Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT

TL;DR

This note addresses various aspects of the correspondence between AdS/CFT correlators with Δ− boundary conditions and the change cUV−cIR of the central charge between the UV and IR fixed points.

Abstract

According to the AdS/CFT dictionary, perturbing the large N boundary theory by a relevant double-trace deformation of the form f O^2 corresponds in the bulk to imposing ``mixed'' boundary conditions for the field dual to O. In this note we address various aspects of this correspondence. The change c_{UV} - c_{IR} of the central charge between the UV ad IR fixed points is known from explicit calculations (hep-th/0210093, hep-th/0212138) to be exactly the same in the bulk and in the boundary theories. By comparing the appropriate bulk and boundary functional determinants, we give a simple ``kinematic'' explanation for this universal agreement. We also clarify the prescription for computing AdS/CFT correlators with Delta_- boundary conditions.

Figures (2)

• Figure 1: The contour and branch cut in Eq. (\ref{['bobservation']}) kirsten04b.
• Figure 2: Numerical value of $R$ in eq. (\ref{['eq:detnumer']}) versus the number of roots included in the computation. The curve converges nicely to the horizontal line (the analytical answer).