Moduli spaces of discrete gravity

Holfter, Alexander; Paschke, Mark W.

doi:10.1016/s0393-0440(02)00176-6

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…Allowing the algebra to be non-commutative is important because it allows a new type of finite-dimensional approximation to a manifold. Staying within the realm of commutative algebras would lead to the algebra of functions on a finite set of points, which is a lattice approximation to a manifold; simple examples of such random commutative spectral triples are studied in [12,18]. The fuzzy spaces are not lattice approximations and so the study of these is complementary to the study of random lattices.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo simulations of random non-commutative geometries

Barrett¹,

Glaser²

2016

J. Phys. A: Math. Theor.

139

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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 of Clifford algebra are investigated using Monte Carlo simulations to compute the integrals. Various qualitatively different types of behaviour of these random Dirac operators are exhibited. Some features are explained in terms of the theory of random matrices but other phenomena remain mysterious. Some of the models with a quartic action of symmetry-breaking type display a phase transition. Close to the phase transition the spectrum of a typical Dirac operator shows manifold-like behaviour for the eigenvalues below a cut-off scale.

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Section: Introductionmentioning

confidence: 99%

Monte Carlo simulations of random non-commutative geometries

Barrett¹,

Glaser²

2016

J. Phys. A: Math. Theor.

139

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“…In order to get a finite-rank Dirac operator one can, on the one hand, truncate the algebra C 1 .M / and the Hilbert space H M in order to get a well-defined measure dD on the space of geometries M, now parametrized by finite, albeit large, matrices. On the other hand, one does not want to fall in the class of lattice geometries [46] nor finite geometries [51]. Fuzzy geometries are finite-dimensional geometries that escape the classification of finite geometries given in [51], depicted in terms of the Krajewski diagrams, and [61].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in contradistinction to lattices, fuzzy geometries are genuinely-and not only in spirit-noncommutative. In particular, the path-integral quantization of fuzzy geometries differs also from the approach in [46] for lattice geometries.…”

Section: Introductionmentioning

confidence: 99%

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

Perez-Sanchez

2022

J. Noncommut. Geom.

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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/ D Tr f .D/ for 2n-dimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial f with f .x/ ! 1 as jxj ! 1 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/ D N tr F C P i tr A i tr B i , being F , A i and B i noncommutative polynomials in 2 2n 1 complex N N matrices that parametrize the Dirac operator D. For arbitrary signature-thus for any admissible KO-dimension-formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F , A i and B i are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of A i and B i simultaneously, for fixed i. Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on.

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“…In order to get a finite-rank Dirac operator one can, on the one hand, truncate the algebra C ∞ (M ) and the Hilbert space H M in order to get a well-defined measure dD on the space of geometries M, now parametrized by finite, albeit large, matrices. On the other hand, one does not want to fall in the class of lattice geometries [HP03] nor finite geometries [Kra98]. Fuzzy geometries are finite-dimensional geometries that escape the classification of finite geometries given in [Kra98], depicted in terms of the Krajewski diagrams, and [PS98].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in contradistinction to lattices, fuzzy geometries are genuinely-and not only in spiritnoncommutative. In particular, the path-integral quantization of fuzzy geometries differs also from the approach in [HP03] for lattice geometries.…”

Section: Introductionmentioning

confidence: 99%

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

Perez-Sanchez¹

2019

Preprint

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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 accommodates familiar fuzzy spaces like spheres and 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) = Trf (D) for 2ndimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial f with f (x) → ∞ as |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 + i tr Ai • tr Bi, being F, Ai and Bi noncommutative polynomials in 2 2n−1 complex N × N matrices that parametrize the Dirac operator D. For arbitrary signature-thus for any admissible KO-dimension-formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F, Ai and Bi are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of Ai and Bi simultaneously, for fixed i. Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on.

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Moduli spaces of discrete gravity

Cited by 4 publications

References 16 publications

Monte Carlo simulations of random non-commutative geometries

Monte Carlo simulations of random non-commutative geometries

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

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

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