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

Abstract

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 $θ_{μν}$, the Noether current $j_μ$ 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 $j_μ$ 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.