Grading of Spinor Bundles and Gravitating Matter in Non-Commutative Geometry

TL;DR

This paper explores how non-commutative geometry can model gravitating matter by formulating a non-commutative Einstein-Hilbert action, leading to new insights into matter-gravity coupling through Dirac operators.

Contribution

It introduces a specific grading choice for spinor bundles in non-commutative geometry, resulting in a nonlinear vector sigma-model coupled to Einstein gravity.

Findings

Derived a non-commutative Einstein-Hilbert action with matter fields

Connected Dirac operator grading to matter field representations

Presented a nonlinear vector sigma-model coupled to gravity

Abstract

The gravitating matter is studied within the framework of the non-commutative geometry. The non-commutative Einstein-Hilbert action on the product of a four dimensional manifold with a discrete space gives the models of matter fields coupled to the standard Einstein gravity.The matter multiplet is encoded in the Dirac operator which yields the representation of the algebra of the universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vector sigma-model coupled to the Einstein gravity.