Yangian symmetry, GKZ equations and integrable Feynman graphs in conformal variables

TL;DR

This paper derives explicit differential equations from Yangian symmetry for conformal Feynman graphs, revealing their relation to GKZ hypergeometric systems and enabling new solution methods in various spacetime dimensions.

Contribution

It provides the first explicit form of Yangian symmetry equations in conformal variables for any dimension, linking them to GKZ hypergeometric operators and exploring their implications.

Findings

Yangian equations expressed in conformal cross-ratios.

Identification of terms with GKZ hypergeometric operators.

Exact relation between certain graphs and GKZ systems.

Abstract

We study the differential equations that follow from Yangian symmetry which was recently observed for a large class of conformal Feynman graphs, originating from integrable `fishnet' theories. We derive, for the first time, the explicit general form of these equations in the most useful conformal cross-ratio variables, valid for any spacetime dimension. This allows us to explore their properties in detail. In particular, we observe that for general Feynman graphs a large set of terms in the Yangian equations can be identified with famous GKZ (Gelfand-Kapranov-Zelevinsky) hypergeometric operators. We also show that for certain nontrivial graphs the relation with GKZ systems is exact, opening the way to using new powerful solution methods. As a side result, we also elucidate the constraints on the topology and parameter space of Feynman graphs stemming from Yangian invariance.