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arXiv:hep-th/0202121v1 19 Feb 2002
Hamiltonian Quantization of Chern-Simons theory with
SL(2,C) Group
E. BUFFENOIR, K. NOUI, Ph. ROCHE
Laboratoire de physique math´ ematique et th´ eorique
Universit´ e Montpellier 2, 34000 Montpellier, France.
February 1, 2008
Abstract
We analyze the hamiltonian quantization of Chern-Simons theory associated to
the real group SL(2,C)R, universal covering of the Lorentz group SO(3,1). The
algebra of observables is generated by finite dimensional spin networks drawn on
a punctured topological surface. Our main result is a construction of a unitary
representation of this algebra. For this purpose we use the formalism of combinato-
rial quantization of Chern-Simons theory, i.e we quantize the algebra of polynomial
functions on the space of flat SL(2,C)R−connections on a topological surface Σ
with punctures. This algebra, the so called moduli algebra, is constructed along the
lines of Fock-Rosly, Alekseev-Grosse-Schomerus, Buffenoir-Roche using only finite
dimensional representations of Uq(sl(2,C)R). It is shown that this algebra admits
a unitary representation acting on an Hilbert space which consists in wave packets
of spin-networks associated to principal unitary representations of Uq(sl(2,C)R).
The representation of the moduli algebra is constructed using only Clebsch-Gordan
decomposition of a tensor product of a finite dimensional representation with a
principal unitary representation of Uq(sl(2,C)R). The proof of unitarity of this
representation is non trivial and is a consequence of properties of Uq(sl(2,C)R) in-
tertwiners which are studied in depth. We analyze the relationship between the
insertion of a puncture colored with a principal representation and the presence of
a world-line of a massive spinning particle in de Sitter space.
I. Introduction
In the pioneering work of [1, 28], it has been shown that there is an “equivalence” be-
tween 2+1 dimensional gravity with cosmological constant Λ and Chern-Simons theory
with a non compact group of the type SO(3,1), ISO(2,1) or SO(2,2) (depending on
the sign of the cosmological constant Λ). A good review on this subject is [15]. As
a result the project of quantization of Chern-Simons theory for these groups has spin-
offs on the program of canonical quantization of 2+1 quantum gravity. However one
should be aware that the two theories, nor in the classical case nor in the quantum
case, are not completely equivalent. These discrepencies arize from various reasons.
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One of them, fully understood by Matschull [20], is that the Chern-Simons formulation
includes degenerate metrics, and the classical phase space of Chern-Simons is therefore
quite different from the classical phase space of 2+1 gravity. Another one comes from
the structure of boundary terms (horizon, observer, particles) which have to be carefully
related in the two models.
In this work, we study Chern-Simons formulation of 2+1 gravity in the case where the
cosmological constant is positive i.e Chern-Simons theory on a 3-dimensional compact
oriented manifold M = Σ × R with the real non compact group SL(2,C)R universal
covering of SO(3,1). This theory has been the subject of numerous studies, the main
contributions being the work of E.Witten [29] using geometric quantization and the
work of Nelson-Regge [21] using representation of the algebra of observables. We will
extend the analysis of Nelson-Regge using the so-called “combinatorial quantization of
Chern-Simons theory” developped in [18, 2, 3, 10, 11]. We first give the idea of the con-
struction when the group is compact. Let Σ be an oriented topological compact surface
of genus n and let us denote by G = SU(2) and g its Lie algebra. The classical phase
space of SU(2)-Chern Simons theory is the symplectic manifold Hom(π1(Σ),G)/AdG.
The algebra of functions on this manifold is a Poisson algebra which admits a quanti-
zation Mq(Σ,G), called Moduli algebra in [2], which is an associative algebra with an
involution ∗. Note that q is taken here to be a root of unity q = ei
coupling constant in the Chern-Simons action. This algebra is built in two stages. One
first defines a quantization of the Poisson algebra F(G2n) endowed with the Fock-Rosly
Poisson structure [18]. This algebra is denoted Lnand called the graph algebra [2]. It is
the algebra generated by the matrix elements of the 2n quantum holonomies around the
non trivial cycles ai,bi. Uq(g) acts on Lnby gauge transformations. The space of invari-
ant elements LUq(g)
by spin networks drawn on Σ. If we define UC to be the quantum holonomy around
the cycle C =?n
In [5, 4] Alekseev and Schomerus have constructed its unique unitary irreducible rep-
resentation acting on a finite dimensional Hilbert space H. This is done in two steps.
They have shown that there exists a unique unitary representation ρ (∗-representation)
of the loop algebra Lnacting on a finite dimensional space H. The algebra generated by
the matrix elements of UCis isomorphic to Uq(g), therefore Uq(g) acts on H. ρ can be
restricted to the subalgebra LUq(g)
The ideal IC is shown to be annihilated, as a result one obtains by this procedure a
unitary representation of the moduli algebra. This representation can be shown to be
unique up to equivalence. Note that this construction is however implicit in the sense
that no explicit formulae for the action of an element of Mq(Σ,G) is given in a basis
of HUq(g). In this brief exposition we have oversimplified the picture: q being a root of
unity the formalism of weak quasi-Hopf algebras has to be used.
We will modify this construction in order to handle the SL(2,C)Rcase. The con-
struction of the moduli algebra in this case is straightforward and is parallel to the
construction in the compact case. One defines the graph algebra Ln, generated by the
π
k+2where k ∈ N is the
n
is a subalgebra of Lnwhose vector space basis is entirely described
i=1[ai,b−1
i], one defines an ideal ICof LUq(g)
n
which, when modded out,
enforces the relation UC= 1. As a result the moduli algebra is Mq(Σ,G) = LUq(g)
n
/IC.
n
and acts on the subspace of invariants HUq(g)= H.
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matrix elements of the 2n quantum SL(2,C)Rholonomies around the non trivial cycles
ai,bi. This is a non commutative algebra on which Uq(sl(2,C)R) acts. We have chosen
q real, in complete agreement with the choice of the real invariant bilinear form on
SL(2,C)Rused to represent the 2+1 gravity action with positive cosmological constant
as a Chern-Simons action. One defines similarly Mq(Σ,SL(2,C)R) = LUq(sl(2,C)R)
which is a non commutative ∗-algebra, quantization of the space of functions on the mod-
uli space of flat-SL(2,C)Rconnections. Although one can generalize the first step of the
construction of [4], i.e constructing unitary representations of Lnacting on an Hilbert
space H, it is not possible to construct a unitary representation of Mq(Σ,SL(2,C)R)
by acting on HUq(sl(2,C)R). Indeed, there is no vector (of finite norm), except 0, in the
Hilbert space H which is invariant under the action of Uq(sl(2,C)R). This is a typical
example of the fact that the volume of the gauge group is infinite (here it comes from
the non compactness of SL(2,C)R). To circumvent this problem we use and adapt the
formalism of [4] to directly construct a representation of Mq(Σ,SL(2,C)R) by acting on
a vector space H. In a nutshell, Mq(Σ,SL(2,C)R) is generated by spin network colored
by finite dimensional representations, whereas vectors in H are integral of spin networks
colored by principal representations of Uq(sl(2,C)R). We give explicit formulae for the
action of Mq(Σ,SL(2,C)R) on H, we endow this space with a structure of Hilbert space
and show that the representation is unitary. Our approach uses as central tools the
harmonic analysis of Uq(sl(2,C)R) and an explicit construction of Clebsch-Gordan co-
efficients of principal representations of Uq(sl(2,C)R), which have been developped in
[12, 14].
Note that Nelson and Regge have previously succeeded to construct unitary repre-
sentation of the Moduli algebra in the case of genus one in [21] and in the genus 2 case
in the SL(2,R) case in [22]. Our method works for any punctured surface of arbitrary
genus and, despite certain technical points which have been mastered, is very natu-
ral. It is a non trivial implementation of the concept of refined algebraic quantization
developped in [7].
n
/IC
II. Summary of the Combinatorial Quantization Formalism:
the compact group case.
Chern-Simons theory with gauge group G = SU(2) is defined on a 3-dimensional com-
pact oriented manifold M by the action
?
where the gauge field A = Aµdxµand Tr is the Killing form on g = su(2). In the sequel
we will investigate the case where Chern-Simons theory has an hamiltonian formulation.
We will therefore assume that the manifold M = Σ × R, where the real line can be
thought as being the time direction and Σ is a compact oriented surface and we will
write A = A0dt+A1dx1+A2dx2. In the action (1) A0appears as a Lagrange multiplier.
Preserving the gauge choice A0= 0 enforces the first class constraint
S(A) =
λ
4π
M
Tr(A ∧ dA +2
3A ∧ A ∧ A) ,(1)
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F12(A) = ∂1A2− ∂2A1+ [A1,A2] = 0. (2)
The space A(Σ,G) of G-connections on Σ is an infinite dimensional affine symplectic
space with Poisson bracket:
{Ai(x)⊗, Aj(y)} =2π
λδ(x − y)ǫijt ,(3)
where t ∈ g ⊗ g is the Casimir tensor associated to the non-degenerate bilinear form
Tr defined by t =?
The constraint (2) generates gauge transformations
a,b(η−1)abTa⊗ Tbwhere Tais any basis of g,ηab= Tr(TaTb) and
i,j ∈ {1,2}.
gA = g A g−1+ dg g−1,∀ g ∈ C∞(Σ,G) .(4)
As a result, the classical phase space of this theory consists of the moduli space of flat
G-bundles on the surface Σ modulo the gauge transformations and has been studied in
[8].
In order that exp(iS(A)) is gauge invariant under large gauge transformations, λ
has to be an integer.
The moduli space of flat connections M(Σ,G) is defined using an infinite dimensional
version of Hamiltonian reduction, i.e M(Σ,G) = {A ∈ A(Σ,G),F(A) = 0}/G where the
group G is the group of gauge transformations. The quantization of this space can
follow two paths: quantize before applying the constraints or quantize after applying
the constraints. The approach of Nelson Regge aims at developping the latter but it
is cumbersome. We can take advantage of the fact that M(Σ,G) is finite dimensional
to replace the gauge theory on Σ by a lattice gauge theory on Σ following Fock-Rosly’s
idea [18]. This method aims at quantizing before applying the constraints but in a finite
dimensional framework.
This framework can be generalized to the case of a topological surface Σ with punc-
tures P1,...,Pp. If A is a flat connection on a punctured surface one denotes by Hx(A)
the conjugacy class of the holonomy around a small circle centered in x. One chooses
σ1,...,σpconjugacy classes in G and defines M(Σ,G;σ1,...,σp) = {A ∈ A(Σ,G),F(A) =
0, HPi(A) = σi}/G where the group G is the group of gauge transformations. The sym-
plectic structure on this space is well analyzed in [6].
II.1. Fock-Rosly description of the moduli space of flat connections.
Functions on M(Σ,G), also called observables, are gauge invariant functions on {A ∈
A(Σ,G),F(A) = 0}. Wilson loops are examples of observables, and are particular ex-
amples of the following construction which associates to any spin-network an observable.
Let us consider an oriented graph on Σ, this graph consists in a set of oriented edges
(generically denoted by l) which meet at vertices (generically denoted by x). Let < be
a choice of an order on the set of oriented edges.
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It is convenient to introduce the notations d(l) and e(l) respectively for the departure
point and the end point of an oriented edge l.
A spin network associated to an oriented graph on Σ consists in two data:
• a coloring of the set of oriented edges i.e each oriented edge l is associated to a
finite dimensional module Vlof the algebra g. We denote by πlthe representation
associated to Vl. For any l and x we define V+
elsewhere, as well as V−
(l,x)= Vlif e(l) = x and V+
(l,x)= C elsewhere.
(l,x)= C
(l,x)= V∗
lif d(l) = x and V−
• a coloring of the vertices i.e each vertex is associated to an intertwiner φx ∈
Homg(⊗<
To each spin network N we can associate a function on M(Σ,G), as follows: let
Ul(A) = πl(
l(V+
(l,x)⊗ V−
(l,x)),C).
←
P exp?
lA) , and define
fN(A) = (⊗xφx)(⊗<
lwith End(Vl).
lUl(A))(5)
where we have identified Vl⊗ V⋆
The Poisson structure on M(Σ,G), can be neatly described in terms of the functions
fNas first understood by Goldman [19]. Given two spin-networks N,N′such that their
associated graphs are in generic position, we have
?
where the graph N ∪xN′is defined to be the union of N and N′with the additional
vertex x associated to the intertwiner P12t12: V ⊗ V′→ V′⊗ V (which can be viewed
as an element of Hom(V ⊗ V′⊗ V⋆⊗ V′⋆;C)) and where the sign ǫx(N;N′) = ±1
is the index of the intersection of the two graphs at the vertex x. Quantizing directly
this Poisson structure is too complicated (see however [21, 26]). We will explain now
Fock-Rosly’s construction and the definition of the combinatorial quantization of the
moduli space Mq(Σ,G).
Finite dimensional representations of G are classified by a positive half integer I ∈
1
2N, and we will denote
V the associated module with representation
Let Σ be a surface of genus n, with p punctures associated to a conjugacy class
σi,i = 1,...,p of G. Fock-Rosly’s idea amounts to replace the surface by an oriented fat
graph T drawn on it and the space of connections on Σ by the space of holonomies on
this fat graph. We assume that the surface is divided by the graph into plaquettes such
that either this plaquette is contractible or contains a unique puncture. Let us denote
T0the set of vertices of the graph, T1the set of edges and T2the set of faces.
The orientation of the surface induces at each vertex x a cyclic order on the set of
edges Lxincident to x.
We can now introduce the space of discrete connections, which is an equivalent name
for lattice gauge field on T . The space of discrete connections A(T ) on the surface Σ is
defined as
{fN, fN′} =2π
λ
x∈N∩N′
fN∪xN′ ǫx(N;N′) ,(6)
I
Iπ .
A(T ) = {U(l) ∈ G ; l ∈ T1}
(7)
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and the group of gauge transformations GT0acts on the discrete connections as follows:
U(l)g= g(e(l)) U(l) g(d(l))−1
,∀g ∈ GT0
. (8)
The discrete connections can be viewed as functionals of the connection A ∈ A(Σ,G)
as U(l) =
f ∈ T2, we denote σf= 1 if f is contractible and σf= σiif f contains the puncture Pi.
The group GT0has a natural Lie-Poisson structure:
←
P exp?
lA. If f ∈ T2, let U(f) be the conjugacy class of
← ?
l∈∂fU(l). For each
{
Ig1(x) ,
Jg2(x)}
Jg2(y)}
=
2π
λ[
IJr12,
Ig1(x)
Jg2(x)] ,(9)
{
Ig1(x) ,=0 if x ?= y ,(10)
where we have used the notations
Ig1(x) =
Ig(x) ⊗
J
1 ,
Jg2(x) =
I
1 ⊗
Jg(x) ,(11)
and
x), r ∈ g⊗2is a classical r-matrix which satisfies the classical Yang-Baxter equation and
r12+ r21= t12.
Fock and Rosly [18] have introduced a Poisson structure on the functions on A(T )
denoted {,}FRsuch that the gauge transformation map
GT0× A(T ) → A(T )
Ig(x) ∈ End(
I
V) ⊗ F(G)x(F(G)xbeing the functions on the group at the vertex
(12)
is a Poisson map.
Note however that this Poisson structure is not canonical and depends on an addi-
tional item (called in their paper a ciliation ), which is a linear order <xcompatible
with the cyclic order defined on the set of edges incident to the vertex x.
We shall give here the Poisson structure on the space of discrete connections in the
case where T is a triangulation:
{
I
U1(l) ,
J
U2(l′)}FR
J
U2(l)}FR
J
U2(l′)}FR
=
2π
λ(
2π
λ(
IJr12
I
U1(l)
J
U2(l′)) , if e(l) = e(l′) = x and l <xl′, (13)
{
I
U1(l) ,=
IJr12
I
U1(l)
J
U2(l) +
I
U1(l)
J
U2(l)
IJr21) ,(14)
{
I
U1(l) ,=0 if l ∩ l′= ∅ ,(15)
the other relations can be deduced from the previous ones using the relation U(−l)U(l) =
1.
The moduli space can be described as: M(Σ,G,σ1,...,σp) = {A(T ),U(f) = σf,f ∈
T2}/GT0. The major result of [18] is that the Poisson structure {,}FR descends to
this quotient, is not degenerate, independent of the choice of the fat graph and on the
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ciliation and is the Poisson structure associated to the canonical symplectic structure
on M(Σ,G;σ1,...,σp).
A quantization of Fock-Rosly Poisson bracket has been analyzed in [2, 3, 10]. In or-
der to give a sketch of this construction we will first recall standard results on quantum
groups.
II.2. Basic notions on quantum groups.
Basic definitions and properties of the quantum envelopping algebra Uq(g) where g =
su(2) are recalled in the appendix A.1. Uq(g) is a quasi-triangular ribbon Hopf-algebra
with counit ǫ :Uq(g) −→ C, coproduct ∆ :
Uq(g) −→ Uq(g). For a review on quantum groups, see [16]. The universal R-matrix
R is an element of Uq(g)⊗2denoted by R =?
the center of Uq(g) and there exists a central element v (the ribbon element) such that
v2= uS(u). We will define the group-like element µ = q2Jz.
Uq(g) −→ Uq(g)⊗2and antipode S :
ixi ⊗ yi= R(+). It is also convenient
iS(yi) xi, uS(u) is into introduce R′=?
iyi ⊗ xi and R(−)= R′−1. Let u =?
Finite dimensional irreducible representations
I
V the associated module. The tensor product
is decomposed into irreducible representations
Iπ of Uq(g) are labelled by I ∈1
Iπ ⊗
2N and
let us define
Jπ of two representations
Kπ
Iπ ⊗
Jπ=
?
K
NIJ
K
Kπ ,
(16)
where the integers NIJ
K∈ {0,1} are the multiplicities. For any representations
we define the Clebsch-Gordan maps ΨK
K
V,V ⊗
Iπ,
Jπ,
J
V,
Kπ,
K
V)
IJ(resp. ΦIJ
K) as a basis of HomUq(g)(
I
V ⊗
(resp. HomUq(g)(
I
J
V)). These basis can always be chosen such that:
NIJ
KΨL
IJΦIJ
K= NIJ
K
δL
KidK
V
,
?
K
ΦIJ
KΨK
IJ= idI
V⊗
J
V.(17)
For any finite dimensional representation I, we will define the quantum trace of an
I
V) as trq(M) = trI
V(
is a central element of Uq(g). For any finite dimensional representation I and for any
irreducible module V associated to the representation π (not necessarily of finite di-
mension), we will denote by ϑIπthe complex number defined by π(cI) = ϑIπidV. For
g = su(2), ϑIJ= ϑIJπ=[(2I+1)(2J+1)]
[2I+1][2J+1]
of Uq(su(2)) and quantum numbers [x] is defined in the appendix.
Iei| i = 1···dim
1···dim
group inherits a structure of Hopf-algebra. It is generated as a vector space by the
element M ∈ End(
Iµ M). The element cI = trq((
Iπ ⊗id)(RR′))
where I,J ∈1
2N label irreducible representations
Let us denote by {
I
V} its dual basis. By duality, the space Polq(G) of polynomials on the quantum
I
V} a particular basis of
I
V and {
Iei
| i =
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coefficients of the representations
Iπ, which will be denoted by
Ig =?
Iga
b=?Iea|
V) ⊗ Polq(G) where
Iπ (.) |
Ieb
?.
To simplify the notations, we define
I
Eb
definitions, we have the fusion relations
a,b
I
Eb
a⊗
V). By a direct application of the
Iga
b∈ End(
I
the elements {
a}a,bis the canonical basis of End(
I
Ig1
Jg2=
?
K
ΦIJ
K
Kg ΨK
IJ, (18)
which imply the exchange relations
I
V) ⊗ End(
The coproduct is
IJ
R12
Ig1
Jg2=
Jg2
Ig1
IJ
R12 , where
IJ
R12= (
Iπ ⊗
Jπ)(R) ∈
End(
J
V).
∆(
Iga
b) =
?
c
Iga
c⊗
Igc
b. (19)
Up to this point, it is possible to give a presentation, of FRT type, of the defining
relations of Uq(g). Let us introduce, for each representation I, the element
I
V) ⊗ Uq(g) defined by
?I
These matrices satisfy the relations:
I
L(±)
∈
End(
I
L(±)= (
Iπ ⊗id)(R(±)). The duality bracket is given by
?
L1(±),
Jg2
=
IJ
R(±)
12
. (20)
I
L1(±)
J
L2(±)
=
?
J
L2(−)
?
K
ΦIJ
K
K
L(±)ΨK
IJ,(21)
IJ
R12(±)
I
L1(+)
J
L2(−)
I
L(±)a
=
I
L1(+)
IJ
R12(±),
I
L(±)a
c .
(22)
∆(
b)=
c
I
L(±)c
b⊗
(23)
The first fusion equation implies the exchange relations
IJ
R12
I
L1(±)
J
L2(±)=
J
L2(±)
I
L1(±)
IJ
R12 .(24)
II.3. Combinatorial Quantization of the moduli space of flat connec-
tions.
We are now ready to define a quantization of the space of flat connections along the
lines of [2]. Because this construction can be shown to be independent of the choice of
ciliated fat graph, we will choose a specific graph, called standard graph, which is shown
in figure 1.
This graph consists in one vertex x , p + 1 2-cells and 2n + p 1-cells. The 2n + p
1-cells are given with the orientation and the order < of the picture.
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.
.
..
.
..
A(1)
B(1)
A(2)
B(2)
A(n)
B(n)
M(n + 1)
M(n + p)
x
Figure 1: Standard Graph.
The space of discrete connections on this graph consists in the holonomies {M(j) |
j = n + 1,··· ,n + p} around the punctures and the holonomies {A(i), B(i) | i =
1,··· ,n} around the handles. We can choose the associated curves in such a way that
they have the same base point x on the surface.
We associate to this graph T a quantization of the Fock-Rosly Poisson structure on
the space of discrete connections on T as follows:
Definition 1 (Alekseev-Grosse-Schomerus)[2] The graph algebra Ln,pis an associative
algebra generated by the matrix elements of (
A(i))i=1,···,n, (
I
V) ⊗ Ln,pand satisfying the relations:
?
IJ
RU1(i)
R−1
U2(j)=
U2(j)
IJ
R′J
B2(i)=
B2(i)
I
I
B(i))i=1,···,n, (
I
M(i))i=n+1,···,n+p
∈ End(
I
U1(i)
IJ
R′J
U2(i)
IJ
R(−)
=
K
J
ΦIJ
K
K
U(i) ΨK
IJ(Loop Equation) ∀ i,(25)
IIJJ IJ
R
I
U1(i)
IJ
R−1
∀ i < j,
∀ i,
(26)
IJ
R
I
A1(i)
J IJ
R
I
A1(i)
IJ
R−1
(27)
where U(i) is indifferently A(i), B(i) or M(i). The relations are chosen in such a way
that the co-action δ:
δ:
Ln,p
I
U(i)a
→ Fq(G) ⊗ Ln,p
?→
c,d
b
?
Iga
cS(
Igd
b) ⊗
I
U(i)c
d=
? Ig
I
U(i) S(
Ig)?a
b
(28)
is a morphism of algebra.
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This last property is the quantum version of the fact that the map (28) is a Poisson
map. Equivalently the coaction δ provides a right action of Uq(g) on Ln,pas follows:
∀a,b ∈ Ln,p,∀ξ ∈ Uq(g),(ab)ξ= aξ(1)bξ(2),
I
U (i)ξ=
U (i)
where U(i) is indifferently A(i),B(i) or M(i).
(29)
Iπ (ξ(1))
I
Iπ (S(ξ(2))), (30)
Let us notice that, from (25),
I
U(i) admits an inverse matrix
I
U(i)−1, see [2, 10].
The space of gauge invariant elements is the subspace of coinvariant elements of Ln,p
i.e Linv
n,p= {a ∈ Ln,p,δ(a) = 1⊗a}. This is an algebra because δ is a morphism of algebra.
In Linv
n,pwe still have to divide out by the flatness condition, i.e the quantum version of
the fact that the holonomies on contractible curves are trivial and that around punctures
they belong to a fixed conjugacy class. Let us define
I
C
=
I
G(1)···
I
A(i)−1
I
G(n)·
I
B(i). The elements
I
M(n + 1)···
I
M(n + p)(31)
where
(25). We will denote by C the subalgebra of Ln,pgenerated by the matrix elements of
I
C,∀ I. It can be shown that trq(
elements of the algebra Linv
We would like first to divide out by the relation
I
G(i) = v2
I
I
A(i)
I
B(i)−1
I
C satisfy the loop equation
I
C) and trq(
I
M (i)),i = n + 1,...,n + p are central
n,p.
I
C= 1.
I
C −1 do not belong to Linv
An annoying fact is that the matrix elements of
result in order to divide out by this relation we have to slightly modify the picture.
n,p. As a
Let I be a finite dimensional representation of Uq(g) and let JI⊂ Ln,p⊗ End(
such that X ∈ JI if and only if X =?
For any Y ∈ JIwe define the invariant element < Y (
Let IC be the ideal of Linv
finite dimensional representation of Uq(g) and Y is any element of JI.
Definition 2 We define Mq(Σ,G,p) to be the algebra Linv
puncture this is the Moduli algebra of [2], and we will write in this case Mq(Σ,G) =
Mq(Σ,G,p = 0).
I
V)
b).
a,bXa
b⊗ Eb
awith δ(Xa
I
C −1) >=?
b) =?
I
C −1) > where I is any
a′,b′ga
IµbbYb
a′Xa′
I
Ca
b′S(gb′
a,b
a(
b− δa
b).
n,pgenerated by the elements < Y (
n,p/IC. When there is no
In the case of punctures a quantization of the coadjoint orbits is necessary. This is
also completely consistent with quantization of Chern-Simons theory, where punctures
are associated to vertical lines colored by representations of the group G.
Definition 3 Let π1,...,πpbe the representations associated to the vertical lines coloring
the punctures. We can define, following [2], the moduli algebra Mq(Σ,G,π1,··· ,πp) =
Mq(Σ,G,p)/{trq(
I
M (n + i)) = ϑIπi,i = 1,...,p,∀I}.
10
Page 11
In order to introduce a generating family of gauge invariant elements we have to
define the notion of quantum spin-network. The definition of this object is the same as
in the classical case except that the coloring of the edges are representations of Uq(g) and
that the coloring of the vertices are Uq(g)-intertwiners. To each quantum spin-network
one associates an element of Mq(Σ,G) by the same equation as (5), the order < is now
essential because it orders non commutative holonomies in the tensor product.
In the following proposition, we will construct an explicit basis of the vector space
Linv
relations defining the moduli algebra, a generating family of this algebra.
We will need the following notations: if L = (L1,...,Lr) and L′= (L′
sequences, we denote LL′to be the sequence (L1,...,Lr,L′
we denote L<j = (L1,...,Lj−1). If L = (L1,...,Lr) is a finite sequence of irreducible
n,plabelled by quantum spin networks. This will provide, after moding out by the
1,...,L′s) are
1,...,L′s). If L is a sequence
representations of Uq(g) we denote V (L) = ⊗r
j=1
Lj
V and we will denote
NL
R(±)=
NL1
R(±)···
NLr
R(±).
For W an irreducible representation of Uq(g) and S = (S3,··· ,Sr) a (r − 2)-uplet
of irreducible representations of Uq(g), we define the intertwiners:
ΨW
L(S) ∈ HomUq(g)(V (L),
W
V) by
ΨW
L(S) = Ψ
W
SrLr···ΨS4
S3L3ΨS3
L1L2
(32)
and ΦL
W(S) ∈ HomUq(g)(
W
V,V (L)) defined by
ΦL
W(S) = ΦL1L2
S3ΦS3L3
S4···ΦSrLr
W. (33)
Definition 4 We will define a “palette” as being a family P = (I,J,N;K,L,U,T,W)
where I,J,K,L (resp. N) (resp. U,T) are n-uplets (resp.p-uplets) (resp. n + p − 2-
uplets ) of irreducible finite dimensional representations of Uq(g) and W is an irreducible
finite dimensional representations of Uq(g). Any palette P defines a unique quantum
spin-network NP associated to the standard graph, precisely: (I,J,N) is coloring of
the non contractible cycles A(i),B(i),M(n + i) and (K,L,U,T,W) is associated to the
intertwiner ΨW
W(LT) coloring the vertex of this spin-network.
IJN(KU) ⊗ ΦIJN
We first define, for i = 1,..,n , θ(i) ∈ Ln,p⊗ Hom(
Li
V,
Ki
V ) by:
θ(i) = ΨKi
JiIi
Ji
B(i)
JiIi
R′
Ii
A(i)
JiIi
R(−)ΦJiIi
Li.
We can now associate to I,J,K,L the element
I,J
θ
(±)
n (K,L) ∈ Ln,p⊗Hom(V (L),V (K))
by
11
Page 12
I,J
θ(±)
n (K,L) =
n
?
j=1
?KjL<j
R(±)θ(j)?. (34)
We associate to N the element of Ln,p⊗ Hom(V (N),V (N))
N
θ(±)
p
=
p
?
j=1
?NjN<j
R(±)Nj
M (n + j)?. (35)
We can introduce the elements of Ln,p⊗ Hom(V (LN),V (KN))
I,J,N
Ω
(±)
n,p(K,L) =
I,J
θ(±)
n (K,L)?
p
?
j=1
NjL
R(±)? N
θ(±)
p (
n+p
?
j=1
(LN)j(LN)<j
R(±)
)−1.(36)
Proposition 1 Let P be a palette labelling a quantum spin network NP associated to
the standard graph. We will define an element of Ln,p
v1/2
K
v1/2
IJ
P
O(±)
n,p=
v1/2
trq(ΨW
KN(U)
I,J,N
Ω
(±)
n,p(K,L) ΦLN
W(T))(37)
where we have defined v1/2
I
= v1/2
I1···v1/2
In.
The elements
P
O(±)
n,p are gauge invariant elements and if ǫ ∈ {+,−} is fixed the nonzero
elements of the family
O(ǫ)
P
n,p is a basis of LUq(g)
n,p
when P runs over all the palettes.
Proof:It is a simple consequence of
δ(
I,J,N
Ω
(±)
n,p(K,L)) = g(KN)
I,J,N
Ω
(±)
n,p(K,L)S(g(LN)))(38)
where g(L) =
L1g1···
Lp
gpand that trq() is invariant under the adjoint action. 2
Remarks.
1. The family
the coefficients of these linear transformations can be exactly computed in terms of 6j
coefficients.
2. The particular normalization of these families has been chosen in order to simplify
the action of the star on these elements (see next section).
P
O(+)
n,p can be linearly expressed in term of the family
P
O(−)
n,p, and
We will denote also by the same notation the image of
space Mq(Σ,G,π1,··· ,πp).
Example.
P
O(±)
n,p in the quantum moduli
12
Page 13
In the case where the surface is a torus with no puncture (n = 1 and p = 0), the
spin-networks are labelled by the colors IJ of the two non-contractible cycles and the
choice of the intertwiner is fixed by a finite dimensional representation W. As a result
the vector space of gauge invariant functions Linv
observables:
1,0is linearly generated by the following
IJW
O 1,0=
v1/2
W
v1/2
I
v1/2
J
trq(ΨW
JI
J
B
JI
R′I
A
JI
R(−)ΦJI
W) , (39)
for all finite dimensional representations IJW. The Moduli algebra is generated as an
algebra by the Wilson loops around the handles in the fundamental representation, i.e.
WA = trq(
I
A) ,WB = trq(
I
B) where I =1
2. (40)
II.4 Alekseev’s Isomorphisms Theorem
The construction of the representation theory of Ln,puses Alekseev’s method [2, 3, 5]: we
first build representations of L0,p(the multi-loop algebra), then we build representations
of Ln,0(the multi-handle algebra) and we use these results to build representations of
the graph algebra Ln,p.
Lemma 1 The algebra L0,pis isomorphic to the algebra Uq(g)⊗p.
The algebra L0,pis generated by the matrix elements of
algebra Uq(g)⊗pis generated by the matrix elements of
label j denotes one of the p copies of Uq(g). An explicit isomorphism in term of these
generators can be constructed as follows:
I
M(i), i = 1,...,p, and the
I
L(j)(±), j = 1,...,p, where the
L0,p ⊗ End(
I
V)
−→
?→
Uq(g)⊗p⊗ End(
I
F(i)
M(i)
I
V)
I
M(i)
I
I
F(i)−1, (41)
where we have defined
I
M(i) =
I
L(i)(+)I
L(i)(−)−1
,
I
F(i) =
I
L(1)(−)···
I
L(i − 1)(−). (42)
Proof:See [5] 2
As an immediate consequence, the representations of the loop algebra L0,pare those
of Uq(g)⊗p. A basis of the irreducible finite dimensional module labelled by J is denoted
Jei| i = 1···dim(
by:
as usual by (
J
V)). The action of the generators on this basis is given
I
L(±)a
b?
Jei=
Jej
IJ
R(±)aj
bi
. (43)
From this relation and the explicit isomorphism of the lemma 1, it is easy to find
out explicit expressions for the representations of L0,pon the module
I1
V ⊗···⊗
Ip
V. In
13
Page 14
particular, the action of
of R-matrices.
I
M(i) on the basis
I1ei1⊗···⊗
Ipeipis given in term of product
The previous theorem can be modified in order to apply to the algebra Ln,0. However,
this algebra can not be represented as a direct product of several copies of Uq(g). An
easy way to understand this point is to consider, for example, the center of each algebras.
The loop algebra L0,1admits a subalgebra generated by the
are central elements. One can show that the center of the handle algebra L1,0is trivial
[2, 3]. To understand the representations of Ln,0, we therefore have to introduce one
more object: the Heisenberg double.
I
W(i) = trq(
I
M(i)) which
Definition 5 Let A be a Hopf algebra (typically Uq(g)) and A⋆its dual. The Heisenberg
double is an algebra defined as a vector space by
H(A) = A ⊗ A⋆;
the algebra law is defined by the following algebra morphisms
A֒→
?→
H(A)
x ⊗ 1
;A⋆֒→ H(A)
f ?→ 1 ⊗ fx
and the exchange relations
xf=(x ⊗ 1)(1 ⊗ f) = x ⊗ f =
?
(x),(f)
?x(1),f(2)
?(1 ⊗ f(1))(x(2)⊗ 1) ,(44)
where we have used Sweedler notation ∆(x) =?
In the case where A = Uq(g), the Heisenberg double may be seen as a quantization
of Fun(T⋆G). So, we can interpret the elements of A⋆as functions and those of A as
derivations.
(x)x(1)⊗ x(2).
Proposition 2 H(A) admits a unique irreducible representation Π realized in the mod-
ule A⋆as follows:
Π : H(A)
A⋆∋ f
−→
?→
?→
End(A⋆),
mf
∇x / ∇x(g) = g(1)
/ mf(g) = fg ,∀g ∈ A∗
A ∋ x
?x,g(2)
?
,∀g ∈ A∗.
In the case where A = Uq(g), H(A) is generated as a vector space by
and the exchange relations (44) take the simple form:
I
L(+)a
b
J
L(−)c
d⊗
Kge
f,
I
L(±)
1
Jg2=
Jg2
I
L(±)
1
IJ
R(±)
12
. (45)
14
Page 15
In order to understand the relation between the multi-handle algebra and the Heisen-
berg double H(Uq(g)), it is convenient to introduce left derivations
H(Uq(g)):
I
˜L ∈ End(
I
V) ⊗
I
˜L = v2
I
Ig
I
L(+)−1I
L(−)Ig−1=
I
˜L(+)
I
˜L(−)−1, (46)
where the last formula corresponds to the Gauss decomposition. As usual, left and right
derivations commute with each other
I
˜L(ǫ)
1
J
L(σ)
2
=
J
L(σ)
2
I
˜L(ǫ)
1
,∀ (ǫ , σ) ∈ {+ , −} , (47)
and realize two independent embeddings of Uq(g) in H(Uq(g)). From the relations of
the Heisenberg double, it is easy to show the following relations:
I
˜L(±)
1
J
˜L(±)
2
=
?
J
˜L(−)
2
K
ΦIJ
K
K
˜L(±)ΨK
IJ,(48)
IJ
R(±)
12
I
˜L(+)
1
I
˜L(±)
J
˜L(−)
2
=
I
˜L(+)
1
IJ
R(±)
12, (49)
IJ
R(±)
12
1
Jg2
=
Jg2
I
˜L(±)
1
.(50)
The action of the elements
I
L(±)
1
,
I
˜L(±)
1
,
Jgthrough representation Π are expressed as:
I
L(±)
1
Ig1?
?
Jg2=
Jg2
IJ
R(±)
12
?
,
I
˜L(±)
1
Kg ΨK
?
Jg2=
IJ
R(±)
12
−1
Jg2 , (51)
Jg2=
Ig1
Jg2=
K
ΦIJ
K
IJ. (52)
The following lemma, due to Alekseev [5], describes the structure of Ln,0:
Lemma 2 The algebra Ln,0is isomorphic to the algebra H(Uq(g))⊗n.
Ln,0 ⊗ End(
I
V)
−→
?→
?→
H(Uq(g))⊗n⊗ End(
I
H(i)
A(i)
I
H(i)
B(i)
I
V)
I
A(i)
I
I
H(i)−1
I
B(i)
I
I
H(i)−1,
where we have defined
I
A(i)=
I
L(i)(+)Ig(i)
I
L(i)(−)−1
,
I
B(i) =
I
L(i)(+)I
L(i)(−)−1
,(53)
I
H(i)=(
I
L(1)(−)
I
˜L(1)(−))···(
I
L(i − 1)(−)
I
˜L(i − 1)(−)) .(54)
15
Page 16
Remark: this lemma can be used to build representations of the the multi-handle
I
A and
B act on the vector space Fq(G) as follows:algebra Ln,0. The two monodromies
I
I
A1?
Jg2=
?
i,K
Ixi1ΦIJ
K
Kg
Kyi ΨK
IJ
IJ
R′12
,
I
B1?
Jg2=
Jg2
IJ
R12
IJ
R′12 ,(55)
where R =?
ixi⊗ yi. In the case of the multi-handle algebra, the action of the mon-
odromies is given in term of product of R-matrices with Clebsh-Gordan maps.
The following lemma shows that the graph algebra Ln,pis isomorphic to Ln,0⊗L0,p.
As a result, from the previous theorems, representations of the graph algebra Ln,pis
constructed from the representations of the multi-loop algebra and the multi-handle
algebra.
Lemma 3 The algebra Ln,p is isomorphic to the algebra H(Uq(g))⊗n⊗ Uq(g)⊗p, the
isomorphism is given by
Ln,p ⊗ End(
I
V)
−→
?→
?→
?→
H(Uq(g))⊗n⊗ Uq(g)⊗p⊗ End(
I
K(i)
A(i)
K(i)−1
I
K(i)
B(i)
K(i)−1,
I
K(n + i)
M(n + i)
I
V)
I
A(i)
I
B(i)
II
II
I
M(n + i)
II
K(n + i)−1,
where we have defined
I
A(i)=
I
L(i)(+)Ig(i)
I
L(n + i)(+)I
I
H(i) ,
K(n + i) =
I
L(i)(−)−1,
I
B(i) =
I
L(i)(+)
I
L(i)(−)−1, (56)
I
M(n + i)=
L(n + i)(−)−1,
I
H(n + 1)
(57)
I
K(i)=
I
I
F(i) ,(58)
where
I
H(i) and
I
F(i) have already been introduced.
Let
I
C as defined by (31), it can easily be shown that:
I
C=
I
C(+)
I
C(−)−1with
n
?
(59)
I
C(±)=
j=1
I
L(±)(j)
I
˜L(±)(j)
p
?
k=1
I
L(±)(n + k).(60)
From the relations (25) satisfied by
L0,1and hence to Uq(g). Let us denote by i : Uq(g) → C the isomorphism of algebra
defined by
I
L(±)−1) =
I
C, one obtains that the algebra C is isomorphic to
i(
I
C(±).
16
Page 17
An important property is that the adjoint action of C on the graph algebra is equivalent
to the action of Uq(g), namely we have:
i(ξ(1)) a i(S(ξ(2))) = aξ,
∀a ∈ Ln,p,∀ξ ∈ Uq(g). (61)
This last property follows easily from the relation
I
C
(±)
1
J
U2(i)
I
C
(±)−1
1
=
IJ
R(±)
12
J
U2(i)
IJ
R(±)
12
−1
where U(i) is indifferently A(i),B(i),M(i). Note that the classical property that the
constraint (2) generates gauge transformation is turned into (61) after quantization.
Finally, the representation theory of the graph-algebra Ln,pis obtained from those
of the quantum group Uq(g) and the Heisenberg double H(Uq(g)). If I = (I1,...,Ip) are
irreducible Uq(g)-modules, Hn,p[I] = Fq(G)⊗n⊗
Ln,pdefing the representation denoted ρn,p[I].
Hn,p[I] is also a Uq(g)-module associated to the representation ρn,p[I]◦i. As a result
the subset of invariant elements is the vector space Hn,p[I]Uq(g)= {v ∈ Hn,p[I],
(
C(±)− 1) ? v = 0}.
Proposition 3 (Alekseev) The representation ρn,p[I] of Ln,prestricted to Linv
Hn,p[I]Uq(g)invariant.
Hn,p[I]Uq(g)= Hn,p[I]. This representation annihilates the ideal IC, therefore one ob-
tains a representation of Mq(Σ,G,p). Moreover ρn,p[I] annihilates the ideals generated
J
M (n + i)) = ϑJπi,i = 1,...,p, where πi=
scends to the quotient, defines a representation denoted ˜ ρn,p[I], of Mq(Σ,G;π1,...,πp).
I1
V ⊗···⊗
Ip
V are irreducible modules of
I
n,pleaves
n,pacting onAs a result one obtains a representation of Linv
by the relations trq(
Iiπ. As a result ρn,p[I] de-
This proposition is a direct consequence of the above constructions and the fact that
J
M (n + i)) is represented by trq((
To complete the construction of combinatorial quantization, the space of states
Hn,p[I] has to be endowed with a structure of Hilbert space and the algebra of ob-
servables Mq(Σ,G;π1,...,πp) has to be endowed with a star structure such that the
representation ˜ ρn,p[I] is unitary.
In the case of Chern-Simons theory with G = SU(2), this last step of the construc-
tion has been fully studied in [2, 3, 4]. We refer to these works for full details but let us
put the emphasis on the following points:
- q is a root of unit which admits the following classical expansion (large λ expansion)
q = 1 + i2π
λ);
- Uq(su(2)) is endowed with a structure of star weak-quasi Hopf: truncation on the spec-
trum of finite dimensional unitary irreducible representations holds and the representa-
tions ˜ ρn,p[I] is the unique finite dimensional irreducible representation of Mq(Σ,G;π1,...,πp).
In the next section we will modify the previous constructions and apply them to the
case of the group SL(2,C)R.
trq(
Jπ ⊗id)(RR′)).
λ+ o(1
17
Page 18
III Combinatorial Quantization in the SL(2,C)Rcase.
III.1. Chern-Simons theory with SL(2,C)Rgroup.
Let G = SU(2), we will denote by GC= SL(2,C) the complex group and by SL(2,C)R
the realification of SL(2,C). The real Lie algebra of SL(2,C)Rdenoted sl(2,C)Rcan
equivalently be described by a star structure on its complexification (sl(2,C)R)C=
sl(2,C) ⊕ sl(2,C).
Chern-Simons theory with gauge group SL(2,C)R is defined on a 3-dimensional
compact oriented manifold M by the action
?
where the gauge field A = Aµdxµis a sl(2,C) 1-form on M and Tr is the Killing form
on sl(2,C).
Following[29], we can always write λ = k+is, with s real and k integer in order that
exp(iS(A)) is invariant under large gauge transformation.
In this paper we will choose the case k = 0 which is selected when one expresses the
action of 2 + 1 pure gravity with positive cosmological constant as a SL(2,C)RChern
Simons action [28].
We shall apply the program of combinatorial quantization in this case. From the
expression of the Poisson structure on the space of flat connections, it is easy to see
that q has to satisfy q = 1 +2π
s) when s is large. As a result we will develop
the combinatorial quantization construction using the Hopf algebra Uq(sl(2,C)R) with
q real.
An introduction to the notion of complexification and realification in the Hopf alge-
bra context can be found in the chapter 2 of [13].
S(A) =
λ
4π
M
Tr(A ∧ dA +2
3A ∧ A ∧ A) +
¯λ
4π
?
M
Tr(¯A ∧ d¯A +2
3
¯A ∧¯A ∧¯A) , (62)
s+ o(1
III.2. Combinatorial Quantization Formalism in the SL(2,C)Rcase: the
algebraic structures.
In this part we describe the modifications that have to be made to construct all the
algebraic structures of the combinatorial quantization formalism in the SL(2,C)Rcase.
In Fock-Rosly construction, we first change G to SL(2,C)R. The Lie algebra g is
changed into sl(2,C)R which is equivalent to the Lie algebra sl(2,C) ⊕ sl(2,C) with
star structure ⋆ defined by (a ⊕¯b)⋆= −(b ⊕ ¯ a). Let † be the star structure on sl(2,C)
selecting the compact form, −† identifies sl(2,C) and sl(2,C) as C-Lie algebras. As a
result we can equivalently describe sl(2,C)Ras being the Lie algebra sl(2,C)⊕ sl(2,C)
with star structure (a⊕b)⋆= (b†⊕a†). We will denote by
representations of dimension 2I +1 of su(2) which are also † representations of sl(2,C),
and let eabe an orthonormal basis of this module. The contragredient representation of
Iπ, denoted
Iπ for I ∈1
2Z+the irreducible
ˇIπ is equivalent to the conjugate representation because it is a † representation.
In the su(2) case it is moreover equivalent to the representation
I
W:
WW−1where
Wa
−b.
Iπ through the intertwiner
ˇIπ=
I
Iπ
I
I
b= (−1)I−aδa
18
Page 19
Finite dimensional irreducible representations of sl(2,C)Rare labelled by a couple
I = (Il,Ir) of positive half integers and we will denote by
I
V=
Il
V ⊗
Ir
V the sl(2,C)⊕sl(2,C)
module labelled by the couple I = (Il,Ir) associated to the representation
These representations, except the trivial one, are not ⋆-representations.
From the action (62) the Poisson bracket on the space of sl(2,C)⊕sl(2,C)-connections
is expressed by
I
Π=
Il
π ⊗
Ir
π .
{Al
{Ar
{Al
i(x)⊗, Al
j(y)}
=
2π
λδ(x − y)ǫijtll
−2π
0
(63)
i(x)⊗, Ar
j(y)}
j(y)}
=
λδ(x − y)ǫijtrr
(64)
i(x)⊗, Ar
=(65)
where tll(resp. trr)is the embedding of t in the l ⊗ l (resp. r ⊗ r) component of
(sl(2,C) ⊕ sl(2,C))⊗2. Note that we have Al
The spin-networks are defined analogously by replacing finite dimensional represen-
tations of g by finite dimensional representations of sl(2,C) ⊕ sl(2,C).
For any representation
Π of sl(2,C)⊕sl(2,C) with I = (Il,Ir), we define
) ⊗ Pol(SL(2,C)R) the matrix of coordinate functions on SL(2,C)R. We have
(ˇIr,ˇIl)
Ga′a
W ⊗
i(x)†= −Ar
j(x) and t††= t.
I
I
G∈ End(
I
V
(Il,Ir)
Gaa′
bb′
⋆=
b′b=?(
Il
Ir
W)
(Il,Ir)
G
(
Il
W
−1⊗
Ir
W
−1)?a′a
b′b.(66)
We denote the holonomy of the sl(2,C) ⊕ sl(2,C) connection in the representation
by
U(l), they satisfy the same relation
I
Π
I
(Il,Ir)
Uaa′
bb′
⋆=?(
Il
W ⊗
Ir
W)
(Il,Ir)
U
(
Il
W−1⊗
Ir
W−1)?a′a
b′b.(67)
We can define a Fock-Rosly structure on them, the Poisson bracket is the same as
(13,14,15) where the classical r matrix of su(2) has been replaced by the r matrix of
sl(2,C)R: rsl(2,C)R= rll
We refer the reader to the article ([12, 13]) for a thorough study of the quantum group
Uq(sl(2,C)R), see also the appendix (A.1) where basic definitions as well as fundamental
results on harmonic analysis are described. It is important to stress that Uq(sl(2,C)R)
admits two equivalent definitions.
The first one is Uq(sl(2,C)R) = Uq(sl(2,C)) ⊗ Uq(sl(2,C)) as an algebra with a
suitable structure of coalgebra and ⋆ structure.
The second one, suitable for the study of harmonic analysis, is Uq(sl(2,C)R) =
D(Uq(su(2))), the quantum double of Uq(su(2)), which is the quantum analog of Iwa-
sawa decomposition.
Uq(sl(2,C)R) is a quasi-triangular ribbon Hopf algebra endowed with a star structure
(see appendix A.1). Finite dimensional representations of Uq(sl(2,C)R) are labelled by
a couple I = (Il,Ir) ∈ (1
12− rrr
21.
2Z+)2= SF. The explicit description of these representations
19
Page 20
is contained in the appendix A.1. The decomposition of the tensor product of these
representations, and the explicit form of the Clebsch-Gordan maps, are described in the
appendix A.2. For any finite dimensional irreducible representation
I
Π, we will define
I
V
the associated module. Let us denote by {
Ir} an orthonormal basis of this vector space, and {
Pol(SLq(2,C)R) of polynomials on SLq(2,C)Ris generated by the matrix elements of
I
GAa
Π (·)|
introduce for each representation I the elements
I
L(±)= (
Π ⊗id)(R(±)) where R is the Uq(sl(2,C)R) R-matrix arising from
the construction of the quantum double.
Uq(sl(2,C)R) = Uq(sl(2)) ⊗R−1 Uq(sl(2)) as a Hopf algebra [12, 13], R is expressed in
term of Uq(sl(2)) R-matrices as R(±)= R(−)
the ribbon elements of Uq(sl(2,C)R) from those of Uq(sl(2)):
Aei(I), i = −A,··· ,A, A = |Il−Ir|,··· ,Il+
Aei(I)} the dual basis. The algebra
the representations
Bb=?Aea(I)|
I
Beb(I)?. As in the previous section, let us
L(±)Aa
Bb
∈ End(
II
V) ⊗ Uq(sl(2,C)R)
defined by
I
Thanks to the factorisation theorem, i.e.
14R(∓)
24R(±)
13R(+)
23. It is therefore easy to obtain
vI = vIl v−1
Ir ,
Iµ=
Il
µ⊗
Ir
µ .
(68)
Let us now study the properties of the star structure on Uq(sl(2,C)R) and, by duality,
on Pol(SLq(2,C)R). In the case of Uq(sl(2,C)R), the star structure, recalled in the
appendix, is an antilinear involutive antimorphism satisfying in addition the condition
∀ a ∈ Uq(sl(2,C)R), (⋆ ⊗ ⋆)∆(a) = ∆(a⋆) . (69)
It is easy to show the following relation between the antipode and the ⋆ :
S ◦ ⋆ = ⋆ ◦ S−1.(70)
The universal R-matrix of Uq(sl(2,C)R) satisfies R⋆⊗⋆= R−1which is compatible with
(69) and is a key property in order to build a star structure on the graph algebra asso-
ciated to Uq(sl(2,C)R).
By duality, Pol(SLq(2,C)R) is endowed with a star structure using the following
definition:
α⋆(a) = α(S−1a⋆) ∀ (α,a) ∈ Pol(SLq(2,C)R) × Uq(sl(2,C)R) .
Let I = (Il,Ir) ∈ SF labelling a finite-dimensional representation of Uq(sl(2,C)R) and
let us define by˜I = (Ir,Il). The following properties of the action of the ⋆ and of
the complex conjugation are proved in the appendix A.3. The explicit action of the ⋆
I
GAa
Bbof Pol(SLq(2,C)R) is:
(71)
involution on the generators
I
G⋆
=
˜I
W
˜I
G
˜I
W−1
(72)
20
Page 21
where we have defined
I
WAa
Bb=
˜I
WAa
Bb= eiπAv−1/2
A
A
wab δA
B. By duality, the action of ⋆
on the generators
I
L(±)Aa
Bbof Uq(sl(2,C)R) is:
I
L(±) ⋆=
˜I
W
˜I
L(±)
˜I
W−1. (73)
We endow the graph-algebra with the following star structure:
Proposition 4 The graph-algebra Ln,pis endowed with a star structure defined on the
generators
A(i),
B(i),
M(i) (denoted generically
I
I
I
I
U(i)), by
I
U(i)⋆=
?
j
v−1
˜I
˜I
W S−1(
˜Ixj)
˜I
U(i)
˜Iyj
˜Iµ
˜I
W−1, (74)
where R =?
definition of this star structure is chosen in order that the coaction δ is a star morphism.
jxj⊗ yj. This star structure is an involutive antilinear automorphism
which in the classical limit gives back the star properties (67) on the holonomies. The
Proof: We just have to prove that it is an involution and that it is compatible with
the defining relations of the graph algebra. These two properties are straightforward to
verify. 2
The star structure on the graph algebra induces a star structure on the algebra Linv
The action of this star structure on the generating family labelled by spin-network is
described in the following proposition:
n,p.
Proposition 5 The action of the star structure on Linv
n,psatisfies:
P
O(±)⋆
n,p
=
˜P
O(∓)
n,p
(75)
where˜P is the spin network deduced from P by turning all the colors I of P into˜I.
Proof: This follows from the action of the star on the monodromies, the commutation re-
lations of the monodromies and the properties with respect to the complex conjugation.
2
Mq(Σ,SL(2,C)R,p) is the algebra defined by Mq(Σ,SL(2,C)R,p) = Ln,p/IC. Let
be an irreducible unitary representation of Uq(sl(2,C)R), labelled by the couple α ∈ SP
we can still define the complex numbers ϑIαwhere I ∈ SF. The explicit formula, which
is proved in the appendix A.3, for ϑIαis
α
Π
ϑIα=[(2Il+ 1)(2αl+ 1)]
[2αl+ 1]
[(2Ir+ 1)(2αr+ 1)]
[2αr+ 1]
. (76)
Let
tures of Σ, the moduli space Mq(Σ,SL(2,C)R;α1,...,αp) is defined by:
α1
Π,...,
αp
Π be irreducible unitary representations of Uq(sl(2,C)R) attached to the punc-
Mq(Σ,SL(2,C)R;α1,...,αp) = Mq(Σ,G,p)/{trq(
I
M(n + i)) = ϑIαi,i = 1,...,p,∀I ∈ SF}.
(77)
21
Page 22
Proposition 6 The star structure on the graph algebra defines a natural star structure
on the algebras Mq(Σ,SL(2,C)R,p) and Mq(Σ,SL(2,C)R;α1,...,αp).
Proof: This is a simple consequence of the fact that ϑIα= ϑ˜Iαfor α ∈ SP and I ∈ SF.
2
The Heisenberg double of Uq(sl(2,C)R) is defined by H(Uq(sl(2,C)R) = Uq(sl(2,C)R)⊗
Pol(SLq(2,C)R).
In order to build unitary representations of the graph algebra and the moduli algebra
in the next chapter we have to study the properties of Alekseev isomorphism with
respect to the star structure. The already defined star structures on Uq(sl(2,C)R) and
Pol(SLq(2,C)R) naturally extend to a star structure on H(Uq(sl(2,C)R), and therefore
to H(Uq(sl(2,C)R)⊗n⊗ Uq(sl(2,C)⊗p
Proposition 7 The Alekseev isomorphism defined in lemma 3
R.
Ln,p
∼
−→ H(Uq(sl(2,C)R))⊗n⊗ Uq(sl(2,C)R)⊗p
is a star-isomorphism.
Proof: Using the star structure on H(Uq(sl(2,C)R))⊗n⊗ Uq(sl(2,C)R)⊗p, we first
show that:
I
B(i)⋆=
?
j
v−1
˜I
˜I
W S−1(
˜Ixj)
˜I
B(i)
˜Iyj
˜Iµ
˜I
W−1. (78)
In order to prove this, we introduce the permutation P12and we have
I
B(i)⋆
= tr2(P12
I
L(−)−1⋆
1
I
L(+)⋆
2
˜I
L(−)−1
1
)
= tr2(P12
I
W1
Iµ−1
1
˜I
L(−)−1
1
Iµ1
I
W−1
1
I
W2
˜I
L(+)
2
I
W−1
2)
=
I
W1 tr2(P12
˜I
L(+)
2
˜I
L(+)
Iµ−1
2)
˜I˜I
R
Iµ1
I
W−1
1
=
I
W1 tr2(P12S−1(
˜I
W S−1(
˜Ixj2)
˜I
B(i)
2
˜I
L(−)−1
1
˜Iyj1
Iµ−1
2)
Iµ1
I
W−1
1
=v−1
˜I
˜Ixj)
˜Iyj
˜Iµ
˜I
W−1.
Then, it is easy to compute that the star acts on
I
K(i) and on its inverse as:
I
K(i)⋆=
˜I
W(i)
˜I
K(i)
˜I
W(i)−1,
I
K(i)−1⋆=
˜I
W(i)
˜Iµ−1
I
K(i)−1
˜Iµ
˜I
W(i)−1.(79)
22
Page 23
Finally, from the previous relations, we have
?
= tr23(
(
I
K(i)
I
B(i)
I
K(i)−1)⋆
= tr23(P12P23
I
K3(i)
I
B2(i)
I
K1(i)−1)
?⋆
I
K1(i)−1⋆
˜I
K1(i)−1
I
B2(i)⋆I
˜Iµ1
W−1
K3(i)⋆)
= tr23(
˜I
1
˜I
K(i)
?˜I
˜Iv−1
2S−1(
˜Ix2(j))
˜I
B2(i)
˜Iy2(j)
˜I
W3
˜I
K3(i))
=
˜Iv−1
˜I
W S−1(
˜Ix(j))
˜Ix(k)S−1(
˜Ix(l))
˜I
B(i)
?˜Iy(j)
˜Iy(l)
˜Iy(k)
˜I
K(i)−1
˜Iy(j)
˜Iµ
˜I
W−1
=
˜Iv−1
˜I
W S−1(
˜Ix(j))
K(i)
˜I
B(i)
˜I
K(i)−1
˜Iµ
˜I
W−1.
As a result, we have shown that the property holds true for the monodromies B(i). The
other cases are proved along the same lines. 2
III. Unitary Representations of the Moduli Algebra in the
SL(2,C)Rcase.
III.1 Unitary representation of the graph algebra.
The reader is invited to read the appendix A.1 where the basic results on harmonic
analysis are recalled.
Proposition 8 H(Uq(sl(2,C)R)) admits a unitary representation acting on the space
Funcc(SLq(2,C)R) endowed with the hermitian form (183) and constructed as:
H(Uq(sl(2,C)R))
Pol(SLq(2,C)R) ∋ f
Uq(sl(2,C)R) ∋ x
Proof:Trivial to check. 2
From the Alekseev isomorphism and the study of harmonic analysis on SLq(2,C)R,
we obtain a simple description of unitary representations of the graph algebra Ln,p:
Proposition 9 Let α1,...,αp∈ SPwe denote by Hn,p[α] = Funcc(SLq(2,C)R)⊗n⊗V(α)
the pre-Hilbert space with sesquilinear form <,>= (⊗n
<,>iis the L2hermitian form (183) on the i-th copy of Funcc(SLq(2,C)R) and <,>n+j
αj
V . Hn,p[α] is endowed with a structure of Ln,pmodule using
the Alekseev isomorphism. This representation is unitary, in the sense that:
−→
?→
?→
End(Funcc(SLq(2,C)R)),
mf
/ mf(g) = fg ,∀g ∈ Funcc(SLq(2,C)R)
∇x / ∇x(g) = g(1)
?x,g(2)
?
,∀g ∈ Funcc(SLq(2,C)R).
i=1<,>i)⊗(⊗p
j=1<,>j+n) where
is the hermitian form on
∀a ∈ Ln,p,∀v,w ∈ Hn,p,< a⋆? v,w >=< v,a ? w > .(80)
Proof:This is a simple consequence of the previous proposition. 2.
We now come to the central part of our work: the construction of a unitary repre-
sentation of the moduli-algebra. We could have hoped to apply the same method as
23
Page 24
in proposition (3), unfortunately this is not possible because there is no normalizable
states in Hn,por in its completion with respect to <,> which are invariant under the
action of Uq(sl(2,C)R) induced from the algebra generated by
the existence of a quantum analogue of Faddeev-Popov procedure to solve this problem
but we were unable to proceed along this path. Instead we will give explicit formulas
P
O
and verify that this representation is unitary. We will enlarge the representation space
of the graph algebra as follows: we will transfer the representation of the graph-algebra
on the dual conjugate space Hn,p[α]∗as follows:
∀φ ∈ Hn,p[α]∗,∀v ∈ V[α],∀a ∈ Ln,p,(a ? φ,v) = (φ,a⋆? v) = (φ,a⋆? v).
Hn,p[α]∗is naturally endowed with a structure of Uq(SL(2,C)R) module which admits,
as we will see, invariant elements. This method is similar in spirit to the concept of
refined algebraic quantization program [7]. In order not to complicate notations, we will
prefer to work with the right module Hn,p[α]∗associated to the anti-representation:
∀φ ∈ Hn,p[α]∗,∀v ∈ V[α],∀a ∈ Ln,p,(φ ? a,v) = (φ,a ? v),
this is of course completely equivalent.We will continue to denote by ρn,p[α] this
antirepresentation.
The construction of a unitary representation of the moduli algebra is exposed through
elementary steps: the p-punctured sphere, the genus-n surface and finally the general
case.
I
C . We cannot exclude
for the action of
(±)
n,p on the space of invariant vectors, with a suitable hermitian form,
(81)
(82)
III.2 Unitary Representation of the moduli algebra
III.2.1. The moduli algebra of a p-punctured sphere
This first subsection is devoted to a precise description of unitary representations of the
moduli-algebra on a sphere with p punctures associated to unitary irreducible represen-
tations α1,··· ,αp.
Let us begin with some particular examples. The three first cases (i.e. p = 1, 2 or 3)
are singular in the sense that the representation of the moduli algebra is one-dimensional
or zero-dimensional.
In the case of the one-punctured sphere, the moduli algebra has the following struc-
ture:
• Mq(S2,SL(2,C)R;Π) = {0} if Π is not the trivial representation
I
Π) = C where I = (0,0).
• Mq(S2,SL(2,C)R;
As a result the representation is non zero-dimensional if and only if the representation
associated to the puncture is the trivial representation. In this case the corresponding
one dimensional representation H0,1is trivially unitarizable.
In the case of the two-punctured sphere, the moduli algebra has the following struc-
ture:
24
Page 25
• Mq(S2,SL(2,C)R;
α1
Π,
α2
Π) = {0} if
α1
Π) = C.
α1
Π is not equivalent to
α2
Π
• Mq(S2,SL(2,C)R;
As a result the representation is non-zero dimensional if and only if the represen-
tations associated to the two punctures are the same, i.e. α1 = α2. The associated
module is (V(α1,α1)∗)Uq(sl(2,C)R)generated by
α1
Π,
αω0,2where α = (α1,α1), such that
<
αω0,2,
Iei(α1)⊗
Jej(α1) > = Ψ0
α1α1
?Iei(α1)⊗
Jej(α1)?
. (83)
We can endow this module with a Hilbert structure < .|. > such that <
1. This is a unitary one-dimensional module H0,2(α) of the moduli algebra because of
the property ϑIΠ= ϑ˜IΠfor unitary representations Π, and I ∈ SF.
αω0,2|
αω0,2>=
In the case of three punctures, the moduli algebra has the following structure:
• Mq(S2,SL(2,C)R;
α1
Π,
α2
Π,
α3
Π) = C if mα1+ mα2+ mα3∈ Z,
α3
Π) = {0} in the other cases.
• Mq(S2,SL(2,C)R;
In the first case, let us denote by α = (α1,α2,α3). The associated module is (V(α)∗)Uq(sl(2,C)R)
generated by
α1
Π,
α2
Π,
αω0,3such that
<
αω0,3,
Iei(α1)⊗
Jej(α2)⊗
Kek(α3) > =?Ψ0
α3α3Ψα3
α1α2
??Iei(α1)⊗
Jej(α2)⊗
Kek(α3)?
αω0,3|
. (84)
We can endow this module with a Hilbert structure < .|. > such that <
This is a unitary one-dimensional module H0,3(α) of the moduli algebra for the same
reasons as before.
αω0,3>= 1.
Definition 6 Let α = (α1,...,αn) ∈ Sn,β = (β3,....,βn) ∈ Sn−2,κ ∈ S, we define
Ψκ
α(β) ∈ HomUq(sl(2,C))R(V(α),
κ
V) as
Ψκ
α(β) = Ψκ
βnαnΨβn
βn−1αn−1···Ψβ4
β3α3Ψβ3
α1α2.(85)
Definition 7 For any family β = (β3,··· ,βp−1) ∈ Sp−3we define the linear form
αω0,p[β] ∈ V(α)∗by:
αω0,p[β] = Ψ0
α(β3,··· ,βp−1,αp). (86)
These linear forms satisfy the constraints
αω0,p[β]?
I
C(±)
0,p
=
αω0,p[β] ,(87)
αω0,p[β] ? trq(
I
M(n + i))=ϑIαi
αω0,p[β] , ∀I ∈ SF,∀i = 1,...,p .(88)
25
Page 26
In particular
αω0,p[β]I
We will now smear
“wave packet.” We have chosen functions which are analytic in the spirit of the Paley-
Wiener theorem.
In the sequel we will use the following notations: we define S = (1
β ∈ Sk, we will denote by ρβ= (ρβ1,...,ρβk) and mβ= (mβ1,...,mβk). Reciprocally m ∈
(1
function on Skwe will denote by fmthe function on Ckdefined by fm(ρ) = f(β(m,ρ)).
αω0,p[β] are invariant elements of V(α)∗. It will be convenient to denote
αω0,p[β],?p
a=<
i=1
αω0,p[β] with a function f sufficiently regular, in order to obtain
Iieai(αi) > .
2Z)2and for
2Z)k,ρ ∈ Ckis associated to a unique element β(m,ρ) ∈ Sk. If f is a complex valued
Definition 8 We define A(k)to be the set of functions f on Sksuch that
• f(β1,...,βr−1,βr,βr+1,...,βk) = f(β1,...,βr−1,βr,βr+1,...,βk), ∀r = 1,...,k,
• fmis a non zero function for only a finite number of m ∈1
• fmis a Laurent series in the variables qiρβ1,...,qiρβk, convergent for all ρβ1,...,ρβk∈
C.
2Zk,
For β ∈ Sp−3we define
ˆΞ0,p[α,β]−1= eiπ(α1+...+αp)ζ(α1,α2,β3)
p−2
?
j=3
ζ(βj,αj,βj+1)ζ(βp−1,αp−1,αp)
p−1
?
j=3
ν1(dβ).
(89)
For I = (I1,...,Ip) we denote νI(α) =?p
Lemma 4
rent polynomial in the variables qiρα1,...,qiραp,qiρβ3,...,qiρβp−1and Q[β] =?p−1
ties: Y (J3,I1,I2) = 1,...,Y (Jp−2,Ip−2,Jp−1) = 1,Y (Ip−1,Jp−1,Ip) = 1 hold. We have
Bj= max{Jj,J ∈ M}.
j=1ν(Ij)(αj).
αω0,p [β]I
aˆΞ0,p[α,β]νI(α) is equal to P[α,β]/Q[β] where P[α,β] is a Lau-
j=3(iρβj−
Bj)2Bj+1 with Bj ∈
1
2N. Let M be the set of J ∈ (1
2N)p−3such that the inequali-
Proof:Trivial application of the proposition (24) of the Appendix A.2. 2
For β ∈ Sp−3we define ξ[β] =
and the Plancherel weight P are defined in the appendix A.1, as well as Ξ0,p[α,β] =
ˆΞ0,p[α,β]ξ[β].
?p−1
j=3ξ(2βl
j+ 1), P[β] =
?p−1
j=3P(βj), where ξ
Proposition 10 We can define a subset of V(α)∗by:
?
H0,p(α) = {
αω0,p(f) =
S(p−3)
P
dβP[β]f(β)Ξ0,p[α,β]
αω0,p[β],with f ∈ A(p−3)}.(90)
The map A(p−3)→ H0,p(α) which sends f to
αω0,p(f) is an injection.
26
Page 27
Proof:In order to show that
ξ[β]Ξ0,p[α,β]
ξ[β] cancels the simple poles of Ξ0,p[α,β]
of the condition on f.
We now prove injectivity of this map. Assume that
have
αω0,p(f) is well defined it is sufficient to show that
ais a Laurent series in qiρβ3,...,qiρβp−1, which is the case because
αω0,p[β]I
a. The sum over mβis finite because
αω0,p[β]I
αω0,p(f) is zero, we would therefore
αω0,p(f)(av) = 0 for all a ∈ Uq(sl(2,C)R)⊗pand v ∈ V[α]. If c is a central element of
Uq(sl(2,C)R) we denote by c(β) its value on the module
are elements of the center of Uq(sl(2,C)R), we take a = ∆(c3)...∆(p−2)(cp) where ∆(k)
are the iterated coproducts, and using the intertwiner property we obtain that
(fg(c3,...,cp−1)) = 0 where g(c3,...,cp−1)(β) =?p−1
satisfy the identity: f(β)Ξ0,p[α,β]
f = 0. This follows from a similar argument exposed in [14] (Th 4) which uses the
asymptotics of the reduced elements. 2
We define, for α ∈ Sp
p−2
?
where M is defined in the appendix A.2, and we denote for α ∈ Sp
Υ0,p[α,β] =|Ξ0,p[α,β]|2
β
V . In particular if c3,...,cp−1
αω0,p
j=3cj(βj). By a similar argument as
the one used in [12] (proof of Th 12 (Plancherel Theorem)) , we obtain that f has to
αω0,p[β] = 0. It remains to show that this implies
P,β ∈ Sp−3
P
,
M0,p[α,β] = M(α1,α2,β3)
j=3
M(βj,αj,βj+1)M(βp−1,αp−1,αp)(91)
P,β ∈ Sp−3
P
M0,p[α,β].(92)
Lemma 5 Υ0,p[α,β] is an analytic function of the real variables ρβ3,...,ρβp−1and there-
fore admits an analytic continuation for ρβ3,...,ρβp−1∈ Cp−3, i.e β3,...,βp−1∈ Sp−3.
Proof:We can extend M to S3as follows [14]:
M(α,β,γ) = ψ1(α,β,γ)
(q−1− q)qmα+mβ+mγ
ν1(dαl)ν1(dαr)ν1(dβl)ν1(dβr)ν1(dγl)ν1(dγr)Θ(α,β,γ)(93)
where
Θ(α,β,γ) =
θ(2αr+ 1)θ(2βr+ 1)θ(2γr+ 1)
θ(αr+βr+γr+2)θ(αr+βr+γr+ 2)θ(αr+βr+γr+2)θ(αr+βr+γr+2)
ϕ(2αr,−2mα−1)ϕ(2βr,−2mβ−1)ϕ(2γr,−2mγ−1)
ϕ(iρα+β+γ,mα+β+γ)ϕ(iρα+β+γ,mα+β+γ)ϕ(iρα+β+γ,mα+β+γ)ϕ(iρα+β+γ,mα+β+γ)
ψ1(α,β,γ) =
.
Moreover |Ξ0,p[α,β]|2can be extended to β3,...,βp−1∈ Sp−3by
?p−1
ψ2(α1,α2,β3)
|Ξ0,p[α,β]|2=
j=3(−1)2mβjq4imβjρβj
?p−2
j=3ψ2(βj,αj,βj+1)ψ2(βp−1,αp−1,αp)
Ξ0,p[α,β]2
(94)
27
Page 28
where
ψ2(α,β,γ)=
q
i
2(ρα+β+γ|mα+β+γ|+ρα+β+γ|mα+β+γ|+ρα+β+γ|mα+β+γ|+ρα+β+γ|mα+β+γ|)
ϕ(−iρα+β+γ,|mα+β+γ|)ϕ(−iρα+β+γ,|mα+β+γ|)ϕ(−iρα+β+γ,|mα+β+γ|)ϕ(−iρα+β+γ,|mα+β+γ|)
.
Due to the following obvious properties,
ϕ2
ϕ(iρα+β+γ,mα+β+γ)ϕ(−iρα+β+γ,|mα+β+γ|)= eiπ
(ν1(dβl)ν1(dβr))2= (dβl)1(dβr)1
(2βr,−2mβ−1)= (−1)2mβ+1
2(|mα+β+γ|−mα+β+γ)
and the fact that Ξ0,p[α,β]2(Θ(α1,α2,β3)
the complex variables ρβ3,...,ρβp−1, Υ0,p[α,β] is an analytic function of the real variables
ρβ3,...,ρβp−1. We will still use the notation Υ0,p[α,β] to denote its analytic continuation
for ρβ3,...,ρβp−1∈ C. 2
?p−2
j=3Θ(βj,αj,βj+1)Θ(βp−1,αp−1,αp))−1is analytic in
Remark: Note that Θ(α,β,γ) is left invariant under permutation of α,β,γ and that
Θ(α,β,γ) is left invariant under the following shifts: Θ(α + s,β,γ) = Θ(α,β,γ) for
s ∈ Z2, and Θ(α,β,γ) = Θ(α + (iπ
lnq,iπ
lnq),β,γ).
Proposition 11 The space H0,p(α) is endowed with the following pre-Hilbert space
structure:
?
<
αω0,p(f) |
αω0,p(g) >=
Sp−3
P
dβP[β] f(β)Υ0,p[α,β] g(β) . (95)
Proof:We use injectivity of the map f ?→
uct is unambiguously defined. The convergence of the integrals in the real variables
ρβ3,...,ρβp−1is ensured by the analyticity of the integrand and the convergence of the
sums in mβ3,...,mβp−1comes from the fact that the wave packets have finite support in
these discrete variables.
The positivity of this hermitian product is due to formulas (92)(204). Showing it is
definite is trivial. 2
αω0,p(f) to show that this hermitian prod-
The major theorem of this section is the result that H0,p(α) is a right unitary module
of the moduli algebra Mq(S2,
Π). This is a non trivial result which is divided in the
following steps. In Lemma 1 we compute the action of an element of Mq(S2,
αω0,p[β]. In particular if β is in SP, the result is a linear combination of
in S. This comes from the fact that the observables are constructed with intertwiners of
finite dimensional representations of Uq(sl(2,C)R) and that the tensor product of finite
dimensional representations and principal representations decomposes as a finite direct
sum of infinite dimensional representations, non unitary in general, of S type as in (187).
As a result the action of an element a of Mq(S2,
α
α
Π) on
αω0,p[γ], with γ
α
Π) on
αω0,p[f] can be defined after the
28
Page 29
α1
α1
N1
N2
N3
α2
α2
α3
α3
α4
α4
β
α4
N4
U3
U4
T3
T4
W
W
Figure 2: Expression of
P
F(+)
0,4.
use of a change of integration in the complex plane, thanks to Cauchy theorem. The
proof of unitarity of this representation is reduced to properties of the kernel Υ0,punder
shifts.
Proposition 12 Let β ∈ Sp−3, the action of
P
O(±)
0,p∈ Linv
?
0,pon
?
αω0,p[β] is given by:
αω0,p[β]?
P
O(±)
0,p=
?
s∈Sp−3
P
K(±)
0,p
α
β,s
αω0,p[β + s] ,(96)
where:
1.the functions
P
K
(±)
0,p
?
α
β,s
?
are non zero only for a finite set of s ∈ Sp−3
according to the selection rules imposed by the palette P. Moreover any element s of this
finite set satisfy sl,sr∈ Zp−3.
2. the functions
K(±)
0,p
β,s
?
C(qiρα1,...,qiραp)[qiρβ3,q−iρβ3,...,qiρβp−1,q−iρβp−1] and Qjis a polynomial with zeroes in
{qn,n ∈1
P
?
α
?
belong to
Ξ0,p[α,β+s]
Ξ0,p[α,β]C(qiρα1,...,qiραp,qiρβ3,...,qiρβp−3),
and more precisely we have:
P
K(±)
0,p
α
β,s
?
Ξ0,p[α,β]
Ξ0,p[α,β+s]=
P[α,β]
?p−1
j=3Qj(q
iρβj)
where P[α,β] ∈
2Z}.
Proof:
(
αω0,p[β]?
P
O(±)
0,p)
Iea(α) =
P
F(±)
0,p
?α
β
?
Iea(α), (97)
where
P
F(±)
0,p
?α
β
?
are elements of HomUq(sl(2,C)R)(
α
V,C). The picture for
?
P
F(+)
0,p
?α
β
?
is shown in figure 2, whereas the picture for
P
F(−)
0,p
?α
β
is the same after having turned
overcrossing colored by couples of finite dimensional representations into the correspond-
ing undercrossing. The picture for p > 4 punctures is a straighforward generalization.
29
Page 30
α1
α1
α2
α2
α3
α3
α4
α4
α4
β
α4
β + s
N1
N2
N3
N4
U3
U4
W
T3
T4
W
W
00
Figure 3: Expression of
P
K(+)
0,4.
When α1,...,αn,β3,...,βp−3are fixed elements of SF, the non zero elements of the
family {
obtain
αω0,p [β + s],s ∈ Sp−3}, form a basis of HomUq(sl(2,C)R)(
α
V,C). As a result we
P
F(±)
0,p
?α
β
?
=
?
s∈Sp−3
P
K(±)
0,p
?
α
β,s
?
αω0,p[β + s] ,(98)
where
P
K(±)
0,p
?
α
β,s
?
can be computed as:
P
K(±)
0,p
?
α
β,s
?
=
P
F(±)
0,p
?α
β
?
αη[β + s] (99)
with
αη[β] = Φα1α2
?
for the figure representing
β3
?
Φβ3α3
β4
···Φβp−1αp−1
αp
Φαpαp
0
.
P
K(+)
0,p
α
β,s
is therefore represented by the picture of figure 3. The same comments
?α
Note that the selection rules in the finite dimensional case, implies that if
then the possible non zero elements
Using the property (22) of factorization of finite dimensional intertwiners, we obtain
that
?
?
expressed in terms of 6j(0) coefficients. As a result it is straightforward to define a
P
K(±)
0,p
β,s
and by replacing where needed 6j(0) by 6j(1) or 6j(3) in (223).
P
F(±)
0,p
β
?
apply here.
αω0,p[β] ?= 0
αω0,p[β + s] are those for s satisfying sl,sr∈ Zp−3.
P
K(±)
0,p
α
β,s
?
?
=
Pl
Kl(±)
0,p
?
αl
βl,sl
?
Pr
Kr(±)
0,p
?
αr
βr,sr
?
(100)
where
Pl
Kl(±)
0,p
αl
βl,sl
?
,
Pr
Kr(±)
0,p
αr
βr,sr
?
are computed (223) in the appendix (B.1) and
continuation of
?
α
?
for α1,...,αp,β3,...,βp−1∈ S,s ∈ Sp−3maintaining (100)
30
Page 31
It can be checked, from this definition, that
Ξ0,p[α,β]
Ξ0,p[α,β+s]
P
K(±)
0,p
?
α
β,s
?
is a rational
function in qiρα1,...,qiραp,qiρβ3,...,qiρβp−3.
We recall that
αω0,p[β]I
aΞ0,p[α,β]νI(α) is an element of C(qiρα1,...,qiραp,qiρβ3,...,qiρβp−3).
From this property we can deduce the more general result that
P
F(±)
0,p
?α
β
?
Iea(α)Ξ0,p[α,β]νI(α)
is also an element of C(qiρα1,...,qiραp,qiρβ3,...,qiρβp−3). The property in qiρβ3,...,qiρβp−3
is a direct consequence of the definition of Ξ0,p[α,β], the only non trivial fact is to show
that this also holds for qiρα1,...,qiραp. This comes from the structure of the graph in
Iα
R in terms of the coefficients ΛBC
N(A)(α)
N(D)(α)ΛBC
As a result we obtain that:
?α
s∈Sp−3
the figure 2, the expression of
AD(α), and the fact that
AD(α) is a Laurent polynomial in qiρα.
Ξ0,p[α,β]νI(α)
P
F(±)
0,p
β
?
I
a−
?
Ξ0,p[α,β]
Ξ0,p[α,β + s]
P
K(±)
0,p
?
α
β,s
??αω0,p[β + s]?I
aΞ[α,β + s]νI(α) ,
is an element of C(qiρα1,...,qiραp,qiρβ3,...,qiρβp−1) which vanishes for an infinite number
of sufficiently large iρα1,...,iραp,iρβ3,...,iρβp−1∈1
is nul and we obtain relation (96).2
0,p, a =?
K(±)
0,p
β,s
β,s
We will now endow H0,p(α) with a structure of right Linv
sense:
2Z+. As a result this rational function
Remark. If a ∈ Linv
?
PλP
?
P
O(±)
?
0,pwe define
a
α
?
=?
PλP
P
K(±)
0,p
α
.
0,pmodule in the following
Proposition 13 Linv
invariant. However the subspace H0,p(α) is in general not invariant. We define the
domain of a ∈ Linv
f belongs to D(a) if and only if for all s ∈ Sp−3the functions
β ?→
β,s
an element
0,pacts on V(α)∗with ρ0,p[α] and leaves the space (V(α)∗)Uq(sl(2,C)R)
0,passociated to ρ0,p[α] to be the subspace D(a) ⊂ A(p−3)defined as:
?
αω0,p(f) ∈ D(a) ⊂ H0,p(α) belongs to H0,p(α), and we have:
αω0,p(f) ? a =
a
K(±)
0,p
?
α
Ξ0,p[α,β]P[β]
Ξ0,p[α,β+s]P[β+s]f(β) are elements of A(p−3). The action of a on
αω0,p(f ? a) ,
with
(f ? a)(β) =
?
s∈Sp−3
f(β − s)
a
K(±)
0,p
?
α
β − s,s
?Ξ0,p[α,β − s]P[β − s]
Ξ0,p[α,β]P[β]
.(101)
In general the domain D(a) is not equal to A(p−3), but it always contains Qa[β]A(p−3)
where Qa[β] =?p−3
j=3((2βl
j+nj)kj)pj((2βr
j+n′
j)k′
j)p′
jfor some nj,n′
jintegers and kj,k′
j,pj,p′
j
non negative integers depending on a.
31
Page 32
Proof:This property follows from the invariance of the sum of integrals under the change
β ?→ β − s, which amounts to replace mβby mβ− msand ρβby ρβ− ρswith ρs∈ iZ
given by iρs= sl+ sr+ 1. The former is a change of index in a sum whereas the latter
is implied by Cauchy Theorem if the function considered is analytic, which follows from
the definition of D(a). 2
It is clear from the properties (88), that the anti-representation ρ0,p[α] on H0,p[α] de-
cends to the quotient, and defines an anti-representation ˜ ρ0,p[α] of Mq(S2,SL(2,C)R;α).
Our main result is that ˜ ρ0,p[α] is unitary.
Theorem 1 The anti-representation ˜ ρ0,p[α] of Mq(S2,SL(2,C)R;α) is unitary:
∀a ∈ Mq(S2,SL(2,C)R,[α]),∀v ∈ D(a⋆),∀w ∈ D(a),< v ? a⋆|w >=< v|w ? a > .
(102)
where < .|. > is the positive sesquilinear form defined by (95).
Proof:The first step amounts to extend to Sp−3the function β ?→ Υ0,p[α,β] entering
in the definition of < .|. >. For this task we use lemma 5. As a result, the exten-
sion of Υ0,p[α,β], denoted by the same notation, is an entire function in the variables
ρβ3,...,ρβp−1. We then compute:
<
?
αω0,p(f)|
?
αω0,p(g)?
P
O(+)
0,p>=
s
Sp−3
P
dβP(β + s)f(β)Υ0,p[α,β]g(β + s)
P
K(+)
0,p
?
α
β + s,−s
?Ξ0,p[α,β + s]
Ξ0,p[α,β]
.
On the other hand we have:
<
αω0,p(f) ? (
?
?
P
O(+)
0,p)⋆|
αω0,p(g) >=<
αω0,p(f)?
˜P
O(−)
˜P
K(−)
0,p|
αω0,p(g) >=
?
?
=
?
s
Sp−3
P
dβP(β + s)f(β + s)Υ0,p[α,β]g(β)
0,p
α
β + s,−s
α
β + s,−s
?Ξ0,p[α,β + s]
Ξ0,p[α,β]
?Ξ0,p[α,β + s]
Ξ0,p[α,β]
=
s
?
Sp−3
P
dβP(β + s)f(β + s)Υ0,p[α,β]g(β)
˜ P
K(−)
0,p
.
Using Cauchy theorem to shift ρβin ρβ+ ρsand simultaneously changing the index of
the sum over the dumb variable mβand s, the last expression is also equal to:
?
As a result, proving unitarity is equivalent to showing the relation:
?
˜P
K(−)
0,p
β,−˜ s
?
s
Sp−3
P
dβP[β]f(β)Υ0,p[α,β + s]g(β + s)
˜P
K(−)
0,p
?
α
β,−˜ s
?
Ξ0,p[α,β]
Ξ0,p[α,β + s].
Υ0,p[α,β]
P
K(+)
0,p
α
β + s,−s
?Ξ0,p[α,β + s]
α
Ξ[α,β]
?
P[β + s] =
Υ0,p[α,β + s]
?
Ξ0,p[α,β]
Ξ0,p[α,β + s]P[β],(103)
32
Page 33
when β ∈ Sp−3
(+)
0,p
β,s
P
. We now make use of the symmetries of the relations satisfied by
P
K
?
?
α
?
proved in the proposition (31) of the appendix A.3.
˜P
K(−)
0,p
α
β,−˜ s
?
=
˜P
K(−)
0,p
?
˜ α
˜β,−˜ s
?
=
P
K(+)
0,p
?
α
β − s,s
?
= ψ0,p[α,β,s]
P
K(+)
0,p
?
α
β + s,−s
?
As a result the unitarity condition is equivalent to prove the following quasi-invariance
under imaginary integer shifts:
Υ0,p[α,β]
|Ξ0,p[α,β]|2P(β)=
Υ0,p[α,β + s]
|Ξ0,p[α,β + s]|2P(β + s)ψ0,p[α,β,s] (104)
which, due to the explicit expression of Υ0,p[α,β], reduces to
ψ1(α1,α2,β3+s3)
?p−2
j=3ψ1(βj+sj,αj,βj+1+sj+1)ψ1(αp−1,βp−1+sp−1,αp)
ψ1(α1,α2,β3)
?p−2
j=3ψ1(βj,αj,βj+1)ψ1(αp−1,βp−1,αp)
= ψ0,p[α,β,s] (105)
which is a trivial fact. 2
III.2.2 The moduli algebra of a genus-n surface
In this subsection, we will construct a unitary representation of the moduli algebra on a
surface of arbitrary genus n. The graph algebra Ln,0is isomorphic to H(Uq(sl(2,C)R))⊗n
and acts on Funcc(SL(2,C)R)⊗n.In order to find invariant vectors, we will apply
the technique developped in the p-punctures case: we will transfer this action on the
dual space (Funcc(SL(2,C)R)⊗n)∗and extract a subspace of invariant elements. We
use the notations of the end of appendix A.1.
?1
From this element we construct an element of (Funcc(SL(2,C)R)⊗n)∗⊗⊗1
defined as ι⊗n(⊗1
action of the lattice algebra on ι(
G (k)) is easily computed by dualization:
Let α ∈ Sn, we define an element
V), where
G (k) = pk(
k=n
αk
G (k) ∈ (Uq(sl(2,C)R)∗)⊗n⊗ ⊗1
is the inclusion of Uq(sl(2,C)R)∗in the k − th copy of (Uq(sl(2,C)R)∗)n.
k=nEnd(
αk
αk
αk
G) where pk
k=nEnd(
αk
V)
k=n
αk
G) =?1
k=nι(
αk
G (k)) where ι is recalled in the appendix A.1 The
αk
ι(
αk
G (k))?
I
L(±)(k) = ι(
αk
G (k))
Iαk
R(±)−1
,ι(
αk
G (k))?
I
L(±)(j) = ι(
αk
G (k)) ∀j ?= k,
αk
G (k)) ∀j ?= k,
αk
G (k))ι(
ι(
αk
G (k))?
I
˜L(±)(k) =
?
Iαk
R(±)ι(
αk
G (k)),ι(
αk
G (k))?
I
˜L(±)(j) = ι(
ι(
αk
G (k))?
I
G(k) =
α′
k∈S(αk,I)
ΦαkI
α′
kι(
α′
G (k))Ψα′
k
k
αkI
,ι(
αk
G (k))?
I
G(j) = ι(
I
G(j)) ∀j ?= k.
(106)
Before studying the general case, it is interesting to give a detailed exposition of the
genus one case which contains all the ideas without being too cumbersome.
33
Page 34
α + s
α + sα + s
α + s
α
I
I
J
W
WW
W
Figure 4: Expression of
P
K1,0.
Given α ∈ S, we can define the character ω1,0(α) ∈ (Funcc(SLq(2,C)))∗of this
representation by ([12]):
?j l
ω1,0(α) =
?
ABC
A B
C
n
??n
C
B A
m i
?
ΛBC
A(α)ι(
A
ki
j⊗
B
Em
l) .(107)
The algebra Linv
have:
1,0acts on the right of (Funcc(SLq(2,C)))∗with ρ1,0and we obviously
ω1,0(α)?
I
C1,0= ω1,0(α). (108)
Lemma 6 The element
IJW
O 1,0acts on the invariant element ω1,0(α) as follows:
IJW
O 1,0=
s∈S
ω1,0(α)?
?
ΛWJ
II[α,s]ω1,0(α + s) (109)
where ΛWJ
II[α,s] is given in the definition (11) .
Proof:Computing the action of
given by the graph (figure 4) which is equal to ΛWJ
IJW
O 1,0on ω1,0(α) amounts to evaluate the intertwiner
II[α,s]. 2
Proposition 14 We define a subset of (Funcc(SLq(2,C)))∗by:
H1,0= {ω1,0(f) =
?
SP
dαP(α)f(α)ω1,0(α), f ∈ A}. (110)
ρ1,0defines a right action of Linv
leave H1,0invariant. To the element
1,0on (Funcc(SLq(2,C)))∗which in general does not
IJW
O 1,0in Linv
O 1,0) if and only if α ?→ ΛWJ
for all s ∈ S. We therefore have:
IJW
O 1,0= ω1,0(f?
1,0we define its domain D(
IJW
O 1,0) ⊂ A
as: f belongs to D(
IJW
II[α,s]
P(α)
P(α+s)f(α) is an element of A
ω1,0(f)?
IJW
O 1,0) (111)
34
Page 35
with
(f?
IJW
O 1,0)(α) =
?
s∈S
f(α + s)ΛWJ
II[α + s,−s]P(α + s)
P(α)
. (112)
The map A → H1,0which sends f to ω1,0(f) is an injection. We can therefore endow
H1,0with the following pre-Hilbert structure
?
< ω1,0(f)|ω1,0(g) >=
SP
dαP(α)2f(α)g(α). (113)
The righ action ρ1,0acting on H1,0descends to the quotient Mq(Σ1,0,SL(2,C)R)) defines
a right action ˜ ρ1,0, which is unitary.
Proof:It is important to note that < ω1,0(α),φ > where φ ∈ Funcc(SLq(2,C)R) is a
Laurent polynomial in qiραbecause ΛBC
result ω1,0(f) is well defined for f ∈ A and the proof of the relation (112) follows from
Cauchy theorem.
The proof of the injection is a straighforward consequence of the proof of Theorem
12 of [12]. Indeed if we have ω1,0(f) = 0, it implies that?
weaker condition on f than f ∈ A the last relation implies f = 0.
To prove the unitarity, we compute
?
AA(α) is itself a Laurent polynomial in qiρα. As a
SPdαP(α)ΛBC
A(α)f(α) = 0,
for every A,B,C. The proof of Plancherel theorem 12 of [12] precisely shows that under
< ω1,0(f)|ω1,0(g)?
IJW
O 1,0>=
?
s∈S
SP
dαP(α)P(α + s)f(α)ΛWJ
II[α + s,−s]g(α + s).
On the other hand we have
< ω1,0(f)?
IJW
O
⋆
1,0|ω1,0(g) >=
?
?
s∈S
?
?
SP
dαP(α)P(α + s)f(α + s)Λ˜ W˜J
˜I˜I[α + s,−s]g(α)
=
s∈S
SP
dαP(α)P(α + s)f(α + s)Λ˜ W˜J
˜I˜I[α + s,−s]g(α).
Using Cauchy theorem to shift ρβand changing the index of the sum, the last expression
is equal to
?
?
s∈S
SP
dαP(α)P(α + s)f(α)Λ˜ W˜ J
˜I˜I[α,−˜ s]g(α + s).
As a result, proving the unitarity is equivalent to showing the relation:
Λ˜ W˜ J
˜I˜I[α,−˜ s] = ΛWJ
II[α + s,−s],
which is consequence of the equality: ΛBC
AD(α) = ΛBC
AD(α). 2
35
Page 36
The simplest non trivial observables are the Wilson loops
handles taken in the representation I ∈ SF. It is straightforward, from the previous
proposition, to compute the action of these observables:
I
WAand
I
WBaround the
(f?
I
WA)(α)=
sl=Il,sr=Ir
?
sl=−Il,sr=−Ir
vα+s
vα
f(α + s) (114)
(f?
I
WB)(α)=ϑIαf(α). (115)
From the representation of the Heisenberg double, WAacts by multiplication in the real
space and by finite difference in the Fourier space whereas WB acts by left and right
derivations in the real space and by multiplication in the Fourier space.
This closes the construction for the genus one. The generalization to the genus n is
similar in spirit although the technical details are much more involved. This is what we
will now develop. We will assume that n ≥ 2.
Let α,β,γ ∈ S and let A ∈ End(
(A ⊗ 1)Φαβ
?
where the sums are finite. A similar conclusion holds true for (1⊗B)Φαβ
+∞.
Definition 9 Let λ = (λ1,...,λn) element of Sn, and σ = (σ3,...,σn),τ = (τ3,...,τn)
elements of Sn−2and κ ∈ S. We define for 2 ≤ n
λ2λ1
R′
α
V) of finite dimensional corank, we can define
γ
∈ Hom(
γ
V,
α
V⊗
β
V) as follows
(A ⊗ 1)Φαβ
γ(v) =
IJij
<
Iei
∗⊗
Jej
∗,Φαβ
γ(v) > (A
Iei)⊗
Jej
(116)
γ with corank(B) <
℘2
=Ψσ3
λ2λ1ι(
∈ (Funcc(SLq(2,C)R)⊗n)∗⊗ Hom(
Ψσi+1
λiσiι(
G(i))
λ2
G(2))
λ2µ−1
λ2µ ι(
λ1
G(1))
λ2λ1
R
(−)Φλ2λ1
τ3
,
τ3
V,
σ3
V),
℘i
=
λi
λiµ−1
λiσi
R′
λiµ ℘i−1
λiσi
R(−)Φλiτi
τi+1,
σi+1
V ), ∀i = 3,...,n ,
∈ (Funcc(SLq(2,C)R)⊗n)∗⊗ Hom(
τi+1
V ,(117)
with the convention τn+1= σn+1= κ.
We can introduce ωn,0[κ,λ,σ,τ] ∈ (Funcc(SLq(2,C)R)⊗n)∗as being:
ωn,0(κ,λ,σ,τ) = trκ
?
℘n
?
.(118)
ωn,0[κ,λ,σ,τ] is well defined because < ωn,0[κ,λ,σ,τ],fn⊗ ... ⊗ f1>= trκ(< ℘n,fn⊗
...⊗f1>) and < ℘n,fn⊗...⊗f1> is, by construction, of finite rank and finite corank.
Proposition 15 ωn,0[κ,λ,σ,τ] are invariant vectors: ωn,0[κ,λ,σ,τ]?
I
Cn,0= ωn,0[κ,λ,σ,τ].
36
Page 37
Proof:Using
ι(
λi
G(i))?
I
C(±)
n,0
−1=
Iλi
R(±)−1ι(
λi
G(i))
Iµ
Iλi
R(±)
Iµ−1,
it is immediate to show, by recursion, that
℘i?
I
C(±)
n,0
−1=
Iσi+1
R
(±)−1℘i
Iµ
Iτi+1
R
(±)
Iµ−1
∀i = 2,...,n − 1.
and then
℘n?
I
C(±)
n,0
−1=
Iκ
R(±)−1℘n
Iµ
Iκ
R(±)
Iµ−1
which allows us to conclude. 2
We will denote for n ≥ 2,
ˆΞn,0[κ,λ,σ,τ]−1
=e2iπ(λ1+...+λn−κ)ν1(dκ)2ζ(λ1,λ2,τ3)
n
?
j=3
ζ(λj,τj,τj+1) ×
×
ζ(λ1,λ2,σ3)
n
?
j=3
ζ(λj,σj,σj+1)
n
?
j=3
ν1(dσj)ν1(dτj).(119)
We have the analogue of the lemma (4) :
Lemma 7 Let f ∈ Funcc(SLq(2,C))⊗n, for every x ∈ κ∪λ∪σ∪τ there exists Ix∈1
such that
ˆΞn,0[κ,λ,σ,τ]
x∈κ∪λ∪σ∪τ
2N,
?
(ν(Ix)(x))2< ωn,0[κ,λ,τ,σ],f >
is a Laurent polynomial in the variables (qiρx),x ∈ κ ∪ λ ∪ σ ∪ τ.
Proof:The proof is analogous to the related lemma of the p-punctures case. 2
We will denote Ξn,0[κ,λ,σ,τ] =ˆΞn,0[κ,λ,σ,τ]ξ[κ]ξ[λ]ξ[τ]ξ[σ].
Proposition 16 We can define a subset of (Funcc(SLq(2,C))⊗n)⋆by:
Hn,0= {ωn,0(f),f ∈ A(3n−3)}
(120)
where
ωn,0(f) =
?
S3n−3
P
dκdλdσdτωn,0[κ,λ,σ,τ]Ξn,0[κ,λ,σ,τ]P[κ,λ,σ,τ]f(κ,λ,σ,τ). (121)
The map A(3n−3)→ Hn,0which sends f to ωn,0(f) is an injection.
37
Page 38
Proof:The proof of the injection is similar to p punctures case. 2
We define for n ≥ 2,
Mn,0[κ,λ,σ,τ]=M(λ1,λ2,τ3)
n
?
n
?
j=3
M(λj,τj,τj+1) ×
×
M(λ1,λ2,σ3)
j=3
M(λj,σj,σj+1) (122)
with τn+1= σn+1= κ.
We define for (κ,λ,σ,τ) ∈ S3n−3
P
the function
Υn,0[κ,λ,σ,τ] =|Ξn,0[κ,λ,σ,τ]|2
Mn,0[κ,λ,σ,τ]
P(λ)
P(κ).
It can be checked that this function is analytic in ρx∈ R for x ∈ κ ∪ λ ∪ σ ∪ τ, and we
will still denote by Υn,0[κ,λ,σ,τ] the analytic continuation to S3n−3.
Proposition 17 The space Hn,0 is endowed with a structure of pre-Hilbert space as
follows:
?
< ωn,0(f)|ωn,0(g) >=
S3n−3
P
dκdλdσdτP(κ,λ,σ,τ)f[κ,λ,σ,τ]Υn,0[κ,λ,σ,τ]g[κ,λ,σ,τ].
(123)
Proof:Same proof as in the p-puncture case: use the injectivity of the map to show that
the scalar product is defined unambiguously and the fact that Υn,0is analytic. 2
Proposition 18 Let (κ,λ,σ,τ) ∈ S3n−3, the action of
given by:
P
O(±)
?
n,0∈ Linv
0,pon ωn,0[κ,λ,σ,τ] is
ωn,0[κ,λ,σ,τ]?
P
O(±)
n,0=
?
(k,ℓ,s,t)∈S3n−3
P
K(±)
n,0
?
κ;k
λ,σ,τ;ℓ,s,t
ωn,0[κ + k,λ + ℓ,σ + s,τ + t]
(124)
where:
1. the functions
P
K(±)
n,0
?
κ;k
λ,σ,τ;ℓ,s,t
?
are non zero only for a finite set of (k,ℓ,s,t) ∈
S3n−3according to the selection rules imposed by the palette P. Moreover any element
(k,ℓ,s,t) of this finite set satisfy (sl,tl),(sr,tr) ∈ Z2n−4.
2. the function
K(±)
n,0
λ,σ,τ;ℓ,s,t
P
?
κ;k
?
belongs toΞn,0[κ+k,λ+ℓ,σ+s,τ+t]
Ξn,0[κ,λ,σ,τ]
C(qiρx)x∈κ∪λ∪τ∪σ.
Proof:Using the expression of the observables and their action on (Funcc(SLq(2,C))⊗n)∗,
we have for f ∈ Funcc(SLq(2,C)⊗n),
n
?
(
i=1
ι(
λi
G(i))?
P
O(±)
n,0)(f) =
?
ℓ1,···,ℓn∈S
trλ1+ℓ1,···,λn+ℓn(
P
F(±)
n,0(λ,ℓ)
n
?
i=1
ι(
λi+ℓi
G (i))(f)),
(125)
38
Page 39
I1
I1
I2
I2
I3
I3
J1
J2
J3
K1
K2
K3
L1
L2
L3
U3
T3
WW
λ1+ ℓ1
λ1+ ℓ1
λ1
λ1
λ1
λ3
λ3
λ3
λ2
λ2
λ2
λ2+ ℓ2
λ2+ ℓ2
λ3+ ℓ3
λ3+ ℓ3
Figure 5: Expression of
P
F(+)
3,0.
where
P
F(±)
n,0(λ,ℓ) are elements of EndUq(sl(2,C)R)(?n
the observable contains multiplication by
after acting on ωn,0[κ,λ,σ,τ], to tensor representations of the principal series with finite
i=1
λi+ℓi
V
⊗
λi
V). Note that the sum
over ℓ does not exist in the p-puncture case. It now appears because the expression of
I
G(with I ∈ SF), which therefore corresponds,
dimensional representations. The picture for
P
F
(+)
n,0(λ,l) is shown in figure 5 (for the
case n = 3), whereas the picture for
crossing colored by couples of finite dimensional representations into the corresponding
undercrossing. The picture for arbitrary genus n is a straighforward generalization.
We will use the same method as in the proposition (12),i.e we first define and prove
everything in the case where (κ,λ,σ,τ) belongs to S3n−3
ation method to extend it to S3n−3
P
. We have
P
F(−)
n,0(λ,l) is the same after having turned over-
F
and then we use the continu-
(ωn,0[κ,λ,σ,τ]?
P
O(±)
n,0)(f) =
?
ℓ
P
F(±)
n,0
?
κ
λ,σ,τ;ℓ
?
(126)
where
P
F(±)
n,0
?
κ
λ,σ,τ;l
?
are elements of (Funcc(SLq(2,C))⊗n)∗and is a linear combina-
tion of the elements ωn,0[κ + k,λ + ℓ,σ + s,τ + t] as follows:
?
k,s,t
?
figure representing
F(±)
n,0(λ,ℓ) apply here.
Note that the selection rules in the finite dimensional case, implies that if ωn,0[κ,λ,σ,τ] ?=
0 then the possible non zero elements ωn,0[κ+k,λ+l,σ +s,τ +t] are those for s and t
satisfying sl,sr,tl,tr∈ Zn−2.
Using the property (22) of factorization of finite dimensional intertwiners, we obtain
P
F(±)
n,0
κ
λ,σ,τ;ℓ
?
?
P
=
?
P
K(±)
n,0
?
κ;k
λ,σ,τ;ℓ,s,t
?
ωn,0[κ + k,λ + ℓ,σ + s,τ + t] .(127)
P
K(+)
n,0
κ;k
λ,σ,τ;ℓ,s,t
is represented by the picture 6, and the same comments for the
39
Page 40
I1
I1
J1
I2
I2
I3
I3
J2
J3
K1
K2
K3
L1
L2
L3
U3
T3
WW
λ1
λ1
λ2
λ2
λ3
σ
λ3
λ1+ℓ1
λ1+ℓ1
λ2+ℓ2
λ2+ℓ2
λ3+ℓ3
λ3+ℓ3
κ
κ+k
κ+k
σ+s
σ+s
τ
Figure 6: Expression of
P
K(+)
3,0.
that
P
K(±)
n,0
?
κ;k
λ,σ,τ;ℓ,s,t
?
Pl
Kl(±)
n,0,
=
Pl
Kl(±)
n,0
?
κl;kl
λl,σl,τl;ℓl,sl,tl
?
Pr
Kr(±)
n,0
?
κr;kr
λr,σr,τr;ℓr,sr,tr
?
(128)
where the functions
Pr
Kr(±)
n,0
are computed in the proposition (32) and expressed
in terms of 6j symbols. The definition of the continuation of
uses these expressions where 6j(0) are replaced by 6j(1) and 6j(3) where needed. 2
a
K(±)
n,0
λ,σ,τ;ℓ,s,t
We will now endow Hn,0with a structure of right Linv
Proposition 19 Linv
invariant. The subspace Hn,0 is in general not invariant. We define the domain of
a ∈ Linv
D(a) if and only if for all (k,ℓ,s,t) ∈ S3n−3the functions
a
K(±)
n,0
λ,σ,τ;ℓ,s,t
Pl
Kl(±)
n,0and
Pr
Kr(±)
n,0
precisely
If a ∈ Linv
n,0we define
?
κ;k
?
by linearity as in the p-puncture case.
n,0module in the following sense:
n,0acts on (Funcc(SLq(2,C))⊗n)∗with ρn,0and leaves (Funcc(SLq(2,C))⊗n)∗)inv
n,0associated to ρn,0to be the subspace D(a) ⊂ A(3n−3)defined as: f belongs to
(κ,λ,σ,τ) ?→
?
κ;k
?
(Ξn,0P)[κ,λ,σ,τ]f(κ,λ,σ,τ)
(Ξn,0P)[κ + k,λ + ℓ,σ + s,τ + t]
are elements of A(3n−3). The action of a on an element ωn,0(f) ∈ D(a) ⊂ Hn,0belongs
to Hn,0, and we have:
ωn,0(f) ? a = ωn,0(f ? a) ,
with
(f ? a)(κ,λ,σ,τ) =
?
(k,ℓ,s,t)∈S3n−3
a
K(±)
n,0
?
κ −k;k
λ−ℓ,σ−s,τ−t;ℓ,s,t
(Ξn,0P)[κ−k,λ−ℓ,σ−s,τ−t]−1(Ξn,0P)[κ,λ,σ,τ]
?
f[κ−k,λ−ℓ,σ−s,τ−t]
.(129)
40
Page 41
Theorem 2 The representation ˜ ρn,0of Mq(Σ,SL(2,C)R) is unitary:
∀a ∈ Mq(Σ,SL(2,C)R),∀v ∈ D(a⋆),∀w ∈ D(a),< v ? a⋆|w >=< v|w ? a > .(130)
where < .|. > is the positive sesquilinear form defined by (123).
Proof:Using Cauchy theorem for the integration in ρxand reindexing the summation
on mxfor x ∈ λ ∪ σ ∪ τ ∪ κ the proof of unitarity of the representation reduces to the
identity:
?
˜P
K(∓)
n,0
λ,σ,τ;−˜ℓ,−˜ s,−˜t
when (κ,λ,σ,τ) ∈ S3n−3
In order to show this relation, we make use of the following identities which are
proved in the appendix:
?
[dλ]
[dλ+ℓ][dκ]
λ − ℓ,σ − s,τ − t;ℓ,s,t
[dλ]
[dλ+ℓ][dκ]
Υn,0(κ,λ,σ,τ)
P
K(±)
n,0
κ +k;−k
λ+ℓ,σ+s,τ+t;−ℓ,−s,−t
?
?(Ξn,0P)(κ+k,λ+ℓ,σ+s,τ+t)
?
Ξn,0(κ,λ,σ,τ)
=
Υn,0(κ+k,λ+ℓ,σ+s,τ+t)
κ;−˜k
(Ξn,0P)(κ,λ,σ,τ)
Ξn,0(κ+k,λ+ℓ,σ+s,τ+t).(131)
P
.
˜P
K(−)
n,0
κ;−˜k
λ,σ,τ;−˜ℓ,−˜ s,−˜t
[dκ+k]
K(+)
?
=
˜P
K(−)
n,0
?
˜ κ;−˜k
˜λ, ˜ σ, ˜ τ;−˜ℓ,−˜ s,−˜t
κ − k;k
?
=
=
P
n,0
?
?
=
= ψn,0(κ,λ,σ,τ;k,ℓ,s,t)
[dκ+k]
P
K(+)
n,0
?
κ + k;−k
λ + ℓ,σ + s,τ + t;−ℓ,−s,−t
?
.
As a result showing unitarity is reduced to showing the quasi-invariance under shifts:
Υn,0[κ,λ,σ,τ]
|Ξn,0[κ,λ,σ,τ]|2P(λ)2P(σ,τ)=
ψn,0(κ,λ,σ,τ,;k,ℓ,s,t)Υn,0[κ + k,λ + ℓ,σ + s,τ + t]
|Ξn,0[κ + k,λ + ℓ,σ + s,τ + t]|2P(λ + ℓ)2P(σ + s,τ + t).
(132)
This is a direct consequence of the fact that Υn,0is expressed in terms of the function
Θ which satisfies: Θ(α + s,β + s,γ) = Θ(α,β,γ) for s ∈ S. 2
III.3. The moduli algebra for the general case
This subsection generalizes the previous ones: we construct a unitary representation
of the moduli algebra on a punctured surface of arbitrary genus n. The graph alge-
bra Ln,pis isomorphic to H(Uq(sl(2,C)R))⊗n⊗ Uq(sl(2,C)R)⊗pand acts on Hn,p(α) =
Funcc(SL(2,C)R)⊗n⊗ V(α) where α = (α1,··· ,αp) ∈ Sp
assigned to the punctures.
Before expressing a theorem for the general case similar to theorems 1 and 2, it
is interesting to study the representation of the moduli algebra on the one punctured
torus.
Pdenotes the representations
41
Page 42
λ + ℓ
λ + ℓλ + ℓ
λ
α
λ
α
W
W
K
N
LN
I
I
J
L
Figure 7: Expression of
P
K(+)
1,1.
Given α,λ ∈ S, we can define
αω1,1(λ) ∈ (Funcc(SLq(2,C)R) ⊗ V(α))∗by:
αω1,1(λ),f ⊗ v >= trλ(<
where f ∈ Funcc(SLq(2,C)R) and v ∈ V(α). The algebra Linv
(Funcc(SLq(2,C)R) ⊗ V(α))∗with ρ1,1and we have:
<
λ
G,f > Ψλ
αλ(v ⊗ id))(133)
1,1acts on the right of
αω1,1(λ)?
I
C1,1
=
αω1,1(λ) (134)
αω1,1(λ) ? trq(
I
M)=ϑIα
αω1,1(λ) . (135)
An observable is given by a palette P = (I,J,N;K,L,W) ∈ S6
v1/2
K
v1/2
IJ
Fas follows:
P
O(±)
1,1=
v1/2
trW(
WµΨW
KN
(I,J)
θ
(K,L)
NL
R(±)
N
M
NL
R(±)−1ΦLN
W)
where
the action of this observable on
(I,J)
θ
(K,L) = ΨK
JI
J
B
JI
R′I
A
JI
R(−)ΦJI
αω1,1(λ) is given by:
?
L. After a direct computation, we can show that
αω1,1(λ)?
P
O(±)
1,1=
ℓ∈S
P
K(±)
1,1
?
α
λ;ℓ
?
αω1,1(λ + ℓ) (136)
where
P
K(+)
1,1
?
α
λ;ℓ
?
is given by the graph (figure 7). The expression for
P
K(−)
1,1
?
α
λ;ℓ
?
is given by the same graph after having turned overcrossing colored by couples of finite
dimensional representations into the corresponding undercrossing.
For α,λ ∈ S, we define
Ξ1,1[α,λ]−1
=
ˆΞ1,1[α,λ]ξ(λ)2
ˆΞ1,1[α,λ]=eiπλζ(λ,λ,α)ν1(dλ)
(137)
(138)
For α,λ ∈ SP, we define M1,1[α,λ] = M(α,λ,λ) where M is defined in the appendix
A.2 and we denote
Υ1,1[α,λ] =|Ξ1,1(α,λ)|2
M1,1[α,λ].(139)
42
Page 43
We can show, as in the lemma 5, that Υ1,1[α,λ] is an analytic function of the real
variable ρλand therefore admits an analytic continuation for λ ∈ S.
Proposition 20 We define a subset of (Funcc(SLq(2,C)R) ⊗ V(α))∗by:
?
H1,1(α) = {
αω1,1(f) =
SF
dλ P(λ)Ξ1,1[α,λ]f(λ)
αω1,1(λ), f ∈ A}. (140)
For the elements f belonging to the domain of
P
O(±)
1,1we have :
αω1,1(f)?
P
O(±)
1,1=
αω1,1(f?
P
O(±)
1,1) (141)
where
(f?
P
O(±)
1,1)(λ) =
?
ℓ∈S
P
K(±)
1,1
?
α
λ + ℓ;−ℓ
?P(λ + ℓ)Ξ1,1[α,λ + ℓ]
P(λ)Ξ1,1[α,λ]
f(λ + ℓ) (142)
The map A −→ H1,1(α) is an injection. We can therefore endow H1,1(α) with the
following pre-Hilbert structure
?
˜ ρ1,1is an antirepresentation of Mq(Σ1,SL(2,C)R;α) which is unitary.
<
αω1,1(f)|
αω1,1(g) >=
SP
dλP(λ)f(λ)Υ1,1[α,λ]g(λ) .(143)
Proof:It is easy to show that
action of ˜ ρ1,1follows from Cauchy theorem. Using the usual method of proof, unitarity
is equivalent to the identity:
?
˜P
K(∓)
1,1
λ;−˜ℓ
when (α,λ) ∈ S2
In order to show this relation, we make use of the following identities which are
proved in the appendix:
?
αω1,1 (f) is well defined and the expression (142) of the
Υ1,1[α,λ]
P
K(±)
1,1
α
λ + ℓ;−ℓ
?Ξ1,1[α,λ + ℓ]
α
Ξ1,1[α,λ]
?
P(λ + ℓ) =
Υ1,1[α,λ + ℓ]
?
Ξ1,1[α,λ]
Ξ1,1[α,λ + ℓ]P(λ)(144)
P.
˜P
K(∓)
1,1
α
λ;−˜ℓ
?
=
P
K(±)
1,1
?
α
λ − ℓ;ℓ
?
(145)
=ψ1,1(α,λ;ℓ)
P
K(±)
1,1
?
α
λ + ℓ;−ℓ
?
.(146)
As a result, showing unitarity reduces to the relation:
Υ1,1[α,λ]
|Ξ1,1[α,λ]|2P(λ)= ψ1,1(α,λ;ℓ)
Υ1,1[α,λ + ℓ]
|Ξ1,1[α,λ + ℓ]|2P(λ + ℓ)
(147)
43
Page 44
which holds. 2
This proposition closes the construction for the torus with one puncture. All the tools
are now ready to construct a right unitary module of the moduli algebra Mq(Σn,SL(2,C)R;α)
.For this reason, we will just describe the representation of the moduli algebra without
giving the technical details.
Proposition 21 Let α = (α1,··· ,αp) ∈ Sp, λ = (λ1,··· ,λn) ∈ Sn, β = (β3,··· ,βp) ∈
Sp−2, σ = (σ3,··· ,σn+1),τ = (τ3,··· ,τn+1) elements of Sn−1and δ ∈ S. We define
αωn,p(β,λ,σ,τ,δ) ∈ (Funcc(SLq(2,C)R)⊗n⊗ V(α))∗by
αωn,p(β,λ,σ,τ,δ),φ ⊗ v >= trσn+1
<
?
< ℘n(λ,σ,τ),φ > Ψτn+1
δ σn+1
?
Ψδ
α(β)
?
(v)
?
(148)
where φ ∈ Funcc(SLq(2,C)R)⊗nand v ∈ V(α).
Ψδ
α(β) ∈ HomUq(sl(2,C)R)(V(α),
(Funcc(SLq(2,C)R)⊗n)∗⊗ Hom(
αωn,p(β,λ,σ,τ,δ) are invariant vectors:
δ
V) has been introduced in the definition 6 and ℘n(λ,σ,τ) ∈
τn+1
V ,
V ) has been introduced in the definition 9.
σn+1
αωn,p(β,λ,σ,τ,δ)?
I
Cn,p=
αωn,p(β,λ,σ,τ,δ) , (149)
and they satisfy the constraints:
αωn,p(β,λ,σ,τ,δ) ? trq(
I
M(n + i)) = ϑIαi
αωn,p(β,λ,σ,τ,δ) , ∀i = 1,...,p .(150)
Proof:The proof of the invariance is a direct consequence of the factorization
I
C(±)
n,p =
C(±)
n,0
M(n + i)) is obtained immediately. 2
I
I
C(±)
0,p. The action of trq(
I
We will define:
ˆΞn,p[α,β,λ,σ,τ,δ]−1=ˆΞn[λ,σ,τ]−1ˆΞc[σn+1,τn+1,δ]−1ˆΞp[α,β,δ]−1,
where
ˆΞn[λ,σ,τ]−1
=eiπ(2λ1+...+2λn−σn+1−τn+1)ζ(λ1,λ2,σ3)
n
?
j=3
ν1(dσj)ζ(λj,σj,σj+1)
×ζ(λ1,λ2,τ3)
n
?
j=3
ν1(dτj)ζ(λj,τj,τj+1)
ˆΞp[α,β,δ]−1
=eiπ(α1+...+αn)ζ(α1,α2,β3)
p−1
?
j=3
ν1(dβj)ζ(αj,βj,βj+1)
×ν1(dδ)ζ(αp,βp,δ)
eiπ(σn+1+δ−τn+1)ν1(dτn+1)ζ(σn+1,τn+1,δ) .
ˆΞc[σn+1,τn+1,δ]−1
=
44
Page 45
From these functions, we also define
Ξn,p[α,β,λ,σ,τ,δ] =ˆΞn,p[α,β,λ,σ,τ,δ]ξ(β)ξ(λ)ξ(σ)ξ(τ)ξ(δ) .
We will also denote Mn,p[α,β,λ,σ,τ,δ] = Mn[λ,σ,τ]Mc[σn+1,τn+1,δ]Mp[α,β,δ]
with:
Mn[λ,σ,τ]=M(λ1,λ2,σ3)M(λ1,λ2,τ3)
n
?
n
?
j=3
(M(λi,σi,σi+1)M(λi,τi,τi+1))
Mp[α,β,δ]=M(α1,α2,β3)M(αp,βp,δ)
j=3
M(αi,βi,βi+1)
Mc[σn+1,τn+1,δ]=M(σn+1,τn+1,δ).
Finally, for α = α1,··· ,αp ∈ Sp
Sp−2
P
, σ = (σ3,··· ,σn+1),τ = (τ3,··· ,τn+1) elements of Sn−1
Υn,p[α,β,λ,σ,τ,δ] =|Ξn,p[α,β,λ,σ,τ,δ]|2
P, λ = (λ1,··· ,λn) ∈ Sn
P, β = (β3,··· ,βp) ∈
and δ ∈ SP, we denote:
P(λ)
P(τn+1)
P
Mn,p[α,β,λ,σ,τ,δ]
. (151)
We can show that Υn,p[α,β,λ,σ,τ,δ] is analytic in ρxfor x ∈ β∪λ∪σ∪τ∪δ and we will
still denote by Υn,p[α,β,λ,σ,τ,δ] the analytic continuation to α,β,λ,σ,τ,δ ∈ S3n+p−3.
Theorem 3 We define a subset of (Funcc(SLq(2,C)R)⊗n)∗⊗ V(α)∗by:
?
f(β,λ,σ,τ,δ)
Hn,p(α) = {
αωn,p(f) =
S3n+p−3
P
dβdλdσdτdδ P(β,λ,σ,τ,δ)Ξn,p[α,β,λ,σ,τ,δ]
αωn,p(β,λ,σ,τ,δ),f ∈ A}
αωn,p(f) is an injection.
(152)
The map A3n+p−3−→ Hn,p(α) which sends f to
The algebra Linv
descends to a right action ˜ ρn,pon Hn,pby:
n,pacts on the right of (Funcc(SLq(2,C)R)⊗n)∗⊗V(α)∗with ρn,pwhich
αωn,p(f)?
P
On,p=
αωn,p(f?
P
On,p) .(153)
The subspace Hn,p(α) is in general not invariant. We define the domain of a ∈ Linv
be the subspace D(a).
We can endow Hn,p(α) with a pre-Hilbert structure as follows:
?
n,pto
<
αωn,p(f)|
αωn,p(f) >=
S3n+p−3
P
dβdλdσdτdδ P(β,λ,σ,τ,δ)Υn,p[α,β,λ,σ,τ,δ]
f(β,λ,σ,τ,δ)g(β,λ,σ,τ,δ) . (154)
The representation ˜ ρn,pof Mq(Σ,SL(2,C)R) is unitary:
∀a ∈ Mq(Σ,SL(2,C)R),∀v ∈ D(a⋆),∀w ∈ D(a),< v ? a⋆|w >=< v|w ? a > . (155)
45
Page 46
Proof:We perform the proof similarly to the proof of the theorems 1 and 2.
First, using usual continuation arguments, we compute the action of
[β,λ,σ,τ,δ] when α = α1,··· ,αp ∈ Sp, λ = (λ1,··· ,λn) ∈ Sn, β = (β3,··· ,βp) ∈
Sp−2, σ = (σ3,··· ,σn+1),τ = (τ3,··· ,τn+1) elements of Sn−1and δ ∈ S. We show that:
?
αωn,p[β + b,λ + ℓ,σ + s,τ + t,δ + d]
?
are non zero only for a set of b,ℓ,s,t,d ∈ S according to the selection rules imposed by
the palette P.
a
K(±)
n,p
β,λ,σ,τ,δ;b,ℓ,s,t,d
cases. The subspace Hn,p(α) is in general not left invariant by the action of Linv
we define the domain D(a) of a such that: f belongs to D(a) if and only if for all
b,ℓ,s,t,d ∈ S the functions
?
Ξn,p[α,β,λ,σ,τ,δ]P[β,λ,σ,τ,δ]
Ξn,p[α,β + b,λ + ℓ,σ + s,τ + t],δ + dP[β + b,λ + ℓ,σ + s,τ + t,δ + d]
are elements of A(3n+p−3). The action ρn,pof Linv
descends to an action ˜ ρn,pon Hn,p[α] defined by:
P
O(±)
n,p on
αωn,p
αωn,p[β,λ,σ,τ,δ]?
P
O(±)
n,p=
b,ℓ,s,t,d∈S
P
K(±)
n,p
?
α
β,λ,σ,τ,δ;b,ℓ,s,t,d
?
(156)
The functions
P
K(±)
n,p
α
β,λ,σ,τ,δ;b,ℓ,s,t,d
?
, defined by the graph in picture (fig 22),
If a ∈ Linv
n,p, we define
?
α
?
by linearity as in the previous
n,p. So,
(α,β,λ,σ,τ,δ) ?→
a
K(±)
n,p
α
β,λ,σ,τ,δ;b,ℓ,s,t,d
?
f(β,λ,σ,τ,δ)
(157)
n,pon (Funcc(SLq(2,C)R)⊗n⊗ V(α))∗
αωn,p(f) ? a =
αωn,p(f ? a) , (158)
with
(f ? a)(β,λ,σ,τ,δ) =
?
b,ℓ,s,t,d∈S
a
K(±)
n,p
?
α
β − b,λ − ℓ,σ − s,τ − t,δ − d;b,ℓ,s,t,d
?
Ξn,p[α,β − b,λ − ℓ,σ − s,τ − t,δ − d]P[β − b,λ − ℓ,σ − s,τ − t,δ − d]
Ξn,p[α,β,λ,σ,τ,δ]P[β,λ,σ,τ,δ]
f(β − b,λ − ℓ,σ − s,τ − t,δ − d).(159)
46
Page 47
Finally, using Cauchy theorem, unitarity reduces to the identity:
?
Ξn,p[α,β + b,λ + ℓ,σ + s,τ + t,δ + d]P[β + b,λ + ℓ,σ + s,τ + t,δ + d]
Ξn,p[α,β,λ,σ,τ,δ]
Υn,p[α,β,λ,σ,τ,δ]
P
K(±)
n,p
α
β + b,λ + ℓ,σ + s,τ + t,δ + d;−b,−ℓ,−s,−t,−d
?
=
Υn,p[α,β + b,λ + ℓ,σ + s,τ + t,δ + d]
P
˜K(∓)
n,p
?
α
β,λ,σ,τ,δ;−˜b,−˜ℓ,−˜ s,−˜t,−˜d
?
Ξn,p[α,β,λ,σ,τ,δ]P[β,λ,σ,τ,δ]
Ξn,p[α,β + b,λ + ℓ,σ + s,τ + t,δ + d]
(160)
where α,β,λ,σ,τ ∈ SP. In order to show this relation, we make use of the identities
related in the proposition 35 given in the appendix and the proof of unitarity reduces
to the relation
Υn,p[α,β,λ,σ,τ,δ]
|Ξn,p[α,β,λ,σ,τ,δ]|2P[β,λ,σ,τ]
P(τn+1)
P(λ)
ψ−1
n,p(α,β,λ,σ,τ,δ;b,ℓ,s,t,d) =
Υn,p[α,β + b,λ + ℓ,σ + s,τ + t,δ + d]
|Ξn,p[α,β + b,λ + ℓ,σ + s,τ + t,δ + d]|2P[β + b,λ + ℓ,σ + s,τ + t,δ + d]
which holds immediately. 2
P(τn+1+ tn+1)
P(λ + ℓ)
IV. Discussion and Conclusion
The major result of our work is the proof that there exists a unitary representation of
the quantization of the moduli space of the flat SL(2,C)Rconnections on a punctured
surface. This is a non trivial result which necessitates to integrate the formalism of
combinatorial quantization and harmonic analysis on SLq(2,C)R. However we have not
studied in details properties of this unitary representation. It would indeed be interesting
to analyze the domains of definition of the operators ρn,p(O) and to study the possible
extensions of ρn,p(O). Related mathematical questions, which are answered positively
in the compact group case, are the following:
• is the representation ρn,pirreducible?
• in this case, is it the only irreducible representation up to equivalence?
• Does this representation provides a unitary representation of the mapping class
group?
What remains also to be done is to relate precisely the quantization of SL(2,C)R
Chern-Simons theory to quantum gravity in de Sitter space.
Discarding the problem of degenerate metrics, the two theories are classically equiv-
alent. As pointed out in [1, 28], three dimensional lorentzian gravity written in the
first order formalism is a gauge theory, specifically a Chern-Simons theory associated
47
Page 48
to the Lorentz group when the cosmological constant Λ is positive. This equivalence
was extensively studied (see for example [15] and the references therein) and it is pos-
sible to relate the observables of Chern-Simons theory with the geometric parameters
associated to the metric solution of Einstein equations. As usual we denote by lP the
Planck length, lP = ?G and the cosmological constant Λ is related to the cosmologi-
cal length l by Λ = l−2. These two length scales define a dimensionless constant lP/l.
The semiclassical regime is obtained when ? approaches zero, and the relation between
Chern-Simons SL(2,C)Rand gravity, imposes q = 1−lP/l+o(lP/l). It is a central ques-
tion to control the other terms of the expansion and the non perturbative corrections.
This issue is not adressed here but could possibly be done by comparing two expectation
value of observables in SL(2,C)R-Chern-Simons theory and in quantum gravity in de
Sitter space. The construction of a unitary representation of the observables, provided
in our work, is a step in this direction.
In the rest of this discussion we will provide a relation between the mass and the spin
of a particle and the parameters (m,ρ) of the insertion of a principal representation.
Let us first give the metric of de Sitter space associated to massive and spinning
particules. In a neighbourhood of a massive spinning particle of mass mpand spin j ,
the metrics takes the form of Kerr-de Sitter solution ([9, 23]):
ds2
=
−(8lPM −r2
+(8lPM −r2
−(r2+ r2
l2+(8lPj)2
l2+(8lPj)2
−)(r2
r2
4r2
)dt2
4r2
)−1dr2+ r2(−8lPj
2r2dt + dφ)2
+− r2)dr2+ r2(dφ +r+r−
=
+− r2)
dt2+
r2
(r2+ r2
−)(r2
r2dt)2
where we have defined
r+= 2l
?
lPM + lP
?
M2+j2
l2
,r−= 2l
?
−lPM + lP
?
M2+j2
l2,
where 8lPM = 1 − 8lPmp.
This metric has a cosmological event horizon located in r = r+. Following [23], this
metric can be conveniently written as
ds2= sinh2R
?r+dt
l
− r−dφ
?2
− l2dR2+ cosh2R
?r−dt
l
+ r+dφ
?2
with r2= r2
is located at R = 0; the exterior of the horizon is described for real R and the interior
for imaginary value of R. To this metric we can associate an orthonormal cotriad ea
with gµν = ea
connection
+cosh2R + r2
−sinh2R. In this coordinate system, the cosmological horizon
µ
µeb
νηaband its spin connection ωµa
b. These data define a flat SL(2,C)R
Aµ= ωa
µJa+1
lea
µPa
48
Page 49
where Ja,Paare the generators of so(3,1) and ωa
connection is given by Park ([23]) in the spinorial representation and a trivial compu-
tation shows that:
?
µ=1
2ǫabcωµab. The explicit value of the
Wcl(A) = 2cosh2πr−− ir+
l
?
(161)
where Wcl(A) is the classical holonomy of the connection along a circle centered around
the world-line of the particle.
In the quantization of Chern-Simons theory that we provided, a puncture is colored
with a principal unitary representation α = (m,ρ). The monodromy
I
Mq(A) around a
I
Mq(A) = ϑIα. puncture belongs to the center of the moduli algebra and it is defined by
In order to compare it to the classical case, one can trivially evaluate the associated
holonomy in the spinorial representation I = (1/2,0):
Wq(A) = qm+iρ+ q−(m+iρ). (162)
At the semi classical level, the comparison between (161, 162), implies the relations:
∓ 2πr+= ρlP
,
±2r−= mlP.(163)
which is equivalent to
32π2lPM =l2
P
l2(ρ2− m2) ,16π2j =
1
π2
l2
p
l2ρm .(164)
From these relations we note that r− is quantized in units of lP whereas r+ has
a continuous spectrum. It is much more delicate to understand what is the physical
meaning of holonomies around several punctures. In particular, the energy of such a
system in de Sitter space is not, a priori, well defined in the absence of boundaries,
even at the classical level. It should be a good chalenge to understand the classical and
quantum behaviour of particules in dS space in the light of Chern-Simons theory on a
p-punctured sphere. We will give an analysis of this problem in a future work.
A comparison, similar to the analysis given above, between the classical geometry
and the quantization in the Chern-Simons approach has been given in [17] for the genus
one case. The generalization to the genus n case is open. Of particular interest is the
construction of coherent states, lying in Hn,pand approaching a classical 3 metric. This
subject is up to now still in its infancy.
49
Page 50
Appendix A: Quantum Lorentz group
A.1 Representations and harmonic analysis
In this work we have chosen q ∈ R,0 < q < 1.
For x ∈ C, we denote [x]q
exp(iπ
2
.
The square root of a complex number is defined by:
=[x]=
qx−q−x
q−q−1,dx
=2x + 1, and v1/4
x
=
2x)q−x(x+1)
∀x ∈ C,√x =
?
|x|eiArg(x)
2
,where x = |x|eiArg(x),Arg(x) ∈] − π,π], (165)
For all complex number z with non zero imaginary part, we define ǫ(z) = sign (Im(z)).
We will define the following basic functions: ∀z ∈ C,∀n ∈ Z,
+∞
?
+∞
?
Let us define the function ξ,θ :
(z)∞= (q2z,q2)∞=
k=0
(1 − q2z+2k) ,(z)n=
(z)∞
(z + n)∞,
ν∞(z) =
k=0
?
1 − q2z+2k,νn(z) =
ν∞(z)
ν∞(z + n).
ξ(z)=(z)∞(1 − z)∞,ξ(z) = ξ(z + iπ
ξ(z)qz2−zeiπz/(1)2
lnq) (166)
θ(z)=
∞,θ(z + 1) = θ(z).(167)
With our choice of square root, we have νn(z) = νn(z). It is also convenient to introduce
the following function ϕ : C × Z −→ {1,i,−i,−1}, defined by
ϕ(z,n) = νn(z − n + 1)ν−n(n − z)q−nz+1
2n(n−1). (168)
We will recall in this appendix fondamental results on Uq(sl(2,C)R). We will give a
summary of the harmonic analysis on SLq(2,C)R, for a complete treatment see [12, 13].
Uq(su(2)), for q ∈]0,1[, is defined as being the star Hopf algebra generated by the
elements J±, q±Jz, and the relations:
q±Jzq∓Jz= 1 ,qJzJ±q−Jz= q±1J±,[J+, J−] =
q2Jz− q−2Jz
q − q−1
.(169)
The coproduct is defined by
∆(q±Jz) = q±Jz⊗ q±Jz,∆J± = q−Jz⊗ J± + J± ⊗ qJz,(170)
and the star structure is given by:
(qJz)⋆= qJz,J⋆
±= q∓1J∓.(171)
50
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