3.6 Symmetries: spacetime vs world volume
The main purpose of this section is to discuss the global symmetries of brane effective actions, the
algebra they close and to emphasise the interpretation of some of the conserved charges appearing in
these algebras before and after gauge fixing of the world volume diffeomorphisms and kappa
symmetry.
- 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,
Summary:
Brane effective actions include the supergravity superisometries
as a subset of their global
symmetries. It is important to stress that kappa symmetry invariance is necessary to define a
supersymmetric field theory on the brane, but not sufficient. Indeed, any on-shell supergravity background
having no Killing spinors, i.e., some superisometry in which fermions are shifted as
,
breaks supersymmetry, and consequently, will never support a supersymmetric action on the
brane.
The derivation discussed above does not exclude the existence of further infinitesimal transformations
leaving the effective action invariant. The question of determining the full set of continuous global
symmetries of a given classical field theory is a well posed mathematical problem in terms of cohomological
methods [50, 51]. Applying these to the bosonic D-string [111] gave rise to the discovery of the existence of
an infinite number of global symmetries [113, 112]. These were also proven to exist for the kappa invariant
D-string action [110].
3.6.1 Supersymmetry algebras
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 [487, 478, 481
]
That this superalgebra is maximal can be argued using the fact that its left-hand side defines a symmetric
tensor with 528 independent components. Equivalently, it can be interpreted as belonging to the adjoint
representation of the Lie algebra of
. The latter decomposes under its subgroup
, the
spacetime Lorentz isometry group, as
The irreducible representations appearing in the direct sum do precisely correspond to the bosonic tensor
charges appearing in the right-hand side: the 11-momentum
, a 2-form charge
, which is
55-dimensional, and a 5-form charge
, which is 462-dimensional.
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.
3.6.2 World volume supersymmetry algebras
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
.
Branes in super-Poincaré:
The above discussion can be made explicit in this case. Consider a
dimensional brane propagating in d dimensional super-Poincaré. For completeness,
let me remind the reader of the full set of transformations leaving the brane actions invariant
where I ignored possible world volume gauge fields. Decomposing the set of bosonic scalar fields
into world volume directions
and transverse directions
, one can now explicitly solve for the preservation of the static gauge slice
, which does globally describe the vacuum choice of a
-brane extending in the first p
spacelike directions and time. This requires some compensating world volume diffeomorphism
inducing the following transformations for the remaining degrees of freedom
The subset of linearly realised symmetries is
. The world volume “Poincaré”
group is indeed
, under which
are scalars, whereas
are fermions, including the standard
spin connection transformation giving them their spinorial nature.
, the transverse
rotational group to the brane is reinterpreted as an internal symmetry, under which
transforms as a
vector. The parameters
describing the coset
are
generically non-linearly realised, whereas the transverse translations
act inhomogeneously on the
dynamical fields
, identifying the latter as Goldstone massless fields, as corresponds to the
spontaneous symmetry breaking of these symmetries due to the presence of the brane in the chosen
directions.
There is a similar discussion for the 32 spacetime supersymmetries
. Before gauge fixing all
fermions
transform inhomogeneously under supersymmetry. After gauge fixing
, the
compensating kappa symmetry transformation
required to preserve the gauge slice in configuration
space will induce an extra supersymmetry transformation for the dynamical fermions, i.e.,
.
On general grounds, there must exist sixteen linear combinations of supersymmetries being
linearly realised, whereas the sixteen remaining will be spontaneously broken by the brane.
There are many choices for
. In [10], where they analysed this aspect for D-branes in
super-Poincaré, they set one of the members of the
fermion pair to zero, leading to fairly simple
expressions for the gauge fixed Lagrangian. Another natural choice corresponds to picking the
projector describing the preserved supersymmetries by the brane from the spacetime perspective.
For instance, the supergravity solution describing M2-branes has 16 Killing spinors satisfying
where the
is correlated with the
flux carried by the solution. If one fixes kappa symmetry
according to
where
stands for the 11-dimensional Clifford algebra matrix, then the physical fermionic degrees of
freedom are not only 3-dimensional spinors, but they are chiral spinors from the internal symmetry
perspective. They actually transform in the
[91
]. Similar considerations would apply for any other
brane considered in this review.
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.
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.