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Non-conformal branes wrapped on a disk

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Pith's one-line read This paper classifies disk solutions from non-conformal D6, D5/NS5, D4, and D2 branes and identifies their common global geometry: a 1/ℓ Euler characteristic, a central monopole, and, in most cases, smeared branes at the boundary, with…

desk verdict Mostly solid taxonomy of non-conformal disk solutions, but the equal-charge D5 sector (§3.2) fails its own flux-quantization condition and should be revised or removed. read the letter →

arxiv 2507.22991 v1 pith:BXJUJU7G submitted 2025-07-30 hep-th

Minwoo Suh This is my paper
classification hep-th
keywords non-conformalbranesdiskorbifoldsspindlesolutionsgaugedsupergravitysmearedmonopolesfluxquantization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper classifies the disk solutions that arise from recently constructed spindle solutions for non-conformal D6, D5/NS5, D4, and D2 branes, organized by which U(1) gauge fields are turned on: no charge, single charge, equal charge, and multi charge. For every sector it writes the gauged-supergravity metric, uplifts it to ten dimensions, and identifies the global geometry of the internal space as a disk with a single $R^2/\mathbb{Z}_\ell$ orbifold point (a mild conical singularity), so the Euler characteristic is $1/\ell$. The common features are a monopole source at the center of the disk and, for the compact solutions that end at $y=0$, smeared brane sources at the boundary—branes whose charge is spread over extra directions. The non-compact solutions that end at $y=\infty$ approach un-smeared branes instead, and the equal-charge D5/NS5 solution has neither a monopole nor a solvable flux-quantization condition. If correct, the paper supplies a complete taxonomy of non-conformal brane-on-disk geometries and shows which global features are universal.

What carries the argument

The load-bearing object is the local spindle solution, truncated to a $U(1)^n$ subsector of the relevant gauged supergravity and written in coordinates $(y,z)$ with a metric function $h(y)$ and gauge fields $A_i = (q_i/(y^2+q_i))\,dz$. A disk is obtained by choosing the radial interval $y \in [0,y_1]$ or $y \in [y_1,\infty)$, imposing the orbifold periodicity $\ell\Delta z/2\pi = 1/E(\cdots)$ so that the $(y,z)$ surface becomes the orbifold $R^2/\mathbb{Z}_\ell$, and computing the Euler characteristic $\chi = (1/4\pi)\int R_\Sigma \mathrm{vol}_\Sigma = 1/\ell$. The ten-dimensional interpretation is produced by uplift formulas: the $D=6$ formula from [41] for D4, the $D=7$ sphere reduction from [49] for D5/NS5, a $D=4$ uplift constructed in appendix C for D2, and the standard $S^2$ reduction for D6. The boundary asymptotics are then matched against the smeared Dp-brane ansatz of appendix A, and the monopole is read off from a jump in the twist function $L(y,\xi)$ at a corner of the base rectangle.

What would settle it

For the equal-charge D5/NS5 disk, solve the pair of conditions (3.31) and (3.33) for integer $\ell$ and real $p$, $\Delta z$; if no solution exists, the background violates Dirac flux quantization and must either be discarded or supplied with a modified quantization rule, and the paper's taxonomy would lose one of its entries.

Watch

Extended reading notes

Core claim

The central discovery is a classification with a common geometric skeleton: each disk solution from a non-conformal Dp-brane spindle has a $(y,z)$ base that is a disk with a single $R^2/\mathbb{Z}_\ell$ orbifold point, which fixes the Euler characteristic at $1/\ell$, and the uplifted metric carries a monopole source at the corner of the base where the circle fibration's twist function jumps. This is verified explicitly for D6, D5/NS5, D4, and D2 branes in the no-charge, single-charge, equal-charge, and multi-charge sectors. The classification also finds two systematic exceptions: non-compact disks with $y \in [y_1, \infty)$ end in un-smeared brane sources rather than smeared ones, and the equal-charge D5/NS5 solution has no monopole structure at all. In that same equal-charge D5/NS5 sector the orbifold periodicity and flux-quantization equations are proportional, so $p$ and $\Delta z$ cannot be solved for in terms of $\ell$ and $p$; the paper nonetheless lists it as a disk solution.

Load-bearing premise

The classification assumes that for every listed charge sector the orbifold periodicity condition and the gauge-field flux quantization can be solved simultaneously; the author explicitly notes that in the equal-charge D5/NS5 case these conditions are proportional and do not fix $p$ and $\Delta z$, so that solution is presented without the standard flux quantization.

Editorial extensions

If this is right

  • For the D6, D4, and D2 families, every charge sector yields a disk background whose local geometry and boundary brane interpretation are now explicit; these can serve as gravity duals of non-conformal field theories on a disk.
  • The universal $1/\ell$ Euler characteristic ties the orbifold order to the global topology and should persist for any further disk solutions built from the same spindle seed.
  • The distinction between compact disks (smeared boundary branes) and non-compact disks (un-smeared boundary branes) is controlled by which root of $h(y)=0$ is chosen, giving a simple rule for where each behaviour occurs.
  • If the equal-charge D5/NS5 disk is accepted without flux quantization, the common features of disk solutions are not universal; if it is rejected, the classification reduces to the sectors where quantization works.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Deriving the missing uplifted flux forms for the $D=6$ and $D=4$ truncations (appendices B and C) and checking their Bianchi identities would independently test the smeared-brane source interpretation, which is currently inferred only from the metric and dilaton asymptotics.
  • The same spindle-seed construction should produce disk×disk and spindle⋉disk products for non-conformal branes, extending the known conformal results; the twist-function jump argument predicts where monopole sources would sit in each factor.
  • The obstruction in the equal-charge D5/NS5 sector may point to a modified quantization condition for NS5-branes; a worldsheet or double-field-theory computation could decide whether that solution has a consistent string-theory completion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper classifies disk solutions obtained by restricting the spindle solutions for non-conformal Dp-branes (D6, D5/NS5, D4, D2) constructed by Boisvert and Ferrero. For each brane type and for each charge sector (no charge, single charge, equal charge, multi charge), the paper gives the explicit gauged-supergravity metric, the orbifold periodicity condition, the Euler characteristic 1/ℓ of the disk, and, when available, the uplifted ten-dimensional metric and dilaton. It also analyzes the global geometry of the uplifted solutions, identifying monopole sources at the disk center and smeared versus un-smeared brane asymptotics at the boundary, with the D5 equal-charge and multi-charge sectors claimed as exceptions. Appendix C constructs a U(1)^3 uplift formula for D=4 ISO(7) gauged supergravity, and Appendix D treats no-charge M5-brane disks.

Significance. If the classification is correct, the paper provides a useful taxonomy of disk solutions for non-conformal branes, with explicit metrics, systematic compact/non-compact classification, and a clear criterion (smeared versus un-smeared boundary behavior) that organizes many examples. The Euler-characteristic computations are explicit and internally consistent, and the paper correctly identifies that several sectors require no fitting because they are obtained by direct restriction of known spindle solutions. The proposed D=4 uplift formula in Appendix C is a concrete new ingredient, though it is not fully checked. However, the completeness of the taxonomy is undermined by unresolved flux-quantization issues in at least one sector explicitly acknowledged by the author, and by missing flux-quantization data in other sectors.

major comments (3)
  1. [§3.2, Eqs. (3.31) and (3.33)] The equal-charge D5 sector is not a flux-quantized disk as presented. Eq. (3.31) fixes ℓΔz/(2π) = 1/[2gp(1±p)], and eq. (3.33) defines the holonomy as p = -gΔz/(2π) p(1±p). Substituting the first into the second gives p = -1/(2ℓ), so the holonomy is fixed by ℓ alone while the gauge-field parameter p and the period Δz remain undetermined. The author's own text after (3.33) acknowledges that one cannot solve for p and Δz in terms of ℓ and p. Since the abstract claims a complete classification and this sector is one of the exceptions to the 'monopoles and smeared branes' rule, the taxonomy is over-inclusive unless this sector is either removed, relabelled as an unquantized family, or supplemented with an additional physical condition that fixes the remaining parameter.
  2. [§5.4 and §4.4] The multi-charge D2 sector in §5.4 presents no flux-quantization equation at all: after the periodicity condition (5.62) and Euler characteristic (5.63), the discussion moves directly to the uplift. For a claimed classification, each gauge field should be quantized; here there are two nonzero gauge fields A1 and A2, so two holonomy conditions are needed. Similarly, in the multi-charge D4 sector of §4.4, eq. (4.66) defines a holonomy p using a symbol q that is not defined in that section, and only one equation is given for two gauge fields A1 and A2; the cross-reference to '(4.44)' is also wrong (it should refer to (4.64)). These sectors are therefore incomplete as classified, and the paper should either provide the missing quantization conditions or explicitly state that they are not imposed.
  3. [Apps. B, C and §§4.1.2, 4.2.2, 5.2.2] For the D4 and D2 uplifts, only the metric and dilaton are presented; the flux fields are not obtained, as the author acknowledges ('Explicit form of the uplift formula for flux fields is not given in [41]' and 'We have not obtained the uplift formula for the three-form potential'). Since the paper's title and abstract present these as brane-wrapped solutions, the ten-dimensional interpretation is incomplete: without the RR or NSNS fluxes one cannot verify the brane charge content or the precise source terms. The smeared/asymptotic analysis based on the metric and dilaton is suggestive but should be labelled as provisional until the fluxes are computed.
minor comments (6)
  1. [Eq. (3.24)] In the equal-charge D5 solution, the scalar expressions read e^{λ1+gρ/5} = e^{λ1+gρ/5} on both sides; the second should involve λ2.
  2. [After Eq. (3.33)] The cross-reference '(3.48)' in the sentence 'Δz p(1±p) is common in both of (3.48) and (3.33)' should be '(3.31)'.
  3. [§3.2, §3.3, §4.1, §5.1] For the non-compact solutions with y ∈ [ya, ∞), the text repeatedly states that the disk has 'the boundary at y=0'; the boundary is at y=∞, which is the side where the metric is analyzed in the corresponding uplifted sections.
  4. [§5.4, text above Eq. (5.64)] The sentence 'For the solution with < y2 < y <∞' is missing the variable y; it should read 'For the solution with y2 < y <∞'.
  5. [§4.4 and §4.3] The flux-quantization equations in these sections contain typographical inconsistencies: eq. (4.66) uses an undefined q, and the sentence after (4.44) refers to solving '(4.64) and (4.44)' where (4.44) is the equation just displayed; the intended references need to be corrected.
  6. [Throughout] There are several duplicated words, e.g., 'the the spindle solution' in §§2.2.1, 3.1.1 and D.1, and 'a single real root, y1, and two complex roots' with 'We find solutions' where the text is clear but would benefit from editing.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the disk solutions are restrictions of the spindle solutions of [14], and the claimed global features are computed from the explicit metrics.

full rationale

The paper's derivation chain is not circular in the sense of the analysis. Each disk solution is obtained by fixing parameters in the spindle solutions previously constructed by Boisvert and Ferrero, e.g., §2.1 sets A=0, q=0 in the spindle solution (3.15) of [14], §3.1 sets q1=1, q2=0 for the solution (4.26) of [14], and the other sectors proceed analogously. The global features are then read off from the explicit metrics: the Euler characteristic χ=1/ℓ is computed from the curvature integral (e.g., eqs. 2.8, 3.10, 4.9), the monopole structure is inferred from the jump of a connection function L on the base space (e.g., eqs. 3.21, 4.29, 5.28), and the smeared or un-smeared brane asymptotics are matched to the independent smeared-brane ansatz in appendix A. None of these conclusions is identical by construction to the input spindle solutions; they are derived properties. Flux quantization is imposed as an additional condition and solved in most sectors (e.g., eqs. 3.11–3.12, 4.22–4.23). In the equal-charge D5 sector, the paper explicitly flags that eqs. (3.31) and (3.33) are proportional, so p and Δz cannot both be determined; this is a correctness gap or limitation in that sector, not a circular step, because the text admits the failure rather than silently using the condition to force the claimed result. The self-citations to the author's earlier disk papers concern known conformal-brane disk solutions and serve as context for the common-features discussion; they are not load-bearing for the non-conformal disk constructions presented here. The paper also explicitly states missing uplift fluxes (appendices B and C) and the absence of a three-form uplift formula, which are acknowledged limitations rather than disguised inputs. Overall, no equation or fitted parameter is renamed as a prediction, and no load-bearing claim reduces to a self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper adds no new particles or forces, and no constants are fitted to data. Its central results rest on inherited truncation and uplift technology, on the implicit assumption that the spindle-to-disk restriction preserves the equations of motion, and on one newly proposed D=4 uplift formula whose consistency is only partially checked.

assumptions (4)
  • domain assumption Validity of the consistent truncations and uplift formulas of D=8, D=7, D=6, D=4 gauged supergravities used to embed the solutions in 10D and 11D.
    The paper relies on uplift formulas from [14,41,49] and appendices B and C without rederiving their consistency; if any uplift is not a consistent truncation, the ten-dimensional metrics are not solutions.
  • domain assumption The local spindle solutions of [14] remain solutions when restricted to disk ranges and charge limits.
    All disk solutions are obtained by setting some q_i to zero or equal and choosing radial intervals y in [0,y1] or [y1,infinity); the paper does not independently verify the equations of motion for each restricted case.
  • ad hoc to paper The proposed D=4 ISO(7) U(1)^3 uplift formula in appendix C is consistent.
    The formula is constructed in this paper and only cross-checked against a known solution when axions are turned off; it is not proven to be a fully consistent truncation and the flux uplift is not given.
  • domain assumption Boundary singularities of the uplifted metrics are correctly identified as smeared or un-smeared Dp-brane sources via the asymptotic matching in appendix A.
    The smeared or un-smeared conclusions follow by comparing the y to 0 or y to infinity limits of the metrics to eq. (A.1); this matching assumes the subleading terms do not change the source interpretation.

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Pith. "Pith review of Non-conformal branes wrapped on a disk." pith.science (2026). https://pith.science/paper/BXJUJU7G

@misc{pith2026250722991,
  author       = {Pith},
  title        = {Pith review of: Non-conformal branes wrapped on a disk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BXJUJU7G}},
  note         = {Machine review of arXiv:2507.22991}
}
abstract

We classify the disk solutions which are obtained from the spindle solutions from non-conformal D$p$-branes recently constructed by Boisvert and Ferrero. Then we study the global geometry of the disk solutions. We discover several common features of disk solutions, $e.g.$, monopoles and smeared branes, in most cases, but there are also exceptions.

Figures

Figures reproduced from arXiv: 2507.22991 by the authors.

Figure 1
Figure 1. A representative solution with g = 2. The solution is regular in the range of 0 < y < y1 = 1. Approaching y → y1, the metric becomes to be ds2 8 ≈ r 3 y 1/2 1 " ds2 1,4 + dr2 r 2 + dρ2 + g 2 4 ρ 2dz2 −y1h ′ (y1) # , (2.6) where we introduced a new parametrization of coordinate, ρ 2 = y1 − y. Then, the ρ − z surface is locally an R 2/Zℓ orbifold if we set ℓ∆z 2π = 2 g , (2.7) where ∆z is the period of coordinate, z, … view at source ↗
Figure 2
Figure 2. A representative solution with g = 1 and q = 2. The solution is regular in the range of 0 < y < y1 = 0.886. Near y → 0 the warp factor vanishes and it is a curvature singularity of the metric, ds2 8 ≈ r 3 q 1/6 y 1/6  ds2 1,4 + dr2 r 2 + y g 2q dy2 + g 2 4 dz2  . (2.16) Approaching y → y1, the metric becomes to be ds2 8 ≈ r 3 y 1/2 1  ds2 1,4 + dr2 r 2 + dρ2 + E(q) 2ρ 2dz2 −y −1 1 (y 2 1 + q) h ′ (y1)  , (2.17) … view at source ↗
Figure 3
Figure 3. A representative solution with g = 1 and p = 0.5. The solution is regular in the range of 0 < y < y1 = 1/3. We plot a representative solution with g = 1 and p = 0.5 in figure 3. The metric functions, f(y), g1(y), and g2(y), are defined by ds2 7 = e 4g 5 ρ f(y) [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: The two-dimensional base space, B2, spanned by y and ξ. where ds2 D5, ΦD5, and F(3) are the metric, dilaton, and RR three-form flux of D5-brane solutions, respectively. We present the metric in the string frame and the dilaton of D5-brane solutions, respectively, ds2 D…
Figure 5
Figure 5. Figure 5: A representative solution with g = 1 and p = 1.5. The solution is regular in the range of y1 = −0.6 < y < ∞. where ∆z is the period of coordinate, z, and ℓ = 1, 2, 3, . . .. The metric spanned by (y, z) has a topology of disk, Σ, with the center at y = ya and the bound…
Figure 6
Figure 6. Figure 6: A representative solution with g = 1, p = 1.5, and q = 1.4. The solution is regular in the range of y1 = −0.553 < y < ∞. where we have E(p, q) 2 ≡ g 2 (ya + q) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: A representative solution with g = 1 and q = 1. The solution is regular in the range of y1 = 4 < y < ∞. Approaching y → y1, the metric becomes to be ds2 6 ≈ y 5/4 1 r 1/2  ds2 1,2 + dr2 r 2 + dρ2 + E(q) 2ρ 2dz2 −y 3 1h(y1)  , (4.6) where we have E(q) 2 ≡ g 2 2 , (4…
Figure 8
Figure 8. Figure 8: A representative solution with g = 1 and q = 1. The solution is regular in the range of 0 < y < y1 = 0.2679. We plot a representative solution with g = 1 and q = 1 in figure 8. The metric functions, f(y), g1(y), and g2(y), are defined by ds2 6 = f(y) r 1/2  ds2 1,2 + …
Figure 9
Figure 9. Figure 9: The two-dimensional base space, B2, spanned by y and ξ. Region II: Monopole We break Dχ1 and complete the square of dz to obtain the metric of ds2 10 = ∆e1/2 r  ds2 1,2 + dr2 r 2 + 1 4y(y 2 + q)h(y) dy2 + 1 g 2y dξ2 + 1 g 2∆e y 2 cos2 ξ [PITH_FULL_IMAGE:figures/full_…
Figure 10
Figure 10. Figure 10: A representative solution with g = 1 and q = 1. The solution is regular in the range of y2 = 3.732 < y < ∞. 0 monopole P2 P1 P3 P4 ⇠ ⇡ 2 shrinks 1 Un-smeared D4-branes L =0 y L ⇠ ` S2 y2 [PITH_FULL_IMAGE:figures/full_fig_p029_10.png]
Figure 11
Figure 11. Figure 11: The two-dimensional base space, B2, spanned by y and ξ. and the dilaton is e Φ ≈ y 3/4 r 3/2 . (4.34) The metric does not get smeared by D4-brane sources. The D4-branes are • extended along the R 1,2 , r, and z directions; • localized at the origin of y, ξ, θ, χ1, and…
Figure 12
Figure 12. Figure 12: A representative solution with g = 1 and q = 0.1. The solution is regular in the range of 0 < y < y1 = 0.157. we introduced a new parametrization of coordinate, ρ 2 = y1 − y. Then, the ρ − z surface is locally an R 2/Zℓ orbifold if we set ℓ∆z 2π = 1 E(q) , (4.42) wher…
Figure 13
Figure 13. Figure 13: The two-dimensional base space, B2, spanned by y and ξ. where we have ∆ = e (y 2 + q) [PITH_FULL_IMAGE:figures/full_fig_p032_13.png]
Figure 14
Figure 14. Figure 14: A representative solution with g = 1 and q = 0.1. The solution is regular in the range of y2 = 3.949 < y < ∞. and the dilaton is e Φ ≈ (q sin ξ) 1/2 y 1/4r 3/2 . (4.53) The metric implies the smeared D4-brane sources. The D4-branes are • extended along the R 1,2 , r, …
Figure 15
Figure 15. Figure 15: The two-dimensional base space, B2, spanned by y and ξ. metric. As y → ∞, the uplifted metric becomes ds2 10 ≈ 1 r  y 3/2  ds2 1,2 + dr2 r 2 + g 2 4 dz2  + 1 g 2y 3/2 h dy2 + y 2  dξ2 + cos2 ξ [PITH_FULL_IMAGE:figures/full_fig_p035_15.png]
Figure 16
Figure 16. Figure 16: A representative solution with g = 1, q1 = 0.1, and q2 = 0.2. The solution is regular in the range of 0 < y < y1 = 0.204. where we have E(q1, q2) ≡ y −1 1 [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]
Figure 17
Figure 17. Figure 17: A representative solution with g = 1 and q = 0.1. The solution is regular in the range of y2 = 3.949 < y < ∞. and the dilaton is e Φ ≈ (q1q2) 1/4 sin1/2 ξ r 3/2y 1/4 . (4.71) The metric implies the smeared D4-brane sources. The D4-branes are • extended along the R 1,2…
Figure 18
Figure 18. Figure 18: A representative solution with g = 1. The solution is regular in the range of y1 = 2 4/3/3 2/3 < y < ∞. where we have E(q) 2 ≡ 1 9 y 5 1h ′ (y1) 2 =  3g 4 2 , (5.7) and we introduced a new parametrization of coordinate, ρ 2 = y1 − y. Then, the ρ − z surface is local…
Figure 19
Figure 19. Figure 19: A representative solution with g = 1, q = 0.5. The solution is regular in the range of y1 = 1.0749 < y < ∞. Near y → ∞ the warp factor diverges and it is a curvature singularity of the metric, ds2 4 ≈ y 7/2 r 1/3  −dt2 + dr2 r 2 + 4 g 2y 5 dy2 + 9g 2 16 dz2  . (5.17…
Figure 20
Figure 20. Figure 20: The two-dimensional base space, B2, spanned by y and ξ. The function, L(y, ξ), is piecewise constant along the sides of y = y1 and ξ = π/2 of the 2d base, B2, L (y, π/2) = 0 , L (y1, ξ) = 4 E(q) = 4 ℓ∆z 2π , (5.28) The jump in L at the corner, (y, ξ) = (y1, π/2), indi…
Figure 21
Figure 21. Figure 21: A representative solution with g = 1, q = 0.5. The solution is regular in the range of 0 < y < y1 = 0.154. and we introduced a new parametrization of coordinate, ρ 2 = y1 − y. Then, the ρ − z surface is locally an R 2/Zℓ orbifold if we set ℓ∆z 2π = 1 E(q) , (5.38) whe…
Figure 22
Figure 22. Figure 22: The two-dimensional base space, B2, spanned by y and ξ. where we have ∆ = e y [PITH_FULL_IMAGE:figures/full_fig_p048_22.png]
Figure 23
Figure 23. Figure 23: A representative solution with g = 1 and q = 0.5. The solution is regular in the range of y2 = 0.856 < y < ∞. and the dilaton is e Φ ≈ r 5/6 [PITH_FULL_IMAGE:figures/full_fig_p050_23.png]
Figure 24
Figure 24. Figure 24: The two-dimensional base space, B2, spanned by y and ξ. metric. As y → ∞, the uplifted metric becomes ds2 10 ≈ r 1/3  y 5/2  −dt2 + dr2 r 2 + 9g 2 16 dz2  + 1 g 2y 5/2 h 4dy2 + y 2  dξ2 + cos2 ξ [PITH_FULL_IMAGE:figures/full_fig_p051_24.png]
Figure 25
Figure 25. Figure 25: Representative solutions with g = 4/3, q1 = 3/10, q2 = 7/10. A solution is regular in the range of 0 < y < y1 = 0.334 and another in y2 = 0.434 < y < ∞. Approaching y → ya, a = 1, 2, the metric becomes to be ds2 4 ≈ y 7/2 a r 1/3  −dt2 + dr2 r 2 + dρ2 + E(q1, q2) 2ρ …

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