# Modular Bootstrap, Elliptic Points, and Quantum Gravity

## TL;DR

This paper extends the modular bootstrap analysis of 2d CFTs by exploring the elliptic point stabilized by ST, deriving a stronger upper bound on the scaling dimension that closely matches the BTZ black hole threshold in AdS3/CFT2.

## Contribution

It introduces a new consistency condition at the elliptic point ST and derives a tighter upper bound on the scaling dimension in the large central charge limit.

## Key findings

- Derived a bound: Δ < (c-1)/12 + 0.092 at τ=exp[2iπ/3]
- Bound is close to the minimal mass threshold of BTZ black holes
- Strengthens the connection between CFT bounds and gravity duals

## Abstract

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.