Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions

广义 theta 函数、射影平坦向量丛与非交换环面

Generalized theta functions, projectively flat vector bundles and noncommutative tori

本文利用非交换环面表示描述二维环面上的 Hermitian-Einstein 向量丛，重建 Matsushima 理论，并将 theta 函数推广为向量值形式以显现 FMN 对偶。

Summation formulae for noncommutative hypergeometric series

We establish several summation formulae for hypergeometric and basic hypergeometric series involving noncommutative parameters and argument. These results were inspired by a recent paper of J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138-8141].

Noncommutative Hypergeometry

A certain special function of the generalized hypergeometric variety is shown to fulfill a host of useful noncommutative identities.

Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work

As the second part of the sequel, we investigate the variation of rearrangement operators (more precisely, the spectral functions behind) arising in the study of modular geometry on noncommutative (two) tori. We initiate a systematic approach by introducing transformations corresponding to basic operations in calculus, like differentiation and integration by parts. As for applications, we extend, in a uniform way, the Connes-Moscovici's functional relations on noncommutative two tori attached to the variation of second heat coefficients to noncommutative tori of arbitrary dimension. Moreover, …

Hypergeometric function and modular curvature

We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation leads to a unified (works for arbitrary dimension) method of computing the modular curvature on toric noncommutative manifolds. We show that the spectral functions that define the quantum part of the curvature have closed forms in terms of hypergeometric functions. As a consequence, we are able to obtained explicit expressions (as functions in the dimension p…

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

We give a new proof of the rearrangement lemma that works for all dimensions and all heat coefficients in the study of modular geometry on noncommutative tori. The building blocks of the spectral functions are landed in a hypergeometric family knowns as Lauricella functions of type $D$. We investigate the differential and recursive relations among the family and obtain a full reduction to Gauss hypergeometric functions. As for applications, we give phase one demonstration on how the hypergeometric features lead to new simplifications in computations involved in the modular geometry.