Let us consider BPS solutions in type IIB supergravity. Such solutions have been well investigated [1] and 7-branes are classified by the Kodaira classification [12]. Various four-dimensional supersymmetric theories are realized on D3-branes probing these solutions [13–15]. 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 and the local R-symmetry group is . Namely, the scalar manifold is locally the two-dimensional homogeneous space . When we discuss the global structure, we also need to take account of the duality group . Quantum numbers of scalar and spinor fields in type IIB supergravity [16] are summarized in Table 2. The gravitino field belongs to the spinor representation of . Namely, has the spacetime vector index and an spinor index which is implicit. The dilatino field has the vector index and an implicit spinor index. It satisfies the -traceless condition (2.1) where are Dirac matrices associated with the orthogonal group . See the appendix for our notation. This condition removes components carrying charge from , and the remaining components in carry charge , as shown in Table 2.

Table 2.

Quantum numbers of scalar and spinor fields in type IIB supergravity.

Due to the existence of the self-dual four-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 vanishing anti-symmetric tensor fields, the equations of motion for the remaining fields are obtained from the Lagrangian (2.2) up to higher-order fermion terms.

The Killing spinor equations are (2.3) (2.4) where is the covariant derivative defined with the spin connection and the connection : (2.5)

We are interested in codimension-2 brane solutions. Let us assume the solution has the eight-dimensional Poincaré invariance along the eight longitudinal directions. We use () and () for longitudinal and transverse coordinates, respectively. We take the ansatz (2.6) for the metric and (2.7) for the scalar fields. We introduce the local frame so that the vielbein has the diagonal components (2.8)

Because we are interested in the rigid supersymmetry on the branes we assume that the supersymmetry parameter depends only on the transverse coordinates: (2.9)

Because is independent of the longitudinal coordinates , the longitudinal components of vanish. For the longitudinal components of the Killing spinor equation of Eq. (2.3), (2.10) to have non-trivial solutions the function must be constant, and without loss of generality we can set .

The covariant derivative in the transverse components of Eq. (2.3) include the connection of , the rotation in the – plane, and that of : (2.11)

For the existence of non-vanishing solutions, the action of two connections on some components of must be pure gauge. To study this condition, it is convenient to decompose the parameter into four parts according to and charges so that (2.12) where both the indices and take values in . We also decompose in the same way into . For distinction we use for and for . We also introduce for the complex basis of vectors, which carry charge . See the appendix for details.

Let us require the solution to be half BPS. Without loss of generality we can assume that and its Majorana conjugate correspond to the unbroken supersymmetries. The other components are set to zero: . Then, the non-vanishing components of are (2.13) and its complex conjugate.

Before proceeding, it would be instructive to check the consistency of the quantum numbers in Eq. (2.13). Let us first consider the quantum numbers. The left-hand side has the lower index . This means the component carries charge (spin) . On the right-hand side, the parameter has lower index which means spin . In addition, has lower index and this component carries spin . Therefore, both left- and right-hand sides carry the same spin . The coincidence of the charge can be confirmed in a similar way. The index is common for the left- and right-hand sides and thus let us focus on the other indices. On the left-hand side we have the lower index and this means it carries charge . On the right-hand side there are the upper index on and the lower index on , which carry charges and , respectively. Therefore, the left- and right-hand sides carry the same charge . The charge counting we have just explained is quite useful when we extract the condition imposed on from Killing spinor equations associated with dilatino fields in different dimensions.

The vanishing of Eq. (2.13) means (2.14)

( is identically zero due to the traceless condition.) We want to solve this with respect to the scalar fields . For this purpose it is convenient to gauge fix the local symmetry so that the matrix is given by (2.15) where for a complex number in the upper half-plane is the following matrix: (2.16)

Then and have the components (2.17)

In this gauge, Eq. (2.14) gives (2.18)

Namely, must be a holomorphic function of .

Now let us turn to the equation . The components including the non-vanishing parameters and are (2.19) and its complex conjugation. For Eq. (2.19) to have solutions with , the net connection must be pure gauge, and we can take the gauge with . The explicit forms of the spin connection and the connection are (2.20) where we used holomorphy of in the last equality. From we obtain (2.21) and this is solved by (2.22) where is an arbitrary real positive constant, which can be absorbed by the coordinate change .

The solution is summarized as follows: (2.23) (2.24) (2.25)0. \end{align}]]>

This solution is specified by the single holomorphic function .

The imaginary part of must be positive, and no globally defined holomorphic function satisfies this condition unless is a constant. For a non-trivial solution must be given as a 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 the Kodaira classification, and we do not give a detailed explanation of this.

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 has been well studied, we only focus on the local structure of solutions.

Let us start the analysis in lower dimensions following the prescription in the last subsection. The scalar and spinor fields in the nine-dimensional supergravity [17] are summarized in Table 3.

Table 3.

The quantum numbers of scalar and spinor fields in the nine-dimensional supergravity.

The fields are subject to the gamma-traceless condition . The Lagrangian is (2.26) where the dots represent terms with gauge fields and four-fermion terms, which play no role in the following analysis. The supersymmetry transformation rules for the spinor fields are (2.27) (2.28) (2.29)

We want to obtain codimension- brane solutions by solving the Killing spinor equations. We use () and () for longitudinal and transverse coordinates, respectively. We take the ansatz (2.30)

In fact, the solution is almost the same as that of the type IIB case. Although we have extra fields and compared to the ten-dimensional case, the condition forces to be constant; (2.31)

Therefore, we can forget about and , and the remaining fields give a set of equations identical to the ten-dimensional case. After some gauge choices the general solution is given by (2.32) (2.33) (2.34)

A solution is specified by a single holomorphic function and a constant vacuum expectation value of . Codimenison- brane solutions appear as singularities of the function .

The scalar and fermion fields in eight-dimensional maximal supergravity [18] are shown in Table 4. Classical -brane solutions with are given in Ref. [19]. Half BPS solutions of ten-dimensional supergravity given in Ref. [20] can be regarded as codimension-2 branes in eight-dimensional supergravity. In the following we construct general 5-brane solutions without assuming a ten-dimensional supergravity description.

Table 4.

Quantum numbers of scalar and spinor fields in eight-dimensional maximal supergravity. The charge of each component of a spinor is proportional to the chirality.

The scalar manifold of the eight-dimensional maximal supergravity is the direct product of two homogeneous spaces: . Each factor can be interpreted geometrically in an appropriate duality frame. The becomes manifest when we regard the theory as compactification of type IIB theory, while can be regarded as the moduli space associated with compactification of M-theory. The S-duality group in the type IIB picture is a subgroup of .

For each factor of the R-symmetry group there is an associated dilatino field. We denote fields associated with and by and , respectively. All fermion fields have implicit spinor indices for all , , and . In addition, and have and vector indices, respectively, and they satisfy the traceless conditions and . Namely, and belong to and , respectively, of .

The Lagrangian is (2.35) where the dots represent four-fermion terms and terms with gauge fields. The supersymmetry transformation laws of fermions are (2.36) (2.37) (2.38)

We are interested in codimension- brane solutions and we use () and () for longitudinal and transverse coordinates, respectively. We take the following ansatz: (2.39)

The covariant derivative contains three connections, , , and , corresponding to , , and , respectively. For the existence of non-trivial solution to , the actions of the three connections to some components of must be pure gauge. For this to be the case, non-vanishing components of connection should be in a certain subgroup of . We can take the gauge such that it is rotation of the plane and (2.40)

After taking this gauge, we have three connections, , , and . As in the ten-dimensional case it is convenient to divide the parameter into components so that (2.41) where all of , , and take values in . For distinction we introduce the notation for , for , and for . We also introduce for the complex basis of an vector and for the basis of an vector that diagonalizes .

The six-dimensional chirality of is given by and as (2.42)

We want to consider a solution in which some of the are preserved. Without loss of generality, we can suppose that and its complex conjugate are non-vanishing. Both of them have positive six-dimensional chirality Eq. (2.42), and they generate six-dimensional supersymmetry.

Let us consider the condition first. The component of depending on is (2.43)

For this to hold for , must vanish. This is the same as Eq. (2.14) in Sect. 2.1, and the solution is given by where is defined in Eq. (2.16) with a holomorphic function .

We can also obtain a similar condition for from . The components of depending on are , and we obtain the following Killing spinor equations: (2.44) (2.45)

Equation (2.44) requires , and combining this with Eq. (2.40) we conclude that is essentially an element. Namely, in an appropriate choice of gauge it is given by (2.46) where is a constant matrix. The condition obtained from Eq. (2.45) requires the function to be a holomorphic function of .

Finally, we can determine the function by using . For this equation to hold for , the sum of three connections , , and must be pure gauge, and we can take the gauge in which (2.47)

This gives the differential equation (2.48) which is solved by (2.49) where is a positive real constant, which can be absorbed by the coordinate change .

The solution is summarized as follows: (2.50) (2.51) (2.52)

This is the general form of BPS solutions. A solution is specified by two holomorphic functions and and constant .

BPS solutions are realized as special cases of this solution. Let us consider the case in which the supersymmetries associated with and its conjugate are also preserved in addition to and its conjugate . Equation (2.42) shows that and have negative six-dimensional chirality and we have supersymmetry in this case. The Killing spinor equations including are (2.53) (2.54) (2.55) (2.56)

We have the additional condition from Eq. (2.53), and this requires to be anti-holomorphic. This means must be a constant. Then the other equations hold.

There is another type of BPS solution with . Equation (2.42) shows that these components have positive six-dimensional chirality, and we obtain supersymmetry in six dimensions. The Killing spinor equations including are (2.57) (2.58) (2.59) (2.60)

Equation (2.59) gives the new condition , and this means is a constant. Then the other conditions are satisfied.

Finally, let us consider the case with and non-vanishing. The Killing spinor equations including are (2.61) (2.62) (2.63) (2.64)

The first gives the additional condition , which requires to be a constant, and the third gives , and this means constant . Then, the solution becomes the trivial flat solution, and all supersymmetries are preserved.

We summarize the non-trivial BPS solutions in Table 5. As shown there, two holomorphic functions and correspond to two types of branes. Namely, singularities of and give -branes with and supersymmetry, respectively. BPS solutions with supersymmetry are regarded as simple superposition of two types of branes.

Table 5.

The non-trivial 5-brane solutions in supergravity and worldvolume supersymmetries.

The most general BPS solutions are embedded in . These two factors are manifest in the type IIB frame. Namely, the factor that is a subgroup of can be associated with the axio-dilaton field in type IIB theory, and the other is associated with the internal space . From the viewpoint of F-theory the BPS solution can be regarded as a compactification of the F-theory in a Calabi–Yau realized as fibration over .

The seven-dimensional maximal supergravity has the field contents in Table 6 [21, 22]. The scalar manifold of seven-dimensional maximal supergravity is . There is no duality frame which manifests the whole of the duality group and the R-symmetry group . When we regard the system as the compactification of M-theory becomes manifest, while compactification of type IIB theory manifests . Combining these we obtain the full symmetry.

Table 6.

The field contents of seven-dimensional maximal supergravity. Anti-symmetric tensor fields are omitted.

The relevant part of the Lagrangian is (2.65) and the supersymmetry transformation rules for fermions are (2.66)

The transformation parameter belongs to the spinor representation of , and the covariant derivative includes the connection .

We are interested in codimension- brane solutions and we use () and () for longitudinal and transverse coordinates, respectively. By assuming the Poincaré invariance in the five dimensions parallel to the brane, we take the following ansatz: (2.67)

Let us first consider the case with the minimum number of unbroken supersymmetries. The supersymmetry parameter belongs to the of 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 . The parameter is decomposed into four irreducible representations and of . ( is the local Lorentz symmetry in the transverse space.) We denote them as (2.68) where represent the charges and and are indices for and , respectively. The fields and are decomposed as (2.69) and the dilatino as (2.70) where a pair of indices in parenthesis are symmetric. If we choose as the component for the unbroken supersymmetry, is broken to . The connection should take its value in . Namely, and the only non-vanishing components are (2.71)

The appear in the supersymmetry transformation as (2.72) (2.73) (2.74) (2.75)

(We omitted the numerical coefficients that are not important here.) If we require for , we obtain and the only non-vanishing components of are (2.76)

(We also have similar conditions for from the equations containing .) Equations (2.71) and (2.76) show that non-vanishing components of and are associated with a subgroup . As we mentioned above, the subgroup of can be realized geometrically if we regard the theory as compactification of M-theory.

We want to give the scalar fields such that and have only the non-vanishing components of Eqs. (2.76) and (2.71). Unfortunately, we have not obtained the answer. To simplify the problem, let us consider a restricted case with . Then takes value in the Cartan part of , and we can take the gauge such that ; then the non-vanishing components are (2.77)

In this case, with an appropriate real basis, the matrices and are block diagonal matrices in the following form: (2.78)

Therefore, the solution reduces to the superposition of two copies of solutions for the scalar manifold. In the same way as in higher dimensions, each part can be expressed in terms of a holomorphic function. Let the two holomorphic functions be and . The solution is given by (2.79) (2.80)

As a special case of this BPS solution we can realize the BPS solution. Let us consider the cases where there is another Killing spinor in addition to . There are two cases.

First, let us consider the case that is also a Killing spinor. In this case, from (2.81) we obtain . Then the only non-vanishing component of is . In this case, just like the case of the BPS solution in eight dimensions, we can show that one of and must be a -independent constant.

If two Killing spinors have the same chirality, Eq. (2.74) requires . Namely, all components of vanish. Because we can choose a gauge with . Therefore, the solution is trivial.