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Tropical Amplitudes

Piotr Tourkine

TL;DR

This work introduces and develops a tropical geometry framework to realize the α'→0 field-theory limit of closed-string amplitudes as a tropical, worldline-like degeneration. By decomposing the string moduli space into domains associated with tropical graphs and employing tropical theta characteristics and the tropical prime form, the authors derive the worldline propagator and the first/second Symanzik polynomials from string theory data. They validate the approach by computing the tropical limit of the genus-2, four-graviton amplitude and confirming agreement with corresponding supergravity results, while clarifying how analytic (counter-term) and non-analytic (tropical) contributions arise and cancel. The work also explains how UV divergences map to weighted vertices (counter-terms) in tropical graphs and discusses the role of maximal supersymmetry in simplifying numerators. Overall, it provides a concrete, geometrically grounded bridge between string perturbation theory and perturbative quantum field theory, offering a practical path to extract Feynman numerators from higher-genus string amplitudes and guiding extensions to higher loops.

Abstract

In this work, we argue that the $α'\to 0$ 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.

Tropical Amplitudes

TL;DR

This work introduces and develops a tropical geometry framework to realize the α'→0 field-theory limit of closed-string amplitudes as a tropical, worldline-like degeneration. By decomposing the string moduli space into domains associated with tropical graphs and employing tropical theta characteristics and the tropical prime form, the authors derive the worldline propagator and the first/second Symanzik polynomials from string theory data. They validate the approach by computing the tropical limit of the genus-2, four-graviton amplitude and confirming agreement with corresponding supergravity results, while clarifying how analytic (counter-term) and non-analytic (tropical) contributions arise and cancel. The work also explains how UV divergences map to weighted vertices (counter-terms) in tropical graphs and discusses the role of maximal supersymmetry in simplifying numerators. Overall, it provides a concrete, geometrically grounded bridge between string perturbation theory and perturbative quantum field theory, offering a practical path to extract Feynman numerators from higher-genus string amplitudes and guiding extensions to higher loops.

Abstract

In this work, we argue that the 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.

Paper Structure

This paper contains 30 sections, 2 theorems, 127 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tropical geometry
  3. Tropical graphs
  4. Homology, forms, Jacobian and divisors
  5. Divisors and theta characteristics
  6. Divisors on graphs
  7. Tropical theta characteristics
  8. Proof.
  9. The tropical moduli space
  10. Classical geometry and the tropical limit
  11. Riemann surfaces and their Jacobians
  12. Riemann surfaces and their moduli spaces $\mathcal{M}_{g,n}$, $\overline{\mathcal{M}}_{g,n}$
  13. Tropicalizing $\mathcal{M}_{g,n}$
  14. Decomposition of the moduli space
  15. Comment on the relation to the minimal area metrics formalism.
  16. ...and 15 more sections

Key Result

Lemma 1

Let $P$ and $Q$ be two points on a tropical graph $\Gamma$, let $\gamma$ be a path joining them and $\mathrm{dist}_{\gamma}(P,Q)$ be the distance between $P$ and $Q$ along $\gamma$. Then, there always exist a tropical theta characteristics $\bm \beta\in \tfrac{1}{2} (\mathbb{Z}/2\mathbb{Z})^{g}$ suc

Figures (19)

  • Figure 1: Examples of tropical graphs (left to right): a $3$-point tropical tree, a once-punctured graph of genus one, a 2-loop tropical graph, a graph of genus $1+w$.
  • Figure 2: Specialization rules as $t\to0$.
  • Figure 3: a) Genus two graph with edges lengths $T_1,T_2,T_3$. b) Image of $\Gamma$ (thick line) by the tropical Abel-Jacobi map in the Jacobian $J(\Gamma)=\mathbb{R}^2/K^{(2)}\mathbb{Z}^2$.
  • Figure 4: Genus-two graphs described in the examples.
  • Figure 5: Example of rational function $f$ on a two-loop graph.
  • ...and 14 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Proposition 1
  • Conjecture 2