Andrzej Sitarz

The Dirac operator on SU(q)(2)

We construct a 3+-summable spectral triple over the quantum group SUq(2) which is equivariant with respect to a left and a right action of The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

Noncommutative differential calculus on the κ-Minkowski space

Abstract Following the construction of the κ-Minkowski space from the bicrossproduct structure of the κ-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study a toy example of 2D κ-Minkowski space and we briefly discuss the main properties of its differential calculi.

Discrete spectral triples and their symmetries

We classify zero-dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure. We investigate also whether such spectral triples admit a symmetry arising from the Hopf algebra structure of the finite algebra. We discuss examples of commutative algebras and group algebras.

The Dirac operator on SU_q(2)

We construct a 3+-summable spectral triple over the quantum group SUq(2) which is equivariant with respect to a left and a right action of The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

Dirac operator on the standard Podles quantum sphere

Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podleś quantum sphere: equivariant representation, chiral grading γ, reality structure J and the Dirac operator D, which has bounded commutators with the elements of the algebra and satisfies the first order condition. Mathematics Subject Classification: Primary 58B34; Secondary 17B37.

The local index formula for SUq(2)

We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.

Noncommutative geometry and gauge theory on discrete groups

We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We study the metric properties of the models and introduce the action functionals for unitary gauge theories. A detailed analysis of two simple models based on Z2 and Z3 follows. Finally we discuss briefly the models with additional symmetries.

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.

Spectral action on noncommutative torus

The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.

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).