Discreetly Conformal: Higher-Order Chiral Scalars and the Discrete Series
With its constant shift symmetry gauged, a massless scalar on two-dimensional flat space defines a conformal field theory whose classical phase space is equivalent to that of a chiral and an anti-chiral Floreanini--Jackiw scalar. We show that this picture generalizes: After gauging a larger set of the shift symmetries of the massless scalar field, it decomposes into a pair of generalized chiral and anti-chiral scalars, described by actions with higher-order spatial derivatives. We spell out their physical and gauge symmetries, and show how these give rise to the hidden conformal symmetry of the scalar with gauged shift symmetries. This lifts to de Sitter space, where we find that the discrete series scalars also decompose into these generalized chiral scalars, providing an explanation for their hidden conformal symmetry.