The example i know does not use 't Hooft's coupling, but i think it may address your question in a more straightforward way (i'll be "loose" with constants and numerical pre-factors, but will keep all the relevant info and details).

Think of a 0-dimensional scalar field (bosonic $D0$-brane) with a quartic potential à la $V(\phi) = \mu\,\phi^2 + \lambda\,\phi^4$. And, if you allow me some *poetic license* with polynomials, let me rewrite this potential in the following form: $V_g(\phi) = \phi^2 + g\,\phi^4$, where $g=\lambda/\mu$, and it should be clear that "large $g$" means "strong coupling", while "small $g$" means "weak coupling".

The Partition Function (Feynman Path Integral) for this function is given by,

$$
\mathcal{Z}_g[J] = \int e^{i\, S_g(\phi)}\,e^{i\,J\,\phi}\,\mathrm{d}\phi < \infty\; ;
$$
where $S_g[\phi] = V_g(\phi)$ is the Action of the system (where i'm keeping the coupling constant explicit), and the requirement is that the integral *converges* (so the Partition Function is "well defined" in some sense).

But, there's a differential version of the above, called the Schwinger-Dyson equation, given by,

$$
\frac{\partial S_g(\phi)}{\partial\phi} = 0 \;\Longleftrightarrow\; \mu\,\phi + \lambda\,\phi^3 = 0 \;\Longleftrightarrow\; \phi\,(1 + g\,\phi^2) = 0\;;
$$
remembering that $\phi\mapsto\partial_J=\partial/\partial J$, giving us the following:

$$
(\partial_J + g\,\partial_J^3)\,\mathcal{Z}(J) = J\;.
$$

From this differential equation you clearly know one thing: there are 3 solutions to the above problem. This implies that each solution admits an integral representation, which is the Partition Function associated with that particular solution.

In fact, each $\mathcal{Z}_g(J)$ is defined within a certain Stokes' wedge, meaning that when you cross a Stokes' line you pick up some non-trivial constant contribution (very much à la wall-crossing phenomena).

Moreover, you can write the solutions to the above in terms of [confluent] hypergeometric functions (or, if you will, in terms of Meijer's G-function, or Fox's H-function) and distinguish its PolyLogarithm contribution, which is related to its singularity (pole) structure, and can be a big deal when talking about perturbation theory.

Anyway, this is just a "crash-core-dump" version of what i believe attacks your question; the point being that $g$ labels your "quantum phases". Note that it's possible to extend the field from scalar-valued to vector-, matrix-, tensor-, and Lie Algebra-valued: all the results mentioned go through with minor modifications (such as tracing appropriate variables, etc).

Hope this helps.

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