7.1 D-branes
The perturbative description of D-branes in terms of opens strings [423] allows one to answer the
question regarding the enhancement of massless modes raised above in a firmer basis, at least at
weak coupling. Consider the spectrum of open strings in the presence of two parallel Dp-branes
separated by a physical distance
. As the latter approaches zero, i.e., it becomes smaller
than the string scale, there is indeed an enhancement in the number of massless modes. Its
origin is in the sector of open strings stretching between D-branes, which is precisely the one
captured by the BIon argument. This enhancement is consistent with an enhancement in the
gauge symmetry from
, corresponding to the two separated D-branes, to
,
corresponding to the overlapping D-branes. The spectrum of massless excitations is then described
by a non-abelian vector supermultiplet in the adjoint representation. To understand how this
comes about, consider the set of massless scalar excitations. These are described by
,
where
labels the transverse directions to the brane, as in the abelian discussion, and the
subindices
label the D-branes where the open strings are attached. This is illustrated in
Figure 10. Since the latter are oriented, there exist
such excitations, which arrange
themselves into a matrix
, with
being generators of
in the adjoint
representation. The conclusion is valid for any number
of D-branes of world volume dimension
[496
].
Super-Yang–Mills action:
The previous discussion identifies the appropriate degrees of freedom to describe
the low energy dynamics of multiple D-branes in Minkowski at weak coupling as non-abelian vector
supermultiplets. Thus, multiple brane effective actions must correspond to supersymmetric non-abelian
gauge field theories in
dimensions. At lowest order in a derivative expansion, these are precisely
super-Yang–Mills (SYM) theories. For simplicity of notation, let me focus on
SYM with
classical action
where the field strength
is the curvature of a
hermitian gauge field
and
is a 16-component Majorana–Weyl spinor
of
. Both fields,
and
, are in the adjoint representation of
. The covariant
derivative
of
is given by
where
is the Yang–Mills coupling constant. This action is also usually written in terms of rescaled
fields, by absorbing a factor of
in both
and
, to pull an overall coupling constant
dependence in front of the full action
where
.
The action (470) is invariant under the supersymmetry transformation
where
is a constant Majorana–Weyl spinor in
, giving rise to 16 independent supercharges.
Classically, this is a well-defined theory; quantum mechanically, it is anomalous. From the string theory
perspective, as explained in Section 3.7, this is just an effective field theory, valid at low energies
and weak coupling
.
Dimensional reduction:
The low energy effective action for multiple parallel Dp-branes in Minkowski is SYM
in
dimensions. This theory can be obtained by dimensional reduction of the ten-dimensional super
Yang–Mills theory introduced above. Thus, one proceeds as described in Section 3.3: assume all fields are
independent of coordinates
. After dimensional reduction, the 10-dimensional gauge field
decomposes into a
-dimensional gauge field
and
adjoint scalar fields
,
describing the transverse fluctuations of the D-branes. The reduced action takes the form
The
dimensional YM coupling
can be fixed by matching the expansion of the square root in
the gauge fixed abelian D-brane action in a Minkowski background (104) and comparing it with Maxwell’s
theory in the field normalisation used in Eq. (473)
Notice also the appearance of a purely non-abelian interaction term in Eq. (475), the commutator
that acts as a potential term. Indeed, its contribution is negative definite since
.
The classical vacuum corresponds to static configurations minimising the potential. This occurs when
both the curvature
and the fermions vanish, and for a set of commuting
matrices, at each point
of the
world volume. In this situation, the fields
can be simultaneously diagonalised, so that
one has
The
diagonal elements of the matrix
are interpreted as the positions of
distinct D-branes in
the
-th transverse direction [496
]. Consider a vacuum describing
overlapping Dp-branes and a
single parallel D-brane separated in a transverse direction
. This is equivalent to breaking the symmetry
group to
by choosing a diagonal matrix for
with
eigenvalue in the first
diagonal entries and
in the last diagonal entry. The off-diagonal components
will
acquire a mass, through the Higgs mechanism. This can be computed by expanding the classical action
around the given vacuum. One obtains that this mass is proportional to the distance
between
the two sets of branes
according to the geometrical interpretation given to the eigenvalues characterising the vacuum. In light of
the open string interpretation, these off-diagonal components do precisely correspond to the open strings
stretching between the different D-branes. The latter allow an alternative description in terms of the
BIon configurations described earlier, by replacing the
Dp-branes by its supergravity
approximation, though the latter is only suitable at large distances compared to the string
scale.
It can then be argued that the moduli space of classical vacua for
-dimensional SYM is
Each factor of
stands for the position of the
D-branes in the
-dimensional
transverse space, whereas the symmetry group
is the residual Weyl symmetry of the gauge
group. The latter exchanges D-branes, indicating they should be treated as indistinguishable
objects.
A remarkable feature of this D-brane description is that a classical geometrical interpretation of D-brane
configurations is only available when the matrices
are simultaneously diagonalisable. This provides a
rather natural venue for non-commutative geometry to appear in D-brane physics at short distances, as first
pointed out in [496].
The exploration of further kinematical and dynamical properties of these actions is beyond the scope of
this review. There are excellent reviews on the subject, such as [424, 472, 320], where the connection to
Matrix Theory [48
] is also covered. If the reader is interested in understanding how T-duality acts
on non-abelian D-brane effective actions, see [471, 221]. It is also particularly illuminating,
especially for readers not used to the AdS/CFT philosophy, to appreciate that by integrating out
overlapping D-branes at one loop, one is left with an abelian theory describing the
remaining (single) D-brane. The effective dynamics so derived can be reinterpreted as describing
a single D-brane in the background generated by the integrated
D-branes, which is
AdS5 × S5 [365].
This is illustrated in Figure 11.
Given the kinematical perspective offered in this review and the relevance of the higher order
corrections included in the abelian DBI action, I want to discuss two natural stringy extensions of the SYM
description
- Keeping the background fixed, i.e., Minkowski, it is natural to consider the inclusion of
higher-order corrections in the effective action, matching the perturbative scattering amplitudes
computed in the CFT description of open strings theory, and
- Allowing to vary the background or equivalently, coupling the non-abelian degrees of freedom
to curved background geometries. This is towards the direction of achieving a hypothetical
covariant formulation of these actions, a natural question to ask given its relevance for the
existence of the kappa invariant formulation of abelian D-branes.
In the following, I shall comment on the progress and the important technical and conceptual difficulties
regarding the extensions of these non-abelian effective actions.
Higher-order corrections:
In the abelian theory, it is well known that the DBI action captures all the
higher-order corrections in
to the open string effective action in the absence of field strength derivative
terms [214].
It was further pointed that such derivative corrections were compatible with a DBI expansion by requiring
conformal invariance for the bosonic string in [1] and for the superstring in [87].
In the non-abelian theory, such distinction is ambiguous due to the identity
relating commutators with covariant derivatives. It was proposed by Tseytlin [482
] that the non-abelian
extension of SYM including higher-order
corrections be given in terms of the symmetrised prescription.
The latter consists of treating all
matrices as commuting. Equivalently, the action is
completely symmetric in all monomial factors of
of the form
. This reproduces
the
and
terms of the full non-abelian action, but extends it to higher orders
The notation
defines this notion of symmetrised trace for each of the monomials appearing in the
expansion of its arguments. For an excellent review describing the history of these calculations, motivating
this prescription and summarising the most relevant properties of this action, see [485].
It is important to stress that, a priori, worldsheet calculations involving an arbitrary number of
boundary disk insertions could determine this non-abelian effective action. Since this is technically hard,
one can perform other consistency checks. For example, one can compare the D-brane BPS spectrum on tori
in the presence of non-trivial magnetic fluxes. This is T-dual to intersecting D-branes, whose spectrum can
be independently computed and compared with the fluctuation analysis of the proposed symmetrised
non-abelian prescription. It was found in [291, 175, 448] that the proposed prescription was breaking down
at order
. Further checks at order
and
were carried over in [103, 346
, 345, 347].
The proposal in [346] was confirmed by a first principle five-gluon scattering amplitude at
tree level in [387]. The conclusion is that the symmetrised prescription only works up to
These couplings were first found in its
form in [266] and in its
form in [482]. For further checks
on Tseytlin’s proposal using the existence of bound states and BPS equations, see the analysis
in [115, 114].
Coupling to arbitrary curved backgrounds:
The above corrections attempted to include higher-order
corrections describing the physics of multiple D-branes in Minkowski. More generally, one is interested in
coupling D-branes to arbitrary closed string backgrounds. In such situations, one would like to achieve a
covariant formulation. This is non-trivial because as soon as the degrees of freedom become
non-abelian, they lose their geometrical interpretation. In the abelian case,
described the
brane location. In the non-abelian case, at most, only their eigenvalues
may keep their
interpretation as the location of the ith brane in the Ith direction. Given the importance and
complexity of the problem, it is important to list a set of properties that one would like such a
formulation to satisfy. These are the D-geometry axioms [186
]. For the case of D0-branes, these
follow.
- It must contain a unique trace since this is an effective action derived from string theory disk
diagrams involving many graviton insertions in their interior and scalar/vector vertex operators
on their boundaries. Since the disk boundary is unique, the trace must be unique.
- It must reduce to N-copies of the particle action when the matrices
are diagonal.
- It must yield masses proportional to the geodesic distance for off-diagonal fluctuations.
Having in mind that we required spacetime gauge symmetries to be symmetries of the abelian brane
effective actions, it would be natural to include in the above list invariance under target space
diffeomorphisms. This was analysed for the effective action kinetic terms in [172]. Instead of
discussing this here, I will discuss two non-trivial checks that any such formulation must satisfy.
- to match the Matrix theory linear couplings to closed string backgrounds, and
- to be T-duality covariant, extending the notion I discussed in Section 3.3.2 for single D-branes.
The first was studied in [473
, 474
] and the second in [395
]. Since the results derived from the latter turned out
to be consistent with the former, I will focus on the implementation of T-duality covariance for non-abelian
D-branes below.
As discussed in Section 3.3, T-duality is implemented by a dimensional reduction. This was already
applied for SYM in Eq. (475). Using the same notation introduced there and denoting the world volume
direction along which one reduces by
, one learns that
, where
is the T-dual
adjoint matrix scalar. Furthermore, covariant derivatives of transverse scalar fields
become
Notice this contribution is purely non-abelian and it can typically contribute non-trivially to the potential
terms in the effective action. To properly include these non-trivial effects, Myers [395
] studied the
consequences of requiring T-duality covariance taking as a starting point a properly covariantised version of
the multiple D9-brane effective action, having assumed the symmetrised trace prescription described
above. Studying T-duality along 9-
directions and imposing T-duality covariance of the
resulting action, will generate all necessary T-duality compatible commutators, which would
have been missed otherwise. This determines the DBI part of the effective action to be [395
]
with
Here
indices stand for world volume directions, and
indices for transverse directions. To deal
with similar commutators arising from the WZ term, one considers [395
]
where the interior product
is responsible for their appearance, for example, as in,
Notice one regards
as a vector field in the transverse space. In both actions (485) and (487),
stands for pullback and it only applies to transverse brane directions since all longitudinal ones are
non-physical. Its presence is confirmed by scattering amplitudes calculations [342, 271, 222]. Some remarks
are in order.
- There exists some non-trivial dependence on the scalars
through the arbitrary bosonic closed
backgrounds appearing in the action. The latter is defined according to
Analogous definitions apply to other background fields.
- There exists a unique trace, because this is an open string effective action that can be derived from
worldsheet disk amplitudes. The latter has a unique boundary. Thus, there must be a unique gauge
trace [186, 188]. Above, the symmetrised prescription was assumed, not only because one is following
Tseytlin and this was his prescription, but also because there are steps in the derivation of T-duality
covariance that assumed this property and the scalar field
dependence on the background
fields (489) is symmetric, by definition.
- The WZ term (487) allows multiple Dp-branes to couple to RR potentials with a form degree greater
than the dimension of the world-volume. This is a purely non-abelian effect whose consequences will
be discussed below.
- There are different sources for the scalar potential:
, its inverse in the first determinant of
the DBI and contributions coming from commutators coupling to background field components in the
expansion (489).
It was shown in detail in [395
], that the bosonic couplings described above were consistent with all the
linear couplings of closed string background fields with Matrix Theory degrees of freedom, i.e., multiple
D0-branes. These couplings were originally computed in [473] and then extended to Dp-branes in [474]
using T-duality once more. We will not review this check here in detail, but as an illustration of the above
formalism, present the WZ term for multiple D0-branes that is required to do such matching
Two points are worth emphasising about this matching:
- There is no ambiguity of trace in the linear Matrix theory calculations. Myers’ suggestion is
to extend this prescription to non-linear couplings.
- Some transverse M5-brane charge couplings are unknown in Matrix theory, but these are absent
in the Lagrangian above. This is a prediction of this formulation.
One of the most interesting physical applications of the couplings derived above is the realisation of the
dielectric effect in electromagnetism in string theory. As already mentioned above, the non-abelian nature of
the degrees of freedom turns on new commutator couplings with closed string fields that can modify the
scalar potential. If so, instead of the standard SYM vacua, one may find new potential minima with
but
. As a toy illustrative example of this phenomenon, consider
D0-branes propagating in Minkowski but in a constant background RR four-form field strength
Due to gauge invariance, one expects a coupling of the form
Up to total derivatives, this can indeed be derived from the cubic terms in the WZ action above. This
coupling modifies the scalar potential to
whose extremisation condition becomes
The latter allows
solutions
having lower energy than standard commuting matrices
It is reassuring to compare the description above with the one available using the abelian formalism
describing a single brane explained in Section 3. I shall refer to the latter as dual brane description. For the
particular example discussed above, since the D0-branes blow up into spheres due to the electric RR
coupling, one can look for on-shell configurations on the abelian D2-brane effective action in the same
background corresponding to the expanded spherical D0-branes in the non-abelian description. These
configurations exist, reproduce the energy
up to
corrections and carry no D2-brane
charge [395]. Having reached this point, I am at a position to justify the expansion of pointlike gravitons
into spherical D3-branes, giant gravitons, in the presence of the RR flux supporting AdS5 × S5
described in Section 5.9. The non-abelian description would involve non-trivial commutators
in the WZ term giving rise to a fuzzy sphere extremal solution to the scalar potential. The
abelian description reviewed in Section 5.9 corresponds to the dual D3-brane description in
which, by keeping the same background, one searches for on-shell spherical rotating D3-branes
carrying the same charges as a pointlike graviton but no D3-brane charge. For a more thorough
discussion of the comparison between non-abelian solitons and their “dual” abelian descriptions,
see [147, 149, 148, 396].
Kappa symmetry and superembeddings:
The covariant results discussed above did not include fermions.
Whenever these were included in the abelian case, a further gauge symmetry was required, kappa symmetry,
to keep covariance, manifest supersymmetry and describe the appropriate on-shell degrees of freedom. One
suspects something similar may occur in the non-abelian case to reduce the number of fermionic degrees of
freedom in a manifestly supersymmetric non-abelian formulation. It is important to stress that at this point
world volume diffeomorphisms and kappa symmetry will no longer appear together. In all the
discussions in this section, world volume diffeomorphisms are assumed to be fixed, in the sense
that the only scalar adjoint matrices already correspond to the transverse directions to the
brane.
Given the projective nature of kappa symmetry transformations, it may be natural to assume that there
should be as many kappa symmetries as fermions. In [79], a perturbative approach to determining such
transformation
was analysed for multiple D-branes in super-Poincaré. The idea was to expand the WZ term in covariant
derivatives of the fermions and the gauge field strength
, involving some a priori arbitrary tensors. One
then computes its kappa symmetry variation and attempts to identify the DBI term in the
action at the same order by satisfying the requirement that the total action variation equals
order by order. In a sense, one is following the same strategy as in [9], determining the different unknown
tensors order by order. Unfortunately, it was later concluded in [76] that such an approach could not
work.
There exists some body of work constructing classical supersymmetric and kappa invariant actions
involving non-abelian gauge fields representing the degrees of freedom of multiple D-branes. This
started with actions describing branes of lower co-dimension propagating in lower dimensional
spacetimes [461, 462, 190]. It was later extended to multiple D0-branes in an arbitrary number of
dimensions, including type IIA, in [411]. Here, both world volume diffeomorphisms and kappa symmetry
were assumed to be abelian. It was checked that when the background is super-Poincaré, the proposed
action agreed with Matrix Theory [48]. Using the superembedding formalism [460
], actions were proposed
reproducing the same features in [40, 44, 42, 41, 43], some of them involving a superparticle propagating
in arbitrary 11-dimensional backgrounds. Finally, there exists a slightly different approach in which, besides
using the superembedding formalism, the world sheet Chan–Paton factors describing multiple D-branes are
replaced by boundary fermions. The actions constructed in this way in [303], based on earlier
work [304], have similar structure to the ones described in the abelian case, their proof of kappa
symmetry invariance is analogous and they reproduce Matrix Theory when the background is
super-Poincaré and most of the features highlighted above for the bosonic couplings described by
Myers.
Relation to non-commutative geometry:
There are at least two reasons why one may expect
non-commutative geometry to be related to the description of multiple D-brane actions:
- D-brane transverse coordinates being replaced by matrices,
- the existent non-commutative geometry description of D-branes in the presence of a
-field
in space-time (or a magnetic field strength on the brane) [187, 146, 444].
The general idea behind non-commutative geometry is to replace the space of functions by a non-commutative
algebra. In the D-brane context, a natural candidate to consider would be the algebra
As customary in non-commutative geometry, the latter does not yet carry any metric information.
Following Connes [145], the construction of a Riemannian structure requires a spectral triple
, which, in addition to
, also contains a Hilbert space
and a self-adjoint
operator
obeying certain properties. It would be interesting to find triples
that
describe, in a natural way, metrics relevant for multiple D-branes, incorporating the notion of
covariance.
Regarding D-branes in the presence of a
-field, the main observation is that the structure of an
abelian non-commutative gauge theory is similar to that of a non-abelian commutative gauge theory. In
both cases, fields no longer commute, and the field strengths are non-linear. Moreover, non-commutative
gauge theories can be constructed starting from a non-abelian commutative theory by expanding around
suitable backgrounds and taking
[443]. This connection suggests it may be possible to
relate the gravity coupling of non-commutative gauge theories to the coupling of non-abelian
D-brane actions to curved backgrounds (gravity). This was indeed the approach taken in [163]
where the stress-tensor of non-commutative gauge theories was derived in this way. In [151],
constraints on the kinematical properties of non-abelian D-brane actions due to this connection were
studied.