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KIAS-P10001 = 2 SCFTs: An M5-brane perspective N 0 1 0 2 n Bin Chen1, EoinO´ Colgaín2, Jun-Bao Wu2 and HosseinYavartanoo2 a J 6 1 DepartmentofPhysics, ] andStateKeyLaboratoryofNuclearPhysicsandTechnology, h t PekingUniversity, - p Beijing100871, P.R.China e h [ 2KoreaInstituteforAdvancedStudy, 1 v Seoul,Korea 6 0 9 0 . 1 0 0 1 : v Xi Abstract r a Inspiredbytherecentlydiscoveredholographicdualitybetween = 2SCFTs N and half-BPS M-theorybackgrounds,westudyprobeM5-branes. Thoughour main focus is supersymmetric M5-branes whose worldvolume has an AdS n factor, we also consider some other configurations. Of special mention is the identification of AdS and AdS probes preserving supersymmetry,with only 5 3 thelattersupportingaself-dualfield strength. 1 Introduction Four-dimensional = 2 supersymmetric theories are truely remarkable. Compared to N = 4 supersymmetric theories, which are finite, they are much richer in physics, but N yet still solvable. Especially, they provide a window to study the nonperturbative aspects of quantum field theories and have been widely studied for more than fifteen years since Seiberg and Witten’s monumental works [1, 2]. However, the surprises they present to us havenot come to the end. Recent developmentson four-dimensional = 2 superconfor- N maltheoriesstartingfrom[3]havedrawnlotsofattention. Thisclassofgeneralizedquiver theories could be constructed geometrically by wrapping M5 branes on Riemann surfaces with genus and punctures. The electric-magnetic duality and the Argyres-Seiberg duality [4] have since been generalized to these theories. It turns out that the gauge couplings of the theory are encoded in the complex structure moduli of the Riemann surface, includ- ing the position of the punctures. More interestingly, it was conjectured in [5] that the NekrasovpartitionfunctionofthesetheorieswithSU(2) gaugegroupscould berelated to theconformalblocksandcorrelationfunctionsoftheLiouvilletheory. Non-localoperators inthesefour-dimensionaltheorieshavebeen studiedin [6, 7, 8, 9,10](See also[11]). The holographic dual of these theories in the large N limit was neatly studied in [12]. The theory on the gravity side of this AdS/CFT correspondence is M-theory on back- groundswhicharetheproductsofAdS spacetimeandsix-dimensionalinternalmanifolds 5 withSU(2) U(1)isometry. Thesegravitybackgroundsbelongtothegeneralgeometries × foundin [13]. Of particularinterest amongthesebackgroundsis theso-called Maldacena- Nuń˜ez(MN) geometry, which was first discovered in [14] by considering the IR limit of M5-branes wrapped on a Riemann surface. In thiscase, thesix-dimensionalinternal man- ifold is simply an S4 fibered over the Riemann surface. On the other side, the field theory corresponding to the MN geometry comes from M5-branes wrapping the same Riemann surface. It is remarkable, that in this case, there are no punctures on the Riemann surface and that the building block of the quiver gauge theory is a strongly coupled superconfor- maltheoryT withthreeSU(N)globalsymmetries,yetwithoutacouplingconstant. One N motivationofthiswork istounderstand thisintriguingT theoryfrom itsgravitydual. N Probebranesplayanimportantroleinthegravitysideofholographiccorrespondences. These branes are intrinsically stringy but accessible. Among other things, they could be dualto local operators [15, 16, 17], loop operators [18, 19, 20], surface operators [21, 22], ordomainwalls[23]inthefieldtheorysideofvariousAdS/CFTcorrespondences. Adding suitablebranes can alsoadd flavortothefield theory [24]. In this paper, we plan to start the search for interesting probe M5-branes in the LLM geometries studied in [12]. We mainly focus on the simplest MN background which we 1 haverederivedin theappendixto includethefluxes. Theformofthesolutionis 1 W−2/3 (dx2 +dy2) ds2 = κ˜2/3 W1/3ds2 + W 11 2 AdS5 4 y2 (cid:18) (cid:20) 2 dx + Wdθ2 +cos2θ(dφ2 +sin2φ dφ2)+2sin2θ dχ+ , (1) 1 1 2 y (cid:18) (cid:19) (cid:21)(cid:19) 1 dx 1cos3θdxdy H = κ˜ [3+cos2θ]sinθcos2θdθ(dχ+ )+ d2Ω, 4 −4W2 y 4 W y2 (cid:18) (cid:19) where φ ,φ parameterize a two-sphere, x,y denote a hyperbolic Riemann surface Σ of 1 2 2 constantnegativecurvature,and W is W = 1+cos2θ. (2) The constant κ˜ denotes an additional scale factor that will be accounted for by ensuring thatthefluxis correctly quantised.1 This geometry may also be obtained from the most general solutionsof eleven dimen- sionalsupergravitypreserving = 2superconformalsymmetry[13]asdescribedin[12]. N Althoughwe focus on the above background, we believe our results can be generalized to moregeneralLLMgeometries. Intheliterature,someBPSprobebraneshavebeenstudied in[12], and half-BPS M2-branedual toloopoperatorhaveappeared in [6]. Starting with Killing spinors of MN geometry and kappa symmetry for M5, one can search for BPS M5-branes. As a first step, one needs to determine the Killing spinors preserved by the MN solution. Luckily, this has already been done for the most general solution [13] and the Killing spinors corresponding to the analytically continued solution correspondingtoMNhaveappearedin[6]. Thelatterappearswithoutderivation,sointhe appendix we validate their claim by following a similar decomposition to that appearing in [13] (see also [25])2. The result of that exercise is that the eleven-dimensional Killing spinors of the MN solution can be expressed in terms of AdS (ψ) and S2 (χ ) Killing 5 + spinorsas ǫ = eλ/2ψ (1+iσ γ )χ e−2iφ0γ10eiχ/2ǫ , 3 (4) + 0 ⊗ ⊗ ⊗ ǫc = eλ/2ψc (1 iσ γ )χ e−2iφ0γ10e−iχ/2γ ǫ , (3) 3 (4) + 7 0 ⊗ − ⊗ ⊗ wherethesuperscriptcdenotes theconjugateand √2cosθ sinθ κ˜2/3W1/3 sinφ = , cosφ = , e2λ = . (4) 0 0 √W −√W 8 1Itisrelatedtotheκin[12]byκ˜ =24Nκ. 2Wealsocuresometyposin[6]. 2 Theconstantspinorǫ satisfiesthefollowingprojectionconditions 0 γ ǫ = ǫ , iγ ǫ = ǫ . (5) 9 0 0 78 0 0 Withpossibledualnon-localobjectsinfieldtheoryinourmind,wepayprincipalatten- tion to M5-branes whose worldvolumes have AdS (2 m 5) factors. The brane with m ≤ ≤ worldvolume AdS S1 has been studied previously in [12]. However, we find that al- 5 × thoughturningonself-dualthree-formfieldstrengthontheworldvolumeinasuitableway doesnotbreak supersymmetry,theequationsofmotionofM5-braneswillnotbesatisfied. This comes as some surprise as the Kaluza-Klein reduction of the two-form potential on the S1(χ) gives rise to a U(1) gauge field in AdS corresponding to a global symmetry 5 rotatingthephaseofadualbifundamentalfield [12]. Moreover, we find BPS M5-branes with an AdS factor. This brane should be dual 3 to some two-dimensional object in the field theory side. However it is not dual to the supersymmetric surface operator studied in [7, 9], since this brane wraps the Riemann surface in the six-dimensional internal space, instead of intersecting with this Riemann surface at a point. In this case, we find that we can turn on a suitable self-dual three- form field strength on the worldvolume such that the BPS condition and the equations of motionarebothsatisfied. WealsofindBPSM5-branesnotembeddedalongtheAdS radial 5 direction that satisfy the equations of motion hinting that there should be non-BPS AdS branestherealso. Inaddition,inspiredbysomeprobeM5-branesinAdS S4[26,27],we 7 × turntosearchingforM5-branesinMNbackgroundwithmorecomplicatedworldvolumes. As a result, we find M5-branes with an AdS S1 and AdS S2 factors which are 3 2 × × completely embedded in the AdS part of the background. We explicitly illustrate that 5 genericallythesebranes arenon-supersymmetric. In the next section, we move to review the M5-brane equations of motion and BPS condition. With these tools at hand, we study various probe M5-branes in section 3. In section4,weexaminemoreexoticembeddingsinAdS beforeconcluding. Sometechnical 5 detailsarelocated intheappendices. 2 M5-brane review Inthissection,wereviewtheM5-branecovariantequationsofmotionsincurvedspacetime and discuss the condition for the M5-brane probe to preserve supersymmetry. For earlier work on various aspects of the M5-brane, see [29, 30, 31, 32, 33, 34] (For a review of M-theorybranes,see[35]). ThissectionechoesthebriefreviewoftheM5-branepresented in[27]and werefer thereader thereforafurtheraccount oftheM5-braneaction. 3 Focusing solely on the bosoniccomponents, we simplyhave two equations of motion: ascalarand atensorequation. Thescalarequationtakes theform Q 1 1 Gmn c = ǫµ1···µ6 Ha + Ha H P c (6) ∇mEn √ g 6! 7µ1···µ6 (3!)2 4µ1µ2µ3 µ4µ5µ6 a − (cid:0) (cid:1) andthetensorequationis oftheform Gmn H = Q−1(4Y 2(mY +Ym)+mYm) . (7) m npq pq ∇ − Here our notation is as follows: indices from the beginning(middle) of the alphabet refer to frame(coordinate) indices, and the underlined indices refer to target space ones. More detailsofourconventionsmay befoundintheappendices. Appearing in the equations of motion, we have the following quantities which are defined intermsoftheself-dual3-form field strengthhon theM5-braneworldvolume k n = h hnpq, (8) m mpq 2 Q = 1 Trk2, (9) − 3 mq = δ q 2k q, (10) p p − p H = 4Q−1(1+2k) qh (11) mnp m qnp Notethath isself-dualwithrespecttoworldvolumemetricbutnotH . Theinduced mnp mnp metricissimply g = a bη (12) mn EmEn ab where a = ∂ zmEa. (13) Em m m Herezm isatargetspacetimecoordinate,whichbecomesafunctionofworldvolumecoor- dinate ξ through the embedding, and Ea is the component of target space vielbein. From m theinduced metric,wecan define anothertensor 2 Gmn = (1+ k2)gmn 4kmn. (14) 3 − Wealsohave P c = δc m c. (15) a a −Ea Em Note that in the scalar equation of motion, the covariant derivative c involvesnot ∇mEn onlytheLevi-CivitaconnectionoftheM5-braneworldvolumebutalsothespinconnection ofthetarget spacetimegeometry. Moreprecisely,onehas c = ∂ c Γp c + a bωc (16) ∇mEn mEn − mnEp EmEn ab 4 where Γp is the Christoffel symbol with respect to the induced worldvolume metric and mn ωc isthespinconnectionofthebackgroundspacetimepulledback to theworldvolume. ab Moreover, there is a background 4-form field strength H4a1···a4 and its Hodge dual 7-formH7a1···a7: H = dC 4 3 1 H = dC + C H (17) 7 6 3 4 2 ∧ TheframeindicesonH andH intheaboveequations(6)and(7)havebeenconvertedto 4 7 worldvolumeindiceswithfactors of c . From thebackgroundfluxes, wecan define Em Y = [4⋆H 2(m⋆H +⋆Hm)+m⋆Hm] , (18) mn mn − where 1 ⋆Hmn = ǫmnpqrsH (19) 4!√ g pqrs − Thefield H is defined by mnp H = dA C , (20) 2 − 3 where A is a 2-form gauge potential and C is the pull-back of the bulk gauge potential. 2 3 Fromitsdefinition,H satisfiestheBianchi identity dH = H (21) − 4 whereH isthepull-backofthetarget space 4-formflux. 4 In general, the supersymmetric embeddings of a probe brane in a background may be determinedfromthekappa-symmetrycondition Γ ǫ = ǫ. (22) κ ± Here,Γ denotesthegammamatrixassociatedtotheprobe,ǫdenotestheKillingspinorof κ thebackgroundand thesignaccounts for thechoice between brane and anti-braneprobes. Theamountofunbroken supersymmetrymaybe determinedby keeping track oftheaddi- tionalprojectionconditionsthat arisefromtheaboveequation. SpecializingtotheMNbackgroundwithM5-braneprobes,thekappasymmetrymatrix Γ maybewrittenfollowing[28] M5 1 ΓM5 = 6!√ gǫj1···j6[Γ<j1···j6> +40Γ<j1j2j3>hj4j5j6]. (23) − Here g is the determinant of the induced worldvolume metric component, h is the j4j5j6 self-dual3-form ontheM5-braneandΓ<j1···jn> isdefined as Γ<j1···jn> = Eja11 ···EjannΓa1···an, (24) 5 whereΓa1···an istheproductoftheGammamatricesin orthonormalframe. We pause here to make a brief comment. Denoting the worldvolume of the M5 by ξa,a = 0, 5, in the case of a simple probe configuration, we may rewrite the above ··· projector(22)as [αΓ +β(Γ Γ )]ǫ = ǫ, (25) 012345 012 345 − ± where α,β denote arbitrary factors. Demanding it to be a projector, it is essential the left hand side squares to unity. In that event, β drops out completely and α2 = 1, meaning thatα = 1. Theimplicationofthisobservation,atleastforthesimpleprobesconsidered ± in this paper, is that if the M5-probe is not supersymmetric, then supersymmetry cannot be restored by introducing h. So the task in the rest of the paper is pretty straightforward: identifysupersymmetricprobesandthenturnonhtoseeifitpreservessupersymmetry. At each stage, itisalso imperitivetoensurethat theequationsofmotionare satisfied. 3 Supersymmetric probes In this section, we focus on the kappa-symmetry condition (22) and isolate probes that will preserve some supersymmetry. We descend in dimension of the part in AdS from 5 d=5 to d=2 and in each case, we enumerate the possibilities. Throughout we differentiate between probes that are AdS i.e. those incorporating the radial direction r of AdS and 5 thoselocatedat afixed r . We beginby examiningtheAdS probes. 3.1 Supersymmetric AdS probes Inthissubsection,wedescendfromAdS toAdS andidentifythesupersymmetricprobes 5 2 (ifany),beforeexaminingtheadditionalconstraintscomingfromtheequationsofmotion. As a warm-up, we begin with theAdS M5-brane probe which received someattentionin 5 [12]. AdS probes 5 In general, one can consider studying the probe brane with worldvolumeAdS in the 5 ×C MNbackground,where denotesacurveinthesix-dimensionalspacetransversetoAdS . 5 C We consider the to be parameterised by σ i.e zm(σ). Using AdS Poincare´ coordinates, 5 C anaturalchoicefortheM5embeddingis ξ = x , ξ = xi, ξ = r, ξ = σ, (26) 0 0 i 4 5 wherei = 1,2,3. 6 Adopting the gauge choice σ = χ, while permitting embeddings of the form x ≡ x(χ),y y(χ),thekappasymmetrymatrixΓ simplifiesto M5 ≡ 1 Γ = Γ Γ , (27) M5 01234 <χ> √g χχ wheretheinducedmetriccomponentis κ˜ 2/3 W (∂ x)2 +(∂ y)2 sin2θ (∂ x) 2 g = χ χ + 1+ χ , (28) χχ W 4 y2 2 y ! (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) andtheinducedgammamatrixis κ˜ 1/3 sinθ (∂ x) W1/2 (∂ x) (∂ y) χ χ χ Γ = 1 1 1+ γ + γ + γ . <χ> 4 2 9 7 8 ⊗ ⊗ W √2 y 2 y y (cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (29) Utilisingρ = i, therequirement forsupersymmetryΓ ǫ = ǫ thenreduces to 01234 M5 ± κ˜1/3W−1/3 sinθ (∂ x) W1/2(∂ x) W1/2(∂ y) 1+ χ γ + χ γ + χ γ ǫ = ǫ , (30) 9 7 8 0 0 √g √2 y 2 y 2 y ± χχ (cid:20) (cid:18) (cid:19) (cid:21) provided φ = π. This means that θ = π and W = 1. In addition, we rquire the probe to 0 2 belocatedat apointontheRiemann surface: ∂ x = ∂ y = 0, (31) χ χ so that the terms proportional to γ and γ disappear. The projection condition γ ǫ = ǫ 7 8 9 0 0 alsosinglesoutthepositivesignaboveindicatingthattheprobeisanM5-braneasopposed toan anti-M5-brane. Therefore,thecurve isexclusivelyalongtheχ-direction. Asnotedin[12],theθ = π C 2 conditioncorrespondstotheS2 shrinking,sothesuperconformalsymmetrySU(2) U(1) × symmetryofthebackgroundispreservedbythisprobe. Also,nosupersymmetryisbroken bythisprobe. Now that we have a supersymmetric probe, we may inquire whether it is possible to turnon self-dualh. As explainedearlier, thisproblemreduces toensuring a(Γ Γ )ǫ = 0, (32) 012 349 − wherewehavedefined a h = ( 012 + 349). (33) 2 E E Again using the decomposition (125) and the relationship ρ = i ρ = iρ , then 01234 34 012 ⇒ itispossibletoshowthatthisconditionis satisfied. 7 Having verified the kappa-symmetry condition is satisfied, it remains to show that the equationsofmotionare satisfied. Theinduced metricmaybewritten κ˜2/3 dx dxµ +dr2 ds2 = µ +dχ2 , (34) ind 2 r2 (cid:20) (cid:21) wherewehaveused Poincare´ coordinates. TheRHSofthetensorequation(7)iszeroasthebackground4-formfluxdoesnotpull back totheM5worldvolume. Thetensorequationis thensimply Gmn H = 0. (35) m npq ∇ As for scalar equation, the RHS vanishestrivially when c = 10. Forthe case with c = 10, 6 the RHS is non-vanishing for general θ due to the vol dθ dχ part of H . How- AdS5 ∧ ∧ 7 ever, the coefficient is proportional to cosθ, so it vanishes when it gets pulled back to the worldvolumeat θ = π. So, neglectingthiscase, thescalarequationissimply 2 Gmn c = 0. (36) ∇mEn Thisequationisquicklyconfirmedtobesatisfiedasitonlyhasonenon-trivialcomponent: κ˜1/3 κ˜1/3 4 = ∂ + = 0. (37) ∇rEr r√2r √2r2 Theansatzweconsiderforhis a h = 012 + 349 2 E E a(cid:0)κ˜ 1 (cid:1) 1 = dtdx dx + dx3drdχ , (38) 4√2 r3 1 2 r2 (cid:18) (cid:19) whereaisafunctionofr. Followingthetreatmentin [27], H maybeexpressed as aκ˜ 1 aκ˜ 1 H = dtdx dx + dx3drdχ. (39) (1+a2)√2r3 1 2 (1 a2)√2r2 − Asthebackground4-formfluxdoesn’tpullback,theBianchiidentity(21)issimplydH = 0. Thismeans that a = constant. (40) (1+a2)r3 Switchingthelocationofdr in theflux ansatzabovewouldmakethis a = constant. (41) (1 a2)r2 − OncetheBianchi issatisfied, onemay return tothetensorequation. HereGmn is diagonal and the only term of interest is H which is not zero unless a is a constant. So, we ∇r rx3χ 8 reachacontradictionandtheconclusionisthatthereisnosupersymmetricAdS M5-brane 5 probewithself-dual 3-formh. AdS probes 4 For AdS probes, a quick look at appendix B reveals that we must mix the spinor ψ with 4 its conjugate ψc. This is because ρ0124η+ and η− both have the same eigenvalue under ρ4. Thus,weconsider ρ ψ = cψc, (42) 0124 where c is a constant. The overall effect of this mixing is that the MN Killing spinor gets related toitsconjugatethroughthekappa-symmetrycondition. AdoptingtheM5embedding ξ = x , ξ = x (i = 1,2), ξ = r, ξ ,ξ M , (43) 0 0 i i 3 4 5 6 { } ⊂ where M denotes the space transverse to AdS , the kappa-symmetry condition may be 6 5 re-writtenas Γ cγ (2)(1+iσ γ )χ e−2iφ0γ10eiχǫ = (1+iσ γ )χ e+2iφ0γ10ǫ , (44) 7 3 (4) + 0 3 (4) + 0 √g ⊗ ⊗ 2 where we have multiplied across by eiχ/2γ and used Γ Γ . One may quickly 7 (2) ≡ <ξ4ξ5> recognizethatanecessary conditionsforsupersymmetryare γ Γ ,σ γ = 0, (45) 7 (2) 3 (4) ⊗ (cid:2) γ7Γ(2),γ10(cid:3) = 0. (46) (cid:8) (cid:9) Thelatterconditionmaybeignoredifφ = π. ThedirectionstransversetotheAdS space 0 5 are the product of a two-sphere with a four-dimensional space M = S2 M . Thus, in 6 4 × general,Γ canbealinearcombinationoftwoanti-symmeterisedgammamatrices,either (2) along S2, along S1 S2 with a direction in M , or along M . The three possibilities for 4 4 ⊂ γ Γ are, respectively 7 (2) iσ γ , σ γ γ γ , 1 γ γ , (47) 3 7 i 7 (4) µ 2 7 µν ⊗ ⊗ ⊗ where i = 1,2 and µ,ν = 7,8,9,10. All three choices fail to satisfy (45), so there is no supersymmetricprobewiththisembedding. AdS probes 3 For AdS probes there is no mixing required between the conjugate MN Killing spinors, 3 soforsimplicity,wesimplyuseǫ = ψ ξ andignoretheconjugate. Referringto(128)and ⊗ 9