The Algebra of the Energy-Momentum Tensor and the Noether Currents in Classical Non-Linear Sigma Models

The recently derived current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is extended to include the energy-momentum tensor. It is found that in two dimensions the energy-momentum tensor θμν , the Noether current jμ associated with the global symmetry of the theory and the composite field j appearing as the coefficient of the Schwinger term in the current algebra, together with the derivatives of jμ and j, generate a closed algebra. The subalgebra generated by the light-cone components of the energy-momentum tensor consists of two commuting copies of the Virasoro algebra, with central charge c=0, reflecting the classical conformal invariance of the theory, but the current algebra part and the semidirect product structure are quite different from the usual Kac-Moody / Sugawara type construction. University of Freiburg THEP 92/24 October 1992 Supported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/2-1 Supported by the Studienstiftung des Deutschen Volkes In a recent paper [1], we have derived the current algebra for classical non-linear sigma models defined on Riemannian manifolds. This algebra is quite simple to write down and yet does not seem to belong to any of the algebras which are well known in mathematical physics, mainly because it involves non-standard (in particular, non-central) extensions of loop algebras [4]. On the other hand, the classical non-linear sigma model in two dimensions is conformally invariant, so its energy-momentum tensor must satisfy the classical version of the standard commutation relations of conformal field theory, that is, under Poisson brackets its light-cone components must generate two commuting copies of the Witt algebra (the Virasoro algebra with vanishing central charge). We shall verify that this is indeed the case. Moreover, we shall derive the Poisson bracket relations between the energy-momentum tensor on the one hand and the Noether currents on the other hand. The resulting total algebra exhibits, in a concrete field-theoretical model with continuous internal symmetries, the possibility of reconciling conformal invariance, expressed through a chiral energy-momentum tensor algebra, with a non-chiral current algebra, at least at the classical level. Thus consider the classical two-dimensional non-linear sigma model, whose configuration space is the space of (smooth) maps φ from a given two-dimensional Lorentz manifold Σ to a given Riemannian manifold M , with metric g, while the corresponding phase space consists of pairs (φ, π) with φ as before and π a (smooth) section of the pull-back φ(T M) of the cotangent bundle of M to Σ via φ. The action, written in terms of isothermal local coordinates x on Σ and of arbitrary local coordinates u on M , reads S = 1 2 ∫ dx η gij(φ) ∂μφ i ∂νφ j , (1) where the ημν are the coefficients of the standard Minkowski metric. Thus using a dot to denote the time derivative and a prime to denote the spatial derivative, we have πi = gij(φ) φ̇ j , (2) and the canonical Poisson brackets are {φ(x) , φ(y)} = 0 , {πi(x) , πj(y)} = 0 , {φ(x) , πj(y)} = δ i j δ(x− y) . (3) The energy-momentum tensor θμν of the theory is most conveniently derived by variation of the Lagrangian with respect to the metric on Σ. (For details, see e.g. [3, p. 64 ff] or [5, p. 504 f].) It reads θμν = gij(φ) ∂μφ i ∂νφ j − 1 2 ημν η κλ gij(φ) ∂κφ i ∂λφ j , (4) and it is obviously traceless: η θμν = 0 . (5)