Reading this [paper](https://arxiv.org/pdf/1704.05852.pdf) of Masahito Yamazaki, I am having trouble turning the physics jargon into mathematical statements.  He is talking about Yang-Mills theory over $\mathbb{R}^4$.

Perturbative expansions are ill-defined if observables suffer from infrared divergences.  For example, in the vacuum energy is computed as:
$$ E(\theta) \sim - \int_0^\infty \frac{d\rho}{\rho^5}(\mu \rho)^{b_1}e^{-8\pi^2/g^2(\mu) } \cos \theta $$
The paper then says this integral is divergence in the infrared scale, which is $\rho \to \infty$.  That range actually has good decay, so he could possibly have meant $\rho \to 0$.  Here $\rho$ is the size of the modulus of the instanton. Introducting an IR cuttoff one obtains:
$$ E(\theta) = \Lambda^4 \cos \theta  $$
which is still wrong because another computation (by [Witten](https://inspirehep.net/record/152297/citations), on a possibly-related QCD theory) shows the vaccum energy is given by:
$$ E(\theta) = \min_{\mathbf{e} \in \mathbb{Z}} \Lambda^4 (\theta - 2\pi \mathbf{e})^2 $$
Having read all of this I still don't know what infrared divergence is (or strong coupling), or how resurgence played a role in any of this.

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After some work I was able to [locate](https://arxiv.org/abs/1703.00501) an action for $SU(N)$ Yang-Mills theory over $\mathbb{R}^4$ 

$$ S = \int d^4 x \, \mathrm{Tr}\left( \frac{-1}{4g^2} F \wedge \ast F + \frac{i\theta}{8\pi^2} F \wedge F \right) $$

and they are asking about the $\theta$ dependence.  The action is periodic in $\theta$ but the vacuum energies they find have severe problems (such as discontinuities at $\theta = 0, \pi$). 

These authors take as a given the definitions of Yang-Mills theory, but also that their eigenvalue estimation procedures work.  So that being un-familiar could have an advantages.