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

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 $\mathrm{\Omega }$-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.