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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.

Daniel Ray coauthored the 1971 analytic torsion paper with Isadore Singer. The original prints D. B. Ray; the author-hosted account by Singer's former student John Lott explicitly identifies him as Daniel Ray.^[ref]

Their construction extracts a geometric invariant from spectra of Laplacians on differential forms. A spectral zeta function makes it possible to define a regularized determinant, which contributes to analytic torsion. Their paper establishes metric-independence under its hypotheses and motivates comparison with a combinatorial invariant.^[ref]

Isadore Singer connected analysis, topology and differential geometry. With Michael Atiyah, he proved the Atiyah-Singer index theorem, linking the analytic index of an elliptic operator to topological data.^[ref]

With Daniel Ray, he introduced analytic torsion using spectral information from Laplacians on differential forms. Their construction uses derivatives of spectral zeta functions to regularize otherwise divergent products of eigenvalues. The original paper proved invariance results and proposed equality with combinatorial torsion; that equality was a conjectural step in their paper.^[ref]