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

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.