Curvature of the Determinant Line Bundle for Noncommutative Tori

Explore the intricacies of differential and conformal geometry in curved noncommutative tori through this 55-minute lecture by Masoud Khalkhali at the Hausdorff Center for Mathematics. Delve into recent advancements, including the computation of spectral invariants like scalar curvature and noncommutative residues. Examine the calculation of the curvature of the determinant line bundle for a family of Dirac operators in noncommutative two tori. Learn about Quillen's original construction for Riemann surfaces and the application of zeta regularized determinants to endow the determinant line bundle with a natural Hermitian metric. Understand the use of Connes' algebra of classical pseudodifferential symbols to compute the curvature form. The lecture covers topics such as perturbed Dolbeault operators, holomorphic determinants, Connes' pseudodifferential calculus, and the Kontsevich-Vishik trace, providing a comprehensive overview of this complex mathematical subject.

Intro
Warm up: zeta regularized determinants
Curved noncommutative tori Ag
Perturbed Dolbeault operator
Scalar curvature for All
What remains to be done
Holomorphic determinants
Cauchy-Riemann operators on An
Quillen's metric on C
Connes' pseudodifferential calculus
Classical symbols
A cutoff integral
The Kontsevich-Vishik trace
Logarithmic symbols
Variations of LogDet and the curvature form
The second variation of log Det
Curvature of the determinant line bundle
A holomorphic determinant a la Quillen