2013/09/30 by Piotr Tourkine · 60 citations
Computer Science · Mathematics · Physics and Astronomy · #Amplitude #Black Holes and Theoretical Physics #Feynman diagram #Genus #Gravitation #Graviton #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Moduli space #Nonlinear Waves and Solitons #Physics #Polynomial and algebraic computation #Propagator #Pure mathematics #Quantum mechanics #Scattering amplitude #String (physics) #String field theory #String theory #Tropical geometry #Worldsheet #hep-th #math.AG
paper · pdf · doi:10.1007/s00023-017-0560-7
published in Annales Henri Poincaré 18(6), 2199-2249 (Birkhäuser) · 44 pages+refs, 19 figures. v2: very significant rewriting, expanded and clarified the discussion. Corrected misuse of denomination "Kontsevich-Penner". Added lemma on tropical theta characteristics, proof of the tropical limit of the prime form, and comment on field theory limit at three loops. Version accepted in Annales Henri Poincaré
arxiv created 2017/01/06 · openalex publication_date 2017/02/25 · arxiv updated 2017/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this work, we argue that the α '→ 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.