# Understanding truncated non-commutative geometries through computer simulations

@article{Glaser2019UnderstandingTN, title={Understanding truncated non-commutative geometries through computer simulations}, author={L. Glaser and Abel B. Stern}, journal={arXiv: Mathematical Physics}, year={2019} }

When aiming to apply mathematical results of non-commutative geometry to physical problems the question arises how they translate to a context in which only a part of the spectrum is known. In this article we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To that end, we numerically investigate the restriction that the higher Heisenberg equation places on a truncated Dirac operator. We find a bounded perturbation of the Dirac… Expand

#### 8 Citations

Reconstructing manifolds from truncations of spectral triples

- Mathematics
- 2021

Abstract We explore the geometric implications of introducing a spectral cut-off on compact Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work… Expand

One-loop corrections to the spectral action

- Physics, Mathematics
- 2021

We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. Our result is based on the perturbative… Expand

Reconstructing manifolds from truncated spectral triples

- Mathematics, Physics
- 2019

We explore the geometric implications of introducing a spectral cut-off on Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral… Expand

On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

- Physics, Mathematics
- 2021

We continue the study of fuzzy geometries inside Connes’ spectral formalism and their relation to multimatrix models. In this companion paper to [arXiv: 2007:10914, Ann. Henri Poincaré, 22:… Expand

Gromov–Hausdorff convergence of state spaces for spectral truncations

- Mathematics, Physics
- 2020

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on… Expand

On Multimatrix Models Motivated by Random Noncommutative Geometry I: The Functional Renormalization Group as a Flow in the Free Algebra

- Mathematics, Physics
- 2020

Random noncommutative geometry can be seen as a Euclidean path-integral approach to the quantization of the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of… Expand

Spectral Truncations in Noncommutative Geometry and Operator Systems

- Mathematics, Physics
- 2020

In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and… Expand

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

- Physics, Mathematics
- 2019

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion was introduced in [J. Barrett, J. Math. Phys. 56, 082301 (2015)] and… Expand

#### References

SHOWING 1-10 OF 29 REFERENCES

Monte Carlo simulations of random non-commutative geometries

- Physics
- 2015

Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space. The cases with the simplest types… Expand

Classification of finite spectral triples

- Mathematics, Physics
- 1998

Abstract It is known that the spin structure on Riemannian manifold can be extended to noncommutative geometry using the notion of spectral triple. For finite geometries, the corresponding finite… Expand

A Short survey of noncommutative geometry

- Mathematics, Physics
- 2000

We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the… Expand

Scaling behaviour in random non-commutative geometries

- Physics
- 2016

Random non-commutative geometries are a novel approach to taking a non-perturbative path integral over geometries. They were introduced in arxiv.org/abs/1510.01377, where a first examination was… Expand

Spectral geometry with a cut-off: topological and metric aspects

- Mathematics, Physics
- 2014

Abstract Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative… Expand

Spectral estimators for finite non-commutative geometries

- Mathematics, Physics
- Journal of Physics A: Mathematical and Theoretical
- 2019

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addreses the question of how to extract… Expand

Elements of Noncommutative Geometry

- Mathematics
- 2000

This volume covers a wide range of topics including sources of noncommutative geometry; fundamentals of noncommutative topology; K-theory and Morita equivalance; noncommutative integrodifferential… Expand

Discrete Approaches to Quantum Gravity in Four Dimensions

- Physics, Biology
- Living reviews in relativity
- 1998

Three major areas of research are reviewed: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation; quantum Regge calculus; and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature. Expand

Moduli Spaces of Dirac Operators for Finite Spectral Triples

- Mathematics, Physics
- 2011

The structure theory of finite real spectral triples developed by Krajewski and by Paschke and Sitarz is generalised to allow for arbitrary KO-dimension and the failure of orientability and Poincare… Expand

The spectral geometry of the equatorial Podles sphere

- Mathematics, Physics
- 2004

Abstract We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the… Expand