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Bootstrapping conformal field theories with the extremal functional method.

Sheer El-Showk1, Miguel F Paulos2

  • 1Institut de Physique Théorique CEA Saclay, CNRS-URA 2306, 91191 Gif sur Yvette, France.

Physical Review Letters
|February 4, 2014
PubMed
Summary

We introduce the extremal functional method (EFM) to determine operator dimensions and OPE coefficients in conformal field theories (CFTs). This numerical approach analyzes boundary solutions, successfully applied to the 2D Ising model.

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Area of Science:

  • Theoretical Physics
  • Quantum Field Theory
  • Conformal Field Theory (CFT)

Background:

  • Positive linear functionals can constrain parameter spaces in CFTs.
  • Extremal functionals at allowed region boundaries may encode operator spectrum and OPE coefficients.

Purpose of the Study:

  • To develop a numerical method, the Extremal Functional Method (EFM), for determining CFT spectra and OPE coefficients.
  • To investigate if extremal functionals contain sufficient information for spectrum reconstruction.

Main Methods:

  • Development of the Extremal Functional Method (EFM), a numerical procedure.
  • Application of EFM to CFTs residing on the boundary of the solution space.
  • Testing EFM by rederiving the spectrum and OPE coefficients of the 2D Ising model.

Main Results:

  • The EFM successfully derives the low-lying spectrum and OPE coefficients of the 2D Ising model.
  • The method requires only the dimension of a single scalar quasiprimary, not the full Virasoro algebra.

Conclusions:

  • The Extremal Functional Method (EFM) is a viable numerical tool for CFT analysis.
  • EFM provides a benchmark for studying more complex and less understood CFTs.
  • This method demonstrates the power of extremal functionals in CFT data extraction.