Positive Geometry for Stringy Scalar Amplitudes

We introduce a new positive geometry, the associahedral grid, which provides a geometric realization of the inverse string theory KLT kernel. It captures the full $α'$-dependence of stringified amplitudes for bi-adjoint scalar $ϕ^3$ theory, pions in the NLSM, and their mixed $ϕ$/$π$ amplitudes, reducing to the corresponding field theory amplitudes in the $α'\to 0$ limit. Our results demonstrate how positive geometries can be utilized beyond rational functions to capture stringy features of amplitudes, such as an infinite resonance structure. The kinematic $δ$-shift, recently proposed to relate field theory $\mathrm{Tr}(ϕ^3)$ and NLSM pion amplitudes, naturally emerges as the leading contribution to the stringy geometry. We show how the connection between $\mathrm{Tr}(ϕ^3)$ and NLSM can be geometrized using the associahedral grid.

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