aainstitutetext: Departamento de Física, Universidad de Santiago de Chile,
Avenida Víctor Jara 3493, Santiago, Chilebbinstitutetext: Université Libre de
Bruxelles and International Solvay Institutes, ULB-Campus Plaine CP231, B-1050
Brussels, Belgiumccinstitutetext: Centro de Estudios Científicos (CECs), Av.
Arturo Prat 514, Valdivia, Chileddinstitutetext: Facultad de Ingeniería,
Arquitectura y Diseño, Universidad San Sebastián, sede Valdivia, General Lagos
1163, Valdivia 5110693, Chile

# Gravity coupled to a scalar field from a Chern-Simons action: describing
rotating hairy black holes and solitons with gauge fields

Marcela Cárdenas b Oscar Fuentealba c,d Cristián Martínez c,d and Ricardo
Troncoso<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Einstein gravity minimally coupled to a scalar field with a two-parameter
Higgs-like self-interaction in three spacetime dimensions is recast in terms
of a Chern-Simons form for the algebra $g^{+}\oplus g^{-}$ where, depending on
the sign of the self-interaction couplings, $g^{\pm}$ can be $so(2,2)$,
$so(3,1)$ or $iso(2,1)$. The field equations can then be expressed through the
field strength of non-flat composite gauge fields, and conserved charges are
readily obtained from boundary terms in the action that agree with those of
standard Chern-Simons theory for pure gravity, but with non-flat connections.
Regularity of the fields then amounts to requiring the holonomy of the
connections along contractible cycles to be trivial. These conditions are
automatically fulfilled for the scalar soliton and allow to recover the
Hawking temperature and chemical potential in the case of the rotating hairy
black holes presented here, whose entropy can also be obtained by the same
formula that holds in the case of a pure Chern-Simons theory. In the conformal
(Jordan) frame the theory is described by General Relativity with cosmological
constant conformally coupled to a self-interacting scalar field, and its
formulation in terms of a Chern-Simons form for suitably composite gauge
fields is also briefly addressed.

††preprint: CECS-PHY-15/04

##  1 Introduction

The formulation of three-dimensional General Relativity as a Chern-Simons
theory Achucarro:1986uwr ; Witten:1988hc  has allowed exploiting time-honored
tools available for gauge fields in order to span a wealth of very interesting
achievements on classical and quantum aspects of gravitation (for a non-
exhaustive list of references, see e.g. Carlip:1995zj ; Coussaert:1995zp ;
Maloney:2007ud ; Kraus:2006wn ; Cotler:2018zff ; Perez:2016vqo ;
Gonzalez:2018jgp ; Melnikov:2018fhb ; Fuentealba:2017omf ; Afshar:2016wfy ;
Cardenas:2021vwo ; Barnich:2013yka ; Grumiller:2019tyl ; Ojeda:2019xih ). Many
of these results also extend to supergravity Achucarro:1989gm ; Howe:1995zm ;
Banados:1996hi ; Henneaux:1999ib ; Giacomini:2006dr ; Barnich:2015sca ;
Fuentealba:2017fck ; Caroca:2018obf ; Caroca:2019dds ; Banerjee:2019lrv ;
Banerjee:2021uxl ; Barnich:2014cwa  as well as for gravitation coupled to
higher-spin fields Blencowe:1988gj ; Bergshoeff:1989ns ; Henneaux:2010xg ;
Campoleoni:2010zq ; Gutperle:2011kf ; Ammon:2011nk ; Castro:2011fm ;
Henneaux:2012ny ; Perez:2012cf ; Campoleoni:2012hp ; Perez:2013xi ;
Henneaux:2013dra ; Bunster:2014mua ; Zinoviev:2014sza ; Fuentealba:2015jma ;
Fuentealba:2015wza ; Henneaux:2015tar ; Grumiller:2016kcp , since both can be
described in terms of a Chern-Simons theory for suitable gauge groups. Ultra
and non-relativistic versions thereof have also been developed in
Aviles:2019xed ; Ravera:2019ize ; Ali:2019jjp ; Concha:2020eam ;
Concha:2021llq ; Caroca:2022byi ; Grumiller:2017sjh .

The case of General Relativity minimally coupled to a real scalar field
appears to be far from that kind of description. Nevertheless, here we show
that in the case of a precise two-parameter Higgs-like self-interaction
potential given by

$V\left(\phi\right)=\frac{\Lambda}{8}\left(\cosh^{6}\phi+\nu\sinh^{6}\phi\right)\,,$
(1)

the theory can be equivalently formulated in terms of a Chern-Simons form that
depends on composite gauge fields, so that the field equations no longer imply
the vanishing of the field strengths. Thus, the theory and its configuration
space can be equivalently described in terms of non-flat connections, so that
many of the gauge theory tools open up to analyze their properties.

The self-interaction potential (1) enjoys some remarkable properties. Indeed,
the first analytic example of a black hole with a minimally coupled scalar
field that circumvents the no-hair conjecture was precisely found for
$V\left(\phi\right)$ in (1) in the range $\Lambda<0$ and $\nu\geq-1$
Henneaux:2002wm . Furthermore, the potential in (1) falls within the class
analyzed in Henneaux:2002wm , for which the scalar field acquires a slow fall-
off at infinity, so that the canonical generators of the asymptotic symmetries
acquire an explicit contribution from the scalar field. Thus, it was shown
that the Brown-Henneaux boundary conditions Brown:1986nw  for gravity with a
localized distribution of matter can be consistently relaxed, so that the
asymptotic symmetries are still given by the conformal group in two dimensions
with the same central extension. As it was also pointed out in Henneaux:2002wm
, for the self-interaction potential (1), once the theory is expressed in the
conformal (Jordan) frame, the matter piece of the action becomes conformally
invariant, and the action reduces to General Relativity with cosmological
constant $\Lambda$ with a conformally coupled scalar field, whose self-
interaction coupling is determined by $\nu$ (see section 6).

The plan of the paper is as follows. In the next section, we show that the
action

$I\left[\phi,g_{\mu\nu}\right]=\frac{8}{\kappa}\int
d^{3}x\sqrt{-g}\left(\frac{R}{16}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V\left(\phi\right)\right)\,,$
(2)

with the self-interaction potential given by (1), whose field equations read

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle 8T_{\mu\nu}\,,$ (3)
$\displaystyle\Box\phi-\frac{dV\left(\phi\right)}{d\phi}$ $\displaystyle=$
$\displaystyle 0\,,$ (4)

with

$T_{\text{$\mu\nu$}}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi-
g_{\text{$\mu\nu$}}V\left(\phi\right)\,,$ (5)

can be expressed in terms of a Chern-Simons form with certain suitable
composite gauge fields. In section 3, the Hamiltonian boundary terms required
to obtain the conserved charges in terms of the connections are worked out,
including the hairy black hole entropy formula in terms of gauge fields. The
rotating extension of the hairy black hole in Henneaux:2002wm  and its global
charges are discussed in section 4. Regularity of hairy black holes and the
scalar soliton in terms of trivial holonomies is addressed in section 5,
including the hairy black hole thermodynamics and the corresponding Cardy
formula for its entropy that depends on the global charges of the scalar
soliton. Finally, section 6 is devoted to some ending remarks as well as a
brief discussion of the formulation of the theory in the conformal frame in
terms of a Chern-Simons form for composite gauge fields.

##  2 Action from a Chern-Simons form with composite gauge fields

Here we show that the action (2), up to a boundary term, can be recast using a
Chern-Simons form. The gauge fields are defined in terms of the direct sum of
the algebras $g^{+}$ and $g^{-}$, where $g^{\pm}$ can be the three-dimensional
(anti-)de Sitter or Poincaré algebras, depending on the signs of
$\Lambda^{+}=\Lambda$ and $\Lambda^{-}=-\nu\Lambda$.

The theory can be formulated through the following composite gauge fields

$\displaystyle A^{+}$ $\displaystyle=$
$\displaystyle\cosh^{2}\left(\phi\right)e^{a}P_{a}^{+}+\omega_{+}^{a}J_{a}^{+}\,,$
(6) $\displaystyle A^{-}$ $\displaystyle=$
$\displaystyle\sinh^{2}\left(\phi\right)e^{a}P_{a}^{-}+\omega_{-}^{a}J_{a}^{-}\,,$
(7)

that depend on the scalar field $\phi$, the dreibein $e^{a}$ and additional
1-forms $\omega_{\pm}^{a}$. The generators $J_{a}^{\pm}$ and
$P_{a}^{{}^{\pm}}$ fulfill the following algebra

$\left[J_{a}^{\pm},J_{b}^{\pm}\right]=\epsilon_{abc}J_{\pm}^{c}\quad,\quad\left[J_{a}^{\pm},P_{b}^{\pm}\right]=\epsilon_{abc}P_{\pm}^{c}\quad,\quad\left[P_{a}^{\pm},P_{b}^{\pm}\right]=-\Lambda^{\pm}\epsilon_{abc}J_{\pm}^{c}\,,$
(8)

so that each copy corresponds to $so(3,1)$ or $so(2,2)$ for positive or
negative signs of $\Lambda^{\pm}$, respectively, or $iso(2,1)$ when
$\Lambda^{+}$ or $\Lambda^{-}$ vanishes. The field strengths associated to the
gauge fields (6) and (7) are given by

$F^{\pm}=dA^{\pm}+(A^{\pm})^{2}=\mathcal{T}_{\text{$\pm$}}^{a}P_{a}^{\pm}+\mathcal{R}_{\pm}^{a}J_{a}^{\pm}\,,$
(9)

where the components along the $so(2,1)$ generators $J_{a}^{\pm}$ read

$\displaystyle\mathcal{R}_{+}^{a}$ $\displaystyle=$ $\displaystyle
R_{+}^{a}-\frac{1}{2}\Lambda^{+}\cosh^{4}\left(\phi\right)\epsilon^{abc}e_{b}e_{c}\,,$
(10) $\displaystyle\mathcal{R}_{-}^{a}$ $\displaystyle=$ $\displaystyle
R_{-}^{a}-\frac{1}{2}\Lambda^{-}\sinh^{4}\left(\phi\right)\epsilon^{abc}e_{b}e_{c}\,,$
(11)

with
$R_{\pm}^{a}=d\omega_{\text{$\pm$}}^{a}+\frac{1}{2}\epsilon^{abc}\omega_{b}^{\pm}\omega_{c}^{\pm}$,
while those along the remaining generators $P_{a}^{\pm}$ are

$\displaystyle\mathcal{T}_{+}^{a}$ $\displaystyle=$
$\displaystyle\cosh^{2}\left(\phi\right)\left(T_{+}^{a}+2\tanh\left(\phi\right)d\phi\,e^{a}\right)\,,$
(12) $\displaystyle\mathcal{T}_{-}^{a}$ $\displaystyle=$
$\displaystyle\sinh^{2}\left(\phi\right)\left(T_{-}^{a}+2[\tanh\left(\phi\right)]^{-1}d\phi\,e^{a}\right)\,,$
(13)

with $T_{\text{$\pm$}}^{a}=de^{a}+\epsilon^{abc}\omega_{b}^{\pm}e_{c}$.

We then consider an action principle defined as a combination of two Chern-
Simons forms for the composite gauge fields $A^{\pm}$, which reads

$I\left[\phi,e,\omega^{+},\omega^{-}\right]=\frac{k^{+}}{4\pi}\int\left\langle
A^{+}dA^{+}+\frac{2}{3}(A^{+})^{3}\right\rangle+\frac{k^{-}}{4\pi}\int\left\langle
A^{-}dA^{-}+\frac{2}{3}(A^{-})^{3}\right\rangle\,,$ (14)

where the nonvanishing components of the invariant bilinear form are given by
$\left\langle J_{a}^{\pm},P_{b}^{\pm}\right\rangle=\eta_{ab}$, with
$\eta_{ab}$ standing for the Minkowski metric, and the levels are defined as
$k^{\pm}=\pm 2\pi/\kappa$.

The field equations are found varying the action (14) with respect to the
dynamical fields $e^{a}$, $\phi$ and $\omega_{\pm}^{a}$

$\displaystyle\delta I$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\intop\left\langle\left(k^{+}F^{+}\frac{\delta
A^{+}}{\delta e^{a}}+k^{-}F^{-}\frac{\delta A^{-}}{\delta e^{a}}\right)\delta
e^{a}+\left(k^{+}F^{+}\frac{\delta A^{+}}{\delta\phi}+k^{-}F^{-}\frac{\delta
A^{-}}{\delta\phi}\right)\delta\phi\right.$ (16)
$\displaystyle+\left.\left(k^{+}F^{+}\frac{\delta
A^{+}}{\delta\omega_{+}^{a}}+k^{-}F^{-}\frac{\delta
A^{-}}{\delta\omega_{+}^{a}}\right)\delta\omega_{+}^{a}+\left(k^{+}F^{+}\frac{\delta
A^{+}}{\delta\omega_{-}^{a}}+k^{-}F^{-}\frac{\delta
A^{-}}{\delta\omega_{-}^{a}}\right)\delta\omega_{-}^{a}\right\rangle\,,$

so that they read

$\displaystyle\cosh^{2}\left(\phi\right)\mathcal{R}_{a}^{+}-\sinh^{2}\left(\phi\right)\mathcal{R}_{a}^{-}=0\,,$
(17)
$\displaystyle\left(\mathcal{R}_{a}^{+}-\mathcal{R}_{a}^{-}\right)e^{a}=0\,,$
(18) $\displaystyle T_{\pm}^{a}+2[\tanh\left(\phi\right)]^{\pm
1}d\phi\,e^{a}=0\,,$ (19)

respectively.

Note that the last field equations (19) are algebraic for $\omega_{\pm}^{a}$,
which can be solved as

$\omega_{\pm}^{a}\left(e,\phi\right)=\omega^{a}\left(e\right)-2[\tanh\left(\phi\right)]^{\pm
1}*\left(e^{a}d\phi\right)\,,$ (20)

where $\omega^{a}$ is the torsionless Levi-Civita spin connection associated
to $e^{a}$, and $*$ stands for the Hodge dual, so that
$*\left(e^{a}d\phi\right)=\epsilon^{abc}\partial_{\nu}\phi e_{b}^{\nu}e_{c}$.

A second order action $I(e^{a},\phi)$ is obtained by replacing (20) in the
Chern-Simons action (14), which reduces to (2) up to a boundary term, where
the self-interaction potential is precisely given by (1). Analogously,
substituting equation (20) in (17) and (18), they reduce respectively to the
Einstein and scalar field equations (3) and (4).

##  3 Global charges and entropy in terms of gauge fields

In order to deal with conserved charges as well as for the hairy black hole
entropy, it is useful to express the action (14) in Hamiltonian form.
Supplementing the action with a boundary term $B$, that is required to ensure
a well-defined variational principle Regge:1974zd , yields to

$I=\frac{k^{+}}{4\pi}\int
dtd^{2}x\epsilon^{ij}\left\langle\dot{A}_{i}^{+}A_{j}^{+}+A_{t}^{+}F_{ij}^{+}\right\rangle+\frac{k^{-}}{4\pi}\int
dtd^{2}x\epsilon^{ij}\left\langle\dot{A}_{i}^{-}A_{j}^{-}+A_{t}^{-}F_{ij}^{-}\right\rangle+\mathcal{B}\,,$
(21)

where $F_{ij}^{\pm}$ are the spatial components of the field strengths
$F^{\pm}$ in (9). Considering that the variations of the gauge fields depend
on the scalar field and the dreiben, and making use of (20), the variation of
the action reads

$\displaystyle\delta I$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa}\int
dtd^{2}x\epsilon^{ij}\left\langle\left[\left(-2F_{tj}^{+}\frac{\delta
A_{i}^{+}}{\delta e^{a}}+F_{ij}^{+}\frac{\delta A_{t}^{+}}{\delta
e^{a}}\right)-\left(-2F_{tj}^{-}\frac{\delta A_{i}^{-}}{\delta
e^{a}}+F_{ij}^{-}\frac{\delta A_{t}^{-}}{\delta e^{a}}\right)\right]\delta
e^{a}\right.$ $\displaystyle+\left.\left[\left(-2F_{tj}^{+}\frac{\delta
A_{i}^{+}}{\delta\phi}+F_{ij}^{+}\frac{\delta
A_{t}^{+}}{\delta\phi}\right)-\left(-2F_{tj}^{-}\frac{\delta
A_{i}^{-}}{\delta\phi}+F_{ij}^{-}\frac{\delta
A_{t}^{-}}{\delta\phi}\right)\right]\delta\phi\right\rangle$
$\displaystyle+\frac{1}{\kappa}\int
dtdS_{i}\epsilon^{ij}\left\langle\left(A_{t}^{+}\frac{\delta A_{j}^{+}}{\delta
e^{a}}-A_{t}^{-}\frac{\delta A_{j}^{-}}{\delta e^{a}}\right)\delta
e^{a}+\left(A_{t}^{+}\frac{\delta A_{j}^{+}}{\delta\phi}-A_{t}^{-}\frac{\delta
A_{j}^{-}}{\delta\phi}\right)\delta\phi\right\rangle+\delta\mathcal{B}\,,$

where the bulk terms are proportional to the field equations (17) and (18).

The variation of the boundary term is then given by

$\displaystyle\delta\mathcal{B}$ $\displaystyle=$
$\displaystyle-\frac{1}{\kappa}\int
dtdS_{i}\epsilon^{ij}\left\langle\left(A_{t}^{+}\frac{\delta A_{j}^{+}}{\delta
e^{a}}-A_{t}^{-}\frac{\delta A_{j}^{-}}{\delta e^{a}}\right)\delta
e^{a}+\left(A_{t}^{+}\frac{\delta A_{j}^{+}}{\delta\phi}-A_{t}^{-}\frac{\delta
A_{j}^{-}}{\delta\phi}\right)\delta\phi\right\rangle$ (22) $\displaystyle=$
$\displaystyle-\frac{1}{\kappa}\int dtd\theta\left\langle A_{t}^{+}\delta
A_{\theta}^{+}-A_{t}^{-}\delta A_{\theta}^{-}\right\rangle\,,$

which agrees with that of a pure Chern-Simons theory. Nonetheless, it should
be emphasized that in our case the variations of the gauge fields are not
fully independent. For the class of configurations that we are interested in,
that fulfill the asymptotic conditions spelled out in Henneaux:2002wm , it can
be also shown that the variation of the surface integral in (22) integrates as

$\mathcal{B}=-\frac{1}{2\kappa}\int dtd\theta\left\langle
A_{t}^{+}A_{\theta}^{+}-A_{t}^{-}A_{\theta}^{-}\right\rangle\,.$ (23)

Intriguingly, if one replaces the explicit form of the composite gauge fields
$A^{\pm}$ in (6) and (7), by virtue of (20), the boundary term (23) completely
gets rid of the scalar field and precisely reduces to that of pure General
Relativity in the standard Chern-Simons formulation, given by

$\mathcal{B}=-\frac{1}{2\kappa}\int dtd\theta\left\langle
A_{t}A_{\theta}\right\rangle\,,$ (24)

where the connection $A$ is defined by setting $\phi=0$ in the definition of
$A^{+}$, i.e.,

$A=\left.A^{+}\right|{}_{\phi=0}=e^{a}P_{a}^{+}+\omega^{a}J_{a}^{+}\,,$ (25)

stands for the gauge field of pure General Relativity Achucarro:1986uwr ;
Witten:1988hc .

Analogously, the black hole entropy formula that applies for a generic Chern-
Simons theory in Bunster:2014mua  (see also deBoer:2013gz ), reduces to that
of General Relativity

$\displaystyle S$ $\displaystyle=\frac{1}{\kappa}\oint d\theta\left\langle
A_{\tau}^{+}A_{\theta}^{+}-A_{\tau}^{-}A_{\theta}^{-}\right\rangle\Big{|}_{r_{+}}$
$\displaystyle=\frac{1}{\kappa}\oint d\theta\left\langle
A_{\tau}A_{\theta}\right\rangle\Big{|}_{r_{+}}\,,$ (26)

where $\tau=-it$ is the Euclidean time and the event horizon is located at
$r=r_{\text{+}}$.

In sum, the boundary term (24) and the entropy (26) are found to exclusively
depend on the pure gravity gauge field $A$ in (25), which noteworthy encodes
all of the relevant information, without requiring the contribution of the
scalar field in an explicit way. Indeed, it has to be stressed that the pure
gravity gauge field (25) in our case is generically no longer flat.

##  4 Rotating hairy black hole

The first analytic example of a black hole solution endowed with minimally
coupled scalar hair was found in Henneaux:2002wm , precisely for the self-
interaction under discussion (1) in the case of $\Lambda<0$ and $\nu\geq-1$.
Some of its properties have been further analyzed in Barnich:2002pi ;
Clement:2003sr ; Gegenberg:2003jr ; Park:2004yk ; Banados:2005hm ;
Myung:2008ze ; Correa:2010hf ; Lashkari:2010ak ; Correa:2011dt ; Hyun:2012bc ;
Aparicio:2012yq ; Xu:2014qaa ; Xu:2014xqa ; Ahn:2015uza ; Aviles:2018vnf
following different approaches, and the rotating extension in the case of
$\nu=0$ was presented in Correa:2012rc .

In this section we extend the rotating hairy black hole solution to the full
allowed range of the self-interaction coupling ($\nu>-1$). It can be readily
obtained from the static one by performing a Lorentz boost parameterized by
$\omega$, with $\omega^{2}<1$, in the $t-\theta$ cylinder111The case of
$\nu=-1$ is excluded since, as pointed out in Henneaux:2002wm , the static
metric is invariant under this kind of boosts. This configuration shares the
causal structure with the massless BTZ black hole, but the null curvature
singularity coincides with the “horizon” (NUT) at $r=0$, possessing vanishing
mass, angular momentum, temperature and entropy, regardless the value of the
integration constant. . The line element then reads

$ds^{2}=-N_{\infty}^{2}N\left(r\right){}^{2}dt^{2}+\frac{dr^{2}}{G\left(r\right){}^{2}}+R\left(r\right)^{2}\left(d\theta+N^{\theta}\left(r\right)dt\right)^{2}\,,$
(27)

with

$\displaystyle N\left(r\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}g\left(r\right){}^{2}}{R\left(r\right){}^{2}}\,,$
(28) $\displaystyle G\left(r\right)^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{H\left(r\right)+2B}{H\left(r\right)+B}\right)^{2}F\left(r\right)^{2}\,,$
(29) $\displaystyle R\left(r\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}-g\left(r\right)^{2}\omega^{2}\ell^{2}}{1-\omega^{2}}\,,$
(30) $\displaystyle N^{\theta}\left(r\right)$ $\displaystyle=$ $\displaystyle
N_{\infty}^{\theta}-\frac{\omega}{\ell}\left(\frac{r^{2}-g\left(r\right)^{2}\ell^{2}}{r^{2}-g\left(r\right)^{2}\omega^{2}\ell^{2}}\right)N_{\infty}\,,$
(31)

where

$g\left(r\right)^{2}=\left(\frac{H\left(r\right)}{H\left(r\right)+B}\right)^{2}F\left(r\right)^{2}\quad,\quad
F\left(r\right)^{2}=\frac{H\left(r\right)^{2}}{\ell^{2}}-\left(1+\nu\right)\left(\frac{3B^{2}}{\ell^{2}}+\frac{2B^{3}}{\ell^{2}H\left(r\right)}\right)\,,$
(32)

and

$H\left(r\right)=\frac{1}{2}\left(r+\sqrt{r^{2}+4Br}\right)\,.$ (33)

Here the cosmological constant is written in terms of the AdS radius as
$\Lambda=-\ell^{-2}$, while $N_{\infty}$ and $N_{\infty}^{\theta}$ stand for
integration constants that turn out to be related to the Hawking temperature
and the chemical potential, respectively (see section 5). The hairy black hole
is dressed with a scalar field given by

$\phi\left(r\right)=\mbox{arctanh}\sqrt{\frac{B}{H(r)+B}}\,,$ (34)

being real provided that $B>0$. In this case, the scalar field is regular
everywhere except at the origin, since it diverges as $\phi_{r\rightarrow
0}=-\frac{1}{4}\log(r)+\cdots$, sourcing a null curvature singularity, as it
is reflected in the behavior of the Ricci scalar near $r=0$, which reads

$R_{r\rightarrow 0}=-\frac{4(\nu+1)B^{5/2}}{\ell^{2}r^{5/2}}+O(r^{-3/2})\,.$
(35)

Singularities in the spacetime metric and scalar field at the origin are
cloaked by the event horizon located at $r=r_{+}=B\Theta_{\nu}$, with
$\Theta_{\nu}$ given by

$\Theta_{\nu}=2\left(z\bar{z}\right)^{2/3}\frac{z^{2/3}-\bar{z}^{2/3}}{z-\bar{z}}\,,$
(36)

and $z=1+i\sqrt{\nu}$.

It is worth highlighting that, unlike the case of the rotating solution in
vacuum, the inner region of the rotating hairy black hole does not possess
neither a region with closed timelike curves to be excised nor an inner
(Cauchy) horizon. Indeed, the analogue of the latter actually corresponds to
the null singularity at the origin. Moreover, the asymptotic behavior of the
rotating hairy black hole does not fulfill the Brown-Henneaux boundary
conditions Brown:1986nw  because the scalar field (34) has a slow fall-off at
infinity, generating a strong backreaction on the metric in the asymptotic
region. Instead, the solution fits within the relaxed asymptotic behavior
described in Henneaux:2002wm .

In order to describe the rotating hairy black hole in terms of gauge fields,
we choose the local frame so that the dreibein reads

$e^{0}=N_{\infty}N\left(r\right)dt\quad,\quad
e^{1}=\frac{dr}{G\left(r\right)}\quad,\quad
e^{2}=R\left(r\right)\left(d\theta+N^{\theta}\left(r\right)dt\right)\,,$ (37)

and hence, the components of the spin connection $\omega^{a}$ are given by

$\displaystyle\omega^{0}$ $\displaystyle=$ $\displaystyle
G(r)R(r)\mbox{${}^{\prime}$}d\theta+\frac{G(r)}{2}\left(2N^{\theta}(r)R(r)\mbox{${}^{\prime}$}+R(r)N^{\theta}(r)\mbox{${}^{\prime}$}\right)dt\,,$
(38) $\displaystyle\omega^{1}$ $\displaystyle=$
$\displaystyle\frac{R(r)N^{\theta}(r)\mbox{${}^{\prime}$}}{2N_{\infty}N\left(r\right)}dr\,,$
(39) $\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle-\frac{G(r)R(r)^{2}N^{\theta}(r)\mbox{${}^{\prime}$}}{2N_{\infty}N\left(r\right)}d\theta-\frac{G(r)}{2}\left(\frac{R(r)^{2}N^{\theta}(r)N^{\theta}(r)\mbox{${}^{\prime}$}}{N_{\infty}N\left(r\right)}-2N_{\infty}N\left(r\right)\mbox{${}^{\prime}$}\right)dt\,.$
(40)

The gauge fields $A^{+}$ and $A^{-}$ in (6), (7) become then fully specified
by virtue of $\omega_{\pm}^{a}$ given by (20). Nevertheless, as pointed out in
the previous section, the pure gravity connection in (25) is enough in order
to evaluate the boundary term in (24), which for the rotating hairy black hole
reduces to

$\mathcal{B}=-(t_{2}-t_{1})\left[N_{\infty}\left(\frac{3\pi\left(1+\nu\right)B^{2}\left(1+\omega^{2}\right)}{\kappa\ell^{2}\left(1-\omega^{2}\right)}\right)-N_{\infty}^{\theta}\left(\frac{6\pi\left(1+\nu\right)B^{2}\omega}{\kappa\ell\left(1-\omega^{2}\right)}\right)\right].$
(41)

Noteworthy, the result is independent of the radial coordinate $r$, and hence
it can be computed even at finite proper distance ($r=r_{0}$) without the need
of taking the limit $r\rightarrow\infty$. The mass $M$ and angular momentum
$J$ can then be recognized from the boundary term (41) in the following way
Regge:1974zd

$\mathcal{B}=(t_{2}-t_{1})\left(-N_{\infty}M+N_{\infty}^{\theta}J\right)\,,$
(42)

so that

$M=\frac{3\pi\left(1+\nu\right)B^{2}\left(1+\omega^{2}\right)}{\kappa\ell^{2}\left(1-\omega^{2}\right)}\qquad\mbox{and}\qquad
J=\frac{6\pi\left(1+\nu\right)B^{2}\omega}{\kappa\ell\left(1-\omega^{2}\right)}\,.$
(43)

The mass and angular momentum in (43) reduce to the result in Henneaux:2002wm
for the static case ($\omega=0$), as well as to that in Correa:2012rc  for
$\nu=0$.

Note that the following bound is fulfilled

$\frac{M}{|J|/\ell}=\frac{1+\omega^{2}}{2|\omega|}\geq 1\,,$ (44)

for $\omega^{2}\leq 1$, being saturated at the extremal case ($\omega^{2}=1$).
The extremal case can be attained from (27) and (34) by first rescaling the
integration constant $B$ according to $b=\frac{B}{\sqrt{1-\omega^{2}}}$, and
then taking the limit $\omega\rightarrow\pm 1$, so that the scalar field
vanishes and the line element reduces to that of the extremal rotating BTZ
black hole, whose (degenerate) horizon locates at $r_{+}^{2}=3b^{2}(1+\nu)$.
It is worth noting that static and stationary extremal cases, respectively for
$M=J=0$, and $M=|J|/\ell$, do not admit scalar hair.

##  5 Regularity and thermodynamics through non-flat connections

Here we carry out the thermodynamic analysis of the rotating hairy black hole
relying on its description in terms of gauge fields in the Euclidean approach,
where Euclidean time $\tau=-it$ is normalized with period equals to 1. The
Euclidean hairy black hole in the non-extremal case possesses the topology of
a solid torus, $\mathbb{R}^{2}\times S^{1}$, where $S^{1}$ is the circle
parametrized by the angular coordinate $\theta$, and $\mathbb{R}^{2}$ stands
for the $r-\tau$ plane described in polar coordinates centered at $r=r_{+}$.
Since thermal cycles around the horizon are contractible, regularity of the
gauge fields at the horizon amounts to require trivial holonomies along them,
allowing to fix the Lagrange multipliers at the boundary, given by
$N_{\infty}$ and $N_{\infty}^{\theta}$, which correspond to the Hawking
temperature and the chemical potential.

In concrete, the regularity condition for the Euclidean rotating hairy black
hole is obtained from demanding that the corresponding holonomies
$\mathcal{H^{\pm}}$ along the thermal cycle at the event horizon to be
trivial, i.e.,

$\mathcal{H^{\pm}}=\exp\left[\intop_{0}^{1}A_{\tau}^{\pm}d\tau\right]_{r_{+}}=\exp\left[A_{\tau}^{\pm}\right]_{r_{+}}=I_{c}\,,$
(45)

where $I_{c}$ is a suitable element of the center of the gauge group.

One possible way to implement the regularity condition corresponds to the
direct diagonalization of the holonomies $\mathcal{H}^{\pm}$ for an
appropriate matrix representation of the full algebra $g^{+}\oplus$ $g^{-}$. A
simpler option is the one spelled out in Matulich:2014hea , being implemented
in two steps:

(i) Finding a group element that allows to gauge away the temporal component
of the dreibein. In our case this condition is already implemented since the
local Lorentz frame has already been chosen, and $e_{\tau}$ vanishes at the
horizon provided that $N^{\theta}(r_{+})=0$, so that the chemical potential
$N_{\infty}^{\theta}$ is fixed in terms of $N_{\infty}$ as

$N_{\infty}^{\theta}=\frac{\omega}{\ell}N_{\infty}\,,$ (46)

and the time component of the gauge fields reduce to
$A_{\tau}^{\pm}(r_{+})=\omega_{\pm\tau}(r_{+})$, with
$\omega_{\pm\tau}^{a}(r_{+})=\omega_{\tau}^{a}(r_{+})$, whose non-vanishing
component is given by

$\omega_{\tau}=N_{\infty}G(r_{+})N\mbox{${}^{\prime}$}(r_{+})J_{2}^{+}\,.$
(47)

Therefore, as naturally expected from the results in section 3, the problem
actually reduces to requiring a trivial holonomy for the pure gravity gauge
field (25), whose time component at the horizon reads
$A_{\tau}=\omega_{\tau}^{a}(r_{\text{+}})J_{a}^{+}$.

(ii) Diagonalizing the remaining components, which in this case reduces to
that of the corresponding $so(2,1)\approx sl(2,\mathbb{R})$ subalgebra. The
basis can be chosen as

$J_{0}^{+}=\frac{1}{2}\left(L_{-1}+L_{1}\right),\quad
J_{1}^{+}=\frac{1}{2}\left(L_{-1}-L_{1}\right),\quad J_{2}^{+}=L_{0}\,,$ (48)

with

$L_{-1}=\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right)\quad,\quad
L_{0}=\left(\begin{array}[]{cc}-\frac{1}{2}&0\\\
0&\frac{1}{2}\end{array}\right)\quad,\quad
L_{1}=\left(\begin{array}[]{cc}0&-1\\\ 0&0\end{array}\right)\,,$ (49)

fulfilling $\left[L_{n},L_{m}\right]=\left(n-m\right)L_{n+m}$. Since the
$sl(2,\mathbb{R})$ generators are given in the fundamental (spinorial)
representation, the suitable element of the center of the group turns out to
be $I_{c}=-\mathbb{I}$.

The diagonalization can be then readily performed, which implies that the
eigenvalues of $\omega_{\tau}$ must be given by $\pm i\pi$, or equivalently,

$tr\left[(\omega_{\tau})^{2}\right]=-2\pi^{2}\,.$ (50)

The latter equation allows to fix the form of $N_{\infty}$, and hence, by
virtue of the former condition in (46), the Lagrange multipliers, related to
the Hawking temperature and the chemical potential, become fixed as

$N_{\infty}=\frac{2\pi\Theta_{\nu}\ell^{2}}{3\left(1+\nu\right)B\sqrt{1-\omega^{2}}}\qquad\mbox{and}\qquad
N_{\infty}^{\theta}=\frac{2\pi\Theta_{\nu}\omega\ell}{3\left(1+\nu\right)B\sqrt{1-\omega^{2}}}\,.$
(51)

Once the Langrange multipliers are fixed as in (51), the pure gravity gauge
field takes the following form at the horizon

$\displaystyle A=$ $\displaystyle R(r_{+})d\theta P_{2}^{+}+\left(2\pi
dt-\frac{G(r_{+})^{2}R(r_{+})^{2}N\mbox{${}^{\prime}$}(r_{+})N^{\theta}\mbox{${}^{\prime}$}(r_{+})}{4\pi
N(r_{+})}d\theta\right)J_{2}^{+}\,,$ (52)

so that the rotating hairy black hole entropy can be directly obtained from
(26), being given by

$S=\frac{4\pi^{2}B\Theta_{\nu}}{\kappa\sqrt{1-\omega^{2}}}=\frac{4\pi^{2}R(r_{+})}{\kappa}=\frac{A_{\textrm{hor}}}{4G},$
(53)

with $\kappa=8\pi G$, which is in full agreement with the Bekenstein-Hawking
area law for the entropy. It is then simple to verify that the first law is
fulfilled in the grand canonical ensemble, $dS=\beta dM-\beta\Omega dJ,$ with
$M$ and $J$ determined by (43), so that the relationship between the Lagrange
multipliers with the Hawking temperature and the angular velocity at the
horizon reads

$\beta=N_{\infty}\qquad\textrm{and}\qquad\beta\Omega=N_{\infty}^{\theta}\,.$
(54)

###  5.1 Soliton mass, regularity and Cardy formula

An analytic soliton solution endowed with a nontrivial scalar field for the
theory under discussion was found in Correa:2010hf . The line element is given
by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\ell^{2}\left(1+\frac{1}{\alpha_{\nu}\left(1+\rho^{2}\right)}\right)^{-2}\times$
$\displaystyle\left[-N_{\infty}^{2}\left[\frac{2\left(1+\rho^{2}\right)}{3c_{\nu}\ell}\right]^{2}dt^{2}+\frac{4d\rho^{2}}{2+\rho^{2}+\frac{c_{\nu}}{1+\rho^{2}}}+\left(\frac{2\rho}{2+c_{\nu}}\right)^{2}\left(2+\rho^{2}+\frac{c_{\nu}}{1+\rho^{2}}\right)d\theta^{2}\right]\,,$

with $c_{\nu}=2\alpha_{\nu}^{-3}\left(1+\nu\right)$ and
$\alpha_{\nu}=\frac{1}{2}\left(\Theta_{\nu}+\sqrt{\Theta_{\nu}^{2}+4\Theta_{\nu}}\right)$,
while the scalar field reads

$\phi\left(\rho\right)=\mbox{arctanh}\left(\sqrt{\frac{1}{1+\alpha_{\nu}\left(1+\rho^{2}\right)}}\right)\,.$
(56)

The radial coordinate has the range $0\leq\rho<\infty$, and the remaining
coordinates range precisely as for the hairy black hole metric, so that the
solitonic configuration is regular everywhere. As in the case of the rotating
hairy black hole, the fall-off at infinity of the soliton also fits that of
the relaxed boundary conditions in Henneaux:2002wm .

Regularity of the soliton at the origin can also be explicitly verified in
terms of its corresponding gauge fields in (6), (7). In order to do that, we
follow the same lines as in the case of the Euclidean hairy rotating black
hole discussed above, adapted to this case. Thus, for the solitonic
configuration, the suitable condition turns out to be requiring the holonomy
of the gauge fields $A^{\pm}$ around a spatial (angular) cycle that encloses
the origin ($\rho=0$) to be trivial, i.e.,

$\mathcal{H^{\pm}}=e^{\oint d\theta
A_{\theta}^{\pm}\Big{|}_{\rho=0}}=\mathbb{I}_{c}\,.$ (57)

The local frame can be suitably chosen as in the case of the hairy black hole,
so that the angular components of the dreibein $e_{\theta}$ automatically
vanish at the origin, without the need of imposing any condition. Thus, the
angular components of the gauge fields now fulfill
$A_{\theta}^{\pm}\Big{|}_{\rho=0}=\omega_{\pm\theta}\Big{|}_{\rho=0}$, with
$\omega_{\pm\theta}^{a}\Big{|}_{\rho=0}=\omega_{\theta}^{a}\Big{|}_{\rho=0}$,
so that the problem again reduces to just requiring the trivial holonomy for
the pure gravity connection in (25), whose angular component at the origin is
given by
$A_{\theta}\Big{|}_{\rho=0}=\omega_{\theta}^{a}J_{a}^{+}\Big{|}_{\rho=0}$. The
remaining components can then be diagonalized by choosing the basis and the
representation as in (48) and (49), respectively, so that the holonomy turns
out to be trivial provided that the condition in (50) with
$\omega_{\tau}\rightarrow\omega_{\theta}$ holds, which automatically does
without the need of requiring any additional condition.

The result is reassuring because the soliton is devoid of integration
constants, and hence, it has no freedom to be adjusted in order to fulfill the
regularity conditions. Besides, as pointed out in Correa:2010hf , this last
feature of the soliton suggests that it can be naturally regarded as a ground
state of the theory for the sector with non-vanishing scalar fields. Indeed,
it is simple to extend the result in Correa:2010hf  to the rotating case, so
that the Euclidean hairy black hole turns out to be diffeomorphic to the
Euclidean soliton provided that the corresponding modular parameters of the
torus are related through S-duality, i.e.,

$\tau_{sol}=-\frac{1}{\tau_{hbh}}$

with $\tau_{hbh}=\frac{i\beta(1-i\Omega)}{2\pi}$, precisely as it occurs for
the Euclidean BTZ black hole Banados:1992wn ; Banados:1992gq  and Euclidean
AdS3 Carlip:1994gc ; Maldacena:1998bw 222Analogue S-duality relationships
between Euclidean three-dimensional black holes and their corresponding
diffeomorphic Euclidean solitons are also know to hold for General Relativity
on AdS3 with boundary conditions of KdV type Perez:2016vqo , as well as for
asymptotically AdS Oliva:2009ip  or asymptotically Lifshitz black holes Ayon-
Beato:2009rgu  and their corresponding solitons in Perez:2011qp  and
Gonzalez:2011nz , respectively in the context of BHT massive gravity
Bergshoeff:2009hq . .

Thus, assuming that the soliton is the ground state of the hairy sector of the
theory, the entropy of the rotating hairy black hole can be seen to be
successfully reproduced by the Cardy formula once expressed in terms of left
and right ground state energies $\tilde{\Delta}_{0}^{\pm}$ instead of the
central charges Correa:2010hf  (see also Correa:2011dt ; Correa:2012rc ). The
entropy then reads

$S=4\pi\sqrt{-\tilde{\Delta}_{0}^{+}\tilde{\Delta}^{+}}+4\pi\sqrt{-\tilde{\Delta}_{0}^{-}\tilde{\Delta}^{-}}\,,$
(58)

where

$\tilde{\Delta}^{\pm}=\frac{1}{2}\left(M\ell\pm J\right)\,,$ (59)

stand for the eigenvalues of the shifted Virasoro operators,
$\tilde{L}_{0}^{\pm}=L_{0}^{\pm}-\frac{c^{\pm}}{24}$, being related to the
mass $M$ and angular momentum $J$ of the hairy black hole in (43), while left
and right energies of the ground state are determined by the soliton mass
according to $\tilde{\Delta}_{0}^{\pm}=\frac{1}{2}\ell M_{sol}$.

The mass of the soliton can then be readily obtained by plugging its
associated pure gravity gauge field (25) into the boundary term in (24), which
yields to

$\mathcal{B}=-\frac{1}{2\kappa}\left(t_{2}-t_{1}\right)\oint
d\theta\left\langle
A_{t}A_{\theta}\right\rangle=\left(t_{2}-t_{1}\right)N_{\infty}\left(\frac{\pi\alpha_{\nu}^{4}}{3\kappa\left(1+\nu\right)\left(1+\alpha_{\nu}\right)^{2}}\right)\,,$
(60)

so that the soliton mass is found to be given by

$M_{sol}=-\frac{\pi\alpha_{\nu}^{4}}{3\kappa\left(1+\nu\right)\left(1+\alpha_{\nu}\right)^{2}}\,,$
(61)

in agreement the the result obtained in Correa:2010hf  through the canonical
approach. Therefore, making use of the precise value of the soliton mass in
(61), it is straightforward to verify that the Cardy formula (58) precisely
reproduces the rotating hairy black hole entropy in (53).

##  6 Conformal frame and ending remarks

As pointed out in the introduction, the action of a minimally coupled scalar
field (1) with the self-interaction potential (1) relates to that of a
conformally coupled self-interacting scalar field. Indeed, changing to the
conformal (Jordan) frame according to
$\hat{g}_{\mu\nu}=\Omega^{-2}g_{\mu\nu}$, with $\Omega=(1-\varphi^{2}),$ and
redefining the scalar field as $\varphi=\tanh\left(\phi\right)$, the action
becomes

$I\left[\hat{g}_{\mu\nu},\varphi\right]=\frac{8}{\kappa}\int
d^{3}x\sqrt{-\hat{g}}\left(\frac{\hat{R}-2\Lambda}{16}-\frac{1}{2}\hat{g}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{16}\hat{R}\varphi^{2}-\lambda\varphi^{6}\right)\,,$
(62)

with $\lambda=\frac{\Lambda\nu}{8}$, so that the matter piece turns out to be
conformally invariant. The field equations then read

$\displaystyle\hat{G}_{\mu\nu}+\Lambda\hat{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle 8\hat{T}_{\mu\nu}\,,$ (63)
$\displaystyle\hat{\Box}\varphi-\frac{1}{8}\hat{R}\varphi-6\lambda\varphi^{5}$
$\displaystyle=$ $\displaystyle 0\,,$ (64)

where the stress-energy tensor

$\hat{T}_{\mu\nu}=\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\hat{g}_{\mu\nu}\partial_{\alpha}\varphi\partial^{\alpha}\varphi-\hat{g}_{\mu\nu}\lambda\varphi^{6}+\frac{1}{8}\left(\hat{g}_{\mu\nu}\hat{\Box}-\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}+\hat{G}_{\mu\nu}\right)\varphi^{2},$
(65)

is traceless by virtue of (64), implying that the Ricci scalar is constant,
$\hat{R}=-6\ell^{-2}$.

The case of a conformally coupled self-interacting scalar field on a fixed
background metric can also be described in terms of a Chern-Simons action
Ricardo-Minas , and here we extend this result to the case of a back-reacting
scalar field with a dynamical metric described by (62).

It is then useful to define $\Lambda^{+}=\Lambda$ and $\Lambda^{-}=-8\lambda$,
as well as the appropriate composite gauge fields as

$\displaystyle A^{+}$ $\displaystyle=$
$\displaystyle\hat{e}^{a}P_{a}^{+}+\omega_{+}^{a}J_{a}^{+}\,,$ (66)
$\displaystyle A^{-}$ $\displaystyle=$
$\displaystyle\varphi^{2}\hat{e}^{a}P_{a}^{-}+\omega_{-}^{a}J_{a}^{-}\,,$ (67)

where according to the signs of $\Lambda^{\pm}$, they take values on the
algebras in (8) that may correspond to $so(3,1)$, $so(2,2)$ or $iso(2,1)$,
precisely as in the case of the minimally coupled scalar field. The field
strengths then read

$F^{\pm}=dA^{\pm}+(A^{\pm})^{2}=\mathcal{T}_{\text{$\pm$}}^{a}P_{a}^{\pm}+\mathcal{R}_{\pm}^{a}J_{a}^{\pm}\,,$
(68)

whose components are given by

$\displaystyle\mathcal{R}_{+}^{a}$ $\displaystyle=$ $\displaystyle
R_{+}^{a}-\frac{1}{2}\Lambda^{+}\epsilon^{abc}\hat{e}_{b}\hat{e}_{c}\,,$ (69)
$\displaystyle\mathcal{R}_{-}^{a}$ $\displaystyle=$ $\displaystyle
R_{-}^{a}-\frac{1}{2}\Lambda^{-}\varphi^{4}\epsilon^{abc}\hat{e}_{b}\hat{e}_{c}\,,$
(70) $\displaystyle\mathcal{T}_{+}^{a}$ $\displaystyle=$ $\displaystyle
T_{+}^{a}\,,$ (71) $\displaystyle\mathcal{T}_{-}^{a}$ $\displaystyle=$
$\displaystyle\varphi^{2}\left(T_{-}^{a}+2\varphi^{-1}d\varphi\hat{e}^{a}\right)\,,$
(72)

where
$R_{\pm}^{a}=d\omega_{\text{$\pm$}}^{a}+\frac{1}{2}\epsilon^{abc}\omega_{b}^{\pm}\omega_{c}^{\pm}$,
and
$T_{\text{$\pm$}}^{a}=d\hat{e}^{a}+\epsilon^{abc}\omega_{b}^{\pm}\hat{e}_{c}$.

Up to a boundary term, the action (62) can then be analogously reformulated in
terms of a Chern-Simons form as in (14), for the composite gauge fields
$A^{\pm}$ in (66), (67) so that
$I=I\left[\varphi,\hat{e},\omega^{+},\omega^{-}\right]$. Varying with respect
to the dynamical fields one obtains

$\displaystyle\frac{\delta I}{\delta\hat{e}^{a}}:$
$\displaystyle\mathcal{R}_{a}^{+}-\varphi^{2}\mathcal{R}_{a}^{-}=0\,,$ (73)
$\displaystyle\frac{\delta I}{\delta\varphi}:$
$\displaystyle\left(\mathcal{R}_{a}^{+}-\mathcal{R}_{a}^{-}\right)\hat{e}^{a}=0\,,$
(74) $\displaystyle\frac{\delta I}{\delta\omega_{+}^{a}}:$ $\displaystyle
T_{+}^{a}=0\,,$ (75) $\displaystyle\frac{\delta I}{\delta\omega_{-}^{a}}:$
$\displaystyle T_{-}^{a}+2\varphi^{-1}d\varphi\hat{e}^{a}=0\,.$ (76)

Since the last field equations (75) and (76) are algebraic for
$\omega_{\pm}^{a}$, they are solved as

$\omega_{-}^{a}\left(\hat{e},\varphi\right)=\omega^{a}\left(\hat{e}\right)-2\varphi^{-1}*\left(\hat{e}^{a}d\varphi\right)\,.$
(77)

and $\omega_{+}^{a}=\omega^{a}\left(\hat{e}\right)$, where
$\omega^{a}\left(\hat{e}\right)$ stands for the (torsionless) spin connection
associated to $\hat{e}^{a}$. Making use of them, the remaining field equations
(73) and (74) reduce to (63) and (64), respectively.

A rotating black hole dressed with a conformally coupled scalar field can then
be readily obtained from the exact solution discussed in section 4, just by
transforming from the Einstein to the conformal frame. In the static case, it
reduces to the black holes discussed in Henneaux:2002wm  and previously in
Martinez:1996gn  in the case of $\nu=0$. As in section 5, the analysis of its
global charges, regularity and thermodynamics can then also be performed in
terms of the gauge fields $A^{\pm}$ in (66), (67) by virtue of the boundary
terms in (23), (24) and the holonomy condition (45). It is also reassuring to
verify that the black hole entropy formula in the Chern-Simons approach
Bunster:2014mua , given by (26), reduces to that in the conformal frame

$S=\left[1-\varphi^{2}\big{(}r_{+}\big{)}\right]\frac{A_{\textrm{hor}}}{4G}\,,$

with $\kappa=8\pi G$, see e.g., Visser:1993nu ; Ashtekar:2003jh , which can
also be reproduced from the Cardy formula (58) when the scalar soliton in the
conformal frame is assumed to be the ground state configuration.

As a final remark, it would be interesting to explore whether the hairy black
holes and similar configurations endowed with a non-trivial scalar field that
have been found for a variety of gravity theories, with different scalar
couplings and self-interactions Hotta:2008xt ; Hortacsu:2003we ; Kwon:2012zh ;
Xu:2013nia ; Zhao:2013isa ; Naji:2014ira ; Mazharimousavi:2014vza ; Xu:2014uha
; Cardenas:2014kaa ; Xu:2014uka ; Gonzalez:2014pwa ; Naji:2014qya ; Ayon-
Beato:2015jga ; Wen:2015xea ; Ayon-Beato:2015ada ; Fan:2015ykb ; Harms:2016pow
; Ozcelik:2016scf ; Erices:2017izj ; Tang:2019jkn ; Harms:2017yko ;
Karakasis:2021lnq ; Bueno:2021krl ; Karakasis:2021ttn ; Arias:2022jax ;
Desa:2022gtw ; Karakasis:2022fep ; Bueno:2022ewf , could also be understood in
terms of suitable composite gauge fields. Nevertheless, the Chern-Simons
formulation for the theory discussed here naively appears to be somewhat
rigid. In fact, the form of the composite gauge fields in (6) and (7) can be
extended as
$A^{\pm}=f_{\pm}^{2}\left(\phi\right)e^{a}P_{a}^{\pm}+\omega_{\pm}^{a}J_{a}^{\pm}$,
so that consistency of the Chern-Simons formulation with a Riemannian
(torsionless) geometry implies that $f_{+}^{2}-f_{-}^{2}=1$, allows to obtain
General Relativity minimally coupled to a scalar field with a non-canonical
kinetic term and a self-interaction that depend on $f_{+}(\phi)$. However, the
scalar field can be appropriately redefined so that in terms of the new scalar
field one recovers the original action in (2) with canonical kinetic term and
precisely the same self-interaction given in (1).

###### Acknowledgements.

We would like to thank Hernán González, Marc Henneaux, Alfredo Pérez, David
Tempo and Jorge Zanelli for useful comments and discussions. This research has
been partially supported by ANID FONDECYT grants N° 11190730, 1201208, 1220862
, 1211226, 1220910, 1221624. The work of O.F. was partially supported by FNRS-
Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by
funds from the Solvay Family.

## References

  * (1) A. Achúcarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-de Sitter Supergravity Theories,” Phys. Lett. B 180, 89 (1986) doi:10.1016/0370-2693(86)90140-1
  * (2) E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B 311, 46 (1988) doi:10.1016/0550-3213(88)90143-5
  * (3) S. Carlip, “Lectures on (2+1) dimensional gravity,” J. Korean Phys. Soc. 28, S447-S467 (1995) [arXiv:gr-qc/9503024 [gr-qc]].
  * (4) O. Coussaert, M. Henneaux and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class. Quant. Grav. 12, 2961-2966 (1995) doi:10.1088/0264-9381/12/12/012 [arXiv:gr-qc/9506019 [gr-qc]].
  * (5) P. Kraus, “Lectures on black holes and the AdS(3) / CFT(2) correspondence,” Lect. Notes Phys. 755, 193-247 (2008) [arXiv:hep-th/0609074 [hep-th]].
  * (6) A. Maloney and E. Witten, “Quantum Gravity Partition Functions in Three Dimensions,” JHEP 02, 029 (2010) doi:10.1007/JHEP02(2010)029 [arXiv:0712.0155 [hep-th]].
  * (7) G. Barnich and H. A. González, “Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity,” JHEP 05, 016 (2013) doi:10.1007/JHEP05(2013)016 [arXiv:1303.1075 [hep-th]].
  * (8) H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Pérez, D. Tempo and R. Troncoso, “Soft Heisenberg hair on black holes in three dimensions,” Phys. Rev. D 93, no.10, 101503 (2016) doi:10.1103/PhysRevD.93.101503 [arXiv:1603.04824 [hep-th]].
  * (9) A. Pérez, D. Tempo and R. Troncoso, “Boundary conditions for General Relativity on AdS3 and the KdV hierarchy,” JHEP 06, 103 (2016) doi:10.1007/JHEP06(2016)103 [arXiv:1605.04490 [hep-th]].
  * (10) O. Fuentealba, J. Matulich, A. Pérez, M. Pino, P. Rodríguez, D. Tempo and R. Troncoso, “Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes,” JHEP 01, 148 (2018) doi:10.1007/JHEP01(2018)148 [arXiv:1711.02646 [hep-th]].
  * (11) J. Cotler and K. Jensen, “A theory of reparameterizations for AdS3 gravity,” JHEP 02, 079 (2019) doi:10.1007/JHEP02(2019)079 [arXiv:1808.03263 [hep-th]].
  * (12) D. Melnikov, F. Novaes, A. Pérez and R. Troncoso, “Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy,” JHEP 06, 054 (2019) doi:10.1007/JHEP06(2019)054 [arXiv:1808.04034 [hep-th]].
  * (13) H. A. González, J. Matulich, M. Pino and R. Troncoso, “Revisiting the asymptotic dynamics of General Relativity on AdS3,” JHEP 12, 115 (2018) doi:10.1007/JHEP12(2018)115 [arXiv:1809.02749 [hep-th]].
  * (14) D. Grumiller and W. Merbis, “Near horizon dynamics of three dimensional black holes,” SciPost Phys. 8, no.1, 010 (2020) doi:10.21468/SciPostPhys.8.1.010 [arXiv:1906.10694 [hep-th]].
  * (15) E. Ojeda and A. Pérez, “Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies,” JHEP 08, 079 (2019) doi:10.1007/JHEP08(2019)079 [arXiv:1906.11226 [hep-th]].
  * (16) M. Cárdenas, F. Correa, K. Lara and M. Pino, “Integrable Systems and Spacetime Dynamics,” Phys. Rev. Lett. 127, no.16, 161601 (2021) doi:10.1103/PhysRevLett.127.161601 [arXiv:2104.09676 [hep-th]].
  * (17) A. Achúcarro and P. K. Townsend, “Extended Supergravities in $d$ = (2+1) as Chern-Simons Theories,” Phys. Lett. B 229, 383-387 (1989) doi:10.1016/0370-2693(89)90423-1
  * (18) P. S. Howe, J. M. Izquierdo, G. Papadopoulos and P. K. Townsend, “New supergravities with central charges and Killing spinors in (2+1)-dimensions,” Nucl. Phys. B 467, 183-214 (1996) doi:10.1016/0550-3213(96)00091-0 [arXiv:hep-th/9505032 [hep-th]].
  * (19) M. Bañados, R. Troncoso and J. Zanelli, “Higher dimensional Chern-Simons supergravity,” Phys. Rev. D 54, 2605-2611 (1996) doi:10.1103/PhysRevD.54.2605 [arXiv:gr-qc/9601003 [gr-qc]].
  * (20) M. Henneaux, L. Maoz and A. Schwimmer, “Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity,” Annals Phys. 282, 31-66 (2000) doi:10.1006/aphy.2000.5994 [arXiv:hep-th/9910013 [hep-th]].
  * (21) A. Giacomini, R. Troncoso and S. Willison, “Three-dimensional supergravity reloaded,” Class. Quant. Grav. 24, 2845-2860 (2007) doi:10.1088/0264-9381/24/11/005 [arXiv:hep-th/0610077 [hep-th]].
  * (22) G. Barnich, L. Donnay, J. Matulich and R. Troncoso, “Asymptotic symmetries and dynamics of three-dimensional flat supergravity,” JHEP 08, 071 (2014) doi:10.1007/JHEP08(2014)071 [arXiv:1407.4275 [hep-th]].
  * (23) G. Barnich, L. Donnay, J. Matulich and R. Troncoso, “Super-BMS3 invariant boundary theory from three-dimensional flat supergravity,” JHEP 01, 029 (2017) doi:10.1007/JHEP01(2017)029 [arXiv:1510.08824 [hep-th]].
  * (24) O. Fuentealba, J. Matulich and R. Troncoso, “Asymptotic structure of $\mathcal{N}=2$ supergravity in 3D: extended super-BMS3 and nonlinear energy bounds,” JHEP 09, 030 (2017) doi:10.1007/JHEP09(2017)030 [arXiv:1706.07542 [hep-th]].
  * (25) R. Caroca, P. Concha, O. Fierro and E. Rodríguez, “Three-dimensional Poincaré supergravity and $N$-extended supersymmetric $BMS_{3}$ algebra,” Phys. Lett. B 792, 93-100 (2019) doi:10.1016/j.physletb.2019.02.049 [arXiv:1812.05065 [hep-th]].
  * (26) N. Banerjee, A. Bhattacharjee, Neetu and T. Neogi, “New $\mathcal{N}$ = 2 SuperBMS3 algebra and invariant dual theory for 3D supergravity,” JHEP 11, 122 (2019) doi:10.1007/JHEP11(2019)122 [arXiv:1905.10239 [hep-th]].
  * (27) R. Caroca, P. Concha, O. Fierro and E. Rodríguez, “On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions,” Eur. Phys. J. C 80, no.1, 29 (2020) doi:10.1140/epjc/s10052-019-7595-5 [arXiv:1908.09150 [hep-th]].
  * (28) N. Banerjee, A. Bhattacharjee, S. Biswas and T. Neogi, “Dual theory for maximally $\mathcal{N}$ extended flat supergravity,” JHEP 05, 179 (2022) doi:10.1007/JHEP05(2022)179 [arXiv:2110.05919 [hep-th]].
  * (29) M. P. Blencowe, “A Consistent Interacting Massless Higher Spin Field Theory in $D$ = (2+1),” Class. Quant. Grav. 6, 443 (1989) doi:10.1088/0264-9381/6/4/005
  * (30) E. Bergshoeff, M. P. Blencowe and K. S. Stelle, “Area Preserving Diffeomorphisms and Higher Spin Algebra,” Commun. Math. Phys. 128, 213 (1990) doi:10.1007/BF02108779
  * (31) M. Henneaux and S. J. Rey, “Nonlinear $W_{infinity}$ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity,” JHEP 12, 007 (2010) doi:10.1007/JHEP12(2010)007 [arXiv:1008.4579 [hep-th]].
  * (32) A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, “Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields,” JHEP 11, 007 (2010) doi:10.1007/JHEP11(2010)007 [arXiv:1008.4744 [hep-th]].
  * (33) M. Gutperle and P. Kraus, “Higher Spin Black Holes,” JHEP 05, 022 (2011) doi:10.1007/JHEP05(2011)022 [arXiv:1103.4304 [hep-th]].
  * (34) M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, “Spacetime Geometry in Higher Spin Gravity,” JHEP 10, 053 (2011) doi:10.1007/JHEP10(2011)053 [arXiv:1106.4788 [hep-th]].
  * (35) A. Castro, E. Hijano, A. Lepage-Jutier and A. Maloney, “Black Holes and Singularity Resolution in Higher Spin Gravity,” JHEP 01, 031 (2012) doi:10.1007/JHEP01(2012)031 [arXiv:1110.4117 [hep-th]].
  * (36) M. Henneaux, G. Lucena Gómez, J. Park and S. J. Rey, “Super- W(infinity) Asymptotic Symmetry of Higher-Spin $AdS_{3}$ Supergravity,” JHEP 06, 037 (2012) doi:10.1007/JHEP06(2012)037 [arXiv:1203.5152 [hep-th]].
  * (37) A. Perez, D. Tempo and R. Troncoso, “Higher spin gravity in 3D: Black holes, global charges and thermodynamics,” Phys. Lett. B 726, 444-449 (2013) doi:10.1016/j.physletb.2013.08.038 [arXiv:1207.2844 [hep-th]].
  * (38) A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, “Towards metric-like higher-spin gauge theories in three dimensions,” J. Phys. A 46, 214017 (2013) doi:10.1088/1751-8113/46/21/214017 [arXiv:1208.1851 [hep-th]].
  * (39) A. Perez, D. Tempo and R. Troncoso, “Higher spin black hole entropy in three dimensions,” JHEP 04, 143 (2013) doi:10.1007/JHEP04(2013)143 [arXiv:1301.0847 [hep-th]].
  * (40) M. Henneaux, A. Pérez, D. Tempo and R. Troncoso, “Chemical potentials in three-dimensional higher spin anti-de Sitter gravity,” JHEP 12, 048 (2013) doi:10.1007/JHEP12(2013)048 [arXiv:1309.4362 [hep-th]].
  * (41) C. Bunster, M. Henneaux, A. Pérez, D. Tempo and R. Troncoso, “Generalized Black Holes in Three-dimensional Spacetime,” JHEP 05, 031 (2014) doi:10.1007/JHEP05(2014)031 [arXiv:1404.3305 [hep-th]].
  * (42) Y. M. Zinoviev, “Hypergravity in AdS3,” Phys. Lett. B 739, 106-109 (2014) doi:10.1016/j.physletb.2014.10.041 [arXiv:1408.2912 [hep-th]].
  * (43) O. Fuentealba, J. Matulich and R. Troncoso, “Extension of the Poincaré group with half-integer spin generators: hypergravity and beyond,” JHEP 09, 003 (2015) doi:10.1007/JHEP09(2015)003 [arXiv:1505.06173 [hep-th]].
  * (44) O. Fuentealba, J. Matulich and R. Troncoso, “Asymptotically flat structure of hypergravity in three spacetime dimensions,” JHEP 10, 009 (2015) doi:10.1007/JHEP10(2015)009 [arXiv:1508.04663 [hep-th]].
  * (45) M. Henneaux, A. Pérez, D. Tempo and R. Troncoso, “Extended anti-de Sitter Hypergravity in $2+1$ Dimensions and Hypersymmetry Bounds,” doi:10.1142/9789813144101_0009 [arXiv:1512.08603 [hep-th]].
  * (46) D. Grumiller, A. Pérez, S. Prohazka, D. Tempo and R. Troncoso, “Higher Spin Black Holes with Soft Hair,” JHEP 10, 119 (2016) doi:10.1007/JHEP10(2016)119 [arXiv:1607.05360 [hep-th]].
  * (47) D. Grumiller, W. Merbis and M. Riegler, “Most general flat space boundary conditions in three-dimensional Einstein gravity,” Class. Quant. Grav. 34, no.18, 184001 (2017) doi:10.1088/1361-6382/aa8004 [arXiv:1704.07419 [hep-th]].
  * (48) L. Ravera, “AdS Carroll Chern-Simons supergravity in 2 + 1 dimensions and its flat limit,” Phys. Lett. B 795, 331-338 (2019) doi:10.1016/j.physletb.2019.06.026 [arXiv:1905.00766 [hep-th]].
  * (49) L. Avilés, J. Gomis and D. Hidalgo, “Stringy (Galilei) Newton-Hooke Chern-Simons Gravities,” JHEP 09, 015 (2019) doi:10.1007/JHEP09(2019)015 [arXiv:1905.13091 [hep-th]].
  * (50) F. Ali and L. Ravera, “$\mathcal{N}$-extended Chern-Simons Carrollian supergravities in $2+1$ spacetime dimensions,” JHEP 02, 128 (2020) doi:10.1007/JHEP02(2020)128 [arXiv:1912.04172 [hep-th]].
  * (51) P. Concha, M. Ipinza, L. Ravera and E. Rodríguez, “Non-relativistic three-dimensional supergravity theories and semigroup expansion method,” JHEP 02, 094 (2021) doi:10.1007/JHEP02(2021)094 [arXiv:2010.01216 [hep-th]].
  * (52) P. Concha, L. Ravera and E. Rodríguez, “Three-dimensional non-relativistic supergravity and torsion,” Eur. Phys. J. C 82, no.3, 220 (2022) doi:10.1140/epjc/s10052-022-10183-6 [arXiv:2112.05902 [hep-th]].
  * (53) R. Caroca, D. M. Peñafiel and P. Salgado-Rebolledo, “Non-relativistic spin-$3$ symmetries in $2+1$ dimensions from expanded/extended Nappi-Witten algebras,” [arXiv:2208.00602 [hep-th]].
  * (54) M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, “Black holes and asymptotics of 2+1 gravity coupled to a scalar field,” Phys. Rev. D 65, 104007 (2002) doi:10.1103/PhysRevD.65.104007 [arXiv:hep-th/0201170 [hep-th]].
  * (55) J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104, 207-226 (1986) doi:10.1007/BF01211590
  * (56) J. de Boer and J. I. Jottar, “Thermodynamics of higher spin black holes in $AdS_{3}$,” JHEP 01, 023 (2014) doi:10.1007/JHEP01(2014)023 [arXiv:1302.0816 [hep-th]].
  * (57) G. Barnich, “Conserved charges in gravitational theories: Contribution from scalar fields,” Ann. U. Craiova Phys. 12, no.III, 14-18 (2002) [arXiv:gr-qc/0211031 [gr-qc]].
  * (58) G. Clément, “Black hole mass and angular momentum in 2+1 gravity,” Phys. Rev. D 68, 024032 (2003) doi:10.1103/PhysRevD.68.024032 [arXiv:gr-qc/0301129 [gr-qc]].
  * (59) J. Gegenberg, C. Martínez and R. Troncoso, “A Finite action for three-dimensional gravity with a minimally coupled scalar field,” Phys. Rev. D 67, 084007 (2003) doi:10.1103/PhysRevD.67.084007 [arXiv:hep-th/0301190 [hep-th]].
  * (60) M. I. Park, “Fate of three-dimensional black holes coupled to a scalar field and the Bekenstein-Hawking entropy,” Phys. Lett. B 597, 237-242 (2004) doi:10.1016/j.physletb.2004.07.023 [arXiv:hep-th/0403089 [hep-th]].
  * (61) M. Bañados and S. Theisen, “Scale invariant hairy black holes,” Phys. Rev. D 72, 064019 (2005) doi:10.1103/PhysRevD.72.064019 [arXiv:hep-th/0506025 [hep-th]].
  * (62) Y. S. Myung, “Phase transition for black holes with scalar hair and topological black holes,” Phys. Lett. B 663, 111-117 (2008) doi:10.1016/j.physletb.2008.03.046 [arXiv:0801.2434 [hep-th]].
  * (63) F. Correa, C. Martínez and R. Troncoso, “Scalar solitons and the microscopic entropy of hairy black holes in three dimensions,” JHEP 01, 034 (2011) doi:10.1007/JHEP01(2011)034 [arXiv:1010.1259 [hep-th]].
  * (64) N. Lashkari, “Holographic Symmetry-Breaking Phases in AdS3/CFT2,” JHEP 11, 104 (2011) doi:10.1007/JHEP11(2011)104 [arXiv:1011.3520 [hep-th]].
  * (65) F. Correa, C. Martínez and R. Troncoso, “Hairy Black Hole Entropy and the Role of Solitons in Three Dimensions,” JHEP 02, 136 (2012) doi:10.1007/JHEP02(2012)136 [arXiv:1112.6198 [hep-th]].
  * (66) S. Hyun, J. Jeong and S. H. Yi, “Fake Supersymmetry and Extremal Black Holes,” JHEP 03, 042 (2013) doi:10.1007/JHEP03(2013)042 [arXiv:1210.6273 [hep-th]].
  * (67) J. Aparício, D. Grumiller, E. Lopez, I. Papadimitriou and S. Stricker, “Bootstrapping gravity solutions,” JHEP 05, 128 (2013) doi:10.1007/JHEP05(2013)128 [arXiv:1212.3609 [hep-th]].
  * (68) W. Xu, J. Wang and X. h. Meng, “A Note on Entropy Relations of Black Hole Horizons,” Int. J. Mod. Phys. A 29, no.18, 1450088 (2014) doi:10.1142/S0217751X14500882 [arXiv:1401.5180 [gr-qc]].
  * (69) W. Xu, “Exact black hole formation in three dimensions,” Phys. Lett. B 738, 472-476 (2014) doi:10.1016/j.physletb.2014.10.026 [arXiv:1409.3368 [hep-th]].
  * (70) B. Ahn, S. Hyun, S. A. Park and S. H. Yi, “Scaling symmetry and scalar hairy rotating AdS3 black holes,” Phys. Rev. D 93, no.2, 024041 (2016) doi:10.1103/PhysRevD.93.024041 [arXiv:1508.06484 [hep-th]].
  * (71) L. Avilés, H. Maeda and C. Martínez, “Exact black-hole formation with a conformally coupled scalar field in three dimensions,” Class. Quant. Grav. 35, no.24, 245001 (2018) doi:10.1088/1361-6382/aaea9f [arXiv:1808.10040 [gr-qc]].
  * (72) F. Correa, A. Faúndez and C. Martínez, “Rotating hairy black hole and its microscopic entropy in three spacetime dimensions,” Phys. Rev. D 87, no.2, 027502 (2013) doi:10.1103/PhysRevD.87.027502 [arXiv:1211.4878 [hep-th]].
  * (73) T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity,” Annals Phys. 88, 286 (1974) doi:10.1016/0003-4916(74)90404-7
  * (74) J. Matulich, A. Pérez, D. Tempo and R. Troncoso, “Higher spin extension of cosmological spacetimes in 3D: asymptotically flat behaviour with chemical potentials and thermodynamics,” JHEP 05, 025 (2015) doi:10.1007/JHEP05(2015)025 [arXiv:1412.1464 [hep-th]].
  * (75) M. Bañados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69, 1849-1851 (1992) doi:10.1103/PhysRevLett.69.1849 [arXiv:hep-th/9204099 [hep-th]].
  * (76) M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D 48, 1506-1525 (1993) [erratum: Phys. Rev. D 88, 069902 (2013)] doi:10.1103/PhysRevD.48.1506 [arXiv:gr-qc/9302012 [gr-qc]].
  * (77) S. Carlip and C. Teitelboim, “Aspects of black hole quantum mechanics and thermodynamics in (2+1)-dimensions,” Phys. Rev. D 51, 622-631 (1995) doi:10.1103/PhysRevD.51.622 [arXiv:gr-qc/9405070 [gr-qc]].
  * (78) J. M. Maldacena and A. Strominger, “AdS(3) black holes and a stringy exclusion principle,” JHEP 12, 005 (1998) doi:10.1088/1126-6708/1998/12/005 [arXiv:hep-th/9804085 [hep-th]].
  * (79) J. Oliva, D. Tempo and R. Troncoso, “Three-dimensional black holes, gravitational solitons, kinks and wormholes for BHT massive gravity,” JHEP 07, 011 (2009) doi:10.1088/1126-6708/2009/07/011 [arXiv:0905.1545 [hep-th]].
  * (80) E. Ayón-Beato, A. Garbarz, G. Giribet and M. Hassaïne, “Lifshitz Black Hole in Three Dimensions,” Phys. Rev. D 80, 104029 (2009) doi:10.1103/PhysRevD.80.104029 [arXiv:0909.1347 [hep-th]].
  * (81) A. Pérez, D. Tempo and R. Troncoso, “Gravitational solitons, hairy black holes and phase transitions in BHT massive gravity,” JHEP 07, 093 (2011) doi:10.1007/JHEP07(2011)093 [arXiv:1106.4849 [hep-th]].
  * (82) H. A. González, D. Tempo and R. Troncoso, “Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes,” JHEP 11, 066 (2011) doi:10.1007/JHEP11(2011)066 [arXiv:1107.3647 [hep-th]].
  * (83) E. A. Bergshoeff, O. Hohm and P. K. Townsend, “Massive Gravity in Three Dimensions,” Phys. Rev. Lett. 102, 201301 (2009) doi:10.1103/PhysRevLett.102.201301 [arXiv:0901.1766 [hep-th]].
  * (84) Ricardo Troncoso and Minas Tsoukalas, “Conformally coupled scalar fields in higher dimensions and a generalization of the Yamabe problem," Preprint CECS-PHY-11/11.
  * (85) C. Martínez and J. Zanelli, “Conformally dressed black hole in (2+1)-dimensions,” Phys. Rev. D 54, 3830 (1996) [gr-qc/9604021].
  * (86) M. Visser, “Dirty black holes: Entropy as a surface term,” Phys. Rev. D 48, 5697-5705 (1993) doi:10.1103/PhysRevD.48.5697 [arXiv:hep-th/9307194 [hep-th]].
  * (87) A. Ashtekar, A. Corichi and D. Sudarsky, “Nonminimally coupled scalar fields and isolated horizons,” Class. Quant. Grav. 20, 3413 (2003) [gr-qc/0305044].
  * (88) M. Hortacsu, H. T. Ozcelik and B. Yapiskan, “Properties of solutions in (2+1)-dimensions,” Gen. Rel. Grav. 35, 1209-1221 (2003) doi:10.1023/A:1024445724029 [arXiv:gr-qc/0302005 [gr-qc]].
  * (89) K. Hotta, Y. Hyakutake, T. Kubota, T. Nishinaka and H. Tanida, “The CFT-interpolating Black Hole in Three Dimensions,” JHEP 01, 010 (2009) doi:10.1088/1126-6708/2009/01/010 [arXiv:0811.0910 [hep-th]].
  * (90) Y. Kwon, S. Nam, J. D. Park and S. H. Yi, “Extremal Black Holes and Holographic C-Theorem,” Nucl. Phys. B 869, 189-215 (2013) doi:10.1016/j.nuclphysb.2012.12.016 [arXiv:1208.4509 [hep-th]].
  * (91) W. Xu and L. Zhao, “Charged black hole with a scalar hair in (2+1) dimensions,” Phys. Rev. D 87, no.12, 124008 (2013) doi:10.1103/PhysRevD.87.124008 [arXiv:1305.5446 [gr-qc]].
  * (92) L. Zhao, W. Xu and B. Zhu, “Novel rotating hairy black hole in (2+1)-dimensions,” Commun. Theor. Phys. 61, no.4, 475-481 (2014) doi:10.1088/0253-6102/61/4/12 [arXiv:1305.6001 [gr-qc]].
  * (93) J. Naji, “Energy Loss of a Heavy Particle near 3D Charged Rotating Hairy Black Hole,” Eur. Phys. J. C 74, 2697 (2014) doi:10.1140/epjc/s10052-013-2697-y [arXiv:1401.4422 [hep-th]].
  * (94) S. H. Mazharimousavi and M. Halilsoy, “Einstein–Born–Infeld black holes with a scalar hair in three dimensions,” Mod. Phys. Lett. A 30, no.33, 1550177 (2015) doi:10.1142/S0217732315501771 [arXiv:1405.2956 [gr-qc]].
  * (95) W. Xu, L. Zhao and D. C. Zou, “Three dimensional rotating hairy black holes, asymptotics and thermodynamics,” [arXiv:1406.7153 [gr-qc]].
  * (96) M. Cárdenas, O. Fuentealba and C. Martínez, “Three-dimensional black holes with conformally coupled scalar and gauge fields,” Phys. Rev. D 90, no.12, 124072 (2014) doi:10.1103/PhysRevD.90.124072 [arXiv:1408.1401 [hep-th]].
  * (97) W. Xu and D. C. Zou, “$(2+1)$ -Dimensional charged black holes with scalar hair in Einstein–Power–Maxwell Theory,” Gen. Rel. Grav. 49, no.6, 73 (2017) doi:10.1007/s10714-017-2237-4 [arXiv:1408.1998 [hep-th]].
  * (98) P. A. González, J. Saavedra and Y. Vásquez, “Three-Dimensional Hairy Black Holes in Teleparallel Gravity,” Astrophys. Space Sci. 357, no.2, 143 (2015) doi:10.1007/s10509-015-2374-8 [arXiv:1411.2193 [gr-qc]].
  * (99) J. Naji and S. Heshmatian, “Novel Rotating Hairy Black Hole in (2+1)-Dimensions and Shear Viscosity to Entropy Ratio,” Int. J. Theor. Phys. 53, no.8, 2579-2586 (2014) doi:10.1007/s10773-014-2056-2
  * (100) E. Ayón-Beato, M. Bravo-Gaete, F. Correa, M. Hassaïne, M. M. Juárez-Aubry and J. Oliva, “First law and anisotropic Cardy formula for three-dimensional Lifshitz black holes,” Phys. Rev. D 91, no.6, 064006 (2015) doi:10.1103/PhysRevD.91.064006 [arXiv:1501.01244 [gr-qc]].
  * (101) Q. Wen, “Strategy to Construct Exact Solutions in Einstein-Scalar Gravities,” Phys. Rev. D 92, no.10, 104002 (2015) doi:10.1103/PhysRevD.92.104002 [arXiv:1501.02829 [hep-th]].
  * (102) E. Ayón-Beato, M. Hassaïne and J. A. Méndez-Zavaleta, “(Super-)renormalizably dressed black holes,” Phys. Rev. D 92, no.2, 024048 (2015) doi:10.1103/PhysRevD.92.024048 [arXiv:1506.02277 [hep-th]].
  * (103) Z. Y. Fan and B. Chen, “Exact formation of hairy planar black holes,” Phys. Rev. D 93, no.8, 084013 (2016) doi:10.1103/PhysRevD.93.084013 [arXiv:1512.09145 [hep-th]].
  * (104) B. Harms and A. Stern, “Spinning $\sigma$-model solitons in $2+1$ anti-de Sitter space,” Phys. Lett. B 763, 401-408 (2016) doi:10.1016/j.physletb.2016.10.075 [arXiv:1608.05116 [hep-th]].
  * (105) H. T. Özçelik, R. Kaya and M. Hortaçsu, “Einstein gravity with torsion induced by the scalar field,” Annals Phys. 393, 132-144 (2018) doi:10.1016/j.aop.2018.04.012 [arXiv:1611.07496 [gr-qc]].
  * (106) B. Harms and A. Stern, “Growing Hair on the extremal $BTZ$ black hole,” Phys. Lett. B 769, 465-469 (2017) doi:10.1016/j.physletb.2017.04.021 [arXiv:1703.10234 [gr-qc]].
  * (107) C. Erices and C. Martínez, “Rotating hairy black holes in arbitrary dimensions,” Phys. Rev. D 97, no.2, 024034 (2018) doi:10.1103/PhysRevD.97.024034 [arXiv:1707.03483 [hep-th]].
  * (108) Z. Y. Tang, Y. C. Ong, B. Wang and E. Papantonopoulos, “General black hole solutions in ( 2+1 )-dimensions with a scalar field nonminimally coupled to gravity,” Phys. Rev. D 100, no.2, 024003 (2019) doi:10.1103/PhysRevD.100.024003 [arXiv:1901.07310 [gr-qc]].
  * (109) T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “Black holes of (2+1)-dimensional $f(R)$ gravity coupled to a scalar field,” Phys. Rev. D 103, no.6, 064063 (2021) doi:10.1103/PhysRevD.103.064063 [arXiv:2101.06410 [gr-qc]].
  * (110) P. Bueno, P. A. Cano, J. Moreno and G. van der Velde, “Regular black holes in three dimensions,” Phys. Rev. D 104, no.2, L021501 (2021) doi:10.1103/PhysRevD.104.L021501 [arXiv:2104.10172 [gr-qc]].
  * (111) T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “(2+1)-dimensional black holes in f(R,$\phi$) gravity,” Phys. Rev. D 105, no.4, 044038 (2022) doi:10.1103/PhysRevD.105.044038 [arXiv:2201.00035 [gr-qc]].
  * (112) P. J. Arias, P. Bargueño, E. Contreras and E. Fuenmayor, “$2+1$ Einstein-Klein-Gordon black holes by gravitational decoupling,” Astronomy 1, no.1, 2-14 (2022) doi:10.3390/astronomy1010002 [arXiv:2203.00661 [gr-qc]].
  * (113) C. Desa, W. Ccuiro and D. Choque, “Exact hairy black holes asymptotically $AdS_{2+1}$,” [arXiv:2210.06421 [hep-th]].
  * (114) T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “Rotating $(2+1)$-dimensional Black Holes in Einstein-Maxwell-Dilaton Theory,” [arXiv:2210.15704 [gr-qc]].
  * (115) P. Bueno, P. A. Cano, J. Moreno and G. van der Velde, “Electromagnetic Generalized Quasi-topological gravities in $(2+1)$ dimensions,” [arXiv:2212.00637 [gr-qc]].


11institutetext: Graduate School of Information Science, Nagoya University,
22institutetext: Information and Communications Headquarters, Nagoya
University,
33institutetext: School of Information Science, Aichi Institute of Technology,
44institutetext: Graduate School of Medicine, Nagoya University,
55institutetext: Department of Endoscopy, Nagoya University Hospital,
66institutetext: National Cancer Center,
77institutetext: Aichi Cancer Center,
88institutetext: International University of Health and Welfare Mita Hospital

Masahiro Oda

# Semi-Automated Virtual Unfolded View Generation Method of Stomach
from CT Volumes

Masahiro Oda 11 Tomoaki Suito 11 Yuichiro Hayashi 22 Takayuki Kitasaka 33
Kazuhiro Furukawa 44 Ryoji Miyahara 44 Yoshiki Hirooka 55 Hidemi Goto 44
Gen Iinuma 66 Kazunari Misawa 77 Shigeru Nawano 88 Kensaku Mori 2211

###### Abstract

CT image-based diagnosis of the stomach is developed as a new way of
diagnostic method. A virtual unfolded (VU) view is suitable for displaying its
wall. In this paper, we propose a semi-automated method for generating VU
views of the stomach. Our method requires minimum manual operations. The
determination of the unfolding forces and the termination of the unfolding
process are automated. The unfolded shape of the stomach is estimated based on
its radius. The unfolding forces are determined so that the stomach wall is
deformed to the expected shape. The iterative deformation process is
terminated if the difference of the shapes between the deformed shape and
expected shape is small. Our experiments using 67 CT volumes showed that our
proposed method can generate good VU views for 76.1% cases.

###### Keywords:

Stomach, virtual unfolding, CT image

##  1 Introduction

In Japan, the mortality rate of stomach cancer is the second highest among
cancer-related mortality [1]. Treatment of stomach cancer in the early stages
is crucial. Gastric roentgenography and gastrofiberscopy are currently
performed as the diagnostic methods of stomach cancer. But these methods are
physically and mentally painful for patients. In recent years, a CT image-
based diagnostic method of the stomach has been developed as an alternative
choice [2] that utilizes virtual gastroscopic views generated from CT images.
Although CT image-based stomach diagnosis systems greatly reduce the
inspection time for patients, physicians need to manually change the
viewpoints and the viewing directions of the virtual gastroscopic views many
times during diagnosis. To reduce this load of physicians, a virtual unfolded
(VU) view of the stomach is suitable for displaying the stomach wall. This
view enables physicians to observe the stomach wall at a glance.

Much research has been reported on the VU view generation of hollow organs.
Most generate VU views of the colon [3, 4]. Because these methods generate
views in real-time, their unfolding processes do not follow the physical
properties of organ deformation. VU view generation methods of the stomach
have been proposed based on surface model [5] and volumetric model [6].
Although these methods simulate realistic deformations, they require the
following complicated manual operations: (1) incision line determination, (2)
unfolding force determination, and (3) the termination of unfolding processes.
Truong et al. [7] automated (2) the unfolding force determination process.
However, the results of their method heavily depend on the results of incision
line determination. Their method requires many trial-and-error corrections in
incision line determination. Suito et al. [8] presented an automated method of
(1) the incision line determination process. These researches only automate
one of the manual processes. Therefore, these VU view generation methods of
the stomach still require complicated manual operations. They are time-
consuming and the quality of their generated VU views heavily depends on the
skill of the user.

In this paper, we propose a semi-automated method for generating VU views of
the stomach from CT volumes. The contribution of this paper is first
presentation of semi-automated method that can drastically reduce manual
operations in generation of VU views of the stomach. Manual operation required
in our method is just specifying two positions on CT images: the cardia and
the pylorus. All other processes are automated in our method. We determine the
incision line using a previous method [8]. Then a stomach wall model is
generated from a stomach wall region extracted from a CT volume. Unfolding
forces are added to the model. We newly introduce a method that automatically
calculates the direction of the unfolding forces for this. The expected
unfolded shape is estimated based on its diameter. Unfolding force is
determined so that the stomach wall is deformed to the expected shape.
Newmark-$\beta$ [9] method is introduced to simulate elastic deformation of
the model. Because the unfolding process is performed iteratively, it must be
terminated after appropriate iterations. For the termination of the unfolding
process, we define a criterion that evaluates the progress of the unfolding
process based on the differences of the shapes between the deformed stomach
shape and the expected unfolded shape of the stomach.

##  2 Method

###  2.1 Preprocessing

In the preprocessing step, we extract an air region in the stomach, a stomach
wall region, and the centerline of the stomach region from abdominal CT
volumes [8]. Then the incision lines are determined using the methods shown in
[8]. Since the incision line determination process [8] requires the positions
of the cardia and the pylorus, these points are manually specified on CT
volumes by mouse-click.

###  2.2 Definition of Stomach Wall Model

We utilize a previously proposed stomach wall model [7] to simulate the
deformation. The model consists of a set of hexahedra covering the stomach
wall region. The center of each hexahedron is a voxel in the stomach wall
region. The length of each hexahedron edge in the $x,y$, and $z$ directions
are $d,d$, and $\hat{d}$ voxels, respectively. Here,
$\hat{d}=d\cdot(\rm{pixel\ spacing})/(\rm{slice\ spacing})$. The hexahedra on
the incision line are removed from the stomach wall model. Each hexahedron is
converted into elastic model by placing mass-points, springs, and dampers on
vertices, edges, and diagonal lines [7].

We define three sets of the hexahedra vertices in the stomach wall model:
$S_{\rm vo}$, $S_{\rm vi}$, and $S_{\rm vb}$. A set of vertices on the outer
and inner surfaces of the stomach wall model is represented as $S_{\rm vo}$
and $S_{\rm vi}$, respectively. The inner surface of the stomach wall model is
a set of the faces of hexahedra in contact with the air region in the stomach
and the incision line. A set of vertices near the incision line, which is
included in both $S_{\rm vo}$ and $S_{\rm vi}$, are defined as $S_{\rm vb}$
(Fig. 1).

Figure 1: Sets of vertices of hexahedra $S_{\rm vo}$, $S_{\rm vi}$, and
$S_{\rm vb}$ on stomach wall model.

###  2.3 Determination of Unfolding Force

The unfolding forces are added to the vertices of hexahedra on the incision
line, which is unfolded to a planar shape by the forces. We first define an
unfolded plane on which the stomach wall model is stretched. Then we obtain
the destination points on the plane, where the vertices on the incision line
must reach after unfolding.

####  2.3.1 Determination of Unfolded Plane

Unfolded plane $\Omega$ is uniquely defined by its normal vector ${\bf
n}_{\Omega}$ and point ${\bf b}_{\Omega}$ on it as

$\displaystyle{\bf n}_{\Omega}$ $\displaystyle=$ $\displaystyle({\bf
u}_{J/2}-{\bf x})/\parallel{\bf u}_{J/2}-{\bf x}\parallel,$ (1)
$\displaystyle{\bf b}_{\Omega}$ $\displaystyle=$ $\displaystyle{\bf
r}^{(0)}_{m},\ m={\rm argmax}_{V_{i}\in S_{\rm vi}}\left({\bf
r}^{(0)}_{i}\cdot{\bf n}_{\Omega}\right),$ (2)

where ${\bf u}_{j}\ (j=1,\ldots,J)$ is a point sequence forming the incision
line. ${\bf x}$ is a point on a line segment, which connects ${\bf u}_{1}$ and
${\bf u}_{J}$, and satisfies $({\bf x}-{\bf u}_{1})\cdot({\bf u}_{J/2}-{\bf
x})=0$. $V_{i}\ (i=1,\ldots,I)$ is the $i$-th vertex in the stomach wall
model. ${\bf r}^{(\alpha)}_{i}$ is the position of $V_{i}$ after $\alpha$
iterations of the unfolding process. The positions of ${\bf n}_{\Omega}$ and
${\bf b}_{\Omega}$ are shown in Fig. 2(a) and (b).

(a)

(b)

(c)

Figure 2: (a) Normal vector ${\bf n}_{\Omega}$, (b) point ${\bf b}_{\Omega}$
on $\Omega$, and (c) radius of stomach $\epsilon_{i^{\prime}}$ at
$V_{i^{\prime}}$.

####  2.3.2 Determination of Destination Point

This process consists of three steps: (Step 1) Calculation of stomach radius,
(Step 2) Calculation of base line, and (Step 3) Determination of destination
point. The thick and thin parts of the stomach wall must be unfolded widely
and narrowly. We compute the width of the unfolded view based on the radius of
the stomach at each point on the incision line. Step 1 calculates the radius
of the stomach. Step 2 computes the base line around which the stomach is
unfolded. Step 3 determines the points where each vertex of the incision line
is forced to be moved.

(Step 1) Calculation of Stomach Radius: We obtain the radius of the stomach
for each vertex existing on the incision line. The point sequence on the
centerline of the stomach is described as ${\bf c}_{k}\ (k=1,\ldots,K)$. ${\bf
c}_{k}$ that is closest to ${\bf u}_{j}$ is obtained as ${\bf c}_{k^{\prime}}$
whose index is calculated by

$k^{\prime}={\rm argmin}_{1\leq k\leq K}\parallel{\bf c}_{k}-{\bf
u}_{j}\parallel.$ (3)

$V_{i^{\prime}}\in S_{\rm vb}\ (i^{\prime}=1,\ldots,I^{\prime})$ is the
$i^{\prime}$-th vertex existing on the incision line on the stomach wall
model. The index of ${\bf u}_{j}$, which is closest to ${\bf
r}^{(0)}_{i^{\prime}}$ is obtained by

$j^{\prime}={\rm argmin}_{1\leq j\leq J}\parallel{\bf
r}^{(0)}_{i^{\prime}}-{\bf u}_{j}\parallel.$ (4)

The radius of the stomach (Fig. 2(c)) at $V_{i^{\prime}}$ is calculated by

$\epsilon_{i^{\prime}}=\parallel{\bf c}_{k^{\prime}}-{\bf
u}_{j^{\prime}}\parallel.$ (5)

(Step 2) Calculation of Base Line: The base line is a long axis of the point
set on the incision line projected on the unfolded plane. Each point ${\bf
u}_{j}$ is projected onto the closest point ${\bf u}^{\prime}_{j}$ on the
unfolded plane. We apply principal component analysis to ${\bf
u}^{\prime}_{j}$ to obtain the first and second eigenvectors of ${\bf
u}^{\prime}_{j}$. The first and second eigenvectors are ${\bf v}^{\prime}_{1}$
and ${\bf v}^{\prime}_{2}$. The direction of the base line is given by

$\displaystyle{\bf v}_{1}=\left\\{\begin{array}[]{ll}{\bf
v}^{\prime}_{1}/\parallel{\bf v}^{\prime}_{1}\parallel,&\ if\ {\bf
v}^{\prime}_{1}\cdot({\bf u}^{\prime}_{J}-{\bf u}^{\prime}_{1})<0,\\\ -{\bf
v}^{\prime}_{1}/\parallel{\bf v}^{\prime}_{1}\parallel,&\
otherwise.\end{array}\right.$ (8)

The base line is represented as a set of points and defined by

$\displaystyle{\bf p}_{j}=\left\\{\begin{array}[]{ll}{\bf
u}^{\prime}_{J/2}-{\bf v}_{1}\sum_{z=j}^{J/2-1}\parallel{\bf
u}^{\prime}_{z+1}-{\bf u}^{\prime}_{z}\parallel,&\ if\ j=1,\ldots,J/2-1,\\\
{\bf u}^{\prime}_{J/2},&\ if\ j=J/2,\\\ {\bf u}^{\prime}_{J/2}+{\bf
v}_{1}\sum_{z=J/2}^{j-1}\parallel{\bf u}^{\prime}_{z+1}-{\bf
u}^{\prime}_{z}\parallel,&\ if\ j=J/2+1,\ldots,J.\end{array}\right.$ (12)

The position of ${\bf p}_{j}$ is shown in Fig. 3(a).

(Step 3) Determination of Destination Point: Destination points ${\bf
g}_{i^{\prime}}$ are determined based on the base line and the radius of the
stomach. ${\bf g}_{i^{\prime}}$ is positioned $\pi\epsilon_{i^{\prime}}$[mm]
away from the base line on the unfolded plane (Fig. 3(b)):

$\displaystyle{\bf g}_{i^{\prime}}=\left\\{\begin{array}[]{ll}{\bf
p}_{j^{\prime}}+\pi\epsilon_{i^{\prime}}{\bf v}_{2},&\ if\ \\{({\bf
u}_{j^{\prime}}-{\bf c}_{k^{\prime}})\times({\bf r}^{(0)}_{i^{\prime}}-{\bf
c}_{k^{\prime}})\\}\cdot({\bf u}_{j^{\prime}}-{\bf u}_{j^{\prime}-1})\geq
0,\\\ {\bf p}_{j^{\prime}}-\pi\epsilon_{i^{\prime}}{\bf v}_{2},&\
otherwise,\end{array}\right.$ (15)

${\bf v}_{2}$ is unit vector perpendicular to the base line obtained by

$\displaystyle{\bf v}_{2}=\left\\{\begin{array}[]{ll}{\bf
v}^{\prime}_{2}/\parallel{\bf v}^{\prime}_{2}\parallel,&\ if\ ({\bf
v}^{\prime}_{2}\times{\bf v}_{1})\cdot{\bf n}_{\Omega}<0,\\\ -{\bf
v}^{\prime}_{2}/\parallel{\bf v}^{\prime}_{2}\parallel,&\
otherwise.\end{array}\right.$ (18)

(a)

(b)

Figure 3: (a) Base line ${\bf p}_{j}$ calculated from projected incision line
${\bf u}^{\prime}_{j}$. (b) Destination point ${\bf g}_{i^{\prime}}$
corresponds to $V_{i^{\prime}}$.

####  2.3.3 Unfolding Force Determination

The direction of the unfolding force for $V_{i^{\prime}}$ calculated as

$\displaystyle{\bf e}^{(\alpha)}_{i^{\prime}}={\bf g}_{i^{\prime}}-{\bf
r}^{(\alpha)}_{i^{\prime}},$ (19)

where $\alpha$ is the number of iterations in the iterative unfolding process.
${\bf e}^{(\alpha)}_{i^{\prime}}$ is updated at each iteration step in the
unfolding process.

###  2.4 Unfolding Process Based on Elastic Deformation

We deform the stomach wall model by Newmark-$\beta$ method [9] with the forces
computed above. The shell-like volumetric model between the inner and outer
surfaces shown in Fig. 1 is deformed here. We employ the elastic deformation
procedure shown in [7]. This method [7] manually terminated the iterative
unfolding process. We propose a criterion to evaluate progress of the
unfolding process in order to determine termination of deformation.

We introduce new metric that measures the average distance between ${\bf
r}^{(\alpha)}_{i^{\prime}}$ and its destination point. The metric is described
as

$\displaystyle D^{(\alpha)}=\frac{1}{N}\sum_{V_{i^{\prime}}\in S_{\rm
vb}}\parallel{\bf r}^{(\alpha)}_{i^{\prime}}-{\bf g}_{i^{\prime}}\parallel,$
(20)

where $N$ is the number of vertices included in $S_{\rm vb}$. If the change of
the average distance is small in the iteration of the unfolding process, the
stomach is adequately stretched. The iterative unfolding process is terminated
if condition

$\parallel D^{(\alpha-1)}-D^{(\alpha)}\parallel\leq\kappa$ (21)

is satisfied.

###  2.5 Unfolded View Generation

After the stomach wall model deformation, we generate an unfolded volume by
using relationship between the stomach wall models of pre- and post-
deformations. The unfolded view is generated by volume-rendering of the
unfolded volume.

##  3 Experiments and Results

We evaluated the proposed method by using 67 CT volumes that were taken from
the distended state of the stomach using a foaming agent at two hospitals. The
followings are the acquisition parameters of the CT volumes: image size:
512$\times$512 pixels, number of slices: 371-644, pixel spacing: 0.625-0.774
mm, slice spacing: 0.4-1.0 mm, slice thickness: 0.5-1.0 mm. Parameter values
$d$ = 8 voxels and $\kappa=0.5$ were experimentally chosen. The quality of the
VU views was visually evaluated by surgeons and engineering researchers by
giving excellent (All region on VU view is visible from a view point), good
(One small failure part (overlapping, bending, or broken) exists on VU view),
fair, and bad (Area of VU view where there are not visible from a view point
by influence of failure parts is over 30% and 50%, respectively)
classifications. The generated VU views were compared with the flattened
pathological specimens of the resected stomachs from the patients. We prepared
ground truth VU views with manual input of the incision lines and the
unfolding forces. The generated VU views are shown in Fig. 4. The numbers of
cases rated as top one (excellent) and two (excellent or good) classifications
were 26 (38.8%) and 51 (76.1%) cases.

Since the quality of the VU views depends on the value of $\kappa$, we
generated VU views by changing $\kappa$ values for 19 cases. Table 1 shows the
numbers of good results and cases that caused overlapping, bending, and broken
parts of the stomach wall in the VU views.

We compared the computation time in VU view generation of the proposed and a
previous [6] methods (Computer: two Intel Xeon 3.33GHz processors, 4GB RAM).
It took about 19 seconds for the manual input and 458 seconds for the
automated process. In the previous method [6], manual processes including the
incision line determination, the unfolding force determination, and the
termination of the unfolding processes took about 900 seconds.

Figure 4: Ground truth VU views, VU views generated by proposed method, and flattened pathological specimens of two cases. Circles indicate positions of a cancer. Table 1: Numbers of good results and numbers of cases that caused overlapping, bending, and broken parts of stomach wall on generated VU views for various $\kappa$. $\kappa$ | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0
---|---|---|---|---|---|---|---|---|---|---
Good | 7 | 10 | 10 | 10 | 10 | 9 | 8 | 7 | 7 | 7
Overlapping | 5 | 4 | 2 | 2 | 2 | 2 | 1 | 0 | 0 | 0
Bending | 10 | 7 | 8 | 7 | 6 | 7 | 7 | 7 | 7 | 7
Broken | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4

##  4 Discussion

From Fig. 4, the VU views generated by our proposed method clearly represent
the shapes of a cancer and the stomach wall. Although the shapes of the views
generated by the proposed method slightly differ from the ground truth VU
views, such surface shapes as the cancer on the stomach wall were well
observed. The surface shape is important for finding lesions. These results
are applicable to other data judged as ”Good”. Therefore, the VU views
generated using our proposed method are applicable for diagnosis.

From Table 1, the numbers of broken parts in the VU views become larger when
large values of $\kappa$ are used. When $\kappa$ is large, the iteration of
the unfolding process is terminated even if the stomach wall model is still
largely being deformed. In such cases, several parts of the stomach wall model
are greatly stretched. Largely stretched parts of the stomach wall model may
cause broken parts in the VU views. The numbers of the overlapping parts on
the VU view becomes larger for smaller $\kappa$. In this case, the iteration
number of the unfolding process becomes large because the iteration continues
until the incision lines of the model reach the destination points. In the
unfolding process [7], forces are added to the vertices on the outer and inner
surfaces of the stomach wall models. The magnitude of these forces increase as
iteration progresses. This cause overlapping of the VU views. To minimize
broken or overlapping parts, $\kappa=0.5$ was selected as an optimal value for
testing 67 cases.

This paper presented a semi-automated VU view generation method of the
stomach. The automation of VU view generation also achieves qualitative stable
VU views. Quality of VU views of the previous method heavily depends on
experience levels of users. Our method generates VU views with minimal
influence of user experience level. It also has potential to explore new
diagnostic method of the stomach like as screening examinations of early
gastric cancers or planning of gastric cancer surgery.

##  5 Conclusion

This paper presented a semi-automated VU view generation method. The
determination of the unfolding forces and the termination of the unfolding
were automated in our method. Experiments using 67 CT volumes showed that our
proposed method can generate VU views with same quality as manually generated
ones. Future work includes the automation of cardia and pylorus detection,
unfolding of twisted stomachs, and evaluation using more samples.

#### Acknowledgments.

Parts of this research were supported by the MEXT, the JSPS KAKENHI Grant
Numbers 21103006, 25242047, and the Kayamori Foundation of Informational
Science Advancement.

## References

  * [1] Isobe, Y., Nashimoto, A., Akazawa, K., Oda, I., Hayashi, K., Miyashiro, I., Katai, H., Tsujitani, S., Kodera, Y., Seto, Y., Kaminishi, M.: Gastric cancer treatment in Japan: 2008 annual report of the JGCA nationwide registry. In: Gastric Cancer, vol.14, no.4, pp.301-316 (2011)
  * [2] Furukawa, K., Miyahara, R., Itoh, A., Ohmiya, N., Hirooka, Y., Mori, K., Goto, H.: Diagnosis of the invasion depth of gastric cancer using MDCT with virtual gastroscopy: comparison with staging with endoscopic ultrasound. Am. J. Roentgenology, 197(4), 867-875 (2011)
  * [3] Wang, G., McFarland, E.G., Brown, B.P., Vannier, M.W.: GI tract unraveling with curved cross sections. IEEE Trans. Med. Imag., 17(2), 318–322 (1998)
  * [4] Bartroli, A.V., Wegenkittl, R., Konig, A., Groller, E.: Virtual colon unfolding. In: IEEE Visualization (2001)
  * [5] Mori, K., Hoshino, Y., Suenaga, Y., Toriwaki, J., Hasegawa, J., Katada, K.: An improved method for generating virtual stretched view of stomach based on shape deformation. In: Proc. CARS2001, pp.425–430 (2001)
  * [6] Mori, K., Oka, H., Kitasaka, T., Suenaga, Y.: A method for generating unfolded views of the stomach based on volumetric image deformation. In: Proc. SPIE Medical Imaging, vol.5746, pp.340–351 (2005)
  * [7] Truong, T.D., Kitasaka, T., Mori, K., Suenaga, Y.: A physically-based method for unfolding the stomach from 3D CT images. In: Proc. IASTED Int. Conf. Computer Graphics and Imaging, pp.231–236 (2008)
  * [8] Suito, T., Oda, M., Kitasaka, T., Iinuma, G., Misawa, K., Nawano, S., Mori, K.: Automated incision line determination for virtual unfolded view generation of the stomach from 3D abdominal CT images. In: Proc. SPIE Medical Imaging, vol.8315, pp.83151M-1–7 (2012)
  * [9] Newmark, N.M.: A method of computation for structural dynamics. In: Journal of Engineering Mechanics, Proc. ASCE, vol.85, no.EM3, pp.67-94 (1959)