The Shadow Formalism of Galilean CFT$_2$

TL;DR

This paper develops the shadow formalism for 2D Galilean conformal field theories, deriving key structures like conformal blocks and exploring new features such as additional branch points affecting OPE convergence.

Contribution

It introduces the shadow formalism for GCFT$_2$, establishes new inversion formulas, and constructs series of bilocal and local actions including the BMS free scalar model.

Findings

Derived the shadow transform for local operators in GCFT$_2$

Identified additional branch points in conformal blocks affecting OPE convergence

Constructed bilocal and local actions, including the BMS free scalar model

Abstract

In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (GCFT$_2$). We define the principal series representation of Galilean conformal symmetry group and find its relation with the Wigner classification, then we determine the shadow transform of local operators. Using this formalism we derive the OPE blocks, Clebsch-Gordan kernels, conformal blocks and conformal partial waves. A new feature is that the conformal block admits additional branch points, which would destroy the convergence of OPE for certain parameters. We establish another inversion formula different from the previous one, but get the same result when decomposing the four-point functions in the mean field theory (MFT). We also construct a continuous series of bilocal actions of MFT, and an exceptional series of local actions, one of which is the BMS free scalar model. We notice that there is an outer automorphism of the Galilean conformal symmetry, and the GCFT$_2$ can be regarded as null defect in higher dimensional CFTs.