$ whose order is $2N'$ (resp. $N'$) if $N'$
is even (resp. odd); when $N'$ is even with even exponent of $2$, then one
can find $j$ such that $j^2\equiv N'$ ($mod\ 2N'$), and the category
is modulo $j$ spin  modular.
\end{rem}

\subsection{The $PU(N,K)$ modular category}
The so called projective  $PSU(N,K)$ modular category was obtained for $N$ and $K$ coprime
by restricting to simple objects in the root lattice \cite{MW,Le,LT}.
 The modular category
$\mathcal{C}^{PU(N,K)}$ (denoted by $\widetilde{\mathrm{H}}^{N,K}$
in \cite{Bhec}) is a generalization
 to the case where
$N$ and $K$ are not required to be  coprime.

Let $N,K\geq 2$. We suppose that 
 in the scalar field
$$\begin{cases}
s \text{ has order $2(N+K)$ if $N+K$ is even,}\\
s \text{ has order $N+K$ if $N+K$ is odd.}
\end{cases}$$ 
Then the Hecke category completed with idempotents and quotiented with
negligible, which we denote by $H^{N,K}$ is semisimple. In addition to simple objects $\lambda\in \Gamma_{N,K}$ there is an invertible simple object $1^N$ and its tensor powers.
 The group of invertible objects is generated by $1^N$ and $(K)$
with the relation $(1^N)^{\otimes K}\approx (K)^{\otimes N}$.
In order to apply the modularization procedure, we have to know
which are the transparent simple objects \cite{Bru}. This depends
on the order $\alpha$ of $(a^Ns)^2$ and  the order $\beta$ of
$(a^Ks^{-1})^2$. The set of isomorphism classes of transparent
simple objects is then the group generated by 
$(1^N)^{\otimes \alpha}$ and $(K)^{\otimes \beta}$.
We choose the framing parameter $a$ in such a way that this group
of transparent objects is as big as possible, and that the modularization
criterion is satisfied.
\begin{thm} \label{reduced}
Set $d=\mathrm{gcd}(N,K)$, $N=dN'$, $K=dK'$, $d=\alpha \beta$ with
$\mathrm{gcd}(\alpha,K')=\mathrm{gcd}(\beta,N')=
\mathrm{gcd}(\alpha,\beta)=1$.\\
Suppose that $a$ satisfies the relations
$$(a^Ns)^\alpha=(-1)^{N+K+1}(a^Ks^{-1})^\beta=(-1)^{(N+K+1)\beta}$$
(such an $a$ exists up to extension of the scalar field).

There exists a modular category
 $\mathcal{C}^{PU(N,K)}$ in which  isomorphism classes of
simple objects
corresponds bijectively with cosets in the quotient of 
$$\dot \Gamma_{N,K}=\{(1^N)^{\otimes j}\otimes \lambda,\ 0\leq j<\alpha,\
\lambda\in \Gamma_{N,K}\}$$
under a free action of the cyclic group of order $N/\alpha$.
\end{thm}
The action of the generator is given by tensor product with $(K)^{\otimes \beta}$
in the completed Homfly category $H$. One has to iterate $\beta$ times
the rule
$$(1^N)^{\otimes j}\otimes\lambda\mapsto (1^N)^{\otimes j'}+(K-\lambda_{N-1},
\lambda_1-\lambda_{N-1},\dots,\lambda_{N-2}-\lambda_{N-1},0)$$
where $j'\equiv j+\lambda_{N-1}\ \mathrm{mod}\ \alpha$.

The quantum dimension and framing coefficient
of a simple object $V=(j^{\otimes N})\otimes\lambda$
 are given by the following formulas.

$$\langle V\rangle=\langle \lambda \rangle=
\prod_{\mathrm{cells}}\frac{[N+cn(c)]}{[hl(c)]}$$


\centerline{\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{-}(.1,0)(.15,.45)(.45,.75)(.7,.5)(.4,.2)(.20,.38)
\pscurve{->}(.14,.62)(.1,.9)(.1,1)
 \put(-.2,0){$V$}
\end{pspicture}
$\ =\ (a^Ns)^{Ni^2+2|\lambda|}a^{|\lambda|^2}s^{N|\lambda|+2\sum_{\mathrm{cells}}cn(c)}\ $
\begin{pspicture}[.5](0,-.5)(1,1)
\psline{->}(.1,0)(.1,1)
 \put(.2,0){$V$}
\end{pspicture}
}
%\end{thm}

We denote by $\widetilde{\V}_{N,K}(\Sigma_g)$
the TQFT vector space, associated with a genus $g$ surface
$\Sigma_g$,
 for the modular category $\mathcal{C}^{PU(N,K)}$ and by
 $\tilde{d}_{N,K}(g)$
 its rank.
\begin{thm}
The rank $\tilde{d}_{N,K}(g)$
is $$\tilde{d}_{N,K}(g)=\frac{d_{N,K}(g)}{{N'}^g}$$  
\end{thm}
\begin{proof}
By Turaev's formula we have the following.
$$\tilde d_{N,K}(g)=
\left(\sum_{V\in\widetilde\Gamma_{N,K}}\langle V\rangle^2\right)
^{g-1}\sum\limits_{V\in\widetilde\Gamma_{N,K}}
\langle V\rangle^{2-2g}$$
Here  $\widetilde\Gamma_{N,K}\subset \dot{\Gamma}_{N,K}$ is a representative set of  the orbits in  $\dot\Gamma_{N,K}$ under the  order $N/\alpha$ free cyclic  action.
 Note that
this action preserves the dimension. We get
$$\tilde d_{N,K}(g)=
\left(\frac{1}{\alpha N'}\sum_{V\in\dot\Gamma_{N,K}}\langle V\rangle^2\right)
^{g-1}\frac{1}{\alpha N'}\sum\limits_{V\in\dot\Gamma_{N,K}}
\langle V\rangle^{2-2g}$$
Write $V=(1^N)^{\otimes i}\otimes \lambda$, $0\leq i<\alpha$ and 
$\lambda\in \Gamma_{N,K}$.
$$\tilde d_{N,K}(g)=
\left(\frac{1}{N'}\sum_{V\in\Gamma_{N,K}}\langle V\rangle^2\right)
^{g-1}\frac{1}{N'}\sum\limits_{V\in\Gamma_{N,K}}
\langle V\rangle^{2-2g}= \frac{d_{N,K}(g)}{{N'}^g}\ .$$
\end{proof}

The following is an integral version
of a reciprocity formula in \cite{rec}.
\begin{thm}[Level-rank duality]
One has
$\tilde{d}_{N,K}(g)=\tilde{d}_{K,N}(g)$.
\end{thm}
\begin{proof}
In the construction arising from Homfly skein theory, the parameters $N$ and $K$ play the same role, so that we can interchange rows and columns in the description of  isomorphisms classes in the modular category $\mathcal{C}^{PU(N,K)}$.
 We will get the same combinatorics as for the modular category  
$\mathcal{C}^{PU(K,N)}$.
 The result follows.
\end{proof}


\subsection{Spin decomposition of the Verlinde formula for $\mathcal{C}^{PU(N,K)}$}
Here we consider the modular category 
$\mathcal{C}^{PU(N,K)}$ in the spin case.
 This means that $d=gcd(N,K)$ is even, and that
 $N'=\frac{N}{d}$ and $K'=\frac{K}{d}$ are both odd.
 We fix the framing parameter $a$ as we did above.
\begin{thm}
Under the above hypothesis,  the category $\mathcal{C}^{PU(N,K)}$
equipped with $\varrho=(K)\otimes (1^N)$
is a  modulo $d$ spin modular category.
\end{thm}
\begin{proof}
In the modular category $\mathcal{C}^{PU(N,K)}$
the object $1^N$ and $(K)$ are  invertible
with respective orders the coprime integers $\alpha$ and $\beta$.  
 It follows that $\varrho$ is invertible with order $\alpha\beta=d$.
 The figure \ref{rho} below shows that the twist coefficient
for $\varrho$ is the product of the two twist coefficients
for $1^N$ and $(K)$ and a {\em braiding coefficient} between
 $1^N$ and $(K)$.  Using 
\cite[Prop. 1.11]{Bhec} we see that the $3$ coefficients are
respectively $(a^Ns)^N=(-1)^\beta$, $(A^Ks^{-1})^K=(-1)^\alpha$, $(a^Ns)^{2NK}=1$. The product is $-1$.

\begin{figure}
\centerline{
\begin{picture}(30,60)
\put(3,5){\includegraphics[height=40mm]{varrho.eps}}
\put(22,25){$=$}
\put(-8,5){$V\otimes W$}
\put(27,5){$V$}
\put(38,5){$W$}
\end{picture}
}
\caption{\label{rho}Framing coefficient for a tensor product}
\end{figure}


\end{proof}


If $\sigma$ is a modulo $d$ spin structure  on the genus $g$ oriented surface
$\Sigma_g$, we denote by $\widetilde{\V}(\Sigma_g,\sigma)$
the corresponding summand and $\tilde d(g,\sigma)$ its dimension.
By applying \ref{spin}, we get.
\begin{thm}\label{spin_Atilde}
a) There exists a splitting of the Verlinde formula
$$\tilde d_{N,K}(g)=\sum\limits_{\sigma\in Spin(\Sigma_g,\mathbb{Z}/d)}
\tilde d_{N,K}(g,\sigma)\ .$$
b) The refined Verlinde formula is the following
\begin{eqnarray*} \tilde d_{N,K}(g,\sigma)&=&
\left((N+K)^{N-1}d\right)^{g-1}
  \sum\limits_{V=(1^N)^{\iota}\otimes\lambda\in\tilde{\Gamma}_{N,K}}\ \prod\limits_{\nu=1}^g
\frac{\epsilon_V(a_\nu(\sigma),b_\nu(\sigma))}
{(\sharp\, \mathrm{orb}(V))^2}\\
& &\times\prod\limits_{1\leq i<j\leq N}
  \left({2\sin{(\lambda_i-i-\lambda_j+j)\frac{\pi}{N+K}}}
\right)^{2-2g} 
\ .\end{eqnarray*}
\end{thm}
Here $\epsilon_V$ and $\sharp\mathrm{orb}(V)$
are defined as before,
 $\tilde{\Gamma}_{N,K}$ is a representative set of the quotient
of $$\dot{\Gamma}_{N,K}=
\{(j^{\otimes N})\otimes\lambda,\ j\in\mathbb{Z}/\alpha, \ \lambda\in\Gamma_{N,K}\}$$ 
under a free action of $\mathbb{Z}/\alpha N'$.
 The formula in b) can be expressed as follows.
\begin{eqnarray*} 
\tilde d_{N,K}(g,\sigma)&=&
\left((N+K)^{N-1}d\right)^{g-1}
\frac{1}{\alpha N'}
\cr & &\times
  \sum\limits_{V=(1^N)^{\iota}\otimes\lambda\in\dot{\Gamma}_{N,K}}\ \prod\limits_{\nu=1}^g
{\epsilon_V(a_\nu(\sigma),b_\nu(\sigma))}
\left(\frac{\alpha N'}{\sharp\, \mathrm{Orb}(V)}\right)^2
\cr & &\times\prod\limits_{1\leq i<j\leq N}
  \left({2\sin{(\lambda_i-i-\lambda_j+j)\frac{\pi}{N+K}}}
\right)^{2-2g} 
\ .\end{eqnarray*}

We consider now  the orbit $\mathrm{Orb}(V)$ under the action of the group
$\mathbb{Z}/\alpha\times\mathbb{Z}/N$
on $\dot{\Gamma}_{N,K}$,
where $(1,0)$ acts by
$$(1^N)^{\otimes\iota}\otimes\lambda\mapsto (1^N)^{\otimes(\iota+1)}\otimes\lambda, $$
and $(0,1)$ acts by
$$(1^N) ^{\otimes \iota}\otimes\lambda\mapsto
(1^N) ^{\otimes(\iota+\lambda_{n-1})}\otimes
((K,\lambda)-\lambda_{N-1}^N)\ .$$

\esp\noindent If $\sharp\mathrm{Orb}(V)/\alpha N'$ is even, then
$$\epsilon_V(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
1\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ are zero modulo $|Stab(V)|$,}\\
0 \ \text{ else.}\end{array}\right.$$
If $\sharp\mathrm{Orb}(V)/\alpha N'$ is odd, then
$$\epsilon_V(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
\frac{1}{2}(-1)^{\frac{2\mathfrak{a}}{|Stab(V)|}\frac{2\mathfrak{b}}{|Stab(V)|}}
\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ 
are  zero modulo $\frac{|Stab(V)|}{2}$,}\\
0\ \text{ else.}\\
\end{array}\right.$$
 
\section{Cohomological decomposition}\label{coho_}
In this section we will establish the decomposition in the cohomological case.


Let $d$ be an integer, and $(\mathcal{C},\varrho)$ be a modulo $d$ cohomological modular category. This means that the object $\varrho$ has order $d$
and  twist coefficient $\theta_\varrho=1$.
We deduce that the quantum dimension of $\varrho$ is $\mathfrak{\mathfrak{d}}=\pm 1$,
and\\[10pt]
\begin{equation}
\begin{pspicture}[.5](0,-0.5)(1,1)
\psline{->}(1,0)(0,1)
\psline{-}(0,0)(.35,.35)
\psline{->}(.65,.65)(1,1)
 \put(0,.3){$\varrho$} \put(.8,.3){$\varrho$}
\end{pspicture}
\ =\ 
\mathfrak{d}\ \ 
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.3,.5)(0,1)
\pscurve{->}(1,0)(.7,.5)(1,1)
 \put(0,.4){$\varrho$} \put(.8,.4){$\varrho$}
\end{pspicture}
\end{equation}

\begin{equation}
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.3,.5)(0,1)
\pscurve{<-}(1,0)(.7,.5)(1,1)
 \put(0,.4){$\varrho$} \put(.8,.4){$\varrho$}
\end{pspicture}
\ =\ \mathfrak{d}\ \ \ 
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{->}(0,0)(.5,.3)(1,0)
\pscurve{<-}(0,1)(.5,.7)(1,1)
 \put(.3,0){$\varrho$} \put(.4,.9){$\varrho$}
\end{pspicture}
\end{equation}
 
After fixing a $d$-th root of unity $\zeta$, the category
is $\Z/d$ graded.
The Kirby color  decomposes according to
this grading.
$$\Omega=\sum_{\lambda\in \Gamma}
\la \lambda \ra \lambda\ =\sum_{j\in \Z/d}\ \Omega_j$$

Using this grading we obtain the theorem below \cite{Bhec,LT}.
\begin{thm}Let $\mathcal{C}$ be a modulo $d$ cohomological modular category,
and 
\mbox{$\Omega=\sum_{j\in \Z/d}\ \Omega_j$} be the graded decomposition of the Kirby
element.
Provided \mbox{$c=(c_1,\dots,c_m)\in \Z^m$} is in the kernel of $B_L\otimes \Z/d$
the formula
$$\tau_{\mathcal{C}}^{\mathrm{coho}}(M,\sigma)=
\frac{{\langle L(\Omega_{c_1},
\dots,\Omega_{c_m})\rangle}}{\langle U_1(\Omega)\rangle^{b_+}
\langle U_{-1}(\Omega)\rangle^{b_-}}
$$
is an invariant of the surgered manifold $M={\bf S}^3(L)$
equipped with the modulo $d$ cohomology class $\sigma$ corresponding to
$c$.
\\
Moreover,
$$\forall M\ \ \tau_{\mathcal{C}}(M)=\sum_{\sigma\in Spin(M;\mathbb{Z}/d)}
\tau_{\mathcal{C}}^{\mathrm{coho}}(M,\sigma)$$
\end{thm}
Following section \ref{spinref} we get the proposition below.
 Note that here the action given by a trivial curve $\gamma$
colored with $\mathfrak{d}\phi_\gamma$ is trivial.
\begin{pro}\label{act_co}
There exists a well defined action
of the group $H_1(\Sigma,\mathbb{Z}/d)$ on $\V_\mathcal{C}(\Sigma)$, 
which maps $x=[\gamma]$ to the operator
$\psi_x=(\mathfrak{d})^{\sharp\gamma}\phi_\gamma$.
\end{pro}
Using this action we get the decomposition theorem below.
\begin{thm}\label{coho}Let $(\mathcal{C},\varrho)$ be a 
modulo $d$ cohomological modular category.\\
a) There exists a splitting of the Verlinde formula
$$ dim(\V_\mathcal{C}(\Sigma_g))=
\sum\limits_{\sigma\in H^1(\Sigma_g,\mathbb{Z}/d)}
dim(\V_\mathcal{C}(\Sigma_g,\sigma))\ .$$
b) The refined Verlinde formula is the following
\begin{eqnarray*} dim(\V_\mathcal{C}(\Sigma_g,\sigma))&=&
\langle \Omega\rangle^{g-1}
  \sum\limits_{\lambda\in\Gamma}\ 
  \la\lambda\ra^{2-2g} \times \prod\limits_{\nu=1}^g
\frac{\epsilon_\lambda(a_\nu(\sigma),b_\nu(\sigma))}
{(\sharp\, \mathrm{orb}(\lambda))^2}
\ .\end{eqnarray*}
\end{thm}
Here $(a(\sigma),b(\sigma))\in (\mathbb{Z}/d)^g\times (\mathbb{Z}/d)^g$
is given by the values of $\sigma$ on a sympleptic basis, and
$$\epsilon_\lambda(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{l}
1\ \text{ if  $\mathfrak{a}$ and $\mathfrak{b}$ are zero modulo $|Stab(\lambda)|$,}\\
0 \ \text{ else.}\end{array}\right.$$
\begin{proof}
The decomposition a) follows from the action given in Proposition \ref{act_co}.

For $\sigma\in  H^1(\Sigma_g,\mathbb{Z}/d)$, we have
$$
dim(\V_\mathcal{C}(\Sigma_g,\sigma))
=\langle \Omega\rangle^{-1-g}
\sum_\lambda \langle \lambda \rangle
B_\lambda\ ,$$
where $B_\lambda$ is the invariant of the colored link
in figure \ref{cborr}.
 Here $(a_1,b_1),\dots,(a_g,b_g)$ are given by the values
of $\sigma$ on the corresponding curves
if $\mathfrak{d}=1$, and this values plus $\frac{d}{2}$
if $\mathfrak{d}=-1$ (in this case $d$ has to be even).

The computation is done as section 3. We have
\begin{equation}\label{blambdac}B_\lambda=\langle \lambda\rangle
\prod\limits_{\nu=1}^g
\sum_{j,j'\in Stab(\lambda)} \frac{\zeta^{ja_\nu+j'b_\nu}}
{\langle \lambda\rangle^2}
\frac{\langle \Omega\rangle^2}{d^2}
\end{equation}
The formula follows.
\end{proof}
Let $d=gcd(N,K)$. If $d$ is odd, or if $d$ is even but
$\frac{NK}{d^2}$ is even, then the category
${\mathcal{C}}^{PU(N,K)}$ is a modulo $d$ cohomological modular category.

If $N=jl$, and $(-a^{N+K})^{Kj^2}=1$, then the category 
$\mathcal{C}^{SU(N,K)}$ equipped with the invertible object $\varrho=(K)^{\otimes j}$ is a 
modulo $l$ cohomological modular category.
In particular, if $N$ divides $K$, and
$N$ is odd or $\frac{K}{N}$ is even, then the category
$\mathcal{C}^{SU(N,K)}$ is a modulo $N$ cohomological modular category.


\section{Some computations}
\label{comp}
We give below some computations obtained with MuPAD \cite{Mupad}.
 Our program implements the Verlinde formulas for the categories
$\mathcal{C}^{SU(N,K)}$. The cardinality of the alcove increases rapidely,
and we obtain results only for small values of $N$,$K$.
 The function Verlinde$(N,K,g)$ gives $d_{N,K}(g)$,
and Spin\_Verl$(N,K,[\dots])$ computes $d_{N,K}(g,\sigma)$,
where  the value of $q_\sigma$ on the standard basis
is the list $[\dots]$. We know \cite{BM} that $d_{2,2}(g,\sigma)$
is $0$ or $1$ according to the Arf invariant of the spin structure.
 It would be interesting to understand the combinatorics
of the formula $d_{N,K}(g,\sigma)$ in the general case.

\begin{verbatim}
Verlinde(2,2,1);
					3
Verlinde(2,2,2);
					10
Spin_Verl(2,2,[[0,0]]);
					1
Spin_Verl(2,2,[[1,1]]);
					0
Spin_Verl(2,2,[[0,0],[1,1]]);
					0
Spin_Verl(2,2,[[1,1],[1,1]]);
					1
Verlinde(2,6,1);
					7
Verlinde(2,6,2);
					84
Spin_Verl(2,6,[[0,0]]);
					2
Spin_Verl(2,6,[[1,1]]);
					1
Spin_Verl(2,6,[[0,0],[1,1]]);
					4
Spin_Verl(2,6,[[0,0],[0,0]]);
					6
Verlinde(4,4,1);
					35
Verlinde(4,4,2);
					4680
Spin_Verl(4,4,[[0,0]]);
					3
Spin_Verl(4,4,[[1,0]]);
					2
Spin_Verl(4,4,[[1,1]]);
					2
Spin_Verl(4,4,[[2,2]]);
					2
Spin_Verl(4,4,[[0,0],[0,0]]);
					24
Spin_Verl(4,4,[[1,0],[0,0]]);
					18
Spin_Verl(4,4,[[1,0],[1,0]]);
					18
Spin_Verl(4,4,[[2,2],[0,0]]);
					20
Verlinde(6,6,1);
					462
Verlinde(6,6,2);
					30660988
Spin_Verl(6,6,[[0,0]]);
					14
Spin_Verl(6,6,[[1,0]]);
					13
Spin_Verl(6,6,[[2,0]]);
					13
Spin_Verl(6,6,[[1,1]]);
					12
Spin_Verl(6,6,[[2,2]]);
					13
Spin_Verl(6,6,[[3,0]]);
					14
Spin_Verl(6,6,[[3,3]]);
					12
Spin_Verl(6,6,[[0,0],[0,0]]);
					23718
Spin_Verl(6,6,[[1,0],[0,0]]);
					23678
Spin_Verl(6,6,[[1,0],[0,0]]);
					23624
Spin_Verl(6,6,[[2,0],[0,0]]);
					23678
Spin_Verl(6,6,[[2,2],[0,0]]);
					23678
Spin_Verl(6,6,[[3,0],[0,0]]);
					23718
Spin_Verl(6,6,[[3,3],[0,0]]);
					23648
\end{verbatim}



\bibliographystyle{amsplain}
\begin{thebibliography}{10}
\bibitem{AM} J. Andersen, G. Masbaum, {\em Involutions on moduli spaces
and refinements of the Verlinde formula}, Math. Ann. 314, No.2, 291-326 (1999).
\bibitem{AP} H. Andersen, J. Paradowski {\em Fusion  category arising from
semisimple Lie algebras}, Comm. Math. Phys. 169, No.3, 563--588 (1995).
\bibitem{A} M. F. Atiyah {\em Riemann surfaces and spin structures},
Ann. Ecole Norm. Sup. (4) 4, 47--62 (1971).
\bibitem{BK} B. Bakalov, A. Kirillov, {\em Lecture on tensor categories and
modular functors}, Univ. Lecture Series No.21, AMS (2001).
\bibitem{Beau} A. Beauville, {\em Conformal blocks,
fusion rules and the Verlinde formula}\/,
Israel Math. Conf. Proceedings Vol. 9, 75--96 (1996).
\bibitem{BL} A. Beauville, Y. Laszlo, {\em Conformal blocks and
generalized theta functions}\/, Comm. Math. Phys. 164, 385--419 (1994).
\bibitem{BB}  A. Beliakova, C. Blanchet, {\em Modular categories
 of types B,C and D}, Comment. Math. Helv. 76 (2001) 467-500.
\bibitem{BiL} J-M. Bismut, F. Labourie, {\em Formules de Verlinde
pour les groupes simplement connexes et g�om�trie sympleptique},
CRAS, t. 325, S�rie I (1997), 1009--1014.
\bibitem{BiL2} J-M. Bismut, F. Labourie, {\em Sympleptic geometry
and the Verlinde formulas}, In Surveys in differential geometry: differential geometry inspired by string theory,
     Int. Press, Boston, MA, 1999,  97--311.
\bibitem{varso} C. Blanchet, {\em Refined quantum invariants for 
three-manifolds with structure}, in Knot Theory, Banach Center Pub. Vol. 42
(1998), 11-22.
\bibitem{Bhec} C. Blanchet, {\em Hecke algebras, modular
categories and $3$-manifolds quantum invariants}, Topology, Vol. 39
(2000), 193--223. 
\bibitem{BHMV} C. {Blanchet}, N. {Habegger}, G. {Masbaum} and
P. {Vogel}, {\em Topological Quantum Field Theories derived from the
Kauffman bracket}\/, Topology~34 No~4 (1995), 883--927.
\bibitem{BM} C. Blanchet, G. Masbaum, {\em Topological quantum field theories for
surfaces with spin structure}, 
Duke Mathematical Journal 82 (1996), 229-267.
\bibitem{Bru} A. Brugui\`eres {Cat\'egories pr\'emodulaires, modularisations et 
invariants des vari\'et\'es de dimension $3$}, Math. Ann. 316, No.2, 215-236 (2000).
\bibitem{Fal} G. Faltings, {\em A proof of the Verlinde formula}\/,
J. Alg. Geometry 3, 347--374 (1994).
\bibitem{J} D. Johnson, {\em Spin structures and quadratic forms on surfaces},
J. London Math Soc. (2) 22 (1980) 365--377.
\bibitem{KT} T. Kohno, T. Takata, {Level-Rank Duality of Witten $3$-manifolds invariants}, Adv. Studies in Pure Math. 24, Progress in Algebraic Combinatorics, 243--264.
\bibitem{Le} T. Le {\em Quantum invariants of $3$-manifolds: integrality, splitting, and perturbative expansion}, preprint math.QA/0004099.
\bibitem{LT} T. Le, V. Turaev, {\em Quantum groups and ribbon $G$-categories},
 J. Pure Appl. Algebra  178  (2003),  no. 2, 169--185.
\bibitem{Li} W.B.R. Lickorish, {\em An Introduction to Knot Theory}, Springer-Verlag, Grad. Texts in Math. 175 (1997).
\bibitem{Macdo} I. G. Macdonald, {\em Symmetric functions and Hall polynomial},
2nd ed. ,
Oxford Science Pub. (1995).
\bibitem{MW} G. Masbaum, H. Wenzl, {\em Integral modular categories
and integrality of quantum invariants at roots of unity of prime order},
J. Reine Angew. Math. 505, 209-235 (1998).
\bibitem{Milnor} J. Milnor, {\em Spin structures on manifolds}
L'Enseignement Math. 9, 198--203 (1963).
 \bibitem{Mu} M. M\"uger, Galois theory for braided tensor categories
 and the modular closure 
Adv. Math. 150, No.2, 151-201.
\bibitem{Mupad} MuPAD, The Open Computer Algebra System,
Sciface Software, www.mupad.de.
\bibitem{rec} W. M. Oxbury, S. M. J. Wilson, {\em Reciprocity laws
in the Verlinde formulae for the classical groups}\/, Trans. AMS
348, No 7 (1996).
\bibitem{PS} C. Kassel, M. Rosso, V. Turaev, {\em Quantum groups
and knots invariants}\/, Panoramas et Synth\`eses No 5, SMF (1997).
\bibitem{So} C. {Sorger}, {\em La formule de Verlinde}\/,
 S\'eminaire Bourbaki 794 (1994).
\bibitem{Sa} S. Sawin, {\em Quantum groups at roots of unity and modularity},
(preprint) arXiv:math.QA/0308281.
\bibitem{Sa2}{S. Sawin}
{\em Jones-Witten invariants for nonsimply connected Lie groups and the geometry of the Weyl alcove}\/,
Adv. Math. 165, No.1, 1--34 (2002).
\bibitem{Sa3}{S. Sawin} {\em Invariants of spin three-manifolds from Chern-Simons theory and finite-dimensional Hopf algebras}\/, Adv. Math. 165, No.1, 35--70 (2002). 
\bibitem{Tu} V.  Turaev, {\em Quantum invariants
of knots and $3$-manifolds}\/, De Gruyter Studies in Math.~18 (1994).
\bibitem{Tu3} V.  Turaev, {\em  Homotopy field theory in dimension $2$
and crossed groups-algebras} preprint math.QA/9910010.
\bibitem{Tu4} V.  Turaev, {\em  Homotopy field theory in dimension $3$
and crossed groups-categories} preprint math.GT/0005291.
\bibitem{TW} V. Turaev, H. Wenzl, {\em Quantum invariants of
$3$-manifolds associated with classical simple Lie algebras},
Int. J. of Math. Vol 4, No 2 (1993) 323--358.
\bibitem{TW2} V. Turaev, H. Wenzl, {\em Semisimple and modular categories
from link invariants}, Math. Ann. 309 (1997), 411--461.
\bibitem{Ver}  E. Verlinde, {\em Fusion rules and modular transformations
in $2d$ conformal field theory}
    Nucl. Physics B 300,
No. 3, 360-376 (1988).
\bibitem{Yo} Y.~Yokota, {\em Skeins and quantum $SU(N)$
invariants of $3$-manifolds}\/, Math. Ann. 307 (1997), 109--138.
\end{thebibliography}
\end{document}
\bibitem{Aiston} A. K.~Aiston, {\em Skein theoretic idempotents
of Hecke algebras
and quantum group invariants}, Thesis, University of Liverpool, 1996.
\bibitem{Aiston2} A. K.~Aiston, {\em A skein theoretic proof of the hook formula for quantum dimension}, (preprint 1997). 
\bibitem{AM} A. K. Aiston, H.R. Morton, {\it Idempotents of Hecke
algebras of type $A$}, (preprint 1997). 
\bibitem{An} H. H. Andersen, {\em Tensor products of quantized
tilting modules}, Comm. Math. Phys. 149 (1991), 149--159.
\bibitem{AM} H. H. Andersen, G. Masbaum {\em Involutions on moduli spaces
and refinements of the Verlinde formula}, preprint 1998.
\bibitem{Bl1} C. Blanchet. {\em Invariants on three-manifolds with spin
stucture}, Comm. Math. Helv. 67(1992), 406-427.
\bibitem{varso} C. Blanchet, {\em Refined quantum invariants for 
three-manifolds with structure},
in Knot Theory, Banach Center Pub. Vol 42,
to appear.
\bibitem{BHMV3} C. {Blanchet}, N. {Habegger}, G. {Masbaum} and
P. {Vogel}, {\em Topological Quantum Field Theories derived from the
Kauffman bracket}\/, Topology~34 No~4 (1995), 883--927.
\bibitem{Bou} N.~Bourbaki, {\it Groupes et alg\`ebres de Lie IV,V,VI},
Masson, Paris (1982).
\bibitem{Brug} A. Brugui\`eres, {\em Tresses et structure enti\`ere
sur la cat\'egorie des repr\'esentations de $\text{Sl}_{N,q}$}
(preprint 1998).
\bibitem{DJ} G. Dipper, G. James, {\em Block idempotents of the Hecke algebra
of the general linear group}, Proc. London Math. Soc.
54 (1987) 57-82.
\bibitem{Erlij} J. Erlijman, {\em New subfactors from braid group
representations}, (preprint 1997).
\bibitem{Geck} M. Geck, {\em Representations of Hecke algebras at
    roots of unity}, S\'eminaire Bourbaki No 836, Nov 1997.
\bibitem{Good} F. M. Goodman, H. Wenzl, {\em Littlewood-Richardson
Coefficients for Hecke algebras at Roots of Unity},
Advances in Math. 82 (1990), 244--265.
\bibitem{Gyo} A. Gyoja, {\em A $q$-analogue of Young symmetrizer},
Osaka J; Math. 23 (1986), 841-852.
\bibitem{homfly} P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish,
K. Millet and A. Ocneanu, {\em A new polynomial invariant of
knots and links}, Bull. AMS 12 (1985), 239--246.
\bibitem {Jones} V.~Jones, {\it Hecke algebra representations of
braid groups and link polynomials}\/, Annals~of~math.~126 (1987), 335--338.
\bibitem{KW} D. Kazhdan, H. Wenzl, {\em Reconstructing monoidal categories},
Adv. in Soviet Math. 16 n. 2 (1993), 111--136.
\bibitem{Ki} R. {Kirby}, {\it A calculus of framed links
in ${\bf S}^3$}\/,
Invent.~Math.~45 (1978), 35--56.
\bibitem{KM} R.  Kirby, P. Melvin. {\sl The $3$-manifold  invariants of
Witten and Reshetikhin-Turaev for $sl(2,{\bf C})$}, Inv. Math.
105, 473-545 (1991).
\bibitem{KT1} T. Kohno, T. Takata, {\em Symmetry for Witten's
 $3$-manifold invariants for $\protect sl(n,\mathbb{C})$},
Journal of Knot Theory
 and its Ram. 2 (1993), 149--169.
 \bibitem{KT2} T. Kohno, T. Takata, {\em Level-rank duality for Witten's
$3$-manifold invariant},
Adv. Studies Pure Math. 24, Progress in Algebraic Combinatorics,
243--264.
\bibitem{LiSamp} W. B. R. Lickorish, {\em Sampling the $SU(N)$
 invariants of three-manifolds}, J. of Knot Theory and Ram. (1997).
\bibitem{LiTQFT}   W. B. R. Lickorish, {\em Skeins, $SU(N)$
three-manifolds invariants and TQFT}, (preprint 1997).
\bibitem{Macdo} I. G. Macdonald, {\em Symmetric functions
and Hall polynomials}, Oxford Math. Monographs (1979).
\bibitem{MW} G. Masbaum, H. Wenzl, {\em Integral modular categories
and integrality of quantum invariants at roots of unity of prime order},
(preprint 1997).
\bibitem{Morton} H.R.Morton, {\it Invariants of links and $3$-manifolds
from skein theory and from quantum groups}\/,
in M. Bozh�y�k, editor, {\it Topics in knot theory}, N.A.T.O.
A.S.I. series C 399, Kluwer (1993), 107--156.
\bibitem {MA} H. R. {Morton} and A. K. {Aiston},
{\it Young diagrams, the Homfly skein of the annulus
and unitary invariants}\/, (preprint 1996).
\bibitem{MT} H. R. Morton and P. Traczyk, {\em Knots
and algebras}, Contribuciones Matematicas en homaje al professor
D. Antonio Plans Sanz de Bremond, E. Martin-Peinador and A. Rodez editors,
University of Saragoza (1990), 201--220.
\bibitem{Mu} H. {Murakami}, {\it Quantum invariants for
$3$-manifolds}\/,
Proc. of the 3rd Korea-Japan School of knots and links. 
\bibitem {MOO} H. {Murakami}, T. {Othsuki} and M. {Okada},
{\it Invariants of three-manifolds derived from linking
matrices of framed links}\/, Osaka~J.~Math.~29 (1992), 545--572.
\bibitem{PT} J. Przytycki and P. Traczyk, {\em Invariants of links of
Conway type}, Kobe J. Math. 4 (1987), 115--139.
\bibitem{RW} A. Ram, H.~Wenzl, {\em Matrix units for centralizer
algebras}, J. of Alg. 145 (1992), 378--395.
\bibitem{RT} N. {Reshetikhin} and V. {Turaev}, 
{\it Invariants of $3$-manifolds via link polynomials and
quantum groups}\/, Invent.~Math.~103 (1991), 547--598.
\bibitem{T1} V. G. Turaev, {\em Operator invariants of tangles,
and $R$-matrices}, Math. USSR Izv. Vol. 35 (1990), No 2, 411--443.
\bibitem{Turaev} V. G. Turaev, {\em Quantum invariants
of knots and $3$-manifolds}\/, De Gruyter Studies in Math.~18 (1994). 
\bibitem{TW1} V. Turaev, H. Wenzl, {\em Quantum invariants of
$3$-manifolds associated with classical simple Lie algebras},
Int. J. of Math. Vol 4, No 2 (1993) 323--358.
\bibitem{TW2} V. Turaev, H. Wenzl, {\em Semisimple and modular categories
from link invariants}, Math. Ann. 309 (1997), 411--461.
\bibitem{Ver}  E. Verlinde, {\em Fusion rules and modular transformations
in $2d$ conformal field theory}
    Nucl. Physics B 300,
No. 3, 360-376 (1988).
\bibitem{Vo} P. {Vogel}, {\em Les invariants r\'ecents des 
vari\'et\'es de dimension $3$}\/, S\'eminaire Bourbaki 799 (1995).
\bibitem{Wenzl} H.~Wenzl, {\em Hecke algebras of type $A_n$
and subfactors}\/, Invent.~Math.~92 (1988), 349--383.
\bibitem{Wenzl2} H.~Wenzl, {\em Braids and invariants of
$3$-manifolds}\/, Invent.~Math.~114 (1993), 235-275.
\bibitem{Yo} Y.~Yokota, {\em Skeins and quantum $SU(N)$
invariants of $3$-manifolds}\/, Math. Ann. 307 (1997), 109--138.
\end{thebibliography}

\end{document}
{\it G. Faltings}
[J. Algebr. Geom. 3, No. 2, 347-374 (1994;
    Zbl. 809.14009)]; {\it A. Beauville} and {\it Y. Laszlo} [Commun. Math.
    Phys. 164, No. 2, 385-419 (1994; Zbl. 815.14015)]; {\it S. Kumar}, {\it
    M. S. Narasimhan} and {\it A. Ramanathan} [Math. Ann. 300, No. 1, 41-75
    (1994; Zbl. 803.14012)];
{\it A. Bertram} and the author [Topology 32, No.
    3, 599-609 (1993; Zbl. 798.14004)]


Our goal is to construct modular categories from the
 Hecke algebras of type $A$ at roots of unity, and to obtain
certain reductions and refinements of them.
Our main results are the followings.
\begin{itemize}
\item We give  a (reasonably self contained)
construction of  modular categories underlying
the known $SU(N)$ quantum invariants.
As expected, at rank-level $(N,K)$, isomorphism classes of simple objects
are indexed by the set of Young diagrams with at most $N-1$ lines and
$K$ columns, whose cardinality is $\frac{(N+K-1)!}{(N-1)!K!}$.
 The dimensions of the TQFT modules are given by the Verlinde
 formula.
\item At rank-level $(N,K)$ we obtain a reduced modular category
$\widetilde{\mathrm{H}}^{N,K}$,
and a reduced invariant $\tilde \tau_{N,K}$. Here the number of
non-isomorphic simple objects is
$\frac{(N+K-1)!}{N!K!}d$, with $d=\mathrm{gcd}(N,K)$.
Except for $\mathrm{gcd}(N,K)=1$, in which case we recover the $PSU(N)$
invariant, this result seems to be new.
\item We use a grading modulo $d=\mathrm{gcd}(N,K)$ 
to define invariants of $3$-manifolds equipped
with what we call a  spin structure
with coefficients modulo $d$, and we prove a decomposition
formula which extends a theorem of H. Murakami \cite{Mu}.
This refinement holds when $d$ is even, and $\frac{N}{d}$,
$\frac{K}{d}$ are odd (the spin case);
in the other cases, we obtain cohomological refinements.
\end{itemize}

The quantum invariants,
predicted by Witten using Chern-Simons theory and path integrals,
were first constructed by Reshetikhin-Turaev \cite{RT} and Turaev-Wenzl
\cite{TW1} using representation theory of quantum groups.
 The work of Turaev \cite{Turaev} shows that a key concept
 in the construction of these invariants, as well as in extending them
 to a  topological quantum field theory (TQFT), is
 that of a modular category.
 
 A modular category is a braided category with some additional
algebraic features (duality, twist, a finite set of simple objects
satisfying a domination property and a non-degeneracy axiom).
The interest of this concept is that it provides
a topological quantum field theory in dimension $3$, and
in particular, invariants of links and $3$-manifolds. However,
in particular examples, it is not  easy to
define precisely the modular category and
to check the required properties.
  For the $SU(N)$ invariants, first constructed using representation
 theory of $U_q(sl(N))$, it is known that underlying modular categories
can be derived from the category of representations of
the quantum group at roots of unity \cite{TW1,MW,An}.
  In section \ref{sectionSUN}, we will give an alternative elementary
  construction of modular   categories producing the same
invariants\footnote
{Two modular categories producing the same invariant should be equivalent.
  A complete treatment of this question has yet to be formed.}.

A skein theoretic construction 
of the $SU(N)$ invariants was obtained by Yokota \cite{Yo};
subsequent developments towards the associated TQFT's were
made by Lickorish \cite{LiSamp,LiTQFT}.
Our main tool here will be this skein method
combined with the structure of the Hecke algebra,
which in our context is obtained as the Homfly skein
module of a cylinder $[0,1]\times\mathbf{D}^2$ with
boundary points.
 This algebra
 has been intensively studied (see \cite{Geck} for a list of
 references). For our purpose, we emphasize the work of
 Jones \cite{Jones}
and Wenzl \cite{Wenzl};
see also \cite{Gyo}, \cite{DJ}.  Our  normalization and description of
idempotents coincide
 with  those of Aiston-Morton
 \cite{Aiston,AM,MA}.

Following the work of Kirby and Melvin for the $SU(2)$
case \cite{KM}, Kohno and Takata have studied symmetry formulas
for the $SU(N)$ quantum invariants \cite{KT1,KT2}.
 This was used by H. Murakami in \cite{Mu}.
 Our reductions and refinements formulas generalize
 these results.
 
 Invariants of $3$-manifolds from  Hecke algebras
 were obtained by Wenzl in \cite{Wenzl2};
 modular and semi-simple categories from unoriented link invariants
(BCD case) are considered by Turaev and Wenzl in \cite{TW2}.

Masbaum and Wenzl \cite{MW} have proved  the integrality of $SU(N)$
quantum invariants at roots of unity of prime order, and
they have shown that this follows from existence of integral
modular categories. This is developed by Brugui\`eres in \cite{Brug}.

\noindent{\bf Acknowledgments.} The author wishes to thanks 
A. Beliakova, A. Brugui\`eres, N. Habegger
G. Masbaum,  V. Turaev  and P. Vogel
for useful discussions and suggestions.

\section{Young idempotents and Homfly skein theory}\label{homflyCAT}
\subsection{The Homfly functor}
Let $M$ be an oriented $3$-manifold.
We denote by $\mathcal{H}(M)$  the
$k$-module freely generated by
isotopy classes of framed links
in $M$ with the Homfly relations 
given in figure \ref{homflyrel}.
 Here a framing  is a trivialization of the normal
 bundle; this gives an orientation of the link.
\begin{figure}[h]
\begin{center}
\vspace{5pt}
$a^{-1}$
\begin{pspicture}[.4](0,0)(1,1)
%\psgrid
\psline{->}(.1,0)(.9,1)
\psline{-}(.9,0)(.6,.4)
\psline{->}(.4,.6)(.1,1)
\end{pspicture}
$\ -\ a$
\begin{pspicture}[.4](0,0)(1,1)
%\psgrid
\psline{->}(.9,0)(.1,1)
\psline{-}(.1,0)(.4,.4)
\psline{->}(.6,.6)(.9,1)
\end{pspicture}
$\ =\ (s-s^{-1})$
\begin{pspicture}[.4](0,0)(1,1)
\pscurve{->}(.1,0)(.4,.5)(.1,1)
\pscurve{->}(.9,0)(.6,.5)(.9,1)
\end{pspicture}\\[10pt]
\begin{pspicture}[.4](0,0)(1,1)
\pscurve{-}(.1,0)(.15,.45)(.45,.75)(.7,.5)(.4,.2)(.20,.38)
\pscurve{->}(.14,.62)(.1,.9)(.1,1)
\end{pspicture}
$\ =\ av^{-1}$
\begin{pspicture}[.4](0,0)(1,1)
\psline{->}(.5,0)(.5,1)
\end{pspicture}
\hspace{2cm}
\begin{pspicture}[.4](0,0)(1,1)
\pscurve{<-}(.1,1)(.15,.55)(.45,.25)(.7,.5)(.4,.8)(.20,.62)
\pscurve{-}(.14,.38)(.1,.1)(.1,0)
\end{pspicture}
$\ =\ a^{-1}v$
\begin{pspicture}[.4](0,0)(1,1)
\psline{->}(.5,0)(.5,1)
\end{pspicture}\\[10pt]
$L\ \cup\ $
\begin{pspicture}[.4](0,0)(1,1)
\pscircle(.5,.5){.4}
\end{pspicture}
$\ =\ {\frac{v^{-1}-v}{s-s^{-1}}}\ L$
\vspace{5pt}
\end{center}
\caption{\label{homflyrel} Homfly relations}
\end{figure}

Here $k$ is an integral domain containing the invertible elements
$a$, $v$, $s$;
we suppose moreover that ${s-s^{-1}}$ is invertible in $k$.
 (For $L\neq \emptyset$, the third equality is a consequence of
the others.)

The Homfly polynomial \cite{homfly,PT} gives an isomorphism
$$\begin{array}{lrcl}
\langle \dots \rangle :&\mathcal{H}({\bf S}^3)&\rightarrow &k\\
&L&\mapsto &\langle L\rangle\end{array}$$
normalized by $\langle \emptyset\rangle=1$.

An embedding $j: M\hookrightarrow N$ gives a well defined operator
$\mathcal{H}(j): \mathcal{H}(M)\rightarrow \mathcal{H}(N)$. 
 This makes $\mathcal{H}$ into a functor from the category
 of $3$-manifolds whose morphisms are isotopy classes of embeddings, to
 the category of $k$-modules.

 To an oriented embedding of a disjoint union of solid tori
$$g=\coprod_{i=1}^mg_i : \ \coprod_{i=1}^m {\bf D}^2_i
\times {\bf S}_i^1\rightarrow {\bf S}^3$$
is associated a multilinear map
$${\mathcal{H} }(g):
\ {\mathcal{H}}({\bf D}^2\times {\bf S}^1)^{\otimes m}
\rightarrow {\mathcal H}({\bf S}^3)\approx k$$
This map only depends  on the isotopy class of the framed link 
$L=(L_1,\dots,L_m)$ underlying $g$.
The image of $x_1\otimes\dots\otimes x_m$ under this map
 is said to be obtained by cabling the components
 $L_i$ with the skein elements $x_i$,
and is denoted by
$\langle L_1(x_1),\dots,L_m(x_m)\rangle$ or
$\langle L(x_1,\dots,x_m)\rangle$.

 
In the case where $M$ has non-empty boundary, we may fix a framed link
$l$ in the boundary of $M$, and define the relative
skein module $\mathcal{H}(M,l)$.
By comparing with the orientation of $\partial M$
(our convention for orientation of the boundary is
outgoing normal vector first), the framing of $l$ gives
an orientation of its components.
% The component is said to
%be positively oriented (or outgoing), if the two orientations coincide;
%else it is said to be negatively oriented (or incoming).

\subsection{The Hecke category}
The {\em $k$-linear Hecke category} ${\mathrm{H}}$ is defined as follows.
An object in this category is a disc $\mathbf{D}^2$ equipped with a framed
link
(note that isotopy is not allowed).
 If $\alpha=(\mathbf{D}^2,l_0)$ and $\beta=(\mathbf{D}^2,l_1)$
are two objects,
 the module $Hom_{\mathrm{H}}(\alpha,\beta)$ is
$\mathcal{H}( \mathbf{D}^2\times[0;1],l_0\times 0\amalg l_1\times 1)$.
 The notation ${\mathrm{H}}(\alpha,\beta)$ and ${\mathrm{H}}_\alpha$ will be used respectively
for $Hom_{\mathrm{H}}(\alpha,\beta)$ and $End_{\mathrm{H}}(\alpha)$.
For composition, we use the {\em covariant} notation
$$\begin{array}{rcl}
{\mathrm{H}}(\alpha,\beta)\times {\mathrm{H}}(\beta,\gamma)&\longrightarrow &{\mathrm{H}}(\alpha,\gamma)\\
(f,g)&\longmapsto &fg
\end{array}$$
When we draw a figure to describe a morphism, the {\it time} parameter
goes upwards, so that the morphism $fg$ is depicted
with $g$ lying above $f$.

\noindent{\bf Note.} The terminology {\em Hecke category} was introduced
by Turaev in \cite{T1}. His definition gives a category
equivalent to an unframed version of ours.

We will use the notation $\hat{f}$ for the closure
in $\mathcal{H}(\mathbf{D}^2\times\mathbf{S}^1)$
of the morphism $f\in {\mathrm{H}}_\alpha$. Let $U_0$ be a $0$-framed unknot
in $\mathbf{S}^3$. We use the notation
$\langle f \rangle$ for $\langle U_0(\hat f) \rangle$
(the quantum trace).

 We will simply denote  by $n$
the object formed by $n$ standard 
 points (along the real axis)
 equipped with the standard
framing $(1,\sqrt{-1})$.

 By using a standard embedding
${\bf D}^2\amalg {\bf D}^2\hookrightarrow {\bf D}^2$, we make
 ${\mathrm{H}}$ into a  monoidal
category\footnote{The associativity
isomorphisms just move the embedded discs along the real axis,
and will be omitted.  Note that the object
$1^{\otimes n}$ is defined up to these associativity isomorphisms,
and is canonically isomorphic to the standard object $n$.}.
 This category has a braiding and a twist operator. We can
define in ${\mathrm{H}}$ a duality rule so that we get a ribbon category
(\cite{Turaev}, see also \cite{Vo}). We proceed as follows.
To an object $\alpha=(\mathbf{D}^2,l)$, we associate
the object $\alpha^*=(\mathbf{D}^2,-\overline l)$, where
$-\overline l$ is the framed link obtained by applying to $l$
the differential of $z\mapsto -\overline z$, and define the morphisms
$b_\alpha\in {\mathrm{H}}(0,\alpha\otimes \alpha^*)$
and $d_\alpha\in {\mathrm{H}}(\alpha^*\otimes\alpha,0)$ according to
the figure below (a copy of $1_\alpha$ is embedded
along the framed arc).
%\begin{figure}[hbt]
%\setlength{\unitlength}{1mm}
\centerline{
        \begin{pspicture}(2.5,2)
\put(-0.8,.9){$b_\alpha=$}
\pscurve{->}(1.5,1.5)(1.47,1.3)(1,.8)(.53,1.3)(.5,1.5)
\put(0.15,1.2){$\alpha$}
\end{pspicture}
\hspace{1cm}
\begin{pspicture}(2.5,2)
\put(-0.2,.9){$d_\alpha=$}
\pscurve{->}(2,.5)(1.97,.7)(1.5,1.2)(1.03,.7)(1,.5)
\put(2.1,.5){$\alpha$}
\end{pspicture}}
%\caption{\label{duality} Duality in Hecke category}
%\end{figure}

Our purpose is to discuss which modular categories
arise from this ribbon Hecke category.
 For this, we need 
  a finite set of  simple objects with nice properties
  (\cite{Turaev} p74).

\subsection{Idempotents in the Hecke algebra}
 The algebra ${\mathrm{H}}_n$
 is isomorphic to the quotient of the algebra
of the braid group $k[B_n]$ by the Homfly relation

\begin{center}
\vspace{5pt}
$a^{-1}$
\begin{pspicture}[.4](0,0)(1,1)
%\psgrid
\psline{->}(.1,0)(.9,1)
\psline{-}(.9,0)(.6,.4)
\psline{->}(.4,.6)(.1,1)
\end{pspicture}
$\ -\ a$
\begin{pspicture}[.4](0,0)(1,1)
%\psgrid
\psline{->}(.9,0)(.1,1)
\psline{-}(.1,0)(.4,.4)
\psline{->}(.6,.6)(.9,1)
\end{pspicture}
$\ =\ (s-s^{-1})$
\begin{pspicture}[.4](0,0)(1,1)
\pscurve{->}(.1,0)(.4,.5)(.1,1)
\pscurve{->}(.9,0)(.6,.5)(.9,1)
\end{pspicture}
\vspace{5pt}
\end{center}
which is the Hecke algebra of type $A_{n-1}$.
 An unframed version of this result was proved independently by
 Morton-Traczyk \cite{MT} and Turaev \cite{T1}.
This algebra is known to be generically
semi-simple.
 It is a deformation of the algebra of the symmetric group,
 and its structure can be obtained by extending
 the classical  Young theory
\cite{Bou,Jones,Wenzl,Gyo,DJ}.
%We could use directly the known results, however we find it
%convenient to give an overview of what is  needeed here.






 Recall that, to a partition of $n$,
 $\lambda=(\lambda_1\geq\dots\geq\lambda_p\geq1)$,
 $\lambda_1+\dots+\lambda_p=n$, is associated a Young diagram
 of size $|\lambda|=n$, which we denote also by $\lambda$. This diagram
 has $n$ cells indexed by
$\{(i,j),\ 1\leq i \leq p,\ 1\leq j\leq \lambda_i\}$.
% This cells are represented with the same line-column
%convention as for matrices.
 


If $c$ is the cell of index $(i,j)$ in a Young diagram,
its hook-length $hl(c)$ and its content $cn(c)$
are defined by
$$hl(c)=\lambda_i+\lambda_j^\vee-i-j+1,\ \ cn(c)=j-i.$$
Here $\lambda^\vee$ is the transposed Young diagram and
$\lambda^\vee_j$ is the length of the $j$-th column
of $\lambda$ (the $j$-th line of $\lambda^\vee$).

For $n\geq0$, the quantum integer $[n]$, and the
quantum factorial $[n]!$ are defined by
$[n]=\frac{s^{n}-s^{-n}}{s-s^{-1}}$ and $[n]!=\prod_{j=1}^n [j]$.\\
For a Young diagram $\lambda$,
we will use the notation
$[hl(\lambda)]$, for the product over all cells
of the quantum
hook-lengths.
$$[hl(\lambda)]=\prod_{\mathrm{\scriptstyle cells}} [hl(c)]$$

\noindent{\bf{Symmetrizers.}}
Let $\sigma_i\in {\mathrm{H}}_n$, $i=1,\dots,n-1$, be represented by
the standard generators of the braid group $B_n$
(the strand numbered $i$ crosses over the strand numbered
$i+1$).
\begin{pro}
If $[n]!$ is invertible in $k$, then there exists a unique idempotent
$f_n\in {\mathrm{H}}_n$ such that $\forall i\ \sigma_if_n=as f_n=f_n\sigma_i$,
 and a unique idempotent
$g_n\in {\mathrm{H}}_n$ such that $\forall i\ \sigma_ig_n=-as^{-1} g_n=g_n\sigma_i$.
\end{pro}
\begin{proof}
It is shown in \cite{Morton,AM}  that the deformation $f_n$
of the Young symmetrizer, given below, satisfies the required condition.
$$f_n=\frac{1}{[n]!}s^{-\frac{n(n-1)}{2}}
\sum_{\pi\in \mathcal{S}_n} (as^{-1})^{-l(\pi)}w_\pi$$
Here $w_\pi$ is the positive braid associated with the
permutation $\pi$, and $l(\pi)$ is the length of $\pi$.

One can also construct $f_n$ recursively using the formulas below (\cite{Yo}).
$$[2]f_2=
s^{-1}\begin{pspicture}[.4](0,0)(1,1)
\pscurve{->}(.1,0)(.4,.5)(.1,1)
\pscurve{->}(.9,0)(.6,.5)(.9,1)
\end{pspicture}
+a^{-1}
\begin{pspicture}[.4](0,0)(1,1)
%\psgrid
\psline{->}(.1,0)(.9,1)
\psline{-}(.9,0)(.6,.4)
\psline{->}(.4,.6)(.1,1)
\end{pspicture}$$
$$[n+1]f_{n+1}=-[n-1]f_n\otimes 1_1 +[2][n]
(f_n\otimes 1_1)(1_{n-1}\otimes f_2)(f_n\otimes 1_1)$$
\noindent(Here, $1_p\in {\mathrm{H}}_p$ is the identity.)

Suppose now that
$f'_n$ satisfies the condition of the theorem. Then
 we have that $f_nf'_n=f'_n=\zeta f_n$, and $\zeta=1$
  by idempotence.

 For $g_n$ we can proceed similarly,
 either with the deformed antisymmetrizer
 $$g_n=\frac{1}{[n]!}s^{\frac{n(n-1)}{2}}
\sum_{\pi\in \mathcal{S}_n} (-as)^{-l(\pi)}w_\pi$$
 or with the recursive formulas
 $$g_1=1_1$$
 $$g_{n+1}=1_1\otimes g_n -\frac{[2][n]}{[n+1]}
(1_1\otimes g_n)( f_2\otimes 1_{n-1})(1_1\otimes g_n)$$
\end{proof}
Note that $1_2=f_2+g_2$; this can be used to obtain
more symmetric recursive formulas for $f_n$ and $g_n$.
Our choice  immediately gives the following  lemma \cite{Yo},
which is useful in Yokota's skein computations.
\begin{lem}
For any two integers $p$, $q$ such that $[p+1]!$ and $[q+1]!$
are invertible, one has
\begin{eqnarray*}
[p+q]f_p\otimes g_q&=&[p+1][q]
(1_p\otimes g_q)( f_{p+1}\otimes 1_{q-1})(1_p\otimes g_q)\\&&
+[p][q+1](f_p\otimes 1_q)(1_{p-1}\otimes g_{q+1})(f_p\otimes 1_q)
\end{eqnarray*}
\end{lem}
\noindent{\bf Aiston-Morton description of Young symmetrizers.}
 For a Young diagram $\lambda$ of size $n$, we denote
by $\square_\lambda$ the object of the category ${\mathrm{H}}$ formed
with one point  for each cell $c$ of $\lambda$; if $c$ has index $(i,j)$
($i$-th line, and $j$-th column),
then the corresponding point in $\mathbf{D}^2$ is
$\frac{j+i\sqrt{-1}}{n}$.
Suppose that $\lambda=(\lambda_1\geq\dots\geq\lambda_p\geq1)$, and
that $\lambda^\vee=(\lambda_1^\vee\geq\dots\geq\lambda_q^\vee\geq1)$
is the transposed Young diagram.\\
Let $F_\lambda$ (resp. $G_\lambda$) be the element
in $ {\mathrm{H}}_{\square_{\lambda}}$ formed with one copy
of $[\lambda_i]!f_{\lambda_i}$ along the line $i$, for $i=1,\dots,p$
(resp. one copy
of $[\lambda^\vee_j]! g_{\lambda_j^\vee}$ along the column $j$,
 for $j=1,\dots,q$).
 Note that these expressions have no denominators.

 In the proposition below, $<$ denotes the lexicographic
 ordering, and $\lambda$ and $\mu$ are Young diagrams with
 the same number of cells.
\begin{pro}\label{minimal}
a) If $\mu<\lambda$, then
$F_\lambda  {\mathrm{H}}(\square_{\lambda},\square_{\mu})
 G_\mu=0$.\\
b) If $\lambda<\mu$, then
$G_\lambda  {\mathrm{H}}(\square_{\lambda},\square_{\mu})
 F_\mu=0$.\\ 
 c) One has $F_\lambda  {\mathrm{H}}_{\square_{\lambda}}  G_\lambda=
kF_\lambda G_\lambda$.
\end{pro}
This proposition is proved in \cite{AM}, section 4.
We outline here the proof given the




In this section we consider the modular category 
$\widetilde{\mathrm{H}}^{N,K}$ in the spin case.
 This means that $N+K$ is even, and that
 $N'$ and $K'$ are both odd.
 We fix $\alpha$ and $\beta$ as we did in the preceding section, 
and we suppose moreover that $\alpha$ is odd.


\begin{tabular}{ccc}
\begin{pspicture}[.5](0,-0.5)(1,1)
\psline{->}(0,0)(1,1)
\psline{-}(1,0)(.65,.35)
\psline{->}(.35,.65)(0,1)
 \put(0,.3){$\varrho$} \put(.8,.3){$\varrho$}
\end{pspicture}
&$\ =\ $&
\begin{pspicture}[.5](0,-0.5)(1,1)
\psline{->}(1,0)(0,1)
\psline{-}(0,0)(.35,.35)
\psline{->}(.65,.65)(1,1)
 \put(0,.3){$\varrho$} \put(.8,.3){$\varrho$}
\end{pspicture}\\
\begin{pspicture}[.5](0,-0.5)(1,1)
\pscurve{-}(.1,0)(.15,.45)(.45,.75)(.7,.5)(.4,.2)(.20,.38)
\pscurve{->}(.14,.62)(.1,.9)(.1,1)
 \put(.2,-.2){$\varrho$}
\end{pspicture}
&$\ =\ $&$-\ $
\begin{pspicture}[.5](0,-.5)(1,1)
\psline{->}(.1,0)(.1,1)
 \put(.2,-.2){$\varrho$}
\end{pspicture}
\end{tabular}}
\begin{figure}
\centerline{
includegraphics{26}{52}{bor}}
\caption{\label{borr} Borromean link}
\end{figure}


\centerline{\begin{tabular}{ccc}
\begin{pspicture}[.5](0,-0.5)(1,1)
\psline{->}(0,0)(1,1)
\psline{-}(1,0)(.65,.35)
\psline{->}(.35,.65)(0,1)
 \put(0,.3){$\varrho$} \put(.8,.3){$\varrho$}
\end{pspicture}
&$\ =\ $&
\begin{pspicture}[.5](0,-0.5)(1,1)
\psline{->}(1,0)(0,1)
\psline{-}(0,0)(.35,.35)
\psline{->}(.65,.65)(1,1)
 \put(0,.3){$\varrho$} \put(.8,.3){$\varrho$}
\end{pspicture}\\

\section{Survey on modular categories derived from Homfly skein theory}
\label{Mod_homfly}

Following Yokota's skein construction
of the $SU(N)$ quantum invariants \cite{Yo}, we have constructed in \cite{Bhec} a version of the $SU(N,K)$
modular category from Homfly skein calculus.
Its relation with the quantum group approach \cite{TW} is explained
in \cite{MW}.
  We denote it by $\mathrm{H}^{SU(N,K)}$.
For each partition 
(or Young diagram)
$\lambda=(\lambda_1,\dots,\lambda_p),\
\lambda_1\leq K\text{ and }p\leq N$, we have defined a simple
object, also denoted by $\lambda$.
A distinguished set of simple objects in the modular
 category $\mathrm{H}^{SU(N,K)}$ is
 $$\Gamma_{N,K}=\{\lambda=(\lambda_1,\dots,\lambda_{N}),\
K\geq\lambda_1\geq\dots\geq \lambda_{N-1}\geq\lambda_N=0\}$$
We have also constructed a reduced modular category
$\widetilde{\mathrm{H}}^{N,K}$. This is a generalization
of the $PSU(N,K)$ modular category to the case where
$N$ and $K$ are not coprime.
In the category $\widetilde{\mathrm{H}}^{N,K}$, 
the set of isomorphism classes of simple
objects is obtained from
$$\dot \Gamma_{N,K}=\{(1^N)^{\otimes i}\otimes \lambda , \
0\leq i< \alpha\text{ and }\lambda\in \Gamma_{N,K}\}$$
quotienting by the free action of a cyclic group
of order $N/\beta$.
Here $d=\mathrm{gcd}(N,K)$, $N=dN'$, $K=dK'$, $d=\alpha \beta$ with
$\mathrm{gcd}(\alpha,K')=\mathrm{gcd}(\beta,N')=
\mathrm{gcd}(\alpha,\beta)=1$. 
 We outline  below the skein construction of these modular categories.


\esp \noindent
{\bf Homfly invariant for framed links}.
The Homfly polynomial for isotopy classes of
framed links $L$ 
in the $3$-sphere, denoted by $\langle L\rangle$,
is uniquely defined by the skein
rules 
given in figure \ref{homflyrel} and the normalization
$\langle\emptyset\rangle=1$.
 Here a framing is a
trivialization of the normal bundle up to homotopy.
This is equivalent to an orientation of the link together with a
non-singular normal vector field up to homotopy.
 In the figures, a preferred convention using the plane gives the framing
(blackboard framing).
\begin{figure}[h]
\begin{center}
\vspace{5pt}
$a^{-1}$
\begin{pspicture}[.4](-.4,0)(1.4,1)
%\psgrid
\psline[linewidth=.01]{-}(0,.1)(-.2,.5)(0,.9)
\psline[linewidth=.01]{-}(1,.1)(1.2,.5)(1,.9)
\psline{->}(.1,0)(.9,1)
\psline{-}(.9,0)(.6,.4)
\psline{->}(.4,.6)(.1,1)
\end{pspicture}
$\ -\ a$
\begin{pspicture}[.4](-.4,0)(1.4,1)
%\psgrid
\psline[linewidth=.01]{-}(0,.1)(-.2,.5)(0,.9)
\psline[linewidth=.01]{-}(1,.1)(1.2,.5)(1,.9)
\psline{->}(.9,0)(.1,1)
\psline{-}(.1,0)(.4,.4)
\psline{->}(.6,.6)(.9,1)
\end{pspicture}
$\ =\ (s-s^{-1})$
\begin{pspicture}[.4](-.4,0)(1.4,1)
\psline[linewidth=.01]{-}(0,.1)(-.2,.5)(0,.9)
\psline[linewidth=.01]{-}(1,.1)(1.2,.5)(1,.9)
\pscurve{->}(.1,0)(.4,.5)(.1,1)
\pscurve{->}(.9,0)(.6,.5)(.9,1)
\end{pspicture}\\[10pt]
\begin{pspicture}[.4](0,0)(1,1)
\psline[linewidth=.01]{-}(-.1,.1)(-.3,.5)(-.1,.9)
\psline[linewidth=.01]{-}(.9,.1)(1.1,.5)(.9,.9)
\pscurve{-}(.1,0)(.15,.45)(.45,.75)(.7,.5)(.4,.2)(.20,.38)
\pscurve{->}(.14,.62)(.1,.9)(.1,1)
\end{pspicture}
$\ =\ av^{-1}$
\begin{pspicture}[.4](-.4,0)(1.4,1)
\psline[linewidth=.01]{-}(0,.1)(-.2,.5)(0,.9)
\psline[linewidth=.01]{-}(1,.1)(1.2,.5)(1,.9)
\psline{->}(.5,0)(.5,1)
\end{pspicture}
\hspace{2cm}
\begin{pspicture}[.4](-.4,0)(1.4,1)
\psline[linewidth=.01]{-}(-.1,.1)(-.3,.5)(-.1,.9)
\psline[linewidth=.01]{-}(.9,.1)(1.1,.5)(.9,.9)
\pscurve{<-}(.1,1)(.15,.55)(.45,.25)(.7,.5)(.4,.8)(.20,.62)
\pscurve{-}(.14,.38)(.1,.1)(.1,0)
\end{pspicture}
$\ =\ a^{-1}v$
\begin{pspicture}[.4](-.4,0)(1.4,1)
\psline[linewidth=.01]{-}(0,.1)(-.2,.5)(0,.9)
\psline[linewidth=.01]{-}(1,.1)(1.2,.5)(1,.9)
\psline{->}(.5,0)(.5,1)
\end{pspicture}\\[10pt]
$\langle L\ \cup\ $
\begin{pspicture}[.4](-.4,0)(1.4,1)
%\psline[linewidth=.01]{-}(0,.1)(-.2,.5)(0,.9)
%\psline[linewidth=.01]{-}(1,.1)(1.2,.5)(1,.9)
\pscircle(.5,.5){.4}
\end{pspicture}
$\rangle\ =\ {\frac{v^{-1}-v}{s-s^{-1}}}\ \langle L\rangle$
\vspace{5pt}
\end{center}
\caption{\label{homflyrel} Homfly relations}
\end{figure}
Here $a$, $v$, $s$, ${s-s^{-1}}$, are non-zero
elements in the scalar field $\fk$.

\esp\noindent
{\bf Hecke category.} The invariant above can be used to construct a quotient of the tangle
category. We found it convenient to consider a version of the tangle
category in which we allow as object any finite set of points in the disc $\mathbf{D}^2$,
equipped with a trivialization of the tangent space at each point
(by comparing with the orientation of the disc,
 we get an orientation of the points).
 
 The quotient kills all {\em negligible} morphisms, i.e.
 those which always contribute zero in the Homfly invariant.

 This category is then completed by adding the idempotents of the category
 as objects.
 In fact we need just enough new simple objects to obtain
 the domination axiom, and those are constructed explicitly.
 If one work with the representation theory of
 $U_qsl(N)$, those simple objects already exist
 as irreducible representations.
 
  At this step, the rank-level specialization plays a role.
The rank $N$ specialization is $v=s^{-N}$, and here level
$K$ means that, in the scalar field $\fk$,
$
s^2$ has order $N+K$.  We have $v=\varepsilon s^K$,
with $\varepsilon=-1$ if $N+K$ is even, and 
$\varepsilon^2=1$ if $N+K$ is odd.
%We suppose that $N+K$ is invertible in $k$.
%Under this conditions
We suppose that the field $\fk$ has characteristic zero.
(In fact the construction works if $N+K$ is non-zero in $\fk$.)
 We have simple objects
denoted by partitions $\lambda=(\lambda_1,\dots,\lambda_p),\
\lambda_1\leq K\text{ and }p\leq N$.

We note that for a fixed rank-level $(N,K)$, the quotient by negligible morphisms 
depends on the value of the framing parameter $a$.
 We denote by $H^{(N,K,a)}$ the category constructed in this way.
 This is a $\fk$-linear ribbon category.
We  use the following notation.
 $$\overline\Gamma_{N,K}=\{\lambda=(\lambda_1,\dots,\lambda_{N}),\
K\geq\lambda_1\geq\dots\geq \lambda_{N-1}\geq\lambda_N\}$$
 $$\Gamma_{N,K}=\{\lambda=(\lambda_1,\dots,\lambda_{N}),\
K\geq\lambda_1\geq\dots\geq \lambda_{N-1}\geq\lambda_N=0\}$$
The key step towards the domination axiom
is the branching formula \cite[Theorem 1.13]{Bhec}.
 Here $\one_\lambda$ denotes the identity morphism of the object $\lambda$.
\begin{thm}[Branching formula]\label{branch}
For any $\lambda\in \Gamma_{N,K}$ one has
$$\one_\lambda\otimes \oneun=
\sum_{\genfrac{}{}{0pt}{2}{\lambda\subset \mu}{|\mu|=|\lambda|+1}}
(\one_\lambda\otimes \oneun)\one_\mu (\one_\lambda\otimes \oneun)$$
%$$\Idone\otimes e_t=\sum_{\check\tau=t}
%(\Idone\otimes e_t)e_\tau (\Idone\otimes e_t)$$
\end{thm}
\noindent We have omitted in this formula the standard
morphisms in $H^{(N,K,a)}(\lambda\otimes 1,\mu)$ and 
$H^{(N,K,a)}(\mu,\lambda\otimes 1)$.
Another key observation is the role of the objects $1^N$ and $(K)$.\\[5pt]
a) They are invertible objects in the category $H^{(N,K,a)}$,
and generate the group of isomorphism classes of invertible objects.\\
b) Let $\mu\in \overline{\Gamma}_{N,K}$ be a
Young diagram of the form $\mu=1^N+\lambda$, then $\mu$ and $1^N\otimes \lambda)$
are isomorphic objects..

\esp Using the branching formula, and the observation b) above, we can show
that the ribbon category $H^{(N,K,a)}$ is semi-simple. By this, we mean
\cite{TW2} that the following infinite set of simple
objects satisfies the Schur lemma and the domination property.
$$\widehat{\Gamma}_{N,K}=
\{(1^N)^{\otimes i}\otimes \lambda, i\in \mathbb{Z} \text{ and }
\lambda\in \Gamma_{N,K}\}$$
In other words, the $\fk$-linear ribbon category $H^{(N,K,a)}$
satisfies the axioms of a pre-modular category,
 except the finiteness of the set of
isomorphism classes of simple objects.

An object  $X$ is called transparent is and only if, for every object $Y$,
the braiding satisfies $R_{Y,X}=R_{X,Y}^{-1}$.
 If $a$ is a root of unity, then the set $T^{(N,K,a)}$ of isomorphism
classes of transparent simple objects is a subgroup
of the group of isomorphism classes of invertible objects.

 The group $T^{(N,K,a)}$  acts on the set of isomorphism classes of simple objects.
Moreover the quotient set modulo this action is finite.
 Following Brugui�res' procedure, we can construct a dominant ribbon functor
from the category $H^{(N,K,a)}$ to a pre-modular category; we can say that 
the category $H^{(N,K,a)}$ is 
pre-modularizable. Furthermore we can study the modularization problem,
and  a slightly extended version of Brugui�res' theorems applies.

 For a given rank-level $(N,K)$,
we  consider two specializations of the parameter $a$.
 The corresponding modularizations can be obtained by applying Brugui�res' method.
 Explicit descriptions of   the modularizations can be found in
 \cite{Bhec}.

\esp\noindent{\bf The modular category $\mathrm{H}^{SU(N,K)}$.}
We consider  
 the $SU(N,K)$ specialization. Here $s= a^{-N}$, and 
$a$ is a $2N(N+K)$th root of unity.
We can define morphisms in the  category
$H^{(N,K,a)}$
 by using colored ribbon graphs (see \cite{Tu,Bhec}).
We  extend the  category $H^{(N,K,a)}$
by allowing  
 incoming or outgoing vertices, colored with $r=1^N$.
Together with the relation of isotopy rel. boundary, the Homfly relations,
and the negligible morphisms, we add the relation given
 by gluing an incoming $r$-colored vertex with an outgoing one.
(Note that the half twist is not trivial; hence the orientations
of the glued ends of the ribbon edges must be respected.)
 We denote by $\mathrm{H}^{SU(N,K)}$ the category whose objects are those of
$H^{(N,K,a)}$, and whose morphisms are generated by extended
ribbon graphs as above with the described relations.
 The object $1^N$ becomes isomorphic to the trivial object, and we have  
Theorem \ref{modcatH} (\cite[Theorem 2.11]{Bhec}).

\esp\noindent{\bf The modular category $\widetilde{\mathrm{H}}^{N,K}$.}
Here $v=s^{-N}$ and 
$$\begin{cases}
s \text{ has order $2(N+K)$ if $N+K$ is even,}\\
s \text{ has order $N+K$ if $N+K$ is odd.}
\end{cases}$$
In both cases $s^2$ has order $N+K$; we have $v=\varepsilon s^K$,
with $\varepsilon=(-1)^{N+K+1}$.\\


We set $d=\mathrm{gcd}(N,K)$, $N=dN'$, $K=dK'$, $d=\alpha \beta$ with
$\mathrm{gcd}(\alpha,K')=\mathrm{gcd}(\beta,N')=
\mathrm{gcd}(\alpha,\beta)=1$.
We choose
the framing parameter $a$ so that
$(a^Ns)^\alpha=\varepsilon(a^Ks^{-1})^\beta=\varepsilon^\beta$.
 In this case, the set of isomorphism classes of transparent simple objects
is the group generared by $(1^N)^{\otimes\alpha}$
and $(K)^{\otimes\beta}$, and we can show that the category is modularizable.
 We denote by $\widetilde{\mathrm H}^{N,K}$ a modularization.
 The  category $\widetilde{\mathrm H}^{N,K}$ can be described 
 as follows. Objects are those of $H^{(N,K,a)}$,
 and the modules of morphisms are obtained from
 those of ${H}^{(N,K,a)}$ by adding as generators
 colored ribbon graphs in which incoming or outgoing
$1$-valent vertices colored with $(1^N)^{\otimes\alpha}$
or $(K)^{\otimes\beta}$ are allowed,
and quotienting by
\begin{verse}
the relations given by gluing
an incoming $(1^N)^{\otimes\alpha}$-colored 
(resp. $(K)^{\otimes\beta}$-colored) vertex
with an outgoing one,\\
the relations given by introducing a copy
of a standard isomorphism 
$y_{N^K}\in H^{(N,K,a)}(K^{\otimes N},(1^N)^{\otimes K})$.
\end{verse}
Set $\dot\Gamma_{N,K}=\{(1^N)^{\otimes i}\otimes \lambda,\ 0\leq i<\alpha
\text{ and }
\lambda \in \Gamma_{N,K}\}$.\\
The category $\widetilde{\mathrm{H}}^{N,K}$ is a  ribbon category,
and $\dot\Gamma_{N,K}$ is a  finite set of dominating simple objects.

There is an action of $\mathbb{Z}/N$ on the set
$\dot \Gamma_{N,K}$ defined as follows.
The generator of $\mathbb{Z}/N$ acts by
$$
(1^N) ^{\otimes i}\otimes\lambda\mapsto
(1^N) ^{\otimes i'}\otimes
((K,\lambda)-\lambda_{N-1}^N)\ ,$$
where $i'\in\{0,\dots,\alpha-1\}$ is congruent to
$i+\lambda_{N-1}$ modulo $\alpha$\ .\\
The idea is that we add to the diagram a row with
$K$ cells, and then each column which has  $N$ cells
is replaced by an added copy of $1^N$.

Restricting this action to the group generated by $\beta$,
we obtain an action of $\mathbb{Z}/{\alpha N'}$.
 By considering the degree mod $\alpha N$, we can see
 that this action is free.
We denote by $\widetilde \Gamma_{N,K}$ a subset of
$\dot\Gamma_{N,K}$ which is a representative set for the orbits, then we have
theorem \ref{reduced} (\cite[Theorem 3.5]{Bhec}).

defined for the generator by
$$(j^N)+\lambda\mapsto (j+\lambda_{N-1})^N+(K-\lambda_{N-1},
\lambda_1-\lambda_{N-1},\dots,\lambda_{N-2}-\lambda_{N-1},0)$$

math0303240.extracted_bib
\begin{thebibliography}{10}
\bibitem{AM} J. Andersen, G. Masbaum, {\em Involutions on moduli spaces
and refinements of the Verlinde formula}, Math. Ann. 314, No.2, 291-326 (1999).
\bibitem{AP} H. Andersen, J. Paradowski {\em Fusion  category arising from
semisimple Lie algebras}, Comm. Math. Phys. 169, No.3, 563--588 (1995).
\bibitem{A} M. F. Atiyah {\em Riemann surfaces and spin structures},
Ann. Ecole Norm. Sup. (4) 4, 47--62 (1971).
\bibitem{BK} B. Bakalov, A. Kirillov, {\em Lecture on tensor categories and
modular functors}, Univ. Lecture Series No.21, AMS (2001).
\bibitem{Beau} A. Beauville, {\em Conformal blocks,
fusion rules and the Verlinde formula}\/,
Israel Math. Conf. Proceedings Vol. 9, 75--96 (1996).
\bibitem{BL} A. Beauville, Y. Laszlo, {\em Conformal blocks and
generalized theta functions}\/, Comm. Math. Phys. 164, 385--419 (1994).
\bibitem{BB}  A. Beliakova, C. Blanchet, {\em Modular categories
 of types B,C and D}, Comment. Math. Helv. 76 (2001) 467-500.
\bibitem{BiL} J-M. Bismut, F. Labourie, {\em Formules de Verlinde
pour les groupes simplement connexes et g�om�trie sympleptique},
CRAS, t. 325, S�rie I (1997), 1009--1014.
\bibitem{BiL2} J-M. Bismut, F. Labourie, {\em Sympleptic geometry
and the Verlinde formulas}, In Surveys in differential geometry: differential geometry inspired by string theory,
     Int. Press, Boston, MA, 1999,  97--311.
\bibitem{varso} C. Blanchet, {\em Refined quantum invariants for 
three-manifolds with structure}, in Knot Theory, Banach Center Pub. Vol. 42
(1998), 11-22.
\bibitem{Bhec} C. Blanchet, {\em Hecke algebras, modular
categories and $3$-manifolds quantum invariants}, Topology, Vol. 39
(2000), 193--223. 
\bibitem{BHMV} C. {Blanchet}, N. {Habegger}, G. {Masbaum} and
P. {Vogel}, {\em Topological Quantum Field Theories derived from the
Kauffman bracket}\/, Topology~34 No~4 (1995), 883--927.
\bibitem{BM} C. Blanchet, G. Masbaum, {\em Topological quantum field theories for
surfaces with spin structure}, 
Duke Mathematical Journal 82 (1996), 229-267.
\bibitem{Bru} A. Brugui\`eres {Cat\'egories pr\'emodulaires, modularisations et 
invariants des vari\'et\'es de dimension $3$}, Math. Ann. 316, No.2, 215-236 (2000).
\bibitem{Fal} G. Faltings, {\em A proof of the Verlinde formula}\/,
J. Alg. Geometry 3, 347--374 (1994).
\bibitem{J} D. Johnson, {\em Spin structures and quadratic forms on surfaces},
J. London Math Soc. (2) 22 (1980) 365--377.
\bibitem{KT} T. Kohno, T. Takata, {Level-Rank Duality of Witten $3$-manifolds invariants}, Adv. Studies in Pure Math. 24, Progress in Algebraic Combinatorics, 243--264.
\bibitem{Le} T. Le {\em Quantum invariants of $3$-manifolds: integrality, splitting, and perturbative expansion}, preprint math.QA/0004099.
\bibitem{LT} T. Le, V. Turaev, {\em Quantum groups and ribbon $G$-categories},
 J. Pure Appl. Algebra  178  (2003),  no. 2, 169--185.
\bibitem{Li} W.B.R. Lickorish, {\em An Introduction to Knot Theory}, Springer-Verlag, Grad. Texts in Math. 175 (1997).
\bibitem{Macdo} I. G. Macdonald, {\em Symmetric functions and Hall polynomial},
2nd ed. ,
Oxford Science Pub. (1995).
\bibitem{MW} G. Masbaum, H. Wenzl, {\em Integral modular categories
and integrality of quantum invariants at roots of unity of prime order},
J. Reine Angew. Math. 505, 209-235 (1998).
\bibitem{Milnor} J. Milnor, {\em Spin structures on manifolds}
L'Enseignement Math. 9, 198--203 (1963).
 \bibitem{Mu} M. M\"uger, Galois theory for braided tensor categories
 and the modular closure 
Adv. Math. 150, No.2, 151-201.
\bibitem{Mupad} MuPAD, The Open Computer Algebra System,
Sciface Software, www.mupad.de.
\bibitem{rec} W. M. Oxbury, S. M. J. Wilson, {\em Reciprocity laws
in the Verlinde formulae for the classical groups}\/, Trans. AMS
348, No 7 (1996).
\bibitem{PS} C. Kassel, M. Rosso, V. Turaev, {\em Quantum groups
and knots invariants}\/, Panoramas et Synth\`eses No 5, SMF (1997).
\bibitem{So} C. {Sorger}, {\em La formule de Verlinde}\/,
 S\'eminaire Bourbaki 794 (1994).
\bibitem{Sa} S. Sawin, {\em Quantum groups at roots of unity and modularity},
(preprint) arXiv:math.QA/0308281.
\bibitem{Sa2}{S. Sawin}
{\em Jones-Witten invariants for nonsimply connected Lie groups and the geometry of the Weyl alcove}\/,
Adv. Math. 165, No.1, 1--34 (2002).
\bibitem{Sa3}{S. Sawin} {\em Invariants of spin three-manifolds from Chern-Simons theory and finite-dimensional Hopf algebras}\/, Adv. Math. 165, No.1, 35--70 (2002). 
\bibitem{Tu} V.  Turaev, {\em Quantum invariants
of knots and $3$-manifolds}\/, De Gruyter Studies in Math.~18 (1994).
\bibitem{Tu3} V.  Turaev, {\em  Homotopy field theory in dimension $2$
and crossed groups-algebras} preprint math.QA/9910010.
\bibitem{Tu4} V.  Turaev, {\em  Homotopy field theory in dimension $3$
and crossed groups-categories} preprint math.GT/0005291.
\bibitem{TW} V. Turaev, H. Wenzl, {\em Quantum invariants of
$3$-manifolds associated with classical simple Lie algebras},
Int. J. of Math. Vol 4, No 2 (1993) 323--358.
\bibitem{TW2} V. Turaev, H. Wenzl, {\em Semisimple and modular categories
from link invariants}, Math. Ann. 309 (1997), 411--461.
\bibitem{Ver}  E. Verlinde, {\em Fusion rules and modular transformations
in $2d$ conformal field theory}
    Nucl. Physics B 300,
No. 3, 360-376 (1988).
\bibitem{Yo} Y.~Yokota, {\em Skeins and quantum $SU(N)$
invariants of $3$-manifolds}\/, Math. Ann. 307 (1997), 109--138.
\end{thebibliography}