3.3 Consistency checks
The purpose of this section is to check the consistency of the kinematic structures governing classical
bosonic brane effective actions with string dualities [312, 495]. At the level of supergravity, these dualities
are responsible for the existence of a non-trivial web of relations among their classical Lagrangians. Here, I
describe the realisation of some of these dualities on classical bosonic brane actions. This will allow us to
check the consistency of all brane couplings. Alternatively, one can also view the discussions below as
independent ways of deriving the latter.
The specific dualities I will be appealing to are the strong coupling limit of type IIA string theory, its
relation to M-theory and the action of T-duality on type II string theories and D-branes. Figure 3
summarises the set of relations between the brane tensions discussed in this review under these
symmetries.
M-theory as the strong coupling limit of type IIA:
From the spectrum of 1/2-BPS states in string theory
and M-theory, an M2/M5-brane in
has a weakly-coupled description in type IIA
- either as a long string or a D4-brane, if the M2/M5-brane wraps the M-theory circle, respectively
- or as a D2-brane/NS5 brane, if the M-theory circle is transverse to the M2/M5-brane world
volume.
The question to ask is: how do these statements manifest in the classical effective action? The answer is by now
well known. They involve a double or a direct dimensional reduction, respectively. The idea is simple. The
bosonic effective action describes the coupling of a given brane with a fixed supergravity background. If the
latter involves a circle and one is interested in a description of the physics nonsensitive to this dimension,
one is entitled to replace the d-dimensional supergravity description by a d-1 one using a Kaluza–Klein
(KK) reduction (see [197] for a review on KK compactifications). In the case at hand, this involves using
the relation between
bosonic supergravity fields and the type IIA bosonic ones summarised
below [409]
where the left-hand side 11-dimensional fields are rewritten in terms of type IIA fields. The above reduction
involves a low energy limit in which one only keeps the zero mode in a Fourier expansion of all background
fields on the bulk S1. In terms of the parameters of the theory, the relation between the M-theory circle
and the 11-dimensional Planck scale
with the type IIA string coupling
and string length
is
The same principle should hold for the brane degrees of freedom
. The distinction between a
double and a direct dimensional reduction comes from the physical choice on whether the brane wraps the
internal circle or not:
- If it does, one partially fixes the world volume diffeomorphisms by identifying the bulk circle
direction
with one of the world volume directions
, i.e.,
, and keeps the
zero mode in a Fourier expansion of all the remaining brane fields, i.e.,
where
. This procedure is denoted as a double dimensional reduction [192
], since
both the bulk and the world volume get their dimensions reduced by one.
- If it does not, there is no need to break the world volume diffeomorphisms and one simply
truncates the fields to their bulk zero modes. This procedure is denoted as a direct reduction
since the brane dimension remains unchanged while the bulk one gets reduced.
T-duality on closed and open strings:
From the quantisation of open strings satisfying Dirichlet boundary
conditions, all D-brane dynamics are described by a massless vector supermultiplet, whose number of scalar
fields depends on the number of transverse dimensions to the D-brane. Since D-brane states are mapped
among themselves under T-duality [160, 424
], one expects the existence of a transformation mapping their
classical effective actions under this duality. The question is how such transformation acts on the action.
This involves two parts: the transformation of the background and the one of the brane degrees of
freedom.
Let me focus on the bulk transformation. T-duality is a perturbative string theory duality [241
]. It says
that type IIA string theory on a circle of radius
and string coupling
is equivalent to
type IIB on a dual circle of radius
and string coupling
related as [121, 122, 240]
when momentum and winding modes are exchanged in both theories. This leaves the free theory
spectrum invariant [337], but it has been shown to be an exact perturbative symmetry when
including interactions [400, 241]. Despite its stringy nature, there exists a clean field theoretical
realisation of this symmetry. The main point is that any field theory on a circle of radius
has
a discrete momentum spectrum. Thus, in the limit
, all non-vanishing momentum
modes decouple, and one only keeps the original vanishing momentum sector. Notice this is
effectively implementing a KK compactification on this circle. This is in contrast with the stringy
realisation where in the same limit, the spectrum of winding modes opens up a dual circle of radius
.
Since Type IIA and Type IIB supergravities are field theories, the above field theoretical realisation
applies. Thus, the
compactification limit should give rise to two separate
supergravity theories. But it is known [388
] that there is just such a unique supergravity theory. In other
words, given the type IIA/B field content
and their KK reduction to
dimensions, i.e.,
and
, the uniqueness of
supergravity guarantees the existence
of a non-trivial map between type IIA and type IIB fields in the subset of backgrounds allowing an S1
compactification
This process is illustrated in the diagram of Figure 4. These are the T-duality rules. When expressed in
terms of explicit field components, they become [82, 388]
These correspond to the bosonic truncations of the superfields introduced in Appendix A.1. Prime and
unprimed fields correspond to both T-dual theories. The same notation applies to the tensor components
where
describe both T-dual circles. Notice the dilaton and the
transformations do capture
the worldsheet relations (56).
Let me move to the brane transformation. A D(
)-brane wrapping the original circle is mapped
under T-duality to a Dp-brane where the dual circle is transverse to the brane [424
]. It must be
the case that one of the gauge field components in the original brane maps into a transverse
scalar field describing the dual circle. At the level of the effective action, implementing the
limit must involve, first, a partial gauge fixing of the world volume diffeomorphisms,
to explicitly make the physical choice that the brane wraps the original circle, and second,
keeping the zero modes of all the remaining dynamical degrees of freedom. This is precisely the
procedure described as a double dimensional reduction. The two differences in this D-brane
discussion will be the presence of a gauge field and the fact that the KK reduced supergravity
fields
will be rewritten in terms of the T-dual ten-dimensional fields using the T-duality
rules (58).
In the following, it will be proven that the classical effective actions described in the previous section are
interconnected in a way consistent with our T-duality and strongly-coupled considerations. Our logic is as
follows. The M2-brane is linked to our starting worldsheet action through double-dimensional reduction.
The former is then used to derive the D2-brane effective by direct dimensional reduction. T-duality
covariance extends this result to any non-massive D-brane. Finally, to check the consistency of the PST
covariant action for the M5-brane, its double dimensional reduction will be shown to match the D4-brane
effective action. This will complete the set of classical checks on the bosonic brane actions discussed so
far.
It is worth mentioning that the self-duality of the D3-brane effective action under S-duality could also
have been included as a further test. For discussions on this point, see [483
, 252].
3.3.1 M2-branes and their classical reductions
In the following, I discuss the double and direct dimensional reductions of the bosonic M2-brane effective
action (40) to match the bosonic worldsheet string action (6) and the D2-brane effective action, i.e., the
version of Eq. (47). This analysis will also allow us to match/derive the tensions of the different
branes.
Connection to the string worldsheet:
Consider the propagation of an M2-brane in an 11-dimensional
background of the form (54). Decompose the set of scalar fields as
, identify one of the
world volume directions
with the KK circle, i.e., partially gauge fix the world volume diffeomorphisms
by imposing
, and keep the zero modes in the Fourier expansion of all remaining scalar
fields
along the world volume circle, i.e.,
. Under these conditions, which
mathematically characterise a double dimensional reduction, the Wess–Zumino coupling becomes
where I already used the KK reduction ansatz (54). Here,
stands for the pull-back of the NS-NS two
form into the surface
parameterised by
. The DBI action is reduced using the identity satisfied
by the induced world-volume metric
Since the integral over
equals the length of the M-theory circle,
where I used Eq. (55),
and absorbed the overall circle length, expressed
in terms of type IIA data, in a new energy density scale, matching the fundamental string
tension
defined in Section 2. The same argument applies to the charge density leading to
.
Altogether, the double reduced action reproduces the bosonic effective action (6) describing the string
propagation in a type IIA background. Thus, our classical bosonic M2-brane action is consistent with the
relation between half-BPS M2-brane and fundamental strings in the spectrum of M-theory and
type IIA.
Connection to the D2-brane:
The direct dimensional reduction of the bosonic M2 brane describes a
three-dimensional diffeomorphism invariant theory propagating in 10 dimensions, with eleven scalars as its
field content. The latter disagrees with the bosonic field content of a D2-brane, which includes a vector field.
Fortunately, a scalar field is Hodge dual, in three dimensions, to a one form. Thus, one expects that by
direct dimensional reduction of the bosonic M2-brane action and after world volume dualisation of
the scalar field
along the M-theory circle, one should reproduce the classical D2-brane
action [439
, 477
, 93
, 480
].
To describe the direct dimensional reduction, consider the Lagrangian [480
]
This is classically equivalent to Eq. (40) after integrating out the auxiliary scalar density
by solving its algebraic equation of motion. Notice I already focused on the relevant case for
later supersymmetric considerations, i.e.,
. The induced world volume fields are
where
Using the properties of
matrices,
where
, the action (62) can be written as
The next step is to describe the world volume dualisation and the origin of the
gauge symmetry
on the D2 brane effective action [480
]. By definition, the identity
holds. Adding the latter to the action through an exact two-form
Lagrange multiplier
allows one to treat
as an independent field. For a more thorough discussion on this point and the
nature of the
gauge symmetry, see [480]. Adding Eq. (69) to Eq. (67), one obtains
Notice I already introduced the same gauge invariant quantity introduced in D-brane Lagrangians
Since
is now an independent field, it can be eliminated by solving its algebraic equation of motion
Inserting this back into the action and integrating out the auxiliary field
by solving its
equation of motion, yields
This matches the proposed D2-brane effective action, since
as a consequence of Eq.s (55) and
(48).
3.3.2 T-duality covariance
In this section, I extend the D2-brane’s functional form to any Dp-brane using T-duality covariance. My
goal is to show that the bulk T-duality rules (58) guarantee the covariance of the D-brane effective action
functional form [453
] and to review the origin in the interchange between scalar fields and gauge fields on the
brane.
The second question can be addressed by an analysis of the D-brane action bosonic symmetries. These
act infinitesimally as
They involve world volume diffeomorphisms
, a
gauge transformation
and global
transformations
. Since the background will undergo a T-duality transformation, by assumption, this
set of global transformations must include translations along the circle, i.e.,
,
,
where the original
scalar fields were split into
.
I argued that the realisation of T-duality on the brane action requires one to study its
double-dimensional reduction. The latter involves a partial gauge fixing
, identifying one world
volume direction with the starting S
bulk circle and a zero-mode Fourier truncation in the
remaining degrees of freedom,
. Extending this functional truncation to the
-dimensional diffeomorphisms
, where I split the world volume indices according to
and the space of global transformations, i.e.,
, the consistency
conditions requiring the infinitesimal transformations to preserve the subspace of field configurations
defined by the truncation and the partial gauge fixing, i.e.,
, determines
where
are constants, the latter having mass dimension minus one. The set of transformations in the
double dimensional reduction are
where
,
and
satisfies
.
Let me comment on Eq. (79).
was a gauge field component in the original action. But in its
gauge-fixed functionally-truncated version, it transforms like a world volume scalar. Furthermore, the
constant piece
in the original
transformation (76), describes a global translation along the
scalar direction. The interpretation of both observations is that under double-dimensional reduction
becomes the T-dual target space direction along the T-dual circle and
describes the
corresponding translation isometry. This discussion reproduces the well-known massless open
string spectrum when exchanging a Dirichlet boundary condition with a Neumann boundary
condition.
Having clarified the origin of symmetries in the T-dual picture, let me analyse the functional dependence
of the effective action. First, consider the couplings to the NS sector in the DBI action. Rewrite the induced
metric
and the gauge invariant
in terms of the T-dual background
and degrees of
freedom
, which will be denoted by primed quantities. This can be
achieved by adding and subtracting the relevant pullback quantities. The following identities hold
It is a consequence of our previous symmetry discussion that
and
, i.e.,
there is no change in the description of the dynamical degrees of freedom not involved in the
circle directions. The determinant appearing in the DBI action can now be computed to be
Notice that whenever the bulk T-duality rules (58) are satisfied, the functional form of the effective action
remains covariant, i.e., of the form
This is because all terms in the determinant vanish except for those in the first line. Finally,
equals the T-dual dilaton coupling
and the original Dp-brane tension
becomes the
D(p-1)-brane tension in the T-dual theory due to the worldsheet defining properties (56) after the
integration over the world volume circle
Just as covariance of the DBI action is determined by the NS-NS sector, one expects the RR sector to do
the same for the WZ action. Here I follow similar techniques to the ones developed in [255, 453
]. First,
decompose the WZ Lagrangian density as
Due to the functional truncation assumed in the double dimensional reduction, the second term vanishes.
The D-brane WZ action then becomes
where
and the following conventions are used
Using the T-duality transformation properties of the gauge invariant quantity
, derived from our DBI
analysis,
it was shown in [453] that the functional form of the WZ term is preserved, i.e.,
,
whenever the condition
holds (the factor
is due to our conventions (91) and the choice of orientation
and
).
Due to our gauge-fixing condition,
, the
components of the pullbacked world volume forms
appearing in Eq. (94) can be lifted to
components of the spacetime forms. The condition (94) is then
solved by
These are entirely equivalent to the T-duality rules (58) but written in an intrinsic way.
The expert reader may have noticed that the RR T-duality rules do not coincide with the ones
appearing in [208
]. The reason behind this is the freedom to redefine the fields in our theory. In particular,
there exist different choices for the RR potentials, depending on their transformation properties under
S-duality. For example, the 4-form
appearing in our WZ couplings is not S self-dual, but transforms as
Using a superindex S to denote an S-dual self-dual 4-form, the latter must be
Similarly,
does not transform as a doublet under S-duality, whereas
does. It is straightforward to check that Eqs. (95) and (96) are equivalent to the ones appearing in [208
]
using the above redefinitions. Furthermore, one finds
, which was not computed in [208
].
In Section 7.1, I will explore the consequences that can be extracted from the requirement of T-duality
covariance for the covariant description of the effective dynamics of
overlapping parallel D-branes in
curved backgrounds, following [395
].
3.3.3 M5-brane reduction
The double dimensional reduction of the M5-brane effective action, both in its covariant [417, 8] and
non-covariant formulations [420
, 420, 78] was checked to agree with the D4-brane effective action. It is
important to stress that the outcome of this reduction may not be in the standard D4-brane action form
given in Eq. (47), but in the dual formulation. The two are related through the world volume dualisation
procedure described in [483, 7].