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

TL;DR

This paper develops a universal algebraic framework for counting and constructing BPS states in N=4 SYM, linking combinatorial group theory with physical observables and giant graviton interpretations.

Contribution

It introduces a universal element in the symmetric group algebra for counting BPS states and constructs a matrix of two-point functions using group-theoretic coefficients, applicable at finite and large N.

Findings

Counting of BPS states refined by global symmetry representations

Explicit construction of the two-point function matrix using symmetric group Clebsch-Gordan coefficients

Interpretation of eigenvalues and eigenvectors in terms of giant gravitons

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.