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I'm interested in the pure gauge (no matter fields) case on Minkowski spacetime with simple gauge groups. It would be nice if someone can find a review article discussing all such solutions

EDIT: I think these are relevant to the physics of corresponding QFTs in the high energy / small scale regime. This is because the path integral for a pure gauge Yang-Mills theory is of the form

$$\int \exp\left(\frac{iS[A]}{ \hbar}\right) \, \mathcal{D}A$$

In high energies we have the renormalization group behavior $g \to 0$ (asymptotic freedom) which can be equivalently described by fixing $g$ and letting $\hbar \to 0$.

EDIT: For the purpose of this question, an "exact" solution is a solution in closed form modulo single variable functions defined by certain ODEs and initial / boundary conditions.

JamalS
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4 Answers4

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Wu and Yang (1968) found a static solution to the sourceless SU(2) Yang-Mills equations, (please, see the following two relatively recent articles containing a rather detailed description of the solution: Marinho, Oliveira, Carlson, Frederico and Ngome The solution constitutes of a generalization of the Abelian Dirac monopole. The vector potential is given by:

$A_i^a=g \epsilon_{iaj}x^j\frac{f(r)}{r^2}$

where $f(r)$ satisfies a nonlinear radial equation (The Wu-Yang equation) obtained from the substitution this ansatz into the Yang-Mills field equations.The Wu-Yang monopole has a singularity at the origin, in which the magnetic energy density diverges. The first article contains references to phenomenological works involving the Wu-Yang monopole.

David Bar Moshe
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There is an old review, Alfred Actor, Classical solutions of SU(2) Yang—Mills theories, Rev. Mod. Phys. 51, 461–525 (1979), that provides some of the known solutions of SU(2) gauge theory in Minkowski (monopoles, plane waves, etc) and Euclidean space (instantons and their cousins). For general gauge groups you can get solutions by embedding SU(2)'s. For instantons the most general solution is known, first worked out by ADHM (Atiyah, Hitchin, Drinfeld, Manin) for the classical groups SU,SO,Sp, and then by Bernard, Christ, Guth, Weinberg for exceptional groups. The latest twist on the instanton story is the construction of solutions with non-trivial holomony: ``Periodic instantons with nontrivial holonomy''. Kraan, van Baal, Nucl.Phys. B533 (1998) 627-659. There is a nice set of lecture notes by David Tong on topological solutions with different co-dimension (instantons, monopoles, vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)'s, as you may find in susy gauge theories.

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Exact solutions could not be the right way to understand infrared behavior of Yang-Mills theory. As we know from quantum field theory, we can start with some approximation (weak coupling). With this in mind, it can be proved that the following holds (see http://arxiv.org/abs/0903.2357) for a gauge coupling going formally to infinity

$$ A_\mu^a(x)=\eta_\mu^a\phi(x)+O(1/\sqrt{N}g) $$

being $\eta_\mu^a$ a set of constants and $\phi(x)$ a solution to the equation

$$ \Box\phi(x)+\lambda\phi(x)^3=0. $$

provided $\lambda=Ng^2$. This is the content of the so called mapping theorem. The relevant aspect of this theorem is that one can provide a set of exact solutions for the scalar field in the form

$$ \phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i) $$

being $\mu$ and $\theta$ constants, sn a Jacobi elliptic function and provided the following dispersion relation holds

$$ p^2=\mu^2\left(\frac{\lambda}{2}\right)^\frac{1}{2}. $$

That is, one has massive classical solutions even if we started from massless equations. So, we can start from these classical approximate solutions to build up an infrared quantum field theory for the Yang-Mills field and displaying in this way a mass gap (see http://arxiv.org/abs/1011.3643).

Jon
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A number of exact solutions (mostly those that are invariant under a certain subgroup of the full symmetry group of the system in question) for the SU(2) Yang Mills equations, including the case of Minkowski signature, can be also found in this survey paper:

R.Z. Zhdanov, V.I. Lahno, Symmetry and Exact Solutions of the Maxwell and SU(2) Yang-Mills Equations, arXiv:hep-th/0405286

Also, if one is willing to keep the Minkowski signature while allowing for a global topology other than $\mathbb{R}^4$, this paper could be of interest:

A.D. Popov, Explicit Non-Abelian Monopoles and Instantons in SU (N) Pure Yang-Mills Theory, Phys. Rev. D77 (2008), 125026, arxiv:0803.3320

just-learning
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