Reading this paper of Masahito Yamazaki and Kazuya Yonekura, 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, 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.

After some work I was able to locate 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.

- From 4d Yang-Mills to 2d $\mathbb{C}P^{N-1}$ model: IR problem and confinement at weak coupling Masahito Yamazaki, Kazuya Yonekura
- Theta, Time Reversal, and Temperature Davide Gaiotto, Anton Kapustin, Zohar Komargodski, Nathan Seiberg

"Of course, Yang-Mills theory on $\mathbb{R}^4$ is strongly-coupled in the IR and hence the perturbative expansion breaks down."so I'd like to understand what is going wrong with perturbation theory here. $\endgroup$ – john mangual Oct 9 '17 at 17:01onlyend in disaster. He's saying the coefficients are infinite so you can't do resurgence at all. And the cutoff prescription gives the wrong answer. $\endgroup$ – john mangual Oct 9 '17 at 18:07