# nLab analytic torsion

### Context

#### Riemannian geometry

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)