Double-trace operators and one-loop vacuum energy in AdS/CFT

TL;DR

This work analyzes double-trace deformations in AdS/CFT as mixed boundary conditions for a tachyonic scalar, identifying UV and IR fixed points corresponding to $\Delta_-$ and $\Delta_+$ via a Legendre transformation at large $N$. It computes the finite one-loop vacuum energy difference $V_+ - V_-$ between these endpoints, deriving an explicit negative result in AdS$_5$ and confirming the sign in general odd dimensions, consistent with a holographic $c$-theorem at one loop. The study further shows that the finite-$f$ vacuum energy difference varies monotonically with the boundary-condition parameter and discusses the implications for an interpolating geometry and central-charge running. Overall, the paper provides a controlled one-loop check of the $c$-theorem beyond classical supergravity and clarifies how multi-trace deformations map to boundary conditions in the bulk.

Abstract

We perform a one-loop calculation of the vacuum energy of a tachyon field in anti de-Sitter space with boundary conditions corresponding to the presence of a double-trace operator in the dual field theory. Such an operator can lead to a renormalization group flow between two different conformal field theories related to each other by a Legengre transformation in the large N limit. The calculation of the one-loop vacuum energy enables us to verify the holographic c-theorem one step beyond the classical supergravity approximation.

Figures (1)

• Figure 1: $i (G_{d-\Delta}-G_\Delta)$ as a function of $\Delta$, for $AdS_6$, $AdS_8$, and $AdS_{10}$ (corresponding to $d=5,7,9$), in units where $L=1$.