Tropical Amplitudes

Abstract

In this work, we argue that the ${\alpha }^{\prime }\to 0$ 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.

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

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Cited by in corpus (32)

• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Scattering Equations and String Theory Amplitudes
• BCJ duality and double copy in the closed string sector
• Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Tree-level amplitudes from the pure spinor superstring
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Monodromy relations from twisted homology
• Superstring loop amplitudes from the field theory limit
• On integrands and loop momentum in string and field theory
• Loop amplitudes monodromy relations and color-kinematics duality
• String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry
• Unitarity Cuts of the Worldsheet
• Infrared divergences and harmonic anomalies in the two-loop superstring effective action
• Two-loop superstring five-point amplitudes III, construction via the RNS formulation: even spin structures
• Local BCJ numerators for ten-dimensional SYM at one loop
• Notes on Biadjoint Amplitudes, $\mathrm{Trop}G(3,7)$ and $X(3,7)$ Scattering Equations
• Exposing the threshold structure of loop integrals
• QFT with Stubs
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Permutation combinatorics of worldsheet moduli space
• Asymptotics of the $D^8{R}^4$ genus-two string invariant
• α'-Expansion of Open String Disk Integrals via Mellin Transformations
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• String theory integrands and supergravity divergences
• Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors
• Tropological Sigma Models
• Feynman Amplitudes in Mathematics and Physics
• One-loop bosonic string and De Sitter space
• Pinching parameters for open (super) strings
• Tropicalized quantum field theory and global tropical sampling
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands