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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
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