Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1 2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare and differentiate our approach to earlier articles addressing the problem of the construction of the Q-operator for the XXX chain. Finally we speculate on the importance of Q-operators for the physical interpretation of recent proposals for the Y-system of AdS/CFT. 3 k=1 S k J k , (3.5)where J ± = J 1 ± iJ 2 and J 3 are the generators of the sl(2) algebraIn the second form of L(z) in (3.5) we used the 2 × 2 spin operators S k appearing in (2.1). Note that for the L-operator (3.5) equation (3.1) holds on the algebraic level in virtue of the commutation relations (3.6). To obtain a specific solution one needs to choose a particular representation of these commutation relations. For further reference define the highest weight representions of sl(2), with highest weight vector v 0 , defined by the conditions 7)

Constraints of the osp(6|4) symmetry on tree-level scattering amplitudes in N = 6 superconformal Chern-Simons theory are derived. Supplemented by Feynman diagram calculations, solutions to these constraints, namely the four-and six-point superamplitudes, are presented and shown to be invariant under Yangian symmetry. This introduces integrability into the amplitude sector of the theory.

We use modern bootstrap techniques to study half-BPS line defects in 4d N = 4 superconformal theories. Specifically, we consider the 1d CFT with OSP(4 * |4) superconformal symmetry living on such a defect. Our analysis is general and based only on symmetries, it includes however important examples like Wilson and 't Hooft lines in N = 4 super Yang-Mills. We present several numerical bounds on OPE coefficients and conformal dimensions. Of particular interest is a numerical island obtained from a mixed correlator bootstrap that seems to imply a unique solution to crossing. The island is obtained if some assumptions about the spectrum are made, and is consistent with Wilson lines in planar N = 4 super Yang-Mills at strong coupling. We further analyze the vicinity of the strong-coupling point by calculating perturbative corrections using analytic methods. This perturbative solution has the sparsest spectrum and is expected to saturate the numerical bounds, explaining some of the features of our numerical results. B Comments on the derivation of the crossing equations 36 C The analytic solutions to the crossing equations 37 D First order perturbation of D 1 D 1 D 1 D 1 (1,0) 41 8 The function θ(O) is included due to the fact that only operators in D 1 × D 1 contribute to the A {1,1,2,2} function.

We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang-Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang-Baxter equation serve as elementary, "partonic" building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider sl(n) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe ansatz techniques.