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 Kawai-Lewellen-Tye kernel. It captures the full α^{'} dependence of stringified amplitudes for biadjoint scalar ϕ^{3} theory, pions in the nonlinear sigma model (NLSM), and their mixed ϕ/π amplitudes, reducing to the corresponding field theory amplitudes in the α^{'}→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 Tr(ϕ^{3}) and NLSM pion amplitudes, naturally emerges as the leading contribution to the stringy geometry. We show how the connection between Tr(ϕ^{3}) and NLSM can be geometrized using the associahedral grid.