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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 –
Page 16
Moreover, the additional terms in the commutator, containing the five-form F5and
the complex three-form G3, vanish if a is chosen to be real.
In order to close the algebra, one also has to produce a general coordinate transforma-
tion with parameter ξµ(2.26), but this exactly cancels with the gauge transformation
of parameter6
Λ
′α
µ1...µ9= Aα
µ1...µ9σξσ
. (4.5)
As a result, the algebra closes without adding any term containing lower-rank forms
in the supersymmetry transformation of eq. (4.2). Correspondingly, this ten-form
doublet is invariant with respect to the gauge transformations of the lower-rank
forms. Without loss of generality, we can fix
a = 1, (4.6)
so that the resulting supersymmetry transformation for the ten-form doublet is
δAα
µ1...µ10= Vα
−¯ ǫγµ1...µ10λ + Vα
+ 20i Vα
+¯ ǫCγµ1...µ10λC
−¯ ǫCγ[µ1...µ9ψµ10]+ 20i Vα
+¯ ǫγ[µ1...µ9ψµ10]C
. (4.7)
4.2 The quadruplet of ten-forms
We consider now a quadruplet of ten-forms Aαβγ
and γ, and satisfying the reality condition
µ1...µ10, completely symmetric in α, β
(A111)∗
µ1...µ10= A222
µ1...µ10
,(A112)∗
µ1...µ10= A122
µ1...µ10
. (4.8)
The most general supersymmetry transformation, compatible with the reality condi-
tion and with U(1) invariance, and consisting of terms that only involve the spinors
and terms containing the lower rank forms and their supersymmetry transformations,
is
δAαβγ
µ1...µ10= a V(α
+ b V(α
+ c A(αβ
+Vβ
+Vγ)
+Vγ)
− ¯ ǫCγµ1...µ10λC+ a∗V(α
−Vβ
−Vγ)
+ ¯ ǫγµ1...µ10λ
−Vγ)
+Vβ
[µ1...µ8δAγ)
− ¯ ǫγ[µ1...µ9ψµ10]C− b∗V(α
µ9µ10]+ d A(α
µ3µ4δAγ)
−Vβ
+ ¯ ǫCγ[µ1...µ9ψµ10]
[µ1µ2δAβγ)
µ3...µ10]+ e A(α
[µ1...µ6Aβ
µ7µ8δAγ)
µ7µ8δAγ)
µ9µ10]
(4.9)
+ f A(α
[µ1µ2Aβ
µ5...µ10]+ g A[µ1...µ4A(α
µ5µ6δAµ7...µ10]+ ik A(α
µ5µ6Aβ
µ9µ10]
+ h A(α
[µ1µ2Aβ
µ3µ4Aγ)
[µ1µ2Aβ
µ3µ4Aγ)
µ5µ6ǫδτAδ
µ7µ8δAτ
µ9µ10].
6For lower rank p-forms these transformations are obtained in the form ξρFρµ1...µp, for p = D
the vanishing of the D + 1-form F corresponds to the cancellation of the two transformations.
This result will be used again in the next subsection, when we will consider ten-forms in other
representations of SU(1,1).
– 15 –
Page 17
We want to analyze the commutator of two such transformations.
We first consider the contribution coming from the fermionic terms, i.e., the first two
lines of eq. (4.9). Those produce the ten-form gauge transformation
δAαβγ
µ1...µ10= 10∂[µ1Λαβγ
=20
µ2...µ10]
+Vγ)
3i∂[µ1(a V(α
+Vβ
− ¯ ǫ2γµ2...µ10]ǫ1C+ a∗V(α
−Vβ
−Vγ)
+ ¯ ǫ2Cγµ2...µ10]ǫ1)(4.10)
together with the terms
20
3a F(αβ
[µ1...µ9
?
Vγ)
+ ¯ ǫ2γµ10]ǫ1C+ Vγ)
− ¯ ǫ2Cγµ10]ǫ1
?
−20ai S(αβ?¯ ǫ2γ[µ1...µ7ǫ1− ¯ ǫ2Cγ[µ1...µ7ǫ1C
?Fγ)
µ8µ9µ10]
, (4.11)
provided that
−20i
3a = b (4.12)
and a is chosen to be imaginary. Without loss of generality, we can fix
a = i(4.13)
from now on. As in the case of the ten-form doublet of the previous subsection, a
general coordinate transformation is automatically produced by means of a compen-
sating gauge transformation of parameter
Λ
′αβγ
µ1...µ9= Aαβγ
µ1...µ9σξσ
.(4.14)
We assume that the ten-form quadruplet transforms non-trivially with respect to the
lower-rank form gauge transformations, and in particular we make the Ansatz
δAαβγ
µ1...µ10= αF(αβ
[µ1...µ9Λγ)
µ10]+ βF(α
[µ1µ2µ3Λβγ)
µ4...µ10]
. (4.15)
We will comment on this choice at the end of this subsection. We now proceed
exactly as in the previous cases, considering the terms in the commutator coming
from the last three lines of eq. (4.9), as well as the two terms in eq. (4.11). Those
have to generate the gauge transformations of eq. (4.15), possibly together with an
additional ten-form gauge transformation. The final result is that the ten-form gauge
transformation of parameter
Λ
′′αβγ
µ1...µ9= −2i
−2
5c A(αβ
[µ1...µ8
?
Vγ)
+ ¯ ǫ2γµ9]ǫ1C+ Vγ)
− ¯ ǫ2Cγµ9]ǫ1
?
5d A(α
[µ1µ2Sβγ)?¯ ǫ2γµ3...µ9]ǫ1− ¯ ǫ2Cγµ3...µ9]ǫ1C
?
(4.16)
is produced, while the coefficients are determined to be
α = −2
3
,β = 32,c = −12
f =21
,
d = 3,e = −63
4
,
4
,
g = −210,h = 105,k =315
8
. (4.17)
– 16 –
Page 18
Summarizing, the supersymmetry transformation of the ten-form quadruplet is
δAαβγ
µ1...µ10= i V(α
+20
− 12 A(αβ
+21
+Vβ
3V(α
+Vγ)
+Vγ)
[µ1...µ8δAγ)
4A(α
+ 105 A(α
− ¯ ǫCγµ1...µ10λC− i V(α
− ¯ ǫγ[µ1...µ9ψµ10]C−20
µ9µ10]+ 3 A(α
µ3µ4δAγ)
−Vβ
−Vγ)
+ ¯ ǫγµ1...µ10λ
−Vγ)
+Vβ
3V(α
µ3...µ10]−63
−Vβ
+ ¯ ǫCγ[µ1...µ9ψµ10]
4A(α
µ7µ8δAγ)
[µ1µ2δAβγ)
[µ1...µ6Aβ
µ7µ8δAγ)
µ9µ10](4.18)
[µ1µ2Aβ
µ5...µ10]− 210 A[µ1...µ4A(α
µ3µ4Aγ)
µ5µ6Aβ
A(α
µ9µ10]
[µ1µ2Aβ
µ5µ6δAµ7...µ10]+315i
8
[µ1µ2Aβ
µ3µ4Aγ)
µ5µ6ǫδτAδ
µ7µ8δAτ
µ9µ10],
while its gauge transformation is
δAαβγ
µ1...µ10= 10∂[µ1Λαβγ
µ2...µ10]−2
3F(αβ
[µ1...µ9Λγ)
µ10]+ 32F(α
[µ1µ2µ3Λβγ)
µ4...µ10]
.(4.19)
Finally, the ten-form gauge transformation parameter appearing in the supersymme-
try algebra is
Λαβγ
µ1...µ9= Aαβγ
µ1...µ9σξσ−2
+24i
[µ1...µ8
3(V(α
Vγ)
+Vβ
+Vγ)
− ¯ ǫ2γµ1...µ9ǫ1C− V(α
+ ¯ ǫ2γµ9]ǫ1C+ Vγ)
−Vβ
?
?
−Vγ)
+ ¯ ǫ2Cγµ1...µ9ǫ1)
5A(αβ
?
− ¯ ǫ2Cγµ9]ǫ1
−6
5A(α
[µ1µ2Sβγ)?¯ ǫ2γµ3...µ9]ǫ1− ¯ ǫ2Cγµ3...µ9]ǫ1C
, (4.20)
as it results from eqs. (4.10), (4.14) and (4.16).
To conclude this subsection, we want to comment on the bosonic gauge transforma-
tion of eq. (4.19). Even though the supersymmetry algebra restricts us in our case
to ten dimensions, it turns out that the bosonic gauge algebra closes for arbitrary
dimension. In particular one can write down an eleven-form field strength that is
gauge invariant with respect to a gauge transformation of the form (4.15):
Fαβγ
µ1...µ11= 11∂[µ1Aαβγ
µ2...µ11]+11
2α A(α
[µ1µ2Fβγ)
µ3...µ11]+11
8β A(αβ
[µ1...µ8Fγ)
µ9µ10µ11]
,(4.21)
where the coefficients α and β have to satisfy the constraint
β = −48α. (4.22)
This relation is in agreement with the values of α and β given in eq. (4.17) and
obtained imposing supersymmetry. This suggests that the bosonic gauge algebra has
an underlying structure that is independent of supersymmetry in ten dimensions7.
4.3 Other ten-forms?
We now want to show that no other ten-forms can be included in the supersymmetry
algebra of IIB supergravity. In order to do this, we consider the most general Ansatz
7This type of gauge algebra is also observed in the doubled fields approach, see [20].
– 17 –
Page 19
for the supersymmetry transformation of a ten-form in a generic representation of
SU(1,1). Without loss of generality, we can limit ourselves to ten-forms with van-
ishing U(1)-charge. The simplest such example is a singlet of SU(1,1), for which the
supersymmetry transformation necessarily is
δAµ1...µ10= ¯ ǫγ[µ1...µ9ψµ10]+ ¯ ǫCγ[µ1...µ9ψµ10]C
.(4.23)
The commutator of two such transformations closes. This is not surprising since
A(10)is the volume form,
Aµ1...µ10∝ ǫµ1...µ10= eµ1
a1...eµ10
a10ǫa1...a10
.(4.24)
This means that there are no independent ten-form singlets in the supersymmetry
algebra of IIB.
One could ask whether additional ten-form doublets could result from objects of the
form Aα1...α2n+1
µ1...µ10
, when 2n SU(1,1) indices are pairwise antisymmetrized. However,
because of the constraint of eq. (2.2) these forms are the same as the one we obtained
in section 4.1, and therefore there is only a single doublet of ten-forms in the theory.
This argument can be iterated, so that for each object with an odd number of
SU(1,1) indices, only the components in the completely symmetric representation
are independent of the ten-forms belonging to lower representations.
Therefore, given a ten-form with n SU(1,1) indices, one has to consider only the
completely symmetric SU(1,1) representation. Let us consider the case n = 2 first.
The most general Ansatz for the fermionic terms is
δAαβ
(10)= a1V(α
+Vβ)
−¯ ǫγ(9)ψ+a2V(α
+Vβ)
−¯ ǫCγ(9)ψC+b1Vα
+Vβ
+¯ ǫγ(10)λC+b2Vα
−Vβ
−¯ ǫCγ(10)λ.
(4.25)
As in the case of the singlet, one can close the algebra on this Ansatz, but again it
is not an independent field. It turns out to be the variation of a composite field:
δ?1
2Sαβǫ(10)
?= δ
?
V(α
+Vβ)
−ǫ(10)
?
= V(α
+Vβ)
−δǫ(10)+ δV(α
+Vβ)
−ǫ(10)+ V(α
+δVβ)
−ǫ(10).
(4.26)
This generalises to ten-forms with n = 2m SU(1,1)-indices, for which we can also
close the algebra, but end up with the variation of the composite field
S(α1β1...Sαmβm)ǫ(10)
.(4.27)
The case of n odd is different, since the requirement of vanishing U(1) charge does
not allow one to write down a volume form. In this case the Ansatz for the fermionic
– 18 –
Page 20
part of the supersymmetry transformation is (we set here n = 2m + 1)
δAα1...α2m+1
µ1...µ10
= a V(α1
+ a∗V(α1
+ b V(α1
− b∗V(α1
+
...Vαm+1
+
...Vαm+1
...Vαm+1
+
...Vαm+1
Vαm+2
−
Vαm+2
+
Vαm+2
−
Vαm+2
+
...Vα2m+1)
−
...Vα2m+1)
+
...Vα2m+1)
−
...Vα2m+1)
+
¯ ǫCγµ1...µ10λC
−−
¯ ǫγµ1...µ10λ
¯ ǫγ[µ1...µ9ψµ10]C
+
−−
¯ ǫCγ[µ1...µ9ψµ10]
. (4.28)
It can be shown that only for the case m = 0, i.e., the doublet that we already
considered, the commutator of two such transformations closes producing just a ten-
form gauge transformation and a general coordinate transformation. As we have
seen already for the quadruplet (m = 1), extra terms are generated that need to
combine with additional terms in eq. (4.28), containing lower-rank forms and their
supersymmetry transformations, to produce bosonic gauge transformations. An ex-
plicit analysis shows that these terms can only be written for the quadruplet. Higher
SU(1,1) representations require introducing additional contributions from the scalars
in these bosonic terms, and the supersymmetry commutator produces derivatives of
these scalars. These contributions can not be identified with any parameter that
appears in the supersymmetry algebra. This suggests that only a doublet and a
quadruplet can be consistently included in the supersymmetry algebra of IIB.
5. The complete IIB transformation rules and algebra
This section collects our results for the SU(1,1)-democratic version of D = 10 IIB su-
pergravity. We present the supersymmetry transformation rules, the transformation
rules of the p-forms under bosonic gauge transformations, the definition of gauge in-
variant curvatures, and finally the results for the commutator of two supersymmetry
transformations. Of course all the transformations and definitions are interdepen-
dent. All results have been derived only up to the quadratic order in the fermions.
The supersymmetry transformation rules in Einstein frame, in the notation of [1, 2],
– 19 –
Page 21
are:
δeµa= i¯ ǫγaψµ+ i¯ ǫCγaψµC
, (5.1)
δψµ= Dµǫ +
δAα
δAµνρσ= ¯ ǫγ[µνρψσ]− ¯ ǫCγ[µνρψσ]C−3i
δλ = iPµγµǫC−
δVα
δVα
δAα
+12?Vα
+40A[µ1...µ4δAα
−15i
δAαβ
+8V(α
i
480Fµν1...ν4γν1...ν4ǫ +
−¯ ǫγµνλ + Vα
1
96GνρσγµνρσǫC−
−¯ ǫCγ[µψν]+ 4iVα
[µνδAβ
ρσ]
3
32GµνργνρǫC
+¯ ǫγ[µψν]C
,(5.2)
µν= Vα
+¯ ǫCγµνλC+ 4iVα
, (5.3)
8ǫαβAα
, (5.4)
i
24Gµνργµνρǫ,(5.5)
+= Vα
−= Vα
−¯ ǫCλ,(5.6)
+¯ ǫλC
−¯ ǫγµ1...µ6λ − iVα
−¯ ǫCγ[µ1...µ5ψµ6]− Vα
µ5µ6]− 20δA[µ1...µ4Aα
2Aα
µ1...µ8= +iV(α
+Vβ)
+21
,(5.7)
µ1...µ6= iVα
+¯ ǫCγµ1...µ6λC
+¯ ǫγ[µ1...µ5ψC µ6]
?
µ5µ6]
[µ1µ2ǫβγAβ
µ3µ4δAγ
µ5µ6]
,(5.8)
−Vβ)
−¯ ǫCγµ1...µ8λ − iV(α
−(¯ ǫγ[µ1...µ7ψµ8]− ¯ ǫCγ[µ1...µ7ψC µ8])
[µ1...µ6δAβ)
−35A(α
−105i
µ1...µ10= Vα
+20i?Vα
δAαβγ
+Vβ)
+¯ ǫγµ1...µ8λC
4A(α
µ7µ8]−7
µ3µ4δAµ5...µ8]+ 70A[µ1...µ4A(α
4A(α
[µ1µ2δAβ)
µ3...µ8]
[µ1µ2Aβ)
8A(α
−¯ ǫγµ1...µ10λ + Vα
+¯ ǫγ[µ1...µ9ψC µ10]+ Vα
µ1...µ10= iV(α
+20
−12A(αβ
−63
−210A[µ1...µ4A(α
+315i
µ5µ6δAβ)
µ7µ8]
[µ1µ2Aβ)
µ3µ4ǫγδAγ
µ5µ6δAδ
µ7µ8]
, (5.9)
δAα
+¯ ǫCγµ1...µ10λC
−¯ ǫCγ[µ1...µ9ψµ10]
?
,(5.10)
+Vβ
3(V(α
+Vγ)
−¯ ǫCγµ1...µ10λC− iV(α
+Vβ
[µ1...µ8δAγ)
4A(α
−Vβ
−Vγ)
+¯ ǫγµ1...µ10λ
−Vγ)
+Vγ)
−¯ ǫγ[µ1...µ9ψC µ10]− V(α
µ9µ10]+ 3A(α
µ7µ8δAγ)
−Vβ
+¯ ǫCγ[µ1...µ9ψµ10])
[µ1µ2δAβγ)
4A(α
µ9µ10]+ 105A(α
µ3...µ10]
[µ1...µ6Aβ
µ9µ10]+21
µ7µ8δAγ)
[µ1µ2Aβ
µ3µ4δAγ)
µ5...µ10]
µ5µ6Aβ
[µ1µ2Aβ
µ3µ4Aγ)
µ5µ6δAµ7...µ10]
8A(α
[µ1µ2Aβ
µ3µ4Aγ)
µ5µ6ǫδτAδ
µ7µ8δAτ
µ9µ10]
.(5.11)
For the bosonic gauge-transformations we find:
δAα
δAµ1...µ4= 4∂[µ1Λµ2µ3µ4]−i
δAα
µ1µ2= 2∂[µ1Λα
µ2]
, (5.12)
4ǫγδΛγ
[µ1Fµ2...µ6]−160
2F(α
,
[µ1Fδ
µ2µ3µ4]
,(5.13)
µ1...µ6= 6∂[µ1Λα
δAαβ
δAα
µ2...µ6]− 8Λα
µ2...µ8]+1
3Fα
2F(α
[µ1µ2µ3Λµ4µ5µ6]
[µ1µ2µ3Λβ)
,(5.14)
µ1...µ8= 8∂[µ1Λ(αβ)
µ1...µ10= 10∂[µ1Λα
δAαβγ
[µ1...µ7Λβ)
µ8]−21
µ4...µ8]
,(5.15)
µ2...µ10]
(5.16)
µ1...µ10= 10∂[µ1Λ(αβγ)
µ2...µ10]−2
3F(αβ
[µ1...µ9Λγ)
µ10]+ 32F(α
[µ1µ2µ3Λβγ)
µ4...µ10]
. (5.17)
– 20 –
Page 22
For the p-form fields we define field-strengths invariant under the bosonic gauge
transformations
Fα
µ1µ2µ3= 3∂[µ1Aα
Fµ1...µ5= 5∂[µ1Aµ2...µ5]+5i
Fα
µ2µ3]
, (5.18)
8ǫαβAα
[µ1µ2Fµ3...µ7]−280
µ2...µ9]+9
µ2...µ11]= 0,
3F(αβ
[µ1µ2Fβ
µ3µ4µ5]
,(5.19)
µ1...µ7= 7∂[µ1Aα
Fαβ
Fα
µ2...µ7]+ 28Aα
3Fα
4F(α
[µ1µ2µ3Aµ4...µ7]
[µ1µ2µ3Aβ)
, (5.20)
µ1...µ9= 9∂[µ1Aαβ
µ1...µ11= 11∂[µ1Aα
Fαβγ
4F(α
[µ1...µ7Aβ)
µ8µ9]−63
µ4...µ9]
, (5.21)
(5.22)
µ1...µ11= 11(∂[µ1Aαβγ
µ2...µ11]−1
[µ1...µ9Aγ)
µ10µ11]+ 4F(α
[µ1µ2µ3Aβγ)
µ4...µ11[) = 0.(5.23)
The duality relations between these field-strengths are:
Fα
Fαβ
µ1...µ7= −i
µ1...µ9= iǫµ1...µ9
3!ǫµ1...µ7µνρSαβǫβγFγ;µνρ
ρ[Vα
,(5.24)
+Vβ
+P∗
ρ− Vα
−Vβ
−Pρ]. (5.25)
The commutator of two supersymmetry transformations, [δ(ǫ1),δ(ǫ2)] must close
on symmetry transformations of the IIB multiplet. In fact, as we saw in previous
sections, this is the way the results of this paper have been obtained. We find the
following contributions to [δ(ǫ1),δ(ǫ2)]:
ξµ= i¯ ǫ2γµǫ1+ i¯ ǫ2Cγµǫ1C
Λα
Λµ1µ2µ3= Aµ1µ2µ3νξν+1
Λα
+40
, (5.26)
µ= Aα
µνξν− 2i[Vα
+¯ ǫ2γµǫ1C+ Vα
−¯ ǫ2Cγµǫ1], (5.27)
4[¯ ǫ2γµ1µ2µ3ǫ1− ¯ ǫ2Cγµ1µ2µ3ǫ1C]
µ1...µ5ρξρ− 2Vα
3A[µ1...µ4Λα
µ1...µ7νξν− 2V(α
+21
, (5.28)
µ1...µ5= Aα
−¯ ǫ2Cγµ1...µ5ǫ1+ 2Vα
3Λ[µ1µ2µ3Aα
+Vβ)
16A(α
+¯ ǫ2γµ1...µ9ǫ1C+ Vα
?
5A(αβ
−6
+¯ ǫ2γµ1...µ5ǫ1C
,
µ5]−40
µ4µ5]
(5.29)
Λαβ
µ1...µ7= A(αβ)
−(¯ ǫ2γµ1...µ7ǫ1− ¯ ǫ2Cγµ1...µ7ǫ1C)
µ7]−21
−¯ ǫ2Cγµ1...µ9ǫ1
16A(α
[µ1...µ6Λβ)
[µ1µ2Λβ)
µ3...µ7]
,(5.30)
Λα
µ1...µ9= −2i?Vα
Λαβγ
?
(5.31)
?
µ1...µ9= A(αβγ)
µ1...µ9νξν−2
+24i
3
V(α
+Vβ
+Vγ)
−¯ ǫ2γµ1...µ9ǫ1C− V(α
+¯ ǫ2γµ9]ǫ1C+ Vγ)
−Vβ
−Vγ)
+¯ ǫ2Cγµ1...µ9ǫ1
[µ1...µ8(Vγ)
5A(α
−¯ ǫ2Cγµ9]ǫ1)
[µ1µ2Sβγ)(¯ ǫ2γµ3...µ9]ǫ1− ¯ ǫ2Cγµ3...µ9]ǫ1C). (5.32)
This concludes the summary of our main results. In the next section we will present
the IIB supergravity multiplet in a real formulation in both Einstein frame and string
frame.
– 21 –
Page 23
6. U(1) gauge fixing and string frame
The results we derived so far were in Einstein frame. To go to string frame we will
first choose a U(1) gauge, so that the dependence on the dilaton becomes explicit.
Our choice is
V1
−∈ R ⇒ V2
+= V1
−
. (6.1)
To preserve this condition the supersymmetry transformations have to be modified
by a field dependent U(1) gauge transformation:
δ′(ǫ) = δ(ǫ) + δU(1)
?
i
2V1
−
(V2
−¯ ǫCλ − V1
+¯ ǫλC)
?
. (6.2)
This modification is only visible on the scalars since on the fermions it gives rise to
terms cubic in fermionic variables. The SU(1,1) transformations are also modified:
the condition (6.1) is preserved under a combination of an SU(1,1) and a U(1)
transformation. On a field χ of U(1)-charge q the required U(1) transformation is
χ → eiqθχ, with e2iθ=α + βz
¯ α +¯β¯ z
, (6.3)
where the coordinate z is defined in (2.16). This is of course visible on all fermions.
To make the dilaton and axion explicit we set
V1
−= V2
+=
1
√1 − z¯ z,
V2
−= (V1
+)∗=
z
√1 − z¯ z
. (6.4)
Using (2.20) and (2.24) we find (we now drop the prime on the redefined supersym-
metry transformation)
δτ = −2ie−φe−2iΛ¯ ǫλC
where
e−2iΛ=1 − iτ
1 + i¯ τ
, (6.5)
. (6.6)
Useful variables are
Pµ= −i
2e−φe−2iΛ∂µτ ,Qµ=1
4eφ
?1 − i¯ τ
1 − iτ∂µτ +1 + iτ
1 + i¯ τ∂µ¯ τ
?
.(6.7)
It is convenient to get rid of the factors of e−2iΛin the supersymmetry transformation
rules [18]. To do this we redefine the fermions by phase factors according to their
U(1) weights:
λ′= e3iΛ/2λ,ψ′
µ= eiΛ/2ψµ,ǫ′= eiΛ/2ǫ. (6.8)
– 22 –
Page 24
In the transformation rules the scalars Vα
nation Vα
±will now occur everywhere in the combi-
±e±iΛ, which are:
V1
−e−iΛ=1
2eφ/2(1 − iτ)
2eφ/2(1 − i¯ τ)
2eφ/2(1 + iτ)
2eφ/2(1 + i¯ τ)
,
V1
+eiΛ=1
,
V2
−e−iΛ=1
, (6.9)
V2
+eiΛ=1
.
Note that interchanging V1↔ V2corresponds to τ ↔ −τ, V+↔ V−to τ ↔ ¯ τ. The
transformation rules for the IIB supergravity multiplet of [1, 2] now become [21]:
δeµa= i(¯ ǫγaψµ) + h.c.
δψµ= Dµǫ −i
−
δA1
(6.10)
4eφǫ∂µℓ +
i
480Fµ1...µ5γµ1...µ5γµǫ
1
192eφ/2(γµγνρσ+ 2γνρσγµ)ǫC(F1− F2+ i¯ τ(F1+ F2))νρσ
µν= 2ieφ/2?(1 − i¯ τ)(¯ ǫγ[µψC ν]−i
+ (1 − iτ)(¯ ǫCγ[µψν]−i
δA2
+ (1 + iτ)(¯ ǫCγ[µψν]−i
δAµνρσ= ¯ ǫγ[µνρψσ]+ h.c. −3i
δλ =1
δℓ = ie−φ(¯ ǫCλ − ¯ ǫλC)
δφ = ¯ ǫCλ + ¯ ǫλC
.
, (6.11)
4¯ ǫCγµνλC)
4ǫγµν]λ)?
4¯ ǫCγµνλC)
4ǫγµν]λ)?
[µνδAβ
, (6.12)
µν= 2ieφ/2?(1 + i¯ τ)(¯ ǫγ[µψCν]−i
,(6.13)
8ǫαβAα
ρσ], (6.14)
2eφγµǫC∂µ¯ τ −
i
48eφ/2γνρσǫ(F1− F2+ i¯ τ(F1+ F2))νρσ
,
, (6.15)
(6.16)
(6.17)
For the higher-rank form fields we present only the transformations to the fermions,
because the contributions containing explicit gauge fields are unchanged by the gauge
fixing and redefinitions and can be read off from (5.8-5.11). We find:
δA1(6)= 6eφ/2?(1 − iτ)(¯ ǫCγ(5)ψ +
−(1 − i¯ τ)(¯ ǫγ(5)ψ +
δA2(6)= 6eφ/2?(1 + iτ)(¯ ǫCγ(5)ψ +
−(1 + i¯ τ)(¯ ǫγ(5)ψC+
δA11(8)= 2eφ?(1 − iτ)(1 − i¯ τ)(¯ ǫγ(7)ψ − ¯ ǫCγ(7)ψC)
+i
δA22(8)= 2eφ?(1 + iτ)(1 + i¯ τ)(¯ ǫγ(7)ψ − ¯ ǫCγ(7)ψC)
+i
δA12(8)= eφ?((1 − iτ)(1 + i¯ τ) + (1 + iτ)(1 − i¯ τ))(¯ ǫγ(7)ψ − ¯ ǫCγ(7)ψC)
+i
i
12¯ ǫγ(6)λ)
12¯ ǫCγ(6)λC)?+ ...
i
12¯ ǫγ(6)λ)
i
12¯ ǫCγ(6)λC)?+ ...
i
, (6.18)
, (6.19)
8((1 − iτ)2¯ ǫCγ(8)λ − (1 − i¯ τ)2¯ ǫγ(8)λC)?+ ..., (6.20)
8((1 + iτ)2¯ ǫCγ(8)λ − (1 + i¯ τ)2¯ ǫγ(8)λC)?+ ...,(6.21)
4((1 − iτ)(1 + iτ)¯ ǫCγ(8)λ − (1 − i¯ τ)(1 + i¯ τ)¯ ǫγ(8)λC)?+ ... , (6.22)
– 23 –
Page 25
δA1(10)= 10ieφ/2?(1 − i¯ τ)(¯ ǫγ(9)ψC−
+(1 − iτ)(¯ ǫCγ(9)ψ −
δA2(10)= 10ieφ/2?(1 + i¯ τ)(¯ ǫγ(9)ψC−
+(1 + iτ)(¯ ǫCγ(9)ψ −
δA111
−(1 − iτ)(¯ ǫCγ(9)ψ +3i
δA222
−(1 + iτ)(¯ ǫCγ(9)ψ +3i
δA112
×(¯ ǫγ(9)ψC+3i
−((1 − iτ)2(1 + i¯ τ) + 2(1 − iτ)(1 + iτ)(1 − i¯ τ)) ×
×(¯ ǫCγ(9)ψ +3i
δA221
×(¯ ǫγ(9)ψC+3i
−((1 + iτ)2(1 − i¯ τ) + 2(1 + iτ)(1 − iτ)(1 + i¯ τ)) ×
×(¯ ǫCγ(9)ψ +3i
i
20¯ ǫCγ(10)λC)
20¯ ǫγ(10)λ)?
i
20¯ ǫCγ(10)λC)
i
20¯ ǫγ(10)λ)?
i
, (6.23)
,(6.24)
(10)=5
6e3φ/2(1 − iτ)(1 − i¯ τ)?(1 − i¯ τ)(¯ ǫγ(9)ψC+3i
20¯ ǫγ(10)λ)?+ ...
(10)=5
20¯ ǫCγ(10)λC)
, (6.25)
6e3φ/2(1 + iτ)(1 + i¯ τ)?(1 + i¯ τ)(¯ ǫγ(9)ψC+3i
20¯ ǫγ(10)λ)?+ ...
5
18e3φ/2?((1 − i¯ τ)2(1 + iτ) + 2(1 − iτ)(1 − i¯ τ)(1 + i¯ τ)) ×
20¯ ǫCγ(10)λC)
20¯ ǫCγ(10)λC)
,(6.26)
(10)=
20¯ ǫγ(10)λ)?+ ...,(6.27)
(10)=
5
18e3φ/2?((1 + i¯ τ)2(1 − iτ) + 2(1 + iτ)(1 + i¯ τ)(1 − i¯ τ)) ×
20¯ ǫCγ(10)λC)
20¯ ǫγ(10)λ)?+ ...,(6.28)
Here the dots stand for the gauge field terms given in (5.8-5.11). So in formulas
(6.10) to (6.28) we have collected the complete set of Einstein frame supersymmetry
transformations in the real formulation.
Let us now review the transformations under SU(1,1) and SL(2,R) transformations.
Consider an SU(1,1) transformation
U =
?α β
¯β ¯ α
?
,α¯ α − β¯β = 1.(6.29)
The field τ transforms under the corresponding SL(2,R) transformation as
τ →aτ + b
a = Re(α − β), d = Re(α + β), b = −Im(α + β), c = Im(α − β)
cτ + d,
δτ →
δτ
(cτ + d)2, ad − bc = 1,
. (6.30)
The redefinition (6.8) modifies the behavior under SU(1,1) transformations. The
compensating U(1) transformation on a field χ of charge q (6.3) is now changed to
χ → eiqξχ,withe2iξ=cτ + d
c¯ τ + d
. (6.31)
For the dilaton one finds
eφ→ eφ(cτ + d)(c¯ τ + d). (6.32)
– 24 –
Page 26
One easily verifies that, e.g., the supersymmetry variation of τ
δτ = −2ie−φ¯ ǫλC
(6.33)
is consistent with these transformations. The bosonic fields with vanishing U(1)
charge still transform in the standard way under SU(1,1).
We will now bring some order into the collection of higher-rank forms (6.18) to
(6.28) by considering certain linear combinations of these. We choose the linear
combinations of the n-forms such that for a given n, each combination has a unique
power of τ in the fermionic terms of the supersymmetry variation. This is motivated
by the fact that the RR-forms come with a prefactor of e−φin the standard string
frame basis, which is proportional to τ − ¯ τ. Thus we make the following definitions:
˜C(2)= A1
(2)
,
˜B(2)= A1
˜C(4)= A(4)
,
˜C(6)= A1
(6)
,
˜B(6)= A1
˜C(8)= A11
˜D(8)= A11
(8)
,
˜D(10)= A1
˜C(10)= A111
˜B(10)= A111
˜E(10)= A111
˜D(10)= A111
(2)− A2
(2)+ A2
(2)
, (6.34)
(6.35)
(6)+ A2
(6)− A2
(8)+ A22
(6)
,(6.36)
(8)+ A22
(8)+ 2A12
(8),
˜B(8)= A11
(8)− 2A12
(8)
, (6.37)
(8)− A22
(10)+ A2
(6.38)
(10),
˜E(10)= A1
(10)+ A221
(10)− A2
?
(10)
?
(10)
?
(10)
?.
(10)
, (6.39)
(10)+ A222
(10)+ 3?A112
(10)− 3?A112
(10)−?A112
(10)+?A112
(10)
,(6.40)
(10)− A222
(10)+ A222
(10)− A222
(10)− A221
(10)+ A221
(10)− A221
, (6.41)
, (6.42)
(6.43)
A nice property, and partial justification why we refer to some of these linear combi-
nations as˜C(n)( RR fields) and˜B(n)( NS-NS fields) is the way these fields transform
into each other under S-duality. The discrete S-duality transformation τ → −1/τ
corresponds to an SL(2,R)-transformation with a = d = 0, b = −c = 1. The
behaviour of the form-fields under S-duality is
˜C(2)→ −i˜B(2)
˜C(4)→˜C(4)
˜C(6)→ −i˜B(6)
˜C(8)→ −˜B(8)
˜D(10)→ −i˜E(10)
˜C(10)→ i˜B(10)
˜D(10)→ i˜E(10)
,
˜B(2)→ −i˜C(2)
,
,
,
˜B(6)→ −i˜C(6)
˜B(8)→ −˜C(8)
˜E(10)→ −i˜D(10)
˜B(10)→ i˜C(10)
˜E(10)→ i˜D(10).
,
,,
˜D(8)→ −˜D(8)
,
,
,
,,
,(6.44)
– 25 –
Page 27
We see that applying S-duality twice gives +1 on τ and on the four- and eight-forms,
but −1 on the two-, six- and ten-forms. That this is indeed right, and that the S-
duality transformation is not its own inverse can be seen easily from translating back
to the SU(1,1) notation via (6.30), in which the S-duality transformation matrix is
given by
U =
?−i 0
0 i
?
(6.45)
so that U2gives a minus on forms with an odd number of SU(1,1)-indices.
Now we are ready to transform to string frame. The basic transformation is e(E)µa=
e−φ/4e(S)µa. We choose to write the variation of the zehnbein in standard form,
which requires a modification of supersymmetry with a λ-dependent local Lorentz
transformation (which we see only on the zehnbein), and a redefinition:
ǫ′= eφ/8ǫ
λ′= e−φ/8λ
ψ′
γ′
,(6.46)
, (6.47)
µ= eφ/8ψµ−i
µ= eφ/4γµ
4γ′
µλ′
C
, (6.48)
.(6.49)
Again we start with the basic supergravity multiplet and then discuss the high-rank
forms. The transformation rules are simplified by writing the complex fermions as
real doublets, i.e. ǫ → (ǫ1, ǫ2), where ǫi are real Majorana-Weyl fermions. This
gives rise to the appearance of Pauli matrices σ0= 1,σ1,iσ2,σ3in the contractions
between such doublets, generically:
¯ ǫCγχ + ¯ ǫγχC→ 2¯ ǫσ3γχ,
¯ ǫCγχ − ¯ ǫγχC→ 2i¯ ǫσ1γχ,
¯ ǫCγχC+ ¯ ǫγχ → 2¯ ǫγχ
¯ ǫCγχC− ¯ ǫγχ → −2i¯ ǫ(iσ2)γχ
, (6.50)
.(6.51)
In addition we redefine λ → λC, or, equivalently, in the real notation
λ → σ3λ. (6.52)
– 26 –
Page 28
We drop all primes in the string frame transformation rules:
δeµa= 2i¯ ǫγaψµ
δψµ= Dµǫ +1
+i
(6.53)
8eφγν∂νℓγµ(iσ2)ǫ −
96eφγνρσγµσ1ǫ(F−+ iℓF+)νρσ
1
480eφγµ1...µ5γµ(iσ2)ǫFµ1...µ5
δ˜Bµν= 8i¯ ǫσ3γ[µψν]
δ˜Cµν= −8e−φ¯ ǫσ1γ[µ(ψν]+i
δ˜Cµνρσ= 2ie−φ¯ ǫ(iσ2)γ[µνρ(ψσ]+i
−3i
δλ =
+1
δℓ = 2e−φ¯ ǫ(iσ2)λ
1
16γνρσ3ǫF+µνρ
−
, (6.54)
, (6.55)
2γν]λ) − iℓδ˜Bµν
4γσ]λ)
,
, (6.56)
16{˜Cδ˜B −˜Bδ˜C}µνρσ
2γµ∂µφǫ −
48eφγνρσσ1ǫ(F−+ iℓF+)νρσ
(6.57)
i
i
48γνρσσ3ǫF+νρσ−i
2eφγµ∂µℓ(iσ2)ǫ
,(6.58)
,(6.59)
δφ = 2¯ ǫλ(6.60)
where we have defined
F+= F1+ F2,F−= F1− F2.(6.61)
For the higher form fields we find:
δ˜C(6)= 24ie−φ¯ ǫσ1γ(5)(ψ +i
6γ(1)λ) + 40˜C(4)δ˜B(2)
4˜B(2)
−20δ˜C(4)˜B(2)−15i
δ˜B(6)= 24e−φ?ℓ¯ ǫσ1γ(5)(ψ +i
+40˜C(4)δ˜C(2)− 20δ˜C(4)˜C(2)−15i
δ˜C(8)= 16ie−φ¯ ǫ(iσ2)γ(7)(ψ +i
−35˜B(2)˜B(2)δ˜C(4)+ 70˜C(4)˜B(2)δ˜B(2)]
−105i
δ˜B(8)= −16ie−φ{ℓ2¯ ǫ(iσ2)γ(7)(ψ +i
+i
+21
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
6γ(1)λ) + e−φ¯ ǫσ3γ(5)(ψ +i
4˜C(2)
?
, (6.62)
3γ(1)λ)?
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
4˜C(6)δ˜B(2)−7
?
, (6.63)
8γ(1)λ) +21
4˜B(2)δ˜C(6)
16˜B(2)˜B(2)
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
?
, (6.64)
8γ(1)λ)
4ℓe−φ¯ ǫγ(8)λ + e−2φ¯ ǫ(iσ2)γ(7)(ψ +3i
4˜B(6)δ˜C(2)−7
+70˜C(4)˜C(2)δ˜C(2)−105i
δ˜D(8)= 16ℓe−φ¯ ǫ(iσ2)γ(7)(ψ +i
+21
−7
+35˜C(4){˜B(2)δ˜C(2)+˜C(2)δ˜B(2)}
−105i
8γ(1)λ)}
4˜C(2)δ˜B(6)− 35˜C(2)˜C(2)δ˜C(4)
16˜C(2)˜C(2)
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
8γ(1)λ) + 2ie−2φ¯ ǫγ(8)λ
?
,(6.65)
8{˜C(6)δ˜C(2)+˜B(6)δ˜B(2)}
8{˜B(2)δ˜B(6)+˜C(2)δ˜C(6)} − 35˜B(2)˜C(2)δ˜C(4)
16˜B(2)˜C(2)
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
?
,(6.66)
– 27 –
Page 29
δ˜D(10)= 40ie−2φ¯ ǫσ3γ(9)(ψ +i
δ˜E(10)= 40e−2φ?ℓ¯ ǫσ3γ(9)(ψ +i
δ˜C(10)= −40
−12˜C(8)δ˜B(2)+ 3δ˜C(8)˜B(2)−63
−210˜C(4)˜B(2)˜B(2)δ˜B(2)+ 105δ˜C(4)˜B(2)˜B(2)˜B(2)
+315
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
δ˜B(10)= +40
+40
−12˜B(8)δ˜C(2)+ 3δ˜B(8)˜C(2)−63
−210˜C(4)˜C(2)˜C(2)δ˜C(2)+ 105δ˜C(4)˜C(2)˜C(2)˜C(2)
+315
?˜C(2)δ˜B(2)−˜B(2)δ˜C(2)
δ˜D(10)= −40
+2˜B(2)δ˜D(8)+˜C(2)δ˜C(8)− 8˜D(8)δ˜B(2)− 4˜C(8)δ˜C(2)
+7
4
−21
−70?˜B2
+315
δ˜E(10)= +40
+80
?ψ +2i
+2˜C(2)δ˜D(8)+˜B(2)δ˜B(8)− 8˜D(8)δ˜C(2)− 4˜B(8)δ˜B(2)
+7
4
−21
−70?˜C2
+315
5γ(1)λ)
5γ(1)λ) − e−φ¯ ǫσ1γ(9)(ψ +3i
i
10γ(1)λ?
4˜C(6)˜B(2)δ˜B(2)+21
, (6.67)
10γ(1)λ)?
, (6.68)
3ie−φ¯ ǫσ1γ(9)
?ψ +(6.69)
4δ˜C(6)˜B(2)˜B(2)
16i˜B(2)˜B(2)˜B(2)
?
, (6.70)
3e−4φ?1 + e2φℓ2?¯ ǫσ3γ(9)
3ℓe−φ?ℓ2+ e−2φ?¯ ǫσ1γ(9)
?ψ +2i
?ψ +
4˜B(6)˜C(2)δ˜C(2)+21
5γ(1)λ?
i
10γ(1)λ?
4δ˜B(6)˜C(2)˜C(2)
16i˜C(2)˜C(2)˜C(2)
?
, (6.71)
9e−2φ¯ ǫσ3γ(9)
?ψ +2i
5γ(1)λ?−40
3ℓe−φ¯ ǫσ1γ(9)
?ψ +
i
10γ(1)λ?
?˜B2
?˜B(2)˜C(6)δ˜C(2)+˜C(2)˜C(6)δ˜B(2)+˜B(2)˜B(6)δ˜B(2)
(2)˜C(4)δ˜C(2)+ 2˜B(2)˜C(2)˜C(4)δ˜B(2)−3
16i?˜B2
3ie−3φ?1
9iℓe−2φ¯ ǫσ3γ(9)
(2)δ˜B(6)+ 2˜B(2)˜C(2)δ˜C(6)
?
4
?
2˜B2
(2)˜C(2)δ˜C(4)
?
(2)˜C2
(2)δ˜B(2)−˜B3
3+ ℓ2e2φ?¯ ǫσ1γ(9)
(2)˜C(2)δ˜C(2)
?ψ +
5γ(1)λ?
?,
i
10γ(1)λ?
?˜C2
?˜B(2)˜B(6)δ˜C(2)+˜C(2)˜C(6)δ˜C(2)+˜C(2)˜B(6)δ˜B(2)
(2)˜C(4)δ˜B(2)+ 2˜B(2)˜C(2)˜C(4)δ˜C(2)−3
16i?˜B(2)˜C3
(2)δ˜C(6)+ 2˜B(2)˜C(2)δ˜B(6)
?
4
?
2˜B(2)˜C2
(2)δ˜C(4)
?
(2)δ˜B(2)−˜B2
(2)˜C2
(2)δ˜C(2)
?.(6.72)
We will now introduce the standard RR and NS-NS fields, and extend this to the
higher rank forms. For this we define:
B(2)=1
C(0)= −1
C(2)= −i
C(4)= 2˜C(4)+ 3C(2)B(2)
2˜B(2)
2ℓ
4˜C(2)
,(6.73)
,(6.74)
,(6.75)
,(6.76)
– 28 –
Page 30
for which we define curvatures
H(3)= 3∂B(2)
G(2n−1)= (2n − 1){∂C(2n−2)−1
,(6.77)
2(2n − 2)(2n − 3)C(2n−4)∂B(2)}.(6.78)
In (6.78) n takes on the values n = 1,2,3, but this will be extended to n ≤ 6 below.
The corresponding bosonic gauge transformations are
δB(2)= ∂Σ
δC(2n−2)= ∂Λ(2n−3)+1
,(6.79)
2(2n − 2)(2n − 3)Λ(2n−5)∂B(2)
. (6.80)
We now rewrite the supergravity multiplet in these variables:
δeµa= 2i¯ ǫγaψµ
δψµ= Dµǫ −1
−1
δBµν= 4i¯ ǫσ3γ[µψν]
δC(0)= −e−φ¯ ǫ(iσ2)λ
δCµν= 2ie−φ¯ ǫσ1γ[µ(ψν]+i
δCµνρσ= 4ie−φ¯ ǫ(iσ2)γ[µνρ(ψσ]+i
δλ =
+ieφ(γ · G(1))(iσ2)ǫ +
δφ = 2¯ ǫλ.
(6.81)
8γνρσ3ǫHµνρ−1
24eφ(γ · G(3))γµσ1ǫ −
4eφ(γ · G(1))γµ(iσ2)ǫ
1
960eφ(γ · G(5))γµ(iσ2)ǫ, (6.82)
,(6.83)
, (6.84)
2γν]λ) + C(0)δBµν
4γσ]λ) + 6C[µνδBρσ]
24(γ · H(3))σ3ǫ
i
12eφ(γ · G(3))σ1ǫ
, (6.85)
,(6.86)
i
2γµ∂µφǫ −
i
, (6.87)
(6.88)
The supersymmetry transformations of the RR fields C can be summarized as (n =
1,2,3, Pn= iσ2for n even, Pn= σ1for n odd):
δC(2n−2)= (2n − 2)ie−φ¯ ǫPnγ(2n−3)(ψ(1)+
+1
i
2n−2γ(1)λ)
2(2n − 2)(2n − 3)C(2n−4)δB(2)
.(6.89)
We will now extend this to the higher-rank forms. We define the following RR fields:
C(6)=1
C(8)=1
C(10)= −3
4˜C(6)+ 5C(4)B(2)
2˜C(8)+ 7C(6)B(2)
4˜C(10)+ 9C(8)B(2)
, (6.90)
, (6.91)
. (6.92)
These combinations transform precisely as (6.89). We have therefore identified the
tower of RR forms, in the same form as in [9]. The S-dual of C(10)is however not
the field B(10)given in [9]. It turns out that B(10)corresponds precisely to our˜D(10).
– 29 –
Page 31
The S-duals of the C(2n−2)should form a tower of NS-NS forms. If one defines that
under S-duality
C(2)→ iS(2),
C(4)→ S(4),
C(6)→ −iS(6),
C(8)→ −S(8),
C(10)→ iS(10),(6.93)
then we find
S(2)=
S(4)= 2˜C(4)+ 6iC(2)S(2),
S(6)=1
S(8)=1
S(10)= −3
i
4˜B(2), (6.94)
(6.95)
4˜B(6)+ 10iC(2)S(4),
2˜B(8)+ 14iC(2)S(6),
4˜B(10)+ 18iC(2)S(8).
(6.96)
(6.97)
(6.98)
For the case ℓ = 0 the supersymmetry variations for S(n)are then described by
δS(2n−2)= (−i)n(2n − 2)e−(n−2)φ¯ ǫPnγ(2n−3)
+i(2n − 2)(2n − 3)S(2n−4)δC(2)
where S(0)= 0 and Pn= σ3for n even and Pn= iσ2for n odd.
?ψ +
n−2
2n−2γ(1)λ?
(6.99)
(6.100)
7. Summary and Discussion
In this work we showed that the standard formulation of IIB supergravity can be
extended to include a doublet and a quadruplet of ten-form potentials. We argued
that no other independent ten-forms can be added to the algebra. We have been using
a “SU(1,1)-democratic” formulation, in which all forms are described together with
their magnetic duals. Furthermore, all forms transform in a given representation
under the duality group SL(2,R). The previously known RR-ten-form potential
C(10)is contained in the quadruplet. The other previously known ten-form (named
B(10)in [9]) is in the doublet and hence not S-dual to C(10) [13].
We have shown that all ten-form potentials have a leading term
δX(10)∼ enφ¯ ǫγ(9)ψ at l = 0 (7.1)
in their supersymmetry transformation in string frame where X(10) represents a
generic ten-form potential.
– 30 –
Page 32
potential in quadrupletpotential in doubletassociated branetension
C(10)
D9-braneg−1
S
D(10)
D(10)
E(10)
solitonic braneg−2
S
E(10)
exoticg−3
S
B(10)
exoticg−4
S
Table 1: Ten-form potentials in string frame, the corresponding branes and their tension
in terms of the string coupling gS.
Such ten-form potentials naturally occur as the leading contribution in Wess-Zumino
terms for space-time filling branes with tension gn
in table 1. These branes and their relevance for theories with sixteen supercharges
will be discussed in some detail in a forthcoming paper [17].
S. The resulting branes can be found
It would be interesting to see how these findings are compatible with the known S-
duality relations between the Heterotic and Type I superstrings. It is well-known that
the (Nambu-Goto part of the) tree-level action of the Type I (Heterotic) superstring
scales with g−1
S
(g−4
is not clear. However, the results presented in Table 3, Appendix B, and the S-
duality assignments of the ten-forms (6.44) open up the possibility to extend this,
consistent with S-duality, to the scaling behaviour g−1
and g−2
S
for the Heterotic superstring such that the Nambu-Goto term at the
Heterotic side contains the more conventional g−2
S) [22]. The interpretation of the g−4
S
term at the Heterotic side
S+g−3
Sfor the Type I superstring
S+ g−4
S
behaviour.
Work on the relation of string- and M-theory with the Kac-Moody algebras E11
[23, 24] and E10[25, 26] has an interesting connection with our results. In [27] it
was pointed out that E10and E11give rise to different IIB ten-form potentials. In
particular, E10does not give rise to ten-forms, whereas E11supports a doublet and
a quadruplet of ten-forms [28]. The latter is in agreement with our results.
It will be worthwile to derive the superspace formulation of our results. Note that,
although ten-form potentials have identically zero field-strengths, this is not true for
the ten-form superpotentials. It would be interesting to calculate the eleven-form
curvatures in flat superspace and to see to which kind of Wess-Zumino terms they
give rise to. This is the first step towards the construction of a kappa-symmetric
Green-Schwarz action for all 9-branes.
– 31 –
Page 33
8. Acknowledgements
We thank Axel Kleinschmidt, Hermann Nicolai and Tomas Ort´ ın for useful remarks.
E.B., S.K. and M. de R. are supported by the European Commission FP6 program
MRTN-CT-2004-005104 in which E.B., S.K. and M. de R. are associated to Utrecht
university. S.K. is supported by a Postdoc-fellowship of the German Academic Ex-
change Service (DAAD). F.R. is supported by a European Commission Marie Curie
Postdoctoral Fellowship, Contract MEIF-CT-2003-500308. The work of E.B. is par-
tially supported by the Spanish grant BFM2003-01090.
A. Conventions
The Levi-Civita symbol used in this paper is a tensor, and therefore includes the
appropriate powers of det e.
Some useful properties of the complex fermions are:
ψµ= −γ11ψµ,
λ = γ11λ,
Dµǫ = (∂µ+1
(A.1)
(A.2)
4ωµabγab−i
2Qµ)ǫ,(A.3)
(¯ χ1γµ1...µnχ2)∗= ¯ χ2γµn...µ1χ1= (−1)n¯ χ1Cγµ1...µnχ2C,
¯ χ1γµ1...µnχ2= (−1)n(n+1)/2¯ χ2Cγµ1...µnχ1C
(A.4)
. (A.5)
In these equations χiare arbitrary spinors, not necessarily Majorana or Weyl.
For the duality transformations of γ-matrices we have:
γµ1...µn= −(−1)
1
2n(n−1)
1
(10−n)!ǫµ1...µ10γµn+1...µ10γ11
(A.6)
The table below gathers the values of the U(1) weights of the different fields. The
zehnbein eµaand all form-fields A(2n)have weight zero.
B. Truncations
We briefly sketch how to apply the heterotic and type I truncations [9] to our IIB
results and give a list of the fields surviving the truncation.
We first express the complex spinor ǫ in terms of two real spinors
ǫ = ǫ1+ iǫ2.(B.1)
– 32 –
Page 34
Vα
+
1G(3)
1A(n)
0
Vα
−
−1
1
2
G(7)
1
ǫǫC
−1
−1
−3
2
¯ ǫ−1
−1
−3
2
¯ ǫC
1
2
ψµ
1
2
ψC µ
2
¯ψµ
2
¯ψCµ
+1
2
λ
3
2
λC
2
¯λ
2
¯λC
3
2
Table 2: Table of U(1) weights
The heterotic truncation is then given by setting
ǫ = ±ǫC. (B.2)
We will work with the ”+” choice. We also need to make a choice of gauge for the
scalars. We make the same choice as in section 6:
V2
+= V1
−. (B.3)
Plugging (B.2) into the SUSY variation of ψ we find
ψ = ψC. (B.4)
Similarly, we use the SUSY variations of the other fields to find how the truncation
acts on all the fields
ψ = ψC,
V2
A1
λ = λC,
V2
(B.5)
+= V1
(2)= A2
A(4)= 0,
A1
A11
A1
A111
−,
−= V1
+, (B.6)
(2),(B.7)
(B.8)
(6)= −A2
(8)= −A22
(10)= A2
(10)= −A222
(6),(B.9)
(8),A12
(8)= 0,(B.10)
(10), (B.11)
(10),A112
(10)= −A122
(10). (B.12)
We also observe that the relations for the scalars (B.6) imply, using the reality prop-
erties of the scalars and (2.16), that z = ¯ z. This implies that the axion is eliminated
by the truncation, using (2.20) and (2.24).
The type I truncation is given by setting
ǫ = ±iǫC
(B.13)
– 33 –
Page 35
where we work with the ”+”-choice again. We choose V2
the SUSY variations
+= V1
−again and find from
ǫ = iǫC, ψ = iψC, λ = −iλC,
V2
A1
(B.14)
+= V1
(2)= −A2
A(4)= 0,
A1
A11
A1
A111
−, V2
−= V1
+
(B.15)
(2), (B.16)
(B.17)
(6)= A2
(8)= −A22
(10)= −A2
(10)= A222
(6), (B.18)
(8), A12
(8)= 0, (B.19)
(10),
(10), A112
(B.20)
(10)= A122
(10). (B.21)
As in the case of the heterotic truncation, the axion is eliminated by the truncation.
We collect the surviving fields of both truncations in table 3.
type I truncationheterotic truncation
φφ
˜C(2)∼ e−φ
˜C(6)∼ e−φ
˜D(8)∼ e−2φ
˜E(10)∼ e−3φ
˜C(10)∼ e−φ
˜E(10)∼ e−3φ
˜B(2)∼ e0φ
˜B(6)∼ e−2φ
˜D(8)∼ e−2φ
˜D(10)∼ e−2φ
˜B(10)∼ e−4φ
˜D(10)∼ e−2φ
Table 3: Field contents of the type I and heterotic truncations. After the tilde we indicate
how the field scales with respect to the dilaton. The entries in every line are S-dual to
each other (up to a factor).
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