ROM2F/2011/13

Electrified branes

Luca Martucci

I.N.F.N. Sezione di Roma “TorVergata” &
Dipartimento di Fisica, Università di Roma “TorVergata”,
Via della Ricerca Scienti ca, 00133 Roma, Italy

Abstract

A geometrical form of the supersymmetry conditions for D-branes on arbitrary
type II supersymmetric backgrounds is derived, as well as the associated BPS
bounds. The treatment is general and allows to consider, for instance, non-
static configurations or D-branes supporting a non-vanishing electric flux,
hence completing previous partial results. In particular, our discussion
clarifies how the notion of calibration can be extended in order to be
applicable to the most general supersymmetric configurations. As an
exemplifying preliminary step, the procedure followed is first applied to
fundamental strings.

e-mail<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Background structures
  3. 3 Fundamental strings
    1. 3.1 Geometric supersymmetry conditions
    2. 3.2 Conserved current and charge
    3. 3.3 Supersymmetry and BPS bound
    4. 3.4 BPS bound and energetics
  4. 4 D-branes
    1. 4.1 Geometric supersymmetry conditions
    2. 4.2 Conserved current and charge
    3. 4.3 Supersymmetry and BPS bound
    4. 4.4 BPS bound and energetics
    5. 4.5 Gauss law, Bianchi identity and equations of motion
  5. 5 Some examples
    1. 5.1 Magnetized D-branes
      1. 5.1.1 Static magnetized D-branes
      2. 5.1.2 Compactifications to ${\rm Mink}_{4}$ and non-static branes
    2. 5.2 Simple D-branes with non-vanishing electric field
      1. 5.2.1 BIons
      2. 5.2.2 Supertubes
  6. 6 Conclusion

##  1 Introduction

String theory is populated by branes of various kinds and their properties are
directly related to the supersymmetric structure of the theory. This deep
interplay has one of its most explicit manifestations in the appearance of
integrable background geometrical structures which are directly related to the
geometry of branes. This relation constitutes an important ingredient in
several applications. For instance, the presence of an integrable complex
structure is naturally associated with holomorphically embedded branes. One
can then often avoid facing the detailed supersymmetric structure, working
directly with its physically relevant geometrical manifestation.

The relation between branes and background geometry can be concretely realized
in terms of calibrations, background forms with particular algebraic and
differential properties – see for instance [1] for a review emphasizing some
of the aspects that are important for applications to string theory. In their
original mathematical definition [2], calibrations are closed forms which
identify volume minimizing submanifolds with special properties. They appeared
in string theory when physicists started studying string theory vacua on
Ricci-flat spaces with special holonomy (i.e. with reduced supersymmetry), and
the associated brane configurations – see for instance [3]. Physically, brane
energy is expected to be subject to a BPS lower bound saturated by
supersymmetric configurations and, in string theory vacua on special holonomy
spaces, the energy of static branes is just given by the volume of the cycle
wrapped by the brane. On the other hand, special holonomy spaces are naturally
equipped with calibrations and indeed these provide a nice geometrical
realization of the expected BPS energetic bound, see e.g. [4].

Clearly, more complicated backgrounds require a modification of the notion of
calibration given in [2]. Still focusing on static settings, the inclusion of
background fluxes which couple minimally to branes has been considered in [5]
and leads to the general receipt that the associated background calibrations
are no-longer closed but rather their exterior derivative is given by the
background flux itself. After this work, several papers followed developing
this idea at various levels of generality, see for instance [6, 7, 8, 9, 10].

On the other hand various branes, and in particular D-branes, can support non-
trivial world-volume fluxes, which contribute to the brane energetics and need
to be taken into account in a proper analysis. The way to incorporate magnetic
world-volume fluxes for static D-branes on static backgrounds has been
identified in [11, 12, 13, 14] and exhibits the relevance of generalized
geometry [15, 16]111For a review for physicists, see for instance [17]. as the
proper framework for describing not only the D-brane geometry but also their
dynamics [18, 19]. Not surprisingly, the results on the D-brane side uniform
well with results the on the closed string sector side [20, 21, 22, 23, 24,
25, 26]. Importantly, it has also become evident that the calibration
structures play a crucial role in proving the integrability properties of
supergravity backgrounds fully coupled to backreacting D-branes and
orientifolds, not only in supersymmetric settings [27] but also in non-
supersymmetric backgrounds which preserve some BPS-like properties [28].

This paper studies the geometrical properties of general supersymmetric
D-branes on general supersymmetric type II backgrounds. In particular,
backgrounds as well as D-branes can be non-static and the latter can support
an electric flux in addition to the magnetic one.222The analogous problem for
M5-branes has been addressed in [29]. The analysis explicitly confirms
generalized geometry as the natural language to be used and indeed we will see
how the background supersymmetry conditions found in [30] within this same
framework are the ones which naturally enter in the geometric characterization
of supersymmetric D-branes and the associated BPS bounds. In particular, our
discussion shows how various notions of calibrations should be extended in
order to uncover non-static configurations and world-volume electric fluxes.
Furthermore, supersymmetry may be broken still preserving the calibrations and
the associated BPS structures, along the lines of what was discussed for
static settings in [28]. In this paper we will not consider this interesting
possibility and the adjectives “BPS” and “supersymmetric” can be considered as
synonymous.

This paper is structured as follows. After a brief summary in section 2 of the
background properties, in section 3 we start by considering the simpler case
of fundamental strings. This will allow us to explain without too many
technical complications the basic conceptual strategy which we will follow
when we will turn to D-branes. These are discussed in section 4 along the same
lines followed in section 3 for fundamental strings. In section 5 we provide a
few simple explicit examples in which the we apply the general results of the
previous sections.

##  2 Background structures

This section briefly reviews the geometric structures characterizing the
background.

Take a supersymmetric type II vacuum, hence characterized by a pair of ten-
dimensional Majorana-Weyl spinors ($\epsilon_{1},\epsilon_{2}$) which satisfy
the supersymmetric Killing spinor equations. In our conventions $\epsilon_{1}$
has positive chirality, $\Gamma_{11}\epsilon_{1}=\epsilon_{1}$, while
$\Gamma_{11}\epsilon_{2}=\mp\epsilon_{2}$ in IIA/IIB respectively. These can
be grouped into the Killing doubled-spinor

$\epsilon:=\left(\begin{array}[]{c}\epsilon_{1}\\\
\epsilon_{2}\end{array}\right)$ (2.1)

The supersymmetric Killing spinor conditions have been analyzed in several
papers, see for instance [6, 7, 9, 31, 30]. For our purposes, the most natural
approach is the one adopted [30], which we follow.

By using $\epsilon$ one can construct several tensorial quantities, which have
special differential properties deriving from the Killing spinor equations.
First of all, one can construct a vector field

$K:=-\frac{1}{2}\bar{\epsilon}\,\Gamma^{M}\epsilon\,\partial_{M}\equiv-\frac{1}{2}(\bar{\epsilon}_{1}\Gamma^{M}\epsilon_{1}+\bar{\epsilon}_{2}\Gamma^{M}\epsilon_{2})\,\partial_{M}$
(2.2)

where we use capital letters from the middle of the alphabet as ten-
dimensional curved indices. By using the Killing spinor equations, $K$ turns
out to be a Killing vector

${\cal L}_{K}g=0$ (2.3)

and actually generates a symmetry of the entire background, hence including
all bosonic supergravity fields and supersymmetric structures. This will be
important for us, since $K$ will identify the naturally selected ‘energy’
which is minimized by the supersymmetric branes. In general $K$ can be either
time-like or null and the overall minus-sign in (2.2) has been introduced in
order for $K$ to be future-pointing.

In addition to the vector $K$, one can introduce a one-form $\omega$ defined
by

$\omega:=-\frac{1}{2}\bar{\epsilon}\,\Gamma_{M}\sigma_{3}\epsilon\,\text{d}x^{M}\equiv-\frac{1}{2}(\bar{\epsilon}_{1}\Gamma_{M}\epsilon_{1}-\bar{\epsilon}_{2}\Gamma_{M}\epsilon_{2})\,\text{d}x^{M}$
(2.4)

Notice that $g(K,K)=-g^{-1}(\omega,\omega)\geq 0$, which follows from the fact
that the vectors $\bar{\epsilon}_{1}\Gamma^{M}\epsilon_{1}$ and
$\bar{\epsilon}_{2}\Gamma^{M}\epsilon_{2}$ are null, so that either $K$ and
$\omega$ are both null or $K$ is time-like and $\omega$ space-like.
Furthermore, $\omega$ is related to $K$ by the differential condition

$\text{d}\omega=-\iota_{K}H$ (2.5)

where $H$ is the (closed) NS-NS three-form which is locally given by the
exterior derivative of the $B$-field, $H=\text{d}B$.

There is one further set of tensors which can be constructed by using
$\epsilon$. These can be organized into a ‘polyform’

$\Psi=\sum_{k}\Psi_{\it(k)}$ (2.6)

with ${\it k}=$even/odd in IIA/IIB and

$\Psi_{\it(k)}=-\frac{1}{{\it k}!}(\bar{\epsilon}_{1}\Gamma_{M_{1}\ldots
M_{k}}\epsilon_{2})\,\text{d}x^{M_{1}}\wedge\ldots\text{d}x^{M_{k}}$ (2.7)

Alternatively, $\Psi$ can be defined by the bi-spinor
$\not{\Psi}=32\epsilon_{1}\otimes\bar{\epsilon}_{2}\Gamma_{11}$.333We use the
supergravity and D-brane conventions described in appendix of the first paper
of Ref. [28] and in [32]. Our conventions and definitions slightly differ from
those of [30]. The dictionary is the following: $H_{\rm here}=-H_{\rm there}$,
$F^{\rm IIA}_{\rm here}=-F^{\rm IIA}_{\rm there}$, $F^{\rm IIB}_{\rm
here}=F^{\rm IIB}_{\rm there}$, $K_{\rm here}=-32K_{\rm there}$, $\omega_{\rm
here}=-32\tilde{K}_{\rm there}$, $\Psi^{\rm IIB}_{\rm here}=-32\Phi^{\rm
IIB}_{\rm there}$, $\Psi^{\rm IIA}_{\rm here}=32\Phi^{\rm IIA}_{\rm there}$.
Notice that $\Psi$ obeys the self-duality relation

$\Psi=*\lambda(\Psi)$ (2.8)

where $\lambda(\Psi_{(k)})=(-)^{\frac{k(k-1)}{2}}\Psi_{(k)}$. Furthermore

$\iota_{K}\Psi+\omega\wedge\Psi=0$ (2.9)

The Killing spinor equations imply that $\Psi$ is linked with the R-R fluxes
by the following differential equations

$\text{d}_{H}(e^{-\phi}\,\Psi)=-\iota_{K}F-\omega\wedge F$ (2.10)

where

$F:=\sum_{\it k}F_{\it(k)}$ (2.11)

is the sum of (even/odd) R-R field strengths (in IIA/IIB) and we have
introduced the $H$-twisted exterior derivative

$\text{d}_{H}:=\text{d}+H\wedge$ (2.12)

which acts on polyforms. Notice that $\text{d}_{H}^{2}=0$ by the Bianchi
identity $\text{d}H=0$. The R-R field strengths are subject to the same self-
duality constraint of $\Psi$: $F=*\lambda(F)$.

Not only the supergravity bosonic fields are invariant under the
diffeomorphism generated by $K$, but also the supersymmetric spinorial
structure. This implies in particular that

${\cal L}_{K}\omega=0\,,\quad{\cal L}_{K}\Psi=0$ (2.13)

as can be derived from (2.5) and (2.10) by using the Bianchi identities.

Notice that the differential conditions reviewed above are necessary but not
generically sufficient for guaranteeing the background to be supersymmetry.
Rather, they must be supplemented by other conditions [30]. However, the
differential conditions above are the only ones that we will need when
discussing fundamental strings and D-branes.

In discussing D-branes, it will be much more natural to use the language of
generalized geometry [15, 16], see [17] for a review. In particular, the
combination

${\cal K}=K+\omega$ (2.14)

can be seen as a generalized vector in $TM\oplus T^{*}M$, known as generalized
tangent bundle of the ten-dimensional manifold $M$. On this there exists a
natural inner product defined by

${\cal I}(X,X)=\iota_{v}\eta\qquad\text{for}\qquad X=v+\eta\in TM\oplus
T^{*}M$ (2.15)

which induces a canonical $O(10,10)$ structure on $TM\oplus T^{*}M$. Polyforms
can be seen as $O(10,10)$ spinors and the Clifford action of a generalized
vector $X=v+\eta$ on them is given by the operator

$X\cdot:=\iota_{v}+\eta\wedge$ (2.16)

Furthermore, by using the ordinary metric $g$, one can introduce the
generalized metric444We are using the SO(10,10) frame in which there are no
off-diagonal elements in ${\cal G}$. They can be obtained by a general
$b$-twist which would however shift $H\rightarrow H-\text{d}b$ – see for
instance [17].

${\cal G}=\left(\begin{array}[]{cc}g&0\\\ 0&g^{-1}\end{array}\right)$ (2.17)

which further reduces the generalized structure group to $SO(1,9)\times
SO(1,9)$.

${\cal K}$ is null in both ${\cal I}$ and ${\cal G}$:

${\cal I}({\cal K},{\cal K})={\cal G}({\cal K},{\cal K})=0$ (2.18)

Furthermore, $\Psi$ can be seen as a spinor in $TM\oplus T^{*}M$ and (2.9) can
be translated by saying that $\Psi$ is annihilated by the Clifford action of
${\cal K}$:

${\cal K}\cdot\Psi=0$ (2.19)

By adopting this viewpoint, one can write (2.10) as

$\text{d}_{H}(e^{-\phi}\Psi)=-{\cal K}\cdot F$ (2.20)

and check that ${\cal K}$ generates a generalized isometry which preserves the
bosonic background as well as the spinorial structure. For instance

$\displaystyle{\cal L}^{H}_{\cal K}(e^{-\phi}\Psi)=0$ (2.21)
$\displaystyle{\cal L}^{H}_{\cal K}F={\cal K}\cdot\text{d}_{H}F=0$ (2.22)

where we have used the Bianchi identity $\text{d}_{H}F=0$ – we will later come
back on the effect of localized sources – and the generalized Lie derivative
on forms is defined by

${\cal L}^{H}_{\cal K}:=\\{\text{d}_{H},{\cal K}\\}$ (2.23)

In conclusion, ${\cal K}$ can be considered as a generalized Killing vector.
${\cal K}$ and $\Psi$ will play an important role when discussing the
energetics of D-branes.

##  3 Fundamental strings

In this section we consider fundamental (F1) strings in general supersymmetric
backgrounds, by using the framework described in section 2. We study the
supersymmetry conditions and we derive the associated BPS bound and its
relation to background calibrations. This section provides also a useful warm-
up for the discussion on D-branes presented in section 4, which will follow a
path which is conceptually identical although slightly more difficult from the
technical point of view. However, this section and section 4 are almost
completely independent and the reader more interested on D-branes may safely
jump directly to section 4.

Let us briefly review some basic facts about the F1 world-volume theory,
basically to fix our notation and conventions. The action is given by

$S_{\rm F1}=-\mu_{\rm F1}\int_{\cal S}\text{d}^{2}\xi\sqrt{-\det g|_{\cal
S}}\,+\mu_{\rm F1}\int_{\cal S}B$ (3.1)

where $\xi^{\alpha}$ are world-sheet coordinates, $\mu_{\rm
F1}=1/(2\pi\alpha^{\prime})$, ${\cal S}$ is the two-dimensional submanifold
wrapped by the F1-string and $g|_{\cal S}$ refers to the pull-back of bulk
metric to ${\cal S}$. In the following we will sometimes use an economic
notation for pulled-back tensors in components, leaving implicit the pull-back
operation $|_{\cal S}$. For instance, $B_{\alpha\beta}\equiv
B_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}\equiv(B|_{\cal
S})_{\alpha\beta}$.

It will result useful to explicitly split the action in terms of the Nambu-
Goto and Wess-Zumino terms: $S=\int\text{d}^{2}\xi\,{\cal
L}=\int\text{d}^{2}\xi({\cal L}_{\rm NG}+{\cal L}_{\rm WZ})$, with

${\cal L}_{\rm NG}=-\mu_{\rm F1}\sqrt{-\det g|_{\cal S}}\,,\qquad{\cal L}_{\rm
WZ}=\frac{\mu_{\rm F1}}{2}\,\epsilon^{\alpha\beta}B_{\alpha\beta}$ (3.2)

###  3.1 Geometric supersymmetry conditions

We now study the conditions that need to be imposed on supersymmetric
fundamental strings and how they can be written in terms of the background
tensors characterizing the bulk supersymmetry.

Let us start from the supersymmetry condition in spinorial form. This reads

$\Gamma_{\rm F1}\epsilon=\epsilon$ (3.3)

where $\epsilon$ is the doubled-spinor introduced in (2.1) and $\Gamma_{\rm
F1}$ is the $\kappa$-symmetry operator [33]

$\Gamma_{\rm F1}=\frac{1}{2\sqrt{-\det g|_{\cal
S}}}\,\epsilon^{\alpha\beta}\Gamma_{\alpha\beta}\otimes\sigma_{3}$ (3.4)

where, according to our general conventions,
$\Gamma_{\alpha\beta}=\Gamma_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}$.
Our aim is to rewrite (3.3) in a more geometrical way. This is readily
obtained just by observing that (3.3) implies that
$\bar{\epsilon}\,\Gamma^{M}\Gamma_{\rm
F1}\epsilon=\bar{\epsilon}\,\Gamma^{M}\epsilon$. In turn, by using the
definitions (2.2) and (2.4), it is easy to see that this latter condition can
be written as

$(\text{d}X^{M}\wedge\omega)|_{\cal S}=K^{M}\sqrt{-\det g|_{\cal
S}}\,\text{d}^{2}\xi$ (3.5)

We then find that (3.5) is a necessary condition for the F1-string to be
supersymmetric. On the other hand, we will show in the following sections that
(3.5) is in fact also sufficient for it. Hence, (3.5) is completely equivalent
to (3.3) and can be taken as defining supersymmetric F1-strings.555Actually,
as we will see, (3.5) is redundant and only one particular ‘component’
provides the minimal necessary and sufficient condition for the F1 to be BPS.

We can write (3.5) in an alternative way. Consider the vector field
$V=V^{M}\partial_{M}$ defined by

$V^{M}=\frac{\mu_{{\rm F}1}}{2\sqrt{-\det g|_{\cal
S}}}\,\epsilon^{\alpha\beta}(\text{d}X^{M}\wedge\omega)_{\alpha\beta}$ (3.6)

$V$ is tangent to the F1 world-sheet, $V\in T{\cal S}\subset(TM)|_{\cal S}$.
Hence the supersymmetry condition (3.5) states that $V$ must be equal to the
Killing vector:

$V=K\qquad\text{(on ${\cal S}$)}$ (3.7)

Cleary, (3.7) implies that $K$ is tangent to the world-sheet ${\cal S}$ of
supersymmetric F1-strings. This means that ${\cal S}$ does not break the
symmetry generated by $K$, as expected. This condition can be alternatively
formulated by saying that the embedding ${\cal S}$ is a stationary
configuration with respect to the ‘time’ $t$ (which can actually be either
time-like or light-like) generated by $K=\partial_{t}$. In particular, (3.5)
implies that for supersymmetric strings

$(\text{d}t\wedge\omega)|_{\cal S}=\sqrt{-\det g|_{\cal S}}\,\text{d}^{2}\xi$
(3.8)

Hence, for supersymmetric F1-strings we can write

$S_{\rm F1}=-\mu_{\rm F1}\int_{\cal S}(\text{d}t\wedge\omega-B)$ (3.9)

On general grounds, supersymmetric configurations are expected to minimize
some BPS-like lower bound on some appropriately defined energy. This is indeed
true, as discussed in the following sections.

###  3.2 Conserved current and charge

We first recall some well known results about symmetries and conserved
currents.

Noether’s theorem states that one can construct a conserved current associated
with any symmetry under which the action is invariant. More explicitly, take a
generic Lagrangian ${\cal L}(\phi^{I},\partial\phi^{I})$ and suppose that it
is invariant, up to a total derivative $\partial_{\alpha}\Lambda^{\alpha}$,
under an infinitesimal deformation $\delta\phi^{I}$:

$\delta{\cal L}=\partial_{\alpha}\Lambda^{\alpha}$ (3.10)

Notice that $\Lambda^{\alpha}$ is defined up to divergence-less additional
terms.

Then, one can construct the current

$j^{\alpha}=-\delta\phi^{I}\,\frac{\partial{\cal
L}}{\partial(\partial_{\alpha}\phi^{I})}+\Lambda^{\alpha}$ (3.11)

which is conserved, $\partial_{\alpha}j^{\alpha}=0$ (on-shell).

Let us now apply Noether’s theorem to our problem. Take the transformation

$\delta_{K}x^{M}=K^{M}$ (3.12)

which corresponds to shifting the fundamental string in the direction of the
Killing vector $K$ defined in (2.2). Consider first ${\cal L}_{\rm NG}$, see
(3.2). Being ${\cal L}_{K}g=0$, we obviously have $\delta_{K}{\cal L}_{\rm
NG}=0$. Hence, we can construct the following associated current

$j^{\alpha}_{\rm NG}=\mu_{\rm F1}\sqrt{-\det g|_{\cal
S}}\,g^{\alpha\beta}K_{\beta}$ (3.13)

Consider now ${\cal L}_{\rm WZ}$. First notice that, by (2.5), we have

$\displaystyle{\cal L}_{K}B$ $\displaystyle=$
$\displaystyle\iota_{K}H+\text{d}(\iota_{K}B)$ (3.14) $\displaystyle=$
$\displaystyle\text{d}(\iota_{K}B-\omega)$ (3.15)

Hence, we can still apply Noether’s procedure but now we have to be careful
since ${\cal L}_{\rm CS}$ is only invariant up to a total derivative.
Comparing with (3.10), we can take

$\Lambda^{\alpha}_{\rm WZ}=\mu_{\rm
F1}\,\epsilon^{\alpha\beta}(\iota_{K}B-\omega)_{\beta}$ (3.16)

Hence, the corresponding current is given by

$\displaystyle j^{\alpha}_{\rm WZ}$ $\displaystyle=$ $\displaystyle-\mu_{\rm
F1}\,\epsilon^{\alpha\beta}(\iota_{K}B)_{\beta}+\Delta^{\alpha}_{\rm WZ}$
(3.17) $\displaystyle=$ $\displaystyle-\mu_{\rm
F1}\,\epsilon^{\alpha\beta}\omega_{\beta}$ (3.18)

Let us stress that, in deriving $j^{\alpha}_{\rm WZ}$, we have crucially used
the background supersymmetry condition (2.5).

We can then conclude that the conserved current associated with the world-
volume symmetry induced by the bulk Killing vector $K$ is given by the sum of
$j^{\alpha}_{\rm NG}$ and $j^{\alpha}_{\rm WZ}$:

$j_{\rm F1}^{\alpha}=j^{\alpha}_{\rm NG}+j^{\alpha}_{\rm WZ}=\mu_{\rm
F1}\big{(}\sqrt{-\det g|_{\cal
S}}\,g^{\alpha\beta}K_{\beta}-\epsilon^{\alpha\beta}\omega_{\beta}\big{)}$
(3.19)

Suppose now to split the world-sheet coordinates as
$\xi^{\alpha}=(\tau,\sigma)$. More precisely, consider world-sheet embeddings
which have structure ${\cal S}=\mathbb{R}\times\Sigma$, where $\mathbb{R}$
parametrizes a time-like world-sheet coordinate $\tau$ whereas $\Sigma$ is a
space-like one-dimensional submanifold parametrized by the space-like
coordinate $\sigma$. Hence, we can use the conserved current $j^{\alpha}_{\rm
F1}$ to define the conserved charge

$\displaystyle\Delta_{\rm F1}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\text{d}\Sigma_{\alpha}\,j^{\alpha}_{\rm
F1}\equiv\int_{\Sigma}\text{d}\sigma\,j^{\tau}_{\rm F1}$ (3.20)
$\displaystyle=$
$\displaystyle-\int_{\Sigma}\big{[}\text{d}\sigma\,g(P,K)+\mu_{\rm
F1}\,\omega\big{]}$ (3.21)

where we have introduced the vector

$P^{M}=-\mu_{\rm F1}\sqrt{-\det g|_{\cal
S}}\,g^{\alpha\tau}\partial_{\alpha}x^{M}\qquad\Rightarrow\qquad j^{\tau}_{\rm
NG}=-g(P,K)$ (3.22)

$P^{M}$ is a vector tangent to ${\cal S}$ and can be identified with the
‘kinematical’ gauge-invariant momentum conjugated to the embedding field
$x^{M}(\xi)$.

###  3.3 Supersymmetry and BPS bound

Experience with supersymmetric theories leads us to expect that the energy
obeys a BPS lower bound which is saturated exactly when the F1-string is
supersymmetric. Standard arguments are usually applied to static situations
and lead to the identification of various kinds of background calibrations,
which naturally realize the BPS bound.

On the other hand, in our analysis we are considering the most generic
supersymmetric background and F1-string configuration. Hence, we need to
extend the usual approach to calibrations. As we now show, such a
generalization is actually very natural. Furthermore, our procedure
straightforwardly generalizes to other branes, as we will explicitly see for
the case of D-branes.

Let us start again from the supersymmetry condition (3.3). By applying the
split ${\cal S}=\mathbb{R}\times\Sigma$ we can write

$\Gamma_{\rm F1}=\frac{1}{\mu_{\rm F1}\sqrt{g|_{\Sigma}}}\not{P}\gamma_{\rm
F1}=-\frac{1}{\mu_{\rm F1}\sqrt{g|_{\Sigma}}}\gamma_{\rm F1}\not{P}$ (3.23)

where $\not{P}=P^{M}\Gamma_{M}$, with $P^{M}$ defined in (3.22), and

$\gamma_{\rm F1}=\frac{1}{\mu_{\rm
F1}\sqrt{g|_{\Sigma}}}\,\Gamma_{\sigma}\otimes\sigma_{3}$ (3.24)

Furthermore, notice that if the string is not collapsed then $C\not{P}$ is a
positive definite matrix, in the sense that

$\chi^{T}C\not{P}\chi\equiv\bar{\chi}\not{P}\chi>0$ (3.25)

for any (non-vanishing) doubled Majorana spinor $\chi$. Indeed, from the
definition (3.22) it is easy to see that $P$ is time-like, $g(P,P)=-\mu_{\rm
F1}^{2}g_{\sigma\sigma}$, whenever the string is not collapsed. In the real
representation that we are using, the charge-conjugation matrix is simply
given by $C=\Gamma^{\underline{0}}$. Being $P$ future pointing by
construction, we can locally choose a frame $e_{\underline{M}}$ such that
$e_{\underline{0}}\propto P$, so that
$C\not{P}=P^{\underline{0}}\mathbbm{1}>0$, which immediately leads to (3.25).
Clearly, such a conclusion cannot depend on the choice of frame and then is
valid in general.

We can now apply the general inequality (3.25) to the spinor

$\chi=(\mathbbm{1}-\Gamma_{\rm F1})\epsilon$ (3.26)

Using $\Gamma^{T}_{\rm F1}=-C^{-1}\Gamma_{\rm F1}C$, a few manipulations allow
to rewrite (3.25) with (3.26) as

$\bar{\epsilon}\not{P}\epsilon\,\geq\,\bar{\epsilon}\not{P}\Gamma_{\rm
F1}\epsilon$ (3.27)

In turn, by taking into account (3.23) and the definitions (2.2) and (2.4),
this can be rewritten in more geometric way as

$-g(P,K)\,\text{d}\sigma\geq\mu_{\rm F1}\,\omega|_{\Sigma}$ (3.28)

which should be read $-g(P,K)\geq\mu_{\rm F1}\omega_{\sigma}$. Recalling
(3.20), we see that (3.28) provides exactly the expected local bound

$\Delta_{\rm F1}\geq 0$ (3.29)

On the other hand, it is clear by construction that (3.29) is saturated if and
only if the F1-string satisfies the supersymmetry condition (3.3). Hence, the
supersymmetry condition can be alternatively written as

$\text{SUSY
F1-string}\qquad\Leftrightarrow\qquad-g(P,K)\,\text{d}\sigma=\mu_{\rm
F1}\,\omega|_{\Sigma}$ (3.30)

We then conclude that $\omega$ can be seen as the proper generalization of the
more standard calibrations for static backgrounds and static world-sheet
configurations. In particular, (3.30) substitutes the usual calibration bound
for minimally coupled static branes [5], where the the l.h.s. simply reduces
to the volume density.

Notice that in order to interpret the algebraic bound (3.29) in terms of a
conserved charge, a key role is played by the construction of the currents
$j^{\alpha}_{\rm NG}$ and $j^{\alpha}_{\rm WZ}$, which in turn uses
differential background conditions. In particular for $j^{\alpha}_{\rm WZ}$ we
have crucially used the bulk supersymmetry condition (2.5). Without such
condition $S_{\rm WZ}$ would not be invariant and no current could be
constructed. In a sense, consistency of the F1 world-sheet theory with
supersymmetry demands that the background fields $\omega$ and $K$ must be
related by the differential condition (2.5).

The discussion followed in this section makes explicit use of the time-plus-
space split, but can be easily related to the covariant condition presented in
(3.5). Indeed, (3.30) can be obtained as the equation resulting from the
contraction of (3.5) with $P$. In terms of the vector $V$ introduced in (3.6),
the BPS condition can be written as $g(P,K)=g(P,V)$ and the BPS bound (3.29)
can be rephrased as

$|g(P,K)|\geq|g(P,V)|$ (3.31)

We then see that, as already anticipated, the conditions encoded in (3.5) are
not just necessary but also sufficient for characterizing BPS F1-strings.
Actually, just the projection of $(\ref{F1form2})$ along $P$ is sufficient and
the remaining components in the orthogonal directions follow from this.

###  3.4 BPS bound and energetics

In order to relate the above analysis to a more standard Hamiltonian
treatment, pick-up a field $B$ such that $\text{d}\iota_{K}B=-\iota_{K}H=0$
or, in other words, ${\cal L}_{K}B=0$. This is consistent with the fact that
${\cal L}_{K}H=0$ and $\iota_{K}H$ is exact, as required by the supersymmetry
condition (2.5). In this case $\delta_{K}{\cal L}_{\rm F1}$ is exactly
invariant and we can apply Noether’s procedure with $\Lambda^{\alpha}_{\rm
CS}\equiv 0$.666This particular choice of $B$ is equivalent to correcting
$\Lambda^{\alpha}_{\rm F1}$ given in (3.16) by a divergence-less term which
make it vanish – see comment after (3.10). It is then easy to see that the
conserved charge is simply given by $-g(\hat{P},K)$, the component along $K$
of the canonical momentum

$\hat{P}_{M}=\frac{\partial{\cal L}_{\rm
F1}}{\partial(\partial_{\tau}x^{M})}=P_{M}+\mu_{\rm F1}(\iota_{M}B)|_{\Sigma}$
(3.32)

which, compared to the gauge-invariant momentum $P_{M}$, incorporates the
effect of the WZ-term too. Hence, we get the conserved energy

$E_{\rm F1}:=-\int_{\Sigma}\text{d}\sigma\,g(\hat{P},K)$ (3.33)

The local bound (3.28) can now be translated into a BPS bound on the energy
$E_{\rm F1}$

$E_{\rm F1}\geq E^{\rm BPS}_{\rm F1}$ (3.34)

where

$E^{\rm BPS}_{\rm F1}:=\int_{\Sigma}(\omega-\iota_{K}B)$ (3.35)

Notice that, thanks to the background supersymmetry condition (2.5), $E^{\rm
BPS}_{\rm F1}$ is a topological quantity which does not change under
continuous deformations of $\Sigma$. Indeed, suppose that $\Sigma$ can be
deformed to $\Sigma^{\prime}=\Sigma+\partial{\cal C}$, for some chain ${\cal
C}$. Hence, by using (2.5) and the fact that $\text{d}\iota_{K}B=-\iota_{K}H$,
it is easy to see that

$\int_{\Sigma^{\prime}}(\omega-\iota_{K}B)=\int_{{\cal
C}}[\text{d}\omega-\text{d}(\iota_{K}B)]+\int_{\Sigma}(\omega-\iota_{K}B)=\int_{\Sigma}(\omega-\iota_{K}B)$
(3.36)

In this sense, $E^{\rm BPS}_{\rm F1}$ can be seen as the central charge which
extremizes the energy canonically associated with the Killing vector $K$. In
fact, by (2.5) we may pick-up $B$ such that $\iota_{K}B=\omega$. With this
choice, $E_{\rm F1}\equiv\Delta_{\rm F1}$ and $E^{\rm BPS}_{\rm F1}\equiv 0$.

By imposing a partial static gauge in which $\tau$ is identified with the
(time- or light-like) coordinate $t$ generated by $K$, $-g(\hat{P},K)$
coincides with the canonical Hamiltonian density. In this case $E_{\rm F1}$
would indeed coincide with the canonical energy, along the lines of what
happens for (generalized) calibrations [5] for static backgrounds and branes.
See also [7, 8, 9] for analogous conclusions obtained by starting from the
local type II supersymmetry algebra.

##  4 D-branes

We now turn to D-branes. The line we will follow is practically identical to
one for F1-strings, up to some technical complication due to the more
sophisticated D-brane world-volume theory.

The effective theory governing the bosonic sector of a D$p$-brane wrapping a
$(p+1)$-cycle ${\cal S}$ is given by Dirac-Born-Infeld (DBI) and Chern-Simons
(CS) terms

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int_{\cal
S}\text{d}^{p+1}\xi\,{\cal L}_{\rm DBI}+\int_{\cal S}\text{d}^{p+1}\xi\,{\cal
L}_{\rm CS}$ (4.1) $\displaystyle=$ $\displaystyle-\mu_{{\rm D}p}\int_{\cal
S}\text{d}^{p+1}\xi e^{-\phi}\sqrt{-\det(g|_{\cal S}+{\cal F})}+\mu_{{\rm
D}p}\int_{\cal S}C\wedge e^{{\cal F}}$ (4.2)

where $\mu_{{\rm D}p}=(2\pi)^{p}(\alpha^{\prime})^{-\frac{p+1}{2}}$ and ${\cal
F}$ is the gauge-invariant world-volume field-strength, which satisfies the
Bianchi identity $\text{d}{\cal F}=H|_{\cal S}$.

As we will see, generalized geometry will naturally enter the description and
will be useful to make more manifest the formal analogy between D-branes and
F1-strings.

###  4.1 Geometric supersymmetry conditions

As in the discussion for fundamental strings, let us start from the
supersymmetry conditions for a D$p$-brane in its spinorial form (see e.g. [34,
32])

$\Gamma_{{\rm D}p}\epsilon=\epsilon$ (4.3)

where $\Gamma_{{\rm D}p}$ acts on type II doubled-spinors as

$\Gamma_{{\rm D}p}=\left(\begin{array}[]{cc}0&\hat{\Gamma}_{{\rm D}p}\\\
\hat{\Gamma}^{-1}_{{\rm D}p}&0\end{array}\right)$ (4.4)

with

$\hat{\Gamma}_{\text{D}p}=\frac{1}{\sqrt{-\det(g|_{\cal S}+{\cal
F})}}\sum_{2l+s=p+1}\frac{\epsilon^{\alpha_{1}\ldots\alpha_{2l}\beta_{1}\ldots\beta_{s}}}{l!s!2^{l}}{\cal
F}_{\alpha_{1}\alpha_{2}}\cdots{\cal
F}_{\alpha_{2l-1}\alpha_{2l}}\Gamma_{\beta_{1}\ldots\beta_{s}}\ .$ (4.5)

and $\hat{\Gamma}^{-1}_{{\rm D}p}=(-)^{p}C^{-1}\hat{\Gamma}^{T}_{{\rm D}p}C$.
As for fundamental strings, we would like now to re-express (4.3) in a more
geometrical way, which highlights its physical implications.

These can be obtained by noticing that the spinorial supersymmetry condition
(4.3) implies in particular $\bar{\epsilon}(\Gamma^{M}\Gamma_{{\rm
D}p})\epsilon=\bar{\epsilon}\,\Gamma^{M}\epsilon$ and
$\bar{\epsilon}(\Gamma_{M}\sigma_{3}\Gamma_{{\rm
D}p})\epsilon=\bar{\epsilon}(\Gamma_{M}\sigma_{3})\epsilon$, which can be
translated as follows in terms of the bulk structures $K$, $\omega$ and
$\Psi$:

$\displaystyle[(\text{d}X^{M}\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm
top}$ $\displaystyle=$ $\displaystyle K^{M}\sqrt{-\det(g|_{\cal S}+{\cal
F})}\,\text{d}^{p+1}\xi$ (4.6) $\displaystyle[(\iota_{M}\Psi)|_{\cal S}\wedge
e^{\cal F}]_{\rm top}$ $\displaystyle=$
$\displaystyle\omega_{M}\sqrt{-\det(g|_{\cal S}+{\cal F})}\,\text{d}^{p+1}\xi$
(4.7)

We then obtain that (4.6) are necessary conditions for the D-brane to be
supersymmetric. In fact, as it will be clear from the discussion of the
following sections, (4.6) are also sufficient and then (4.6) provides a
complete, actually redundant, set of conditions which a supersymmetric D-brane
must satisfy.

With the help of generalized geometry one can rewrite (4.6) in more inspiring
way. First take the following generalized vector in $(TM\oplus T^{*}M)|_{\cal
S}$:

${\cal V}=\frac{\epsilon^{\alpha_{0}\ldots\alpha_{p}}}{\sqrt{-\det{\cal
M}}}\big{\\{}[(\text{d}x^{M}\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{\alpha_{0}\ldots\alpha_{p}}\partial_{M}+[(\iota_{M}\Psi)|_{\cal S}\wedge
e^{\cal F}]_{\alpha_{0}\ldots\alpha_{p}}\text{d}x^{M}\big{\\}}$ (4.8)

where we have introduced the shorthand notation

${\cal M}:=g|_{\cal S}+{\cal F}$ (4.9)

This is the D-brane counterpart of the vector $V$ introduced in (3.6) for
F1-strings. It is easy to see that ${\cal V}$ is tangent to $({\cal S},{\cal
F})$ in the generalized sense, i.e. it takes values in the D-brane generalized
tangent bundle $T_{({\cal S},{\cal F})}$, which is defined as [16]

$T_{({\cal S},{\cal F})}=\\{v+\eta\in T{\cal S}\oplus T^{*}M|_{\cal
S}\,:\,\eta|_{\cal S}=\iota_{v}{\cal F}\\}$ (4.10)

Then, it is easy to see that the supersymmetry condition (4.6) is equivalent
to

${\cal V}={\cal K}$ (4.11)

where ${\cal K}$ is given in (2.14). It is clear that (4.11) necessary
requires that

${\cal K}\in T_{({\cal S},{\cal F})}$ (4.12)

This condition can be interpreted as the condition that the D-brane is
stationary with respect to ${\cal K}$. More in detail, this incorporates the
following two conditions. First, $K$ is tangent to ${\cal S}$, $K\in T{\cal
S}$, and then ${\cal S}$ is invariant under the symmetry generated by $K$.
Second, $K$ identifies an naturally selected electric flux component ${\cal
E}=\iota_{K}{\cal F}$, which is completely fixed in terms of background one-
form $\omega$:

${\cal E}=\omega|_{\cal S}$ (4.13)

In particular, this implies that there cannot be supersymmetric electric field
${\cal E}=\iota_{K}{\cal F}\neq 0$ if the background supersymmetry has
$\omega=0$.777As an aside remark, let us mention that, although so far we have
treated D-branes as probes, the same conditions must hold in the case of
backreacting D-branes too. Notice in particular that the condition (4.12) for
all localized RR sources in the background is equivalent to the requirement
that ${\cal K}\cdot j_{\rm loc}=0$, where $j_{\rm loc}\simeq\delta({\cal
S})\wedge e^{-{\cal F}}$ is the current associated with backreacting D-branes.
The same current appears on the r.h.s of the R-R Bianchi identity
$\text{d}_{H}F=j_{\rm loc}$, implying (2.22) even in presence of
supersymmetric back-reacting D-branes. From the Bianchi identity
$\text{d}{\cal F}=H|_{\cal S}$ and the bulk equation (2.5) it immediately
follows that ${\cal F}$ is stationary too: ${\cal L}_{K}{\cal F}=0$.

Finally, notice that by introducing the ‘time’ $t$ (which, actually, can be
light-like) defined by $K=\partial_{t}$, the first of (4.6) implies the
following value for the D-brane action evaluated on the BPS configuration

$S^{\rm BPS}_{{\rm D}p}=-\mu_{{\rm D}p}\int_{\cal S}(\text{d}t\wedge
e^{-\phi}\Psi-C)\wedge e^{{\cal F}}$ (4.14)

In the next sections we will discuss the bounds associated with the BPS
condition.

###  4.2 Conserved current and charge

In order to discuss the the BPS bound associated with the supersymmetry
conditions we must identify the proper conserved current and charge that enter
the bound. Of course, these quantities are associated with the Killing vector
$K$.

The bulk symmetry generated by $K$ induces a symmetry of would-volume theory.
The action of such a symmetry on the embedding world-volume fields
$x^{M}(\xi)$ is given by

$\delta_{K}x^{M}=K^{M}$ (4.15)

Being ${\cal L}_{K}g=0$, the pulled-back metric $g|_{\cal S}$ is clearly
invariant under (4.15). On the other hand, $B$ is not in general invariant
under the symmetry generated by $K$, but rather we have (3.14). Hence ${\cal
F}$, which can be locally written as $\text{d}A+B|_{\cal S}$, is generically
not automatically invariant under (4.15). On the other hand, it should be
clear from (3.14) that we can compensate (4.15) by a transformation of the
world-volume gauge-field

$\delta_{K}A=(\omega-\iota_{K}B)|_{\cal S}$ (4.16)

so that $\delta_{K}{\cal F}=0$.

With this definition of $\delta_{K}A$, the DBI lagrangian ${\cal L}_{\rm DBI}$
in (4.1) is manifestly invariant under the symmetry generated by $K$:
$\delta_{K}{\cal L}_{\rm DBI}=0$. By following Noether’s procedure, recalled
in section 3.2, we can then construct the following current

$j^{\alpha}_{\rm DBI}=\mu_{{\rm D}p}\,e^{-\phi}\sqrt{-\det{\cal
M}}\,\big{(}{\cal M}^{(\beta\alpha)}K_{\beta}+{\cal
M}^{[\beta\alpha]}\omega_{\beta}\big{)}$ (4.17)

Take now the CS Lagrangian ${\cal L}_{\rm CS}$ in (4.1). By using (2.10), it
is possible to see that ${\cal L}_{\rm CS}$ tranforms into a total derivative
$\delta_{K}{\cal L}_{\rm CS}=\partial_{\alpha}\Lambda^{\alpha}_{\rm CS}$ under
(4.15) and (4.16), with

$\Lambda^{\alpha}_{\rm CS}=\frac{\mu_{{\rm
D}p}}{p!}\,\epsilon^{\alpha\beta_{1}\ldots\beta_{p}}[({\cal K}\cdot
C-e^{-\phi}\Psi)|_{\cal S}\wedge e^{{\cal
F}}\big{]}_{\beta_{1}\ldots\beta_{p}}$ (4.18)

where ${\cal K}$ is the generalized Killing vector defined in (2.14) and
${\cal K}\cdot$ acts as in (2.16). Then, Noether’s procedure gives the current

$j^{\alpha}_{\rm CS}=-\frac{\mu_{{\rm
D}p}}{p!}\,\epsilon^{\alpha\beta_{1}\ldots\beta_{p}}[(e^{-\phi}\Psi)|_{\cal
S}\wedge e^{{\cal F}}\big{]}_{\beta_{1}\ldots\beta_{p}}$ (4.19)

Notice that the bulk differential condition (2.10) has been crucial in
deriving this expression.

The total conserved current is then given by

$j^{\alpha}_{{\rm D}p}=j^{\alpha}_{\rm DBI}+j^{\alpha}_{\rm CS}$ (4.20)

Consider now a time-plus-space split ${\cal S}=\mathbb{R}\times\Sigma$, with
adapted coordinate $(\tau,\sigma^{a})$, $a=1,\ldots,p$, where $\Sigma$ is the
space-like surface (alias space-volume) spanning ${\cal S}$, and let us define
the split

${\cal F}=\text{d}\tau\wedge{\cal E}+{\cal F}_{\rm
mg}\quad\Rightarrow\quad{\cal F}_{\rm mg}:={\cal F}|_{\Sigma}$ (4.21)

in terms of the electric field ${\cal E}$ and the magnetic component ${\cal
F}_{\rm mg}$.

We can then construct the conserved charge

$\displaystyle\Delta_{{\rm D}p}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\text{d}\Sigma_{\alpha}\,j^{\alpha}_{{\rm
D}p}\equiv\int_{\Sigma}\text{d}^{p}\sigma\,j^{\tau}_{{\rm D}p}$ (4.22)
$\displaystyle=$ $\displaystyle-\int_{\Sigma}\text{d}^{p}\sigma\,{\cal
G}({\cal P},{\cal K})-\mu_{{\rm
D}p}\int_{\Sigma}(e^{-\phi}\Psi)|_{\Sigma}\wedge e^{{\cal F}_{\rm mg}}$ (4.23)

Here we have used the generalized metric defined in (2.17). Furthermore, in
the last line we have introduced the ‘generalized momentum’, which is a
generalized vector defined by

${\cal
P}:=P^{M}\partial_{M}+g_{MN}\partial_{\alpha}x^{N}\Pi^{\alpha}\text{d}x^{M}$
(4.24)

where

$\displaystyle P^{M}$ $\displaystyle:=$ $\displaystyle-\mu_{{\rm
D}p}\,e^{-\phi}\sqrt{-\det{\cal M}}\,({\cal
M}^{-1})^{(\alpha\tau)}\partial_{\alpha}X^{M}$ (4.25)
$\displaystyle\Pi^{\alpha}$ $\displaystyle:=$ $\displaystyle-\mu_{{\rm
D}p}\,e^{-\phi}\sqrt{-\det{\cal M}}\,({\cal M}^{-1})^{[\alpha\tau]}$ (4.26)

can be considered as the kinematical gauge-invariant momenta associated with
$x^{M}$ and $A_{\alpha}$. It is easy to check that ${\cal P}$ belongs to the
generalized tangent space $T_{({\cal S},{\cal F})}$ defined in (4.10) and,
moreover, for non collapsed D-branes it is always ‘time-like’ with respect to
the generalized metric ${\cal G}$:

${\cal G}({\cal P},{\cal
P})=P^{M}P_{M}+h_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}=-\mu^{2}_{p}\,e^{-2\phi}\det
m$ (4.27)

where

$m:=g|_{\Sigma}+{\cal F}_{\rm mg}$ (4.28)

The identity (4.27) corresponds to the superhamiltonian constraint originating
from the world-volume time-like diffeomorphism invariance.888The supermomentum
constraint, which is associated with the space-like diffeomorphism invariance,
can be equally elegantly written as ${\cal P}\in T^{\perp}_{(\Sigma,{\cal
F}_{\rm mg})}$, where $T^{\perp}_{(\Sigma,{\cal F}_{\rm mg})}$ is the
orthogonal complement of the generalized tangent bundle $T_{(\Sigma,{\cal
F}_{\rm mg})}$ in $(TM\oplus T^{*}M)|_{\cal S}$ with respect to the
generalized metric ${\cal G}$. Furthermore,

${\cal I}({\cal P},{\cal P})=P_{\alpha}\Pi^{\alpha}=0$ (4.29)

consistently the fact that ${\cal P}$ belongs to $T_{({\cal S},{\cal F})}$,
which is maximally isotropic with respect to ${\cal I}$.

We see that by using the language of generalized geometry the conserved charge
$\Delta_{{\rm D}p}$ has a structure very similar to the corresponding
conserved charge for F1-strings found in (3.20). As explained in section 3.3,
$\omega$ can be interpreted as a calibration for fundamental strings. Indeed,
a similar interpretation holds for $e^{-\phi}\Psi$ too, as we will presently
explain.

###  4.3 Supersymmetry and BPS bound

We start by considering the following operator acting on bi-spinors

$\not{{\cal
P}}:=P^{M}\Gamma_{M}+\Pi^{\alpha}\Gamma_{\alpha}\sigma_{3}=\left(\begin{array}[]{cc}P^{M}\Gamma_{M}+\Pi^{\alpha}\Gamma_{\alpha}&0\\\
0&P^{M}\Gamma_{M}-\Pi^{\alpha}\Gamma_{\alpha}\end{array}\right)$ (4.30)

The notation has been chosen on purpose, in order to highlight the analogy
with the analysis for F1-strings. By using (4.27) and (4.29), one can
immediately conclude that the vector fields
$P^{M}\pm\Pi^{\alpha}\partial_{\alpha}x^{M}$ are time-like and future-ponting
which, by the same reasoning presented in section 3.3, implies that

$\chi(C\not{{\cal P}})\chi>0$ (4.31)

for any non-vanishing doubled-spinor $\chi$. We can now apply this inequality
to

$\chi=(1-\Gamma_{\rm Dp})\epsilon$ (4.32)

in order to obtain a bound which is saturated exactly by supersymmetric
D-branes, i.e. when (4.3) is satisfied.

The resulting bound can be written in physically more meaningful form by
considering a time-plus-space split ${\cal S}=\mathbb{R}\times\Sigma$, with
adapted coordinates $\xi^{\alpha}=(\tau,\sigma^{a})$, $a=1,\ldots,p$. Observe
that

$\Gamma_{{\rm D}p}=\frac{1}{\mu_{{\rm D}p}\,e^{-\phi}\sqrt{\det
m}}\,\not{{\cal P}}\,\gamma_{{\rm D}p}=\frac{(-)^{p}}{\mu_{{\rm
D}p}\,e^{-\phi}\sqrt{\det m}}\,\gamma_{{\rm D}p}\,\not{{\cal P}}$ (4.33)

where

$\displaystyle\gamma_{\text{D}p}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&\hat{\gamma}_{{\rm D}p}\\\
(-)^{p+1}\hat{\gamma}^{-1}_{{\rm D}p}&0\end{array}\right)$ (4.36) with
$\displaystyle\hat{\gamma}_{\text{D}p}=\frac{1}{\sqrt{\det
m}}\sum_{2l+s=p}\frac{\epsilon^{a_{1}\ldots a_{2l}b_{1}\ldots
b_{s}}}{l!s!2^{l}}{\cal F}_{a_{1}a_{2}}\cdots{\cal
F}_{a_{2l-1}a_{2l}}\Gamma_{b_{1}\ldots b_{s}}\ .$ (4.37)

One can then rewrite the inequality (4.31) with $\chi$ given in (4.32) as

$\bar{\epsilon}\,\not{{\cal P}}\epsilon\geq\bar{\epsilon}\,\not{{\cal
P}}\Gamma_{{\rm D}p}\epsilon$ (4.38)

which, by (4.33), in turn translates into the following bound involving the
world-volume fields and the background polyform $\Psi$

$-{\cal G}({\cal P},{\cal K})\geq\frac{\mu_{{\rm D}p}}{p!}\epsilon^{\tau
a_{1}\ldots a_{p}}[(e^{-\phi}\Psi)|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{a_{1}\ldots a_{p}}$ (4.39)

or, in more intrinsic notation,

$-{\cal G}({\cal P},{\cal K})\,\text{d}^{p}\sigma\geq\mu_{{\rm
D}p}[(e^{-\phi}\Psi)|_{\Sigma}\wedge e^{{\cal F}_{\rm mg}}]_{(p)}$ (4.40)

This bound is saturated if and only if the D-brane is supersymmetric

$\text{SUSY D-brane}\quad\Leftrightarrow\quad-{\cal G}({\cal P},{\cal
K})\,\text{d}^{p}\sigma=\mu_{{\rm D}p}[(e^{-\phi}\Psi)|_{\Sigma}\wedge
e^{{\cal F}_{\rm mg}}]_{(p)}$ (4.41)

Recalling (4.22) we see that (4.39) translates into the lower bound on the
conserved charge $\Delta_{{\rm D}p}$ associated with the symmetry $K$:

$\Delta_{{\rm D}p}\geq 0$ (4.42)

This is exactly what we expected from the discussion of section 4.2! D-branes
preserving the supersymmetry $\epsilon$ are exactly those that minimize the
conserved charge $\Delta_{{\rm D}p}$. This is in line with what we expect from
our experience on static backgrounds. In particular $e^{-\phi}\Psi$ can be
seen as an extension of the generalized calibrations of [11, 12, 13, 14] to
the most general supersymmetric setting.

Let us stress once again that the bulk differential condition (2.10) has been
crucial for deriving this current. As it happened for fundamental strings, a
consistent D-brane world-volume theory seems to ‘know’ about (at least part
of) the bulk differential conditions.

Finally, notice that the supersymmetry condition (4.41) is obtained by
projecting (4.6) along ${\cal P}$ by using the metric ${\cal G}$. This proves
the statement anticipated in section 4.1 that the conditions (4.6) are not
only necessary but actually also sufficient for a D-brane to be
supersymmetric. Moreover, the reformulation in terms of the generalized vector
${\cal V}$ defined in (4.8) allows us to write the local bound (4.41) as

$|{\cal G}({\cal P},{\cal K})|\geq|{\cal G}({\cal P},{\cal V})|$ (4.43)

whose saturation reproduces the supersymmetry condition in the form (4.11).

###  4.4 BPS bound and energetics

As done for fundamental strings, in order to give a Hamiltonian interpretation
of the above results, let us pick-up potential forms $B$ and $C$ that preserve
the symmetry generated by $K$, namely ${\cal L}_{K}B=0$ and ${\cal L}_{K}C=0$.
In this case the complete Lagrangian ${\cal L}_{{\rm D}p}$ is exactly
invariant under the symmetry generated by $K$ and the conserved charge density
is $-K^{M}\hat{P}_{M}$, with

$\displaystyle\hat{P}_{M}:=\frac{\partial{\cal L}_{{\rm
D}p}}{\partial(\partial_{\tau}x^{M})}$ $\displaystyle=$ $\displaystyle
P_{M}-\mu_{{\rm D}p}e^{-\phi}\sqrt{-\det{\cal M}}\,({\cal
M}^{-1})^{[\alpha\tau]}B_{MN}\partial_{\alpha}X^{N}$ (4.45)
$\displaystyle+\frac{\mu_{{\rm
D}p}}{p!}\,\epsilon^{\tau\alpha_{1}\ldots\alpha_{p}}\big{[}\iota_{M}(C\wedge
e^{B})\wedge e^{{\cal F}-B}\big{]}_{\alpha_{1}\ldots\alpha_{p}}$

The conserved charge is then given by the energy

$E_{{\rm D}p}=-\int_{\Sigma}\text{d}^{p}\sigma\,g(\hat{P},K)$ (4.46)

and the bound (4.42) translates into

$E_{{\rm D}p}\geq E^{\rm BPS}_{{\rm D}p}$ (4.47)

with

$E^{\rm BPS}_{{\rm
D}p}=\int_{\Sigma}\text{d}^{p}\sigma\,\hat{\Pi}^{a}(\omega-\iota_{K}B)_{a}+\mu_{{\rm
D}p}\,\int_{\Sigma}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge e^{{\cal F}_{\rm
mg}}$ (4.48)

where we have introduced the momenta conjugated to the gauge field

$\hat{\Pi}^{\alpha}:=\frac{\partial{\cal L}_{{\rm
D}p}}{\partial(\partial_{\tau}A_{\alpha})}=\Pi^{\alpha}+\frac{\mu_{{\rm
D}p}}{(p-1)!}\,\epsilon^{\tau\alpha\beta_{1}\ldots\beta_{p-1}}\big{[}C\wedge
e^{{\cal F}_{\rm mg}}\big{]}_{\beta_{1}\ldots\beta_{p-1}}$ (4.49)

These satisy the constraint $\hat{\Pi}^{\tau}\equiv 0$ and the Gauss law
constraint
$\partial_{\alpha}\hat{\Pi}^{\alpha}\equiv\partial_{a}\hat{\Pi}^{a}=0$.

Remarkably, $E^{\rm BPS}_{{\rm D}p}$ is a topological quantity, which is
invariant under continuous deformations of the D-brane configuration. Let us
discuss this point in some detail.

First consider the first term on the r.h.s. of (4.48) and rewrite it as

$\mu_{\rm F1}\int_{\Sigma}(\omega-\iota_{K}B)|_{\Sigma}\wedge\rho_{\rm F1}$
(4.50)

where we have introduced the $(p-1)$-form $\rho_{\rm F1}$ defined by

$(\rho_{\rm F1})_{a_{1}\ldots
a_{p-1}}=2\pi\alpha^{\prime}\,\hat{\Pi}^{b}\,\epsilon_{ba_{1}\ldots a_{p-1}}$
(4.51)

As the notation suggests, $\rho_{\rm F1}$ corresponds to the density of
fundamental strings ‘dissolved’ on the world-space $\Sigma$. By the Gauss law
constraint $\partial_{a}\hat{\Pi}^{a}=0$, $\rho_{\rm F1}$ is closed and
defines a cohomology class on $\Sigma$ which, once integrally quantized, is
Poicaré dual to the space-like one-cycle $\Sigma_{\rm F1}\subset\Sigma$
wrapped by the dissolved fundamental strings. Hence, we allow for continuous
deformations of $\hat{\Pi}^{a}$ which does not change the homology class of
the dual strings, i.e. which correspond to deformations of $\rho_{\rm F1}$ by
an exact piece. By (2.5) and ${\cal L}_{K}B=0$, the one-form
$\omega-\iota_{K}B$ is closed and then we can write (4.51) as

$\mu_{\rm F1}\int_{\Sigma_{\rm F1}}(\omega-\iota_{K}B)$ (4.52)

Comparing with (3.35), we see that it clearly reproduces the contribution of a
BPS F1-string wrapping $\Sigma_{\rm F1}$. Repeating the arguments following
(3.35), (4.52) is clearly invariant under continuous deformations of
$\Sigma_{\rm F1}$ (as well as of its hosting cycle $\Sigma$) within the same
homology class.

Consider now the second term on the r.h.s. of (4.48) and notice that by ${\cal
L}_{k}C=0$ and $\text{d}_{H}C=F$ one can easily check that $\text{d}_{H}({\cal
K}\cdot C)=-{\cal K}\cdot F$ and then, by (2.10), we also have
$\text{d}_{H}(e^{-\phi}\Psi-{\cal K}\cdot C)=0$. This immediately implies that
by continuously deforming $(\Sigma,{\cal F}_{\rm mg})$ the second term on the
r.h.s. of (4.48) does not change. Indeed, consider a $(p+1)$-dimensional chain
$\tilde{\Sigma}$ supporting $\tilde{\cal F}$ and interpolating between two
configurations $(\Sigma,{\cal F}_{\rm mg})$ and $(\Sigma^{\prime},{\cal
F}^{\prime}_{\rm mg})$, i.e. such that $\partial{\cal
C}=\Sigma-\Sigma^{\prime}$, $\tilde{\cal F}|_{\Sigma}={\cal F}_{\rm mg}$ and
$\tilde{\cal F}|_{\Sigma^{\prime}}={\cal F}^{\prime}_{\rm mg}$. Then

$\displaystyle\int_{\Sigma}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge e^{{\cal
F}_{\rm mg}}$ $\displaystyle=$
$\displaystyle\int_{\Sigma^{\prime}}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge
e^{{\cal F}^{\prime}_{\rm mg}}+\int_{{\cal C}}\text{d}[(e^{-\phi}\Psi-{\cal
K}\cdot C)\wedge e^{\tilde{\cal F}}]$ (4.53) $\displaystyle=$
$\displaystyle\int_{\Sigma^{\prime}}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge
e^{{\cal F}^{\prime}_{\rm mg}}$ (4.54)

where in the last step we have used the Bianchi identity $\text{d}\tilde{\cal
F}=H|_{\cal S}$ and the bulk equation $\text{d}_{H}(e^{-\phi}\Psi-{\cal
K}\cdot C)=0$. Hence, the r.h.s. of (4.48) depends just on the generalized
homology class [13] of $(\Sigma,{\cal F}_{\rm mg})$ and is, in this sense,
topological.

In conclusion, we find that the energy of D-branes is (globally) minimized by
the supersymmetric configurations.

###  4.5 Gauss law, Bianchi identity and equations of motion

So far we have focused on the supersymmetry conditions for D-branes. On the
other hand, general on-shell D-brane configurations must satisfy a certain set
of equations of motion and Bianchi identities. Here we would like to discuss
which equations are automatic consequences of the supersymmetry condition and
which ones must be independently added.

The equations of motion associated with the embedding scalar fields are
automatically satisfied thanks to the energetics arguments implied by the BPS
bound discussed above. On the other hand, the story is not that simple when we
turn to the equations associated with the world-volume field-stregth ${\cal
F}$. These are the Bianchi identity999The delta-like $k$-form
$\delta^{(k)}(M)$ on a $n$-dimensional manifold $M$ associated with a
$(n-k)$-dimensional submanifold $N\subset M$ is defined by the identity
$\int_{M}\alpha\wedge\delta^{(k)}(M)\equiv\int_{N}\alpha$ for any $(n-k)$
smooth differential form $\alpha$ on $M$.

$\text{d}{\cal F}=H|_{\cal S}+\delta^{(3)}(\partial\tilde{\cal S})$ (4.55)

where $\partial\tilde{\cal S}\subset{\cal S}$ is the boundary of the world-
volume $\tilde{\cal S}$ of D$(p-2)$-branes ending on ${\cal S}$, and the
equations of motion

$\partial_{\beta}\left(\frac{\partial{\cal L}_{{\rm D}p}}{\partial{\cal
F}_{\alpha\beta}}\right)=-\frac{\mu_{\rm
F1}}{p!}\,\epsilon^{\alpha\beta_{1}\ldots\beta_{p}}\,\delta^{(p)}(\partial{\cal
S}_{\rm F1})_{\beta_{1}\ldots\beta_{p}}$ (4.56)

where now $\partial{\cal S}_{\rm F1}\subset{\cal S}$ is the world-line which
is the boundary of the world-sheet ${\cal S}_{\rm F1}$ of F1-strings ending on
${\cal S}$.

For completeness, we have indicated the possible delta-like contributions from
localized sources, neglected so far. The localized sources contributing to the
the Bianchi identity are given by D$(p-2)$-branes ending on a codimension-
three submanifold of ${\cal S}$. The localized sources contributing to (4.56)
are given by the end-points of fundamental strings attached to the D-brane.
Notice that the overall sign of the localized contributions depends on the
orientation of open branes and strings ending on the hosting D-brane.

Let us now use the natural time-plus-space split introduced by the Killing
vector $K=\partial_{t}$, by choosing the partial static gauge $\tau=t$. The
split (4.21) in this partial static gauge reads

${\cal F}=\text{d}t\wedge{\cal E}+{\cal F}_{\rm mg}$ (4.57)

Analogously, we can split $H=\text{d}t\wedge\iota_{K}H+H_{\rm mg}$. Remember
now the supersymmetry condition (4.13). By using (2.5), it directly implies
that the Bianchi identity for the electric field ${\cal E}$ is automatically
implied by the supersymmetry condition. Hence, it cannot receive any
contribution from localized sources, which is consistent with the fact that
the possible D$(p-2)$-branes wrapping $\tilde{\cal S}$ must be supersymmetric
and then tangent to $K$ as well, hence giving a vanishing localized
contribution $\iota_{K}\delta^{(3)}(\partial\tilde{\cal S})\equiv 0$.

On the other hand, it remains to independently impose by hand the Bianchi
identity for ${\cal F}_{\rm mg}$. By splitting $\tilde{\cal
S}=\mathbb{R}\times\tilde{\Sigma}$, with
$\partial\tilde{\Sigma}\subset\Sigma$, we have

$\text{d}{\cal F}_{\rm mg}=H_{\rm
mg}|_{\Sigma}+\delta^{(3)}(\partial\tilde{\Sigma})$ (4.58)

Consistency requires that the r.h.s. of (4.58) is exact. Hence, if $H_{\rm
mg}|_{\Sigma}$ is cohomologically non-trivial then $\Sigma$ must host the
boundary of an appropriate number of D$(p-2)$-branes.

The situation is reversed for the equations of motion (4.56). By splitting
${\cal S}_{\rm F1}=\mathbb{R}\times\Sigma_{\rm F1}$, the time component of
(4.56) gives rise to the Gauss law constraint

$\partial_{a}\hat{\Pi}^{a}=-\frac{\mu_{\rm F1}}{p!}\,\epsilon^{b_{1}\ldots
b_{p}}\,\delta^{(p)}(\partial\Sigma_{\rm F1})_{b_{1}\ldots b_{p}}$ (4.59)

which must be independently imposed, in addition to the BPS conditions. It is
easier to describe the localized source terms entering (4.59) by using the
dual description of $\hat{\Pi}^{a}$ provided by the $(p-1)$-form $\rho_{\rm
F1}$ defined in (4.51). $\rho_{\rm F1}$ is closed up to localized
contributions coming from the end-points of F1-strings

$\text{d}\rho_{\rm F1}=-\delta^{(p)}(\partial\Sigma_{\rm F1})$ (4.60)

Notice that, by (4.49), $\rho_{\rm F1}$ receives a contribution from the R-R
potentials, which could be not well defined globally. However, we can split
$\rho_{\rm F1}=\tilde{\rho}_{\rm
F1}-\frac{1}{(2\pi\ell_{s})^{p}}\,[C|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p-1)}$, with $\ell_{s}:=\sqrt{\alpha^{\prime}}$, where now
$\tilde{\rho}_{\rm F1}$ is globally defined and is given by the same formula
(4.51) with $\Pi^{a}$ instead of $\hat{\Pi}^{a}$. Hence, (4.60) can be written
as

$\text{d}\tilde{\rho}_{\rm
F1}=\frac{1}{(2\pi\ell_{s})^{p}}\,[F|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p)}-\delta^{(p)}(\partial\Sigma_{\rm F1})$ (4.61)

Being $\tilde{\rho}_{\rm F1}$ globally defined on $\Sigma$, consistency
requires the r.h.s. of this equation to be exact. For instance, this is the
reason why a D0-brane in a massive IIA background with $F_{(0)}=k$ requires
$k$ attached F1-strings.

The fact that Bianchi identity and Gauss law must be imposed by hand in
addition to the supersymmetry conditions should be also clear from the
argument leading to the BPS energy bound, which uses them as external
ingredients. On the other hand, the BPS energy bound itself guarantees that
the remaining equations are automatically fulfilled.

Finally notice that, by using the source-modified equations (4.58) and (4.60),
one can extend the description of D-brane networks given in [13] to our more
general framework while incorporating fundamental strings too.

##  5 Some examples

We now discuss a few examples of applications of the above general results.

###  5.1 Magnetized D-branes

In this section we consider examples of D-branes with at most pure magnetic
flux ${\cal F}_{\rm mg}$ on static backgrounds. In the following two
subsections, we focus on two subcases.

In the first subcase, discussed in section 5.1.1, $K$ is timelike and we
identify it with the generator of the time characterizing the static
background $K\equiv\partial_{t}$. In this case all supersymmetric branes,
being tangent to $K$, must be static themselves and the energetics governing
these configurations can be characterized in terms of the generalized
calibrations described in [11, 12, 13, 14]. These are a direct extension of
the calibrations of [2] and [5] which allows to incorporate the effect of
${\cal F}_{\rm mg}$. We then show how to recover these cases from our general
analysis.

In the second subcase $K$ is null and then, in principle, supersymmetric
D-branes can be either static (filling the 1+1 directions spanned by
$\partial_{t}$ and $K$) or travel at the speed of light along $K$. In
particular, $K$ is null in the interesting case of ${\cal N}=1$ (static)
compactifications to four dimensions and, in section 5.1.2, we focus for
concreteness on this important class of backgrounds.

####  5.1.1 Static magnetized D-branes

Here we consider the case in which $K$ is time-like and the space is static.
Hence, introducing the time $t$ defined by $K=\partial_{t}$, we can split the
ten-dimensional space as $\mathbb{R}\times M_{9}$, with metric

$\text{d}s^{2}_{(10)}=-e^{2A(y)}\text{d}t^{2}+g_{mn}(y)\text{d}y^{m}\text{d}y^{n}$
(5.1)

Furthermore, since we want to focus on magnetized branes, we impose the
condition $\omega=0$, which implies that $H$ has only internal legs,
$\iota_{t}H=0$. We can then split

$\Psi=\psi+\text{d}t\wedge\tilde{\psi}$ (5.2)

where $\iota_{t}\psi=0$ and $\text{d}t\wedge\tilde{\psi}=*\lambda(\psi)$ by
(2.8). The supersymmetry condition (2.10) splits into

$\text{d}_{H}\psi=-\iota_{t}F\,,\qquad\text{d}_{H}\tilde{\psi}=0$ (5.3)

Consider now a D$p$-brane wrapping ${\cal S}=\mathbb{R}\times\Sigma$, with
$\Sigma\subset M_{9}$, possibly supporting a non-trivial magnetic flux ${\cal
F}_{\rm mg}$, such that $\text{d}{\cal F}_{\rm mg}=H|_{\Sigma}$. By defining
$m=g|_{\Sigma}+{\cal F}_{\rm mg}$, we have $\sqrt{-\det{\cal
M}}=e^{A}\sqrt{\det m}$ and

$P^{M}=\mu_{{\rm D}p}\,e^{-A-\phi}\sqrt{\det
m}\,\delta^{M}_{t}\,,\qquad\Pi^{\alpha}\equiv 0$ (5.4)

Hence, $-{\cal G}(P,{\cal K})=\mu_{{\rm D}p}\,e^{A-\phi}\sqrt{\det m}$ and the
BPS bound $\Delta_{{\rm D}p}\geq 0$ given by (4.40) becomes

$e^{A}\sqrt{\det m}\,\text{d}^{p}\sigma\,\geq\,[\psi|_{\Sigma}\wedge e^{{\cal
F}_{\rm mg}}]_{(p)}$ (5.5)

This is saturated, i.e. the D$p$-brane is supersymmetric, exactly when
$(\Sigma,{\cal F}_{\rm mg})$ is calibrated by $\psi$. This reproduces the
result of the discussion for static configurations presented in [14].

Of course, in the limit in which one turns off all background and world-volume
fluxes, $e^{A}\sqrt{\det m}\,\text{d}^{p}\sigma$ is just the volume form,
$\psi$ reduces to a calibration of [2] and (5.5) gives the ordinary
calibration bound on the volume of the cycle $\Sigma$.

####  5.1.2 Compactifications to ${\rm Mink}_{4}$ and non-static branes

A particularly important class of static backgrounds is provided by the ${\cal
N}=1$ compactifications to four-dimensional Minkowski space with
SU(3)$\times$SU(3) structure considered in [20], whose supersymmetry
conditions can be completely interpreted in terms of (static) calibrations
[12]. Concerning the background structures, the relation between the general
ten-dimensional approach and this particular sub-case is discussed in detail
in [30].

In this case the space has structure $\mathbb{R}^{1,3}\times M_{6}$, the
general metric has the form

$\text{d}s^{2}=e^{2A(y)}\text{d}x^{\mu}\text{d}x_{\mu}+\hat{g}_{mn}(y)\text{d}y^{m}\text{d}y^{n}$
(5.6)

and all fields preserve the four-dimensional Poincaré invariance. We can split
the ten-dimensional gamma-matrices as follows

$\Gamma_{\mu}=e^{A}\gamma_{\mu}\otimes\mathbbm{1}\,,\qquad\Gamma_{m}=\gamma_{5}\otimes\hat{\gamma}_{m}$
(5.7)

where $\gamma_{\mu}$ are the four-dimensional gamma matrices for Minkowski
space. By using the real representation in which $C=\Gamma^{\underline{0}}$,
the four independent Killing spinors (2.1) have the form

$\epsilon_{1}=\zeta\otimes\eta_{1}+\text{c.c.}\,,\quad\epsilon_{2}=\zeta\otimes\eta_{2}+\text{c.c.}$
(5.8)

where $\zeta$ is an arbitrary four-dimensional spinor of positive chirality
$\gamma_{5}\zeta=\zeta$, while the internal spinors have chiralities
$\hat{\gamma}_{7}\eta_{1}=\eta_{1}$ and $\hat{\gamma}_{7}\eta_{2}=\mp\eta_{2}$
in IIA/IIB. We denote the norms of the internal spinors by
$|a|^{2}=\eta^{\dagger}_{1}\eta_{1}$ and $|b|^{2}=\eta^{\dagger}_{2}\eta_{2}$.

One can readily compute that $K^{m}=\omega_{m}=0$ while

$K^{\mu}=-e^{-A}\,\bar{\zeta}\gamma^{\mu}\zeta\,(|a|^{2}+|b|^{2})\,,\qquad\omega_{\mu}=-e^{A}\,\bar{\zeta}\gamma_{\mu}\zeta\,(|a|^{2}-|b|^{2})$
(5.9)

Furthermore, (2.3) implies that $\partial_{m}[e^{-A}(|a|^{2}+|b|^{2})]=0$,
while (2.5) implies that $\partial_{m}[e^{A}(|a|^{2}-|b|^{2})]=0$ since by
Poincaré invariance $H$ has only internal legs.

By (4.13) the background is compatible with an electric flux (along $K$) only
if $|a|^{2}\neq|b|^{2}$. On the other hand, in ordinary flux compactifications
O-planes are usually necessary and they preserve the same supersymmetry of
non-fluxed D-branes, hence requiring $|a|^{2}=|b|^{2}$. Focusing on this
latter case, we have $\omega=0$ and we can choose $|a|^{2}=|b|^{2}=e^{A}$, so
that

$K^{\mu}=-2\,\bar{\zeta}\gamma^{\mu}\zeta$ (5.10)

We can use the real representation of the four-dimensional gamma matrices in
which

$\gamma_{0}={\rm
i}\sigma_{2}\otimes\mathbbm{1}_{2}\,,\quad\gamma_{1}=\sigma_{1}\otimes\mathbbm{1}_{2}\,,\quad\gamma_{2}=\sigma_{3}\otimes\sigma_{3}\,,\quad\gamma_{3}=\sigma_{3}\otimes\sigma_{1}$
(5.11)

and the four-dimensional chirality operator is $\gamma_{5}:={\rm
i}\gamma^{0123}=\mathbbm{1}\otimes\sigma_{2}$, and pick up a particular
$\zeta$

$\zeta=\frac{1}{2}\left(\begin{array}[]{c}1\\\
0\end{array}\right)\otimes\left(\begin{array}[]{c}1\\\ {\rm
i}\end{array}\right)$ (5.12)

We can then easily compute

$K=\frac{\partial}{\partial t}+\frac{\partial}{\partial x^{1}}$ (5.13)

We have used the notation $(t,x^{i})\equiv x^{\mu}$ in order to emphasize the
distinguished role of time. In turn, the polyform $\Psi$ becomes

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle(\text{d}x^{1}-\text{d}t)\wedge\,e^{2A}{\rm Im\hskip
1.00006pt}\Phi_{1}+(\text{d}x^{1}-\text{d}t)\wedge\text{d}x^{2}\wedge\text{d}x^{3}\wedge\,e^{4A}{\rm
Re\hskip 1.00006pt}\Phi_{1}$ (5.15)
$\displaystyle+(\text{d}x^{1}-\text{d}t)\wedge\,e^{3A}{\rm Im\hskip
1.00006pt}\big{[}(\text{d}x^{2}+{\rm i}\text{d}x^{3})\wedge\Phi_{2}\big{]}$

where we have introduced the internal pure spinors defined, through the
Clifford map, by

$\Phi_{1}:=-8{\rm
i}e^{-A}\,\eta_{1}\otimes\eta_{2}^{\dagger}\,,\qquad\Phi_{2}:=\mp\ 8{\rm
i}e^{-A}\,\eta_{1}\otimes\eta_{2}^{T}$ (5.16)

the upper/lower sign in $\Phi_{2}$ is for IIB/IIA respectively.

On the right-hand side of (5.15) one can recognize the calibrations ${\rm
Re\hskip 1.00006pt}\Phi_{1}$, ${\rm Im\hskip 1.00006pt}\Phi_{1}$ and ${\rm
Re\hskip 1.00006pt}(e^{{\rm i}\theta}\Phi_{2})$ (for some constant $e^{{\rm
i}\theta}$) on $M_{6}$ which were introduced in [12] to describe static space-
filling, string-like and domain-wall D-branes respectively.

For instance, a static D$p$-brane which appears as a string in
$\mathbb{R}^{1,3}$ and preserves (5.12) must be stretched along $x^{1}$ and
wrap an internal magnetized $(p-1)$-cycle $(\hat{\Sigma},{\cal F}_{\rm mg})$
in $M_{6}$ calibrated by ${\rm Im\hskip 1.00006pt}\Phi_{1}$:

$[{\rm Im\hskip 1.00006pt}\Phi_{1}|_{\hat{\Sigma}}\wedge e^{{\cal F}_{\rm
mg}}]_{(p-1)}=\sqrt{\det m}\,\text{d}^{p-1}\sigma$ (5.17)

where $m:=\hat{g}|_{\hat{\Sigma}}+{\cal F}_{\rm mg}$.

More generically, if we want to consider more complicated static D-branes or
D-brane networks, analogously to the case of time-like $K$ considered in
section 5.1.1 we could actually write

$\Psi=\psi+*\lambda(\psi)$ (5.18)

where

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\text{d}x^{1}\wedge\,e^{2A}{\rm Im\hskip
1.00006pt}\Phi_{1}+\text{d}x^{1}\wedge\text{d}x^{2}\wedge\text{d}x^{3}\wedge\,e^{4A}{\rm
Re\hskip 1.00006pt}\Phi_{1}$ (5.20)
$\displaystyle+\text{d}x^{1}\wedge\,e^{3A}{\rm Im\hskip
1.00006pt}\big{[}(\text{d}x^{2}+{\rm i}\text{d}x^{3})\wedge\Phi_{2}\big{]}$

is the calibration for static (possibly magnetized) branes on
$\mathbb{R}^{3}\times M_{6}$ which was already considered in [13,
14].101010See in particular [13] for a discussion on how D-brane networks can
be described in this framework.

On the other hand, differently from the assumptions of section 5.1.1, here $K$
is null and this opens the possibility of having non-static branes. A simple
example of non-static supersymmetric D-branes is obtained as follows. Consider
a D-brane which appears as a (two-dimensional) string in $\mathbb{R}^{1,3}$
and wraps a internal magnetized cycle $(\hat{\Sigma},{\cal F}_{\rm mg})$
calibrated by ${\rm Im\hskip 1.00006pt}\Phi_{1}$ as in (5.17). If $(\tau,\xi)$
are the two world-sheet coordinates along the string in $\mathbb{R}^{1,3}$,
take an embedding of the form

$t=\tau\,,\quad x^{1}=\tau+f^{1}(\xi)\,,\quad x^{2}=f^{2}(\xi)\,,\quad
x^{3}=f^{3}(\xi)$ (5.21)

where $f^{1,2,3}$ are arbitrary functions. Such an embedding describes a
string of arbitrary shape which travels at the speed of light in the $x^{1}$
direction. One can readily verify that $\sqrt{-\det{\cal
M}}=|\partial_{\xi}f^{1}|\sqrt{\det m}$ and then, by using the formula (4.6),
one can easily check that all these configurations are supersymmetric.
Alternatively, one can compute ${\cal P}$ from (4.25) and obtain that

$-{\cal G}({\cal P},{\cal K})=-g(P,K)=\mu_{{\rm
D}p}\,e^{-\phi}\,|\partial_{\xi}f^{1}|\sqrt{\det m}$ (5.22)

On the other hand, we have that the $p$-dimensional world-space is given by
$\Sigma=\gamma\times\hat{\Sigma}$, where $\gamma$ is the curve in
$\mathbb{R}^{3}$ described by $x^{i}=f^{i}(\xi)$. Hence

$[\Psi|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p)}=(\partial_{\xi}f^{1})\text{d}\xi\wedge[{\rm Im\hskip
1.00006pt}\Phi_{1}|_{\hat{\Sigma}}\wedge e^{{\cal F}_{\rm mg}}]_{(p-1)}$
(5.23)

and by (5.17) the supersymmetry condition in the form (4.41) is satisfied by
appropriately choosing the orientation of $\xi$.

Similar supersymmetric configurations are known in the literature and the
peculiar mechanism which allows them to be supersymmetric is somewhat
analogous to the one behind the existence of giant gravitons [35], as
discussed in [36]. Indeed, giant gravitons can be equally well described in
our general formalism, although we refrain from explicitly discussing them in
the present paper.111111A proposal of calibration for giant gravitons appears
in [7], which is consistent with the results of our approach.

###  5.2 Simple D-branes with non-vanishing electric field

In the literature, there are several examples of BPS D-branes supporting an
electric field. In this section we revisit a few simple known examples by
exploiting the perspective offered by our general approach. We will consider
the BIon solution [37] and the supertube solutions of [38, 39, 40, 41]. Our
discussion will be purposely slightly more general than in the original works,
in order to highlight the power of our framework.

These examples can be seen as describing threshold bound states of D-branes
and fundamental strings. Hence, this class of examples can be obtained by
selcting backgrounds whose supersymmetry is compatible with supersymmetric
F1-strings. A simple, still quite general class of backgrounds of this kind
have the form $\mathbb{R}^{1,1}\times M_{8}$ with $M_{8}$ any eight-
dimensional (Ricci-flat) special holonomy space. We can write the ten-
dimensional metric as

$\text{d}s^{2}_{M_{10}}=-\text{d}t^{2}+\text{d}x^{2}+\hat{g}_{mn}(y)\text{d}y^{m}\text{d}y^{n}$
(5.24)

These backgrounds preserve at least two supersymmetries. We are interested in
the particular supersymmetry which is mutually supersymmetric with a
fundamental string stretching along $\mathbb{R}^{1,1}$. We can describe it
quite explicitly, by using a two-plus-eight split $x^{M}=(t,x,y^{m})$ and
accordingly decomposing the gamma matrices (in the real representation) as
follows

$\Gamma_{t}={\rm
i}\sigma_{2}\otimes\mathbbm{1}\,,\quad\Gamma_{x}=\sigma_{1}\otimes\mathbbm{1}\,,\quad\Gamma_{m}=\sigma_{3}\otimes\hat{\gamma}_{m}$
(5.25)

and $\hat{\gamma}_{(8)}=\hat{\gamma}_{1\ldots 8}$ as eight-dimensional chiral
operator. Then, it is easy to see that the preserved supersymmetry has the
form

$\epsilon_{1}=\left(\begin{array}[]{c}1\\\
0\end{array}\right)\otimes\eta_{1}\,\quad\epsilon_{2}=\left(\begin{array}[]{c}0\\\
1\end{array}\right)\otimes\eta_{2}$ (5.26)

for $\eta_{1}$ and $\eta_{2}$ two Killing spinors on $M_{8}$, with
$\gamma_{(8)}\eta_{1}=\eta_{1}$ and $\gamma_{(8)}\eta_{2}=\pm\eta_{2}$ in
IIA/IIB respectively. We can then easily compute the tensors characterizing
this supersymmetry:

$K=\frac{\partial}{\partial
t}\,,\quad\omega=\text{d}x\,,\quad\Psi=(1-\text{d}t\wedge\text{d}x)\wedge\psi$
(5.27)

where $\psi$ is the calibration for static magnetized D-branes (in the sense
of [11, 12, 13, 14] recalled in section 5.1) for D-branes filling the time
direction $t$ and wrapping a cycle along $M_{8}$. Take for instance IIA and
$M_{8}$ to be a Spin(7) space, with Majorana-Weyl Killing spinor $\eta$,
$\eta^{T}\eta=1$. Hence we can take $\eta_{1}=\eta_{2}=\eta$ and
$\psi=1+\Omega_{4}+\rm{vol}_{8}$, where $\Omega_{4}$ is the Cayley four-form,
which is the calibration for the so-called Cayley cycles.

####  5.2.1 BIons

In the space $\mathbb{R}^{1,1}\times M_{8}$ described above, consider a
D$p$-brane with the following profile

$t=\tau\,,\quad x=X(\sigma)\,,\quad y^{m}=Y^{m}(\sigma)$ (5.28)

More precisely, we take $\sigma^{a}$ to parametrize an internal cycle
$\Sigma\subset M_{8}$, defined by the embedding $Y^{m}(\sigma)$, and allow for
a possible displacement of the embedding in the flat $x$ direction.

From our general result (4.13) we can immediately conclude that the electric
field is fixed to be

${\cal E}=\partial_{a}X\,\text{d}\sigma^{a}$ (5.29)

We then have

${\cal M}_{\alpha\beta}:=(g|_{\cal S}+{\cal
F})_{\alpha\beta}=\left(\begin{array}[]{cc}-1&\nabla_{b}X\\\
-\nabla_{a}X&\hat{m}_{ab}+\nabla_{a}X\nabla_{b}X\end{array}\right)$ (5.30)

where $\hat{m}=\hat{g}|_{\Sigma}+{\cal F}_{\rm mg}$. One then easily gets
$\det{\cal M}=-\det\hat{m}$. On the other hand, by using (5.27) it is also
easy to see that

$\displaystyle[(\text{d}t\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+1)}=[(\iota_{x}\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+1)}=\text{d}t\wedge[\psi|_{\Sigma}\wedge e^{{\cal F}_{\rm mg}}]_{(p)}$
(5.31) $\displaystyle[(\text{d}x\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm
top}=[(\text{d}y^{m}\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm top}$ (5.32)
$\displaystyle=[(\iota_{t}\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm
top}=[(\iota_{m}\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm top}=0$ (5.33)

Take now $\Sigma\subset M_{8}$ to be calibrated by $\psi$:

$[\psi|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p)}=\sqrt{\det\hat{m}}\,\text{d}^{p}\sigma$ (5.34)

With this prescription on $\Sigma$, one can readily verify that the
supersymmety conditions in the form (4.6) are satisfied. In IIA with $M_{8}$
Spin(7) – see comment after (5.27) – we have that $\Sigma$ can be a point
(D2-brane), or a Cayley four-cycle (D6-brane), or a $M_{8}$-filling eight-
cycle (D8-brane), or a more complicated configuration including a possible
world-volume flux ${\cal F}_{\rm mg}$ too [42].

What remains is to impose the space-like Bianchi identity $\text{d}{\cal
F}_{\rm mg}=0$ and the Gauss law constraint $\partial_{a}\hat{\Pi}^{a}=0$. The
canonical momentum turns out to be given by

$\hat{\Pi}^{a}=\frac{\mu_{{\rm
D}p}}{g_{s}}\,\sqrt{\det\hat{m}}\,\hat{m}^{(ab)}\partial_{b}X$ (5.35)

where $g_{s}\equiv e^{\phi}$ is constant and $\hat{m}^{(ab)}$ is the inverse
of $\hat{m}_{(ab)}$. Now, the Gauss law constraint (4.59) can be rewritten as
the condition

$\mu_{{\rm
D}p}\partial_{a}(\sqrt{\det\hat{m}}\,\hat{m}^{(ab)}\partial_{b}X)=\sqrt{\det\tilde{g}}\,\tilde{\nabla}^{2}X=-\mu_{\rm
F1}\delta^{(p)}(\partial\Sigma_{\rm F1})_{1\ldots p}$ (5.36)

where $\tilde{\nabla}_{a}$ is computed by using the world-volume metric

$\tilde{g}_{ab}=e^{2B}(\hat{g}_{ab}-{\cal F}_{ac}\hat{g}^{cd}{\cal F}_{db})$
(5.37)

with $e^{2B}=[(\det\hat{m})/(\det\hat{g}|_{\Sigma})]^{\frac{2}{2-p}}$. Hence,
$X$ must be harmonic with respect to the metric $\tilde{g}$, up to localized
sources. If we specialize to a flat internal space $M_{8}=\mathbb{R}^{8}$,
with the D$p$-brane just spanning $p$ flat directions, we obtain the funnel-
shaped solution $X\sim{\ell^{p-1}_{s}}/r^{p-2}$, which describes a fundamental
string ending on the D-brane. This is the BIon of [37] and the solution
described above is just a generalization of this kind of configuration.

Recall that, in terms of the $(p-1)$ form $\rho_{\rm F1}$ introduced in
(4.51), the first term on the r.h.s. of (4.48) can be written as

$\int_{\Sigma}\text{d}^{p}\sigma\,\hat{\Pi}^{a}\omega_{a}=\mu_{\rm
F1}\int_{\Sigma}\omega\wedge\rho_{\rm F1}$ (5.38)

On the other hand, $\omega|_{\Sigma}=\text{d}X(\sigma)$. Let us now impose
boundary conditions corresponding to $N$ F1 endpoints $\hat{\sigma}_{(i)}$ on
$\Sigma$, $\partial\Sigma_{\rm F1}=\sum^{N}_{i=1}\hat{\sigma}_{(i)}$. Then
$\text{d}\rho_{\rm F1}=-\sum^{N}_{i=1}\delta^{p}(\hat{\sigma}_{(i)})$ and by
integrating by part we obtain that (5.38) becomes

$\mu_{\rm
F1}\sum_{i}\int_{\Sigma}X(\sigma)\,\delta^{p}(\hat{\sigma}_{(i)})-\mu_{\rm
F1}\int_{\partial\Sigma}X\rho_{\rm F1}$ (5.39)

If we can approximate $X$ on $\partial\Sigma$ to a constant $X_{\infty}$, by
observing that $\int_{\partial\Sigma}\rho_{\rm F1}=N$, we see that (5.39)
reduces to

$\mu_{\rm F1}\sum^{N}_{i=1}\Delta X_{(i)}\quad\quad\text{with }\quad\Delta
X_{(i)}\equiv X(\hat{\sigma}_{(i)})-X_{\infty}$ (5.40)

which is just the energy of the $N$ strings ending on the D$p$-brane and
stretching along the $x$-direction with lengths $\Delta X_{(i)}$.

In conclusion, the formula (4.48) for the energy reduces to

$E^{\rm BPS}_{\rm BIon}=\mu_{\rm F1}\sum^{N}_{i=1}\Delta X_{(i)}+{\mu_{{\rm
D}p}}\int_{\Sigma}e^{-\phi}\psi\wedge e^{{\cal F}_{\rm mg}}$ (5.41)

which, as expected, is just the sum of the F1 energies and the energy
associated with a purely magnetized D-brane.

####  5.2.2 Supertubes

In the same background $\mathbb{R}^{1,1}\times M_{8}$ described above, take
now a D$(p+2)$-brane of the form ${\cal S}=\mathbb{R}^{1,1}\times\Sigma$.
Moreover, we take $\Sigma$ to be of the form $I\times\hat{\Sigma}$, where $I$
is an interval parametrized by the coordinate $\chi$ and
$\hat{\Sigma}=\hat{\Sigma}(\chi)$ is a one-parameter family of $p$-cycles
calibrated by $\psi$. Clearly, this is possible only if $\hat{\Sigma}$ is
deformable in $M_{8}$. If there are different possible deformations, we can
take an arbitrary one.121212The simplest possibility [38, 39, 40, 41] is to
take $p=2$, i.e. a D2-brane, which implies that $\hat{\Sigma}$ is just a
point. If for instance $M_{8}$ is Spin(7) – see comment after (5.27) – a point
is obviously calibrated by $\psi$ and then always deformable to any other
point.

We work in partial static gauge by choosing world-volume coordinates
$\xi^{\alpha}=(t,x,\chi,\sigma^{a})$, $a=1,\ldots,p$. Hence the embedding
$\hat{\Sigma}$ is described by

$y^{m}=Y^{m}(\chi,\sigma)$ (5.42)

and the calibration condition on $\hat{\Sigma}$ translates into

$\psi|_{\hat{\Sigma}}=\sqrt{\det\hat{g}|_{\hat{\Sigma}}}\,\text{d}^{p}\sigma$
(5.43)

Take then a world-volume flux of the form

${\cal F}=\text{d}t\wedge\text{d}x+b(\chi)\text{d}x\wedge\text{d}\chi$ (5.44)

The first term on the r.h.s. corresponds to the electric field ${\cal
E}=\text{d}x$ required by (4.13).

It is then easy to see that $\det{\cal
M}=-|b|^{2}{\det\hat{g}|_{\hat{\Sigma}}}$ while by using (5.27) it is
immediate to check that

$\displaystyle[(\text{d}t\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=[(\iota_{x}\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=b\,\text{d}t\wedge\text{d}x\wedge\text{d}\chi\wedge\psi|_{\hat{\Sigma}}$
(5.45) $\displaystyle[(\text{d}x\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=[(\text{d}y^{m}\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{(p+3)}=0$
(5.46) $\displaystyle[(\iota_{t}\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=[(\iota_{m}\Psi)|_{\cal S}\wedge e^{\cal F}]_{(p+3)}=0$ (5.47)

Hence, by combining these simple results, one can readily verify that the
supersymmetry conditions (4.6) are satisfied for any $b(\chi)$ and for any
choice any family of calibrated submanifolds $\hat{\Sigma}(\chi)$.

What remains is to impose the space-like Bianchi identity and Gauss law. Both
are automatically satisfied. The former because $b$ depends just on $\chi$.
The latter because $\hat{\Pi}^{a}$ has as only non-trivial component
$\hat{\Pi}^{x}$, which manifestly fulfills $\partial_{x}\hat{\Pi}^{x}=0$ since
$\hat{\Pi}^{x}\equiv\hat{\Pi}^{x}(\chi,\sigma)$ does not depend on $x$.

By using $\rho_{\rm F1}$ as defined in (4.51), the total energy is given by

$E_{\text{STube}}=\mu_{\rm F1}\int_{\Sigma}\text{d}x\wedge\rho_{\rm
F1}+\mu_{{\rm D}(p+2)}\int_{\Sigma}{\cal F}_{\rm mg}\wedge e^{-\phi}\psi$
(5.48)

The first term provides the contribution coming from a number of dissolved
F1-strings wrapping a curve $\Sigma_{\rm F1}$ stretched along the
$x$-direction. The second term corresponds to the energy associated with a
number of dissolved D$p$-branes wrapping an internal cycle $\Sigma_{{\rm D}p}$
belonging to the family $\hat{\Sigma}(\chi)$, hence calibrated by $\psi$, and
appearing as points in $\mathbb{R}^{1,1}$. In other words, $\rho_{\rm F1}$ and
$(2\pi\ell_{s})^{-2}{\cal F}_{\rm mg}$ are cohomology classes Poincarè dual to
$\Sigma_{\rm F1}\subset\Sigma$ and $\Sigma_{{\rm D}p}\subset\Sigma$ and then

$E_{\text{STube}}=\mu_{\rm F1}\int_{\Sigma_{\rm F1}}\text{d}x+\mu_{{\rm
D}p}\int_{\Sigma_{{\rm D}p}}e^{-\phi}\psi$ (5.49)

On the other hand, there is no contribution associated with the supporting
D$(p+2)$-brane, which appears as ‘effectively tensionless’, consistently with
the fact that $\hat{\Sigma}(\chi)$ is arbitrary – see [38, 39, 40, 41] for
more discussions.

##  6 Conclusion

In this paper we have obtained a geometric characterization of the
supersymmetry conditions for fundamental strings and D-branes. The analysis is
completely general concerning both background and brane configurations. In
particular, D-branes can support the most general world-volume flux compatible
with supersymmetry.

The various equations for F1-strings and D-branes look very similar and come
in pairs, with the difference that D-branes ‘see’ a generalized geometry
rather than an ordinary one: the covariant supersymmetry conditions are
provided by (3.5) and (4.6) [or equivalently (3.7) and (4.11)] respectively;
the non-covariant minimal BPS conditions are (3.30) and (4.41) respectively;
the local BPS bounds saturated by supersymmetric configurations are (3.29) and
(4.42) [or equivalently (3.31) and (4.43)] respectively.

Furthermore, we have shown that the total energy satisfies lower bounds
saturated by its BPS-saturating values $E_{\rm F1}^{\rm BPS}$ and $E_{{\rm
D}p}^{\rm BPS}$ respectively – see (3.35) and (4.48). These are defined by
integrals which are invariant under continuous deformations of the brane
configuration. Hence $E_{\rm F1}^{\rm BPS}$ and $E_{{\rm D}p}^{\rm BPS}$ can
be considered as ‘topological’ and provide the natural central charge
associated with the preserved supersymmetry in presence of F1-strings and
D-branes. In particular, our formalism takes into account the effect of the
most general world-volume flux on D-branes and can be applied to study
D-branes and F1-strings networks extending the approach of [13].

The results obtained here can be useful in several contexts, for instance in
the study of black holes or more general black brane configurations. In
particular, it would be important to study their implications to fully coupled
bulk-plus-branes systems. With regard to this point, the relation between bulk
structures and brane energetics highlighted in this paper should play a key
role as it happens for static settings [27, 28]. Furthermore, it would be
interesting to understand whether one can break the bulk and brane
supersymmetry while preserving (part of) the brane BPS bounds. This could
allow better control over the fully coupled system in non-supersymmetric
settings as well, as for instance it is realized in the non-supersymmetric
flux compactifications considered in [28].

Acknowledgments

I would like to thank M. Bianchi, F. Fucito, P. Koerber and specially A.
Tomasiello for useful discussions. This work is partially supported by the ERC
Advanced Grant n.226455 “Superfields”, by the Italian MIUR-PRIN contract
20075ATT78 and by the NATO grant PST.CLG.978785.

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