JHEP10(2019)198 ∗ Springer October 1, 2019 August 18, 2019 October 18, 2019 : : string theory is : 0 Received Accepted Published string. In the IIA dual, this 0 -flux can be realised in string . Further T-dualities then give 2 H T Published for SISSA by https://doi.org/10.1007/JHEP10(2019)198 string, with the Kaluza-Klein monopoles 0 , with the 3-torus fibred over a line. T-dualising 3 T [email protected] , . 3 1907.04040 The Authors. c p-branes, String Duality, Flux compactifications, Superstring Vacua

A recently constructed limit of K3 has a long neck consisting of segments, , [email protected] The Blackett Laboratory, Imperial CollegePrince London, Consort Road, London SW7 @AZ, U.K. E-mail: Open Access Article funded by SCOAP Keywords: ArXiv ePrint: These are all domainthe wall natural configurations, home dual for toof D8-branes, these the and branes. D8-brane. we The dualise Kaluza-Klein TypeT-duals this monopoles of I to arise this in find give the string exotic IIA theory branes string homes on on for the non-geometric degenerate each spaces. K3. dualised to NS5-branes ortheory exotic as branes. an A NS5-branegives 3-torus wrapped a with on 4-dimensional hyperk¨ahlermanifoldcan be which viewed is as anon-geometric a Kaluza-Klein nilfold spaces monopole fibred fibred wrapped over over on a a line line, and which can be regarded as wrapped exotic branes. dual to theorientifold D8-branes planes. and At the strongplane, coupling, Tian-Yau so each spaces O8-plane that providing can there emitphenomenon can a a be occurs geometric D8-brane up at dual to to weak givein 18 to coupling an the D8-branes and the O8 dual in there geometry. O8 the can We type be consider I up further duals to in 18 which Kaluza-Klein the monopoles Kaluza-Klein monopoles are Abstract: each of which ismonopoles. a The nilfold neck is fibredhyperk¨ahlerand asymptotic capped over to at a a either nilfold line, fibred end overon by that a a this line. are Tian-Yau degeneration We space, joined show of that which together K3 the is type is with non-compact, IIA dual Kaluza-Klein string to the type I N. Chaemjumrus and C.M. Hull Degenerations of K3, orientifolds and exotic branes JHEP10(2019)198 ] and 9 – 6 ] and references 5 – 3 12 ], the reviews [ 2 , 1 13 6 – 1 – 11 22 17 13 16 11 5 string theory 6 0 16 string 19 10 0 24 14 18 string 0 1 ]. One of our purposes here is to seek new natural set-ups in string theory 25 11 , 10 6.1 Duals of6.2 the type I Moduli spaces and dualities 5.1 The D8-brane5.2 and its duals The type I 3.1 The NS5-brane3.2 solution Nilfold background 3.3 T-fold and3.4 R-fold backgrounds Exotic branes and T-folds sponding non-geometric backgrounds, and that there is a relation between the brane and were associated with non-geometric spaces thattions, are in U-folds, [ with U-duality transitionthat func- give riseconstruct to complete consistent consistent non-geometric configurations stringstring of backgrounds vacua that exotic with arise branes. chains from of acting on dualities. Another related We find aim that is exotic to branes naturally live in corre- 1 Introduction String theory can be defined intime non-geometric manifolds backgrounds that equipped are not withtherein. conventional space- tensor Exotic fields branes — arise see from [ conventional branes after a chain of dualities [ 10 Non-geometric duals 11 Discussion 7 Dual configurations 8 A degeneration of K3 9 Matching dual moduli spaces 6 Duals and moduli spaces 4 Single-sided domain walls and5 Tian-Yau spaces The D8-brane and type I 3 Domain walls from the NS5-brane Contents 1 Introduction 2 The nilfold and its T-duals JHEP10(2019)198 theory to 0 string theory 0 ]. T-dualising 21 ]. T-dualising in – 20 [ 19 τ -flux and the essentially ] (a space with T-duality spaces. In the nomencla- Q 1 combined with a reflection L F ]. T-duality on one circle turns 1) of the line [ 14 − τ string theory to a dual configuration by ( 0 -flux [ 4 T H known as a nilfold, but further T-dualities 2 essentially doubled ] are dual to D7-branes. Type I -flux, the T-fold has T ] and the subsequent T-dualities lead to exotic f – 2 – 11 , 13 10 ]. The simplest case is that in which these spaces are to the type IIA string on K3 while taking the D8-branes 18 3 T ] with D8-branes between two O8 orientifold planes. The next 12 -flux with a metric and dilaton depending on ], these are referred to as -flux. However, the 3-torus with flux and its duals do not define world- H 16 R theory [ 0 ]. In [ 15 ] the nilfold has geometric flux or -flux and gives a circle bundle over 17 theory compactified on ]. We aim here to understand these backgrounds and their implications for string H 1 0 [ These configurations can arise in string theory, however, as the fibres in string back- Another motivation for our project is as follows. The starting point for a much-studied Compactifying type I string theory on a four-torus and T-dualising on all four circles The exotic branes that we will consider here can be obtained from D8-branes by a chain τ in one direction giveswhich a has conformal constant field theoryanother direction on gives the the product product ofon of a T-fold the with line a with linetheory with a better. the 3-torus, moduli again depending sheet conformal field theories and so cannot be directly usedgrounds as which string have theory a backgrounds. using bundle the structure, adiabatic and argumner thenfibred of these over [ are a related line. byin There fibrewise which is dualities, the a hyperk¨ahlermetric nilfold on moduli the depend product on of the the nilfold coordinate with a line transition functions) and ation further T-duality which is is argued notgeometry to a [ give conventional a space non-geometric eventure configura- locally of [ but candoubled be space represented has by a doubled branes in a non-geometric background. chain of T-dual backgrounds isoff the the 3-torus with give non-geometric spaces. T-dualising the nilfold gives a T-fold [ and ON-planes. Wethe shall directions be that interested are inNS5-brane dual to the to a the result Kaluza-Klein original of monopolebranes. 4-torus. T-dualising [ We From this shall here in see thea 1,2,3 that first geometric or the T-duality background 4 first takes and a T-duality of we leads shall to argue Kaluza-Klein that monopoles the as part subsequent of dualities lead to exotic gives an orientifold ofto IIB the string theory. type I Thethree T-dualities first take T-duality the of D8-branestakes and the O8-planes this type to to I D5-branes an and theoryin orbifold O5-planes. takes of the S-duality it IIB four string toroidal theory coordinates, on and takes the D5-branes and O5-planes to NS5-branes the D8-branes to thethat exotic is branes the takes natural the hometype type I for I the exotic branes.to One Kaluza-Klein of monopoles. thecertain dualities It hyperk¨ahlerspaces in and also the gives chain takes an takes the interesting the picture orientifold of planes strings of moving on the K3. type I geometric string backgrounds, such as the IIA string onof K3. T- and S-dualities, whereasprovides those of a [ consistent string theorya home line for interval, with D8-branes, orientifold and planes has at 16 the D8 ends. branes moving The on same chain of dualities that takes the background. Exploring these issues also gives interesting insights into conventional JHEP10(2019)198 ], to ] as 13 3 8 T , giving 2 T × R , Kaluza-Klein R × -brane and in [ 3 2 2 T -flux fibred over a line. H ] as a 5 9 -flux can be thought of as an H and then to a Kaluza-Klein monopole and then to domain walls separating 3 string. This is the T-dual of the type I 3 T 0 T × R ] requires an isometry and typically leads to – 3 – 24 , and independent of the torus coordinates. Then S- -flux on 23 H R ]. A final T-duality gives another exotic brane, which corresponds to with a nilfold. The D8-branes are domain walls separating regions with 11 , R ] is specified by a linear function on the one-dimensional transverse space. 10 22 )-branes with ‘transverse space’ given by a the product of a T-fold with a line. 2 2 , which is a linear function on )-brane. The solution with a T-fold fibred over a line is then seen as a smeared 2 3 2 T We will examine in some detail the dualities that map the D8-brane wrapped on The D8-brane is a domain wall separating regions with different values of the Romans T-duality via the Buscher rules [ A T-duality on a transverse circle takes the NS5-brane to a Kaluza-Klein monopole [ The product of the line with a 3-torus with constant , × with a Gibbons-Hawking metric whichthe is product a of circle bundledifferent over values a of base the space regions Romans with mass, different and valuesdifferent these values of of are the the mapped degree (first to Chern domain class) walls of the separating nilfold. The O8 orientifold planes of a string theory set-uping for branes. each These ofmonopoles include the with the duals ‘transverse NS5-branes ofexotic space’ with the (5 given transverse torus by space with a flux the and product the ofthe correspond- NS5-brane a with transverse nilfold space given with by a line, and rated in a stringstring theory compactified background in on the a type circle,that I and is is the an product orientifoldO8 of orientifold of a plane the line at type interval either IIA withpositions end string nine-dimensional of on with Minkowski the the space. interval a interval. and vacuum There 16 Then is D8 branes following an distributed the at chain arbitrary of dualities discussed above provides supergravity solutions in thisthat way we can will give be interesting particularly string interested backgrounds in and here. itmass. is The these various dualsdepending of on this a considered single above transverse are coordinate. then The all D8-branes domain can wall be solutions consistently too, incorpo- branes smeared over the isometryHowever, dimensions, the and smeared such branes smeared can branesisometric be are direction, resolved often obtained by singular. by going going toperiodic to a array the solution of covering localised that space sources, is of and localised then an in periodically isometric the identifying. circle, Resolving taking the a smeared gives what has beena termed (5 an exotic brane,version referred of to the in exotic [ was brane. given An in interpretation [ the of configuration the with unsmeared an exotic essentially brane doubled as space a fibred T-fold over a line. R dualising gives an NS5-brane smeared over theof three 6-dimensional torus directions, Minkowski and space this with is the the product desired 3-torus with and so takes thesolution smeared with NS5-brane the to nilfold a smeared fibred Kaluza-Klein over monopole, a which line. is the A further T-duality on a transverse circle NS5-brane smeared over three directions.obtained as The follows. corresponding Starting supergravity with solutionthree a can torus D8-brane directions be wrapped gives a on D5-braneity a smeared 3-torus solution over and [ the T-dualising 3-torus. inT-dualising the The gives D8-brane supergrav- a D5-brane determined by a harmonic function on the transverse space JHEP10(2019)198 ] ∗ 25 to an theory 3 0 T 9 and dualises − n string on 0 , we discuss duals of the type 7 , we review the nilfold and its 2 and 6 planes. The limiting form of the K3 is the , we review the explicit construction [ ∗ 8 – 4 – ]. The usual representation of type I 28 coincident D8-branes has charge string, 18 Kaluza-Klein monopoles can be seen at n 0 -flux, a T-fold and an essentially doubled space. In . The relation between the IIA string on the degenerate theory. In sections n H 0 theory [ − 0 plane with ] reconciles these two pictures, providing confirmation of our planes dualising to Tian-Yau spaces and the D8-branes dualising ]. The Tian-Yau caps then provide smooth geometries that can ∗ 25 ∗ 27 ]. These Tian-Yau caps can be viewed as the duals of the regions string involves an S-duality, so that whereas 18 D8-branes can only be 26 0 ] these are resolved to give localised Kaluza-Klein monopoles. 25 ], giving 18 D8 branes and two O8 , we review the type I 5 29 , we discuss the string theory solutions obtained by fibring these spaces over a , , we the discuss single-sided ‘end of the world’ branes that give supergravity solu- are mapped to ON5-planes and then to a gravitational version. This gives a picture 3 4 28 0 The plan of this paper is as follows. In section The classification of Tian-Yau spaces gives the maximum number of Kaluza-Klein On the other hand, the same chain of dualities takes the type I theory and their moduli spaces. In section 0 tions corresponding to orientifoldKaluza-Klein planes. monopole, For we discuss the the single-sidedIn resolution brane of section corresponding the singularity to with the aI Tian-Yau space. of a limit of K3 and discuss how this resolves the singular supergravity solution obtained weak coupling in the type IIA string, T-dualisation to asection 3-torus with line, to obtainsection smeared NS5-brane, KK monopole and exotic brane configurations. In dual of this, with theto O8 KK monopoles. An O8 to a Tian-Yau space of degreeK3 9 and the type I seen at strong coupling in the type I monopoles on the degeneratebranes K3 possible as in 18,has the which 16 type is D8 I precisely branescoupling the each and O8 maximum two plane number O8 canplane of orientifold emit [ a D8 planes, further but D8 it brane to has leave been what has argued been that called at an strong O8 domain walls of thealso smoothed supergravity out solution in obtainedduals the by of K3 dualising geometry the D8-brane asgeometry D8 solutions Kaluza-Klein of branes [ monopole are would geometries. have The been naive smeared Kaluza-Klein monopoles, but in the Tian-Yau spaces [ around the ON or orientifoldgeometries, planes similar and to it the is realisationAtiyah-Hitchin remarkable of spaces that certain these [ other are duals realisedbe of as regarded orientifold smooth planes as as the smooth gravitational duals of the orientifold planes. Moreover, the singular apparently different representations of whatwork should on be a the same limitapproach. dual. of Remarkably, recent K3 There [ isdevelops a a long region neckwith near which Kaluza-Klein is the monopoles inserted locally boundary in ofwith that of the space. hyperk¨ahlerspaces K3 form The asymptotic ends of moduli of to the space the the product long in neck of product are which a capped of the nilfold a with K3 nilfold a line, with a line, known as of KK monopole domainversion of walls the distributed orientifold along planes at a either line end interval oforientifold with the of interval. some the gravitational typethe IIB K3 string geometry and and then the to KK the monopole type domain IIA walls distributed string along theory a on line K3. give Then two type I JHEP10(2019)198 , = 9 are 11 10 R (2.1) (2.2) H and a radius ]. It is a circle x, z 25 , ) theory. In section 2 , which is the degree 15 0 m πτ dz , π ) ∧ + 2 = 0 (2.3) 2 while the fibre coordinate is dy + 2 πτ , z H is an integer. 1 ∧ z x, z πτ + 2 m ∼ and again N 2 . The metric and 3-form flux mdx + 2 , dz z, z ) m dz 1 x as = ∧ ( dz + πτ . This viewpoint is useful in considering H π z ∼ 2 x ∧ πz, z ) (2.4) dB ) ) + 2 mdx = + 2 + 2 , y theory with those of its duals. In section x, z y 2 ] the nilfold 1 ( H πR, z 0 – 5 – mxdz m xdy for the 2-torus with coordinates ∼ dz 20 π, y πτ 2 − + 2 = + iτ with ]. , 2 + 2 + 2 dy B ) + x x x, y 20 B 1 dy ( ( ( τ + ( π y π, z + 2 ∼ ∼ ∼ 2 ) ) ) + 2 dx + 2 dx x -flux given by an integer x = ( = H ∼ 2 N direction gives [ directions, resulting in either case in a fibration over the circle x, y, z x, y, z x, y, z 3 ∼ x ( ( ( 2 T y ) ]. ds z ds 15 [ or x, z ( x y and base circle parameterised by = 1; the generalisation of the results presented here to general values of these y, z 2 , τ = 0 The nilfold can also be viewed as a 2-torus bundle over a circle, with a 2-torus param- A complex structure modulus Choosing the 2-form potential 1 for the circle fibre can be introduced by choosing the identifications . Here the first Chern class is represented by , τ moduli is straightforward; see e.g. [ eterised by T-duality in the parameterised by To simplify our formulae,1 we will here display results for the simple case in which R so that for the nilfold we have of the Heisenberg groupbundle by over a a cocompact 2-torus, discrete wherey subgroup; the see 2-torus e.g. has [ coordinatesof the nilfold, is required to be an integer. and T-dualising in the This is a compact manifold that can be constructed as a quotient of the group manifold Here flux quantisation requires that (in our conventions) Consider the 3-torus with with periodic coordinates we match the moduliwe spaces extend of our the discussion typepresents to further I the discussion. non-geometric duals involving exotic branes.2 Section The nilfold and its T-duals from dualising the multi-D8 brane configuration arising in the type I JHEP10(2019)198 . (3.6) (3.1) (3.2) (3.3) (3.4) (3.5) (2.5) ) are the ,p 1 R ( dz 2 ∧ ds ] with metric and + 1, and so has a dy essentially doubled x ), 2 30 direction gives a con- ] from identifications d ) , ], we will refer to such x R → 20 ( ) is a harmonic function 16 14 i 2 mx , x x mx 1 ( [ ds V , 1 + ( T ) 5 , = 1 and so is not locally geometric but R B ( x 2 . (Here V V. 4 ]. Following [ m ds R is given by . , ∂ = 1 ) 31 d 2 g lm , − ) + = 1. The dilaton is δ V. 4 √ dz 15 ) = 0 dV R i = 2Φ 4 ( ijkl + x  2 ∗ 1234 − ( 2 – 6 –  e 2Φ − ds V = e ) dy 2 = i transverse to the world-volume of the NS5-brane. ( = x H ∇ 2 d 4 ( ) 2 ijk R − V -direction gives a T-fold e H z mx 1 = , respectively.) The function ,p 10 2 direction. A further T-duality in the ) T-duality transformation under 1 1 + ( Z x R ds , 2; + d , 2 R (2 dx O = 2 T-Fold ds is the alternating symbol with are coordinates of the transverse space is the Hodge dual on the i 4 ijkl x -flux is ∗  H These examples are instructive but have the drawback of not defining a CFT and so T-dualising the nilfold in the -field given by Therefore, the T-duality invariant scalar density where Explicitly, this gives where standard flat metrics on satisfying Laplace’s equation The The NS5-brane supergravity solution has metric where discuss these and their T-duals in the following section. 3 Domain walls from3.1 the NS5-brane The NS5-brane solution solutions in which these backgrounds appearacting as fibres on over some the base, fibres. relatedline, by a defining The T-duality a simplest solution casecases that is with is that a sometimes in 3-torus referred whichof or to suitable these nilfold as NS5-brane solutions fibred a or are over KK-monopole domain a fibred solutions, wall line over and background. were a are obtained The dual in to [ D8-brane solutions. We T-duality monodromy in the figuration with explict dependence on ahas dual coordinate a ˜ well-defined doubled geometrynon-geometric given spaces in with [ explicit dependence on dual coordinates as not giving a solution of string theory. However, these examples can arise in string theory in B which changes by an JHEP10(2019)198 ) π (3.8) (3.9) (3.7) + 2 (3.13) (3.10) (3.11) y → y , then the smeared ) is independent of 2 τ ( R 2 -flux fibred over a line ) (3.12) ) V 2 , the NS5-brane is said ω H satisfying 4 which satisfies = dz R 3 + 2 R + R V dy 2 ( , 1 dy ) directions and can be dualised in , . − ( 5 ) , ∂τ ∂w 2 5 1 V , directions is the same as T-dualising 1 w R − y, z ( z R -flux on the 3-torus with coordinates 2 + ) + V ( = ) is a function on 2 2 2 H V ds a 1-form on V and ds dz ~ ∇ a function on ∂x ω τ, x ∂V y ( + ) + = w is independent of one of the coordinates ) + V 2 ) + Φ = constant 2 ~ω and – 7 – X = GH dx ( V direction and gives , ( 3 dx 2 and ) then represents the product of flat 6-dimensional 2 + -flux) fibred over a line. V z R 2 ~ + ∇ × , ds ∂x 2 ∂w ds R 3.5 R 2 = dτ = = = 0 ( dτ to be periodic (i.e. we can identify under 2 ( V 2 10 y H ds V ∂τ ∂V ) in the ds , so that ), and ( ) = 3.7 ) = 3.3 y, z X directions, and there is GH ( ( ), ( 2 2 direction to obtain the KK-monopole solution ]. Successive T-dualities will then take the 3-torus with flux fibred ds 3.1 y ds , which can then all be taken to be periodic. The NS5-brane is then 20 [ x, y, z say. We take τ y x, y, z ) is a Gibbons-Hawking metric ), ) a harmonic function on ) a harmonic function on with the 4-dimensional space given by a 3-torus with GH ( is independent of one or more of the coordinates of is a four-dimensional space with metric 5 is independent of 2 , τ, x τ, x, z 1 ( ( V X ds V R τ, x, y, z . The solution ( V V If Consider first the case in which We shall be particularly interested in the case in which If = ( i with the KK-monopole solution ( where The metric is hyperk¨ahler. NS5-brane solution can be takenone to or be both periodic directions. in T-dualising the in both the with and where to an essentially doubled space (with x and T-dualise in the 3 coordinates, smeared over the x, y, z space with coordinate over a line first to a nilfold fibred over a line, then to a T-fold fibred over a line and finally to be smeared inwe those can directions. then These T-dualise directionsthat in can emerge them. then in be In thissupersymmetry. taken what to way. follows, be we In periodic shall and each review case, the various we dual get spaces a string background preserving half the JHEP10(2019)198 R as × 3 (3.14) (3.15) (3.16) (3.17) (3.19) (3.20) (3.21) (3.23) (3.24) (3.18) T , while 0 x, y, z , one could V — the metric R ) as ) to be a harmonic τ ) = log . This 4-dimensional ( τ 3.13 ( σ is quantized; we adopt ( V σ = w m → V τ ] (3.22) for some constant Φ 2 ), then we can take τ 0 . dz ( dz .  V ∧ ) + 0 2 g 2 Φ = w dy √ dz. , dy / ∧ g + → 0 V V 2 c, ∧ . mx. g 2 + √ w ) Φ + 2 2 τ V p dy = ( 2Φ + 2 V mdx  x − dx 2 V mτ ) w − e log( V τ ) implies the form of + , = ( log + w – 8 – τ = ≡ 0 1 2 2 , so that ( 0 and take Φ = Φ µν = ) = 2 2 1 1 d V g 2Φ 0 τ 2 V dτ − V e ( g + − V = )[ e → V 0 τ = p dV x, y, z ( w = 4 = Φ + µν ∗ B V 0 g g Φ = = √ is Φ = Φ 3 2 H T X ds . The geometry is a 3-torus with flux fibred over 3 is an integer. By changing coordinates T × m -flux on ) we have R = 0, it must be a linear function. The simplest case is to take H is independent of 00 3.8 are constant. The form V V c , R is also harmonic, and and ) we have w m 3.12 Finally, if The usual flux quantisation conditionconventions implies in that which the coefficient arrange for the dilaton tospace have has linear topology dependence on the coordinate together with the and dilaton Then for the NS5-brane solution we obtain the following conformally flat metric on periodic and can T-dualise infunction one, on two or three directions. For where so the dilaton for the space For the metric ( for ( is invariant under T-duality, so that if under T-duality we have that To find the dilaton, we note that so that JHEP10(2019)198 ] . i τ 20 -flux = (3.27) (3.28) (3.29) (3.30) (3.25) (3.26) H τ remains m = H ) and the 3 T = 0 corresponds to R ) inserted at 1 are given in terms 3.26 3 τ is piece-wise constant . A full string solution T 5 is given by τ r > . This can be thought of 0 n n τ for ) with ( 2 ≤ τ r n m, m c , . . . τ 3.5 dz. ≤ . 2 2 . r , but the flux 1 n ∧ < τ 0 0 τ , τ τ τ ) 1 1 τ < τ . ≤ dy − ≤ . +1 1 n . The solution can be understood [ < τ < τ 1 n r r 0 ) with 6-dimensional Minkowski space = ∧ 1 . m < τ < τ m m 3 τ − 1 n T τ, τ ) with respect to − 3.30 − − 0 τ, τ > τ τ r Mdx × ( = 0. The singularity at m m mτ, τ > +1 +1 ), ( , τ > τ ) and dilaton ( − m r V n R τ is the integer + + τ, τ – 9 – τ,τ, τ τ , τ m +1 ,, τ < τ τ = m r c c n 3.1 1 2 + ( 1 2 n n τ 3.22 1 = ( r m + ) of m m m m . . . m m − c r τ + ( ], as we will discuss in section + +                    : +1 0 N = r dV ) = 2 n n 1 τ 4 22 V c . . . c c c τ ∗ ( +1 ) =                    r with NS5-branes (smeared over the ≡ τ V c = ( 5 ) , 1 τ H ) = M ( R τ ( M × , and for continuity the constants V i 3 can be taken to be piecewise linear, e.g. T , m 1 V × = 0 separating two ‘phases’ with fluxes c R τ by i given by 3 gives a space , m Taking the product of the solution ( A multi-brane solution with domain walls at More generally, 1 T 5 c , 1 on R The transverse space for the NS5-branes is The solution is then given by the metric ( Note that the derivative away from the domain wall points The brane charge of the domain wall at for some constants of This is continuous but nota differentiable domain at wall at as a brane that hasas a a tension dual proportional of to theis D8-brane solution then [ obtained by introducing O8-planes in the D8-brane solution and dualising. constant and is quantised. of the 3-torus and the dilaton depend on the coordinate JHEP10(2019)198 . π /V = x, z τ or for (3.34) (3.35) (3.31) (3.32) (3.33) +1 i = 0, fibres. τ 2 T is zero, but the < τ < τ 2 ) dB i τ = xdz ) τ H . ( dz dz dx. V ∧ ∧ M ∧ , and the Taub-NUT space is is a line interval. The dilaton 3 I + , so that the Gibbons-Hawking dτ R dx dτ 2 ) ) . ) τ τ τ dy T , smeared over the ( ( ( ( directions. Performing T-duality in i ) τ × V V V z τ where 1 ( R , and the geometry can be thought of as V ) + ) + ) + and the torus base has coordinates 2 and . Here, we are obtaining a version of this T , with a metric dependent on the coordinate × N y xdz xdz ) + xdz I ) ) ) 2 x, y : the circumference of the circle fibre is 1 × N Φ = constant τ τ τ ( ×N ( ( V dz – 10 – R R M M M + 2 , + + + ; we will discuss the resolution of these singularities i . In the region between walls dx of the hyperk¨ahlerwall solution with 6-dimensional τ . The space is singular at the end points dy dy dy = 0 ˆ τ ( ( ( N 5 + , = 1 2 ∧ ∧ ∧ H τ R dτ dτ dx dz )( τ = = = ( it has the topology N × 1 2 3 ) we have a multi-domain wall solution and we will refer to this V J J J ) has isometries in the = . This has a transverse space that is 3.26 5 2 , 1 ds 3.22 R < τ < π , the geometry is a nilfold, which can be viewed as a circle bundle over a n with a Kaluza-Klein monopole at each τ τ 5 , depending linearly on a single coordinate. It preserves half the supersymmetry or 1 1 R V of the form ( , so that it is nilfold fibred over the real line. The metric is -direction gives a background that is R Taking the product For fixed V y × N × of < τ < τ a circle bundle over thisKaluza-Klein space monopole (with whose a transverse pointcircle space removed). bundle is We over shall the be transversesmeared interested space over later the is in coordinates the ofR the transverse while the circumference of each of the circles of theMinkowski 2-torus space is gives amonopoles. space which The can usual Kaluza-Klein beNUT space monopole viewed with is as given a by background the with product Kaluza-Klein of self-dual Taub- and the domain wallin positions later sections. 2-torus, where the circleThe fibre geometry has is coordinate warped by the factor of For as the hyperk¨ahlerwall solution 0 is constant and themetric antisymmetric depends tensor on the gauge coordinate field strength This 4-dimensional metric canfunction be viewed as aand so Gibbons-Hawking is metric hyperk¨ahler.The with three a complex harmonic structures are given by and the other fields are trivial: The background ( the τ 3.2 Nilfold background JHEP10(2019)198 . 5 ]. , R 1 R 32 × , (3.36) (3.37) (3.38) 1 1 ]. The with a S through 5 13 , the space 1 τ ×N × R -flux. R +1 ] as a T-fold, i R with smeared , ) 5 11 , , 2 1 × T × R 10 dz R × < τ < τ + . i 3 2 2 . The second T-duality τ T R )-brane. In the notation fibred over a line [ R dy 2 ( 3 × T . Then the first T-duality × 2 . , and the solution is given by ) and the harmonic function 2 3 R R x . , so that the transverse space 2 T R 3 ) with transverse space 3 R  τ ) was thought of as R 1 T ( 2 ) × S ) × M x τ 2 ] as a (5 3.32 ) ( 1 8 T τ S V ( + ( ), ( . In the same spirit, the T-fold solu- ) 2 i ) with the fields depending on ) M τ τ ( τ , then a further T-duality is possible. The ( 3.31 2 2.5 V + ( V T 2 ) dz + τ ( – 11 – 2 ∧ ) ) was interpreted as direction gives an essentially doubled space which V and the solution is given by a harmonic function with transverse space )-brane. This was interpreted in [ dy dx x  1 2 leads to a monodromy round each source. A further 2 2 3.30 )( S ). In the region between walls 2 ) 2 τ , T x log )-brane smeared over ( ) × R 2 ), ( 2.5 1 2 τ -direction results in the T-fold V 2 2 ( ] fibred over a line. This will be discussed further elsewhere. z , . x i R with the T-fold ( + ) τ M 3.24 τ 2 and the geometry ( 31 I Φ = ( , ) ] as a (5 i . T-dualising on one circle gives a KK-monopole smeared over τ 8 + ( ), ( 15 2 M dτ 2 -brane. This is an essentially doubled solution with = R ) can be thought of as the T-fold background ) )( 3 2 τ τ τ 3.22 ( ( . V 3.38 V -brane or (5 V 2 2 = = ), ( 2 B ds 3.37 -brane and in [ we obtain the T-fold ( 2 2 and the harmonic function is a linear function on ), ( τ R . The harmonic function in × 3.36 2 ], this would be a 5 ] as a 5 The background ( Here we are interested in an NS5-brane smeared over If the NS5-brane solution is smeared over Finally, a further T-duality in the 3 9 R 9 T NS5-branes inserted at with a smeared Kaluza-Kleintion ( monopole at each smeared exotic brane at each is gives a KK-monopole smeared over gives a an exotic 5 A third T-duality gives anof exotic [ brane referred to in [ a circle with transverseon space T-duality on a transversein circle [ gives what haswith been a T-duality termed monodromy an round exotic each brane, source in referred the to transverse NS5-brane is smeared oversmeared the NS5 solutions transverse are circle. given by Then a harmonic both functiontransverse the on space KK-monopole of the and NS5a the brane harmonic is function now on taken to be 3.4 Exotic branesT-dualising and the T-folds NS5-brane on atransverse transverse space circle gives of adetermining the Kaluza-Klein the monopole NS5 solution [ brane is is taken to taken be to independent be of the circle coordinate, so that the is the product of thethe interval warp by factor is not locally geometric but whichdoubled has configurations a of well-defined [ doubled geometry which is given by the while the dilaton is For fixed Performing T-duality along the The metric and B-field of this background are given by 3.3 T-fold and R-fold backgrounds JHEP10(2019)198 ] , . 4 ]. b ]. 4 R 33 x mτ .A 21 and 25 2 (4.2) (4.1) + × N ), this c is a del CP R , where b = 0 [ 4.1 ,... = 2 M τ 0 resulting V  , ) for a nilfold of the T-dual 1 ]. The T-dual τ >  4 , 38 x 3.31 , where – given by ( D 35 = 0 V \ m M ) with 3.31 ]; see also [ . 2 0 with the half-line 34 0, with a singular wall at CP ) for integers τ < ≥ | τ τ | del Pezzo surface can be constructed from ) is a non-zero linear function τ. τ m b πRm ( it approaches the metric ( 2 + , V τ 0 c submanifold (which is a smooth anti-canonical → − , that is invariant under the reflection – 12 – 0 2 = τ , T m V is a non-compact space that is asymptotic to . In the asymptotic region, the Tian-Yau metric can b ) where D = \ b can be constructed by blowing up a point in the del Pezzo . Summing the contributions from the sources as in [ 3.31 , and there are restrictions on the positions of the points ] is a smooth four-dimensional hyperk¨ahlermanifold fibred m b 2 M 26 πR ] is a complete non-singular non-compact hyperk¨ahlerspace = . That is, a degree CP is a certain b 26 +1 + 2 = 0 of charge b X M 4 ) of the transverse coordinate. Taking τ τ M x 0 such that for large ( ⊂ of degree ∼ V b D +1, 4 points in τ > M b x b direction is a line interval with a single-sided domain wall at either end. − τ 9. The del Pezzo surface of degree nine is the complex projective space . This can be constructed by taking a periodic array of NS5-branes on and the result of blowing up one point in 3 is a nilfold of degree 1 R b ,..., CP × N 2 ]. The Tian-Yau space [ , 1 × The Tian-Yau space The Tian-Yau space [ For the case of the nilfold fibred over a line with metric ( For the T-duality of the NS5-brane on a transverse circle, it is not necessary to assume S 21 1 = 1 that can be blown up.CP There are two types of del Pezzo surfacewhere of degree eight, whichbe are approximated by the metric ( Pezzo surface and divisor). The delb Pezzo surfaces are complexdel surfaces Pezzo surface classified bysurface their of degree degree blowing up 9 in [ over the half-line fibred over a line, so that it can bethat regarded is as asymptotic to a a resolution nilfold of bundle the over single a sided line. brane It [ is of the form in a single-sided domain wallFor solution the defined applications for to stringin theory which the in the next section, we will beorbifold singularity interested has in a the remarkable resolution case to give a smooth manifold, as was proposed gives a domain wall at Quotienting by this reflection identifies the half-line 4 Single-sided domain wallsConsider and a Tian-Yau spaces domain wallgiven of by the a function kind discussed in the previous subsections, with a profile The T-duality of thissolution solution is has essentially been doubled,circle. discussed depending This then in explicitly gives [ a ondependence modification on the of the the coordinate dual Kaluza-Klein monopole coordinate. ˜ geometry with explicit that the NS5-brane is smeared overon the circle. Instead, onelocated can take at an NS5-brane points localised arrangedthen identifying on a linegives a (0 harmonic function defining the NS5-brane solution that depends explicitly on JHEP10(2019)198 ]. 9. 40 with , the (5.1) (5.2) (5.3) (5.4) (5.5) (4.3) 2 . It is of the 1 / ,..., I 3 b 2 ) , × CP 1 3 , mτ → T = ( = 0 M s : b f . ) is piecewise linear. 2 τ ) ( V ) bxdz 1 dτ − + dτ ∧ ) 2 τ 8 ∧ while the size of each 1-cycle of ( dy . Here dτ ( 8 τ 3 2 dx V 3 / / ( / 1 1 dx 2 d : − − V s s ∧ s . ), so that the Romans mass parameter 8 ) 5 4 τ ∧ · · · ∧ ) + − ( ) + dx 8 ) 1 , 9 can arise. 2 ∧ · · · ∧ 3.29 1 τ M 1 of the nilfold is given by the degree ( is a zero-form, dx R dz − ( V ] has string-frame metric dx 2 ∧ m + ,..., = – 13 – 2 Vol = ∧ (10) 2 22 ∧ · · · ∧ ds dt , 2 F 1 ) ) Φ / )Ω (0) dx e τ 1 τ ( τ gdt F ( ( dx 3 − = 1 string theory ( 2 / − fibres falls off as 0 2 V ∧ M V M m s √ 1 with an interval. Changing variables to = dt − − S + = 2 ]. It will be convenient to refer to this case as a Tian-Yau 2 = = = m . . The degree 3 ds Vol ds 39 / [ 1 2 Ω (10) s 4 m F → ∞ M 9 τ ∼ is a non compact space that is asymptotic to a cylinder gives the mass parameter in the massive type IIA supergravity [ as g D (0) \ , the size of the F s M → ∞ 6= 0 case gives a generalisation of the ALH case which is asymptotic to the ) τ ( m V = 0 zero degree space has the structure of an elliptic fibration, b The Starting from a del Pezzo surface of degree nine and blowing up nine points gives a This zero-form For our solution it isis piece-wise different constant on as either in sideand ( of a a domain one-dimensional wall. transverse There space is with an coordinate 8+1 dimensional longitudinal space where is the volume form. The Hodge dual of with dilaton and RR field strength 5 The D8-brane and5.1 type I The D8-braneThe and D8-brane its solution duals of the IIA string [ Then for large the 2-torus base grows as the fiber being an ellipticand curve the or space 2-torus. Forthen zero degree, an the ALH nilfold gravitational reduces instanton. to a 3-torus product of a nilfoldmetric of takes degree the following asymptotic form for large del Pezzo surface, so only degrees rational elliptic surface space of zero degree,This so that we can extend the range of the degree to so that JHEP10(2019)198 . ), n , τ (5.6) (5.9) 3.22 (5.12) (5.11) (5.10) . The ··· π , 3 1 ≤ T τ τ . bundle over a = 0 while the ≤ 3 τ 10 T , 0 at . ) (5.7) τ 0 2 2 and ) located at arbitrary m 2 dz 9 dτ is the RR field strength 2 Z − / + arising from the quotient 1 2 (7) m V I F dy , string with 16 D8-branes has 2 dz. 0 + )] + ∧ 2 dτ 5 , ) (5.8) 2 1 1 / dx dy 1 R − with coordinate ( ) V ∧ + 2 τ I ) (5.13) 2 ( 5 4 ds τ dx V ( ) + − ) dτ ( 8 ) ( + , τ d 2 1 τ ( M 2 / ( ∧ R 1 − ( M V dz 5 V 2 = − – 14 – + = dx ds 2 = 2 Φ ) + (0) ). The Hodge dual of / 5 e 1 , F dy 1 (7) − , with a transverse space given by a F R + 3.26 V ( ∗ 3 2 ∧ · · · ∧ 2 ) gives a D8-brane of charge = 1 = 1 dx ds 2 [ 2 − dx 2 / 3.25 ) (3) / ), or ( ds 1 τ 1 ∧ F − ( − ) is a multi-brane solution with D8-branes at positions V dt V directions gives a D5-brane IIB solution smeared over 3.25 V = = = = 3.26 with O8 orientifold planes introduced at the fixed points, and with 16 2 2 2Φ string (7) ). 2 ds e x, y, z τ ds F 0 Z ( 0 / ) given in section 1 V S ≡ 3.30 ], which arises from compactifying the type I string on a circle and T-dualising. ) ) is a harmonic function on the interval τ τ is of the form ( 12 ( ( with O8-planes at the end points, and with 16 D8-branes located at arbitrary points V V M , the metric can be written as 2 ), and ( The supergravity solution corresponding to the type I If three of the transverse dimensions are compactified to a 3-torus with coordinates Z / string [ 1 3.24 0 and the RR field strength is given by where string frame metric where dilaton is given by D8-branes (together with their mirrorlocations. images under It the can actionS also of be viewedon as the a interval. theory on the interval The multi-D8 brane solution hasa a way dilaton that depending on thecertain the dilaton regions. transverse becomes coordinate A well-behaved in large solution such I and of hence string theory the with stringThis D8-branes can becomes arises be in viewed strongly as the an coupled type in orientifold a in of theory the type on IIA string compactified on a circle, resulting Next, S-duality gives a smeared( NS5-brane solution, which isline. precisely the We will solution ( discuss further T-duals of5.2 this background in sections The type I where three-form, which is given by T-dualising in the multi-brane solution ( x, y, z Taking it to be of the form ( JHEP10(2019)198 . ) ]. i τ τ for ( 29 i V , (5.16) (5.17) (5.18) (5.14) (5.15) jumps ) with m 28 V D8-branes 3.29 +1 n theory [ D8-branes at 0 N i N . The function π = = 0 and . The gradient of i = 8, so that the orientifold I +1 τ 16 τ τ n 8 emits a D8-brane of charge 2 τ π the slope jumps from , while the 16 D8-branes are 1 17 τ ≤ N − τ π 1 ≤ ≤ m 16 ≤ − − ≤ 2 τ π. r = 1 < τ + τ i ≤ ≤ < τ < τ τ 1 τ < τ = 8 corresponding to ≤ − 9 and 1 i 15 16 = 0 and . = n 0 τ < τ r . +1 < τ < τ τ − and τ < τ < τ n ≤ i r 1 15 16 ··· − ]. While at weak coupling there are 16 D8 τ, τ 0 = 16 r . In general this leads to domain walls at m = ,...N c = 0 and = 9) i i ) and is given by 2 29 , , τ +1 − , i τ N τ τ,τ, τ τ = 8 7 − m ( – 15 – ,, τ τ τ, τ, τ , m +1 0 ,N i 28 i − − . . . 7 8 ) +1 7 8 8 at the end-points of between 1 +1 =0 τ r 0 +1 V i + 7 + 8 then X n − r                    N 9 [ − − + ( N m c 16 = ≡ π 1 2 i 16 17 − c c c . . . c . . . c ) − i = ) = = τ τ < τ < τ r                                τ ( (8 τ ( i N , . . . , τ − M τ 1 + 1 so that M ) = τ i = is τ ( for 1 m V V m ) with r = + +1 i 3.26 i plane of charge m m ∗ D8-branes coincident with the O8-plane at ). with positive charges 8, to 0 n − i N 3.29 = coincident branes with 1 , . . . τ is ( is given by ( r 2 m , τ V M 1 < τ < τ τ At strong coupling, new effects can arise. Each orientifold plane can emit a further For The orientifold planes are located at 1 = 16, = − i D8-brane at strong coupling, so+1 that to an leave O8 an planebranes, of O8 at charge strong coupling therethe can gauge be symmetry, 17 e.g. or to 18 SU(18) D8 that branes; is this non-perturbative leads in to the a type enhancement I of Then while τ τ If there are coincident with the O8-plane at The slope of the function with located at arbitraryis points, piecewise linear and,n for general positionsplanes of are the treated as D8-branes, sources itby of +1 is charge at given each by D8-brane. ( Then JHEP10(2019)198 . i 8 9 i ), T Γ X ,X 6.2 11 (6.5) (6.4) (6.1) (6.2) (6.3) 8 which ,X 89 = Γ , with an 7 in the i R . The O8 π t i π Ω . ,X T / ≤ 6 = 9 X . An orientifold 6789 9 X 2 where IIB R direction acts as a Z X L ≤ Ω / i 1 S i X 6 S t . This ambiguity gives T L −−→ F → which is supersymmetric. L 1) L F . However, this leads to a = 0 and in the L S when acting on fermions, the 9 − 1) F i i 9 R T − T 8 1) ] ( j X R IIA . − T 12 ) ( 789 direction and T-dualising gives the i R 6= = 89 R 9 j circle becomes R Ω ,X T . 89 X 9 i i Ω i L R / T 7 R X 9 X T −−→ Ω R − Ω ( IIA IIB L Ω → F ] results from the fact that the T-dualities → is identified under the action of the reflection where = 1) – 16 – 0 I = ) 42 :Ω I − π i 89 R i ( can be viewed as the interval 0 IIB T R 89 ] for further discussion. 2 ,X + 2 R i Z L 9 44 / Ω , it follows that – X 1 and T-dualising might be expected to give, from ( , this reflection acts through X j ( S L 42 8 8 T 6= S ∼ string −−→ X i . This results from the fact that the T-duality 0 9 9 9 8 (in units in which a single D8-brane has charge +1) and this R X X − IIA Ω ≡ 0 ] I 41 so that after the quotient, the 9 T 9 −−→ X anticommute for Ω IIB j is reflection in t → − 9 ≡ do not commute in the superstring. The T-duality string. This is now a quotient of the IIA string [ 9 R I 0 9 and X T In this way, we obtain the standard chain of supersymmetric orientifolds by succes- Next, compactifying on i : t 9 sive T-dualities for thedirections: type I string compactified on the 4-torus in the As result of two T-dualities istwo only possible determined ways up of taking to two ais T-duals factor of not of the ( type supersymmetric, IHere string: and and one the in way gives each other IIB casepreserves gives that supersymmetry. follows, IIB See we [ will define T-duality to be the transformation that and reflection on the left-moving bosonic world-sheet fields: On the left-moving spin-fields orientifold plane at either end of the interval. an orientifold of theproblem, IIB as this string orientifold bysupersymmetry. is Ω not The supersymmetric, resolution but of T-duality this is [ expected to preserve R O8-plane is introduced atplanes each each have of charge theis two cancelled by fixed the points, chargeof of the the type 16 D8-branes I arising string. from The the space T-dual of the 16 D9-branes where direction acts as [ The periodic coordinate where Ω is theof world-sheet the parity O9-plane. operator. Compactifyingtype on This I a has circle 16 in D9-branes the to cancel the charge 6.1 Duals ofThe the type type I I string can be obtained as an orientifold of the type IIB string 6 Duals and moduli spaces JHEP10(2019)198 4 in T 8 with (6.6) (6.7) E π theory , which ) where × 2 2 ≤ 8 and 16 concident 8 Z Z τ 6789 − / E / p 4 fixed points p R 2 : ≤ 4 T-dualities, T / L T 2 p ( F Z ≤ / 1) × 4 p can be resolved by p − T ( − 2 / 4 = 4, there is a single Z . As the type I string T p / 2 4 Z T / . After , which is cancelled by 16 4 . k . This has 2 p 4) by performing a T-duality T p is the interval 0 2 / I ≤ ...i 6789 becomes 1 IIA i 16 ...X p 1, together with 16 D5-branes to R 4 ( j R − − p T 6 , and the length of the interval with T T ,X where 1 −−→ i c L 8 , F X ] 1 = 1) R 6789 45 V direction gives the supersymmetric orbifold − × R ( . L 6 I F IIB X → 10 ) orientifold planes. For . – 17 – 1) p 4 / − ]. :Ω 1 ( − ) and 1 S 46 c 9 , . The 16 D5-branes become 16 NS5-branes and the 16 S ( ]. −−→ O(9 π ]. Each orbifold singularity of 45 49 is a constant, p 6789 13 – = is an orientifold of type IIB compactified on R 6789 V L 47 L IIB R theory there are two orientifold planes of charge F 6789 0 Ω 1) string is defined on ) we have the duality between the type I string compactified on ) is R 0 − Ω ( 6.7 5.1 / / is identified under the reflection p circles, the locally charge-cancelling configuration has 16 ...i )-plane at each fixed point of charge 1 i p p ) and ( − is identified under reflections so that the /R 6.5 denotes a reflection in the directions p p T SO(16) gauge symmetry perturbatively, which can be enhanced to , resulting in the IIA string compactified on the K3 orbifold T )-branes at each of the 2 )-branes. × = ij...k p p 6789 R 2 − − The general configuration for each case is obtained by moving in the corresponding In this case, the function For the theory obtained from the type I string on Finally, acting with a T-duality in the The next case IIB Z and the type IIA string compactified on the K3 orbifold /R -torus / p 4 p D(9 D5-brane at each of the 16with fixed each points, of and the S-dualising gives 16 a ON-planes. single NS5-brane coincident moduli space. The type I SO(16) the non-perturbative theory [ respect to the metric ( on each of the branes and orientifold planes andcase in there general is the dilaton ais is particular constant. non-constant. configuration However, where In inD8-branes each the the of charge type charge cancels I +1,total locally charge so and at that the each if dilaton plane there is are zero 8 and the D8-branes dilaton at is each constant. orientifold plane, This configuration the has an can be thought of asT-duals dual will to be the discussed ALE in spaces sections glued6.2 in to resolve the Moduli singularities. spaces Further In and the dualities orientifolds considered in the previous subsection, the total charge cancels between Combining ( T is dual to theand heterotic the type string, IIA this stringglueing gives in on the an K3 Eguchi-Hansen duality [ space, between and the the heterotic 16 NS5-branes string of on the IIB O5-planes become 16 ON-planes [ IIA has 16 O5-planes at thecancel 16 the fixed charge. points, Acting each with of S-duality charge takes [ giving the orbifold IIB a T with an O(9 D(9 Here JHEP10(2019)198 5 , 1 R (6.8) (6.9) -field (6.11) (6.10) × B , while 3 4 and the T T 2 Z × periodically 16 positions / I 4 × ) is the auto- T Z x, y, z 20; . , + (4 R O ) with × . ) + p 5.3 R 4) by performing a T-duality . + . With the charge-cancelling ), ( × ) ≤ 3 ]; here R p p ( and O8-planes at the end-points ) acts as T-duality on T (16 + 5.2 ( 50 × Z O O p (20) 16 moduli of constant metrics and RR )[ ), ( T × O × 2 20; ) and the dilaton zero-mode. The du- (17) p , ) × 5.1 p p 6.11 ( (4 ( L /O , . . . , τ O 1 (4) /O τ /O , the ) ) 17) 2 /O p , Z (17) identified under the action of the discrete p, p / (1 – 18 – ( p 20) O , /O O T \ 16 + directions, the supergravity solution transforms ac- \ ) (4 ) p, Z 17) O Z ( , ; \ O ) (1 17; \ parameters that can be interpreted as determining the string compactified on ) subgroup of Z , x, y, z ) p, p O Z 0 ( is Z and the dilaton zero-mode. The moduli space for metrics p (1 ; ]) with constant dilaton. For general positions of the D8- 20; O 4; p 2 O p , , T )-branes on Z , π (4 p / (4 p O O ). The 18 moduli can be viewed as consisting of the 16 D8-brane − 16 + T Z ) (corresponding to the T-duality symmetry of the heterotic string p, ( Z , the length of the interval 17; ) acts on all 17 of the coset moduli. O , 1 16 17; circles, the moduli space is S (1 , p O (1 , with 16 D8-branes at positions O , . . . , τ τ 1 = 4, we obtain is the interval [0 τ p . It has an 18-dimensional moduli space I , π Applying T-dualities in the For IIA on K3, the moduli space is again ( For For the theory obtained from the type I string on = 0 (where branes, the corresponding supergravity solution isidentified. ( cording to the Buscher rules, taking each D8-brane to a D5-brane that is smeared over the K3 manifolds. 7 Dual configurations We will startconfiguration with of the 8 D8-branes type coincident I with each O8-plane, the geometry is the remaining transformations mix the NS5-brane positions with torusmorphism moduli. group of theand K3 CFT mirror and transformations. containsblows large Moving up diffeomorphisms, in the shifts singularities, the of and the moduli generic space points away in from the the moduli orbifold space point correspond to smooth After S-duality, this becomes the 81-dimensionalof moduli NS5-branes, space consisting 16 of 4 modulidilaton for zero-mode. metrics An and NS-NS 2-form gauge fields on and 2-form gauge-fields on The moduli consistpositions of of the the 16 16 2-form D(9 gauge-fields on positions ality group compactified on on each of the τ which includes the cosetduality space group coordinate JHEP10(2019)198 , , . . ) 2 π τ N ( Z 2 |  / (7.1) 2 V 4 ≤ T π/ µ x < τ < π − ≤ τ | has periodic can be repre- 3 but to capped at the . ) consisting of ). The domain ) 2 T 2 z with 0 R Z Z , was constructed / − / 3.30 τ 3.31 β × 4 1 y, 3 T S has 4 periodic coordi- ), ( − T . As each point with 4 are fixed points. It can and the µ x, T x , the metric is given to a π 3.24 π − with identified under the action β with 8 fixed points on each 3 where there are single-sided = π ≤ π, 2 ), ( ( T τ τ Z → − , π +2 / ], with an explicit understanding → ≤ 3 µ τ 3.22 ) x T 25 , so that we can take 0 ∼ ), we get a harmonic function ). This suggests that the O8-planes π , each point in for generic points τ π . 3 5.9 = 0 and ]. For large π + 2 ≤ T µ τ to be large, then the naive supergravity , π ≤ capped off by suitable smooth geometries. π, x, y, z τ ), ( x ( L τ acting to take τ, x, y, z R ≤ ( ∼ 5.8 2 ≤ V µ Z x – 19 – ), ( , N × ) , so that z 5.7 . There is then a further quotient at the end points τ − π with 0 is the interval with 0 y, I I − → − with the fibre a ). This then implies that K3 should have a limit in which τ x, I , the fibre degenerates to − gives a boundary of the K3 moduli space in which the K3 , . 3.32 where , π (0 3 , π is the circle with coordinate ), ( can be realised in a similar fashion. The T = 0 → 2 2 = 0 → ∞ ) τ × . Applying the standard Buscher T-duality rules takes the solu- Z ) has singularities at the end points 0 Z ) to the solution ( . However, this picture is too naive: under T-duality, the transverse I 3.31 τ is identified under the τ / ) each identified with / β 3 1 4 3 each with period 2 transforms not to the product of the dual ‘fibred’ over T 5.3 S T 3.31 T 2 ( 3 , x, y, z Z . This then is a space with a long neck of the form ), ( T × ˆ / N (0 I 1 , π is identified with a point with 0 τ, x, y, z x, y, z 5.2 S π acting as a reflection 2 × = ( = 0 ), ( 2 3 µ τ ≤ Z . If we take the length of the interval T x ], a family of hyperk¨ahlermetrics on K3, labelled by a parameter 5.1 2 Dualising the D8-brane configuration then gives a configuration that, for The quotient The orbifold Z , so that 25 / . In other words, instead of getting a D5-brane localised at a point on the 4-dimensional I 2, is well approximated by the supergravity solution ( 3 3 In [ in which the limit collapses to the one-dimensionalgood line approximation segment at [0 generic pointswall by solution the multi-domain wall metric ( of the geometries neededas to we cap discuss off in the the neck next and section. to resolve the domain8 wall singularities, A degeneration of K3 the three-torus with fluxpoints fibred over a line,two ends. but this A willgiving further need the T-duality modification solution takes near ( theit the degenerates fibres end to from a long aRemarkably, such neck 3-torus a of limit with the of form K3 flux has to recently been a found [ nilfold T solution can be a goodto be approximation a modified long near way away from the end points,π/ but will need Thus we have but at the end points Then this isπ identified < under τ thesented reflection by a acting point as in coordinates of space of the be represented by the line interval nates transverse space with adepending harmonic function only on tion ( could behave like negativesmeared tension over D8-branes the under T-duality, transforming to O5-planes T JHEP10(2019)198 . . i  i of τ m by i R m I = 0 i − m (8.3) (8.4) τ b 0 τ + X , and so i c + b = for some i V ] resolves these m . The degree 25 mτ ) (8.2)  ) with + − ) of K3 projecting to c i  i  3.31 R = ( , τ − 1  is chosen in each segment so 9. These are each asymptotic − n and one of degree β + V ] (8.5) − c F 1 − ≤ 2 − b i , π , with the degree jumping at  τ ]. It will be convenient to let − n b −   . ) is approximately a Tian-Yau space , π ] (8.1) ) with π ≤ = ( i +   < τ < τ [0 + − } with a nilfold of degree , π m S  , but the K3 metric of [ 1 n , . . . , τ = ( ( + i  i  < τ < τ [0 3.31 2 1 − τ 1 − R + } ⊂  − + β i , τ − → S +1 1 1 n F i , π τ ) is approximately a Tian-Yau space 3 +1, the region n ) projecting to m −  – 20 – K + =  , τ > τ S : S = ( ) τ ( 1 i , τ β +1 ,, τ < τ τ 1 ,  − 1 2 n n β , . . . , τ F , . . . , n N −  < τ < τ β 1 F m . . . m m m F . + , τ = [0                    = 1 0 4 1 ), are some integers 0 i { − −  −  i S τ  S ) = = b ( : τ 1 ( S τ − β { M F = be the interval i  i  , where R and the region R − 1, there exists a continuous surjective map from K3 to the interval b + −  . Then for each b  β . Let π = , by Tian-Yau spaces. The region is continuous. The domain wall connecting two segments is realised as a smooth 4 +1 n of degree V τ For the end regions For large The smooth K3 geometry is obtained by glueing together a number of hyperk¨ahler + b the singularities ofsection the single-sided domainX walls at thenegative end degree points areto resolved, the as product in of a line with a nilfold, one of degree and jumps across the ‘domain walls’ at The degree of the nilfold fibres is piecewise constant: for some small is diffeomorphic to the productThe of metric the is interval approximately given by the hyperk¨ahlermetric ( and a discrete set ofand points of the nilfold jumpsthat between segments, andGibbons-Hawking the space constant corresponding to multiplethe Kaluza-Klein monopoles. geometry At is eithergeometry, capped end as with discussed in a section Tian-Yau space, resolving the single-sided domain wall singularities to give a smooth geometry. spaces, and these thenlong give neck approximate consisting metrics ofwith for a a different number line regions of interval, of segments, with K3. with metric each There segment of is a the a product form of ( a nilfold domain walls and at the domain wall locations JHEP10(2019)198 , . i i i π + N N N b = (8.6) (8.7) (8.8) + = i τ m with +1 ], making of degrees n i  to 25 is zero. + m i . There is a b Kaluza-Klein i  × S i X (0) 2 N F and × S τ < τ T = 0 these are ALH 2 − and b T  for − − b b i ) on X = m 1 = 0 for this case. This K3 m n x, z, τ ( V sources on 18 if all domain wall charges fibration over the space given by i configuration, in which one end of 1 N . ≤ are asymptotically cylindrical ALH 0 we take S ) .  + n − 0 these are Tian-Yau spaces asymptotic b b 18 +1 + n  ≤ > + X i ≤ R −  i b , τ b , ) is taken to be approximately a Gibbons- 1  N i  and − = S R i i n ( – 21 – + =1 τ i 1 b X N − and a line interval, while for β 9. For X ≤ = ( F n . This can be constructed from a Gibbons-Hawking =1  i i ≤ 0  i X  b S Kaluza-Klein monopoles inserted. For  b × S + . A K3 surface can be constructed by glueing the two b 2 3 ≤ T T + = 0, both − , the degree of the nilfold jumps from i b τ + b = ] is that these hyperk¨ahlerspaces can be glued together to give a τ = ) specified by a harmonic function 25 − 3.8 with a doubly periodic array of sources, with contributions summed as b points from = 0 there is an O8-plane and 8 D8-branes, while at the other end, i  i of domain walls then satisfies 0 τ N n × S ], and these various hyperk¨ahlermetrics provide good approximate metrics for 2 ]. This means there are no domain walls, so that . The smooth K3 geometry is constructed by glueing together the Tian-Yau 25 i R 51 [ which are integers with 0 3 ], the geometry in the region ], by taking the quotient by a lattice to obtain T + In the general case we have a geometry capped by two spaces For the case The result of [ Consider now the interval 25 τ > τ 33 , b × − monopole bubbles at for spaces, the product ofshown the in nilfold [ with athe line corresponding interval and regions the of Gibbons-Hawking the spaces K3. as b to the product of aspaces nilfold asymptotic of to degree theneck product region of which can a be 3-torusinterval thought and and of a a as nilfold line a with Gibbons-Hawking interval. space These on are the product joined of by a a line cylindrical ends of two ALHR space together with asurface long cylindrical is neck dual region to ofthe the the interval form locally at charge-cancelling typethere is I also an O8-plane and 8 D8-branes, so that the RR field strength K3 manifold with aspaces, smooth the nilfold hyperk¨ahlermetric. fibred The over agood model line approximate geometries and metrics the — in Kaluza-Klein the each monopole of Tian-Yau spaces — the then corresponding provide regions. spaces, with fibres given by in [ ‘bubbling limit’ in which thisprecise region the is sense mappedregion in to is a which then Gibbons-Hawking the said space to Gibbons-Hawking [ have space Gibbons-Hawking is or KK an monopole approximation. bubbles. The neck In [ Hawking metric ( sources. This gives aremoving space hyperk¨ahler the which is an space on The number are positive. Then the sum of the charges is and so is an integer in the range to match with the solutions projecting to JHEP10(2019)198 ) = 8 (8.9) 3.26 − b orbifold = 1 ) with D8- = 9, there − + k b ) for generic 5.1 + A b given by ( 5.1 moduli space is 8 and is dual to 0 V = that led from the or to the quotient − − 6 2 n b i Z fibration over a line or m / ) with 4 3 − 8 and 16 D8-branes then T T − 3.31 +1 i m = i . The equivalence of these theories 2 with 8 D8-branes at each O8-plane, Z / 8 4 , string theory into the moduli space of the 1 ,N T 0 R + b × gauge symmetry. With = – 22 – I 1 string moves the 16 D8-branes away from the O8- 0 − +1 (17), so that moving in the type I k n A /O 7. This corresponds to blowing up a point of the original 17) 8. Moving a further D8 brane to the O8 plane increases , − , m ). string on − (1 n − (17) of the type I 0 b n ) with smeared NS5-branes and a O 8.6 on the interval, corresponding to the solution ( \ − ) /O i τ Z = 3.22 17) 1 . The geometry discussed in the previous section then provides a , 17; i m ) of smeared KK-monopoles with a nilfold fibration over a line. The , τ satisfy ( (1 i (1 7 and corresponds in moving a Kaluza-Klein monopole to the Tian-Yau of type IIB compactified on O configuration with two O8-planes of charge \ 3.31 N O D8 branes coincident with an O8-plane has charge ) − 0 Z n n 6789 R 17; L , F (1 string theory to the IIA string on K3. Starting from the local charge-cancelling 1) 0 O − of the Kaluza-Klein monopoles are coincident, the K3 surface has an ( The type I The domain wall supergravity solutions provide a guide as to how this should work. In the following sections, the duals of the IIA string compactified on this degenerate A set of The geometry is smooth if all of the Kaluza-Klein monopoles are at distinct locations, The form of the solution away from the domain walls is ( / k the same locations non-singular K3 geometry thatsolution, resolves and the its singularities existence of supports the the picture domain-wall arising supergravity fromcorresponds duality to arguments. the K3 geometry with end-caps given by Tian-Yau spaces with Moving in the moduli space ofplanes the to type generic I points points away from the locationsbranes of to the the branes. solutionto Dualising ( the takes solution the ( solutionlocations ( of domain walls arising from the smeared NS5-branes or KK-monopoles are at space IIA string on K3for and all points of in thedual IIB to quotient, a and corresponding movement give in an the equivalence moduli between space the of type theories II on K3. type I configuration of thedualising type took I usIIB to the typeat IIA one string point in on moduli the space K3 then, orbifold in principle, should give an embedding of the moduli K3 will be considered. 9 Matching dual moduli spaces In this section, we revisit the chain of dualities discussed in section a Tian-Yau space ofthe degree charge to space, changing the degree to del Pezzo surface. If singularity and thereare is 18 a Kaluza-Klein resulting monopolesgauge symmetry. and if these are coincident, there is a resulting SU(18) with and the charges so that the geometry is approximately that of self-dual Taub-NUT near each monopole. JHEP10(2019)198 0 0 . 3 8, T − and of the + , which n 2 they are , π Z − 2 / 4 Z = 0 / T 4 = 8 τ ]. In particular, -plane of charge T ∗ + 8, and it would be b 52 , − 44 = 9, which would lead to D8-branes at the O8-plane ) with a nilfold fibred over + − has mapped strong coupling b n with one point blown up. One 6 3.31 2 ( CP ˆ N string, up to 16 D8-branes are possible = 0 and 0 9 and 18 D8-branes. The configuration and τ − 1 = 9 and/or CP ] is constructed by glueing a number of + with flux fibred over a line. b × 25 3 theory, they should be dual to the region around 1 – 23 – T 0 = 9 and 18 Kaluza-Klein monopoles corresponds CP − b = Kaluza-Klein monopoles, this corresponds in the type I + b ) with a 3-torus with flux fibred over a line segment. The -planes of charge + ∗ b theory. D8-branes distributed over the interval. 0 3.22 with an identification of the 3-tori at the ends − with 16 ON-planes and 16 NS5-branes smeared over the + , with an O5-plane at each fixed point. S-dualising gives the 3 b 2 − and dualising on all three toroidal directions gave 16 D5-branes b T Z − / 3 6789 − string to weak coupling physics in the IIA string on K3. 3 × and 16 O5-planes. The IIB theory is an orientifold on − T R 0 b T D8-branes at the O8-plane at I L 3 F − T = 8, there are two distinct Tian-Yau spaces, corresponding to the two − 1) n b − string to two O8 ( with 16 / 0 − string on ]. Then the K3 with n 0 , with 16 − 29 π , = 28 The degenerating K3 geometry of [ Consider now the type IIB dual of the K3 compactification. Compactifying the weakly- For degree However, the K3 geometry also allows = 8 τ 9 [ − a line segment dualisesTian-Yau to caps ( do not have the requiredHowever, from isometries the and so duality do with not thethe have type conventional 8 I T-duals. ON-planes. Theso Tian-Yau their caps duals are should asymptotic be to asymptotic the to nilfold a fibred over a line, and related by a T-duality.be a However, conventional the T-duality relationrequires at the between generic geometry the to points IIA have in an and the isometry IIB and moduli a pictures space, smooth cannot ashyperk¨ahlersegments. K3 does Buscher not The T-duality have segment any isometries. with geometry coupled I smeared over the can be regarded as interval to become quotient IIB This should be dual to the IIA string on K3, and for the K3 orbifold whether there is a relationit to seems the that variant O8 the planes IIAstrongly discussed theory coupled in at e.g. dual weak [ coupling type is I revealing some interesting structure in the distinct del Pezzo surfaces ofof degree these 8, Tian-Yau spaceswhile is the other presumably is dual presumably dualinteresting to to to a the understand variant standard of this this, further; O8 also in with plane particular, charge with it charge would be interesting to understand (together with a further U(1) factor).at weak For the coupling type and I 17 orthe 18 K3 D8-branes geometries, are the only possibleand IIA at in string strong particular theory coupling. on 17 However,coupling. K3 for or can 18 The be S-duality KK-monopoles taken in andphysics at the the of weak chain gauge the IIA of type group string dualities I SU(18) coupling in can section arise at weak string at strong coupling for an− O8-plane to emit a D8-branein leaving an the O8 type I in which the 18the Kaluza-Klein 18 monopoles D8-branes are are coincident, coincident and corresponds either picture to gives the an enhanced one gauge in group which SU(18) b string to having at up to 17 or 18 Kaluza-Klein monopoles. This corresponds to the possibility in the type I and with 16 Kaluza-Klein monopoles distributed over the interval. If JHEP10(2019)198 , i ). R N 6789 × (9.1) x, z 3 /R . This T 2 points in Z . This can / i I 4 . N T I × directions and 3 × T -branes replaced 2 p ] (which is in turn T x, z localised D5-branes 13 i [ 4 N . A single Kaluza-Klein T I ) with harmonic function ) is independent of ( , one takes a superposition × i 2 τ 3.22 is realised in the K3 geometry T sources on coordinates and gives the GH = i fibre of this Gibbons-Hawking i NS5-branes on τ of τ, x, z τ 1 i ( N realised as I = S x, z of the IIB string on N i V given by . The smeared NS5-brane domain × τ 3 N I 2 to ). It is independent of the coordinate | R R at 0 × 6789 I r i 9.1 2 direction gives the NS5-brane on R 1 N L − T y F . N × r | I 1) ), so that × − + ) in . At generic points in the moduli space of con- ( 2 – 24 – p , arising as a Gibbons-Hawking metric with c x, z / I Z − 8 / = and T-dualising in all four torus directions and then 4 T τ, x, z 4 V T N × T . T-dualising on the given again by ( I ]. Then S-dualising to D5-branes, this would lead to the 4 ) in terms of the Gibbons-Hawking form of the Taub-NUT × R string, this becomes a duality between a IIB configuration of 34 -branes, we would then have the smeared D 2 0 3.7 p . T-dualising in the T ) a harmonic function on R direction. For the segment near ). T-dualising gives the NS5-brane ( y × 2 , as in [ T I τ, x, z ( τ, x, z Kaluza-Klein monopoles on × ] allows periodic identification of the V i map to the KK-monopole domain walls with Gibbons-Hawking metric 3 is given by ( 33 = ( N T 3 3 r T R ) with ) on the transverse . T-dualising to D direction. Taking a periodic array of such solutions in the I 3.8 y local sources on the transverse Kaluza-Klein monopoles on × sources giving the Gibbons-Hawking solution with i i 3 i The first T-duality works well, as has been discussed in the preceding sections. At the At the level of supergravity solutions, the D8-brane domain wall supergravity solutions The segment near the domain wall of charge N T N τ, x, y, z N that is T-dual to the Gibbons-Hawking fibre coordinate and so the solution is smeared ( figurations dual to the typeNS5-branes I and ON-planes and the IIAof string moduli on space a in smooth whichsmooth K3 the manifold K3 near K3 has the becomes boundary no aform isometries, long of this the neck duality capped duality between with is IIA Tian-Yau not on spaces. properly K3 As and a the a T-duality heterotic but string on instead a dual taking the S-dual givescan the then quotient be IIB T-dualisedconfigurations. in one, two, three or four directions, leadinglocally-charge-cancelling to orbifold new point, string the theory of T-duality takes the this IIA to string the on orbifold the IIA K3 orbifold by 10 Non-geometric duals Compactifying the type I string on metric with local sourceswalls at can also points be ( the replaced transverse by NS5-braneD5-brane solutions domain with wall T-dual sources to localised aon D8-brane at of charge of wrapped on smeared over two transverseWe directions have ( seen that these singular domain walls are resolved to give a Gibbons-Hawking y in the summing as in [ solution localised on smeared over one of the metric ( with 3-vector V as sources on the basespace space takes the be understood by first lookingmonopole at the in covering space JHEP10(2019)198 I × ) to with 3 I T 2.3 × 3 T ] provides a good approxi- 25 in three directions, taking an 6789 R direction takes the nilfold ( L z F 1) in two directions, taking an NS5-brane to , and these are presumably non-geometric. − 3 ( ) with the nilfold fibred over a line to the / T 6789 × 3.31 R I – 25 – -brane. This takes the configuration L 3 2 F ∼ 2 1) Z . T-dualising in two or more directions takes the NS5- − / ( I 4 / T ). However, in the long neck region, the geometry is well 4 of N × T )-brane or 5 2 2 Z 3 )-brane. In the last section, we have seen how the naive T-duality / , 2 3 2 T , ), and takes the solution ( ) with a T-fold fibred over a line. If this is subsumed into a proper string 2.5 -brane or (5 2 2 3.36 Similar remarks apply to T-dualising IIB Consider first T-dualising IIB The first T-duality takes NS5-branes to KK-monopoles, with NS5-branes on -flux to the configuration with R-flux that is not geometric even locally. It cannot be approximate metric can be verymate useful. metric for The K3 constructionline. near of a [ The boundary K3 metric of isperk¨ahlermetric moduli provides obtained space an by glueing at approximate together K3 which somelong metric the hyperk¨ahlerspaces, neck and in space which each the is degenerates hy- divided relevant to into region. a segments, The each of K3 which has is a approximated by the hyperk¨ahler of this configuration will be discussed elsewhere. 11 Discussion Strings propagating on athe spacetime absence of with an a explicit K3 metric for factor K3 have makes some been issues much hard studied. to analyse, and However, for these an NS5-brane to an exotic (5 H formulated as a conventionalwith background explicit but dependence on can the be coordinates dual formulated to as string a winding. doubled The doubled geometry, geometry with bubbles or insertionsnilfold of fibred exotic over a branes,the line just neck with should as Kaluza-Klein be the monopole cappedgeometries K3 off bubbles that neck by cap or configurations off can insertions. the that befixed degenerate can The points K3 thought be of ends or thought the of as of ends of as the as double a the T-dual duals of of the the ON-planes on the by the T-fold fibred over aof line. a More T-fold precisely, it fibred should overdual consist a of to segments line each the with consisting different supergravitysmeared charges solution, for exotic the the branes; T-fold segments thesestring for are theory each are separated dual. segment. expected In by The to the neck domain configuration become walls can localised that be thought exotic are of branes as in a T-fold the fibred full over a line an NS5-brane configuration.the T-fold A ( T-dualitysolution in ( the theory duality, then this impliesthe that nilfold the fibred degenerate over K3 a solution line with is a dual long to neck a given non-geometric by configuration with a long neck given branes to exotic branes, sonow will explore result this in in string more theory detail. in a non-geometrican background. exotic We 5 between the supergravity solutions representing KK-monopoledomain domain walls walls and becomes NS5-brane a proper string theory duality between a smooth K3 geometry and approximated by a Gibbons-Hawkingconfiguration space of and NS5-branes, the so T-dual dualisinggood the of guide corresponding this to supergravity how gives solutions the the gives duality appropriate a works. mapped to KK-monopoles on dual to the type I string on JHEP10(2019)198 ], which 56 – Spin(7) are , 2 53 ,G SU(4) , – 26 – string, with the Kaluza Klein monopole bubbles dual 0 ], the duals of these spaces are constructed and the extension of the 58 ]. In [ 57 A central role in this paper has been played by the hyperk¨ahlerwall solution that is a Earlier string theory roles for del Pezzo surfaces have been discussed in [ This K3 limit in turn is dual to a IIB configuration which can be viewed as having a Acknowledgments We are grateful toof Amihay Hanany CH and is Costas supportedConsolidated Bachas by Grant for the ST/L00044X/1. helpful EPSRC discussions. Programme Grant The EP/K034456/1, work and by the STFC nilfold fibred over a line,analogues with of SU(2) this holonomy. in Remarkably,to there which are give a higher-dimensional a higher special dimensionalgiven holonomy version in space. of [ the Examples nilfold withresults is holonomy of fibred SU(3) this over paper a to line these examples is explored. these surfaces as the startingprovide point geometric for duals the of construction orientifoldprovides of planes, Tian-Yau and a spaces. so classification the These of classification inthere of a can turn del be class Pezzo a of surfaces link between orientifold these planes. various occurrences of It del would Pezzo surfaces be in interesting string to theory. see if that are ON-planes. Furthera dualities line replace and the replace neckleading the segments to with exotic Kaluza duals T-folds Klein of fibred monopole the over bubbles orientifold planes. withinclude exotic suggestions brane of a bubbles, mysterious duality while involving them. Here we have seen a new role for to the D8-branes andthe the orientifold Tian-Yau planes caps is dualto interesting and to be allows the addressed strong O8-planes. atO8* coupling weak planes. properties This coupling of geometric in the dual the O8-plane to dual type IIAlong neck theory, which such is as a 3-torus the with emergence H-flux fibred of over a line with NS5-brane bubbles and caps by Kaluza Klein monopolesand — the the Gibbons-Hawking ends metric ofspaces arises the — in neck there a are is bubblingbackground limit a capped arises — bubbling in with a limit spaces regionthe of which of moduli the are the space end moduli of approximated region the space by dual that of Tian-Yau type type gives I II a strings Tian-Yau on space. K3 which This matches metric on a nilfold fibred over a line. These are joined by regions which are approximated JHEP10(2019)198 B 07 566 (2007) (2005) ]. 24 JHEP (1995) 109 10 , B 525 (2013) 65 SPIRE Phys. Rev. Lett. (2019) 1 Nucl. Phys. IN Phys. 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