New modular representations and fusion algebras from quantized SL(2,R) Chern-Simons theory

17 Pages·0.17 MB·English

by  C. Imbimbo

GEF-TH 2/1993
NEW MODULAR REPRESENTATIONS AND
FUSION ALGEBRAS FROM QUANTIZED
SL(2,R) CHERN-SIMONS THEORIES
CAMILLO IMBIMBO
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9 I.N.F.N., Sezione di Genova
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Via Dodecaneso 33, I-16146 Genova, Italy

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ABSTRACT
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We consider the quantum-mechanical algebra of observables generated by canonical
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quantization of SL(2,R) Chern-Simons theory with rational charge on a space manifold
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3 with torus topology. We produce modular representations generalizing the representations
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/ associated to the SU(2) WZW models and we exhibit the explicit polynomial representa-
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t
- tions of the corresponding fusion algebras. The relation to Kac-Wakimoto characters of
p
e highest weight sl(2) representations with rational level is illustrated.
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GEF-TH 2/1993
January 1993
1. INTRODUCTION
Three-dimensional topological Chern-Simons gauge theory with gauge group G is
described by the following action [1]:
k 2
S = η AadAb + f AaAbAc (1)
ab abc
2π Z 3
M3
where f are the structure constants of the Lie algebra of G, η is the invariant Killing
abc ab
metric on it, and Aa are the gauge field 1-forms.
Chern-Simons theory with G compact was solved non-pertubatively by means of
“holomorphic” canonical quantization methods which uncovered its relation to the two-
dimensional Wess-Zumino-Witten model on the group manifold G [2]-[4]. In this talk we
will discuss the much less well understood Chern-Simons theory with non-compact gauge
group G = SL(2,R)[5]-[8]. Since the meaning of the SL(2,R)WZW model as a conformal
quantum field theory is far from clear [9], the relation of SL(2,R) Chern-Simons theory
to two-dimensional conformal field theory, if it exists, must be of a novel type. In what
follows we will describe various features (such as modular properties and fusion algebras)
of the yet unknown two-dimensional counterpart of SL(2,R) Chern-Simons theory.
Let us first briefly review the main aspects of the holomorphic canonical quantization
of the action (1) to point out why it does not extend to theories with non-compact gauge
groups. In the Hamiltonian formalism, the three-dimensional space-time manifold M is
3
the product Σ R1 of a two-dimensional compact surface Σ and of the time axisR1. Going
×
to the A = 0 gauge, one obtains a free gauge-fixed action
0
k
S = dt ǫijη AaA˙bd2x (2)
2π Z Z ab i j
Σ
where ǫij is the anti-symmetric tensor on the two-dimensional space manifold Σ. The
constraint
ǫijFa = 0 (3)
ij
associated to the gauge-fixing encodes the non-linearity of the theory. The “Gauss law”
(3) states that the classical physical phase space is the space of flat G-connections on
M
the two-dimensional surface Σ.
In the holomorphic quantization, one selects a complex structure on Σ which deter-
mines a complex structure on the space of G-connections on Σ. States, in the “quantize-
first” approach, are then described by holomorphic wave-functionals Ψ(Aa) which depend
z
2
only on the holomorphic components of the gauge field 1-forms Aa = Aadz + Aadz¯ and
z z¯
which are normalizable in the scalar product
< Ψ1,Ψ2 >= [DAzDAz¯]e−2kπ ΣηabAazAbz¯Ψ¯1(Az¯)Ψ2(Az). (4)
Z R
Physical states Ψ(Aa) are normalizable solutions of the functional equation which is
z
the quantum version of the “Gauss law” constraint (3):
δ k
Dab Ψ = ∂¯AaΨ. (5)
z δAb 2π z
z
Eqs.(5) are identical to the Ward Identities for the generating functional of current
correlators of the two-dimensional WZW model on the Riemann surface Σ. This shows
that for G compact, the vector space of physical states of the Chern-Simons theory (1) is
isomorphic to the space of current blocks of the two-dimensional Gˆ current algebra. The
central task of canonical quantization of Chern-Simons theory is to prove that the unitary
structure (4) on the vector space of quantum states of the Chern-Simons theory is in fact
the same as the natural unitary structure on the current blocks of the WZW theory for
which modular transformations are represented by unitary matrices. For G compact, this
has been proven explicitly for Σ of genus zero and one [4],[10].
When G is non-compact, η is not positive-definite, and the scalar product (4) is
ab
not, even formally, well defined. In this case, the holomorphic gauge-invariant polarization
Ψ(Aa) does not define a genuine positive-definite K¨ahler structure on the space of G
z
connections on Σ. If G is a non-compact but complex group, one can find a family of
real gauge-invariant polarizations which is preserved by the reparametrizations of Σ [11].
The existence of a reparametrization invariant family of gauge-invariant polarizations is
the essential condition that makes it possible to discuss topological invariance at quantum
level.
When G = SL(2,R) a reparametrization invariant family of gauge-invariant (positive-
definite) polarizations does not exist [11]. This is the main difficulty in quantizing the
theory in a topological invariant way, or, equivalently, in proving that the mapping class
group is implemented unitarily on the Hilbert space of quantum states. The difficulty is
analogous to the one that is met when quantizing the Heisenberg algebra [x ,p ] = iη
µ ν µν
with a Lorentzian type of metric η . Choosing a real polarization, one obtains states
µν
represented by wave-functions ψ(x ), and p = i∂ . This gives a perfectly unitary (but
µ µ µ
−
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not highest-weight) representation of the Heisenberg algebra. This representation is, how-
ever, not equivalent to the (heighest-weight but not positive-definite) Fock representation
defined by the creation and annihilation operators a† = x +ip ,a = x ip . The point
µ µ µ µ µ− µ
is that though dependence on the polarization is a ‘fact of life’ of the quantization process,
it potentially jeopardizes quantum topological invariance of the Chern-Simons theory.
Becauseofthisfundamental difficultywhichaffectsanyattemptto“first-quantize” the
full spaceoftwo-dimensional SL(2,R)connections onΣ andtoimposethe“Gauss-law” (3)
as an operatorial constraint on the physical states, we will work in the so-called “constrain-
first” approach [2],[10] in which one quantizes directly the physical classical phase space
. Since is finite-dimensional, the canonical quantization problem actually has a
M M
finite number of degrees of freedom. However, the topology of is, for a generic Σ,
M
quite intricate. For this reason, we will restrict ourselves to the case when Σ has the torus
topology; such a limitation has been sufficient to unravel the underlying two-dimensional
current algebra structure in the case of the compact gauge group SU(2) [10].
When Σ is a torus, the problem of quantizing is reduced to the problem of quan-
M
tizing the moduli space of flat-connections of an abelian gauge group [4]. This makes the
computation for genus one drastically simpler than for higher genus, where non-abelian
Chern-Simons theory appears to be vastly more complex than abelian. On the other hand,
the factorization properties of 2-dimensional conformal field theories suggest that the torus
topology already contains most, if not all, of the complexities of higher genus. The solution
of this apparent paradox is that for a torus is almost the space of flat connections of an
M
abelian group, but not quite: it is the space of abelian flat connections modulo the action
of a discrete group whose fixed points give rise to orbifold singularities. It is only here
that the quantization of non-abelian Chern-Simons theory with compact gauge group for
genus one differs from the computationally trivial abelian case. The projection associated
to such a discrete group is responsible for the emergence of a non-abelian structure for the
SL(2,R) Chern-Simons theory as well.
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2. QUANTIZATION OF
M
The difficulties of the “quantize-first” approach for SL(2,R) gauge group have their
counterpart in the “constrain-first” method in the singular geometry of . Corresponding
M
to the three types of inequivalent Cartant subgroups of SL(2,R),there exist three different
“branches” of which have a non-vanishing intersection: = , with i =
M M i=1,2,3Mi
S
1,2,3. It is interesting to notice that similarly “branched” phase spaces appear also in
the context of two-dimensional topological theories based on the gauged SL(2,R) WZW
model and related to solvable string theories [12].
Flat SL(2,R) connections on a torus correspond to pairs (g ,g ) of commuting
1 2
SL(2,R) elements modulo overall conjugation in SL(2,R). The elements g and g repre-
1 2
sent the holonomies of the flat connections around the two non-trivial cycles of the torus.
The branch is made of flat connections whose holonomies can be simultaneously
1
M
brought by conjugation into the compact U(1) subgroup of SL(2,R). Therefore,
1
M ≈
T(1), the two dimensional torus.
is the branch of flat connections whose holonomies, when represented by 2 2 real
2
M ×
exi 0
unimodular matrices, can be conjugated into a diagonal form: i.e. g = ± ,
i (cid:18) 0 e−xi (cid:19)
±
with i = 1,2. However, one can still conjugate diagonal holonomies by an element of
the gauge group which permutes the eigenvalues, mapping x onto x . Therefore,
i i 2
− M
consists of four copies of R(2)/Z whose originsare attached to the four pointsof which
2 1
M
correspond to flat connections with holonomies in the center of the gauge group SL(2,R).
Finally, isthebranch offlat connections withholonomies which canbe conjugated
3
M
into anupper triangularformwithunits onthe diagonal. Conjugationallowsone torescale
the (non-vanishing) elements in the upper right corner by an arbitrary positive number.
Thus, S1, the real circle. Being odd-dimensional, S1 cannot be a genuine non-
3
M ≈
degenerate symplectic space. In fact, when pushed down to , the symplectic form on
3
M
the space of flat connections coming from the Chern-Simons action vanishes identically.
represents a “null” direction for the symplectic form of the SL(2,R) Chern-Simons
3
M
theory,reflectingtheindefinitenessoftheSL(2,R)Killingform. Since isadisconnected
3
M
piece of the total phase space , it is consistent to consider the problem of quantizing
M
independently of .
1 2 3
M ∪M M
There areno “rigorous”waystoquantizea phasespace consisting ofdifferent branches
with a non-zero intersection. The strategy adopted both in [8] and [12] is to consider the
smooth, non-compact manifold / obtained by deleting the intersection
1 2
M N N ≡ M ∩M
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of the two branches of . / consists of disconnected smooth components / and
1
M M N M N
/ , which, upon quantization, give rise to Hilbert spaces of wave functions
M2 N HM1/N
and . It seems reasonable to think of a wave function on the union
HM2/N M1 ∪ M2
as a pair (ψ ,ψ ) of wave functions, with ψ and ψ , “agreeing” in
1 2 1 ∈ HM1/N 2 ∈ HM2/N
some sense on the intersection . The proposal of [8] is that ψ and ψ , when represented
1 2
N
by holomorphic functions, should have the same behaviour around the points in . This
N
implies that the pair (ψ ,ψ ) should be determined uniquely by ψ and that most of the
1 2 1
states ψ in the infinite-dimensional should be discarded. The conclusion of the
2 HM2/N
analysis in [8] is that the quantization of produces a Hilbert space which is a
CS
M H
subspace of with definite parity under the conjugation operator C,
HM1/N
C : (g ,g ) (g−1,g−1). (6)
1 2 → 1 2
The space coming from the quantization of / is the representation space
HM1/N M1 N
of the ’t Hooft algebra [13]:
AB = µBA, (7)
whereµisaphaserelatedtothecouplingconstantappearingintheSL(2,R)Chern-Simons
action (1) through the equation:
iπ
µ = e k . (8)
The quantum operators A and B are the quantum versions of the classical holonomies g
1
and g . Unitary, irreducible representations of (7)are finite-dimensional when k is rational:
2
2k = 2s/r = p/q, (9)
with s,r and p,q coprime integers.
SinceweareinterestedininvestigatingtheconnectionbetweenSL(2,R)Chern-Simons
theory and two-dimensional rational conformal field theories, we will restrict ourselves to
k rational as in Eq.(9), and we will denote the corresponding ’t Hooft algebra by .
p/q
O
Modular transformations are external automorphisms of the algebra :
p/q
O
A B−1 A A A A−1
S : (cid:26)B → A T : (cid:26)B → µ−1/2AB C : (cid:26)B → B−1. (10)
→ → →
One can verify that S,T,C satisfy the modulargroup relations, S2 = C and (ST)3 = 1 and
that the conjugation operator C commutes with the modular group generators, SC = CS,
TC = CT.
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The requirement that the automorphisms S,T,C be represented unitarily on the re-
presentation space of the ’t Hooft algebra (7) selects a unique (up to equivalence)
HM1/N
unitary, irreducible representation of with dimension p:
p/q
O
(A) = ( 1)pqµNδ
MN N,M
− (11)
(B) = ( 1)pqδ , M,N = 0,1,...,p 1.
MN M,N+1
− −
The corresponding unitary representation of the modular group is:
(S)MN = 1 e2πipqMN
√p
(T)MN = ( 1)Npqe2πi2qpN2−2πiθ(q;p)/3δN,M (12)
−
(C) = δ M,N = 0,1,...,p 1,
MN N+M,0
−
where the phase θ(q;p) is determined by the SL(2,Z) relation (ST)3 = 1 and can be
written as a generalized Gauss sum:
p−1
e2πiθ(q;p) = 1 ( 1)pqne2πi2qpn2. (13)
√p −
nX=0
An explicit formula for θ(q;p) has been found in [8].
The modular invariant representations (11) of the ’t Hooft algebra (7) admit concrete
realizations in terms of holomorphic functions only if 2k is integer (i.e., if q = 1). When
q = 1, a holomorphic realization of (11) involves rather q-multiplets of holomorphic func-
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tions [14],[7]. Geometrically this can be understood by noting that the compact is
1
M
quantizable, in the sense of geometric quantization [15], only if 2k is an integer. However,
the non-compact / is quantizable for any real k since holomorphic wave functions
1
M N
might have non-trivial monodromies around loops surrounding points of . For 2k = p/q
N
rational, monodromies are represented by q q matrices, and we can think of a holomor-
×
phic wave function on / with non-trivial monodromy around the points of as a
1
M N N
q-multiplet of wave functions holomorphic on [8].
1
M
The holomorphic representation of is better understood by considering the fol-
p/q
O
lowing isomorphisms of ’t Hooft algebras
, (14)
pq q/p p/q
O ≈ O ×O
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˜
with and commuting among themselves. In fact, denoting by A and B, A and
q/p p/q
O O
B˜, Aˆ and Bˆ, the generators respectively of the algebras , and , the following
p/q q/p pq
O O O
relations hold:
A = Aˆq, B = Bˆq
A˜ = Aˆp, B˜ = Bˆp (15)
Aˆ = A˜m¯An¯, Bˆ = B˜m¯Bn¯,
where m¯ and n¯ are integers determined by the conditions:
1 = m¯p+n¯q
(16)
0 m¯ q 1, 0 n¯ p 1.
≤ ≤ − ≤ ≤ −
The modular invariant representation of can be realized on holomorphic theta
pq
O
functions. When pq is even, the holomorphic realization of (11) is:
Ψ (τ;z) = θ (τ;z), λ = 0,1,...,pq 1, (17)
λ λ,pq/2
−
where the θ (τ;z) (n integer modulo 2m) are level m SU(2) theta functions [16]:
n,m
θn,m(τ;z) e2πimτ(j+2nm)2+2πimz(j+2nm).
≡
X
j∈Z
If pq is odd, the holomorphic, modular invariant realization of (11) is instead:
Ψ (τ;z) = ( 1)λ(θ (τ;z/2) θ (τ;z/2)). (18)
λ 2λ+pq,2pq 2λ−pq,2pq
− −
Because of the algebra decomposition (14), the representations (17) and (18) decom-
pose into q copies of the representation (11) of . Defining the indices N and α through
p/q
O
λ = qN +pα, 0 N p 1, 0 α q 1, (19)
≤ ≤ − ≤ ≤ −
one obtains the q-components holomorphic representation of :
p/q
O
(Ψ (τ;z))α = θ (τ;z/q) (20)
N qN+pα,pq/2
if pq is even, and
(Ψ (τ;z))α = ( 1)λ θ (τ;z/2q) θ (τ;z/2q) (21)
N q(2N+p)+2pα,2pq q(2N−p)+2pα,2pq
− −
(cid:0) (cid:1)
if pq is odd.
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The algebra decomposition (14) implies as well that the group of external automor-
phisms of also factorizes into two copies of the modular group commuting among
pq
O
themselves and acting independently on and . In particular, the center of
q/p p/q pq
O O C
the group of external automorphisms factorizes: = . Thus, the conjugation
pq q/p p/q
C C ×C
operator C of the algebra , which in the representation (17),(18) acts as follows
pq pq pq
∈ C O
C : λ λ, (22)
pq
→ −
satisfies the equation:
C = C C = C C , (23)
pq q/p p/q p/q q/p
where C and C are the conjugation operators of the algebras and with
p/q q/p p/q q/p
O O
action given by
¯
C : λ λ qN +pα
p/q
→ ≡ − (24)
¯
C : λ λ.
q/p
→ −
SincebothC andC areinthecenter , itispossibletoprojecttheholomorphic
p/q q/p pq
C
representation (17), (or (18)) onto subrepresentations with definite values of C and/or
p/q
C , each one carrying a unitary representation of the modular group. It is somewhat
q/p
remarkable that in this way one obtains the characters of the A and D diagonal series
of integrable representations of SU(2) current algebra, the Kac-Wakimoto characters of
admissiblerepresentationsofSL(2,R)current algebrawithfractionallevel, andtheRocha-
Caridi characters of the completely degenerate representations of the discrete Virasoro
series.
From the point of view of the quantization of , the relevant projection is the one
M
onto the subspace − ( + ) of with C = 1 (C = 1). For reasons
HM1/N HM1/N HM1/N p/q − p/q
which are still rather mysterious [4],[10], when pq is even (odd) the projection onto +
HM1/N
( − ) does not lead to characters related to two-dimensional conformal field theories.
HM1/N
Therefore, in what follows, we will take as Chern-Simons space the subspace − ,
HCS HM1/N
if pq is even, and + , if pq is odd.
HM1/N
The C -odd (or even) combinations of the multi-component wave functions (20)
p/q
spanning turnouttobe(thenumeratorsof)theKac-Wakimotocharactersχ (z;τ)
CS j(n,k);m
H
[17],[18] of the irreducible, highest weight representations of SL(2,R) current algebra with
level m t/u and spin j(n,k) = 1/2(n k(m + 2)), with n = 1,2,...,2u + t 1 and
≡ − −
k = 0,1,...,t 1.
−
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When p is even and q odd, the explicit relation between Kac-Wakimoto characters
(−)
and the C -odd combinations Ψ (τ;z) of the Chern-Simons multi-component wave
p/q N
functions (20) is:
(Ψ(−))α χ (τ;z) if α q−1
N > N = j(N,2α);m ≤ 2 (25)
| ≡ Π(τ;z) (cid:26) χ (τ;z) if α q+1,
− j(p/2−N,2α−q);m ≥ 2
where N = 1,...,p/2 1, and the level m of the current algebra is related to the Chern-
−
Simons coupling constant k through the equation
m+2 = k. (26)
Π(τ;z) is the Kac-Wakimoto denominator,
Π(τ;z) = θ (τ,z) θ (τ,z), (27)
1,2 −1,2
−
which is holomorphic and non-vanishing on / . Therefore, the wave functions
1
M N
(−)
Ψ (τ;z) and the wave functions
N
(−)
Ψ (τ;z)
Ψ′ (τ;z) = N
N Π(τ;z)
appearing in (25), describe equivalent wave functions on / , related to each other by
1
M N
a K¨ahler transformation.
If p is odd and q even, Eqs.(25) and (26) are replaced by
(Ψ(−))α χ (τ;z) if α q/2 1
N > N = j(2N,α);m ≤ − (28)
| ≡ Π(τ;z) (cid:26) χj(p−2N,α−q/2);m(τ;z) if α q/2,
− ≥
with N = 1,..., p−1 and the level m given by
2
m+2 = 4k. (29)
Finally, if both p and q are odd, Eq.(29) is true and the relation between C -even
p/q
wave functions and characters becomes:
( 1)λ(Ψ(+))α
N > − N = χ (τ;z/2)+χ (τ;z/2), (30)
j(p+2N,α);m j(p−2N,α);m
| ≡ Π(τ;z)
with N = 0,..., p−1.
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