Coulomb gas integrals, classical conformal blocks and Richardson-Gaudin models

Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional CFTs with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs' classical blocks. The latter are special functions which have a wide spectrum of awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, N=2 gauge theory and quantum integrable systems (QIS) are just some of contexts and topics, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko--Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the "Dotsenko--Fateev matrix model action". The latter is the "DF action" evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). The relationship with integrable models is our main motivation for this research line. ... .