Anomalous dimensions from boson lattice models

TL;DR

This paper maps the problem of calculating anomalous dimensions of certain operators in N=4 super Yang-Mills theory to a lattice boson model, including previously neglected terms, to better understand large N dynamics.

Contribution

It introduces a novel boson lattice model approach that incorporates all relevant terms for diagonalizing the dilatation operator in the SU(3) sector.

Findings

Lowest energy states have evenly distributed momenta among giants.

Including neglected terms refines the spectrum of anomalous dimensions.

The lattice boson model captures large N dynamics beyond planar approximation.

Abstract

Operators dual to strings attached to giant graviton branes in AdS$_5\times$S$^5$ can be described rather explicitly in the dual ${\cal N} = 4$ super Yang-Mills theory. They have a bare dimension of order $N$ so that for these operators the large $N$ limit and the planar limit are distinct: summing only the planar diagrams will not capture the large $N$ dynamics. Focusing on the one-loop $SU(3)$ sector of the theory, we consider operators that are a small deformation of a ${1\over 2}-$BPS multi-giant graviton state. The diagonalization of the dilatation operator at one loop has been carried out, but explicit formulas for the operators of a good scaling dimension are only known when certain terms which were argued to be small, are neglected. In this article we include the terms which were neglected. The diagonalization is achieved by a novel mapping which replaces the problem of diagonalizing the dilatation operator with a system of bosons hopping on a lattice. The giant gravitons define the sites of this lattice and the open strings stretching between distinct giant gravitons define the hopping terms of the Hamiltonian. Using the lattice boson model, we argue that the lowest energy giant graviton states are obtained by distributing the momenta carried by the $X$ and $Y$ fields evenly between the giants with the condition that any particular giant carries only $X$ or $Y$ momenta, but not both.