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).
Highlights
The AdS/CFT correspondence [1,2,3] allows us to use the physics of N = 4 U(N ) super Yang-Mills to find new properties of type IIB string theory on AdS5 × S5
The question we are asking can be posed as: how does the orientifold projection operation of string theory act on the Young diagram operators of the U(N ) half-BPS sector to produce the Young diagram operators of the SO(N ) theory? Surprisingly, we find that this question, in the case of the half-BPS sector, has a simple and elegant answer in terms of a classic concept in the combinatorics of Young diagrams, called plethysms of Young diagrams
The second main result of this paper is to develop the analogous picture for the counting of SO(N ) gauge invariant operators made from two matrices, i.e. the free field quarter-BPS sector of N = 4 SYM
Summary
Aperiodic multi-traces (appearing in the right box of the figure) are in one to one correspondence with the free monoid generated by two letters x, y This builds on known results concerning Lyndon words [34], which play an important role in the field of “combinatorics on words”, an area with diverse applications in mathematics [35, 36]. The second main result of this paper is to develop the analogous picture for the counting of SO(N ) gauge invariant operators made from two matrices, i.e. the free field quarter-BPS sector of N = 4 SYM This involves defining an analogue of the notion of Lyndon words,.
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