Twisted analytic torsion and adiabatic limits

Mickler, Ryan

doi:10.1090/proc/12673

Twisted analytic torsion and adiabatic limits
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Proc. Amer. Math. Soc. 143 (2015), 5455-5469 Request permission

Abstract:

We study an analogue of the analytic torsion for elliptic complexes that are graded by $\mathbb {Z}_2$, orignally constructed by Mathai and Wu. A particular example of a $\mathbb {Z}_2$-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 $d_H = d+H$, where $H$ 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. $H=0$) 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 $\mathbf{\chi}’(d_H)$, as defined by Bismut and Zhang.

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