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, $\mathfrak{g}$ 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.

Related concepts

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

• theta angle

References

The term originates with

• L. Baulieu, Isadore Singer, Topological Yang-Mills symmetry, Nuclear Physics B - Proceedings Supplements Volume 5, Issue 2, December 1988, Pages 12-19 (doi:10.1016/0920-5632(88)90366-0)

following

• Edward Witten, Topological quantum field theory, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. (euclid:1104161738)

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

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

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)