Journal ArticleDOI

# AdS 3 × w (S 3 × S 3 × S 1 ) solutions of type IIB string theory

Aristomenis Donos
25 Feb 2009-Classical and Quantum Gravity (IOP Publishing)-Vol. 26, Iss: 6, pp 065009

AbstractWe analyse a recently constructed class of local solutions of type IIB supergravity that consist of a warped product of AdS 3 with a seven-dimensional internal space. In one duality frame the only other non-vanishing fields are the NS 3-form and the dilaton. We analyse in detail how these local solutions can be extended to give infinite families of globally well-defined solutions of type IIB string theory, with the internal space having topology S 3 × S 3 x S 1 and with properly quantized 3-form flux.

Topics: String duality (66%), Dilaton (56%), String theory (55%), Supergravity (52%), Duality (optimization) (52%)

## Summary (2 min read)

### 1 Introduction

• Starting with the work of [1], general characterisations of the geometries underlying such solutions, using G-structure techniques [2, 3], have been achieved for various d and for various amounts of supersymmetry [4]–[23].
• The solutions are again locally determined by a six-dimensional Kähler metric and a choice of a closed, primitive (1, 2)-form on the Kähler space.
• After two T-dualities on the two-torus it was also shown that these explicit solutions give type IIB AdS3 solutions with non-vanishing dilaton and RR three-form flux only.
• Furthermore the authors need to check that the three-form flux is properly quantised.

### 2.1 The local solutions

• The authors start with the explicit class of AdS3 solutions of section 4.3 of [26].
• After a further S-duality transformation the authors obtainAdS3 solutions with only NS fields non-vanishing, but they will continue to work with the above solution.
• The two generators are clearly the two copies of S3, at a fixed point on the other copy.

### 2.5 A quotient of M6 and integral three-forms

• Such as the suitably normalised RR three-form, has integral periods.the authors.
• 7We thank Dominic Joyce for suggesting this approach.the authors.the authors.

### 3.1 Electric and magnetic charges

• The authors next turn to the magnetic three-form charge.
• In the previous subsection the authors gave a prescription for performing such integrals by instead calculating integrals on submanifolds of the quotient space M̂6.
• This will illuminate and confirm many of their observations about the topology in the previous section.
• In the present case, however, there is a much simpler way to impose flux quantisation.
• Here the authors are not distiguishing between Φ and Π∗Φ. where G(3) is the three-dimensional Newton’s constant and RAdS3 is radius of the AdS3 space.

### 3.2 Computing periods using coordinate patches

• This provides a nice cross-check on various calculations carried out so far.
• Recall from section 2.5 that instead of considering the circle bundle L over M5 with total space M6 the authors should consider the circle bundle L̂ = L (p+q)q with total space M̂6 = M6/Z(p+q)q.
• This is because the azimuthal coordinate φ degenerates at the poles of the base S2.
• Consider first the overlap of U1N with U2N .

### 3.3 Taking the limit Q→ 0

• In particular, the solutions, with the internal manifold having topology S3 × S3 × S1, and all other parameters fixed, are smoothly connected with each other as Q varies and they all have the same central charge.
• While this may in fact be the correct interpretation, it seems difficult to draw this conclusion based on the results of this paper.
• The authors also need to check what happens to the global identifications they have made on the coordinates, and also the quantisation of the flux.
• The problem arises in taking the limit of the v circle fibration.
• In this limit the supergravity approximation is clearly breaking down and one needs to analyse string corrections before any definite conclusions can be drawn.

### 4 More general identifications

• In this section the authors will generalise the class of solutions that they have already constructed.
• Similarly imposing p = −q is not compatible with (2.22). of the last two sections into larger families.
• The authors will not need the explicit details.
• After completing the square in the metric on M6 the authors obtain an expression for what will become the globally defined angular one-form corresponding to the v′ circle fibration and it has the form dv − Av′Dψ − k(y)Dw′ (4.6) for some smooth function k(y) that can easily be determined.
• Note that, in general, the solutions of this section are not exactly marginal deformations of those in section 3: for example, in section 3 the authors saw that the magnetic three-form flux quantum numbers were constrained to be of the form (3.36), whereas here there is no such constraint.

### 5 Final Comments

• The authors have analysed in detail some local supersymmetric AdS3 solutions of type IIB supergravity, first found in [26], that have non-vanishing dilaton and RR three-form flux.
• The authors have shown that the parameters can be chosen and coordinates identified in such a way that the solutions extend to give rich classes of globally defined solutions of the form AdS3 ×w (S3 × S3 × S1) with properly quantised flux.
• The authors have shown that the solutions depend on continuous parameters and are hence dual to continuous families of SCFTs in two spacetime dimensions with (0, 2) supersymmetry.
• This will lead to larger families of solutions that would be worth exploring.
• Looking at equation (A.16) of [38] the authors see that the β-deformation activates a non-trivial dilaton and three-form.

### B More on the topology of M5

• This in turn defines a circle bundle with Chern number a, using (2.34).
• One can do similar computations for the three-submanifolds E1 and E2, with similar conclusions.
• 17Changing the sign of the last term leads to type IIB bubble solutions, as explained in [26].
• When Q 6= 0, one will also need to check the flux quantisation conditions and this will require generalising the techniques that the authors have used in this paper.

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arXiv:0810.1379v3 [hep-th] 27 Feb 2009
Imperial/TP/2008/JG/03
DESY 08-147
3
×
w
(S
3
× S
3
× S
1
) Solution s
of Type IIB String Theory
Aristomenis Donos
1
, Jerome P. Gauntlett
2
and James Sparks
3
1
DESY Theory Group, DESY Hamburg
Notkestrasse 85, D 22603 Hamburg, Germany
2
Theoretical Physics Group, Blackett Laboratory,
Imperial College, London SW7 2AZ, U.K.
2
The Institute for Mathematical Sciences,
Imperial College, London SW7 2PE, U.K.
3
Mathematical Institute, University of Oxford,
24-29 St Giles’, Oxford OX1 3LB, U.K.
Abstract
We analyse a recently constructed class of local solutions of type IIB
sup ergravity that consist of a warped product of AdS
3
with a seven-
dimensional internal space. In one duality frame the only other non-
vanishing ﬁelds are the NS three-form and the dilaton. We analyse in
detail how these local solutions can be extended to give inﬁnite fa milies
of globally well-deﬁned solutions o f type IIB string theory, with the in-
ternal space having topology S
3
× S
3
× S
1
and with properly quantised
three-form ﬂux.

Contents
1 Introduction 1
3
solutions 4
2.1 The local solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 B
4
= S
2
× S
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 M
5
= S
3
× S
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 M
6
= S
3
× S
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 A quotient of M
6
and integral three-forms . . . . . . . . . . . . . . . 13
3 Flux Quantisation and the Central Charge 18
3.1 Electric and magnetic charges . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Computing periods using coordinate patches . . . . . . . . . . . . . . 20
3.3 Taking the limit Q 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 More general ident iﬁcations 27
5 Final Comments 33
A U(1) bundles over Lens spaces 35
B More on the topology of M
5
36
C More general AdS
3
solutions 38
1 Introductio n
Supersymmetric solutions of string or M-theory that contain AdS
d+1
factors are dual
to supersymmetric conformal ﬁeld theories in d spacetime dimensions. Starting with
the work o f [1], general characterisations of the geometries underlying such solutions,
using G-structure techniques [2, 3], have been achieved for vario us d and fo r various
amounts of supersymmetry [4]–[23]. With a few exceptions, mostly with sixteen
sup ersymmetries, many of these geometries are still poorly understood, and it has
proved diﬃcult to ﬁnd explicit solutions.
One notable exception is the class of AdS
3
solutions of type IIB string theory
with non-vanishing ﬁve-form ﬂux, dual to d = 2 conformal ﬁeld theories with (0, 2)
sup ersymmetry, that were classiﬁed in [7]. It was shown that the seven-dimensional
internal space has a Killing vector which is dual to the R-symmetry of the dual SCFT.
1

The Killing vector deﬁnes a foliation and the solution is completely determined,
locally, by a ahler metric on the six-dimensional leaf space whose Ricci tensor
satisﬁes an additional diﬀerential condition. Moreover, a rich set of explicit solutions
have been constructed in [24, 14, 2 5] and the corresponding central charges of the
dual SCFTs have also been calculated.
More recently, it was understood how to generalise this class of type IIB AdS
3
so-
lutions to also include three-form ﬂux [26]. The solutions are again locally determined
by a six-dimensional ahler metric and a choice of a closed, primitive (1, 2)-form on
the ahler space. Once again additional explicit solutions were constructed with
the six-dimensional ahler space having a two-torus factor and the three-form ﬂux
being parametrised by a real parameter Q. After two T-dualities on the two-torus
it was also shown that these explicit solutions give type IIB AdS
3
solutions with
non-vanishing dilaton and RR three-form ﬂux only. After an additional S-duality
the solutions only involve NS ﬁelds.
In [26] these explicit solutions were examined in more detail for the special case
of Q = 0. It was shown that the parameters and ranges of the coordinates could be
chosen to give globally deﬁned supergravity solutions consisting of a warped product
3
with a seven-dimensional internal manifold that is diﬀeomorphic to S
2
×
S
3
× T
2
. It was shown that the solutions, with properly quantised three-form ﬂux,
are speciﬁed by a pair of positive coprime integers p, q.
The purpose of this paper is to carry out a similar analysis when we switch on
the parameter Q. We will ﬁnd that we are led to inﬁnite classes of solutions, with
the seven-dimensional internal space being diﬀeomorphic to S
3
× S
3
× S
1
and with
properly quantised ﬂuxes.
While the ﬁnal topology of the solutions is simple, it is not easy to see this in
the local coordinates in which the solutions are presented. When Q = 0 the S
2
× S
3
factor is realised in a manner very similar to the Y
p,q
Sasaki-Einstein spaces [27].
When Q 6= 0 one of the circles in the T
2
factor is ﬁbred over the S
2
×S
3
and we need
to carefully check that the circle ﬁbration is globally well-deﬁned, leading to S
3
×S
3
.
Furthermore we need to check that the three-form ﬂux is properly quantised. This is
not straightforward since it is not clear “where” the two S
3
factors are in the local
coordinates. After some false starts we developed a workable prescription for ensuring
that the three-form is properly quantised, as we shall explain.
The plan of the paper is as follows. In section 2, we begin by recalling the local
solutions of [26] and then discuss how, after suitable choices of parameters and periods
for the coordinates, the seven-dimensional internal manifold has topology S
3
× S
3
×
2

S
1
. We discuss some aspects of the topology in detail, leading to a prescription for
carrying out ﬂux quantisation which is dealt with in section 3. Our method uses a
quotient construction, which is explained in section 2, as well a s explicit coordinate
patches. In sections 2 and 3, the solutions depend on a pair of coprime positive
integers p, q, the electric three-form ﬂux, n
1
, the magnetic three-form ﬂux through
each of the two S
3
factors, M
1
and M
2
, and the parameter Q. For these solutions,
it turns out that M
1
and M
2
are not independent and are given by M
1
= M(p + q)
2
and M
2
= Mq
2
, where M is an integer. We calculate the centra l charge and show
that it is given by the simple fo r mula
c = 6n
1
(M
1
M
2
)M
2
M
1
. (1.1)
In particular it is independent of Q, and since the solutions with Q 6= 0 and other
parameters ﬁxed are all smoothly connected with each other, we conclude that when
Q 6= 0, the parameter Q corresponds to an exactly marginal deformation in the dual
(0, 2) SCFT. It is interesting to observe that the value of the central charge is precisely
the same as for the Q = 0 solutions studied in [2 6]. However, as we shall explain in
section 3.3 taking the limit Q 0 does not smoothly lead to the Q = 0 solutions and
so it is not clear whether or not t he Q 6= 0 solutions correspond to exactly marginal
deformations of the Q = 0 solutions.
In section 4, we generalise our construction by making more general identiﬁcations
on the coor dinates, obtaining solutions that involve more parameters. We show that
the central charge has exactly the same form as in (1.1), but now, however, the
integers M
1
and M
2
labelling the three-form ﬂux through the two S
3
’s are no longer
constrained. Thus not all of these more general solutions correspond to exactly
marginal deformations of those that we consider in sections 2 and 3. We conclude in
section 5.
We noted above that the S
2
× S
3
factor in the AdS
3
solutions constructed in
[26], with Q = 0, is realised in a similar way to the Y
p,q
Sasaki-Einstein spaces
found in [27]. In particular, in bo th cases the metrics on S
2
× S
3
are cohomogeneity
one. Given that the Y
p,q
metrics can be generalised to cohomogeneity two Sasaki-
Einstein metrics L
a,b,c
on S
2
× S
3
[28] (see also [29]), it is natura l to suspect that
there are analogous AdS
3
solutions, with ﬁve-form ﬂux only, with internal space
having topology S
2
× S
3
× T
2
and with the metric on the S
2
× S
3
factor having
cohomogeneity two. This is indeed possible, and moreover it is also possible to ﬁnd
generalisations with non-zero three-form ﬂux and with the internal manifold having
topology S
3
× S
3
× S
1
. We will present such solutions in appendix C, but we will
3

leave a detailed analysis of the regularity and ﬂux quantisation conditions fo r future
work.
3
solution s
2.1 The local solutions
3
solutions of section 4.3 of [26]. The string
frame metric is g iven by
1
L
2
ds
2
=
β
y
1/2
[ds
2
3
) + ds
2
(X
7
)] (2.1)
where
ds
2
(X
7
) =
β
2
1 + 2y Q
2
y
2
4β
2
Dz
2
+
U(y)
4(β
2
1 + 2y Q
2
y
2
)
Dψ
2
+
dy
2
4β
2
y
2
U(y)
+
1
4β
2
ds
2
(S
2
) + (du
1
Qy
2β
[(1 g)Dψ Dz])
2
+ (du
2
)
2
, (2.2)
where β, Q ar e positive constants, L is an arbitrary length scale and
U(y) = 1
1
β
2
(1 y)
2
Q
2
y
2
. (2.3)
2
(S
2
) is the standard
1
metric on a two-sphere, ds
2
(S
2
) =
2
+sin
2
θ
2
,
and we have deﬁned
Dψ = dψ + P (2.4)
with
dP = Vol(S
2
) = sin θ J . (2.5)
Note that P is only a locally deﬁned o ne-form on S
2
. In fact, more precisely, P is a
connection one-form on the U(1 ) principal bundle associated to the tangent bundle
of S
2
. The two-form J int roduced in (2.5) may be regarded as a ahler form on S
2
.
We also have
Dz = dz g(y)Dψ (2.6)
with
g(y) =
y(1 Q
2
y)
β
2
1 + 2y Q
2
y
2
. (2.7)
1
Note that we have rescaled the metric on S
2
appearing in [26] by a factor of 4.
4

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