# Fast Conformal Bootstrap and Constraints on 3d Gravity

## TL;DR

This paper introduces a fast algorithm for conformal bootstrap that yields precise spectral bounds in 2d CFTs, extends to large central charge, and provides new constraints on 3d gravity black hole spectra.

## Contribution

It develops a rapid computational method for modular bootstrap, enabling high-precision spectral gap calculations and new bounds on 3d gravity black holes.

## Key findings

- Spectral gap bound: Δ₁ ≲ c/9.1 at large c
- Computed scaling dimensions for over a thousand operators
- Extended bootstrap to large central charge regime

## Abstract

The crossing equations of a conformal field theory can be systematically truncated to a finite, closed system of polynomial equations. In certain cases, solutions of the truncated equations place strict bounds on the space of all unitary CFTs. We describe the conditions under which this holds, and use the results to develop a fast algorithm for modular bootstrap in 2d CFT. We then apply it to compute spectral gaps to very high precision, find scaling dimensions for over a thousand operators, and extend the numerical bootstrap to the regime of large central charge, relevant to holography. This leads to new bounds on the spectrum of black holes in three-dimensional gravity. We provide numerical evidence that the asymptotic bound on the spectral gap from spinless modular bootstrap, at large central charge $c$, is $\Delta_1 \lesssim c/9.1$.