Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Masoud Khalkhali

Abstract

We extend the canonical trace of Kontsevich and Vishik to the algebra of non-integer order classical pseudodifferntial operators on noncommutative tori. We consider the spin spectral triple on noncommutative tori and prove the regularity of eta function at zero for the family of operators $e^{th/2}De^{th/2}$ and the couple Dirac operator $D+A$ on noncommutative $3$-torus. Next, we consider the conformal variations of $\eta_{D}(0)$ and we show that the spectral value $\eta_D(0)$ is a conformal invariant of noncommutative $3$-torus. Next, we study the conformal variation of $\zeta'_{|D|}(0)$ and show that this quantity is also a conformal invariant of odd noncommutative tori. This the analogue of the vanishing of the conformal anomaly of $\LogDet$ in odd dimensions in commutative case. We also consider $\eta_{D+A}(0)$ for the coupled Dirac operator $D+A$ on noncommutative 3-torus and compute a local formula for the variation of $\eta_{D+A}(0)$ with respect to the vector potential $A$. In the second part, we consider a family of elliptic first order differential operators $\bar{\partial}_A$ on noncommutative two torus which are the noncommutative analogues of Cauchy-Riemann operators on a closed Riemann surface. We consider the Quillen determinant line bundle associated to this family and by using the machinery of the canonical trace, we compute the second variation of the $\zeta_{\Delta}'(0)$ where $\Delta=\bar{\partial}_A^2$ is the Dolbeault Laplacian. This gives the analogue of the Quillen's computations for the curvature form of the determinant line bundle in commutative case.

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