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7 International Centre for Theoretical Physics IC/93/173 -n I 'r.2_. \ INTERNATIONAL CENTRE FOR 7 THEORETICAL PHYSICS IIK;IIKK SPIN EXTENSIONS OF THE VIRASORO ALCE1IRA, AREA-PRESERVING AL(;EURAS AND THE MATRIX ALGEBRA M. Zakkuri INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE IC/93/173 Infinite dimensional algebras have played a central role in the development of string theories and two dimensional conformal field theory. These symmetries have been shown International Atomic Energy Agency to be related to the Virasoro algebra, its supersymmetric extensions and the Kac-Moody and one. Together, they are generated by conformal spin ,s currents with s <2. United Nations Educational Scientific and Cultural Organization However, some years ago, Zamolodchikov discovered a new extension, W3, involving besides the usual spin 2 conformal current, a conformal spin 3 current [1]. The obtained INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS structure is called non-linear Lie algebra. Recently, much interest has been made in the understanding of these higher conformal spin extensions of the Virasoro algebra [2.3,4}. In this paper, we study these algebraic structures using the infinite matrix representa- tion. Actually the present work may be viewed as a generalization of the infinite matrix HIGHER SPIN EXTENSIONS OF THE VIRASORO ALGEBRA, realization of the Virasoro algebra obtained by Kac et al. [5|. Some results have been AREA-PRESERVING ALGEBRAS AND THE MATRIX ALGEBRA described in our first papers [6,7]. However, we give here some precisions on the generality of above results. Moreover, we extend the above construction to include the algebra of area preserving diffeomorphism on the 2-dimensional torus [8], Actually, this result is k 1 expected to be generalizable for higher dimensions [10], namely for the torus T , k > 1. M. Zakkari Among our results is the obtainment of a generating functional of higher integer conformal International Centre for Theoretical Physics, Trieste, Italy. spin extensions of the Virasoro algebra, and an embedding of the area preserving algebra the torus T2 in the Lie algebra a,*, [5]. By fioo we denote the Lie algebra of infinite matrices having a finite number of nonva- nishing diagonals (5). It is generated by the set {dk. k £ Z) with ABSTRACT d*(a) = £>(*.« +fc)£,-.i+*, (1) The generating functional of Kac-Moody, Virasoro algebra and area preserving diffeomor- ;eZ 2 phism on the torus T is given. Realization of higher integer spin symmetries is discussed. where a(i,i-\-k) are for instance arbitrary complex numbers and E,; are the usual (//(cso, C) The correspondence between the area preserving diffeomorphism on the torus T1* and the generators satisfying algebra a«, is pointed out. ijEk.i = 6j),E,i . (2) In Ref. [6[, we have constructed a class of subalgebras of aM obtained by solving the constraint equations implied by the following requirement [dk[a),dt(a)\=s{kj)dk+((a). (3) These constraint equations read as MFRAMARE - TRIESTE July 1993 s(i, k)e{i + k,t)- s(i, ()s(i + L k) = s(k, ()s(i, k + i) &(k,l) =-s(t,k) (4) together with a(i,i + k)a(i + k,i + k + t)- a(i, I + C)a(i + l,i + k + t) = s(i, k)a(i, • (5) We have then shown the solutions of Eqs. (4-5) are given by [6,7j s(k,l) = a(k--t)exp{.\ke) 'Permanent address: L.P.T., Facultedes Sciences de Rabat, B.P. 1014 Rabat, Morocco. (6) b{i,k) = [ai + 0k + -v)exp(AiJfc) , 1 -*. — m •-• where a, A, /*, ,3 and 7 have been shown to obey where the composite conformal spin 3 mode generators are given by 2 ti = A = 0 X(kJ) = \kLn - l(k + + [(t + t) + a = a, 7 = " jff + y3" = a . T( (L2) Thus the free parameters in (6) are a, ii - 3- and 7. Therefore, we obtained two kir of proper subalgebras of a,x. The first one. correspond to take a = 0 is nothing but 1 In the next paragraph, we give the generating functional of higher integer conforma! spin U[l) affiiie algebra. In this case the matrix components of the (/(I) generators are gi\ extensions of the Virasoro algebra. Consider than the new object [6]: by b{i,k) = 1. The second subalgebra, which corresponds to a ^ 0 is just the Viras< algebra with a trivial central charge. The free remaining parameters have been shown ••* (13) ptay a central role at the level of the representation theory of the a*, on the Dirac wee space [5,9], In fact, the parameter 7 can be set to zero without loss of generality. Inde where u is an arbitrary parameter. it is easy to see that 7^0 leads to a global shift of the eigenvalues of the zero me Using Eqs. (2), one calculates the denning relations of the algebra of the 7\(u)'s. they operator d0 when acting on certain vector space V (e.g. such that End V = gt(ao, C read as follows The generators of the f/(l) bac-Moody and the Virasoro algebras read then in the infin [Tk(u),Tt(v)] = (exp(M -exp(£u))7V<(« + u) • (14) matrix representation as follows It is easy to check that the generators V^ . V't defined by •eZ •ez describe respectively the L'[\.) affine and the Virasoro algebras. Furthermore, one can Using Eq. (S), one expects that higher integer conformal spin extensions of the Viras. define the basic objects in the higher integer conforma! spin matrix representation E(|. algebra should have the generators of the form [6]. (9) as follows if^T'.Ml^ (16) d (s) = £(i!~' + aj('"-•*• + •-.+ rt,_,A-'"')£..,+* k At this stage, one notes that 0u = djdu behaves as a conformal spin 1 operator. i€Z Naturally, we are led to invesiigate the central extension of the above algebras. This where a2 a,_, are arbitrary complex parameters. The set of generators {dk(s),l is obtained by extending the relation of Ref. [0] giving the two cocycle of the U(\) Kac- Z.s < ;V}, N fiexd integer, describe then higher conformal integer spin extensions 5 < Moody and Virasoro algebra. Thfi latter are constructed out of the iy((oo,C) two cocycle, of the Virasoro algebra. In the case N = 3. they are given by namely C{El}.Eji) = -C(£•,,.£,,) = L if 1 < Oand j > 1 , (L7) and zero elsewhere. Therefore, the generating functional of the central extension of the algebra (14) is given by where at = |, a? = •; and /3 = 1. Their commutation relations are given by C0(Tk(u),Tf{v)) = 6k+r (18) This leads to a C = —2 Virasoro algebra generated by the Yk s of Eq. (15). However, [\k,Jt\ = -2f one can see that C = 1 Virasoro algebra is generated by the functional (19) The corresponding central charge formula is given by (20) where the choice 7 = —7 has been done. This result generalizes the known central charge showing thus that q is a yVth root of unity. Plugging Eq, (2<) in Eq. (26) one derives the of a = 3 conformal algebra namely expression of the /r's, namely C{k) = Akh + fit3 + Ck- (21) ^ 2 ' ' with A,B and C fixed numerical coefficients. In what follows, we show that the construction Eq.(3) is in fact a particular one and leading to a general t(m,n) than the one given by Eq. (26) that the solutions Eq. ^>) are valid at first order, thus one may then find other solutions. s(m,n) = <7nAm - q-n*"> . (30) Therefore, the system of Eq. (6) has an infinite set of solutions, leading then to infinite set of subalgebras of ax. To start, consider the following, simple, generalization of Eq. The generators dmimj(a) and their defining commutation relations are given by (3): + r •m, (31) T (32) where mi,mj,ni,n2 € Z, a (/t)i,mj,i) are arbitrary complex numbers subject to satisfy certain conditions we will give below. The integer r is a fixed given parameter, and we Taking then q = exp ^J as JV is odd. Eq. (32) becomes are restricting the indices i and j in E,3 to range over the interval [~'\r\. In this case, if the index i + m3 lies outside [—'",'•) it is to !>e " umklapped" back into the interval by the (:J3) addition of an appropriate multiple of .V = 2r + 1. The equality (23) leads then to the following consistency conditions Defining the rescaled generators Lnumi = ^dm,Ml one can show that in the limit .V ;(m. 11) = — i(n, m) (24a) +oo the i'satisfy the following commutation relation „, L,n] - n A f34) £{n,p)s(m,n + }>) — i(n.m)i(p, m -f '() = ;"(/'. .m + p) (24b) which is precisely the algebra of local area preserving difTcomorphissns of the two-dimeu.sional tors T2 [Sj. Moreover, the limit A' —> +x coi'sespond to r —> +oc>. this establishes that r m2) - <i ( n.i) , (25) the aw contains the area preserving dilfeoniorpliisms of the torus. Again, one can con- where m = (mum?), n s (ns, n?). and in + a = [mx + "[•»>•> + '*?)• struct a generating functional of the generators Eq. (31). This corresponds to rescaling r 2 The first remark to do is that Eq. (24) are similar in form to those given by Eqs. {4) the 7*(«), given by Eq. (13). by the terms (?""< +"«>/ ) and taking u = i~. So that the modulo a detailed structure of the variables. An obvious solution is then study developed after Eq. (13) can be performed again. In the end of this letter, we note that one can generalize the constructions Eqs.
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