Hypergeometric function and Modular Curvature I. --Hypergeometric functions in Heat Coefficients

Abstract

We give a three-fold upgrade to the rearrangement lemma (pioneer by Connes-Tretkoff-Moscovici) base on the functional analytic ground laid by Lesch. First, we show that the building blocks of the spectral functions appeared in the lemma belong to a class of multivariable hypergeometric functions called Lauricella functions of type D, which includes Gauss hypergeometric functions and Appell's F1 functions as one and two variable subclass respectively. Second, we extend previous results from dimension two to arbitrary dimensions by absorbing the dimension into the parameters of those hypergeometric functions. Third, the highlight of this paper is a reduction formula for the multivariable functions to the Gauss hypergeometric functions using iterated divided differences and differentiations. As for applications, we take perturbed Laplacian operator (the degree zero part that acts on functions) in Connes-Moscovici's paper as an example and extend the computation of the associated modular curvature (functional density of the second heat coefficient) two noncommutative tori of arbitrary dimension. The two spectral functions are written at first as linear combinations of Gauss and Appell's hypergeometric functions. Based on the reduction to Gauss hypergeometric functions, we are able to further derive theirs explicit expressions as functions in the dimension parameter. At last, symbolic verification shows that the two spectral functions fulfill the corresponding Connes-Moscovici type functional relation, even when the dimension parameter takes real-value, which provides a strong confirmation for the validation of our computation.

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,…”

“…Thus, up to unitary equivalence, we recover the conformally deformed Laplacian k ´1∆k ´1 of . This operator was also considered in several subsequent papers (see, e.g., [6,10,11,12,13,14,15,16,23,24,25]).…”

“…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.…”