A farewell to unimodularity

Abstract. We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.

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“…We refer to [35] for finer points concerning the lifting of diffeomorphisms preserving the given spin structure.…”

Section: The Adjoint Representation and The Gauge Symmetriesmentioning

confidence: 99%

Gravity and the standard model with neutrino mixing

Chamseddine¹,

Connes²,

Marcolli³

2007

Advances in Theoretical and Mathematical Physics

385

893

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“…This only holds for complex algebras B. For real algebras (including the one describing the noncommutative Standard Model [Con96,CCM07]) one needs to impose unimodularity by hand (see also [LS01] and references therein).…”

Section: The Gauge Groupmentioning

confidence: 99%

On globally non-trivial almost-commutative manifolds

Boeijink

Dungen

2014

Journal of Mathematical Physics

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Within the framework of Connes' noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almostcommutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a 'principal module', which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of 'gauge modules', and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.

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“…3 Noncommutative geometry (in a general sense) has been previously used to build a theory beyond Standard Model, see e.g., [28]. Recently within the Connes formulation, the unimodularity condition have been used to obtain the hyper-chrages for the fermions [29]. However, these models are based on a very different approach than ours, where the fields evolve in almost commutative spaces (the space-time is commutative with a minimal noncommutativity in the internal space).…”

Section: Introductionmentioning

confidence: 99%

Non-commutative standard model: model building

et al. 2003

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A noncommutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: 1) although we can have noncommutative U (n) (which we denote by U ⋆ (n)) gauge theory we cannot have noncommutative SU (n) and 2) the charges in noncommutative QED are quantized to just 0, ±1. We show how the latter problem with charge quantization, as well as with the gauge group, can be resolved by taking U ⋆ (3) × U ⋆ (2) × U ⋆ (1) gauge group and reducing the extra U (1) factors in an appropriate way. Then we proceed with building the noncommutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the noncommutative Standard Model (NCSM). In addition, among several peculiar features of our model, we address the inherent CP violation and new neutrino interactions.

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A farewell to unimodularity

Abstract: We interpret the unimodularity condition in almost commutative geometries as central extensions of spin lifts. In Connes' formulation of the standard model this interpretation allows to compute the hypercharges of the fermions.

Cited by 38 publications

References 7 publications

Gravity and the standard model with neutrino mixing

Gravity and the standard model with neutrino mixing

On globally non-trivial almost-commutative manifolds

Non-commutative standard model: model building

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