  Rydberg chains provide an appealing platform for probing conformal field
theories (CFTs) that capture universal behavior in a myriad of physical
settings. Focusing on a Rydberg chain at the Ising transition separating charge
density wave and disordered phases, we establish a detailed link between
microscopics and low-energy physics emerging at criticality. We first construct
lattice incarnations of primary fields in the underlying Ising CFT including
chiral fermions -- a nontrivial task given that the Rydberg chain Hamiltonian
does not admit an exact fermionization. With this dictionary in hand, we
compute correlations of microscopic Rydberg operators, paying special attention
to finite, open chains of immediate experimental relevance. We further develop
a method to quantify how second-neighbor Rydberg interactions tune the sign and
strength of four-fermion couplings in the Ising CFT. Finally, we determine how
the Ising fields evolve when four-fermion couplings drive an instability to
Ising tricriticality. Our results pave the way to a thorough experimental
characterization of Ising criticality in Rydberg arrays, and can inform the
design of novel higher-dimensional phases based on coupled critical chains.
