Momentum Space Correlation Functions in 2D Galilean Conformal Algebra

TL;DR

The paper tackles the problem of defining and computing momentum-space correlators in the 2D Galilean Conformal Algebra, addressing the challenge that real-space correlators grow exponentially and their Fourier transforms do not exist. By deriving momentum-space Ward identities and performing an analytic continuation of the boost eigenvalues ($\xi \to -i\xi$), it obtains well-defined two- and three-point functions that exactly match the Fourier transforms of the corresponding position-space correlators. The work confirms these momentum-space results through a nonrelativistic limit of the relativistic CFT two-point function as an independent check, and demonstrates consistency between position- and momentum-space approaches for correlators in $GCA_2$. These results lay the groundwork for a momentum-space bootstrap program in nonrelativistic conformal theories and potentially shed light on holographic dualities involving BMS$_3$ and flat-space contexts.

Abstract

Galilean Conformal Algebra (GCA) arises as a controlled nonrelativistic limit of the relativistic conformal algebra. In this paper, we initiate the study of momentum space correlation functions in two-dimensional GCA. We derive and solve momentum space Ward identities to obtain two-point and three-point functions. However, relating them to position space correlation functions presents a challenge as Fourier transforms of the latter do not exist. This is resolved by analytically continuing the boost eigenvalues to imaginary values. In this regime, the Fourier transform of the position space two-point and three-point functions exist and match exactly with the momentum space two-point and three-point function obtained by solving the Ward identities.