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PyCFTBoot: A flexible interface for the conformal bootstrap

Connor Behan

TL;DR

PyCFTBoot delivers a flexible, open-source interface for performing SDP-based conformal bootstrap calculations, integrating SDPB and the SymEngine backend to automate block generation, convolution, and crossing-consistency checks. It introduces efficient derivative computation for conformal blocks via meromorphic representations and Casimir-based recurrences, and provides structured objects (ConformalBlockTable, ConvolvedBlockTable, SDP) to build and solve bootstrap constraints. The paper demonstrates extensive usage through simple and advanced examples (identical scalars, mixed correlators, global symmetry, and SCFTs) and a longer scan exploring the Wilson-Fisher line between $d=3$ and $d=4$, including island formation and OPE-coefficient bounds. It discusses practical considerations such as even vs odd dimensions, bilinear-orthonormal polynomial bases, XML generation for SDPB, and strategies for reusing work across multi-correlator runs, highlighting the software’s potential to accelerate nonperturbative CFT studies while acknowledging limitations related to non-unitarity and complex spectra.

Abstract

We introduce PyCFTBoot, a wrapper designed to reduce the barrier to entry in conformal bootstrap calculations that require semidefinite programming. Symengine and SDPB are used for the most intensive symbolic and numerical steps respectively. After reviewing the built-in algorithms for conformal blocks, we explain how to use the code through a number of examples that verify past results. As an application, we show that the multi-correlator bootstrap still appears to single out the Wilson-Fisher fixed points as special theories in dimensions between 3 and 4 despite the recent proof that they violate unitarity.

PyCFTBoot: A flexible interface for the conformal bootstrap

TL;DR

PyCFTBoot delivers a flexible, open-source interface for performing SDP-based conformal bootstrap calculations, integrating SDPB and the SymEngine backend to automate block generation, convolution, and crossing-consistency checks. It introduces efficient derivative computation for conformal blocks via meromorphic representations and Casimir-based recurrences, and provides structured objects (ConformalBlockTable, ConvolvedBlockTable, SDP) to build and solve bootstrap constraints. The paper demonstrates extensive usage through simple and advanced examples (identical scalars, mixed correlators, global symmetry, and SCFTs) and a longer scan exploring the Wilson-Fisher line between and , including island formation and OPE-coefficient bounds. It discusses practical considerations such as even vs odd dimensions, bilinear-orthonormal polynomial bases, XML generation for SDPB, and strategies for reusing work across multi-correlator runs, highlighting the software’s potential to accelerate nonperturbative CFT studies while acknowledging limitations related to non-unitarity and complex spectra.

Abstract

We introduce PyCFTBoot, a wrapper designed to reduce the barrier to entry in conformal bootstrap calculations that require semidefinite programming. Symengine and SDPB are used for the most intensive symbolic and numerical steps respectively. After reviewing the built-in algorithms for conformal blocks, we explain how to use the code through a number of examples that verify past results. As an application, we show that the multi-correlator bootstrap still appears to single out the Wilson-Fisher fixed points as special theories in dimensions between 3 and 4 despite the recent proof that they violate unitarity.

Paper Structure

This paper contains 19 sections, 86 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Conformal Blocks
  3. Rational approximations
  4. Further processing
  5. Even dimensions
  6. Overall structure
  7. Working with SDPs
  8. Writing XML files
  9. Some examples
  10. Identical scalars
  11. A faster approach
  12. The next scalar
  13. Imposing another gap
  14. OPE maximization
  15. Global symmetry
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Plot of allowed anomalous dimensions in $d \in \{ 3, 3.25, 3.5, 3.75 \}$. For each $d$, there is a closed region of points that cannot be excluded using the constraints of the conformal bootstrap on the correlators $\left < \sigma \sigma \sigma \sigma \right >$, $\left < \sigma \sigma \epsilon \epsilon \right >$, $\left < \epsilon \epsilon \epsilon \epsilon \right >$. Their positions have good agreement with the dimensions of the Wilson-Fisher fixed points calculated with the $\varepsilon$-expansion.