Conformal correlators as simplex integrals in momentum space

TL;DR

The paper develops a general framework for conformal correlators in momentum space by expressing scalar $n$-point functions as integrals over an $(n-1)$-simplex with an arbitrary function of momentum-space cross ratios. It proves conformal invariance via multiple routes, introduces mesh integrals and their recursive structure, and then specializes to holographic (contact) correlators, obtaining explicit cross-ratio functions through star-mesh duality and convolution methods. A key payoff is that the $n$-point simplex representation reduces the number of integrations needed for the cross-ratio function to $\mathcal O(n-2)$ in the holographic case, offering a more efficient description than Mellin approaches for $n>4$. The results provide a versatile tool for momentum-space CFT analyses, with potential applications to cosmology, bootstrap, and higher-point holographic correlators, while opening avenues for further mathematical connections and generalizations to spins and exchanges.

Abstract

We find the general solution of the conformal Ward identities for scalar $n$-point functions in momentum space and in general dimension. The solution is given in terms of integrals over $(n-1)$-simplices in momentum space. The $n$ operators are inserted at the $n$ vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where $n$-point functions are built in terms of $(n-1)$-point functions. To illustrate our discussion, we derive the simplex representation of $n$-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves $(n-2)$ integrations, which is an improvement (when $n>4$) relative to the Mellin representation that involves $n(n-3)/2$ integrations.

Figures (4)

• Figure 1: Representation of the 4-point function as the 3-simplex or tetrahedral integral \ref{['kint4']}. Each of the internal lines represents a generalised propagator in \ref{['Den3']}, and the dots denote operator insertions carrying ingoing momenta $\boldsymbol{p}_i$.
• Figure 2: The decomposition of the 5-point mesh $M_5$. The solid internal lines on the right-hand side of the figure represent the 4-point mesh $M_4$ evaluated with ingoing momenta $\boldsymbol{p}_j - \boldsymbol{q}_j$.
• Figure 3: Equivalent electrical networks of resistors under star-mesh duality, where the conductivities and currents are related as given in \ref{['starmesh1']}. The external currents flowing into the corresponding dotted nodes and the overall power dissipation are equal.
• Figure 4: In \ref{['newparam']}, we exchange the Schwinger parameters $t_i$ where $i=1,\ldots,n$ for a new set consisting of $z_{23}$ and $z_{1i}$ for $i=2,\ldots n$. These correspond to the solid legs on the diagram above, shown for the case $n=5$.