Ludwik Dąbrowski

Strong Connections and Chern-Connes Pairing in the Hopf-Galois Theory

Abstract: We reformulate the concept of connection on a Hopf–Galois extension B⊆P in order to apply it in computing the Chern–Connes pairing between the cyclic cohomology HC2n (B) and K0 (B). This reformulation allows us to show that a Hopf–Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz–Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern–Connes pairing for the super line bundles associated to a super Hopf fibration.

Dirac operators on all Podles quantum spheres

We construct spectral triples on all Podles quantum spheres S 2 qt . These noncom- mutative geometries are equivariant for a left action of Uq(su(2)) and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S 2 . There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.

Spinors and diffeomorphisms

We discuss the action of diffeomorphisms on spinors on an oriented manifoldM. To do this, we first describe the action of the diffeomorphism groupD(M) on the set Π =H1 (M,Z2) of inequivalent spin structures and show that it is affine. We argue that in the presence of spinors the gauge group of gravity is a certain double cover ofD(M) which depends on the spin structure. We explicitly compute the action ofD(M) on Π whenM is a closed Riemann surface; Π is seen to consist of exactly two orbits, corresponding to even and odd spin structures.

Curved noncommutative torus and Gauss–Bonnet

We study perturbations of the flat geometry of the noncommutative two-dimensional torus Tθ2 (with irrational θ). They are described by spectral triples (Aθ,H,D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra Aθ of Tθ. We show, up to the second order in perturbation, that the ζ-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.

An Asymmetric Noncommutative Torus

We introduce a family of spectral triples that describe the curved noncommuta- tive two-torus. The relevant family of new Dirac operators is given by rescaling one of two terms in the flat Dirac operator. We compute the dressed scalar curvature and show that the Gauss{Bonnet theorem holds (which is not covered by the general result of Connes and Moscovici).

The κ‐Minkowski Spacetime: Trace, Classical Limit and Uncertainty Relations

Starting from a discussion of the concrete representations of the coordinates of the κ‐Minkowski spacetime (in 1+1 dimensions, for simplicity), we explicitly compute the associated Weyl operators as functions of a pair of Schrodinger operators. This allows for explicitly computing the trace of a quantised function of spacetime. Moreover, we show that in the classical (i.e. large scale) limit the origin of space is a topologically isolated point, so that the resulting classical spacetime is disconnected. Finally we show that there exist states with arbitrarily sharp simultaneous localisation in all the coordinates; in other words, an arbitrarily high energy density can be transferred to spacetime by means of localisation alone, which amounts to say that the model is not stable under localisation.

Instantons on the Quantum 4-pheres S4q

Abstract: We introduce noncommutative algebras Aq of quantum 4-spheres S4q, with q∈ℝ, defined via a suspension of the quantum group SUq(2), and a quantum instanton bundle described by a selfadjoint idempotent e∈ Mat4(Aq), e2=e=e*. Contrary to what happens for the classical case or for the noncommutative instanton constructed in [8], the first Chern–Connes class ch1(e) does not vanish thus signaling a dimension drop. The second Chern–Connes class ch2(e) does not vanish as well and the couple (ch1(e), ch2(e) defines a cycle in the (b,B) bicomplex of cyclic homology.

Twisted Reality Condition for Dirac Operators

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones, we propose a new twisted reality condition for the Dirac operator.

The Standard Model in noncommutative geometry: fundamental fermions as internal forms

Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra.

Towards a noncommutative Brouwer fixed-point theorem

Abstract We present some results and conjectures on a generalization to the noncommutative setup of the Brouwer fixed-point theorem from the Borsuk–Ulam theorem perspective.