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Superstring dualities and p-brane bound states

We show that the M-theory/IIA and IIA/IIB superstring dualities together with the diffeomorphism invariance of the underlying theories require the presence of certain p-brane bound states in IIA and IIB superstring theories preserving 1/2 of the spacetime supersymmetry. We then confirm the existence of IIA and IIB supergravity solutions having the appropriate p-brane bound states interpretation.


Introduction
Much evidence have been gathered in the past year to support the conjecture that the strong coupling limit of IIA superstring theory is an 11-dimensional supersymmetric theory called M-theory [1,2]. More precisely, the IIA superstring theory is the compactification of M-theory on a S 1 with the IIA string coupling constant λ related to the compactification radius R as λ = R 2/3 . This interpretation of the IIA superstring theory requires the presence of non-perturbative states in IIA some of which should carry Ramond-Ramond (RR) charges. The first indication that such states were present in IIA superstrings was the existence of solutions of D=10 IIA supergravity carrying the appropriate charges [3,4], sub-try and therefore this equivalence between IIA and IIB superstrings can be verified to all orders in string perturbation theory. As in the M-theory/IIA duality, the IIA p-branes transform under IIA/IIB T-duality to the IIB p-branes and vice-versa. These BPS-saturated states are below threshold and have the interpretation of pbrane bound states preserving 1/2 of the spacetime supersymmetry; we shall list these p-brane bound states in section 2. The presence of these p-brane bound states † See [11] for the T-duality transformation on the D-branes from the superstring point of view.
in IIA and IIB superstring theory is required by the M-theory/IIA and IIA/IIB dualities together with the diffeomorphism invariance of the underlying theories.
This association of p-brane bound states to the M-theory/IIA and IIA/IIB dualities is similar to the association of KK modes to the KK reduction. Therefore, the existence of these p-brane bound states in IIA and IIB superstrings is necessary for the consistency of these duality conjectures. Some of the p-brane bound states that we shall consider in this paper have already been investigated either in the context of D-branes [10,11,12,13,14] or from the effective theory point of view [15,16,17,18].
They are also required for the consistency of other superstring duality conjectures [19,20].
We shall describe the above p-brane bound states from the macroscopic point of view. However in certain cases we shall also comment on their microscopic properties. The evidence that we shall present for the existence of these p-brane bound states and for their interpretation is derived from consideration of the solutions of the associated effective supergravity theories with the analogous interpretation. To derive the solutions in IIA and IIB supergravity with the desirable interpretation, we shall start from the solution of D=11 supergravity with the interpretation of a membrane/fivebrane bound state preserving 1/2 of the spacetime supersymmetry [15]. This solution will be reduced to solutions of IIA supergravity using the 'standard' reduction along either a worldvolume or a transverse direction of the configuration [16,17]. The reduced solutions have a IIA interpretation as bound states of two IIA p-branes preserving 1/2 of the spacetime supersymmetry. Then starting from these IIA solutions and using the T-duality rules [21], we shall construct 'standard' T-duality chains as in [7], i.e. T-duality transformations along worldvolume or transverse directions of these configurations, to find most of the required p-brane bound state solutions in IIA and IIB supergravity theories.
These p-brane bound states will also be derived using KK reductions and T-duality transformations from the p-brane solutions of D=11, IIA and IIB supergravities but this time these operations will be taken along a generic direction in spacetime. In this way, we shall establish that the origin of these bound states is due to the M-theory/IIA and IIA/IIB dualities and the diffeomorphism invariance of the underlying theories.
The organization of this paper is as follows: In section two, we shall present the bound states of IIA and IIB superstring theories expected from the reduction of M-theory to IIA and from the IIA/IIB T-duality. In section three, we shall give the IIA supergravity solutions with the interpretation of p-brane bound states that are expected from the reduction of M-theory to IIA superstring. In section four, we shall give the IIA and IIB supergravity solutions with the interpretation of p-brane bound states that are expected from the IIA/IIB duality. In section five, we shall present an M-theory interpretation of some of these solutions, and in section six we shall give our conclusions.

p-brane bound states
The reduction and T-duality operations do not commute with spacetime diffeomorphisms. Because of this there are many ways to choose a direction in spacetime to perform a reduction or a T-duality transformation. However, as in [18], here we shall consider only the following two cases: (i) In the first case, we shall perform the reduction and the T-duality transformation along a spatial spacetime direction which can be decomposed as a linear combination of a spatial worldvolume and a transverse direction of a p-brane, i.e. this direction intersects the p-brane at an angle α. We shall denote the corresponding operations with R α and T α , and we shall refer to them as reduction at an angle and T-duality at an angle, respectively.
For α = 0 or α = π/2, R α and T α become the standard reduction and T-duality transformations along a worldvolume or a transverse directions of the p-brane, respectively. Therefore R α (T α ) is a linear combination of a standard reduction (T-duality transformation) along a worldvolume direction and a standard reduction (T-duality transformation) along a transverse direction of a p-brane. (ii) In the second case, we shall perform the reduction and the T-duality transformation along a direction which is transverse to a p-brane and with the p-brane moving with relativistic velocity v in this direction, i.e. the p-brane is 'boosted' in this direction.
We shall denote the corresponding operations with R v and T v , and we shall refer to them as reduction along a boost and T-duality along a boost, respectively. In what follows, we shall use R and T to denote the 'standard' reduction and the 'standard' T-duality transformation along either a worldvolume or a transverse direction of a p-brane. We shall also use, when it is necessary in order to avoid ambiguities, and respectively, where 0 w denotes the plane wave and 0 m denotes the KK monopole. Now since R α is equivalent to a simultaneous double and direct reduction, reducing the M-p-branes at an angle leads to IIA (p-1)-brane/p-brane bound states.
It is convenient to use the notation (r|p, q) to denote a solution representing an r-brane intersection of a p-brane with a q-brane; in the special case where p = r, (p|p, q) denotes a p-brane within a q-brane representing a bound state of a p-brane with a q-brane. In this notation, the reduction of M-branes at an angle is described Similarly, the reduction of the D=11 plane wave and the D=11 KK monopole at an angle leads to the IIA bound states where (0 w |0) A denotes a bound state of a plane wave with a 0-brane, and (0 m |6) A denotes the bound state of a KK monopole with a 6-brane. The (0 w |0) A bound state is a 'boosted' 0-brane.
Next, the reduction of the M-branes, the D=11 plane wave and the D=11 KK monopole along a boost leads to IIA bound states that always involve the IIA D-0brane. This is because from the D=10 perspective the (quantised) momentum in the compactifying direction becomes the mass of the KK modes of the compactification. But as we have mentioned in the introduction, these are identified with the IIA D-0-branes. So this observation together with the fact that we reduce along a transverse direction of the configurations lead to the following IIA p-brane bound states: (2.7) We remark that it has been argued in [11] that there is not a IIA BPS D-0-brane/D-6-brane bound state. We shall return to this point when we discuss the (0|0, 6) A solution of IIA supergravity in the next section. As in (2.6), the (0 w |0) A bound state is a 'boosted' 0-brane.
We now turn to the p-brane bound states required by the IIA/IIB superstring duality. As we have already mentioned T-duality transforms a (IIA or IIB) D-(p+1)-brane either as depending on whether the T-duality transformation is performed along a worldvolume or a transverse direction of the D-(p+1)-brane, respectively, where the p-brane and the (p+2)-brane are (IIB or IIA) D-branes. Similarly, the IIA or IIB fundamental string, solitonic 5-brane, plane wave and KK monopole transform under T-duality as follows: T-duality at an angle to the (IIA or IIB) fundamental string, solitonic 5-brane, plane wave and KK monopole leads to the following bound states: (2.12) The bound state (0 w |1 F ) is simply a boosted fundamental string.
Next, the p-brane bound states expected from applying T-duality along a boost to the IIA and IIB p-branes will always involve the fundamental (IIA or IIB) string. (2.14) We remark that, as we shall see in section 4, the supergravity solution associated with the (1|1 F , 5 S ) bound state has similar qualitative features as the solution that is interpreted as a (0|0, 6) A bound state.
So far we have listed the bound states that we expect to find in IIA and IIB superstring theories due to the M-theory/IIA and IIA/IIB dualities and to the diffeomorphism invariance of the underlying theories. Evidence for the existence of all these bound states will be given in the next two sections by finding the solutions of the corresponding supergravity theories with the appropriate interpretation. We have been able to carry out this computation for all the required p-brane bound states. The resulting IIA and IIB p-brane bound state solutions will be expressed in the string frame. All of them have a parameter α which interpolates between the two constituent objects that form the bound states.

M-theory/IIA duality and bound states
As where E n is the Euclidean space of dimension n, M n is the Minkowski space of dimension n, H is a harmonic function on E 10−p , ǫ(M) is the volume form of a manifold M, and the star is the Hodge star in E 5 . We also write for later use the (purely gravitational) D=11 plane wave solution where H is a harmonic function on E 9 , and the D=11 KK monopole solution where ds 2 (G/H) = H −1 (dρ + ω) 2 + Hds 2 (E 3 ) is the Gibbons-Hawking metric [25], H is a harmonic function on E 3 and dω = * dH. Both the plane wave [26,27] and the KK monopole preserve 1/2 of the spacetime supersymmetry; for the latter case this follows from the fact that ds 2 (G/H) is hyper-Kähler. To choose the compactifying direction u 11 in D=11, we use a D=11 diffeomorphism that mixes the spatial worldvolume and the transverse space coordinates of the M-branes. An example of such diffeomorphism is u 10 where ρ is the coordinate along a spatial worldvolume direction, y is the coordinate along a transverse direction and A is a real invertible 2 × 2 matrix. Let us choose where α is the angle that appears as a parameter in the (2|2, 5) M solution above.
The solutions of IIA supergravity that are constructed by reducing the M-branes along u 11 [18] are identical to the solutions found by the duality chains (3.2) and respectively. The (0 w |0) A solution may be found by extending the T-duality chain where (0 w |0) A is where dω = * dH and the Hodge duality is with respect to the Euclidean metric in E 3 . Both the (0 w |0) A and the (0 m |6) A solutions can also be obtained from the D=11 plane wave and the KK monopole by first rotating the solutions using (3.7) and (3.8), and then reducing them to D=10 along u 11 . For the (0 w |0) A solution this was done in [18].
Next we turn to the IIA supergravity solutions associated with the bound states Observe that this metric has an off-diagonal term which is reminiscent to a similar term in the Taub-NUT metric. This solution appears to have the appropriate non-vanishing supergravity field strength to be interpreted as a 0-brane/6-brane bound state. However although for α = 0 we find the familiar 6-brane solution of IIA supergravity, for α = π 2 we find the 0-brane solution but with negative ADM mass. In addition, the solution, apart from the singularities at the centres of the harmonic function H, has another singularity atH(r 0 ) = 0 which occurs at small string coupling whenever sin α = 0. Note that in order to boost the KK monopole one needs to 'decompactify' the coordinate ρ along the killing direction in order to impose the periodicity in the compactifying u 11 coordinate. This re-introduces the NUT singularities of the KK monopole that were resolved by the periodic identification of ρ. It is not clear though that there is a direct relation between the NUT singularities of the KK monopole and theH(r 0 ) = 0 ones since in any case they occur at different points. It is more likely that the NUT singularities of the KK monopole are related to the singularities of the D=10 solution that occur at the centres of H. The above properties of the solution obscure its interpretation as a 0-brane/6-brane bound state. This appears to be in agreement with the result of [11] that there does not exist such a BPS bound state.

IIA/IIB T-duality and bound states
where ϕ is the IIB dilaton. We parameterise M 2 with the coordinates (t, ρ), and E 8 with the coordinates {y 1 , . . . , y 8 } and set y = y 1 . Then we change coordinates from (ρ, y) to (u 10 , u 11 ) as in (3.7) and (3.8), and perform a T-duality transformation along u 11 . The resulting IIA (0|0, 2) A solution is in agreement with [18], where φ is the IIA dilaton. Similarly, the bound states (0 w |1 F ) and (0 m |5 S ) expected from applying T-duality at an angle to the (IIA or IIB) fundamental string, the solitonic 5-brane, the plane wave and the KK monopole are as follows: The (0 w |1 F ) solution is and it can be interpreted as a boosted fundamental string so it is diffeomorphic to the fundamental string solution [29] (up to a rescalling of H − 1). The (0 m |5 S ) solution is where ω is as in (3.17).
Next we turn to examine the solutions corresponding to the p-brane bound states which are expected from applying a T-duality along a boost to the (IIA or IIB) D-p-branes. As we have explained in section 2, these bound states always involve the (IIA or IIB) fundamental string. The corresponding solutions can be found from the T-duality chains and It turns out that these bound states can also be derived by first using the D=10 diffeomorphism (3.15) together with (3.16), and then performing a T-duality transformation along u 11 . Thus confirming our interpretation in section 2 for the origin of these bound states. We shall not present all these solutions here since they can be easily derived by applying the T-duality rules [21]. In any case, some of these solutions are already known like for example the (1|1 F , 4) A and the (1|1 F , 1 D ) B solutions; the latter is associated with the IIB (p,q) strings [19,20]. Nevertheless, we present the (1|1 F , 6) A solution as another example, where ω is given as in (3.17). Next applying T-duality along a boost to the (IIA or IIB) fundamental string , we do not find a new bound state.
To be more precise, the D=10 solution that one finds can be brought into the form of that of the (IIB or IIA) fundamental string up to a D=10 diffeomorphism.
Similarly, applying T-duality along a boost to the plane wave we get the (0 w |1 F ) solution as in eqn. (4.5). The (0 m |1 F ) bound state solution associated with a boosted solitonic (IIA or IIB) 5-brane is where ω is as in (3.17). Finally, applying T-duality along a boost to the KK monopole yields the (1|1 F , 5 S ) solution whereH and ω is given as in (3.17). The interpretation of this solution as a fundamental string/solitonic 5-brane bound state is obscured for similar reasons to that of the interpretation of the solution (3.17) as a 0-brane/6-brane bound state.
We shall not repeat the argument here. solution that yields (0 m |5 S ) or (4|4, 6) A after reduction has the interpretation of a D=11 fivebrane/KK monopole bound state and it can be expressed as

Concluding remarks
It is well known that apart from the M-brane solutions of D=11 supergravity, there are other solutions that have the interpretation of intersecting M-branes [30].
The argument that we have used to propose the existence of p-brane bound states in IIA and IIB superstring theories also applies in D < 10. Apart from the p-brane bound states that are obtained by reducing the D=10 p-brane bound states that we have found in section 2 to lower dimensions, there should also exist additional p-brane bound states in D < 10. This will be the case whenever a duality between two superstring theories breaks some of their reparameterisation invariance.

Note added in the proof
During the preparation of this paper we have received [36] in which the solutions interpreted as D-p-brane/D-(p+2)-brane bound states are constructed from the Dbrane solutions using rotations and T-duality transformations. This paper overlaps with some of our material in section 4.