Chiral conformal field theories are mathematical objects which are related to but distinct from two-dimensional conformal field theories. They transform under a single copy of the Virasoro algebra $\mathcal{V}$ and depend on $z$ but not $\bar z$. Notable examples include bosons compactified on any self-dual Euclidean lattice, and the Monster CFT.
See these notes by Andre Henriques for a mathematical perspective.