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:
mesonspion ($u d$)
rho-meson ($u d$)
omega-meson ($u d$)
kaon ($q_{u/d} s$)
eta-meson (u u + d d + s s)
B-meson ($q b$)
baryonsproton $(u u d)$
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.