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

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

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

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

• charged black hole

• Einstein-Hilbert action

• Einstein-Maxwell 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]

Solutions in Einstein-Yang-Mills theory:

• Robert Bartnik, John McKinnon, Particle-like solutions of the Einstein-Yang-Mills equations, Phys. Rev. Lett., 61, pp. 141-144 (1988) [doi:10.1103/PhysRevLett.61.141 bibcode:1988PhRvL..61..141B]

• Joel Smoller, Arthur Wasserman, Shing-Tung Yau, J. Bryce McLeod, Smooth static solutions of the Einstein-Yang/Mills equation, Bull. Amer. Math. Soc. (N.S.) 27, pp. 239-242 (1992) [arXiv:math/9210226 doi:10.1007/BF02097002]

• Joel Smoller, Arthur Wasserman, Existence of infinitely-many smooth, static global solutions of the Einstein-Yang-Mills equations, Commun. Math. Phys. 51, pp. 303-325 (1993) [doi:10.1007/BF02096771 pdf]

• Joel Smoller, Arthur Wasserman, Shing-Tung Yau, Existence of black hole solutions for the Einstein-Yang-Mills equations, Commun. Math. Phys. 154, pp. 377-401 (1993) [doi:10.1007/BF02097002 pdf]

• Joel Smoller, Arthur Wasserman, Regular solutions of the Einstein Yang-Mills equations, J. Math. Phys. 36, pp. 4301-4323 (1995) [doi:10.1063/1.530963]

• Joel Smoller, Arthur Wasserman, Uniqueness of extreme Reissner-Nordstrom solution in SU(2) Einstein-Yang-Mills theory for spherically symmetric space-time, Phys. Rev., D 52, pp. 5812-5815 (1995) [doi:10.1103/PhysRevD.52.5812]

• Joel Smoller, Arthur Wasserman, Investigation of the Interior of Colored Black Holes and the Extendability of Solutions of the Einstein-Yang/Mills Equations, Commun. Math. Phys. 194 pp. 707-732 (1998) [arXiv:gr-qc/9706039 doi:10.1007/s002200050375]

• Mark Fisher, Todd Oliynyk, There are no magnetically charged particle-like solutions of the Einstein Yang-Mills equations for Abelian models, Commun. Math. Phys. 312, pp. 137-177 (2012) [arXiv:1104.0449 doi:10.1007/s00220-011-1388-5]