What is called analytic torsion or Ray-Singer torsion (Ray-Singer 73) is the invariant of a Riemannian manifold given by a product of powers of the functional determinants of the Laplace operators of the manifold acting on the space of differential p-forms:
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).
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 is expressed by Reidemeister torsion
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.
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
Revised on June 25, 2015 17:16:45
by Urs Schreiber