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Large $c$ Virasoro Blocks from Monodromy Method beyond Known Limits

Yuya Kusuki

TL;DR

This work analytically proves Cardy-like large-$c$ asymptotics for Virasoro blocks using an extended Zamolodchikov monodromy method focused near $x=1$, revealing a sharp transition in ABBA blocks when external dimensions reach $h_A=c/32$ or $h_B=c/32$. By solving the monodromy problem with a twisted hypergeometric reduction and examining the resulting accessory parameters, the authors derive explicit piecewise leading behaviors for ABBA and AA blocks and connect these to Cardy-like growth in the Zamolodchikov recursion. The results provide an analytic underpinning for the Cardy-like structure of recursion solutions and offer a bulk interpretation as a mass-dependent collision transition in AdS, bridging boundary monodromy calculations with bulk gravitational dynamics. Additionally, the work extends the regime of validity to very heavy intermediate states, showing consistency with high-low temperature bootstrap and yielding insights into the asymptotics and non-perturbative corrections of large-$c$ conformal blocks.

Abstract

In this paper, we study large $c$ Virasoro blocks by using the Zamolodchikov monodromy method beyond its known limits. We give an analytic proof of our recent conjectur, which implied that the asymptotics of the large $c$ conformal blocks can be expressed in very simple forms, even if outside its known limits, namely the semiclassical limit or the heavy-light limit. In particular, we analytically discuss the fact that the asymptotic behavior of large $c$ conformal blocks drastically changes when the dimensions of external primary states reach the value $\frac{c}{32}$, which is conjectured by our numerical studies. The results presented in this work imply that the general solutions to the Zamolodchikov recursion relation are given by Cardy-like formula, which is an important conclusion that can be numerically drawn from our recent works. Mathematical derivations and analytical results imply that, in the bulk, the collision behavior between two heavy particles may undergo a remarkable transition associated with their masses.

Large $c$ Virasoro Blocks from Monodromy Method beyond Known Limits

TL;DR

This work analytically proves Cardy-like large- asymptotics for Virasoro blocks using an extended Zamolodchikov monodromy method focused near , revealing a sharp transition in ABBA blocks when external dimensions reach or . By solving the monodromy problem with a twisted hypergeometric reduction and examining the resulting accessory parameters, the authors derive explicit piecewise leading behaviors for ABBA and AA blocks and connect these to Cardy-like growth in the Zamolodchikov recursion. The results provide an analytic underpinning for the Cardy-like structure of recursion solutions and offer a bulk interpretation as a mass-dependent collision transition in AdS, bridging boundary monodromy calculations with bulk gravitational dynamics. Additionally, the work extends the regime of validity to very heavy intermediate states, showing consistency with high-low temperature bootstrap and yielding insights into the asymptotics and non-perturbative corrections of large- conformal blocks.

Abstract

In this paper, we study large Virasoro blocks by using the Zamolodchikov monodromy method beyond its known limits. We give an analytic proof of our recent conjectur, which implied that the asymptotics of the large conformal blocks can be expressed in very simple forms, even if outside its known limits, namely the semiclassical limit or the heavy-light limit. In particular, we analytically discuss the fact that the asymptotic behavior of large conformal blocks drastically changes when the dimensions of external primary states reach the value , which is conjectured by our numerical studies. The results presented in this work imply that the general solutions to the Zamolodchikov recursion relation are given by Cardy-like formula, which is an important conclusion that can be numerically drawn from our recent works. Mathematical derivations and analytical results imply that, in the bulk, the collision behavior between two heavy particles may undergo a remarkable transition associated with their masses.

Paper Structure

This paper contains 7 sections, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Standard Monodromy Method
  3. Monodromy Method at $x=1$
  4. Conformal Blocks at $x=1$
  5. General Solutions to the Recursion Relation
  6. Conformal Blocks with Heavy Intermediate States
  7. Discussion

Figures (2)

  • Figure 1: Since the operator $\mathcal{O}_p$ arises from the OPE between $\mathcal{O}_1$ and $\mathcal{O}_2$, the monodromy of $\Psi_p$ around $\mathcal{O}_p$ is given by encircling both points $z=\{0,x \}$ along a blue contour.
  • Figure 2: To obtain the monodromy matrix, we have to evaluate the monodromy around $z=x$, while excluding $z=1$. However, as long as we are restricted to the solution of the ODE (\ref{['eq:ODEatz=1']}), we cannot distinguish the points $z=\{x,1\}$. Therefore, we have to see the solution to the ODE in another limit $1-x \ll 1$ with $\frac{z-x}{1-x} \gg 1$, which is shown in (\ref{['eq:monoeq2']}). The matrices displayed in this figure correspond to the matrices (\ref{['eq:M12']}), (\ref{['eq:M11']}), and the whole circle corresponds to (\ref{['eq:MMMM']}).