I shall proceed in order of increasing complexity, starting with the M2-brane effective
action, which is purely geometric, continuing with D-branes and their one form gauge
potentials and finishing with M5-branes including their self-dual three form field
strength^{13}.

- Geometrically, branes are hypersurfaces propagating in a fixed background with metric . Thus, their effective actions should account for their world volumes.
- Physically, all branes are electrically charged under some appropriate spacetime gauge form . Thus, their effective actions should contain a minimal coupling accounting for the brane charges.

Both requirements extend the existent effective action describing either a charged particle or a string . Thus, the universal description of the purely scalar field brane degrees of freedom must be of the form

where and stand for the brane tension and charge densityThe relevance of the minimal charge coupling can be understood by considering the full effective action involving both brane and gravitational degrees of freedom (17). Restricting ourselves to the kinetic term for the target space gauge field, i.e., , the combined action can be written as

Here stands for the -dimensional spacetime, whereas is a -form whose components are those of an epsilon tensor normal to the brane having a -function support on the world volumeSince M2-branes do not involve any gauge field degree of freedom, the above discussion covers all its bosonic degrees of freedom. Thus, one expects its bosonic effective action to be

in analogy with the bosonic worldsheet string action. If Eq. (40) is viewed as the bosonic truncation of a supersymmetric M2-brane, then . Besides its manifest spacetime covariance and its invariance under world-volume diffeomorphisms infinitesimally generated by this action is also quasi-invariant (invariant up to total derivatives) under the target space gauge transformation leaving supergravity invariant, as reviewed in Eq. (551) of Appendix A.2. This is reassuring given that the full string theory effective action (17) describing both gravity and brane degrees of freedom involves both actions.

The DBI action is a natural extension of the NG action for branes, but it does not capture all the relevant physics, even in the absence of acceleration terms, since it misses important background couplings, responsible for the WZ terms appearing for strings and M2-branes. Let me stress the two main issues separately:

- The functional dependence on the gauge field in a general closed string background. D-branes are hypersurfaces where open strings can end. Thus, open strings do have endpoints. This means that the WZ term describing such open strings is not invariant under the target space gauge transformation due to the presence of boundaries. These are the D-branes themselves, which see these endpoints as charge point sources. The latter has a minimal coupling of the form , whose variation cancels Eq. (44) if the gauge field transforms as under the bulk gauge transformation. Since D-brane effective actions must be invariant under these target space gauge symmetries, this physical argument determines that all the dependence on the gauge field should be through the gauge invariant combination .
- The coupling to the dilaton. The D-brane effective action is an open string tree level action, i.e., the self-interactions of open strings and their couplings to closed string fields come from conformal field theory disk amplitudes. Thus, the brane tension should include a factor coming from the expectation value of the closed string dilaton . Both these considerations lead us to consider the DBI action where stands for the D-brane tension.
- The WZ couplings. Dp-branes are charged under the RR potential . Thus, their effective actions should include a minimal coupling to the pullback of such form. Such coupling would not be invariant under the target space gauge transformations (525). To achieve this invariance in a way compatible with the bulk Bianchi identities (523), the D-brane WZ action must be of the form where stands for the corresponding pullbacks of the target space RR potentials to the world volume, according to the definition given in Eq. (521). Notice this involves more terms than the mere minimal coupling to the bulk RR potential . An important physical consequence of this fact will be that turning on non-trivial gauge fluxes on the brane can induce non-trivial lower-dimensional D-brane charges, extending the argument given above for the minimal coupling [185]. This property will be discussed in more detail in the second part of this review. For a discussion on how to extend these couplings to massive type IIA supergravity, see [255].

Putting together all previous arguments, one concludes the final form of the bosonic D-brane action
is:^{17}

- One natural option is to give-up Lorentz covariance and work with non-manifestly Lorentz invariant actions. This was the approach followed in [420] for the M5-brane, building on previous work [213, 295, 441].
- One can introduce an infinite number of auxiliary (non-dynamical) fields to achieve a covariant formulation. This is the approach followed in [384, 502, 375, 177, 66, 98, 99, 100].
- One can follow the covariant approach due to Pasti, Sorokin and Tonin (PST-formalism) [416, 418], in which a single auxiliary field is introduced in the action with a non-trivial non-polynomial dependence on it. The resulting action has extra gauge symmetries. These allow one to recover the structure in [420] as a gauge fixed version of the PST formalism.
- Another option is to work with a Lagrangian that does not imply the self-duality condition but allows it, leaving the implementation of this condition to the path integral. This is the approach followed by Witten [497], which was extended to include non-linear interactions in [140]. The latter work includes kappa symmetry and a proof that their formalism is equivalent to the PST one.

In this review, I follow the PST formalism. This assigns the following bosonic action to the M5-brane [417]

As in previous effective actions, all the dependence on the scalar fields is through the bulk fields and their pullbacks to the six-dimensional world volume. As in D-brane physics, all the dependence on the world volume gauge potential is not just simply through its field strength , but through the gauge invariant 3-form The physics behind this is analogous. describes the ability of open strings to end on D-branes, whereas describes the possibility of M2-branes to end on M5-branes [469, 479]Besides being manifestly invariant under six-dimensional world volume diffeomorphisms and ordinary abelian gauge transformations , the action (49) is also invariant under the transformation

Given the non-dynamical nature of , one can always fully remove it from the classical action by gauge fixing the symmetry (53). It was shown in [417] that for an M5-brane propagating in Minkowski, the non-manifest Lorentz invariant formulation in [420] emerges after gauge fixing (53). This was achieved by working in the gauge and . Since is a world volume vector, six-dimensional Lorentz transformations do not preserve this gauge slice. One must use a compensating gauge transformation (53), which also acts on . The overall gauge fixed action is invariant under the full six-dimensional Lorentz group but in a non-linear non-manifestly Lorentz covariant way as discussed in [420].As a final remark, notice the charge density of the bosonic M5-brane has already been set equal to its tension .

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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