Supersymmetric quantum theory and non-commutative geometry

TL;DR

This paper develops a supersymmetric framework for non-commutative geometry, extending Connes' spectral triples to encompass various geometric structures like Riemannian, symplectic, and Kähler geometries, with detailed examples.

Contribution

It introduces a supersymmetric spectral data approach to non-commutative geometry, generalizing existing frameworks and providing new insights into non-commutative spaces.

Findings

Generalized Connes' spectral triples with supersymmetry

Extended non-commutative geometries including Kähler and symplectic cases

Detailed analysis of non-commutative torus and 3-sphere

Abstract

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.