arXiv subjects

Pierre Martinetti

Publications and source records attributed to Pierre Martinetti.

At least 19 recordsLinked to original sources

arXiv2026

Twisted Standard Model and its Krein structure -- in memoriam Manuele Filaci

We review the contributions of Manuele Filaci - a PhD student from the university of Genova prematurely deceased a little more than a year ago - to the description of the Standard Model in noncommutative geometry. Building on Manuele's discovery that there exist various ways to minimally twist the spectral triple of the Standard Model, we study in a systematic way the inner product induced by the twist. Under loose assumptions, this product turns the Hilbert space of the spectral triple into a Krein space. For the Standard Model, the group of unitary with respect to the twisted product contains the symmetry group of twistors as a subgroup.

math-ph↗

arXiv2024

Torsion and Lorentz symmetry from Twisted Spectral Triples

By twisting the spectral triple of a riemannian spin manifold, we show how to generate an orthogonal and geodesic preserving torsion from a torsionless Dirac operator. We identify the group of twisted unitaries as the generator of torsion with co-exact three form. Through the fermionic action, the torsion term identifies with a Lorentzian energy-momentum 4-vector. The Lorentz group turns out to be a normal subgroup of the twisted unitaries. We also investigate the spectral action related to this model.

math-ph↗

arXiv2023

A critical survey of twisted spectral triples beyond the Standard Model

We review the applications of twisted spectral triples to the Standard Model. The initial motivation was to generate a scalar field, required to stabilise the electroweak vacuum and fit the Higgs mass, while respecting the first-order condition. Ultimately, it turns out that the truest interest of the twist lies in a new -- and unexpected -- field of 1-forms, which is related to the transition from Euclidean to Lorentzian signature.

math-ph↗

arXiv2021

Minimal twist of almost commutative geometries

We classify the twists of almost commutative spectral triples that keep the Hilbert space and the Dirac operator untouched. The involved twisting operator is shown to be the product of the grading of a manifold by a finite dimensional operator, which is not necessarily a grading of the internal space. Necessary and sufficient constraints on this operator are listed.

math-ph↗

arXiv2021

Twisted Spectral Triples without the First-Order Condition

We extend twisted inner fluctuations to twisted spectral triples that do not meet the twisted first-order condition, following what has been done in [6] for the non-twisted case. We find a similar non-linear term in the fluctuation, and work out the twisted version of the semi-group of inner perturbations.

math-ph↗

arXiv2020

Real Part of Twisted-by-Grading Spectral Triples

After a brief review on the applications of twisted spectral triples to physics, we adapt to the twisted case the notion of real part of a spectral triple. In particular, when one twists a usual spectral triple by its grading, we show that - depending on the $KO$ dimension - the real part is either twisted as well, or is the intersection of the initial algebra with its opposite. We illustrate this result with the spectral triple of the standard model.

math-ph↗

arXiv2020

Twisted Standard Model in noncommutative geometry I: the field content

Noncommutative geometry provides both a unified description of the Standard Model of particle physics together with Einstein-Hilbert action (in euclidean signature) and some tools to go beyond the Standard Model. In this paper, we extend to the full noncommutative geometry of the Standard Model the twist (in the sense of Connes-Moscovici) initially worked out for the electroweak sector and the free Dirac operator only. Namely, we apply the twist also to the strong interaction sector and the finite part of the Dirac operator. To do so, we are forced to take into account a violation of the twisted first-order condition. As a result, we still obtain the extra scalar field required to stabilise the electroweak vacuum and fit the Higgs mass, but it now has two chiral components. We also get the additive field of 1-forms already pointed out in the electroweak model, but with a richer structure. Finally, we obtain a pair of Higgs doublets, which are expected to combine into a single Higgs doublet in the action formula, as will be investigated in the second part of this work.

math-ph↗

arXiv2020

Actions for twisted spectral triple and the transition from the Euclidean to the Lorentzian

This is a review of recent results regarding the application of Connes' noncommutative geometry to the Standard Model, and beyond. By twisting (in the sense of Connes-Moscovici) the spectral triple of the Standard Model, one does not only get an extra scalar field which stabilises the electroweak vacuum, but also an unexpected 1-form field. By computing the fermionic action, we show how this field induces a transition from the Euclidean to the Lorentzian signature. Hints on a twisted version of the spectral action are also briefly mentioned.

math-ph↗

arXiv2019

Lorentzian fermionic action by twisting euclidean spectral triples

We show how the twisting of spectral triples induces a transition from an euclidean to a lorentzian noncommutative geometry, at the level of the fermionic action. More specifically, we compute the fermionic action for the twisting of a closed euclidean manifold, then that of a two-sheet euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in euclidean signature. We obtain the Weyl and the Dirac equations in lorentzian signature (and in the temporal gauge). The twisted fermionic action is then shown to be invariant under an action of the Lorentz group. This permits to interprete the field of 1-form that parametrizes the twisted fluctuation of a manifold as the (dual) of the energy momentum 4-vector.

math-ph↗

arXiv2018

A dual formula for the spectral distance in noncommutative geometry

In noncommutative geometry, Connes's spectral distance is an extended metric on the state space of a C*-algebra generalizing Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a supremum. We present a dual formula - as an infimum - generalizing Beckmann's "dual of the dual" formulation of the Wasserstein distance. We then discuss some examples with matrix algebras, where such a dual formula may be useful to obtain upper bounds for the distance.

math.OA↗

arXiv2018

Connes distance and optimal transport

We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states.

math-ph↗

arXiv2017

Lorentz signature and twisted spectral triples

We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twist and Wick rotation.

hep-th↗

arXiv2017

Gauge transformations for twisted spectral triples

We extend to twisted spectral triples the fluctuations of the metric, as well as their gauge transformations. The former are bounded perturbations of the Dirac operator that arise when a spectral triple is exported between Morita equivalent algebras; the later are obtained by the action of the unitary endomorphisms of the module implementing the Morita equivalence. It is shown that the twisted gauged Dirac operators, previously introduced to generate an extra scalar field in the spectral description of the standard model of elementary particles, in fact follow from Morita equivalence between twisted spectral triples. The law of transformation of the gauge potentials turns out to be twisted in a natural way. In contrast with the non-twisted case, twisted fluctuations do not necessarily preserve the self-adjointness of the Dirac operator. For a self-Morita equivalence, some conditions are obtained in order to maintain self-adjointness, that are solved explicitly for the minimal twist of a Riemannian manifold.

math-ph↗

arXiv2016

Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand Symmetry

Grand symmetry models in noncommutative geometry have been introduced to explain how to generate minimally (i.e. without adding new fermions) an extra scalar field beyond the standard model, which both stabilizes the electroweak vacuum and makes the computation of the mass of the Higgs compatible with its experimental value. In this paper, we use Connes-Moscovici twisted spectral triples to cure a technical problem of the grand symmetry, that is the appearance together with the extra scalar field of unbounded vectorial terms. The twist makes these terms bounded and - thanks to a twisted version of the first-order condition that we introduce here - also permits to understand the breaking to the standard model as a dynamical process induced by the spectral action. This is a spontaneous breaking from a pre-geometric Pati-Salam model to the almost-commutative geometry of the standard model, with two Higgs-like fields: scalar and vector.

hep-th↗

arXiv2016

On twisting real spectral triples by algebra automorphisms

We systematically investigate ways to twist a real spectral triple via an algebra automorphism and in particular, we naturally define a twisted partner for any real graded spectral triple. Among other things we investigate consequences of the twisting on the fluctuations of the metric and possible applications to the spectral approach to the standard model of particle physics.

math-ph↗

arXiv2016

From Monge to Higgs: a survey of distance computations in noncommutative geometry

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, and the comparison of Connes distance with the minimal length that emerges in various models of quantum spacetime. Links with other areas of mathematics are studied, in particular the horizontal distance in sub-Riemannian geometry. The interpretation of Connes distance as a noncommutative version of the Monge-Kantorovich metric in optimal transport is also discussed.

math-ph↗

arXiv2015

Beyond the Standard Model with noncommutative geometry, strolling towards quantum gravity

Noncommutative geometry, in its many incarnations, appears at the crossroad of various researches in theoretical and mathematical physics: from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory, from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime, from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions. We list several of these applications, emphasizing also the original point of view brought by noncommutative geometry on the nature of time. This text serves as an introduction to the volume of proceedings of the parallel session "Noncommutative geometry and quantum gravity", as a part of the conference "Conceptual and technical challenges in quantum gravity" organized at the University of Rome "La Sapienza" in September 2014.

hep-th↗

arXiv2015

Twisted spectral geometry for the standard model

The Higgs field is a connection one-form as the other bosonic fields, provided one describes space no more as a manifold M but as a slightly non-commutative generalization of it. This is well encoded within the theory of spectral triples: all the bosonic fields of the standard model - including the Higgs - are obtained on the same footing, as fluctuations of a generalized Dirac operator by a matrix-value algebra of functions on M. In the commutative case, fluctuations of the usual free Dirac operator by the complex-value algebra A of smooth functions on M vanish, and so do not generate any bosonic field. We show that imposing a twist in the sense of Connes-Moscovici forces to double the algebra A, but does not require to modify the space of spinors on which it acts. This opens the way to twisted fluctuations of the free Dirac operator, that yield a perturbation of the spin connection. Applied to the standard model, a similar twist yields in addition the extra scalar field needed to stabilize the electroweak vacuum, and to make the computation of the Higgs mass in noncommutative geometry compatible with its experimental value.

hep-th↗