Representations of minimal model primary fields in the Coulomb-gas Formalism
This question is a cross-post from MO (link).
Is it known how to construct the primary field operators of the unitary minimal models $\mathcal{M}(m+1,m)$ in the Coulomb gas formalism? As far as I can tell, the original work of <PERSON> and <PERSON> [2] is limited to the 4-point functions and does not give a direct definition of the field operators (as honest operators, on some not-necessarily "minimal" Hilbert space(s)) themselves (though, at a physicist's level of rigor, it should fix the N-point functions via the conformal bootstrap, which ought then determine the field operators via a reconstruction theorem). The (physics?) paper [3] of <PERSON> seems to me to strongly suggest that such a direct definition is possible. But, as far as I can tell, it itself falls a bit short of a rigorous definition.
A related question question is: for which primary fields in which minimal models are all $N$-point functions known in some closed form? All $N$-point functions are known for all fields in the critical Ising model, but is that it (besides the free field)? Are there any other nontrivial fields in any other minimal models for which the $N$-point functions are known in this way? (Of course, the $N$-point functions should be determined via the $3$-point functions via the conformal bootstrap, but this shouldn't count as being in closed form.)
Here is my incomplete attempt at making sense of <PERSON>'s definition in the simplest interesting case, the spin field in the critical Ising model $\mathcal{M}(4,3)$. Hopefully this will indicate what I have in mind, but unfortunately it requires a bit of explanation. First, some generalities about vertex operators. I will use the notation of [3][4]. 324
In [3], we have defined some Verma models $\mathcal{F}{\alpha,\alpha_0}$ of a central extension of the Heisenberg algebra that also come equipped with representations of the Virasoro algebra. Here, $\alpha \in \mathbb{R}$ is arbitrary and $\alpha_0$ is some fixed number. For each $z\in \mathbb{C}$ and $\beta \in \mathbb{R}$, we have a formal vertex operator $V{\beta}(z)$ which is defined as a binary form $$\mathcal{F}{\alpha+\beta,\alpha_0}\times\mathcal{F}{\alpha,\alpha_0} \to \mathbb{C}$$ (see pg. 14 of [5]). In [5] is constructed a nested 1-parameter family ${\mathcal{H}{\alpha,\alpha_0,r}}{r>0}$ of completions of $\mathcal{F}{\alpha,\alpha_0}$, and then it can be shown that, whenever $r<|z|<R$, $V\beta(z)$ defines a Hilbert-Schmidt (in particular, bounded) map $\mathcal{H}{\alpha,\alpha_0,r}\to \mathcal{H}{\alpha+\beta,\alpha_0,R}$ depending analytically on $z$. Then, for $z_1,\ldots,z_N$ with $r<|z_N|<\cdots<|z_1|<R$ we have a composition $V_{\beta_1}(z_1)\cdots V_{\beta_N}(z_N)$ which is well-defined in the sense that it does not depend on which intermediate $\mathcal{H}{\bullet,\alpha_0,\rho}$'s we use. 616