Shift operators and momentum-space conformal field theory

TL;DR

This paper develops a momentum-space conformal field theory framework using shift operators and hypergeometric functions, revealing new relations between n-point functions across dimensions and simplifying conformal Ward identities.

Contribution

It introduces determinant-based shift operators connecting n-point functions in different dimensions and constructs novel weight-shifting operators for conformal and Witten diagrams.

Findings

Expressed graph polynomials via Laplacian minors

Derived inverse parametrisation using Cayley-Menger determinants

Constructed new weight-shifting operators for diagrams

Abstract

A momentum-space approach to conformal field theory offers a new perspective on cosmological correlators and better reveals the underlying connections to scattering amplitudes. This thesis explores the interplay between integral representations and shift operators. A representation for the general $n$-point function of scalar operators was recently proposed in the form of a Feynman integral with the topology of an $(n-1)$-simplex, featuring an arbitrary function of momentum-space cross ratios. We show the graph polynomials for this integral can all be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of shift operators expressible as determinants that connect $n$-point functions in spacetime dimensions differing by two. Furthermore, they enable the validity of the conformal Ward identities to be established directly without recourse to recursion in the number of points. We then analyse the representation of conformal and Feynman integrals through a class of multivariable hypergeometric functions proposed by Gelfand, Kapranov \& Zelevinsky. Among other advantages, this formalism enables the systematic construction of highly non-trivial weight-shifting operators known as ``creation'' operators. We discuss these operators emphasising their close connection to the spectral singularities that arise for special parameter values, and their relationship to the Newton polytope of the integrand. Via these methods we construct novel weight-shifting operators connecting contact Witten diagrams of different operator and spacetime dimensions, as well as exchange diagrams with purely non-derivative vertices.