Differential operators and W-algebra

Physics Letters B 274 (1992) 317-322 North-Holland

PHYSICS LETTERS 13

Differential operators and W-algebra I. Vaysburd Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

and A. R a d u l Institute of Cybernetics Problems, Academy of Sciences, SU.117 901 Moscow, USSR Received 2 September 1991

The connection between W-algebras and the algebra of differential operators is conjectured. The bosonized representation of the differential operator algebra with c = - 2n and all the subalgehras are examined. The degenerate representations and null-state classifications for c = - 2 are presented.

1. Introduction This paper is to add more to many attempts to understand the nature of the W-symmetry [1 ] in 2D field theories. This is not a Lie-algebraic symmetry which is supposed to be related to some coset construction. The modern point of view claims Wn to be a gauged sl(n)k Kac-Moody symmetry [ 2 ]. The basic point of these considerations is the hamiltonian reduction of Drinfeld and Sokolov which converts the Kirillov-Kostant Poisson bracket on some coisotropic subspace of sl(n)~ into the Gelfand-Dikii Poisson bracket on the properly defined manifold of differential operators. The Gelfand-Dikii algebra is the classical limit of the quantum W-algebra. The usual quantization of the above-mentioned gauge fixing process yields the W,-algebra [ 2-7 ]. There is another way to relate W, to some Lie algebra. That is the n ~ o o limit [8-10]. This enables us to provide a geometrical meaning for the W-symmetry. The theories possessing W~ gauge symmetry are called W~-gravities. The gauge symmetry of Einstein gravity is the algebra of the vector fields (reparametrizations), differential operators of order one. The underlying symmetry of the W~o-gravity is the whole algebra of the differential operators which is the main subject of the present paper. Elsevier Science Publishers B.V.

2. The algebra of classical pseudodifferential symbols on the circle Consider the manifold of the pseudodifferential symbols DOP(S ~)

= {L(x, ~)= ,=+~oo-o~u"(x)~"' u"(x)eCo~(S' ) } ' (1) C°°(S t ) is the ring of smooth functions on S I. The product of symbols L(u) and M(u) can be defined by the following rule:

10kL OkM LoM= k>~o ~ k! O~k Oxk '

(2)

[L,M]_ =LoM-MoL.

(3)

This rule is very natural, because it coincides with the way of multiplying the differential operators N

E= ~ en(x)O",

(4)

n=O

if 0 is substituted by ~. We define the residue of a symbol: 317

Volume 274, number 3,4 ResL(~,x)=u_l(x)

PHYSICS LETTERSB ,

(5)

the Tr-functional: YrL= [ ResLdx,

(6)

Now we are ready to define the hamiltonian map T [ ~ T L which determines the hamiltonian structure on the manifold .~¢/,: VL: X ~ VL ( x ) = L ( X L ) + - ( L X ) + L ,

and the pairing of two symbols: (A, B ) = f dxRes(AoB) = T r ( A o B ) .

(7)

S1

One can prove that .

(8)

The algebra of pseudodifferential symbols is a Lie algebra with the bracket (3).

{f g}L = ( VL(dr), d g ) .

W= w,,_ ~0n- ~+ ...+ Wo,

[[W,Z]]F(L)

.g. = {L = & + u , _ l ( x ) 0 "-~ +... + u, ( x ) 0 + Uo,

In fact

u~eC~(S')}.

(9)

The tangent space m the point L ( u ) , TL(~#~), is formed by the operators a L = v . _ , (x)O"-' +...+Vo(X) .

(10)

The cotangent space consists of the symbols Yf(x)¢-' t

,

(ll)

where the pairing (7) was used. The differential of the function F: ~ ¢ l ~ C is the one-form on the manifold ~'., (12)

Vd[l'~g } :

[[

Vd[ ~

Vdg]]

.

(18)

(19)

Sometimes we shall denote Vect (.~') as the G D algebra also. The classical W.-algebra is the algebra of vector fields Vect (Ms~(.~), where Msj(~) c M., (20)

Some restrictions on the one-forms should be imposed as well. If they are taken into account, we get the G D (sl ( n ) ) algebra coinciding with the classical W,. For example, a one-form o n Msl(2 ) looks as follows: X = f ( x ) ~ - ' + ½Of(x) ~ - - 2 . The commutator of two one-forms spans the Virasoro algebra [X,, )(21 = F ( x ) ~ - 1+ ½OF(x) ~-2 ,

(21)

with F ( x ) = Of, ( x ) f 2 ( x ) - f z ( x ) Of2(x).

and, of course, obeys the relation 8F=Tr(~L dF),

= [0e,, 0t2 ] F ( L + ~ I W+e2g)It,.c2=o •

Ms,(,~ = {L(u), u,,_, = 0 } .

!

gF d F = ~ ~ - i - i 8u,(x) '

(17)

is defined by the natural relation

Consider the manifold of differential operators of nth order:

X=

(16)

The Gelfand-Dikii (GD~) algebra is the PoissonLie algebra defined by (16). We emphasize the useful connection [4 ] between the G D , algebra and the algebra of the vector fields on o#, - Vect(J[). The commutator of the two vector fields

Z = z , _ l O " - J +...+Zo, 3. G e l f a n d - D i k i i algebra [11]

T[(~)--

( 15 )

where " + " denotes keeping of the differential part of the symbol. This map provides the functions on Jgn with the Poisson bracket

S1

(A,B)=(B,A)

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(13) 4. The map D O P ( S I ) ~ G D .

where n--I

5L= ~ aui~'. i=l

318

(14)

There is a homomorphism which maps the Lie algebra of differential operators DOP (S z) into the algebra of vector fields on ~/gn [ 12,13 ],

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H:E-oV~eL-,)_ ( L ) = L E - ( L E L - ~ ) + L ,

(22)

[IV(EL-l)_ , V(FL-I)_ ]]=V([E.F]L-,)_ .

(23)

The kernel of this homomorphism consists of the operators of the form AoL, where A s D O P ( S ~). Of course, if we want to get a map into G D ( s l ( n ) ) or classical W., we have to impose

V
(24)

For example, the sl(2, R) restriction on E =

e2(x)OZ+et (x)O+eo(x) looks as follows: e2u-eo = ½0xe~.

(25)

The projective connection u(x) parametrizes the manifold Msl~2) = {L; L = 0 2 "~-U }. All these facts are valid if the differential operators E do not depend on the point L ~ # . . If 8E/SU.v~O, one has to improve the commutator in DOP(S ~) in order to save the homomorphism: {E, F}= [E, F] + V
WE(L)

16 January 1992

= V(EL-I)_

,

(32)

5. The central extention of DOP(S ~) In the previous section we told about the close connection between the classical W and the differential operator algebra DOP (S ~). It is worth examining the central extension of DOP(S ~) [12] in order to observe some relation to quantum W. One of the authors (A.R.) proved the uniqueness of the following cocycle on DOP(S ~):

c(f(x)O",g(x)O") - ( m +m!n! n+l)!

I

f(n)g~m+l)dx"

(33)

$1

We shall construct the quantum fields corresponding to the extended algebra. The generators of the classical algebra are

(26)

Vx(A(L ) ) = 8,A(L +~Vx(L ) ) I,=o •

em,n=XmOn .

The factor algebra o f D O P (S t ), {, }/Ker H, is isomorphic to GD~. That is the main statement of the connection between G D . and D O P ( S ~). To get a better understanding of what our subalgebra does look like, consider a tangent space TM (Lo=0"). It is obvious that all the operators K = AoLo [A~DOP(S t ) ] belong to the kernel of the map H and form a subalgebra of the differential operators of the >i n order D O P ~ . ( S ~). Roughly speaking GD. =DOP(S')/DOP>..(S*).

(28)

The construction described above possesses a very nice geometrical interpretation [ 12,13 ]. If one deforms the solution of the differential equation

Lf=O

+or) Em,n

(35)

Using the commutation relation of E .... [E . . . . E ..... ] E Ckl . . . . Inl Ek./+ c(e,,,,,,e .... , ) , k,I

(36)

one can get the fusion rules for the infinite set of currents J~(x),

J,,(x)J~(y) = Y" [ n ( n - 1 ) . . . ( n - a + 1 )Jm+n--a(Y) a>~l

- ( - 1 ) " m ( r n - 1 ) . . . ( m - a + 1 )J,,+,,_a(x) ] (30)

and looks for a corresponding deformation of L,

L--.L+EWE(L) ,

Let us denote the corresponding generators of the quantum algebra as Em,n and construct the following operators:

(29)

by means of the differential operator E,

f~f+eEf ,

(34)

(27)

(31)

preserving (29) up to second order in e and obeying We(L) ~TL(M), one finds

X ( x - - Y ) -(a+l)+ (__ 1) m + l (~c)m.n. 1 I I (X__y)m+n+2

(37)

The same fusion rules are provided by the bosonized construction for c = - 2 :

1

J~(x)= ~-~ : [0+I(x)]n+l: 1 ,

(38) 319

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where I ( x ) is the chiral bosonic U(1 ) current. For Jl(X) (38) reads Ji(x)=½:12:

+½I,, •

(39)

So the Virasoro algebra formed by Jt (x) is contained in DOP (S ~) with central extension.

6. The algebra of the symmetric polynomials in the ncurrent module It was an idea of A. Zamolodchikov, Fateev and Lukyanov to represent the W~-symmetry in the tensor product of n - 1 Fock modules [2 ]. The polynomials present an exact representation of the W,-generators. In the W~-case we would need the tensor product of an infinite number of Fock modules to represent the algebra precisely. If we represent W~ (differential operators) using only n scalar fields we get some quantum analog of the classical construction described in section 4. In fact, in this case the generators Ju with N > n can be expressed as some differential polynomials of the lower ones:

J N - P( JI , J2

.....

Jn, OJ,, OJ2.... ) = 0 .

16 January 1992

For example, if n = 2 , J3 = : (O+Jl),/2 : - ~: (O+ Jl )3 : 1. The analogy with the classical case is straightforward. The basic point of both constructions is the action of some symmetry generated by the infinite set of currents on the space parametrized by some finite set of classical (Mg~{,) or Ms~n~) or quantum , ~F ( k ) 1/2) functions respectively. The orbits of ( ® , k= --do= the action of these symmetries can be treated as reduced ones if equipped with the proper algebraic structure. Particularly, W~ +n is the result of the W~ +oo (differential operators) reduction described above. The reason that the resulting algebra is nonlinear is that the corrected currents ,In =J~ - P ( J ) do not form an ideal in the universal enveloping. That is why the orbit we are going to associate with W~ +~ cannot be equipped with a Lie algebraic structure.

7. Verma module over DOP(S~). DOP>~.(S ~) as the symmetry of the one-current Fock module Let us define the Verma module over this algebra. The highest weight vector of the Verma module obeys the condition

(40) J~.o IIa)=~2n If a ) ,

The LHS of (40) belongs to the kernel of the homomorphism

J .... I ~ ) = 0 ,

UDoe(s, ) --' Uw,+, •

where

(41)

fa= (12~, ~2, ...),

m<0,

(45) (46)

In the representation considered above the currents Jk look as follows:

J .... = + J , , ( x ) x . . . . . d x .

J k = ~1 [: (0+&p~ )k: 1 + : (0"~0~02)k: 1 +...

The Verma module over DOP(S ~) is created by Jn.m ( m > 0 ) acting on the vacuum Ig2). The problem of the spectrum of degenerate dimensions seems to be rather complicated and we are going to study only the bosonized representation with c = - 2 . It should be mentioned that Fock modules with highest weight vectors

+ : (0+0~0n)k: 1 ] .

(42)

The operators defined in (42) form a set of generating elements in the ring of symmetric polynomials and any symmetric vector in the tensor product ofn Fock modules

090", ,J2 ..... jn) =09Us~ ,Js..... ,J~.) (j~ = 0 ~ ) ,

320

10) = exp (a~p)]0)

(48)

present only a small class of representations with (43)

can be expressed through the Jk currents and their derivatives

og=q( J~., OJk, O2Jk, ...).

(47)

S1

(44)

O = (c~, ½(o~2- c~) .... ).

(49)

Nevertheless, it seems to be useful to study this representation. DO~P(S ~) contains the Heisenberg subalgebra generated by the current Jo(x). That is why

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PHYSICS LETTERS B

Table 1 n

Level 1

Level 2

(a-1)(a-½) (a-3)(a-2)(a-1) 2 (a-4)(a-3)(a-2)z(a-~)(a-l) (a--5)(a-4)(a--3)z(a-2)3(a-1) (a-6)(a--5)(a--4)z(a--3)Z(a--~)(a-2)(a--1) (a--7)(a--6)(a-5)2(a--4)2(a--3)2(a--2)(a--l)

1

~-1

2

(a-1)(a-2)

3 4 5 6

(a-1)(a-2)(a-3) (a-1)(a-2)(a-3)(a-4) (a-l)(a-2)(a-3)(ot-4)(a-5) (a-l)(a-2)(a-3)(ot-4)(a-5)(a-6)

all the representations of type (49) are irreducible. It is worth considering the subalgebras of D O P (S ~) : D'~P >/, (S l ), which are generated by the set of currents Jk(x) (k>_,n). The explicit calculations show that the number of a-values corresponding to the modules with null-states on each level increases proportionally to n. Table 1 illustrates the dependence on n of the polynomials for "degenerate" a (first two levels). Each root a = a ' of the polynomial corresponds to the null-state in the Fock module with highest weight vector I~9,, ) = exp ( a ' tp) I 0 ) . The multiplicity of the root equals the dimension of the null-subspace on the corresponding level. This spectrum of roots can be explained in terms o f Felder's construction of the screening operators [ 14]. For the Virasoro algebra, generated by J~ = : I z: + Ix, there are two screening operators

Jl(w)exp[a+~o(z)]--

½ a + ( a + - - 1 ) = 1 =~ a + = 2 , --1 .

(5o)

(52)

Jn(w)exp[a+~o(z)]=OzR(z,w),

n>l,

(53)

only a = - 1 survives. So S= f exp[-~0(z) ]

(54)

is the only screening operator for D~P>~(S~). D~P>_.n(S t ) does not contain Vir, therefore the screening operators for D ~ P >_..(S ~) are not necessarily zero-dimensional. The operators St must obey the relations

R(.~?(u)

J . ( w ) S ~ ( z ) - (w"_z)~ + (w_z)k+, + .... k>~2.

The integration contour encloses the point where some null-state is placed. Dim [exp(a_+~o) ] = 1, so the St, $2 operators commute with the generators of the Virasoro algebra if there are no boundary terms. In order to obey the above-mentioned condition we have to take care of the closure of the integration contour. Then we shall get an operator (which is called the BRST-operator in Felder's paper) which maps the Fock modules so as to map null-states into null-states. We are going to construct such an operator for the algebra D O P >~t containing the Virasoro algebra, and the subalgebras D~'P >~.($1 ). The screening operator for D ~ P >/t (S t ) should be at least the screening operator of Vir(Sl): exp[a±~0(z) ],

(51)

If we impose the condition

R(°)tw) $1,2= ~ dzexp[a+_ ~o(z) ] .

e x p ( a + ~o) + . . . , ( z _ w)2

(55)

These relations are equivalent to (53) in the case of Vir (S~). The results of the calculations for different subalgebras of D'~P (S t ) are collected in table 2. Each null-vector from table 1 can be represented in the following way:

Table 2 Algebra

Screening operators

Heisenberg Vir(S ~) DOP~j(S *)

e2~ e - ~

DOP~.(S *) i

e-~ e - ~ z e - ~ ... z n - | e - ~

321

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" t r u l y m a r g i n a l " operators in such theories. T h e operators

£2~.N= f d z z "° : e x p [ - ( p ( z ) ] : X j dZl z ? ' : e x p [ - q ~ ( Z l ) ]

...

Jn,0=

z

X j dzk z~? :exp[--~o(zk) ] z

X :exp[(fl+k+

N=fl(k+l)-

1)q~(0)]: ,

~c~,>~l,

c~
(56)

fJn(z)zndz,

8,$2= Ank f2 .

T h e Lie algebra o f the differential operators can be u n d e r s t o o d as a q u a n t i z e d v e r s i o n o f the area preserving d i f f e o m o r p h i s m s [8]. T h e latter is d e f i n e d by the P o i s s o n bracket

References

[A(x, ~), B(x, ~ ) ] = A o B - B o A ,

(58)

A o B = A B + 0cA 0xB.

(59)

To c o n v e r t it into D O P ( S ~) o n e has to a d d c o u n t e r t e r m s to the R H S o f ( 5 9 ) : (60)

T h i s algebra generalizes the r e p a r a m e t r i z a t i o n symmetry. I n s t e a d o f the usual generators o f the coordin a t e t r a n s f o r m a t i o n 8, = e 8,. g e n e r a t e d by the s t r e s s energy t e n s o r we proceed to the t r a n s f o r m a t i o n (61)

generated by the set o f the c u r r e n t s i n t r o d u c e d in section 5. It is n a t u r a l to call this s y m m e t r y W ~ - c o v a r i ance a n d to a s s u m e that this is the s y m m e t r y o f W ~ gravity. It is also a h i d d e n s y m m e t r y o f the theories d e f i n e d o n the m a n i f o l d o f the differential operators. T h e m a i n hope o f the a u t h o r s is to discover this symm e t r y in topological gravity. T h e c u r r e n t s Jk renorrealized by the Liouville field are s u p p o s e d to be

322

k = l ..... o o ,

(63)

I2e ( H i l b e r t space o f the t h e o r y ) . O n the subspace o f null-states eqs. ( 6 3 ) are s i m p l i f i e d drastically, yielding

8. Discussion

,

(62)

form the c e n t e r o f the DO~P ~>l (S ~) algebra a n d can play the role o f h a m i l t o n i a n s o f the n o n l i n e a r q u a n t u m K d V - t y p e p r o b l e m described by the set o f equations

8,kf2=[Jk,o,12],

A o B = A B + 8¢A O,-B+ ½ 8~A 82B+ ....

n>_-i ,

(57)

T h e i n t e g r a t i o n is over the c o n t o u r s chosen according to F e l d e r ' s prescription.

6,1,,2,.. . = ~10-'}- ~ 2 0 2 " { - . . .

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( 64 )

[ 1 ] A.B. Zamolodchikov, Teor. Mat. Fiz. 65 ( 1985 ) 347. [2] V.A. Fateev and S. Lukyanov, Intern. J. Mod. Phys. A 3 (1988) 507. [3] V.G. Drinfeld and V.V. Sokolov, J. Sov. Math. 30 (1985) 1975. [4] T. Khovanova, Funct. Anal. Appl. 20 (1986) 89. [ 5 ] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 ( 1989 ) 49. [6] A. Gerasimov et al., WZW model as a theory of free fields, ITEP preprint NN 64, 70, 72, 74 (1989). [ 7 ] A. Belavin, in: Proc. Kyoto Conf. ( 1987 ) (Springer, Berlin, 1988). [8] I. Bakas, Phys. Lett. B 228 (1989) 57. [9] C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B 238 (1990) 173. [ 10 ] C.N. Pope, L.J. Romans and X. Shen, A brief history of W~, Texas A&M preprint CTP TAMU-89/90. [ 11 ] I.M. Gelfand and L.A. Dikii, Funct. Anal. Appl. 10 (1976) 4. [ 12] A.O. Radul, JETP Lett. 50 (1989) 371. [ 13] A.O. Radul, Funct. Anal. Appl. ( 1991 ), to appear. [14] G. Felder, BRST approach to minimal models, Zurich preprint ( 1988 ). [ 15 ] 1. Bakas and E. Kiritsis, Bosonic realization of a universal W-algebra and Z~ parafermions, preprint LBL-28714, UCBPTH-9018, UMD-PP90-160.