Modular bootstrap agrees with 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 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.

24 pages, 3 figures