$\mathcal{A}$-hypergeometric functions and creation operators for Feynman and Witten diagrams

Abstract

Both Feynman integrals and holographic Witten diagrams can be represented as multivariable hypergeometric functions of a class studied by Gel'fand, Kapranov & Zelevinsky known as GKZ or $\mathcal{A}$-hypergeometric functions. Among other advantages, this formalism enables the systematic construction of highly non-trivial weight-shifting operators known as 'creation' operators. We derive these operators from a physics perspective, highlighting their close relation to the spectral singularities of the integral as encoded by the facets of the Newton polytope. Many examples for Feynman and Witten diagrams are given, including novel weight-shifting operators for holographic contact diagrams. These in turn allow momentum-space exchange diagrams of different operator dimensions to be related while keeping the spacetime dimension fixed. In contrast to previous constructions, only non-derivative vertices are involved.