JHEP11(2016)064 Springer -corrected ′ α f space-time. August 10, 2016 October 31, 2016 : ). On the other : ′ November 10, 2016 ravity where the α : ( ngs O Received 10.1007/JHEP11(2016)064 Accepted KK5-brane and the exotic sfies the self-duality condi- Published doi: -brane solution, which satis- l and the gauge NS5-branes. 2 2 -corrected T-duality transfor- bits the non-geometric nature. the brane world-volumes. The ′ α Published for SISSA by [email protected] -brane solutions are investigated by the , 2 2 -brane solution is a T-fold at least at 2 2 . 3 -brane solution is not a T-fold but show unusual structures o 1608.01436 2 2 The Authors. c p-branes, String Duality, Superstrings and Heterotic Stri

-corrections are present. By performing the We study T-duality chains of five-branes in heterotic superg ′ , α [email protected] 2) monodromy structures of the 5 , -brane solutions associated with the symmetric, the neutra (2 Department of Physics, National2, University Science of Drive Singapore, 3, Singapore 117542, Singapore E-mail: Department of Physics, KitasatoSagamihara University, 252-0373, Japan 2 2 ArXiv ePrint: Keywords: O We find that thetion Yang-Mills in gauge the field three- in and these two-dimensional solutions transverse spaces sati to hand, the gauge 5 generalized metric. Our analysis shows that the symmetric 5 mations of the heterotic5 NS5-brane solutions, we obtain the fies the standard embedding condition, isWe a also T-fold find and that it the exhi neutral 5 first order Open Access Article funded by SCOAP Abstract: Shin Sasaki and Masaya Yata Non-geometric five-branes in heterotic supergravity JHEP11(2016)064 1 d 7 ]. 1 2 7 ]. 11 19 19 6 15 17 19 T 3 , ions 5 mations -brane, 2 2 ) BPS point R ( 8(8) ]. BPS branes in perstring theories, 4 E – 2 s known as the 5 eory compactified on erstring theories. The U- e brane world-volume [ dinary branes. These states s. However, there are states of these point particles is the ume. As its notation suggests, has T-duality symmetry group nder the U-duality group. For , we have the n is known as exotic branes [ d 8 T T . Here, the ordinary branes are waves, 8 T -corrected Buscher rule ′ ) in lower dimensions [ α is the string coupling constant. Therefore the R – 1 – s ( ) g d ( d E ]. The most tractable duality in string theory is the where in the sense that they are obtained by the T-duality transfor 17 2 2 2 – − s 7 g -brane is obtained by performing the T-duality transformat 2 2 -branes 2 2 ], which makes non-trivial connections among consistent su 1 ). The exotic 5 Z -brane has two isometries in the transverse directions to th 2 2 Among other things, an exotic brane in type II string theorie In this paper, we use the notation 5 A.1 Defect gaugeA.2 NS5-brane solution Smeared gauge KK5-brane solution 3.1 Heterotic KK5-branes 3.2 Heterotic 5 d, d, 1 ( has the U-duality symmetry group along the transverse directionsthe to 5 the NS5-brane world-vol O has been studied intensively [ are called exotic states and their higher dimensional origi F-strings, D-branes, NS5-branes,whose Kaluza-Klein higher (KK) dimensional brane origin does not trace back to the or twice on the heterotic NS5-branes. T-duality. Type II string theories compactified on superstring theories formexample, lower-dimensional when multiplets we consider u M-theory compactified on relates various branes in each theory. It was shown that M-th particle multiplet. Theordinary higher branes dimensional wrapped/unwrapped on origin cycles of in parts Its tension is proportional to Extended objects such asduality branes [ play an important role in sup 1 Introduction 5 Conclusion and discussions A Smeared solutions for gauge type 3 T-duality chains of five-branes and 4 Monodromy and T-fold Contents 1 Introduction 2 Heterotic NS5-brane solutions JHEP11(2016)064 . ]. ]. M 21 32 A – 2). 30 ] of the , 26 (2 -corrected ′ ]. We will , O α ]. When one 33 25 – 20 , 30 ]. Therefore they -brane in type II 2 2 19 23 he theory, which is ]. The most famous , -corrected generalized ion, we introduce the 29 ′ 22 – -corrected Buscher rule. ic string theories. Com- α ′ n study the monodromy 27 α -duality group c supergravity are the viel- 2) T-duality group for the ) solutions associated with ality transformations on the field living in the space-time he branes changes according ransverse two directions and , ral and gauge types [ jects [ -dimensional heterotic super- r background fields are gener- n remains geometric. Section imension two objects in string the upergravity [ on-geometry becomes healthy S5-branes by the (2 m of the smeared gauge KK5- se directions to the NS5-brane s into the space-time action as d heterotic string theories have c superstring theory. Heterotic O e heterotic NS5-brane solutions. dified [ on is called U-fold. An important hey are called non-geometric [ and the Yang-Mills gauge field = 1 gravity multiplet coupled with the ]. In particular, the 5 MN 24 B N -corrections, the Buscher rule [ ′ α -brane solutions. In section 4, we examine the – 2 – ], exhibit specific properties [ 2 2 -brane solutions by the -field 2 2 18 B , the NS-NS φ -corrections. Due to the ′ α , the dilaton A M e The purpose of this paper is to study exotic branes in heterot The organization of this paper is as follows. In the next sect = 1 vector multiplet. The relevant bosonic fields in heteroti -brane is a solitonic object of co-dimension two. These co-d 2 2 theory, sometimes called defect branes [ heterotic NS5-brane solutions known as the symmetric, neut In section 3, we introduceworld-volumes and the perform isometries the along T-duality the transformation transver by T-duality transformation for heterotic supergravityextended is objects mo in heteroticThere supergravity are theories are three th distinctperform NS5-brane the T-duality solutions transformations in to heterotic these s heterotic N However when the monodromygenerically is the given U-duality group bycandidate in the of string solutions symmetry to theory, group string then theory.nature of the This of n t kind the of exotic soluti branesare is called that non-geometric they are branes non-geometric or ob Q-branes [ goes around the center ofcome the back co-dimension to two the branes original into point, the the t the non-trivial background monodromy. geometry Therefore ofically the t governed metric by and multi-valued othe functions. In this sense, t string theories is a T-fold, whose monodromy is given by the T 5 In this section, wegravity introduce which the is NS5-brane the solutions low-energysupergravity in effective consists ten theory of of theN heteroti ten-dimensional bein symmetric and the neutral5 solutions is while devoted the to gaugebrane conclusion solutio solution and is discussions. found The in explicit appendix. for 2 Heterotic NS5-brane solutions metric. We show that the monodromy is given by the structures of these solutions. Buscher rule and obtain new five-brane solutions. We will the We obtain the heteroticthe three Kaluza-Klein types five-brane of solutions. (KK5-brane KK5-branes We and then perform write the downmonodromy second the T-du structure exotic of 5 the heterotic 5 pared with type II stringbeen theories, poorly exotic understood. branes in This typein is I due these an to theories. the non-Abelian Notably,the gauge the first Yang-Mills gauge order field enter JHEP11(2016)064 ′ . in α i C The M (2.2) (2.6) (2.3) (2.5) (2.1) (2.4) e ) MN 2 g ]. PC . ) terms is 35 e ′  .
, Q α ) ( ∂ CB + 34 9 are the local [ O − ω M ) is constructed ′ ( ω ω α ( QC ,..., where the metric in e CD R 1 are the gravitational η P , B φ ∂ MNAB Yang-Mills gauge field 10 N ]. ( M . AD g e R is defined by emann curvature square ) . = 0 35 ) A N BQ , φ∂  + ω e dimensional metric M ω 34 M MN and e ( − ∂ AP B (3) MNP me order in e is taken over the matrices of 10 e AB L+ MNP P ˆ , H κ η cing [ CB ∂ + 4 onic fields at the first order in Ω − A, B, . . . N ) AB = + MNAB − and its inverse: ω A M (3) R M -flux A e MNP ˆ where CD is in the adjoint representation of the PM H MN + H N M η ′ g (3) B e 1). The Ricci scalar ∂ YM MNP ± α M ˆ N Ω H AD − MN ∂ A  = AB ′ A M F -correction. The famous anomaly cancellation N + ,..., ′ α ω 2 10 M e 2 10 1 g – 3 – (3) MNP α κ . We employ the convention such that the gauge , ω 2 + + M MN 8 ˆ 1 NP H ∂ , F = E -field is defined by which enters into the action in the ( ) 1 3 − B AB B ω Tr × M ( − (3) 8 BN M MNP AB ∂ AB ′ ) e ( ω E H M α M ω 2 1 MN N − ( and the modified ± ± : ∂ + = = diag( ) ω of the R ω = AB B  − M AB φ R e AB 2 (3) M MNP B η N − (3) MNP AB ˆ ω (3) MNP (3) N ∂ MNP H ˆ N e H ge H H − ω A − PB e B M M √ 9 are the curved space indices while N ∂ e e x NA 10 M e d ∂ ) = ) = which is SO(32) or ,..., ( = ω ω Z ( ( G AN = 0 R 2 10 AB e 1 κ h MN (3) 2 M 1 2 ˆ AB H = = M,N, R A remarkable property of heterotic supergravity is that the S The authors would like to thank Tetsuji Kimura for his introdu 2 AB M ω contributes to the action as the first order The spin connection is expressed by the vielbein mechanism and the supersymmetryterm which completion involves result higher in derivative the corrections Ri in the sa field is represented by anti-hermitianthe matrices fundamental representation. and the trac from the spin connection gauge group Lorentz indices. The Yang-Mills gauge field the local Lorentz frame is ten-dimensional heterotic supergravity action for the bos The modified spin connection Here we employ the convention such that Here the field strength is given by and the gauge coupling constants in ten dimensions. The ten- Here defined as where the string frame is defined through the vielbein as JHEP11(2016)064 ) 2.5 our- rder (2.9) (2.7) (2.8) asso- ) and (2.11) (2.10) (2.13) (2.12) + Q . , ω 2 (  ) ) and ( symmetric, 4 x FG 2.4 ] P ) are defined as ABMN + ( + 2 R ω ) ) is determined by ) 2.5 3 r In the following we + ( is x CD ω ( H N N + ( + . , 2 ω ) dx N ear in ( ) otion derived from the ac- associated with the gauge 2 ′ 2 M K ]. They are 1/2 BPS config- ∧ α x k ABMN is the structure constant for A AB +[ ( 32 hese heterotic NS5-branes are M he other is the charge R ω ey satisfy the following ansatz: – M O + ( J dx Mills gauge field: JK 2 30 , DF I ) A ] η ) + 1 . f F MN x R JK (3) F BC I ∧ 4 represent the world-volume and the ∧ η ˆ 1 , 2! f H = ( , is the Levi-Civita symbol. The 1/2 BPS F 3 3 R 2 , + AG S = obeys the following Bianchi identity: 2 mn η Z , Tr Tr[ , ˜ 9)-valued curvature 2-form. The component I M F 2 3 ′ F , mnpq (3)  − α , r A ] Z ε = ) 2 ˆ + 1 = 1 P H N . The harmonic function 2 r – 4 – F π ( ∂ π A 2 pq 1 mn ∧ CD H N ] − , − F 32 F P F m, n n A -flux I N − + . Note that the terms = log M H dx [ A 9, ω is the SO(1 G (Tr = mnpq 2 1 , Q ′ m A N ε M 8 N k α ∂ 1 2 2 3 ∂ , = dx ] stands for the anti-symmetrization of indices with weight 7 dx n = , = + = AB -flux is iteratively defined through the relations ( 6 ∧ ] mn are the gauge indices and I , M M δ -flux: P (3) H M 5 ) , φ mn G vanish. Here the Hodge dual field strength in the transverse f ) +[ , A r ˆ ) are higher derivative corrections to the ordinary second o ˜ H H F ··· ( MN r i ω dx N d ( 2 F ∂ A H 2.1 dim = 0 H AD M M q MN + [ η 1 . The modified ∂ ′ j associated with A i, j M α AB ). There are three distinct NS5-brane solutions known as the BC ′ ,...,  dx G η R α i mnpq ( 1  2! ε dx = 1 O 2 1 = ij η − = 3! = 3!Tr is the asymptotic three-sphere surrounding the NS5-brane. ) at AB = = in the action ( 3 R 2 S I, J, K 2.1 The heterotic NS5-brane solutions satisfy the equation of m L+ MNP YM MNP ds (3) mnp Ω Ω ˆ . Note that the modified H MNP L+ ! 1 n and other components dimensions is defined by condition leads to the self-duality condition for the Yang- Ω transverse directions to the NS5-branes and Here tion ( the Lie algebra where the indices briefly introduce the three distinct NS5-brane solutions. the neutral and the gauge types in heterotic supergravity [ derivative terms. urations and preserve a half of the sixteen supercharges. Th order by order in follows: Here the symbol [ of the Yang-Mills gauge field strength 2-form the Yang-Mills gauge field configurationscharacterized of by the two solutions. charges. T One is the topological charge The Yang-Mills and the Lorentz Chern-Simons terms which app where Here instantons: where the integral is defined in the transverse four-space. T ciated with the modified JHEP11(2016)064 . ′ ]. ), ay 36 nα auge 2.10 (2.18) (2.16) (2.19) (2.15) (2.17) (2.20) (2.14) √ 0 φ in ( − auge [ ) becomes e eral BPST = 2.8 ) cancel out. (3) mnp . written in the ρ ˆ H        2.5 1 − tion in terms of the , ) and ), is obtained through 4 -flux ( r erse directions. If the r ( m ( . H δ 1 0 0 0 H n 0 0 0 0 0 1 0 0 1 0 0 3) H I − and the explicit solution is , eld which satisfies the self- , δ 2        Z instanton size , on G − = ∈ 4 n 3 = 1 δ . m I pq . . δ ,T B I pq ab , n + n η        0 I, J, K m ∂ 4 ( φ 1 + 2 − ω − mn n , mnpq I , e -field and the Bianchi identity ( ′ 1 0 ′ mnpq x ε ε = K 2 ε 1 2 B − r – 5 – 2 1 T = nα nα mn ab ) σ = + = + ). A remarkable fact about the symmetric solution m 1 0 00 1 0 0 0 0 0 0 0 0 I mn IJK 2 0 A ε I mn        r φ H , n ( 2 2 η m − e = − , η ∂ of the gauge field are identified with the indices of the I 2 = are defined as T ) = ) = (1 ] = I m r a, b J ( T I mn A ,T η H ,T k, Q        I The most tractable NS5-brane solution in heterotic theory m = 1 T [ ) is given by an instanton solution in four-dimensions. The g − indices mn σ (2) algebra G 2.11 1 0 0 ) is nothing but the BPST one-instanton in the non-singular g − su is the asymptotic value of the dilaton. Compared with the gen ⊂ -flux only comes from the ], 0 1 0 0 0 0 0 10 0 0 0 0 H 2.14 φ ) holds, the Chern-Simons terms in the modified 32        , is the self-dual part of the SO(4) Lorentz generator. This is : is the ’t Hooft symbol which satisfies the self-duality condi = 31 2.19 1 mn I mn m, n T σ η = 0. From the Bianchi identity and the relation between -field is determined through the following condition: B (3) ˆ H indices This solution has charges ( is that the dilaton configuration, hence the harmonic functi The anti-hermitian matrices Here the SU(2) where Therefore the d the The constant the standard embedding ansatz: field takes value ingiven the by SU(2) [ subgroup of the gauge group SU(2) subgroup ofrelation the ( local Lorentz group SO(4) in the transv instanton solution, the symmetric solution has a fixed finite following form, They satisfy the be the so-called symmetricduality solution. condition ( The Yang-Mills gauge fi Symmetric solution. The solution ( where JHEP11(2016)064 ) ) ], ]. ear 37 32 and ′ 2.22 2.21 , (2.21) (2.22) ]. The α in [ 31 -flux in ), ( 32 H )[ , ]. Indeed, ally found nts, which ]. e , n 31 32 2.10 32 0) worldsheet , , , 31 [ 31 ′ ) = (0 . From the Bianchi = (4 α ton. As in the case F N k, Q ∧ he gauge NS5-brane in ). Therefore the Lorentz is not fixed for the gauge F -corrections [ ′ ′ ρ α 0) worldsheet sigma model Tr α -flux is neglected as it is a this is a solution to type II ( ′ , ne. Therefore the instanton α orders in ) is a solution to the equation H O = 0. With the solution ( ion. It is remarkable that the l and the harmonic function is der . = = (4 2 2.14 (3) 2 ) 2 (3) ρ ˆ N H ρ ˆ . H d ′ is order d + + 2 2 r and the functional forms ( 2 2 nα r r ′ ) with ( ( α ′ , + ), the string sigma-model analysis indicates 2 α ′ 0 n ) becomes a higher order correction in ρ MNAB 2.10 φ α x , 2 ), the size modulus ( R e 2.5 + + 8 0, a gauge multiplet on the brane world-volume – 6 – O mn 4) sigma model description and it is protected 8). Similar to the neutral solution, the gauge 2 0 , , = 0 σ r φ 2.14 → -corrections in heterotic theories [ 2 ) = ′ − e r M ρ ) at α ( -flux ( A = (4 = H H 2.1 ) = ) = (1 ) for heterotic supergravity is valid at the first order in m r N ( A . Indeed, the gauge solution has the ′ H k, Q 2.21 α ), are the same as the ones of the symmetric solution in the lin The neutral solution is the one with charges ( ) with ( The last is the so called gauge solution which has been origin 2.20 . However, the Yang-Mills gauge field still contributes to th 6= 0 resides in the core of the NS5-brane. It has been discussed 2.10 ′ α ρ -field is given by that of the type II NS5-brane and its compone B . We note that the neutral solution has the ′ -corrections. ′ α α ]. The Yang-Mills gauge field is again given by the BPST instan 30 . For this solution, we find the curvature ′ in [ solution is obtained by a perturbative series of identity, the harmonic function is determined to be negligible. Then the Bianchi identity again becomes becomes massless and the SU(2) gauge symmetry is enhanced. T Chern-Simons term in the modified that the symmetric solution is an exact solution valid at all Since the gauge fieldsupergravities. does not Indeed, appear this inneutral is the solution the neutral ( type solution, II NS5-brane solut solution and it has charges ( position of the instantonwith corresponds finite to size thatwhen of the the instanton NS5-bra shrink to zero size are valid at the first order in Gauge solution. of the neutral solution, the curvature term in the modified at hand, the order in α For this solution, the Yang-Millsgiven gauge by field becomes trivia of motion derived from the action ( the symmetric solution has the description and receives higher order higher order in Although the symmetric NS5-brane solution ( the gauge solution and the Bianchi identity becomes Compared with the symmetric solution ( are determined by ( sigma model description and is expected to receive higher or Neutral solution. against JHEP11(2016)064 - ′ To ns. α 3 (3.2) (3.1) (2.23) , heterotic , specify an ˆ 8 M ′ yy I y ′ y ′ yy E G A G G nds in type II ) the heterotic × ˆ 8 N... − , 3.2 E ,  ˆ ˆ are turned off, the M ) M, ′ yy  y N −→ − ˆ G N B ) as the first order -field is not excited in y A AB I y g and B M 2.1 ) with ( M ˆ M y + A ′ y ω 3.1 -field take a constant value: theory and vice versa. G −→ − 8 Tr( B the brane world-volumes. tion ( ′ yy erotic NS5-brane solutions, we m contributes to the modified E ,A ˆ tic supergravity in the family of − -corrections to the Buscher rule I y ), we have defined the following G M ′ supergravity should be modified × . ) y A α 8 − ]. This is given by N 3.1 ˆ E M 0 is related to the D5-brane in type -volume direction of the NS5-branes in the ˆ + ′ yy N ′ y 28 ′ y ω . We call ( G G → , G -corrected Buscher rule M ). The indices ′ n the ,B ρ 2 ˆ + yy ′ − M ) yy ). Therefore the α y ′ ω NBA g G ˆ (constant) g I M ˆ α + ˆ My ( M, y ω A ′ yy Tr( B ), the actions of SO(32) and , O  coincide. Note that when the Yang-Mills gauge G ′ ˆ – 7 – N AB −→ = ( 1 ′ y α 3.1 mn − S M −→ G ˆ M + ]. The first order yy N + 2 ′ y − ω ˆ -field for the gauge NS5-brane solution. A careful = Θ I M 26 G B , = ˆ ˆ Ny MN M mn ′ y 25 B N B , g B ˆ + G M ˆ − ω M ′ y yy ,A ′ y | 2 g -branes, we first write down the heterotic KK5-brane solutio G = 0 at least at M ′ yy G ( 2 2  ′ yy + G MN yy ω g G ′ yy 1 | ′ yy g 1 G = G (3) MNP + log + H + ˆ 2 1 ˆ M ′ yy N ′ MN ˆ y N ˆ G g − G ˆ M M B g φ ) reduces to the ordinary Buscher rule for the NS-NS backgrou and the higher derivative corrections coming from −→ −→ − −→ −→ 3.1 I ˆ ˆ ˆ φ N N M M y ˆ ˆ M M g A g B We make a comment on the If we perform the T-duality transformations along the world 3 -flux and we find relation ( the gauge NS5-brane solution. Then, the components of the where we have defined Tr where we have decomposed the indices analysis reveals that onlyH the Yang-Mills Chern-Simons ter 3 T-duality chains of five-branes and field isometry and non-isometry directions respectively. In ( correction, the T-duality transformationfrom in the the standard heterotic Buscher rule [ Before going to the exotic 5 supergravities. 3.1 Heterotic KK5-branes quantity, Buscher rule. Under the transformation ( In order to perform theintroduce T-duality a transformation U(1) for the isometry het along the transverse direction to I theory by the S-duality. the T-duality chains. Since the gauge field enters into the ac In this section, we derive new five-brane solutions in hetero SO(32) heterotic theory, we obtain the identical solutions i in heterotic supergravity have been written down in [ the SO(32) heterotic theory in the zero size limit supergravities dimensionally reduced on JHEP11(2016)064 ) ). 3. = , = 0 φ 2 3.1 2 (3.3) (3.4) (3.5) (3.6) , 2.11 H e   = 1 harge for , ole whose 2 ′ ) vanishes. . i 2 ′ ) m 3.2 3 m x 0, we introduce dx governed by the ′ + ( m → 2 β ) ] where 4 2 4 + ons along the isometry x R 4 ,A ) is approximated by the 4 t have been written down ′ and consider the periodic terotic Buscher rule ( + ( dx lution where the standard ane of co-dimension three m 4 h originate from the periodic 2 3.3 self-duality condition ( ) 1 A ) is quite simplified. This is R ormations of the NS5-brane 1 − eads to the condition x th the three types of the NS5- is determined by the following 3.1 + [ H ′ 4 = ( by solving the Laplace equation m + 0 but this is an artifact of the smearing monopoles or smeared NS5-branes. A . β 2 ′  ′ C p , 2 → m ) β r ∂ 4 ′ 3 C 2 n R ]: , ∂ dx = ′ + ) and ) holds and the last term in ( , r ]. We now perform the T-duality transfor- p 0, the sum in ( ′ 3) ′ 40 0 2 p [ N , n ′ + ( φ ) ′ 40 3.4 2 2 n → s 1 A 2 , m e 4 ) F 4 S ε ′ M – 8 – ) which finite 2 p R ′ A × πR = 1 = n 3.4 dx ) = ′ 2 ′ ′ 3 r m ( H R m − ′ ε + ( nα ( H 1 2 4 m 2 -direction with the radius ) x ∂ 4 ) = Tr( 1 is given in ( x ) on N r + ( dx ( , + H ( 2 For the symmetric solution where the standard embedding ω  H r = 0 M would diverge in the limit H s + 1 X ω + − 4 MN j R + ′ 0 dx nα φ is well-defined. On the other hand, the Yang-Mills monopole c i 2 e = Q dx ) is satisfied, the heterotic Buscher rule ( ,B ij C ]. We note that the smeared symmetric NS5-brane is an H-monop η . We call this procedure as smearing. The result is ) = s 38 r 2.19 = = 0 ( 2 φ H ]. By performing the T-duality transformation of the soluti ds 40 , = 0. The periodic array of the symmetric NS5-brane solution is We perform the T-duality transformation by utilizing the he The constant 38 4 H following harmonic function procedure. We can find the harmonic function ( Taking the compactification radius small  the U(1) isometry to the heterotic NS5-brane solutions. The array of the NS5-branes. Then by taking the small radius limi in three dimensions. arrays of thein symmetric, [ the neutral and the gauge solutions Therefore the resulting solutions are calledThe the explicit heterotic forms of the smeared NS5-brane solutions which where the harmonic function The corresponding solutiondiscussed is in the [ smeared symmetric NS5-br relation: mation on the smeared symmetricembedding NS5-brane ( solution. Forbecause a the so relation Tr( Then we obtain the symmetric KK5-brane solution: quantized charge integral over Symmetric KK5-brane. condition is satisfied, the ansatz and the Bianchi identity l direction, we obtain thebranes. KK5-brane solutions associated In wi thesolutions and following, derive we the KK5-brane calculate solutions. the T-duality transf reduces to the monopole equation this end, we first compactify the the smeared solution is not defined anymore [ JHEP11(2016)064 l . 5 ′ α (3.7) (3.8) (3.9) ) still 2.19 , , associated , ,     , 3 3 in disguise. , 3 3 +   is obtained as T 3 ω in the right hand T 3 ) I ) T I y HT T mn HT ) H 3 ) M A H 3 ˜ 3 = 0. Again, the F ∂ 3 H ∂ A ∂ 3 H ∂ 3 3 − 1 3 I β , this belongs to the 1 ∂ β β ∂ 2 β M 2 − = β mn − − β A ˜ − 2 F 2 ometric solution but the + t the leading order in T + H T − . ]. mn H ) ld 2 = 0. Therefore the neutral ) 2 theories. H F . H auge field H 39 = I 1 H ) satisfies the anti-monopole 1 2 = 0 2 H∂ he smeared neutral NS5-brane M H∂ ∂ i on ∂ = 0 3 H∂ 3.7 mn 3 A + H∂ ad of the anti-self-duality condition. i ( β + ( ( F ). Applying the heterotic Buscher β + ( d in [ comes from the choice of the freedom for 2 − − 2 ) for the neutral solution. Therefore − − ′ 3.2 T 2 T 2 mn ) α H ) ˜ H ( F ). As we have claimed in the previous , ω 1 ,A 1 H  H  O HT 2 3 φ. HT 2 3 2 3.4 i ∂ 2 H∂ ∂ ∂ 1 H∂ ∂ 1 D 2 β 2 HT β β HT − + ( 3 β + ( 3 + ∂ + 1 ), the modified spin connection ∂ − – 9 – 1 = − T 1 H − is in the i T ). If we choose another sign in front of 1 H ) − 3 3.6 ) T 2 3 B T 2 + ) H 3.1 ) H ω 1 1 H∂ H H∂ ∂ H ( HT ∂ 1 ( HT 3 1 2 3 -field and the dilaton for the neutral KK5-brane solution ∂ 2 β ∂ ∂ − 2 β − ∂ 2 ) satisfies the equation B β 1 + β 1 -field vanish but the Yang-Mills gauge field remains non- − + − ), we find that the standard embedding condition ( 3.7 − 1 B − 1 H H HT 2 HT 3.9 2 1 H 1 H 3 ∂ HT We next study the neutral KK5-brane solution. For the neutra 3 ∂ HT 1 1 H∂ 1 1 H∂ ∂ β ( in the heterotic Buscher rule ( β ( ∂ ) is calculated to be H∂ H∂ − − ( − − ( − −     ) and (     3.5 2 2 2 2 2 2 2 2 ′ MN 1 1 1 1 3.7 H H H H 1 1 1 1 G H H H H ) and the gauge field remains vanishing 2 2 2 2 2 2 2 2 3.5 ======2 4 1 3 +1 +4 +2 +3 A A A A ), the gauge field satisfies the self-duality condition inste ω ω ω ω 3.1 Meanwhile, by using the relation ( The flip of the sign in the Hodge dualized field strength 5 smearing procedure is applicablesolution to is the neutral governed by solution. the T harmonic function ( NS5-brane solution, we have the trivial Yang-Mills gauge fie We find that the Yang-Mills gauge field configuration ( with the geometry ( Indeed, this is nothing but the KK5-brane solution in type II general class of spherically symmetric solutions discusse it is negligible in side of ( section, the modified spin connection rule, we find that the metric, the holds for the symmetric KK5-brane solution. Neutral KK5-brane. (Bogomol’nyi) equation KK5-brane is a purely geometric Taub-NUT solution at least a the overall sign in the heterotic Buscher rule ( are given by ( This equation is nothing but the anti-self-duality conditi Since the field configuration ( The dilaton and the NS-NS From the results ( trivial. The symmetricgeometry is KK5-brane dressed up solution with the is Yang-Mills gauge not fields. a The g purely ge JHEP11(2016)064 ∼ c- 4 along ) cor- of the (3.10) (3.13) (3.11) (3.12) (3.14) where (3) mn ˆ f ˜ r ρ H . r 3.10 φ -instanton ]. In order 2 k e pq q nents as F 0 limit of the tion and they ∂ = 1 + mn , → f 2 F ) mnpq 4 for the 4 ε en as R x 2 2 1 . ) -fields in the smeared i mnpq 4 2 x − ) based on the smeared ε B + ( ρ m − δ on of the smeared gauge 2 x = ( n ) -flux behaves like Tr[ y the I 3 ′ 3.13 also found in the heterotic δ n is obtained by the famous 2 , ( H α x 3 =1 n in contrast to the monopole ng, we find (3) auge solution, we perform the k i mnp + ˆ + ( 4 H P ) = n for the smeared gauge NS5-brane = Θ 2 r δ ) ( 2 m 34 H I x , H δ f. 2  ) ). Then the gauge field becomes + ( − ) = 1 + ρ ,B r 2 4 log = 0 is not satisfied for the gauge solution. 2 2 ( ) + ˜ ˜ n ρ 1 f 2.22 φ mn r ∂ 2 x I ( e ε 2 = Θ mn r  = ( ′ vanishes. This is in contrast to the symmetric σ = – 10 – 24 α 2 2 = ¯ Q I mn is a rescaled size modulus. Then the smeared gauge ¯ η − m 4 ,B 2 0 A 1 R , ρ φ 2 , n , r 2 I e n ) = dx 2 T x = Θ = ρ ρ m = 0 and given by φ 14 I mn mn ]. The harmonic function + 2 η dx ¯ σ f B 2 2 41 and ˜ r  = ρ mn ( ). This periodic array of the instantons is just the calorons For the gauge NS5-brane solution, the periodic harmonic fun 1 f 2 2 δ 2 ) r φ ]. We also note that the smeared gauge NS5-brane does not ex- − 3 , e mn 3.4 2 ) and the charge x ) e ¯ σ 40 ρ = ν are the positions of the instantons. Note that the solution ( + + ( + ˜ m j i 2 ˜ ρx ) and the Bianchi identity indicates → ∞ -field is trivial in the gauge solution, we can set these compo r ) A satisfies x ( 2 = 1. Using this fact, we can perform the smearing procedure fo − dx ]. B 2 r i f x ( r k 2.10 40 is the antisymmetric and anti-self-dual matrix and is writt dx q 6 + ( are constants. The other components do not appear in the solu r x mn ij 2 4 η σ i ) does not work as a solution since ) mn 1 σ = = ¯ x 3.4 mnq 2 Since the m ˜ ρǫ = ( ′ ds A α 2 4 where ¯ the line of obtaining ( As in the case of the gauge NS5-brane solution, we find that the to find a solution ofsingular co-dimension gauge three transformation associated of with the the g solution ( calorons, namely, the smeared instantons. After the smeari hibit the H-monopole property. Indeed, we find the modified instanton does not have a finitesolution monopole in charge. [ This is agai responds to Harrington-Shepard type [ NS5-brane solution is found to be gauge NS5-brane solution becomesmonopole trivial. solution [ This property is r The ansatz ( is determined by the Laplace equation whose source is given b solution. Here tion ( Gauge KK5-brane. This is the BPST instanton’t in Hooft the ansatz: singular gauge. The solutio The function where Θ can be set to zero. Now we perform the T-duality transformati and the neutral solutions. We also comment that the solution JHEP11(2016)064 to 3 x (3.16) (3.18) (3.15) (3.17) , , , . Since the    ) ) ) 3 3 3 T T T , → ∞ 3 3 3 2 ) x x x ′ m + + + ) is approximated by 2 2 2 is obtained by the peri- s property is similar to dx . ′ , T T T 2 2 2 2 2 3.16 m ation is performed on the lls gauge field satisfies the ) ality transformation along x x x ] T = 0 s 3 i he transverse direction × + + + 23 , + Θ 2 , 1 1 1 2 4 πR 0 whose physical interpretation ) R ω, T T T 1 2 22 ρ 1 1 1 1 -field does so. The only non-zero ,A − 3 dx becomes negative at a finite value ∂ ) x x x ( 2 − , B ( ( ( + ˜ 3 R ˜ ′ ρ − n < ) in 1 2 3 3 r H µ H r T r 0 x ( 3 Θ Θ Θ ( = φ 2 nα 0 0 0 x 4 r φ φ φ log H ′ − + ( 0, the sum in ( 2 2 2 H + e 2 α 2 σ − − − 2 2 2 e e e r → + T − ′ . − 2 p 3 0 n 2 x ) + ) + ) + 0 ) h R , the NS-NS – 11 – 3 1 2 φ 2 s -brane solutions associated with the symmetric, dx 2 2 + 2 2 ′ X x T T T e ω, ∂ 2 3 1 1 m , x 1 2 2 ) = x x x 1 T = r ∂ + ( 1 ( x dx + − − − ′ 2 x H = is determined by the following relations: ) n 0 H ( 2 3 1 ′ 1 φ ) 2 ω T T T m x -direction. After calculations, we find the following gauge H r 3 1 2 e δ 4 1 x x x = , x ∂ + diverges in the large cutoff scale limit Λ H ( ( ( = ( 2 2 ). Notably, the function H    0 34 ) = 2 ρ + 0 ρ ) ) ) r r h ( The harmonic function φ is a constant which specifies the region where the solution is B j ρ ρ ρ ( 3.8 2 4 r − H µ + ˜ + ˜ + ˜ dx 2 e i ˜ ˜ ˜ ρ ρ ρ 0 r r r ( ( ( φ -branes -field dx 2 2 2 2 log with the cutoff scale Λ. The result is [ 2 2 r r r and ij − B 2 2 2 2 1 e η s 4 ′ -brane. R ======2 2 3 πα 1 2 3 4 2 ]. 2 φ R A A A A ds 40 = ]. The constant σ 42 -direction, we obtain the exotic 5 3 x . The solution is ill-defined near the center of the brane. Thi r the neutral and the gauge solutions. Symmetric 5 odic array of the KK5-brane. This is given by Taking the compactification radius small the KK5-brane world-volumes. Bythe performing the second T-du the symmetric and neutralis NS5-brane unclear solutions [ with the integral over of harmonic function only depends on The solution seems complicated butanti-monopole one equation finds ( that the Yang-Mi where 3.2 Heterotic 5 NS5-brane solution along the valid [ KK5-brane solutions. We introduce another isometry along t Now we are in a position where the second T-duality transform component of the KK5-brane solution: JHEP11(2016)064 - (3.19) (3.22) (3.20) (3.21) ) to two erotic vor- 2.11 , -brane solution: -brane solution.  , 2 2 2 2 2 )  , 4 2  is the angular coor- T 2 ndition ( dx ) θ T ) H 1 + ( rd embedding condition, H tains the explicit angular 2 . 2 ) unction is characteristic to 2 ω∂ 3 ge field remains non-trivial. where ω , ω∂ + 3 dx θ + , − ( B H 3  2 2 sin T H H ) 1 H K . r ] = ) are the same with the ones in type . This result leads to the fact that -field in the solution are not single- ¯ H∂ ϕ H = ) on which the harmonic function is  θ + = 1 B H∂ 2 1  ∂ ( ϕ, 2 ). By performing the second T-duality + ( x x 3.20 [ ( 2 3.5 ) 1 −  ] where the harmonic function behaves as 2 H 1 3.1 T θ, x 1 1 2 ) 44 − dx T ,K , ) , − H cos 2 ω H + ( K tan 43 r = – 12 – 1 -field in ( = 0 2 σ ω∂ 2 − ) = B 1 ϕ − ω∂ − = 1 2 x dx + = ( H 34 iD  1 ω is proportional to ,A H 3 2 − H ω H∂ T ϕ ( ) + 1 H∂  ( j H ,B D  2 -direction, we obtain the following symmetric 5 3 ∂ dx H K ) is a complexified adjoint scalar field. The vortex-like equa 1 i ( x 4 HK 1 2 is found to be H HK dx 1 . iA log 2 2 is calculated as − ij ω 1 2 is not a single-valued function also in the symmetric 5 η I + = = = = 0 -plane, then ) satisfies the vortex-like equation in two dimensions: m 3 M = = i 2 3 1 4 ) the smeared symmetric KK5-brane solution. Indeed, the het A x A 2 A A φ A A A ( 1 , 3.21 2 x ds ω 1 3.17 √ ) is obtained by dimensionally reducing the self-duality co = ϕ 3.22 Since the heterotic KK5-brane solution satisfies the standa dinate in the the space-time metric, the dilaton and the NS-NS valued anymore. The logarithmicco-dimension behaviour two of objects. the harmonic We f call the solution ( If we employ the coordinate system where dimensions. We notecoordinate that in since the Yang-Mills gauge field con II theory. However,The in gauge heterotic field theory, the Yang-Mills gau tex and the domain walllogarithmic are and discussed linear in functions. [ transformation along the replaced by ( The metric, the dilaton and the NS-NS The explicit form of We find that ( this again simplify the heterotic Buscher rule ( tion ( JHEP11(2016)064 are MN F (3.24) (3.25) (3.23) MN olutions MN F F , .               MN F 1 0 -corrections. We ′ − ) is calculated as α , . 1 0 0 0 2 2 1 0 0 0 1 0 0 0 0 0 0 0 0 , − 0 0 00 0 0 00 0 0 00 1 0 T T 3.20 T 1 , 34

             
T -corrections are expected . 34 ′ N )
H H H SO(4)) invariant quantity 2 1 ) α 1 + N H ) = ) = rom the ) ω 2 H n. This is similar to the situ- ) is satisfied up to the gauge ( variant quantity Tr 14 34 2 H ωH∂ ωH∂ 1 ωH∂ N N ω∂ 2 ( ( 2 2.19 ωH∂ ω∂ ) does not hold for the symmetric − + 2 − − MNBA + + 2 , , H R H H H 1 ) 1 2 ]. We therefore conclude that the het- 2 2.19 H ] where divergences in Tr               H ∂ ∂ + ∂ 2 1 2 2 45 2 ω 38 H∂ ∂ ( ( 2 H H H H∂ 1 ( 1 0 0 H − − − − 1 − 1 0 0 0 − − ) and find K 0 0 0 0 0 0 1 0 0 0 0 0 H H H – 13 – MNAB − 0 0 00 0 0 00 1 0 00 0 1 2 + 2 K − H ∂ ∂ ∂ R ω               1 2 2 2 3 ( + are identified up to a gauge transformation. associated with the solution ( ∂ ω ω ω . = T 2 3 is the generator of the SO(4) Lorentz group. More ) + + ω ) = ) = T 3 3 2 2 2 3 ω ω ) H 13 24 − − − 1 MN 2 3 aB MNBA H ∂ δ N N − 2 F K K ( K MNBA ( ( R and ∂ ) bA K ( R H H 1 1 H H 1 1 + δ 2 2 m MN 2 2 H H ω 1 1 F A ( , , − 2 2 − − + +               Tr = = = = MNAB bB ) is exact for solutions of the symmetric type. The family of s δ R +4 +1 +2 +3 3.1 MNAB aA ω ω ω ω δ R 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = − − and               AB ) MN ) = ) = ab F 23 12 N N N When the standard embedding condition is satisfied, all the The modified spin connection ( ( MN F -brane solution. However we find that the condition ( 2 2 to be canceled in the T-duality transformations [ in the action is canceledation by discussed the in standard the embedding multi-monopole conditio solutions [ At first sight, the standard embedding condition ( related by the T-duality transformations seem not to suffer f erotic Buscher rule ( 5 transformation. To seeTr this, we calculate the gauge (and the This result indicates that canceled in the term Here ( explicitly, they are given by note that the multi-valuedness which appears in the gauge in JHEP11(2016)064 - 2 2 − = 0 , 0 ˜ 2 h scher g the ˜ h
(3.26) ) is an ) = , , term in   f ) ) and 2 + r/µ 3.20 2 1 . However, ). This is a ω ′ 2 T T ) 2 α 2 2 2 log( ˜ σ x x x ′ 3.20 2 ˜ σ α + − − + ( 1 2 expansion. and, unfortunately, 0 ′ T T 2 , ˜ h ) 1 1 α rder in ] , 1 . However, since Θ is a 2 x x 3 2 ) x ω -plane. In the gauge 5 m the second T-duality r 4 T 2 2 etric solution ( he heterotic five-branes is
ne, again the x on this property in section ) dx 1 = ( ne. The calculation is the ious subsection. After some − x ) + Θ( ) + Θ( 2 0 1 2 tes to the Buscher rule. After + ( -brane solutions, there was the r/µ φ r 2 2 2 T T 2 lve gauge field anymore and it is e ) 2 2 tion in the 3 becomes negative at a finite value x x log( 1 = -field are given by ( ) which is a reminiscent of the self- I 2 ˜ dx σ − + I ˜ B σx [( 2 1 . One obtains the function − 2 2 T T 3.22 ). Here 0 1 1 T -field is non-trivial. As discussed before, = Θ. The explicit form of the smeared ˜ h x x , B ( ( × + Θ I 2 0 0 3.12  34 2 0 φ φ r 2 φ 2 2 4 B R 4 e e -brane solution: e 2 2   – 14 – = ) ) + Θ I 2 2 2 ) in 0 ] + φ 2 4 ) e 2 + Θ + Θ 3.17 0 0  dx ,A φ φ -brane solution is completely determined by single-valued are constants. For the smeared gauge KK5-brane of co- 4 4 3 2 2 e e log 3 T ( ( + ( ρ


2˜ 1 2 2 πR ) ) ) ) = 1 = ˜ ˜ σ σ r/µ r/µ r/µ dx σ [( For the neutral KK5-brane, we again find that the heterotic Bu 2 ). In the symmetric and the neutral 5 I , φ ], ˜ We introduce another isometry along the same way of obtainin log( log( log( -field is governed by a parameter Θ instead of 2 2 2 2 ˜ ˜ ˜ σ σ σ + ˜ σx B /µ j Λ) in the ’t Hooft ansatz ( 2.11 − Λ − − − . 3 + Θ r/ 1 Θ dx 0 0 0 ( i 0 ˜ ˜ ˜ h h h φ πR O 4 dx -brane. 2 2 2 e which contains explicit angular coordinate on the 2 2 r r r -brane. . ij + 4 4 4 2 2 η − ω log[4  = = = 0 = = = r µ 3 in the heterotic Buscher rule is negligible as it is a higher o ˜ i ρ 2 4 1 3  34 πR A . This is the same situation in the case of the gauge KK5-brane A A A ds B r log MN ′ 2 ˜ σ the Yang-Mills gauge field remainscalculations, we non-trivial find and the contribu following gauge 5 G 1 + Gauge 5 brane solution, the Harrington-Shepard calorons forsame the with smeared the gauge harmonic NS5-bra function ( gauge KK5-brane solution istransformation found of in the appendix. gauge solution. Now we For perfor the gauge KK5-bra dimension two, only the one component of the duality equation ( conceivable result since the neutrala solution solution does in not type invo exact II solution, theory. the We neutral stress solution that is although a perturbative the solu symm this is just a constant and we choose this function The gauge field satisfies the vortex-like equation ( rule is simplified due tocalculations, the the same metric, reason the discussed dilaton in and the the prev NS-NS Neutral 5 constant parameter, the gauge 5 functions and it is a geometric solution. The function of its physical meaning is still5. obscure. A We summary will offound make the in a smearing figure comment and the T-duality relations for t JHEP11(2016)064 . In ′ (4.1) (4.2) is the gauge α ) itself matrix 8 ider the ]. Their G d E d, d 2 49 ( emarkably, × × O )[ 8 d R E -branes 2 2 d, d, 5 ]. This is given by ( ) where dim -corrections. This is ′ O 49 R [ α rtan subgroup. In this ]. with Wilson lines have five-branes of various co- on-Abelian gauge group r-dimensions, there are d , gonal parts of the Yang- ) takes the same form in G, 48  d T ) -brane solutions. The T- T ν T-dual p [ 4.1 2 2 ction of the metric, the NS- A µ on lt is true in all orders in + dim otic theories is determined by , A # ) reduces to d, d Tr( ( B R 1 smear along smear O − B − G, ) 1 ) is characteristic to heterotic theories. ν − + BG in type II supergravity is a 2 G ω 4.2 µ − − ) compactified on H + dim Smeared KK5-branes KK5-branes + G ω 2.1 Heterotic five-branes 1 – 15 – d, d 1 − ( -field. Although, the T-duality group Tr( −  O ′ B 4 G α BG T-dual -field, the Yang-Mills gauge field plays an important " , the generalized metric receives B ′ + 2 = α µν H g ]. The Wilson line fields break the SO(32) or = smear along smear 47 , ) covariance of field configurations, it is convenient to cons µν G 46 d, d ( . The generalized metric Smeared NS5-branes NS5-branes -corrected Buscher rule of the T-duality transformation. R O ′ H α are the isometry directions. The generalized metric ( is defined by 3 2 4 . The smearing and the T-duality relations for the heterotic G -field and the Yang-Mills gauge field that correspond to the Ca µ, ν On the other hand, when the Wilson line fields are absent, the n codimensions B duality symmetry in heterotic string theories compactified case, the T-duality group has been determined to be number of the U(1) sector associated with the Cartan subgrou been studied in detail [ 4 Monodromy and T-fold In this section, we study the monodromy of the heterotic 5 dimensions. order to clarify the Here where type II supergravities but the second term in ( is not broken and thestudy T-duality relies group on the S-matrix analysis of strings and the resu group down to a CartanMills subgroup gauge of the field gauge areU(1) group. gauge Higgsed The fields off-dia and which becomesNS originate massive. from the Kaluza-Klein In redu the lowe Figure 1 generalized metric in addition to the metric and the role in heterotic theories.utilizing the The heterotic supergravity generalized action metric ( in heter and defined through the metric and the does not change for all orders in analogous to the JHEP11(2016)064 ) is 2.1 ). (4.3) (4.5) (4.6) (4.4) ) and ′ 4.2 α ( O .        2 ase. Therefore ω -brane does not 1 2 2 ) is canceled. We − ) 4.2 ω 1 HK 0 0 − e II theory. For the solu- H + ( m in ( 1 . − ) in heterotic theories. 2 fore the monodromy becomes ′ ). However, since all the fields ic for example in Θ = 0 gauge. . 1 ′ α 0 ( α 2) φ HK ( , 2 O 2 − O in the two-dimensional base space, (2 2) T-duality transformation. There- e ω , O 1 θ O, = ]. In the following, we investigate the − (2 ∈ ) ) at (0) O 49 # ω -brane, since the standard embedding con- 4.2 1 H 2 t 2 2 HK − , then the generalized metric is evaluated as iτ 0 O π H − ,G 2 + ( − – 16 – # 2 -branes by using the generalized metric ( 1 -brane, the bulk gauge field is trivial and we can 2 ) = 2 2 1 → − 2 2 0 G π -brane and come back to the original point, namely iτ -brane, the situation is different. The gauge field 2 2 2 2 1 πσ (2 − 2 − = 0 0 = 0. Again, the modified spin connection is H " G θ ) has been introduced as it enters into the action ( ω HK M " 1 = K -brane solution is non-geometric, it is a T-fold and a con- A 1 − 2 2 4.2 0 0 0 0 = − O H ) and to the generalized metric. Then the generalized metric H H − -brane is a T-fold at least at For the symmetric 5 2 2 (0). Therefore we concludes that the gauge 5 4.2 ω K H 1 1 For the neutral 5 0 0 − − For the gauge 5 H H ) =        π -brane. (2 2 2 ) = H θ ) and its monodromy structure is the same with the symmetric c -brane. ( 2 2 -brane. H 2 2 4.3 ), this is given by 3.20 dition is satisfied, we can choose a gauge where the second ter it does not contribute to ( contributes to the generalized metric through ( Gauge 5 in the gauge solutionthey do do not not inherit dependtrivial. multi-valuedness of on This the the can geometry. be angle There In seen this by evaluating gauge, the we generalized have metr When we go around the center of the 5 if the angular position changes as always choose the gauge where given by ( we find that the neutral 5 tion ( in the same waymonodromy as structures the of the Yang-Mills heterotic gauge 5 field [ Symmetric 5 find that the generalized metric is the same with the one in typ This implies that the monodromyfore, is although given by the the symmetricsistent 5 solution to heterotic string theories. Neutral 5 The spin connection term in ( This implies where exhibit non-geometric nature. JHEP11(2016)064 - - ′ 2 2 α ric. auge pe II where s of the F -brane is -field and 2 2 -branes in ∧ 2 2 B F like equation Tr ′ α = utions and perform -flux. . d the U(1) isometry 1 H -brane solutions, they (3) ition is satisfied up to 2 2 ˆ H metric nature due to the d n to the NS5-brane world- on in type II theory. We ow the nature of a T-fold. ndition is satisfied, the ntons. The resulting gauge -branes. We calculated the ified hat the neutral 5 n heterotic supergravity. A 2 2 in three dimensions. For the lting metric, the dius limit. The dilaton and mptotic region. rane core after the smearing mmetric and the neutral 5 gical charge of instantons of andard embedding condition s modified by the corrections. olution. For the gauge KK5- ion of these solutions. Due to -field and the dilation are given en by the KK5-brane in type II ons are the exotic 5 ty for the smeared caloron. The s gauge sector which appears in B -brane solution, we found that the fields 2 2 . The three different half BPS five-brane ′ α 2) T-duality transformation. Therefore they , ], the symmetric solution seems to be exact in (2 – 17 – 45 O -field. For the gauge 5 -corrections. There are also higher derivative correction ′ B α ). The solution is given by the purely geometric Taub-NUT met ′ -field is not excited which is the same with the KK5-brane in ty -corrections. The result is summarized in table α ′ ( order are known. They are the symmetric, the neutral and the g B α ) are determined through the Bianchi identity ′ O ′ α -brane in type II theory. The gauge field satisfies the vortex- α 2 2 ( O . On the other hand, for the neutral and the gauge 5 ′ α -corrections in heterotic supergravity, the Buscher rule i ′ We then introduce another U(1) isometry to the KK5-brane sol We introduced the U(1) isometry along a transverse directio Following the general discussion in [ A few comments are in order about the solutions. We introduce We next studied the monodromies of the three different 5 α For the gauge solution, theprocedure. geometry This is property ill-defined isbrane near carried solution, the the over b to the T-dualized s neutral solution, we find thattheory the at T-dualized least solution at is giv the dilaton are nothingdemonstrated but that the the Yang-Mills onesagain gauge for and field it the satisfies is KK5-brane given the by soluti st the solution to the monopole equation theory. However, the geometry is well-defined only at the asy corrections in the modified Buscher rule cancel out. The resu curvature square term in the same order in NS5-brane solutions. TheseYang-Mills are gauge distinguished field and by the the charge topolo associated with the mod generically receive For the symmetric solution, where the standard embedding co volume and explicitly performedthe the T-duality transformat solutions in this terms of In this paper wespecific studied feature the of T-duality heterotic chainsthe supergravity of linear is five-branes order the i in Yang-Mill the 5 Conclusion and discussions heterotic theory. For the symmetric solution, the metric, the second T-duality transformation. The resulting soluti by that of the 5 in two dimensions.a We gauge found transformation. that the For the standard neutral embedding solution, cond we found t generalized metric for these solutions. We found that the sy branes have a monodromyare given by T-folds. the On the other hand, the gauge solution does not sh to the gauge NS5-brane byfield the is smearing just procedure of thethe the Harrington-Shepard metric insta at calorons in the small ra the same with themulti-valuedness one of in the type II theory. They exhibit a non-geo the right-hand side is given by the topological charge densi do not show the multi-valuedness and they remain geometric. JHEP11(2016)064 ) 4.2 ic ( ]. Similar to ] in heterotic 23 67 , ]. The heterotic – ]. Comprehensive ] and the inclusion 22 63 65 ]. There are several 49 53 ]. 52 useful discussions and ixed-symmetric tensor 50 , er 1 FRC Grant R-144- iversity Research Grant c nature of string theory ]. It is also interesting to 51 indicate any inconsistency ooft-Polyakov type instead ]. We believe that globally 64 I theory [ ite down asymptotically flat o objects. Analogous to the of the heterotic supergravity ) specifies the “cutoff” point 42 ic branes [ is is true [ s [ -branes. 2 2 -corrections [ avity [ ′ related property of the solutions e. We stress that there is another upergravity [ 3.17 α -field behave well-defined near the Notes standard embedding Type II solution ill-defined near the center B -branes discussed in section 3.2 seem in ( 2 2 ′ µ -brane is not well-defined as the stand- α 2 2 -branes of BPS monopole type would show 2 2 – 18 – ) ) ′ ′ α α -branes of all types. This is characteristic to the co- -exact ( ( ′ 2 2 O Valid order in α O Indeed, the scale . Properties of the heterotic 5 ]. In the gauge NS5-brane solution of co-dimension three 6 40 , ) multiplet of 7-branes in type IIB string theory [ 38 Z , Table 1 ]. Although the spin connection term in the generalized metr geometric Geometry T-fold T-fold 57 – 54 -brane solutions exist even in heterotic theories. 2 2 Gauge Type Symmetric Neutral -corrections [ We note that the most tractable way to study the non-geometri ′ An example is the SL(2 6 α -brane is expected to be a source of the non-geometric flux or m 2 2 solutions is the double field theory construction of supergr The authors would likecomments. to The thank work T. of Kimurafor S. Young and S. Researchers. S. is The supported Mizoguchi work000-316-112. in for of part M. Y. by is Kitasato supported Un by NUS Ti study the world-volume effectivetheory. action We for will the come non-geometr back to these issues in futureAcknowledgments studies. is a conjectural one,and the the supersymmetry double transformation field law theorystudies analysis are strongly presented suggest in that5 the th generalized geometry with where the effect ofwell-defined the 5 next duality branes is not negligible [ studies about double field theory formulationof of heterotic s the gauge solutions of the smeared caloron type, the 5 based on the BPS monopole type, the metric and the to be ill-defined atof asymptotic the region. solutions but HoweverD7-brane the this in general does type property not IIB of string co-dimension theory, tw the exotic 5 alone object. We needglobally other well-defined co-existing solutions. branes in order to wr core of the five-brane.better The physical gauge interpretation KK5-and of 5 is T-dualized the solutions. logarithmic A behaviour of the 5 dimension two objects and found also in the solution in type I resulting geometry is ill-defined near theco-dimension center three of the solution bran based onof the the BPS monopole smeared of caloron ’t [ H and they are in the T-duality multiplets in lower-dimension JHEP11(2016)064 (A.2) (A.1) , along the -direction  4 , , ) x ). When we 2 I 2 , , q
-field is taken 3 ) T 2 ∂ 2
T B A.1 ) x 2 r/µ
) + mnpq r/µ ε 1 2 ) along the same way 1 2 log( T ˜ σ r/µ , -direction in the gauge ′ 2 1 2 ˜ σ − 2 log( 3 ] ˜ x α σ ′ 3 x 2 ˜ 2.22 -brane and it is obtained σ 1 = − heterotic T-duality trans- log( α solution is obtained by the Θ( g e T-duality transformation 0 dx 2 ˜ σ − 0 ˜ vant non-zero component of σx ˜ h φ on ( olution before we write down 0 2 (3) the smeared gauge KK5-brane mnp smeared gauge KK5-brane. It − 2 ˜ h . − ˆ + Θ r 0 H ) e 2 2 2 4 ˜ h r T 2 2 , − dx 2 -direction instead of the ). The result is ) + [ r I 3 x 0 2 0 4 − φ x φ T 2 0 -smeared gauge NS5-brane. The result- 4 + 1 e 3.13 log 3 φ = − 1 x 2 x e 2 2 1 e = T I − 1 I = = x 1 + ( T ′ I 2 2 n
– 19 – ,A x ) , 3 ( dx -smeared gauge KK5-brane. ) ′ T  4 I , φ 2 2 m x r/µ 0


φ , ) ) ) 2 dx n ′ − log( n e ′ ˜ σ dx r/µ r/µ r/µ 2 ˜ σ -direction to the m 4 m δ x 2 − log( I log( log( log( dx n 0 2 1 ˜ σ ˜ σx ˜ 2 σ 2 2 ˜ ˜ ˜ h σ σ + . In the gauge solution, since the non-zero components of the ˜ σx mn ). The explicit form is as follows: − j 2 = 2 δ − − − ) r I come from the Yang-Mills Chern-Simons term, the 2 -direction on the defect gauge NS5-brane solution ( 0 dx 0 0 4 i 4 x ˜ h ˜ ˜ h h 3.26 0 + x φ dx j 2 2 2 2 mnp + ( r r r , φ ij − = Θ. When we perform the heterotic T-duality transformation ˆ 4 4 4 dx η e 2 H i ) 34 1 = = = = = 0 mn dx x B ¯ σ 1 3 2 4 -direction, we obtain the ij -flux 3 − η A A A ds MN x = ( H = = B 2 2 r m -field is ds A B -direction for the defect NS5-brane solution, we can obtain 4 KK5-brane solution ( A.1 Defect gaugeThe NS5-brane defect solution gauge NS5-braneby is smearing the the co-dimension two directions two of gauge the NS5 gauge NS5-brane soluti As we mentioned above,formation the along solution the is obtained by taking the solution shown below. A.2 Smeared gaugeThe KK5-brane smeared solution gauge KK5-brane solution is obtained by smearin x take the heterotic T-duality transformation with the to obtain the smeared gauge NS5-brane solution ( modified along the ing solution is a brane of co-dimension two. By performing th to be a constant.the For the defect NS5-brane solution, the rele where In this appendix, weis introduce convenient the first explicit to solutions introducethe of the smeared the defect KK5-brane gauge solution. NS5-branesmearing The s procedure defect along gauge the NS5-brane A Smeared solutions for gauge type JHEP11(2016)064 ] ). ]. 3.26 ] ommons SPIRE -direction (1995) 109 3 IN , x ][ ]. , B 438 , arXiv:1602.08606 The solution is dif- [ SPIRE (1999) 113 for nongeometric , arXiv:1503.08635 IN , , arXiv:1410.8403 ][ [ 318 -brane geometry 2 2 ]. 5 (2016) 013 redited. Nucl. Phys. ]. duality along the (2016) 060 , , arXiv:1506.05005 ]. [ 02 ]. ]. ]. 05 (2015) 1 -brane solution as we see in ( SPIRE SPIRE 2 2 IN Algebraic aspects of matrix theory on IN SPIRE Phys. Rept. ][ SPIRE SPIRE SPIRE JHEP ][ IN , , ]. 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