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arXiv:hep-th/0205099v1 10 May 2002

NEIP-02-004

hep–th/0205099

Branes with fluxes wrapped on spheres

Rafael Hern´ andez1

and Konstadinos Sfetsos2

1Institut de Physique, Universit´ e de Neuchˆ atel

Breguet 1, CH-2000 Neuchˆ atel, Switzerland

rafael.hernandez@unine.ch

2Department of Engineering Sciences, University of Patras

26110 Patras, Greece

sfetsos@mail.cern.ch, des.upatras.gr

Abstract

Following an eight-dimensional gauged supergravity approach we construct the most gen-

eral solution describing D6-branes wrapped on a K¨ ahler four-cycle taken to be the product

of two spheres of different radii. Our solution interpolates between a Calabi–Yau four-fold

and the spaces S2× S2× S2× IR2or S2× S2× IR4, depending on generic choices for

the parameters. Then we turn on a background four-form field strength, corresponding to

D2-branes, and show explicitly how our solution is deformed. For a particular choice of

parameters it represents a flow from a Calabi–Yau four-fold times the three-dimensional

Minkowski space-time in the ultraviolet, to the space-time AdS4× Q1,1,1in the infrared.

In general, the solution in the infrared has a singularity which within type-IIA super-

gravity corresponds to the near horizon geometry of the solution for the D2-D6 system.

Finally, we uncover the relation with work done in the eighties on Freund–Rubin type

compactifications.

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Branes wrapped on supersymmetric cycles provide a natural path to obtain gravity du-

als of field theories with low supersymmetry. These field theories are twisted since preserv-

ing some supersymmetry after wrapping the brane, requires the identification (expressed

better, the relation) of the spin connection on the cycle and some external R-symmetry

gauge fields [1]. Therefore, the dual supergravity solutions can be naturally constructed in

an appropriate gauged supergravity, and are eventually lifted to ten or eleven dimensions.

This approach was started in [2], and has been further developed for a wide variety of

branes wrapped on diverse supersymmetric cycles [3]-[21].

The case of D6-branes is of special interest because they lift to pure geometry in eleven

dimensions. This fact allows to argue how compactifications of M-theory on manifolds with

reduced holonomy arise as the local eleven dimensional description of D6-branes wrapped

on supersymmetric cycles in manifolds of lower dimension with a different holonomy group

[22]. This extends the work of [23], where D6-branes wrapping the three-cycle in the de-

formed conifold were shown to be described in eleven dimensions as a compactification on a

seven manifold with G2holonomy. These lifts to eleven dimensions for D6-branes wrapping

various cycles were explicitly constructed using eight dimensional gauged supergravity [24]

in [7, 11, 13, 19].

However these purely gravitational geometries are deformed when background fluxes

are included. In [13] M-theory on a Calabi-Yau four-fold was shown to arise as the eleven

dimensional description of D6-branes wrapped on K¨ ahler four-cycles inside Calabi-Yau

three-folds. The deformation of this background by a four-form field strength along the

unwrapped coordinates was recently considered in [21], where it was shown to induce a

flow from E2,1× CY4at ultraviolet to AdS4× Q1,1,1in the infrared limit.

The four-cycle in [13, 21] was taken to be a product of two two-spheres of the same

radius so that the metric was Einstein. In this paper we will eliminate the Einstein condi-

tion on the four-cycle allowing the spheres to have different radii and will also introduce a

four-form flux. When lifted to eleven dimensions, and in the absence of flux, our solution

will represent M-theory on a Calabi–Yau four-fold. We will find a three parameter family

of metrics in which the conical singularity of the four-fold is generically resolved by being

replaced by a bolt or nut singularity which is removable [25]. Then we turn on a background

four-form field strength, corresponding to D2-branes, which provides another mechanism

for resolving the singularity. After determining the most general supersymmetry preserv-

ing solution we discuss its behavior for various choices for the parameters. A special choice

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of parameters leads to an eleven-dimensional solution that flows from a Calabi–Yau four-

fold times the three-dimensional Minkowski space-time in the ultraviolet, to the space-time

AdS4×Q1,1,1in the infrared, where Q1,1,1is the seven-manifold coset space SU(2)3/U(1)2

that is supersymmetric [26]. While this is similar to [21], in the general case the singular-

ity persists and is the same as in the near horizon metric for the D2-D6 system. Finally,

we end the paper by making a precise connection of our work with Freund–Rubin type

compactifications of eleven-dimensional supergravity to four dimensions.

Before constructing our solution we will briefly review some relevant facts about gauged

supergravity in eight dimensions which was constructed by Salam and Sezgin [24] through

Scherk–Schwarz compactification of eleven-dimensional supergravity [27] on an SU(2)

group manifold. The field content of the theory consists of the metric gµν, a dilaton

Φ, five scalars given by a unimodular 3 × 3 matrix Li

an SU(2) gauge potential Aµ, all in the gravity sector, and a three-form coming from

reduction of the eleven dimensional three-form.1In addition, on the fermion side we have

αin the coset SL(3,IR)/SO(3) and

the pseudo–Majorana spinors ψµand χi.

The Lagrangian density for the bosonic fields is given, in κ = 1 units, by

L =

1

4R −1

g2

16e−2Φ(TijTij−1

4e2ΦFα

µνFµν βgαβ−1

2T2) −1

4Pµ ijPµ ij−1

2(∂µΦ)2

−

48e2ΦGµνρσGµνρσ,(1)

where Fα

µνis the Yang–Mills field strength. Supersymmetry will be preserved by bosonic

solutions to the equations of motion of eight dimensional supergravity if the supersymmetry

variations for the gaugino and the gravitino vanish. These are, respectively, given by

δχi =

1

2(Pµ ij+2

g

8e−Φ(Tij−1

3δij∂µΦ)ˆΓjΓµǫ −1

4eΦFµν iΓµνǫ

1

144eΦGµνρσˆΓiΓµνρσǫ = 0 ,

−

2δijT)ǫjklˆΓklǫ −

(2)

and

δψγ

= Dγǫ +1

g

288e−ΦǫijkˆΓijkΓγTǫ −1

1Reduction of the eleven-dimensional three-form also produces a scalar, three vector fields and three

two-forms. However, we will set all these fields to zero.

24eΦFi

µνˆΓi(Γµν

γ

− 10δµ

γΓν)ǫ

−

96eΦGµνρσ(Γµνρσ

λ

− 4δµ

λΓνρσ)ǫ = 0 ,(3)

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where the covariant derivative is

Dµǫ = ∂µǫ +1

4ωab

µΓabǫ +1

4Qµ ijˆΓijǫ,(4)

while Pµijand Qµijare, respectively, the symmetric and antisymmetric quantities entering

the Cartan decomposition of the SL(3,IR)/SO(3) coset, defined through

Pµij+ Qµij≡ Lα

i(∂µδβ

α− g ǫαβγAγ

µ)Lβ j,(5)

and Tijis the T-tensor defining the potential energy associated to the scalar fields,

Tij≡ Li

αLj

βδαβ, (6)

with T ≡ Tijδij, and

Li

αLα

j= δi

j,Li

αLj

βδij= gαβ,Li

αLj

βgαβ= δij. (7)

As usual, curved directions are labeled by greek indices, while flat ones are labeled by

latin, and µ,a = 0,1,...,7 are space-time coordinates, while α,i = 8,9,10 are in the

group manifold. Note also that upper indices in the gauge field, Aα

µ, are curved, and that

the field strength in eight dimensional curved space is defined as

Gµνρσ= e−4Φ/3ea

µeb

νec

ρed

σFabcd.(8)

The 32 × 32 gamma matrices in eleven dimensions can be represented as

Γa= γa× 1 12,

ˆΓi= γ9× τi,(9)

where the γa’s denote the 16 × 16 gamma matrices in eight dimensions and as usual

γ9 = iγ0γ1...γ7, so that γ2

9= 1 1, while τiare Pauli matrices. It will prove useful to

6iǫijkˆΓijk= −iˆΓ1ˆΓ2ˆΓ3= γ9× 1 12.

Let us now present the system under study in this paper and construct a solution de-

introduce Γ9≡

1

scribing this configuration. We will consider a D2-D6 brane system, with the D6-branes

wrapped on a K¨ ahler four-cycle inside a Calabi–Yau three-fold, and the D2-branes along

the unwrapped directions. Keeping some supersymmetry unbroken involves an identifi-

cation of the spin connection of the supersymmetric cycle and the gauge connection of

the structure group of the normal bundle. When we wrap the D6-branes on the four-

cycle, the SO(1,6) × SO(3)R symmetry group of the branes in flat space is broken to

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SO(1,2) × SO(4) × U(1)R. The breaking of the R-symmetry takes place because there

are two normal directions to the D6-branes that are inside the Calabi–Yau three-fold; the

R-symmetry is therefore broken to the U(1)Rcorresponding to them. The twisting will

amount to the identification of this U(1)Rwith a U(1) subgroup in one of the SU(2) fac-

tors in SO(4) ≃ SU(2) × SU(2). The remaining scalar after the twisting, together with

the vector and two fermions preserved by the diagonal group of U(1) × U(1)R, determine

the field content of N = 2 three-dimensional Yang–Mills. In the absence of D2-branes the

lift to eleven-dimensions corresponds to M-theory on a Calabi–Yau four-fold [22, 13].

We will choose the four-cycle to be a product of two two-spheres of different radii,

S2×¯S2. The deformation on the world-volume of the D6-branes will then be described

by a metric of the form

ds2

8= e2fds2

1,2+ dρ2+ α2dΩ2

2+ β2d¯Ω2

2,(10)

where ds2

1,2is the three-dimensional Minkowski metric, the line elements for the two-spheres

are

dΩ2

2= dθ2+ sin2θdφ2,d¯Ω2

2= d¯θ2+ sin2¯θd¯φ2, (11)

and f, α and β depend only on the radial variable ρ. The same will be true for all

additional fields that we will turn on. The four-form flux implied by the D2-branes along

the unwrapped directions will be

Gx0x1x2ρ= Qe−2Φ+3f

α2β2

,(12)

where in the above x0,x1,x2 are curved directions, Q is a dimensionfull constant and

the specific functional dependence is uniquely fixed by the equation of motion for the

three-form potential.

Turning now to the Killing spinor equation we should observe that it is quite useful to

introduce the triplet of Maurer–Cartan 1-forms on S2

σ1= sinθdφ ,σ2= dθ ,σ3= cosθdφ .(13)

We note that they obey the conditions dσi=1

of Maurer–Cartan forms on S3, but obviously only two of them are the independent ones.

We also introduce a similar triplet ¯ σidefined on the other sphere¯S2.

2ǫijkσj∧σj, so that they resemble the triplet

Consistency of the Killing spinor equations after splitting of the four-cycle into the

product of spheres in (10) requires turning on only one of the components of the gauge

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