Mathematical physicist Walter van Suijlekom uses noncommutative geometry to explore the laws of physics. Read more on http://www.ru.nl/research/mathematics (animation by Bruno van Wayenburg)

# Public lecture Alain Connes, Nijmegen

# Grand Unification in the Spectral Pati-Salam Model

Today we (Chamseddine-Connes-van Suijlekom) posted a preprint on grand unification in the spectral Pati–Salam model which I summarize here.

The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed how to obtain inner fluctuations of the metric without having to assume the order one condition on the Dirac operator. Moreover the original argument by classification of finite geometries \(F\) that can provide the fine structure of Euclidean space-time as a product \(M\times F\) (where \(M\) is a usual 4-dimensional Riemannian space) has now been replaced by a much stronger uniqueness statement. This new result shows that the algebra

\(

M_{2}(\mathbb{H})\oplus M_{4}(\mathbb{C})

\)

where \(\mathbb{H}\) are the quaternions, appears uniquely when writing the higher analogue of the Heisenberg commutation relations. This analogue is written in terms of the basic ingredients of noncommutative geometry where one takes a spectral point of view, encoding geometry in terms of operators on a Hilbert space \(\mathcal{H}\). In this way, the inverse line element is an unbounded self-adjoint operator \(D\). The operator \(D\) is the product of the usual Dirac operator on \(M\) and a `finite Dirac operator’ on \(F\), which is simply a hermitian matrix \(D_{F}\). The usual Dirac operator involves gamma matrices which allow one to combine the momenta into a single operator. The higher analogue of the Heisenberg relations puts the spatial variables on similar footing by combining them into a single operator \(Y\) using another set of gamma matrices and it is in this process that the above algebra appears canonically and uniquely in dimension 4.

This leads without arbitrariness to the Pati–Salam gauge group \(SU(2)_{R}\times SU(2)_{L}\times SU(4)\), together with the corresponding gauge fields and a scalar sector, all derived as inner perturbations of \(D\). Note that the scalar sector can not be chosen freely, in contrast to early work on Pati–Salam unification. In fact, there are only a few possibilities for the precise scalar content, depending on the assumptions made on the finite Dirac operator.

From the spectral action principle, the dynamics and interactions are described by the *spectral action,*

\(

\mathrm{tr}(f(D/\Lambda))

\)

where \(\Lambda\) is a cutoff scale and \(f\) an even and positive function. In the present case, it can be expanded using heat kernel methods,

\(\mathrm{tr}(f(D/\Lambda))\sim F_{4}\Lambda^{4}a_{0}+F_{2}\Lambda^{2}%

a_{2}+F_{0}a_{4}+\cdots

\)

where \(F_{4},F_{2},F_{0}\) are coefficients related to the function \(f\) and \(a_{k}\) are Seeley deWitt coefficients, expressed in terms of the curvature of \(M\) and (derivatives of) the gauge and scalar fields. This action is interpreted as an effective field theory for energies lower than \(\Lambda\).

One important feature of the spectral action is that it gives the usual Pati–Salam action with unification of the gauge couplings. Indeed, the scale-invariant term \(F_{0}a_{4}\) in the spectral action for the spectral Pati–Salam model contains the terms

\(

\frac{F_{0}}{2\pi^{2}}\int\left( g_{L}^{2}\left( W_{\mu\nu L}^{\alpha

}\right) ^{2}+g_{R}^{2}\left( W_{\mu\nu R}^{\alpha}\right) ^{2}%

+g^{2}\left( V_{\mu\nu}^{m}\right) ^{2}\right) .

\)

Normalizing this to give the Yang–Mills Lagrangian demands

\(

\frac{F_{0}}{2\pi^{2}}g_{L}^{2}=\frac{F_{0}}{2\pi^{2}}g_{R}^{2}=\frac{F_{0}%

}{2\pi^{2}}g^{2}=\frac{1}{4},

\)

which requires gauge coupling unification. This is very similar to the case of the spectral Standard Model where there is unification of gauge couplings. Since it is well known that the SM gauge couplings do not meet exactly, it is crucial to investigate the running of the Pati–Salam gauge couplings beyond the Standard Model and to find a scale \(\Lambda\) where there is grand

unification:

\(

g_{R}(\Lambda)=g_{L}(\Lambda)=g(\Lambda).

\)

This would then be the scale at which the spectral action is valid as an effective theory. There is a hierarchy of three energy scales: SM, an intermediate mass scale \(m_{R}\) where symmetry breaking occurs and which is related to the neutrino Majorana masses (\(10^{11}-10^{13}\)GeV), and the GUT scale \(\Lambda\).

In the paper, we analyze the running of the gauge couplings according to the usual (one-loop) RG equation. As mentioned before, depending on the assumptions on \(D_{F}\), one may vary to a limited extent the scalar particle content, consisting of either composite or fundamental scalar fields. We will not limit ourselves to a specific model but consider all cases separately. This leads to the following three figures:

In other words, we establish grand unification for all of the scenarios with unification scale of the order of \(10^{16}\) GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of \(D_{F}\).

# 1st COST QSPACE call for Short Term Scientific Missions

Two months ago a new European network within the COST framework was inaugurated:

MPNS COST Action MP1405

**Quantum structure of spacetime (QSPACE)**

Up to now 25 European countries (plus Japan as partner) have signed up so far. We shall soon have a dedicated website, but for now, further information can be found at http://www.cost.eu/COST_Actions/mpns/Actions/MP1405. The network does not fund positions but workshops, training schools, visits etc. Among its various activities, the so-called `short term scientific missions’ (STSMs) play a central role. These are visits of a researcher from one participating country to a colleague in another, for 5-90 days (180 days if PhD was < 8 yrs ago).

Here now comes the first call for such STSMs within our COST Action. So please spread among your colleagues the attached call, since anyone from a COST member country may participate. For your information, I also attach a presentation of the COST rules. We hope for a good number of qualified applications.

###### Attachments:

# Hausdorff Trimester Program “Noncommutative Geometry and Applications” – VIDEO’S

During the summer school and the first two workshops that are part of the Hausdorff Trimester Program on “Noncommutative Geometry and Applications” several talks and lectures have been recorded. The Youtube Channel HIM Lectures contains them and is a great source for looking back or as an update on recent progress in the field.The schedule for the school and the workshops can be found here. With many thanks to the IT-Support team from the Hausdorff Institute for Mathematics in Bonn.

# Two PhD defenses in noncommutative geometry

Early September two of my PhD students will defend their PhD thesis at the Radboud University Nijmegen.

On Friday September 5 my PhD student **Thijs van den Broek** (supervised together with Wim Beenakker and promotor Ronald Kleiss) will defend his thesis **“Supersymmetry and the Spectral Action: On a geometrical interpretation of the MSSM”**. Thijs worked on the intersection between supersymmetry and noncommutative geometry, searching for a theory arising from noncommutative geometry that describes the MSSM, or something alike. More details on the defense can be found here, the contents of the thesis will appear soon on the arXiV.

**Update:** The full PhD thesis of Thijs van den Broek can be found online at http://arxiv.org/abs/1409.6751, and the corresponding arXiv-papers at http://arxiv.org/abs/1409.5982 , http://arxiv.org/abs/1409.5983 , http://arxiv.org/abs/1409.5984.

On Thursday September 11 my PhD student **Jord Boeijink** (promotor Klaas Landsman) will defend his thesis **“Dirac operators, gauge systems and quantisation”**. Jord worked on two subjects: one was the problem whether quantization commutes with reduction for gauge systems. More specifically, he analyzed the quantization of the cotangent bundle to a compact Lie group \(G\) with symmetries given by the adjoint action of \(G\). The second subject that Jord worked on was the extension of almost-commutative manifold to the topologically non-trivial case, and already appeared as the preprint arXiv:1405.5368. More details on the defense can be found here.

# Book “Noncommutative Geometry and Particle Physics

# NCG at Frontiers of Fundamental Physics

Here is an update (from a noncommutative geometry point of view) of the talks at the conference Frontiers of Fundamental Physics. The conference started off with a great welcome reception at the Fort Ganteaume, enjoying a great view on the fireworks for the 14th of July.

On Monday morning there was a nice overview on the status of HEP after LHC run 1 by Paraskesas Sphicas, were especially the experimental finding of spin 0 for the Higgs boson is interesting for NCG and applications where it naturally appears as a scalar boson.

In one of the so-called `parallel plenary’ sessions Pierre Bieliavsky gave an overview of deformation quantization using the deformation of a matrix algebra as an interesting 0-dimensional toy model.

There were two interesting contributions on causal structures in noncommutative geometry, addressing some of the first questions on the way towards a Lorentzian version of spectral triples. Fabien Besnard introduces so-called \(I^*\)-algebras that translates to the \(C^*\)-algebraic level the causal structure on the state space of that \(C^*\)-algebra. A reference is

http://arxiv.org/abs/1312.2442

Mickal Eckstein presented some of his recent work with Nicolas Franco on Lorentzian spectral triples, arriving at a different notion of causality in the context of spectral triples. Some of the examples he discussed were two-sheeted spacetime for which he derived a causal relation between the two sheets when the distance in the continuous direction was larger than the distance in the discrete direction. The corresponding paper is in preparation.

# Perturbations of the metric and Weyl’s Eichinvarianz

Thanks to David Broadhurst for stressing the following point during my lecture at a summer school in Les Houches.

Whilst introducing gauge fields from noncommutative spin manifolds (aka spectral triples) I first explained how the Dirac operator can be seen as a metric on a (possibly noncommutative) space described via Connes’ distance formula. Then the action of a unitary in the algebra of coordinates was given as a gauge transformation on the Dirac operator, generating a pure gauge field.

What David noticed was that this is in compelling agreement with Weyl’s old idea of gauge invariance. Indeed, the term *Eichinvarianz *was preceded by *Maβstabinvarianz* in the original work (see Yang’s review below). This is precisely the notion captured by noncommutative geometry: a gauge transformation actually acts on the metric (the Dirac operator) but leaves the distance function invariant.

# Re-launch of noncommutativegeometry.nl

After a transition period it is time to re-launch this website. It has been transformed from a website dedicated to the workshop at the Lorentz Center in Leiden in October 2013 to a more general repository and discussion platform for noncommutative geometry and its applications to particle physics. Everyone is welcome to share his or her thoughts here.

For now, let me mention some good video excerpts from a lecture that Alain Connes gave in Nijmegen, see the documents section for that.

Also, allow me to advertise my upcoming book “Noncommutative Geometry and Particle Physics” which is due to appear this summer with Springer:

This textbook provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. It is intended for graduate students in mathematics/theoretical physics who are new to the field of noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in the physical applications of noncommutative geometry. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a “light” approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model.