On Galilean Conformal Bootstrap II: $ξ=0$ sector

Abstract

In this work, we continue our work on two dimensional Galilean conformal field theory (GCFT 2 ). 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 four-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.

Intrinsic Approach to $1+1$D Carrollian Conformal Field Theory

“…It was shown in [22] that the correlation function, first derived in [23], between two fields corresponding to states (by the assumption of state-operator correspondence) transforming under such a 'singlet' representation (as defined in [21]) can arise from a non-local action. The conclusion that this correlator can originate only from a non-local action can be reached by taking a Fourier transformation of the same: the momentumspace form of the Carrollian correlator makes it manifest that the equation of motion, of which this correlator is a Green's function, must contain spatial derivatives of negative order whenever ξ = 0, irrespective of ∆ (notations are as used in [21] whose ξ = 0 counterpart is [24]). This conclusion also applies for any higher rank multiplets [21] with ξ = 0.…”