New Properties of Large-$c$ Conformal Blocks from Recursion Relation

TL;DR

This paper explores new properties of large central charge conformal blocks using recursion relations, revealing asymptotic behaviors, phase transitions at specific dimensions, and implications for holography and bootstrap methods.

Contribution

It introduces numerical methods to analyze large $c$ conformal blocks outside known limits, uncovering new asymptotic formulas and dimension-dependent behaviors.

Findings

Discovered a Cardy-like asymptotic formula for large $c$ blocks with light intermediate states.

Identified a drastic change in behavior when external dimensions reach $c/32$.

Found simple dependence of large $c$ blocks on heavy intermediate dimensions.

Abstract

We study large $c$ conformal blocks outside the known limits. This work seems to be hard, but it is possible numerically by using the Zamolodchikov recursion relation. As a result, we find new some properties of large $c$ conformal blocks with a pair of two different dimensions for any channel and with various internal dimensions. With light intermediate states, we find a Cardy-like asymptotic formula for large $c$ conformal blocks and also we find that the qualitative behavior of various large $c$ blocks drastically changes when the dimensions of external primary states reach the value $c/32$. And we proceed to the study of blocks with heavy intermediate states $h_p$ and we find some simple dependence on heavy $h_p$ for large $c$ blocks. The results in this paper can be applied to, for example, the calculation of OTOC or Entanglement Entropy. In the end, we comment on the application to the conformal bootstrap in large $c$ CFTs.