• Non-commutative Geometry and its Applications
• Workshop: Quantum physics and non-commutative geometry

• Schedule
• Participants

The framework of Connes' noncommutative geometry provides a generalisation of ordinary Riemannian spin manifolds to noncommutative manifolds. Within this framework, the special case of a (globally trivial) almost-commutative manifold has been shown to describe a (classical) gauge theory over a Riemannian spin manifold, which ultimately led to a description of the full Standard Model of high energy physics, including the Higgs mechanism and neutrino mixing.

These gauge theories are, by construction, topologically trivial in the sense that the corresponding principal bundles are globally trivial bundles. We adapt the framework in order to allow for globally non-trivial gauge theories as well. We pay special attention to the form of the gauge groups and to the construction of gauge theories from these almost-commutative manifolds.

Based on joint work with Koen van den Dungen.

Connes' notion of non-commutative geometry (NCG) generalizes Riemannian geometry and yields a striking reinterepretation of the standard model of particle physics, coupled to Einstein gravity.  We suggest a simple reformulation with two key mathematical advantages: (i) it unifies many of the traditional NCG axioms into a single one; and (ii) it immediately generalizes from non-commutative to non-associative geometry.  Remarkably, it also resolves a long-standing problem plaguing the NCG construction of the standard model, by precisely eliminating from the action the collection of 7 unwanted terms that previously had to be removed by an extra, non-geometric, assumption.  With this problem solved, the NCG algorithm for constructing the standard model action is significantly tighter and more explanatory than the traditional one based on effective field theory.

We give an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes—Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group . We do so by proposing an abstract definition for such spectral triples, where noncommutativity is entirely governed by a class in the second group cohomology of the Pontrjagin dual of , and then showing that such spectral triples are well-behaved under further Connes—Landi deformation, thereby allowing for both quantisation from and dequantisation to -equivariant commutative spectral triples. If time permits, we will also discuss a discrete analogue of the Connes—Dubois-Violette splitting homomorphism, which then allows us to conclude that sufficiently well-behaved rational Connes—Landi deformations of commutative spectral triples are almost-commutative in the general, topologically non-trivial sense.

We compute the scalar curvature and prove the Gauss-Bonnet formula for families of Dirac operators on a noncommutative 2-torus, which are not (a priori) conformally related to "flat" Dirac operators.

Motivated by Dirac operators on Lorentzian manifolds, we propose a new framework to deal with non-symmetric and non-elliptic operators in noncommutative geometry. We provide a definition for indefinite spectral triples, and show that these correspond bijectively with certain pairs of spectral triples. Examples also include Euclidean spacetimes with harmonic oscillator potential. Next, I will show how a special case of indefinite spectral triples can be obtained from a Dirac-Schrödinger-type construction. In particular, this construction provides a convenient setting to study the Dirac operator on a foliated spacetime. This talk is based on ongoing joint research with Adam Rennie.

The asymptotic expansion of the spectral action at large energies is powerful tool for building models of fundamental interactions. For a suitable almost-commutative geometry it encodes the full lagrangian of the Standard Model minimally coupled to gravity. However, beyond the almost-commutative setting even the existence of an asymptotic expansion is obscure. In my talk I will formulate sufficient conditions for the existence of an asymptotic expansion of the spectral action using the tools of Laplace and Mellin transforms. I will give a precise formula, in terms of an asymptotic series, for the bosonic spectral action on general spectral triples. It contains terms logarithmic as well as oscillating with the energy scale. I will also discuss the convergence of the asymptotic expansion.

The preceding talk described a reformulation of Connes' non-commutative geometry (NCG), and some of its consequences for the NCG construction of the standard model of particle physics. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra gauge symmetry, and a single extra complex scalar field sigma, which is a singlet under but has . This field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction.

Lorentzian spectral triples are an attempt to adapt noncommutative geometry to Lorentzian signature using the notion of Krein space and fundamental symmetry. From the data given by the Dirac operator and the fundamental symmetry, a causal structure can be defined on the space of states. For a commutative Lorentzian spectral triples, this causal structure corresponds exactly to the one given by the Lorentzian metric. We will construct the causal structure of toy models based on almost-commutative algebras. We will see that the result is non-trivial with the existence of a "speed of light" constraint for the finite noncommutative algebra. Then, we will show that this approach could be adapted to recover the complete metric information. This is a joint work with Micha Eckstein.

Inspired by previous work by S. Brain, G. Landi and W. van Suijlekom, we study functorial deformations of algebras and modules based on actions of Abelian locally compact groups. We consider the case of , provide an explicit form for the deformation and show how functoriality can be used to prove invariance of periodic cyclic cohomology under the deformation (for the particular case of generalized crossed products). This talk is based on work in progress in collaboration with R. Meyer.

In this talk, we present the philosophy and the basic concepts of Noncommutative Supergeometry, i.e. Hilbert superspaces, C*-superalgebras and quantum supergroups. Then, we give examples of these structures coming from deformation quantization and we expose an application to renormalizable quantum field theory on the Moyal space.

The Yang-Mills instantons have several natural extensions to higher dimensions. This talk concerns one such instance, namely contact instantons, which arise in the study of a super Yang-Mills theory on 5-dimensional contact manifolds. The geometry transverse to the Reeb foliation turns out to be important in understanding the moduli space and in particular, the dimension of the moduli space is given by the index of a transverse elliptic complex. This is joint work with David Baraglia.

I will give examples of spectral triples constructed using the algebra of Toeplitz operators on smoothly bounded strictly pseudoconvex domains in Cn, or the star product for the Berezin-Toeplitz quantization. The main tool is the theory of generalized Toeplitz operators on the boundary of such domains, due to Boutet de Monvel and Guillemin. (joint work with Kevin Falk and Miroslav Englis)

It is known that there exists a strong connection between noncommutative geometry and gauge-invariant theories, due to the fact that gauge theories are naturally induced by the spectral triples. Thus it is reasonable to try to insert in the setting of noncommutative geometry also procedures which have been developed for the analysis of gauge theories. One of these is the BV approach to the BRST quantization of non-abelian gauge theories. In this talk we present a possible method to incorporate this approach in the framework of noncommutative geometry. We restrict to a U(2)-gauge invariant matrix model: through the introduction of a so-called BV-spectral triple we describe the minimally-extended theory, obtained by inserting the minimal number of ghost fields. An interesting aspect of this approach is that it gives a "geometric interpretation" for all the physical properties of the ghost fields such as their bosonic or fermionic character, which have a natural translation in terms of the spectraltriple itself.

The spectral action principle plays a predominant role in applications of noncommutative geometry to fundamental physics. In this talk I shall survey our recent preprint arXiv:1407.5972 [math-ph].  We use pseudodifferential calculus and heat kernel techniques to prove a conjecture of  Chamseddine and Connes on rationality of the coefficients of the polynomials in the cosmic scale factor a(t) and its higher derivatives. These coefficients describe the general Seeley-De Witt terms a2n in the expansion of the spectral action for general Robertson-Walker metrics. We also compute the terms up to a12 in the expansion of the spectral action by our method. As a byproduct, we verify that our computations agree with the terms up to a10 that were previously computed by Chamseddine and Connes by a different method. This is joint work with Farzad Fathizadeh and Asghar Ghorbanpour.

We generalize the notion, due to H. Cartan, of an operation of a Lie algebra in a graded differential algebra. Firstly, for such an operation we give a natural extension to the universal enveloping algebra of the Lie algebra and analyze all of its properties. Building on this we define the notion of an operation of a general Hopf algebra H in a graded differential algebra. We then introduce for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra as the universal initial object of the category of H-operations with connections.

The spectral approach to the standard model based on noncommutative geometry pionered by Connes and collaborators is now confronting itself with the new data coming from experiments. In particular it is now possible to check its predicitive power (with all due caveats) in view of recent experiments. I will discuss the aspects connected with the symmetries, the field content and the mass of the Higgs. In particular the role of higher symmetries beyond the standard model is treated.

K-Theory and its twisted version has appeared in different areas of theoretical physics, e.g. in quantum field theory via T-duality and recently in condensed matter physics as was pointed out by Kitaev. During the first part of the talk I would like to review some these applications. Twisted K-Theory is most elegantly expressed in terms of non-commutative geometry as K-Theory of section algebras of locally trivial bundles of compact operators. However, from the point of view of algebraic topology, this setup just captures a small portion of the possible twists. In the second half, I would like to report on joint work with Marius Dadarlat, in which we generalize the classical theory to a C*-algebraic model, which captures all possible twists of K-Theory as predicted by stable homotopy theory. It is based on strongly self-absorbing C*-algebras, which play a fundamental role in Elliott's classification program.

I'll demonstrate a class of models, to illustrate a principle of evolution for 3-dimensional noncommutative geometries, determined exclusively by

a spectral action. One particular case is a model, which allows evolution of noncommutativeness (deformation parameter) itself for a specific choice of the metric

Alain Connes' noncommutative geometry allows to unify the classical Yang-Mills-Higgs theory and General relativity in a single geometrical framework, so called almost-commutative geometries. This unification implies restrictions for the couplings of the Standard Model at a given cut-off energy which reduce the degrees of freedom compared to the classical Standard Model.

I will give an introduction to the basic ideas of almost-commutative model building and present models beyond the Standard Model that may be phenomenologically interesting. These models include extensions of the fermionic and the gauge sector as well as extensions of the scalar sector.

Starting with an algebra, we define a semigroup which extends the group of invertible elements in that algebra. As we will explain, this semigroup describes inner perturbations of noncommutative manifolds, and has applications to gauge theories in physics. We will present some elementary examples of the semigroup associated to matrix algebras, and to (smooth) functions on a manifold. Joint work with Ali Chamseddine and Alain Connes.

The analysis of heat trace asymptotics on NC torus performed by Gayral, Iochum and D.V. (CMP 273 (2007) 415) shows that the heat kernel coefficients are obtained from their standard "commutative" expressions by exchanging the trace to a new one. The same rule is valid for one-loop counterterms. Here we check whether this also applies to higher order diagrams in phi-4 theory on NC 4-torus and discuss implications for renormalizability. (Joint work with D. D'Ascanio and P. Pisani.)

Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form (D + sW) f = 0 are studied, where D is a normal or prenormal hyperbolic differential operator on Minkowski spacetime, s  is a coupling constant, and W is a regular integral operator with compactly supported kernel. In particular, W can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small s, the hyperbolic character of the PDE is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in s, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of W on the space of solutions of D.

It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces. This is joint work with Gandalf Lechner (see arXiv:1307.1780).

Together with Harald Grosse we showed that the quartic matrix model with an external matrix is exactly solvable in terms of the solution of a non-linear equation and the eigenvalues of that matrix. The self-coupled scalar model on Moyal space is of this type, and our solution leads to a non-perturbative construction of Schwinger functions.  After taking a suitable limit, the model satisfies growth properties, covariance and symmetry, and there is numerical evidence for reflection positivity of the 2-point function for a certain range of the coupling constant.