Resolution of overlapping branes

We obtain singularity resolutions for various overlapping brane conﬁgurations, including those of two heterotic 5-branes, type II 5-branes or D4-branes. In these solutions, the “harmonic” function H for each brane component depends only on the associated four-dimensional relative transverse space. The resolution is achieved by replacing these transverse spaces with Eguchi–Hanson or Taub-NUT spaces, both of which admit a normalisable self-dual (or anti-self-dual) harmonic 2-form. Due to the manner in which the interaction terms for the form ﬁelds modify their Bianchi identities or equations of motion, these normalisable harmonic 2-forms provide regular sources for the branes. We also obtain resolved 5-branes and D4-branes wrapped on S 1 , which is ﬁbred over the transverse Eguchi–Hanson or Taub-NUT spaces. The T-duality invariance of the NS– NS 5-brane is retained after the resolution. The resolved 5-branes and D4-branes provide regular supergravity duals of certain supersymmetric Yang–Mills theories in ﬁve and four dimensions.


Introduction
BPS branes play an important role in string and M-theory. Certain supergravity solutions provide an explicit demonstration of various properties in dual gauge theories. However, in most cases such a solution exhibits a singularity at the origin of the brane, which imposes a severe restriction on the range of validity. It is expected that higher-order stringy terms or non-E-mail address: honglu@umich.edu (H. Lü). 1 Research supported in full by DOE grant DE-FG02-95ER40899. 2  perturbative effects can resolve these singularities, while maintaining a fixed mass/charge ratio. A typical p-brane or intersecting p-brane solution makes use of only the kinetic terms of the form fields, with zero contribution from the interacting terms. The interactions between form fields modify their Bianchi identities and/or equations of motion, e.g., (1) and/or d * F (n) = F (s) ∧ F (t ) .
In many cases, the inclusion of such a contribution can resolve brane singularities at the level of supergravity. An early example of this is the heterotic 5brane constructed in [1], where the 5-brane source is the matter SU(2) Yang-Mills instanton living in the four-dimensional Euclidean transverse space. This construction makes use of the Bianchi identity dF (3) = 1 2 F i (2) ∧ F i (2) of the heterotic string theory. The solution is expected to be valid at all orders of string perturbation. This construction can also be extended to the case where the transverse space is K3, since one can equate the SU(2) Yang-Mills fields to the self-dual spin connection of the K3 [1][2][3]. An alternative resolution makes use of the well-known fact that the K3 manifold can be constructed from the orbifold T 4 /Z 2 , with the sixteen fixed points each blown up using the Eguchi-Hanson metric [4,5]. The Eguchi-Hanson metric has one normalisable self-dual 2-form, which can be set equal to a U(1) 2-form field strength associated with one of the sixteen Cartan generators of the heterotic string. The resulting 5-brane on the K3 is then regular everywhere [6].
Recently, a resolution procedure which makes use of interacting terms in the Bianchi identity or equations of motion has been extensively discussed for the D3-brane solutions of type IIB theory [7][8][9][10][11][12] and branes in type IIA and M-theory [6,[13][14][15][16][17][18]. These non-singular examples may provide important supergravity dual solutions of supersymmetric field theories on the worldvolume of the branes, with the conformal symmetry typically broken by the resolution.
So far, the study of brane resolution has focused on a single brane or, more precisely, on coincidental branes of the same type. Branes can also intersect with each other [19][20][21]. For standard intersections, the harmonic functions corresponding to each brane depend on the overall transverse space. For non-standard intersections, which are also called overlapping branes, the harmonic functions depend only on the relative transverse spaces [22][23][24].
In this Letter, we find that, for the overlap of two heterotic 5-branes as well as that of two D4branes, the two relative transverse spaces are fourdimensional Ricci-flat spaces, and hence can be replaced by the Eguchi-Hanson or Taub-NUT metrics. Since the Eguchi-Hanson and Taub-NUT metrics both support one normalisable self-dual (or anti-self-dual, depending on the orientation) 2-form, we can resolve these two overlapping branes by making use of the corresponding Bianchi identities. We obtain resolved overlapping heterotic 5-brane and D4-brane solutions in Sections 2 and 3, respectively. In Section 4, by performing the T-duality on the D4/D4 system, we obtain a resolved 5-brane overlap in type II theories. In addition, we obtain regular 5-brane, D5-brane and D4-brane wrapped on an S 1 , which is fibred over the transverse Eguchi-Hanson or Taub-NUT spaces. We conclude our Letter in Section 5. In Appendix A, we present certain properties of Eguchi-Hanson and Taub-NUT metrics that are used extensively throughout the Letter.
It should be emphasised that, although results are presented here mostly using the Eguchi-Hanson metric, the construction also works when it is replaced with the Taub-NUT metric of an appropriate orientation.

Overlapping heterotic 5-branes
The Lagrangian for the bosonic sector of tendimensional heterotic supergravity is given by Consider the solution describing a non-standard intersection of two heterotic 5-branes [22,25] for the case in which the Yang-Mills fields A i (1) are set to zero: and ( ) is taken over the y i (ỹ i ) directions. The solution can be represented diagrammatically (see Diagram 1).
The isotropic solution is given by H = 1 + Q/r 2 and H = 1 + Q/r 2 , with r 2 = y i y i andr 2 =ỹ iỹi .

Diagram 1
The overlapping of two 5-branes t w y 1 y 2 y 3 y 4ỹ1ỹ2ỹ3ỹ4 The solution has three singularities. The first one corresponds to rr → 0, which is also a horizon. The other two are naked, corresponding to r/r → 0 orr/r → 0 while leaving rr held fixed. These singularities can be resolved by introducing two sets of SU(2) Yang-Mills instantons living in the Euclidean 4-spaces dy 2 i and dỹ 2 i [26]. Here we demonstrate that it can also be resolved by a gravitational instanton together with matter U(1) contributions. Since the solution (4) requires only that dy 2 i and dỹ 2 i are Ricci-flat, we can replace each of them with an Eguchi-Hanson metric, which we discuss in Appendix A. Since the Eguchi-Hanson metric admits a self-dual (or anti-self-dual) normalisable harmonic 2-form, we can turn on the matter U(1) fields. The solution now becomes (2) , where Ω (4) and Ω (4) are the volume forms for the metric ds 2 EH and ds 2 EH , respectively. Here we have turned on two U(1) Cartan field strengths, labeled as F 1 (2) and F 2 (2) , living on ds 2 EH and ds 2 EH , respectively. Now the Bianchi identity dF (3) where η 2 = 1 =η 2 are the orientation parameters of the Eguchi-Hanson metrics (see Appendix A). The equations of motion for F (3) and F i (2) are straightforwardly satisfied. The dilaton equation and the Einstein equation imply that η = 1 =η. In other words, although the equations of motion and Bianchi identity for the form fields imply that the L 2 (2) and L 2 source terms can contribute both negatively or positively, depending on whether they are self-dual or anti-self-dual, they are restricted to a positive contribution due to the dilaton equation and Einstein equation. Thus, the Eguchi-Hanson instanton has to be such that its normalisable harmonic 2-form is self-dual. The generic solution for H and H has a logarithmic divergent term, which vanishes for appropriate integration constants [6], giving where a andã are the sizes of the two Eguchi-Hanson instantons. Since the coordinates r andr run from a andã to infinity respectively, it follows that the functions H and H are regular everywhere for nonvanishing a andã.
We have seen that the orientation of the Eguchi-Hanson metric has a significant consequence in the resolution of the 5-branes. The positive orientation with η = 1 leads to resolution whilst the negative orientation does not solve all of the equations of motion. We could have replaced dy 2 i and dỹ 2 i by a Taub-NUT metric, in which case we would require the negative orientation with η = −1 for resolution, since the normalisable harmonic 2-form has to be self-dual. The corresponding harmonic functions for the regular solution become [6] (9) For vanishing m or m, the solution reduces to the resolution of the heterotic 5-brane obtained in [6]. One can add a string along the common worldvolume of the above configuration. The corresponding "harmonic" function K satisfies the equation [27] (10) which holds in our case as well. The general solution of K depends on H and H . A natural special solution is K = hh, where h andh are the harmonic functions of the two relative four-dimensional transverse spaces respectively. If the transverse spaces are Euclidean, for the limit in which the gravitational instanton sizes a andã vanish, then the three-component solution has the near-horizon structure AdS 3 × S 3 × S 3 × E 1 [27]. The non-vanishing entropy of this configuration disappears from the metric contribution once the gravita-tional instanton is present. This suggests a phase transition associated with the vanishing gravitational instanton; this is analogous to the one associated with the Yang-Mills instantons for overlapping heterotic 5branes, discussed in [26]. Note that it is also possible to add a pp-wave component [26].
The present resolution of overlapping 5-branes incorporates Yang-Mills fields, which are available only for heterotic string theory. On the other hand, unresolved overlapping 5-branes exist also in type II theories. While the resolution for the NS-NS and R-R overlapping 5-branes of type II theories is also possible, there are subtleties involved which will be discussed in Section 4.
The solution for a non-standard intersection of two D4-branes is given by As with the heterotic 5-brane overlap, we replace the Euclidean 4-spaces dy 2 i and dỹ 2 i by Eguchi-Hanson metrics ds 2 EH and ds 2 EH , respectively. By making use of the Bianchi identity, dF (4) (2) , we can consider the following ansatz F (2) = mL (2) + m L (2) . L (2) is a normalisable self-dual harmonic 2-form in ds 2 EH , which hence has positive orientation η = 1, and L (2) is a normalisable anti-self-dual harmonic 2-form in ds 2 EH , which hence has negative orientation η = −1. Since we have where * 4 and * 4 are the Hodge duals in ds 2 EH and ds 2 EH , respectively, the Bianchi identity for F (4) implies that It should be mentioned that the different choice of sign in F (3) and F (2) ensures that the cross term in F (3) ∧ F (2) cancels out. Due to the minus sign of L (2) in F (3) , it contributes to H in the same way that L (2) contributes to H , even though L (2) is anti-self-dual and L (2) is self-dual.
To check the rest of the equations of motion, it is useful to present the following: (4) ∧ L (2) + H H −1 dt ∧ Ω (4) ∧ L (2) , We verify that all of the equations of motion for the form fields and the dilaton are satisfied. We did not verify the Einstein equation due to its complexity for these cases, although past experience leads us to believe that it is satisfied. The regular solution for (17) is given by Each of the Eguchi-Hanson metrics can also be replaced by a Taub-NUT metric with proper orientation such that it has the required normalisable self-dual or anti-self-dual harmonic 2-forms. The resulting regular solutions for H and H are given by (20) .
If m or m vanishes, then the solution becomes that of a single resolved D4-brane with the transverse space ds 2 5 = ds 2 4 + dz 2 , where ds 2 4 is a Ricci-flat manifold that admits normalisable self-dual or antiself-dual harmonic 2-forms L (2) . In this case, we have F (2) = mL (2) and F (3) = m * 5 L (2) . Thus, we see that F (3) ∧ F (2) always contributes positively to the Bianchi identity for the F (4) . Both orientations of Eguchi-Hanson instanton or Taub-NUT can be used to resolve the D4-brane. This is very different from the case of the heterotic 5-brane, in which only one orientation can be used to resolve the brane. In order to cancel out the cross term in F (3) ∧ F (2) , we find that ds 2 4 and ds 2 4 must have opposite orientations such that they admit normalisable self-dual and anti-self-dual harmonic 2forms, respectively.

Overlapping type II 5-branes
Overlaps of type II NS-NS or R-R 5-branes share the same metric structure as that of overlapping heterotic 5-branes, illustrated in Diagram 1. However, the resolution of the heterotic 5-brane is rather unique, since it makes use of multiple matter Yang-Mills fields which are absent in the type II theories. For this reason, the resolution of the type II 5-brane was previously unknown. A regular solution of the 5-brane wrapped around S 2 was obtained in [28] by lifting the four-dimensional SU(2) gauged black hole [29]. This solution can apply for both type II and heterotic 5branes. In this section, we obtain a resolved type II 5brane overlap by performing T-duality on the D4/D4 system of the previous section. By turning off one component, we obtain a resolved 5-brane wrapped on S 1 .
The resolved NS-NS overlapping 5-brane can be easily obtained by performing the S-duality of the type IIB theory, with the F RR (3) replaced by F NS (3) and the sign of the dilaton changed. Since the solution involves only the metric, dilaton and the 3-form field strength, it is also valid for resolving the 5-branes in type IIA or heterotic strings (without a Yang-Mills source). It is interesting to note that there is a twist along the direction z. The topology of a spatial slice of the solution can be viewed as a U(1) fibration of the product space of two Eguchi-Hanson instantons with opposite orientations. The solution describes that a regular effective string, as common worldvolume of two 5-branes, wraps on the fibre circle z. Note that unlike the previous examples this resolution makes use of only the interaction between the gravity and the 3-form field strength, instead of the interactions associated with the Bianchi identities or the equations of motion of the form fields.
It is important to note that the resolved NS-NS 5brane as well as the resolved overlap of two NS-NS 5branes are invariant under a T-duality transformation along the z direction. This is rather different from the usual situation where T-duality would untwist the fibration [30]. (Analogous phenomenon was observed in the NS-NS dyonic string [31].) Invariance under T-duality is a property of NS-NS 5-branes. Since Tduality is a symmetry at all orders of perturbative string theory, we would expect that the invariance should hold for the resolved solution, and that is indeed the case.
Since this solution describes D4-brane wrapped on S 1 , which is fibred over the transverse space ds 2 EH + dz 2 , it has an effective four-dimensional world-volume. Thus we find a new well-behaved supergravity solution dual to a certain N = 2, D = 4 supersymmetric Yang-Mills theory.

Conclusions
The resolution of singularities in supergravity BPS brane solutions provides a convenient way of extending the validity of these solutions. In this Letter, we have considered the resolutions of two overlapping heterotic 5-branes, type II 5-branes or D4-branes. The relative transverse spaces in these overlapping solutions are all four-dimensional and hence can be replaced by either Eguchi-Hanson or Taub-NUT spaces, both of which admit normalisable self-dual or antiself-dual harmonic 2-forms, depending on the orientation. Terms corresponding to interactions between form fields modify Bianchi identities and equations of motion. It follows that these normalisable harmonic 2forms provide regular sources for the branes.
When each of the two relative transverse Euclidean 4-spaces is replaced by the Eguchi-Hanson or the Taub-NUT instanton, half of the supersymmetry is broken. Introducing a brane configuration will not break the supersymmetry any further. Thus our resolved overlapping brane solutions preserve 1 4 of the supersymmetry.
We have also obtained resolved 5-branes and D4branes wrapped on S 1 , which is fibred over the transverse Eguchi-Hanson or Taub-NUT spaces. This provides a regular supergravity dual to a certain D = 5 and D = 4 super Yang-Mills theory.
The resolved solution are regular everywhere in the spacetime, with a stable dilaton and hence a stable string coupling constant. Furthermore, unlike a typical brane solution that requires a brane source term that is beyond supergravity, the resolved ones are complete purely within supergravity. Not all the BPS branes can be resolved at the level of supergravity. It is interesting to find those that can be resolved and study the special role they play in string and M-theory.
The radial coordinate r lies in the range a r ∞, and ψ has period 2π [33] to ensure regularity at r = a. Thus, the metric is asymptotically locally Euclidean (ALE), with the periodicity condition on ψ implying that constant r surfaces are RP 3 = S 3 /Z 2 [33]. The coordinate ψ can have two orientations of the fibration, corresponding to η = ±1. The metric is Kähler with the vielbein basis It is anti-self-dual for η = 1 and self-dual for η = −1. The metric is also admits a harmonic 2-form, given by [6] (A.4) L (2) = 2 r 4 e 0 ∧ e 3 + ηe 1 ∧ e 2 , which is self-dual for η = 1 and anti-self-dual for η = −1. The square of L (2) is given by L 2 (2) = 16/r 8 , and hence this 2-form is normalisable. Note that we have (A.5) L (2) ∧ L (2) = 1 2 ηL 2 (2) Ω (4) , where Ω (4) is the volume form of the metric (A.1). It follows that the sign of η is not merely a choice of convention but rather it has a non-trivial physical consequence.