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arxiv: 2105.11639 · v2 · pith:5PT46PZPnew · submitted 2021-05-25 · ✦ hep-th

Interpolating between multi-center microstate geometries

Masaki Shigemori This is my paper

Pith reviewed 2026-05-24 13:32 UTC · model grok-4.3

classification ✦ hep-th
keywords microstate geometriesLunin-Mathur geometriesmulti-center solutionsinterpolating solutionshelical profilescharge delocalizationspectral flow3-charge black holes
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The pith

A helical-profile Lunin-Mathur geometry interpolates between two multi-center microstate geometries as a 2-center solution with a codimension-2 source.

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

The paper constructs an interpolating solution between two multi-center microstate geometries that represent Lunin-Mathur geometries with circular profiles. This solution is itself a Lunin-Mathur geometry but with a helical profile, expressed as a two-center solution that includes a codimension-2 source. As the interpolation parameter varies, some charges delocalize and transfer from the codimension-2 center to the codimension-3 center. The family of solutions undergoes spectral flow and is speculated to be relevant for describing general microstates of three-charge black holes.

Core claim

The interpolating solution between two multi-center microstate geometries in 4d/5d that represent Lunin-Mathur geometries with circular profiles is a Lunin-Mathur geometry with a helical profile, represented by a 2-center solution with a codimension-2 source. The interpolating 2-center solution exhibits charge delocalization and transfer from the codimension-2 center to the codimension-3 center as the interpolation proceeds, along with spectral flow of the entire process.

What carries the argument

The 2-center solution with a codimension-2 source that realizes the helical-profile Lunin-Mathur geometry.

If this is right

  • Charges delocalize in the interpolating geometry.
  • Charges transfer from the codimension-2 center to the codimension-3 center during interpolation.
  • The interpolation process is accompanied by spectral flow.
  • The solutions may help describe general microstates of 3-charge black holes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The charge transfer indicates that distinctions between centers can blur continuously along families of solutions.
  • Spectral flow during interpolation may connect microstates related by duality transformations.
  • Such interpolations suggest that the space of microstate geometries contains connected components linking different center configurations.

Load-bearing premise

A helical-profile Lunin-Mathur geometry can be realized as a regular 2-center solution with a codimension-2 source without introducing new singularities or violating the equations of motion outside the centers.

What would settle it

An explicit check showing that the proposed interpolating metric fails to satisfy the supergravity equations of motion or develops singularities at some intermediate value of the interpolation parameter.

Figures

Figures reproduced from arXiv: 2105.11639 by Masaki Shigemori.

Figure 1
Figure 1. Figure 1: The R 3 profile and D4 charges of the “helical” solution, as we change the parameters a, b. (a): the south pole limit (a > 0, b = 0). The profile is a point at the south pole of an ellipsoid (dashed ellipse). The purple dot represent the D4 charge at Σ = 0, while the green dot the r = 0 center. The D4 charge is shown next to each center in units of Q5Ω/2. (b) as we make b nonzero, the profile goes off the … view at source ↗
Figure 2
Figure 2. Figure 2: Toroidal coordinates (on the y2 = 0 plane). where 1 ≤ u < ∞, φ ∼= φ + 2π, σ ∼= σ + 2π. The ring sits at u = ∞, while u = 1, σ = 0 corresponds to r = |y| → ∞. φ is the angle along the ring while σ is the angle going around the ring. See [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Gaussian surfaces. The large gray circle represents S 2 at infinity, which can be deformed into two disconnected surfaces: S 2 0 , enclosing the origin, and S 2 1 , surrounding the disk D2 (red zigzag line) whose boundary is the ring (blue dots). The difference of a quantity X across D2 is defined to be [X] = Xout − Xin, where “out” and “in” are defined as shown here, for the southern case. In the “sou… view at source ↗
Figure 4
Figure 4. Figure 4: Repeated interpolations in the bulk. L0 J 3 L jL N 0 N − 2 N 2 3N − 2 3N 2 5N − 2 5N 2 |++i 1 |++i 1,−1 |++i 1,−2 |++i 1,−3 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Repeated interpolations in CFT to reach [ [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An example of interpolation between [|++i k,n] N/k and [|++i k 0 ,n0] N/k0 with n 6= n 0 . 6 Discussions In this note, we studied the Lunin-Mathur geometry interpolating the states [|++i k ] N/k and [|−+i k 0] N/k0 (6.1) in the framework of harmonic solutions in 4d/5d. Although the geometries dual to (6.1) are codimension-3 solutions, the interpolating solution is of codimension two, because of the puffed-… view at source ↗
read the original abstract

We study interpolation between two multi-center microstate geometries in 4d/5d that represent Lunin-Mathur geometries with circular profiles. The interpolating solution is a Lunin-Mathur geometry with a helical profile, and is represented by a 2-center solution with a codimension-2 source. The interpolating 2-center solution exhibits interesting features such as some of the charges being delocalized, and some of the charges getting transferred from the codimension-2 center to the other, codimension-3 center as the interpolation proceeds. We also discuss the spectral flow of this entire process and speculate on the relevance of such solutions to understanding general microstates of 3-charge black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript constructs an interpolating solution between two multi-center microstate geometries in 4d/5d that represent Lunin-Mathur geometries with circular profiles. The interpolating solution is identified as a Lunin-Mathur geometry with a helical profile, realized explicitly as a 2-center solution containing one codimension-2 source. The solution exhibits charge delocalization and progressive transfer of charges from the codimension-2 center to the codimension-3 center during interpolation; the paper also examines the spectral flow of the process and speculates on its relevance to general 3-charge black hole microstates.

Significance. If the central identification holds, the work supplies an explicit, continuous family connecting distinct multi-center microstate geometries and demonstrates concrete mechanisms of charge delocalization and transfer within a 2-center ansatz. This could aid in mapping the space of smooth horizonless solutions and in understanding spectral flow operations relevant to the microstate program. The explicit 2-center representation with a codimension-2 source is a concrete technical contribution that, if verified, would be useful for further studies of helical Lunin-Mathur profiles.

major comments (1)
  1. [Abstract / construction paragraph] Abstract and the paragraph describing the interpolating solution: the central claim equates the helical Lunin-Mathur geometry to a regular 2-center solution with a codimension-2 source. This identification requires that the metric and fields satisfy the 5d supergravity equations identically outside the two centers. No explicit substitution into the equations of motion, Bianchi-identity verification, or curvature analysis at the codimension-2 locus is supplied, leaving the absence of new singularities or hidden sources unconfirmed. This step is load-bearing for the representation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recognizing the potential utility of the explicit 2-center interpolating solution. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / construction paragraph] Abstract and the paragraph describing the interpolating solution: the central claim equates the helical Lunin-Mathur geometry to a regular 2-center solution with a codimension-2 source. This identification requires that the metric and fields satisfy the 5d supergravity equations identically outside the two centers. No explicit substitution into the equations of motion, Bianchi-identity verification, or curvature analysis at the codimension-2 locus is supplied, leaving the absence of new singularities or hidden sources unconfirmed. This step is load-bearing for the representation.

    Authors: We agree that an explicit verification strengthens the central claim. The interpolating solution is obtained by a continuous deformation of the profile function within the Lunin-Mathur construction; by definition this guarantees that the 5d supergravity equations and Bianchi identities are satisfied away from the sources. Nevertheless, to directly confirm that the 2-center ansatz with the codimension-2 source introduces neither additional singularities nor hidden sources, we will add in the revised manuscript an explicit substitution of the metric and fields into the equations of motion outside the centers, together with a verification of the Bianchi identities and a curvature analysis at the codimension-2 locus. This material will appear as a new subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction presented as explicit without reduction to inputs by definition or self-citation.

full rationale

The abstract and description frame the interpolating solution as an explicit 2-center ansatz with codimension-2 source that realizes a helical Lunin-Mathur geometry, with features like charge delocalization and transfer arising from the interpolation parameter. No quoted step shows a parameter fitted to the output quantity, a self-citation supplying the uniqueness or ansatz, or a renaming of a known result as a derivation. The regularity assumption outside centers is an explicit premise rather than a self-definitional closure, leaving the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of explicit free parameters, axioms, or invented entities; the construction appears to rely on the standard Lunin-Mathur ansatz and multi-center supergravity solutions without introducing new postulated entities beyond the codimension-2 source already named in the abstract.

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