An analytic torsion for graded D-branes

I consider the semiclassical approximation of the graded Chern-Simons fieldtheories describing certain systems of topological A type branes in the largeradius limit of Calabi-Yau compactifications. I show that the semiclassicalpartition function can be expressed in terms of a certain (differential)numerical invariant which is a version of the analytic torsion of Ray andSinger, but associated with flat graded superbundles. I also discuss a`twisted' version of the Ray-Singer norm, and show its independence of metricdata. As illustration, I consider graded D-brane pairs of unit relative gradewith a scalar condensate in the boundary condition changing sector. For theparticularly simple case when the reference flat connections are trivial, Ishow that the generalized torsion reduces to a power of the classicalRay-Singer invariant of the base 3-manifold.Comment: 28 pages, no figures; v2: added a footnote and one reference, corrected a typ