nLab
analytic torsion
Contents
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 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 . (…)

Relation to Selberg and Ruelle zeta function
The special value? of a Ruelle zeta function at $s= 0$ is expressed by Reidemeister torsion

(Fried 86 )

Relation to Chern-Simons theory
Analytic torsion appears as one factor in the perturbative path integral quantization of Chern-Simons field theory . See there at Quantization – Perturbative – Path integral quantization .

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 )
Discussion for hyperbolic manifolds in terms of the Selberg zeta function /Ruelle zeta function is due to

David Fried , Analytic torsion and closed geodesics on hyperbolic manifolds , Invent. math. 84, 523-540 (1986) (pdf )
with further developments including

Ulrich Bunke , Martin Olbrich , Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one (arXiv:dg-ga/9407012 )

Jinsung Park, Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps , Journal of Functional Analysis Volume 257, Issue 6, 15 September 2009, Pages 1713–1758

Last revised on June 25, 2015 at 21:16:45.
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