Codimension-2 brane solutions of maximal supergravities in 9, 8, and 7 dimensions

We construct codimension-2 BPS brane solutions in $D=9,8,7$ maximal supergravities by solving Killing spinor equations. We assume the Poincare invariance along the worldvolume and vanishing gauge fields, and determine the metric and the scalar fields. The solution in $D=9$ is essentially the same as the ten-dimensional one, which is specified by a holomorphic function in the transverse space. For $D=8$, the solution is specified by two holomorphic functions, and regarded as $T^2\times T^2$ compactification of F-theory. For $D=7$, we find that the solution can be interpreted as M-theory on Calabi-Yau, and under an additional assumption a solution is specified by two holomorphic functions.

1 Introduction by explicitly solving Killing spinor equations. Namely, BPS solutions of codimension-2 branes always have geometric realization in M/F-theory for D = 9, 8, 7. Maximal supergravities in various dimensions have common structure. The scalar manifolds of these theories have the form G/H, where G is the classical global symmetry and H is the local symmetry group, which is the maximal compact subgroup of G. See Table 1 for G and H in dimensions D = 10, . . . , 4 [11]. The scalar fields are coordinates of this manifold, and  E 7(7) E 7(7) (Z) SU (8) represented as a matrix L ∈ G with left action of G and right action of H. The U-duality group G Z is the integral form of G. For the theories in seven or higher dimensions G are all SL type and H are all SO type, and they can be dealt in similar ways. In this paper we investigate these theories, and theories in D ≤ 6 are left for future work. A maximal supergravity contains The scalar fields appear in the action and the supersymmetry transformation laws through 1-form fields P and Q, which are defined as the traceless symmetric and anti-symmetric parts of the Maurer-Cartan form: (1.1) Under H transformation P transforms homogeneously as the symmetric matrix representation of H, while Q transforms inhomogeneously and plays the role of H-connection.
To obtain BPS solutions we solve the Killing spinor equations for the gravitino ψ M and dilatino λ i . In the next section we first look at 10d case to explain basic prescription to solve the Killing spinor equations and then we move on to lower dimensional cases.
2 Solving Killing spinor equations 2.1 D = 10 Let us consider BPS solutions in type IIB supergravity. Such solutions have been well investigated [1] and 7-branes are classified by Kodaira classification [12]. Various 4d N = 2 supersymmetric theories are realized on D3-branes probing these solutions [13,14,15]. A purpose of this subsection is to review how we can obtain BPS solutions in ten dimensions by solving the Killing spinor equations. The derivations in lower dimensions are parallel.
The classical global symmetry of type IIB supergravity is G = SL(2, R) and the local R-symmetry group is H = SO(2) R . Namely, the scalar manifold is locally the two-dimensional homogeneous space SL(2, R)/SO(2) R . When we discuss the global structure, we also need to take account of the duality group G Z = SL(2, Z). Quantum numbers of scalar and spinor fields in type IIB supergravity [16] are summarized in Table 2. The gravitino field ψ M belongs to the spinor representation of H. Namely, ψ M has the spacetime Table 2: Quantum numbers of scalar and spinor fields in type IIB supergravity vector index M and an SO(2) R spinor index which is implicit. The dilatino field λ i has the SO(2) R vector index i and an implicit SO(2) R spinor index. It satisfies the ρ-traceless condition where ρ i are Dirac matrices associated with the orthogonal group H = SO(2) R . See appendix for our notation. This condition removes components carrying SO(2) R charge ±1/2 from λ i , and the remaining components in λ i carry SO(2) R charge ±3/2 as is shown in Table 2.
Due to the existence of the self-dual 4-form field it is difficult to write down the full Lagrangian of the type IIB supergravity. However, it is easy to give the Lagrangian of the subsector which is relevant to us. If we assume the vanishing anti-symmetric tensor fields the equations of motion for the remaining fields are obtained from the Lagrangian up to higher order fermion terms. The Killing spinor equations are where D M is the covariant derivative defined with the spin connection ω and the SO(2) R connection Q: We are interested in codimension-2 brane solutions. Let us assume the solution has the eight dimensional Poincare invariance along the eight longitudinal directions. We use x µ (µ = 0, 1, . . . , 7) and x m (m = 8,9) for longitudinal and transverse coordinates, respectively. We take the ansatz for the metric and for the scalar fields. We introduce the local frame so that the vielbein has the diagonal components Because we are interested in the rigid supersymmetry on the branes we assume the supersymmetry parameter ǫ depends only on the transverse coordinates: Because L α i is independent of the longitudinal coordinates x µ , the longitudinal components of Q vanish. For the longitudinal components of the Killing spinor equation (2.3) to have non-trivial solutions the function f must be constant, and without loss of generality we can set f = 1. The covariant derivative in the transverse components of (2.3) include the connection of SO(2) 89 , the rotation in the 8-9 plane, and that of H = SO(2) R : For the existence of non-vanishing solutions, the action of two connections on some components of ǫ must be pure gauge. To study this condition, it is convenient to decompose the parameter ǫ into four parts ǫ s,r according to SO(2) 89 and SO(2) R charges so that where both the indices s and r take values in {+ 1 2 , − 1 2 }. We also decompose λ i in the same way into λ i sr . For distinction we use s = {↑, ↓} for SO(2) 89 and r = {+, −} for SO(2) R . We also introduce i = {⊕, ⊖} for complex basis of SO(2) R vectors, which carry SO(2) R charge ±1. See appendix for detail.
Let us require the solution to be half BPS. Without loosing generality we can assume that ǫ ↑+ and its Majorana conjugate ǫ ↓− correspond to the unbroken supersymmetries. The other components are set to be zero: ǫ ↓+ = ǫ ↑− = 0. Then, the non-vanishing components of δλ are and its complex conjugate. Before proceeding, it would be instructive to check the consistency of the quantum numbers in (2.13). Let us first consider the SO(2) 89 quantum numbers. The left hand side has the lower index ↓. This means the component carries SO(2) 89 charge (spin) −1/2. On the right hand side, the parameter ǫ has lower index ↑ which means SO(2) 89 spin +1/2. In addition, P has lower index z * and this component carries SO(2) 89 spin −1. Therefore, both left and right hand sides carry the same SO(2) 89 spin −1/2. The coincidence of the SO(2) R charge can be confirmed in a similar way. The index i is common for left and right hand sides and thus let us focus on the other indices. On the left hand side we have lower − index and this means it carries SO(2) R charge −1/2. On the right hand side there are the upper ⊕ index on P and the lower + index on ǫ, which carry SO(2) R charges −1 and +1/2, respectively. Therefore, the left and right hand sides carry the same SO(2) R charge −1/2. The charge counting we have just explained is quite useful when we extract condition imposed on P from Killing spinor equations associated with dilatino fields in different dimensions.
The vanishing of (2.13) mean (2.14) (P ⊖⊕ z * is identically zero due to the traceless condition.) We want to solve this with respect to the scalar fields L α i . For this purpose it is convenient to gauge fix the local SO(2) R symmetry so that the matrix L is given by where K(τ ) for a complex number τ in the upper half plane is the following 2 × 2 matrix: Then P and Q have the components In this gauge the equation (2.14) gives Namely, τ must be a holomorphic function of z. Now let us turn to the equation δψ m = D m ǫ = 0. The components including the non-vanishing parameters ǫ ↑+ and ǫ ↓− are and its complex conjugation. For (2.19) to have solutions with ǫ ↑+ = 0, the net connection ω 89 + Q 12 must be pure gauge, and we can take the gauge with ω 89 + Q 12 = 0. The explicit form of the spin connection and the SO(2) R connection are where we used holomorphy of τ in the last equality. From ω 89 + Q 12 = 0 we obtain and this is solved by where c is an arbitrary real positive constant, which can be absorbed by the coordinate change cx m → x m . The solution is summarized as follows.
This solution is specified by the single holomorphic function τ (z). The imaginary part of τ (z) must be positive, and no globally defined holomorphic function satisfy this condition unless τ (z) is a constant. For nontrivial solution τ (z) must be given as multi-valued solution with singularities. These singularities are regarded as branes, and the monodromies associated with the multi-valueness specify the charges of the branes. It is well-known that these singularities are classified by Kodaira classification, and we do not give detailed explanation about it.
In the following we will construct solutions in lower dimensions, and find that they are also described by holomorphic functions with positive imaginary part. Because the classification of the singularity can be done in a similar way to the ten-dimensional case, and it is well studied, we only focus on the local structure of solutions.

D = 9
Let us start the analysis in lower dimensions following the prescription in the last subsection. The scalar and spinor fields in the nine-dimensional N = 2 supergravity [17] are summarized in Table 3. The fields λ i are subject to the gamma-traceless condition ρ i λ i = 0. The Lagrangian is where the dots represent terms with gauge fields and four-fermi terms, which play no role in the following analysis. The supersymmetry transformation Table 3: The quantum numbers of scalar and spinor fields in the nine- rules for the spinor fields are We want to obtain codimension 2 brane solutions by solving the Killing spinor equations. We use x µ (µ = 0, 1, . . . , 6) and x m (m = 7, 8) for longitudinal and transverse coordinates, respectively. We take the ansatz In fact, the solution is almost the same as that of type IIB case. Although we have extra fields ϕ and λ compared to the ten-dimensional case, the condition δ λ = 0 forces ϕ to be constant; Therefore, we can forget about λ and ϕ, and remaining fields give the set of equations identical to the ten-dimensional case. After some gauge choices the general solution is given by A solution is specified by a single holomorphic function τ (z) and a constant vacuum expectation value of ϕ. Cosimenison 2 brane solutions appear as singularities of the function τ (z).

D = 8
The scalar and fermion fields in 8d maximal supergravity [18] are shown in in Table 4. Classical p-brane solutions with p = 0, 1, 3, 4 are given in [19]. Half Table 4: Quantum numbers of scalar and spinor fields in 8d maximal supergravity. The SO(2) R charge of each component of a spinor is proportional to the chirality.
BPS solutions of 10 dimensional supergravity given in [20] can be regarded as codim-2 branes in 8 dimensional supergravity. In the following we construct general 5-brane solutions without assuming 10 dimensional supergravity description.
The scalar manifold of the eight dimensional maximal supergravity is the direct product of two homogeneous spaces: SL(2, Z)/SO(2) R ×SL(3, Z)/SO(3) R . Each factor can be interpreted geometrically in an appropriate duality frame. The SL(2, Z)/SO(2) R becomes manifest when we regard the theory as T 2 compactification of type IIB theory, while SL(3, Z)/SO(3) R can be regarded as the moduli space associated with T 3 compactification of M-theory. The S-duality group in the type IIB picture is a subgroup of SL(3, Z).
The Lagrangian is where the dots represent four-fermion terms and terms with gauge fields. The supersymmetry transformation laws of fermions are We are interested in codimension 2-brane solutions and we use x µ (µ = 0, 1, . . . , 5) and x m (m = 6, 7) for longitudinal and transverse coordinates, respectively. We take the following ansatz: The covariant derivative D M ǫ contains three connections ω, Q, and Q, corresponding to SO(2) 67 , SO(2) R , and SO(3) R , respectively. For the existence of non-trivial solution to δψ m = 0, the actions of three connections to some components of ǫ must be pure gauge. For this to be the case, nonvanishing components of SO(3) R connection Q should be in a certain SO(2) subgroup of SO(3) R . We can take the gauge such that it is rotation of 1 2 plane and (2.40) After taking this gauge, we have three SO(2) connections ω 67 , Q 12 and Q 1 2 . As in the 10 dimensional case it is convenient to divide the parameter ǫ into components ǫ sr r so that 1 2 Γ 67 ǫ sr r = isǫ sr r , 1 2 ρ 12 ǫ sr r = irǫ sr r , 1 2 ρ 12 ǫ sr r = i rǫ sr r , where all of s, r, and r take values in {+ 1 2 , − 1 2 }. For distinction we introduce the notation s ∈ {↑, ↓} for SO(2) 67 , r ∈ {+, −} for SO(2) R , and r ∈ { +, −} for SO(3) R . We also introduce {⊕, ⊖} for the complex basis of SO(2) R vector and { ⊕, ⊖, 3} for the basis of SO(3) R vector that diagonalize SO(2) 1 2 .
The 6d chirality of ǫ sr r is given by s and r as γ 7 ǫ sr r = sign(s r)ǫ sr r . (2.42) We want to consider solution in which some of ǫ sr r are preserved. Without loss of generality, we can suppose that ǫ ↑+ + and its complex conjugate ǫ ↓− − are non-vanishing. Both of them have positive 6d chirality (2.42), and they generate six-dimensional N = (1, 0) supersymmetry.
Let us consider the condition δλ = 0 first. The component of δλ depending on ǫ ↑+ + is For this to hold for ǫ ↑+ + = 0, P ⊕⊕ z * must vanish. This is the same as (2.14) in Section 2.1, and the solution is given by L = K(τ ) where K(τ ) is defined in (2.16) with a holomorphic function τ (z).
We can also obtain similar condition for L form δ λ = 0. The components of δ λ depending on ǫ ↑+ + are δ λ i ↓+ ± , and we obtain the following Killing spinor equations.
The equation (2.44) requires P i 3 = 0, and combining this with (2.40) we conclude that L is essentially SL(2) element. Namely, in an appropriate choice of gauge it is given by where L 0 ∈ SL(3, R) is a constant matrix. The condition P i ⊕ z * = 0 obtained from (2.45) requires the function τ be a holomorphic function of z.
Finally, we can determine the function g by using δψ m = D m ǫ = 0. For this equation to hold for ǫ ↑+ + = 0, the sum of three connections ω, Q and Q must be pure gauge, and we can take the gauge in which ω m67 + Q m12 + Q m 1 2 = 0.
(2.47) This gives the differential equation which is solved by where c is a positive real constant, which can be absorbed by the coordinate change cx m → x m . The solution is summarized as follows. (2.51) This is the general form of 1/4 BPS solutions. A solution is specified by two holomorphic functions τ (z) and τ (z) and constant L 0 ∈ SL(3, R). 1/2 BPS solutions are realized as special cases of this solution. Let us consider the case in which the supersymmetries associated with ǫ ↑− + and its conjugate ǫ ↓+ − are also preserved in addition to ǫ ↑+ + and its conjugate ǫ ↓− − . (2.42) shows that ǫ ↑− + and ǫ ↓+ − have negative 6d chirality and we have N = (1, 1) supersymmetry in this case. The Killing spinor equations including ǫ ↑− + are We have additional condition P −− z * = 0 from (2.53), and this requires τ to be anti-holomorphic. This means τ must be a constant. Then the other equations hold.
There is another type of 1/2 BPS solutions with ǫ ↑+ − , ǫ ↓− + = 0. (2.42) shows that these components have positive 6d chirality, and we obtain N = (2, 0) supersymmetry in six dimensions. The Killing spinor equations including ǫ ↑+ − are The first gives the additional condition P ⊕⊕ z = 0, which requires τ to be a constant, and the third gives P i ⊕ z = 0, and this means constant τ . Then, the solution becomes trivial flat solution, and all supersymmetries are preserved.
We summarize non-trivial BPS solutions in Table 5. As is shown in Table 5 two holomorphic functions τ and τ correspond to two types of branes. Namely, singularities of τ and τ give 5-branes with N = (2, 0) and N = (1, 1) supersymmetry, respectively. 1/4 BPS solutions with N = (1, 0) supersymmetry is regarded as simple superposition of two types of branes.
The most general 1/4 BPS solution are embedded in SL(2) × SL(2) ⊂ SL(2) × SL(3). These two SL(2) factors are manifest in the type IIB frame. Namely, the SL(2) factor which is a subgroup of SL(3) can be associated with the axio-dilaton field in type IIB theory, and the other SL(2) is associated with the internal space T 2 . From the viewpoint of F-theory the 1/4 BPS solution can be regarded as a compactification of the F-theory in a Calabi-Yau realized as T 4 fibration over C.
The scalar manifold of 7d maximal supergravity is SL(5)/SO (5). There is no duality frame which manifests whole of the duality group SL(5, Z) and the R-symmetry group SO(5) R . When we regard the system as the T 4 compactification of M-theory SL(4)/SO(4) becomes manifest, while T 3 compactification of type IIB theory manifests SL(3)/SO(3) × SL(2)/SO (2). Combining these we obtain the full symmetry.
The relevant part of the Lagrangian is The transformation parameter ǫ belongs to the spinor representation of H = SO(5), and the covariant derivative D M ǫ includes the connection Q M ij . We are interested in codimension 2-brane solutions and we use x µ (µ = 0, 1, . . . , 4) and x m (m = 5, 6) for longitudinal and transverse coordinates, respectively. By assuming the Poincare invariance in the five dimensions parallel to the brane, we take the following ansatz.
Let us first consider the case with minimum number of unbroken supersymmetries. The supersymmetry parameter ǫ belongs to the 4 of SO(5) R symmetry, and in the minimum case we have only one non-vanishing component. Then the R-symmetry is broken to It is convenient to consider the intermediate subgroup SU(2) l ×SU(2) r ∼ SO(4) ⊂ SO(5) R . The parameter ǫ is decomposed into four irreducible representation (2, 1)
(2.76) (We also have similar conditions for P z from the equations containing ǫ ↓2 ∼ (ǫ ↑1 ) * .) (2.71) and (2.76) show that non-vanishing components of P and Q are associated with a subgroup SO(4) ⊂ SO(5) R . As we mentioned above SO(4) subgroup of SO(5) R can be realized geometrically if we regard the theory as T 4 compactification of M-theory. We want to give the scalar fields L such that P and Q have only nonvanishing components (2.76) and (2.71). Unfortunately, we have not obtained the answer. To simplify the problem, let us consider a restricted case with P (12) = 0. Then F (Q) = P ∧P takes value in the Cartan part of SU(2)×U(1), and we can take the gauge such that Q (11) = Q (22) = 0, then non-vanishing components are Q (12) , Q (12) , P (11) (11) , P (22) (11) . (2.77) In this case, with an appropriate real basis, the 5 × 5 matrices P and Q are block diagonal matrices in the following form.
Therefore, the solution reduces to the superposition of two copies of solutions for SL(2, R)/SO(2) scalar manifold. In the same way as in higher dimensions, each SL(2, R) part can be expressed in terms of a holomorphic function. Let the two holomorphic functions be τ ′ and τ ′′ . The solution is given by As a special case of this 1/4 BPS solution we can realize 1/2 BPS solution. Let us consider the cases there is another Killing spinor in addition to ǫ˙1. There are two cases.
First, let us consider the case that ǫ 1 is also a Killing spinor. In this case, from we obtain P z * (a2)(ȧḃ) = 0. Then the only non-vanishing component of P z * is P z * (11) (11) . In this case, just like the case of 1/2 BPS solution in 8d, we can show that one of τ ′ and τ ′′ must be z-independent constant.
If two Killing spinor have the same SO(4) R chirality, the equation (2.74) require P z * (ab)(ȧḃ) = 0. Namely, all components of P vanish. Because F (Q) = P ∧ P = 0 we can choose a gauge with Q = 0. Therefore, the solution is trivial.

Conclusions
In this paper we investigated codimension-2 BPS solutions in maximal supergravities in 9, 8, and 7 dimensions. We assumed the Poincare invariance along branes and vanishing of various gauge fields.
The scalar manifold of 9d maximal supergravity is (SL(2, R)/SO(2))×R. In a BPS solution the scalar field associated with the factor R must be constant and play no role. Therefore, the solutions are essentially the same as those of type IIB supergravity in 10d, and simply interpreted as the double dimensional reduction of type IIB 7-branes.
In 8d, the scalar manifold consists of two factors SL(2, R)/SO(2) and SL(3, R)/SO(3). Killing spinor equations associated with these factors decouple, and we can solve them one by one. From the SL(2, R)/SO(2) part we obtain 1/2 BPS branes on which six-dimensional N = (2, 0) supersymmetry is realized while from the SL(3, R)/SO(3) part we obtain 1/2 BPS solutions on which six-dimensional N = (1, 1) supersymmetry is realized. The latter is always embedded in SL(2, R)/SO(2) ⊂ SL(3, R)/SO(3). These two types of solutions have essentially the same structure as the 7-brane solution in 10d. Namely, each type of classical solution is specified by a holomorphic function and singularities of the function give branes. We also found 1/4 BPS solutions, which are specified by two holomorphic functions and are regarded as simple superposition of two types of 1/2 BPS solutions. If we regard the 8d supergravity as the T 2 compactification of type IIB theory, the two copies of SL(2, R)/SO(2) are associated with the complex moduli of the torus and the axio-dilaton field in type IIB theory, and both are geometrically realized in F-theory.
In 7d, a generic BPS solution is 1/4-BPS. We showed that such solution can be embedded in SL(4, R)/SO(4) ⊂ SL(5, R)/SO (5). This means the solution can be realized as a geometric compactification of M-theory. We could not solve the Killing spinor solutions in the general situation. We introduced one additional restriction to simplify the problem, and then the solution is factorized into two copies of solutions associated with SL(2, R)/SO (2). Again, similarly to the 8d case, the solution can be regarded as a simple superposition of two 1/2 BPS solutions.
Although our original motivation of this work was to find essentially new BPS branes, all solutions we found have simple geometric realization in Mor F-theory. upper indices like (ρ i ) a b . For components of spinors we define the SO(2) R charge as eigenvalues of the generator (−i/2)ρ 12 . This means the upper and the lower components of spinors carry the charges +1/2 and −1/2, respectively. To specify these components of a spinor χ we use the notation χ + and χ − , respectively.
For the analysis of the Killing spinor equations it is convenient to the complex basis for vectors. For example, for a vector v i (i = 1, 2) we define With this representation the lower index ⊕ and upper index ⊖ carry SO(2) R charge +1, while the lower ⊖ and upper ⊕ carry SO(2) R charge −1. This is checked by looking at the non-vanishing components of ρ i . For example the non-vanishing component of ρ ⊕ is (ρ ⊕ ) − + , and the total charge of this component must be zero. The statement above about SO(2) R charge is consistent to this.
For the local rotation symmetry in the transverse space to branes we use up and down for the SO(2) charge (spin) ±1/2. With the choice of the Dirac matrices, the lower indices z and z * carry spin +1 and −1, respectively.
In 8d we also deal with SO(3) R symmetry. The notation is basically the same as the SO(2) R case except we put tildes on variables and indices for distinction from SO(2) R objects. We specify components of spinors by eigenvalues of the Cartan generator (−i/2) ρ 1 2 . χ ± carry the charge ±1/2, and a vector v has three components v ⊖ = v ⊕ , v ⊕ = v ⊖ , and v 3 = v 3 that carry the Cartan charge +1, −1, and 0, respectively.
In the following we give relations of fields in this paper and those in references. We will not give detailed explanations for normalization of fields, spinor conventions, etc., because they are not important in our analysis of the Killing spinor equations. We focus on giving rough correspondence between fields used in this paper and those in references. D = 10 Ten dimensional supergravity is given in [16]. The global symmetry SL(2, R) is isomorphic to SU(1, 1). In [16] the scalar fields are expressed as the matrix V a ± , which is defined with a complex basis natural for SU(1, 1).
The real matrix L α i used in this paper is related with V by The dilatino field λ in [16] is defined as the field with U(1) R charge ±3/2, while we denote this as a field with vector and spinor indices. They are related by Due to the ρ-traceless condition λ ⊕ + = λ ⊖ − = 0. D = 9 The 9-dimensional maximal supergravity is given in [17]. Two dilatino fields in [17] are renames as λ i and λ to to match the fields in the other dimensions. SO(2) Dirac matrices are denoted by τ i in [17] while we use ρ i for them. D = 8 The 8-dimensional maximal supergravity is given in [18]. The dilatino field χ i in [18] does not satisfy the ρ-traceless condition, and we decompose it into the traceless part λ i and the trace part λ I . To make the SL(2, R)/SO(2) structure manifest we combine the scalar fields φ and B in [18] into the matrix L A I . With the gauge choice like (2.15) these are related by L = K(τ ) with τ = −2B + ie 2φ . D = 7 The 7-dimensional maximal supergravity is given in [22]. In the reference the scalar matrix is denoted by Π instead of L. Notation for other fields is similar to ours.