Tate tame symbol and the joint torsion of commuting operators

TL;DR

This paper connects the analytic joint torsion of commuting operators with the Tate tame symbol, providing a local formula and extending classical tame symbols to complex analytic curves, with applications to Toeplitz operators.

Contribution

It introduces a local formula for the analytic joint torsion involving the Tate tame symbol and extends tame symbols to complex analytic curves.

Findings

Derived a local formula for joint torsion using the tame symbol.

Extended the tame symbol concept to complex analytic curves.

Applied results to Toeplitz operators on Hardy spaces.

Abstract

We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc.