On Galilean conformal bootstrap

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 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.

On Higher-dimensional Carrollian and Galilean Conformal Field Theories

“…It is worth noticing that the Casimir operators (2.5) acting on the multiplet representations have nonvanishing kernels. This also happens in 2d GCFT with the multiplets [25],…”

“…There is no such strangely behaved invariants for the insertions in the Galilean spacetime. It can be checked that for general d > 2 GCA, the invariants are similar with 2d GCA [25]. For 4-pt insertions in 4d GCA, the two invariants are: (2.25)…”

“…The finite dimensional representations give logarithmic multiplets, e.g. [55] and boost multiplets [25] respectively. For CFT d , with d ≥ 3, the rotation group SO(d) or SO(d − 1, 1) is semisimple, and finite dimensional representations are completely reducible.…”

“…In [23,24], the Bondi-Metzner-Sachs(BMS) bootstrap was studied. In [25][26][27], two-dimensional Galilean conformal bootstrap has been initiated.…”

Selected Topics in Analytic Conformal Bootstrap: A Guided Journey

“…One can also consider theories with modified conformal symmetries. Examples include nonrelativistic theories with Schrödinger, Carrollian and Galilean symmetries [320][321][322][323][324][325][326][327][328][329][330].…”