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Non-saturation of Bootstrap Bounds by Hyperbolic Orbifolds

Alex Radcliffe

TL;DR

This work investigates whether bootstrap bounds on Laplace eigenvalues and related OPE-like data for hyperbolic orbifolds are exactly saturated by explicit orbifolds such as the [0; 2, 3, 7] orbifold and the Bolza surface. By combining crossing relations for holomorphic differentials with an extremal-functional framework and a novel connection to Rankin-Cohen brackets of modular forms, the authors derive explicit OPE coefficients and show that the asymptotic bounds are not saturated (e.g., a gap of about $4.1\times 10^{-11}$ for $S_{12}$ and $1.7\times 10^{-8}$ for $\lambda_1$ on the [0; 2, 3, 7] orbifold, plus $5.6\times 10^{-6}$ for the Bolza surface). The analysis relies on expressing OPE data as Petersson norms of Rankin-Cohen brackets on punctured spheres and Bolza via a uniformization map, enabling explicit numerical estimates and a clear obstruction to saturation. The results highlight that near-saturation observed in bootstrap bounds does not guarantee exact saturation by simple geometric models, guiding future exploration of more elaborate correlator setups and higher-weight modular forms. The methodology offers a pathway to quantify non-saturation in related spectral-geometry bootstrap problems and informs the interpretation of near-saturated CFT bootstrap bounds.

Abstract

In recent years the conformal bootstrap has produced surprisingly tight bounds on many non-perturbative CFTs. It is an open question whether such bounds are indeed saturated by these CFTs. A toy version of this question appears in a recent application of the conformal bootstrap to hyperbolic orbifolds, where one finds bounds on Laplace eigenvalues that are exceptionally close to saturation by explicit orbifolds. In some instances, the bounds agree with the actual values to 11 significant digits. In this work we show, under reasonable assumptions about the convergence of numerics, that these bounds are not in fact saturated. In doing so, we find formulas for the OPE coefficients of hyperbolic orbifolds, using links between them and the Rankin-Cohen brackets of modular forms.

Non-saturation of Bootstrap Bounds by Hyperbolic Orbifolds

TL;DR

This work investigates whether bootstrap bounds on Laplace eigenvalues and related OPE-like data for hyperbolic orbifolds are exactly saturated by explicit orbifolds such as the [0; 2, 3, 7] orbifold and the Bolza surface. By combining crossing relations for holomorphic differentials with an extremal-functional framework and a novel connection to Rankin-Cohen brackets of modular forms, the authors derive explicit OPE coefficients and show that the asymptotic bounds are not saturated (e.g., a gap of about for and for on the [0; 2, 3, 7] orbifold, plus for the Bolza surface). The analysis relies on expressing OPE data as Petersson norms of Rankin-Cohen brackets on punctured spheres and Bolza via a uniformization map, enabling explicit numerical estimates and a clear obstruction to saturation. The results highlight that near-saturation observed in bootstrap bounds does not guarantee exact saturation by simple geometric models, guiding future exploration of more elaborate correlator setups and higher-weight modular forms. The methodology offers a pathway to quantify non-saturation in related spectral-geometry bootstrap problems and informs the interpretation of near-saturated CFT bootstrap bounds.

Abstract

In recent years the conformal bootstrap has produced surprisingly tight bounds on many non-perturbative CFTs. It is an open question whether such bounds are indeed saturated by these CFTs. A toy version of this question appears in a recent application of the conformal bootstrap to hyperbolic orbifolds, where one finds bounds on Laplace eigenvalues that are exceptionally close to saturation by explicit orbifolds. In some instances, the bounds agree with the actual values to 11 significant digits. In this work we show, under reasonable assumptions about the convergence of numerics, that these bounds are not in fact saturated. In doing so, we find formulas for the OPE coefficients of hyperbolic orbifolds, using links between them and the Rankin-Cohen brackets of modular forms.
Paper Structure (18 sections, 79 equations, 3 figures)

This paper contains 18 sections, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary
  3. Numerical Bootstrap and OPE Coefficients
  4. Setup and Bounds
  5. Single-correlator bounds
  6. Multi-correlator bounds
  7. OPE bounds
  8. Definitions of the OPE coefficients
  9. OPE coefficients from Rankin-Cohen brackets
  10. Rankin-Cohen Algebra on Punctured Spheres
  11. Results
  12. Non-saturation of $\widetilde{S}^\infty_{12}$ by the $[0; 2, 3, 7]$ Orbifold
  13. Non-saturation of $\widetilde{\lambda}_1^\infty$ by the $[0; 2, 3, 7]$ Orbifold
  14. Non-saturation of $\widetilde{\lambda}_1^\infty$ by the Bolza Surface
  15. Conclusions
  16. ...and 3 more sections

Figures (3)

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  • Figure 3: