Parity and the modular bootstrap

Abstract

We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation P P . Combining P P with modular inversion S S leads to a continuous family of fixed points of the SP S P 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 β L β R = 4 π 2 . We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large- c c CFT _2 2 with a suitably sparse low-lying spectrum matches that of AdS _3 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 ( c − 1 ) / 12 when c>1 c > 1 . At large c c we show that the operator dimension of the first excited primary lies in a region in the (h,\overline{h}) ( h , h ¯ ) -plane that is significantly smaller than h+\overline{h} h + 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.