nLab
Yang-Mills field

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Differential cohomology

Fields and quanta

Contents

Idea

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

It is modeled by a cocycle F^H(X,BU(n) conn)\hat F \in \mathbf{H}(X, \mathbf{B}U(n)_{conn}) in differential nonabelian cohomology. Here BU(n) conn\mathbf{B} U(n)_{conn} is the moduli stack of U(n)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 XX is represented by

  • a cover {U iX}\{U_i \to X\}

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

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

  • such that on double overlaps

    A j=Ad g ijA i+g ijgg 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 ijg jk=g ik. g_{i j} g_{j k} = g_{i k} \,.

Examples

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

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

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

Revised on January 10, 2013 20:07:47 by Urs Schreiber (89.204.153.52)