# Supersymmetric solitons and a degeneracy of solutions in AdS/CFT

Andrés Anabalón(1) and Simon F. Ross(2)

(1)Departamento de Ciencias, Facultad de Artes Liberales,
Universidad Adolfo Ibáñez, Avenida Padre Hurtado 750, Viña del Mar, Chile.

(2)Centre for Particle Theory, Department of Mathematical Sciences,
Durham University, South Road, Durham DH1 3LE, U.K

###### Abstract

We study Lorentzian supersymmetric configurations in $D=4$ and $D=5$ gauged
$\mathcal{N}=2$ supergravity. We show that there are smooth $1/2$ BPS
solutions which are asymptotically AdS4 and AdS5 with a planar boundary, a
compact spacelike direction and with a Wilson line on that circle. There are
solitons where the $S^{1}$ shrinks smoothly to zero in the interior, with a
magnetic flux through the circle determined by the Wilson line, which are AdS
analogues of the Melvin fluxtube. There is also a solution with a constant
gauge field, which is pure AdS. Both solutions preserve half of the
supersymmetries at a special value of the Wilson line. There is a phase
transition between these two saddle-points as a function of the Wilson line
precisely at the supersymmetric point. Thus, the supersymmetric solutions are
degenerate, at least at the supergravity level. We extend this discussion to
one of the Romans solutions in four dimensions when the Euclidean boundary is
$S^{1}\times\Sigma_{g}$ where $\Sigma_{g}$ is a Riemann surface with genus
$g>0$. We speculate that the supersymmetric state of the CFT on the boundary
is dual to a superposition of the two degenerate geometries.

##  1 Introduction

There is a long-standing interest in smooth soliton solutions in gravity.
There are many examples where some $S^{1}$ direction shrinks smoothly to zero,
starting with the bubble of nothing instability of the Kaluza-Klein vacuum
[1]. A similar solution with asymptotically locally AdS boundary conditions
was dubbed the AdS soliton [2]. These solutions are easily constructed; they
are simply double analytic continuations of black hole solutions (in these
examples, uncharged black holes in asymptotically flat or asymptotically AdS
space). In the AdS case, the soliton is a time-independent solution, with a
flat conformal boundary. The spin structure of a spacetime with a contractible
$S^{1}$ has antiperiodic boundary conditions on the $S^{1}$. In the
holographic AdS/CFT correspondence, the dual theory thus has a spatial circle
with antiperiodic boundary conditions for fermions; if the CFT was
supersymmetric this choice of boundary conditions breaks the supersymmetry.
The AdS soliton is dual to the ground state in this theory. It has a negative
energy, which is identified with the Casimir energy of the dual CFT; the
antiperiodic boundary conditions for fermions spoil the cancellation between
the bosons and fermions.

A simple generalization of these boundary conditions in the CFT is to add a
Wilson line around the non-trivial cycle, giving a non-trivial holonomy for
the bulk gauge field. A family of bulk solutions satisfying these boundary
conditions can be obtained by a double analytic continuation of electrically
charged black holes. After the analytic continuation, the solutions have a
magnetic flux through the circle; in the bulk the Wilson line is related to
the total amount of magnetic flux through this cycle. These are AdS analogues
of the Melvin fluxtube solutions in flat space [3], and were previously
studied in [4, 5, 6]. For simplicity, we study the solutions primarily in four
bulk dimensions with a flat boundary, but we also consider generalisations in
Euclidean signature to a boundary which is $S^{1}\times\Sigma_{g}$ for some
Riemann surface of genus $g>1$, and to five bulk dimensions, which may be
interesting for its relation to $\mathcal{N}=4$ SYM.

Surprisingly, there is a solution of vanishing energy in this family which is
supersymmetric. This is surprising, as the solutions still have the same
structure as the AdS soliton, with a smooth origin which enforces antiperiodic
boundary conditions for the fermions on the shrinking $S^{1}$, which we
normally think of as breaking the supersymmetry. However, the holonomy around
the circle can compensate for the change in boundary conditions. The
supersymmetry of these solutions has been recognised in various guises before.
They are double analytic continuations of a set of supersymmetric naked
singularities identified in four dimensions in [7, 8] and in five dimensions
in [9]. They were also studied in four dimensional Euclidean space very
recently in [10] as an extension of the supersymmetric black holes dual to the
twisted supersymmetric index. We add to these previous discussions by
emphasizing the previously unnoticed fact that these solutions are perfectly
well behaved Lorentzian configurations. Indeed, these configurations can be
thought as a supersymmetric extension of the AdS soliton of [2]. We provide a
detailed discussion of the role of its spin structure; we will explicitly
construct the antiperiodic Killing spinors in the bulk spacetime.

In addition to these smooth soliton solutions, there are also solutions with
these boundary conditions where the $S^{1}$ does not have a smooth origin in
the bulk. In the simple case with flat boundary conditions, this solution is
simply AdS in Poincaré coordinates, with one of the spatial directions
identified. For $S^{1}\times\Sigma_{g}$ boundary conditions, the solution is
an analytic continuation of the magnetically charged black hole solution of
Romans, with an AdS${}_{2}\times\Sigma_{g}$ near-horizon region [7, 11, 12]. A
solution with a Wilson line is easily obtained by adding a constant gauge
potential in the bulk, without changing the geometry. These solutions are
supersymmetric for integer-quantized values of the Wilson line, with periodic
Killing spinors for even $n$ and antiperiodic Killing spinors for odd $n$.

Thus, for a range of values of the Wilson line on the $S^{1}$, there are two
bulk solutions with these boundary conditions: the magnetic flux tube solution
and the solution dressed with a constant gauge field.111As noted in [6], the
magnetic flux solution exists only for a finite range of values of the
holonomy. We will show that there is a phase transition between these two
families of solutions precisely at the supersymmetric point with antiperiodic
Killing spinors; the ground state of the CFT with these boundary conditions is
the soliton for smaller values of the Wilson line, and the constant gauge
field solution for larger values.

The path integral for the CFT on an $S^{1}\times\Sigma_{g}$ with the
supersymmetry-preserving choice of Wilson line computes the twisted
supersymmetric index [13, 14, 15].222See [16] for a review and more
references. In this context, both of these three-dimensional configurations
have been considered before: the solution with the constant gauge field was
studied in [12] as the bulk dual of this twisted partition function. It was
noted in [10] that there is a supersymmetric solution with an
$\mathbb{R}^{2}\times\Sigma_{g}$ near-horizon region; our magnetic flux tube.
However, it was not previously appreciated that these solutions compete.

This is as far as we know the first example where there is a degeneracy
between bulk solutions in a supersymmetric partition function, and it would be
fascinating to understand its implications for the field theory. It would be
interesting to study the one-loop determinant of the bulk fields on both
solutions to see if this lifts the degeneracy. We conjecture that it does not,
and the supersymmetric ground state in field theory is dual to a superposition
of the two geometries. If so, this would be the first example we know where a
field theory ground state has such an interpretation. It would then be very
interesting to consider the observables in this state, such as correlation
functions, whose calculation in the bulk will be sensitive to the relative
phase between the two solutions. It is also interesting that there is
qualitative difference between even and odd integer-quantized Wilson lines; in
the even case, with periodic Killing spinors, there is a unique supersymmetric
solution in the bulk, while in the odd case there is this degeneracy.

In the first part of the paper, we will discuss the case with flat boundary in
AdS4 in some detail. We set up the bulk theory in section 2.1, and describe
the solutions in section 2.2. We give a careful analysis of supersymmetry in
section 2.3, and describe the phase structure in section 2.4. We then briefly
discuss the extension to $S^{1}\times\Sigma_{g}$ boundaries in section 3. We
generalize the analysis to AdS5 for flat boundaries in section 4.

##  2 Four dimensions

###  2.1 Gauged $\mathcal{N}=2$ supergravity

Our explicit discussion of solutions in four dimensions is carried out in the
context of minimal gauged $\mathcal{N}=2$ supergravity in four dimensions.
This theory was originally constructed in terms of the physical fields [17,
18], whose supersymmetry transformations only close under commutation up to
equations of motion. Subsequently two alternative constructions were presented
based on the superconformal multiplet calculus [19, 20, 21]. This theory is a
universal subsector of a range of four-dimensional supersymmetric theories.
The bosonic sector is the familiar Einstein-Maxwell-AdS theory, with action

$S\left(g,A\right)=\int
d^{4}x\sqrt{-g}\left[\frac{R}{2}-\frac{1}{8}F_{\mu\nu}F^{\mu\nu}+\frac{3}{\ell^{2}}\right],$
(1)

where $F(A)_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,.$The field
equations are

$\displaystyle\partial_{\mu}\big{(}\sqrt{-g}F^{\mu\nu}\big{)}=0\,\,,$
$\displaystyle
R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R-\tfrac{1}{2}\big{[}F_{\mu\rho}\,F_{\nu}{}^{\rho}-\tfrac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\big{]}-\frac{3}{\ell^{2}}\,g_{\mu\nu}=0\,.$
(2)

When supplemented by the fermionic sector, the supergravity theory is
invariant under supersymmetry transformations of all the fields. In our
calculations we will only make use of the transformation of the chiral Rarita-
Schwinger fields $\psi_{\mu}{\\!}^{i}\,$:

$\displaystyle\delta\psi_{\mu}{\\!}^{i}=\;$ $\displaystyle
2\,\mathcal{D}_{\mu}\epsilon^{i}-\tfrac{1}{4}F(A)_{\rho\sigma}\gamma^{\rho\sigma}\gamma_{\mu}\,\varepsilon^{ij}\,\epsilon_{j}+\ell^{-1}\,\varepsilon^{ij}\,t_{j}{}^{k}\,\gamma_{\mu}\epsilon_{k}\,,$
(3) $\displaystyle\delta\psi_{\mu\,i}=\;$ $\displaystyle
2\,\mathcal{D}_{\mu}\epsilon_{i}-\tfrac{1}{4}F(A)_{\rho\sigma}\gamma^{\rho\sigma}\gamma_{\mu}\,\varepsilon_{ij}\,\epsilon^{j}+\ell^{-1}\,\varepsilon_{ij}\,t^{j}{}_{k}\,\gamma_{\mu}\epsilon^{k}\,,$
(4)

where $\gamma^{5}=-\mathrm{i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$,
$\gamma^{5}\psi_{\mu}^{i}=\psi_{\mu}^{i}$ and $\gamma^{5}\psi_{\mu
i}=-\psi_{\mu i}$ and $i=1,2$ and we shall pick $t^{i}{}_{j}=$
$\mathrm{i}\sigma_{3}\Longrightarrow t_{i}{}^{j}=$ $-\mathrm{i}\sigma_{3}$.
The covariant derivatives of the supersymmetry parameters are given by

$\displaystyle\mathcal{D}_{\mu}\epsilon^{i}=$
$\displaystyle\;\big{(}\partial_{\mu}+\tfrac{1}{4}\omega_{\mu}{\\!}^{ab}\gamma_{ab}\big{)}\epsilon^{i}-\tfrac{1}{2\ell}A_{\mu}\,t^{i}{\\!}_{j}\,\epsilon^{j}\,,$
(5) $\displaystyle\mathcal{D}_{\mu}\epsilon_{i}=$
$\displaystyle\;\big{(}\partial_{\mu}+\tfrac{1}{4}\omega_{\mu}{\\!}^{ab}\gamma_{ab}\big{)}\epsilon_{i}-\tfrac{1}{2\ell}A_{\mu}\,t_{i}{}^{j}\,\epsilon_{j}\,.$
(6)

In our calculations we find that is convenient to work with the spinor

$\chi\equiv\epsilon^{1}+\epsilon_{2}\,,$ (7)

such that the Killing spinor equation is

$2\,\mathcal{D}_{\mu}\chi+\tfrac{1}{4}F(A)_{\rho\sigma}\gamma^{\rho\sigma}\gamma_{\mu}\gamma^{5}\,\chi-\ell^{-1}\mathrm{i}\,\gamma_{\mu}\gamma^{5}\chi=0\,,$
(8)

where the covariant derivative of $\chi$ follows from (5),

$\mathcal{D}_{\mu}\chi=\big{(}\partial_{\mu}+\tfrac{1}{4}\omega_{\mu}{\\!}^{ab}\gamma_{ab}-\tfrac{1}{2\ell}\,\mathrm{i}\,A_{\mu}\big{)}\chi\,.$
(9)

We recall that all fermionic fields have been suppressed on the right-hand
side of equation (9), because we will be dealing with purely bosonic
backgrounds. In the following sections we will consider a class of soliton
solutions that can be partially supersymmetric. Their possible supersymmetry
will be investigated by analyzing the equation (8).

The Lagrangian (1) can be obtained from the compactification of eleven
dimensional supergravity over the seven sphere with the ansatz [22]

$\displaystyle ds_{11}^{2}$
$\displaystyle=ds_{4}^{2}+4\ell^{2}\sum_{i}\left(d\mu_{i}^{2}+\mu_{i}^{2}\left(d\phi_{i}+\frac{1}{4\ell}A\right)^{2}\right).$
(10) $\displaystyle F_{4}$
$\displaystyle=-\frac{3}{\ell}\epsilon_{4}-2\ell^{2}\sum_{i}\mu_{i}d\mu_{i}\wedge\left(d\phi_{i}+\frac{1}{4\ell}A\right)\wedge\ast_{4}dA$
(11)

where $\ast_{4}$ is the Hodge dual with respect to the four-dimensional metric
$ds_{4}^{2}$ and $\epsilon_{4}$ its volume form. The $\phi_{i}$ are $2\pi$
periodic angular coordinates parametrizing the four independent rotations on
$S^{7}$. We will be interested in considering the higher-dimensional
interpretation of some of our solutions using this uplift.

###  2.2 Planar solitons

The solutions we consider all have metric

$ds^{2}=\frac{r^{2}}{\ell^{2}}\left(-dt^{2}+dz^{2}\right)+\frac{dr^{2}}{f(r)}+f(r)d\phi^{2}\,.$
(12)

When considering supersymmetry we will also need a corresponding set of
vierbeine, which we take to be

$\displaystyle e^{0}$ $\displaystyle=\frac{r}{\ell}dt\,,$ $\displaystyle
e^{1}$ $\displaystyle=\frac{\,dr}{\sqrt{f(r)}}\,,$ $\displaystyle e^{2}$
$\displaystyle=\sqrt{f(r)}d\phi\,,$ $\displaystyle e^{3}$
$\displaystyle=\frac{r}{\ell}dz.$ (13)

The simplest solution has

$f(r)=\frac{r^{2}}{\ell^{2}}\,.$ (14)

If $\phi$ is not periodically identified, this is simply pure
$\mathrm{AdS}_{4}$ in Poincaré coordinates. However, we are interested in
considering solutions with $\phi$ periodically identified in the boundary. For
Poincaré-AdS, we can impose this identification on the bulk as a quotient. We
will postpone a full discussion of this quotient until we have discussed the
soliton solution.

The soliton solution is

$\displaystyle f(r)$
$\displaystyle=\frac{r^{2}}{\ell^{2}}-\frac{\mu}{r}-\frac{Q^{2}}{r^{2}}\,,$
(15) $\displaystyle A$
$\displaystyle=\left(\frac{2Q}{r}-\frac{2Q}{r_{0}}\right)d\phi\text{ ,}$ (16)

where $r_{0}$ is the largest root of the equation $f(r_{0})=0$.333Note that
$f$ always has at least one positive root, for all real values of $\mu,Q$.
This solution can be obtained by a double analytic continuation from an
electrically charged black hole solution, where the analytic continuation also
involves analytically continuing the charge $Q$. We have
$r\in\left[r_{0},\infty\right)$, and regularity of the metric at $r=r_{0}$
requires that $\phi$ is periodic with period

$\Delta\phi=\frac{4\pi\ell^{2}r_{0}^{3}}{3r_{0}^{4}+Q^{2}\ell^{2}}.$ (17)

We have added a constant contribution to the gauge potential to ensure
regularity at $r=r_{0}$; there is then a non-trivial holonomy of the gauge
field at infinity. This is related to the net magnetic flux along the $z$
axis,

$\Phi=-\oint A_{\phi}\left(r=\infty\right)d\phi=-\frac{1}{2}\int
F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}=\frac{2Q}{r_{0}}\Delta\phi.$ (18)

The soliton solution is determined by two parameters; from the bulk
perspective the natural parametrization is $\mu,Q$ or $r_{0},Q$. From the
boundary perspective, the natural parameters are $\Delta\phi,\Phi$. In the
usual holographic dictionary, we fix the boundary geometry and the boundary
value of $A_{\mu}$, which is interpreted as a background gauge field coupled
to a global $U(1)$ symmetry of the CFT. Recall that as discussed in the
introduction, for the soliton solutions the bulk spin structure is
antiperiodic on the $\phi$ circle, as this contracts smoothly in the interior.

Thus, we consider the CFT on a background

$ds_{Boundary}^{2}=-dt^{2}+dz^{2}+d\phi^{2},$ (19)

with $\phi$ taken to be periodic with period $\Delta\phi$, and an antiperiodic
spin structure for the fermions, and a Wilson line (holonomy) on the $\phi$
circle with value $\Phi$, and we look for bulk solutions satisfying these
boundary conditions. We obtain a suitable bulk soliton solution by solving for
$\mu,Q$ as functions of $\Delta\phi,\Phi$. We have

$\mu=\frac{r_{0}^{4}-Q^{2}\ell^{2}}{r_{0}\ell^{2}},\quad Q=\frac{\Phi
r_{0}}{2\Delta\phi},$ (20)

and inverting (17) gives [6]

$r_{0}=\frac{2\pi\ell^{2}}{3\Delta\phi}\left(1\pm\sqrt{1-\frac{\Phi^{2}}{\Phi_{max}^{2}}}\right),$
(21)

where $\Phi_{max}=\frac{4\pi}{\sqrt{3}}\ell$. We see that there are two
solutions for $r_{0}$ for $\Phi\in[0,\Phi_{max}]$. At $\Phi=0$, the minus
branch has $r_{0}=0$, giving $\mu=Q=0$, so it reduces to the Poincaré-AdS
solution. The plus branch has $Q=0$, and

$\mu=\left(\frac{4\pi\ell}{3\Delta\phi}\right)^{3}\ell,$ (22)

which is just the AdS soliton. The two branches coalesce at $\Phi=\Phi_{max}$.

The parameters $\mu,Q$ control the subleading parts of the metric and gauge
field asymptotically, so they can be interpreted as vevs of the corresponding
operators in the field theory. The gauge field gives a vev for the current in
the boundary theory,

$\langle J_{\phi}\rangle=2Q,$ (23)

and the metric gives a vev for the stress tensor,

$\left\langle T_{tt}\right\rangle=-\frac{\mu}{2\ell^{2}}\text{
},\qquad\left\langle T_{zz}\right\rangle=\frac{\mu}{2\ell^{2}}\text{
},\qquad\left\langle T_{\phi\phi}\right\rangle=-\frac{\mu}{\ell^{2}}\text{ }.$
(24)

We see that the energy density of the soliton solutions is negative when $\mu$
is positive. We can write explicitly [6]

$\left\langle
T_{tt}\right\rangle=-\left(\frac{4\pi\ell}{3\Delta\phi}\right)^{3}\frac{1}{4\ell}\left[1-\frac{3\Phi^{2}}{2\Phi_{max}^{2}}\pm\left(1-\frac{\Phi^{2}}{\Phi_{max}^{2}}\right)^{3/2}\right].$
(25)

We see that the minus branch always has $\left\langle T_{tt}\right\rangle\geq
0$, and the plus branch has $\left\langle T_{tt}\right\rangle<0$ for
$\Phi\in[0,\Phi_{S})$, where

$\Phi_{S}=\frac{\sqrt{3}}{2}\Phi_{max}=2\pi\ell.$ (26)

In addition to the soliton solutions, another solution with these boundary
conditions is simply to take the Poincaré-AdS solution, with a constant gauge
field $A=-\frac{2Q}{r_{0}}d\phi$. If $\phi$ were not periodic, this is just
pure AdS in Poincaré coordinates, and the gauge field is pure gauge. Taking
$\phi$ to be periodic with period $\Delta\phi$, the gauge field has a constant
Wilson loop $\Phi=-\frac{2Q}{r_{0}}\Delta\phi$ for all $r$, which can’t be set
to zero by a gauge transformation.

It is useful to think of this solution as a quotient of AdS${}_{4}\times
S^{7}$. For $\Phi=0$, the quotient has fixed points at $r=0$, so this is not a
smooth solution. Furthermore, for the antiperiodic boundary conditions we
consider, this solution is unstable as a solution of string theory, as a
string wrapped around the $\phi$ circle will become tachyonic for sufficiently
small $r$. This solution decays by tachyon condensation, whose likely endpoint
is the AdS soliton [23, 24].

However, for $\Phi\neq 0$, the identification involves a shift of the angular
coordinates on the sphere. The solution with a constant gauge field uplifts as
in (10). Since the gauge potential is constant, if $\phi$ is not periodically
identified we can eliminate the gauge field by coordinate redefinitions
$\tilde{\phi}_{i}=\phi_{i}+\frac{A_{\phi}}{4\ell}\phi$. The periodic
identification of $\phi$ at fixed $\phi_{i}$ then acts as
$(\phi,\tilde{\phi}_{i})\sim(\phi+\Delta\phi,\tilde{\phi}_{i}+\Phi/4\ell)$.
This is now a smooth quotient, with no fixed points,444More carefully, the
identification is free of fixed points so long as $\Phi\neq 8\pi\ell n$ for
integer $n$, so that the action on the $\phi_{i}$ coordinates is globally non-
trivial. and the circle in the higher-dimensional space has a minimum size
$\Phi/8$, so for $\Phi$ larger than the string scale the tachyon is lifted,
and this is a physical solution. It is then interesting to compare this
solution to the soliton. We will discuss the phase structure in section 2.4,
but we first discuss in more detail the supersymmetric solutions at the
special value $\Phi=\Phi_{S}$.

###  2.3 Supersymmetric solutions

The configuration with zero energy, $\mu=0$, which occurs on the plus branch
of solitons at $\Phi=\Phi_{S}=2\pi\ell$, is supersymmetric. At this value of
$\Phi$, the Poincaré-AdS solution is also supersymmetric. The supersymmetry of
the soliton in the Euclidean section has been noted before [10], but we give a
slightly more careful analysis taking into account the antiperiodic boundary
conditions for the fermions on the $\phi$ circle. Our remark is that the
supersymmetric soliton solutions are perfectly well behaved in the Lorentzian
section. We will explicitly exhibit the Killing spinors for this solution, and
see how they arise from the higher-dimensional perspective.

For the solitons with $\mu=0$, we can write $r_{0}\vskip 12.0pt plus 4.0pt
minus 4.0pt=\sqrt{\ell\left|Q\right|}$. The form of the solution can be
simplified by a coordinate transformation

$r=r_{0}\vskip 12.0pt plus 4.0pt minus 4.0pt\sqrt{\cosh(\rho)},$ (27)

where the configuration reads

$ds^{2}=\frac{\ell^{2}}{4}d\rho^{2}+\frac{|Q|}{\ell}\left[\cosh\left(\rho\right)\left(-dt^{2}+dz^{2}\right)+\frac{\sinh\left(\rho\right)^{2}}{\cosh(\rho)}d\phi^{2}\right],$
(28) $A=\frac{2Q}{r_{0}}\left(\frac{1}{\sqrt{\cosh\rho}}-1\right)d\phi.$ (29)

We use a Majorana basis for the Clifford algebra

$\gamma^{0}=-\mathrm{i}\left(\begin{array}[c]{cc}0&\sigma_{2}\\\
\sigma_{2}&0\end{array}\right),\quad\gamma^{1}=-\left(\begin{array}[c]{cc}\sigma_{3}&0\\\
0&\sigma_{3}\end{array}\right),\quad\gamma^{2}=\mathrm{i}\left(\begin{array}[c]{cc}0&-\sigma_{2}\\\
\sigma_{2}&0\end{array}\right),\quad\gamma^{3}=\left(\begin{array}[c]{cc}\sigma_{1}&0\\\
0&\sigma_{1}\end{array}\right).$ (30)

There are two independent solutions to the Killing spinor equation on this
background:

$\chi_{1}=\frac{\exp\left(-\mathrm{i}\pi\frac{\phi}{\Delta\phi}\right)}{\cosh\left(\rho\right)^{1/4}}\left(\begin{array}[c]{c}\sinh\frac{\rho}{2}\\\
-\cosh\frac{\rho}{2}\\\ \mathrm{i}\cosh\frac{\rho}{2}\\\
\mathrm{i}\sinh\frac{\rho}{2}\end{array}\right),$ (31)
$\chi_{2}=\frac{\exp\left(-\mathrm{i}\pi\frac{\phi}{\Delta\phi}\right)}{\cosh\left(\rho\right)^{1/4}}\left(\begin{array}[c]{c}-\cosh\frac{\rho}{2}\\\
\sinh\frac{\rho}{2}\\\ \mathrm{i}\sinh\frac{\rho}{2}\\\
\mathrm{i}\cosh\frac{\rho}{2}\end{array}\right).$ (32)

The two spinors $\chi_{1}$ and $\chi_{2}$, can be projected over four chiral
spinors and therefore the soliton solution is 1/2 BPS. They close on the
following Killing vectors:

$\chi_{2}^{\dagger}\gamma^{0}\gamma^{\mu}\chi_{2}\partial_{\mu}=\chi_{1}^{\dagger}\gamma^{0}\gamma^{\mu}\chi_{1}\partial_{\mu}=\frac{2\ell}{r_{0}}\partial_{t},$
(33)
$\chi_{1}^{\dagger}\gamma^{0}\gamma^{\mu}\chi_{2}\partial_{\mu}=-\frac{2\ell}{r_{0}}\left(\partial_{\phi}+\mathrm{i}\partial_{z}\right)\text{
}.$

There is still the issue that the spinors $\chi_{1}$ and $\chi_{2}$ seem to be
ill-defined at $\rho=0$, as they have an explicit dependence on $\phi$,
however this dependence corresponds exactly to the form of the Killing spinors
of Minkowski in cylindrical coordinates:

$\left.\chi_{1}\right|_{\rho=0}=\exp\left(-\mathrm{i}\pi\frac{\phi}{\Delta\phi}\right)\left(\begin{array}[c]{c}0\\\
-1\\\ \mathrm{i}\\\ 0\end{array}\right),$ (34)
$\chi_{2}=\exp\left(-\mathrm{i}\pi\frac{\phi}{\Delta\phi}\right)\left(\begin{array}[c]{c}-1\\\
0\\\ 0\\\ \mathrm{i}\end{array}\right).$ (35)

Hence, our Killing spinors interpolate smoothly between half of the Killing
spinors of Minkowski and half of the Killing spinors of AdS4. Indeed, we see
that we have solutions of the Killing spinor equation with non-trivial
dependence on $\phi$, whereas we would normally expect the Killing spinors to
be constant along these flat directions. This satisfies the Killing spinor
equation (8) due to a cancellation between the derivative $\partial_{\phi}$
and the gauge potential $A_{\phi}$ in the covariant derivative
$\mathcal{D}_{\phi}$.

For $\Phi=\Phi_{S}=2\pi\ell$, the Poincaré -AdS solution is also
supersymmetric. This follows trivially from the fact that it gives the large
$r$ asymptotics of the supersymmetric soliton solution; if the soliton
preserves some supersymmetry, the metric it approaches at large $r$ must
preserve at least the same amount of supersymmetry. Taking the large $\rho$
limit gives

$\chi_{1}=\exp\left(-\mathrm{i}\pi\frac{\phi}{\Delta\phi}\right)e^{\frac{\rho}{4}}\left(\begin{array}[c]{c}1\\\
-1\\\ \mathrm{i}\\\ \mathrm{i}\end{array}\right),$ (36)
$\chi_{2}=\exp\left(-\mathrm{i}\pi\frac{\phi}{\Delta\phi}\right)e^{\frac{\rho}{4}}\left(\begin{array}[c]{c}-1\\\
1\\\ \mathrm{i}\\\ \mathrm{i}\end{array}\right).$ (37)

On pure AdS4, the theory has four Killing spinors; if we introduce a Wilson
loop on this background with $A=A_{\phi}d\phi$, we have that the local
solutions to the Killing spinor equation are

$\chi_{1}^{AdS}=\exp\left(\frac{\mathrm{i}A_{\phi}}{2\ell}\phi\right)r^{1/2}\left(\begin{array}[c]{c}1\\\
-1\\\ 0\\\
0\end{array}\right)\text{,}\qquad\chi_{2}^{AdS}=\exp\left(\frac{\mathrm{i}A_{\phi}}{2\ell}\phi\right)r^{1/2}\left(\begin{array}[c]{c}0\\\
0\\\ 1\\\ 1\end{array}\right)\text{ ,}$ (38)
$\chi_{3}^{AdS}=\exp\left(\frac{\mathrm{i}A_{\phi}}{2\ell}\phi\right)\left(\begin{array}[c]{c}r^{1/2}\left(t+\phi\right)\\\
-r^{1/2}\left(t+\phi\right)\\\ -r^{1/2}z-\ell^{2}r^{-1/2}\\\
-r^{1/2}z+\ell^{2}r^{-1/2}\end{array}\right)\text{,}\qquad\chi_{4}^{AdS}=\exp\left(\frac{\mathrm{i}A_{\phi}}{2\ell}\phi\right)\left(\begin{array}[c]{c}-r^{1/2}z+\ell^{2}r^{-1/2}\\\
r^{1/2}z+\ell^{2}r^{-1/2}\\\ r^{1/2}\left(t-\phi\right)\\\
r^{1/2}\left(t-\phi\right)\end{array}\right)\,.$ (39)

The second two spinors, $\chi_{3}^{AdS}$ and $\chi_{4}^{AdS}$, are not
invariant under the identification which makes $\phi$ periodic, so this breaks
at least half the supersymmetry. The first two, $\chi_{1}^{AdS}$ and
$\chi_{2}^{AdS}$, are invariant (up to sign) if $A_{\phi}=2\pi
n\ell/\Delta\phi$, so $\Phi=2\pi n\ell$, for integer $n$. Thus, the identified
Poincaré-AdS solution with $\phi$ periodic and $\Phi=2\pi\ell$ has the same
supersymmetry as the soliton solution. The preserved supersymmetries
correspond to the Poincaré supersymmetries of the boundary theory, while the
broken ones correspond to superconformal symmetries of the boundary theory,
which are broken as the periodic identification introduces a choice of scale,
breaking the conformal symmetry.

It is also instructive to understand how the dependence on $\phi$ arises from
a higher-dimensional perspective. Thought of as solutions on AdS${}_{4}\times
S^{7}$, the Killing spinors corresponding to Poincaré supersymmetries have the
structure

$\chi\sim
F(r)G(\mu_{i})e^{-\frac{\mathrm{i}}{2}(\tilde{\phi}_{1}+\tilde{\phi}_{2}+\tilde{\phi}_{3}+\tilde{\phi}_{4})}\chi_{0},$
(40)

where $\chi_{0}$ is a constant spinor. The dependence on the
$\tilde{\phi}_{i}$ is fixed by the fact that these are contractible cycles on
$S^{7}$, so all fermions are antiperiodic around them. Thus, under the
quotient action
$(\phi,\tilde{\phi}_{i})\sim(\phi+\Delta\phi,\tilde{\phi}_{i}+\Phi/4\ell)$
with $\Phi=\Phi_{S}=2\pi\ell$, these spinors are antiperiodic,
$\chi\rightarrow-\chi$, and they survive the identification, as we are keeping
precisely the sector of spinors which are antiperiodic under the quotient
action.555Note we have considered a quotient action which acts in the same way
on all the $\phi_{i}$ because we are focusing for simplicity on the minimal
gauged supergravity in four dimensions, but this supersymmetry argument
requires only that the sum of the shifts
$\sum_{i}\Delta\tilde{\phi}_{i}=2\pi$. Another way to say this is to rewrite
$\tilde{\phi}_{i}=\phi_{i}+\frac{\pi}{2}\frac{\phi}{\Delta\phi}$, which gives

$\chi\sim
F(r)G(\mu_{i})e^{-\mathrm{i}\pi\frac{\phi}{\Delta\phi}}e^{-\frac{\mathrm{i}}{2}(\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4})}\chi_{0}.$
(41)

Dimensionally reducing over the $\phi_{i}$ then gives the Killing spinors with
a Wilson line written above.

Thus, for $\Phi=\Phi_{S}$, we have two supersymmetric solutions; the soliton
with $\mu=0$ and the quotient of AdS. We next turn to the comparison between
these two solutions.

###  2.4 Phase diagram

We now consider the phase transitions between the different solutions. We
consider the standard AdS boundary conditions fixing the leading asymptotic
falloff of the metric and gauge field $A$. The boundary conditions are then
parametrized by $\Delta\phi$ and $\Phi$. We note that the $U(1)$ gauge field,
and hence $\Phi$, take values in a circle. This is most evident from the
perspective of the uplift (10); a shift of $\Phi\to\Phi+8\pi\ell n$ for
integer $n$ shifts the $\phi_{i}$ by $2\pi n$. The range of physically
inequivalent values is thus $\Phi\in[-4\pi\ell,4\pi\ell)$. We can restrict
attention to positive $\Phi$, as we have done so far; corresponding results
for negative $\Phi$ are obtained by reversing the sign of $Q$ for the soliton.
For $\Phi\in[0,\Phi_{max}]$, there are three different solutions for each
choice of $\Delta\phi$ and $\Phi$; the two branches of solitons and Poincaré-
AdS with the appropriate identification on $\phi$. For
$\Phi\in(\Phi_{max},4\pi\ell]$, we just have Poincaré-AdS.

In Lorentzian signature, these solutions should all be dual to some states in
the dual CFT. For $\Phi=0$, the solution on the lower soliton branch is the
AdS soliton, and this is conjectured to be dual to the ground state of the CFT
[2]. It is reasonable to assume that the lowest-energy solution remains dual
to the ground state as we turn on $\Phi$. The energy for the soliton solutions
is plotted in figure 1. For $\Phi\in[0,\Phi_{S})$, the lowest energy solution
is the lower soliton branch. At $\Phi=\Phi_{S}$, the two supersymmetric
solutions discussed in the previous section both have zero energy, so they are
degenerate. For $\Phi\in(\Phi_{S},4\pi\ell]$, Poincaré-AdS is the lowest-
energy solution.

Figure 1: The rescaled energy density,
$Y=\frac{27\Delta\phi^{3}}{8\pi^{3}\ell^{2}}\langle T_{tt}\rangle$, vs the
rescaled magnetic flux, $X=\Phi/\Phi_{max}$, for the two branches of soliton
solutions. The lower branch crosses the axis at
$X_{S}=\Phi_{S}/\Phi_{max}=\sqrt{3}/2$.

As for the AdS soliton, the energy $\langle
T_{tt}\rangle=-\frac{\mu}{2\ell^{2}}$ should be interpreted as a Casimir
energy of the ground state. We see that turning on a Wilson line initially
increases (reduces the magnitude of) the Casimir energy, presumably by
deforming the spectrum of fields charged under this $U(1)_{R}$ symmetry. It is
surprising that for $\Phi>\Phi_{S}$ the energy is zero, independent of the
Wilson line $\Phi$. The cancellation between bosons and fermions is restored
at the supersymmetric point, but it’s unexpected that this continues for
higher values of the Wilson line. It would be interesting to investigate this
directly from the field theory perspective.

The other mystery is the precise nature of the ground state at
$\Phi=\Phi_{S}$. In our supergravity analysis, we have two degenerate
solutions. This is the first example we know of with distinct supersymmetric
solutions with the same boundary conditions, which are degenerate in energy.
It is possible that this degeneracy is lifted by subleading effects, for
example by considering the Casimir energy for bulk fields on these two
geometries. This is certainly an interesting issue to investigate, but it
seems unlikely that this will lift the degeneracy: the bulk solutions are
supersymmetric, so the natural guess is that contributions to the bulk Casimir
from bosonic and fermionic fields will cancel. Also, there is a phase
transition between the two solutions which is at least very near the
supersymmetric point; it would be surprising if it were then not precisely at
this point. The alternative possibility is that the supersymmetric ground
state in the field theory is dual to a superposition of these two geometries.
This would be very interesting if true: the calculation of observables, such
as correlation functions, in this ground state, would then involve
contributions from both geometries. If these observables could be calculated
on the CFT side, they would give a unique probe of relative phase information
in the superposition of bulk geometries.

It is also interesting to note that the Poincaré-AdS solution is
supersymmetric for $\Phi=2\pi n\ell$. Since $\Phi$ is periodic with period
$8\pi\ell$, there are four physically inequivalent values giving
supersymmetric solutions, which we can take to be $n=-1,0,1,2$. For odd $n$,
the unbroken Killing spinors in Poincaré-AdS are antiperiodic on the $\phi$
circle, while for even $n$, they are periodic. For the odd cases, there is a
degenerate soliton solution, but for the even cases Poincaré-AdS is the only
supersymmetric solution. It would be interesting to understand this difference
between odd and even $n$ from the field theory perspective.

###  2.5 Euclidean partition function

It is also interesting to consider the phase structure from a Euclidean
perspective, to make contact with previous work on supersymmetric partition
functions [13, 14, 16]. Previous work on this area has focused on the
Euclidean theory with an $S^{1}\times\Sigma_{g}$ boundary, where $\Sigma_{g}$
is a Riemann surface of genus $g>1$, but this was extended to consider genus 1
in [10], corresponding to the case we have considered so far. We will comment
on the extension of our discussion to higher genus in section 3.

The Euclidean continuation of our solutions are solutions with an
$S^{1}\times\mathbb{R}^{2}$ boundary. These are usually treated thinking of
the $S^{1}$ as the Euclidean time circle, so $\Phi$ is a chemical potential
for the R-charge of the theory and the boundary path integral is interpreted
as the partition function $Z=\mathrm{Tr}\,e^{i\Phi J_{R}}e^{-\Delta\phi\,H}$
of the CFT on $\mathbb{R}^{2}$. For $\Phi=2\pi\ell$, this partition function
becomes a supersymmetric index.

In the saddle-point approximation, $Z\approx e^{-S_{E}}$, where $S_{E}$ is the
action of the Euclidean solutions. For the solitons, the Wick rotated metric
is

$ds^{2}=\frac{r^{2}}{\ell^{2}}\left(d\tau^{2}+dz^{2}\right)+\frac{dr^{2}}{f(r)}+f(r)d\phi^{2}\,,$
(42)

with $f(r)=\frac{r^{2}}{\ell^{2}}-\frac{Q^{2}}{r^{2}}-\frac{\mu}{r}$ as
before. The Euclidean action is

$\displaystyle\frac{S_{E}}{V}$
$\displaystyle=-\lim_{R\rightarrow\infty}\left[\left(\int_{r_{0}}^{R}dr\sqrt{g_{E}}\frac{R}{2}-\frac{1}{8}F_{\mu\nu}F^{\mu\nu}+\frac{3}{\ell^{2}}\right)+\left(K\sqrt{h}-\frac{2}{\ell}\sqrt{h}\right)_{r=R}\right]$
$\displaystyle=-\frac{\mu}{2\ell^{2}},$ (43)

where $V$ is the coordinate volume element on the boundary, $V=\int d\tau
d\phi dz$. The Poincaré-AdS solution with a constant Wilson line always has
zero action. Since the Euclidean action is proportional to $\mu$, the phase
structure is the same as in our Lorentzian discussion above; for
$\Phi\in[0,\Phi_{S})$, the dominant saddle-point is the lower soliton branch,
for $\Phi\in(\Phi_{S},4\pi\ell]$ it is Poincaré-AdS. There is a phase
transition at $\Phi=\Phi_{S}$.

For the supersymmetric case $\Phi=\Phi_{S}$, the partition function is an
index, which can be computed by supersymmetric localization. For the present
case, the CFT calculation gives an answer which vanishes to leading order in
$N$ [12], in agreement with the bulk calculation. This calculation has
previously been matched to the bulk result from Poincaré-AdS; we observe that
both bulk saddles match the CFT result, and as discussed previously it will be
interesting to understand further what the precise bulk dual of the index is.

An interesting issue here is the calculation of the expectation value of the
$R$ charge in the ensemble. In the CFT, we can calculate the partition
function exactly only at $\Phi=\Phi_{S}$, so we can’t extract a value for the
R-charge from this; thermodynamically, to calculate the R-charge we need to
consider the variation $\frac{\partial Z}{\partial\Phi}$. Holographically, we
can evaluate the partition function for general $\Phi$, but at the
supersymmetric point the degeneracy prevents us from evaluating the R-charge.
The soliton saddle has $\langle J_{\phi}\rangle=2Q$, corresponding to a non-
zero R-charge density, while Poincaré-AdS has $\langle J_{\phi}\rangle=0$.
Since they are degenerate at $\Phi=\Phi_{S}$, we don’t know what value to use.
Thermodynamically, the phase transition implies $Z$ is not a smooth function
of $\Phi$, so $\frac{\partial Z}{\partial\Phi}$ is ill-defined at
$\Phi=\Phi_{S}$.

###  2.6 Fixed charge

Since we are in four bulk dimensions, we can also define a holographic
correspondence with alternative boundary conditions, where we fix
$F^{\nu}\equiv\sqrt{h}n_{\mu}F^{\mu\nu}$ at the boundary (where $n_{\mu}$ is
the unit normal to the boundary), instead of fixing $A_{\mu}$ [25]. This
corresponds to fixing $\Delta\phi$ and $Q$. With these boundary conditions,
Poincaré-AdS is not a solution. For the soliton solutions, we can solve (17)
to determine $r_{0}$ as a function of $\Delta\phi$ and $Q$. We can see that
$\Delta\phi$ is small at both $r_{0}\to 0$ and $r_{0}\to\infty$, with a
maximum at $r_{0}^{4}=Q^{2}\ell^{2}$, which implies $\mu=0$, where
$\Delta\phi=\Delta\phi_{max}=\pi\sqrt{\frac{\ell^{3}}{Q}}$. Thus, the
supersymmetric solution in this case corresponds to the maximum value of
$\Delta\phi$.

In the Euclidean action, we need to add a boundary term associated with the
change in boundary conditions [25]

$\frac{S_{\partial}}{V}=-\lim_{R\rightarrow\infty}\left(\frac{1}{2}\sqrt{h}N_{\mu}A_{\nu}F^{\mu\nu}\right)_{r=R}=-\frac{2Q^{2}}{\ell^{2}r_{0}},$
(44)

so

$S^{\prime}=S+S_{\partial}=-\left(\frac{\mu}{2\ell^{2}}+\frac{2Q^{2}}{\ell^{2}r_{0}}\right)V.$
(45)

In figure 2, we plot the action for these solutions as a function of
$\Delta\phi$ at fixed $Q$. We see that there are again two branches of
solutions, coalescing at the supersymmetric solution at
$\Delta\phi=\Delta\phi_{max}$. The lower branch corresponds to the solutions
with $\mu<0$, which are always the dominant saddle for these boundary
conditions.

Figure 2: The action $\epsilon=\frac{\ell}{2\pi\left|Q\right|}S^{\prime}/V$ vs
$\Delta=\frac{\sqrt{\left|Q\right|\ell}}{4\pi\ell^{2}}\Delta\phi$, at fixed
$Q$. There are two solitons for each boundary condition for
$\Delta\phi<\Delta\phi_{max}$. The supersymmetric solution is at the maximum
value $\Delta\phi=\Delta\phi_{max}$.

##  3 Generalization to higher genus

It is straightforward to extend this discussion to the Euclidean partition
function of the field theory on $S^{1}\times\Sigma_{g}$, where $\Sigma_{g}$ is
a Riemann surface of genus $g>1$, studied in [13, 14, 12, 10]. The non-linear
form of Euclidean gauged $\mathcal{N}=2$ supergravity was constructed in [26].
The relevant solutions are obtained from the analytic continuation of a black
hole with horizon $\Sigma_{g}$, carrying both electric and magnetic charges,

$ds^{2}=r^{2}ds^{2}_{\Sigma_{g}}+\frac{dr^{2}}{f(r)}+f(r)d\phi^{2},$ (46)

with

$f(r)=\frac{r^{2}}{\ell^{2}}-1-\frac{\mu}{r}+\frac{P^{2}-Q^{2}}{r^{2}},$ (47)

where $ds^{2}_{\Sigma_{g}}$ is the unit constant negative curvature metric on
$\Sigma_{g}$, and we have a gauge field

$F=2P\epsilon_{\Sigma_{g}}+\frac{2Q}{r^{2}}d\phi\wedge dr,$ (48)

where $\epsilon_{\Sigma_{g}}$ is the volume form of $ds^{2}_{\Sigma_{g}}$.
Note that in the analytic continuation, we have analytically continued the
electric charge $Q$ but not the magnetic charge $P$, hence the difference in
sign in the metric function. We want to restrict the parameters so that $f(r)$
has at least one real positive root; we take as before $r_{0}$ to be the
largest positive root, $f(r_{0})=0$.

The solution considered in [12] has $\mu=Q=0$, and $P=\pm\ell/2$. This is a
supersymmetric black hole, first studied in [7]; the contributions of the
field strength and the spin connection on $\Sigma_{g}$ in (8) cancel, to allow
us to find two chiral Killing spinors, namely the solution is $1/4$ BPS. For
these parameters, $f(r)$ has a double root at $r_{0}=\ell/\sqrt{2}$,

$f(r)=\left(\frac{r}{\ell}-\frac{\ell}{2r}\right)^{2}.$ (49)

The geometry near $r=r_{0}$ is thus $H^{2}\times\Sigma_{g}$; in Lorentzian
signature, this is an extremal black hole, with a near-horizon $AdS_{2}$
region.

The solution is regular for $\phi\in\mathbb{R}$, but we can obtain solutions
with periodic $\phi$ by quotienting this solution, as in our previous
discussion of Poincaré-AdS. As before, we can consider adding a Wilson line
along the $\phi$ circle in the quotient; this corresponds to a chemical
potential for the R-charge in the partition function. This quotient will have
unbroken supersymmetry if $\Phi=2\pi n\ell$ for integer $n$. Recall that
$\Phi$ is periodic with period $8\pi\ell$, so there are four physically
inequivalent values giving supersymmetric solutions, which we can take to be
$n=-1,0,1,2$. For odd $n$, the unbroken Killing spinors are antiperiodic on
the $\phi$ circle, while for even $n$, they are periodic.

In [10], the solutions with $\mu=0$ but non-zero $Q$ were considered;

$f(r)=\left(\frac{r}{\ell}-\frac{\ell}{2r}\right)^{2}-\frac{Q^{2}}{r^{2}}$
(50)

with

$A=A_{\Sigma_{g}}+\left(\frac{2Q}{r}-\frac{2Q}{r_{0}}\right)d\phi,$ (51)

where $A_{\Sigma_{g}}$ is a gauge potential on the Riemann surface giving the
magnetic part of $F$. The largest root of $f$ is at

$r_{0}=\frac{\ell}{\sqrt{2}}\sqrt{1+|Q|/\sqrt{2}\ell}.$ (52)

This is now a single root, so the geometry near $r=r_{0}$ is
$\mathbb{R}^{2}\times\Sigma_{g}$, and we need to choose $\phi$ periodic with
period

$\Delta\phi=\frac{4\pi}{f^{\prime}(r_{0})}=\frac{\pi\ell r_{0}}{|Q|}.$ (53)

There is a boundary Wilson line $\Phi=\frac{2Q}{r_{0}}\Delta\phi=\pm
2\pi\ell$, and the solution is supersymmetric, with antiperiodic Killing
spinors.

Thus, as in the earlier discussion, there are two supersymmetric solutions
with antiperiodic Killing spinors for $\Phi=\pm 2\pi\ell$; the quotiented
solution with $Q=0$ and the soliton-like solution with non-zero $Q$. The
action of these solutions is equal [10],

$S_{E}=-\frac{\sqrt{2}\pi}{3}(g-1)N^{3/2}.$ (54)

This reproduces the value of the twisted topological index calculated in the
field theory using localization methods [14]. It is in principle
straightforward to extend this analysis to more general solutions with
scalars, although explicitly constructing the soliton solutions is challenging
[10].

As in the genus one case, it would be very interesting to understand further
the degeneracy between these two solutions. It would also be interesting to
understand the difference between odd $n$ (where we have this degeneracy) and
even $n$ (where the $Q=0$ solution is the unique supersymmetric solution) from
the field theory perspective.

If we want to consider different values of $\Phi$, we can detune this solution
by considering the geometry with $\mu$ also non-zero,

$f(r)=\left(\frac{r}{\ell}-\frac{\ell}{2r}\right)^{2}-\frac{\mu}{r}-\frac{Q^{2}}{r^{2}}.$
(55)

We leave a complete understanding of the space of possible smooth non-
supersymmetric solutions for future work; it is not trivial in this case to
solve for $\mu,Q$ in terms of the natural boundary parameters
$\Delta\phi,\Phi$, or to establish the range of parameters for which there is
a positive root $r_{0}$.

##  4 Five dimensions

###  4.1 Minimal gauged supergravity

The discussion so far has been in four bulk dimensions; it is easy to
generalise the construction to higher dimensions. Here we will consider in
particular Einstein-Maxwell-AdS in five bulk dimensions constructed in several
papers [27, 28, 29, 30, 31, 32, 33, 34]. We follow [9, 22]. The action
principle is

$S\left(g,A\right)=\frac{1}{8\pi G}\int
d^{5}x\left(\sqrt{-g}\left[\frac{R}{2}+\frac{6}{\ell^{2}}-\frac{1}{8}F_{\mu\nu}F^{\mu\nu}\right]+\frac{1}{24\sqrt{3}}F\wedge
F\wedge A\right)\text{ },$ (56)

with field equations

$\displaystyle\partial_{\mu}\big{(}\sqrt{-g}F^{\mu\nu}\big{)}=0\,\,,$
$\displaystyle
R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R-\tfrac{1}{2}\big{[}F_{\mu\rho}\,F_{\nu}{}^{\rho}-\tfrac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\big{]}-\frac{6}{\ell^{2}}\,g_{\mu\nu}=0\,.$
(57)

where we have used that our solutions shall satisfy $F\wedge F=0$. In our
conventions, the Killing spinor equations can be written in terms of a single
complex spinor, $\Psi$, as follows

$\big{(}\partial_{\mu}+\tfrac{1}{4}\omega_{\mu}{\\!}^{ab}\gamma_{ab}-\frac{\mathrm{i}}{8\sqrt{3}}\gamma_{\mu\lambda\alpha}F^{\lambda\alpha}+\frac{\mathrm{i}}{2\sqrt{3}}F_{\mu\alpha}\gamma^{\alpha}\big{)}\Psi+\ell^{-1}\left(\frac{\mathrm{i}}{2}\gamma_{\mu}-\frac{\sqrt{3}}{2}A_{\mu}\,\right)\gamma^{4}\Psi^{\ast}=0\text{
.}$ (58)

where $\gamma^{4}=\mathrm{i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ is the
gamma matrix along the fifth dimension and $\Psi^{\ast}$ is the complex
conjugate of $\Psi$.

The theory we considered here is obtained by a dimensional reduction of type
IIB supergravity over the five-sphere with the ansatz [22]

$\displaystyle ds_{10}^{2}$
$\displaystyle=ds_{5}^{2}+\ell^{2}\sum_{i=1}^{3}d\mu_{i}^{2}+\mu_{i}^{2}(d\phi_{i}+\frac{1}{\sqrt{3}\ell}A)^{2}$
(59) $\displaystyle F_{5}$ $\displaystyle=G_{5}+\ast_{5}G_{5}$ (60)
$\displaystyle G_{5}$
$\displaystyle=-\frac{4}{\ell}\epsilon_{5}+\frac{\ell^{2}}{\sqrt{3}}\sum_{i=1}^{3}\mu_{i}d\mu_{i}\wedge(d\phi_{i}+\frac{1}{\sqrt{3}\ell}A)\wedge\ast_{5}dA$
(61)

where $\ast_{5}$ is the Hodge dual with respect to the five-dimensional metric
$ds_{5}^{2}$ and $\epsilon_{5}$ is its volume form.

###  4.2 Soliton solutions

By double analytic continuation of the electrically charged black hole
solutions with a flat boundary, we obtain a solution

$ds^{2}=\frac{r^{2}}{\ell^{2}}(-dt^{2}+dy^{2}+dz^{2})+\frac{dr^{2}}{f(r)}+f(r)d\phi^{2},$
(62)

where

$f(r)=\frac{r^{2}}{\ell^{2}}-\frac{\mu}{r^{2}}-\frac{q^{2}}{r^{4}}.$ (63)

This is a solution of Einstein-Maxwell with the gauge field

$A=\left(\frac{\sqrt{3}q}{r^{2}}-\frac{\sqrt{3}q}{r_{0}^{2}}\right)d\phi,$
(64)

where $r_{0}$ is the largest root of $f(r)$, $f(r_{0})=0$. We can write

$\mu=\frac{(r_{0}^{6}-q^{2}\ell^{2})}{\ell^{2}r_{0}^{2}}.$ (65)

The solution is smooth at $r=r_{0}$ if $\phi$ has period

$\Delta\phi=\frac{4\pi}{f^{\prime}(r_{0})}.$ (66)

The flux through the $\phi$ circle at $r\rightarrow\infty$ is $\Phi=\oint
A=\frac{\sqrt{3}q}{r_{0}^{2}}\Delta\phi$. The boundary data is $\Delta\phi$
and $\Phi$, so it is convenient to re-express the bulk parameters in terms of
these; we have

$q=\frac{r_{0}^{2}}{\sqrt{3}}\frac{\Phi}{\Delta\phi},$ (67)

with

$r_{0}=\frac{\pi\ell^{2}}{2\Delta\phi}\left(1\pm\sqrt{1-\frac{\Phi^{2}}{\Phi_{max}^{2}}}\right),$
(68)

where $\Phi_{max}=\sqrt{\frac{3}{2}}\pi\ell$. We see that for
$\Phi<\Phi_{max}$, there are two branches of solutions. The $+$ branch
approaches the AdS soliton as $\Phi\rightarrow 0$, while the $-$ branch
approaches Poincaré - AdS, and they coalesce at $\Phi=\Phi_{max}$. We have
$\mu=0$ when $r_{0}^{6}=q^{2}\ell^{2}$, that is when

$r_{0}^{2}=\frac{\Phi^{2}\ell^{2}}{3\Delta\phi^{2}}\quad\Rightarrow\quad
2-3\frac{\Phi^{2}}{\Phi_{max}^{2}}\pm
2\sqrt{1-\frac{\Phi^{2}}{\Phi_{max}^{2}}}=0,$ (69)

which is satisfied on the $+$ branch at
$\Phi=\Phi_{S}=\frac{2\sqrt{2}}{3}\Phi_{max}$, that is
$\Phi_{S}=\frac{2\pi}{\sqrt{3}}\ell$.

###  4.3 Supersymmetric solutions

To solve the Killing spinor equation at the supersymmetric point $\mu=0$, we
introduce the change of coordinates

$r=\left|q\ell\right|^{2/3}\cosh(\rho)^{1/3},$ (70)

in terms of which the metric and gauge field read

$\displaystyle ds^{2}$
$\displaystyle=\frac{\ell^{2}}{9}d\rho^{2}+\frac{r_{0}^{2}}{\ell^{2}}\left[\cosh\left(\rho\right)^{2/3}\left(-dt^{2}+dy^{2}+dz^{2}\right)+\frac{\sinh\left(\rho\right)^{2}}{\cosh(\rho)^{4/3}}d\phi^{2}\right],$
(71) $\displaystyle A$
$\displaystyle=\frac{\sqrt{3}q}{r_{0}^{2}}\left(\frac{1}{\cosh(\rho)^{2/3}}-1\right)d\phi,$
(72)

where in this section we take $r_{0}=\left|q\ell\right|^{1/3}$ and
$\Delta\phi=\frac{2\pi\ell^{5/3}}{3q^{1/3}}$. The calculation shall be carried
in the same basis as in the $D=4$ with the extra vielbein

$e^{4}=\frac{r_{0}}{\ell}\cosh(\rho)^{1/3}dy\text{ .}$ (73)

There are four independent solutions to the Killing spinor equation on this
background:

$\displaystyle\Psi_{1}$
$\displaystyle=\cosh(\rho)^{-1/3}\left(\begin{array}[c]{c}s_{\rho}c_{\phi}+\mathrm{i}c_{\rho}s_{\phi}\\\
-c_{\rho}c_{\phi}+\mathrm{i}s_{\rho}s_{\phi}\\\
c_{\rho}s_{\phi}-\mathrm{i}s_{\rho}c_{\phi}\\\
s_{\rho}s_{\phi}+\mathrm{i}c_{\rho}c_{\phi}\end{array}\right)\text{
},\qquad\Psi_{2}=\cosh(\rho)^{-1/3}\left(\begin{array}[c]{c}c_{\rho}c_{\phi}+\mathrm{i}s_{\rho}s_{\phi}\\\
-s_{\rho}c_{\phi}+\mathrm{i}c_{\rho}s_{\phi}\\\
-s_{\rho}s_{\phi}+\mathrm{i}c_{\rho}c_{\phi}\\\
-c_{\rho}s_{\phi}-\mathrm{i}s_{\rho}c_{\phi}\end{array}\right),$ (82)
$\displaystyle\Psi_{3}$
$\displaystyle=\cosh(\rho)^{-1/3}\left(\begin{array}[c]{c}s_{\rho}s_{\phi}-\mathrm{i}c_{\rho}c_{\phi}\\\
-c_{\rho}s_{\phi}-\mathrm{i}s_{\rho}c_{\phi}\\\
-c_{\rho}c_{\phi}-\mathrm{i}s_{\rho}s_{\phi}\\\
-s_{\rho}c_{\phi}+\mathrm{i}c_{\rho}s_{\phi}\end{array}\right)\text{
},\qquad\Psi_{4}=\cosh(\rho)^{-1/3}\left(\begin{array}[c]{c}c_{\rho}s_{\phi}-\mathrm{i}s_{\rho}c_{\phi}\\\
-s_{\rho}s_{\phi}-\mathrm{i}c_{\rho}c_{\phi}\\\
s_{\rho}c_{\phi}+\mathrm{i}c_{\rho}s_{\phi}\\\
c_{\rho}c_{\phi}-\mathrm{i}s_{\rho}s_{\phi}\end{array}\right)\text{ ,}$ (91)

where $s_{\rho}=\sinh\left(\frac{\rho}{2}\right)$,
$c_{\rho}=\cosh\left(\frac{\rho}{2}\right)$,
$s_{\phi}=\sin\left(\frac{\pi}{\Delta\phi}\phi\right)$ and
$c_{\phi}=\cos\left(\frac{\pi}{\Delta\phi}\phi\right)$, which implies that the
solution is $1/2$ BPS. As a cross-check we verify the formulae which relate
Killing spinors and Killing vectors as given in section 2 of [9]. The most
general Killing spinor for this solution is

$\vskip 12.0pt plus 4.0pt minus 4.0pt\Psi=\sum_{i=1}^{4}c_{i}\Psi_{i}\text{
.}$ (92)

where the $c_{i}$ are real constants. It statisfies that

$\displaystyle\Psi^{T}\left(\gamma^{4}\right)^{\dagger}\gamma^{0}\Psi$
$\displaystyle=0\text{ ,}$ (93)
$\displaystyle\Psi^{T}\left(\gamma^{4}\right)^{\dagger}\gamma^{0}\gamma^{\mu}\gamma^{4}\Psi^{\ast}$
$\displaystyle=K^{\mu}\text{ ,}$ (94)

where the components of the Killing vector are

$\displaystyle K^{t}$
$\displaystyle=\frac{2\ell}{r_{0}}\left(c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+c_{4}^{2}\right)\text{
,}$ (95) $\displaystyle K^{\phi}$
$\displaystyle=\frac{4\ell}{r_{0}}\left(c_{3}c_{4}+c_{1}c_{2}\right)\text{ ,}$
(96) $\displaystyle K^{z}$
$\displaystyle=-\frac{4\ell}{r_{0}}\left(c_{1}c_{4}-c_{3}c_{2}\right)\text{
,}$ (97) $\displaystyle K^{y}$
$\displaystyle=-\frac{2\ell}{r_{0}}\left(c_{1}^{2}-c_{2}^{2}+c_{3}^{2}-c_{4}^{2}\right)\text{
.}$ (98)

Let us note that in an asymptotic expansion, the leading term of the spinors
is

$\displaystyle\lim_{\rho\rightarrow\infty}e^{-\frac{\rho}{6}}\Psi_{1}$
$\displaystyle=\left(\begin{array}[c]{c}\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\mathrm{i}\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
\mathrm{i}\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\end{array}\right),\qquad\lim_{\rho\rightarrow\infty}e^{-\frac{\rho}{6}}\Psi_{2}=\left(\begin{array}[c]{c}\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
\mathrm{i}\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\mathrm{i}\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\end{array}\right),$
(107) $\displaystyle\lim_{\rho\rightarrow\infty}e^{-\frac{\rho}{6}}\Psi_{3}$
$\displaystyle=\left(\begin{array}[c]{c}-\mathrm{i}\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\mathrm{i}\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\end{array}\right),\qquad\lim_{\rho\rightarrow\infty}e^{-\frac{\rho}{6}}\Psi_{4}=\left(\begin{array}[c]{c}-\mathrm{i}\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
-\mathrm{i}\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
\exp\left(\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\\\
\exp\left(-\mathrm{i}\frac{\pi}{\Delta\phi}\phi\right)\end{array}\right)\text{.}$
(116)

On pure AdS5, the theory has four Killing spinors; if we introduce a Wilson
loop on this background with $A=A_{\phi}d\phi$. Indeed, a basis for the local
solutions to the Killing spinor equation in the Poincaré patch are

$\Psi_{1+}^{AdS}=\Delta_{+}\left(\begin{array}[c]{c}0\\\ 1\\\ 0\\\
0\end{array}\right),\text{
}\Psi_{1-}^{AdS}=\Delta_{-}\left(\begin{array}[c]{c}-1\\\ 0\\\ 0\\\
0\end{array}\right),\text{
}\Psi_{2+}^{AdS}=\Delta_{+}\left(\begin{array}[c]{c}0\\\ 0\\\ 0\\\
1\end{array}\right),\text{
}\Psi_{2-}^{AdS}=\Delta_{-}\left(\begin{array}[c]{c}0\\\ 0\\\ 1\\\
0\end{array}\right)\text{,}$ (117)

$\displaystyle\Psi_{3+}^{AdS}$
$\displaystyle=\Delta_{+}\left(\begin{array}[c]{c}0\\\ \phi+t\\\
\ell^{2}r^{-1}\\\ z+\mathrm{i}y\end{array}\right),\text{
}\Psi_{3-}^{AdS}=\Delta_{-}\left(\begin{array}[c]{c}-\phi-t\\\ 0\\\
z-\mathrm{i}y\\\ -\ell^{2}r^{-1}\end{array}\right),$ (126)
$\displaystyle\Psi_{4+}^{AdS}$
$\displaystyle=\Delta_{+}\left(\begin{array}[c]{c}\ell^{2}r^{-1}\\\
z-\mathrm{i}y\\\ 0\\\
t-\phi\end{array}\right),\Psi_{4-}^{AdS}=\Delta_{-}\left(\begin{array}[c]{c}-z-\mathrm{i}y\\\
\ell^{2}r^{-1}\\\ -\phi+t\\\ 0\end{array}\right)\text{ }.$ (135)

where
$\Delta_{\pm}=\sqrt{r}\exp\left(\pm\mathrm{i}\frac{\sqrt{3}A_{\phi}}{2\ell}\phi\right)$.
The most general solution to the Killing spinor equation in this basis is

$\Psi^{AdS}=\sum_{i=1}^{4}\alpha_{i}\Psi_{i+}^{AdS}+\sum_{i=1}^{4}\alpha_{i}^{\ast}\Psi_{i-}^{AdS}$
(136)

where the constants $\alpha_{i}^{\ast}$ are the complex conjugate of
$\alpha_{i}$. Again we can see from the Killing spinors that the
identification in $\phi$ breaks half of the supersymmetries of AdS5.

For $\Phi=\Phi_{S}=\frac{2\pi}{\sqrt{3}}\ell$, we can also obtain a
supersymmetric solution from Poincaré-AdS with a constant Wilson loop along
the $\phi$ circle in the bulk. As in four dimensions, this is easily seen from
the uplift of the theory to ten dimensions. The theory we considered here is
obtained by a dimensional reduction [22]

$ds_{10}^{2}=ds_{5}^{2}+\ell^{2}\sum_{i=1}^{3}d\mu_{i}^{2}+\mu_{i}^{2}(d\phi_{i}+\frac{1}{\sqrt{3}\ell}A)^{2};$
(137)

The coordinates $\phi_{i}$ are $2\pi$ periodic. The ten-dimensional metric is
thus invariant under shifts of the Wilson loop by
$\Phi\rightarrow\Phi+2\pi\sqrt{3}\ell$, making this a periodic variable.
Quotienting Poincaré-AdS with a Wilson loop on the $\phi$ circle corresponds
to a quotient of $AdS_{5}\times S^{5}$ by
$(\phi,\tilde{\phi}_{i})\sim(\phi+\Delta\phi,\tilde{\phi}_{i}+\Phi/\sqrt{3}\ell)$,
so the ten-dimensional Killing spinors will be invariant under the quotient if
$\Phi=\frac{2\pi}{\sqrt{3}}n\ell$ for integer $n$. Given the periodicity of
$\Phi$, there are three physically inequivalent cases, $n=-1,0,1$. For
$n=-1,1$, we have a degeneracy between the Poincaré-AdS solution and the
supersymmetric soliton. The main qualitative difference between the four- and
five-dimensional cases is that we only have one case with a unique
supersymmetric solution here.

##  5 Conclusions

We have shown that the simplest $D=4$ and $D=5$ gauged supergravities
theories, which allow for charged solutions, contain in its spectrum of
supersymmetric solutions Lorentzian planar solitons. These solitons can be
seen as a supersymmetric extension of the rather well-known AdS soliton of
Horowitz and Myers [2]. The existence of these supersymmetric solitons poses
an holographic puzzle. One usually expects that for given boundary conditions,
at least with sufficient supersymmetry, the gravitational solution would be
completely fixed. This solution would then be dual to a supersymmetric state
in the dual CFT. However, as we have explictly shown this is not the case. One
possible way out would be if the solutions were supersymmetric with respect to
different realizations of the supersymmetry algebra in $AdS_{4}$ [35]. However
in the different realizations, for the same supersymmetry and same boundary
conditions for the metric, the form of the Killing spinors is different. Our
explicit construction of the Killing spinors shows that the soliton Killing
spinors are asymptotically the Killing spinors of a locally AdS spacetime.
Hence, these configurations are indeed completely indistiguishable from the
boundary point of view. Moreovoer, the Killing spinors for the solitons
provide an smooth interpolation between some Killing spinors of MinkowskiD and
AdSD, for $D=4$ and $D=5$.

The simplicity of the susy AdS soliton make us conjecture that these $1/2$ BPS
solutions can be generalized to have different charges by the introduction of
running scalars in $D=4$ and $D=5$ gauged supergravity [22, 36, 37, 38].
Furthermore, we believe that this susy AdS soliton should also be part of the
spectrum of any gauged supergravity in any dimension that contains a
consistent supersymmetric truncation with a $U(1)$ gauge field. Therefore the
degeneracy of susy states we presented here, it is very likely, a generic
feature of these boundary conditions.

The structure of the soliton seems to be well suited to be interpreted in the
context of holographic superconductivity [39]. Indeed, from the holographic
point of view, the boundary of the soliton is a cylinider with a current
flowing along the circle. The current is flowing without any electromagnetic
field on the surface of the cylinder and there is a magnetic field piercing
the cylinder. It would be worth understanding if the periodic oscillation of
the critical temperature of the superconductor takes place holographically as
expected due to the Little-Parks effect [40].

An interesting question to be further explored is whether other boundary
conditions can yield a degeneration in the spectrum of supersymmetric
solutions. Asymptotically locally AdS, everywhere regular supersymmetric
solutions in the pure Einstein-Maxwell theory with a negative cosmological
constant exist in four dimensions but with a much more complex structure than
the one discussed here [41].

## Acknowledgements

We thank Bernard de Wit, Jerome Gauntlett and Mario Trigiante for enlightening
discussions and comments. We would like to thank the support of Proyecto de
cooperación internacional 2019/13231-7 FAPESP/ANID. The research of AA is
supported in part by the Fondecyt Grants 1210635, 1181047 and 1200986. SFR is
supported in part by STFC through grant ST/T000708/1.

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