Kac-Moody and Virasoro Symmetries of Principal Chiral Sigma Models

TL;DR

This paper reveals that the principal chiral model's symmetry is larger than previously thought, extending to  G  G, and explicitly constructs its Virasoro-like symmetry with detailed algebraic relations.

Contribution

It demonstrates the full  G  G symmetry extension and provides an explicit realization of the Virasoro-like symmetry in the principal chiral model.

Findings

The symmetry extends to  G  G, not just  G  G.

Explicit Virasoro-like generators obey Sugarawara-type relations.

The Virasoro-like symmetry commutes with the Kac-Moody algebra.

Abstract

It is commonly asserted that there is a \hat G\times G centreless Kac-Moody extension of the manifest G\times G global symmetry of the two-dimensional principal chiral model (PCM) for the group manifold G. Here, we show that the symmetry is in fact larger, namely \hat G\times \hat G, the full centreless Kac-Moody extension of the entire manifest G\times G global symmetry. Extending previous results in the literature, we also obtain an explicit realisation of the Virasoro-like symmetry of the PCM, generated by K_n=L_{n+1} - L_{n-1} for both positive and negative n. We show that these generators obey Sugarawara-type commutation relations with the two commuting copies of the Kac-Moody algebra \hat G.