The Ricci curvature for noncommutative three tori

Dong, Rui; Ghorbanpour, Asghar; Khalkhali, Masoud

doi:10.1016/j.geomphys.2020.103717

Journal of Geometry and Physics

2020

DOI: 10.1016/j.geomphys.2020.103717

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The Ricci curvature for noncommutative three tori

Rui Dong

Asghar Ghorbanpour

Masoud Khalkhali

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“…For the computational aspect, the analytical technical tools, such as Connes's pseudo-differential calculus and the rearrangement lemmas go back to [Con80,CT11], see also [FK12]. Various computation of this type on (low dimensional) noncommutative tori has been carried out [KMS16,FGK17,DGK18,FGK16]. An non-conformal perturbation of the flat metric was studied in [DS15].…”

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confidence: 99%

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

Liu¹

2018

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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.

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confidence: 99%

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

Liu¹

2018

Preprint

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The Ricci curvature for noncommutative three tori

Cited by 1 publication

References 13 publications

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

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

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