A Novel Long Range Spin Chain and PlanarN=4 Super Yang-Mills

We probe the long-range spin chain approach to planar N=4 gauge theory athigh loop order. A recently employed hyperbolic spin chain invented byInozemtsev is suitable for the SU(2) subsector of the state space up to threeloops, but ceases to exhibit the conjectured thermodynamic scaling propertiesat higher orders. We indicate how this may be bypassed while neverthelesspreserving integrability, and suggest the corresponding all-loop asymptoticBethe ansatz. We also propose the local part of the all-loop gauge transfermatrix, leading to conjectures for the asymptotically exact formulae for alllocal commuting charges. The ansatz is finally shown to be related to astandard inhomogeneous spin chain. A comparison of our ansatz to semi-classicalstring theory uncovers a detailed, non-perturbative agreement between thecorresponding expressions for the infinite tower of local charge densities.However, the respective Bethe equations differ slightly, and we end by refiningand elaborating a previously proposed possible explanation for thisdisagreement.Comment: 48 pages, 1 figure. v2, further results added: discussion of the relationship to an inhomogeneous spin chain, normalization in sec 3 unified, v3: minor mistakes corrected, published versio