The Curvature of the Determinant Line Bundle on the Noncommutative Two Torus

TL;DR

This paper calculates the curvature of the determinant line bundle associated with Dirac operators on a noncommutative two torus, extending classical geometric concepts to noncommutative geometry using zeta regularization and pseudodifferential calculus.

Contribution

It provides the first explicit computation of the curvature form of the determinant line bundle in the setting of noncommutative geometry, adapting Quillen's construction to the noncommutative two torus.

Findings

Curvature form expressed via second variation of the zeta-regularized determinant.

Extension of classical determinant line bundle concepts to noncommutative spaces.

Application of Kontsevich-Vishik trace in noncommutative pseudodifferential calculus.

Abstract

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation $\delta_{w}\delta_{\bar{w}}\log\det(\Delta)$.