BPS operators in $$ \mathcal{N}=4 $$ SO(N) super Yang-Mills theory: plethysms, dominoes and words

Abstract

Abstract Permutations and associated algebras allow the construction of half and quarter BPS operators in maximally supersymmetric Yang Mills theory with U(N), SO(N) and Sp(N) gauge groups. The construction leads to bases for the operators, labelled by Young diagrams and associated group theory data, which have been shown to be orthogonal under the inner product defined by the free field two-point functions. In this paper, we study in detail the orientifold projection map between the Young diagram basis for U(N) theories and the Young diagram basis for SO(N) (and Sp(N)) half-BPS operators. We find a simple connection between this map and the plethystic refinement of the Littlewood Richardson coefficients which couple triples of Young diagrams where two of them are identical. This plethystic refinement is known to be computable using an algorithm based on domino tilings of Young diagrams. We discuss the domino combinatorics of the orientifold projection map in terms of giant graviton branes. The permutation construction of SO(N) operators is used to find large N generating functions for both the half and quarter BPS sectors. The structure of these generating functions is elucidated using the combinatorics of words, organised according to the number of periods. Aperiodic words in the U(N) theory are Lyndon words and an SO(N ) analogue of Lyndon words is defined using a minimally periodic condition. We calculate the normalization factor for the orthogonal basis of Young diagram operators in the baryonic sector of SO(N).