1. Introduction
Dualities among superstring theories play important roles to reveal both perturbative and non-perturbative aspects of superstring theories. Especially, type IIA superstring theory is related to type IIB superstring theory by T-duality, which interchanges Kaluza–Klein modes (KK modes) and winding modes of a compactified circle direction [
1,
2]. In the low energy limit, massive modes in the type II superstring theories are decoupled, and the effective actions are well described by corresponding type II supergravity theories [
3,
4]. The T-duality transformations of background massless fields are well known as Buscher rule [
5,
6].
When the superstring theories are toroidally compactified on
, the duality transformation can be generalized to
duality [
7,
8]. Actually it is argued in ref. [
9] that, by assuming all fields depend only on a time coordinate, NS-NS sector in the low energy effective action, which consists of a graviton, a dilaton and Kalb–Ramond B field (B field), can be rewritten in manifestly
invariant expression. In addition,
invariance of the NS-NS sector in general background was confirmed in refs. [
10,
11]. Furthermore, it is also proven that
invariance can be extended to all orders in
corrections to the low energy effective action [
12].
transformation of the R-R sector has been investigated in refs. [
13,
14,
15,
16,
17,
18,
19,
20]. One approach is to note that R-R potentials fill up a spinor representation of
duality transformation [
13,
14]. The spinor representation of R-R potentials combined with B field was explicitly constructed when the compactified space was
[
15], and a general case of
compactification was completed in ref. [
16]. Another approach was investigated by Hassan in refs. [
17,
18,
19,
20], where the consistency of the duality transformation with local supersymmetry transformation is imposed. In this approach, the
transformations of dilatinos and gravitinos are explicitly written in terms of 10 dimensional forms, and those of R-R potentials are derived in a bispinor form. In the type II superstring theories, the formulation of superspace that is compatible with T-duality was discussed in ref. [
21], and inclusion of R-R fields and an application to AdS background were investigated in refs. [
22,
23]. Generalization of ref. [
20] to non-abelian T-duality was performed in refs. [
24,
25].
Although the type II supergravities possess
duality invariance, forms of the action are not manifestly invariant in terms of 10 dimensional fields. There are two formalisms to improve this point. The first one is a double field theory, which treats internal coordinates of winding modes and KK modes simultaneously [
26,
27,
28].
transformation is realized as a rotation among these
coordinates, and fields are generalized to behave as tensors under this coordinate transformation.
invariant forms of the type II supergravities are discussed in the framework of the double field theory in refs. [
29,
30,
31]. The second one is a generalized geometry, which treats tangent and cotangent bundles of compactified manifold on equal footing [
32,
33,
34]. Lie brackets of two vector fields are also modified to Courant brackets to incorporate B field transformation with the general coordinate one.
invariant forms of the type II supergravities are discussed in the framework of the generalized geometry in ref. [
35].
The double field theory or the generalized geometry played important roles to reveal the
invariant structure, however, it is not so clear to derive such structure within the framework of the type II supergravities. In this paper, we revisit the
subgroup of the duality transformation discussed in ref. [
20] to construct
invariants within the framework of the type II supergravities. We review that
transformations of NS-NS fields and fermionic fields are completely written in terms of 10 dimensional fields and construct
invariants by evaluating these. The actions of the type II supergravities are completely written by combinations of these building blocks, which are consistent with ones obtained in refs. [
31,
35].
This paper is organized as follows. In
Section 2, we review the
duality transformations of fields shown in ref. [
20]. Especially, we show that these transformations can be written by using 10 dimensional fields
1. In
Section 3, we construct
duality invariants for NS-NS fields and fermionic ones. We also check these duality invariants in the background of fundamental strings and wave solutions, or NS5-branes and KK monopoles. In
Section 4, we construct NS-NS bosonic terms in the type II supergravities by using the duality invariants. In
Section 5, we construct fermionic bilinear terms in the type II supergravities by duality invariants.
Section 6 is devoted to conclusions and discussions. In
Appendix A, we review the actions of the type II supergravities for the NS-NS sector and fermionic bilinear terms.
2. Brief Review of Transformations
In this section, we briefly review
transformations of massless fields in the type II supergravities. NS-NS fields of the type II supergravities consist of the graviton
, the Kalb–Ramon field
and the dilaton
. First, we take into account these fields to show a standard dimensional reduction of 10 dimensional supergravity action to
non-compact dimensions [
11]. The reduced action is written in a manifestly
invariant form, and
transformations of reduced fields can be obtained by using
matrix notation. Among
transformations of NS-NS fields,
transformations are non-trivial if we ignore general linear coordinate transformations and shift of the B field [
12]. Thus, we focus on the transformations of
, and it is possible to express the duality transformations in terms of the original 10 dimensional fields. Then, we review the
transformations of fermionic fields, two gravitinos and two dilatinos, which are compatible with local supersymmetry transformations in the type II supergravities.
We denote the 10 dimensional spacetime indices as
. Non-compact spacetime directions are labeled by
and compact
d dimensions are done by
. On the other hand, local Lorentz indices are denoted as
. Non-compact local Lorentz indices are labeled by
, and those for compact
d dimensions are noted by
. The explanation in this section is based on ref. [
20], but some of the transformations are not written in 10 dimensional fields there, which are repaired below.
The bosonic part of the action for NS-NS fields is common to both type II supergravities, and the explicit form is written as
where
and
is the gravitational constant in 10 dimensions.
R is a scalar curvature and
is a three-form field strength of the B field. Now, we consider the dimensional reduction of the above action on
d dimensional torus. The dimensional reduction of the metric is given by
where
and
. Here,
is a metric,
are
gauge fields and
are scalars for non-compact spacetime directions. Note that all fields are assumed to be dependent on
directions but not on
ones. The dimensional reduction of the three-form field strength is a little bit complicated, and it is easier to consider in the local Lorentz frame. By using a vielbein
in 10 dimensions, the three-form field in the local Lorentz frame is defined as
, and the dimensional reduction of each component is written as
Note that
since all fields are dependent only on
. Here,
are gauge field strengths. Gauge fields, which originate from B field, are defined as
, and
are corresponding gauge field strengths. The dimensional reduction of the dilaton field is defined as
is a dilaton field in non-compact directions. Substituting Equations (
2)–(
4) into the 10-dimensional action (
1), we obtain the action for the non-compact directions of the form
where
is a volume of the
d dimensional torus and
r is a scalar curvature constructed out of
. The indices for the compact directions are expressed by the matrix notation, as will be explained below.
Since
transformations act on indices for compactified directions, fields only with non-compact directions,
,
and
, are invariant under
transformations. The first line of action (
5) consists of kinetic terms of
and
, so this line is invariant under
transformation. In the second line, scalar fields with compact spatial indices
and
are gathered into
and the
transformation
for massless NS-NS fields is defined by [
9]
Here,
and
are
matrices and
acts on the gauge indices of
and
. It is obvious that the first and second terms in the second line of Equation (
5) are invariant under these transformations. As for the third term in the second line,
contains
,
and their field strengths, which transform under
transformation, but still
is
invariant.
There are
elements for the duality transformations of
, but some of them are trivial in the sense that these do not mix NS-NS fields. Actually,
elements of general linear coordinate transformation
and
elements for the shift of the B field are trivial. The remaining
elements are non-trivial, and these construct a subgroup of
. The
subgroup is expressed as [
12]
The case of
corresponds to a part of general linear coordinate transformation.
From Equation (
7), it is possible to extract duality transformations of dimensionally reduced fields. These are then gathered into duality transformations of the original 10 dimensional fields. Below, we summarize
transformations of fields in 10 dimensions [
18]. By introducing
matrices as
the 10-dimensional inverse metric transforms as
Since the duality invariant, which includes the dilaton field, is written by
, the duality transformation of the dilaton field is given by
The ± sign originates from actions to the world-sheet left and right moving modes, respectively. From Equation (
10), it is possible to define
transformation of the vielbein as
Notice that
are related by local Lorenz transformation of
Thus, the local Lorentz frame of the left moving sector is obtained by twisting that of the right moving sector by
. Therefore, invariants under local Lorentz transformation, which are constructed out of
, can always be written in terms of
.
Since two kinds of vielbein can be used after the duality transformation, the three-form field strength
also transforms in two ways as [
20]
Here,
are connections defined by using torsionless spin connection
as
and the duality transformations are calculated as
Notice that
are constructed out of
, respectively. Similarly,
are connections defined by using affine connection
as
and the duality transformations are derived as
Since the vielbein is not used in Equation (
17), there are no
subscriptions in the above.
Next, let us summarize duality transformations of gravitinos
and dilatinos
. In ref. [
20], these transformations are derived so as to be consistent with the local supersymmetry (
A4). It is easy to check this for the gravitino
, and the result is
To derive the above, we used
. This holds because the derivatives of fields with respect to the compact directions are zero. For the gravitino
, the duality transformation
is defined by using
and the susy transformation becomes
where
is a gamma matrix in 10 dimensions. In the above, we ignored R-R fields and used local Lorentz transformation to change
to
.
is a spinor representation of the local Lorentz transformation of
and satisfies
. Equation (
20) is compatible with the duality transformation if we define
Finally, we consider
duality transformations of dilatinos. As in the case of the gravitinos, the duality transformations are derived so as to be consistent with the local supersymmetry (
A4).
In the second equality, we used Equation (
14) and employed the fifth line of Equation (
32). Thus, the duality transformation of the dilatino
is compatible with the local supersymmetry if we define
As in the case of the gravitino, the duality transformation
is defined by using
. By taking into account the local Lorentz transformation, we obtain
5. Construction of Fermionic Bilinear Terms in Type II Supergravities via Duality Invariants
Let us construct fermionic bilinear terms in the type II supergravities by using duality invariants. First, we consider bilinear terms of the dilatinos. Since the duality invariant forms of the dilatinos are given by
, we would like to construct duality invariants that partially contain
These are not duality invariants nor scalars under local Lorentz transformation. In order to recover the latter covariance, we add the connection
as follows.
Here, we used
for Majorana fermions, and
is a covariant derivative with respect to the connection of
. In this case,
. Thus, the terms in Equation (
50) are scalars under local Lorentz transformation. Furthermore, these are
duality invariant, as we show below. The dual theory is written by
for the vielbein, and the dual of the above is written as
In the third equality, we used local Lorentz covariance for the + mode, such as
. Thus, the terms of Equation (
50) are
invariant.
Next, we consider two derivative terms, which consist of
and
. The duality invariants should partially contain
These are not duality invariants nor scalars under local Lorentz transformation. In order to make scalars under local Lorentz transformation, we need to add the connection term to the above.
Furthermore, these are duality invariants, as we show below.
Thus, the terms of Equation (
53) are
invariant.
Finally, let us investigate two derivative terms that are bilinear of Majorana gravitinos. These should partially contain the following terms.
These are not duality invariants nor scalars under local Lorentz transformation. In order to recover the latter covariance, we add connection terms
and
as follows.
Note that
. These are scalars under local Lorentz transformation. The first two terms are similar to Equation (
51), so the transformations under
are also similar. One difference is in the derivative of
, which is written as
On the other hand, the
transformations of the connections
are given by Equation (
18), and the third term in Equation (
56) transforms as
In the second equality, we used
. Thus, we see that the last term in Equation (
57) is cancelled by the last term in Equation (
58). The combinations of Equation (
56) are
invariant.
So far, we constructed
invariants of (
50), (
53) and (
56). Then, up to overall factor, the Lagrangian is expressed as
In the last equality, if we choose
and
, it is possible to express the derivative of the Majorana gravitinos as field strengths of
up to partial integral. Since this prescription is important to realize local supersymmetry, we employ these values. Then, the
invariant action of the fermionic bilinear is uniquely determined as
Of course, a linear combination of these terms is consistent with the type II supergravities. Thus, we showed that fermionic bilinears without R-R fields can be written in terms of the duality invariants within the framework of the type II supergravities. Invariant forms of fermionic bilinears with R-R fluxes are obtained in the framework of the double field theory [
31] or generalized geometry [
35].
6. Conclusions and Discussion
In this paper, within the framework of the type II supergravities, we have constructed
duality invariants of Equations (
31), (
33) and (
35) by examining
transformations of three-form H field, dilaton and dilatino. These invariants are checked in the background of fundamental strings and wave solutions, or NS5-branes and KK monopoles. By using these duality invariants, we reconstructed the actions of type II supergravities in a manifestly
invariant form in
Section 4 and
Section 5. Since these actions are also invariant under linear
transformation and shift of the B field, these are exactly
invariant. As for the kinetic terms on R-R fields,
invariant construction was already discussed within the framework of the type II supergravities in ref. [
20].
As we have checked the duality invariants in the background of strings and wave solutions, or NS5-branes and KK monopoles, it is easy to apply to other non-geometric backgrounds [
37,
38,
39,
40]. It is interesting to see corrections to the non-geometric background, which was studied from the viewpoint of world-sheet instantons [
41]. It is also interesting to investigate
-twisted solutions of the double field theory [
42] by evaluating
invariants in this paper.
Since we have constructed
duality invariants within the framework of the type II supergravities, it is natural to generalize these formulations to higher derivative corrections in the type II superstring theories. However, this is not a simple task, and it is shown that higher derivative corrections in bosonic or heterotic string theory cannot be written in terms of generalized metric [
43,
44]. We should take into account total derivative terms and field redefinitions, which consist of dimensionally reduced fields. Constraint on
terms via cosmological ansatz was investigated in ref. [
45], was executed via T-duality in refs. [
46] and was performed via
duality in ref. [
47]. In our formalism, the difficulty can be seen by duality transformation of the Riemann tensor (
46), which is calculated as
where
. If we consider
, which exists as a part of higher derivative terms in bosonic string theory, the duality transformation of this term contains
. However, this cannot be cancelled by other terms even if we consider total derivatives and field redefinitions of 10 dimensional fields.
Although we should decompose Equation (
61) in terms of dimensionally reduced fields,
,
and
are invariant under
transformations. Thus, we should only take care of
. It is also useful to consult a frame formalism of the double field theory [
48]. If we find nice structure on total derivatives and field redefinitions in terms of these fields, it will be possible to apply our
construction to higher derivative terms such as
terms [
49,
50,
51,
52,
53,
54,
55].