Feynman integrals and iterated integrals on moduli spaces of curves of genus zero

TL;DR

The paper develops a comprehensive, algorithmic framework for computing period integrals on the genus-zero moduli spaces $\mathcal{M}_{0,n}$ via iterated integrals and the bar-de Rham complex, and demonstrates how these techniques yield exact symbolic results for a broad class of Feynman-type integrals. Central to the approach are the symbol/unshuffle maps, the Gauss–Manin connection, and a transformation from Schwinger parameters to cubical coordinates, enabling systematic primitives and regularised limits that express results in terms of multiple zeta values. The methods are implemented (e.g., in Maple) and shown to handle cellular integrals, hypergeometric expansions, and high-loop Feynman graphs, with explicit examples including $I=20\zeta(5)$ and Appell-type expansions. This work provides a unified, geometry-driven pathway to exact, automatable evaluation of integrals common in quantum field theory and related areas, bridging moduli-space geometry, iterated integrals, and high-precision symbolic computation.

Abstract

This paper describes algorithms for the exact symbolic computation of period integrals on moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points, and applications to the computation of Feynman integrals.

Figures (2)

• Figure 2.1: On the left is a picture of $\mathcal{M}_{0,5}$ in cubical coordinates $(x_1,x_2)$, and two paths going from the origin to $(1,1)$. On the right-hand side is the space obtained by blowing up the point $(1,1)$. The exceptional divisor is $\mathcal{E}\cong \mathbb{P}^1$. There are two tangential base points defined over $\mathbb{Z}$ which lie above $(1,1)$, which are based at $z_1$ and $z_2$. The inverse image of the two paths end at the point $z_1$, or $z_2$ respectively.
• Figure :

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4
• Example 5
• Example 6
• Example 7
• Example 8
• Example 9
• Example 10