Differential Reductions and Cosmological Correlations

Abstract

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 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 trough the lens of o-minimality and Pfaffian complexity, as well as some of the limitations of this perspective.