Bounds on CFTs with $W_3$ algebras and AdS${}_3$ higher spin theories

Abstract

The scaling dimension of the first excited state in two-dimensional conformal field theories (CFTs) satisfies a universal upper bound. Using the modular bootstrap, we extend this result to CFTs with $W_3$ algebras which are generically dual to higher spin theories in AdS${}_3$. Assuming unitarity and modular invariance, we show that the conformal weights $h$, $\bar{h}$ of the lightest charged state satisfy $h<c/12+O(1)$ and $\bar{h}<\bar{c}/12+O(1)$ in the limit where the central charges $c$, $\bar{c}$ are large. Furthermore, we show that in this limit any consistent CFT with $W_3$ currents must contain at least one state whose $W_3$ charge $w$ obeys $|w|>4|h-c/24|/\sqrt{10\pi c}+O(1)$. We discuss hints on the existence of stronger bounds and comment on the interpretation of our results in the dual higher spin theory.

8 pages, 1 figure; v2: added references, added bound on mass-to-charge ratio, corrected several equations in Section III and Appendix B that leave results unchanged, matches published version

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• Constraints on Higher Spin CFT${}_2$
• Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit
• Conformal Bootstrap Deformations
• Aspects of the S transformation Bootstrap
• Higher spin wormholes from modular bootstrap