## Parity and the modular bootstrap

Tarek Anous, Raghu Mahajan, Edgar Shaghoulian

SciPost Phys. 5, 022 (2018) · published 11 September 2018

- doi: 10.21468/SciPostPhys.5.3.022
- Submissions/Reports

### 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.

### Cited by 26

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Tarek Anous, -
^{2}^{3}Raghu Mahajan, -
^{4}Edgar Shaghoulian

^{1}University of British Columbia [UBC]^{2}Institute for Advanced Study [IAS]^{3}Princeton University^{4}Cornell University [CU]