# nLab Yang-Mills field

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

## Application to gauge theory

#### Fields and quanta

field (physics)

standard model of particle physics

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadron (bound states of the above quarks)

solitons

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

The Yang–Mills field is the gauge field of Yang-Mills theory.

It is modeled by a cocycle $\hat F \in \mathbf{H}(X, \mathbf{B}U(n)_{conn})$ in differential nonabelian cohomology. Here $\mathbf{B} U(n)_{conn}$ is the moduli stack of $U(n)$-principal connections, the stackification of the groupoid of Lie-algebra valued forms, regarded as a groupoid internal to smooth spaces.

This is usually represented by a vector bundle with connection.

As a nonabelian Čech cocycle the Yang-Mills field on a space $X$ is represented by

• a cover $\{U_i \to X\}$

• a collection of $Lie(U(n))$-valued 1-forms $(A_i \in \Omega^1(U_i, Lie(U(n))))$;

• a collection of $U(n)$-valued smooth functions $(g_{i j} \in C^\infty(U_{i j}, U(n)))$;

• such that on double overlaps

$A_j = Ad_{g_{i j}} \circ A_i + g_{i j} g g_{i j}^{-1} \,,$
• and such that on triple overlaps

$g_{i j} g_{j k} = g_{i k} \,.$

# Examples

• For $U(n) = U(1)$ this is the electromagnetic field.

• For $U(n) = SU(2) \times U(1)$ this is the “electroweak field”;

• For $U(n) = SU(3)$ this is the strong nuclear force field.

Last revised on August 5, 2015 at 03:55:38. See the history of this page for a list of all contributions to it.