Mathematical models for noncommutativity in physics and quantum spacetime

02.11.2017 - 03.11.2017 | Warsaw

Abstracts & Slides


Joakim Arnlind, A noncommutative catenoid
The concept of a minimal surface has started to make its way into noncommutative geometry. A quite natural approach, as taken by some authors, is to consider a variational problem and its corresponding minimizers. In contrast, our approach focuses on the equations that the embedding coordinates of a minimal surface have to satisfy; for instance, in the case of a surface embedded in Euclidean space, the embedding coordinates have to be harmonic. An advantage of this approach is that one can easily construct noncommutative analogues of many classical minimal surfaces. In this talk, I will consider a particular example; namely, a noncommutative analogue of the catenoid. It turns out to have many interesting properties which in some sense are similar to that of the noncommutative torus. A derivation based calculus will be introduced together with a Levi-Civita connection and its corresponding curvature. At the end, I will present classes of bimodules and connections of constant curvature.

Fabien Besnard, The signature problem in noncommutative geometry
Alain Connes' spectral triples theory is a far-reaching generalization of riemannian geometry, which throws light on the Standard Model of particle physics, if only for its euclidean version. The search for a semi-riemannian generalization is still work in progress, and we will review some aspects of it, focusing on the seemingly simple question: if we are to define lorentzian noncommutative geometry, what is the meaning of 'lorentzian' ?"

Ludwik Dabrowski, On noncommutative geometry of the Standard Model
Acccording to Connes the topology and metric structure of a spin manifold can be encoded in terms of the canonical spectral triple, of which the crucial ingredient is the Dirac operator on Dirac spinors. Another natural spectral triple on a manifold incorporates the Hodge-de Rham operator acting on differential forms. Both of them can be characterised in terms of certain Morita equivalence, which involves the Clifford algebra. These notions admit a generalization to noncommutative algebras, and in particular we can establish of which type is the internal finite spectral triple in the almost commutative formulation of the Standard Model of fundamental particles and their interactions.

Daniele Guido, Spectral triples for (noncomutative) fractals
We review some possible ways of constructing spectral triples on self-similar fractals as direct sums of suitable building blocks, discussing how and when some classical results on fractals can be recovered from the spectral triples. For the discrete triple on the Sierpinski gasket, previous construction can be formulated from a functional point of view, giving an algebraic description of self-similarity in terms of Bratteli diagrams. This formulation provides a way to quantize the gasket, namely to produce a self-similar noncommutative C*-algebra containing the continuous functions on the gasket as a sub-algebra. The representations of this algebra are studied, it is shown that a noncommutative Dirichlet form can be dened, which restricts to the classical energy form on the gasket, and a spectral triple is proposed. Such triple reconstructs in particular the Dirichlet form. Work in progress with F.Cipriani, T.Isola e J-L.Sauvageot.

Giovanni Landi, Noncommutative products of Euclidean spaces
We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are parametrised by a two-dimensional sphere. Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". There is invariance for an action of $SU(2) \times SU(2)$ in parallel with the action of $U(1) \times U(1)$ on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.

Shahn Majid, Some finite quantum Riemannian geometries in a bimodule approach
I outline a formulation of quantum Riemannian geometry using bimodule connections. I give some examples in the finite case where it is possible for a fixed differential algebra to  construct the moduli of all possible quantum Riemannian geometries on it, as a step towards quantum gravity. In particular, this will be solved for the geometry of a square graph, including a reasonable Einstein-Hilbert action for the quantum metric as given by lengths assigned to the edges

Alexander Schenkel,  Categorical techniques for NC geometry and gravity
To test whether an algebra is non-commutative, one has to compare its product with its opposite product, i.e. one studies the commutator. Forming the opposite product, however, depends on the choice of a “rule” (called braiding) to exchange the order of two tensor factors. I will show that many examples of NC algebras can be reinterpreted as braided-commutative algebra objects in suitable braided monoidal categories. This point of view enables us to develop techniques for doing geometry on such algebras by working internally in the relevant categories. As an example, I will explain in more detail the theory of connections on suitable bimodule objects and in particular how to lift connections to tensor products of bimodules. I will also sketch some applications to NC Cartan geometry and vielbein gravity.

Andrzej SitarzQuotients of noncommutative tori and smoothness.
Quotient spaces of actions of finite groups on the tori provide interesting examples of flat manifolds and orbifolds. In some cases the actions could also be defined on the noncommutative tori and the fixed point subalgebras give their noncommutative counterparts. I'll discuss the three-dimensional examples (noncommutative Bieberbach manifolds) and two-dimensional (noncommutative pillow) adressing the question whether the resulting objects are manifolds or orbifolds. Based on joint work with P.Olczykowski (Bieberbachs) and T.Brzezinski (pillow).


Adam Skalski, On constructions of compact quantum groups.
Initial developments in the theory of topological quantum groups were motivated on one hand by the desire to extend the classical Pontryagin duality for locally compact abelian groups to a wider class of objects and on the other by the idea of replacing the study of a space by the investigation of the algebra of functions on it. In 1980s the theory was given a big boost by the discovery of a big class of examples arising as deformations of classical compact Lie groups and the resulting conceptual progress, mainly due to Woronowicz. In this talk we will describe this background and then finally present two approaches to constructions of quantum groups developed in the last decade, leading to so-called quantum symmetry groups and liberated quantum groups. They turn out to produce very interesting examples and offer connections to noncommutative geometry and free probability. We will also discuss some examples of questions motivated purely by `topological' aspects of the theory.

Harold Steinacker, Fuzzy spaces and space-times from Yang-Mills matrix models
I will first recall how fuzzy spaces arise as solutions of Yang-Mills matrix models.  The fluctuation modes are discussed, and the effective metric is identified.  In the second part, we focus on covariant 4-dimensional fuzzy spaces, including  candidates for cosmological space-times featuring a "Big Bang" and finite density of microstates.

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