# Toward Null State Equations in $d>2$

**Authors:** Kuo-Wei Huang  
**arXiv:** [2308.03229](https://arxiv.org/abs/2308.03229)  
**Date:** 2023-08-06  
**Summary page:** https://www.summarizepaper.com/fr/paper/2308.03229v1/

## Abstract

In two-dimensional CFTs with a large central charge, the level-two BPZ equation governs the heavy-light scalar four-point correlator when the light probe scalar has dimension $h= - {1\over 2}$; the corresponding linear ordinary differential equation can be recast into a schematic form $x^2 u_{xx}+u=0$. In this paper, we make an observation that in a class of four-dimensional CFTs with a large central charge, the heavy-light scalar correlator in the near-lightcone limit obeys a similar equation, $x^3 u_{xxxy}+u=0$, when the light scalar has dimension $\Delta=-1$. We focus on the multi-stress tensor sector of the theory and also discuss the corresponding equations for the cases with $\Delta = -2, -3$. The solutions to these linear partial differential equations in higher dimensions are shown, after a suitable change of variables, to reproduce the near-lightcone correlators previously obtained via holography and conformal bootstrap.