arXiv:hep-th/0205099v1 10 May 2002
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
2Department of Engineering Sciences, University of Patras
26110 Patras, Greece
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
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 . 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 , and has been further developed for a wide variety of
branes wrapped on diverse supersymmetric cycles -.
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
. This extends the work of , 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 
in [7, 11, 13, 19].
However these purely gravitational geometries are deformed when background fluxes
are included. In  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 , 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 . 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
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 . While this is similar to , 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  through
Scherk–Schwarz compactification of eleven-dimensional supergravity  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
4Pµ ijPµ ij−1
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
144eΦGµνρσˆΓiΓµνρσǫ = 0 ,
= Dγǫ +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.
λΓνρσ)ǫ = 0 ,(3)
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