= Analytic torsion
{wiki}
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>, define
$$
\zeta_q(s)=\sum_{\lambda>0}\lambda^{-s},\qquad
\det{}'\Delta_q=\exp[-\zeta_q'(0)],
$$
using <analytic continuation> to $s=0$; the prime omits zero <eigenvalues>. The Ray-Singer analytic torsion is
$$
T(M,E)=\exp\!\left(\frac12\sum_q(-1)^q q\,\zeta_q'(0)\right)
=\prod_q(\det{}'\Delta_q)^{(-1)^{q+1}q/2}.
$$
This packages the form spectra into a regularized <determinant>, supplying an analytic counterpart of <Reidemeister torsion>.<R-torsion and the Laplacian on Riemannian manifolds by Daniel Ray and Isadore Singer>[Definition 1.6, page 149]{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 $k\ne0$ in degrees zero and one. Their zeta regularization gives $T=L$, illustrating that dependence.<R-torsion and the Laplacian on Riemannian manifolds by Daniel Ray and Isadore Singer>[Section 2 for acyclic invariance; Section 3 for the nonacyclic construction]{ref}
<Turaev torsion> refines the combinatorial side's sign and cellular-lift choices; comparing analytic and combinatorial constructions requires matching those conventions.
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