Volume 194, number 1 PHYSICS LETTERS B 30 July 1987 SCALE INVARIANT SIGMA MODELS ON HOMOGENEOUS SPACES E. GUADAGNINI Dipartimento di Fisica dell'Universita di Pisa and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, 1-56100 Pisa, Italy M. MARTELLINI Dipartimento di Fisica dell'Universit~ di Milano, 1-20133 Milan, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, 1-2 7100 Pavia, Italy and M. MINTCHEV lstituto Nazionale di Fisica Nucleare, Sezione di Pisa and Dipartimento di Fisica dell'Universiti~ di Pisa, 1-56100 Pisa, Italy Received 11 March 1987 A class of two-dimensional scale invariant a-models on homogeneous spaces is presented. We discuss the symmetry properties of the models and show that they are finite. We compute at two loops the central charge of the Virasoro algebra associated with the energy-momentum tensor. Introduction. The classical approach (see e.g. ref. [ 1 ]) to construct a string theory, based on a free two- dimensional field theory (a-model with fiat target space), has been extended on group manifolds in refs. [ 2-6 ]. The key points of this extension are the existence of two commuting (analytic and antianalytic) Kac-Moody current algebras and a Sugawara representation [ 7] of the energy-momentum tensor in terms of the above currents. The study of the propagation of a string on a curved background (a-model with curved target space) is important mainly for two reasons. Firstly, it can be considered as a starting point for defining a string theory. Secondly, it provides insight on the compactification problem. In relation with the last point, it becomes rel- evant to consider more general target spaces than group manifolds. A natural step after group manifold com- pactifications is to construct a scale invariant theory on a homogeneous space. The first problem which arises is if a scale invariant model of this kind actually exists. As proven by Mukhi [ 8], a sufficient condition for the vanishing of the//-functions (rigid scale invariance) is the parallelizability of the target manifold. Unfortunately we cannot use this mechanism for homogeneous spaces, because apart from the seven-sphere, all parallelizable manifolds are groups. A second point to keep in mind is that the existence of a Kac-Moody algebra implies [ 9 ] that the target manifold is parallelizable. So for a generic homogeneous space the standard approach to the string theory based on Kac-Moody currents [3,6] is no more applicable. In this paper we construct a class of models on homogeneous spaces. We show that (i) they are finite, i.e., the fl-functions vanish and in this sense there is rigid scale invariance; (ii) they admit two partial Kac-Moody algebras, where by partial we mean that the number of left (right) currents is not equal to the dimension of the manifold; (iii) the energy-momentum tensor is not of the Sugawara form with respect to the currents mentioned in (ii). Nevertheless, as a consequence of (i), there is a Virasoro algebra associated with the energy-momentum tensor; 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 69