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4.0

Jun 30, 2018
06/18

by
M. Englis; K. Falk; B. Iochum

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We give examples of spectral triples, in the sense of A. Connes, constructed using the algebra of Toeplitz operators on smoothly bounded strictly pseudoconvex domains in $C^n$, or the star product for the Berezin-Toeplitz quantization. Our main tool is the theory of generalized Toeplitz operators on the boundary of such domains, due to Boutet de Monvel and Guillemin.

Topics: Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1402.3061

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3.0

Jun 28, 2018
06/18

by
B Iochum

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This is a report on a joint work [12] with D. Essouabri, C. Levy and A. Sitarz. The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined as far as the difficulties to go beyond. Some results on holomorphic continuation of series of holomorphic functions are presented.

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1512.07077

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31

Sep 19, 2013
09/13

by
B. Iochum; T. Krajewski; P. Martinetti

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Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in the finite commutative case which corresponds to a metric on a finite set, and also give some examples of computations in both commutative and noncommutative cases.

Source: http://arxiv.org/abs/hep-th/9912217v1

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34

Sep 21, 2013
09/13

by
B. Iochum; C. Levy; D. Vassilevich

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The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, $p\to\infty$, the action decays as $1/p^4$ in any even dimension.

Source: http://arxiv.org/abs/1108.3749v2

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30

Sep 18, 2013
09/13

by
B. Iochum; C. Levy; A. Sitarz

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The spectral action on the equivariant real spectral triple over \A(SU_q(2)) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S^3.

Source: http://arxiv.org/abs/0803.1058v2

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Sep 22, 2013
09/13

by
V. Gayral; J. M. Gracia-Bondía; B. Iochum; T. Schücker; J. C. Varilly

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Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces $\R^{2N}$ endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes--Lott functional action, are given for these noncommutative hyperplanes.

Source: http://arxiv.org/abs/hep-th/0307241v3

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42

Sep 19, 2013
09/13

by
J. M. Gracia-Bondia; B. Iochum; T. Schucker

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The link between chirality in the fermion sector and (anti-)self-duality in the boson sector is reexamined in the light of Connes' noncommutative geometry approach to the Standard Model. We find it to impose that the noncommutative Yang-Mills action be symmetrized in an analogous way to the Dirac-Yukawa operator itself.

Source: http://arxiv.org/abs/hep-th/9709145v1

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Sep 18, 2013
09/13

by
D. Essouabri; B. Iochum; C. Levy; A. Sitarz

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The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.

Source: http://arxiv.org/abs/0704.0564v2

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92

Jul 20, 2013
07/13

by
B. Iochum; T. Masson; Th. Schücker; A. Sitarz

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The aim of the paper is to answer the following question: does $\kappa$-deformation fit into the framework of noncommutative geometry in the sense of spectral triples? Using a compactification of time, we get a discrete version of $\kappa$-Minkowski deformation via $C^*$-algebras of groups. The dynamical system of the underlying groups (including some Baumslag--Solitar groups) is used in order to construct \emph{finitely summable} spectral triples. This allows to bypass an obstruction to...

Source: http://arxiv.org/abs/1107.3449v1

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40

Sep 20, 2013
09/13

by
V. Gayral; B. Iochum; D. V. Vassilevich

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The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing...

Source: http://arxiv.org/abs/hep-th/0607078v1