Instanton algebras and quantum 4-spheres

We study some generalized instanton algebras which are required to describe `instantonic complex rank 2 bundles'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with q\in\IC, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.Comment: 10 pages. Minor changes; final version for the journa

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 Podle\'s quantum sphere: equivariant representation, chiral grading $\gamma$, reality structure $J$ and the Dirac operator $D$, which has bounded commutators with the elements of the algebra and satisfies the first order condition.Comment: 10 pages, LaTeX, to appear in Banach Center Publicatio

Dirac operator on spinors and diffeomorphisms

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor fields on $M$ and a Hilbert space \HH_{\sigma, g}= L^2(S_{\sigma, g},\vol{M}{g}) of $L^2$-spinors of $S_{\sigma, g}$. The group \diff{M} of orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback) and on $[\sigma]$ (by a suitably defined pullback $f^*\sigma$). Any f\in \diff{M} lifts in exactly two ways to a unitary operator $U$ from \HH_{\sigma, g} to \HH_{f^*\sigma,f^*g}. The canonically defined Dirac operator is shown to be equivariant with respect to the action of $U$, so in particular its spectrum is invariant under the diffeomorphisms.Comment: 13 page

Spinors and Theta Deformations

The construction due to Connes and Landi of Dirac operators on theta-deformed manifolds is recalled, stressing the aspect of spin structure. The description of Connes and Dubois-Violette is extended to arbitrary spin structure.Comment: 10 pages, based on talks at "NCGOA", Nashville (US), May 2007 and "Geometry and Operators Theory", Ancona (Italy), September 200

A Finite Quantum Symmetry of M(3,C)

The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2,C))->A(SL_q(2))->A(F), q^3=1, is studied as a finite quantum group symmetry of the matrix algebra M(3,C), describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra H,investigated in a recent work by Robert Coquereaux, is established and used to define a representation of H on M(3,C) and two commuting representations of H on A(F).Comment: Amslatex, 17 pages, only Reference [DHS] modifie