KUNS-2982

YITP-23-101

End of the World Branes from Dimensional Reduction

Shigeki Sugimoto†,∗,⋆ and Yu-ki Suzuki∗

† Department of Physics, Kyoto University, Kyoto 606-8502, Japan

∗Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

⋆Kavli Institute for the Physics and Mathematics of the Universe, University
of Tokyo, Kashiwano-ha, Kashiwa, Chiba 277-8582, Japan

We consider dimensional reduction of cigar geometries which are obtained by a
Wick rotation of black hole solutions. Originally the cigar geometry is smooth
around the tip, but after the dimensional reduction along the Euclidean time
direction, there appears an end-of-the-world brane (ETW brane). We derive the
tension of the brane by two methods: bulk equations of motion and boundary
equations of motion.

In particular, for AdS7-soliton $\times$S4 and AdS4-soliton $\times$S7
backgrounds in M-theory, we find that the tension of the emerging ETW branes
behaves as $t(\Phi)\sim e^{-3\Phi}$ in the string frame. This indicates the
existence of such ETW branes in the strongly coupled regime of type 0A string
theory.

###### Contents

  1. 1 Introduction and Summary
  2. 2 ETW branes from dimensional reduction
    1. 2.1 Action and EOM
    2. 2.2 Brane Tension from the bulk EOM via $\mathbf{Z}_{2}$ orbifold
    3. 2.3 Brane Tension from the boundary EOM
  3. 3 Examples
    1. 3.1 AdSd+1 soliton
    2. 3.2 AdSd+1 soliton $\times$ Sn
    3. 3.3 ETW branes in type 0A string theory
  4. 4 The O8-D8 brane system
  5. 5 Conclusion and Discussion

##  1 Introduction and Summary

The end-of-the-world brane (ETW brane) is a codimension one brane at which the
space-time ends. It appears in various theories with gravity defined in a
space-time with boundaries. It could couple with gravity as well as other
fields, and some of the degrees of freedom may be localized on it. In string
theory, such an object has been studied for long time since it can be obtained
as an orientifold-plane or a fixed plane of a ${\bf Z}_{2}$ orbifold and
played important roles in understanding various dualities in string theory. A
well-known example is an M-theory lift of the $E_{8}\times E_{8}$ Heterotic
string theory, which is obtained by compactifying the 11-dimensional space-
time on an interval $S^{1}/{\bf Z}_{2}$ [1]. The ETW branes are located at the
boundary of the interval, and each of them carries an $E_{8}$ gauge theory on
it. This system is related by S and T-dualities to type IIA string theory with
two orientifold 8-planes (O8-planes)111See e.g. [2] for a review, which is
also called type I’ or type IA string theory. This O8-plane is also an example
of the ETW brane realized in string theory.

More recently, the ETW branes are studied in the context of AdS/BCFT
correspondence[3, 4, 5], which is a holographic duality between conformal
field theory (CFT) with boundaries called boundary confomral field theory
(BCFT)222See [6] for a nice review of BCFT. and theory of gravity in
asymptotically anti-de Sitter (AdS) space-time with ETW branes. It has been
applied to many physical problems such as the Kondo effect [7], the
information paradox [8, 9, 10, 11] and the measurement-induced phase
transition [12, 13], and provided holographic explanations of various
phenomena including relation to quantum information theory [14]. In most of
these works, however, the ETW branes are introduced by hand in bottom up
models of the AdS/BCFT correspondence. It would be nice to have top down
realizations of the ETW branes in string theory.

In this paper, we pursue novel examples of the ETW brane. Usually, in the
bottom up models, the tension of the ETW brane as well as other interaction to
the bulk fields are assumed and dynamics of the brane is determined by solving
the equations of motion (EOM). Unlike such an approach, we start with a smooth
geometry without boundaries (except for the asymptotic boundary), and find
that the ETW brane emerges through dimensional reduction. The tension of the
ETW brane is determined from consistency, as explained below in detail.

To be more concrete, consider a cigar geometry obtained by the Wick rotation
of a static black hole (brane) solution. There is a $U(1)$ isometry acting as
a constant shift of the Euclidean time coordinate $\tau$ and an orbit of the
$U(1)$ action form an $S^{1}$ whose radius shrinks to a zero size at the
horizon. A two-dimensional surface parametrized by $\tau$ and the radial
coordinate $r$ looks like a cigar as depicted in the left panel of figure 1.1.
The tip of the cigar corresponds to the horizon in the original black hole
(brane) geometry. The AdS-soliton geometry, obtained by the Wick rotation of
the AdS-Schwarzschild solution, and Wick rotated black D-brane solutions are
important examples of such kind, which are used to obtain the holographic dual
descriptions of non-supersymmetric gauge theories.333See [15] for a review.
Before the dimensional reduction, the cigar geometry is smooth everywhere.

Figure 1.1: Dimensional reduction of the cigar geometry. The left figure
represents the cigar geometry. This geometry is smooth everywhere and there
are no boundaries (except for the asymptotic boundary). The right figure
describes the geometry after dimensional reduction along $\tau$. Accordingly,
there emerges a physical boundary (ETW brane), which is represented by the
blue dot.

However, the geometry after the dimensional reduction along the Euclidean time
direction has a boundary (ETW brane) at the tip of the cigar as depicted in
the right panel of figure 1.1. Taking into account that through dimensional
reduction, a part of metric component give rise to the dilaton, our statement
is summarized as:

Einstein gravity (with some matter fields) on the $(d+1)$-dimensional cigar
geometry induces the Einstein-dilaton gravity (with some matter fields) with a
$(d-1)$-dimensional ETW brane in $d$-dimensions upon dimensional reduction
along the $\tau$ direction.

Note that the ETW brane obtained in this way is a co-dimension one brane,
which couples with the dilaton field as well as gravity. Our main focus is on
calculating the dilaton dependent tension of the ETW brane via two apparently
different methods: one of them makes use of the bulk EOM and the other uses
the boundary EOM. For the method using bulk EOM, we first note that the metric
and the dilaton field satisfy the EOM for the Einstein-dilaton theory obtained
by the dimensional reduction, because the background cigar geometry we used is
a solution of the Einstein equation in $(d+1)$ dimensions. Then, we introduce
a $\mathbf{Z}_{2}$ orbifold as a guide to evaluate the tension of the ETW
brane so that it can be a source of gravity and the dilaton field consistent
with the bulk solution.444This method was recently used in [16] for the study
of codimension one branes in string theory. For the other method, the tension
of the ETW brane is fixed by solving the boundary EOM as in [17, 4, 5]. We can
see that the energy-momentum tensor on the brane contributes to the boundary
EOM, enabling us to evaluate the tension as well. As a consistency check, we
also apply these methods to a system with an O8-plane and D8 branes in type
IIA superstring theory, and show that the known values of the tensions are
reproduced.

For example, we carry out the calculation in AdS-soliton backgrounds with or
without internal geometies (spheres). For the purely $d+1$-dimensional AdS-
soliton background, the tension of the resulting ETW brane is proportional to
$t(\Phi)=-de^{-\sqrt{\frac{d-1}{2(d-2)}}\Phi}$, where $\Phi$ is the dilaton
field. Note that only for $d=3$ the coefficient of $\Phi$ in the exponent
becomes an integer. As an application to string theory, we also consider the
AdS${}_{7}\times$S4 and the AdS${}_{4}\times$S7 soliton solutions in M-theory
and take the dimensional reduction along the $\tau$ direction. Since the
fermions of the system obey anti-periodic boundary condition along the
Euclidean time direction, the supersymmetry is completely lost and the theory
is conjectured to be equivalent to type 0A string theory [18]. Therefore, the
ETW branes obtained by the dimensional reduction are expected to exist in type
0A string theory. In both backgrounds, it turns out that the tension of the
ETW branes behaves as $t(\Phi)\sim e^{-3\Phi}$ in the string frame, which does
not appear in the perturbative string coupling expansion series. However, this
is not a contradiction because the supergravity background in 10 dimensions is
singular at the ETW brane and the perturbative calcultions in string theory
cannot be trusted. It would be interesting to have further evidence of the
existence of such ETW branes.

This paper is organized as follows. In section 2, first we fomulate the
general action after the dimensional reduction and derive the EOM. Then, we
explain how to fix the tension from the bulk EOM via $\mathbf{Z}_{2}$ orbifold
and boundary EOM. We show that these two methods give the same answer. In
section 3, we present our calculations in various examples. We firstly
consider AdSd+1 soliton geometry and then also extend to AdS${}_{4}\times$S7
and AdS${}_{7}\times$S4 geometries. In particular, we obtain a peculiar
dilaton dependence of the ETW branes obtained via the dimensional reduction.
In section 4, we apply our method to determine the tension from bulk EOM and
boundary EOM to the O8-D8 brane systems. We can see that the correct value of
the tension is reproduced. Section 5 is for conclusion and comments on
possible application to the replica manifold [19] and connection to the cosmic
brane [20].

##  2 ETW branes from dimensional reduction

###  2.1 Action and EOM

In this section, we first explain our set-up and prepare the action and
equations of motion (EOM) that will be used in later sections.

Our starting point is Einstein gravity theory with an $(n-1)$-form gauge field
in $D=d+n+1$ dimensions. The action is

$S=\frac{1}{16\pi G_{N}^{(D)}}\int
d^{D}x\sqrt{-G_{(D)}}\left(R_{(D)}-\frac{1}{2}|f_{n}|_{G}^{2}-2\Lambda\right),$
(2.1)

where $G_{N}^{(D)}$ is the Newton’s constant, $G_{(D)}=\det{(G_{AB}})$ is the
determinant of the $D$-dimensional metric $G_{AB}$ ($A,B=0,\cdots,D-1$),
$R_{(D)}$ is the Ricci scalar curvature, $\Lambda$ is the cosmological
constant, $f_{n}=\frac{1}{n!}f_{A_{1}\cdots A_{n}}dx^{A_{1}}\wedge\cdots\wedge
dx^{A_{n}}$ is the field strength of the $(n-1)$-form gauge field and

$|f_{n}|_{G}^{2}=\frac{1}{n!}f_{A_{1}\cdots A_{n}}f_{B_{1}\cdots
B_{n}}G^{A_{1}B_{1}}\cdots G^{A_{n}B_{n}}.$ (2.2)

We consider a background parametrized by $(x^{A})=(x^{\mu},\theta^{i},r,\tau)$
($A=0,1,\cdots,D-1$; $\mu=0,1,\cdots,d-2$; $i=1,2,\cdots,n$), where $r$ is a
radial coordinate bounded from below as $r\geq r_{h}$, $\tau$ is a periodic
coordinate of period $\beta$, and $(\theta^{i})$ are coordinates of the unit
$n$-sphere $S^{n}$. The background configuration of the $n$-form field
strength $f_{n}$ is assumed to be proportional to the volume-form of this
$S^{n}$, and the metric is of the form

$ds_{(D)}^{2}=G_{AB}dx^{A}dx^{B}=W_{1}(r)(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+f(r)d\tau^{2})+W_{2}(r)dr^{2}+W_{3}(r)d\Omega_{n}^{2},$
(2.3)

where $W_{1,2,3}(r)$ and $f(r)$ are non-negative functions of $r$,
$\eta_{\mu\nu}$ is the $d-1$ dimensional Minkowski metric and
$d\Omega_{n}^{2}$ is the line element of the unit $S^{n}$. The key assumption
here is that the function $f(r)$ vanishes at $r=r_{h}$, while $W_{1,2,3}$ are
non-zero there. The period $\beta$ is chosen so that the geometry is smooth
everywhere including $r=r_{h}$. Therefore, the radius of the $S^{1}$
parametrized by $\tau$ shrinks to zero size at $r=r_{h}$ and the geometry
contains a structure of the cigar depicted in the left panel of figure 1.1.
This kind of geometry is realized as the (double) Wick rotation of a black
$(d-1)$-brane solution, in which $\tau$ is the Euclidean time coordinate and
$r$ is the radial coordinate transverse to the brane. The tip of the cigar at
$r=r_{h}$ corresponds to the horizon of the black brane solution.

The next step is the dimensional reduction along the $\tau$ direction.555Here,
we focus on the zero modes of the Fourier decomposition of the fields along
the $\tau$ direction, assuming that the contribution from the non-zero modes
can be neglected. In some examples considered in the next section, the radius
of the $S^{1}$ diverges at the asymptotic boundary $r\rightarrow\infty$, where
the dimensional reduction cannot be justified. These backgrounds can be
regarded as the near horizon geometry of asymptotically flat backgrounds in
which the radius of $S^{1}$ approaches a constant at $r\rightarrow\infty$.
Since we are mainly interested in the physics around the tip of the cigar, we
do not consider compactness seriously here. The geometry after the dimensional
reduction has a new co-dimension one boundary at $r=r_{h}$, which is regarded
as an ETW brane.

Let us decompose the metric as

$ds_{(D)}^{2}=e^{-\alpha\Phi}\hat{g}_{MN}dx^{M}dx^{N}+e^{(d+n-2)\alpha\Phi}d\tau^{2},$
(2.4)

where $(x^{M})=(x^{\mu},\theta^{i},r)$ ($M=0,1,\cdots,d+n-1$) are the
$D-1=d+n$ dimensional coordinates, $\hat{g}_{MN}$ is the $D-1=d+n$ dimensional
metric, $\Phi$ is the dilaton field and

$\alpha=\sqrt{\frac{2}{(d+n-1)(d+n-2)}}=\sqrt{\frac{2}{(D-2)(D-3)}}\ .$ (2.5)

Here, we have neglected the off-diagonal $(M,\tau)$ components of the metric.
Similarly, we only keep the components of $f_{n}$ without the $\tau$ index.
Then, the theory is reduced to the Einstein-dilaton graviy with $(n-1)$-form
gauge field in a $D-1$ dimensional space-time with the ETW brane at $r=r_{h}$.
Inserting the metric (2.4) into the action (2.1) the bulk action is obtained
as

$\displaystyle S_{\rm bulk}=$ $\displaystyle\frac{1}{16\pi
G_{N}}\int_{r>r_{h}}d^{d-1}x\,dr\,d^{n}\theta\sqrt{-\hat{g}}\left(\hat{R}-\frac{1}{2}\hat{g}^{MN}\partial_{M}\Phi\partial_{N}\Phi-v(\Phi)-\frac{1}{2}w(\Phi)|f_{n}|^{2}\right)\
,$ (2.6)

where $G_{N}=G_{N}^{(D)}/\beta$, $\hat{g}=\det(\hat{g}_{MN})$, $\hat{R}$ is
the Ricci scalar with respect to the $D-1$ dimensional metric $\hat{g}_{MN}$,

$|f_{n}|^{2}=\frac{1}{n!}f_{M_{1}\cdots M_{n}}f_{N_{1}\cdots
N_{n}}\hat{g}^{M_{1}N_{1}}\cdots\hat{g}^{M_{n}N_{n}}$ (2.7)

and

$v(\Phi)=2\,e^{-\alpha\Phi}\Lambda,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ w(\Phi)=e^{(n-1)\alpha\Phi}\ .$ (2.8)

In this paper, we analyze the ETW branes in two ways. One of them is to treat
it as a boundary of the space-time, and add a boundary action that consists of
Gibbons-Hawking term and the world-volume action with the dilaton dependent
tension. This treatment will be discussed in section 2.3. The other one is to
regard the space-time as a ${\bf Z}_{2}$ orbifold with the ETW brane obtained
as the fixed plane of the ${\bf Z}_{2}$ orbifold action. To be more explicit,
we introduce a coordinate $y\in{\bf R}$ of the covering space related to $r$
as $|y|=r-r_{h}$, on which the ${\bf Z}_{2}$ acts as a sign flip
$y\rightarrow-y$.666We actually take the orbifold fixed point to be slightly
away from $r=r_{h}$, since $r=r_{h}$ is singular after the dimensional
reduction. See section 2.2 for details. Then, working in the covering space,
the ETW brane is a brane with dilaton dependent tension placed at $y=0$. The
action for this system is

$\displaystyle S=$ $\displaystyle\frac{1}{16\pi G_{N}}\int
d^{d-1}x\,dy\,d^{n}\theta\sqrt{-\hat{g}}\left(\hat{R}-\frac{1}{2}\hat{g}^{MN}\partial_{M}\Phi\partial_{N}\Phi-v(\Phi)-\frac{1}{2}w(\Phi)|f_{n}|^{2}-\frac{t(\Phi)}{\sqrt{\hat{g}_{yy}}}\delta(y)\right),$

where $t(\Phi)$ is the dilaton dependent tension of the ETW brane to be
determined.

To obtain the background, we assume the following ansatz for the $D-1$
dimensional metric:

$ds^{2}=\hat{g}_{MN}dx^{M}dx^{N}=e^{2A(r)}g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+e^{2B(r)}dr^{2}+e^{2F(r)}d\Omega_{n}^{2}.$
(2.10)

The $d-1$ dimensional metric $g_{\mu\nu}(x)$, functions $A(r)$, $B(r)$, $F(r)$
and the dilaton field $\Phi(r)$ for the background (2.3) are given by

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}\ ,$ (2.11) $\displaystyle
e^{(D-3)\alpha\Phi(r)}=W_{1}(r)f(r)\ ,$ (2.12) $\displaystyle
e^{2A(r)}=e^{\alpha\Phi(r)}W_{1}(r)\ ,$ (2.13) $\displaystyle
e^{2B(r)}=e^{\alpha\Phi(r)}W_{2}(r)\ ,$ (2.14) $\displaystyle
e^{2F(r)}=e^{\alpha\Phi(r)}W_{3}(r)\ .$ (2.15)

The $n$-form field strength $f_{n}$ is assumed to be of the form

$f_{n}=\frac{1}{n!}f_{i_{1}\cdots i_{n}}d\theta^{i_{1}}\wedge\cdots\wedge
d\theta^{i_{n}}=\frac{c}{n!}\epsilon_{i_{1}\cdots
i_{n}}d\theta^{i_{1}}\wedge\cdots\wedge d\theta^{i_{n}},$ (2.16)

where $\theta^{i}$ ($i=1,2,\cdots,n$) are coordinates of the unit $S^{n}$ and
$\epsilon_{i_{1}\cdots i_{n}}$ is the Levi-Civita symbol on $S^{n}$. Since the
flux $\int_{S^{n}}f_{n}$ should be independent of $r$, the coefficient $c$ is
a constant. Then, we obtain

$\displaystyle|f_{n}|^{2}$ $\displaystyle=\frac{1}{n!}f_{i_{1}\cdots
i_{n}}f^{i_{1}\cdots i_{n}}=c^{2}e^{-2nF}.$ (2.17)

Substituting these ansatz (2.10) and (2.16) into the action (LABEL:action2)
and varying it with respect to $A$, $B$, $F$ and $\Phi$, we obtain the
following EOM:

$\displaystyle\frac{\delta S}{\delta A}=0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ $
$\displaystyle(1-2/(d-1))e^{-2A+2B}R+n(n-1)e^{-2F+2B}-2(d-2)(A^{\prime\prime}-A^{\prime}B^{\prime})$
$\displaystyle-2n(F^{\prime\prime}-F^{\prime}B^{\prime})-(d-1)(d-2)(A^{\prime})^{2}-2n(d-2)A^{\prime}F^{\prime}-n(n+1)(F^{\prime})^{2}$
$\displaystyle=\frac{1}{2}(\Phi^{\prime})^{2}+e^{2B}v(\Phi)+\frac{c^{2}}{2}e^{-2nF+2B}w(\Phi)+e^{B}t(\Phi)\delta(y)\
,$ (2.18) $\displaystyle\frac{\delta S}{\delta B}=0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ $ $\displaystyle e^{-2A+2B}R+n(n-1)e^{-2F+2B}$
$\displaystyle-\left((d-1)(d-2)(A^{\prime})^{2}+n(n-1)(F^{\prime})^{2}+2ndA^{\prime}F^{\prime}\right)+\frac{1}{2}(\Phi^{\prime})^{2}$
$\displaystyle=e^{2B}\left(v(\Phi)+\frac{c^{2}}{2}e^{-2nF}w(\Phi)\right)\ ,$
(2.19) $\displaystyle\frac{\delta S}{\delta F}=0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ $ $\displaystyle
e^{-2A+2B}R+(n-1)(n-2)e^{-2F+2B}-2(n-1)(F^{\prime\prime}-F^{\prime}B^{\prime})$
$\displaystyle-2(d-1)(A^{\prime\prime}-A^{\prime}B^{\prime})-n(n-1)(F^{\prime})^{2}-2(d-1)(n-1)A^{\prime}F^{\prime}-d(d-1)(A^{\prime})^{2}$
$\displaystyle=\frac{1}{2}(\Phi^{\prime})^{2}+e^{2B}v(\Phi)-\frac{c^{2}}{2}e^{-2nF+2B}w(\Phi)+e^{B}t(\Phi)\delta(y)\
,$ (2.20) $\displaystyle\frac{\delta S}{\delta\Phi}=0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ $
$\displaystyle\Phi^{\prime\prime}+((d-1)A^{\prime}+nF^{\prime}-B^{\prime})\Phi^{\prime}$
$\displaystyle=e^{2B}\left(\frac{dv(\Phi)}{d\Phi}+\frac{c^{2}}{2}e^{-2nF}\frac{dw(\Phi)}{d\Phi}\right)+e^{B}\frac{dt(\Phi)}{d\Phi}\delta(y),$
(2.21)

where prime stands for the derivative with respect to $y$, e.g.
$A^{\prime}=\frac{\partial A}{\partial y}$,
$A^{\prime\prime}=\frac{\partial^{2}A}{\partial y^{2}}$, etc., and $R$ is the
Ricci scalar for the $d-1$ dimensional metric. These EOM can also be obtained
directly from the EOM derived from the action (LABEL:action2). Though we do
not write down explicitly, the EOM for the metric $g_{\mu\nu}$ should also be
imposed.

###  2.2 Brane Tension from the bulk EOM via $\mathbf{Z}_{2}$ orbifold

The next task is to determine $t(\Phi)$ for given $A,B,F$ and $\Phi$. In this
section we apply the $\mathbf{Z}_{2}$ orbifold method to the above bulk EOM
(2.18)–(2.21) to fix the tension. To do that, as mentioned in the previous
section, we introduce the coordinate $y\in\mathbf{R}$ related to $r$ as

$r=\begin{cases}r_{c}+y,&\text{for \quad}y>0\ ,\\\ r_{c}-y,&\text{for
\quad}y<0\ ,\end{cases}$ (2.22)

where $r_{c}$ is a cut-off satisfying $r_{c}>r_{h}$. In this coordinate $y$,
the geometry restricted to $r\geq r_{c}$ and its copy are glued together at
$y=0$. We will eventually take $r_{c}\rightarrow r_{h}$ limit. According to
this, the functions $A(r),B(r)$, etc. in (2.10) can be redefined as a function
of $y$ as

$A(y)=\begin{cases}A|_{r=r_{c}+y},&\text{for \quad}y>0\ ,\\\
A|_{r=r_{c}-y}.&\text{for \quad}y<0\ .\\\ \end{cases}$ (2.23)

Taking the derivative with respect to $y$, we obtain

$A^{\prime}:=\partial_{y}A(y)=\begin{cases}\partial_{r}A|_{r=r_{c}+y},&\text{for}\leavevmode\nobreak\
y>0\ ,\\\ -\partial_{r}A|_{r=r_{c}-y},&\text{for}\leavevmode\nobreak\ y<0\
,\end{cases}$ (2.24)

which leads to the discontinuity at $y=0$:

$[A^{\prime}]^{+0}_{-0}=2\partial_{r}A|_{r=r_{c}}\ .$ (2.25)

Integrating the EOM (2.18)-(2.21), we obtain

$\displaystyle{[}-2(d-2)A^{\prime}-2nF^{\prime}{]}^{+0}_{-0}=e^{B(0)}t(\Phi(0))\
,$ (2.26)
$\displaystyle{[}-2(n-1)F^{\prime}-2(d-1)A^{\prime}{]}^{+0}_{-0}=e^{B(0)}t(\Phi(0))\
,$ (2.27)
$\displaystyle{[}\Phi^{\prime}{]}^{+0}_{-0}=e^{B(0)}\frac{dt(\Phi(0))}{d\Phi}\
,$ (2.28)

where we pick up the contribution from the delta fucntion. Here, we have
assumed $g^{\prime}_{\mu\nu}$ is continuous at $y=0$. From (2.26) and (2.27),
we obtain

$\displaystyle{[}A^{\prime}{]}^{+0}_{-0}={[}F^{\prime}{]}^{+0}_{-0}\ ,$ (2.29)
$\displaystyle-2(d+n-2)[A^{\prime}]^{+0}_{-0}=e^{B(0)}t(\Phi(0))\ .$ (2.30)

When there is no sphere part (the case with $n=0$), the relevant EOM are
(2.28) and (2.30). (2.28) and (2.30) imply

$\frac{d}{d\Phi}\log
t(\Phi)=-\frac{1}{2(d+n-2)}\frac{[\Phi^{\prime}]^{+0}_{-0}}{[A^{\prime}]^{+0}_{-0}}\
.$ (2.31)

The tension $t(\Phi)$ of the ETW brane can be determined by solving these
equations as we will demonstrate in section 3.

###  2.3 Brane Tension from the boundary EOM

It is also possible to determine the tension of the ETW brane by using the
boundary EOM. Here, we explain how to do this and show that it leads to the
same conditions as those found in the previous subsection.

Let us consider the space-time for $y>0$ with the ETW brane at $y=0$. The
action consists of the bulk and boundary contributions as

$S=S_{\rm bulk}+S_{\rm bdy}\ ,$ (2.32)

where $S_{\rm bulk}$ is the bulk action given in (2.6) integrated in the $y>0$
region. The boundary action $S_{\rm bdy}$ includes the Gibbons-Hawking term
$S_{\rm GH}$ and the brane action $S_{\rm brane}$ given by

$\displaystyle S_{\rm bdy}$ $\displaystyle=S_{\rm GH}+S_{\rm brane}\ ,$ (2.33)
$\displaystyle S_{\rm GH}$ $\displaystyle=\frac{1}{8\pi
G_{N}}\int_{y=0}d^{d-1}x\,d^{n}\theta\sqrt{-h}\,K\ ,$ (2.34) $\displaystyle
S_{\rm brane}$ $\displaystyle=-\frac{1}{32\pi
G_{N}}\int_{y=0}d^{d-1}x\,d^{n}\theta\sqrt{-h}\,t(\Phi)\ ,$ (2.35)

where $h$ is the determinant of the induced metric $h_{ab}$
($a,b=0,\cdots,d+n-2$) on the brane at $y=0$:

$ds^{2}|_{y=0}=h_{ab}(x)dx^{a}dx^{b}=e^{2A(0)}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2F(0)}d\Omega_{n}^{2}\
,$ (2.36)

and $K=h^{ab}K_{ab}$ is the trace of the extrinsic curvature $K_{ab}$. Note
that the coefficient of $t(\Phi)$ in the brane action (2.35) is a half of that
in (LABEL:action2). This is because (LABEL:action2) is the action for the
covering space of the $\mathbf{Z}_{2}$ orbifold.777The tension $T$ in the
original reference [4] is related with the tension $t(\Phi)$ in (2.35) via
$T=\frac{t(\Phi)}{4}$.

When the background metric is given by

$ds^{2}=dz^{2}+h_{ab}(x,z)dx^{a}dx^{b}$ (2.37)

(Gaussian normal coordinates), the extrinsic curvature for the co-dimension
one surface at $z=0$ is given by

$K_{ab}=\frac{1}{2}\partial_{z}h_{ab}|_{z=0}.$ (2.38)

Using this formula, the extrinsic curvature for the background with metric
(2.10) can be computed as

$\displaystyle K_{\mu\nu}$ $\displaystyle=$ $\displaystyle
A^{\prime}(0)e^{2A(0)-B(0)}g_{\mu\nu}(x)\ ,$ (2.39) $\displaystyle K_{ij}$
$\displaystyle=$ $\displaystyle F^{\prime}(0)e^{2F(0)-B(0)}g_{ij}(\theta)\ ,$
(2.40) $\displaystyle K$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}K_{\mu\nu}+g^{ij}K_{ij}=e^{-B(0)}(A^{\prime}(0)d+F^{\prime}(0)n)\ ,$
(2.41)

where $g_{ij}(\theta)$ is the metric of the unit $S^{n}$:

$\displaystyle d\Omega_{n}^{2}=g_{ij}d\theta^{i}d\theta^{j}\ .$ (2.42)

We impose the Neumann type boundary condition for the metric and the dilaton
fields. The variation of the action with respect to the metric gives the
boundary terms as

$\delta S=(\mbox{bulk terms})+\frac{1}{16\pi
G_{N}}\int_{y=0}d^{d-1}x\,d^{n}\theta\sqrt{-h}\,\delta
h^{ab}\left(K_{ab}-Kh_{ab}+8\pi G_{N}T_{ab}\right)\ ,$ (2.43)

where $T_{ab}$ is the energy momentum tensor for the brane at the boundary

$T_{ab}=\frac{2}{\sqrt{-h}}\frac{\delta S_{\rm brane}}{\delta
h^{ab}}=-\frac{1}{32\pi G_{N}}t(\Phi)h_{ab}\ .$ (2.44)

Therefore, the boundary equations of motion is

$K_{ab}-Kh_{ab}+8\pi G_{N}T_{ab}=0\ ,$ (2.45)

which implies

$\displaystyle-4((d-2)A^{\prime}+nF^{\prime})|_{y=0}$ $\displaystyle=$
$\displaystyle e^{B(0)}t(\Phi)\ ,$ (2.46)
$\displaystyle-4((d-1)A^{\prime}+(n-1)F^{\prime})|_{y=0}$ $\displaystyle=$
$\displaystyle e^{B(0)}t(\Phi)\ .$ (2.47)

These equations are equivalent to (2.26) and (2.27).

Next, consider the variation with respect to $\Phi$. The boundary terms for
the variation of the action is

$\delta S=(\mbox{bulk terms})+\frac{1}{16\pi
G_{N}}\int_{y=0}d^{d-1}x\,d^{n}\theta\sqrt{-h}\,\delta\Phi\left(\frac{1}{\sqrt{g_{yy}}}\partial_{y}\Phi-\frac{1}{2}\frac{dt(\Phi)}{d\Phi}\right),$
(2.48)

which yields

$2\Phi^{\prime}(0)=e^{B(0)}\frac{dt(\Phi(0))}{d\Phi}\ .$ (2.49)

This agrees with (2.28).

##  3 Examples

###  3.1 AdSd+1 soliton

Let us start with the case with $n=0$ and a negative cosmological constant
parametrized as

$\Lambda=-\frac{d(d-1)}{2L^{2}}\ ,$ (3.1)

for which the action (2.1) is

$S=\frac{1}{16\pi G_{N}^{(d+1)}}\int
d^{d+1}x\sqrt{-G}\left(R-\frac{d(d-1)}{L^{2}}\right)\ .$ (3.2)

We consider the $(d+1)$-dimensional AdS-soliton background [21, 22] given by

$ds^{2}=\frac{r^{2}}{L^{2}}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+f(r)d\tau^{2}\right)+\frac{L^{2}}{r^{2}}\frac{dr^{2}}{f(r)}\
,$ (3.3)

where $\mu,\nu=0,1,\cdots,d-2$ and

$f(r)=1-\frac{r_{h}^{d}}{r^{d}}\ .$ (3.4)

This is a solution of the EOM obtained from (3.2).

In order to analyze the geometry near the tip $r=r_{h}$, we introduce
dimensionless coordinates $\tilde{x}^{\mu}$, $\tilde{\tau}$ and $z$ defined by

${\tilde{x}}^{\mu}=\frac{r_{h}x^{\mu}}{L^{2}}\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{\tau}=\frac{r_{h}\tau}{L^{2}}\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{r^{d}}{r_{h}^{d}}=1+z^{2}\ .$
(3.5)

For convenience, we omit the tilde of these coordinates and set $L=1$ in the
following. Then, the metric becomes

$\displaystyle ds^{2}$ $\displaystyle=G_{AB}dx^{A}dx^{B}$
$\displaystyle=(1+z^{2})^{\frac{2}{d}}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{z^{2}}{1+z^{2}}d\tau^{2}\right)+\frac{4dz^{2}}{d^{2}(1+z^{2})}\
,$ (3.6)

where $A,B=0,\cdots,d$. Then, the metric near $z=0$ becomes

$ds^{2}\simeq\eta_{\mu\nu}dx^{\mu}dx^{\nu}+z^{2}d\tau^{2}+\frac{4}{d^{2}}dz^{2}\
,$ (3.7)

from which we find that the geometry is smooth at the tip $z=0$ if we choose
the period of $\tau$ to be $\beta=4\pi/d$.

Using the ansatz (2.4) and (2.10), the metric is written as

$\displaystyle ds^{2}$
$\displaystyle=e^{-\sqrt{\frac{2}{(d-1)(d-2)}}\Phi}\left(e^{2A}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2B}dz^{2}\right)+e^{\sqrt{\frac{2(d-2)}{d-1}}\Phi}d\tau^{2}\
,$ (3.8)

with

$\displaystyle e^{2A}$
$\displaystyle=(1+z^{2})^{\frac{1}{d}}z^{\frac{2}{d-2}}\ ,$ $\displaystyle
e^{2B}$
$\displaystyle=\frac{4}{d^{2}}(1+z^{2})^{-\frac{1}{d}-1}z^{\frac{2}{d-2}}\ ,$
$\displaystyle e^{\sqrt{\frac{2(d-2)}{d-1}}\Phi}$
$\displaystyle=z^{2}(1+z^{2})^{\frac{2}{d}-1}\ .$ (3.9)

Here, we have assumed $d>2$.

Following the procedure explained in section 2.2, we introduce a coordinate
$-\infty<y<\infty$ related to $z$ with $z=z_{c}+|y|$, where $z_{c}$ is a
cutoff that will be sent to $z_{c}\rightarrow 0$ at the end. Then, near the
orbifold fixed point $y=0$, $A$ and $\Phi$ behave as

$\displaystyle A$ $\displaystyle\simeq\frac{1}{d-2}\log(z_{c}+|y|)\ ,$
$\displaystyle B$
$\displaystyle\simeq\frac{1}{d-2}\log(z_{c}+|y|)+\log(4/d^{2})\ ,$
$\displaystyle\Phi$
$\displaystyle\simeq\sqrt{\frac{2(d-1)}{d-2}}\log(z_{c}+|y|)\ ,$ (3.10)

which lead to the discontinuity

$\displaystyle[A^{\prime}]_{-0}^{+0}=\frac{1}{d-2}\frac{2}{z_{c}}\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
[\Phi^{\prime}]_{-0}^{+0}=\sqrt{\frac{2(d-1)}{d-2}}\frac{2}{z_{c}}\ .$ (3.11)

Using this and (2.31), we obtain

$\frac{d}{d\Phi}\log
t(\Phi)=-\sqrt{\frac{d-1}{2(d-2)}}\quad\Rightarrow\leavevmode\nobreak\
t\propto e^{-\sqrt{\frac{d-1}{2(d-2)}}\Phi}.$ (3.12)

Substituting this into (2.30), we can also fix the coefficient as

$t(\Phi)=-2d\,e^{-\sqrt{\frac{d-1}{2(d-2)}}\Phi}\ .$ (3.13)

Note that only for $d=3$ the coefficient of $\Phi$ in the exponent becomes
integer.

Note that the tension (3.13) diverges at $z=0$, since $\Phi\rightarrow-\infty$
as $z\rightarrow 0$. Indeed, the $d$-dimensional theory obtained via the
dimensional reduction is singular at $z=0$. However, this singularity is
resolved in the original $d+1$ dimensional theory. The situation is similar to
a D6-brane in type IIA string theory and its M-theory lift. Note also that
although the tension (3.13) is negative, the system is stable, since the AdS-
soliton background is believed to be stable.[21]

###  3.2 AdSd+1 soliton $\times$ Sn

It is straightforward to generalize the discussion in the previous subsection
to the case with $n>0$. The $D=d+n+1$ dimensional theory (2.1) admits an
AdSd+1 soliton $\times S^{n}$ background with

$ds^{2}_{(D)}=\frac{r^{2}}{L^{2}}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+f(r)d\tau^{2}\right)+\frac{L^{2}}{r^{2}}\frac{dr^{2}}{f(r)}+\tilde{L}^{2}d\Omega_{n}^{2}$
(3.14)

with

$f(r)=1-\frac{r_{h}^{d}}{r^{d}}\ ,$ (3.15)

where $\mu,\nu=0,1,\cdots,d-2$, and $L$, $\tilde{L}$ and $r_{h}$ are positive
constants. This metric takes the form (2.3) with

$W_{1}=\frac{r^{2}}{L^{2}}\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
W_{2}=\frac{L^{2}}{r^{2}f(r)}\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
W_{3}=\tilde{L}^{2}\ .$ (3.16)

The configuration of the dilaton field $\Phi$ and the metric (2.10) for the
$D-1=d+n$ dimensional theory obtained via dimensional reduction along the
$\tau$ direction is obtained from the relations (2.12)–(2.15) as

$\displaystyle e^{(D-3)\alpha\Phi(r)}=\frac{r^{2}}{L^{2}}f(r)\ ,$ (3.17)
$\displaystyle
e^{2A(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{\frac{1}{D-3}}\frac{r^{2}}{L^{2}}\
,$ (3.18) $\displaystyle
e^{2B(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{-\frac{D-4}{D-3}}\ ,$ (3.19)
$\displaystyle
e^{2F(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{\frac{1}{D-3}}\tilde{L}^{2}\
,$ (3.20)

where $\alpha$ is defined in (2.5). Then, near the tip of the cigar $r=r_{h}$,
$\Phi$ and $A$ behave as

$\displaystyle\Phi$ $\displaystyle\simeq\frac{1}{(D-3)\alpha}\log f(r)\ ,$
(3.21) $\displaystyle A$ $\displaystyle\simeq\frac{1}{2(D-3)}\log f(r)\ ,$
(3.22)

with which (2.31) implies

$\frac{d}{d\Phi}\log
t(\Phi)=-\frac{1}{(D-3)\alpha}=-\sqrt{\frac{D-2}{2(D-3)}}\ ,$ (3.23)

and

$t(\Phi)\propto e^{-\sqrt{\frac{D-2}{2(D-3)}}\Phi}\ .$ (3.24)

Substituting this to (2.30), we obtain

$t(\Phi)=-2d\frac{r_{h}}{L^{2}}\,e^{-\sqrt{\frac{D-2}{2(D-3)}}\Phi}\ .$ (3.25)

###  3.3 ETW branes in type 0A string theory

In this section, we consider AdS soliton backgrounds (AdS4-soliton $\times$S7
and AdS7-soliton $\times$S4) in M-theory and take the dimensional reduction
along the $\tau$ direction. Such backgrounds are obtained as supergravity
solutions corresponding to M2 or M5-branes wrapped on a circle, along which
the anti-periodic boundary condition is imposed on the fermions. This theory
is conjectured to be dual to the type 0A string theory, because supersymmetry
is completely broken by this boundary condition [18]. Correspondingly, the tip
of the cigar in the AdS-soliton in M-theory is expected to become ETW brane in
type 0A string theory. The tension of these objects can be calculated by using
the results obtained in the previous subsection.

The low energy effective theory of M-theory is described by 11 dimensional
supergravity, in which the massless bosonic fields are the metric and a 3-form
gauge field. The action is given by (2.1) with $D=11$, $n=4$, $d=6$ and
$\Lambda=0$. The electromagnetic dual description obtained by taking the Hodge
dual of the 4-form field strength is $D=11$, $n=7$ and $d=3$. It is known that
it admits AdS4-soliton $\times$S7 and AdS7-soliton $\times$S4 backgrounds with
(2.16) and (3.14). In both of these backgrounds, (2.5) implies $\alpha=1/6$
and (3.17)–(3.20) become

$\displaystyle e^{\Phi(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{3/4}\ ,$
(3.26) $\displaystyle
e^{2A(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{\frac{1}{8}}\frac{r^{2}}{L^{2}}\
,$ (3.27) $\displaystyle
e^{2B(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{-\frac{7}{8}}\ ,$ (3.28)
$\displaystyle
e^{2F(r)}=\left(\frac{r^{2}}{L^{2}}f(r)\right)^{\frac{1}{8}}\tilde{L}^{2}\ .$
(3.29)

The tension (3.25) is

$t(\Phi)=-2d\frac{r_{h}}{L^{2}}e^{-\frac{3}{4}\Phi}\ ,$ (3.30)

with $d=3$ and $d=6$ for AdS4-soliton $\times$S7 and AdS7-soliton $\times$S4
backgrounds, respectively.

Note that these results are for the Einstein frame. If we write the brane
action (2.35) in the string frame as

$S_{\rm brane}=-\frac{1}{32\pi G_{N}}\int d^{9}x\sqrt{-h_{\rm st}}\,t_{\rm
st}(\Phi)\ ,$ (3.31)

where $h_{\rm st}$ is the determinant of the induced metric in the string
frame ($h^{\rm st}_{ab}=e^{-\Phi/2}h_{ab}$), the tension is

$t_{\rm st}(\Phi)=-2d\frac{r_{h}}{L^{2}}e^{-3\Phi}\ .$ (3.32)

The peculiar dilaton dependence in (3.32) suggests that this ETW brane is a
non-perturbative object in type 0A string theory. In fact, in perturbative
string theory, the scattering amplitudes depend on the dilaton as
$e^{-\chi\Phi}$, where $\chi$ is the Euler characteristic of the string world-
sheet. Since the Euler characteristic $\chi$ is given by $\chi=2-2g-b-c$,
where $g$, $b$ and $c$ are the number of handles, boundaries and crosscaps of
the world-sheet, respectively, $\chi$ is at most $2$ and it is not possible to
realize $t(\Phi)\sim e^{-3\Phi}$ found in (3.32) in the perturbative
expansion. This is not a contradiction, because our calculation is based on
M-theory description, which corresponds to the strongly coupled regime in
string theory. Note that the dilaton dependence $e^{-3\Phi}$ has been
observed, for example, in the tension of the S-dual of a D7-brane in type IIB
string theory.

##  4 The O8-D8 brane system

In section 2, we explained how to determine the tension of the ETW brane via
bulk EOM and boundary EOM. In this section, as a consistency check, we apply
these methods to a system with an O8-plane888Here, we consider a negative
tension orientifold 8-plane called O8--plane. and D8-branes in type IIA string
theory, and show that it reproduces the correct value of the tension of these
objects.

Let us consider type IIA string theory in ${\bf R}^{9}\times I$, where
$I=[0,\ell]$ is an interval parametrized by $u$, with an O8-plane at each
boundary $u=0,\ell$ and 16 parallel D8-branes localized along $I$.999This
system is obtained as the T-dual of type I string theory and is also called as
type I’ or type IA string theory. We place $N$ D8-branes at $u=0$ and $16-N$
D8-branes at $u=\ell$. The corresponding supergravity solution [23, 24, 25] in
the Einstein frame reads

$\displaystyle ds_{\rm E}^{2}$ $\displaystyle=g_{MN}dx^{M}dx^{N},$
$\displaystyle=H(u)^{1/8}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H(u)^{9/8}du^{2}$
$\displaystyle=:e^{2A}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2B}du^{2}\ ,$ (4.1)

($M,N=0,1,\cdots,9$; $\mu,\nu=0,1,\cdots,8$) with

$\displaystyle e^{2A}$ $\displaystyle=H(u)^{1/8}\ ,$ (4.2) $\displaystyle
e^{2B}$ $\displaystyle=H(u)^{9/8}\ ,$ (4.3) $\displaystyle H(u)$
$\displaystyle=a+\frac{g_{s}(8-N)}{2\pi l_{s}}u\ ,$ (4.4)

and

$e^{\Phi}=g_{s}H(u)^{-\frac{5}{4}}\ ,$ (4.5)

where $g_{s}$ is the string coupling constant, $l_{s}$ is the string length,
and $a$ is a constant. Below, we set $g_{s}=1$ for simplicity.

The interval $I$ can be regarded as $S^{1}/{\bf Z_{2}}$, where ${\bf Z_{2}}$
is the orientifold action. Here, $S^{1}$ is parametrized by
$u\in[-\ell,+\ell]$, where $u=-\ell$ and $u=+\ell$ are identified, and the
${\bf Z_{2}}$ acts on $u$ as $u\rightarrow-u$. In this case, $u$ in (4.4) is
replaced with $|u|$ and hence the derivatives of $A$, $B$ and $\Phi$ with
respective to $u$ have discontinuity at $u=0,\ell$. Then, substituting (4.3)
and (4.5) into (2.30) and (2.31), the tension $t(\Phi)$ for the ETW brane at
$u=0$ is obtained as

$t(\Phi)=\frac{N-8}{\pi l_{s}}e^{\frac{5}{4}\Phi}\ .$ (4.6)

This is the dilaton dependent tension of the O8-plane with $N$ D8-branes at
$u=0$ in the Einstein frame appeared in (2.35):

$S_{\text{O8-D8}}=-\frac{1}{32\pi G_{N}}\int d^{9}x\,du\sqrt{-h}\,t(\Phi)\ ,$
(4.7)

where the Newton constant $G_{N}$ is given by

$G_{N}=\frac{(2\pi l_{s})^{8}}{32\pi^{2}}\ .$ (4.8)

Then, the action (4.7) reproduces the correct tension of the O8-plane and
D8-branes in the string frame:

$\displaystyle S_{\text{O8-D8}}$
$\displaystyle=-\frac{N-8}{(2\pi)^{8}l_{s}^{9}}\int
d^{9}x\,e^{-\Phi}\sqrt{-h_{\rm st}}$ $\displaystyle=-(N-8)\,T_{\rm D8}\int
d^{9}x\,e^{-\Phi}\sqrt{-h_{\rm st}},$ (4.9)

where $T_{\rm D8}=1/((2\pi)^{8}l_{s}^{9})$ is the tension of the D8-brane and
$h_{\rm st}$ is the determinant of the induced metric on the ETW brane at
$u=0$ in the string frame.

We can also confirm that the boundary equations of motion are satisfied. The
dilaton EOM obtained from (2.48) is satisfied as

$\frac{\partial_{u}\Phi}{\sqrt{g_{uu}}}=-\frac{5}{4}H^{\prime}H^{-\frac{25}{16}}=\frac{N-8}{2\pi
l_{s}}\frac{5}{4}e^{\frac{5}{4}\Phi}=\frac{1}{2}\frac{dt}{d\Phi}\ .$ (4.10)

Similarly, using the explicit expression for the extrinsic curvature

$K_{\mu\nu}=\frac{1}{16}H^{\prime}H^{-\frac{23}{16}}\eta_{\mu\nu}=\frac{8-N}{32\pi
l_{s}}H^{-\frac{23}{16}}\eta_{\mu\nu}\ ,$ (4.11)

obtained from (2.39), the boundary EOM for the metric (2.45) can be checked as

$K_{\mu\nu}-Kh_{\mu\nu}=-\frac{8-N}{4\pi
l_{s}}H^{-\frac{23}{16}}\eta_{\mu\nu}=-\frac{8-N}{4\pi
l_{s}}e^{\frac{5}{4}\Phi}e^{2A}\eta_{\mu\nu}=-\frac{1}{4}t(\Phi)h_{\mu\nu}=-8\pi
G_{N}T_{\mu\nu}\ .$ (4.12)

##  5 Conclusion and Discussion

In this paper, we have considered the dimensional reduction of the cigar
geometries, where Euclidean time direction shrinks at the tip. The key
observation was that the tip of the cigar became an ETW brane after the
dimensional reduction. To derive the tension of the ETW brane, we used a trick
to regard the ETW brane as the fixed plane on a $\mathbf{Z}_{2}$ orbifold. We
also gave another derivation of the tension using boundary EOM, which leads to
the same results. As a check of our calculation, we applied our methods to the
O8-D8 brane system and confirmed that the known value of the tension of the
O8-plane with $N$ D8-branes is reproduced consistently.

In particular, we considered the AdS7-soliton $\times$S4 and the AdS4-soliton
$\times$S7 backgrounds in M-theory and took the dimensional reduction along
the Euclidean time direction $\tau$, which is regarded as the M-theory
direction $x^{11}$. This theory is conjectured to be type 0A string theory
[18] with the ETW brane corresponding to the tip of the cigar of the AdS-
soliton geometry. We calculated the tension of the ETW brane by the above
methods and found that the tension is proporthional to $t(\Phi)\sim
e^{-3\Phi}$ in the string frame in both cases. This dilaton dependence does
not appear in a perturbative expansion series with respect to the string
coupling constant, and hence it should be considered as a non-perturbative
object in string theory.

Although we mainly focused on the AdS-soliton geometries in the examples
considered in section 3, our methods can be applied to other cigar geometries.
One interesting geometry with the cigar structure is the Euclidean
Schwarzschild black hole geometry where $\tau$ is the Euclidean time direction
and $r_{h}$ is the horizon. It would be extremely interesting if the ETW brane
obtained via the dimensional reduction play some role in the black hole
thermodynamics.

As another possible application of the dimensional reduction of the cigar
geometry, let us make a few comments on a replica manifold in gravitational
path integral [19] which have been used for the derivation of the Ryu-
Takayanagi formula in AdS/CFT [26, 27]. This is an Euclidean path integral and
the replicated manifold can be described by the cigar geometry. In general,
the $n$-replicated manifold is related via holographic dual to the Rényi-$n$
entropy $S_{n}$ in CFT. This geometry is described by the cigar geometry with
a cosmic brane with tension $T_{n}=\frac{n-1}{4nG}$ [20] at the fixed point of
the replica symmetry. At this stage, our set up and this geometry is quite
different since in this case, it contains a co-dimension two cosmic brane even
in the background geometry (even before the dimensional reduction). However,
we can take the $n\rightarrow 0$ limit and the replica manifold has a boundary
at the position of the fixed point of the $\mathbf{Z}_{N}$ symmetry. In the
gravity side, the co-dimension two cosmic brane reduces to the co-dimension
one ETW brane and the tension of the brane is fixed as $T_{0}=-\lim_{n\to
0}\frac{1}{4nG}$, which has infinitely negative value as in our previous
results. Although we naively analytically continued the tension $n\rightarrow
0$ here, we expect that this tension can be reproduced as in our explicit
calculation of dimensional reduction especially in AdS3 case. For the CFT
side, $n\rightarrow 0$ limit of the Rényi entropy is called Hartly entropy
$S_{0}=\log|\mathcal{A}|$, where $\mathcal{A}$ is the dimension of the Hilbert
space of the subsystem (See [28] for a recent work on field theory analysis).
This is dual to the area of the cosmic brane and especially ETW brane for
$n\rightarrow 0$. Therefore, we do not expect the tension of the brane itself
is related with the dimension of the Hilbert space, but it will be interesting
to study in more detail along this direction.

## Acknowledgement

We thank T.Kawamoto, T.Takayanagi and S.Terashima for stimulating discussions.
We also thank the participants of the 18th Kavli Asian Winter School, where a
part of the work was presented. The work of SS is supported by JSPS KAKENHI
(Grant-in-Aid for Scientific Research (B)) grant number JP19H01897 and MEXT
KAKENHI (Grant-in-Aid for Transformative Research Areas A "Extreme Universe")
grant number 21H05187. YS is supported by Grant-in-Aid for JSPS Fellows
No.23KJ1337.

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