Large conformal invariance and duality in self-dual gravity and (2,1) heterotic string theory

A system of gravity coupled to a 2-form gauge field, a dilaton and Yang-Mills fields in 2n dimensions arises from the (2,1) sigma model or string. The field equations imply that the curvature with torsion and Yang-Mills field strength are self-dual in four dimensions, or satisfy generalised self-duality equations in 2n dimensions. The Born-Infeld-type action describing this system is simplified using an auxiliary metric and shown to be classically Weyl invariant only in four dimensions. A dual form of the action is found (no isometries are required). In four dimensions, the dual geometry is self-dual gravity without torsion coupled to a scalar field. In D>4 dimensions, the dual geometry is hermitian and determined by a D−4 form potential K, generalising the Kähler potential of the four dimensional case, with the fundamental 2-form given by J̃=i∗∂∂̄K. The coupling to Yang-Mills is through a term K∧tr (F∧F) and leads to a Uhlenbeck-Yau field equation J̃ijFij=0.