# Romans type IIA theory and the heterotic strings

###### Abstract:

In this paper we study compactification of six-dimensional massive type IIA supergravity in presence of Ramond-Ramond background fluxes. The resulting theory in four dimensions is shown to possess duality symmetry. It is shown that specific elements of this symmetry relate massive type IIA compactified on (fluxes along ) to the ordinary type IIA compactified on (fluxes along ) . In turn, this relationship is exploited to relate Romans theory to heterotic strings. The D8-brane (domain-wall) wrapped on is found to correspond to pure gravity heterotic solution which is a direct product of six-dimensional flat space and a four-dimensional Taub-NUT instanton.

^{†}

^{†}preprint: hep-th/0201206

## 1 Introduction

Recently a lot of attention is given to the studies of gauged and massive supergravity theories for the reason that these theories play a significant role in AdS/CFT analysis [1] and also because they might be useful in string phenomenology [2]. Massive supergravities are regarded as closed counterparts of gauged supergravities. In massive theories some of the vector or tensor fields become massive upon eating other fields in the spectrum, analogous to a Higgs type mechanism. In this procedure the total degrees of freedom remain unaltered and so do the number of supercharges, however it turns out that we need to sacrifice some of the global symmetries, like dualities. But it is not always so, certain fraction of these symmetries, at least the perturbative ones, could be restored. This is the goal of our investigation in this paper to look for these unbroken global symmetries.

A well studied and unique example of a massive theory of gravity is the massive type IIA supergravity in ten dimensions constructed by Romans [3]. In string theory, massive supergravities can typically be constructed through a Scherk-Schwarz reduction [4], in which some field strength is given a non-trivial background value (flux) along the compact directions [5]. Established criterion for turning on such background fluxes with consistency is that the potential appears only through its field strength in the action, or equivalently in the field equations. There has been several works along these lines in recent past [5, 6, 11, 13, 14, 7, 8, 10, 9, 12, 16, 17, 15], for more latest works see [18, 19, 20, 21, 22, 23].

Amongst massive theories, massive type IIA supergravities can be easily characterised solely by there field content in which the NS-NS 2-rank tensor field is massive. In massive type IIA theories, in which tensor fields are massive, can also be obtained through generalised and toroidal compactifications of the ordinary (massless) type IIA supergravities with Ramond-Ramond (RR) fluxes turned on [18, 21]. On the other hand massive heterotic supergravities with massive 1-form fields could be obtained by generalised Kaluza-Klein compactification [10, 16]. Thus there is no way out we can make a connection between massive type IIA theories and massive heterotic strings because different type of fields carry masses [18], though there is a duality relationship in involving ordinary type II and heterotic supergravities, for review see [27]. Therfore unless there is a machenism by which we can trade massive 2-rank tensor fields into a massive vector fields we cannot help us out. Fortunately there is such a relationship in four dimensions where a massive tensor field carries same degrees of freedom as a massive vector field carries [24]. So in four dimensions it would be possible to dualize a massive 2-rank tensor field into massive 1-form field. Hence the study of massive theories in is very crucial for duality relationship between massive type IIA and heterotic strings. Though, we will not show this duality relationship between massive tensors and massive vector fields explicitely, we shall be adopting an equivalent tool of compactifications and use duality symmetries, which do this job implicitely.

Paper is organized in the following way. We work out a generalized compactification of the six-dimensional massive type IIA supergravity [18] with RR 2-form fluxes turned on. Our goal is to investigate the fate of the perturbative duality symmetry. We then are interested in relating this massive type IIA compactified on to the ordinary type IIA compactified on with fluxes and eventually relate it to heterotic string theory compactified on . In section 2 we briefly recall massive type IIA sugra and work out its compactification on with RR fluxes. We discover that provided RR 2-form fluxes are turned on the resulting four-dimensional theory can be presented in a manifestly covariant form. In section 3 we provide a mechanism to relate Romans theory compactified on (with fluxes) to the ordinary type IIA compactified on (with fluxes) which is in turn related to heterotic strings. In section 4 we study the vacuum solutions of this massive type IIA supergravity and relate them to the solutions of ordinary IIA by using the elements of duality group. We then use standard heterotic-type IIA duality in six dimensions to relate corresponding ordinary IIA vacua to heterotic solutions. Particularly the D8-brane solution is shown to correspond to a pure gravity heterotic vacua in ten dimensions which contains a domain-wall-type instanton line element. Thus perturbative duality symmetry relates the vacua of massive type IIA and ordinary type IIA theories in six dimensions provided we have isometries. This property also allows us to further relate massive type IIA vacua to six-dimensional heterotic solutions. We have summarized our results in the section 5.

## 2 Toroidal compactification

The compactification of Romans type IIA supergravity [3] on with RR fluxes was provided in [18]. The resulting bosonic composition of massive type II theory is given by

(1) | |||||

where we have adopted the notation that
every product of forms is understood
as a wedge product.^{1}^{1}1
The signature of the metric is
and for a -form we use the convention
while the Poincare dual is given by

(2) |

There are eighty scalar fields those parameterize coset matrix which satisfies

(3) |

where is metric. Thus above six dimensional action has manifest invariance [18]. Once is vanishing, above action represents ordinary type IIA compactified on .

We shall now work out the compactification of above theory on in presence of RR fluxes. To recall, compactification of Romans type IIA theory has been worked out in detail in [21], so we shall omit various fine details here. Thus for the 6-dimensional sechsbein we take the standard toroidal ansatz

(4) |

where coordinates are tangent to the tori. The internal metric on tori is given by while the 4-dimensional spacetime metric is . are the Kaluza-Klein gauge fields and we define 1-forms .

The standard toroidal ansätze for the dilaton, tensor field and the moduli matrix are taken to be

(5) |

where with being a 2-form in four spacetime dimensions, are two vector fields and the represents scalar fields antisymmetric in indices. With these ansätze the NS-NS part of the action (1) reduces to [25]

(6) |

with

(7) |

Next, for twenty-four 1-forms
we would consider a generalized
Kaluza-Klein ansatz where RR fluxes along are included.^{2}^{2}2From
here onwards we switch to a notation where an
vector like is represented simply by an
overhead vector, like .
This generalization is possible since
appear in the action (1)
only through derivatives, therefore an appropriate background value
can be consistently turned on.
We take an ansatz
where scalar fields are allowed to retain a
dependence on the coordinates of the torus. The consistency of toroidal
reduction requires it can at most be
a linear dependence on the torus coordinates,
we define

(8) |

where new constants are antisymmetric in indices. This gives us

(9) |

where various -derivatives are defined as

(10) |

Note that various forms are distinguishable by the symbols and the internal indices they carry. Thus through above generalized ansatz we have effectively introduced 24 new parameter in the form of fluxes. This generalization has been possible only because ’s appears in the action covered with derivative and has 2-cycle along which an appropriate background flux could be turned on.

After compactification the bosonic spectrum of four-dimensional theory consists of the graviton , dilaton , 2-form , 4 scalars from the components of metric and tensor field, and 4 1-forms in the NS-NS sector. From the R-R sector we have scalars , and twenty-four 1-forms whose field strength are (anti)self-dual in . Also we have two sets of 24 constant parameters in the form of fluxes and masses . This is the precisely the bosonic field content of type II supergravity theory in . In the massless case these fields fit in various representation of the T-duality group . Here too various fields combine into the representations as

(11) |

where indices belong to first while indices belong to the second group. Note that the mass and flux parameters also fit into a fundamental representation of .

In order to obtain the action involvng low energy modes of this theory we substitute the ansatz (4)-(9) into the action (1). The resulting four-dimensional bosonic action reads in the kinetic part

(12) | |||||

where the Chern-Simon part of the action is

(13) |

Various field strengths in the above action are

(14) |

The indices and can be raised or lowered by the use of two metrics and , respectively, which are given by

The uni-modular matrices which belong to two cosets are given by

(15) |

and they satisfy We shall take and we have defined , . Under the transformations which act upon indices

(16) |

, alongwith the transformation for which is yet to be determined. The second group acts only on indices in a similar way. The action (12) has manifest invariance from the beginning as all indices are contracted with metric .

Note that the kinetic terms in the action (12) except the terms involving 24 two-form field strength remain invariant under the action of above T-duality group. It remains to be seen if the field equations and the Bianchi identity for 1-form potential transform covariantly. From the 4-dimensional action in (12) the field equations for are

(17) |

while the Bianchi Identities are

(18) |

We now define the dual field strengths as , then field equations (17) and the Bianchi identities for form a covariant set of equations provided and its dual transform as a vector under transformations. Thus symmetry mixes with its dual. One can also combine and its dual into a field strength

(19) |

where .

This completes our analysis of the four-dimensional massive type II supergravity action which we have shown to possess an explicit duality symmetry at the level of field equations provided the fluxes on , , and the masses, , which come from -compactification, transform as doublet.

The action (12) possesses Stueckelberg gauge invariances [3] which is obvious from the investigation of the field strengths in eq.(14). Through these gauge invariaces, the vector fields can eat the scalars and can become massive. Similarly the tensor field can eat one of the two vector fields and can become massive. However, this process of swellowing in of the fields will break the duality symmetry explicitely.

## 3 Massive IIA and heterotic strings

As we have seen in the previous section, the restoration of the T-duality
symmetry of the massive II theories in has been a direct
consequence of our generalised flux-type
ansatz in (8). This tells us that in this framework a wide class
of type II theories with various RR
fluxes on and/or , in fact, get
unified. Specifically under the elements of
this duality symmetry the
-masses (or fluxes) and the
-fluxes are mixed up and rotated. We now discuss a particular case
which is of interest in the rest of this paper.
Consider a compactification of massive II theory (1)
on without fluxes (i.e. ),
the compactified four dimensional massive theory will then be characterisd
by the mass vector
.^{3}^{3}3Here reprsents 24-dimensional
vector with zero entries. Similarly if we compactify a
ordinary type IIA theory (that means in (1)) on 2-torus with
fluxes, the resulting four-dimensional theory will be chracterised by
a different mass vector . These two
four-dimensional
theories obtained in two different ways can
simply related by the following element

(20) |

In many ways this is analogous to the identification between 2-form RR flux and the Romans’ mass parameter when massive IIA is compactified on and a ordinary type IIA compactified on with RR flux [21]. Other elements of above group mix two types of fluxes which could be used to generate new background configurations. In the rest of this paper our aim is to relate massive type IIA backgrounds to heterotic string backgrounds.

Having achieved this relationship between massive IIA theory and the ordinary ordinary type IIA, both compactified on 2-torus, former without fluxes and the latter with fluxes, it is now straight forward to achieve the heterotic connection via following six-dimensional S-duality relations between type IIA compactified on and heterotic string theory compactified on [27]

(21) |

In this approach Romans theory compactified on is mapped to the ordinary type IIA compactified on which in turn is related by duality (21) to heterotic string compacified on . In the next section we shall take an explicit examples of domain-wall solutions and display this duality chain.

## 4 D8-brane vs heterotic instanton

The ten-dimensional massive IIA supergravity theory has D8-brane (domain-wall) solutions which preserve sixteen supercharges [5]. In the string frame metric this solution is given by

(22) |

where is a harmonic function of only the transverse coordinate and all other fields have vanishing background values, refers to the location of the domain-wall. We compactify this solution on by wrapping four of its world-volume directions, say . The corresponding six-dimensional domain-wall solution of the action (1) can be written down as [18]

(23) |

with mass vector .^{4}^{4}4The last entry in
the mass vector
represents the mass parameter of Romans theory.
Clearly this vacuum configuration corresponds to the situation when
there is no background flux along .
The solution (4) is left with unbroken supersymmetries.
Further compactification of this on gives us a solution of the
action (12)

(24) |

Now, by applying transformations (16) on the fields in (4) the solutions with non-trivial R-R fluxes can be generated. Let us consider the specific case where transformation is given by (20). Inserting and the configuration (4) in (16) we get

(25) |

while four-dimensional metric and the dilaton remain invariant. The transformed mass vector implies that the new configuration is a solution of a ordinary IIA compactified on with 2-form fluxes given by . Lifting the rotated solution (25) to six dimensions, we get the following ordinary type IIA configuration (we write new fields with a prime)

(26) |

This is in accordance with our ansatz in (8) and corresponds to
swiching on the flux.
Since this solution is obtained by incorporating T-duality rotation
(16) the number of preserved supersymmetries will remain
unchanged.
Thus by making an transformation
we have transformed domain-wall solution (4) of 6D massive IIA
theory
into a domain-wall solution (4) of ordinary type IIA
which is supported by a non-trivial 2-form flux.
Thus, the four-dimensional perturbative duality
interpolates between vacua of massive type IIA
and ordinary type IIA. It is parallel to
the situation encountered
in the case of massive type II duality in
[21].

Heterotic instanton: Since ordinary type IIA theory on is equivalent to heterotic theory on , in order to relate solutions of massive IIA theory to six dimensional heterotic string vacua we need first to map them to the vacua of ordinary IIA by using the element (20) and then use the relations (21). Let us consider for definitness the configuration in eq.(4) which is already a ordinary IIA background and can therefore be mapped to heterotic side using (21). After, some straight forward calculation we get the following six-dimensional ordinary heterotic solution (in string frame)

(27) |

where the harmonic function . Note that heterotic string theory compactified on has the T-duality group and belongs to the vector representation of this group [25]. It could be easily seen that the vector field in (4) corresponds to a constant field strength in directions which are along . A compactification of heterotic strings along these coordinates with such background fluxes gives rise to masses in four dimensions, see [10, 16]. On the other hand when (4) is oxidised to ten dimensions, as in ordinary toroidal cases, we obtain following 10D heterotic vacua

where is one of the coordinates along on which heterotic string is compactified. and are also periodic but are along . This pure gravity heterotic vacua preserves only 8 supersymmetries and has the geometry which is a product of a 4-dimensional Taub-NUT instanton, , and a 6-dimensional Minkowski space, . Properties of these Taub-NUT type instanton line element are discussed in detail [28, 21].

Thus D8-brane wrapped on emerges
from purely geometrical
configuration of the
heterotic strings such that it involves ‘domain-wall-instanton’ (4).
Compare (4) with M-theory instanton which is also related via duality
to the D8-brane wrapped on and is given by
[21]^{5}^{5}5
11-dimensional solutions similar to
(29) originally appear in [6, 8]. The line element in
(29) differs only in
the structure of
the vector field from previous occasions.

(29) | |||||

where is the coordinate of 11-dimensional circle . This solution however preserves 16 supersymmetries [21]. This is quite consistent as Heterotic theory is obtained by orbifolding of M-theory. These triad of solutions (22), (4) and (29) thus represent the same duality web involving chain. It has been shown [21] that M-theory compactifications on correspond to massive IIA compactification on 2-torus. Here we have presented an evidence that M-theory compactifications on orbifold of should corresspond to heterotic compactification on .

## 5 Summary

To summarize in this work we have studied the compactification of six-dimensional massive type IIA theory [18] with Ramond-Ramond background fluxes corresponding to 2-form field strength. We have found that the resulting four-dimensional theory has global symmetries, same as the perturbative duality symmetries which appear in ordinary compactifications. The mass and flux parameters transform under accordingly. Thus the perturbative T-duality survives at the massive level, though in a different form that it requires appropriate masses and fluxes to be switched on. Next we have shown that the elements of this duality symmetry relate Romans theory compactified on (with RR fluxes along ) with ordinary type IIA compactified on (with RR fluxes along the ). This relationship between ordinary and massive IIA theories compactified on fluxes has led us to provide a heterotic string interpretation for massive IIA theory. As an example we have shown that the wrapped D8-brane solution of massive type IIA turns out to be dual of the solution of ordinary type IIA theory with flux, which in turn is related to pure gravity vacua of heterotic string theory which is a direct product of Minkowski spacetime and a Ricci-flat instanton. The instanton line element is a domain-wall generalization of Taub-NUT instantons [26]. We recall that in [21] we have shown that D8-brane are also related to the compactifications of M-theory invloving such instantons and these solutions have 16 supersymmetries intact. While the Heterotic solution (4) has only 8 supercharges intact. This is entirely consistent given the fact that heterotic strings are orbifolds of M-theory.

###### Acknowledgments.

I am grateful to J. Louis for many interesting discussions and for carefully reading the draft of this paper. I would also like to thank A. Micu for useful discussions. This work is supported by AvH (the Alexander von Humboldt foundation).## References

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