AdS 3 integrability, the Sine-Gordon model, and fractional supersymmetry

The massless S matrix of the pure Ramond-Ramond sector of string theory on three-dimensional Anti de Sitter times a four-torus is very similar but also crucially distinct from Fendley-Intriligator’s model famous N = 2 S -matrix, in turn related to Sine-Gordon model taken at a special coupling. In this article we review the reason why supersymmetry emerges but with a fractional statistics of the particles. We then use this to obtain an interpolating S matrix between massless AdS 3 and Sine-Gordon model, and further find two interpolating S matrices between these and the mixed-flux relativistic S matrix, solving the Yang-Baxter equation all the way in between. They are all relativistic invariant and braiding unitary, can be written in terms of particles with fractional statistics, and embed all the AdS 3 models into a parent integrable system. This also achieves the reformulation of all the AdS 3 integrable scattering problems in the domain of fractional statistics.