To argue this, analyse the field content of these vector supermultiplets. These include a set of
scalar fields
and a gauge field
in
dimensions, describing
physical polarisations.
Thus, the total number of massless bosonic degrees of freedom is
Since Dp-branes propagate in 10 dimensions, any covariant formalism must involve a set of 10 scalar
fields , transforming like a vector under the full Lorentz group
. This is the same
situation we encountered for the superstring. As such, it should be clear the extra bosonic gauge symmetries
required to remove these extra scalar fields are
dimensional diffeomorphisms describing the
freedom in embedding
in
. Physically, the Dirichlet boundary conditions used in the
open string description did fix these diffeomorphisms, since they encode the brane location in
.
What about the fermionic sector? The discussion here is entirely analogous to the superstring one. This is because spacetime supersymmetry forces us to work with two copies of Majorana–Weyl spinors in 10 dimensions. Thus, matching the eight on-shell bosonic degrees of freedom requires the effective action to be invariant under a new fermionic gauge symmetry. I will refer to this as kappa symmetry, since it will share all the characteristics of the latter for the superstring.
First, target space covariance requires the background to allow a superspace formulation in
dimensions10.
Such formulation involves a single copy of
Majorana fermions, which gives rise to a pair of
Majorana–Weyl fermions, matching the superspace formulation for the superstring described in
Section 2.
Majorana spinors have
real components, which are further reduced to 16
due to the Dirac equation. Thus, a further gauge symmetry is required to remove half of these fermionic
degrees of freedom, matching the eight bosonic on-shell ones. Once again, kappa symmetry will be required
to achieve this goal.
What about the M5-brane? The fermionic discussion is equivalent to the M2-brane one. The bosonic one
must contain a new ingredient. Indeed, geometrically, there are only five scalars describing the transverse
M5-brane excitations. These do not match the eight on-shell fermionic degrees of freedom. This is reassuring
because there is no scalar supermultiplet in dimensions with such number of scalars.
Interestingly, there exists a tensor supermultiplet in
dimensions whose field content
involves five scalars and a two-form gauge potential
with self-dual field strength. The
latter involves 6-2 choose 2 physical polarisations, with self-duality reducing these to three
on-shell degrees of freedom. To keep covariance and describe the right number of polarisations,
the
theory must be invariant under
gauge transformations for the 2-form
gauge potential. I will later discuss how to keep covariance while satisfying the self-duality
constraint.
p + 1 | X | X | X | X |
1 | 1 | 2 | 4 | 8 |
2 | 1 | 2 | 4 | 8 |
3 | 1 | 2 | 4 | 8 |
4 | 2 | 4 | ||
5 | 4 | |||
6 | 4 | |||
Concerning vector supermultiplets with scalars in
dimensions, the results are
summarised in Table 2. Note that the last column describes the field content of all Dp-branes, starting from
the D0-brane
and finishing with the D9 brane
filling in all spacetime. Thus, the field
content of all Dp-branes matches with the one corresponding to the different vector supermultiplets in
dimensions. This point agrees with the open string conformal field theory description of D
branes.
p + 1 | X | X | X | X |
1 | 2 | 3 | 5 | 9 |
2 | 1 | 2 | 4 | 8 |
3 | 0 | 1 | 3 | 7 |
4 | 0 | 2 | 6 | |
5 | 1 | 5 | ||
6 | 0 | 4 | ||
7 | 3 | |||
8 | 2 | |||
9 | 1 | |||
10 | 0 | |||
Finally, there is just one interesting tensor multiplet with scalars in six dimensions,
corresponding to the aforementioned M5 brane, among the six-dimensional tensor supermultiplets listed in
Table 3.
p + 1 | X | X |
6 | 1 | 5 |
By construction, an effective action written in terms of these on-shell degrees of freedom can neither be
spacetime covariant nor invariant (in the particular case when branes propagate in
Minkowski, as I have assumed so far). Effective actions satisfying these two symmetry requirements involve
the addition of both extra, non-physical, bosonic and fermionic degrees of freedom. To preserve their
non-physical nature, these supersymmetric brane effective actions must be invariant under additional gauge
symmetries
Branes carry energy, consequently, they gravitate. Thus, one expects to find gravitational configurations
(solitons) carrying the same charges as branes solving the classical equations of motion capturing the
effective dynamics of the gravitational sector of the theory. The latter is the effective description provided
by type IIA/B supergravity theories, describing the low energy and weak coupling regime of closed strings,
and supergravity. The purpose of this section is to argue the existence of the same
world-volume degrees of freedom and symmetries from the analysis of massless fluctuations of these solitons,
applying collective coordinate techniques that are a well-known notion for solitons in standard,
non-gravitational, gauge theories.
In field theory, given a soliton solving its classical equations of motion, there exists a notion of
effective action for its small excitations. At low energies, the latter will be controlled by massless
excitations, whose number matches the number of broken symmetries by the background soliton [243]12.
These symmetries are global, whereas all brane solitons are on-shell configurations in supergravity, whose
relevant symmetries are local. To get some intuition for the mechanism operating in our case, it is
convenient to review the study of the moduli space of monopoles or instantons in abelian gauge theories.
The collective coordinates describing their small excitations include not only the location of the
monopole/instanton, which would match the notion of transverse excitation in our discussion given the
pointlike nature of these gauge theory solitons, but also a fourth degree of freedom associated with the
breaking of the gauge group [431, 288]. The reason the latter is particularly relevant to us is because,
whereas the first set of massless modes are indeed related to the breaking of Poincaré invariance,
a global symmetry in these gauge theories, the latter has its origin on a large gauge
transformation.
This last observation points out that the notion of collective coordinates can generically be associated with large gauge transformations, and not simply with global symmetries. It is precisely in this sense how it can be applied to gravity theories and their soliton solutions. In the string theory context, the first work where these ideas were applied was [127] in the particular set-up of 5-brane solitons in heterotic and type II strings. It was later extended to M2-branes and M5-branes in [332]. In this section, I follow the general discussion in [6] for the M2, M5 and D3-branes. These brane configurations are the ones interpolating between Minkowski, at infinity, and AdS times a sphere, near their horizons. Precisely for these cases, it was shown in [236] that the world volume theory on these branes is a supersingleton field theory on the corresponding AdS space.
Before discussing the general strategy, let me introduce the on-shell bosonic configurations to be analysed below. All of them are described by a non-trivial metric and a gauge field carrying the appropriate brane charge. The multiple M2-brane solution, first found in [198], is
Here, and in the following examples, Let me first sketch the argument behind the generation of massless modes in supergravity
theories, where all relevant symmetries are gauge, before discussing the specific details below.
Consider a background solution with field content , where
labels the field, including its
tensor character, having an isometry group
. Assume the configuration has some fixed
asymptotics with isometry group
, so that
. The relevant large gauge transformations
in our discussion are those that act non-trivially at infinity, matching a broken global
transformation asymptotically
, but differing otherwise in the bulk of the background geometry
In the following, I explain the origin of the different bosonic and fermionic massless modes in the world volume supermultiplets discussed in Section 3.1 by analysing large gauge diffeomorphisms, supersymmetry and abelian tensor gauge transformations.
Symmetry |
|
M2 |
M5 |
D3 |
Reparametrisations: |
|
|
|
|
Local supersymmetry: |
|
|
|
|
Tensor gauge symmetry: |
|
|
|
|
|
|
|
|
|
where indices stand for the chirality of the fermionic zero modes. In particular, for the
M2 brane it describes negative eight dimensional chirality of the 11-dimensional spinor
,
while for the M5 and D3 branes, it describes positive six-dimensional and four-dimensional
chirality.
Thus, using purely effective field theory techniques, one is able to derive the spectrum of massless excitations of brane supergravity solutions. This method only provides the lowest order contributions to their equations of motion. The approach followed in this review is to use other perturbative and non-perturbative symmetry considerations in string theory to determine some of the higher-order corrections to these effective actions. Our current conclusion, from a different perspective, is that the physical content of these theories must be describable in terms of the massless fields in this section.
http://www.livingreviews.org/lrr-2012-3 |
Living Rev. Relativity 15, (2012), 3
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