Holomorphic structure of massive scalar fields in $\text{(A)dS}_2$

Scalar field theories in $\text{(A)dS}_{2}$ with integer scaling dimensions $Δ= k+1$ are characterised by the existence of a pair of (anti-)holomorphic higher-spin currents. We explore the consequences of this to describe their quantisation and subsets of their linear and non-linear symmetries, taking care to treat $\text{AdS}_{2}$ and $\text{dS}_{2}$ separately. In particular, we point out that the theories admit mode expansions reminiscent of standard two-dimensional conformal field theories in complex coordinates, with which we are able to construct operators implementing global conformal and Virasoro symmetry. We further leverage holomorphicity of the currents to show that the full set of symmetries of theories with $k>0$ is captured by a chiral algebra, which is a subalgebra of the one in the $k=0$ (massless) theory. This allows us to identify integrable deformations for $k \in \{0,1,2\}$. We finally observe that a lack of integrable deformations for $k>2$ is a consequence of a known conjecture.