Their tension is related to the different vacuum expectation values (VEV’s)
for the gluino condensate at both sides of the domain wall. In subsection 6.4
we will study a candidate to be domain wall in the MN background. It is a
D5-brane wrapping in the far IR the nontrivial three-cycle $S^{3}$.

There exist more checks which show up that some of the properties of an ${\cal
N}=1$ SYM theory are encoded by the Maldacena-Núñez background. Reviews on
this topic can be found in [10, 75, 76, 77, 78]. Recall that an ${\cal N}=1$
SYM theory is confining. The computation of the tension of a fundamental
string in the MN background was done in [34] finding a finite value, what
confirms that the dual gauge theory confines. The tension of a $q$-string in
backgrounds dual to four-dimensional $SU(N_{c})$ ${\cal N}=1$ gauge theories
was computed in [55], with the result displayed in (1.2.65). The computation
of the $\beta$-function on the gravity side of the duality was performed in
[74]. They found, neglecting subleading exponential corrections, the exact
perturbative NSVZ $\beta$-function in the Paulli-Villars renormalization
scheme for ${\cal N}=1$ SYM theory. The glueball spectrum of the theory was
analysed in [79] and a formula for the mass spectrum of the mesons in the
quenched approximation (the limit of the gauge theory with $N_{f}\ll N_{c}$)
was given in [71]. In [80] a supergravity background dual to a four-
dimensional ${\cal N}=1$ SQCD with quartic superpotential was proposed and
some consistency checks which support the field theory interpretation were
studied.

###  1.3 D-branes in supergravity backgrounds

According to the gauge/gravity extensions of the AdS/CFT correspondence, the
chiral operators (the gauge invariant operators which have the lowest possible
conformal dimension for a given R-charge) of an ${\cal N}=1$ SCFT are in one-
to-one correspondence with the modes of type IIB supergravity on
$AdS_{5}\times X^{5}$, where $X^{5}$ is a five-dimensional Sasaki-Einstein
manifold. However, the massive string modes correspond to (non-chiral)
operators in long multiplets whose dimensions diverge for the large ’t Hooft
coupling limit that is taken in the low energy version of the AdS/CFT
conjecture. Thus, in the limit $g_{YM}^{2}N_{c}\rightarrow\infty$ the stringy
nature of the dual theory is obscured by the decoupling of the non-chiral
operators, which incidentally constitute the majority of possible gauge
invariant operators. If we depart from the limit of infinite ’t Hooft
coupling, then all non-chiral operators do not decouple at all and the
spectrum of the gauge theory is presumably related to type IIB string theory
on $AdS_{5}\times X^{5}$. Even in the very large ’t Hooft coupling limit, it
is possible to demonstrate the stringy nature of the dual theory by including
extra D-branes wrapped on nontrivial cycles in the $X^{5}$ manifold. Although
in a different context, Witten showed in [8] that an ${\cal N}=4$
supersymmetric $SO(2N_{c})$ gauge theory (which is dual to type IIB strings on
$AdS_{5}\times RP^{5}$) possesses chiral operators of dimension $N_{c}$, the
Pfaffians, whose dual interpretation is provided by a D3-brane wrapping a
three cycle of the manifold $RP^{5}$. This shows that the dual theory cannot
be simple supergravity but it must contain D-branes. In principle, one would
expect that branes wrapped on nontrivial cycles correspond to states in the
conformal field theory that are nonperturbative from the point of view of the
$1/N_{c}$ expansion. In [81, 82] it was shown that wrapped branes could be
interpreted indeed as soliton-like states in the large $N_{c}$ gauge theory
for certain ${\cal N}=1$ theories.

The kind of solitonic-like state in the field theory strongly depends on the
dimension of the D-branes as an object in the $AdS_{5}$ space. Consider then a
Dp-brane in an $AdS_{5}\times X^{5}$ background of type IIB supergravity. The
first thing that one has to do is to study the homology groups of the internal
manifold $X^{5}$. This will give us the different ways in which a Dp-brane can
wrap a nontrivial q-cycle ($q\leq p$). When the cycle is calibrated, the dual
state will not spoil supersymmetry while it will do in the opposite case. Let
us consider first the case in which the Dp-brane is not extended along the
holographic (radial) direction and does not fill either the gauge theory
directions completely. From the field theory point of view, the solitonic-like
state dual to this brane is a extended $(p-q)$-dimensional object. For
instance, baryonic operators are particles, strings are one-dimensional
objects whereas domain walls are two-dimensional defects in the gauge theory
dual. Let us analyse more in detail each of the three cases and give some
evidences of the statement:

  * •

A vertex connecting $N_{c}$ fundamental strings –known as the baryon vertex–
can be identified with a baryon built out of external quarks, since each
string ends on a charge in the fundamental representation of $SU(N_{c})$. Such
an object can be constructed by wrapping a D5-brane over the whole five-
dimensional compact manifold $X^{5}$ [8]. The argument is that a D5-brane
wrapping $X^{5}$ captures $N_{c}$ units of the RR five-form flux $F_{5}$ of
the background. There is a $U(1)$ gauge field on the D5-brane worldvolume (see
eq. (1.3.1)) which couples to $F_{5}$ and takes $N_{c}$ units of charge. Since
the total charge of a $U(1)$ gauge field must cancel in a closed universe,
there must be $-N_{c}$ unit of charges coming from another source. Such source
are $N_{c}$ elementary strings that end on the D5-brane since each end point
is electrically charged with respect to the $U(1)$ field, with charge $+1$ or
$-1$ depending on the orientation of each fundamental string. In order to
cancel the charge, all must have the same orientation.

Another object of particular interest in quantum field theories that arises
when D3-branes are placed at conical singularities (${\cal N}=1$ SCFT’s) is
given by D3-branes wrapped on supersymmetric three-cycles; these states are
dual to dibaryons built from chiral fields charged under two different gauge
groups of the resulting quiver theory [8, 81, 82, 83]. The argument given in
this identification is the matching of the conformal dimension of the dibaryon
operators with the mass of wrapped D3-branes using general rules of the
AdS/CFT correspondence. We will see explicit examples of this matching as well
as of the baryon vertex configuration in chapters 2 and 3.

  * •

Domain walls and strings (flux tubes) in the field theory side can be
introduced holographically as Dp-branes wrapping q-cycles of the internal
geometry [8, 81, 82] with $p-q=2$ or $p-q=1$ respectively. Actually, the
superalgebras of ${\cal N}=1$ supersymmetric theories admit central charges
associated with objects extended in two or one space directions (codimension
one and two, respectively). For instance, in $SU(N_{c})$ ${\cal N}=1$ SYM
there are 1/2-BPS domain walls which interpolate between the inequivalent
$N_{c}$ vacua which come from the spontaneous breaking of the
${\mathbb{Z}}_{2N_{c}}$ symmetry (the non-anomalous subgroup of the
$U(1)_{R}$) to ${\mathbb{Z}}_{2}$ by the gaugino condensate. There can be also
BPS codimension two objects, namely strings which have been studied in the
context of different ${\cal N}=1$ theories, see [84] and references therein.
The physics of such objects turns out to be quite rich, including for instance
the phenomenon of enhanced (supersized) supersymmetry, also present for domain
walls [85]. We will find potential dual objects to domain-walls and flux-
strings mainly in chapters 2, 3 and 6.
In [81] it was also argued that a D3-brane wrapping a nontrivial two-cycle in
$X^{5}$ gives rise to a non BPS object called “fat” string. We will also find
examples of the explicit configuration of that object in chapters 2 and 3.

Moreover, one can think of modifying the theory by introducing supersymmetric
defects of codimension one or two (regions of space-time where some fields are
localised), which break the $SO(1,3)$ Lorentz invariance. In particular, on
the field theory side one can add spatial defects which reduce the amount of
supersymmetry but nevertheless preserve conformal invariance [86], giving rise
to the so-called “defect conformal field theories” (dCFT). Since this modifies
the lagrangian of the field theory, we expect, on general grounds, that the
string theory setup should be modified at infinity. Therefore, the defects
should be dual to D-branes extending infinitely in the holographic direction
but without filling completely the gauge theory directions. It is important to
point out that the effective gauge coupling (in the Minkowski directions that
they fill) of these branes is zero since they are extended along a non-compact
direction. Therefore, from the point of view of the dual gauge theory, these
objects give rise to the addition of fundamental multiplets to a region
(defect) of the spacetime.

A holographic dual of four-dimensional ${\cal N}=4$ super Yang-Mills theory
with a three-dimensional defect was proposed in ref. [87] by Karch and
Randall, who conjectured that such a dCFT can be realised in string theory by
means of a D3-D5 intersection. In the near-horizon limit the D3-branes give
rise to an $AdS_{5}\times S^{5}$ background, in which the D5-branes wrap an
$AdS_{4}\times S^{2}$ submanifold. It was argued in ref. [87] that the AdS/CFT
correspondence acts twice in this system and, apart from the holographic
description of the four dimensional field theory on the boundary of $AdS_{5}$,
the fluctuations of the D5-brane should be dual to the physics confined to the
boundary of $AdS_{4}$. The defect conformal field theory associated with the
D3-D5 intersection corresponds to ${\cal N}=4$, $d=4$ super Yang-Mills theory
coupled to ${\cal N}=4$, $d=3$ fundamental hypermultiplets localised at the
defect [88]. These hypermultiplets arise as a consequence of the strings
stretched between the D3- and D5-branes.

The defect field theories corresponding to other intersections have also been
studied in the literature. For example, from the D1-D3 intersection one gets a
four-dimensional CFT with a hypermultiplet localised on a one-dimensional
defect [89]. Moreover, the D3-D3 intersection gives rise to a two-dimensional
defect in a four-dimensional CFT [90, 91].

In chapters 2 and 3 we will study embeddings of D-branes which are suitable to
introduce defects of codimension one and two in ${\cal N}=1$ SCFT’s.
Extensions of this analysis to more realistic theories where the conformal
symmetry is broken (for instance ${\cal N}=1$ SYM) will be performed in
chapter 6.

Finally, it deserves special attention the D-branes which fill completely the
gauge theory directions and wrap a cycle of the internal manifold. According
to the original proposal of ref.[92], these spacetime filling configurations
can be used as flavour branes, i.e. as branes whose fluctuations can be
identified with the dynamical mesons of the gauge theory (see refs.[71, 91,
93, 94, 95, 96] for the analysis of the meson spectrum in different theories).
These flavour branes must extend along the radial direction from an infinity
value since the addition of flavour may modify the lagrangian of the dual
field theory. It could be that they reach a minimum $r_{0}$. It was argued in
[92] that the dual interpretation of this mass scale is given by the mass of
the quarks introduced in the gauge theory. It may be possible to set that mass
scale to zero and to deal with massless quarks by just taking
$r_{0}\rightarrow 0$, namely extending the flavour brane completely along the
radial direction. Since they are extended along a non-compact direction, the
gauge theory which support has vanishing four-dimensional effective coupling
on the Minkowski directions. Thus, the gauge symmetry on the flavour branes is
seen as a flavour symmetry by the four-dimensional gauge theory of interest.
In chapters 2 and 3 we will also look for embeddings of flavour branes which
are suitable to accommodate dynamical quarks in the field theory dual. The
construction of supergravity backgrounds dual to ${\cal N}=1$ field theories
which include flavour branes will be carry out in chapters 4 and 5.

In the absence of a string theory formulation on backgrounds with Ramond-
Ramond forms, the final goal in order to extract valuable information about
the stringy spectrum would be to introduce extra D-branes in a supergravity
background and to take into account, not only the effects that the D-branes
feel coming from the background fields, but also the backreaction undergone by
the supergravity background due to the presence of these extra D-branes. The
techniques developed so far have only been applied to study backreacted
flavour branes, as we will explain in subsection 1.3.3. A generalisation to
other kind of D-branes may be possible. However, as a first approach we can
neglect the backreaction undergone by the supergravity background. This can be
achieved by considering D-brane probes of various dimensions as we will
explain in subsection 1.3.2.

####  1.3.1 Effective Dp-brane action

In this subsection we want to introduce one of the main tools that we will use
along this thesis. It is the effective action that describes the low energy
dynamics of small bosonic fluctuations around a classical supergravity Dp-
brane solution. This action can be obtained if one requires that the non-
linear sigma model describing the propagation of an open string with Dirichlet
boundary conditions (and therefore fixed to a Dp-brane) in a general
supergravity background is conformally invariant. The constraints in the
fields coming from this invariance are the same as the equation of motion
derived from the following effective action (in string frame):

$S_{D_{p}}\,=\,-T_{p}\int_{\Sigma_{p+1}}d^{p+1}\xi\,e^{-\phi}\sqrt{-\rm{det}(g+{\cal
F})}\,\,.$ (1.3.1)

This is the Dirac-Born-Infeld (DBI) action for a Dp-brane. Here
$g_{\mu\nu}=\partial_{\mu}X^{M}\partial_{\nu}X^{N}\,G_{MN}$ is the pullback of
the spacetime metric $G_{MN}$ on the worldvolume $\Sigma_{p+1}$, where greek
indices $\mu,\nu\ldots$ are worldvolume indices and $M,N\ldots$ are target
spacetime indices. In addition, ${\cal F}=B+2\pi\alpha^{\prime}dA$, with $B$
being the pullback of the NSNS two-form,
$B_{\mu\nu}=\partial_{\mu}X^{M}\partial_{\nu}X^{N}\,B_{MN}$, and $A$ is a
worldvolume $U(1)$ gauge field with field strength $F=dA$. $T_{p}$ is the
tension of a Dp-brane (1.1.5) and $\phi$ is the dilaton field.

The DBI action is an abelian $U(1)$ gauge theory which reduces, to leading
order in $\alpha^{\prime}$, to Yang-Mills (YM) in $p+1$ dimensions with $9-p$
scalar fields when the target space is flat. Let us set to zero the NSNS two-
form $B$ and perform the expansion:

$S_{D_{p}}\,\approx\,-\int_{\Sigma_{p+1}}d^{p+1}\xi\,e^{-\phi}\sqrt{-\rm{det}g}\Big{(}\frac{1}{4g^{2}_{YM}}F_{\mu\nu}F^{\mu\nu}\,+\,\frac{2}{(2\pi\alpha^{\prime})^{2}}\partial_{\mu}X^{i}\partial^{\mu}X_{i}\,+\,{\cal
O}(F^{4})\Big{)}\,\,,$ (1.3.2)

where the index $i$ stands for spacetime indices perpendicular to the Dp-
brane. The YM coupling constant $g_{YM}$ is related to the string parameters
by

$g^{2}_{YM}\,=\,(2\pi)^{p-2}\alpha^{\prime\frac{p-3}{2}}g_{s}\,\,,$ (1.3.3)

reproducing the relation written in (1.1.1) in the particular case of
D3-branes. By including fermionic superpartners, the low energy action for the
D-branes becomes that of SYM theory in $p+1$ dimensions.

Moreover, Dp-branes are objects charged under RR potentials and their action
should contain a term of coupling to these fields. This term must fulfil
certain requirements. It must be invariant under worldvolume diffeomorphisms
and it should be classically equivalent when the brane propagates in two
target space configurations related by a target space gauge symmetry. We will
not go into details but one can show that the term

$\int_{\Sigma_{p+1}}C\wedge e^{{\cal F}}\,\,,$ (1.3.4)

where $C$ denotes the pullback of the sum of the RR background121212Along this
thesis the RR scalar $\chi$ will also be denoted by $C^{0}$. fields,
$C=\sum_{r=0}^{8}C^{(r)}$ and $C^{(r)}$ is a $r$-form, fulfils the demanded
requirements. This topological term is called the Wess-Zumino (WZ) action.

If the Dp-brane has RR charge $q_{D_{p}}$, the total action will be the sum of
the DBI and WZ part in the form

$S_{D_{p}}\,=\,-T_{p}\int_{\Sigma_{p+1}}d^{p+1}\xi\,e^{-\phi}\sqrt{-\rm{det}(g+{\cal
F})}\,+\,q_{D_{p}}\int_{\Sigma_{p+1}}C\wedge e^{{\cal F}}\,\,.$ (1.3.5)

Extreme branes satisfy the BPS bound $T_{p}=\mid q_{D_{p}}\mid$ and their
action will be given by

$S_{D_{p}}\,=\,-T_{p}\int_{\Sigma_{p+1}}d^{p+1}\xi\,e^{-\phi}\sqrt{-\rm{det}(g+{\cal
F})}\,\pm\,T_{p}\int_{\Sigma_{p+1}}C\wedge e^{{\cal F}}\,\,,$ (1.3.6)

where the sign $+$ stands for branes and the sign $-$ for antibranes. It will
also be useful to write down the action in Einstein frame:

$S_{D_{p}}\,=\,-T_{p}\int_{\Sigma_{p+1}}d^{p+1}\xi\,e^{\frac{p-3}{4}\phi}\sqrt{-\rm{det}(g^{(E)}+e^{-\phi/2}{\cal
F})}\,\pm\,T_{p}\int_{\Sigma_{p+1}}C\wedge e^{{\cal F}}\,\,,$ (1.3.7)

where $g^{(E)}_{\mu\nu}=e^{-\phi/2}g_{\mu\nu}$ denotes the metric in Einstein
frame.

It is possible to generalise the above action and consider a stack of N Dp-
branes since they are BPS objects and they can remain statically at any
distance of each other. However, this is beyond the scope of the work we want
to present in the forthcoming sections.

In short, with the action (1.3.6) we can describe, in the low energy limit,
the dynamics of the extra D-branes that we are going to place in some
supersymmetric supergravity backgrounds dual to four-dimensional ${\cal N}=1$
gauge theories. We will be interested in not to break completely supersymmetry
since ultimately we will be concerned about the study of some features of a
four-dimensional ${\cal N}=1$ supersymmetric field theory on the supergravity
side. Therefore an additional constraint is imposed on the way of placing
D-branes in a supergravity background. They must satisfy a local fermionic
symmetry called “kappa symmetry”, which we explain in the next subsection.

####  1.3.2 The probe approximation and the kappa symmetry analysis

As a first approach to understand on the gravity side some features of the
field theory which need the introduction of additional D-branes, we can
neglect the backreaction undergone by the supergravity background. This can be
achieved by considering a number of D-branes much smaller than the number of
branes which generate the background. This simplification, whose dual
counterpart is known as the quenched approximation, is called the probe
approximation on the gravity side. Let us explain a bit more in detail the
physical meaning of this limit. Let us call $N_{D_{p}}$ to the number of extra
$D_{p}$ branes we are introducing on the gravity side in order to study some
feature of the field theory. We will use $N_{c}$ to denote the number of
branes that engineer the supergravity background. The quenched/probe
approximation ($N_{D_{p}}\ll N_{c}$) consists of neglecting all the effects of
order $N_{D_{p}}/N_{c}$ on both sides of the duality. On the gravity side this
can be done by considering the $N_{D_{p}}$ extra branes as brane probes. The
meaning of a brane probe is that it does not modify the background
configuration but it does couple to the background fields. In consequence we
can study its dynamics by means of the effective action displayed in eq.
(1.3.6). A particular case where the probe limit has a well-known dual
interpretation is when we deal with a number $N_{f}$ of flavour branes. When
the ratio $N_{f}/N_{c}$ is very small, we are neglecting the effects that
include the running of fundamentals in internal loops. Even when these
fundamentals are massless, their effects while running in loops are suppressed
by a factor of ${\cal O}(N_{f}/N_{c})$. In the strict ’t Hooft limit [7], if
the number of flavours is kept fixed, the corrections due to the quantum
dynamics of quarks exactly vanish [92].

We are interested in bosonic configurations of the brane probe since we want
to keep the classical limit. Hence we set to zero the fermionic fields of the
background. However, if we wish that the brane probe does not break
supersymmetry completely, we should look at the coupling of the fermionic
degrees of freedom of the Dp-branes to the bosonic background fields. A
crucial ingredient (in the Green-Schwarz formulation) is then a local
fermionic symmetry of the worldvolume theory called kappa symmetry [97, 98].
The role played by this symmetry is to eliminate the extra fermionic degrees
of freedom which appear when the target space supersymmetry becomes manifest.
A nice review of this topic can be found in [99]. The idea is to obtain kappa
symmetric actions for Dp-branes embedded in a given background. This
determines the form of the kappa symmetry matrix $\Gamma_{\kappa}$:

$\Gamma_{\kappa}\,=\,\frac{1}{\sqrt{-\rm{det}(g+{\cal
F})}}\sum_{n=0}^{\infty}\frac{1}{2^{n}n!}\gamma^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}{\cal
F}_{\mu_{1}\nu_{1}}\ldots{\cal F}_{\mu_{n}\nu_{n}}\,J^{(n)}_{(p)}\,\,,$
(1.3.8)

where $g$ is the induced metric, $\gamma_{\mu_{1}\mu_{2}\ldots}$ is the
antisymmetric product of induced worldvolume Dirac matrices and ${\cal
F}=B+2\pi\alpha^{\prime}F$ as in eq. (1.3.1). In eq. (1.3.8), $J^{(n)}_{(p)}$
is the following matrix:

$J_{(p)}^{(n)}\,=\,\left\\{\begin{aligned}
&(\Gamma_{11})^{n+(p-2)/2}\,\Gamma_{(0)}&\qquad&\text{(IIA)}\\\
&(-1)^{n}(\sigma_{3})^{n+(p-3)/2}\,i\sigma_{2}\otimes\Gamma_{(0)}&\qquad&\text{(IIB)}\,\,,\end{aligned}\right.$

where IIA and IIB stand for type IIA and type IIB string theory respectively,
$\Gamma_{11}$ is the chiral matrix in ten dimensions and $\Gamma_{(0)}$ is
defined by

$\Gamma_{(0)}\,=\,\frac{1}{(p+1)!}\epsilon^{\mu_{1}\ldots\mu_{(p+1)}}\gamma_{\mu_{1}\ldots\mu_{(p+1)}}\,\,.$
(1.3.9)

Recall that in the type IIB string theory the spinor $\epsilon$ is composed by
two Majorana-Weyl spinors which can be arranged as a two-component vector. In
order to write the expression of $\Gamma_{\kappa}$, it is convenient sometimes
to decompose the complex spinor $\epsilon$ in its real and imaginary parts as
$\epsilon\,=\,\epsilon_{1}+\,i\,\epsilon_{2}$. It is straightforward to find
the following rules to pass from complex to real spinors:

$\epsilon^{*}\,\leftrightarrow\,\sigma_{3}\,\epsilon\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i\epsilon^{*}\,\leftrightarrow\,\sigma_{1}\,\epsilon\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i\epsilon\,\leftrightarrow\,-i\sigma_{2}\,\epsilon\,\,.$
(1.3.10)

We can write an induced worldvolume Dirac matrix in terms of the ten-
dimensional constant gamma matrices $\Gamma_{\underline{M}}$. We need to
choose a frame basis $e^{\underline{M}}$ in the background geometry where
$G_{MN}=\eta_{MN}\,e^{\underline{M}}\,e^{\underline{N}}$. The
$e^{\underline{M}}$ one-forms are related to the coordinates chosen in the
geometry by means of the vielbein coefficients $E_{N}^{\underline{M}}$,
namely:

$e^{\underline{M}}\,=\,E_{N}^{\underline{M}}\,dX^{N}\,\,.$ (1.3.11)

Then, the induced Dirac matrices on the worldvolume are defined as

$\gamma_{\mu}\,=\,\partial_{\mu}\,X^{M}\,E_{M}^{\underline{N}}\,\,\Gamma_{\underline{N}}\,\,.$
(1.3.12)

We mentioned that we must fix the local kappa symmetry in order to remove the
extra fermionic degrees of freedom of the worldvolume theory of the Dp-brane.
The extra bosonic degrees of freedom are removed by choosing the “static
gauge”. In this gauge the worldvolume general coordinate invariance is used to
equate $p+1$ of the target space coordinates with the worldvolume coordinates,
$X^{\mu}=\xi^{\mu}\,(\mu=0,\cdots,p)$.

The fact that we are interested in bosonic configurations of D-branes that
preserve a fraction of the spacetime supersymmetry imposes projections on the
Killing spinor of the background of the kind:

$\Gamma_{\kappa}\epsilon=\pm\epsilon\,,$ (1.3.13)

where $+$ stands for branes and $-$ for anti-branes. In what follows, we will
concentrate on the kappa symmetry condition applied to D-branes. The equation
(1.3.13) can be used to determine the supersymmetric configuration of a brane
probe. $\Gamma_{\kappa}$ is a matrix which depends on the embedding of the
probe and the condition (1.3.13) gives rise to a set of first-order BPS
equations which fix, up to integration constants, the embedding of the brane
probe. In addition, one has to check that these first-order equations fulfil
the second-order equations of motion for the worldvolume bosonic fields and,
actually, we will see that they saturate a bound for the energy, as it usually
happens in the case of worldvolume solitons [100]. Moreover, eq. (1.3.13) can
also be applied to determine the fraction of spacetime supersymmetry preserved
by a bosonic D-brane. For brane probes, eq. (1.3.13) is the only supersymmetry
condition that arises. However, for supergravity configurations with branes as
sources, we have the D-brane action coupled to supergravity. Thus, the
supersymmetry condition (1.3.13) must be complemented with the supersymmetric
transformations of the supergravity theory. Therefore, the fraction of
supersymmetry preserved by a D-brane is determined by the number of solutions
of (1.3.13), where $\epsilon$ is the Killing spinor of the background.
Obviously, to apply the technique sketched above one has to know first the
Killing spinor (see the first part of section 1.2).

In chapters 2, 3 and 6 we will apply the kappa symmetry technique in order to
find, in a systematic way, supersymmetric configurations of brane probes which
preserve some amount of supersymmetry in different supersymmetric backgrounds.
Moreover, in those chapters, we will give a field theory interpretation of
these configurations supported with some gauge/gravity checks.

####  1.3.3 Introducing backreacting D-branes: the smearing procedure

It is interesting to go beyond this ‘quenched’ or ‘non-backreacting’ probe
approximation and see what happens when one adds a large number of D-branes,
of the same order of magnitude as the number of colour branes which generate
the geometry, and the backreaction effects of the D-branes are considered.
Indeed, many phenomena that cannot be captured by the quenched approximation
might be apparent when a string backreacted background is found. For instance,
if we consider a number of flavour branes in a given background of the order
of the colour branes, we are working on the so called Veneziano’s topological
expansion [101]. New physics (beyond the ’t Hooft limit [7] where the number
of flavours is kept fixed and much smaller than the number of colours) is
captured by Veneziano’s proposal: we will be able to see this in chapters 4
and 5 that consider the backreaction of the flavour branes in the Klebanov-
Witten and Klebanov-Strassler models respectively.

Although all the procedure we are going to carry out below may be generalised
for the case of dealing with any kind of D-brane introduced in a given
background, we will just pay attention in this introductory chapter to the
particular case of flavour D7-branes in the Klebanov-Witten background and its
extensions to the $AdS_{5}\times Y^{p,q}$ and $AdS_{5}\times L^{a,b,c}$. The
explanation of what we will call “smearing procedure” is particularly simple
and well motivated from a geometrical point of view in those cases. Extensions
of the smearing procedure to all kind of D-branes, besides the flavour branes,
may be a simple generalisation of the ideas that we will discuss in this
subsection.

The procedure that we will follow is inspired mostly by the papers [80, 102]
and more recently [103, 104]. In those papers flavours are added into the
dynamics of the dual background via the introduction of $N_{f}$ spacetime
filling flavour branes, whose dynamics is given by a Dirac-Born-Infeld action
(1.3.6). This dynamics is intertwined with the usual Einstein action of type
IIB supergravity.

To illustrate the way in which flavour branes will be added, let us start by
considering the background of type IIB supergravity studied in subsection
1.2.1. For the sake of brevity, in the following we will take units is which
$g_{s}=1$, $\alpha^{\prime}=1$.

We will add $N_{f}$ spacetime filling D7-branes to this geometry, in a way
that preserves some amount of supersymmetry. This problem was studied in [105,
106] for the conformal case and in [94, 107] for the cascading theory. These
authors found calibrated embeddings of D7-branes which preserve (at least some
fraction of) the supersymmetry of the background. We will choose to put two
sets of D7-branes on the surfaces parameterised by

$\displaystyle\xi^{\alpha}_{1}$
$\displaystyle=\\{x^{0},x^{1},x^{2},x^{3},r,\theta_{2},\varphi_{2},\psi\\}\qquad\theta_{1}=\text{const.}\qquad\varphi_{1}=\text{const.}\;,$
$\displaystyle\xi^{\alpha}_{2}$
$\displaystyle=\\{x^{0},x^{1},x^{2},x^{3},r,\theta_{1},\varphi_{1},\psi\\}\qquad\theta_{2}=\text{const.}\qquad\varphi_{2}=\text{const.}\;.$
(1.3.14)

Note that these two configurations are mutually supersymmetric with the
background. Moreover, since the two embeddings are non-compact, the gauge
theory supported on the D7-branes has vanishing four-dimensional effective
coupling on the Minkowski directions; therefore the gauge symmetry on them is
seen as a flavour symmetry by the four-dimensional gauge theory of interest.
The two sets of flavour branes introduce a $U(N_{f})\times U(N_{f})$ symmetry,
the expected flavour symmetry with massless flavours. The configuration with
two sets (two branches) can be deformed to a single set, shifted from the
origin, that represents massive flavours, and realises the explicit breaking
of the flavour symmetry to the diagonal vector-like $U(N_{f})$. Our
configuration (eq. (1.3.3)) for probes is nothing else than the $z_{1}=0$
holomorphic embedding of [106].

We will then write an action for a system consisting of type IIB supergravity
(1.2.1) plus D7-branes described by their Dirac-Born-Infeld action (in
Einstein frame, see eq, (1.3.7)):

$\begin{split}S&=\frac{1}{2\kappa_{10}^{2}}\int
d^{10}x\,\sqrt{-G}\Big{[}R-\frac{1}{2}\partial_{M}\phi\partial^{M}\phi-\frac{1}{2}e^{2\phi}F_{1}^{2}-\frac{1}{4}F_{5}^{2}\Big{]}+\\\
&\qquad-T_{7}\sum^{N_{f}}\int
d^{8}x\,e^{\phi}\Big{[}\sqrt{-\hat{G}_{8}^{(1)}}+\sqrt{-\hat{G}_{8}^{(2)}}\Big{]}+T_{7}\sum^{N_{f}}\int
C_{8}\;.\end{split}$ (1.3.15)

Notice that we did not excite the worldvolume gauge fields, but this is a
freedom of the approach that we adopted. Otherwise one may need to find new
suitable kappa symmetric embeddings.

These two sets of D7-branes are localised in their two transverse directions,
hence the equations of motion derived from (1.3.15) will be quite complicated
to solve, due to the presence of source terms (Dirac delta functions).

But we can take some advantage of the fact that we are adding lots of
flavours. Indeed, since we will have many ($N_{f}\sim N_{c}$) flavour branes,
we might think about distributing them in a homogeneous way on their
respective transverse directions. This ‘smearing procedure’ boils down to
approximating

$\displaystyle T_{7}\sum^{N_{f}}\int
d^{8}x\,e^{\phi}\sqrt{-\hat{G}_{8}^{(i)}}\quad$
$\displaystyle\rightarrow\quad\frac{T_{7}N_{f}}{4\pi}\int
d^{10}x\,\sin\theta_{i}\sqrt{-\hat{G}_{8}^{(i)}}\,\,,$ $\displaystyle
T_{7}\sum^{N_{f}}\int C_{8}\quad$
$\displaystyle\rightarrow\quad\frac{T_{7}N_{f}}{4\pi}\int
d^{10}x\,\Big{[}Vol(Y_{1})+Vol(Y_{2})\Big{]}\wedge C_{8}\;,$ (1.3.16)

with $Vol(Y_{i})=\sin\theta_{i}\,d\theta_{i}\wedge d\varphi_{i}$ the volume
form of the $S^{2}$’s.

This effectively generates a ten-dimensional action

$\begin{split}S&=\frac{1}{2\kappa_{10}^{2}}\int
d^{10}x\,\sqrt{-G}\Big{[}R-\frac{1}{2}\partial_{M}\phi\partial^{M}\phi-\frac{1}{2}e^{2\phi}F_{1}^{2}-\frac{1}{4}F_{5}^{2}\Big{]}+\\\
&\qquad-\frac{T_{7}N_{f}}{4\pi}\int
d^{10}x\,e^{\phi}\sum_{i=1,2}\sin\theta_{i}\,\sqrt{-\hat{G}_{8}^{(i)}}+\frac{T_{7}N_{f}}{4\pi}\int\Big{[}Vol(Y_{1})+Vol(Y_{2})\Big{]}\wedge
C_{8}\;.\end{split}$ (1.3.17)

We can derive in the smeared case the following equations of motion, coming
from the action (1.3.17):

$\displaystyle
R_{MN}-\frac{1}{2}G_{MN}R=\frac{1}{2}\Big{(}\partial_{M}\phi\partial_{N}\phi-\frac{1}{2}G_{MN}\partial_{P}\phi\partial^{P}\phi\Big{)}+\frac{1}{2}e^{2\phi}\Big{(}F_{M}^{(1)}F_{N}^{(1)}-\frac{1}{2}G_{MN}F_{1}^{2}\Big{)}+$
$\displaystyle\qquad+\frac{1}{96}F_{MPQRS}^{(5)}F_{N}^{(5)PQRS}+T_{MN}\,\,,$
$\displaystyle\frac{1}{\sqrt{-G}}\partial_{M}\Big{(}G^{MN}\,\sqrt{-G}\,\partial_{N}\phi\Big{)}=e^{2\phi}F_{1}^{2}+\frac{2\kappa_{10}^{2}T_{7}}{\sqrt{-G}}\frac{N_{f}}{4\pi}e^{\phi}\sum_{i=1,2}\sin\theta_{i}\,\sqrt{-\hat{G}_{8}^{(i)}}\,\,,$
$\displaystyle d\Big{(}e^{2\phi}\star F_{1}\Big{)}=0\,\,,$ $\displaystyle
dF_{1}=-2\kappa_{10}^{2}T_{7}\frac{N_{f}}{4\pi}\Big{[}Vol(Y_{1})+Vol(Y_{2})\Big{]}\,\,,$
$\displaystyle dF_{5}=0\,\,.$ (1.3.18)

The modified Bianchi identity comes from the WZ part of the action (1.3.17).
The contribution to the stress-energy tensor coming from the two sets of
$N_{f}$ D7 flavour branes is given by

$\begin{split}T^{MN}&=\frac{2\kappa_{10}^{2}}{\sqrt{-G}}\frac{\delta
S^{flavour}}{\delta
G_{MN}}=-\frac{N_{f}}{4\pi}\frac{e^{\phi}}{\sqrt{-G}}\sum_{i=1,2}\sin\theta_{i}\frac{1}{2}\sqrt{-\hat{G}_{8}^{(i)}}\hat{G}_{8}^{(i)\alpha\beta}\delta_{\alpha}^{M}\delta_{\beta}^{N}\,\,,\end{split}$
(1.3.19)

where $\alpha,\beta$ are coordinate indices on the D7-brane. In chapter 4 we
will solve the equations of motion (1.3.18)-(1.3.19) and we will propose that
this type IIB background is dual to the Klebanov-Witten field theory when two
sets of $N_{f}$ flavours are added for each gauge group. We will actually find
BPS equations for the purely bosonic background, by imposing that the
variations of the dilatino and gravitino (1.2) vanish. We will verify that
these BPS first-order equations solve all the equations of motion (1.3.18).

After explaining the strategy that we will adopt to add flavours, let us
discuss the significance and effect on the dual gauge theory of the ‘smearing
procedure’ introduced above. It is clear that we smear the flavour branes just
to be able to write a ten-dimensional action that will produce ordinary
differential equations without Dirac delta functions source terms.

The results that we will show in chapters 4 and 5 state that many properties
of the flavoured field theory are still well captured by the solutions
obtained following this procedure. It is not clear what important phenomena on
the gauge theory we are losing in smearing, but see below for an important
subtlety.

One relevant point to discuss is related to global symmetries. Let us go back
to the weak coupling ($g_{s}N_{c}\rightarrow 0$) limit, in which we have
branes living on a spacetime that is the product of four Minkowski directions
and the conifold. When all the flavour branes of the two separate stacks
(1.3.3) are on top of each other, the gauge symmetry on the D7’s worldvolume
is given by the product $U(N_{f})\times U(N_{f})$. When we take the decoupling
limit for the D3-branes $\alpha^{\prime}\rightarrow 0$, with fixed
$g_{s}N_{c}$ and keeping constant the energies of the excitations on the
branes, we are left with a solution of type IIB supergravity that we propose
is dual to the Klebanov-Witten field theory with $N_{f}$ flavours for both
gauge groups [106]. In this case the flavour symmetry is $U(N_{f})\times
U(N_{f})$, where the axial $U(1)$ is anomalous. When we smear the $N_{f}$
D7-branes, we are breaking $U(N_{f})\rightarrow U(1)^{N_{f}}$ (see Fig. 1.2).

Figure 1.2: We see on the left side the two stacks of $N_{f}$ flavour-branes
localised on each of their respective $S^{2}$’s (they wrap the other $S^{2}$).
The flavour group is clearly $U(N_{f})\times U(N_{f})$. After the smearing on
the right side of the figure, this global symmetry is broken to
$U(1)^{N_{f}-1}\times U(1)^{N_{f}-1}\times U(1)_{B}\times U(1)_{A}$.

One can also think about the smearing procedure in the following way: usually
(unless they are D9 branes) the “localised” flavour branes break part of the
isometries of the original background dual to the unflavoured field theory. On
the other hand, the “smeared” flavour branes reinstate these isometries, which
are global symmetries of the field theory dual. In some sense the flavour
branes are ‘deconstructing’ these dimensions (or these global groups) for the
field theory of interest. In the case in which we have a finite number of
flavours, these manifolds become fuzzy, while for $N_{f}\rightarrow\infty$, we
recover the full invariance.

Another point that is worth elaborating on is whether there is a limit on the
number of D7-branes that can be added. Indeed, since a D7-brane is a
codimension-two object, its gravity solution will generate a deficit angle;
having many seven branes, will basically “eat-up” the transverse space. This
led to the conclusion that solutions that can be globally extended cannot have
more than a maximum number of twelve D7-branes [108] (and exactly twenty-four
in compact spaces). We are adding a number $N_{f}\rightarrow\infty$ of
D7-branes, certainly larger that the bound mentioned above. However, the
smearing procedure distributes the D7-branes all over a two-dimensional
compact space, in such a way that the equation for the axion-dilaton is not
the one in the vacuum at any point. This avoids the constraint on the number
of D7-branes, which came from solving the equation of motion for the axion-
dilaton outside sources.

It is possible to extend the smearing procedure of the D7-brane, which was
formulated above for the particular case of the $AdS_{5}\times T^{1,1}$ space,
to the more general case of a geometry of the type $AdS_{5}\times M_{5}$,
where $M_{5}$ is a five-dimensional compact manifold. Of course, the
requirement of supersymmetry restricts greatly the form of $M_{5}$. Actually,
we will verify that, when $M_{5}$ is Sasaki-Einstein, the formalism can be
easily generalised. The five-dimensional manifolds described in subsections
1.2.4 and 1.2.5 are particular cases of Sasaki-Einstein space where the
generalisation could be applied, as we will see in chapter 4. As a result of
this generalisation we will get a more intrinsic formulation of the smearing,
which eventually could be further generalised to other types of D-branes in
different geometries.

Following the line of thought that led to the action (1.3.17), let us assume
that, for a general geometry, the effect of the smearing on the WZ term of the
D7-brane action can be modelled by means of the substitution:

$S_{WZ}\,=\,T_{7}\,\,\sum_{N_{f}}\,\,\int_{{\cal
M}_{8}}\,\,C_{8}\,\,\rightarrow\,\,T_{7}\,\,\int_{{\cal M}_{10}}\,\Omega\wedge
C_{8}\,\,,$ (1.3.20)

where $\Omega$ is a two-form which determines the distribution of the RR
charge of the D7-brane in the smearing and ${\cal M}_{10}$ is the full ten-
dimensional manifold. Notice that a well defined $\Omega$ not only must be
closed (which is charge conservation) but also exact. Moreover the
supersymmetry of this class of solutions forces $\Omega$ to be a real
(1,1)-form with respect to the complex structure (as we will see below). For a
supersymmetric brane one expects the charge density to be equal to the mass
density and, thus, the smearing of the DBI part of the D7-brane action should
be also determined by the form $\Omega$. Let us explain in detail how this can
be done. First of all, let us suppose that $\Omega$ is decomposable, i.e. that
it can be written as the wedge product of two one-forms. In that case, at an
arbitrary point, $\Omega$ would determine an eight-dimensional orthogonal
hyperplane, which we are going to identify with the tangent space of the
D7-brane worldvolume. A general two-form $\Omega$ will not be decomposable.
However, it can be written as a finite sum of the type:

$\Omega\,=\,\sum\nolimits_{i}\Omega^{(i)}\;,$ (1.3.21)

where each $\Omega^{(i)}$ is decomposable. At an arbitrary point, each of the
$\Omega^{(i)}$’s is dual to an eight-dimensional hyperplane. Thus, $\Omega$
will determine locally a collection of eight-dimensional hyperplanes. In the
smearing procedure, to each decomposable component of $\Omega$ we associate
the volume form of its orthogonal complement in ${\cal M}_{10}$. Thus, the
contribution of every $\Omega^{(i)}$ to the DBI action will be proportional to
the ten-dimensional volume element. Since energy is additive, the DBI action
is obtained by summing the moduli of each decomposable piece (and not just
taking the modulus of $\Omega$). We simply sum the separate contributions
because of supersymmetry: the D7-branes do not interact among themselves due
to the cancellation of attractive/repulsive forces. Accordingly, let us
perform the following substitution:

$S_{DBI}=-T_{7}\;\sum_{N_{f}}\int_{{\cal
M}_{8}}d^{8}\xi\,\sqrt{-\hat{G}_{8}}\;\;e^{\phi}\quad\rightarrow\quad-
T_{7}\int_{{\cal
M}_{10}}d^{10}x\,\sqrt{-G}\;\;e^{\phi}\;\sum\nolimits_{i}\big{|}\,\Omega^{(i)}\,\big{|}\;,$
(1.3.22)

where $\big{|}\,\Omega^{(i)}\,\big{|}$ is the modulus of $\Omega^{(i)}$ and
represents the mass density of the $i^{th}$ piece of $\Omega$ in the smearing.
There is a natural definition of $\big{|}\,\Omega^{(i)}\,\big{|}$ which is
invariant under coordinate transformations. Indeed, let us suppose that
$\Omega^{(i)}$ is given by:

$\Omega^{(i)}\,=\,{1\over 2!}\,\,\sum_{M,N}\,\,\Omega^{(i)}_{MN}\,dx^{M}\wedge
dx^{N}\,\,.$ (1.3.23)

Then, $\big{|}\,\Omega^{(i)}\,\big{|}$ is defined as follows:

$\big{|}\,\Omega^{(i)}\,\big{|}\,\equiv\,\sqrt{{1\over
2!}\,\Omega^{(i)}_{MN}\,\Omega^{(i)}_{PQ}\,G^{MP}\,G^{NQ}}\,\,.$ (1.3.24)

Notice that $\Omega$ acts as a magnetic source for the field strength $F_{1}$.
Actually, from the equation of motion of $C_{8}$ one gets that $\Omega$ is
just the violation of the Bianchi identity for $F_{1}$, namely:

$dF_{1}\,=\,-\,\Omega\,\,.$ (1.3.25)

For a supersymmetric configuration the form $\Omega$ is not arbitrary. Indeed,
eq. (1.3.25) determines $F_{1}$ which, in turn, enters the equation that
determines the Killing spinors of the background. On the other hand, $\Omega$
must come from the superposition (smearing) of kappa symmetric branes. When
the manifold $M_{5}$ is Sasaki-Einstein, we will show in subsection 4.2.1 of
chapter 4 that $\Omega$ can be determined in terms of the Kähler form of the
Kähler-Einstein base of $M_{5}$ and that the resulting DBI+WZ action is a
direct generalisation of the result written in (1.3.17).

The last step is to provide a well defined and coordinate invariant way of
splitting the charge distribution $\Omega$ in decomposable pieces. It turns
out that the splitting in the minimal number of pieces131313The minimal number
of decomposable pieces needed to write a general two-form is half of its rank
as a matrix. compatible with supersymmetry is almost unique.

In our setup, $\Omega$ lives on the internal six-dimensional manifold, which
is complex and $SU(3)$-structure. This means that the internal geometry has an
integrable complex structure $\mathcal{I}$ and a non-closed Kähler form
$\mathcal{J}$ compatible with the metric:
$\mathcal{J}_{ab}=g_{ac}\mathcal{I}_{b}^{\phantom{b}c}$. We can always find a
vielbein basis that diagonalizes the metric and block-diagonalizes the Kähler
form:

$\displaystyle g$ $\displaystyle=\sum\nolimits_{a}e^{a}\otimes e^{a}\,\,,$
(1.3.26) $\displaystyle\mathcal{J}$ $\displaystyle=e^{1}\wedge
e^{2}+e^{3}\wedge e^{4}+e^{5}\wedge e^{6}\;.$

This pattern is invariant under the structure group $SU(3)$, as it is also
clear by expressing them in local holomorphic basis: $e^{z_{i}}\equiv
e^{2i-1}+i\,e^{2i}$, $\bar{e}^{\bar{z}_{i}}\equiv e^{2i-1}-i\,e^{2i}$, with
$i=1,2,3$. One gets the canonical expressions:
$g=\sum_{i}e^{z_{i}}\otimes_{S}\bar{e}^{\bar{z}_{i}}$ and
$\mathcal{J}=\frac{i}{2}e^{z_{i}}\wedge\bar{e}^{\bar{z}_{i}}$.

In our class of solutions, the supersymmetry equations force the charge
distribution to be a real $(1,1)$-form with respect to the complex structure
(see [109]). Notice that such a property is shared with $\mathcal{J}$. The
dilatino equation is $e^{\phi}\bar{F}_{1}^{(0,1)}=i\bar{\partial}\phi$ (which
without sources amounts to the holomorphicity of the axion-dilation
$\tau=C_{0}+i\,e^{-\phi}$). From this one gets

$\Omega=-dF_{1}=2i\,e^{-\phi}\big{(}\partial\phi\wedge\bar{\partial}\phi-\partial\bar{\partial}\phi\big{)}\;.$
(1.3.27)

It is manifest that $\Omega$ is $(1,1)$ and $\Omega^{*}=\Omega$. Going to
complex components
$\Omega=\Omega_{l\bar{k}}e^{z_{l}}\wedge\bar{e}^{\bar{z}_{k}}$, the reality
condition translates to the matrix $\Omega_{l\bar{k}}$ being anti-hermitian.
Thus it can be diagonalized with an $SU(3)$ rotation of vielbein that leaves
(1.3.26) untouched, and the eigenvalues are imaginary.

Going back to real vielbein and summarizing, there is always a choice of basis
which satisfies the diagonalizing condition (1.3.26) and in which the charge
distribution can be written as the sum of three real (1,1) decomposable
pieces:

$\Omega=-\lambda_{1}\,e^{1}\wedge e^{2}-\lambda_{2}\,e^{3}\wedge
e^{4}-\lambda_{3}\,e^{5}\wedge e^{6}\;.$ (1.3.28)

Supersymmetry forces the eigenvalues $\lambda_{a}$ to be real and, as we will
see in chapters 4 and 5, positive. Moreover, as inferred by the previous
construction, the splitting is unique as long as the three eigenvalues
$\lambda_{a}$ are different, while there are ambiguities for degenerate
values, but different choices give the same DBI action.

We conclude noticing that, in order to extract the eigenvalues
$|\lambda_{k}|=|\Omega^{(k)}|$ it is not necessary to construct the complex
basis: one can simply compute the eigenvalues of the matrix
$(\Omega)_{MP}g^{PN}$ in any coordinate basis. But in order to compute the
stress-energy tensor, the explicit splitting into real (1,1) decomposable
pieces is in general required.

##  Chapter 2 Supersymmetric Branes on ${\bf AdS_{5}\times Y^{p,q}}$

In this chapter we perform a systematic classification of supersymmetric
branes in the $AdS_{5}\times Y^{p,q}$ geometry and we study their field
theoretical interpretation. Geometrical aspects of the $Y^{p,q}$ manifold, as
well as the type IIB supergravity background $AdS_{5}\times Y^{p,q}$ and its
field theory dual were reviewed in subsection 1.2.4. It is worth mentioning
that the spectrum of type IIB supergravity compactified on $Y^{p,q}$ is not
known due to various technical difficulties including the general form of
Heun’s equation [110]. Therefore, leaving aside the chiral primaries, very
little is known about the gravity modes dual to protected operators in the
field theory. As we explained in section 1.3, our study of supersymmetric
objects in the gravity side is a way to obtain information about properties of
these operators in the gauge theory. Important aspects of this duality,
relevant in the context of this chapter, have been further developed in [81,
83, 111]. They comprise interesting physical objects of these theories such as
the baryon vertex, wall defects, the introduction of flavour, “fat” strings,
etc. It is very remarkable that we are able to provide precise information
about operators with large conformal dimension that grows like $N_{c}$.
Moreover, we can also extract information about excitations of these
operators.

The main technique that we employ to determine the supersymmetric embeddings
of D-brane probes in the $AdS_{5}\times Y^{p,q}$ background is the kappa
symmetry of the brane probe studied in subsection 1.3.2. The configurations
found by solving the kappa symmetry condition also solve the equations of
motion derived from the Dirac-Born-Infeld action of the probe and, actually,
we will verify that they saturate a bound for the energy, as it usually
happens in the case of worldvolume solitons [100].

###  2.1 Supersymmetric probes on $AdS_{5}\times Y^{p,q}$

In the remainder of this chapter we will consider D-brane probes moving in the
$AdS_{5}\times Y^{p,q}$ background. To write the kappa symmetry matrix (see
eq.(1.3.8))111We will use a simpler notation for the determinant of the
induced metric, namely $\det g\equiv g$. we will assume that the worldvolume
gauge field $A$ is zero:

$\Gamma_{\kappa}\,=\,{1\over(p+1)!\sqrt{-g}}\,\epsilon^{\mu_{1}\cdots\mu_{p+1}}\,(\sigma_{3})^{{p-3\over
2}}\,i\sigma_{2}\,\otimes\,\gamma_{\mu_{1}\cdots\mu_{p+1}}\,\,.$ (2.1.1)

This assumption is consistent with the equations of motion of the probe as far
as there are no source terms in the action (1.3.6) which could induce a non-
vanishing value of $A$. These source terms must be linear in $A$ and can only
appear in the Wess-Zumino term of the probe action, which is responsible for
the coupling of the probe to the Ramond-Ramond fields of the background. In
the case under study only $F^{(5)}$ is non-zero and the only linear term in
$A$ is of the form $\int A\wedge F^{(5)}$, which is different from zero only
for a D5-brane which captures the flux of $F^{(5)}$. This only happens for the
baryon vertex configuration studied in subsection 2.5.5. In all other cases
studied in this chapter one can consistently put the worldvolume gauge field
to zero. Nevertheless, even if one is not forced to do it, in some cases we
can switch on the field $A$ to study how this affects the supersymmetric
embeddings.

As we discuss in subsection 1.3.2, the kappa symmetry condition
$\Gamma_{\kappa}\,\epsilon=\epsilon$ imposes a new projection to the Killing
spinor $\epsilon$. In general, it will not be compatible with those already
satisfied by $\epsilon$ (see eq.(1.2.99)). This is so because the new
projections involve matrices which do not commute with those appearing in
(1.2.99). The only way of making these two conditions consistent with each
other is by requiring the vanishing of the coefficients of those non-commuting
matrices, which will give rise to a set of first-order BPS differential
equations. By solving these BPS equations we will determine the supersymmetric
embeddings of the brane probes that we are looking for. Notice also that the
kappa symmetry condition must be satisfied at any point of the probe
worldvolume. It is a local condition whose global meaning, as we will see in a
moment, has to be addressed a posteriori. This requirement is not obvious at
all since the spinor $\epsilon$ depends on the coordinates (see eqs. (1.2.97)
and (1.2.106)). However this would be guaranteed if we could reduce the
$\Gamma_{\kappa}\,\epsilon=\epsilon$ projection to some algebraic conditions
on the constant spinor $\eta$ of eqs.(1.2.97) and (1.2.106). The counting of
solutions of the algebraic equations satisfied by $\eta$ will give us the
fraction of supersymmetry preserved by our brane probe.

###  2.2 Supersymmetric D3-branes on $AdS_{5}\times Y^{p,q}$

Let us now apply the methodology just described to find the supersymmetric
configurations of a D3-brane in the $AdS_{5}\times Y^{p,q}$ background. The
kappa symmetry matrix in this case can be obtained by putting $p=3$ in the
general expression (2.1.1):

$\Gamma_{\kappa}\,=\,-{i\over
4!\sqrt{-g}}\,\epsilon^{\mu_{1}\cdots\mu_{4}}\,\gamma_{\mu_{1}\cdots\mu_{4}}\,\,,$
(2.2.1)

where we have used the rule (1.3.10) to write the expression of
$\Gamma_{\kappa}$ acting on complex spinors. Given that the $Y^{p,q}$ space is
topologically $S^{2}\times S^{3}$, it is natural to consider D3-branes
wrapping two- and three-cycles in the Sasaki-Einstein space. A D3-brane
wrapping a two-cycle in $Y^{p,q}$ and extended along one of the spatial
directions of $AdS_{5}$ represents a “fat” string. We will study such type of
configurations in section 2.5 where we conclude that they are not
supersymmetric, although we will find stable non-supersymmetric embeddings of
this type.

In this section we will concentrate on the study of supersymmetric
configurations of D3-branes wrapping a three-cycle of $Y^{p,q}$. These objects
are pointlike from the gauge theory point of view and, on the field theory
side, they correspond to dibaryons constructed from the different
bifundamental fields. In what follows we will study the kappa symmetry
condition for two different sets of worldvolume coordinates, which will
correspond to two classes of cycles and dibaryons.

####  2.2.1 Singlet supersymmetric three-cycles

Let us use the global coordinates of eq. (1.2.102) to parameterise the
$AdS_{5}$ part of the metric and let us consider the following set of
worldvolume coordinates:

$\xi^{\mu}=(T,\theta,\phi,\beta),$ (2.2.2)

and the following generic ansatz for the embedding:

$y=y(\theta,\phi,\beta),\qquad\psi(\theta,\phi,\beta).$ (2.2.3)

The kappa symmetry matrix in this case is:

$\Gamma_{\kappa}\,=\,-iL\,{\cosh\varrho\over\sqrt{-g}}\,\Gamma_{T}\,\gamma_{\theta\phi\beta}\,\,.$
(2.2.4)

The induced gamma matrices along the $\theta$, $\phi$ and $\beta$ directions
can be straightforwardly obtained from (1.3.12), namely:

$\displaystyle{1\over L}\gamma_{\theta}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-c\,y}}{\sqrt{6}}\,\Gamma_{3}+\frac{1}{3}\psi_{\theta}\,\Gamma_{5}-{1\over\sqrt{6}\,H}\,y_{\theta}\,\Gamma_{1},$
$\displaystyle{1\over L}\gamma_{\phi}$ $\displaystyle=$
$\displaystyle\frac{cH\cos\theta}{\sqrt{6}}\,\Gamma_{2}+\frac{\sqrt{1-c\,y}}{\sqrt{6}}\sin\theta\,\Gamma_{4}+\frac{1}{3}\left(\psi_{\phi}+(1-c\,y)\cos\theta\right)\,\Gamma_{5}-{1\over\sqrt{6}\,H}\,y_{\phi}\,\Gamma_{1},$
$\displaystyle{1\over L}\gamma_{\beta}$ $\displaystyle=$
$\displaystyle-\frac{H}{\sqrt{6}}\,\Gamma_{2}+\frac{1}{3}\left(\psi_{\beta}+y\right)\,\Gamma_{5}-{1\over\sqrt{6}\,H}\,y_{\beta}\,\Gamma_{1},$
(2.2.5)

where the subscripts in $y$ and $\psi$ denote partial differentiation. By
using this result and the projections (1.2.99) the action of the
antisymmetrised product $\gamma_{\theta\phi\beta}$ on the Killing spinor
$\epsilon$ reads:

$-{i\over
L^{3}}\,\gamma_{\theta\phi\beta}\,\epsilon\,=\,[\,a_{5}\,\Gamma_{5}\,+\,a_{1}\Gamma_{1}\,+\,a_{3}\Gamma_{3}\,+\,a_{135}\,\Gamma_{135}\,]\,\epsilon\,\,,$
(2.2.6)

where the coefficients on the right-hand side are given by:

$\displaystyle a_{5}\,=\,{1\over
18}\,\Bigg{[}\,(y+\psi_{\beta})\,[\,(1-cy)\,\sin\theta\,+\,c\,y_{\theta}\cos\theta]\,+$
$\displaystyle\qquad\qquad+\,[\,\psi_{\phi}\,+\,(1-cy)\,\cos\theta\,]\,y_{\theta}\,-\,\psi_{\theta}y_{\phi}\,-\,c\cos\theta\psi_{\theta}y_{\beta}\,\Bigg{]}\,\,,$
$\displaystyle a_{1}\,=\,-{1-cy\over
6\sqrt{6}}\,\,\sin\theta\,\big{[}\,{y_{\beta}\over H}\,-\,iH\,\big{]}\,\,,$
$\displaystyle a_{3}\,=\,-{\sqrt{1-cy}\over
6\sqrt{6}}\,\big{[}\,y_{\phi}\,+\,c\cos\theta y_{\beta}\,-\,i\sin\theta
y_{\theta}\,\big{]}\,\,,$ $\displaystyle a_{135}\,=\,{\sqrt{1-cy}\over
18}\,\,\Bigg{[}\,{\sin\theta\over
H}\,\big{[}\psi_{\theta}y_{\beta}\,-\,(y+\psi_{\beta})\,y_{\theta}\,\big{]}\,+\,H\big{[}\psi_{\phi}\,+\,(1+c\psi_{\beta})\cos\theta\,\big{]}\,\,+$
$\displaystyle\qquad\qquad+\,\,{i\over
H}\,\Big{[}(\psi_{\phi}\,+\,(1-cy)\,\cos\theta\,)y_{\beta}\,-\,(y+\psi_{\beta})\,y_{\phi}\,\Big{]}\,-\,iH\sin\theta\psi_{\theta}\,\,\Bigg{]}\,\,.$
(2.2.7)

As discussed at the end of section 2.1, in order to implement the kappa
symmetry projection we must require the vanishing of the terms in (2.2.6)
which are not compatible with the projection (1.2.99). Since the matrices
$\Gamma_{1}$, $\Gamma_{3}$ and $\Gamma_{135}$ do not commute with those
appearing in the projection (1.2.99), it follows that we must impose that the
corresponding coefficients vanish, i.e.:

$a_{1}=a_{3}=a_{135}=0\,\,.$ (2.2.8)

Let us concentrate first on the condition $a_{1}=0$. By looking at its
imaginary part:

$H(y)=0\,\,,$ (2.2.9)

which, in the range of allowed values of $y$, means:

$y=y_{1}\leavevmode\nobreak\ ,\qquad{\rm or}\qquad y=y_{2}\leavevmode\nobreak\
.$ (2.2.10)

If $H(y)=0$, it follows by inspection that $a_{1}=a_{3}=a_{135}=0$. Notice
that $\psi$ can be an arbitrary function. Moreover, one can check that:

$\left.\sqrt{-g}\right|_{BPS}\,=\,L^{4}\,\cosh\varrho\,\left.{a_{5}}\right|_{BPS}\,\,.$
(2.2.11)

Thus, one has the following equality:

$\left.\Gamma_{\kappa}\,\epsilon\right|_{BPS}\,=\,\Gamma_{T}\Gamma_{5}\,\epsilon\,\,,$
(2.2.12)

and, therefore, the condition $\Gamma_{\kappa}\epsilon=\epsilon$ becomes
equivalent to

$\Gamma_{T}\Gamma_{5}\,\epsilon\,=\,\epsilon\,\,.$ (2.2.13)

As it happens in the $T^{1,1}$ case [105], the compatibility of (2.2.13) with
the $AdS_{5}$ structure of the spinor implies that the D3-brane must be placed
at $\varrho=0$, i.e. at the center of $AdS_{5}$. Indeed, as discussed at the
end of section 2.1, we must translate the condition (2.2.13) into a condition
for the constant spinor $\eta$ of eq. (1.2.106). Notice that
$\Gamma_{T}\Gamma_{5}$ commutes with all the matrices appearing on the right-
hand side of eq. (1.2.106) except for $\Gamma_{\varrho}\gamma_{*}$. Since the
coefficient of $\Gamma_{\varrho}\gamma_{*}$ in (1.2.106) only vanishes for
$\varrho=0$, it follows that only at this point the equation
$\Gamma_{\kappa}\,\epsilon=\epsilon$ can be satisfied for every point in the
worldvolume and reduces to:

$\Gamma_{T}\Gamma_{5}\,\eta\,=\,\eta\,\,.$ (2.2.14)

Therefore, if we place the D3-brane at the center of the $AdS_{5}$ space and
wrap it on the three-cycles at $y=y_{1}$ or $y=y_{2}$, we obtain a
$\frac{1}{8}$ supersymmetric configuration which preserves the Killing spinors
of the type (1.2.106) with $\eta$ satisfying (1.2.98) and the additional
condition (2.2.14).

The cycles that we have just found have been identified by Martelli and Sparks
as those dual to the dibaryonic operators $\det(Y)$ and $\det(Z)$, made out of
the bifundamental fields that, as the D3-brane wraps the two-sphere whose
isometries are responsible for the global $SU(2)$ group, are singlets under
this symmetry [25]. For this reason we will refer to these cycles as singlet
(S) cycles. Let us recall how this identification is carried out. First of
all, we look at the conformal dimension $\Delta$ of the corresponding dual
operator. Following the general rule of the AdS/CFT correspondence (and the
zero-mode corrections of ref. [83]), $\Delta=LM$, where $L$ is given by
(1.2.68) and $M$ is the mass of the wrapped three-brane. The latter can be
computed as $M=T_{3}\,V_{3}$, with $T_{3}$ being the tension of the D3-brane
($1/T_{3}\,=\,8\pi^{3}(\alpha^{\prime})^{2}g_{s}$) and $V_{3}$ the volume of
the three-cycle. If $g_{{\cal C}}$ is the determinant of the spatial part of
the induced metric on the three-cycle ${\cal C}$, one has:

$V_{3}\,=\,\int_{{\cal C}}\sqrt{g_{{\cal C}}}\,\,\,d^{3}\xi\,\,.$ (2.2.15)

For the singlet cycles ${\rm S}_{i}$ at $y=y_{i}$ ($i=1,2$) and
$\psi$=constant, the volume $V_{3}$ is readily computed, namely:

$V_{3}^{{\rm S}_{i}}\,=\,{2L^{3}\over
3}\,(\,1-cy_{i}\,)\,|\,y_{i}\,|\,(2\pi)^{2}\,\ell\,\,.$ (2.2.16)

Let us define $\lambda_{1}=+1$, $\lambda_{2}=-1$. Then, if $\Delta_{i}^{{\rm
S}}\,\equiv\,\Delta^{{\rm S}_{i}}$, one has:

$\Delta_{i}^{{\rm S}}\,=\,{N_{c}\over
2q^{2}}\,\Big{[}\,-4p^{2}+3q^{2}+2\lambda_{i}\,pq\,+\,(2p-\lambda_{i}\,q)\,\sqrt{4p^{2}\,-\,3q^{2}}\,\,\Big{]}\,\,.$
(2.2.17)

As it should be for a BPS saturated object, the R-charges $R_{i}$ of the ${\rm
S}_{i}$ cycles are related to $\Delta_{i}^{{\rm S}}$ as $R_{i}={2\over
3}\,\Delta_{i}^{{\rm S}}$. By comparing the values of $R_{i}$ with those
determined in [26] from the gauge theory dual (see Table 1.1 in chapter 1) one
concludes that, indeed, a D3-brane wrapped at $y=y_{1}$ ($y=y_{2}$) can be
identified with the operator $\det(Y)$ ($\det(Z)$) as claimed. Another piece
of evidence which supports this claim is the calculation of the baryon number,
that can be identified with the third homology class of the three-cycle ${\cal
C}$ over which the D3-brane is wrapped. This number (in units of $N_{c}$) can
be obtained by computing the integral over ${\cal C}$ of the pullback of a
$(2,1)$ three-form $\Omega_{2,1}$ on $CY^{p,q}$:

${\cal B}({\cal C})\,=\,\pm i\int_{{\cal
C}}\,P\big{[}\,\Omega_{2,1}\,\big{]}_{{\cal C}}\,\,,$ (2.2.18)

where $P[\cdots]_{{\cal C}}$ denotes the pullback to the cycle ${{\cal C}}$ of
the form that is inside the brackets. The sign of the right-hand side of
(2.2.18) depends on the orientation of the cycle. The explicit expression of
$\Omega_{2,1}$ has been determined in ref. [33]:

$\Omega_{2,1}\,=\,K\,\Big{(}\,{dr\over
r}\,+\,\frac{i}{L}\,e^{5}\,\Big{)}\wedge\omega\,\,,$ (2.2.19)

where $e^{5}$ is the one-form of our vielbein (1.2.71) for the $Y^{p,q}$
space, $K$ is the constant

$K\,=\,{9\over 8\pi^{2}}\,(p^{2}-q^{2})\,\,,$ (2.2.20)

and $\omega$ is the two-form:

$\omega\,=\,-{1\over(1-cy)^{2}\,L^{2}}\,\,\Big{[}\,e^{1}\wedge
e^{2}\,+\,e^{3}\wedge e^{4}\,\Big{]}\,\,.$ (2.2.21)

Using $(\theta,\phi,\beta)$ as worldvolume coordinates of the singlet cycles
${\rm S}_{i}$,

$P\big{[}\,\Omega_{2,1}\,\big{]}_{{\rm S}_{i}}\,=\,-i\,{K\over
18}\,{y_{i}\over 1-cy_{i}}\,\sin\theta\,d\theta\wedge d\phi\wedge d\beta\,\,.$
(2.2.22)

Then, changing variables from $\beta$ to $\alpha$ by means of (1.2.76), and
taking into account that $\alpha\in[0,2\pi\ell]$, one gets:

$\int_{{\rm S}_{i}}\,P\big{[}\,\Omega_{2,1}\,\big{]}_{{\rm
S}_{i}}\,=\,-i\,{8\pi^{2}\over 3}\,{K\ell y_{i}\over 1-cy_{i}}\,\,.$ (2.2.23)

After using the values of $y_{1}$ and $y_{2}$ displayed in (1.2.75), we arrive
at:

$\displaystyle{\cal B}({\rm S}_{1})\,=\,-i\int_{{\rm
S}_{1}}\,P\big{[}\,\Omega_{2,1}\,\big{]}_{{\rm S}_{1}}\,=\,p-q\,\,,$
$\displaystyle{\cal B}({\rm S}_{2})\,=\,i\int_{{\rm
S}_{2}}\,P\big{[}\,\Omega_{2,1}\,\big{]}_{{\rm S}_{2}}\,=\,p+q\,\,.$ (2.2.24)

Notice the perfect agreement of ${\cal B}({\rm S}_{1})$ and ${\cal B}({\rm
S}_{2})$ with the baryon numbers of $Y$ and $Z$ displayed in Table 1.1.

####  2.2.2 Doublet supersymmetric three-cycles

Let us now try to find supersymmetric embeddings of D3-branes on three-cycles
by using a different set of worldvolume coordinates. As in the previous
subsection it is convenient to use the global coordinates (1.2.102) for the
$AdS_{5}$ part of the metric and the following set of worldvolume coordinates:

$\xi^{\mu}=(T,y,\beta,\psi)\leavevmode\nobreak\ .$ (2.2.25)

Moreover, we will adopt the ansatz:

$\theta(y,\beta,\psi)\leavevmode\nobreak\
,\qquad\phi(y,\beta,\psi)\leavevmode\nobreak\ .$ (2.2.26)

The kappa symmetry matrix $\Gamma_{\kappa}$ in this case takes the form:

$\Gamma_{\kappa}=-iL\,{\cosh\varrho\over\sqrt{-g}}\,\Gamma_{T}\,\gamma_{y\,\beta\,\psi}\,\,,$
(2.2.27)

and the induced gamma matrices are:

$\displaystyle{1\over
L}\gamma_{y}=-\frac{1}{\sqrt{6}H}\Gamma_{1}+\frac{cH\cos\theta}{\sqrt{6}}\phi_{y}\Gamma_{2}+\frac{\sqrt{1-cy}}{\sqrt{6}}\left(\theta_{y}\Gamma_{3}+\phi_{y}\sin\theta\,\Gamma_{4}\right)+\frac{1-c\,y}{3}\cos\theta\phi_{y}\,\Gamma_{5}\,\,,$
$\displaystyle{1\over
L}\gamma_{\beta}=\frac{H}{\sqrt{6}}\left(-1+c\cos\theta\,\phi_{\beta}\right)\,\Gamma_{2}+\frac{\sqrt{1-c\,y}}{\sqrt{6}}\,\theta_{\beta}\,\Gamma_{3}+\frac{\sqrt{1-c\,y}}{\sqrt{6}}\sin\theta\,\phi_{\beta}\,\Gamma_{4}$
$\displaystyle\qquad\qquad+{1\over
3}\Big{(}\,y+(1-c\,y)\cos\theta\,\phi_{\beta}\Big{)}\,\Gamma_{5}\,\,,$
(2.2.28) $\displaystyle{1\over
L}\gamma_{\psi}=\frac{cH\cos\theta}{\sqrt{6}}\,\phi_{\psi}\,\Gamma_{2}+\frac{\sqrt{1-c\,y}}{\sqrt{6}}\,\left(\theta_{\psi}\,\Gamma_{3}+\sin\theta\,\phi_{\psi}\,\Gamma_{4}\right)+{1\over
3}\left(1+(1-c)\,\cos\theta\phi_{\psi}\right)\,\Gamma_{5}\,\,.$

By using again the projections (1.2.99) one easily gets the action of
$\gamma_{y\,\beta\,\psi}$ on the Killing spinor

$-{i\over
L^{3}}\,\gamma_{y\,\beta\,\psi}\,\epsilon\,=\,\big{[}\,c_{5}\,\Gamma_{5}\,+\,c_{1}\,\Gamma_{1}\,+\,c_{3}\,\Gamma_{3}\,+\,c_{135}\,\Gamma_{135}\,\big{]}\,\epsilon\,\,,$
(2.2.29)

where the different coefficients appearing on the right-hand side of (2.2.29)
are given by:

$\displaystyle c_{5}={1\over
18}\,\Bigg{[}-1-\cos\theta(\phi_{\psi}-c\phi_{\beta})+(1-cy)\sin\theta\,\Big{[}\theta_{y}(\phi_{\beta}-y\phi_{\psi})-\phi_{y}(\theta_{\beta}-y\theta_{\psi})\,\Big{]}\,\Bigg{]}\,,$
$\displaystyle c_{1}\,=\,-{1-cy\over
6\sqrt{6}}\,\sin\theta\,\Big{[}\,{\theta_{\beta}\phi_{\psi}-\theta_{\psi}\phi_{\beta}\over
H}\,+\,iH\,(\theta_{y}\phi_{\psi}\,-\,\theta_{\psi}\phi_{y})\,\Big{]}\,\,,$
$\displaystyle c_{3}\,=\,-{\sqrt{1-cy}\over
6\sqrt{6}}\,\Big{[}\,\theta_{\psi}\,-\,c\cos\theta\,(\theta_{\psi}\phi_{\beta}\,-\,\theta_{\beta}\phi_{\psi})\,+\,i\sin\theta\phi_{\psi}\,\Big{]}\,\,,$
$\displaystyle c_{135}=-{\sqrt{1-cy}\over 18}\,\Bigg{[}{\sin\theta\over
H}(\phi_{\beta}-y\phi_{\psi})+H\bigg{(}\theta_{y}+\cos\theta\bigg{[}\theta_{y}(\phi_{\psi}-c\phi_{\beta})-\phi_{y}(\theta_{\psi}-c\theta_{\beta})\bigg{]}\bigg{)}$
$\displaystyle\qquad\qquad+iH\sin\theta\,\phi_{y}-{i\over
H}\,\Big{[}\theta_{\beta}-y\theta_{\psi}+(1-cy)\cos\theta(\theta_{\beta}\phi_{\psi}-\theta_{\psi}\phi_{\beta})\Big{]}\Bigg{]}\,.$
(2.2.30)

Again, we notice that the matrices $\Gamma_{1}$, $\Gamma_{3}$ and
$\Gamma_{135}$ do not commute with the projections (1.2.99). We must impose:

$c_{1}=c_{3}=c_{135}=0\,\,.$ (2.2.31)

From the vanishing of the imaginary part of $c_{3}$ we obtain the condition:

$\sin\theta\,\phi_{\psi}\,=\,0\,\,.$ (2.2.32)

One can solve the condition (2.2.32) by taking $\sin\theta\,=\,0$, i.e. for
$\theta=0,\pi$. By inspection one easily realises that $c_{1}$, $c_{3}$ and
$c_{135}$ also vanish for these values of $\theta$ and for an arbitrary
function $\phi(y,\beta,\psi)$. Therefore, we have the solution

$\theta=0,\pi\,\,,\qquad\phi=\phi(y,\beta,\psi)\,\,.$ (2.2.33)

Another possibility is to take $\phi_{\psi}=0$. In this case one readily
verifies that $c_{1}$ and $c_{3}$ vanish if $\theta_{\psi}=0$. Thus, let us
assume that both $\phi$ and $\theta$ are independent of the angle $\psi$. From
the vanishing of the real and imaginary parts of $c_{135}$ we get two
equations for the functions $\theta=\theta(y,\beta)$ and $\phi=\phi(y,\beta)$,
namely:

$\displaystyle\theta_{y}\,+\,{\sin\theta\over
H^{2}}\,\phi_{\beta}\,+\,c\cos\theta\,(\phi_{y}\,\theta_{\beta}\,-\,\theta_{y}\,\phi_{\beta})\,=\,0\,\,,$
$\displaystyle\theta_{\beta}\,-\,H^{2}\,\sin\theta\phi_{y}\,=\,0\,\,.$
(2.2.34)

If the BPS equations (2.2.34) hold, one can verify that the kappa symmetry
condition $\Gamma_{\kappa}\,\epsilon\,=\,\epsilon$ reduces, up to a sign, to
the projection (2.2.13) for the Killing spinor. As in the case of the S three-
cycles studied in subsection 2.2.1, by using the explicit expression (1.2.106)
of $\epsilon$ in terms of the global coordinates of $AdS_{5}$, one concludes
that the D3-brane must be placed at $\varrho=0$. The corresponding
configuration preserves four supersymmetries.

In the next subsection we will tackle the problem of finding the general
solution of the system (2.2.34). Here we will analyze the trivial solution of
this system, namely:

$\theta\,=\,{\rm constant}\,\,,\qquad\qquad\phi\,=\,{\rm constant}\,\,.$
(2.2.35)

This kind of three-cycle was studied in ref. [33] by Herzog, Ejaz and Klebanov
(see also [26]), who showed that it corresponds to dibaryons made out of the
$SU(2)$ doublet fields $U^{\alpha}$. In what follows we will refer to it as
doublet (D) cycle. Let us review the arguments leading to this identification.
First of all, the volume of the D cycle (2.2.35) can be computed with the
result:

$V_{3}^{D}\,=\,{L^{3}\over 3}\,(2\pi)^{2}\,(y_{2}-y_{1})\,\ell\,\,.$ (2.2.36)

By using the values of $y_{1}$ and $y_{2}$ (eq.(1.2.75)), $L$ (eq.(1.2.68))
and $\ell$ (eq.(1.2.78)) we find the following value of the conformal
dimension:

$\Delta^{D}\,=\,N_{c}\,{p\over
q^{2}}\,\Big{(}\,2p\,-\,\sqrt{4p^{2}-3q^{2}}\,\,\Big{)}\,.$ (2.2.37)

By comparison with Table 1.1 in chapter 1, one can verify that the
corresponding R-charge, namely $2/3\,\Delta^{D}$, is equal to the R-charge of
the field $U^{\alpha}$ multiplied by $N_{c}$. We can check this identification
by computing the baryon number. Since, in this case, the pullback of
$\Omega_{2,1}$ is:

$P\big{[}\,\Omega_{2,1}\,\big{]}_{{\rm D}}\,=\,i\,{K\over
3(1-cy)^{2}}\,dy\wedge\,d\alpha\wedge d\psi\,\,,$ (2.2.38)

we get:

${\cal B}({\rm D})\,=\,-i\int_{{\rm D}}\,P\big{[}\,\Omega_{2,1}\,\big{]}_{{\rm
D}}\,=\,-p\,\,,$ (2.2.39)

which, indeed, coincides with the baryon number of $U^{\alpha}$ written in
Table 1.1.

##### General integration

Let us now try to integrate in general the first-order differential system
(2.2.34). With this purpose it is more convenient to describe the locus of the
D3-brane by means of two functions $y=y(\theta,\phi)$,
$\beta=\beta(\theta,\phi)$. Notice that this is equivalent to the description
used so far (in which the independent variables were $(y,\beta)$), except for
the cases in which $(\theta,\phi)$ or $(y,\beta)$ are constant. The
derivatives in these two descriptions are related by simply inverting the
Jacobian matrix, i.e.:

$\begin{pmatrix}y_{\theta}&y_{\phi}\cr\beta_{\theta}&\beta_{\phi}\end{pmatrix}\,=\,\begin{pmatrix}\theta_{y}&\theta_{\beta}\cr\phi_{y}&\phi_{\beta}\end{pmatrix}^{-1}\,\,.$
(2.2.40)

By using these equations the first-order system (2.2.34) is equivalent to:

$\beta_{\theta}\,=\,{y_{\phi}\over
H^{2}\sin\theta}\,\,,\qquad\qquad\beta_{\phi}\,=\,c\cos\theta\,-\,{\sin\theta\over
H^{2}}\,y_{\theta}\,\,.$ (2.2.41)

These equations can be obtained directly by using $\theta$ and $\phi$ as
worldvolume coordinates. Interestingly, in this form the BPS equations can be
written as Cauchy-Riemann equations and, thus, they can be integrated in
general. This is in agreement with the naive expectation that, at least
locally, these equations should determine some kind of holomorphic embeddings.
In order to verify this fact, let us introduce new variables $u_{1}$ and
$u_{2}$, related to $\theta$ and $y$ as follows:

$u_{1}=\,\log\,\bigg{(}\,\tan\frac{\theta}{2}\,\bigg{)},\qquad\qquad
u_{2}=\,\log\,\bigg{(}\,\frac{(\sin\theta)^{c}}{f_{1}(y)}\,\bigg{)}.$ (2.2.42)

By comparing the above expressions with the definitions of $z_{1}$ and $z_{2}$
in eq. (1.2.86), one gets:

$u_{1}-i\phi\,=\,\log z_{1}\,\,,\qquad\qquad u_{2}-i\beta\,=\,\log z_{2}\,\,.$
(2.2.43)

The relation between $u_{1}$ and $\theta$ leads to
$du_{1}=d\theta/\sin\theta$, from which it follows that:

${\partial u_{2}\over\partial u_{1}}\,=\,c\cos\theta-{\sin\theta\over
H^{2}}\,y_{\theta}\,\,,\qquad\qquad\qquad{\partial\beta\over\partial
u_{1}}\,=\,\sin\theta\,\beta_{\theta}\,\,,$ (2.2.44)

and it is easy to demonstrate that the BPS equations (2.2.41) can be written
as:

$\frac{\partial u_{2}}{\partial
u_{1}}=\frac{\partial\beta}{\partial\phi}\,,\qquad\qquad\frac{\partial
u_{2}}{\partial\phi}=-\frac{\partial\beta}{\partial u_{1}}\,,$ (2.2.45)

these being the Cauchy-Riemann equations for the variables $u_{2}-i\beta=\log
z_{2}$ and $u_{1}-i\phi=\log z_{1}$. Then, the general integral of the BPS
equations is

$\log z_{2}\,=\,f(\log z_{1})\,\,,$ (2.2.46)

where $f$ is an arbitrary (holomorphic) function of $\log z_{1}$. By
exponentiating eq. (2.2.46) one gets that the general solution of the BPS
equations is a function $z_{2}=g(z_{1})$, in which $z_{2}$ is an arbitrary
holomorphic function of $z_{1}$. This result is analogous to what happened for
$T^{1,1}$ [105]. The appearance of a holomorphic function in the local complex
coordinates $z_{1}$ and $z_{2}$ is a consequence of kappa symmetry or, in
other words, supersymmetry. But one still has to check that this equation
makes sense globally. We will come to this point shortly. The simplest case is
that in which $\log z_{2}$ depends linearly on $\log z_{1}$, namely

$\log z_{2}\,=\,n(\log z_{1})\,+\,{\rm const.}\leavevmode\nobreak\ ,$ (2.2.47)

where $n$ is a constant. By exponentiating this equation we get a relation
between $z_{2}$ and $z_{1}$ of the type:

$z_{2}\,=\,{\cal C}\,z_{1}^{n}\leavevmode\nobreak\ ,$ (2.2.48)

where ${\cal C}$ is a complex constant. If we represent this constant as
${\cal C}=Ce^{-i\beta_{0}}$, the embedding (2.2.48) reduces to the following
real functions $\beta=\beta(\phi)$ and $y=y(\theta)$:

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle n\phi\,+\,\beta_{0}\,\,,$
$\displaystyle f_{1}(y)$ $\displaystyle=$ $\displaystyle
C\,\,{\big{(}\,\sin\theta\,\big{)}^{c}\over\Big{(}\tan{\theta\over
2}\Big{)}^{n}}\,\,.$ (2.2.49)

This is a nontrivial embedding of a D3-brane probe on $AdS_{5}\times Y^{p,q}$.
Notice that in the limit $c\rightarrow 0$ one recovers the results of [105].
For $c\neq 0$, a key difference arises. As we discussed earlier, $z_{2}$ is
not globally well defined in $CY^{p,q}$ due to its dependence on $\beta$. As a
consequence, eqs.(2.2.48)-(2.2.49) describe a kappa-symmetric embedding for
the D3-brane on $Y^{p,q}$ but it does not correspond to a wrapped brane. The
D3-brane spans a submanifold with boundaries.222In this respect, notice that
it might happen that global consistency forces, through boundary conditions,
the D3-brane probes to end on other branes. The only solution corresponding to
a probe D3-brane wrapping a three-cycle is $z_{1}={\rm const.}$ which is the
one obtained in the preceding subsection.

In order to remove $\beta$ while respecting holomorphicity, we seem to be
forced to let $z_{3}$ enter into the game. The reason is simple, any
dependence in $\beta$ disappears if $z_{2}$ enters through the product
$z_{2}z_{3}$. This would demand embeddings involving the radius that we did
not consider. In this respect, it is interesting to point out that this is
also the conclusion reached in [63] from a different perspective: there, the
complex coordinates corresponding to the generators of the chiral ring are
deduced and it turns out that all of them depend on $z_{1}$, $z_{2}z_{3}$ and
$z_{3}$. It would be clearly desirable to understand these generalised wrapped
D3-branes in terms of algebraic geometry, following the framework of ref.
[111] which, in the case of the conifold, emphasizes the use of global
homogeneous coordinates. Unfortunately, the relation between such homogeneous
coordinates and the chiral fields of the quiver theory is more complicated in
the case of $CY^{p,q}$.

####  2.2.3 The calibrating condition

Let us now verify that the BPS equations we have obtained ensure that the
three-dimensional submanifolds we have found are calibrated. Given a three-
submanifold in $Y^{p,q}$ one can construct its cone ${\cal D}$, which is a
four-dimensional submanifold of $CY^{p,q}$. The calibrating condition for a
supersymmetric four-submanifold ${\cal D}$ of $CY^{p,q}$ is just:

$P\Big{[}\,{1\over 2}\,J\wedge J\,\Big{]}_{{\cal D}}\,=\,{\rm Vol}({\cal
D})\,\,,$ (2.2.50)

where ${\rm Vol}({\cal D})$ is the volume form of the divisor ${\cal D}$ and
$J$ is the Kähler form of $CY^{p,q}$ (1.2.93). Let us check that the condition
(2.2.50) is indeed satisfied by the cones constructed from our three-
submanifolds. In order to verify this fact it is more convenient to describe
the embedding by means of functions $y=y(\theta,\phi)$ and
$\beta=\beta(\theta,\phi)$. The corresponding BPS equations are the ones
written in (2.2.41). By using them one can verify that the induced volume form
for the three-dimensional submanifold is:

$vol\,=\,{1\over 18}\,\Big{|}\,(1-cy)\sin\theta\,+\,c\cos\theta
y_{\theta}\,+\,\beta_{\theta}y_{\phi}\,-\,y_{\theta}\beta_{\phi}\,\Big{|}_{BPS}\,\,d\theta\wedge
d\phi\wedge d\psi\leavevmode\nobreak\ .$ (2.2.51)

By computing the pullback of $J\wedge J$ one can verify that the calibrating
condition (2.2.50) is indeed satisfied for:

${\rm Vol}({\cal D})\,=\,-r^{3}\,dr\wedge vol\,\,,$ (2.2.52)

which is just the volume form of ${\cal D}$ with the metric
$ds^{2}_{CY^{p,q}}$ having a particular orientation. Eq. (2.2.50) is also
satisfied for the cones constructed from the singlet and doublet three-cycles
of subsections 2.2.1 and 2.2.2. This fact is nothing but the expression of the
local nature of supersymmetry.

####  2.2.4 Energy bound

The dynamics of the D3-brane probe is governed by the Dirac-Born-Infeld
lagrangian that, for the case in which there are not worldvolume gauge fields,
reduces in Einstein frame (see eq. (1.3.7)) to:

${\cal L}\,=\,-\sqrt{-g}\,\,,$ (2.2.53)

where we have taken the D3-brane tension equal to one. We have checked that
any solution of the first-order equations (2.2.34) or (2.2.41) also satisfies
the Euler-Lagrange equations derived from the lagrangian density (2.2.53).
Moreover, for the static configurations we are considering here the
hamiltonian density ${\cal H}$ is, as expected, just ${\cal H}=-{\cal L}$. We
are now going to verify that this energy density satisfies a bound, which is
just saturated when the BPS equations (2.2.34) or (2.2.41) hold. In what
follows we will take $\theta$ and $\phi$ as independent variables. For an
arbitrary embedding of a D3-brane described by two functions
$\beta=\beta(\theta,\phi)$ and $y=y(\theta,\phi)$ one can show that ${\cal H}$
can be written as:

${\cal H}\,=\,\sqrt{{\cal Z}^{2}\,+\,{\cal Y}^{2}\,+{\cal W}^{2}}\,\,,$
(2.2.54)

where ${\cal Z}$, ${\cal Y}$ and ${\cal W}$ are given by:

$\displaystyle{\cal Z}$ $\displaystyle=$ $\displaystyle{L^{4}\over
18}\,\Bigg{[}\,(1-cy)\sin\theta\,+c\cos\theta
y_{\theta}+y_{\phi}\beta_{\theta}-y_{\theta}\beta_{\phi}\,\Bigg{]}\,\,,$
$\displaystyle{\cal Y}$ $\displaystyle=$ $\displaystyle{L^{4}\over
18}\,\sqrt{1-cy}\,\,H\,\Bigg{[}\,\beta_{\phi}\,-\,c\cos\theta\,+\,{\sin\theta\over
H^{2}}\,y_{\theta}\,\Bigg{]}\,\,,$ $\displaystyle{\cal W}$ $\displaystyle=$
$\displaystyle{L^{4}\over
18}\,\sqrt{1-cy}\,\,H\,\Bigg{[}\,\sin\theta\,\beta_{\theta}\,-\,{y_{\phi}\over
H^{2}}\,\Bigg{]}\,\,.$ (2.2.55)

Obviously one has:

${\cal H}\geq\big{|}\,{\cal Z}\,\big{|}\,\,.$ (2.2.56)

Moreover, since

${\cal Y}_{\big{|}\,BPS}\,=\,{\cal W}_{\big{|}\,BPS}\,=\,0\,\,,$ (2.2.57)

the bound saturates when the BPS equations (2.2.41) are satisfied. Thus, the
system of differential equations (2.2.41) is equivalent to the condition
${\cal H}=\big{|}\,{\cal Z}\,\big{|}$ (actually ${\cal Z}\geq 0$ if the BPS
equations (2.2.41) are satisfied). Moreover, for an arbitrary embedding ${\cal
Z}$ can be written as a total derivative, namely:

${\cal Z}\,=\,{\partial\over\partial\theta}\,{\cal
Z}^{\theta}\,+\,{\partial\over\partial\phi}\,{\cal Z}^{\phi}\,\,.$ (2.2.58)

This result implies that ${\cal H}$ is bounded by the integrand of a
topological charge. The explicit form of ${\cal Z}^{\theta}$ and ${\cal
Z}^{\phi}$ is:

$\displaystyle{\cal Z}^{\theta}$ $\displaystyle=$ $\displaystyle-{L^{4}\over
18}\,\Big{[}\,(1-cy)\cos\theta\,+\,y\,\beta_{\phi}\,\Big{]}\,\,,$
$\displaystyle{\cal Z}^{\phi}$ $\displaystyle=$ $\displaystyle{L^{4}\over
18}\,y\,\beta_{\theta}\,\,.$ (2.2.59)

In this way, from the point of view of the D3-branes, the configurations
satisfying eq. (2.2.41) can be regarded as BPS worldvolume solitons.

####  2.2.5 BPS fluctuations of dibaryons

In this subsection we study BPS fluctuations of dibaryon operators in the
$Y^{p,q}$ quiver theory. We start with the simplest dibaryon which is singlet
under $SU(2)$, let us say $\det Y$. To construct excited dibaryons we should
replace one of the $Y$ factors by any other chiral field transforming in the
same representation of the gauge groups. For example, replacing $Y$ by
$YU^{\alpha}V^{\beta}Y$, we get a new operator of the form

$\displaystyle\epsilon_{1}\epsilon^{2}(YU^{\alpha}V^{\beta}Y)Y\cdots Y\,\,,$
(2.2.60)

where $\epsilon_{1}$ and $\epsilon^{2}$ are abbreviations for the completely
anti-symmetric tensors for the respective $SU(N_{c})$ factors of the gauge
group. Using the identity

$\displaystyle\epsilon^{a_{1}\cdots a_{N_{c}}}\epsilon_{b_{1}\cdots
b_{N_{c}}}=\sum_{\sigma}(-1)^{\sigma}\delta^{a_{1}}_{\sigma(b_{1})}\cdots\delta^{a_{N_{c}}}_{\sigma(b_{N_{c}})}\,\,,$
(2.2.61)

the new operator we get can factorise into the original dibaryon and a single-
trace operator

$\displaystyle{\rm Tr}(U^{\alpha}V^{\beta}Y)\;{\rm det}\,Y\,\,.$ (2.2.62)

Indeed for singlet dibaryons, a factorisation of this sort always works. This
fact seems to imply, at least at weak coupling, that excitation of a singlet
dibaryon can be represented as graviton fluctuations in the presence of the
original dibaryon.

For the case of a dibaryon with $SU(2)$ quantum number the situation is
different. Consider, for simplicity, the state with maximum $J_{3}$ of the
$SU(2)$

$\displaystyle\epsilon_{1}\epsilon^{2}(U^{1}\cdots U^{1})={\rm det}\,U^{1},$
(2.2.63)

we can replace one of the $U^{1}$ factors by $U^{1}\,{\cal O}$, where $\cal O$
is some operator given by a closed loop in the quiver diagram. As in the case
of a singlet dibaryon, this kind of excitation is factorisable since all
$SU(2)$ indices are symmetric. So this kind of operator should be identified
with a graviton excitation with wrapped D3-brane in the dual string theory.
However, if the $SU(2)$ index of the $U$ field is changed in the excitation,
i.e. $U^{1}\rightarrow U^{2}\,{\cal O}$, then the resulting operator cannot be
written as a product of the original dibaryon and a meson-like operator.
Instead it has to be interpreted as a single particle state in $AdS$. Since
the operator also carries the same baryon number, the natural conclusion is
that the one-particle state is a BPS excitation of the wrapped D3-brane
corresponding to the dibaryon [83].

Figure 2.1: Loops in the $Y^{4,2}$ quiver representing mesonic operators in
the chiral ring. There are short loops such as $UVY$, $VUY$ or $YUZU$ (upper),
longest loops as $VUVUZUZU$ (middle) and long loops like $YUYYYU$ (bottom).
The representative of each class in the chiral ring is, respectively, ${\cal
O}_{1}$, ${\cal O}_{2}$ and ${\cal O}_{3}$.

In order to classify all these BPS excitations of the dibaryon, we have to
count all possible inequivalent chiral operators $\cal O$ that transform in
the bifundamental representation of one of the gauge group factors of the
theory. In $Y^{p,q}$ quiver gauge theory, these operators correspond to loops
in the quiver diagram just like the mesonic chiral operators discussed in
[112]. The simplest ones are operators with R-charge 2. They have been
thoroughly discussed in [113]. They are given by short loops of length 3 or 4
in the quiver, precisely as those operators entering in the superpotential
(1.2.107). They are single trace operators of the form (in what follows we
omit the trace and the $SU(2)$ indices) $UVY$, $VUY$ or $YUZU$ (see the upper
quiver in Fig. 2.1). Since they are equivalent in the chiral ring, we can
identify them as a single operator ${\cal O}_{1}$. It transforms in the spin
$\frac{1}{2}\otimes\frac{1}{2}=0\oplus 1$ representation of the global
$SU(2)$. The scalar component vanishes in the chiral ring. Thus, we end up
with a spin 1 chiral operator with scaling dimension $\Delta=3$. Its
$U(1)_{F}$ charge vanishes.

There are also two classes of long loops in the quiver. The first class, whose
representative is named ${\cal O}_{2}$, has length $2p$, winds the quiver from
the left to the right and is made of $p$ $U$ type operators, $q$ $V$ type
operators and $p-q$ $Z$ type operators. For example, in $Y^{4,2}$, a long loop
of this class is $VUVUZUZU$ (middle quiver in Fig. 2.1). It transforms in the
spin $\frac{1}{2}\otimes...\otimes\frac{1}{2}=\frac{p+q}{2}\oplus\dots$
representation of $SU(2)$. The dots amount to lower dimensional
representations that vanish in the chiral ring. The resulting operator, ${\cal
O}_{2}$, has spin $\frac{p+q}{2}$. There is another class of long loops which
has length $2p-q$, running along the quiver in the opposite direction, build
with $p$ $Y$ type operators and $p-q$ $U$ type operators. We name its
representative as ${\cal O}_{3}$. In the case of $Y^{4,2}$, it is an operator
like $YUYYYU$ (bottom quiver in Fig. 2.1). $SU(2)$ indices, again, have to be
completely symmetrised, the spin being $\frac{p-q}{2}$. Long loops wind around
the quiver and this leads to a nonvanishing value of $Q_{F}$ [112]. The
baryonic charge vanishes for any of these loops. We summarise in Table 2.1 the
charge assignments for the three kinds of operators ${\cal O}_{i}$ [112].

$\begin{array}[]{|c|c|c|c|}\hline\cr{\rm Operator}&Q_{R}&Q_{F}&{\rm Spin}\\\
\hline\cr\hline\cr&&&\\\\[-6.45831pt] {\cal O}_{1}&2&0&1\\\\[4.30554pt]
\hline\cr&&&\\\\[-6.45831pt] {\cal
O}_{2}&p+q-\frac{1}{3\ell}&p&\frac{p+q}{2}\\\\[4.30554pt]
\hline\cr&&&\\\\[-6.45831pt] {\cal
O}_{3}&p-q+\frac{1}{3\ell}&-p&\frac{p-q}{2}\\\\[4.30554pt]
\hline\cr\end{array}$ Table 2.1: Charges assignments for the mesonic operators
${\cal O}_{i}$ that generate the chiral ring.

We can see that these operators satisfy the BPS condition $\Delta={3\over
2}\,Q_{R}$. In fact, they are the building blocks of all other scalar BPS
operators. The general BPS excitation corresponds to operators of the form

$\displaystyle{\cal O}=\prod_{i=1}^{3}{\cal O}_{i}^{\,\,n_{i}}\,\,.$ (2.2.64)

It is interesting to notice that the spectrum of fluctuations of a dibaryon
must coincide with the mesonic chiral operators in the $Y^{p,q}$ quiver
theory. This would provide a nontrivial test of the AdS/CFT correspondence. We
show this result explicitly via an analysis of open string fluctuation on
wrapped D3-branes.

Now we are interested in describing the excitations of dibaryon operators from
the dual string theory. For those excitations that are factorisable, the dual
configurations are just the multi-particle states of graviton excitations in
the presence of a dibaryon. The correspondence of graviton excitation and
mesonic operator were studied in [112, 114]. What we are really interested in
are those non-factorisable operators that can be interpreted as open string
excitations on the D-brane. This can be analyzed by using the Dirac-Born-
Infeld action of the D3-brane. In what follows we will focus on the dibaryon
made of $U$ fields, which corresponds to the three-cycle D studied in
subsection 2.2.2 which, for convenience, we will parameterise with the
coordinates $(y,\psi,\alpha)$. The analysis of the dibaryon made of $V$ field
is similar. For our purpose we will use, as in eq. (1.2.102), the global
coordinate system for the $AdS_{5}$ part of the metric and we will take the
$Y^{p,q}$ part as written in eq. (1.2.80). We are interested in the normal
modes of oscillation of the wrapped D3-brane around the solution corresponding
to some fixed worldline in $AdS_{5}$ and some fixed $\theta$ and $\phi$ on the
transverse $S^{2}$. For such a configuration, the induced metric on the
dibaryon is:

$\displaystyle
L^{-2}ds^{2}_{ind}=-dT^{2}+\frac{1}{wv}dy^{2}+\frac{v}{9}d\psi^{2}+w(d\alpha+fd\psi)^{2}\,\,,$
(2.2.65)

where the functions $v(y)$, $w(y)$ and $f(y)$ have been defined in eq.
(1.2.81) (in what follows of this subsection we will take $c=1$).

The fluctuations along the transverse $S^{2}$ are the most interesting, since
they change the $SU(2)$ quantum numbers and are most readily compared with the
chiral primary states in the field theory. Without lost of generality, we
consider fluctuations around the north pole of the $S^{2}$, i.e.
$\theta_{0}=0$. Instead of using coordinates $\theta$ and $\phi$, it is
convenient to go from polar to Cartesian coordinates:
$\zeta^{1}=\theta\sin\phi$ and $\zeta^{2}=\theta\cos\phi$. As a further
simplification we perform a shift in the coordinate $\psi$. The action for the
D3-brane is:

$S\,=\,-T_{3}\,\int d^{4}\xi\sqrt{-\det g}\,+\,T_{3}\,\int P[C_{(4)}]\,\,.$
(2.2.66)

Let us expand the induced metric $g$ around the static configuration as
$g=g_{(0)}+\delta g$, where $g_{(0)}$ is the zeroth order contribution. The
corresponding expansion for the action takes the form:

$S\,=\,S_{0}\,-\,{T_{3}\over 2}\,\int d^{4}\xi\sqrt{-\det g_{(0)}}\,\,{\rm
Tr}\,\big{[}g_{(0)}^{-1}\,\delta g\big{]}\,+\,T_{3}\,\int P[C_{(4)}]\,\,,$
(2.2.67)

where $S_{0}=-T_{3}\,\int d^{4}\xi\sqrt{-\det g_{(0)}}$. Note that the
determinant of the induced metric at zeroth order is a constant: $\sqrt{-{\rm
det}(g_{(0)})}=\frac{1}{3}L^{4}$. The five-form field strength is

$\displaystyle F_{5}=(1+*)\,4\sqrt{{\rm det}(G_{Y^{p,q}})}L^{4}d\theta\wedge
d\phi\wedge dy\wedge d\psi\wedge d\alpha\,\,.$ (2.2.68)

Moreover, using that ${\rm det}(G_{Y^{p,q}})$ is the determinant of the metric
of the $Y^{p,q}$ manifold and $\sqrt{{\rm
det}(G_{Y^{p,q}})}=\frac{1-y}{18}\sin\theta$, we can choose the four-form
Ramond-Ramond field to be

$\displaystyle C_{4}=\frac{2}{9}(1-y)L^{4}(\cos\theta-1)\,d\alpha\wedge
dy\wedge d\psi\wedge d\phi\,,$ (2.2.69)

which is well defined around the north pole of $S^{2}$. At quadratic order,
the four form $C_{4}$ is

$\displaystyle C_{4}=-\sqrt{-{\rm
det}\,g_{(0)}}\,\,\frac{1-y}{3}\epsilon_{ij}\,\zeta^{i}\,d\zeta^{j}\wedge
d\alpha\wedge dy\wedge d\psi\,\,.$ (2.2.70)

The contribution from the Born-Infeld part of the effective action is:

${\rm Tr}\,\big{[}g_{(0)}^{-1}\,\delta
g\big{]}=G_{ij}\;g_{(0)}^{\mu\nu}\,(\partial_{\mu}\zeta^{i}\partial_{\nu}\zeta^{j})+2g^{\mu\nu}_{(0)}\;G_{\mu
i}\,\partial_{\nu}\zeta^{i},$ (2.2.71)

where $G$ is the metric of the background, $i,j$ denote the components of $G$
along the $\zeta^{1,2}$ directions and the indices $\mu,\nu$ refer to the
directions of the worldvolume of the cycle. The non-vanishing components of
$G$ are:

$G_{ij}=\frac{1-y}{6}L^{2}\delta_{ij},\quad G_{\psi
i}=-\frac{1}{2}\bigg{(}wf^{2}+\frac{v}{9}\bigg{)}L^{2}\epsilon_{ij}\,\zeta^{j},\quad
G_{\alpha i}=-\frac{wf}{2}L^{2}\epsilon_{ij}\,\zeta^{j}\,\,.$ (2.2.72)

Using these results one can verify that the effective Lagrangian is
proportional to:

$\displaystyle\sum_{i}L^{2}\frac{1-y}{6}g_{(0)}^{\mu\nu}(\partial_{\mu}\zeta^{i}\partial_{\nu}\zeta^{i})+2g^{\mu\nu}_{(0)}\;G_{\mu
i}\,\partial_{\nu}\zeta^{i}+\frac{2(1-y)}{3}\epsilon_{ij}\,\zeta^{i}\,\partial_{T}\zeta^{j}\,\,.$
(2.2.73)

The equations of motion for the fluctuation are finally given by

$\displaystyle{L^{2}\over
6}\,\partial_{\mu}\bigg{(}(1-y)\,g^{\mu\nu}_{(0)}\,\partial_{\nu}\zeta^{i}\,\bigg{)}+2\partial_{\nu}(g^{\mu\nu}_{(0)}\,G_{\mu
i})-\frac{2(1-y)}{3}\epsilon_{ij}\,\partial_{T}\zeta^{j}=0\,\,.$ (2.2.74)

Introducing $\zeta^{\pm}=\zeta^{1}\pm i\zeta^{2}$, the equations of motion
reduce to

$\displaystyle\bigg{(}\nabla^{2}-\frac{1-y}{6}\partial_{T}^{2}\,\bigg{)}\zeta^{\pm}\pm
i\frac{2(1-y)}{3}\partial_{T}\zeta^{\pm}\pm
i\partial_{\psi}\zeta^{\pm}=0\,\,,$ (2.2.75)

where $\nabla^{2}$ is the laplacian along the spatial directions of the cycle
for the induced metric $g_{(0)}$. The standard strategy to solve this equation
is to use separation of variables as

$\displaystyle\zeta^{\pm}=\exp(-i\omega
T)\exp\bigg{(}i\frac{m}{\ell}\alpha\bigg{)}\exp(in\psi)\,Y^{k\pm}_{mn}(y)\,\,.$
(2.2.76)

Plugging this ansatz into the equation of motion, we find

$\displaystyle\frac{1}{1-y}\frac{d}{dy}\bigg{[}(1-y)w(y)v(y)\frac{d}{dy}Y^{k\pm}_{mn}(y)\bigg{]}$
$\displaystyle=$
$\displaystyle\bigg{[}\bigg{(}\frac{9f^{2}(y)}{v(y)}+\frac{1}{w(y)}\bigg{)}\frac{m^{2}}{\ell^{2}}-\frac{18f(y)}{v(y)}\frac{m}{\ell}n+\frac{9}{v(y)}n^{2}-\omega(\omega\pm
4)\pm\frac{6n}{1-y}\bigg{]}Y^{k\pm}_{mn}(y)\,\,.$

The resulting equation has four regular singularities at $y=y_{1},y_{2},y_{3}$
and $\infty$ and is known as Heun’s equation (for clarity, in what follows we
omit the indices in $Y$) [110]:

$\frac{d^{2}}{dy^{2}}Y^{\pm}+\bigg{(}\sum_{i=1}^{3}\frac{1}{y-y_{i}}\bigg{)}\frac{d}{dy}Y^{\pm}+q(y)Y^{\pm}=0,$
(2.2.78)

where, in our case

$\displaystyle q(y)$ $\displaystyle=$ $\displaystyle\frac{2}{{\cal
Q}(y)}\bigg{[}\mu-\frac{y}{4}\omega(\omega\pm 4)-{1\over
2}\,\sum_{i=1}^{3}\frac{\alpha_{i}^{2}{\cal
Q}^{\prime}(y_{i})}{y-y_{i}}\bigg{]}\,\,,$ $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{3}{32}(\frac{m}{\ell}+2n)(\frac{m}{\ell}-6n)+\frac{1}{4}\omega(\omega\pm
4)\mp\frac{3n}{2}\,\,,$ (2.2.79)

with ${\cal Q}(y)$ being the function defined in eq. (1.2.74). Now, given that
the R-symmetry is dual to the Reeb Killing vector of $Y^{p,q}$, namely
$2\partial/\partial\psi\,-\,{1\over 3}\,\partial/\partial\alpha$, we can use
the R-charge $Q_{R}=2n-m/3\ell$ instead of $n$ as quantum number. The
exponents at the regular singularities $y=y_{i}$ are then given by

$\displaystyle\alpha_{i}=\pm\frac{1}{2}\frac{(1-y_{i})(m/\ell+3Q_{R}\,y_{i})}{{\cal
Q}^{\prime}(y_{i})}.$ (2.2.80)

The exponents at $y=\infty$ are $-\frac{\omega}{2}$ and $\frac{\omega}{2}+2$
for $Y^{+}$, while $-\frac{\omega}{2}+2$ and $\frac{\omega}{2}$ for $Y^{-}$.
We can transform the singularity from $\\{y_{1},y_{2},y_{3},\infty\\}$ to
$\\{0,1,b=\frac{y_{1}-y_{3}}{y_{1}-y_{2}},\infty\\}$ by introducing a new
variable $x$, defined as:

$\displaystyle x=\frac{y-y_{1}}{y_{2}-y_{1}}.$ (2.2.81)

It is also convenient to substitute

$\displaystyle
Y=x^{|\alpha_{1}|}(1-x)^{|\alpha_{2}|}(b-x)^{|\alpha_{3}|}\,h(x)\,\,,$
(2.2.82)

which transforms equation (2.2.78) into the standard form of the Heun’s
equation

$\frac{d^{2}}{dx^{2}}h(x)+\bigg{(}\frac{C}{x}+\frac{D}{x-1}+\frac{E}{x-b}\,\bigg{)}\frac{d}{dx}h(x)+\frac{ABx-k}{x(x-1)(x-b)}h(x)=0.$
(2.2.83)

Here the Heun’s parameters are given by

$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{\omega}{2}+\sum_{i=1}^{3}|\alpha_{i}|\,,\quad\quad
B=\frac{\omega+4}{2}+\sum_{i=1}^{3}|\alpha_{i}|\,\,,$ $\displaystyle C$
$\displaystyle=$ $\displaystyle 1+2|\alpha_{1}|\,,\quad\quad
D=1+2|\alpha_{2}|\,,\quad\quad E=1+2|\alpha_{3}|\,,$ (2.2.84)

and

$\displaystyle k$ $\displaystyle=$
$\displaystyle(|\alpha_{1}|+|\alpha_{3}|)(|\alpha_{1}|+|\alpha_{3}|+1)-|\alpha_{2}|^{2}$
$\displaystyle+$ $\displaystyle
b\bigg{[}(|\alpha_{1}|+|\alpha_{2}|)(|\alpha_{1}|+|\alpha_{2}|+1)-|\alpha_{3}|^{2}\bigg{]}-{\tilde{\mu}}\,\,,$
$\displaystyle{\tilde{\mu}}$ $\displaystyle=$
$\displaystyle-\frac{1}{y_{1}-y_{2}}\bigg{(}\mu-\frac{y_{1}}{4}\omega(\omega+4)\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{p}{q}\bigg{[}\frac{1}{6}(1-y_{1})\omega(\omega+4)-\frac{3}{16}Q_{R}\,\bigg{(}Q_{R}+\frac{4m}{3\ell}\bigg{)}-\frac{1}{2}\bigg{(}Q_{R}+\frac{m}{3\ell}\bigg{)}\bigg{]}\,\,,$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}1+\frac{\sqrt{4p^{2}-3q^{2}}}{q}\bigg{)}\,\,.$
(2.2.85)

We only presented the equation for $Y^{+}$; the corresponding equation for
$Y^{-}$ can be obtained by replacing $\omega$ with $\omega-4$ and changing the
sign of the last term in (2.2.5).

Now let us discuss the solutions to this differential equation. For quantum
number $Q_{R}=2N_{c}$ (which implies $m=0$), we find all $\alpha_{i}$ equal to
$N_{c}/2$. If we set $\omega=3N_{c}$, the Heun’s parameters $A$ and $k$ both
vanish. The corresponding solution $h(x)$ is a constant function. Similarly if
$\omega=-3N_{c}-4$, then $B$ and $k$ vanish which also implies a constant
$h(x)$. The complete solution of $\zeta^{\pm}$ in these two cases is given by

$\displaystyle\zeta_{1}^{\pm}$ $\displaystyle=$ $\displaystyle e^{\pm
i(-3N_{c}T+N_{c}\psi)}\prod_{i=1}^{3}(y-y_{i})^{N_{c}/2}\,\,,$
$\displaystyle\zeta_{2}^{\pm}$ $\displaystyle=$ $\displaystyle e^{\pm
i((3N_{c}+4)T+N\psi)}\prod_{i=1}^{3}(y-y_{i})^{N_{c}/2}\,\,.$ (2.2.86)

These constant solutions represent ground states with fixed quantum numbers
and, since they have the lowest possible dimension for a given R-charge, they
should be identified with the BPS operators. Indeed, in the solutions (2.2.86)
the energy is quantised in units of $3L^{-1}$, and $3$ is precisely the
conformal dimension of ${\cal O}_{1}$. This provides a perfect matching of
AdS/CFT in this setting.

The situation for quantum numbers $Q_{R}=N_{c}(p\pm q\mp 1/3\ell)$ and $m=\pm
N_{c}$ is similar to the case we have just discussed. The solutions for $h(x)$
are constant with

$\omega=\frac{N_{c}p}{2}\bigg{(}3\pm\frac{2p-\sqrt{4p^{2}-3q^{2}}}{q}\bigg{)}\,\,,$
(2.2.87)

and

$\omega=-\frac{N_{c}p}{2}\bigg{(}3\pm\frac{2p-\sqrt{4p^{2}-3q^{2}}}{q}\bigg{)}-4.$
(2.2.88)

We can see that the conformal dimension satisfies $\Delta=\frac{3}{2}Q_{R}$.
So all these solutions are BPS fluctuations which should correspond to the
operators ${\cal O}_{2}$ and ${\cal O}_{3}$.

An interesting comment is in order at this point. Notice that the dibaryon
excitations should come out with the multiplicities associated to the SU(2)
spin (see Table 2.1) of the ${\cal O}_{i}$ operators. However, in order to
tackle this problem, we would need to consider at the same time the
fluctuation of the D3-brane probes and the zero-mode dynamics corresponding to
their collective motion along the sphere with coordinate $\theta$ and $\phi$
(see ref. [83] for a similar discussion in the conifold case). This is an
interesting problem that we leave open.

###  2.3 Supersymmetric D5-branes in $AdS_{5}\times Y^{p,q}$

In this section we will study the supersymmetric configurations of D5-branes
in the $AdS_{5}\times Y^{p,q}$ background. First of all, notice that in this
case $\Gamma_{\kappa}$ acts on the Killing spinors $\epsilon$ as:

$\Gamma_{\kappa}\,\epsilon\,=\,{i\over
6!\,\sqrt{-g}}\,\,\epsilon^{\mu_{1}\cdots\mu_{6}}\,\gamma_{\mu_{1}\cdots\mu_{6}}\,\epsilon^{*}\,\,,$
(2.3.1)

where we have used the relation (1.3.10) to translate eq. (2.1.1). The
appearance of the complex conjugation on the right-hand side of eq. (2.3.1) is
crucial in what follows. Indeed, the complex conjugation does not commute with
the projections (1.2.99). Therefore, in order to construct an additional
compatible projection involving the $\epsilon\rightarrow\epsilon^{*}$
operation we need to include a product of gamma matrices which anticommutes
with both $\Gamma_{12}$ and $\Gamma_{34}$. As in the D3-brane case just
analyzed, this compatibility requirement between the
$\Gamma_{\kappa}\,\epsilon=\epsilon$ condition and (1.2.99) implies a set of
differential equations whose solutions, if any, determine the supersymmetric
embeddings we are looking for.

We will carry out successfully this program only in the case of a D5-brane
extended along a two-dimensional submanifold of $Y^{p,q}$. One expects that
these kinds of configurations represent wall defects in the field theory. When
we allow the D5-brane to extend infinitely in the holographic direction, we
get a configuration dual to a defect conformal field theory. In the remainder
of this section we will find the corresponding configurations of the D5-brane
probe. Moreover, in section 2.5 we will find, based on a different set of
worldvolume coordinates, another embedding of this type preserving the same
supersymmetry as the one found in the present section and we will analyze the
effect of adding flux of the worldvolume gauge fields. In section 2.5 we will
also look at the possibility of having D5-branes wrapped on a three-
dimensional submanifold of $Y^{p,q}$. This configuration looks like a domain
wall in the field theory dual and although these configurations are not
supersymmetric, we have been able to find stable solutions of the equations of
motion. The case in which the D5-brane wraps the entire $Y^{p,q}$ corresponds
to the baryon vertex. In this configuration, studied also in section 2.5, the
D5-brane captures the flux of the RR five-form, which acts as a source for the
electric worldvolume gauge field. We will conclude in section 2.5 that this
configuration cannot be supersymmetric, in analogy with what happens in the
conifold case [105].

####  2.3.1 Wall defect solutions

We want to find a configuration in which the D5-brane probe wraps a two-
dimensional submanifold of $Y^{p,q}$ and is a codimension one object in
$AdS_{5}$. Accordingly, let us place the probe at some constant value of one
of the Minkowski coordinates (say $x^{3}$) and let us extend it along the
radial direction. To describe such an embedding we choose the following set of
worldvolume coordinates for a D5-brane probe

$\xi^{\mu}\,=\,(t,x^{1},x^{2},r,\theta,\phi)\,\,,$ (2.3.2)

and we adopt the following ansatz:

$y=y(\theta,\phi)\,\,,\qquad\beta\,=\,\beta(\theta,\phi)\,\,,$ (2.3.3)

with $x^{3}$ and $\psi$ constant. The induced Dirac matrices can be computed
straightforwardly from eq. (1.3.12). From the general expression (2.3.1) one
readily gets that the kappa symmetry matrix acts on the spinor $\epsilon$ as:

$\Gamma_{\kappa}\,\epsilon\,=\,{i\over\sqrt{-g}}\,{r^{2}\over
L^{2}}\,\Gamma_{x^{0}x^{1}x^{2}r}\,\gamma_{\theta\phi}\,\epsilon^{*}\,\,.$
(2.3.4)

By using the complex conjugate of the projections (1.2.99) one gets:

${6\over
L^{2}}\,\gamma_{\theta\phi}\,\epsilon^{*}\,=\,\big{[}\,b_{I}\,+\,b_{15}\,\Gamma_{15}\,+\,b_{35}\,\Gamma_{35}\,+\,b_{13}\,\Gamma_{13}\,\big{]}\,\epsilon^{*}\,\,,$
(2.3.5)

where the different coefficients are:

$\displaystyle
b_{I}\,=\,-i\Big{[}\,(1-cy)\sin\theta\,+\,c\cos\theta\,y_{\theta}\,+\,y_{\phi}\beta_{\theta}\,-\,y_{\theta}\beta_{\phi}\,\Big{]}\,\,,$
$\displaystyle b_{15}\,=\,-\sqrt{{2\over 3}}\,\,{1\over
H}\Big{[}\,(1-cy)\cos\theta\,y_{\theta}\,+\,y\,(\beta_{\phi}\,y_{\theta}-\beta_{\theta}\,y_{\phi})\,\Big{]}\,-\,i\sqrt{{2\over
3}}\,H\,\cos\theta\,\beta_{\theta}\,\,,$ $\displaystyle
b_{35}\,=\,\sqrt{{2\over
3}}\sqrt{1-cy}\,\Big{[}\,(1-cy)\cos\theta\,+\,y\beta_{\phi}\,\Big{]}\,+\,i\sqrt{{2\over
3}}\sqrt{1-cy}\,\,y\sin\theta\beta_{\theta}\,\,,$ $\displaystyle
b_{13}\,=\,\sqrt{1-cy}\,\Big{[}\,{y_{\phi}\over
H}\,-\,H\beta_{\theta}\,\sin\theta\,\Big{]}\,+\,i\sqrt{1-cy}\,\Big{[}\,{\sin\theta\over
H}\,y_{\theta}\,-\,H\,(c\cos\theta\,-\,\beta_{\phi})\,\Big{]}\,\,.\qquad$
(2.3.6)

As discussed above, in this case the action of $\Gamma_{\kappa}$ involves the
complex conjugation, which does not commute with the projections (1.2.99).
Actually, the only term on the right-hand side of (2.3.5) which is consistent
with (1.2.99) is the one containing $\Gamma_{13}$. Accordingly, we must
require:

$b_{I}\,=\,b_{15}\,=\,b_{35}\,=\,0\,\,.$ (2.3.7)

From the vanishing of the imaginary part of $b_{15}$ we get:

$\beta_{\theta}=0\,\,,$ (2.3.8)

while the vanishing of the real part of $b_{15}$ leads to:

$\beta_{\phi}\,=\,-{1-cy\over y}\,\cos\theta\,\,.$ (2.3.9)

Notice that $b_{35}$ is zero as a consequence of equations (2.3.8) and (2.3.9)
which, in particular imply that:

$\beta=\beta(\phi)\,\,.$ (2.3.10)

Moreover, by using eq. (2.3.8), the condition $b_{I}=0$ is equivalent to

$(1-cy)\sin\theta\,+\,(c\cos\theta\,-\,\beta_{\phi})y_{\theta}\,=\,0\,\,,$
(2.3.11)

and plugging the value of $\beta_{\phi}$ from (2.3.9), one arrives at:

$y_{\theta}\,=\,-(1-cy)\,y\tan\theta\,\,.$ (2.3.12)

In order to implement the kappa symmetry condition at all points of the
worldvolume the phase of $b_{13}$ must be constant. This can be achieved by
requiring that the real part of $b_{13}$ vanishes, which for
$\beta_{\theta}=0$ is equivalent to the condition $y_{\phi}=0$, i.e.:

$y=y(\theta)\,\,.$ (2.3.13)

The equation (2.3.12) for $y(\theta)$ is easily integrated, namely:

${y\over 1-cy}\,=\,k\cos\theta\,\,,$ (2.3.14)

where $k$ is a constant. Moreover, by separating variables in eq. (2.3.9), one
concludes that:

$\beta_{\phi}=m\,\,,$ (2.3.15)

where $m$ is a new constant. Plugging (2.3.14) into eq. (2.3.9) and using the
result (2.3.15) one concludes that the two constants $m$ and $k$ must be
related as:

$km=-1\,\,,$ (2.3.16)

which, in particular implies that $k$ and $m$ cannot vanish. Thus, the
embedding of the D5-brane becomes

$\displaystyle\beta=m\phi+\beta_{0}\,\,,$ $\displaystyle y=-{\cos\theta\over
m-c\cos\theta}\,\,.$ (2.3.17)

Notice that the solution (2.3.17) is symmetric under the change
$m\rightarrow-m$, $\theta\rightarrow\pi-\theta$ and $\phi\rightarrow
2\pi-\phi$. Thus, from now on we can assume that $m\geq 0$.

It is now straightforward to verify that the BPS equations are equivalent to
impose the following condition on the spinor $\epsilon$:

$\Gamma_{x^{0}x^{1}x^{2}r13}\epsilon^{*}\,=\,\sigma\epsilon\,\,,$ (2.3.18)

where $\sigma$ is:

$\sigma\,=\,{\rm sign}\,\Big{(}\,{\cos\theta\over y}\,\Big{)}\,=\,-{\rm
sign}\Big{(}\,m\,-\,c\cos\theta\,\Big{)}\,\,.$ (2.3.19)

Obviously, the only valid solutions are those which correspond to having a
constant sign $\sigma$ along the worldvolume. This always happens for $m/c\geq
1$. In this case the minimal (maximal) value of $\theta$ is $\theta=0$
($\theta=\pi$) if $|m-c||y_{1}|>1$ ($|m-c||y_{2}|>1$). Otherwise the angle
$\theta$ must be restricted to lie in the interval
$\theta\in[\theta_{1},\theta_{2}]$, where $\theta_{1}$ and $\theta_{2}$ are
given by:

$\theta_{i}\,=\,\arccos\Big{[}{my_{i}\over
cy_{i}-1}\Big{]}\,\,,\qquad\qquad(i=1,2)\,\,.$ (2.3.20)

Notice that, similarly to what we obtained in the previous section,
eq.(2.3.17) implies that the configuration we arrived at does not, in general,
correspond to a wrapped brane but to a D5-brane that spans a two-dimensional
submanifold with boundaries.

Let us now count the number of supersymmetries preserved by our configuration.
In order to do so we must convert eq. (2.3.18) into an algebraic condition on
a constant spinor. With this purpose in mind let us write the general form of
$\epsilon$ as the sum of the two types of spinors written in eq. (1.2.101),
namely:

$e^{{i\over 2}\psi}\,\epsilon\,=\,r^{-{1\over 2}}\,\eta_{+}\,+\,r^{{1\over
2}}\,\Big{(}\,{\bar{x}^{3}\over
L^{2}}\,\Gamma_{rx^{3}}\,\eta_{+}\,+\,\eta_{-}\,\Big{)}\,+\,{r^{{1\over
2}}\over L^{2}}\,x^{p}\,\Gamma_{rx^{p}}\,\eta_{+}\,\,,$ (2.3.21)

where $\bar{x}^{3}$ is the constant value of the coordinate $x^{3}$ in the
embedding and the index $p$ runs over the set $\\{0,1,2\\}$. By substituting
eq. (2.3.21) on both sides of eq. (2.3.18), one can get the conditions that
$\eta_{+}$ and $\eta_{-}$ must satisfy. Indeed, let us define the operator
${\cal P}$ as follows:

${\cal P}\,\epsilon\,\equiv\,i\sigma
e^{i\psi_{0}}\,\Gamma_{rx^{3}}\,\Gamma_{13}\,\epsilon^{*}\,\,.$ (2.3.22)

Then, one can check that eq. (2.3.18) is equivalent to:

$\displaystyle{\cal P}\,\eta_{+}\,=\,\eta_{+}\,\,,$ $\displaystyle(1\,+\,{\cal
P}\,)\,\eta_{-}\,=\,-{2\bar{x}^{3}\over
L^{2}}\,\Gamma_{rx^{3}}\,\eta_{+}\,\,.$ (2.3.23)

As ${\cal P}^{2}=1$, we can classify the four spinors $\eta_{-}$ according to
their ${\cal P}$-eigenvalue as: ${\cal
P}\,\eta_{-}^{(\pm)}\,=\,\pm\eta_{-}^{(\pm)}$. We can now solve the system
(2.3.23) by taking $\eta_{+}=0$ and $\eta_{-}$ equal to one of the two spinors
$\eta_{-}^{(-)}$ of negative ${\cal P}$-eigenvalue. Moreover, there are other
two solutions which correspond to taking a spinor $\eta_{-}^{(+)}$ of positive
${\cal P}$-eigenvalue and a spinor $\eta_{+}$ related to the former as:

$\eta_{+}\,=\,{L^{2}\over\bar{x}^{3}}\,\Gamma_{rx^{3}}\,\,\eta_{-}^{(+)}\,\,.$
(2.3.24)

Notice that, according to the first equation in (2.3.23), the spinor
$\eta_{+}$ must have positive ${\cal P}$-eigenvalue, in agreement with eq.
(2.3.24). All together this configuration preserves four supersymmetries, i.e.
one half of the supersymmetries of the background, as expected for a wall
defect.

####  2.3.2 The calibrating condition

For any two-dimensional submanifold $\tilde{L}$ of $Y^{p,q}$ one can construct
its three-dimensional cone ${\cal L}\subset CY^{p,q}$. The holomorphic $(3,0)$
form $\Omega$ of $CY^{p,q}$ can be naturally used to calibrate such
submanifolds. Indeed, ${\cal L}$ is called a special Lagrangian submanifold of
$CY^{p,q}$ if the pullback of $\Omega$ to ${\cal L}$ is, up to a constant
phase, equal to the volume form of ${\cal L}$, namely:

$P\big{[}\,\Omega\,\big{]}_{{\cal L}}\,=\,e^{i\lambda}\,{\rm Vol}\,({\cal
L})\,\,,$ (2.3.25)

where $\lambda$ is constant on ${\cal L}$. If the cone ${\cal L}$ is special
Lagrangian, its base $\tilde{L}$ is said to be special Legendrian. It has been
argued in ref. [115] that the supersymmetric configurations of a D5-brane
extended along a two-dimensional submanifold $\tilde{L}$ of a Sasaki-Einstein
space are those for which ${\cal L}$ is special Lagrangian. Let us check that
this is indeed the case for the embeddings (2.3.17). First of all, we notice
that the expression of $\Omega$ written in (1.2.89) can be recast as:

$\Omega\,=\,e^{i\psi}\,r^{2}\,\Omega_{4}\,\wedge\big{[}\,dr+\,i\,{r\over
L}\,e^{5}\,\big{]}\,\,,$ (2.3.26)

where $\Omega_{4}$ is the two-form:

$\Omega_{4}\,=\,{1\over
L^{2}}\,\,\big{(}\,e^{1}+ie^{2}\,\big{)}\wedge\big{(}\,e^{3}-ie^{4}\,\big{)}\,\,.$
(2.3.27)

In eqs. (2.3.26) and (2.3.27) $e^{1}$, $\cdots$, $e^{5}$ are the vielbein one-
forms of (1.2.71). Moreover, the volume form of ${\cal L}$ can be written as:

${\rm Vol}\,({\cal L})\,=\,r^{2}dr\wedge{\rm Vol}\,(\tilde{L})\,\,.$ (2.3.28)

For our embeddings (2.3.17) one can check that:

${\rm Vol}\,(\tilde{L})\,=\,\,{H\over 6}\,\Big{|}\,{\cos\theta\over
y}\,\Big{|}\,\sqrt{1-cy}\,\Bigg{[}\,1\,+\,(1-cy)\,{y^{2}\over
H^{2}}\,\tan^{2}\theta\,\Bigg{]}\,d\theta\wedge d\phi\,\,.$ (2.3.29)

It is now straightforward to verify that our embeddings (2.3.17) satisfy
(2.3.25) with $e^{i\lambda}\,=\,-i\sigma e^{i\psi}$, where $\sigma$ is the
constant sign defined in (2.3.19) (recall that in our ansatz (2.3.3) the angle
$\psi$ is constant). Thus, we conclude that $\tilde{L}$ is special Legendrian,
as claimed. Moreover, one can check that:

$P\big{[}\,J\,\big{]}_{{\cal L}}\,=\,0\,\,.$ (2.3.30)

####  2.3.3 Energy bound

Let us consider a generic embedding $y=y(\theta)$, $\beta=\beta(\phi)$ and let
us define the following functions of $\theta$ and $y$

$\Delta_{\theta}\equiv-y(1-cy)\tan\theta\,\,,\qquad\Delta_{\phi}\equiv-{1-cy\over
y}\,\,\cos\theta\,\,.$ (2.3.31)

In terms of these functions the BPS equations (2.3.9) and (2.3.12) are simply
$y_{\theta}=\Delta_{\theta}$ and $\beta_{\phi}=\Delta_{\phi}$. We have checked
that any solution of this first-order equations also solves the Euler-Lagrange
equations derived from the Dirac-Born-Infeld lagrangian (2.2.53). Moreover,
the hamiltonian density ${\cal H}=\sqrt{-g}$ satisfies a BPS bound as in
(2.2.56), where ${\cal Z}$ is a total derivative. To prove this statement, let
us notice that ${\cal H}$ can be written as:

$\displaystyle{\cal H}\,=\,{r^{2}\over
6}\,{H\over\sqrt{1-cy}}\,\Big{|}\,{y\over\cos\theta}\,\Big{|}\,\sqrt{\Delta_{\phi}^{2}\,+\,(1-cy)\,{\cos^{2}\theta\over
y^{2}H^{2}}\,y_{\theta}^{2}}\,\,\times$
$\displaystyle\qquad\times\sqrt{(c\cos\theta\,-\,\beta_{\phi})^{2}\,+\,{\cos^{2}\theta\over
H^{2}y^{2}(1-cy)}\,\Delta_{\theta}^{2}\,+\,{2y^{2}\over
3H^{2}}\,(\beta_{\phi}\,-\,\Delta_{\phi})^{2}}\,\,.$ (2.3.32)

Let us now rewrite ${\cal H}$ as ${\cal H}=|{\cal Z}|+{\cal S}$, where

${\cal Z}\,=\,{r^{2}\over
6}\,{H\over\sqrt{1-cy}}\,{y\over\cos\theta}\,\Big{[}\,{\cos^{2}\theta\over
y^{2}H^{2}}\,\Delta_{\theta}\,y_{\theta}\,-\,(c\cos\theta\,-\,\beta_{\phi})\Delta_{\phi}\,\Big{]}\,\,.$
(2.3.33)

One can check that $|{\cal Z}|_{|BPS}\,=\,\sqrt{-g}_{|BPS}$. Moreover, for
arbitrary functions $y=y(\theta)$ and $\beta=\beta(\phi)$, one can verify that
${\cal Z}$ is a total derivative, namely:

${\cal Z}\,=\,{\partial\over\partial\theta}\,{\cal
Z}^{\theta}\,+\,{\partial\over\partial\phi}\,{\cal Z}^{\phi}\,\,.$ (2.3.34)

In order to write the explicit expressions of ${\cal Z}^{\theta}$ and ${\cal
Z}^{\phi}$, let us define the function $g(y)$ as follows:

$g(y)\,\equiv\,-\int{\sqrt{1-cy}\over H(y)}\,dy\,\,.$ (2.3.35)

Then one can verify that eq. (2.3.34) is satisfied for ${\cal Z}^{\theta}$ and
${\cal Z}^{\phi}$ given by:

$\displaystyle{\cal Z}^{\theta}$ $\displaystyle=$ $\displaystyle{r^{2}\over
6}\,\sin\theta\,g(y)\,\,,$ $\displaystyle{\cal Z}^{\phi}$ $\displaystyle=$
$\displaystyle{r^{2}\over
6}\,\Big{[}\,-\cos\theta\,g(y)\,\phi\,+\,H(y)\,\sqrt{1-cy}\,\,(c\phi\cos\theta\,-\,\beta)\,\Big{]}\,\,.$
(2.3.36)

One can prove that ${\cal H}\geq\big{|}\,{\cal Z}\,\big{|}$ is equivalent to:

$\displaystyle{\cos^{2}\theta\over
y^{2}(1-cy)}\,\Big{[}\,\Delta_{\phi}\,\Delta_{\theta}\,+\,(1-cy)\,(c\cos\theta\,-\,\beta_{\phi})\,y_{\theta}\,\Big{]}^{2}\,+$
$\displaystyle\qquad\qquad{2y^{2}\over
3}\,\Big{[}\,\Delta_{\phi}^{2}\,+\,{(1-cy)\cos^{2}\theta\over
y^{2}H^{2}}\,y_{\theta}^{2}\,\Big{]}\,[\,\beta_{\phi}\,-\,\Delta_{\phi}\,]^{2}\,\geq\,0\,\,,$
(2.3.37)

which is always satisfied. Moreover, by using that
$(c\cos\theta\,-\,\beta_{\phi})_{|BPS}\,=\,\cos\theta/y$, one can prove that
this inequality is saturated precisely when the BPS differential equations are
satisfied.

###  2.4 Supersymmetric D7-branes in $AdS_{5}\times Y^{p,q}$

For a D7-brane the kappa symmetry matrix (2.1.1) takes the form:

$\Gamma_{k}=-\frac{i}{8!\sqrt{-g}}\epsilon^{\mu_{1}\ldots\mu_{8}}\gamma_{\mu_{1}\ldots\mu_{8}},$
(2.4.1)

where, again, we have used the rules of eq. (1.3.10) to write the expression
of $\Gamma_{k}$ acting on complex spinors. The D7-branes which fill the four
Minkowski spacetime directions and extend along some holographic non-compact
direction can be potentially used as flavour branes, i.e. as branes whose
fluctuations can be identified with the dynamical mesons of the gauge theory.
In this section we will find a family of these configurations which preserve
four supersymmetries. In section 2.5 we will determine another family of
supersymmetric spacetime filling configurations of D7-branes and we will also
demonstrate that there are embeddings in which the D7-brane wraps the entire
$Y^{p,q}$ space and preserve two supersymmetries.

####  2.4.1 Spacetime filling D7-brane

Let us choose a system of worldvolume coordinates motivated by the spacetime
filling character of the configuration that we are trying to find, namely:

$\xi=(t,x^{1},x^{2},x^{3},y,\beta,\theta,\phi).$ (2.4.2)

The ansatz we will adopt for the embedding is:

$\psi=\psi(\beta,\phi),\qquad r=r(y,\theta).$ (2.4.3)

In this case the general expression of $\Gamma_{\kappa}$ (eq. (2.4.1)) reduces
to:

$\Gamma_{\kappa}\,=\,-i\,{r^{4}\over L^{4}\sqrt{-g}}\,\Gamma_{x^{0}\cdots
x^{3}}\,\gamma_{y\beta\theta\phi}\,.$ (2.4.4)

In order to implement the $\Gamma_{\kappa}\,\epsilon\,=\,\epsilon$ condition
we require that the spinor $\epsilon$ is an eigenvector of the matrix
$\Gamma_{*}$ defined in eq. (1.2.96). Then, according to eq. (1.2.101),
$\Gamma_{*}\epsilon=-\epsilon$, i.e. $\epsilon$ is of the form $\epsilon_{-}$
and, therefore, it satisfies:

$\Gamma_{x^{0}\cdots x^{3}}\,\epsilon_{-}\,=\,i\epsilon_{-}\,\,.$ (2.4.5)

Moreover, as $\epsilon_{-}$ has fixed ten-dimensional chirality, the condition
(2.4.5) implies:

$\Gamma_{r5}\epsilon_{-}\,=\,-i\epsilon_{-}\,\,.$ (2.4.6)

By using the projection (2.4.5), one immediately arrives at:

$\Gamma_{\kappa}\,\epsilon_{-}\,=\,{r^{4}\over
L^{4}\sqrt{-g}}\,\gamma_{y\beta\theta\phi}\,\epsilon_{-}\,\,.$ (2.4.7)

After using eqs. (1.2.99) and (2.4.6), the action of
$\gamma_{y\beta\theta\phi}$ on $\epsilon$ can be written as:

${1\over
L^{4}}\,\gamma_{y\beta\theta\phi}\,\epsilon_{-}\,=\,\big{[}\,d_{I}\,+\,d_{15}\,\Gamma_{15}\,+\,d_{35}\,\Gamma_{35}\,+\,d_{13}\Gamma_{13}\,\big{]}\,\epsilon_{-}\,\,,$
(2.4.8)

where the different coefficients are given by:

$\displaystyle d_{I}\,=\,{1-cy\over 36}\,\sin\theta\,+\,{1-cy\over
18}\,\sin\theta\,(y+\psi_{\beta})\,{r_{y}\over r}\,-\,{1\over
18}\,\big{[}\,(1+c\psi_{\beta})\cos\theta+\psi_{\phi}\,\big{]}\,{r_{\theta}\over
r}\,\,,$ $\displaystyle d_{15}\,=\,i\,{1-cy\over
6\sqrt{6}}\,H\,\sin\theta\,\Big{[}\,{r_{y}\over r}\,-\,{y+\psi_{\beta}\over
3H^{2}}\,\Big{]}\,\,,$ $\displaystyle d_{35}\,=\,-i\,{\sqrt{1-cy}\over
6\sqrt{6}}\,\Big{[}\,\sin\theta\,{r_{\theta}\over r}\,+\,{1\over
3}\,\big{(}\,(1+c\psi_{\beta})\cos\theta\,+\,\psi_{\phi}\,)\,\Big{]}\,\,,$
$\displaystyle d_{13}\,=\,{{\sqrt{1-cy}\over
18}}\,H\,\Big{[}\,\sin\theta\,{y+\psi_{\beta}\over H^{2}}\,{r_{\theta}\over
r}\,+\,\big{(}\,(1+c\psi_{\beta})\,\cos\theta\,+\,\psi_{\phi}\big{)}\,{r_{y}\over
r}\,\Big{]}\,\,.$ (2.4.9)

As the terms containing the matrices $\Gamma_{15}$, $\Gamma_{35}$ and
$\Gamma_{13}$ give rise to projections which are not compatible with those in
eq. (1.2.99), we have to impose that:

$d_{15}\,=\,d_{35}\,=\,d_{13}\,=\,0\,\,.$ (2.4.10)

From the vanishing of $d_{15}$ and $d_{35}$ we obtain the following first-
order differential equations

$r_{y}\,=\,\Lambda_{y}\,\,,\qquad\qquad r_{\theta}\,=\,\Lambda_{\theta}\,\,,$
(2.4.11)

where we have defined $\Lambda_{y}$ and $\Lambda_{\theta}$ as:

$\displaystyle\Lambda_{y}\,=\,{r\over
3H^{2}}\,\big{(}y+\psi_{\beta}\big{)}\,,$
$\displaystyle\Lambda_{\theta}\,=\,-{r\over
3\sin\theta}\,\Big{[}\,(1+c\psi_{\beta})\,\cos\theta\,+\,\psi_{\phi}\,\Big{]}\,\,.$
(2.4.12)

Notice that the equations (2.4.11) imply that $d_{13}=0$. One can also check
that $r^{4}\,d_{I}\,=\,\sqrt{-g}$ if the first-order equations (2.4.11) hold
and, therefore, one has indeed that
$\Gamma_{\kappa}\epsilon_{-}=\epsilon_{-}$. Thus, any Killing spinor of the
type $\epsilon=\epsilon_{-}$, with $\epsilon_{-}$ as in eq. (1.2.101),
satisfies the kappa symmetry condition if the BPS equations (2.4.11) hold.
Therefore, these configurations preserve the four ordinary supersymmetries of
the background and, as a consequence, they are 1/8 supersymmetric.

##### Integration of the first-order equations

Let us now obtain the general solution of the system (2.4.11). Our first
observation is that, according to (2.4.3), the only dependence on the
coordinates $\beta$ and $\phi$ appearing in eqs. (2.4.11) and (2.4.12) comes
from the derivatives of $\psi$. Therefore, for consistency with the assumed
dependence of the functions of the ansatz (2.4.3), $\psi_{\phi}$ and
$\psi_{\beta}$ must be constants. Thus, let us write:

$\psi_{\phi}\,=\,n_{1}\,\,,\qquad\qquad\psi_{\beta}\,=\,n_{2}\,\,,$ (2.4.13)

which can be trivially integrated, namely:

$\psi\,=\,n_{1}\,\phi\,+\,n_{2}\,\beta\,+\,{\rm constant}\,\,.$ (2.4.14)

It is now easy to obtain the function $r(\theta,y)$. The equations to
integrate are:

$r_{y}\,=\,{r\over 3H^{2}}\,(y+n_{2})\,\,,\qquad r_{\theta}\,=\,-{r\over
3\sin\theta}\,\Big{[}\,(1+cn_{2})\cos\theta\,\,+\,n_{1}\,\Big{]}\,\,.$
(2.4.15)

Let us first integrate the equation for $r_{\theta}$ in (2.4.15). We get:

$r(y,\theta)\,=\,{A(y)\over\Big{[}\sin{\theta\over
2}\Big{]}^{{1+n_{1}+cn_{2}\over 3}}\,\,\Big{[}\cos{\theta\over
2}\Big{]}^{{1-n_{1}+cn_{2}\over 3}}}\,\,,$ (2.4.16)

with $A(y)$ a function of $y$ to be determined. Plugging this result in the
equation for $r_{y}$ in (2.4.15), we get the following equation for $A$:

${1\over A}\,{dA\over dy}\,=\,{1\over 3}\,{y+n_{2}\over H^{2}}\,\,,$ (2.4.17)

which can be integrated immediately, namely:

$A^{3}(y)\,=\,C\,\Big{[}f_{1}(y)\Big{]}^{n_{2}}\,f_{2}(y)\,\,,$ (2.4.18)

with $C$ a constant and $f_{1}(y)$ and $f_{2}(y)$ being the functions defined
in (1.2.87). Then, we can write $r(y,\theta)$ as:

$r^{3}(y,\theta)\,=\,C{\Big{[}f_{1}(y)\Big{]}^{n_{2}}\,f_{2}(y)\over\Big{[}\sin{\theta\over
2}\Big{]}^{1+n_{1}+cn_{2}}\,\,\Big{[}\cos{\theta\over
2}\Big{]}^{1-n_{1}+cn_{2}}}\,\,.$ (2.4.19)

Several comments concerning the solution displayed in eqs. (2.4.14) and
(2.4.19) are in order. First of all, after a suitable change of variables it
is easy to verify that for $c=0$ one recovers from (2.4.14) and (2.4.19) the
family of D7-brane embeddings in $AdS_{5}\times T^{1,1}$ found in ref. [105].
Secondly, the function $r(y,\theta)$ in (2.4.19) always diverges for some
particular values of $\theta$ and $y$, which means that the probe always
extends infinitely in the holographic direction. Moreover, for some particular
values of $n_{1}$ and $n_{2}$ there is a minimal value of the coordinate $r$,
which depends on the integration constant $C$. This fact is important when one
tries to use these D7-brane configurations as flavour branes, since this
minimal value of $r$ provides us with an energy scale, which is naturally
identified with the mass of the dynamical quarks added to the gauge theory. It
is also interesting to obtain the form of the solution written in eqs.
(2.4.14) and (2.4.19) in terms of the complex variables $z_{i}$ defined in
(1.2.86). After a simple calculation one can verify that this solution can be
written as a polynomial equation of the form:

$z_{1}^{m_{1}}\,z_{2}^{m_{2}}\,z_{3}^{m_{3}}\,=\,{\rm constant}\,\,,$ (2.4.20)

where the $m_{i}$’s are constants and $m_{3}\not=0$.333 It is natural to
expect a condition of the form $f(z_{1},z_{2},z_{3})=0$, where $f$ is a
general holomorphic function of its arguments. However, in order to be able to
solve the problem analytically we started from a restrictive ansatz (2.4.3)
that, not surprisingly, leads to a particular case of the expected answer. The
relation between the $m_{i}$’s of (2.4.20) and the $n_{i}$’s of eqs. (2.4.14)
and (2.4.19) is:

$n_{1}\,=\,{m_{1}\over m_{3}}\,\,,\qquad n_{2}\,=\,{m_{2}\over m_{3}}\,\,.$
(2.4.21)

Notice that when $n_{2}=m_{2}=0$ the dependence on $\beta$ disappears and the
configuration is reminiscent of its analog in the conifold case [105]. When
$n_{2}\not=0$ the D7-brane winds infinitely the $\psi$-circle.

####  2.4.2 Energy bound

As it happened in the case of D3- and D5-branes, one can verify that any
solution of the first-order equations (2.4.11) also solves the equations of
motion. We are now going to check that there exists a bound for the energy
which is saturated by the solutions of the first-order equations (2.4.11).
Indeed, let $r(y,\theta)$ and $\psi(\beta,\phi)$ be arbitrary functions. The
hamiltonian density ${\cal H}=\sqrt{-g}$ in this case can be written as:

${\cal H}\,=\,{r^{2}\over
6}\,\sin\theta\,\sqrt{\Bigg{(}r_{\theta}^{2}\,+\,(1-cy)\,\Big{[}H^{2}\,r_{y}^{2}\,+\,{r^{2}\over
6}\Big{]}\Bigg{)}\,\Bigg{(}\Lambda_{\theta}^{2}\,+\,(1-cy)\,\Big{[}H^{2}\,\Lambda_{y}^{2}\,+\,{r^{2}\over
6}\Big{]}\Bigg{)}}\,\,,$ (2.4.22)

where $\Lambda_{y}$ and $\Lambda_{\theta}$ are the functions displayed in eq.
(2.4.12). Let us rewrite this function ${\cal H}$ as ${\cal Z}+{\cal S}$,
where ${\cal Z}$ is given by:

${\cal Z}\,=\,{r^{2}\over
6}\,\sin\theta\,\Bigg{[}\,r_{\theta}\Lambda_{\theta}\,+\,(1-cy)\,\Big{(}H^{2}\,r_{y}\,\Lambda_{y}\,+\,{r^{2}\over
6}\Big{)}\Bigg{]}\leavevmode\nobreak\ .$ (2.4.23)

One can prove that ${\cal Z}$ is a total derivative:

${\cal Z}\,=\,\partial_{\theta}\,{\cal Z}^{\theta}\,+\,\partial_{y}\,{\cal
Z}^{y}\,\,,$ (2.4.24)

where ${\cal Z}^{\theta}$ and ${\cal Z}^{y}$ are:

$\displaystyle{\cal Z}^{\theta}\,=\,-{r^{4}\over
72}\,\Big{[}\,\psi_{\phi}\,+\,(1+c\psi_{\beta})\,\cos\theta\,\Big{]}\,\,,$
$\displaystyle{\cal Z}^{y}\,=\,{r^{4}\over
72}\,(1-cy)\,(y+\psi_{\beta})\sin\theta\,\,.$ (2.4.25)

Moreover, when ${\cal Z}$ is given by (2.4.23), one can demonstrate the bound
(2.2.56). Actually, one can show that the condition ${\cal H}\geq|{\cal Z}|$
is equivalent to the inequality:

$(r_{\theta}-\Lambda_{\theta})^{2}\,+\,H^{2}\,(1-cy)\,(r_{y}-\Lambda_{y})^{2}\,+\,{H^{2}\over
r^{2}}\,(r_{\theta}\,\Lambda_{y}-r_{y}\,\Lambda_{\theta})^{2}\,\geq 0\,\,,$
(2.4.26)

which is always satisfied and is saturated precisely when the BPS equations
(2.4.11) are satisfied. Notice also that ${\cal Z}_{|BPS}$ is positive.

###  2.5 Other interesting possibilities

Let us now look at some other configurations of different branes and cycles
not considered so far. We first consider D3-branes extended along one of the
Minkowski coordinates and along a two-dimensional submanifold of $Y^{p,q}$.
These configurations represent “fat” strings from the point of view of the
gauge theory. We verify in subsection 2.5.1 that an embedding of this type
breaks completely the supersymmetry, although there exist stable non-
supersymmetric “fat” strings. In subsection 2.5.2 we find a new configuration
of a D5-brane wrapping a two-dimensional submanifold, whereas in subsection
2.5.3 we add worldvolume flux to the wall defect solutions of section 2.3. In
subsection 2.5.4 we consider the possibility of having D5-branes wrapping a
three-cycle. We show that such embeddings cannot be supersymmetric, even
though stable solutions of the equations of motion with these characteristics
do exist. In subsection 2.5.5 we analyze the baryon vertex configuration and
we verify that such embedding breaks supersymmetry completely. In subsection
2.5.6 we explore the existence of spacetime filling supersymmetric
configurations of D7-branes by using a set of worldvolume coordinates
different from those used in section 2.4. Finally, in subsection 2.5.7 we show
that a D7-brane can wrap the whole $Y^{p,q}$ space and preserve some fraction
of supersymmetry. It can be thought of as a codimension two defect in the
gauge theory dual.

####  2.5.1 D3-branes on a two-submanifold

Let us take a D3-brane which is extended along one of the spatial directions
of the worldvolume of the D3-branes of the background (say $x^{1}$) and wraps
a two-dimensional cycle. The worldvolume coordinates we will take are

$\xi^{\mu}\,=\,(x^{0},x^{1},\theta,\phi)\,\,,$ (2.5.1)

and we will look for embeddings with $x^{2}$, $x^{3}$, $r$ and $\psi$ constant
and with

$y\,=\,y(\theta,\phi)\,\,,\quad\qquad\beta\,=\,\beta(\theta,\phi)\,\,.$
(2.5.2)

In this case the kappa symmetry matrix acts on $\epsilon$ as:

$\Gamma_{\kappa}\,\epsilon\,=\,-{i\over\sqrt{-g}}\,{r^{2}\over
L^{2}}\,\Gamma_{x^{0}x^{1}}\,\gamma_{\theta\phi}\,\epsilon\,\,.$ (2.5.3)

The expressions of $\gamma_{\theta}$ and $\gamma_{\phi}$ are just those
calculated in section 2.3 and $\gamma_{\theta\phi}\,\epsilon$ can be obtained
by taking the complex conjugate of eq. (2.3.5):

${6\over
L^{2}}\,\gamma_{\theta\phi}\,\epsilon\,=\,\big{[}\,b_{I}^{*}\,+\,b_{15}^{*}\,\Gamma_{15}\,+\,b_{35}^{*}\,\Gamma_{35}\,+\,b_{13}^{*}\,\Gamma_{13}\,\big{]}\,\epsilon\,\,,$
(2.5.4)

where the $b$’s are given in eq. (2.3.6). Since now the complex conjugation
does not act on the spinor $\epsilon$, the only possible projection compatible
with those of the background is the one originated from the term with the unit
matrix in the previous expression. Then, we must require:

$b_{15}=b_{35}=b_{13}=0\,\,.$ (2.5.5)

The conditions $b_{15}=0$ and $b_{35}=0$ are equivalent and give rise to eqs.
(2.3.8) and (2.3.9), which can be integrated as in eq. (2.3.17). Moreover, the
condition $b_{13}=0$ leads to the equation:

${y\over H^{2}}\,y_{\theta}\,=\,\cot\theta\,\,.$ (2.5.6)

The integration of this equation can be straightforwardly performed in terms
of the function $f_{2}(y)$ defined in eq. (1.2.87) and can be written as:

${1\over\sqrt{a-3y^{2}+2cy^{3}}}\,=\,k\sin\theta\,\,,$ (2.5.7)

with $k$ being a constant of integration, which should be related to the
constant $m$ in eq. (2.3.17). However, the dependence of $y$ on $\theta$
written in the last equation does not seem to be compatible with the one of
eq. (2.3.17) (even for $c=0$). Thus, we conclude that there is no solution for
the kappa symmetry condition in this case.

If we forget about the requirement of supersymmetry it is not difficult to
find solutions of the Euler-Lagrange equations of motion of the D3-brane
probe. Indeed, up to irrelevant global factors, the lagrangian for the
D3-brane considered here is the same as the one corresponding to a D5-brane
extended along a two-dimensional submanifold of $Y^{p,q}$. Thus, the
embeddings written in eq. (2.3.17) are stable solutions of the equations of
motion of the D3-brane which represent a “fat” string from the gauge theory
point of view.

####  2.5.2 More D5-branes wrapped on a two-cycle

Let us consider a D5-brane wrapped on a two-cycle and let us choose the
following set of worldvolume coordinates:
$\xi^{\mu}\,=\,(x^{0},x^{1},x^{2},r,\theta,y)$. The embeddings we shall
consider have $x^{3}$ and $\psi$ constant and $\phi=\phi(\theta,y)$,
$\beta=\beta(\theta,y)$. For this case, one has:

$\Gamma_{\kappa}\,\epsilon\,=\,{i\over\sqrt{-g}}\,{r^{2}\over
L^{2}}\,\,\Gamma_{x^{0}x^{1}x^{2}r}\,\,\gamma_{\theta y}\,\epsilon^{*}\,.$
(2.5.8)

where

${6\over L^{2}}\,\gamma_{\theta
y}\,\,\epsilon^{*}\,=\,\Big{(}\,f_{I}\,+\,f_{15}\Gamma_{15}\,+\,f_{35}\Gamma_{35}\,+\,f_{13}\Gamma_{13}\,\Big{)}\,\epsilon^{*}\,\,,$
(2.5.9)

and the different coefficients are given by:

$\displaystyle
f_{I}\,=\,-i\Big{(}\,(1-cy)\,\sin\theta\,\phi_{y}\,-\,c\cos\theta\phi_{\theta}\,+\,\beta_{\theta}\,\Big{)}\,\,,$
$\displaystyle f_{15}\,=\,\sqrt{{2\over 3}}\,{1\over
H}\,\Big{(}\,y\beta_{\theta}\,+\,(1-cy)\cos\theta\phi_{\theta}\,\Big{)}\,+\,i\sqrt{{2\over
3}}\,H\,\cos\theta\,\Big{(}\,\beta_{y}\,\phi_{\theta}\,-\,\beta_{\theta}\,\phi_{y}\,\Big{)}\,\,,$
(2.5.10) $\displaystyle f_{35}\,=\,\sqrt{{2\over