String correlators on $\text{AdS}_3$: Analytic structure and dual CFT
Andrea Dei, Lorenz Eberhardt
Abstract
We continue our study of string correlators on Euclidean \text{AdS}_3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mtext mathvariant="normal">AdS</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> </mml:math> with pure NS-NS flux. The worldsheet and spacetime correlators have a rich analytic structure, which we analyse completely for genus 0 four-point functions. We show that correlators exhibit a simple behaviour near their singularities. The spacetime correlators are meromorphic functions in the \mathrm{SL}(2,\mathbb{R}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> -spins, whose pole structure is shown to agree with the prediction of a recent proposal for the dual \text{CFT}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mtext mathvariant="normal">CFT</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . Moreover, we also compute the residues of the spacetime correlators for some of the poles exactly and find again a perfect match with the proposal for the dual \text{CFT}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mtext mathvariant="normal">CFT</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , thereby checking the duality for some non-trivial four-point functions exactly. Our computations simplify drastically in the tensionless limit of \mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mstyle mathvariant="normal"> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> <mml:mi>S</mml:mi> </mml:mstyle> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> </mml:mstyle> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>𝕋</mml:mi> </mml:mstyle> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> where the behaviour near the poles gives in fact the exact answer. This paper is the third in a series with several installments.