The Ricci Curvature for Noncommutative Three Tori

Abstract

We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed. Using Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second term of the short time heat kernel expansion for the perturbed Laplacians on functions and on 1-forms. The Ricci curvature is defined by localizing heat traces suitably. Equivalerntly, it can be defined through special values of localized spectral zeta functions. We also compute the scalar curvatures and compare our results with previous calculations in the conformal case. Finally we compute the classical limit of our formulas and show that they coincide with classical formulas in the commutative case.

Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals

“…The paper of Connes-Tretkoff sparked a vast surge of research activity on applying pseudodifferential techniques to the differential geometry study of noncommutative tori. The main directions of research include reformulations of the Gauss-Bonnet and Hirzebruch-Riemann-Roch theorems for noncommutative tori and similar noncommutative manifolds [18,20,21,32,33,48,49], constructions of scalar and Ricci curvatures for conformal deformations of noncommutative tori [16,25,31,34,36,38,54,56,57], and construction and study of noncommutative residue, zeta functions and log-determinants of elliptic operators [17,30,35,37,51,52,75]. There is also a construction of a Ricci flow for noncommutative 2-tori [6].…”

Pseudodifferential calculus on noncommutative tori, II. Main properties

“…This paper is the 2nd part of a two-paper series whose aim is to give a thorough account on the pseudodifferential calculus on noncommutative tori of Connes [5] (see also [2]). Following the seminal paper of Connes-Tretkoff [11], this pseudodifferential calculus has been used in numerous recent papers (see [4,9,10,13,14,18,21,22,23,24,25,26,27,28,29,38,39,42,44,46,48,49,65]). However, a detailed description of this calculus is still missing.…”

Laplace–Beltrami operators on noncommutative tori

“…Such metrics have been considered in various subsequent papers as well (see, e.g., [6,10,11,12,13,14,15,16,23,24,25]). As in [18] we may also consider conformal deformations of more general flat metrics,…”

“…Here I mj is the m j ˆmj -identity matrix. Similar kind of metrics have been considered in [5,10,8,22].…”

“…For instance, noncommutative 2-tori arise from actions on circles by irrational rotations. Following the seminal work of Connes-Tretkoff [7] and Connes-Moscovici [6], a very active current trend in noncommutative geometry is the building of a differential geometric apparatus on noncommutative tori that take into account the non-triviality of their modular automorphism groups (see, e.g., [5,6,7,8,10,11,12,13,14,15,16,23,24,25,29]). So far the main focus has been mostly on conformal deformations of the flat Euclidean metric or products of such metrics.…”