Pseudo-Riemannian Spectral Triples for the Standard Model

Arkadiusz Bochniak

doi:10.1051/epjconf/201920109002

Abstract

We present the importance of the pseudo-Riemannian structure in the spectral triple formalism that is used to describe the Standard Model of Particle Physics. The filnite case is briefly described and its role in the context of leptoquarks is presented. The proposal for the reverse engineering program for the Standard Model is also described, together with recent results.

Highlights

The approach to the Standard Model of Particle Physics based on the Connes’ idea of the Noncommutative Geometry (NCG) allows for the geometrical analysis of the structure of the Standard Model (SM), reveals the origin of the Higgs mechanism and, using spectral action methods and renormalization group techniques, produces numbers that can be compared with experimental data
The main idea of NCG is based on the Connes’ reconstruction theorem [8] from which it is known that the whole metric and spin structure of a closed, orientable Riemannian spinc manifold M can be encoded in a system consisted of a commutative ∗-algebra C∞(M) of smooth complex-valued functions on M, a Hilbert space H of square-integrable spinors and the Dirac operator DM that acts on sections of the spinor bundle over M
We presented the approach to the Standard Model based on the Noncommutative Geometry methods

Summary

Introduction

The approach to the Standard Model of Particle Physics based on the Connes’ idea of the Noncommutative Geometry (NCG) allows for the geometrical analysis of the structure of the Standard Model (SM), reveals the origin of the Higgs mechanism and, using spectral action methods and renormalization group techniques, produces numbers that can be compared with experimental data. This is the concept of a spectral triple - a system (A, H, D, γ, J) with a ∗-algebra A represented in a faithful way on a Hilbert space H, Z/2Z-grading γ† = γ commuting with A, antilinear isometry J such that [Ja∗ J−1, b] = 0 for all a, b ∈ A and (essentially) self-adjoint operator D, called Dirac operator, with compact resolvent They are supposed to satisfy few compatibility conditions [11],[12],[16], for example DJ = JD, J2 = id and Jγ = γJ, where the choice of signs , , = ±1 defines the so-called KO-dimension that is an integer modulo 8. In [5] we proposed a new point of view on the lepton-quark symmetry based on the existence of an additional Z/2Z-grading that distinguishes between these sectors and is a shadow of a pseudo-Riemannian structure on the finite spectral triple for the SM The goal is to characterize possible pseudo-Riemannian spectral triples or their modifications that can describe the SM together with all its hidden structures or symmetries

Finite Pseudo-Riemannian Spectral Triples and Leptoquarks

Reverse Engineering for the Standard Model

Summary