### SO(20) : twenty years of spectral obstinacy

**Have A. Chamseddines and A. Connes opened the way of a new wisdom for physics towards quantum gravity since 1996?**

The spectral action principle that epitomizes the geometric unification of the electroweak and strong interactions with the gravitational one in a noncommutative formalism was put forward for the first time in an article submitted the 3rd of June 1996. Exactly 20 years later, its first author Ali Chamseddine, a former Ph. D. student of Abdus Salam choses to celebrate the memory of his mentor with a synthetic summary of the last developments of this joint program. Here are a few extracts:

The work I am reporting in this presentation is the result of a long-term collaboration with Alain Connes over a span of twenty years starting in 1996[3] [4] [5] [6] [7] [8]. In the latest work on volume quantization we were joined by Slava Mukhanov [9] [10]. On the inner fluctuations of the Dirac operator over automorphisms of the noncommutative algebra times its opposite, we were joined by Walter van Suijlekom [11] [12] [13]...In this setting, the four dimensional manifold emerges as a composite of the inverse maps of the product of two spheres of Planck size. The manifoldM_{4}which is folded many times in the product, unfolds to macroscopic size. The two different spheres, associated with the two Clifford algebras can be considered as quanta of geometry which are the building blocks to generate an arbitrary oriented four dimensional spin-manifold. We can show that the manifold M_{4}, the two spheres with their maps Y, Y' and their associated Clifford algebras define a noncommutative space which is the basis of unification of all fundamental interactions, including gravity...The phase space of coordinates and Dirac operator defines a noncommutative space of KO dimension 10.The symmetries of the algebras defining the noncommutative space turn out to be those ofSU(2)_{R}×SU(2)_{L}×SU(4)_{C}known as the Pati-Salam models. Connections along discrete directions are the Higgs fields... The action has a very simple form given by a Dirac action for fermions and a spectral action for bosons. The 16 fermions (per family) are in the correct representations with respect to Pati-Salam symmetries or the Standard Model symmetries.There are many consequences of the volume quantization condition which could be investigated. For example imposing the quantization condition through a Lagrange multiplier would imply that the cosmological constant will arise as an integrating constant in the equations of motion. One can also look at the possibility that only the three volume (space-like) is quantized. This can be achieved provided that the four-dimensional manifold arise due to the motion of three dimensional hypersurfaces, which is equivalent to the 3+1 splitting of a four-dimensional Lorentzian manifold.Then three dimensional space volume will be quantized, provided that the field X that maps the real line have a gradient of unit norm g[21] [22]._{µν}∂_{µ}X∂_{ν}X=1. It is known that this condition when satisfied gives a modified version of Einstein gravity with integrating functions giving rise to mimetic dark matter

(Submitted on 3 Jun 2016)

//addition on June 8, 2016:

**A Salam's student slow drift from superstring theory to noncommutative geometry ...**

... and a roughly 40 year old cyclic unfolding time from one paper to another:

In 1973 I got a scholarship from government of Lebanon to pursue my graduate studies at Imperial College, London. Shortly after I arrived, I was walking through the corridor of the theoretical physics group, I saw the name Abdus Salam on a door. At that time my information about research in theoretical physics was zero, and since Salam is an Arabic name, and the prime minister in Lebanon at that time was also called Salam, I knocked at his door and asked him whether he is Lebanese. He laughed and explained to me that he is from Pakistan.Hethenasked me why I wanted to study theoretical physics. I said the reason is that I love mathematics.He smiled and told me that I am in the wrong department.In June 1974, having finished the Diploma exams I asked Salam to be my Ph.D. advisor and he immediately accepted and gave me two preprints to read and to chose one of them as my research topic. The first paper was with Strathdee [1] on the newly established field of supersymmetry (a word he coined), and the other is his paper with Pati [2] on the first Grand Unification model, now known as the Pati-Salam model. Few days later I came back and told Salam that I have chosen supersymmetry which I thought to be new and promising. Little I knew that the second project will come back to me forty years later from studying the geometric structure of space-time, as will be explained in what follows. In this respect, Salam was blessed with amazing foresight.

Id.

Grand uniﬁed theories provide an attractive mechanism to unify the weak, strong and electromagnetic interactions and put order into the representations of quarks and leptons. At present, the simplest models are based on SU(5)[1] and SO(10) gauge theories[2]. The second class of models has the advantage of including all the fermions (plus a right handed neutrino) in one representation. This advantage does not translate itself into a more predictive theory, because there are many possibilities to break SO(10) down to SU(3)×U(1)em requiring many diﬀerent and often complicated Higgs representations [3].What is clearly needed in grand uniﬁed theories is a principle to put order into the Higgs sector. During the last few years, much effort has been directed towards this problem by studying uniﬁed theories as low-energy limits of the heterotic string . Although this is an attractive strategy, it has proven to be a difficult one, due to the fact that one must search for good models among the very large number of string vacua. We shall follow, instead, a different strategy.

It has been shown by Connes [4-5] and Connes and Lott [6-7] that the ideas of non-commutative geometry can be applied to, among other things, model building in particle physics. In particular, the Dirac operator, deﬁned on the one-particle Hilbert space of quarks and leptons, is used to construct the standard SU(3)×SU(2)×U(1) model with the Higgs ﬁeld uniﬁed with the gauge ﬁelds.The space-time used in this construction is a product of a Euclidean four-dimensional manifold by a discrete two-point space. If, in coming years, an elementary Higgs ﬁeld is observed experimentally, one can turn the argument around and view it as an indication that space-time has the product structure proposed by Connes.

SO(10) Unification in Non-Commutative GeometryA. H. Chamseddine, J. Fröhlich(Submitted on 7 Apr 1993)

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