Comments on $T \bar T$ double trace deformations and boundary conditions

Abstract

We study the UV dynamics of $μT \bar T$ deformed conformal field theories formulated as a deformation of generating functions. We explore the issue of non-perturbative completion of the $μ$ expansion by deriving an integral expression using the Fourier/Legendre transform technique, and show that it is more natural to impose Neumann, as opposed to the Dirichlet, boundary condition, for the metric at the cut-off surface recently proposed by McGough, Mezei, and Verlinde. We also comment on interesting connection to boundary conformal field theories.

Figures (2)

• Figure 1: Illustration of locally $AdS_{d+1}$ bulk spacetime $N$ bounded by boundary components $M$ and $Q$. (a) is a reproduction of figure 1 of Takayanagi:2011zk. (b) is the embedding of (a) inside global anti de Sitter geometry as was illustrated in figure 6 of Karch:2000ct. The spacetime of boundary field theory from the $T \bar{T}$ deformed CFT point of view is $Q$. Here, we are taking the effective cosmological constant to be negative so that the geometry on $Q$ is $AdS_d$. From the boundary field theory point of view, $M$ is an auxiliary structure which emerges as a dual representation of large number of degrees of freedom that the original CFT contained. Operators are inserted on $M$ and are to be interpreted as boundary observables for the gravity theory on $AdS_d$, or as $S$-matrix elements in the limit where $AdS_d$ approaches flat $d$-dimensional space-time.
• Figure 2: An example of non-single valued field configuration which are allowed in Nambu-Goto theory. Similar issue arises in the "tree stump" configuration of BIons illustrated in figure 2 of Callan:1997kz.