## 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
and world volume superconnection , where stands for the new world
volume fermionic coordinates, then the pullback of the bosonic component can be expanded as
The constraint consists in demanding
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 [460]. 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
superparticle, [47] for the superstring and supermembrane, [306] for superbranes and [305] for the
M5-brane.
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.

##### Blackfolds:

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 AdS_{5} × S^{5} 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 AdS_{5} × S^{5}
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 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 SYM along
with an abelian (2,0) supermultiplet. This connection to 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.