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

TL;DR

This paper explores the analogy between collapsing conformal field theories and geometric limits of Riemannian manifolds, proposing a new framework of non-commutative Riemannian geometry linked to spectral triples.

Contribution

It introduces the concept of non-commutative Riemannian geometry inspired by the analogy with geometric limits and relates it to spectral triples, extending the mathematical framework of QFTs.

Findings

Proposes non-commutative Riemannian geometry as a new framework.

Connects quantum geometry with spectral triples in dimension one.

Discusses deformation theory of Quantum Field Theories.

Abstract

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 In the Appendix based on the unpublished joint work with Maxim Kontsevich we discuss deformation theory of Quantum Field Theories as well as an approach to QFTs in the case when the space-time is an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.