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Holographic complexity of charged Taub-NUT-AdS black holes

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Pith's one-line read For charged Taub-NUT-AdS black holes, the late-time complexity-equals-action growth rate receives a contribution from the Misner-string singularities, and the electric charge that enters is the total charge of the black hole.

desk verdict A careful CA calculation for charged Taub-NUT-AdS whose headline result depends on an unproven tube regularization of the Misner strings. read the letter →

arxiv 1908.06565 v1 pith:LD3TLKBD submitted 2019-08-19 hep-th

Jie Jiang , Banglin Deng , Xiao-Wei Li This is my paper · ORCID
classification hep-th
keywords holographiccomplexityequalsactionTaub-NUT-AdSblackholesMisnerstringselectromagneticdualityWheeler-DeWittpatchgrowthrateLloydbound
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

This paper asks whether the 'complexity equals action' formula for holographic complexity can detect the unusual topology of Taub-NUT-AdS black holes, which contain line singularities called Misner strings in addition to the usual two Reissner-Nordstrom-type horizons. It argues that it can: at late times the complexity growth rate acquires an explicit term built from the Misner potential and the Misner charge, alongside the usual horizon terms, and the electric charge appearing in the rate is the total charge rather than a horizon-local charge. The paper also shows that the original action conjecture breaks electromagnetic duality for these dyonic solutions, and that adding a Maxwell boundary term with coefficient $\gamma=1/2$ restores the duality while leaving the Misner-string term unchanged. If these calculations are right, holographic complexity is sensitive to spacetime topology in a way that horizon-only data cannot capture.

What carries the argument

The load-bearing device is the treatment of the Misner string singularities as boundaries of the Wheeler-DeWitt patch. Because the strings lie on the polar axes at $\theta=0$ and $\theta=\pi$, the authors surround each string by an infinitesimally thin tube of radius $\varepsilon$ and compute the action variation on null segments whose boundary is the union of the usual meeting-point spheres, the asymptotic sphere at infinity, and the two Misner tubes, taking $\varepsilon\to 0$ at the end. The tube integrals assemble into the compact identity $K^{(-)}(r_-)-K^{(+)}(r_+)=\psi(N^{(-)}-N^{(+)})$, which turns the complicated string contribution into the product of Misner potential and Misner charge. The calculation also uses a Noether-charge/Stokes decomposition of the bulk action, rewriting the gravitational and Maxwell bulk contributions as boundary integrals, and the Maxwell boundary term $I_{\mu Q}$ with free coefficient $\gamma$, which is fixed to $\gamma=1/2$ by the requirement of electromagnetic duality.

What would settle it

Compute the WDW action for the same charged Taub-NUT-AdS spacetime with a different regularization of the Misner strings—for instance, replacing each string by a conical defect or a codimension-2 brane with its own action—and compare the coefficient of $\psi(N^{(-)}-N^{(+)})$ in the late-time rate; if it changes sign or disappears, the tube-boundary prescription is not scheme-independent. Alternatively, compute the complexity-equals-volume rate for this spacetime: a genuinely topological Misner-string contribution should appear there too, while an action-boundary artifact would not.

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Extended reading notes

Core claim

On the paper's own terms, the central result is the late-time limit of the CA complexity growth rate for a charged Taub-NUT-AdS black hole with two RN-type horizons, $\lim_{t\to\infty} \frac{dC_A}{dt} = \frac{1}{\pi\hbar}[(\varphi_e^{(-)} - \varphi_e^{(+)}) Q_e + \psi(N^{(-)} - N^{(+)})]$, where $\varphi_e^{(\pm)}$ are the electric potentials at the outer and inner horizons, $Q_e$ is the total electric charge, and $\psi$ and $N^{(\pm)}$ are the Misner potential and Misner charge associated with the string singularities. Unlike ordinary black holes, the rate is not fixed by horizon quantities alone; the Misner-string data contribute directly. A corollary the authors stress is that the electric charge entering the formula is the total charge, not a charge evaluated on a horizon. A further corollary is that the original CA rate is independent of the magnetic charge and violates electromagnetic duality; adding the Maxwell boundary term $I_{\mu Q}=\frac{\gamma}{4\pi}\int_{\partial M} G\wedge A$ changes only the proportion between electric and magnetic terms and, at $\gamma=1/2$, restores the duality and makes the rate sensitive to the magnetic charge, while the Misner term $\psi(N^{(-)}-N^{(+)})$ is unchanged.

Load-bearing premise

The calculation assumes that a Misner string singularity can be regularized by an infinitesimally thin tube whose boundary belongs to the WDW action with a chosen orientation; if that tube prescription is not the right way to handle the singular spacetime, the central formula would change.

Editorial extensions

If this is right

  • The late-time CA complexity growth rate for charged Taub-NUT-AdS black holes is determined by outer/inner horizon data together with Misner-string data through the term $\psi(N^{(-)}-N^{(+)})$.
  • The electric charge entering the late-time rate is the total charge of the solution, so horizon-local charge measurements alone do not determine the complexity growth.
  • The original CA conjecture violates electromagnetic duality for these dyonic solutions; the duality-sensitive version requires the Maxwell boundary term, and $\gamma=1/2$ is the special value that restores the duality.
  • With $\gamma=1/2$, the late-time rate is bounded by half the Lloyd bound, and the full-time rate can be kept below the bound by choosing the counterterm scale $\ell_{ct}$ appropriately.
  • The time dependence of the rate follows the known RN-AdS pattern, including approach from above and Lloyd-bound violation for small charges, so the qualitative behavior familiar from ordinary charged black holes survives in the Taub-NUT-AdS family.

Reading between the lines

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

  • If the tube prescription is the correct way to regularize the WDW action on a singular spacetime, the same Misner-string term should show up in other holographic complexity proposals; for example, a complexity-equals-volume computation for this spacetime would provide a cross-check, and its absence there would suggest the tube term is an artifact of the action boundary treatment.
  • The invariance of the Misner term under changes of $\gamma$ hints that this contribution is purely gravitational or topological, independent of the electromagnetic field; one could test this by taking the neutral limit and checking whether $\psi(N^{(-)}-N^{(+)})$ survives.
  • The appearance of the total electric charge rather than horizon charges may mean the boundary dual sees the Misner strings as global charge sources; the paper does not develop this dual interpretation, but it is a natural next step.
  • The special value $\gamma=1/2$ could be probed in other dyonic or NUT-charged solutions to see whether it is a universal requirement for duality-invariant complexity rather than a coincidence of this family.
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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 / 4 minor

Summary. The paper studies the 'complexity equals action' (CA) conjecture for charged Taub-NUT-AdS black holes in four-dimensional Einstein-Maxwell theory. Using the Wheeler-DeWitt patch method, the authors compute the time-dependent complexity growth rate and its late-time limit. Their central result, Eq. (67), states that the late-time growth rate contains, in addition to the usual horizon terms, a contribution ψ(N(−)−N(+)) proportional to the Misner potential and Misner charge, and that the electric term involves the total asymptotic charge Qe rather than horizon charges. They then add a Maxwell boundary term (68) and claim that for γ=1/2 the late-time rate satisfies electromagnetic duality, becoming sensitive to magnetic charges, and that this modification leaves the Misner term unchanged. The paper also presents numerical plots of the time dependence and discusses the Lloyd bound.

Significance. If the central formula is reliable, the paper is a meaningful step: it shows that non-trivial spacetime topology (Misner strings) can enter holographic complexity, going beyond the usual statement that late-time CA growth is fixed by outer and inner horizon data. The n→0 limit reproduces the RN-AdS result, which is a useful consistency check. The γ=1/2 Maxwell boundary-term discussion connects to the recent 'which action?' literature and is of topical interest. The paper is clearly written and the main algebra is substantially explicit. These strengths are, however, conditional on the tube regularization used to define the action in the presence of Misner string singularities.

major comments (3)
  1. [§III, Eqs. (38), (43)–(45), and (50)] The Misner-string contribution to the central late-time formula (67) rests on a regularization prescription that is not justified. Replacing the string singularities by infinitesimal tubes at θ=ε and θ=π−ε and using ∂N = TN + S∞ − TS − C treats the tubes as boundaries, but the paper does not include the corresponding GHY surface terms or corner terms at the tube endpoints, and the tube integrals (43) and (45) are quoted without derivation. Since the tubes are timelike, the affine-null-boundary argument used in (25) does not apply to them. The identity (50) is then only asserted. This is load-bearing: if a different tube shape or an alternative regulator (for example, a Noether-charge computation) changes the K(±) terms, the claimed dependence of the late-time rate on ψ(N(−)−N(+)) is an artifact. The n→0 limit in Sec. III only tests the vanishing of the tube terms, not their finite value for n≠0.
  2. [§IV, around Eq. (77)] The claim that γ=1/2 restores electromagnetic duality is not demonstrated. The sentence 'It is not difficult to verify' is insufficient. One needs an explicit transformation of Eq. (77) under the stated duality e↔−2ng, 2ng↔e, showing how the electric term (1/2)(φ_e^− − φ_e^+)Qe and the magnetic term (1/2)(φ_m^− Q_m^− − φ_m^+ Q_m^+) map into each other, including the r-dependent charges Q_m^± and the Misner term. Without this check, the duality-restoration result in the abstract and conclusion is unsupported.
  3. [§III, Eqs. (46)–(50)] The derivation of the key K(±) functions in Eq. (47) and of the identity (50) is not shown. In particular, the step from the tube integrals (43) and (45) to the compact form (47), and then the comparison with the Misner charge N(±) in Eq. (20) leading to (50), are essential for the final result. These steps should be presented, or at least collected in an appendix, so that a reader can check that no factor of 2 or sign error enters the Misner term.
minor comments (4)
  1. [§IV, Eq. (77)] The displayed formula for the late-time rate is missing a '+' between the electric and magnetic terms; as printed, the expression '(1−γ)(φ_e^− − φ_e^+)Qe γ(φ_m^−Q_m^− − φ_m^+Q_m^+)' is ambiguous and should read '(1−γ)(φ_e^− − φ_e^+)Qe + γ(φ_m^−Q_m^− − φ_m^+Q_m^+)'.
  2. [§III, after Eq. (60)] The word 'ralation' should be 'relation'.
  3. [§IV and §V] The name 'Lloyd' is misspelled as 'Llyod' in the sentence before Eq. (79) and in the concluding paragraph.
  4. [§II, Eq. (20)] The notation N(±) for the Misner charge is introduced in Eq. (20) without explaining its physical normalization; a brief comment connecting N(±) to the string contribution to the Komar integral would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Misner term is obtained from an explicit action calculation, not fitted or defined into existence.

full rationale

The paper performs a direct computation of the Wheeler-DeWitt action for the charged Taub-NUT-AdS solution under the CA conjecture. The parameters appearing in the final late-time rate (67), such as e, g, n, m, and the horizon radii, are solution parameters fixed by the metric; none are fitted to the complexity result. The Misner potential ψ and Misner charge N are taken from prior thermodynamic literature, but they enter only after the tube integrals are computed, through the algebraic identity (50), which the paper verifies by comparing its independently derived expression (47) with the definitions (20). This is a check of a derived equality, not an input used to construct the result. The choice γ=1/2 for the Maxwell boundary term is a free parameter of that added term, selected to restore electromagnetic duality, and the paper explicitly shows how the final formula depends on γ; this is a parameter choice, not a fitted prediction. Self-citations, such as Ref. [6] for the general action-growth formalism and Ref. [13] for the null-boundary action, provide standard ingredients, but the horizon and Misner-string contributions are calculated in this paper rather than imported as the conclusion. The tube regularization around the Misner strings is a prescription and a possible correctness risk, but it is not circular: no target result is used to define the regulator, the tube orientation, or the value of K(±). A regulator-dependent prescription can be wrong without being circular. Accordingly, no load-bearing step reduces to its own inputs or to a self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the CA conjecture and on the thermodynamic definition of Misner charges from prior literature. No new physical entities are introduced; the only hand-picked number is gamma=1/2 in the modified action.

free parameters (1)
  • gamma = 1/2
    Coupling constant of the additional Maxwell boundary term. Chosen by hand to restore electromagnetic duality in the late-time complexity growth rate; not derived from a variational principle or data.
assumptions (5)
  • domain assumption The CA conjecture: boundary complexity equals WDW patch action divided by pi hbar.
    Assumed in Eq. (1) and throughout; this is the foundational conjecture under test.
  • domain assumption The WDW patch action is well-defined in a spacetime with Misner string singularities using the tube regularization described in Sec. III.
    The paper cuts out the string singularities with thin tubes of radius epsilon and takes epsilon to zero; the validity of this procedure for a singular action is assumed, not proven.
  • domain assumption The first law of thermodynamics (21) including Misner potential psi and Misner charge N(+/-) from Ref. [69] is the correct thermodynamic description.
    The final complexity rate is expressed in terms of these thermodynamic charges; the paper relies on the extended thermodynamics literature for their definitions.
  • standard math The line element (9) is the correct charged Taub-NUT-AdS solution of Einstein-Maxwell theory.
    The solution is taken from Refs. [70,71]; the paper does not re-derive it.
  • domain assumption The mass M=m from the conformal method [72].
    Used when normalizing the complexity growth rate relative to the Lloyd bound in Eq. (79).

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Pith. "Pith review of Holographic complexity of charged Taub-NUT-AdS black holes." pith.science (2026). https://pith.science/paper/LD3TLKBD

@misc{pith2026190806565,
  author       = {Pith},
  title        = {Pith review of: Holographic complexity of charged Taub-NUT-AdS black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LD3TLKBD}},
  note         = {Machine review of arXiv:1908.06565}
}
abstract

In this paper, we investigate the holographic complexity in the charged Taub-NUT-AdS black holes with Misner strings present in the Einstein-Maxwell gravity. We show that differing from the normal black holes, where the late-time complexity growth rate is only determined by the quantities at outer and inner ``Reissner-Nordstrom''-type (RN-type) horizons, here the quantities (the Misner potential and Misner charge) related to the Misner strings also play an important role in CA complexity. Similar to the case of the normal electromagnetic black hole, the late-time rate for the original CA conjecture is independent on the magnetic charges. However, disparate with common results of the dyonic solutions, the electric charge appeared here is the total charge of this black hole. Besides, we found that the result in this original CA conjecture also violates the electromagnetic duality. And this duality can be restored by adding the Maxwell boundary term with the proportional constant $\g=1/2$. In this case, the late-time rate is sensitive to the magnetic charge. Moreover, we also found that the additional term only changes the proportion between the electric and magnetic charges, and it does not affect the Misner term appeared in the late-time rate. Finally, we studied the time-dependence of the complexity growth rate and found that they share similar behaviors with that in RN-AdS black holes.

Figures

Figures reproduced from arXiv: 1908.06565 by the authors.

Figure 1
Figure 1. FIG. 1. Wheeler-DeWitt patch of a charged Taub-NUT-AdS black holes with two RN-type horizons, where the dashed lines denote the cut-off [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The null segment [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The time dependence of the complexity growth rate in the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The time dependence of the complexity growth rate in the context of CA conjecture with Maxwell boundary term. In the left panel, we [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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