Continuity of the spectrum of Dirac operators of spectral triples for the spectral propinquity

Latrémolière, Frédéric

doi:10.1007/s00208-023-02659-x

Cited by 3 publications

(1 citation statement)

References 47 publications

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“…Thus, in some sense, this would recover a theory of Euclidean quantum gravity. The limits of finite approximations of spectral triples is an active area of study [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

Khalkhali,

Pagliaroli

2024

J. High Energ. Phys.

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We compute the first twenty moments of three convergent quartic bi-tracial 2-matrix ensembles in the large N limit. These ensembles are toy models for Euclidean quantum gravity originally proposed by John Barrett and collaborators. A perturbative solution is found for the first twenty moments using the Schwinger-Dyson equations and properties of certain bi-colored unstable maps associated to the model. We then apply a result of Guionnet et al. to show that the perturbative and convergent solution coincide for a small neighbourhood of the coupling constants. For each model we compute an explicit expression for the free energy, critical points, and critical exponents in the large N limit. In particular, the string susceptibility is found to be γ = 1/2, hinting that the associated universality class of the model is the continuous random tree.

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“…Thus, in some sense, this would recover a theory of Euclidean quantum gravity. The limits of finite approximations of spectral triples is an active area of study [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

Khalkhali,

Pagliaroli

2024

J. High Energ. Phys.

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Convergence of inductive sequences of spectral triples for the spectral propinquity

Farsi,

Latrémolière,

Packer

2024

Advances in Mathematics

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Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

Bassi

Conti

Farsi

et al. 2023

Journal of London Math Soc

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In this paper, we study the groups of isometries and the set of bi‐Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

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Continuity of the spectrum of Dirac operators of spectral triples for the spectral propinquity

Cited by 3 publications

References 47 publications

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

Convergence of inductive sequences of spectral triples for the spectral propinquity

Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

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