Threeforms in supergravity and flux compactifications
Eur. Phys. J. C
Threeforms in supergravity and flux compactifications
Fotis Farakos 0
Stefano Lanza 0
Luca Martucci 0
Dmitri Sorokin 0
0 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova & I.N.F.N. Sezione di Padova , Via F. Marzolo 8, 35131 Padova , Italy
We present a duality procedure that relates conventional fourdimensional mattercoupled N = 1 supergravities to dual formulations in which auxiliary fields are replaced by field strengths of gauge threeforms. The duality promotes specific coupling constants appearing in the superpotential to vacuum expectation values of the field strengths. We then apply this general duality to type IIA string compactifications on CalabiYau orientifolds with RR fluxes. This gives a new supersymmetric formulation of the corresponding effective fourdimensional theories which includes gauge threeforms. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Threeform multiplets in supersymmetry . . . . . . . 2.1 Single threeform multiplets . . . . . . . . . . . 2.2 Double threeform multiplets . . . . . . . . . . . 2.3 Double threeform multiplets and nonlinear dualization . . . . . . . . . . . . . . . . . . . . . 3 Threeform multiplets in N = 1 supergravity . . . . . 3.1 Variant minimal supergravities from duality . . . 3.1.1 Single threeform supergravity . . . . . . . 3.1.2 Double threeform supergravity . . . . . . 3.2 Threeform mattercoupled supergravities . . . . 4 Application to type IIA flux compactifications . . . . 4.1 Dualization to the threeform effective theory . . 4.2 Back to the original theory . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . A Component structure of N = 1 superfields . . . . . . B Note on threeforms, scalar potentials and boundary terms . . . . . . . . . . . . . . . . . . . . . . . . . . C Properties of the Kähler potential and superpotential of type IIA compactifications with RR fluxes . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

Contents
1 Introduction
The physical role of gauge threeforms in fourdimensional
field theories has been studied for several decades. For
instance, constant fourform fluxes of these fields may effect
the value of the cosmological constant directly or via
couplings of the threeforms to membranes (see e.g. [
1–15
]). A
possible role of threeforms in the solution of the strong CP
problem was discussed e.g. in [
16–21
] and in inflationary
models in [
22–27
]. In the context of fourdimensional global
and local supersymmetric theories, threeform gauge fields
can be naturally incorporated as auxiliary fields of
supermultiplets, as e.g. in [
8,12–15,28–41
].
Furthermore, effective field theories with gauge
threeforms can find a natural application in the context of string
compactifications [
25,42–44
]. In particular, the effective
fourdimensional theories describing flux compactifications
of type IIA and IIB string theories should allow for a
supersymmetric formulation including gauge threeforms, whose
field strengths are dual to the fluxes threading the internal
compactified space. In [25] it was suggested that threeforms
coming from the dimensional reduction of type II
supergravities could be associated with auxiliary fields of chiral and
gravity multiplets. However, this idea does not seem to be
realizable within any of the fourdimensional
supersymmetric models constructed so far.
This problem motivated us to revisit the role of gauge
threeforms in fourdimensional rigid and local
supersymmetry, focusing on the minimal N = 1 case and looking for
supergravitymatter models in which the results of [
25
] could
fit. More specifically, we will address the following general
question, suggested by the somewhat universal structure of
the fourdimensional effective theories describing string flux
compactifications. Consider a supersymmetric theory with
a set of chiral superfields A and a superpotential of the
form
W = eA
A + m AGAB ( ) B + Wˆ ( ),
(1.1)
where eA and m A are real constants, and Wˆ ( ) and
GAB ( ) are arbitrary holomorphic functions which, even
if not explicitly indicated, can possibly depend on
additional chiral superfields. The question is then: does there
exist an alternative supersymmetric formulation of the
effective theory with a set of pairs of gauge threeforms
( A3A, A˜3A) in which the coupling constants eA and m B
are promoted to vacuum expectation values of the field
strengths F4A = d A3A and F˜4A = d A˜3A? Note that this
procedure is a certain kind of duality transformation that
trades coupling constants for gauge threeforms, which do
not carry propagating degrees of freedom in four
dimensions.
In this paper we will provide a positive answer to this
question. The new formulation will be obtained by a
supersymmetric duality transformation, which modifies the structure
of the chiral multiplets A, substituting their scalar complex
auxiliary fields F A or just the real parts thereof with a
combination of the field strengths F4A and F˜4A. Furthermore, this
procedure naturally generalizes to the locally
supersymmetric case when one of the scalar superfields A (e.g. 0) is
considered to be the compensator of the superWeylinvariant
formulation of supergravity. After gaugefixing the
superWeyl symmetry, the duality transformation involves also the
auxiliary field of the old minimal supergravity multiplet.
Before arriving at the detailed discussion of the general
dualization procedure outlined above, we will first consider
the simpler subcases in which GAB is constant. In these
subcases, our dualization explicitly relates the three known types
of chiral multiplets: the conventional one with the complex
scalar as the auxiliary field, the single threeform multiplet
in which the complex auxiliary field is a sum of a real scalar
and the Hodge dual of the field strength of a real gauge
threeform, and the double threeform multiplet in which the
auxiliary field is the field strength of a complex gauge threeform.
In particular, the single threeform multiplets arise when the
matrix ImGAB is degenerate, as for instance in the extreme
case ImGAB ≡ 0.
In the case of constant GAB the relation between the
conventional chiral and the dual threeform multiplet is linear.
This is no longer true for a general GAB ( ) in which case
the duality relation is nonlinear and might not allow for a
general explicit superfield solution. However, it turns out to
be tractable if we assume that GAB ( ) is identified with the
second derivative of a homogeneous “prepotential” G( ) of
degree 2. In fact, this is what happens in string flux
compactifications.
In the course of the study of the dual formulations with
threeform multiplets we will encounter a subtlety
regarding the presence of boundary terms in the Lagrangian. The
necessity to take into account appropriate boundary terms in
the theories with gauge threeforms, either supersymmetric
or not, is well known (see e.g. [
4,6,7,38
]). As we will show,
our dualization procedure automatically produces the correct
boundary terms, which then do not need to be introduced by
hand.
As a concrete nontrivial example, we will perform the
duality transformation of the supersymmetric effective
theory associated with type IIA orientifold string
compactifications on Calabi–Yau spaces with Ramond–Ramond (RR)
fluxes. This effective theory has a superpotential of the form
(1.1) with GAB ( ) = ∂A∂B G( ) and G( ) being
homogeneous of degree 2. In this superpotential the constants eA and
m B are identified with the quanta of the internal RR fluxes
threading the compactification space and A with a
combination of the Kähler moduli and the superWeyl compensator
superfields. As we will see, the field strengths F4A and F˜4A
produced by the duality procedure perfectly match the field
strengths obtained by direct dimensional reduction of the IIA
RR field strengths in [
25
]. For simplicity, we will work under
the assumption that the internal NSNS flux vanishes, which
allows us to ignore the tadpole cancellation condition. For
more general type IIA, as well as type IIB flux
compactifications, the tadpole condition must be appropriately taken
into account. Furthermore, the dual formulation with gauge
threeforms should allow for a natural incorporation of the
openstring sector into the effective theory, as in [
42–44
]. We
leave these interesting developments for the future.
The paper is organized as follows. In Sect. 2 we
introduce the duality procedure in rigid supersymmetric theories.
We first discuss simpler cases with constant GAB ,
reviewing the structure of the corresponding known types of
chiral threeform multiplets. We then generalize the dualization
procedure to a general GAB ( ), which leads to a nonlinear
duality relation.
In Sect. 3 we extend the duality procedure to
supergravity. We first apply it to pure old minimal N = 1 supergravity
in its superWeylinvariant formulation, producing the
threeform formulations thereof. In particular, this shows how the
different formulations are related to each other by duality
transformations of the corresponding superWeyl
compensators. Then we consider models with chiral multiplets
coupled to supergravity and apply to them the nonlinear
duality transformation put forward in the rigid case. The duality
acts simultaneously on matter superfields and the superWeyl
compensator. In the resulting dual formulation the auxiliary
fields of the chiral and gravity multiplets are expressed in
terms of the gauge threeforms and the scalar fields.
In Sect. 4 we apply the duality transformation to the
effective fourdimensional theory associated with orientifold type
IIA string compactifications with RR fluxes. We also
provide the explicit relation between field strengths of the
fourdimensional theory and the tendimensional RR fields.
In Appendix A we give the component content of the
different fourdimensional N = 1 superfields which are used
in the main text. In Appendix B we show how the
dualization procedure works for a simple bosonic field theory and
then consider an instructive example which explains how the
bosonic boundary terms can be obtained as components of a
superspace defined Lagrangian. Appendix C contains useful
expressions for the applications to type IIA flux
compactifications.
We mainly use notation and conventions of [
45
].
2 Threeform multiplets in supersymmetry
In this section we explain how the dualization procedure
works in the case of rigid N = 1 supersymmetric theories.
In the simplest case of constant GAB in (1.1), it will produce
known variants of offshell chiral multiplets, whose
auxiliary fields are replaced by the field strength of one or two
gauge threeforms. We will refer to these chiral multiplets
as single and double threeform multiplets, respectively. As
we will see, in the case of generic GAB ( ), the dualization
will provide a generalization of these offshell threeform
multiplets.
2.1 Single threeform multiplets
(2.1)
(2.2)
(2.3)
(2.4)
Consider a rigid supersymmetric theory for a set of chiral
superfields
F A
4 = d A3A.
A = ϕ A +
√2θ ψ A + θ 2 F A,
with a superpotential of the form (1.1) in the simplest case in
which ImGAB = 0. In such a case, since GAB is holomorphic,
ReGAB is necessarily constant and then the Lagrangian takes
the form
L =
where r A ≡ eA + m B ReGAB are real constants.
To dualize the Lagrangian (2.2), we promote the constants
r A to chiral superfields X A and introduce real scalar
superfields U A as Lagrange multipliers. The Lagrangian (2.1) gets
substituted by
L =
The chirality of X A (D¯ α˙ X A = 0 = Dα X¯ A) then implies
that X A = r A, with r A being real constants. Plugging this
solution back into (2.3) we get the initial Lagrangian (2.2).
To find the formulation of the theory in terms of
threeform multiplets we vary (2.3) with respect to X A subject to
the boundary conditions
δ X Abd = 0,
which gives
A = Y A,
with
Y A
i
≡ 4
D¯ 2U A.
The superfields Y A differ from ordinary chiral superfields
only in their θ 2components
Y A = y A +
√2θ χ A + θ 2(∗F4A + i D A),
where D A are real auxiliary scalar fields and
Hence the realpart of the ordinary scalar auxiliary fields is
substituted by the field strengths F4A of the gauge threeform
A A, which are part of the U A multiplets (see Appendix A).
3
The threeform fields appear only inside their field strengths
because of the invariance of (2.7) under the gauge
transformations
U A
→
U A + L A,
where L A are arbitrary real linear superfields D2 L A =
D¯ 2 L A = 0. This superspace gauge symmetry incorporates
the bosonic gauge symmetry
A3A → A3A + d 2A,
and mods out the redundant components of U A, which do
not survive the chiral projection 4i D¯ 2.
We will refer to the chiral superfields Y A as single
threeform multiplets. This kind of scalar multiplet was introduced
in [
31
] and studied in detail in [
38
]. For instance [
38
], studied
the relation of these multiplets with other multiplets, in
particular, with conventional chiral multiplets. The above
simple duality argument explicitly shows how the conventional
chiral multiplets and threeform multiplets are related in a
manifestly supersymmetric way.
To complete the dualization procedure, we should also
take into account the equations of motion of A obtained
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
d4θ K (Y, Y¯ ) +
d2θ Wˆ (Y ) + c.c. + Lbd, (2.13)
Combining this equation with the (anti)chirality of its
components, it follows that
where r A can be identified with the real constants appearing
in (2.2).
Finally, let us present the explicit form of the bosonic
sector of the dual Lagrangian:
L
bos
= K AB¯
D A − i ∂m Am A
D B + i ∂n An B
+ i Wˆ A
D A − i ∂m A Am
+ c.c. + Lbbdos,
(2.18)
from (2.3) with δ A subject to the boundary condition
δ Abd = 0. These give the expression for X A in terms
of Y A
1
X A = 4 D¯ 2 K A(Y ) − Wˆ A(Y ),
where K A ≡ ∂A K and WA ≡ ∂A W .
Upon plugging (2.6) and (2.12) into (2.3) we get the dual
Lagrangian describing the dynamics of the superfields Y A
(2.12)
with
bos
Lbd = −∂m i Am A K B A¯ − K AB¯ D B
+ Am A K B A¯ + K AB¯ ∂n ABn
−∂m
Am A Wˆ A + Am A W¯ˆ A¯ ,
Lˆ =
where
(2.17)
(2.19)
(2.20)
(2.21)
A¯ )
(2.22)
(2.23)
where A Am ≡ 31 εmnlp AnAlp = (∗A3A)m . Notice that the
!
boundary term automatically guarantees a consistent
variational principle.
2.2 Double threeform multiplets
Let us now consider the dualization of a Lagrangian with a
slightly more general superpotential (1.1) in which GAB is
a generic constant matrix and its imaginary part is
invertible, det(ImGAB) = 0. Hence we can introduce the arbitrary
complex constants
cA ≡ eA + GAB m B ,
and rewrite the Lagrangian in the form
is a total derivative and hence a boundary term. Notice that
in (2.13) there is no r A A term in the superpotential.
Furthermore, in general the boundary term (2.14) gives a
nonvanishing contribution to the Lagrangian and hence cannot
be neglected.1
The Lagrangian (2.13) has been studied at length in
reference [
38
], to which we refer for further details. In [
38
]
the boundary term has been identified by requiring a
consistent variational principle (for previous discussions in
nonsupersymmetric settings see e.g. [
4,6
]). On the other hand,
the boundary term is automatically produced by our
duality procedure, once we fix the form of the Lagrangian (2.3).
The only apparent ambiguity, related to the choice of the
form i d4θ (X A − X¯ A)U A of the Lagrange multiplier term
in (2.3), is completely fixed by the following criterion: (2.4)
must be produced without having to impose specific
boundary conditions for the gauge superfield U A. Combined with
the boundary condition δ Abd = 0, this implies that in the
dual theory (2.13) we need only impose the gaugeinvariant
boundary condition
δY A
i
bd = 4
D¯ 2δU A
bd = 0.
As a simple consistency check of the equivalence between
the Lagrangians (2.13) and (2.2) we calculate the variation
of (2.13) with respect to U A, which results in an equation of
motion of the form
Im
1
− 4 D¯ 2 K A + Wˆ A
= 0.
1 Note that the Lagrangians (2.3) and (2.13) are gauge invariant under
(2.10) provided X A satisfy the boundary conditions X Abd = rA, where
rA are (at least, classically) arbitrary real constants which characterize
the asymptotic vacuum of the theory. From (2.12) these boundary
conditions translate into corresponding boundary conditions for Y A.
As in Sect. 2.1, we can promote the constants to chiral
superfields X A by adding appropriate Lagrange multiplier
terms to the Lagrangian. The modified Lagrangian is
L =
see Eq. (A.12) for the component expansion of . This
constraint is explicitly solved in terms of a general Weyl spinor
superfield αA as
By integrating out αA from (2.22) we get the condition
Dα X A = 0, which, combined with the chirality of X A,
implies that
where cA are arbitrary constants. Inserting (2.25) into (2.22)
one gets back the Lagrangian (2.21). On the other hand, we
can integrate out X A by imposing their equations of motion
and get
A = S A
≡ − 41 D¯ 2 ¯ A,
S A = s A +
√2θ λA + θ 2 ∗G4A.
Here ∗G4A are Hodge duals2 of the field strengths
G4A = dC3A,
of complex 3form gauge fields C3A. The Hodge duals of
C3A are complex vector components of the complex linear
superfields A (see Appendix A).
We call the chiral superfields S A double threeform
multiplets. These kinds of multiplets were introduced in [
32
] and
considered in more detail in [
36
] but, in contrast to the
single threeform multiplets Y A of Sect. 2.1, they have attracted
much less attention in the literature. The bosonic gauge
transformation C3A → C3A + d 2A (where 2A is a complex
twoform) are part of the gauge superfield transformation
A + L1A + i L2A,
(2.24)
(2.25)
(2.27)
(2.28)
where S A are chiral superfields with the following θ
expansion:
In (2.32) X A should be replaced by its expression obtained
from (2.22) as the equation of motion of A, namely
Note that the Lagrange multiplier term in (2.22) is singled
out by a criterion analogous to the one introduced at the end
of Sect. 2.1. Namely, it leads to (2.25) without the need for
any specific boundary condition on the gauge superfield αA
and it directly gives back the original Lagrangian, without
involving possible boundary terms. As a consequence, the
dual Lagrangian describing the dynamics of the superfields
S A is also completely fixed, including the appropriate
boundary term. Indeed, by plugging (2.26) back into (2.22), we get
the dual Lagrangian
Lˆ =
→
→
where L1A and L2A are real linear superfields.
It is easy to see that (2.29) leaves S A invariant. The
counterparts of the gauge transformations (2.29) acting on the
‘prepotential’ α are
W = ra a + c p
p + W˜ ( ),
where
A
α +
α + Dβ βAα,
A
(2.30)
ra ≡ (eA + m B ReGAB )uaA, c p ≡ eAv p + mq vqAGAB v p
A B
A
where D¯ β˙ α = 0 and
A
βα =
αAβ .
2 In our conventions, the fourdimensional Hodge dual of a pform ω
is defined by (∗ω)m1...m4−p = p1! εm1...m4−pn1...n p ωn1...n p , where ε0123 =
−ε0123 = 1.
3 The free Lagrangian Lˆfree = d4θ SS¯ was briefly discussed in [
32
].
The component form of (2.31) with K = δAB¯ S A S¯ B¯ and Wˆ (S) =
m AB S A S B + gABC S A S B SC but without the boundary term was
considered in [
36
].
1
X A = 4 D¯ 2 K A − WA.
An example of the component field form of the boundary
term which one gets from (2.32) is given in Appendix B.3
Let us now turn to the case of constant GAB with
noninvertible imaginary part ImGAB. If A, B = 1, . . . , n, then
the matrix ImGAB has a rank r < n. This implies that there
are n − r > 0 vectors u A, a = 1, . . . , n − r , such that
a
ImGAB uaB = 0. We can complete them with r vectors vqA,
q = 1, . . . , r , which together with uaA form a basis of Rn .
We can use this basis to reorganize the chiral superfields as
follows:
A
=
a u A
a +
q A
vq .
(2.29)
and, analogously, m A = ma uaA + m pv pA. Then the
superpotential (1.1) takes the form
(2.33)
(2.34)
(2.35)
(2.36)
are, respectively, arbitrary real and complex constants and
W˜ ≡ Wˆ + ma uaAGAB v pB p.
(2.37)
d2θ D¯ 2
A MAB (X A−X¯ A) +c.c. (2.40)
Lˆ =
d2θ Wˆ (S) + c.c. +Lbd, (2.46)
Dα(MAB ImXB) = 0.
Notice that the variation of (2.40) with respect to A does
not involve any boundary terms and the Lagrange multiplier
term in (2.40) satisfies the criterion discussed in the previous
sections. The general solution of (2.41) is
X A = eA + GAB ( )m B ,
with eA and m B being arbitrary real constants.5 Hence, by
plugging (2.42) back into (2.40) one obtains the original
Lagrangian (2.39).
Alternatively, we get the dual description by integrating
out X A in (2.39). This results in the following expression for
the chiral superfields A:
(2.41)
(2.42)
(2.43)
(2.44)
(2.45)
We can then proceed by dualizing a to single threeform
multiplets Y a as in Sect. 2.1 and q to double threeform
multiplets Sq as in the present section.4
2.3 Double threeform multiplets and nonlinear
dualization
We are now ready to consider the more general case of
nonconstant holomorphic matrix GAB ( ), still in the case of rigid
supersymmetry. Even though not explicitly indicated, the
following discussion allows for the inclusion of additional chiral
multiplets in the theory, which can enter GAB ( ) and Wˆ ( )
in (1.1), but which are not subject to the dualization
procedure. For instance, extra chiral multiplets T p will explicitly
appear in Sect. 4, in which we will apply our construction to
type IIA flux compactifications.
For convenience we define the matrices
NAB = ReGAB ,
MAB = ImGAB.
(2.38)
We will assume that, for generic values of the chiral fields
A, the matrix MAB is invertible. We will briefly come back
to the degenerate case det(MAB ) = 0 at the end of the
section. Furthermore, for simplicity, we assume that GAB ( ) is
symmetric, although most of the discussion holds for
nonsymmetric GAB ( ). This symmetry is automatic if we regard
GAB ( ) as the second derivative of a holomorphic
prepotential G( ), as we will assume in the local supersymmetry
case.
Our starting point is the Lagrangian
L =
d4θ K ( , ¯ )
+
d2θ eA
A + m AGAB ( ) B +Wˆ ( ) +c.c. .
The strategy followed in the previous sections is then
generalized by replacing (2.39) with the following Lagrangian:
L =
d4θ K ( , ¯ )+
4 Notice that the choice of the vectors uaA is not unique, as we could
raemdbefiignuaeitvyqAin→ducveqAs+thαeqarueaAdewfiinthitiαoqansbeinag →arbitraary−reαaqal coqnsatnadntsc.qTh→is
cq + αq ra, which mix the two kinds of dual threeform multiplets.
5 Indeed, from (2.41) and its complex conjugate one gets MAB ImXB =
mA, with m A being arbitrary real constants. We can then write X A =
ReX A + iIm GAB m B ≡ Re(X A − GAB m B ) + GAB m B . This
equation is compatible with the chirality of X A and GAB only if Re(X A −
GAB m B ) = eA are constant. We thus arrive at (2.42).
A = S A,
where
S A
≡ 41 D¯ 2
MAB ( B − ¯ B ) .
The chiral superfields S A provide a generalization of the
double threeform multiplets encountered in Sect. 2.2. Note that,
once we impose (2.43), MAB depends on S A. Then, in
general, Eq. (2.44) is nonlinear and cannot be explicitly solved
for S A as a function of A. However, this does not necessarily
create complications in specific applications, as for instance
to type IIA flux compactifications discussed in Sect. 4.
The above formulation in terms of A, which contains
gauge threeforms, is invariant under the following gauge
transformations which generalize (2.29):
A
→
A + L˜ A + GAB L B ,
where L˜ A and L B are arbitrary real linear superfield
parameters. This gauge symmetry guarantees that the gauge
threeforms enter (2.43) via their gaugeinvariant field strengths
only. We will discuss the component structure of the relation
(2.44) in the supergravity case in Sect. 3.2.
If we substitute the solution (2.43) back into the
Lagrangian (2.40) we obtain
where the boundary term is now given by the total derivative
contribution
Lbd =
d2θ
d2θ¯+ 41 D¯ 2
X AMAB ( B − ¯ B ) +c.c.,
(2.47)
in which X A is expressed via A on account of the equation
of motion of A, as in Sect. 2.2. We will give the explicit
expression of the boundary term in the supergravity case in
the next section.
The Lagrangian (2.31) provides us with the dual
formulation of the considered theory in terms of the double
threeform multiplet (2.44), with the ‘reduced’
superpotential Wˆ (S) and the appropriate boundary term. The
information as regards the form of the matrix GAB ( ) appearing in
the superpotential of the original theory is encoded in the
form of the matrix MAB which enters the definition (2.44)
of the double threeform multiplet. On the other hand, as in
the previous sections, the constant parameters eA and m A got
dualized into the expectation values of the field strengths of
the gauge threeforms.
Before passing to the locally supersymmetric case, let us
briefly discuss the situation in which Im GAB , with A, B =
1, . . . , n, is degenerate of rank r < n. Then there should
exist n − r > 0 real vectors uaA, a = 1, . . . , n − r , such that
Im GAB ( )uaB = 0 and hence GAB ( )uaB = G¯AB ( ¯ )uaB .
Taking into account the holomorphicity of GAB ( ), this
condition is quite strong and puts strong constraints on the form
of GAB ( ). Suppose, for instance, that the vectors uaA are
constant, as at the end of Sect. 2.2. This would imply that
GAB ( )uaA is constant too. We could then proceed as in Sect.
2.2, rewriting the superpotential as follows:
W = ra a + e p + mq Gqp( )
p + Wˆ ( ),
(2.48)
where ea ≡ eAuaA, Gqp ≡ uqAGAB u Bp , ra is as in (2.36) and
Wˆ is as in (2.37). One can then dualize a to single
threeform multiplets Y a and p to double threeform multiplets
S p. We expect similar combinations of different dualizations
to be possible in more general cases.
3 Threeform multiplets in N = 1 supergravity
We now extend the duality procedure described in Sect. 2 for
rigid supersymmetry to mattercoupled N = 1
supergravity. The extension is rather natural if we use a
superWeylinvariant approach [
46
]. Before proceeding let us recall that
the old minimal formulation of supergravity [
28
] describes
the interactions of the gravitational multiplet
(3.1)
The physical fields are the vielbein eam and the gravitino ψmα ,
whereas the auxiliary fields are the real vector ba and the
complex scalar M .
We will construct threeform mattercoupled supergravity
by dualizing a superWeylinvariant formulation. The curved
superspace supervielbeins transform as follows under the
superWeyl transformations [
46
]:
E M → eϒ+ϒ¯ E aM ,
a
E αM → e2ϒ¯ −ϒ
i
E αM − 4 E aM σaαα˙ D¯ α˙ ϒ¯ ,
where (a, α) are flat superspace indices, M = (m, μ) are
curved indices and ϒ is an arbitrary chiral superfield
parameterizing the superWeyl transformation. We will focus on a
theory for n + 1 chiral multiplets Z A, A = 0, . . . , n, that
transform as follows under superWeyl transformations:
Z
A → e−6ϒ Z A.
The chiral superfields Z A comprise, in a democratic way, a
superWeyl compensator and n physical multiplets.
The ordinary old minimal formulation of supergravity
is obtained by choosing a superWeyl compensator Z , e.g.
Z ≡ Z0, and subject it to a gaugefixing condition using the
superWeyl invariance. On the other hand, we will perform
the duality transformation of the conventional chiral
multiplets Z A to threeform multiplets before gaugefixing the
superWeyl invariance. In this way, the procedure will work
exactly as in the rigid supersymmetry case, but will involve
the superWeyl compensator in addition to the physical
chiral superfields. Gaugefixing the superWeyl symmetry
afterwards will produce a Lagrangian describing the coupling of
threeform multiplets to a supergravity multiplet with one or
two gauge fields substituting the scalar auxiliary fields.
In the next section we will focus on pure supergravity and
its threeform variants. The inclusion of additional physical
chiral multiplets and a general superpotential of the form
(1.1) will be considered in Sect. 3.2. The following
discussion can include additional ‘spectator’ matter or gauge
multiplets, which will not be explicitly indicated for notational
simplicity.
3.1 Variant minimal supergravities from duality
We start by considering the minimal theory, in which the old
minimal supergravity multiplet is coupled just to the
superWeyl compensator Z , which transforms as in (3.3). Then,
up to a complex constant rescaling of Z , the most general
superWeylinvariant Lagrangian has the form
in which E denotes the Berezinian superdeterminant of the
supervielbein, d2 2E is a chiral superspace measure [
45
]
and c is an arbitrary complex number which gives rise to the
gravitational cosmological constant and the gravitino mass.
Under (3.2), the superspace measures rescale as
E → e2(ϒ+ϒ¯ ) E , d2
E → e6ϒ d2
E .
(3.5)
Hence the superWeyl invariance of the supergravity
Lagrangian is manifest. We can now follow Sect. 2, distinguishing
two cases.
3.1.1 Single threeform supergravity
We first proceed along the lines of Sect. 2.1, setting c ≡ ir ,
with real r , and promoting r to a chiral multiplet X by adding
an appropriate Lagrange multiplier.6 Consider the modified
Lagrangian
where U is a scalar real superfield and R is the chiral
superfield curvature whose leading component is the auxiliary field
M = − 61 R of the gravity multiplet. Notice that (3.6) is
superWeyl invariant if we impose the requirement that
U → e−2(ϒ+ϒ¯ )U,
under superWeyl transformations, since D¯ 2 − 8R →
e−4ϒ (D¯ 2 − 8R)e2ϒ¯ .
Integrating U out of (3.6) by imposing its equation of
motion implies that X must be an arbitrary real constant r
and then one goes back to (3.4). Instead, integrating out X
gives
Z ≡ Y,
1
Y ≡ − 4
where the chiral superfield
D¯ 2 − 8R U
is the natural generalization of the rigid single threeform
multiplets discussed in Sect. 2.1. In particular, the bosonic
threeform A3 is contained in the component
6 We choose a purely imaginary c in order to obtain the single
threeform supergravity in its most common form, as used for instance in
[
8,13–15,32,33,35
]. Clearly, by a simple redefinition Z → −i Z one
can make c purely real.
− 81 σ¯mα˙ α[Dα, D¯ α˙ ]U ≡ (∗A3)m ,
of U . The bosonic gauge transformation A3 → A3 + d 2
is contained in the superfield gauge transformation U →
U + L, where L is an arbitrary linear multiplet. This gauge
invariance allows one to write the superfield U in an
appropriate WZ gauge U  = 0, which we have already used in
(3.10).
By integrating out Z one gets the equation
and by plugging (3.8) and (3.11) back into (3.6) one obtains
the dual Lagrangian
By recalling the definition of Y given in (3.9) and its
expansion (2.8) and skipping the dependence on the fermions, the
lowest component of this equation gives Y  = 1 while the
1
highest component − 4 D2Y  = 0 gives ImM + ∗dA3 = 0, so
that the conventional scalar auxiliary field of the supergravity
multiplet has the form M = ReM − i ∗F4, as proposed in [
32
]
and discussed in detail in [
8
]. Hence, the component fields
of the supergravity multiplet of this formulation are
Note that Lbd is indeed a total derivative. Y transforms as Z
under superWeyl transformations (Y → e−6ϒ Y ) and plays
the role of the superWeyl compensator.
It is well known that different offshell formulations of
fourdimensional N = 1 supergravity can be obtained
from its superconformal version by choosing different
compensator fields [
47–49
]. Here the use of Y as a
compensator in the superWeylinvariant formulation leads, as was
shown in [
35
], to the threeform minimal supergravity [
8,13–
15,32,33
], in which the imaginary part of the old minimal
auxiliary field M is substituted by the Hodge dual of a real
field strength F4 = d A3.
In order to see this, we can use the superWeyl symmetry
to set
3.1.2 Double threeform supergravity
In order to arrive at the minimal double threeform
supergravity [
28
] we must promote the entire arbitrary constant c
to a dynamical chiral field X and proceed as in the previous
examples. This can be done by starting from the Lagrangian
where = D¯ ¯ is a complex linear superfield, the locally
supersymmetric generalization of the complex linear
superfield introduced in Sect. 2.2. The components of in the
appropriate WZ gauge are
21 σ¯mα˙ α[Dα, D¯ α˙ ]  = −i Cm ,
D2D¯ 2 ¯  = 8 ∗G¯ 4 + 16M¯ s,
with G4 ≡ dC3 and Cm ≡ (∗C3)m . One can go to this gauge
because of the invariance of the construction under the
superfield gauge transformation of the form (2.29)–(2.30).
The action (3.16) is invariant under superWeyl
transformations if α, and eventually , transform as follows [
48
]:
α → e−3ϒ
α,
→ e−2(ϒ+ϒ¯ ) .
As in the previous examples, by integrating out α one gets
back (3.4). On the other hand, by integrating out X and Z
one finds
1
Z = S ≡ − 4
1
X = − 4
where S is a double threeform multiplet which plays the role
of the superWeyl compensator. Note that X and S in (3.20)
are given by (3.19), and that the second term in (3.20) is the
boundary term.
where the kinetic potential (Z, Z¯ ) and the superpotential
W(Z) have the following homogeneity properties:
2
(λZ, λ¯Z¯ ) = λ 3 (Z, Z¯ ),
W(λZ) = λW(Z). (3.26)
(3.21)
(3.22)
(3.23)
(3.24)
One can then gaugefix the superWeyl invariance by
putting S = 1 and find that
where C3 is a complex threeform. Therefore we refer to this
formulation as double threeform supergravity. The bosonic
sector of this minimal supergravity theory follows from the
Lagrangian (3.20) and has the following form:
If we integrate out C3 by inserting (3.24) into the Lagrangian
(3.23) we find the standard supergravity theory with a
negative cosmological constant. Notice that (3.23) has a
welldefined variation with respect to C3 thanks to the presence
of the boundary term. As in the previous sections, this is
guaranteed by our duality procedure once one appropriately
chooses the form of the Lagrange multiplier term in (3.16).
3.2 Threeform mattercoupled supergravities
In the previous section we obtained known minimal
threeform supergravities with the use of the locally
supersymmetric counterpart of the duality procedure described in Sect. 2.
We now pass to the considerably more general case outlined
at the beginning of this section. We consider a
superWeylinvariant supergravity theory coupled to n + 1 chiral
superfields Z A which transform as in (3.3). We stress once again
that, even if not explicitly indicated for notational
simplicity, additional spectator chiral and vector multiplets may be
included without difficulties (as in the example discussed in
Sect. 4).
The general form of the superWeylinvariant Lagrangian
d4θ E
(Z, Z¯ ) +
d2
2E W(Z),
(3.25)
is
where Z0A( ) is a set of functions of the physical chiral
multiplets i (i = 1, . . . , n), which are inert under the
superWeyl transformations. Clearly, the split (3.27) has a large
arbitrariness and one may redefine
Z → e− f ( ) Z ,
Z0A( ) → e f ( )Z0A( ).
The kinetic potential (Z, Z¯ ) can be written as follows:
(Z, Z¯ ) = Z  23 e− 31 K ( , ¯ ),
where K ( , ¯ ) ≡ −3 log (Z0( ), Z¯0( ¯ )) is the
ordinary Kähler potential. Note that the possibility of making the
redefinition (3.28) corresponds to the invariance under
Kähler transformations K ( , ¯ ) → K ( , ¯ ) − f ( ) − f¯( ¯ ).
The conventional superpotential W ( ) is singled out by
using the split (3.27) and defining
W(Z) = Z W ( ),
where W ( ) ≡ W(Z0( )). Under the redefinition (3.28) W
transforms as follows: W ( ) → e f ( )W ( ). The
conventional formulation can then be obtained by gaugefixing the
superWeyl invariance, e.g. by putting
Before discussing the duality procedure, let us briefly recall
how this formulation is related to the more standard
supergravity formulation. First, one singles out a superWeyl
compensator Z as follows:
Z = 1.
In order to perform the duality procedure, let us come back
to the superWeylinvariant Lagrangian (3.25) and consider
the superpotential of the form
W(Z) ≡ eAZ A + m B GB A(Z)Z A + Wˆ (Z).
The homogeneity condition (3.26) requires that GAB (λZ) =
GAB (Z) and Wˆ (λZ) = λWˆ (Z). Though the construction
under consideration can be applied to generic GAB , we will
restrict ourselves to the case in which
GAB (Z) ≡ ∂A∂B G(Z),
with G(Z) being a (possibly locally defined) homogeneous
prepotential of degree 2 G(λZ) = λ2G(Z) defining a local
special Kähler space parametrized by homogeneous
coordinates Z A, A = 0, 1, . . . , n.7 As we will see, string flux
com7 The minimal supergravities considered in Sect. 3.1 correspond to the
simplest subcases with n = 0, Z0 = Z and c ≡ e0 + G00m0, where G00
pactifications have superpotentials of this kind with (eA, m B )
representing appropriately quantized units of fluxes.
We would like to make the 2n + 2 constants (eA, m A) in
(3.32) dynamical, i.e. to replace them with the field strengths
of 2n +2 threeforms. This is achieved by dualizing the chiral
fields Z A, easily adapting the procedure introduced in Sect.
2.3 for the rigid supersymmetric case. As in that section, we
assume that MAB defined as in (2.38) is invertible. (The
case of degenerate MAB can be addressed as outlined in
Sect. 2, combining dualizations to single and double
threeform multiplets.) First, we substitute the chiral superspace
integral of the superpotential term (3.32) with
LX = 2
d2
E
X AZ A
where, as in the rigid supersymmetry case, MAB is inverse
of MAB = Im GAB , X A are chiral superfields and A are
complex linear superfields A ≡ D¯ α˙ ¯ αA˙. Upon integrating
out αA one gets back (3.25). On the other hand, by integrating
out X A and Z A one finds
Z
where the chiral superfields S A are double threeform
multiplets, defined by the generalization of (2.44),
S A = 41 (D¯ 2 − 8R) MAB ( B − ¯ B ) ,
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
(3.35)
(3.36)
(3.38)
1
X A = −Wˆ A + 4 (D¯ 2 − 8R)
×
A +
∂MBC
∂ S A
The Lagrangian then reads
X B − X¯ B
C − ¯ C
. (3.37)
Lˆ = − 3
d4θ E (S, S¯) +
d2 2E Wˆ (S) + c.c. + Lbd,
in which the boundary term is given by the X dependent
part of (3.34) once one replaces Z A with S A and X A with
(3.37). Note that, as in the rigid supersymmetry case, the
dual Lagrangian does not have the part of the superpotential
that depended on eA and m A. We thus end up with a theory
in which the only independent superfields are the complex
linear multiplets A.
Footnote 7 continued
is necessarily constant by homogeneity. In particular, the single
threeform minimal supergravity of Sect. 3.1.1 is obtained by setting G00 = 0
and redefining Z → i Z .
The double threeform multiplets S A are defined by (3.36),
in which MAB should be considered as a function of S A and
S¯ A. Hence (3.36) is nonlinear and so is not generically
solvable for S A as functions of A. However, it turns out to be
tractable for superfield components. For simplicity, we will
restrict ourselves to the bosonic ones setting the fermionic
components equal to zero. Using the local symmetry (2.45)
we can impose the Wess–Zumino gauge in which, in
particular, A = 0. Then the remaining bosonic components of
A are
D
2
A = −4s¯A,
21 σ¯mα˙ α[Dα, D¯ α˙ ] A = − A˜ Am − GAB AmB ,
D2D¯ 2 ¯ A = 8i Dm
A˜mA + G¯AB ABm
+ 16M¯ sA,
where AmA ≡ (∗A3A)m and A˜ Am ≡ (∗A˜3A)m .
From (3.36) it follows that the scalar component sA, with
lower indices, appearing in (3.39) is related to s A ≡ S A,
with upper indices, by the inverse metric MAB . Since S A ≡
Z A ≡ z A, we can use z A instead of s A and write this relation
as follows:
z A
= MAB (z, z¯) sB .
(3.40)
In general it is not possible to explicitly invert the above
expression and express z A in terms of the scalar fields sA
of the complex linear superfield ¯ A. Hence, in what follows
it will be more convenient to use z A as independent scalar
fields in the component Lagrangians which we will shortly
present. The θ 2component of (3.36) is then
F A 1 i
S ≡ − 4 D2 S A = M¯ z A + 2 M
AB
×
∗F˜4B + G¯BC ∗F4C + 2Re G¯BC D F¯SD z¯C
,
(3.41)
where F˜4A = d A˜3A, F4A = d A3A and GABC ≡ ∂AGBC . Now,
taking into account that z AGABC = 0 by homogeneity, we
reduce Eq. (3.41) to
FSA = M¯ z A + 2i MAB ∗F˜4B + G¯BC ∗F4C .
To fix the superWeyl invariance it turns out to be convenient
to choose one of the superfields S A ( A = (0, i )), say S0, as
the superWeyl compensator and impose
S0 = 1.
The superspace condition (3.43) implies the component field
conditions z0 = 1 and F 0
S = 0. The bosonic relations (3.42)
(3.42)
(3.43)
(3.39)
+
d2
2E Wˆ (S) + c.c.
split as follows:
i
M¯ = − 2 M0B (z, z¯) ∗F˜4B + G¯BC (z¯) ∗F4C ,
FSi = M¯ zi + 2i Mi B (z, z¯) ∗F˜4B + G¯BC (z¯) ∗F4C ,
where zi ≡ Si  and FSi ≡ − 41 D2 Si  are the lowest and
highest scalar components of the threeform multiplets Si
(i = 1, . . . , n).
After having gaugefixed the superWeyl symmetry, the
Lagrangian describing the coupling of the Si superfields to
supergravity takes the form
(3.44)
Note that in this Lagrangian the scalar auxiliary fields of the
gravity and matter multiplets are defined by (3.44) (ignoring
fermions).
4 Application to type IIA flux compactifications
As an application of the above dualization procedure, we will
now consider an example of type IIA flux compactifications
of string theory on a Calabi–Yau threefold C Y3 in the
presence of O6planes. In particular, we will focus on the effective
theory obtained by turning on RR fluxes in the internal C Y3
space. For simplicity, we will also set the internal NSNS flux
H3 to zero, so that the tadpole condition just requires that the
O6 charge is canceled by the presence of D6planes, without
involving the RR fluxes.
We will focus on the closed string scalar spectrum. The
relevant terms in the effective N = 1 supergravity for these
kinds of compactifications can be found in [
50
]. The closed
string moduli vi (x ) and bi (x ), i = 1, . . . , h1−,1(C Y3), are
obtained by expanding the Kähler form J and the NSNS
twoform B2 in a basis of orientifoldodd integral harmonic
2forms ωi ∈ H 2 (X ; Z)
−
J = vi ωi ,
B2 = bi ωi .
These moduli, together with their supersymmetric partners,
combine into n ≡ h1−,1(C Y3) chiral superfields i with
lowest components
i  = ϕi = vi − i bi .
Furthermore, the complex structure, the dilaton, the internal
RR threeform moduli and the associated supersymmetric
partners combine into additional chiral superfields T q , q =
(4.1)
(4.2)
1, . . . , h2,1(C Y3) + 1. The effective supergravity theory is
characterised by the following Kähler potential 8:
K( , ¯ , T , T¯ ) = K ( , ¯ ) + Kˆ (T , T¯ ).
In the following we will not need the explicit form of
Kˆ (T , T¯ ), but we will just use the fact that it satisfies the
condition
Kˆ r¯q Kˆ q Kˆr¯ = 4,
where Kˆ q ≡ ∂∂TKˆq , Kˆ qr¯ ≡ ∂T∂q2∂KˆT¯ q¯ , …, and Kˆ r¯q is the
∂ K ,
inverse of the Kähler metric Kˆ qr¯ . Similarly, Ki ≡ ∂ i
Kij¯ ≡ ∂ ∂i2∂K¯ j¯ , …, and K j¯i is the inverse of the Kähler
metric Kij¯.
The Kähler potential K ( , ¯ ) is given by
K ( , ¯ ) = − log
1
3 ki jk (Re i )(Re j )(Re k ) ,
!
where ki jk are the intersection numbers
CY3
ki jk =
ωi ∧ ω j ∧ ωk .
Notice that K ( , ¯ ) depends only on the real combinations
i + ¯ i , so that we can make the identification Ki ≡ Kı¯.
We will also use the fact that K ( , ¯ ) satisfies the noscale
condition
K j¯i Ki Kj¯ − 3 = 0.
The fluxinduced superpotential is of the form introduced in
[
51–53
] and depends only on the chiral superfields i 9
1
W = e0 +i ei i − 2 ki jk m
i j k
i
+ 6 m0ki jk
i j k , (4.8)
where e0, ei , mi and m0 represent the flux quanta of the
internal RR fields.
4.1 Dualization to the threeform effective theory
The effective theory described above has exactly the same
structure as the theories considered in Sect. 3.2, up to the
explicit presence of a spectator sector given by the chiral
fields T r . In order to make this similarity manifest, we rewrite
8 The formulas of [
50
] are valid in the large volume and constant
warping approximation, which then neglects the backreaction of the fluxes
and branes on the underlying Calabi–Yau geometry. The backreaction
of fluxes and branes is expected to break the split structure of (4.3).
9 In what follows we will tend to use notation close to that of [
25,44
].
F i ≡ −i FSi
In the following it will also be convenient to use
instead of FSi , such that − 41 D2 i = F i .
Upon setting to zero the fermions, the independent bosonic
components of these superfields are given by (3.40) and
this theory in a superWeylinvariant form by adding a
superWeyl compensator Z and combining it with the chiral fields
i into n + 1 chiral superfields Z A = (Z0, Zi ) such that
Z0 ≡ Z
and
Zi ≡ i Z i ,
which transform as in (3.3) under the superWeyl
transformations. Then it is easy to see that the superpotential (4.8)
gets transformed into (3.30) of the form
W(Z) = eAZ A + 2Z1 0 mi ki jk Z j Zk − 6(Z10)2 m0ki jk Zi Z j Zk .
This clearly satisfies the homogeneity condition (3.26) and
can be written in the form (3.32) with Wˆ (Z) = 0 and GAB =
∂A∂B G(Z), where
1 i j k
G(Z) = 6 0 ki jk Z Z Z .
Z
We are now in a position to apply the duality
transformation described in Sect. 3.2. After dualization and gaugefixing
the superWeyl symmetry by setting
Z = S0 = 1,
the final result is a Lagrangian of the form (3.45) with Wˆ = 0
and a Kähler potential which is modified by a contribution
of the ‘spectators’ T r
Moreover, the superpotential has completely disappeared
from the dual effective theory, since it is now encoded in
the structure of the constrained superfields (3.36).
Notice that because of the definition (4.10), after
dualization and gaugefixing we have i = −i Si and we can
identify the lowest components as follows:
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.15)
(4.16)
(3.44). The latter take the following form in the case under
consideration10
1
Re M = 2 ∗F40,
1
Im M = −2eK ∗F˜40 − 2 Ki ∗F4i ,
Re F i = 41 ∗F40 vi − eK (K i j − 2vi v j ) ∗F˜4 j ,
Im F i = 2eK ∗F˜40 vi + 21 ( ∗F4i + vi K j ∗F4j ),
where the fourforms F4A and F˜4 A are defined in terms of the
field strengths F A
4 = d A3A and F˜4 A = d A˜ 3 A as follows:
F40 = −F40, F4i = −F4i + bi F40,
1
4 − 2 ki j k b j bk F40,
F˜4i = F˜4i + ki j k b j F k
1 1
4 − 6 ki j k bi b j bk F 0.
F˜40 = F˜40 + bi F˜4i + 2 ki j k bi b j F k 4
Note that the fourforms F4A and F˜4 A have exactly the same
structure as the fourforms obtained in [
25, 44
] upon
dimensional reduction of the type IIA RR field strengths,11 which
is quite encouraging. To convince ourselves that this is not
a mere coincidence, in the following section we will
compute the onshell values of (4.18) by solving the equations
of motion of A3A and A˜ 3 A, which follow from the dual
Lagrangian (4.14). As we will see, these onshell values
perfectly match those obtained by tentofourdimensional
reduction [
25, 44
].
The bosonic part of the dual Lagrangian (4.14) can be
computed by setting to zero fermionic component fields,
integrating over the Grassmann variables and integrating out the
supergravity auxiliary vector field. Finally, one can go to the
Einstein frame by performing a Weyl rescaling of the
vielbeins, the dual fourform field strengths and the component
fields in (4.17) as follows:
1
eam → eam e 6 (K + Kˆ ),
F i → F i e− 23 (K +Kˆ ),
2
M → M e− 3 (K + Kˆ ),
FTq → FTq e− 23 (K +Kˆ ). (4.19)
The result is the following bosonic Lagrangian:
1
e−1Lbos = − 2 R − Kij¯(ϕ, ϕ¯ ) ∂ ϕi ∂ ϕ¯ j¯ − Kˆ qr¯ (t , t¯) ∂ t q ∂ t¯r¯
(4.17)
(4.18)
+e−1 L3form,
(4.20)
10 To arrive at these relations we have used the specific form of the
Kähler and superpotenional associated to the type IIA compactification
in question given in Appendix C.
11 The structure of these field strengths reflects the B2twisting of the
tendimensional RRfields in the so called Abasis of the democratic
formulation of [
54
], which provides a dualitysymmetric description of the
type IIA supergravity theory, whose E11 origin was revealed in [
55,56
].
in which t q ≡ T q  and L3form contains the threeform sector
encoded in M and F i as in (4.17) and the auxiliary fields FTq
of the T q multiplets
q r
e−1L3form = e−K K ij¯ F i F¯ j¯ + e−K Kˆ qr¯ FT F¯T¯
− 31 e−K M + Kı¯ F¯ ı¯ + Kˆ q¯ F¯Tq¯
×
M¯ + Ki F i + Kˆ q FTq
where we recall that K ≡ K +Kˆ . With the help of the noscale
condition (4.4), we can easily integrate out the auxiliary fields
FTq by solving their equations of motion, whose solution is
r
Kˆ qr¯ F¯T¯ = −
M + Kı¯ F¯ ı¯ Kˆ q .
(4.22)
Substituting it back into the Lagrangian (4.21) and using
(4.17) we obtain the following Lagrangian for the gauge
threeforms:
eKˆ e−1L3form =
e−K
+ 4
with the boundary term
j
Ki j ∗F4i ∗F4 + 4eK
e−K
16
Lbd = −2∂m e A˜0m 4eK −Kˆ ∗F˜40 + e A˜im eK −Kˆ
(4.24)
where we recall that AmA ≡ (∗ A3A)m and A˜ Am ≡ (∗ A˜ 3 A)m .
This boundary term is directly extracted by writing the
superspace Lagrangian (3.34) in components.
The Lagrangian (4.20)–(4.24) provides a nontrivial
example of the effect of the nonlinear dualization procedure put
forward in this paper. We explicitly see that it does not
depend on the constants eA and m A appearing in the
original Lagrangian and does not contain any potential for the
scalar fields. Rather, as we will discuss in the next section, it
is generated dynamically by the gauge threeforms A3A and
A˜ 3 A.
4.2 Back to the original theory
Let us show how the bosonic Lagrangian of the original
theory is reproduced from the bosonic Lagrangian (4.20). This is
done by integrating out the gauge threeforms A3A and A˜3A,
which enter F4A and F˜4A as defined in (4.18). Indeed, the
integration of the equations of motion which follow from
(4.23) produces the following expressions involving 2n + 2
integration constants which, for obvious reasons, we call eA
and m A:
− 4e−Kˆ eK ∗F˜40 = m0,
−e−Kˆ eK K i j ∗F˜4 j = mi − mbi ≡ pi ,
− 41 e−(K +Kˆ ) Ki j ∗F4j = ei + ki jk b j mk − 21 ki jk b j bk m0 ≡ ρi ,
− 116 e−(K +Kˆ ) ∗F40 = e0 + bi ei + 21 ki jk bi b j mk
These are exactly (modulo some conventions) the onshell
values of the fourforms obtained in [
25,44
] by
dimensionally reducing the tendimensional Hodge duality relations
between the type IIA RR field strengths.
Substituting (4.25) back into the bosonic Lagrangian
(4.23) and (4.24), one obtains the scalar potential of the
original theory which coincides with the wellknown form of the
type IIA RR flux potential [
50,57
],
(4.26)
V = −e−1 L3formonshell
2
= eKˆ 16eK ρ0 + 4eK K i j ρi ρ j
+e−K Ki j pi p j + 41 (m0)2e−K .
Note that upon this substitution the term (4.24) is no more
a total derivative. Without the contribution of this term, the
effective scalar potential would have a wrong (negative) sign.
This is why we have kept track of the boundary terms in our
construction all the time.
Note also that, if we substitute the onshell values (4.25)
of the fourforms F4A and F˜4A into the boundary Lagrangian
(4.24), while still keeping the potentials Am A and A˜mA
offshell, upon some algebra we get
Lˆbd = 2e ρ0 ∗F40 + ρi ∗F4i + pi ∗F˜4i + m0 ∗F˜40
= 2∂m
e m A A˜mA − eA A Am
.
(4.27)
This boundary term is the same as the linear term of
the effective Lagrangian obtained in [
25
] by the
dimensional reduction of the democratic type IIA pseudoaction
of [
54
]. It is a total derivative because of the use of the
tendimensional Hodge duality relations between the
lowerand higherform RR field strengths, which, as we have
already mentioned, are equivalent to the onshell expressions
(4.25) for the fourforms (see [
25
] for details). To perform
the offshell dimensional reduction one could use the
fullfledged dualitysymmetric action of type IIA supergravity
constructed in [
58
]. In this way, in principle, one should get
the fourdimensional Lagrangian (4.23) with the boundary
term (4.24), which produces the constants eA and m A after
one integrates out the 3forms.
5 Conclusion
In this paper we have shown how to construct globally and
locally supersymmetric models with gauge threeforms, by
dualizing more conventional theories with standard chiral
multiplets and a superpotential of the form (1.1). In the
dualization process, the coupling constants eA, m B are promoted
to (appropriate combinations of) expectation values of the
field strengths F4A = d A3A, F˜A4 = d A˜ A3 associated with
the threeform gauge fields A A, A˜ A3. The dual theory is
3
manifestly supersymmetric and is constructed in terms of
threeform multiplets which contain a complex scalar and
one or two gauge threeforms as bosonic components, the
latter replacing scalar auxiliary fields of the conventional chiral
multiplets.
As an application, we applied our duality procedure to
the fourdimensional effective theory describing the closed
string sector of type IIA orientifold compactifications on
Calabi–Yau threefolds with RR fluxes. In particular, we
discussed the explicit form of the bosonic action for the scalar
and threeform fields. By solving the equations of motion
for the threeform fields we found the same onshell values
of their field strengths as those obtained by direct
dimensional reduction [
25
] and the correct potential for the scalar
fields [
50
].
Even though our approach is quite general, the
application to more general string compactifications requires further
work. First of all, in the type IIA models considered in Sect.
4 the tadpole condition does not directly concern the internal
fluxes that are involved in the dualization. In more general
IIA compactifications, for instance with a nontrivial H3flux,
the tadpole condition would become relevant for the
dualization procedure. The same is true for type IIB orientifold
compactifications, which have a fluxinduced superpotential
[
51–53,59,60
] compatible with our general framework too.
Also in these cases a nontrivial tadpole condition should be
appropriately taken into account.
Another aspect that deserves further study is the inclusion
of the openstring sector in the effective theory, which may be
naturally incorporated in a threeform formulation [
42–44
].
It would be interesting to revisit this point in the manifestly
supersymmetric framework provided in the present paper.
Related questions concern its applications to Mtheory and
Ftheory compactifications, which can be considered as strong
coupling limits of type IIA and IIB compactifications with
backreacting branes; see for instance [
61,62
] for reviews.
In four dimensions, gauge threeforms couple to
membranes that appear as domainwalls generating jumps of the
 = ϕ,
− 41 D2  = F,
where the vertical line means that the quantity is evaluated
at θ = θ¯ = 0. The real scalar multiplet U has the following
component structure:
value of the corresponding field strength, as e.g. in [
4
]. In
the context of string/Mtheory compactifications, these
membranes correspond to higherdimensional branes wrapping
various cycles in the internal space and are crucial for the
mechanisms of dynamical relaxation of the cosmological
constant discussed, for instance, in [
9,10
]. Our formulation
should be the starting point for revisiting these aspects at
the level of a fourdimensional effective theory with
manifest linearly realized supersymmetry, generalizing the results
of [
8,12–14
]. Furthermore, in this same context and in the
presence of spontaneously broken supersymmetry, our
formulation should be related, at low energies, to models with
nonlinearly realized local supersymmetry as the ones
introduced in [
15
]. It would be interesting to elucidate this
relation. More generically, it would be worth using this general
framework to construct and study supersymmetric extensions
of various models based on gauge threeforms, as for instance
those discussed in [
1,3–7,11,16–19,22–26
].
Acknowledgements We thank Igor Bandos, Massimo Bianchi, Sergei
Kuzenko and Irene Valenzuela for useful discussions and comments.
Work of F.F., S.L. and L.M. was partially supported by the Padua
University Project CPDA144437. Work of D.S. was supported in part by the
Australian Research Council Project No. DP160103633 and by the
Russian Science Foundation grant 144200047 in association with Lebedev
Physical Institute.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
A Component structure of N = 1 superfields
In this appendix we collect some useful formulas on the
component structure of the multiplets considered in the present
paper. We mostly follow notation and conventions of [
45
].
The chiral multiplet is defined by the condition
D¯ α
˙
= 0.
Its component expansion is
√2θ ψ + θ 2 F + i θ σ m θ¯∂m ϕ
= ϕ +
− √i θ 2∂m ψ σ m θ¯ + 41 θ 2θ¯2 ϕ, (A.2)
2
where ϕ and F are complex scalar fields and ψ is a
Weyl spinor. The independent bosonic components of are
defined as follows:
(A.4)
(A.6)
(A.7)
(A.8)
(A.9)
2 2
−θ θ¯
1
D + 4
u ,
− 81 σ¯mα˙ α[Dα, D¯ α˙ ]U  = Am ,
U  = u,
i
D2U  = ϕ¯,
4
where u and D are real scalar fields, ϕ is a complex scalar
field, Am is a real vector field and χ and λ are Weyl spinors.
The independent bosonic components of U are defined as
follows:
D¯ 2
Here σ and s¯ are complex scalars, ρ, ψ and ξ are Weyl
spinors and C m is a complex vector which is Hodge dual to
the threeform
C m
In this appendix we illustrate the dualization procedure with
two simple examples: first we will consider a purely bosonic
Lagrangian of a single gauge threeform and then we will
examine the case of a Lagrangian with a single complex linear
multiplet.
Let us consider a real threeform with couplings described
by the Lagrangian
1
L = K (ϕ) 3!
1
+W (ϕ) 3!
∂m εmnpq Anpq
∂m εmnpq Anpq ,
2
where K (ϕ) and W (ϕ) are real functions of the scalar fields
ϕ, denoted in this way to be reminiscent of the structure of
supersymmetric chiral field models. To further simplify the
formulas, let us replace Anpq with its Hodgedual vector field
Am
1 εmnpq Anpq ,
= 3!
so that (B.1) becomes
L = K (ϕ) ∂m Am 2 + W (ϕ) ∂m Am .
Note that the gauge invariance of the threeform becomes
an invariance of the action under the transformation of the
oneform A1 → A1 + ∗d 2, where A1 = Am dx m .
We wish to integrate out Am to find the contribution to
the scalar potential. To perform a consistent variation of the
action with respect to the threeform, one should introduce
an appropriate boundary term of the form [
4,6
]
(A.11)
Then the equations of motion for the threeform (which are
unaffected by the boundary terms) give
(A.12)
(B.1)
(B.2)
(B.3)
Lbd = −∂m (W
+ 2K ∂n An) Am .
(B.4)
α = −2K F − W , F = ∂m Am .
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
∂m Am = − W2K+ r ,
L = −
(r + W )2
4K
.
where r is a real integration constant. Substituting (B.5) into
(B.3) + (B.4) we get the following Lagrangian which provides
the potential for the scalar fields ϕ:
There is an alternative way to integrate out the threeform
without the need to introduce the boundary terms. We can
rewrite (B.3) by using a Lagrange multiplier scalar α and an
auxiliary field F
L = K F 2 + W F + α F + Am ∂m α .
By varying α in (B.7) with the boundary condition δαbd =
0 we get F = ∂m Am and then back (B.3). Now Am is a
Lagrange multiplier and we can consistently integrate it out
without the need of additional boundary terms thus getting
α = r,
where r is a real constant related to the onshell value of
F4 = d A3. Now we have
L = K F 2 + (r + W )F,
and once we integrate out the scalar F we find (B.6), which
produces a positive definite contribution to the scalar
potential (if K > 0).
On the other hand, this dualization procedure provides
a systematic way to get the boundary term (B.4), which is
necessary to make the variation of the Lagrangian (B.1)
consistent. To do this we should reverse the dualization
procedure starting from the Lagrangian (B.7). The variation of the
Lagrangian (B.7) with respect to the auxiliary field gives
δαL = δα F − ∂m Am
+ ∂m ( Am δα),
δF L = 2K F + α + W
δ F.
Imposing the boundary conditions
δαbd = 0, δ F bd = 0,
and setting the variations to zero we get
Here we give some useful expressions that we used for the
analysis of the effective fourdimensional theory associated
with the example of type IIA flux compactification in Sect. 4.
The K part (4.5) of the Kähler potential (4.3) of the model
under consideration is
K = −log 8k,
where
k = 31 ki jk vi v j vk , ki j ≡ ki jk vk , ki ≡ ki jk v j vk
!
and ki jk is the triple intersection number of the C Y3 manifold.
Defining Ki ≡ ∂∂ϕKi and Ki j ≡ ∂ϕ∂i2∂Kϕ¯ j , we have
K i j ≡
and
ki
Ki = − 4k = −2ki eK ,
1
Ki j = − 4k
ki j −
Ki j −1 = −4k ki j − 2k
ki k j
4k
vi v j
,
,
Plugging (B.12) back into the Lagrangian (B.7), we get
L = K (∂m Am )2 + W ∂m Am − ∂m Am (W
with c being a complex constant. In order to make the
auxiliary field F of dynamical, we promote the complex
constant c to a chiral superfield X and add a new term which
contains the complex linear superfield
L =
+
d4θ
¯ +
d2θ
− 41 D¯ 2
d2θ (X
+ W ( )) + c.c.
X
¯
+ c.c. .
(B.15)
Using the expansions of the superfields in component fields
given in Appendix A and focusing on the bosonic
components only, we get from (B.15) the following part of the
component Lagrangian which contains the auxiliary fields
F and F¯ :
LF = F F¯ +
i
W F + α F + 2 C m ∂m α + c.c. ,
(B.16)
where α = X  and, as usual, the vector field C m is the dual of
a threeform. This is a complexified version of the Lagrangian
(B.7).
To obtain the dual Lagrangian for the fields C m we vary
(B.16) with respect to α and F , and get the equations of
motion
i
F = 2 ∂m C m ,
α = −F¯ − W .
Plugging them back into (B.16), we get
1
L = 4
∂nC n
i
∂m C¯ m + 2
W ∂nC n + c.c.
with the boundary term Lagrangian having the required form
i
Lbd = 2 ∂m
i
2 ∂nC n − W
From (4.12), upon gaugefixing Z0 = 1, we get the
following components of the imaginary and the real parts of the
holomorphic matrix GAB (2.38):
M00 = −2k + ki j bi b j ,
Mi j = ki j ,
M0i = −ki j b j = −Mi j b j ,
(C.5)
1
N00 = 3 ki jk bi b j b j − ki bi ,
1
N0i = 2
Ni j = ki jk bk .
The inverse matrix MAB has the following components:
M
00
1
= − 2k , M
0i
1 i
= − 2k b , M
i j = ki j
ki − ki jk b j bk ,
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