Supersymmetric quantum theory and (non-commutative) differential geometry

TL;DR

This paper explores the connection between supersymmetric quantum mechanics and differential geometry, extending it to non-commutative settings to aid in quantum gravity research.

Contribution

It introduces a supersymmetry-based framework for non-commutative geometry, generalizing Connes' approach and defining non-commutative manifolds and phase spaces.

Findings

Classified differential geometry by supersymmetries.

Extended non-commutative geometry with supersymmetry.

Proposed mathematical tools for quantum gravity and superstring vacua.

Abstract

We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in algebraic data consisting of an algebra of functions on a manifold and a family of supersymmetry generators represented on a Hilbert space. We show that known types of differential geometry can be classified in terms of the supersymmetries they exhibit. Replacing commutative algebras of functions by non-commutative *-algebras of operators, while retaining supersymmetry, we arrive at a formulation of non-commutative geometry encompassing and extending Connes' original approach. We explore different types of non-commutative geometry and introduce notions of non-commutative manifolds and non-commutative phase spaces. One of the main motivations underlying our work is to construct mathematical tools for novel formulations of quantum gravity, in particular for the investigation of superstring vacua.