NS Three-form Flux Deformation for the
Critical Non-Abelian Vortex String

A. Yung${}^{\,\,a,\,b}$

aNational Research Center “Kurchatov Institute”, Petersburg Nuclear Physics
Institute, Gatchina, St. Petersburg 188300, Russia
b Higher School of Economics, National Research University, St. Petersburg
194100, Russia

Abstract

It has been shown that non-Abelian solitonic vortex string supported in four-
dimensional (4D) ${\mathcal{N}}=2\;$supersymmetric QCD (SQCD) with the U(2)
gauge group and $N_{f}=4$ quark flavors becomes a critical superstring. This
string propagates in the ten-dimensional space formed by a product of the flat
4D space and an internal space given by a Calabi-Yau noncompact threefold,
namely, the conifold. The spectrum of low lying closed string states in the
associated type IIA string theory was found and interpreted as a spectrum of
hadrons in 4D ${\mathcal{N}}=2\;$SQCD. In particular, the lowest string state
appears to be a massless BPS baryon associated with the deformation of the
complex structure modulus $b$ of the conifold. In the previous work the
deformation of the 10-dimensional background with nonzero Neveu-Schwarz 3-form
flux was considered and interpreted as a switching on a particular choice of
quark masses in 4D SQCD. This deformation was studied to the leading order at
small 3-form flux. In this paper we study the back reaction of the nonzero
3-form flux on the metric and the dilaton introducing ansatz with several warp
factors and solving gravity equations of motion. We show that 3-form flux
produces a potential for the conifold complex structure modulus $b$, which
leads to the runaway vacuum. At the runaway vacuum warp factors disappear,
while the conifold degenerates. In 4D SQCD we relate this to the flow to the
U(1) gauge theory upon switching on quark masses and decoupling of two
flavors.

##  1 Introduction

Non-Abelian vortices were first found in 4D ${\mathcal{N}}=2\;$SQCD with the
gauge group U$(N)$ and $N_{f}\geq N$ flavors of quarks [1, 2, 3, 4]. The non-
Abelian vortex string is 1/2 Bogomolny-Prasad-Sommerfeld (BPS) saturated and,
therefore, has ${\mathcal{N}}=(2,2)\;$supersymmetry on its world sheet. In
addition to four translational moduli of the Abrikosov-Nielsen-Olesen (ANO)
strings [5], the non-Abelian string carries orientational moduli, as well as
the size moduli if $N_{f}>N$ [1, 2, 3, 4] (see [6, 7, 8, 9] for reviews).

It was shown in [10] that the non-Abelian solitonic vortex string in
${\mathcal{N}}=2\;$supersymmetric QCD (SQCD) with the U(N=2) gauge group and
$N_{f}=4$ flavors of quark hypermultiplets becomes a critical superstring. The
dynamics of the internal orientational and size moduli of the non-Abelian
vortex string for the case $N=2$, $N_{f}=4$ is described by the so-called two-
dimensional (2D) weighted CP sigma model, which we denote as
$\mathbb{WCP}(N=2,N_{f}-N=2)$.

For $N_{f}=2N$ this world sheet sigma model becomes conformal. Moreover, for
$N=2$ the number of the orientational and size moduli is six and they can be
combined with four translational moduli to form a ten-dimensional (10D) space
required for a superstring to become critical [10, 11]. In this case the
target space of the world sheet sigma model on the non-Abelian vortex string
is $\mathbb{R}^{4}\times Y_{6}$, where $Y_{6}$ is a non-compact six
dimensional Calabi-Yau (CY) manifold, the conifold [12, 13]. Moreover, the
theory of the critical vortex string at hand was identified as the superstring
theory of type IIA [11]. This allows one to apply the string theory for the
calculation of the spectrum of string states and identify it with a spectrum
of hadrons in 4D ${\mathcal{N}}=2\;$SQCD [11]. Since Non-Abelian vortex
strings are topologically stable and cannot be broken (see [8] for a review)
we focus on the closed strings and consider Kaluza-Klein reduction of 10D
string theory associated with the non-Abelian vortex to 4D.

A version of the string-gauge duality for 4D SQCD was proposed [10]: at weak
coupling this theory is in the Higgs phase and can be described in terms of
quarks and Higgsed gauge bosons, while at strong coupling hadrons of this
theory can be understood as closed string states formed by the non-Abelian
vortex string. We call this approach ”solitonic string-gauge duality”.

The first step of the above program, namely, finding massless string states
was carried out in [11, 14] using supergravity approximation. It turns out
that most of massless modes have non-normalizable wave functions over the non-
compact conifold $Y_{6}$, i.e. they are not localized in 4D and, hence, cannot
be interpreted as dynamical states in 4D SQCD. In particular, the 4D graviton
and unwanted vector multiplet associated with deformations of the Kähler form
of the conifold are absent. However, a single massless BPS hypermultiplet was
found at the self-dual point at strong coupling. It is associated with
deformations of a complex structure of the conifold and was interpreted as a
composite 4D baryon $b$ 111 The definition of the baryonic charge is non-
standard and will be given below in Sec. 2..

Later low lying massive non-BPS 4D states were found in [15, 16] using the
little string theory approach, see [17] for a review.

In the previous work [18] a study of possible flux deformations of the 10D
background for non-Abelian vortex string was initiated. The goal is to look
for flux deformations of the string background which do not destroy
${\mathcal{N}}=2\;$supersymmetry in 4D and interpret them in terms of certain
deformations in SQCD. Fluxes generically induce a potential for CY moduli
lifting flat directions, see, for example, [19] for a review. It is known that
for type IIA CY compactifications the potential for the Kähler form moduli
arise from Ramond-Ramond (RR) even-form fluxes, while the potential for
complex structure moduli is induced by the Neveu-Schwarz (NS) 3-form flux
$H_{3}$ [20, 21]. Since for the conifold case at hand the only modulus
associated with a physical state is the complex structure modulus $b$ we focus
on the NS 3-form flux. It does not break ${\mathcal{N}}=2\;$supersymmetry in
4D theory [20].

In [18] the NS 3-form flux $H_{3}$ was interpreted as switching on quark
masses in 4D SQCD. The reason is that the only scalar potential deformation,
which is allowed in SQCD by ${\mathcal{N}}=2\;$supersymmetry is the mass term
for quarks. Field theory arguments were used to find a particular choice of
nonzero quark masses associated with $H_{3}$.

The flux deformation was studied in [18] to the leading order at small $H_{3}$
which translates into small values of quark masses. In this paper we study the
back reaction of the nonzero 3-form flux on the metric and dilaton. We
introduce ansatz with several warp factors and solve gravity equations of
motion for arbitrary value of $H_{3}$. This allows us to switch on large
masses for certain flavors in 4D SQCD and consider the decoupling limit.

We show that 3-form flux produces a potential for the conifold complex
structure modulus $b$, which leads to the runaway vacuum. At the runaway
vacuum warp factors disappear, while the deformed conifold degenerates. In 4D
SQCD we relate this to the flow to U(1) gauge theory upon switching on quark
masses and decoupling of two flavors.

Note that we assume that the conifold complex structure modulus $b$ is large
enough to make sure that the curvature of the conifold is everywhere small.
This justify the gravity approximation.

The paper is organized follows. In Sec. 2 we briefly review 4D
${\mathcal{N}}=2\;$SQCD and the world sheet sigma model on the non-Abelian
string. Next we review massless baryon $b$ as a deformation of the complex
structure of the conifold. In Sec. 3 we introduce the metric ansatz and solve
gravity equations of motion with nonzero 3-form $H_{3}$ in the limit of large
radial coordinate of the conifold. In Sec. 4 we solve gravity equations for
the deformed conifold and calculate the potential for the complex structure
modulus $b$ in the large $b$ limit. In Sec. 5 we interpret $H_{3}$-form in
terms of quark masses in 4D SQCD. We also discuss the degeneration of the
conifold at the runaway vacuum as a flow of 4D SQCD to
${\mathcal{N}}=2\;$supersymmetric QED (SQED) upon decoupling of two quark
flavors. Sec. 6 summarizes our conclusions.

##  2 Non-Abelian critical vortex string

###  2.1 Four-dimensional ${\mathcal{N}}=2\;$ SQCD

As was already mentioned, non-Abelian vortex-strings were first found in 4D
${\mathcal{N}}=2\;$SQCD with the gauge group U$(N)$ and $N_{f}\geq N$ quark
flavors supplemented by the Fayet-Iliopoulos (FI) term [22] with parameter
$\xi$ [1, 2, 3, 4], see for example, [8] for a detailed review of this theory.
Here, we just mention that at weak coupling $g^{2}\ll 1$, this theory is in
the Higgs phase in which the scalar components of the quark multiplets
(squarks) develop vacuum expectation values (VEVs). These VEVs break the
U$(N)$ gauge group Higgsing all gauge bosons. The Higgsed gauge bosons combine
with the screened quarks to form long ${\mathcal{N}}=2\;$multiplets with mass
$m_{G}\sim g\sqrt{\xi}$.

The global flavor SU$(N_{f})$ is broken down to the so called color-flavor
locked group. The resulting global symmetry is

${\rm SU}(N)_{C+F}\times{\rm SU}(N_{f}-N)\times{\rm U}(1)_{B},$ (2.1)

see [8] for more details.

The unbroken global U(1)B factor above is identified with a baryonic symmetry.
Note that what is usually identified as the baryonic U(1) charge is a part of
our 4D theory gauge group. “Our” U(1)B is an unbroken by squark VEVs
combination of two U(1) symmetries; the first is a subgroup of the flavor
SU$(N_{f})$, and the second is the global U(1) subgroup of U$(N)$ gauge
symmetry.

As was already noted, we consider ${\mathcal{N}}=2\;$SQCD in the Higgs phase:
$N$ squarks condense. Therefore, non-Abelian vortex strings confine monopoles.
In the ${\mathcal{N}}=2\;$4D theory these strings are 1/2 BPS-saturated;
hence, their tension is determined exactly by the FI parameter,

$T=2\pi\xi\,.$ (2.2)

However, as we already mentioned, non-Abelian strings cannot be broken,
therefore monopoles cannot be attached to the string end points. In fact, in
the U$(N)$ theories confined monopoles are junctions of two distinct
elementary non-Abelian strings [23, 3, 4] (see [8] for a review). As a result,
in four-dimensional ${\mathcal{N}}=2\;$SQCD we have monopole-anti-monopole
mesons in which the monopole and anti-monopole are connected by two confining
strings. In addition, in the U$(N)$ gauge theory we can have baryons appearing
as a closed “necklace” configurations of $N\times$(integer) monopoles [8]. For
the U(2) gauge group the massless BPS baryon $b$ found from string theory in
[11] consists of four monopoles [24].

Below we focus on the particular case $N=2$ and $N_{f}=4$ because, as was
mentioned in the Introduction, in this case 4D ${\mathcal{N}}=2\;$SQCD
supports non-Abelian vortex strings which behave as critical superstrings
[10]. Also, for $N_{f}=2N$ the gauge coupling $g^{2}$ of the 4D SQCD does not
run; the $\beta$ function vanishes. However, the conformal invariance of the
4D theory is explicitly broken by the FI parameter $\xi$, which defines VEV’s
of quarks. The FI parameter is not renormalized.

Both stringy monopole-antimonopole mesons and monopole baryons with spins
$J\sim 1$ have masses determined by the string tension, $\sim\sqrt{\xi}$ and
are heavier at weak coupling $g^{2}\ll 1$ than perturbative states with masses
$m_{G}\sim g\sqrt{\xi}$. Thus, they can decay into perturbative states
222Their quantum numbers with respect to the global group (2.1) allow these
decays, see [8]. and in fact at weak coupling we do not expect them to appear
as stable states.

Only in the strong coupling domain $g^{2}\sim 1$ we expect that (at least some
of) stringy mesons and baryons become stable. These expectations were
confirmed in [11, 15] where low lying string states in the string theory for
the critical non-Abelian vortex were found at the self-dual point at strong
coupling.

Below in this paper we introduce quark masses $m_{A}$, $A=1,...4$ assuming
that two first squark flavors with masses $m_{1}$ and $m_{2}$ develop VEVs.

###  2.2 World-sheet sigma model

The presence of the color-flavor locked group SU$(N)_{C+F}$ is the reason for
the formation of non-Abelian vortex strings [1, 2, 3, 4]. The most important
feature of these vortices is the presence of the orientational zero modes. As
was already mentioned, in ${\mathcal{N}}=2\;$SQCD these strings are 1/2 BPS
saturated and preserve ${\mathcal{N}}=(2,2)\;$supersymmetry on the world
sheet.

Let us briefly review the model emerging on the world sheet of the non-Abelian
string [8].

The translational moduli fields are described by the Nambu-Goto action and
decouple from all other moduli. Below we focus on internal moduli.

If $N_{f}=N$ the dynamics of the orientational zero modes of the non-Abelian
vortex, which become orientational moduli fields on the world sheet, are
described by 2D ${\mathcal{N}}=(2,2)\;$supersymmetric ${\mathbb{CP}}(N-1)$
model.

If one adds additional quark flavors, non-Abelian vortices become semilocal –
they acquire size moduli [25]. In particular, for the non-Abelian semilocal
vortex in U(2) ${\mathcal{N}}=2\;$SQCD with four flavors, in addition to the
complex orientational moduli $n^{P}$ (here $P=1,2$), we must add two complex
size moduli $\rho^{K}$ (where $K=3,4$), see [4, 1, 25, 26, 27, 28].

The effective theory on the string world sheet is a two-dimensional
${\mathcal{N}}=(2,2)\;$supersymmetric $\mathbb{WCP}(2,2)\;$model, see review
[8] for details. This model can be defined as a low energy limit of the U(1)
gauge theory [29]. The fields $n^{P}$ and $\rho^{K}$ have charges $+1$ and
$-1$ with respect to the U(1) gauge field. The target space of the
$\mathbb{WCP}(2,2)\;$model is defined by the $D$-term condition

$|n^{P}|^{2}-|\rho^{K}|^{2}={\rm Re}\,\beta,\qquad P=1,2,\qquad K=3,4.$ (2.3)

The number of real bosonic degrees of freedom in the model
$\mathbb{WCP}(2,2)\;$is $8-1-1=6$. Here 8 is the number of real degrees of
freedom of $n^{P}$ and $\rho^{K}$ fields and we subtracted one real constraint
imposed by the the $D$ term condition in (2.3) and one gauge phase eaten by
the Higgs mechanism. As we already mentioned, these six internal degrees of
freedom in the massless limit can be combined with four translational moduli
to form a 10D space needed for a superstring to be critical.

The global symmetry of the world sheet $\mathbb{WCP}(2,2)\;$model is

${\rm SU}(2)\times{\rm SU}(2)\times{\rm U}(1)_{B}\,,$ (2.4)

i.e. exactly the same as the unbroken global group in the 4D theory at $N=2$
and $N_{f}=4$. The fields $n$ and $\rho$ transform in the following
representations:

$n:\quad\left(\textbf{2},\,\textbf{1},\,\frac{1}{2}\right),\qquad\rho:\quad\left(\textbf{1},\,\textbf{2},\,\frac{1}{2}\right)\,.$
(2.5)

Here the global “baryonic” U(1)B group rotates $n$ and $\rho$ fields with the
same phase, see [11] for details.

Twisted masses of $n^{P}$ and $\rho^{K}$ fields coincide with quark masses of
4D SQCD and are given respectively by $m_{P}$ and $m_{K}$, $P=1,2$ and
$K=3,4$, see [8]. Non-zero twisted masses $m_{A}$ break each of the SU(2)
factors in (2.4) down to U(1).

The 2D coupling constant ${\rm Re}\,\beta$ can be naturally complexified to
the complex coupling constant $\beta$ if we include the $\theta$ term in the
action [29]. At the quantum level, the coupling $\beta$ does not run in this
theory. Thus, the $\mathbb{WCP}(2,2)\;$model is superconformal at zero masses
$m_{A}=0$. Therefore, its target space is Ricci flat and (being Kähler due to
${\mathcal{N}}=(2,2)\;$supersymmetry) represents a non-compact Calabi-Yau
manifold, namely the conifold $Y_{6}$, see [13] for a review.

The $\mathbb{WCP}(2,2)\;$model with $m_{A}=0$ was used in [10, 11] to define
the critical string theory for the non-Abelian vortex at hand.

Typically solitonic strings are ”thick” and the effective world sheet theory
has a series of unknown high derivative corrections in powers of
$\partial/m_{G}$. The string transverse size is given by $1/m_{G}$, where
$m_{G}\sim g\sqrt{\xi}$ is a mass scale of the gauge bosons and quarks forming
the string. The string cannot be thin in a weakly coupled 4D SQCD because at
weak coupling $m_{G}\sim g\sqrt{T}$ and $m_{G}^{2}$ is always small in the
units of the string tension $T$, see (2.2).

A conjecture was put forward in [10] that at strong coupling in the vicinity
of a critical value $g_{c}^{2}\sim 1$ the non-Abelian string in the theory at
hand becomes thin, and higher-derivative corrections in the world sheet theory
are absent. This is possible because the low energy $\mathbb{WCP}(2,2)\;$model
already describes a critical string and higher-derivative corrections are not
required to improve its ultra-violet behavior, see [30] for the discussion of
this problem. The above conjecture implies that $m_{G}(g^{2})\to\infty$ at
$g^{2}\to g_{c}^{2}$. As expected the thin string produces linear Regge
trajectories even for small spins [16].

It was also conjectured in [11] that $g_{c}$ corresponds to the value of the
2D coupling constant $\beta=0$. The motivation for this conjecture is that
this value is a self-dual point for the $\mathbb{WCP}(2,2)\;$model. Also
$\beta=0$ is a natural choice because at this point we have a regime change in
the $\mathbb{WCP}(2,2)\;$model. The resolved conifold defined by the $D$ term
condition (2.3) develops a conical singularity at this point. The point
$\beta=0$ corresponds to $\tau_{SW}=1$ in the 4D SQCD, where $\tau_{SW}$ is
the complexified inverse coupling,
$\tau_{SW}=i\frac{8\pi}{g^{2}}+\frac{\theta_{4D}}{\pi}$, where $\theta_{4D}$
is the 4D $\theta$ angle [24].

As we already mentioned in the Introduction a solitonic string-gauge duality
proposed in [10, 11] for 4D SQCD imply that at weak coupling this theory is in
the Higgs phase and can be described in terms of quarks and Higgsed gauge
bosons, while at strong coupling hadrons of this theory can be understood as
closed string states in the string theory on $\mathbb{R}^{4}\times Y_{6}$.

Nonzero twisted masses $m_{A}\neq 0$ define a mass deformation of the
superconformal CY theory on the conifold. Generically quark masses break the
world sheet conformal invariance. The $\mathbb{WCP}(2,2)\;$model with nonzero
$m_{A}$ can no longer be used to define a string theory for the non-Abelian
vortex in the massive 4D SQCD.

###  2.3 Massless 4D baryon

In this section we briefly review the only 4D massless state found in the
string theory of the critical non-Abelian vortex in the massless limit [11].
It is associated with the deformation of the conifold complex structure. As
was already mentioned, all other massless string modes have non-normalizable
wave functions over the conifold. In particular, 4D graviton associated with a
constant wave function over the conifold $Y_{6}$ is absent [11]. This result
matches our expectations since we started with ${\mathcal{N}}=2\;$SQCD in the
flat four-dimensional space without gravity.

We can construct the U(1) gauge-invariant “mesonic” variables

$w^{PK}=n^{P}\rho^{K}.$ (2.6)

These variables are subject to the constraint

${\rm det}\,w^{PK}=0.$ (2.7)

Equation (2.7) defines the conifold $Y_{6}$. It has the Kähler Ricci-flat
metric and represents a non-compact Calabi-Yau manifold [12, 13, 29]. It is a
cone which can be parametrized by the non-compact radial coordinate

$\widetilde{r}^{\,2}={\rm Tr}\,\bar{w}w\,$ (2.8)

and five angles, see [12]. Its section at fixed $\widetilde{r}$ is
$S_{2}\times S_{3}$.

At $\beta=0$ the conifold develops a conical singularity, so both spheres
$S_{2}$ and $S_{3}$ can shrink to zero. The conifold singularity can be
smoothed out in two distinct ways: by deforming the Kähler form or by
deforming the complex structure. The first option is called the resolved
conifold and amounts to keeping a non-zero value of $\beta$ in (2.3). This
resolution preserves the Kähler structure and Ricci-flatness of the metric. If
we put $\rho^{K}=0$ in (2.3) we get the $\mathbb{CP}(1)$ model with the sphere
$S_{2}$ as a target space (with the radius $\sqrt{\beta}$). The resolved
conifold has no normalizable zero modes. In particular, the modulus $\beta$
which becomes a scalar field in four dimensions has non-normalizable wave
function over the $Y_{6}$ and therefore is not dynamical [11].

If $\beta=0$ another option exists, namely a deformation of the complex
structure [13]. It preserves the Kähler structure and Ricci-flatness of the
conifold and is usually referred to as the deformed conifold. It is defined by
deformation of Eq. (2.7), namely,

${\rm det}\,w^{PK}=b\,,$ (2.9)

where $b$ is a complex parameter. Now the sphere $S_{3}$ can not shrink to
zero, its minimal size is determined by $b$.

The modulus $b$ becomes a 4D complex scalar field. The effective action for
this field was calculated in [11] using the explicit metric on the deformed
conifold [12, 31, 32],

$S_{{\rm kin}}(b)=T\int
d^{4}x|\partial_{\mu}b|^{2}\,\log{\frac{\widetilde{R}_{\rm IR}^{2}}{|b|}}\,,$
(2.10)

where $\widetilde{R}_{\rm IR}$ is the maximal value of the radial coordinate
$\widetilde{r}$ introduced as an infrared regularization of the
logarithmically divergent $b$-field norm. Here the logarithmic integral at
small $\widetilde{r}$ is cut off by the minimal size of $S_{3}$, which is
equal to $|b|$.

To avoid confusion we note that in AdS/CFT correspondence the radial
coordinate of internal dimensions has an interpretation of energy. The large
values of this coordinate correspond to the ultraviolet region. In our
approach it is vise-verse. The radial coordinate $\widetilde{r}$ measures
absolute values of products $n^{P}\rho^{K}$ and since $\rho$’s are vortex
string size moduli [25] $\widetilde{r}$ has a 4D interpretation as a distance
from the string axis. In particular, large $\widetilde{r}$ corresponds to the
infrared region.

We see that the norm of the modulus $b$ turns out to be logarithmically
divergent in the infrared. The modes with the logarithmically divergent norm
are at the borderline between normalizable and non-normalizable modes. Usually
such states are considered as “localized” in the 4D. We follow this rule. This
scalar mode is localized near the conifold singularity in the same sense as
the orientational and size zero modes are localized on the vortex string
solution, see [28].

The field $b$ being massless can develop a VEV. Thus, we have a new Higgs
branch in 4D ${\mathcal{N}}=2\;$SQCD which is developed only for the critical
value of the 4D coupling constant $\tau_{SW}=1$ associated with $\beta=0$.

In [11] the massless state $b$ was interpreted as a baryon of 4D
${\mathcal{N}}=2\;$QCD. Let us explain this. From Eq. (2.9) we see that the
complex parameter $b$ (which is promoted to a 4D scalar field) is a singlet
with respect to both SU(2) factors in (2.4), i.e. the global world-sheet
group.333Which is isomorphic to the 4D global group (2.1) for $N=2$,
$N_{f}=4$. What about its baryonic charge? From (2.5) and (2.9) we see that
the $b$ state transforms as

$({\bf 1},\,{\bf 1},\,2).$ (2.11)

In particular it has the baryon charge $Q_{B}(b)=2$.

In type IIA superstring compactifications the complex scalar associated with
deformations of the complex structure of the Calabi-Yau space enters as a 4D
${\mathcal{N}}=2\;$BPS hypermultiplet, see [19] for a review.

On the field theory side we know that if we switch on generic quark masses in
4D SQCD the $b$-baryon becomes massive. Since it is a BPS state its mass is
dictated by its baryonic charge [24],

$m_{b}=|m_{1}+m_{2}-m_{3}-m_{4}|.$ (2.12)

To conclude this section let us present the explicit metric of the singular
conifold (with both $\beta$ and $b$ equal to zero), which will be used in the
next section. It has the form [12]

$ds^{2}_{6}=dr^{2}+\frac{r^{2}}{6}(e_{\theta_{1}}^{2}+e_{\varphi_{1}}^{2}+e_{\theta_{2}}^{2}+e_{\varphi_{2}}^{2})+\frac{r^{2}}{9}e_{\psi}^{2},$
(2.13)

where

$\displaystyle e_{\theta_{1}}=d\theta_{1},\qquad
e_{\varphi_{1}}=\sin{\theta_{1}}\,d\varphi_{1}\,,$ $\displaystyle
e_{\theta_{2}}=d\theta_{2},\qquad
e_{\varphi_{2}}=\sin{\theta_{2}}\,d\varphi_{2}\,,$ $\displaystyle
e_{\psi}=d\psi+\cos{\theta_{1}}d\varphi_{1}+\cos{\theta_{2}}d\varphi_{2}\,.$
(2.14)

Here $r$ is another radial coordinate on the cone while the angles above are
defined at $0\leq\theta_{1,2}<\pi$, $0\leq\varphi_{1,2}<2\pi$,
$0\leq\psi<4\pi$.

The volume integral associated with this metric is

$({\rm Vol})_{Y_{6}}=\frac{1}{108}\int
r^{5}\,dr\,d\psi\,d\theta_{1}\,\sin{\theta_{1}}d\varphi_{1}\,d\theta_{2}\,\sin{\theta_{2}}d\varphi_{2}\,.$
(2.15)

The radial coordinate, $\widetilde{r}$ defined in terms of matrix $w^{PK}$,
see (2.8) is related to $r$ in (2.13) via [12]

$r^{2}=\frac{3}{2}\,\widetilde{r}^{4/3}\,.$ (2.16)

##  3 Gravity equations in the large $r$ limit

Below we switch on NS 3-form flux $H_{3}$ and study its back reaction on the
metric and the dilaton solving gravity equations of motion. As we already
mentioned in the Introduction $H_{3}$ flux produces a potential lifting the
flat direction associated with the conifold complex structure modulus $b$. We
confirm the result obtained in [18] for this potential.

In this section we start with the large $r$ limit and show that the geometry
is smooth and metric warp factors do not develop singularities at
$r\to\infty$. Large $r$ limit means that $r\gg|b|^{1/3}$ (see (2.16)) so for
$H_{3}=0$ we can use the metric of the singular conifold (2.13).

###  3.1 The setup

The bosonic part of the action of the type IIA supergravity in the Einstein
frame is given by

$\displaystyle S_{10D}=\frac{1}{2\kappa^{2}}\,\int
d^{10}x\sqrt{-G}\left\\{R-\frac{1}{2}G^{MN}\,\partial_{M}\Phi\,\partial_{N}\Phi\right.$
$\displaystyle-\left.\frac{e^{-\Phi}}{12}\,H_{MNL}H^{MNL}\right\\},$ (3.1)

where $G_{MN}$ and $\Phi$ are 10D metric and dilaton, the string coupling
$g_{s}=e^{\Phi}$. We also keep only NS 2-form $B_{2}$ with the field strength
$H_{3}=dB_{2}$. We do not consider RR forms here, in particular, the RR 3-form
potential $C_{3}$. For compact CYs the mass term for complex structure moduli
can be generated via topological term $\int\frac{1}{2}\,H_{3}\wedge
C_{3}\wedge dC_{3}$ in the action [20]. However, it was shown in [18] that for
the noncompact case of the conifold this mechanism does not work due to the
non-normalizability of the 4D part of $C_{3}$.

Einstein’s equations of motion following from action (3.1) have the form

$R_{MN}=\frac{1}{2}\,\partial_{M}\Phi\,\partial_{N}\Phi+\frac{e^{-\Phi}}{4}\,H_{MAB}H_{N}^{AB}-\frac{e^{-\Phi}}{48}\,G_{MN}\,H_{3}^{2},$
(3.2)

while the equation for the dilaton reads

$G^{MN}D_{M}\Phi D_{N}\Phi+\frac{e^{-\Phi}}{12}\,H_{3}^{2}=0.$ (3.3)

Finally the equation for the NS 3-form is

$d(e^{-\Phi}\ast H_{3})=0,$ (3.4)

where $\ast$ denotes the Hodge star.

We will see below that we need to introduce four warp factors to solve
Einstein equations. Our ansatz for the metric is

$ds^{2}_{10}=T\,h^{-1/2}_{4}(r)\,\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\,g_{mn}dx^{m}dx^{n},$
(3.5)

where $\mu,\nu=0,...,3$ are indices of the 4D space and $\eta_{\mu\nu}$ is the
flat Minkowski metric with signature $(-1,1,1,1)$, while $m,n=5,...10$ are
indices of the 6D internal space. Here internal coordinates $x^{m}$ defined to
be dimensionless to match the dimension of scalar fields in the world sheet
$\mathbb{WCP}(2,2)\;$model. We also introduced the string tension $T$ (see
(2.2)) in (3.5) to fix dimensions.

The internal space has a conifold metric deformed by three warp factors

$g_{mn}dx^{m}dx^{n}=h^{1/2}_{6}(r)\left\\{a(r)\,dr^{2}+\frac{r^{2}}{6}(e_{\theta_{1}}^{2}+e_{\varphi_{1}}^{2}+e_{\theta_{2}}^{2}+e_{\varphi_{2}}^{2})+\frac{r^{2}}{9}\omega(r)\,e_{\psi}^{2}\right\\},$
(3.6)

see (2.13) and we assume that warp factors $h_{4}$, $h_{6}$, $a$ and $\omega$
depend only on the radial coordinate $r$. If $H_{3}=0$ all warp factors are
equal to unity and the 10D space has the structure $\mathbb{R}^{4}\times
Y_{6}$.

###  3.2 NS 3-form at large $r$

We will see below that solution of gravity equations of motion in the large
$r$ limit can be expanded in powers of $\mu^{2}/r^{4}$, where $\mu$
parametrize the $H_{3}$ flux. To find its behavior we can use a perturbation
theory in powers of the above parameter. At the first step we solve equations
of motion for $H_{3}$ form using undeformed conifold metric. This was done in
[18].

Let us define two real 3-forms on $Y_{6}$,

$\alpha_{3}\equiv\frac{dr}{r}\wedge\left(e_{\theta_{1}}\wedge
e_{\varphi_{1}}-e_{\theta_{2}}\wedge e_{\varphi_{2}}\right)$ (3.7)

and

$\beta_{3}\equiv e_{\psi}\wedge\left(e_{\theta_{1}}\wedge
e_{\varphi_{1}}-e_{\theta_{2}}\wedge e_{\varphi_{2}}\right)$ (3.8)

They are both closed [33, 34],

$d\alpha_{3}=0,\qquad d\beta_{3}=0,$ (3.9)

Moreover, using the conifold metric (2.13) to the leading order one can check
that their 10D-duals are given by

$\displaystyle\ast\alpha_{3}\approx-\frac{T^{2}}{3}\,dx^{0}\wedge dx^{1}\wedge
dx^{2}\wedge dx^{3}\wedge\beta_{3},$ $\displaystyle\ast\beta_{3}\approx
3\,T^{2}\,dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}\wedge\alpha_{3}.$
(3.10)

The above relations ensure that both 10D-dual forms are also closed.

$d\ast\alpha_{3}=0,\qquad d\ast\beta_{3}=0.$ (3.11)

Two solutions for $H_{3}$-form found in [18] are

$H_{3}\approx\mu_{1}\alpha_{3}+\frac{\mu_{2}}{3}\,\beta_{3},$ (3.12)

where $\mu_{1}$ and $\mu_{2}$ are two independent real parameters, while the
factor $\frac{1}{3}$ is introduced for convenience. This $H_{3}$-form satisfy
both Bianchi identity and equations of motion (3.4), where the dilaton is
considered as a constant to the leading order in $\mu^{2}/r^{4}$.

3-Forms (3.7) and (3.8) form a basis similar to the simplectic basis of
harmonic $\alpha$ and $\beta$ 3-forms for compact CYs, see for example review
[19]. In particular,

$\int_{Y_{6}}\alpha_{3}\wedge\alpha_{3}=\int_{Y_{6}}\beta_{3}\wedge\beta_{3}=0,$
(3.13)

while

$\int_{Y_{6}}\alpha_{3}\wedge\beta_{3}\sim-\int\frac{dr}{r}\sim-\log{\frac{R_{\rm
IR}^{3}}{|b|}}.$ (3.14)

Here $R_{\rm IR}$ is the maximal value of the radial coordinate $r$ introduced
to regularize the infrared logarithmic divergence, while at small $r$ the
integral is cut off by the minimal size of $S_{3}$ which is equal to $|b|$.
Note that this logarithm is similar to the one, which determines the metric
for the $b$-baryon in (2.10) 444Note that $R_{\rm
IR}^{3}\sim\widetilde{R}_{\rm IR}^{2}$, see (2.16)..

###  3.3 Warp factors at large $r$

For Minkowski indices $\mu,\nu=0,1,2,3$ Einstein’s equations (3.2) read

$R_{\mu\nu}=-\frac{\eta_{\mu\nu}}{48}\,\frac{e^{-\Phi}}{h_{4}^{1/2}}\,H_{3}^{2},$
(3.15)

where Ricci components for the ansatz (3.5), (3.6) can be calculated using
results of [35]

$R_{\mu\nu}=\frac{\eta_{\mu\nu}}{4ah_{4}^{1/2}h_{6}^{1/2}}\left\\{\frac{1}{h_{4}}\Delta
h_{4}+\frac{h_{6}^{\prime}h_{4}^{\prime}}{h_{6}h_{4}}-2\frac{(h_{4}^{\prime})^{2}}{h_{4}^{2}}-\frac{1}{2}\,\frac{a^{\prime}h_{4}^{\prime}}{ah_{4}}+\frac{1}{2}\,\frac{\omega^{\prime}h_{4}^{\prime}}{\omega
h_{4}}\right\\}.$ (3.16)

Here prime denotes the derivative with respect to $r$ and $\Delta$ is the
Laplacian calculated using the conifold metric (2.13).

Using expression in (3.16) we can compare Einstein’s equations for Minkowski
indices (3.15) with the dilaton equation (3.3). Rewriting the latter one as

$\Delta\Phi+\left(\frac{h_{6}^{\prime}}{h_{6}}-\frac{h_{4}^{\prime}}{h_{4}}-\frac{1}{2}\,\frac{a^{\prime}}{a}+\frac{1}{2}\,\frac{\psi^{\prime}}{\omega}\right)\Phi^{\prime}=-\frac{e^{-\Phi}}{12}\,ah_{6}^{1/2}\,H_{3}^{2}$
(3.17)

it is easy to see that it is identical to the equation (3.15) upon
substitution

$\Phi=\Phi_{0}+\ln{h_{4}},$ (3.18)

where $\Phi_{0}$ is a constant value of the dilaton present at $H_{3}=0$.

Let us now continue studying the equation (3.15). At the first non-trivial
order in the parameter $\mu^{2}/r^{4}$ all non-linearities in the expression
in (3.16) can be neglected and it reduces simply to

$R_{\mu\nu}\approx\frac{\eta_{\mu\nu}}{4}\,\Delta h_{4}.$ (3.19)

This gives for the Minkowski part of Einstein’s equations

$\Delta h_{4}\approx-\frac{e^{-\Phi_{0}}}{12}\,H_{3}^{2},$ (3.20)

where $H_{3}^{2}$ can be calculated using the conifold metric and we used only
the constant part of the dilaton $\Phi_{0}$ at this order. We have

$e^{-\Phi_{0}}\,H_{3}^{2}=3!\,72\,\frac{\mu_{1}^{2}+\mu_{2}^{2}}{g_{s}}\,\frac{1}{r^{6}}=2^{4}\,3^{3}\,\frac{\mu_{1}^{2}+\mu_{2}^{2}}{g_{s}}\,\frac{1}{r^{6}}$
(3.21)

where $g_{s}=e^{\Phi_{0}}$, while $72/r^{4}$ say, for the first solution for
$H_{3}$ (proportional to $\mu_{1}$ in (3.12)) comes from
$g^{\theta_{1}\theta_{1}}g^{\varphi_{1}\varphi_{1}}$ and
$g^{\theta_{2}\theta_{2}}g^{\varphi_{2}\varphi_{2}}$.

Then equation (3.20) gives

$h_{4}=1+\frac{9}{g_{s}}\,\frac{\mu_{1}^{2}+\mu_{2}^{2}}{r^{4}}\,\log{\frac{r}{|b|^{1/3}}}+O(\mu^{4}/r^{8}).$
(3.22)

up to a non-logarithmic term proportional to $\mu^{2}/r^{4}$ which we set to
zero.

Consider now Einstein’s equations with internal indices. Let index $\alpha$
($\beta$) denote differentials
$e_{\theta_{1}},e_{\varphi_{1}},e_{\theta_{2}},e_{\varphi_{2}}$. Then we can
calculate Christoffel symbols with $r$ indices, namely

$\displaystyle\Gamma^{r}_{\alpha\beta}=-\frac{g^{(c)}_{\alpha\beta}}{a}\left(\frac{1}{r}+\frac{1}{4}\,\frac{h_{6}^{\prime}}{h_{6}}\right),\qquad\Gamma^{r}_{\psi\psi}=-\frac{g^{(c)}_{\psi\psi}}{a}\left(\frac{1}{r}+\frac{1}{4}\,\frac{h_{6}^{\prime}}{h_{6}}+\frac{1}{2}\omega^{\prime}\right)\,,$
$\displaystyle\Gamma^{\beta}_{r\alpha}=\Gamma^{\beta}_{\alpha
r}=\delta^{\beta}_{\alpha}\left(\frac{1}{r}+\frac{1}{4}\,\frac{h_{6}^{\prime}}{h_{6}}\right),\qquad\Gamma^{\psi}_{r\psi}=\Gamma^{\psi}_{\psi
r}=\frac{1}{r}+\frac{1}{4}\,\frac{h_{6}^{\prime}}{h_{6}}+\frac{1}{2}\omega^{\prime}\,,$
$\displaystyle\Gamma^{r}_{rr}=\frac{1}{2}\,\frac{a^{\prime}}{a}+\frac{1}{4}\,\frac{h_{6}^{\prime}}{h_{6}},\qquad\Gamma^{n}_{rr}=\Gamma^{r}_{rn}=\Gamma^{r}_{nr}=0,\qquad
n\neq r,$ (3.23)

where $g^{(c)}_{\alpha\beta}$ and $g^{(c)}_{\psi\psi}$ denote the conifold
metric (2.13).

Using these formulas we find nonzero Ricci components at the leading order in
$\mu^{2}/r^{4}$. We have

$\displaystyle R_{\alpha\beta}\approx
g^{(c)}_{\alpha\beta}\left\\{\frac{4}{r^{2}}\,(a-1)-\frac{1}{4}\Delta
h_{6}-\frac{1}{r}\,h_{6}^{\prime}+\frac{1}{r}\,h_{4}^{\prime}+\frac{1}{2r}\,a^{\prime}-\frac{1}{2r}\,\omega^{\prime}\right\\},$
$\displaystyle R_{\psi\psi}\approx
g^{(c)}_{\psi\psi}\left\\{\frac{4}{r^{2}}\,(a-1)-\frac{1}{4}\Delta
h_{6}-\frac{1}{r}\,h_{6}^{\prime}+\frac{1}{r}\,h_{4}^{\prime}+\frac{1}{2r}\,a^{\prime}-\frac{2}{r}\,\omega^{\prime}-\frac{1}{2}\,\omega^{\prime\prime}\right\\},$
$\displaystyle R_{rr}\approx-\frac{1}{4}\Delta
h_{6}-h_{6}^{\prime\prime}+h_{4}^{\prime\prime}+\frac{5}{2r}\,a^{\prime}-\frac{1}{r}\,\omega^{\prime}-\frac{1}{2}\,\omega^{\prime\prime}\,.$
(3.24)

Here we used that Ricci tensor is zero if all warp factors are equal to unity
and the dependence on $h_{4}$ can be found using formulas in [35].

Now calculating r.h.s.’s for Einstein’s equations (3.2) we get for the first
solution proportional to $\mu_{1}$ in (3.12)

$\displaystyle
R_{\alpha\beta}=\frac{1}{48}\,g^{(c)}_{\alpha\beta}e^{-\Phi_{0}}\,(H_{3}^{(1)})^{2},$
$\displaystyle
R_{\psi\psi}=-\frac{g^{(c)}_{\psi\psi}}{48}\,e^{-\Phi_{0}}\,(H_{3}^{(1)})^{2}$
$\displaystyle R_{rr}=\frac{1}{16}\,e^{-\Phi_{0}}\,(H_{3}^{(1)})^{2},$ (3.25)

where $(H_{3}^{(1)})^{2}$ is given by (3.21) with $\mu_{2}=0$.

For the second solution in (3.12) (proportional to $\mu_{2}$ ) we have

$\displaystyle
R_{\alpha\beta}=\frac{1}{48}\,g^{(c)}_{\alpha\beta}e^{-\Phi_{0}}\,(H_{3}^{(2)})^{2},$
$\displaystyle
R_{\psi\psi}=\frac{g^{(c)}_{\psi\psi}}{16}\,e^{-\Phi_{0}}\,(H_{3}^{(2)})^{2}$
$\displaystyle R_{rr}=-\frac{1}{48}\,e^{-\Phi_{0}}\,(H_{3}^{(2)})^{2},$ (3.26)

where $(H_{3}^{(2)})^{2}$ is given by (3.21) with $\mu_{1}=0$.

Above equations together with expressions (3.24) and solution for $h_{4}$
(3.22) determine three warp factors $h_{6}$, $a$ and $\omega$ at the leading
order. For the first solution for $H_{3}$ we have

$\displaystyle
h_{6}^{(1)}=1+\frac{9}{g_{s}}\,\frac{\mu_{1}^{2}}{r^{4}}\,\log{\frac{r}{|b|^{1/3}}}-\frac{9}{5}\,\frac{1}{g_{s}}\,\frac{\mu_{1}^{2}}{r^{4}}+\cdots,$
$\displaystyle
a^{(1)}=1-\frac{9}{10}\,\frac{1}{g_{s}}\,\frac{\mu_{1}^{2}}{r^{4}}+\cdots,$
$\displaystyle\omega^{(1)}=1+\frac{9}{2}\,\frac{1}{g_{s}}\,\frac{\mu_{1}^{2}}{r^{4}}+\cdots,$
(3.27)

where dots stand for sub-leading terms of order of $\mu^{4}/r^{8}$. Warp
factors for the second solution for $H_{3}$ have the form

$\displaystyle
h_{6}^{(2)}=1+\frac{9}{g_{s}}\,\frac{\mu_{2}^{2}}{r^{4}}\,\log{\frac{r}{|b|^{1/3}}}+\frac{9}{5}\,\frac{1}{g_{s}}\,\frac{\mu_{2}^{2}}{r^{4}}+\cdots,$
$\displaystyle
a^{(2)}=1+\frac{9}{10}\,\frac{1}{g_{s}}\,\frac{\mu_{2}^{2}}{r^{4}}+\cdots,$
$\displaystyle\omega^{(2)}=1-\frac{9}{2}\,\frac{1}{g_{s}}\,\frac{\mu_{2}^{2}}{r^{4}}+\cdots.$
(3.28)

Finally solutions (3.18) and (3.22) give for the dilaton

$e^{(\Phi-\Phi_{0})}=1+\frac{9}{g_{s}}\,\frac{\mu_{1}^{2}+\mu_{2}^{2}}{r^{4}}\,\log{\frac{r}{|b|^{1/3}}}+\cdots$
(3.29)

We see that warp factors and dilaton have smooth behavior at large $r$ and can
be found order by order in the parameter $\mu^{2}/r^{4}$ using perturbation
theory in gravity equations. The region of validity of the above solutions is

$r\gg|b|^{1/3}\gg\mu^{1/2}.$ (3.30)

To conclude this section we would like to comment on a subtlety in solving
equations (3.25) and (3.26). In fact these equations do not determine
coefficients in non-logarithmic terms proportional to $1/r^{4}$ for $h_{6}$
and $a$ separately. Denoting these coefficients $\chi$ and $A$ respectively we
find that the first and the third equations in (3.25) and (3.26) give the same
conditions for them, namely

$\chi^{(1)}+\frac{A^{(1)}}{2}=-\frac{9}{4}\,\frac{\mu_{1}^{2}}{g_{s}},\qquad\chi^{(2)}+\frac{A^{(2)}}{2}=\frac{9}{4}\,\frac{\mu_{2}^{2}}{g_{s}}$
(3.31)

for (3.25) and (3.26) respectively. The resolution of this puzzle is related
to the possibility of redefinition of the conifold radial coordinate $r$. Let
us put $H_{3}=0$ so the metric is reduced to the conifold one in (2.13).
However, we can redefine $r$ at the relevant order,

$r=f(r^{\prime})=r^{\prime}\left(1+\frac{\alpha}{r^{\prime 4}}\right),$ (3.32)

where $\alpha$ is a constant. This gives

$r^{2}\approx r^{\prime 2}\left(1+\frac{2\alpha}{r^{\prime 4}}\right),\qquad
dr^{2}\approx dr^{\prime 2}\left(1-\frac{6\alpha}{r^{\prime 4}}\right)$ (3.33)

which in terms of the new coordinate $r^{\prime}$ imply nontrivial warp
factors

$h_{6}=1+\frac{4\alpha}{r^{\prime 4}},\qquad a=1-\frac{8\alpha}{r^{\prime 4}}$
(3.34)

or nonzero coefficients

$\chi=4\alpha,\qquad A=-8\alpha.$ (3.35)

Now we see that the combination which enters Eqs. (3.31) is zero on this
solution,

$\chi+\frac{A}{2}=0.$ (3.36)

Thus, nontrivial solutions of the above equation are related to the
possibility of $r$ redefinition.

To fix the definition of $r$ we require that the orthogonal combination to the
one which enters (3.36) should be zero, namely

$\frac{\chi}{2}-A=0.$ (3.37)

This condition together with equation (3.31) gives coefficients

$\displaystyle\chi^{(1)}=-\frac{9}{5}\,\frac{\mu_{1}^{2}}{g_{s}},\qquad
A^{(1)}=-\frac{9}{10}\,\frac{\mu_{1}^{2}}{g_{s}},$
$\displaystyle\chi^{(2)}=\frac{9}{5}\,\frac{\mu_{2}^{2}}{g_{s}},\qquad
A^{(2)}=\frac{9}{10}\,\frac{\mu_{2}^{2}}{g_{s}}$ (3.38)

for two solutions for $H_{3}$ respectively, which we presented in (3.27) and
(3.28).

###  3.4 The scalar potential

To find the scalar potential induced by 3-form flux $H_{3}$ we substitute the
solution of the gravity equations found above into the 10D action (3.1). The
trace of the Einstein’s equations (3.2) reads

$R-\frac{1}{2}G^{MN}\,\partial_{M}\Phi\,\partial_{N}\Phi-\frac{e^{-\Phi}}{12}\,H_{3}^{2}=0.$
(3.39)

Substituting this into Eq. (3.1) we get the action calculated on the solution,

$S_{10D}=\frac{1}{2\kappa^{2}}\,\int
d^{10}x\sqrt{-G}\left\\{-\frac{e^{-\Phi}}{24}\,H_{3}^{2}\right\\},$ (3.40)

where $2\kappa^{2}=(2\pi)^{3}g_{s}^{2}$ in our conventions.

This leads to the potential for $b$-baryon (complex structure modulus $b$ of
the conifold) in 4D SQCD,

$V(b)=\frac{T^{2}}{(2\pi)^{3}g_{s}^{2}}\,\int
d^{6}x\sqrt{g_{6}}\;\frac{e^{-\Phi}}{24}\,H_{3}^{2},$ (3.41)

where the string tension $T$ appears due to our normalization of the Minkowski
part of the metric, see (3.5). Here the integral is taken over the internal 6D
space and $g_{6}$ is the determinant of the 6D metric. To the leading order we
can neglect warp factors and calculate the above integral using the conifold
metric (2.13). Using Eqs. (2.15) and (3.21) we get

$V(b)=\frac{4}{3}\,\frac{T^{2}}{g_{s}^{3}}(\mu_{1}^{2}+\mu_{2}^{2})\int\frac{dr}{r}=\frac{4}{9}\,\frac{T^{2}}{g_{s}^{3}}(\mu_{1}^{2}+\mu_{2}^{2})\,\log{\frac{R^{3}_{IR}}{|b|}},$
(3.42)

where $R_{IR}$ is the infrared cutoff for the radial coordinate $r$, while
modulus $b$ plays a role of the ultraviolet cutoff at small $r$, cf. (3.14).
This potential was calculated in [18]. Note, that the same infrared logarithm
determines the metric (2.10) for the $b$-baryon. If we take into account warp
factors in the integrand in (3.41) this would give finite corrections to the
potential of order of

$T^{2}\,\frac{\mu^{4}}{|b|^{4/3}},$ (3.43)

which are negligible as compared to the logarithmic term.

We see that the Higgs branch for $b$ is lifted by $H_{3}$ flux deformation and
we have a runaway vacuum with VEV

$\langle|b|\rangle\to R_{\rm IR}^{3}\to\infty.$ (3.44)

However, our solution of gravity equations is found in this section using the
metric of the singular conifold and therefore is valid at $r\gg|b|^{1/3}$.
Thus, the potential (3.42) cannot be trusted at $|b|\sim R_{\rm IR}^{3}$ where
the logarithm becomes small. In the next section we consider the region of
$r\sim|b|^{1/3}$ and confirm our conclusion in (3.44) that the VEV of the
baryon $b$ tends to infinity.

##  4 Gravity equations for the deformed conifold

The result for the potential (3.42) suggests that we have a runaway vacuum and
VEV of $b$ becomes infinitely large. To confirm this in this section we study
gravity equations with nonzero $H_{3}$-flux on the deformed conifold assuming
that the radial coordinate $r\sim|b|^{1/3}$. Anticipating the runaway behavior
(3.44) we still keep the second condition in (3.30),

$\mu\ll|b|^{2/3}$ (4.1)

###  4.1 Metric of the deformed conifold

In this section we briefly review the metric of the deformed conifold. It has
the form [12, 31, 32]

$\displaystyle ds_{6}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,|b|^{2/3}\,K(\tau)\left\\{\frac{1}{3K^{3}(\tau)}\left(d\tau^{2}+e_{\psi}^{2}\right)+\cosh^{2}{\frac{\tau}{2}}\,\left(g_{3}^{2}+g_{4}^{2}\right)\right.$
(4.2) $\displaystyle+$
$\displaystyle\left.\sinh^{2}{\frac{\tau}{2}}\,\left(g_{1}^{2}+g_{2}^{2}\right)\right\\},$

where angle differentials are defined as

$\displaystyle g_{1}=-\frac{1}{\sqrt{2}}\,(e_{\phi_{1}}+e_{3}),\qquad
g_{2}=\frac{1}{\sqrt{2}}\,(e_{\theta_{1}}-e_{4}),$ $\displaystyle
g_{3}=-\frac{1}{\sqrt{2}}\,(e_{\phi_{1}}-e_{3}),\qquad
g_{4}=\frac{1}{\sqrt{2}}\,(e_{\theta_{1}}+e_{4}),$ (4.3)

while

$e_{3}=\cos{\psi}\sin{\theta_{2}}\,d\varphi_{2}-\sin{\psi}\,d\theta_{2},\qquad
e_{4}=\sin{\psi}\sin{\theta_{2}}\,d\varphi_{2}+\cos{\psi}\,d\theta_{2},$ (4.4)

see also (2.14).

Here

$K(\tau)=\frac{(\sinh{2\tau}-2\tau)^{1/3}}{2^{1/3}\sinh{\tau}}$ (4.5)

and the new radial coordinate $\tau$ is defined as

$\widetilde{r}^{2}=|b|\,\cosh{\tau}=\left(\frac{2}{3}\right)^{\frac{3}{2}}\,r^{3}.$
(4.6)

In the limit of large $\tau$ the metric (4.2) reduces to the metric (2.13) of
the singular conifold.

Results of the previous section show that we have a runaway vacuum with
$|b|\sim R_{IR}^{3}$ so we are interested in the metric (4.2) in the limit of
small $\tau$, $\tau\ll 1$. In this limit the metric of the deformed conifold
takes the form

$\displaystyle ds_{6}^{2}|_{\tau\to 0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,|b|^{2/3}\left(\frac{2}{3}\right)^{\frac{1}{3}}\,\left\\{\frac{1}{2}\,d\tau^{2}+\frac{1}{2}\,e_{\psi}^{2}+g_{3}^{2}+g_{4}^{2}\right.$
(4.7) $\displaystyle+$
$\displaystyle\left.\frac{\tau^{2}}{4}\,\left(g_{1}^{2}+g_{2}^{2}\right)\right\\}.$

The last term here corresponds to the collapsing sphere $S_{2}$, while the
sphere $S_{3}$ associated with three angular terms in the first line has a
fixed radius in the limit $\tau\to 0$ [12, 32]. The radial coordinate $r$
approaches its minimal value with

$r^{3}|_{min}=\left(\frac{3}{2}\right)^{\frac{3}{2}}\,|b|$ (4.8)

at $\tau=0$.

The square root of the determinant of the metric

$\sqrt{g_{6}}\sim|b|^{2}\,\cosh^{2}{\frac{\tau}{2}}\sinh^{2}{\frac{\tau}{2}}|_{\tau\to
0}\sim|b|^{2}\,\tau^{2}$ (4.9)

vanishes at $\tau=0$, which shows the degeneration of the conifold metric.

###  4.2 NS 3-form at small $\tau$

We will see below that leading non-trivial contributions to warp factors are
proportional to $\mu^{2}\tau^{2}/|b|^{4/3}$. At the first step of the
perturbation theory we can neglect them and look for solutions for $H_{3}$
flux using the metric of the deformed conifold summarized in the previous
section and a constant dilaton, $\Phi\approx\Phi_{0}$.

One solution was found in [18] using the ansatz suggested in [32] for the type
IIB flux compactification on the deformed conifold. The ansatz reads

$H_{3}=p^{\prime}\,d\tau\wedge g_{1}\wedge g_{2}+k^{\prime}\,d\tau\wedge
g_{3}\wedge g_{4}-\frac{1}{2}\,(p-k)\,e_{\psi}\wedge(g_{1}\wedge
g_{3}+g_{2}\wedge g_{4}),$ (4.10)

where $p$ and $k$ are functions of the radial coordinate $\tau$. Here primes
denote derivatives with respect to $\tau$. The 3-form above is closed so the
Bianchy identity is satisfied.

At large $\tau$ $p^{\prime}\approx k^{\prime}\to\mu_{1}/3$ and using the
identity [32]

$e_{\theta_{1}}\wedge e_{\varphi_{1}}-e_{\theta_{2}}\wedge
e_{\varphi_{2}}=g_{1}\wedge g_{2}+g_{3}\wedge g_{4}.$ (4.11)

it is easy to show that this solution tends to the first solution for $H_{3}$
(proportional to $\mu_{1}$) in (3.12).

For small $\tau$ equation of motion (3.4) for $H_{3}$ was solved in [18] at
the leading order using the metric of the deformed conifold and a constant
dilaton. The result is $k\approx\mu_{1}\tau$ and $p\approx-\mu_{1}\tau^{5}/80$
so the solution takes the form

$H_{3}^{(1)}\approx\mu_{1}\gamma_{3}$ (4.12)

up to an overall constant, where we introduced a 3-form

$\gamma_{3}=d\tau\wedge g_{3}\wedge g_{4}-\frac{\tau^{4}}{16}\,d\tau\wedge
g_{1}\wedge g_{2}+\frac{\tau}{2}\,e_{\psi}\wedge(g_{1}\wedge g_{3}+g_{2}\wedge
g_{4}).$ (4.13)

Now let us find another solution which at large $\tau$ tends to the second
solution in (3.12) (proportional to $\mu_{2}$). To do so we use the ansatz,

$H_{3}=l(\tau)\,e_{\psi}\wedge g_{1}\wedge g_{2}+n(\tau)\,e_{\psi}\wedge
g_{3}\wedge g_{4}+q(\tau)\,d\tau\wedge(g_{1}\wedge g_{3}+g_{2}\wedge g_{4}),$
(4.14)

where $l$, $n$ and $q$ are functions of $\tau$. Using identity (4.11) and [32]

$d(g_{1}\wedge g_{3}+g_{2}\wedge g_{4})=e_{\psi}\wedge(g_{1}\wedge
g_{2}-g_{3}\wedge g_{4})$ (4.15)

we calculate

$dH_{3}=l^{\prime}\,d\tau\wedge e_{\psi}\wedge g_{1}\wedge
g_{2}+n^{\prime}\,d\tau\wedge e_{\psi}\wedge g_{3}\wedge
g_{4}-q(\tau)\,d\tau\wedge e_{\psi}\wedge(g_{1}\wedge g_{2}-g_{3}\wedge
g_{4}).$ (4.16)

Now Bianchy identity $dH_{3}=0$ leads to

$l^{\prime}-q=0,\qquad n^{\prime}+q=0.$ (4.17)

A solution to these equations with $q=0$, $l=n=\mu_{2}/3$ corresponds to the
second solution in (3.12) at large $\tau$. Let us find the extrapolation of
this solution to small $\tau$. For nonzero $q$ we have
$l^{\prime}=-n^{\prime}$ and setting the integration constant to zero we get
$l=-n$. The ansatz for $H_{3}$ acquires the form

$H_{3}=l\,\left(e_{\psi}\wedge g_{1}\wedge g_{2}-e_{\psi}\wedge g_{3}\wedge
g_{4}\right)+l^{\prime}\,d\tau\wedge(g_{1}\wedge g_{3}+g_{2}\wedge g_{4}).$
(4.18)

Calculating 10D dual of (4.18) using metric in (4.7) we get

$\displaystyle\ast H_{3}$ $\displaystyle=$ $\displaystyle-T^{2}\,dx^{0}\wedge
dx^{1}\wedge dx^{2}\wedge dx^{3}\wedge\left\\{\frac{4l}{\tau^{2}}\,d\tau\wedge
g_{3}\wedge g_{4}\right.$ (4.19) $\displaystyle-$
$\displaystyle\left.\frac{l\tau^{2}}{4}\,d\tau\wedge g_{1}\wedge
g_{2}+l^{\prime}\,e_{\psi}\wedge(g_{1}\wedge g_{3}+g_{2}\wedge
g_{4})\right\\}.$

Then the equation of motion (3.4) reads

$\displaystyle d\ast H_{3}=T^{2}\,dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge
dx^{3}\wedge$
$\displaystyle\left\\{-\left(\frac{2l}{\tau^{2}}+\frac{l\tau^{2}}{8}-l^{\prime\prime}\right)\,d\tau\wedge
e_{\psi}\wedge(g_{1}\wedge g_{3}+g_{2}\wedge g_{4})\right\\}=0$ (4.20)

where we used the identity [32]

$d(g_{1}\wedge g_{2}-g_{3}\wedge g_{4})=-e_{\psi}\wedge(g_{1}\wedge
g_{3}+g_{2}\wedge g_{4}).$ (4.21)

Equation (4.20) gives

$l^{\prime\prime}-\frac{2l}{\tau^{2}}=0,$ (4.22)

where we neglect $\tau^{2}$-term at small $\tau$.

Eq. (4.22) gives $l\approx\mu_{2}\tau^{2}/4$ up to a constant and we write
down the second solution for $H_{3}$ in the form

$H_{3}^{(2)}\approx\mu_{2}\delta_{3},$ (4.23)

where

$\delta_{3}=\frac{\tau^{2}}{4}\,e_{\psi}\wedge\left(g_{1}\wedge
g_{2}-g_{3}\wedge g_{4}\right)+\frac{\tau}{2}\,d\tau\wedge(g_{1}\wedge
g_{3}+g_{2}\wedge g_{4}).$ (4.24)

Both 3-forms $\gamma_{3}$ and $\delta_{3}$ are closed. Moreover, their 10D
duals are given by (see [18] and (4.19))

$\displaystyle\ast\gamma_{3}\approx T^{2}\,dx^{0}\wedge dx^{1}\wedge
dx^{2}\wedge dx^{3}\wedge\delta_{3},$
$\displaystyle\ast\delta_{3}\approx-T^{2}\,dx^{0}\wedge dx^{1}\wedge
dx^{2}\wedge dx^{3}\wedge\gamma_{3}.$ (4.25)

The above relations ensure that both 10D-dual forms are also closed.

$d\ast\gamma_{3}=0,\qquad d\ast\delta_{3}=0.$ (4.26)

Much in the same way as forms (3.7) and (3.8) 3-forms (4.13) and (4.24)
satisfy relations

$\int_{Y_{6}}\gamma_{3}\wedge\gamma_{3}=\int_{Y_{6}}\delta_{3}\wedge\delta_{3}=0,$
(4.27)

while

$\int_{Y_{6}}\gamma_{3}\wedge\delta_{3}\sim\int d\tau\tau^{2}$ (4.28)

at small $\tau$.

To conclude this section, we note that at $\tau=0$ the first solution solution
(4.12) tends to a constant

$H_{3}^{(1)}(\tau=0)=\mu_{1}\,d\tau\wedge g_{3}\wedge g_{4},$ (4.29)

which we impose as boundary conditions at $S_{3}$, which does not shrinks at
$\tau=0$. These boundary conditions ensure a non-zero solution for
$H^{(1)}_{3}$.

Similarly for the second solution (4.23) we fix its derivative with respect to
$\tau$ as boundary conditions at $S_{3}$ at $\tau=0$,

$\frac{\partial}{\partial\tau}\,H_{3}^{(2)}(\tau=0)=\frac{\mu_{2}}{2}\,d\tau\wedge(g_{1}\wedge
g_{3}+g_{2}\wedge g_{4}).$ (4.30)

###  4.3 Warp factors at small $\tau$

In this section we study the back reaction of the two solutions for $H_{3}$
flux found above on the metric and dilaton to the leading order in
$\mu^{2}\tau^{2}/|b|^{4/3}$. Our ansatz for the metric is given by (3.5) where
$h_{4}$ now is a function of $\tau$, while

$\displaystyle g_{mn}dx^{m}dx^{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,|b|^{2/3}\left(\frac{2}{3}\right)^{\frac{1}{3}}\,\left\\{h_{1}^{1/2}(\tau)\,\left(\frac{1}{2}\,a(\tau)\,d\tau^{2}+\frac{1}{2}\,e_{\psi}^{2}+g_{3}^{2}+g_{4}^{2}\right)\right.$
(4.31) $\displaystyle+$
$\displaystyle\left.h_{2}^{1/2}(\tau)\,\frac{\tau^{2}}{4}\,\left(g_{1}^{2}+g_{2}^{2}\right)\right\\},$

where the metric of the deformed conifold (4.7) is further deformed with
another three warp factors $h_{1}$, $h_{2}$ and $a$, which are assumed to be
functions of $\tau$. Here we also assume the limit of small $\tau$, $\tau\ll
1$.

For Minkowski indices $\mu,\nu=0,1,2,3$ Einstein’s equations (3.2) has the
form (3.15), where using results from [35] we calculate

$R_{\mu\nu}=\frac{\eta_{\mu\nu}\,g^{\tau\tau}_{c}}{4ah_{4}^{1/2}h_{1}^{1/2}}\left\\{\frac{1}{h_{4}}\Delta
h_{4}+\frac{1}{2}\frac{h_{1}^{\prime}h_{4}^{\prime}}{h_{1}h_{4}}+\frac{1}{2}\frac{h_{2}^{\prime}h_{4}^{\prime}}{h_{2}h_{4}}-2\frac{(h_{4}^{\prime})^{2}}{h_{4}^{2}}-\frac{1}{2}\,\frac{a^{\prime}h_{4}^{\prime}}{ah_{4}}\right\\},$
(4.32)

where $\Delta$ is the Laplacian calculated using metric (4.7). Here and below
$g^{mn}_{c}$, $g_{mn}^{c}$ denote the deformed conifold metric (4.7), for
example

$g^{\tau\tau}_{c}\approx\frac{2^{5/3}3^{1/3}}{|b|^{2/3}}.$ (4.33)

At the first order all non-linearities in (4.32) can be neglected and
Einstein’s equations (3.15) reduce to

$\Delta h_{4}\approx-\frac{e^{-\Phi_{0}}\,g_{\tau\tau}^{c}}{12}\,H_{3}^{2},$
(4.34)

where $H_{3}^{2}$ can be calculated using the deformed conifold metric and we
used only the constant part of the dilaton $\Phi_{0}$ at this order. We have

$e^{-\Phi_{0}}\,H_{3}^{2}\approx
2^{4}\,3^{3}\,\frac{\mu_{1}^{2}+\mu_{2}^{2}}{g_{s}}\,\frac{1}{|b|^{2}}\,\left[1+O(\tau^{2})\right]$
(4.35)

where, say, for the first solution for $H_{3}$ in (4.12) only first and the
last terms in $\gamma_{3}$ contribute at the leading order in $\tau$.

Then equation (4.34) gives

$h_{4}=1-\frac{3^{2/3}}{2^{2/3}}\,\frac{\mu_{1}^{2}+\mu_{2}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}\left[1+O(\tau^{2})\right].$
(4.36)

Much in the same way as in the large $r$ limit it is easy to see that the
dilaton equation reduces to the equation (3.15) on the solution (3.18).

Consider now Einstein’s equations with internal indices. Let index $a$ ($b$)
denote differentials $e_{\psi},g_{3},g_{4}$, while index $i$ ($j$) denote
$g_{1},g_{2}$. Then we can calculate leading contributions to Christoffel
symbols with $\tau$ indices at small $\tau$, namely

$\displaystyle\Gamma^{\tau}_{ij}=-g^{(c)}_{ij}\,\frac{g^{\tau\tau}_{c}\,h_{2}^{1/2}}{ah_{1}^{12}}\left(\frac{1}{\tau}+\frac{1}{4}\,\frac{h_{2}^{\prime}}{h_{2}}\right),\qquad\Gamma^{\tau}_{ab}=-g^{(c)}_{ab}\,\frac{g^{\tau\tau}_{c}}{a}\,\frac{1}{4}\,\frac{h_{1}^{\prime}}{h_{1}}\,,$
$\displaystyle\Gamma^{j}_{\tau
i}=\Gamma^{j}_{i\tau}=\delta^{j}_{i}\left(\frac{1}{\tau}+\frac{1}{4}\,\frac{h_{2}^{\prime}}{h_{2}}\right),\qquad\Gamma^{b}_{\tau
a}=\Gamma^{b}_{a\tau}=\delta^{b}_{a}\,\frac{1}{4}\,\frac{h_{1}^{\prime}}{h_{1}}\,,$
$\displaystyle\Gamma^{\tau}_{\tau\tau}=\frac{1}{2}\,\frac{a^{\prime}}{a}+\frac{1}{4}\,\frac{h_{1}^{\prime}}{h_{1}},\qquad\Gamma^{n}_{\tau\tau}=\Gamma^{\tau}_{\tau
n}=\Gamma^{\tau}_{n\tau}=0,\qquad n\neq r,$ (4.37)

Then nonzero components of the Ricci tensor to the leading order in $\tau$
take the form

$\displaystyle R_{ij}\approx
g^{(c)}_{ij}\,g^{\tau\tau}_{c}\left\\{\frac{1}{\tau^{2}}\,\left(\frac{ah_{1}^{1/2}}{h_{2}^{1/2}}-1\right)-\frac{1}{4}\Delta
h_{2}-\frac{1}{2\tau}\,h_{2}^{\prime}-\frac{1}{2\tau}\,h_{1}^{\prime}+\frac{1}{\tau}\,h_{4}^{\prime}+\frac{1}{2\tau}\,a^{\prime}\right\\},$
$\displaystyle R_{ab}\approx
g^{(c)}_{ab}\,\,g^{\tau\tau}_{c}\left\\{-\frac{1}{4}\Delta h_{1}\right\\},$
$\displaystyle R_{\tau\tau}\approx-\frac{1}{2}\Delta h_{2}+\frac{1}{4}\Delta
h_{1}-h_{1}^{\prime\prime}+h_{4}^{\prime\prime}+\frac{1}{\tau}\,a^{\prime}\,.$
(4.38)

Here again we used that Ricci tensor is zero if all warp factors are equal to
unity and the dependence on $h_{4}$ is found using formulas in [35].

For the first solution in (4.12) r.h.s.’s of Einstein’s equations (3.2) take
the form

$\displaystyle
R_{ij}=\frac{1}{2^{4}\,3^{2}}\,g^{(c)}_{ij}e^{-\Phi_{0}}\,(H_{3}^{(1)})^{2},$
$\displaystyle
R_{ab}=\frac{5}{2^{4}\,3^{2}}\,g^{(c)}_{ab}\,e^{-\Phi_{0}}\,(H_{3}^{(1)})^{2}$
$\displaystyle
R_{\tau\tau}=\frac{1}{2^{4}\,3^{2}}\,g^{(c)}_{\tau\tau}e^{-\Phi_{0}}\,(H_{3}^{(1)})^{2},$
(4.39)

where $(H_{3}^{(1)})^{2}$ is given by (4.35) with $\mu_{2}=0$.

Solutions to these equations are given by

$\displaystyle
h_{1}^{(1)}=1-\frac{5}{2^{2/3}\,3^{1/3}}\,\frac{\mu_{1}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}+\cdots$
$\displaystyle
h_{2}^{(1)}=1-\frac{5}{13}\,\frac{3^{2/3}}{2^{2/3}}\,\frac{\mu_{1}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}+\cdots$
$\displaystyle
a^{(1)}=1+\frac{5}{13}\,\frac{2^{1/3}}{3^{1/3}}\,\frac{\mu_{1}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}+\cdots\,,$
(4.40)

where dots stand for corrections in powers of $\tau$ and powers of
$\mu^{2}/|b|^{4/3}$.

For the second solution (4.23) r.h.s.’s of Einstein’s equations (3.2) has the
form

$\displaystyle
R_{ij}=\frac{5}{2^{4}\,3^{2}}\,g^{(c)}_{ij}e^{-\Phi_{0}}\,(H_{3}^{(2)})^{2},$
$\displaystyle
R_{ab}=\frac{1}{2^{4}\,3^{2}}\,g^{(c)}_{ab}\,e^{-\Phi_{0}}\,(H_{3}^{(2)})^{2}$
$\displaystyle
R_{\tau\tau}=\frac{5}{2^{4}\,3^{2}}\,g^{(c)}_{\tau\tau}e^{-\Phi_{0}}\,(H_{3}^{(2)})^{2},$
(4.41)

where $(H_{3}^{(2)})^{2}$ is given by (4.35) with $\mu_{1}=0$.

For this case solutions take the form

$\displaystyle
h_{1}^{(2)}=1-\frac{1}{2^{2/3}\,3^{1/3}}\,\frac{\mu_{2}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}+\cdots$
$\displaystyle
h_{2}^{(2)}=1-\frac{3^{2/3}}{2^{2/3}}\,\frac{\mu_{2}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}+\cdots$
$\displaystyle
a^{(2)}=1+\frac{2^{1/3}}{3^{1/3}}\,\frac{\mu_{2}^{2}}{g_{s}}\,\frac{\tau^{2}}{|b|^{4/3}}+\cdots\,.$
(4.42)

Note that much in the same way as for the large $r$ case the first and the
third Einstein’s equations in (4.39) and (4.41) coincides and give rise to
conditions for the same combination $(3c_{2}-2\tilde{A})$, where $c_{2}$ and
$\tilde{A}$ are coefficients in front of $\tau^{2}$ for $h_{2}$ and $a$. As we
explained before this is due to the possibility of redefinition of the radial
coordinate ($\tau$ in the present case), see Sec. 3.3. To fix the definition
of $\tau$ we require that the orthogonal combination to the one above is zero,
$(2c_{2}+3\tilde{A})=0$. This gives warp factors $h_{2}$ and $a$ presented in
(4.40) and (4.40).

We see that warp factors in (4.40) and (4.42) as well as $h_{4}$ (4.36) and
the dilaton (3.18) have smooth behavior at small $\tau$ and do not develop
singularities provided $\mu^{2}\ll|b|^{4/3}$. They can be found order by order
at $\mu^{2}\ll|b|^{4/3}$ using perturbation theory in gravity equations.

###  4.4 The scalar potential at large $|b|$

To find the scalar potential for the complex structure modulus $b$ we
substitute solutions found above in this section into Eq. (3.41). At the
leading order in $\mu^{2}/|b|^{4/3}$ we can neglect warp factors and use
metric of the deformed conifold together with leading order expression (4.35).
Using (4.9) at small $\tau$ we get

$V(b)={\rm
const}\,(\mu_{1}^{2}+\mu_{2}^{2})\,\frac{T^{2}}{g_{s}^{3}}\,\tau_{{\rm
max}}^{3},$ (4.43)

where $\tau_{{\rm max}}$ is the infrared cutoff with respect to the radial
coordinate $\tau$ related to $R_{\rm IR}$ as follows

$|b|\cosh(\tau_{{\rm max}})=\left(\frac{2}{3}\right)^{\frac{3}{2}}\,R_{\rm
IR}^{3},$ (4.44)

see (4.6). This potential was obtained in [18] for the first solution for
$H_{3}$ proportional to $\mu_{1}$.

As we already explained, we expect that in our runaway vacuum $b$ is large,
close to $R_{\rm IR}$, therefore $\tau_{{\rm max}}$ is small. Expanding
$\cosh{\tau}$ at small $\tau$ we get

$\tau_{{\rm max}}\sim\sqrt{\frac{\left(\frac{2}{3}\right)^{\frac{3}{2}}R_{\rm
IR}^{3}-|b|}{|b|}}.$ (4.45)

This gives the potential for the baryon $b$ at large $|b|$

$V(b)={\rm
const}\,\left(\mu_{1}^{2}+\mu_{2}^{2}\right)\,\frac{T^{2}}{g_{s}^{3}}\,\left[\frac{\left(\frac{2}{3}\right)^{\frac{3}{2}}R_{\rm
IR}^{3}-|b|}{|b|}\right]^{\frac{3}{2}}.$ (4.46)

We see that to minimize the potential above $|b|$ becomes large and approaches
the infrared cutoff,

$\langle|b|\rangle=\left(\frac{2}{3}\right)^{\frac{3}{2}}\,R_{\rm
IR}^{3}\to\infty.$ (4.47)

As we expected earlier in Sec. 3.4, we get a runaway vacuum.

The corrections to the potential (4.46) arise from taking into account higher
powers of $\tau$ in the deformed conifold metric as well as from warp factors
and go in powers of $\tau_{{\rm max}}$ and in powers of $\mu^{2}/|b|^{4/3}$
respectively. Both type of corrections disappear at the runaway vacuum (4.47).

In fact, $\tau_{{\rm max}}^{3}$ which enters (4.43) is the volume of the three
dimensional cone bounded by the sphere $S_{2}$ of the conifold with the
maximum radius $\tau_{{\rm max}}$. It shrinks to zero as $b$ tends to its VEV
(4.47). To avoid singularities we can regularize the size of $S_{2}$
introducing small non-zero $\beta$, which makes the conifold ”slightly
resolved”, see (2.3). We take the limit $\beta\to 0$ at the last step. Then
the value of the potential and all its derivatives vanish in the vacuum (4.47)
at $|b|=\langle|b|\rangle$, for example

$V(b)|_{|b|=\langle|b|\rangle}={\rm
const}\,\left(\mu_{1}^{2}+\mu_{2}^{2}\right)\,\frac{T^{2}}{g_{s}^{3}}\,\frac{\beta^{3}}{R_{\rm
IR}^{9/2}}\to 0.$ (4.48)

In particular, the mass term for $b$ is zero.

Absence of warp factors and vanishing of the potential $V(b)$ together with
all its derivatives at the runaway vacuum confirms that
${\mathcal{N}}=2\;$supersymmetry is not broken in 4D SQCD.

To summarize, the $H_{3}$-form flux produces following effects.
(i) The Higgs branch of the baryon $b$ in 4D SQCD is lifted.
(ii) The vacuum is of the runaway type $\langle|b|\rangle\to\infty$.
(ii) At the runaway vacuum warp factors tend to unity and the geometry becomes
that of the deformed conifold.
(iii) At the runaway vacuum the radial coordinate $\tau$ and the sphere
$S_{2}$ of the conifold degenerates, while the radius of the sphere $S_{3}$
tends to infinity.

We will interpret this degeneration in terms of ${\mathcal{N}}=2\;$SQCD in the
next section.

##  5 Interpretation in terms of 4D SQCD

###  5.1 3-form flux in terms of quark masses

As we already mentioned in the Introduction $H_{3}$-form flux was interpreted
in terms of 4D SQCD in [18] as switching on quark masses. The motivation is
that the only scalar potential deformation allowed in 4D SQCD by
${\mathcal{N}}=2\;$supersymmetry is the mass term for quarks. Field theory
arguments were used in [18] to find a particular choice of nonzero quark
masses associated with $H_{3}$. In this section we briefly review this
interpretation. For $N_{f}=4$ we have four complex mass parameters. However, a
shift of the complex scalar $a$, a superpartner of the U(1) gauge field,
produces an overall shift of quark masses. Thus, in fact we have three
independent complex mass parameters in our 4D SQCD. For example, we can choose
three mass differences

$m_{1}-m_{2},\qquad m_{3}-m_{4},\qquad m_{1}-m_{3}$ (5.1)

as independent parameters.

On the string theory side our solution (3.12) for the 3-form $H_{3}$ is
parametrized by two real parameters $\mu_{1}$ and $\mu_{2}$. Thus, we expect
that non-zero $H_{3}$-flux can be interpreted in terms of a particular choice
of quark masses, subject to two complex constraints.

One constraint follows from (2.12). We have seen in the previous subsection
that $H_{3}$ does not produces a mass term for the $b$-baryon. This ensures
that

$m_{1}+m_{2}-m_{3}-m_{4}=0.$ (5.2)

Another constraint is

$m_{1}m_{2}-m_{3}m_{4}=0.$ (5.3)

It is imposed to avoid infinite VEV of $\sigma$ (a scalar superpartner of the
U(1) gauge field), which would costs an infinite energy in the world sheet
$\mathbb{WCP}(2,2)\;$model at large $b$, see [18] for details.

Solving two constraints above leads to two options for the choice of the quark
masses

$m_{3}=m_{1},\qquad m_{4}=m_{2}$ (5.4)

and

$m_{3}=m_{2},\qquad m_{4}=m_{1}.$ (5.5)

These two options are essentially the same, up to permutation of quarks
$q^{3}$ and $q^{4}$. Let us choose the first option in (5.4).

The arguments above lead to the conclusion that the $H_{3}$-flux can be
interpreted in terms of the single mass difference $(m_{1}-m_{2})$. We define
a complex parameter $\mu$ and identify [18]

$\mu\equiv\mu_{1}+i\mu_{2}={\rm
const}\,\sqrt{\frac{g_{s}^{3}}{T}}\,(m_{1}-m_{2}),\qquad m_{3}=m_{1},\qquad
m_{4}=m_{2}.$ (5.6)

The potential (3.42) calculated at large $r$, $r\gg|b|^{1/3}$ takes the form

$V(b)={\rm const}\,T\,|m_{1}-m_{2}|^{2}\,\log{\frac{R^{3}_{IR}}{|b|}}.$ (5.7)

Similar substitution can be done for the large-$b$ potential (4.46).

###  5.2 Degeneration of the conifold and flow to SQED

Since our solution to the gravity equations is valid at

$\frac{|\mu|^{2}}{|b|^{4/3}}\sim\frac{|m_{1}-m_{2}|^{2}}{T\,|b|^{4/3}}\ll 1$
(5.8)

and VEV of $b$ goes to infinity ( see (4.47)) we can use our solution at
arbitrary large fixed values of $(m_{1}-m_{2})$. In particular, if we take
$|m_{1}-m_{2}|\gg\sqrt{\xi}$ in 4D SQCD keeping the constraint (5.4) non-
Abelian degrees of freedom decouple and U(2) gauge theory flows to
${\mathcal{N}}=2\;$supersymmetric QED with the gauge group U(1) and $N_{f}=2$
quark flavors. Off-diagonal gauge fields together with two quark flavors
acquire large masses $\sim|m_{1}-m_{2}|$ 555In addition to masses $m_{G}\sim
g\sqrt{\xi}$ due to the Higgs mechanism, see [8] for a review. and decouple.

What happens to the non-Abelian vortex string upon this decoupling? The string
survives, but transforms into an Abelian string. To see this note, that if we
say, increase masses $m_{2}=m_{4}$ keeping $m_{1}=m_{3}=0$ fields $n^{2}$ and
$\rho^{4}$ decouple in the world sheet $\mathbb{WCP}(2,2)\;$model on the
string and it flows into $\mathbb{WCP}(1,1)\;$model. The $D$-term condition
(2.3) now reads

$|n^{1}|^{2}-|\rho^{3}|^{2}={\rm Re}\,\beta\,.$ (5.9)

The number of real degrees of freedom in $\mathbb{WCP}(1,1)\;$model is
$4-1-1=2$ where 4 is the number of real degrees of freedom of $n^{1}$ and
$\rho^{3}$ and we subtract 2 due to the $D$-term constraint (5.9) and the U(1)
phase eaten by the Higgs mechanism.

Physically $\mathbb{WCP}(1,1)\;$model describes an Abelian semilocal vortex
string supported in ${\mathcal{N}}=2\;$supersymmetric U(1) gauge theory with
$N_{f}=2$ quark flavors. This vortex has no orientational moduli, but it has
one complex size modulus $\rho^{3}$, see [25, 26, 27]. Thus, we see that upon
switching on $(m_{1}-m_{2})$ a non-Abelian string flows to an Abelian one.

The low energy $\mathbb{WCP}(1,1)\;$model is also conformal. Moreover, it was
shown in [36] that in the non-linear sigma model formulation it flows to a
free theory on $\mathbb{R}^{2}$ in the infrared. Thus, in fact, switching on
$(m_{1}-m_{2})$ with constraint (5.4) does not break the conformal invariance
on the world sheet. It just reduces the number of degrees of freedom
transforming a non-Abelian string into an Abelian one. The string theory which
one would associate with the $\mathbb{WCP}(1,1)\;$model is non-critical.

The field theory physics described above supports our interpretation of the
$H_{3}$-form flux on the conifold in terms of quark masses. On the string
theory side switching on $(m_{1}-m_{2})$ is reflected in the degeneration of
the conifold, which effectively reduces its dimension. Also in the limit
$|b|\to\infty$ the radius of the sphere $S_{3}$ of the conifold becomes
infinite and it tends to a flat three dimensional space. This matches the
field theory result [36] that $\mathbb{WCP}(1,1)\;$model flows to a free
theory in the infrared. It would be tempting to interpret the extra coordinate
of the sphere $S_{3}$ of the conifold in the limit $|b|\to\infty$ as a
Liouville coordinate for a non-critical string associated with the
$\mathbb{WCP}(1,1)\;$model. This is left for a future work.

We also note that Eq. (2.10) suggests that massless stringy baryon $b$
acquires infinitely strong interactions at the runaway vacuum (4.47) and the
associated physics is no longer under analytic control.

##  6 Conclusions

In this paper we considered a deformation of the string theory for the
critical non-Abelian vortex supported in ${\mathcal{N}}=2\;$SQCD with gauge
group U(2) and $N_{f}=4$ quark flavors with NS 3-form flux building on the
results of our previous paper [18]. Using supergravity approach we found a
solution for the 3-form $H_{3}$ and its back reaction on the conifold metric
and the dilaton at the first non-trivial order in the parameter
$\mu^{2}/|b|^{4/3}$. The non-zero 3-form $H_{3}$ generates a potential for the
complex structure modulus $b$ of the conifold, which is interpreted as a
massless BPS baryonic hypermultiplet in 4D SQCD at strong coupling. This
potential lifts the Higgs branch formed by VEVs of $b$ and leads to a runaway
vacuum for $b$, $\langle|b|\rangle\to\infty$. The warp factors disappear at
this runaway vacuum.

Following [18] we interpret the 3-form $H_{3}$ as a quark mass deformation of
4D SQCD. Field theory arguments are used to relate the 3-form $H_{3}$ to the
quark mass difference $(m_{1}-m_{2})$, subject to the constraint (5.4), see
(5.6).

At the run away vacuum the conifold degenerates to lower dimensions. This
qualitatively matches with a flow to the $\mathbb{WCP}(1,1)\;$model on the
string world sheet, expected if one switches on the mass difference
$(m_{1}-m_{2})$ and decouples one $n$-field and one $\rho$-field. In 4D SQCD
this corresponds to a flow to ${\mathcal{N}}=2\;$supersymmetric QED with two
charged flavors.

## Acknowledgments

The author is grateful to P. Gavrylenko, A. Gorsky, E. Ievlev, A. Marshakov
and M. Shifman for very useful and stimulating discussions. This work was
supported by Skolkovo Institute of Science and Technology and by the
Foundation for the Advancement of Theoretical Physics and Mathematics ”BASIS”,
Grant No. 22-1-1-16-1.

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