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Diffeomorphism Group of a Closed Surface*) LAPP-TH-254/89 . June :;/•. .u Diffeomorphism Group of a Closed Surface*) 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 : <oc(X,Y)= J<X,dY> (D 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<p = ]£ 3i <pi dui A...A dun = 0 (4) 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 (<Pi ^i Y) «l - ànn (5) J i=l M 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 diffeo- morphism <p on M 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,<p) or (cosO,(p) for example - (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,<p). So, choosing as an example: h(8,(p) = YJJJ (0,<p) (13) and for X and Y : 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 decomposes as the sum of an exact form and an harmonic one. Choosing the complex variables t and w (ItI = IwI = I) as the coordinates on each of the two circles making up the torus, the general Hodge decomposition of <p reads (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 <p reads 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.
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