All loop BMN state energies from matrices

Abstract

We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S 3 . For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S 5 which we identify with the sphere on the dual geometry AdS 5 × S 5 . We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N = 4 SYM planar spin chain model.

Anomalous dimensions of heavy operators from magnon energies

“…when l 2 > l 1 + 1. 6 In the situation that the magnons are adjacent, we find Two simple checks of this result are 1. We see that R 12 R 21 = 1.…”

“…Once the central charge of the spin-chain/worldsheet excitations have been determined, their spectrum and constraints on their two body scattering are determined. A powerful conclusion argued for in [7] using the physical picture developed in [6] is that there is a natural geometric interpretation for these central charges in the classical string theory. This geometric interpretation also proved useful in the analysis of maximal giant gravitons in [5].…”

“…We are using the non-local notation of [4]. Using the representation introduced above 6) so that k k = 2 a k b k . An identical argument using the S a α supercharges gives k * k = 2 c k d k .…”

Commuting quantum matrix models

“…[15]- [18]. The corresponding saddle point equation is given by: 22) comparing this to equation (2.6) in ref.…”

“…[10] for the case ξ = 0. For ξ = 0 one can show that the ground state wave function is given by [10]: 15) which is the naive generalisation to higher dimensions of the wave function (2.30). Note that this is an exact result at any N .…”