Does anyone know of a (most likely heuristic) derivation of the use of the string sigma model action to model the soft gluonic interactions between color charges? I'm familiar with the classic literature, like 't Hooft's explanation using the dual superconductivity and the chromoelectric flux tube threading the QCD vacuum. But these papers haven't addressed questions that I have in mind. In particular, is there a way to arrive at the tension of the string action from the coupling constant of the underlying Yang-Mills theory?
Edit: So the route that this derivation might take is the following. 't Hooft paints a picture in the large $N_c$ limit with large expansion parameter $\lambda = g^2 N_c$. The interaction between two heavy sources isn't dominated by one gluon exchange but instead by a dense "net" of many, many gluonic excitations interacting with the sources and more importantly with one another. In the $\lambda \gg 1$ limit, the Fock states aren't appropriate and it's not surprising that a very different model (like a string) must be used to describe the physics of this gluonic smear between the color charges. I'm curious to know if there is any way to match the heuristic picture of a dense net of gluons onto the effective string theory, and specifically match the coupling constant of the former onto the string tension of the latter.