Proof of Zamolodchikov conjecture for semi-classical conformal blocks on the torus

TL;DR

This paper rigorously proves Zamolodchikov's conjecture for semi-classical Liouville conformal blocks on a torus, establishing their exponential structure and linking to classical integrable models.

Contribution

It provides the first rigorous proof of the conjecture for torus conformal blocks and connects the semi-classical limit to classical elliptic integrable systems.

Findings

Confirmed exponential structure of semi-classical conformal blocks on a torus.

Derived a closed-form solution for the Lamé equation.

Linked the accessory parameter to the classical action of elliptic Calogero-Moser model.

Abstract

In 1986, Zamolodchikov conjectured an exponential structure for the semi-classical limit of conformal blocks on a sphere. This paper provides a rigorous proof of the analog of Zamolodchikov conjecture for Liouville conformal blocks on a one-punctured torus, using their probabilistic construction and show the existence of a positive radius of convergence of the semi-classical limit. As a consequence, we obtain a closed form expression for the solution of the Lam\'e equation, and show a relation between its accessory parameter and the classical action of the non-autonomous elliptic Calogero-Moser model evaluated at specific values of the solution.