Universal Lowest-Twist in CFTs from Holography

TL;DR

This work investigates heavy-light four-point functions in $d\ge 2$ CFTs using holography and higher-derivative gravity. The authors isolate the lowest-twist multi-stress-tensor contributions by solving the bulk scalar equation in a black hole background and performing a near-boundary expansion, revealing a universal set of OPE coefficients that depend only on $\Delta_L$ and the combination $f_0 \sim \Delta_H/C_T$; higher-twist coefficients, in contrast, retain sensitivity to bulk couplings. A reduced field equation in general dimensions and a recursion relation enable all-orders computation of these universal lowest-twist coefficients, suggesting a higher-dimensional Virasoro-like structure near the lightcone. While the planar (high-temperature) limit yields clear universality, the spherical case is explored in appendices with stronger conjectures; finite-$C_T$ corrections and odd dimensions remain important directions for future work. Overall, the paper provides a robust holographic mechanism for universal lowest-twist OPE data in higher-dimensional CFTs and highlights deep connections between bulk geometry, near-boundary data, and conformal blocks.

Abstract

We probe the conformal block structure of a scalar four-point function in $d\geq2$ conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal blocks are insensitive to the horizon boundary condition. The main result of this paper is that the lowest-twist operator product expansion (OPE) coefficients of the multi-stress tensor conformal blocks are universal: they are fixed by the dimension of the light operators and the ratio between the dimension of the heavy operator and the central charge $C_T$. Neither supersymmetry nor unitary is assumed. Higher-twist coefficients, on the other hand, generally are not protected. A recursion relation allows us to efficiently compute universal lowest-twist coefficients. The universality result hints at the potential existence of a higher-dimensional Virasoro-like symmetry near the lightcone. While we largely focus on the planar black hole limit in this paper, we include some preliminary analysis of the spherical black hole case in an appendix.

Figures (6)

• Figure 1: ${\cal V}_d$ in $d=4$ and $\Delta=-1$; eq(\ref{['eq:CV4DeltaMinusOne']}).
• Figure 2: Ratio $r_n(\Delta)$ for $d=2,4,6$ (left, middle, right) at various $n$: $n=10$ (red, dashed), $n=50$ (blue, dotted), $n=100$ (purple, dot-dashed), and $n=300$ (black, solid). The curves appear to flatten as $n$ increases and approach a constant value $\lim_{n\rightarrow \infty} r_n(\Delta) = 9.87, -15.756$, and $50.302$ at infinite $n$ in $d=2,4$ and $6$, respectively. The curves do not all intersect at a single point, though they appear to do so by eye; in the limit $n\rightarrow \infty$, $r_{n}(\Delta)$ and $r_{n-1}(\Delta)$ intersect at $\Delta= 1,{5\over 4},{7\over 4}$ for $d=2,4,6$, respectively.
• Figure 3: Ratio $r_n(\Delta)$ for $d=2,4, 6$ (left, middle, right) at various $\Delta$: $\Delta=-0.75$ (red, dashed), $\Delta=1.25$ (blue, dotted), $\Delta=7.25$ (purple, dot-dashed), and $\Delta=26.25$ (black, solid). The curves appear to converge to the same value at infinite $n$.
• Figure 4: Ratio $r_8(\Delta)$ for $d=4$ zoomed in near $\Delta=-1$.
• Figure 5: Power $p(\Delta)$ for the large $n$ behavior of the coefficients $a_{n, {d-2\over 2}n} \approx A r_{\infty}^{-n} n^{p}$, for $d=2,4,6,8$. For $d=2$, $p(\Delta) = \Delta-1$, whereas for $d=4,6,8$, $p(\Delta)= 2 \Delta- {d+1\over 2}$.