Interpolating Boundary Conditions on $AdS_2$

TL;DR

This paper investigates boundary conditions for massless scalars in AdS2 that interpolate between Dirichlet and Neumann types, analyzing their conformal invariance and holographic correlation functions in the context of ABJM theory.

Contribution

It introduces and studies boundary conditions interpolating between Dirichlet and Neumann in AdS2, assessing their conformal invariance through holographic correlators in a string theory setup.

Findings

Only one boundary condition preserves full conformal invariance.

The other boundary condition is covariant under translations and rescalings but not special conformal transformations.

Holographic correlators reveal the impact of boundary conditions on conformal symmetry.

Abstract

We consider two instances of boundary conditions for massless scalars on $AdS_2$ that interpolate between the Dirichlet and Neumann cases while preserving scale invariance. Assessing invariance under the full $SL(2;\mathds{R})$ conformal group is not immediate given their non-local nature. To further clarify this issue, we compute holographically 2- and 4-point correlation functions using the aforementioned boundary conditions and study their transformation properties. Concretely, motivated by the dual description of some multi-parametric families of Wilson loops in ABJM theory, we look at the excitations of an open string around an $AdS_2\subset AdS_4\times\mathbb{CP}^3$ worldsheet, thus obtaining correlators of operators inserted along a $1$-dimensional defect in ${\cal N}=6$ super Chern-Simons-matter theory at strong coupling. Of the two types of boundary conditions analyzed, only one leads to the expected functional structure for conformal primaries; the other exhibits covariance under translations and rescalings but not under special conformal transformations.