Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universality in the OPE Coefficients of Holographic 2d CFTs

Ben Michel

TL;DR

This work probes when universal OPE asymptotics in holographic 2d CFTs extend beyond the classic Cardy regime to $\Delta\sim c$ at large central charge. By adapting the covariant Hartman–Keller–Stoica framework to covariant OPE data and employing pillow and genus-twisted constructions, the authors obtain extended-regime results for averaged OPE coefficients and primary-state densities under sparseness conditions such as $\rho(\Delta)\lesssim e^{2\pi\Delta}$ up to $\Delta\lesssim c/12+\varepsilon$. They derive explicit extended-regime formulas for $\overline{C_{HLL}^2}$, $\overline{C_{HHL}^2}$, $\overline{C_{HHL}}$, and $\rho_p$, and discuss their bulk interpretations in terms of BTZ black holes and entropy $S_{BH}(\Delta)$, while noting obstacles for permutation orbifolds and for averages over primaries. The notes also address genus-two (vacuum-content) contributions and outline the conditions under which vacuum-block dominance may hold, highlighting the need for further understanding of conformal blocks in the $\Delta\sim c$ regime. Overall, the results constrain non-vacuum block contributions in holographic CFTs and provide bounds that connect modular properties, sparseness, and bulk black-hole physics.

Abstract

The thermodynamic stability of large AdS$_3$ black holes implies that Cardy's $Δ\rightarrow\infty$ formula for the density of states remains approximately valid when $Δ\sim c$ in holographic 2d CFTs, constraining their light spectra. Averaged OPE coefficients take a similarly universal asymptotic form, and black hole arguments again imply an extended regime of validity. In this note we study conditions under which the OPE asymptotics extend to $Δ\sim c$ at large central charge. Some of the conditions found are stronger than required by an extended Cardy regime and are violated by permutation orbifolds, such as the D1-D5 system at zero coupling. Our results suggest new bounds on non-vacuum block contributions to correlation functions in holographic CFTs.

Universality in the OPE Coefficients of Holographic 2d CFTs

TL;DR

This work probes when universal OPE asymptotics in holographic 2d CFTs extend beyond the classic Cardy regime to at large central charge. By adapting the covariant Hartman–Keller–Stoica framework to covariant OPE data and employing pillow and genus-twisted constructions, the authors obtain extended-regime results for averaged OPE coefficients and primary-state densities under sparseness conditions such as up to . They derive explicit extended-regime formulas for , , , and , and discuss their bulk interpretations in terms of BTZ black holes and entropy , while noting obstacles for permutation orbifolds and for averages over primaries. The notes also address genus-two (vacuum-content) contributions and outline the conditions under which vacuum-block dominance may hold, highlighting the need for further understanding of conformal blocks in the regime. Overall, the results constrain non-vacuum block contributions in holographic CFTs and provide bounds that connect modular properties, sparseness, and bulk black-hole physics.

Abstract

The thermodynamic stability of large AdS black holes implies that Cardy's formula for the density of states remains approximately valid when in holographic 2d CFTs, constraining their light spectra. Averaged OPE coefficients take a similarly universal asymptotic form, and black hole arguments again imply an extended regime of validity. In this note we study conditions under which the OPE asymptotics extend to at large central charge. Some of the conditions found are stronger than required by an extended Cardy regime and are violated by permutation orbifolds, such as the D1-D5 system at zero coupling. Our results suggest new bounds on non-vacuum block contributions to correlation functions in holographic CFTs.

Paper Structure

This paper contains 12 sections, 37 equations.

Table of Contents

  1. Introduction
  2. Covariant HKS
  3. Extended regimes
  4. $\overline{C_{HLL}^2}$: plane four-point function
  5. $\overline{C_{HHL}^2}$: torus two-point function
  6. $\overline{C_{HHL}}$: torus one-point function
  7. Density of primary states
  8. Further remarks
  9. $\overline{C_{HHH}^2}$: genus two partition function
  10. Averages over primary states
  11. Vacuum block dominance
  12. Results at fixed spin