[Paper Review] Conformal Field Theory and Geometry of Strings

Jürg Fröhlich, Krzysztof Gawędzki|ArXiv.org|Oct 28, 1993

Black Holes and Theoretical Physics4 citations

TL;DR

This paper explores the geometric foundations of conformal field theory (CFT) in string theory, linking global CFT structures like duality and mirror symmetry to Alain Connes' non-commutative geometry. It proposes that non-commutative geometry provides a unifying framework for understanding quantum geometry in string theory, offering a conceptual bridge between algebraic structures in CFT and geometric invariants in quantum field theory.

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

Motivation & Objective

• To investigate the role of quantum geometry in conformal field theory and string theory.
• To understand how global geometric structures in CFT—such as duality and mirror symmetry—can be interpreted through the lens of non-commutative geometry.
• To provide a conceptual and mathematical framework connecting algebraic structures in CFT with geometric invariants using Connes' non-commutative geometry.
• To extend the understanding of string geometry beyond classical Riemannian geometry by incorporating non-commutative structures.
• To present a synthesis of global CFT properties within a non-commutative geometric formalism, offering new insights into quantum gravity and string compactifications.

Proposed method

• Utilizes the framework of Connes' non-commutative geometry to reinterpret global structures in conformal field theory.
• Analyzes duality and mirror symmetry in CFT as manifestations of non-commutative geometric invariants.
• Applies algebraic and topological tools from non-commutative geometry to classify and interpret CFT partition functions and correlation functions.
• Draws analogies between non-commutative spaces and the moduli spaces of conformal field theories.
• Employs the operator algebraic approach to CFT, focusing on representations and symmetries in terms of non-commutative algebras.
• Integrates insights from mathematical physics and differential geometry to unify geometric and algebraic structures in CFT.

Experimental results

Research questions

• RQ1 How can non-commutative geometry serve as a unifying framework for understanding quantum geometry in conformal field theory?
• RQ2 In what way do duality and mirror symmetry in CFT emerge from non-commutative geometric principles?
• RQ3 What is the role of non-commutative spaces in describing the moduli space of conformal field theories?
• RQ4 How do global topological and algebraic structures in CFT relate to non-commutative invariants?
• RQ5 Can the geometric structures of string compactifications be naturally described using non-commutative geometry?

Key findings

• Duality and mirror symmetry in conformal field theory are shown to arise naturally from non-commutative geometric structures.
• The paper establishes a conceptual link between the algebraic data of CFT—such as operator product expansions and modular invariance—and non-commutative spaces.
• Non-commutative geometry provides a natural language for describing quantum geometric phenomena in string theory beyond classical Riemannian geometry.
• The authors demonstrate that global invariants in CFT, such as partition functions, can be interpreted as traces over non-commutative algebras.
• The framework suggests that quantum geometry in string theory may be inherently non-commutative, with CFT serving as a testing ground for such ideas.
• The extended lectures present a coherent picture where non-commutative geometry offers a deeper understanding of the geometric underpinnings of string theory.