Intergrability in giant graviton dynamics

Abstract

In this thesis we will study a system of giant gravitons with strings stretched between them. These strings can assume an excited state and these excitations can be described as particles known as magnons. We discuss this system in the AdS/CFT context, where we focus on the SU(2) and SU(3) subsectors of the N = 4 super Yang-Mills theory. We consider the anomalous dimensions of restricted Schur polynomials constructed using n ∼ O(N) complex adjoint scalars Z and m complex adjoint scalars Y . These operators belong to the SU(2) sector of the theory. We fix m n so that our operators are almost half BPS. At leading order in m n this system corresponds to a dilute gas of m free magnons. Adding the first correction of order m n to the anomalous dimension, which arises at two loops, we find nonzero magnon interactions. The form of this new operator mixing is studied in detail for a system of two giant gravitons with four strings attached. We also consider computing the anomalous dimensions for operators belonging to the SU(3) sector of the theory, and with bare dimension of order N. These operators are built out of n Z, m Y and p X complex adjoint scalars. n, m and p all scale as N in the large N limit, and we fix n p+m and m p ∼ 1 so that our operators are again almost half BPS. For these operators the large N limit and the planar limit are distinct and summing only the planar diagrams will not capture the large N dynamics. Previous studies have computed the spectrum of anomalous dimensions for this class of operators, but neglected terms which were argued to be small. Dropping these terms and diagonalizing the dilatation operator reduces to diagonalizing a set of decoupled oscillators. We compute explicitely the terms which were neglected and show that diagonalizing the dilatation operator still reduces to diagonalizing a set of decoupled oscillators. Our key insight is that the problem is equivalent to bosons hopping on a lattice.