Comments on two-loop Kac-Moody algebras

Physics Letters B 274 ( 1992) 65-7 1 North-Holland

PHYSICS

LETTERS

B

Comments on two-loop Kac-Moody algebras L.A. Ferreira

1,2,J.F. Gomes ‘TV,A.H. Zimerman

’

Instituto de Fisica Tedrica, UNESP, Rua Pamplona 145, 01405 gao Paul0 SP, Brazil

and A. Schwimmer

4s

SISSA and INFN, Sezione de Trieste, 2-4 via Beirut, I-34014 Trieste, Italy Received

11 October

199 1

It is shown that the two-loop Kac-Moody algebra is equivalent to a two-variable-loop algebra and a decoupled P-y system. Similarly WZNW and CSW models having as algebraic structure the Kac-Moody algebra are equivalent to an infinity of versions of the corresponding ordinary models and decoupled abelian fields.

Infinite-dimensional extensions of the chiral algebras appearing in conformal field theories received a lot of attention recently. The general feature of these extensions is the dependence of the currents on an additional variable besides z, the chiral coordinate on the 2D world sheet. These new algebras [ 1 ] include the different versions of W, [ 2,3] and the two-loop Kac-Moody (KM ) [ 4-8 1. One version of the two-loop KM algebra was introduced in ref. [ 41 in the context of a WZNW type of model having as target space an infinite-dimensional group manifold which, under a hamiltonian reduction procedure, gives rise to the conformal affme Toda model (CAT) [ 91. Under a similar line of reasoning a Chern-Simons theory associated to the same algebraic structure was also considered in ref. [ 7 1. For the W, type algebra it was pointed out by Witten [ lo] that there is a natural geometric interpretation in terms of symplectic diffeomorphisms of the cotangent bundle T*C of a Riemann surface Z. The second variable is therefore intrinsically linked to z.

Such an interpretation is lacking for two-loop KM algebras. We suspect that in this case the two variables can be decoupled. The main result of this paper is to show that this is indeed true. The two-loop KM algebra can be decomposed by a redefinition of currents into a two-variable-loop algebra, with two central extensions, and a P-r system. As a reflection of this algebraic fact the WZNW and CSW models based on KM algebras are shown to be equivalent to an infinite number of copies, labeled by an additional variable, of the ordinary (based on Lie algebra) versions of the corresponding models and decoupled sets of abelian fields. Let us start with the algebraic treatment. The twoloop KM algebra is defined by the following commutation relations [ 41:

’ Supported in part by CNPq. * E-mail address: 47553::LAF. 3 E-mail address: 47553::JFG. a On leave from Department of Physics, Weizmann Rehovot 76 100, Israel. 5 Supported by FAPESP-Brazil and BSF.

[D(x), J,“(Y) 1=mJ,m(JJV(x-Y) ,

(2)

[c(x), w)

(3)

0370-2693/92/$05.00

Institute,

0 1992 Elsevier Science Publishers

+w)~(x-Ykd&z,-.

1=i % h4x-~)

[C(X),J,“(Y)l=O, B.V. All rights reserved.

7

(1)

,

(4) 65

PHYSICS LETTERS B

Volume 274. number 1

where f fib are the structure constants of a finite semiSiIIIpk! Lie algebra 3, while g&, is the Killing form of B (g,,=Tr( T,Tb), T, being the generator of 9) and n, mEH. We will consider ( 1 )- (4) as the algebra of equal time commutators of the currents with the spatial variable x being on a circle of radius 1, and the currents satisfying the periodic boundary conditions J,“(x+2n) =J,“(x), C(x+2n) = C(x) and D(x+ 272) =D(x). Using the fact that C(x) commutes with J,“(x) we can define a new set of currents by

J:(x)=J:(x)

exp T 1 Ge-i’X r+~

(

where C, are the modes of C(x), C(x)=;

I

=g

co

,

r

>

i.e.

Cre-‘“.

the operators, i.e., acts like a second central extension in the algebra of 7s. i-,, For certain classes of representations one of the central extensions can be removed as we now show. In order to do that we have to make specific assumptions about the groups and classes of representations. We study algebras based on a compact Lie group and we limit ourselves to unitary, highest weight representations for the J:(x) currents (we will discuss the possibility for the decoupled D-C system afterwards ). Decomposing Jz (x) in its modes as Jz (x) =(1/2rr)Cz__,J::“e-‘“,weseefrom(8)thatwe can require the highest weight state to be annihilated by all the operators corresponding to (m, r) in a half plane limited by mCo + rk> 0. Then we can use a slight extension of the standard argument [ 111 leading to the quantization of k. Consider the SU (2) subalgebras generated by ( Weyl-Cartan basis)

(6) El.0

a

The zero mode Co is not included in the transformation (5 ) in order to respect the periodicity properties of J:(x). Notice that the transformation (5) is defined on a given representation and not on the abstract algebra. Using ( 1 )-( 4) and the identity f(x)

a,d(x-Y)

=f(~)

a,J(x-y)

- a,f(j+0-Y)

(7)

one obtains [J;(x),

+

2

a(y)]

=if&,&?+n(X)&X--y)

&x-.Y)&bmh-H,

[Do,_lh”(x)]

(8)

,

=me(x)

2 January 1992

,

E:k”,

$

((r.HOTO+k) ,

(11)

E:,‘,

$

(cvHO~O+Co) .

(12)

and Eo.1 o1 ,

The unitary, highest weight representations of the algebra ( 8 )- ( 10 ) will break into unitary, highest weight representations of these subalgebras. Since (2/cr2) x (cY.H”,o+k) and (~/cx~)(~~.H~*~+C~) play the role of 2T,, their eigenvalues must be integers. But since for a given weight ,u, ~~.cY/cx’ is also an integer, one concludes that k and Co must be integers (when the squared length of the highest root is normalized to 2). Such an integral Co can be removed from the algebra, as we now show. Consider the relabeling of the pair of indices (m, r) defined by

(9) (13)

[&P(x)l=O,

r#O,

(10)

where D, are the modes of D(x), i.e., D(x)= Do is the operator measur(1/2rc)XE_, D,e-‘“. ing the “momenta” m in a usual KM algebra [ 111. Formulae (8 )- ( 10 ) show that the C-D system is decoupled from the 7s. Introducing X as a conjugate variable to m, 7(x, X) obeys an algebra (eq. (8) ) completely symmetric in x, I Co commutes with all 66

where M= (F $), a, 6, c, dsz and det M= 1. The last condition is needed in order to make the transformation one to one. We define then new currents by ?m,r= Jm, ,r, J‘, a . Using (8) algebra

and

(14) ( 13) the new currents

satisfy the

PHYSICS LETTERS B

Volume 274, number 1

2 January 1992

where (Y is the positive Lie algebra, and +CbgnbmL+.,O&+s,o

(15)

root of the underlying

j~,(n)I(n(~)}>=n,(n)I{n(Y>))

y

finite

(20)

with (Co

k’)=(C,

k)M

(16)

and where we have used the fact that S,. +n.,o&+s’,o =bm+n.o&.+s,O (if detM#O). We discuss separately the two possibilities: ( 1) k is a divisor of Co. So one can choose a = 1, b=O, c= -Co/k and d= 1. Then k’=k and C&=0. Analogously if Co is a divisor of k one can choose a=k/C&b=l,c=1 and d= 0. Then one gets k’= Co and again C& = 0. (2) If Co/k is rational, i.e., Co=& and k=lq with (p, q) relative primes, one chooses a=q and c= -p with b and d satisfying dq+ bp= 1, which always has a solution. One then gets k’= 1 and Cb = 0. Consequently the last term on the RHS of ( 15 ) can always be transformed away reducing ( 15 ) to a very similar structure as given in ref. [ 5 1. Introducing the currents (17) one can then write ( 15 ) as liX%A(Y)

I (18)

=8(X-g) where 6(x-y)

= &

=g eim(x--9)= .=g, m cm

6(X-y+27cn).

This indicates that the algebra is decomposed into an infinity of decoupled ordinary KM algebras and a decoupled D-C system (except for their zero modes which are still coupled to the ordinary KM’s, having the role mentioned above). The highest weight representation of ( 18) is constructed by applying j: (X) on the highest weight state IA(X) ) characterized by .UX)I~(y)>=O, j3)

In(y) > =0 ,

r>O,

where the j&(X) correspond to the Cartan subalgebra generators. The “distribution of weights” I(X) is arbitrary, being limited by the condition

I

n(n) dx=a

(21)

where A is a weight of the ordinary Lie algebra. For the decoupled D-C system we have two options: ( 1) A unitary representation, which however will not be highest weight, in terms of the modes of a scalar field 9, C(X) being identified with a,~, and D(x) with l7, (the conjugate momentum). (2) A nonunitary highest weight representation in terms of a P-7 system [ 121, C(x) being identified with 3,p and D(x) with y. We remark that the delta function normalization in ( 18) shows that irreducible representations (irreps) of that algebra cannot be constructed by just taking infinite products of ordinary KM irreps. One can convince oneself of that by considering the algebra ( 15) (with Cb =O) with the integers m and n varying from -N to N and the sum modulo 2N+ 1. Making the discrete version of the transformation ( 17) one obtains ( 18) with a Kronecker delta and central term k’/ (2N+ 1). Therefore by taking the limit N-co one sees that ( 15 ) is equivalent to an infinity of ordinary KM algebras with vanishingly small central extension. We now show that the algebraic decomposition discussed above can be implemented in the WZNW model having as current algebra the two-loop KM [ 41. The arguments are not entirely rigorous but show the essential features of the factorization. The ordinary Lie algebra underlying the standard WZNW model is replaced by a KM algebra with commutation relations [Tr,

El =if~bTi!‘+n+&gabm&+n,O,

[a, T,“] =mT," , [&?]=O,

(19)

)

[C, T,m]=O,

(22) (23) (24)

and trace form 67

Volume 274, number

Tr(T::T”b)=gab, Tr(d&

1

PHYSICS

LETTERS

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2 January

1992

element g can be factorized, using the Baker-Campbell-Hausdorff theorem as follows:

hn+.,oW~~)=l ,

=Tr(&?) =Tr(G”F)

=Tr(&2)=0.

(25)

g=exp

i U

dX TU(X)cya(R)+i@+i$ >

Defining formally a group element 2 by =exp g=exp i C T:a”, ( m

+i@+$

(26)

>

+J-rr(g-’a+w a-8

s B

Tr(g-’

.

dS)3

(27)

>

Since the global properties of the group defined by (26 ) are not known it is not clear at this stage if k should be quantized and what are the conditions on the zero modes of ff, /I and y which would make 2 single valued on the two-dimensional manifold X. The action (27) leads to the conservation of the left- and right-handed currents a_@-’ and g-‘a+g. Their Poisson brackets lead to the “two-loop affrne algebra” for J(z)=2 m,n,bgabJ,“(z)T;m +D(z)c”+ C(z)awherezcC. In order to facilitate some of the manipulations which will follow it is convenient to present the algebra as equal time commutators with spatial variable x being on a circle of radius 1 (i.e. z=e’“) and the discrete index m with its conjugate variable XE (0,2x). This amounts to defining J,(x, 2)~

-&

=g_ J,“(x) m co

eeima.

exp($E+iyd) > (33)

>

where p“(X) is the composite parameter, in the ordinary Lie group sense (at each point P separately) of the parameters un ( X) and rz”(X- r), d being the solution of the differential equation

I

+ k

dX T,(X)p’(X)

* ^ =g,g2

the WZNW action is defined by ~(g(~+,~_))=k

i (s

(28)

a,-ff(x) =

M(X)

a(x)

y( 1 -e”(+))

(34)

with the matrix M(X) being defined as =_,-&,ab(.%) .

M;(x)

(35)

We do not reproduce the rather involved expression of Bin terms of (Y,/3, y, which is not essential for our argument. We use now the Polyakov-Wiegmann identity [ 131 s(g) =S(&?,) +

k

s x

+G?,)

a+&,) .

Tr(a_g,g,Lg,l

(36)

Notice that a _&g: ’ is a linear combination of the d and 2 generators. Although 2, is an exponential of a linear combination of r, (n)‘s only, gy * a+ 2, contains terms also in the direction of ?. So, according to (25), the last term on the RHS of (36) apparently does not vanish. However, we can make it vanish as follows. Write

The two-loop Kh4 algebra ( 1)-( 4) will be then

igl’

[JJX, X), Jb(Y,Y) I =if-kJc(&

Since r,, is a vector in 2D one can write it as

+ikg,, +

a&x-Y)s(n-g)

c(x)&x---Ykb

[WX),

r,=a,A+f,, h(x-~)

J,(Y, Y) I =ia,J,(Y,

[C(X), WY) [C(x),

I =W&x-y)

J,(Y, 9) I =0 .

We return now to the action 68

W(-x-YM(~-Y)

>

YV(x-Y) ,

,

Redefininggin

(30)

& =e

(31)

one gets

(32)

is;- I a,g, =ig;’

i(A--B)i‘

= (apB+t,,

(37)

av3.

(29)

(27). A general group

.

a,g, =r,?+...

(38)

(33) as&g,,& g,^

,

=&&

g;,,-iL-B)E

g2

Y

with (39)

a,g, - a,(~--B)e a”B)?+...

.

(40)

Volume 274, number

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1

It then follows that S’,- ’ 8+&Y,-f’ &,g’, + g,- ’ a, g’, does not contain any term in the direction of E. Consequently the overlap term in (36) will vanish. Moreover due to the properties of the trace form (25), S(g’, ) does not get any contribution of terms in the direction of E and therefore it could be equivalently generated by an algebra of T,” with E (and also 8) put to zero (so-called loop algebra, and which is consistent with the Jacobi identities being a direct product of decoupled ordinary Lie algebras at each point x). Finally S(&) is simply

LETTERS

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1992

d3x dX epvp A”, &A; + $&A;AbyA; MBS’

+B P(a

(44)

where M is a three-dimensional manifold. The gauge fields A”,(x, X) should therefore be single valued on M@S’. In the generating functional Z, which is given by (45)

(41) the integration i.e., a pair of scalar fields coupled off-diagonally. The current algebra that the two-loop decoupled actions ( 36 ) would lead to is therefore li,(x, X),j,(Y,

Y) 1

K(X), d(v) 1=i&&=Y)

.

- a,C,)

(46)

This implies that

c,=a,a+ +, ,

(42)

all other commutators, including [b, j,] , vanishing. Now, since (27) and (36) represent the same model, we expect the algebras (29)-(32) and (42) to be equivalent. As we mentioned before the above argument is not rigorous since the measure is not under control and the zero modes may be shared between 8, and & leading to a failure of the factorization. Let us now study the factorization from the point of view of Chern-Simons-Witten (CSW) lagrangians. As discussed in refs. [ 14,151 a three-dimensional CSW lagrangian based on a Lie algebra 3 has a gauge invariant Hilbert space related to the Hilbert space of the two-dimensional WZNW model built on 9 and therefore to the representation of the current algebra @. Motivated by this connection, in ref. [ 71 a CSW lagrangian based on the KM algebra (22)(24) was proposed. The gauge fields associated with T:, E and d respectively are (A,)“,, B, and C,. It is convenient to replace the discrete index m by a continuous periodic variable XE (0, 2n) through A”,(x, X) = mGz [A,(x)]%

s(a,c,

over B leads to the delta function

e”“‘.

Then the CSW action proposed rewritten as

(47)

where f3, are constants and (11(x) is single valued on M. Inside the generating functional (45) the action kS takes the form

+a

(48

Performing [A;(x)]%

now the transformation = [A,(x)]$

ema@)

we obtain for the generating Z=

s

tidc9eiks /

xexp

t

s

functional

J

(45 )

dBdc

I-

ik

(49

\ d3xc~uPB,(aY~p-aP~:Y)

,

(50)

where cfl stands for the C, field whithout the zero mode 0, and where we have reinserted the integration over B,,

(43) in ref. [ 71 can be

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PHYSICS LETTERS B

e;,=(N-y;e,.

(54)

Using this transformation, 0, can be oriented along one of the directions, e.g. 8, # 0, 0, = 0, = 0. We next define a transformation of coordinates Notice that the jacobian of the transformation (49) is one. We have therefore shown that under the above transformations the generating functional factorizes into two parts, one containing the B-c system and the other the A-0 system. Let us now consider the transformation properties of Sunder the gauge transformation

A;(x, X)-g-‘(x,

X)A;g(x,

(;J=(;

;)(;),

xf2=x2,

x13=x3,

with integral parameters a, b, c, d and unit determinant. If k and 8, have a maximal common divisor I, i.e. k=Kand 0, =16, with &and & being relatively primes, we define K,(x,,X)=~,(x;,f’),

X)

A;,,(&,

(52) Actually the action (5 1) is automatically invariant under the infinitesimal form of (52). Under the full finite transformation one gets the additional terms

(55)

@

=A2,3(x;,

(56) n’)

(57)

>

with a=& b= 6, and c, d are integral solutions of Ed-& c= 1, which always exist. Then the action (5 1) can be rewritten as S+-jd3xd%

e”‘(A’ap a,J;

+ -&&A’:&)

, (58)

If g represents a nontrivial mapping from M to the Lie group G the first integral does not vanish and in order that the path integral will be invariant under G, we need to have k quantized, i.e. keH. In a similar way choosing M = S’ (xi ) 6_3Mz3if g(x,, X) is a nontrivial mapping from the manifold M2,@S’ (X) to G we also need to quantize 8,, i.e. 8,EZ. Therefore k and 0, should be integers. The situation becomes particularly symmetric on a torus. We shall now proceed in steps. First we perform a linear reparametrization involving only coordinates xfl by a matrix N with integral coefficients and determinant one. Under such a transformation the first two terms in (5 1) are invariant, while the third term induces a transformation on e,, i.e.

70

i.e. the standard CSW model with X labeling decoupled gauge fields. A strange feature of the action ( 58) is the summation over the coupling constant 1 due to its implicit dependence on the 8, which are dynamical degrees of freedom. The decoupled nature of (58) proves again our conjecture, namely, the gauge invariant Hilbert space corresponding to (48 ) is related to the Hilbert space of decoupled WZNW models. Therefore the Hilbert space of the KM WZNW model in its original formulation (44) is isomorphic to the Hilbert space of decoupled ordinary WZNW models. Finally let us comment on the possible applications of the above constructions to the study of conformal models. As shown in ref. [ 41 the CAT model can be obtained by hamiltonian reduction from the WZNW model based on KM algebras. It will be very interesting to investigate the consequences of the factorization of the two-loop KM algebra to the CAT model. This could perhaps lead to a better understanding of CAT models and its relationship to ordinary Toda field theories. One of us (A.S. ) would like to thank IFT-UNESP for the warm hospitality extended to him.

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References [ 1 ] E. Floratos and J. Iliopoulos, Phys. Lett. B 201 ( 1988) 237. [2] I. Bakas, Phys. Lett. B 228 (1989) 57. [3] C. Pope, L. Romans and X. Shen, Phys. Lett. B 236 (1990) 173;B245 (1990) 72. [4] H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Phys. Lett. B 254 ( 1991) 372. [5] J.M. Maillet, Phys. Lett. B 167 (1986) 401. [ 61 A. Gerasimov, D. Lebedev and A. Morozov, Intern. J. Mod. Phys. A 6 (1991) 977; A. Morozov, Sov. J. Nucl. Phys. 52 (1990) 755. [7] L. Bonora, M. Martellini and Y.-Z. Zhang, Phys. Lett. B 253 (1991) 373; L. Bonora, M. Martellini and Y.-Z. Zhang, Affine CS theories

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and WZNW models, preprint SISSA/ISAS-7919 1/EP, BONN-HE-9 l-08. [ 81 H. Aratyn, C.P. Constantinidis, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, preprint IFI-P.O23/9 1. [ 910. Babelon and L. Bonora, Phys. L&t. B 244 ( 1990) 220. [ 10 ] E. Witten, Lecture Strings 90, Texas A&M preprint IASSNSHEP-90/37 (1990). [ 111 P. Goddard and D. Olive, Intern. J. Mod. Phys. A 1 ( 1986) 303. [ 121 D. Friedan, E. Martinet and S. Shenker, Nucl. Phys. B 27 1 (1986) 93. [ 131 A.M. Polyakov and P.B. Wiegmann, Phys. Lett. B 141 (1984) 223. [ 141 E. Witten, Commun. Math. Phys. 121 (1989) 351. [ 151 S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326 (1989) 108.

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