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A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance

John Cardy, Alexander Maloney, Henry Maxfield

TL;DR

<3-5 sentence high-level summary> This work develops a universal, large-$c$ framework to constrain OPE data in two-dimensional CFTs from genus-two modular invariance. By recasting higher-genus partition functions as twist-operator correlators in a $\mathbb{Z}_n$ orbifold and applying Zamolodchikov’s monodromy method, the authors derive explicit asymptotics for heavy OPE coefficients in terms of genus-two conformal blocks and a dimension-ratio dependent constant $\mathcal{F}_0$. The main results include a genus-two average for squared three-point coefficients with exponential suppression controlled by left- and right-moving dimensions and a precise block asymptotic formula, linking high-energy OPE data to low-energy input. These findings illuminate the higher-genus modular bootstrap program and bear holographic interpretation in AdS$_3$ gravity via multi-black-hole saddle geometries.

Abstract

We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.

A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance

TL;DR

<3-5 sentence high-level summary> This work develops a universal, large- framework to constrain OPE data in two-dimensional CFTs from genus-two modular invariance. By recasting higher-genus partition functions as twist-operator correlators in a orbifold and applying Zamolodchikov’s monodromy method, the authors derive explicit asymptotics for heavy OPE coefficients in terms of genus-two conformal blocks and a dimension-ratio dependent constant . The main results include a genus-two average for squared three-point coefficients with exponential suppression controlled by left- and right-moving dimensions and a precise block asymptotic formula, linking high-energy OPE data to low-energy input. These findings illuminate the higher-genus modular bootstrap program and bear holographic interpretation in AdS gravity via multi-black-hole saddle geometries.

Abstract

We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.

Paper Structure

This paper contains 17 sections, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Note:
  3. CFT partition functions and twist operator OPE
  4. CFT partition function as twist operator correlation function
  5. Crossing symmetry as modular symmetry
  6. An asymptotic formula for OPE coefficients
  7. The general strategy
  8. Genus one: the density of states
  9. Genus two: three point coefficients
  10. Asymptotics of higher genus blocks
  11. The monodromy method
  12. For higher genus blocks
  13. The large dimension WKB approximation
  14. Stokes phenomena:
  15. Pinching limits
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Surfaces constructed by gluing together two spheres with cylinders. The partition function is then a sum over states labelled $i,j,k$ propagating along the cylinders, times the square of a two- or three-point function on the sphere.
  • Figure 2: Branch points and cuts, and cycles $\gamma_k$ in the $z$ and $y$ planes. Dashed lines are branch cuts on the $z$ plane, and dotted lines branch cuts on the $y$ plane. The blue loops are the cycles $\gamma_k$ around which we fix the monodromy. The two sheets of the $y$ plane correspond to the inside and outside of the dotted circle in the $z$ plane, with the points $z=\frac{1}{2}(1+e^{\pm i\theta})$ mapping to $y=u_k,v_k$, and the three sheets of the $z$ plane correspond to the three wedges of the $y$ plane separated by the dashed lines.