IL NUOVO CIMENTO
VOL. 105 B, N. 10
Ottobre 1990
Abelian Chern-Simons Theory and CFT of Rational Torus.
MIAO LI(*)
ICRA-International Center for Relativistic Astrophysics - 00185 Roma, Italia
Dipartimento di Fisica, Universita <,La Sapienza,, - 00185 Roma, Italia
(ricevuto il 60ttobre 1989)
Summary. - - We use the loop formulation to study Abelian Chern-Simons theory.
It is Shown that the Hflbert space can be identified with the conformal blocks of
rational torus.
PACS 02.40 - Geometry, differential geometry, and topology.
There has been much progress in understanding conformal field theories in 2
dimensions [1]. Inspired by a work of Verlinde [2], Moore and Seiberg developed an
axiom system for rational CFT's in [3], and finally proved Verlinde's conjecture. While
Witten, using the Chern-Simons theories in 2 + 1 dimensions in understanding Jones
polynomial[4], found that the Chern-Simons theories are closely related to WZW
models. He actually observed that the Hilbert space of the Chern-Simons theory on a
Riemann surface is isomorphic to the space of conformal blocks of the W Z W model.
This is based on the geometric formalism of Friedan and Shenker [5], where conformal
blocks are formulated as sections of a flat holomorphic bundle over moduli space.
We can construct the Hilbert space by directly quantizing the Chern-Simons
theory. To do this, one must first solve the constraint equations and then work on the
reduced phase space. The constraint here means that the phase space is the space of all
flat connections modulo gauge transformations. Usually this space is complicated, so
quantization based on this space appears hopeless. However, one can draw some
interesting information from this scheme. First, since the phase space is compact, one
cannot use normal polarization. In fact, given a conformal structure on the Riemann
surface, one can naturally give a complex structure on the phase space. The symplectic
form then is K~ihler. One then identifies the Hilbert space with the space of all
holomorphic sections of a holomorphic line bundle. The first Chern class is the given
Kahler form. When the complex structure varies on moduli space, one then has a flat
vector bundle with the Hilbert spaces as fibres on the moduli space [4].
(*) Permanent address: Institute of Theoretical Physics, Academia Sinica, Beijing, P.R. China.
Address after October 1, 1990: Department of Physics, University of California, Santa Barbara.
1113
1114
MIAO LI
In this short paper, inspired by the work of Rovelli and Smolin on loop formulation
of quantum gravity [6], we use the quantum loop algebra to study the Abelian ChernSimons theory. This is the first result in studying WZW models via (non)-Abelian
Chern-Simons theories. We find that the Hilbert space is actually made of conformal
blocks of rational toms. We notice that Abelian Chern-Simons has been studied in path
integral formalism in [7] and also by Witten in [4].
We begin with the Chern-Simons action
I=kfAAdA.
(1)
In the canonical form
(2)
,:r.f(
k
.
.
1A is~
2 2~0 ~ r~s).
.
.
We find the Poisson bracket
(3)
{Ai(x), As(x') ) = - 2._~ r ~2(x _ x')
k
and the constraint F~ = 0.
Consider the Chern-Simons theory defined on Z x R, Z as a Riemann surface.
Before discussing loop algebra, we first note some facts in direct quantization by first
imposing constraints. In this simplest case, the reduced phase is tractable. A point in
the reduced phase space is the holonomy group of a given fiat connection. Since the
gauge group is Abelian, the whole holonomy group is characterized by its elements
which are obtained along loops in the first homology group of Z. Now there is a basis
which consists of 2g elements (a~, b~). It is the usual canonical basis. The phase space is
the tensor space of g tori T 2, each one is associated with a pair of (ai, bi). The
symplectic form restricted on these tori is the k multiple of the fundamental Kiihler
form, which is the first Chern class of the line bundle of holomorphic 1-form. The
relevant line bundle on each torus is the k-th power of this line bundle and has kindependent holomorphic sections. We then have k a holomorphic sections of the
relevant line bundle on the reduced phase space.
WedefineWilsonloopbyT(~,)=Pexp[JA]
for a loop y on2L Here P denotes path
ordering, thus, any loop ~, is also marked with an orientation. The Wilson loops are
gauge invariant. Using the fundamental Poisson bracket (3), we find the Poisson
bracket between two given Wilson loops:
(4)
{T(y1), T(y2)} -- - ~ T(y1) T(y2) e(Y1 (~ Y2),
where ~(Y1• ~'2) is the intersection number between y1 and ]'2. For a canonical basis
(ai, hi), we have ~(al (~ bi) = ~is. From (4) we see that Wilson loops are closed under the
Poisson bracket. Next, we use the constraint Fij = 0 in reducing the number of Wilson
loops. The variation of Wilson loop with respect to the variation of • is proportional to
Fis, as well as zero. Thus, we find that T(y) depends only on the homotopy class of y.
ABELIAN CHERN-SIMONS THEORY AND
CFT
OF RATIONAL TORUS
1115
Our problem is to convert this classical algebra into a quantum one. One cannot
simply replace the above formula by multiplying by a factor i on the right-hand side. It
is necessary that we use some well-defined ordering. The unique way to do so is to put
(5)
IT(r1), T()2)] =
iak(Y1 n
)2)[T(]'l) T()2) -[- T()2) T(~I)] 9
We do not expect that the numerical factor ak is (Tr/k)~(]'~ n ]'2). However, at the limit
of large k, we expect that ak ~ (=/k)e(r~ n ]'2), since this corresponds to the weakcoupling case.
Solving (5) gives
V(]'l ) Y()2) = (1 ~- iak(]'l (~)2))/(1 -- io~k()l (~ y2)) T()2)
T(rl).
This form looks like the relations among scattering data in quantum soliton theory. To
find the coefficients ak, we must use the consistent condition on the product of three
loops. Since T ( ) 1 , ]'2) is proportional to T()~)T(],2) (and not equal to, as we will see),
multiplying T()3), we find the following relation:
(6)
Ok(]'l ('~)3) -~- 0k()2 ~ )3) ~- Ok(()l * )2) ('~)3),
where we have defined exp[i0k] = (1 + iak)/(1- i~). The simplest solution to (6) is
0k(]'l n ]'2)~ ~(]'1 n ]'2). To agree with the limit at large k, we must have
Ok(]'1 (~ ]'2) ~---~
$(]'1 (~ ]'2),
(7)
ak(]'l N ]'2) = t g ( ~ e(]'l n ]'2)) 9
The Hilbert space on torus can be k dimensional, only when we choose this solution in
the operator algebra.
Usually, we have not T(]'I 9 ]'2) = T(]'I) T(ye). This can be seen in the case of torus, a
and b commute as elements of the fundamental group, however, this does not mean
T(a) and T(b) commute. We assume the following ansatz:
T(]'I 9 ]'2) = exp [ir
]'2)] T(]'I)
T(]'2) 9
Associativity implies that ~ is a cocycle, namely
(8)
$(]'1, ]'2 * ]'3) -- $(]'1 * ]'2, ]'3) -[- ~(]'2, ]'3) -- ~(]'1, ]'2) = 0 .
These ~'s will be proved to be closely related to the modular transformation property
of the Hilbert space. We find that a simple solution to this is ~()1, ]'2) - 0k(]'l n ]'2). The
last condition can be used in determining the proportional relation
l~ aibia~l b~ 1= 1.
i
Using it, we derive
dri, r2) = - ~ ~(r~n r2).
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MIAOLI
We are now ready to discuss the Hilbert space. The Hilbert space furnishes a finite
representation of loop algebra. The first nontrivial case is the torus. We have
T(a)T(b) = exp [i27r/k] T(b) T(a). Choosing a vector which satisfies T(a)lO) = t0), one
may assume a phase factor in the definition, it can be absorbed into the definition of
T(a), without changing the commutation relation. The whole Hilbert space is spanned by {T"(b)10)}. Usually, this space is of infinite dimensional if there are no restrictions. We have in mind that we should have a k-dimensional space, thus we
put rk(b)10)=]0). The series (T~(b)I0)) terminates at Tk-l(b)10). With
T(a) Tk(b) = Tk(b) T(a), our definition is consistent. Without choosing the appropriate
Ok, this definition is unavailable. Again one can find the Okby this definition. We define
a natural inner product as <rain) = ~m~. Note that T(a)ln) = exp [i2rcn/k]]n).
The modular transformations can be recovered in this context. The modular group
PSL(2,Z) for genus one is generated by two elements T and S, subject to the
constraint (ST) 3 = 1. Under transformation T: a--~ a, b--* b 9 a. Thus, the new creation
operator is T(b') = T(b 9 a) = exp [irdk ] T(b) T(a) and the new states are
I >'--exp
(9)
Remember that there is an overall phase factor which we cannot determine at present.
Recall that when the Hilbert space is identified with conformal blocks of some CFT,
under T, one must have In)--* exp [i2=(h~- c/24)]ln ). Thus, we have h~ = n2/2k, this
result is in accordance with that which has been obtained by the path integral in [4].
The second step is to determine the transformations under S, by which one can
calculate the fusion rules. Under S:a-->b, b---~a-1. The new states In)' satisfy
T(b)ln)' =exp[i2=n/k]ln)',
T(a-1)[n) '= In+ 1)'
The solutions are
(10)
In>
Again there may be an overall phase. But now this phase must be 1, since the entry S O
of matrix S must be positive. Witten shows that the partition function of the ChernSimons theory on S 3 must be this number. We conclude that the partition function on
S ~ is 1/V~. We use Verlinde's formula
/'~ oi
n
~oj o n
s,J 0
in finding the fusion rules
(11)
N~j=I,
iff i + j = l m o d k
to be the same as calculated by Moore and Seiberg[3].
Using another Verlinde formula dimVg= Y~ IS~t-2~--1), where Vg is the space of
conformal blocks on g-genus Riemann surface, we find dim Vg = kg as we noted before.
ABELIAN CHERN-SIMONS THEORY AND
CFT
OF RATIONAL TORUS
1117
On higher genus, we have the following relations:
(12)
T(a~)T(bj) = exp [ ~ j
T(b~)T(a~).
The representation space is built from 10} satisfying T(a~)lO} = t0}. Thus the Hilbert
space is isomorphic to the k-th tensor product space of what we find for genus one. It is
indeed k g dimensional. A basis is (T~l(bl)... T~(bg)lO>). Since modular transformations
are generated by Dehn twists, we can find the modular transformations of the basis.
Once the modular transformations are given, it is simple to construct the flat bundle
associated to conformal blocks. Thus, all known results can be obtained once again. We
must stress here that, in constructing the Hilbert space, conformal structures on the
Riemann surface has not been used (compare what was done in [8]). But the global
information is sufficient in drawing local formulae for the rational torus.
Lastly, we would like to point out that it is possible to generalize the construction
within this paper to the case of Riemann surfaces with punctures, provided we
incorporate the corrected constraints [4]. By careful study on the sphere with four
punctures, one might derive the duality matrices [3]. In [9], we find that to generalize
the construction to a non-Abelian case, one must introduce operators which are just
those operators studied firstly by Verlinde [2].
After the completion of this paper, we received [8] in which a different method was
used to investigate the same problem. The commutation relations among Wilson loops
are derived in a different way. A 3-dimensional loop formalism is developed in [10].
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preprint, IASSNS-HEP-88/39.
[4] E. WITTEN: Quantum Field Theory and the Jones Polynomial, IAS preprint, LASSNSHEP-88/33, Lecture at the IAMP Congress (Swansea, 1988); E. WITTEN: Nucl. Phys. B,
311, 46 (1988/89).
[5] D. FRIEDAN and S. SHENKER: Nucl. Phys. B, 281, 509 (1987).
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325 (1989).
[8] M. BOS and V. P. NAIR: Columbia University, preprint (1988).
[9] M. LI and M. Yu: Modular transformations in WZW models on Riemann surfaces of
higher genus (1989).
[10] M. AWADA:Imperial College, preprint, ImperialfrP/88-89/5.