Feigin-Fuchs representation of generalized parafermions

Volume 252, number l

PHYSICS LETTERS B

6 December 1990

Feigin-Fuchs representation of generalized parafermions ¢r K a t s u s h i Ito 1

YukawaInstitutefor TheoreticalPhysics,Kyoto University,Kyoto606,Japan Received 2 August 1990; revised manuscript received l September 1990

We study the Feigin-Fuchs representations of the generalized parafermions associated with arbitrary affine Lie algebras. The formulas for the generating parafermionic currents are obtained by factorizing the Caftan part from the Kac-Moody currents.

Rational conformal field theories are characterized by the chiral algebras. An important problem is to explicitly formulate the chiral algebras and to construct their representations. For this purpose the Feigin-Fuchs representation [ l ] is a useful approach because the generating chiral currents are expressed in terms of free bosons, which makes the computations o f correlation functions and singular vectors explicit. Since a wide class of rational conformal field theories can be realized as coset models, the most fundamental model seems to be the W e s s - Z u m i n o - W i t t e n model based on a Lie group G. The Feigin-Fuchs constructions o f the affine Lie algebras are currently studied by several groups [ 2-6 ]. On the other hand the general coset models are not yet understood systematically because of the lack o f our knowledge o f the chiral algebra structure. We notice however that the coset models are decomposed into the generalized parafermions and the free bosons coupled to the world sheet curvature [ 7 ]. The parafermionic algebra generated by the fractional spin chiral currents is recognized to form a fundamental class o f the chiral algebras [ 8 ]. Hence it is important to study the structure of the parafermionic current algebras in order to further understand the properties o f the coset conformal field theories. In this note we will construct the Feigin-Fuchs representations of the generalized parafermions associated with the affine Lie algebras. The generalized parafermions are realized by the coset models G / U (1)n [ 9,10 ], where n is the rank o f a complex simple Lie group G. The present construction will be performed by factorizing the Caftan part from the K a c - M o o d y currents in the Feigin-Fuchs representation which is systematically obtained by K o m a t a and the author [ 5 ] based on the geometrical point of view [ 2 ]. We start from the Feigin-Fuchs representations of affine Lie algebras. If we take the Chevalley basis, the operator product expansions between the currents J~,(z) (a~ A) and Hi(z) ( i = l, ..., n), where A is a set of roots and n is the rank of a Lie algebra g, are expressed as follows:

J~(z) Jp(w)= N~'pJ~÷p(w) +... f o r a + p e A ,

2k/a 2 2a.H(w)/a 2 J~(z) J_~(w)= ( z _ w ) ~ + +...

Z--W

k,~ij Hi(z) Ja(w)= ot'J,~(W)z_w +"" ' Hi(z) HJ(w)= ( z - w ) 2 + "'" .

Z--W

'

(1)

In the Feigin-Fuchs representations the K a c - M o o d y currents are expressed as the first order differential operators on the infinite dimensional flag manifolds associated with the affine Lie algebras [2,4,5 ]. The field y~,(z) ~r Worksupported in part by the Grant in Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan No. 02952037. Fellow of the Japan Society for the Promotion of Science. z Bitnetaddress: [email protected] 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

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Volume 252, number

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1990

represents the coordinates of the flag manifold and Pa(z) represents the conjugate differential operators, where asA+ and A+ is a set of positive roots. Here/Im(z) and am are the commuting ghosts with conformal dimensions 1 and 0, respectively. The correlation function is .

(Bol(z) Y&(W)) =&,,,/(z-w)

(2)

In addition we need the fields q’(z) (i= 1, .... n) for the characterization of the line bundle. the currents is given in ref. [ 51. For the negative roots the Kac-Moody currents result in

of

cE*+ N-,,,-,-...-a ...N-Bn,-aY~,...YBnPB,+...+Bn+a(3) N-a,,-~Y~,P8,+~+~~2~~, .,,p . . 3,”

J-a(z)=LL+f I+ where the numbers

The derivation

3

& related to the Bernoulli

&,=(-l)“-‘B,,,

&,+,=O

numbers

B, are defined as

(n>l).

(4)

For the positive simple roots the currents take the form

where (Y, = Jk+g and g is the dual Coxeter number of g. The constant expansion between J, and J_, and is given by a

a

=k+l

cc2 2,%,PI

C

-aeA+

a, is determined

by the operator product

(6)

N-~,,a~-,~,+a,-a.

The Cartan part becomes H’(z)=-ia++‘(

1

a$&.

(7)

asA+

Using the Sugawara construction

we have the energy-momentum

tensor

where p is half the sum of positive roots. It is now checked explicitly that the central charge c is equal to k dim g/ (k+g), as is well known for the Wess-Zumino-Witten model. The parafermionic algebra associated with the afline Lie algebra g is generated by the fields I,V~which are obtained by factorizing the Cartan part from the Kac-Moody currents:

J,(z) =

2k

lr

T;zy,(z) exp[ia.@(z)/@]

, H’(z)=i&aQi(z)

.

Notice that the cocycle factors which appear in the usual definitions of the parafermions the definition of va. The generating parafermionic fields !Y~have conformal dimensions obey the operator product expansions: l&(z) W_JW)=(Z-w)-z+a2’k l&(z)

70

V/s(W)=(Z-W)-‘-a.p’k

(I+...)

(9) have been included in 1- a2/2k and should

) CY‘p’

&,py/a+~(w)+... . >

2(a+P12

(10)

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6 December 1990

In order to get the fully bosonized expressions for generalized parafermions, we bosonize the commuting ghosts (fl~, y,,) as follows [ 11 ]: fl,(z)=-exp(-O~)

0K(z),

y~(z)=exp(O~,) q,~(z)

(ot~A+) ,

(ll)

and q~ = e x p ( - i z ~ ) ,

~,, =exp(iz,,).

(12)

The correlators among the fields q~, and Z,~ are given by =-6~,.~, l n ( z - w ) ,

=-6,~.,~, I n ( z - w ) .

(13)

Following ref. [ 12 ], it is convenient to adopt the orthogonal basis by introducing new fields q~, q3, ~, L and g, through ~=

iot.~ k + g a . ~ v/~ + - - -g- - x//-~

ia+ ~7~a'~v+f,~,

kot.~ ia+ g~+-~ot'~V+iga,

ixa-

ot+~ ia+q3 (p=_--~-+-~---+~V. (14)

In this basis the Cartan part takes the desired diagonal form as written in (9). We note that 0c,,} and {g,,} are not all independent. They satisfy the relations 2

aL=0,

~

ag~=0,

f~(z)fp(w)=g~(z)g~(w)=-

6~,p--2a" p I n ( z - w ) .

(15)

Inserting the bosonized expressions of (p,~, y~) into the Kac-Moody currents and factorizing the Cartan part, we get the Feigin-Fuchs representations of the generating parafermions: ot z Via=

+

.

2ia+ot.0~

a2(_20~_i0x~)_½

a2

aaO(Oa"}-lXa)+

/~,

•

y.

°t 2 ( - 0 ~ , )

fll ~A+, fll ~ot

2a'fl,

~ N - B , , a 10X#, - a + n~>2 ~-I ~ B~...,B,,~a+ Bi,Bl --aeA+

or2

] N-P--a2 . . . . . #.... N - # . - l , - # . i0X#l +...+ #. ]

(

or.q3 ) X exp \ ~ - + L + ig~ , for the simple root a , ~v_,, = Xexp -

iSx,, + ½ E

flleA+

~-

+L+ig,,

N_p,,_,,iSzz, +o,+ E ~

)]

n>~2

(16) 2

N_p,,_a2...... ... N_p.,_,,i0xp,+...+a.+,,

• flt,...,flneA+

, fora~A+ .

It is evident that the Cartan part bosons @ are absent in ~v_,,(z). On the other hand we must check if the derivative terms in ~v,,do not contain q~ since the terms proportional to q~ apparently arise from q~, and ~0.From (14) and ( 16 ), the decoupling of this Cartan factor requires that the constant a,, should be

2k 2[g -- ~,( c~2+ ~-i,

a,~= ~-~ + ~-i

~ B~A+

)]

(a'fl) 2

•

(17)

This is simplified to 2k a,~=~+

g--o~ 2

a2

,

(18)

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by virtue of the formula flea+

( a'fl) (fl'7)=g( a'7 ) .

(19)

This result must be consistent with (6). From (6) and ( 18 ), therefore, one should have the identity

g-°t2 O[2

!

~

N_a~,~N_a,+,~,_,~ .

(20)

-- 2 .Ol,fll--ot~A+

We can prove this relation for any Lie algebra by inserting the value of the structure constants N~,a. In the Chevalley basis the structure constants N,~,a satisfy the properties [ 13 ] N.,p

Np,r _ N~,. 72 = c~2 - fl: , forot, f l , ~ A a n d o t + f l + ~ , = 0 .

N~,B=-N_,,_p,

(21)

Moreover in the Chevalley basis N,,p is equal to + ( r + 1 ), where f l - ra ..... fl+pa belong to an or-series offl (r, p are non-negative integers). Using (21 ) we can show that the RHS of (20) is equal to 1

Z fllz )2 (N-P,.-) 2 pl,pl--~+ ( i l l - a

Explicit computation confirms that this is equivalent to the LHS of (20) for any Lie algebra, although we do not know the simple proof of this identity. Let us now discuss the structure of the screening operators. The screening currents for the affine Lie algebras, which correspond to the simple roots or, are

S~,(z)=(fl,~-½

+ F~ ~i-

Z

,a~A+

F~

N-p,-.TpflP+.

N_p,_p2 ...... ...N_p._~yp,...),~.pp,+...+p.+~

exp(ia a.~0).

(22)

Though these operators commute with the parafermionic currents, these do not contain the q3 charge and are not qualified as the screening fields to analyze the parafermionic Fock modules. The appropriate screening operators can be expressed formally as follows:

S + (z) =([3~ - ½ Z \

#cA+

N-a,-,~Tafl.a+.

-t- ~ --'7Z N-Bt,-B2 . . . . . . . . . n ) 2 H! fll,...,,OnS~+

/~--Bn,--offfll"'" ?~P,BBI+...+P,+~

exp(ia+a.~0),

(23)

where a is a simple root. The meaning of the negative power - a 2 will become clear after introducing the bosonized expression of the (fl,~, y,) system [ 12 ]. We note that the ghost fields t/, (oteA+) also commute with the parafermionic currents, and hence become the screening operators. These screening operators S~ (z) and ~/a(z) characterize the parafermionic Fock modules. We expect that the BRST cohomology which is constructed from these screening operators leads to the explicit calculations of the string functions and the correlation functions on the sphere and the torus. These problems will be discussed in a subsequent paper. The parafermionic currents ~u~ can be defined for every a on the root lattice. The discrete symmetry is generated by the shift along the simple root vectors; ~u~+m,= ~u~,where ot~ is a simple root. This discrete symmetry is a natural generalization of the Zg symmetry in Zamolodchikov-Fateev parafermions. We note that the present representation has an outer-automorphism symmetry of the Lie algebra. As a generalization of the Zk parafermions there is another class of parafermionic algebra with the dihedral 72

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s y m m e t r y D k (ref. [ 8 ], appendix A ] ). The Feigin-Fuchs representations of the parafermionic algebra are not yet studied. We conjecture that this parafermionic algebra will be expressed as a sum of SO(k) parafermions and the Feigin-Fuchs bosons. Finally we mention the integrable models corresponding to the generalized parafermionic system. For the Zk parafermionic case Tsvelick studied the Zk invariant statistical models by using the Bethe ansatz technique [ 14 ]. He found that the perturbation by a relevant operator with a conformal dimension 2 / ( k + 2 ) makes the parafermionic theory massive. This observation was extended in part to the generalized parafermions corresponding to simply laced affine Lie algebras [ 14,15 ]. The Feigin-Fuchs representations of the generalized parafermions will give a new insight for the analysis of the off-critical structure for these models.

The authors would like to thank S.-K. Yang for valuable comments and carefully reading the manuscript.

References [ 1 ] B.L. Feigin and D.B. Fuchs, Representations of Virasoro algebra, in: Representations of infinite-dimensional Lie groups and Lie algebras (Gordon and Breach, New York, 1986 ). [ 2] B.L. Feigin and E.V. Frenkel, Russ. Math. Surv. 43 ( 1989 ) 221; in: Physics and mathematics of strings, eds. L. Brink et al. (World Scientific, Singapore, 1990) p. 271; Commun. Math. Phys. 128 (1990) 161; Lett. Math. Phys. 19 (1990) 307. [ 3 ] A. Gerasimov, A. Marshakov, A. Morozov, M. Olshanetsky and S. Shatashvili, Intern. J. Mod. Phys. A 5 (1990) 2495. [4] P. Bouwknegt, J. McCarthy and K. Pilch, MIT preprint CTP# 1797 (October 1989). [5] K. Ito and S. Komata, Kyoto preprint YITP/K-861 (June 1990). [6] M. Kuwahara and H. Suzuki, Phys. Lett. B 235 (1990) 52; M. Kuwahara, N. Ohta and H. Suzuki, Osaka preprint OS-GE 05-89 October 1989). [7] D. Kastor, E. Martinec and Z. Qiu, Phys. Lett. B 200 (1988) 434. [8] A.B. Zamolodchikov and V.A. Fateev, Soy. Phys. JETP 62 (1985) 215. [9] M. Ninomiya and K. Yamagishi, Phys. Lett. B 183 (1987) 323. [ 10] D. Gepner, Nucl. Phys. B 290 [FS20] (1987) 10. [ 11 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 ( 1986 ) 93. [ 12] K. Ito and Y. Kazama, Mod. Phys. Lett. A 5 (1990) 215. [ 13 ] J.E. Humphreys, Introduction to Lie algebras and representation theory (Springer, Berlin, 1972 ). [ 14] A.M. Tsvelick, Nucl. Phys. B 305 [FS23] (1988) 675. [ 15 ] M.J. Martins, preprint UCSBTH-90 (February 1990).

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