  We consider a heterotic version of six-dimensional Kodaira-Spencer gravity
derived from the heterotic superpotential. We compute the one-loop partition
function and find it can be expressed as a product of holomorphic Ray-Singer
torsions. We discuss its topological properties and potential gauge and
gravitational anomalies. We show these anomalies can be cancelled using
Green-Schwarz-like counter-terms. We also discuss the dependence on the
background geometry, and in particular the choice of hermitian metric needed
for quantisation. Given suitable topological constraints, this dependence may
again be cancelled by the addition of purely background-dependent
counter-terms. We also explain how our methods provide the one-loop partition
functions of a large class of more general holomorphic field theories in terms
of holomorphic Ray-Singer torsions.
