nLab topological Yang-Mills theory



\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



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


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.


The term originates with


Review emphasizing the relation to Chern-Simons theory is

  • P van Baal, An introduction to topological Yang-Mills theory, Acta Phys.Polon. B21 (1990) 73 (spire:280417, 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

On the Gribov ambiguity in topological Yang-Mills theory:

  • D. Dudal, C. P. Felix, O. C. Junqueira, D. S. Montes, A. D. Pereira, G. Sadovski, R. F. Sobreiro, A. A. Tomaz, Gribov problem in topological Yang-Mills theories (arXiv:1907.05460)

See also

  • O. C. Junqueira, A. D. Pereira, G. Sadovski, R. F. Sobreiro, A. A. Tomaz, Absence of radiative corrections in 4-dimensional topological Yang-Mills theories, Phys. Rev. D 98, 021701 (2018) (arXiv:1805.01850)

Last revised on August 1, 2019 at 12:49:11. See the history of this page for a list of all contributions to it.