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 quasiprimary 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.

In this paper, we study the Carrollian and Galilean conformal field theories (CCFT and GCFT) in d > 2 dimensions. We construct the highest weight representations (HWR) of Carrollian and Galilean conformal algebra (CCA and GCA). Even though the two algebras have different structures, their HWRs share similar structure, because their rotation subalgebras are isomorphic. In both cases, we find that the finite dimensional representations are generally reducible but indecomposable, and can be organized into the multiplets.Moreover, it turns out that the multiplet representations in d > 2 CCA and GCA carry not only the simple chain structure appeared in logCFT or 2d GCFT, but also more generally the net structures. We manage to classify all the allowed chain representations. Furthermore we discuss the two-point and three-point correlators by using the Ward identities.We mainly focus on the two-point correlators of the operators in chain representations.Even in this relative simple case, we find some novel features: multiple-level structure, shortage of the selection rule on the representations, undetermined 2-pt coefficients, etc.. We find that the non-trivial correlators could only appear for the representations of certain structure, and the correlators are generally polynomials of time coordinates for CCFT (spacial coordinates for GCFT), whose orders depend on the levels of the correlators.

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.

In this work, we continue our work on two dimensional Galilean conformal field theory (GCFT2). Our previous work (2011.11092) focused on the ξ ≠ 0 sector, here we investigate the more subtle ξ = 0 sector to complete the discussion. The case ξ = 0 is degenerate since there emerge interesting null states in a general ξ = 0 boost multiplet. We specify these null states and work out the resulting selection rules. Then, we compute the ξ = 0 global GCA blocks and find that they can be written as a linear combination of several building blocks, each of which can be obtained from a sl(2, ℝ) Casimir equation. These building blocks allow us to give an Euclidean inversion formula as well. As a consistency check, we study 4-point functions of certain vertex operators in the BMS free scalar theory. In this case, the ξ = 0 sector is the only allowable sector in the propagating channel. We find that the direct expansion of the 4-point function reproduces the global GCA block and is consistent with the inversion formula.

In this paper, we study the Carrollian and Galilean conformal field theories (CCFT and GCFT) in d>2d>2 dimensions. We construct the highest weight representations (HWR) of Carrollian and Galilean conformal algebra (CCA and GCA). Even though the two algebras have different structures, their HWRs share similar structure, because their rotation subalgebras are isomorphic. In both cases, we find that the finite dimensional representations are generally reducible but indecomposable, and can be organized into the multiplets. Moreover, it turns out that the multiplet representations in d>2d>2 CCA and GCA carry not only the simple chain structure appeared in logCFT or 2d2d GCFT, but also more generally the net structures. We manage to classify all the allowed chain representations. Furthermore we discuss the two-point and three-point correlators by using the Ward identities. We mainly focus on the two-point correlators of the operators in chain representations. Even in this relative simple case, we find some novel features: multiple-level structure, shortage of the selection rule on the representations, undetermined 2-pt coefficients, etc.. We find that the non-trivial correlators could only appear for the representations of certain structure, and the correlators are generally polynomials of time coordinates for CCFT (spacial coordinates for GCFT), whose orders depend on the levels of the correlators.