Modular bootstrap agrees with path integral in the large moduli limit

TL;DR

This paper rigorously confirms that the modular bootstrap predictions for Liouville Conformal Field Theory match the asymptotic behavior derived from the path integral approach on the torus as the imaginary part of the modulus becomes large.

Contribution

It provides the first rigorous derivation of the asymptotic behavior of the toric correlation function, confirming bootstrap predictions in the large moduli limit.

Findings

Asymptotic decay rate of (Im τ)^{-3/2} for the 1-point toric correlation function.

Identification of the derivative of the DOZZ formula in the large moduli limit.

Agreement between bootstrap formalism and path integral results in Liouville CFT.

Abstract

Based on the rigorous path integral formulation of Liouville Conformal Field Theory initiated by David-Kupiainen-Rhodes-Vargas on the Riemann sphere and David-Rhodes-Vargas on the torus of modulus $\tau$, we give the exact asymptotic behaviour of the 1-point toric correlation function as $\mathrm{Im}\:\tau\to\infty$. In agreement with formulae predicted within the bootstrap formalism of theoretical physics, our results feature an $(\mathrm{Im}\:\tau)^{-3/2}$ decay rate and we identify the derivative of DOZZ formula in the limit.