Automorphic Symmetries, String integrable structures and Deformations

Abstract

We address the novel structures arising in quantum and string integrable theories, as well as construct methods to obtain them and provide further analysis. Specifically, we implement the automorphic symmetries on peri- odic lattice systems for obtaining integrable hierarchies, whose commutativity along with integrable transformations induces a generating structure of inte- grable classes. This prescription is first applied to 2-dim and 4-dim setups, where we find the new sl2 sector, su(2) ⊕ su(2) with superconductive modes, Generalised Hubbard type classes and more. The corresponding 2- and 4-dim R matrices are resolved through perturbation theory, that allows to recover an exact result. We then construct a boost recursion that allows to address the systems, whose R-/S-matrices exhibit arbitrary spectral dependence, that also is an apparent property of the scattering operators in AdS integrability. It is then possible to implement the last for Hamiltonian Ansätze in D = 2, 3, 4, which leads to new models in all dimensions. We also provide a method based on a coupled differential system that allows to resolve for R matrices ex- actly. Importantly, one can isolate a special class of models of non-difference form in 2-dim case (6vB/8vB), which provides a new structure consistently arising in AdS3 and AdS2 string backgrounds. We prove that these classes can be represented as deformations of the AdS{2,3} models. We also work out that the latter satisfy free fermion constraint, braiding unitarity, crossing and exhibit deformed algebraic structure that shares certain properties with AdS3 × S3 × M4 and AdS2 × S2 × T 6 models. The embedding and mappings of known AdS{2,3} models to 6vB/8vB deformations are demonstrated, along with a discussion on the associated candidates of sigma models.