Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
Fields and quanta
The Yang–Mills field is the gauge field of Yang-Mills theory.
It is modeled by a cocycle in differential nonabelian cohomology. Here is the moduli stack of -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 is represented by
a collection of -valued 1-forms ;
a collection of -valued smooth functions ;
such that on double overlaps
and such that on triple overlaps
For this is the electromagnetic field.
For this is the “electroweak field”;
For this is the strong nuclear force field.