Positive Geometry for Stringy Scalar Amplitudes

TL;DR

The paper develops a novel positive-geometry framework, the associahedral grid, to geometrize stringy scalar amplitudes via the inverse KLT kernel $m_n^{\alpha'}$. By tessellating kinematic space with an infinite lattice of ABHY associahedra, it captures the full $\alpha'$-dependence for BAS, NLSM pions, and mixed amplitudes, and recovers field-theory limits as $\alpha'\to 0$. It provides explicit constructions for diagonal and off-diagonal matrix elements, demonstrates a geometric recursion, and explains how the $\delta$-shift and various $\alpha'$-shifts map to subgrids that yield NLSM and mixed amplitudes. The work extends positive geometry beyond rational functions to genuinely stringy structures and points to further connections with stringy Galileon amplitudes and gravity. Overall, the associahedral grid offers a unifying, geometric description of a broad class of string-inspired amplitudes and their intricate pole/resonance structure.

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 $α'$-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.

Figures (4)

• Figure 1: Four-point geometries: (a) ${\text{Tr}(\phi^3)}$ amplitude ($\mathcal{A}_4$), (b) diagonal matrix element $m_4^{\alpha'}\!(\mathds{1}|\mathds{1})$\ref{['invKLTex']} ($\mathcal{A}_4^{\alpha'}$), (c) off-diagonal matrix element $m_4^{\alpha'}\!(\mathds{1}|1243)$\ref{['exKLToffdiag']}, (d) stringy NLSM \ref{['NLSM4dform']} ($\mathcal{A}_4^{\text{NLSM},\alpha'}$).
• Figure 2: Associahedral grid for $m_5^{\alpha'}\!(\mathds{1}|\mathds{1})$\ref{['invKLTex']} (left), and the off-diagonal element $m_5^{\alpha'}\!(\mathds{1}|13452)$\ref{['exKLToffdiag']} (right), where we use $c=c_{14}+c_{24}$.
• Figure 3: Associahedral subgrid for the stringy mixed amplitude $M_5^{1\pi,\alpha'}\!(\pi\phi\phi\phi\phi)$\ref{['M1pialpha']} (left), and stringy NLSM at 6-points \ref{['NLSM6alpha']} (right).
• Figure 4: The 5-point associahedron $\mathcal{A}_5$ is triangulated by the chambers and fibers in equation \ref{['eq:A5-chamber-fiber']}.