nLab
analytic torsion
Context
Riemannian geometry
Contents
Idea
What is called analytic torsion or Ray-Singer torsion (Ray-Singer 73 ) is the invariant $T(X,g)$ of a Riemannian manifold $(X,g)$ given by a product of powers of the functional determinants $det_{reg} \Delta|_{\Omega^p}$ of the Laplace operators $\Delta|_{\Omega^p}$ of the manifold acting on the space of differential p-forms :

$T(X,g)
\coloneqq
\underset{p}{\prod}
\left(det_{reg} \Delta|_{\Omega^p}\right)^{-(-1)^p \frac{p}{2}}
\,.$

This analytic torsion is an analogue in analysis of the invariant of topological manifolds called Reidemeister torsion . The two agree for the compact Riemannian manifolds (Cheeger 77 ).

Properties
Relation to Iwasawa theory
According to (Morishita 09 ) the relation between Reidemeister torsion and analytic torsion is analogous to that between Iwasawa polynomials and zeta functions obtained by adelic integration . (…)

References
D. Ray, Isadore Singer , Analytic torsion for complex manifolds , Ann. Math. 98 , 1 (1973), 154–177.

Jeff Cheeger , Analytic torsion and Reidemeister torsion , Proc. Natl. Acad. Sci. USA 74 , No. 7, pp. 2651-2654 (1977), pdf

Wikipedia, Analytic torsion

A.A. Bytsenko, A.E. Goncalves, W. da Cruz, Analytic Torsion on Hyperbolic Manifolds and the Semiclassical Approximation for Chern-Simons Theory (arXiv:hep-th/9805187 )

Review of the role played in the perturbative quantization of Chern-Simons theory includes

M. B. Young, section 2 of Chern-Simons theory, knots and moduli spaces of connections (pdf )

Revised on October 1, 2014 09:10:22
by

Urs Schreiber
(185.26.182.27)