Jhep07(2020)086

Home , Chris Hull, Gauged supergravity

JHEP07(2020)086 , Springer July 14, 2020 June 18, 2020 April 14, 2020 : : : and then further 4 b T Published Received Accepted [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP07(2020)086 and Stefan Vandoren b [email protected] , 0) in ﬁve dimensions. The eﬀective supergravity theory is , Koen Stemerdink 2 , b 4 , . 3 = 6 2003.11034 N The Authors. c Black Holes in String Theory, Supergravity Models, Supersymmetry Breaking

Eric Marcus, We consider 5D supersymmetric black holes in string theory compactifications , a [email protected] Utrecht University, 3508 TD Utrecht, The Netherlands E-mail: [email protected] The Blackett Laboratory, Imperial CollegePrince London, Consort Road, London SW7Institute 2AZ, for U.K. Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, b a Open Access Article funded by SCOAP ArXiv ePrint: gravitational Chern-Simons terms inhole the entropy. action and the resulting corrections to the black Keywords: mass parameters. Infrom this various theory, ten-dimensional weduality brane study twist configurations. black holes to with Forsupersymmetric be three each solution a of charges black transformation the that holediscuss twisted that descend the we theory quantum preserves corrections choose with arising the partially the from broken solution, the supersymmetry. twist so to We the that pure gauge it and mixed remains gauge- a that partially breakcompactify supersymmetry. on We a compactifysupersymmetry circle type with ( IIB a on given duality by twist a to Scherk-Schwarz reduction givemoduli with space, Minkowski a and Scherk-Schwarz vacua supergravity the preserving potential lift on partial of the this to string theory imposes a quantization condition on the Abstract: Chris Hull, Black holes in string theory with duality twists JHEP07(2020)086 4 51 33 4 T 39 12 10 44 24 27 42 35 47 – i – 37 49 (5) 4 so 42 31 11 ∼ = 46 23 (4) 27 14 26 52 35 51 53 usp 5) and its algebra , 18 20 9 47 19 7 1 44 C.1 The D1-D5C.2 system The F1-NS5C.3 system The D3-D3 system B.1 The groupB.2 SO(5 The isomorphism 6.1 Quantization of6.2 the twist parameters Orbifold picture and modular invariance 4.3 Preserving further black holes by tuning mass5.1 parameters Corrections to5.2 Chern-Simons terms Corrections to5.3 black hole entropy One-loop calculation of the Chern-Simons coefficients 3.8 Gauged supergravity3.9 and gauge group Kaluza-Klein towers 4.1 The D1-D5-P4.2 system Dual brane configurations 3.3 Massive field3.4 content Mass matrices 3.5 5D scalars 3.6 5D tensors 3.7 Conjugate monodromies 2.2 6D scalars 2.3 6D tensors 3.1 Monodromies3.2 and masses Supersymmetry breaking and massless field content 2.1 Ansätzefor reduction to 6D C Scalar and tensor masses after Scherk-Schwarz reduction A Conventions and notation B Group theory 6 Embedding in string theory 7 Conclusion 5 Quantum corrections 4 Five-dimensional black hole solutions 3 Scherk-Schwarz reduction to five dimensions Contents 1 Introduction 2 Duality invariant formulation of IIB supergravity on JHEP07(2020)086 4 0 ]. × , T 4 2 , , 3 4 [ , 1 Spin(5); S Spin(5) 5) duality = 6 × / , × 5) 3 N , 4) CFTs on the CY , ) provides a set-up 1 ]. S 1 × ] in an M-theory setting 3 8 K (or = 4 supergravity). The entropy 1 S N ] or in F-theory on × 2 [ 4 3 T CY 4) CFT dual to the near horizon geometry , – 1 – ], which are the lifts to string theory of Scherk-Schwarz Spin(5) local symmetry that is introduced in some for- 5 × 3) as they have less supersymmetry [ ]. Such compactifications allow for partial supersymmetry 7 K , ] and references therein. IIB supergravity compactified on 6 17 (or – 2) supergravity in six dimensions which has a Spin(5 4 , 9 , = 8 supergravity (or 1/4 BPS in T 7 – N 5 ]. This system has U-dual formulations in terms of F1 and NS5-branes, = (2 1 N Scherk-Schwarz reduction of supergravity theories has been extensively studied in the This work is a follow-up to the ideas proposed earlier in [ In this paper, we consider a different way to reduce supersymmetry, namely string com- It is interesting to consider extensions of this to black holes in theories with less su- 4) CFT. These CFTs are considerably more complicated than the (4 , symmetry. The maximal compactwe subgroup shall of refer this to global thisnot symmetry as be is the confused R-symmetry Spin(5) with group. themulations (Note Spin(5) of that the this theory.) global The R-symmetrySpin(5). 6D should scalar fields We take values will in be the coset interested Spin(5 in the Scherk-Schwarz reduction of this theory to five some supersymmetry. The currentbreaking work and studies the string macroscopic vacuamicroscopic with supergravity description description partial of supersymmetry of the black dual holes CFTs in is such left vacua. forliterature; future The see study. e.g. [ gives maximal possible to consider the effectthe of microscopic the aspects twist of on these the black corresponding holes. CFT and soin to which investigate supersymmetry isa completely broken. well controlled Completely broken situation, supersymmetry and is for not that reason we will focus on the twists preserving choose the twist inducing theleaves supersymmetry the breaking to original be system auntwisted duality theory of transformation remains branes that a invariant, solutionof and of branes so are the invariant the twisted under theory.massless 5D the and The twist, black the fields so hole same sourcing that solution solution the the remains fields system of appearing as the a in the solution solution of remain the twisted theory. This makes it reductions in supergravity [ breaking and include stringbe vacua preserving the focus no in supersymmetry thissupersymmetry. at work). all We This (though investigate gives this 5D riseto won’t supersymmetric to 10D black 5D Minkowski holes systems vacua in preserving of these branes theories in that lift the string theory picture. An important point is that we In these cases, the microscopic(0 field theory dual to thesymmetric black product hole of horizon geometry is a 2D pactifications with a duality twist [ of the black hole [ and in terms of intersecting D3-branes. persymmetry. Black holesdimensions in can compactifications be preserving constructed eight in supersymmetries M-theory on in five The D1-D5-P system in typefor IIB the string study theory of on macroscopically. BPS It black can holes be in describedhole solution as five of an spacetime 5D asymptotically dimensions, flatcan both three-charge be 1/8 microscopically computed BPS microscopically and black from a 2D (4 1 Introduction JHEP07(2020)086 ] ) ) p p 0 5 , Z Z Z Z 2 = 8 , ). If 5; 4 , Z , on the ]. N 6 5; 9 , , ). A key re- = 8 πR/p are zero, then Z ]. Such a twist i N 5 5; with a U-duality m , 0 vacua [ , 1 2 S , 4 × , ) is a local element of followed by a reduction 4 6 z ( , subgroup of Spin(5 4 T , which requires that it is g 4 p T ]. This uses ansätzeof the Z T = 8 5) on the theory, but only the , N 12 , asymmetric orbifold. 5) supergravity duality symmetry 11 of the parameters p , , the fields pick up a monodromy ]. r Z 20 which become mass parameters in the πR 4 combined with a shift by 2 , which requires that it is in a GL(4; + 2 4 coordinate and 4 , m , where the theory is quotiented by the 3 T T z 5 1 T S , m → – 2 – = 4 is the untwisted reduction to 5D 2 has a Spin(5 ]. Here we draw on these and the results of [ ]. The moduli space is the scalar coset space ), then this constructs a T-fold background, while z 4 r 18 , m Z ), in which there is a Scherk-Schwarz potential for ), imposing a ‘quantization’ condition on the twist 17 T 1 , µ is the Z 5; x ]. These have been worked out in detail for compacti- m 16 , ( which is required to be an element of Spin(5 z 5 5; ] identified under the action of Spin(5 , ψ 2 Z M / that are equal to zero: if = 0 is the twisted reduction that breaks all supersymmetry). i r ) where m µ x ), then this corresponds to compactification of the IIB string on a 5). Such a reduction gives a consistent truncation to a 5D gauged ( Spin(5)) , Z ψ ] (where the case ) is a symmetry. Reduction of the theory on a circle with a duality ]. ]. The monodromy will then generate a ) × 5; Z ) is compact, i.e. it is an element of the R-symmetry group, then the 5 z , 12 . Furthermore, at this critical point the construction becomes a z . If the monodromy acts as a T-duality of ( p 19 ( 5; acting on the IIB string on 1 g , )[ , g Spin(5 5 S Z ) subgroup of Spin(5 [ Z M i ∈ 5; ) = , ) m 4; ) by quantum corrections [ , [(Spin(5) , z Z / µ πR supersymmetry is preserved. This yields 5D supergravities with x 5; 5) 5). On going round the circle (2 ( r , , , g Spin(5 ˆ ψ A point in the scalar coset will be a minimum of the scalar potential giving a stable There is still an action of the continuous group Spin(5 The lift of these supergravity reductions to full compactifications of string theory in- If the twist bundle over = = 2 = 8 supergravity to 4D with four mass parameters and 4 M ∈ for some integer generalized orbifold of IIBgenerated string by theory on circle. When the monodromy is a T-duality, this is a subgroup of Spin(5 T in an SO(4 for general U-duality monodromies this is a U-fold [ Minkowski vacuum if and only if it is a fixed point under the action of the monodromy parameters subgroup Spin(5 twist introduces a monodromy the monodromy acts as a diffeomorphism of that, on the levelSpin(5 of the fullSpin(5 string theory, is brokenquirement to for there the to discretelies be U-duality in subgroup a the lift U-duality group to Spin(5 string theory is that the Scherk-Schwarz monodromy volves a number of subtlefications features of [ IIA stringon theory a on circle K3 or withfor the a our heterotic duality construction, string twist which on in istwist IIB [ around string the theory circle. compactified on Type IIB on Minkowski vacua [ supergravity, and the case This reduction is aN straightforward generalization of the Scherk-Schwarz reduction of 5D potential is non-negative and has stablecan five-dimensional be Minkowski vacua specified [ by fourreduced theory. parameters The amount of supersymmetrythe number that of is parameters preserved in theN vacuum depends on type Spin(5 M supergravity theory for thethe fields scalar fields and mass terms are generated for all fields charged under the monodromy. dimensions, which has been considered previously in [ JHEP07(2020)086 ] ]. ), 5 Z = 4 = 4 28 = 4 , 4; ], the 5) [ , N N 27 , N 31 ]) that the at the four- lies either in 25 Spin(5 R R ∈ ∧ k R ∧ A 1) BPS supermultiplets, must be accompanied by , z Spin(5) subgroup. For 2) theory but not in the × , Spin(5) and 2) or (1 , × 1) to distinguish these two ]. Put differently, the quotient intro- , ]. For non-perturbative monodromies, = 4 (0 23 = 2 theory the supersymmetrization is , 17 [ N N 22 Spin(5) , ; the results are relevant for understanding M = 6 supersymmetry to be preserved, three of ∈ 17 2 2) supergravity. We construct mass matrices , 2) and (1 – 3 – , = 2 supersymmetry to be preserved, one of the N R lies in an SU(2) subgroup of the R-symmetry. N 5), the monodromy is conjugate to an R-symmetry , R in fact lies in an SU(2) , we work out which choices of Scherk-Schwarz twist at the two-derivative level and ]. Similar calculations have been done in different setups, SU(2) subgroup (with one SU(2) factor in each Spin(5)). 4 R . For i for some × F 28 is a perturbative symmetry (i.e. a T-duality) in Spin(4 m , 1 ∧ ]. Lastly, for − = 6 theory. In the 27 M F 21 N ] lead to phases dependent on other brane wrapping numbers. kRk ∧ A 24 = , 17 M conjugate to a given monodromy is specified by four angles, which are , we perform the Scherk-Schwarz reduction to five dimensions, starting 3 R term can be supersymmetrized in the ]. ]. Our results are in agreement with these claims. That is, we find corrections . While the general techniques and results are known in the literature [ = 2 supergravity [ 4 R 26 T 30 N , ∧ 29 In section As a by-product of our analysis, we present in detail the supergravity reduction of type Quantum effects can lead to corrections to the coefficients of the five-dimensional As mentioned above, if Regarded as an element of Spin(5 R 1) theory, nor in the ∧ , from the maximally supersymmetricand 6D decompose (2 thetruncating 5D to field the contentdimensional massless into theories. sector, massless we and In can massive section embed multiplets. known BPS By black simply holes in these five see [ IIB on explicit relation between the 10D andedge. 6D fields We has not present been thiswhich given, calculation to black the in holes best survive section of which our knowl- twist in subsequent sections. to the Chern-Simons coefficientsand modify therefore also the the black entropy.coefficients hole We compute and solutions the the in quantum resulting corrections supergravityand to modifications [ the the Kaluza-Klein to Chern-Simons modes the from blackfor the 5D circle hole compactification, entropies using from results in supergravity the literature derivative level. ThereA have been(1 indications in theknown literature [ (see e.g.only in [ the cases where supersymmetric Chern-Simons terms are expected. The corrections duces phases dependent on bothon the the momentum and charges the of winding the number statethe on under arguments the the of circle, action [ and of Chern-Simons terms the theory in thebe Minkowski modular vacuum invariant, and is further anFor modifications perturbative asymmetric are monodromies, needed orbifold. the to shift in achievea In modular the shift general circle in invariance. coordinate this the coordinate will of not the T-dual circle [ a Spin(5) subgroup or aThese SU(2) two options leadmassive to theories sectors. that have Wetheories, the use reflecting same the whether massless the notation sector,using massive but (0 the states differ terminology are in of in their the [ (0 parameters must be zero so that transformation: The rotation given by the fourparameters parameters must be zero sosupersymmetry that to be preserved, two of the parameters must be zero so that JHEP07(2020)086 ) m 1 ∗ (2.4) (2.1) (2.2) (2.3) , y ˆ µ x 2

= (ˆ (10) 5 M F

X 1 4 4 − . 2 T of the twist.

i (10) , 2 m (10) 3 C ! F d

ˆ m µ ∧ Φ A e mn 1 2 g (10) 2 mn 5) duality symmetry group. The g , − B 5) symmetry are manifest. We do 1 2 2 , | ˆ a n ν + . 4. We now present the ans¨atzethat d A | ˆ m µ (10) 2 2Φ (10) 5 A e ,..., B F 1 2 d ∗ ˆ n ν , mn A ∧ g − = = 1 – 4 – 2 + (10) 2

mn m (10) ˆ ν 2 g (10) B 5 ˆ µ C d (10) g 3 F 1 2 a 4 H /

, 1 − − , Φ − 5 and 4 g − (10) 3 e (10) (10) (10) 2 2 4

F 1 2 B C C , we discuss how to embed our supergravity model in string ,..., = ∧ − 6 2) supersymmetry and a Spin(5 = d = d = d 2 | , = 0 MN (10) 3 g µ H dΦ (10) (10) (10) 3 3 5 | , we study one-loop effects by integrating out the massive fields. We = (2 ∧ 1 2 F F H 9, ˆ 5 − N (10) 4 C ,..., (10) 1 2 R = 0 − = M In our compactification to six dimensions, the coordinates split up as IIB L with we use in ourten-dimensional reduction. metric as In order to arrive in Einstein frame in 6D, we decompose the supplemented by the self-duality constraint The superscripts (10) indicate that the fields live in 10 dimensions. The field equations are where the field strengths are given by We start from type IIBLagrangian supergravity. read Written in Einstein frame, the bosonic terms in the goal of this sectionIIB is origin to of write the thisthis six-dimensional supergravity explicitly fields theory for and in the the scalar a Spin(5 and form tensor in fields. 2.1 which both the type Ansätzefor reduction to 6D theory and discuss the quantization conditions on the parameters 2 Duality invariant formulation ofReducing type IIB IIB supergravity supergravity on on aThis four-torus gives theory six-dimensional has maximal supergravity. By tuning the mass parameters,solution we (e.g. can we find find twists thatholes). the preserve twists more In that than section preserve onecompute both black the the hole corrections D1-D5-P and toholes. the the F1-NS5-P Chern-Simons Finally, black terms in and to section the entropy of 5D BPS black preserve the D1-D5-P black hole, the F1-NS5-P black hole and the D3-D3-P black hole. JHEP07(2020)086 (2.8) (2.9) (2.7) (2.5) (2.6) decomposes as n y d (10) 2 ∧ ) does not properly B n m 2.1 y = d m m < n m > n . parametrizes the volume of mn 4 B , we parametrize in terms of , To find the fields that descend φ for for for , 1 2 n , . mn y g d + , , . 1 2 (10) (10) 2 2 ∧ m , C C y kn 2 m d d 1 2 1 2 1 2 1 d A y √ , , , ∧ ∧ d ∧ 2 2 2 − km 1 1 1 ˆ 2 µ , √ √ √ ) A x mn 2 (10) (10) 2 2 , , ) by 1 d ~ − φ , so the scalar B · 2 2 √ km B B , are given by 4 k 1 1 1 2 b 2 φ ~ A √ √ − ˆ µm – 5 – 2 e 1 ( m , , √ e = d = d b − B + ~ ~ φ N 2 2 · , , m < n 1 1 k = m ( + 2 2 √ √ b X ~ X k

ˆ µ mn . H 2 ,mn . We ﬁnd the action

(6) and 1 b mp (for H (6) g 1

(6) 3 P Φ

and , P ,mn − 4 (6) ˆ µmn 3 a e . φ (6) 2 2 F , in terms of ﬁelds that we call H 1 2 ∗ − B ,rs 1 (6) (recall that the metric on 2! e 3 d mn − (6) 3; 3 F 1 2 2 C 12 | = P ,rs and ∗ (6) . By substituting these expressions , 3 − C F qs mn 2 (6) 3 ∗ mn

g g − ,mn and ˆ (6) µmn P 3 d B pr | ∗ F (6) H g , 2 ,mn 1 2 . These ﬁeld equations are quite unwieldy, (6) (6) 1 3; 2 C 34, we ﬁnd the equations of motion for the F P mn d −

, ˆ µmn (10) 4 ) for three of the six field strengths results in A ) for . Here the first subscript indicates that these 2 mnpq Φ | 12 24 H C , – 7 – ε e 4 13 , i , B 1 4 (6) 2! φ indices, each component of this equation contains 1 2 3 φ 2.14 2.14 g d F + s | − √ = 1 4 = 23 2 1 2 = 2 | (6) 2; 1 | a − and R = d (6) and their duals 2 3; 3 | mn r | ,mn P (6) 1 = d 2Φ ,mn (6) 3 dΦ (6) H e 3 | | P F ) for 1 2 (6) 1 2 3; 1 and P − − 14 , 2.13 (6) 3 = . F s L 2.3 = purely in terms of the (dual) field strengths 1 − (6) (6) e 3; 2 respectively. (6) 2 P 34. For this, we again use the reduced self-duality constraint. The relevant )). Consequently, solving ( R , , is then one of the two-forms that arise from compactifying the ten-dimensional (6) 2; 3 12 ). For example, 2.5 24 , R (6) , 3 (6) 2; 1 F R 2.13 and = Next, we solve the six equations in ( First, we introduce a new notation for the field strengths that we plan on retaining: = 23 (6) (6) 2; 2 3; 1 Note that the absoluteon the values apply torus. both For to example, the 6D Lorentz indices and to the indices The field content ofof maximal their six-dimensional origin supergravity infor contains type these 25 IIB, scalar scalars. these fields are byThis In Φ, using yields terms the methods and ansätzedescribed in the previous section. down this awkward versionversion of of the the action action forfields the here. in six-dimensional subsection tensor Instead, fields and we their write interactions with down2.2 scalar a more 6D elegant scalars terms of the fieldin strengths the components oftensor ( fields that don’t descend from thebut RR with four-form some careful bookkeeping they can be integrated to an action. We will not write where 4-form. Similar expressions canR be found for sometimes drop this labelfor when we these are field talkingfrom strengths about ( all in three terms of of them). the The corresponding expressions two-form fields can be deduced unwieldy expressions. We choose notgive to a write step-by-step outline down these of expressions the here, way we but use insteadP them to to find anare action three-forms, for and the the 6D second tensors. subscript labels the three distinct field strengths (we will components are Due to the summations overa the linear combination ofgiven all by the ( dual field strengths mn JHEP07(2020)086 ]. V × U 5) 5) , , 31 V → 5) in (5 , (2.17) (2.18) (2.19) 5) and so , is written H Spin(5 ). The exact Spin(5)) [ is a conjugation × as X a V ∈ B.1 H a T . . # is the vielbein that is used under global Spin(5 i ) exp (Spin(5) T is written with upper indices: H pq Spin(5) symmetry. In its / i × U B H φ × 5) [Tanii] # d ]. The matrix 5) and it transforms as , H U , 4 32 =1 the transformation matrices i U X . mn C , see appendix mn 2 τ C T + H where 0 − ˆ µ mn H → Spin(5). We now define the Spin(5) H X ∂ pq C 1 5) that span the subspace of × Φ , C − -frame, " + d τ . The inverse of (5 H [Tanii] ˆ B µ Spin(5)). The precise expressions for these B , so mn exp mn D ∂ T XU × ) in B T × Spin(5) mn ( from the 25 scalar fields so that the two La- = Tr – 8 – U mn B ! ∈ 1 8 V V d 2.19 B ) 5) and a local Spin(5) a CD mnpq x = , 4 A (ˆ ε mn H s . All the scalar fields appear under the same name as − , ≤ T 1 8 (Spin(5) (for the definition of C W L / A τ mn 1 mn is the vielbein that is used in [ + mn X U , that transforms as . B.1 5) 5) indices, but we will need them later on. With indices, m

) and ( V 2.16 × = ’s and the 2.16 = exp . V Spin(5) groups appear (unless mentioned otherwise). T ), with V AB × ). are also written in x 5). and it transforms as (ˆ , H ], and W = Because we construct our vielbein ( We now specify the way we build In order to do this, we construct a generalized vielbein (or coset representative) W AB 2.16 In this section we suppress Spin(5 For the convenience of the reader, it might be useful to mention how this convention is related to those 31 1 2 1 H V − -frame, i.e. it satisfies as H of other authors. Thein following [ relations hold: matrix that is defined in appendix in ( and Spin(5) Here the that generates the cosetgenerators Spin(5 are given in appendix τ construction is as follows: In this formulation,Spin(5 the Lagrangian is manifestly invariant undergrangians the ( U-duality group from the scalar fields.U This vielbein isSpin(5) an invariant element field of Spin(5 transformations. These 25 scalar fieldsThe together action parametrize above the hascurrent coset a form, Spin(5 global these Spin(5 symmetriesthat are makes not both visible, symmetries so manifest. we will now write this action in a form strengths in ( JHEP07(2020)086 ) = 0 2.21 ,a (2.23) (2.20) (2.21) (2.22) (2.24) (2.25) (6) 3 . G , we can ) 1 = 0, where − ) c d a b b ,A (6) 3 − a G = ] )( 5), and their field V d 32 − , . . c ,..., 31 ,b 5) (6) 3 , +( , . The advantage of ( 1 G = 1 − ) ∧ (6) 2 b a 2.1 ( ,a B 5 blocks as + (6) 3 Spin(5 − a ,a G × (6) 2 , . ∈ )( ab A a d (6) . 2 B L + (6) 3 A c 1 2 ! ˜ ,C G a (( ,a − ∧ 2 1 (6) 3 (6) 3 (6) 2; 3 ,b ˜ ,a G G (6) 3 = (6) 3 ] to construct an action that realizes the full ,R G

G – 9 – ,U 33 1 2 = (6) 2; 2 ,L ∧ ∗ ,B − ) (6) 3 ,A 1 ,a (6) 3 ,R 5) is a symmetry of the action. The full symmetry (6) 3 = G − , G ) t G B b (6) 2; 1 L A ab − R as U a K )( Spin(5 ab 1 2 = d so that we can write the Lagrangian in the more compact form L → − ⊂ ,a − ,b (6) 2 . The Lagrangian for these fields reads [ c (6) 3 = ( It is a common feature of supergravity actions in even dimensions A ,a ,A G and t 2 (6) (6) 3 − 1 A G L ab ab ) is exactly equal to the one that we find by explicit reduction from − L ) K are functions of the scalar fields. We define a set of dual field strengths b = d + 2.21 + ,b ab ,a (6) 3 a 3 (6) L G )( G d ∗ 5) duality symmetry acts on this ten-component vector as = 0. We can combine these in the more compact notation d , + ab and a c K (6) 3 (( ab ˜ 1 2 G = K is defined as When we decompose our coset representative in 5 a = ,A (6) 3 (6) 3 K ˜ that only a subgroup ofcan the use duality the group so-called issymmetry doubled a group. formalism symmetry In of [ order theform-valued to fields action. do as In this, well such as onecontain a cases, needs more constraint one to that degrees introduce makes of twice sure freedom the that than the original the doubled amount original theory of does theory. not the Lagrangian ( type IIB supergravity usingis the that ansätzegiven in we have subsection made the duality symmetryDoubled manifest. formalism. Now, by making the identification Only the subgroup GL(5) group is only manifest on the level of thewrite field the equations. matrices The Spin(5 In this notation, we writeand the Bianchi d identities and theG equations of motion as d Here G The field content ofgauge maximal fields. supergravity in Collectively, six westrengths by dimensions denote contains these five fields 2-form by tensor 2.3 6D tensors JHEP07(2020)086 (3.3) (3.1) (3.2) (2.26) (2.27) (2.28) , dimensions. 5) , D . These doubled to field strengths a a (6) 2 is sometimes called Spin(5 (6) 3 ˜ A ˜ G ∈ M B = d A a . (6) 3 ,B ) are invariant under the full ˜ , -dimensional theory. For more G (6) 3 ,U ) D G µ . ,B 2.27 x = 0. Furthermore, these fields are (6) 3 that satisfy the Bianchi identities ( on the circle, which has periodicity ,C + 1)-dimensional theory that trans- ∧ ∗ G ψ (6) 3 z ,A + 1)-dimensional supergravity theory ,B B . D G ,A , (6) 3 (6) 3 that is compactified to (6) 3 1 A 1 ∗ D G G − G − U G g ∗ πR Mz BC 2 → AB M H AB in the ( g gMg H ,A H (6) 3 ˆ . – 10 – 1 4 ψ = = AB T τ 0 0 − ) . The Lie algebra element 1 = M . This ansatz is not periodic around the circle, but M = − G ,G ) = exp G U ,A B ∈ , z (6) 3 D µ ) and the constraint ( G = ( x M 1 ( e (for scalars, this is typically a non-linear realization, while T − ˆ ψ 2.26 (doubled) − t U G = U L ∈ ] and references therein. A conjugate mass matrix M CD g 34 H , a dependence on the coordinate C 17 A ˆ with – ψ ˆ 9 T ψ , − g 7 – U , given by 5 → are now treated as independent fields. We write down the doubled Lagrangian as , gives a conjugate monodromy lies in the Lie algebra of ˆ a ψ πR G → ). Both the action ( 5) duality group. This can be seen directly from the way that these transforma- (6) 2 ∈ M = 0 and the equations of motion d , ˜ + 2 A AB g We apply this formalism to our 6D tensor fields. We promote the z ,A 2.21 H (6) 3 ' G with where picks up a monodromy the mass matrix becausedetails, it see appears [ in mass terms in the forms as some fields such asansatz the then metric gives inz Einstein frame will be invariant). The Scherk-Schwarz 3 Scherk-Schwarz reduction to fiveIn dimensions a Scherk-Schwarz reduction,with one a considers global a ( symmetryThe difference given between by ‘ordinary’ acompactification Kaluza-Klein Lie ansatz. and group Consider Scherk-Schwarz a reduction field lies in the where we use the notation By imposing this constraintthat on correspond the to field theof equations, undoubled ( action. we see ThusSpin(5 that we they have reduce found a totions proper work the on doubled ones the version fields: In this formulation wed have ten fieldsubject strengths to the self-duality constraint that correspond to the doubledfields fields, i.e. we write them as JHEP07(2020)086 4 (3.8) (3.5) (3.6) (3.7) (3.4) . The . = U(1) πR . R , T 5) R + 2 , z ' USp(4) . z and the coordinate Spin(5 × USp(4) ] for further details. ! µ L ∈ ⊂ 5 µ x 34 , , 5 A R φ (4) R 5 14 2 φ / usp R 2 3 . However, our goal is to / USp(4) 3 √ G M of a maximal torus e ⊂ USp(4) √ e ¯ ]. See [ Spin(5) ]. T × M 5), so in principle we can choose 11 × L , 35 , 5 ν L L ¯ A M ∈ 5 µ A 5 is then specified by four angles, which we USp(4) 5) is gauged. The gauge group contains an = 8 supergravity theory in five dimensions φ , 4 , USp(4) = Spin(5 2 / 5 ν . Writing R R N 3 ∈ ˜ = Spin(5) π G A to an element M ∈ √ 2 – 11 – 5 e (4) R φ ˜ < 2 M = U(1) is the corresponding gauge field. For each twist, the + / usp L i 3 USp(4) 5). We take then a monodromy USp(4) 5 µ , T m , µν √ M A × g × e 5) ≤ 5 USp(4) , L L φ 6 × / , 1 L is canonically normalized [ √ Spin(5 5 − USp(4) (4) φ e ∈ ∈

usp R ; we take 0 USp(4) 4 = M ]. An important feature is that reducing self-dual 2-form gauge fields ˆ , ν ˆ , m µ , g 3 . The compact coordinate is periodic with periodicity 12 g (4) of a maximal torus , by a field redefinition, so that it defines an equivalent theory. Thus the , h z 1 1 − usp L ¯ , m 11 − g 2 M M 8 effective field theory. This reduction from 6D to 5D has been considered h M ˜ M ¯ , m < g M 1 ]. h = m = 5 N ¯ = M The result of our reduction is a gauged In our case, we reduce from 6D to 5D on a circle with a Scherk-Schwarz twist. We M ˜ M The element denote By a further conjugation, weof can the bring R-symmetry group USp(4) introduction, we restrict our twist to be conjugate to an element of the R-symmetry group that preserves the identity in Spin(5 3.1 Monodromies and masses In six dimensions thethe global mass symmetry matrix is obtain to a be Minkowski vacuum any with element partially broken of supersymmetry, the so, as Lie discussed algebra in of the important U(1) subgroup for which theory has a vacuumby (partially) an breaking thepreviously supersymmetry in where [ itin can 6D be can result described in massive self-dual 2-form fields in 5D [ The factors in the exponentssions are and chosen the so scalar that field we arrive in Einsteinin frame which in a five non-semi-simple dimen- subgroup of Spin(5 on the circle by metric (in Einstein frame) isKaluza-Klein inert metric under ansatz: the duality group, so we choose the conventional monodromy possible Scherk-Schwarz reductions aregroup classified [ by the conjugacy classes of thedenote duality the coordinates on the five-dimensional Minkowski space by This conjugated monodromy gives a massive theory that is related to the one for the JHEP07(2020)086 (3.9) (3.13) (3.10) (3.12) (3.11) trans- where + χ . πR ) and dilatini 2 , , , . 2 ) ) ) ) , − µ R 2 1 2 2 1 iµz/ ψ , , , , , e 2 1 1 1 1 , , , , , , 3 2 1 2 1 1 , , , , σ 4 1 1 1 1 USp(4) and dilatini m , )+( ) and the anti-self-dual e ) of each field under ) × + µ 1 1 4 ) will then be an eigen- 1 )+( )+( )+( )+( , , ψ L 4 ⊗ , 2 2 1 1 2 5 , e 1 3 , , , , , 3 , e , σ 2 2 2 2 1 3 -dependence 3 1 , , , , , , e . z , m 2 1 1 2 1 , e 1 1 , , , , e 2 USp(4) , , . , e 2 1 1 1 πR 1 , e ) ) ) = 2 )+( e 1 5 4 5 / ⊂ e 1 | , , , )+( )+( )+( )+2( )+( , , , (4) µ 1 1 4 2 2 1 2 2 1 ) ) | 1 . SU(2) subgroup of USp(4) for both the , , , , , , R 5 4 usp R i 1 1 1 2 1 2 , , × , , , , , , m 5 4 ) + ( ) + ( ) + ( M i 2 2 1 1 2 1 , , , , , e 1 1 4 and will have , , , 2 2 2 1 1 transform in the ( SU(2) 5 4 5 4 =1 )+( i + iµ 2 X × 1 – 12 – , USp(2)): )+( )+( )+( )+( )+( , B 1 3 = 1 2 1 1 1 1 R σ , , , , , ∼ = , 2 µ ) decompose into the following representations under 2 2 2 1 1 1 m , , , , , , ). e 2 1 2 1 2 2 ]): 5 , , , , , 3.11 scalars : ( ⊗ , vectors : ( SU(2) tensors : ( dilatini : ( ). The positive chiral gravitini 2 2 2 2 2 +( ( ( ( ( ( 4 gravitini : ( 3 5 31 , × σ , 1 2 1 → → → → → 12 m L . A field with charges ( ) ) ) e ) respectively, and the negative chiral gravitini R 5 4 5 4 , , , ): ) and ( = ) ) , 1 1 4 4 5 4 5 , SU(2) , , 3.9 (4) 1 5 4 conjugation. × )+( )+( )+( USp(4) usp L 1 1 4 1 R , , , L × M 5 4 5 ) and ( L 1 , 4 SU(2) transform in the ( subgroup ( USp(4) 4 − is the usual Pauli matrix. Other choices of the monodromy are related to this by 2 × USp(4) 3 B L σ ⊂ scalars : ( vectors : ( tensors : ( dilatini : ( 4 These representations determine the charges ( The six-dimensional supergravity fields fit into the following representations under the gravitini : ( transform in the ( − 3.2 Supersymmetry breakingThe and R-symmetry massless representations ( field content the SU(2) The resulting mass for the field will turn out to be form in the ( χ U(1) vector of the mass matrix with eigenvalue We have an equal numbernumber of of self-dual and fermions anti-self-dual of 2-formresentations positive tensor above, fields, the and and self-dual negative an tensors tensors chirality. equal In terms of the R-symmetry rep- where USp(4) R-symmetry group (see e.g. [ left and right factors (note that SU(2) We can then take, for example, we can take the monodromies to be in the SU(2) JHEP07(2020)086 × L (3.15) , with πR . 2 / | 1) 0 Minkowski i that are zero. ]. , i 2 m , | 36 , 0 ) (3.14) m 4 , 2 , 1 , 6 1 , ). , 6= 0 corresponding to , 2 i , e 3.9 = 8 1 = 0, or with a non-chiral = 0. Both types of twists 2 3 N 0) + (0 m , m ) + ( subgroup: each doublet gives 1 1 = , = 4 2 , 1 1 ). The eight gravitini (symplectic , 1 m 2 m , , 1 3.12 with with 2 ) + ( 2 R 2 R 1) + (0 , = 0, where all fields remain massless. Apart is the number of parameters 1 , , i under the U(1) r 0 – 13 – 1 of the SU(2) factors in ( , i m , 0 supersymmetry in the Minkowski vacuum and 0 e r , 2 , 2 1 − , 4 and SU(2) and SU(2) where , = 6 supergravity, we take only one of the four mass ) representation of the R-symmetry group USp(4) ) + ( 1 6 2 r 4 1 , L R , , N ]. We see that our massless spectrum consists of the gravity 1 2 = 2 0) + ( , = 8 , of unbroken supersymmetries is then given by the number of 37 1 1 , N 2 ) + ( N N give rise to 5D supergravities with ( , 1 0 , r , 4 → 1 = 4 supergravity by twisting in two SU(2) groups, with two mass ) 4 , N = 4 theory, the gravity multiplet contains the graviton, 4 gravitini, 6 4 ) = ( ( 4 N , e 3 4. The number 1 and each singlet gives charge 0. For example, the sixteen vector fields in the decompose into four pairs, each of which has a different mass , e , = 6 theory. = 4 = 6 = 8 2 3 R , 6= 0, become massive in 5D. Below we give the massless field content of reductions , e 2 and 6 scalars, although as weIn shall the see, they resultvectors, in 4 different dilatini and massive a spectra. 4 single dilatini scalar and field, 5 and scalars the [ multiplet vector coupled multiplet contains to 1 one vector, vector multiplet. We obtain parameters zero. Thisa can chiral be twist, done say in intwist, two for SU(2) qualitatively example different in ways: SU(2) result in either the with same massless spectrum: the graviton, 4 gravitini, 7 vectors, 8 dilatini parameters to be non-zero soThe that massless we spectrum twist in fromvectors, only 20 such one of a dilatini the reduction andN four contains 14 SU(2) subgroups. a scalars. graviton, These 6 fields gravitini, form 15 the gravity multiplet of the 42 scalars (all massless). Asof expected, maximal these 5D fields supergravity. make up a single gravity multiplet In order to end up with We start with thefrom untwisted the case, 5D graviton, the spectrum contains 8 gravitini, 27 vectors, 48 dilatini and 1 i , N N N In general, all fields that are charged, with at least one of the e • • • ( m = 1 check that they fit into the relevant supermultiplets of 5D supergravities [ massless gravitini, which is The different values of vacua, corresponding to twisting in 4 an with twists that preserve These charges then determineWeyl the spinors) masses through in ( theUSp(4) ( i charges have charges This then determines the four charges JHEP07(2020)086 ) ) R N p, q p, q = (3.17) (3.16) /g ). For a ). For = 0. As a N i e p, q ] and corresponds . The spectrum of 12 1 , ) . In order to find the actual q | µ | USp(2 of each 5D field is equal to 1 . q 5 µ × ) A ) preserving the central charge, where p N with covariant derivatives of the form iq g even), the R-symmetry is USp( 5 1 ) determine the massive spectrum for the re- A − USp(2 N µ ] and are labeled by two integers ( 3.13 – 14 – ∂ × 21 = . µ , and the charge D πR SU(2) /R × = 1 g SU(2) ) following from ( 4 , e 3 ), i.e. the subgroup of USp( = 8 five-dimensional supergravity. , e q 2 N , e 1 e = 2 supersymmetry, the gravity multiplet contains the graviton, 2 gravitini, . The nomenclature was chosen such that a massless supermultiplet of ( ) massive supermultiplets in five dimensions. The physical states of a ( USp(2 N N × p, q = 0 = 2 = ) has been previously derived from Scherk-Schwarz reduction in [ q p break all supersymmetry. The onlythe fields graviton that and are the not charged singletsresult, under the which such massless are a spectrum completely twist are in uncharged,Note 5D with that consists all all of fermions the become graviton, massive. 3 vectors and 2 scalars. field. Thus, the field content thatcoupled we find to from two this vector reduction forms multiplets. a gravity multiplet By twisting in all four SU(2) groups, with all four mass parameters non-zero, we We end up withsubgroups, minimal with 5D just one supergravityafter such of by a the twist twisting contains mass in the graviton, parameters threeFor 2 zero. gravitini, of 3 the vectors, The 4 fourand dilatini massless and 1 SU(2) field 2 vector scalars. content field, and the vector multiplet contains 1 vector, 2 dilatini and 1 scalar 1 N N The massive multiplets we find are all BPS multiplets in five dimensions; these mul- We now give the supermultiplet structure of the massive spectra that follow from the ) massive multiplet, the choice of central charge breaks the R-symmetry to a subgroup • • + 2 p p, q 2 supersymmetry in six-dimensions has, afterfit reducing into on ( a circle,massive Kaluza-Klein multiplet modes in that five dimensions then fit into representations of tiplets were analyzed andsupersymmetries classified in in five [ dimensions( (with USp(2 times its mass. Becausereduction the have massive to fields are combine charged, intothe the complex number real fields. of fields In complex thatmass the follow fields of from spectra (unless a the that stated field wemass, otherwise). or give this collection below, needs Furthermore, we of to when list be fields we divided we give by only 2 the write down various twists preserving different amountsalso of become supersymmetry. charged All under fields the that acquire graviphoton mass Here the gauge coupling is The charges ( duced theory in five dimensions.table This spectrum is summarized into table a gauging of 3.3 Massive field content JHEP07(2020)086 ∗ 1) , im (3.18) 2) and = , 2 that there 0)) theory. ) denotes 16 B , 4 , 3 3.2 ) m supersymmetry supersymmetry πR

indicates that both indices, so that e.g. N 4 2 N ,

2) (or (2 1 m = 4 supersymmetry: i,j 4 4 ij , 1) theory. The chiral , m

, 3 m m 4

, N , 0 , 0 m 3 2 4 3

2 4 m m m m 2 4 4 3 2 , , , , m m 1 3 2 3 1 m 0 m m m m m 2 1 3 2 , 1 1 3 m m m 2 ) representation consist of two with ,

m m m 1 1 1 signs and the , SU(2) that arise include (3

m m 1 5

, × . The notation m 2

B πR

2 ∗ = Mass (multiplied by 2 / | | ) µ i im | for the 5D fields coming from the different types m − | ( ) – 15 – i µ = | and one with mass 0. m 2 ( | µ 2 B | d m ) ) ) ) ) ) ) 2) for massive vector fields. The representation (3 ) 2. The massless states are in the , 4 1 5 1 4 4 5 − , , , , , , , , 4 1 5,5 4 5 1 4 1 5 4 , ( ( ( ( ( ( ( ( m | = 6 Representation SO(4) is the little group for massive representations in five dimen- N ∼ 1) multiplets and we will refer to this as the (1 , 2) supermultiplets and we will refer to this as the (0 , SU(2) , two with mass Fields ), but their massive spectra are different. The non-chiral twist gives massive | Vectors Tensors Dilatini Scalars 3) representation corresponds to the anti-self dual case with d Gravitini 2 × , occur. There is no correlation between the m ) denotes 4 different combinations of mass parameters, and ( 3.2 2 j + m . This table gives the value of m 1 m | 1 and 3) for massive gravitini and (2 m . In the following, we consider the cases in which the Scherk-Schwarz reduction breaks , i 2 are two qualitatively differenta twists that chiral result one in andsubsection a a theory non-chiral with one.fields fitting Both into theories (1 havetwist leads the to same (0 massless spectrum (see while the (1 B the supersymmetry to representations given in therepresentations previous of subsection, the massive and fields. we It now was already give pointed the out in subsection sions. The representations of(2 the little group SU(2) corresponds to a massivecondition self-dual two-form field satisfying the five-dimensional duality m ( different combinations. For example,mass the 5 tensors in the ( where SU(2) Table 1 of 6D fields. The mass of the field is then JHEP07(2020)086 | , ] | 3 1 21 m and = 0. m | | + (3.25) (3.22) (3.23) (3.24) (3.19) (3.20) (3.21) 4 2 1 m m , m | 3 . + 1 ). ), 4 dilatini m 5) , m | 3.2 3.18 , one of the two 1; 2 2 , . m 6= 0 and 2) 2 , = 0 in order obtain the = 4) + (1 , . 4 , m , 1; 2 1 1 , m , m m . . 1; 2 2 , . multiplets, one with mass m USp(2) USp(4) 3 2 2; 5) 1; 5) 2) + (1 , , × × , 6= 0 and the other three parameters 5) + (2 1 USp(4) , 2; 1 m , × 6= 0 and 2) spin- 2; 1 = 4 theory (see subsection , 3 , USp(2) USp(2) 2; 4) + (1 1; 4) + (1 , , N × × 1) vector multiplets, one with mass , m , 1) + (1 1 SU(2) 2) tensor multiplets with masses , , – 16 – m × 4) + (1 , 1; 2 SU(2) SU(2) , 2; 1) + (2 1; 1) + (2 2; 1 × × , , , . Each consists of 1 gravitino, 4 vectors and 5 dilatini, . Each consists of 1 vector, 4 dilatini and 4 scalars [ SU(2) | | 2 (3 (3 3 2) BPS supermultiplet with the representations , 1) + (2 m m | = 6, we twist with just one of the four mass parameters , SU(2) SU(2) − 1) + (2 1 , N 2; 1 , m | 1; 2 (2 2) theory by taking chiral twist with , , ], fitting in the representations 21 . This is a (1 | 1 1) + (3 . Each of these contains one self-dual 2-form satisfying ( 1) 2) , | m , , | 2 m 2; 1 , − (3 1) theory. There are two massive (1 = 4 (1 = 4 (0 = 6 1 , m corresponding to a representation of given by For the non-chiral twist, we(1 choose and one with mass We note at this point thatby a part tuning of the the massive mass spectrum(complex) above parameters. tensor can be multiplets That made becomes massless is, massless.multiplets in if This the we gives massless choose two sector additional of real the vector Furthermore, we find two massive (0 | and 5 scalars [ From the reduction we findand two massive the (0 other with mass which are in the representations We obtain the (0 The physical states will then fall in representations of The massive field content from4 such vectors, a 13 twist dilatini containsequal and 1 to gravitino, 10 2 scalars. self-dual All tensors, these fields are complex, and their mass is In order to breaknon-zero. to Without loss of generality,equal we to take zero. The physical states will then fall into representations of N N N • • • JHEP07(2020)086 , | 1) 2 , ): m | , and | (3.33) (3.31) (3.26) (3.27) (3.28) 1 3.28 m 1) theory | , . , 1) 2) , , consisting of 1 | 1; 2 1; 1 3 field: 1 gravitino, , , consist of 1 vector m | 3 2 . This is reducible, 3 | 3 m 2 m | 2) + (1 2) + (1 m and one with mass 2 , , , | 1 1 = 2 case with massive (0 1 . m 2; 2 1; 2 | m , , . | m N | 2; 1) (3.32) 1; 1) . , , is in the representation multiplets, one with mass , one of the massive vector multiplets | 1) + (2 1) + (1 3 1 USp(2) 1; 2)2; 2) (3.29) (3.30) , , 3 2 3; 2) , , , m m × | 2; 1 1; 1 , , 2; 2) + (1 in the following representation of ( 1; 2) + (1 , = | , 2 1 ). Note that, even though the massive tensor 1) spin- SU(2) , m m 1; 1) + (1 2; 1) + (1 – 17 – 3; 1) + (1 × , , = 0 to obtain the 2) + (1 1) + (2 , 3.2 , , 4 (2 (2 1 (2 m 2; 1) + (2 m 1; 1) + (2 2; 1 2; 2 | , , , , SU(2) is in the representation (3 (3 | multiplets, one with mass 3 3 2 m | 1) + (2 2) + (2 6= 0 and , , 3 2) theory and the massive vector multiplet of the (1 ) and one multiplet consisting of 1 gravitino and 2 anti-self-dual . Each consists of 1 gravitino, 2 (anti-)self-dual tensors, 2 vectors, , 1; 2 3; 1 | , m , , 3 2 3.29 m | , m 1 2) theory, we can tune the mass parameters in such a way that a part of , m 1) + (3 1) + (1 , , 3; 1 2; 1 , , (2 (3 = 2 2 anti-self-dual tensors, 1giving dilatino one and massive 2representation hypermultiplet scalars ( with consisting mass oftensors 1 in dilatino the and representation: 2 scalars with the There are also twocontaining spin- 1 gravitino, 2 vectors and 1 dilatino in the representation. We also find another multiplet containing a spin- representation. Furthermore, wedilatini, find 1 scalar) two with tensor masses multiplets (1 self-dual tensor, 2 representation. The four vector multipletsand with 2 masses dilatini in the multiplets in representations of There are four massive hypermultipletscomplex with dilatino masses and 2 complex scalars in the of the theory (againmultiplet see of subsection the (0 contain different fields, they give the same field content in the massless limit. We choose As in the (0 this spectrum becomes massless. For becomes massless, and so we get two more real vector multiplets in the massless sector one with mass 5 dilatini and 2 scalars. The one with mass and the one with mass In addition, there are two massive (1 N • JHEP07(2020)086 . ) 2 4 σ M m ⊕ − is the (3.34) (3.35) (3.36) (3.40) (3.37) (3.38) (3.39) 2 → σ = 2 † BA m Ω , M ( ) − so that one of σ and B · ), we can bring = . 3 3 A 4 m m n M 3.9 . ! , . ( 2 AB 3 4 0 ) σ → σ      ) are the exponentials 2 4 m ) 1 2 (4) 0 ) 2 σ m m 2 ⊕ 3.8 m usp R m

) ⊕ m σ M would make either a tensor 3 in ( = · , = sin( σ 4 matrices subgroup ( 3 3 R 3 ), where Ω 1      4 n − × 4 matrix representation AB m 2 ( m m ) 0 3 Ω BA × ) 0 1 0 m = exp( = 1 m . − M m = m USp(4) (4) = (4) 1 ! 2 = , 2 , ∈ 0 0 0 sin( m usp R 1 usp 1 R (4) ! − − m ) (4) AB M 2 usp R 2 ) 0 σ ) 0 cos( of the SU(2) 1 0 2 0 0 0 2 · usp R or M m − h 2 m m 2 1 M n

– 18 – , ,M ( 0 0 0 00 0 m ) 3 m 2 = σ ,M for more details. In a basis in which Ω = is the 3-vector of Pauli matrices. m (4) 2 and      ) 0 ) ) 0 cos( 1 ) 0 1 AB = σ σ m L usp = L σ · that can be found by replacing Ω m B.2 m 1 · 0 cos( 0 sin( ⊕ 2 M 1 (4) 3 m n (4) 0 ( n σ sin( cos( 2 ( is symmetric ( usp 1 L usp R 1 . Here USp(4)      i m m B M m M n ), the mass matrices are given by ∈ = exp( ), by an element = C ⊕ =

) M (4) (4) 3.7 (4) σ = 3.10 (4) ) is block diagonal, we have the 4 · usp L AC usp L usp L 1 usp (4) L n 3.9 = 4 theories, we can tune the mass parameters in order to obtain extra M ( M M M usp L 1 = Ω N m M AB = M (4) usp L massless ﬁelds. Choosing multiplet or a vector multipletvector massless. multiplets. Both Another of choicethe these would massive would hypermultiplets be give becomes two to massless. real set massless As for the M The Lie algebra of USp(4) consists of anti-hermitian 4 The monodromy for the above mass matrix is given by and a similar expression for In this basis, we can take for example and the subgroup ( However, for our purposes, it will be useful to have mass matrices in a basis in which for any four unit 3-vectors such that symplectic invariant; see appendix By conjugating, as inthis ( to the form The monodromies of mass matrices in the Lie algebra of USp(4): For the monodromies ( 3.4 Mass matrices JHEP07(2020)086 to 1 πR (5) 2 R so (3.43) (3.44) (3.41) (3.42) ∼ = (see ap- 4 . (4) m        ) usp and SO(5) ) 2 → , 2 L m 2 . m        − ) m − 5) 2 1 , 1 m m (5 in such a way that the m (5)). The corresponding and − so R 5) metric takes the form so 1 3 sin( , M m ∈ m ∼ = − ( − ! → b (4) and 0 0 ) 0 ) 0 cos( ab a 2 2 1 ) ) L 2 usp R R m m m , M m M M − − ! − 1 1 5 − + 0 in the algebras of SO(5) 1 1 m m L L . m R 5 1 M M 0 ) 0 0 0 ( 1 ( M . We can use the isomorphism 2 −

b ) 0 0 0 m – 19 – (4) = ) 0 0 0 ab 2 a ) = 2 ) and + usp R m 0 0 0 0 R R (4) m 1 L + AB M M M m + usp L τ 1 M ( where we replace 1 − + m − M L L m R 2 M here and in the next subsection where we reduce the 6D M M sin( ( m ( 5) mass matrix is given by but , π 1

000 0 0 0 0 0 0 0 0 0 − + 2 1 (5 2 1 ) 1 ) cos( 2 2 = so = m = m m L        R B + + M A = 000 0 0 0 cos( sin( 0 1 0 1 1 ) this L M m m , we consider various brane configurations that result in five-dimensional M 5) mass matrix. In the basis in which the SO(5 gives 4 , B.1 sin( cos( ) to the corresponding generator in the Lie algebra of SO(5). This yields π (5        for more information on the isomorphism so = 3.39 = 2 L B.2 m In section We can use the mass matrices M = 1 In this section, weis go to through compute the theconvenience, reduction mass we of that set the eachtensors. 6D of scalar Consequently, the fields the 25that in masses scalar can detail. that be fields we reinstated obtains The compute by in goal checking here the 5D. carry mass an dimensions. For ‘invisible’ notational factor black holes. Forfields each that of charge these theones systems, black hole given we remain here. choose masslessmass in In spectrum. 5D. particular, All they of all these have are the3.5 conjugate same to eigenvalues the and 5D so scalars give the same It is this matrix that appearssubsections. explicitly in the bosonic action, as we shall see in the following (see appendix The USp(4) monodromy is ofm course a double cover of the SO(5) monodromy:create an taking e.g. and a similarpendix expression for SO(5) monodromy is given by to map ( and there is a similar expression for JHEP07(2020)086 3 ), we (3.50) (3.45) (3.47) (3.48) (3.49) (3.46) 5). This 3.45 , . is given by j and neglect the σ Spin(5 H j i π 1 2 ∈ Q is a diagonal matrix ) . Tables are provided U = = . i 2.2 4 ]; consequently, a global diag σ R 5 m H . with ) . M T H 1 ( T U . − , V M H H z , where H T − j . T H l ˆ U µ M 25, and compute the mass matrix as M ∂ Q + e =0 1 H k ) + µ = 0. − H σ µ 2

diag kl D H x V j ,..., 1 ( ˆ µ M m M − H → ∂ ). By substituting this ansatz in ( H V k ∂σ 5 µ 2 i i = 1 H z Tr ∂ µ i Tr Q M – 20 – 3.44 ∂σ 5 e D 1 8 φ = 3 = 0. We find such a minimum by putting all 25 = / = ij H − A in 5D. The covariant derivative on Tr 8 ) = s µ ij V , with ˆ µ 1 5 1 8 m √ i ∂ . By realizing that our mass matrix is anti-symmetric, x L m − A σ (ˆ 1 1 = e = − (6) s H . 1 4 = e H ij L δ µ H 1 D ) = − (5) e H (0) = ( ij V g ) is given by 5) transformations act as , 3.47 with ]. Just like in the previous subsection, we set j σ . µ 34 ∂ , i C , we immediately see that this gives σ 11 µ is the mass matrix defined in ( ∂ M ) k − is a conjugation matrix built from an orthonormal basis of eigenvectors. In this way, M σ ( = Q We have computed these masses explicitly for the mass matrices that preserve the We now compute the masses of the scalar fields in this minimum. We denote the The scalar Lagrangian in six dimensions reads (see subsection ij The kinetic term of the sigma model is diagonal at the minimum of the potential, i.e. it takes the form g 3 T 1 2 In this section wefollowing work [ out theKaluza-Klein reduction towers for of notational the convenience. six-dimensional tensor fields− in detail, we find the mass that corresponds to each ofvarious the 6D redefined black fields string ˜ in configurations appendix that we consider in section 3.6 5D tensors We diagonalize this mass matrixand as minimum can be foundscalar fields by to solving zero,M so that collection of all 25 scalar fields by The potential in ( For an R-symmetry twist, such potentials must be non-negative [ Matter that is charged undercorresponding the to monodromy the becomes graviphoton charged under the U(1) symmetry where find the five-dimensional Lagrangian The global Spin(5 leads us to the following Scherk-Schwarz ansatz: JHEP07(2020)086 B A M (3.57) (3.54) (3.55) (3.56) (3.51) (3.52) (3.53) . After ,β (5) 1 so that the A d ,B β , (5) 1 α 5 1 , ) A 1 d − ∧ A B 5 1 ˙ M A α A , ( ) (5) 2 1 + − − G z , . ,α M − . d . (5) 2 ( ), so we choose our Scherk- ˙ ˙ α α , , A ,B − (5) 2 (5) 1 ∧ = 0 = 0 . This is not possible, however, (6) 3 ) A A 2.28 . , ,A , G ,A → µ 5 1 5 1 (5) 2 . (5) 1 5 1 x ,B ,β ( A A (5) 2 (5) 2 = d = d ,α ,C ∧ ∗ are non-zero, this splitting goes like (5) 2 (6) 3 ∧ A ∧ A A ,B ˙ ˙ A ∧ A α α (5) 2 , , 4 A → G ,A β (5) 3 (5) 2 B ,A ,B (6) 3 G ,A α ∗ m (5) 2 A (5) 2 (5) 2 G ,A G G (5) 2 M M G BC takes the other eight values of the original A 3 ) + + AB + . + − µ H α ˙ m α H + d x ,B ,α , ,A ,A ( (5) 3 (5) 1 (5) 1 – 21 – (5) 1 (5) 2 1 4 AB ,A transform as in ( A A ,B G τ A A (5) 3 ,G − (5) 3 and ) now separates into 5 1 is the index that corresponds to the row and column G ,A B = G 2 = (6) 3 d i = d = d A = d = d m G ,A ˙ . For example, for the reduction of the D1-D5 system α B ∧ A 3.54 ,α , (6) 3 M ,A ,A R (5) 2 (5) 2 A (5) 3 (5) 2 G ,α is not invertible. We therefore need to diagonalize − G G 1 M (5) 2 z G G ,G . The index B m G M ,A } , where (doubled) ,β t (5) 2 A e 9 (5) 2 } − , and G L A M 4 d ,α L +5 β (5) 2 α ) = M ∈ { A ˆ µ i, i ) would lose their dependence on are five-dimensional field strengths that are independent of the circle M x α (ˆ ,A = d = ∈ { ,A (5) 2 3.54 is invertible, so now we can shift (6) 3 α ,α ,α = 0. We find G β (5) 3 (5) 2 G α ,A G G (6) 3 . As usual for self-dual tensor fields, we don’t compactify the Lagrangian of M and z G ) with ˙ ,A ˙ α )) we have ˙ (5) 3 . The second equation in ( G α, A 4.8 ( The Lagrangian for the six-dimensional tensor fields reads → The six-dimensional field strengths are subject to the self-duality constraint The matrix this shift, the five-dimensional field strengths read A that we set to(see zero ( in index Normally at this point, wefield would strengths like in to ( shift because our mass matrix and split the indices thatcase correspond where to zero the and combinations non-zero eigenvalues. In the most general corresponding two-form and one-form potentials: identities d From these we deduce expressions for the five-dimensional field strengths in terms of the where coordinate the theory but rather its field equations. We start by reducing the six-dimensional Bianchi The ten three-form fieldSchwarz strengths ansatz to be JHEP07(2020)086 . ) 2 ˆ B AB ) as ˆ d ) as δ 3.59 carry (3.62) (3.61) (3.58) (3.59) (3.60) = = 3.57 ,α 3 3.46 (5) 2 ˆ H . A AB H ]). They read ,B (5) 2 38 . , G ˙ β , (5) (5) 1 = 0 ∗ A are determined by the 5 d ˙ α φ , ,α ∗ 3 (5) 1 (5) 2 / ˙ β ), we find 2 A A ˙ . α d √ with field strength τ 3.4 ∗ − ,C 2 . (5) 2 e − ˆ d B 3 G ˆ = − H ∗ . ˙ α , 5 1 (5) 2 5 1 C BC A A = A H d , with raised indices is the inverse of the 3 . As it turns out, there is a difference in + + ˆ ,β H z 5 z (5) 2 AB d ˆ ∗ AB τ d A 5 H ∗ φ ∧ ∧ 3 , β are non-zero). The self-duality constraint ( – 22 – / α ,B 2 ,γ ,B 4 and an analogous expression for the dotted indices. (5) 3 (5) 2 . Consequently, we use the inverse of ( (5) 2 √ , m G A G M − γβ and makes sure that the massive tensors e ) to zero. In particular, this means that τ δ ∗ (5) 2.2 ˙ α + , γ = 0 3 ∗ − M (5) 2 αγ β 3 τ 5 τ m 3.59 ,B A We now pay some extra attention to the interactions be- = ˆ φ (5) 3 H M 3 = ˆ d / − G β ,A 2 (5) 3 β and α = τ . By diagonalizing this matrix, we find the mass corresponding α √ G 2 τ e β (6) − ∗ m α ,α B B (5) 2 = M A A 1 A In order to find the mass spectrum of the fields that descend from ,α τ d (5) 2 z z m ∗ M A M M τ d e e d − = = . These masses can be found in appendix as defined in subsection ,A 4 (6) 3 H G (6) , we put all other fields in ( Consider a six-dimensional (anti-)self-dual tensor field Just as for the scalar fields, we have computed these masses explicitly for the mass ma- ∗ ,A (6) 3 tensor. We illustrate this difference with two simple examples. (here hats denote 6Dfield quantities). read The field equations and self-duality constraint for this in section Graviphoton interactions. tween the graviphoton andThey the will prove vector to and bethe tensor very result important fields that in we that section find we for find the in reduction of this a subsection. self-dual 6D tensor and an anti-self-dual 6D only half their usual degrees of freedom.mass The matrix masses of the fields to each field. trices that we use for the reduction of the D1-D5 system and the dual brane configurations So in 5D, wefor end the up case where witheliminates eight the massive tensors massless tensors and two massless vectors (again, this is where we use the notation These are massive andcan massless deduce the five-dimensional equations self-duality of conditions. motion for the From corresponding these, fields we (following [ Scherk-Schwarz ansatz. Mass spectrum. G We find This result allows us to write down the 5D self-duality constraint that follows from ( Recall for the derivationmatrix of this result that star to five dimensions. By using the metric decomposition ( We now compactify this constraint. First, we need to reduce the six-dimensional Hodge JHEP07(2020)086 R in 2 . To 1 B (3.63) (3.67) (3.66) (3.64) (3.65) Spin(5) ), and so . × → −A 5) L , 3.65 1 A ), we will need to Spin(5 , ⊂ , which is the point in 6.1 ) R R 1 ), and by using straight- R given by the monodromy, 1 , A 2 A ψ . + H + z z ∧ USp(4) Spin(5) USp(4) = 0 (d 2 → M (d 2 × × × ∧ H ∧ ψ B L L L 2 ∧ 2 ∗ H 1 H A + im is flipped with respect to ( + 3 1 2 3 2 USp(4) USp(4) Spin(5) H ¯ H / B 2 ∈ # 5) 2 = B ˜ , z M ∈ 3 – 23 – H ∧ ˆ ! H 1 m ∧ ∗ A 0 , h 2 5) transformation − , 1 , im H − effectively interchanges the result for a self-dual and an 0 5) m is complex so that we also have the complex conjugate h 2 1 , 1 − ¯ 2 M − " 2 B h B = in order to protect the sign in the covariant derivative. The d Spin(5 → −A 2 L 5) ¯ 1 ∈ B = exp , M → A 3 ˆ ↔ H 2 B , g 1 . We see that a self-dual and an anti-self-dual tensor give a Chern-Simons 1 sign indicates the difference between the result for the reduction of a self- − g B ˜ M g = d ). By flipping the sign of the graviphoton, we effectively switch the particle and 2 = H However, for the embedding in string theory (see subsection Now take a real doublet of (anti-)self-dual tensor fields, that we Scherk-Schwarz reduce 3.65 M sign in the mass term of the equation for the origin. consider monodromies that areby related an to element an of Spin(5 R-symmetry transformation by conjugation and as we haveMinkowski vacuum. seen Conjugating this by point an element is of a the minimum R-symmetry group of the Scherk-Schwarz potentialwill giving then a preserve the fixed point in the moduli space and the minimum will remain at We have so farpreserving considered the monodromies identity in inthe the the moduli coset space R-symmetry at Spin(5 which group allpoint Spin(5) scalar under fields the vanish. action Then of this the point Spin(5 in moduli space is a fixed the anti-particle we see that redefining anti-self-dual tensor. 3.7 Conjugate monodromies Again, the dual and an anti-self-dualthe tensor interaction with from the six graviphoton,see dimensions. but this, we recall Now, can that still thisof flip the sign ( it field is by not redefining in front of (apart from the ansatz andto the the fact previous that we one). arefive Going considering dimensions a through subject doublet this to this reduction the set-up gives self-duality is a equation similar complex massive tensor from 6D to 5D with the ansatz forward reduction techniques andLagrangian: the conventions of this paper, we find the following 5D with interaction term with the graviphoton with an opposite sign. By decomposing this field (strength) as JHEP07(2020)086 2 4). 8), 4)- , , aris- (3.68) where A µ SO(4). ,..., A αβ × 5). There ) , = 1 AB 8 labelling the γ α ( is the SO(4 SO(4) / AB 4) ( 4) 4) twist with mass ,..., AB , of Spin(5 N , , η 4 1 4), which has a U(1) , = 1 16 . This was of course to = , ˜ A ˜ z M ) are the gauge fields for a αβ T where 5 µ N from the metric and a vector B AB , 5 µ CB 5 µ N η A = 8 supergravity theory in which A 5). The mass matrix acts on the C 4) (with of this U(1) factor commutes with , , , A ] = t A µ N N B . For the supergravity theory, this A , and this is now the location of the 4). These combine with the degrees of } ,T = gψ , R A T of SO(4 [ AB → of Spin(5 8 N ψ respectively. After the field redefinitions given 10 of SO(4 Spin(5) 4) symmetry. There is a 6D metric, B-field and – 24 – , ˜ z , 8 × B in the as the mass matrix is in the Lie algebra of SO(4 ,T T L ˆ z A µ ]. In 5D, there are then 10 gauge fields: eight B A BA ,T which is given as usual by A 14 N A N β T − α 5). Consider first the bosonic NS-NS sector of the ten- Spin(5) 4)) and scalars in the coset space SO(4 , N = ] = , . A further U(1) factor can be obtained by dualising the 2- ]. At this point, the kinetic terms of the various fields are transforming as a Weyl spinor of Spin(4 ˜ z A ∈ 5 α ˆ µ AB ,T ,T h z z C | N T ] T [ by a transformation gh duality symmetry of the ungauged 5D theory is promoted to a gauge 14 { 6 ˜ M 4) of Spin(5 , ] = g was given in detail in [ ,[ gives a 6D theory with SO(4 g 4 B A to give an extra gauge field and the generator T from the reduction of the 6D B-field. Then ( N 5 µ µν ] to obtain tensorial fields transforming covariantly under duality transformations, b B Next, consider reintroducing the R-R sector. In six dimensions, there are a further We start with the case in which the twist is a T-duality transformation in the T-duality 14 which combine with the 8are NS-NS one-form also gauge a fields to furtheranti-self form dual 4 the ones two-form that gauge transformfreedom fields, as of an which the NS-NS splitspinor 2-form into representation to through four form self-dual the ones and four This then represents a gauging ofsubgroup a generated 10-dimensional subgroup by of SO(4 form all other generators. 8 one-form gauge fields with all other commutatorsinvariant vanishing. metric, so Here that ing from the 6D vectorfield fields, the graviphoton vector field gauge group with 10in generators [ the gauge algebra is [ fying on dilaton, together with 8 vector fields vector representation of SO(4 The Scherk-Schwarz compactification of thismatrix on a circle with an SO(4 The result of the Scherk-Schwarz reductiona is subgroup a of gauged the E symmetry. In this subsection we discuss this gauged supergravitysubgroup and Spin(4 its gauge group. dimensional supergravity theory, consisting of the metric, B-field and dilaton. Compacti- not conventionally normalized.precisely On the bringing ones these given to earlierbe standard for expected: the form, a theory the field with redefinition masses monodromy cannot become change physical3.8 parameters such as Gauged masses. supergravity and gauge group with monodromy is just a fieldconsequences redefinition for giving the ancontaining embedding equivalent in theory, but string asminimum theory. we of shall the The see potential fixed later [ point this is has now at the coset This change of monodromy can be thought of as the result of acting on the theory twisted JHEP07(2020)086 i a m A (3.69) (3.70) (3.71) Spin(4) × 4) triality. , . ) comes from z 5 1 T A , , ˜ . For special values of the z ˜ ) and will give isomorphic z t T T come from the gauge transfor- αβ ab 4.2 ) and a N M α T , the two-form gauge fields in the i . and ,T 4) can become invariant under the ] = m ] = , A β b ! T β 3.7 ,T ,T α 0 α a = ( N T T [ [ a B . For special values of the parameters of SO(4 T A 0 , t 8 ˜ z N – 25 – , , (if the NS-NS two-form is dualized) with possible

,T b β t z T (corresponding to the graviphoton has no zero eigenvalues, then the vector fields ]. The generators T = T b z β b b a a α T a 32 , M all become massive, while the gauge fields corresponding N M M 31 a T ] = ] = a come from the gauge transformation that correspond to the α t ,T ,T z z remain massless. Then the gauge group is spontaneously broken T 4) T-duality group. The other twists we will consider are all [ T is the symmetric charge conjugation matrix. The structure in the [ , , t and ˜ z ˜ z αβ , we will consider a twist of this kind in the compact Spin(4) T ,T η . The generator z t T 4.2 subgroup generated by and . For generic values of the parameters 3 has some zero eigenvalues, there will be more massless gauge fields and the and 6 subgroup generated by b ˜ z γβ a 3 η T 4) become massive, while the 5D NS-NS two-form remains massless and can γ , M α N = In subsection One can argue what part of the matter content is charged under each of these gener- In the generic case in which The gauge algebra can now be written . The generators of SO(4 αβ a 6D tensor field thattensor is field, a so after singletformations reduction under no the matter twist. becomes6D diffeomorphisms charged This in under transformation the the circle actswe resulting direction. find only 5D By that on explicit trans- all reduction this fields of that these become diffeomorphisms, massive in 5D carry U(1) charge under ators of the gaugeseeing group how by the Scherk-Schwarz 5D reducingtransformations fields the can be transform 6D found under gauge inmations these transformations [ corresponding reduced and to transformations. thethe 16 The 6D vector 6D tensors, fields gauge so in theT 6D. 5D descendants These of transform these the fields can 6D become vectors charged and under generators subgroup of therelated Spin(4 to this onegauge by groups. conjugation (see subsection corresponding to the generators to the generators to the U(1) such that unbroken gauge group will be larger. There is a U(1) further U(1) factors coming in if some of the R-R two-forms remain massless. with all other commutators vanishing, where subgroup of E 8 again be dualized toparameters, give a some further of U(1) thetwist factor and two-forms with so in generator become the massless as well. These can be dualized to give further U(1) factors. spinor representation is relatedThe to gauge algebra that then in gains the the terms vector representation by SO(4 to give an 18-dimensional gauge group. This corresponds to gauging an 18-dimensional N JHEP07(2020)086 ] 4 , π [0 (3.74) (3.75) (3.72) (3.73) ∈ shifts all i . The full i m r 4 combination . of the form 1 5 1 S A and so corresponds r + . As an example, we z − , it is an eigenvector of d n = 0 mode. This gives a . i 1 , e ) n ) µ ∧ → µ for any integers x ) x ( n i µ ( n , we see that the x n ψ µ ( πr ψ , )’th one while leaving the sum ,n r (5) 2 Z ) to the z R ) unchanged. For this reason, there G by 2 − ∈ ’s further by realizing that all eigen- i inz/R i has charges n n e 3.73 m m 3.73 ψ ) + , so that Z + . Consequently, we can take µ ∈ π µ X n x π 2 µ ( 2 iµ ψ ,n − (5) 3 , n ). i = 1

G π – 26 – πR 2 Mz 2 n R , → Mψ [0 exp + π µ 2 ), Z ∈ inz/R ∈ i e µ X can be absorbed into a shift n πR , then each ﬁeld picks up an inﬁnite Kaluza-Klein tower. 2 m Z r 3.12 ) is extended to 1

∈ ) = exp S X n ) = , z 3.52 sign in front of them. By taking into account two towers of , z µ µ x z x ( , we see that shifting the ( πR ψ M 1 2 ψ ) for each field can be read off from table ’th KK-mode is given by given by ( i n m iµ by an integer ( ’th Kaluza-Klein mode to the ( µ n π µ 2 as a schematic notation for any field in the theory that transforms in some ), one of them with a minus sign in front of ψ ) = exp ˆ µ x 3.73 (ˆ appear with a (6) 3 , where we discuss the full string theory. by an integer and so leaves the above sum ( iµ G 6 π µ The mass of the The Scherk-Schwarz ansatz is a truncation of ( Without loss of generality, we can restrict the 2 Of course, even this is not the whole story. There are also Kaluza-Klein modes that come from the with eigenvalue 4 reduction from 10D tosection 6D on the four-torus, plus stringy degrees of freedom. We will return to these in and the value of check this for thethe reduction Scherk-Schwarz of ansatz ( the 6D tensors. If the whole tower is taken into account, for e.g. determining which twistsstring preserve theory which requires brane keeping configuration allfreedom in these from section modes, branes together wrapping with the stringy internal modes space. and degrees of the form ( of these towers iswithout unchanged loss under of generality. consistent truncation to a gauged five-dimensional supergravity theory, which is sufficient unchanged. From table the is no loss of generality in taking values Clearly, shifting to changing the where we use representation of the R-symmetryM group. Then if In the previous subsections, wesive have constructed fields 5D from theories Scherk-Schwarz withconsider a both reduction. compactification massless on and However, mas- We this choose is Scherk-Schwarz ansätzeincluding Kaluza-Klein not towers the on the whole story: if we 3.9 Kaluza-Klein towers JHEP07(2020)086 ]. = 8 16 , (3.76) (3.77) N 5 are chosen i m will be massless. As ) to see that the masses N 5) indices are suppressed − , 3.12 = . 8 supergravity describing the n . < Z = 0 N ∈ 4 given by ( ) the Spin(5 µ π , m 3.75 = in this ansatz; from now on, we will keep it 3 with , n ). R iµ , m – 27 – 1 3.74 is to 2 π R in , then the mode with M = 3.6 + 2 N 5) duality group, but will be preserved by a stabilizing m = , πR M = 0 that are kept in the Scherk-Schwarz reduction are not ) 2 = i n π 1 m 2 2) supergravity in six dimensions. This solution will not be ( , m µ fermions that become massless, which form a hypermultiplet of 1 2 , we can see which additional massless fields arise. In this case, there are 1 is an integer, ) i as we used it in subsection π m 2 ( µ M Primarily, we focus on the D1-D5 system, but later in this section we also consider the Note that the modes with = 2 supergravity. For further discussion of such accidental massless modes, see [ dual F1-NS5 and D3-D3 systems. 4.1 The D1-D5-PThe system D1-D5 system,D1-branes, sometimes D5-branes more and accurately waves carrying called momentum. the The D1-D5-P ten-dimensional configuration system, consists of Scherk-Schwarz twist are the onesconsequence, that the are black hole trivialIndeed, will (zero) also it in be the descends BPS black fromtwist and hole a preserves preserving solution. the BPS (at supercharges black As least)black and a string four hole Killing solution supercharges. as spinors in a of solution. six the dimensions, truncated theory and that the has duality the supergravity in five dimensions.twist on This the reductionhole circle. can solution be will If modified remain theSchwarz a by reduction, solution duality including and of twist of a the its ismassless gauged duality truncation supergravity in sector. to resulting an the from effective This stabilizing the is Scherk- subgroup, because the the same only black fields that become massive as a result of the black holes in 5D.a black We do string this solutioninvariant in of under (2 two the steps. wholesubgroup. Spin(5 First, we If reduce we the thenthe brane do black a configuration string standard wrapped to (untwisted) along compactification the of circle, this we on obtain a a BPS circle black with hole solution of N 4 Five-dimensional black holeIn solutions this section, we consider several 10D brane configurations that we compactify to give an example of this, we can choose By using table four scalars and two spin- We can now use that theof eigenvalue of the Kaluza-Klein modes are given by ( necessarily the lightest modes inso the that tower. In particular, if the parameters manifest in all our equations.for clarity. Furthermore, in It ( canmatrix be seen directly that this extended ansatz essentially changes the mass Note that we have restored the circle radius JHEP07(2020)086 (4.5) (4.2) (4.3) (4.4) (4.1) , which have no 1 4 − 5 . (10) 2 H 2 3 2 3 1 4 C 1 2 K r H dΩ Q dΩ 2 2 r r ) to 6D using the { = 4 y + + 4 4.1 / 2 2 = 1 + 1 4 r r g d 3 d . K y 3 4 5 1 2 4 5 . ) so we find that }| H T = 0 H 4 1 4 2 1 1 2 ) denotes a pointlike direction, 1 3 4.1 y · = 1 + φ H H ), we can solve for the individual , . K + ,..., = 2 + 2 1 2.6 2 ϕ − − − − ϕ y 2 2 ························ φ d ) = 1 d ) z z ∧ = ∧ d d 2 4 1 1 1 1 y z − − − − ϕ S − ϕ t ,H d t d + d 5 θ given in ( 2 (d θ , m (d 2 { z}|{ z 2 r 2 2 3 , is diagonal in ( Q 1 4 K – 28 – m y ϕ K is the metric on the three-sphere written in Hopf b − ~ 5 cos mn cos + 2 2 , φ + 5 g 5 H + d 1 2 ϕ 1 2 2 1 4 Q 1 z d 2 Q 2 We compactify the metric in ( z = 1 + − 5 4 y θ , H + 5 + 1 2 ). The dilaton Φ and the R-R two-form H }| + d z + d 1 2 z R = 1 + d d 2 d 2 2.4 t ~ φ t H 2 1 · ∧ d ············ ∧ r θ ϕ d y + cos m t t b = d ~ 2 1 − e − 1 2 4 ϕ 1 4 t φ 1 4 − − − z q 1) d 1 2 d ,H 1) d e − 5 1 2 − − 5 5 θ − 1 5 2 − 2 ) denote a direction in which the brane or wave is smeared. − H − 5 r H H Q H 1 4 1 1 1 3 4 1 2 H 1 2 − 1 P − ··· 1 1 2 − H 1 1 D1 D5 ) denotes an extended direction, a dot ( − q 1 1 H + sin H H H − + H H 2 θ = 1 + . We find that only one of them is non-zero in the 6D solution: = = ( = = ( = = i 1 φ Φ + e = d H ). The metric on the torus, 2 (6) (6) φ 2 (10) 2 (10) 2 s e 2 3 s C C d d 2.4                             Here a line ( We start from the ten-dimensional solution and reduce it to 5D with the ansätzethat The rest of theis reduction related is straightforward. tois the The incorporated ten-dimensional six-dimensional in Einstein one the framenon-zero by ansatz components metric a on ( Weyl the rescaling torus, so with they reduce trivially. The result reads By using the expressionsscalar for fields the vectors Reduction to sixansatz dimensions. ( where dΩ coordinates. The harmonic functionsthe corresponding momentum to modes the can be D1-branes, written the in D5-branes terms and of their total charges as frame reads and multiple dots ( are given in previous sections. The D1-D5 solution of type IIB supergravity in Einstein is as follows: JHEP07(2020)086 ). = a (4.7) (4.8) (4.6) 3.44 (6) 3 maps ˜ G B.2 .      4 , . 0 m is given in ( −               B ) ) 3 2 4 A , . 0 0 m m m M 2 2 − to zero, this reduces to ϕ ϕ − − 4 d d (5) of appendix 1 3 + 0 0 0 m ∧ ∧ so m m φ ) we can rewrite the solution ( ( 1 1 3 ∼ = ϕ ϕ − − 0 0 0 00 0 m d d 2.3 0 0 0 0 θ θ (4) (4) mass matrices:      2 2 The last step is to Scherk-Schwarz 4 2 = m m 1). Hence, we find that the doubled usp usp cos cos + Φ). This solution describes a black , 5) mass matrix 5 1 − − 4 , + (4) φ φ Q Q 3 1 (5 ( 2 usp R 2 m m + + √ 1 so √ ) 0 0 0 ) 0 0 0 z z , e 2 4 d d 1 = , m m – 29 – ∧ ∧ 1 t t + , + + ,M 0 0 00 0 0 0 0 φ 1 3      1) d 1) d m m 2 ( ( − − 0 m − − 1 1 − 4 2 − − 1 5 ). The isomorphism = diag (1 1 m m H H 3.9 0 0 m ab 00 0 000 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 ) so the total number of degrees of freedom of the fields that + + are real mass parameters, each corresponding to one SU(2) in (5) matrices in the − = ( = ( 1 3 K 4 so 2 m m 2.27 0 0 0 m (6) (6) 2 2 m . By putting all scalar fields except               ˜ C C ,b with 1 (6) 3 = = 0 0 0 0 0 0 and m ,b G L R (6) 3 3      ab M M G m L = ∗ , 2 + ab (5) mass matrices. We find (4) m ,b K so , (6) 3 usp L 1 G = M m ∗ For the D1-D5 system, we choose the following When we use the doubled formalism (see subsection a ab 3 (6) ˜ The embedding of these Here the R-symmetry subgroup ( these to reduce the six-dimensional black string solution,sions. which results We in choose a the blackgroup, hole twist i.e. in matrices the five to subgroup dimen- of befields the in that R-symmetry the that are preserves stabilizing non-constant the in subgroup solution. the of As black the a hole R-symmetry result, remain all massless. the In the doubled formalism,self-duality the constraint degrees ( ofthe freedom black of string couples both to these remain fields unchanged. Scherk-Schwarz are reduction halved to by five the dimensions. G tensors to which the black string solution couples are given by string in six dimensions. above in terms offields the to doubled the tensor black string fields.K solution, To recall derive that the the dual contributions field of strengths these are defined doubled as Here, we have defined the scalar field JHEP07(2020)086 ) (6) 2 ˜ (4.9) C ] (4.10) (4.11) (4.12) , yield 39 (6) 2 . We now and ˜ C . (5) 1 (6) 2 ˜ C C (5) 2 , and , C + (5) 2 3 d 2 (6) φ ˜ 2 ∗ C . C dΩ , + 2 K φ . In particular, we find r . (5) 1 2 N C √ + 5 to determine the masses (5) N e , C.1 2 N 5 G πR r 1 ), where we only take along φ (5) 4 2 d 3.6 3 N to the black hole solution in C / 1 3 = 2 p 3.59 ) (5) √ 2 π K and K e ˜ C H = 2 5 = 3.5 H K and 1 (5) , c 1 Q ˜ s H C 5 g (5) d 0 2 Q α 1 C + ( 2 Q 2 3 = t – 30 – d 5 p q K 2 3 , H (5) N 2 − 1 6 ) ) are presented in appendix t t t . π G (5) 1 2 2 K 2 q ˜ , c C are integers and the basic charges are given by [ 4.8 q − 5 H 1) d 1) d 1) d d i 5 = − 5 s R H ∗ g − − − N H 1 6 0 H (5) N 1 1 1 1 + (5) N 1 2 ) to a 5D black hole is the same as in the untwisted case. q φ A − − − G 5 K 1 H πα G q − 2 1 1 ( 4 4 H H H 4.5 √ − H H − = = e , where = ( = ( = = = = ( 1 i 5 c 5 5 1 φ are full vector fields, meaning that they are not subject to a self- + BH N φ (5) 2 (5) (5) φ 3 1 1 i A S e / s e c ˜ C C 2 d (5) 1 = ˜ √ C                              e i Q = and (5) 1 C (5) 1 d C This three-charge black hole has been well studied in the literature. Its charges are The two self-dual tensors that charge the black string solution, We can now follow the techniques of subsection The entropy of thiswhich black yields hole can be computed with the Bekenstein-Hawking formula, quantized as Here duality constraint andcompactification carry can the be generalized usual byto adding number the angular 10D of momentum branes in degrees to directions give of transverse a freedom. rotating black hole Note in five that dimensions. this five-dimensional black hole solution is then given by We use these toterms of write the the dual contributions one-forms. of In doing so, we move to an undoubled formalism. The full two tensors and two vector fieldsconsider in the 5D. self-duality We conditions denotefields these for by that these are fields non-zero from inby ( the the 5D relations black hole solution. We find that they are pairwise dual that each of thecalculations scalar for the and mass matrices tensor ( that fields the acquires fields due that appear todo in not this the become twist. six-dimensional massive black in Thereduction string this of solution results Scherk-Schwarz reduction, ( the of as solution required. these ( This means that the JHEP07(2020)086 ) 4.1 . We (4.14) (4.13) (4.16) (4.15) 2 B 2 3 dΩ . After reduction , 2 r ) with the replace- 4        y (10) 2 + , . 4.5 2 B 2 2 ) 0 r 2 ϕ d ϕ 3 → d d y m , 3 4 N ∧ ∧ − N 2 (10) H 0 0 1 2 1 1 r 4 1 2 F ϕ Q C ϕ y d m d H ( , θ θ 2 2 − 2 + 1 ϕ 2 − − − − -direction. This system is related to y 2 d = 1 + ············ cos Φ and cos z ) m z ∧ N F N d − 1 2 4 z Q Q , the twist is chosen to preserve − − y 1 ϕ 2 → − − . In the doubled formalism, the black string d + + C t m θ 2 z z (6) 2 + d ) 0 0 0 2 (d d d ϕ 2 2 3 B – 31 – K y ∧ ∧ m cos ,H t t 1 → + + N F 2 0 0 0 0 + d 2 r 1 Q Q (6) z 2 2 2 1) d 1) d y m instead of C + ( − − 2 + d z − 1 1 + d d 2 B ········ − − r θ ϕ F N t 2 2 1 = 1 + ∧ d H H y m We now study the F1-NS5-P system, which consists of F1 and t F d Φ) and − t 0 000 0 0 0 0 0 0 0 − − + H = ( = ( 1 4 − 1 4 1 1) d 4 − N 1 2 − N N m φ (6) (6) 2 2 − H ( ˜ H H        B B 1 4 1 2 F 1 2 3 4 1 F1 − √ F NS5 − − = H F F H L = H H + ). − M = = ( = (5) matrices φ Φ 2.27 so e → (10) 2 (10) 2 s is as before. Note that this solution can be obtained from the D1-D5 solution ( B + d φ K , we obtain a very similar six-dimensional solution, given by (                  4 H To reduce to five dimensions, we need to specify the mass matrices. We choose the Again there are waves with momentum in the T constraint ( Scherk-Schwarz twist to bethe in F1-NS5 the system stabilizing couples subgroupchoose to the of the R-symmetry group. Since Again these fields carry only half their usual degrees of freedom due to the self-duality and by an S-duality transformation, whichon sends Φ ments couples to the two-forms where we have the harmonic functions the D1-D5 systemsupergravity via solution S-duality. in ten As dimensions. in It can the be previous written in case, Einstein we frame start as by considering the The F1-NS5-P system. NS5-branes arranged as follows 4.2 Dual brane configurations JHEP07(2020)086 , B.2 (4.21) (4.17) (4.19) (4.20) (4.18) , 4 2 3 y d dΩ ∧ 2 r 3 4 . y y + d . 2      ∧ r 2 2 5) d 3 z m , m y 2 d 0 . + 1 2 . 2 3 1 − ) with the field redefinitions 1 ∧ 0 m H        3 2 t m 1 2 r 2 3 − 2 4 Q y 4.10 Spin(5 y H 2 1) d 2 ∈ m + m − 2 + d + 1 2 00 0 0 1 − − − · · · · · · 1 y 2 3 2 · · · · · · − − 1 = 1 + 0 ) m y − 3 0 m 0 0 10 0 0 0 00 0 1 0 0 1 0 1 0 00 0 0 1 0 0 0 z d 3 − H d        z 1 2 -direction. We start with the supergravity − − ,C 2 0 − 2 z − 1 3 = + ( m t . By using the isomorphism in appendix m − 2 + 4 H 2 2 2 + 1 y (d 1 2 C 1 ϕ 3 m d m – 32 – . It is not surprising that these black holes are m K H − ,H ∧ → (5) 1 2 + , c 1 + 3 2 2 ˜ 2 2 y D1-D5 B 2 r m Q ! z m d 2 2 m + c M 0 2 2 . 0 0 0 0 y + 1 → ∧ 1 m C c + d 0 z m − and + d d 2 (5) 2

C.2 ········ = r θ ϕ t = 1 + 2 3 ˜ 1      C ∧ d Finally, we consider the reduction of the D3-D3-P system of y 3 = t m is given by d = − t H − − C 0 1 2 C → 3 and 1 2 1) d (4) F1-NS5 0 1 − H 3 0 − 1 2 usp M (5) L 2 m H 1 − 3 1 2 B D3 D3 M − 3 − H 3 (4) generators of the form H → H + usp = = ( (5) 2 C , 2 (10) (10) 4 similar with s − C d φ R          M As before, we also have momentum in the The resulting five-dimensional black hole is given by ( → + where the harmonic functions are given by solution in ten dimensions, in Einstein frame it can be written as branes. We specify the brane configuration: Essentially this conjugation matrix interchangesthe the ninth fourth and and tenth row fifth and row column andThe in column the and D3-D3-P mass systems. matrix (and monodromy). related by field redefinitions.U-duality, and the After corresponding all, mass matrices the are D1-D5 related by and conjugation F1-NS5 systems are relatedThis by conjugation matrix The masses of thematrices scalar are and given tensor in fields appendix that follow from the reductionφ with these mass these map to and JHEP07(2020)086 . i in (5) m and so (4.24) (4.22) (4.23) (10) 4 ). The (6) 2; 1 C periodic, R 4 4.10 switches the → C (6) 2 , . . . , y , C 1 y .        ) ,      2 . ) 2 2 2 m 2 ϕ ϕ m m d d − + 2 2 − 1 ∧ 1 1 in the solution ( ∧ m 1 m m 1 ( i ( ϕ ϕ (5) 2; 1 d − ) d 2 ˜ 2 R θ ) 0 θ m 2 m 2 2 2 − + 2 00 0 0 1 → m 1 cos m cos m 0 ( + 3 3 0 0 i − (5) 2 1 Q Q ˜ (4) this mass matrix reads ), except now the matrix ) C 2 2 m + + ( m m usp 2 − z z − 4.18 + 2 by taking the coordinates 1 and 1 d d 2 m 4 – 33 – m ∧ ∧ ( m T i − t t (5) 2; 1 0 0 0 ) − R 2 2 1 1) d 1) d m m m → + 2 2 0 0 0 0 − − − 1 1 2 1 1 m m 0 ( (5) 2 − − m 3 3 i C H H +      , 1 1 = = ( = ( φ m 00 00 00 00 0 0 0 0 0 0 0 (4) (6) (6) 2; 1 2; 1 . The resulting five-dimensional black hole is given by making the → remaining massless. In ˜        usp R R L + = φ M (6) 2; 1 C.3 L R M ) by a field redefinition. We find this field redefinition as . In the doubled formalism, the six-dimensional black string arising from the 4.5 1 φ Here, we treat the special cases in which the mass parameters can be tuned in such a In the last step of the reduction we need to ensure the fields that are non-trivial in → + that the fields thatConsequently, source the the black black hole hole remains aremass a parameters. left valid solution unchanged of by the the 5D Scherk-Schwarz theory twist. for allway values that, of in the addition to the original black hole, other black hole solutions are also preserved relation is similar tofirst the and F1-NS5 fourth rows result and in columns ( instead4.3 of the fourth and Preserving fifth further ones. In black the holes previous by subsection, we tuningFor chose mass each twist black matrices parameters hole with four solution arbitrary (D1-D5, real F1-NS5, parameters D3-D3), we chose this matrix in such a way The scalar and tensor massesgiven that in follow appendix from thefield reduction redefinitions with these massmass matrices matrices are are again conjugate to those of the dual D1-D5 and F1-NS5 solutions. The which results in such systems. All of these systems are relatedthe by T-duality. black hole solution remainmass massless matrices in of 5D. the form For this twisted reduction we choose Different D3-D3 systems can betorus. constructed These by would arranging be the charged D3-branes under differently on the the two-forms coming from the reduction of this brane configuration isthe related black string to solution thesystem for D1-D5 ( the system D3-D3 by systemφ T-duality. can be This obtained meansD3-D3 from that system that couples for to the the D1-D5 two-forms On compactifying to six dimensions on JHEP07(2020)086 2) , = 0. . 4 (4.25) (4.26) 2 m , m = 4 (0 3 = m 1 N m 2) theory and in , ). These are exactly in this reduction to 6= 0 and For this example, we 2 2 C.2 m = 4 (0 , m 1 in the example above, we − N 2). 2 m , = m 1 (5) = 2 m ˜ C 1 , = 4 (0 m o (5) 2 ) that preserve the F1-NS5 black hole N (5) 2;3 ,C ˜ R 4.16 , , we choose (5) 2;3 2 + Φ) By taking 2) theory, we choose 4 – 34 – m , ,R φ ( 3 = 2 φ 4) subgroup of the duality group. From the perspec- = 0, we see that each field either becomes massive 2). 1 1 , n , √ 4 m = 4 (0 2) case in detail below. = , , m N 3 + φ m = 4 (0 = 4 (0 N N or remains massless. It is therefore straightforward to tune the mass | , but instead other solutions are preserved. Now the fields coupling to 2 m (5) 2;3 . By setting R − 1 C.2 m | 2). This choice does not preserve the D1-D5 solution and the D3-D3 solution , = 0 theory. Since we are interested mostly in partial supersymmetry breaking, we Suppose now that, instead of We thus see that the D1-D5 solution can be preserved in a reduction to Suppose that, in addition to the F1-NS5 solution, we also want to preserve the D1-D5 N = 4 (0 the D1-D5 and F1-NS5this black reduction. holes, but also one of the D3-D3N black holes is preservedcharged in under massless than just the ones that charge the D1-D5 solution. In particular, the fields also remain massless (as can bethe checked from fields the that tables in are appendix non-trivial in one of the three possible D3-D3 black holes. So not only this elsewhere. Other possibilities in managed to keep the fieldspreserve that this couple particular to solution. the D1-D5 It black turns hole out, massless, and however, that so we this could choice kept more fields tive of the fullsymmetry, we string can theory in this principleperturbative work is string out a theory the explicitly. T-duality corresponding On orbifold twist.D1-D5 the compactification black other of hole, Since hand, the the the T-duality microscopic D1-D5possible is description CFT, to of a is study the understood perturbative this particular reasonablystring reduction well. theory, thoroughly Therefore, and both it from from should the the be perspective perspective of of the the full black hole microscopics. We will return to with the twist matrix thatby was originally taking proposed the to two preserveteresting the mass possibilities. F1-NS5 parameters solution, On to simply the besolution one lies equal. hand, in we the This note perturbative particular that SO(4 example the twist offers that some preserves in- the F1-NS5 have to remain massless as well.in The masses appendix of these fields forwith this mass twist matrix can be found parameters in such a way that all of these fields remain massless by taking solution. In order to twist to the solution with this twist. Then the fields NS5 black holes. Asthe it turns out, thistreat can an only example be of done the in the Preserving D1-D5 withconsider T-duality mass twist matrices in of the form given in ( by the same twist. For example, we consider twists that preserve both the D1-D5 and F1- JHEP07(2020)086 2 m = (4.27) SU(2) 1 ) of the × 0 m s, s . In the supergravity 0 s . 6= o s (5) 2;2 ˜ R ) again. If we take , . Subsequently, we compute the (5) 2;2 4.16 4 2) results in the preservation of two ,R , 2 φ The only other theory in which we can n as they are in representations ( = 0. = 4 (0 . For each case, we find that taking either ) that are non-trivial in the D1-D5 black hole N N = 0. and 4.2 – 35 – 4.25 chiral N o and (5) 2;1 ˜ R 4.1 , = 0 case. Now all four mass parameters are non-zero. As (5) 2;1 N ,R 1 in the reduction to φ = 0 modes of the Kaluza-Klein towers and identified black hole 2) theory. The same game can be played, however, with the other n 2 , n m SU(2), and we will refer to them as chiral if − × = = 4 (0 1 in this reduction, all fields ( fields that contribute to these parity-violating terms. This is because in five N m 4 m or 2 = 3 m massive m In this section, we consider corrections to the coefficients of the 5D Chern-Simons Under certain conditions, which we discuss in the next section, the black hole solutions = 1 5.1 Corrections toIn Chern-Simons five terms dimensions massive fieldslittle group can SU(2) be theory we have been discussing, the chiral massive field content consists of the gravitino quantum corrections in afor general the setting and entropy then ofquantum discuss the corrections to their black the origins holes Chern-Simons andfields. terms solutions from consequences of This integrating out section is massive ofstringy supergravity course modes; not these will the be full considered story: elsewhere. there are in principle further corrections from modifications of the black hole solutions and hence to quantum correctionsterms to their that entropy. result fromit integrating is out thedimensions massive massive spectrum. fields can be Itand in so is chiral can representations a contribute of to little the the little unusual parity-violating group that Chern-Simons SU(2) terms. First, we consider these solutions in the massless sector after this truncation. we have been consideringthe lift effective to supergravity solutions theoryquantum of corrections receives the to the quantum full coefficients corrections. string of the theory. 5D In Chern-Simons In terms particular, which the in there string turn are lead theory, to 5 Quantum corrections So far, we havemassive considered fields. five-dimensional For supergravity the purpose theories of(consistently) with finding black to both hole solutions massless the in these and theories, we truncated an example, let’s considerand the geometric F1-NS5 twistsolution ( remain massless. Consequently, thecan D1-D5 be solution worked is out preserved. for similar Other reductions examples to m additional black hole solutions. Preserving further black holespreserve in several black holebreak solutions all supersymmetry: by the tuning mass parameters is the one in which we These are all the possibilitiestwist for to preserving the multiple blacktwist hole matrices solutions given with a in T-duality subsection the two other D3-D3 black holes remain massless: JHEP07(2020)086 ) I 2 ], A 26 > p, q (5.2) N ]. There ) (5.1) 25 R denotes the ∧ fermions) con- R 1 2 R Tr ( = 2 (and 0) theories. ∧ 1) theory, nor for the , I N A 1) representation and the term are fixed for I , k F terms present however. For term exists and is known [ Z = 4 (1 ∧ F 2 , ). There are also Chern-Simons R g F π N , these massive fields fit into ( ∧ corresponding to the three gauge ∧ (5) − 1 ∧ 48 ˜ 3.6 F R C 3.3 A d = ∧ ∧ ∧ A A I, J, K running over the number of massless vec- (5) 1 ARR I fermions) and dilatini (spin- C d 3 2 ∧ – 36 – denotes the gauge coupling and 5 1 ,S A g K F Z term is generated by quantum corrections, leading to ∧ 2 (5) for the indices J 1 ]. The origin of this lies in parity: since the Chern-Simons term for the non-chiral R κ ]. In principle, one would need to integrate out the entire 3 F 2 . Here terms. There are g 40 2 ∧ R , with the index I 40 π ∧ I k 2 (5) 4 R R ∧ I 1) representation (together with their anti-chiral counterparts κ A . , , A ∧ I ∧ R − k = d A ∧ R , IJK = IJK I k k ∧ A F IJK 2)). As we have seen in subsection A Z , k IJK k 2 3 g π − 48 ). This Chern-Simons term (and other similar terms) can be found from the 2) theory a = , 5.2 2) representation, the self-dual two-form field in the (3 3) and (1 , , = 2, the supersymmetric completion of the AF F S Quantum corrections to the Chern-Simons terms are only allowed for certain amounts Consider first the Chern-Simons terms in the classical 5D supergravity obtained by The non-abelian gauge symmetry of the 5-dimensional gauged supergravity is spon- From the fields that we obtain in our duality-twisted compactification of 6D supergrav- N = 4 (0 3), (1 , For but this is notN the case forthe theories conjecture with that more ais supersymmetry. no supersymmetric such completion However, quantum of in this the term chiral should exist [ fields in ( reduction of the 6Dterms tensor coming fields from (following the subsection reduction of the 6D vectors. of unbroken supersymmetry. Thesupersymmetry, coefficients so corrections of to the this terms are only allowed in the hole, we find the term so that we have to the coefficients Scherk-Schwarz reduction from maximalthat 6D there supergravity. are By no explicitexample, reduction, in the we find reduction with the Scherk-Schwarz twist that preserves the D1-D5 black for some coefficients curvature two-form. Integrating out the chiral massive fields yields quantum corrections taneously broken to anwith abelian field subgroup strengths with masslesstor abelian fields in gauge the fieldterms theory. one-forms involving The these pure fields gauge are and of the the mixed form gauge-gravitational Chern-Simons as massive stringy modes. We focus on the supergravityity, only fields the here. self-dual tensors, gravitinitribute (spin- to the Chern-Simons terms.yield Chern-Simons Integrating couplings out [ otherterms types violate of parity, massive they fields can does only not be generated by integrating out parity-violating fields. spin-half dilatino in the(2 (2 BPS supermultiplets. By integratingthe out 5D this Chern-Simons chiral terms matter,chiral [ we massive can spectrum; obtain the corrections fields to that we found in our supergravity calculation, as well in the (3 JHEP07(2020)086 ). ∧ R 4 = 2 ARR ∧ k 5 N A term and and 5 and A 5 d AF F A k ∧ = 4 theory to an d USp(2) subgroup 5 = 2 and the chiral ∧ × 5 A N term only for d N A = 2 supergravity with d ∧ ). These general results F ∧ 5 N 5 ∧ 2) theory. As discussed in 5.1 A , A F ∧ terms. A USp(2) subgroup so that all the = 4 (0 R × ∧ N R 2) theories are the only ones for which , that the corrections that we find from ∧ 5 = 2 supergravity models, with the specific term for the graviphoton. As a result, both A 5.3 5 N = 4 (0 term is induced only for A term. Neither of these terms are present in the – 37 – d and R N R 5 ∧ ∧ 5 ∧ for the coefficient of the A d for the gauge-gravitational Chern-Simons term. In A R R d ∧ ∧ . This is because the black holes that we consider couple ∧ . As it turns out, both the coefficients ∧ 5 AF F 5 5 A 4 5 k = 2 and ARR A A A k d A 2) theory. By integrating out the massive field content we N , ∧ 5 A = 4 (0 N = 2 supergravity. We can consistently truncate this have no classical contributions and arise only from quantum corrections. N ] general BPS black hole solutions were found for ARR 28 k 2) theory. , ] for , 6 theories. We will see in subsection 27 , 28 and for the coefficient of the , Consider the We can also apply this to the black holes in the In [ We introduce the notation For our purposes, we will focus on the Chern-Simons terms 27 = 8 = 2 theory by decomposing fields into representations of an USp(2) = 4 (0 that involve the graviphoton ARR AF F N of the R-symmetry groupof and these removing USp(2)’s. allcorresponding fields For choice that each of transform of the non-triviallyfields the embedding under that black are of one non-trivial hole the in solutions USp(2) the black we hole have solution considered, survive the we truncation. make As a a result, the corrections to the Chern-Simons coefficientsin are which allowed, we and find so these corrected are black the hole only solutions. theories obtain a non-zeroorder coefficient to compute corrected BPSof black [ hole solutions in this theory, we use the framework values of the Chern-Simons coefficients obtainedBPS in the black next holes subsection. are In preservedwhich particular, by we these compute four the supersymmetries, entropy. and these are the blackthe holes previous for subsection, the affect the black holeby solutions. the In corrections particular, to these the coefficients. entropy of theseboth black holes pure gauge is and modified gauge-gravitationalthen Chern-Simons give terms BPS ( black hole solutions for our k 5.2 Corrections toWe black now hole study the entropy effectholes that the that corrections we to studied the in Chern-Simons section terms have on the black to consider couplings of this chiralto matter the to the coefficients graviphoton; of these the then lead to corrections k classical theory — there is no R only to the graviphotonThe and chiral massive to field vectors contentcharged descending that under we from the find the gauge fromtensors, symmetries 6D so duality-twisted corresponding for tensors compactification the to is (see purposes not the of section studying vectors corrections that to descend the black from hole 6D solutions we only need integrating out the massive6D fields supergravity that (including come the fromin Kaluza-Klein our agreement towers duality-twisted with from compactification the the of supersymmetry above: circle and compactification) we a are find quantum correctionsN to the N JHEP07(2020)086 ), 5 A (5.9) (5.5) (5.6) (5.7) (5.8) (5.3) (5.4) . that couple 2 ] will also be , ) for general I ). The values ! 2 A 28 5 5.3 , 5.1 K 2 Q 5 ˆ 1 27 Q 2 N K Q 1 ˆ N ]. N 3 3 (5) N with the basic charges ] 3 28 πR G i ! , 3 4 28 N AF F 5 , i 27 5 . k 2 K Q c 2 K ˆ N 1 I Q 1 3 27 ˆ 1 AF F N k Q = k N + i 3 term given in [ + ) and compute ( (5) N Q . + 2 . 5 G πR 5 2 , K R 4 N I 5.4 AF F K K 2 ˆ 1 Q N ∧ ˆ . K k 1 Q Q N Q ARR R Q 5 X k ∧ AF F (5) N J = 0 this expression for the black hole Q 3 1 + direction is shifted k (5) N ARR 1 A π AF F X (5) N k G z πR r Q I g G 4 k 1+ G πR 2 + X 4 p ARR 2) theory that we have been considering here. r , k K 1 + = 1 + + AF F (5) N 2 N – 38 – k IJK K = 1+ r π K G k = 2

X Q 1+ 6 π J = 4 (0 1 + K = = r AF F ˆ X = N N k are the Chern-Simons coeﬃcients from ( K 2 BH S 1+ ˆ Q S IJK K times the basic charges as

k IJK ˆ i 2 N k 1 2 5 (to the Chern-Simons terms that contain the graviphoton N K N − ˆ Q 1 5 N ARR Q k 1 v u u u u u u t Q π ). This yields and v u u u u u u u u t = 2 2 4.11 (5) N π BH G AF F 2 4 S k are the (rescaled) moduli corresponding to the three gauge fields = I X BH S We now apply this to our setup. When we solve ( We now briefly review the procedure to compute the entropy of BPS black holes in as given in ( i where the shifted momentum charge number is given by Just as was done forcharges in the terms uncorrected of expression integers c for the entropy, we can express the three It can easily beentropy reduces checked to that the uncorrected for result Here the charge arising from momentum in the More comprehensive studies of these solutions can be foundcoefficients in [ we find the entropycharges to of be the corrected D1-D5-P black hole solution in terms of its three where to the black hole charges and of these moduli innear-horizon the limit solution is are found by solving the attractor equation, which in the solutions of the quantum-corrected these quantum corrected theories. It is given by the formula [ black hole solutions of the effective theory with an JHEP07(2020)086 . 2 (5.10) (equal field c . χ q 3 q χ 2 AF F ) and the second term c χ k c 1 8 O 4.12 dilatino that come from integrating Chern-Simons coupling. The + 0 and absorb any minus signs F 2 3 ARR µ ≥ ∧ k ) ψ q q 5 2 F 3 p ψ K N q as ( ∧ c 1 ˆ 5 1 and ]. N ψ c 8 A N A 19 . The diagrams that contribute to the 40 5 ]. As an example, we show the diagram AF F − 1 AF F k ) 40 gravitino k . 3 5 1 p AF F ( A k 5 1 2 A – 39 – fermions). The relevant diagrams for corrections k π 12 B 1 2 . We always take + . ) q 3 1 q K p term in figure q ( ˆ B N 5 1 AF F c ]. The results of these computations are shown in table 5 B k F A c 4 N 40 1 ∧ − N F . q ∧ π self-dual tensor field A c and the charge = 2 under the graviphoton ) have been computed in [ ) can be expanded for small q field 5.1 c BH ARR AF F 5.8 k k S fermions) and dilatini (spin- 3 2 term can be found in [ . This diagram generates corrections to the . The coefficients of the Chern-Simons couplings that are induced by integrating out the R ∧ 1) that depends on the field’s representation under the massive little group, and the In order to find the corrections to the Chern-Simons terms that are induced by the We see that the contribution of a massive field to each of the Chern-Simons couplings Contributions are only obtained from integrating out massive self-dual tensors, gravi- R ∧ Figure 1 external lines represent the graviphotontensor, whilst gravitino the or solid dilatino internal running lines in represent the a loop. massive self-dual into the corresponding to field’s U(1) charge massive spectra of our 5Dfields: theories, the we sign need of to know two things about each of the massive to the couplings ( that contributes to the A consists of three parts: a prefactor that depends on the field type, a constant ity. While this is acontributions well-defined from calculation, the some chiral caution spectrumThe is of coefficients needed stringy since that modes there we will to compute also the here be Chern-Simons come coefficients. purely fromtini the (spin- supergravity modes. is the correction to first order in 5.3 One-loop calculationIn of this section the we Chern-Simons compute the coefficients out contributions chiral to massive fields arising from the duality-twisted compactification of 6D supergrav- The expression ( The first term is equal to the uncorrected black hole entropy ( Table 2 different types of massive chiral fields, as computed in [ JHEP07(2020)086 1, − (5.11) (5.13) (5.14) (5.15) (5.12) = . field c 1 60 ] ] ] ] ] ] 4 3 4 4 4 4 + m m m m m 2 m + + − + − − 1 6 , , . 3 2 2 3 3 3 + 1) 2 m m m m m m k [ ( 3 2 − − + + − s k = +1 = +1 = +1 1 2 1 1 1 + 1) + − B ψ χ m m m m m − ] [ [ [ [ [ c c c k , 4 3 3 3 3 3 ( s s s s s 1 6 k m − − − + ): ): ): ] + ] ] ] + ] + 2 . + 1) 4 3 4 4 4 5 4 5 3

, , , ] k m m m m m π m 4 1 1 4 [ n m + 1) 2 ( ( ( , 3 − + − + − 3 m k s [ + k ( 3 2 2 2 3 3

3 ) k s − i π m m m m m + 1)(2 ] m − 2 π − 2 m k

2 + + + − − ( ] + ( j m 3 2 1 1 1 1 µ k – 40 –

+ 1) ≡ m − m m m m m + 1) [ [ [ [ [ [ 1 3 k 3 3 3 3 3 k , , ,

k s −∞ s s s s s ∞ 1 1 1 π m (2 X = = +1. In terms of the six-dimensional R-symmetry [ (2 m 2 − − − 3 − − − 3 , n

] + ] ] ] ] + ] + ] s of the corresponding five-dimensional massive fields are that each 6D field produces a Kaluza-Klein tower of 5D . 2 3 3 4 4 4 4 π π = = = x m m field 2 2 m m m m m m m + 3 B ψ χ c

] + [ 3.9 c c c 2 3 ﬁeld ], the regularized expressions are + − + − + − c s = = m 2 2 2 3 3 3 41 3 −

+ m m m m m m ] ): ): ): n n 1 1 1 1 4 + − − + + − , , , m + + m 1 1 1 1 2 2 [ [ 5 4 5 3 3 ( ( ( π π m m m m m m s s m m [ [ [ [ [ [ 2 2 ) that are generated by integrating out our massive ﬁve-dimensional 3 3 3 3 3 3

− + s s s s s s 5.1 − − − + + + −∞ −∞ ∞ ∞ X X = = = 4 n n is the integer part of c ] = ] = AF F x b k m m [ [ 1 3 We now have all the information that we need to compute the corrections to the Chern- We know from subsection The conventions in this work are such that 5D tensors that descend from 6D self-dual s s where Simons terms ( spectra. Now, forcorrection to a the general pure twist gauge term (i.e. all twist parameters are turned on) we ﬁnd the Here we use the notation We need to regularize suchof sums the (and charges). similar Following sums [ in which we take the sum of the cube ﬁelds for which the sum of the charges is given by tensors and 5D fermionswhile that tensors descend descending from from 6Dnegative 6D anti-self-dual positive chiral tensors chiral fermions and fermions have representations, fermions have the descending signs from of 6D JHEP07(2020)086 6= 0 . 2 3 (5.19) (5.18) (5.17) (5.16) , s , m 1 1 s from inte- ] m 3 ] ] ] ] . , 4 4 4 4 ] m ARR 2 m m m m = 0, as can be ) for | k + 2 m − + − + 2 m 2 3 3 3 ] − 5.17 m 4 ARR and 1 m m m m − k m − 1 m + + − − [ 1 − 1 m 1 1 1 2 | AF F s m 3 [ k m m m m 3 2 does not. For such a twist, 3 2 1 [ [ [ [ m 1 1 1 1 s [ . s s s s 1 − 3 = 0 and s | − − ]

2 − ] 2 ARR = 4 theory, say with π 3 ] ] + ] + ] + k m m 2 4 4 4 4 m ] +

m N 4 AF F m m m m + + k m − 1 1 + − + − ] = 2 + m 2 2 3 3 m | [ m m 3 [ 1 ] m m m m 3 2 3 s 4 m + [ + − − + 3 2 1 1 m [ 1 1 1 2 | − s ) equal to zero. We work out some interesting 1 vanishes but m 2 − [ s m m m m ] 1 [ [ [ [ – 41 – m , s 2 ] + 1 1 1 1 | s − 5.16 2 s s s s

= 0 modes to be ] m ) with the simpler expressions ( . AF F − [ 3 m π − − k ) are the quantum corrections for general values of n 1 ] reduce to m + 3 2 3 s ] ] ] + ] + ] m |

− [ 3 4 4 4 4 3 1) 5.16 1 1 m s 1 , ) and ( s 5.16 m m m m m m | m ] = + [ ] + 3 − − + − + − 3 1 2 1 m ] and 2 2 3 3 3 s [ 5.15 2 m 1 m 1 ) and ( = 4 (1 [ m m m m m s − m π s 1 1 [ + ] ) and ( 2 s 1 − − + + − 2 N 1 ). This is consistent with expectations based on supersymmetry s 5.15 1 1 1 2 2 = m m [ 5.15 + 3 m m m m m 1 [ [ [ [ [ = 0 modes and not the whole KK-towers, and on restricting to the ] + + 5.16 1 1 1 1 1 s 1 2 1 = 0, we find that 1 s s s s s n ARR 4 m − = [ m k − − + + + [ 1 = 6 and m 2) 1 s 1 8 , s = N ) and ( 5 2 ARR + − 3 k = m 5.15 = 4 (0 = 8, and by taking into account the Kaluza-Klein towers as well we find For the case where we chooseand a chiral twist to the we find the correction from the By twisting to anychecked of straightforwardly these by cases setting the wein appropriate find ( mass that parameters equaland to chirality, as zero was explained earlier in this section. ARR N N k The expressions ( The above formulae give the contributions from summing over all Kaluza Klein modes • • = 0 modes the functions the mass parameters. Thetaking results certain for parameters twists in that ( preservecases supersymmetry below. can be found by Then the quantum corrections tograting the out Chern-Simons only coefficients the massive modesreduction of are the given 5D by supergravity that ( arises from Scherk-Schwarz arising from the reduction fromity 6D to keeps 5D. only The the Scherk-Schwarzn reduction to 5D supergrav- and the correction to the mixed gauge-gravitational term JHEP07(2020)086 ) ]. 26 = 0 (6.2) (6.1) 5.15 (5.20) (5.21) (5.22) n . 5) ]. , they simplify 19 , 5 m ], and the Scherk- Spin(5 = . term is known [ 18 ] ⊂ 3 , )[ m R

R m , Z [3 π ] ∧ 1 m 2 5; = s m

, R 1 8 2 [3 9 4 ∧ 3 USp(4) m s − = A ] × − . = . We now investigate the lifts of m ] i L ) that we found in this subsection are 1 [2 m Z m ARR 1 m s [2 5; 3 , ARR 3 4 s k USp(4) − ] , however, is beyond the scope of this paper ] + 6 and m , k = 0 modes and the Kaluza-Klein towers read Spin(5 [ ˜ 1 m M ∈ 3 [ n s – 42 –

ARR 3 k s π AF F 8 , m 2 can be computed from the general formulas ( 33 k M ∈

15 5) + , and − ARR 1 6 13 24 k = 36 = 0 and the other parameters non-zero. The general ex- AF F = = Spin(5 k 4 and AF F m ∈ k ARR AF F k k = 2 theory corrections to both the Chern-Simons coefficients are AF F , g k N 1 − g ˜ M 1) ) by taking g , = 5.16 M ]. = 2 (0 17 substantially. For this choicemodes of are mass parameters the corrections due to the and the corrections due to both the In the minimal allowed, and the supersymmetricThe coefficients extension of the and ( pressions are quite unwieldy, but if we take , The monodromy is conjugate to an R-symmetry The monodromy is a U-duality N We then have three conditions on the monodromy, similar to the three conditions The expressions for the coefficients • 16 2. 1. symmetry is broken to theSchwarz discrete monodromy U-duality has symmetry to Spin(5 be restricted to be inin this [ discrete subgroup [ 6.1 Quantization ofThe the twist Scherk-Schwarz reductions parameters weM-theory) have only been for considering specialScherk-Schwarz have reductions values lifts of to to the full string parameters string theory theory (or constructions. The supergravity duality 6 Embedding in stringSo theory far we have consideredScherk-Schwarz reduced a theory. supergravity In setup somestring in cases, theory which Scherk-Schwarz as we reductions compactifications studied can with be BPS a lifted black duality to holes twist. in We a study such lifts in this section. A more thorough calculationThe would embedding into be string neededcalculation theory to of is include the discussed coefficients all inand the the left next stringy for section. future modes study. The as full well. string theory computed from the supergravity fields that come from the duality-twisted compactification. JHEP07(2020)086 5) 5). M , , (6.5) (6.6) (6.7) (6.8) (6.9) (6.3) (6.4) . (6.11) (6.10) i m Spin(5 Spin(5 ∈ , so that , p ) g . Z ) Z ’th roots of unity, 5; p . , SL(2; . , ! × i 5) , and this is the point at ) i , m Z m Spin(5 M are all continuous group i sin . ⊂ USp(4) cos , for some integer ) parameterised by four angles ) ) im − × 4 Spin(5 SL(2; R e 1 Z i i 3.7 , × ⊂ m m of the = ) ] of the group element , p g ,..., 4) Z g 1 SL(2; , sin cos SL(2; − , π i M USp(4) × k

= 1 generated by × π 3 ) ) ∈ 2 ) p i SL(2; , R Z Z ) is an element of an elliptic conjugacy class m ) = Spin(4 2 π i ( × ]. The corresponding low energy supergravity Z , 5 ) . R ⊂ m 3 π satisfies ( i Z – 43 – , SL(2; k SL(2; USp(4) is zero, so that the monodromy is conjugate to a 2) , i 0 3.7 , i i M SL(2; × = ∼ = × i ) m p ∈ SL(2; ∈ πn Z 2) i 2 M i , ∈ Spin(2 M m ,R 4 = SU(2) ) i × ]. As a result, the phases M SL(2; R 5 m [ 2) × Spin(2 × ˜ , p 3 M ∈ ) Z is then conjugate to a rotation: Z M SL(2; i × ∈ M Spin(2 2 i . This can be thought of as a quantization of the parameters k SL(2; i M n × must each take one of the values 1 i M ∈ ]. Each M 5 m = )[ . Note that the conjugation is by an element Z i M ) is a fixed point under the action of the This condition ensures that some supersymmetry is preserved. in fact conjugate to anm element of a maximal torus ( subgroup of the R-symmetry This ensures that there is a Minkowski vacuum and implies that the monodromy is At least one of the parameters The general solution to these three requirements is not known. Consider, however, the The point in the moduli space given by the coset [ Conditions (1) and (2) imply that 6.2 3. The angles there are solutions in whichof each SL(2; where Then taking This subgroup arises from considering and the isomorphism which the scalar potential has itsdescription minimum is [ as described in subsection special case in which so that for some integers in ( generates a cyclic group JHEP07(2020)086 , ]: 4 T 23 , , and where 6 (the , 6 or 12 p 4 /R which is 22 , given by 0 , , Z 4 α 3 , n quotiented 1 , 17 3 1 S 2 , , S is accompanied . At this point, × 4 z M T ] for further discussion. 17 then generates a being one of 1 ]. However, this asymmetric i n and then further compactified M 23 4 , ; see [ then provides a condition on the T 22 for a particular integer M i 4) and so is equal to 2 m /pR 0 ,..., ) with each ]. Z combined with a shift on the πnα = 1 20 i M ( + 2 – 44 – i ˜ z n ). The monodromy 1 → z = under the action of ). i i 1 n ), the corrections to the coefficients of the Chern-Simons e ) i = , but for a U-duality twist this gives a U-fold, which is a 1 subgroup of SL(2; m 6.11 generated by the U-duality transformation ( ]. i S ), and when the monodromy is a T-duality, this orbifold con- M 5 n R p Z Z Z ]. This can also be understood in momentum space. The quotient 5; , 23 , 17 given by ( undergoes a shift ˜ i Spin(5 . z bundle over m ]. The full string construction is then IIB string theory on ⊂ generates a 4 5 i [ generated by the monodromy ) T = 8 supergravity and the 10D fields of type IIB supergravity on a four-torus πR/p Z 1 M ]. p S Z 4; N with a duality twist along the circle. If the twist is with a diffeomorphism of , + 2 24 × , 1 4 z We have given the relations between the 6D fields of the duality-invariant formulation Acting on this asymmetric orbifold with a U-duality transformation will take the mon- The T-duality subgroup of the U-duality group is a particular embedding of The quantization condition on the parameters S 17 T is the least common multiple of the → on this gives non-geometric generalization of this bundle [ of 6D in [ 7 Conclusion In this paper we haveOur studied duality set-up twists was and type their effect IIB on string black holes theory in compactified string on theory. in its Minkowski vacuum [ odromy to aIt conjugate will U-duality then monodromy takebrane that the wrapping numbers, will phase giving depending in a non-perturbative on general construction the similar not winding to be the number ones a to given a T-duality. phase depending on its coordinate ˜ determined as in [ introduces phases dependent on both theand momentum and dependent the on winding number theThis on the charges then circle, gives an exact conformal field theory formulation of the duality twisted theory Spin(4 struction becomes a conventional asymmetricorbifold orbifold is [ not modularmodular invariant invariance in can general. beby achieved The a if remedy shift the is in shift straightforward the in [ coordinate the of circle the coordinate T-dual circle. The T-dual circle has radius the construction can be realizedon as a generalized orbifoldby of IIB the string theory compactified z values of the terms satisfy the appropriate quantization conditions. 6.2 Orbifold pictureThe and point modular in moduli invariance space at whichunder there the is a action minimum of of the the scalar potential is a fixed point lowest number such that p (excluding the trivial case corrections to the coefficients of the Chern-Simons terms. We have checked that for the and each JHEP07(2020)086 = 8 supergravity, with N ). As a consequence, the Z 5; ] and it will be interesting to , 42 to a 6D solution and then chose the twist 4 T – 45 – 0 supersymmetry. The amount of supersymmetry , 2 , 4 , = 6 N Another open question is the computation of the Chern-Simons coefficients in the full Several interesting directions for follow-up research remain. One is to investigate the Our reduction scheme yielded a rich spectrum of massive modes. In 5D, massive BPS One of our main objectives was to study black holes in this set-up with partially broken This Scherk-Schwarz reduction in supergravity can be embedded in string theory as string theory. That is,view (such including as modes winding modes). thatout For we twists in don’t that lie detail see in as the from an T-dualitywe the group, asymmetric have this supergravity orbifold can seen point be compactification worked of in of perturbative this string work, theory. it As is possible to choose a T-duality twist that preserves the the D1-D5 CFT has beenmost studied practical the option most for inbreaking this. the patterns literature In and arise general, thereforee.g. in seems one the to (4,2), might be dual expect (2,2) the thatbackgrounds superconformal or with similar CFT, (4,0) a supersymmetry from duality supersymmetry.apply twist (4,4) the have supersymmetry D-branes results been found to discussed and there in their to [ the world-volume configurations theories discussed in here. gauge-gravitational Chern-Simons terms andblack these hole terms solutions led and to in modifications particular of modifies the the BPS expressionmicroscopic side for of the the black macroscopic hole storythe laid entropy. effects out in of this the work. duality This twist would on involve studying the CFT dual of the black holes in our set-up. Of these, partially broken supersymmetry. multiplets can be chiral.chiral fields For gives quantum twists corrections that to thegravitational yield coefficients of Chern-Simons chiral the BPS terms. pure gauge multiplets, and This integrating mixed gauge- gives out an the EFT with both pure gauge and mixed supersymmetry. Here we consideredD3-D3 — several that brane result in configurations five-dimensionalreduction. — black holes D1-D5, In after each F1-NS5 standard (untwisted) case, and dimensional in we such compactified a on way thatensures all that the the original fields black that hole source solution the remains 6D a solution solution remain of massless the in twisted 5D. theory with This orbifold of the type IIB stringstringy theory quantum and corrections so has can anthe be exact twist CFT calculated description. is exactly. In a thisIIB For non-perturbative case, more string the symmetry, theory general the in twists result which in is it which a is quotiented generalized by orbifold a of U-duality the symmetry. type a compactification with amust duality be twist. an For such elementtwist an of embedding parameters the to were discrete exist, U-dualitythought constrained the group of monodromy to Spin(5 as take beingcompactifications certain is quantized. discrete a values, fixed Thetwist point and is minimum under a could of the T-duality, hence the action the of theory be Scherk-Schwarz the arising potential at monodromy. in the When such minimum the of duality the potential is an asymmetric For this reduction weindependent have twist chosen parameters. a This monodromyMinkowski reduction vacua in yields preserving gauged the 5D R-symmetry,that depending is on preserved four depends on the number of the twist parameters that are equal to zero. explicitly. We then reduced this six-dimensional theory on a circle with a duality twist. JHEP07(2020)086 d (A.1) (A.2) . p − d ν x d ∧ 9 5 4 4 ]. ... 43 ∧ ,..., ,..., ,..., 1 1 ,..., 1 1 , ν is the rank of the form and , , . x p d = 1 3 = 0 p = 0 = 0 Indices , ...µ 1 ) 2) theory. In this reduction it may µ d , ( ˆ ν, . . . , where A ) m, n, . . . p L d ˆ µ, ν, . . . µ, ( p − M,N,... d A Z ) = 4 (0 ...ν d 1 2 ( ν 1 ) m p κ N – 46 – 2 = 1, and we work in the ‘mostly plus’ convention , etc. In all dimensions, we normalize Lagrangians , z , y ) ...µ +). The notations that we use for the coordinates ˆ µ µ B d 1 ) more often to indicate the number of spacetime = ( ˆ x x µ k d µ m e ) ε x y d , ) ( = ) = ( d = ,..., d S ( ( ~ ˆ µ g + Coordinate M R ˆ x , = p − X c )! -dimensional Newton’s constant. p d − 1 4 d = 6 = 5 = 10 = diag( T !( p is the D D Space D µν ) η d = ( N ) d ( p πG A . This table summarizes our notational conventions for coordinates and indices. ∗ = 8 ) d 2 ( κ Table 3 In general, we denote form-values fields as Finally, it would be interesting to extend this work to the study of four-dimensional such that the corresponding actions are given by where We use the subscriptdimensions or where necessary, superscript e.g. ( Throughout this work, we set for the metric, i.e. and indices in various dimensions are summarized in table is the dimension in which it lives. We define the Hodge star operator on forms as the D-ITP consortium, a program(NWO) of that the is Netherlands funded Organization byand for the by Scientific Dutch the Research Ministry FOM of programme Education, “Scanning Culture New Horizons”. and Science (OCW), A Conventions and notation Acknowledgments It is our pleasureR. to Minasian thank and C. M. Couzens, Trigiante N. for Gaddam, useful T. discussions. Grimm, This H. work het is Lam, supported K. Mayer, in part by black holes in stringexample the compactifications four-charge with D2-D6-NS5-P black dualityconfiguration. hole twists. of Another type possibility IIA For would string be this,reduction theory to to one or study four a five-dimensional could dimensional black dual take rings brane multi-center and for black their hole solutions [ D1-D5-P black hole inbe reductions possible to to the combinecalculation. the detailed study of the full string theory with the microscopic JHEP07(2020)086 5) , . For (B.1) (B.2) (B.3) (B.4) (A.3) a (6) 3 ˜ 5) can be G -frame (as , τ is the matrix and ), and therefore X ,a η 5). We construct (6) 3 , , which yields the G g 5). It is clear from the 5) in the main text. , , where , 10 matrix, satisfying the . × gX 1 (instead of , SO(5 . − B.1 τ ! X ! 5 5 for indices transforming under 5 V ∈ 1 0 ) in terms of ˜ 0 = 1 . − g 5 . 0 5 B.1 1 ! ,..., 0 , 1 5 -frame of SO(5 -frame, a generator of the Lie algebra

! 5 5) indices that transform in τ η 1

a ! , 1 -frame. Furthermore, we build an explicit ,a = 1 c = − (6) 3 τ = (6) 3 5 5 ˜ G T G a b 1 1 b

of the (dual) field strengths – 47 – 2 = a, b, . . . a 1 = is unconstrained. √ preserve the matrix 5). In the 5) is represented by a 10 ,A b (6) 3 , , g M τ , τ = η , η G 5 blocks as = X = × ˜ g SO(5 -frame and the η τ ∈ 5). For example, in 6D we have ten tensor fields (subject to η g T 10 to denote Spin(5 , T g ˜ g . We can now rewrite ( g T ), and we use 5) and its algebra , X ,..., Spin(5 -frame of SO(5 B.1 = η 5). In particular, we construct two bases in which SO(5 = 1 ⊂ , 1 − (5 X so are antisymmetric and = c A, B, . . . X 5) can be written in 5 ) = 1. Henceforth, we refer to group elements satisfying these conditions as , g and (5 a so There is another (isomorphic) way of writing down the group SO(5 Canonically, an element We use ∈ We see that the conjugatedwe matrices refer ˜ to these matrices as being written in the Note that following conditions on the conjugated group elements: where this other basis by conjugating the group elements as ˜ and det( being written in the M basis for the algebra that we use to constructconditions a vielbein B.1 The groupIn SO(5 this appendix we discuss someand details its and our algebra conventions concerningwritten the down; group we SO(5 call these the The GL(5) subgroup works onmore the information index on how this subgroup works, see appendix B Group theory explained in appendix the subgroup GL(5) a self-duality constraint), whose field strengths we write as JHEP07(2020)086 n A H (B.7) (B.5) (B.6) -frame -frame, τ τ = 0. We = 0 and C C . = = 5 matrices, and together fill the        B B × + : n A, nm -frame can be found , we build a basis of H η E M = 0. Furthermore, for 1 0 0 0 0 C , 0 0 00 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 − ). The ) by taking 5) =        , , , , . -frame is of the form = τ B B.5 0) 0) 0) 1) , , , − 1) − 0 0 0 SO(5 n < m , , A, ). 12 , , , . 1 1 0 0 1 E matrices for the ⊂ , , − − ! − A B.3 0 1 , , , A T , 0 0 − 0 1 5) in the , , A 5 and , , , 0 0 − − , , (5 GL(5) AB C so ,..., and 5) that is embedded diagonally in the – 48 – ∈ fill the lower-triangular part. They do so in such

, diag ( diag (0 diag (0 given in ( ), they all have Using the general form of 2 2 2 − diag (0 diag (0 are antisymmetric. = 1 = ! 1 1 1 X 1 1 2 √ √ √ 1 2 A, nm B.5 ˜ C − SO(5 M        E ) = = = = = 0 5). that the two frames are isomorphic. The general block T n, m ⊂ 1 2 3 4 0 , ( P and H H H H H (5 − gX A A A A A B 0 ( 1 . For example, we have A, P nm 1 0 0 0 so − − and the E −

A, X nm 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A E 5) has rank five, we have five Cartan generators, denoted by =        , and = g (5 = + ) with the matrix T so A, + nm A, 12 E + B.6 E A, (5) generator. By exponentiating, we find the corresponding group element nm A E gl 4). We choose the Cartan subalgebra to be block-diagonal in the . Generators of GL(5) can be represented by unconstrained 5 is an invertible five by five matrix. The embedding in the g is unconstrained and ,..., P A There is a subgroup GL(5) = 0 n Apart from these Cartandenote generators, them there by areupper 20 triangular root part generators of with a way that A basis forgenerators. the algebra Since ( so that whenconvenience written we in choose the the following form form for ( the where by conjugating ( matrices ˜ these can be embedded diagonallyequal in to the block the structure ( to be of the form structure for generators of the Lie algebra Here conjugation relation ˜ JHEP07(2020)086 := C B nm nm E E F (B.9) ij (B.12) (B.10) (B.11) ~ T . are cer- = 0 and + A, B mn T E gives a complete . that defined o C B nm etc. (B.8) E C nm A 23 , − E ,E        − = : satisfying , , B nm T T T , but now we have B ) T T A ,E A B 12 nm 1 B nm M C C 14 15 − . E E − M = 0, and ten root generators E − E E ( A, g nm 1 0 0 0 0 below we always mean ! 0 0 10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 − − − C = − 2 2 , , , ,E = 1 × × satisfying A        mn T = 2 C C 2 34 35 + − ) 1 g 0 1 E − A, Ω = A nm A 13 is equal to the matrix 2 ,E ,E Ω 2 E B 13 × M C C ( 24 25 g ,E 2 4 matrices × E C 2 n nm 1 − E E Ω Ω with used in the text. These matrices 0 × E , H − − − – 49 – n (5) C , , F 13

ij ,B B nm T A A E 24 25 have 4 matrices so E = ,        − 1 × M, , ,E ,E ∼ = B nm − AB − C A A g 12 34 35 E Ω = = (4) ,E ,E ,E † † C A A 23 14 15 g 1 0 0 0 0 M C A 45 45 usp 0 0 00 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 E E E − (5,5) algebra described above. In particular if we let E E        (5,5). When we mention so = = = = = that corresponds to so = b a A C B ij ij ij are constructed in the same way as T T (4) is represented by 4 ~ ~ ~ C T B T T 12 E C nm ), then we have the following definitions for usp B E F 34 = 0. The generators B ,...,T = F 13 A The set of matrices defined above Let us now discuss the notation ,T F 6= 0. The matrix 12 T The Lie algebra where Ω is the symplectic metric, given by the block matrix B.2 The isomorphism The group USp(4) is the group of 4 tain generators of( the in the construction above. Note that this implies that basis of generators of The generators C with Finally, there are ten root generators JHEP07(2020)086 4 = ] × AB 44 . ) (B.16) (B.17) (B.18) (B.13) (B.14) (B.15) a a , ). Using ˜ Γ 4 are the      B.12 1 0 ,..., − reads . To prove the 1 0 0 ba ab = 1 − Γ . 0 00 0 0 1 0 1 0 0 0 − ac Γ      . A, B = 1 bd = δ − ). Furthermore, using the (5) algebra. We conclude ab 2 3 Ω Γ so a Spin(5). Using these gamma 1 − − B.12 ). The antisymmetry follows ∼ = . ad Ω . Γ ab C = Ω Γ ab bc . B.17 , δ B satisfy the conditions ( B T A ) M . ) δ ΩΓ      ∗ 1 ab B i 1 ) 0 =      + 2 ab − A . Using this, we deduce that − δ ) i (Γ ab a bc 0 0 Ω (Ω B ˜ ab − Γ ab Γ (Γ a A Γ ∗ i − ) and the Hermitian property of the Dirac 0 0 0 = 2 M ˜ ad Γ δ M i = B − M M Ω 0 0 0 0 0 0 0 1 0 0 1 0 00 0 0 00 1 0 1 0 − A – 50 – T B.15 } and (Γ = Tr ) + 2           b Tr Tr Tr a (5) consists of real antisymmetric matrices. We 1 2 T Γ B ˜ 1 2 1 2 1 2 bd Γ ) = = , so A − a Γ a 2 5 − − − satisfies the conditions ( ˜ Γ Γ M ac Γ Γ = 1 { δ = = = ab Spin(5) can be made explicit by introducing five 4 i.e. ( − 2 ab ∗ 5 ) ]. From ( ∼ = − b M ab Γ ), is given by = (Ω = , , M , a ( T ) form a set of generators of USp(4)      cd [Γ      a B.13 1 i 1 2 Γ ab 0 , − (Γ − ab = i 0 0 Γ 1 0 0 5 are the indices corresponding to Spin(5), and i 0 0 0 ab − i 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 1 0 − ,...,           , are antisymmetric, B = = = 1 C 4 1 ) a Γ Γ a, b (Γ The isomorphism USp(4) This property is used in what follows, but it is not generally true for other choices of Ω and Γ 5 AC can check these propertiesimmediately from for the the antisymmetryreality found in condition generators we the use ( gamma thatthese matrices both constraints we Γ find that the ten matrices Γ matrices the explicit form of the isomorphism between the algebras can be derived [ The special orthogonal Lie algebra matrices, it follows directly thatClifford Γ algebra, it is straightforward to check that the commutator of Γ This is exactly the commutator of the basis elements of the Ω Hence, the symplectic metric Ω actsWe on the now gamma define matrices as Γ a charge conjugation matrix. It can easily be checked that the gamma matrices with upper indices, defined as ( gamma matrices, that satisfy the Euclidean Clifford algebra Here indices corresponding to USp(4).matrices, that An satisfies ( explicit basis of (Hermitian and traceless) gamma JHEP07(2020)086 (B.19) | | | | | | | | | | | | | | | | 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 m m m m m m m m m m m m m m m m + − − + + − + − + − + − − + − + | | | | | | | | 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 4 4 2 2 4 2 4 m m m m m m m m m m m m m m m m m m m m m m m m 0 − + − + − + + − + − − + − + + − + + − − + + − − 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 3 3 1 1 3 1 3 Mass . m m m m m m m m m m m m m m m m m m m m m m m m | | | | | | | | B − − + + + + − − − − + + + + − − A ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 m m m m m m m m m m m m m m m m ab | | | | | | | | | | | | | | | | (Γ ab (4): ) ) ) ) ) ) ) ) ), is a real antisymmetric matrix, and there- M 23 34 34 34 34 23 23 23 – 51 – usp 1 4 C C C C C C C C ) ) 3 3 + ) ) ) ) ) ) = B.17 + − − + − + + φ φ ) ) 14 34 34 24 24 23 23 2 2 2 2 ) ) 12 12 12 12 14 14 14 B i C b b φ φ B B B B B B √ √ A C C C C C C C σ (5) to + Φ) + + − + − + − + − + − − 13 24 + − + − + − + 13 24 4 M 1 1 so a a 23 ). The scalar masses are as follows: C C A A 12 12 13 13 14 14 Φ + Φ ( ( φ φ φ 34 34 34 34 23 23 23 ( 2 2 ( ( A B B B B B B 1 1 Field ˜ 2 − − A A A A A A A 2 2 ( ( ( ( ( ( √ √ 1 1 1 4.8 √ 2 2 2 2 2 2 + 4 4 √ √ 1 1 1 1 1 1 + + − − + + − φ φ √ √ √ √ √ √ ( ( 14 12 12 12 12 14 14 14 1 2 1 2 A A A A A A A A ( ( ( ( ( ( ( − ( , as given in ( 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ) and ( ab M 4.7 ) which maps B.17 Here we show thematrices masses shown in of ( the fields for the D1-D5 set-up, corresponding to the mass C Scalar and tensor masses afterC.1 Scherk-Schwarz reduction The D1-D5 system fore a suitable generatorisomorphism of ( SO(5). For completeness we also mention the inverse of the Thus we find that JHEP07(2020)086 ). 4.17 ) and ( 4.16 | | | | | | | | | | | | | | | | 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 m m m m m m m m m m m m m m m m + − − + + − + − + − + − − + − + | | | | | | | | 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 4 4 2 2 4 4 2 m m m m m m m m m m m m m m m m m m m m m m m m | | | | | | | | 0 − + − + − + + − + − − + − + + − + + − − + + − − 2 4 2 4 2 4 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 Mass m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m 0 0 | | | | | | | | − − + + + + − − − − + + + + − − − − + + + + − − 1 3 1 3 1 3 1 3 Mass 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 m m m m m m m m | | | | | | | | m m m m m m m m m m m m m m m m | | | | | | | | | | | | | | | | ) ) ) ) ) ) ) ) 23 34 34 34 34 23 23 23 – 52 – B (5) (5) (5) (5) (5) (5) (5) (5) 2; 1 2; 1 2; 2 2; 2 2; 3 2; 3 2 2 B B B B B B B ) ) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ,A B B R R R R R R 3 3 + (5) 2 ) ) ) ) ) ) + − − + − + + φ φ A ) ) + − + − + − + − 14 2 2 34 34 24 24 23 23 (5) (5) 2 2 2 2 ) ) 12 12 12 12 14 14 14 i Φ) ˜ B b b φ φ C C C C C C √ √ C C (5) (5) 2 2 B B B B B B B (5) (5) (5) (5) (5) (5) 2; 1 2; 1 2; 2 2; 2 2; 3 2; 3 σ − + + − + − + − + − + − B B − R R R R R R 13 24 13 24 + − + − + − + Field 4 1 1 a a 23 2 2 A A B B 12 12 13 13 14 14 ( ( 2 2 2 2 2 2 φ φ φ 1 1 34 34 34 34 23 23 23 1 1 1 1 1 1 ( 2 2 ( ( √ √ √ √ √ √ √ √ A C C C C C C 1 1 Field ˜ 2 + Φ + + Φ A A A A A A A 2 2 ( ( ( ( ( ( √ √ 1 1 1 √ 2 2 2 2 2 2 4 4 + √ √ 1 1 1 1 1 1 + + − − + + − √ √ √ √ √ √ φ φ ( ( 14 12 12 12 12 14 14 14 1 2 1 2 A A A A A A A A ( ( ( ( ( ( ( − ( 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 The tensor masses are given by: The scalar masses are: C.2 The F1-NS5For system the reduction of the F1-NS5 system, we chose mass matrices as in ( JHEP07(2020)086 | | | | | | | | | | | | | | | | 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 , we use the mass matrices m m m m m m m m m m m m m m m m + − − + + + − − + + − − + + − − | | | | | | | | 2 4 4 2 2 4 2 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4.2 m m m m m m m m m m m m m m m m m m m m m m m m | | | | | | | | 2 4 2 4 2 4 2 4 0 + + − − + + − − − + − + + − + − + − + − + − + − m m m m m m m m 1 3 3 1 1 3 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Mass 0 0 − − + + + + − − m m m m m m m m m m m m m m m m m m m m m m m m | | | | | | | | 1 3 1 3 1 3 1 3 Mass − − + + − + + − − + + − + − − + m m m m m m m m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | | | | | | | | m m m m m m m m m m m m m m m m | | | | | | | | | | | | | | | | ) ) ) ) 13 13 13 13 – 53 – ) ) ) ) (5) (5) (5) (5) (5) (5) (5) (5) 2; 1 2; 1 2; 2 2; 2 2; 3 2; 3 2 2 B B B B ) ) ) ) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ,A 34 34 34 34 C C R R R R R R 4 4 4 4 (5) 2 ) ) ) ) ) ) ) ) − − + + φ φ φ φ A A A A A + − + − + − + − 34 34 23 23 24 24 34 34 (5) (5) 2 2 + − − + 24 24 24 24 + + − − i ˜ B B A A A A C C B B (5) (5) 2 3 3 2 2 2 (5) (5) (5) (5) (5) (5) B B B B 2; 1 2; 1 2; 2 2; 2 2; 3 2; 3 σ 12 12 12 12 φ φ φ φ − + − + − + − + C C R R R R R R 1 23 14 24 13 + − + − A A A A Field 2 2 2 2 φ 2 2 C C B B 12 12 14 14 13 13 12 12 2 2 2 2 2 2 1 1 √ √ √ √ 14 14 14 14 1 1 1 1 1 1 − + − + √ √ √ √ √ √ √ √ C C A A A A B B Field ˜ C C C C ( ( a a a a ( ( ( ( ( ( − − 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 − + + − + − − + √ √ √ √ √ √ √ √ (Φ (Φ (Φ + (Φ + b b b b 23 23 23 23 1 2 1 2 1 2 1 2 ( ( ( ( 1 2 1 2 1 2 1 2 C C C C ( ( ( ( 1 2 2 1 1 2 1 2 ). The scalars acquire the following masses: 4.24 ) and ( 4.23 The tensor masses are: given in ( C.3 The D3-D3For system the D3-D3 brane set-up that we consider in subsection JHEP07(2020)086 153 12 Phys. , Phys. , JHEP , (1999) 010 05 (1998) 207 ]. Nucl. Phys. B (2003) 054 2 , Supergravity 09 JHEP SPIRE = 8 ]. , IN N ][ JHEP , SPIRE IN | | | | | | | | 4 2 4 2 4 2 4 2 ][ m m m m m m m m 0 0 − − + + − − + + Rholography, Black Holes and 3 1 3 1 3 1 3 1 Mass F-Theory, Spinning Black Holes and m m m m m m m m | | | | | | | | ]. Black hole entropy in M-theory Adv. Theor. Math. Phys. ]. , arXiv:1509.00455 SPIRE [ Spontaneously Broken IN SPIRE – 54 – ][ (5) (5) (5) (5) (5) (5) (5) (5) 2; 3 2; 3 2; 2 2; 2 2 2 2 2 IN arXiv:1412.7325 ]. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ,A R R R R B B C C [ (5) 2 [ A − + − + − + − + (5) (5) 2; 1 2; 1 ˜ R R (5) (5) (5) (5) 2 2 2 2 (5) (5) (5) (5) 2; 3 2; 3 2; 2 2; 2 (2016) 009 SPIRE C C B B R R R R Field IN ]. 2 2 2 2 01 2 2 2 2 ), which permits any use, distribution and reproduction in 1 1 1 1 1 1 1 1 The Odd story of massive supergravity √ √ √ √ (1979) 60 √ √ √ √ How to Get Masses from Extra Dimensions Spontaneous Breaking of Supersymmetry Through Dimensional ][ Microscopic origin of the Bekenstein-Hawking entropy ]. ]. ]. Duality twists, orbifolds and fluxes (2015) 058 82 SPIRE 06 hep-th/9601029 JHEP IN [ , SPIRE SPIRE SPIRE [ IN IN IN ]. CC-BY 4.0 ][ ][ ][ JHEP , This article is distributed under the terms of the Creative Commons (1996) 99 SPIRE hep-th/9711053 (1979) 83 Phys. Lett. B IN [ , [ Black holes and Calabi-Yau threefolds 84 379 hep-th/9901045 hep-th/9711067 hep-th/0210209 Scherk-Schwarz Lett. B [ (1979) 61 Reduction [ Multi-string Branches [ Lett. B (1997) 002 The tensors masses are: E. Cremmer, J. Scherk and J.H. Schwarz, J. Scherk and J.H. Schwarz, J. Scherk and J.H. Schwarz, N. Gaddam, A. Gnecchi, S. Vandoren and O. Varela, B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, A. Dabholkar and C. Hull, A. Strominger and C. Vafa, J.M. Maldacena, A. Strominger and E. Witten, C. Vafa, N. Kaloper and R.C. Myers, [9] [6] [7] [8] [4] [5] [1] [2] [3] [10] Attribution License ( any medium, provided the original author(s) and source are credited. References Open Access. JHEP07(2020)086 ] , , , JHEP , ] (1987) ] (1995) 109 (2003) 034 term in 288 3 2 10 438 K R (1998) 027 supergravities with hep-th/0004086 (2005) 065 [ ]. ]. 11 D ]. JHEP 5 10 , ]. hep-th/0611329 hep-th/0406018 ]. SPIRE [ SPIRE [ Nucl. Phys. B JHEP IN SPIRE IN , , JHEP IN BPS Black Holes and Rings ][ ][ (2000) 019 Nucl. Phys. B ]. SPIRE , supergravity from , SPIRE ][ origins of IN 06 IN = 5 ][ ]. D = 5 ]. ][ 6 (2007) 533 Four-dimensional black hole entropy D SPIRE (2004) 018 D ]. IN Second quantized mirror symmetry 06 JHEP ][ Black Holes and Strings with Higher SPIRE 117 ]. , SPIRE IN D IN SPIRE ][ ][ Exploring IN No-scale JHEP arXiv:0902.4032 arXiv:1303.2661 SPIRE [ , ]. [ [ IN Supersymmetric Completion of an Asymmetric Orbifolds Asymmetric orbifolds: Path integral and operator arXiv:1906.02165 – 55 – ][ [ arXiv:1710.00853 arXiv:1808.05228 [ Heterotic/type-II Duality and Non-Geometric SPIRE [ Flux compactifications of string theory on twisted tori Gauge symmetry, T-duality and doubled geometry Non-geometric backgrounds, doubled geometry and theories IN hep-th/0703087 (1991) 163 [ ][ (2009) 014 (2013) 124 Non-geometric Calabi-Yau Backgrounds and Unity of superstring dualities = 6 Prog. Theor. Phys. Compactifications with S duality twists hep-th/0503114 , hep-th/9505162 (2019) 214 Near-Horizon Analysis of 09 [ 05 D 356 [ ]. ]. ]. ]. (2017) 084 10 (2019) 037 11 (2007) 007 arXiv:0910.4907 01 JHEP JHEP SPIRE SPIRE SPIRE SPIRE [ , , 06 IN IN IN IN JHEP (1995) 59 (2009) 862 , ][ ][ ][ ][ JHEP arXiv:0711.4818 JHEP 57 , 361 [ Nucl. Phys. B , ]. JHEP , Massive string theories from M-theory and F-theory A Geometry for non-geometric string backgrounds BPS supermultiplets in five-dimensions , (2010) 056 ]. ]. ]. 02 SPIRE IN [ (2008) 043 SPIRE SPIRE SPIRE IN IN hep-th/9410167 hep-th/9811021 hep-th/0406102 hep-th/0308133 IN Derivatives JHEP from F-theory Chern-Simons terms Five-dimensional Supergravity [ formulations Phys. Lett. B [ 551 [ [ [ automorphisms Compactifications Fortsch. Phys. 08 generalised T-duality [ Scherk-Schwarz reduction of [ B. de Wit and S. Katmadas, T.W. Grimm, H. het Lam, K. Mayer and S. Vandoren, K. Hanaki, K. Ohashi and Y. Tachikawa, A. Castro, J.L. Davis, P. Kraus and F. Larsen, 5 K.S. Narain, M.H. Sarmadi and C. Vafa, S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, F. Bonetti, T.W. Grimm and S. Hohenegger, C.M. Hull, K.S. Narain, M.H. Sarmadi and C. Vafa, C.M. Hull, C.M. Hull, C. Hull, D. Israel and A. Sarti, Y. Gautier, C.M. Hull and D. Israël, C.M. Hull and P.K. Townsend, C.M. Hull and R.A. Reid-Edwards, C.M. Hull and R.A. Reid-Edwards, L. Andrianopoli, S. Ferrara and M.A. Lledó, C.M. Hull and R.A. Reid-Edwards, C.M. Hull and A. Catal-Ozer, [28] [29] [26] [27] [23] [24] [25] [21] [22] [19] [20] [16] [17] [18] [14] [15] [12] [13] [11] JHEP07(2020)086 ] , , 523 ]. (2006) Phys. , 02 factors: SPIRE IN [ Supergravity SCFTs from U(1) JHEP , 4) arXiv:1305.1929 Nucl. Phys. B = 6 , ]. [ , D (0 ]. (1984) 197 SPIRE ]. ]. IN black holes 145 ]. SPIRE F-theory with ][ ]. (2013) 115 IN D D ]. 4 ][ SPIRE SPIRE 6 07 IN IN SPIRE SPIRE IN ][ Black Holes and IN SPIRE Selfduality in Odd Dimensions ][ ]. JHEP IN no-scale model from generalized ]. , [ Phys. Lett. B ]. , = 4 One-loop Chern-Simons terms in five Dualization of dualities. 1. hep-th/0406185 , Ph.D. Thesis, Princeton U. (1996) (1984) 443] [ , (1980) [ [ SPIRE The Gaugings of Maximal black rings and N IN D – 56 – ][ 137 arXiv:1904.05361 Effective action of (1985) 617 Maxwell-Einstein Supergravity in Five-dimensions and [ ]. arXiv:1903.04947 arXiv:1302.2918 255 = 4 [ (2004) 055 [ ]. N SPIRE 07 , Centre Emile Borel, Institut Henri Poincaré(2000) The Minimal IN (2019) 043 ]. ][ SPIRE Noncommutative gauge theories on D-branes in non-geometric 08 (2019) 051 IN Addendum ibid. JHEP (2013) 043 arXiv:0712.4277 [ , ][ SPIRE [ 09 Nucl. Phys. B 07 IN , JHEP ][ Black holes in string theory Supergravity in Six-dimensions Scherk-Schwarz Reductions of Effective String Theories in Even Dimensions 3, Supergravities in 5 Dimensions JHEP K (1984) 38 JHEP = 8 , , hep-th/9710119 (2008) 068 Gauging ¨ N Kaluza-Klein Theory [ Ozer, ]. 136 03 hep-th/0504126 [ SU(2) SPIRE IN hep-th/9607235 http://people.physics.tamu.edu/pope/ihplec.pdf https://open.metu.edu.tr/handle/123456789/13736 023 dimensional reduction Rational sections make Chern-Simons[ terms jump backgrounds Lett. B [ dimensions [ Its (1998) 73 Ph.D. Thesis, Middle East[ Technical U. (2003), Type IIB on JHEP D. Gaiotto, A. Strominger and X. Yin, 5 G. Villadoro and F. Zwirner, T.W. Grimm, A. Kapfer and J. Keitel, C. Hull and R.J. Szabo, J.M. Maldacena, F. Bonetti, T.W. Grimm and S. Hohenegger, E. Cremmer, M. Awada and P.K. Townsend, P.K. Townsend, K. Pilch and P. van Nieuwenhuizen, A. Çatal- C. Pope, Y. Tanii, E. Bergshoeff, H. Samtleben and E. Sezgin, E. Cremmer, B. Julia, H. Lüand C.N. Pope, C. Couzens, H. het Lam, K. Mayer and S. Vandoren, [43] [44] [41] [42] [39] [40] [36] [37] [38] [34] [35] [31] [32] [33] [30]