nLab
Yang-Mills field

Context

Physics

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Surveys, textbooks and lecture notes

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Differential cohomology

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Higher abelian differential cohomology

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Higher nonabelian differential cohomology

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Fiber integration

Application to gauge theory

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Fields and quanta

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Exotica

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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 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 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.

Last revised on August 5, 2015 at 03:55:38. See the history of this page for a list of all contributions to it.