Semi-Chiral Operators in 4d ${\cal N}=1$ Gauge Theories

We discuss the properties of quarter-BPS local operators in four dimensional ${\cal N}=1$ supersymmetric Yang-Mills theory using the formalism of holomorphic twists. We study loop corrections both to the space of local operators and to algebraic operations which endow the twisted theory with an infinite symmetry algebra. We classify all single-trace quarter-BPS operators in the planar approximation for $SU(N)$ gauge theory and propose a holographic dual description for the twisted theory. We classify perturbative quarter-BPS operators in $SU(2)$ and $SU(3)$ gauge theories with sufficiently small quantum numbers and discuss possible non-perturbative corrections to the answer. We set up analogous calculations for some theories with matter.