nLab
topological Yang-Mills theory

Idea

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

Definition

For $X$ a 4-dimensional smooth manifold, $𝔤$ a Lie algebra with Lie group $G$ and $⟨-,-⟩\in W(𝔤)$ a binary invariant polynomial on $𝔤$, topological Yang-Mills theory is the quantum field theory defined by the action functional

$$S:G{\mathrm{Bund}}_{\nabla }(X)\to ℝ$$S : G Bund_\nabla(X) \to \mathbb{R}

on the groupoid of $G$-principal bundles with connection on a bundle that sends a connecton $\nabla$ to the integral of the curvature 4-form $⟨F_{\nabla }\wedge F_{\nabla }⟩$ of the corresponding Chern-Simons circle 3-bundle:

$$S:\nabla \mapsto {\int }_X⟨F_A\wedge F_A⟩.$$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 $⟨F_{\nabla }\wedge \star F_{\nabla }⟩$. 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

• Edward Witten, Fivebranes and Knots (arXiv:1101.3216)