Defect branes
Introduction
Branes are a fundamental ingredient of string theory. Prime examples of their many applications are the calculation of the entropy of certain black holes [1] and the AdS/CFT correspondence [2]. The properties of branes crucially depend on two quantities: the scaling of the brane tension with the string coupling constant in the string frame and the number T of transverse directions. The first quantity can be characterized by a number α such that It turns out that α is a non-positive number.1 Branes with are called Fundamental, Dirichlet, Solitonic, etc. The second quantity T naturally splits the branes into two classes: the standard branes with and the non-standard ones with . Only the standard branes are asymptotically flat. The non-standard branes require special attention. For instance, the non-standard branes with are space-filling branes which can only be defined consistently in combination with an orientifold. The ones with are domain walls. The potentials coupling to these domain walls are dual to constants such as mass parameters or gauge coupling constants. By T-duality, these domain walls need orientifolds as well [3].
In this paper we wish to focus on non-standard branes with . We call such branes “defect branes” since branes with co-dimension 2, like the D7-brane or 4D cosmic strings, are not asymptotically flat and can have non-trivial deficit angles at spatial infinity. A prime example of a Dirichlet defect brane is the ten-dimensional D7-brane [4] whose solution has been discussed in [5], [6], [7]. It is well known that the single D7-brane solution has no finite energy [5], [6]. To obtain such a finite-energy solution one should construct a multiple brane solution which includes orientifolds. In this paper we will only consider single defect branes and assume that finite energy solutions can be obtained by applying the same techniques as for the D7-brane.
Defect branes couple to -form potentials. These potentials are dual to the scalars that parametrize the non-linear coset of the corresponding maximal supergravity theory.2 It turns out that the number of -form potentials is not equal to the number of coset scalars, i.e. , see Table 1. The reason of this is that the -form potentials transform in the adjoint representation of the duality group G. Their -form field strengths are essentially the Hodge duals of the Noether current 1-forms associated to the global invariance under G [8], which transform in the adjoint representation of G. The Noether currents are constrained by relations [9] and, therefore, the -form potentials describe as many physical degrees of freedom as the coset scalars. These constraints, however, do not lead to algebraic relations among the potentials themselves and therefore do not play a role in the present discussion.
To determine whether we are dealing with a supersymmetric defect brane we will use a criterion that is based on the construction of a gauge-invariant Wess–Zumino (WZ) term that describes the coupling of the defect brane to a given -form potential [10], [11]. This WZ term should contain worldvolume fields that precisely fit into a half-supersymmetric vector or tensor multiplet. This supersymmetric brane criterion leads to a full classification of supersymmetric defect branes in dimensions . It turns out that the number of supersymmetric defect branes in any dimension is less than the number of -form potentials, i.e. . This means that not all potentials correspond to supersymmetric branes, see Table 1. This is different from the standard branes where the number of potentials always equals the number of supersymmetric branes. The number of all non-standard branes have been recently derived in dimension higher than five in [12] using the method of [11], and in all dimensions in [13] using an approach based on [14] and the observation that imaginary roots do not lead to supersymmetric branes [15]. As far as the number of defect branes is concerned, we will give yet another derivation of this number using a different method, see Section 2. The final result can be found in Table 1. This Table also shows that in the number of supersymmetric defect branes is not equal to the number of coset scalars, i.e. . It is just a coincidence that these two numbers are the same in ten dimensions.
The lower-dimensional branes with can all be seen to arise as dimensional reductions of branes and a set of generalized KK monopoles in ten dimensions [16], [17]. The generalized KK monopoles can be schematically represented by the introduction of mixed-symmetry fields in ten dimensions, provided that one applies a restricted dimensional reduction rule when counting the branes in the lower dimension: given a mixed-symmetry field with , indicating a Young tableaux consisting of a column of length m and a column of length n, one requires that the n indices have to be internal and parallel to n of the m indices [16], [17].3 Here we generalize this result, and we determine all the ten-dimensional mixed-symmetry fields that are required to generate all the defect branes for any value of α using the restricted reduction rule. We also derive the eleven-dimensional origin of these fields.
Remarkably, all the solutions corresponding to the generalized KK monopoles that we introduce here were already determined in [18], and as we will show the restricted reduction rule automatically translates into the dictionary used in [18] to classify these solutions. The mixed-symmetry fields we introduce can all be seen as generalized duals [19], [20] of the graviton and the other potentials in the ten- or eleven-dimensional theory.4 This means that at least at the linearized level one can impose a duality relation, which can be used to predict the behavior of the fields in the various solutions. By explicitly writing down some of the explicit solutions of [18], we will show that this predicted behavior is indeed correct.
The organization of this paper is as follows. In Section 2 we derive the expression for the number of supersymmetric defect branes given in Table 1. In Section 3 we give the string and M-theory origin of these defect branes in terms of branes and a set of “generalized” Kaluza–Klein (KK) monopoles. Furthermore, we discuss the relation between the generalized KK monopoles and mixed-symmetry fields of a certain type. In Section 4 we show how these mixed-symmetry fields classify all defect brane solutions. As an example we give the string and M-theory monopole solutions that give rise to all the defect branes. We also show in Section 5 how the linearized duality relations between these mixed-symmetry fields and the propagating forms determine the behavior of the fields in the various solutions. This is compared with the explicit known results in all cases. In Section 6 we explain why the number of supersymmetric defect branes is, for each dimension D, equal to twice the number of corresponding central charges in the supersymmetry algebra. In the final section we give our conclusions.
Section snippets
Supersymmetric defect branes
At first sight one might think that the number of supersymmetric defect branes is equal to the number of dual -form potentials. However, this is not the case. A prime example is ten-dimensional IIB string theory where the 8-forms are in the 3 of and we only have a supersymmetric D7-brane and its S-dual, i.e. [10]. The reason why we only have two supersymmetric seven-branes can be seen as follows: using an notation the WZ terms for the three candidate seven-branes
String and M-theory origin
In this section we wish to consider the string and M-theory origin of the defect branes of the previous Section, see Table 2.
The string-theory origin of the Fundamental defect branes is the IIA/IIB Fundamental string supplied with the fundamental wrapping rule This means that the IIA/IIB fundamental string, upon applying the wrapping rule (8) leads to the numbers of fundamental defect branes given in Table 2. We can represent the Fundamental string by the
Mixed-symmetry fields and monopole solutions
In this section we show how the mixed-symmetry fields, together with the restricted reduction rule, are in one to one correspondence with the classification of generalized KK monopole solutions of [18].9 We are going to use the following notation: an extended object of D-dimensional string theory with mass proportional to , T transverse dimensions, p spacelike
Duality relations and explicit solutions
In this section we want to show that the linearized duality relations that the mixed-symmetry fields satisfy can be used to deduce the behavior of the fields of the corresponding solution. We consider as a first example the ten-dimensional field together with all its generalized duals , and . For each of these fields, there is a solution in which the field can be considered to be electric, that is with non-zero components along the worldvolume directions and along the
Central charges
It is well known that there is a 1–1 correspondence between standard branes and the central charges in the supersymmetry algebra in type II string and M-theory. The standard branes, for dimensions have a universal behavior with respect to T-duality. For each dimension they are given by a singlet and vector of Fundamental branes, a chiral spinor of D-branes and antisymmetric tensors of solitonic branes. On top of this we have in each dimension a pp-wave and a standard KK
Conclusions
In this work we have discussed some basic properties of branes with co-dimension 2, i.e. defect branes. Requiring the existence of a supersymmetric gauge-invariant WZ term we gave a full classification of these branes, see Table 2. Their string and M-theory origin as seven-branes and a set of generalized KK monopoles was determined. These included monopoles with two inequivalent isometry directions. We explained why the number of supersymmetric defect branes does not equal the number of
Note added
During the course of this work the paper [13] appeared which has some overlap with this work. In particular, Section 3 of [13] discusses defect brane solutions from an -point of view.
Acknowledgements
E.B. wishes to thank Axel Kleinschmidt for a useful correspondence. T.O. and F.R. wish to thank the Center for Theoretical Physics of the University of Groningen for its hospitality and financial support. The work of T.O. has been supported in part by the Spanish Ministry of Science and Education grant FPA2009-07692, the Comunidad de Madrid grant HEPHACOS S2009ESP-1473, and the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042. T.O. wishes to thank M.M. Fernández for her unfaltering
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