Current Algebra of Classical Non-Linear Sigma Models

The current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is analyzed. It is found that introducing, in addition to the Noether current jμ associated with the global symmetry of the theory, a composite scalar field j, the algebra closes under Poisson brackets. University of Freiburg THEP 91/10 August 1991 ∗Address after September 1, 1991: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge MA 02138 / USA It is well known that for the classical non-linear sigma models, the Poisson brackets between the Noether currents associated with the global symmetry of the theory involve Schwinger terms which, in general, are field dependent. Explicit expressions have been written down, e.g., for the principal chiral models (sigma models on compact Lie groups) [1, 2] and for the spherical models (O(N)-invariant sigma models on spheres) [3, 4], but the structure of the resulting algebra has, to our knowledge, not yet been analyzed in full generality. In the following, we want to show that, for classical non-linear sigma models on arbitrary Riemannian manifolds, introducing the coefficient of the Schwinger term appearing in the Poisson bracket between the time component and the space component of the Noether current jμ as a new composite field j leads to an algebra which closes under Poisson brackets: this is what we propose to call the current algebra for these models. To this end, consider the classical two-dimensional non-linear sigma model on an arbitrary Riemannian manifold M with metric g. The configuration space of the theory consists of (smooth) maps φ from a fixed two-dimensional Lorentz manifold Σ (typically two-dimensional Minkowski space) to M , while the corresponding phase space consists of pairs (φ, π) of fields, π being a smooth section of the pull-back φ(T M) of the cotangent bundle of M to Σ via φ. In terms of local coordinates u on M , φ and π are represented by multiplets of ordinary functions φ and πi on Σ; then the action reads S = 1 2 ∫ dx gij(φ) ∂ φ ∂μφ j , (1) 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) . (2) Denoting the time derivative by a dot and the spatial derivative by a prime, we have πi = gij(φ) φ̇ j . (3) Under local coordinate transformations u → u , the component fields φ, ∂μφ i and πi transform according to φ → φ , ∂μφ i → ∂μφ ′k = ∂φ ∂φi ∂μφ i , πi → π ′ k = ∂φ ∂φ′k πi , (4) from which it can be checked that the action (1) and the canonical commutation relations (2) are invariant. Next, let G be a (connected) Lie group acting on M by isometries. Then every generator X in the Lie algebra g of G is represented by a fundamental vector field XM