Quarter-BPS states, multi-symmetric functions and set partitions

We give a construction of general holomorphic quarter BPS operators in $ \mathcal{N}=4$ SYM at weak coupling with $U(N)$ gauge group at finite $N$. The construction employs the M\"obius inversion formula for set partitions, applied to multi-symmetric functions, alongside computations in the group algebras of symmetric groups. We present a computational algorithm which produces an orthogonal basis for the physical inner product on the space of holomorphic operators. The basis is labelled by a $U(2)$ Young diagram, a $U(N)$ Young diagram and an additional plethystic multiplicity label. We describe precision counting results of quarter BPS states which are expected to be reproducible from dual computations with giant gravitons in the bulk, including a symmetry relating sphere and AdS giants within the quarter BPS sector. In the case $n \leq N$ ($n$ being the dimension of the composite operator) the construction is analytic, using multi-symmetric functions and $U(2)$ Clebsch-Gordan coefficients. Counting and correlators of the BPS operators can be encoded in a two-dimensional topological field theory based on permutation algebras and equipped with appropriate defects.