nLab analytic torsion

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

• John Lott, The Ray-Singer torsion [arXiv:2309.05688]

• Wikipedia, Analytic torsion

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

• 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)

• 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

In relation to the topological string and black hole entropy:

• Cumrun Vafa, Ray-Singer Torsion, Topological Strings and Black Holes [arXiv:2401.12816]

See also:

• Matthias Blau, Mbambu Kakona, George Thompson, On the Evaluation of the Ray-Singer Torsion Path Integral [arXiv:2402.14437]