Gravity with de Sitter and Unitary Tangent Groups

TL;DR

This paper explores extending Einstein gravity as a gauge theory with larger tangent groups, specifically de Sitter and unitary groups, revealing new unification and matter coupling features.

Contribution

It demonstrates how Einstein gravity can be formulated with larger tangent groups, leading to unification of fields and new matter coupling insights.

Findings

De Sitter tangent group unifies 4D vectors and scalars as 5D vectors.

Complex tangent space yields Einstein-Strauss Hermitian gravity.

Spinors require complexification or doublets, akin to N=2 supersymmetry.

Abstract

Einstein Gravity can be formulated as a gauge theory with the tangent space respecting the Lorentz symmetry. In this paper we show that the dimension of the tangent space can be larger than the dimension of the manifold and by requiring the invariance of the theory with respect to 5d Lorentz group (de Sitter group) Einstein theory is reproduced unambiguously. The other possibility is to have unitary symmetry on a complex tangent space of the same dimension as the manifold. In this case the resultant theory is Einstein-Strauss Hermitian gravity. The tangent group is important for matter couplings. We show that in the de Sitter case the 4 dimensional space time vector and scalar are naturally unified by a hidden symmetry being components of a 5d vector in the tangent space. With a de Sitter tangent group spinors can exist only when they are made complex or taken in doublets in a way similar to N=2 supersymmetry.