JHEP03(2021)168 - 16 D8 ]. When 1 Springer plane, but a,c − March 17, 2021 February 3, 2021 February 4, 2021 : December 3, 2020 O8 : : : plane with sources [ − O8 O8 Revised Published Accepted Received , and Gabriele Lo Monaco plane or to an Published for SISSA by https://doi.org/10.1007/JHEP03(2021)168 a + O8 plane. We comment on the possible ways of − O8 Mariana Graña [email protected] b , [email protected] , . 3 2010.05936 The Authors. c Superstring Vacua, D-, Flux compactifications

G. Bruno De Luca, We analyze the stability of four-dimensional de Sitter vacua constructed by , a [email protected] E-mail: [email protected] Orme des Merisiers, 91191 Gif-sur-YvetteStanford Institute Cedex, France for ,382 Stanford Via University, Pueblo Mall, Stanford,Department CA of Physics, 94305, U.S.A. StockholmAlbaNova, University, 10691 Stockholm, Sweden Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: the second one couldbranes correspond on to top. either We an to find the that D8 this branes latter movingdistinguishing solution away between has from these a the sources. tachyonic instability, corresponding Abstract: compactifying massive Type IIA supergravityembedded in in the presence Theory of the two first source has a clear interpretation as an Iosif Bena, Oh, wait, O8 de Sitter may be unstable! JHEP03(2021)168 1 ]. 12 – 3 ]. However, the 2 5 ]. -branes 8 16 – 13 – 1 – 6 7 4 ]. 21 – 11 backgrounds with D 18 1 solutions 13 1 ] can be realized in as orientifold planes. As a first 1 17 10 -branes D0 , section 2] for a recent review of orientifold singularities in . 22 Solutions with explicitly localized orientifold sources are on much firmer physical A way to bypass the complications inherent to the construction of de Sitter solutions See [ 3.1 The bosonic3.2 and fermionic action Approaching the singularities 1 ground, but constructing them istions more point-wise challenging on since the itversion. compactification requires , solving Furthermore, and the localized not field orientifold just equa- planes the source integrated singular (averaged) supergravity solutions no-go theorems [ approximation, one canthe consider internal the directions. orientifoldinvolution, Even planes this if, approximation to by may be definition, bemore justified orientifold complete smeared and planes solutions serve along sit [ as a (some at guide fixed of) to loci the construction of of some cosmological constant cannot be constructed [ using effective four-dimensional theoriesto and work to directly verify in theto ten validity the (or of ten-dimensional eleven) stress-energy these dimensions. tensor, conjectures Upon the is restricting negative-energy to sources classical required contributions to evade metastable de Sitter vacua.four-dimensional An efficient theory way to that constructembedding incorporates such of vacua String-Theory these is ingredients ingredients to innontrivial, use the [ and an full have effective String opened Theory aIn and rich their parallel debate interactions about to therein the theseconjectures are validity of top-down suggesting these investigations, that constructions there [ metastable also String-Theory exist compactifications bottom-up with arguments a and positive 1 Introduction One of the most challenging open problems in String Theory is determining whether it has A The fermionic action B Probe 3 Probing the CDLT 4 Discussion Contents 1 Introduction 2 Review of CDLT JHEP03(2021)168 z ), 1 (1.1) , with + O8 3 , The existence of symmetry acting ) gauge . and a source with 2 3 4 . ) Z 2 ). 2 M n 1 ds = 0 3 Sp(2 z U 2 e ] in String Theory (CDLT + 1 2 and CDLT 2 M 1 D8-branes on top of it, while ds 2 plane is at n U , where two O8 sources are located. 2 } + e 0 + , z O8 2 0 -dimensional Einstein spaces; at least one z { i d ]. , with negative charge and tension, is the standard ( 5 source, which has negative mass and charge. − = (positive) mass and charge, the singularity is , in this class of solutions the metric takes the – 2 – W ]. − 2 z O8 2 + − 17 e O8 + (as well as the ) depend on the coordinate 4 i 2 dS U are compact ds gauge group on a stack of i ) W and n M 2 e W = SO(2 plane is at the other fixed point (see figure 1 must have negative scalar curvature. There is a − 3 ], Córdova, Tomasiello and one of the authors have constructed de Sitter O8 2 CDLT M 22 supergravity solution with singularities coming from localized orientifold ds and any ]. We therefore look for instabilities in the open string sector, which can be revealed using 2 planes, whose negative tension violates the standard energy conditions that are 2 27 M ± ] and in [ – 1 23 O8 This is not the situation for the O8 As remarked above, near the orientifold sources the supergravity approximation breaks As we are going to review in section In [ The purpose of this paper is to analyze the solutions of [ Hence, Computations of the full KK spectrum of warped non-supersymmetric AdS compactificationsWe have use the standard convention where an This happens for example with anti-D3 branes [ 4 2 3 only been feasible recentlytheory and [ in veryprobe specific D-branes. situations by applying the techniques oforientifold giving exceptional rise field topositive an charge and tension equal to that of eight D8-branes, gives rise to an Very near this source the solutionreview is below, both the strongly region coupled and where strongly the curved. supergravity As approximation we is will not valid anymore can incompatible with a de Sitter solution [ down. However, atrelatively the mild: source approaching it, with discontinuity all in the their physical derivative. quantities Hence, remainnear this finite, normal singularity is with D8 of only branes. the a same finite type one encounters on the circle withMore precisely, the a two source fixedthe with loci charges the at of charges an ofthis an de Sitter solutionthese crucially depends on the presence of sources with the charges of where the warp factors parametrizing a circlebetween and and to see whether theyCDLT backgrounds, suffer looking from for any instabilities. intask, the Despite mostly full the because Kaluza-Klein relative spectrum of simplicity is the of a presence the prohibitive of a warpform factor with a non-trivial profile. present there exist other methods tobut assess for the de validity of Sitter the solutions supergravity these approximations, methods do notbackgrounds work. with localized O8 anddiscussed O8-O6 above. sources We respectively, are which going have to the refer properties to them as CDLT for the orientifold plane, thereproblems. could still be subleading divergences that may signal deeper sources has to befledged understood string-theory as solution only a away good from these approximation singular of loci. a When would-be corresponding is full- and even when the leading-order divergence of the supergravity fields match those expected JHEP03(2021)168 ]. ] of 33 30 deviates 5 plane with source with background solutions, in . Hence, our − 1 7 O8 O8 3.1 ]. -branes, that one -branes on top. If 35 16 D8 16 D8 6 source and this indicates that background can be thought of plane and the , Footnote 6]. Similar related models 1 1 + O8 plane with planes with O8 − source with positive charge and mass. − O8 source, and focus on the O8 O8 − -branes, destabilizing the backgrounds [ configuration that one obtains by T-dualizing ], except that the does not come out from D8 – 3 – − 32 O8 source with positive charge and mass corresponds , ]. Furthermore, in this realization, it appears rather - background can be thought as just the compactified + 31 1 38 – O8 O8 ], and it is unclear whether they can be trusted at the source, which we will discuss in section -branes have also been studied from an effective field theory point 36 p − 29 , 22 solutions [ , 4 1 plane, the subleading behavior of the supergravity fields -branes moving away from the source. This instability is similar to the brane-jet instability [ plane could emit a D8 brane, especially because there is no String- − D8 + O8 -brane from emerging from the O8 D8 -brane sources polarize into -planes and mobile D 6 p solution is unstable when embedded in String Theory. plane. If this possibility is realized the CDLT 1 ]. Some of these models display other forms of perturbative instabilities [ + 34 O8 ]. branes have the same charge and tension and all the bulk supergravity fields behave Unfortunately supergravity is unable to distinguish between the two interpretations of The second possibility is that The first possibility is that it corresponds to an The key question raised by our investigation is whether there is any physics that may We find that at the supergravity level the non-supersymmetric CDLT 28 At higher codimension these could be divergent,Such instabilities much exist like also it in happens related near models, anti-D3 as brane mentioned in singulari- [ -brane from this 5 6 ties [ with pairs of O of view [ 16D8 identically when approaching thedistinguishable two in singularities. the full TheseIt String two would Theory, configurations taking be into do interestingmanifest account appear to itself also in understand the the if open low-energy there string supergravity description. sector. is any way in which this difference could the Dabholkar-Park background [ unlikely that the Theory object in which it could decay, at least perturbatively. the orientifold source with positive charge. Both the simply to one ofthe the CDLT to an as the compactified version of the string-theory object corresponding to this source. this possibility is realized, theversion CDLT of the veryobtains familiar by configuration T-dualizing of Type-I two String Theory. If so, the instability we found corresponds also responsible for the decaywhich of case an D infinite class of non-supersymmetric AdS prevent such a To address this question it is important to point out that there are two possibilities for the positive charge and mass, where supergravity is more trustworthy. with mobile D8-branes suffersD8 from an instabilitynon-supersymmetric that AdS corresponds tobehind the a emission horizon of but a rather from a supergravity singularity. An analogous mechanism is one approaches the O8 from the flat-space onesupergravity [ level. Besides these conceptualnumerical issues, solution there near are the alsoinvestigation O8 technical will issues mostly with steer the away from the O8 be made parametrically small but never made to disappear completely. Furthermore, as JHEP03(2021)168 k (2.1) (2.2) ), on which solution. In 1 solutions. Here 1 1 is a coordinate , ) z 3 ] which only contain 2 M . 1 3 . The two fixed points ds z 3 M U 2 background: in particular, e vol ∼ − 1 ∧ + z 2 solution [ dz 2 M 1 plane. Evaluating the equations 2 U ds − 2 and four-form RR field strength 2 source is approached. We conclude − U 0 3 we review the CDLT 2 O8 U F e 2 O8 . All the functions depend only on the +3 + 0 2 W ). In particular a positive four-dimensional 6 z plane gives a constraint on the signs of the ∼ − d 2.1 ( e solution is: 0 + 4 z W – 4 – 1 f probes in the CDLT 2 2 ]. − = e plane and an 22 4 D8 + + with 4 ] 0 O8 2 dS z 2 ] describing the intenrnal topology of the CDLT ds ,F , 1 s singularities. W 4 [0 2 πl ± e solutions 2 ∈ O8 correspond to the two halves of the circle (see figure 1 = z 0 = 1 F 0 F , are two O8-sources. 0 CDLT 2 z ds = ) is an Einstein manifold of dimension two(three) and z 3 involution acting as the antipodal identification summarizing the results and suggesting some candidate ways to distinguish 2 M ( Z 4 2 and we discuss the behavior of a . A picture taken from [ M 3 . = 0 The integrated Bianchi identities require the total charge in the internal space to The metric of the most general CDLT The paper is organized as follows. In section z = 1 s vanish, and thus the twoof orientifold sources a have compact oppositethese charge. solution two Similarly, requires objects the are also existence thoseof the of motion an total on top tensioncurvatures of of to the the various vanish, source Einstein and factors with in an so ( O8 the charges of Here the two values of there is a at where parameterizing a circle circle coordinate. The solution has a Romans mass In this section we reviewO8 the sources. main properties More of details, theboundary CDLT including conditions the can full be set found of in equations [ of motion and an analysis of in section between the two possible 2 Review of CDLT section we provide the bosonicprobes and and we fermionic comment action on describing its the stability fluctuations when an of a stack of Figure 1 l JHEP03(2021)168 4 F ) as (2.6) (2.7) (2.3) (2.4) (2.5) ). 2.1 2.1 -function δ charge has − 8 ). Flux quan- , ]. 4 / 39 2.2 [ 5 z − Λ 2 r )–( / 1 ≈ 2.1 N : φ . N ) → 5 , which solves the Einstein | , section 4.4]. . 2 M r ]. This can be understood as | φ 22 e , e [ b ds 44 4 5 2 / – 0 + U r 1 , it is possible to generate other 2 -branes d − 40 a e ∼ r 8 N. . N r = c + √ = 1 4 , z = i 2 e and the four-form RR field strength → + 4 H z U 4 2 = 5 d F  φ F ( 3 ≡ 3 U 3 7 N Ne 2 M W M M -function source. In our more complicated 2 W ≡ × δ 4 × ds via − 1 3 1 → − e – 5 – S , e S U i e + R N Z U + 2 3 be related to an integer, µν 2 = 3 4 g − 2 M ) 4 2 2 ) s / 2 dS s F U ds 1 1 πl ds πl N + ) by rescaling (2 , e W (2 4 2 → W e 2.3 2 dS 2 backgrounds with D ] is related to e 1 = µν ds 2 1 is sourced by a / g  1 2 10 r plane) in flat space or in AdS [ , in [ 1 N ds c , becomes large, the curvature becomes parametrically small, and √ − = 0 N → plane with flat worldvolume. The discontinuity of the first derivative ≈ r − W − 2 e . This behavior is the same as near a source with negative D8 charge and of the harmonic function 2 O8 z 0 ds − 0 → z a ≡ r It is worth noticing that there exists a simple subclass obtained by tasking the two As discussed in the Introduction, near the orientifold sources the solution is generically There is a discrete rescaling that can be done on the solution ( The rescaling constant, One can construct analytic solutions valid everywhere as a formal expansion in 7 8 3 Probing the CDLT In this section we investigate the physics of probe D8-branes in the background ( internal Einstein spacessolutions with the to warp factors have arevanishes. the equal The same 5 (negative) dimensional internal Einstein space constant. is Einstein, and In the these metric of the form equations for an O8 of this functionsolution at the warping is notis a responsible simple for harmonic the function same asymptotic anymore, but behavior near the same where tension (such as an O8 the limit numerically, but analytic expansion around the limiting points aresingular. available. Near the sourceencounters when with approaching positive D8 chargenegative branes. the mass singularity and However, is since negative mild, charge, the the similar plane metric to with and the O8 the one dilaton behave as When the total flux, the solution is weakly coupled. This rescaling acts on the warp factors in the metric ( increasing the size of the circle. The solutions to the full Einstein equations were obtained Starting from a solutionsolutions with where quantized flux ( has negative curvature. tization requires that the integral of cosmological constant (dS) is allowed only if at least one of the internal Einstein spaces JHEP03(2021)168 - λ D8 (3.3) (3.4) (3.5) (3.6) (3.1) (3.2) -field to W . − B φ + e D8 z , | αβ ) ] D8 of a single z F F ∆ ( λ 0 f , D8 αβ e n z )Φ + z ) F F 3 ∂ ∧ 10 λ ≡ D8 f ( ) at second order in C = (including the warping [ ∆ 10 ) O + , describing the positions P n 1 4 3.1 ( can be expanded in Taylor D8 αβ + Φ is the brane tension while D8 f g Z z D8  , f 9 z ∆ 2

) ) Φ + z D8 s φ α ∂ τ . )Φ + ( D8 πl 2 D D8 ± 1 λ 10 with U τ D8 (2 Φ ) f vol 2 z 2

α z 9 − λ F c π/ ∂ 3 n  ) + D ( λ U 9 Φ , expanding ( d 3 + c αβ 4 D8 − g = 2 Tr F = e + ] + ∆ D8 n W g center-of-mass degrees of freedom decouple from the [ 2 W ∆ λ ≡ D8 D8 D8 2 ) 2 − z P τ n e ( − ∂ ( D8 – 6 – vol e ( (∆  U(1) 2 YM f 2 1 2 1 2 1 det ∧ = λ + + in absence of warping. Observe that worldvolume =0 − ∞  i z X , g 5 αβ d q ) φ Φ (0) M 10 − g D8 , and each polynomial term in the series corresponds to a f ⇒ 0 γ × 1 − z YM ( ξ e . ≡ 4 ] g 9 · q d , ξ n 10 -branes, the gauge group of the worldvolume theory becomes but the Abelian ≡ 9 ) α ) Z d A e k Φ D8 D8 [ i z Z D8 9 ,F U( k τ − φ − − Tr − α z 2 ( ∂ W = λ ) − n ≡ ) 2 ( D8 D8 U B f α τ worldvolume scalar field. Besides this scalar field, the worldvolume theory ( D8 +2 D S − =0 ∞ 3 i X U 3 = is the worldvolume field strength and we already assumed the NSNS is the worldvolume gauge coupling. Each function of is the metric on AdS U(1) e ) 2 ) = s and the action is F B z ≡ ( D8 non-Abelian sector. (0) ( πl ) g S ) It is convenient to use canonically normalized scalar fields, which we can obtain by f k The full gauge group is D8 k 9 = 2 ∆ and to use a normalized worldvolume metric: SU( factors) and defining: where we have defined and indices are raised and lowered with the worldvolume metric Neglecting for the momentgives: the contribution of λ series around a reference point, polynomial interaction of the worldvolumeof adjoint the scalar D8 field, branes: where be vanishing. In the previous expression, The bosonic actionperspective of of the the D8 worldvolumebrane D8-brane branes is theory has the interval ahas position DBI a gauge and field. aSU( For Wess-Zumino part. From the 3.1 The bosonic and fermionic action JHEP03(2021)168 ) D8 3.3 F (3.7) (3.8) (3.9) (3.13) (3.11) (3.12) (3.10) branes. , . ) 4 D8 2 Φ dS , 2 B 2 } vol ] m ∧ , the action ( , χ − 2 Φ M [Φ αβ . The parameters → F , remains finite and can vol YM . ), which acts as D8 4 e Φ γ αβ ) f i g YM 2.5 F , . g − W − ) −  ] )–( = φ 0 2 YM 1 χ e g F 4 , the boson mass and the force α 0 2.4 4 ) W F , γ ?F − YM 10 α φ + g f Φ+ − ∧ F ∧ F e z A . α 5 ∂ D8 + = 2 C D + [ / + ( 6 ) exactly equals the Yang-Mills coupling. 3 Φ YM D8 F backgrounds. When the four-form is non- χ solutions can be explored using α g N D8 Tr log ∆ 1 1 F D 3.11 ∆ z Z 1 2 m ]. 2 z ∂ → 2 – 7 – log ∆ ( ∂ λ z ( 49 + 1 ∂ – D8 Φ+ χ YM − D8  D8 ∆ τ g α 45 ∆ D8 , YM W − ∇ 2 D8 2 F g W α e 30 4 8 = e λτ W D i γ 4 ) − − = e { ∆ B ( D8 χ F − = = ( m δS 2 B D8 D8 γ m D8 F γ √ ξ − 9 d √ ξ 9 Z d 1 2 . The final result is: we also show that the fermionic action is not affected by the presence of the Z = A A Tr ) is the potential for the dual six-form − F is a nine-dimensional adjoint (Majorana) spinor with mass can be identified as the force acting on the stack of brane and as the mass of the ( D8 5 = S χ C B ) B m The orientifold singularities of the CDLT The fermionic action can be also computed. The details of the computation are given ( D8 S First, we observe that, even ifthe the unique probe coupling approaches entering a the stronglybe bosonic coupled (and and made curved) fermionic arbitrarily region, action, small using the rescaling symmetry ( 3.2 Approaching the singularities The dynamics of probeground. D-branes Quite can often, unveil such thenon-perturbative pathologies decay presence manifest channels of themselves [ as pathologies perturbative of instabilities a or given back- Observe that the Wess-Zumino couplingIn in appendix ( four-form flux. where where in appendix Let us stress thatare the finite on effective the Yang-Millsvanishing, whole one coupling should interval also in take into the account the CDLT topological term coming from the WZ term: and worldvolume bosonic field, and are given by: becomes: where the worldvolume indices are now raised and lowered using Performing the previous redefinitions and renaming for clarity JHEP03(2021)168 z ) − has 3.9 O8 . As 50 (3.14) D8 tot DBI F F F ). 40 2 30 ) where the four-form flux is ]. This discrepancy arises 20 2.7 44 0 – z 40 could naively seem not to pose a → 10 z − is the result of a huge cancellation O8 D8 should be vanishing. Evaluating ( has to be taken with a grain of salt, since for F tot DBI 0 F F -3 -3 -3 z in the effective action remain finite on the D8 2 F ) ∼ F z 10 x 3 10 x 2 10 x 1 ) to provide a meaningful effective description z 1 we find the force plotted in figure ( – 8 – − , m , the force drops and appears to reach a non-vanishing 0 B 2 z 3.11 10 z 52 ( m )–( orientifolds such as [ , ∼ 50 ∝ 0 − 3.7 z YM , the total force appears to have a finite value at the g O8 2 as 40 el, grav 0 background, F z 1 = ). As a consequence, it is quite questionable how much the numer- approaches z 30 can be trusted in the strongly curved region, where the cancellations z 2.6 D8 F 20 solution, a more severe conceptual problem must be taken into account. While worldvolume dynamics, the force 1 10 . The left figure (orange) gives the total force on a probe D8, computed as the sum of the D8 Furthermore, even if one could reach an infinite precision in evaluating the D8 action in From the first plot in figure The analysis is qualitatively the same also for the particular solution ( D8 10 F the CDLT the divergence of the dilatonsevere as problem one as approaches the the couplings vanishing. where we have used ( ical evaluation of are very large and the numerical noise is important. a strong dependence onentirely subleading reliable: components. first,between We the the thus WZ non-trivial believe and profile the thatwhich DBI of this (electric diverge and close result dilato-gravitational) to is forces acting not on the probe, singularity. This would besupersymmetric a qualitative solutions difference with withdespite the the common fact behavior that observed the in However, dilaton while and a the possible metric orientifold havecomponents source the can of right be the behavior identified at warp by that factors just singularity. and looking the at dilaton the close leading to the singularity, the force However, in order for theof actions ( the numerically for the CDLT negative value (dashed), but wefrom believe the that cancellation this of residualfigure, result the which is two gives most diverging theexplained likely contributions. a ratio in numerical between the This artifact the maincorrections can text, total to be the the force seen probe behavior and from action near the are the gravitational not right negligible attraction: (red) in the strongly-coupled region. Figure 2 DBI and WZ forces. As JHEP03(2021)168 2 − Z O8 (3.15) ]), one 44 – and an 40 plane, and a + + ]. It is possible . The asymptotic O8 O8 51 1 should be computed plane. We know that  source, which one can D8 + N F + O8 O8 ] and is dual to M-theory on 38 plane and an – ]. A similar phenomenon might − 36 52 , ) O8 4 ) provides a good approximation, but z ( . In the usual supergravity regime, the 3.1 φ O e + ≈ 3 source is the following: s + g – 9 – α z branes on top, or as an O8 = D8 . The singularity is of exactly the (much milder) type D8 F = 0 close to singularity, and thus the full force z source is tachyonic. − planes and 16 mobile D8 branes are consistent. The former is branes, the solution is tachyonic, and the corresponds D8 + F − O8 D8 O8 O8 plane with 16 − branes on top. O8 D8 ], the loop corrections to the probe potential end up canceling to all loops, plane with 16 50 -branes are actually unstable, and their worldvolume scalar parameterizing their − a constant. The probe D8 feels no force next to the D8 O8 planes, it is clear that the end point of this tachyonic instability will involve a α Our analysis indicates that if the orientifold source with positive charge corresponds As we explained in the Introduction, the key question is whether this probe D8 brane On the other hand, we have a much better control in the region close to the orientifold There is however a silver lining. In supersymmetric solutions (such as [ − plane with 16 to an to the D8 branesO8 moving away from this source. Given the fact that the solution has two D8 charge as an in flat un-warped space bothconfiguration a with configuration two with an the T-dual of the Typethe I’ Klein Dabholkar-Park configuration bottle. [ The latterthe is supergravity the regime T-dual it of appears vanilla impossible Type I to String distinguish Theory. between Furthermore, an in probe distance away from the tachyon signals an instability ofof the background the or solution. is just This an hinges interesting tangential on feature the interpretation of the orientifold source with positive with attribute to the fact thatthe in D8 branes flat are space immediately they repelled preserve if the an same infinitesimal . displacement occurs. However, Hence, the source with a positive charge, one encounters near D8 branes:quantities, it the manifests dilaton as is discontinuities finite of andbehavior the can of be derivatives of the made physical arbitrarily total small force if that this cancelation issame related as to that the of fieldmake the content the maximally of DBI+WZ supersymmetric the theory approximation underlyingeven [ of theory, in the which the D8 is strongly-coupling the brane regime. action describe the physics correctly using the full genus expansion that we do notcan have invoke at supersymmetry our protection disposal. proximation and well obtain beyond correct its physicssolution using validity [ regime. the DBI+WZ Furthermore, ap- giving for a anti-D3 flat branes potential in despite the the KS fact that supersymmetry is broken [ in a more generalhigher-genus expansion contributions in can be powers neglected of in and a ( strongly coupled regionrelevant. such corrections cannot Therefore, be we consideredthe cannot sub-leading situation and rely close become on to the the usual effective probe action. This is exactly whole interval, we have to remember that the DBI and WZ actions are only the first terms JHEP03(2021)168 + O8 stack 16D8 ]. If such D8 54 , configuration. de Sitter solu- solution if the 53 plane, which by 1 1 -branes sitting at + D8 plane. This solution k +16 D8 + − plane, it does not appear O8 2O8 + WZ action, we computed the planes, one of which has O8 + − and an O8 − O8 plane may be related by dualities to an + plane with a stuck D3-brane [ − O8 – 10 – source. Accessing this region, where the action configuration or in the − ± O8 O8 -branes are good probes close to the orientifold source with D8 solutions are sourced by two . We observed that this action, despite its finiteness everywhere, branes. These two objects have the same charge, and there exist 1 D8 solutions are sourced by an z -branes. Starting from the usual DBI with 16 D8 branes on top, and since these two possibilities cannot be planes and the other half is on the other. On the other hand, if the 1 D8 − − 16 D8 O8 solution gives interesting physics, but unfortunately does not help us solve O8 1 Or the CDLT does not have anynot mobile affect D8 it. brane, and hence the probe D8 brane instability does Either the CDLT branes on top. This solution is unstable. plane with The third is to understand whether an There are several ways one can try to break the tie between these possibilities. The Since the orientifold source with positive D8 charge can correspond either to an On the other hand, − • • O8 other known situations whereorientifold orientifold planes planes with the with same stuck totalSeiberg-type D-branes dualities charge. can are One be such related related example to to an is O3 other an O3 first is to understandcosmological if constant, in there either the isThe any second inconsistency is upon to turning understandtwo whether configurations. on other a A probe preliminary finite branes explorationin worldvolume can of the help the CDLT distinguish action between ofour the other conundrum. types We of present probe some branes of these results in appendix B plane or to an distinguished using just the supergravity solution, our work leaves open two possibilities. positive D8 charge, where thearbitrarily singularity small is by milder: increasing inis particular, the tachyonic, the units and dilaton of can thisorientifold four-form be would source flux. made signal with positive an The D8 instability position charge of of had the the the possibility de to Sitter emit CDLT D8 branes. becomes harder to trustby when vanishing the warp stack factorbecomes of near strongly D8 the coupled, branes approachesinteresting requires to the some construct singularity alternative in caused the effective future. description that would be 4 Discussion In this note we havetions investigated using the probe orientifold singularitiesbosonic of and the fermionic CDLT action describingsome the fluctuations reference of point a stack of on one of the orientifold source with positivepossible charge for corresponds this to planeD8 an to brane perturbatively probes emit we D8 found branes, does not and correspond hence to the an instability instability of if the the solution. symmetric configuration, most likely the flat-space solution where half the D8 branes are JHEP03(2021)168 . ) = . P ( α + o Γ θ (A.3) (A.1) (A.2) + θ i + (2) + θ i − E , while the ten- For a D8 brane, α (1) , , the emission of a Ξ ) ) P P − E ( α ( β -fixing such that β . Γ Γ κ 16 D8 ˘ 0 0 ∇ αβ ) F F ) A, B, . . . α , β , . . . P φ φ 1 ( α e e − -matrices is denoted by Γ ): first this action is already 5 8 1 8 Γ planes avoids the introduction M αβ − − ( ) − A.1 h 1 = = 1 − plane with − D8 α ˘ M Γ − (2) ( Ξ h + E θ O8 + − θ n ) F – 11 – and the ] + + g backgrounds. [ 1 P O8 ( while the pullback of the , det αβ configuration, which would be fascinating to figure out. φ , A − F A + configuration would correspond to a non-perturbative tachy- ∇ λ ∂ q . A 0 φ A + ) are: O8 Γ x - − Γ t β αβ − 1 2 ] iθ ∂ A.1 the ten-dimensional chirality matrix. The action is written in terms ξ e g 11 [ ] on the CDLT O8 = backgrounds, where the presence O6 = = +16 D8 +1 P p θ 2 (10) β 55 − supergravity is kept: in particular, we choose a d Γ = ˘ (1) ∇ E Θ Z O8 sources. αβ 2 λ − M , with 2 Dp τ )Θ = (10) ) F The worldvolume coordinates are labeled by the indices Its general form is: Given the importance of orientifold planes in the construction of explicit de Sitter ( Dp In our conventions, S 11 (1+Γ -fixed, in such a way that only one chiral component of the type IIA ten-dimensional 1 2 of the matrix dimensional space-time coordinates arethe labeled other quantities by in the ( indices Let us explain theκ meaning of the variousMajorana quantities spinor in ( The fermionic action ofaction D8 proposed in branes [ in flux backgrounds can be computed evaluating the inflationary cosmology collaboration) andGLM by is a supported by Simons the Investigator Swedish award. Research Council The grant work numberA of 2015-05333. The fermionic action Tomasiello for valuable discussions.the We draft. also The thank work Ivan ofdS-String Garozzo IB ANR-16-CE31-0004-01, for and the MG useful ERC work comments Grants“QBH was 772408 on partially Structure” “String supported and landscape” by the the andis John ANR 787320 supported grant Templeton in Black- Foundation part grant by 61149. the Simons The Foundation work Origins of of GBDL the Universe Initiative (modern richer class of CDLT of singular O8 Acknowledgments We would like to thank Emilian Dudaş, Miguel Montero, and Alessandro D8 brane from a onic instability of the backgrounds of String Theory,objects. it Hence is another crucial natural extension to of understand the in present detail work would the be physics to of investigate the these a duality relation exists between the JHEP03(2021)168 , = 12 λ ] + . · 8 ) , + 1 (A.9) (A.4) (A.5) (A.6) (A.7) (A.8) θ α (A.10) (A.11) (A.12) iσ  , to an A ( . . . γ z [ θ splitting i 0 ⊗ Γ γ + I −  + 0 α = θ , F ∂ 9 + 1 0 z φ z , obtaining the Γ e . , γ -matrices, that can λ →  )Γ γ ) 2 0 α − σ α λ F ( ∂ , ⊗ φ φ (Γ ) O I z z e  ] λ ∂ ∂ ( = + + + W W 8 θ O e e z D8 , γ 1 2 (10) Γ + ; consistency with T-duality − , 3 ] , Γ · α σ ) z + α , , α [Φ . The final result is: α ∇ Γ ⊗ 1 ] , log ∆ z · E σ (0) α W A 3 z , ( ∂ [ (7) i = = Γ − z ∂ ∇ σ α α γ ⊗ ∂ [Φ ie Γ Γ α I W z ⊗ = (0) D8 = , e W ˘ i ∇ Γ W I − e Γ  = γ e z ! ]  0 1 1 2 Γ + = z -matrices with Minkowskian signature. = θ α

1 2 γ Γ 9 + -matrices can be chosen such that α Γ z as spin connection. At the lowest order in z ⊗ Γ γ Γ + α + , , β θ χ Ω ω α α α 2 α -matrices that make manifest the – 12 – ... ∇ σ A Ω Γ = [ ∇ α 0 , i 1 2 α ⊗ Γ , the non-trivial components of the spin connection β + Γ  − α α A θ A +  γ = Γ ω + E , γ + , θ 0 ) θ β ) = 1 = λ λ Γ D8 Γ ( ( α γ D8 β ∆ O Γ α β O ∆ α Ω + (0) + ) for a single D8 brane to the lowest order in ω (0) g 1 4 g α αβ and: q g A.1 q + ξ z ξ 9 ∂ 9 α = = Γ d d ∂ ) = Z P α β Z z ( α = 2 2 Γ ∇ M λ α λ ], the non-Abelian action can be obtained promoting the fermion, ∇ 2 D8 2 D8 is the covariant derivative with τ is nothing but the worldvolume spin connection in the absence of warping. As 56 τ are 9-dimensional purely-imaginary β (0) = = α α ) γ ω ∇ ) F F , introducing the additional coupling of the form ( D8 ( D8 ) A way to construct them, for instance, is to start from 7-dimensional Euclidean S S λ are: ; in this basis. Hence, a chiral Majorana spinor can be decomposed as 12 ( I i be chosen to be purely imaginary. Then, one can construct With these conventions, the 10d chirality matrix can be written as: where we used the fact that the 9-dimensional We can now introduce aof smart the choice ten-dimensional of space: where requires an additional companion term of the form Following [ adjoint (Majorana) fermion and performingO the usual covariantization We can now evaluate ( following Abelian action: where we can use the following approximations: Ω where a consequence, If we denote the 10d vielbein with JHEP03(2021)168 (B.1) . The (A.17) (A.13) (A.14) (A.15) (A.16) 2 / ) W . However, 8 , ] 00 − g g branes, whose ] . φ [ ( 3 , − e } ] P ] M 4 D0 √ F1 strings via the D8 / / φ τ F , χ 1 det vol 0 − √ z n e − − [Φ λ Γ R α q . It is easy to see that: 0 − YM Γ 2 ξ 4 D λ α d 2 τ i g t / . F Γ − d − 3 α 3 −  γ λ ] 0 M [Γ Z backgrounds the four-form flux χ z F +3 / 1 χ χ , α Γ 1 vol F W by a factor W z τ + 6 − = , γ Γ + θ 0 φ χ − α θ α n e θ e φ 3 8 A 4 e z [Γ M − + f D8 branes creates ] z Γ g + [ Γ 0 / [ θ 00 vol D8 ≡ n + . g χ branes in our solution is: P + θ λ 4 − F θ e 0 4 f e f det √ χ , m log ∆ φ – 13 – − α z − = = 2 + ∂ q χ γ  t e χ φ + + solution. The most interesting are i α d planes as well), these D0 branes come with F1 strings θ θ − 1 ] ] W 3 4 ∇ 2 2 Z = ξ e α / e M F O8 9 0 θ d D / i γ − τ vol ≡ α { α Z . As we did for the bosonic action, it is best to normalize Γ − − F 2 χ Γ θ 0 4 3 . We recall that in the CDLT λ = m γ / t F M 1 D8 ]. 2 D8 α F γ / iχ τ 58 S mnpq vol [Γ √ 0 z Γ ξ [3 = = 9 F Γ + d -matrices. This establishes that the non-trivial four-form flux does not 4 + χ θ ] but this does not indicate which D8 branes the F1 strings terminate + Γ ), and define for simplicity θ F 4 -branes mnpq Z e 0 f 57 F D 1 2 δS 2.2 1 4! S = D0 = = ) = F is a 9d Majorana spinor. Using such decomposition of the spinors, it is straight- 0 ( 4 0 D8 F D / S χ F S Hence, the complete action describing D no strings attached, but passingHanany-Witten them effect through [ action in this background naivelyin the contains presence only of a D8 DBIattached. branes term: (and One way toRomans see mass this [ ison. from imposing A tadpole simpler cancelation way in is the to presence realize of that a in a region with no Romans mass, D0 branes have affect our computation at the lowest order in B Probe Besides the action of probe D8have interesting branes, physics it in is the instructive CDLT to see whether other probe D-branes However, this combination vanishes for a ten-dimensionalproperties chiral of spinor, given the the symmetry where has the form ( Finally, let us comment aboutfact, a the possible fermionic action contribution in coming principle from contains the a four-form Lagrangian flux. term In of the form: canonically normalized action is then: where the fermion mass is defined as: forward to show that the following relations hold: where, as before, the fermions and in particular to rescale the spinor where JHEP03(2021)168 (B.2) Phys. on the , 1 − -dimensional is such that ]. = 10 γ zz g SPIRE 00 IN , g ]. ][  ) z ] = D0 g branes are stable on both z SPIRE [ IN − P D0 0 . The surface ][ grows. z 0 ( 1.2 ) strings attached. We plot this F 0 N 1 s F ) can be thought as the interval- as πl ]. + De Sitter vacua in string theory 2 0 B.2 ) z arXiv:1812.04147 D0 ≡ [ z and that det ( SPIRE Classical de Sitter solutions of 0 φ ) n IN 0.8 − s ) ][ πl arXiv:0912.3519 D0 z – 14 – [ ( (2 attached strings ending on the orientifold plane branes with (F / W 0 0 On the existence of meta-stable vacua in 0 e . It can be seen numerically that the unstable equilibrium n D  τ (2019) 091601 brane and a D8 brane (or an O8 plane) sourcing the D0 dt V 0.4 = 0 Z (2010) 087 122 1 ), which permits any use, distribution and reproduction in 0 F the position of the D0 brane along the interval: the tadpole τ D and we defined 09 hep-th/0301240 τ and that it gets closer to [ t D0 − z felt by the D = 6= 0 0 D0 JHEP 4 D V 0 , F V CC-BY 4.0 F D0 . It is interesting to observe that the S 3 This article is distributed under the terms of the Creative Commons . In a configuration of minimal energy, it is natural to expect that the Phys. Rev. Lett. 0 z , (2003) 046005 = The D0-brane potential, 68 ) can be written as: z − . background. The integrand in equation ( B.1 1 supergravity Rev. D Klebanov-Strassler S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, I. Bena, M. Graña and N. Halmagyi, C. Córdova, G.B. De Luca and A. Tomasiello, [2] [3] [1] any medium, provided the original author(s) and source are credited. References dependent potential potential in figure orientifold planes, and also have an unstable equilibriumOpen point. Access. Attribution License ( where we used theCDLT fact that Romans mass. Let uscancellation call requires the presencelocated of at attached string extends onlyaction along ( the interval transverse to the O8 planes, so that the point only exists when where we already assumed that thethe NS potential temporal is vanishing in direction thethe by background, we string denoted extends between the D Figure 3 JHEP03(2021)168 , 02 3 Class. Class. Phys. , 11 ]. , , , , ] JHEP orientifold (2019) , Int. J. Mod. 4 Fortsch. , SPIRE JHEP , 67 , IN ]. ][ Phys. Rev. Lett. ]. , (2018) 570 SPIRE IN SPIRE ][ arXiv:1106.6165 arXiv:2009.03914 785 [ IN ]. , Fortsch. Phys. ][ ]. , ]. The backreaction of anti-D hep-th/0007018 ]. ]. SPIRE [ IN On supersymmetric AdS ]. SPIRE ]. SPIRE (2013) 060 IN ][ IN SPIRE ][ SPIRE ]. Phys. Lett. B Kähler moduli stabilization from ten ][ 06 IN De Sitter space and the swampland IN Micromanaging de Sitter holography De Sitter vacua from ten dimensions , SPIRE Giant tachyons in the landscape ]. 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