CORE JHEP05(2015)130 brought to you by by you to brought Open Access Repository Access Open by provided Springer May 7, 2015 May 26, 2015 : April 29, 2015 or monopole- : : December 15, 2014 : Revised Accepted Published 10.1007/JHEP05(2015)130 Received doi: icture is completed by com- this single solution of Excep- n of Exceptional Field Theory. ng Theory, School of Physics, symptotic behavior at the core their bound states, in ten- and The 1/2 BPS brane spectrum, con- Published for SISSA by [email protected] , . 3 1412.2768 The Authors. c p-branes, M-Theory, String Duality

It has been shown that membranes and fivebranes are wave-like ,

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Queen Mary University ofMile London, End Centre for Road, London, Research in E1 Stri 4NS, England E-mail: Exceptional Field Theory Strings, branes and the self-dual solutions of Open Access Article funded by SCOAP Abstract: David S. Berman and Felix J. Rudolph ArXiv ePrint: eleven-dimensional supergravity may alltional be Field extracted Theory. from Theand solution’s at properties infinity such are as investigated. its a Keywords: like solutions in somebining higher the dimensional wave and theory. monopoleThis solutions Here solution into the solves a the p single twisted solutio sisting self-duality constraint. of fundamental, solitonic and Dirichlet branes and View metadata, citation and similar papers at core.ac.uk at papers similar and citation metadata, View JHEP05(2015)130 4 7 1 3 8 26 27 30 31 32 10 12 13 15 16 20 21 23 24 25 29 13 the context of supersym- seems to imply a lack of string theory. A duality — is manifest. n for the hidden symmetry es — once found, immediately = 11 D = 10 D = 4 super Yang-Mills theory and in the N – 1 – EFT = 10 7 E D 4.1.1 From4.1.2 wave to membrane The membrane/fivebrane bound state Certainly since 1995, this idea has been very successfull in B.1 Wave, membrane, fivebraneB.2 and monopole Wave, in String, FivebraneB.3 and Monopole D-Branes in in A.1 The TypeA.2 IIA theory The Type IIB theory 4.2 Type IIA4.3 solutions Type IIB solutions 4.1 Supergravity solutions 2.1 Basics of2.2 the Embedding supergravity into EFT 3.1 Interpreting the3.2 solution Twisted self-duality provokes a set ofunderstanding questions. of the The theory;and very one perhaps presence discover hopes of a to theory the discover in duality the which this reaso dualitymetric symmetry field theories. The S-duality present in 1 Introduction Duality symmetries have been atthe the presence of heart a of hidden developments symmetry in or relation between theori A Embedding the Type II theories into EFT B Glossary of solutions 5 Wave vs. monopole 6 Discussion 4 The spectrum of solutions 3 A self-dual solution in EFT Contents 1 Introduction 2 Exceptional Field Theory JHEP05(2015)130 ]. 9 se. cycles b or a which determines rus and the reduced . This then produces ven dimensions or less. R e for a self-dual theory. iguity in the description s of the gauge fields of planation of the esoteric any new dimensions and ne and string solutions of sional theory is equipped ries in the extended space ost simply this is achieved s to all the 1/2 BPS objects different perspectives in the dius perturbative regime we must n numerous applications and ge and the wave momentum. by the complex structure of = 4 theory that in turn forms ity. hen just a consequence of the s. The string itself is self-dual is field comes from a self-dual can form a small dimensionless ime as embedded in this theory rom the dimensional reduction an wrap either the e a supergravity analogue. be a single object in Exceptional N wave and monopole with a single f the geometric description of the . eminal work of Cremmer, Julia and Pope [ R ]. physical section condition ]. The SL(2) duality symmetry is then just 4 2 – 2 – = 2 super Yang-Mills theory is explained by the ]. This has led to a profound exploration of field N 3 – 1 ] was developed as a theory to make manifest the U- 8 – 5 2) theory to explain all manner of field theory properties , It will also be geometrically self-dual in the following sen 2) supersymmetry in six dimensions (in M-theory terms, the , 1 This object obeys a twisted self-duality constraint in term Exceptional Field Theory [ Central to these explanations is the fact that the six-dimen The more interesting thing happens when we compactify on a to The twisted self-duality equation was first described in the s 1 Exceptional Field Theory. five-dimensional Yang-Mills with coupling given by The solution is heuristically speakingfree a quantized superposition parameter of which a gives both the monopole char becomes ambiguous. Thus theof duality the is reduction. a Now consequnce whatsupergravity of of all 1/2 transform an BPS under amb states U-duality.Field in We Theory will the and descri theory? show Bra howin the ambiguity supergravity in that its transform reduction into lead each other under U-dual duality groups of M-theory.comes equipped The with theory something lives knownhow in as one the a may space carry with outThe a m U-duality reduction groups of enter whenthere the are due theory different to back ways to down the do to this presence reduction ele of and isomet usual spacet from relabelling the cycles onand the has torus no that duality property; thereduction the string of duality wrap the is theory. emergent Thisdirections. based idea on has In been what used follows, and we studied will i describe something lik parameter with which weby can compactifying do the perturbative six calculations. dimensional theory M on a circle of ra with a self-dual three-formstring. field strength. The The coupling sourceThus for in there is th this no theory notionintroduce of is a perturbation scale of theory. into order To the generate theory one a so that as at given it energies must we b theories that have used the (0 the torus. The S-dualitymodular in invarince of the the four-dimensional torus, theory i.e.torus. a is The trivial t self-dual consequence strings o of the six dimensional theory c theory is four-dimensional Yang-Mills with coupling given theory of the M-theory fivebrane) [ low energy effective description of the realization that these theories canof be a described single as theory, coming with f (0 of the torus. This describes the 1/2 BPS spectrum of the a representation of the SL(2) duality group [ duality of the Geometric Langlands program [ in lower dimensions and has even gone as far as providing an ex JHEP05(2015)130 est es to (2.1) (2.2) ]. In 51 , 9 a sufficent outline to S spectrum and the sition of a wave and program of West and s in some sense it is a split. That is one takes 11 n to the various branes ]. Then we describe the E 8 ar look at the difference views of DFT and related D – iscussion on the absence of elling in terms of this single 5 × c infinity towards the core of first discussed in [ ) n of eleven-dimensional super- e [ appearence of the exceptional hat the wave propogates along anes and D-branes). We also es in supergravity. We analyze D y no means complete sample one ]. And there is now a whole host m the reduced perspective which D the previous works on truncated ] through introducing novel extra E − . ] is to make the exceptional sym- 36 26 5 – – D dim 21 21 M M ]. The steps towards an M-theory equiv- × × 20 D – D − − 17 11 11 – 3 – M M = ]. 11 −→ 50 M 11 – 46 M ]. This object is the analogue of the self-dual string in the so called “internal” directions with additional coordinat 10 D ]; the primary work of Siegel in constructing a duality manif 13 ]. All of this is related to the long standing – 45 11 – ]; the work outlining DFT [ 37 16 – 14 In this paper we will begin with an EFT primer that should give We will analyze this solution and make explicit its reductio For the relevant literature the reader may consult recent re 2) theory. Its reduction provides us with the complete 1/2 BP , in supergravity (such asshow how fundamental the strings, EFT solitonicmonopole) solution changes br its (which character maybe asthe thought one solution. moves of from as asymptoti a superpo theories given in [ collaborators, see for example [ self-dual object. action of the duality group on the BPS spectrum is just a relab Then one supplements the the eleven dimensions of supergravity to be dimensions and leads to gravity but with no reduction or truncation into an (11 2 Exceptional Field Theory The primary idea behindmetries Exceptional of Field eleven-dimensional supergravity Theorygroups manifest. [ in dimensionally The reducedExceptional Field supergravity Theory theories one first was performes a decompositio of interesting works in thismay field, start for with a [ representative but b singularities of the EFT solution. alent to DFT for the truncated theory are given in [ solution and subsequently its reductionthe to behavior the various of bran between the the core self-dual and solution asymptotic regions in and EFT conclude with and a d in particul (0 linearly realize the exceptionaltheories symmetries. that realize This the follows exceptional duality groups [ some circle then this willis give rise equal to to an electric thelift charge magnetic of fro charge a self-dual coming KK-dyon from [ the monopole. Thu We call this self-dual because if one reduces this object so t follow the results presented here. For the original works se formalism [ JHEP05(2015)130 ]). /H 56 D 7 can , E SU(8). / d for the 7 of = 6 ] where all E symmetry. 7 D D which may be E b a η b time. The four di- ν e a presence of isometries directions. Its tangent n internal directions of µ roups become related to e the extended space. This theory which allows for a ]. acetime. When there are s fully covariant under the = inuous local etry that only occurs in the was previously restricted to . This exceptional extended . The combination of p-form ons and thus there naturally 54 ed internal space has coordi- ero. Furthermore, coordinate closely following [ 7 d easily be repeated for other , nded. This allows for eleven- µν ternal” metric was taken to be ates. E g 45 wrapping tail for the case of DFT in [ oduces a canonical choice of how ent choices of spacetime associated 5 [ , ). This “exceptional extended geom- representation of the exceptional group D = 4 which parametrizes the coset 2 E . D 10 and metric MN µ M x ] (a supersymmetric extension for – 4 – 8 since it has all the complexities that we wish to – that provides a constraint in EFT that restricts the 7 6 ] the 56-dimensional exceptional extended geometry group and the corresponding EFT. We will give a E 7 30 8 [ , , E 7 , 26 = SL(5) it is the , ] for the equivalent discussion for DFT). Crucially however 4 = 6 25 E 55 , D 8. For other groups a different representation might be neede 23 , EFT – 7 , for 7 21 E D = 6 = 4 where E ]) and more recently also for D D is a coset manifold that comes equipped with the coset metric 53 D , is the dimension of the fundamental E physical section condition 52 D dim which are in the fundamental representation of E M Exceptional Field Theory lives in a 4 + 56-dimensional space M 7 Y E Exceptional Field Theory provides the full, non-truncated In this paper we focus on the It is worthwhile at this stage to describe how the U-duality g From earlier work [ and This is true for 2 D explore. It is expectedchoices that of the duality group. narrative of this paper coul 2.1 Basics of the construction, e.g. for dependence was restricted to the internal extended coordin dependence on all coordinates,dimensional supergravity external, to internal be andexceptional embedded groups exte into a theory that i etry” has been constructedtruncations for of several the eleven-dimensional U-duality theoryflat groups where and but off-diagonal the terms “ex (the “gravi-photon”) were set to z (where H is the maximally compact subgroup of mensional external space has coordinates space is equipped with a generalized metric The is known. Thisthe space KK-decomposition can with be the seen M2-, as M5- the and combination D6-brane of the seve expressed in terms ofnates a vierbein. The 56-dimensional extend the details can be found. We choose there is an ambiguityambiguity in is essentially how the one origin of identifies U-dualityto with the U-duality differ related submanifold descriptions. in (This is discussed in de there is also a coordinate dependence of the fieldsappears to a a physical subset of submanifoldno the isometries which dimensi present we this identify sectionspacetime condition as is constraint usual embedded pr sp in the extended space. However, in the the embedding of the elevengauge dimensions in transformations the and extended diffeomorphism space give rise to a cont E be found in [ where dim This however is not U-dualitypresence which of is isometries. a (See global [ discrete symm brief overview of the most important concepts of the theory, JHEP05(2015)130 144 h is (2.8) (2.4) (2.7) (2.6) (2.3) µν M B as ,..., 7 . E b and , EFT also µν ρ . = 1 e g ⊃  α . For more on N M MN 7 ∂ ∂ µν α top ρ E M µν a B L e g 1 − M M which takes the form e ∂ of Sp(56) Ψ = 0 (2.5) µν can be derived from the N F ) + MN MN ∂ ucial for the consistency of NK MN µν Φ built from internal extended . ] + M M SU(8). The third term is a M M an be formulated in terms of , g ω 1 4

which are used to define the / V [ ndition which picks a subspace ∂ L b 7 pergravity. Here TM ∂ a n in terms of the spin connection − 6 E MN M g MN µν µ Λ µν M KL N M Ω R A ⊕ ( ∂ ,B M 1 ≡ V MN − M M b ∗ ∂ − a µν α gg M T ν and a pair of two-forms µν N 5 M ˆ ,B MN D R ∂ Λ µν and the generalized metric 1 M M F µ M ⊕ − – 5 – µ ) MN g 1 2 µν A e A g M , M + ∗ µν M MN µ exceptional field theory is T , F 2 D MN 7 KL M ]. For the main part of this paper they will both be zero 56 the fundamental representation of where Λ E µν 4 1 7 M MN g M ⊕ , − N 1 Φ = 0 48 M µν ∂ ,..., b g 1 8 N a TM  + ∂ MN KL − and the invariant symplectic form Ω µν = 1 ˆ M R ˆ R M ∂  M . The fourth term is the “potential” . The last term is a topological Chern-Simons-like term whic ν M N µ b MN M (with determinant µν ∂ e ) D ∂ Y e MN g g µ α a ) t a 56 µ α M t d e MN ∂ ee ( x 1 4 − = M = det d g ˆ M 1 2 R 2 1 48 Z e ∂ e − is the seven-space. − Ψ stand for any field and gauge parameter. = = = , = generators ( S g M 7 EH V The equations of motion describing the dynamics of the fields Thus, the field content of the In addition to the external metric L E of the vierbein covariant derivatives the nature of these two-forms see [ labels the adjoint and derivatives following action of a non-linear gauged sigma model with target space Yang-Mills-type kinetic term for the gauge vectors to describe all degrees of freedom of eleven-dimensional su required for consistency. where Φ where All these fields are thenof subjected the to exceptional the physical extendedthe section space. co This section condition c The second term is a kinetic term for the generalized metric and not play a rolethe in theory. what follows though they are of course cr requires a generalized gauge connection The first term is a covariantizedω Einstein-Hilbert term give space is given by where JHEP05(2015)130 (2.9) M (2.11) (2.12) (2.13) (2.10) µ A erivative ]. M 7 V , K L we will consider a µ ] . ν 3 A . . The final ingredient A K d internal coordinates. L N ∂ µν N µνρ N ∂ . V λ we note that the bosonic itself when the coordinate B 2 K H K M Thus any derivative of the ]. The Bianchi identity for ven by the generalized dif- µνρ M ection µ 2 µ [ 7 − [ µν ates. For more on the novel H ction, defined as nes. This self-duality relation kelberg-type couplings to the A MN ns. In order to form a prop- A MN 1 iant derviative for a vector of F  56 EFT gauge vectors Ω N field theory see [ Ω ρσ K ∂ 1 2 + 1 2 ed many years ago in the seminal and  KL F M − − Ω µ KL NK A Ω MN µνρ α µν α K M M Ω ] µνρ α ∂ H B ν MN K H − N A MN ∂ N V N Ω ∂ + Ω KL as follows ∂ + ) – 6 – MN N α ) KL MN M µ t µνρσ [ α ) ) ( t V α α eǫ A in the field strengths µν M t t 2 K 1 2 ( B MN 12( ∂ ) µν − 12( = K α allows for the theory to be formulated in a manifestly − , vanishes trivially. Furthermore, our solution comes with B MN t µ − ) M ] and M M M α EFT and is essential for the results presented here. In fact M ν ∂ = t 24( µ µν 7 µν  A − A F A µ M E F [ 1 2 24( ] µν α M  ∂ νρ B = 2 − V 1 2 F µ M ≡ µ ∂ [ + µν appear in the non-abelian gauge structure of the covariant d D M = 3 F µν M M ∂ F V µ D ]. is given by 9 λ To conclude this brief overview of exceptional field theory, The gauge connection An immediate simplification to the above equations presents All fields in the action depend on all the external and extende For a detailed derivation and explanation of this we refer to The associated non-abelian field strength of the gauge conne and together with the two-forms features of the generalized diffeomorphisms in exceptional gauge symmetries uniquely determinefeomorphisms the of theory. the They external are and gi extended internal coordin which also defines the three-form field strengths dependence of fields andsolution gauge of parameters EFT which isinternal only restricted. extended coordinates, depends In on external s coordinates. this generalized field strength is The derivatives is a crucial property of the of the theory are the twisted self-duality equations for the which relate the 28 “electric” vectors to the 28 “magnetic” o this sort of twisted self-dualitywork equation of has [ been describ invariant way under generalizedweight Lie derivatives. The covar erly covariant object wecompensating extend two-forms the field strength with St¨uc is not covariant with respect to vector gauge transformatio JHEP05(2015)130 . mn M g , the µν of the (2.16) (2.17) = 0. ], here M B ] 7 ν V becomes M , reduce to A 6 F m µ µ [ µ ∂ D is included in ]. A 57 mnp C found in [ ! ] closely). Applying 7 e on internal extended mn (7) g mn m 0 0 g GL ar part µ luza-Klein coordinate split. d to a decomposition of the luza-Klein decomposition of wing [ mpensating two-form and the internal metric A → → riant derivatives ing the coordinate dependence internal coordinates . The KK-vector extended space [ l. The four-dimensional external mn ng form is simply given by 2 reduces to the usual d mn mn e purely external three-form part ry one and the potential ) (2.15) g n µν MN M ν mn n ν µν A M , y A F m , ∂ ,B m µ 0 0 ) to the EFT produces the dynamics of mn , y A g EFT, we proceed by showing how eleven- → → 2.5 + mn 7 m m E . , y µν – 7 – g µν coordinates. We thus have m M 7 + 21 + 7 + 21 (2.14) y

µ m , thus simplifying the gauge structure further. The A y = → of supergravity are also decomposed under the 4 + 7 , ∂ ,B = ( is antisymmetric. We thus have indeed 7+21+7+21 = 0 0 6 ! under its maximal subgroup 56 M C µν M → → µn 7 mn Y mn B ˆ g ˆ g E and mn mn ∂ is carried over to the EFT. The seven-dimensional internal µν mν 3 µν and ˆ g ˆ g C B

µν g = µν α ˆ ν B ˆ , the generalized field strength µ ˆ µ g 7 and the pair ∂ ,..., = 1 -component of the EFT vector m which lives in the external sector. The purely internal scal m We will comment in future work on how to reinstate a dependenc The appropriate solution to the section condition is relate The gauge potentials The complete procedure to embed supergravity into EFT can be y µνρ generalized metric vanishes. Finally the Bianchi identity ordinary partials covariantized Einstein-Hilbert term reduces to the ordina coordinate split. Starting withC the three-form, there is th where which translates to the following splitting of the extended dimensional supergravity can be embedded in it (again follo coordinates and thus localize solutions in the exceptional the upshot of this is a drastic simplification of the theory: cova where the second line is the necessary consequence for the co we will focus on thosethe aspects eleven-dimensional spacetime relevant to metric our takes results. the followi The Ka supergravity with its fields rearranged according to a 4fundamental + representation 7 of Ka 2.2 Embedding supergravityHaving into EFT outlined the main features of the a specific solution of the section condition ( 56 coordinates. Theof section fields condition and is gauge solved parameters by to restrict the zero two-form fields where hatted quantities and indicessector are eleven-dimensiona with its metric sector is extended to the 56-dimensional exceptional space becomes a building block of the generalized metric JHEP05(2015)130 A nal with (3.1) (2.18) (2.19) i w . 5 is dualized on , the four-index 5 ...m 1 i sector to be four- mn ...m g 1 ions into EFT. The d Theory to eleven- µ m r h n terms of a harmonic C mn,kl µ m 5 d properly) are encoded where the gauge poten- g C 1 y in ten dimensions. This . The remaining two-form = det tion condition by a simple ...m − ing next. 1 he one-form type under the M g ntent and equations, we can EFT can therefore be simply , g µ r parts or other mixed index extracted by a suitable choice quires a different, inequivalent ) = 1 + quations. We are looking for a s solution needs to satisfy the A r . Similarly for the six-form, the mn . The remaining components of ( mn m g ǫ 1 M 1 − µ 5! µν M A , g and three spacelike directions B = t ,H of the external sector is carried over; the i mn . The one-form mn,kl µ ij µν δ g , g A 2 MN / -component of 1 mn – 8 – M g mn h ,H y , and similarly for the inverse; and the compontents 2 / n -component of 1 ] l diag − g 2 µ mn mn . k are / [ H ]. Both Type II embeddings are presented in appendix ). is the y 1 C 7 m − g M is part of g h = µ µν M 2.13 6 = µ mn A ) = B , is related to the dual graviton and has no appearence in the C ]. is some constant (which will be interpreted later). ...m of the extended internal sector is given in terms of the inter 7 µ mn 1 mn h g = diag m µ m A and mn,kl ( g C A MN µν g and MN M µν α j , B M w m i µ w A ij gets encoded in compensating two-form δ by = = m mn 3. The external metric is that of a point-like object, given i 2 µ , g r µν m . The one-form part 2 A , C In the next section we will work with supergravity solutions Now consider the following set of fields. We take the external It is also possible to embed the Type II theories in ten dimens We are now equipped with the tools to relate Exceptional Fiel with a mixed index structure (some of which need to be dualize MN 6 = 1 objects are defined by tials only have a singleabove non-zero coordinate component split. which willcomponents. There be will The of above not embedding t be ofsummarized supergravity any as fields internal follows. into scala The spacetime metric metric generalized metric where the determinant of the internal metric is denoted by of the EFT vector potential in the two-forms C i function of the transvere coordinates by supergravity picture, see [ purely internal scalar part dimensional spacetime with one timelike direction where now consider specific field configurationssolution which from solve which these the e known supergravityof solutions section. can be Furthermore, astwisted argued self-duality equation in ( the introduction, thi will be useful when analyzing the EFT solution3 we are present A self-dual solutionHaving in introduced Exceptional EFT Field Theory with its field co Type IIA embedding followsreduction from on the a above circle. solution Insolution to contrast, to the the the Type sec section IIB condition embedding [ re dimensional supergravity and the Type IIA and Type IIB theor the internal space and forms the M The final component, JHEP05(2015)130 ). (3.5) (3.2) (3.4) (3.3) 2.15 . . Then olution. ˜ z i which we M 15 ) points in δ δ satisfies the M 2 / a with its dual = M 1 given in ( µ ernal sector are − beys a BPS-like M M A a a M ,H 6 Y . δ denotes the direction 2 . , is a diagonal matrix / in the desired direction form a SO(56) rotation 1 3.2 , ents, reflecting the split M N rection for a L . M M MN H. inates ,H a ˜ a R k 6 t minus signs and factors of er of the 56 entries of course i nic function that appears in 3.2 δ ∂ M cterizes the solution together his direction as the direction 2 A k KL / . The first and last one appear and we have ˜ section ij 1 on = e given respectively by ǫ

. This sense of duality between − M z 2 1 2 K / K M 3 ˜ components of the potential. The a i ,H = − M 2 A ] / M j R NK 3 i ,H A − 2 i A = [ M / SO(56) rotates 1 ∂ ,H − ∈ MN according to ′ MN 15 and δ Ω R ,H 2 – 9 – 2 M / / 1 or ∼ MN 1 , one can immediately check that i ,H M M M A 3 ,H 6 a a , 2 δ / 1 2 . 3 N and magnetic H ) is / and ). Loosly speaking, the duality on the external spacetime a 1 − M ~ H H ∇ of our solution has “electric” and “magnetic” components − . For completeness, the full expression for the generalized N M a  H 2 t H 2.15 = / M M 2.13 3 A ,H µ ~ R 6 = − A δ A 2 H = M / into 28 “electric” and 28 “magnetic” components. t 1 acts on the extended internal space and swaps = ~ ∇ × M A ′ M labels the 56 vectors, only two of which are non-zero for our s ˜ z -dimensional Kronecker delta. a . The dual direction is denoted by ˜ ,H ˜ µ z n 2 M / A MN 3 Mz µ M δ A H in the extended space (a scalar form a spacetime point of view h = in and M only work out if both actions on the external and extended int given approximately by M a 2 / a denotes an M is a potential of the magnetic field. The magnetic potential o exchanges electric H 3 M n i = diag a δ H A , i.e. = µνρσ and at the same time transforms MN z To get the fields for any other direction, one simply has to per For definiteness, let’s fix the coordinate system and pick a di The generalized metric of the extended internal sector, Using the relation between The 56-dimensional extended internal sector uses the coord ǫ Here . If one goes through the calculation carefully, one sees tha zz 3 M M ′ M ˜ (from the four-dimensional spacetime perspective) that ar a carried out simultaneously. We will show this explicitly in metric for the coordinates in ( in the extended space. The rotation matrix powers of with the choice of direction for once each, the other two appeardepends 27 on times a each. coordinate The choice, precise but ord once this is fixed it chara condition where its curlthe is metric given by the gradient of the harmo a symplectic form Ω M where directions of the extended space will be formalized in secti with just four different entries, via The EFT vector potential The second 28 componentsof are the the EFT inverse vector of the first 28 compon The vector one of the 56of extended propagation directions. of Later a we wave or will momentum interprete mode. t The dual vector ˜ dual to call The index twisted self-duality equation ( JHEP05(2015)130 , ) 1 S di- 3.4 2 µν M , y B rection, 1 which in y uation and ) and ( H and 3.2 µν α ), ( B 3.1 ified by two pieces of , vanishes for our solu- fields a given solution in both bration — where the onic function to the wave’s momentum. µνρ e interesting and provide reted. A wave whose mo- ns to some linear abelian of EFT is that there is no n-trivial Hopf fibration, i.e. t gives the direction along as monopole or as a wave. C his solution therefore is de- y? Before we do this let us fluxes such as those on the view of the KK-graviphoton. y and the solution thought of fact EFT). As such they are . There is thus no coordinate t winding direction, e.g. if the e. In theory, EFT can handle i ions. It is an interesting open upergravity solutions obtained ial, y relation is simply the Kaluza- re charged with respect to some w under such a transformation, the in extended space. Our intuition 1 S ]. as given in equations ( 58 , 56 MN – 10 – M that appears in the harmonic function. The vector describes a membrane extended over the h 12 y and coordinates, even the extended ones. We leave this for M µ A all then the solution describes a fivebrane. Thus in the extended , µν 12 g y direction describes a brane associated with that winding di and the constant )) form our solution to EFT. They satisfy the self-duality eq M a 3.3 winding Note that all fields directly or indirectly depend on the harm The remaining fields of the theory, namely the two-form gauge To recap, the fields Now because of the truncation it was not possible to describe Let us look at the moduli of the solution. The solution is spec mentum is in a future work. 3.1 Interpreting theHow solution do we interpretreturn this to solution how in solutions Exceptional in Field Theor the truncated theory may be interp turn only depends on the external transverse coordinates dependence on any oflocalized the internal and or smeared extended overquestion coordinates. all to the T look internal at extendedcoordinate solutions direct dependencies localized in on the extended spac fibre is a winding directionfibre describes of the the S-dual monopole brane is to tha (but truncated) theory branesThese statements can were have the either conclusions of a [ description rections. A monopole-like solution — by which we mean a Hopf fi Klein description of a solutionit that has is both simultaneously momentum electric andAs no and magnetic commented from in thetheory the point but introduction, of full these solutionsexact are to self-dual not solutions the to just gravitational theU(1) solutio theory non-linear symmetry theory (or that though in is a given by the existence of the ways within the sametruncation description and of so spacetime. such The things are key possible. point The self-dualit (together with ( their respective equations of motion. should be shaped by thisas experience simultaneously with a Kaluza-Klein wave and theor a monopole whose charge isdata, equal the vector e.g. a wave with momentum along fields can freely be rotated in the extended space. Since the action and the self-duality equation is invariant a technical challenge toexternal repeat space. this paper but include other tion. This will eventuallyfrom restrict this somewhat EFT the solution. possible s Dropping these restrictions would b are trivial. Also the external part of the three-form potent specifies the direction the wave is propagating in. That is, i JHEP05(2015)130 (3.9) (3.6) (3.8) (3.7) ) but (3.10) Z ; magnetic ived some M , the radial ( . 2 r 4 H m. m numbers are = n these solutions. The and then the our theory also requires 1 ure, whose fibre is in the f the circle by S tion then implies that this s paper this fibration may ual direction to the fibre is s is given by tail in section tions like the NS5-brane are . localized solutions where one s or the Type IIA and Type . | . d whose base is in the external e the next section) then reveals = 1 and 5 m M . at is we can interpret this single ). Thus for the smeared branes resting insights when we analyze ˜ a R | ] for a discussion of the localized m Z Z e ; Z R = 1 57 R ∈ e ∈ S R m ( = m ). For the smeared solution these two n 1 R Z ; H . (See [ n/m × M with with ( h ) 3 Z thus in the harmonic function of the solution is then H ; (and thus the radii) is a function of and – 11 – 2 h ⇒ m e S H = ( n vectors that specifiy the solution we can determine R 2 4 / | 7 mR 3 H = M E − m . Let us examine this quantitatively. e = a q | 1 q H ) = m S = q = Z = that one chooses determines how one interprets the solution e ; e q 1 m R S M a × ,R 2 4 S / ( ]. 3 3 is however an important open question that has recently rece 60 H H 3 , = H 59 e , as R related radii are duals and the electric and magnetic quantu 38 m 7 is integral. This is essentially Dirac quantization but now R E h To give a non-trivial first Chern class the fibre must be an In addition, the solution comes with a monopole-like struct Finally let us a add a comment about the topological nature of The actual direction The electric charge of the solution is related to the radius o and e self-duality which in turn impliesquantized. that The the presence momentum of indirection quantized the itself d momentum must also in be this an direc non-smeared solution.) So the Calculating these norms using the metric of the solution (se charge which there is momentum. The constant R in terms of the varioussolution usual supergravity in descriptions. terms Th ofIIB the brane brane solutions solutions in ten in dimensions. eleven We dimension will show this in de and the magnetic charge is related to the radius of the fibre by proportional to the amount of momentum carried. From examining the norms of the Now the twisted self-duality relaltion implies attention [ there is no issue.has The a question genuine of the global structure of the equal. Note that the harmonic function are related since coordinate of the external spacetime.the This solution will close lead to to inte its core or far away from it in section direction dual to the directionspacetime. of propagation of In the the wavebe an case classified of by the its smeared first Chern solution class studied which in is thi more mathematically minded reader willnot classified note by that the brane first solu Chern class which in cohomolgy term instead by the Dixmier-Douady class, i.e. JHEP05(2015)130 that which (3.16) (3.15) (3.12) (3.13) (3.11) (3.14) . This M ˜ z a H M δ = 4 | / and ˜ 3 µν H g M K KL s of the equation a = det M . The unit vectors Ha | M l L M Mz ˆ ˜ a ∂ tors gin by computing the ˜ a ˜ a = 2 . MN K − 2 Hδ K and the spacetime vector ˜ a e i . and ! . Now we can look at the M H ng the indices on the four- ∂ λτ I ˜ 0 z p K 2 H M gives F lt I − M Mz NK 0 F = tric F lation introduces extra factors. sional one. In the next step we δ − H | 4 NK Hδ M

/ N M k 3 = NK ˜ a ˜ a ∂ M − = | k MN M M ) as H ij a Ω = ǫ MN Kz
MN = 1) δ 2 Ω MN M / = Ω 2.13 1 ˆ ˜ a − Ω M ). στ NK M 1 a tt g H ), recalling the simplifications our solution ˜ g a − 3.4 ] ρλ M − j kl , g H g A ( 2.11 – 12 – i i K ) and has determinant [ kl MN ∂ ˆ ˜ a and ∂ δ ijkt µνρσ Ω 2 ǫ − . The duality relation between them can be made 3.1 ǫ
/ 2 2 1 NK / M / = 2 = H 1 i − 1 given in ( k M A H M ∂ M H presented above satisfies the twisted self-duality equa- H ] ] k j t KL 1 2 = ij MN A A MN M tijk ǫ i i M ǫ and = [ [ µ as given in ( M 2 M L 2 M ∂ ∂ / A / ij a a 1 M M = Ω is given in ( 3 t M F K ). − H µ µν = 2 = 2 a A M µν F H − A ˆ a g 3.3 M M p it = = ij of ) for the = F F M | M ij N M 3.12 µν a F a | F given in ( = i ). This can be checked explicitly by looking at the component A M ˆ a 2.13 Let’s now turn to the self-duality of the field strength. We be First though, we will look at the relation between the two vec indeed satisfy ( where the extra minus signdimensional in epsilon the which is first then line turned comes into from a permuti three-dimen If the vectors are not normalizedFor the the specific metric in directions the given duality above re we have ˆ and inserting for the spacetime metric and the field strength components of the equation. Starting with can be used to rewrite the self-duality equation ( potential The spacetime metric are related via the symplectic form Ω by provides. There are two components which read field stregth and making use of the relation between the harmonic function where the spacetime metric is used to lower the indices on precise by normalizing the vectors using the generalized me tion ( The EFT gauge potential 3.2 Twisted self-duality define the directions of JHEP05(2015)130 ) tra 3.13 (3.20) (3.19) (3.18) (3.17) given in ( K ˜ a M ] µ q A A M p or by an extra minus [ f an intersecting brane appropriate solution to ∂ electric charge, this has . 2 Ha M i onal supergravity point of ˜ a K ∂ ] mponent of the self-duality ty fields will be rearranged j 2 NK pq − litonic and Dirichlet p-branes, A . This choice would then be d the Type IIA and Type IIB F i . direction of the exceptional ex- [ M n gives rise to the full spectrum H ∂ . NK r Kaluza-Klein decomposition can = ˜ z MN atisfy an anti-self-duality equation. 2 = 2 K Ω M / δ 2 3 lq that are picked out by the summation zz / − δ 3 2 MN NK M / H H M ˜ Ω z 1
˜ z ˜ z M − lq Mz M g Mz H Ω δ M ] ] δ jp ˜ z MN j j kp g – 13 – M δ Ω − A A i i 2
] (of course the choice of self-dual or anti-self-dual [ [ Ω / k itjk 7 ∂ ∂ 1 in reverse, the supergravity fields can be extracted ǫ H H 2 2 A 2 − i i 2 2 j / / / ∂ ∂ 1 ∂ H 3 3 2 2 2.2 / / jk − − H gives the expected result 1 1 i 2 1 tijk ǫ − − H H ǫ 2 2 H H M = / = = / 1 1 − − M − M H it = = ij 2 1 H for easy referal. F F − − M B it . The above calculation then works exactly the same but the ex = = F ). M M ˜ a i it ) and the components of Ω and 3.13 A F − 3.3 = M i A Thus the components of the field strength of the EFT vector to match with ( minus sign ensures thatconsistent with the the orginal field EFT strength paper [ is anti-self-dual 4.1 Supergravity solutions We start by lookig atview. the Using EFT the solution resultsfrom from of an the section eleven-dimensi EFT solution.according to a Recall 4 that + 7 the Kaluza-Klein resulting coordinate supergravi split. the section condition and rotatingtended our space solution leads in to a the specific the wave solution, KK-branes the which fundamental, are so solution. extended All monopoles, these extracted and solutionsbe an together found with example in thei o appendix 4 The spectrum ofThe solutions self-dual EFT solution presentedof in the 1/2 previous BPS sectio solutionstheories in in eleven-dimensional supergravity ten an dimensions. We will now show how applying the sign, is ultimately related to how supersymmetry is represented) satisfy the self-duality condition. ItIf is the also magnetic possible charge tothe s of effect our of solution modifying is the taken to magnetic be component minus of the the EFT vect over indices are substituted Again substituting for Ω and make use of ( Going through the same steps as before leads to equation reads and we obtained the expected result. Similarly, the other co JHEP05(2015)130 ) z , it and 3.1 z form (4.2) given . See M entary z = 6 a in ( M 1 C = ˜ µ y membrane µν 1 A g y and e our self-dual . is 3 e membrane is for the M2 and . If our solution , the  z C 2 3 ic of the internal 67 g six directions of mn and so on as given , δ C y 5 , the fourth possible M δ tum in the extended δ m . If the procedure of 1 or mn y − y = low. e direction of the vector , where we now identify H MN mn  M m y rdinary direction rection to a y which gives their duals, i.e. M solution of supergravity can he extra exceptional aspects Mz (essentially swapping as n. 1), becomes the cross-term in deed the case. If the direction δ or M diag m i 3 − is like a dual graviton and does ), one can work out the seven- y / = n supergravity. Since we are now 12 A 1 2 i − M ed to the EFT solution as presented A M H δ 2.18 H a ] (4.1) pp-wave ( 6 = = along which encodes the = − ) and extracting the internal metrics for ˜ z i M M M mn = H, δ a ) to ( t a A 3.5 z -type, A t . This result is obtained by an accompanying 3.4 5 m A y y – 14 – , g are decomposed into = diag[  5 and ], we know that a wave in an exceptional extended M ] that the wave in EFT along , and thus ˜ 4 ) directions, the , δ mn ˜ z ). Finally, the external spacetime metric 58 2 Y g 56 δ , y , m M 1 3 B.1 y δ − 56 2.19 , y . The components of the EFT vector potential = 2 H for the M5. We will explain this procedure of obtaining the  mn , y 3 M g 1 a C y , the internal metric is given by diag , for the latter, the fivebrane is strechted along the complim 3 2 / , of our solution in ( MN y 1 , i.e. 7 H M y MN and = -type, e.g. M 1 y and m mn solutions of supergravity are recovered. For the former, th y 6 for the supergravity wave decomposed under a 4+7 split. Sinc g y , i.e. with the direction of the 3 B.1 is a Kronecker delta of dimension six. These are the remainin ). Then by comparing the expression for the generalized metr 6 for the M5. The magnetic potential is given by , the first of the ordinary for the M2 and the δ ) can be related to the KK-vector of the decomposition and the 1 is of the 6 6 y fivebrane We have previously speculated [ As shown in previous work [ As mentioned before, the EFT solution is characterized by th First, the extended coordinates and a corresponding ordering in the diagonal entries of C C 2.15 3.2 M . The “electric” part of the EFT vector, a M m the membrane and fivebrane from the EFT solution in more detail be directions to and rotation of the generalized metric according to ( is rotated to propagate in those directions, e.g. stretched along the y where extracting a supergravity solution just describedin is appli section The masses and chargesdirections. of The the electric branes potential are is provided given by by the momen appendix the supergravity metric. The “magentic” part the M2 and the M5 (cf. appendix geometry can also propagate along the novel dimensions such EFT solution is interpreted as a wave now propagating in the o is simply carried over to the 4-sector of thea KK-decompostio extended space, dimensional internal metric be extracted. From are removed. is not too surprising to recover the supergravity wave once t direction, should correspond to aworking monopole-like with solution a i self-dual solution,of we can show that this is in fields respectively according to ( not appear in the supergravity picture. Note that the dual di in ( in ( with JHEP05(2015)130 . , 12 12 y M (4.5) (4.6) (4.4) (4.3) δ tes and = M ′ a given there and . ) and are all the i direction, say M , 10 a mn at the same time by δ g 10 mn 2 ( δ / y 2 1 / FT solution all have the a 1 mple of such a rotation. MN − ,H takes the values 3 to 7) is obtained by rotating the he coordinate system given M 1) now has the nature of a cture. This is the opposite pected in relating the wave 10 ,H δ a s that reflect the new duality . of the solution. 2 − sions). The four solutions only ifferent section of the extended a 10 / ectric-magnetic duality of these be extracted i 1 1 δ 1 and M 2 y ij − − a / δ 1 . 2 M H / ↔  ( µ ,H has to be rotated into 1 6 , becomes part of the KK-monopole 2 a ,H − i A / 1 1 2 , δ 3 / A ,H 1 M 3 − = 2 δ − / , y ˜ = z 1 t H ,H ,H and their corresponding components in the − = ab  z 5 5 A i δ δ y H 12 2 2 M A y / / − – 15 – a 1 1 ↔ h with the choice for the vector − is obtained. Again performing the corresponding ab 3 ,H = diag ). 2 with ,H δ ) for a fixed coordinate system directly reduces to the under the KK-decomposition, 2 2 1 , y δ mn 3.5 / y 2 = diag g 2 1 3.4 µν / , the KK-vector of the decomposition and of course the y − 1 g µν H H g mn in ( ↔ h g 2 y . This should not come as a surprise since here momentum and MN KK-monopole M 12 12 = diag , the frame change also swaps the following pairs of coordina 12 MN y ↔ M M ↔ 1 1 1 y M )) After the rotation, the generalized metric reads (still in t Besides We now want to demonstrate how this can also give the M2-brane As explained above, if the EFT solution is propagating along The four supergravity solutions we have extracted from our E 2.15 of the pp-wave), the M ˜ by ( the corresponding components of the metric (here the index frame. These are simple exchanges between dual pairs of coordinate winding directions are exchangedand which the is membrane exactly via duality. what is ex same in EFT, up to an SO(56) rotation of the direction metric, i.e. 4.1.1 From waveLet’s to membrane pause here brieflyThe EFT and solution take presented a in section closer look at a specific exa C-fields. But these elements are just rearranged in simply picking a different duality frame,space that to is give choosing the afields d physical of spacetime. the This solution new according to duality ( frame the generalized metric This has the effect of exchanging which has the character ofdiffer a point-like in object the (in four internal dimen metric it gives the membrane. Thus the vector pp-wave in eleven dimensions. same external spacetime metric a scenario to the pp-wave describedtwo above, solutions. underlining the el dual graviton and does not contribute in the supergravity pi metric in supergravity. The “electric” part The “magnetic” part of the EFT vector, rotation of the generalized metric, the internal metric can JHEP05(2015)130 , 1 = y 67 and with M m δ 12 mn = M y 5 are the δ into , M 4 = ′′ m , a y 7 M ′ y = 3 a α y A is useful to see how 6 ) and transverse (a –– oordinate labels and y ◦ he external or internal rdinate with ] and then interpreted ucture thus unifies the -wave or the membrane 5 with rane instead of the wave. 61 y [ A direction, i.e he (pure) fivebrane. Above y 4 2 BPS branes of supergravity A 67 y y ey are transverse directions for y internal 3 ––––– y nd provides the so-far missing link ), the M2-brane strechted along 4.2 2 y 7 are transverse directions to both the a , y 1 ◦ ◦ ◦ ◦ ◦ ◦ ◦ y = 6 α 3 – 16 – denotes a worldvolume direction of the brane while a w ◦ ]. As an illustrative example we will show how the and their correspondig components in the metric, i 2 with 62 ] can be obtained from our EFT solution. shows the worldvolume (a circle w w ) to read off the internal metric in the reduced, eleven- 67 α y 63 1 y 2 are the two worldvolume directions of the membrane 1 , 2.18 –––––– external w ), the new frame rotation exchanges the membrane direction in this kind of duality transformation. = 1 ◦ ◦ t 4.6 a mn y with a . Again it is very natural to exchange a membrane coordinate y membrane fivebrane sector coordinate 67 67 . The coordinates of the membrane and fivebrane are either in t ↔ M ). Here indicates a transverse direction. ) directions of the membrane and fivebrane together with the c . Similar rotation procedures can be applied to relate the pp 2 − − y Starting from the membrane frame of the previous subsection Before we find this bound state of a membrane and a fivebrane, it In what follows, it will be useful to split the index of the coo Our self-dual EFT wave solution with attached monopole-str with the fivebrane direction 12 12 12 a, A, α the generalized metric in ( then we get the fivebrane. sector under the KK-decomposition. we have just seen howIf to rotate we the rotate frame further to to get the have (pure) the memb solution propagate in the y by Papadopoulos and Townsend indyonic [ M2/M5-brane solution of [ to pick a duality frame such that the EFT solution reduces to t dimensional picture as described above, and gives ( dash M a fivebrane coordinate dash This can now be compared to ( and Table 1 sector of the KK-decomposition. A circle but also bound states. Such solutions were first mentioned in in the duality web of exceptionally extended solutions. 4.1.2 The membrane/fivebrane boundThe state self-dual EFT solution does not only give the standard 1/ four classic eleven-dimensional supergravity solutions a ( to the fivebrane and the monopole. remaining three worldvolume directions ofthe the membrane. fivebrane, th And finally the membrane and fivebrane. Table or two of the worldvolume directions of the fivebrane. The JHEP05(2015)130 ξ ]. 12 63 M (4.8) (4.9) (4.7) δ ), but (4.10) AB,CD tential, δ = e. 2 4.7 / 1 ), one finds − M (M2) a H 4.8 . In most cases brane from our = , , H 2 i / 2 2 the membrane is 3 to be applied to the / es. To see how they − 3 π/ e same as in ( AB CD ,H ,H = 67 components where = been exchanged as well, ). Inserting the rotated ut introduce a parameter ne frame to the fivebrane 10 10 M pletely. One can think of 2/M5 bound state of [ ξ irections and fivebrane di- δ δ purely magnetic M5-brane. 5-brane (here 4.2 y frame. This time though, 2 2 αβ rdinates. / / 1 1 and )andtheM5in( − . M (M5) , ,H 4.6 ,H AB ξ a 10 BC aα δ δ y y 10 ! 2 2 δ / ξ ξ / 1 2 = 12 and 1 / ↔ ↔ 1 − ) differ by a factor of H + cos sin cos A ab aα ξ = ,H ,H 4.7 ξ 2 2 M (M2) δ δ 2 2 cos 2 — might be counterintuitive but has been made to AB . Since we are dealing with a dyonic solution, / / sin ξ a 1 1 , y ξ – 17 – − , y π/ − M

,H (together with their duals) of the membrane and BC αβ -dependent rotation now introduces factors of sin 5 ,H y y , the frame change also swaps some other pairs of = ξ δ 6 = sin 5 2 2 δ 67 or cos C / 2 ↔ ↔ y 1 R / ξ − 1 M5) A ab / = y y H H h M (M2 αβ a y and the 3 ↔ = diag C . The partial, or vice versa, e.g. 2 12 / ) then gives the corresponding C-form field as explained abov 2 1 y from above). If this vector is inserted into the EFT vector po / MN 1 ± — and not one shifted by 3.2 − = 67 ξ H M ]. H M ab 63 δ y Therefore a vector of the form into ( in [ = 4 into the metric components and generates off-diagonal entri ξ M ′′ ξ becomes a = 0 the transformation gives the fivebrane whereas for M (M5) becomes 2 a / ξ Having found the new EFT vector, the above rotation now needs The coordinate pairs which get rotated into each other are th Now that it is clear how to obtain both the M2-brane and the M5- Besides 2 This choice of 3 / 4 ± 1 which interpolates between a purely electric M2-brane and a H generalized metric. Comparing the metric for the M2 in ( and cos are exchanged. The only exceptions are for the which can be reduced to give the internal fivebrane metric ( arise, one has to consider the effect of the rotation on the coo points the EFT solution in the direction which gives the M2/M each pair as a 2-vector acted on by now superpositons are formed instead of exchanging them com that the components which get exchanged in ( we do not rotate theξ frame all the way into the fivebrane frame b vector self-dual EFT solution, weTo can achieve attempt this, to we obtain will the again dyonic start M from the membrane dualit fivebrane, each modulated by sin both an electric and a magnetic potential are expected. H frame. Once theit corresponding reads components of the metric have The first line containsrections. further exchanges The between second membrane line d is a result of going from the membra coordinates. This can now be neatly written as one obtains both the and match the For recovered. JHEP05(2015)130 = A ′ y (4.11) (4.12) (4.15) (4.13) (4.14) (4.16) . gives for ! 2 ξ ξ R 2 ]. . These terms nternal C-field cos 56 cos   ξ ξ 2 mnk / rd coset form of a 1 C 1 cos . H − 2 1) sin ξ + R ! − . ξ A occur in the bound state sin Ξ ! 2 ! H BC 2 1 are ( lled out in [ ). Then the new coordinate / ξ y 2 he form − otated into each other. The 1 A Ξ sin 2 n for this is that the original indices are surpressed. The / BC s actually possible to rewrite to components of the C-field ξ y e by using some trigonometric y − H 1 2 al terms, i.e. without a C-field. 3.5 y R 0 / s in the generalized metric. In − e corresponding components of cos Ξ 1 CDEF

ndex structure works out, e.g. 2 − H / in ( − 1 M − + cos − R ξ . ξ A ξ H BC H 2 2 and AB 0 − ξ y ξ y cos cos M cos i ξ 2 blocks of the metric with ξ 2 !

sin sin H / 2 2 1 bβ × aα

+ − R y y ξ H cos ξ = 1) sin

H – 18 – ξ = 2 − , 2 cos ! + ! ] and the appropriate reduction ansatz (which has ξ ξ H ! A 2 ( sin BC y 30 2 y ab 2 αβ / sin EF y 1 sin y 1)

Ξ = sin 1 ′ A − 2 CDEF − H 2

− ′ / Ξ 1 H H R , H M ( M − H Ξ ! = 2

B / ab αβ 1 1 + ! y = h y − ′ AB A CD 1 1 ′ ′

′ BC H y − − 2 y given by, for example M − Ξ M . R

5 2

/ BC ! 1 y generalized metric with cross-terms due to non-vanishing i 2 / H 7 2 submatrix of the full rotation matrix 1 0   − E ABC × H ξ ǫ 2 / 1 0 + cos H A

The next step is thus to bring the above matrix into the standa More formally, one can introduce epsilon symbols so that the i 2 ξ y 5 R sin The other coordinate pairs which are acted on by copies of These rotations have quitethe non-trivial consequences generalized for metric. th Conjugating the 2 Then the new metric components read result, which is the same for all the blocks, is and similarly for allessential the other action metric of components this which rotation are r becomes clearest when the our example the ordinary supergravity picturein these the extra internal terms sector of reduce the KK-decomposition which are of t This transformation produces additional off-diagonal term the same form as the metric) for our concretegeneralized metric scenario or was a KK-reduction spe ansatzidentities which can and be introducing don the shorthand pair is schematically which is the 2 was constructed in general form in [ which is of thethe desired transformed form. metric It in a ismatrix coset was interesting already form. to in coset The see form, underlying that just reaso it without any i off-diagon are not present forsolution. the pure The membrane and fivebrane, they only JHEP05(2015)130 . ) j 67 w i , y w (4.19) (4.20) (4.17) (4.18) 12 ij y δ which is . But this = 2 µν / wo matrices 2 1 g that does not r  H ) . B 3 ] the solution is V / y = 2 d BC 63 − y A 67 ABC , Ξ where (and its determinant y one needs to define allows for a complete 3 d i 67 hat need to be trans- / z to extract the (com- ξ ǫ α ! 1 . In [ ding metric component w and mn y AB α M g H δ difference is for ( A y cos  the determinant all blocks ternal metric y n ansatz — adapted to our = ξ 3 o / and CD,EF n are given by 2 3 and g 2 sin / − eference a multi-brane solution / tion presented above. Similarly, 2 , g CD,EF 3 A he dyonic M2/M5-brane solution 1 Ξ l metric . − g  3 y i H / ACD Ξ − Ξ 1 β 3 αβ ] C / y = 1 H H , δ d 63 α 12 − H + , y AB i EFB δ 12 d = = b 1 C y g − αβ M d , , the only component of this δ in the metric. Ξ a i , ABC y 1 + EFB ij d – 19 – ab C − ] for more details. Since the only non-zero com- j δ , i.e. CD,EF ABC δ C 2 . Simply delocalizing it in ab 1 g H w / 30 C δ . H A 1 − d i y is + H w  ACD ,H 2 3 abαβ d CD,EF t 2 C C / , see [ g d ij ) here only depends on the three ξǫ 1 since it is constructed in eight dimensions and then lifted 3 δ r + − − ( CD,EF h AB diag h A δ δ ...n 3 3 cos y H H 3 3 3 1 / / / / / AB ξ 1 n 1 − 2 4 1 ). Reversing the KK-decomposition, finally gives the eleven- g = Ξ Ξ C h − Ξ Ξ 3 3 3

sin 3 3 / / Ξ 2 / H / 1 1 1 3 4.15 / 2 2 ...n / 1 − 1 Ξ H 1 − − H g n H 4 H H H = diag = + is the determinant of the internal metric. Comparing these t − = = = ...m µν 1 = mn 2 mn g g m s g AB ǫ d g 1 3! ) there is an extra factor of abαβ CD,EF 67 V = g = det , y 4 g 12 y ...m Also note that for some cross-terms in the reduction anstatz Once the transformation of each index pair and the correspon Now this matrix can be compared to a suitable reduction ansat The same procedure also works for the other pairs of indices t 1 m ) recovered form the generalized metric together with the ex with Ξ as defined in ( It is smeared over the remaining transverse coordinates just carried over from the external sector of the EFT solutio The harmonic function V together with the reduction to supergravityis is obtained performed, in t the usual 4+7 Kaluza-Klein split. Its interna factor is just carried through andfor does ( not affect the calcula g where only delocalized in theto three eleven dimensions by including the dimensional spacetime metric of the solution as in [ ponent (in the internal sector) of ponents of) the metriccoordinates and here the — C-field takes in the supergravity. form Such a vanish is where the metric has an extra factor of identification with the solution here. Furthermore, in the r formed and their corresponding metric components. The only leads to the followinghave components to (note be that taken of into course account, not to just find those corresponding t JHEP05(2015)130 ) ) A.3 2.19 (4.21) . Having tly be em- . The com- mn y M an check that he generalized µ or A of the bound state m ) and we can thus y contains the dyonic or Ξ = 1 reproduces )). one external and two A.1 H Type IIA. The general- A.4 . Using the ansatz ( -dimensional exceptional onal Type IIA case, the the EFT solution is made z r ( w choice of coordinates, its andard ones are the objects imensional theory. Applying s gave rise to the four differ- = m a Type IIA point of view. , the internal extended coor- only give the standard super- t. e EFT vector potential ( ) such as ns is possible, thus giving rise 1 s of the Type II theories in ten ed. The result can of course be nt. y irections ( ABC e orientation of the EFT solution 2.15 ξ ǫ ) respectively. , say ¯ m cos ab = 0, the three-form potential only has B.5 y ξ ξ ξ ǫ αβ ) respectively. The third component above sin 2 and thus either Ξ = 1 sin ξ ǫ B.7 π/ 1 − Ξ – 20 – − cos H H i H − A = 0 or sin ) and ( to 0 or ξ = = = ξ B.5 tab iαβ C ABC C C were obtained from the EFT vector potential ) and the pure M2-brane ( µ mn B.7 C it is shown how the ten-dimensional Type IIA theory can direc ] they are given in terms of their field strengths). Again one c which is entirely in the internal sector was extracted from t . Together they read A 63 was decomposed into four distinct subsets ( MN is defined as before. These are exactly the C-field components mnk M i C M Y A In summary, it has been shown that the self-dual EFT solution Let us first obtain the WA-solution, the pp-wave spacetime in In the case of extracting the eleven-dimensional solutions The components of the three-form gauge potential which have It can be checked that setting the pure M5-brane ( a single component as given in ( solution (in [ (together with the eight types of components of theized EFT metric vecto has to beprecise slightly form reshuffled can to be accommodate foundto our in propagate ne the along appendix. one To of obtain the the ordinary wave, directions dinate providing the KK-vector andgeneralized coordinate C-fields. splits Now intoexpect eight in to separate get the eight sets different ten-dimensi of solutions, one d for each possibl vanishes in the two pure cases. in the pure cases where either cos where the EFT wave propagating alongent those solutions four in kinds supergravity of with direction the four components of th extended to take several identical brane sources into accou is constructed whereas here only a single source is consider metric In appendix obtained by pointing theextended vector coordinate along space. one Butto of any dyonic the combination bound of axes states directio of ofdimensions branes. our are Furthermore, also 56 the included. solution We will look4.2 at this aspect nex Type IIA solutions M2/M5-brane solution. Therefore, the EFTgravity solution branes does but not in fact also the brane bound states. The st ponent bedded into EFT without anthis intermediate step procedure to in the reverse, eleven-d the EFT solution can be viewed fro internal indices, i.e. JHEP05(2015)130 1) − , the gives vides 1 θ i y − A H ( = − ˜ z i 1) is the dual sional internal = A ric of the EFT . These are the − z θ t it couples to can 1 y A − 1 an all be extracted C H he generalized metric ompact circle ( and wave. The generalized − are obtained. ated in the appropriate rm ¯ ent is m is derived from the other ) which gives five possible 5 ially the KK-monopole of the Type IIB theory into solution propagates along = y nes with the internal and e and string, the solitonic 7 C e naturally by applying the z provides the RR-seven-form and other Type II solutions t C A.6 s of components in the EFT rresponding solutions can be ave. The other component of have to be rearranged to ac- of the WA-solution under the imensional theory arise. The A ¯ n ¯ m MN g , the D2- and D4-branes with the ). Comparing the Type IIB ansatz ¯ n for the string and vice versa for the M ¯ m . B.2 y 6 1 B C and ¯ n and RR-five-form ¯ m 3 y – 21 – as the direction, the EFT vector C z . The dual seven-form , are rotated in a specific direction. Depending on θ = ˜ t M 1 A µ y A is related to the dual graviton which does not appear in the and the internal 6-metric ¯ . and hence provide the solutions dual to WA and D0, that is i φ θ A 2 y B.2 e = directions respectively. The corresponding EFT vector pro and dual NSNS-six-form ˜ z i 2 and ¯ mθ A B y ¯ m y ) to the (rotated) generalized metric leads to the six-dimen . and is now split into five distinct sets according to ( ˜ θ i A.7 M ¯ A mθ )). y Y in Type IIA and comparing it to the (rotated) generalized met in ( A.9 and the vector potential MN MN We have thus outlined how eight different Type IIA solutions c The picture should be clear by now. The EFT solution, that is t The last two directions the EFT solution can be along are As before, the entries of the generalized metric If instead the EFT solution is chosen to propagate along the c M M it couples to together with the dual one-form MN 7 graviton for that solution.C For the D6-brane, the EFT vector the KK-vector for the ten-dimensional metric and the dual found at the end of this section. 4.3 Type IIBAlong solutions the same linesEFT as allows above, using for the furthercoordinate ansatz solutions for to embedding be extracted from the EFT from a single self-dualmonopole solution and in fivebrane, and EFT.Type the The four IIA fundamental p-even solution wav D-branes todirection. all the aris A section summery condition of to all the the possible EFT orientations wave and rot co the KK6A-brane and the D6-brane.the For Type the IIA KK6A-brane, theory, essent if we choose corresponding set of dual RR-three-form dual directions to component, be extracted from the EFT vector ten-dimensional picture. The KK-decompositioncan of be the found wave in appendix fivebrane. Similarly, if the directions are 4 + 6 KK-decomposition. The corresponding EFT vector compon which provides the KK-vectorexternal of metrics the to decomposition form thethe and vector ten-dimensional combi potential, metric of the w same procedure as above leads to the D0-brane. The RR-one-fo directions to align thevector EFT ( solution (together with five type for solution gives the dilaton M the NSNS-two-form the nature of thatF1-string direction, and NS5-brane different solution solutionone can in of be the the extracted ten-d if the EFT for commodate the choice of coordinates (see appendix JHEP05(2015)130 = M µ and A type, ¯ 6 m a t ) upon ¯ m B i encoded A y A 4 transverse C = ¯ three m a i the internal metric A e solitonic monopole xternal sector but in irections, so it cannot l four-form MN lution along one of the ng and D1-brane. They om a Type IIB point of e EFT solution, they are e. They couple to a six- ontributes the KK-vector S- and RR-part, M rane. time plus s is not as straightforward from the EFT solution by to distinguish between the e implies, does not have a back to the physical space, n (the D(-1)-brane) and its f the EFT vector otating the fields appropri- th the remaining transverse f the Type IIB theory which a t of the exceptional extended vanishes for this solution) can up. , together with the orientation e same as before with the d the dual graviton which plays 0 oes not work for the D7-brane. 2 C gives rise to the other S-duality . Therefore the EFT solution has ( φ 2 ¯ m a e y . If the direction of choice is of the ab γ – 22 – , gives the dual solution, that is the KK6B-brane, provides the two-form (and also the dual six-form). ¯ m y ¯ m a . From the generalized metric 2 µ C A containing the dilaton ab γ and the RR-field 2 together with the SL(2) matrix B , leads to the self-dual D3-brane together with its self-dua 1) (and the dual two-form is encoded in the magnetic part . ¯ n ¯ k ¯ k ¯ ¯ n m ¯ n − ¯ g m ¯ m 1 y − . This produces the Type IIB S-duality doublet of the F1-stri and the SL(2) matrix µ Finally, having the EFT solution along the fifth direction fr The issue for the D7-brane is that it only has two transverse d As in the Type IIA theory, the fundamental wave and string, th In theory it should also be possible to obtain the D-instanto Similarly, the EFT solution along one of the The dual choice of direction, i.e. A more interesting choice of direction is to rotate the EFT so Both of these reasons are not fundamental shortcomings of th respectively. The six-form is encoded in the electric part o ¯ H A n ( 6 ¯ ¯ m a m ¯ dualization on the internal coordinate. − and fivebrane, and threeapplying p-odd the D-branes, Type can IIB allately. solution be All to extracted the the obtained section solutions condition are and summarized r in table to be set upthe in internal sector such of a thespace way KK-decomposition. it that can Then the be being time rotated par leaving and coordinate a “removed” solution is when without taking not a the in section time the direction. e dual, the D7-brane, from theas EFT for solution. all The the reasontime other why direction, D-branes thi it is is a that ten-dimensional the Euclidean instanton, solution as the nam in the EFT solution, i.e. its direction of propagation. view, metric ¯ fully be accommodated by our KK-decomposition which places C the WB-solution can be extracted.is identical This to is the the WA-solution. pp-spacetime The o procedure is exactly th be extracted. The EFT vector form which also carries an SL(2) index to distinguish the NSN doublet of the Type IIB theory, the NS5-brane and the D5-bran providing the KK-vector for the ten-dimensionalno metric role). (an couple to a two-form which carries an additional SL(2) index NSNS-field the KK-monopole of the Type IIBand theory. dual Again graviton. the EFT The vector KK6B-brane c is identical to they KK6A-b direction in the externalbits, sector if and there the are the any) world in volume the (wi internal sector.just This technical clearly issues d arising from the way we set everything g JHEP05(2015)130 , is M are r µ (5.1) A M µ A 3 6 2 where C C C 6 2 6 3 7 5 1 3 4 M j i / / ⊕ C C C C C C C B B w A 6 2 6 i B B C w ij δ = ng in 4+56 dimensions ! 2 r it? 6 2 6 MN nsional metric goes to zero or infintiy, i.e. C t to analyze it further. The C C tituents of the solution, the en dimensions discussed in this 2 6 3 6 1 3 7 5 4 M r MN tion of a membrane direction t / / M ⊕ C C C C C C C B B ernal sector. This leads to the and the vector potential ole-structure in EFT and shown 2 6 A 3 M ) with M µ r B B C and the vector potential ( KK-vector dual graviton KK-vector dual graviton KK-vector dual graviton A termines the nature of the components MN dual graviton KK-vector dual graviton KK-vector dual graviton KK-vector H ong which the EFT solution propagates MN M MN ¯ k M ¯ ¯ n n ¯ n ¯ ¯ ¯ ¯ ¯ ¯ M m mθ m m m m a ¯ ¯ ¯ ¯ ¯ m m m m mθ m a mn θ N and N µ µ µ µ µ µ µ µ mn µ µ m µ µ θ µ µ µ µ µ ν ν EFT A A A A A A A A A A A A A A A A A A µν vector A g – 23 – M MN µ A M ¯ k + ¯ ¯ ¯ ¯ θ n n n ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ m m m m m a mn θ m m mθ m m a m m m θ mn µν ∗ ∗ . Therefore the EFT vector in that hybrid direction gives both y y y y y y y y y y y y y y y y y g

mn y orientation = ˆ N ˆ M H since it is a dyonic solution. F1 D3 D0 D2 D6 D4 M5 M2 , the extended internal metric WA WB 6 NS5 WM KK7 KK6B KK6A C M2/M5 µν solution F1 / D1 NS5 / D5 g and the 3 and a fivebrane direction = 10 = 10 = 11 . This table shows all supergravity solutions in ten and elev C The orientation of the M2/M5 bound state requires a superposi theory D D D mn To carry out the analysis, the solution is not treated as livi Type IIB Type IIA ∗ y the combined in the usual Kaluza-Klein fashion to form a 60-dime but as a truelyexternal 60-dimensional metric solution. Thus the three cons what happens close to the core of the solution or far away from the radial coordinate ofimmediate the question tranverse of directions what in the happens ext to the solution when are all expressed in terms of the harmonic function Table 2 5 Wave vs. monopole Having constructed this self-dual wave solutionhow with monop it relates tofields the known of solutions the of solution, supergravity, that we wan is the metrics section. The orientation indicatesto the give rise type to ofof each direction the of al EFT the vector supergravity in solutions. the It supergravity also picture. de JHEP05(2015)130 2 / 1 . ± r (5.2) (5.5) (5.3) (5.4) H vanishes) region. It i r A heory. The space ) to be . 3.4 ], the nature of singu- magine asymptotically i . section condition gives 2 64 ansverse dimensions and / in the small d some of this work. In structure of supergravity    lock diagonal form simply 3 Taub-NUT), also with 27 − 27 0 0 us electromagnetism (and a δ ! mes then one would imagine becomes large and the wave ome with prefactors of between large and small y frame that one uses. Given i ,H llows various singular solutions and M-theory when thought of ¯ 1 A ave dimension 29 and 31 respectively B H 27 1 ¯ ry Kaluza-Klein theory that will − mono A δ 6 − 2 H d through a duality transformation. H / 0 1 2 H is taken from ( / , − 1
j −    ,H A MN H i is small, 27 0 27 δ A will be close to one (and thus δ r j M 2 2 1 0 / − A wave AB 1 H 1 − 0 0 H H , H 0 − – 24 – r 2 ,H H + / ) and H 2 1 / 1 2 ij 3 H δ 3.2 − − 0 0 H

h H H H . in ( = ¯ A       ˆ M N µ ˆ = = M and = diag A H A ¯ B ), , wave AB ¯ mono A ˆ MN M H 3.1 H M As is well known from the works of Duff and others [ is given in ( 7 µν g ) it is interesting to see that there is a natural split into a b Close to the core of the solution, where A good way to think about DFT or EFT is as a Kaluza-Klein-type t 5.3 The resulting object has dimension 60 whereas the two blocks h We are grateful to Michael Duff for discussions on this issue. 6 7 geometry dominates. Far away for large either description is valid and theHowever different choices what relate is curious is the dominance of the wave solution and neither the monopole nor wave dominates. Thus one would i 6 Discussion Here we wish toparticular enter we into wish some tosolutions. speculation comment that on has the motivate issue of the singularity appears according to this analysisas that solutions all in branes EFT in are string wave solutions in the core. with opposite power, so the geometry will change distinctly We then insert the EFT monopole/wave solution to find In ( that EFT provides athat formalism solutions unifying in different EFT duality maybe fra be singularity free at the core. larities at the core of brane solutions depends on the dualit by composing the fields `ala Kaluza-Klein. These two blocks c as indicated by the indices where to top leftthe block bottom is right simply block thetransverse metric is dimensions of the a metric wave of with a 27 monopole tr (euclidean where be useful for ourscalar intuition. field which will The not reduced be theory relevant here). is One gravity typically pl a is extended andsupergravity. the Let’s reduction recall of some the basic properties theory of through ordina use of the JHEP05(2015)130 of singularities at ed into EFT. The ]. ing avatar DFT. The s aspects of this work. ip. 65 make such a statement different solutions of the 1 “String Theory, Gauge ecome removed when one lyakov monopole where in ace is also not something become smoothed out by uld be singularity free in the core of the monopole solutions smooths out the the Riemann tensor is not uce a total space describing at we wish to envoke when s in a more rigorous fashion to U(1) and the monopole is ear the core of the monopole y but to see that one must are simply the result of some me bigger non-abelian group, nd the full non-abelian theory t the naive singularities in the ting around the KK-circle and eneralized geometry in analogy gularities and magnetic sources one by examining amongst other extended geometries we can only l theory [ ] which discusses this extensively for brane 64 – 25 – . Thus the singularities inherent from the abelian charges b 3 So can one show that the EFT solutions described here are free A similar process happens when one considers the ’tHooft-Po S A Embedding the Type IIIn theories this into appendix EFT difference we between show Type how IIAsection and the condition Type Type to IIB the II arises EFT from theories equations. applying can be embedd We thank Michael Duff andDSB Malcolm is Perry supported for by commentsTheory the on and STFC variou Duality” consolidated and grant FJR ST/J000469/ is supported by an STFC studentsh say that the solutionsto are the suggestive of Riemannian a case.in singularity It future free is work. g hoped to explore these question Acknowledgments solutions. In EFT,with we full do confidence. not haveunambiguously Even at defined the this and idea point geodesic ofyet all a motion explored. the generalized in tools notion the Thus of to extendedrelativity, until sp although we have the developed more solutions analytic tools presented for here wo which also are singular.singularities. The Then electric thethe charges Kaluza-Klein magnetic are charges lift come just of from wavesan fibering these propaga the KK-circle to prod such as electric sources which have delta function-type sin and the singularity isthinking removed. about DFT This andthe intuition embedding its is in M-theory exactly a counterpart. bigger wh reduction theory, and U(1) Solutions the charges singularities in are particular non-existent in the ful the low energy effective fielda theory normal the Dirac gauge group monopole is (withhowever, broken a the singularity low at energy the effective origin).becomes description breaks N relevant. down, a The non-abelian interactions smooth out in this case five-dimensional diffeomorphisms. considers the full theory and the U(1) is just a subgroup of so their core? Thewave and question monopole drivescarry are at out singularity the a reasonabley free heart sophisticatedsolution in of analysis are general purely and EFT coordinate show relativit and singularities. tha things its This geodesic maybe str d motion. See for example [ JHEP05(2015)130 , 2 θ B µ A (A.2) (A.1) (A.4) (A.3) nternal i ¯ n cise nummeri- ¯ m e how the self- ¯ g 1 − . ¯ g are encoded in φ o embedding the Type the coordinate of the 7 , e by simply splitting the ¯ l ¯ mθ ero components are the θ C . ¯ k µ ting this ansatz into the is just the KK-vector of ¯ n, eleven-dimensional super-  A 2.2 = ¯ ¯ θ , m ¯ t for each of the directions m ) gives contain the NSNS-fields well using the same solution tween the form fields in the ¯ ¯ ¯ g and θ n , m µ as before. The vector is also 1 r KK-decomposition. y ¯ ¯ n m 5 e not considering any internal A − ¯ , y ¯ mθ µ m i M 2.18 ¯ C g ¯ n µ as before (barred quantities are ¯ g 3 2 µ A ¯ , m φ , A 3 ¯ n φ/ A φ/ − 4 ¯ , y C m − 6 and ¯ θ µ θ , e g , , e l ¯ , e 1 A and ¯ ¯ k n , y 1 , ¯ C ¯ ¯ n, m m − ¯ m ¯ g ,..., ¯ ¯ ¯ g m mθ 6 , y 2 ¯ g µ µ 2 φ/ . Instead of a 4 + 7 coordinate split in the φ/ ¯ A mθ = 1 3 A − φ/ , e − , y 2.2 h – 26 – , e ¯ m ¯ mθ n 2 , e ¯ m ¯ n µ φ/ ¯ m 3 A of our solution is diagonal, no KK-vector will arise , y ¯ g , θ , e 1 ¯ n = diag ¯ n − , y ) with ¯ ¯ mn m ¯ θ ¯ g m ¯ g m ¯ µ g mn y , y h g A  ¯ m , y θ are defined in terms of ¯ = µ diag l ¯ 2 ¯ k A M / = ( encode a component of a field from the Type IIA theory except , ¯ 1 n, respectively where the latter two have to be dualized on the i Y ¯ g ¯ m m m M µ g y µ µ θ A ) = ¯ A A n and ¯ , φ ¯ n l ¯ ¯ ¯ and k m ) g ¯ n, ¯ n (¯ ¯ ¯ is the string theory dilaton of the Type IIA theory and the pre m m g A.1 µ φ MN , ¯ 2 A , where again the second one has to be dualized. It is nice to se g e , 6 which relates to the dual graviton. The first one, M ¯ n B Under the split ¯ ¯ m m µ µ from this decomposition. We thus simply have cal powers have beengeneralized chosen metric to for embedding be supergravity in into the EFT Einstein ( frame. Inser where Noting that the internal metric circle, the corresponding generalized coordinates read split under the abovegiven decomposition in ( resulting in a componen six-dimensional space. The remaining two, A the original 4 + 6 decomposition. The RR-fields duality of the EFTType vector IIA contains theory. all the known dualities be All these parts of and A components of the RRone-form or parts NSNS gauge which potentials. are in The the only EFT non-z vector potential six-dimensional). As in the eleven-dimensional case, we ar where ¯ internal seven-dimensional sector into 6 + 1 by doing anothe KK-decomposition we now haveIIA a fields into 4 EFT + can 6 be split. obtained from The the dictionary results of for section The ten-dimensional Type IIAgravity on theory a is circle. a Itto simple is the thus reduction possible section of to condition embed it given into in EFT as section A.1 The Type IIA theory JHEP05(2015)130 2 0 ) C /C 2 A.6 (A.7) (A.8) (A.9) B denotes B the same as i ¯ n ¯ m ives the duality ¯ g not 1 rom the internal e. Here . Loosly speaking − direction in ( ntal representation ¯ ¯ g + 2 split where the m , y ab is related to the eleven- γ φ ¯ is the metric on the torus n xternal four-dimensional 1) (A.5) − t of ¯ . , m M-theory on a torus. This he Type IIB theory except w from the solution to the ab ie ¯ g corresponds to the self-dual . The seven dimensions are γ

1 ¯ k ¯ + − ) (A.6) m ¯ n ¯ g sional index is 0 ¯ ¯ internal coordinates m is the KK-vector of the original m µ , C ¯ q 2) + (6 l ¯ contain the SL(2) doublets µ A , y ¯ , m ¯ n , vity. There is another, inequivalent = µ A ¯ p, ¯ ¯ k m a A ¯ m a ¯ m a ¯ m µ µ SL(2) and we have , y ¯ g above) and ¯ k A 1 A 1) + (6 ¯ × n − , , , τ ¯ ¯ ¯ m k g ¯ , n and ! mn,kl (6) , y ¯ m ab g τ γ µ 1 ¯ – 27 – GL ¯ ¯ n m a m a Re A ¯ m 2) + (20 , µ , y , ¯ g τ 2 ¯ | , m A ¯ τ n ¯ 2 is an SL(2) index. The middle component is totally y m a | (the “axio-dilaton”) given in terms of the RR-scalar Re , ¯ m µ ¯ g τ

= ( h A 1) + (6 τ = 1 . We will come back to this setup at the end of this section. , , φ M 1 ¯ a (in analogy to m 2 (6 Im e ] µ Y diag q | 2 A p / = →  ¯ 1 g | g l | 6 and ab 56 k γ ¯ g | ) = ¯ n the dual RR-field. The component [ ,..., ab m C g , γ ¯ n = 1 = ¯ ¯ m . Again it can be seen that the self-duality of the EFT vector g g m ], the generalized metric (again without any contribution f (¯ 4 where the latter one needs to be dualized on the internal spac 7 ] which is related to a different decomposition of the fundame C 6 7 MN mkp,nlq /C g which relates to the dual graviton. As always, 6 M . The relevant maximal subgroup is B 7 Let us conclude by checking how the Type IIB theory on a circle From [ The EFT vector is also decomposed and has a component for each ¯ (a single component) is reinterpreted as the sixth componen m E µ ab dimensional theory onspacetime a in torus. common, we Since will both only look theories at have the the internal e sector a NSNS-field and and relations between the form fields in the Type IIB theory. is made precise at the end of this section. this comes from the fact that Type IIB on a circle is related to four-form where again ¯ in the 6 + 1y Type IIA decompostion above. Here we rather have a 5 which translates to the following splitting of the extended 4+6 decomposition. The components antisymmetric in all three indices. Note that the six-dimen components of the form fields) for this case is given by of where ¯ with the complex torus parameter As before, these parts eachA encode a component of a field from t Unlike the Type IIA theory,section the condition that Type gives IIB eleven-dimensional supergra theorysolution does [ not follo A.2 The Type IIB theory and the string theory dilaton JHEP05(2015)130 = 2. nt. , the ¯ n ¯ m which = 1 (A.11) (A.14) (A.10) (A.13) (A.12) g a 12 , 12 γ 5 and ,   ab ˙ q , , these coordinates ˙ ab,cd ˙ ) to give p γ q ,..., . ˙ γ 1 p ˙ 12 n, ˙ − y ∆ ˙ n,  m ˙ g ˙ = 1 ˙ − 2.18 m g  ab ˙ 1 g  2∆ , e m − , y ∆ 2∆ ab − dimensional metric ˙ g ab e − γ ˙ 1 γ e ma , e the discussion here, such that ˙  2∆ 1 − n . This ansatz can be inserted 2∆ ˙ ) ab ˙ g − , y − m s. o EFT ( γ , ˙ ˙ 2∆ ˙ , e n g g ) 2∆ ˙ ˙ above. Now turn to the seven- n n , e . , ˙ − ˙ , e ) where ˙ ˙ n m ab q ˙ ˙ ˙  ) ) q ) ) i ˙ m m a ˙ p ˙ ab m p γ ˙ ˙ , e a a g g m ab , y = ˙ ˙ ˙ γ 12 12 n n, n ˙ 1 1 n, , y a γ , y ˙ ˙ , y ˙ | , y has only one component ˙ ˙ m m − − m ˙ m , y ab,cd m , y n ∆ 2∆ ˙ ˙ ˙ ˙ ˙ , y g g g g g ˙ ). With this in mind, the components ab b g ˙ ˙ ˙ ˙ ˙ γ y ] ma m      − m m ma ˙ γ , e m d ∆ e ∆ y y y y y ˙ n γ ∆ 1 e , y ˙ c just like the coordinates. The five parts of = ( e , e ab,cd 1 [ m − | ˙ ˙ a γ g = ( = ( = ( = ( = ( ˙ − diag diag diag diag diag ab g g ab m ˙  γ , g γ ¯ k y 2 ∆ ˙ ¯ ¯ , y , n m m 2 2 / ¯ n e ¯ ¯ ˙ ma ma – 28 – y = / / y m ab 2∆ 2 2 ¯ ˙ m y y ˙ ma g / / is det ∆ ∆ 3∆ γ = y ˙ , e n ∆ − ∆ − − ∆ , y ˙ n g ˙ e e e e e ab ˙ = diag m − n ˙ ab,cd ) and similar for the dual coordinates to make contact ˙ m g g e ˙ ˙ γ m 1) : 2) : 1) : 1) : 2) : g = = = = = 1 1 12 , , , , , mn − − ∆ ¯ q , y ¯ ¯ n n ˙ ˙ l ¯ g g e g , y ab ab (6 (6 (6 (6 ). These identifications here are not obvious, but can be a ¯ ¯ ¯ n m m h ˙ γ γ (20 m ¯ g ¯ g p, ¯ n ¯ , y n y ¯ 1 k ¯ ˙ A.6 ¯ m m m ¯ − m 2 sector ¯ g g diag y g ¯ g 1 = ( 2  1 / × − ). There the torus metric is conformal and has unit determina 1 ¯ − m g . Again omitting a KK-vector for cross-terms, the ansatz for = g g y A.8 = ˙ mn M g ), the generalized coordinates then decompose as 1) : ¯ 2) : ¯ 1) : ¯ 2) : ¯ 1) : ¯ has only a single component (by antisymmetry), Y ] and determinant ¯ , , , , , MN ab 2∆ (6 (6 (6 (6 y − 2.15 (20 M , e ˙ n ˙ m ). We thus have g as given in ( ab A.6 γ diag[ ˙ 2 / ∆ e the generalized metric thus read justifying the presence of the six-dimensional index ¯ of the generalized metric can be repackaged in terms of a six- dimensional metric with ( can be repackaged into evaluates to 1 (and similarly for the inverse which is in agreement with ( checked by an explicit calculation of individual component For completeness, we include a volumethe factor determinant for of the torus the in 2 It is easy to check that the object By noting that decomposition is (a dot denotes a five-dimensional quantity into the generalized metric for embedding supergravity int split into 5+2 such that the coordinates are Starting from ( with JHEP05(2015)130 ˆ ν ). ˆ µ g and 7 in , the have (B.1) (B.2) 2 ˆ ν 2.18 . The ˆ µ B g all M ,..., µ . The form A = 1 to remain in µν 4 + 6 and the p g mn → g according to ( dimensional external scaled ven ones to the Type mn damental, solitonic and g umber of external dimen- s of the harmonic function ion in the main text. It also dimensional internal sector. ! za-Klein coordinate split. It 4 + 7 or 10 three of the transverse direc- the internal sector is six- or ed in terms of the metric ˆ ons of each solution. Then we lutions are the metric ˆ ompostion. potentials with and takes the form mn orld volume directions together → p g ravities as they can be found in which are duals of each other. In is called the KK-vector and will mn C m is a constant which can be set to zero if 6 g lutions, such as that they . µ m 0 C A µ φ µν g A 2 in Type IIB also enter the generalized / mn and ] was an invaluable source), but also to 1 g | τ 3 where n 66 C ν 0 n mn φ ν g A 2 e A m µ – 29 – det mn A g + → | µν µν g ) are either split into 11 g

m constitute the components of the EFT vector , x = m µ ˆ ν µ x ) respectively. ˆ µ ˆ and the two-form and six-form Kalb-Ramond potentials g A is constructed from the internal metric = ( φ A.7 2 ˆ µ e MN x 8 M ) and ( in Type IIA or the axio-dilaton 2 in our case. / A.3 φ has to be rescaled by the determinant of the internal metric µν g The eleven-dimensional supergravity solutions are specifi The coordinates ˆ Each solution is presented with its full field content in term From an EFT point of view, the external metric is simply the re The constant part of the dilaton is denoted by which again are duals. In the RR-sector we have the which has a functional dependence on the transverse directi 8 6 string theory dilaton the NSNS-sector, the fields of the ten-dimensional Type II so and the three-form and the six-form potentials where hatted quantities areseven-dimensional. ten- The or off-diagonal eleven-dimensional or and cross-term metric mostly be zero except for the wave and the monopole. The four- corresponding KK-decomposition takes the form The power of the determinantsions in and the is 1 rescaling depends on the n the Einstein frame. This is crucial for comparing solutions convenient. perform the explained KK-decomposition bytions picking time to and be in the four-dimensionalwith external the sector and remaining the transverse w ones to be in the six- or seven- The dilaton metric as in ( H highlights some interesting similarities betweenthe these same so four-dimensional external spacetime under the dec present them with their fieldsis the rearranged decomposed according fields to that a are Kalu extracted from the EFT solut The purpose ofDirichlet this solutions appendix of is ten-any and not standard eleven-dimensional only text superg to book collect (for all us Ortin’s the book fun [ B Glossary of solutions B this paper. The oddIIB theory. ones belong to the Type IIA theory and the e fields and the KK-vector generalized metric JHEP05(2015)130 (B.4) (B.5) (B.3) (B.6) (B.8) (B.7) 3. , 2 , .
= 1 1 i, 3 − 3 / 2 / (9) 2 2 1 1 (5) (9) 1 ~y i for − − ~y ~y i i d H A 2 (8) H H 3 y A ~y + d / = 11 + d is the “special” direction of = = 2 d = 2 = = 2 H − 3 8 z z  H i / y D 5 t d 1 mn 7 y = = sverse directions in the external w mn y g + 4 g 6 H H z i y iy t 1)d 5 mn + + ’s denote these three directions. y det 2

g (5) − det 4 i z w over those transverse directions in the ~x 1 iy d (5) (2) .
2 t . − (2) det i , ~y , ~y i ~x , (6) + d ,C H ij , world volume directions of a p-brane and  ij ( 1)d δ
2  (2) (5) 5 δ t 2 p . 1 2 2 + d z, ~y ~x ~x / − ,A − d / , δ 1 3 ,C and the ] 2 , δ | 1 smeared − 2 z − t . 5 δ ij j H 1) d 1 h = = = h d δ 2 – 30 – 1 δ (5) ,H 6  1 w − 3 | ,H 2 2 ~y − − i − / | 2 m m m / / − h 1 H h / 1 H 1 1 w x x x (8) + 2( H 1 3 − H . − ~y −  ij / 2 | + −  is the time coordinate, 2 δ t 7 ,H − H H H 2 | 2 t − H t ( and and and transverse directions, the first three of which are usually / denotes the , = − h = = 1 + = d 1 − (9) ] diag 2)d − H h )


1 p diag 2 6 h 3 ~y 5 − 2 p | 3 r / − ( x s H − = 1 + = = / − 1 (3) (3) (3) 4 H ~x d 2 x H 2 2 1 3 H H, δ − H x s H H t, ~y t, ~y t, ~y x − 1 − d 2 with = = diag x tx = = diag = 1 + = = ( | 1 D = = = C r 2 µν | tx mn µν s H µ µ µ g mn g C g = diag[ = diag[ d x x x g h/ µν mn g g = 1 + H the remaining ) p − 1 A final note on the notation: − D ( The wave — WM. B.1 Wave, membrane, fivebrane and monopole in taken to be in the external sector as explained above, i.e. The membrane — M2. KK-decomposition: KK-decomposition: The fivebrane — M5. the wave and the monopole, KK-decomposition: internal sector so that itsector, is i.e. only localized in the three tran As part of the decomposition the solution is ~y JHEP05(2015)130 (B.9) (B.15) (B.12) (B.13) (B.11) (B.10) (B.14) j y d i 3 y / 2 (3) ξ 2 d . ~z ξ . −

d 2 j Ξ 3 1 cos 3 / A /
2 i . − cos 1 1 1 2 . − (8) 0 A − 1 i φ 2 H ~y Ξ H − − 2 3 − H A e 1 / = + H ξ 1 1 − H + d = = ξ − i H 2 z H 2 + mn sin H −  i H A t g i mn + = 10 ij
g 2 = = − (8) = δ A = 2 ~y z ¯ 1)d (5) n (3) D det 8 φ t = d = ~y y 2 ¯ m det 4 7 H . , ~z − d 3 3 g Ξ = sin / , 2 y 3 (3) z z 0 1 1 6 / 2  ,A 2 +
~y (2) φ
y − 1 x 3 x 2 5 d H i z 1 det ¯ 1 δ e y Ξ , ~y (6) z (5) z H 1 d ij 4 3 H 2 5 i + ( δ / − z = y ,A iy (2) y 2 1  1 4 + Ξ / z, ~x z, ~y − ~x i d 2 φ 1 , i tx iy H 2 2 x . ij z 2  A δ i i d H. , + = = = 1 2 – 31 – ,H , δ  k y ij /  2 2 − + d i ∂ 1 d ¯ δ 6 / m m m ξ δ ,B i H k 2 2 1 1 H x x x 2 t / (2) , δ , e A , e ij − 1) 1 − ,H 1 d + ~x ǫ 6 cos 2 6 | − H + 2 | 2 1 − H / − + 2 ξ t 1  , ,H  and and and h ξ, C z 1 − 2 H (8) h d + d ] 2 (8) − 4 =  h d z ~y − / 1 5


/ ~y 2 | sin  1 ] | d 3 t − H sin j H 1 1 1 − (3) (3) (3) d − diag (
− A H − − − 3 h H, δ i 1 h H / [ H − 3 − , 1 t, ~y t, ~y t, ~y H H / − | − = diag = diag 1 Ξ 1)Ξ = 1 + = = h = 1 + = + + 1 3 ξ, C Ξ (3) h 2 / = = = 2 − − 3 tx µν 1 ~y s H mn / s H | 2 2 g µ µ µ (6) (6) d 2 g B = diag[ = diag H , ∂ d cos H H x x x ~x ~x − | ( i ¯ n µν = diag = ¯ − A H − m (3) h g + d + d ¯ g ~y | µν = 1 + = = = = 2 2 mn t t g 3 5 2 2 g d d z y x s H 2 4 1 d − − z iy tx 1 z = 1 + = = C C C 2 s H d The wave — WA/B. KK-decomposition: B.2 Wave, String, Fivebrane and Monopole in KK-decomposition: KK-decomposition: The M2/M5 bound state. The monopole — KK7. The fundamental string — F1. JHEP05(2015)130 (B.22) (B.18) (B.19) (B.17) (B.21) (B.16) (B.20) . 0 2 p 2 φ 1 / 2 . 4 (3) − . / 2 2 1 (4) 3 − 0 ~y 1 i e ~y − φ d p 2 H A d rane. ) H H 2 − H p 4 i H 3 / i − = = h the F1-string. This means = = 3 A He = 2 + H (9 A nd their dilatons are inverses z ¯ n ¯ ¯ n n i H ~y ¯ n 2 ¯ = = ¯ ¯ m m m  = d ¯ = m i g g g 4 + φ g φ p y 2 8 +1 iy
i 2 − d p ) 9 i p det ¯ det ¯ det ¯ 2 (5) det ¯ H − A
~x ...y . ,A 4
4 (6
. , 0 . + . + i  i iy φ i , ~y , y i i (5) (5) z p 2 + d ,B ij ) ij ) e ij d ij − δ p , p δ
2 2 , (5) ( δ (  6 δ 2 t 2 1  , e 2  / 2 z, ~x x, ~y ~x ~x / = ~x 1 d 1 / 1 / 5 , δ 2 1 , e 1 − | , 1 − p φ − p 5 2 δ , δ ,C , 1 h = = = = h δ H + d – 32 – − ,H 1 1 (4) ,H
 1 − 4 ,H 7 ,H 2 2 ~y 2 − − | ¯ ¯ ¯ ¯ 1 5 / 2 | / 2 m m m m + / − t ) 1 / 1 / H 1 p x x x x d h H H 1 1 , δ − − − H −   − 2 1 − (5) −  − 1 , e − H − (9 h H ~x H − | H H ~y and and and and 7 | − H − = = 1 + = 6. 8 − H − − diag diag  h (3) h h



diag p + d h h 5 4 2 ~y 4 / | x s H 8 2 +1 / 1 (3) (3) (3) (3) 4 H − t p d 3 x d 3 H ,..., H = = 1 + = H x t, ~y t, ~y t, ~y t, ~y − 2 = diag = diag 2 p = = diag x = = diag = diag = s H = 10 1 = 1 + = ¯ n ¯ n = = = = = 0 d ¯ ...x n ¯ n µν µν tx ¯ 2 ¯ 1 m µν m µν ¯ g ¯ µ µ µ µ g m s H D m p ¯ g ¯ g g g B tx ¯ g x x x x ¯ g d C Note: in Type IIB thethey D1-brane are forms an identical S-duality solutions doubletof wit up each to other. an The SL(2) same transformation applies a for the D5-brane and the NS5-b The monopole — KK6A/B. KK-decomposition: KK-decomposition: KK-decomposition: KK-decomposition: The Dp-brane for The solitonic fivebrane — NS5. B.3 D-Branes in JHEP05(2015)130 ] , ack , ] ommons ]. ]. ]. (1995) 211 ]. , SPIRE ]. SPIRE (1993) 2826 IN IN [ SPIRE ][ IN , arXiv:0904.4664 B 455 [ SPIRE ][ SPIRE , D 48 ]. IN , hep-th/9806106 [ IN [ ][ ]. covariant Form of M-theory , redited. , ]. SPIRE ]. ]. 8(8) IN 6(6) 7(7) SPIRE Supergravity (2009) 099 ]. Nucl. Phys. E ][ Phys. Rev. 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