nLab
topological Yang-Mills theory

Context

\infty-Chern-Simons theory

Ingredients

Definition

Examples

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  • For targets

  • For discrete targets

  • For targets

    • coupled to
  • For targets extending the

    (such as the , the )

    • Chern-Simons-

  • for higher abelian targets

  • for targets

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  • for the L L_\infty-structure on the of the closed :

Quantum field theory

Contents

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Physics

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Surveys, textbooks and lecture notes

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    • Axiomatizations

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    • Structural phenomena

    • Types of quantum field thories

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Contents

Idea

Topological Yang-Mills theory is a gauge theory topological quantum field theory .

Definition

For XX a 4-dimensional smooth manifold, 𝔤\mathfrak{g} a Lie algebra with Lie group GG and ,W(𝔤)\langle-,-\rangle \in W(\mathfrak{g}) a binary invariant polynomial on 𝔤\mathfrak{g}, topological Yang-Mills theory is the quantum field theory defined by the action functional

S:GBund (X) S : G Bund_\nabla(X) \to \mathbb{R}

on the groupoid of GG-principal bundles with connection on a bundle that sends a connecton \nabla to the integral of the curvature 4-form F F \langle F_\nabla \wedge F_\nabla \rangle of the corresponding Chern-Simons circle 3-bundle:

S: XF AF A. S : \nabla \mapsto \int_X \langle F_A \wedge F_A\rangle \,.

Relation to other models

The ordinary kinetic term of Yang-Mills theory differs from this by the fact that the Hodge star operator appears F F \langle F_\nabla \wedge \star F_\nabla \rangle. In full Yang-Mills theory both terms appear.

The topological Yang-Mills action also appears in the generalized Chern-Simons theory given by a Chern-Simons element in a Lie 2-algebra, where it is coupled to BF-theory. See Chern-Simons element for details.

References

A general account emphasizing the relation to Chern-Simons theory is

  • P van Baal, An introduction to topological Yang-Mills theory , Acta Physica Polonica Vol. B21 (pdf)

The relation to Chern-Simons theory on the boundary in an ambient string theoretic context is indicated in section 2 (starting around p. 21) of

Last revised on September 21, 2016 at 14:41:25. See the history of this page for a list of all contributions to it.