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 K\"{a}hler potential of the four dimensional case, with the fundamental 2-form given by $\tilde J= i*\partial \bar \partial K$. The coupling to Yang-Mills is through a term $K\wedge tr (F\wedge F)$ and leads to a Uhlenbeck-Yau field equation $\tilde J^{ij}F_{ij}=0$.