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The scaling dimension of the first excited state in two-dimensional conformalfield theories (CFTs) satisfies a universal upper bound. Using the modularbootstrap, we extend this result to CFTs with $W_3$ algebras which aregenerically dual to higher spin theories in AdS$_3$. Assuming unitarity andmodular invariance, we show that the conformal weights $h$, $\bar{h}$ of thelightest 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 $W_3$ currentsmust 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 boundsand comment on the interpretation of our results in the dual higher spintheory.

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