Modular Bootstrap Revisited

TL;DR

This paper uses modular bootstrap to numerically constrain the spectrum of 2D conformal field theories with central charge c > 1, refining previous bounds and exploring implications for known theories and deformations.

Contribution

It provides improved numerical bounds on operator dimensions in 2D CFTs using semi-definite programming, extending previous results and analyzing extremal spectra.

Findings

Unitary CFTs with c < 8 must have relevant deformations.

Bounds on scalar primary gaps are established for c < 25.

Extremal spectra can be numerically determined to maximize degeneracy.

Abstract

We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how "extremal spectra" which maximize the degeneracy at the gap can be determined numerically.