GCA in 2d

TL;DR

This paper explores the quantum realizations of the infinite-dimensional Galilean Conformal Algebra in two dimensions, derived from non-unitary 2d CFTs with large, oppositely signed central charges, developing tools for its analysis.

Contribution

It introduces the first detailed quantum mechanical framework for 2d GCA, including representation theory, Ward identities, and correlation functions, based on limits of non-unitary 2d CFTs.

Findings

Constructed explicit GCA four-point functions from differential equations.

Developed a nonrelativistic Kac table for GCA representations.

Validated the solutions against consistency checks.

Abstract

We make a detailed study of the infinite dimensional Galilean Conformal Algebra (GCA) in the case of two spacetime dimensions. Classically, this algebra is precisely obtained from a contraction of the generators of the relativistic conformal symmetry in 2d. Here we find quantum mechanical realisations of the (centrally extended) GCA by considering scaling limits of certain 2d CFTs. These parent CFTs are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We therefore develop, in parallel to the usual machinery for 2d CFT, many of the tools for the analysis of the quantum mechanical GCA. These include the representation theory based on GCA primaries, Ward identities for their correlation functions and a nonrelativistic Kac table. In particular, the null vectors of the GCA lead to differential equations for the four point function. The solution to these equations in the simplest case is explicitly obtained and checked to be consistent with various requirements.