One-cycle negative geometries in N=4 SYM have singularities only at z=-1, 0, and infinity to all loop orders.
Unwinding the Amplituhedron in Binary
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized 'winding number'. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of 'sign flips' of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural 'dual' of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar N = 4 SYM as a differential form on the space of physical kinematical data.
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Four-loop four-point correlator integrand in planar N=4 SYM decomposes into chamber forms identical to three loops times local integrands, with leading singularities as linear combinations of those forms and a diagonalized pure-function representation including one pure elliptic integrand.
Explicit three-loop computation of negative geometries for F(g,z) with all-loop resummation of one-cycle diagrams and extraction of the cusp anomalous dimension via z-integration.
citing papers explorer
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Landau Analysis of One-Cycle Negative Geometries
One-cycle negative geometries in N=4 SYM have singularities only at z=-1, 0, and infinity to all loop orders.
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Leading singularities and chambers of Correlahedron
Four-loop four-point correlator integrand in planar N=4 SYM decomposes into chamber forms identical to three loops times local integrands, with leading singularities as linear combinations of those forms and a diagonalized pure-function representation including one pure elliptic integrand.
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Multi-Loop Negative Geometries
Explicit three-loop computation of negative geometries for F(g,z) with all-loop resummation of one-cycle diagrams and extraction of the cusp anomalous dimension via z-integration.