Resummation of multistress tensors in higher dimensions

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> CFTs at large central charge. Solving these equations in a near-light-cone expansion yields solutions that match the minimal-twist multistress 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi><c:mo>=</c:mo><c:mn>2</c:mn></c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>d</e:mi><e:mo>=</e:mo><e:mn>2</e:mn></e:math> correlator and the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>d</g:mi><g:mo>=</g:mo><g:mn>4</g:mn></g:math> near-light-cone correlator involving light scalars with the same conformal dimension. The resummed <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>d</i:mi><i:mo>=</i:mo><i:mn>4</i:mn></i:math> correlator takes a simple form in the complex frequency domain. Unlike the Virasoro vacuum block, the resummation in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>d</k:mi><k:mo>=</k:mo><k:mn>4</k:mn></k:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>d</m:mi><m:mo>=</m:mo><m:mn>2</m:mn></m:math>, these modes are related to the Virasoro algebra.