Modular Bootstrap, Elliptic Points, and Quantum Gravity

The modular bootstrap program for 2d CFTs could be seen as a systematic exploration of the physical consequences of consistency conditions at the elliptic points and at the cusp of their toruspartition function. The study at $\tau=i$, the elliptic point stabilized by the modular inversion $S$, was initiated by Hellerman, who found a general upper bound for the most relevant scaling dimension $\Delta$. Likewise, analyticity at $\tau=i\infty$, the cusp stabilized by the modular translation $T$, yields an upper bound on the twist gap. Here we study consistency conditions at $\tau=\exp[2i\pi/3]$, the elliptic point stabilized by $S T$. We find a much stronger upper bound in the large-c limit, namely $\Delta<\frac{c-1}{12}+0.092$, which is very close to the minimal mass threshold of the BTZ black holes in the gravity dual of $AdS_3/CFT_2$ correspondence.