Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

Journal of noncommutative geometry

article

Carlos I. Pérez-Sánchez¹ 

[1] University of Warsaw, Poland, Ruprecht-Karls-Universität Heidelberg

关键词: Noncommutative geometry;    random geometry;    spectral action;    spectral triples;    matrix models;    fuzzy spaces;    chord diagrams;    noncommutative polynomials;    free probability;   

DOI  :  10.4171/jncg/482

学科分类：神经科学

来源: European Mathematical Society

【 摘 要 】

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion incorporates familiar examples like fuzzy spheres and fuzzy tori. In the framework of random noncommutative geometry, we use Barrett’s characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action S(D)=Tr⁡f(D){S(D)= \operatorname{Tr} f(D)}S(D)=Trf(D) for 2n2n2n-dimensional fuzzy geometries. In contrast to the original Chamseddine–Connes spectral action, we take a polynomial fff with f(x)→∞f(x)\to \inftyf(x)→∞ as ∣x∣→∞\vert x\vert \to\infty∣x∣→∞ in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type S(D)=N⋅tr⁡F+∑itr⁡Ai⋅tr⁡Bi{S(D)=N \cdot \operatorname{tr} F+\sum_i \operatorname{tr}A_i \cdot \operatorname{tr} B_i }S(D)=N⋅trF+∑i​trAi​⋅trBi​, being FFF, Ai\smash{A_i}Ai​ and BiB_iBi​ noncommutative polynomials in 22n−1\smash{2^{2n-1}}22n−1 complex N×NN\times NN×N matrices that parametrize the Dirac operator DDD. For arbitrary signature—thus for any admissible KO-dimension—formulas for 222-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 444-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials FFF, Ai\smash{A_i}Ai​ and BiB_iBi​ are obtained via chord diagrams and satisfy: independence of NNN; self-adjointness of the main polynomial FFF (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of Ai\smash{A_i}Ai​ and BiB_iBi​ simultaneously, for fixed iii. Collectively, this favors a free probabilistic perspective for the large-NNN limit we elaborate on.

【 授权许可】

CC BY