Long-range psu(2,2\4) Bethe ansatze for gauge theory and strings

We generalize various existing higher-loop Bethe ansatze for simple sectors of the integrable long-range dynamic spin chain describing planar N = 4 super-Yang-Mills theory to the full psu(2, 2 vertical bar 4) symmetry and, asymptotically, to arbitrary loop order. We perform a large number of tests of our conjectured equations, such as internal consistency, comparison to direct three-loop diagonalization and expected thermodynamic behavior. In the special case of the su(1 vertical bar 2) subsector, corresponding to a long-range t-J model, we are able to derive, up to three loops, the S-matrix and the associated nested Bethe ansatz from the gauge theory dilatation operator. We conjecture novel all-order S-matrices for the su(1 vertical bar 2) and su(1, 1 vertical bar 2) subsectors, and show that they satisfy the Yang-Baxter equation. Throughout the paper, we muse about the idea that quantum string theory on AdS(5) x S-5 is also described by a psu(2, 2 vertical bar 4) spin chain. We propose asymptotic all-order Bethe equations for this putative "string chain", which differ in a systematic fashion from the gauge theory equations. (c) 2005 Published by Elsevier B.V.