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.

 

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.