hep-th/9912152 HUTP-99/A073 NUB-3207

RR Flux on Calabi-Yau and Partial

Supersymmetry Breaking

Tomasz R. Taylor

Department of Physics, Northeastern University

Boston, MA 02115, USA

Cumrun Vafa

Jefferson Laboratory of Physics

Harvard University, Cambridge, MA 02138, USA

We show how turning on Flux for RR (and NS-NS) field strengths on non-compact Calabi-Yau 3-folds can serve as a way to partially break supersymmetry from to by mass deformation. The freezing of the moduli of Calabi-Yau in the presence of the flux is the familiar phenomenon of freezing of fields in supersymmetric theories upon mass deformations.

December 1999

1. Introduction

Type II strings compactified on Calabi-Yau threefolds give rise to theories in 4 dimensions. The geometry of Calabi-Yau threefold and its moduli space provides a deep insight into the dynamics of gauge theories. It is thus natural to ask if the simple operation of breaking supersymmetry from to (say by addition of mass terms) has a Calabi-Yau counterpart. If so, this may provide insight into the dynamics of gauge theories.

A particular approach to breaking supersymmetry in the context of Type II compactification on Calabi-Yau threefolds was taken in [1] where Ramond-Ramond fields strengths were turned on. It was shown, however, that one either preserves all supersymmetries (and freeze the moduli of Calabi-Yau to make it correspond to singular limits such as the conifold) or one breaks the supersymmetry completely. Furthermore it was argued in [2] that this is a general result.

On the other hand it was found in [3] that in the context of quantum field theories it is possible to add FI terms, and break the supersymmetry to . These constructions were generalized to the local case in [4] (see also [5]). There seemed, therefore, to exist a conflict between the results coming from considerations of type II compactifications on Calabi-Yau threefolds which suggested finding supersymmetric theories by turning on fluxes is not possible, whereas field theory arguments suggested that some such deformations should be possible.

We will see in this paper that indeed we can obtain partial supersymmetry breaking by considering non-compact Calabi-Yau manifolds with fluxes turned on. The way this avoids the no-go theorem in [2] is by taking a certain decompactification limit, which renders some fields non-dynamical. In other words, it would have corresponded to a theory with no supersymmetric vacua in the compact situation, and where it not for making some fields non-dynamical, we could not have obtained partial supersymmetry breaking. However, as far as geometric engineering of theories are concerned the non-compactness of the Calabi-Yau is a perfectly acceptable condition, and this is already the case for geometric engineering of theories.

The organization of this paper is as follows: In section 2 we show how supersymmetry can be partially broken by considering a simple generalization of models of [3] where we include two vector multiplets. In section 3 we consider type II compactifications on Calabi-Yau threefolds and show why turning on RR fluxes (and in addition NS flux for type IIB) is equivalent to turning on FI terms in the supersymmetric theory. We also review the no-go theorem of [2] and show how it may be avoided in certain non-compact limits. In section 4 we briefly review aspects of geometric engineering and show how RR flux can give mass to the adjoint field breaking the theory to and freezing some of the Calabi-Yau moduli.

Main results of this paper have also been obtained by Peter Mayr [6]. Related ideas have also been considered in [7].

2. Partial Supersymmetry Breaking and Mass Generation

In this section, we present a simple generalization of the model discussed in [3] which exhibits partial supersymmetry breaking with mass generation for multiplets. It involves two vector multiplets, and , with the prepotential

where is a real constant, and with the superpotential (which in general can be taken to be a linear combination of “periods”)

where and are complex constants and . The corresponding Lagrangian is supersymmetric. The constants and correspond to electric and magnetic Fayet-Iliopoulos terms, respectively. In the manifestly supersymmetric notation of Ref.[3]:

The superpotential of Eq.(2.2) gives rise to the following potential for the scalars and :

In the above equation, we neglected an irrelevant, additive constant term. The potential has a zero-value minimum at . For generic values of and , both supersymmetries are broken spontaneously. There are, however, two special configurations of these parameters:

for which supersymmetry is broken partially to . In this case, both scalars and , as well as the fermionic component of the chiral multiplet acquire an equal mass of . The simplest way to prove formally that such a partial breaking does indeed occur is to follow the method of [3] and examine the supersymmetry variations of fermions. In this way, one can identify the goldstino as one of the two fermionic components (gauginos) of the multiplet. In fact, the full vector multiplet and the vector component of remain massless while the chiral multiplet acquires a mass.

The above model can be generalized to more complicated prepotentials, of the form

As in the previous case, the potential has a minimum at . However, there is also another minimization, with respect to , which yields two solutions

similar to (2.5). It is easy to see that the above equation is exactly the condition for partial supersymmetry breaking. Hence we conclude that an supersymmetric vacuum exists also in the general case. In particular, the mass is generated again for the chiral multiplet .

So far we have been discussing the case of global supersymmetry. In the context of string theory we of course have local supersymmetry. In such a case to obtain global limit we have to take some particular limit, where gravity decouples, say by taking in the type II context weak limit of string coupling constant, and perhaps some other limits for other fields. In this context we can break to in an even simpler way. Set , so that the prepotential is just

This would have given a singular kinetic term for in the global case, but it is perfectly fine in the local case. We can think of for example as the “heterotic string coupling constant”. We now turn on FI term . We take the limit where the vev of becomes large (i.e. weak coupling heterotic string limit). In this limit becomes non-dynamical. And the superpotential term (where is related to by a shift) simply gives mass to the scalar , breaking to . It is this realization of partial supersymmetry breaking that we will find applicable in the Calabi-Yau context later in this paper.

3. Type IIB on Calabi-Yau 3-fold with -flux

Consider compactification of type IIB on a Calabi-Yau threefold. We would like to consider turning on flux for NS and R threeform field strengths and . This is a case already considered in [2] following the work of [1] and more recently from the viewpoint of F-theory in [8,9]. The theory has vector multiplets and hypermultiplets in addition to the gravitational multiplet, where denotes Hodge numbers of Calabi-Yau. The relevant modification to the effective action due to turning on -flux is in interactions with the vector multiplets. Let denote the holomorphic threeform on the Calabi-Yau. We write the effective Lagrangian we obtain in 4 dimensions in an supersymmetric framework. The net effect of turning on -flux is to add a superpotential of the form

in the 4-dimensional effective theory, where denotes the complexified coupling constant of type IIB strings. Note that and are dual to some integral 3-cycles and and the above formula can also be written as

To see how (3.1) arises note that if we consider a five brane (NS or R) wrapped around a 3-cycle in the Calabi-Yau, it corresponds to a domain wall in 3+1 dimensional theory, whose BPS bound for tension should be given by across the domain wall. On the other hand the tension of the 5-brane should be (times in the case of NS 5-brane). Since the 5-brane wrapped around changes the flux across the domain wall by a 3-form dual to the cycle we see that this gives the expected change . This argument was discussed in [8] in the context of F-theory on 4-folds, and type IIB on Calabi-Yau 3-folds is a special case of it.

We can also write (3.1) explicitly if we choose a basis for , given by , , with and . Sometimes we refer to as the electric cycles and as the magnetic cycles. This clearly is a basis dependent definition. Let

Moreover denote the dual 3-cycle to the H-fluxes by

where

and the complex vectors and are defined as:

The superpotential (3.1) can be written explicitly as

As is well known there is a prepotential , a homogeneous function of weight in in terms of which

Thus the FI terms are realized by fluxes in type IIB string compactification on Calabi-Yau threefolds.

3.1. Type IIA version

The same analysis can be done in the type IIA language (for the case of type IIA on Calabi-Yau 4 folds see [10]). In fact mirror symmetry already tells us what the story will be in the type IIA case. The story is much simpler in the context of just turning on the flux. In this case the mirror corresponds to turning on and fluxes which are dual to and cycles on Calabi-Yau 3-fold. The analog of (3.1) is now

where denotes the Kähler class on the Calabi-Yau threefold and denotes the quantum of flux. The above formula receives world sheet instanton correction as is well known, and in fact by mirror symmetry one can recover the instanton corrected superpotential on the Calabi-Yau.

3.2. Scalar Potential

Our next step is to obtain the scalar potential corresponding to the superpotential (3.4). We would like to maintain a manifest supersymmetry, however this is not possible in the locally supersymmetric case because the superpotential is a genuinely quantity. Instead of turning to the fully-fledged supersymmetric formalism [11] (like in Ref.[2]), we can try to obtain the potential by compactifying the 10-dimensional action. Alas, this is not so simple in view of the absence of a fully covariant, off-shell formulation of type IIB supergravity. The best we can do is to start from the “non-self-dual” (NSD) action [12] employing a 4-form field strength which is not self-dual. The equations of motion of IIB supergravity follow from the NSD action after imposing the self duality constraint at the level of field equations. Using the NSD action to determine the scalar potential is somewhat questionable, nevertheless it is interesting to compare the result with the superpotential (3.4). In fact, this method will provide an independent derivation of (3.4).

In order to parameterize the -fluxes, we will use the basis , dual to the basis of , with . The fluxes can be written as

In the presence of the fluxes, the 10-dimensional kinetic terms give rise to the potential:

The integration over the Calabi-Yau manifold can be performed by using standard properties of basis (see e.g. [13,14]), with the result

where is the period matrix [14] while and are the complex vectors defined in (3.3). The potential can be rewritten as

where the constant term

As we will discuss in the next subsection and should be chosen so that is zero (for cancellation of 3-brane tadpoles), which we will assume is the case.

In order to relate the above potential with the superpotential (3.4), we first use the identity [14]:

where is the Kähler potential of the vector multiplet moduli , , and is the inverse metric on the vector moduli space. The above expression contains the Kähler covariant derivatives:

By using the relations

we can rewrite the potential as

where is the superpotential (3.4). The dilaton Kähler potential is and accordingly,

Eq.(3.13) is very similar to the standard supergravity
formula for the potential. However,
it is not exactly the same: for instance, the term is missing.
This apparent discrepancy has a simple explanation.
Although the superpotential does not depend on hypermultiplets,
except on the dilaton , the potential receives contributions
from the Kähler covariant derivatives^{†}^{†} For simplicity, we assume here
that the quaternionic hypermultiplet manifold is Kähler.
with respect to chiral components of all hypermultiplets,
including the Calabi-Yau volume etc. All these contributions are proportional
to and must cancel the term. The coefficient
is related to the fact that the 4d coupling is rescaled
by the volume of the internal Calabi-Yau threefold.

Now we consider the rigid supersymmetry limit of (3.13) and (3.4). The Weyl rescaling of the metric that restores the Planck mass in the action introduces the factors in front of the Ricci scalar and scalar kinetic terms; the potential acquires a factor of . Gravity decouples in the limit and the only scalars surviving as dynamical fields are those with the Kähler metric ; all other scalars “freeze” and can be treated as constant parameters. In order to recover the globally supersymmetric models of the type discussed in Section 2 and in Ref.[3] from type IIB theory with -fluxes, we scale the holomorphic sections as and , . The Kähler potential scales then as , so the vector moduli survive as dynamical fields in the limit while and other hypermultiplets decouple and can be treated as (complex) parameters. Furthermore, we set and adjust the remaining Fayet-Iliopoulos parameters so that the superpotential (3.4) scales as . In this way [11], we obtain from (3.13) a finite potential corresponding to the rigid superpotential [3]. The procedure of taking the rigid limit can be further refined to treat some vector moduli in a special way (such as the field discussed in the previous section), in order to freeze them in a way similar to . This will be useful in the context of geometric engineering of theories, to be discussed in the next section.

3.3. Supersymmetric Vacua

Now we analyze supersymmetric solutions with superpotential given by (3.4). The condition for getting a supersymmetric solution with background in this context has been studied by [2] with the conclusion that either there are no supersymmetric vacua or that the is preserved at the vacua. In particular no supersymmetric vacua were found in this way. Let us review these results in the setup that we are considering. The condition for finding supersymmetric vacua in background is that

where denotes the derivative of with respect to all chiral fields. In the context of compact Calabi-Yau, considered in [2], turning on both and can preserve supersymmetry only if

Otherwise these fluxes induce anti-3-brane charge in the uncompactified spacetime (proportional to ) and to cancel it we will necessarily break supersymmetry. If and satisfy (3.15) then we can choose both of them be dual to some A-cycles (i.e. m=0). If we denote the dual three cycles by and with periods the superpotential will take the form,

The condition that in terms of physical fields , is equivalent to in terms of the variables. Since and are independent fields, we see that condition has no solutions, and so supersymmetry is completely broken.

So we see that if we consider smooth Calabi-Yau manifolds there are no supersymmetric solutions. However, near singular Calabi-Yau manifolds the low energy effective Lagrangian description breaks down and one could have additional light fields. The particular case of conifold was studied in [1]. In that case, say we have a vanishing cycle, and we turn on a flux dual to that cycle. This means we have a superpotential

where and . Let us set . If is shrinking we have in addition a light wrapped D3 brane which in language corresponds to chiral fields and with charge respectively under the gauge field whose supersymmetric scalar partner has vev . So the actual Lagrangian superpotential should be modified to

Now the condition that has a solution and is given by

In fact it is possible to check that this actually preserves the full supersymmetry. Even though some other singularities of Calabi-Yau manifolds have also been considered in [2] none has been shown to lead to unbroken supersymmetry (though a full no-go theorem is not available in this context).

3.4. How to obtain supersymmetry?

It thus seems difficult to obtain an supergravity solution with -flux turned on for compact Calabi-Yau manifolds. How could we possibly relax some conditions to make this possible? The hint comes from considering supersymmetric Yang-Mills theories. In these cases one expects to have a mass gap with some number of vacua given by the dual Coxeter number of the group. Moreover one can assign a superpotential to each vacuum given by

where and is an root of unity. The meaning of this superpotential is that the domain walls stretched going from one vacuum to the other will have a central terms for their tension given by the difference of the corresponding values of the superpotential. In the usual Yang-Mills the coupling constant is not a field but a parameter. But we can actually promote it to a chiral field whose vev is given by the coupling constant. If we do that, then we will also have to consider for finding supersymmetric ground states, otherwise we would get a positive energy given by

where is the inverse to the Kähler metric for . However, has no solutions, which means that we have no supersymmetric vacuum (or any vacuum in this case). But clearly we can embed the usual gauge theory in this theory, simply by taking the kinetic term for field to be very large and thus effectively freezing it (this corresponds to making vanishing and thus giving no energy to the vacuum). Note in this case that even if is treated as a parameter the vacuum has an energy and so the supersymmetric background makes sense if we decouple the gravity, by taking . This is in fact how we will generate QFT’s by turning on -fluxes; namely, as we will see later, the field will play the role of an extra field, whose dynamics we will freeze in the limit of interest and concentrate on a decoupling limit where gravity is irrelevant. This will in particular avoid the no-go theorem of [2] for obtaining solutions .

4. Geometric Engineering for Theories and Their Partial Breaking to

In preparation of our discussion for turning on RR-fluxes and breaking theories to we first review some relevant aspects of geometric engineering for gauge theories, in the context of type IIA compactification on Calabi-Yau threefold and its type IIB mirror (see [15-18] for more detail). Instead of being general, consider a simple example: Let us review how the Yang-Mills is geometrically engineered: We consider in type IIA a local CY geometry where a is fibered over another . The simplest possibility is . In the limit the goes to zero we obtain an enhanced gauge symmetry from the singularity. To suppress the gravity effects we take . The area of , , determines the coupling constant for the gauge theory:

We can identify the size of , , with the vev of a scalar in the adjoint of ; more precisely, we have the classical relation that

One considers the regime where is large and is small. This is the same as taking a finite size and zero size in the string frame in the limit where (the effective mass of branes is given by and where the and denote the volume of the ’s in the string frame). In this limit the field becomes non-dynamical and plays only the role of a parameter in the field theory. The prepotential in this case has the following structure

for some constants . Note that can be absorbed into a redefinition of . In fact this is the limit one is taking in the geometry, namely we take and keeping the combination fixed. The way this expression is obtained is to use mirror symmetry to compute the worldsheet instanton corrections in this type IIA background by relating it to complex structure variation in a type IIB background. Physically, the net result is that the apparent instanton corrections in (4.1) can be related to one loop corrections to the prepotential summed over all D2 branes wrapped around the geometry. In other words

and the rich structure of the instanton corrections gets mapped to an intricate structure of wrapped BPS D2 branes in this geometry, in the limit we are taking.

4.1. Adding the Flux and breaking

Now we are ready to add the flux. We choose the flux to be an RR 2-form flux in the direction dual to the 2-cycle . This means according to our discussion in section 3, generating a superpotential given by

In the context of type IIB it means turning on a specific RR H-flux dual to the 3-cycle representing . In that context is a classical computation of periods. Now, it has been shown [19] that

where , and the classical result receives quantum correction exactly as given by . In particular, not just , but also itself is among the periods of the type IIB Calabi-Yau geometry (this was a crucial fact for extracting gauge theory implication of type IIB geometry [15]). We thus have a superpotential

This can be viewed in the gauge theory language as a mass deformation (giving mass to the scalar partner of the vector multiplet) breaking . To find whether there are any supersymmetric vacua one will have to analyze it, exactly as was done originally in the field theory context by Seiberg and Witten [20]. Namely one finds near the points where there is a massless dyon, a modification of the superpotential, by including the light states. In this case we get (somewhat analogous to the conifold case)

and one finds an supersymmetric solution

In the type IIB context this corresponds to freezing some complex moduli of Calabi-Yau threefold and rendering some other moduli non-dynamical.

4.2. Generalizations

It is clear that this generalizes to a large number of gauge models engineered in [18] (at least for the cases where the beta function is not zero). Just as was done above for the case, one turns on in the type IIA context independent RR 2-form flux in the bases whose sizes control the coupling constants of various gauge groups. By going over to its type IIB mirror, the RR flux turns into a particular flux, which again serve to freeze the moduli in the type IIB side, exactly as was done for the case above.

Acknowledgements We have benefited from discussions with S. Ferrara, K. Hori, P. Mayr, M. Porrati and A. Strominger. The work of T.R.T. was supported in part by NSF grant PHY-99-01057 and that of C.V. was supported in part by NSF grant PHY-98-02709.

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