Parity and the modular bootstrap

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 βLβR = 4π2. We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large-c CFT2 with a suitably sparse low-lying spectrum matches that of AdS3 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, h)-plane that is significantly smaller than 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.