Friday, December 25, 2020

Fields Academy Graduate Courses

 Starting in January 2021, the Fields Institute in Toronto is going to run various advanced graduate courses in collaboration with Ontario universities. Students from all over the world can either take these courses, or just audit them. For details  see  here


The Fields Academy has an undergraduate training component as well. For details  check


Monday, November 2, 2020

The Metric Nature of Matter, Guest Post byJohannes Aastrup and Jesper Grimstrup

In a recent series of papers [1-3], we have discovered a connection between non-perturbative quantum field theory and the (noncommutative) geometry of configuration spaces. In this blog-post, we would like to outline our findings in a non-technical manner intended for readers, who are familiar with the basics of noncommutative geometry and high-energy physics.

The noncommutative standard model

Chamseddine and Connes and co-workers have shown that the standard model of particle physics coupled to general relativity can be formulated in terms of an almost-commutative spectral triple [4-8]. Their seminal work, which renders key aspects of the standard model in a completely new light, raises two fundamental questions:

  1. where does the almost-commutative algebra, that underlies Chamseddine’s and Connes’ work, originate from? Is there a deep reason why Nature chose this algebraic structure?
  2. what role should quantum field theory play in this approach to fundamental physics? As it stands now, perturbative quantum field theory is applied to Chamseddine’s and Connes’ construction as something like an afterthought and only to the standard model part: gravity is not quantized.

Embedded within this last question lies also the question of whether gravity should be quantized. 

In our recent work, we propose a novel answer to these two questions. It turns out that a geometrical construction over a configuration space of gauge connections gives rise to a non-perturbative quantum field theory on a curved background — including both bosonic and fermionic degrees of freedom — which produces an almost-commutative algebra similar to that of Chamseddine and Connes. Thus, what we propose is that the standard model of particle physics originates from a geometrical framework intimately related to non-perturbative quantum field theory. 

For another possible answer to the first question see also the work of Connes, Mukhanov, and Chamseddine [9,10]. 

The geometry of moving stuff around

The starting point of our construction is an algebra called the HD algebra [11,12], which is generated by parallel transports along flows of vector fields in a three-dimensional manifold. That is, the HD algebra encodes how tensor-degrees of freedom are moved around in space. The HD algebra, which comes with a high degree of canonicity, is a non-commutative algebra of functions over a configuration space of gauge connections.

The next step is to formulate an infinite-dimensional Bott-Dirac operator on the configuration space of gauge connections. The construction of this operator, which resembles an infinite-dimensional Bott-Dirac operator that Kasparov and Higson constructed in 2001 [13], is severely restricted by the requirement of gauge-covariance.

A key feature of the Bott-Dirac operator is that its square produces the Hamilton operator of a Yang-Mills theory coupled to a fermionic sector as well as a topological Yang-Mills term together with higher-order terms — all on a curved background. Furthermore, the Bott-Dirac operator and its interaction with the HD algebra reproduce the canonical commutation relations of a quantized gauge and fermionic fields. 

Thus, the formulation of a noncommutative geometry on a configuration space of gauge connections gives rise to the basic building blocks of a non-perturbative Yang-Mills-Dirac theory on a curved background. 

The origin of fermionic quantum field theory

One interesting feature of this construction is the role fermions play. The Bott-Dirac operator requires an infinite-dimensional Clifford algebra — just as an ordinary Dirac operator requires a finite-dimensional Clifford algebra — and it is this Clifford algebra that gives us the CAR algebra and the fermionic Fock space. Furthermore, when we compute the square of the Bott-Dirac operator then a certain commutator turns up, which gives us precisely the Dirac Hamilton operator. Thus, the Dirac Hamiltonian can be understood as a quantum fluctuation of the bosonic theory.

All this shows that the fermionic degrees of freedom play an intrinsically geometrical role in this framework; they are intimately related to the geometry of the underlying configuration space. 

A link to the standard model

In a semiclassical limit, the HD-algebra will give rise to a matrix algebra. If we choose a configuration space of spin connections then the HD-algebra will give us a three-by-three matrix algebra in a classical limit. The construction of the Bott-Dirac operator gives, however, rise to additional structure, which means that in a classical limit we find an almost-commutative algebra with a matrix factor given by: 

M_3(C) + M_3(C) + M_2(C)

which looks surprisingly similar to the matrix factor, which Chamseddine and Connes have found in the standard model [4,5]:

C + H + M_3(C)

where H is the quaternions. Could there be a connection here? More analysis is required to determine whether this link can be substantiated, but for now, we are encouraged by our findings.

Dynamical gauge-covariant regularisation

A key question is whether a Hilbert space representation of the HD algebra and the Bott-Dirac operator exists. We have shown that such a representation (strongly continuous, separable) exists in the special case where the gauge-symmetry is broken [3], but we do not have a proof in the gauge-covariant case. We are, however, confident that such representations do exist.

A key feature of the representation, which we have found, is a UV-regularisation in the form of a Sobolev norm that dampens degrees of freedom beyond a certain scale. In order to obtain a gauge-covariant representation, it is natural to make this Sobolev norm gauge-covariant. This is possible but in doing so we change the entire construction. 

Why is that? Well, normally a UV-regularisation is a computational artifact, that should ultimately be removed. This is the origin of renormalization theory. There are of course several reasons why a regularisation must be removed, one being that there are countless ways of regularising and no way to choose between them. But this is no longer the case with a gauge-covariant regularisation because it will have a time-evolution, i.e. it will be dynamical. This is what we call a dynamical gauge-covariant regularisation [1].

What we find is that the UV-regularisation should be understood not as a computational artifact but as a physical feature: the UV-regularisation is part of the metric data of the configuration space. One thing that makes this possible is that from a perturbative QFT perspective this UV-regularisation will simply look like higher-order derivative terms. The non-locality, that the dynamical regularisation introduces, will only be seen non-perturbatively.

Note that the concept of a dynamical regularisation is almost inevitable in a non-perturbative gauge theory since any Hilbert space representation will likely require some kind of UV-regularisation. The requirement that this regularisation is gauge-covariant automatically makes it dynamical.

It is widely believed that a theory of quantum gravity will give rise to a Planck-scale screening, which in turn will impact the Lorentz symmetry and the causal structure of space-time. If, however, the Planck-scale screening originates from a framework of non-perturbative quantum field theory, as we suggest, then it would make a theory of quantum gravity obsolete. The regime, where such a theory would otherwise reign, would be screened off.  We find this idea intriguing: perhaps the reason why a theory of quantum gravity has eluded theoretical physicists for so long is that it does not exist?

We believe that the idea of a dynamical UV-regularisation deserves much attention. After all, there exist very few concrete ideas as to how a Planck-scale screening may arise in Nature. This is one such idea.

Open questions

In our recent work, we have shown that essentially all the key building blocks of modern high-energy physics — bosonic and fermionic quantum gauge theory, gravity, an almost-commutative algebraic structure — emerges from a simple geometrical framework on a configuration space. This work raises many questions, some of the most important ones are:

  • does a gauge-covariant Hilbert space representation exist?
  • what does the construction look like when we use the Levi-Civita connection on the configuration space (we know that the Levi-Civita connection exists).
  • what does the time-evolution of the UV-regularisation look like?
  • we assume that the non-locality will be restricted to scales beyond the Planck scale. What happens to causality? The Lorentz symmetry?
  • can the conjectured connection to the standard model be substantiated? In particular, does it offer an answer to the question of why there are three particle generations?

With this, we end this blog-post; 

Johannes Aastrup 

Jesper Møller Grimstrup


[1] J. Aastrup and J. M. Grimstrup, “The Metric Nature of Matter” arXiv: 2008.09356.

[2] J. Aastrup and J. M. Grimstrup, “Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces," arXiv:1910.01841.

[3] J. Aastrup and J. M. Grimstrup, “Representations of the Quantum Holonomy-Diffeomorphism Algebra," arXiv:1709.02943.

[4] A. Connes, “Gravity coupled with matter and the foundation of noncommutative geometry," Commun. Math. Phys. 182 (1996) 155.

[5] A. H. Chamseddine and A. Connes, “Universal formula for noncommutative geometry actions: Unification of gravity and the standard model," Phys. Rev. Lett. 77 (1996) 4868.

[6] A. Connes, ”Noncommutative geometry and the standard model with neutrino mixing,'' JHEP 11 (2006), 081.

[7] A. H. Chamseddine, A. Connes and M. Marcolli, ”Gravity and the standard model with neutrino mixing,’' Adv. Theor. Math. Phys. 11 (2007) no.6, 991-1089.

[8] A. H. Chamseddine and A. Connes, “Why the Standard Model,’' J. Geom. Phys. 58 (2008), 38-47.

[9] A. H. Chamseddine, A. Connes and V. Mukhanov, “Geometry and the Quantum: Basics,’' JHEP 12 (2014), 098.

[10] A. H. Chamseddine, A. Connes and V. Mukhanov, “Quanta of Geometry: Noncommutative Aspects,’' Phys. Rev. Lett. 114 (2015) no.9, 091302.

[11]  J. Aastrup and J. M. Grimstrup, “C*-algebras of Holonomy- Diffeomorphisms and Quantum Gravity II”, J. Geom. Phys. 99 (2016) 10. 

[12]  J. Aastrup and J. M. Grimstrup, “The quantum holonomy-diffeomorphism algebra and quantum gravity,’' Int. J. Mod. Phys. A 31 (2016) no.10, 1650048.

[13] N. Higson and G. Kasparov, "E-theory and KK-theory for groups which act properly and isometrically on Hilbert space", Inventiones Mathematicae, vol. 144, issue 1, pp. 23-74.

Wednesday, October 14, 2020

Global Noncommutative Geometry Seminar on YouTube

 With YouTube and Zoom playing a much bigger role now in disseminating knowledge across the globe, it is only natural to have a YouTube channel dedicated to posting talks in an ongoing global noncommutative geometry seminar. The seminar is a collective effort with three nodes in Europe, Americas and Asia. There are many details that still has to be worked out and currently many people are working on that. Meanwhile we are glad that at least the YouTube channel is up and running!  Here it is


Tuesday, September 15, 2020


 We all learned with immense sorrow that Vaughan Jones died on Sunday September 6th.

I met him in the late seventies when he was officially a student of André Haefliger but contacted me as a thesis advisor which I became at a non-official level. I had done in my work on factors the classification of periodic automorphisms of the hyperfinite factor and Vaughan Jones undertook the task of classifying the subfactors of finite index of the hyperfinite factor among which the fixed points of the periodic automorphisms give interesting examples.
By generalizing an iterative construction which I had introduced he was first able to show that the indices of subfactors form the union of a discrete set with a continuum exactly as in conformal field theory. But his genius discovery was when he understood the link between his theory of subfactors and knot theory which is the geometry of knots in three space!
This is really a fantastic discovery that led to a new invariant of knots : the Jones polynomials!
This discovery was afterwards dressed using functional integrals but the real breakthrough is indisputably due to Vaughan Jones. 
To me his discovery is one of the great jewels of the unity of mathematics where a seemingly remote problem such as the classification of subfactors of finite index turned out to be deeply related to a fundamental geometric problem! 
For this reason I do not hesitate to affirm that Vaughan Jones' discovery is one of pure genius and that his work has all characteristic features that grant it immortality.

Friday, September 4, 2020

The Noncommutative Geometry Seminar

The Noncommutative Geometry Seminar

This is a virtual seminar on topics in noncommutative geometry, which is open to anyone anywhere interested in noncommutative geometry.

Starting September 22 we meet on Tuesdays at 15:00-16:00 (CET) via zoom (link will be distributed via mail)

Organizers: Giovanni Landi, Ryszard Nest, Walter van Suijlekom, Hang Wang

More infomation on the schedule, and also how to join the mailing please visit the website

Hope to see many of you soon!

Monday, February 3, 2020

Advances in Noncommutative Geometry

While noncommutative geometry is entering into its fifth decade, we are sure some of this blog's readers were thinking would be appropriate to have a volume of articles by experts looking at its main developments in the past 40 years and looking ahead.The book, Advances in Noncommutative Geometry, dedicated to Alain Connes' 70th birthday, published by Springer late last year, certainly fills this gap! Please take a look by checking Springer's book website linked above. See also further down the page for a brief introduction.  (Posted by Caterina Consani and Masoud Khalkhali). 

From the foreword to the book:

"Deeply rooted in the modern theory of operator algebras and inspired by two of the most influential mathematical discoveries of the 20th century, the foundations of quantum mechanics and the index theory, Connes' vision of noncommutative geometry echoes the astonishing anticipation of Riemann that ''it is quite conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry'' and accordingly ''we must seek
the foundation of its metric relations outside it, in binding forces which act upon it. The radically new paradigm of space proposed by Connes in order to achieve such a desideratum is that of a spectral triple, encoding the (generally non-commuting) spatial coordinates in an algebra of operators in Hilbert space, and its metric structure in an analogue of the Fermion propagator viewed as ``line-element.''

For the analytic treatment of such spaces, Connes devised the quantized calculus, whose infinitesimals are the compact operators, and where the role of the integral is assumed by the Dixmier trace. On the differential-topological side, Connes has invented a far-reaching generalization of de Rham's theory, cyclic cohomology which, in conjunction with KK-theory, provides the key tool for a vast extension of index theory to the realm of noncommutative spaces. Besides the wealth of examples of noncommutative spaces coming from physics (including space-time itself with its fine structure), from discrete groups, Lie groups (and smooth groupoids), with their rich K-theory, a whole class of new spaces can be handled by the methods of noncommutative geometry and in turn lead to the continual enrichment of its toolkit. They arise from a general principle, which first emerged in the case of foliations. It states that difficult quotients such as spaces of leaves are best understood using, instead of the usual commutative function algebra, the noncommutative convolution algebra of the associated equivalence relation.
An important such new space is the space of adele classes of a global field that Connes has introduced to give a geometric interpretation of the Riemann--Weil explicit formulas as a trace formula. The set of points of the simplest Grothendieck toposes are typically noncommutative spaces in the above sense and the adele class space itself, for the field of rationals, turns out to be the set of points of the scaling site, a Grothendieck topos which provides the missing algebro-geometric structure as a structure sheaf of tropical nature.

The pertinence and potency of the new concepts and methods are concretely illustrated in the contributions which make up this volume. They cover a broad spectrum of topics and applications, shedding light on the fruitful interactions between noncommutative geometry and a multitude of areas of contemporary research, such as operator algebras, K-theory, cyclic homology, index theory, spectral theory, geometry of groupoids and in particular of foliations. Some of these contributions stand out as groundbreaking forays into more seemingly remote areas, namely high energy physics, algebraic geometry, and number theory."