Positive geometry beyond planar N=4 super-Yang-Mills

4 problems

• Positive geometry for de Sitter correlators of massless fields

  Open

  A geometric shape, the cosmological polytope, gives the quantum state of the early universe for a simple toy field. Extending it to the fields that matter, such as massless particles and gravitons in an expanding universe, is open.

• Positive geometry for graviton scattering amplitudes

  Open

  Gravity amplitudes lack the cyclic ordering of particles that the gauge-theory shapes rely on. No shape whose volume-like form gives graviton scattering is known.

• Positive geometry for nonplanar $N=4$ super-Yang-Mills loop integrands

  Open

  The amplituhedron covers only the planar limit, where the number of colors is infinite. Finite numbers of colors require nonplanar diagrams, and no geometric shape for them is known.

• Geometric origin of the non-supersymmetric Yang-Mills loop integrand

  Open

  For gluons without supersymmetry, a 2024 construction using curves drawn on surfaces gives a natural loop integrand (the function integrated over loop momenta) in the limit of many colors. Whether it is the volume-like form of an actual geometric shape is not known.