Jul 18, 1994 - We would like to thank David Olive for many helpful discussions in the ... [11] M. Chu, P. Goddard, I. Halliday, D. Olive, A. Schwimmer, Phys.
DAMTP-94-42 hep-th/9407108 QUANTISATION OF THE SU(N) WZW MODEL AT LEVEL k
arXiv:hep-th/9407108v1 18 Jul 1994
Meifang Chu and Peter Goddard D.A.M.T.P University of Cambridge Silver Street, Cambridge, CB3 9EW United Kingdom
ABSTRACT The quantisation of the Wess-Zumino-Witten model on a circle is discussed in the case of SU (N ) at level k. The quantum commutation of the chiral vertex operators is described by an exchange relation with a braiding matrix, Q. Using quantum consistency conditions, the braiding matrix is found explicitly in the fundamental representation. This matrix is shown to be related to the Racah matrix for Ut (SL(N )). From calculating the four-point functions with the Knizhnik-Zamolodchikov equations, the deformation parameter t is shown to be t = exp(iπ/(k + N )) when the level k ≥ 2. For k = 1, there are two possible types of braiding, t = exp(iπ/(1 + N )) or t = exp(iπ). In the latter case, the chiral vertex operators are constructed explicitly by extending the free field realisation a la Frenkel-Kac and Segal. This construction gives an explicit description of how to chirally factorise the SU (N )k=1 WZW model.
18 July, 1994
Introduction Rational conformal field theory[1] has been very useful and instructive for many aspects of two-dimensional physics. This is largely due to the nice properties that follows from its symmetry algebras[1,2] . The generally accepted definition of rational conformal field theory[3] is that the symmetry algebras are formed by two commuting chiral algebras, ¯ and each algebra contains a copy of Virasoro algebra which ensures the conformal A × A, symmetry. In general, it also contains other chiral algebras such as the Kac-Moody algebras in the case of Wess-Zumino-Witten models. Consequently, the Hilbert space can be ¯ decomposed into a finite number of highest weight representations of A × A. H=
M λ
Hλ ⊗ H λ
(1)
In the Euclidean formulation, there is a state/vertex operator correspondence which says that the highest weight state in Hλ ⊗ H λ can be obtained by applying the corresponding primary field at the origin to the vacuum, Φλ,λ (0, 0)|0i. Under conformal transformations, the behaviour of each primary field is determined by its conformal weight (∆λ , ∆λ ). One important consequence of these properties is that one can calculate all the correlation functions in the theory from those of the primary fields using the Ward identities derived from the two conserved chiral currents. Since the two chiral algebras commute, these correlation functions are decomposed into a sum of products of holomorphic and antiholomorphic functions, which are called the conformal blocks[1] . This suggests that the primary field Φλ,λ (z, z) is also decomposed into a sum of products of holomorphic primary fields, Uaλ (z), and antiholomorphic primary fields, Vaλ (z), Φλ,λ (z, z) =
X
λ,λ λ Ca,a Ua (z)Vaλ (z).
(2)
a,a
From this point of view, conformal blocks are formed by the correlation functions of the chiral primary fields and one should be able to derive the braiding and fusion properties[3] of the conformal blocks from the local properties of these operators. In particular, they are nonlocal operators and obey braiding relations of the following type, U (z) ⊗ U (w) = (1 ⊗ U (w)) (U (z) ⊗ 1) Q(z, w),
V (z) ⊗ V (w) = Q(z, w) (1 ⊗ V (w)) (V (z) ⊗ 1) .
(3)
They generate the entire field content of the chiral models and the chiral Hilbert space can be determined completely from them. The chiral primary fields satisfying these descriptions are called the chiral vertex operators[3−8] . (In particular, we are refering to the verter-chiral vertex operators discussed in [4].) It is therefore natural to ask how one can formulate the chiral conformal field theory from these chiral vertex operators such that their correlation functions reproduce the conformal blocks. Such a formulation is especially desirable from a phenomenological point 2
of view when one discusses the compactification of string theories. Whilst there has been a number of studies[3−11] indicating that a quantum group related symmetry acts in the chiral theory, a description of precisely how the factorisation occurs and how the physical states in the chiral Hilbert space transform under the quantum group is lacking. This paper is a continuation of the investigation of this problem using the approach of canonical quantisation begun in [11] and briefly reviewed below. The aim is to quantise the chiral primary fields in the Wess-Zumino-Witten (WZW) models associated with a group G and compactified on a circle. As explained in [10-12], these chiral primary fields obey a quantum exchange relation given by a braiding matrix, Q. An explicit solution for Q was obtained for SU (2). In this paper, we solve for the braiding matrix in the fundamental representation for the model associated with SU (N ) at level k. Because the braiding matrix can be related to the Racah matrix for the quantum group Ut (SL(N )), this implies indirectly that Ut (SL(N )) is acting on the chiral Hilbert space. Instead of using the conformal scaling argument in [11], we show that the quantum value of the deformation parameter t can be determined using the Knizhnik-Zamolodchikov (KZ) equations. For k ≥ 2, our results agree with the usual observation that t = exp(iπ/(k + N )). In the levelone case, we find that Ut (SL(N )) with t = eiπ also describes the additional symmetry in the quantised chiral theory. Notice that when the deformation parameter is a root of unity, the tensor products of the irreducible representations of Ut (SL(N )) are truncated to contain only the integrable representations. These tensor products can not be made co-associative for all irreducible representations[13,14] . Consequently, the corresponding symmetry algebra should be a quasi quantum group, i.e. a quasi-Hopf algebra[15] . For the simplest models with the level k = 1, the chiral symmetry algebras, i.e. level-one Kac-Moody algebras, has an unitary realisation in terms of free scalar fields a la Frenkel-Kac and Segal[16] . In this paper, we extend this construction to give the chiral vertex operators such that their local properties can be examined directly. This provides an explicit description of the chiral Hilbert space acted by two symmetry algebras, SU (N )k=1 Kac-Moody algebra and the quantum group U−1 (SU (N )) (co-associative when t = −1). This construction not only exemplifies the results obtained from the canonical quantisation, but it also provides a useful insight for formulating the corresponding chiral WZW models. Canonical Structures of the WZW Models Let us first briefly review the Poisson structures of the WZW models derived in [11]. Consider the defining field g(τ, x) which takes values in a compact simple Lie group G of rank r and x is the spatial coordinate for a circle of length 2π such that g(τ, x + 2π) = g(τ, x). Let ψ be the longest root and {ta } ≡ {H i , E α } denote a basis for the generators of the Lie algebra, G. We shall normalise these generators so that T r(ta tb ) = N δ ab . The action of the WZW model has an infra-red fixed point and it can be written as Z Z 2 3 2 kψ 2 −1 −1 µ λµν −1 −1 −1 tr g ∂µ gg ∂ g d x − ǫ tr gˆ ∂λ gˆgˆ ∂µ gˆgˆ ∂ν gˆ d x , A[g] = − 16πN 3 B M (4) where B is a three-dimensional manifold whose boundary is the cylinder M and gˆ(τ, x, y) ∈ G is defined on B such that it maps to g(τ, x) at y = 0. In order for the action to be well 3
defined, k must be an integer so that eA is single valued. The equations of motion, i.e. the Euler-Lagrange equations obtained from (4), can be written in terms of two chirally conserved currents: ∂+ (g −1 ∂− g) = 0,
∂− (∂+ gg −1 ) = 0,
x± ≡ τ ± x.
(5)
In the gauge fixed approach[11] , a general solution of the equations of motion can be written in terms of the periodic coset variables u ˜ ∈ LG/H and v˜ ∈ H\LG: +
g(x, t) = u ˜(x+ )eiqν ·H ei2(x
+x− )ν·H
v˜(x− ).
(6)
Substituting the parameterisation of (6) into the symplectic two form, we can invert it and obtain the Poisson brackets. In terms of the new chiral variables, +
u(x+ ) ≡ u ˜(x+ )ei2x
−
v(x− ) ≡ eiqν ·H ei2x
ν·H iqν ·H
e
ν·H
,
(7)
v˜(x− ),
the Poisson brackets are listed as follows. 1 ij βδ , {˜ u1 (x+ ), v˜2 (x− )} = 0, {˜ u(x+ ), ν j } = 0, {˜ v(x− ), ν j } = 0, 2 + {u1 (x+ ), v2 (y − )} = (x+ − y − )βu1 (x+ )Hv2 (y − ) + {˜ u1 (x+ ), iqν H2 }ei2x νH1 eiqν H1 v2 (y − )
{qνi , ν j } =
+ u1 (x+ )eiqν H2 ei2y {u1 (x+ ), u2 (y + )} = u1 (x+ )u2 (y + ) r(x+ − y + ),
−
νH2
{iqν H1 , v˜2 (y − )},
{v1 (x− ), v2 (y − )} = r(x− − y − ) v1 (x− )v2 (y − ),
(8) j=1 H ⊗ H and β = 4π/ψ k. The classical r-matrices in the last
Pr
where H ≡ H1 · H2 = two brackets of (8) are
j
j
2
+ β βX 1 η(x+ )H + i e−iα·νη(x ) E α ⊗ E −α , 2 2 α sin(α · ν) − βX 1 β eiα·νη(x ) E α ⊗ E −α , r(x− ) = η(x− )H − i 2 2 α sin(α · ν)
r(x+ ) =
(9)
with η(x) = 2[x/2π] + 1 and [y] denotes the maximal integer less than y. Notice that the Poisson bracket, {u1 , v2 } in (8), is different from the one given in Eq. (36) of [11] because {˜ u, qν } and {˜ v , qν } do not vanish in general. They can be made to vanish in a natural way for u ˜ and v˜ at the identity but will be present away from it. Their precise forms will depend on the way the coset representations u ˜ and v˜ are chosen. We have not specified them explicitly and they are not needed for the discussion in the rest of this paper because each chiral sector can be quantised separately. The fact that the Poisson bracket in (36) of [11] was not consistent was drawn to our attention by G. Papadopoulos. One can easily verify that the Poisson brackets in (8) satisfy the Jacobi’s identities. 4
Classically, we can use these Poisson brackets to verify that u (v) transforms as a primary field of the right-moving (left-moving) Kac-Moody algebra, i.e. a {J+ (x+ ), u(y +)} = ita u(y + )δ(x+ −y + ),
a {J− (x− ), v(y − )} = iv(y − )ta δ(x− −y − ). (10)
The Kac-Moody currents can be written in terms of the chiral group elements u(x+ ) and v(x− ) as J+ (x+ ) = (ik/4π)∂+ uu−1 , J− (x− ) = (ik/4π)v −1 ∂− v. (11) Quantisation and the Braiding Matrix Let us proceed to quantise these Poisson brackets. The first four brackets in (8) can simply be replaced by Dirac commutators, [qνi , ν j ] = i
β~ ij δ , 2
[˜ u1 (x+ ), v˜2 (y − )] = 0,
[˜ u(x+ ), ν j ] = 0,
[˜ v (x− ), ν j ] = 0. (12)
The quantum correction to the constant β~ will be determined later from consistency conditions. Following [11], we quantise the last two brackets in (8) by the following exchange relation, u1 (x+ )u2 (y + ) = u2 (y + )u1 (x+ )Q(x+ − y + ), −
−
v1 (x )v2 (y ) = Q
−1
−
−
−
−
(x − y )v2 (y )v1 (x ).
(13a) (13b)
where Q(x) is the braiding matrix which can be determined from the consistency of quantisation as follows. Unitarity,
Q† (x) = Q∗T (x) = Q−1 (x)
(14a)
Antisymmetry property, Monodromy property,
Q(−x) = PQ(x)−1 P Q(x + 2π) = (M (ν)−1 ⊗ I)Q(x)(M (ν2 ) ⊗ I)
(14b) (14c)
Classical limit, Locality condition, Jacobi identities,
= (I ⊗ M (ν1 ))Q(x)(I ⊗ M (ν)−1 ) Q(x) = I ⊗ I + i~r(x) + O(~2 )
[Q(x), eiqν ·(H1 +H2 ) ] = 0 with xij ≡ xi − xj ,
(14d) (14e) (14f )
Q23 (ν1 , x23 )Q13 (ν, x13 )Q12 (ν3 , x12 ) = Q12 (ν, x12 )Q13 (ν2 , x13 )Q23 (ν, x23 ), (14g) The subscript of ν denotes a shift, e.g. ν2 = ν + 21 ~βH2 , due to the commutation relation, 1 [u(x), ν] = − β~u(x) · H. 2
(15)
These six requirements are fairly obvious except the locality condition, (14f), which is added to ensure that two g’s at a space-like separation commute with each other. 5
In the following, we solve for the braiding matrix in the N-dimensional fundamental representation for SU (N ) at level k. Let Φ denotes the space of roots and ~λa for a = 1, 2, ..N be the weights in the fundamental representation whose highest weight is λ1 . The normalisation of these vectors is chosen to be ~λa · ~λb = δab − 1/N . Let eab denote a N × N matrix which has only one non-zero element, 1, P located at the ath row and the bth ~ = N ~λa eaa and the step-operator column. Then, the Cartan-Weyl basis is given by H a=1 ~ ~ Eαab = eab for α ~ ab = λa − λb ∈ Φ. With this normalisation, N = 1 and the length of the longest root is ψ 2 = 2. In order to solve for Q in (14), we exponentiate the classical r-matrix in (9) and replace its coefficients by their quantum values: namely, χ, θ, and γ. (
Q(x) = exp iχη(x)H − =t
−1 )η(x) ( NN
+t
1 −N
α∈Φ
1⊗1−
η(x)
X
α∈Φ 1
X
− t− N η(x)
X
α∈Φ
θ(α · ν)e−iγα η(x) Eα ⊗ E−α
X
α∈Φ
Eα E−α ⊗ E−α Eα
)
,
!
t ≡ eiχ .
cos θ(α · ν)Eα E−α ⊗ E−α Eα
(16)
e−iγα η(x) sin θ(α · ν)Eα ⊗ E−α ,
Without specifying the values of these coefficients, (16) already satisfies most of the requirements in (14) except the monodromy property and the Jacobi’s identities. These two properties impose constraints on the coefficients in Q and they can be solved as follows. γα = α · ν,
sin θ(α · ν) =
sin(χ) , sin(α · ν)
χ=
1 π β~ −→ + O(~). 2 k
(17)
Therefore, the braiding matrix is determined completely up to the braiding parameter t. The quantum value of this parameter will be determined in the next section when we study the monodromy properties of the four-point functions by solving the KnizhnikZamolodchikov equations. Notice that this braiding matrix can be related to the Racah matrix of Ut (SL(N )) and consequently indicates that Ut (SL(N )) is an additional symmetry of the chiral theory. This identification comes from the following observation. From the conformal scaling argument given in the appendix, we know that α · ν/χ ∈ Z because the monodromy takes values of ν/χ = Λ + δ for some highest weight Λ and δ denotes one half of the sum of all positive roots. Subsequently, when t 6= ±1, the braiding matrix Q in (16) is a N 2 × N 2 matrix related to the deformed Racah matrix[17−19] for Ut (SL(N )) up to some normalisation: Qmn,m′ n′ (ν) ∝ δ(λm + λn − λm′ − λn′ )
6
λ1 λ1
Λ Λ − λm − λn
Λ − λn′ Λ − λm
RW
q=t2
.
(18)
This identification is valid only when all the entries in the bracket are highest weights. In 1 order to avoid confusion, we have denoted the quantum group by t = q 2 rather than q such as in [17,18]. In the limit when t2 = 1, Q can be related to the undeformed Racah matrix of SU (N ). √ For example, when N = 2 and t 6= ±1, we can substitute 2ν/χ = (2j + 1) for j ≥ 1 and obtain the following 4×4 matrix in terms of the deformed-Racah matrix of Ut (SL(2)), 1 RW n′ j j − ℓ+L−j−J (Cj +CJ −Cℓ −CL ) 2 2 Qmn,m′ n′ (ν) = δm+n,m′ +n′ (−1) t , 1 j − m+n j−m q=t2 2 2 2 with Cj = j(j + 1),
ℓ=j−
n′ , 2
L=j−
m , 2
J =j−
m+n ; ∀m, n, n′ = ±1. 2
(19) In the limit when t = eiπ , this gives the Racah({6j})-matrix for SU (2) according to 1 n′ n−n′ mn j j − 2 2 . (20) Qmn,m′ n′ (ν) = δm+n,m′ +n′ (−1) 2 (−1) 2 1 j − m+n j−m 2 2 2 As explained in the next section, the quantum value of χ is typically given by π divided by a positive integer. This raises a concern that (16) and (17) become singular when α·ν/π ∈ Z. However, for the WZW models which we are studying, only the primary fields in the integrable representations[20] contribute to correlation functions, i.e their highest weights satisfy 0 < λ0 · ψ < k + 1. Restricted to these representations only[4] , the monodromy momentum takes the value of ν/χ = λ0 + δ and it satisfies 0 < α · ν/π < 1 when χ = π/(k + N ). When the level k = 1, the coefficient sin θ(α · ν) in (17) becomes a finite integer in the limit of χ = π for α · ν/π ∈ Z. Therefore, the braiding matrix in (16) is regular for the physical representations considered in the chiral SU (N )k WZW models. Braiding Matrix and Monodromy Matrix In order to determine the quantum value of the braiding parameter, t = eiχ , one can follow the conformal scaling argument given in [11]. We give the result of that argument for SU (N ) in the appendix. The drawback of this argument is that it does not apply to the case when k = 1. In this section, we give a new derivation of the braiding parameter based on quantum consistency, i.e. the Knizhnik-Zamolodchikov (KZ) equations. Since Q in (16) describes the braiding relation of two chiral primary fields in the fundamental representation, we shall consider the four-point functions in the representations: Λ1 , Λ, Λ, Λ4 ; namely, the two vertex operators in the middle are in the fundamental representation, Λ. Let us denote the weights in these representations by aj , bj and define the following four-point functions, Fab11ab22ba33ba44 (z1 , z2 , z3 , z4 ) ≡ h0|UaΛ11b1 (z1 )UaΛ2 b2 (z2 )UaΛ3 b3 (z3 )UaΛ44b4 (z4 )|0i.
(21)
By exchanging the vertex operators at z2 and z3 according to (13a), we find the following braiding relation between the four-point functions, X z2 b1 b2 b3 b4 b1 r3 r2 b4 ,t . (22) Fa1 a2 a3 a4 (z1 , z2 , z3 , z4 ) = Fa1 a3 a2 a4 (z1 , z3 , z2 , z4 )Qr2 r3 ,b2 b3 z3 r ,r 2
3
7
The main idea is to solve for the four-point functions using the Ward-identities and then determine the braiding parameter t from (22). As shown in [2], these four-point functions satisfy the following set of differential equations, 4 X j=1
zjn zj ∂zj + (n + 1)∆j F {b} = 0;
4 X F {b} = 0, tα j
n = −1, 0, 1
(23a)
(23b)
j=1
(k + N )∂zi F
{b}
=
4 X
j=1,j6=i
ti · tj {b} F , zi − zj
(23c)
where {tj } denote the SU (N ) generators in Λj representation and ∆j denotes the conformal weight of the j-th vertex. These weights are given by their highest weight vectors according to ∆j = hΛj , Λj + 2δi/2(k + N ). For example, ∆(Λ) = (N 2 − 1)/2N (k + N ) for the fundamental representation. Before solving the KZ equations, let us first recall that, following from (23a), the correlation functions depend only on the coordinate differences, zij ≡ zi − zj . Once a pre-factor independent of the indices {a} and {b} has been extracted, the rest will depend only on the cross-ratio, y ≡ (z12 z34 )/(z13 z24 ), Y −γ {b} {b} F{a} (z1 , z2 , z3 , z4 ) = zij ij f{a} (y), i |w|,
When η(x) = 1, R(x) can be written as N N N X X X 1 −N −1 R(t) = t t err ⊗ err + ers ⊗ ers + (t − t ) ers ⊗ esr , r=1
(62)
t = eiπ .
r>s=1
r6=s=1
(63) Thus, we can identify e R(x) as the R-matrix in the fundamental representation of Ut (SU (N )) with t = eiπ . Note that we have replaced Ut (SL(N )) by Ut (SU (N )) since t = −1 ∈ R. This braiding parameter agrees with the one in (36b) obtained from solving the KZ equations. 1 iN η(x)
[17]
However, the braiding relation in (62) is given by the R-matrix rather than the braiding matrix Q in (16). In order not to alter the defining properties of chiral primary fields, we ˆ (z) by a N -dimensional matrix, A, which commutes can multiply the free-field vertex U with the Kac-Moody currents. In particular, let the matrix A depend on the momentum, p− and denote Aˆ ≡ eiHq− A(p− ) such that × × −iqφ H ˆ −iq− H ˆ U (z) ≡ eiφ(z)H e C e A. (64) × ×
Then, the braiding of U will be given by Q provided that Aˆ satisfies the following equation, R(t)Aˆ1 Aˆ2 = Aˆ2 Aˆ1 Q(ν, t).
(65)
It is sufficient to show that (65) holds for η(x) = 1 because the same result will follow automatically for other x. In fact, this requirement gives the operator identity for the IRFVertex transformation[4,9,17,24] for Ut (SL(N )) when Aˆ is identified as the Wigner-matrix which relate states in different irreducible representations of Ut (SL(N )). For Ut (SL(2)), this operator identity has been realised explicitly in [24] in terms of two pairs of harmonic oscillators. In particular, we take their result in the limit of [Y ]t = Y tY −1 for Y ∈ Z when t = eiπ and obtain ! r (ν+p ) −iπ √ − σ3 σ 2 ν − p− √3 q− √ i q i cos θ −e sin θ e 2 ν, sin θ ≡ . (66) Aˆ ≡ e 2 (ν+p− ) √ iπ 2ν 2 e sin θ cos θ 16
Notice that Aˆ depends on two pairs of harmonic oscillators, (q− , p− ) and an additional pair (qν , ν) where ν is the monodromy momentum in the braiding matrix Q. These zero modes are independent degrees of freedom in the sense that they commute with the Kac-Moody current. We will come back to the discussion of the monodromy in a short while. Thus, we have completed the construction of the chiral vertex operator, U (z) in (64). Its unitarity is manifest because both Cˆ and Aˆ are unitary. According to [24], Aˆ transforms covariantly under U−1 (SU (2)), i.e. 1
̺ (A+b , A−b ) = (A+b , A−b ) (Π 2 ⊗ id)∆(̺),
∀ ̺ ∈ U−1 (SU (2)),
(67)
1
where Π 2 denotes the spin 21 representation and the coproduct ∆(̺) can be obtained from those of the quantum group generators, ∆(tS3 ) = tS3 ⊗ tS3 ,
∆(S± ) = t∓S3 ⊗ S± + S± ⊗ t±S3 .
(68)
Since t2 = 1, the symmetry algebra U−1 (SU (2)) is isomorphic to SU (2). Its generators can be realised in terms of the harmonic oscillators as s 2 2 √ ν 1 p− 1 ν ∓√ 3 ± ±i 2q− √ S ≡ √ p− , . (69) t 2 − √ ± S ≡e 2 2 2 2 The transformation laws of (67) can be verified directly with (66) and (69). Moreover, the quadratic Casimir operator of the algebra is given by ( √ν2 )2 − 14 . Thus, its irreducible
representations, V j , are labelled by the eigenvalues of ( √ν2 − 12 ) = j for j = 0, 12 , 1, 23 , .. ˆ consequently the chiral vertex operator U , takes a state in V j The Wigner operator A, to another state in V j± 12 of U−1 (SU (2)). Because this algebra commutes with the chiral affine algebras, it plays the role of an internal symmetry in the original WZW model.
Let us now discuss the monodromy momentum, ν. From its definition given in (7), the chiral vertex operator has the following monodromy, Urs (zei2π ) = Urs (z)ei2πν·λs .
(70)
Comparing with the explicit construction of U in (64), we need to impose a constraint between ν and the free-field momentum pφ such that ei2πpφ ·λr = ei2πν·λs ,
1 ∀λr , λs = ± √ . 2
(71)
This condition is manifestly satisfied when the spins of U−1 (SU (2)) and the spin of the Kac-Moody representation differ by an integer. Applying the chiral vertex operator and the Kac-Moody current successively on the vacuum, we obtain all the states in the chiral Hilbert space. It can be decomposed as follows. M H 21 ⊗ (⊕j= 21 , 32 ,.. V j ) . (72) HR = H0 ⊗ (⊕j=0,1,2,3,..V j ) 17
Hilbert Space for the Chiral SU (N )k=1 WZW Model The results obtained in the last section for the chiral vertex operators in the fundamental representation of SU (2) can be generalised to other basic representations of SU (N ). The chiral Hilbert space is decomposed as HR =
M
λ=0,minimal
Hλ ⊗ ⊕ξ∈Λ˜ λ V ξ ,
(73)
where Hλ ’s are the minimal representations of the right-moving SU (N )k=1 Kac-Moody ˜ λ denotes the space of all the highest weights of U−1 (SU (N )) contained in algebra and Λ the weight lattice, Λλ . Then, the monodromy constraint in (71) will hold for any two weights λr and λs in the same representation. The Hilbert space of the left-moving sector can be constructed in a similar way using the left-moving free scalar fields and (N − 1) additional pairs of harmonic oscillators. HL =
M
λ=0,minimal
⊕ξ∈Λ˜ λ Vξ ⊗ H λ .
(74)
These new degrees of freedom do not appear in the original WZW models and they should be gauged away when we “join” the left-moving and the right-moving sectors together. In particular, we should recover the original defining field g(z, z) in (53) and also the original Hilbert space M (75) HW ZW = Hλ ⊗ H λ . λ=0,minimal
One has to make this gauging process more precise in order to obtain the Lagrangian formulation of the chiral WZW model. We hope to investigate this question in the near future. From the free field construction of the chiral vertex operators, we obtain an explicit description of how to factorise the SU (N )k=1 WZW model into the right-moving and the left-moving sectors. In contrast to the previous treatments[3−11,14] , the chiral Hilbert space acquires an additional symmetry given by U−1 (SU (N )) instead of Ut (SL(N )) with t = eiπ/(1+N) . From the study of the KZ equations, see (36), we learn that these correspond to the only two possible braidings which describe the local properties of the chiral vertex operators in the level-one chiral models. However, an explicit realisation of the quasiquantum group symmetry Ut (SL(N )) with t = eiπ/(1+N) is still lacking. When k ≥ 2, we know from the braiding matrix that there is a unique chiral facotrisation with the additional symmetry given by the quasi-quantum group Ut (SL(N )) with t = exp(iπ/(k+N )). We are currently investigating its implementation in the chiral WZW models of higher levels. At the moment, the different nature of the chiral factorisation in the level-one WZW models remains somewhat mysterious.
18
Acknowledgements We would like to thank David Olive for many helpful discussions in the early stages of this work. We are also grateful to Fedor Smirnov for helpful discussions about the IRFVertex relation. This work is supported by the Science and Engineering Research Council under the grant GR/H57929. Appendix: Conformal Scaling and the Braiding Parameter In this appendix, we use the scaling argument[9] to determine the braiding parameter t for the SU (N )k WZW model when k ≥ 2. Let us denote ζ = ei2πx/l . For some constant µ, ζ µ u(ζ) transforms as a primary field under the Sugawara stress tensor, L=
N X 1 × a a× J J . × 2(k + N ) a=1 ×
(A1)
The conformal weight of u(ζ) in the [Λ] representation is given by h = a scaling, ζ 7→ sζ, with s = e2πi , u transforms according to
2(k+N) .
sL0 uΛ (ζ)s−L0 = sh−µ uΛ (ζ)e2iν·H .
Under
(A2)
The left-hand side comes from the monodromy of the u-vertex. Consider the matrix elements of (A2) between two states hΛL , ρL | and |ΛR , ρR i in the highest weight representations ΛL and ΛR . We find that the non-zero elements of this matrix for urs satisfy ρL = ρR + λr ,
ΛL = ΛR + λs .
(A3)
For k > 1, we can consider ΛL,R in the range 0 < ΛL ·ψ < k and 0 < ΛR ·ψ < k. Then, with ν|ΛR , ρR i = νR |ΛR , ρR i and similarly for νL , (A2) gives the following constraints for all λs in the fundamental representation, (ΛL − ΛR , ΛR + δ) + 12 (ΛL − ΛR )2 1 νR · λs − + h − µ ∈ Z; π k+N 1 ∈ Z. (νL − νR ) · λs − ~βλ2s 2
(A4)
By choosing νR in the positive Weyl chamber and excluding the extreme representations such as ΛL · ψ = 0, k and ΛR · ψ = 0, k, we find ( π π νR = (ΛR + δ) k+N , νL = (ΛL + δ) k+N , (A5) 1 νL − νR = 2 ~β(ΛL − ΛR ). Subsequently, we are lead to the following result: χ=
π 1 ~β = , 2 k+N
i.e.
π
t ≡ eiχ = ei k+N , 19
for
k ≥ 2.
(A6)
References [1] A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333. [2] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83. [3] G. Moore, N. Seiberg, Comm. Math. Phys. 123 (1989) 177; Phys. Lett. B. 212 (1988) 451. [4] G. Moore, N. Reshetikhin, Nucl. Phys. B328 (1989) 557. [5] B. Schroer, Nucl. Phys. B295 (1988) 4. [6] K. -H. Rehren Commun. Math. Phys. 116 (1988) 675. [7] L. Alvarez-Gaume, C. Gomez, G. Sierra, Phys. Lett. B220 (1989) 142. [8] A. Tsuchiya, Y. Kanie, Lett. Math. Phys. 13 (1987) 303, Advanced Studies in Pure Mathematics, 16 (1988) 297. [9] G. Felder, K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 130 (1990) 1. [10] L.D. Faddeev, Commun. Math. Phys. 132 (1990) 131. [11] M. Chu, P. Goddard, I. Halliday, D. Olive, A. Schwimmer, Phys. Lett. B266 (1991) 71. [12] A. Alekseev, S. Shatashvili, Commun. Math. Phys. 133 (1990) 353. [13] G. Mack, V. Schomerus, Nucl. Phys. B370 (1992) 185. [14] M. R. Garberdiel, “An explicit construction of the quantum group in chiral WZWmodels”, preprint DAMTP-94-51. [15] V.G. Drinfel’d, Leningrad Math. J. 1 (1990) 1419. [16] I.B. Frenkel, V.G.Kac, Invent. Math. 62 (1980) 23; G. Segal, Commun. Math. Phys. 80 (1981) 301. [17] N. Yu. Reshetikhin, “Quantised Universal Enveloping Algebras, The Yang-Baxter Equation and Invariants of Links, I, II”, LOMI preprint E-4-87. [18] A.N. Kirillov, N. Yu. Reshetikhin, “Infinite-dimensional Lie algebras and groups” edited by V. G. Kac, World Scientific, Singapore, (1989) pp 285. [19] F. Pan, Jour. Phys. A. Math. Gen. 26 (1993) 4621. [20] P. Goddard, D. Olive, Int. Jour. Mod. Phys. A1 (1986) 303. [21] See Eq. (15.3.7) in “Handbook of Mathematical Functions” edited by M. Abramowitz, I. A. Stegun, Dover Publications, Inc., New York. [22] Private communication with M. Gaberdiel. [23] P. Goddard, W. Nahm, D. Olive, A. Schwimmer, Commun. Math. Phys. 107 (1986) 179. 20
[24] M. Chu, P. Goddard, “Quantisation of a particle moving on a group manifold”, preprint DAMTP-94-41.
21
