Modular bootstrap agrees with the path integral in the large moduli limit

Abstract

Based on the rigorous path integral formulation of Liouville Conformal Field Theory initiated by David-Kupiainen-Rhodes-Vargas [6] on the Riemann sphere and David-Rhodes-Vargas [11] 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.