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Canonical Differential Equations Beyond Genus One

Claude Duhr, Franziska Porkert, Sven F. Stawinski

TL;DR

This work extends the canonical differential equation program from elliptic to hyperelliptic (genus-two) Feynman integral geometries by focusing on three- and four-parameter Lauricella functions that model maximal cuts of genus-two curves. It presents an explicit rotation to an ε-factorized canonical form, leveraging twisted cohomology intersection matrices to constrain new, non-periodic functions and to organize the differential equation matrix into modular blocks. A key outcome is the emergence of Siegel modular forms in the canonical matrix, lifting the familiar elliptic modular structure to higher genus, while revealing a genuine new antisymmetric function a(λ) that can spoil full modularity. Overall, the paper provides substantial evidence that a canonical basis exists for higher-genus Feynman integrals (at least for maximal cuts) and lays the groundwork for future work toward full higher-genus formalisms and their modular structures.

Abstract

We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known to involve a hyperlliptic curve of genus two. We consider specifically three- and four-parameter Lauricella functions, each associated to a hyperelliptic curve of genus two, and construct their canonical differential equations. Whilst core steps of this construction rely on existing methods $\unicode{x2014}$ that we show to be applicable in the higher-genus case $\unicode{x2014}$ we use new ideas on the structure of the twisted cohomology intersection matrix associated to the integral family in canonical form to obtain a better understanding of the appearing new functions. We further observe the appearance of Siegel modular forms in the $\varepsilon$-factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.

Canonical Differential Equations Beyond Genus One

TL;DR

This work extends the canonical differential equation program from elliptic to hyperelliptic (genus-two) Feynman integral geometries by focusing on three- and four-parameter Lauricella functions that model maximal cuts of genus-two curves. It presents an explicit rotation to an ε-factorized canonical form, leveraging twisted cohomology intersection matrices to constrain new, non-periodic functions and to organize the differential equation matrix into modular blocks. A key outcome is the emergence of Siegel modular forms in the canonical matrix, lifting the familiar elliptic modular structure to higher genus, while revealing a genuine new antisymmetric function a(λ) that can spoil full modularity. Overall, the paper provides substantial evidence that a canonical basis exists for higher-genus Feynman integrals (at least for maximal cuts) and lays the groundwork for future work toward full higher-genus formalisms and their modular structures.

Abstract

We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known to involve a hyperlliptic curve of genus two. We consider specifically three- and four-parameter Lauricella functions, each associated to a hyperelliptic curve of genus two, and construct their canonical differential equations. Whilst core steps of this construction rely on existing methods that we show to be applicable in the higher-genus case we use new ideas on the structure of the twisted cohomology intersection matrix associated to the integral family in canonical form to obtain a better understanding of the appearing new functions. We further observe the appearance of Siegel modular forms in the -factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.

Paper Structure

This paper contains 54 sections, 234 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review: Feynman Integrals, Maximal Cuts and Differential Equations
  3. Feynman Integrals and Their Differential Equations
  4. Maximal Cuts of Feynman Integrals
  5. Warm-up: Elliptic Hypergeometric ${}_2F_1$-function
  6. Step 1: Derivative Basis.
  7. Step 2: Semi-simple Rotation.
  8. Step 3: Rescaling with $\varepsilon$.
  9. Step 4: Integrating out.
  10. Discussion.
  11. Background on Hyperelliptic Curves
  12. (Hyperelliptic) Lauricella Functions
  13. Geometry of Hyperelliptic Curves
  14. Canonical Cycles for a Hyperelliptic Curve
  15. Abelian Differentials on the Hyperelliptic Curve
  16. ...and 39 more sections

Figures (2)

  • Figure 1: The non-planar crossed box.
  • Figure 2: A canonical basis of cycles for an even hyperelliptic curve of genus $g$.

Theorems & Definitions (7)

  • Example 1: The non-planar crossed box
  • Example 2: Abelian Differentials at Genus One
  • Example 3: Abelian Differentials for Odd Hyperelliptic Curves of Genus Two
  • Example 4: Abelian Differentials for Even Hyperelliptic Curves of Genus Two
  • Example 5: Period Matrix and Bilinear Relations for Hyperelliptic Curves of Genus One
  • Example 6: Period Matrix and Bilinear Relations for Hyperelliptic Curves of Genus Two
  • Example 7: Hypergeometric ${}_2F_1$-function