From counting to construction of BPS states in N=4 SYM

Abstract

We describe a universal element in the group algebra of symmetric groups, whose characters provides the counting of quarter and eighth BPS states at weak coupling in N=4 SYM, refined according to representations of the global symmetry group. A related projector acting on the Hilbert space of the free theory is used to construct the matrix of two-point functions of the states annihilated by the one-loop dilatation operator, at finite N or in the large N limit. The matrix is given simply in terms of Clebsch-Gordan coefficients of symmetric groups and dimensions of U(N) representations. It is expected, by non-renormalization theorems, to contain observables at strong coupling. Using the stringy exclusion principle, we interpret a class of its eigenvalues and eigenvectors in terms of giant gravitons. We also give a formula for the action of the one-loop dilatation operator on the orthogonal basis of the free theory, which is manifestly covariant under the global symmetry.

41 pages + Appendices, 4 figures; v2 - refs and acknowledgments added

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• Quarter-BPS states, multi-symmetric functions and set partitions
• Multi-matrix models and Noncommutative Frobenius algebras obtained from symmetric groups and Brauer algebras
• Flavour singlets in gauge theory as Permutations
• Coherent state operators, giant gravitons, and gauge-gravity correspondence
• CFT4 as SO(4,2)-invariant TFT2
• Coherent state excitations and string-added coherent states in gauge-gravity correspondence
• SO(N) restricted Schur polynomials
• Restricted Schurs and correlators for SO(N) and Sp(N)
• Free BMN Correlators With More Stringy Modes