virasoro_blocks: Software for the 2d bootstrap

Abstract

We introduce virasoro_blocks, software that computes Virasoro conformal blocks at any central charge in a form suitable for the numerical bootstrap. At each finite order in the elliptic nome, Zamolodchikov's recursion gives the block as a positive prefactor times a rational function of the exchanged weight. However, its many Kac poles make it difficult to use in semidefinite programming approaches to the bootstrap. In this software we use optimal interpolation to reduce the number of poles while keeping high accuracy. The software works in arbitrary precision and handles singular central charges, including the minimal models and $c=1$. It also computes global blocks, $N=1$ superconformal blocks, torus characters, and torus one-point blocks. We illustrate it with two applications. Assuming only scalar Virasoro primaries, we compute bounds on the scalar gap that approach the window of Liouville solutions between $(c-1)/16$ and $(c-1)/12$. We also bound the average exchanged dimension in four-point functions of identical scalars at several central charges. These bounds have a rich structure, and many known 2d CFTs lie on or near their boundary.