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Twisted analytic torsion and adiabatic limits

Author: Ryan Mickler
Journal: Proc. Amer. Math. Soc. 143 (2015), 5455-5469
MSC (2010): Primary 58J52; Secondary 58J40
Published electronically: May 22, 2015
MathSciNet review: 3411159
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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 $ \bm \chi '(d_H)$, as defined by Bismut and Zhang.

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Ryan Mickler
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Received by editor(s): November 26, 2013
Received by editor(s) in revised form: November 3, 2014, and November 12, 2014
Published electronically: May 22, 2015
Communicated by: Varghese Mathai
Article copyright: © Copyright 2015 American Mathematical Society

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