nLab
analytic torsion

Contents

Idea

What is called analytic torsion or Ray-Singer torsion (Ray-Singer 73) is the invariant T(X,g)T(X,g) of a Riemannian manifold (X,g)(X,g) given by a product of powers of the functional determinants det regΔ| Ω pdet_{reg} \Delta|_{\Omega^p} of the Laplace operators Δ| Ω p\Delta|_{\Omega^p} on the manifold acting on the space of differential p-forms:

T(X,g)p(det regΔ| Ω p) (1) pp2. 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).

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)

Revised on September 2, 2014 19:34:25 by Urs Schreiber (82.136.246.44)