Parity and the modular bootstrap

TL;DR

This work shows that combining parity with modular inversion in 2D CFTs yields a continuous SP-fixed locus, enabling a parity-invariant auxiliary partition function to apply bootstrap techniques even in non-parity theories. It proves the Hartman–Keller–Stoica conjecture on AdS$_3$ free energy for large central charge and sparse spectra, and extends the modular bootstrap to SP-invariant settings, deriving new constraints and improving bounds on the low-lying operator spectrum and the twist gap. The results unify gravity-duality insights with CFT spectral constraints, providing Cardy-regime generalizations and a framework that extends to theories with $U(1)$ symmetry. The approach relies on positivity along the SP-fixed locus and yields a robust toolkit for bounding operator dimensions across all temperatures and angular potentials. Extensions to time-reversal and internal symmetries indicate broad applicability of parity-modular techniques in higher- and lower-dimensional conformal systems.

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 $β_L β_R = 4π^2$. 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.

Figures (3)

• Figure 1: Fundamental domain and its images for $SL(2,\mathbb{Z})$ extended by parity $P$. New boundaries arising from the parity transformation are delineated by the dashed red curves.
• Figure 2: In any CFT, a state should exist within the blue shaded region. The dashed red lines delimit the existence region derived in Hellerman:2009bu whose upper bound asymptotes to $h+\overline{h}=\frac{c}{6}+0.473695$ as $c\rightarrow \infty$. Along the line $h=\overline{h}=\Delta/2$, the tip of the shaded region asymptotes to $\Delta=\frac{c}{6}+0.951160$ at large-$c$. Interestingly, the $h$ and $\overline{h}$ intercepts of the upper boundary of the shaded region are at $c/12$ at leading order in $c$.
• Figure 3: Deriving the twist gap. The shaded regions are where the functional \ref{['eq:twistfunctional']} is negative, for $c=2$. The expected upper bound $(c-1)/24$ is denoted by the dashed red line. We have taken $q=24$ and $\beta_L = 100\pi$. There must be a state in the shaded regions with $h$ and $\overline{h}$ positive. We have shown the unphysical quadrants with negative $h$ and $\overline{h}$ to exhibit a more complete picture of the functional.