Reductions of GKZ Systems and Applications to Cosmological Correlators

TL;DR

This paper introduces an algorithm to simplify GKZ systems associated with Feynman integrals and cosmological correlators, enabling more efficient solutions by reducing system complexity especially in resonant parameter cases.

Contribution

The paper presents a novel reduction algorithm for GKZ systems that simplifies their structure, applicable without advanced D-module theory knowledge.

Findings

The algorithm successfully reduces GKZ systems in cosmological correlator calculations.

Application to a tree-level correlator demonstrates the relation between locality and system reduction.

Simplified systems facilitate solving complex differential equations in physics.

Abstract

A powerful approach to computing Feynman integrals or cosmological correlators is to consider them as solution to systems of differential equations. Often these can be chosen to be Gelfand-Kapranov-Zelevinsky (GKZ) systems. However, their naive construction introduces a significant amount of unnecessary complexity. In this paper we present an algorithm which allows for reducing these GKZ systems to smaller subsystems if a parameter associated to the GKZ systems is resonant. These simpler subsystems can then be solved separately resulting in solutions for the full system. The algorithm makes it possible to check when reductions happen and allows for finding the associated simpler solutions. While originating in the mathematical theory of D-modules analyzed via exact sequences of Euler-Koszul homologies, the algorithm can be used without knowledge of this framework. We motivate the need for such reduction techniques by considering cosmological correlators on an FRW space-time and solve the tree-level single-exchange correlator in this way. It turns out that this integral exemplifies an interesting relation between locality and the reduction of the differential equations.