Proof of Zamolodchikov conjecture for semi-classical conformal blocks on the torus
arXiv:2407.05839
The paper rigorously proves a conjectured exponential form for the semi‑classical limit of Liouville conformal blocks on a one‑punctured torus, establishing convergence and linking the result to solutions of the Lamé equation and an 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 Lamé 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.
72 pages. Major revision. Corrected the proof of the main theorem, major modifications to the methodology