# Positive Geometry for Stringy Scalar Amplitudes

## TL;DR

This paper introduces the associahedral grid, a positive geometry that geometrically encodes string amplitudes' full alpha' dependence, revealing stringy features and resonance structures beyond rational functions.

## Contribution

It presents the associahedral grid as a new positive geometry capturing string amplitudes, extending geometric methods beyond rational functions to include stringy effects.

## Key findings

- The associahedral grid encodes the inverse string theory KLT kernel.
- It captures the full alpha' dependence of various string-inspired amplitudes.
- The geometry naturally incorporates the kinematic delta-shift relating field theory and NLSM amplitudes.

## Abstract

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.