Kac–Moody and Virasoro symmetries of principal chiral sigma models

It is commonly asserted that there is a Gˆ×G centreless Kac–Moody extension of the manifest G×GG×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 Gˆ×Gˆ, the full centreless Kac–Moody extension of the entire manifest G×GG×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 Kn=Ln+1−Ln−1Kn=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 Gˆ.