- before gauge fixing, the field theory will be invariant under the full superisometry of the background where the brane propagates. This is a natural extension of the super-Poincaré invariance when branes propagate in Minkowski. As such, the algebra closed by the brane conserved charges will be a subalgebra of the maximal spacetime superalgebra one can associate to the given background.
- after gauge fixing, only the subset of symmetries preserved by the brane embedding will remain linearly realised. This subset determines the world volume (supersymmetry) algebra. In the particular case of brane propagation in Minkowski, this algebra corresponds to a subalgebra of the maximal super-Poincaré algebra in dimensions.

To prove that background symmetries give rise to brane global symmetries, one must first properly define the notion of superisometry of a supergravity background. This involves a Killing superfield satisfying the properties

denotes the Lie derivative with respect to , is either the or Minkowski metric on the tangent space, depending on which superspace we are working on and are the different M-theory or type IIA/B field strengths satisfying the generalised Bianchi identities defined in Appendix A. Notice these are the superfield versions of the standard bosonic Killing isometry equations. Invariance of the field strengths allows the corresponding gauge potentials to have non-trivial transformations for some set of superfield forms .The invariance of brane effective actions under the global transformations

was proven in [94]. The proof can be established by analysing the DBI and WZ terms of the action separately. If the brane has gauge field degrees of freedom, one can always choose its infinitesimal transformation where stands for pullback to the world volume, i.e., . This guarantees the invariance of the gauge invariant forms, i.e., . Furthermore, the transformation of the induced metric vanishes because of Eq. (160). This establishes the invariance of the DBI action. On the other hand, the WZ action is quasi-invariant by construction due to Eqs. (163) and (164). Indeed,

Since spacetime superisometries generate world-volume global symmetries, Noether’s theorem [406, 407] guarantees a field theory realisation of the spacetime (super)symmetry algebra using Poisson brackets. It is by now well known that such (super)algebras contain more bosonic charges than the ones geometrically realised as (super)isometries. There are several ways of reaching this conclusion:

- Grouped theoretically, the anticommutator of two supercharges defines a symmetric matrix belonging to the adjoint representation of some symplectic algebra , whose order depends on the spinor representation . One can decompose this representation into irreducible representations of the bosonic spacetime isometry group. This can explicitly be done by using the completeness of the basis of antisymmetrised Clifford algebra gamma matrices as follows where the allowed values of depend on symmetry considerations. The right-hand side defines a set of bosonic charges that typically goes beyond the spacetime bosonic isometries.
- Physically, BPS branes in a given spacetime background have masses equal to their charges by virtue of the amount of supersymmetry they preserve. This would not be consistent with the supersymmetry algebra if the latter would not include extra charges, the set introduced above, besides the customary spacetime isometries among which the mass (time translations) always belongs to. Thus, some of the extra charges must correspond to such brane charges. The fact that these charges have non-trivial tensor structure means they are typically not invariant under the spacetime isometry group. This is consistent with the fact that the presence of branes breaks the spacetime isometry group, as I already explicitly discussed in super-Poincaré.
- All brane effective actions reviewed above are quasi-invariant under spacetime superisometries, since the WZ term transformation equals a total derivative (169). Technically, it is a well-known theorem that such total derivatives can induce extra charges in the commutation of conserved charges through Poisson brackets. This is the actual field theory origin of the group theoretically allowed set of charges .

Let me review how these structures emerge in both supergravity and brane effective
actions. Consider the most general superPoincaré algebra in 11 dimensions. This is spanned
by a Majorana spinor supercharge satisfying the anti-commutation
relations^{25} [487, 478, 481]

The above is merely based on group theory considerations that may or may not be realised in a given physical theory. In 11-dimensional supergravity, the extra bosonic charges are realised in terms of electric and magnetic charges, the Page charges [410], that one can construct out of the 3-form potential equation of motion, as reviewed in [467, 466]

The first integral is over the boundary at infinity of an arbitrary infinite 8-dimensional spacelike manifold , with volume . Given the conserved nature of this charge, it does not depend on the time slice chosen to compute it. But there are still many ways of embedding in the corresponding ten-dimensional spacelike hypersurface . Thus, represents a set of charges parameterised by the volume element 2-form describing how is embedded in . This precisely matches the 2-form in Eq. (171). There is an analogous discussion for , which corresponds to the 5-form charge . As an example, consider the M2 and M5-brane configurations in Eqs. (20) and (22). If one labels the M2-brane tangential directions as 1 and 2, there exists a non-trivial charge computed from Eq. (173) by plugging in Eq. (20) and evaluating the integral over the transverse 7-sphere at infinity. The reader is encouraged to read the lecture notes by Stelle [467] where these issues are discussed very explicitly in a rather general framework including all standard half-BPS branes. For a more geometric construction of these maximal superalgebras in AdS × S backgrounds, see [211] and references therein.The above is a very brief reminder regarding spacetime superalgebras in supergravity. For a more thorough presentation of these issues, the reader is encouraged to read the lectures notes by Townsend [481], where similar considerations are discussed for both type II and heterotic supergravity theories. Given the importance given to the action of dualities on effective actions, the reader may wonder how these same dualities act on superalgebras. It was shown in [96] that these actions correspond to picking different complex structures of an underlying superalgebra.

Consider the perspective offered by the M5-brane effective action propagating in superPoincaré. The latter is invariant both under supersymmetry and bulk translations. Thus, through Noether’s theorem, there exist field theory realisations of these charges. Quasi-invariance of the WZ term will be responsible for the generation of extra terms in the calculation of the Poisson bracket of these charges [165]. This was confirmed for the case at hand in [464], where the M5-brane superalgebra was explicitly computed. The supercharges are

where , and are the variables canonically conjugate to , and . As in any Hamiltonian formalism, world volume indices were split according to . Notice that the pullbacks of the forms and appearing in for the M5-brane in Eq. (169) do make an explicit appearance in this calculation. The anti-commutator of the M5 brane world volume supercharges equals Eq. (171) with where all integrals are computed on the 5-dimensional spacelike hypersurface spanned by the M5-brane. Notice the algebra of supercharges depends on the brane dimensionality. Indeed, a single M2-brane has a two dimensional spacelike surface that cannot support the pullback of a spacetime 5-form as a single M5-brane can (see Eq. (178)). This conclusion could be modified if the degrees of freedom living on the brane would be non-abelian.Even though my discussion above only applies to the M5-brane in the super-Poincaré background, my conclusions are general given the quasi-invariance of their brane WZ action, a point first emphasised in [165]. The reader is encouraged to read [165, 168] for similar analysis carried for super -branes, [281] for D-branes in super-Poincaré and general mathematical theorems based on the structure of brane effective actions and [438, 437], for superalgebra calculations in some particular curved backgrounds.

Once the physical location of the brane is given, the spacetime superisometry group is typically broken into

The first factor corresponds to the world volume symmetry group in (-dimensions, i.e., the analogue of the Lorentz group in a supersymmetric field theory in -dimensions, whereas the second factor is interpreted as an internal symmetry group acting on the dynamical fields building -dimensional supermultiplets. The purpose of this subsection is to relate the superalgebras before and after this symmetry breaking process [328].The link between both superalgebras is achieved through the gauge fixing of world volume diffeomorphisms and kappa symmetry, the gauge symmetries responsible for the covariance of the original brane action in the GS formalism. Focusing on the scalar content in these theories , these transform as

The general Killing superfield was decomposed into a supersymmetry translation denoted by and a bosonic Killing vector fields . World volume diffeomorphisms were denoted as . At this stage, the reader should already notice the inhomogeneity of the supersymmetry transformation acting on fermions (the same is true for bosons if the background spacetime has a constant translation as an isometry, as it happens in Minkowski).Locally, one can always impose the static gauge: , where one decomposes the scalar fields into world volume directions and transverse directions . For infinite branes, this choice is valid globally and does describe a vacuum configuration. To diagnose which symmetries act, and how, on the physical degrees of freedom , one must make sure to work in the subset of symmetry transformations preserving the gauge slice . This forces one to act with a compensating world volume diffeomorphism

The latter acts on the physical fields giving rise to the following set of transformations preserving the gauge fixed action There are two important comments to be made at this point- The physical fields transform as proper world volume scalars [3]. Indeed, induces the infinitesimal transformation for any preserving the dimensional world volume. Below, the same property will be checked for fermions.
- If the spacetime background allows for any constant isometry, it would correspond to an inhomogeneous symmetry transformation for the physical field . In field theory, the latter would be interpreted as a spontaneous broken symmetry and the corresponding would be its associated massless Goldstone field. This is precisely matching our previous discussions regarding the identification of the appropriate brane degrees of freedom.

There is a similar discussion regarding the gauge fixing of kappa symmetry and the emergence of a subset of linearly realised supersymmetries on the -dimensional world volume field theory. Given the projector nature of the kappa symmetry transformations, it is natural to assume as a gauge fixing condition, where stands for some projector. Preservation of this gauge slice, determines the kappa symmetry parameter as a function of the background Killing spinors

When analysing the supersymmetry transformations for the remaining dynamical fermions, only certain linear combinations of the original supersymmetries will be linearly realised. The difficulty in identifying the appropriate subset depends on the choice of .

Having established the relation between spacetime and world volume symmetries, it is natural to close our discussion by revisiting the superalgebra closed by the linearly realised world volume (super)symmetries, once both diffeomorphisms and kappa symmetry have been fixed. Since spacetime superalgebras included extra bosonic charges due to the quasi-invariance of the brane WZ action, the same will be true for their gauge fixed actions. Thus, these -dimensional world volume superalgebras will include as many extra bosonic charges as allowed by group theory and by the dimensionality of the brane world spaces [81]. Consider the M2-brane discussed above. Supercharges transform in the representation of the bosonic isometry group. Thus, the most general supersymmetry algebra compatible with these generators, , is [81]

stands for a 3-dimensional one-form, the momentum on the brane; transforms in the under the R-symmetry group , or equivalently, as a self-dual 4-form in the transverse space to the brane; is a world volume scalar, which transforms in the of , i.e., as a 2-form in the transverse space. The same superalgebra is realised on the non-abelian effective action describing coincident M2-branes [415] to be reviewed in Section 7.2. Similar structures exist for other infinite branes. For example, the M5-brane gives rise to the superalgebra [81] Here is an index of , the natural Lorentz group for spinors in dimensions, is an index of , which is the double cover of the geometrical isometry group acting on the transverse space to the M5-brane and is an invariant antisymmetric tensor. Thus, using appropriate isomorphisms, these superalgebras allow a geometrical reinterpretation in terms of brane world volumes and transverse isometry groups becoming R-symmetry groups. The last decomposition is again maximal since stands for 1-form in (momentum), transforms as a self-dual 3-form in and a 2-form in the transverse space and as a 1-form both in and in the transverse space. For an example of a non-trivial world volume superalgebra in a curved background, see [152]. I would like to close this discussion with a remark that is usually not stressed in
the literature. By construction, any diffeomorphism and kappa symmetry gauge fixed
brane effective action describes an interacting supersymmetric field theory in
dimensions.^{27}
As such, if there are available superspace techniques in these dimensions involving the relevant brane
supermultiplet, the gauge fixed action can always be rewritten in that language. The matching between
both formulations generically involves non-trivial field redefinitions. To be more precise, consider the
example of supersymmetric abelian gauge theories coupled to matter fields. Their kinetic
terms are fully characterised by a Kähler potential. If one considers a D3-brane in a background breaking
the appropriate amount of supersymmetry, the expansion of the gauge fixed D3-brane action must match
the standard textbook description. The reader can find an example of the kind of non-trivial bosonic field
redefinitions that is required in [321]. The matching of fermionic components is expected to be
harder.

Living Rev. Relativity 15, (2012), 3
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