The Bethe-Ansatz for N=4 Super Yang-Mills
J. A. Minahan, K. Zarembo
TL;DR
This work maps the one-loop mixing matrix of scalar operators in N=4 SYM to the Hamiltonian of an integrable SO(6) vector-spin chain and demonstrates that the spectrum of anomalous dimensions can be extracted via the Bethe ansatz. It provides explicit BMN results for two impurities, extends to many impurities with first-order 1/J corrections, and leverages Reshetikhin's construction to obtain exact large-dimension results, establishing a square-root relation between anomalous dimensions and string level in the weak-coupling regime. The study both confirms known BMN outputs and broadens the operator classes accessible to exact Bethe-ansatz analysis, signaling strong integrability structures in the gauge/string duality. It also highlights how large-excitation analyses yield insights into the connection between gauge theory spectra and string-theoretic expectations, including the square-root scaling of the full dimension with the level.
Abstract
We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.