3.1 Degrees of freedom and world volume supersymmetry

In this section, I focus on the identification of the physical degrees of freedom describing a single brane, the constraints derived from world volume symmetries to describe their interactions and the necessity to introduce extra world volume gauge symmetries to achieve spacetime supersymmetry and covariance. I will first discuss these for Dp-branes, which allow a perturbative quantum open string description, and continue with M2 and M5-branes, applying the lessons learnt from strings and D-branes.

Dp-branes:
Dp-branes are p + 1 dimensional hypersurfaces Σp+1 where open strings can end. One of the greatest developments in string theory came from the realisation that these objects are dynamical, carry Ramond–Ramond (RR) charge and allow a perturbative worldsheet description in terms of open strings satisfying Dirichlet boundary conditions in p + 1 dimensions [423Jump To The Next Citation Point].The quantisation of open strings with such boundary conditions propagating in 10-dimensional ℝ1,9 Minkowski spacetime gives rise to a perturbative spectrum containing a set of massless states that fit into an abelian vector supermultiplet of the super-Poincaré group in p + 1 dimensions [425Jump To The Next Citation Point, 426]. Thus, any physical process involving open strings at low enough energy, √ -- E α′ ≪ 1, and at weak coupling, gs ≪ 1, should be captured by an effective supersymmetric abelian gauge theory in p + 1 dimensions. Such vector supermultiplets are described in terms of U (1) gauge theories to achieve a manifestly ISO (1,p) invariance, as is customary in gauge theories. In other words, the formulation includes additional polarisations, which are non-physical and can be gauged away. Notice the full ISO (1,9) of the vacuum is broken by the presence of the Dp-brane itself. This is manifestly reflected in the spectrum. Any attempt to achieve a spacetime supersymmetric covariant action invariant under the full ISO (1,9) will require the introduction of both extra degrees of freedom and gauge symmetries. This is the final goal of the GS formulation of these effective actions.

To argue this, analyse the field content of these vector supermultiplets. These include a set of 9 − p scalar fields XI and a gauge field V1 in p + 1 dimensions, describing p − 1 physical polarisations. Thus, the total number of massless bosonic degrees of freedom is

Dp -brane: 10 − (p + 1) + (p − 1) = 8 .
Notice the number of world volume scalars I X matches the number of transverse translations broken by the Dp-brane and transform as a vector under the transverse Lorentz subgroup SO (9 − p), which becomes an internal symmetry group. Geometrically, these modes XI (σ) describe the transverse excitations of the brane. This phenomena is rather universal in brane physics and constitutes the essence in the geometrisation of field theories provided by branes in string theory.

Since Dp-branes propagate in 10 dimensions, any covariant formalism must involve a set of 10 scalar fields Xm (σ ), transforming like a vector under the full Lorentz group SO (1,9). 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 p + 1 dimensional diffeomorphisms describing the freedom in embedding Σp+1 in ℝ1,9. Physically, the Dirichlet boundary conditions used in the open string description did fix these diffeomorphisms, since they encode the brane location in ℝ1,9.

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.

M-branes:
M-branes do not have a perturbative quantum formulation. Thus, one must appeal to alternative arguments to identify the relevant degrees of freedom governing their effective actions at low energies. In this subsection, I will appeal to the constraints derived from the existence of supermultiplets in p + 1 dimensions satisfying the geometrical property that their number of scalar fields matches the number of transverse dimensions to the M-brane, extending the notion already discussed for the superstring and Dp-branes. Later, I shall review more stringy arguments to check the conclusions obtained below, such as consistency with string/M theory dualities. Let us start with the more geometrical case of an M2-brane. This is a 2 + 1 surface propagating in d = 1 + 10 dimensions. One expects the massless fields to include 8 scalar fields in the bosonic sector describing the M2-brane transverse excitations. Interestingly, this is precisely the bosonic content of a scalar supermultiplet in d = 1 + 2 dimensions. Since the GS formulation also fits into a scalar supermultiplet in d = 1 + 1 dimensions for a long string, it is natural to expect this is the right supermultiplet for an M2-brane. To achieve spacetime covariance, one must increase the number of scalar fields to eleven Xm (σ ), transforming as a vector under SO (1,10 ) by considering a d = 1 + 2 dimensional diffeomorphic invariant action. If this holds, how do fermions work out?

First, target space covariance requires the background to allow a superspace formulation in d = 1 + 10 dimensions10. Such formulation involves a single copy of d = 11 Majorana fermions, which gives rise to a pair of d = 10 Majorana–Weyl fermions, matching the superspace formulation for the superstring described in Section 2. d = 11 Majorana spinors have [11∕2] 2 = 32 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 d = 6 dimensions with such number of scalars. Interestingly, there exists a tensor supermultiplet in d = 6 dimensions whose field content involves five scalars and a two-form gauge potential V2 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 d = 1 + 5 theory must be invariant under U (1 ) gauge transformations for the 2-form gauge potential. I will later discuss how to keep covariance while satisfying the self-duality constraint.

Brane scan:
World volume supersymmetry generically constrains the low energy dynamics of supersymmetric branes. Even though our arguments were concerned with M2, M5 and D-branes, they clearly are of a more general applicability. This gave rise to the brane scan programme [3Jump To The Next Citation Point, 196Jump To The Next Citation Point, 193, 191]. The main idea was to classify the set of supersymmetric branes in different dimensions by matching the number of their transverse dimensions with the number of scalar fields appearing in the list of existent supermultiplets. For an exhaustive classification of all unitary representations of supersymmetry with maximum spin 2, see [468]. Given the importance of scalar, vector and tensor supermultiplets, I list below the allowed multiplets of these kinds in different dimensions indicating the number of scalar fields in each of them [73]. Let me start with scalar supermultiplets containing X scalars in d = p + 1 dimensions, the results being summarised in Table 1. Notice, we recover the field content of the M2-brane in d = 3 and X = 8 and of the superstring in d = 2 and X = 8.


Table 1: Scalar multiplets with X scalars in p + 1 worldvolume dimensions.
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 X scalars in d = p + 1 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 (p = 0) and finishing with the D9 brane (p = 9) filling in all spacetime. Thus, the field content of all Dp-branes matches with the one corresponding to the different vector supermultiplets in d = p + 1 dimensions. This point agrees with the open string conformal field theory description of D branes.


Table 2: Vector multiplets with X scalar degrees of freedom in p + 1 worldvolume dimensions.
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 X = 5 scalars in six dimensions, corresponding to the aforementioned M5 brane, among the six-dimensional tensor supermultiplets listed in Table 3.


Table 3: Tensor multiplets with X scalar degrees of freedom in p + 1 world volume dimensions.
p + 1 X X
6 1 5

Summary:
All half-BPS Dp-branes, M2-branes and M5-branes are described at low energies by effective actions written in terms of supermultiplets in the corresponding world-volume dimension. The number of on-shell bosonic degrees of freedom is 8. Thus, the fermionic content in these multiplets satisfies

8 = 1-M 𝒩 , (19 ) 4
where M is the number of real components for a minimal spinor representation in D spacetime dimensions and 𝒩 the number of spacetime supersymmetry copies. These considerations identified an 𝒩 = 8 supersymmetric field theory in d = 3 dimensions (M2 brane), 𝒩 = (2,0 ) supersymmetric gauge field theory in d = 6 (M5 brane) and an 𝒩 = 4 supersymmetric gauge field theory in d = 4 (D3 brane), as the low energy effective field theories describing their dynamics11. The addition of interactions must be consistent with such d dimensional supersymmetries.

By construction, an effective action written in terms of these on-shell degrees of freedom can neither be spacetime covariant nor ISO (1,D − 1) 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

3.1.1 Supergravity Goldstone modes

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 𝒩 = 1 d = 11 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 U(1) 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

2 − 2 μ ν 1 p q ds = U 3ημνdx dx + U 3δpqdy dy , 1 −1 μ ν ρ A3 = ± -- U 𝜀μνρdx ∧ dx ∧ dx . (20 ) 3!
Here, and in the following examples, μ x describe the longitudinal brane directions, i.e., μ = 0,1,2 for the M2-brane, whereas the transverse Cartesian coordinates are denoted by yp, p = 3,...10. The solution is invariant under ISO (1,2) × SO (8) and is characterised by a single harmonic function U in ℝ8
( R )6 2 p q U = 1 + -- , r = δpqy y . (21 ) r
The structure for the M5-brane, first found in [273], is analogous but differs in the dimensionality of the tangential and transverse subspaces to the brane and in the nature of its charge, electric for the M2-brane and magnetic for the M5-brane below
ds2 = U − 13η dxμdx ν + U 23δ dymdyn, μν mn R = dA = ± -1 δmn∂ U 𝜀 dyp ∧ dyq ∧ dys ∧ dys. (22 ) 4 3 4! m npqsu
In this case, μ = 0,1 ...,5 and p = 6,...,10. The isometry group is ISO (1,5) × SO (5) and again it is characterised by a single harmonic function U in 5 ℝ
( )3 U = 1 + R- , r2 = δpqypyq . (23 ) r
The D3-brane, first found in [195Jump To The Next Citation Point], similarly has a non-trivial metric and self-dual five form RR field strength
1 1 ds2 = U − 2ημνdxμdx ν + U 2δmndymdyn, 1 F5 = ± --(δmn ∂mU 𝜀npqstudyp ∧ dyq ∧ dys ∧ dyt ∧ dyu 5! − 1 m μ ν ρ σ +5 ∂mU 𝜀 μνρσdy ∧ dx ∧ dx ∧ dx ∧ dx ), (24 )
with isometry group ISO (1, 3) × SO (6). It is characterised by a single harmonic function U in ℝ6
( )4 R- 2 p q U = 1 + r , r = δpqy y . (25 )
All these brane configurations are half-BPS supersymmetric. The subset of sixteen supercharges being preserved in each case is correlated with the choice of sign in the gauge potentials fixing their charges. I shall reproduce this correlation in the effective brane action in Section 3.5.

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 φ(i0), where i labels the field, including its tensor character, having an isometry group G′. Assume the configuration has some fixed asymptotics with isometry group G, so that ′ G ⊂ G. The relevant large gauge transformations p ξi(y ) in our discussion are those that act non-trivially at infinity, matching a broken global transformation asymptotically 𝜖i, but differing otherwise in the bulk of the background geometry

lim ξ (y) = 𝜖 . (26 ) r→∞ i i
In this way, one manages to associate a gauge transformation with a global one, only asymptotically. The idea is then to perturb the configuration (0) φi by such pure gauge, δξiφi and finally introduce some world volume dependence on the parameter 𝜖i, i.e., μ 𝜖i(x ). At that point, the transformation δξiφi is no longer pure gauge. Plugging the transformation in the initial action and expanding, one can compute the first order correction to the equations of motion fixing some of the ambiguities in the transformation by requiring the perturbed equation to correspond to a massless normalisable mode.

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.

Scalar modes:
These are the most intuitive geometrically. They correspond to the breaking of translations along the transverse directions to the brane. The relevant gauge symmetry is clearly a diffeomorphism. Due to the required asymptotic behaviour, it is natural to consider p s ¯p 𝜖 = U ϕ, where ¯p ϕ is some constant parameter. Notice the dependence on the harmonic function guarantees the appropriate behaviour at infinity, for any s. Dynamical fields transform under diffeomorphisms through Lie derivatives. For instance, the metric would give rise to the pure gauge transformation

hmn = ℒ 𝜖g(m0n). (27 )
If we allow ¯p ϕ to arbitrarily depend on the world volume coordinates μ x, i.e., ¯p p μ ϕ → ϕ (x ), the perturbation hmn will no longer be pure gauge. If one computes the first-order correction to Einstein’s equations in supergravity, including the perturbative analysis of the energy momentum tensor, one discovers the lowest-order equation of motion satisfied by ϕp is
∂μ∂μϕp = 0, (28 )
for s = − 1. This corresponds to a massless mode and guarantees its normalisability when integrating the action in the directions transverse to the brane. Later, we will see that the lowest-order contribution (in number of derivatives) to the gauge-fixed world-volume action of M2, M5 and D3-branes in flat space is indeed described by the Klein–Gordon equation.

Fermionic modes:
These must correspond to the breaking of supersymmetry. Consider the supersymmetry transformation of the 11-dimensional gravitino Ψm

δΨ = &tidle;D ζ , (29 ) m m
where &tidle;D is some non-trivial connection involving the standard spin connection and some contribution from the gauge field strength. The search for massless fermionic modes leads us to consider the transformation ζ = Us ¯λ for some constant spinor ¯λ. First, one needs to ensure that such transformation matches, asymptotically, with the supercharges preserved by the brane. Consider the M5-brane, as an example. The preserved supersymmetries are those satisfying δΨm = 0. This forces s = − 172 and fixes the six-dimensional chirality of ¯λ to be positive, i.e., λ¯+. Allowing the latter to become an arbitrary function of the world volume coordinates λ+(xμ), δΨm becomes non-pure gauge. Plugging the latter into the original Rarita–Schwinger equation, the linearised equation for the perturbation reduces to
Γ μ∂μλ+ = 0 . (30 )
The latter is indeed the massless Dirac equation for a chiral six-dimensional fermion. A similar analysis holds for the M2 and D3-branes. The resulting perturbations are summarised in Table 4.

Vector modes:
The spectrum of open strings with Dirichlet boundary conditions includes a vector field. Since the origin of such massless degrees of freedom must be the breaking of some abelian supergravity gauge symmetry, it must be the case that the degree form of the gauge parameter must coincide with the one-form nature of the gauge field. Since this must hold for any D-brane, the natural candidate is the abelian gauge symmetry associated with the NS-NS two-form

δB = dΛ . (31 ) 2 1
Proceeding as before, one considers a transformation with Λ1 = U k ¯V1 for some number k and constant one-form ¯V1. When ¯V1 is allowed to depend on the world volume coordinates, the perturbation
k μ δB2 = dU ∧ V1(x ), (32 )
becomes physical. Plugging this into the NS-NS two-form equation of motion, one derives dF = 0 where F = dV1 for both of the four-dimensional duality components, for either k = ±1. Clearly, only k = − 1 is allowed by the normalisability requirement.

Tensor modes:
The presence of five transverse scalars to the M5-brane and the requirement of world volume supersymmetry in six dimensions allowed us to identify the presence of a two-form potential with self-dual field strength. This must have its supergravity origin in the breaking of the abelian gauge transformation

δA3 = dΛ2 , (33 )
where indeed the gauge parameter is a two-form. Consider then Λ2 = Uk ¯V2 for some number k and constant two form ¯V2. When ¯V2 is allowed to depend on the world volume coordinates, the perturbation
k μ δA3 = dU ∧ V2(x ), (34 )
becomes physical. Plugging this into the A3 equation of motion, we learn that each world volume duality component ⋆ F = ±F x 3 3 with F = dV 3 2 satisfies the bulk equation of motion if dF3 = 0 for a specific choice of k. More precisely, self-dual components require k = 1, whereas anti-self-dual ones require k = − 1. Normalisability would fix k = − 1. Thus, this is the origin of the extra three bosonic degrees of freedom forming the tensor supermultiplet in six dimensions. The matching between supergravity Goldstone modes and the physical content of world volume supersymmetry multiplets is illustrated in Figure 5View Image. Below, a table presents the summary of supergravity Goldstone modes


Table 4: Summary of supergravity Goldstone modes.

Symmetry

 

M2

M5

D3

Reparametrisations:

𝜀m =

U− 1ϕ¯m

U −1¯ϕm

U −1¯ϕm

Local supersymmetry:

ζ =

− 2∕3¯ U λ−

−7∕12¯ U λ+

−5∕8¯ U λ+

Tensor gauge symmetry:

Λ =

U −1¯V(2)

U −1¯V(1)

(⋆ ¯H = ¯H )

(i⋆ ¯F = F¯)


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 λ, 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.


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