Spectral action on noncommutative torus

Essouabri, Driss; Iochum, Bruno; Lévy, Cyril; Sitarz, Andrzej

doi:10.4171/jncg/16

Cited by 26 publications

(43 citation statements)

References 35 publications

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“…Note, that direct calculations of the spectral action without use of the heat kernel are very cumbersome (though possible, cf. [17]). Some examples of spectral actions will be briefly discussed below.…”

Section: Application: Spectral Action Principlementioning

confidence: 99%

Heat Trace Asymptotics on Noncommutative Spaces

Vassilevich¹

2007

SIGMA

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Section: Application: Spectral Action Principlementioning

confidence: 99%

Heat Trace Asymptotics on Noncommutative Spaces

Vassilevich¹

2007

SIGMA

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“…This construction, using the spectral action principle, predicts certain relations between the coupling constants, that can only hold at very high energies of the order of the unification scale. The spectral action principle is the simple statement that the physical action is determined by the spectrum of the Dirac operator D. This has now been tested in many interesting models including Superstring theory [6], noncommutative tori [30], Moyal planes [34], 4D-Moyal space [37], manifolds with boundary [12], in the presence of dilatons [10], for supersymmetric models [5] and torsion cases [38]. The additivity of the action forces it to be of the form Trace f (D/Λ) .…”

Section: Introductionmentioning

confidence: 99%

Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Part I.

Chamseddine

Connes

2010

Fortschritte der Physik

132

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We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of dimension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU (2) × SU (2) × SU (4) to U (1) × SU (2) × SU (3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat kernel expansion are evaluated. We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions: (i) The number of fermions per family is 16. There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent. We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.

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“…Let us recall a few definitions; see [12], [13], [18], [19], [31]: Definition 3.1. A 1-form A is a finite sum of operators like a 1 OED; a 2 where a i 2 A.…”

Section: Tadpoles In Spectral Triplesmentioning

confidence: 99%

“…For examples of spectral action in the real noncommutative setting, see [9], [6], [37] for the case of almost commutative instances which pops up in particle physics, [21] for the Moyal plane (and few points for non-compact manifolds [22]), [23], [19] for the noncommutative torus, and [33] for the quantum group SU q .2/. In the latter case there are tadpoles.…”

Section: Introductionmentioning

confidence: 99%