The Spectral Action for Dirac Operators with Skew-Symmetric Torsion

Hanisch, Florian; Pfäffle, Frank; Stephan, Christoph A.

doi:10.1007/s00220-010-1135-3

Cited by 23 publications

(30 citation statements)

References 21 publications

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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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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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“…The large Λ expansion of S l is given by the heat trace asymptotics those structure differs considerably for the standard case of spin Dirac operators with torsion, cf. [10,11]. Note that in four dimensions the spectral action for the spin Dirac operator is restricted by the chiral symmetry [11], that is not present in the symplectic case.…”

Section: The Distancementioning

confidence: 99%

Spectral geometry of symplectic spinors

Vassilevich

2015

Journal of Mathematical Physics

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“…If the twist bundle has a chiral asymmetry terms known from Loop Quantum Gravity ([Ro04], [Th07]) arise. The situation with purely anti-symmetric torsion has been examined before (in [HPS10], [ILV10], [PS12]), and there also are results on Dirac operators with scalar perturbations (in [SZ11]).…”

Section: Connes' Spectral Action Principle and Chiral Projectionsmentioning

confidence: 99%

“…Remark 3.2 In the case of totally anti-symmetric torsion, i.e. V ≡ 0, formulas for these Seeley-deWitt coefficients have been given in [Go80], [Ob83], [Gr86], [HPS10], [ILV10], [PS12].…”

Section: The Bosonic Spectral Actionmentioning

confidence: 99%

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Chiral Asymmetry and the Spectral Action

Pfäffle

Stephan

2012

Commun. Math. Phys.

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We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.

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The Spectral Action for Dirac Operators with Skew-Symmetric Torsion

Cited by 23 publications

References 21 publications

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

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

Spectral geometry of symplectic spinors

Chiral Asymmetry and the Spectral Action

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