Conformal integrals in all dimensions as GKZ hypergeometric functions and Clifford groups

TL;DR

The paper shows that Euclidean conformal integrals with any number of points in any dimension can be expressed as GKZ hypergeometric functions by embedding the conformal group into the Möbius action via Clifford algebras. It derives a toric (GKZ) structure from cross-ratio invariants, obtaining explicit Gamma-series solutions that form a basis for the germs of the integrals and recovers permutation symmetry through monodromy-invariant Hermitian forms. The approach yields concrete results for N=4 and N=5, including Appell $F_4$ representations for N=4 and a full 11-term Gamma-series structure for N=5, while clarifying reductions to lower-point configurations. The work highlights the interplay between Clifford-algebraic Möbius transformations, toric geometry, and GKZ systems for conformal blocks, and discusses limitations related to independent cross-ratio counts and possible Lorentzian generalizations. Overall, it provides a unified, dimension-agnostic framework for conformal integrals with potential applications in conformal field theory and related computational methods.

Abstract

Euclidean conformal integrals for an arbitrary number of points in any dimension are evaluated. Conformal transformations in the Euclidean space can be formulated as the Moebius group in terms of Clifford algebras. This is used to interpret conformal integrals as functions on the configuration space of points on the Euclidean space, solving linear differential equations, which, in turn, is related to toric GKZ systems. Explicit series solutions for the conformal integrals are obtained using toric methods as GKZ hypergeometric functions. The solutions are made symmetric under the action of permutation of the points, as expected of quantities on the configuration space of unordered points, using the monodromy-invariant unique Hermitian form. Consistency of the solutions among different number of points is shown.