Holographic solutions from 5D $SO(2)\times ISO(3)$ $N=4$ gauged supergravity

Technical Summary: Holographic Solutions from 5D SO(2) × ISO(3) N = 4 Gauged Supergravity

Problem and Context
The paper investigates holographic solutions within the framework of five-dimensional $N=4$ gauged supergravity coupled to three vector multiplets, specifically with an $SO(2) \times ISO(3)$ gauge group. This theory arises from a consistent truncation of eleven-dimensional supergravity on $H_2 \times S^4$, which itself is derived from the maximal $N=8$ gauged supergravity in seven dimensions compactified on a Riemann surface of genus $g > 1$ ($H_2$). The primary physical motivation is to explore the holographic duals of deformations of the $N=2$ superconformal field theory (SCFT) in four dimensions, which corresponds to the supersymmetric $AdS_5$ vacuum of this gauged supergravity. This vacuum is dual to M5-branes wrapped on $H_2$. The study aims to map out the landscape of supersymmetric solutions, including Renormalization Group (RG) flows, conformal interfaces (Janus solutions), and black objects (strings and holes), to understand the phase structure of the dual field theories and their dimensional reductions.

Methodology
The authors employ the embedding tensor formalism to construct the bosonic Lagrangian and fermionic supersymmetry transformations for the $N=4$ gauged supergravity with the specified gauge group. The methodology involves:

1. Truncation and Symmetry Analysis: Identifying consistent truncations of the scalar manifold $SO(5,3)/SO(5) \times SO(3)$ by imposing specific residual symmetries ($SO(2) \times SO(3)$, $SO(2) \times SO(2)$, and $SO(2)_{\text{diag}}$). This reduces the number of active scalar fields to a manageable set of singlets.
2. BPS Equations: Deriving first-order BPS equations from the vanishing of fermionic variations ($\delta \psi_\mu, \delta \chi, \delta \lambda$) under specific Killing spinor projectors. These equations govern the radial evolution of the scalars and the warp factor of the metric.
3. Topological Twists: For solutions involving compactification on curved manifolds (Riemann surfaces $\Sigma$ or 3-manifolds $M_3$), the authors implement topological twists by turning on specific gauge fields to cancel the spin connection, thereby preserving supersymmetry in the lower-dimensional effective theory.
4. Analytic and Numerical Solutions: While some RG flows admit analytic solutions (particularly those driven by relevant operators or specific symmetry limits), the authors resort to numerical integration for flows involving irrelevant operators, Janus interfaces, and black hole/string interpolations.
5. Uplift and Physicality Checks: Solutions are uplifted to eleven dimensions using the known consistent truncation ansatz on $H_2 \times S^4$. The physical acceptability of singularities in the IR is assessed using standard criteria (boundedness of the scalar potential and behavior of the uplifted metric components).

Key Contributions and Results

• Holographic RG Flows:

  • The paper identifies RG flows between the $N=2$ SCFT (UV) and non-conformal phases.
  • Relevant Deformations: Solutions driven by a dimension $\Delta=2$ operator (dual to the dilaton $\Sigma$) describe flows from the SCFT to a non-conformal phase in the IR. The IR singularity is shown to be physically acceptable.
  • Irrelevant Deformations: Solutions driven by dimension $\Delta=6$ operators (dual to scalars $\phi_3, \phi_5$) describe flows where the $N=2$ SCFT acts as an IR fixed point, approached from a UV non-conformal phase.
  • Supersymmetry Breaking: Flows preserving only four supercharges ($N=1$ in 4D) are found, driven by irrelevant operators of dimension $\Delta=5$ and $\Delta=6$.
  • UV Identification: The UV limits of these non-conformal phases are identified as decompactification limits of the six-dimensional $N=(0,2)$ SCFT on M5-branes wrapped on $H_2$.

• Supersymmetric Janus Solutions:

  • The authors search for $AdS_4$-sliced domain walls representing conformal interfaces.
  • Result: No regular Janus solutions interpolating between two $AdS_5$ vacua were found within the considered truncation.
  • Non-Conformal Interfaces: Regular solutions interpolating between non-conformal phases of the $N=2$ SCFT were found numerically. These solutions preserve four supercharges, while the asymptotic non-conformal phases preserve eight.
  • Singular Interfaces: Solutions interpolating between a non-conformal phase and a singularity, or between two singularities, were also identified, potentially describing conformal boundaries.

• Supersymmetric Black Strings:

  • Solutions interpolating between the $AdS_5$ vacuum and $AdS_3 \times \Sigma$ geometries are constructed via topological twists on a Riemann surface $\Sigma$.
  • Twist Types:
    • An $SO(2)$ twist yields an $AdS_3 \times H_2$ near-horizon geometry preserving four supercharges in the flow (enhanced to eight in the IR).
    • An $SO(2) \times SO(2)$ twist yields both $AdS_3 \times H_2$ and $AdS_3 \times S^2$ near-horizon geometries, preserving two supercharges in the flow (enhanced to four in the IR).
  • Dual Field Theories: These solutions describe RG flows from the 4D $N=2$ SCFT to 2D $N=(2,2)$ or $N=(0,2)$ SCFTs.
  • Scalar Effects: By turning on vector multiplet scalars, the authors find flows starting from non-conformal 4D phases (rather than the SCFT) flowing to the 2D SCFTs, as well as flows with intermediate $AdS_5$ regions.

• Supersymmetric Black Holes:

  • Solutions interpolating between the $AdS_5$ vacuum and $AdS_2 \times M_3$ geometries are found via an $SO(3)$ twist on a 3-manifold $M_3$.
  • Geometry: The only valid near-horizon geometry is $AdS_2 \times H_3$, preserving four supercharges.
  • Dual Field Theory: These describe RG flows from the 4D $N=2$ SCFT (or non-conformal phases) to superconformal quantum mechanics in the IR.
  • Entropy: The Bekenstein-Hawking entropy for these black holes is computed in terms of the rank $N$ of the dual theory and the genus of the Riemann surface.

Significance and Claims
The paper claims to provide a comprehensive classification of holographic solutions within this specific $N=4$ gauged supergravity model. Its significance lies in:

1. Completing the Solution Space: It systematically maps out RG flows, Janus interfaces, and black objects, identifying the specific operator deformations (relevant, marginal, irrelevant) that drive these flows.
2. Novelty of Janus Solutions: It presents the first examples of supersymmetric Janus solutions in five-dimensional gauged supergravities that interpolate between non-conformal phases rather than conformal fixed points.
3. M-Theory Embedding: All solutions are explicitly shown to be consistent truncations of M-theory on $H_2 \times S^4$, providing a robust gravitational dual for the proposed field theory deformations and dimensional reductions.
4. Physicality of Singularities: The paper rigorously demonstrates that the IR singularities appearing in the RG flows and black hole solutions are physically acceptable according to established criteria, validating the holographic interpretation of these flows to lower-dimensional theories.

The authors conclude that while the near-horizon geometries of the black strings and holes often coincide with those of pure $N=4$ gauged supergravity (when scalars vanish), the inclusion of vector multiplet scalars allows for a richer set of interpolating solutions connecting non-conformal phases to lower-dimensional SCFTs. They suggest future work could extend this to line defects and solutions involving spindle geometries.