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

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