Chiral Modular Bootstrap

TL;DR

This paper uses modular bootstrap to establish bounds on the conformal weights of primary fields in 2D conformal field theories, linking these bounds to properties of black holes in 3D gravity via AdS/CFT.

Contribution

It demonstrates that the lightest primary fields' conformal weights are bounded by c/12 plus corrections, depending on their spin, revealing a connection between CFT spectra and black hole properties.

Findings

Lightest primary fields have conformal weight h₁ ≤ c/12 + O(1).

Bounds depend on the spin-to-dimension ratio of primary fields.

Large spin primaries imply extremal or near extremal BTZ black holes.

Abstract

Using modular bootstrap we show the lightest primary fields of a unitary compact two dimensional conformal field theory(with $c, \bar{c}>1$) has a conformal weight $h_1\le \frac{c}{12}+\mathcal{O}(1)$.This implies that the upper bound on the dimension of the lightest primary fields depends on their spin. In particular if the set of lightest primary fields includes extremal or near extremal states whose spin to dimension ratio $\frac{j}{\Delta}\approx 1$, the corresponding dimension is $\Delta\le \frac{c}{12}+\mathcal{O}(1)$. From AdS/CFT correspondence, we obtain an upper bound on the spectrum of black hole in three dimensional gravity. Our results show that if the first primary fields have large spin, the corresponding three dimensional gravity has extremal or near extremal BTZ black hole.