# The Conformal Manifold of S-folds
in String Theory

Nikolay Bobev,1 Friðrik Freyr Gautason,2 and Jesse van Muiden3,4
1KU Leuven, Instituut voor Theoretische Fysica
Celestijnenlaan 200D, B-3001 Leuven, Belgium
2University of Iceland, Science Institute
Dunhaga 3, 107 Reykjavík, Iceland
3SISSA
Via Bonomea 265, I 34136 Trieste, Italy
4INFN - Sezione di Trieste
Via Valerio 2, I-34127 Trieste, Italy

###### Abstract

We continue the holographic exploration of the conformal manifold of 3d
$\mathcal{N}=2$ S-fold SCFTs constructed by gauging the flavor symmetry of the
Gaiotto-Witten $T[U(N)]$ theory. We show how to uplift the two-parameter
family of AdS4 vacua dual to this conformal manifold to 10d backgrounds of
type IIB supergravity. We use these uplifted solutions to shed new light on
the mysterious nature of the infinite distance limit on the conformal manifold
and to study probe strings and D3-branes. This analysis uncovers an intriguing
structure of the $S^{3}$ partition function of the S-fold SCFTs which
resembles the giant graviton expansion of the superconformal index of 4d
$\mathcal{N}=4$ SYM. We also show how to each member of the family of
supersymmetric AdS4 vacua one can associate a consistent truncation to 4d
$\mathcal{N}=2$ gauged supergravity and use this result, in conjunction with
holography, to calculate the large $N$ partition function of the 3d S-fold
SCFT on compact Euclidean manifolds. Finally, we generalize the supersymmetric
AdS4 vacua to a four-parameter family of non-supersymmetric AdS4 solutions.

###### Contents

  1. 1 Introduction and summary
  2. 2 S-fold conformal manifold in type IIB
    1. 2.1 Review of the five-dimensional solution
    2. 2.2 One parameter family in type IIB
    3. 2.3 Three parameter family in type IIB
    4. 2.4 Infinite distances on the conformal manifold
  3. 3 Non-perturbative corrections to the $S^{3}$ partition function
  4. 4 Holographic line operators
    1. 4.1 Gravity
    2. 4.2 Field theory
  5. 5 4d $\mathcal{N}=2$ supergravity and S-fold black holes
  6. A The IIB background for $Y\rightarrow 0$
  7. B Four parameter family of AdS4 solutions

## 1 Introduction and summary

In this work we explore an interesting example of the AdS4/CFT3 correspondence
in type IIB supergravity that arises from $N$ D3-branes on $S^{1}$ with an
$\mathop{\rm SL}(2,\mathbf{Z})$ S-duality twist along the circle. The
backreaction of the branes produces an AdS4 solution of type IIB supergravity
that preserves $\mathcal{N}=4$ supersymmetry and has an internal space of the
schematic form $S^{1}\times S^{5}$ with an $\mathop{\rm SL}(2,\mathbf{Z})$
monodromy of the axio-dilaton along the $S^{1}$ [1, 2]. The 3d CFT
holographically dual to this AdS4 vacuum can be obtained by starting with the
$T[U(N)]$ theory of Gaiotto-Witten [3, 4] and gauging its
$\mathop{\rm{}U}(N)\times\mathop{\rm{}U}(N)$ flavor symmetry with a vector
multiplet and adding a Chern-Simons term at level $k$ for the gauge field
together with a specific superpotential [2]. Despite the non-geometric
features of the string theory background and the strongly coupled nature of
the dual SCFT, this model can be studied quantitatively with various
techniques and admits interesting generalizations and extensions.

The $\mathcal{N}=4$ AdS4 solution of [1, 2] belongs to a two-dimensional
family of supersymmetric AdS4 vacua of type IIB supergravity that preserve
$\mathcal{N}=2$ supersymmetry. These solutions can be constructed using the 4d
$\mathcal{N}=8$ gauged supergravity with $[\mathop{\rm SO}(6)\times\mathop{\rm
SO}(1,1)]\ltimes\mathbf{R}^{12}$ gauging, see [5, 6]. Alternatively, one can
use the 5d $\mathcal{N}=8$ $\mathop{\rm SO}(6)$ gauged supergravity and
construct these supersymmetric AdS4 solutions as limits of gravitational
solutions describing Janus interfaces [7, 8]. Both of these perspectives and
the consistent truncation results for the 4d and 5d gauged supergravity
theories in [1] and [9], respectively, guarantee that the AdS4 vacua can be
uplifted to full-fledged backgrounds of type IIB string theory. Performing
these uplifts explicitly is an arduous exercise which we perform in detail
below and obtain the full 10d uplift of the two-parameter family of AdS4
$\mathcal{N}=2$ supersymmetric vacua of [6]. In addition to serving as rare
explicit examples of non-geometric backgrounds in string theory, this family
of solutions is of interest in the context of the AdS/CFT correspondence.

The ten-dimensional type IIB backgrounds we construct should be thought of as
arising from a stack of $N$ D3-branes wrapped on $S^{1}$ with an $\mathop{\rm
SL}(2,\mathbf{Z})$ monodromy along the circle. The background takes the
schematic form

$w\text{AdS}_{4}\times S^{1}_{\mathfrak{J}}\times\tilde{S}^{5}\,,$ (1.1)

where $\mathfrak{J}$ denotes the $\mathop{\rm SL}(2,\mathbf{Z})$ element,
$\tilde{S}^{5}$ is a squashed five sphere, and there is warp factor in front
of the AdS4 part of the metric that depends on the coordinates on the sphere.
We note that this type IIB background is non-geometric since
$S^{1}_{\mathfrak{J}}$ is made compact by using the $\mathop{\rm
SL}(2,\mathbf{Z})$ symmetry of type IIB string theory. The ten-dimensional
background has all NS-NS and R-R fluxes turned on and the squashed metric on
$\tilde{S}^{5}$ preserves only $\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$
isometry. This isometry corresponds to the $\mathop{\rm{}U}(1)_{R}$ R-symmetry
and $\mathop{\rm{}U}(1)_{F}$ flavor symmetry preserved along the conformal
manifold in the dual 3d $\mathcal{N}=2$ SCFT. The number of D3-branes, as
determined by the flux quantization of the five-form flux, and the periodicity
of the circle $S^{1}_{\mathfrak{J}}$ determine the rank $N$ of the gauge group
and the corresponding CS level $k$ in the SCFT, respectively. There are two
special members of the family of supersymmetric AdS4 backgrounds that enjoy
enhanced symmetry. One of these solutions has $\mathop{\rm
SU}(2)\times\mathop{\rm SU}(2)$ isometry on $\tilde{S}^{5}$ and
$\mathcal{N}=4$ supersymmetry and corresponds to the background discussed in
[1, 2], while the other has $\mathop{\rm SU}(2)\times\mathop{\rm{}U}(1)$
isometry, $\mathcal{N}=2$ supersymmetry, and is the same as the AdS4 vacuum
presented in [5, 7].

These features of the family of ten-dimensional backgrounds mirror the
properties of the conformal manifold in the dual S-fold SCFT. A convenient way
to think about this conformal manifold is to remember that upon gauging the
$\mathop{\rm{}U}(N)\times\mathop{\rm{}U}(N)$ flavor symmetry symmetry of the
$T[U(N)]$ $\mathcal{N}=4$ SCFT the conformal symmetry is broken and an RG flow
ensues. As argued in [2] the IR theory is conformal and furthermore there is a
freedom in the choice of IR superpotential that can be expressed schematically
as, see [6],

$W_{\text{IR}}=-\frac{2\pi}{k}\text{Tr}~{}(\mu_{H}\mu_{C})+\lambda\,\text{Tr}~{}(\mu_{H}\mu_{C})\,,$
(1.2)

where $\mu_{H}$ and $\mu_{C}$ are the two moment map operators associated with
the $\mathop{\rm{}U}(N)\times\mathop{\rm{}U}(N)$ flavor symmetry of the
$T[U(N)]$ theory. Here $k$ is the CS level of the gauge group and $\lambda$ is
a complex parameter that acts as an exactly marginal coupling preserving
$\mathcal{N}=2$ supersymmetry. For generic values of $\lambda$ the symmetry of
the $\mathcal{N}=2$ SCFT is
$\mathop{\rm{}U}(1)_{R}\times\mathop{\rm{}U}(1)_{F}$. At $\lambda=0$ one finds
enhancement to $\mathcal{N}=4$ superconformal symmetry and its associated
$\mathop{\rm SU}(2)\times\mathop{\rm SU}(2)$ R-symmetry. Despite the strongly-
coupled nature of this family of SCFTs, it is possible to use supersymmetric
localization to calculate the $S^{3}$ partition function of the theory. The
result is remarkably simple and reads, see [2, 10, 11],

$\mathcal{Z}_{N}=\frac{q^{N^{2}/2}}{(q)_{N}}\,,$ (1.3)

where $q=\mathrm{e}^{-T}$, $k=2\cosh T$, and $(q)_{N}$ is the $q$-Pochhammer
symbol, see Section 3 below for further details. As emphasized in [6] this
partition function is independent of the marginal coupling $\lambda$ and is
therefore the same over the whole conformal manifold. Given this, and the
scarcity of other QFT techniques to study the properties of conformal
manifolds in 3d $\mathcal{N}=2$ SCFTs, one can resort to the supergravity
description to find further insights into the properties of this model.

Using the 4d maximal gauged supergravity construction of the family of
$\mathcal{N}=2$ AdS4 vacua it was shown in [6] that the Zamolodchikov metric
of the conformal manifold takes the form

$\mathrm{d}s_{Z}^{2}=\frac{4-Y^{2}}{2}\left(\frac{\mathrm{d}Y^{2}}{Y^{2}(2-Y^{2})}+d\chi^{2}\right)\,.$
(1.4)

Here $Y$ and $\chi$ are conveniently chosen combinations of the massless 4d
supergravity scalar fields and this result should be viewed as a supergravity
prediction for the large $N$ limit of the Zamolodchikov metric. The values of
$Y$ are constrained by the structure of 4d supergravity to lie in the interval
$[0,\sqrt{2}]$, while a priori the supergravity scalar $\chi$ can take any
real value. A highly non-trivial calculation of the type IIB supergravity KK
spectrum as a function of $\chi$ can be performed using techniques from
Exceptional Field Theory [12]. The details of this analysis reveal that $\chi$
should be periodically identified and should lie in the range $[0,2\pi/T)$
with $T$ defined below (1.3), see [13, 14]. A similar KK spectroscopy analysis
can be performed along the entire two-dimensional space of vacua parametrized
by $Y$ and $\chi$ but it does not reveal any structure that may suggest
additional periodic identifications [6, 15]. We note that in this
parametrization the $\mathcal{N}=4$ AdS4 vacuum is at $Y=1$ and $\chi=0$,
while the $\mathcal{N}=2$ AdS4 solution with enhanced $\mathop{\rm
SU}(2)\times U(1)$ symmetry is at $Y=\sqrt{2}$ and $\chi=0$.

The supergravity analysis above therefore suggests that at large $N$, the
conformal manifold has the topology of a cylinder and moreover one can show
that the metric in (1.4) features unbounded geodesics. This may lead to the
puzzling conclusion that the conformal manifold is non-compact. As discussed
in [16, 17], non-compact conformal manifolds are usually equipped with an
infinite distance locus where the CFT at hand becomes free. The analog of such
a locus in the S-fold conformal manifold discussed above is at $Y\to 0$.
Indeed, this appears to be a singular limit in the 5d gauged supergravity
construction of the family of AdS4 solutions. However, the KK spectrum of
supergravity excitations does not reveal the emergence of higher spin
conserved currents associated with a free sector in the CFT. This is
furthermore supported at finite $N$ by the $S^{3}$ partition function in (1.3)
which does not have the structure of the partition function of free 3d
$\mathcal{N}=2$ SCFTs. We are thus led to the conclusion that either the
conformal manifold is non-compact with unbounded geodesics but without a free
locus, or there is a tower of light higher-spin modes that arises in the $Y\to
0$ locus and remains inaccessible in supergravity. The 10d uplift of the
family of AdS4 vacua that we present in this work sheds new light on this
mysterious feature of the conformal manifold. We find that the singular
behavior of the 5d gauged supergravity in the limit $Y\to 0$ becomes much more
mild in the 10d background. As $Y\to 0$ the $w$AdS4 and $\tilde{S}^{5}$ parts
of the 10d metric, see (1.1), remain regular and finite size, while the radius
of the non-geometric circle $S^{1}_{\mathfrak{J}}$ blows up as $1/Y^{2}$. We
do not know how to resolve this singularity and have not completed the study
of the spectrum of light string or D-brane modes in the $Y\to 0$ limit and so
the question whether this conformal manifold is compact or whether it has a
free locus remains open. Nevertheless, we cannot resist to make a few
speculations and observations. It appears that in the $Y\to 0$ limit the
period of the S-fold circle blows up. Taking this as a starting point, we note
that since the periodicity of the circle is related to the CS level $k$ (as
described above), one can conclude that this limit can be offset by an
appropriate change of the CS level $k$. Perhaps this points to a dual
description of the SCFT in terms of a different theory with a large CS level
but we were unable to come up with a concrete proposal for such a theory. It
will be very interesting to explore this question further and understand
better the global properties of this conformal manifold of S-fold SCFTs.

While the main focus of our work is on the two-parameter family of
$\mathcal{N}=2$ supersymmetric AdS4 vacua described above, as a byproduct of
our calculations we also present a new four-parameter family of non-
supersymmetric AdS4 solutions. As discussed in [18, 19, 5, 7, 20, 21] one can
use 4d and 5d maximal gauged supergravity to construct S-fold AdS4 solutions
that preserve only $\mathcal{N}=1$ supersymmetry or break it completely.
Furthermore, it was shown in [21] that for S-fold AdS4 solutions with
continuous global symmetry, one can introduce new parameters that act as a
marginal deformations that potentially break supersymmetry. We show below how
to use 4d maximal gauged supergravity in combination with the method of [21]
to construct a four-parameter family of such AdS4 solutions. For general
values of the parameters, the solutions break supersymmetry completely but on
a two-dimensional subspace the solutions reduce to the family of ${\cal N}=2$
AdS4 S-folds described above. One could reasonably wonder whether the non-
supersymmetric AdS4 solutions we construct are stable. A setup similar to ours
was studied in [22], where it was argued that a similar family of non-
supersymmetric AdS4 S-fold solutions is stable against many perturbative and
non-perturbative potential decay channels. Much of the reasoning [22] relies
on the fact that the non-supersymmetric vacua are continuously connected
through 10d diffeomorphisms to the supersymmetric vacua. This feature is also
exhibited by the four-parameter family of solutions we construct here and thus
we could import the conclusions of [22]. It will of course be very interesting
to study in more detail the perturbative and non-perturbative stability of our
solutions but we leave this important question to future work.

The $S^{3}$ partition function (1.3) of the S-fold SCFTs we study has a simple
form that reveals an interesting structure. We take inspiration from the well-
known relation between the 4d $\mathcal{N}=1$ superconformal index and the 3d
$\mathcal{N}=2$ $S^{3}$ partition function for certain classes of
supersymmetric QFTs discussed in [23, 24, 25, 26] to think of the result in
(1.3) as a limit of some type of a 4d partition function. Given the D3-brane
origin of the S-fold SCFTs we discuss, it is natural to conjecture that this
is the partition function of the 4d $\mathcal{N}=4$ SYM theory on
$S^{1}_{\mathfrak{J}}\times S^{3}$ where one performs an S-duality twist,
specified by the $\mathop{\rm SL}(2,\mathbf{Z})$ element ${\mathfrak{J}}$, of
the 4d theory as one goes along the circle. To the best of our knowledge such
S-duality “twisted indices” have not been studied in the literature and it
will be very interesting to understand whether they can be properly defined
and calculated. Undeterred by the lack of a proper QFT treatment of these
putative new indices we show that indeed (1.3) can be rewritten in a way
reminiscent of the 1/2-BPS superconformal index of the 4d $\mathcal{N}=4$ SYM.
We do this by exploiting the so-called giant graviton expansion of the 4d
$\mathcal{N}=4$ SYM index which is tailored to holography and organizes the
index as a series of contributions from BPS operators associated to D3-branes,
see [27, 28, 29, 30]. We show how the $S^{3}$ partition function of the S-fold
theories in (1.3) can be written in a similar fashion. Moreover, we study
probe supersymmetric D3-branes in the 10d type IIB supergravity solutions we
construct and show that their on-shell action agrees with the structure of the
“giant graviton expansion” of the $S^{3}$ partition function. This result is a
non-trivial test of the holographic correspondence for the S-fold SCFT and
their interpretation in terms of D3-branes. Moreover, it underscores the
possible relation between the S-fold $S^{3}$ partition function and a new
S-duality “twisted index” of the 4d $\mathcal{N}=4$ SYM theory.

As an additional check of the holographic duality in our setup we also explore
supersymmetric line operators in the S-fold SCFTs. We first study probe
$(p,q)$-strings with an AdS2 worldvolume in the family of AdS4 type IIB
solutions we construct. We compute their regularized on-shell action and show
that it is independent of the parameters $Y$ and $\chi$. This implies that
these probe strings should be dual to supersymmetric line operators in the
S-fold theory on $S^{3}$. The vacuum expectation values of such line operators
have not been studied in the literature, although it should be possible to do
so using supersymmetric localization. We eschew a detailed supersymmetric
localization analysis in favor of a simple Gaussian matrix model argument with
which we are able to estimate the expectation value of a supersymmetric Wilson
loop along the equator of $S^{3}$ in the fundamental representation of the
gauge group. Despite this crude matrix model approximation we find very good
agreement with the probe string analysis on the gravity side.

It is natural to wonder whether other supersymmetric observables in the S-fold
SCFTs can be computed using supersymmetric localization. While there are some
results on the 3d superconformal index available in the literature, see [31,
32], this question is largely unexplored due to the difficulties presented by
the non-Lagrangian nature of the $T[U(N)]$ theory. In the large $N$ limit,
supergravity can again be brought to bear on this problem and leads to
interesting results. We show that for every allowed value of the parameters
$Y$ and $\chi$ one can find a consistent truncation of type IIB supergravity
to the 4d $\mathcal{N}=2$ minimal gauged supergravity which has a bosonic
sector consisting of the metric and a $\mathop{\rm{}U}(1)$ Maxwell field. This
simple supergravity theory is expected to describe holographically the
universal large $N$ dynamics of the energy-momentum multiplet in 3d
holographic $\mathcal{N}=2$ SCFT. This result can be used to compute the large
$N$ behavior of supersymmetric partition functions of the S-fold SCFTs on a
variety of Euclidean 3-manifolds $\mathcal{M}_{3}$. The calculation leverages
available results in the literature for the regularized on-shell action of
Euclidean gauged supergravity solutions that can be viewed as smooth fillings
of $\mathcal{M}_{3}$. Using this approach we present explicit results for the
large $N$ squashed sphere partition function, topologically twisted index, and
superconformal index of the S-fold SCFTs. As expected from supersymmetric
localization, our results do not depend on the exactly marginal couplings and
are thus valid along the entire conformal manifold. Our analysis also reveals
the existence of asymptotically AdS4 black hole solutions in the S-fold
compactifications of type IIB supergravity and provide simple expressions for
their Bekenstein-Hawking entropy.

We continue in the next section with a summary of the gauged supergravity
description of the family of AdS4 vacua of interest as well as its explicit
uplift to 10d solutions of type IIB supergravity. In Section 3 we show that
the $S^{3}$ partition function of the S-fold SCFT can be written in a
suggestive way resembling the giant graviton expansion familiar from 4d
$\mathcal{N}=4$ SYM and support this interpretation with a probe D3-brane
calculation. In Section 4 we study probe strings in the type IIB supergravity
background and discuss their dual SCFT interpretation as supersymmetric line
operators. We discuss how to construct a consistent 4d minimal gauged
supergravity truncation associated to each of the AdS4 vacua we study in
Section 5 and explore the consequences of this result for holography. In the
two appendices we provide more details on the construction of the
supersymmetric family of AdS4 vacua, its non-supersymmetric generalization, as
well as the infinite distance limit of the uplifted 10d supergravity
solutions. Along with this paper we include an ancillary Mathematica file that
contains some details on the 10d type IIB supergravity backgrounds we
construct.

## 2 S-fold conformal manifold in type IIB

To construct the IIB supergravity solutions of interest we start from the
five-dimensional realization of the one parameter family presented in [8] and
apply the uplift formulae in [9]. In ten-dimensions it is then straight-
forward to add two additional parameters as locally they can be expressed as
coordinate transformations [13]. This will result in a three-parameter family
of 10d solutions of which there is a two-parameter sub-family of type IIB AdS4
solutions that preserve ${\cal N}=2$ supersymmetry. This two-parameter family
of supersymmetric solutions will form the backbone of the analysis in the rest
of the paper. An alternative way to construct this family of supersymmetric
solutions would be to directly uplift the two-parameter family of
supersymmetric solutions in four-dimensional supergravity found in [6] using
the uplift formulae in [1]. Presumably this alternative method will lead to
the same results as the ones we present below.

### 2.1 Review of the five-dimensional solution

The five-dimensional realization of a one-parameter family of AdS4 solutions
was constructed in [8] by starting with a ten-scalar subsector of the maximal
$\mathop{\rm SO}(6)$ gauged supergravity theory in five dimensions. This
$\mathcal{N}=8$ theory has $42$ scalar fields which often presents
difficulties when trying to construct explicit solutions. For this reason one
usually employs a convenient consistent truncation to a subset of the scalar
fields. In [33], see also [34], such a ten-scalar consistent truncation was
found by keeping only 5d supergravity fields that are invariant under a
$\mathbf{Z}^{3}_{2}$ subgroup of the $\mathop{\rm SO}(6)\times\mathop{\rm
SL}(2,\mathbf{R})$ symmetry of the maximal supergravity theory. The bosonic
field content of this truncated theory consists of the five-dimensional
metric, together with two real scalars; $\beta_{1},\beta_{2}$ and four complex
scalar field $z^{i}$ with $i=1,\dots,4$. In [8] a further six-scalar
truncation of this model was considered where

$z^{2}=\bar{z}^{2}=-z^{4}=-\bar{z}^{4}\,,\qquad\beta_{2}=0\,.$ (2.1)

Writing the remaining scalar fields in terms of real variables

$\begin{split}z^{1}&=\tanh\Big{[}\frac{1}{2}(2\alpha_{1}+\alpha_{3}+\varphi-2i\phi_{1}+2i\phi_{4})\Big{]}\,,\\\
z^{2}&=\tanh\Big{[}\frac{1}{2}(\alpha_{3}-\varphi)\Big{]}\,,\\\
z^{3}&=\tanh\Big{[}\frac{1}{2}(2\alpha_{1}-\alpha_{3}+\varphi-2i\phi_{1}-2i\phi_{4})\Big{]}\,,\end{split}$
(2.2)

the one-parameter family of AdS4 vacua of interest here can be found by
imposing $\alpha_{1}=\alpha_{3}=0$ and

$\mathrm{e}^{-12\beta_{1}}=2\tan^{2}2\phi_{4}\equiv Y^{2}\,,\qquad\cos
4\phi_{4}+2\cos 4\phi_{1}=1\,.$ (2.3)

Furthermore the dilaton is linear along the fifth direction of space-time and
can therefore, by a simple coordinate transformation, be used as a convenient
coordinate itself. Explicitly the metric takes the form

$\mathrm{d}s_{5}^{2}=L^{2}\Big{(}\frac{Y^{2/3}}{2}\mathrm{d}s_{\text{AdS}_{4}}+\frac{1}{Y^{4/3}}\mathrm{d}\varphi^{2}\Big{)}\,,$
(2.4)

where we have slightly changed notation compared to [8]111In [6] these
solutions were studied from a four-dimensional perspective. The relation of
$Y$ to the parameter used there is
$\varphi_{\text{\cite[cite]{[\@@bibref{}{Bobev:2021yya}{}{}]}}}=\sqrt{2-Y^{2}}/Y$.
This parameterization should not be confused with the dilaton in this paper.
and in particular we work with mostly plus signature for the metric. Note that
here $L=2/g$ where $L$ is the length scale of the maximally symmetric AdS5
vacuum solution and $g$ is the five-dimensional gauge coupling.

The solution preserves ${\cal N}=2$ supersymmetry and should therefore be
holographically dual to a family of 3d $\mathcal{N}=2$ SCFT with its
associated $\mathop{\rm{}U}(1)_{R}$ symmetry. In addition to this R-symmetry
the family of AdS4 vacua exhibits an additional $\mathop{\rm{}U}(1)_{F}$
flavor symmetry. The parameter $Y$ is unfixed and does not show up in the
supergravity potential. This parameter is holographically dual to an exactly
marginal operator in the field theory. Regularity of the five-dimensional
metric and scalars implies that we must constrain $Y$ to lay in the range
$[0,\sqrt{2}]$.

A priori, the dilaton and the coordinate direction it parametrizes seem to be
non-compact. When the dilaton grows without bound however this would
ultimately lead to an inconsistent supergravity solution since the string
coupling blows up. We must therefore impose that the dilaton remains finite
and that $\varphi$ describes a compact direction. To do this we impose that
the coordinate direction parametrized by the dilaton is compact and
simultaneously apply an $\mathop{\rm SL}(2,\mathbf{Z})$ action to glue the
dilaton together resulting in an $S$-fold string theory background. The
$\mathop{\rm SL}(2,\mathbf{Z})$ transformation acts on the five-dimensional
scalars coset as, see [7],

$\mathcal{V}(\varphi+T)=\mathcal{V}(\varphi)\mathfrak{J}\,,\quad\text{where}\quad\mathfrak{J}=\left(\begin{array}[]{cc}\mathrm{e}^{-T}&0\\\
0&\mathrm{e}^{T}\end{array}\right)\,,$ (2.5)

and $T$ is the period of the $S^{1}_{\mathfrak{J}}$ circle parametrized by
$\varphi$. As written $\mathfrak{J}$ is an element of $\mathop{\rm
SL}(2,\mathbf{R})$ and not $\mathop{\rm SL}(2,\mathbf{Z})$. This can be
remedied by employing a similarity transformation such that the element
$\mathfrak{J}\in\mathop{\rm SL}(2,\mathbf{Z})$ is

$\mathfrak{J}_{k}=\left(\begin{array}[]{cc}k&1\\\ -1&0\end{array}\right)\,,$
(2.6)

where $k=2\cosh T\in{\bf Z}$. As discussed in the introduction, this integer
is identified with a CS level in the dual three-dimensional field theory.
Regularity of the background imposes that $k$ is bounded from below $k>2$.

In [8, 6] the on-shell action of this family of AdS4 solutions was shown to be
independent of $Y$ consistent with the fact that this parameter is dual to an
exactly marginal deformation. The value of the on-shell action is222This holds
for the full two-dimensional family of vacua studied in [6].

$S_{\text{on-shell}}=\frac{N^{2}T}{2}\,,$ (2.7)

which matches the leading order term in the large $N$ limit of the field
theory supersymmetric partition function in equation (1.3).

At two special values of the parameter $Y$ there is enhanced symmetry. At
$Y=1$ the solutions preserves $\mathcal{N}=4$ supersymmetry and the global
symmetry is enhanced to $\mathop{\rm SO}(4)$, this is the original ${\cal
N}=4$ S-fold background found in [1, 2]. The second point of interest is
$Y=\sqrt{2}$ where the $\mathop{\rm{}U}(1)_{F}$ flavor symmetry is enhanced to
$\mathop{\rm SU}(2)$. This flavor symmetry enhanced point plays an important
role in our discussions since its ten-dimensional uplifted solution was
already constructed in [7]. The new type IIB solution discussed below share
many features with the solution in [7] and will act as an important
consistency check of our calculations.

The limit $Y\rightarrow 0$ corresponds to taking an infinite distance limit on
the space of massless supergravity scalars which in turn corresponds to an
infinite distance limit on the conformal manifold of the dual 3d
$\mathcal{N}=2$ SCFT. This point is of special interest in this work and will
be discussed further below. Here we note that the 5d metric in (2.4) becomes
singular as $Y\rightarrow 0$. One of the results we present below is that this
singular behavior becomes more mild in the 10d uplift of this family of
solutions dimensions.

### 2.2 One parameter family in type IIB

To uplift the five-dimensional family of AdS${}_{4}\times
S^{1}_{\mathfrak{J}}$ vacua we use the techniques of exceptional field theory
described in [9]. The only input needed for this calculation is the 5d
solution in (2.1), (2.3), and (2.4).333We follow the conventions described in
[35] for the explicit uplift. As discussed in the beginning of this section,
the family presented in Section 2.1 is a one-dimensional subset of the full
two-dimensional family of solutions first found in [6]. First we uplift the
one-parameter family controlled by $Y$, which will already prove to be
formidable, and then in Section 2.3 we discuss how to include two additional
parameters based on symmetry considerations. We will see that for generic
values of the three parameters, all supersymmetries are broken. However, by
setting one of the three parameters to zero, we find the ten-dimensional
uplift of the set of supersymmetric solutions described in [6].444In Appendix
B we provide a four dimensional extension of the solution in [6] using four-
dimensional gauged supergravity. We do not uplift this extended solution to
ten-dimensions.

Because the solutions of interest generically only preserve a
$\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$ isometry in the internal space,
the expressions for the 10d fields will be rather involved. Nevertheless, we
are able to write the full solution analytically as a function of $Y$. For
convenience we have included a Mathematica notebook with the full type IIB
solution as an ancillary file along with the arXiv submission.

We start with the embedding coordinates used to describe the five-sphere

$\displaystyle\mathcal{Y}^{1}+\mathrm{i}\mathcal{Y}^{3}=\mathrm{e}^{-\mathrm{i}(\xi_{1}+\xi_{2})/2}\cos\theta\cos\xi_{3}\,,$
(2.8)
$\displaystyle\mathcal{Y}^{2}+\mathrm{i}\mathcal{Y}^{4}=\mathrm{i}\mathrm{e}^{-\mathrm{i}(\xi_{1}-\xi_{2})/2}\cos\theta\sin\xi_{3}\,,$
$\displaystyle\mathcal{Y}^{5}+\mathrm{i}\mathcal{Y}^{6}=\mathrm{e}^{-\mathrm{i}\phi}\sin\theta\,.$

In these coordinates the metric on the round five-sphere is given by

$\mathrm{d}s_{S^{5}}^{2}=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}+\cos^{2}\theta(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})\,,$
(2.9)

where

$\sigma_{1}+\mathrm{i}\sigma_{2}=-\mathrm{e}^{-\mathrm{i}\xi_{2}}(\mathrm{i}\mathrm{d}\xi_{3}-\frac{1}{2}\sin
2\xi_{3}\mathrm{d}\xi_{1})\,,\quad\sigma_{3}=-\frac{1}{2}(\mathrm{d}\xi_{2}+\cos
2\xi_{3}\mathrm{d}\xi_{1})\,.$ (2.10)

The $\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$ isometry of the solution is
generated by translations along the angles $\xi_{1}$ and $\xi_{2}$. Later on
we will determine which of the two $\mathop{\rm{}U}(1)$’s is mapped to the
R-symmetry and which one to the flavor symmetry in the QFT. The ranges of the
coordinates are

$(\theta,\xi_{3})\in[0,\pi/2]\,,\qquad(\phi,\xi_{1},\xi_{2})\in[0,2\pi]\,.$
(2.11)

We now introduce the following functions

$\displaystyle X=$ $\displaystyle\,\sqrt{2-Y^{2}}\,,$ (2.12) $\displaystyle
a_{1}^{\pm}=$ $\displaystyle\,Y^{2}(2\pm X\cos 2\phi)\,,$ $\displaystyle
a_{2}^{\pm}=$ $\displaystyle\,X(\cos 2\xi_{3}+(1\pm X)(\cos 2\xi_{3}-X\cos
2\phi)\sin^{2}\theta)\,,$ $\displaystyle a_{3}^{\pm}=$ $\displaystyle\,(2\pm
X\cos 2\xi_{3})\cos^{2}\theta\mp X^{2}\cos 2\phi\sin^{2}\theta-4\,,$
$\displaystyle b^{\pm}_{X}=$ $\displaystyle\,(2\pm X)(2+X\cos 2\xi_{3})\,,$
$\displaystyle v_{0}=$ $\displaystyle\,-\sin 2\theta\cos\theta\sin\phi\,,$
$\displaystyle v_{1}=$ $\displaystyle\,X(X\cos 2\xi_{3}\cos^{2}\theta+\cos
2\phi\sin^{2}\theta)\,,$ $\displaystyle v_{2}=$ $\displaystyle\,X(\cos
2\xi_{3}\cos^{2}\theta+XY^{2}\cos 2\phi\sin^{2}\theta)\,.$

The full 10d metric in Einstein frame is given by

$\mathrm{d}s_{10}^{2}=\frac{\Delta^{1/4}}{2Y^{2/3}}(\mathrm{d}s_{5}^{2}+Y^{2/3}\mathrm{d}\tilde{s}^{2}_{S^{5}})$
(2.13)

where $\mathrm{d}s_{5}^{2}$ is given in (2.4), the warp factor reads

$\displaystyle\Delta=$ $\displaystyle
16\big{(}4(\cos^{2}\theta+Y^{2}\sin^{2}\theta)-\cos^{2}\theta(v_{1}\cos
2\xi_{3}-v_{2}\cos 2\phi)-v_{2}\cos 2\phi\big{)}\,,$ (2.14)

and finally the metric on the internal squashed five-sphere is given by

$\displaystyle\mathrm{d}\tilde{s}_{5}^{2}=$
$\displaystyle\frac{64}{g^{2}\Delta}\Big{[}\tfrac{1}{2}Y^{2}\cos^{2}\theta\Big{(}(1+\sin^{2}\theta)\frac{1}{2}\left[\mathrm{d}\xi_{1}^{2}+\mathrm{d}\xi_{2}^{2}+2\cos
2\xi_{3}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right]$ (2.15) $\displaystyle-X\cos
2\phi\sin^{2}\theta\frac{1}{2}\left[\cos
2\xi_{3}(\mathrm{d}\xi_{1}^{2}+\mathrm{d}\xi_{2}^{2})+2\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right]$
$\displaystyle+\frac{1}{4}\sin^{2}2\xi_{3}\cos^{2}\theta\left[\mathrm{d}\xi_{1}^{2}+\mathrm{d}\xi_{2}^{2}(\frac{4}{Y^{2}}-1)\right]\Big{)}$
$\displaystyle+2\cos^{2}\theta\Big{(}(X^{2}+(1-X^{2})\cos^{2}\xi_{3})\cos^{2}\theta+Y^{2}(1+\tfrac{1}{4X}\cos
2\xi_{3}(a_{3}^{-}-a_{3}^{+}))\Big{)}\mathrm{d}\xi_{3}^{2}$
$\displaystyle+\sin^{2}\theta\left(4\cos^{2}\phi(1+\sin^{2}\theta)+a_{3}^{+}\cos
2\phi\right)\mathrm{d}\phi^{2}$ $\displaystyle+X\sin 2\xi_{3}\sin
2\phi\sin^{2}2\theta\,\mathrm{d}\xi_{3}\mathrm{d}\phi$
$\displaystyle+\Big{(}2+\left(\frac{1}{2}\cos 2\phi-\frac{2}{X}\cos
2\xi_{3}\right)\cos^{2}\theta(a_{3}^{-}-a_{3}^{+})+2(1-X^{2})\sin^{2}\theta$
$\displaystyle-\cos
2\xi_{3}\left[\frac{1}{X}v_{2}\sin^{2}\theta+\cos^{2}\theta\cos
2\xi_{3}(4\cos^{2}\theta-(1+X^{2})\sin^{2}\theta)\right]\Big{)}\mathrm{d}\theta^{2}$
$\displaystyle-\frac{X}{1-X^{2}}\sin 2\theta\Big{[}\sin
2\xi_{3}(Xv_{1}-v_{2}+2(1-X^{2})\cos 2\phi)\mathrm{d}\xi_{3}$
$\displaystyle-\sin 2\phi(v_{1}-\frac{1}{X}v_{2}+2(1-X^{2})\cos
2\xi_{3})\mathrm{d}\phi\Big{]}\mathrm{d}\theta\Big{]}\,.$

The complex axio-dilaton reads

$\tau=C_{0}+\mathrm{i}\mathrm{e}^{-\Phi}=\frac{\Lambda_{22}\hat{\tau}+\Lambda_{21}}{\Lambda_{12}\hat{\tau}+\Lambda_{11}}\,,$
(2.16)

where

$\hat{\tau}=\frac{(1-X^{2})\sin
2\phi\sin^{2}\theta+\mathrm{i}\sqrt{\Delta}/4}{a_{3}^{+}+2(X^{2}+1)\cos^{2}\phi\sin^{2}\theta}\mathrm{e}^{-2\varphi}\,,\quad\Lambda=\frac{\mathrm{e}^{-T/2}}{\sqrt{2\sinh
T}}\left(\begin{array}[]{cc}{\mathrm{e}^{T}}&-1\\\
-1&{\mathrm{e}^{T}}\end{array}\right)\,.$ (2.17)

Using the same $\mathop{\rm SL}(2,\mathbf{R})$ element, the RR and NSNS two-
forms are given by

$\left(\begin{array}[]{c}B_{2}\\\
C_{2}\end{array}\right)=\Lambda\cdot\left(\begin{array}[]{c}\hat{B}_{2}\\\
\hat{C}_{2}\end{array}\right)\,,$ (2.18)

with

$\displaystyle\hat{B}_{2}=$
$\displaystyle\frac{16\mathrm{e}^{\varphi}}{g^{2}\Delta}\partial_{\phi}(v_{0})\Big{[}\frac{1}{4}\sin
2\xi_{3}[(1+X)(b^{+}_{-X}\cos^{2}\theta+a_{1}^{-}\sin^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
(2.19) $\displaystyle\hskip
110.96556pt+(1-X)(b_{-X}^{-}\cos^{2}\theta+a_{1}^{+}\sin^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})]\wedge\mathrm{d}\xi_{3}$
$\displaystyle\hskip
42.67912pt-\frac{1}{2}\tan\phi[-(1+X)\cos^{2}\xi_{3}(a_{3}^{+}+4X\cos^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
93.89418pt+(1-X)\sin^{2}\xi_{3}(a_{3}^{+}-4X\cos^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})]\wedge\mathrm{d}\phi$
$\displaystyle\hskip 42.67912pt-\frac{1}{\sin
2\theta}[-(1+X)\cos^{2}\xi_{3}(a_{1}^{-}\sin^{2}\theta+(2-a_{2}^{-})\cos^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
85.35826pt+(1-X)\sin^{2}\xi_{3}(a_{1}^{+}\sin^{2}\theta+(2-a_{2}^{+})\cos^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})]\wedge\mathrm{d}\theta\Big{]}\,,$
$\displaystyle\hat{C}_{2}=$
$\displaystyle\frac{16\mathrm{e}^{-\varphi}}{g^{2}\Delta}v_{0}\Big{[}-\frac{1}{4}\sin
2\xi_{3}[(1-X)(b^{+}_{X}\cos^{2}\theta+a_{1}^{-}\sin^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
(2.20) $\displaystyle\hskip
93.89418pt+(1+X)(b_{X}^{-}\cos^{2}\theta+a_{1}^{+}\sin^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})]\wedge\mathrm{d}\xi_{3}$
$\displaystyle\hskip
42.67912pt+\frac{1}{2}\cot\phi[(1-X)\cos^{2}\xi_{3}(a_{3}^{-}-4X\cos^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
85.35826pt-(1+X)\sin^{2}\theta_{3}(a_{3}^{-}+4X\cos^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})]\wedge\mathrm{d}\phi$
$\displaystyle\hskip 42.67912pt+\frac{1}{\sin
2\theta}[-(1-X)\cos^{2}\xi_{3}(a_{1}^{-}\sin^{2}\theta+(2+a_{2}^{+})\cos^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
85.35826pt+(1+X)\sin^{2}\xi_{3}(a_{1}^{+}\sin^{2}\theta+(2+a_{2}^{-})\cos^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})]\wedge\mathrm{d}\theta\Big{]}\,.$

Finally, the four-form with legs along the internal five-sphere is given by

$\displaystyle C_{4}^{\text{int}}=$
$\displaystyle\frac{64}{g^{4}}\Big{[}\frac{\cos^{2}\theta\sin 2\xi_{3}\sin
2\theta}{\Delta}\Big{(}(-v_{1}\sin 2\xi_{3}\mathrm{d}\phi+v_{2}\sin
2\phi\mathrm{d}\xi_{3})\wedge\mathrm{d}\theta$ (2.21)
$\displaystyle\,+\frac{1}{4}\sin 2\theta\left(2(Y^{2}-1)-(v_{1}\cos
2\xi_{3}-v_{2}\cos
2\phi)\mathrm{d}\xi_{3}\wedge\mathrm{d}\phi\right)\wedge\mathrm{d}\xi_{1}\wedge\mathrm{d}\xi_{2}\Big{)}+\omega_{4}\Big{]}\,,$

where $\omega_{4}$ is the integrated volume form on $S^{5}$

$\omega_{4}=\frac{1}{16}\sin(2\xi_{3})\cos^{4}\theta\mathrm{d}\xi_{1}\wedge\mathrm{d}\xi_{2}\wedge\mathrm{d}\xi_{3}\wedge\mathrm{d}\phi\,.$
(2.22)

To relate the 5d gauge coupling $g$ to the number of D3-branes sourcing the
10d background we compute the integral of $F_{5}$ to find that

$N=\int_{S^{5}}\left.F_{5}\right|_{S^{5}}=\int_{S^{5}}\mathrm{d}C_{4}^{\text{int}}=\frac{4}{\pi
g^{4}\ell_{s}^{4}}\,,$ (2.23)

where we projected the 5-form to its $S^{5}$ components. For convenience we
also provide the four-form gauge potential with legs along the AdS4 space.
This can be found using the self-duality constraint in type IIB supergravity

$F_{5}=\star F_{5}\,.$ (2.24)

Explicitly, working out the Hodge-dual of the five-form flux on the internal
space and integrating over the $\varphi$ direction one finds that the four-
form along the AdS space is given by

$C_{4}^{\text{AdS}_{4}}=\frac{4}{g^{4}}\left(6\varphi+2X\cos
2\xi_{3}\cos^{2}\theta+(1+X^{2})\cos
2\phi\sin^{2}\theta\right)\text{vol}_{\text{AdS}_{4}}\,.$ (2.25)

One can show that 10d background presented above solves the type IIB equations
of motion, for which we have used the conventions of [35].

Since this background is rather involved we present an additional consistency
check by studying the flavor symmetry enhanced point $Y=\sqrt{2}$ which can be
compared to the 10d solution presented in [7]. Indeed, taking $Y=\sqrt{2}$ the
background presented above greatly simplifies. The metric becomes555Here we
correct a mistake in [7]. To get the correct expression presented here one
should perform the rescaling $g=2^{\frac{1}{32}}g_{\text{there}}$.

$\mathrm{d}s_{10}^{2}=\frac{2(4w)^{1/4}}{g^{2}}\left(\mathrm{d}\varphi^{2}+\mathrm{d}s_{\text{AdS}_{4}}^{2}+\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}+\cos^{2}\theta\left(\sigma_{3}^{2}+\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{w}\right)\right)\,,$
(2.26)

with the $\mathfrak{su}(2)$ one-forms defined in (2.10), and
$w=1+\sin^{2}\theta$. The form potentials on the internal space equal

$\displaystyle\hat{C}_{2}=$
$\displaystyle\,\frac{2\sqrt{2}\mathrm{e}^{-\varphi}}{g^{2}}\frac{\cos\theta\sin\phi}{w}\left(\sin
2\theta\sigma_{1}\wedge\sigma_{2}-w(\mathrm{d}\theta-\frac{1}{2}\cot\phi\sin
2\phi\mathrm{d}\phi)\wedge\sigma_{3}\right)\,,$ (2.27)
$\displaystyle\hat{B}_{2}=$
$\displaystyle\,\frac{2\sqrt{2}\mathrm{e}^{-\varphi}}{g^{2}}\frac{\cos\phi\cos\theta}{w}\left(\sin
2\theta\sigma_{1}\wedge\sigma_{2}-w(\mathrm{d}\theta-\frac{1}{2}\tan\phi\sin
2\theta\mathrm{d}\phi)\wedge\sigma_{3}\right)\,,$ $\displaystyle C_{4}=$
$\displaystyle\,-\frac{2}{g^{4}w}\cos^{2}\theta\sin^{2}2\theta\,\mathrm{d}\phi\wedge\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}\,.$

Finally, the axion and dilaton equal

$\hat{C}_{0}+\mathrm{i}\mathrm{e}^{-\hat{\Phi}}=\mathrm{e}^{-2\varphi}\frac{\mathrm{i}\sqrt{1+\sin^{2}\theta}+\frac{1}{2}\sin
2\phi\sin^{2}\theta}{1+\sin^{2}\phi\sin^{2}\theta}\,.$ (2.28)

These expressions agree with the background presented in [7].

### 2.3 Three parameter family in type IIB

So far we described the type IIB background dual to a one-dimensional slice of
the two-dimensional supersymmetric conformal manifold.666In Appendix B we show
that the full parameter space of AdS4 vacua is four-dimensional, of which only
a two-dimensional subspace is supersymmetric. In this section we construct a
three-dimensional subspace of vacua directly in type IIB supergravity. In [21]
it was shown how one can turn on a special class of exactly massless scalar
fields additional in the four-dimensional $\mathop{\rm SO}(6)\times\mathop{\rm
SO}(1,1)$ gauged supergravity truncation. In the five-dimensional gauged
supergravity description these massless scalars arise from Wilson loop
deformations turned on along $S^{1}_{\mathfrak{J}}$ [36], breaking the
supergravity gauge symmetry to its Cartan subalgebra. Since our supergravity
solutions preserve $\mathop{\rm{}U}(1)_{R}\times\mathop{\rm{}U}(1)_{F}$
symmetry there are two such Wilson loops we can turn on. In type IIB
supergravity these deformations are simply described by the coordinate
transformation

$\xi_{1}\rightarrow\xi_{1}+\chi_{1}\varphi\,,\quad\xi_{2}\rightarrow\xi_{2}+\chi_{2}\varphi\,,$
(2.29)

which is well-defined only locally due to the $\mathop{\rm SL}(2,\mathbf{Z})$
monodromy along $S^{1}_{\mathfrak{J}}$. One can show that such a deformations
applied to the type IIB supergravity background described in the previous
section indeed solves the equations of motion. For generic values of
$\chi_{i}$ the solution breaks all supersymmetry, and thus we find a
holographic description of a three-dimensional conformal manifold of non-
supersymmetric CFTs. This raises the important question whether the
corresponding AdS4 vacua are stable. In [22] a two-dimensional non-
supersymmetric slice of this three-dimensional family of vacua was studied,
and it was argued that indeed they are perturbatively stable. Furthermore, it
was argued that neither higher derivative corrections in the gravitational
theory, nor known non-perturbative effects can spoil the stability of the
vacua. Much of the reasoning presented in [22] relies on the fact that the
non-supersymmetric vacua are continuously connected through diffeomorphisms to
the $\mathcal{N}=4$ preserving vacuum, which in our coordinates is found at
$(Y,\chi_{1},\chi_{2})=(1,0,0)$. Since all the type IIB vacua we presented
above are connected to the $\mathcal{N}=4$ supersymmetric one, in a similar
fashion, we find that the results in [22] imply that our full three-
dimensional family of type IIB vacua is stable as well, regardless of whether
supersymmetry is broken. We leave a more detailed and rigorous analysis of the
very interesting question of the stability of these vacua for future work.

The $\mathcal{N}=2$ supersymmetric AdS4 vacua are of special interest. For
these solutions we can compute physical observables and hope to compare to
known exact QFT results obtained via supersymmetric localization. This will be
of particular interest in the following sections and thus we now study which
two-dimensional subset of vacua preserve at least $\mathcal{N}=2$
supersymmetry. To preserve supersymmetry only a linear combination of the
$\chi_{i}$ parameters that leaves the $\mathop{\rm{}U}(1)_{R}$ isometry
unchanged can be turned on. To determine this linear combination one can
simply take the limit $Y\rightarrow\sqrt{2}$, in which case the background
corresponds to the one described in Section 3.3 in [7], which has an isometry
enhancement to $\mathop{\rm SU}(2)_{F}\times\mathop{\rm{}U}(1)_{R}$. The full
solution in this limit is given in (2.26), (2.27), and (2.28) above. The
$\mathop{\rm SU}(2)_{F}$ isometry is described by the left invariant one-
forms, the $\mathop{\rm{}U}(1)_{F}\subset\mathop{\rm SU}(2)_{F}$ preserved
away from the enhanced point is generated by $\mathrm{d}\xi_{1}$. The
$\mathop{\rm{}U}(1)_{R}$ on the other hand is described by a rotation between
$\sigma_{1}$ and $\sigma_{2}$, which is in turn described by a shift in
$\xi_{2}$. We thus see that the $\chi_{i}$ deformation preserving
supersymmetry, leaving the $\mathop{\rm{}U}(1)_{R}$ untouched, is along
$\mathrm{d}\xi_{1}$ in equation (2.29). We therefore conclude that the
deformation parametrized by $\chi_{1}$ in (2.29) preserves $\mathcal{N}=2$
supersymmetry. In summary, the 10d uplift of the two-parameter family of
$\mathcal{N}=2$ supersymmetric AdS4 vacua found in [6] is given by the 10d
background above parametrized by $Y$ and $\chi_{1}$ with vanishing $\chi_{2}$.

### 2.4 Infinite distances on the conformal manifold

As mentioned in the introduction, previous holographic studies of the
$\mathcal{N}=2$ conformal manifold of S-folds showed the manifold to be non-
compact. The Zamolodchikov metric of the manifold was computed to equal [6]

$\mathrm{d}s_{Z}^{2}=\frac{4-Y^{2}}{2}\left(\frac{\mathrm{d}Y^{2}}{Y^{2}(2-Y^{2})}+d\chi^{2}\right)\,,$
(2.30)

which has unbounded geodesic curves. In this parametrization the infinite
distance is reached when $Y\rightarrow 0$, and in previous studies it was
shown that the five-dimensional solution is badly singular in this limit since
the full AdS4 space vanishes [8], which can be directly seen from equation
(2.4).

In this section we study the behavior of the full ten-dimensional solution in
the limit where $Y\rightarrow 0$. As can be directly seen from (2.13) we find
that the singular behavior of the metric is improved in ten dimensions. To
leading order in a small $Y$ expansion the metric becomes

$\displaystyle\mathrm{d}s_{10}^{2}=\frac{(\Delta^{(0)})^{1/4}}{g^{2}}\left[\left(2^{4/3}\mathrm{d}\varphi^{2}\right)\frac{1}{Y^{2}}+\left(2^{7/3}\mathrm{d}s_{\text{AdS}_{4}}^{2}+\frac{1}{2}(\mathrm{d}s_{5}^{2})^{(0)}\right)+\mathcal{O}(Y)\right]\,,$
(2.31)

where $(\cdot)^{(p)}$ denotes the leading order non-trivial term in the
expansion of $(\cdot)$, which is of order $Y^{p}$. It turns out that to zeroth
order the remaining type IIB field content becomes independent of the
parameter $Y$

$\displaystyle\mathrm{e}^{\hat{\Phi}}=(\mathrm{e}^{\hat{\Phi}})^{(0)}+\mathcal{O}(Y)\,,\quad\hat{C}_{0}=\,(\hat{C}_{0})^{(0)}+\mathcal{O}(Y)\,,$
(2.32) $\displaystyle
B_{2}=(\hat{B}_{2})^{(0)}+\mathcal{O}(Y)\,,\quad\hat{C}_{2}=(\hat{C}_{2})^{(0)}+\mathcal{O}(Y)\,,\quad
C_{4}=(C_{4})^{(0)}+\mathcal{O}(Y)\,.$

The full expressions for the leading order terms are rather involved functions
depending on the internal coordinates, and we have summarized the details of
their explicit form in Appendix A. We thus find that to leading order in $Y$
the only dependence on $Y$ in the IIB background is given by the metric
component that determines the radius of the S-fold circle $\mathrm{d}\varphi$.
Importantly, as we take $Y\rightarrow 0$ the size of the circle blows up. This
property of the 10d type IIB supergravity background may explain some of the
features of the KK spectrum of this family of AdS4 solutions that was computed
in [15] using tools from exceptional field theory, see [12] and references
thereof. Indeed, it was found in [15] that the masses of certain unprotected
KK modes do not increase as one takes $Y\rightarrow 0$, i.e. they stay “light”
in this limit.

In view of the CFT distance conjecture discussed in [16], see also [17], one
may expect to find a tower of massless higher-spin fields, corresponding to
conserved higher-spin operators in the dual CFT, emerging in the spectrum as
we take $Y\rightarrow 0$. We could not find any evidence for the existence of
such modes and given the results above and the ones in [15] it is quite clear
that if these modes exist they should not arise from the supergravity fields
but rather from excitations associated with fundamental strings or D-branes.
It will be most interesting to understand whether the locus $Y\rightarrow 0$
provides a counterexample to the CFT distance conjecture and it does not to
identify the origin of the infinite tower of light higher-spin modes.

Before closing this section we would like to note that the increase in the
radius of the $\varphi$ circle in the metric as $Y\rightarrow 0$ may have an
interesting interpretation in the dual SCFT. From (2.4) and (2.13) it is clear
that changing $Y$ simply changes the size of the circle parametrized by
$\varphi$. On the other hand, as discussed below (2.4), the periodicity of
that same S-fold circle is given by $T$. In the dual 3d S-fold SCFT $Y$ and
$T$ are mapped to the exactly marginal coupling $\lambda$ and CS level $k$,
respectively. In the formal superpotential that specifies the gauging of the
$T[U(N)]$ theory the dependence on these parameters is given by [6]

$W_{\text{IR}}=-\frac{2\pi}{k}\text{Tr}~{}(\mu_{H}\mu_{C})+\lambda\,\text{Tr}~{}(\mu_{H}\mu_{C})\,.$
(2.33)

We therefore observe that a change in $\lambda$ can be reabsorbed by a change
in the CS level $k$ in the superpotential. This analysis seems to suggest that
at infinite distance on the conformal manifold, where $Y\rightarrow 0$, the
SCFT may have an equivalent description as an S-fold SCFT with the
superpotential above in the limit of infinite CS level $k$. It will be very
interesting to study this analogy further and understand whether it can be
made more precise.

## 3 Non-perturbative corrections to the $S^{3}$ partition function

As reviewed in the Introduction, the $S^{3}$ partition function of the
$\mathcal{N}=4$ S-fold SCFT was computed in [2, 10, 11] using supersymmetric
localization. The result can be written as a matrix model of the following
form777The $\sinh$ term one would expect from the vector multiplet
contribution cancels agains a term in the $T[\mathop{\rm{}U}(N)]$
contribution, greatly simplifying the matrix model. Note also that the same
matrix model has appeared in [37] in a different context.

$\displaystyle\mathcal{Z}_{N}=$
$\displaystyle\,\frac{1}{N!}\int\mathrm{d}^{N}\sigma\,Z_{\text{C}S}(\sigma)Z_{\text{v}ec}(\sigma)Z_{T[\mathop{\rm{}U}(N)]}(\sigma,-\sigma)$
(3.1) $\displaystyle=$ $\displaystyle\frac{1}{N!}\sum\limits_{\tau\in
S_{N}}(-1)^{\tau}\int\mathrm{d}^{N}\sigma\,\mathrm{e}^{i\pi
k\sum_{j=1}^{N}\sigma_{j}^{2}\,-\,2\pi
i\sum_{j=1}^{N}\sigma_{j}\sigma_{\tau(j)}}\,,$

where $k$ is the Chern-Simons level. Defining the quantities

$q=\mathrm{e}^{-T}\,\qquad\text{and}\quad k=2\cosh T\,,$ (3.2)

this matrix model can be solved analytically for any $N$ and $k$ to find

$\mathcal{Z}_{N}=\frac{q^{N^{2}/2}}{(q)_{N}}\,,$ (3.3)

where $(q)_{N}$ is the $q$-Pochhammer symbol

$(q)_{N}=\prod\limits_{m=1}^{N}(1-q^{m})\,.$ (3.4)

Taking the large $N$ limit we can find the leading order term in the $S^{3}$
free energy

$F_{S^{3}}=-\log\mathcal{Z}_{N}=\frac{N^{2}T}{2}+\ldots\,.$ (3.5)

This result agrees with the on-shell action of the dual AdS4 solution, see
(2.7). Moreover, in 3d SCFTs exactly marginal deformations are exact with
respect to the localization supercharged used for the calculation of the
$S^{3}$ partition function and therefore, as discussed in [7, 6], the result
in (3.5) is valid along the full two-dimensional conformal manifold.

Inspired by some recent observations in [29, 30, 38] we can proceed to rewrite
the $S^{3}$ partition function (3.3) in a suggestive way. First we note that
the infinite extension of the Pochhammer symbol
$(x;q)_{\infty}=\prod\limits_{n=0}^{\infty}(1-xq^{n})$ can be used to derive
the following identity

$\frac{(q)_{\infty}}{(q)_{N}}=(q^{N+1};q)_{\infty}\,.$ (3.6)

Importantly, the right hand side above can be rewritten as a sum using the
relation [39]

$(x;q)_{\infty}=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n-1)/2}}{(q)_{n}}x^{n}\,.$
(3.7)

Using the identities above it is easy to show that the $S^{3}$ path integral
of the S-fold SCFT of interest (3.3) can be written as

$\mathcal{Z}_{N}=\frac{q^{N^{2}/2}}{(q)_{\infty}}\left(1+\sum\limits_{n=1}^{N}(-1)^{n}\frac{q^{n(n+1)/2}}{(q)_{n}}q^{nN}\right)\,,$
(3.8)

or equivalently

$\mathcal{Z}_{N}=\frac{q^{N^{2}/2}}{(q)_{\infty}}\left(1+\sum\limits_{n=1}^{\infty}(-1)^{n}\mathcal{Z}_{n}q^{n(N+1/2)}\right)\,.$
(3.9)

This is a rather intriguing result since we find that the finite $N$ partition
$S^{3}$ partition function can be written as an infinite series of terms each
of which is determined again by the same supersymmetric partition function
$\mathcal{Z}_{n}$. Such a structure is reminiscent of the giant graviton
expansion [27, 28, 29, 30] discussed in the context of the superconformal
index, or $S^{3}\times S^{1}$ partition function, of 4d $\mathcal{N}=1$ SCFTs.
Remarkably, the expansion in (3.8) is identical to the non-perturbative series
of giant gravitons in the $1/2$-BPS superconformal index of ${\rm U}(N)$
$\mathcal{N}=4$ SYM, see equation (1.9) in [30]). The difference between the
two expressions comes from the fact that the fugacity in the $\mathcal{N}=4$
index is replaced by the parameter $q$ in (3.2) determined by the parameter
$T$ that controls the periodicity of the S-fold circle, or equivalently the CS
level of the 3d SCFT. It is tempting to ponder the reason for this close
similarity between two seemingly different supersymmetric observables. A
possible explanation may arise through a proper definition of a new type of 4d
supersymmetric index computed by the path integral of $\mathcal{N}=4$ SYM on
$S^{3}\times S^{1}_{\mathfrak{J}}$, where the notation $S^{1}_{\mathfrak{J}}$
indicates that one should perform an S-duality twist determined by
$\mathfrak{J}$ in (2.6) as one goes around the circle. One could then try to
find a relation between the ordinary superconformal index defined as a path
integral on $S^{3}\times S^{1}$ and the S-duality twisted index that will
explain the relation above. Indeed, the fact that in the sum (3.8) the
numerator contains only integer powers of $q$ resembles the structure of an
index. Moreover, note that the $q^{N^{2}/2}$ prefactor on the right hand side
in (3.9) resembles the supersymmetric Casimir energy on $S^{3}\times S^{1}$
which is determined by the conformal anomalies of the 4d SCFT, see [40, 41].
Stripping off this prefactor then naturally leads to the conjecture that the
factor of $1/(q)_{\infty}$ should be interpreted as the supergravity
contribution to the putative S-duality twisted index. It will be most
interesting to explore these questions further and study the existence and
properties of such S-duality twisted indices of 4d SCFTs.

Regardless of the existence of a properly defined S-duality twisted index we
can adopt a string theory perspective to shed more light on the structure of
the partition function in (3.9). The form of this expression suggests that in
string theory this partition function can be thought of as a leading order
supergravity piece coming from the large $N$ stack of D3-branes wrapping
$S^{3}\times S^{1}_{\mathfrak{J}}$, accompanied by an infinite sum of non-
perturbative corrections coming from D3-branes wrapping $\tilde{S}^{3}\times
S^{1}_{\mathfrak{J}}$. Notably, the two three spheres are embedded differently
in the type IIB supergravity background. The $S^{3}$ is simply given by the
conformal boundary of the Euclidean AdS4 while the $\tilde{S}^{3}$ is embedded
in the squashed internal $S^{5}$ spanned by the coordinates
$(\theta,\phi,\xi_{1,2,3})$ used in Section 2. The explicit 10d supergravity
solution constructed in Section 2 indeed allows us to check this intuition in
detail.

To this end let us study a probe Euclidean D3-brane that wraps the three
angular directions parametrized by $\xi_{1,2,3}$, and the S-fold circle
parameterized by $\varphi$. To compute the on-shell action of this D3-brane we
first have to minimize the brane action that consists of the DBI action,
coupling the brane to the NS-NS sector, and the WZ terms that determine the
coupling to the RR forms. It turns out that for the D3-brane configurations
described above it is sufficient to minimize the DBI action, and furthermore
that the RR forms and the $B_{2}$-field do not contribute either to the
extremization problem or to the on-shell action of the brane. With this in
mind the D3-brane action is simply controlled only by the Einstein frame
metric and reads

$S_{\text{D3}}=\frac{2\pi}{(2\pi\ell_{s})^{4}}\int\mathrm{d}^{4}\sigma\,\sqrt{\det
G_{ab}}\,.$ (3.10)

Using static gauge we simply have to extremize this action with respect to the
transverse coordinates, which shows that in the internal space the probe
D3-brane sits at the points

$\sin 2\theta=0\,,\quad\text{and}\quad\sin 2\phi=0\,.$ (3.11)

The pulled back metric on the D3-brane in this configuration becomes888The
$\text{SU}(2)$ one-forms are defined in (2.10).

$\displaystyle\mathrm{d}s_{\text{D3}}^{2}=$
$\displaystyle\frac{4\Delta_{\text{D3}}^{1/4}}{g^{2}}\Big{(}\frac{\mathrm{d}\varphi^{2}}{Y^{2}}+\sigma_{1}^{2}+\sigma_{2}^{2}+\frac{1}{\Delta_{\text{D3}}}\left(2+(2-Y^{2})(2-\cos^{2}\xi_{3})\right)\sigma_{3}^{2}$
(3.12)
$\displaystyle\qquad-\frac{2-Y^{2}}{2\Delta_{\text{D3}}}\left(\mathrm{d}\xi_{1}^{2}+\mathrm{d}\xi_{2}^{2}+(2+\sin^{2}2\xi_{3})\cos
2\xi_{3}\,\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right)\Big{)}\,,$

where

$\Delta_{\text{D3}}=4-(2-Y^{2})\cos^{2}2\xi_{3}\,.$ (3.13)

The on-shell action can be easily calculated to find the simple result

$S_{\text{D3}}=\frac{4T}{\pi g^{4}\ell_{s}^{4}}=TN\,,$ (3.14)

where we have used (2.23) to relate $g$ and $\ell_{s}$ to the number $N$ of
D3-branes that produce the AdS4 supergravity solution. We would like to stress
that this probe brane analysis is valid for any value of $Y$, i.e. for the
full family of AdS4 supergravity solutions discussed in Section 2. This is
compatible with the expectation that the probe D3-brane with the embedding
described above preserves supersymmetry.

Given the probe brane analysis above, one should expect that the non-
perturbative contributions of such probe D3-branes to the full string theory
partition function on the AdS4 S-fold background takes the form

$\mathcal{Z}_{\text{np}}=\sum\limits_{n=0}^{\infty}Z^{\text{D3}}_{n}\mathrm{e}^{-nS_{\text{D3}}}=\sum\limits_{n=0}^{\infty}Z^{\text{D3}}_{n}q^{nN}\,,$
(3.15)

where in the second identity we have used (3.2) and (3.14). This result is in
remarkable agreement with the $S^{3}$ partition function of the dual SCFT as
presented in (3.8) and (3.9). The off-set by $1/2$ in the $q^{n(N+1/2)}$ term
of (3.9) could perhaps be explained by a finite $N$ correction to the Newton’s
constant, similar to what was found in [42]. Nevertheless, as noted above,
this shift in $N$ is crucial when interpreting the results in (3.8) as a
putative 4d SCFT index.

Note that the D3 brane metric in (3.12) is topologically that of $S^{1}\times
S^{3}$, however the three sphere is equipped with a squashed metric. It is
thus rather remarkable that the non-perturbative corrections the probe
D3-branes contribute to the full string partition function takes the same form
as the one of the $N$ backreacted D3-branes that produce the supergravity
background since these $N$ branes wrap a round sphere. A potential explanation
for this fact could be that we are computing a supersymmetric quantity and
that the squashing deformation of the $S^{3}$ are $Q$-exact. A hint in support
of this interpretation is provided by the observation that the $S^{3}$
squashing is controlled by the parameter $Y$, which is dual to an exactly
marginal $Q$-exact deformation of the S-fold partition function.

It will be interesting to use string theory to reproduce the
$Z_{n}^{\text{D3}}$ contribution in (3.15). One way to do this is to study
small fluctuations of the fields on the worldvolume of the D3-brane and
compute their superdeterminants. This calculation may proceed similarly to
what was done recently for the fundamental string wrapping Euclidean two-
cycles in various holographic backgrounds in [43], which was subsequently
generalized to M-theory and fluctuating M2-branes in [44, 45]. The D3-brane
one-loop partition function has been computed with such methods in a different
setup in [46]. Another way to go about this may be to compute the
$\tilde{S}^{3}$ partition function of the S-fold theory which lives on the
Euclidean probe D3-brane wrapped in the internal $\tilde{S}^{3}$, similarly to
what was done in [28] for Euclidean D3-branes wrapping an $S^{3}$ in the
internal space of AdS${}_{5}\times S^{5}$ exploiting the giant graviton
expansion in the bulk. It will be very interesting to explore this question
further.

## 4 Holographic line operators

In this section we will use the new ten-dimensional background presented in
Section 2 to compute the expectation values of line operators in the dual CFT.
The CFT line operators we consider are dual to fundamental strings, D1-branes,
or more general $(p,q)$-strings wrapping the equator of $S^{3}$ and extending
into the bulk geometry. We also study the vacuum expectation value of the
Wilson loop using the results from supersymmetric localization in the dual 3d
CFT and find good schematic agreement with the supergravity analysis.

### 4.1 Gravity

We consider the on-shell action of a $(p,q)$-string that wraps the equator of
$S^{3}$ on the boundary of Euclidean AdS4. The string extends into the bulk
geometry in such a way that its worldsheet spans an $\text{AdS}_{2}$ subspace
of $\text{AdS}_{4}$. We assume that the string sits at a fixed location along
the remaining directions of the 10d background. This is done in order to
retain the isometries of AdS2 as symmetries of the string. The classical
action of the $(p,q)$-string takes the form

$S_{(p,q)}=\frac{1}{2\pi\ell_{s}^{2}}\int\left[\mathrm{d}^{2}\sigma\sqrt{\mathbf{p}\cdot\mathcal{M}\cdot\mathbf{p}^{T}\,\det[G_{ab}]}+i\mathbf{p}\cdot
A_{2}\right]\,,$ (4.1)

where $\mathbf{p}=(p,q)$ determines the charge of the string,
$A_{2}=(C_{2},B_{2})$ gives the coupling to the background type IIB two-forms,
and the matrix $\mathcal{M}$ encodes the axio-dilaton, see (2.16)

$\mathcal{M}=\mathrm{e}^{\Phi}\left(\begin{array}[]{cc}\tau\bar{\tau}&C_{0}\\\
C_{0}&1\end{array}\right)\,.$ (4.2)

The fundamental string has charge $\mathbf{p}=(0,1)$, which corresponds to a
Wilson line in the fundamental representation in the dual CFT, while any other
charge corresponds to ’t Hooft or dyonic line operators in the QFT.
Extremizing the on-shell action of the $(p,q)$-string leads to a number of
different solutions. Notably, only one of them is independent of the
supergravity parameter $Y$ that is dual to the marginal coupling in the dual
SCFT. Since the expectation value of a supersymmetric Wilson loop operators in
3d $\mathcal{N}=2$ theories are independent of exactly marginal couplings we
expect that the on-shell action of the supersymmetric string should be
independent of $Y$. We therefore assume that the other extrema for the on-
shell action of the probe string break supersymmetry and we do not consider
them further. The extremum we are interested in satisfies

$\cos\theta=1\,,\quad\cos
2\xi_{3}=0\,,\quad\mathrm{e}^{2\varphi-T}=-\frac{p+\mathrm{e}^{-T}q}{p+\mathrm{e}^{T}q}\,.$
(4.3)

The on-shell action of such a $(p,q)$-string is easily computed after using
the fact that the regularized volume of the unit-radius Euclidean AdS2 is
$-2\pi$. Using (2.23) and the holographic dictionary we find that the
expectation value of the $(p,q)$ line operator scales reads

$\log{\cal W}_{(p,q)}=-S_{(p,q),{\rm on-shell}}^{1}=\sqrt{\frac{2\pi N}{\sinh
T}(p^{2}+q^{2}-pq\cosh T)}\,.$ (4.4)

In particular for the fundamental Wilson loop we find

$\log{\cal W}=\sqrt{\frac{2\pi N}{\sinh T}}\,.$ (4.5)

The results (4.4) and (4.5) should be viewed as predictions for the
holographic 3d $\mathcal{N}=2$ SCFT. Although we have not been able to
independently and rigorously compute these line operator expectation values
from the dual SCFT, we nevertheless were able to estimate the WL vev using the
matrix model obtained by supersymmetric localization and the Wigner semi-
circle law. We now proceed to briefly describe this calculation.

### 4.2 Field theory

To properly calculate the vev of a supersymmetric Wilson loop for the S-fold
SCFT on $S^{3}$ one needs to apply supersymmetric localization and derive the
analog of the matrix model result in (3.1) for the $S^{3}$ partition function
of the SCFT. We will not pursue this calculation here but will rather draw on
the results of [2] and the calculation of the WL vev in the ABJM theory
studied in [47] to make an educated guess about the result.

We will assume that the Wilson loop that preserves the large amount of
supersymmetry wraps the great circle on $S^{3}$. Following [47] we can write
this WL vev as999In this section we temporarily absorb a factor of $i$ in the
definition of $\sigma^{2}$ as compared to the matrix model in (3.1) in order
to follow standard Gaussian matrix model conventions.

$\langle W_{R}\rangle=\langle\text{Tr}_{R}\,\mathrm{e}^{2\pi\sigma}\rangle\,,$
(4.6)

where $R$ is the representation of the Wilson loop with respect to the gauge
group and $\sigma$ is same variable as in the matrix model in (3.1). We hasten
to add that this expression is a guess and should be treated with care
especially since the S-fold SCFT of interest is non-Lagrangian. We proceed
further with a crude approximation, namely we assume that in the large $N$
limit the eigenvalues $\sigma$ lay on a Wigner semi-circle distribution
specified by the eigenvalue density

$\rho(\sigma)=\frac{2}{\pi\mu^{2}}\sqrt{\mu^{2}-\sigma^{2}}\,,\quad\text{where}\quad-\mu<\sigma<\mu\,,$
(4.7)

where $\mu$ is a positive real number and we have normalized the eigenvalue
density such that

$\int_{-\mu}^{\mu}\rho(\sigma)d\sigma=1\,.$ (4.8)

We stress that this assumption for the eigenvalue density is non-trivial since
the matrix model at hand is not exactly Gaussian. With these assumptions at
hand we can determine the value $\mu$ by computing the expectation value of
$\sigma^{2}$ at large $N$ in two alternative ways. First, we can take two
derivatives of the matrix model partition function in (3.1) with respect to
the coupling. The role of coupling is played by the CS level $k$ which we
promote temporarily to a continuous variable to find

$\langle\sum\limits_{j}\sigma_{j}^{2}\rangle=-\frac{1}{\pi}\frac{\partial}{\partial
k}\log\mathcal{Z}_{N}[T]\approx\frac{N^{2}}{4\pi\sinh T}\,,$ (4.9)

where in the second relation on the right hand side we have taken the large
$N$ limit. Alternatively we can first take the large $N$ limit and then use
the eigenvalue density in (4.7) to find

$\langle\sigma^{2}\rangle=N\int\limits_{-\mu}^{\mu}\mathrm{d}\sigma\rho(\sigma)\sigma^{2}=\frac{\mu^{2}N}{4}\,,$
(4.10)

where the additional factor of $N$ arises from changing from a sum over
eigenvalues to an integral. Comparing the two alternative approaches we
conclude that

$\mu=\sqrt{\frac{N}{\pi\sinh T}}\,.$ (4.11)

To calculate the Wilson line vev in the fundamental representation we now
simply have to evaluate the expectation value of $\mathrm{e}^{2\pi\sigma}$
using the eigenvalue density $\rho(\sigma)$ and the result for $\mu$ above.
The result is simple and reads

$\langle
W_{F}\rangle=\int\limits_{-\mu}^{\mu}\mathrm{d}\sigma\rho(\sigma)\mathrm{e}^{2\pi\sigma}=\frac{1}{2\mu}I_{1}(2\pi\mu)\,,$
(4.12)

where $I_{1}$ is the modified Bessel function of the first kind. Taking a
logarithm and using the large $N$ limit we find that

$\lim\limits_{N\rightarrow\infty}\log\langle W_{F}\rangle\approx
2\pi\mu=\sqrt{\frac{4\pi N}{\sinh T}}\,.$ (4.13)

Despite the guesses we made this schematic calculation leads to a result that
is of the same form as the one obtained by the probe string calculation on the
gravity side and differs only by a simple numerical factor, see (4.5). We
consider this to be a good qualitative check of holography in this setup. It
will certainly be very nice to put this calculation of the WL vev on a more
solid footing by a careful application of supersymmetric localization in the
S-fold SCFT.

## 5 4d $\mathcal{N}=2$ supergravity and S-fold black holes

The goal of this section is to show that for each member of the family of
supersymmetric AdS4 vacua discussed in Section 2 and Appendix B there is a
consistent truncation of the maximal 4d $\mathcal{N}=8$ $[\mathop{\rm
SO}(6)\times\mathop{\rm SO}(1,1)]\ltimes\mathbf{R}^{12}$ gauged supergravity
theory to the minimal 4d $\mathcal{N}=2$ gauged supergravity.

To restrict ourselves to the supersymmetric family of AdS4 vacua we set

$z=-1\,,\quad\chi_{2}=0\,.$ (5.1)

The scalar manifold is then spanned by only two real scalar fields, $x$ and
$\chi_{1}$, that parametrize the coset element

$\mathcal{V}=\,\mathrm{e}^{\chi_{1}\mathfrak{h}_{1}}\cdot\mathrm{e}^{-\log\sqrt{2}\mathfrak{t}_{1}}\cdot\mathrm{e}^{\frac{1}{2}\log\frac{3+x}{4}\mathfrak{t}_{2}}\cdot\mathrm{e}^{\frac{2}{x-3}\frac{\sqrt{1-x}}{\sqrt{3+x}}[\mathfrak{r}_{2},\mathfrak{r}_{3}]}\cdot\mathrm{e}^{-\mathfrak{r}_{2}}\cdot\mathrm{e}^{-\frac{\sqrt{1-x}}{\sqrt{3+x}}\mathfrak{r}_{3}}\,.$
(5.2)

In addition to these two scalar fields the other bosonic fields in the
truncation are the metric and two Abelian vector fields associated to the $\rm
SO(6)$ generators $g_{1}$ and $g_{2}$ in (B.8). Using this information and the
embedding tensor in (B.1) one can utilize the framework described in [48] to
write down the full Lagrangian of the truncated theory. To simplify the end
result we define a complex scalar $\zeta$ such that

$x=\frac{8}{1+\cosh(2\text{Re}\,\zeta)}-3\,,\quad\text{and}\quad\chi_{1}=\text{Im}\,\zeta\,.$
(5.3)

The truncated bosonic Lagrangian then reads

$\mathcal{L}=R-h_{IJ}F^{I}_{\mu\nu}F^{\mu\nu
J}-(\partial_{\zeta}\partial_{\bar{\zeta}}\mathcal{K})\zeta^{\prime}\bar{\zeta}^{\prime}+6\,,$
(5.4)

where the $F^{I}$ are two Maxwell fields and we have rescaled four-dimensional
coordinates to absorb the length scale of the four-dimensional AdS vacuum
$y\mapsto yL_{\text{AdS}}$ where $L_{\text{AdS}}=\sqrt{2}/g$. We have also
defined

$\begin{split}\mathcal{K}&=|\zeta|^{2}-2\log\cosh\text{Re}\,\zeta\,,\quad
h=\left(\begin{matrix}{}1/4&\\\
&\rho(\zeta,\bar{\zeta})/2\end{matrix}\right)\,,\\\
\rho(\zeta,\bar{\zeta})&=\frac{1+\cosh 2\text{Re}\,\zeta}{(1+\cosh
2\text{Re}\,\zeta)\cosh
2\text{Re}\,\zeta+8(\text{Im}\,\zeta)^{2}}\,.\end{split}$ (5.5)

There are two important features of the Lagrangian in (5.6). First, there is
no potential for the scalar fields. This is compatible with the fact that they
are exactly massless and reflects their role as gravitational duals to the
exactly marginal couplings in the dual S-fold SCFT. Second, the kinetic term
of the $F^{1}_{\mu\nu}$ Maxwell field is independent of the scalar fields.
These properties of the truncated Lagrangian ensure that it is consistent with
the equations of motion to set $F^{2}_{\mu\nu}=0$ and the scalars $x$ and
$\chi_{1}$ to an allowed constant value that corresponds to any of the two-
parameter family of supersymmetric AdS4 vacua. Therefore, we conclude that for
any value of $\zeta$ we have a consistent truncation to the following
gravitational action

$S=-\frac{L^{2}_{\text{AdS}}}{16\pi
G_{N}}\int\star_{4}\Big{(}R-\frac{1}{4}F_{\mu\nu}^{1}F^{1\mu\nu}+6\Big{)}\,.$
(5.6)

This is simply the action of the 4d Einstein-Maxwell theory with a negative
cosmological constant which in turn is the bosonic sector of 4d
$\mathcal{N}=2$ minimal gauged supergravity. The prefactor
$L^{2}_{\text{AdS}}$ appears because of the coordinate rescaling performed
above. The existence of this consistent supergravity truncation is compatible
with the general expectation, see [49, 50] and references thereof, that every
supersymmetric $\mathcal{N}=2$ AdS4 vacuum of string or M-theory should give
rise to a consistent truncation to 4d $\mathcal{N}=2$ minimal gauged
supergravity which describes holographically the universal dynamics of the
energy-momentum tensor in the dual 3d $\mathcal{N}=2$ SCFT. The novel feature
of the construction above is that since we have a two-parameter family of AdS4
supersymmetric vacua we are led to a two-parameter space of consistent
truncations to 4d $\mathcal{N}=2$ minimal gauged supergravity.

We can exploit the existence of this 4d $\mathcal{N}=2$ minimal gauged
supergravity truncation in the context of holography and learn about some of
the properties of the dual 3d $\mathcal{N}=2$ family of S-fold SCFTs. In
particular, we find that the free energy of the SCFT placed on a compact
smooth Euclidean 3-manifold $\mathcal{M}_{3}$, $F_{\mathcal{M}_{3}}=-\log
Z_{\mathcal{M}_{3}}$, can be written as

$F_{\mathcal{M}_{3}}=S_{\text{on-shell}}\equiv\frac{\pi
L^{2}_{\text{AdS}}}{2G_{N}}\mathcal{F}\,.$ (5.7)

Here $\mathcal{F}$ is determined by the regularized on-shell action of an
appropriate Euclidean gravitational solution of the equations of motion
obtained from (5.6) which is given by a smooth filling of $\mathcal{M}_{3}$
with an accompanying Maxwell field. Using the same arguments as in [51] we can
evaluate the prefactor to be $\pi L^{2}_{\text{AdS}}/2G_{N}=N^{2}T/2$. Using
this, we find the following prediction for the dual SCFT free energy on
Euclidean 3-manifolds

$F_{\mathcal{M}_{3}}=\frac{N^{2}T}{2}\mathcal{F}\,.$ (5.8)

There are a number of important Euclidean solutions of the 4d $\mathcal{N}=2$
minimal gauged supergravity theory that can be utilized in this context, see
[52, 53] for a recent summary and a more extensive bibliography, together with
the explicit form of the metric and Maxwell field as well as the regularized
on-shell action $\mathcal{F}$.

Below we briefly summarize the results for three examples of particular
interest associated with well-known supersymmetric partition functions.
Notably, the results below are independent of the exactly marginal couplings
of the SCFT which suggests that these deformations should be $Q$-exact with
respect to the localization supercharge.

  * •

One can take $\mathcal{M}_{3}$ to be a particular squashed metric on $S^{3}$
that preserves supersymmetry. The corresponding supersymmetric AdS-Taub-NUT
solution can be found in [53]. Using the results above we then find that 3d
S-fold SCFT free energy on this supersymmetric background reads

$F_{S^{3}_{b}}=\frac{(b+b^{-1})^{2}}{4}\frac{N^{2}T}{2}\,,$ (5.9)

where the right hand side should be understood as the leading term in the
large $N$ limit. One can then use a superconformal Ward identity to obtain the
dynamical coefficient, $C_{T}$, that controls the two-point-function of the
energy momentum tensor of the SCFT at large $N$

$C_{T}=\frac{32}{\pi^{2}}\frac{\partial^{2}F_{S^{3}_{b}}}{\partial
b^{2}}|_{b=1}=\frac{32}{\pi^{2}}N^{2}T\,.$ (5.10)

  * •

Another interesting choice for $\mathcal{M}_{3}$ is the manifold
$S^{1}\times\Sigma_{\mathfrak{g}}$ where $\Sigma_{\mathfrak{g}}$ is a smooth
compact Riemann surface of genus $\mathfrak{g}$. The partition function of the
SCFT on this background is referred to as the topologically twisted index,
reflecting the fact that there is a particular background flux for the
$\mathop{\rm{}U}(1)_{R}$ R-symmetry that implements Witten’s topological twist
on $\Sigma_{\mathfrak{g}}$. Employing the results above we conclude that the
large $N$ topologically twisted index for these S-fold SCFTs should read

$F_{S^{1}\times\Sigma_{\mathfrak{g}}}=(1-\mathfrak{g})\frac{N^{2}T}{2}\,.$
(5.11)

The 4d gravitational solutions dual to this index is the Euclidean Romans
solution which admits a Lorentzian interpretation, when $\mathfrak{g}>1$, as
the extremal supersymmetric Reissner-Nordström black hole in AdS4, see [53]
for more details. The topologically twisted index serves as the counting
function that accounts for the Bekenstein-Hawking entropy of this black hole.

  * •

The final example we consider is the supersymmetric limit of the Euclidean
Kerr-Newman black hole. The 3d SCFT partition function in this case
corresponds to the so-called superconformal index which counts particular
supersymmetric operators in the SCFT. This index can also be computed via a
path integral of the 3d theory on the manifold $S^{1}\times_{\omega}S^{2}$
where $\omega$ is a real number associated with the angular momentum fugacity
on $S^{2}$. Using the known regularized on-shell action for this gravitational
background, see [53] for a summary, we arrive at the following prediction for
the large $N$ superconformal index of the family of 3d $\mathcal{N}=2$ S-fold
SCFTs discussed in this work

$F_{S^{1}\times_{\omega}S^{2}}=\frac{(\omega+1)^{2}}{2\omega}\frac{N^{2}T}{2}\,.$
(5.12)

This result for the index, after an appropriate choice of ensemble and a
Legendre transformation, can also be used to obtain the Bekenstein-Hawking
entropy of the supersymmetric Kerr-Newman black hole.

It will be very interesting to revisit the supersymmetric localization
analysis in [2] and understand whether the path integral of the S-fold SCFT on
these and other Euclidean 3-manifolds can be computed at large $N$. This will
open the door to exploring precision holography in this context and give
access to the observables discussed above beyond the leading order in the
large $N$ approximation.

Acknowledgements

We are grateful to Chris Beem, Junho Hong, Krzysztof Pilch, Valentina Puletti,
and Valentin Reys for fruitful discussions. The research of NB is supported by
the FWO projects G003523N and G094523N, and Odysseus grant G0F9516N, as well
as by the KU Leuven C1 grant ZKD1118 C16/16/005. FFG is supported by the
Icelandic Research Fund under grant 228952-052 and is partially supported by
grants from the University of Iceland Research Fund. JvM is supported by the
ERC-CoG grant NP-QFT No. 864583 “Non-perturbative dynamics of quantum fields:
from new deconfined phases of matter to quantum black holes” and by INFN
Iniziativa Specifica ST&FI.

##  Appendix A The IIB background for $Y\rightarrow 0$

To understand the properties of the conformal manifold of $\mathcal{N}=2$
S-folds we study the geometry in the limit $Y\rightarrow 0$. First we analyze
the limit of the functions defined in (2.12) and find that the useful
quantities that remain are

$\displaystyle(a_{2}^{\pm})^{(0)}=$ $\displaystyle\sqrt{2}(\cos
2\xi_{3}+(1\pm\sqrt{2})(\cos 2\xi_{3}-\sqrt{2}\cos
2\phi)\sin^{2}\theta)+\mathcal{O}(Y)\,,$ (A.1)
$\displaystyle(a_{3}^{\pm})^{(0)}=$ $\displaystyle-4+(2+\pm\sqrt{2}\cos
2\xi_{3})\cos^{2}\theta\mp 2\cos 2\phi\sin^{2}\theta+\mathcal{O}(Y)\,.$

The ten-dimensional warp factor becomes

$\displaystyle\Delta^{(0)}=-32\cos^{2}\theta(\cos^{2}2\xi_{3}\cos^{2}\theta+\sqrt{2}\cos
2\xi_{3}\cos 2\phi\sin^{2}\theta-2)+\mathcal{O}(Y)\,.$ (A.2)

For the metric on the internal space we find that

$\displaystyle(\mathrm{d}s_{5}^{2})^{(0)}=$
$\displaystyle\frac{64}{g^{2}\Delta^{(0)}}\Big{[}2\cos^{4}\theta\Big{(}\tfrac{1}{4}\sin^{2}2\xi_{3}\mathrm{d}\xi_{2}^{2}+(1+\sin^{2}2\xi_{3})\mathrm{d}\xi_{3}^{2}\Big{)}$
(A.3) $\displaystyle+\sin^{2}\theta(2+\sqrt{2}\cos 2\xi_{3}\cos
2\phi\cos^{2}\theta+2\sin^{2}2\phi\sin^{2}\theta)\mathrm{d}\phi^{2}$
$\displaystyle+\sqrt{2}\sin 2\xi_{3}\sin
2\phi\sin^{2}2\theta\,\mathrm{d}\xi_{3}\mathrm{d}\phi$ $\displaystyle+\sin
2\theta\Big{(}\cos^{2}\theta\sin 2\xi_{3}(\cos 2\xi_{3}-\sqrt{2}\cos
2\phi)\mathrm{d}\xi_{3}$ $\displaystyle-\sqrt{2}\sin 2\phi(\sqrt{2}\cos
2\phi\sin^{2}\theta-\cos
2\xi_{3}(1+\sin^{2}\theta))\mathrm{d}\phi\Big{)}\mathrm{d}\theta$
$\displaystyle+2\cos^{2}\theta\Big{(}1-\tfrac{1}{\sqrt{2}}\cos 2\xi_{3}\cos
2\phi(3\sin^{2}\theta+1)$ $\displaystyle\hskip
56.9055pt+(\cos^{2}2\xi_{3}+\cos^{2}2\phi)\sin^{2}\theta\Big{)}\mathrm{d}\theta^{2}\Big{]}+\mathcal{O}(Y)\,.$

The full ten-dimensional metric is then given by

$\mathrm{d}s_{10}^{2}=\frac{(\Delta^{(0)})^{1/4}}{g^{2}}\left[\left(2^{4/3}\mathrm{d}\varphi^{2}\right)\frac{1}{Y^{2}}+\left(2^{7/3}\mathrm{d}s_{\text{AdS}_{4}}^{2}+\frac{1}{2}(\mathrm{d}s_{5}^{2})^{(0)}\right)+\mathcal{O}(Y)\right]\,.$
(A.4)

The important point here is that the only place where the parameter $Y$ shows
up is in front of the S-fold circle parameterized by $\varphi$. For the axion
and dilaton we find that they also become independent of $Y$ in this limit:

$\displaystyle\mathrm{e}^{\hat{\Phi}^{(0)}}=$
$\displaystyle\,\frac{4\mathrm{e}^{2\varphi}(2-\sqrt{2}\cos
2\xi_{3}\cos^{2}\theta-2\cos^{2}\phi\sin^{2}\theta)}{\sqrt{\Delta^{(0)}}}+\mathcal{O}(Y)\,,$
(A.5) $\displaystyle\hat{C}_{0}^{(0)}=$
$\displaystyle\,\frac{\mathrm{e}^{-2\varphi}\sin
2\theta_{4}\sin^{2}\theta}{(2-\sqrt{2}\cos
2\xi_{3}\cos^{2}\theta-2\cos^{2}\phi\sin^{2}\theta)}+\mathcal{O}(Y)\,.$

For the R-R and NS-NS forms we find

$\displaystyle\hat{C}_{2}^{(0)}=$
$\displaystyle\frac{16\mathrm{e}^{-\varphi}}{g^{2}}w\Big{[}(\sqrt{2}+\cos
2\xi_{3})\cos^{2}\theta\sin 2\xi_{3}\mathrm{d}\xi_{2}\wedge\mathrm{d}\xi_{3}$
(A.6)
$\displaystyle+\frac{1}{2}\cot\phi\Big{(}-(1-\sqrt{2})\cos^{2}\xi_{3}((a_{3}^{-})^{(0)}-4\sqrt{2}\sin^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
28.45274pt+(1+\sqrt{2})\sin^{2}\xi_{3}((a_{3}^{-})^{(0)}+4\sqrt{2}\sin^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})\Big{)}\wedge\mathrm{d}\phi$
$\displaystyle+\frac{1}{\sin
2\theta}\Big{(}-(1-\sqrt{2})\cos^{2}\xi_{3}\cos^{2}\theta((a_{2}^{+})^{(0)}+2)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
28.45274pt+(1+\sqrt{2})\cos^{2}\theta\sin^{2}\theta_{3}((a_{2}^{-})^{(0)}+2)\mathrm{d}(\xi_{1}-\xi_{2})\Big{)}\Big{]}+\mathcal{O}(Y)\,,$

$\displaystyle\hat{B}_{2}^{(0)}=$
$\displaystyle\frac{16\mathrm{e}^{\varphi}}{g^{2}}\partial_{\theta_{4}}w\Big{[}-(\sqrt{2}-\cos
2\xi_{3})\cos^{2}\theta\sin 2\xi_{3}\mathrm{d}\xi_{2}\wedge\mathrm{d}\xi_{3}$
(A.7)
$\displaystyle+\frac{1}{2}\tan\phi\Big{(}-(1+\sqrt{2})\cos^{2}\xi_{3}((a_{3}^{+})^{(0)}+4\sqrt{2}\cos^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle\hskip
28.45274pt+(1-\sqrt{2})\sin^{2}\xi_{3}((a_{3}^{+})^{(0)}-4\sqrt{2}\cos^{2}\phi\sin^{2}\theta)\mathrm{d}(\xi_{1}-\xi_{2})\Big{)}\wedge\mathrm{d}\phi$
$\displaystyle+\frac{1}{2}\cot\theta\Big{(}-(1+\sqrt{2})\cos^{2}\xi_{3}(2-(a_{2}^{-})^{(0)})\mathrm{d}(\xi_{1}+\xi_{2})$
$\displaystyle+(1-\sqrt{2})\sin^{2}\theta_{3}(2-(a_{2}^{+})^{(0)})\mathrm{d}(\xi_{1}-\xi_{2})\Big{)}\wedge\mathrm{d}\theta\Big{]}+\mathcal{O}(Y)\,.$

And finally, the four form is

$\displaystyle C_{4}^{(0)}=$ $\displaystyle\frac{16\cos^{2}\theta\sin
2\xi_{3}\sin 2\theta}{g^{4}}\Big{[}\sin 2\theta(\sqrt{2}\cos 2\xi_{3}\cos
2\phi\cos 2\theta+2$ (A.8) $\displaystyle\hskip
28.45274pt-2\cos^{2}2\xi_{3}\cos^{2}\theta)\mathrm{d}\xi_{3}\wedge\mathrm{d}\phi-2\Big{(}\sqrt{2}\cos
2\phi\cos 2\xi_{3}\sin^{2}\theta\mathrm{d}\phi$ $\displaystyle\hskip
28.45274pt-\cos 2\xi_{3}\cos^{2}\theta(\sqrt{2}\sin
2\phi\mathrm{d}\xi_{3}+2\sin
2\xi_{3}\mathrm{d}\phi)\Big{)}\wedge\mathrm{d}\theta\Big{]}\wedge\mathrm{d}\xi_{1}\wedge\mathrm{d}\xi_{2}+\mathcal{O}(Y)\,.$

##  Appendix B Four parameter family of AdS4 solutions

In this appendix we further extend the family of supergravity solutions, dual
to the conformal manifold, to include a fourth independent parameter. As shown
in Section 2.1, the three parameter family of backgrounds is already rather
involved. It is outside the scope of this paper to provide a type IIB
supergravity description of the four-dimensional family of solutions, instead
we will utilize four-dimensional gauged supergravity to construct the full
four-dimensional manifold of solutions. In particular, we will use
$[\mathop{\rm SO}(6)\times\mathop{\rm SO}(1,1)]\ltimes\mathbf{R}^{12}$ gauged,
$\mathcal{N}=8$ supergravity in four dimensions. The gauging is specified by
an embedding tensor ${X_{\mathbb{M}\mathbb{N}}}^{\mathbb{R}}$ [48] where
$\mathbb{M}=1,\cdots,56$. The embedding tensor takes the explicit form

$\displaystyle
X_{[AB][CD]}^{\phantom{[AB][CD]}[EF]}=-X_{[AB]\phantom{[EF]}[CD]}^{\phantom{[AB]}[EF]}=-8\delta_{[A}^{[E}\theta_{B][C}\delta_{D]}^{F]}\,,$
(B.1) $\displaystyle
X^{[AB]\phantom{[EF]}[EF]}_{\phantom{[AB]}[CD]}=-X^{[AB][EF]}_{\phantom{[AB][EF]}[CD]}=-8\delta_{[C}^{[A}\xi^{B][E}\delta_{D]}^{F]}\,,$

where $A,B,\ldots$ are $\mathfrak{sl}(8)$ indices running from $1$ to $8$. For
the $\mathop{\rm SO}(6)\times\mathop{\rm SO}(1,1)\ltimes\mathbf{R}^{12}$
gauging, the $8\times 8$ matrices $\theta$ and $\xi$ take the form

$\theta=g\,\text{diag}(1,1,1,1,1,1,0,0)\,,\quad\xi=g\,\text{diag}(0,0,0,0,0,0,-1,1)\,,$
(B.2)

with $g$ the gauge coupling constants. The scalar sector of this theory
consists of 70 scalars, describing the coset $E_{7(7)}/\mathop{\rm SU}(8)$. We
use the real $\mathop{\rm SL}(8)$ basis of $\text{E}_{7(7)}$ as in [54, 55,
56, 21]. For a given $\mathfrak{sl}(8)$ matrix ${\Lambda_{A}}^{B}$ and four-
form $\Sigma_{ABCD}$, the corresponding $\mathfrak{e}_{7(7)}$ generator can be
constructed as

$\mathfrak{X}_{\mathbb{M}}^{\phantom{\mathbb{M}}\mathbb{N}}(\Lambda,\Sigma)=\left(\begin{matrix}{}2\Lambda_{[A}^{\phantom{[A}[C}\delta_{B]}^{D]}&\Sigma_{[AB][CD]}\\\
\Sigma^{[AB][CD]}&-2\Lambda_{[C}^{\phantom{[C}[A}\delta_{D]}^{B]}\end{matrix}\right)\,.$
(B.3)

Using these generators, a generic element ${\cal V}$ of the coset space can be
parametrized by a product of exponentials. The metric on the coset manifold is
then given by

$\mathcal{M}=\mathcal{V}\cdot\mathcal{V}^{T}\,.$ (B.4)

Using the technology in [48] one can readily compute the Lagrangian associated
to this truncation. In particular, the potential on the scalar manifold takes
the form

$V=\frac{1}{672}\mathcal{M}^{\mathbb{M}\mathbb{P}}X_{\mathbb{M}\mathbb{N}}^{\phantom{\mathbb{M}\mathbb{N}}\mathbb{R}}X_{\mathbb{P}\mathbb{Q}}^{\phantom{\mathbb{P}\mathbb{Q}}\mathbb{S}}(\mathcal{M}^{\mathbb{N}\mathbb{Q}}\mathcal{M}_{\mathbb{R}\mathbb{S}}+7\delta_{\mathbb{S}}^{\mathbb{N}}\delta_{\mathbb{R}}^{\mathbb{Q}})\,,$
(B.5)

and the family of vacua we are after are extrema of this potential, with value

$V=-3g^{2}\,.$ (B.6)

A two-dimensional parameter space of such vacua was found in [6] by using a
$\mathbf{Z}_{2}^{3}$ truncation on the scalar coset. We will use a slightly
different construction here which allows us to extend this solution to a four-
dimensional family of solutions. We start by noting that the form of the
embedding tensor specified in (B.2) implies that the $\mathop{\rm SO}(6)$
gauge group is embedded in $\mathop{\rm SL}(8)$ in the upper left block

$\mathop{\rm SL}(8)\supset\begin{bmatrix}\mathop{\rm SO}(6)&\\\ &\mathop{\rm
SO}(1,1)\end{bmatrix}\,.$ (B.7)

We look for $\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$ invariant critical
points of the potential, where the two $\mathop{\rm{}U}(1)$’s are embedded in
$\mathop{\rm SO}(6)$ as specified by the following generators

$g_{1}=\frac{1}{2}\begin{bmatrix}0&0&0&1&0&0\\\ 0&0&1&0&0&0\\\ 0&-1&0&0&0&0\\\
-1&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{bmatrix}\,,\qquad
g_{2}=\frac{1}{2}\begin{bmatrix}0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ -1&0&0&0&0&0\\\
0&-1&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{bmatrix}\,.$ (B.8)

As discussed in [6] the $\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$
truncation of the four-dimensional theory contains 18 scalar fields
parametrizing the coset

${\cal M}=\frac{\mathop{\rm
SL}(2,\mathbf{R})}{\mathop{\rm{}U}(1)}\times\frac{\mathop{\rm
SO}(4,4)}{\mathop{\rm SO}(4)\times\mathop{\rm SO}(4)}\,.$ (B.9)

The four-dimensional solution we are after is contained in this scalar
manifold, but is difficult to construct analytically. Fortunately there is a
simplification we can employ. Two out of the four parameters come ‘for free’
using the result of [21] because we impose
$\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$ invariance. The results of [21]
state that for any critical point ${\cal V}_{0}$ which is invariant under some
$H\subset\mathop{\rm SO}(6)$ there are $\text{rk}(H)$ flat deformation of the
seed solution ${\cal V}_{0}$ that leave the potential invariant. The flat
deformation can be constructed as follows. Let $g$ be a element of the Cartan
subalgebra of $H$ and embed that into $\mathop{\rm SL}(8)$ as a two-form, then
let

$\Sigma^{(g)}=\mathrm{d}x^{7}\wedge\mathrm{d}x^{8}\wedge g\,,$ (B.10)

here we use $x^{i}$ as coordinates on the auxiliary $\mathbf{R}^{8}$ where the
$\mathop{\rm SL}(8)$ forms live. The $\text{E}_{7(7)}$ generator
$\mathfrak{h}=\mathfrak{X}(0,\Sigma^{(g)})$ is a flat direction in the sense
that ${\cal V}=\mathrm{e}^{\chi\mathfrak{h}}\cdot{\cal V}_{0}$ is a critical
point with the same value of the potential for any $\chi$. For our
$\mathop{\rm{}U}(1)^{2}$ symmetry there will be two such flat directions we
can utilize to simplify our construction. In particular we can impose
additional discrete symmetry and construct a two-dimensional subspace of
solutions, and then generate the remaining two directions as described. We use
the $\mathbf{Z}_{2}$ symmetry embedded in $\mathop{\rm SO}(6)$ as follows

$S=\begin{bmatrix}-1&0&0&0&0&0\\\ 0&-1&0&0&0&0\\\ 0&0&1&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&0&0&-1&0\\\ 0&0&0&0&0&-1\end{bmatrix}\,.$ (B.11)

Truncating further to the invariant subspace of $S$ the scalar manifold
simplifies

$\frac{\mathop{\rm SO}(4,4)}{\mathop{\rm SO}(4)\times\mathop{\rm
SO}(4)}\to\frac{\mathop{\rm SO}(3,3)}{\mathop{\rm SO}(3)\times\mathop{\rm
SO}(3)}\times\mathop{\rm O}(1,1)\,.$ (B.12)

Notice that the $\mathop{\rm SL}(2,\mathbf{R})/\mathop{\rm{}U}(1)$ factor in
(B.9) is left invariant by the discrete symmetry. However we find that the
condition for finding a critical point sets all parameters of that factor to
zero and so we only have to specify the location within the scalar manifold on
the right-hand-side of (B.12).

Even though this manifold is ten-dimensional, the family of critical points
can be specified in terms of only six generators. These are

$\displaystyle\mathfrak{t}_{1}=\mathfrak{X}\big{(}\text{diag}(0,0,0,0,0,1,0,-1),0\big{)}\,,\qquad\mathfrak{t}_{2}=\mathfrak{X}\big{(}\text{diag}(0,0,0,0,0,-1,1,0),0\big{)}\,,$
(B.13)
$\displaystyle\mathfrak{r}_{1}=\mathfrak{X}\big{(}0,g_{1}\wedge\mathrm{d}x^{6}\wedge\mathrm{d}x^{7}\big{)}^{T}\,,\qquad\mathfrak{r}_{2}=\mathfrak{X}\big{(}0,g_{2}\wedge\mathrm{d}x^{6}\wedge\mathrm{d}x^{7}\big{)}^{T}\,,$
$\displaystyle\mathfrak{r}_{3}=\mathfrak{X}\big{(}0,g_{1}\wedge\mathrm{d}x^{5}\wedge\mathrm{d}x^{7}\big{)}^{T}\,,\qquad\mathfrak{r}_{4}=\mathfrak{X}\big{(}0,g_{2}\wedge\mathrm{d}x^{5}\wedge\mathrm{d}x^{7}\big{)}^{T}\,.$

To simplify the expression for the scalar vielbein we define two field-
dependent generators:

$\begin{split}\mathfrak{A}&=-\frac{\sqrt{(1+x)(1-z)}\,\mathfrak{r}_{1}+\sqrt{(1-x)(1+z)}\,\mathfrak{r}_{2}}{\sqrt{4+2x+2z}}\,,\\\
\mathfrak{B}&=\frac{-\sqrt{1-z^{2}}\,\mathfrak{r}_{3}+\sqrt{1-x^{2}}\,\mathfrak{r}_{4}}{\sqrt{(2+x+z)(4+x+z)}}\,.\end{split}$
(B.14)

The two-parameter seed solution can now be specified by the 56-bein

$\displaystyle\mathcal{V}_{0}=$
$\displaystyle\,\mathrm{e}^{-\log\sqrt{2}\mathfrak{t}_{1}}\cdot\mathrm{e}^{\frac{1}{2}\log\frac{4+x+z}{4}\mathfrak{t}_{2}}\cdot\mathrm{e}^{\frac{2}{-2+x+z}[\mathfrak{A},\mathfrak{B}]}\cdot\mathrm{e}^{\mathfrak{A}}\cdot\mathrm{e}^{\mathfrak{B}}\,.$
(B.15)

We have verified that this is a critical point by evaluating the gradients of
the potential in all 70 directions of the maximal supergravity theory. Another
way to verify criticality would be to construct the potential of the
$\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)\times\mathbf{Z}_{2}$ truncation
and demonstrate that the potential is extremized by the above solution.

Before extending this solution to a four-parameter family let us point out a
few of its features. First note that interchanging $x$ and $z$ leads to the
same solution and is therefore a symmetry. The special line of solutions $x=z$
has enhanced $\mathop{\rm SU}(2)\times\mathop{\rm{}U}(1)$ continuous symmetry
which gets enhanced to $\mathop{\rm SO}(5)\times\mathop{\rm SO}(1,1)$ at
$x=z=1$ and it gets enhanced to ${\cal N}=4$ supersymmetry with $\mathop{\rm
SO}(4)$ R-symmetry at $x=z=-1$. Except for this one point the line $x=z$ is
non-supersymmetric. A one-dimensional ${\cal N}=2$ supersymmetric family first
identified in [6] can be obtained by setting

$z=-1\,,\qquad\text{or}\qquad x=-1\,.$ (B.16)

For $z=-1$ the ${\cal N}=2$ R-symmetry is generated by $g_{1}$ whereas for
$x=-1$ it is $g_{2}$ that generates the R-symmetry. Along the ${\cal N}=2$
supersymmetric lines we have two enhancement points. One is the ${\cal N}=4$
supersymmetric point at $x=z=-1$, the other is located at $x=-1$ and $z=1$ or
$x=1$ and $z=-1$. In Section 2.1 we uplifted this supersymmetric family in
addition to the two simple flat directions discussed below. In order to make
contact with section 2.1 the parameters used in this appendix are related to
the parameters used there as follows

$x=2Y^{2}-3\,,\qquad z=-1\,.$ (B.17)

In Figure 1 we display the allowed range of parameters $x$, $z$ and also
indicate the symmetry enhancement points.

\begin{overpic}[width=284.52756pt]{conformalmfd}
\put(53.0,53.0){\Large$\bullet$} \put(55.5,49.0){${\mathcal{N}}=4$}
\put(96.5,96.5){\Large$\bullet$}
\put(101.0,96.0){$\text{SO}(5)\times\text{SO}(1,1)$}
\put(53.0,96.5){\Large$\bullet$} \put(42.0,95.0){$\text{SU}(2)$}
\put(96.5,53.0){\Large$\bullet$} \put(101.0,50.0){$\text{SU}(2)$}
\end{overpic} Figure 1: The allowed region for the two parameters $x$ and $z$.
The solution is symmetric under the interchange of $x$ and $z$ and the dark-
gray diagonal line is the axis of symmetry. Along this axis the continuous
symmetry is enhanced from $\text{U}(1)^{2}$ to
$\text{SU}(2)\times\text{U}(1)$. The ${\mathcal{N}}=2$ supersymmetric lines
are at $z=-1$ or $x=-1$ – both represent the same solution. Other symmetry
enhanced points are marked on the figure.

In order to extend the above solution to a full four-parameter family, we use
the procedure outlined above and write

${\cal
V}=\mathrm{e}^{\chi_{1}\mathfrak{h}_{1}}\cdot\mathrm{e}^{\chi_{2}\mathfrak{h}_{2}}\cdot{\cal
V}_{0}\,,$ (B.18)

where the generators $\mathfrak{h}_{1,2}$ are defined by

$\mathfrak{h}_{1}=\mathfrak{X}\big{(}0,g_{1}\wedge\mathrm{d}x^{7}\wedge\mathrm{d}x^{8}\big{)}\,,\qquad\mathfrak{h}_{2}=\mathfrak{X}\big{(}0,g_{2}\wedge\mathrm{d}x^{7}\wedge\mathrm{d}x^{8}\big{)}\,.$
(B.19)

Even though $x$ and $z$ may be chosen such that ${\cal V}_{0}$ exhibits
enhanced symmetry, the deformation in (B.18) generically brakes the bosonic
symmetry to $\mathop{\rm{}U}(1)\times\mathop{\rm{}U}(1)$. In order to maintain
supersymmetry the parameter $\chi_{i}$ should be set to zero for the
$\mathfrak{h}_{i}$ that corresponds to the R-symmetry. Note that these
$\chi_{i}$ parameters can be directly identified with those introduced in ten
dimensions in Section 2.3.

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