# Summer School “Coarse Index Theory”, September 26-30, 2016, Freiburg

Topic:    Index theory is a prime example of fruitful interaction between analysis, geometry, topology and operator algebras.

The index is associated to a global differential operator and is computed from the set of solutions of the associated differential equation. It turns out, however, that the index has remarkable stability properties and can often be computed a priori without solving the differential equation. This uses index theorems and the underlying topology. On the other hand, the most interesting operators are tied to the geometry and the geometry determines the set of solutions. The most powerful implementations of this idea that the relevant operators lie in operator algebras which are specific to the situation at hand. The indices are then naturally defined as elements in K-theory groups of these operator algebras.

It turns out that a particularly useful setup uses the ideas of “coarse  geometry”. The basic idea is to study (non-compact) metric spaces; but  considering only their large scale features. A lot of this can be captured  in appropriately associated C*-algebras; the coarse C*-algebras of the space (often called Roe algebra). This tool also applies to compact spaces, by passing first to their universal covering.

The corresponding manifestation of index theory in this context is “coarse   index theory” or “large scale index theory” and has many interesting  properties and applications.

The summer school will explain the relevant general background in index theory, operator algebras; and then focus on large scale geometry and index theory and its numerous applications.

Program: There will be lecture series by

• John Roe: Coarse geometry and index theory
• Thomas Schick: (Secondary) Coarse index and applications
• Rudolf Zeidler: K-theory of C*-algebras

along with daily exercise and discussion sessions in the afternoon.

Schedule: Preliminary schedule

Funding: As a general rule, you are supposed to arrive with your own funding but there are also some limited funds available.

Contact: Please send an informal email to enroll until September 1, 2016, latest, to Mrs. Ursula Wöske, coarse16@math.uni-freiburg.de

Speakers: John Roe, Thomas Schick, Rudolf Zeidler

Poster: Poster Summer School 2016

# School and Conference on Noncommutative Geometry

### Villa de Leyva, Colombia,  June 20 to July 1, 2016

Noncommutative Geometry is a growing and active field whose roots and branches intertwine with various areas of mathematics and physics. The school and conference on Noncommutative Geometry, Noncommutative Geometry 2016, aims at introducing advanced students and young researchers to various results and techniques from noncommutative geometry giving special attention to those which play a key role in applications. The lectures will focus on some of the main themes lying at the core of the current developments of the theory while the mini courses will provide both background and context for those themes.

#### Lecture Series:

Walter van Suijlekom (Radboud University Nijmegen)
Noncommutative geometry and particle physics

Heat kernel methods and local geometric invariants of noncommutative spaces

Özgür Ceyhan (University of Luxembourg)
Feynman integrals, associated arrangements and their motives

Gunther Cornelissen (Utrecht University)
Zeta functions in number theory and differential geometry in the light of noncommutative geometry

Bram Mesland (Leibniz Universität Hannover)
A categorical approach to spectral triples and KK-theory

#### Mini Courses:

Carolina Neira (Universidad Nacional de Colombia)
Hochschild cohomology and residues

Elements of K-theory

Andrés Reyes (Universidad de los Andes)
Basics of quantum field theory

Characteristic classes

Spin manifolds and Dirac operators

Alexander Cardona (Universidad de los Andes)
Index theorems

Scientific committee:
Matilde Marcolli (California Institute of Technology)
Özgür Ceyhan (University of Luxembourg)

Organising committee:

# Masterclass on unbounded KK-theory in Copenhagen

At the end of a summer there will be a master class in Copenhagen on unbounded KK-theory and the more analytic aspects of non commutative geometry. During the master class there will be lecture series:
• Matthias Lesch (University of Bonn)
• Adam Rennie (University of Wollongong)
“Applications of KK-theory in non-commutative geometry and physics”
• Fedor Sukochev (University of New South Wales)
“Introduction to Double Operator Integration and Quantum Differentiability of Essentially Bounded Functions on Euclidean Space”
The master class will take place 22-26/8 at the mathematics department in Copenhagen. More information can be found on:
There is funding for local expenses and possibilities for participants to contribute with talks. The deadline for registration is August 1st. If you want to stay in the shared accommodation booked by the department it is July 1st.

# Public lecture Alain Connes, Nijmegen

On April 4 (18:00) Alain Connes will give a public lecture on ‘Geometry and the Quantum’, as part of the conference ‘Gauge Theory and Noncommutative Geometry’ at Radboud University Nijmegen.

More details can be found here and on the conference website

# 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:

Running of coupling constants for the spectral Pati–Salam model with composite Higgs fields

Running of coupling constants for the spectral Pati–Salam model with fundamental Higgs fields

Running of coupling constants for the left-right symmetric spectral Pati–Salam model

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.

# 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

Please allow me to present my book “Noncommutative Geometry and Particle Physics” that just arrived by mail:

# 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.