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Extended conformal algebras generated by a multiplet of primary fields
Volume 259, number 4
PHYSICS LETTERS B
2 May 1991
Extended conformal algebras generated by a multiplet of primary fields H.G. Kausch Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and Department of Applied Mathematics and Theoretical Physics ~, University of Cambridge, Silver Street, Cambridge CB3 9EW,, UK
Received 22 January 1991
Extended conformal algebras generated by a single multiplet of primary fields of integer or half-integer spin are studied. Closed W-algebraic structures are obtained for a finite set of values of the central charge. For odd spin a series of algebras with SO (3) symmetry is found. They are present in the ( l, q) Virasoro minimal models. A free field construction for these algebras is discussed.
I. Introduction Infinite dimensional symmetry algebras play an important r61e in the study of two-dimensional conformal field theories. The study of the Virasoro algebra, which is always present in any conformal field theory, lead to the classification of all unitary, modular invariant conformal field theories with 0 < c < 1. Extended conformal algebras, or W-algebras, provide a promising approach to extend this classification to rational conformal field theories with c>~ 1. W-algebras also appear classically in the context of integrable systems o f KdV types [ 1,2 ]. Recently Walgebras have received interest in the context o f twodimensional quantum gravity [ 3-5 ]. Extended conformal algebras are operator product algebras which contain the Virasoro algebra as a subalgebra. They close on normal ordered products of the fields and their derivatives. W-algebras were introduced intuitively in ref. [6] and have since then been studied by several authors [ 1,2,7-17 ]. Many W-algebras can be found in coset models [ 18 ] based on finite dimensional Lie algebras [ 8,9 ] and have been constructed using free fields [2,7,8,10]. W-algebras can also be obtained from WZW-models and Toda theories [ l, 11 ] by hamiltonian reduction [ 19-21 ]. 1 Permanent address. 448
However, little is known about the general structure of W-algebras. The guiding principle in their study is the requirement o f associativity of the operator product algebra which is equivalent to the Jacobi identity for the commutators of the modes. W-algebras generated by a single primary field have been studied in refs. [ 1 2 - 1 5 ] . An extensive study o f Walgebras involving more than one primary field has been started in refs. [ 16,17 ]. Here we continue this study by considering multicomponent W-algebras, that is a W-algebra generated by a multiplet of primary fields. This letter is organised as follows. First we specialise the approach of ref. [ 16 ] and review the cases of spin less than or equal to 3. We then investigate the Jacobi identities and discuss a free field construction for a series o f solutions. We conclude with a summary of the results and c o m m e n t on the relation to general W-algebras.
2. Multicomponent W-algebras In this letter we will consider W-algebras generated by the stress-energy tensor and a multiplet o f primary fields W i ( z ) , i = 1..... N, of integer or half-integer spin ~. The algebra is fixed by specifying the commutators of the W ~with themselves in terms of normal ordered products of the W-fields. A special
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feature of the case where all the W-fields have the same spin/1 is that, because composite fields made out of the W ~have at least spin 2/1, the only primary fields appearing in the commutators are the identity and W( In the bosonic case the algebra is thus given by the operator product expansion
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Separating out, as above, the contributions from derivative fields and Virasoro descendent fields we can write the consistency constraint as
1 Wk(¢) Wi(z)WJ(~)= ~ o (z-~) 2~ +fijk ( z - ~ ) ~ + descendants.
( 1)
The structure constants f ok are antisymmetric for A odd and symmetric for/1 even. In the commutator [W~,, W~] we have to consider Virasoro descendent fields. The contributions from derivative fields can be summed up and we are left with a sum over quasi-primary fields. Denoting by ~//U and t~ k ' N the quasi-primary fields of weight N in the conformal family of the identity and W k, respectively, we can write the commutator as 2zI-- 1
[W~, W~]= Z p~f~(m,n) N=0
x (| C v n~,,,N +CO* V/ , Pnaa~k.u ~ / , IJgt W m + n J 'Z Am+nl k ~J t/IN
)Ck'N
) /
"
~,
X (~r+s,/ll+J2-Ll3-lcrS
r
(3)
where
c,s=(-)"
(233 -- 1 )!r!s! -2)!
( 3 1 "~/12 " [ - 3 3
X(/1'--A2--/13)( - A I r
+/12--/13) " s
where
p~M(m, n,p)=pj'M(m, n + p ) p ~ ( n , p ) .
(6)
~¢i, ~M contain contributions from the Virasoro descendants of the identity and the W q field, respectively, appearing as intermediate fields in the double commutator. They depend only on the conformal weights of the fields and can be calculated as a sum over quasi-primary fields ~,, q / a n d Z, Z' of weight M in the conformal family of the identity and W q, respectively, /"W ~
t-g.,t ,
~,W'
r,s>~O
xfm+/1,- l)(n+/1;- 1),
(5)
(2)
The coefficients fl~ are fixed by conformal invariance and can be calculated perturbatively. The pj~2 (m, n) are universal polynomials given by [ 17 ]
p~a~2(m, n ) =
=o,
.:(X'
~-x-
~-z' ,
where we denote by D the inner product matrix. The calculation of ff and ~ can easily be automated using the explicit basis of quasi-primary states introduced in ref. [ 16 ]. The analysis is particularly simple for /1~<2 as there are no Virasoro descendants, apart from the stress tensor. In these cases we obtain a Lie algebra. They have been studied in ref. [6], here we briefly review the results. For A= 1 we obtain a Kac-Moody algebra ~.
[ W~,,, W~] =fOkWkm+,, +km6°Sm+.. (4,
Provided the Jacobi identities are satisfied the W-algebra can be consistently constructed. As argued in refs. [ 16,17 ] we only have to check that the coefficient of W t in the Jacobi identity for WiWJW k vanishes. All other Jacobi identities are then automatically satisfied.
(7)
(8)
The f ok- are the structure constants of the finite dimensional Lie algebra g and the constraint (5) is simply the Jacobi identity
f oqf klq+ f ikqf Oq+ f Uqfjkq= O.
(9)
The algebra is consistent for all values ofc. For 3 = 2 the Jacobi identity (5) yields the constraint on the couplings 449
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f itqf jkq_f ikqf Oq= _ 4 ( ~ilfilk-- 6 ik~ 0 ) .
( 10 )
In this case it is natural to include the Virasoro generators as W m0 =- - Lm. The c o m m u t a t o r algebra is then given by
[ WI,~, W~] = (m--n)biJkWk, n+ . +~iJ.~cm(rn2-- 1 )tim+,,
(11)
where b~Jk= _~f~jk,bZJ°= ~tj. After a change of basis one obtains ( N + 1 ) commuting copies of the Virasoro algebra. In the fermionic case there is no coupling of three W-fields and the operator product expansion is simply c
1
Wg(z) W J ( ¢ ) = ~ c~~j~( z _ ¢ ) 2 + d e s c e n d a n t s .
(12)
In this case the associativity constraint is
-
(
~M ~'Jc~klP~(m, n,P)+fi'k~Op~(n,P, m)
AM=I
+ fiil~jkp~(p, m, n ) ) =0.
(13)
For A = ½the algebra is given by N free fermions: {We, WJs}=2CfiaO~r+s.
(14)
For A >/3 descendants of the identity, such as the stress tensor, appear in the anticommutator of two W-fields. If we can choose two different external fields, i = j # k = l say, the constraint ( 13 ) reduces to 2J-- 1
~, ~ , p ~ ( m ,
M=I
n, p) = 0 .
However, since the polynomials p ~ (m, n, p) are linearly independent, this implies that ~M has to vanish for all M~<2A--1. For M = 1 there is no quasi-primary state, while for M = 2 there is only the stress tensor, which yields f#2 = 2A2/c. Therefore there are no fermionic multicomponent W-algebras for d >/3. Algebras with a single field W are possible and have been studied in refs. [ 6,17 ]. The same argument implies that there are no bo-
sonic multicomponent W-algebras with vanishing structure constantsf'Jk for A >i 2. Multicomponent W-algebras consistent for generic 450
2 M a y 1991
c exist only for the cases A = ½, 1, 2, which have been detailed above. This can be seen as follows. Taking all four external fields equal we obtain the same associativity condition as for a W-algebra generated by the stress tensor and a single additional field of spin A. It has been shown in refs. [12,17] that such Walgebras are consistent for a continuous range of cvalues only for A = ½, 1, 3, 2, 3, 4, 6. We examined the cases A = 3, 4, 6 and found that multicomponent Walgebras are consistent at special values o f c only. We restrict attention now to bosonic algebras with A >/3 and non-vanishing structure constants. In this case we have to consider the general constraint (5). The polynomials appearing in there are not all independent. In fact, the cyclic permutations p ~ (n, p, m ) and p ~ (p, m, n) are linear combinations of the original polynomials p ~ ( m , n, p). This allows us to reduce the constraint (5) to a set o f 2 A - 1 equations on f ijqf klq, f ikqf ljq and f ,lqf jkq. (A similar approach can be found in ref. [22] where these polynomials are related to SU (2) Racah coefficients and a general associativity constraint is derived from crossing symmetry of the four-point functions. ) The resulting overdetermined system is only consistent for a finite set of values o f c for which these equations become linearly dependent. We investigated the cases of spin A = 3 , ..., 8. For N > 1 we found consistent constraints only for
(i) A=3, c = - 3 0 , (ii) A=4, c = l , - ~6, (iii) a series of odd spins, in particular there are no multicomponent W-algebras for A = 6, 8. The case of fields of spin-3 has also been studied in ref. [23], however, with a different result. We first discuss the case A = 3 , c = - 3 0 . Here (5) reduces to the single constraint filqfjkq--fikqfljq=2(2(~ij~kl--(~ikfilj--~il(~jk)
.
(15)
This is equivalent to g~jk~being totally antisymmetric, where
g,skl = f ikqf jlq + 2 ( c~itfiJk-- fi ijfikt) .
( 16 )
f a n d g transform as tensors under the S O ( N ) given by changes of basis on the W i. f i s the antisymmetrised cube o f the vector representation and can only be invariant for S O ( 3 ) . In this case g is zero and ftjk=x//~ ~tjk. For N = 2 the structure constants van-
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ish and the constraint cannot be satisfied. For N > 3 the tensor f i s no longer an invariant tensor which makes the analysis much harder.
2 May 1991
Table 1 J
c
2
3 5
-2 - 7
-~ ,76_~,
7
25 --~-
--
3. The spin 4 algebra For 3 = 4 the associativity constraint is only consistent for two values ofc. The structure constants have to satisfy f ijqf klq=•( __6ij~k/_}. (~ik~lj w c~it~)jk) ,
( 17 )
with either c = 1, 2 = ~ or c = _~_6, 2 = ~ . The constraint ( 17 ) can only be satisfied for at most two fields W i, which can easily be seen by introducing vectors ( Ui ) q= f iiq, ( uij ) qm f ijk, i
ui.v"=O,
v".va=2~ "a,
(18)
where a, fl label pairs of indices (ij). We have ½N(N-1) orthogonal vectors v" and at least one other vector u ~ orthogonal to all the v s. Since these vectors are N-dimensional we can have at most two fields. The solutions are then given by [ 16 ] U=l:
fill=
N=2:
flll=--f122=K+,
(19)
+N/~ , f222=--fllZ=K-,
K + 2 "[-/~-- 2 =/~ .
(20)
Note that both values o f c can be found in the minimal series for D4 for (p, q) = (8, 7 ) and ( 11, 6). The W D 4 algebra is generated by two primary fields of spin-4 and one field of spin-6. Presumably precisely at these two c-values the spin-6 field becomes composite.
4. Series with SO(3) symmetry All other values o f c and J for which the associativity constraint is consistent fall into a series of algebras with an SO (3) symmetry. In each case the condition on the couplings is f ijqf klq = 2 ( ~ ik(~lj -- s ilsjk ) .
(21)
For the cases investigated we have the values shown in table 1. To investigate solutions to (21) we intro-
61347 14
duce v e c t o r s (l)ij)q=f ijq, 1 <~i
(22)
We thus need ½N(N-1) orthogonal vectors of dimension N. This is only possible for N~< 3. It immediately follows that the only solutions are N=I"
fiJk=0,
(23)
N=3:
fiJk=q-X/~.Ok.
(24)
More information about these solutions is obtained by noticing that they can be found as non unitary minimal models for the Virasoro algebra. In particular, for the ( 1, q) models we have c=l
6(q--l) 2 - - , q
Ai,2n+l = n [ ( n + l ) q - -
1] .
(25)
Table 1 is reproduced for q = 2 , 3, 4 with the fields W j given by the fields W i given by the field ~,3. The fields ~l.s, s odd, form a closed subset o f integer spin fields with su(2) fusion rules. In particular for q~l,3 we have [ 24 ] 01,3 X~I,3 :(~1,1 "]- ~1,3 "~-01,5 -
(26)
However, ~J,s does not appear in the singular part of the operator product expansion since its spin is too high. This suggests the existence of a whole series of W-algebras based on the ~1,3 field in the ( 1, q) minimal models. This is indeed the case and we will give a construction below.
5. Free field construction A free field construction for the Virasoro minimal models has been proposed in ref. [25 ] and has been proven in ref. [26 ]. However, for the (p, q) minimal model, these papers concentrate on the finite opera451
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tor product algebra given by the fields 0~,s with 1 ~
[am, a.] =mOm+n.
(28)
and has central charge c=l_12p2=l
6 ( q - 1) 2 q
The vertex operators V(2, z) =exp[i2X< (z)] exp(i2q)z xa° Xexp[i2X> (z)] ,
(32)
where Xe (z) = - ~ , e o ( a n / n ) z - ' , for momenta 2 is the rescaled root lattice of su(2), A q = Z x / ~ , create integer spin representations of su(2). Specifically V " ( z ) = V( - n v / ~ , z) creates n units of momentum and has conformal weight An=n[ ( n + 1 ) s - 1 ]. Applied to the vacuum it creates the highest weight state for a spin n representation of su (2). To obtain the other fields of the su(2) representation we need a charge lowering operator which leaves the conformal weight unchanged. The vertex operator V - l ( z ) has conformal weight 1. Integrating it along a suitable contour gives us the required lowering operator J-=
~
V-~(co).
(33)
The full su (2) representation is obtained by repeatedly acting with J - on the highest weight field. This leads to the screened vertex operators
(27)
The space of states of the model is taken as the direct sum of bosonic Fock spaces ~ generated by the action of the creation operators a,, n < 0, from highest weight vectors 12), satisfying a o [ 2 ) = 2 1 2 ) and a. [ 2 ) = 0 , for n > 0. The Virasoro algebra is implemented by
L~=½ ~ :am_~ak:-p(m+l)am,
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v . , r ( z ) = f dm~ dt-°r V-l((.Ol)...V-l((A)r)Vn(Z), 2zri "'" 2rri (34) where r = 0 , ..., 2n. For r > 2n the conformal weight of the highest weight state An_r would be greater than the weight of V ".r which therefore has to vanish. To make the correspondence to the W-algebra explicit we write the W-algebra in a Cartan-Weyl basis, H = W 3, E -+= W ~_+i W 2, which yields the fusion rules c 1 H ( z ) H ( ( ) = A (Z--() 4q-2 +descendants,
(35)
E-+(() H ( z ) E ± (() = + x (z_()2q_ , +descendants,
(36)
(29)
where q--1 p= x//~.
(30)
The momentum eigenstates are Virasoro highest weight states with Lo 12> = ½ 2 ( 2 - 2 p ) 1 2 ) •
(31)
E ± ( z ) E ~- ( ( ) -
+ descendants,
452
(37)
where x2= - 2 . The correspondence of these fields to the vertex operators is then E +,-~V l'°,
#' This explains the entries in table 4.4 ofref. [ 17] with (p, q) not coprime.
c 1 H(() +_to 3 ( z - ~ W -2 ( z - ( ) 2~-1
H ~ V la,
E - ~ V 1'2.
(38)
To fix the normalisation we calculate the two-point
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functions. Explicitly evaluating the integrals over the screening operators yields ( V 1'1 ( z ) g 1"1 ( I x ) ) : - ( V I ' 0 ( Z )
V l'2(ix) )
(39)
N,
-- (Z__ Ix)4q-2 ,
where the constant Nq is given by a Wigner 3j-symbol Nq=
2q
( 2 q - 1)! x(2qo1
~/(6q-3)! (2q)!
2q--lo
=(-1) u
2q02)
2 (3q--2)! 2 q - 1 ( q - 1)! 3"
(40)
The three-point function relevant for the calculation of the structure constants is
This agrees with table 1. We have thus a series of Walgebras with SO(3) symmetry generated by three fields of odd spin. By applying the screened vertex operators to the vacuum we obtain a state consisting of a finite expression in oscillators on a momentum highest weight state. By the isomorphism between states and fields we can thus express any of the fields in the Walgebra as a product of a polynomial expression in the spin-1 momentum field P ( z ) = ~ , , a , , z - " - ~ and a vertex operator V m ( z ) for the momentum highest weight state. As an example consider H ( z ) which maps ~ into itself. This means that we can express the state IH ) = limz~0H(z) 10) in terms of the oscillators alone. The corresponding field is then simply the polynomial in P ( z ) specified by this state. Explicitly taking the limit we can derive the general expression 1
( VI'0(Z 1 ) V IA (z2) V I ' 2 ( z 3 ) ) =
2 May 1991
IH) - - ( 2 q - 1 )!
Cq (-71 - - Z 2 ) 2 q - I ( z I --Z3) 2 q - l ( Z 2 - Z 3 ) 2q-I '
(41)
×02_.q-'{exp[ix/~X<(z)]}z=o[O)
(45)
.
The first few cases are
where
q=l,
4q-2 Cq=(2q_
(42)
l )N q .
The first few of these are shown in table 2. We can now identify the fields in the W-algebra (35) as H=N/~
d=l,
IH) =a_~10), q=2,
A=3,
c=-2:
[ n ) = Y ~2l ~, ~j ~t ~ - 3 + 3Z~~, :, : " - 2a- - 1 + a 3 1) 1 0 ) ,
q=3,
VI'I,
c=l:
A=5,
c=-7:
IH) =3Q(3a_sa_2+~Qa_3a_2+~Qa_4a_,
E+=I " ~
VI'° ,
E-=I '£-~q
V 1'2 .
(43)
From this we obtain the general formula for the structure constant C C 2 ___(_l)qc
2-- ~Nq3
(4q--2)!(q--1)! 3 2 ( 3 q - 2 ) ! ( 2 q - 1)! 4.
(44)
+5a2
~
-2--1
q=4,
+lo~
A=7,
.,2
5
3---3,.-1+sQa-2a-l
3
+a5-1)[O)
,
25.
c=-~-.
H"
32 ~z45 -- t°5"a a +63"a a ) =3-i3{d{,57a-7-t-l-~gd - 4 - 3 ~[,~ -5 - 2 lOS 2 + 1 6 2 . ~ Q a _ 6 a - T±35~2 37-a- 3a -2 1 -V ¢*_ 3 t* _ 1 / 105fl~3 ~ --63 2
+ ~a_aa_ 2a_ 1 " I - ~ - ~ 1 . 4
--2"--
l
+ T a _ sa
_ 1
105 2 A_ 105/q~ ~3 -- 105 2 3 + T K Q a _ 3a_ 2a _ l T~-~._41a-_ 1-,r--g-a_2a_ l
Table 2 q
Nq
1
-2
2 3 4
16 -252 4800
+35 a a4 21 5 +aT_~) 10) 'a- -3 -L + y Q a _ 2 a _ ~
Cq -4
320 -63504 16473600
where Q = x / ~ . The existence of these polynomial expressions for the 0~.3 states was anticipated in ref.
[3o]. 453
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6. Discussion
Acknowledgement
In this letter we considered W-algebras generated by a multiplet of primary fields. The only fermionic algebra is given by a set of free fermions, The only bosonic m u l t i c o m p o n e n t algebras consistent for generic c are K a c - M o o d y algebras and tensor products of the Virasoro algebra. For spins greater than two the algebras are consistent for special values o f c only a n d have necessarily n o n - v a n i s h i n g structure constants. Apart from the case A = 3 , c = - 3 0 , for which we have no construction so far, we found an algebra generated by two fields of spin 4, which we conjecture to be a reduction of the WD4 Casimir algebra, and a series of algebras of odd spin. Using a free field construction this series can be constructed as a deform a t i o n of the usual vertexo ~ r a t o r construction for the K a c - M o o d y algebra s u ( 2 ) . An analogous construction of the ( 1, q) m i n i m a l models for other Lie algebras might also produce m u l t i c o m p o n e n t W-algebras with integer weight fields. Note that c = - 3 0 is the c-value for the ( 1, 3 ) m i n i m a l model of su (3). The system of constraints becomes more over-determ i n e d with increasing spin. Any further solutions not based upon such m i n i m a l models seem therefore unlikely. It can happen that for a higher rank W-algebra some of the basic fields become composite at specific values of the central charge. The two solutions c = 1, _ ~ 6 of the W(4.4 ) algebra are contained in the minimal series of D4 at (p, q) = (7, 8) and (6, 11 ). Apart from this c = 1 solution we have found no other unitary m u l t i c o m p o n e n t algebra of spin higher than 2. This places restrictions on W-algebras consistent for generic c. If the basic fields of such an algebra contains several fields of the same spin then there can be no unitary representation for which only these fields and the stress-energy tensor are basic, with the exception of the WD4 algebra. Note that D2n are the only semi-simple Lie algebras with two i n d e p e n d e n t Casimir operators of the same dimension, 2n. For W D 4 these basic fields have spins 2, 4, 4, 6 and at c = 1 the spin-6 fields becomes composite. The next case in this series is W D 6 which has basic fields of spin 2, 4, 6, 6, 8, 10. The results of this letter imply that it is not possible for the fields of spin 4, 8 and 10 to become composite at the same time.
I would like to thank P. Bowcock, G. Felder, P. Goddard, A. Kent and G. Watts for stimulating discussions. I would like to thank the Gottlieb Daimleru n d Karl Benz-Stiftung for a research scholarship. This research was supported in part by the National Science F o u n d a t i o n u n d e r G r a n t No. PHY89-04035 at the University of California at Santa Barbara.
454
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[20] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 ( 1989 ) 49. [ 21 ] J.M. Figueroa-O'Farrill, Leuven University preprint KULTF-90/12 (1990). [22] P. Bowcock, Enrico Fermi Institute preprint EFI-90-54 (1990). [23] S.A. Apikyan, Mod. Phys. Lett. A 2 (1987) 317. [24] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333.
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[25] V.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [FS12] (1984) 312; B 251 [FS13] (1985) 691; Phys. Lett. B 154 (1985) 291. [26] G. Felder, Nucl. Phys. B 317 (1989) 215. [ 27 ] M. Wakimoto, Commun. Math. Phys. 104 (1990) 605. [28] D. Bernard and G. Felder, Commun. Math. Phys. 127 (1990) 145. [29] H.G. Kausch, in preparation. [30] P. Bouwknegt, A. Ceresole, J.G. McCarthy and P. van Nieuwenhuizen, Phys. Rev. D 40 (1989) 415.
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Extended conformal algebras generated by a multiplet of primary fields H.G. Kausch Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and Department of Applied Mathematics and Theoretical Physics ~, University of Cambridge, Silver Street, Cambridge CB3 9EW,, UK
Received 22 January 1991
Extended conformal algebras generated by a single multiplet of primary fields of integer or half-integer spin are studied. Closed W-algebraic structures are obtained for a finite set of values of the central charge. For odd spin a series of algebras with SO (3) symmetry is found. They are present in the ( l, q) Virasoro minimal models. A free field construction for these algebras is discussed.
I. Introduction Infinite dimensional symmetry algebras play an important r61e in the study of two-dimensional conformal field theories. The study of the Virasoro algebra, which is always present in any conformal field theory, lead to the classification of all unitary, modular invariant conformal field theories with 0 < c < 1. Extended conformal algebras, or W-algebras, provide a promising approach to extend this classification to rational conformal field theories with c>~ 1. W-algebras also appear classically in the context of integrable systems o f KdV types [ 1,2 ]. Recently Walgebras have received interest in the context o f twodimensional quantum gravity [ 3-5 ]. Extended conformal algebras are operator product algebras which contain the Virasoro algebra as a subalgebra. They close on normal ordered products of the fields and their derivatives. W-algebras were introduced intuitively in ref. [6] and have since then been studied by several authors [ 1,2,7-17 ]. Many W-algebras can be found in coset models [ 18 ] based on finite dimensional Lie algebras [ 8,9 ] and have been constructed using free fields [2,7,8,10]. W-algebras can also be obtained from WZW-models and Toda theories [ l, 11 ] by hamiltonian reduction [ 19-21 ]. 1 Permanent address. 448
However, little is known about the general structure of W-algebras. The guiding principle in their study is the requirement o f associativity of the operator product algebra which is equivalent to the Jacobi identity for the commutators of the modes. W-algebras generated by a single primary field have been studied in refs. [ 1 2 - 1 5 ] . An extensive study o f Walgebras involving more than one primary field has been started in refs. [ 16,17 ]. Here we continue this study by considering multicomponent W-algebras, that is a W-algebra generated by a multiplet of primary fields. This letter is organised as follows. First we specialise the approach of ref. [ 16 ] and review the cases of spin less than or equal to 3. We then investigate the Jacobi identities and discuss a free field construction for a series o f solutions. We conclude with a summary of the results and c o m m e n t on the relation to general W-algebras.
2. Multicomponent W-algebras In this letter we will consider W-algebras generated by the stress-energy tensor and a multiplet o f primary fields W i ( z ) , i = 1..... N, of integer or half-integer spin ~. The algebra is fixed by specifying the commutators of the W ~with themselves in terms of normal ordered products of the W-fields. A special
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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feature of the case where all the W-fields have the same spin/1 is that, because composite fields made out of the W ~have at least spin 2/1, the only primary fields appearing in the commutators are the identity and W( In the bosonic case the algebra is thus given by the operator product expansion
2 May 1991
Separating out, as above, the contributions from derivative fields and Virasoro descendent fields we can write the consistency constraint as
1 Wk(¢) Wi(z)WJ(~)= ~ o (z-~) 2~ +fijk ( z - ~ ) ~ + descendants.
( 1)
The structure constants f ok are antisymmetric for A odd and symmetric for/1 even. In the commutator [W~,, W~] we have to consider Virasoro descendent fields. The contributions from derivative fields can be summed up and we are left with a sum over quasi-primary fields. Denoting by ~//U and t~ k ' N the quasi-primary fields of weight N in the conformal family of the identity and W k, respectively, we can write the commutator as 2zI-- 1
[W~, W~]= Z p~f~(m,n) N=0
x (| C v n~,,,N +CO* V/ , Pnaa~k.u ~ / , IJgt W m + n J 'Z Am+nl k ~J t/IN
)Ck'N
) /
"
~,
X (~r+s,/ll+J2-Ll3-lcrS
r
(3)
where
c,s=(-)"
(233 -- 1 )!r!s! -2)!
( 3 1 "~/12 " [ - 3 3
X(/1'--A2--/13)( - A I r
+/12--/13) " s
where
p~M(m, n,p)=pj'M(m, n + p ) p ~ ( n , p ) .
(6)
~¢i, ~M contain contributions from the Virasoro descendants of the identity and the W q field, respectively, appearing as intermediate fields in the double commutator. They depend only on the conformal weights of the fields and can be calculated as a sum over quasi-primary fields ~,, q / a n d Z, Z' of weight M in the conformal family of the identity and W q, respectively, /"W ~
t-g.,t ,
~,W'
r,s>~O
xfm+/1,- l)(n+/1;- 1),
(5)
(2)
The coefficients fl~ are fixed by conformal invariance and can be calculated perturbatively. The pj~2 (m, n) are universal polynomials given by [ 17 ]
p~a~2(m, n ) =
=o,
.:(X'
~-x-
~-z' ,
where we denote by D the inner product matrix. The calculation of ff and ~ can easily be automated using the explicit basis of quasi-primary states introduced in ref. [ 16 ]. The analysis is particularly simple for /1~<2 as there are no Virasoro descendants, apart from the stress tensor. In these cases we obtain a Lie algebra. They have been studied in ref. [6], here we briefly review the results. For A= 1 we obtain a Kac-Moody algebra ~.
[ W~,,, W~] =fOkWkm+,, +km6°Sm+.. (4,
Provided the Jacobi identities are satisfied the W-algebra can be consistently constructed. As argued in refs. [ 16,17 ] we only have to check that the coefficient of W t in the Jacobi identity for WiWJW k vanishes. All other Jacobi identities are then automatically satisfied.
(7)
(8)
The f ok- are the structure constants of the finite dimensional Lie algebra g and the constraint (5) is simply the Jacobi identity
f oqf klq+ f ikqf Oq+ f Uqfjkq= O.
(9)
The algebra is consistent for all values ofc. For 3 = 2 the Jacobi identity (5) yields the constraint on the couplings 449
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f itqf jkq_f ikqf Oq= _ 4 ( ~ilfilk-- 6 ik~ 0 ) .
( 10 )
In this case it is natural to include the Virasoro generators as W m0 =- - Lm. The c o m m u t a t o r algebra is then given by
[ WI,~, W~] = (m--n)biJkWk, n+ . +~iJ.~cm(rn2-- 1 )tim+,,
(11)
where b~Jk= _~f~jk,bZJ°= ~tj. After a change of basis one obtains ( N + 1 ) commuting copies of the Virasoro algebra. In the fermionic case there is no coupling of three W-fields and the operator product expansion is simply c
1
Wg(z) W J ( ¢ ) = ~ c~~j~( z _ ¢ ) 2 + d e s c e n d a n t s .
(12)
In this case the associativity constraint is
-
(
~M ~'Jc~klP~(m, n,P)+fi'k~Op~(n,P, m)
AM=I
+ fiil~jkp~(p, m, n ) ) =0.
(13)
For A = ½the algebra is given by N free fermions: {We, WJs}=2CfiaO~r+s.
(14)
For A >/3 descendants of the identity, such as the stress tensor, appear in the anticommutator of two W-fields. If we can choose two different external fields, i = j # k = l say, the constraint ( 13 ) reduces to 2J-- 1
~, ~ , p ~ ( m ,
M=I
n, p) = 0 .
However, since the polynomials p ~ (m, n, p) are linearly independent, this implies that ~M has to vanish for all M~<2A--1. For M = 1 there is no quasi-primary state, while for M = 2 there is only the stress tensor, which yields f#2 = 2A2/c. Therefore there are no fermionic multicomponent W-algebras for d >/3. Algebras with a single field W are possible and have been studied in refs. [ 6,17 ]. The same argument implies that there are no bo-
sonic multicomponent W-algebras with vanishing structure constantsf'Jk for A >i 2. Multicomponent W-algebras consistent for generic 450
2 M a y 1991
c exist only for the cases A = ½, 1, 2, which have been detailed above. This can be seen as follows. Taking all four external fields equal we obtain the same associativity condition as for a W-algebra generated by the stress tensor and a single additional field of spin A. It has been shown in refs. [12,17] that such Walgebras are consistent for a continuous range of cvalues only for A = ½, 1, 3, 2, 3, 4, 6. We examined the cases A = 3, 4, 6 and found that multicomponent Walgebras are consistent at special values o f c only. We restrict attention now to bosonic algebras with A >/3 and non-vanishing structure constants. In this case we have to consider the general constraint (5). The polynomials appearing in there are not all independent. In fact, the cyclic permutations p ~ (n, p, m ) and p ~ (p, m, n) are linear combinations of the original polynomials p ~ ( m , n, p). This allows us to reduce the constraint (5) to a set o f 2 A - 1 equations on f ijqf klq, f ikqf ljq and f ,lqf jkq. (A similar approach can be found in ref. [22] where these polynomials are related to SU (2) Racah coefficients and a general associativity constraint is derived from crossing symmetry of the four-point functions. ) The resulting overdetermined system is only consistent for a finite set of values o f c for which these equations become linearly dependent. We investigated the cases of spin A = 3 , ..., 8. For N > 1 we found consistent constraints only for
(i) A=3, c = - 3 0 , (ii) A=4, c = l , - ~6, (iii) a series of odd spins, in particular there are no multicomponent W-algebras for A = 6, 8. The case of fields of spin-3 has also been studied in ref. [23], however, with a different result. We first discuss the case A = 3 , c = - 3 0 . Here (5) reduces to the single constraint filqfjkq--fikqfljq=2(2(~ij~kl--(~ikfilj--~il(~jk)
.
(15)
This is equivalent to g~jk~being totally antisymmetric, where
g,skl = f ikqf jlq + 2 ( c~itfiJk-- fi ijfikt) .
( 16 )
f a n d g transform as tensors under the S O ( N ) given by changes of basis on the W i. f i s the antisymmetrised cube o f the vector representation and can only be invariant for S O ( 3 ) . In this case g is zero and ftjk=x//~ ~tjk. For N = 2 the structure constants van-
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ish and the constraint cannot be satisfied. For N > 3 the tensor f i s no longer an invariant tensor which makes the analysis much harder.
2 May 1991
Table 1 J
c
2
3 5
-2 - 7
-~ ,76_~,
7
25 --~-
--
3. The spin 4 algebra For 3 = 4 the associativity constraint is only consistent for two values ofc. The structure constants have to satisfy f ijqf klq=•( __6ij~k/_}. (~ik~lj w c~it~)jk) ,
( 17 )
with either c = 1, 2 = ~ or c = _~_6, 2 = ~ . The constraint ( 17 ) can only be satisfied for at most two fields W i, which can easily be seen by introducing vectors ( Ui ) q= f iiq, ( uij ) qm f ijk, i
ui.v"=O,
v".va=2~ "a,
(18)
where a, fl label pairs of indices (ij). We have ½N(N-1) orthogonal vectors v" and at least one other vector u ~ orthogonal to all the v s. Since these vectors are N-dimensional we can have at most two fields. The solutions are then given by [ 16 ] U=l:
fill=
N=2:
flll=--f122=K+,
(19)
+N/~ , f222=--fllZ=K-,
K + 2 "[-/~-- 2 =/~ .
(20)
Note that both values o f c can be found in the minimal series for D4 for (p, q) = (8, 7 ) and ( 11, 6). The W D 4 algebra is generated by two primary fields of spin-4 and one field of spin-6. Presumably precisely at these two c-values the spin-6 field becomes composite.
4. Series with SO(3) symmetry All other values o f c and J for which the associativity constraint is consistent fall into a series of algebras with an SO (3) symmetry. In each case the condition on the couplings is f ijqf klq = 2 ( ~ ik(~lj -- s ilsjk ) .
(21)
For the cases investigated we have the values shown in table 1. To investigate solutions to (21) we intro-
61347 14
duce v e c t o r s (l)ij)q=f ijq, 1 <~i
(22)
We thus need ½N(N-1) orthogonal vectors of dimension N. This is only possible for N~< 3. It immediately follows that the only solutions are N=I"
fiJk=0,
(23)
N=3:
fiJk=q-X/~.Ok.
(24)
More information about these solutions is obtained by noticing that they can be found as non unitary minimal models for the Virasoro algebra. In particular, for the ( 1, q) models we have c=l
6(q--l) 2 - - , q
Ai,2n+l = n [ ( n + l ) q - -
1] .
(25)
Table 1 is reproduced for q = 2 , 3, 4 with the fields W j given by the fields W i given by the field ~,3. The fields ~l.s, s odd, form a closed subset o f integer spin fields with su(2) fusion rules. In particular for q~l,3 we have [ 24 ] 01,3 X~I,3 :(~1,1 "]- ~1,3 "~-01,5 -
(26)
However, ~J,s does not appear in the singular part of the operator product expansion since its spin is too high. This suggests the existence of a whole series of W-algebras based on the ~1,3 field in the ( 1, q) minimal models. This is indeed the case and we will give a construction below.
5. Free field construction A free field construction for the Virasoro minimal models has been proposed in ref. [25 ] and has been proven in ref. [26 ]. However, for the (p, q) minimal model, these papers concentrate on the finite opera451
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tor product algebra given by the fields 0~,s with 1 ~
[am, a.] =mOm+n.
(28)
and has central charge c=l_12p2=l
6 ( q - 1) 2 q
The vertex operators V(2, z) =exp[i2X< (z)] exp(i2q)z xa° Xexp[i2X> (z)] ,
(32)
where Xe (z) = - ~ , e o ( a n / n ) z - ' , for momenta 2 is the rescaled root lattice of su(2), A q = Z x / ~ , create integer spin representations of su(2). Specifically V " ( z ) = V( - n v / ~ , z) creates n units of momentum and has conformal weight An=n[ ( n + 1 ) s - 1 ]. Applied to the vacuum it creates the highest weight state for a spin n representation of su (2). To obtain the other fields of the su(2) representation we need a charge lowering operator which leaves the conformal weight unchanged. The vertex operator V - l ( z ) has conformal weight 1. Integrating it along a suitable contour gives us the required lowering operator J-=
~
V-~(co).
(33)
The full su (2) representation is obtained by repeatedly acting with J - on the highest weight field. This leads to the screened vertex operators
(27)
The space of states of the model is taken as the direct sum of bosonic Fock spaces ~ generated by the action of the creation operators a,, n < 0, from highest weight vectors 12), satisfying a o [ 2 ) = 2 1 2 ) and a. [ 2 ) = 0 , for n > 0. The Virasoro algebra is implemented by
L~=½ ~ :am_~ak:-p(m+l)am,
2 May 1991
v . , r ( z ) = f dm~ dt-°r V-l((.Ol)...V-l((A)r)Vn(Z), 2zri "'" 2rri (34) where r = 0 , ..., 2n. For r > 2n the conformal weight of the highest weight state An_r would be greater than the weight of V ".r which therefore has to vanish. To make the correspondence to the W-algebra explicit we write the W-algebra in a Cartan-Weyl basis, H = W 3, E -+= W ~_+i W 2, which yields the fusion rules c 1 H ( z ) H ( ( ) = A (Z--() 4q-2 +descendants,
(35)
E-+(() H ( z ) E ± (() = + x (z_()2q_ , +descendants,
(36)
(29)
where q--1 p= x//~.
(30)
The momentum eigenstates are Virasoro highest weight states with Lo 12> = ½ 2 ( 2 - 2 p ) 1 2 ) •
(31)
E ± ( z ) E ~- ( ( ) -
+ descendants,
452
(37)
where x2= - 2 . The correspondence of these fields to the vertex operators is then E +,-~V l'°,
#' This explains the entries in table 4.4 ofref. [ 17] with (p, q) not coprime.
c 1 H(() +_to 3 ( z - ~ W -2 ( z - ( ) 2~-1
H ~ V la,
E - ~ V 1'2.
(38)
To fix the normalisation we calculate the two-point
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functions. Explicitly evaluating the integrals over the screening operators yields ( V 1'1 ( z ) g 1"1 ( I x ) ) : - ( V I ' 0 ( Z )
V l'2(ix) )
(39)
N,
-- (Z__ Ix)4q-2 ,
where the constant Nq is given by a Wigner 3j-symbol Nq=
2q
( 2 q - 1)! x(2qo1
~/(6q-3)! (2q)!
2q--lo
=(-1) u
2q02)
2 (3q--2)! 2 q - 1 ( q - 1)! 3"
(40)
The three-point function relevant for the calculation of the structure constants is
This agrees with table 1. We have thus a series of Walgebras with SO(3) symmetry generated by three fields of odd spin. By applying the screened vertex operators to the vacuum we obtain a state consisting of a finite expression in oscillators on a momentum highest weight state. By the isomorphism between states and fields we can thus express any of the fields in the Walgebra as a product of a polynomial expression in the spin-1 momentum field P ( z ) = ~ , , a , , z - " - ~ and a vertex operator V m ( z ) for the momentum highest weight state. As an example consider H ( z ) which maps ~ into itself. This means that we can express the state IH ) = limz~0H(z) 10) in terms of the oscillators alone. The corresponding field is then simply the polynomial in P ( z ) specified by this state. Explicitly taking the limit we can derive the general expression 1
( VI'0(Z 1 ) V IA (z2) V I ' 2 ( z 3 ) ) =
2 May 1991
IH) - - ( 2 q - 1 )!
Cq (-71 - - Z 2 ) 2 q - I ( z I --Z3) 2 q - l ( Z 2 - Z 3 ) 2q-I '
(41)
×02_.q-'{exp[ix/~X<(z)]}z=o[O)
(45)
.
The first few cases are
where
q=l,
4q-2 Cq=(2q_
(42)
l )N q .
The first few of these are shown in table 2. We can now identify the fields in the W-algebra (35) as H=N/~
d=l,
IH) =a_~10), q=2,
A=3,
c=-2:
[ n ) = Y ~2l ~, ~j ~t ~ - 3 + 3Z~~, :, : " - 2a- - 1 + a 3 1) 1 0 ) ,
q=3,
VI'I,
c=l:
A=5,
c=-7:
IH) =3Q(3a_sa_2+~Qa_3a_2+~Qa_4a_,
E+=I " ~
VI'° ,
E-=I '£-~q
V 1'2 .
(43)
From this we obtain the general formula for the structure constant C C 2 ___(_l)qc
2-- ~Nq3
(4q--2)!(q--1)! 3 2 ( 3 q - 2 ) ! ( 2 q - 1)! 4.
(44)
+5a2
~
-2--1
q=4,
+lo~
A=7,
.,2
5
3---3,.-1+sQa-2a-l
3
+a5-1)[O)
,
25.
c=-~-.
H"
32 ~z45 -- t°5"a a +63"a a ) =3-i3{d{,57a-7-t-l-~gd - 4 - 3 ~[,~ -5 - 2 lOS 2 + 1 6 2 . ~ Q a _ 6 a - T±35~2 37-a- 3a -2 1 -V ¢*_ 3 t* _ 1 / 105fl~3 ~ --63 2
+ ~a_aa_ 2a_ 1 " I - ~ - ~ 1 . 4
--2"--
l
+ T a _ sa
_ 1
105 2 A_ 105/q~ ~3 -- 105 2 3 + T K Q a _ 3a_ 2a _ l T~-~._41a-_ 1-,r--g-a_2a_ l
Table 2 q
Nq
1
-2
2 3 4
16 -252 4800
+35 a a4 21 5 +aT_~) 10) 'a- -3 -L + y Q a _ 2 a _ ~
Cq -4
320 -63504 16473600
where Q = x / ~ . The existence of these polynomial expressions for the 0~.3 states was anticipated in ref.
[3o]. 453
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2 May 1991
6. Discussion
Acknowledgement
In this letter we considered W-algebras generated by a multiplet of primary fields. The only fermionic algebra is given by a set of free fermions, The only bosonic m u l t i c o m p o n e n t algebras consistent for generic c are K a c - M o o d y algebras and tensor products of the Virasoro algebra. For spins greater than two the algebras are consistent for special values o f c only a n d have necessarily n o n - v a n i s h i n g structure constants. Apart from the case A = 3 , c = - 3 0 , for which we have no construction so far, we found an algebra generated by two fields of spin 4, which we conjecture to be a reduction of the WD4 Casimir algebra, and a series of algebras of odd spin. Using a free field construction this series can be constructed as a deform a t i o n of the usual vertexo ~ r a t o r construction for the K a c - M o o d y algebra s u ( 2 ) . An analogous construction of the ( 1, q) m i n i m a l models for other Lie algebras might also produce m u l t i c o m p o n e n t W-algebras with integer weight fields. Note that c = - 3 0 is the c-value for the ( 1, 3 ) m i n i m a l model of su (3). The system of constraints becomes more over-determ i n e d with increasing spin. Any further solutions not based upon such m i n i m a l models seem therefore unlikely. It can happen that for a higher rank W-algebra some of the basic fields become composite at specific values of the central charge. The two solutions c = 1, _ ~ 6 of the W(4.4 ) algebra are contained in the minimal series of D4 at (p, q) = (7, 8) and (6, 11 ). Apart from this c = 1 solution we have found no other unitary m u l t i c o m p o n e n t algebra of spin higher than 2. This places restrictions on W-algebras consistent for generic c. If the basic fields of such an algebra contains several fields of the same spin then there can be no unitary representation for which only these fields and the stress-energy tensor are basic, with the exception of the WD4 algebra. Note that D2n are the only semi-simple Lie algebras with two i n d e p e n d e n t Casimir operators of the same dimension, 2n. For W D 4 these basic fields have spins 2, 4, 4, 6 and at c = 1 the spin-6 fields becomes composite. The next case in this series is W D 6 which has basic fields of spin 2, 4, 6, 6, 8, 10. The results of this letter imply that it is not possible for the fields of spin 4, 8 and 10 to become composite at the same time.
I would like to thank P. Bowcock, G. Felder, P. Goddard, A. Kent and G. Watts for stimulating discussions. I would like to thank the Gottlieb Daimleru n d Karl Benz-Stiftung for a research scholarship. This research was supported in part by the National Science F o u n d a t i o n u n d e r G r a n t No. PHY89-04035 at the University of California at Santa Barbara.
454
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[20] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 ( 1989 ) 49. [ 21 ] J.M. Figueroa-O'Farrill, Leuven University preprint KULTF-90/12 (1990). [22] P. Bowcock, Enrico Fermi Institute preprint EFI-90-54 (1990). [23] S.A. Apikyan, Mod. Phys. Lett. A 2 (1987) 317. [24] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333.
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