nLab Einstein-Yang-Mills theory

Idea

What is called Einstein-Yang-Mills theory in physics is the theory/model 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,

$$S_{G+YM} \; \colon \; (e, \nabla) \mapsto \int_{X} R(e) vol(e) + \int_X \langle F_\nabla \wedge \star_e F_\nabla\rangle \,,$$

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,

• $vol(e)$ 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 $e$.

• $\langle -,-\rangle$ is the given invariant polynomial on the Lie algebra of the gauge group.

Related concepts

• charged black hole

• Einstein-Hilbert action

• Einstein-Maxwell theory

• Einstein-Yang-Mills 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
$L =$	$R(e) vol(e) +$	$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$	$(\psi , D_{(e,\nabla)} \psi) vol(e) +$	$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

References

Section Prequantum gauge theory and Gravity in

• geometry of physics

On Yang-Mills monopoles:

• Jefferson Bjoraker, Yutaka Hosotani, Monopoles, Dyons and Black Holes in the Four-Dimensional Einstein-Yang-Mills Theory, Phys. Rev. D 62 043513 (2000) [arXiv:hep-th/0002098, doi:10.1103/PhysRevD.62.043513]