Infinite-Dimensional Symmetries of Two-Dimensional Coset Models

It has long been appreciated that the toroidal reduction of any gravity or supergravity to two dimensions gives rise to a scalar coset theory exhibiting an infinite-dimensional global symmetry. This symmetry is an extension of the finite-dimensional symmetry G in three dimensions, after performing a further circle reduction. There has not been universal agreement as to exactly what the extended symmetry algebra is, with different arguments seemingly concluding either that it is $\hat G$, the affine Kac-Moody extension of G, or else a subalgebra thereof. Exceptional in the literature for its explicit and transparent exposition is the extremely lucid discussion by Schwarz, which we take as our starting point for studying the simpler situation of two-dimensional flat-space sigma models, which nonetheless capture all the essential details. We arrive at the conclusion that the full symmetry is described by the Kac-Moody algebra G, although truncations to subalgebras, such as the one obtained by Schwarz, can be considered too. We then consider the explicit example of the SL(2,R)/O(2) coset, and relate Schwarz's approach to an earlier discussion that goes back to the work of Geroch.