<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> </mml:math> integrability, the Sine-Gordon model, and fractional supersymmetry

The massless <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> </a:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi mathvariant="script">N</c:mi> <c:mo>=</c:mo> <c:mn>2</c:mn> </c:math> <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>S</f:mi> </f:math> -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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>S</h:mi> </h:math> matrix between massless <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi>AdS</j:mi> <j:mn>3</j:mn> </j:msub> </j:math> and Sine-Gordon model, and further find two interpolating <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>S</l:mi> </l:math> matrices between these and the mixed-flux relativistic <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>S</n:mi> </n:math> 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 <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msub> <p:mi>AdS</p:mi> <p:mn>3</p:mn> </p:msub> </p:math> models into a parent integrable system. This also achieves the reformulation of all the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>AdS</r:mi> <r:mn>3</r:mn> </r:msub> </r:math> integrable scattering problems in the domain of fractional statistics.