This work examines the deformed fuzzy sphere, as an example of a fuzzy space that can be described through a spectral triple, using computer visualizations. We first explore this geometry using an analytic expression for the eigenvalues to examine the spectral dimension and volume of the geometry. In the second part of the paper we extend the code from Glaser and Stern [J. Geom. Phys. 159, 103921 (2021)], in which the truncated sphere was visualized through localized states. This generalization allows us to examine finite spectral triples. In particular, we apply this code to the deformed fuzzy sphere as a first step in the more ambitious program of using it to examine arbitrary finite spectral triples, like those generated from random fuzzy spaces, as show in Barrett and Glaser [J. Phys. A: Math. Theor. 49, 245001 (2016)].

1.
Glaser
,
L.
and
Stern
,
A. B.
, “
Reconstructing manifolds from truncations of spectral triples
,”
J. Geom. Phys.
159
,
103921
(
2021
).
2.
Barrett
,
J. W.
and
Glaser
,
L.
, “
Monte Carlo simulations of random non-commutative geometries
,”
J. Phys. A: Math. Theor.
49
,
245001
(
2016
).
3.
Connes
,
A.
,
Noncommutative Geometry
(
Academic Press
,
San Diego
,
1994
), ISBN: 978-0-12-185860-5.
4.
Barrett
,
J. W.
, “
Matrix geometries and fuzzy spaces as finite spectral triples
,”
J. Math. Phys.
56
(
8
),
082301
(
2015
).
5.
Glaser
,
L.
, “
Scaling behaviour in random non-commutative geometries
,”
J. Phys. A: Math. Theor.
50
(
27
),
275201
(
2017
).
6.
Barrett
,
J. W.
,
Druce
,
P.
, and
Glaser
,
L.
, “
Spectral estimators for finite non-commutative geometries
,”
J. Phys. A: Math. Theor.
52
(
27
),
275203
(
2019
).
7.
D’Arcangelo
,
M.
, “
Numerical simulation of random Dirac operators
,” Ph.D. thesis,
University of Nottingham
,
Nottingham
,
2022
.
8.
Azarfar
,
S.
and
Khalkhali
,
M.
, “
Random finite noncommutative geometries and topological recursion
,” arXiv:1906.09362 [hep-th, physics:math-ph] (
2019
).
9.
Khalkhali
,
M.
and
Pagliaroli
,
N.
, “
Phase transition in random noncommutative geometries
,”
J. Phys. A: Math. Theor.
54
,
035202
(
2020
).
10.
Hessam
,
H.
,
Khalkhali
,
M.
, and
Pagliaroli
,
N.
, “
Bootstrapping Dirac ensembles
,”
J. Phys. A: Math. Theor.
55
,
335204
(
2022
).
11.
Pérez-Sánchez
,
C. I.
, “
On multimatrix models motivated by random noncommutative geometry I: The functional renormalization group as a flow in the free algebra
,”
Ann. Henri Poincare
22
(
9
),
3095
3148
(
2021
).
12.
Khalkhali
,
M.
and
Pagliaroli
,
N.
, “
Spectral statistics of Dirac ensembles
,”
J. Math. Phys.
63
,
053504
(
2022
).
13.
Hessam
,
H.
,
Khalkhali
,
M.
, and
Pagliaroli
,
N.
, “
Double scaling limits of Dirac ensembles and Liouville quantum gravity
,”
J. Phys. A: Math. Theor.
56
(
22
),
225201
(
2023
).
14.
Verhoeven
,
L.
, “
Geometry in spectral triples: Immersions and fermionic fuzzy geometries
,” Electronic Thesis and Dissertation Repository,
Radboud University
,
2023
, https://ir.lib.uwo.ca/etd/9561.
15.
Ambjørn
,
J.
,
Jurkiewicz
,
J.
, and
Loll
,
R.
, “
The spectral dimension of the universe is scale dependent
,”
Phys. Rev. Lett.
95
(
17
),
171301
(
2005
).
16.
Stern
,
A. B.
, “
Finite-rank approximations of spectral zeta residues
,”
Lett. Math. Phys.
109
,
565
(
2018
).
17.
Glaser
,
L.
(
2023
). “
Deformed fuzzy sphere visualisation
,” GitHub. https://github.com/LisaGlaser/deformed_sphere_visualisation
18.
Chamseddine
,
A. H.
,
Connes
,
A.
, and
Mukhanov
,
V.
, “
Geometry and the quantum: Basics
,”
J. High Energy Phys.
2014
(
12
),
98
; arXiv:1411.0977.
19.
Schneiderbauer
,
L.
and
Steinacker
,
H. C.
, “
Measuring finite quantum geometries via quasi-coherent states
,”
J. Phys. A: Math. Theor.
49
(
28
),
285301
(
2016
); arXiv:1601.08007.
20.
Virtanen
,
P.
et al, “
SciPy 1.0: Fundamental algorithms for scientific computing in Python
,”
Nat. Methods
17
,
261
272
(
2020
).
21.
Grosse
,
H.
and
Prešnajder
,
P.
, “
The Dirac operator on the fuzzy sphere
,”
Lett. Math. Phys.
33
(
2
),
171
181
(
1995
).
22.
Barrett
,
J. W.
and
Gaunt
,
J.
, “
Finite spectral triples for the fuzzy torus
,” arXiv:1908.06796 [math.QA] (
2019
).
23.
Glaser
,
L.
(
2023
). “
Deformed fuzzy spheres
,” Zenodo, version v2 https://doi.org/10.5281/zenodo.7864066
You do not currently have access to this content.