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A natural supersymmetricextension (dG)~ is defined of the current (= affine Kac-Moody Lie) algebra d(7; it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in s uperstring models. All unitary irreducible positive energy representations of (d~)~ are cons~tructed. They extend to unitary representations of the semidirect sum S~(G) of (dG)~ with the superconformal algebra of NeveuSchwarz, for x = 1⁄2, or of Ramond, for ~c = 0. O. Introduction The semidirect sum of the Virasoro algebra Wc and the algebra dG of left (or right) currents for a compact Lie group G arises naturally in both conformal invariant 2dimensional QET models [1-3] and in the general study of infinite dimensional Lie algebras [4-7] (see also [8, 9]). Its supersymmetric extension which is implicit in recent work on superstrings 1-10-12] also admits a local field interpretation (partly exploited in [13, 14] as a development of the QFT approach of [15]). The objective of this note is two-fold: (a) to set a mathematical framework in which the supercurrent and string superalgebras arise naturally; (b) to classify all hermitian ( = unitary) positive energy representations of these algebras. A remark is also included, concerning the unitarity of the discrete series of representations of the super Virasoro algebra (with central charge c < ~-). In the theory of infinite dimensional Lie algebras a chiral current algebra dG (~lled an affine Kac-Moody algebra) arises as a central extension of the loop algebra dG generated by tensor products of elements of the finite dimensional Lie algebra dG with Laurent polynomials of a complex variable t. The supersymmetric extensions * On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria 338 V. G. Kac and I. T. Todorov A (dG)~ discussed in this paper are obtained by simply adding a Grassmann variable 0 to the argument of the polynomials. We construct a "minimal representation" of the arising algebra which allows us to reduce the representation theory of (d~)~ (and of its super Virasoro extension S~(G)) to the known classification of unitary highest weight irreducible representations (UHWIRs) of d~. 1. S u p e r c o n f o r m a l Current A l g e b r a s Let G be a compact Lie group and dG be its Lie algebra equipped with the (negative definite) Killing form (x, y). The super loop algebra d'G is defined as dG=dG@~C[t , t -1;O] , teC×(= {teC;t ~=O}), 82=0 , (1.1) regarded as an infinite Lie superalgebra with bracket [x®P(t,t-1;O), y®Q(t,t-1;O)] = [x,y]®P(t,t-1;O)Q(t,t-l;O), (1.2) where P and Q are any (linear in 0) polynomials and Ix, y] is the Lie bracket of dG. We introduce a 1⁄27/-gradation on dG setting d e g d G = 0 , d e g t = l , deg0=tce1⁄2Z; (1.3) the corresponding graded algebra will be denoted by (dG)~. The general even central extension (dG), of (dG), is obtained by adding a cocycle ~k(x ® P(t, t l; 0), y ® Q(t, t l; 8)) = (x, y) f ( (dP)Q), (1.4a) to the right-hand side of(1.2), wherefis a linear functional on 1-forms that vanishes on exact and on odd (in 0) forms: dt f (Podt PiOdO + PzdO -P3Odt) = ~ (otP o + fit 2~ip~) (lAb) Itl = 1 2rci ' where Pk = Pk(t, t l ) are polynomials and we assume that ~ and fl are positive numbers 1. (The powers are chosen in such a way that deg ~ = 0.) P r o p o s i t i o n 1. The most general graded odd and even differentiations D~(e = 1, O) satisfying D~f({d(Po + OP1)}Q): =f({dD~(Po + OPO}Q) + f({d(Po + ( 1)~OP~)}D~Q) = 0 (1.5) (P0.1 = P0,1(t, t i) , Q = Qo(t, t l ) + OQl(t, t l ) ) are multiples of \.~lot 08 ~[ flOOt ] \~/-£-~0 t X/ fl O ~ J ' (l'6a) ~ [ D . _ ~ , D ~ ] + = -t" t + + ~c 0 ( n e Z ) . (1.6b) We shall sketch the proof for the odd generators. Setting D i = RoO(~/~t ) + If aft < 0, then the energy operator Lo, constructed below, would have negative spectrum

Superconformal current algebras and their unitary representations