The shadow formalism of Galilean CFT2

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.

Automated summary

In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (GCFT(2)), establishing a relation between the principal series representation of Galilean conformal symmetry group and the Wigner classification. We also derive various important quantities such as OPE blocks, Clebsch-Gordan kernels, conformal blocks, and conformal partial waves using this formalism. Another significant finding is the introduction of additional branch points in the conformal block, potentially affecting the convergence of OPE for specific parameters. Additionally, we propose a new inversion formula and identify a continuous series of bilocal actions and an exceptional series of local actions within the mean field theory.