Abstract
In this work, we argue that the alpha 'rightarrow 0 limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes at higher genus. An important technical input from tropical geometry is the use of tropical theta functions with characteristics to rigorously derive the worldline limit of the worldsheet propagator. This enables us to perform a non-trivial computation at two loops: we derive the tropical form of the integrand of the genus-two four-graviton type II string amplitude, which matches the direct field theory computations. At the mathematical level, this limit is an implementation of the correspondence between the moduli space of Riemann surfaces and the tropical moduli space.
Highlights
It is well accepted that the field theory limit1 of string theory scattering amplitudes reproduces the usual perturbative expansion of quantum field theory
Besides the study of the α → 0 limit of string amplitudes, our approach sheds a new light on the geometry of field theory amplitudes: they are integrals over the tropical moduli space
We identify Da as the maximally non-analytic domain and Dc as the analytic domain. Since this decomposition is rather special (as (a). We leave this problem for future investigations, and on focus on the type II four-graviton string amplitude restricted to Da, in order to compute the tropical limit of the integrand
Summary
It is well accepted that the field theory limit of string theory scattering amplitudes reproduces the usual perturbative expansion of quantum field theory. The aim of this work is computational: it is to develop methods based on tropical geometry to extract the field theory limit of higher genus closed string theory amplitudes. 4, we formulate the field-theory limit of closed string theory amplitudes in the context of tropical geometry. Besides the study of the α → 0 limit of string amplitudes, our approach sheds a new light on the geometry of field theory amplitudes: they are integrals over the tropical moduli space. We want to be able to take a string theory amplitude, expressed in its compact form as a single moduli space integral, and extract field theory graphs out of it, in the spirit of the Bern–Kosower rules. In the second version of this paper, the author added a comment on the three-loop amplitude of [55] at the end of Sect. 5
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