Noncommutative Geometry and Physics
Seminar series and special issue of Journal of Physics A

Noncommutative geometry is a very general mathematical paradigm arising from quantum mechanics. As such, it permeates different branches of mathematics and physics.

The series of monthly talks accompanies the special issue of Journal of Physics A and is intended to present the many facets of the emergence of noncommutativity in physics.

The seminar series is organized monthly, last Mondays of a month at 17:00 CEST (Central European Summer Time).

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PAST TALKS

Anna Pachoł
Twisted differential geometry and dispersion relations in κ-noncommutative cosmology. One of the most studied possible phenomenological effect of quantum gravity is the modifications in wave dispersion. Thanks to the noncommutative deformations of wave equations in curved backgrounds we can investigate the propagation of waves in noncommutative cosmology and consider the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. In the talk, I will follow the twisted differential geometry approach, give an overview of this framework and then focus on the results obtained by the Jordanian twist. The corresponding noncommutative spacetime is kappa-Minkowski considered in the presence of Friedman-Lemaitre-Robertson-Walker (FLRW) cosmological background.
The Euclidean contour rotation in quantum gravity. The talk will discuss the rotation of the contour of functional integration in quantum gravity from Lorentzian geometries to Euclidean geometries. In the usual framework of metric tensors, the functional integral does not have a good definition and so the formulas are necessarily heuristic. However, it is hoped that these formulas will provide exact mathematical results when applied to theories that are constructed with a fundamental Planck scale cut-off.
John Barrett
The Euclidean contour rotation in quantum gravity. The talk will discuss the rotation of the contour of functional integration in quantum gravity from Lorentzian geometries to Euclidean geometries. In the usual framework of metric tensors, the functional integral does not have a good definition and so the formulas are necessarily heuristic. However, it is hoped that these formulas will provide exact mathematical results when applied to theories that are constructed with a fundamental Planck scale cut-off.
Alessandra Frabetti
Noncommutative renormalization Hopf algebras. In pQFT, the renormalization group acts on the Lagrangian as a group of formal diffeomorphisms in the powers of the coupling constant, by substitution of the bare coupling and multiplication by some renormalization factors built on the counterterms of divergent Feynman graphs. For scalar theories, such groups are proalgebraic (functorial on the coefficients algebra) and are represented by Faà di Bruno types of Hopf algebras on graphs, called renormalization Hopf algebras. In this talk I review Connes-Kreimer's settings and comment on the improvements expected for the BPHZ formula which computes the counterterms of the graphs. For non-scalar theories, Feynman graphs have matrix-valued amplitudes: even if the counterterms are scalar-valued, the renormalization group cannot be represented by a Hopf algebra in a functorial way, because associativity fails for the composition of series with non-commutative coefficients. Both commutative and noncommutative renormalization Hopf algebras can be defined, with different meanings. In this talk I explain in which sense the first ones are not functorial (hence not universal) and how the second ones require a functorial extension of proalgebraic groups to non-commutative algebras which can only be done as "non-associative" groups. The talk is based on Connes-Kreimer's results (2000), on joint works with Christian Brouder (2000-2006) and on the recent paper doi.org/10.1016/j.aim.2019.04.053

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Raimar Wulkenhaar
From noncommutative field theory towards topological recurrence. Finite-dimensional approximations of noncommutative quantum field theories are matrix models. They often show rich mathematical structures: many of them are exactly solvable or even related to integrability, or they generate numbers of interest in enumerative or algebraic geometry. For many matrix models, it was possible to prove that they are governed by a universal combinatorial structure called Topological Recursion. The probably most beautiful example is Kontsevich's matrix Airy function which computes intersection numbers on the moduli space of stable complex curves. The Kontsevich model arises from a $\lambda\Phi^3$-model on noncommutative geometry. The talk addresses the question which structures are produced when replacing $\lambda\Phi^3$ by $\lambda\Phi^4$. The final answer will be that $\lambda \Phi^4$ obeys an extension of topological recursion. We encounter numerous surprising identities on the way.

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Richard Szabo
A new look at symmetries in noncommutative field theory. I will describe a new class of noncommutative field theories, building on many older works in the literature, which possess 'braided gauge symmetries'. Their construction is motivated by recent attempts to relieve the constraints imposed by conventional star-gauge symmetries and their tension with twisted diffeomorphisms, and by the modern perspective on classical field theories based on homotopy algebras. I will review all of the necessary background, focusing on the case of diffeomorphism invariant theories for illustration. As an example, I will show how these considerations lead to a new theory of noncommutative gravity in four dimensions within the Einstein-Cartan-Palatini formalism.

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Emil Prodan
Topological Insulators at Strong Disorder. Topological insulators display two remarkable properties. Firstly, they are genuine thermodynamic phases, i.e. they are separated by sharp phase boundaries where Anderson’s localization length diverges. Secondly, when two distinct topological phases are interfaced, wave propagation is enabled along the interface, which cannot be suppressed by disorder. In the first part of the talk, I will exemplify these phenomena with exactly solvable models, of which one with disorder, as well as with numerical simulations. In the second part of the talk, I will show how index theorems generated with Alain Connes’ quantized calculus explain both remarkable properties mentioned above.

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Stefan Waldmann
Convergence of Star Products: Examples and Concepts. In usual formal deformation quantization one considers formal deformations of the algebra of functions on a Poisson manifold viewing them as observables of a mechanical system whose quantum version one is interested in. However, there is yet another interpretation in terms of noncommutative geometry: the noncommutative products can be viewed as models of noncommutative manifolds, which, in turn, can be used for describing space-time geometry at small distances etc. While formal deformation quantization has very general existence and classification results by Kontsevich's formality theorem, it lacks the immediate applicability to physical problems: the deformation parameter (e.g. Planck's constant or the Planck length etc.) are formal only. Thus the understanding of the (non-) convergence of the formal series is one of the most important issues if one is interested in finding more realistic models beyond an "infinitesimal" deformation. Here in the recent years several classes of examples have been discussed. In my talk I will report on some of these examples illustrating the underlying geometry as well as some of the quite involved functional-analytic questions.

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Shahn Majid
Quantum groups and quantum spacetime models at the Planck scale. What can quantum groups and noncommutative geometry plausibly tell us about Planck scale physics? I will review key ideas and the current state of the art as I see it. In general terms, since x,p do not commute in quantum mechanics, and covariant momenta do not commute on a curved space, likewise by `Born reciprocity’ we should not expect the positions $x$ to commute either. This suggests the `quantum spacetime hypothesis’ that quantum gravity effects are better modelled by allowing spacetime to have noncommutative or `quantum’ coordinates. Early models in the late 1980s were based on quantum groups either as self-dual paradigms with observable-state/quantum-Born reciprocity, or as Poincare symmetries of quantum Minkowski spacetimes (leading to the bicrossproduct family of quantum groups). By now, there is a general framework of quantum Riemannian geometry which allows a new generation of models, with discrete and/or curved quantum spacetimes and including cosmological and black hole models. I will outline the formalism and some of the common features and issues to date, as well as first applications to baby models of quantum gravity itself.

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