Kac-Moody and Virasoro Symmetries of Principal Chiral Sigma Models

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.