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Reductions of GKZ Systems and Applications to Cosmological Correlators

Thomas W. Grimm, Arno Hoefnagels

TL;DR

The paper presents an algorithm to reduce GKZ differential systems, arising from integrals in quantum field theory and cosmology, to smaller, solvable subsystems when a parameter is resonant. By introducing reduction operators and a resonance/face/pyramid criterion, it shows how to systematically obtain partial solution bases and reconstruct the full solution space from simpler pieces. The single-exchange cosmological integral serves as a detailed benchmark, where all reduction operators are derived and used to obtain a complete basis of solutions in terms of physical variables, with locality emerging as a consequence of the reduction structure. The work connects GKZ reducibility with locality/Twists and offers a practical framework that can generalize to other Feynman integrals and period-type problems, potentially simplifying complex calculations across physics and geometry.

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.

Reductions of GKZ Systems and Applications to Cosmological Correlators

TL;DR

The paper presents an algorithm to reduce GKZ differential systems, arising from integrals in quantum field theory and cosmology, to smaller, solvable subsystems when a parameter is resonant. By introducing reduction operators and a resonance/face/pyramid criterion, it shows how to systematically obtain partial solution bases and reconstruct the full solution space from simpler pieces. The single-exchange cosmological integral serves as a detailed benchmark, where all reduction operators are derived and used to obtain a complete basis of solutions in terms of physical variables, with locality emerging as a consequence of the reduction structure. The work connects GKZ reducibility with locality/Twists and offers a practical framework that can generalize to other Feynman integrals and period-type problems, potentially simplifying complex calculations across physics and geometry.

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.
Paper Structure (68 sections, 243 equations, 4 figures)

This paper contains 68 sections, 243 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation -- cosmological correlators
  3. Correlators and the wave function of the universe.
  4. The toy model.
  5. The four particle single exchange integral.
  6. Comments on locality.
  7. GKZ systems and the single exchange integral
  8. A brief introduction to GKZ systems
  9. The matrix defining a GKZ system.
  10. Differential equations from the matrix $\mathcal{A}$.
  11. Structure of the solutions.
  12. The single exchange integral
  13. Reducing GKZ systems
  14. Reduction operators
  15. Obtaining reduction operators.
  16. ...and 53 more sections

Figures (4)

  • Figure 1: The general in-in diagram corresponding to $n$ external particles.
  • Figure 2: The 4-point single exchange diagram. Here $X_1=\vert \vec{k}_1\vert+\vert\vec{k}_2\vert$, $X_2=\vert \vec{k}_3\vert+\vert\vec{k}_4\vert$ and $Y=\vert \vec{k}_1+\vec{k}_2\vert=\vert \vec{k}_3+\vec{k}_4\vert$.
  • Figure 3: The diagrammatical interpretation of the integrals in equation \ref{['eq:Qactions']}. In particular, we have that (a) corresponds to the left-ordered part of the propagator, (b) to the right-ordered part and (c) to the non-time-ordered part. Finally, (d) corresponds to the collapsed propagator.
  • Figure 4: Acting with the reduction operators makes it possible to either contract or remove an edge from the diagram.