Conceptual explanation for the algebra in the noncommutative approach to the standard model

The purpose of this Letter is to remove the arbitrariness of the ad hoc choice of the algebra and its representation in the noncommutative approach to the standard model, which was begging for a conceptual explanation. We assume as before that space-time is the product of a four-dimensional manifold by a finite noncommmutative space F. The spectral action is the pure gravitational action for the product space. To remove the above arbitrariness, we classify the irreducible geometries F consistent with imposing reality and chiral conditions on spinors, to avoid the fermion doubling problem, which amounts to have total dimension 10 (in the K-theoretic sense). It gives, almost uniquely, the standard model with all its details, predicting the number of fermions per generation to be 16, their representations and the Higgs breaking mechanism, with very little input.     

Property T and asymptotically invariant sequences

A countable group Γ has property T of Kazhdan if and only if no measure preserving ergodic action of Γ has non-trivial asymptotically invariant sets.     

The hyperring of adèle classes

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We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc.     

Transgression and the Chern character of finite-dimensional K-cycles

It is shown that the [JLO] entire cocycle of a finitely summable unbounded Fredholm module can be retracted to a periodic cocycle. Moreover, the retracted cocycle admits a zero-temperature limit, which provides the extension of the transgressed cocycle of [CM1] from the invertible case to the general case.Dedicated to Huzihiro ArakiResearch supported in part by NSF Grant DMS-9101557     

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.     

Caractère multiplicatif d'un module de Fredholm

L'objectif essential de cet article est de définir un accouplement

Hecke algebras,type III factors and phase transitions with spontaneous symmetry breaking in number theory

In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings provides the desiredC*-dynamical system, which admits the function as partition function and the Galois group Gal(^cycl/) of the cyclotomic extension ^cycl of as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature =1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).     

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.     

High resolution vibration-rotation spectra of carbon suboxide: Molecular constants for the ground state and 2ν70

The infrared spectrum of C₃O₂ was recorded with the vacuum Fourier transform interferometer of Laboratoire Aimé Cotton at a resolution of 0.005 cm^?1. The ground state molecular constants were calculated from lower state combination relations in a simultaneous analysis of six ground state transitions situated in the region 3000 to 5000 cm^?1. Through the analysis of a difference band we established that $\text{2ν}{}_7{}^0$ is 60.7022 ± 0.0005 cm^?1 above the ground vibrational state. Accurate molecular constants were also determined for this vibrational level.