nLab
topological Yang-Mills theory

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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

Revised on September 17, 2014 05:52:32 by Urs Schreiber (185.26.182.30)