3.2 Bosonic actions
After the identification of the relevant degrees of freedom and gauge symmetries governing brane
effective actions, I focus on the construction of their bosonic truncations, postponing their
supersymmetric extensions to Sections 3.4 and 3.5. The main goal below will be to couple brane
degrees of freedom to arbitrary curved backgrounds in a world volume diffeomorphic invariant
way.
I shall proceed in order of increasing complexity, starting with the M2-brane effective
action, which is purely geometric, continuing with D-branes and their one form gauge
potentials and finishing with M5-branes including their self-dual three form field
strength.
Bosonic M2-brane:
In the absence of world volume gauge field excitations, all brane effective actions must
satisfy two physical requirements
- Geometrically, branes are
hypersurfaces
propagating in a fixed background with
metric
. Thus, their effective actions should account for their world volumes.
- Physically, all branes are electrically charged under some appropriate spacetime
gauge
form
. Thus, their effective actions should contain a minimal coupling accounting for
the brane charges.
Both requirements extend the existent effective action describing either a charged particle
or a string
. Thus, the universal description of the purely scalar field
brane degrees of freedom must be
of the form
where
and
stand for the brane tension and charge
density.
The first term computes the brane world volume from the induced metric
whereas the second WZ term
describes the pullback of the target space
gauge field
under which the brane is charged
At this stage, one assumes all branes propagate in a background with Lorentzian metric
coupled
to other matter fields, such as
, whose dynamics are neglected in this approximation. In string
theory, these background fields correspond to the bosonic truncation of the supergravity multiplet and their
dynamics at low energy is governed by supergravity theories. More precisely, M2 and M5-branes
propagate in
supergravity backgrounds, i.e.,
, and they are electrically
charged under the gauge potential
and its six-form dual potential
, respectively
(see Appendix A for conventions). D-branes propagate in
type IIA/B backgrounds
and the set
correspond to the set of RR gauge potentials in these theories, see
Eq. (521).
The relevance of the minimal charge coupling can be understood by considering the full effective action
involving both brane and gravitational degrees of freedom (17). Restricting ourselves to the kinetic
term for the target space gauge field, i.e.,
, the combined action can be written as
Here
stands for the
-dimensional spacetime, whereas
is a
-form whose
components are those of an epsilon tensor normal to the brane having a
-function support on the world
volume.
Thus, the bulk equation of motion for the gauge potential
acquires a source term whenever a brane
exists. Since the brane charge is computed as the integral of
over any topological
-sphere
surrounding it, one obtains
where the equation of motion was used in the last step. Thus, minimal WZ couplings do capture the brane
physical charge.
Since M2-branes do not involve any gauge field degree of freedom, the above discussion
covers all its bosonic degrees of freedom. Thus, one expects its bosonic effective action to be
in analogy with the bosonic worldsheet string action. If Eq. (40) is viewed as the bosonic truncation of a
supersymmetric M2-brane, then
. Besides its manifest spacetime covariance and its invariance
under world-volume diffeomorphisms infinitesimally generated by
this action is also quasi-invariant (invariant up to total derivatives) under the target space gauge
transformation
leaving
supergravity invariant, as reviewed in Eq. (551) of
Appendix A.2. This is reassuring given that the full string theory effective action (17) describing both
gravity and brane degrees of freedom involves both actions.
Bosonic D-branes:
Due to the perturbative description in terms of open strings [423
], D-brane effective
actions can, in principle, be determined by explicit calculation of appropriate open string disk amplitudes.
Let me first discuss the dependence on gauge fields in these actions. Early bosonic open string calculations
in background gauge fields [1
], allowed to determine the effective action for the gauge field, with purely
Dirichlet boundary conditions [214
] or with mixed boundary conditions [354], gave rise to a non-linear
generalisation of Maxwell’s electromagnetism originally proposed by Born and Infeld in [108]:
I will refer to this non-linear action as the Dirac–Born–Infeld (DBI) action. Notice, this is an exceptional
situation in string theory in which an infinite sum of different
contributions is analytically
computable. This effective action ignores any contribution from the derivatives of the field
strength
, i.e.,
terms or higher derivative operators. Importantly, it was shown
in [1
] that the first such corrections, for the bosonic open string, are compatible with the DBI
structure.
Having identified the non-linear gauge field dependence, one is in a position to include the dependence
on the embedding scalar fields
and the coupling with non-trivial background closed string fields.
Since in the absence of world-volume gauge-field excitations, D-brane actions should reduce to Eq. (35), it
is natural to infer the right answer should involve
using the general arguments of the preceding paragraphs. Notice, this action does not include any contribution from
acceleration and higher derivative operators involving scalar fields, i.e.,
terms and/or higher derivative
terms.
This proposal has nice properties under T-duality [24
, 77
, 16
, 75
], which I will explore in detail in
Section 3.3.2 as a non-trivial check on Eq. (43). In particular, it will be checked that absence of
acceleration terms is compatible with T-duality.
The DBI action is a natural extension of the NG action for branes, but it does not capture all the
relevant physics, even in the absence of acceleration terms, since it misses important background couplings,
responsible for the WZ terms appearing for strings and M2-branes. Let me stress the two main issues
separately:
- The functional dependence on the gauge field
in a general closed string background. D-branes are
hypersurfaces where open strings can end. Thus, open strings do have endpoints. This means that the
WZ term describing such open strings is not invariant under the target space gauge transformation
due to the presence of boundaries. These are the D-branes themselves, which see these
endpoints as charge point sources. The latter has a minimal coupling of the form
,
whose variation cancels Eq. (44) if the gauge field transforms as
under the bulk gauge transformation. Since D-brane effective actions must be invariant
under these target space gauge symmetries, this physical argument determines that all the
dependence on the gauge field
should be through the gauge invariant combination
.
- The coupling to the dilaton. The D-brane effective action is an open string tree level action, i.e., the
self-interactions of open strings and their couplings to closed string fields come from conformal field
theory disk amplitudes. Thus, the brane tension should include a
factor coming from the
expectation value of the closed string dilaton
. Both these considerations lead us to consider the
DBI action
where
stands for the D-brane tension.
- The WZ couplings. Dp-branes are charged under the RR potential
. Thus, their effective
actions should include a minimal coupling to the pullback of such form. Such coupling would not be
invariant under the target space gauge transformations (525). To achieve this invariance in a way
compatible with the bulk Bianchi identities (523), the D-brane WZ action must be of the form
where
stands for the corresponding pullbacks of the target space RR potentials
to the world
volume, according to the definition given in Eq. (521). Notice this involves more terms than the mere
minimal coupling to the bulk RR potential
. An important physical consequence of this
fact will be that turning on non-trivial gauge fluxes on the brane can induce non-trivial
lower-dimensional D-brane charges, extending the argument given above for the minimal
coupling [185
]. This property will be discussed in more detail in the second part of this
review. For a discussion on how to extend these couplings to massive type IIA supergravity,
see [255
].
Putting together all previous arguments, one concludes the final form of the bosonic D-brane action
is:
If one views this action as the bosonic truncation of a supersymmetric D-brane, the D-brane charge density
equals its tension in absolute value, i.e.,
. The latter can be determined from first principles
to be [423
, 24
]
Bosonic covariant M5-brane:
The bosonic M5-brane degrees of freedom involve scalar fields and a world
volume 2-form with self-dual field strength. The former are expected to be described by similar
arguments to the ones presented above. The situation with the latter is more problematic given
the tension between Lorentz covariance and the self-duality constraint. This problem has a
fairly long history, starting with electromagnetic duality and the Dirac monopole problem in
Maxwell theory, see [105] and references therein, and more recently, in connection with the
formulation of supergravity theories such as type IIB, with the self-duality of the field strength of
the RR 4-form gauge potential. There are several solutions in the literature based on different
formalisms:
- One natural option is to give-up Lorentz covariance and work with non-manifestly Lorentz
invariant actions. This was the approach followed in [420
] for the M5-brane, building on
previous work [213, 295, 441].
- One can introduce an infinite number of auxiliary (non-dynamical) fields to achieve a covariant
formulation. This is the approach followed in [384, 502, 375, 177, 66, 98, 99, 100].
- One can follow the covariant approach due to Pasti, Sorokin and Tonin (PST-formalism) [416
,
418
], in which a single auxiliary field is introduced in the action with a non-trivial
non-polynomial dependence on it. The resulting action has extra gauge symmetries. These
allow one to recover the structure in [420
] as a gauge fixed version of the PST formalism.
- Another option is to work with a Lagrangian that does not imply the self-duality condition but
allows it, leaving the implementation of this condition to the path integral. This is the approach
followed by Witten [497], which was extended to include non-linear interactions in [140]. The
latter work includes kappa symmetry and a proof that their formalism is equivalent to the PST
one.
In this review, I follow the PST formalism. This assigns the following bosonic action to the
M5-brane [417
]
As in previous effective actions, all the dependence on the scalar fields
is through the bulk fields and
their pullbacks to the six-dimensional world volume. As in D-brane physics, all the dependence on the world
volume gauge potential
is not just simply through its field strength
, but through the gauge
invariant 3-form
The physics behind this is analogous.
describes the ability of open strings to
end on D-branes, whereas
describes the possibility of M2-branes to end on
M5-branes [469, 479].
Its world volume Hodge dual and the tensor
are then defined as
The latter involves an auxiliary field
responsible for keeping covariance and implementing the
self-duality constraint through the second term in the action (49). Its auxiliary nature was
proven in [418, 416], where it was shown that its equation of motion is not independent from
the generalised self-duality condition. The full action also includes a DBI-like term, involving
the induced world volume metric
, and a WZ term, involving the
pullbacks
and
of the 3-form gauge potential and its Hodge dual in
supergravity [11].
Besides being manifestly invariant under six-dimensional world volume diffeomorphisms and ordinary
abelian gauge transformations
, the action (49) is also invariant under the transformation
Given the non-dynamical nature of
, one can always fully remove it from the classical action by gauge
fixing the symmetry (53). It was shown in [417
] that for an M5-brane propagating in Minkowski, the
non-manifest Lorentz invariant formulation in [420
] emerges after gauge fixing (53). This was achieved by
working in the gauge
and
. Since
is a world volume vector, six-dimensional
Lorentz transformations do not preserve this gauge slice. One must use a compensating gauge
transformation (53), which also acts on
. The overall gauge fixed action is invariant under the full
six-dimensional Lorentz group but in a non-linear non-manifestly Lorentz covariant way as discussed
in [420
].
As a final remark, notice the charge density
of the bosonic M5-brane has already been set equal
to its tension
.