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arXiv:hep-th/0506013v3 17 Jan 2006
Preprint typeset in JHEP style - HYPER VERSION
DAMTP-2005-47
UG-05-04
IIB Supergravity Revisited
Eric A. Bergshoeff, Mees de Roo, Sven F. Kerstan
Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG
Groningen, The Netherlands
E.A.Bergshoeff, M.de.Roo, S.Kerstan@phys.rug.nl
Fabio Riccioni
DAMTP, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, UK
F.Riccioni@damtp.cam.ac.uk
Abstract: We show in the SU(1,1)-covariant formulation that IIB supergravity
allows the introduction of a doublet and a quadruplet of ten-form potentials. The
Ramond-Ramond ten-form potential which is associated with the SO(32) Type I
superstring is in the quadruplet. Our results are consistent with a recently proposed
E11symmetry underlying string theory.
For the reader’s convenience we present the full supersymmetry and gauge transfor-
mations of all fields both in the manifestly SU(1,1) covariant Einstein frame and in
the real U(1) gauge fixed string frame.
Keywords: Extended Supersymmetry, Supergravity Models, Field Theories in
Higher Dimensions.
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Contents
1. Introduction2
2. The SU(1,1)-covariant formulation4
3. Six-forms and eight-forms7
3.1Six-forms7
3.2Eight-forms 10
4. Ten-forms 14
4.1 The doublet of ten-forms14
4.2The quadruplet of ten-forms15
4.3Other ten-forms?17
5. The complete IIB transformation rules and algebra19
6. U(1) gauge fixing and string frame22
7. Summary and Discussion 30
8. Acknowledgements32
A. Conventions32
B. Truncations32
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1. Introduction
IIB supergravity [1, 2, 3] is the low energy effective action of type-IIB superstring the-
ory. Its scalar sector describes the coset manifold SL(2,R)/SO(2) ≃ SU(1,1)/U(1),
whose isometry SL(2,R) is a symmetry of the low energy theory. Since the isometry
acts non-trivially on the dilaton, the full perturbative string theory does not preserve
the symmetry, but the conjecture is that non-perturbatively an SL(2,Z) subgroup
of the full symmetry group of the low energy action survives [4].
The particular feature of type-IIB string theory with respect to the other theories
of closed oriented strings is that it is symmetric under the orientation reversal of
the fundamental string. Ten-dimensional type-I string theory is obtained from type-
IIB through an orientifold projection [5] that gauges this symmetry, and tadpole
cancellation requires the introduction of an open sector, corresponding to D9-branes.
The standard supersymmetric projection gives rise to the type-I superstring, with
gauge group SO(32) [6], while a non-supersymmetric, anomaly-free projection gives
rise to a model with gauge group USp(32) [7], in which supersymmetry is realized
on the bulk and spontaneously broken on the branes [8].
In the low-energy effective action, the closed sector of type-I strings is obtained
by performing a consistent Z2 truncation of the IIB supergravity, while the open
sector corresponds to the first order in the low-energy expansion of the D9-brane
action in a type-I background. In [9] it was shown that the Z2symmetry responsible
for this truncation can be performed in two ways, and in a flat background, with
all bulk fields put to zero, the D9-brane action reduces in one case to the Volkov-
Akulov action [10], and in the other case to a constant. In [11] these results were
extended to a generic background, showing that also in the curved case there are two
possibilities of performing the truncation. In one case one gets a dilaton tadpole and
a RR tadpole plus goldstino couplings, which is basically the one-brane equivalent
of the Sugimoto model, while in the other case the goldstino couplings vanish and
one is left with a dilaton and a RR tadpole, which is the one-brane equivalent of
the supersymmetric model. In order to truncate the theory in the brane sector, the
“democratic formulation” of IIB supergravity was derived [9, 12]. This amounts to
an extension of the supersymmetry algebra, so that both the RR fields and their
magnetic duals appear on the same footing. The closure of the algebra then requires
the field strengths of these fields to be related by duality conditions. The result
is that, together with the RR forms C(2n), n = 0,...,4 associated with D-branes
of non-vanishing codimension, the algebra naturally includes a RR ten-form C(10),
with respect to which the spacetime-filling D9-branes are electrically charged. This
field does not have any field strength, and correspondingly an object charged with
respect to it can be consistently included in the theory only when one performs a
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type-I truncation, so that the resulting overall RR charge vanishes. The analysis
of [9] also showed that an additional ten-form B(10)can be introduced in the algebra,
and this form survives a different Z2truncation, projecting out all the RR-fields. In
the string frame, the tension of a spacetime-filling brane electrically charged with
respect to B(10)would scale like g−2
action for this object can not be obtained performing an S-duality transformation
on the D9-brane effective action [13]. We are therefore facing a problem, since two
ten-forms are known in IIB supergravity, but they do not form a doublet with respect
to SL(2,R).
S, instead of g−4
S, thus implying that the brane
In this paper we will clarify this issue. We want to obtain all the possible independent
ten-forms that can be added to 10-dimensional IIB supergravity, with their assign-
ment to representations of SL(2,R). In order to perform this analysis, we express
the theory in a “SU(1,1)-democratic formulation”, in which all the forms, not only
the RR ones, and their magnetic duals are described in a SU(1,1)-covariant way. We
use the notation of [1, 2], so that the scalars parametrize the coset SU(1,1)/U(1),
while the two two-forms, as well as their duals, form a doublet of SU(1,1). The
eight-forms, dual to the scalars, transform as a triplet of SU(1,1), with the field
strengths satisfying an SU(1,1) invariant constraint [14, 15]. Eventually, we find
that the algebra includes a doublet and a quadruplet of ten-forms1, and the dilaton
dependence of the supersymmetry transformation of these objects shows that the RR
ten-form belongs to the quadruplet. We claim that no other independent ten-forms
can be added to the algebra. In summary, we find the following bosonic field content:
ea
µ,Vα
+,Vα
−,Aα
(2),A(4),Aα
(6),A(αβ)
(8),Aα
(10),A(αβγ)
(10),(1.1)
where ea
an SU(1,1) index and the subindex (n) indicates the rank of the potential.
µis the zehnbein, (Vα
+,Vα
−) parametrizes the SU(1,1)/U(1) coset, α = 1,2 is
This paper will be devoted to the construction and the properties of the extended IIB
supergravity theory (1.1). Clearly the properties of the dual forms and ten-forms have
implications for the structure of the brane spectrum, dualities, etc. These aspects of
this work will be addressed in a forthcoming paper [17].
The structure of the paper is as follows.
transformation rules and algebra of the extended IIB-supergravity theory in the
SU(1,1)/U(1) formulation, are given in section 5. In section 6 these results are
rewritten in a U(1) gauge in the Einstein frame and in the string frame. In this
section we also recover the Ramond-Ramond “harmonica” of [9] and then extend it
to the Neveu-Schwarz forms. We also list the action of S-duality on all form fields.
The preceding sections lead up to these results and sketch the derivation. In section
The main result, the supersymmetry
1Gauge fields of maximal rank have been explored in the literature [16].
– 3 –
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2 we review the SU(1,1)-covariant notation of [1, 2]. In section 3 we introduce in
the algebra the six- and the eight-forms dual to the two-forms and the scalars re-
spectively. Section 4 contains the analysis of the ten-forms. We finally conclude with
a summary of our results and a discussion. Some basic formulas and truncations to
N = 1 supergravity can be found in the Appendices.
2. The SU(1,1)-covariant formulation
In this section we review the notation and the results of [1, 2].
The theory contains the graviton, two scalars, two two-forms and a self-dual four-form
in the bosonic sector, together with a complex left-handed gravitino and a complex
right-handed spinor in the fermionic sector. We will use the mostly-minus spacetime
signature convention throughout the paper. The two scalars parametrize the coset
SU(1,1)/U(1), that can be described in terms of the SU(1,1) matrix (α,β = 1,2)
U = ( Vα
− Vα
+), (2.1)
satisfying the constraint
Vα
−Vβ
+− Vα
+Vβ
−= ǫαβ
, (2.2)
with (V1
charge, and ǫ12= ǫ12= 1. From the left-invariant 1-form
−)∗= V2
+, where α = 1,2 is an SU(1,1) index and + and − denote the U(1)
U−1∂µU =
?−iQµ Pµ
P∗
µ
iQµ
?
(2.3)
one reads off the U(1)-covariant quantity
Pµ= −ǫαβVα
+∂µVβ
+
,(2.4)
that has charge 2, and the U(1) connection
Qµ= −iǫαβVα
−∂µVβ
+
.(2.5)
Note that
PµVα
P∗
−= DµVα
+
, (2.6)
µVα
+= DµVα
−
, (2.7)
where the derivative D is covariant with respect to U(1). The two-forms are collected
in an SU(1,1) doublet Aα
µνsatisfying the constraint
(A1
µν)∗= A2
µν
.(2.8)
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The corresponding field strengths
Fα
µνρ= 3∂[µAα
νρ]
(2.9)
are invariant with respect to the gauge transformations
δAα
µν= 2∂[µΛα
ν]
. (2.10)
The four-form is invariant under SU(1,1), and varies as
δAµνρσ= 4∂[µΛνρσ]−i
4ǫαβΛα
[µFβ
νρσ]
(2.11)
under four-form and two-form gauge transformations, so that the gauge-invariant
five-form field-strength is
Fµνρστ= 5∂[µAνρστ]+5i
8ǫαβAα
[µνFβ
ρστ]
.(2.12)
This five-form satisfies the self-duality condition
Fµ1...µ5=1
5!ǫµ1...µ5ν1...ν5Fν1...ν5
.(2.13)
It is convenient to define the complex three-form
Gµνρ= −ǫαβVα
+Fβ
µνρ
,(2.14)
that is an SU(1,1) singlet with U(1) charge 1. Finally the gravitino ψµis complex
left-handed with U(1) charge 1/2, while the spinor λ is complex right-handed with
U(1) charge 3/2.
In [2] the field equations for this model were derived by requiring the closure of
the supersymmetry algebra. All these equations can be derived from a lagrangian,
imposing eq. (2.13) only after varying [18]2. It is interesting to study in detail the
kinetic term for the scalar fields,
Lscalar=e
2P∗
µPµ
. (2.15)
The complex variable
z =V2
−
V1
−
(2.16)
2A lagrangian formulation for self dual forms has been developed in [19], and then applied in [15]
to the ten-dimensional IIB supergravity. It corresponds to the introduction of an additional scalar
auxiliary field, and the self-duality condition results from the gauge fixing (that can not be imposed
directly on the action) of additional local symmetries.
– 5 –
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is invariant under local U(1) transformations, and so it is a good coordinate for the
scalar manifold. Under the SU(1,1) transformation
?V1
V2
−
−
?
→
?α β
¯β ¯ α
??V1
−
V2
−
?
, (2.17)
that is an isometry of the scalar manifold, z transforms as
z →¯ αz +¯β
βz + α
. (2.18)
The variable z parametrizes the unit disc, |z| < 1, and the kinetic term assumes the
form
Lscalar= −e
2
The further change of variables
z =1 + iτ
1 − iτ
maps the disc in the complex upper-half plane, Imτ > 0, and in terms of τ the
transformations (2.17) become
∂µz∂µ¯ z
(1 − z¯ z)2
. (2.19)
(2.20)
τ →aτ + b
cτ + d
,(2.21)
where
?a b
c d
?
∈ SL(2,R), (2.22)
while the scalar lagrangian takes the form
Lscalar= −e
8
∂µτ∂µ¯ τ
(Imτ)2
. (2.23)
Expressing τ in terms of the RR scalar and the dilaton,
τ = ℓ + ie−φ
(2.24)
and performing the Weyl rescaling g(E)µν→ e−φ/2g(S)µνone ends up with the stan-
dard form of the kinetic term of the scalars in IIB supergravity in the string frame.
The supersymmetry transformations that leave the field equations of [2] invariant
are
δeµa= i¯ ǫγaψµ+ i¯ ǫCγaψµC
,
δψµ= Dµǫ +
δAα
δAµνρσ= ¯ ǫγ[µνρψσ]− ¯ ǫCγ[µνρψσ]C−3i
δλ = iPµγµǫC−
δVα
,
δVα
.
i
480Fµν1...ν4γν1...ν4ǫ +
−¯ ǫγµνλ + Vα
1
96GνρσγµνρσǫC−
−¯ ǫCγ[µψν]+ 4iVα
[µνδAβ
3
32GµνργνρǫC
+¯ ǫγ[µψν]C
,
,
µν= Vα
+¯ ǫCγµνλC+ 4iVα
,
8ǫαβAα
ρσ]
i
24Gµνργµνρǫ,
+= Vα
−= Vα
−¯ ǫCλ
+¯ ǫλC
(2.25)
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where we denote with ΨC the complex (Majorana) conjugate of Ψ. The commu-
tator [δ1,δ2] of two supersymmetry transformations of (2.25) closes on all the local
symmetries of the theory, provided one uses the fermionic field equations and the
self-duality condition of eq. (2.13). To lowest order in the fermions, the parameters
of the resulting general coordinate transformation, four-form gauge transformation
and two-form gauge transformation are3
ξµ= i ¯ ǫ2γµǫ1+ i ¯ ǫ2Cγµǫ1C
Λα
Λµνρ= Aµνρσξσ−1
−3
,
µ= Aα
µνξν− 2i[Vα
+¯ ǫ2γµǫ1C+ Vα
−¯ ǫ2Cγµǫ1],
4[¯ ǫ2γµνρǫ1− ¯ ǫ2Cγµνρǫ1C]
?Vβ
8ǫαβAα
[µν
+¯ ǫ2γρ]ǫ1C+ Vβ
−¯ ǫ2Cγρ]ǫ1
?
. (2.26)
In the next section we will extend the algebra in order to include the magnetic duals
of the scalars and of the two-form, in such a way that the supersymmetry algebra still
closes, once the proper duality relations are used. Once we obtain the supersymmetry
transformation of the six- and the eight-forms that are compatible with the algebra
obtained from eq. (2.25), we will include in Section 4 all the possible independent
ten-forms that this algebra allows.
3. Six-forms and eight-forms
In this section we show how the algebra of eq. (2.25) is extended introducing the
forms magnetically dual to the scalars and the two-forms. As anticipated, closure of
the supersymmetry algebra requires the field strengths of these forms to be related
to Pµand the field strengths of the two-forms by suitable duality relations. Gen-
eralizing what happens for the four-form (see eqs. (2.11) and (2.12)), we will see
that the gauge transformations of these fields involve the gauge parameters of all the
lower rank forms, and the gauge invariant field strengths will therefore contain lower
rank forms as well. After introducing our Ansatz for these field strengths and gauge
transformations, the supersymmetry transformations of these fields will then be de-
termined requiring the closure of the supersymmetry algebra. As in the previous
section, we will not consider terms higher than quadratic in the fermi fields.
3.1 Six-forms
We want to obtain the gauge and supersymmetry transformations for the doublet of
six-forms Aα
µ1...µ6, which are the magnetic duals of the two-forms and thus satisfy the
3We only present the parameters of translations and the two- and four-form gaugetransforma-
tions. The parameters of other local symmetries, namely supersymmetry, local Lorentz and local
U(1) are not used in the analysis of the next sections, and are given in [1, 2].
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reality condition
(A1)∗
µ1...µ6= A2
µ1...µ6
. (3.1)
Generalizing what one obtains for the four-form, we expect the supersymmetry trans-
formation of the six-forms to contain terms involving only spinors and terms con-
taining forms of lower rank. The condition of eq. (3.1), as well as the requirement
that all the terms must have vanishing local U(1) charge, fixes the most general
transformation of the doublet to be
δAα
µ1···µ6= a Vα
+ b Vα
+ c A[µ1···µ4δAα
+ d δA[µ1···µ4Aα
+ ie ǫβγδAβ
−¯ ǫγµ1...µ6λ + a∗Vα
−¯ ǫCγ[µ1...µ5ψµ6]− b∗Vα
+¯ ǫCγµ1...µ6λC
+¯ ǫγ[µ1...µ5ψµ6]C
µ5µ6]
µ5µ6]
[µ1µ2Aγ
µ3µ4Aα
µ5µ6]
.(3.2)
We want to consider the commutator [δ1,δ2] of two such transformations, to lowest
order in the fermi fields.
We first take into account the terms involving the spinors, i.e., the first two lines in
eq. (3.2). Those terms produce the gauge transformation for the six-forms
δAα
µ1...µ6= 6∂[µ1Λα
= −12i∂[µ1(a Vα
µ2...µ6]
+¯ ǫ2γµ2...µ6]ǫ1C+ a∗Vα
−¯ ǫ2Cγµ2...µ6]ǫ1)(3.3)
if the constraint
12ia∗= b(3.4)
is imposed, while the other terms that are produced are
20iaFα
−1
[µ1µ2µ3(¯ ǫ2Cγµ4µ5µ6]ǫ1C− ¯ ǫ2γµ4µ5µ6]ǫ1)
6aǫµ1...µ6σµνρSαβǫβγFγ;µνρξσ
,(3.5)
where we have defined
Sαβ= Vα
−Vβ
++ Vα
+Vβ
−
(3.6)
and we have assumed that a is imaginary. Note that Sαβsatisfies
SαβǫβγSγδǫδǫ= δα
ǫ
. (3.7)
Observe that there are no terms involving the five-form field strength. Without loss
of generality, we fix
a = i (3.8)
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from now on. In order for the last term in (3.5) to produce a general coordinate
transformation with the right coefficient as dictated by eq. (2.26), we impose the
duality relation4
Fα
µ1...µ7= −i
µ2...µ7]+... are the field strengths of the six-forms, and the dots
stand for terms involving lower rank forms that we will determine in the following.
Note that the second term of eq. (3.5) contains, together with a general coordinate
transformation, a gauge transformation of parameter
3!ǫµ1...µ7µνρSαβǫβγFγ;µνρ
, (3.9)
where Fα
µ1...µ7= 7∂[µ1Aα
Λ′α
µ1...µ5= Aα
µ1...µ5σξσ
.(3.10)
The SU(1,1)-invariant quantities
Gµ1...µ7= −ǫαβVα
+Fβ
µ1...µ7,G∗
µ1...µ7= ǫαβVα
−Fβ
µ1...µ7
,(3.11)
which have U(1) charge +1 and −1 respectively, satisfy
G(7)
µ1...µ7=
i
3!ǫµ1...µ7µνρGµνρ,G∗
µ1...µ7= −i
3!ǫµ1...µ7µνρG∗µνρ
. (3.12)
In order to proceed further, in analogy with eq. (2.12) we make the following Ansatz
for the seven-form field strengths:
Fα
µ1...µ7= 7∂[µ1Aα
µ2...µ7]+ αAα
[µ1µ2Fµ3...µ7]+ βFα
[µ1...µ3Aµ4...µ7]
.(3.13)
For these forms to be gauge invariant, the must transform non-trivially with respect
to the two-form and four-form gauge transformations. The result is
δAα
µ1...µ6= −2
7αΛα
[µ1Fµ2...µ6]+4
7βFα
[µ1...µ3Λµ4...µ6]
, (3.14)
and gauge invariance requires
β = −10
3α.(3.15)
Now we come back to the commutator. The terms that are left are the ones coming
from the last three lines in eq. (3.2), together with the first line in eq. (3.5) and the
terms coming from (3.13) in the second line of eq. (3.5). All these terms have to
produce gauge transformations according to (3.14), with parameters given from eqs.
(2.26), possibly together with additional gauge transformations. The end result is
that one produces the additional gauge transformations
Λ′′α
µ1...µ5= −2i
−1
3c A[µ1...µ4(Vα
6d Aα
+¯ ǫ2γµ5]ǫ1C+ Vα
−¯ ǫ2Cγµ5]ǫ1)
[µ1µ2(¯ ǫ2γµ3...µ5]ǫ1− ¯ ǫ2Cγµ3...µ5]ǫ1C), (3.16)
4Note that this duality relation induces field equations for the potentials.
– 9 –
Page 11
while all the coefficients are uniquely determined to be
c = 40,d = −20,e =15
2
,α = 28. (3.17)
Summarizing, we get that the supersymmetry transformations of the six-forms are
δAα
µ1···µ6= i Vα
+ 12 Vα
+ 40 A[µ1···µ4δAα
− 20 δA[µ1···µ4Aα
+15i
−¯ ǫγµ1...µ6λ − i Vα
−¯ ǫCγ[µ1...µ5ψµ6]− 12 Vα
+¯ ǫCγµ1...µ6λC
+¯ ǫγ[µ1...µ5ψµ6]C
µ5µ6]
µ5µ6]
µ3µ4Aα
2ǫβγδAβ
[µ1µ2Aγ
µ5µ6]
. (3.18)
The doublet of seven-form field strengths is
Fα
µ1...µ7= 7∂[µ1Aα
µ2...µ7]+ 28Aα
[µ1µ2Fµ3...µ7]−280
3Fα
[µ1...µ3Aµ4...µ7]
.(3.19)
This is gauge invariant with respect to the transformations of the two-forms, the
four-form and the six-forms, where
δAα
µ1...µ6= 6∂[µ1Λα
µ2...µ6]− 8Λα
[µ1Fµ2...µ6]−160
3Fα
[µ1...µ3Λµ4...µ6]
.(3.20)
Moreover, the six-form gauge transformation parameter resulting from the commu-
tator of two supersymmetry transformations is
Λα
µ1...µ5= Aα
µ1...µ5σξσ+ 2(Vα
−80i
+10
+¯ ǫ2γµ1...µ5ǫ1− Vα
+¯ ǫ2γµ5]ǫ1C+ Vα
−¯ ǫ2Cγµ1...µ5ǫ1C)
3A[µ1...µ4(Vα
3Aα
−¯ ǫ2Cγµ5]ǫ1)
[µ1µ2(¯ ǫ2γµ3...µ5]ǫ1− ¯ ǫ2Cγµ3...µ5]ǫ1C), (3.21)
as results from eqs. (3.3), (3.10) and (3.16). Finally, a comment is in order. At
first sight, the Ansatz we made for the field strengths in eq. (3.13) does not seem
to be the most general one, since one could in principle include a term of the form
iǫβγAa
ways reabsorb such a term by performing a redefinition of the six-forms of the type
Aα
freedom will be used to constrain the form of the field strengths of the eight-forms
as well, as we will see in the next subsection.
[µ1µ2Aβ
µ3µ4Fγ
µ5...µ7]. The reason why we did not include it is that one can al-
µ1...µ6→ Aα
µ1...µ6+ γAα
[µ1µ2Aµ3...µ6], and choose γ so that this term vanishes. This
3.2 Eight-forms
The eight-forms are the magnetic duals of the scalars. As we reviewed in Section 2,
the scalars are described in terms of the left-invariant 1-form of eq. (2.3), transform-
ing in the adjoint of SU(1,1), and propagating two real degrees of freedom because
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Page 12
of local U(1) invariance. One therefore expects a triplet of eight-forms (as observed
in [14, 15])5, that we denote by Aαβ
reality condition
µ1...µ8, symmetric under α ↔ β, and satisfying the
(A11)∗
µ1...µ8= A22
µ1...µ8
,(A12)∗
µ1...µ8= A12
µ1...µ8
. (3.22)
The fact that only two scalars propagate will result in a constraint for the field
strengths of these eight-forms [20, 15]. This is exactly what we are going to show
in this subsection. Following the same arguments as in the previous subsection, we
write the most general supersymmetry transformations for the eight-forms, compati-
ble with the reality condition and with U(1) invariance, consisting of terms that only
involve the spinors and terms containing the lower rank forms and their supersym-
metry transformations. The result is
δAαβ
µ1...µ8= a Vα
+ b V(α
+ cA(α
+Vβ
+Vβ)
[µ1...µ6δAβ)
+¯ ǫγµ1...µ8λC+ a∗Vα
− ¯ ǫγ[µ1...µ7ψµ8]− b∗V(α
µ7µ8]+ dA(α
−Vβ
−¯ ǫCγµ1...µ8λ
+Vβ)
µ3...µ8]+ ieA(α
µ5µ6δAβ)
− ¯ ǫCγ[µ1...µ7ψµ8]C
[µ1µ2δAβ)
[µ1µ2Aβ)
µ3µ4ǫγδAγ
µ5µ6δAδ
µ7µ8]
+ fA(α
[µ1µ2Aβ)
µ3µ4δAµ5...µ8]+ gA[µ1...µ4A(α
µ7µ8]
.(3.23)
We first consider the contributions coming from the first two lines of eq. (3.23), in
order to get a relation between a and b. We obtain the gauge transformation
δAαβ
µ1...µ8= 8∂[µ1Λαβ
= −4ia∂[µ1
µ2...µ8]
?Sαβ(¯ ǫ2γµ2...µ8]ǫ1− ¯ ǫ2Cγµ2...µ8]ǫ1C)?
(3.24)
together with the terms
28ia(V(α
+ ¯ ǫ2γ[µ1...µ5ǫ1− V(α
−4a(V(α
−aǫµ1...µ8στξσ(Vα
− ¯ ǫ2Cγ[µ1...µ5ǫ1C)Fβ)
µ6...µ8]
+ ¯ ǫ2γ[µ1ǫ1+ V(α
− ¯ ǫ2Cγ[µ1ǫ1C)Fβ)
µ2...µ8]
,
+Vβ
+P∗τ− Vα
−Vβ
−Pτ)(3.25)
provided that
8ia = b (3.26)
and a is chosen to be imaginary. Fixing, without loss of generality,
a = −i,(3.27)
one finds that the last term in eq. (3.25) contains the correct general coordinate
transformation, plus an gauge transformation of parameter
Λ′αβ
µ1...µ7= Aαβ
µ1...µ7σξσ
(3.28)
5A similar observation was made for the curvatures in [20].
– 11 –
Page 13
provided the duality relation
Fαβ
µ1...µ9= iǫµ1...µ9
σ[Vα
+Vβ
+P∗
σ− Vα
−Vβ
−Pσ] (3.29)
holds, where Fαβ
lower rank forms. From the field strengths of the eight-forms, one can define the
SU(1,1) invariant quantity
µ1...µ9= 9∂[µ1Aαβ
µ2...µ9]+ ..., and the dots stand for terms involving
Gµ1...µ9= ǫαγǫβδVα
+Vβ
+Fγδ
µ1...µ9
, (3.30)
with U(1) charge +2, and its complex conjugate
G∗
µ1...µ9= ǫαγǫβδVα
−Vβ
−Fγδ
µ1...µ9
. (3.31)
In terms of these objects, the duality relation of eq. (3.29) becomes
Gµ1...µ9= −iǫµ1...µ9σPσ,G∗
µ1...µ9= iǫµ1...µ9σP∗σ
. (3.32)
One can define a third nine-form,
˜Gµ1...µ9= ǫαγǫβδVα
+Vβ
−Fγδ
µ1...µ9
, (3.33)
with vanishing U(1) charge, but the duality relation (3.29) implies that this nine-
form vanishes identically [15], thus determining an SU(1,1) invariant constraint.
Therefore only two eight-forms are actually independent.
We now come to our choice for the field strengths, for which the most general general
expression is
Fαβ
µ1...µ9= 9∂[µ1Aαβ
µ2...µ9]+ αF(α
+iδǫγδAγ
[µ1...µ7Aβ)
µ6µ7Aβ)
µ8µ9]+ βF(α
[µ1...µ3Aβ)
µ4...µ9]+ γF[µ1...µ5A(α
µ5...µ7Aβ)
µ8µ9]
µ6µ7Aβ)
µ8µ9]
[µ1µ2Fδ
µ3...µ5A(α
µ8µ9]+ ξA[µ1...µ4F(α
. (3.34)
The freedom of redefining the eight-form, A8→ A8+ A6A2+ A4A2A2, can be used
to put to zero the coefficients ξ and δ in (3.34). It turns out that defining the gauge
transformation of the eight-forms as
δAαβ
µ1...µ8= 8∂[µ1Λαβ
µ2...µ8]+2
9αF(α
[µ1...µ7Λβ)
µ8]+2
3βF(α
[µ1...µ3Λβ)
µ4...µ8]
, (3.35)
the field strengths of eq. (3.34) are gauge invariant if the coefficient γ vanishes as
well, and if the coefficients α and β are related by
β = −7α.(3.36)
To summarize, we have obtained
Fαβ
µ1...µ9= 9∂[µ1Aαβ
δAαβ
µ2...µ9]+ αF(α
[µ1...µ7Aβ)
9αF(α
µ8µ9]− 7αF(α
[µ1...µ7Λβ)
µ1...µ3Aβ)
µ4...µ9]
, (3.37)
µ1...µ8= 8∂[µ1Λαβ
µ2...µ8]+2
µ8]−14
3αF(α
[µ1...µ3Λβ)
µ4...µ8]
. (3.38)
– 12 –
Page 14
We now consider the terms in the commutator coming from the last two lines of eq.
(3.23), as well as the first two terms in eq. (3.25) and the part of the third containing
lower rank forms. All these terms have to produce the gauge transformations of
eq. (3.38) with the parameters given in eqs. (2.26) and (3.21), plus possibly a
gauge transformation. The end result is that one produces the additional gauge
transformation
Λ
′′αβ
µ1...µ7= −4ic
+ 12d
?
A(α
A(α
[µ1...µ6(Vβ)
+ ¯ ǫ2γµ7]ǫ1+ Vβ)
− ¯ ǫ2Cγµ7]ǫ1C)
?
?
[µ2µ3(Vβ)
+ ¯ ǫ2γµ4...µ7]ǫ1− Vβ)
− ¯ ǫ2Cγµ4...µ7]ǫ1C)
?
,(3.39)
and the algebra closes provided that the coefficients are fixed to be
c =21
4
,d = −7
4
,e = −105
α =9
8
,
f = −35,g = 70,
4
. (3.40)
In conclusion, the supersymmetry transformation for the eight-forms is
δAαβ
µ1...µ8= −i Vα
+ 8 V(α
+Vβ
+Vβ)
4A(α
− 35A(α
+¯ ǫγµ1...µ8λC+ i Vα
− ¯ ǫγ[µ1...µ7ψµ8]− 8 V(α
[µ1...µ6δAβ)
−Vβ
−¯ ǫCγµ1...µ8λ
+Vβ)
− ¯ ǫCγ[µ1...µ7ψµ8]C
8A(α
µ5µ6δAβ)
+21
µ7µ8]−7
µ3µ4δAµ5...µ8]+ 70A[µ1...µ4A(α
4A(α
[µ1µ2δAβ)
µ3...µ8]−105i
[µ1µ2Aβ)
µ3µ4ǫγδAγ
µ5µ6δAδ
µ7µ8]
[µ1µ2Aβ)
µ7µ8]
, (3.41)
while the gauge invariance of the field strengths
Fαβ
µ1...µ9= 9∂[µ1Aαβ
µ2...µ9]+9
4F(α
[µ1...µ7Aβ)
µ8µ9]−63
4F(α
[µ1...µ3Aβ)
µ4...µ9]
(3.42)
requires
δAαβ
µ1...µ8= 8∂[µ1Λαβ
µ2...µ8]+1
2F(α
[µ1...µ7Λβ)
µ8]−21
2F(α
[µ1...µ3Λβ)
µ4...µ8]
.(3.43)
Finally, the seven-form gauge parameter that appears in the commutator is
Λαβ
µ1...µ7= Aαβ
−21i
−21
µ1...µ7σξσ−1
8Aα
8Aα
2Sαβ[¯ ǫ2γµ1...µ7ǫ1− ¯ ǫ2Cγµ1...µ7ǫ1C]
[µ1...µ6(Vβ
[µ1µ2(Vβ
+¯ ǫ2γµ7]ǫ1+ Vβ
+¯ ǫ2γµ3...µ7]ǫ1− Vβ
−¯ ǫ2Cγµ7]ǫ1C)
−¯ ǫ2Cγµ3...µ7]ǫ1C),(3.44)
as one obtains from eqs. (3.24), (3.28) and (3.39).
– 13 –
Page 15
4. Ten-forms
The construction of ten-forms differs in an essential way from that of the six- and
eight-forms: they do not have a field strength and therefore they cannot be dual
to some other form within the IIB theory. They do not have propagating degrees
of freedom, since the charge associated to them must vanish. Therefore there is no
a priori limit on the number of ten-forms one could introduce. Also the SU(1,1)
representations cannot be guessed from the duality relations with lower rank forms.
However, their supersymmetry transformations are well defined. We therefore pro-
ceed as before, determining the independent ten-forms from the requirement that the
supersymmetry algebra must close. We want to determine the most general super-
symmetry transformations for the ten-forms, compatible with U(1) invariance, for
a given SU(1,1) representation. We first prove that both a doublet and a quadru-
plet of ten-forms are allowed, and then we discuss the claim that these are the only
possible ten-forms that are compatible with all the symmetries of IIB supergravity.
4.1 The doublet of ten-forms
We want to determine the supersymmetry transformations of a doublet of ten-forms
Aα
µ1...µ10satisfying the reality condition
(A1)∗
µ1...µ10= A2
µ1...µ10
. (4.1)
As we have seen already in the previous sections, the supersymmetry transformation
of any form consists of terms containing spinors, plus possibly terms containing
lower-rank forms and their supersymmetry transformations. In the case of the ten-
form doublet, U(1) invariance requires that the most general fermionic part in the
supersymmetry transformation of the ten-form doublet is
δAα
µ1...µ10= a Vα
+ b Vα
−¯ ǫγµ1...µ10λ + a∗Vα
−¯ ǫCγ[µ1...µ9ψµ10]− b∗Vα
+¯ ǫCγµ1...µ10λC
+¯ ǫγ[µ1...µ9ψµ10]C
. (4.2)
The commutator of two such transformations contains the ten-form gauge transfor-
mation
δAα
µ1...µ10= 10∂[µ1Λα
= −20i∂[µ1
µ2...µ10]
?a Vα
+¯ ǫ2γµ2...γ10]ǫ1C+ a∗Vα
−¯ ǫ2Cγµ2...γ10]ǫ1
?
, (4.3)
provided that the coefficients a and b satisfy
b = 20ia∗
.(4.4)
– 14 –
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