Kac–Moody and Virasoro symmetries of principal chiral sigma models

Abstract

It is commonly asserted that there is a G × G centreless Kac-Moody extension of the manifest G × 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 G × G, the full centreless Kac-Moody extension of the entire manifest G × 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 G.