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

5+2 pages, 4 figures; v2: minor mistake in End Matter corrected, matches published version

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Cited by in corpus (1)

• The Geometry of BCFW for ABJM Loop Integrands