Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields

Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres

The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes non-commutative. Here we explore the quantization of world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study the example of open strings in the SU(2) Wess-Zumino-Witten model, and establish a relation with fuzzy spheres or certain (non-associative) deformations thereof. These findings could be of direct relevance for D-branes in the presence of Neveu-Schwarz 5-branes; more importantly, they provide insight into a completely new class of world-volume geometries.Comment: 19 pages, LaTeX, 1 figure; some explanations improved, references adde