Photo: Cristiano de Nobili

## Program

**ABSTRACTS**

**Paolo Antonini**

Index theory on manifolds with boundary and noncommutative geometry

Index theory on manifolds with boundary and noncommutative geometry

I will discuss the theory of boundary value problems for Dirac operators with special focus towards its noncommutative generalizations. As a special noncommutative instance I will present my results concerning the index (and signature formula) on foliated manifolds with boundary.

**Francesco D’Andrea**

The Standard Model in Noncommutative Geometry and Morita equivalence

The Standard Model in Noncommutative Geometry and Morita equivalence

After a brief review of the spectral action approach to the Standard Model of particle physics, I will discuss some properties of the finite-dimensional spectral triple describing the internal degrees of freedom of elementary parti- cles. On a Riemannian spin manifold M, an algebraic characterization of the module of Dirac spinors (sections of the spinor bundle) is as the Morita equiv- alence bimodule between the algebra of continuous functions on M and the Clifford algebra bundle. In the case of Hodge-Dirac operator on a oriented Riemannian manifold, on the other hand, the module of Hodge spinors can be characterized as self Morita equivalence bimodule for the Clifford algebra bundle. Both conditions admit a natural generalization to non-commutative manifolds and impose some constraints on the form of the Dirac operator. I will report on a recent work with L. Dabrowski and A. Sitarz, where we investigate such constraints for the spectral triple of the Standard Model of elementary particles.

**Paolo Aschieri**

Symmetries, observables and dispersion relations in noncommutative spacetimes

Symmetries, observables and dispersion relations in noncommutative spacetimes

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are the generators of infinitesimal transformations and are identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator.

This general noncommutative geometry construction is then exemplified in the case of κ-Minkowski spacetime. The d’Alembert operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta, i.e. infinitesimal quantum translations, are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum dispersion relations are deduced and found undeformed for massless particles. This is no more the case when we turn on a nontrivial metric background.

**Fabien Besnard**

Spacetimes and Noncommutative Geometry

Spacetimes and Noncommutative Geometry

In this talk we will propose, after many others, an extension of the notion of even spectral triple to the antilorentzian (mostly minus) signature. Our approach will differ only on a few points from previous proposals but these will turn out to have important consequences. In particular we will require the fundamental symmetry to be a 1-form, and not require it to commute with the algebra. This last point will allow us to put forward an elementary example which is not covered by existing frameworks. On the other hand it forces us to drop the C*-structure of the algebra as a basic object of the theory. If some "reconstructibility condition" is met, a whole family of C*-structures can be recovered, corresponding in the classical case to the different foliations of the manifold by timelike curves (congruences of observers). A finite-dimensional and noncommutative example which is related to the discretization of the spinor bundle will be given, in which the reconstructibility condition can be shown to be equivalent to the existence of a covariantly constant timelike vector field.

**Michel Dubois-Violette**

Exceptional quantum geometry and particle physics

Exceptional quantum geometry and particle physics

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the ``finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. More generally it is is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of ``the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras. We formulate the corresponding definition of connections on Jordan modules.

**Shane Farnsworth**

Jordan Algebras, Particle Physics, and Non-commutative Geometry

Jordan Algebras, Particle Physics, and Non-commutative Geometry

In this talk I will discuss the generalization of Connes' non-commutative geometry (NCG) to non-associative geometry. I will focus in particular on Jordan algebras, and motivate a reformulation of the finite space of the NCG standard model of particle physics in terms of Jordan algebras.

**Nicolas Franco**

Physical models from noncommutative causality

Physical models from noncommutative causality

We introduced few years ago a new notion of causality for noncommutative spacetimes directly related to the Dirac operator and the concept of Lorentzian spectral triple. This notion of causality corresponds to the usual one for commutative spectral triples and could be extended in order to get a full Lorentzian metric. We explored the noncommutative causal structure of several toy models as almost commutative spacetimes and Moyal-Weyl spacetime. From those models, we discovered some unexpected physical interpretations as a geometrical explanation of the ‘Zitterbewegung' trembling motion of a fermion and geometrical constraints on translations and energy jumps of wave packets on Moyal spacetime.

**Harald Grosse**

Exact solution of quantum field theory toy models

Exact solution of quantum field theory toy models

Matrix models share all interesting features of quantum field theory: A graphical description with Feynman rules, power counting dimension, regularisation and renormalisation, divergence of the perturbation series.

We report on models, where much more is possible: We give exact non-perturbative formulae for all renormalised correlation functions. We describe a map, which projects these matrix correlation functions to Schwinger functions of an ordinary quantum field theory. The Schwinger 2-point functions satisfies in some of these models the Osterwalder-Schrader axioms.

These results were obtained in joint work with Raimar Wulkenhaar and partly with Akifumi Sako.

**Tajron Jurić**

Effects of Noncommutativity on the Black Hole Entropy and QNM

Effects of Noncommutativity on the Black Hole Entropy and QNM

The topic of this talk is based on probing the BTZ black hole geometry with a noncommutative field which obeys the κ-Minkowski algebra. The entropy of the BTZ black hole can be obtained using the brick wall method. It will be argued that the correction to the black hole entropy can be interpreted as arising from the renormalization of the Newton's constant due to the effects of the noncommutativity.

I will also present an expression for the quasinormal modes of a noncommutative massless scalar field in the background of a spinless BTZ black hole up to the first order in the deformation parameter. It can be shown that the equations of motion governing these quasinormal modes are identical in form to the equations of motion of a commutative massive scalar field in the background of a fictitious massive spinning BTZ black hole. This results hints at a duality between the commutative and noncommutative systems in the background of a BTZ black hole.

The fermionic quasinormal modes of the BTZ black hole in the presence of spacetime noncommutativity will also be discussed. Our analysis exploits a duality between a spinless and spinning BTZ

black hole, the spin being proportional to the noncommutative deformation parameter. Using the

AdS/CFT correspondence we show that the horizon temperatures obtained from the dual CFT pick

up noncommutative contributions. We demonstrate the equivalence between the quasinormal and

non-quasinormal modes for the noncommutative fermionic probes, which provides further evidence

of holography in the noncommutative setting.

**Sasa Krešić-Jurić**

Differential calculus on quantum spaces via deformation theory

In this talk we discuss realizations of Lie superalgebras as formal power series in a Weyl superalgebra.

Differential calculus on quantum spaces via deformation theory

We use these realizations to construct a bicovariant differential calculus on noncommutative spaces

of the Lie algebra type as a deformation of the standard differential caluclus on the Euclidean space.

Examples are given for the kappa-deformed space.

**Giovanni Landi**

Line bundles : commutative and noncommutative

Line bundles : commutative and noncommutative

**Fedele Lizzi**

Dimensional Deception from Noncommutative Tori

Dimensional Deception from Noncommutative Tori

While it may appear evident that we live in four dimensions, renormazability of field theories with gravity work in two dimensions, and this has given rise to anisotrpic models, lile the one proposed by Horava. This talk, work in collaboration with Pinzul, presents a model based on the noncommutative torus, for which the number of dimensions is four in the infrared, but only two in the ultraviolet.

**Pierre Martinetti**

Gauge theory from twisted spectral triples

Gauge theory from twisted spectral triples

Recently, it has been shown how a twist (in the sense of Connes, Moscovici) of the spectral triple of the standard model

yields a model « beyond the standard model », including the extra-scalar field required to stabilize the electroweak vacuum

and obtain the correct mass for the Higgs boson. We shall show how this example actually fits a generic procedure

to twist any graded spectral triple, and how gauge transformations are implemented in the twisted context.

**Chiara Pagani**

Twist deformation of quantum principal bundles

Twist deformation of quantum principal bundles

Principal bundles are described in noncommutative geometry in terms of algebra extensions that satisfy the condition to be Hopf-Galois. In this setting, the groups of symmetries of the bundle (structure group, groups of bundle automorphisms,...) are described via Hopf algebras and their coactions.

I will report on a joint work with P. Aschieri, P. Bieliavsky and A. Schenkel in which we study the behaviour of Hopf-Galois extensions under Drinfeld twist deformation and prove

we can canonically deform Hopf-Galois extensions into new Hopf-Galois extensions by using twists on different Hopf algebras of symmetries. These results also extend to the case of sheaves of Hopf-Galois extensions.

**Timothé Poulain**

Involutive representations of coordinate algebras and quantum spaces

Involutive representations of coordinate algebras and quantum spaces

We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently represented by $SO(3)$-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for $SU(2)$. Selecting a subfamily of $^*$-representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra $\mathfrak{su}(2)$. We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.

**Amilcar Queiroz**

Matrix Model for QCD: Color States are Mixed

Matrix Model for QCD: Color States are Mixed

Abstract: Gribov's observation that global gauge fixing is impossible has led to suggestions that there may be a deep connection between gauge fixing and confinement. We propose to approximate QCD by a rectangular matrix model that captures the essential topological features of the gauge bundle, and demonstrate the impure nature of coloured states explicitly. Our matrix model also allows the inclusion of the QCD theta-term, as well as to perform explicit computations of low-lying global masses, which we show is gapped. Since an impure state cannot evolve to a pure one by a unitary transformation, our result shows that the solution to the confinement problem is fundamentally quantum information nature.

**Mairi Sakellariadou**

Group Field Theory Condensate Cosmology

Group Field Theory Condensate Cosmology

I will first introduce Group Field Theory (GFT), a nonperturbative approach to quantum gravity, and then presentaspects of Group Field Theory Condensate Cosmology.

In particular, I will discuss the impact of nonlinear effective interactions on GFT quantum gravity condensates and then present results on the relational evolution of such condensates. I will show the emergence of a cyclic universe and the onset of an early era of accelerated expansion without having to introduce an inflaton filed with a fine tuned potential.

**Andrzej Sitarz**

Gravity on almost noncommutative spaces

Gravity on almost noncommutative spaces

I'll discuss gravity action on almost noncommutative manifolds (described by algebra of functions valued in finite-dimensional noncommutative algebras).

**Z. Škoda:**

Gluing quantum principal bundles

Gluing quantum principal bundles

A number of examples of principal and associated bundles with the gauge group replaced by a quantum group is known in the literature.

They may carry a noncommutative analogue of gauge fields. Quantum bundles, possibly with connection, can be glued and appropriate analogue of Cech cocycles will be presented.

I will present several algebraic variants of gluing noncommutative bundles, using noncommutative localization, algebraic ideals as well as via unifying apparatus of corings.

**Koen van den Dungen**

Families of spectral triples and foliations of space(time)

Families of spectral triples and foliations of space(time)

We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces. First, we consider a spacetime given by a family of spacelike hypersurfaces (M,g_t) parametrised by the real line. We then construct a 'product spectral triple' from the corresponding family of canonical spectral triples over M and the lapse function. This product spectral triple is closely related to the internal Kasparov product of the family of spectral triples over M with the standard spectral triple over the real line, and it reproduces the canonical spectral triple of the product manifold equipped with the 'Wick rotated' Riemannian metric. The canonical Lorentzian Dirac operator can then be obtained as the 'reverse Wick rotation' of the Riemannian Dirac operator obtained from our product construction.

For the noncommutative scenario, we show that the construction of the product spectral triple also works in the case of a family of abstract spectral triples parametrised by the real line. Motivated by the classical case, we can then construct abstract 'Lorentzian spectral triples' as the reverse Wick rotation of such product spectral triples.

**Thomas Weber**

Obstructions for Twist Star Products

Obstructions for Twist Star Products

In this talk we will discuss obstructions for star products that can be induced by Drinfel’d twists on Poisson manifolds. Examples include the symplectic two sphere and the higher genus Riemann surfaces. One aim of Deformation Quantization is to replace the commutative pointwise product on the Poisson algebra of smooth functions by a star product, i.e. a noncommutative product on its formal power series. This can be interpreted as a quantization of the classical observables of a physical system. Thus we require the noncommutative product to coincide with the original product in the classi- cal limit. Furthermore, the correspondence principle ensures that the first order of the commutator with respect to the new product coincides with the Poisson bracket of the system. This is motivated by the canonical commutation relations. In the 1980’s Drin- fel’d introduced a special kind of Deformation Quantization of a physical system via its symmetries: a Drinfel’d twist on a Hopf algebra does not only provide a noncocommu- tative comultiplication but also a Deformation Quantization of any module algebra of the Hopf algebra. Moreover, the star product can explicitly be written in terms of the twist. As a consequence, the Poisson bivector is the image of a solution of the classical Yang-Baxter equation under a Lie algebra action. However, it is the goal of the talk to show that this desirable situation can not be obtained in general. In fact, it is quite rare. We state conditions on Poisson manifolds to inhabit twist star products and give concrete obstructions, e.g. for the symplectic two sphere and the higher genus Riemann surfaces. These results were obtained in joint work with P. Bieliavsky, C. Esposito and S. Waldmann.