Constructing Non-Relativistic AdS$_5$/CFT$_4$ Holography

Technical Summary: Constructing Non-Relativistic AdS5/CFT4 Holography

Problem Statement
The paper addresses the challenge of incorporating the non-relativistic limit into the standard Maldacena $AdS_5/CFT_4$ correspondence. While the relativistic duality between Type IIB string theory on $AdS_5 \times S^5$ and $\mathcal{N}=4$ Super Yang-Mills (SYM) is well-established, it is not guaranteed that taking a non-relativistic limit preserves this equivalence. Limits are generally non-invertible operations, and the order in which they are applied (e.g., near-horizon/decoupling vs. non-relativistic) can alter the final physical theory. Furthermore, non-relativistic holography is intrinsically "non-AdS" and non-Lorentzian, necessitating a re-evaluation of the holographic principle's scope. The authors aim to determine if a consistent holographic duality exists between non-relativistic string theory and a non-relativistic gauge theory, specifically investigating whether the near-horizon limit and the non-relativistic limit commute.

Methodology
The authors employ a dual-perspective approach, analyzing the limits from both the gravity (string theory) and gauge theory sides to ensure consistency.

1. Gravity Perspective:

   • Starting with a stack of $N$ coincident black D3-branes in Type IIB string theory, the authors derive the spacetime metric.
   • They systematically apply the near-horizon limit ($\alpha' \to 0$) and the non-relativistic limit ($c \to \infty$) in both possible orders.
   • The non-relativistic limit is defined as "stringy," involving the rescaling of both a time-like and a space-like coordinate by a parameter $c$. This transforms the Lorentzian geometry into a String Newton-Cartan (SNC) geometry, characterized by a longitudinal metric $\tau_{\mu\nu}$ and a transverse tensor $h_{\mu\nu}$.
   • To handle divergences, the string is coupled to a critical closed Kalb-Ramond $B$-field before taking the limit.
   • The authors utilize the Penrose procedure to identify the conformal boundary of the resulting SNC $AdS_5 \times S^5$ geometry.

2. Gauge Theory Perspective:

   • The authors analyze the Dirac-Born-Infeld (DBI) action for a stack of D3-branes in flat space, which governs the open string sector.
   • They apply the decoupling limit ($\alpha' \to 0$) and the non-relativistic limit ($c \to \infty$) in both orders.
   • Crucially, they explore two distinct non-relativistic rescaling schemes:
     • Scheme A: Rescaling transverse coordinates by $1/c$ and gauge fields by $1/c^2$.
     • Scheme B: Rescaling longitudinal coordinates by $c$ (an alternative limit).
   • They expand the action to leading order in the limits to derive the effective field theories.

3. Symmetry Matching:

   • To test the proposed duality, the authors compute the global symmetries (Killing vectors) of the non-relativistic string action on SNC $AdS_5 \times S^5$.
   • These are compared against the symmetries of the derived non-relativistic gauge theories.

Key Contributions and Results

• Commutativity of Limits: The paper demonstrates that the near-horizon/decoupling limit commutes with the non-relativistic limit. Whether one takes the near-horizon limit first or the non-relativistic limit first, the final result is the same:

  • Gravity Side: The geometry converges to SNC $AdS_5 \times S^5$. The $S^5$ sphere "flattens" into a 5-dimensional Euclidean space in the limit.
  • Gauge Side: The theory converges to Galilean Electrodynamics (GED) supplemented by 5 uncharged massless free scalar fields.
  • The conformal boundary of the SNC bulk is identified as Newton-Cartan 4d Minkowski spacetime (NC Mink$_4$), where the dual gauge theory resides.

• Symmetry Analysis:

  • The authors systematically solve the SNC Killing equations for the bulk geometry. They find an infinite-dimensional isometry algebra containing an $sl(2, \mathbb{R})$ subalgebra (generated by $H, D, K$) corresponding to $AdS_2$ isometries, and infinite towers of translations dependent on light-cone coordinates.
  • They match these symmetries with the symmetries of the abelian GED theory with 5 scalars. The bulk Killing vectors evaluated at the boundary ($z=0$) match the generators of the GED theory (Galilean conformal algebra) and the infinite-dimensional symmetries acting on the scalar fields.

• Correction and Refinement (Note Added):

  • Initially, the authors proposed the duality between SNC string theory and the abelian GED with free scalars.
  • However, a "Note Added" (August 2025) corrects this claim based on recent literature [43]. The authors acknowledge that the abelian theory with free scalars possesses more symmetries than the bulk Killing vectors.
  • The paper concludes that the correct holographic dual is actually Galilean Yang-Mills (GYM) theory with 5 interacting (non-abelian) scalar fields. The symmetries of this non-abelian theory match the bulk Killing vectors in a one-to-one correspondence. This resolves the discrepancy where the "flattening" of the sphere in the gravity limit was initially thought to imply an abelian gauge theory.

• Alternative Limit: The paper also derives a Galilean Yang-Mills theory via an alternative rescaling (Scheme B), which preserves the non-abelian structure ($U(N)$) and includes a potential for the scalars. While this theory is derived, the authors do not provide a holographic description for it in the main text, though the "Note Added" suggests it is the relevant dual.

Significance and Claims
The paper claims to construct a new type of holographic correspondence: a duality between non-relativistic string theory in SNC $AdS_5 \times S^5$ and a non-relativistic gauge theory (specifically Galilean Yang-Mills with interacting scalars) on the NC Mink$_4$ boundary.

• Uniqueness: The authors emphasize that this is the first holographic duality involving a non-relativistic string theory with a relativistic world-sheet. Previous works on non-relativistic holography often involved non-relativistic world-sheets (e.g., Spin Matrix Theory).
• Consistency: The primary evidence for the duality is the rigorous proof that the decoupling and non-relativistic limits commute, yielding a unique geometry and field theory, and the subsequent matching of global symmetries between the bulk and the boundary.
• Modesty: The authors acknowledge that this is a "first test" based on symmetry matching. They explicitly state that quantitative tests (such as matching semiclassical string energies to conformal dimensions of operators) are the necessary next step and have not yet been performed. They also note that the construction currently covers only the bosonic sector, with supersymmetry being a subject for future work.

In summary, the paper establishes a formal framework for non-relativistic $AdS_5/CFT_4$ holography, demonstrating the compatibility of limits and identifying the correct non-abelian gauge theory dual, while deferring quantitative verification to future research.