Deforming and dissecting AdS$_3$ with matter

TL;DR

This work analyzes two holographic deformations of the HMTZ AdS$_3$ setup with a bulk scalar: the $T\overline{T}$ deformation implemented via asymptotic mixed boundary conditions (MBC) on the metric, and a Dirichlet-type deformation defined on a finite bulk cutoff. It shows that the $T\overline{T}$ deformation corresponds to MBC in the presence of bulk matter, while the Dirichlet deformation yields a field-theory operator containing $T\overline{T}$ plus additional irrelevant terms, with the exact form depending on the scalar source flow. The authors compute deformed energy spectra from both bulk and boundary perspectives, confirming agreement once the scalar-source flow is properly accounted, and they highlight the essential difference introduced by bulk matter. The results illuminate how bulk matter modifies holographic $T\overline{T}$ structures, clarify the relation between finite-cutoff holography and $T\overline{T}$, and connect with broader proposals in the literature about Weyl generators and matter counterterms.

Abstract

We study deformations of the model by Henneaux, Martínez, Troncoso and Zanelli [arXiv:hep-th/0201170] which features asymptotically AdS$_3$ black hole solutions that incorporate the exact backreaction of a scalar field. The presence of bulk matter causes the $T \overline T$ deformation of the (putative) dual CFT$_2$ to differ from the deformation defined in the bulk by imposing Dirichlet boundary conditions at finite radius. We work out both of these deformations explicitly and verify that $T \overline T$-deforming the boundary theory corresponds to imposing mixed boundary conditions on the metric at the conformal boundary, whereas the bulk "Dirichlet deformation" gives rise to a field theory deforming operator that includes $T \overline T$ as well as other irrelevant terms. We check our results by calculating the deformed energy spectrum for either case using both the bulk and boundary prescriptions, finding agreement after taking into account additional terms coming from the flow of the scalar source. We interpret our explicit results and compare them with the predictions of similar proposals in the literature.

Figures (3)

• Figure 1: Geometric representation on the $(\mu,\gamma^{[\cdot]}_{ij})$ plane of the auxiliary flow \ref{['eq:Zmu2Z0']} compared to the standard field theory flow, represented respectively by the curved (green) and straight (orange) trajectories. Along the auxiliary flow, the background geometry $\gamma_{ij}^{[\mu]}$ is non-constant, flowing from $\gamma_{ij}^{[0]}$ to $\gamma_{ij}$ according to $\gamma_{ij}^{[\mu]} = \gamma_{ij}^{[0]} - 2 \mu \braket{\hat{T}^{[0]}_{ij}} + \mu^2 \braket{\hat{T}^{[0]}_{ik}} \braket{T^{[0]}_{jl}} \gamma^{kl}_{[0]}$. The short arrows indicate the corresponding $\mu$-derivatives, illustrating their relation to the $\langle \mathcal{O}_TTbar\xspace \rangle$ operator.
• Figure 2: A schematic representation for the holographic interpretation of the TTbar deformation in terms of the asymptotic mixed boundary conditions (MBC), applied in particular to the HMTZ solution. On the left, we have an equal-time slice of the HMTZ solution \ref{['metriclinelement']} with asymptotic Dirichlet boundary conditions (DBC) on the metric, dual to the holographic $\text{CFT}_2$ discussed in \ref{['subsection:2.1']}. Deforming this $\text{CFT}_2$ with TTbar is holographically dual to changing the asymptotic DBC to the asymptotic MBC \ref{['MBC']}, as depicted on the right. The deformation can be interpreted in terms of the flow of the state parameter ($B\rightarrow \tilde{B}$) labeling the corresponding bulk solution.
• Figure 3: A schematic representation of the holographic interpretation of the Dirichlet deformation, applied to the HMTZ solution. On the left, we have the HMTZ solution \ref{['metriclinelement']} with asymptotic DBC on the metric, dual to the holographic $\text{CFT}_2$ discussed in \ref{['subsection:2.1']}. Deforming this $\text{CFT}_2$ with $f(J)(TTbar\xspace + \dots)$, as prescribed by the flow \ref{['partialrhoflow']}, is holographically dual to changing the asymptotic DBC to a DBC at finite cutoff \ref{['DBC']}, as depicted on the right. Due to our convenient choice of deformation coupling \ref{['tilderhor']}, the state parameter of the HMTZ solution is unchanged.