Classical conformal blocks from TBA for the elliptic Calogero-Moser system

TL;DR

This paper derives new expressions for classical 4-point conformal blocks on the sphere using integrable systems and gauge theory relations, advancing the understanding of uniformization and connections to supersymmetric gauge theories.

Contribution

It introduces novel representations of 4-point classical blocks linking uniformization, elliptic Calogero-Moser systems, and supersymmetric gauge theories.

Findings

Relation between accessory parameter and elliptic Calogero-Moser functional

Representation of classical blocks via AGT and Nekrasov-Shatashvili formulas

Connection to Seiberg-Witten prepotential for Nf=4 SU(2) gauge theory

Abstract

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 $N_f=4$, ${\sf SU(2)}$ twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the $N_f=4$, ${\sf SU(2)}$ supersymmetric gauge theory from the classical block.