CaltechAUTHORS: Article
https://feeds.library.caltech.edu/people/Fathizadeh-F/article.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 21 May 2024 19:25:07 -0700Scalar curvature for noncommutative four-tori
https://resolver.caltech.edu/CaltechAUTHORS:20150903-144355330
Year: 2015
DOI: 10.4171/JNCG/198
In this paper we study the curved geometry of noncommutative 4-tori T^4_θ. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of T^4_θ. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of T^4_θ. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action.https://resolver.caltech.edu/CaltechAUTHORS:20150903-144355330On the scalar curvature for the noncommutative four torus
https://resolver.caltech.edu/CaltechAUTHORS:20150720-094414972
Year: 2015
DOI: 10.1063/1.4922815
The scalar curvature for noncommutative four tori T^4_Θ, where their flat geometries are conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p∈C∞(T^4_Θ) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by Connes and Moscovici, J. Am. Math. Soc. 27(3), 639-684 (2014), unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated.https://resolver.caltech.edu/CaltechAUTHORS:20150720-094414972Spectral Action for Bianchi Type-IX Cosmological Models
https://resolver.caltech.edu/CaltechAUTHORS:20151005-140911430
Year: 2015
DOI: 10.1007/JHEP10(2015)085
A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors w_1(t), w_2(t), w)3(t), and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients a_0 ,a_2 ,a_4 of the spectral action by using heat kernel methods and parametric pseudodifferential calculus. An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki's noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article. The advantages of the new method are discussed, which combined with symmetries of the Bianchi type-IX metric, yield an elegant proof of the rationality result.https://resolver.caltech.edu/CaltechAUTHORS:20151005-140911430On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C^*-Dynamical Systems
https://resolver.caltech.edu/CaltechAUTHORS:20160324-085850986
Year: 2016
DOI: 10.3842/SIGMA.2016.016
The analog of the Chern-Gauss-Bonnet theorem is studied for a C^∗-dynamical system consisting of a C^∗-algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A ⊂ A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.https://resolver.caltech.edu/CaltechAUTHORS:20160324-085850986Periods and motives in the spectral action of Robertson-Walker spacetimes
https://resolver.caltech.edu/CaltechAUTHORS:20170712-091141446
Year: 2017
DOI: 10.1007/s00220-017-2991-x
We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson–Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes.https://resolver.caltech.edu/CaltechAUTHORS:20170712-091141446Motives and periods in Bianchi IX gravity models
https://resolver.caltech.edu/CaltechAUTHORS:20181101-083114372
Year: 2018
DOI: 10.1007/s11005-018-1096-6
We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat-kernel expansion of the Dirac–Laplacian on SU(2) Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.https://resolver.caltech.edu/CaltechAUTHORS:20181101-083114372Modular forms in the spectral action of Bianchi IX gravitational instantons
https://resolver.caltech.edu/CaltechAUTHORS:20170712-105237127
Year: 2019
DOI: 10.1007/JHEP01(2019)234
We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.https://resolver.caltech.edu/CaltechAUTHORS:20170712-105237127