PPrepared for submission to JHEP
Black Holes and the Swampland: the Deep Throatrevelations
Yixuan Li a a Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique,91191, Gif-sur-Yvette, France.
E-mail: [email protected]
Abstract:
Multi-centered bubbling solutions are black hole microstate geometries thatarise as smooth solutions of 5-dimensional N = 2 Supergravity. When these solutionsreach the scaling limit, their resulting geometries develop an infinitely deep throat and lookarbitrarily close to a black hole geometry. We depict a connection between the scaling limitin the moduli space of Microstate Geometries and the Swampland Distance Conjecture.The naive extension of the Distance Conjecture implies that the distance in moduli spacebetween a reference point and a point approaching the scaling limit is set by the properlength of the throat as it approaches the scaling limit. Independently, we also compute adistance in the moduli space of 3-centre solutions, from the Kähler structure of its phasespace using quiver quantum mechanics. We show that the two computations of the distancein moduli space do not agree and comment on the physical implications of this mismatch. a r X i v : . [ h e p - t h ] F e b ontents AdS throat in terms of the scaling parameter 93.2 The AdS throat and Kaluza-Klein modes 113.3 M2 branes at the bottom of the throat 12 The fact that black holes are statistical objects with temperature and entropy raises twokey issues. First, how to describe the microstates accounting for the statistical entropy?Second, how does the black hole restore the information that falls in it? String Theory’shistorical answer to the first question is to describe the microstates at low string coupling,where all the possible open strings that stretch between brane bound states have an entropythat matches the statistical entropy of the black hole. As the string coupling constant istuned to larger values to a regime where gravity is dominant, the branes expand in size, soone could expect that the microstates differ from the black hole at horizon-scale.The Fuzzball paradigm [1, 2] proposes that black hole microstates do possess a horizon-size structure that differs from the classical black hole. Within this approach, it is expectedthat the black hole evaporation is similar to the burning of a star or a piece of coal [3],differing from Hawking’s calculation which leads to the Information Paradox. Within theFuzzball paradigm, the Microstate Geometries programme [4, 5] endeavours to describethese black hole microstates within the Supergravity approximation of String Theory bysmooth horizonless solutions. If one succeeds in finding a large number — hopefully e S – 1 – igure 1 . Schematic picture of a classical black hole (left) and a microstate geometry (right).From the asymptotic observer’s point of view, instead of the event horizon lying at the bottom ofa throat of infinite length, a microstate geometry would replace it with a smooth cap lying at thebottom of a throat of finite length. In the scaling limit, its throat length increases to infinity whilethe cap’s geometry stays constant. The figures are from [6]. — of them, then we have answered to the question “What do black hole microstates looklike?”.In all classes of microstate geometries, the infinitely-long throat of the black hole isreplaced by a cap at the end of a long, but finite throat [7–9] (See Fig. 1). The procedureto construct a large number of Supergravity microstates is the following: Take a black holewith given charges and angular momenta. Supergravity admits a large number of solutionswith finite throat length, with charges and angular momenta equal to those of the blackhole. In the moduli space (of a particular superselection sector, if any) , each of thesesolutions admits a limit — called the scaling limit — where, from the perspective of anobserver at infinity, they become more and more similar to the black hole; in particular,their throat length increases to infinity in the scaling limit, while the size of the cap remainsfixed.In the moduli space of solutions, the scaling limit point plays a particular role, for thefollowing reasons:(1) The scaling limit lies at the boundary of moduli space where the throat length increasesto infinity.(2) As we approach the scaling limit, global symmetries of the black hole, who obeys the no-hair theorem, are restored. For instance, microstate geometries do not generically possessthe SO(3) -rotational symmetry of the black hole. In some models of microstate geometries, as in Multi-centered bubbling models, there are families ofsolutions labeled by the fluxes Γ i wrapping the bubbles (see Section 2). Inside each of these families, or superselection sectors , there are still real parameters left to characterize the solutions, defining a modulispace. There are restrictions on the bubble fluxes (and on the superselection sectors) to admit a scalinglimit; but here, we consider one superselection sector which does. – 2 –3) Everywhere in moduli space, energy excitations at the bottom of the throat of deepmicrostates are gapped. Their gap matches that of their dual CFT states [7, 10]. However,in the scaling limit, the mass gaps of these modes decrease to zero, because of the increasingredshift due to the lengthening of the throat.Taking the limit to a boundary point of moduli space is reminiscent of the SwamplandDistance Conjecture [11], that we are reformulating hereinbelow. Consider an effectivefield theory consistent with quantum gravity, with an arcwise-connected moduli space — apoint in the moduli space fixes the expectation value of the scalar fields of the EFT. TheSwampland Distance Conjecture states that:(1) The moduli space is not bounded in terms of its geodesic distance d . In other words,given p a point in the bulk of the moduli space, there exists a family of arcwise-connectedpoints { p } going from p to an infinite geodesic distance with respect to p .(2) Global symmetries are restored at infinite distance in moduli space. [12](3) Given the point p and the path of points { p } defined in (1), there exists α > andthere exists an infinite tower of states with an associated mass scale M ( p ) such that M ( p ) ∼ d ( p ,p ) →∞ M ( p ) e − αd ( p ,p ) . (1.1)Originally, the distance in moduli space was defined according to the kinetic terms ofthe scalar fields in the EFT in the following sense: Consider a d -dimensional EFT whoseaction in the d -dimensional Einstein frame is written as S = (cid:90) d d x √− g (cid:20) R − g ij (cid:0) φ i (cid:1) ∂φ i ∂φ j + ... (cid:21) . (1.2)Then g ij defines a metric on the moduli space of effective field theories. Following [13],it has been proposed to generalise the Swampland Distance Conjecture — about modulispaces of scalar fields — to a space of metrics. A notion of distance can be defined on atransverse-traceless metric g µν of a spacetime M of volume V M = (cid:82) M √ g [14] ∆ generalized = c (cid:90) τ f τ i (cid:32) V M (cid:90) M √ g tr (cid:34)(cid:18) g − ∂g∂τ (cid:19) (cid:35)(cid:33) d τ . (1.3)This distance boils down to the moduli space of the scalar fields in the case of Calabi-Yaucompactifications on 4-dimensional Minkowski space [15].Thanks to this notion of distance between two metrics, it was argued in [13] that thevanishing limit of the negative cosmological constant, Λ , in an AdS vacuum in String The-ory leads to an infinite tower of light states — for instance the tower of Kaluza-Klein modesof some decompactifying parts from the internal manifold.To extend the Swampland Distance Conjecture for metrics to the scaling limit of mi-crostate geometries, we would like to show that the infinite tower of gapped modes on topof any microstate geometry collapses, and the masses of all these modes decrease exponen-tially. In this paper, we study a class of bubbling microstate geometries , that descend in4 dimensions to multicentered solutions [16, 17]. In the scaling limit, different microstate– 3 –eometries approach the BMPV black hole [18]. These microstate geometries possess an AdS × S throat. We will show that, in the scaling limit, the mass the Kaluza-Klein modesof the S measured by an observer at spacial infinity decrease exponentially with respectto the length of the throat. At the bottom of this throat lie also non-trivial two-cycles; themass of the M2 branes wrapping these two-cycles decrease exponentially in the same fashionin the scaling limit. By reading off the expression inside the exponential, one can infer thedistance in moduli space ∆ exponential that would be in agreement with the extension of theSwampland Distance Conjecture; one thus expects this distance to be proportional to thelength of the AdS throat, which is becoming infinite in the scaling limit. As a result, ourstudy quite possibly extends the Swampland Distance Conjecture in a rather unusual way.In addition, we will also compare this distance with another notion of distance in themoduli space of solutions, whose computation is independent of Swampland notions. Outof a Lagrangian theory characterizing a set of fields φ A , the symplectic form, Ω , of thetheory can be defined, from the Crnković-Witten-Zuckerman formalism, as an integral overa Cauchy surface Σ [19, 20] Ω = (cid:90) d Σ l δ (cid:18) ∂L∂ ( ∂ l φ A ) (cid:19) ∧ δφ A . (1.4)If Ω is closed and non-degenerate, the m -dimensional solution-space manifold is reinter-preted as the phase space , whose symplectic volume (in units of h m ) gives the number ofmicroscopic ground states. When the phase-space manifold is furthermore endowed withan integrable complex structure, J , and if Ω( · , J · ) is a Riemannian metric, the manifold isKähler and one can define a distance, ∆ phase , on the moduli space of solutions using theKähler metric Ω( · , J · ) . Luckily, the solution space of three-centered multicenter solutions one constructs as Microstate geometries is a Kähler manifold [21]; so we will measure thedistance to the scaling limit with respect to this Kähler metric.Surprisingly, we find that with respect the “canonical” ∆ phase that would be in agree-ment with computations in [21], the scaling limit lies at finite distance in moduli space, intension with the distance ∆ exponential . However, the computation of ∆ phase is performed atweak string coupling regime using quiver quantum mechanics, and one can wonder whetherthis computation is still reliable in the regime where Supergravity dominates. However,in [21], the authors argue that the reduced symplectic form does not vary with the stringcoupling constant thanks to a non-renormalization theorem, and further conclude that Su-pergravity is breaking down because of large quantum fluctuations in scaling geometries,and hence could not be a good description of these geometries. From the weak-couplingregime, they also infer that the scaling limit, which was perfectly in reach within Supergrav-ity, is actually prohibited if one accounts for quantum effects, which prevent the quantumwave functions to populate the region of classical moduli space close to the scaling limit.Thus, if the correct normalization of distance on moduli space is given by ∆ exponential andnot by ∆ phase , then the breakdown of Supergravity due to quantum effects prescribed in[21] would be softened. – 4 –he organisation of the paper is the following. In Section 2, we review smooth multi-centered bubbling solutions in five dimensions (and their M-theory uplift) of [7]. In Section3, we first compute how the throat lengths of bubbling solutions behave in the scaling limit.We then find an exponential decrease of the S Kaluza-Klein mass tower, consistent witha naive extension of the Swampland Distance Hypothesis to this system. In Section 4, westudy the moduli space of three-centre bubbling solutions, independently of the Swamplandprogramme. Using the results of [21], we determine the metric on moduli space of solutionscoming from the symplectic form. We show that with this distance, ∆ phase , the scalinglimit lies at finite distance in moduli space and that all the moduli space is bounded. Insection 5, we discuss the tension between the two distances on moduli space and share someinsight about the ability of Supergravity to describe black hole microstates with arbitrarilydeep throats. Upon compactifying maximal eleven-dimensional Supergravity on Calabi-Yau threefold, theresulting five-dimensional N = 2 supergravity coupled to n V vector multiplets with n V ≤ contains the following bosonic fields:• a gravitational field, g ,• n V + 1 U(1) vector gauge fields, A Iµ , whose field strengths are denoted F I = d A I ,• n V + 1 scalars, X I .This theory is described by the action (16 πG ) S = (cid:90) d x √− g R − Q IJ (cid:90) (cid:0) F I ∧ (cid:63) F J − d X I ∧ (cid:63) d X J (cid:1) + C IJK (cid:90) A I ∧ F J ∧ F K , (2.1)where C IJK are the structure constants satisfying the fixed-volume constraint C IJK X I X J X K = 1 = ⇒ X I = 16 C IJK X J X K , (2.2)and the couplings Q IJ depend on the scalars via Q IJ = 92 X I X J − C IJK X K . (2.3)The action admits the following Einstein-Maxwell-scalar equations of motion R µν + Q IJ (cid:18) ∂ µ X I ∂ ν X J + F Iµρ F Jν ρ − g µν F Iρσ F J ρσ (cid:19) = 0 ,d (cid:0) Q IJ (cid:63) F J (cid:1) + 14 C IJK F J ∧ F K = 0 , − d (cid:63) d X I + (cid:18) C IJK X L X K − C ILJ (cid:19) (cid:0) F L ∧ (cid:63) F J − dX L ∧ (cid:63) dX J (cid:1) = 0 . (2.4)– 5 –he most general supersymmetric solution to N = 2 five-dimensional Supergravitycoupled to n V extra gauge fields with structure constant C IJK , admitting a time-like Killingvector ∂ t are characterized by n V + 1 electric warp factors Z I , n V + 1 magnetic self-dualtwo-forms Θ I , an angular momentum one-form ω , and a space-like hyper-Kähler manifold B . The metric and the field strengths are stationary, and are split in the following way[22, 23]: ds = − (cid:18) C IJK Z I Z J Z K (cid:19) − ( dt + ω ) + (cid:18) C IJK Z I Z J Z K (cid:19) ds ( B ) ,F I = d A I = d (cid:0) Z − I ( dt + ω ) (cid:1) + Θ I . (2.5)In terms of these new data, the Einstein-Maxwell-scalar equations of motion (2.4) arerewritten as the so-called BPS equations (cid:63) Θ I = Θ I , with d Θ I = 0 , (2.6) ∇ V ≡ (cid:63) d (cid:63) d Z I = 12 C IJK (cid:63) (cid:0) Θ J ∧ Θ K (cid:1) , (2.7) d ω + (cid:63) d ω = Z I Θ I . (2.8)The first set of n V + 1 equations (2.6) ( I = 1 , . . . , n V + 1 ) determine the the magnetictwo-forms. The second set of n V + 1 equations (2.7) determine the electric warp factors,sourced by the magnetic fields. The fact that magnetic fluxes source a net electric chargeis made possible thanks to the Cherns-Simons term in the five-dimensional Supergravityaction (2.1); this is essential in the construction of smooth solitonic solutions in Supergrav-ity. The last equation (2.8) tells that the angular momentum ω is sourced by electric andmagnetic fields, recalling the Poynting vector in electromagnetism.We now consider B to be a four-dimensional Gibbons-Hawking space. The Gibbons-Hawking space is made of multiple centers of Kaluza-Klein monopoles. The Gibbons-Hawking space possesses non-trivial two-cycles called bubbles, defined by the shrinkingof the coordinate ψ fibered along any line running between a pair of Gibbons-Hawkingpoints in R . The spatial part of the metric in (2.5) is thus an S fibered along R ; it isdetermined by a harmonic function V in R ( ∇ V ≡ (cid:63) d (cid:63) d V = 0 ) and a one-form A (with ∇ A ≡ (cid:63) d A = d V ): ds ( B ) = V − ( dψ + A ) + V (cid:2) dρ + ρ (cid:0) dϑ + sin ϑ dφ (cid:1) (cid:3) . (2.9)The potential V is sourced by a set of n Gibbons-Hawking centres labeled by j , of charge q j : V ( (cid:126)ρ ) = h ∞ + n (cid:88) j =1 q j ρ j , A = n (cid:88) j =1 q j cos ϑ j dφ j , (2.10)where ( ρ j , ϑ j , φ j ) are the shifted spherical coordinates around the j th center. The potential V is a harmonic function on R . The Gibbons-Hawking space pinches off smoothly around– 6 –ach center j : the geometry is a flat R modded by Z | q j | along ψ , where q j ∈ Z . Besides, R is asymptotically modded by Z (cid:80) | q j | , so it is convenient to subsequently impose (cid:80) j | q j | = 1 to have an asymptotic R .We will consider solutions that are independent of ψ . With this assumption, the othersolution data — Z I , Θ i and ω — are all given in terms of harmonic functions on R .The n V + 1 self-dual magnetic two-forms Θ I are of the form Θ I = ∂ a (cid:0) V − K I (cid:1) Ω a , (2.11)where (Ω , Ω , Ω ) is a basis of self-dual (in 4 dimensions) two-forms and K I are harmonicfunctions on R of the form K I = k I ∞ + n (cid:88) j =1 k Ij ρ j . (2.12)The number k Ii − k ij is the magnetic flux on the two-cycle between centres i and j .The n V + 1 warp factors Z I are Z I = L I + C IJK K J K K V (2.13)where L I is a harmonic function on R is L I = l I ∞ + n (cid:88) j =1 l Ij ρ j . (2.14)From the 5-dimensional Supergravity perspective, l Ij is the electric charge of L I at the j th center.Finally, the angular-momentum one-form can be decomposed along the U(1) ψ -fiber: ω = (cid:18) M + K I L I V + C IJK K I K J K K V (cid:19) ( dψ + A ) + (cid:36) ≡ µ ( dψ + A ) + (cid:36) , (2.15)where (cid:36) is a one-form on R and M — the harmonic conjugate of ω in (2.8) — is a harmonicfunction on R of the form M = m ∞ + n (cid:88) j =1 m j ρ j . (2.16)To put it in a nutshell, the multicenter bubbling solutions are characterized by theharmonic functions Γ = (
V, K , . . . , K n v +1 ; L , . . . , L n v +1 , M ) on R . Schematically, wecan write Γ = Γ ∞ + n (cid:88) j =1 Γ j ρ j . (2.17)One can define a symplectic product on R n V +4 : for A = ( A , A , . . . , A n v +1 ; A , . . . , A n v +1 , A ) and B = ( B , B , . . . , B n v +1 ; B , . . . , B n v +1 , B ) , (cid:104) A, B (cid:105) ≡ A B − A B + A I B I − A I B I . (2.18)– 7 –he absence of Dirac-Misner strings in the multicenter bubbling solutions then leadsto conditions on the relative positions of the Gibbons-Hasking centres, the so-called bubbleequations , or Denef integrability equations [16, 24]: n (cid:88) j =1 (cid:104) Γ i , Γ j (cid:105) ρ ij = (cid:104) Γ ∞ , Γ i (cid:105) , for i = 1 , . . . n . (2.19) The requirement that the five-dimensional geometry be asymptotically flat R , constrainsthe asymptotic values of the harmonic functions h ∞ , l ∞ and k ∞ such that V = n (cid:88) j =1 q j ρ j , L I = 1 + n (cid:88) j =1 l Ij ρ j , K I = n (cid:88) j =1 k Ij ρ j , M = m ∞ + n (cid:88) j =1 m j ρ j . (2.20)Besides, requiring the resulting geometry to be smooth in five-dimensions amounts to con-straining the values of the electric and momentum charges in terms of the magnetic andKaluza-Klein monopole charges [25, 26]: l Ij = − C IJK k Jj k Kj q j , m j = 112 C IJK k Ij k Jj k Kj q j . (2.21)These conditions allow each centre to preserve 16 supercharges; the overall solution,made of several centres, preserves 4 supercharges as the BMPV black hole [18]. It can beshown that this solution is equivalent to multiple stacks of D3-branes at angles in a T-dualframe [26].The N = 2 five-dimensional Supergravity coupled to n V = 2 extra vector fields has ametric and field strength (2.5) that simplify to three-charge solutions: ds = − ( Z Z Z ) − ( dt + µ ( dψ + A ) + (cid:36) ) + V − ( Z Z Z ) ( dψ + A ) + V ( Z Z Z ) (cid:20) dρ + ρ (cid:0) dϑ + sin ϑ dφ (cid:1) (cid:21) ,F I = d (cid:0) Z − I ( dt + ω ) (cid:1) + Θ I . (2.22)This class of horizonless solutions have the same asymptotic geometry as the 4-superchargefive-dimensional rotating BMPV black holes [18], which have a macroscopic horizon andare described by the harmonic functions V = 1 ρ , L I = 1 + Q I ρ , K I = 0 , M = J L ρ . (2.23)Indeed, asympototically, these bubbling solutions behave like a BMPV black hole withcharges Q I , and left angular momentum J L : Q I = n (cid:88) j =1 l Ij + C IJK n (cid:88) ( i,j )=1 k Ji k Kj ,J L = 12 n (cid:88) j =1 m j + 12 n (cid:88) ( i,j )=1 l Ii k Ij + C IJK n (cid:88) ( i,j,k )=1 k Ii k Jj k Kk . , with ˆ ρ ij ≡ (cid:126)ρ i − (cid:126)ρ j | (cid:126)ρ i − (cid:126)ρ j | . (2.24)– 8 –n addition, the bubbling solutions have a right angular momentum J R : j ≡ J R = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i AdS throat in terms of the scaling parameter It has been mentioned in the Introduction that in the scaling limit, the throat of microstategeometries deepens. We would like to estimate the length of the throat of a near-scalingsolution presented in section 2.2, in terms of the scaling parameter λ . We will computethe length of the throat of bubbling solutions approaching the scaling limit, and compareits behaviour with respect to λ with the logarithmic divergence of the throat length of theBMPV black hole. The position of the centres in bubbling solutions is arbitrary (insofar– 9 –s they satisfy the bubble equations), and different centre configurations will modify thethroat length; however, we will show that this modification is set by the (coordinate) sizeof the region containing the centres.For a BMPV black hole, the (radial) throat length is infinite, with the divergence beinglogarithmic. In other terms, let us fix a coordinate ρ M not too far at infinity ( ρ M < Q I );its distance to a near-horizon cut-off ρ is L BMPVthroat ( ρ , ρ M ) = (cid:90) ρ M ρ V / ( Z Z Z ) / d ρ = ρ → ( Q Q Q ) / ln (cid:18) ρ M ρ (cid:19) + F ( ρ M ) . (3.1)The correction F ( ρ M ) is of order O (cid:16) ρ M Q I (cid:17) and induced by the constant term in Z I . At firstorder in ρ M , it is equal to ρ M − ρ Q har , where Q har is the harmonic mean of ( Q , Q , Q ) . Thiscorrection behaves like a constant as ρ approaches 0.Now, consider a family of smooth multi-center bubbling solution in 5D approaching thescaling limit. The Gibbons-Hawking centres are at a coordinate distance ρ ij ( λ ) = λd ij ofeach other, where d ≡ max d ij is of order 1. We choose the origin of the coordinates suchthat all the GH centres are within a radius of ρ = λd . We want to know how the throatlength scales with the scaling parameter λ . The throat length shall be computed from thesame ρ M < Q I to a region at the bottom of the throat at coordinate ρ ( λ ) > λd . We shalldefine for instance ρ ( λ ) = 2 max i,j ρ ij ( λ ) = 2 λd , (3.2)so that we are looking at the distance between the asymptotics and the blob of GH centres.The point is to keep some distance with respect to each individual GH centres. The lengthof the throat in question is L throat ( ρ ( λ ) , ρ M ) = (cid:90) ρ M ρ ( λ ) V / ( Z Z Z ) / d ρ . (3.3)As the scaling parameter λ is sent to zero, the metric of a bubbling solution approachesthat of a BMPV black hole. The more we are away from the bottom of the throat, thebetter the BMPV black hole approximation to the bubbling solution is. More precisely, inthe integration domain of the integral (3.3), ρ j = ρ + O ( λd ) , and ρ j = ρ (cid:16) O (cid:16) λdρ (cid:17)(cid:17) ; sothe function Z I V approximates to Z I V = Q I + O ( λd × charges) ρ + 1 ρ . (3.4)The integrand of (3.3) is then V / ( Z Z Z ) / = ( Q Q Q ) / ρ (cid:20) O (cid:18) ρQ I (cid:19) + O (cid:18) λd charges (cid:19)(cid:21) . (3.5)The first correction to the logarithm comes from the asymptotic behaviour dominated bythe /ρ term in (3.4), and is exactly the same one as for the BMPV black hole. Thesecond correction comes from the fact that the centres are arbitrarily distributed in a– 10 –egion of radius λd . Therefore, integrating the dominant term and its corrections leads tothe following reorganization of terms : L throat ( ρ ( λ ) , ρ M ) = λ → − ( Q Q Q ) / ln (cid:18) dλρ M (cid:19) + F ( ρ M ) + O (cid:18) λd charges (cid:19) ln (cid:18) dλρ M (cid:19) . (3.6)Although the positions of the Gibbons-Hawking centres are arbitrary, they lie in a smallregion inside ρ < λd . So in the scaling limit, they give rise to the geometry of the BMPVblack hole outside of the blob region ( ρ ≥ λd ), only up to small corrections. These aredominated by a λ ln( λ ) term whose limit is zero. It was important to know that thiscorrection’s limit is zero, so that inverting equation (3.6) gives ρ ( λ ) ∼ λ → ρ M exp (cid:18) − L throat ( ρ ( λ ) , ρ M ) − F ( ρ M )( Q Q Q ) / (cid:19) . (3.7) AdS throat and Kaluza-Klein modes In this section we compute the mass scale of the S Kaluza-Klein towers. The five-dimensional metric (2.22) asymptotes to the AdS × S metric in the throat region. Ofcourse, near the Gibbons-Hawking centres, the geometry differs, but as long as we do notapproach the GH centres too closely (for example ρ ≥ dλ ), Z I = Q I ρ (1 + O ( ρ )) , V = 1 ρ (1 + O ( ρ )) , (3.8)so that we get the metric of an AdS × S up to ds = − (cid:104) ( Q Q Q ) − ρ + O ( ρ ) (cid:105) ( dt + ω ) + (cid:20) ( Q Q Q ) ρ + O (cid:18) ρ (cid:19)(cid:21) dρ + (cid:104) ( Q Q Q ) + O ( ρ ) (cid:105) (cid:104) ( dψ + A ) + dϑ + sin ϑ dφ (cid:105) . (3.9)At the location where ρ = ρ ( λ ) , which, can be understood being roughly the “bottom ofthe throat”, there is an infinite tower of Kaluza-Klein modes on the S , with the lightestmass measured at the bottom of the throat being m KK ( ρ ( λ )) ∝ R S ( ρ ( λ )) ≈ Q Q Q ) . (3.10)The mass of the n th Kaluza-Klein mode measured at infinity gets redshifted to M n = n m KK √ g tt | ρ = ρ ( λ ) = n m KK ( Z Z Z ) − / | ρ = ρ ( λ ) ≈ n ρ ( λ )( Q Q Q ) / . (3.11)Injecting (3.7) into (3.11), we deduce that the tower of Kaluza-Klein states have massesthat scale like M n ( L throat ) ≈ λ → n G ( ρ M )( Q Q Q ) / exp (cid:18) − L throat ( Q Q Q ) / (cid:19) , (3.12) Note that we could have taken ρ M arbitrarily big. The important point is that integrating the O (cid:16) ρQ I (cid:17) term in (3.5) gives exactly the function F ( ρ M ) appearing in (3.1). – 11 –here G ( ρ M ) = ρ M exp (cid:16) F ( ρ M )( Q Q Q ) / (cid:17) . This decreasing exponential mass is consistentwith the extension of the Swampland Distance Conjecture to this system that we discussedin the Introduction. More generally, any locus in the cap verifies the approximation ρ i (cid:28) Q I , so although g tt isnot constant in the cap, its dependence with respect to the scaling parameter λ is the sameeverywhere in the cap, leading to the same redshift behavior √ g tt | ρ i = λd i ∼ λ → f ( d i ) λ . (3.13)The proportionality factor, f ( d i ) , depends on the location of the point in the cap and is setby the charges Γ j .As a result, M2-branes wrapping the two-cycle linking two Gibbons-Hawking centreswill experience a redshift that globally scales like λ in the scaling limit, so using (3.7) anddropping the proportionnality constant gives M M2 ∼ λ → exp (cid:18) − L throat ( Q Q Q ) / (cid:19) . (3.14)We meet again the same exponential mass decrease for the tower of M2 branes.As we go into the scaling limit and the throat becomes longer and longer, the M2branes become also exponentially light.Our system can be used to extend the Swampland Distance hypothesis. In our example,we move in the moduli space of metrics. In the scaling limit, the asymptotic geometry isunchanged. Besides, the size of the cap remains constant, as well as the inter-center physicalproper distances [27], up to order O ( λ ) : the geometry of the cap remains also fixed. In thescaling limit, the only modulus we are moving is the throat length which grows to infinity.Let p a point in moduli space (a reference point), characterising a solution that pos-sesses a throat region; and { p ( λ ) } λ ∈ (0 ,λ ] the set of points in moduli space approaching thescaling limit ( λ → ) from p ( λ ) = p . By reading off the argument inside the exponential,one possible conclusion is that the distance in moduli space between p and p ( λ ) should beproportional to the length of p ( λ ) ’s throat, L throat ( λ ) : α ∆ exponential ( p , p ( λ )) ∼ λ → L throat ( ρ ( λ ) , ρ M )( Q Q Q ) / , (3.15)where α corresponds to the mass decay rate of the Swampland Distance Conjecture in (1.1).The distance to the scaling limit is therefore infinite .Note that, instead of having n BPS Gibbons-Hawking centres coming closer to reachthe scaling limit, the limit of n coincident BPS black holes in N = 1 Supergravity mergingtogether lies also at infinite distance in moduli space [28]. The computation leading to (3.12) The approximation sign ≈ in (3.12) is here for the factor n G ( ρ M )( Q Q Q ) / in front of the exponential, butthe exponential is exact. – 12 –oes not require the horizonless regularity conditions (2.21) at each centre, and Gibbons-Hawking centres with a horizon going to the scaling limit are actually merging black holes.Our results thus agree with the infinite distance in moduli space in [28]. However, it is notclear that those two computations should give the same result. Indeed, the bubble equations(2.19) constrains the relative position of the “Denef black holes” (the Gibbons-Hawking cen-tres with a horizon) from one another to be dependent of the charges Γ i ; whereas there isno such a constraint on the relative position of the “Michelson-Strominger black holes” of[28].Interestingly, ( Q Q Q ) / is approximately the radius of the 3-sphere in the regime ρ ( λ ) < ρ < Q I . Indeed, the radius of the 3-sphere is ( Q Q Q ) / up to corrections oforder O (cid:16) ρQ I (cid:17) near ρ ∼ Q I , and corrections of order the magnitude of the charges Γ j in thevicinity of ρ ∼ ρ ( λ ) ; so the throat looks very much like a cylinder with an S base. Let usdefine the aspect ratio R of the throat to be the throat length divided by the radius of the S base. Then α ∆ exponential ( p , p ( λ )) ∼ λ → L throat ( ρ ( λ ) , ρ M ) R S = R ( λ ) . (3.16)Note that reading off the argument of the decreasing exponential gives only the distancein moduli space in the vicinity of the scaling limit, and only in the direction towards thescaling limit; we do not have any piece of information about how the distance behaves near p . The dimensional reduction of the smooth five-dimensional Supergravity solutions of Sec-tions 2 and 3 along the ψ -fiber leads to the four-dimensional multi-centered solutions [17].Describing these centres at equilibrium separations from each other (2.19) from Supergrav-ity at g s N (cid:29) is related to the quiver description of wrapped D-branes at g s N (cid:28) [29].As mentioned in the Introduction, one can compute the symplectic form from the quiverdescription, and, when possible, use the compatible complex structure to define a distanceon moduli space, ∆ phase .In this section, we wish to check whether the distance ∆ phase coincides with the distanceobtained by reading off the exponential decrease. Given L , the Lagrangian governing the dynamics of n -centered bubbling solutions of four-dimensional N = 2 Supergravity, coupled to n V gauge fields, and given φ A a basis of thefields apprearing in the Lagrangian, the symplectic form of the Supergravity-solution spaceis defined by Ω ≡ (cid:90) d Σ l δ (cid:18) ∂L∂ ( ∂ l φ A ) (cid:19) ∧ δφ A , (4.1)– 13 –here Σ is a Cauchy surface (in the 4-dimensional spacetime). We consider ˜Ω the restrictionof the symplectic form Ω to the space of multicentered solutions (which verify the bubbleequations). This consists of changing and restricting the variable fields φ A , such that thenew fields φ (cid:48) I define the n − -dimensional configuration of the n GH centres.The symplectic form of BPS solutions in Supergravity is difficult to compute for multi-centre solutions through the Supergravity action. Nevertheless, in [21], the authors com-puted the symplectic form in the open string description, valid when the centres do notbackreact ( g s N (cid:28) ). Thanks to a non-renormalization theorem in a similar spirit as[29], this symplectic form is independent of g s and equal to the symplectic form of BPSSupergravity solutions.Indeed, the authors of [16, 21, 29] argue that the open string dual of n GH centersin Supergravity is described by supersymmetric vacua of a (0+1)-dimensional quiver gaugetheory, whose Coulomb branch consists — after integrating out the massive bifundamentals— of n abelian vector multiplets. Each of the vector multiplets comprises three scalars ( x , x , x ) which characterize the positions of the D6 branes in R , one auxiliary field, D ,and one gauge field, A , which corresponds to the spatial components of the 4D gauge field A in Supergravity. The effective action of the vector multiplets in the Coulomb branch isdetermined by the Lagrangian L quiver = n (cid:88) p =1 ( − U p D p + A p · ˙ x p ) + fermions + higher-order terms , (4.2)where U p is found to be U p = (cid:104) Γ p , H p ( x p ) (cid:105) ≡ (cid:42) Γ p , θ + (cid:88) q (cid:54) = p Γ q | x p − x q | (cid:43) . (4.3)The symplectic form can be extracted from L quiver .Applying (4.1) to L quiver , the authors of [21] obtain the symplectic form to be of the form (cid:80) p δx p ∧ δA p . The restriction to BPS solutions corresponds, in the open string language,to restricting the solution space to (cid:84) p { U p = 0 } . In terms of the Supergravity data, therestricted symplectic form becomes ˜Ω = 12 (cid:88) p δx ip ∧ (cid:104) Γ p , δ A id ( x p ) (cid:105) . (4.4)After calculations detailed in [21], the infinitesimal variations of the field δ A id ( x p ) in (4.4)can be replaced by infinitesimal variations of the locations of the GH centres δ x p , such that ˜Ω = 14 (cid:88) p (cid:54) = q (cid:104) Γ p , Γ q (cid:105) (cid:15) ijk ( δ ( x p − x q ) i ∧ δ ( x p − x q ) j ) ( x p − x q ) k | x p − x q | . (4.5)Because the GH centres satisfy the bubble equations (2.19), acting on the positions x p of asolution with SO(3) rotations gives another configuration satisfying the bubble equations.Thus, if we impose the variations of the positions of the GH centres to be an infinitesimal– 14 –otation along the n -axis as δx ip = (cid:15) iab n a x bp , and call X n the vector field corresponding tothe rotation, then the reduced symplectic form satisfies ˜Ω( X n , · ) = n i δJ i , (4.6)where J i are the components of the angular momentum vector J i = 14 (cid:88) p (cid:54) = q (cid:104) Γ p , Γ q (cid:105) x ip − x iq | x p − x q | . (4.7)Using equation (4.6), it is possible to deduce the whole reduced symplectic form for twoand three GH centres. Furthermore, the reduced symplectic form (4.5) is closed, so the (2 n − -dimensional solution space can be viewed as a phase space. We now specialize in a solution with three Gibbons-Hawking centres. In this superselec-tion sector, the moduli space of solutions, which is also the phase space, has n − dimensions. Here, we have already set the centre of mass of the three GH points to be atthe origin of R . The total angular momentum vector J of the three-centre system (4.7)is described by its norm, j , and its direction — parameterized by the ( θ, φ ) angles in S .Now, rotating the triangle formed by the GH centres around the axis of J does not modifythe angular momentum vector, so the forth real variable that we call σ characterizes this U(1) rotational symmetry.In a nutshell, ( j, θ, φ, σ ) are the coordinates on the four-dimensional phase space. Oncethe charges on each GH centre are fixed, the intersection products (cid:104) h, Γ p (cid:105) and (cid:104) Γ p , Γ q (cid:105) arealso fixed. Given the length of two sides of the triangle of the GH centres, the third one isdetermined by the bubble equations (2.19). In other terms, for a given size of the triangle,its shape is determined. And what controls the size of triangle in these coordinates is theangular momentum j through j = 12 (cid:115) − (cid:88) p In this paper, we have shown that the scaling limit of the class of microstate geometries— the bubbling solutions — lies at a corner of the moduli space of solutions. As oneapproaches the scaling limit, bubbling solutions develop a throat whose depth is increasingto infinity. Besides, the deepening of the throat makes the redshift from the cap to thespatial asymptotics stronger and stronger, so the energy of all excitations lying at thebottom of the throat decrease to zero.This decrease of energy excitations at the bottom of the throat is independent of thetype of excitation we consider, as the redshift affecting them, set by √ g tt , is the same. InSection 3, we have proved that the redshift decreases the energy excitations by a factorof exp (cid:16) − L throat ( Q Q Q ) / (cid:17) . Thus, one may argue that our model is a new instance of the Since the distance to the scaling limit is finite here, we are measuring distances from the moving point p ( j ) to the scaling limit p (0) ; while in Section 3, as the distance to the scaling limit was infinite, we wereconsidering distances from the moving point p ( j ) to a reference point p ( j ) at finite angular momentum. – 19 –wampland Distance Conjecture for metrics. If it turns out to be true, one can extract,from the mass decay, a notion of distance in moduli space, ∆ exponential ( p ( λ ) , p ( λ )) , from areference solution p ( λ ) to a solution p ( λ ) approaching the scaling limit ( λ → ).As discussed in Section 3.3, the Michelson-Strominger derivation of the distance inmoduli space [28] shows that the merging of n BPS black holes happens as well at an infi-nite moduli space distance from the bulk, and seems to support the ∆ exponential distance.However, it is not clear that these two distances — one involving black holes with uncon-strained positions, and the other involving charged Gibbons-Hawking centres/black holeswhose positions satisfy the Denef integrability equations — should agree.A second notion of distance, ∆ phase , can be derived from the Kähler metric of thephase space of three-centre solutions. This distance is a priori computed in the weakcoupling regime. The first question is whether one can extrapolate this distance up tostrong string coupling regime. The non-renormalization theorem of [21] shows that thereduced symplectic form, ˜Ω , (4.4) remains the same in the Supergravity regime up to anormalization factor. Nevertheless, the potential g P (4.14) used to compute the integrablestructure J is not unique (4.18) — and is so in all regimes of the string coupling. As aresult, in order to assert that the complex structure, J , and the metric of the moduli spaceare invariant under the tuning of the string coupling, one must show that the effects of h in (4.18) on the metric on moduli space are dominated by those of the canonical potential, g P . We have shown that there exists a tension between the “canonical” distance accordingto the phase-space computation, ∆ phase , whose distance to the scaling limit is finite, and ∆ exponential . Now, there is only one correct normalization of the distance on the modulispace of bubbling solutions at strong string coupling: the one from the variations of theeffective Supergravity action. Thus, we have the following possibilities: (1) Neither ∆ phase nor ∆ exponential give the correct normalization. (2) Only the canonical ∆ phase gives the correct distance on moduli space, even in thestrong string coupling regime. If the Swampland Distance Hypothesis for metrics iscorrect, then it will not apply to our metrics. (3) The Swampland Distance Hypothesis applies to our solutions, and ∆ exponential givesthe correct normalization of the distance to the scaling limit. The use of canonical ∆ phase is not reliable in the strong coupling regime.If possibility (2) is correct, then our computation gives an explicit example of a metricon moduli space which blows up at all points on the boundary of moduli space, but whereall of the boundary points lie at finite distance in moduli space. In particular, the scalinglimit of bubbling solutions — at which global symmetries of the Black hole are restored —is within finite -distance reach from any other point in the moduli space. Besides, the massdecay of the tower of Kaluza-Klein modes does not behave like a decreasing exponentialwith respect to the moduli space distance ∆ phase between p ( λ ) and p ( λ ) .– 20 –lthough there is an infinite tower of states whose mass is decaying to zero, the decayis due to an universal redshift in a fixed-warp region of space-time, and thus does not intro-duce any singularities. Besides, the three-sphere at the bottom of the throat on which theKaluza-Klein modes live is macroscopic and is part and parcel of the five-dimensional Super-gravity solution. Therefore, the example we provide here differs from the usual Swamplandpicture, in which going at a corner in moduli space implies the appearance of singularities(for instance the shrinking of Calabi-Yau cycles in [32]), which involve the breakdown ofthe effective field theory.As a result, the Swampland conjectures would not forbid the scaling limit to be acces-sible from the bulk moduli space. However, in possibility (2), as argued in [21], quantummechanics, by virtue of the uncertainty principle, will imply the breakdown of Supergravityat the scaling limit.If possibility (3) is correct, then one cannot extend the “canonical” ∆ phase to the strongcoupling regime, because the integrable complex structure, J , is not invariant under theshift of g s , or because one has to take into account the effect of the additional potential h at weak coupling in the first place. However, the authors of [21] computed J in the openstring picture with the canonical potential g P , and used it at strong coupling regime. Inparticular, the probability distribution e −K of quantum wave functions in the phase spaceof bubbling solution they derive depends on the Kähler potential K , whose value will shiftif one considers the potential h in addition of the canonical potential g P .Therefore, if the canonical ∆ phase somehow gives the wrong normalization of distance,some of the conclusions in [21] could be revisited. The relative coordinate positions ofthe centers −→ ρ ij define solutions in Supergravity. In particular, near the scaling limit, thecoordinate positions −→ ρ ij need to be arbitrarily precise. However, because of the form ofthe symplectic form (4.5) computed from the open string sector, these coordinates do notcommute; hence, it is not possible to localize the positions −→ ρ ij with arbitrarily good pre-cision in coordinate space . In the closed string sector, the fully back-reacted solution doesnot require its Gibbons-Hawking centres to be localized with arbitrarily good precision inthe geometry in terms of proper distance , as the cap keeps its shape in the scaling limit.However, if one follows the logic of [21], the uncertainty about positions in coordinate spaceat weak string coupling is transported unto the phase space of three-centered solutions inSupergravity: one cannot localize any classical Supergravity bubbling solution with arbi-trary high precision in phase space. Instead, each classical Supergravity bubbling solutionis defined with some inherent quantum uncertainty and must be coarsed-grained with a“droplet” of solutions around it in a volume h m in the phase space. How far in the phasespace one should apply the coarse-graining depends on the metric/distance in moduli spacearound that particular classical Supergravity solution p .On the one hand, when the components of the moduli-space metric have small values,as one schematically moves away from p within the coarse-graining droplet region of p ,one can reach solutions that are very different from p . In particular, according to the Actually, this argument does not depend on assuming possibility (2). – 21 – anonical distance on moduli space that [21] used — where the metric behaves like /J R in the vicinity of the scaling limit — the scaling limit lies at finite distance to any otherpoint in the moduli space, so the coarse-graining of a solution p close to the scaling limitpoint contains solutions { p } which possess throats that have very different macroscopicphysical lengths. Therefore, Heisenberg’s uncertainty principle prevents classical solutionsfrom Supergravity to be a good description of black hole microstates.On the other hand, when the components of the moduli-space metric are large aroundthe solution p , the solutions { p } reached within a distance ∼ √ h look much more like p . In particular, with a metric that, in the vicinity of the scaling limit, scales like /J R as advocated by the Swampland Distance Hypothesis, the distance in moduli space tothe scaling limit is infinite . Wandering around p within a distance √ h along the angular-momentum coordinate J R (or equivalently the scaling-parameter coordinate λ ) cannot givesolutions { p } whose physical throat length is arbitrarily long. Instead, with ∆ exponential ,one deduces from (3.16) that the variation of the length of the throat in the set of solutions { p } will be of order ∆ L throat = α R S . (5.1)Whether quantum fluctuations in p ’s coarse-graining droplet are negligible or too largedepends on the value of the mass decay rate α of (3.16). If α (cid:28) , the geometries describedby Supergravity are reliable and well-defined. If α is of order one or bigger however —as in the context of Calabi-Yau compactifications [33–35] —, quantum fluctuations of thethroat length of each bubbling solution have macroscopic size, so describing those arbitrarilydeep geometries with Supergravity is still not reliable. Nonetheless, in both instances, thebreakdown (should it happen) of Supergravity here is milder than the one from the canonicalphase space distance of [21].In a nutshell, regardless of the value of α , coarse-graining droplets defined using ∆ exponential contain a much smaller range of solutions than those using the canonical ∆ phase .While with the canonical ∆ phase as the correct normalization of distance on the modulispace, the coarse-graining droplet of a deep-throat bubbling solution could contain thescaling limit point; the droplet derived from the ∆ exponential normalization only containssolutions with similar throat lengths. Therefore, with ∆ exponential , the breakdown of Super-gravity at the scaling limit is softened.If extending the Swampland Distance Hypothesis to our model is possible, then wehave drawn a parallel between (i) travelling within Planckian field range in field space toavoid the breakdown of the EFT in the context of the Swampland Distance Hypothesisand (ii) travelling a distance of √ h around a classical solution in phase space within theregion of its quantum fluctuations. While the Swampland Distance Hypothesis only consid-ers field ranges that are isotropic in moduli space, the fundamental quantity on the phasespace side is the coarse-grained volume h m , and the symplectic form defines an anisotropic“droplet” around a classical solution. In our example, the Swampland Distance Hypothesiscould be interpreted as a consequence of the symplectic form establishing Heisenberg’s un- This gives the logarithmic dependence of the distance on λ in (3.6) – 22 –ertainty principle. Whether or not this interpretation is legitimate is a question to explore.As for Black Hole physics, both possibilities (2) and (3) entail at least some breakdownof Supergravity as the description of arbitrarily-deep-throat bubbling solutions. As oneapproaches the scaling limit from a bubbling solution, the resulting geometry enters intoa new phase, whose precise description may require other tools, for instance perturbativeString Theory — that one uses in the microstate solutions of [36–40]. However, the extent ofthe breakdown, depending on which one of possibility (2) or (3) is correct, is very different.With possibility (2), the bubbling solution acquires a critical maximal throat lengthafter which supergravity completely breaks down. Thus, the new phase can possess a throatthat is not arbitrarily deep, like in the instances of [41–44] and [37–40].With possibility (3), when the Supergravity description of a geometry with a very longthroat (of length L throat ) becomes unreliable, one has to scramble the initial Supergravitysolution with solutions whose throat length is between L throat − α R S and L throat + α R S .Therefore, the new phase should still possess a throat which can be tuned to be arbitrarilydeep. Thus, any complete description of bubbling solutions up to the scaling limit shouldstill capture the presence of a cap and an arbitrarily deep throat. Finding such a descriptionbeyond Supergravity of those geometries would then be an interesting direction for thefuture. Acknowledgments I would like to thank Iosif Bena, Guillaume Bossard, Mariana Graña, Álvaro Herráez, EliasKiritsis, Severin Lüst, Daniel Mayerson, Ruben Monten, Cumrun Vafa and Nick Warnerfor useful discussions. This work was partially supported by the ERC Consolidator Grant772408-Stringlandscape and the ANR grant Black-dS-String ANR-16-CE31-0004-01. A Appendix: Boundedness of the moduli space of 3-centre solutionsfrom the phase space distance To show that the entire moduli space is bounded using the canonical ∆ phase , we will probethe asymptotic behaviour of the metric at all the different facets and vertices of the polytope:the vertex at (0 , , ( j + , j + ) and ( j + , − j + ) , and the facets at x = j + , x − y = 0 and x + y = 0 . The vertex at (0 , (the scaling limit). This instance has already been studied inthe subsection 4.2. The facets x − y = 0 and x + y = 0 . Given x F ∈ (0 , j + ) , we approach any point M F ( x F , ± x F ) on the facets by a straight horizontal line from a point M ( x , ± x F ) in thebulk. On the path, at the point M ( x, ± x F ) , G xx ∼ M → M F j + x F − x F j + − x F ) x F x − x F = 12( x − x F ) (A.1)– 23 –iving a square-root behaviour to the path distance ∆( M F , M ) ∼ M → M F (cid:112) x − x F ) . (A.2)Therefore the distance in moduli space is finite.Note that this computation does not take into account the scaling-limit point, as weused x F (cid:54) = 0 in our equations. The facet x = j + . Given y F ∈ ( − j + , j + ) , we approach any point M F ( j + , y F ) on thefacet by a straight horizontal line from a point M ( x , y F ) in the bulk. On the path, at thepoint M ( x, y F ) , G xx ∼ M → M F j + − x ) (A.3)gives a square root behaviour to the path distance ∆( M F , M ) ∼ M → M F (cid:112) j + − x ) . (A.4)Therefore the geodesic distance in moduli space is finite. The vertex at ( j + , j + ) . We approach this limit from the point M of coordinates ( j + − r cos α, j + − r sin α ) , with α ∈ ( π/ , π/ . 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On each facet of the polytope, one of the angles σ or φ becomes degenerate, that is to say the cycle they parameterize shrinks to zero size.Indeed, on the x − y = 0 and x + y = 0 facets, φ becomes degenerate, whereas on the x − j − = 0 and j + − x = 0 facets, σ becomes degenerate.We consider the instance j − = 0 where scaling solutions are admitted. The polytopeof the ( x, y ) moduli space is defined by 3 inequalities defining a triangle: l ( x, y ) = − x + j + ≥ l ( x, y ) = x − y ≥ l ( x, y ) = x + y ≥ . (4.12)The scaling limit lies at (0 , in the ( x, y ) -plane. The scaling limit loci lie at (0 , , σ, φ ) .As both φ and σ become degenerate at the scaling point (0 , , this shows the scaling limitboils down to one point, at the boundary of moduli space.The phase space is a convex toric manifold endowed with a closed symplectic form,and the vertices lie on integer coordinates (so our polytope is a 2-dimensional Delzantpolytope ). Consequently, the symplectic form is compatible with an integrable complexstructure, so that our phase space is a Kähler manifold [30]. Thus, we can use the followingresult for Kähler toric manifolds to determine the Kähler metric: Further details on Kähler toric manifolds and definition of Delzant polytopes can be found in AppendixB of [21] – 16 –ith the set of p inequalities characterizing the (two-dimensional) polytope P of theKähler toric manifold, l i ( x ) = c xi x + c yi y − λ i ≥ , (4.13)one can define the “canonical potential” of the polytope Pg P ( x ) = 12 p (cid:88) r =1 l r ( x ) log l r ( x ) (4.14)whose Hessian G = (cid:16) ∂ g P ∂x i ∂x j (cid:17) i,j in the “action” coordinates determines the complex Kählerstructure in the action-angle coordinates [30, 31] J = (cid:32) − G − G (cid:33) . (4.15)Thus the Riemannian Kähler metric in the action-angle coordinates is ˜Ω( · , J · ) = G ij d x i ⊗ d x j + ( G − ) ij d θ i ⊗ d θ j = (cid:32) G G − (cid:33) . (4.16)Consequently, we apply this result and deduce that the moduli space metric in thesymplectic coordinates is of the form (4.16) with G = 1 x − y (cid:32) xj + − x − y j + − x ) − y − y x (cid:33) , G − = 12 j + − x (cid:32) x ( j + − x ) 2 y ( j + − x )2 y ( j + − x ) 2 xj + − x − y (cid:33) . (4.17)We immediatly see that the G part of the Riemannian metric blows up on all the facetsof the triangle. Thus, the metric is defined only in the interior of the triangle. In thisregard, the scaling limit is not part of the bulk moduli space, but in its boundary.Actually, the symplectic form on the Kähler toric manifold does not define a unique“potential” determining J . In fact, J can be defined by any potential g of the form g = g P + h , (4.18)where h is a smooth function on the whole polytope P satisfying the requirements that [30]: (1) the Hessian G of g is positive definite on the interior P ◦ of P , and (2) the determinant of G is of the form det( G ) = γ ( x ) (cid:32) p (cid:89) r =1 l r ( x ) (cid:33) − , (4.19)with γ being a smooth and strictly positive function on the whole P .The Hessian G = (cid:16) ∂ g∂x i ∂x j (cid:17) i,j defines the compatible toric complex structure J and Rie-mannian Kähler metric ˜Ω( · , J · ) the same way as in using (4.15) and (4.16).Nevertheless, we will continue our computations with the metric defined by the “canon-ical potential” g P , as did [21]. – 17 – .3 The distance to the scaling limit The volume of the moduli space of the bosonic sector of BPS, classical configurations ofSupergravity (which is our phase space) naively counts, in units of the Planck constant h n − (where n is the number of GH centres), the number of quantum states in a particularsuperselection sector. Indeed, V phase = (cid:126) (cid:90) d x d y d σ d φ (cid:113) det( GG − = h j +2 (4.20)in the instance where j − = 0 . When j − (cid:54) = 0 , the number of states is j +2 − j − . This naivecounting, which does not include the fermionic degrees of freedom, matches neverthelesswith the result in [21]. From the symplectic-form derivation, the volume of the entire modulispace is finite.Imposing that the volume of the entire moduli space is finite has a consequence on theshape of the vicinity of scaling limit. Indeed, if the length to the scaling limit was infinite,then the area of its orthogonal directions should shrink at a rate such that the volumeremains finite. Then, for a parametrically small angular momentum, j , in the classicalregime, the density of quantum states at that given phase-space hypersurface (defining agiven throat length) would be parametrically small. Each superselection sector’s vicinityto the scaling limit would have the shape of a spike of infinite-length and finite volume.Now we wish to assess whether the geodesic distance in solution space between thescaling limit and any point in the bulk moduli space is infinite. The symplectic coordinatesof the solution space are bounded, and the metric is not singular in its bulk, so any twopoints in the bulk solution space are at finite distance of each other. The only place wherethe distance could be infinite is at the facets of the polytope.The coordinate values of ( σ, φ ) are chosen in [0 , π ] and their metric is bounded by j + from above. We will therefore only consider the metric from the ( x, y ) coordinates.Consider the straight path between the scaling limit −→ , and the point −→ r =( x , y ) = r (cos α, sin α ) , with α ∈ [ − π/ , π/ . The distance of this path is given by ∆ phase ( −→ , −→ r ) = (cid:90) r (cid:114) G ab d x a d r d x b d r d r . (4.21)In the vicinity of the scaling point, although the metric blows up ( G ab d x a d r d x b d r ∼ r → αr ), tocompute the distance we integrate its square root: ∆ phase ( −→ , −→ r ) ∼ r → √ cos α √ r = 2 √ x . (4.22)Then the distance to the scaling limit ∆ phase ( −→ , −→ r ) is finite ; therefore the geodesic dis-tance in moduli space is finite too. This contradicts the naive extension of the SwamplandDistance Conjecture.With similar reasoning, we can show that although the metric is blowing up on thefacets of polytope in the ( x, y ) -plane, the entire moduli space is bounded. The details are– 18 –n the Appendix.We wish now to relate the distance on moduli space with the masses S Kaluza-Kleinmodes. Recall that the angular momentum j is proportional to the scaling parameter λ : j = 12 (cid:115) − (cid:88) a