Short Boundary RG Flows in 3D Supergravity

• Short boundary RG flows are spatially homogeneous domain wall solutions in 3D supergravity, characterized by first-order dynamical equations and a dual CFT interpretation.
• The choice of Dirichlet, Neumann, or mixed boundary conditions directly influences whether flows are driven by explicit deformations or spontaneous VEVs.
• Analytic and numerical analyses reveal rich phase diagrams where superpotential expansions and target space curvature dictate transitions between UV AdS fixed points and IR endpoints.

Short boundary renormalization group (RG) flows, in the context of three-dimensional $\mathcal{N}=(2,0)$ gauged supergravity coupled to a nonlinear sigma model with hyperbolic target space $H^2=SU(1,1)/U(1)$, refer to spatially homogeneous domain wall solutions whose dual two-dimensional conformal field theory (CFT) interpretation is controlled by the structure of scalar field boundary conditions at the asymptotic AdS boundary. These flows are characterized both by their dynamical realization as solutions to the Einstein–scalar coupled system and by their dual CFT meaning via the AdS/CFT correspondence. The defining feature is that the dynamics, spectrum, and phase structure of these RG flows are strongly sensitive to the choice of boundary condition (e.g. Dirichlet, Neumann, or mixed) and to the expansion type of the fake (or actual) superpotential near the boundary.

1. Dynamical Framework: Domain Walls and Superpotentials

The bosonic sector is truncated to a real scalar $\phi$ and the metric is assumed in domain-wall form,

$ds^2 = e^{2A(r)} (-dt^2 + dx^2) + dr^2,$

where $r$ is interpreted holographically as the RG scale. The scalar potential $V(\phi)$ is determined by the gauged supergravity structure, specifically,

$$V(\phi) = \frac{a^2}{4} [W'(\phi)]^2 - \frac{1}{2} W(\phi)^2,$$

for some (possibly "fake") superpotential $W(\phi)$.

The RG flow equations are recast into a first-order dynamical system,

$$\frac{dA}{dr} = -W, \qquad \frac{d\phi}{dr} = a^2 W'(\phi),$$

so that solutions correspond to curves in the $(\phi,A)$ or, after a change of variables, in the scale-invariant $(Z,X)$ plane with $Z = 1/(1+e^\phi)$ and $X = \frac{d\phi/dr}{dA/dr}$.

In this framework, fixed points where $W'(\phi^*)=0$ yield AdS vacua, corresponding to conformal boundary theories, while flows connecting such extrema or approaching singular points represent boundary RG flows.

2. Boundary Conditions and Holographic Dictionary

The holographic interpretation of the RG flow crucially depends on the boundary conditions imposed on the fluctuating scalar field. These are determined by the asymptotic (near-boundary) expansions,

$$\phi(r,x) \sim e^{-\Delta_{-} r} \phi_{-}(x) + e^{-\Delta_{+} r} \phi_{+}(x), \quad \Delta_{\pm}=1 \pm |1-2a^2|.$$

Three main types arise:

• Dirichlet: $\phi_-$ is fixed. This corresponds to specifying a source for a dual operator of dimension $\Delta_+$, triggering a deformation of the dual CFT.
• Neumann: The canonically conjugate momentum $\hat{p}_+$ is fixed (Neumann quantization), so it is $\phi_+$ that acts as the boundary data. Now, the dual operator has dimension $\Delta_-$, and flows can be driven by VEV rather than deformation.
• Mixed: A boundary condition of the form $\delta \hat{p}_+ + f''(\phi_-) \delta\phi_- = 0$. This realizes multitrace deformations in the dual CFT and interpolates between Dirichlet and Neumann.

The choice of boundary condition directly affects the RG interpretation: for example, a Dirichlet boundary can represent a flow triggered by an explicit operator insertion (deformation), while a Neumann or mixed condition can correspond to a flow triggered by spontaneous symmetry breaking (VEV flow).

3. Superpotential Expansion and Classification of Flows

The quadratic expansion of the superpotential near the UV critical point $\phi^*$ yields two branches,

$$W_\pm(\phi) = -2m \left[1 + \frac{1}{2a^2} _\pm (\phi-\phi^*)^2 + \ldots\right].$$

• The $W_+$ branch leads to RG flows where the subleading term (i.e., the VEV) is generically nonzero for Dirichlet boundary conditions and zero for Neumann; the RG flow is then interpreted as VEV-driven.
• Conversely, the $W_-$ branch (when realized) corresponds to a flow generated by an explicit single-trace deformation.

This distinction reflects the standard holographic UV asymptotics, with the normalizable versus nonnormalizable mode relating to the operator VEV and source, respectively.

4. Dynamical System Structure and Phase Diagrams

The RG system is written in terms of $(Z,X)$ variables: $$\frac{dZ}{dA} = X Z (Z-1), \qquad \frac{dX}{dA} = \left(\frac{X^2}{a^2} - 2\right)\left(X + \frac{a^2}{2} \frac{V'(\phi)}{V(\phi)}\right).$$ The phase portraits in the $(Z,X)$ plane reveal the topology of RG flows: the number and character (node, saddle, focus) of fixed points depend on the parameter $a^2$, which sets the curvature of the target hyperbolic space. For $a^2 \leq 1/2$ there is a single extremum; for $1/2 < a^2 < 1$, multiple extrema arise, allowing for a richer structure of fixed points and the possibility of more exotic RG trajectories ("bouncing" flows, etc.)

These diagrams illustrate which flows connect UV AdS fixed points to IR AdS vacua or to singular endpoints, and how the boundary condition types manifest in the global structure.

5. Numerical Analysis and Behavior of Flows

Numerical integration complements the analytic structure, providing explicit flows for various values of $a^2$ and different initial conditions. The solutions for $\phi$ and the scale factor $A$ along the RG trajectories can be classified by their near-boundary expansion, and the corresponding fake superpotentials $W(\phi)$ can be plotted to confirm their asymptotic form and to distinguish between $W_+$ and $W_-$ behaviors.

The numerics robustly confirm the existence of short RG flows with the expected UV and IR properties, including both supersymmetric and nonsupersymmetric branches.

6. Holographic Interpretation and Physical Consequences

The boundary data and superpotential expansion types map, through the holographic dictionary, to precise statements in the dual two-dimensional CFT. For Dirichlet boundary conditions, a $W_+$ (VEV) flow indicates the IR is reached by spontaneous symmetry breaking; a $W_-$ (deformation) flow signals an explicit breaking via relevant operator insertion.

Mixed boundary conditions, corresponding to multitrace deformations, can interpolate between these. The mode structure of the scalar and the expansion of the fake superpotential encode physical quantities such as operator scaling dimensions, sources, and VEVs.

This rigorous connection allows one to chart which boundary sources or VEVs drive the RG flows, how the flows terminate, and how the space of boundary RG trajectories is globally organized. Moreover, this framework is applicable to probe more intricate deformations, the effect of multi-extremum potentials, and the emergence of nonperturbative or oscillatory RG flows.

7. Summary Table of Key Features

Boundary Condition	Leading Term Fixed	Dual CFT Driver	Flow Type (Superpotential)
Dirichlet	$\phi_-$	Source	$W_-$ (deformation)
Neumann	$\hat{p}_+$	VEV	$W_+$ (VEV)
Mixed	Both	Multitrace; interpolates	$W_{\pm}$, depending on $f''(\phi_-)$

This table summarizes the relationship between boundary condition, dual operator content, and the analytic behavior of the RG flow.

Conclusion

Short boundary RG flows in three-dimensional $\mathcal{N}=(2,0)$ gauged supergravity with $SU(1,1)/U(1)$ target space are realized as domain wall solutions whose nature—deformation-driven or VEV-driven—is governed by near-boundary expansions of the fake superpotential and the choice of boundary condition for the scalar. The rich phase diagram, modulated by the curvature parameter $a^2$, displays both standard and exotic RG trajectories, each with a clear dual CFT interpretation through holographic renormalization methods. The analytic and numerical characterization of these flows, as well as their translation into dual operator phenomena, provides a comprehensive geometric and physical picture of short boundary RG flows in holographic settings (Arkhipova et al., 18 Feb 2024).