A spectral invariant of a closed Riemannian manifold (possibly with a flat vector bundle) defined as a weighted product of zeta-regularized determinants of the Hodge Laplacians on differential forms; it refines topological Reidemeister torsion by analytic means and is sensitive to the spectrum of Laplacians across degrees.

Principle

Construct zeta functions ζ_k(s) for the Laplacians on k-forms, form the regularized determinants det' Δ_k = exp(-ζ_k'(0)), and combine them with alternating degree weights to produce the analytic torsion T = ∏_k (det' Δ_k)^{(-1)^{k} k/2}; under favorable conditions this analytic torsion equals Reidemeister torsion.

Demonstration

On a closed odd-dimensional manifold with an acyclic flat bundle, compute ζ_k(s) from the nonzero eigenvalues of Δ_k, regularize to get det' Δ_k, and assemble T; for lens spaces and some spherical space forms this produces explicit values matching combinatorial torsion up to normalization.

Misapplication

Applying analytic torsion without removing zero eigenvalues or ignoring that the bundle representation is nonunitary leads to ill-defined phases or divergent factors; treating the invariant as purely metric-independent also misstates its subtle dependence and required hypotheses for equality with Reidemeister torsion.

Consequence

Analytic torsion provides a bridge between spectral geometry and combinatorial topology: when the Cheeger–Müller theorem applies, analytic torsion equals Reidemeister torsion, yielding analytic proofs of topological invariants and contributing to refined invariants in quantum field theory and Arakelov geometry.

Reversal

The reversal is the combinatorial viewpoint: reconstructing spectral determinants from a given Reidemeister torsion requires extra analytic input; conceptually, replacing analytic spectral data by purely combinatorial chain-level data loses the immediate access to spectral geometry.

Boundary

Defined for closed manifolds or for manifolds with boundary when appropriate boundary conditions are imposed; requires control over zero modes (acyclicity simplifies definitions) and is subtle for nonunitary flat bundles or in presence of continuous spectrum.

Semantic Tension

Sits between spectral zeta-regularized determinants and combinatorial (Reidemeister) torsion: the tension is whether one treats the torsion as an analytic spectral object or as a combinatorial topological invariant, and what extra hypotheses ensure their equality.

Synthesis

Analytic torsion packages regularized spectral determinants of Laplacians across degrees into a single invariant that captures refined topological information; under the right hypotheses it coincides with Reidemeister torsion, thus linking analysis, topology, and geometry.