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Extended Kac-Moody Algebras '• • and Applications \j LAPP-TH-335/91 Avril 9l I 1 Extended Kac-Moody algebras '• • and Applications E. Ragoucy and P. Sorba Laboratoire Physique Théorique ENSLAPP" - Groupe d'Annecy Chemin de Bellevue BP 110, F-74941 Annecy-le- Vieux Cedex, France Abstract : * We extend the notion of a Kac-Moody algebra defined on the S1 circle to super Kac-Moody algebras defined on M x GN, M being a smooth closed compact manifold of dimension greater than one, and G/v the Grassman algebra with N generators. We compute all the central extensions V-, »! of these algebras. Then, we determine for each such algebra the derivation algebra constructed : from the M x <7/v diffeomorphisms. The twists of such super Kac-Moody algebras as well as , the generalization to non-compact surfaces are partially studied. Finally, we apply our general construction to the study of conformai and superconformai algebras, as well as area-preserving j diffeomorphisms algebra and its supersymmetric extension. Wf A review to be published in Int. Jour, of Mod. Phys. A "URA 14-36 du CNRS, associée à l'Ecole Normale Supérieure de Lyon, et au Laboratoire d'Annecy-le-Vieux de Physique des Particules. ••• \ ; Contents 8 Appli< 8.1 (, 1 Introduction. 8.2 I- 8.3 h-. FUNDAMENTALS 6 O Cone U The usual Kac-Moody algebras. G 2.1 Loop algebra 6 A Deter •****< 2.2 Twists of loop algebras 7 A.I h 2.3 Central extension of a loop algebra 7 A.2 C, 2.4 Kac-Moody algebras and Hochschilq (cyclic) cohomology 8 A.3 C 2.5 Derivation algebra S B KM a 2.6 The Sugawaraand GKO constructions 10 B.I K B.2 h Central extensions for generalized loop algebraB. 12 3.1 Generalized loop algebras 12 C Parmi 3.2 Determination of the conditions to be verified by the central extension 14 3.3 Explicit form of the cocycles 15 D The r 3.4 First properties of the central extensions 17 E JV = 3 3.5 Twists of extended Kac-Moody algebras: an introduction 17 3.6 Divergence free vector fields and Kac superalgebras 20 F JV = 2 Diffeomorphisms compatible with a central extension. 20 4.1 What is a compatible algebra ? 20 4.2 The compatibility equation and its consequences 21 4.3 Compatibility equations for T>\} 23 Generalization to non-compact surfaces (without boundaries) 25 5.1 Coming back to compact surfaces with no supersymmetry 25 5.2 Generalization to non-compact surfaces 26 5.3 Supersymmetrization 27 •"I II APPLICATIONS 28 6 Application to the superconformai case. 28 28 6.1 The superconformai algebras SC(N) 28 28 6.2 Are the SC(TV) algebras compatible algebras ? 29 29 6.3 N=I case : the RNS algebra 30 30 6.4 The N = 2 case : t/(l)dSC(2) 34 34 6.5 Relation with complex structures 37 37 6.6 JV > 2 : New super conformai algebras 40 10 7 Application to (super) membrane and the area-preserving diffeomorphisms. 42 42 7.1 The aera-preserving diffeomorphisms algebra 42 •12 7.2 The bosonic case 44 44 7.3 The supersymmetric case 46 16 ; • ! t , 8 Applii 8 Application to the Krichever-Novikov algebras 50 8.1 i. 8.1 Case of the cylinder: the Virasoro algebra 50 8.2 I 8.2 Punctured surfaces of genus g> 0: the Krichever-Novikov algebras 51 8.3 S. 8.3 Supersymmetrization 53 8 Conrii 9 Conclusion and outlooks. 56 A Deter it A Determination of the cocycles of generalized KM algebras. 56 A.I It (i A.I Demonstration of the general theorem 58 A.2 C, 7 A.2 Classes of the cocycles 60 A.3 C 7 A.3 Covariance of the cocycles 60 8 B KM a 9 B KM algebras, fields and OPE's 62 B.I K 0 B.I Kac Moody algebras and current algebras 62 B.2 h B.2 Superfields and N = 1 Kac Moody algebras 63 2 C Paran 2 C Parametrization of two-dimensionnal surfaces. 65 4 D Thee 1 r> D The compatible algebras related to G^(S x G1). 66 .7 E /V = 3 E N = 3 Kac-Moody algebra defined on 51 leading to SC{2) algebra 68 ,7 F N = 2 JO F TV = 2 Kac-Moody algebras defined on M compatible with SDiJT(M) 70 10 20 21 23 25 25 26 27 28 28 28 29 30 34 37 10 42 42 44 46 1 Introduction. Vect(5')ot the circle v algebra Q. The importance of Kac-Moody algebras has been increasing since 1984 because of the essential KM algebi role they play in two dimensional conforma] theories[l], themselves covering a rather wide domain in theoretical physics. Indeed in string theory, the string develops a two-dimensional surface in the Therefo space in which it is embedded, i.e. the world sheet. Conformai transformations on the world sheet natural to a leave the string action invariant. In statistical mechanics, conformai invariant quantum field theory can be achi describes the critical behaviour of systems at 2nd order phase transitions. Conformai invariance i) super is also present in integrable systems. As a (good) example, Toda field theories associated with for the Gra simple Lie algebras are both integrable and conformally invariant. Their conformai structure can ii) enlar be derived from their integrability properties, and directly shows up in the A-matrix solution of iii) imp< the classical Yang-Baxter equation [2] with dimA' In their fundamental paper [3], Belavin, Polyakov and Zamolodchikov have in particular shown As soon that two-dimensional conformai field theories (CFT) can be treated from a group theoretical point of or iii) a sin view. The basic ingredient is the two-dimensional conformai algebra, which is infinite dimensional, show up. and its central extension, called the Virasoro algebra V. The infinite dimensional algebras directly case, the m constructed from simple Lie algebras Q as (central extensions of) the set of maps from the circle infinite. Tit S1 into Ç, and called Kac Moody (KM) algebras are directly related to the Virasoro algebra. central ex te acting as a They appear in two-dimensional current algebras when considering a massless spinor (or scalar) a subalgebr Held, such a model leading to an exactly solvable CPT. They are the two-dimensional expression sum with i' of gauge symmetry in string theory : let us remember that in the heterotic string model, string states are classified in a representation of the Kac-Moody algebra of the gauge group EH x EH or These Spin(32)/Z2 [4j. The best example of the deep connection between a Kac-Moody algebra and the applicatioi Virasoro algebra is without doubt given by the Wess-Zumino-Witten (WZW) models [5], where naturally a both algebras are symmetries of snch field theories. general con performed Prom an algebraic point of view, for each KM algebra <7(5') - where S1 is the one-dimensional on the alg circle and the'denotes the central extension of the loop algebra G(S{), see below for more precise by an algeb definitions — one can construct the semi-direct sum : Q(S1 )OV, the Virasoro generators acting diffeomorpi as Lie derivatives on the elements of G(S1). An illustration of this property might be given by manifold: the Sugawara construction, which allows a realization of the Virasoro algebra from a Kac-Moody order to IP one. It is in part because of the semi-direct structure and of the Sugawara construction that one general the can determine, through the famous Goddard-Kent-Olive (GKO) construction, the minimal unitary appendix I representations of V from the unitary representations of some G(S1) algebras [6]. In Par! Now, we would like to point out the two following properties. First, most of the results obtained (section 6 up to now in two dimensions conformai theories need for G only the simple algebra SU(2) (we ignore seen as the now the string models with 2?« or spin(32) unification groups) : examples can be given by the However it A.D.E. classification [7] of modular invariant partition functions for "minimal" conformai theories, et al. alget) or by the GKO coset construction [6] of minimal theories. The use of bigger G algebras, that is For N = 2 algebras containing SU(2), opens new roads : the W11 algebras, whose interest increased recently, one in sue', in particular with their appearance [8] in Toda field theories [9] (based on groups G, bigger than us from co SU(2)!). completen to constrii' As a second remark, the Virasoro algebra which is actually the central extension of the algebra method, v Wct(S') o! Vect(S') of the diffeomorphisms group on the S1 circle (see section 2.5 below) acts effectively on the circle v the circle variable of the KM algebra C(S*) and its action is independent of the structure of the algebra Q. algebra Q. This remark may look naive, but shows explicitly the role of the continuous part of the KM algebr . KM algebra. Therefo Therefore, in the same way the "fundamental" SU(2) algebra can be enlarged, it is rather natural to a natural to ask oneself what can be learned when increasing this continuous z-part. Such a program can be achi can be achieved in different ways, for example by : i) super, i) supersymmetrizing the S1 circle, i.e. replacing it by the supercircle S1 x GN where GN stands for the Gra fur the Grassmann algebra generated by N Grassmann variables 0,,, a = \,...,N.
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