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

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We consider unitary, modular invariant, twodimensional 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 rightmoving 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 lowlying 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 $(c1)/12$ 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 parityinvariant partition function.