  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.
