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I have been watching the lecture https://youtu.be/lrikIt9MXpQ from the school LACES 2020 by INFN.

The $\mathcal{N}=1$ SYM is investigated, with the action: $$ \mathcal{L} = \frac{1}{32\pi} \mathrm{Im} \left[\tau \int d^2 \theta \ W^{\alpha} W_\alpha\right] $$ And there is claim, that the holomorphy constraints the dependence of coupling constant on the holomorphic scale $\bar{\Lambda}$ to be: $$ \tau(\mu) = \underbrace{\frac{b_1}{2 \pi} \log \frac{\bar{\Lambda}}{\mu}}_{\text{1-loop}} + \underbrace{\sum_n c_n \left(\frac{\bar{\Lambda}}{\mu} \right)^{n b_1}}_{\text{instantons}} $$ Where holomorphic scale $\bar{\Lambda}$ is defined as: $$ \bar{\Lambda} = \Lambda e^{i \theta / b_1} \qquad \Lambda = \mu_0 e^{-8 \pi^2 / b_1 g^2 (\mu)} $$ It claimed (approx 6:10 on the video) that there can be no corrections beyond 1-loop, because instanton dependence on $g^2 (\mu)$ has to be via $\tau$, and the higher loop corrections would come a powers of $g^2 (\mu)$ and they have to be combined with the corresponding power of $\theta$, but these terms cannot be generated, because $\theta$ enters the Lagrangian only as a term, proportional to the derivative of something ($F \wedge \tilde{F} = d(A \wedge d A + \frac{2}{3} A \wedge A \wedge A)$), and does not contribute to the perturbation theory.

I do not understand, how the instanton dependence is connected with the reasoning on the perturbation theory, instanton contributions are in some sense, by definition, non-perturbative phenomena. How does this constrain the perturbative expansion? I guess, that the crucial part in the reasoning should be some holomorphy properties, but I do not see how, they emerge in the present case.

I would be grateful for comments and clarification

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The point is that the $U(1)_{R}$ current and the conformal current, at the classical level, lives in the same super-multiplet, so an anomaly of one current is related to the other. Indeed at classical level we have

$$ \partial_{m}j^{m} = \sigma^{m}_{\alpha\dot\alpha}\bar S_{m}^{\dot\alpha} = \sigma^{m}_{\alpha\dot\alpha}S_{m}^{\alpha}=T^{m}\,_{m}=0 $$

where $j^{m}$ is the $U(1)_{R}$ current, $(S_{m}^{\alpha},\bar S_{m}^{\dot\alpha})$ is the supersymmetry current and $T_{mn}$ is the energy-momentum tensor. Conformal symmetry means $T^{m}\,_{m}=0$.

Since the anomaly of the $U(1)_{R}$ current is one-loop exact we know how this symmetry is broken: it is broken by instanton effects. This is so because the anomaly is of the form

$$ dj \propto F\wedge F \implies (Q_{2}-Q_{1})=\int_{\partial \Sigma} j = \int_{\Sigma}F\wedge F $$

where $j$ is the $3$-from current, $F$ is the $2$-form field strength, $(Q_{2}-Q_{1})$ is the charge violation and $\Sigma$ is the history of the space-time from time $1$ to time $2$. Since, by definition, instantons are the only thing that couples with $\int _{\Sigma}F\wedge F$, the only thing that couples to the $U(1)_{R}$ anomaly are the instantons.

Once we know how the $U(1)_{R}$ is broken, by supersymmetry you can learn about how the conformal symmetry is broken and this is why instantons are the only non-perturbative objects that shows up. The relation between these two anomalies is not so straightforward so I will not discuss it here.

You can see more about it in Supersymmetry and the Adler-Bardeen theorem - M. T. Grisaru, P. C. West and also in Supersymmetry and Nonperturbative beta Functions - Nathan Seiberg.

(particularly in the first paragraph bellow equation $13$ of the last reference).

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