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 Möbius 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 ≤ 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.

我们给一般全纯季度的结构BPS在运营商\（\ mathcal {N} \） =在与U（弱耦合4 SYM Ñ在有限的）规范群Ñ。该构造将Möbius求逆公式用于集划分，并应用于多对称函数，同时还要对对称组的组代数进行计算。我们提出了一种计算算法，该算法为全纯算子空间上的物理内积产生了正交基础。该基础由U（2）杨图，U（N）年轻图和一个附加的plethystic多重性标签。我们描述了四分之一BPS状态的精确计数结果，这些结果有望通过大量巨型引力子的双重计算而重现，包括在四分之一BPS部门内的对称关联球体和AdS巨人。在这种情况下Ñ ≤ Ñ（Ñ是该复合操作者的尺寸）的结构解析，使用多对称函数和U（2）克莱布希-高登系数。BPS算子的计数和相关器可以在基于置换代数的二维拓扑场论中进行编码，并具有适当的缺陷。