Group theoretical evaluation of the action of the dilatation operator in the large N limit

Abstract

Restricted Schur polynomials can be used to describe large N, non-planar limits of N = 4 super Yang-Mills theory. The R-symmetry generators commute with the dilatation operator. For small deformations of 1 2-BPS operators, the matrix elements of these generators have been computed and a set of recursion relations for the matrix elements of the dilatation operator are obtained from this commutation relation. Together with the knowledge that the smallest eigenvalues of the dilatation operator (corresponding to BPS operators) vanish, these recursion relations can be used to determine the matrix elements of the dilatation operator. Studies up to now have computed the matrix elements of the su(2) generators in the displaced corners approximation. Our first novel result is the computation of the exact su(2) generators. We obtain the matrix elements for the su(3) generators in the displaced corners approximation and exactly, for the first time. This is the first step to computing exact matrix elements of the dilatation operator.