IIB Supergravity Revisited

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 $E_{11}$ symmetry underlying string theory. For the reader's convenience we present the full supersymmetry and gauge transformations of {\it all} fields both in the manifestly SU(1,1) covariant Einstein frame and in the real U(1) gauge fixed string frame. Comment: 36 pages; additional comments in section 7, typos corrected in formulae in sections 5 and 6, references added

Full text

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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
3.1 Six-forms7
3.2Eight-forms10
4. Ten-forms14
4.1The doublet of ten-forms14
4.2 The 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 Discussion30
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].
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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)
– 4 –