Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories

TL;DR

The paper addresses universal bounds on the lightest operator dimensions in unitary 2D CFTs with discrete spectra by leveraging modular invariance of the torus partition function. It extends Hellerman’s modular bootstrap approach to include theories with chiral currents, deriving bounds of the form $Δ_1 \le \frac{c_{\rm tot}}{12} + O(1)$ and, under suitable conditions, $Δ_n \le \frac{c_{\rm tot}}{12} + O(1)$ for higher primaries. The method uses derivative constraints encoded in polynomials $f_p$ and $b_p$, analyzes ratios of modular constraints, and establishes large-$c_{\rm tot}$ asymptotics alongside numerical constants (e.g., $0.4755...$ for $Δ_1$ and $0.5531...$ for $Δ_2$). The results have gravitational implications via AdS$_3$/CFT$_2$, predicting bounds on the lightest bulk masses and a minimal growth rate of state degeneracy, and are tested against concrete CFTs such as $u(1)_k$, $su(2)_k$, and free-fermion theories to confirm compliance with the proposed bounds.

Abstract

We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d conformal field theories with discrete spectra using modular invariance, including CFTs with chiral currents. We derive a bound on the conformal dimensions $Δ_1$ and $Δ_2$ going as $c_{\rm tot}/12 + O(1)$. The bound is of the same form found for CFTs without chiral currents in arXiv:0902.2790 and arXiv:1312.0038. We then prove the inequality $Δ_n \leq c_{\rm tot}/12 + O(1)$ for large $c_{\rm tot}$ and with appropriate restrictions on $n$. Using the AdS$_3$/CFT$_2$ correspondence, our bounds correspond to upper bounds on the masses of the lightest few states and a lower bound on the number of states. We conclude by checking our results against several candidate conformal field theories.