Conformal Mapping of Non-Lorentzian Geometries in SU(1,2) Conformal Field Theory

TL;DR

This paper constructs an explicit conformal mapping between state and operator pictures in (2+1)-dimensional non-Lorentzian field theories with SU(1,2) symmetry, establishing a concrete state-operator correspondence.

Contribution

It introduces a novel geometric mapping between state and operator descriptions in non-Lorentzian conformal field theories with explicit symmetry realization.

Findings

Derived a conformal mapping between state and operator pictures.

Established a concrete state-operator correspondence in non-Lorentzian CFTs.

Connected null reduction of 4D CFTs to 3D non-Lorentzian geometries.

Abstract

We realize an explicit conformal mapping between the state and operator pictures in a class of (2+1)-dimensional non-Lorentzian field theories with SU(1,2)$\times$U(1) conformal symmetry. The state picture arises from null reducing four-dimensional relativistic conformal field theories on a three-sphere, yielding a non-Lorentzian geometry with the conformal Killing symmetry group SU(1,2). This is complementary to the operator picture recently studied by Lambert et al., where the geometry acquires an $\Omega$-deformation. We then use the geometric mapping between the two pictures to derive a correspondence between the generators. This provides a concrete realization of the state-operator correspondence in non-Lorentzian conformal field theories.