A Carroll Limit of AdS/CFT: A Triality with Flat Space Holography?

Technical Summary: A Carroll Limit of AdS/CFT: A Triality with Flat Space Holography?

Problem Statement
The paper addresses the challenge of formulating holographic dualities for spacetimes that are not Anti-de Sitter (AdS), specifically flat space holography. While flat space holography admits formulations on the celestial sphere or Carrollian manifolds, a direct derivation of flat space holography from the standard AdS$_5$/CFT$_4$ correspondence via a large radius limit presents a fundamental inconsistency. As the bulk radius $R \to \infty$, the bulk geometry becomes 10-dimensional Minkowski space (Mink$_{10}$), but the boundary of Mink$_{10}$ is not the 4-dimensional Carrollian manifold required for the Carrollian limit of the boundary gauge theory. Specifically, $\partial \text{Mink}_{10} \neq \text{Carroll}_4$. The authors seek a consistent limit where the Carrollian structure emerges on both the bulk (gravity) and boundary (gauge) sides simultaneously, preserving the holographic duality.

Methodology
The authors construct a novel holographic duality by incorporating the Carroll limit ($c \to 0$) directly into Maldacena's AdS/CFT construction. The methodology relies on a specific rescaling of fields and parameters on both sides of the correspondence, ensuring that the near-horizon limit (Maldacena's decoupling limit) commutes with the Carroll limit.

1. Gravity Side (Bulk): Starting with the metric of a stack of $N$ D3-branes, the authors first take the near-horizon limit to obtain the standard AdS$_5 \times S^5$ geometry. They then apply a Carrollian rescaling to the coordinates and the radius: $z \to z/c$, $x^i \to x^i/c$, and $R \to R/c$. To maintain consistency with the definition of the radius $R^2 \sim \sqrt{g_s N} \alpha'$, they simultaneously rescale the string coupling and the number of branes as $g_s \to c g_s$ and $N \to N/c$. This ensures $g_s N$ remains finite. Taking $c \to 0$ yields a Carrollian AdS$_5 \times S^5$ geometry. The authors verify that taking the limits in the reverse order (rescaling first, then taking the near-horizon limit) yields the same result. The resulting bulk theory corresponds to a magnetic Carroll string action with a target space defined by the Carrollian metric.
2. Gauge Theory Side (Boundary): On the gauge theory side, the authors analyze the non-abelian Dirac-Born-Infeld (DBI) action describing open strings on the D3-branes. They implement a Carrollian rescaling of the world-volume coordinates where the time coordinate scales as $X^0 \to c X^0$ while spatial coordinates remain unscaled ($X^i \to X^i$). This specific scaling is chosen because it commutes with the low-energy limit ($\alpha' \to 0$), unlike the alternative scaling where spatial coordinates are rescaled. Combined with the rescaling of the string coupling ($g_s \to c g_s$), this limit yields the bosonic sector of a magnetic Carroll $\mathcal{N}=4$ Super Yang-Mills (SYM) theory.
3. Symmetry Analysis: The authors perform a detailed analysis of the symmetries of the resulting Carroll $\mathcal{N}=4$ SYM action. They identify the spacetime symmetries as the Conformal Carroll Algebra (CCA) and derive a large set of internal symmetries acting on the fields and Lagrange multipliers introduced by the Hubbard-Stratonovich transformation.

Key Contributions and Results

• Carroll AdS/CFT Duality: The paper establishes a duality between string theory in a Carrollian AdS$_5 \times S^5$ bulk geometry and magnetic Carroll $\mathcal{N}=4$ SYM on the boundary. This is achieved by demonstrating that the Carroll limit can be consistently applied to both sides of the correspondence.
• Proposed Triality: Based on the connection between Carrollian limits and flat space holography, the authors propose a "triality" connecting three theories:
  1. String theory in the Carroll limit of AdS$_5 \times S^5$.
  2. Relativistic string theory in flat spacetime (specifically Mink$_5 \times X_5$ after compactification).
  3. Carroll $\mathcal{N}=4$ SYM.
     This framework suggests that Carroll string theory and relativistic string theory in flat space are dual to the same boundary Carroll gauge theory, potentially unifying different holographic webs.
• Symmetry Structure: The analysis reveals that the symmetries of the Carroll $\mathcal{N}=4$ SYM include the infinite-dimensional Conformal Carroll Algebra (CCA) and a rich set of internal symmetries. Crucially, the authors find that the full algebra of symmetries does not close under ordinary Lie brackets for certain generators, particularly those involving Lagrange multipliers. The commutators of internal symmetries with conformal generators often produce terms that are not symmetries of the action, suggesting a non-linear realization or the need for modified brackets (e.g., Dirac brackets) to handle constraints.
• Boundary Geometry: The authors show that the Penrose conformal boundary of the Carroll AdS$_5 \times S^5$ geometry is indeed the Carroll geometry of 4-dimensional flat spacetime, resolving the initial inconsistency regarding the boundary of Mink$_{10}$.

Significance and Claims
The paper claims to provide a "new perspective" on the problem of flat space holography by embedding the Carroll limit within the standard AdS/CFT framework. The significance lies in:

• Offering a concrete setting where infrared-divergent observables in flat space might be computed in a Carroll-AdS context, where such divergences are naturally tamed.
• Highlighting the necessity of developing a rigorous formulation of Carroll string theory, particularly regarding the equivalence of Polyakov and Nambu-Goto actions in the Carroll limit and the coupling of fermions in magnetic Carroll supersymmetry.
• Proposing that the non-closure of the symmetry algebra in the boundary theory may be linked to the non-closure of supersymmetry commutators in magnetic Carroll limits, suggesting a deep structural connection.

The authors remain modest regarding the immediate verification of the duality. They explicitly state that due to the current lack of a developed Carrollian Type IIB supergravity, they cannot yet verify that the bulk symmetries (Killing vectors) match the boundary symmetries. Furthermore, they note that the proposed triality requires further investigation into the behavior of the internal compact space $X_5$ under the Carroll limit and the specific duality maps (such as T-duality analogues) between Carroll and relativistic strings. The work serves as a foundational proposal and a motivation for systematic future study of Carroll string theory and related holographic dualities.