Conformal Field Theory and Geometry of Strings

TL;DR

The paper argues that quantum geometry of strings can be probed through conformal field theory in the framework of Connes' non-commutative geometry. It develops a program built from toroidal sigma-models, Wess-Zumino-Witten (WZW) theories and coset constructions, and their supersymmetric extensions, to extract finite non-commutative effective targets from low-energy (zero-mode) sectors. It demonstrates key dualities, notably T-duality for circle targets and mirror symmetry for complex tori, and shows how WZW/coset data yield deformed, dilatonic geometric structures that reduce to ordinary geometry in suitable limits. It further connects these constructions to Calabi-Yau geometry via N=2 superconformal symmetry, chiral rings, and mirror pairs, illustrating how non-commutative geometry provides a natural language for stringy geometric data beyond classical Riemannian geometry. Overall, the work lays a structural bridge between string geometry, non-commutative techniques, and Calabi-Yau dualities, advancing a program for understanding quantum geometry at large scales.

Abstract

What is quantum geometry? This question is becoming a popular leitmotiv in theoretical physics and in mathematics. Conformal field theory may catch a glimpse of the right answer. We review global aspects of the geometry of conformal fields, such as duality and mirror symmetry, and interpret them within Connes' non-commutative geometry. Extended version of lectures given by the 2nd author at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4 to 8, 1993