nLab Yang-Mills field

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.