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 work, we study two-dimensional Galilean field theories with global translations and anisotropic scaling symmetries. We show that such theories have enhanced local symmetries, generated by the infinite dimensional spin-l Galilean algebra with possible central extensions, under the assumption that the dilation operator is diagonalizable and has a discrete and non-negative spectrum. We study the Newton-Cartan geometry with anisotropic scaling, on which the field theories could be defined in a covariant way. With the well-defined Newton-Cartan geometry we establish the state-operator correspondence in anisotropic Galilean conformal field theory and determine the two-point functions of primary operators.

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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 work, we study the tensionless (super)string in the formalism of path-integral quantization. We introduce BMS bc and βγ ghosts intrinsically by accounting for the Faddeev-Popov determinants appeared in fixing the gauges. We then do canonical quantization and obtain the critical dimensions for different tensionless strings. We find that among four kinds of tensionless superstrings, the N = 2 homogeneous and inhomogeneous doublet tensionless superstrings have the same critical dimension as the usual superstrings. Taking the BMS bc and βγ ghosts as new types of BMS free field theories, we find that their enhanced underlying symmetries are generated by BMS-Kac-Moody algebras, with the Kac-Moody subalgebras being built from a three-dimensional non-abelian and nonsemi-simple Lie algebra.