Correlators on non-supersymmetric Wilson line in $$ \mathcal{N}=4 $$ SYM and AdS2/CFT1

Abstract

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title> <jats:p>Correlators of local operators inserted on a straight or circular Wilson loop in a conformal gauge theory have the structure of a one-dimensional “defect” CFT. As was shown in arXiv:1706.00756, in the case of supersymmetric Wilson-Maldacena loop in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{N}=4 $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> </jats:alternatives> </jats:inline-formula> SYM one can compute the leading strong-coupling contributions to 4-point correlators of the simplest “protected” operators by starting with the AdS<jats:sub>5</jats:sub> × <jats:italic>S</jats:italic> <jats:sup>5</jats:sup> string action expanded near the AdS<jats:sub>2</jats:sub> minimal surface and evaluating the corresponding tree-level AdS<jats:sub>2</jats:sub> Witten diagrams. Here we perform the analogous computations in the non-supersymmetric case of the standard Wilson loop with no coupling to the scalars. The corresponding non-supersymmetric “defect” CFT<jats:sub>1</jats:sub> should have an unbroken SO(6) global symmetry. The elementary bosonic operators (6 SYM scalars and 3 components of the SYM field strength) are dual respectively to the <jats:italic>S</jats:italic> <jats:sup>5</jats:sup> embedding coordinates and AdS<jats:sub>5</jats:sub> coordinates transverse to the minimal surface ending on the line at the boundary. The SO(6) symmetry is preserved on the string side provided the 5-sphere coordinates satisfy Neumann boundary conditions (as opposed to Dirichlet in the supersymmetric case); this implies that one should integrate over the <jats:italic>S</jats:italic> <jats:sup>5</jats:sup> expansion point. The massless <jats:italic>S</jats:italic> <jats:sup>5</jats:sup> fluctuations then have logarithmic propagator, corresponding to the boundary scalar operator having dimension <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \Delta =\frac{5}{\sqrt{\lambda }}+\dots $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:msqrt> <mml:mi>λ</mml:mi> </mml:msqrt> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mo>…</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> at strong coupling. The resulting functions of 1d cross-ratio appearing in the 4-point functions turn out to have a more complicated structure than in the supersymmetric case, involving polylog (Li<jats:sub>3</jats:sub> and Li<jats:sub>2</jats:sub>) functions. We also discuss consistency with the operator product expansion which allows extracting the leading strong coupling corrections to the anomalous dimensions of the operators appearing in the intermediate channels.</jats:p>