Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Algebraic Approach to the Analytic Bootstrap

Luis F. Alday, Alexander Zhiboedov

TL;DR

The paper develops an algebraic, Casimir-operator–driven approach to the analytic bootstrap in CFTs, recasting large-spin sums into recursive problems in the conformal spin J and exploiting a kernel-based convolution. This yields an efficient, largely kernel-free method to compute corrections to anomalous dimensions and OPE coefficients from a crossed-channel primary and its descendants, establishing that the resulting large-spin series are typically asymptotic but Borel-summable. The authors apply the framework to the critical O(N) model, recovering known two-loop results for higher-spin currents and making new predictions across representations and for small N, while clarifying the role of accumulation points and infinite towers in maintaining crossing symmetry. The approach thus links analytic bootstrap, perturbation theory, and numerical bootstrap methods, providing a versatile tool for probing CFT data in and beyond the O(N) universality class.

Abstract

We develop an algebraic approach to the analytic bootstrap in CFTs. By acting with the Casimir operator on the crossing equation we map the problem of doing large spin sums to any desired order to the problem of solving a set of recursion relations. We compute corrections to the anomalous dimension of large spin operators due to the exchange of a primary and its descendants in the crossed channel and show that this leads to a Borel-summable expansion. We analyse higher order corrections to the microscopic CFT data in the direct channel and its matching to infinite towers of operators in the crossed channel. We apply this method to the critical $O(N)$ model. At large $N$ we reproduce the first few terms in the large spin expansion of the known two-loop anomalous dimensions of higher spin currents in the traceless symmetric representation of $O(N)$ and make further predictions. At small $N$ we present the results for the truncated large spin expansion series of anomalous dimensions of higher spin currents.

An Algebraic Approach to the Analytic Bootstrap

TL;DR

The paper develops an algebraic, Casimir-operator–driven approach to the analytic bootstrap in CFTs, recasting large-spin sums into recursive problems in the conformal spin J and exploiting a kernel-based convolution. This yields an efficient, largely kernel-free method to compute corrections to anomalous dimensions and OPE coefficients from a crossed-channel primary and its descendants, establishing that the resulting large-spin series are typically asymptotic but Borel-summable. The authors apply the framework to the critical O(N) model, recovering known two-loop results for higher-spin currents and making new predictions across representations and for small N, while clarifying the role of accumulation points and infinite towers in maintaining crossing symmetry. The approach thus links analytic bootstrap, perturbation theory, and numerical bootstrap methods, providing a versatile tool for probing CFT data in and beyond the O(N) universality class.

Abstract

We develop an algebraic approach to the analytic bootstrap in CFTs. By acting with the Casimir operator on the crossing equation we map the problem of doing large spin sums to any desired order to the problem of solving a set of recursion relations. We compute corrections to the anomalous dimension of large spin operators due to the exchange of a primary and its descendants in the crossed channel and show that this leads to a Borel-summable expansion. We analyse higher order corrections to the microscopic CFT data in the direct channel and its matching to infinite towers of operators in the crossed channel. We apply this method to the critical model. At large we reproduce the first few terms in the large spin expansion of the known two-loop anomalous dimensions of higher spin currents in the traceless symmetric representation of and make further predictions. At small we present the results for the truncated large spin expansion series of anomalous dimensions of higher spin currents.

Paper Structure

This paper contains 27 sections, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Anomalous dimensions of higher spin operators: an algebraic approach
  3. The sum rule
  4. Algebraic approach
  5. The Casimir operator and recurrence relations for ${\cal F}^{(n)}(v)$
  6. The Casimir operator and recurrence relations for ${\cal F}^{(n)}(v)$
  7. Application: contribution from a primary plus all its descendants
  8. The problem and summary of the results
  9. Results for general $\Delta_\epsilon$
  10. Analytic results for $\Delta_\epsilon=2m$
  11. Exchange of operators with spin
  12. General picture
  13. Example: the critical $O(N)$ model at large $N$
  14. Known results
  15. Microscopic interpretation
  16. ...and 12 more sections

Figures (1)

  • Figure 1: Asymptotic behaviour of the coefficients $c_k$ for $d=3$, $\Delta=0.518151$ and $\Delta_\epsilon=1.41264$. The data is consistent with the behaviour $|\frac{c_{k+1}}{c_k}| \sim k^2$ for large $k$.