arXiv subjects

Arno Hoefnagels

Publications and source records attributed to Arno Hoefnagels.

5 recordsLinked to original sources

arXiv2026

Resonance and Differential Reduction of Feynman Integrals

Feynman integrals may be viewed as generalized hypergeometric functions, and specifically as solutions of GKZ systems of partial differential equations that typically exhibit resonance. Resonance is a type of non-genericity implying reducibility to subsystems. We use this resonance to construct reduction operators, which are differential operators that can contract edges of Feynman graphs. Correspondingly, their action is naturally compatible with cuts of Feynman graphs. Reduction operators may be used to close the system of differential equations for a given integral. The remaining GKZ data lead to algebraic relations identifying a smaller system that is fully reduced to master integrals. We develop the construction for one-loop, sunrise and banana graphs and discuss restrictions to physical kinematics. While reduction operators can generally shift both propagator powers and spacetime dimension, certain combinations isolate a pure dimension shift together with contraction of a chosen edge.

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arXiv2025

Differential Reductions and Cosmological Correlations

The study of cosmological correlators, and more generally Feynman integrals, is greatly aided by considering them as solutions to differential equations. Often, such systems of differential equations are reducible, which, broadly speaking, implies that the differential system is composed of various subsystems. Studying such decompositions and subsystems can greatly aid in solving the full differential system, as well as bring to light a substantial amount of structure. In this PhD thesis, we study reducibility for a particular system of differential equations known as GKZ (Gelfand, Kapranov and Zelevinsky) systems. We show how reducibility manifests itself in the differential equations through the existence of certain special operators, reduction operators, and explain their properties. Furthermore, we apply this framework to cosmological correlators, exemplifying how these reduction operators can be used to obtain and understand the structure within the system. Here the amount of structure seems remarkably large, and we leverage this structure to obtain many algebraic and permutative identities within the space of solutions to the differential equations. Interestingly, these include various cut and contraction relations between diagrams. We show how to obtain all such relations and how they reduce the full solution set to a certain, remarkably small, subset. Finally, we explain how such simplifications can be understood through the lens of o-minimality and Pfaffian complexity, as well as some of the limitations of this perspective.

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arXiv2025

Reductions of GKZ Systems and Applications to Cosmological Correlators

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.

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arXiv2025

A Reduction Algorithm for Cosmological Correlators: Cuts, Contractions, and Complexity

Cosmological correlators are fundamental observables in an expanding universe and are highly non-trivial functions even at tree-level. In this work, we uncover novel structures in the space of such tree-level correlators that enable us to develop a new recursive algorithm for their explicit computation. We begin by formulating cosmological correlators as solutions to GKZ systems and develop a general strategy to construct additional differential operators, called reduction operators, when a GKZ system is reducible. Applying this framework, we determine all relevant reduction operators, and show that they can be used to build up the space of functions needed to represent the correlators. Beyond relating different integrals, these operators also yield a large number of algebraic relations, including cut and contraction relations between diagrams. This implies a significant reduction in the number of functions needed to represent each tree-level cosmological correlator. We present first steps to quantify the complexity of our reduction algorithm by using the Pfaffian framework. While we focus on tree-level cosmological correlators, our approach provides a blueprint for other perturbative settings.

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arXiv2024

Structure and Complexity of Cosmological Correlators

Cosmological correlators capture the spatial fluctuations imprinted during the earliest episodes of the universe. While they are generally very non-trivial functions of the kinematic variables, they are known to arise as solutions to special sets of differential equations. In this work we use this fact to uncover the underlying tame structure for such correlators and argue that they admit a well-defined notion of complexity. In particular, building upon the recently proposed kinematic flow algorithm, we show that tree-level cosmological correlators of a generic scalar field theory in an FLRW spacetime belong to the class of Pfaffian functions. Since Pfaffian functions admit a notion of complexity, we can give explicit bounds on the topological and computational complexity of cosmological correlators. We conclude with some speculative comments on the general tame structures capturing all cosmological correlators and the connection between complexity and the emergence of time.

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