On supersymmetric D7-branes in the warped deformed conifold
Heng-Yu Chen
1
Peter Ouyang
1
Gary Shiu
0
1
0
PH-TH Division
, CERN, CH-1211 Geneva 23,
Switzerland
1
Department of Physics, University of Wisconsin
,
Madison, WI 53706, U.S.A
We study the supersymmetric properties of D7-branes in the warped deformed conifold. We consider the -symmetry conditions on D7-branes in this specific warped background, taking into account the background NS-NS 2-form flux. While any holomorphic embedding defines a supersymmetric D7-brane in the absence of background H-flux, most of the D7-brane embeddings considered in the literature do not preserve supersymmetry for the warped deformed conifold without also including brane worldvolume flux. For the simplest such embedding, we construct numerically the worldvolume flux necessary to restore supersymmetry. We also comment on the dual field theory descriptions in terms of cascading N = 1 supersymmetric gauge theories with flavors. Finally, we discuss some possible applications of our results to moduli stabilization and vacuum energy uplifting, gauge/gravity duality, and string inflationary model building.
Contents
1 Introduction
2 Review of the warped deformed conifold
3 Supersymmetry conditions
4 Supersymmetric D7 embeddings in the conifold 4.1 z-embedding 4.2 w-Embedding
4.3 Restoring supersymmetry with worldvolume flux
4.3.1 Boundary conditions and finiteness
4.3.2 Large radius factorization limit
4.3.3 Large radius interpolating limit
4.3.4 Consistent boundary conditions and subleading asymptotics
4.3.5 Numerical solution
4.3.6 Summary of results
4.4 Other embeddings
5 Field theory remarks
5.1 Cascade pattern for z embedding and its generalizations
5.2 The vacua for Kuperstein-like embeddings 5.3 w-embedding and its generalizations
6 Discussion
A Deformed conifold coordinates
B Generalized z-embeddings
C Karch-Katz embedding
D Comparison with Beninis proposal
1 Introduction
One of the major developments in string theory of the last decade has been the emergence
of warped spacetimes as a setting for a rich variety of physics. To date, perhaps the
most important application of such warped geometries has been in the gauge theory/string
theory correspondence [13]. In this setting, noncompact spacetimes which are warped
by the presence of background flux have been conjectured to give a dual description of
gauge theories. In many cases, this duality can relate strongly coupled field theories to
weakly coupled gravity theories, and as such offers new tools for studying field theories
when traditional perturbative methods fail.
Another application of warped geometry has been in the study of string theory as a
theory of particle physics. When one compactifies string theory from ten dimensions to
four dimensions, the compactification manifold can support nontrivial background fluxes,
and these fluxes will generically have a backreaction on the geometry, resulting in warping.
The gravitational redshift generated by a strong warp factor can lead to a hierarchy of
scales, thus offers a geometrical explanation for the huge disparity between the electroweak
and the Planck scale [4, 5] (see also [69] for string theory realizations). For the same
reason, warped geometries have also been invoked in supersymmetry breaking scenarios in
string theory, both to lower the scale of supersymmetry breaking [10, 11] and to control
the amount of sequestering [12]. The low energy effective supergravity theory describing
strongly warped backgrounds is challenging to derive, though continued progress has been
made [10, 11, 1317].
Warping has also found application in inflationary cosmology. A key issue in
inflationary cosmology is to understand the underlying dynamics which allows for a sufficient
number e-folds of expansion. It turns out that warping can be fruitfully used in stringy
cosmological models to construct suitable potentials for slow roll inflation as well as for
motivating new inflationary mechanisms (see [1823] and the references therein).
Strikingly, there is a particular warped geometry, the warped deformed conifold in
type IIB supergravity [24, 25], which is simple enough to allow detailed study, but is
also rich enough to have been used in all three of these areas. It has almost anti-de Sitter
asymptotics, and in gauge/gravity duality it is dual to an interesting confining gauge theory.
If one embeds the deformed conifold in a compact Calabi-Yau, and turns on appropriate
three-form fluxes, one finds that the resulting background has a natural and stable hierarchy
of scales [9]. And in cosmology, interesting inflationary models have been also constructed
in the conifold [2629].
In this paper, we study a particularly interesting variation of the warped deformed
conifold, in which we add a certain number of probe D7-branes to the background. These
branes fill four noncompact directions and wrap a four-cycle 4 in the transverse space. In
the gauge theory dual, these branes add fundamental matter to the theory [30, 31], (...truncated)