# Parity and the modular bootstrap

## TL;DR

This paper explores the implications of parity invariance in 2D conformal field theories, proving a conjecture about free energy matching with AdS3 gravity, and deriving new spectral constraints via a generalized modular bootstrap.

## Contribution

It introduces a fixed locus of the combined parity and modular inversion transformation, leading to proofs of free energy equivalence and new spectral bounds in large central charge CFTs.

## Key findings

- Proves the Hartman-Keller-Stoica conjecture on free energy matching.
- Derives novel constraints on the operator spectrum.
- Shows the twist gap is smaller than (c-1)/12 for c>1.

## Abstract

We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation $P$. Combining $P$ with modular inversion $S$ leads to a continuous family of fixed points of the $SP$ transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying $\beta_L \beta_R = 4\pi^2$. We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large-$c$ CFT$_2$ with a suitably sparse low-lying spectrum matches that of AdS$_3$ gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than $(c-1)/12$ when $c>1$. At large $c$ we show that the operator dimension of the first excited primary lies in a region in the $(h,\overline{h})$-plane that is significantly smaller than $h+\overline{h}<c/6$. Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.