The elliptic scattering theory of the 1/2-XYZ and higher order Deformed Virasoro Algebras

Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framework by exploiting the equivalent Non-Linear Integral Equations. Of course, these bound states go to the sine-Gordon breathers in the suitable limit and therefore the scattering factors between them are explicitly computed by inspecting the corresponding Non-Linear Integral Equations. As a consequence, abstracting from the physical model the Zamolodchikov-Faddeev algebra of two $n$-th elliptic breathers defines a tower of $n$-order Deformed Virasoro Algebras, reproducing the $n=1$ case the usual well-known algebra of Shiraishi-Kubo-Awata-Odake \cite{SKAO}.Comment: Latex version, 13 page

Beyond cusp anomalous dimension from integrability in SYM4_4

We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. This term is still governed by a linear integral equation which we study in the weak and strong coupling regimes. In the strong coupling regime we find agreement with the string theory computationsComment: 5 pages, contribution to the proceedings of the workshop Diffraction 2010, Otranto, 10th-15th September, talk given by M.Rossi; v2: references adde

Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops

Moving from Beisert-Staudacher equations, the complete set of Asymptotic Bethe Ansatz equations and $S$-matrix for the excitations over the GKP vacuum is found. The resulting model on this new vacuum is an integrable spin chain of length $R=2\ln s$ ($s=$ spin) with particle rapidities as inhomogeneities, two (purely transmitting) defects and $SU(4)$ (residual R-)symmetry. The non-trivial dynamics of ${\cal N}=4$ SYM appears in elaborated dressing factors of the 2D two-particle scattering factors, all depending on the 'fundamental' one between two scalar excitations. From scattering factors we determine bound states. In particular, we study the strong coupling limit, in the non-perturbative, perturbative and giant hole regimes. Eventually, from these scattering data we construct the $4D$ pentagon transition amplitudes (perturbative regime). In this manner, we detail the multi-particle contributions (flux tube) to the MHV gluon scattering amplitudes/Wilson loops (OPE or BSV series) and re-sum them to the Thermodynamic Bubble Ansatz.Comment: 103 pages; typos corrected, references added: journal versio

On the scattering over the GKP vacuum

By converting the Asymptotic Bethe Ansatz (ABA) of ${\cal N}=4$ SYM into non-linear integral equations, we find 2D scattering amplitudes of excitations on top of the GKP vacuum. We prove that this is a suitable and powerful set-up for the understanding and computation of the whole S-matrix. We show that all the amplitudes depend on the fundamental scalar-scalar one.Comment: final version, 14 pages, to appear in Physics Letters

On the finite size corrections of anti-ferromagnetic anomalous dimensions in N=4{\cal N}=4 SYM

Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formul\ae and compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations.Comment: 38 pages, Latex revised version: draft formulae indicator deleted, one reference added, typos corrected, few minor text modification

Mal'cev classes of left-quasigroups and Quandles

In this paper we investigate some Mal'cev classes of varieties of left-quasigroups. We prove that the weakest Mal'cev condition for a variety of left-quasigroup is having a Mal'cev term. Then we specialize to the setting of quandles for which we prove that the meet semidistributive varieties are those which have no finite models