The dynamics of two interacting giant

Abstract

In this thesis, the large N limit of the anomalous dimension of operators in N = 4 super Yang-Mills theory described by restricted Schur Polynomials are studied. The operators studied in this thesis are labelled by Young Di- agrams which have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with n 6= 2 columns. The planar approximation does not does not capture the large N dynamics. The dilata- tion operator is explicitly evaluated for 2, 3, and 4 impurities. In all three cases, for a certain limit, the dilatation operator is a discretized version of the second derivative de ned on a lattice emerging from the Young Diagram itself. The dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of 8g2 Y M and there are interesting degeneracies in the spectrum. The spectrum obtained in this thesis for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. The equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian de ned by the one loop large N dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both the planar and the non-planar diagrams.