A Reconstruction Theorem for Almost-Commutative Spectral Triples

Ćaćić, Branimir

doi:10.1007/s11005-011-0534-5

Cited by 26 publications

(36 citation statements)

References 25 publications

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“…By following the steps in the previous section it is not difficult to see that the gauge theory (P B , ω B ) obtained from this spectral triple (Γ ∞ (M, B), L 2 (M, B ⊗ S), ic(∇ B ⊗ 1 + 1 ⊗ ∇ S ), J, γ ) is isomorphic to (P, ω). This is in accordance with the approach to almost commutative manifolds taken in [18]. …”

Section: From the Spectral Triple To Principal Bundlessupporting

confidence: 89%

The noncommutative geometry of Yang–Mills fields

Boeijink

Suijlekom

2011

Journal of Geometry and Physics

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a b s t r a c tWe generalize to topologically non-trivial gauge configurations the description of the Einstein-Yang-Mills system in terms of a noncommutative manifold, as was done previously by Chamseddine and Connes. Starting with an algebra bundle and a connection thereon, we obtain a spectral triple, a construction that can be related to the internal Kasparov product in unbounded KK-theory. In the case that the algebra bundle has typical fiber M N (C), we construct a PSU(N)-principal bundle for which it is an associated bundle.The so-called internal fluctuations of the spectral triple are parametrized by connections on this principal bundle and the spectral action gives the Yang-Mills action for these gauge fields, minimally coupled to gravity. Finally, we formulate a definition for a topological spectral action.

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Section: From the Spectral Triple To Principal Bundlessupporting

confidence: 89%

The noncommutative geometry of Yang–Mills fields

Boeijink

Suijlekom

2011

Journal of Geometry and Physics

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“…This observation formed the basis of our earlier work on almost-commutative spectral triples [3], where we obtained a reconstruction theorem for a more general, manifestly global-analytic notion of almost-commutative spectral triple based on general Dirac-type operators; in particular, we were able to refine Connes's reconstruction theorem into a precise noncommutative-geometric characterisation of Dirac-type operators on compact oriented Riemannian manifolds.…”

mentioning

confidence: 85%

“…We have already argued in [3] for a more general, indeed, manifestly global analytic notion of almost-commutative spectral triple, which does not require the base manifold to be spin, accommodates "non-trivial fibrations" already present in the literature, and is stable under inner fluctuation of the metric. To write down this definition succinctly, it will be convenient to give the following definitions: Definition 1.13.…”

Section: Preliminariesmentioning

confidence: 95%

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Real Structures on Almost-Commutative Spectral Triples

Ćaćić

2013

Lett Math Phys

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Abstract. We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying in the process both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almostcommutative spectral triple algebraically encodes the commutative * -algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KOdimension. Moreover, we define a notion of smooth family of real finite spectral triples and of the twisting of a concrete real commutative spectral triple by such a family, with interesting KK-theoretic and gauge-theoretic implications.The famed Gel'fand-Naȋmark duality allows for a contravariant functorial identification of the theory of C * -algebras as a theory of noncommutative topological spaces. Analogously, Connes's reconstruction theorem for commutative spectral triples [6] suggests a partial identification, at least at the level of objects, of the theory of spectral triples as a theory of noncommutative manifolds. However, since there is no canonical choice of commutative spectral triple for a compact oriented manifold, it has become traditional in the noncommutative-geometric literature to restrict attention to the case of compact spin manifolds, which admit a canonical Dirac-type operator, the Dirac operator, and hence a canonical commutative spectral triple. Indeed, the influence of this example has been profound on the development of the theory of spectral triples, both implicitly through the traditional forms of the definitions of commutative and almost-commutative spectral triples, and explicitly through the focus on real spectral triples; the ubiquity of real spectral triples in the literature, together with the explicit use of real spectral triples in applications to theoretical high energy physics, would seem to justify this restriction.There is, however, another way to approach this issue: every compact oriented manifold admits a Riemannian metric, and a compact oriented Riemannian manifold X admits a canonical spectral triple, namely, the. This, then, suggests that the theory of spectral triples might be fruitfully viewed as a theory of noncommutative Riemannian manifolds. Indeed, Lord-Rennie-Várilly have developed a full noncommutative generalisation of the Hodge-de Rham spectral triple qua candidate notion of noncommutative Riemannian manifold [10]. On the other hand, in the (almost-)commutative context, one might observe that the Hodge-de Rham spectral triple of a compact oriented Riemannian manifold X, or more generally, the spectral triple 2010 Mathematics Subject Classification. Primary 58B34, Secondary 46L87, 81T75.

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“…In particular, in the noncommutative geometry models of particle physics the Higgs field arises because of the presence of a nontrivial noncommutative space F , and an almost commutative geometry X ×F (the product of the spacetime manifold X and F , or more generally a nontrivial fibration as analyzed in [5]). The Higgs field is described geometrically as the inner fluctuations of the Dirac operator in the noncommutative direction F .…”

Section: 4mentioning

confidence: 99%

Coupling of gravity to matter, spectral action and cosmic topology

Ćaćić¹,

Marcolli²,

Teh³

2014

J. Noncommut. Geom.

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Abstract. We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat-kernel computation. Contents

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A Reconstruction Theorem for Almost-Commutative Spectral Triples

Cited by 26 publications

References 25 publications

The noncommutative geometry of Yang–Mills fields

The noncommutative geometry of Yang–Mills fields

Real Structures on Almost-Commutative Spectral Triples

Coupling of gravity to matter, spectral action and cosmic topology

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