A study of giant graviton dynamics in the restricted schur polynomial basis

Abstract

Anomalous dimensions are calculated for a certain class of operators in the restricted Schur polynomial basis in the large N limit. A new computationally simple form of the dilatation operator is derived and used in this dissertation. The class of operators investigated have bare dimension of O(N). Thus the calculation necessarily sums non-planar Feynmann diagrams as the planar approximation has broken down for operators of this size. The operators investigated have two long columns and the operators mix under the action of the dilatation operator, however the mixing of operators having a different number of columns is suppressed and can be neglected in the large N limit. The action of the one loop dilatation operator is explicitly calculated for the cases where the operators have two, three and four impurities and it is found that in a particular limit the action of the one loop dilatation operator reduces to that of a discrete second derivative. The lattice on which the discretised second derivative is defined is provided by the Young tableaux itself. The one loop dilatation operator is diagonalised numerically and produces a surprisingly simple linear spectrum, with interesting degeneracies. The spectrum can be understood in terms of a collection of harmonic oscillators. The frequencies of the oscillators are all multiples of 8g2Y M and can be related to the set of Young tableaux acted upon by the dilatation operator. This equivalence to harmonic oscillators generalises on previously found results in the BPS sector, and suggests that the system is integrable. The work presented here is based primarily on research carried out by R.de Mello Koch, V De Comarmond, and K. Jefferies in [1].