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Self-duality from twisted cohomology

Claude Duhr, Franziska Porkert, Cathrin Semper, Sven F. Stawinski

TL;DR

This work establishes a natural self-duality for maximal cuts by embedding them in twisted cohomology. It proves that, once both the differential equations and their duals are brought to a canonical ε-factorised form, the cohomology intersection matrix is constant, linking canonical bases to a Lie-algebra representation where the dual is equivalent to the original. For maximal cuts, this representation is irreducible and self-dual, constraining the differential equation matrix via a constant, basis-independent object. The analysis reconciles earlier persymmetric observations with a broader, basis-invariant framework, while also exposing potential extra symmetries in even-top-sector cases and clarifying the role of Galois-type phenomena arising from basis choices. Overall, the paper connects canonical bases, intersection theory, and Lie-algebra representations to provide a robust, rationally defined notion of self-duality in Feynman integral differential equations.

Abstract

Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an ε-factorised differential equation is persymmetric. It was observed that the rotation to this special basis may introduce a Galois symmetry relating different integrals. We argue that the proposed notion of self-duality for maximal cuts stems from a very natural notion of self-duality from twisted cohomology. Our main result is that, if the differential equations and their duals are simultaneously brought into canonical form, the cohomology intersection matrix is a constant. Furthermore, we show that one can associate quite generically a Lie algebra representation to an ε-factorised system. For maximal cuts, this representation is irreducible and self-dual. The constant intersection matrix can be interpreted as expressing the equivalence of this representation and its dual, which in turn results in constraints for the differential equation matrix. Unlike the earlier proposal, the most natural symmetry of the differential equation matrix is defined entirely over the rational numbers and is independent of the basis choice.

Self-duality from twisted cohomology

TL;DR

This work establishes a natural self-duality for maximal cuts by embedding them in twisted cohomology. It proves that, once both the differential equations and their duals are brought to a canonical ε-factorised form, the cohomology intersection matrix is constant, linking canonical bases to a Lie-algebra representation where the dual is equivalent to the original. For maximal cuts, this representation is irreducible and self-dual, constraining the differential equation matrix via a constant, basis-independent object. The analysis reconciles earlier persymmetric observations with a broader, basis-invariant framework, while also exposing potential extra symmetries in even-top-sector cases and clarifying the role of Galois-type phenomena arising from basis choices. Overall, the paper connects canonical bases, intersection theory, and Lie-algebra representations to provide a robust, rationally defined notion of self-duality in Feynman integral differential equations.

Abstract

Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an ε-factorised differential equation is persymmetric. It was observed that the rotation to this special basis may introduce a Galois symmetry relating different integrals. We argue that the proposed notion of self-duality for maximal cuts stems from a very natural notion of self-duality from twisted cohomology. Our main result is that, if the differential equations and their duals are simultaneously brought into canonical form, the cohomology intersection matrix is a constant. Furthermore, we show that one can associate quite generically a Lie algebra representation to an ε-factorised system. For maximal cuts, this representation is irreducible and self-dual. The constant intersection matrix can be interpreted as expressing the equivalence of this representation and its dual, which in turn results in constraints for the differential equation matrix. Unlike the earlier proposal, the most natural symmetry of the differential equation matrix is defined entirely over the rational numbers and is independent of the basis choice.
Paper Structure (25 sections, 5 theorems, 170 equations)

This paper contains 25 sections, 5 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Feynman integrals and their cuts
  3. Feynman integrals and their differential equations
  4. Maximal cuts and self-duality
  5. Twisted cohomology and self-duality
  6. Twisted cohomology
  7. Self-duality from twisted cohomology
  8. $\varepsilon$-factorised differential equations and their properties
  9. $\varepsilon$-factorised and canonical differential equations
  10. Linear independence of iterated integrals
  11. Interpretation for Feynman integrals
  12. Interpretation in terms of Lie algebras
  13. Maximal cuts and self-dual systems
  14. Self-dual systems
  15. Symmetries of differential equations from self-duality
  16. ...and 10 more sections

Key Result

Theorem 1

We assume the notations and conventions from above hold. Then, the following statements are equivalent:

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • proof