On Galilean conformal bootstrap

TL;DR

This paper develops the conformal bootstrap framework for Galilean conformal field theories, including the computation of global blocks, harmonic analysis, and the explicit construction of a generalized Galilean free theory, highlighting unique features like non-unitarity.

Contribution

It introduces the conformal bootstrap approach for GCFTs, computes global blocks for multiplets, and constructs the first explicit generalized Galilean free theory.

Findings

Global blocks of multiplets are computed.

An inversion formula for Galilean conformal symmetry is derived.

The generalized Galilean free theory (GGFT) is explicitly constructed.

Abstract

In this work, we develop conformal bootstrap for Galilean conformal field theory (GCFT). In a GCFT, the Hilbert space could be decomposed into quasiprimary states and its global descendants. Different from the usual conformal field theory, the quasi-primary states in a GCFT constitute multiplets, which are block-diagonized under the Galilean boost operator. More importantly the multiplets include the states of negative norms, indicating the theory is not unitary. We compute global blocks of the multiplets, and discuss the expansion of four-point functions in terms of the global blocks of the multiplets. Furthermore we do the harmonic analysis for the Galilean conformal symmetry and obtain an inversion formula. As the first step to apply the Galilean conformal bootstrap, we construct generalized Galilean free theory (GGFT) explicitly. We read the data of GGFT by using Taylor series expansion of four-point function and the inversion formula independently, and find exact agreement. We discuss some novel features in the Galilean conformal bootstrap, due to the non-semisimpleness of the Galilean conformal algebra and the non-unitarity of the GCFTs.