Resolving spacetime singularities in flux compactifications & KKLT
RResolving spacetime singularities in fluxcompactifications & KKLT
Federico Carta and Jakob Moritz Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK Department of Physics, Cornell University, Ithaca, NY 14853, USA
Abstract
In flux compactifications of type IIB string theory with D3 and seven-branes, thenegative induced D3 charge localized on seven-branes leads to an apparently patho-logical profile of the metric sufficiently close to the source. With the volume modulusstabilized in a KKLT de Sitter vacuum this pathological region takes over a signif-icant part of the entire compactification, threatening to spoil the KKLT effectivefield theory. In this paper we employ the Seiberg-Witten solution of pure SU ( N )super Yang-Mills theory to argue that wrapped seven-branes can be thought of asbound states of more microscopic exotic branes. We argue that the low-energyworldvolume dynamics of a stack of n such exotic branes is given by the ( A , A n − )Argyres-Douglas theory. Moreover, the splitting of the perturbative (in α (cid:48) ) seven-brane into its constituent branes at the non-perturbative level resolves the appar-ently pathological region close to the seven-brane and replaces it with a region of O (1) Einstein frame volume. While this region generically takes up an O (1) fractionof the compactification in a KKLT de Sitter vacuum we argue that a small flux su-perpotential dynamically ensures that the 4d effective field theory of KKLT remainsvalid nevertheless. a r X i v : . [ h e p - t h ] F e b ontents Introduction
A promising and conceptually simple avenue towards understanding the nature of darkenergy in quantum gravity is the pursuit of explicit de Sitter solutions in string theory[1–15]. It was argued long ago by Kachru, Kallosh, Linde and Trivedi (KKLT) that suchsolutions should exist in the landscape of type IIB Calabi-Yau (CY) orientifolds withthreeform fluxes, D3 and seven-branes [3]. Specifically, KKLT showed that if the classicalflux superpotential W can be tuned to very small values, volume moduli are genericallystabilized at large values, and the resulting vacuum is AdS . If moreover, the very samefluxes generate strongly warped and weakly curved Klebanov-Strassler throats [16, 17], ananti-D3 brane at the bottom of the throat can lift the vacuum energy to small positivevalues, provided that W can be finely tuned against the throat hierarchy.Various aspects of the KKLT proposal have been scrutinized extensively in the past:some of the earliest work has focused on establishing that all K¨ahler moduli can in prin-ciple be stabilized non-perturbatively [18–20]. Only more recently, it was shown thatflux vacua with exponentially small flux superpotential can be found systematically [21],and furthermore that the required solutions hosting both long warped throats and smallflux superpotentials simultaneously can likewise be found [22, 23]. Another part of therecent literature has focused on the consistency and meta-stability of the warped anti-D3-brane [24–33], and its surrounding throat region [34–37]. In particular, [31] explains nicelyhow the apparently pathological near-brane behavior of supergravity fields is resolved bybrane polarization. In yet another line of work the 10d uplift of gaugino condensation effectsin the 4d EFT was studied [38–49]. Specifically, after proper inclusion of quartic gauginoterms and contributions to the stress-tensor from the (non-local) volume dependence ofthe gaugino bi-linear, one arrives at a consistent picture [44–47, 49] (see however [48]).In contrast, the aim of this paper is to resolve (literally) a parametric control problemas posed in [47, 50] (see also [51]) which we summarize in Section 3. Briefly stated, theproblem is that an efficient competition between stabilizing potential and a meta-stableanti-D3 uplift forces the overall volume modulus to take values of the same order as theoverall D3-charge hosted in the warped throat. In such a regime, the backreaction radii ofboth the sources of positive and negative D3 charges turn out to be of the same order asthe typical length scale of the compact CY. Since the square of the conformal factor of thecompact threefold is driven to unbounded negative values near localized sources of negativeD3 charge with real co-dimension larger than one (see Section 2) this implies that at theKKLT minimum an entire O (1) fraction of the compact threefold can no longer be described2y a semi-classical solution. This is to be contrasted with the large volume regime whereonly the regions very close to localized sources of negative D3 charge cannot be describedby the semi-classical solution which is of course entirely expected, and irrelevant for thePhysics at length-scales larger than the string scale.The aim of this paper is to understand in detail how string theory resolves this semi-classical singularity via effects that are non-perturbative in the α (cid:48) expansion. In order tomake progress, we will make the important simplifying assumption that all negative D3charge is hosted on seven-branes wrapping K3 surfaces. While such orientifold vacua arenot generic, many exist (see e.g. [52]). Then, since the low-energy world-volume dynamicsof the seven-branes accidentally respects N = 2 SUSY in four dimensions this will allowus to understand the relevant non-perturbative bulk effects in terms of instanton effects inthe gauge theory. These in turn are known by virtue of the Seiberg-Witten solution of thelow-energy limit of N = 2 pure Yang-Mills theory [53–57]. Using this, we will argue forthe following proposals:(a) The wrapped seven-branes hosting negative D3 charge must properly be thought of asbound states of more elementary co-dimension two exotic branes of d N = (2 , SUGRA separated by a distance that is non-perturbatively small in the α (cid:48) -expansion. (b) As the overall volume modulus takes values of order of the D3 tadpole, the exoticbranes forming the perturbative defects get split over a distance comparable to the sizeof the compact threefold. This splitting of branes stops the running of the conformalfactor before it turns negative, thus resolving the apparent pathology. (c)
The ’inside region’ left behind by the spreading of exotic defects has O (1) Einsteinframe curvature.
More precisely, we will resolve the singularity of the conformal factor much in the same waythat F-theory [58–60] resolves the apparent singularity of the axio-dilaton in the vicinity ofan O7 plane (where formally 1 /g s <
0) by splitting the perturbative O7 plane into a pairof mutually non-local defects [61, 62]. This well-known effect can be seen either directly using F-theory [61] or alternatively by probing the background with a D3 brane realizingthe SU (2) Yang-Mills theory as in [62]. Similarly, we can realize the SU ( N ) Yang-Millstheory by wrapping N D7 branes on K3, thus giving rise to − N units of induced D3charge [63]. We argue that the non-perturbative gauge theory effects of the Yang-Mills Note that this is also morally similar to the resolution of the anti-D3 brane singularity at the bottomof a warped throat by polarization into a spherical NS5-brane as discussed in [31]. By comparing withthe known singularities in the moduli space of SU ( N ) Yang-Mills theory [55–57, 64] weconclude that a stack of n such exotic branes hosts the Argyres-Douglas (AD) theory oftype ( A , A n − ). Thus, if we assume that a significant fraction of the D3 tadpole resideson a stack of seven-branes wrapping a K3 surface, we can establish our claim (a).Using (a) it follows immediately that the exotic branes are separated by a distance oforder the strong coupling scale Λ (in appropriate units). Thus, as the UV gauge coupling(the K¨ahler modulus) is driven towards stronger coupling, the exotic branes spread out-wards in their transverse space. Since the UV coupling is naturally defined at distances oforder the size of the compact threefold our claim (b) follows. Furthermore, in this regimethe strong coupling scale Λ is of the same order as R-symmetry breaking spurions Λ /R in-duced by the compact bulk threefold. Therefore, the above regime is naturally associatedwith poor control over the low energy expansion in Λ / Λ /R . Furthermore, we will use the Seiberg-Witten solution of SU ( N ) Yang-Mills theory toargue that the ’inside region’ is indeed strongly curved, i.e. (c). More precisely, we will showthat the origin of the Coulomb branch of the gauge theory approaches a wall of marginalstability in the large N limit. We will conclude from this that there is no (parametrically)preferred string with small tension that can be stretched in the interior region and thusEinstein frame volumes must be of O (1) in the large N limit.As the resolution of semi-classical singularities can be understood entirely in terms ofgauge theory effects known to be present in the 4d EFT we view this as strong evidencethat the EFT employed by KKLT remains well-controlled even though an O (1) fraction ofthe bulk CY has O (1) Einstein frame curvature. More precisely, in Sections 5.2 and 6 wewill argue that(d) A small classical flux superpotential ensures control over the euclidean D3 (ED3)instanton expansion even if an O (1) fraction of the ED3 worldvolume probes thestrongly curved region. (e) A small classical flux superpotential also ensures that the non-perturbative spreadingof defects that ensures the resolution of singularities does not extend far enough tosignificantly affect the local warped throat region. As in [63] the stretched D5 string realizes a magnetic monopole. Note that sub-leading corrections in Λ / Λ /R are expected to respect only four supercharges.
4e argue for (d) by noting that if the leading ED3 instanton wrapped on some divisor D has disappeared entirely into the strongly curved inside region its euclidean action must beof O (1). Since a small W ensures that its action is of order log( | W | − ) (cid:29) D remains in the weakly curved outside region. Furthermore, we argue that ifa significant fraction of positive D3 charge is hosted in a warped throat, the final stages ofthe small volume limit correspond to the exotic branes ’walking’ down the warped throatleading to recombination of D3 charge (see Section 5.2). Thus, if D does not reach all theway to the bottom of the throat, a large ED3 action dynamically ensures that the warpedthroat is not significantly affected by the strongly coupled effects in the bulk, i.e. (e).Since the KKLT proposal only requires 1) tunable control over the instanton expansionand 2) a tunable uplift we conclude that the presence of even a large region of O (1)curvature in the bulk should not jeopardize the EFT employed for moduli stablization anduplift. Nevertheless, our analysis suggests (as in [47, 50]) that a considerable fraction ofthe bulk CY has O (1) Einstein frame curvature in KKLT de Sitter vacua. This should betaken into account in phenomenological model building based on the KKLT proposal.This paper is organized as follows: We start by reviewing basics about the overallvolume modulus and the local behavior of the conformal factor near sources of D3 charge(Section 2). In Section 3 we review the potential control problem of KKLT discussedin [47, 50], related to fitting throats into compact bulk CYs and avoiding large singularregions in the bulk. In Section 4 we explain how non-perturbative effects cure the apparentsingularities via the splitting of perturbative D7 branes into elementary monodromy defectsof 6 d N = (2 ,
0) SUGRA bound together by a potential. In Section 5 we comment on thenature of the small volume limit in a global model 5.1, and relate its final stages witha process of recombination of D3 charges 5.1, 5.2. Finally, in Section 6 we argue thatKKLT de Sitter vacua are generically safe from significant alterations of the EFT due tothe appearance of regions of strong bulk curvature. We conclude with Section 7.
As a starting point we consider the (classical) GKP solutions [17]: tree level vacua oftype IIB string theory on an O3/O7 orientifold B of a Calabi-Yau (CY) threefold X withD7 branes, D3 branes and threeform fluxes, or more generally F-theory on an elliptically5bered CY fourfold with base B and four-form fluxes. Let { D i } be a basis of H +4 ( X, Z ) = H +2 , ( X, Z ), i.e. the orientifold-even four-cycles. Then, at the classical level, the h , ( X )K¨ahler moduli T i remain exactly flat directions, while the h , − complex structure moduli z a and the dilaton τ are frozen by the fluxes. Below the mass-scale of the K¨ahler modulithe classical (in the α (cid:48) expansion) effective N = 1 superpotential is a constant W , W cl ( T i ) = W := (cid:114) π (cid:28)(cid:90) X ( F − τ H ) ∧ Ω( z ) (cid:29) + O ( e πi (cid:104) τ (cid:105) ) , (2.1)where the first term is the Gukov-Vafa-Witten (GVW) flux superpotential [65], and the O ( e πiτ ) D ( −
1) instanton corrections can be computed in F-theory.As the discrete axionic shift symmetries T i → T i + ic i , c i ∈ Z , cannot be broken byperturbative effects, the superpotential receives only non-perturbative corrections in theK¨ahler moduli [66] W ( T i ) = W cl ( T i ) + W np ( T i ) ≡ W + (cid:88) (cid:126)n ∈ Γ A (cid:126)n e − π (cid:80) i n i T i , (2.2)where the n i are rational numbers, and the coefficients A (cid:126)n are in general hard to compute.The 10d metric takes the form [17] ds = e A ( y ) t dx + e − A ( y ) g int ij dy i dy j , (2.3)with warp factor e A ( y ) varying over the 6d internal Calabi-Yau orientifold B (or moregenerally an F-theory base) parameterized by local coordinates y i . The metric g int ij is aRicci flat metric (or more generally an F-theory solution) normalized to unit volume, (cid:90) B d y (cid:112) g int = 1 , (2.4)and dx denotes the 4d Minkowski line element. The overall volume modulus t is definedas [67] t := (cid:90) B d y (cid:112) g int e − A ( y ) . (2.5)The physical metric of B differs from g int ij by the crucial factor e − A which by virtue of the We work in 10d Einstein frame, and set the 10d reduced Planck mass M P, d = 4 π . This amountsto setting l s ≡ (2 π ) α (cid:48) = 1. Moreover, we use our freedom to Weyl-rescale the 4d components of the 10dmetric to also set the 4d reduced Planck mass to M P, d = 4 π . e − A asthe conformal factor .In terms of the 10d metric and four form potential C the K¨ahler moduli are defined as T i := (cid:90) D i e − A J ∧ J − iC , (2.6)where J is the K¨ahler form of the unit-volume orientifold or F-theory base B .A GKP solution [17] is given by a choice of flux quanta [ F ] , [ H ] ∈ H ( X, Z ) stabilizingthe complex structure moduli of B and the axio-dilaton τ := C + ie − φ at values such that ∗ int G = iG , G := F − τ H . (2.7)The self-dual five form field strength F becomes F = (1 + ∗ ) de A ∧ d x , (2.8)and the conformal factor is a solution to the electro-static problem − ∇ e − A = ρ D , (2.9)with D3 charge density ρ D with overall net zero charge, and ∇ is the Laplacian associatedwith the metric g int ij . The fluxes carry positive D3 charge density F ∧ H , mobile D3 branesand O3 planes have localized charge +1 respectively − , while seven-branes wrapped on asurface S carry negative D3-charge smeared across their world-volume proportional to theEuler characteristic χ ( S ).Eq. (2.9) clearly has a modulus e − A ( y ) → e − A ( y ) + δt corresponding to shifts of theoverall volume modulus t → t + δt in eq. (2.5). The reason for this terminology is the factthat in the large t limit we get e − A ( y ) → t = const (except near singularities) so the 10dmetric approaches the form ds t →∞ = t − dx + t g int ij dy i dy j . (2.10)Therefore, in this limit the physical volume of B becomes Vol( B ) → t (cid:29)
1. In the vicinityof singular sources the conformal factor behaves as it would in flat space: near a stack of N D3 branes we get e − A ( r ) = N π r + c , (2.11)7here r is a radial distance measuring radial distance from the source in the unit volumemetric g int ij , and c is an integration constant. Shifts c → c + δc correspond to shifts t → t + δc ,and at distances much smaller than the compactification scale the integration constant c is meaningless because it can be absorbed by a redefinition of the radial coordinate: (cid:114) N π r + c ( dr + r d Ω S ) ≡ (cid:114) N π ˜ r + 1( d ˜ r + ˜ r d Ω S ) , (2.12)with ˜ r := c r .Near a (real) co-dimension two source wrapped on a surface S the conformal factorscales as e − A ( r ) = − ρ D , ⊥ π log( r/r ∗ ) , (2.13)where ρ D , ⊥ is the (assumed constant) D3-charge density along the surface S , and with dynamically generated radial scale r ∗ . Rescalings r ∗ → r ∗ e πρD , ⊥ δt correspond to shifts in thevolume modulus, t → t + δt . Thus, for negative D3 charge density, and for large volume t ,one formally gets a negative conformal factor below an exponentially small radial distance r ≤ r ∗ ∼ e − πt | ρD , ⊥| = e − π | Q | Vol( S ) , (2.14)where Q is the integrated D3 charge on the surface S . This formula of course alreadysuggests that the negative conformal factor regime is cured by non-perturbative effects inthe α (cid:48) -expansion, which will indeed turn out to be true.Now, as in (the appendix of) [47], let us consider the case where all positive D3 chargeis localized at real co-dimension six (a stack of N D3 branes), and all negative D3 charge islocalized on a complex surface S at real co-dimension two (a seven-brane stack with inducedD3 charge − N ). As usual, by inspecting the D3 stack one sees that its backreaction turnsit into an AdS × S throat of radius N / . This throat can be glued into a weakly curvedbulk only when the physical bulk-volume is large enough,Vol( B ) (cid:38) N . (2.15)This geometrical argument can be verified at the technical level via inspection of thesolution near the seven-branes: in the regime Vol( S ) (cid:29) | Q | = N , where the throat isparametrically smaller than the bulk, the conformal factor is negative only exponentiallyclose to the brane stack. In contrast, in the critical regime Vol( S ) ∼ N , where the throat8ts into the bulk only marginally, the exponential suppression gets lost and the vanishinglocus of the conformal factor has moved into generic position [47, 50]. This latter point hasbeen emphasized and discussed in detail particularly in [50]. To our knowledge the regime where the conformal factor is formally negative in an O (1) fraction of the CY has remained largely unstudied so it has been standard practiceto impose eq. (2.15) with some reasonable control factor [47, 51, 68]. Part of the aim ofthis paper is to relax this requirement and see where it takes us. In the KKLT scheme of moduli stabilization one stabilizes the K¨ahler modulus T (weassume h , ( X ) = 1) using a small classical flux superpotential W (cid:28) α (cid:48) expansion ∼ e − π c Re( T ) , where c ∈ N is the dual-Coxeter number ofa confining seven-brane gauge theory or c = 1 for a euclidean D3 brane (ED3) instantonwrapping the generator of H ( X, Z ) [3]. This leads to moduli stabilization at large volumeRe( T ) ≈ c π log( | W | − ) , (3.1)if the classical flux superpotential is small, | W | (cid:28)
1, and the scalar potential is of order V bulk ∼ − e − π Re( T ) /c ∼ −| W | < . (3.2)For simplicity, let us set c = 1, so we consider moduli stabilization from euclidean D3brane instantons. The negative scalar potential from the bulk moduli stabilization mustbe compensated for by a small ‘uplifting’ potential energy from an anti-D3 brane at thebottom of a warped throat. Its contribution to the scalar potential is gravitationallyredshifted by a factor [3, 16, 17] a := e A | IR e A | UV ∼ e − π KgsM = e − π Q throat gsM (cid:28) , (3.3)where K and M are flux quanta, g s is the string coupling, and Q throat := KM is the D3charge of the fluxes generating the throat. By appropriate fine-tuning of the IR warp factor It is not essential to the argument what the general charge configuration is as long as we keep negativeand positive D3 charge seperated from each other, to avoid recombination or significant screening of branecharges. A detailed discussion of this can be found in [50]. ≈ | W | ∼ e − π Re( T ) , i.e. g s M ∼ Q throat Re( T ) , (3.4)one can find vacua with small positive cosmological constant [3]. Furthermore, for thethroat to fit into the bulk CY the r.h. side of (3.4) should be smaller than unity, so weobtain a constraint Re( T ) (cid:38) Q throat → g s M (cid:46) . (3.5)However, for a confident prediction of the meta-stability of the anti-D3 brane [69] (seealso [35, 36]) one would like to stay at weak string coupling g s (cid:28) R IR ∼ g s M (cid:29)
1, so g s M = ( g s M ) g s (cid:29) . (3.6)The (parametric) tension between eq. (3.6) and eq. (3.5) suggests a difficulty to ’fit’an appropriately red-shifted and weakly curved throat into a likewise weakly curved bulkgeometry with moduli stabilization in place, as was noted in [47]. Upon relaxing thecondition (3.5) one has to deal with the fact that the classical vanishing locus of theconformal factor e − A is no longer exponentially close to the position of the seven-branes.In fact, the region where formally e − A < O (1) fraction of theCY [47, 50]. In this section we would like to explain our proposal that seven-branes wrapped on K3split into elementary monodromy defects of 6 d N = (2 ,
0) supergravity, argue that stacksof these host Argyres-Douglas SCFTs of type ( A , A n ), and explain how this resolves theapparent singularities of the conformal factor (see Section 4.1). Furthermore, we will arguethat after this is taken into account, the bulk region where semi-classically e − A < O (1) Einstein frame curvature (in Section 4.2).But first, as a proof of principle, let us exhibit an N = 1 O7 orientifold of a CY threefoldsuch that the entire D3 tadpole is generated by seven-branes wrapping K3 surfaces. Many Note that the problem becomes more severe when the uplift scale is decreased, i.e. for AdS vacuawith small SUSY breaking, the prospects of which have recently been studied in more generality in [70].We note that the results of this paper may also be used to argue for the possibility of small SUSY breakingin AdS using KKLT. P × P which has two K¨ahler moduli associatedwith the hyperplane classes of P and P (and it is a K3 fibration over P ). The orientifoldinvolution inherited from the ambient space involution I : P × P → P × P , ([ x : x ] , [ y : ... : y ]) (cid:55)→ ([ − x : x ] , [ y : ... : y ]) (4.1)gives rise to two non-intersecting O7 planes at the intersection of { x = 0 } respectively { x = 0 } with the hypersurface. Both arise as an anti-canonical hypersurface in P whichis K3. Placing four D7 branes on each O7 plane gives rise to gauge algebra so (8) (whichcan be enhanced to so (16) by moving all eight seven branes onto the same O7 plane) andinduced D3 charge −
12 that can be canceled by fluxes and/or mobile D3 branes. In thefollowing we will discuss the resolution of singularities of the conformal factor in termsof the Seiberg-Witten solution of SU ( N ) Yang-Mills theory, but the discussion for gaugegroup SO (2 N ) is analogous. We will start by making an analogy to F-theory. The technical point that has to beaddressed is the apparent singularity where the conformal factor turns negative in theproximity of seven-brane stacks with negative D3 charge, e − A ( r ) = | ρ D | π log( r/r ∗ ) < , for r < r ∗ . (4.2)It turns out that such apparent pathologies are completely analogous to the behavior ofthe dilaton near a perturbative (in g s ) O7 plane: e − φ ( r ) = 42 π log( r/r ∗ ) . (4.3)Here, e − φ is the radially dependent inverse string coupling, and the coefficient in frontof the logarithm arises because an O7 plane has − bulk string coupling the singularity is exponentially close to the O-plane, i.e. at distances O ( e − πgs ), but at g s ∼ To define this one uses the fact that the negative D7 charge of the O7 is canceled by other seven-branesscreening the charge at large radii. At yet larger radii the dilaton profile is constant, and this constant isa modulus that we call 1 /g s . p, q ) seven-branes with monodromy charges [61](2 , −
1) and (0 , , (4.4)separated from each other at non-perturbative distance ∼ e − πgs .Another way to describe the same phenomenom is to consider the setup from thepoint of view of a probe D3 brane. From this perspective, instanton effects in the gaugetheory living on the D3 worldvolume will resolve the dilaton singularity, as first notedin [62]. Concretely, a probe D3 brane has gauge coupling πg = g s and in the vicinity ofan O7 plane the U (1) gauge group on its worldvolume enhances to U Sp (2) (cid:39) SU (2). Theposition modulus of the D3 brane in the transverse space is identified with the vacuumexpectation value (vev) of the Coulomb branch (CB) operator of the worldvolume theory,which is 4d N = 2 pure SU (2). The low energy dynamics of this theory is determined bythe Seiberg-Witten solution [53, 54]. In particular, in the interior of the CB there are two monodromy defects where dyons carrying electric-magnetic charges (2 , −
1) respectively(0 ,
1) become massless. By identifying the W-boson as the stretched F-string, and themagnetic monopole as the D-string, one sees that the defects indeed correspond to the( p, q ) seven-branes listed in eq. (4.4). Note that the D3 brane is a particularly clean probeof the axio-dilaton profile: the gauge coupling corresponds to the axio-dilaton which issourced only by the background O7 plane and not by the probe itself.Given the qualitative and even technical similarity of the apparent singularity of theconformal factor near loci of negative D3 charge (4.2) and the dilaton singularity (4.3),and given the well known resolution of the latter problem in terms of a probe D3 branedynamics [62], we find it natural and promising to consider very analogous reasoning inorder to discuss the resolution of the former.In order for this to work, one would like to relate the conformal factor e − A to theYang-Mills coupling of wrapped branes. Thus, we should consider a stack of N wrappedD7 branes on a surface S . Upon moving away a single D7 brane in the transverse direc-tion, its effective U (1) coupling probes the backreaction of the remaining seven-branes.This deformation is a flat direction only if the normal bundle of the surface S is trivial:classically, the Coulomb branch is explored by simply moving D7 branes away from thestack in the transverse directions. If the normal bundle N were non-trivial, there would either not exist a Coulomb branch at all (the divisor is rigid) or a D7 brane deformed awayfrom the main stack still intersects it along a curve leading to massless bifundamental mat-12er. Since c ( N ) = − c ( S ) via the adjunction formula, we avoid this if we consider N D7branes wrapping a K3 surface in a CY(-orientifold) X . The holographic correspondencebetween the one-loop running of the gauge coupling and the solution for the conformalfactor has been observed in [63] which also identifies the W-bosons as stretched F-stringsand the magnetic monopoles as stretched effective strings from D5 branes wrapped on K3.Indeed, by evaluating the probe brane actions on the supergravity background sourced bythe seven-branes, one recovers the field theory central charges of W-bosons and magneticmonopoles (see e.g. our Appendix A).The precise holographic dictionary between the 7-brane gauge coupling and the confor-mal factor is τ D := ˆ τ − τ , ˆ τ := (cid:90) K C + i (cid:90) K d y (cid:113) g int K e − A , (4.5)where g int K is the induced metric of the K3 surface obtained from the unit volume bulkmetric g int . The negative correction by the axio-dilaton τ is due to the O ( α (cid:48) ) curvaturecorrection to the gauge-kinetic term living on the brane stack. Indeed, the log-coefficientof ˆ τ − τ is given by the difference of D7 and D3 brane charges ( Q D , Q D ) = ( N, − N ), i.e. τ D ( z ) = − N πi log( z/z ∗ ) + ... , (4.6)where z is the complex transverse coordinate. This matches with the beta-function coeffi-cient 2 N of SU ( N ) Yang-Mills theory. Naturally, the distance scale z ∗ corresponds to thestrong coupling scale Λ of the gauge theory. Thus, in order to describe what happens at | z | < | z ∗ | one can again invoke the Seiberg Witten solution of pure SU ( N ) N = 2 gaugetheory.Ideally, one would like to determine the supergravity solution by probing N backgroundD7 branes with a further D7 brane that itself does not source the gauge coupling τ D . How-ever, unlike the D3 brane as a probe of a single O7-plane solution a probe D7 brane itselfalso sources τ D . Therefore, one has to be a bit careful in identifying the Coulomb branchof the gauge theory with the physical transverse space of the brane stack. Happily, we willsee that from the behavior of SU (2) Yang-Mills theory one can quite readily understandhow a single D7-brane splits into more elementary monodromy defects. The generalization Strictly speaking, in eq. (4.5) we should replace τ by the axio-dilaton averaged over the K3 surface. The correction by Re( τ ) = C follows immediately from the well-known α (cid:48) -corrected D-brane CSaction [71, 72]. The correction by Im( τ ) can most easily be seen from the fact that a gauge-instanton withaction 2 π Im( τ D ) can be thought of as a wrapped euclidean D3 brane: the DBI action of a D3 branereceives a correction δS = − π (cid:82) K Im( τ )Tr ( R ∧ ∗R ) = − π Im( τ ) [73–75]. SU ( N ) turns out to be straightforward.Just as F-theory describes ’elementary’ mutually non-local SL (2 , Z ) monodromy defectsin 10d, i.e. ( p, q )-seven-branes, one should be able to describe the corresponding defects inour situation as monodromy defects of 6d N = (2 ,
0) SUGRA: type IIB string theory onK3 as in [77–80]. The U-duality group and spectrum of strings is much larger in 6d thanit is in 10d [81], U = O (5 , Z ) , (4.7)and we get (cid:126)p -strings with (cid:126)p ∈ Z from 10d ( p, q ) strings, ( p, q ) 5-branes wrapped onK3, and D3 branes wrapped on the 22 two-cycles of K3, coupling to the (anti-)self-dual2-forms in 6d transforming in the of the monodromy group U . As in ten dimensions,there are cosmic defects on which (some) strings can end. We consider configurations ofmonodromy defects such that the monodromy transformations on overlapping patches arecontained in a U (cid:48) := O (2 , Z ) ⊂ U subgroup acting only on the ( p, q ) strings and wrapped( p, q ) five-branes transforming in the , as in [78]. The tension of a (cid:126)p -string, with (cid:126)p ∈ Z ,in 6 d Planck units is given by [82] T (cid:126)p = √ π Im( τ ) − Im(ˆ τ ) − | p τ + p ˆ τ + p − p τ ˆ τ | . (4.8) U (cid:48) is generated by two commuting SL (2 , Z ) sub-groups acting on the modular parameters( τ, ˆ τ ), as well as a Z subgroup interchanging ˆ τ ↔ τ [78]. The latter can morally bethought of as ’four T-dualities’ because the string-frame K3 volume gets inverted. Indeed,Im( τ ) = C and Im(ˆ τ ) = (cid:82) K C so the RR potentials transform according to the standardrules of T-dualities.Configurations of cosmic defects whose monodromies are contained in U (cid:48) include inparticular configurations of ( p, q ) seven-branes wrapped on K3 and D3 branes on points inK3. Since O (2 , Z ) ⊂ O (2 , Z ) which is the monodromy group of the complex structuremoduli space of a K3 surface, one may encode the axio-dilaton, conformal factor and C profiles in a K3 surface fibered over the base P [77, 78]. This approach has been termed G-theory by analogy to F-theory [78]. As a consequence, each such brane configuration isdual to type IIB on a certain K3-fibered CY threefold.We will now invoke the Seiberg-Witten solution of SU ( N ) gauge theory living on a stackof N D7 branes wrapped on K3, in order to understand the resolution of the singularity For an exposition of exotic defects in string theory, see e.g. [76] A cosmic defect is defined by an element of the U-duality group M ∈ U . A necessary condition for a (cid:126)p -string to be able to end on it is that its tension is invariant under M .
14f the conformal factor, in the spirit of [62]. First, let us recall some well-known factsabout the Seiberg-Witten solution. Pure N = 2 SU ( N ) Yang-Mills theory has an N − U (1) N − and an over-complete set of coordinates on moduli space is givenby the periods of an auxiliary Riemann surface Σ N − of genus N −
1. The latter can betaken to be the hypersurface in C (cid:51) ( x, y ) specified by [53–57] (we follow the conventionsof [57]) y = W ( x ; { u k } ) − Λ N , (4.9)where Λ is the strong coupling scale of the UV theory and W ( x ; { u k } ) is the characteristicpolynomial W ( x ) := det ( x I − Φ) ≡ x N − N (cid:88) k =2 u k x N − k , (4.10)and the Casimirs u k are the invariant polynomials u k = k Tr (cid:0) Φ k (cid:1) + O ( u k − , ..., u ). Onecan think of the curve Σ N − as the double cover of the complex x -plane with branch cutsrunning between the N roots e + i of W + := W − Λ N and the roots e − i of W − := W + Λ N .One defines a holomorphic one-form (the Seiberg-Witten form or differential ) λ ( u ) := 12 πi x∂ x W y dx , (4.11)and the period vector is defined as the integral of λ over a basis of one-cycles. In thesingular classical limit Λ /u k k → e ± i → e i , (4.12)where the e i are the roots of W ( x ). It follows that at weak coupling there is a preferredset of N shrinking cycles α i that encircle each coinciding pair of roots in the x -plane. Inthis limit the residue theorem implies that the associated period components are equal tothe roots, a i := (cid:90) α i λ −→ e i , i = 1 , ..., N . (4.13)We have (cid:80) i a i = 0 because (cid:80) i [ α i ] = 0 ∈ H (Σ N − , Z ), so we can use N − a i as coordinates on moduli space in the weak coupling patch. In our context of realizingthe gauge theory with N D7 branes wrapped on K3 we can think of the roots e i as thepositions of the N perturbative D7 branes in the transverse plane parameterized by x ∈ C .15ndeed, the charged W-bosons stretching between the i -th and j -th D7 brane have centralcharge Z ij = a i − a j , (4.14)so the mass of the corresponding BPS particle is proportional to the distance between thebranes in flat space parameterized by a flat coordinate a , which matches the mass of astretched F-string (see Appendix A). Away from the classical limit each root e i splits intwo, e i → ( e i + , e i − ) , (4.15)separated from each other at a non-perturbative distance scale. Thus, even away fromthe classical limit we can think of the a i ’s as the center of mass positions of the N D7branes, but each D7 brane should be interpreted as a bound state of a pair of elementarymonodromy defects at positions ( e i + , e i − ). The two constituents are interpreted as elemen-tary monodromy defects that are separated from each other by a non-perturbative distancescale in the transverse plane.For simplicity, let us first consider the case N = 2: for two D7 branes we have W ( x ) = x − u with u := Tr (Φ ) a gauge invariant coordinate on the Coulomb branch. As famouslyshown in [53] the classical singularity at u = 0 is resolved into a pair of singularities at u = ± Λ where magnetic monopoles become massless. At each of these two points oneof the two defects forming the first perturbative D7 brane coincides with one of the twodefects forming the other one, see Figure 1. As shown in [63], the magnetic monopole isa stretched string obtained from wrapping a D5 brane on K3. Since this is the dyon thatbecomes massless at one of the two singularities in moduli space we learn that a wrappedD5 string can end on one of the two elementary monodromy defects (or exotic branes) of6d N = (2 ,
0) supergravity that constitute a perturbative D7 brane.The monodromies of ( τ, ˆ τ ) can be determined using G-theory. In fact, the appropriateK3 fibrations over P with monodromies only in U (cid:48) = O (2 , Z ) have been describedin [78], and the relevant monodromy matrices have been worked out in [82], albeit in avery different context. Here, we present a simplified argument to get to the right result:first, one notices that the perturbative monodromy around a D7 brane acts as τ → τ + 1and ˆ τ → ˆ τ −
1, in particular leaving the combination ˆ τ + τ invariant. Let us make thenatural assumption that the microscopic monodromies also leave ˆ τ + τ invariant. Then, We are indebted to Andreas Braun for pointing out and explaining ref. [82] to us. In an earlier versionof this paper we had given the monodromies under the incorrect assumption that they act like a standard SL (2 , Z ) transformation on τ D . W + ( x ) (blue) and W − ( x ) in the complex x -plane. A pair of roots of W + W − that coincides as Λ → u → +Λ . Right: roots near the other singularity u → − Λ .the only allowed monodromies are generated by { T , ( − } with T : τ D (cid:55)→ τ D + 1, and( −
1) : τ D (cid:55)→ − τ D . Using that the D5 string (with tension ∝ | τ D | ) can end on the firstdefect, as in [63], it follows that M = ( −
1) : τ D (cid:55)→ − τ D , M = ( − · T − : τ D (cid:55)→ − τ D + 2 . (4.16)These are indeed the monodromies found in [82] but now reinterpreted as the monodromiesaround the constituent defects forming a perturbative D7 brane. In the decompactificationlimit Vol( K → ∞ the constituents merge and one recovers the elementary D7 brane inten flat dimensions (this limit is dual to the degeneration limit of the K3 fiber consideredin [82] in the geometric context).Note that M and M differ from each other only by a choice of base point: we have M = T − · M · T . (4.17)As a D5-string can end on one of the two defects forming a perturbative D7 brane, thestring that can end on the other one (let us call it the (cid:102)
D5 string) differs from a D5 stringby one unit of F-string charge, see Figure 2. Indeed, a D5 string ending on the first defectpicks up one unit of F-string charge upon rotating half-way around the transverse plane ofa D7 brane due to the non-trivial C and C profile around it, C → C − , (cid:90) K C → (cid:90) K C + 12 . (4.18)17igure 2: Splitting of perturbative D7 brane into two cosmic defects (red and blue points).A D5-string (blue) can end on one of the defects, and a D5 with a negative unit of electriccharge (red) ( (cid:102) D5) can end on the other. The F-string (purple) can end on the entireconfiguration via a string junction.This induces F-string charge on the D5-brane due to its CS-coupling, S CS ⊃ π (cid:90) Σ B (cid:18)(cid:90) K C − C (cid:19) , (4.19)see Figure 3 for a pictorial representation. After splitting off an F-string, which can endon the pair of defects via a string junction, we get the (cid:102) D5 string that can end on thesecond defect. Of course there exists an entire tower of monopoles with arbitrary integerelectric charge, corresponding to D5 strings winding around the pair of defects, eventuallyending on one. By analogy with ( p, q )-strings in 10d we will refer to ( p (cid:48) , q (cid:48) )-strings asthe strings with charges ( − q (cid:48) , q (cid:48) , p (cid:48) , ∈ Z such that the (1 , , SU (2) Yang-Mills theory withelectric-magnetic charges (2 p (cid:48) , q (cid:48) ) can be thought of as a stretched ( p (cid:48) , q (cid:48) )-string.The typical distance separating the two defects is non-perturbative in the 7-brane gaugecoupling and thus of order Λ eff ∼ r e πi τ D ( r ) , with arbitrary radial scale r . Below thisdistance scale, the logarithmic running of the conformal factor terminates because thereis no remaining D3 charge at smaller radii. It is apparent that this is what keeps the > π/ (cid:102) D5 string can end on the second defect. conformal factor from turning negative too close to the location of negative D3-charge .It does not take too much imagination to anticipate what happens in the case of N D7 branes wrapped on K3 (at the origin of the Coulomb branch): non-perturbatively, theinstanton corrections to the holographic running of the gauge coupling become importantat a distance scale Λ ∼ r e πi N τ D ( r ) , (4.20)and the N perturbative (in the α (cid:48) expansion) D7-branes split into N pairs of exotic cosmicdefects: each pair can be thought of as a bound state realizing a perturbative D7 brane as ineq. (4.16) and (4.17). At the origin of the Coulomb branch (i.e. u k = 0) the characteristicpolynomial W ( x ) reads W ( x ) | u k =0 = x N → W ± ( x ) = x N ∓ Λ N , (4.21)and the entire Z N R-symmetry group is unbroken. There, the 2 N defects align along acircle of radius Λ, see Figure 4, W + ( x ) W − ( x ) = 0 → x = Λ e πi N k , k ∈ { , ..., N − } . (4.22)Since the R-symmetry of the gauge theory corresponds geometrically to the rotationalsymmetry around the stack of seven-branes, we see that the non-perturbative splitting ofthe perturbative D7 branes into 2 N defects is the bulk version of the explicit symmetrybreaking U (1) R → Z N by instantons in the gauge theory. The bulk monodromies aroundthe 2 N defects (with loops γ n taken as depicted on the left in Figure 5) take the simple19igure 4: The 2 N monodromy defects forming N perturbative D7 branes, depicted on thetransverse plane, at the origin of the Coulomb branch. Locally, each defect looks the same,but the ones drawn in blue are mutually non-local compared to the ones drawn in red.Figure 5: Left: Loops γ n encircling the defects arranged in a circle from the ’outside’.Right: Different choice of loops γ ∗ n encircling the defects from the ’inside’.20orm M n = T − ( n − · M · T n − , (4.23)and with M and T as in eq. (4.16) and (4.17). Thus, the monodromy M n leaves thetension of a D5 string with − n + 1 units of induced F-string charge invariant. The 2 N elementary defects look the same locally because the monodromies differ from each otheronly by a choice of base point as is manifest in eq. (4.23).Thus again, the splitting of the perturbative D7-brane into the more elementary mon-odromy defects halts the running of the coupling at the non-perturbative scale Λ andprevents regions of negative conformal factor from appearing. It is also clear that thedefects corresponding to the roots of (say) W + ( x ) can be stacked on top of each other bytuning the moduli u i , W + ( x ) = N (cid:89) i =1 ( x − e + i ) → x N (cid:89) i =2 ( x − e + i ) → ... → x N , (4.24)i.e. they are mutually local . The effective strings that can end on them are the D5 stringswith minimal induced F-string charge. Correspondingly, the monodromy matrices corresponding to suitably defined loops γ ∗ n (as shown on the right in Figure 5) around the roots of W + ( x ) satisfy: M ∗ n = M ∗ ≡ M n ∈ Z + 1 M ∗ ≡ M · M · M − n ∈ Z . (4.25)Thus, each perturbative D7 brane splits non-perturbatively into two mutually non-local defects (that look the same locally due to M = T − M T ). Let us call them the A-typeand B-type defect respectively. All the A-type (B-type) defects are mutually local with Note that while the tension of a string that can end on a monodromy defect is invariant under themonodromy transformation, its charge vector (cid:126)p ∈ Z ⊂ Z is mapped to − (cid:126)p in our case, while strings with invariant charge vector are not allowed to end on the defect. Otherwise, there would be unwanted extramassless particle states e.g. from D3 branes wrapped on curves in K3 at singular loci of the gauge theory.The mathematical reason for this in G-theory was pointed out to us by A. Braun: at the location of anelementary monodromy defect, a curve γ with self-intersection γ = − γ ] → [ γ ] + γ [ γ ] = − [ γ ] under the monodromy. Note however that the dyons from the set of stretched D5 strings between three or more mutuallylocal cosmic defects are non-local in the sense that the pairwise Dirac-products are generally non-vanishing[56, 64]. (cid:15) is dual to thedeformed conifold. Right: two defects stacked on top of each other are dual to a singularconifold.respect to each other, so a D5 string that can end on any of the A-type defects can alsoend on any of the other A-type defects as long as it stretches along the ’interior’ region ofthe circle of defects. Indeed, such configurations give rise to BPS dyons at the origin ofthe Coulomb branch [83]. The field theory that lives on a stack of n A-type (or n B-type)defects is the Argyres-Douglas (AD) CFT of type ( A , A n − ) because these theories arewell-known to arise when multiple roots of W + ( x ) or W − ( x ) coincide [64, 84–86]. Indeed,the most generic singularity is obtained by colliding two defects, and a single dyon becomesmassless. At such a point in moduli space the dual K3 fibered CY threefold should developa conifold singularity because this is the most generic singularity (at finite distance) thatarises in its complex structure moduli space, see Figure 6.For n ∈ Z we can form n/ n conifolds on the geometric side. Each conifold hosts a single massless particle from aD3 brane wrapped on the shrinking A-cycle [87], which is dual to the D5 string stretchedbetween two colliding monodromy defects. Furthermore, there exists a basis of n/ − n/ − n conifolds. Stacking all n monodromy defects on top of each other corresponds to colliding the n conifolds, see Figure7. Indeed, this is well known to geometrically engineer the AD CFT of type ( A , A n − ) intype IIB string theory [88,89]. Consider for example geometrically engineering the ( A , A )22igure 7: Left: two pairs of two coincident defects, separated from each other a distance δ corresponds to two conifolds sharing a compact B-cycle with volume δ . Right: colliding thetwo pairs is dual to shrinking the conifold B-cycle and leads to the geometrically engineered( A , A ) AD CFT.CFT from type IIB on the non-compact CY threefold embedded into C as f ( x ) + y + u + v = 0 , with f ( x ) := (cid:18) x + δ (cid:19) (cid:18) x − δ (cid:19) + (cid:15) δ (cid:18) x + δ (cid:19) + (cid:15) δ (cid:18) x − δ (cid:19) . (4.26)In the regime | (cid:15) i | (cid:28) | δ | we get two conifolds with deformation parameters (cid:15) i near the loci { x = ± δ/ , y = u = v = 0 } , and δ is a measure for the distance between them. In thelimit ( (cid:15) i , δ ) → (cid:15) i /δ →
0) we get f ( x ) → x which realizes the unbroken( A , A ) SCFT. The extended Coulomb branch of such SCFT is parametrized by the vevof the CB operator (cid:104)O (cid:105) , a mass parameter m , and a coupling µ , where the subscriptsdenote conformal dimension. The geometric parameters ( (cid:15) i , δ ) correspond to the SCFTparameters as µ = − δ , m = (cid:15) + (cid:15) δ , (cid:104)O (cid:105) = δ
16 + (cid:15) − (cid:15) , (4.27)In other words, the ( A , A ) point realized by the two conifolds arises at generic pointsalong the locus (cid:110) (cid:104)O (cid:105) = µ , m = 0 (cid:111) in the extended Coulomb branch of the ( A , A )theory.As an aside, it would be very interesting to understand the Higgs branch of the( A , A n − ) theory with n even in terms of its realization by stacks of cosmic defects in6 d supergravity. Note however that in the fully geometric duality frame one can under-stand the Higgs branch as a resolution of conifolds. Indeed, as mentioned above, for n evenone can find k := n/ k corresponding shrinking A -cycle three-spheres con-nected by k − A , A n − ) AD theories is always U (1) (for n ≥ n even) but for n = 4 it gets enhanced to SU (2). We leave these interesting questions to further studies.Finally, we expect that the moduli space of local non-compact CY threefolds dual (viaan element in O (5 , Z )) to a stack of N D7 branes wrapped on K3 can be embeddedlocally in C (cid:51) ( x, y, u, v ) via y + u + v = W ( x ; { u k } ) − Λ N , (4.28)such that the co-dimension two slice u = v = 0 is the Seiberg-Witten curve. This non-compact model indeed geometrically engineers the pure SU ( N ) gauge theory [88, 95].Let us distill what we have learned into a rather simple effective model that should bevalid when the negative D3 charge is large. We encode the non-perturbative splitting ofN D7 branes at the origin of the Coulomb branch as follows: the 2 N monodromy defectsare smeared along a circle of radius r ∗ ∼ Λ, giving rise to an effective circular defect of real co-dimension one that hosts all the negative D3 charge. In this case, the conformalfactor runs logarithmically at large radii r ≥ r ∗ ∼ Λ and remains constant at smaller radii r ≤ r ∗ ∼ Λ, see Figure 8. There is only one number that this effective description requiresas input from a more microscopic analysis: the precise radius r ∗ at which the effective co-dimension one membrane lives, or equivalently the value of the K3-volume in the interiorregion r ≤ r ∗ .Because the interior region can be probed by the individual strong coupling defectsforming a perturbative D7 brane (see Figure 9) it is natural to propose that the gaugecoupling (K3 volume) in the interior region is truly strongly coupled. More precisely, weclaim that τ D | r =0 = ∞ (cid:88) k =0 ω k N − k , (4.29)with numerical O (1) constants ω k . In particular, the l.h. side approaches a universal finite value ω as N → ∞ . More precisely, we claim that ω cannot be dualized to somethingweakly coupled, i.e. 0 < Im( ω ) < ∞ . In the next section, we will confirm this byinspecting the Seiberg-Witten solution of SU ( N ) Yang-Mills theory near the origin of theCoulomb branch. The reader not interested in how we arrive at this technical conclusion This can be clearly seen at least in two ways. Either from the computation of the superconformalindex [92], or from the topological symmetry enhancement of their 3d N = 4 mirror theory [93, 94], whichis U (1) with N f = n/ N f = 2. N defects forming the N perturbative D7 branes along theangular circle we obtain a simplified effective model, where the conformal factor runs at one-loop in the exterior region r ∗ ≤ r ≤ r and remains constant in the interior region r ≤ r ∗ .Also, we assume that the negative D3 charge on the D7 branes is screened by positive D3brane charge at r = r resulting in a constant profile at r ≥ r which corresponds to thevolume modulus.Figure 9: By taking u ,...,N − = 0 and u N ≤ Λ N N mutually local exotic defects move intothe interior region until they collide at the origin for u N = Λ N realizing the ( A , A N − )AD theory. 25ay skip Section 4.2.The crucial consequence of our claim is that as the UV-gauge coupling Im( τ D ) UV isdialed to values (cid:46) N the non-perturbatively resolved seven-branes start to explore theentire CY threefold , leaving behind a region with O (1) Einstein frame curvature. Thedefects that resolve the apparent negative conformal factor singularity live at real co-dimension two, so we expect to find singularities in the 4d EFT only for isolated valuesof τ D where one of the defects collides with (say) a mobile D3 brane. Therefore, at leastwhen all negative D3 charge comes from a stack of wrapped seven-branes, we expect the4d effective field theory to generically remain regular even when Im( τ D ) UV (cid:28) N . Now we would like to consider quantitatively the SU ( N ) gauge theory engineered by D7branes wrapped on K3 in order to substantiate the claim of eq. (4.29). To this end, we willanalyze the Seiberg Witten solution near the origin of the Coulomb branch u ,...,N = 0.First, at weak coupling | u k | (cid:29) τ D to suitable compo-nents of the gauge coupling matrix τ Y M of the effective U (1) N − gauge theory. This goesas follows: the W-bosons and magnetic monopoles correspond to F-strings and D5-stringsstretched between the two D7 branes, see Figure 3. Thus, for a pair of D7 branes realizingan SU (2) ⊂ SU ( N ) Yang-Mills theory its effective U (1) coupling τ eff is equal to ∂ Z M /∂ Z W where ( Z M , Z W ) are the central charges of the magnetic monopole and the W-boson [53].In other words, it measures how fast the mass of a magnetic monopole grows in relationto that of a W-boson upon moving the two D7-branes apart. This identifies τ eff with thebulk coupling τ D at a point between the two D7-branes. In the same way, one can repro-duce the semi-classical monodromy of the SU (2) Yang-Mills theory from the semi-classicalmonodromies of the bulk field τ D that we discussed in the last section (see Appendix A).In regions of stronger coupling | u k | (cid:38) τ D (cid:39) τ eff . At even stronger coupling | u k | (cid:46) q e , q m ) = (2 , −
1) and(0 , The co-dimension one surfaces in moduli space across From the gauge theory perspective one can think of this as the insertion of R-symmetry breakingspurions with characteristic scale of order the strong coupling scale Λ. In this sense the 7-brane gaugetheory would become truly strongly coupled. Note that this does not imply that the F-string itself becomes unstable. Rather, the F-string compo- SU (2) theory. Right: weakly coupledchamber where the W-boson is BPS and realized via a string junction system ending onthe defects. Left: strongly coupled chamber where the W-boson has decayed into a two-particle state corresponding to the two BPS dyons with electric-magnetic charges (2 , − , walls of marginal stability [53, 54]. In chambersof moduli space where W-bosons are not BPS, the precise relation between the bulk andYang-Mills couplings becomes even less clear.Nevertheless, one can use the gauge theory solution to argue for genuine strong bulkcoupling τ D in the interior region near the origin of the Coulomb branch. First, considerthe SU (2) solution of Seiberg and Witten [53]. Along the real imaginary line u ≡ u = it with t ∈ R + the two dyons with charges (0 ,
1) and (2 , −
1) have equal mass. For t ≥ t c ≈ .
860 the W-boson is BPS and at the intersection of the imaginary line with the wall ofmarginal stability u = it c we have a = a D and thus m (0 , = m (2 , − = 12 m (2 , , (4.30)and all three dyons are mutually BPS. At this point in moduli space the W-boson is aboutto decay into the two dyons (of equal mass), and thus a small F-string localized in theinterior must have comparable tension to either of the two D5-strings. We conclude that τ D | interior = O (1). Finally, let us assume that there is a duality frame in which the bulkcoupling is actually weak. Then, there would exist a ( p (cid:48) , q (cid:48) ) string with parametricallysmaller tension than the one of the D5 and F-strings. If moreover q (cid:48) /p (cid:48) > − ,
1) and (2 , − , , −
1) are mutually BPS. Thus, such a parametrically light string does not exist. If nent of the junction system shrinks to zero size. q (cid:48) (0 ,
1) + p (cid:48) (2 , −
1) and q (cid:48) /p (cid:48) > q = 4 and p = 3) could reduceits mass by producing a ( p (cid:48) , q (cid:48) − p (cid:48) )-string (green) in the middle if this string were para-metrically light. But this cannot be the case because (0 ,
1) and (2 , −
1) are mutually BPS. q (cid:48) /p (cid:48) ≤ − N limit. Ignoring this for now, we conclude that τ D | interior = O (1) (4.31)in any duality frame. In the rest of this section we will show that the origin of the Coulombbranch for SU ( N ) Yang-Mills theory approaches a wall of marginal stability of the sametype in the large N limit N → ∞ . The claim of eq. (4.29) then follows.The Seiberg-Witten solution is determined in terms of the Seiberg-Witten curve of eq.4.9, and the quantum periods are integrals of the Seiberg-Witten form of eq. 4.11 overone-cycles of the Riemann surface Σ N − . First, let us specify a standard symplectic basisof H (Σ N − , Z ), following [57]. As explained in the previous section we identify Σ N − withthe double cover of the complex x -plane branched over cuts between the N roots e i + of W + ( x ) and the N roots e i − of W − ( x ), as depicted in Figure 12.Each pair ( e i + , e i − ) can be thought of as a perturbative D7 brane wrapped on K3.Let cycles α i , i = 1 , .., N , encircle each pair of roots ( e i + , e i − ) counter-clockwise, anddefine [ ˆ α i ] := (cid:80) ij =1 [ α j ]. Furthermore, we let the cycles β i , i = 1 , ..., N −
1, connectthe i -th and ( i + 1)-th D7 brane as depicted in Figure 12. The set of cycle classes { [ ˆ α ] , ..., [ ˆ α N − ] , [ β ] , ..., [ β N − ] } is a symplectic basis of middle homology H (Σ N − , Z ),i.e. ˆ α i ∩ β j = δ ji , ˆ α i ∩ ˆ α j = β i ∩ β j = 0 . (4.32)In order to evaluate the periods at the origin of the Coulomb branch we can use the factthat the α -cycles are mapped into each other by discrete rotations in the x -plane, i.e. bya Z N ⊂ Z N R-symmetry group. This implies that the periods a l := (cid:82) α l λ can be obtained28igure 12: We show (for N = 10) two sheets of the complex x -plane marked by the rootsof W + (blue) and the roots of W − (red), with branch cuts running between them (blackdots), at the origin of the Coulomb branch u ,...,N = 0. The cycles α i (red) run along theupper sheet, while the cycles β i (blue) run into the branch cuts and continue along thelower sheet (dotted blue). 29igure 13: Same setup as in Figure 12 but instead of the β -cycles we depict the γ -cycles(in green) which are obtained from the α -cycles via action of the R-symmetry.via analytic continuation in the small- u patch as a l ( { u k } k ) = e πiN ( l − a ( { e − πiN k ( l − u k } k ) , l = 1 , ..., N , (4.33)and ˆ a l = (cid:80) lm =1 a m . Similarly, we can obtain the periods a iD . However, since they are notsimply mapped into each other by the R-symmetry group it turns out to be useful to definea further set of cycles γ ,...,N drawn in Figure 13 that are better adapted to the symmetryof the problem than the β -cycles. These can be expressed as[ γ ] l = − [ β i ] − ( − l [ α l e +1 ] , l = 1 , ..., N − , (4.34)where l e is the even part of l . The γ i can also be obtained from α via the action of theR-symmetry group so the corresponding periods π lγ := (cid:82) γ l λ satisfy π lγ ( { u k } k ) = e πi N (2 l − a ( { e − πi N k (2 l − u k } k ) , l = 1 , ..., N . (4.35)We can now express the periods a lD as a lD = − π lγ − ( − l a l e +1 , l = 1 , ..., N − , (4.36)30igure 14: Left: Two D5-strings stretching between neighboring D7 branes forming BPSdyons with central charges Z d and Z d . Right: State with central charge Z W = Z d + Z d is marginally BPS in the large N limit according to eq. (4.38). Upon slightly deformingaway from the origin of the Coulomb branch this state should become finitely bound dueto the formation of a small F-string (purple).and use the continuation formulae 4.33 and 4.36 to compute the periods in terms of a := a ,though we will not explicitly make use of them.In order to show that the origin of the Coulomb branch approaches a wall of marginalstability as N → ∞ we consider the ratio of central chargesΩ( N ) := Z d Z W , (4.37)with Z W = a − a , and Z d = a − π γ , as a function of N . Using our continuation formulaein eq. (4.33) and eq. (4.36) it follows thatΩ( N ) = 11 + e πi N = 12 − i (cid:0) π N (cid:1) (cid:0) π N (cid:1) →
12 as N → ∞ . (4.38)Thus, in the large N limit the origin of the Coulomb branch of the SU ( N ) theory iscompletely analogous to the special point u = it c on the wall of marginal stability of the SU (2) theory. By reasoning that we applied for the SU (2) theory in the beginning of thissection a strong bulk coupling as claimed in eq. (4.29) can be derived: the three relevant(almost-)BPS states are the W-boson with central charge Z W and the two dyons (which areexactly BPS [83]) with central charges Z d and Z d = −Z d + Z W , depicted in Figure 14. Asa consequence, the bulk gauge coupling τ D must take a universal O (1) value at a point inthe interior region close to its boundary, as well as its 2 N − i -th pair of D7 branes, i.e. with central charges Z Wi = a − a i +1 , and exactly-BPSstretched dyons with central charges Z d i = (cid:80) il =1 ( a l − π lγ ), and Z d i = −Z d i + Z Wi , aswell as their images under the R-symmetry group to probe essentially the entirety of theinterior region (see Figure 15 for the case i ∼ N/ i ( N ) := Z d i Z Wi ≡ Ω( N ) . (4.39)Not only does this confirm that the typical gauge coupling in the interior region is strong,but it also gives further justification for our effective model (cf. Figure 8) treating the bulkcoupling as essentially constant throughout the interior region, at large N .Finally, we can also close the loophole that we left in the beginning of this section:one would like to exclude light ( p (cid:48) , q (cid:48) ) strings in the interior region also for q (cid:48) /p (cid:48) ≤ −
1. Ifwe consider a pair of D7-branes as in Figure 15 the relative separation of two D7-branesdiverges in the large- N limit. Then, if we assume that a ( p (cid:48) , q (cid:48) ) string exists that becomestensionless in the large- N limit, we can build a dyon from the dyons with central charges Z d i and Z d i that can form this light string in the interior as in Figure 11. The contributionto the mass of this dyon from the junction system is finite in the large N limit whilethe contribution to its mass from the light string in the interior grows at most as N α with α <
1. Thus, at sufficiently large N the mass of this dyon is smaller than the mass ofthe BPS dyons we have used as building blocks, so the new dyon should be BPS. Butsuch a state is not part of the BPS-spectrum [83] so indeed no ( p (cid:48) , q (cid:48) ) string can becometensionless in the interior region as we take the large- N limit. In units where the BPS dyon connecting neighboring mutually local defects has mass equal to one. Comments
We have described how the non-perturbative gauge theory effects lead to the splitting ofwrapped D7 branes into more elementary cosmic defects and thus prevent regions of nega-tive conformal factor to occur. We have described this rather quantitatively under certainsimplifying assumptions: 1) seven-branes wrap a K3 surface, 2) Classical R-symmetrybreaking effects from the compact CY as well as other sources of D3 charge can be ne-glected locally. This means that positively charged D3 branes screen the negative chargeon the seven-branes at distances r much bigger than the strong coupling scale Λ.In this section we would like to sketch plausible outcomes upon dropping our simplifyingassumption 2): In 5.1 we consider type IIB on an N = 2 orientifold of T × K K × K K (cid:46) N the D3 charges recombine dueto the non-perturbative splitting of seven-branes sweeping up positive D3 charge in thebulk. In 5.2 we consider the final stages of recombination against the D3 charge hosted by awarped throat. We argue that recombination of D3 charge can be understood as a loweringof the UV cutoff of the KS gauge theory dual the throat. This leads to a reduction of theexponential hierarchy between the IR scale and the UV scale, or in gravitational termsa reduction of warp factor hierarchy. In this extreme regime, the effective field theory ofKKLT must be significantly modified (but we will argue in Section 6 that this regime turnsout to be dynamically avoided in KKLT). We consider the N = 2 O7 orientifold of type IIB on T × K T factor with four fixed points (as analyzed in [96]). This is a weakcoupling description of F-theory on K × K K SO (2 N ) × U Sp (2 M ) with N ≤
16 D7 branes and M ≤
24 D3branes for D3/D7 branes on O7 planes, and their subgroups U ( N ) × U ( M ) for D3/D7branes away from O7 planes.In the unitary case, we can engineer precisely the situation analyzed in the previoussection: a stack of N D7 branes surrounded by N D3 branes at radius r , with Λ (cid:28) r .The classical K3-volume is defined as a modulus at scales r ≥ r . As we take Λ → r δN D3 branes towards the center means sending the massof δN flavors to zero. This reduces the beta-function coefficient of the seven-brane gaugetheory 2 N → N − δN . (5.1)As a consequence, strong coupling effects become important at smaller K3 volume Vol( K − g s ∼ N − δN . Moreover, once δN = N charge-screening D3 branes have been sent tothe origin r → U ( N ) × U ( N ) gauge theory from N D7branes and N D3 branes with a massless bi-fundamental hyper and another hypermultipletin the (1 , Adj). The 7-brane stack is still asymptotically free, but this is now only due tothe running of the dilaton while the conformal factor remains constant. At a radial scalecorresponding to the new strong coupling scale of the 7-brane stack we get Im(ˆ τ ) = Im( τ )and the holomorphic 7-brane coupling vanishes. Below this radial scale the string frameK3 volume is smaller than unity so we should ’T-dualize’ ˆ τ ↔ τ . (5.2)This corresponds to Seiberg duality on the field theory side [98], and sends the N D7 branesand N D3 branes to a stack of N D3 branes. Indeed, for SU ( N c ) gauge theory, Seibergduality makes sense when there are at least as many flavors (D3 branes) as N c [99]. In thedual description, the dilaton stays constant, and the conformal factor approaches the oneof the AdS × S throat, with typical length scale of order N .As we further decrease the K3 volume as defined in the outside region r ≥ r the radialscale below which one should T-dualize increases further due to the running of the dilatontowards strong coupling at large r until the negative D7 charge of the bulk has been sweptup as well. Beyond this, the entire compactification should instead be described by theT-dual frame. In the absence of local charges, i.e. with unbroken (conformal) D7-D3 gaugesector ( SO (8) × U Sp (12)) , T-duality simply maps the orientifold to itself [100], and the K3-volume is uniquely defined independent of radial scale. For a generic brane configuration itis natural to define the K3-volume and dilaton τ zero modes (ˆ τ , τ ) at some generic position z on the T / Z (cid:39) P . As we send ˆ τ − τ → P and the spreading of branes should lead to recombination of D3/D7 charges Note that this is not a monodromy transformation. τ D , ) ≡ Im(ˆ τ − τ ) = 0.Therefore, in the range 0 ≤ Im( τ D , ) (cid:46) N ≤ , (5.3)we expect Im( τ D , ) to be a measure of how much un-screened D3/D7 charge remains. Wewill say more about this once we consider charge recombination against a warped throatin Section 5.2. Finally, we expect that the above discussion carries over directly at leastfor genuine N = 1 compactifications on O7 orientifolds with seven-branes wrapped on K3surfaces. If moreover the CY threefold is K3- fibered and the orientifold involution acts onlyon the base of the fibration (with fixed points), one would hope to be able to describe suchvacua non-perturbatively in fiber volumes and field-strengths along the fiber by an N = 1version of [77–80]. In Section 5.1 we have collected evidence indicating that the non-perturbative resolutionof singularities of the conformal factor effectively localizes the negative D3 charge on areal co-dimension one locus that becomes macroscopic as Vol( X ) / → | Q D | , and sweepsout the entire CY threefold as Vol( X ) →
0. It appears unavoidable that the amountof positive D3 charge, hosted in fluxes or mobile D3 brane swept up by the membraneeffectively recombines with the negative charge on the membrane thus reducing the overallamount of unscreened D3 charge Q D . So far, we have argued for this phenomenon fromthe perspective of the gauge theories residing on the negatively charged defects, but onecan argue for the same result from the perspective of a region carrying overall positive D3charge:Specifically, we consider a warped throat with total D3 charge N := | Q D | . As usual,the D3 charge is spread out along a radial direction such that the total integrated D3charge up to radial distance r is given by N eff ( r ) := N + 32 π g s M log( r/r UV ) , (5.4)for r ≤ r UV [16, 101]. From the local solution of the conformal factor, one notices that if One might expect that such recombination of negative D3 charges on seven-branes against bulk fluxeschanges the classical flux superpotential. However, the size of the strongly curved region is exponentiallydependent on the vev’s of the K¨ahler moduli, so there is no room for additional corrections to the 4dsuperpotential beyond the standard non-perturbative expansion. We thank Daniel Junghans for a usefuldiscussion about this point. t (cid:46) N the effective membraneof defects pinches off the throat at a radial scale r ∗ s.t. N eff ( r ∗ ) ∼ t .this running is cut-off at intermediate radial distance r ∗ (cid:28) r UV due to e.g. the spreading ofseven-branes down into the throat, the apparent volume of the throat at the cut-off valueis still of order t ∼ (Vol(throat)( r ∗ )) ≈ (cid:18) (cid:19) N eff ( r ∗ )4 π , (5.5)and the warp factor log-hierarchy between that point and the infra-red end is given by A ( r IR ) − A ( r UV ) =: log( a − ) ≈ π N eff ( r ∗ ) g s M . (5.6)Since the overall volume modulus corresponds to an additive constant to e − A we seethat for sufficiently small value of the K¨ahler modulus the semi-classical singularity at thelocus where e − A = 0 starts to walk down the throat, effectively pinching of the throat atsmaller radii. At least when the negative charge comes from seven-branes wrapping K3we can argue that the pinching of the throat is non-perturbatively replaced by a set ofelementary monodromy defects, see Figure 16. In this regime, one can think of the systemas a Randall-Sundrum throat with total D3 charge N eff ( r ∗ ) and a Planck-brane residing at r = r ∗ carrying − N eff ( r ∗ ) units of charge. If this occurs at radial position r ∗ , the typicallength scale of the setup is given by eq. (5.5), and we can think of all the positive D3charge residing at larger radii as having been absorbed by the effective membrane carryingthe negative charge.As a consequence, the throat hierarchy starts to become exponentially sensitive tothe value of the K¨ahler modulus due to the screening of charges above the dynamically36djusting UV-cutoff r ∗ ,log( a − ) −→ π N eff ( r ∗ ) g s M ≡ π (cid:94) Vol( K g s M /c , c = O (1) . (5.7)Here, the O (1) constant depends on the precise relation between the throat volume mea-sured near the radial scale r ∗ and the value of the real part of the holomorphic modulusˆ τ − χ ( S )24 τ (cid:39) (cid:94) Vol( K Crucially, the validity of the standard KKLT EFT starts to breakdown once a significant fraction of the throat fluxes that induce the running of the warpfactor are swept up by the effective membrane carrying the negative D3 charge. This isbecause the uplift starts to become exponentially sensitive to the volume modulus t . Asargued in [47, 50] this regime begins once t (cid:46) N D . However, and this will become crucialin Section 6 when we discuss the KKLT uplift in more detail, in the regime t (cid:46) N D we alsono longer have a relation S ED ∼ t where S ED is the action of the leading ED3 instantoncontributing to the superpotential. Now consider again the KKLT scenario and again for simplicity we set h , = 1 and assumethat the K¨ahler modulus T is stabilized by an ED3 instanton wrapping some divisor D .Furthermore, in order for the ED3 instanton to contribute to the superpotential even in theabsence of worldvolume and background fluxes, we will assume that D intersects the O7plane transversally along some curve and that D is rigid, i.e. h i ( O D , D ) = 0 for i = 1 , Q denotesthe negative D3 charge spread along the seven-branes.We will also (for now) set the K¨ahler modulus T to a (very) large value. As shown inSection 2 in the vicinity of the seven-branes the conformal factor will vary logarithmically,matching the one-loop running of the gauge theory living on the seven-brane stack, while If they were exactly the same we would get c = π (cid:0) (cid:1) − . D supporting an ED3 instanton responsiblefor moduli stabilization. We depict the large volume regime where the D7-branes can betreated semi-classically, i.e. they are all stacked on top of each other. The bulk CY isweakly curved everywhere except exponentially close to the seven-branes.away from the sources it can be treated as a constant. The (complexified) ED3 action is S ED = 2 π (cid:90) D (cid:18) e − A J ∧ J − iC (cid:19) . (6.1)If we assume that D is the generator of the cone of effective divisors this is equal to thereal part of our K¨ahler modulus T . We may write the unit volume K¨ahler form J as J = (cid:18) κ (cid:19) [ D ] , (6.2)with triple intersection number κ , where [ D ] is the (harmonic) Poincar´e dual two-form tothe divisor D . Using that dJ = 0 we can relate the divisor volume Re( T ) with the overallvolume modulus t (defined in eq. 2.5),Re( T ) = (cid:90) D e − A J ∧ J = (cid:90) X e − A J ∧ J ∧ δ (2) D = (cid:18) κ (cid:19) t − (cid:90) X d ( e − A ) ∧ ω ∧ J ∧ J , (6.3)38igure 18: Same CY as in Figure 17 but at small value of the volume modulus t ∼ N . Thenon-perturbative splitting of seven-branes has eaten up almost the entire bulk CY but hasnot yet entered the warped throat. The inside region (on the left) that has been swept upby the spreading of seven-branes is strongly curved, while the outside region (on the right)is still weakly curved.with δ -function two-form δ (2) D = [ D ] + dω , for some one-form ω , and we have integratedby parts and used that [ D ] ∧ [ D ] ∧ [ D ] = κ √ g CY d y .In the large volume limit we can treat e − A as a constant and we get a simple relationbetween the overall volume modulus and Re( T )Re( T ) → (cid:18) κ (cid:19) t as t → ∞ , (6.4)but this relation breaks down once warping becomes significant because the second term in(6.3) cannot be neglected. In particular, in a situation where we have engineered a warpedthroat carrying a large amount of positive D3 charge N D , we can decrease the overallvolume modulus into the critical regime t ∼ N D where the classical vanishing locus of theconformal factor e − A has swept up almost the entire bulk CY but has not yet crept intothe warped throat (see Figure 18). If the divisor D does not significantly reach into thethroat, it has been swept up by the vanishing locus of the conformal factor as well. Inparticular, as we have argued that volumes are O (1) in the interior region, it follows thatat this point in moduli space the ED3 action is of order one, i.e.Re( T ) = O (1) . (6.5)39evertheless, as argued in [47, 50] the overall volume modulus t is of order N . This isconsistent with eq. (6.5) because the relation of eq. (6.4) holds only at very large volume t (cid:29) N . Since the leading contribution to the non-perturbative superpotential scales as W np ∼ e − πT = O (1) , (6.6)we see that the instanton expansion is poorly controlled at this point in moduli space, andthe scale of the stabilizing potential becomes large. Since the warped throat still carriessignificant D3 charge of order N , an anti-brane uplift gives a contribution to the scalarpotential that is much smaller than that of the non-perturbative superpotential. In otherwords, the uplift is too small to reach a de Sitter vacuum. But as argued in [47], in theregime where the singularities of the conformal factor can be completely neglected, i.e. t (cid:29) N D , the uplift is too large giving rise to a runaway towards large volume. Hence, bycontinuity, there must exist an intermediate regime with t (cid:46) N D , < Re( T ) < N D (6.7)such that the uplift precisely competes with the stabilizing potential (depicted in Figure19). If this point actually corresponds to the minimum of the scalar potential shoulddepend only on the value of the classical flux superpotential W , i.e. e − π Re( T ) ∼ | W | (cid:28) . (6.8)Indeed, the AdS KKLT minimum arises via a competition of the classical flux superpoten-tial with the leading non-perturbative correction and is thus very robust against correctionsto the K¨ahler potential, as already pointed out in [3]. Even if significant non-perturbativecorrections would render the K¨ahler potential hard to compute it is difficult to imaginehow this could drive the KKLT minimum to strong coupling T = T s = O (1). This wouldrequire a quite remarkable fine-tuning | ∂ T K − π | T = T s (cid:46) | W | (cid:28) W . Interestingly, even though the non-perturbative expansion of the superpotential is con-trolled by virtue of a small classical flux superpotential, the gauge theory hosted on theseven-branes that dominate the D3 tadpole is strongly coupled in the sense that classicalR-symmetry breaking spurions take values of order the strong coupling scale of the gauge We thank Daniel Junghans for a useful discussion over this point. t (cid:46) N . The non-perturbative splitting of seven-branes has eaten up an O (1) fraction ofthe bulk CY but leaves enough room for a large physical volume of the divisor D , i.e.Vol( D ) ∼ log( | W | − ) (cid:29) O (1) fraction of the bulk CY is now strongly curved, one might be worriedabout the stability of the complex structure since their potential is (in practice) computedusing the large volume approximation. However, since the size of the strongly curved re-gion depends explictly on the K¨ahler moduli, its presence can enter the superpotential onlynon-perturbatively in T . Since a small W ensures that non-perturbative corrections arenegligible we can still rely on the classical approximation of W in order to compute the F-term potential for the CS moduli and the dilaton. Again, obtaining a good approximationof the K¨ahler potential may be difficult. However, if all moduli have a steep potential atlarge volume, and if W is small, the F-terms are again independent of the K¨ahler potentialto good approximation, D a W = ∂ a W + ∂ a KW ≈ ∂ a W . If some of the CS moduli havemasses of order W one has to be a bit more careful, but at least the subset of solutions ofthe supergravity F-terms of the light moduli which can be deformed to nearby solutions of ∂ a W = 0 should survive because they again rely on a competition between different termsin the superpotential (the solutions of [21–23] are of this type). As a consequence we donot see how the presence of the large strongly curved region could jeopardize the schemeof moduli stabilization. Finally, in order to argue for an actual dS vacuum one has to alsoensure that the anti-brane uplift potential can be computed reliably. But up to the irrel-evant overall e K factor in the F-term potential, the uplift potential is determined by the41ocal Physics of the IR region of the throat which remains in its supergravity regime nearthe AdS KKLT minimum. Thus, it appears that even the KKLT dS solution is genericallysafe from corrections induced by the strongly curved region in the bulk.Nevertheless, the fact that generically an O (1) fraction of the bulk CY is stronglycurved in KKLT should be kept in mind in the construction of phenomenological modelsbased on the KKLT scenario. Finally, since the local divisor volume (of K3) is not single-valued upon encircling the elementary defects that we have described, we see that at theKKLT minimum a significant part of the full bulk CY may be best thought of as a U -foldor T -fold (see e.g. [102] for a review of non-geometric backgrounds in string theory). In this paper we have revisited a potential control problem for the KKLT scenario of modulistabilization [3]: as argued in [47, 50], when K¨ahler moduli take values near a meta-stableKKLT de Sitter vacuum the backreaction radii associated with D3 charges from fluxes andbranes can no longer be neglected anywhere in the compactification. In particular, thesingular near-brane behavior of the supergravity fields in the vicinity of sources of negative
D3 charge extends over an O (1) fraction of the bulk CY.We have restricted ourselves to compactifications where negative D3 charge is sourcedby seven-branes wrapped on K3 surfaces, realizing (at low energies) N = 2 pure Yang-Millssectors. In Section 4 we have argued that the singular near-brane behavior of supergravityfields is resolved by non-perturbative effects in the α (cid:48) -expansion that can be understoodquantitatively in terms of instanton effects in the gauge theory. In a way that is analo-gous to the resolution of dilaton singularities near perturbative (in g s ) O7 planes [61, 62]the non-perturbative corrections in the Seiberg-Witten solution of the IR-behavior of thegauge theory can be understood to split a perturbative (in α (cid:48) ) D7 brane into a boundstate of two mutually non-local defects collectively carrying the negative D3 charge. Thisnon-perturbative splitting of branes then stops the perturbative running of supergravityfields before a singularity forms. Concretely, N wrapped D7 branes split into 2 N defectsdistributed along a circle around the perturbative location of the branes, leaving behind an’inside region’ with O (1) local K3-volume (in Einstein frame). Each of the 2 N defects canbe thought of as a cosmic defect in 6 d N = (2 ,
0) supergravity, realized by compactifyingtype IIB string theory on K3. We have described the monodromies of the conformal factor We thank Alexander Westphal for a helpful discussion about this. SU ( N )Yang-Mills theory we have concluded that a stack of n mutually-local such defects hoststhe Argyres-Douglas (AD) SCFT ( A , A n − ).Because the resolution of singularities can be described entirely in terms of the lowenergy degrees of freedom already known to be present in the 4d EFT we have concludedthat even a macroscopic strongly curved ’inside region’ would not invalidate the KKLTEFT employed in [3] to discuss the stabilization of K¨ahler moduli. Moreover, we haveargued in Section 6 that a small flux superpotential should ensure that a sufficiently largepart of the bulk CY remains weakly curved such that the leading ED3 instanton has largeaction, thus dynamically preventing significant alterations of the 4d EFT treatment of theanti-brane uplift.It would be interesting to understand in detail the resolution of the singularity of theconformal factor also for other sources of negative D3 charge. Arguably, O3 planes areeasiest to understand because the singularity of the conformal factor can be related viaT-duality to the dilaton singularity of an O7 plane wrapped on T which is resolved inF-theory [61]. Also, it appears plausible that one can understand the case of seven-braneswrapped on rigid divisors because they realize N = 1 pure Yang-Mills theories which canbe thought of as massive deformations of the N = 2 theory that we have considered.Perhaps the most mysterious case is that of wrapped seven-branes on surfaces with manynormal bundle deformations. The gauge theory is IR-free so explaining the resolution of thenear-brane singularity in terms of gauge theory instantons seems difficult. Furthermore, itwould be interesting to turn things around and ask what other AD SCFTs can be realizedby stacks of exotic branes in N = (2 ,
0) supergravity. Finally, one might be able to useour results to improve control over the K¨ahler uplifting scenario of [5] where a confiningseven-brane theory is at most marginally weakly coupled at the UV-cutoff. Acknowledgments:
We thank Michael Douglas, Daniel Junghans, Manki Kim, LiamMcAllister, Thomas Van Riet, Alexander Westphal, Raffaele Savelli and Eva Silversteinfor useful discussions and/or comments on a draft. We are particularly grateful to AndreasBraun for explaining ref. [82] to us and for useful discussions surrounding it. The work ofJ.M. was supported by the Simons Foundation Origins of the Universe Initiative. F.C. issupported by STFC consolidated grant ST/T000708/1. We thank Alexander Westphal for useful discussions about this. Bulk equations of motion
We would like to show that the non-perturbatively corrected 10d metric still solves thetree level equations of motion (but with different singular source terms). We will zoom-inclose to the stack of D7 branes so we approximate the total space as C × K
3. First, letus forget about D3 brane charges. It is well known that the inclusion of 7-brane chargesmodifies the Ricci flat 10d metric as (see e.g. [103]) ds → dx + Im( τ ( a b )) | da b | + ds K , (A.1)where a b is a complex coordinate parameterizing the transverse complex plane. As in [17],the inclusion of D3 charges further modifies the metric as ds → e A ( a, ¯ a ) dx + e − A ( a b , ¯ a b ) (Im( τ ( a b )) | da b | + ds K ) , (A.2)where e − A satisfies an electro-static equation − ∇ e − A = ρ D , (A.3)with ρ D the D3-charge density and ∇ the 6d Laplacian of Im( τ ( a b )) | da b | + ds K . Here, wehave normalized ds K to be the unit volume metric on K3. In the approximation where allD3-charges are smeared over K3 the 6d electro-static problem reduces to a two-dimensionalone over the transverse space parameterized by a b . Since the conformal factor Im( τ ) dropsout in two dimensions, away from sources the equation of motion reduces to ∂ a b ¯ ∂ ¯ a b e − A = 0which is indeed true when e − A ( a b , ¯ a b ) = Im(ˆ τ ( a b )) for ˆ τ ( a b ) holomorphic.At leading order in the α (cid:48) expansions, τ and ˆ τ are indeed the gauge couplings on probeD3 and D7 branes as is seen from expanding the DBI action to leading order in the branevelocities, S D ⊃ − π (cid:90) d x (cid:18)
12 Im( τ ) | ∂a D | + 12 g K ij ∂φ i ∂φ j + ... (cid:19) , (A.4) S D ⊃ − π (cid:90) d x (cid:18)
12 Im(ˆ τ ) | ∂a D | + ... (cid:19) , (A.5)where ds K ≡ g K ij dφ i dφ j is the unit volume K3 metric, and a D /D denotes the positionof the D3 respectively D7 brane in the transverse plane. Inspecting the CS term of aD7-brane one sees that the α (cid:48) correction leads to the replacement ˆ τ → τ D := ˆ τ − χ ( K τ
44n (A.5). This is indeed what is required by N = 2 supersymmetry: the kinetic terms ofthe scalars in vector multiplets are the imaginary parts of the gauge couplings and the fieldspace metric for the scalars in hypermultiplets is hyperk¨ahler because K3 is hyperk¨ahler.It remains to be shown that Re(ˆ τ ) is identified with the axion (cid:82) K C . For this, wesimply plug in the solution for the warp factor into the solutions of [17] F =(1 + ∗ ) de A ∧ d x = de A ∧ d x + ω K ∧ i ( ∂ a b − ¯ ∂ a b ) e − A ( a b , ¯ a b ) (A.6)= d (cid:0) e A ∧ d x + ω K Re(ˆ τ ( a b )) (cid:1) ≡ dC , (A.7)where ω K is the volume form on K3. Here, we have used that e − A ( a b , ¯ a b ) = Im(ˆ τ ( a b ))and i ( ∂ − ¯ ∂ )Im( f ) = d Re( f ) for a holomorphic function f . So indeed, it follows thatRe(ˆ τ ) = (cid:82) K C .Finally, the action of a (energy-minimizing) F-string stretched between two D7 branesat points ( a (1) b , a (2) b ) is given by S F = − π (cid:90) Σ d σ √ g Σ (cid:112) Im( τ ) = − π (cid:90) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) a (2) b a (1) b da (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − π | a (1) b − a (2) b | (cid:90) dt , (A.8)where Σ denotes the worldsheet and g Σ is the induced Einstein frame metric on theworldsheet. Therefore, defining a := a (1) b − a (2) b we get the central charge of a W-boson inthe SU (2) theory realized by the two D7 branes. Since the semi-classical monodromy inthe field theory corresponds to exchanging the two D7 branes we get indeed that a → − a , (A.9)as in [53]. Likewise, the tree-level DBI action of a stretched (energy-minimizing) stringfrom a D5 brane wrapped on K3 is given by S D = − π (cid:90) Σ d σ √ g Σ (cid:112) Im( τ ) = − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) da b e − A ( a b , ¯ a b ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) dt , (A.10)which is corrected due to induced D1 and F-string charge from the K3-curvature and (cid:82) K C -profile respectively to give S D = − π | a D | (cid:90) dt , with a D := (cid:90) a (2) b a (1) b da b τ D . (A.11)45nder the field theory monodromy at infinity the central charge of the magnetic monopole(as realized by a D5 string stretched between two D7 branes) a D indeed transforms as a D → − a D + 2 a , (A.12)as in [53]. The minus sign again comes from the fact that the two D7s are interchanged,and the additive term is generated because the stretched D5 string is rotated half-wayaround each of the two D7 branes thus picking up the monodromy of the bulk field τ D asdescribed in Section 4.1. References [1] E. Silverstein, (A)dS backgrounds from asymmetric orientifolds , Clay Mat. Proc. (2002) 179, [ hep-th/0106209 ].[2] A. Maloney, E. Silverstein, and A. Strominger, De Sitter space in noncritical stringtheory , in
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