Analytic Torsion for Complex Manifolds

Genus one polyhedral surfaces, spaces of quadratic differentials on tori and determinants of Laplacians

“…When the flat metric m is everywhere nonsingular formula (3.2) reduces to the well-known Ray-Singer result [18].…”

“…The similar question for smooth conformal metrics and arbitrary holomorphic bundles was very popular in the eighties and early nineties being motivated by the string theory. Among the most notable results one can mention the Ray-Singer calculation of the determinant of the Laplacian in arbitrary flat line bundle over flat tori [18], an explicit formula for the determinant of Laplacian in the Arakelov metric found by Dugan and Sonoda [7], the D'Hoker-Phong formula relating the determinant of the Laplacian in the Poincaré metric to Selberg's zeta-function [6], the Zograf-Takhtajan formula for variation of the determinant of Laplacian in the Poincaré metric with respect to moduli of the Riemann surface [22], Fay's formula for variation of the determinant of Laplacian under arbitrary (not necessarily conformal) variation of the metric [9].…”

The Mathematical Work of V. K. Patodi

“…In another direction, Ray and Singer also defined analytic torsion for the Dolbeault complex on complex manifolds in [106], which we now call the Ray-Singer holomorphic torsion. While this holomorphic torsion is no longer a topological invariant, it has played important roles in complex geometry, arithmetic geometry and mathematical physics.…”

The geometry of determinant line bundles in noncommutative geometry

“…Recall that the noncommutative torus A  is the universal C -algebra generated by two unitaries U , V satisfying the Weyl commutation relations U V D e i V U for fixed  2 R. There is a natural smooth subalgebra A is holomorphic for Re.s/ 0 and is precisely the -function considered in the proof of Theorem 4.1 of [13], where it arose in a completely different context. Using the results from there, it follows that .s/ has a meromorphic continuation to C, with no pole at s D 0, showing in particular that the spectral triple that we started out with on the noncommutative torus is a regular spectral triple such that zero is not in the dimension spectrum.…”