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 periodic lattice systems for obtaining integrable hierarchies, whose commutativity along with integrable transformations induces a generating structure of integrable classes. This prescription is first applied to 2-dim and 4-dim setups, where we find the new sl 2 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 exactly. 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 AdS 3 and AdS 2 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 AdS 3 × S 3 × M 4 and AdS 2 × S 2 × 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.

Infinite-dimensional R-matrices for the relativistic scattering of massless modes on AdS2

“…The change of variables -also used in AdS 3 [56][57][58][59] -actually admits a generalisation to massive modes, which partially puts the S-matrix in difference form [57][58][59]. Second, the construction of an S-matrix that interpolates between AdS 2 and AdS 3 [60], 1 based itself on previous work [61][62][63][64][65], and the determination of the associated symmetry superalgebra.…”