Diagonal multi-matrix correlators and BPS operators in N=4 SYM (0711.0176v3)

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.

Diagonal Multi-Matrix Correlators and BPS Operators in $=4$ SYM

The paper "Diagonal Multi-Matrix Correlators and BPS Operators in $\mathcal{N}=4$ SYM" provides a detailed exploration of the structure and properties of gauge-invariant operators within the framework of $\mathcal{N}=4$ super Yang-Mills theory. The authors develop a comprehensive basis for multi-trace, multi-matrix operators, which achieves a diagonalization in their two-point correlators when evaluated at zero Yang-Mills coupling. This approach generalizes the well-known diagonalization achieved by Schur polynomials in the case of single matrix theories to encompass multiple matrices, intertwining the gauge group and global symmetries through the Clebsch-Gordan coefficients associated with symmetric groups.

Key Methodology and Results

• Complete Diagonal Basis: The central achievement of this paper is the construction of a complete diagonal basis for multi-trace multi-matrix operators. This is achieved through the use of symmetric group representations and their tensor products. A novel transformation utilizing Clebsch-Gordan coefficients enables the basis states to exhibit orthogonality, thereby simplifying the paper of BPS operators in weak coupling limits.
• Gauge Theory and Giant Gravitons: By focusing on the $U(3)$ subgroup of the full symmetry group, the authors elucidate the connection between quarter and eighth-BPS operators in $\mathcal{N}=4$ SYM theory and giant gravitons within $AdS_5 \times S^5$ spacetime. This equivalence is rooted in AdS/CFT correspondence and is facilitated by the diagonalization framework provided.
• Theoretical Implications: The diagonal basis provides essential insights into the comparison of operators transforming within short representations. This characterization aids in identifying the metric space for short representations, thus providing a glimpse into the mechanics and interactions of giant gravitons at strong coupling.
• Counting and Extending to Fermions: The authors deftly handle the counting of operators by leveraging symmetric group theory and Littlewood-Richardson coefficients, ensuring an accurate description of operator dimensions and correlators. Additionally, the paper extends its methodology to incorporate fermionic fields within operators, thereby broadening its applicability across different supergroup symmetries.

Implications for Future Research

The framework established by the authors for diagonalizing multi-matrix operators in $\mathcal{N}=4$ SYM provides a foundation upon which further exploration into the dynamics of supersymmetric states can be built. Future research may explore non-renormalization properties of extremal correlators, explore connections to emergent quantum geometries, and investigate the implications of these findings within reduced matrix models, mapping out novel conserved quantities. Additionally, there is scope to broaden the application of these techniques to other gauge groups beyond $U(N)$, and potentially uncover relationships with topological field theories.

In conclusion, this paper's contribution lies in its systematic approach to diagonalizing complex operator correlators in a high-dimensional symmetry setting, facilitating greater understanding of the intricate relationships between field theory operators and the corresponding holographic descriptions in string theory, while embedding its methodologies in a firm mathematical framework.