Diagonal multi-matrix correlators and BPS operators in N=4 SYM

TL;DR

This work constructs a complete diagonal basis for holomorphic multi-trace, multi-matrix operators in free $U(N)$ gauge theories with a global $U(M)$ symmetry, extending Schur‑polynomial diagonalisation to multiple matrices via Schur–Weyl duality and symmetric group Clebsch–Gordan data. Operators are labeled by $(\Lambda,\mu,R,\beta,\tau)$ and organized through a projection that yields a diagonal two-point function at zero coupling and finite $N$, with explicit formulas for the metric and a finite-$N$ projector on traces. The formalism naturally extends to fermions (via $U(M_1|M_2)$), and to extremal higher-point correlators, providing a framework that connects to the chiral ring, giant gravitons, and reduced multi-matrix quantum mechanics. These results offer a concrete basis suitable for comparing gauge-invariant operators with AdS$_5\times$S$^5$ giant gravitons across coupling regimes and finite-$N$ effects, and they propose a path toward understanding the spacetime interpretation of the diagonal basis. Overall, the paper delivers a robust group-theoretic construction that diagonalises holomorphic operator correlators and relates them to holographic duals in a finite-$N$ setting.

Abstract

We present a complete basis of multi-trace multi-matrix operators that has a diagonal two point function for the free matrix field theory at finite N. This generalises to multiple matrices the single matrix diagonalisation by Schur polynomials. Crucially, it involves intertwining the gauge group U(N) and the global symmetry group U(M) with Clebsch-Gordan coefficients of symmetric groups S_n. When applied to N=4 super Yang-Mills we consider the U(3) subgroup of the full symmetry group. The diagonalisation allows the description of a dual basis to multi-traces, which permits the characterisation of the metric on operators transforming in short representations at weak coupling. This gives a framework for the comparison of quarter and eighth-BPS giant gravitons of AdS_5 x S^5 spacetime to gauge invariant operators of the dual N=4 SYM.

Figures (7)

• Figure 1: two-point function for covariant operator
• Figure 2: two-point function for the gauge-invariant operator
• Figure 3: Allowed shape for the Young tableau of the representations $\Lambda$ of $U(M_1|M_2)$.
• Figure 4: Diagram for three-point function
• Figure 5: Diagram for four-point function