Differential Equations for Moving Hyperplane Arrangements

TL;DR

The paper develops a purely combinatorial method to produce annihilating holonomic $D$-ideals for Mellin-type integrals associated to moving hyperplane arrangements. By combining operators from individual hyperplanes, circuits, and syzygies, it constructs a left $D$-ideal that annihilates the correlator $\phi(c)$ for all twisted cycles, and shows this ideal is holonomic with a singular locus closely tied to discriminantal arrangements. The approach is illustrated through numerous plane- and space-configuration examples, including cosmological setups, demonstrating both the rank behavior and the geometric structure of singularities. The work opens pathways to relate these $D$-ideals to GKZ systems, discriminantal geometry, and potential physical applications, with computational tools implemented in existing computer algebra systems.

Abstract

We investigate Mellin integrals of products of hyperplanes, raised to an individual power each. We refer to the resulting functions as combinatorial correlators. We investigate their behavior when moving the hyperplanes individually. To encode these functions as holonomic functions in the constant terms of the hyperplanes, we aim to construct a holonomic annihilating $D$-ideal purely in terms of the hyperplane arrangement.

Theorems & Definitions (13)

• lemma 1
• Remark 3.1
• lemma 2
• Theorem 3.2
• Theorem 3.3: circuit
• Proposition 3.4
• proof
• lemma 3
• proof
• lemma 4