Modular bootstrap, elliptic points, and quantum gravity

The modular bootstrap program for two-dimensional conformal field theories could be seen as a systematic exploration of the physical consequences of consistency conditions at the elliptic points and at the cusp of their torus partition function. The study at tau = i, the elliptic point stabilized by the modular inversion S, was initiated by Hellerman [S. Hellerman, J. High Energy Phys. 08 (2011) 130], who found a general upper bound for the most relevant scaling dimension Delta. Likewise, analyticity at tau = i infinity, 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 ST. We find a much stronger upper bound in the large-c limit, namely, Delta < c-1/12 + 0.092, which is very close to the minimal mass threshold of the Banados-Teitelboim-Zanelli black holes in the gravity dual of three-dimensional anti-de Sitter/two-dimensional conformal field theory correspondence.