Supersymmetric domain walls in maximal 6D gauged supergravity I
SSupersymmetric domain walls inmaximal 6D gauged supergravity I
Parinya Karndumri a and Patharadanai Nuchino b String Theory and Supergravity Group, Department of Physics, Faculty ofScience, Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok10330, ThailandE-mail: a [email protected]: b [email protected] Abstract
We find a large class of supersymmetric domain wall solutions fromsix-dimensional N = (2 ,
2) gauged supergravity with various gauge groups.In general, the embedding tensor lives in c representation of the globalsymmetry SO (5 , − and − representations of GL (5) ∼ R + × SL (5) ⊂ SO (5 ,
5) leading to
CSO ( p, q, − p − q ) and CSO ( p, q, − p − q ) (cid:110)R s gauge groups, respectively.These gaugings can be obtained from S reductions of seven-dimensionalgauged supergravity with CSO ( p, q, − p − q ) and CSO ( p, q, − p − q )gauge groups. As in seven dimensions, we find half-supersymmetric domainwalls for purely magnetic or purely electric gaugings with the embeddingtensors in − or − representations, respectively. In addition, fordyonic gauge groups with the embedding tensors in both − and − representations, the domain walls turn out to be -supersymmetric as inthe seven-dimensional analogue. By the DW/QFT duality, these solutionsare dual to maximal and half-maximal super Yang-Mills theories in fivedimensions. All of the solutions can be uplifted to seven dimensions andfurther embedded in type IIB or M-theories by the well-known consistenttruncation of the seven-dimensional N = 4 gauged supergravity. a r X i v : . [ h e p - t h ] F e b Introduction
Supersymmetric domain walls in gauged supergravities in various space-time di-mensions have provided a useful tool for studying various aspects of the AdS/CFTcorrespondence since the original proposal in [1], see also [2, 3]. In particular,these solutions play an important role in the so-called DW/QFT correspondence[4, 5, 6], a generalization of the AdS/CFT correspondence to non-conformal fieldtheories. They are also useful in studying some aspects of cosmology, see forexample [7, 8, 9]. Due to their importance in many areas of applications, manydomain wall solutions in gauged supergravities have been found in different space-time dimensions [10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26]. Asystematic classification of supersymmetric domain walls in various space-timedimensions can also be found in [27].In this paper, we are interested in maximal N = (2 ,
2) six-dimensionalgauged supergravity with SO (5 ,
5) global symmetry. Compared to other dimen-sions, supersymmetric solutions to this six-dimensional gauged supergravity havenot been systematically studied since the original construction of the ungauged N = (2 ,
2) supergravity long ago in [28]. The first six-dimensional gauged su-pergravity with SO (5) gauge group has been constructed in [29] by performingan S reduction of the SO (5) maximal gauged supergravity [30]. More recently,the most general gaugings have been constructed and classified in [31] using theembedding tensor formalism. From the results of [31], there are two particularlyinteresting classes of gaugings under GL (5) and SO (4 ,
4) subgroups of SO (5 , S reduction of seven-dimensionalmaximal gauged supergravity while the latter can be truncated to half-maximal N = (1 ,
1) gauged supergravity. In this paper, we will consider only gaugingsin the first class with the embedding tensor in − and − representations of GL (5). These gaugings have known seven-dimensional origins via an S reductionand can also be embedded in string/M-theory using the truncations to maximalgauged supergravity in seven dimensions. The fact that there does not exist an N = 4 superconformal symmetry in five dimensions [32] is in agreement withthe recent classification of maximally supersymmetric AdS vacua given in [33].This implies that there is no AdS /CFT duality in the case of 32 supercharges.Therefore, maximally supersymmetric vacuum solutions of the N = (2 ,
2) gaugedsupergravity are expected to be half-supersymmetric domain walls.It has been shown recently that maximally supersymmetric Yang-Millstheory in five dimensions plays an important role in the dynamics of (conformal)field theories in both higher and lower dimensions via a number of dualities, seefor example [34, 35, 36, 37, 38, 39]. In particular, this theory could even be usedto define the less known N = (2 ,
0) superconformal field theory in six dimen-sions compactified on S . The latter is well-known to describe the dynamics ofstrongly coupled theory on M5-branes. Accordingly, we expect that supersym-metric domain walls of the maximal gauged supergravity in six dimensions could2e useful in studying various aspects of the maximal super Yang-Mills theory infive dimensions via the DW/QFT correspondence. A simple domain wall solutionwith SO (5) symmetry has already been given in [29] for SO (5) gauging. In thispaper, we extend this study by including a large class of supersymmetric domainwalls with different unbroken symmetries in N = (2 ,
2) gauged supergravity withvarious gauge groups.The paper is organized as follows. In section 2, the construction of six-dimensional maximal gauged supergravity in the embedding tensor formalism isreviewed. Supersymmetric domain wall solutions from gaugings in − , − ,and ( + ) − representations are respectively given in sections 3, 4, and 5.Conclusions and discussions are given in section 6. Branching rules for rele-vant SO (5 ,
5) representations under GL (5) are given in appendix A. The conven-tions on symplectic-Majorana-Weyl Spinors in six-dimensional space-time usedthroughout this work are collected in appendix B. Finally, consistent truncationansatze for seven-dimensional SO (5) gauged supergravity on S giving rise to SO (5) maximal gauged supergravity in six dimensions are reviewed in appendixC. N = (2 , gauged supergravity in six dimen-sions We begin by giving a brief review of six-dimensional N = (2 ,
2) gauged super-gravity in the embedding tensor formalism constructed in [31]. We will mainlycollect relevant formulae for constructing the embedding tensor and finding su-persymmetric domain wall solutions. For more details, we refer the reader to theoriginal construction in [31].As in other dimensions, N = (2 ,
2) maximal supersymmetry in six di-mensions allows only a unique graviton supermultiplet with the following fieldcontent (cid:0) e ˆ µµ , B µνm , A Aµ , V Aα ˙ α , ψ + µα , ψ − µ ˙ α , χ + a ˙ α , χ − ˙ aα (cid:1) . (2.1)Most of the conventions are the same as in [31]. Curved and flat space-time indicesare respectively denoted by µ, ν, . . . = 0 , , . . . , µ, ˆ ν, . . . = 0 , , . . . ,
5. Lowerand upper m, n, . . . = 1 , . . . , GL (5) ⊂ SO (5 , A, B, . . . = 1 , . . . , SO (5 ,
5) duality symmetry. We also notethat according to this convention, the electric two-form potentials B µνm transformas under GL (5) while the vector fields A Aµ transform as c under SO (5 , SO (5) × SO (5) symmetry,are symplectic-Majorana-Weyl (SMW) spinors, see appendix B for more detail onthe convention. Indices α, . . . = 1 , . . . , α, . . . = ˙1 , . . . , ˙4 are respectively twosets of SO (5) spinor indices in SO (5) × SO (5). Similarly, vector indices of the3wo SO (5) factors are denoted by a, . . . = 1 , . . . , a, . . . = ˙1 , . . . , ˙5. We use ± to indicate space-time chiralities of the spinors. Under the local SO (5) × SO (5)symmetry, the two sets of gravitini ψ + µα and ψ − µ ˙ α transform as ( , ) and ( , )while the spin- fields χ + a ˙ α and χ − ˙ aα transform as ( , ) and ( , ).In ungauged supergravity, only the electric two-forms B µνm appear in theLagrangian while the magnetic duals B µν m transforming in representation of GL (5) are introduced on-shell. The electric and magnetic two-forms are combinedinto a vector representation of the full global symmetry group SO (5 ,
5) denotedby B µνM = ( B µνm , B µν m ). Therefore, only the subgroup GL (5) ⊂ SO (5 ,
5) is amanifest off-shell symmetry of the theory. On the other hand, the full SO (5 , GL (5) symmetry. Moreover, the magnetic two-forms can also appearin the gauged Lagrangian via topological terms.In N = (2 ,
2) supergravity, there are 25 scalar fields parametrizing thecoset space SO (5 , / ( SO (5) × SO (5)). In chiral spinor representation, we candescribe the coset manifold by a coset representative V Aα ˙ β transforming underthe global SO (5 ,
5) and local SO (5) × SO (5) by left and right multiplications,respectively. The inverse elements ( V − ) α ˙ βA will be denoted by V Aα ˙ β satisfyingthe relations V Aα ˙ β V Bα ˙ β = δ BA and V Aα ˙ β V Aγ ˙ δ = δ αγ δ ˙ β ˙ δ . (2.2)In vector representation, the coset representative is given by a 10 ×
10 matrix V M A = ( V M a , V M ˙ a ) with A = ( a, ˙ a ). This is related to the coset representative inchiral spinor representation by the following relations V M a = 116 V Aα ˙ α (Γ M ) AB ( γ a ) α ˙ αβ ˙ β V Bβ ˙ β , (2.3) V M ˙ a = − V Aα ˙ α (Γ M ) AB ( γ ˙ a ) α ˙ αβ ˙ β V Bβ ˙ β . (2.4)In these equations, (Γ M ) AB and (Γ MA ) B = (( γ a ) α ˙ αβ ˙ β , ( γ ˙ a ) α ˙ αβ ˙ β ) are respectively SO (5 ,
5) gamma matrices in non-diagonal η MN and diagonal η AB bases, see ap-pendix A for more detail.The inverse will be denoted by V MA satisfying the following relations V Ma V M b = δ ab , V M ˙ a V M ˙ b = δ ˙ a ˙ b , V Ma V M ˙ a = 0 , (2.5)and V M a V Na − V M ˙ a V N ˙ a = δ NM . (2.6)In these equations, we have explicitly raised the SO (5) × SO (5) vector index A = ( a, ˙ a ) resulting in a minus sign in equation (2.6).4he most general gaugings of six-dimensional N = (2 ,
2) supergravitycan be efficiently described by using the embedding tensor Θ
AMN . This tensorintroduces the minimal coupling of various fields via the covariant derivative D µ = ∂ µ − gA Aµ Θ AMN t MN (2.7)where g is a gauge coupling constant. The embedding tensor identifies generators X A = Θ AMN t MN of the gauge group G ⊂ SO (5 ,
5) with particular linear combi-nations of the SO (5 ,
5) generators t MN . Supersymmetry requires the embeddingtensor to transform as c representation of SO (5 , AMN canbe parametrized in term of a vector-spinor θ AM of SO (5 ,
5) asΘ
AMN = − θ B [ M (Γ N ] ) BA ≡ (cid:0) Γ [ M θ N ] (cid:1) A (2.8)with θ AM subject to the constraint(Γ M ) AB θ BM = 0 . (2.9)With the SO (5 ,
5) generators in vector and spinor representations givenby ( t MN ) P Q = 4 η P [ M δ QN ] and ( t MN ) AB = (Γ MN ) AB (2.10)in which η MN is the off-diagonal SO (5 ,
5) invariant tensor given in (A.1), thecorresponding gauge generators take the forms( X A ) M N = 2 (cid:0) Γ M θ N (cid:1) A + 2 (cid:0) Γ N θ M (cid:1) A and ( X A ) BC = (cid:0) Γ M θ N (cid:1) A (Γ MN ) BC . (2.11)For consistency, the gauge generators must form a closed subalgebra of SO (5 , X A , X B ] = − ( X A ) BC X C . (2.12)In term of θ AM , the quadratic constraint reduces to the following two conditions θ AM θ BN η MN = 0 , θ AM θ B [ N (Γ P ] ) AB = 0 . (2.13)It follows that any θ AM ∈ c satisfying this quadratic constraint defines a con-sistent gauging of the theory.To identify possible gaugings, we first decompose θ AM under a given sub-group of SO (5 , GL (5) subgroup of SO (5 ,
5) is ofparticular interest since this is the symmetry of the ungauged Lagrangian. Asgiven in [31], θ AM ∈ c decomposes under GL (5) ⊂ SO (5 ,
5) as c → +3 ⊕ +7 ⊕ − ⊕ − ⊕ − ⊕ − ⊕ +3 . (2.14)The explicit form of all the seven irreducible components can be found in ap-pendix A.4. In this case, determining consistent gaugings is to find the irreduciblecomponents satisfying the quadratic constraint (2.13).5y decomposing the SO (5 ,
5) vector index under GL (5), we can write θ AM = ( θ Am , θ Am ) with θ Am and θ Am containing the following irreducible compo-nents θ Am = +3 ⊕ − ⊕ − ⊕ − , (2.15) θ Am = +3 ⊕ +7 ⊕ − ⊕ − ⊕ +3 . (2.16)It is easily seen that the first equation in (2.13) is automatically satisfied forpurely electric or purely magnetic gaugings that involve only θ Am or θ Am compo-nents. We note that as pointed out in [31], gaugings triggered by θ Am are electricin the sense that only electric two-forms participate in the resulting gauged the-ory while magnetic gaugings triggered by θ Am involve magnetic two-forms togetherwith additional three-form tensor fields. Comparing (2.15) and (2.16) to (2.14),we immediately see that gaugings in − ⊕ − and +7 ⊕ − ⊕ +3 repre-sentations are respectively purely electric and purely magnetic whereas those in +3 ⊕ − representation correspond to dyonic gaugings involving both electricand magnetic two-forms. Other dyonic gaugings can also arise from combina-tions of various electric and magnetic components leading to many possible gaugegroups. Apart from the minimal coupling implemented by the covariant derivative(2.7), gaugings also lead to hierarchies of non-abelian vector and tensor fields ofvarious ranks. However, since we are only interested in domain wall solutionswhich only involve the metric and scalar fields, we will, from now on, set allvector and tensor fields to zero. It is straightforward to verify that this is indeeda consistent truncation. With only the metric and scalars non-vanishing, thebosonic Lagrangian of the maximal N = (2 ,
2) gauged supergravity takes theform e − L = 14 R − P a ˙ aµ P µa ˙ a − V , (2.17)and supersymmetry transformations of fermionic fields are given by δψ + µα = D µ (cid:15) + α + g γ µ T α ˙ β (cid:15) − ˙ β , (2.18) δψ − µ ˙ α = D µ (cid:15) − ˙ α − g γ µ T β ˙ α (cid:15) + β , (2.19) δχ + a ˙ α = 14 P µa ˙ a ˆ γ µ ( γ ˙ a ) ˙ α ˙ β (cid:15) − ˙ β + 2 g ( T a ) β ˙ α (cid:15) + β − g T α ˙ α ( γ a ) αβ (cid:15) + β , (2.20) δχ − ˙ aα = 14 P µa ˙ a ˆ γ µ ( γ a ) αβ (cid:15) + β + 2 g ( T ˙ a ) α ˙ β (cid:15) − ˙ β + g T α ˙ α ( γ ˙ a ) ˙ α ˙ β (cid:15) − ˙ β . (2.21)The covariant derivatives of supersymmetry parameters, (cid:15) + α and (cid:15) − ˙ α , aredefined by D µ (cid:15) + α = ∂ µ (cid:15) + α + 14 ω µνρ ˆ γ νρ (cid:15) + α + 14 Q abµ ( γ ab ) αβ (cid:15) + β , (2.22) D µ (cid:15) − ˙ α = ∂ µ (cid:15) − ˙ α + 14 ω µνρ ˆ γ νρ (cid:15) − ˙ α + 14 Q ˙ a ˙ bµ ( γ ˙ a ˙ b ) ˙ α ˙ β (cid:15) − ˙ β (2.23)6ith ˆ γ µ = e ˆ µµ ˆ γ ˆ µ . Matrices ˆ γ ˆ µ are space-time gamma matrices, see the conventionin appendix B. For simplicity, we will suppress all space-time spinor indices.The scalar vielbein P a ˙ aµ and SO (5) × SO (5) composite connections, Q abµ and Q ˙ a ˙ bµ , are given by P a ˙ aµ = 14 ( γ a ) αβ ( γ ˙ a ) ˙ α ˙ β V Aα ˙ α ∂ µ V Aβ ˙ β , (2.24) Q abµ = 18 ( γ ab ) αβ Ω ˙ α ˙ β V Aα ˙ α ∂ µ V Aβ ˙ β , (2.25) Q ˙ a ˙ bµ = 18 Ω αβ ( γ ˙ a ˙ b ) ˙ α ˙ β V Aα ˙ α ∂ µ V Aβ ˙ β (2.26)in which Ω αβ and Ω ˙ α ˙ β are the two U Sp (4) symplectic forms whose explicit formscan be found in (A.23). These definitions can be derived from the followingrelation V Aα ˙ α ∂ µ V Aβ ˙ β = 14 P a ˙ aµ ( γ a ) αβ ( γ ˙ a ) ˙ α ˙ β + 14 Q abµ ( γ ab ) αβ Ω ˙ α ˙ β + 14 Q ˙ a ˙ bµ Ω αβ ( γ ˙ a ˙ b ) ˙ α ˙ β . (2.27)The scalar potential is given by V = g θ AM θ BN V M a V N b (cid:104) V Aα ˙ α ( γ a ) αβ ( γ b ) βγ V Bγ ˙ α (cid:105) = − g (cid:2) T α ˙ α T α ˙ α − T a ) α ˙ α ( T a ) α ˙ α (cid:3) (2.28)where we have introduced the T-tensors defined by( T a ) α ˙ α = V M a θ AM V Aα ˙ α , ( T ˙ a ) α ˙ α = −V M ˙ a θ AM V Aα ˙ α (2.29)with T α ˙ α ≡ ( T a ) β ˙ α ( γ a ) βα = − ( T ˙ a ) α ˙ β ( γ ˙ a ) ˙ β ˙ α . (2.30) − representation In this section, we consider gauge groups arising from the embedding tensor in − representation. These are purely magnetic gaugings with the correspondingembedding tensor given by θ Am = T An Y nm . (3.1)The matrix T An is the inverse of the transformation matrix T An given in (A.59)and Y mn is a symmetric 5 × θ Am = 0, the embedding tensor θ AM =( 0 , T An Y nm ) automatically satisfies the quadratic constraint (2.13). Therefore,7very symmetric tensor Y mn defines a viable gauging in − representation. Asin [40], we can use SL (5) ⊂ GL (5) symmetry to bring Y mn to the form Y mn = diag(1 , .., (cid:124) (cid:123)(cid:122) (cid:125) p , − , .., − (cid:124) (cid:123)(cid:122) (cid:125) q , , .., (cid:124) (cid:123)(cid:122) (cid:125) r ) (3.2)where p + q + r = 5.Under GL (5), the gauge generators transforming as a spinor s of SO (5 , X A = T Am X m + T mnA X mn + T ∗ A X ∗ . (3.3)For the embedding tensor in − representation, the only non-vanishing gaugegenerators are given by X mn = 2 Y p [ m t pn ] (3.4)with t pn being GL (5) generators. In vector representation, the explicit form of X mn is given by ( X mn ) P Q = − δ Q [ m Y n ] p δ pP + η P [ m Y n ] q η qQ ) . (3.5)These generators satisfy the commutation relations[ X mn , X pq ] = ( X mn ) pqrs X rs (3.6)in which ( X mn ) pqrs = 2( X mn ) [ p [ r δ s ] q ] . Therefore, the corresponding gauge group isdetermined to be G = CSO ( p, q, r ) = SO ( p, q ) (cid:110) R ( p + q ) · r . (3.7)These gaugings arise from an S reduction of seven-dimensional maximal gaugedsupergravity with the same gauge groups. In the case of SO (5) gauge group ( p = 5and q = r = 0), the complete reduction ansatz has already been constructed in[29]. In order to find supersymmetric domain wall solutions, we take the space-timemetric to be the standard domain wall ansatz ds = e A ( r ) η ¯ µ ¯ ν dx ¯ µ dx ¯ ν + dr (3.8)where ¯ µ, ¯ ν = 0 , , . . . ,
4, and A ( r ) is a warp factor depending only on the radial co-ordinate r . To parametrize the coset representative of SO (5 , / ( SO (5) × SO (5)),we first identify the corresponding non-compact generators of SO (5 ,
5) in the ba-sis with diagonal SO (5 ,
5) metric η AB . These are given byˆ t a ˙ b = M aM M ˙ bN t MN (3.9)8here M AM = ( M aM , M ˙ aM ) is the inverse of the transformation matrix M givenin (A.50).We then split these generators into two parts that are symmetric andantisymmetric in a and ˙ b indices as followsˆ t a ˙ b = ˆ t + a ˙ b + ˆ t − a ˙ b (3.10)with ˆ t + a ˙ b = 12 (cid:0) ˆ t a ˙ b + ˆ t b ˙ a (cid:1) and ˆ t − a ˙ b = 12 (cid:0) ˆ t a ˙ b − ˆ t b ˙ a (cid:1) . (3.11)It is now straightforward to check that symmetric generators ˆ t + a ˙ b are given by ( t mn + t nm ) which are non-compact generators of GL (5). Accordingly, thescalars corresponding to these generators parametrize the submanifold GL (5) /SO (5).The antisymmetric generators ˆ t − a ˙ b correspond to the shift generators s mn . There-fore, the 25 non-compact generators decompose into25 (cid:124)(cid:123)(cid:122)(cid:125) ˆ t a ˙ b → (cid:124) (cid:123)(cid:122) (cid:125) ˆ t + a ˙ b + 10 (cid:124)(cid:123)(cid:122)(cid:125) s mn . (3.12)We can also separate the trace part of ˆ t + a ˙ b , corresponding to the dilatonscalar field ϕ in GL (5) /SO (5) ∼ R + × SL (5) /SO (5) scalar coset. This generatoris the R + (cid:39) SO (1 ,
1) generator defined in (A.4). In terms of ˆ t + a ˙ b , this is given by d = ˆ t +1˙1 + ˆ t +2˙2 + ˆ t +3˙3 + ˆ t +4˙4 + ˆ t +5˙5 . (3.13)The remaining generators can be identified as the fourteen non-compact genera-tors corresponding to scalar fields { φ , ..., φ } in the SL (5) /SO (5) coset. Thesegenerators are given by the symmetric traceless part˜ t a ˙ b = ˆ t + a ˙ b − d δ a ˙ b (3.14)satisfying δ a ˙ b ˜ t a ˙ b = 0.The other ten scalars denoted by { ς , ..., ς } correspond to the shift gen-erators s mn . These will be called the axions or shift scalars in this work. Thedecomposition in equation (3.12) is in agreement with that in [29] in which theconsistent circle reduction of seven-dimensional SO (5) gauged supergravity giv-ing rise to SO (5) gauged theory in six dimensions is performed. From a higher-dimensional perspective, the fourteen scalars are the seven-dimensional scalarsparameterizing the SL (5) /SO (5) coset in seven dimensions while the dilaton andshift scalars descend from the reduction of seven-dimensional metric and vectorfields, respectively, see appendix C for more detail.By this decomposition of the scalar fields, we can rewrite the kinetic termsof the scalars in (2.17) and obtain the following bosonic Lagrangian e − L = 14 R − G IJ ∂ µ Φ I ∂ µ Φ J − V (3.15)9n which G IJ is a symmetric matrix depending on scalar fields denoted by Φ I = { ϕ, φ , . . . , φ , ς , . . . , ς } with I, J = 1 , . . . , δψ +¯ µα : A (cid:48) ˆ γ ˆ r (cid:15) + α + 12 gT α ˙ α (cid:15) ˙ α − = 0 , (3.16) δψ − ¯ µ ˙ α : A (cid:48) ˆ γ ˆ r (cid:15) − ˙ α − gT ˙ αα (cid:15) α + = 0 . (3.17)In these equations, we have used the notation A (cid:48) = dAdr . We will use a prime todenote an r -derivative throughout the paper.Multiply the first equation by A (cid:48) ˆ γ ˆ r and use the second equation or vice-versa, we find the following consistency conditions A (cid:48) δ αβ = 14 g T α ˙ α Ω ˙ α ˙ β T γ ˙ β Ω βγ = W δ αβ , (3.18) A (cid:48) δ ˙ α ˙ β = 14 g T ˙ αα Ω αβ T β ˙ γ Ω ˙ β ˙ γ = W δ ˙ α ˙ β (3.19)in which we have introduced the “superpotential” W . We then obtain the BPSequations for the warped factor A (cid:48) = ±W . (3.20)Using this result in equations (3.16) and (3.17) leads to the following projectorson the Killing spinorsˆ γ ˆ r (cid:15) + α = P α ˙ α (cid:15) ˙ α − and ˆ γ ˆ r (cid:15) − ˙ α = P ˙ αα (cid:15) α + (3.21)with P α ˙ α = − g T α ˙ α A (cid:48) and P ˙ αα = 12 g T ˙ αα A (cid:48) (3.22)satisfying P α ˙ α P ˙ αβ = δ αβ and P ˙ αα P α ˙ β = δ ˙ α ˙ β . The conditions δψ + rα = 0 and δψ − r ˙ α determine the Killing spinors as functions of the radial coordinate r as usual.Using these projectors in δχ + a ˙ α = 0 and δχ − ˙ aα = 0 equations, we even-tually obtain the BPS equations for scalars. These equations are of the formΦ I (cid:48) = ∓ G IJ ∂ W ∂ Φ J (3.23)in which G IJ is the inverse of the scalar metric G IJ defined in (3.15).In addition, the scalar potential can also be written in term of W as V = 2 G IJ ∂ W ∂ Φ I ∂ W ∂ Φ J − W . (3.24)10t is well-known that the BPS equations of the form (3.20) and (3.23) satisfy thesecond-order field equations derived from the bosonic Lagrangian (3.15) with thescalar potential given by (3.24), see [41, 42, 43, 44, 45, 46] for more detail.As in other dimensions, we will follow the approach introduced in [47] toexplicitly find supersymmetric domain wall solutions involving only a subset ofthe 25 scalars that is invariant under a particular subgroup H ⊂ G to make theanalysis more traceable. SO (5) symmetric domain walls We first consider supersymmetric domain walls with the maximal unbroken sym-metry SO (5) ⊂ CSO ( p, q, − p − q ). The only gauge group containing SO (5)as a subgroup is SO (5) with Y mn = δ mn . In this case, only the dilaton ϕ corre-sponding to the non-compact generator (3.13) is invariant under SO (5). Thus,the coset representative can be written as V = e ϕ d . (3.25)We recall that this coset representative is a 16 ×
16 matrix with an index structure V AB . To compute the T-tensor, we need to write the SO (5) × SO (5) index asa pair of SO (5) spinor indices resulting in the coset representative of the form V Aα ˙ α . To achieve this, we use the transformation matrices p introduced in (A.35)so that V Aα ˙ α and its inverse V Aα ˙ α are given by V Aα ˙ α = V AB p Bα ˙ α and V Aα ˙ α = ( V − ) BA p Bα ˙ α . (3.26)With all these, it is now straightforward to find the T-tensor T α ˙ β = 52 √ e ϕ Ω αβ δ ˙ ββ = 2 g W Ω αβ δ ˙ ββ (3.27)from which the superpotential is given by W = 5 g √ e ϕ . (3.28)The resulting scalar potential reads V = − g e ϕ (3.29)which does not admit any stationary points.The general analysis given above leads to the BPS equation for the warpfactor A (cid:48) = 5 g √ e ϕ (3.30)11nd the following projector ˆ γ r (cid:15) ± = (cid:15) ∓ . (3.31)For definiteness, we have chosen a particular sign choice in the A (cid:48) equation andthe γ ˆ r projector. The condition δψ ± r = 0 gives the standard solution for theKilling spinors (cid:15) ± = e A ( r )2 (cid:15) ± (3.32)with the constant spinors (cid:15) ± satisfying ˆ γ r (cid:15) ± = (cid:15) ∓ . Accordingly, the solution ishalf-supersymmetric.The BPS equation for the dilaton can be found from the condition δχ ± =0 with the projector (3.31). This results in a simple equation ϕ (cid:48) = − g √ e ϕ . (3.33)All of these equations can be readily solved to obtain the solution A = 5 ln (cid:18) gr √ − C (cid:19) , ϕ = − ln (cid:18) gr √ − C (cid:19) . (3.34)The integration constant C can be removed by shifting the radial coordinate r . We have also neglected an additive integration constant for A since it canbe absorbed by rescaling the coordinates x ¯ µ . This is the SO (5) domain walloriginally found in [29]. In order to recover the same form of the solution, weredefine the radial coordinate as r → √ g (cid:104) C + (3 √ gr + C ) − (cid:105) and set ϕ = √ σ . SO (4) symmetric domain walls We now look for more complicated solutions with SO (4) symmetry. The gaugegroups that contain SO (4) as a subgroup are SO (5), SO (4 , CSO (4 , , Y mn = diag(1 , , , , κ ) (3.35)with κ = 1 , , − SO (5), CSO (4 , , SO (4 ,
1) gaugegroups, respectively.There are two SO (4) singlet scalars. The first one is the dilaton corre-sponding to the non-compact generator (3.13), and the other one comes from the SL (5) /SO (5) coset corresponding to the non-compact generator Y = ˆ t +1˙1 + ˆ t +2˙2 + ˆ t +3˙3 + ˆ t +4˙4 − t +5˙5 . (3.36)Using the coset representative V = e ϕ d + φ Y , (3.37)12e find that the T-tensor is given by T α ˙ β = 12 √ e ϕ − φ (4 + κe φ ) Ω αβ δ ˙ ββ = 2 g W Ω αβ δ ˙ ββ . (3.38)This leads to the superpotential and the scalar potential of the form W = g √ e ϕ − φ (4 + κe φ ) , (3.39) V = − g e ϕ − φ (cid:0) κe φ − κ e φ (cid:1) . (3.40)Using the projector (3.31), we find the BPS equations A (cid:48) = g √ e ϕ − φ (4 + κe φ ) , (3.41) ϕ (cid:48) = − g √ e ϕ − φ (4 + κe φ ) , (3.42) φ (cid:48) = g √ e ϕ − φ (1 − κe φ ) . (3.43)The resulting solutions for the dilation ϕ and the warp factor A as functions of φ are given by ϕ = − φ + C + 116 ln (cid:0) − κe φ (cid:1) , (3.44) A = − ϕ = 5 φ − C −
516 ln (cid:0) − κe φ (cid:1) . (3.45)To obtain the solution for φ , we change r to a new radial coordinate ρ defined by dρdr = e ϕ +6 φ . The solution for φ is then given by e φ = 1 √ κ tanh (cid:104) √ κ ( √ gρ + C ) (cid:105) (3.46)for an integration constant C . It is useful to note that for κ = −
1, the solutionfor φ can be written as e φ = tan (cid:104) √ gρ + C (cid:105) . (3.47)For κ = 0, the solution is simply given by e φ = √ gρ + C . (3.48) SO (3) × SO (2) symmetric domain walls We now consider SO (3) × SO (2) residual symmetry, which is possible only for SO (5) and SO (3 ,
2) gauge groups. In this case, we write the embedding tensoras Y mn = diag(1 , , , κ, κ ) (3.49)13ith κ = 1 and κ = − SO (5) and SO (3 , SO (3) × SO (2) symmetry is generated by X ij , i, j = 1 , ,
3, and X . There arethree singlet scalars corresponding to the dilaton and the following non-compactgenerators Y = 2 ˆ t +1˙1 + 2 ˆ t +2˙2 + 2 ˆ t +3˙3 − t +4˙4 − t +5˙5 , Y = s . (3.50)With the coset representative V = e ϕ d + φ Y + ς Y , (3.51)we find the scalar potential V = − g e ϕ − φ ) (1 + 4 κe φ ) . (3.52)The superpotential reads W = g √ e ϕ − φ (cid:112) (3 + 2 κe φ ) + 8 κ ς e φ (3.53)which can be found from the T-tensor given by T α ˙ β = 12 √ e ϕ − φ (3 + 2 κe φ ) Ω αβ δ ˙ ββ − √ κςe ϕ +12 φ δ α ˙ β . (3.54)In this case, it turns out that consistency of the supersymmetry conditionsfrom δχ ± requires ς = 0. Therefore, in order to find a consistent set of BPSsolutions, we need to truncate the axion out. With ς = 0, the superpotential isgiven by W = g √ e ϕ − φ (3 + 2 κe φ ) . (3.55)With the projector (3.31), we find the following BPS equations A (cid:48) = g √ e ϕ − φ (3 + 2 κe φ ) , (3.56) ϕ (cid:48) = − g √ e ϕ − φ (3 + 2 κe φ ) , (3.57) φ (cid:48) = g √ e ϕ − φ (1 − κe φ ) . (3.58)It can be verified that all these equations satisfy the corresponding field equationsas expected.With a new radial coordinate ρ given by dρdr = e ϕ +2 φ , we obtain the domainwall solution ϕ = − φ C + 116 ln (cid:0) − κe φ (cid:1) , (3.59) A = − ϕ = 15 φ − C −
516 ln (cid:0) − κe φ (cid:1) , (3.60) e φ = 1 √ κ tanh (cid:104) √ κ ( √ gρ + C ) (cid:105) . (3.61)14 .5 SO (3) symmetric domain walls We now move to domain wall solutions with SO (3) symmetry. Many gauge groupscontain SO (3) as a subgroup with the embedding tensor parameterized by Y mn = diag(1 , , , κ, λ ) (3.62)for κ, λ = 0 , ±
1. With this embedding tensor, the SO (3) symmetry is generatedby X mn , m, n = 1 , ,
3. In addition to the dilaton, there are four singlet scalarscorresponding to the following non-compact generators Y = 2 ˆ t +1˙1 + 2 ˆ t +2˙2 + 2 ˆ t +3˙3 − t +4˙4 − t +5˙5 , Y = ˆ t +4˙5 , Y = ˆ t +4˙4 − ˆ t +5˙5 , Y = s . (3.63)With the only exception for κ = λ = 0 corresponding to CSO (3 , ,
2) gaugegroup, we need to truncate out the scalar corresponding to s generator in orderto find a consistent set of BPS equations as in the previous case. For the moment,we will set this shift scalar to zero and consider the particular case of κ = λ = 0afterward.For vanishing shift scalars, the coset representative is given by V = e ϕ d + φ Y + φ Y + φ Y (3.64)giving rise to the superpotential and the scalar potential of the form W = ge ϕ − φ √ (cid:2) e φ (( κ + λ ) cosh 2 φ cosh 4 φ − ( κ − λ ) sinh 4 φ ) (cid:3) , (3.65) V = − g e ϕ − φ ) (cid:20) e φ (( κ + λ ) cosh 2 φ cosh 4 φ − ( κ − λ ) sinh 4 φ )+ e φ (cid:18) κ + 10 κλ + λ − (3 κ − κλ + 3 λ ) cosh 8 φ − κ + λ ) cosh 4 φ cosh φ + 4( κ − λ ) cosh 2 φ sinh 8 φ (cid:19)(cid:21) . (3.66)We also note the matrix G IJ in this case G IJ = 160 diag(6 , , φ ,
15) (3.67)for Φ I = { ϕ, φ , φ , φ } with I, J = 1 , , , Q r and Q ˙4˙5 r appearingin δψ ± r = 0 conditions. In more detail, there are additional terms involving( γ ) αβ (cid:15) + β and ( γ ˙4˙5 ) ˙ α ˙ β (cid:15) − ˙ β in the covariant derivative of the supersymmetry pa-rameters, see equations (2.22) and (2.23). According to this, we modify the ansatzfor the Killing spinors to (cid:15) + = e A ( r )2 + B ( r ) γ (cid:15) and (cid:15) − = e A ( r )2 + B ( r ) γ ˙4˙5 (cid:15) − (3.68)15here B ( r ) is an r -dependent function, and (cid:15) ± are constant symplectic-Majorana-Weyl spinors satisfying ˆ γ r (cid:15) ± = (cid:15) ∓ .Using this ansatz for the Killing spinors satisfying the projector (3.31), wefind the following set of BPS equations from the supersymmetry transformationsof fermions A (cid:48) = ge ϕ − φ √ (cid:2) e φ (( κ + λ ) cosh 2 φ cosh 4 φ − ( κ − λ ) sinh 4 φ ) (cid:3) , (3.69) ϕ (cid:48) = − ge ϕ − φ √ (cid:2) e φ (( κ + λ ) cosh 2 φ cosh 4 φ − ( κ − λ ) sinh 4 φ ) (cid:3) , (3.70) φ (cid:48) = ge ϕ − φ √ (cid:2) − e φ (( κ + λ ) cosh 2 φ cosh 4 φ − ( κ − λ ) sinh 4 φ ) (cid:3) , (3.71) φ (cid:48) = − g √ e ϕ +12 φ ( κ + λ ) sinh 2 φ sech 4 φ , (3.72) φ (cid:48) = ge ϕ +12 φ √ κ − λ ) cosh 4 φ − ( κ + λ ) cosh 2 φ sinh 4 φ ) (3.73)together with B (cid:48) = − g √ e ϕ +12 φ ( κ + λ ) sinh 2 φ tanh 4 φ . (3.74)To find explicit solutions, we will separately discuss various possible values of κ and λ . CSO (4 , , and CSO (3 , , gauge groups For λ = 0 and κ (cid:54) = 0, the gauge groups are given by CSO (4 , ,
1) and
CSO (3 , , κ = 1 and κ = −
1, respectively. Using a new radial coordinate ρ defined by dρdr = e ϕ +12 φ , a domain wall solution to the BPS equations can be obtained φ = 14 ln (cid:20) g ρ (1 + 2 C ) + 2(1 + C ) g ρ (1 + 2 C ) + 2 C (cid:21) , (3.75) φ = 18 ln (cid:20) e φ − C e φ + C + 1 e φ + C e φ − C − (cid:21) , (3.76) φ = 120 ln (cid:34) κ (cid:0) C ( e φ − (cid:1)(cid:112) (1 − e φ ) ( C e φ − ( C + 1) ) (cid:35) , (3.77) ϕ = φ C −
116 ln (cid:20) C ( e φ −
1) + 2 e φ − (cid:21) , (3.78) A = − ϕ = − φ − C + 516 ln (cid:20) C ( e φ −
1) + 2 e φ − (cid:21) (3.79)together with B = 14 sin − C (cid:115) e φ − C + 1 + 14 tan − (cid:34)(cid:115) (1 − e φ )( C + 1) C e φ − ( C + 1) (cid:35) . (3.80)16e have chosen integration constants for φ and B to be zero for simplicity. SO (4 , gauge group In SO (4 ,
1) gauge group with κ = − λ = 1, the BPS equations give φ (cid:48) = B (cid:48) = 0.Accordingly, we can set B = 0 and φ = 0. We can readily verify that this is aconsistent truncation. Taking φ = 0 and redefining the radial coordinate r to ρ as given in the CSO (4 , ,
1) and
CSO (3 , ,
1) gauge groups, we obtain a domainwall solution φ = 12 tanh − (cid:104) tan (cid:104) √ gρ + C (cid:105)(cid:105) , (3.81) φ = 120 ln (cid:2) e φ ( C + 1) + C e − φ (cid:3) , (3.82) ϕ = φ C + 116 ln (cid:20) e φ + 12 ( C + e φ ( C + 1)) (cid:21) , (3.83) A = − ϕ = − φ − C −
516 ln (cid:20) e φ + 12 ( C + e φ ( C + 1)) (cid:21) . (3.84) SO (5) and CSO (3 , gauge groups For κ = λ = ± SO (5) and SO (3 ,
2) gauge groups. we find thefollowing domain wall solution φ = 14 ln (cid:34) − e √ gκρ + e √ gκρ + e √ gκρ +2 C e √ gκρ + e √ gκρ + e √ gκρ +2 C (cid:35) , (3.85) φ = 18 ln (cid:20) e φ + e φ + C − e C e φ − e φ + C + e C (cid:21) , (3.86) φ = φ
10 + 120 ln (cid:34) κe − φ (cid:0) C ( e φ − − (cid:1)(cid:112) e φ − e C ( e φ − (cid:35) , (3.87) ϕ = φ C −
116 ln (cid:2) e φ − (cid:3) + 116 ln[ C ( e φ − − , (3.88) A = − ϕ = − φ − C + 516 ln (cid:2) e φ − (cid:3) + 116 ln[ C ( e φ − −
2] (3.89)in term of the new radial coordinate ρ defined previously. The function B ( r )appearing in the Killing spinors is given in term of φ as B = −
18 tan − (cid:34) − e φ + 2 e − C (cid:112) e φ − C − ( e φ − (cid:35) −
18 tan − (cid:34) e φ (1 + 2 e − C ) − (cid:112) e φ − C − ( e φ − (cid:35) (3.90)in which the integration constant has been set to zero.17 .5.4 Domain walls in CSO (3 , , gauge group In the case of
CSO (3 , ,
2) gauge group with κ = λ = 0, supersymmetry al-lows a non-vanishing axion corresponding to Y generator. We write the cosetrepresentative as V = e ϕ d + φ Y + φ Y + φ Y + ς Y (3.91)and find a simple scalar potential V = − g e ϕ − φ ) . (3.92)We also note that this potential does not depend on ς and can be obtained from(3.66) by setting κ = λ = 0. This potential can also be written in the form (3.24)using the superpotential W = 3 g √ e ϕ − φ (3.93)and the symmetric matrix G IJ = ς − ς φ ς − ς ς (3.94)for Φ I = { ϕ, φ , φ , φ , ς } , I, J = 1 , , , , A (cid:48) = 3 g √ e ϕ − φ , ϕ (cid:48) = − g √ e ϕ − φ , φ (cid:48) = g √ e ϕ − φ ,φ (cid:48) = φ (cid:48) = 0 , ς (cid:48) = − g √ e ϕ − φ ς . (3.95)Except for an additional equation for ς , these are the BPS equations obtainedfrom (3.69) to (3.73) by setting κ = λ = 0. Furthermore, φ and φ can beconsistently truncated out since the scalar potential (3.92) is independent of φ and φ . With all these, we find a domain wall solution φ = 14 ln (cid:20)
25 ( √ gρ + C ) (cid:21) , (3.96) ϕ = C −
18 ln (cid:20)
25 ( √ gρ + C ) (cid:21) , (3.97) ς = C e − φ = C (cid:0) ( √ gρ + C ) (cid:1) , (3.98) A = − ϕ = − C + 58 ln (cid:20)
25 ( √ gρ + C ) (cid:21) (3.99)18n which ρ is a new radial coordinate defined by dρdr = e ϕ − φ . It should also benoted that the axion ς can also be truncated out. SO (2) × SO (2) symmetric domain walls As a final example of domain wall solutions in − representation, we consideran SO (2) × SO (2) unbroken symmetry. In this case, the embedding tensor forall possible gauge groups takes the form Y mn = diag(1 , , κ, κ, λ ) (3.100)for λ = 0 , ± κ = ±
1. These gauge groups are SO (5) ( κ = λ = 1), SO (4 , κ = − λ = 1), SO (3 ,
2) ( κ = − λ = − CSO (4 , ,
1) ( κ = 1 , λ = 0) and CSO (2 , ,
1) ( κ = − , λ = 0).There are five scalars invariant under SO (2) × SO (2) generated by X and X . As usual, one of these is the dilaton and the other four are associatedwith the following non-compact generators Y = ˆ t +1˙1 + ˆ t +2˙2 − t +5˙5 , Y = ˆ t +3˙3 + ˆ t +4˙4 − t +5˙5 , Y = s , Y = s . (3.101)As in many previous cases, we need to truncate out the axions corresponding tothe shift generators s and s in order to find a consistent set of BPS equationsthat are compatible with the field equations. We then take the coset representa-tive of the form V = e ϕ d + φ Y + φ Y . (3.102)The resulting scalar potential reads V = − g e ϕ − φ + φ )) (cid:2) κ (2 + λe φ +8 φ ) + λe φ +12 φ (4 − λe φ +8 φ ) (cid:3) (3.103)which can be written in terms of the superpotential W = g √ e ϕ (2 e − φ + 2 κe − φ + λe φ + φ ) ) (3.104)using G IJ = − −
110 320 (3.105)for Φ I = { ϕ, φ , φ } , I, J = 1 , ,
3. 19sing the projector (3.31) together with the Killing spinors (3.32), wefind the following BPS equations A (cid:48) = g √ e ϕ (2 e − φ + 2 κe − φ + λe φ + φ ) ) , (3.106) ϕ (cid:48) = − g √ e ϕ (2 e − φ + 2 κe − φ + λe φ + φ ) ) , (3.107) φ (cid:48) = g √ e ϕ (3 e − φ − κe − φ − λe φ + φ ) ) , (3.108) φ (cid:48) = g √ e ϕ (3 κe − φ − e − φ − λe φ + φ ) ) . (3.109)Solving these BPS equations gives a domain wall solution φ = − √ gρ
10 + 320 ln (cid:104) κe √ gρ + C (cid:105) −
120 ln (cid:104) C e − √ gρ + λ (cid:105) , (3.110) φ = − φ −
112 ln (cid:104) C e − √ gρ + λ (cid:105) , (3.111) ϕ = − φ − √ gρ
24 + C −
148 ln (cid:104) C e − √ gρ + λ (cid:105) , (3.112) A = − ϕ = 5 φ √ gρ − C + 548 ln (cid:104) C e − √ gρ + λ (cid:105) (3.113)in which ρ is the new radial coordinate defined by the relation dρdr = e ϕ − φ .For domain walls preserving smaller residual symmetries such as SO (2) diag ⊂ SO (2) × SO (2) and SO (2), there are many more scalars, and the analysis is muchmore involved without any possibility for complete analytic solutions. We willnot consider these cases in this work. − represen-tation In this section, we consider gaugings in which the irreducible part of the em-bedding tensor transforms in − representation. These gauged theories areobtained from a consistent circle reduction of the maximal seven-dimensional CSO ( p, q, − p − q ) gauged supergravity constructed in [40].In six dimensions, gaugings in − representation are purely electric andtriggered by θ Am = T Anp U np,m (4.1)where U mn,p = U [ mn ] ,p satisfying U [ mn,p ] = 0. With θ AM = ( T Anp U np,m , U mn,r U pq,s ε mnpqt = 0 . (4.2)20his condition can be solved by setting U mn,p = v [ m w n ] p (4.3)in which v m is a GL (5) vector and w mn is a symmetric tensor, w mn = w ( mn ) .To classify possible gauge groups, we follow [40] by using the SL (5) sym-metry to further fix v m = δ m and split the index m = ( i, i = 1 , ..,
4. Forsimplicity, we also restrict to cases with w i = w = 0. The remaining SL (4)residual symmetry can be used to diagonalize the 4 × w ij of the form w ij = diag(1 , .., (cid:124) (cid:123)(cid:122) (cid:125) p , − , .., − (cid:124) (cid:123)(cid:122) (cid:125) q , , .., (cid:124) (cid:123)(cid:122) (cid:125) r ) (4.4)with p + q + r = 4. From the decomposition in (3.3), we find that in this case,only X ij and X i gauge generators are non-vanishing. The generators X ij aregiven in terms of the GL (5) generators while X i only involve the shift generators.Explicitly, these generators are given by X ij = 1 √ ε ijkm w kl t ml and X i = w ij s j . (4.5)It is now straightforward to show that the gauge generators satisfy thefollowing commutation relations[ X i , X j ] = 0 , [ X ij , X k ] = ( X ij ) lk X l , [ X ij , X kl ] = ( X ij ) klmn X mn (4.6)in which ( X ij ) klmn = 2( X ij ) [ k [ m δ n ] l ] . This implies that the corresponding gaugegroup is of the form G = CSO ( p, q, − p − q ) (cid:110) R s = SO ( p, q ) (cid:110) (cid:0) R ( p + q )(4 − p − q ) × R s (cid:1) . (4.7)The CSO ( p, q, − p − q ) factor and the four-dimensional translation group fromthe shift symmetries R s are respectively generated by X ij and X i .We should note here that the corresponding gauge group in seven dimen-sions is just CSO ( p, q, − p − q ). After an S reduction, this gauge group isaccompanied by a translation group R s . As pointed out in [31], the completeoff-shell symmetry group of the maximal six-dimensional gauged supergravity is GL (5) (cid:110) − , with − being shift symmetries of scalar fields. The gauge groupgiven in (4.7) is embedded in GL (5) (cid:110) − as CSO ( p, q, − p − q ) ⊂ GL (5) and R s ⊂ − . We also note that in vector representation of SO (5 , X i ) P Q = 4 η P [ j δ Qk ] δ j w ki , ( X ij ) P Q = √ ε ijkm w kl ( δ mP δ Ql − η mP η lP ) . (4.8)By splitting the SO (5) × SO (5) vector indices as a = ( i,
5) and ˙ a =(˙ i, ˙5), we find the following decomposition for non-compact generators of SL (5) ⊂ GL (5) ⊂ SO (5 ,
5) under SL (5) → SL (4) × SO (1 , t a ˙ b → (cid:0) ˜ t i ˙ j , ˜ t i ˙5 , ˜ t (cid:1) . (4.9)21ince the SL (5) generators ˜ t a ˙ b are traceless, the generator ˜ t is related to thetrace part of ˜ t i ˙ j according to ˜ t + ˜ t + ˜ t + ˜ t = − ˜ t . It is then convenienceto define new non-compact generators t i ˙ j as t i ˙ j = ˜ t i ˙ j − t δ i ˙ j (4.10)which are symmetric traceless. The nine scalar fields corresponding to these gen-erators then parametrize an SL (4) /SO (4) coset. The other four scalars associatedwith ˜ t i ˙5 = ˆ t + i ˙5 are nilpotent scalars and will be denoted by b i as in seven dimen-sions. In addition, there are also ten axions corresponding to the antisymmetricshift generators as in the previous section.As in the previous section, we will systematically find supersymmetricdomain walls invariant under some residual symmetries of the CSO ( p, q, − p − q )factor in the gauge group. SO (4) symmetric domain walls We first consider domain walls with the largest possible unbroken symmetrynamely SO (4). The only gauge group containing SO (4) as a subgroup is SO (4) (cid:110)R s with the embedding tensor parametrized by w ij = δ ij . The SO (4) symmetryis generated by X ij , i, j = 1 , , ,
4, generators.There are two SO (4) singlet scalars given by the dilaton ϕ and anotherdilatonic scalar corresponding to the SO (1 ,
1) factor in SL (4) × SO (1 , ⊂ SL (5).The latter is given by the non-compact generator (cid:101) Y = ˆ t +1˙1 + ˆ t +2˙2 + ˆ t +3˙3 + ˆ t +4˙4 − t +5˙5 (4.11)and will be denoted by φ .The coset representative can be written as V = e ϕ d + φ (cid:101) Y (4.12)leading to the T-tensor given by T α ˙ β = e ϕ − φ ( γ ) αβ δ ˙ ββ = 2 g W ( γ ) αβ δ ˙ ββ (4.13)with the superpotential W = g e ϕ − φ . (4.14)The appearance of γ rather than other SO (5) gamma matrices is due to thespecific choice of v m = δ m for the tensor U mn,p . The scalar potential can also bedirectly computed and is given by V = − g e ϕ − φ . (4.15)22he Killing spinors are given by the same ansatz as in (3.32) but in thiscase subject to the following projectorˆ γ r (cid:15) ± = γ (cid:15) ∓ . (4.16)because of the appearance of γ in the T-tensor. With this new projector, it isnow straightforward to derive the following BPS equations A (cid:48) = g e ϕ − φ , ϕ (cid:48) = − g e ϕ − φ , φ (cid:48) = g e ϕ − φ . (4.17)These equations are solved by the solution ϕ = 4 C −
15 ln (cid:104) gr C (cid:105) , (4.18) φ = C + 15 ln (cid:104) gr C (cid:105) , (4.19) A = − ϕ = ln (cid:104) gr C (cid:105) − C . (4.20) SO (3) symmetric domain walls We now look for more complicated solutions with SO (3) symmetry. Gauge groupswith an SO (3) subgroup are SO (4) (cid:110) R s , SO (3 , (cid:110) R s , and CSO (3 , , (cid:110) R s which are collectively described by the symmetric tensor w ij = diag(1 , , , κ ) (4.21)for κ = 1 , − ,
0, respectively.The residual symmetry SO (3) is generated by the generators X ˆ i withˆ i = 1 , ,
3. Apart from the two dilatons, there are three additional SO (3) singletscalars, one from the SL (4) /SO (4) coset and the other two from symmetric andantisymmetric axions denoted by b and ς . These three singlets correspond to thefollowing SO (5 ,
5) non-compact generators (cid:101) Y = ˆ t +1˙1 + ˆ t +2˙2 + ˆ t +3˙3 − t +4˙4 , (cid:101) Y = ˆ t +4˙5 , (cid:101) Y = s . (4.22)Using the coset representative of the form V = e ϕ d + φ (cid:101) Y + φ (cid:101) Y + b (cid:101) Y + ς (cid:101) Y , (4.23)we find the scalar potential and the T-tensor given by V = − g e ϕ − φ +3 φ )) (cid:0) κe φ + (9 e φ + κ ) cosh 2 b (cid:1) (4.24)and T α ˙ β = 14 e ϕ − φ +3 φ ) (cid:104) (3 e φ + κ ) cosh b ( γ ) αβ δ ˙ ββ + (3 e φ − κ ) sinh b ( γ ) αβ δ ˙ ββ (cid:105) . (4.25)It turns out that consistency of the BPS equations from δχ ± conditions requiresvanishing symmetric axion b unless κ = 0 corresponding to CSO (3 , , (cid:110) R s gauge group. 23 .2.1 Domain walls without the symmetric axion With b = 0, the scalar potential and superpotential read V = − g e ϕ − φ +3 φ )) (3 e φ + 6 κe φ − κ ) , (4.26) W = g e ϕ − φ +3 φ ) (3 e φ + κ ) . (4.27)Imposing the projector (4.16) on the Killing spinors (3.32), we can derivethe following set of BPS equations A (cid:48) = g e ϕ − φ +3 φ ) (3 e φ + κ ) , (4.28) ϕ (cid:48) = − g e ϕ − φ +3 φ ) (3 e φ + κ ) , (4.29) φ (cid:48) = g e ϕ − φ +3 φ ) (3 e φ + κ ) , (4.30) φ (cid:48) = − g e ϕ − φ +3 φ ) (3 e φ − κ ) , (4.31) ς (cid:48) = − ge ϕ − φ +3 φ ) ς . (4.32)From these equations, we can find the solutions for A , ϕ and φ as functions of φ of the form φ = φ C −
120 ln( e φ − κ ) , (4.33) ϕ = − φ C − C + 120 ln( e φ − κ ) , (4.34) A = − ϕ = φ − C + 5 C −
14 ln( e φ − κ ) . (4.35)With the new radial coordinate ρ defined by dρdr = e ϕ − φ + φ ) , the solutions for φ and ς are given by e φ = √ κ tanh (cid:2) √ κ ( gρ + C ) (cid:3) and ς = C csch (cid:2) √ κ ( gρ + C ) (cid:3) . (4.36)In particular, for κ = − κ = 0, we find respectively e φ = − tan (cid:2) √ κ ( gρ + C ) (cid:3) , ς = C csc ( gρ + C ) (4.37)and e φ = 1( gρ + C ) , ς = C . (4.38)24 .2.2 Domain walls with the symmetric axion For κ = 0 corresponding to CSO (3 , , (cid:110) R s gauge group, it is possible to findsolutions with the symmetric axion b non-vanishing. With κ = 0, the scalarpotential and the T-tensor are given by V = − g e ϕ − φ +4 φ ) cosh 2 b (4.39)and T α ˙ β = 34 e ϕ − φ − φ ) (cid:104) cosh b ( γ ) αβ δ ˙ ββ + sinh b ( γ ) αβ δ ˙ ββ (cid:105) . (4.40)By the general procedure given in section 3.1, we find the superpotential and ˆ γ r projectors on the Killing spinors W = 3 g e ϕ − φ +4 φ √ cosh 2 b (4.41)and ˆ γ r (cid:15) + α = Ω αβ √ cosh 2 b (cid:2) cosh b ( γ ) βγ δ ˙ αγ + sinh b ( γ ) βγ δ ˙ αγ (cid:3) (cid:15) − ˙ α , (4.42)ˆ γ r (cid:15) − ˙ α = − Ω ˙ α ˙ β √ cosh 2 b (cid:104) cosh b ( γ ) αβ δ ˙ ββ + sinh b ( γ ) αβ δ ˙ ββ (cid:105) (cid:15) + α . (4.43)It should be noted that these projectors are not independent. Therefore, theresulting solutions will preserve half of the supersymmetry. Moreover, we caneasily see that these projectors reduce to that given in (4.16) for b = 0.With all these, we find the following set of BPS equations A (cid:48) = 3 g e ϕ − φ +4 φ √ cosh 2 b, B (cid:48) = − ge ϕ − φ +4 φ tanh 2 b √ cosh 2 b , (4.44) ϕ (cid:48) = − g e ϕ − φ +4 φ √ cosh 2 b, φ (cid:48) = − ge ϕ − φ +4 φ (cosh 4 b − / b , (4.45) φ (cid:48) = − ge ϕ − φ +4 φ (cosh 4 b + 7)64 cosh / b , b (cid:48) = − ge ϕ − φ +4 φ sinh 2 b √ cosh 2 b , (4.46)together with ς (cid:48) = 0. Since the scalar potential does not depend on ς , we canconsistently truncate ς out by setting ς = 0. The domain wall solution to the25bove BPS equations is then given by √ sinh 2 b ( 3 gρ C b ) = F ( 14 , , , − b ) , (4.47) ϕ = C − b
10 + 120 ln(1 − e b ) , (4.48) φ = C − b −
120 ln(1 − e b ) + 116 ln(1 + e b ) , (4.49) φ = C − b
24 + 112 ln(1 − e b ) −
116 ln(1 + e b ) , (4.50) A = − ϕ = − C + b −
14 ln(1 − e b ) , (4.51) B = 12 tan − (cid:0) e b (cid:1) + C B (4.52)in which the new radial coordinate ρ is defined by dρdr = e ϕ − φ +4 φ , and F is thehypergeometric function. SO (2) × SO (2) symmetric domain walls Domain walls preserving SO (2) × SO (2) symmetry can be found in SO (4) (cid:110) R s and SO (2 , (cid:110) R s gauge groups described by the embedding tensor with w ij = diag(1 , , κ, κ ) , κ = 1 , − . (4.53)In addition to the two dilatons, there are three SO (2) × SO (2) singletscalars corresponding to the following SO (5 ,
5) non-compact generators (cid:98) Y = ˆ t +1˙1 + ˆ t +2˙2 − ˆ t +3˙3 − ˆ t +4˙4 , (cid:98) Y = s , (cid:98) Y = s . (4.54)In this case, a consistent set of BPS equations can be found only when the scalarscorresponding to (cid:98) Y and (cid:98) Y generators vanish.With the coset representative V = e ϕ d + φ (cid:101) Y + φ (cid:98) Y , (4.55)the scalar potential and superpotential are given by V = − g κe ϕ − φ and W = g e ϕ − φ + φ ) ( e φ + κ ) . (4.56)With all these and the usual Killing spinors (3.32) subject to the projector(4.16), the resulting BPS equations read A (cid:48) = g e ϕ − φ + φ ) ( e φ + κ ) , (4.57) ϕ (cid:48) = − g e ϕ − φ + φ ) ( e φ + κ ) , (4.58) φ (cid:48) = g e ϕ − φ + φ ) ( e φ + κ ) , (4.59) φ (cid:48) = − g e ϕ − φ + φ ) ( e φ − κ ) . (4.60)26sing a new radial coordinate ρ defined by dρdr = e ϕ − φ , we find a domain wallsolution φ = φ C −
120 ln (cid:0) e φ − κ (cid:1) , (4.61) ϕ = − φ C − C + 120 ln (cid:0) e φ − κ (cid:1) , (4.62) A = − ϕ = φ − C + 5 C −
14 ln (cid:0) e φ − κ (cid:1) , (4.63) e φ = √ κ tanh (cid:2) √ κ ( gρ + C ) (cid:3) . (4.64) SO (2) symmetric domain walls As a final example in this case, we consider SO (2) symmetric domain walls. Thereare many gauge groups admitting an SO (2) subgroup. They are collectivelycharacterized by the following component of the embedding tensor w ij = diag(1 , , κ, λ ) . (4.65)Together with the two dilatons, there are additional nine SO (2) singletscalars. Three of them are in the SL (4) /SO (4) coset corresponding to non-compact generators Y = ˆ t +1˙1 + ˆ t +2˙2 − ˆ t +3˙3 − ˆ t +4˙4 , Y = ˆ t +3˙4 , Y = ˆ t +3˙3 − ˆ t +4˙4 . (4.66)The remaining ones consist of two nilpotent scalars associated with Y = ˆ t +3˙5 , Y = ˆ t +4˙5 (4.67)and four shift scalars corresponding to Y = s , Y = s , Y = s , Y = s . (4.68)However, dealing with all eleven scalars turns out to be highly complicated, sowe perform a subtruncation by setting the shift scalar corresponding to s andthe two nilpotent scalars to zero. It is straightforward to verify that this is aconsistent truncation and still gives interesting solutions. We now end up witheight singlet scalars with the coset representative V = e ϕ d + φ (cid:101) Y + φ Y + φ Y + φ Y + ς Y + ς Y + ς Y . (4.69)Consistency of the resulting BPS equations requires vanishing of the shift scalar ς unless κ = λ = 0 corresponding to CSO (2 , , (cid:110)R s gauge group. In what follows,we will for the moment set ς = 0 and separately consider the CSO (2 , , (cid:110) R s gauge group with ς (cid:54) = 0. 27ith ς = 0, we can compute the scalar potential and the superpotentialof the form V = − g e ϕ − φ + φ )) (cid:20) κ + 10 κλ + λ − κ + λ ) cosh 4 φ cosh φ − (3 κ − κλ + 3 λ ) cosh 8 φ + 16( κ − λ ) e φ sinh 4 φ + 4( κ + λ ) cosh 2 φ (cid:0) e φ cosh 4 φ − ( κ − λ ) sinh 8 φ (cid:1)(cid:21) , (4.70) W = g e ϕ − φ + φ ) (cid:2) e φ + ( κ + λ ) cosh 2 φ cosh 4 φ + ( κ − λ ) sinh 4 φ (cid:3) . (4.71)This scalar potential can be written in term of the superpotential according to(3.24) using G IJ = (cid:18) G ij G iy G xj G xy (cid:19) (4.72)where G ij = 140 diag(4 , , , φ , , G xj = (cid:32) ς − ς − ς ς e φ (1+ e φ ) ς ς − ς − ς ς e φ (1+ e φ ) ς (cid:33) , (4.73)and G xy = (cid:32) ς + ς e φ sech φ ς ς (1+4 e φ + e φ )(1+ e φ ) ς ς (1+4 e φ + e φ )(1+ e φ ) ς + ς e φ sech φ (cid:33) . (4.74)Here, we have denoted Φ I = { ϕ, φ , φ , φ , φ , ς , ς } = { Φ i , Φ x } for i, j = 1 , , . . . , x, y = 6 ,
7. Note also that the scalar potential for
CSO (2 , , (cid:110) R s gaugegroup with κ = λ = 0 vanishes identically leading to a family of Minkowski vacua.Imposing the projector (4.16) on the Killing spinors of the form (cid:15) + = e A ( r )2 + B ( r ) γ (cid:15) and (cid:15) − = e A ( r )2 + B ( r ) γ ˙3˙4 (cid:15) − , (4.75)we obtain the following set of BPS equations A (cid:48) = g e ϕ − φ + φ ) (cid:2) e φ + ( κ + λ ) cosh 2 φ cosh 4 φ + ( κ − λ ) sinh 4 φ (cid:3) , (4.76) ϕ (cid:48) = − g e ϕ − φ + φ ) (cid:2) e φ + ( κ + λ ) cosh 2 φ cosh 4 φ + ( κ − λ ) sinh 4 φ (cid:3) , (4.77) φ (cid:48) = g e ϕ − φ + φ ) (cid:2) e φ + ( κ + λ ) cosh 2 φ cosh 4 φ + ( κ − λ ) sinh 4 φ (cid:3) , (4.78) φ (cid:48) = − g e ϕ − φ + φ ) (cid:2) e φ − ( κ + λ ) cosh 2 φ cosh 4 φ − ( κ − λ ) sinh 4 φ (cid:3) , (4.79) φ (cid:48) = − g e φ − φ + φ ) ( κ + λ ) sinh 2 φ sech 4 φ , (4.80) φ (cid:48) = − g e φ − φ + φ ) (( κ + λ ) cosh 2 φ sinh 3 φ + ( κ − λ ) cosh 4 φ ) (4.81)28ogether with B (cid:48) = − g e φ − φ + φ ) ( κ + λ ) sinh 2 φ tanh 4 φ , (4.82) ς (cid:48) = − ge ϕ +4 φ e φ + φ ) [ ς ( κ − λ + ( κ + λ ) cosh 2 φ ) + ς ( κ + λ ) sinh 2 φ sech 4 φ ] , (4.83) ς (cid:48) = − ge ϕ − φ e φ + φ ) [ ς ( κ + λ ) sinh 2 φ sech 4 φ − ς ( κ − λ − ( κ + λ ) cosh 2 φ )] . (4.84)We are unable to completely solve these equations for arbitrary values of theparameters κ and λ . However, the solutions can be separately found for eachvalue of κ and λ . SO (3 , (cid:110) R s gauge group In this case, we set κ = − λ = 1, and the BPS equations give B (cid:48) = φ (cid:48) = 0. Wecan again truncate φ out and set the constant B = 0. As a result, we find adomain wall solution φ = 12 φ −
18 ln (cid:2) C (1 + e φ ) (cid:3) , (4.85) φ = C + 110 φ −
120 ln(1 + e φ ) + 140 ln (cid:2) C (1 + e φ ) (cid:3) , (4.86) ϕ = C − C − φ + 120 ln(1 + e φ ) −
140 ln (cid:2) C (1 + e φ ) (cid:3) , (4.87) A = − ϕ = 5( C − C ) + 12 φ −
14 ln(1 + e φ ) + 18 ln (cid:2) C (1 + e φ ) (cid:3) , (4.88) φ = 14 ln tan( C − gρ ) , (4.89) ς = C sec( C − gρ ) , (4.90) ς = C csc( C − gρ ) (4.91)with ρ defined by dρdr = e ϕ − φ + φ ) . 29 .4.2 Domain walls in CSO (3 , , (cid:110) R s and CSO (2 , , (cid:110) R s gaugegroups For λ = 0 and κ = ± CSO (3 , , (cid:110) R s and CSO (2 , , (cid:110) R s gauge groups, the domain wall solution is given by φ = 116 ln (cid:2) ( e φ − e C + e C − e C +4 φ ) (cid:3) −
18 ln(4 − e φ ) , (4.92) φ = 14 ln (cid:20) e C ) + (1 + 2 e C ) g ρ e C + (1 + 2 e C ) g ρ (cid:21) , (4.93) φ = 18 ln (cid:20) ( e φ − e C + e C +2 φ )1 + e C + e φ − e C +4 φ (cid:21) , (4.94) φ = C − φ + 140 ln(1 − e φ ) −
140 ln (cid:2) e C + e C − e C +4 φ (cid:3) , (4.95) ϕ = C − φ , (4.96) A = − ϕ (4.97)together with B = C B + 14 sin − (cid:34) e C (cid:114) e φ −
11 + 2 e C (cid:35) + 14 tan − (cid:34)(cid:114) ( e φ − e C ) e C + e C − e C +4 φ (cid:35) . (4.98)In this solution, we have defined the coordinate ρ by dρdr = e − φ − φ and set theintegration constant for φ solution to be C = e C ) in order to simplify theexpression for the solution. We also note that the two gauge groups have exactlythe same domain wall solution since the parameter κ does not appear anywherein the solution. In more detail, κ appears in φ solution as g κ ρ , but this termis simply given by g ρ for κ = ± ς and ς , we are not able to analytically findtheir solutions. We can instead perform a numerical analysis to find these solu-tions, but we will not pursue any further along this direction. In any case, thesescalars can be consistently truncated out since they do not appear in the scalarpotential. SO (4) (cid:110) R s and SO (2 , (cid:110) R s gauge groups In this case, we set κ = λ = ± SO (4) (cid:110) R s and SO (2 , (cid:110) R s gauge groups. As in the previous case, the resulting BPS solutions are verycomplicated to find explicit solutions. Therefore, we will set ς = ς = 0 and find30he domain wall solution for the remaining fields as follows φ = 116 ln (cid:2) e φ − e C ( e φ − (cid:3) −
18 ln(2 − e φ ) , (4.99) φ = 14 ln (cid:20) − e gκρ + e gκρ + 4 e C e gκρ + e gκρ + 4 e C (cid:21) , (4.100) φ = 18 ln (cid:20) e φ − e C + e C +4 φ e φ + e C − e C +4 φ (cid:21) , (4.101) φ = C − φ −
120 ln( e φ −
1) + 14 ln (cid:2) e φ − e C + e φ +2 C (2 − e φ ) (cid:3) , (4.102) ϕ = C − φ , (4.103) B = C B −
18 tan − (cid:20) e − C ( e φ − e C + 2 e C +4 φ )2 √ e φ − e C + 2 e C +4 φ − e C +8 φ (cid:21) −
18 tan − (cid:34) e − C (1 + 2 e C + 2 e C +4 φ )2 (cid:112) e φ (1 + 2 e C ) − e C − e C +8 φ (cid:35) , (4.104) A = − ϕ (4.105)with dρdr = e ϕ − φ + φ ) . CSO (2 , , (cid:110) R s gauge group Finally, we consider the case of κ = λ = 0 corresponding to CSO (2 , , (cid:110) R s gauge group. Using the coset representative (4.69), we find the T-tensor given by T α ˙ β = 12 e ϕ − φ − φ ) (cid:104) ( γ ) αβ δ ˙ ββ + 2 ς ( γ ) αβ δ ˙ ββ (cid:105) . (4.106)By the general procedure given in section 3.1, we find the superpotential W = g e ϕ − φ − φ ) (cid:113) ς + 1 (4.107)and the following projectorsˆ γ r (cid:15) + α = Ω αβ (cid:104) ( γ ) αβ δ ˙ ββ + 2 ς ( γ ) αβ δ ˙ ββ (cid:105)(cid:112) ς + 1 (cid:15) − ˙ β , (4.108)ˆ γ r (cid:15) − ˙ α = − Ω ˙ α ˙ β (cid:104) ( γ ) αγ δ ˙ βγ + 2 ς ( γ ) αγ δ ˙ γβ (cid:105)(cid:112) ς + 1 ] (cid:15) + α . (4.109)As expected for half-supersymmetric solutions, these projectors are not indepen-dent. In addition, for ς = 0, they reduce to a simpler projector given in (4.16).At this point, it is useful to note that for this gauge group, the scalar potentialvanishes as previously mentioned, so there exists a six-dimensional Minkowski31acuum for this gauge group. However, the superpotential (4.107) does not haveany stationary points, so this Minkowski vacuum is not supersymmetric.With the following ansatz for the Killing spinors (cid:15) + = e A ( r )2 + B ( r ) γ (cid:15) and (cid:15) − = e A ( r )2 − B ( r ) γ ˙3˙4 (cid:15) − , (4.110)we can obtain the BPS equations A (cid:48) = g e ϕ − φ − φ ) (cid:113) ς + 1 , ϕ (cid:48) = − ge ϕ − φ − φ ) (1 + 20 ς )20 (cid:112) ς + 1 ,φ (cid:48) = ge ϕ − φ − φ ) (cid:112) ς + 1 , φ (cid:48) = − ge ϕ − φ − φ ) (cid:112) ς + 1 , φ (cid:48) = φ (cid:48) = 0 ,ς (cid:48) = − ge ϕ − φ − φ ) ς ς (cid:112) ς + 1 , ς (cid:48) = − ge ϕ − φ − φ ) ς ς (cid:112) ς + 1 ,ς (cid:48) = − ge ϕ − φ − φ ) ς (cid:113) ς + 1 , B (cid:48) = ge ϕ − φ − φ ) (cid:112) ς + 1 . (4.111)With a new radial coordinate ρ defined by dρdr = e ϕ − φ , the corresponding solutionis given by A = −
14 ln ς , B = C B −
12 tan − ς , (4.112) ϕ = C + 120 ln ς + 110 ln(1 + 4 ς ) , (4.113) φ = C + 120 ln ς −
140 ln(1 + 4 ς ) , (4.114) φ = C + 14 ln ς −
18 ln(1 + 4 ς ) , (4.115) ς = C (cid:113) ς , ς = C (cid:113) ς , (4.116) ς = 1 gρe C + C . (4.117) ( + ) − rep-resentation We now consider gaugings with non-vanishing components of the embedding ten-sor in both − and − representations. These gaugings are dyonic with theembedding tensor containing both electric and magnetic parts. The full embed-ding tensor is given by θ AM = ( θ Am , θ Am ) with θ Am = T Anp U np,m and θ Am = T An Y nm (5.1)32or Y mn = Y ( mn ) and U mn,p = U [ mn ] ,p satisfying U [ mn,p ] = 0.However, for dyonic gaugings, the first condition in the quadratic con-straint (2.13) is not automatically satisfied. For the embedding tensor given in(5.1), we find that this constraint imposes the following condition U np,m Y qm = 0 . (5.2)To solve this condition, we follow [40] and split the GL (5) index as m = ( i, x ).By choosing a suitable basis, we can take Y mn to be Y ij = diag(+1 , ..., +1 , − , ..., −
1) and Y xy = 0 . (5.3)The constraint (5.2) then implies that only the components U xy,z and U ix,y = U i ( x,y ) are non-vanishing. As a result, the embedding tensor is parametrized bythe following tensors Y ij , U i ( x,y ) , U xy,z . (5.4)We now consider different possible gauge groups with rank Y = 0 , , ...,
5. Thereare two trivial cases for rank Y = 5 with U mn,p = 0 and rank Y = 0 with all Y mn =0. These correspond respectively to gaugings in − and − representationsand have already been considered in the previous two sections.For rank Y = 4, only U i , can be non-vanishing, but another conditionfrom the quadratic constraint (2.13) requires U i , = 0. Accordingly, the corre-sponding gauge groups are given by CSO (4 , , CSO (3 , ,
1) and
CSO (2 , , Y = 3 and rank Y = 2. Gaugings in these cases areexpected to arise from a circle reduction of seven-dimensional maximal gaugedsupergravity with the embedding tensor in both and representations of SL (5). Similar to the seven-dimensional solutions given in [15], we will find thatin these gaugings, the domain walls are -BPS preserving eight supercharges. Forthe case of rank Y = 1, the second condition from the quadratic constraint (2.13)is much more complicated to find a non-trivial solution for U i ( x,y ) and U xy,z . Werefrain from discussing this case here. -BPS domain walls for rank Y = 3 We first consider the case of rank Y = 3 with i, j = 1 , ,
3. The second conditionfrom the quadratic constraint (2.13) becomes ε ijk U jx,z ε zw U kw,y = 1 √ Y ij U jx,y (5.5)which can be solved by U ix,y of the form U ix,y = − √ ε xz (Σ i ) zy (5.6)33here (Σ i ) xy are 2 × i , the quadratic constraint(5.5) can be rewritten as [Σ i , Σ j ] = 2 ε ijk Y kl Σ l . (5.7)As pointed out in [40], a real, non-vanishing solution for U ix,y is possible only for Y ij = diag(1 , , −
1) (5.8)with the explicit form of Σ i given in terms of Pauli matrices asΣ = σ , Σ = σ , Σ = iσ . (5.9)The constraint (5.7) is then the Lie algebra of a non-compact group SO (2 , U xy,z is not constrained by this condition,so it can be parametrized by an arbitrary two-component vector u x as U xy,z = ε xy u z . (5.10)We now consider the corresponding gauge algebra spanned by the follow-ing gauge generators X x = − √ ε yz (Σ i ) zx s iy + ε yz u x s yz , (5.11) X ij = 2 Y k [ i t kj ] + 2 √ ε ijk u x t kx − ε ijk (Σ k ) zx t zx , (5.12) X ix = Y ik t kx + 12 ε ijk (Σ j ) xz t kz . (5.13)To determine the form of the corresponding gauge group, we explicitly evaluatethese generators in vector representation and find the following commutationrelations [ X x , X y ] = 0 , [ X ij , X x ] = ( X ij ) yx X y , [ X ix , X y ] = 0 , (5.14)[ X ix , X jy ] = 0 , [ X ij , X kx ] = − X ij ) kxly X ly , (5.15)[ X ij , X kl ] = − ( X ij ) klpq X pq − X ij ) klpx X px . (5.16)Redefining the X ij generators as (cid:101) X ij = X ij − √ ε ijk η kl X lx (5.17)with η ij = diag(+1 , +1 , − (cid:101) X ij generate an SO (2 ,
1) subgroup withthe Lie algebra (cid:104) (cid:101) X ij , (cid:101) X kl (cid:105) = − ( (cid:101) X ij ) klpq (cid:101) X pq . (5.18)34he remaining generators X ix and X x , which transform non-trivially under SO (2 , X ix generators. With all these, the resulting gauge group is then given by G = SO (2 , (cid:110) (cid:0) R × R s (cid:1) (5.19)in which R s is the translation group from the shift symmetries generated by X x .As also pointed out in [40], we see that the vector u x does not change the gaugealgebra, so we can set u x = 0 for simplicity.We now look for supersymmetric domain wall solutions invariant under SO (2) ⊂ SO (2 ,
1) generated by X . There are five SO (2) singlet scalars corre-sponding to the non-compact generators Y d = ˆ t +1˙1 + ˆ t +2˙2 + ˆ t +3˙3 + ˆ t +4˙4 + ˆ t +5˙5 , (5.20) Y = 2 ˆ t +1˙1 + 2 ˆ t +2˙2 + 2 ˆ t +3˙3 − t +4˙4 − t +5˙5 , (5.21) Y = ˆ t +1˙1 + ˆ t +2˙2 − t +3˙3 , (5.22) Y = s , (5.23) Y = s . (5.24)Using the coset representative of the form V = e ϕ Y d + φ Y + φ Y + ς Y + ς Y , (5.25)we find the scalar potential V = g e ϕ − φ +2 φ ) ( e φ +6 ) . (5.26)Consistency of the BPS equations from δχ ± conditions requires ς = 0. Aftertruncating out ς , we find the T-tensor T α ˙ β = 2 g e ϕ − φ − φ (cid:104) W ( δ α δ ˙ β − δ α δ ˙ β ) + W ( δ α δ ˙ β − δ α δ ˙ β ) (cid:105) (5.27)with W = g √ e ϕ − φ − φ (3 − e φ ) , W = g √ e ϕ − φ − φ (1 − e φ ) . (5.28)It turns out that only W gives rise to the superpotential in term of which thescalar potential can be written.With the superpotential given by W , the unbroken supersymmetry cor-responds to (cid:15) ± and (cid:15) ± . Therefore, we set (cid:15) ± = (cid:15) ± = 0 in the following analysis.Alternatively, we can implement this by imposing an additional projector of theform γ (cid:15) ∓ = (cid:15) ∓ . (5.29)35y the same procedure as in the previous cases together with the projector(3.31), we obtain the BPS equations, with ς = ς , A (cid:48) = g √ e ϕ − φ − φ (3 − e φ ) , (5.30) ϕ (cid:48) = − g √ e ϕ − φ − φ (3 − e φ ) , (5.31) φ (cid:48) = g √ e ϕ − φ − φ (3 − e φ ) , (5.32) φ (cid:48) = g √ e ϕ − φ − φ (3 + e φ ) , (5.33) ς = − g √ e ϕ − φ − φ (3 − e φ ) ς . (5.34)Introducing a new radial coordinate ρ via dρdr = e ϕ − φ +2 φ , we find a domain wallsolution e φ = (cid:114)
32 tan( √ gρ + C ) , (5.35) φ = C + 25 φ −
120 ln(3 + 2 e φ ) , (5.36) ς = C e − φ (3 + 2 e φ ) , (5.37) ϕ = C − C − φ −
380 ln(3 + 2 e φ ) , (5.38) A = − ϕ = − C + 154 C + 32 φ + 316 ln(3 + 2 e φ ) . (5.39) -BPS domain walls for rank Y = 2 In this case, i, j = 1 ,
2, we have Y ij = diag(1 , ± U xy,z , x, y, . . . = 3 , , × u xy as U xy,z = 12 √ ε xyt u tz (5.40)with u xx = 0. The non-vanishing gauge generators read X x = 12 √ (cid:15) yz,t u tx s yz , (5.41) X = 2 Y k [1 t k + 12 u xy t xy , (5.42) X ix = Y ij t jx − ε ij u xy t jy (5.43)with the commutation relations given by[ X x , X y ] = 0 , [ X x , X iy ] = 0 , [ X ix , X jy ] = 0 , (5.44)[ X , X x ] = ( X ) yx X y , [ X , X ix ] = − X ) ixjy X jy . (5.45)36 x and X ix commute with each other and separately generate two translationgroups R s and R which transform non-trivially under X . The single X gen-erator in turn leads to a compact SO (2) or a non-compact SO (1 ,
1) group for Y ij = diag(1 ,
1) or Y ij = diag(1 , − SO (2) (cid:110) ( R × R s ) or SO (1 , (cid:110) ( R × R s ). SO (2) (cid:110) ( R × R s ) gauge group To find solutions with a non-trivial residual symmetry, we will consider SO (2) (cid:110) ( R × R s ) gauge group with Y ij = δ ij . In vector representation, the X generatoris given by ( X ) mn = (cid:18) i ( σ ) ij × × u xy (cid:19) . (5.46)Accordingly, we choose the matrix u xy to be u xy = − λ λ (5.47)with λ ∈ R . The SO (2) subgroup is then embedded diagonally with only X and X non-vanishing. Thus, the corresponding gauge group, in this case, is given by SO (2) (cid:110) ( R × R s ).There are five SO (2) singlets corresponding to the following non-compactgenerators commuting with X Y d = ˆ t +1˙1 + ˆ t +2˙2 + ˆ t +3˙3 + ˆ t +4˙4 + ˆ t +5˙5 , (5.48) Y = 3 ˆ t +1˙1 + 3 ˆ t +2˙2 − t +3˙3 − t +4˙4 − t +5˙5 , (5.49) Y = − t +3˙3 + ˆ t +4˙4 + ˆ t +5˙5 , (5.50) Y = s , (5.51) Y = s . (5.52)With the coset representative V = e ϕ Y d + φ Y + φ Y + ς Y + ς Y , (5.53)it turns out that the scalar potential vanishes identically. The T-tensor is givenby T α ˙ β = e ϕ − φ √ (cid:2) λ ( γ ) αβ + 2 Ω αβ + 2 ς (cid:2) λ ( γ ) αβ − γ ) αβ (cid:3)(cid:3) δ ˙ ββ (5.54)or explicitly T α ˙ β = e ϕ − φ √ λ + 2) ς λ + 2) 00 2( λ − ς − ( λ − − ( λ + 2) 0 2( λ + 2) ς
00 ( λ −
2) 0 2( λ − ς . (5.55)37his leads to two superpotentials W = g √ e ϕ − φ ( λ + 2) (cid:113) ς , (5.56) W = g √ e ϕ − φ ( λ − (cid:113) ς . (5.57)Unlike the previous rank Y = 3 case, both of these give a valid superpotential interm of which the scalar potential can be written. As in the previous case, half ofthe supersymmetry is broken by choosing any one of these two possibilities whichagain corresponds to imposing an additional γ projector of the form γ (cid:15) ± = (cid:15) ± or γ (cid:15) ± = − (cid:15) ± (5.58)for W = W or W = W , respectively. Together with the usual ˆ γ r projectorsˆ γ r (cid:15) + α = Ω αβ T α ˙ β A (cid:48) (cid:15) − ˙ β , ˆ γ r (cid:15) − ˙ α = − Ω ˙ α ˙ β T α ˙ β A (cid:48) (cid:15) + α , (5.59)the resulting solutions will preserve only eight supercharges or of the originalsupersymmetry.With the following ansatz for the Killing spinors (cid:15) + = e A ( r )2 + B ( r ) γ (cid:15) and (cid:15) − = e A ( r )2 − B ( r ) γ ˙1˙2 (cid:15) − , (5.60)for (cid:15) ± satisfying the projectors (5.58) and (5.59), we obtain the following BPSequations A (cid:48) = g √ e ϕ − φ ( λ ± (cid:113) ς , B (cid:48) = gς e ϕ − φ ( λ ± (cid:112) ς , (5.61) ϕ (cid:48) = − ge ϕ − φ ( λ ± ς )20 (cid:112) ς , φ (cid:48) = ge ϕ − φ ( λ ± (cid:112) ς , (5.62) φ (cid:48) = 0 , ς (cid:48) = g √ ς e ϕ − φ ( λ ± (cid:113) ς , (5.63) ς (cid:48) = g √ ς e ϕ − φ ( λ ± (cid:113) ς . (5.64)The choices of plus or minus signs in these equations are correlated with the plusor minus signs of the two projectors given in (5.58).We can consistently set φ = 0 and find a domain wall solution A = −
14 ln ς , B = C B −
12 tan − ς , (5.65) ϕ = C + 120 ln ς + 110 ln(1 + 4 ς ) , (5.66) φ = C + 110 ln ς −
120 ln(1 + 4 ς ) , (5.67) ς = 12 tan (cid:104) √ e − C ( λ ± gρ + C (cid:105) , (5.68) ς = C ς (5.69)38here ρ is the new radial coordinate defined by dρdr = e ϕ − φ . CSO (2 , , (cid:110) R s gauge group From the previous result, there are special values of λ = ± SO (2) (cid:110) ( R × R s ) gauge group reduces to SO (2) (cid:110) ( R × R s ) ∼ CSO (2 , , (cid:110) R s . Thetwo choices are equivalent, so we will choose λ = 2 for definiteness.In this case, there are nine scalars invariant under the residual SO (2)symmetry generated by X . They are given by the five scalars associated withthe non-compact generators given in (5.48) to (5.52) together with additional twosymmetric and two shift scalars respectively corresponding to Y = ˆ t +1˙4 + ˆ t +2˙5 , Y = ˆ t +1˙5 − ˆ t +2˙4 , (5.70) Y = s + s , Y = s − s . (5.71)However, with this large number of scalar fields, the analysis is highly compli-cated. To make things more manageable, we will further truncate the nine scalarsto the previous five singlets together with each of the two sets of axionic scalarsseparately.Turning on two shift scalars, denoted by ς and ς , corresponding to Y and Y generators, we find the solution given in equations (5.65) to (5.69) to-gether with the solutions for ς and ς of the form ς = C (cid:113) ς and ς = C (cid:113) ς . (5.72)More interesting solutions are obtained by including the scalars corre-sponding to Y and Y generators. With the coset representative V = e ϕ Y d + φ Y + φ Y + φ Y + φ Y + ς Y + ς Y , (5.73)we find two superpotentials. One of them vanishes identically while the non-trivial one is given by W = g √ e ϕ − φ (cid:113) cosh φ cosh φ + ( ς − ς + cosh 2 φ cosh 2 φ ( ς + ς )) . (5.74)Only the supersymmetry corresponding to this superpotential is unbroken. Thisagain amounts to imposing a γ projector of the form (5.58). Furthermore, con-sistency of the BPS equations from δχ ± requires ς = ς = ς . It is useful to notethe explicit form of the T-tensor for ς = ς = ς which is given by T α ˙ β = e ϕ − φ √ φ cosh 2 φ (cid:2) ( γ ) αβ + Ω αβ + 2 ς (cid:0) ( γ ) αβ − ( γ ) αβ (cid:1)(cid:3) δ ˙ ββ . (5.75)39sing the Killing spinors (5.60) subject to the projectors in (5.59) andthe first projector in (5.58), we can derive the following BPS equations A (cid:48) = g √ e ϕ − φ cosh 2 φ cosh 2 φ (cid:112) ς , (5.76) B (cid:48) = 2 ge ϕ − φ cosh 2 φ cosh 2 φ ς √ ς , (5.77) ϕ (cid:48) = − ge ϕ − φ cosh 2 φ cosh 2 φ (1 + 20 ς )5 √ ς , (5.78) φ (cid:48) = ge ϕ − φ (cosh φ cosh φ + 5) sech 2 φ sech 2 φ √ ς , (5.79) φ (cid:48) = ge ϕ − φ (cosh φ cosh φ −
1) sech 2 φ sech 2 φ √ ς , (5.80) φ (cid:48) = − √ ge ϕ − φ sinh 2 φ ) sech 2 φ √ ς , (5.81) φ (cid:48) = − √ ge ϕ − φ cosh 2 φ ) sinh 2 φ √ ς , (5.82) ς (cid:48) = − ge ϕ − φ cosh 2 φ cosh 2 φ ς (cid:112) ς . (5.83)Introducing a new radial coordinate ρ via dρdr = e ϕ − φ √ ς , we eventually find adomain wall solution φ = C −
112 ln (cid:0) e φ + 1 (cid:1) + 124 ln (cid:2) e C (1 − e φ + e φ ) − e φ (cid:3) , (5.84) φ = 14 ln (cid:34) e √ gρ + e √ gρ + 4 e C − e √ gρ + e √ gρ + 4 e C (cid:35) , (5.85) φ = 14 ln (cid:20) e φ − e C + e C +4 φ e φ + e C − e C +4 φ (cid:21) , (5.86) φ = C + φ −
110 ln( e φ −
1) + 110 ln( e φ + 1) , (5.87) ϕ = C + 120 ln( e φ −
1) + 110 ln (cid:2) e C (1 − e φ + e φ ) − e φ (cid:3) (5.88) −
18 ln (cid:2) e φ − e C (1 − e φ + e φ ) + 4 e C (1 − e φ + e φ ) (cid:3) , (5.89) A = 18 ln (cid:20) e φ − e C (1 − e φ + e φ ) + 4 e C (1 − e φ + e φ )( e φ − (cid:21) , (5.90) ς = e C ( e φ − (cid:112) e C (1 − e φ + e φ ) − e C (1 − e φ + e φ ) − e φ . (5.91)We end this section by noting that a domain wall solution with ς = 0 cansimilarly be obtained with the coordinate ρ defined by dρdr = e ϕ − φ . In this case,40he solutions for the dilaton and warp factor are given by ϕ = C + 120 ln( e φ − −
140 ln (cid:2) e φ − e C (1 − e φ + e φ ) (cid:3) , (5.92) A = − ϕ (5.93)while solutions for the remaining scalars are the same as given above. We have constructed the embedding tensors of six-dimensional maximal N =(2 ,
2) gauged supergravity for various gauge groups with known seven-dimensionalorigins via an S reduction. These gaugings are triggered by the embedding tensorin − and − representations of GL (5) ⊂ SO (5 ,
5) duality symmetry. In − representation, the corresponding gauge group is CSO ( p, q, − p − q ) which isthe same as its seven-dimensional counterpart. On the other hand, for gaugingsin − representation, additional translation groups R n s associated with the shiftsymmetries on the scalar fields appear in the gaugings resulting in CSO ( p, q, − p − q ) (cid:110) R s gauge group. This is also the case for gaugings in ( + ) − representation with gauge groups SO (2 , (cid:110) ( R × R s ), SO (2) (cid:110) ( R × R s ), and CSO (2 , , (cid:110) R s .We have also studied supersymmetric domain wall solutions and founda large number of half-supersymmetric domain walls from purely magnetic andpurely electric gaugings in − and − representations, respectively. In addi-tion, we have given -supersymmetric domain walls for dyonic gaugings involvingthe embedding tensor in both − and − representations. These are similarto the seven-dimensional solutions and in agreement with the general classifi-cation of supersymmetric domain walls in [27] in which the existence of -BPSdomain walls has been pointed out.Apart from solutions with seven-dimensional analogues, we have alsofound solutions that are not uplifted to seven-dimensional domain walls due tothe presence of axionic scalars leading to non-vanishing vector fields in seven di-mensions. This can be explicitly seen from the truncation ansatz collected inappendix C. Although this ansatz has originally been given only for SO (5) gaugegroup, a similar ansatz with possibly suitable modifications in the tensor fieldcontent is also applicable for other gauge groups. In particular, the fact thata truncation of seven-dimensional vectors leads to axionic scalars in six dimen-sions is still true. Therefore, domain wall solutions with non-vanishing axionicscalars obtained in this work cannot be obtained from an S reduction of anydomain wall solutions in seven dimensions. Accordingly, these solutions are gen-uine six-dimensional domain walls without seven-dimensional analogues. As afinal comment, we note that there is no SO (5) symmetric domain wall in sevendimensions since there is no SO (5) singlet scalar in SL (5) /SO (5) coset. The41ix-dimensional SO (5) symmetric domain wall, on the other hand, arises form an S reduction of the supersymmetric AdS vacuum by the general result of [48].The seven-dimensional origin of all the gaugings considered in this workcan also be embedded in ten or eleven dimensions, so the six-dimensional domainwall solutions can be embedded in string/M-theory via the corresponding seven-dimensional truncations. Accordingly, the solutions given here are hopefully use-ful in the study of DW /QFT duality for maximal supersymmetric Yang-Millstheory in five dimensions from both six-dimensional framework and string/M-theory context. It is interesting to explicitly uplift the domain wall solutions toseven dimensions and subsequently to ten or eleven dimensions using the trunca-tion ansatze given in [49, 50, 51, 52, 53].Constructing truncation ansatz of string/M-theory to six dimensions us-ing SO (5 ,
5) exceptional field theory given in [54] is also of particular interest.This would allow uplifting the six-dimensional solutions directly to ten or elevendimensions. In this paper, we have considered only gaugings with the embeddingtensor in − and − representations. It is natural to extend this study byperforming a similar analysis for the embedding tensors in other GL (5) repre-sentations as well as finding supersymmetric domain walls. Unlike the solutionsobtained in this paper, these solutions will not have seven-dimensional counter-parts since these gaugings are not simply related to seven-dimensional gaugingsvia an S reduction.It is also interesting to construct the embedding tensors for various gaug-ings under SO (4 , ⊂ SO (5 , N = (1 ,
1) supergravity coupled to four vector multiplets in whichsupersymmetric
AdS vacua are known to exist in the presence of both conven-tional gaugings and massive deformations [55, 56, 57]. Finding supersymmetricsolutions from these gauge groups could be useful in the study of AdS /CFT correspondence. Finally, finding supersymmetric curved domain walls with non-vanishing vector and tensor fields as in seven-dimensional maximal gauged su-pergravity in [58, 59] is worth considering. This type of solutions can describeconformal defects or holographic RG flows from five-dimensional N = 4 superYang-Mills theories to lower dimensions. Along this line, examples of solutionsdual to surface defects from N = (1 ,
1) gauged supergravity have appeared re-cently in [60].
Acknowledgement
This work is supported by the Second Century Fund (C2F), Chulalongkorn Uni-versity. P. K. is supported by The Thailand Research Fund (TRF) under grantRSA6280022. 42 GL (5) Branching rules
In this appendix, we collect all of the SO (5 , → GL (5) branching rules usedthroughout the paper. Relevant decompositions have already been given in [31],but in order to construct the embedding tensor, we need a concrete realization.Therefore, we will determine the decompositions for various representations of SO (5 ,
5) in terms of GL (5) representations using explicit matrix forms. A.1 Vector
A vector or fundamental representation of SO (5 ,
5) decomposes under GL (5) ⊂ SO (5 ,
5) as and , i.e., V M = ( V m , V m ). The SO (5 ,
5) vector index M = 1 , .., SO (5 ,
5) invariant metric in thelight cone or off-diagonal basis η MN = η MN = (cid:18) (cid:19) (A.1)in which n is an ( n × n ) identity matrix. For example, V M = η MN V N = ( V m , V m ).In vector representation, the SO (5 ,
5) algebra[ t MN , t P Q ] = 4( η M [ P t Q ] N − η N [ P t Q ] M ) (A.2)is realized by SO (5 ,
5) generators , t MN = t [ MN ] , of the form( t MN ) P Q = 4 η P [ M δ QN ] (A.3)where δ MN = . Defining an R + (cid:39) SO (1 , ⊂ GL (5) ∼ R + × SL (5) generatorby d = t mm = t + t + t + t + t , (A.4)we find the explicit form of the R + generator in vector representation given by d = 2 (cid:18) − (cid:19) . (A.5)With an SO (5 ,
5) vector decomposing as V M = ( V m , V m ), we obtain the commu-tation relations [ d , V m ] = +2 V m and [ d , V m ] = − V m . (A.6)These imply that we can assign the R + weights ± and representationsof SL (5) ⊂ GL (5). Therefore, the branching rule for a vector representation reads (cid:124)(cid:123)(cid:122)(cid:125) V M → +2 (cid:124)(cid:123)(cid:122)(cid:125) V m ⊕ − (cid:124)(cid:123)(cid:122)(cid:125) V m . (A.7)43 .2 Adjoint The decomposition of adjoint representation follows from the branching rule ofvector representations. Using (A.7), we can decompose the SO (5 ,
5) generatorsas t MN → ( t mn , t mn , t mn ) (A.8)with t mn = − t nm . The 25 generators t mn of GL (5) consist of the R + generatordefined in (A.4) and the SL (5) generators given by τ mn = t mn − d δ mn (A.9)with τ mm = 0.We denote the shift and hidden symmetries by s mn = t mn and h mn = t mn ,respectively. In vector representation, the SO (5 ,
5) generators can be written as( t MN ) P Q = (cid:18) τ mn h mn s mn − τ mn (cid:19) . (A.10)From the SO (5 ,
5) algebra, we can derive the following commutation relations[ d , d ] = 0 , [ d , τ mn ] = 0 , (A.11)[ d , s mn ] = − s mn , [ d , h mn ] = +4 h mn , (A.12)[ s mn , s pq ] = 0 , [ h mn , h pq ] = 0 , (A.13)[ τ mn , τ pq ] = 2( δ pn τ mq − δ mq τ pn ) , (A.14)[ τ mn , s pq ] = 2( δ mq s np − δ mp s nq + 25 δ mn s pq ) , (A.15)[ τ mn , h pq ] = 2( δ pn h mq − δ qn h mp − δ mn h pq ) , (A.16)[ s mn , h pq ] = 2( δ pm τ qn − δ qm τ pn − δ pn τ qm + δ qn τ pm ) − d δ [ pm δ q ] n = 2( δ pm t qn − δ qm t pn − δ pn t qm + δ qn t pm ) (A.17)in which δ [ pm δ q ] n = ( δ pm δ qn − δ qm δ pn ). In the second line of ( ?? ), we have used (A.9)to rewrite the commutation relation in terms of the GL (5) generators. Note alsothat (A.14) is the SL (5) algebra. It follows that the GL (5) branching rule foradjoint representation is given by (cid:124)(cid:123)(cid:122)(cid:125) t MN → (cid:124)(cid:123)(cid:122)(cid:125) d ⊕ (cid:124)(cid:123)(cid:122)(cid:125) τ mn ⊕ − (cid:124) (cid:123)(cid:122) (cid:125) s mn ⊕ +4 (cid:124) (cid:123)(cid:122) (cid:125) h mn (A.18)where the R + weights are determined from the relations (A.11) and (A.12).44 .3 Spinor Unlike the vector, decomposition of SO (5 ,
5) spinor representation under GL (5) isnot straightforward. To describe this branching rule, we begin with the followingtwo sets of U Sp (4) (cid:39) SO (5) gamma matrices satisfying { γ a , γ b } = 2 δ ab , δ ab = diag(+ , + , + , + , +) , (A.19) { γ ˙ a , γ ˙ b } = 2 δ ˙ a ˙ b , δ ˙ a ˙ b = diag(+ , + , + , + , +) (A.20)where a, b, ... = 1 , .., a, ˙ b, ... = ˙1 , .., ˙5 are two sets of SO (5) vector indicesraised and lowered with δ ab and δ ˙ a ˙ b , respectively. For both sets of SO (5) gammamatrices, we will use the following explicit representation γ = − σ ⊗ σ , γ = ⊗ σ , γ = ⊗ σ ,γ = σ ⊗ σ , γ = σ ⊗ σ (A.21)where { σ , σ , σ } are the usual Pauli matrices given by σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . (A.22)Each gamma matrix is a 4 × γ a ) αβ and ( γ ˙ a ) ˙ α ˙ β . Indices α, β, ... = 1 , .., α, ˙ β, ... = ˙1 , .., ˙4 are two sets of U Sp (4)fundamental or SO (5) spinor indices raised and lowered through two identical U Sp (4) symplectic forms Ω αβ = Ω ˙ α ˙ β = ⊗ iσ (A.23)satisfying Ω βα = − Ω αβ , Ω αβ = (Ω αβ ) ∗ , Ω αβ Ω βγ = − δ γα , (A.24)and similarly for Ω ˙ α ˙ β . Therefore, the matrices ( γ a ) αβ = Ω βγ ( γ a ) αγ satisfy( γ a ) βα = − ( γ a ) αβ , Ω αβ ( γ a ) αβ = 0 , (( γ a ) αβ ) ∗ = Ω αγ Ω αδ ( γ a ) γδ , (A.25)and similarly for ( γ ˙ a ) ˙ α ˙ β = Ω ˙ β ˙ γ ( γ ˙ a ) ˙ α ˙ γ .The 32-dimensional SO (5 ,
5) gamma matrices, ˜ Γ A = ( ˜ Γ a , ˜ Γ ˙ a ) with A =1 , ...,
10, satisfying the Clifford algebra (cid:110) ˜ Γ A , ˜ Γ B (cid:111) = 2 η AB (A.26)with η AB = diag(+ , + , + , + , + , − , − , − , − , − ) can be constructed from the SO (5)gamma matrices as˜ Γ a = ( σ ⊗ ⊗ γ a ) and ˜ Γ ˙ a = ( iσ ⊗ γ ˙ a ⊗ ) . (A.27)45he matrices A , B , and C , which respectively realize Dirac, complex, and chargeconjugation, have the following defining properties( ˜ Γ A ) † = − A ˜ Γ A A − , ( ˜ Γ A ) ∗ = − B ˜ Γ A B − , ( ˜ Γ A ) T = − C ˜ Γ A C − . (A.28)In our explicit representation, the matrices A and B are given by A = ˜ Γ ˜ Γ ˜ Γ ˜ Γ ˜ Γ and B = ⊗ Ω ⊗ Ω . (A.29)The charge conjugation matrix C can be obtained from A and B through therelation C = B T A . (A.30)The SO (5 ,
5) chirality matrix takes the following diagonal form˜ Γ ∗ = ˜ Γ ... ˜ Γ = diag( , − ) . (A.31)Therefore, as seen from their definition (A.27), SO (5 ,
5) gamma matrices arechirally decomposed as˜ Γ a = (cid:18) ⊗ γ a ⊗ γ a (cid:19) and ˜ Γ ˙ a = (cid:18) γ ˙ a ⊗ − γ ˙ a ⊗ (cid:19) . (A.32)Elements of the 16 × SO (5) gamma matrices ⊗ γ a and γ ˙ a ⊗ are denotedby the following index structure ⊗ γ a = ( γ a ) α ˙ αβ ˙ β = ( γ a ) αβ δ ˙ β ˙ α and γ ˙ a ⊗ = ( γ ˙ a ) α ˙ αβ ˙ β = δ βα ( γ ˙ a ) ˙ α ˙ β . (A.33)On the other hand, we can split a 32-dimendional SO (5 ,
5) spinor indexinto A = ( A, A (cid:48) ) for
A, B, ... = 1 , ..,
16 and A (cid:48) , B (cid:48) , ... = 17 , ...,
32 so that( ˜ Γ A ) AB = (cid:32) A ) AB (cid:48) (Γ A ) A (cid:48) B (cid:33) . (A.34)We can then relate these two decompositions of SO (5 ,
5) spinor indices into
A, A (cid:48) and a pair of
U Sp (4) indices ( α ˙ α ) by using the following transformation matrices p α ˙ αA = δ αA δ ˙ α + δ α +4 A δ ˙ α + δ α +8 A δ ˙ α + δ α +12 A δ ˙ α , p Aα ˙ α = δ Aα δ α + δ Aα +4 δ α + δ Aα +8 δ α + δ Aα +12 δ α . (A.35)These matrices satisfy the relations p α ˙ αA p Bα ˙ α = δ BA and p α ˙ αA p Aβ ˙ β = δ αβ δ ˙ α ˙ β . (A.36)We can now write chiral SO (5 ,
5) gamma matrices in terms of the SO (5) ones as(Γ a ) AB (cid:48) = p α ˙ αA ( γ a ) α ˙ αβ ˙ β p B (cid:48) β ˙ β , (Γ ˙ a ) AB (cid:48) = p α ˙ αA ( γ ˙ a ) α ˙ αβ ˙ β p B (cid:48) β ˙ β , (A.37)(Γ a ) A (cid:48) B = p α ˙ αA (cid:48) ( γ a ) α ˙ αβ ˙ β p Bβ ˙ β , (Γ ˙ a ) A (cid:48) B = − p α ˙ αA (cid:48) ( γ ˙ a ) α ˙ αβ ˙ β p Bβ ˙ β . (A.38)46o raise and lower the spinor indices A and A (cid:48) , we use the charge conjugationmatrix which in this basis takes the form of C = (cid:18) ⊗ Ω − Ω ⊗ Ω 0 (cid:19) . (A.39)Its elements can be explicitly expressed as C AB = (cid:18) c AB (cid:48) c A (cid:48) B (cid:19) and C AB = (cid:18) c AB (cid:48) c A (cid:48) B (cid:19) . (A.40)The 16 ×
16 matrix c A (cid:48) B is antisymmetric, c A (cid:48) B = − c BA (cid:48) . Similarly, the matrix c AB (cid:48) satisfying the relations c AC (cid:48) c C (cid:48) B = − δ BA and c A (cid:48) C c CB (cid:48) = − δ B (cid:48) A (cid:48) (A.41)is also antisymmetric c A (cid:48) B = − c BA (cid:48) .By raising and lowering the SO (5 ,
5) spinor index, we can define gammamatrices with all upper or lower indices( ˜ Γ A ) AB = C AC ( ˜ Γ A ) CB = (cid:18) (Γ A ) AB
00 (Γ A ) A (cid:48) B (cid:48) (cid:19) , (A.42)( ˜ Γ A ) AB = ( ˜ Γ A ) AC C CB = (cid:18) (Γ A ) AB
00 (Γ A ) A (cid:48) B (cid:48) (cid:19) (A.43)in which (Γ A ) AB = c AC (cid:48) (Γ A ) C (cid:48) B , (Γ A ) A (cid:48) B (cid:48) = c A (cid:48) C (Γ A ) C B (cid:48) , (Γ A ) AB = (Γ A ) AC (cid:48) c C (cid:48) B , (Γ A ) A (cid:48) B (cid:48) = (Γ A ) A (cid:48) C c CB (cid:48) . (A.44)In terms of the U Sp (4) symplectic forms Ω αβ and Ω ˙ α ˙ β which can be used to raiseor lower U Sp (4) fundamental indices, we can write the matrices c A (cid:48) B and c A (cid:48) B as c A (cid:48) B = p α ˙ αA (cid:48) p β ˙ βB Ω αβ Ω ˙ α ˙ β and c A (cid:48) B = p A (cid:48) α ˙ α p Bβ ˙ β Ω αβ Ω ˙ α ˙ β . (A.45)With all these, we can eventually find the following relations(Γ a ) AB = p Aα ˙ α p Bβ ˙ β ( γ a ) α ˙ α,β ˙ β , (Γ ˙ a ) AB = − p Aα ˙ α p Bβ ˙ β ( γ ˙ a ) α ˙ α,β ˙ β , (Γ a ) A (cid:48) B (cid:48) = p A (cid:48) α ˙ α p B (cid:48) β ˙ β ( γ a ) α ˙ α,β ˙ β , (Γ ˙ a ) A (cid:48) B (cid:48) = p A (cid:48) α ˙ α p B (cid:48) β ˙ β ( γ ˙ a ) α ˙ α,β ˙ β , (Γ a ) AB = p α ˙ αA p β ˙ βB ( γ a ) α ˙ α,β ˙ β , (Γ ˙ a ) AB = p α ˙ αA p β ˙ βB ( γ ˙ a ) α ˙ α,β ˙ β , (Γ a ) A (cid:48) B (cid:48) = p α ˙ αA (cid:48) p β ˙ βB (cid:48) ( γ a ) α ˙ α,β ˙ β , (Γ ˙ a ) A (cid:48) B (cid:48) = − p α ˙ αA (cid:48) p β ˙ βB (cid:48) ( γ ˙ a ) α ˙ α,β ˙ β (A.46)with ( γ a ) α ˙ α,β ˙ β = Ω αδ Ω ˙ α ˙ δ ( γ a ) δ ˙ δβ ˙ β = ( γ a ) αβ Ω ˙ α ˙ β , ( γ ˙ a ) α ˙ α,β ˙ β = Ω αδ Ω ˙ α ˙ δ ( γ ˙ a ) δ ˙ δβ ˙ β = Ω αβ ( γ ˙ a ) ˙ α ˙ β , ( γ a ) α ˙ α,β ˙ β = Ω βδ Ω ˙ β ˙ δ ( γ a ) α ˙ αδ ˙ δ = ( γ a ) αβ Ω ˙ α ˙ β , ( γ ˙ a ) α ˙ α,β ˙ β = Ω βδ Ω ˙ β ˙ δ ( γ ˙ a ) α ˙ αδ ˙ δ = Ω αβ ( γ ˙ a ) ˙ α ˙ β . (A.47)47e now transform all these results to the basis with off-diagonal η MN given in (A.1). Denoting SO (5 ,
5) gamma matrices in this basis by ˜ Γ M , we canwrite the corresponding Clifford algebra as (cid:110) ˜ Γ M , ˜ Γ N (cid:111) = 2 η MN . (A.48)From [31], the relation between diagonal and off-diagonal η is given by η MN = M M A M N B η AB (A.49)with M = 1 √ (cid:18) − (cid:19) . (A.50)We can then find the following relation between these two sets of gamma matrices˜ Γ M = M M A ˜ Γ A (A.51)with the same chiral decomposition of the form( ˜ Γ M ) AB = (cid:32) M ) AB (cid:48) (Γ M ) A (cid:48) B (cid:33) . (A.52)Moreover, we can still raise and lower the chirally decomposed spinorindices with the charge conjugation matrix given in (A.39) such that( ˜ Γ M ) AB = ( ˜ Γ M ) AC C CB = (cid:18) (Γ M ) AB
00 (Γ M ) A (cid:48) B (cid:48) (cid:19) (A.53)with (Γ M ) AB = (Γ M ) AC (cid:48) c C (cid:48) B and (Γ M ) A (cid:48) B (cid:48) = (Γ M ) A (cid:48) C c CB (cid:48) . We will see in thefollowing analysis that (Γ M ) AB play an important role in determining specificforms of the embedding tensor.In spinor representation, the SO (5 ,
5) generators satisfying (A.2) aregiven by ( t MN ) AB = (Γ MN ) AB . (A.54)In 32 ×
32 representation, we can write( ˜ Γ MN ) AB = 12 (cid:16) ( ˜ Γ M ) AC ( ˜ Γ N ) CB − ( ˜ Γ N ) AC ( ˜ Γ M ) CB (cid:17) (A.55)= (cid:32) (Γ MN ) AB
00 (Γ MN ) A (cid:48) B (cid:48) (cid:33) with (Γ MN ) AB = 12 (cid:104) (Γ M ) AC (cid:48) (Γ N ) C (cid:48) B − (Γ N ) AC (cid:48) (Γ M ) C (cid:48) B (cid:105) , (A.56)(Γ MN ) A (cid:48) B (cid:48) = 12 (cid:104) (Γ M ) A (cid:48) C (Γ N ) C B (cid:48) − (Γ N ) A (cid:48) C (Γ M ) C B (cid:48) (cid:105) . (A.57)48t should be noted that the SO (5 ,
5) generators in spinor representation given in(A.54) also decompose according to (A.18) and satisfy the same algebra given in(A.11) to ( ?? ) for vector representation.As pointed out in [31], the branching rules for spinor and conjugate spinorrepresentations of SO (5 ,
5) are respectively given by s → +3 ⊕ − ⊕ − and c → − ⊕ +1 ⊕ +5 . (A.58)To find the corresponding decompositions of spinor indices, we define the followingtransformation matrices T Am = 12 √ m ) AB p Bαβ Ω αβ , (A.59) T mnA = 14 √ mn ) AB p Bαβ Ω αβ , (A.60) T A ∗ = 110 (Γ mm ) AB p Bαβ Ω αβ . (A.61)In these equations, p Aαβ matrices are defined in the same way as p Aα ˙ β in (A.35).We can now decompose an SO (5 ,
5) spinor in s representation asΨ A = T Am Ψ m + T mnA Ψ mn + T A ∗ Ψ ∗ , (A.62)where Ψ mn = Ψ [ mn ] . The commutation relations between these components andthe R + generator are given by[ d , T Am Ψ m ] = +3 T Am Ψ m , (A.63)[ d , T mnA Ψ mn ] = − T mnA Ψ mn , (A.64)[ d , T A ∗ Ψ ∗ ] = − T A ∗ Ψ ∗ . (A.65)in accord with the branching rule s (cid:124)(cid:123)(cid:122)(cid:125) Ψ A → +3 (cid:124)(cid:123)(cid:122)(cid:125) Ψ m ⊕ − (cid:124) (cid:123)(cid:122) (cid:125) Ψ [ mn ] ⊕ − (cid:124)(cid:123)(cid:122)(cid:125) Ψ ∗ . (A.66)The inverse matrices of T A are simply given by their complex conjugation T A = ( T A ) − = ( T A ) ∗ satisfying T Am T An = δ mn , T Amn T pqA = δ [ pm δ q ] n , T A ∗ T A ∗ = 1 , T Am T Anp = 0 , T Am T A ∗ = 0 , T Amn T A ∗ = 0 (A.67)together with T Am T Bm + T Amn T mnB + T A ∗ T ∗ B = δ AB . (A.68)In addition, we also note that a complex conjugation of the SO (5 ,
5) gammamatrices is related to raising the indices ((Γ M ) AB ) ∗ = (Γ M ) AB . We can thensimilarly decompose a conjugate spinor of SO (5 ,
5) transforming in c as followsΨ A = T Am Ψ m + T Amn Ψ mn + T ∗ A Ψ ∗ . (A.69)49he following commutation relations (cid:2) d , T Am Ψ m (cid:3) = − T Am Ψ m , (A.70) (cid:2) d , T Amn Ψ mn (cid:3) = + T Amn Ψ mn , (A.71)[ d , T ∗ A Ψ ∗ ] = +5 T ∗ A Ψ ∗ (A.72)imply the branching rule c (cid:124)(cid:123)(cid:122)(cid:125) Ψ A → − (cid:124)(cid:123)(cid:122)(cid:125) Ψ m ⊕ +1 (cid:124) (cid:123)(cid:122) (cid:125) Ψ [ mn ] ⊕ +5 (cid:124)(cid:123)(cid:122)(cid:125) Ψ ∗ . (A.73) A.4 Vector-spinor
The vector-spinor of SO (5 ,
5) we are interested in is given by θ AM ∈ c , whichparameterizes the embedding tensor. It transforms according to (cid:2) t MN , θ AP (cid:3) = − ( t MN ) QP θ AQ − ( t MN ) BA θ BP . (A.74)Here, θ AM is a (16 ×
10) matrix subject to(Γ M ) AB θ BM = 0 (A.75)which is the linear constraint required by supersymmetry, reducing 160 compo-nents of the θ AM to 144 in c representation.To determine the decomposition of the vector-spinor representation under GL (5), we first split the SO (5 ,
5) vector index M as θ AM = ( θ Am , θ Am ). Then,with the inverse of the transformation matrices T A given in (A.59) to (A.61), θ Am and θ Am can be further decomposed into the following six components θ Am = T An ( ϑ ) nm + T Anp ( ϑ ) np,m + T A ∗ ( ϑ ) m , (A.76) θ Am = T An ( ϑ ) nm + T Anp ( ϑ ) npm + T A ∗ ( ϑ ) m . (A.77)It is straightforward to show that their commutation relations with the R + gen-erators in both vector and spinor representations are given by (cid:2) d , T An ( ϑ ) nm (cid:3) = − T An ( ϑ ) nm , (cid:2) d , T An ( ϑ ) nm (cid:3) = − T An ( ϑ ) nm , (cid:2) d , T Anp ( ϑ ) np,m (cid:3) = − T Anp ( ϑ ) np,m , (cid:2) d , T Anp ( ϑ ) npm (cid:3) = +3 T Anp ( ϑ ) npm , (cid:2) d , T A ∗ ( ϑ ) m (cid:3) = +3 T A ∗ ( ϑ ) m , (cid:2) d , T A ∗ ( ϑ ) m (cid:3) = +7 T A ∗ ( ϑ ) m . (A.78)The branching rule for c representation is then given by c (cid:124) (cid:123)(cid:122) (cid:125) θ AM → +3 (cid:124)(cid:123)(cid:122)(cid:125) J m ⊕ +7 (cid:124)(cid:123)(cid:122)(cid:125) K m ⊕ − (cid:124) (cid:123)(cid:122) (cid:125) Z mn ⊕ − (cid:124) (cid:123)(cid:122) (cid:125) Y mn ⊕ − (cid:124) (cid:123)(cid:122) (cid:125) S mn ⊕ − (cid:124) (cid:123)(cid:122) (cid:125) U mn,p ⊕ +3 (cid:124) (cid:123)(cid:122) (cid:125) W npm (A.79)in agreement with that given in [31]. We now explicitly construct a number ofpossible embedding tensors arising from various components of the above decom-position. 50 .4.1 24 − representation With only ϑ (cid:54) = 0, we have θ Am = 0, and θ Am is parametrized by a 5 × ϑ ) nm . Therefore, the embedding tensor is given by θ AM = (cid:0) T An ( ϑ ) nm , (cid:1) . (A.80)The linear constraint (A.75) implies that ( ϑ ) nm is traceless or( ϑ ) nm = S nm (A.81)for S mm = 0. This leads to an embedding tensor in − representation of GL (5)given by θ AM − = (cid:0) T An S nm , (cid:1) . (A.82) A.4.2 15 − representation For θ Am = 0, θ Am is parametrized by a 5 × ϑ ) mn which can further bedecomposed in terms of symmetric and antisymmetric parts, Y mn = Y ( mn ) and Z mn = Z [ mn ] , as θ Am = T An ( ϑ ) nm = T An ( Y nm + Z nm ) . (A.83)The linear constraint (A.75) requires Z mn = 0, so the embedding tensor is givenby θ AM − = (cid:0) , T An Y nm (cid:1) (A.84)in − representation of GL (5). A.4.3 40 − representation With only ( ϑ ) np,m non-vanishing, we have θ Am = 0 and θ Am given by θ Am = T Anp ( ϑ ) np,m . (A.85)The tensor ( ϑ ) np,m can in turn be parametrized as( ϑ ) np,m = U np,m + 12 ε mnpqr ζ qr . (A.86) U mn,p = U [ mn ] ,p with U [ mn,p ] = 0 and ζ mn = ζ [ mn ] correspond to − and − representations, respectively. The condition (A.75) requires ζ qr = 0 resulting inthe embedding tensor in − representation of the form θ AM − = (cid:0) T Anp U np,m , (cid:1) . (A.87)51 .4.4 10 − representation Turning on − irreducible part of both ( ϑ ) nm and ( ϑ ) np,m by setting Y mn = 0and U mn,p = 0, we find the embedding tensor of the form θ AM − = θ AM + θ AM = (cid:18) T Anp ε mnpqr ζ qr , T An Z nm (cid:19) . (A.88)The condition (A.75) is satisfied for ζ mn = √ Z mn . (A.89)Therefore, the embedding tensor in − representation is given by θ AM − = θ AM + θ AM = (cid:18) √ T Anp ε mnpqr Z qr , T An Z nm (cid:19) . (A.90) A.4.5 45 +3 representation In this case, we consider non-vanishing ( ϑ ) npm which can be decomposed into +3 and +3 irreducible representations of the form( ϑ ) npm = W npm + J [ n δ p ] m (A.91)with W npm = W [ np ] m satisfying W nmm = 0. The linear constraint (A.75) requires J m = 0 leading to the embedding tensor in +3 representation given by θ AM +3 = (cid:0) , T Anp W npm (cid:1) . (A.92) A.4.6 5 +3 representation We now consider non-vanishing +3 components from both ( ϑ ) m and ( ϑ ) m interms of which the embedding tensor is given by θ AM +3 = θ AM + θ AM = (cid:0) T A ∗ ξ m , T Anp J [ n δ p ] m (cid:1) . (A.93)This satisfies the linear constraint (A.75) for ξ m = − √ J m . (A.94)Therefore, we find the embedding tensor in +3 representation given by θ AM +3 = (cid:32) − √ T A ∗ J m , T Anp J [ n δ p ] m (cid:33) . (A.95)52 .4.7 5 +7 representation Finally, we consider ( ϑ ) m corresponding to +7 representation. This can beparameterized by an arbitrary GL (5) vector of the form ( ϑ ) m = K m . Thecorresponding embedding tensor is also in +7 representation and takes the form θ AM +7 = (cid:0) , T A ∗ K m (cid:1) (A.96)which automatically satisfies the condition (A.75).We end this appendix by giving the full parametrization of the embeddingtensor θ AM = ( θ Am , θ Am ) under GL (5) θ Am = T An S nm + T Anp ( U np,m + 13 √ ε mnpqr Z qr ) − √ T A ∗ J m , (A.97) θ Am = T An ( Y nm + Z nm ) + T Anp ( W npm + J [ n δ p ] m ) + T A ∗ K m . (A.98) B Symplectic-Ma jorana-Weyl spinors in six di-mensions
In this appendix, we collect conventions and fundamental relations involving ir-reducible spinors in six-dimensional space-time used throughout this work. Insix dimensions, there exist Dirac spinors with 32 real components. The Diracspinors are reducible and can be decomposed into two irreducible Weyl spinorsof opposite chirality with 16 real components each. The six-dimensional Cliffordalgebra is defined by the relation { ˆ γ ˆ µ ˆ γ ˆ ν + ˆ γ ˆ ν ˆ γ ˆ µ } = 2 η ˆ µ ˆ ν . (B.1)Here, ˆ γ ˆ µ are 8 × η ˆ µ ˆ ν = diag( − , +1 , +1 , +1 , +1 , +1) withˆ µ, ˆ ν, ... = 0 , , ..., γ = σ ⊗ iσ ⊗ , ˆ γ = σ ⊗ ⊗ σ , ˆ γ = σ ⊗ σ ⊗ , ˆ γ = σ ⊗ ⊗ σ , ˆ γ = σ ⊗ ⊗ σ , ˆ γ = σ ⊗ σ ⊗ . (B.2)In this representation, the Dirac, complex, and charge conjugation matrices arerespectively given byˆ A = ˆ γ , ˆ B = − i ˆ γ ˆ γ , ˆ C = i ˆ γ ˆ γ ˆ γ . (B.3)They satisfy the following relations(ˆ γ ˆ µ ) † = − ˆ A ˆ γ ˆ µ ˆ A − , (ˆ γ ˆ µ ) ∗ = − ˆ B ˆ γ ˆ µ ˆ B − , (ˆ γ ˆ µ ) T = − ˆ C ˆ γ ˆ µ ˆ C − (B.4)53ogether with the identitiesˆ B T = ˆ C ˆ A − , ˆ B ∗ ˆ B = − , ˆ C T = − ˆ C − = − ˆ C † = ˆ C . (B.5)The chirality operator ˆ γ ∗ can be defined asˆ γ ∗ = ˆ γ ˆ γ ˆ γ ˆ γ ˆ γ ˆ γ = diag( , − ) with ˆ γ ∗ ˆ γ ∗ = . (B.6)The diagonal form of ˆ γ ∗ implies that a Dirac spinor Ψ can be chirally decomposedas Ψ = Ψ + + Ψ − with P ± Ψ ± = ± Ψ ± (B.7)where the projection operators are given by P ± = 12 ( ± ˆ γ ∗ ) . (B.8)Therefore, we can define two irreducible four-dimensional Weyl spinors ψ + and χ − from the Dirac spinor Ψ byΨ = (cid:18) ψ + χ − (cid:19) , Ψ + = (cid:18) ψ + (cid:19) , Ψ − = (cid:18) χ − (cid:19) . (B.9)Although the second property in (B.5) implies that a reality conditioncannot be imposed on the Dirac or Weyl spinors, we can define a symplectic-Majorana-Weyl spinor of the formΨ + α = (cid:18) ψ + α (cid:19) and Ψ − ˙ α = (cid:18) χ − ˙ α (cid:19) . (B.10)Ψ + α and Ψ − ˙ α satisfy the pseudo-reality condition given byΨ α + = (Ψ + α ) ∗ = Ω αβ ˆ B Ψ + β and Ψ ˙ α − = (Ψ − ˙ α ) ∗ = Ω ˙ α ˙ β ˆ B Ψ − ˙ β . (B.11)Ω αβ and Ω ˙ α ˙ β are symplectic forms of the two U Sp (4) factors in the R-symmetry
U Sp (4) × U Sp (4) under which Ψ + α and Ψ − ˙ α transform separately. C Truncation ansatze
In this appendix, we collect some useful formulae for a consistent truncationof seven-dimensional SO (5) gauged supergravity on a circle ( S ), giving rise to SO (5) gauged supergravity in six dimensions. This truncation has been con-structed in [29]. 54he truncation ansatze for the seven-dimensional metric, scalars, vector,and tensor fields are respectively given by d ˆ s = e σ √ ds + e − σ √ ( dz + A (1) ) , (C.1)ˆΠ iI ( x µ , z ) = Π I i ( x µ ) , (C.2)ˆ B J (1) I = B (1) I J + B (0) I J ( dz + A (1) ) , (C.3)ˆ S (3) I = S (3) I + S (2) I ( dz + A (1) ) (C.4)in which hatted quantities refer to seven-dimensional fields. x µ are six-dimensionalspace-time coordinates, and z is the coordinate on S . Here, I, J, ... = 1 , ..., SO (5) while i, j, ... = 1 , ..., SO (5) c .There are (1 + 14 + 10) scalars denoted by { σ, Π I i , B (0)
I J } in the six-dimensional theory. These are given by the dilaton scalar field from the matricansatz (C.1), fourteen scalars parametrizing SL (5) /SO (5) c coset, and ten ax-ionic scalar fields from the truncation ansatz of vector fields (C.3). There arealso (10 + 1) vectors { B (1) I J , A (1) } together with five two-form potentials S (2) I .The five three-form potentials S (3) I do not contribute to the Lagrangian of the SO (5) gauged theory in six dimensions since the seven-dimensional self-dualitycondition allows to eliminate them in favor of the two-form potentials.For supersymmetric domain wall solutions considered in this work, wecan set A (1) = B (1) I J = S (2) I = 0. Using the domain wall ansatz for the matricfrom (3.8) together with ϕ = σ √ , we find that (C.1) becomes e A ds , + d ˆ r = e A +2 ϕ ds , + e − ϕ dz + dr = e A ( ds , + dz ) + dr (C.5)In the second line, we have substituted ϕ = − A from the domain wall solutionsgiven here. This is also necessary for ds , and dz to form a six-dimensional flatspace-time matching ds , on the left hand side. We can also see the relationsbetween the warp factors ˆ A = A and the radial coordinates ˆ r = r .The ansatz (C.2) implies that the scalars parametrizing SL (5) /SO (5) c coset in seven- and six-dimensional supersymmetric domain walls are the samesince they are independent of z and depend only on the corresponding radialcoordinates ˆΠ iI (ˆ r ) = Π I i ( r ) . (C.6)For ten axionic scalars B (0) I J , which are called shift scalars in this work, wecan see from (C.3) that they give rise to non-vanishing vector fields in sevendimensions ˆ B J (1) I = B (0) I J dz. (C.7)Therefore, domain wall solutions with non-vanishing axionic scalars obtained inthis work cannot be obtained from an S reduction of any domain wall solutionsin seven dimensions. 55 eferences [1] J. M. Maldacena, “The large N limit of superconformal field theories andsupergravity”, Adv. Theor. Math. Phys. (1998) 231-252, arXiv: hep-th/9711200.[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlatorsfrom non-critical string theory”, Phys. Lett. B (1998) 105-114, arXiv:hep-th/9802.109.[3] E. Witten, “Anti De Sitter Space and holography”, Adv. Theor. Math. Phys. (1998) 253-291, arXiv: 9802150.[4] H.J. Boonstra, K. Skenderis and P.K. Townsend, “The domain-wall/QFTcorrespondence”, JHEP (1999) 003, arXiv: hep-th/9807137.[5] T. Gherghetta and Y. Oz, “Supergravity, Non-Conformal Field Theories andBrane-Worlds”, Phys. Rev. D (2002) 046001, arXiv: hep-th/0106255.[6] Ingmar Kanitscheider, Kostas Skenderis and Marika Taylor, “Precisionholography for non-conformal branes”, JHEP (2008) 094, arXiv:0807.3324.[7] K. Skenderis and P. K. Townsend, “Hidden supersymmetry of domainwalls and cosmologies”, Phys. Rev. Lett. (2006) 191301, arXiv: hep-th/0602260.[8] Kostas Skenderis and Paul K. Townsend, “Hamilton-Jacobi method for Do-main Walls and Cosmologies”, Phys. Rev. D74 (2006) 125008, arXiv: hep-th/0609056.[9] Kostas Skenderis, Paul K. Townsend and Antoine Van Proeyen, “Domain-wall/Cosmology correspondence in AdS/dS supergravity”, JHEP 08 (2007) , arXiv: 0704.3918.[10] E. Bergshoeff, M. de Roo, M. B. Green, G. Papadopoulos and P. K.Townsend, “Duality of type II 7 branes and 8 branes”, Nucl. Phys.
B470 (1996) 113, arXiv: hep-th/9601150.[11] P. M. Cowdall, “Novel Domain Wall and Minkowski Vacua of D = 9 Maximal SO (2) Gauged Supergravity”, Nucl. Phys. B600 (2001) 81, arXiv: hep-th/0009016.[12] E. Bergshoeff, U. Gran and D. Roest, “Type IIB seven-brane solutions fromnine-dimensional domain walls”, Class. Quant. Grav. (2002) 4207, arXiv:hep-th/0203202. 5613] N. Alonso Alberca, E. Bergshoeff, U. Gran, R. Linares, T. Ortin and D.Roest, “Domain walls of D = 8 gauged supergravities and their D = 11origin”, JHEP 06 (2003) , arXiv: hep-th/0303113.[14] E. Bergshoeff, U. Gran, R. Linares, M. Nielsen, T. Ortin and D. Roest,“The Bianchi classification of maximal D = 8 gauged supergravities”, Class.Quant. Grav. (2003) 3997, arXiv: hep-th/0306179.[15] P. Karndumri and P. Nuchino, “Supersymmetric domain walls in 7 D maxi-mal gauged supergravity”, Eur. Phys. J. C79 (2019) 648, arXiv: 1904.02871.[16] P. M. Cowdall, H. Lu, C. N. Pope, K. S. Stelle and P. K. Townsend, “Domainwalls in massive supergravities”, Nucl. Phys.
B486 (1997) 49, arXiv: hep-th/9608173.[17] I. Bakas, A. Brandhuber and K. Sfetsos, “Domain walls of gauged super-gravity, M-branes, and algebraic curves”, Adv. Theor. Math. Phys. (1999)1657-1719, arXiv: hep-th/9912132.[18] E. Bergshoeff, M. Nielsen and D. Roest, “The Domain Walls of GaugedMaximal Supergravities and their M-theory Origin”, JHEP 07 (2004) ,arXiv: hep-th/0404100.[19] E. A. Bergshoeff, A. Kleinschmidt and F. Riccioni, “Supersymmetric DomainWalls”, Phys. Rev. D86 (2012) 085043, arXiv: 1206.5697.[20] M Cvetic, S. S. Gubser, H. Lu and C. N. Pope, “Symmetric Potentials ofGauged Supergravities in Diverse Dimensions and Coulomb Branch of GaugeTheories”, Phys. Rev.
D62 (2000) 086003, arXiv: hep-th/9909121.[21] C. M. Hull, “Domain Wall and de Sitter Solutions of Gauged Supergravity”,JHEP 11 (2011) , arXiv: hep-th/0110048.[22] H. Singh, “New Supersymmetric Vacua for N=4, D=4 Gauged Supergrav-ity”, Phys. Lett.
B429 (1998) 304-312, arXiv: hep-th/9801038.[23] P. Karndumri, “Domain walls in three dimensional gauged supergravity”,JHEP 10 (2012) , arXiv: 1207.1227.[24] P. Karndumri, “ -BPS Domain wall from N = 10 three dimensional gaugedsupergravity”, JHEP 11 (2013) , arXiv: 1307.6641.[25] T. Ortiz and H. Samtleben, “ SO (9) supergravity in two dimensions”, JHEP01 (2013) , arXiv: 1210.4266.[26] A. Anabalon, T. Ortiz and H. Samtleben, “Rotating D0-branes and consis-tent truncations of supergravity”, Phys. Lett. B727 (2013) 516-523, arXiv:1310.1321. 5727] E. A. Bergshoeff, A. Kleinschmidt and F. Riccioni, “Supersymmetric DomainWalls”, Phys. Rev. D (2012) 085043, arXiv: 1206.5697.[28] Y. Tanii, “ N = 8 supergravity in six dimensions”, Phys. Lett. B145 (1984)197.[29] P.M. Cowdall, “On gauged maximal supergravity in six-dimensions”, JHEP06 (1999) , arXiv: hep-th/9810041.[30] M. Pernici, K. Pilch, and P. van Nieuwenhuizen, “Gauged maximally ex-tended supergravity in seven-dimensions”, Phys. Lett. B (1984) 103.[31] E. Bergshoeff, H. Samtleben, and E. Sezgin, “The Gauging of Maximal D=6Supergravity”, JHEP (2020) 015, arXiv:0712.4277.[32] W. Nahm, “Supersymmetries and their representations”, Nucl. Phys. B135 (1978) 149-166.[33] S. Lust, P. Ruter, and J. Louis, “Maximally supersymmetric AdS solutionsand their moduli spaces”, JHEP (2018) 019.[34] M. R. Douglas, “On D = 5 super Yang-Mills theory and (2 ,
0) theory”,JHEP 02 (2011) , arXiv: 1012.2880.[35] N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, “M5-Branes,D4-Branes and Quantum 5D super-Yang-Mills”, JHEP 01 (2011) , arXiv:1012.2882.[36] Y. Tachikawa, “On S-duality of 5d super Yang-Mills on S ”, JHEP 11 (2011) , arXiv: 1110.0531.[37] N. Lambert, H. Nastase and C. Papageorgakis, “5D Yang-Mills instantonsfrom ABJM Monopoles”, Phys. Rev. D85 (2012) 066002, arXiv: 1111.5619.[38] J. A. Minahan, A. Nedelin and M. Zabzine, “5D super Yang-Mills theoryand the correspondence to AdS /CFT ”, J. Phys. A46 (2013) 355401, arXiv:1304.1016.[39] D. Bak and A. Gustavsson, “One dyonic instanton in 5d maximal SYMtheory,” JHEP 07 (2013) , arXiv: 1305.3637.[40] H. Samtleben, and M. Weidner, “The maximal D=7 supergravities”, Nucl.Phys. (2005): 383-419, arXiv: hep-th/0506237.[41] M. Cvetic, F. Quevedo and S. Rey, “Stringy domain walls and target-spacemodular invariance”, Phys. Rev. Lett. (1991) 1836.5842] M. Cvetic, S.Griffies and S. Rey, “Static Domain Walls in N=1 Supergrav-ity”, Nucl. Phys. B381 (1992) 301-328, arXiv: hep-th/9201007.[43] M. Cvetic, S.Griffies and S. Rey, “Non-perturbative stability of supergrav-ity and superstring vacua”. Nucl. Phys.
B389 (1993) 3-24, ArXiv: hep-th/9206004.[44] W. Boucher, “Positive energy without supersymmetry”, Nucl. Phys.
B242 (1984) 282-296.[45] P.K. Townsend, “Positive Energy and the Scalar Potential in Higher Dimen-sional (Super)gravity Theories”, Phys. Lett.
B148 (1984) 55-59.[46] K. Skenderis and P.K. Townsend, “Gravitational Stability andRenormalization-Group Flow”, Phys. Lett.
B468 (1999) 46-51, arXiv:hep-th/9909070.[47] N. P. Warner, “Some New Extrema of the Scalar Potential of Gauged N = 8Supergravity”, Phys. Lett. B (1983): 169.[48] H. Lu, C. N. Pope and P. K. Townsend, “Domain Walls from Anti-de SitterSpacetime”, Phys. Lett. B391 (1997) 39-46, arXiv: hep-th/9607164.[49] K. Pilch, P. van Nieuwenhuizen, and P. K. Townsend, “Compactification of d = 11 Supergravity on S (4) (Or 11 = 7 + 4, Too)”, Nucl. Phys. B (1984): 377–392.[50] H. Nastase, D. Vaman, and P. van Nieuwenhuizen, “Consistent nonlinearKK reduction of 11 − d supergravity on AdS (7) × S (4) and selfduality in odddimensions”, Phys. Lett. B (1999): 96–102, arXiv: hep-th/9905075.[51] H. Nastase, D. Vaman, and P. van Nieuwenhuizen, Consistency of the AdS × S reduction and the origin of self-duality in odd dimensions, Nucl. Phys. B (2000): 179–239, arXiv: hep-th/9911238.[52] M. Cvetic, H. Lu, C. N. Pope, A. Sadrzadeh and T. A. Tran, “ S and S reductions of type IIA supergravity, Nucl. Phys. B (2000): 233–251,arXiv: hep-th/0005137.[53] E. Malek and H. Samtleben, “Dualising consistent IIA /IIB truncations”,JHEP 12 (2015) , arXiv: 1510.03433.[54] A. Abzalov, I. Bakhmatov and E. T. Musaev, “Exceptional field theory: SO (5 , , arXiv: 1504.01523.[55] A. Brandhuber and Y. Oz, “The D4-D8 brane system and five dimensionalfixed points”, Phys. Lett. B460 (1999) 307-312, arXiv: hep-th/9905148.5956] P. Karndumri, “Holographic RG flows in six dimensional F(4) gauged super-gravity”, JHEP 01 (2013) , Erratum-ibid. JHEP 06 (2015) , arXiv:1210.8064.[57] P. Karndumri and J. Louis, “Supersymmetric
AdS vacua in six-dimensional N = (1 ,
1) gauged supergravity”, JHEP 01 (2017) , arXv: 1612.00301.[58] P. Karndumri and P. Nuchino, “Supersymmetric solutions of 7D maximalgauged supergravity,” Phys. Rev.
D101 (2020) 086012, arXiv: 1910.02909.[59] P. Karndumri and P. Nuchino, “Twisted compactifications of 6D field theo-ries from maximal 7D gauged supergravity,” Eur. Phys. J.
C80 (2020) 201,arXiv: 1912.04807.[60] G. Dibitetto and N. Petri, “Surface defects in the D4-D8 brane system”,JHEP 01 (2019)193