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Split attractor flows and the spectrum of BPS D-branes on the Quintic

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Frederik Denef, Brian Greene, Mark Raugas

High Energy Physics - Theory · Submitted 20 Jan 2001 · 45 references

Pith's verdict: The analytic results — the exact q2/q6 = 3 threshold, the moduli-independent area bound, the monodromy-stability criterion — are solid and will outlive the headline spectrum predictions, which rest on an unquantified lattice-wide mass bound.

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CitationBrian Greene, Frederik Denef, Mark Raugas. "Split attractor flows and the spectrum of BPS D-branes on the Quintic." arXiv:hep-th/0101135 (2001). https://pith.science/paper/Y7R6WH6R
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The paper

The paper claims

The four-dimensional supergravity of a compactified string theory, taken seriously as a flow on moduli space, predicts which BPS D-brane charges exist, where they decay, and into what.

The reading

This paper asks whether the low-energy supergravity description of a compactified string theory — the attractor mechanism and its multicentre 'split flow' generalization — can be used as a practical calculator of the BPS D-brane spectrum, rather than merely as a description of black hole horizons. The authors work on the quintic Calabi-Yau in type IIA (equivalently type IIB on its mirror), where the moduli space is one complex dimensional and the periods are known exactly as Meijer functions, and they run extensive numerical flows to read off which charges survive in which regions. The payoff is a set of concrete predictions: a specific charge that is a black hole at large radius but is absent at the Gepner point, a charge that exists only as a composite of two black holes and never as a single one, and a moduli-independent lower bound on the core area of any BPS solution. If the picture is right, the effective supergravity theory carries far more information about the string spectrum than its apparent simplicity suggests, and it does so in a form that can be checked charge by charge.

What carries the argument

The load-bearing object is the split attractor flow: the gradient flow of $|Z(Γ)|$ on moduli space, which may break at a surface of $(Γ_1,Γ_2)$-marginal stability, where $Z(Γ_1)$ and $Z(Γ_2)$ become parallel, each branch then flowing to its own attractor point. The existence criteria are that the total-charge flow must cross the marginal stability surface at positive flow parameter, $r_{ms} = 1/τ_{ms} = 2⟨Γ_1,Γ_2⟩|Z_1+Z_2| / (|Z_1|^2|Z_2|^2)$ evaluated at spatial infinity being positive, and that both branches must exist afterwards. Conifold monodromy $Γ → Γ + ⟨Γ,V⟩V$ is what generates multiple basins of attraction and forces the creation of $V$-charge branches when a flow is dragged past the singularity. Numerically the authors use two complementary schemes: step-by-step steepest descent on $|Z|$ with monodromy matrices applied as cuts are crossed, and the integrated BPS equation $2e^{-U}Im[e^{-iα}Ω̃] = -Γτ + const$, solved as the algebraic system $f(ℓ_i,z) = f(ℓ_i,z_0)μ$ for charges $ℓ_i$ mutually local with $Γ$.

Extended reading notes

Core claim

The central claim is that the BPS spectrum is essentially the spectrum of attractor flows, split flows included, on moduli space. Concretely for the quintic, the authors predict that the charge $(2,-1,-2,2)$ exists at large complex structure as a single-centre black hole but is absent from the spectrum at the Gepner point, where it decays into $2(1,0,0,0) + (0,-1,-2,2) = 2(1,0,0,0) + (-1,0,-4,0)_2$; the support is a numerical single-flow check plus a screening of $6{,}765{,}200$ candidate two-constituent charges with $-25 < q_i < 25$, all of which fail the existence criteria of section 3.1. A second prediction is that the charge $(0,3,9,-8)$ has a large-radius realization as a composite of two regular black holes, of charges $(1,0,4,0)T[0]^{-1}$ and $(1,0,4,0)T[0]T[1]$, but no single-centre black hole realization, together with an alternative four-leg realization whose constituents are all of empty-hole or $D0$-$D2$ type. The paper also reports a moduli-independent lower bound on the core area of every BPS solution, black hole or not, with the smallest value $A_{min} = 19.664719$ attained by the lightest regular black hole, of charge $(1,0,4,0)$ and mass $M_{min} = 1.250947$.

Load-bearing premise

The absence claims rest on truncating the candidate constituent charges at $|q_i| < 25$, which is licensed only by the argument of Appendix A that the single-flow mass spectrum is discrete, has no accumulation points, and grows roughly in proportion to charge — an argument the authors themselves call plausible rather than proved.

Editorial extensions

If this is right

  • Charge $(2,-1,-2,2)$ is a BPS state at large radius but is absent at the Gepner point, giving a sharp qualitative distinction between large-radius and small-radius physics for a single charge.
  • Charge $(0,3,9,-8)$ is realized at large radius only as a composite of two regular black holes, never as a single-centre black hole, so its entropy has no single-centre supergravity reading.
  • The lightest regular BPS black hole has charge $(1,0,4,0)$ and mass $M_{min} \approx 1.250947$ at $ψ \approx 0.375603$, with about $15.44\%$ of its mass coming from worldsheet instanton corrections.
  • Every BPS solution satisfies a moduli-independent core-area bound: $A_{min} \approx 19.66$ for $(1,0,4,0)$, and $A_* \approx 43.92\,N^2$ for $N$ units of pure $D6$ empty-hole charge.
  • Monodromy stability restricts bound states of the $D6$ and its $Z_5$ cousin to $\frac{1}{λ}n_2 \le n_1 \le λ n_2$ with $λ = \frac{1}{2}(5+\sqrt{21}) \approx 4.79129$ on the Gepner side, a condition that reproduces the known spectrum in the $SU(2)$ Seiberg-Witten case at $κ = \pm 2$.

Reading between the lines

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

  • If the split-flow spectrum really matches the microscopic BPS spectrum, the flow picture becomes a cheap algorithm for the D-brane spectrum at arbitrary moduli, and its predictions for charges like $(2,-1,-2,2)$ and $(0,3,9,-8)$ could be checked directly against boundary CFT or quiver stability computations on the quintic.
  • The area bound holds even for empty holes with no horizon, which suggests it is really a statement about the length of the flow in moduli space rather than about horizon geometry; a natural extension would be to ask whether the bound is saturated only in the large complex structure limit and how it deforms under worldsheet instanton corrections.
  • The unresolved monodromy-stability paradox of section 6.7 — a flow that crashes on a would-be marginal stability line when transported around a conifold point — points to a missing ingredient, plausibly the multi-pronged-string s-rule or a more careful treatment of multicenter solutions carrying $D6$ charge, and is a concrete target for a follow-up.
  • The gap between the exact critical ratio $q_2/q_6 = 3$ and the large-complex-structure value $25/12 \approx 2.08333$ indicates that instanton corrections move the boundary of the single-flow existence region substantially; a systematic expansion in $1/t$ near the critical point would turn this numerical observation into a controlled statement.
Original abstract

We investigate the spectrum of type IIA BPS D-branes on the quintic from a four dimensional supergravity perspective and the associated split attractor flow picture. We obtain some very concrete properties of the (quantum corrected) spectrum, mainly based on an extensive numerical analysis, and to a lesser extent on exact results in the large radius approximation. We predict the presence and absence of some charges in the BPS spectrum in various regions of moduli space, including the precise location of the lines of marginal stability and the corresponding decay products. We explain how the generic appearance of multiple basins of attraction is due to the presence of conifold singularities and give some specific examples of this phenomenon. Some interesting space-time features of these states are also uncovered, such as a nontrivial, moduli independent lower bound on the area of the core of arbitrary BPS solutions, whether they are black holes, empty holes, or more complicated composites.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.62 · compares with prior results 0.17 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We investigate the spectrum of type IIA BPS D-branes on the quintic from a four dimensional supergravity perspective and the associated split attractor flow picture. We obtain some very concrete properties of the (quantum corrected) spectrum, mainly based on an extensive numerical analysis, and to a lesser extent on exact results in the large radius approximation. We predict the presence and absence of some charges in the BPS spectrum in various regions of moduli space, including the precise location of the lines of marginal stability and the corresponding decay products. We explain how the generic appearance of multiple basins of attraction is due to the presence of conifold singularities and give some specific examples of this phenomenon. Some interesting space-time features of these states are also uncovered, such as a nontrivial, moduli independent lower bound on the area of the core of arbitrary BPS solutions, whether they are black holes, empty holes, or more complicated composites.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the spectrum of type IIA BPS D-branes on the quintic from a four dimensional supergravity perspective and the associated split attractor flow picture. contribution 0.97 0.90 0.08 asserted 0.80
s1 We obtain some very concrete properties of the (quantum corrected) spectrum, mainly based on an extensive numerical analysis, and to a lesser extent on exact results in the large radius approximation. method 0.68 0.96 0.12 asserted 1.00
s2 We predict the presence and absence of some charges in the BPS spectrum in various regions of moduli space, including the precise location of the lines of marginal stability and the corresponding decay products. result 0.98 0.94 0.36 asserted 0.87
s3 We explain how the generic appearance of multiple basins of attraction is due to the presence of conifold singularities and give some specific examples of this phenomenon. interpretation 0.66 0.87 0.09 asserted 1.00
s4 Some interesting space-time features of these states are also uncovered, such as a nontrivial, moduli independent lower bound on the area of the core of arbitrary BPS solutions, whether they are black holes, empty holes, or more complicated composites. result 1.00 0.91 0.13 asserted 0.99

What kind of paper the classifier reports this is: simulation study 0.68 · evidence from simulation or computation 1.00.

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Figures
Figure 1. Sketch of some single charge attractor flows in moduli space, for different values z (i) 0 of the moduli at spatial infinity. The attractor point is indicated by zm. 2.2 BPS configurations 2.2.1 Single charge type: single flows Static, spherically symmetric BPS configurations [24, 32, 33, 34] with charge Γ ∈ H3 (X, Z) at the origin of space have a spacetime metric of the form ds2 = −e 2U dt2 + e −2U dxi dxi , (2.6) with U a function of the radial coordinate distance r = |x|, or equivalently of the inverse radial coordinate τ = 1/r. We will take space to be asymptotically fla
Figure 2. Example of a (Γ1, Γ2)-multicenter configuration with total charge 8 Γ1 + 7 Γ2. the given charge in a vacuum where Z vanishes would imply the existence of a massless particle there, which in turn is expected to create a singularity in moduli space at the zero, contradicting the assumption of regularity of that point. Or, from a geometric point of view: if a supersymmetric wrapped brane exists at the zero, its volume is zero, so we must have a vanishing cycle in the Calabi-Yau, leading to a singularity in moduli space. 3. |Zm| = 0 and zm is a singular (or boundary) point in moduli…
Figure 3. Sketch of some split flows for different moduli values z (i) 0 at spatial infinity, in the case of two different constituent charges Γ1 and Γ2, with attractor points z1 resp. z2. The line labeled “MS” is (part of) the (Γ1, Γ2)-marginal stability line. and (2.11) elegantly generalizes to 2 e −U Im[e −iαΩ] = ˜ H , (2.15) ∗dω = hdH, Hi, (2.16) with H(x) an H3 (X)-valued harmonic function (on flat coordinate space R 3 ), and ∗ the flat Hodge star operator on R 3 . For N charges Γp located at coordinates xp, p = 1, . . . , N, in asymptotically flat space, one has: H = − X N
Figure 4. Sketch of the image of z(x) in moduli space for a (Γ1, Γ2)-multicenter solution, with attractor points z1 resp. z2, and modulus at spatial infinity z0. The line labeled “MS” is a (Γ1, Γ2)-marginal stability line. space through the moduli fields z a (x), with the positions of the charges mapped to their respective attractor points in moduli space. It turns out that the splitting points of the flows have to lie on a surface of marginal stability in moduli space [25]; more precisely, a Γ-flow can only split in a Γ1- and a Γ2-flow at a surface of (Γ1, Γ2)-marginal stability, that is,…
Figure 5. Left: A conifold singularity (the red dot), with vanishing cycle V , as source for different basins of attraction. The flows to the left and the right of the basin boundary (labeled “bb”) have different attractor points, due to the monodromy. The horizontal wiggly line is a cut. Right: This is what really happens when a flow is continuously moved from 1 to 2: it transforms into a split flow with the new branch ending on the singularity. The red line labeled “ms” is the corresponding (hΓ, V iV, Γ − hΓ, V iV )-marginal stability line. The key observation is that jumps in the basin of…
Figure 6. Sketch of the “fundamental wedge” 0 < arg ψ < 2π/5 in the ψ-plane, a sec￾tion of Teichm¨uller space unambiguously parametrized by z = ψ −5 . Attractor flows have definite charge and are continuous on Teichm¨uller space, but if we want to represent them exclusively on the fundamental domain, the assigned charge jumps at the wedge boundaries as q → q.M, with M the appropriate monodromy matrix. The labels T[0] and T[∞] along each branch cut in the picture indicate the monodromy matrix to be applied for |z| < 1 resp. |z| > 1. Indicated is an attractor flow line which undergoes a…
Figure 7. The w-plane is a 5-fold cover of moduli space, and is especially convenient for plotting flows. The five fat purple lines are period cuts starting at the conifold point copies w = e 2niπ/5 and running to the large complex structure limit. The black dot in the middle is the Gepner point ψ = w = 0. Since all periods can be taken to be continuous around w = 0, we only indicate cuts running to w = ∞. Our conventional path for interpolating between ψ = 0 and ψ = ∞ is the line ψ = e iπ/5R +. We will mostly focus on the analysis of states with low charges, and use super￾gravity language to
Figure 8. From top to bottom: (1): spectrum of the lightest single flows originating at ψ = 0. The horizontal axis gives the mass in four dimensional Planck units. The state with pure D2 charge (and its Z5 images) has the lowest mass, followed by the D0, the D6 (which is an empty hole), and five mixed (D2, D0) states. The lightest black hole comes next, with charge (1, 0, 4, 0). (2): Same as (1), excluding the regular black holes. (3): Like (1), but starting at the point ψ = 0.0851 − 0.3997i, i.e. the “crash point” of fig. 10. Here the state (1, 0, 0, 0)−1 is the lightest. Note that the…
Figure 9. Masses Mmin evaluated at the attractor points for a set of charges (1, q4, q2, 0), with flows starting at ψ = 0. A black bar of zero height indicates that the flow does not exist. indicates that the masses of the BPS states indeed tend to grow with charge (see also fig. 9). This is not trivial, since it is not true for the mass of arbitrary candidate BPS charges. At the Gepner point, the BPS mass of a charge q is given by M(q) = c
Figure 10. Decay of a BPS state with charge q = (2, −1, −2, 2) when moving from large complex structure w = ∞ to the Gepner point w = 0 (see section 6.1 for the relation between w and ψ). The state decays into 2(1, 0, 0, 0) + (0, −1, −2, 2) = 2(1, 0, 0, 0) + (−1, 0, −4, 0)2 . The green and brown lines 1 to 8 are flows with charge q, respectively starting at w1 = ∞, w2 = 8 e iπ/5 , w3 = 4 e iπ/5 , w4 = 3 e iπ/5 , w5 = 2.35 e iπ/5 , w6 = 1.6e iπ/5 , w7 = 1.2e iπ/5 , and w8 = 0 (note that the points w1, w2 and w3 are not in the picture). The blue square is a regular attractor point, and
Figure 11. Potentials W for test particle with charge ǫ(1, 0, 0, 0) in the backgrounds corre￾sponding to the flows in fig. 10. As a convenient radial coordinate, we use the gravitational redshift factor e U . locus in space
Figure 12. Split flow with regular constituents, which exists at large complex structure, but not at the Gepner point. The total charge is Γ = (0, 3, 9, −8), decaying into Γ2 + Γ1 = (−1, 1, 4, −1) + (1, 2, 5, −7) = (1, 0, 4, 0).T[0]−1 + (1, 0, 4, 0).T[0].T[∞]. The purple line extending the incoming branch at the split point is the would-be single flow, crashing on a regular zero at the orange triangle. particular, we expect this charge to be absent from the BPS spectrum at the Gepner point, with a somewhat lower level of confidence though (it becomes slightly less unlikely that more…
Figure 13. Alternative split flow to fig. 12: the initial branch (1) of charge (0, 3, 9, −8) splits at the MS line ms23 in a branch (2) of charge (3, 3, 10, −11) = (0, 0, 1, 3)1 and a branch (3) of charge (−3, 0, −1, 3) = (0, 0, −1, 2)−1 . Branch (2) then splits on ms45 as (4): 3(1, 0, 0, 0)1 + (5): (0, 0, 1, 4) and similarly (3) on ms67 as (6): 3(−1, 0, 0, 0) + (7): (0, 0, −1, 3). Note that the split flows (245) and (367) can be created from single flows (of D0 − D2 type) by the conifold branch creation mechanism of section 3.3. On the full Teichm¨uller space, there will in general be…
Figure 14. The charge q = (0, −3, 10, 18) = (1, −3, −5, −1)1 has two different basins of attraction in the fundamental domain 0 < arg ψ < 2π/5, separated by the blue basin boundary labeled “bb”. Attractor point a has |Z∗| = 6.045, while attractor point a ′ has |Z∗| = 5.603, so the corresponding near horizon geometries and entropies of the black holes are not equal. The green (brown) lines 1, 2, 3 (1′ , 2 ′ , 3 ′ ) are a continuous family of flows attracted to a (a ′ ), with 3 and 3′ the split flows obtained upon crossing the basin boundary. Those split flows decay at the marginal stabilit
Figure 15. Comparison of numerical results with the LCSL approximation for charge (q6, 0, q2, 0), q6 > 0, where q2/q6 is varied continuously over the x-axis. The lines hitting zero give the relative attractor mass M∗/q6 for the given charge, with the zero corresponding to the critical point for existence of a single flow. The lines diverging at those zeros indicate the mass of the “dual” charge defined in (5.5), with the divergence corresponding to reaching a boundary of Teichm¨uller space. Note that the exact M∗ curve is much steeper near its critical point than the LCS curve near its…
Figure 16. Two possible split flows with charge q = (3, 0, 8, 0), near the Gepner point. Left: two-legged realization, with one leg (2) of charge (−2, −2, −7, 7) = −(0, 0, 1, 2)1 and the other one (3) of charge (5, 2, 15, −7) = (0, 0, −1, 1)3 . Note that when the starting point is moved up sufficiently, branch (3) gets an additional conifold leg, with marginal stability line ms3. Right: For starting points above the real w-axis, we get an alternative split flow of the form (3, 0, 8, 0) → (1, 0, 0, 0)3 +[(0, 0, −2, −3)−1 → (1, 0, 0, 0)2 + (0, 0, −2, −1)2 ], with MS lines ms+ 1 resp. ms+ 2 . F
Figure 17. The split flows on the left and the right are related by continuous variation of the starting point. By charge conservation, we have Γ = Γ1 +nV = Γ1 + (m+k)V , and by the Picard-Lefshetz theorem k = hΓ1, V i. The purple wiggly line is the cut corresponding to the conifold monodromy. The red line ms is the (Γ1, nV )-marginal stability line; the pink line ms is the “conjugate” (Γ1, −nV )-marginal stability line (as Z(V ) flips sign upon crossing the conifold singularity). stability is crossed, or, in the split flow picture, when the incoming branch shrinks to zero size. However, a…
Figure 18. When the starting point of the split flow (7, 1, 3, −5) → 6(1, 0, 0, 0)+(1, 0, 0, 0)1 is continuously transported as indicated, it ceases to exist: it crashes on the would￾be line of marginal stability. The transport is counterclockwise with respect to the conifold point (whereas in fig. 17, it was clockwise), so we have here m = 6, n = 6 + h(1, 0, 0, 0)1 ,(1, 0, 0, 0)i = 6 − 5 = 1, and nm = 6, which does not satisfy (6.13). So we are facing a paradox again. One could contemplate the possibility that the BPS state just decays into other BPS particles at the basin boundary, but this…
Figure 19. Split flow spectrum at the Gepner side of the MS ellipsoid for flows with a charge n1 leg on (1, 0, 0, 0) and a charge n2 leg on (1, 0, 0, 0)1 . The orange dots (n1n2 < 0) indicate charges that do not have a split flow realization, the yellow dots are the charges that have a split flow realization, but one that is unstable under repeated monodromy around the MS ellipsoid, and the green and blue dots are the monodromy stable split flows, with the blue dots the charges related to pure (1, 0, 0, 0) or the pure (1, 0, 0, 0)1 charge. Support for this possibility is given by the strong…

Figures reproduced from arXiv:hep-th/0101135 by the authors.

References
How later papers used it

65 later papers cite it

Evidence: some. 63 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 7 of the 128 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 9 papers with this label
  • When an attractor flow line enters a branch cut, we follow [ref] and find a path in the $u$-plane which connects the point at which the attractor flow line enters the branch cut (shown from above in Fig. [ref]) and leaves it (shown from below).

    Special geometry, quasi-modularity and attractor flow for BPS structures (2023) · Comment on branch cuts · 1.00

How the literature uses it

65 arXiv papers cite this in 128 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 69 sentences without a confident label

Uses its method or tool 9 papers
Compares with it 2 papers
Cites it as support 4 papers
  • However, thanks to the work done in [ref] , one knows that the conditions of existence for quiver BPS multi-particle solutions are exactly the same as the ones for BPS multicenter solutions in the supergravity picture even if the supergravity solution is not reliable.

    The full space of BPS multicenter states with pure D-brane charges · Introduction · 0.96

  • In particular, it follows from [ref] that the flow cannot cross the same wall more than once, and that BPS states do not decay as one follows the attractor flow in reverse [ref] .

    BPS Dendroscopy on Local $P^2$ · Exact attractor flow · 0.99

  • It is often possible [ref] to dial the moduli while keeping $R$ positive and finite, from a marginal to an anti-marginal stability wall, as illustrated in fig.

    Bound state transformation walls · Puzzle · 0.97

Cites it as background 29 papers
  • The SAFC posits a mostly-combinatorical formula for the index $Ω_σ(γ)$ for a given charge vector $γ$ and $Π$-stability condition $σ$ as a sum over `attractor flow trees' [ref] .

    BPS Dendroscopy on Local $\mathbb{P}^1\times \mathbb{P}^1$ · Split Attractor Flow Conjecture · 0.98

  • The Split Attractor Flow Conjecture, originally proposed in [ref] and sharpened in [ref] , amounts to the statement that for any $z∈Π$ and $γ∈Γ$, there exists only a finite number of decompositions $γ=∑_i γ_i$ such that $_z({γ_i})$ is non-empty; the rational index $_z(γ)$ is then obtained by summing over all attractor flow trees,

    BPS Dendroscopy on Local $P^2$ · Attractor flows and Split Attractor Flow Conjecture · 0.95

  • A general class of stationary BPS solutions with total electromagnetic charge $γ=(p^Λ,q_Λ)$ ($Λ=1,…,r$) can be written as [ref] (in units where the Newton constant $G_4=1$) s^2 = - e^2U(t+ω)^2 +e^-2U x⃗^2 , A = 2 e^U (e^-αZ) (t+ω) + A_d, where $A=(A^Λ, A_Λ)$ denotes the symplectic vector of the electric and magnetic gauge fields.

    Attractor flow trees, BPS indices and quivers · Multi-centered black holes in $=2$ supergravity · 0.97

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