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Determine the εT–εB relation for cosmological correlahedra at general multiplicity

Determine the functional relation between the top and bottom shaving parameters εT and εB that is required to produce the correct combinatorics for the cosmological correlahedron at arbitrary n.

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Background

The cosmological correlahedron is proposed as a higher-dimensional positive geometry interpolating between the associahedron (top facet) and the cosmohedron (bottom facet). Its realization involves inequalities with shaving parameters assigned to rays associated with the top and bottom directions.

While explicit checks confirm correct combinatorics for small n (e.g., n=6) under certain parameter choices, a general relation between εT and εB ensuring correctness at all n is unknown.

References

For general $n$, we expect an interesting relation between $\epsilon_{T,B}$ to produce the correct combinatorics. We leave an exploration of this question, as well as the systematics of extracting the correlator from the geometry, to future work.

— Cosmohedra (2412.19881 - Arkani-Hamed et al., 27 Dec 2024) in Section “Cosmological correlahedra”