Moduli space of Conformal Field Theories and non-commutative Riemannian geometry

We discuss the analogy between collapsing Conformal Field Theories and measured Gromov-Hausdorff limit of Riemannian manifolds with non-negative Ricci curvature. Motivated by this analogy we propose the notion of non-commutative (``quantum") Riemannian d-geometry. We explain how this structure is related to Connes’s spectral triples in the case d=1. In the Appendix based on the unpublished joint work with Maxim Kontsevich we discuss deformation theory of Quantum Field Theories as well as an approach to QFTs in the case when the space-time is an arbitrary compact metric space.