Physics Letters B 704 (2011) 367–372 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Branes and wrapping rules ∗ Eric A. Bergshoeff a, Fabio Riccioni b, a Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands b INFN Sezione di Roma, Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy article info abstract Article history: We show that the branes of ten-dimensional IIA/IIB string theory must satisfy, upon toroidal compactifi- Received 6 September 2011 cation, specific wrapping rules in order to reproduce the number of supersymmetric branes that follows Accepted 12 September 2011 from a supergravity analysis. The realization of these wrapping rules suggests that IIA/IIB string theory Available online 16 September 2011 contains a whole class of generalized Kaluza–Klein monopoles. Editor: M. Cveticˇ © 2011 Elsevier B.V. All rights reserved. Keywords: Branes Duality Supersymmetry 1. Introduction that will lead us to suggest a class of generalized Kaluza–Klein (KK) monopoles in string theory. It is by now well-understood that branes form a crucial ingredi- In this Letter we will consider the supersymmetric branes of ent of string theory. For instance, they have been used to calculate IIA/IIB string theory compactified on a torus, which couple to the the entropy of certain black holes [1] and they are at the heart of fields of the corresponding maximal supergravities. As mentioned the AdS/CFT correspondence [2]. Often, the presence of a p-brane above these fields do not only include the physical potentials, in string theory can be deduced from the presence of a rank p + 1- i.e. the p-forms with 0 p D −2 but also the un-physical poten- form potential in the corresponding supergravity theory. It is a tials, i.e. D − 1-forms (which are dual to constant parameters) and relatively new insight that the potentials of a given supergravity D-forms (that have no field strength). In [6] we distinguished be- theory are not only the ones that describe the physical degrees tween standard branes, i.e. branes of co-dimension higher than 2, of freedom of the supermultiplet. It turns out that the supersym- and non-standard branes, i.e. branes of co-dimension 2, 1 and 0. metry algebra allows additional high-rank potentials that do not While standard branes are automatically classified because their describe any degree of freedom but, nevertheless, play an impor- number coincides with the dimension of the U-duality representa- tant role in describing the coupling of branes to background fields. tion of the corresponding field, this is in general not true for the For maximal supergravity theories, the allowed U-duality repre- non-standard branes. A prototype example are the 7-branes of IIB sentations of these “un-physical” potentials have been classified string theory: although the supersymmetry algebra closes on an in [3–5]. SL(2, R) triplet of 8-forms,1 only two of them are actually associ- A distinguishing feature of the un-physical potentials is that, ated to supersymmetric branes [7]: the D7-brane and its S-dual. At when considered in different dimensions, they are not related to present, it has not been worked out what the number of super- each other by toroidal compactification. This is unlike the “phys- symmetric non-standard branes is in a given dimension. ical” potentials, including the dual potentials, whose numbers are Recently, a step forward in this direction was performed in fixed by the representation theory of the supersymmetry algebra. [8,9]. The strategy of these papers was to analyze the structure Indeed, all physical potentials are related by toroidal compactifica- of the gauge-invariant Wess–Zumino (WZ) terms and to introduce tion. Supergravity is therefore not complete in the sense that the the following brane criterion: a potential can be associated to a su- lower-dimensional supergravity theories, including the un-physical persymmetric brane if the corresponding gauge-invariant WZ term potentials, do not follow from the reduction of the ten-dimensional requires the introduction of worldvolume fields that fit within the supergravity theory. It is this incomplete nature of supergravity 1 Actually, the situation in this case is slightly more subtle since the triplet of 9- * Corresponding author. form curvatures of these potentials satisfies a non-linear constraint. This is a general E-mail addresses: [email protected] (E.A. Bergshoeff), property of branes of co-dimension 2 which does not play a role in the present [email protected] (F. Riccioni). discussion. 0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2011.09.043 368 E.A. Bergshoeff, F. Riccioni / Physics Letters B 704 (2011) 367–372 bosonic sector of a suitable supermultiplet. Decomposing in each Table 1 dimension D = 10 − d the U-duality representations in terms of Forms with α =−3 in any dimension. All representations are T-duality representations as meant to be irreducible, and the T-duality vector indices AB... are always meant to be antisymmetrized. The a, a˙ denote chiral + and anti-chiral T-duality spinor indices. U-duality ⊃ SO(d, d) × R (1) (D − 2)-form E D−2,a˙ (D − 1)-form E − ˙ + E − one can deduce how the tension T of each brane scales with the D 1,Aa D 1,a D-form E D,ABa˙ + E D,Aa + 3E D,a˙ string coupling constant gS in terms of a number α α = = T ∼ (gS ) . (2) (n 0, 1, 2, 3, 4) [9].Heren 0 corresponds to the NS–NS 5- brane and n = 1 to the standard KK monopole. Formally, one can + The value of α follows from the R -weight of the corresponding associate to these generalized KK monopoles the following mixed- potential. The analysis of the fields as T-duality representations for symmetry fields: each value of α reveals a remarkable recurrence [8,9] at least for the highest values of α. The fundamental fields, that is the fields IIA/IIB: D6+n,n, n = 0, 1, 2, 3, 4. (4) = with α 0, are in all cases a 1-form and a 2-form, which trans- The field D is the magnetic dual of the NS–NS 2-form B , while 2 6 2 form respectively as a vector and a singlet under T-duality. The RR the field D , which is the dual of the graviton, is associated to =− 7,1 fields, which have α 1, are in all dimensions T-duality spinors the standard KK monopole. Although this dual graviton field D =− 7,1 of alternating chirality. Finally, the solitonic fields, with α 2, can only be introduced consistently at the linearized level, it can belong to T-duality representations corresponding to antisymmet- still be considered as a tool to determine all the lower-dimensional ric tensors of rank zero to four (see [9] for the details). standard solitons by dimensional reduction. Solutions correspond- While the fundamental branes, the D-branes and the standard ing to more general mixed-symmetry fields have been considered solitons are in all cases in correspondence with their potentials, in e.g. [10,11]. The whole set of supersymmetric solitons in any di- the same is not true for the non-standard solitonic branes, and mensions can be obtained from these mixed-symmetry fields by indeed the analysis of [9] reveals that only some components of imposing a restricted reduction rule which states that a supersym- the representations of the solitonic fields actually lead to super- metric brane is only obtained when the n indices on the right of symmetric branes. The overall result can be nicely summarized by the comma in D6+n,n are internal and along directions that coin- introducing a set of wrapping rules that give the number of fun- cide with n of the indices on the left of the comma. damental branes (F), D-branes (D) and solitons (S) in dimension D The only ten-dimensional supersymmetric brane which is left + 3 from those in dimension D 1 [6]: aside by this analysis4 is the S-dual of the D7-brane of the IIB − theory. The tension of this brane scales like (g ) 3 in the string wrapped → doubled, S F frame. In any dimension below ten, one can deduce the T-duality unwrapped → undoubled, =− representations of the α 3 fields by simply looking at the ta- wrapped → undoubled, bles in Ref. [9]. This leads to the remarkable result that also for D unwrapped → undoubled, these fields the pattern of T-duality representations is universal, see Table 1. wrapped → undoubled, S (3) In this Letter we will analyze the structure of the WZ terms cor- unwrapped → doubled. responding to the fields in Table 1, in order to determine which of them correspond to supersymmetric branes. As we will see, for the This means that all the branes in a given dimension can be ob- (D − 1)- and the D-forms only the highest dimensional irreducible tained by a simple counting rule starting from the ten-dimensional representation corresponds to a supersymmetric brane. Moreover, ones. we will discover that only a subset of the components of the repre- The wrapping rule for fundamental branes and D-branes can sentations of these fields actually corresponds to a supersymmetric be easily understood. For fundamental branes, the doubling upon brane. The final result will lead to yet another wrapping rule: wrapping corresponds to the fact that after compactification on a circle there is an extra fundamental 0-brane resulting from the wrapped → doubled, reduction of a pp-wave, while for D-branes the wrapping rule sim- → ply means that the ten-dimensional D-branes (of either IIA or IIB) unwrapped doubled.