# nLab topological Yang-Mills theory

### Context

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

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

## Definition

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

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

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 $\langle F_\nabla \wedge F_\nabla \rangle$ of the corresponding Chern-Simons circle 3-bundle:

$S : \nabla \mapsto \int_X \langle F_A \wedge F_A\rangle \,.$

## Relation to other models

The ordinary kinetic term of Yang-Mills theory differs from this by the fact that the Hodge star operator appears $\langle F_\nabla \wedge \star F_\nabla \rangle$. 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 21, 2016 14:41:25 by Anonymous Coward (71.93.119.239)