8 Related Topics

There are several topics not included in previous sections that are also relevant to the subjects covered in this review. The purpose of this last section is to mention some of them, mentioning their main ideas and/or approaches, and more importantly, referring the reader to some of the relevant references where they are properly developed and explained.

Superembedding approach:
The GS formulation consists in treating the bulk spacetime as a supermanifold while keeping the bosonic nature of the world volume. The superembedding formalism is a more symmetric formulation, in which both bulk and world volume are described as supermanifolds. As soon as the world volume formulation is extended into superspace, it incorporates extra degrees of freedom, which are non-physical. There exists a geometrically natural interpretation for the set of constraints, first discussed in [463], imposed to remove them. Given a target space supervielbein M a α E (Z ) = (E , E ) and world volume superconnection A a α e (σ, η) = (e , e ), where η stands for the new world volume fermionic coordinates, then the pullback of the bosonic component can be expanded as

Ea (Z (σ, η)) = ebEab + eαEaα. (513 )
The constraint consists in demanding
Eaα(Z(σ,η )) = 0. (514 )
This means that at any world volume point, the tangent space in the Grassmann directions forms a subspace of the Grassmann tangent space in the bulk. There are many results in this subject, nicely reviewed in [460Jump To The Next Citation Point]. It is worth mentioning that some equations of motion for supersymmetric objects in different numbers of dimensions were actually first derived in this formalism rather than in the GS one, including [220] for the d = 10 superparticle, [47] for the superstring and supermembrane, [306] for superbranes and [305] for the M5-brane50. It is particularly relevant to stress the work done in formulating the M5-brane equations of motion covariantly [307, 308] and their use to identify supersymmetric world volume solitons [301, 302], and in pointing out the relation between superembeddings and non-linear realisations of supersymmetry [5].

MKK-monopoles and other exotic brane actions:
This review was focused on the dynamics of D-branes and M-branes. It is well known that string and M theory have other extended objects, such as KK-monopoles or NS5-branes. There is a nice discussion regarding the identification of the degrees of freedom living on these branes in [311]. Subsequently, effective actions were written down to describe the dynamics of its bosonic sectors in [83, 80, 208, 209]. In particular, it was realised that gauged sigma models are able to encapsulate the right properties for KK monopoles. The results obtained in these references are consistent with the action of T-duality and S-duality. Of course, it would be very interesting to include fermions in these actions and achieve kappa symmetry invariance.

The blackfold approach is suitable to describe the effective world volume dynamics of branes, still in the probe approximation, having a thermal population of excitations. In some sense, it describes the dynamics of these objects on length scales larger than the brane thickness. This formalism was originally developed in [201, 202] and extended and embedded in string theory in [203]. It was applied to the study of hot BIons in [261, 262], emphasising the physical features not captured by the standard Dirac–Born–Infeld action, and to blackfolds in AdS [20].

Non-relativistic kappa invariant actions:
All the branes described in this review are relativistic. It is natural to study their non-relativistic limits, both for its own sake, but also as an attempt to identify new sectors of string theory that may be solvable. The latter is the direction originally pursued in [246, 161] by considering closed strings in Minkowski. This was extended to closed strings in AdS5 × S5 in [244]. At the level of brane effective actions in Minkoswki, non-relativistic diffeomorphism and kappa symmetry invariant versions of them were obtained in [245] for D0-branes, fundamental strings and M2-branes, and later extended to general Dp-branes in [247]. The consistency of these non-relativistic actions under the action of duality transformations was checked in [330]. This work was extended to non-relativistic effective D-brane actions in AdS5 × S5 in [119, 436].

Multiple M5-branes:
It is a very interesting problem to find the non-abelian formulation of the (2,0) tensor multiplet describing the dynamics of N M5-branes. Following similar ideas to the ones used in the construction of the multiple M2-brane action using 3-algebras, some non-abelian representation of the (2,0) tensor supermultiplet was found in [351]. Their formulation includes a non-abelian analogue of the auxiliary scalar field appearing in the PST formulation of the abelian M5-brane. Closure of the superalgebra provides a set of equations of motion and constraints. Expanding the theory around a particular vacuum gives rise to d = 5 SYM along with an abelian (2,0) d = 6 supermultiplet. This connection to d = 5 SYM was further studied in [352]. Some further work along this direction can be found in [299]. Some of the BPS equations derived from this analysis were argued to be naturally reinterpreted in loop space [414]. There has been a different approach to the problem involving non-commutative versions of 3-algebras [275], but it seems fair to claim that this remains a very important open problem for the field.

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