Correlation functions for non-conformal D$p$-brane holography

TL;DR

This work computes holographic two- and three-point functions for local operators in the strongly coupled, large-$N$ limit of maximally supersymmetric YM theories on D$p$-branes, by exploiting a scaling similarity that yields an auxiliary AdS structure with fractional dimension $ eta=rac{(3-p)^2}{5-p}$. The authors postulate that $(p{+}1)$-dimensional correlators can be obtained by integrating standard CFT$_{p{+}1+ eta}$ correlators over the $ eta$-dimensional auxiliary space, and they analytically derive the scalar 2pt function as a shifted-CFT form with dimension $\\\Delta_ eta=\\Delta- eta/2$, while the scalar 3pt function is given by a sum of four Appell $F_4$ functions of the conformal cross-ratios, $ abla o 0$ recovering the conventional CFT result. The work validates these results through several consistency checks, including permutation symmetry, extremal and OPE limits, and the $ eta o0$ limit, providing a nontrivial prediction for the kinematic structure of non-conformal holographic correlators. It lays a foundation for future bulk Witten-diagram calculations, KK-spectrum analyses, and potential cross-checks with localization and lattice methods.

Abstract

We use holography to study correlation functions of local operators in maximally supersymmetric Yang-Mills theories arising on the world-volume of D$p$-branes in the large-$N$ and strong-coupling limit. The relevant supergravity backgrounds obtained from the near-horizon limit of the D$p$-branes enjoy a scaling similarity, which leads to an auxiliary AdS space of fractional dimension. This suggests that holographic correlation functions in this setup can be computed by integrating standard CFT correlators over the auxiliary extra dimensions. We apply this prescription to analytically compute two- and three-point correlators of scalar operators. The resulting two-point functions take a familiar CFT form but with shifted conformal dimensions, while the three-point correlators have a much more involved position dependence which we calculate explicitly in terms of a sum of Appell functions.

Figures (4)

• Figure 1: The regions of the $(A,B)$ plane relevant for our analysis.
• Figure 2: The analytically continued 3pt-function viewed as a function of $A$ and $B$ for $p=2$ and two specific choices of scalar operators.
• Figure 3: The 3pt-function for $p=0$ and KK levels $\ell_i = (6,4,4)$. Note that without loss of generality we have chosen $y=0$, $z=1$, and $0<x<1$.
• Figure 4: The profile of the 3pt-function \ref{['eq:3pt_Appell']} along the curve $\sqrt{A}+\sqrt{B}=1$ that separates the physical and unphysical regions of the unit square, for the values of the parameters $p=2$ and $\ell_i=(2,2,2)$. The blue curve was evaluated using the defining series of the $F_4$ Appell function \ref{['eq:Appell']}, valid in the unphysical region, while for the orange curve we used the analytic continuation discussed in this appendix, valid in the physical one. We have used the parametrization $A=x^2$ and $B = (1-x)^2$.