Toward Null State Equations in $d>2$

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 $h=-\frac{1}{2}$; 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 the conformal bootstrap.

arXiv comment27 pages, v2: minor corrections and references added, v3: notation improved

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