Resummation of Multi-Stress Tensors in Higher Dimensions

TL;DR

This paper derives exact solutions for higher-dimensional null-state equations in holographic CFTs, revealing connections between 2D and 4D correlators, and providing insights into multi-stress tensor contributions and their holographic duals.

Contribution

It introduces exact solutions to higher-dimensional null-state equations, linking 2D and 4D correlators, and explores their holographic interpretations involving black holes.

Findings

Resummation of correlators yields the Virasoro vacuum block in 2D.

In 4D, resummation leads to simple forms with essential singularities.

Correlators can be reconstructed via modes related to the Virasoro algebra.

Abstract

In the context of holographic conformal field theories (CFTs), a system of linear partial differential equations was recently proposed to be the higher-dimensional analog of the null-state equations in $d=2$ CFTs at large central charge. Solving these equations in a near-lightcone expansion yields solutions that match the minimal-twist multi-stress tensor contributions to a heavy-light four-point correlator (or a thermal two-point correlator) computed using holography, the conformal bootstrap, and other methods. This paper explores the exact solutions to these equations. We begin by observing that, in an expansion in terms of the ratio between the heavy operator's dimension and the central charge, the $d=2$ correlator involving the level-two degenerate scalars at each order can be represented as a Bessel function; the resummation yields the Virasoro vacuum block. We next observe a relation between the $d=2$ correlator and the $d=4$ near-lightcone correlator involving light scalars with the same conformal dimension. The resummed $d=4$ correlator takes a simple form in the complex frequency domain. Unlike the Virasoro vacuum block, the resummation in $d=4$ leads to essential singularities. Similar expressions are also obtained when the light scalar's dimension takes other finite values. These CFT results correspond to a holographic computation with a spherical black hole. In addition, using the differential equations, we demonstrate that the correlators can be reconstructed via certain modes. In $d=2$, these modes are related to the Virasoro algebra.