# nLab Einstein-Yang-Mills theory

### Context

#### Gravity

gravity, supergravity

## Quantum theory

#### Differential cohomology

differential cohomology

# Contents

## 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,

${S}_{G+\mathrm{YM}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left(e,\nabla \right)↦{\int }_{X}R\left(e\right)\mathrm{vol}\left(e\right)+{\int }_{X}⟨{F}_{\nabla }\wedge {\star }_{e}{F}_{\nabla }⟩\phantom{\rule{thinmathspace}{0ex}},$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

standard model of particle physics and cosmology

theory:Einstein-Yang-Mills-Dirac-Higgs
gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field $e$principal connection $\nabla$spinor $\psi$scalar field $H$
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield strength squared + potential density
$L=$$R\left(e\right)\mathrm{vol}\left(e\right)+$$⟨{F}_{\nabla }\wedge {\star }_{e}{F}_{\nabla }⟩+$$\left(\psi ,{D}_{\left(e,\nabla \right)}\psi \right)\mathrm{vol}\left(e\right)+$$\nabla \overline{H}\wedge {\star }_{e}\nabla H+\left(\lambda {\mid H\mid }^{4}-{\mu }^{2}{\mid H\mid }^{2}\right)\mathrm{vol}\left(e\right)$

## References

Section Prequantum gauge theory and Gravity in

Revised on January 14, 2013 18:21:35 by Urs Schreiber (203.116.137.162)