Classical conformal blocks from TBA for the elliptic calogero-moser system

The so-called Poghossian identities connecting the toric and spherical blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for the elliptic Calogero-Moser Yang’s (eCMY) functional are used to derive certain expressions for the classical 4-point block on the sphere. The main motivation for this line of research is the longstanding open problem of uniformization of the 4-punctured Riemann sphere, where the 4-point classical block plays a crucial role. It is found that the obtained representation for certain 4-point classical blocks implies the relation between the accessory parameter of the Fuchsian uniformization of the 4-punctured sphere and the eCMY functional. Additionally, a relation between the 4-point classical block and the N f  = 4, SU(2) twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the N f  = 4, SU(2) supersymmetric gauge theory from the classical block.