Modular Bootstrap Revisited

TL;DR

This work advances the modular bootstrap program for two-dimensional unitary, compact CFTs with $c>1$ by leveraging full modular invariance and spin information to derive rigorous bounds on operator gaps and degeneracies via semidefinite programming. It introduces spin-dependent bounds, refines the Hellerman–Friedan–Keller bounds, and reveals sharp connections to known CFTs through kinks saturated by WZW models and Narain lattices; it also establishes the existence and structure of extremal spectra when degeneracy at the gap is saturated. The results include a bound on the twist gap $t_{ m gap} o (c-1)/12$, a scalar-gap bound $ riangle^{s=0}_{ m mod}$ with notable kinks at $c=1,4,8$, and explicit extremal spectra that reproduce spectra of theories such as $SU(2)_1$, $SU(3)_1$, $G_2{}_1$, $SO(8)_1$, and $E_8$-based constructions. The study highlights the rich interplay between modular invariance, operator degeneracies, and extremal spectra, and discusses open questions and future directions, including large-$c$ behavior and integrating higher-genus modular data into the bootstrap.

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.

Figures (20)

• Figure 1: Top: The colored curves are the upper bound on the twist gap as a function of the central charge for increasing derivative order (up to $43$) from green to red. The results are consistent with the convergence to the bound on the twist gap $t_{\rm mod} = {c-1\over 12}$ predicted by the analytic argument in the infinite-$N$ limit. Bottom: The bound on the twist gap $t^{(N)}_{\rm mod}$ as a function of inverse derivative order for $c=5$ and $c=100$.
• Figure 2: The gap bound $\Delta_{\rm HFK}^{(N)}(c)$ obtained using HFK functionals up to derivative order $N=175$ for central charge $c=20$ (left) and up to $N=183$ for $c=60$ (right).
• Figure 3: Left: The green to red curves are plots of $\Delta_{\rm HFK}^{(N)}(c)$ with increasing $N$ ranging from 7 to 183, while the black curve represents the numerically extrapolated bound $\Delta_{\rm HFK}(c)=\Delta_{\rm HFK}^{(\infty)}(c)$. Right: The slope of $\Delta_{\rm HFK}(c)$ as a function of $c$, obtained from taking the derivative of the Bezier fitting function of the extrapolated $\Delta_{\rm HFK}^{(\infty)}(c)$, over a range of $c$ where the numerical extrapolation appears reliable. The slope drops below ${1\over 9}$ for $c>75.5$.
• Figure 4: The bound on the dimension gap $\Delta_{\rm mod}$ as a function of central charge $c$, obtained by extrapolating $\Delta_{\rm mod}^{(N)}$ to $N=\infty$. The numerical extrapolation is performed by fitting $31\leq N\leq 55$ bounds with a quadratic polynomial in $1/N$.
• Figure 5: $\Delta_{\rm HFK}-\Delta_{\rm mod}^{(N)}$ as a function of central charge $c$, for $c\in [4,22]$ and increasing $N$. The numerical values of the HFK bound are obtained by a linear extrapolation of $\Delta_{\rm HFK}^{(N)}$ to $1/N \to 0$ using bounds for $137\leq N\leq 175$.