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

TL;DR

This paper explores the construction and combinatorics of BPS operators in $SO(N)$ super Yang-Mills theory, revealing connections to plethystic refinement, domino tilings, and Lyndon words, with implications for large N generating functions.

Contribution

It introduces a detailed analysis of the orientifold projection map for $SO(N)$ operators and links it to domino tilings and Lyndon words, advancing the combinatorial understanding of these operators.

Findings

Established a connection between the orientifold projection and domino tilings.

Derived large N generating functions using permutation constructions.

Calculated normalization factors for orthogonal basis in the baryonic sector.

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. The structure of these generating functions is elucidated using the combinatorics of words, organised according to their periodicity. Aperiodic words in the $U(N)$ theory are Lyndon words and an $SO(N)$ analogue of Lyndon words is defined using a minimal periodicity condition. We calculate the normalization factor for the orthogonal basis of Young diagram operators in the baryonic sector of $SO(N)$.