Abstract
The massless ๐ 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-Intriligatorโs model famous ๐ฉ =2 ๐-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 ๐ matrix between massless AdS3 and Sine-Gordon model, and further find two interpolating ๐ matrices between these and the mixed-flux relativistic ๐ 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 AdS3 models into a parent integrable system. This also achieves the reformulation of all the AdS3 integrable scattering problems in the domain of fractional statistics.