nLab
Yang-Mills field

Idea

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

It is modeled by a cocycle $\hat{F}\in H(X,BU(n{)}_{\mathrm{conn}})$ in differential nonabelian cohomology. Here $BU(n{)}_{\mathrm{conn}}$ is the moduli stack of $U(n)$-principal connections, the stackification pf 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 $\mathrm{Lie}(U(n))$-valued 1-forms $(A_i\in {\Omega }^1(U_i,\mathrm{Lie}(U(n))))$;

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

• such that on double overlaps

  $$A_j={\mathrm{Ad}}_{g_{ij}}\circ A_i+g_{ij}g{g}_{ij}^{-1},$$A_j = Ad_{g_{i j}} \circ A_i + g_{i j} g g_{i j}^{-1} \,,

• and such that on triple overlaps

  $$g_{ij}{g}_{jk}=g_{ik}.$$g_{i j} g_{j k} = g_{i k} \,.

Examples

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

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

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