Infinite-dimensional symmetries of two-dimensional coset models

Abstract

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 G ˆ , the affine Kac–Moody extension of G , or else a subalgebra thereof. We take the very explicit approach of Schwarz 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 ˆ , whilst the subalgebra obtained by Schwarz arises as a gauge-fixed truncation. 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.