# Three-dimensional Yang-Mills Chern-Simons theory from D3-brane background
with D-instantons

###### Abstract

By constructing the configuration of D3-branes with D(-1)-branes as
D-instantons, we study the three-dimensional Yang-Mills Chern-Simons theory in
holography. Due to the presence of the D-instantons, the D7-branes with
discrepant embedding functions are able to be introduced in order to include
the fundamental fermions (as flavors) and the Chern-Simons term (at very low
energy) in the dual theory. The vacuum structure at zero temperature is
studied in the soliton background and it illustrates the topological phase
transition in the presence of instantons. Moreover, since the
confinement/deconfinement phase transition could be holographically identified
as the Hawking-Page transition in the bulk, we accordingly calculate the
critical temperature of the deconfinement phase transition by collecting the
bulk onshell action as the thermodynamical free energy. On the other hand, we
evaluate the difference of the entanglement entropy in slab configuration by
using the RT formula since the confinement may also be characterized by the
entanglement entropy. Altogether we find the behavior of the critical
temperature is in qualitative agreement with the behavior of the critical
length determined by the entanglement entropy which implies the entanglement
entropy could indeed be a character of the confinement in our setup and the
D3-D(-1) system would be a remarkable approach to study the three-dimensional
gauge theory.

Si-wen Li111Email<EMAIL_ADDRESS>Sen-kai Luo222Email:
<EMAIL_ADDRESS>Mu-zhi Tan333Email<EMAIL_ADDRESS>
_Department of Physics, School of Science,_

_Dalian Maritime University,_

_Dalian 116026, China_

###### Contents

  1. 1 Introduction
  2. 2 Three-dimensional theory from confining geometry
  3. 3 Flavor and Chern-Simons brane
    1. 3.1 Adding flavors
    2. 3.2 Embedding of the CS D7-brane
  4. 4 Vacuum structure of the dual theory
  5. 5 Entanglement entropy and confinement
  6. 6 Summary and discussion

##  1 Introduction

In the past decade, a specific class of Chern-Simons (CS) theory involving
fundamental matters with $N_{f}$ flavors attracts many interests and the large
$N$ ’t Hooft limit of such theories with a $U\left(N\right)$ gauge symmetry
has been studied in detail [1, 2, 3, 4, 5, 6, 7]. It is conjectured there
would be a conformal field theory living in the limitation of vanishing flavor
mass. Along this direction, some evidence was found which may be supportable
to the conjecture e.g. boson/fermion duality [8], level/rank duality [9]. On
the other hand, the large $N$ field theory can be analyzed holographically by
using string theory according to gauge/gravity duality or AdS/CFT
correspondence which has become very famous nowadays [10, 11]. So
investigation of CS theory in holography naturally becomes a remarkable topic.

The most simple way to build a holographic duality for three dimensional (3d)
non-supersym-metric theory is to construct the configuration of coincident
$N_{c}$ D3-branes, i.e. to compactify one of the spatial directions of the
D3-brane on a supersymmetry breaking circle $S^{1}$ [12, 13]. Such
configuration of $N_{c}$ D3-branes at large $N_{c}$ limit is described by IIB
supergravity and it has been used to study the 3d Yang-Mills Chern-Simons YMCS
theory or 3d QCD (QCD3) in holography e.g. the vacuum structure [14, 15] and
the quantum Hall Effect [16]. However, the D3-brane background does not
include a dynamical D-instanton, or the axion field in terms of the field
theory, which plays an important role to involve the topological properties in
the dual theory. As it is known, instantons in quantum chromodynamics (QCD)
are topologically nontrivial excitations of the vacuum which contribute to the
thermodynamics of the theory and related to chiral symmetry breaking [17, 18].
Based on gauge/gravity duality, a holographic way to include instantons is to
take into account the configuration of coincident $N_{c}$ D3-branes with
$N_{\mathrm{D}}$ D(-1)-branes as D-instantons smeared homogeneously in the
worldvolume of the D3-branes [19, 20, 21]. The system of D3-branes with
dynamical D-instantons succeeds to describe the 4d QCD-like theories and the
presence of instantons could reflect some topological properties of the theory
[22, 23, 24, 25].

Therefore in this note we focus on constructing the D3-D-instanton
configuration to study the 3d confining Yang-Mills or QCD-like theory with a
CS term by following the standard technique [12, 13] in string theory. Since
the soliton AdS solution is chosen, it will lead to an area law in the dual
theory due to the asymptotics of the Wilson loop, so confinement is expected
to exhibit in the dual theory at very low temperature. To further include
matters, the flavors are introduced by embedding a stack of probe D7-branes
[26] and the spontaneous breaking of chiral symmetry can be identified as the
separation of $N_{f}$ flavor branes in IR region depicted by the IIB
supergravity as the holographic description of [27]. Due to the presence of
the D-instantons, additional D7-branes as CS branes with a discrepancy
embedding function can be put into the background, accordingly at very low
energy, the dual theory is expected to be a pure CS theory. We specifically
analyze the behaviors of the embedding functions of the various D7-branes and
evaluate the associated free energy density by including both flavors and CS
term. The result illustrates the topological phase transition which is
enhanced by the D-instantons and it seems to be qualitatively consistent with
that the presence of CS term topologically contributes to the mass parameter
[28] and the D0-D4 approach in 4d [29, 30, 31, 32].

Moreover, we evaluate the critical temperature of the deconfinement phase
transition in this holographic setup since the dual theory is expected to
exhibit the confinement. While the deconfining geometry in holography is less
clear, the phase transition at a critical temperature can be identified as the
bubble/black brane transition, or namely Hawking-Page transition, in the bulk
which is very suggestive of the deconfinement transition [33, 34, 35, 36].
Following this idea, we collect the total onshell action as the holographic
free energy consisted of the bulk fields (as the color sector), the flavor and
the CS branes, in the soliton and black brane background respectively. By
comparing the free energy, we find the charge density of D-instantons indeed
contributes to the thermodynamics and the critical temperature is decreased by
the D-instantons through the flavor and CS branes. However, at large-$N_{c}$
limit, the critical temperature becomes independent on the D-instantons since
the backreaction of the CS branes is not included in this setup. To parallely
examine whether our analysis for the deconfinement transition is consistent,
we on the other side calculate the entanglement entropy holographically in
this system since the entanglement entropy may also be a characteristic tool
to detect the confinement in the dual theory [37, 38, 39, 40]. Using the RT
formula [41] with the slab geometry, the holographic entanglement entropy can
be obtained and it exhibits a first order phase transition at a critical
length who behaves similarly as the critical temperature evaluated by the
thermodynamics. In this sense, we believe the configuration of D3-branes with
D-instantons would be a remarkable approach to study 3d gauge theory.

The outline of this note is as follows. In Section 2, we construct the black
D3-D(-1) solution to obtain a confining geometry and examine the dual theory
by a probe D3-brane. Afterwards, we analyze the embedding function of the
flavor and CS branes, compute the free energy by including the flavor and CS
term in Section 3 then evaluate the vacuum structure of the dual theory in
Section 4. In Section 5, we investigate the deconfinement phase transition by
comparing the free energy of this model thermodynamically and compute the
variation of the entanglement entropy as a parallel verification. Summary and
comments are given in the last section.

##  2 Three-dimensional theory from confining geometry

In this section, we will briefly review the system of $N_{c}$ D3-branes with
$N_{\mathrm{D}}$ D-instantons i.e. the D(-1)-branes, then construct the
background geometry for a confining dual theory at large-$N_{c}$ limit.

The D3-D(-1) brane system is geometrically represented by a deformed D3-brane
solution with a nontrivial Ramond-Ramond (R-R) scalar field $C_{0}$ which is
recognized as a marginal “bound state” of D3-branes with $N_{\mathrm{D}}$
smeared D(-1)-branes. We denote the $N_{c}$ D3-branes as color branes. In the
large $N_{c}$ limit, the 10 dimensional (10d) type IIB supergravity action, as
the effective action, describes the low-energy dynamics of this system which
in string frame is given as,

$S_{\mathrm{IIB}}=\frac{1}{2\kappa_{10}^{2}}\int
d^{10}x\sqrt{-g}\left[e^{-2\Phi}\left(\mathcal{R}+4\partial\Phi\cdot\partial\Phi\right)-\frac{1}{2}\left|F_{1}\right|^{2}-\frac{1}{2}\left|F_{5}\right|^{2}\right].$
(2.1)

Here $2\kappa_{10}^{2}=\left(2\pi\right)^{7}l_{s}^{8}$ is the 10d gravity
coupling constant, $l_{s},g_{s}$ is respectively the length and the coupling
constant of string and $F_{1,5}$ is the field strength of the R-R zero and
four form $C_{0,4}$. The near-horizon solution of non-extremal D3-branes with
a non-trivial $C_{0}$ in string frame reads,

$\displaystyle ds^{2}$
$\displaystyle=e^{\frac{\phi}{2}}\left\\{\frac{r^{2}}{R^{2}}\left[-f_{T}\left(r\right)dt^{2}+d\mathbf{x}\cdot
d\mathbf{x}\right]+\frac{1}{f\left(r\right)}\frac{R^{2}}{r^{2}}dr^{2}+R^{2}d\Omega_{5}^{2}\right\\},$
$\displaystyle e^{\phi}$
$\displaystyle=1+\frac{Q}{r_{H}^{4}}\ln\frac{1}{f\left(r\right)},\
f_{T}\left(r\right)=1-\frac{r_{H}^{4}}{r^{4}},\
F_{5}=dC_{4}=g_{s}^{-1}\mathcal{Q}_{3}\epsilon_{5},$ $\displaystyle F_{1}$
$\displaystyle=dC_{0},\ C_{0}=-ie^{-\phi}+i\chi,\ \phi=\Phi-\Phi_{0},\
e^{\Phi_{0}}=g_{s},$ (2.2)

where $\epsilon_{5}$ is the volume element of a unit $S^{5}$ and

$R^{4}=4\pi g_{s}N_{c}l_{s}^{4},\ \mathcal{Q}_{3}=4R^{4},\
Q=\frac{N_{\mathrm{D}}}{N_{c}}\frac{\left(2\pi\right)^{4}\alpha^{\prime
2}}{V_{4}}\mathcal{Q}_{3}.$ (2.3)

This solution describes that the D-instanton charge $N_{\mathrm{D}}$ is
smeared over the worldvolume $V_{4}$ of the coincident black $N_{c}$ D3-branes
homogeneously with a horizon at $r=r_{H}$. And it implies
$N_{\mathrm{D}}/N_{c}$ must be fixed since the backreaction of the
D-instantons has been involved in the bulk action. The dual theory of this
system is conjectured as the 4d $\mathcal{N}=4$ super Yang-Mills theory (SYM)
in a self-dual gauge field background or with a dynamical axion at finite
temperature characterized by the order parameter $Q$. In order to obtain a
confining or QCD-like dual theory, let us follow the discussion in [12, 13].
Specifically we first take one of the three spatial dimensions $x^{i}$ of the
D3-branes to be compactified on a circle $S^{1}$ with a period $x^{i}\sim
x^{i}+\delta x^{i}$. Hence below the Kaluza-Klein energy scale defined as
$M_{KK}=2\pi/\delta x^{i}$, the dual theory becomes effectively three-
dimensional. Then we are going to get rid of all massless particles other than
the gauge fields. The most simple way to achieve this is to impose the anti-
periodic and periodic boundary condition on fermion and bosonic fields
respectively along $S^{1}$. So the supersymmetric fermions and scalars in the
dual theory acquire mass of order $M_{KK}$ which is accordingly decoupled in
the low-energy dynamics. Next we perform a double Wick rotation on the
D(-1)-D3 brane background i.e. $t\rightarrow-ix^{i},x^{i}\rightarrow-it$ to
identify the bulk gravity solution as its holographic correspondence. Without
loss of generality, let us denote the direction along $S^{1}$ as $x^{i}=x^{3}$
throughout this manuscript, thus the confining solution of non-extremal
D3-branes with smeared D-instantons is obtained as,

$\displaystyle ds^{2}$
$\displaystyle=e^{\phi/2}\left\\{\frac{r^{2}}{R^{2}}\left[\eta_{ab}dx^{a}dx^{b}+f\left(r\right)\left(dx^{3}\right)^{2}\right]+\frac{1}{f\left(r\right)}\frac{R^{2}}{r^{2}}dr^{2}+R^{2}d\Omega_{5}^{2}\right\\},$
$\displaystyle f\left(r\right)$ $\displaystyle=1-\frac{r_{KK}^{4}}{r^{4}},\
a,b=0,1,2,$ (2.4)

where the solution of dilaton $\Phi$ and R-R fields $C_{0,4}$ remains. Since
the warp factor $e^{\phi/2}\frac{r^{2}}{R^{2}}$ never goes to zero, the
solution (2.4) defined for $r>r_{KK}$ does not have a horizon. And it would
lead to an area law in the dual theory according to the asymptotics of the
Wilson loop in this geometry. Namely below the energy scale $M_{KK}$, the dual
field theory should exhibit confinement. To avoid the conical singularities in
the region of $r>r_{KK}$, we have to further require,

$M_{KK}=\frac{2r_{KK}}{R^{2}}.$ (2.5)

Afterwards, the dual theory can be examined by taking into account the action
of a probe D3-brane which is expected to be a 3d Yang-Mills (YM) plus Chern-
Simons (CS) theory at $r\rightarrow\infty$ as,

$\displaystyle S_{\mathrm{D3}}$ $\displaystyle=-\mu_{3}\int
d^{4}xe^{-\phi}\mathrm{Str}\sqrt{-\det\left(g+\mathcal{F}\right)}+\mu_{3}\int
C_{4}+\frac{1}{2}\mu_{3}\mathrm{Tr}\int C_{0}\mathcal{F}\land\mathcal{F}$
$\displaystyle\simeq-\frac{1}{2g_{YM}^{2}}\mathrm{Tr}\int
d^{4}xF_{\mu\nu}F^{\mu\nu}-\frac{1}{4\pi}\mathrm{Tr}\int
dC_{0}\omega_{3}+\mathcal{O}\left(F^{4}\right)$
$\displaystyle=-\frac{1}{2g_{3d,YM}^{2}}\mathrm{Tr}\int
d^{3}xF_{ab}F^{ab}+i\frac{k_{b}}{4\pi}\mathrm{Tr}\int_{\mathbb{R}^{1,2}}\omega_{3},$
(2.6)

where $\mathcal{F}=2\pi\alpha^{\prime}F$ is the gauge strength,
$\mu_{p}=\left(2\pi\right)^{-p}l_{s}^{-p-1}$ refers to the D-brane charge and
$\omega_{3}$ is the Chern-Simons 3-form,

$\displaystyle\omega_{3}$ $\displaystyle=A\wedge dA+\frac{2}{3}A\wedge A\wedge
A.$ (2.7)

By imposing the background solution, it leads to

$dC_{0}\big{|}{}_{r\rightarrow\infty}=-ik_{b}M_{KK}\delta\left(x^{3}-\bar{x}^{3}\right)dx^{3},g_{3d,YM}^{2}=\frac{g_{YM}^{2}M_{KK}}{2\pi},$
(2.8)

where we have assumed that $\omega_{3}$ is independent on $x^{3}$ and does not
have components along $x^{3}$. So (2.6) represents the YM-CS action located at
$x^{3}=\bar{x}^{3}$, which means $C_{0}\big{|}_{r\rightarrow\infty}=0$ if
$x^{3}\in\left(0,\bar{x}^{3}\right)$; $C_{0}\big{|}_{r\rightarrow\infty}\neq
0$ if $x^{3}\in\left(\bar{x}^{3},2\pi M_{KK}^{-1}\right)$. In this case, we
have to slightly modify the supergravity solution for $C_{0}$ in (2.2) as,

$\chi=1-k_{b}M_{KK}\Theta\left(x^{3}-\bar{x}^{3}\right),$ (2.9)

where $\Theta\left(x^{3}-\bar{x}^{3}\right)$ is the step function.

##  3 Flavor and Chern-Simons brane

In this section, let us discuss the embedding of flavor and CS brane in the
D3-brane background with D-instantons (2.4) in holography.

###  3.1 Adding flavors

According to the dictionary of AdS/CFT, introducing flavors into the
holographic background is to add fundamental matter in the dual theory [26].
So follow the discussion of D3/D7 approach, we put a stack of $N_{f}$
D7-branes as probes, as $N_{f}$ copies of fundamental flavors, into our
background (2.4) and the configuration of various D-branes is illustrated in
Table 1.

| -1 | 0 | 1 | 2 | (3) | 4 (r) | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---|---
D(-1)-branes | - |  |  |  |  |  |  |  |  |  |
Color D3-branes |  | - | - | - | - |  |  |  |  |  |
Flavor D7-branes |  | - | - | - |  | - | - | - | - | - |
CS D7-branes |  | - | - | - |  |  | - | - | - | - | -
Table 1:  The configuration of various D-branes. “-” represents the D-branes
extend along this direction. Note that “-1” is vertical to all the directions
of bulk spacetime.

Note that in this configuration the suppersymmetry is completely broken even
in an extremal D3-brane background since the leftover direction $x^{9}$ is
transverse to both flavor D7- and color D3-branes which leads to 6 mixed
Neumann-Dirichlet boundary conditions. The bare mass of flavors corresponds to
the separation between color and flavor branes at the UV boundary, which means
the scalar field respected to $x^{9}$ on the worldvolume of the D7-branes is
the mass operator $\bar{\psi}\psi$ in the dual field theory.

Since the directions $x^{4}...x^{9}$ transverse to the $N_{c}$ D3-branes are
usually described by spherical coordinates , for convenience we introduce a
new radius coordinate $\rho$ as,

$r\left(\rho\right)=\left(\rho^{2}+\frac{r_{KK}^{4}}{4\rho^{2}}\right)^{1/2},\rho>\frac{r_{KK}}{\sqrt{2}},$
(3.1)

thus the metric (2.4) on coordinate $\rho$ can be written as,

$ds^{2}=e^{\phi/2}\left\\{\frac{r^{2}}{R^{2}}\left[\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}+f\left(r\right)\left(dx^{3}\right)^{2}\right]+\frac{R^{2}}{\rho^{2}}\left(d\rho^{2}+\rho^{2}d\Omega_{5}^{2}\right)\right\\}.$
(3.2)

Then let us choose the spherical coordinates $\lambda,\Omega_{4}$ to
reparametrize the directions $x^{4}...x^{8}$ which are part of the worldvolume
of flavor branes and separate transverse coordinate $x^{9}\equiv u$ with
$\rho^{2}=\lambda^{2}+u^{2}$. Afterwards the metric on
$\left\\{x^{a},x^{3},\lambda,\Omega_{4},u\right\\}$ takes the form as,

$ds^{2}=e^{\phi/2}\left\\{\frac{r^{2}}{R^{2}}\left[\eta_{ab}dx^{a}dx^{b}+f\left(r\right)\left(dx^{3}\right)^{2}\right]+\frac{R^{2}}{\rho^{2}}\left(d\lambda^{2}+\lambda^{2}d\Omega_{4}^{2}+du^{2}\right)\right\\},$
(3.3)

where $r=r\left(\rho\right)$. Embedding the flavor brane into
$\left\\{x^{a},\lambda,\Omega_{4}\right\\}$ at a constant $x^{3}$ with
$u=u\left(\lambda\right)$, the induced metric on the flavor D7-brane becomes,

$ds_{\mathrm{D}7}^{2}=e^{\phi/2}\left\\{\frac{r^{2}}{R^{2}}\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}+\frac{R^{2}}{\rho^{2}}\left[\left(1+u^{\prime
2}\right)d\lambda^{2}+\lambda^{2}d\Omega_{4}^{2}\right]\right\\}.$ (3.4)

Note that we need to impose the following boundary condition
($\frac{du}{d\lambda}\equiv u^{\prime}$),

$u^{\prime}\big{|}_{\lambda=0}=0,u\big{|}_{\lambda=\lambda_{\infty}}=u_{\infty}.$
(3.5)

We have use $\lambda_{\infty}$ to denote the UV boundary or UV cutoff in the
dual field theory. So for a single D7-brane, its action can be collected as,

$S_{\mathrm{D}7}=-T_{\mathrm{D}7}\int d^{8}xe^{-\phi}\sqrt{-g_{\mathrm{D7}}},$
(3.6)

where $T_{\mathrm{D}7}=g_{s}^{-1}\mu_{p}$ is the tension of the Dp-brane.
Plugging the solution (2.2) into (3.6), the action of a probe flavor brane is
obtained as,

$S_{\mathrm{D}7}=-T_{\mathrm{D}7}V_{3}V_{S^{4}}R^{2}\int d\lambda
e^{\phi\left(\rho\right)}\left(\rho^{2}+\frac{r_{KK}^{4}}{4\rho^{2}}\right)^{3/2}\frac{\lambda^{4}}{\rho^{5}}\sqrt{1+u^{\prime
2}},$ (3.7)

where $V_{3},V_{S^{4}}$ refers to the Minkowskian worldvolume of D3-brane and
the volume of a unit $S_{4}$. By varying the D7-brane action respected to
$u\left(\lambda\right)$, the associated equation of motion is,

$\displaystyle\frac{d}{d\lambda}\left[e^{\phi}\left(r_{KK}^{4}+4\rho^{4}\right)^{3/2}\frac{\lambda^{4}}{8\rho^{8}}\frac{u^{\prime}}{\sqrt{1+u^{\prime
2}}}\right]$ $\displaystyle=$
$\displaystyle-e^{\phi}\left[r_{KK}^{4}+\rho^{4}-\frac{1}{8}\rho\frac{d\Phi}{d\rho}\left(r_{KK}^{4}+4\rho^{4}\right)\right]\left(r_{KK}^{4}+4\rho^{4}\right)^{1/2}\frac{\lambda^{4}u}{\rho^{10}}\sqrt{1+u^{\prime
2}}.$ (3.8)

In order to obtain the embedding function $u\left(\lambda\right)$, we have to
solve (3.8) with (3.5). So let us analyze massless and massive embedding of
the flavor brane respectively.

#### Massless case

First let us investigate the case of $r_{KK}\rightarrow 0$ which corresponds
to the extremal D3-D(-1) solution. The equation of motion (3.8) comes to

$\frac{d}{d\lambda}\left[e^{\phi}\frac{\lambda^{4}}{\rho^{2}}\frac{u^{\prime}}{\sqrt{1+u^{\prime
2}}}\right]=-\frac{2\lambda^{4}u}{\rho^{4}}e^{\phi}\left(1-\frac{1}{2}\rho\frac{d\phi}{d\rho}\right)\sqrt{1+u^{\prime
2}},$ (3.9)

where

$\phi\rightarrow 1+\frac{Q}{r^{4}}.$ (3.10)

It is clear that at $\lambda=0$ the right-hand side of (3.9) is not vanished
unless $u\left(\lambda\right)=0$ is the solution. We expect
$u\left(\lambda\right)=0$ to be an unstable solution as it is discussed in the
D3/D7 approach [14, 15] since this solution is invariant under the parity
transformation $u\left(\lambda\right)\rightarrow-u\left(\lambda\right)$.

Then let us investigate the case of $r_{KK}>0$. In the massless, we need to
choose $u_{\infty}=0$ in (3.5) since there is a parity transformation
$u\left(\lambda\right)\rightarrow-u\left(\lambda\right)$ in the massless case
and $u_{\infty}$ corresponds to the bare mass of the flavors. In order to
search for an analytical solution, we use the following ansatz for
$u\left(\lambda\right)$ as,

$u\left(\lambda\right)=\begin{cases}\pm\sqrt{\frac{r_{KK}^{2}}{2}k-\lambda^{2}},&0\leq\lambda\leq\sqrt{\frac{k}{2}}r_{KK},\\\
0,&\lambda>\sqrt{\frac{k}{2}}r_{KK},\end{cases}$ (3.11)

where $k\equiv k\left(q\right)$ is a constant dependent on $q=Q/r_{KK}^{4}$
only. Notice that $r\in\left(r_{KK},\infty\right)$ so that $k\geq 1$. Plugging
(3.11) into (3.8), it leads to a constraint equation which determines the
relation of $k$ and $q$ as,

$3+3k^{4}-2k^{2}\left(3+8q\right)-3q\left(k^{2}-1\right)^{2}\ln\left[\left(\frac{k^{2}-1}{k^{2}+1}\right)^{2}\right]=0.$
(3.12)

This equation can be numerically solved and the relation of $k$ and $q$ is
illustrated as in Figure 1.

Figure 1:  Numerical solution of $k$ and $q$.

The solution (3.11) has two branches which refers to a pair of D7-branes
wrapping the upper and lower half-five-sphere with various numbers of
D-instantons represented by $q$. The flavor branes wrapping the upper and
lower half-five-sphere have opposite parity as it is illustrated in Figure 2.

Figure 2:  The maximal embedding of flavor D7-branes in the $u,\lambda$ plane
with various $q$.

Since the relation of $k$ and $q$ smoothly returns to the case without
D-instantons i.e. $k=1,q=0$, the solution (3.11) corresponds to the maximal
embedding of the flavor branes as it is in the D3/D7 approach which refers to
the embedded flavor branes have maximal energy among all possible solutions to
(3.8) with $u_{\infty}=0$ in the boundary condition (3.5).

In order to find a more general configuration of $u\left(\lambda\right)$, let
us take a look at the asymptotic behaviors of (3.8). In the region of
$\lambda\rightarrow 0$, we have $u^{\prime}\left(\lambda\right)\rightarrow
0,\rho\rightarrow u,$ so (3.8) leads to a solution as,

$u\left(\lambda\right)=\pm
u_{0}\mp\left[\frac{4\left(r_{KK}^{4}+u_{0}^{4}\right)}{5u_{0}\left(r_{KK}^{4}+4u_{0}^{4}\right)}-\frac{1}{10}\phi^{\prime}\left(u_{0}\right)\right]\lambda^{2}+\mathcal{O}\left(\lambda^{4}\right).$
(3.13)

Note that $u_{0}>0$ and

$\phi^{\prime}\left(u_{0}\right)\equiv\frac{\partial}{\partial\rho}\phi\left(\rho\right)\bigg{|}_{\rho=u_{0}}=\frac{64Qr_{KK}^{4}u_{0}^{3}}{\left(r_{KK}^{4}-4u_{0}^{4}\right)\left(r_{KK}^{4}+4u_{0}^{4}\right)\left[r_{KK}^{4}+Q\ln\frac{\left(r_{KK}^{4}+4u_{0}^{4}\right)^{2}}{\left(r_{KK}^{4}-4u_{0}^{4}\right)^{2}}\right]}<0,$
(3.14)

due to $u_{0}>\frac{\sqrt{2}}{2}r_{KK}$. The second derivative of
$u\left(\lambda\right)$ takes the opposite sign to
$u\left(\lambda\right)\Big{|}_{\lambda=0}$. On the other hand, in the region
of $\lambda\rightarrow\infty$, we have $u\left(\lambda\right)\rightarrow
u_{\infty},\rho\rightarrow\lambda,$ so the equation (3.8) becomes,

$\frac{d}{d\lambda}\left(\lambda^{2}u^{\prime}\right)=-2u.$ (3.15)

Accordingly, the asymptotic behavior of $u\left(\lambda\right)$ at large
$\lambda$ takes the general form as,

$u\left(\lambda\right)=\pm\sqrt{\frac{\mu^{3}}{\lambda}}\sin\left(\frac{\sqrt{7}}{2}\ln\frac{\lambda}{\lambda_{\infty}}\right),$
(3.16)

by imposing the boundary condition $u\left(\lambda_{\infty}\right)=0$ where
$\mu,\lambda_{\infty}>0$ are the integration constants. This solution also has
two branches thus it implies the global signs of $u\left(\lambda\right)$ and
$u^{\prime}\left(\lambda\right)$ are opposite as well. Keeping this in mind,
we numerically evaluate the minimal embedding solution (without any nodes) of
(3.8) with various charge density of D-instantons represented by $q$ and the
results are illustrated as in Figure 3.

Figure 3:  The minimal embedding function $u\left(\lambda\right)$ of flavor
brane with various $q$ and the parameter is set to be
$\lambda_{\infty}=5.73r_{KK}$.

Our results show that $q=0$ corresponds to the minimal embedding among various
value of $q$ respected to the solutions with zero node and this is consistent
with that the D3-D-instanton solution describes the dual theory in an excited
background. The equation (3.8) also includes solutions with $n$ nodes and let
us denote it as $u_{n}\left(\lambda\right)$, so the minimal solutions are
identified as $u_{0}\left(\lambda,q\right)$ now. We also show the numerical
relation of $u_{n=0,1,2}\left(\lambda,q\right)$ for a fixed $q$ and
$u_{1,2}\left(\lambda,q\right)$ with various $q$ as in Figure 4.

Figure 4:  The upper figures illustrate the relation of
$u_{0,1,2}\left(\lambda\right)$ with $q=0,0.5$, the lower figures show the
relation of $u_{1,2}$ respected to various $q$.

The numerical calculation implies that the associated energy of the embedded
flavor brane is a monotonically increased function of the number of nodes for
any $q$, and this is numerically verified as in Figure 5.

Figure 5:  The ratios of $E_{n}$ and $E_{0}$. $E_{n}$ refers to the energy of
a flavor brane with $n$-nodes embedding function $u_{n}\left(\lambda\right)$.

In this sense, the solution of maximal embedding presented in (3.11) can be
treated as having infinity nodes. On the other hand, we can find the total
energy of flavor brane is always minimized at $q=0$. To verify this conclusion
quantitatively, we could in particular evaluate the energy density of a
maximal embedded flavor brane since it would be semi-analytical. Plugging
(3.11) into (3.6), we could obtain the energy density of a maximal embedded
flavor brane as,

$E_{\mathrm{D7}}^{max}=-\frac{1}{V_{3}}S_{\mathrm{D7}}^{max}=T_{\mathrm{D7}}V_{S^{4}}R^{2}\left[\frac{\lambda_{\infty}}{3}+b_{max}\left(q\right)r_{KK}^{3}\right].$
(3.17)

Here $b_{max}\left(q\right)$ is a constant dependent on $q$ which can be
expressed by the combination of generalized hypergeometrical functions. We
plot out the numerical values of $b_{max}\left(q\right)$ as in Figure 6

Figure 6:  The relation of $b_{max}$ and $q$. The numerical value shows
$b_{max}\left(0\right)\simeq 1.026$ which exactly returns to the result in
[15].

and it indeed shows $q=0$ corresponds to the the flavor brane of the lowest
energy. In this sense, the vacuum with non-zero $q$ could be recognized as the
metastable vacua of flavors in the presence of instantons in the dual theory
which is in agreement of [14, 15].

Since we are usually interested in comparing energies with the same boundary
condition between different solutions, the maximal energy (3.17) should be
subtracted as the regularization of the total energy. Let us denote the
onshell action with $n$-nodes embedding function $u_{n}\left(\lambda\right)$
as $S_{\mathrm{D7}}^{n}$, so the total energy of the flavor brane is redefined
as,

$E_{f}^{n}\left(q\right)=-\frac{1}{V_{3}}\left(S_{\mathrm{D7}}^{n}-S_{\mathrm{D7}}^{max}\right).$
(3.18)

For the minimal embeddings of two parity-related flavor branes, (3.18) could
be evaluated as,

$E_{f}^{0}\left(q\right)\simeq-
N_{c}\left(g_{s}N_{c}\right)\left[b_{0}\left(q\right)M_{KK}^{3}+a_{0}\left(q\right)M_{\mu}^{3}\right],$
(3.19)

where $b_{0}\left(q\right)$ and $a_{0}\left(q\right)$ depend on the
D-instanton charge $q$. The energy scale $M_{\mu}$ is related to the length
scale $\mu=M_{\mu}R^{2}/2$ which comes from the duality of holographic
radius/energy relation [42]. The difference between the energy density at
large $N_{c}$ should be relevant to the potential barrier of the instanton
vacuum in the dual theory. Besides, the general configuration of $N_{f}$
flavor branes can also be obtained by (3.19). Let us consider $p$ of $N_{f}$
flavor branes located in the upper $u,\lambda$ plane while the other $N_{f}-p$
flavor branes located the lower plane with minimal embedding. Since the energy
of each flavor brane is equivalent, the total energy of these flavor branes
should be,

$E_{f,tot}^{0}=pE_{f}^{0}\left(q\right)+\left(N_{f}-p\right)E_{f}^{0}\left(q\right)=N_{f}E_{f}^{0}\left(q\right).$
(3.20)

#### Massive case

Let us turn to the massive case by considering the inclusion of a bare mass of
the quarks or fermions in the dual theory. The bare mass in this model can be
viewed as a source for the condensate operator
$\left\langle\bar{\psi}\psi\right\rangle$ of fermions. Since the bare mass of
fermions is identified as the spatial separation between D3- and D7-branes
along the transverse direction $u$ in the UV region and we have seen in the
last subsection

$u\left(\lambda\right)\rightarrow\frac{1}{\sqrt{\lambda}},$ (3.21)

at large $\lambda$, we can set

$\lim_{\lambda\rightarrow\lambda_{\infty}}\sqrt{\frac{\lambda}{\mu}}u\left(\lambda\right)=2\pi
l_{s}^{2}m.$ (3.22)

This is also equivalent to set $u_{\infty}\simeq 2\pi l_{s}^{2}m$ while this
boundary condition breaks the parity symmetry. So the parity transformation
$u\rightarrow-u$ implies the flavor branes with up/down embedding have to take
positive/negative mass which succeeds the discussion in the D3/D7 approach
consistently. It means the fermion mass leads to the degeneracy between the
upper and lower embedding of the flavor branes. Then in order to obtain the
energy of the flavor brane in the massive case, we can start from the massless
case by performing a small variation in the boundary condition,

$\delta u_{\infty}=\sqrt{\frac{\mu}{\lambda_{\infty}}}2\pi l_{s}^{2}\delta m.$
(3.23)

So the associated variation in the onshell action of the flavor brane becomes,

$\displaystyle\delta S_{\mathrm{D7}}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial u^{\prime}}\delta
u\bigg{|}_{\lambda=0}^{\lambda=\lambda_{\infty}}=-T_{\mathrm{D7}}V_{3}V_{S^{4}}R^{2}\left[e^{\phi}\left(r_{KK}^{4}+4\rho^{4}\right)^{3/2}\frac{\lambda^{4}}{8\rho^{8}}\frac{u^{\prime}}{\sqrt{1+u^{\prime
2}}}\delta u\right]\bigg{|}_{\lambda=0}^{\lambda=\lambda_{\infty}},$ (3.24)

which leads to

$\displaystyle\delta E_{f}$
$\displaystyle=T_{\mathrm{D7}}V_{S^{4}}R^{2}\lambda_{\infty}^{2}u^{\prime}\left(\lambda_{\infty}\right)\delta
u_{\infty}e^{\phi\left(\lambda_{\infty}\right)}\equiv\mp
c\left(q,\lambda_{\infty}\right)\delta m,$ $\displaystyle
c\left(q,\lambda_{\infty}\right)$
$\displaystyle=e^{\phi\left(\lambda_{\infty}\right)}\frac{N_{c}\sqrt{g_{s}N_{c}}}{24\pi^{5/2}}M_{\mu}^{2},$
(3.25)

where we have used $u^{\prime}\left(0\right)=0$ and the equation of motion for
$u\left(\lambda\right)$. Since the embedding function can have both signs as
in the massless case characterized by $u_{\infty}^{\prime}$, the
positive/negative mass $m$ determines the sign of $u_{\infty}^{\prime}$ as it
is preferred. As the massless case, the energy of the flavor brane can be
obtained by using (3.18) which consists of the massless part
$E_{f}^{0}\left(q\right)$ plus a small variation $\delta E_{f}$ as,

$\displaystyle E_{f}^{0}\left(q,m\right)$
$\displaystyle=E_{f}^{0}\left(q\right)+\delta E_{f}$ $\displaystyle\simeq
E_{f}^{0}\left(q\right)\mp c\left(q,\lambda_{\infty}\right)m.$ (3.26)

This result shows that the fermion condensate is negative/positive for
positive/negative mass due to

$\left\langle\bar{\psi}\psi\right\rangle=\frac{dE_{f}^{0}\left(q,m\right)}{dm}=\mp
c\left(q,\lambda_{\infty}\right)\mathrm{sign}\left(m\right).$ (3.27)

And it would be slightly modified by the presence of the D-instantons in the
quadratic order of the fermion mass.

To close this subsection, let us evaluate the total energy of $N_{f}$ flavor
branes with a bare mass. As before, we consider the configuration that $p$ of
$N_{f}$ branes wraps the upper half-five-sphere separated from the other
$N_{f}-p$ branes wrapping the lower half-five-sphere with a common mass $m$.
So the total energy is a collection of upper and lower branes which is,

$\displaystyle E_{f,tot}^{0}\left(q,m\right)$
$\displaystyle=p\left[E_{f}^{0}\left(q\right)-c\left(q,\lambda_{\infty}\right)m\right]+\left(N_{f}-p\right)\left[E_{f}^{0}\left(q\right)+c\left(q,\lambda_{\infty}\right)m\right]$
$\displaystyle=N_{f}E_{f}^{0}\left(q\right)+\left(N_{f}-2p\right)c\left(q,\lambda_{\infty}\right)m.$
(3.28)

It would be obvious that, for any $q$, the minimal energy occurs at $p=N_{f}$
for $m>0$ and $p=0$ for $m<0$. And (3.28) reduces to (3.20) if $m\rightarrow
0$ so that the degeneracy of upper and lower embeddings is regained.

###  3.2 Embedding of the CS D7-brane

Since the R-R flux $C_{0}$ is non-vanished in our D3-D(-1) background, there
should also be a magnetic source for $C_{0}$. And the source could be provided
by $n_{b}$ CS D7-branes as probes coupled to $C_{0}$ magnetically. The
configuration of the CS brane is illustrated in Table 1. Since the number of
CS branes should be an integer, the CS level is automatically quantized in
holography.

In the D3-brane approach, the CS brane can be set located at $r=r_{KK}$ in
order to minimize their energy density, however this does not work in the
presence of D-instantons because the energy density of a single CS brane is
evaluated as,

$\displaystyle S_{\mathrm{D}7}^{\mathrm{CS}}$
$\displaystyle=-T_{\mathrm{D}7}\int
d^{8}xe^{-\phi}\sqrt{-g_{\mathrm{D7}}}-\mu_{7}\int
C_{8}=-T_{\mathrm{D}7}V_{3}V_{S^{5}}R^{2}r^{3}e^{\phi}-N_{d},$ (3.29)

where $C_{8}$ is the dual form of $C_{0}$ defined as $dC_{8}=^{\star}dC_{0}$.
This action is divergent at $r=r_{KK}$ which leads to an IR divergence in the
dual field. While this is not important when we are interested in comparing
the difference of the energy, the position of the CS brane would be less
clear. To figure out this problem we require that our discussion should reduce
to the D3-brane approach if $q\rightarrow 0$. In this sense, we assume that
the location of the CS brane $r=r_{KK}$ is shifted by the presence of
D-instantons which becomes $r=r_{Q}>r_{KK}$. And we furthermore treat the
solution (3.11) describing a CS D7-brane embedding at $r=r_{Q}$ according to
the embedding equation (3.8), so that

$r_{Q}=\frac{1}{\sqrt{2}}\left(k+\frac{1}{k}\right)^{1/2}r_{KK}.$ (3.30)

Hence for a fixed $q$, the maximal embedding of a flavor D7-brane can be
identified as an embedding function of CS D7-brane as it is done in the
D3-brane approach. The positive and negative level of the CS brane corresponds
to the orientation of counterclockwise and clockwise respectively in the
$u,\lambda$ plane. Therefore the total energy of a CS brane can be evaluated
as,

$E_{\mathrm{CS}}\left(q\right)=-\frac{S_{\mathrm{D7}}^{\mathrm{CS}}}{V_{3}}=\mu_{7}V_{S^{5}}R^{2}r_{Q}^{3}e^{\phi\left(r_{Q}\right)}=\frac{g_{s}N_{c}^{2}M_{KK}^{3}}{64\pi^{2}}\left[\frac{1}{2\sqrt{2}}\left(k+\frac{1}{k}\right)^{3/2}e^{\phi\left(r_{Q}\right)}+\frac{1}{2}q\right],$
(3.31)

which indeed reduces to the D3-brane approach when $q\rightarrow 0$. We plot
out
$G\left(q\right)=E_{\mathrm{CS}}\left(q\right)/E_{\mathrm{CS}}\left(0\right)$
and $c\left(q,\lambda_{\infty}\right)/c\left(0,\lambda_{\infty}\right)$ as a
function of $q$ in Figure 7.

Figure 7:  The relation of $c\left(q,\lambda_{\infty}\right)$ (left),
$G\left(q\right)=E_{\mathrm{CS}}\left(q\right)/E_{\mathrm{CS}}\left(0\right)$
(right) and $q$. For $q=0$, it returns to the D3-brane approach without any
D-instantons.

The numerical result also illustrates $q=0$ corresponds to the CS brane with
minimal energy and the limitation of $r_{Q}\rightarrow r_{KK}$.

At low energy, the CS brane reduces to a 3d $U\left(n_{b}\right)$ gauge theory
and leads to a Chern-Simons action at level $-N$ due to the Wess-Zumino term
of the D-brane action which is,

$S_{C_{4}}=\frac{1}{2\left(2\pi\right)^{5}l_{s}^{4}}\int_{\mathrm{D7}}C_{4}\land\mathrm{Tr}\left(F\wedge
F\right)=-\frac{1}{2\left(2\pi\right)^{5}l_{s}^{4}}\int_{S^{5}}F_{5}\int_{\mathbb{R}^{1,2}}\mathrm{Tr}\omega_{3}=-\frac{N_{c}}{4\pi}\int_{\mathbb{R}^{1,2}}\mathrm{Tr}\omega_{3}.$
(3.32)

All excitations on the CS branes will decouple at very low energy scale except
this CS term, so we can obtain the level/rank duality
$SU\left(N\right)_{n_{b}}\leftrightarrow U\left(\left|n_{b}\right|\right)_{N}$
through this holographic system as the QFT expectations.

##  4 Vacuum structure of the dual theory

In this section, let us analyze the vacuum structure in the dual theory at
large-$N_{c}$ expansion. Since the vacuum of the dual theory in general may
include both flavor and CS branes, we are going to take into account the
configuration with two kinds of the D7-branes.

The effective CS level in the dual theory is required as,

$i\int_{S^{1}}F_{1}\big{|}_{r\rightarrow\infty}=k_{eff}.$ (4.1)

However in order to define the CS level $\kappa=k_{b}-N_{f}/2$, the CS level
must depend on $p$. To find the result, we can straightforwardly count the
number of the orientation in the $u,\lambda$ plane. Defining the number of
D7-branes with counterclockwise/clockwise orientation is positive/negative,
let us consider the configuration of that, in the $u,\lambda$ plane $k_{0}$
counterclockwise CS branes live in $\mathcal{R}_{0}$, $p$ flavor branes live
in $\mathcal{R}_{+}$ and $N_{f}-p$ flavor branes live in $\mathcal{R}_{-}$
where $\mathcal{R}_{+},\mathcal{R}_{-},\mathcal{R}_{0}$ respectively refers to
the regions of the $u,\lambda$ plane which are above, between and below the
flavor branes as it is illustrated in Figure 8.

Figure 8:  The configuration of flavor and CS branes on $u,\lambda$ plane. The
flavor branes are represented by the orange line and the CS brane are
represented by the blue line.
$\mathcal{R}_{+},\mathcal{R}_{-},\mathcal{R}_{0}$ respectively refers to the
regions above, between and below the flavor branes.

We only consider the minimal embedding of the flavor branes since the concern
is the vacuum structure. Requiring $\kappa=k_{eff}$ at the UV boundary, we
have [15],

$k_{eff}=\begin{cases}\kappa-\frac{N_{f}}{2},&\mathrm{in}\ \mathcal{R}_{+},\\\
\kappa+p-\frac{N_{f}}{2},&\mathrm{in}\ \mathcal{R}_{0},\\\
\kappa+\frac{N_{f}}{2},&\mathrm{in}\ \mathcal{R}_{-},\end{cases}$ (4.2)

and $k_{0}=\kappa+p-N_{f}/2$ which is what we desire in the dual field theory.
The interpretation of such D-brane configuration at low energy is that, the
flavor symmetry $U\left(N_{f}\right)$ is broken spontaneously to
$U\left(p\right)\times U\left(N_{f}-p\right)$ which creates
$2p\left(N_{f}-p\right)$ Goldstone bosons and their target space is Grassmann,

$\mathrm{Gr}\left(p,N_{f}\right)=\frac{U\left(N_{f}\right)}{U\left(p\right)\times
U\left(N_{f}-p\right)}.$ (4.3)

The CS branes leads to a level/rank duality of
$U\left(\left|\kappa+p-N_{f}/2\right|\right)_{N}\leftrightarrow
SU\left(N\right)_{k+p-N_{f}/2}$. So the low-energy dynamics of a $p$ sector
would have the symmetry,

$\mathrm{Gr}\left(p,N_{f}\right)\times SU\left(N\right)_{k+p-N_{f}/2},$ (4.4)

in which the $N_{f}+1$ sectors describe the vacuum of the dual theory
holographically. To analyze the phase structure of the vacuum, the minimal
value of the (free) energy is necessary. Since the total energy of the $p$
sector consists of flavor plus the CS part and the flavor energy has be
obtained in (3.28), we need to include the energy of the CS brane which is the
number of CS branes times the energy density $E_{\mathrm{CS}}\left(q\right)$
given in (3.31). Therefore the total free energy density is collected as
($\kappa\geq 0,0\leq p\leq N_{f}$) [15],

$E\left(p,q\right)=N_{f}E_{f}^{0}\left(q\right)+\left(N_{f}-2p\right)c\left(q,\lambda_{\infty}\right)m+\left|\kappa+p-N_{f}/2\right|E_{\mathrm{CS}}.$
(4.5)

Minimize (4.5) then compare the free energy, the associated free energy
density is obtained as, for $\kappa>N_{f}/2$,

$\displaystyle E_{vac}$ $\displaystyle=\begin{cases}\left(\kappa-
N_{f}/2\right)E_{\mathrm{CS}},&m<m^{*},\\\
\left(\kappa+N_{f}/2\right)E_{\mathrm{CS}}-2N_{f}cm,&m>m^{*},\end{cases}\begin{array}[]{c}SU\left(k-N_{f}/2\right),\\\
SU\left(k+N_{f}/2\right),\end{array}$ (4.8)

where $SU\left(k\pm N_{f}/2\right)$ refers to the corresponding topological
phase in the dual theory. And for $\kappa<N_{f}/2$, the minimized free energy
density and associated topological phase are collected as,

$\displaystyle E_{vac}$
$\displaystyle=\begin{cases}\left(N_{f}/2-\kappa\right)E_{\mathrm{CS}},&m<-m^{*},\\\
2\left(\kappa-
N_{f}/2\right)c\left(q,\lambda_{\infty}\right)m,&-m^{*}<m<m^{*},\\\
\left(N_{f}/2+\kappa\right)E_{\mathrm{CS}}-2N_{f}c\left(q,\lambda_{\infty}\right)m,&m>m^{*},\end{cases}\begin{array}[]{c}SU\left(k-N_{f}/2\right),\\\
\mathrm{Gr}\left(p,N_{f}\right),\\\ SU\left(k+N_{f}/2\right),\end{array}$
(4.12)

where the critical mass $m^{*}$ is defined as,

$m^{*}=\frac{E_{\mathrm{CS}}\left(q\right)}{2c\left(q,\lambda_{\infty}\right)}.$
(4.13)

Since $c\left(q,\lambda_{\infty}\right)$ never goes to zero, the derivative
with respect to $m$ both in (4.8) and (4.12) is discontinuous which means
there remains to be a first order phase transition at $m=\pm m^{*}$ in the
presence of the D-instantons. And the vacua would be degenerate at the
critical point. However in our holographic approach, the mass $m^{*}$
additionally depends on the charge density $q$ of the D-instantons. So we
numerically evaluate $m^{*}$ as a function of $q$ as in Figure 9.

Figure 9:  Relation of $m^{*}\left(q\right)$ and $q$. The critical mass for
the phase transition is increased by the presence of the D-instantons.

According to the numerical calculation, while the order parameter
$c\left(q,\lambda_{\infty}\right)$ in the UV limitation
$\lambda_{\infty}\rightarrow 0$ is almost unchanged, the critical mass $m^{*}$
is increased by the presence of the D-instantons which shifts the phase
transition point for $\kappa>N_{f}/2$ thus enhance the phase of
$\mathrm{Gr}\left(p,N_{f}\right)$ for $\kappa<N_{f}/2$. Accordingly, the
D3-D(-1) approach implies the phase transition point is also determined by the
D-instanton charge. This could be interpreted as the topological effect in the
dual theory which is similar as the topological contribution to the mass in
the presence of the CS term [28].

##  5 Entanglement entropy and confinement

Since the entanglement entropy is expected to be a tool to characterize the
confinement/deconfinement phases of the dual theory [37, 38, 39, 40], in this
section we will compute the quantum entanglement entropy between two
physically disjoint spatial regions in the bulk, then compare the results with
the analysis of the free energy.

Before the holographic calculation, we first take into account the simplest
geometry: region $A$ is the product of $\mathbb{R}^{2}\times I_{l}$ where
$I_{l}$ is a line interval of length $l$ and region $B$ is the complement of
$A$. According to the AdS/CFT dictionary, the quantum entanglement entropy
between region $A$ and $B$ relates to the surface $\gamma$ in bulk whose
boundary coincides with the boundary of $A$. Supposing we are discussing the
correspondence of AdSd+2/CFTd+1, the classical area of surface $\gamma$ is
given as,

$S_{\gamma}=\frac{1}{4G_{N}^{d+2}}\int_{\gamma}d^{d}x\sqrt{g_{\mathrm{ind}}},$
(5.1)

where $G_{N}^{d+2}$ is the $d+2$ dimensional Newton constant and
$g_{\mathrm{ind}}$ refers to the induced metric on $\gamma$. Notice $\gamma$
has to be spatial like to represent the entanglement entropy at a fixed time.
The (5.1) can also be generalized into non-conformal situations. For example,
in 10d geometry of D-branes, (5.1) could be naturally modified as,

$S_{A}=\frac{1}{4G_{N}^{10}}\int_{\gamma}d^{8}x\sqrt{g_{\mathrm{ind}}}.$ (5.2)

We will use (5.2) to evaluate the the quantum entanglement entropy in our
holographic model.

The most convenient way to begin the calculation is to write the 10d metric
as,

$ds^{2}=\alpha\left(r\right)\left[\beta\left(r\right)dr^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right]+g_{mn}dy^{m}dy^{n},$
(5.3)

where $\mu=0,1,...d$, $m=d+2,...9$ parametrize $\mathbb{R}^{d+1}$ and $8-d$
internal directions respectively, $r$ refers to the holographic radial
coordinate. Using (5.2) with formula (5.3), the minimized action is given as,

$S_{A}=\frac{V_{d-1}}{2G_{N}^{10}}\int_{r_{*}}^{r_{\infty}}dr\frac{\sqrt{\beta\left(r\right)}H\left(r\right)}{\sqrt{H\left(r\right)-H\left(r_{*}\right)}},$
(5.4)

where

$\displaystyle l\left(r_{*}\right)$
$\displaystyle=2\sqrt{H\left(r_{*}\right)}\int_{r_{*}}^{\infty}dr\frac{\sqrt{\beta\left(r\right)}}{\sqrt{H\left(r\right)-H\left(r_{*}\right)}},$
$\displaystyle H\left(r\right)$
$\displaystyle=e^{-4\phi}V_{\mathrm{int}}^{2}\alpha^{d}\left(r\right),$
$\displaystyle V_{\mathrm{int}}$
$\displaystyle=\int\prod_{m=1}^{8-d}dy^{m}\sqrt{\det g}.$ (5.5)

The minimal surface has distinct features for small and large $l$ according to
the definition of region $A$ and $B$. The minimal surface extends into the
bulk up to the radial position $r_{*}>r_{KK}$ as a connected surface for small
$l$ while the minimal surface becomes two disconnected pieces and extends in
the bulk all the way up to $r_{KK}$ for large $l$. In order to characterize
the phase transition, we need to compare the entanglement entropy of connected
with disconnected configuration of the minimal surface. While the entanglement
entropy itself may be divergent, its difference $\Delta S$ could be finite
which according to (5.4) could be written as,

$\displaystyle\Delta S$
$\displaystyle\equiv\frac{2G_{N}^{10}}{V_{d-1}}\left(S_{A}^{\mathrm{conf}}-S_{A}^{\mathrm{dic}}\right)$
$\displaystyle=\int_{r_{*}}^{\infty}dr\sqrt{\beta\left(r\right)H\left(r\right)}\left\\{\left[1-\frac{H\left(r_{*}\right)}{H\left(r\right)}\right]^{-1/2}-1\right\\}-\int_{r_{KK}}^{r_{*}}dr\sqrt{\beta\left(r\right)H\left(r\right)}.$
(5.6)

Plugging (2.4) into (5.3) - (5.6), we can numerically calculate the relation
of $\Delta S$ and $l$, $l$ and $r_{*}$ as illustrated in Figure 10.

Figure 10:  Left: Relation of $l$ and $r_{*}$ for $q=0,0.5,1,2,3,4,5$ (lower
to upper), $l_{\mathrm{max}}$ is increased by $q$. Right: Relation of $\Delta
S$ and $l$ for $q=0,0.5,1,2,3,4,5$ (left to right) as a typical swallow-tail
behavior. The critical length $l_{c}$ is also increased by $q$.

As we can see, the critical length $l_{c}$ ($\Delta S=0$) is increased by the
density of the D-instanton charge denoted by $q$, which implies there would be
a first order phase transition at scale $l_{c}$ and it is enhanced in the
presence of the D-instantons. Since the entanglement entropy takes order of
$\mathcal{O}\left(N_{c}^{0}\right),\mathcal{O}\left(N_{c}^{2}\right)$
respectively for $l>l_{c}$ and $l<l_{c}$, this phase transition may probably
relates to the deconfinement phase transition in a QCD-like theory. So it
would be, on the other hand, very interesting to evaluate the critical
temperature of the deconfinement phase transition to examine whether it is
consistent with the analyses of the entanglement entropy in holography.

To obtain the critical temperature of the deconfinement in the dual theory, we
should compute the associated free energy $F$ in holography which is the
summary of the Euclidean version of the onshell action (2.1) denoted as
$S_{\mathrm{IIB}}^{E}$, Gibbons-Hawking term $S_{\mathrm{GH}}$ and holographic
counterterm $S_{\mathrm{ct}}^{\mathrm{bulk}}$ in bulk, since the deconfinement
phase transition is suggested to be identified as the Hawking-Page transition
in the bulk [33, 34, 35, 36]. In Einstein frame, they are given as [43, 44],

$\displaystyle S_{\mathrm{bulk}}$
$\displaystyle=S_{\mathrm{IIB}}^{E}+S_{\mathrm{GH}}+S_{\mathrm{ct}}^{\mathrm{bulk}},$
$\displaystyle S_{\mathrm{IIB}}^{E}$
$\displaystyle=-\frac{1}{2\kappa_{10}^{2}}\int
d^{10}x\sqrt{g}\left[\mathcal{R}-\frac{1}{2}\partial\Phi\cdot\partial\Phi-\frac{1}{2}e^{2\Phi}\left|F_{1}\right|^{2}-\frac{1}{2}\left|F_{5}\right|^{2}\right],$
$\displaystyle S_{\mathrm{GH}}$ $\displaystyle=-\frac{1}{\kappa_{10}^{2}}\int
d^{9}xe^{-2\Phi_{0}}\sqrt{h}\left(K-K_{0}\right),$ $\displaystyle
S_{\mathrm{ct}}^{\mathrm{bulk}}$ $\displaystyle=-\frac{1}{\kappa_{10}^{2}}\int
d^{9}xe^{-2\Phi_{0}}\frac{5}{R}\sqrt{h},$ (5.7)

where $h$ is the determinant of the boundary metric i.e. the slice of the 10d
metric (2.4) in Einstein frame at fixed $r=r_{\infty}$ with
$r_{\infty}\rightarrow\infty$. $K$ is the trace of the extrinsic curvature at
the boundary and $K_{0}$ arises from the standard transformation of the
gravity action from Einstein to string frame and they are given as,

$K=-\frac{1}{\sqrt{g}}\partial_{r}\sqrt{h}\big{|}_{r\rightarrow\infty},\
K_{0}=\frac{9}{4}\sqrt{g^{rr}}\partial_{r}\phi\big{|}_{r\rightarrow\infty}.$
(5.8)

Then to include the contribution of the flavors and CS level, we additionally
need to evaluate the Euclidean onshell action of the flavor and CS brane with
respect to background (2.2) and (2.4). For flavor D7-brane, it is embedded at
$x^{3},u=\mathrm{const}$ and the onshell action and the holographic
counterterm $S_{\mathrm{ct}}^{f}$ could be chosen as [44, 45, 46],

$\displaystyle S_{f}$ $\displaystyle=S_{\mathrm{DBI}}+S_{\mathrm{ct}}^{f},$
$\displaystyle S_{\mathrm{DBI}}$ $\displaystyle=-N_{f}T_{\mathrm{D}7}\int
d^{8}xe^{\phi}\sqrt{g},$ $\displaystyle S_{\mathrm{ct}}^{f}$
$\displaystyle=\frac{R}{3}T_{\mathrm{D7}}\int d^{7}x\sqrt{h_{\mathrm{D}7}}.$
(5.9)

We note that the metric presented in (5.7) - (5.9) refers to the metric (2.2)
and (2.4) in Einstein frame which is defined as
$g_{\mathrm{string}}=g_{\mathrm{Einstein}}e^{\phi/2}$. For the CS brane, it is
embedded at $x^{3}=\mathrm{const}$ and $r=r_{Q},r_{H}$ with respect to (2.2)
and (2.4). The onshell action of a CS brane evaluated in the confining
background has been given in (3.31) while it vanishes in the black brane
background (2.2) calculated by using (5.9). Therefore the total onshell action
including the bulk part, flavor part and CS part is,

$S_{\mathrm{onshell}}=S_{\mathrm{bulk}}+S_{f}+S_{\mathrm{CS}}.$ (5.10)

Afterwards using the AdS/CFT dictionary,

$\left\langle e^{-F}\right\rangle=e^{S_{\mathrm{onshell}}},$ (5.11)

with the solution (2.2) (2.4) we can obtain the free energy $F$ respectively,

$\displaystyle F_{d}$
$\displaystyle=-\frac{1}{8}N_{c}^{2}\pi^{2}T^{4}V_{3}\beta_{T}-\frac{36\pi}{\lambda_{t}}N_{c}N_{\mathrm{D}}+\frac{1}{18}N_{c}N_{f}T^{3}V_{3}\lambda_{t}\left[\left(\pi+\ln
64\right)q_{T}-2\right],$ $\displaystyle F_{c}$
$\displaystyle=-\frac{M_{KK}^{4}N_{c}^{2}V_{3}\beta_{3}}{128\pi^{2}}-\frac{36\pi}{\lambda_{t}}N_{c}N_{\mathrm{D}}+\frac{\lambda
M_{KK}^{3}N_{c}k_{0}V_{3}}{64\pi^{2}}G\left(q\right)+\frac{\lambda_{t}M_{KK}^{3}N_{c}N_{f}V_{3}\left[2+q\left(8+\pi-\ln
4\right)\right]}{512\sqrt{2}\pi^{3/2}\Gamma\left(\frac{7}{4}\right)^{2}},$
$\displaystyle\beta_{T}$
$\displaystyle=1/T,q_{T}=Q/r_{H}^{4},\beta_{3}=2\pi/M_{KK},$ (5.12)

where $\lambda_{t}$ is the ’t Hooft coupling constant defined as
$\lambda_{t}=g_{s}N_{c}$ and $F_{d,c}$ refers to the free energy evaluated in
the background (2.2) and (2.4) respectively. $k_{0}$ is the number of CS brane
and we have assumed $k_{0}$ and $N_{f}$ take same order at large-$N_{c}$ due
to the flavor and CS branes as probes. $G\left(q\right)$ is a function defined
as
$G\left(q\right)=E_{\mathrm{CS}}\left(q\right)/E_{\mathrm{CS}}\left(0\right)$
whose behavior has been numerically illustrated in Figure 7. Follow the most
discussion in gauge/gravity duality [33, 34, 35, 36], the black brane (2.2)
and soliton (bubble) solution (2.4) respectively corresponds to the
deconfinement and confinement phase in the dual theory, so the phase
transition can be obtained by compare their free energy which identifies the
confinement/deconfinement phase transition in the field theory as the Hawking-
Page transition in the bulk. According to (5.12), we can find the D-instantons
as D(-1)-branes negatively increase the bulk free energy as a contribution of
$\mathcal{O}\left(N_{c}^{2}\right)$ because at large-$N_{c}$ limit,
$N_{d}/N_{c}$ must be fixed otherwise the backreaction of the D-instantons in
bulk vanishes. The critical temperature $T_{c}$ of the phase transition can be
obtained by comparing the free energy at $F_{d}=F_{c}$ which is evaluated as,

$T_{c}=\frac{M_{KK}}{2\pi}-\frac{\lambda_{t}M_{KK}}{6\pi^{2}}\frac{k_{0}}{N_{c}}G\left(q\right)-\frac{\lambda_{t}M_{KK}}{432\pi^{2}}\frac{N_{f}}{N_{c}}\left(C_{1}+C_{2}q\right)+\mathcal{O}\left(N^{-2}\right),$
(5.13)

where $C_{1,2}$ are two constants given as,

$\displaystyle C_{1}$
$\displaystyle=64\pi+\frac{9\sqrt{2}\pi^{5/2}}{\Gamma\left(\frac{7}{4}\right)^{2}}\simeq
464.66,$ $\displaystyle C_{2}$
$\displaystyle=\frac{9\sqrt{2}\pi^{5/2}\left(8+\pi-\ln
4\right)}{2\Gamma\left(\frac{7}{4}\right)^{2}}-32\pi\left(\pi+\ln
64\right)\simeq 551.815,$ $\displaystyle q$
$\displaystyle=\frac{Q}{r_{KK}^{4}}=\frac{128\pi^{2}}{\lambda_{t}M_{KK}^{3}V_{3}}\frac{N_{\mathrm{D}}}{N_{c}}.$
(5.14)

Notice in the large-$N_{c}$ limit, $q$ is fixed thus $G\left(q\right)$ is also
fixed. So the critical temperature is not affected at
$\mathcal{O}\left(N_{c}^{0}\right)$ while it decreases at
$\mathcal{O}\left(N_{c}^{-1}\right)$ by the presence of the D-instantons
through the flavor and CS branes due to $G\left(q\right)>0$. Since the
behavior of $T_{c}$ is qualitatively consistent with the behavior of
$l_{c}\sim T^{-1}$ obtained by evaluating the entanglement entropy, we may
conclude that the entanglement entropy is indeed able to characterize the
deconfinement phase transition.

##  6 Summary and discussion

In this work, by compactifying on the supersymmetry breaking $S^{1}$, we
construct the supergravity solution for $N_{c}$ black D3-branes with dynamical
$N_{\mathrm{D}}$ D-instantons, i.e. D(-1)-branes, to obtain a 3d confining
Yang-Mills in holography. To exhibit flavors and the CS term in the dual
theory, we also add flavor and CS branes as probe into the bulk geometry hence
the dual theory is expected to be a 3d YMCS with matters or CS QCD-like
theory. The low-energy regime of the 3d dual theory is analyzed by the IIB
supergravity solution which geometrically shows the spontaneous breaking of
the chiral symmetry $U\left(N_{f}\right)$ down to $U\left(p\right)\times
U\left(N_{f}-p\right)$, $p\in\mathbb{Z}$. And at very low-energy, D-instantons
could reduce to a pure CS theory. Due to the presence of the dynamical
D-instantons, the embedding function of the flavor branes depends on the non-
zero charge density of the D-instantons which is realized to be metastable
vacua of instantons in the dual theory. Then we further evaluate the vacuum
structure of the dual theory by including both flavor and CS branes which
leads to a topological phase transition determined by the order parameter
$m^{*}$ in the large-$N_{c}$ limit and $m^{*}$ is increased by the presence of
D-instantons as it is expected. This behavior of $m^{*}$ can be interpreted as
the topological contribution from the CS term in the dual theory, similarly as
the topological contribution to mass in the CS theory. Moreover, we
additionally evaluate the entanglement entropy and total free energy in
holography to investigate the critical length $l_{c}$ and critical temperature
$T_{c}$ which is expected to be the characters of the deconfinement phase
transition. The behavior of $T_{c}$ is in qualitative agreement with the
behavior of $l_{c}$ which implies the quantum entanglement entropy could
indeed be a tool to determine confinement/deconfinement in this holographic
approach.

We would like to give some comments to close this work. First, we notice that
the discrepancy between topological phases characterized by $m^{*}$ becomes
vanished if $E_{\mathrm{CS}}\rightarrow 0$. And in the black brane background
(2.2), the CS brane is excepted to be embedded at $r=r_{H}$ to minimize its
energy which leads to a vanished $E_{\mathrm{CS}}$. Since the black brane
background corresponds to a dual theory at finite temperature, the topological
structure of the vacuum may therefore becomes vanished. So in this sense, our
model might provide a holographic interpretation of that why the topological
aspects of hot QCD by instantons is quite difficult to be measured in
experiment [47, 48, 49, 50].

Second, it is expected the topological phase transition is second order [1, 2,
3, 4, 5, 6, 7] if the number of the CS brane is
$\mathcal{O}\left(N_{c}\right)$. This can be achieved by taking into account
the backreaction of CS branes. However the number of CS brane is given by
$\int_{S^{1}}F_{1}$ which relates to the boundary value of $C_{0}$ in our
current setup. So the bulk dynamic could not involve the backreaction of CS
branes in this work. The valid way to include the backreaction of CS brane is
to solve the IIB supergravity action with a fluctuation of $C_{0}$ sourced by
the CS branes then the next-to-the-leading-order contribution in the
large-$N_{c}$ limit to the vacuum structure would be able to analyze in this
sense. However, we would like to leave this for the future study.

Last but not least, since the topological entanglement entropy is defined as
the finite part of the entanglement [51, 52] which could be the measure of the
topological order, $\Delta S$ should relate to the topological entanglement
entropy. So the critical length $l_{c}$ seemingly shows the transition between
the phases with different topological entanglement entropy. Thus if the
entanglement entropy can characterize the deconfinement phase transition,
$T_{c}$ may also reflect some properties of the topological order in the
theory. However our result also shows, in the large-$N_{c}$ limit, $T_{c}$
becomes nearly independent on the instantons while the behavior of $l_{c}$
remains to be determined by the instantons. Accordingly it seems the
entanglement entropy is more sensitive to the topological properties of the
theory than the critical temperature. And we expect it could be an instructive
way to study the topological structure of YMCS theory.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 12005033, the research startup foundation of Dalian
Maritime University in 2019 under Grant No. 02502608 and the Fundamental
Research Funds for the Central Universities under Grant No. 3132021205.

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