Bootstrapping Conformal Field Theories with the Extremal Functional Method (1211.2810v1)

Abstract: The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and OPE coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the Extremal Functional Method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the 2d Ising model based solely on the dimension of a single scalar quasi-primary -- no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.

• The paper demonstrates the utility of the Extremal Functional Method by extracting the 2D Ising model’s operator spectrum and OPE coefficients with precision up to six decimal places.
• The authors leverage crossing symmetry and positivity constraints on conformal blocks to delineate the boundaries of viable CFT spectra without relying on full algebraic structures.
• The results imply that the EFM offers a powerful, extendable approach for investigating higher-dimensional CFTs and testing additional constraints like superconformal symmetry.

Bootstrapping Conformal Field Theories with the Extremal Functional Method

The paper by Sheer El-Showk and Miguel F. Paulos examines an advanced numerical technique called the Extremal Functional Method (EFM) to explore the parameter space of conformal field theories (CFTs). This work leverages crossing symmetry and positivity of linear functionals to derive the spectra and operator product expansion (OPE) coefficients within CFTs, focusing on those situated specifically on the boundary of the solution space.

Summary of the Main Concepts

The research starts by asserting that for a CFT with a scalar field of dimension $\Delta_\sigma$, crossing symmetry of the four-point function dictates certain functional constraints. These constraints thrive in having a linear form and rely on the infinite-dimensional space defined by conformal blocks. By employing the EFM, the authors numerically explore the spectrum of two-dimensional CFT models without relying on the full algebraic structure typically required, such as the Virasoro algebra.

Methodology Overview

EFM harnesses the idea that at the boundaries of the allowed parameter regions in CFTs, the extremal functional should theoretically be able to pinpoint the exact spectra and OPE coefficients. This approach is validated through its application to the two-dimensional Ising model. By considering specific facets of the so-called polyhedral cone, the paper utilizes the presence of crossing symmetry to set bounds—effectively identifying valid or invalid regions for certain operator dimensions and spins.

Numerical Results and Implications

Strong numerical results are a highlight, particularly the rederivation of the low-lying spectrum and OPE coefficients of the 2D Ising model. For example, the central charge and dimensions of several primary operators were found to match theoretical predictions to six decimal places. Such meticulous numerical verification serves as a testament to the EFM’s precision and robustness.

Theoretical and Practical Implications

The implications of this work are manifold. Theoretically, the EFM proposes a pragmatic route to bootstrap constraints without relying heavily on algebraic structures traditionally considered necessary. Practically, this opens up a powerful computational method to scrutinize less explored CFTs and further test the boundaries set by analytic predictions in various dimensions.

Future Prospects

Future applications might include exploring three-dimensional CFTs such as the 3D Ising model. Moreover, the potential for imposing additional theoretical constraints (such as superconformal symmetry) or using higher-dimensional conformal blocks systematically to examine more complex CFTs remains a tantalizing prospect.

In conclusion, the paper extends both the theoretical and numerical frontier of CFT analysis, marking a significant contribution to the understanding and utility of the conformal bootstrap in examining the enigmatic domain of quantum field theories. While this work predominantly illustrates the capability of EFM within a two-dimensional framework, its potential utility in higher-dimensional theories poses an exciting avenue for upcoming investigations.