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 $\alpha'$-dependence of stringified amplitudes for bi-adjoint scalar $\phi^3$ theory, pions in the NLSM, and their mixed $\phi$/$\pi$ amplitudes, reducing to the corresponding field theory amplitudes in the $\alpha'\to 0$ 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 $\delta$-shift, recently proposed to relate field theory $\mathrm{Tr}(\phi^3)$ and NLSM pion amplitudes, naturally emerges as the leading contribution to the stringy geometry. We show how the connection between $\mathrm{Tr}(\phi^3)$ and NLSM can be geometrized using the associahedral grid.