Covariant operators and higher-spin conformal algebras

Nuclear Physics B344 (1990) 196—206 North-Holland

COVARIANT OPERATORS AND HIGHER-SPIN CONFORMAL ALGEBRAS Qinmou WANG *, Prasanta K. PANIGRAHI, Uday SUKHATME and Wai-Yee KEUNG Physics Department, UniL’ersily of Illinois at Chicago, IL 60680, USA Received 29 January 1990 (Revised 1 May 1990)

We construct the most general operators transforming covariantly under the action of DiffS’. These operators are used as Lax operators to study the second hamiltonian structure of the Lax equations. The classical spin-4 algebra is obtained explicitly. It contains the spin-3 algebra as a proper subalgebra. The associated nonlinear equations are also investigated.

1. Introduction Conformal field theories (CFT) have been studied quite extensively in recent times in connection with string theories and statistical mechanic models with second-order phase transitions [1]. The goal is to construct all conformal-invariant solutions of the two-dimensional euclidean quantum field theory, thereby classifying all possible types of critical and multicritical behavior. The large group of conformal transformations has at its root the Virasoro algebra, an infinite-dimensional Lie algebra. The solutions that are so far known possess additional symmetries besides this conformal invariance; such as Superconformal [2] and Kac—Moody symmetries [3]. Recently, Zamolodchikov introduced a conserved spin-3 current, which forms an associative algebra (W3 algebra), with the energy—momentum tensor, enveloping the Virasoro algebra [4]. These theories have a central charge 2+p), p=4,S,6,..., (1) c~=2—24/(p and all the conformal weights of the primary fields are known [51.Introduction of still higher spin fields leads to theories with c <3, 4 After the complete classification of the c < 1 theories [61,the higher-spin algebras have enabled us to analyze theories with c <2, 3 Recently, a deep connection of the CFT with another well-studied subject, namely the integrable models is becoming clear [7]. The much studied Korteweg— *

Permanent address: Physics Department, Anhui Normal University, Wuhu, Anhui, China.

0550-3213/90/$03.50 © 1990

—

Elsevier Science Publishers B.V. (North-Holland)

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de Vries (KdV) equation u,=u~+6uu~, (u~~au,/ax),

(2)

and the related hierarchy of equations can be written as hamiltonian equations, u~ {u, ~

(3)

=

in two distinct ways. The two hamiltonians 27-

~P’i=~f

(2u3—u~)dx

(4)

and ~r~=f2~u2dx give rise to eq. (2) if the respective Poisson brackets (1

(5) =

n) are defined as

{u(x), u(y)}( 1)

=

~x

—y),

1 {u(x),u(y)}(2)=~[aX+4UaX+2uXjo(x—y). 113

( )

The two Poisson brackets in eq. (6) can be recognized as the abelian current algebra and the Virasoro algebra (in the sense that i{j —s [,]),respectively. Recently, this connection has been explored in the context of the higher-spin algebras [81.Mathieu has used an operator transforming covariantly under Diff S’ as a Lax operator and derived the spin-3 algebra from the corresponding second Poisson bracket structure [9]. Bakas has used the standard form of the Lax operators and the Drinfeld—Sokolov construction [101 in deriving the spin-3 [11] and spin-4 [12] algebras. Fateev and Lykyanov also gave a prescription for the construction of higher-spin algebras [13]. In refs. [11—13]these operators are not of the covariant type. The covariant form of the Lax operators, as will be seen later, separates the fields into those of definite conformal weights and hence enables one to determine the algebras directly. It is worth mentioning that although the stress—energy tensor T(z) transforms anomalously under a conformal transformation, all the higher-spin fields are primary fields. Both these aspects come out naturally if one uses covariant operators. The field identification carried out in refs. [11, 121, although simple for lower-spin cases, becomes progressively more complicated. This can be easily seen for the spin-4 case in the text. These higher-spin algebras have been shown to be associated with the Toda field theory (for a recent review and relevant references see ref. [14]) and the KP [15] equation. In particular, the Kyoto group shows [161how the KdV and the Boussinesq sequences emerge as special cases of the KP hierarchy. The connection with the Boussinesq equation and the SL(3, R)

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symmetry has also been pointed out by the Kyoto group. This clarifies the origin of the spin-3 algebra in the hamiltonian reduction of the SL(3, R) symmetry in the papers of Bakas [11, 12]. The purpose of this note is twofold. First, we construct and use the covariant operators (including higher-spin fields) as Lax operators and study the resulting second hamiltonian structure. In particular, the classical spin-4 algebra is explicitly obtained, containing the spin-3 algebra as a proper subalgebra. We propose a generalization for the higher-spin cases. In this generalization, instead of u(x) the variable t’(x) appearing in the modified KdV equation plays a fundamental role for spins greater than four. Secondly, we list the sequences of nonlinear equations associated with the spin-3 algebra following the Lax formalism. This procedure yields the well known Boussinesq sequence. We also give a simpler proof of the covariance properties of the above-mentioned class of operators.

2. Covariant operators and J4~algebra Recently, Scherer [17] has constructed a class of operators, which transform covariantly under the reparametrizations of S’. in general, they are defined as L~(v)=[D—(k—1)v][D—(k—3)v]...[D+(k—3)v][D+(k—1)v}, (7) for k ~ 2 together with the trivial case L~11~ 1 and L~’~D defined by the Miura transformation =

u(x)

=

=

d~.The field v(x) is

v~(x) — L2(x)

(8)

which appears in the modified KdV equation. The above operators connect densities of weight (1 k)/2 to those of weight (1 + k)/2, i.e. L~(v): Q( 2 1 k)/ ~ it has been shown that L~ can be rewritten explicitly in terms of u only, with v eliminated by means of eq. (8), —

L~(u) =L(<1(v) The function u(x) transforms like the energy—momentum tensor in a CFT. This can be easily deduced by analyzing the Hill operator, L~2~(v)=(D—v)(D+v)=D2+u(x)=I/2~(u).

(9)

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x

—~z=z(x),

Coi’ariant operators

199

Under the transformation

L(2)(u) ~

u(x) —~ü(x)=z~U(z(x)) +~S7-(x), where

z~=

3z/3x and

(10)

S7-(x) is the schwartzian derivative [18], S7-(x) z~/z~ =

—

A simpler proof of the covariance property can be given as follows. First, we define t(x)

(11)

=~&X/~X.

(A similar definition appears in the study of the Painlevé property of the modified KdV equation [191.)Under a reparametrization, i7(x) =z~t(z(x))+ ~

(12)

Using eq. (8), it can be checked in a straightforward manner that ii(x) =z~u(z(x))

+

(13)

~S~(x).

Eq. (12) indicates that v(x) transforms as a density of weight one with an anomalous term. From the viewpoint of CFT, the Miura transformation is the realization of u(x) in terms of a rank-one field, in the presence of a background charge. Now D

+

2jv

=

and hence 29~(x~) —(k— 1)/2

L(U(

where ~

=

~)

=

~/ax

~(k+ 1)/2(~~-la)~(k_

l)/2

=

and x~, ax/0~.Under x =

(14)

(x,~)~k+ —~

L~’<~(v) ~ az =

ax

Bz

(k+1)/2

L~

—(k—I)/2 .

(15)

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Here

ax

az ~

ax

—

L(k)_ -

ax

—(k+1)/2

ak+1

a~(z(x))

~

16

a~(z(x))

Comparing eq. (16) with eq. (14), L(k)

~(x)

=

=

L~(iT)

~

+

~

(17)

thereby proving the assertion that L~ connects ~ to 12(1 ±k)/2~ Some of the covariant operators L~(U) can be explicitly written down,

=

1,

i7~ =

+ U,

=

L~3~D3

+

4uD

=

D4

+

lOuD2

+

10u~D+ 9u2

=

D5

+

20uD3

+

30u~D2+ 18u~~D + 64u2D

=

+

2u~, +

~ +

~

+

64uu~.

(18)

The operators L~1~ and L~3~ appear in the context of the current algebra and the Virasoro algebra respectively and hence in the first and second Poisson brackets of the KdV sequence. As we will see later, L~3~ and L~5~ appear in the Poisson bracket structure of the Boussinesq sequence and the W 3 algebra. The primary fields Wh of weights h transform as Wh(z)

=

and they can be included directly in the covariant operators. The following

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operators have been checked to be covariant: }~4(2)

L~2~(v)

M~3~=L~3~(v) + J’V~, M~4~L~4~(v) + DW

2t(D+ 3i) +DW~+ W

=

3

W3D

+

+ W4

=

(D

3t)M~

—

3D + W4,

2)[D~i~(m 1)t’] +DU’~ —

M(m)= [D—(m

1)L]M(m

—

1+ ft~ 1D+ J4’~, (m~4).

(19)

Formal differential operators are used in the works of Bakas, and Fateev and Lykyanov, in connection with the J’V~~algebras. They are of the form 7 ~(n)

=

+

u~

D’

2+ u~ 2D’~

3+

...

+u

3D’~

1D

+ u~.

(20)

Comparing these operators with those in eq. (19), we are able to identify the primary fields in terms of the variables in eq. (20). Some examples are given below, (a)

.~~‘~2):U =

(b)

~

(U

U

(c)

~4)•

=

~W3

=

=

~

~(u, U0

—

=

-

~U1~

=

=

(d)

—

=

w~ =

u

U0

+

~

—

—

U~ —

(21)

~ =

U0

—

~U1~

2vU —

+

~U2~

2~+ 2v~u2+

—

~

+

©~UiUs~ — 3t)~U3~.

4vu3~~—

As mentioned earlier, L’(x) is more fundamental than that using U(x) in this approach. For the case of M~3)and ~ all the operators can be expressed in terms of U(x) only. However, this situation fails for ~ Hence, when one

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202

determines the Poisson bracket structure, the algebra involving v, W3, W4 and W~ is obtained first, from which the corresponding algebra in terms of u(x) will follow from the Miura transformation. The field definitions in eq. (21) agree with that of Bakas [12] up to W4. Bakas has adopted in the Lax formalism to obtain the spin-4 algebra. However, the covariant operators in eq. (19) allow a much more straightforward determination of the second Poisson brackets. Now, we proceed to find the W4 algebra explicitly using the covariant operators as Lax operators [20]. We define p4(n) ( M(n)F) + ~,j(n) p4(n)( FM’”~)+ , (22) 2f t~~ in eq. (19) has been arranged where F D~f11 D 1 +of derivatives +D~f~_1 and~ M D’1 + ~ according to the +powers as u,D’. Also, ( )~ denotes the differential part (non-negative derivatives only) of the pseudo-differential operator. Comparing eq. (22) with ~

=

=

—

...

=

(u~)~= ~

(23)

j = (I

we find the Poisson brackets as (u~(x),UJ(y))(

(24)

2)= ~1~(x—y).

For the spin-4 case, we obtain 3~(x-y), {u(x),u(y)} =~L~ {u(x),W 3(y)} ~2DW3 + W3D)5(x—y), =

{W3(x),u(y)} =~(2W3D+DW3)~(x—y), {u(x),W4(y)}

=

~~(3DW4+ W4D)6(x—y),

{W4(x),u(y)} =~(3W4D+DW4)6(x—y),

(W3(x), W3(y))

=

~

+ W4D)]6(x -y), 2W 2 ~(5D 3 + 5D 3D + 3DW3D + W 3 + 54DuW 3D 3 + 36uW3D + l4uDW3)~(x y), 3 + 5DW 2 + 3D2 W ~(5W3D 3D 3D 3W +D 3 + 54uW3D + 36DuW3 + l4W3Du)~(x—y), 7) + ~(D3W 2W 2+W 3 [~iJ 4 + 2D 4D + 2DW4D 4D +28uW 4D + 28DuW4 30W3DW3)]~(x—y). (25)

[-

+

~(DW4

3W

(W3(x), W4(y)}

=

—

—

{W4(x), W3(y)}

=

{W4(x), W4(y)}

=

—

—

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Here, the covariant rank-7 operator is given by L~7~D7 + 56uD5 =

+

240u~D4+ (168u~~ + 784U2)D~

+(112u+~~+ 2352UU~)D2+ +

~

(40u~~~~+ 14O8UU~~+ 1180u~+2304u3)D (26)

+ 7O8U~U~~ + 312Uu~~~ + 3456U2U~.

Several remarks are in otder at this point. First of all from the Poisson brackets of u with W 3 and U”4, it is evident that these fields transform like densities of weight three and four respectively. Secondly, when W~algebra appears as a subalgebra of W4, in addition to L~other terms involving U’4 arise on the side. side. This 3~ appears onright-hand the right-hand is different from the Virasoro algebra, where L~ It is worth mentioning that, using Mt3~ one obtains the W 3 algebra [9], where 3~(x-y), {u(x),u(y)} =~L~ {U(x),W 3(y)} =~(2DW3+W3D)~(x-y), {W3(x),U(y)} =~(2W3D+DW3)6(x-y), {W3(x),W3(y)}=-~L~(x-y).

(27)

5kU). However, {W Note that W3} involves only L~ U” 4, W4} involves not only 7~(u), but {W3, also other terms containing L~ 4, W3 and u. In this sense, the W4 algebra differs from the W3 algebra. So far, for fields of both odd weights (i.e. h ~-,i,...) and even weights (h 1,2,3,...), the Poisson bracket structures have given rise to covariant operators. These algebras are the limiting cases of the corresponding quantum algebras. For proper quantization, some coefficients have to be deformed in addition to i( , } —s [,] in order that the Jacobi identity is satisfied. Our U’4 algebra agrees with that of Bakas [12], when the field definitions given in eq. (21) are used. =

=

3. Example on the Boussinesq sequence In31(u, this W section we proceed to show the nonlinear equations arising from M~ 3). Given a Lax operator ~ the corresponding Lax equations are given by a1~~)= ~

(28)

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The KdV sequence arises naturally from (29) Here we will study the sequence arising from 3~(U,W

.../~3)=M~

3 3)

+

4uD

+

2u~+ W

D

=

3.

(30)

For k 1, it gives the trivial translation equations, Ut u~,W3~ W3~.For k the corresponding simultaneous nonlinear equations are =

=

=

—

2,

(31)

U1”~W3~,

W3~=

=

~

~UUX.

—

Eliminating W3 in the above equations yields the well known Boussinesq equation [211, (32)

~

~1=

In connection with the W3 algebra, these equations have been obtained previously by alternative methods [22]. The higher-order equations can be similarly computed. Choosing k 3 does not give any evolution equation. For k 4, one gets =

=

4 + ~uD2 (.~z~3))~~ =

+

(~w

D

2,

3+

+ 4W~V+ ~

~UX)D

(33)

+ ~U

and the corresponding nonlinear equations are +

U~=

~uW 3~ + ~u~W3,

W3,

=

(34)

.!4!UU~~~

—

~

—

—

~

—

+

~-W3W3~.

For k 5, we get the next section of simultaneous equations in the Boussinesq sequence, =

2UX

W

=

—

3~

=

+

~ —

—

~V.~XXXX

10

~W3W3~ ~

—

—

—

—

~

—

~U

—

2)W —

~

+ ~U

3~

(35)

16(1

-~--UU~W3.

It is worth noting that in eq. (35), if we let W3

=

0, the corresponding fifth-order

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equation is the Kuperschmidt equation [22]. Like the KdV equation, the Boussinesq equations can be formulated as hamiltonian equations with a bi-hamiltonian structure. The connection with the spin-3 algebra becomes clear if we recast eqs. (31) and (35) (corresponding to k 2, 5 respectively) as =

U

(36)

~

and ~2(D

(37)

~)‘

(u,U)

(W~,u)

(U,U’

3)

(W3,~)

with the reduced bracket defined by the relation (A, B) eq. (27). Here the hamiltonian is H=fdx(~W32_~(u3_3U~)).

(38) =

(A, B)t~(x—y) in

(39)

The other hamiltonians in the sequence can be obtained by the method [23] used for the KdV equations. We would like to thank D. Chang, A. Kumar and C. Zachos for useful discussions. This research work was supported by the US Department of Energy.

References [1] A.A. Belavin, AM. Polyakov and A.B. Zamolodchikov, Mud. Phys. B241 (1984) 333; For a comprehensive review, see; MB. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge Univ. Press, Cambridge, 1987) [2] D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B151 (1985) 33; M. Bershadsky, V. Knizhnik and M. Tietelman, Phys. Lett. B151 (1985) 21; M. Eichenherr, Phys. Lett. B151 (1985) 26 [3] V. Knizhnik and A.B. Zamolodchikov, NucI. Phys. B247 (1984) 83 [4] A.B. Zamolodchikov, Theor. Math. Phys. 65 (1986) 1205 [5] V.A. Fateev and A.B. Zamolodchikov, NucI. Phys. B280 [FS18] (1987) 644 [61 D. Friedan, Z. Qui and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; A. Capelli, C. Itzykson and J.B. Zuber, Mud. Phys. B280 [FS18] (1987) 445; J. Cardy, NucI. Phys. B270 [FS16] (1986) 186; P. Goddard, A. Kent and D. Olive, Commun. Math. Phys. 103 (1986) 105 [7] iL. Gervais and A. Neveu, NucI. Phys. B209 (1982) 125; J.L. Gervais, Phys. Lett. B160 (1985) 277, 279; BA. Kuperschmidt, Phys. Lett. A109 (1985) 417; K. Tanaka, Phys. Rev. D40 (1989) 4092; See also: A. Das, Integrable models (World Scientific, Singapore, 1989)

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[8] A.A. Belavin, in Quantum string theory, ed. N. Kawamoto and T. Kugo (Springer, Berlin, 1988); S.U. Park and B.H. Cho, Phys. Lett. B218 (1989) 200 [9] P. Mathieu, Phys. Lett. B208 (1988) 101 [101 V.G. Drinfeld and V.V. Sokolov, J. Soy. Math. 30 (1985) 1975 [11] 1. Bakas, Phys. Lett. B219 (1989) 283 [12] 1. Bakas, Phys. Lett. B213 (1988) 313; See also: T.G. Khovanova, Funct. Anal. AppI. 20 (1986) 332 [13] V.A. Fateev and S.L. Lykyanov, lot. J. Mod. Phys. A3 (1988) 507 [14] A. Bilal, W-algebra extended conformal theories, cosets and integrable lattice models, preprint CERM-TH-5567-89 (1989) [15] K. Yamagishi, Phys. Lett. B205 (1988) 466 [161 E. Date, M. Jimbo, M. Kashiwara and T. Miwa, PubI. MIS 18 (1982) 1011, 1077; J. Phys. Soc. 50 (1982) 3806; Physica D4 (1982) 343; in Mon-linear integrable system, ed. M. Jimbo and T. Miwa (World Scientific, Singapore. 1983); M. Jimbo and T. Miwa, PubI. RIMS 19 (1983) 943; M. Kashiwara and T. Miwa, Proc. J. Acad. 57A (1981) 342, 387 [17] W. Scherer, Univ. of Rochester preprint UR-1046 (1988) [18] 1. Bakas, Mud. Phys. B302 (1988) 189 [19] J. Weiss, J. Math. Phys. 25 (1984) 13 [20] BA. Kuperschmidt and G. Wilson, Invent. Math. 62 (1981) 403 [21] AS. Fokas and R.L. Anderson, J. Math. Phys. 23 (1982) 1066; J. Weiss, J. Math. Phys. 26 (1985) 258 and references therein [22] S.U. Park, B.H. Cho and Y.S. Myung, J. Phys. A21 (1988) 1167; BA. Kuperschmidt and P. Mathieu, Phys. Lett. B227 (1989) 245 [23] A. Das, Phys. Lett. B207 (1988) 429