---
tag: MZT9D34B0A
kind: concept
name: Analytic torsion
msc: [58J52, 57Q10]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_torsion
same_as:
  wikidata: Q4751142
  wikipedia: Analytic torsion
---

## Summary

Analytic torsion, or Ray–Singer torsion, is a spectral invariant of a closed Riemannian manifold with an orthogonal or unitary representation of its fundamental group.
It is an alternating product of [[zeta-regularized determinants|YT3ZQ2F7FG]] of the Laplacians on forms with values in the associated flat bundle.
Ray and Singer introduced it as an analytic counterpart of Reidemeister torsion, a combinatorial invariant that can distinguish homotopy equivalent lens spaces.
The [[Cheeger–Müller theorem|C9DT4Y63CY]] says that the two torsions agree.

## Explanations

### L1 human

Homology counts holes, and when all homology with twisted coefficients vanishes, a finer invariant called torsion still remains.
Reidemeister torsion computes it from a triangulation.
Analytic torsion computes it instead from the vibration frequencies of the manifold, the eigenvalues of Laplacians on forms.
On a circle with a twist by the unit complex number $e^{2\pi i a}$, both give $2\sin(\pi a)$.