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Refined Analytic Torsion

Maxim Braverman·Thomas Kappeler
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Abstract

For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. The refined analytic torsion is a holomorphic function of the representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. In particular, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the eta-invariant.

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Maxim Braverman, Thomas Kappeler. 2005-12-20. Refined Analytic Torsion. https://arxiv.org/abs/math/0505537

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