I will give an overview of recent progress in understanding string theory in AdS3 backgrounds with a mixture of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three-form flux. Such theories are integrable, but provide many features not encountered in the more familiar case of pure Ramond-Ramond flux. In this talk I will explore the analytic structure of the dispersion relation of the world-sheet excitations and how it relates to the crossing equations of the two-particle S matrix. Determining the dressing phases of the mixed flux S matrix is the next major step in using integrability to the AdS3/CFT2 correspondence. -------- Part of London Integrability Journal Club. New participants please register at integrability-london.weebly.com to receive the link.

The ABJM model is a superconformal Chern-Simons theory with N = 6 supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension. We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.