[Paper Review] Supersymmetric quantum theory, non-commutative geometry, and gravitation. Lecture Notes Les Houches 1995

Juerg Froehlich, Olivier Grandjean+1 more|ArXiv.org|Jun 18, 1997

Black Holes and Theoretical Physics91 references13 citations

TL;DR

This paper develops a framework unifying supersymmetric quantum theory, non-commutative geometry, and gravity by extending differential topology and geometry to non-commutative spaces, using tools like the non-commutative torus and M(atrix) models. It demonstrates how non-commutative geometry provides a geometric language for quantum gravity and string theory, offering new insights into the structure of spacetime at Planck scales.

ABSTRACT

This is an expanded version of the notes to a course taught by the first author at the 1995 Les Houches Summer School. Constraints on a tentative reconciliation of quantum theory and general relativity are reviewed. It is explained what supersymmetric quantum theory teaches us about differential topology and geometry. Non-commutative differential topology and geometry are developed in some detail. As an example, the non-commutative torus is studied. An introduction to string theory and $M$(atrix) models is provided, and it is outlined how tools of non-commutative geometry can be used to explore the geometry of string theory and conformal field theory.

Motivation & Objective

• To reconcile quantum theory and general relativity through the lens of supersymmetric quantum theory and non-commutative geometry.
• To develop a non-commutative differential topology and geometry framework applicable to quantum gravity.
• To explore the geometric structure of string theory and conformal field theory using non-commutative tools.
• To investigate how M(atrix) models and non-commutative tori can model quantum spacetime geometry.
• To provide a conceptual and mathematical bridge between supersymmetry, non-commutative geometry, and gravitational theories.

Proposed method

• Adapts supersymmetric quantum theory to extract topological and geometric invariants of manifolds.
• Constructs a non-commutative differential geometry formalism, generalizing classical differential geometry to non-commutative algebras.
• Analyzes the non-commutative torus as a prototype example of non-commutative geometry with applications to string compactifications.
• Introduces M(atrix) models as a non-perturbative formulation of string theory, linking them to non-commutative geometry.
• Applies tools from non-commutative geometry to study the geometry of conformal field theories and string backgrounds.
• Uses K-theory and cyclic cohomology to describe topological invariants in non-commutative spaces.

Experimental results

Research questions

• RQ1 How can supersymmetric quantum theory inform the differential topology and geometry of spacetime?
• RQ2 What is the role of non-commutative geometry in describing quantum spacetime structures?
• RQ3 How do M(atrix) models and non-commutative tori provide a geometric framework for string theory?
• RQ4 In what way can non-commutative differential geometry unify quantum field theory and general relativity?
• RQ5 What are the topological and geometric invariants of non-commutative spaces relevant to quantum gravity?

Key findings

• Non-commutative geometry provides a natural framework for describing quantum spacetime, with the non-commutative torus serving as a key example.
• Supersymmetric quantum theories yield deep insights into the differential topology of manifolds, including index theorems and topological invariants.
• The interplay between non-commutative geometry and M(atrix) models reveals new geometric structures underlying string theory.
• Cyclic cohomology and K-theory emerge as essential tools for classifying non-commutative spaces relevant to quantum gravity.
• The formalism successfully generalizes classical geometric concepts to non-commutative algebras, enabling a new approach to quantum gravity.
• The framework suggests that spacetime at Planck scales may be inherently non-commutative, with profound implications for unification.