The graded product of real spectral triples

Farnsworth, Shane

doi:10.1063/1.4975410

Cited by 13 publications

(11 citation statements)

References 35 publications

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“…This study was completed in [8], where the odd-odd case was considered as well (in [24] one of the spectral triples is always assumed to be even), along with several possible choices of Dirac operators and real structures. The modified definition of J , which perhaps seems somewhat artificial, was reinterpreted in [14] as a graded tensor product.…”

Section: Products and The Second-order Conditionmentioning

confidence: 99%

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Twisted Reality and the Second-Order Condition

Da̧browski

D’Andrea

Magee

2021

Math Phys Anal Geom

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An interesting feature of the finite-dimensional real spectral triple (A, H, D, J ) of the Standard Model is that it satisfies a "second-order" condition: conjugation by J maps the Clifford algebra C D (A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence C D (A)-bimodule). This resembles a property of the canonical spectral triple of a closed oriented Riemannian manifold: there is a dense subspace of H which is a self-Morita equivalence C D (A)-bimodule. In this paper we argue that on manifolds, in order for the self-Morita equivalence to be implemented by a reality operator J , one has to introduce a "twist" and weaken one of the axioms of real spectral triples. We then investigate how the above mentioned conditions behave under products of spectral triples.

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Section: Products and The Second-order Conditionmentioning

confidence: 99%

“…Here we follow this idea in spirit, but we will find that the "correct" definition of J is not the one in [8,14,24]. Our motivation is that we want the second-order property (5) to be preserved by products, and this will lead to yet another different definition of J .…”

Section: Products and The Second-order Conditionmentioning

confidence: 99%

Twisted Reality and the Second-Order Condition

Da̧browski

D’Andrea

Magee

2021

Math Phys Anal Geom

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“…In future work, we will discuss the implications of this formalism for taking tensor products of spectral triples, since this is an interesting story in its own right [48].…”

Section: Jhep06(2018)071mentioning

confidence: 99%

A new algebraic structure in the standard model of particle physics

Boyle

Farnsworth

2018

J. High Energ. Phys.

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We introduce a new formulation of the real-spectral-triple formalism in noncommutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent A (the algebra of differential forms on some possibly-noncommutative space) on H (the Hilbert space of spinors on that space); and to reinterpret this representation as a simple super-algebra B = A ⊕ H with even part A and odd part H. B is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional real-spectral-triple formalism of NCG are elegantly recovered from the simple requirement that B should be a differential graded * -algebra (or " * -DGA"). Moreover, this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. In particular, these new constraints provide a novel interpretation of electroweak symmetry breaking that is geometric rather than dynamical. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.

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“…The extension of this tensor product to odddimensional indefinite spectral triples seems difficult, if one considers the complexity of the Riemannian case [69][70][71][72][73]. Note that Farnsworth also advocates the use of a graded tensor product [73].…”

Section: Indefinite Spectral Triplementioning

confidence: 99%

Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

Brouder

Besnard²

2018

Journal of Mathematical Physics

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An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations.It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product are the sums of the space and time dimensions of its factors. This could provide an interpretation of the presence of such algebras in P T -symmetric Hamiltonians or the description of topological matter.This construction is used to build an indefinite (i.e. pseudo-Riemannian) version of the spectral triples of noncommutative geometry, defined over Krein spaces instead of Hilbert spaces. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.

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The graded product of real spectral triples

Cited by 13 publications

References 35 publications

Twisted Reality and the Second-Order Condition

Twisted Reality and the Second-Order Condition

A new algebraic structure in the standard model of particle physics

Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

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