 Analytic torsion

Let $M$ be a closed Riemannian manifold and $E$ a flat Hermitian vector bundle with unitary holonomy. On $E$-valued differential forms of degree $q$, let ${\Delta }_q$ be the nonnegative Hodge Laplacian. For its positive eigenvalues, defineusing analytic continuation to $s=0$; the prime omits zero eigenvalues. The Ray-Singer analytic torsion isThis packages the form spectra into a regularized determinant, supplying an analytic counterpart of Reidemeister torsion.^[ref]

For acyclic unitary coefficients on a closed manifold, this scalar is independent of the Riemannian metric. With nonzero cohomology, the scalar alone can depend on the metric; the full construction also tracks the norm of cohomology bases. For a circle of length $L$ with trivial coefficients, the positive eigenvalues are $(2\pi k/L{)}^2$ for $k\ne 0$ in degrees zero and one. Their zeta regularization gives $T=L$, illustrating that dependence.^[ref]

Turaev torsion refines the combinatorial side's sign and cellular-lift choices; comparing analytic and combinatorial constructions requires matching those conventions.

Analytic torsion is a concept in mathematical analysis, particularly in the fields of differential geometry and topology, relating to the behavior of certain types of Riemannian manifolds. It arises in the context of studying the spectral properties of differential operators, especially the Laplace operator.