Twisted analytic torsion and adiabatic limits

We study an analogue of the analytic torsion for elliptic complexes that are graded by Z2\mathbb {Z}_2, orignally constructed by Mathai and Wu. A particular example of a Z2\mathbb {Z}_2-graded complex was given by Rohm and Witten in 1986 when they studied the complex of forms on an odd-dimensional manifold equipped with a twisted differential dH=d+Hd_H = d+H, where HH is a closed odd-dimensional form. We show that the Ray-Singer metric on the determinant line of this twisted operator is equal to the untwisted (i.e. H=0H=0) Ray-Singer metric when the determinant lines are identified using a canonical isomorphism. We also study another analytical invariant of the twisted differential, the derived Euler characteristic χ′(dH)\mathbf {\chi }’(d_H), as defined by Bismut and Zhang.