We begin a study of higher-loop corrections to the dilatation generator of N = 4 SYM in non-compact sectors. In these sectors, the dilatation generator contains infinitely many interactions, and therefore one expects very complicated higher-loop corrections. Remarkably, we find a short and simple expression for the two-loop dilatation generator. Our solution for the non-compact su(1, 1|2) sector consists of nested commutators of four O(g 1 ) generators and one simple auxiliary generator. Moreover, the solution does not require the planar limit; we conjecture that it is valid for any gauge group. To obtain the two-loop dilatation generator, we find the complete O(g 3 ) symmetry algebra for this sector, which is also given by concise expressions. We check our solution using published results of direct field theory calculations. By applying the expression for the two-loop dilatation generator to compute selected anomalous dimensions and the bosonic sl(2) sector internal S-matrix, we confirm recent conjectures of the higher-loop Bethe ansatz of hep-th/0412188. 1 Integrability in four-dimensional Yang-Mills theory was first observed by Lipatov [5].
The complete spin chain representation of the planar N = 4 SYM dilatation generator has long been known at one loop, where it involves leading nearest-neighbor 2 → 2 interactions. In this work we use superconformal symmetry to derive the unique solution for the leading L → 2 interactions of the planar dilatation generator for arbitrarily large L. We then propose that these interactions are given by the scattering operator that has N = 4 SYM tree-level scattering amplitudes as matrix elements. We provide compelling evidence for this proposal, including explicit checks for L = 2, 3 and a proof of consistency with superconformal symmetry.
Bethe ansatz equations have been proposed for the asymptotic spectral problem of AdS 4 /CF T 3 . This proposal assumes integrability, but the previous verification of weak-coupling integrability covered only the su(4) sector of the ABJM gauge theory. Here we derive the complete planar two-loop dilatation generator of N = 6 superconformal Chern-Simons theory from osp(6|4) superconformal symmetry. For the osp(4|2) sector, we prove integrability through a Yangian construction. We argue that integrability extends to the full planar two-loop dilatation generator, confirming the applicability of the Bethe equations at weak coupling. Further confirmation follows from an analytic computation of the two-loop twist-one spectrum.Note the symmetry between the supercharge actions on barred and unbarred states, only differing by a minus sign and appropriate interchanges of Gothic and Latin indices. Finally, the action of the J (independent of R andR indices) is the same for barred and unbarred states. Representing both φ a andφ a with φ, and ψ a andψ a with ψ, we have J 11 φ (n) = (n + 1 2 )(n + 1) φ (n+1) , J 11 ψ (n) = (n + 1)(n + 3 2 ) ψ (n+1) , J 22 φ (n) = (n − 1 2 )n φ (n−1) , J 22 ψ (n) = n(n + 1 2 ) ψ (n−1) , J 12 φ (n) = (n + 1 4 ) φ (n) , J 12 ψ (n) = (n + 3 4 ) ψ (n) .(2.31)
Strong evidence indicates that the spectrum of planar anomalous dimensions of N = 4 super Yang-Mills theory is given asymptotically by Bethe equations. A curious observation is that the Bethe equations for the psu(1, 1|2) subsector lead to very large degeneracies of 2 M multiplets, which apparently do not follow from conventional integrable structures. In this article, we explain such degeneracies by constructing suitable conserved nonlocal generators acting on the spin chain. We propose that they generate a subalgebra of the loop algebra for the su(2) automorphism of psu(1, 1|2). Then the degenerate multiplets of size 2 M transform in irreducible tensor products of M two-dimensional evaluation representations of the loop algebra.
We develop algebraic methods for finding loop corrections to the N = 4 SYM dilatation generator, within the noncompact psu(1, 1|2) sector. This sector gives a 't Hooft coupling λ-dependent representation of psu(1, 1|2) × psu(1|1) 2 . At first working independently of the representation, we present an all-order algebraic ansatz for the λ-dependence of this Lie algebra's generators. The ansatz solves the symmetry constraints if an auxiliary generator, h, satisfies certain simple commutation relations with the Lie algebra generators. Applying this to the psu(1, 1|2) sector leads to an iterative solution for the planar three-loop dilatation generator in terms of leading order symmetry generators and h, which passes a thorough set of spectral tests. We argue also that this algebraic ansatz may be applicable to the nonplanar theory as well.While AdS/CFT provides a powerful weak-strong duality, finding the weak-to strongcoupling interpolation of unprotected physical quantities generically remains a very difficult problem. However, for planar N = 4 SYM and its string theory dual, integrability [1] leads to great simplifications. Here finding anomalous dimensions of single-trace local operators is equivalent to the spectral problem of an integrable spin chain [1][2][3]. Due to integrability, the spectral problem at weak and strong coupling can be reduced to solving a system of Bethe equations [4][5][6][7]. In fact, superconformal symmetry fixes the asymptotic Bethe equations up to an overall phase [8,9], which is constrained by crossing symmetry [10]. Following a proposal for the phase at large λ [11], an all-order solution for the phase was found [12] simultaneously and completely consistently with a four-loop gauge theory calculation of the cusp anomalous dimension [13].Now through an integral equation [14,12], the asymptotic Bethe equations apparently give the planar cusp anomalous dimension's interpolation from weak to strong coupling [15]. The asymptotic Bethe equations also pass multiple tests in the near-flat-space limit [16]. Furthermore, the asymptotic spectrum of BPS bound states [17] is consistently reflected by the analytic structure of the phase [18]. Finally, recent work has focused on the scaling function for the minimal anomalous dimensions of long operators with Lorentz spin growing exponentially with twist [19]. At strong coupling, the scaling function (in a more specialized limit) can be computed using a relation to the O(6) sigma model [20], as has been checked at two loops [21]. From the asymptotic Bethe equations, [22] derived a generalized integral equation for the scaling function, which interpolates from weak to strong coupling [23] in perfect agreement with the previous results. For additional related work see [24].Despite these impressive results, there are questions that remain challenging even for the Bethe ansatz approach. Integrability is an assumption, and it seems that other methods will be required to verify that integrability is preserved by quantum corrections for all values of λ. Also,...
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