A generalization of analytic torsion via differential forms on spaces of metrics (Q6630807)

The author makes use of results introduced in his thesis, see [\textit{P. Andreae}, Analytic torsion, the eta invariant, and closed differential forms on spaces of metrics. Ann Arbor, MI: ProQuest LLC (PhD Thesis) (2016)], to explore the concept of multi-torsion, which plays the role of a higher torsion invariant for compact manifolds satisfying specific local properties, giving a bi-grading on differential forms. This, metric-independent new notion, corresponding to a spectral invariant generalization of Ray-Singer analytic torsion, is viewed then as a regularization that leads to an (multi)-asymptotic properties analysis of heat kernel and development of multi-zeta functions. A new proof of Schwarz's theorem, generalizing Ray-Singer's groundbreaking result for de Rham torsion is given in Theorem 3.2, see [\textit{D. B. Ray} and \textit{I. M. Singer}, Adv. Math. 7, 145--210 (1971; Zbl 0239.58014); \textit{A. S. Schwarz}, Commun. Math. Phys. 67, 1--16 (1979; Zbl 0429.58015)]. The fourth, of the six sections, handles multi-torsion. Its main results (Theorems 4.1--4.3) analyses, respectively, multi-admissibility of the corresponding trace, which is proved in the fifth section, and multi-torsion nullity. The vanishing of constant terms in multi-asymptotic expansions is also handled in the fifth section (Proposition 5.1). The paper concludes with a study of the independence of multi-torsion on the local product metric (proof of Theorem 4.3).