BMN Gauge Theory as a Quantum Mechanical System

TL;DR

This paper derives an effective quantum mechanical Hamiltonian from N=4 gauge theory in the BMN limit, simplifying the calculation of anomalous dimensions and extending to higher genus corrections.

Contribution

It introduces a quantum mechanical reformulation of gauge theory that reproduces known results and facilitates computations of higher-genus corrections.

Findings

Reproduces genus one anomalous dimension with a one-line calculation

Calculates non-zero genus two correction efficiently

Suggests potential extension to higher loops and more impurities

Abstract

We rigorously derive an effective quantum mechanical Hamiltonian from N=4 gauge theory in the BMN limit. Its eigenvalues yield the exact one-loop anomalous dimensions of scalar two-impurity BMN operators for all genera. It is demonstrated that this reformulation vastly simplifies computations. E.g. the known anomalous dimension formula for genus one is reproduced through a one-line calculation. We also efficiently evaluate the genus two correction, finding a non-vanishing result. We comment on multi-trace two-impurity operators and we conjecture that our quantum-mechanical reformulation could be extended to higher quantum loops and more impurities.