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

TL;DR

This paper extends the current algebra in classical non-linear sigma models to include the energy-momentum tensor, revealing a closed algebra structure with two Virasoro algebras at zero central charge, highlighting classical conformal invariance.

Contribution

It introduces a novel algebraic structure combining the energy-momentum tensor and Noether currents in classical sigma models, differing from traditional Kac-Moody/Sugawara frameworks.

Findings

The energy-momentum tensor and currents form a closed algebra in 2D.

The light-cone components generate two commuting Virasoro algebras with zero central charge.

The algebraic structure differs from standard Kac-Moody/Sugawara constructions.

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 $\theta_{\mu\nu}$, the Noether current $j_\mu$ 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 $j_\mu$ 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.