Holographic Derivation of BPZ-Type Null State Equations in Higher Dimensional CFTs

Referee: Integration-constant argument after stripping ∂̄^{n+1}: for Δ=−2,−3,−4 the two universal inputs (identity + single-stress block) cannot, by counting alone, fix all undetermined functions of z. Please make explicit which homogeneous mode space is allowed and why no nonzero element survives.

Authors: The referee is right that the manuscript glosses over this step and we will expand it. The point is that the undetermined functions are not arbitrary functions of z to be matched against g_0 and g_1 only: they are constrained order-by-order in z̄ by the bulk near-boundary expansion (13) itself, before any reference to the stripped equation. Concretely, after stripping ∂̄^{n+1}, the residual object is a polynomial of degree n in z̄ whose coefficients are functions of z. The bulk expansion (13)–(19), once written in (z,z̄) and expanded as in (23), populates each power (μz̄)^k with a definite source built from g_0,…,g_{k-1} via the lower-order bulk equations. The homogeneous-mode space allowed by the stripped operator at order (z̄)^k therefore intersects the image of (13) only along directions consistent with the recursion, and the recursion is initialised by g_0 = 1 and the Ward-identity-fixed g_1. We will (i) state the homogeneous mode space explicitly for n=2,3,4, (ii) show that compatibility with the bulk recursion at the first n+1 orders in z̄ — not just identity and single-stress block — forces all residual functions to vanish, and (iii) make clear that this is a structural consequence of (13), not an extra assumption. We thank the referee for prompting this clarification. revision: yes

Referee: Lightcone scaling v = μz̄ fixed: argue more explicitly that no contributions from horizon boundary conditions / double-trace operators or from higher-twist mixing leak into the equations after this scaling.

Authors: We agree this deserves an explicit comment. The cleanness of the separation has two ingredients, both already implicit in the construction. First, the near-boundary expansion (13) is a Frobenius expansion organised in inverse powers of r at the conformal boundary; its coefficients Φ_{2n} are determined recursively by Φ_0 with no reference to the horizon, and the data sensitive to horizon boundary conditions (i.e. the double-trace OPE coefficients, as discussed in [14]) enter only through the second independent solution of the radial equation, which is not part of (13). Hence at every order in 1/r the equations we obtain are blind to horizon data. Second, the scaling v = μz̄ fixed, z̄ → 0 isolates the leading τ = τ_min behaviour of the d=4 conformal block (18): higher-twist blocks contribute at relative order z̄^{(τ−τ_min)/2} and are therefore dropped uniformly, order by order in (μz̄)^k. Because μ enters the bulk equation only through f(r) = 1 − μ/r^4, the rescaling commutes with the 1/r expansion, so the suppression of higher-twist contributions and the suppression of horizon-sensitive modes are preserved at every order. We will add a short paragraph stating exactly this around Eqs. (18)–(19). revision: yes

Referee: Pattern discontinuity at Δ=−4 in Eq. (29): (i) is it an artefact of the prefactor (zz̄)^Δ Φ^{τmin}_{0,Δ=−4} = (1/z^5 z̄^4) Q, i.e., could a different stripping restore the pattern? (ii) state precisely which orders k of (23) and to what order in z were verified against [17, 48, 49].

Authors: Both points are well taken. (i) The prefactor 1/(z^5 z̄^4) is chosen so that Q^{τmin}_{Δ=−4} admits the universal expansion (23) with g_0 = 1 and the same Ward-identity normalisation of g_1 used at Δ = −1, −2, −3; this is the natural choice for comparison with [57]. We have checked that simple alternative strippings (shifts of the powers of z, z̄, or factoring out (1−z), (1−z̄)) do not restore the regular ∂̄ pattern of (24), (26), (28): the extra μ^2 term and the mixed-derivative structure 56μ(1−z)^3 ∂^3 ∂̄ + (1−z)^6 ∂^6 ∂̄^2 are intrinsic to Δ = −4 in the minimal-twist limit and cannot be absorbed into a redefinition of Q. We will state this explicitly. (ii) Regarding the cross-check: using (23) with g_i(0)=0 for i ≠ 0, we solved (29) recursively and matched g_2(z) and g_3(z) against the closed-form resummed double- and triple-stress-tensor contributions of [17, 48, 49]; the match was performed analytically for g_2 and as a series expansion in z up to order z^{12} for g_3 (well beyond the orders required to fix the rational/hypergeometric ansatz). We will state these orders explicitly in the revised text. revision: yes