Reading a Paper 2026, 06 - A Holographic Constraint on Scale Separation

Back to my roots. I never got around to looking into holography, and while I would like to find a good 500-page tome on it one of these days, I might as well try to get a sense of where the topic is currently. (A Holographic Constraint on Scale Separation)[https://arxiv.org/abs/2512.11031v1] sounds like it might not be too terribly difficult to get. It basically presents a condition for compatibility between effective field theories and a holographic description. I've heard general relativity, so hopefully that will give me half of what this paper consists of.

Specifically, the paper centers the AdS/CFT correspondence which is used in strongly coupled QFT and also shows up in quantum gravity and M-theory (Section 1, par. 1). It sets out with the scalar field φ_i of the holographic theory in AdS, which is dual to operators with dimensions Δ_i, which decomposes into Δ_i = Δ_j + Δ_k in general. Assuming the φ's have cubic interactions in the AdS bulk, divergences in 3-point Witten diagrams occur. This problem arises in paradigmatic holographic duality between type IIB string theory and 4d N = 4 SYM theory (Section 1, par. 2). The holographic constraint may at first appear of limited utility. Large classes arising from scale separated string theory constructions create a new constraint. The paper shows in simplest (non)supersymmetric AdS₄ DGKT vacuo based on a T⁶/ℤ²₃ IIA orientifold compactification with fluxes, that cubic extremal couplings vanish after non-trivial cancellation (Section 1, par. 4). A 2-derivative gravitational effective theory in d + 1 dims coupled to a finite number of matter fields (spin ≤ 3/2), then write

(Eq. 1). The UV cutoff scale in the EFT is assumed to follow from Newton's constant G, with normalization η = (16πG_N)^-1, which then enables expansion of the Lagrangian around the AdS vacuum to cubic order. The central charge of the CFTs described by the UV-completed EFT is defined through 2pt-functions of the stress tensor, and proportional to the gravitational coupling c (Section 2, par. 1). Holography posits that for all single-particle bulk field φ_i corresponds to a dual operator O^(s)_φi with classical conformal dimension Δ⁽⁰⁾_i. The dimension is determined by the mass. The 2pt-functions scale with 1/c, and the 3pt-functions scale with 1/c². At large c, the dimension expands Δ_i = Δ⁽⁰⁾_i + γ_i/c + ... (Eq. 6), where γ is a real number independent of c (Section 2, Bulk). The scalar operator spectrum consists of single-trace operators and multi-trace operators. They are distinguished by their correlations with c (Section 2, Field Theory). Holography relates the two kinds of operators by a generic situation. If the theory allows extremal arrangements of three different bulk fields with masses so that Δ⁽⁰⁾_i + Δ⁽⁰⁾_j = Δ⁽⁰⁾_k, an ambiguity is included in the identification of the operator O^(s)_φk due to degeneracy of the CFT spectrum. The single-trace (k) operator and the double-trace (i, j) operator have classical dimension Δ⁽⁰⁾_k enables mixing, the details of which depends on the specific CFT. The SPO is orthogonal to all multi-trace operators in the context of N = 4 SYM theory. The holographic 3pt functions of the SPOs includes a pole, which cancels one order of 1/c in the conformal dimensions. The definition of the SPO can also be found for the following scaling of the same correlator with c. By non-vanishing bulk extremal cubic coupling implies that the SPO for k breaks down due to β < 0. The gravitational EFT described by the bulk effective Lagrangian has a dual CFT, failing to obey large-N factorization. As such either the holographic dual for the EFT in AdS needs to be redefined, or all cubic couplings vanish. Then, the extremal 3pt-functions must also vanish (Section 2). In some type IIA string theory, the extremal 3pt-functions are explicitly non-zero. The arising spectrum of conformal dimensions is not specific to the choice of internal CY orientifold, arising universally for other DGKT vacua constructed from CY orientifolds. In these, the 3pt-function coefficients are zero. This constitutes the additional constraint, realized through non-trivial cancelling in the general case (Section 3).