nLab
Einstein-Yang-Mills theory

Idea

What is called Einstein-Yang-Mills theory in physics is the theory/model (in theoretical physics) describing gravity together with Yang-Mills fields such as the electroweak field or the strong nuclear force of quantum chromodynamics. For the special case that the gauge group is the circle group this reproduces Einstein-Maxwell theory.

Einstein-Yang-Mills theory is a local Lagrangian field theory defined by the action functional which is the Einstein-Hilbert action plus the Yang-Mills action functional involving the given metric,

$$S_{G+\mathrm{YM}}:(e,\nabla )\mapsto {\int }_{X}R(e)\mathrm{vol}(e)+{\int }_X⟨F_{\nabla }\wedge {\star }_e{F}_{\nabla }⟩,$$S_{G+YM} \; \colon \; (e, \nabla) \mapsto \int_{X} R(e) vol(e) + \int_X \langle F_\nabla \wedge \star_e F_\nabla\rangle \,,

where

• $X$ is the compact smooth manifold underlying spacetime,

• $e$ is the vielbein field which encodes the field of gravity

• $\nabla$ is the $G$-principal connection which encodes the Yang-Mills field,

• $\mathrm{vol}(e)$ is the volume form induced by $e$;

• $R(e)$ is the scalar curvature of $e$;

• $F_{\nabla }$ is the field strength/curvature differential 2-form of $\nabla$;

• ${\star }_e$ is the Hodge star operator induced by $e$.

• $⟨-,-⟩$ is the given invariant polynomial on the Lie algebra of the gauge group.

Related concepts

• Einstein-Hilbert action

• Einstein-Maxwell theory

• Einstein-Yang-Mills theory

• Einstein-Yang-Mills-Dirac theory

• Einstein-Yang-Mills-Dirac theory

• Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory

standard model of particle physics and cosmology

theory:	Einstein-	Yang-Mills-	Dirac-	Higgs
	gravity	electroweak and strong nuclear force	fermionic matter	scalar field
field content:	vielbein field $e$	principal connection $\nabla$	spinor $\psi$	scalar field $H$
Lagrangian:	scalar curvature density	field strength squared	Dirac operator component density	field strength squared + potential density
$L=$	$R(e)\mathrm{vol}(e)+$	$⟨F_{\nabla }\wedge {\star }_e{F}_{\nabla }⟩+$	$(\psi ,D_{(e,\nabla )}\psi )\mathrm{vol}(e)+$	$\nabla \overline{H}\wedge {\star }_e\nabla H+(\lambda {\mid H\mid }^4-{\mu }^2{\mid H\mid }^2)\mathrm{vol}(e)$

References

Section Prequantum gauge theory and Gravity in

• geometry of physics