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The formalism of momentum twistors realises the kinematic space of 4-d planar N=4 SYM amplitudes as a configuration space of lines in P^3. All massless kinematics corresponds to consecutive lines intersecting which in turn is interpreted as a configuration of points in P^3, while massive kinematics corresponds to generic configurations of lines. I’ll give an overview of some current work in progress which aims to understand and relate cluster structures on each of these configuration spaces with an eye towards computing amplitudes using the symbol bootstrap. A key idea is to use “partial abelianiziation” of framed PGL_4 local systems on a surface to produce GL_2 local systems on a two fold spectral cover of the surface

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