Diffeomorphism Group of a Closed Surface*)

LAPP-TH-254/89 . June :;/•. .u

L. Frappât, E. Ragoucy, P. Sorba, F. Thuillier

LAPP, Chemin de Bellevue, BP 110, F-74941 ANNECY-LE-VIEUX cedex and

H. H0gâsen Universitetet i Oslo, P.O. 1048 Blindern, N-0316 OSLO 3

Abstract

We extend the notion of loop algebra ^(S1) to algebras C(M), M being a closed surface with dim M > 1. We point out and discuss the correspondence between the central extensions of such generalized Kac-Moody algebras and the volume preserving diffeomorphism algebra on M.

*) Work supported in part by D JÎJE.T. under contract n0 88/1334/DRET/DS/SR.

L.Â.P.P. Bf III) l-74')4l \N\I ( VU-Wri \ ( I.!>1 \ • 11-1 ITl K)M 50.2.'.32 4^ • « 1 i.l.l \ 1»^ 1«« I- • HIIXOIMI 5O27')4')S 1. Introduction

Quite recently, it has been shown that the symmetries of a closed bosonic membrane W M form the area preserving subgroup SDiff(M) of the diffeomorphism Diff(M) Pl. Explicit realizations of the corresponding Lie algebra when M is either the torus T^ = S1XS1 or the two dimensional sphere S2 have been given. Some efforts have recently been brought to a better understanding of such a symmetry and of its properties : among themselves a research of the Virasoro algebra from SDiff(T2) 131, the determination of the central extensions of SDiff(M) t2»4] and the study of the possible connections between this algebra and the SU(°°) group (51. It is on the eventual existence of (generalized) Kac-Moody algebras behind the already determined symmetry of a membrane theory that we would like to think in this letter.

It is today well recognized that in two dimensional conformai field theory (CFT) the algebra of Diff(SJ) or Virasoro algebra I^aad the Kac-Moody algebras*) ^(S1) are intimately related. Among the numerous examples one can think about, let us first remind that in the heterotic string model, the string states form a representation of the Kac-Moody algebra of the gauge group EgxE$ or spin (32)/Z2 ^. Kac-Moody algebras immediately show up in two dimensional current algebra when studying massless spinor or scalar field, such a model leading to exactly soluble CFT. The Sugawara construction allows to realize V once given an algebra Q (S1) I7I. The WZW models may be the best example of the existing link between a Kac-Moody algebra and the algebra of S'-diffeomorphisms, both being symmetries of such field theories BH. Algebraically, for each Q (S1) can be constructed in a canonical way the semi-direct sum : Q (S1) D *K the Virasoro generators acting as Lie derivatives on the elements of ^(S1) - such algebras being sometimes called conformai current algebras Pl. It is in part because of this structure and of the Sugawara construction that one can determine, through the famous GKO construction (10I, the minimal unitary representations of 1U from the unitary representation of some ^(S1) algebras.

Therefore, one can wonder whether the area preserving diffeomorphisms group on a compact manifold M = T^ = S1XS1 (two dimensional torus) or S2 (two dimensional sphere), can be related to Kac-Moody algebras Q (M), defined as central extensions of the algebra of smooth maps from M onto the (semi-simple) group G with Lie algebra Q. Such

*) or affine extensions of the loop algebra C(S1) of smooth maps from S1 into a simple Lie group G of Lie algebra Ç. C(S1) is usually written as : C[W1] ®c Q where C[Lr1] is the ring of Laurent polynomials in t tul algebras are direct generalizations of loop algebras and of their affine extensions. Let us add that (7(1TV) with TV = S1 x ... x S1 v times have already been called quasi-simple Lie algebras t12].

If, in the case M = S1, the algebra C(S1) admits, up to a multiplicative constant, only one central extension, the situation appears completely different when dim M > 1. The general two cocycle construction for C(M) is presented in Section 2, in which a direct correspondence between central extensions of ÇÇM) and elements of SDiff(M) is shown. Explicit construction in the cases M = S2 and T2 are also given. In Section 3, we study the compatibility condition that an element of Diff(M) has to satisfy in order to be a derivation of Q^ (M) where <0 is the cocycle related to the central extension under consideration*). Again a detailed study of the two dimensional sphere on one hand and torus on the other hand will be performed. Contrarily again to the S1 case where the whole diffeomorphism algebra is compatible with the unique non trivial cocycle, now to each cocycle co will be associate only a subalgebra of Diff(M). A general discussion follows in Section 4 where a particular attention is given to the connection between volume preserving diffeomorphisms and Q (M) cocycles.

2. Central extensions of Q (M)

2.1. General results

Following prop.(4.2.8) of Ref. [13], the complex central extensions of the algebra C(M), that is the algebra of smooth maps from a smooth compact manifold M of dimension n into the (semi-)simple Lie algebra Q, are given by the two-cocycles eoc : C(M) x C(M) -» C defined by :

n where X,Y e £(M), dY = ]j£ 3jY dui with (ui,...,un) parametrizing the manifold M, < , > denotes the bilinear symmetric Killing form on Ç, and C is a closed one chain Ci

Equation (1) can be rewritten as:

*) the suffix co on Q^ (M) being now necessary due to the non uniqueness of co for dim M > 1. COc(X1Y) = J XdY A (p (2) M where q> is the (n-1) closed current - i.e. (n-1) closed form with distributional coefficients - associated to C. Note that the Killing form is implicit. Setting*):

A...A duj A...dun (3)

(where the symbol v expresses that the term below has been taken away). The closure condition reads:

n d

It follows that to any central extension of C(M) corresponds a (n-1) closed form (p on M, the cocycle being given by:

n f d A A CO9(X1Y) = X X (

n with Y. 9i (pi =0

(see also ex.7.9 p.91 of Ref. [11] for the case M = V). In the following of our study, we will restrict the (pi to be functions.

Now, let us note that Eq. (6) expresses also the condition for infinitesimal diffeomorphism

Ui -> Ui(ui,...,Un) = Uj + (pi(Ul,...,Un)

*) The elements of differential geometry necessary in these section can be found in Ref. [14]. to preserve the volume : indeed such a constraint implies the Jacobian of the transformation Jij = -J-; to be of determinant equal to one, or infinitesimally to be traceless, that is to satisfy Eq. (6). In other words, and coming back to the beginning of this section, a volume preserving diffeomorphism on M determines a (n-1) closed current - or a central extension of Q (M) - and vice-versa.

2.2 Applications to the sphere S2 and the torus T2

On the two dimensional sphere, any closed one-form (p is an exact form, that is:

(p = dh (7) with h function on S2, the de Rham cohomology H1 (S2) being trivial. Then, Eq. (2) takes the ïemarkable form

(Oh(X5Y) = - J X {h,Y }U1U2 duidu2 (8) S2 with {.,.} being the usual Poisson bracket with respect to the coordinates (ui,U2) -which are (0,

(X, Y) U1U2 = 3U1X 9U2Y - 3U2X dui Y (9)

Another interesting expression for (Oh can be given, integrating by part Eq. (2) with :

XdY A dh = hdXAdY-d(hXdY) (10) and using the Stokes formula J drj = J Tj : then the absence of boundary on the sphere D 3D leads to the formula

coh(X,Y)= JhdXAdY (11) S2 that is also

û)h(X,Y)= Jh{X,Y)ulU2duidu2 (12) S2 One can present a more explicit form of Eq. (8), using the property of any function defined on the sphere to be developed in terms of the spherical harmonics Y** (9,

h(8,(p) = YJJJ (0,

X a Y e < Xa> Y 2>b = Y 2 9 Yb 14 m' ~ m ( ' P>® ' m m ( ^ ® ( > with Xa,Ybs Q, the relations:

Ml y*2 1 = - i e1*31"3 /V3 V (15)

involving the coefficients g»31"3 . defined in Ref. [15], and:

Jd(cos9) d(p Yj1 (Y1*)* = ô^-î' Smm- (16) S2 lead to the simple formula

|X, Y) = ff d(cos9d(cos9) d(d(pp Y /Y, Y mo I mi m2 J mo 1 mi |

(17)

On the two dimensional torus T2, any closed one-form

(p = dh + aY+P^ (18) h being a periodic function on T2, and a and p arbitrary constants.

It follows that the cocycle associated to

COp(X,Y) = |h{X,Y}tjWdtdw+ J X (-^9WY + ^Y^dtdw (19) T2 T2

where we have used for the exact form part the result given in Eq. (11). Developing h(t,w) in Fourier modes:

h(t,w) =—Lj ^h-J t'w* (20) r.seZ

one finally obtains:

a n b 2 wP ® X , t w

2.3 Miscellaneous remarks

It has been shown in the determination of central extensions of SDiff(M)[4l, that an exact form h on M is associated to a trivial central extension of this diffeomorphism subalgebra. Let us note that this is not the case for Kac-Moody cocycles : indeed it is not the de Rham cohomology of the manifold M which is now involved but the cyclic cohomologyf 16I on M, and an exact form h on M does correspond to a non trivial two cocycle Cj(M).

n ine It might be also amusing to remark that the basis: z Xa = e Xa, with a = l,..,dim Q and n e Z, of the loop algebra Q (S1) involves all the one dimensional unitary representations einQ, n e Z, of the U(I) or SO(2) group. In the same way all the unitary representations of the SO(3) group - resp. U(I) x U(I) group - appear in the chosen basis of -resp Cj(T2). 3. Diffeomorphism algebra associated to a generalized Kac-Moody algebra

3.1 Diffeomorphism algebra compatible with a central extension

When the algebra Cj(M) is extended to a Kac-Moody algebra £a>(M) withcocycle co, the former Lie bracket [.,.] becomes [.,.]„,, that is

[X1Y]11, = [X,Y] + to[X,Y] I (22) for any couple X,Ye ^r(M), I standing for the generator of the central extension. Let L be a n ^ vector field on M £=X AKuIv1Un)BUi ; L, will be said compatible with ^©(M) if it acts i=l on the Kac-Moody algebra as a derivative i.e.

(23)

Since L is naturally a derivation on

UX, Y] = [IX.Y] + \X,LY] (24) the compatibility condition writes:

co[ZX,Y] + a>(X,IY] =0 VX1Ye §(M) (25) which, in other words, asserts that the cocycle co is invariant under the transformation generated by L. We will denote by Diff( ^0(M)) the algebra of vector fields £ on M compatible with ^0(M).

2 2 3.2 Application : the algebra DJfT(^0(M)) with M = S or T

Let us start by applying condition (25) on a cocycle related to an exact one-form (Eq. (11) or Eq. (12)):

2 COh(X5Y)= J hfX.YJduMu = JhdX A dY M M One has, after integration by parts:

Cu(ZX, Y) + co(X,£Y) = J(Lh) {X,Y} M

= J XdY Ad(Xh) =0 VX1Ye £(M) (26) M

which leads to : Zh = c (27)

c being a constant. Setting

L = A3i + B32 (28)

the general solution of L is

A = ^a21) (32h9i + 3ih32) (29)

the existence of the generator A being submitted to the condition

aih.32h * 0 (30)

By direct calculation, one gets the commutation relations

L^] = L^ with V = T{U,h} - U{T,h}

[A, Lj] = L^ with W » hi3iT - T3ihi + h232T - T32h2 if hi = ^ (i=l,2) (31)

It is worth noticing that the generators L11 appears as the product of the preserving area diffeomorphism generator related to h by the dilatation T(ui,U2) : let us postpone these remarks for the next paragraph. In the case of M = S2, considering again the example of Eq. (13) :

o h(9,(p) = Y£ a (9,(P) (32)

a basis for the LT generators is given by:

T. =Y(* !"JLy*0 —- â—y*0 —1 (33) m m •* m [19: c mo acosQ acose ° aJ

leading to the quite heavy commutation relations:

[

-i mu,m-m>Lx,m.m j A-Jl'\ (34)

In the torus case, for a cocycle related to an exact form, the explicit form of is much simpler to handle with if one chooses

h(t,(0) = t1110©"10 (35)

Associating to this h the derivation generators

and 1 0 0 A = I " Gf" (n013t + mo to 3«) (36)

One can easily obtain: [Lmn, LpqJ = (mo(n-q) - no(m-p)) Lmo+m+ptno+n+q

[A, Lmn] = (mo(n+no) + no(m+mo)) Lm-mo+i^^+i (37)

1 as well as the action of the Lmn and P on the ^to (S ) generators:

m Xj1n = t CO" ® Xa (38)

2 (semi direct sum Q^ (T ) D Diff (^0

IA, XJ= (pno + qmo) X (39)

the X^1n satisfying among themselves the commutation relation :

LXmn' Xpq-" = ^c Xm+p,n+q + ^m(l~nP) °mo+m+m "no+n+q (40)

Finally, considering a cocycle related to an harmonic form, that is, for example :

d CO(X1Y) = Jf dt »~ X9tY (41) T2 the compatibility condition implies L of the form

£=A(t,co)9t (41)

Choosing as a basis of generators

m+l Lmn =-t (iPdt (42)

2 One obtains for Diff(^0 (T )) a "Virasoro-like" algebra L L mn»LpqJ = (m-p) Lm+P)n+q (43)

10 acting on the XJL, themselves satisfying:

»• Xb> mn ' as a [Lmn,X pq]=-pX^+nn+q (45)

Yd© X9WY the result will be of the same form, with the change toco.

4. Comments on the algebras Diff(£® (M)) and SDiff(M)

4.1 A generalization of the Virasoro algebra

From the results of Section 3, one sees that, up to the A generator (see Eq. (31)), the diffeomorphism algebra stabilizing COh defined from an exact form h, is made of generators :

LJ" = T.[O!h)a2 - @2h)ai], T varying (46)

= T.Lh

with Lh = (3ih)92 - (32h)3i belonging to SDiff(M)

SDiff(M) = { Lk, k varying } that is the preserving area diffeomorphism algebra of M, and T appearing as a dilatation. In fact, this is also the case for a cocycle related to an harmonic form (in the torus case) : for example, the generator LA = A(t,w)9t can be rewritten as (- - A(t,w)) it9t = T(t,w)Pt where

Pt = it8t is one of the two "translation" generators of Ref.[2] - the second one being Pw = iwdw. Note that is exactly the generalization of the Virasoro algebra seen as the 1 m+1 diffeomorphism algebra on the circle S : any generator Lm = z dz appears as die product m+1 of a translation 9Z - or length preserving transformation - and a dilatation z . At this point,

11 it might be useful to remark that the only cocycle for a Kac-Moody algebra Q (S *) is given by [13]:

w(X,Y) •= JI (X,3ZY) dz (47)

and involves the operator 3Z. In the same way on M = S2OrT2, the cocycle:

(Bh(X,Y) » dui du2 (48)

involves the area-preserving differential operator Lh. Let us add that for T(uiU2) = T(h(uiU2)), the operator Lj can be rewritten as:

Lu (49) with U(h) such that: ^jJ-= T and therefore belongs to SDiff(M).

One easily deduces that the algebras Diff(^œh)) and Diff(^Wf(h)) differ only by their P generator. Moreover one can prove that the area preserving diffeomorphisms compatible with a cocycle Cun form the Abelian subalgebra of Diff(^Wh)

SDiff(£o)h) = { Lf(h) ; f varying, h fixed }

4.2 Action of SDiff(M) on Q (M) cocycles (the case dim M = 2)

We remark that the generators Lj1 with T and h varying generate the whole Diff(M) algebra, the general commutation relation being:

. ifl. L^ + LWW+L]Ji, (50) and the particular position of the subalgebra SDiff(M) with commutation relation :

12 [Lh,Lk]=L{hik) (51)

acting on Diff(M) as:

[Lh, (52)

From the results of Section 3, it is clear that SDiff(M) cannot be associated to a particular cocycle. But the group of area preserving diffeomorphisms on M can be seen as acting on the set of cocycles GDh. Even more, the action of SDiff(M) on COh can be induced from the action of SDiff(M) on ^h- Indeed, considering the group element e k, one has:

Lk Lk Lk Lk û>h(e X,e Y) = j h { e X , e Y } duidu2 (53> M

VX1Ys that is also:

= jheLk{X,Y} duidu2 (54) M

and using integration by parts

Lk = j(e' h){X,Y}duidu2 (55) M

It then follows that :

Lk Lk (e X, e Y) (X1Y). (56) a>e_Lk(h)

We may add that such a property can be seen as a consequence of the action of SDiff(M) on itself if one remembers the correspondence between SDiff(M) and the set of cocycles which can be associated to

13 L.A.P.P. ii I- mi i -.CMi WM ( N-i I \ H I \ < I •)! \ • H I I riiuNI si) :< i; I* • « n i i \

4.3 The case dim(M) > 2

Up to now we have concentrate our attention on examples associated to a two dimensional sphere or torus (membrane case). Of course, following Section 2.1 the connection between SDiff(M) and the set of (7 (M) cocycles is valid whatever the (finite) dimension p (p-brane case) of the smooth compact manifold M under consideration as well as the property expressed by Eq.(56).

Taking as an illustration the case M = T3 = S1XS1XS1 and following the same lines, the general C(T3) cocycle reads :

(X,Y) = |(h {x,Y}tw + k {X,Y}WZ + 1 {x,Y}zt) dtdwdz (57)

the functions h,k,l depending of the three complex variables t,w,z of module 1 which parametrize T3.

The volume preserving diffeomorphism related to Cûhjçj is :

LhJcJ = (dvM - dth9w) + @zkdw - 9wk32) + OA - dzldt) (58)

and it is straightforward to verify that any element in Diff(T3) of the form

= T 1 (59)

with T(t,z,w) varying and hjc,l fixed, leaves the cocycle C0hjc,l invariant. Moreover, one obtains the CR. :

(60)

with W = T Lh,k,i(V) - V UkI(T) and Lh,k,i(V) = {h,V)wt + {k,V}zw + {l,V}t2 and generalizing to three dimensions the commutation relations of Eq. (31).

14 Conclusion

In this short note, we have emphasize on the correspondence between cocycles of generalized Kac-Moody algebra Ç& (M) and the algebra SDiff(M) of volume preserving diffeomorphisms on a smooth compact manifold M of dimension greater than one. We have also remarked the particular role of SDiff(M) inside Diff(M). A generalization of these properties to the supersymmetric case does not seem to present any difficulty (see for example Ref. [17]). More informations on the representation theory of generalized Kac-Moody algebras may also be useful.

Acknowledgements

One of us (P.S.) is endebted to E.G. Floratos for interesting discussions.

15 References

[I] P.A.M. Dirac, Proc. R. Soc. A268 (1962) 57 RA. Collins, R.W. Tucker, Nucl. Phys. B112 (1976) 150 [2] E.G. Floratos, J. Iliopoulos, Phys. Lett. B201 (1988) 237 [3] I. Antoniadis, P. Ditsas, E.G. Floratos, J. liiopoulos, Nucl. Phys. B300 [FS22] (1988) 549 [4] I. Bars, CN. Pope, E. Sezgin, Phys. Lett. B210 (1988) 85 A. Arakelyan, G.K. Sawidy, Phys. Lett. B214 (1988) 350 and Fermilab Pub-88/203-T [5] CN. Pope, K.S. Stelle, preprint CTP-TAMU 07/89 D.B. Fairlie, CK. Zachos, preprint ANL-HEP-PR 89/08 [6] D. Gross, J. Harvey, E. Martinec, R. Rohm , Nucl. Phys. B256 (1985) 253 and Nucl. Phys. B267 (1986) 75 [7] T. Banks, "Lectures on conformai field theory", lectures presented at the 1986 Theoretical Advanced Study Institute in Particle Physics, Santa Cruz, California, 1987 [8] E. Witten, Comm. Math. Phys. 92 (1984) 455 V.G. Knizhnik, A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83 [9] V.G. Kac, "Highest weight representations of conformai current algebras", MIT preprint [10] P. Goddard, A. Kent, D. Olive, Comm. Math. Phys. 103 (1986) 115 [II] V.G. Kac, "Infinite dimensional Lie algebras", Cambridge : Cambridge University Press (1985) [12] R. H0egh-Krohn, B. Torrésani, preprint CPT-86/P-1935, J. Funct. Anal, to appear [13] A. Pressley, G. Segal, "Loop groups", Oxford : Clarendon Press Oxford, 1986 [14] T. Eguchi, P.B. Gilkey, AJ. Hanson, Phys. Rep. 66 (1980) 213 [15] J. Hoppe, MIT Ph.D. thesis 1982, preprint Aachen PITHA86/24 [16] J.L. Loday, D. Quillen, Comments Math. HeIv. 59 (1984). [17] R. Coquereaux, L. Frappât, E. Ragoucy, P. Sorba, preprint LAPP-TH-246/89 and CPT- 89/PE-2269

16 Appendix

Hereafter we present a direct calculation of the general two cocycle for an algebra ÇÇI2). It is a direct generalization of the method used in Ref.[13] - p.40 and 41 - to determine the central extensions of a loop algebra Q(S1).

The antisymmetric two form CD is supposed invariant under the group G of the Lie algebra Q. Writing:

(0 (tP zP' ® Xa, W zfl' ® Xb) = coPj' (X«, Xb) (A.1) then OUP^ is a G invariant bilinear map : QQ X QQ -> C : £ being semi-simple, col:H is therefore symmetric, proportional to the killing form of Q and thus :

<* - -

The Jacobi identity implies:

d)pl+q'>r + û)q>+r>p' + tor'+p>q' = 0 (A.3)

Now we have just to choose particular values for the Z elements p,q,r and p',q',r' to recover the results of Eq.(21).

Taking q = r = q' = r* = 0, we get:

Choosing p' = q' = r1 = 0 we are in the conditions to be satisfied by a Q (S1) cocycle (see again Ref.[13]) and thus we obtain:

\j Il O O O /A C\ S A 5 %q =P p+q ^-I ^ - ) same result for

17 «y -v -i (A.6)

Then, putting r = n-p-q and r1 = n'-p'-q' we have:

(A.7)

that is with n' = p' and n = p :

(A.8)

so that 1 = p'q - pq w p+q 3*0 p+q 0 p+q P'q - pq' Qp'+a'-l p'+q' * 0 p'+q' l P+q-l

p = -q and p' = -q' (A.9)

Using Eq. (A.7) we have also:

(A.10)

Then, defining:

r 1 hp'+q 1 cjOp'+q'-i p+q ^O * p'+q' Vl " (p+q)(p'+q') VqO with .0 1_ 1-1 p'+qVO P+q p+q ^) P+q

p'+q'^O (A. 11)

and : a = oy _j and

18 the most general solution satisfying (A.2) and (A.3) is :

1 1 flag* = (oq - (Jq ) Op+1,5p.+q. - (pq - p'q) h£j" (A.12) which is exactly the result expressed by Eq.(21).

Last remark : in Eq.(21) the h^1 are the Fourier modes of the function h(Q,t), while in

Eq1(A. 12) they represent any coefficient. Therefore, Eq.(A.12) includes also the distribution case, since, on the torus, every distribution admits a Fourier decomposition.