hep-th/yymmnnn

KUL-TF-06/02

D-branes on general backgrounds:

superpotentials and D-terms

Luca Martucci

Institute for Theoretical Physics, K.U. Leuven,

Celestijnenlaan 200D, B-3001 Leuven, Belgium

Abstract

We study the dynamics governing space-time filling D-branes on Type II flux backgrounds preserving four-dimensional supersymmetry. The four-dimensional superpotentials and D-terms are derived. The analysis is kept on completely general grounds thanks to the use of recently proposed generalized calibrations, which also allow one to show the direct link of the superpotentials and D-terms with BPS domain walls and cosmic strings respectively. In particular, our D-brane setting reproduces the tension of D-term strings found from purely four-dimensional analysis. The holomorphicity of the superpotentials is also studied and a moment map associated to the D-terms is proposed. Among different examples, we discuss an application to the study of D7-branes on -structure backgrounds, which reproduces and generalizes some previous results.

e-mail:

###### Contents

- 1 Introduction
- 2 D-calibrated vacua
- 3 The four-dimensional point of view
- 4 Superpotential for D-branes on vacua
- 5 Superpotential from domain walls
- 6 Fayet-Iliopoulos terms and cosmic strings
- 7 Holomorphicity of the superpotential
- 8 Reduced configuration and moduli spaces
- 9 D-flatness and moment map
- 10 Examples and applications for D-branes in -structure vacua
- 11 Concluding remarks
- A Deformations of D-branes on non-trivial field

## 1 Introduction

The study of string theory compactifications to four dimensions with non-trivial fluxes is not only interesting by itself, but seems to be necessary if we hope to use string theory to describe realistic scenarios. Moreover, backgrounds with fluxes arise naturally also in the context of the gauge-gravity correspondence. D-branes play a central role in several aspects of these constructions and thus the study of their properties on nontrivial flux backgrounds is of both formal and phenomenological interest.

In this paper we study the dynamics of space-time filling D-branes in the most general Type II backgrounds preserving four-dimensional Poincaré invariance and supersymmetry. The aim is that of presenting a unified analysis that automatically includes a large class of cases, having background supersymmetry as the unique requirement. This analysis obviously includes as special subcases the backgrounds that are obtained by turning off the Ramond-Ramond (RR) fields, and in particular the limit in which the internal space reduces to a standard Calabi-Yau space.

D-brane dynamics in compactifications on standard Calabi-Yau three-folds constitute an active past and present topic of research (for reviews and complete lists of references see for example [1, 2]). One approach, that we will follow in the present paper, is to consider D-branes filling the four flat directions and wrapping some internal cycle, describing the system by an effective four-dimensional theory. The well-known geometrical properties of the underlying Calabi-Yau spaces allow one to employ a series of familiar technics. Many problems can be addressed systematically by using the two integrable structures of the Calabi-Yau, i.e. the complex and symplectic structures, and for example the associated twisted topological theory constitutes an efficient way to inspect the holomorphic sector of the theory [3].

In general, the reduction of the background supersymmetry to , obtained by giving non-trivial expectation value to the internal fluxes, can drastically change the geometry of the internal space^{1}^{1}1For a review on flux
compactifications see for example [4].. In particular, the
symplectic and complex structures cease to be defined
in general and, even in cases when they are both defined, they may not be
simultaneously integrable. However, as discussed in [5]
for a wide class of vacua, the minimal supersymmetry still
imposes an integrable structure on the internal
manifold that can be described as a generalized complex structure by using the language of generalized complex geometry [6, 7]. The complex and
symplectic structures are substituted by a pair of pure spinors
(that are particular kinds of polyforms - formal sums of forms of different degrees) of definite and opposite
parity, that are associated to corresponding generalized almost
complex structures. The background supersymmetry conditions are
written in terms of these two pure spinors and imply that one of the
associated generalized almost complex structures is actually integrable, while the RR
background fluxes provide an obstruction to the integrability of the
other.

In this paper we will consider the most general class of
backgrounds admitting supersymmetric static D-brane
configurations. These backgrounds constitute a subclass of the vacua
analyzed in [5] and we will refer to them as D-calibrated backgrounds. The name is justified by the fact that,
as shown in [8], these supersymmetric backgrounds can be
completely characterized in terms of a new kind of generalized
calibrations associated to the possible supersymmetric static
D-brane configurations (i.e. filling two, three or all four
space-time directions)^{2}^{2}2Similar generalized calibrations
were introduced in [9] for the subclass of
backgrounds obtained by switching off the RR fields.. The
generalized calibrations are essentially given by the real and
imaginary parts of the background pure spinors, and provide an elegant physical
interpretation for them.

Introducing a space-time filling D-brane wrapping some internal generalized cycle (defined as cycle with a world-volume field-strength on it) on these D-calibrated backgrounds, the effective four-dimensional description should admit an structure. Indeed, we will show how it is possible to identify superpotentials and D-terms that can be written in a completely general form in terms of the underlying generalized calibrations (and then of the background pure spinors). The associated F-flatness and D-flatness conditions are equivalent to the supersymmetry/calibration condition found in [8].

Regarding the superpotential, we will see how it only involves the background integrable pure spinor and the associated F-flatness condition requires that the D-brane must wrap a generalized complex submanifold, as defined in [7]. This result can be seen as an extension of the “decoupling statement” of [10], that in the present context can be rephrased as the requirement that the superpotential only ‘sees’ the underlying (integrable) generalized complex structure. The superpotentials we find may be adopted for the topological branes [11, 12, 13, 14, 15] of the associated topological models [11, 16, 5]. Our superpotentials generalize known superpotentials for D-branes on Calabi-Yau manifolds, as studied for example in [3, 17, 18, 19]. They are also in agreement with previous results for D7-branes with world-volume and/or background fluxes [20, 21]. We will discuss the holomorphic properties of the superpotentials and shall see how they can be addressed in a unified way, again generalizing previous results for D-branes on Calabi-Yau spaces (see for example the discussion of [22, 23]).

It is well known that the tension of a possible BPS domain wall in an theory is expressed uniquely in term of the superpotential. This relation has been used for example in [24, 25] for deriving flux induced superpotentials for the closed string moduli. Using the underlying generalized calibrations, we will see how the same approach can also be used to give an alternative and more physical derivation of the D-brane superpotentials, thus obtaining a non-trivial consistency check of our results.

Regarding the D-flatness condition, in the standard Calabi-Yau case, it can be seen as a deformed Hermitian-Yang-Mills equation for the holomorphic connection on the holomorphic B-cycles, while for Lagrangian A-branes it corresponds to the additional “speciality” conditions (a discussion and more references can be found in [1]). These conditions are equivalent to the vanishing of a moment map associated to the gauge symmetry on the D-brane through an appropriate symplectic structure on the configuration space [22, 23]. The vanishing of the moment map provides a transversal slicing for the imaginary extension of the gauge group action, whose complexification is a symmetry of the superpotential. An extension of this approach to the case of -structure backgrounds has been discussed in [26]. We will propose a symplectic form that generalizes the known symplectic structures of the above mentioned particular subcases to our more general setting. Using this, the gauge symmetry on the wrapped cycle is associated to a moment map whose vanishing condition is equivalent the our general D-flatness condition.

The insight given by the generalized calibrations characterizing these backgrounds allows us to derive another interesting physical result. Namely, the D-term turns out to be strictly related to the BPS cosmic strings obtained by wrapping D-branes filling only two flat space-time directions around internal generalized cycles. First, we will discuss how the D-flatness condition can be satisfied only if a certain topological constant vanishes. This constant can be identified with the Fayet-Iliopoulos term in the effective four-dimensional description. Then, we will see how the BPS cosmic string tension computed from our D-brane setting matches precisely the BPS cosmic string tension obtained from supergravity [27, 28, 29], which should describe a D-brane pair. This result provides a non-trivial check of the identification proposed in [29] between these D-term supergravity string solitons and the effective cosmic strings obtained by wrapping D-branes.

The plan of the paper is as follows. In section 2 we review the basic results of [8], i.e. the D-brane supersymmetry conditions and the associated generalized calibrations. In section 3 we show how, starting from the Dirac-Born-Infeld (DBI) plus Chern-Simons (CS) action for D-branes, we can organize the four-dimensional potential in an explicit form, recognizing the supersymmetry/calibration conditions in the F- and D-flatness conditions of the four-dimensional description. In section 4 we introduce a superpotential that gives rise to the F-flatness condition. This can be written in a universal way by using the underlying integrable pure spinor. In section 5 we give an alternative derivation of the superpotential by using domain wall D-brane configurations. Cosmic string D-brane configurations are considered in section 6, stressing their relation with the Fayet-Iliopoulos contribution to the D-term and giving a general nontrivial argument in favor of their identification with the supergravity cosmic strings constructed in [29]. The holomorphicity properties of the superpotentials are studied in section 7, where an almost complex structure is introduced on the D-brane configuration space from the structure of the underlying background. In section 8 we discuss the reduction of this almost complex structure to the superpotential critical subspace. In section 9 we turn to the D-flatness condition and see how it can be interpreted as the vanishing of a moment map associated to the world-volume gauge symmetry by an appropriately defined symplectic structure. Section 10 is dedicated to some explicit examples in the more specific case of backgrounds with internal -structure [30]: we will consider D3, D5, D6 and D7-branes, with particular attention being paid to the last case, for which our general analysis reproduces results present in the literature (see for example [31, 33, 32, 20, 21, 34]). Finally, in section 11 we present our concluding remarks. Appendix A contains a more detailed discussion on our parametrization of the infinitesimal deformations of generalized cycles.

## 2 D-calibrated vacua

In this paper we consider the most general Type II backgrounds with four-dimensional Poincaré invariance which admit possible supersymmetric D-branes filling one or more flat space directions and wrapping some internal cycle. As discussed in [8], all the backgrounds satisfying these conditions consists of a subclass of the family of vacua analyzed in [5], that we refer to as D-calibrated since they can be completely characterized by the existence a kind of generalized calibrations [8] as we are going to review in this section.

Let us discuss briefly the main properties of the D-calibrated backgrounds, following the conventions of [8]. The ten dimensional metric can be written in the general form

(2.1) |

where , label the four-dimensional flat space, and , the internal space. The -form field-strength can have legs only along internal directions, while the generalized RR field-strengths

(2.2) |

are allowed to have the restricted form

(2.3) |

All the fields appearing in this ansatz (including the dilaton ) can depend only on the internal coordinates . Note also that the usual electric-magnetic Hodge duality relating lower and higher degree RR field-strength translates into the relation between their internal components.

Starting from this bosonic ansatz, the supersymmetry imposes that there exist four independent ten dimensional Killing spinors that can be written in terms of an arbitrary four-dimensional constant spinor of positive chirality and two internal six-dimensional spinors and . The resulting Killing equations give strong constraints on the background bosonic ansatz. The important result proved in [5] is that these supersymmetry constraints on the background fields can be nicely written in terms of the following two polyforms of definite parity

(2.4) |

corresponding via the usual Clifford map to the bispinors^{3}^{3}3See [35] for a previous analysis using these bispinors in pure NS backgrounds.

(2.5) |

can be seen to be pure spinors in the context of the generalized complex geometry and define corresponding generalized almost complex structures^{4}^{4}4More explicit expressions for the pure spinors can be found in [5]. The case of D-calibrated -structure vacua, which include the standard compactifications on flux-less Calabi-Yau’s as a subcase, will be discussed in detail in section 10.. As we will presently recall, in the vacua we are interested in, only one of these two generalized almost complex structure will actually be integrable [5].

Note that not all the possible solutions with 4d Poincaré invariance can be studied in these terms. Indeed there are some pure NS vacua [36, 37, 38] that cannot be incorporated in these class of backgrounds, since in these cases one of the two internal spinors vanishes and then both vanish as well, spoiling of any mean the above approach. However, as discussed in [8], the only backgrounds that can admit supersymmetric (static) D-branes filling one or more flat space directions are those whose internal spinors have the same norm

(2.6) |

This means that the cases we are interested in can be completely covered by the description given in [5] and the condition (2.6) allows also to characterize this class of backgrounds as D-calibrated. More explicitly, taking into account the additional requirement given in (2.6), the supersymmetry conditions for the backgrounds can be split in two parts. One relates the warp factor to the norm of the internal spinor

(2.7) |

for some constant . This relation is a direct consequence of the 4d supersymmetry as it is equivalent to require that (here is the 10d Killing spinor doublet) must be a (constant) Killing vector generating the 4d spacetime translations.

The other supersymmetry conditions involve the two pure spinors characterizing our backgrounds^{5}^{5}5See [8] for the conventions we are using and how they are related
to the ones adopted in [5].

(2.8) | |||||

(2.9) |

where

(2.10) |

is the -twisted differential (such that ) and for Type IIA we have

(2.11) |

while for Type IIB

(2.12) |

As proved in [8], the equations given in (2.8) can be completely characterized in terms of properly defined generalized calibrations associated to possible static supersymmetric D-branes. These are given by D-branes filling two (strings), three (domain walls) or all four flat space-time directions, and wrapping some internal generalized cycle , i.e. a cycle with a possible general world-volume field strength on it (which satisfies the Bianchi identity ). The associated generalized calibrations are given by -closed formal sums of (real) forms of definite degree parity which are written in terms of the pure spinors and are properly energy minimizing when combined with the world-volume field strength . More explicitly, for any generalized cycle ,

(2.13) |

where refers to the energy density of the D-brane wrapping . Note that, if on one hand the form of the generalized calibration is completely general for wrapped cycles of any dimension, on the other hand it does depend explicitly on the number of flat space-like directions filled by the D-brane^{6}^{6}6This effect comes directly from the supersymmetry of the background (that, for example, implies a nontrivial warp factor). In the limit reached by turning off the RR fields, the form of the generalized calibrations acquires an arbitrary phase [9] and does not depend on the filled flat directions.. If we introduce the normalized pure spinors

(2.14) |

the generalized calibrations for space-time filling D-branes are given by [8]

(2.15) |

where with even in Type IIB and odd in Type IIA, and are potentials for , such that . The generalized calibrations for four-dimensional strings and domain walls are given by

(2.16) |

The calibration for the domain walls contain an a priori arbitrary phase specifying the preserved half of the four-dimensional supersymmetry. The inequality (2.13) for each of the generalized calibrations comes from completely algebraic considerations while the differential requirement that they are -closed is completely equivalent to the background supersymmetry conditions given in (2.8). Note also that the generalized calibrations for the space-time filling and four-dimensional string D-branes involve the non-integrable pure spinor, while the generalized calibration for four-dimensional domain wall D-branes involves the integrable pure spinor.

A supersymmetric D-brane configuration can be completely characterized as a D-brane wrapping a generalized calibrated cycle, i.e. a generalized cycle which saturates in each point the upper bound in (2.13). As discussed in [8], this condition can be split in an equivalent pair of conditions. In the case of the space-time filling D-branes (on which we focus from now on), these are given by^{7}^{7}7Calibrated strings are alternatively defined by the conditions and , while domain walls by and .

(2.17) | |||

(2.18) |

The reason why we have used the names F-flatness and D-flatness will be the focus of the following discussions. For the moment, let us only recall that the F-flatness imposes that the D-brane must wrap a generalized complex submanifold [8] and then specifies the generalized complex geometry of the supersymmetric D-branes. In the -structure cases, where the internal manifold is either complex (IIB) or symplectic (IIA), this requirement is completely equivalent [7] to require that the D-brane must be holomorphically embedded with in Type IIB, and must wrap Lagrangian or more general coisotropic [39] generalized cycles in Type IIA. This is completely analogous to what happens in the flux-less Calabi-Yau case, where the above geometrical conditions must be supplemented by a stability condition which can be read as a deformed Hermitian-Yang-Mills equation for B branes and the “speciality” condition for Lagrangian A-branes and their coisotropic generalization [40, 41]. As discussed in a series of paper by Douglas and collaborators (see e.g. [1] for a review), this stability condition can be seen as a D-flatness condition, obtained by imposing the vanishing of an associated moment map. In the following sections we will discuss in detail the above F-flatness and D-flatness in our general setting considering backgrounds, trying to clarify their meaning and their relation with the results already known in the Calabi-Yau case.

## 3 The four-dimensional point of view

In this section, using the results of [8] reviewed in section 2, we would like to pass to a four-dimensional description of the dynamics of the space-time filling D-branes, which should be ultimately described by a four-dimensional effective theory.

Let us start by deriving a form which depends explicitly on the pure spinors for the potential associated to a space-time filling D-brane wrapping the generalized cycle . Consider a D-brane wrapping an -dimensional cycle and introduce a complex F-term vector density , a real D-term density and the scalar density in the following way

(3.1) | |||||

(3.2) | |||||

(3.3) |

Note that, if we are not “too far” from a supersymmetric configuration (which has also an appropriate orientation on ), we can assume that . From the discussion presented in [8], we can argue that

(3.4) |

The complete four-dimensional potential for the D-brane is then given by

(3.5) |

This potential contains the full nonlinear (static) interactions governing the D-brane. We want now to consider the expansion of such a potential around a supersymmetric vacuum configuration , which is characterized by the condition . Then, we can consider very small and and expand the square root in the potential (3.5). As a result, at the quadratic order in and we obtain the following potential

(3.6) | |||||

(3.7) |

where in the last step we have used the -closedness of the generalized calibration defined in (2.15).

In order to better identify this potential with the standard potential of gauge theories, we must introduce metrics on the spaces^{8}^{8}8We indicate with the space of sections of a vector bundle and with the space of real functions on a manifold . (that can be identified with the Lie algebra of the four-dimensional gauge group ) and .
Let us consider first two world-volume functions . We define

(3.8) |

This can be easily seen to be the natural metric for the Lie algebra of the gauge group by expanding the DBI action to write the kinetic term for the four-dimensional field-strength. Secondly, we introduce the following metric on

(3.9) |

that on the other hand defines the natural metric on the space of four-dimensional scalars (this will become more evident from the following discussions). The above metrics are non-degenerate for generalized cycles not “too far” from the supersymmetric ones, for which

(3.10) |

We can now consider the densities and as belonging to the dual of and by using the natural pairing given by the ordinary integration (if for example and is a dual density, ). Thus, we can write the potential (3.6) in the form

(3.11) |

where the ’s have been considered as the components of the formal object . As it will be clear from the following sections, we can really think to as a differential of a proper superpotential . Thus, in the expansion of given in (3.11) one can recognize a contribution , which can be seen as a zero point energy, plus a term that is formally identical to the standard potential of the theories, given by the sum of the squares of the D- and F-terms.

In order to identify more explicitly and as the actual F- and D-terms of the four-dimensional description for space-time filling D-branes, we can give a look at the form of the supersymmetry transformations of the world-volume fermions. This can be obtained by gauge fixing the -symmetry of the D-brane superactions [42, 43]. Here we will use the conventions of [44], consistently with [8]. In particular, we will use the covariant -fixing explicitly discussed in [44], where the second Majorana-Weyl spinor is put to zero. Then, the D-brane fermionic degrees of freedom are described by a single ten dimensional Majorana-Weyl spinor and a background Killing spinors induces a corresponding supersymmetry transformation for . When we specialize to the supersymmetry transformations of an vacuum configuration, i.e. with constant fields preserving Poincaré symmetry, these are given by

(3.13) | |||||

where is an arbitrary constant four-dimensional spinor of positive chirality generating the supersymmetry (see [8] for more details on the notation). If we now consider and very small (around a supersymmetric configuration), the above supersymmetry transformation becomes at leading order

(3.14) |

where and are the duals associated to and by using the metrics (3.8) and (3.9) respectively. In order to give a four-dimensional interpretation of the above supersymmetry transformation, let us first have a look at the bosonic field content living on the D-brane.

By starting from a fixed world-volume field strength configuration on the internal cycle , the world-volume gauge field fluctuations split in two parts: , with indices along the four flat directions, and with indices along the internal directions wrapped by the cycle. On the other hand, the fluctuations of the brane can be described by sections of the orthogonal bundle. Note that all the fields depend on , since we are not doing a real dimensional reduction. So, we can think of them as containing an infinite set of four-dimensional fields. In particular, gives rise to an infinite set of four-dimensional abelian vector multiplets, once completed with the corresponding gaugini that we indicate schematically with . On the other hand and should combine to form the bosonic content of an infinite number of chiral multiplets whose fermionic components we indicate with .

We can now consider more closely the structure associated to the internal manifold . Each factor gives a different reduction of the structure group of , and is an independent symmetry acting separately on and (see e.g. the discussion on [45]). Since we are adding to such a background a space-time filling D-brane wrapping an -dimensional internal cycle, the symmetry of the four-dimensional part is unbroken, while the two possible reductions of the internal structure group are generally spontaneously broken. Note that the chiral multiplets are the only ones who transform under the structure groups of the internal manifold, while the gauge multiplets are singlets with respect to it.

Turning to fermions, before fixing the -symmetry in the superaction of the D-brane, the fermionic content is given by a pair of ten dimensional Majorana-Weyl fermions and of opposite/same chirality in Type IIA/IIB, each transforming under one of the two ’s structure groups. The covariant -fixing used here consists in imposing the condition , keeping only as physical degrees of freedom (for which the supersymmetry transformations take the form (3.13)). Note that this type of -fixing select the associated to , that we call from now on, as the natural one to be used to classify the world-volume fields. Of course, we could have made the alternative (still covariant) choice , but this would have been a little less natural since the physical fermion has different chirality for Type IIA and IIB, leading to a somehow less mirror-symmetric description. This natural selection of the structure group will emerge again in the following discussions.

Now, the sixteen components of the ten dimensional fermion splits in the following way under the full structure group

(3.15) |

The four-dimensional vector fields transform as singlets under the internal structure group, while the scalar fields transform in the representation. Then, the fermions in the sector must be clearly included in the vector multiples and identified with the gaugini , while the sector is given by the fermionic fields of the chiral multiplets. Since a base for the sector is given by , while a base for the sector is given by , we can extract and from the following splitting of

(3.16) |

where we have indicated explicitly the dependence on the flat and internal world-volume coordinates and . The normalizations in (3.16) have been fixed by requiring that and must have canonical kinetic term, using the internal metrics (3.8) and (3.9) for and respectively. Indeed, the explicit form of the quadratic fermionic terms on a general background was found in [46] and using this it is easy to see that the kinetic term for and are given by

(3.17) | |||||

(3.18) |

Thus, from (3.14) we obtain the standard supersymmetry transformations (for constant field configurations)

(3.19) |

Even if we have not computed precisely the dimensional reduction and appropriately organized all the tower of KK fields in supersymmetric multiplets, we can nevertheless conclude that from the four-dimensional point of view one can indeed consider as a D-term and as an F-term, motivating the names used to label the supersymmetry conditions (2.17).

Let us stress again that the above analysis was done in the ‘linearized’ approximation where and are small and we expect the theory to be well described by a standard theory. Note that for the full DBI theory the vanishing of the D-term alone is not enough to insure the vanishing of the gaugini supersymmetry transformations (see (3.13)). On the other hand, the vanishing of the F-term alone is enough to insure that vanishing of supersymmetry transformation of the fermions in the chiral multiplets. We will see in the next section that the F-terms do indeed have a clear non-linear validity, by constructing the explicit complete superpotential generating them. A nonlinear interpretation of the D-term will arise in section 9.

## 4 Superpotential for D-branes on vacua

In section 2 we have recalled how supersymmetric D-branes in D-calibrated backgrounds can be seen as calibrated D-branes with respect to properly defined generalized calibrations. This condition is in turn equivalent (up to an appropriate orientation choice) to the pair of conditions given in (2.17) that, as discussed in section 3, may be seen as the F-flatness and D-flatness conditions in the language of the four-dimensional formulation. In this section we show how the F-flatness condition in (2.17) can be further extracted from a corresponding superpotential. This provides a generalization and, in some sense, a reformulation in a unified language, of previous superpotentials obtained in the Calabi-Yau case along the lines of the two-cycle case considered in [17] (see for example [22] for a general discussion in the Calabi-Yau case). Our superpotential is obviously applicable also in the limiting case of backgrounds with only nontrivial NS fields and their simplest subcase in which the internal manifold reduces to a standard Calabi-Yau.

Let us start by discussing the space of relevant degrees of freedom. We take as configuration space the space of all the generalized cycles quotiented by the group of internal world-volume diffeomorphisms . The space can be properly identified with the space of the scalar fields in the four-dimensional description of the system. The world-volume gauge transformations that depends only on the -coordinates (and not on the ones filling the four flat directions) corresponds to an infinite family of abelian rigid symmetries of the scalar field space , that are gauged in the full theory. The tangent space to should describe the infinitesimal deformations of the embedded submanifolds and of the world-volume field strength on them.

We first describe the deformations of the field strength due to the deformations of the world-volume gauge field, while keeping the embedded submanifold fixed. Since must satisfy the generalized Bianchi identity , an infinitesimal variation of must be of the form , where is a globally defined one-form on . As we have said, the infinitesimal gauge transformations , with , can be considered as the rigid transformations in the four-dimensional description of the system, that are gauged by the coupling to the four-dimensional vector fields. Secondly, we consider the general class of deformations of the submanifold in generated by a section of the bundle restricted to . Note that such a a deformation induces also a corresponding infinitesimal transformation on the world-volume field-strength. A more detailed discussion on the infinitesimal deformations of the generalized cycle is contained in appendix A.

Now, not all these infinitesimal deformations are physically distinguishable since some could be related by an infinitesimal -diffeomorphism. At the infinitesimal level, a -diffeomorphism can be identified by a vector field . Then, by associating to its push-forward in , we obtain that the infinitesimal transformations of the form

(4.1) |

must be considered as non-physical, and must be quotiented out. Then we must consider the following “gauge” equivalence between two infinitesimal deformations of the generalized cycle

(4.2) |

Such identifications can be appropriately described in generalized geometry terms, by recalling the definition of generalized tangent bundle of a generalized cycle given in [7]:

(4.3) |

From (4.3) it is clear that the tangent space of at a “point” can be identified with the space of sections of the vector bundle , that we call the generalized normal bundle of .

We are now ready to rewrite the F-flatness condition in (2.17) in a form that can be more immediately recognized as coming from a superpotential. In order to do this, let us start by splitting the F-flatness condition in two, by projecting it in the orthogonal and tangent directions using the metric structure of the background. We consider first an arbitrary vector field orthogonal to . If we consider the projection of the F-flatness condition in (2.17) along , we obtain

(4.4) |

Secondly, we consider an arbitrary section of the tangent bundle of . Projecting the F-flatness condition along gives the equation

(4.5) |

where is the one-form canonically associated to through the world-volume metric . It is easy to see that it is possible to write this equation in the following equivalent way (we use internal world-volume coordinates )

(4.6) |

Thus, since is non-degenerate for non-degenerate brane configurations, we can rewrite the F-flatness condition in (2.17) as the following pair of conditions

(4.7) | |||

(4.8) |

where now is an arbitrary section of . Note that, by using the first of the F-flatness conditions (4.7), the second can in fact be though as was actually a section the canonical normal bundle , since it is left invariant if we substitute with for any .

We can now present the superpotential generating the F-flatness conditions (4.7), postponing to section 7 the discussion of how it can be actually considered as holomorphic. We want to define a superpotential as functional of the pair defining the internal configuration of the four-dimensional space-time filling D-brane. In order to define such a functional, we need to introduce a fixed reference generalized cycle which is smoothly related to . More precisely, we require that is in the same generalized homology class of , that is there must exist a chain and a field strength on it (satisfying ) such that^{9}^{9}9In the case in which has zero homology class we can take an empty and the conditions (4.9) can be simplified to the pair of conditions and .

(4.9) |

Then the superpotential whose critical points are given by the F-flatness conditions (4.7) can be defined by

(4.10) |

The formula (4.10) defines the superpotential up to an additive constant, whose indeterminacy comes from the arbitrary choice of and also by the possible nontrivial topology of the background^{10}^{10}10If for example the homology group is non-zero, there are possible non-homologous choices of (for fixed boundary conditions). The choice of a different class in gives a shift of by a constant.. We will see in the next section how we can give to and defined in (4.9) a clear physical interpretation.

It is immediate to obtain (4.7) as critical point conditions for the superpotential (4.10). Indeed, consider any generalized normal vector , associated to the representative . Then, the infinitesimal variation of defined by is given by

(4.11) |

Note that clearly the above infinitesimal variation is invariant under the substitution , for any , and thus it is well defined for the equivalence class . From (4.11) it is clear that the superpotential critical points are defined by the conditions (4.7). Note also that the two terms (4.11) can be directly identified with the left hand side of (4.4) and (4.6) by choosing a gauge with orthogonal to and making the identification

(4.12) |

This provides an explicit identification of with , which uses in an essential way the background metric. In the following we will often use this identification, which will allow us to introduce an almost complex and a symplectic structure on .

Note that, as the F-flatness conditions in the form (4.7), the superpotential does not depend on the full structure (which contains also the metric structure) characterizing the internal manifold of the backgrounds we are considering, but it involves only the integrable pure spinor. This result could be seen as a generalization of the “decoupling statement” presented in [10], which asserts that the superpotentials governing D-branes in Calabi-Yau spaces depend only on the background complex structure and not on the Kähler structure for B-branes, and vice-versa for A-branes. The same superpotential may be used to describe also topological D-branes [11, 12, 13, 14, 15] for the underlying topological model [11, 16, 5], since its form is clearly valid for any generalized Calabi-Yau structure, as defined by Hitchin in [6]. Namely, for any generalized Calabi-Yau manifold defined by a -closed pure spinor , we can introduce a variational problem to characterize the generalized complex submanifolds as the extrema of the functional

(4.13) |

where and are defined as for the specific case of the backgrounds considered before.

The above superpotentials can be written directly in terms of the generalized cycle by using the -closedness of (or analogously of in (4.13)). Indeed, we can locally write , where is again a polyform, and then

(4.14) |

Note that the expression (4.10) for the superpotential is completely analogous to the CS term of the D-brane action and like that it is meaningful even if the ‘potential’ polyform is not generally globally defined.

To close this section, let us stress that till now we have deliberately ignored the tension of the D-brane we are considering (i.e., we have fixed ). The tension should be of course reintroduced to have the correct dependence on the fundamental quantities and . The canonically normalized superpotential which includes the correct dependence on the tension is given by

(4.15) |

as follows directly from the form of the potential (3.11), since the canonically normalized potential and metric are given by and respectively, where is defined in (3.9). As we will see in the following section, the superpotential (4.10) can be derived from an argument involving domain walls, which also gives an alternative consistency check of the above normalization of the superpotential. In the following we will re-intruduce the correct dependence on the tension only when needed, continuing to neglect it in most of the discussions.

## 5 Superpotential from domain walls

In the previous section we have shown how to obtain the F-flatness conditions in (2.17) or equivalently (4.7) as the conditions defining the critical points of the superpotential (4.10). In this section we use a physical argument that leads directly to the above superpotential, confirming its validity from a more physical point of view. This can be seen as a generalization to the D-brane context of the standard Gukov-Vafa-Witten argument used to derive the superpotential governing supergravity compactifications with fluxes [24, 25]. In particular we will see how the domain wall generalized calibration given in (2.16), being naturally related to the integrable generalized complex structure of the background, is also naturally related to the F-term associated to the space-time filling D-brane. Along the way, it will allow to check the canonical normalization of the superpotential given in (4.15).

For a given space-time filling D-brane, consider two supersymmetric configurations and that belong to the same generalized homology class. These can be seen as vacua of the effective four-dimensional supersymmetric theory governing the D-brane dynamics. Then, on general grounds, we expect that a domain wall interpolating between the two vacua can exist. Such a domain wall configuration can be constructed in the following way. Take a D-brane filling the half of space-time with positive third space coordinate, , and wrapping the supersymmetric generalized cycle , and another D-brane (of the same kind) filling the other half of space-time with and wrapping the other supersymmetric generalized cycle . These two D-brane configurations with boundary can be glued together in a consistent way by filling the common boundary with another D-brane (again, of the same kind) wrapping a generalized cycle defined by a chain with boundary such that and a world-volume field-strength such that and . The choice of the field-strength is the right one to glue together the three D-brane configurations with boundaries in such a way that the usual anomaly terms coming from the boundaries of each D-brane [47, 48, 49] cancel each other. In order to see it, let us write the complete set of Ramond-Ramond potentials in the form , where is odd in Type IIA and even in Type IIB, and consider the general gauge transformation , where . The CS term in the action of the two half space-time filling D-branes transforms in the following way

(5.1) | |||||

(5.2) |

Then the gauge symmetry is broken by the boundary terms if we consider the two half space-time filling D-branes alone. However, the introduction of the domain wall D-brane located at as described above provides the necessary counterterm to reabsorb the undesired terms in (5.1). Indeed, the domain wall D-brane action contains the CS term

(5.3) |

and it is easy to see that its variation under the gauge transformation exactly cancels the two terms in (5.1).

Now, from general arguments in supersymmetric field theories (see e.g. [50]), it is known that the tension of a BPS domain wall is simply given by

(5.4) |

where is the superpotential difference of the two different vacua and define a constant phase related to the preserved half of supersymmetry. On the other hand, from our D-brane construction the field theory domain wall tension should be exactly given by the effective tension of a supersymmetric configuration for the D-brane domain wall introduced above. But, from the general discussion of [8] reviewed in section 2, we know that such a supersymmetric domain wall D-brane must wrap a generalized cycle calibrated with respect to the generalized calibration written in (2.16). From this, we immediately obtain that the tension of the BPS D-brane domain wall is given by

(5.5) |

where again defines the preserved supersymmetry. Comparing this expression with the one given in (5.4), one can immediately extract the the form of the superpotential as written in (4.10) (again defined up to an additive constant). Furthermore, by reintroducing the neglected tension in front of the right hand side of (5.5), we obtain the canonically normalized superpotential (4.15). Note that, from the general analysis of [8], the fact that the domain wall D-brane is calibrated with respect to the generalized calibration of (2.16) implies also that . Thus, as in field theory, the phase in (5.5) is directly related to the phase of superpotential difference, i.e. , so that

(5.6) |

## 6 Fayet-Iliopoulos terms and cosmic strings

In the previous section we have seen how the well known relation between the superpotential of an theory and supersymmetric domain walls can be exactly reproduced in our D-brane context by using the calibration defined in (2.16) for D-branes filling only three flat space-time directions.

It this section we will discuss how, on the other hand, our D-terms are related to the another possible solitonic objects allowed by an theory, namely cosmic strings^{11}^{11}11Using this name, we implicitly refer to cosmological scenarios obtained from flux compactifications. In the context of the gauge/gravity correspondence, these effective string configurations can be also seen as proper solitonic objects of rigid supersymmetric theories.. There has been a lot of recent activity focused on the embedding of these kind of solitons into string theory (for a review see for example [51]). In particular, in [29] it has been stressed how the only allowed supersymmetric cosmic string solutions of four-dimensional supergravity must have a vanishing F-term and can exist thanks to D-terms with a non-vanishing constant Fayet-Iliopoulos (FI) term. Furthermore the authors of [29] proposed an identification of the four-dimensional supergravity they started from with the effective supergravity theory describing some main features of a D-brane pair filling the four flat space-time dimensions and wrapping some internal cycle (see also the related discussions in [52, 53]) . Our formalism allows to give a non-trivial explicit argument in favor of this proposal and a direct D-brane derivation of some of the results of [29] (see also [55, 54]).

Let us start by considering a single space-time filling D-brane wrapping an internal -dimensional generalized cycle . The crucial observation is that the D-flatness condition [the D-term is defined in (3.1)] can be satisfied only if . By recognizing in the presence of the string generalized calibration written in (2.16), which is -closed, we immediately see that this condition is topological, i.e. does not change if we continuously deform . Then, from the analysis of section 3, it is natural to identify the constant (reintroducing the tension of the D-brane)

(6.1) |

with the FI term of the lowest Kaluza-Klein four-dimensional gauge field. Indeed, the corresponding gauge group has no associated charged chiral fields and thus the necessary requirement for having a supersymmetric vacuum is that . Note that even if was identified as a D-term expanding the action around a supersymmetric configuration, the fact that defined in (6.1) is constant for any configuration supports the idea that its identification with an effective FI term should indeed be more general. This will be confirmed by the following analysis.

Take a space-time filling D-brane wrapping a generalized cycle such that . As we have said, this cannot admit a supersymmetric configuration (at least considering only classical deformations). However, we can add an anti -brane wrapping the same internal generalized cycle . As a consequence, the resulting spectrum on the branes includes now also a complex tachyon which is charged under the combination of the two gauge fields and living on the two branes. Thus, from the discussion of the previous paragraph, it seems reasonable to conclude that the lowest Kaluza-Klein mode of the diagonal gauge group under which the tachyon is charged has as non-vanishing FI term. The (unstable) system then admits a vortex solution [56] that can be identified with a D–brane filling only two flat space-time directions and wrapping the internal -cycle, thus leaving an effective cosmic string. From the analysis of [8], we can immediately conclude that the resulting cosmic string is supersymmetric if and only if it is calibrated with respect to the generalized calibration . This implies that the cosmic sting tension is given by

(6.2) |

On the other hand, since we are considering backgrounds, the D-brane system should be described by a four-dimensional low energy effective theory. Moreover, since we consider BPS cosmic strings, their tension computed in (6.2) using a probe D–brane should be reproduced by the four-dimensional results of [29]. Indeed, to recognize the perfect agreement it is enough to remember that in the description given in section 3 we have used fluctuating fields with the dimension of a length. The standard dimensions for the fields are obtained by simply rescaling them by . This induces a corresponding rescaling of the FI term. Thus, in terms of the proper dimensional FI term, the cosmic string tension reads , which is exactly reproduced by the effective supergravity calculation [27, 28, 29].

Our argument also allows one to obtain from a purely D-brane setting the observation of [29] that for BPS cosmic strings of an four-dimensional supergravity the F-term must vanish identically. Indeed, from the discussion of [8] it follows that the calibration condition on the generalized cycle wrapped by the D-brane forming a BPS cosmic string implies also that must be a generalized complex submanifold, i.e. the F-term must vanish identically so that the superpotential (4.10) is extremized everywhere.

Let us stress another outcome of our approach. The system constituted by a D-brane pair added to an background should be described by an effective supergravity theory like the one considered in [29]. As is clear from the above analysis, we can obtain an effective cosmic string as a tachyonic vortex on a D-brane pair only if these space-time filling branes wrap an internal generalized cycle that cannot be deformed in such a way that the two D-branes, taken singularly, become supersymmetric. In few words, we must start from a pair of non-supersymmetric space-time filling D-branes if we want to create a cosmic string from tachyon condensation. Vice-versa, if we start from supersymmetric D-branes, then tachyon condensation cannot give rise to any supersymmetric D–brane configuration wrapping a generalized cycle homologous to . This is an immediate consequence of the fact that in general parallel D- and D–branes do not separately preserve any common supersymmetry (but generally form a proper bound state). In our case, the supersymmetry of the background implies that a generalized -cycle cannot be contemporary homologous to generalized calibrated cycles for both D- and D-branes.

This last conclusion cannot be extended to the particular subcases where the RR fields are switched off and the background preserves supersymmetry. Indeed, in these cases we have an arbitrary phase entering the generalized calibrations (that can be adjusted giving a different preserved internal supersymmetry) and the condition for a generalized cycle to be calibrated does not depend on the number of filled flat directions [40, 9]. However, a supersymmetric D-brane wrapping a generalized -cycle preserves exactly the supersymmetry that is broken by a D-brane wrapping the same generalized -cycle. The associated non-linearly realized supersymmetry on the world-volume of the D-branes constituting the D-brane pair should then be associated to a FI term in a four-dimensional description of the system, as happens for backgrounds. Then, the above analysis for backgrounds can be repeated with no changes giving again . It would be interesting to understand better the relation between the D-brane picture and a complete four-dimensional supergravity description of one-half BPS cosmic strings, like for example the one presented in [57].

## 7 Holomorphicity of the superpotential

We can now pass to the discussion of the holomorphic structure of the superpotential introduced in section 4. More precisely, we will introduce an almost complex structure on the space of the generalized cycles with respect to which the superpotential is holomorphic, i.e. it is annihilated by the vectors on . Since the space of possible deformations is infinite dimensional, we will work quite at the formal level treating it as finite dimensional, neglecting possible related subtleties. Furthermore, we shall not worry about the integrability of the almost complex structures introduced. Such an issue is already present for example in the study of Lagrangian submanifolds [22], but is not so crucial for the following discussion.

Let us start by recalling that the internal manifold has an integrable generalized complex structure associated to the integrable pure spinor . It is clearly not sufficient by itself to induce an almost complex structure (integrable or not) on . However, it does define a natural almost complex structure, in the sense of an endomorphism of the tangent bundle that squares to minus one, if we restrict to the subspace of the generalized complex submanifolds. As we have seen in section 3, can be characterized as the space of critical points of the superpotential (4.10). Indeed, by definition a generalized cycle is complex if the associated tangent bundle is stable under . As a consequence, defines a natural almost complex structure on the generalized normal bundle