Kac-Moody and Virasoro symmetries of principal chiral sigma models

It is commonly asserted that there is a GxG centreless Kac-Moody extension of the manifest GxG 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 GxG, the full centreless Kac-Moody extension of the entire manifest GxG global symmetry. Extending previous results in the literature, we also obtain an explicit realisation of the Virasoro-like symmetry of the PCM, generated by Kn=Ln+1-Ln-1 for both positive and negative n. We show that these generators obey Sugawara-type commutation relations with the two commuting copies of the Kac-Moody algebra G.