$\mathcal{N}=2$ $\,\textrm{CFT}_{3}\textrm{'s}\,$ from $\,\mathcal{N}=4\,$
gauged supergravity

Miguel Chamorro-Burgos$\,{}^{a}$ , Adolfo Guarino$\,{}^{a,b}$ and Colin
Sterckx$\,{}^{c,a}$

a Departamento de Física, Universidad de Oviedo,

Avda. Federico García Lorca 18, 33007 Oviedo, Spain.

b Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias
(ICTEA)

Calle de la Independencia 13, 33004 Oviedo, Spain.

c Université Libre de Bruxelles (ULB) and International Solvay Institutes,

Service de Physique Théorique et Mathématique,

Campus de la Plaine, CP 231, B-1050, Brussels, Belgium.

###### Contents

  1. 1 Introduction
  2. 2 $\textrm{ISO}(3)\times\textrm{ISO}(3)\,$ half-maximal supergravity
    1. 2.1 A crash course on $\,\mathcal{N}=4\,$ gauged supergravity
    2. 2.2 From $\,[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$ to $\,\textrm{ISO}(3)\times\textrm{ISO}(3)\,$
    3. 2.3 Deforming $\,\textrm{ISO}(3)\times\textrm{ISO}(3)\,$ half-maximal supergravity
    4. 2.4 The gauge algebra of the deformed $\,\mathcal{N}=4\,$ models
  3. 3 $\mathbb{Z}_{2}^{2}$-invariant sector
    1. 3.1 The $\,\mathcal{N}=1\,$ seven-chiral model
    2. 3.2 Warming up: the $\varphi$-family of $\,\mathcal{N}=2\,$ AdS4 solutions of [1]
    3. 3.3 A $\,(\varphi\,;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$ AdS4 solutions
    4. 3.4 $\mathfrak{osp}(2|4)\,$ superconformal multiplets
    5. 3.5 Special loci
    6. 3.6 General axion deformations of the exotic $\mathcal{N}=4\,$ CFT3
  4. 4 $\textrm{U}(1)_{\textrm{R}}$-invariant sector
    1. 4.1 The $\,\mathcal{N}=2\,$ three-vector and two-hyper model
    2. 4.2 Warming up: the $(\varphi,\chi)$-family of $\,\mathcal{N}=2\,$ AdS4 solutions of [1]
    3. 4.3 A $\,(\varphi,\chi\,;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$ AdS4 solutions
    4. 4.4 $\mathfrak{osp}(2|4)\,$ superconformal multiplets
    5. 4.5 Special loci
    6. 4.6 On the conformal manifold of $\,\mathcal{N}=2\,$ CFT3’s
  5. 5 Final remarks
  6. A $\textrm{ISO}(3)\times\textrm{ISO}(3)$ gaugings of half-maximal supergravity
    1. A.1 Quadratic constraints and algebra structure
    2. A.2 $\mathcal{N}=1\,$ superpotentials
    3. A.3 Back to our model

##  1 Introduction

Conformal field theories in three space-time dimensions (CFT3’s) have played
and continue to play a central role in string and M-theory. When endowed with
enough supersymmetry, various non-renormalisation theorems have been stated
making superconformal field theories (SCFT3’s) very interesting from the
viewpoint of the AdS4/CFT3 correspondence. Interacting superconformal field
theories in three space-time dimensions are special because gauge fields are
described by (an up to $\,\mathcal{N}=3\,$ supersymmetric extension of) the
topological Chern–Simons (CS) Lagrangian111The standard $\,F^{2}\,$ term is
dimension $\,4\,$ and therefore is not compatible with scale invariance.

$\mathcal{L}_{\textrm{CS}}=\frac{k}{4\pi}\textrm{Tr}\left[\epsilon^{\mu\nu\rho}\left(A_{\mu}\partial_{\nu}A_{\rho}+\tfrac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right)\right]\
,$ (1.1)

where $\,k\in\mathbb{Z}\,$ is the so-called CS level. It was then shown in [2]
that gauge fields can be consistently coupled to matter fields (scalars and
fermions) carrying new degrees of freedom in a way compatible with
$\,\mathcal{N}=1,2\,$ supersymmetry as well as with conformal symmetry. The
result is a superconformal CS-matter theory with
$\,\textrm{OSp}(\mathcal{N}|4)\,$ superconformal symmetry.

### CS-matter theories in three dimensions and holography

One of the motivations to investigate superconformal CS-matter theories in
three dimensions was to provide a holographic dual to the maximally
supersymmetric $\,\textrm{AdS}_{4}\times\textrm{S}^{7}\,$ solution of eleven-
dimensional (11D) supergravity that describes a stack of $\,N\,$ M2-branes on
flat space. An important step towards constructing such an $\,\mathcal{N}=8\,$
CFT3 was the BLG theory [3, 4, 5, 6] with gauge group
$\,\textrm{G}=\textrm{SO}(4)\sim\textrm{SU}(2)_{k}\times\textrm{SU}(2)_{-k}\,$
and where $\,k\in\mathbb{Z}\,$ is the CS level of the corresponding gauge
factor (see [7] for a study of the realisation of the $\,\textrm{OSp}(8|4)\,$
superconformal symmetry in the BLG theory). However, there was no large
$\,N\,$ limit to be taken on the BLG theory and, therefore, no gravitational
interpretation of the CS level $\,k\,$. These questions cleared up with the
advent of the $\,\mathcal{N}=6\,$ ABJM theory [8]: a CS-matter theory with
gauge group $\,\textrm{G}=\textrm{U}(N)_{k}\times\textrm{U}(N)_{-k}\,$. In the
large $\,N\,$ limit, it describes an
$\,\textrm{AdS}_{4}\times\textrm{S}^{7}/\mathbb{Z}_{k}\,$ orbifold solution of
11D supergravity and is conjectured to feature $\,{\mathcal{N}=8}\,$
supersymmetry enhancement at low CS levels $\,k=1,2\,$ (see [9] for a
generalisation of ABJM theory and its geometrical realisation in 11D
supergravity). Other interesting SCFT3’s with $\,\mathcal{N}=1,2,3\,$ have
appeared in the context of ten-dimensional (10D) massive type IIA supergravity
with gauge groups
$\,\textrm{G}=\textrm{SU}(N)_{k_{1}}\times\textrm{SU}(N)_{k_{2}}\,$ [10] and
$\,\textrm{G}=\textrm{SU}(N)_{k}\,$ [11] including charged matter fields
transforming in different representations of G . On the gravity side, they
correspond to $\,\textrm{AdS}_{4}\times\mathbb{CP}_{3}\,$ and
$\,\textrm{AdS}_{4}\times\textrm{S}^{6}\,$ solutions of massive IIA
supergravity and the Romans mass parameter is related to the CS levels
$\,k_{1}+k_{2}\neq 0\,$ and $\,k\,$, respectively.

SCFT3’s have also been extensively investigated in the context of type IIB
string theory. A precursor of these theories was the three-dimensional planar
interface in four-dimensional $\,{\mathcal{N}=4}\,$ super Yang–Mills [12] dual
to the simple non-supersymmetric Janus solution of type IIB supergravity
constructed in [13]. Supersymmetric $\,\mathcal{N}=1,2,4\,$ three-dimensional
interfaces were constructed in [14] and shown to admit a superconformal limit.
The dual type IIB supersymmetric Janus solutions were constructed in [15, 16],
[17] and [18], respectively. Motivated by the $\,\mathcal{N}=4\,$
superconformal interface and its type IIB dual of the form
$\,\textrm{AdS}_{4}\times\textrm{S}^{2}\times\textrm{S}^{2}\times\Sigma\,$ –
with $\Sigma\,$ being a Riemann surface with coordinates
$\,\eta\in(-\infty,\infty)\,$ and $\,\alpha\in[0,\frac{\pi}{2}]\,$ having the
topology of the infinite strip –, a general class of half-maximal
$\,\mathcal{N}=4\,$ SCFT3’s was put forward in [19, 20] for which brane setups
were provided in terms of D$3$-branes suspended between NS$5$-branes and
D$5$-branes [21]. These brane setups are dual to linear quivers which can be
made circular by taking the $\,\eta\,$ direction to be an $\,\textrm{S}^{1}$,
namely, by changing the topology of $\Sigma\,$ to that of the annulus. In this
context, mirror symmetry [22] between two theories is interpreted as an
equivalence relation between their brane setups: D5-branes and NS5-branes are
exchanged by the inversion element
$\,\mathcal{S}\in\textrm{SL}(2,\mathbb{Z})$. This can be generalised to other
elements of $\,\textrm{SL}(2,\mathbb{Z})$. For example, acting $\,k\,$ times
with the unit-translation element, i.e.
$\,\mathcal{T}^{k}\in\textrm{SL}(2,\mathbb{Z})\,$, leaves a D5-brane invariant
and transforms an NS$5$-brane into a $\,(1,k)$ $5$-brane. This is interpreted
as turning on a CS level $\,k\,$ in the corresponding quiver [23, 20]. Amongst
the class of quivers with a known type IIB brane intersection, the mirror
symmetric $\,\textrm{T}[\textrm{U}(N)]\,$ linear quiver plays a prominent role
[20]. This quiver has an infrared (IR) SCFT3 fixed point, the
$\,\textrm{T}[\textrm{U}(N)]\,$ theory, with a
$\,\textrm{U}(N)\times\textrm{U}(N)\,$ global symmetry, although only one
$\,\textrm{U}(N)\,$ is present in the ultraviolet (UV) Lagrangian description
of the theory. Gauging the diagonal $\,\textrm{U}(N)\,$ subgroup of the
$\,\textrm{U}(N)\times\textrm{U}(N)\,$ global symmetry using an
$\,\mathcal{N}=4\,$ vector multiplet and turning on a CS coupling $\,k$, it
was argued in [24] that a novel class of $\,\mathcal{N}=4\,$ CS-matter
theories, dubbed S-fold CFT3’s, emerges in the IR.

### Holographic duals of S-fold CFT3’s

The gravity dual of the simplest $\,\mathcal{N}=4\,$ S-fold CFT3 was
constructed in [25] upon uplift of an AdS4 solution originally reported in
[26]. It is a type IIB supergravity solution of the form
$\,\textrm{AdS}_{4}\times\textrm{S}^{1}_{\eta}\times\textrm{S}^{5}\,$ where
the $\,\textrm{SO}(6)\,$ isometry group of the $\,\textrm{S}^{5}\,$ is broken
to the $\,\textrm{SO}(3)\times\textrm{SO}(3)\sim\textrm{SO}(4)_{\textrm{R}}\,$
isometry group222We have attached an R-symmetry label $\,{}_{\textrm{R}}\,$ to
$\,\textrm{SO}(4)_{\textrm{R}}\,$ in order to highlight the
$\,\mathcal{N}=4\,$ supersymmetry emerging in the infrared S-fold CFT3. of the
$\,\textrm{S}^{2}\times\textrm{S}^{2}\subset\textrm{S}^{5}$. Therefore, the
S-fold geometry lies within the class of
$\,\textrm{AdS}_{4}\times\textrm{S}^{2}\times\textrm{S}^{2}\times\Sigma\,$
(local) solutions of [18] with an annulus topology for $\,\Sigma\,$. However,
when looping around the $\,\textrm{S}^{1}_{\eta}\subset\Sigma$, the
$\,\mathcal{N}=4\,$ S-fold solution of [25] comes along with a non-trivial
$\,\textrm{SL}(2,\mathbb{Z})\,$ monodromy of hyperbolic type, i.e.
$\,J_{k}=-\mathcal{S}\mathcal{T}^{k}$, for the type IIB supergravity fields
transforming under S-duality (hence the term S-fold). This
$\,\textrm{SL}(2,\mathbb{Z})\,$ monodromy renders the type IIB S-fold solution
non-geometric. Interestingly, the $\,\mathcal{N}=4\,$ S-fold of [25] was first
obtained as a half-maximal AdS4 solution ($16$ supercharges) of an effective
four-dimensional maximal supergravity ($32$ supercharges) with a dyonic
gauging of the group

$\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\subset\textrm{E}_{7(7)}\
,$ (1.2)

where $\,\textrm{E}_{7(7)}\,$ is the duality group of maximal supergravity
[27]. After the AdS4 solution is found in four dimensions, it is
straightforward (but tedious) to uplift it to ten dimensions by implementing a
generalised Scherk-Schwarz reduction [28] of the $\,\textrm{E}_{7(7)}\,$
exceptional field theory ($\textrm{E}_{7(7)}$-ExFT) [29], and then using the
dictionary between $\textrm{E}_{7(7)}$-ExFT and type IIB fields [30]. This
bottom-up procedure has proven very successful and various other type IIB
S-folds have been constructed in this manner [31, 32, 33, 34].

Of particular interest for this work is the $\,\mathcal{N}=2\,$ S-fold with
$\,\textrm{SU}(2)\times\textrm{U}(1)_{\textrm{R}}\,$ symmetry put forward in
[32]. Both the $\,\mathcal{N}=4\,$ and $\,\mathcal{N}=2\,$ S-folds have the
same AdS4 radius and were shown in [1] to be connected by a scalar modulus
$\,\varphi\in\mathbb{R}\,$, namely, a non-compact flat direction in the scalar
potential of the maximal gauged supergravity. In light of the AdS4/CFT3
correspondence, these two solutions are dual to two different points in a
conformal manifold of $\,\mathcal{N}=2\,$ S-fold CFT3’s with the scalar
modulus $\,\varphi\,$ being dual to an exactly marginal deformation [35, 1].
Such a conformal manifold of $\,\mathcal{N}=2\,$ S-fold CFT3’s was shown in
[1] to actually be two-dimensional, with the additional marginal deformation
being dual to another flat direction $\,\chi\,$ of the scalar potential
originally identified in [32]. The Zamolodchikov metric on the conformal
manifold (CM) of $\,\mathcal{N}=2\,$ CFT3’s was computed in [1] using
holography and reads

$ds_{\textrm{CM}}^{2}=\dfrac{1+2\varphi^{2}}{2(1+\varphi^{2})^{2}}\big{(}d\varphi^{2}+(1+\varphi^{2})\,d\chi^{2}\big{)}\
.$ (1.3)

The type IIB uplift of the modulus $\,\chi\,$ in the AdS4 solutions at
$\,\varphi=0\,$ ($\mathcal{N}=2\,$ S-fold) and $\,\varphi=1\,$
($\mathcal{N}=4\,$ S-fold) showed that $\,\chi\,$ is a compact parameter [35,
36]. However, the apparent non-compactness of the modulus $\,\varphi\,$ [1,
37] remains a puzzle and poses some challenges to the CFT distance conjecture
[38]. Further insights on this issue could come from the ten-dimensional
uplift of the $\,\varphi\,$ modulus which has not been worked out yet. Lastly,
$\,\mathcal{N}=1\,$ [39] as well as non-supersymmetric [33] conformal
manifolds of S-fold CFT3’s have also been investigated.

### Plan of this work

We will continue using the effective four-dimensional supergravity approach to
holographically explore the landscape of $\,\mathcal{N}=2\,$ CFT3’s. This
approach allows to compute the conformal dimensions of all the operators in
the $\,\mathcal{N}=2\,$ CFT3’s dual to supergravity excitations of the
corresponding AdS4 solutions. According to the standard AdS4/CFT3
correspondence, the relation between the normalised mass $\,mL\,$ of a
supergravity field of spin $\,[j]\,$ in a given AdS4 solution with radius
$\,L\,$ and the conformal dimension $\,\Delta\,$ of the dual operator in the
CFT3 is given by

$\begin{array}[]{lll}[\frac{3}{2}]&:&mL=\Delta-\frac{3}{2}\ ,\\\\[8.53581pt]
[1]&:&m^{2}L^{2}=(\Delta-2)(\Delta-1)\ ,\\\\[8.53581pt]
[\frac{1}{2}]&:&mL=\Delta-\frac{3}{2}\ ,\\\\[8.53581pt]
[0]&:&m^{2}L^{2}=\Delta(\Delta-3)\ ,\end{array}$ (1.4)

with the graviton $\,[j]=[2]\,$ being massless. Moreover, using three-
dimensional $\,\mathcal{N}=2\,$ representation theory, the set of operators
can be arranged into superconformal multiplets of the
$\,\mathfrak{osp}(2|4)\,$ superconformal algebra.

We will concentrate on the holographic study of $\,\mathcal{N}=2\,$ CFT3’s
with a potential type IIB gravity dual of the form
$\,\textrm{AdS}_{4}\times\textrm{S}^{2}\times\textrm{S}^{2}\times\Sigma\,$.
Firstly, in order to be general but still keep control over the effective
supergravity, we will consider four-dimensional supergravity Lagrangians
preserving half-maximal supersymmetry ($16$ supercharges). Secondly, in order
to be able to recover the conformal manifold of $\,\mathcal{N}=2\,$ S-fold
CFT3’s constructed in the context of maximal supergravity [1], we will couple
the half-maximal supergravity multiplet to six vector multiplets so that the
field content of the half-maximal supergravity forms a subset of the (unique)
field content of maximal supergravity. Thirdly, in order to be compatible with
a potential uplift to a type IIB solution of the form
$\,\textrm{AdS}_{4}\times\textrm{S}^{2}\times\textrm{S}^{2}\times\Sigma\,$,
the gauging must contain an $\,\textrm{SO}(3)\times\textrm{SO}(3)\,$ factor.
It becomes then natural to investigate gaugings of the group

$\textrm{G}_{\textrm{half-
max}}=\textrm{ISO}(3)\times\textrm{ISO}(3)\subset\textrm{SL}(2)\times\textrm{SO}(6,6)\
,$ (1.5)

with $\,\textrm{ISO}(3)=\textrm{SO}(3)\ltimes\mathbb{R}^{3}\,$, as well as
their possible group-theoretical embeddings (1.5) into the
$\,\textrm{SL}(2)\times\textrm{SO}(6,6)\,$ duality group of half-maximal
supergravity coupled to six vector multiplets. The most general such
embeddings turns out to depend on eight parameters. However, a full analysis
of the eight-parameter family of gaugings in (1.5) goes beyond the scope of
this work.

In this work we will simplify the setup by choosing the same group-theoretical
embedding for the two $\,\textrm{ISO}(3)\,$ factors in (1.5). This reduces the
number of embedding parameters down to four. Turning on three out of the four
embedding parameters produces new multi-parametric families of AdS4 solutions,
all of them with the same AdS4 radius. Interestingly, for any value of the
three embedding parameters, the AdS4 solutions still feature two scalar moduli
$\,(\varphi,\chi)\in\mathbb{R}\,$ associated with non-compact flat directions
in the scalar potential. By adjusting the embedding parameters and the scalar
moduli $\,(\varphi,\chi)\,$ in the AdS4 solutions, these can preserve
$\,\mathcal{N}=2\,$ ($8$ supercharges), $\,\mathcal{N}=3\,$ ($12$
supercharges) or $\,\mathcal{N}=4\,$ ($16$ supercharges) supersymmetry. Via
the gauge/gravity duality, these AdS4 solutions are conjectured to be dual to
new classes of strongly-coupled $\,\mathcal{N}=2\,$, $\,\mathcal{N}=3\,$ or
$\,\mathcal{N}=4\,$ CFT3’s provided an embedding in string theory (yet to be
worked out) exists.

Our results point at the existence of a web of $\,\mathcal{N}=2\,$ CFT3’s
containing a special “line” of $\,\mathcal{N}=3\,$ supersymmetry enhancement
which, in turn, contains isolated “points” where supersymmetry gets enhanced
to $\,\mathcal{N}=4\,$. We will characterise this web of $\,\mathcal{N}=2\,$
CFT3’s by arranging their set of low lying operators into superconformal
multiplet of $\,\mathfrak{osp}(2|4)$, and by further discussing the phenomenon
of supermultiplet shortening that occurs at the special loci where
supersymmetry gets enhanced to $\,\mathcal{N}=3\,$ or $\,\mathcal{N}=4\,$.
Regarding the latter, two isolated points describing $\,\mathcal{N}=4\,$
CFT3’s are identified. The first one describes the $\,\mathcal{N}=4\,$ S-fold
CFT3 of [24, 25] dual to the $\,\mathcal{N}=4\,$ AdS4 solution of the maximal
supergravity with gauge group (1.2).333Note that $\,\textrm{G}_{\textrm{half-
max}}\subset\textrm{G}_{\textrm{max}}\,$. This makes it possible to rediscover
an AdS4 solution of a maximal supergravity as an AdS4 solution of a half-
maximal supergravity provided certain relations between the couplings in the
half-maximal supergravity Lagrangian hold (see eq.(2.8)). The second one
describes a novel $\,\mathcal{N}=4\,$ CFT3 – we will refer to it as the exotic
$\,\mathcal{N}=4\,$ CFT3 – dual to an $\,\mathcal{N}=4\,$ AdS4 solution of a
half-maximal supergravity with gauge group (1.5).444This $\,\mathcal{N}=4\,$
AdS4 solution is “maximally” supersymmetric within the theory it belongs to:
it preserves the $16$ supercharges of the half-maximal supergravity where it
lives. This AdS4 solution appeared originally in [40] where it was argued to
be a non-geometric solution still admitting a locally geometric type IIB
description. We now see that it is actually connected to the locally geometric
$\,\mathcal{N}=4\,$ S-fold solution of type IIB supergravity [25], at least at
the effective four-dimensional supergravity level. As a by-product, we will
also present an additional set of generically non-supersymmetric marginal
deformations of the exotic $\,\mathcal{N}=4\,$ CFT3 which resemble (without
being the same) the axion-like deformations of S-folds introduced in [35, 36,
41, 34].

The paper is organised as follows. In Section 2 we review half-maximal gauged
supergravity in four dimensions and present a simple class of
$\,\textrm{ISO}(3)\times\textrm{ISO}(3)\,$ gaugings which depends on a
specific deformation parameter $\,\tilde{\varphi}\,$. In Section 3 we
construct the $\mathbb{Z}_{2}^{2}$-invariant sector of the theory and present
a $\,(\varphi;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$ AdS4 solutions
that incorporates the $\,\varphi\,$ modulus dual to one of the two marginal
operators spanning the conformal manifold of $\,\mathcal{N}=2\,$ S-fold
CFT3’s. These AdS4 solutions are holographically conjectured to describe a web
of $\,\mathcal{N}=2\,$ CFT3’s whose spectrum of low lying operators is
arranged into superconformal multiplets of $\,\mathfrak{osp}(2|4)$.
Supersymmetry as well as flavour symmetry enhancements are discussed together
with the corresponding shortening of superconformal multiplets. In Section 4
we construct the $\textrm{U}(1)_{\textrm{R}}$-invariant sector of the theory
in order to also incorporate the modulus $\,\chi\,$ dual to the second
marginal operator compatible with $\,\mathcal{N}=2\,$ supersymmetry in the
dual CFT3’s. We present a $\,(\varphi,\chi;\tilde{\varphi})$-family of
$\,\mathcal{N}=2\,$ AdS4 solutions that generalises the results of [1] to the
context of half-maximal supergravity, and arrange the spectrum of low lying
operators of the dual $\,\mathcal{N}=2\,$ CFT3’s into superconformal
multiplets of $\,\mathfrak{osp}(2|4)$. After a suitable treatment of vector
fields and gauge redundancies in the $\textrm{U}(1)_{\textrm{R}}$-invariant
sector, the Zamolodchikov metric on the conformal manifold of such
$\,\mathcal{N}=2\,$ CFT3’s is shown to still be (1.3) for arbitrary values of
the half-maximal deformation parameter $\,\tilde{\varphi}\,$. We conclude in
Section 5 with some implications and potential applications of the results
presented in this work. Appendix A discusses more general gaugings of
$\,\textrm{ISO}(3)\times\textrm{ISO}(3)\,$ in half-maximal supergravity.

##  2 $\textrm{ISO}(3)\times\textrm{ISO}(3)\,$ half-maximal supergravity

As stated in the introduction, our starting point is the four-dimensional
maximal ($\mathcal{N}=8$) supergravity with gauge group

$\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\subset\textrm{E}_{7(7)}\
,$ (2.1)

put forward in [25], and shown to accommodate various classes of
$\,\textrm{AdS}_{4}\,$ solutions uplifting to S-fold backgrounds of type IIB
supergravity [25, 31, 32, 33, 34]. Following the prescription in [42], we will
mod out this maximal supergravity by a $\,\mathbb{Z}_{2}\,$ discrete group to
produce a very specific half-maximal ($\mathcal{N}=4$) supergravity with gauge
group

$\textrm{G}_{\textrm{half-
max}}=\textrm{ISO}(3)\times\textrm{ISO}(3)\subset\textrm{SL}(2)\times\textrm{SO}(6,6)\
.$ (2.2)

Lastly, we will introduce a specific deformation of such a half-maximal
supergravity – which we parameterise in terms of a continuous parameter
$\,\tilde{\varphi}\in\mathbb{R}\,$ – and characterise it from an algebraic
perspective.

###  2.1 A crash course on $\,\mathcal{N}=4\,$ gauged supergravity

We frame our work within the context of half-maximal $\,\mathcal{N}=4\,$
gauged supergravity in four dimensions [43]. The ungauged theory features a
global duality group $\,\mathcal{G}=\textrm{SL}(2)\times\textrm{SO}(6,n)\,$
where $\,n\,$ is the number of vector multiplets to which the supergravity
multiplet is coupled. Promoting a subgroup $\,\textrm{G}\subset\mathcal{G}\,$
from global to local, i.e. performing a so-called gauging, the duality group
$\,\mathcal{G}\,$ is generically broken and only the local gauge symmetry G is
left in the gauged supergravity. Still, the commutant (if any) of G inside
$\,\mathcal{G}\,$ remains as a global symmetry of the theory after the gauging
procedure.

In this work we will consider the case $\,n=6\,$. This is the largest value
for which the duality group of half-maximal supergravity can be embedded into
the one of maximal supergravity, i.e.
$\,\textrm{SL}(2)\times\textrm{SO}(6,6)\subset\textrm{E}_{7(7)}\,$, and the
half-maximal supergravity Lagrangian can be embedded into the maximal one
provided certain quadratic constraints on the couplings in the theory hold
(see eq.(2.8) below). We investigate a specific class of gaugings of the group

$\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\ ,$ (2.3)

which is embedded in the duality group as

$\textrm{G}\,\,\subset\,\,\textrm{SO}(3,3)_{1}\times\textrm{SO}(3,3)_{2}\,\,\subset\,\,\textrm{SL}(2)\times\textrm{SO}(6,6)\
,$ (2.4)

where we have attached labels $\,{}_{1}\,$ and $\,{}_{2}\,$ in order to keep
track of each independent $\,\textrm{ISO}(3)\,$ and $\,\textrm{SO}(3,3)\,$
factor in (2.3) and (2.4). General classes of gaugings of
$\,\textrm{G}\subset\textrm{SO}(3,3)_{1}\times\textrm{SO}(3,3)_{2}\,$ in half-
maximal supergravity [44, 45, 46] have been extensively investigated in the
past, for example, with the aim of charting the landscape of flux
compactifications [47, 40].

In the duality-covariant formulation of [43], the bosonic field content of the
half-maximal supergravity consists of the metric $\,g_{\mu\nu}\,$, $\,12\,$
(electric) plus $\,12\,$ (magnetic) vector fields $\,A_{\mu}{}^{\alpha M}$,
and $2+36$ scalar fields $\,\phi\textrm{'s}\,$ serving as coordinates in the
coset space geometry

$\mathcal{M}_{\textrm{scal}}=\frac{\textrm{SL}(2)}{\textrm{SO}(2)}\times\frac{\textrm{SO}(6,6)}{\textrm{SO}(6)\times\textrm{SO}(6)}\
,$ (2.5)

and being parameterised by a coset representative $\,\mathcal{V}(\phi)\,$.
Already at this level we have introduced a fundamental $\,\textrm{SL}(2)\,$
index $\,\alpha=\pm\,$ as well as a fundamental $\,\textrm{SO}(6,6)\,$ index
$\,M$. These are raised/lowered using the $\,\epsilon_{\alpha\beta}\,$ and
$\,\eta_{MN}\,$ invariant tensors of $\,\textrm{SL}(2)\,$ and
$\,\textrm{SO}(6,6)$, respectively.

Having set the bosonic field content of the theory, all the interactions
compatible with $\,{\mathcal{N}=4}\,$ supersymmetry and induced by gaugings of
the type555The gauge group
$\,\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ is a particular
example within the larger class of
$\,\textrm{G}=\textrm{CSO}(p,q,r)\times\textrm{CSO}(p^{\prime},q^{\prime},r^{\prime})\,$
gaugings investigated in [46, 48]. The case of $\,\textrm{ISO}(3)\,$
corresponds to $\,{(p,q,r)=(3,0,1)}\,$ or $\,{(p,q,r)=(0,3,1)}\,$ (and
equivalently for the primed factor). (2.4) are encoded in a so-called
embedding tensor. This tensor comes along with an index structure $\,f_{\alpha
MNP}=f_{\alpha[MNP]}\,$ and thus lives in the $\,(\textbf{2},\textbf{220})\,$
irreducible representation (irrep) of $\,\mathcal{G}\,$. It also specifies the
non-Abelian gauge structure of the half-maximal gauged supergravity, namely,

$[\,T_{\alpha M}\,,\,T_{\beta N}\,]=f_{\alpha MN}{}^{P}\,T_{\beta P}\ ,$ (2.6)

where $\,T_{\alpha M}\,$ are the generators of $\,\mathcal{G}\,$ that couple
to the vector fields $\,A_{\mu}{}^{\alpha M}\,$ in the gauge connection.
Consistency of the gauging procedure requires a set of quadratic constraints
on the embedding tensor of the form [43]

$(\mathbf{3},\mathbf{495}):\,\,\,f_{\alpha[MN}{}^{R}\,f_{\beta PQ]R}=0\hskip
22.76219pt\textrm{ and }\hskip
22.76219pt(\mathbf{1},\mathbf{66}+\mathbf{2079}):\,\,\,\epsilon^{\alpha\beta}\,f_{\alpha
MN}{}^{R}\,f_{\beta PQR}=0\ ,$ (2.7)

where we have included the irrep of $\,\mathcal{G}\,$ where each constraint
lives. In addition, there are two additional constraints given by

$(\mathbf{3},\mathbf{1}):\,\,\,f_{\alpha MNP}\,f_{\beta}{}^{MNP}=0\hskip
22.76219pt\textrm{ and }\hskip
22.76219pt(\mathbf{1},\mathbf{462^{\prime}}):\,\,\,\left.\epsilon^{\alpha\beta}\,f_{\alpha[MNP}\,f_{\beta
QRS]}\right|_{\textrm{SD}}=0\ ,$ (2.8)

where SD refers to the self-dual projection of the $\,\textrm{SO}(6,6)\,$ six-
form. These two additional constraints (2.8) are not required by
$\,\mathcal{N}=4\,$ supersymmetry but must hold for the half-maximal
Lagrangian to be embeddable into an $\,\mathcal{N}=8\,$ maximal gauged
supergravity [42]. Whenever these two additional constraints are satisfied,
the half-maximal supergravity with gauging (2.3) can be viewed as a subsector
of the maximal gauged supergravity with gauging

$\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\subset\textrm{E}_{7(7)}\
.$ (2.9)

This maximal supergravity has recently received a lot of attention due to its
connection to S-fold solutions in type IIB string theory as originally noticed
in [25].

As a consequence of the gauging procedure, the fermions in the theory develop
scalar-dependent mass terms and supersymmetry requires to introduce a non-
trivial scalar potential. This is given by

$\begin{array}[]{lll}V&=&\frac{1}{64}\,f_{\alpha MNP}\,f_{\beta
QRS}M^{\alpha\beta}\left[\frac{1}{3}\,M^{MQ}\,M^{NR}\,M^{PS}+\left(\frac{2}{3}\,\eta^{MQ}-M^{MQ}\right)\eta^{NR}\eta^{PS}\right]\\\\[8.53581pt]
&-&\frac{1}{144}\,\epsilon^{\alpha\beta}\,f_{\alpha MNP}\,f_{\beta
QRS}\,M^{MNPQRS}\ ,\end{array}$ (2.10)

where

$M_{\alpha\beta}=\frac{1}{\textrm{Im}z_{7}}\left(\begin{array}[]{cc}|z_{7}|^{2}&\textrm{Re}z_{7}\\\
\textrm{Re}z_{7}&1\end{array}\right)\in\textrm{SL}(2)\ ,$ (2.11)

encodes a complex scalar $\,z_{7}\,$ spanning the
$\,\textrm{SL}(2)/\textrm{SO}(2)\,$ factor in (2.5). Together with this, the
potential depends on additional scalars spanning the
$\,\textrm{SO}(6,6)/(\textrm{SO}(6)\times\textrm{SO}(6))\,$ factor in (2.5).
These are $\,36\,$ real scalars which can be assembled in a matrix

$M_{MN}=\left(\begin{array}[]{cc}G^{-1}&-G^{-1}\,B\\\
B\,G^{-1}&G-B\,G^{-1}\,B\end{array}\right)\in\textrm{SO}(6,6)\ ,$ (2.12)

where $\,G\,$ and $\,B\,$ are arbitrary symmetric and anti-symmetric
$\,6\times 6\,$ matrices accounting for $\,21\,$ and $\,15\,$ scalars,
respectively. For the class of gaugings in (2.3)-(2.4), the kinetic terms for
the scalar fields serving as coordinates on the scalar geometry (2.5) are
constructed using standard coset techniques and read

$\mathcal{L}_{\textrm{kin}}=\frac{1}{8}\,\partial_{\mu}M_{\alpha\beta}\,\partial^{\mu}M^{\alpha\beta}+\frac{1}{16}\,D_{\mu}M_{MN}\,D^{\mu}M^{MN}\
,$ (2.13)

where $\,M^{\alpha\beta}\,$ and $\,M^{MN}\,$ are the inverse of
$\,M_{\alpha\beta}\,$ in (2.11) and $\,M_{MN}\,$ in (2.12), respectively. In
this work we will focus on maximally symmetric AdS4 solutions of the theory so
that vector fields will be set to zero in the general covariant derivatives

$D_{\mu}M_{MN}=\partial_{\mu}M_{MN}+2\,A_{\mu}{}^{\alpha P}\,f_{\alpha
P(M}{}^{Q}\,M_{N)Q}\ .$ (2.14)

Lastly, the scalar potential (2.10) depends on a specific
$\,\textrm{SO}(6,6)\,$ six-form666Due to the $\,\epsilon_{mnpqrs}\,$ tensor
with $\,{m,n,\ldots=1,\ldots,6}\,$ in the definition of the
$\,\textrm{SO}(6,6)\,$ six-form (2.15), the components
$\,\mathcal{V}_{M}{}^{n}\,$ entering (2.15) must be extracted from the coset
representative $\,\mathcal{V}_{M}{}^{N}\,$ using a Lorentzian basis (for the
column index $\,N\,$) of $\,\textrm{SO}(6,6)\,$ where
$\,\eta_{MN}=\textrm{diag}(-\mathbb{I}_{6},\mathbb{I}_{6})\,$.

$M_{MNPQRS}=\epsilon_{mnpqrs}\,\mathcal{V}_{M}{}^{m}\,\mathcal{V}_{N}{}^{n}\,\mathcal{V}_{P}{}^{p}\,\mathcal{V}_{Q}{}^{q}\,\mathcal{V}_{R}{}^{r}\,\mathcal{V}_{S}{}^{s}\
,$ (2.15)

that is constructed from the
$\,\textrm{SO}(6,6)/(\textrm{SO}(6)\times\textrm{SO}(6))\,$ coset
representative $\,\mathcal{V}_{M}{}^{N}\,$ such that
$\,{M_{MN}=(\mathcal{V}\,\mathcal{V}^{t})_{MN}}\,$ (see [43] for more
details).

The scalar potential (2.10) is a complicated function of the $\,2+36\,$
scalars spanning (2.5). Trying to chart the landscape of $\,\mathcal{N}=2\,$
supersymmetric AdS4 solutions by direct extremisation of (2.10) is out of
computational reach, so we will resort to two simpler setups where only
$\,2+12\,$ scalars are kept and the rest are set to zero. Following the
original idea in [49], this will be done in a group-theoretical consistent
manner by retaining only those scalars that are invariant (singlets) under
specific residual symmetry groups. In particular, we will consider discrete
$\,\mathbb{Z}_{2}^{2}\,$ and continuous $\,\textrm{U}(1)_{\textrm{R}}\,$
subgroups of G as such residual symmetry groups. In this manner, whenever an
extremum is found in the simplified setup, it is guaranteed that it
corresponds to an actual extremum of the scalar potential of the full theory.
Moreover, although we will find extrema of the scalar potential in the setups
with $\,2+12\,$ scalars, we will provide the full mass spectrum for all the
bosonic and fermionic fields in half-maximal supergravity using [43, 50]. This
supergravity spectrum maps to the spectrum of operators in the would-be dual
CFT3’s.

###  2.2 From
$\,[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$ to
$\,\textrm{ISO}(3)\times\textrm{ISO}(3)\,$

Starting from the
$\,\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$
gauged maximal supergravity of [25], and modding it out by a discrete subgroup
$\,\mathbb{Z}_{2}\subset\textrm{G}_{\textrm{max}}\,$ [42], one is left with a
very specific gauging

$\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\ ,$ (2.16)

of half-maximal supergravity. In order to describe the resulting half-maximal
supergravity, it will prove convenient to first perform a light-cone splitting
$\,M=(m,\bar{m})\,$ with respect to the $\,\textrm{SO}(6,6)\,$ invariant
metric

$\eta_{MN}=\left(\begin{array}[]{cc}0&\delta_{m\bar{n}}\\\
\delta_{\bar{m}n}&0\end{array}\right)\hskip 22.76219pt\textrm{ with }\hskip
22.76219ptm=1,\ldots,6\hskip 11.38109pt\textrm{ , }\hskip
11.38109pt\bar{n}=\bar{1},\ldots,\bar{6}\ ,$ (2.17)

and then a further splitting $\,m=(a,i)\,$ and $\,\bar{m}=(\bar{a},\bar{i})\,$
with $\,a=1,3,5\,$ and $\,i=2,4,6\,$. In this manner, the original
$\,\textrm{SO}(6,6)\,$ fundamental index $\,M\,$ has a decomposition

$\begin{array}[]{ccccccc}\textrm{SO}(6,6)&\supset&\textrm{SO}(3,3)_{1}&\times&\textrm{SO}(3,3)_{2}\\\\[5.69054pt]
M&\rightarrow&(\,a\,,\,\bar{a}\,)&\oplus&(\,i\,,\,\bar{i}\,)\end{array}$
(2.18)

and the $\,\textrm{ISO}(3)_{1,2}\,$ factors in (2.16) are embedded into
$\,\textrm{SO}(3,3)_{1,2}\sim\textrm{SL}(4)_{1,2}\,$, respectively. An
explicit computation of the resulting embedding tensor $\,f_{\alpha MNP}\,$
specifying the half-maximal supergravity yields (using conventions in [40])

$\begin{array}[]{lclclc}f_{+\bar{a}bc}=2\,g&\hskip 14.22636pt,&\hskip
14.22636ptf_{-abc}=\pm\,2\,g\,c&,\\\\[5.69054pt]
f_{-i\bar{j}\bar{k}}=2\,g&\hskip 14.22636pt,&\hskip
14.22636ptf_{+\bar{i}\bar{j}\bar{k}}=\pm\,2\,g\,c&,\end{array}$ (2.19)

where $\,g\,$ is the gauge coupling and $\,c\,$ is a parameter encoding the
dyonic nature of the gauging, and with all the other components vanishing. In
what follows we are assuming a cyclic structure in all the embedding tensor
components of the same type, i.e.
$\,f_{+\bar{1}35}=f_{+\bar{3}51}=f_{+\bar{5}13}=2\,g\,$, etc., so we are
intentionally omitting epsilon symbols in (2.19) to lighten the notation.
Lastly, since the embedding tensor in (2.19) is the result of halving the
$\,\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$
gauging of maximal supergravity, it automatically satisfies the extra
constraints in (2.8) for a half-maximal supergravity to be embeddable in a
maximal supergravity.

###  2.3 Deforming $\,\textrm{ISO}(3)\times\textrm{ISO}(3)\,$ half-maximal
supergravity

Following [48] (in the conventions of [40]), we will deform the
$\,\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gauging
specified by (2.19) while preserving the $\,\mathcal{N}=4\,$ supersymmetry of
half-maximal supergravity. We are doing so by activating two additional
embedding tensor components

$f_{+abc}\hskip 22.76219pt\textrm{ and }\hskip
22.76219ptf_{-\bar{i}\bar{j}\bar{k}}\ .$ (2.20)

As discussed in [48, 40], turning on these two components modifies how the
gauge group $\,{\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}}\,$ is
embedded into the duality group, see (2.4). We will change this embedding in a
parametrically controlled manner yielding a one-parameter generalisation of
the gauging in (2.19).

Let us denote $\,\tilde{\varphi}\,$ the new parameter entering the embedding
tensor, which now has components

$\begin{array}[]{lclclc}f_{+\bar{a}bc}=\frac{2\,\sqrt{2}\,g}{\sqrt{1+\tilde{\varphi}^{2}}}&\hskip
5.69054pt,&\hskip 5.69054ptf_{-abc}=\pm
2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}}{\sqrt{1+\tilde{\varphi}^{2}}}&\hskip
5.69054pt,&\hskip
5.69054ptf_{+abc}=-2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}^{2}-1}{\tilde{\varphi}^{2}+1}&,\\\\[11.38109pt]
f_{-i\bar{j}\bar{k}}=\frac{2\,\sqrt{2}\,g}{\sqrt{1+\tilde{\varphi}^{2}}}&\hskip
5.69054pt,&\hskip 5.69054ptf_{+\bar{i}\bar{j}\bar{k}}=\pm
2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}}{\sqrt{1+\tilde{\varphi}^{2}}}&\hskip
5.69054pt,&\hskip
5.69054ptf_{-\bar{i}\bar{j}\bar{k}}=-2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}^{2}-1}{\tilde{\varphi}^{2}+1}&,\end{array}$
(2.21)

and, as we will see in a moment, accommodates a rich structure of new AdS4
vacua. The class of gaugings in (2.21) automatically solves the quadratic
constrains in (2.7) required by half-maximal supersymmetry. However, an
explicit computation of the additional quadratic constraints in (2.8) yields

$f_{\alpha MNP}\,f_{\beta}{}^{MNP}=0\hskip 28.45274pt\textrm{ and }\hskip
28.45274pt\left.\epsilon^{\alpha\beta}\,f_{\alpha[MNP}\,f_{\beta
QRS]}\right|_{\textrm{SD}}\propto
g^{2}c\,\dfrac{\tilde{\varphi}^{2}-1}{\left(\tilde{\varphi}^{2}+1\right)^{\frac{3}{2}}}\
.$ (2.22)

As a result, due to the violation of the constraint living in the
$\,(\mathbf{1},\mathbf{462^{\prime}})\,$ irrep, the deformed theories do not
admit an uplift to maximal supergravity unless $\,\tilde{\varphi}^{2}=1\,$.
Note that, at $\,\tilde{\varphi}^{2}=1\,$, the embedding tensor in (2.21)
consistently reduces to the one in (2.19) and the theory becomes a
$\,\mathbb{Z}_{2}$-invariant subsector of the
$\,\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$
maximal supergravity.

###  2.4 The gauge algebra of the deformed $\,\mathcal{N}=4\,$ models

It is instructive to take a closer look at how the precise $\,\mathcal{N}=4\,$
gauging specified by (2.21) is realised at an algebraic level in terms of the
generators

$T_{\alpha M}=\left(\,T_{\alpha a}\,,\,T_{\alpha\bar{a}}\,\,\,;\,\,\,T_{\alpha
i}\,,\,T_{\alpha\bar{i}}\,\right)$ (2.23)

entering the commutation relations (2.6). We will show that each of the
$\,\textrm{ISO}(3)_{1,2}\,$ factors is at a different $\,\textrm{SL}(2)\,$
angle in the spirit of [51]. Moreover, each $\,\textrm{ISO}(3)_{1,2}\,$ factor
by itself involves a non-trivial $\,\textrm{SO}(3,3)_{1,2}\,$ angle (both
unbar/bar generators are present) in the spirit of [48].

Before discussing its gauge structure in detail, it is also worth noticing
that the class of half-maximal
$\,\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings
specified by the embedding tensor (2.21) depends on three arbitrary parameters
$\,(g,c,\tilde{\varphi})\,$. However the most general class of gaugings of
$\,\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ in half-maximal
supergravity involves eight parameters (up to gauge fixing) and is discussed
in detail in Appendix A. The study of the structure of AdS4 solutions in this
more general class of models goes beyond the scope of this work and is
postponed for the future.

The algebraic realisation of the
$\,\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings
specified by (2.21) involves a set of $\,12\,$ independent generators. For
example, choosing them to be $\,(T_{+a},T_{+\bar{a}})\,$ and
$\,(T_{-i},T_{-\bar{i}})\,$, the antisymmetry of the brackets (2.6) further
sets

$T_{-a}=c\,\tilde{\varphi}\,\,T_{+\bar{a}}\hskip 14.22636pt,\hskip
14.22636ptT_{+\bar{i}}=c\,\tilde{\varphi}\,\,T_{-i}\hskip 14.22636pt,\hskip
14.22636ptT_{-\bar{a}}=T_{+i}=0\ ,$ (2.24)

and the independent generators satisfy non-trivial commutation relations of
the form

$\begin{array}[]{llll}\left[\,T_{+a},T_{+b}\,\right]&=&2\sqrt{2}\,g\,\left(\dfrac{1}{\sqrt{1+\tilde{\varphi}^{2}}}\,\epsilon_{ab}{}^{c}\,T_{+c}+c\,\dfrac{1-\tilde{\varphi}^{2}}{1+\tilde{\varphi}^{2}}\,\epsilon_{ab}{}^{\bar{c}}\,\,T_{+\bar{c}}\right)\
,\\\\[11.38109pt]
\left[\,T_{+a},T_{+\bar{b}}\,\right]&=&2\sqrt{2}\,g\,\dfrac{1}{\sqrt{1+\tilde{\varphi}^{2}}}\,\epsilon_{a\bar{b}}{}^{\bar{c}}\,T_{+\bar{c}}\
,\\\\[11.38109pt] \left[\,T_{+\bar{a}},T_{+\bar{b}}\,\right]&=&0\
,\end{array}$ (2.25)

for the $\,\textrm{ISO}(3)_{1}\,$ factor in the gauge group and, similarly,

$\begin{array}[]{llll}\left[\,T_{-\bar{i}},T_{-\bar{j}}\,\right]&=&2\sqrt{2}\,g\,\left(\dfrac{1}{\sqrt{1+\tilde{\varphi}^{2}}}\,\epsilon_{\bar{i}\bar{j}}{}^{\bar{k}}\,T_{-\bar{k}}+\,c\,\dfrac{1-\tilde{\varphi}^{2}}{1+\tilde{\varphi}^{2}}\,\epsilon_{\bar{i}\bar{j}}{}^{k}\,T_{-k}\right)\
,\\\\[11.38109pt]
\left[\,T_{-\bar{i}},T_{-j}\,\right]&=&2\sqrt{2}\,g\,\dfrac{1}{\sqrt{1+\tilde{\varphi}^{2}}}\,\epsilon_{\bar{i}j}{}^{k}\,T_{-k}\
,\\\\[11.38109pt] \left[\,T_{-i},T_{-j}\,\right]&=&0\ ,\end{array}$ (2.26)

for the $\,\textrm{ISO}(3)_{2}\,$ factor. There is the limiting case
$\,\tilde{\varphi}\rightarrow\pm\infty\,$ for which the embedding tensor
(2.21) stays regular and the $\,\textrm{ISO}(3)_{1,2}\,$ factors become
nilpotent. More concretely, they reduce to the nilpotent algebra denoted
$\,n(3.5)\,$ in Table $4$ of [52]. The drastic change in the four-dimensional
algebra structure at $\,\tilde{\varphi}\rightarrow\pm\infty\,$ suggests a
drastic change in the interpretation of the corresponding supergravity
solutions as well as of their possible uplifts to ten or eleven dimensions.

##  3 $\mathbb{Z}_{2}^{2}$-invariant sector

We will look at the $\,\mathbb{Z}_{2}^{2}$-invariant sector of half-maximal
supergravity. Being an invariant sector with respect to a (in this case
discrete) $\,\mathbb{Z}_{2}^{2}\,$ subgroup of G , an extremum of the scalar
potential in this simplified setup automatically implies an extremum of the
full scalar potential.

###  3.1 The $\,\mathcal{N}=1\,$ seven-chiral model

The $\,\mathbb{Z}_{2}^{2}$-invariant sector of half-maximal supergravity was
investigated in [53]. It can be recast as a minimal $\,\mathcal{N}=1\,$
supergravity coupled to seven chiral multiplets. We will denote $\,z_{I}\,$,
with $\,I=1,\ldots,7\,$, the seven complex scalars in the chiral multiplets.
One of them, we choose it to be $\,z_{7}\,$, is the one parameterising the
$\,M_{\alpha\beta}\in\textrm{SL}(2)\,$ element in (2.11). The remaining six
complex fields specify the $\,G\,$ and $\,B\,$ matrices in (2.12) from which
the $\,M_{MN}\in\textrm{SO}(6,6)\,$ element is constructed. More concretely,

$G=\left(\begin{array}[]{ccc}G_{1}&0&0\\\ 0&G_{2}&0\\\ 0&0&G_{3}\\\
\end{array}\right)\hskip 22.76219pt\textrm{ and }\hskip
22.76219ptB=\left(\begin{array}[]{ccc}B_{1}&0&0\\\ 0&B_{2}&0\\\ 0&0&B_{3}\\\
\end{array}\right)$ (3.1)

are block-diagonal matrices with components

$G_{i}=\frac{\textrm{Im}z_{i+3}}{\textrm{Im}z_{i}}\left(\begin{array}[]{cc}1&\textrm{Re}z_{i}\\\
\textrm{Re}z_{i}&|z_{i}|^{2}\end{array}\right)\hskip 11.38109pt,\hskip
11.38109ptB_{i}=\left(\begin{array}[]{cc}0&\textrm{Re}z_{i+3}\\\
-\textrm{Re}z_{i+3}&0\end{array}\right)\hskip 11.38109pt,\hskip
11.38109pti=1,2,3\ ,$ (3.2)

that depend on the complex scalars $\,z_{1},\ldots,z_{6}\,$. The scalar
kinetic terms for this sector of the theory take the form

$\mathcal{L}_{\textrm{kin}}=-\frac{1}{4}\,\sum_{I=1}^{7}\left(\,d\varphi_{I}^{2}+e^{2\varphi_{I}}\,d\chi_{7}^{2}\,\right)\hskip
22.76219pt\textrm{ with }\hskip 22.76219ptz_{I}=-\chi_{I}+i\,e^{-\varphi_{I}}\
.$ (3.3)

The scalar manifold invariant under the $\,\mathbb{Z}_{2}^{2}\,$ discrete
symmetry is therefore identified with the special Kähler (SK) factorised
geometry

$\mathcal{M}^{{}^{\mathbb{Z}_{2}^{2}}}_{\textrm{scal}}=\left[\frac{\textrm{SL}(2)}{\textrm{SO}(2)}\right]^{7}\subset\,\frac{\textrm{SL}(2)}{\textrm{SO}(2)}\times\frac{\textrm{SO}(6,6)}{\textrm{SO}(6)\times\textrm{SO}(6)}\
.$ (3.4)

In the $\,\mathbb{Z}_{2}^{2}$-invariant sector of half-maximal supergravity,
the scalar potential takes a lengthy but more tractable expression in terms of
the seven complex scalars $\,z_{I}\,$. We will find families of
$\,\mathcal{N}=2\,$ supersymmetric AdS4 extrema analytically in this setup.

###  3.2 Warming up: the $\varphi$-family of $\,\mathcal{N}=2\,$ AdS4
solutions of [1]

The half-maximal gauged supergravity specified by the undeformed
($\tilde{\varphi}=\pm 1$) embedding tensor (2.19) possesses a one-parameter
$\varphi$-family of $\,\mathcal{N}=2\,$ supersymmetric AdS4 solutions.777These
solutions were originally constructed in [1] within the context of
$\,\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$
maximal gauged supergravity. Two additional moduli fields $\,\chi_{1,2}\,$
dual to marginal operators and dubbed axion-like flat deformations in [41]
were also identified for this family of AdS4 solutions. In an appropriate
basis, the combination $\,\chi\equiv\chi_{1}-\chi_{2}\,$ preserves
$\,\mathcal{N}=2\,$ supersymmetry whereas the orthogonal combination breaks
supersymmetry completely. We will come back to the axion-like flat deformation
$\,\chi\,$ in Section 4 when exploring a
$\,\textrm{U}(1)_{\textrm{R}}$-invariant sector of the theory. This
$\varphi$-family of solutions lies at the loci

$z_{1}=-\bar{z}_{3}=i\,c\,\sqrt{\frac{1+\varphi^{2}}{2}}\hskip
5.69054pt,\hskip 5.69054ptz_{2}=i\,c\hskip 5.69054pt,\hskip
5.69054ptz_{4}=-\bar{z}_{6}=\dfrac{-\varphi+i}{\sqrt{1+\varphi^{2}}}\hskip
5.69054pt,\hskip 5.69054ptz_{5}=z_{7}=\dfrac{\mp 1+i}{\sqrt{2}}\ ,$ (3.5)

where the $\,\mp\,$ sign in (3.5) is correlated with the $\,\pm\,$ sign in
(2.19). Having $\,\textrm{Im}z_{1,2,3}>0\,$ then requires $\,c>0\,$. The
vacuum energy turns out to be independent of $\,\varphi\,$ and given by

$V_{0}=-3\,g^{2}\,c^{-1}\ .$ (3.6)

The AdS4 solutions in (3.5) preserve $\,\mathcal{N}=2\,$ supersymmetry and a
$\,\textrm{U}(1)_{\textrm{R}}\,$ residual symmetry at generic values of
$\,\varphi\,$ within half-maximal supergravity888$\mathcal{N}=2$ supersymmetry
and $\,\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)_{\textrm{F}}\,$ symmetry
in the maximal theory without the $\mathbb{Z}_{2}$ projection. To set up
notation, we have attached labels ${}_{\textrm{R}}$ and ${}_{\textrm{F}}$ to
identify the corresponding R-symmetry and flavour symmetry groups in the
would-be dual SCFT3’s. However, and even though the vacuum energy in (3.6)
does not depend on $\,\varphi\,$, there are two special values of
$\,\varphi\,$ at which (super) symmetry enhancements occur:

  * •

Point $\,\varphi=0\,$: At this value the AdS4 solution preserves
$\,\mathcal{N}=2\,$ supersymmetry and a
$\,\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)_{\textrm{F}}\,$ symmetry
within half-maximal supergravity.999$\mathcal{N}=2$ supersymmetry and
$\,\textrm{U}(1)_{\textrm{R}}\times\textrm{SU}(2)_{\textrm{F}}\,$ symmetry in
the maximal theory without the $\mathbb{Z}_{2}$ projection.

  * •

Point $\,\varphi=\pm 1\,$: At these values the AdS4 solution preserves
$\,\mathcal{N}=3\,$ supersymmetry and an $\,\textrm{SO}(3)_{\textrm{R}}\,$
symmetry within the half-maximal supergravity.101010$\mathcal{N}=4$
supersymmetry and $\,\textrm{SO}(4)_{\textrm{R}}\,$ symmetry in the maximal
theory without the $\mathbb{Z}_{2}$ projection.

The AdS4 solutions at $\,\varphi=0\,$ and $\,\varphi=\pm 1\,$ uplift to the
$\,\mathcal{N}=2\,$ and $\,\mathcal{N}=4\,$ type IIB S-folds constructed in
[32] and [25], respectively. From a four-dimensional perspective, the
parameter $\,\varphi\,$ appears to be non-compact. From a higher-dimensional
perspective, the compactness of $\,\varphi\,$ remains an open question since
the type IIB uplift of the entire $\,\varphi$-family of AdS4 solutions in
(3.5) has not been constructed yet.

#### Marginal deformation and $\mathfrak{osp}(2|4)\,$ superconformal
multiplets

Being a flat direction in the scalar potential, $\,\varphi\,$ was identified
with a marginal deformation specifying a direction in an $\,\mathcal{N}=2\,$
conformal manifold of S-fold CFT3’s [1].111111The apparent non-compactness of
$\,\varphi\,$ (see [37] for a KK approach to this question) poses some
challenges to the CFT Distance Conjecture [38]. Interestingly, there are
unprotected operators in the $\,\mathcal{N}=2\,$ S-fold CFT3’s whose conformal
dimensions depend on $\,\varphi\,$.

According to the AdS4/CFT3 correspondence, the mass spectrum of the full set
of half-maximal supergravity fields at the $\,\mathcal{N}=2\,$ AdS4 solutions
in (3.5) can be arranged into multiplets of the $\,\mathfrak{osp}(2|4)\,$
superconformal symmetry of the dual $\,\mathcal{N}=2\,$ CFT3’s. Following the
notation121212For the $\,\mathcal{N}=2\,$ supermultiplets in three dimensions,
our conventions for the Lorentz and R-symmetry Dynkin labels differ from the
one in [54]:
$\,j=\frac{1}{2}j_{\tiny{\cite[cite]{[\@@bibref{}{Cordova:2016emh}{}{}]}}}\,$
and $\,R=r_{\tiny{\cite[cite]{[\@@bibref{}{Cordova:2016emh}{}{}]}}}\,$. of
[54] for a superconformal multiplet $\,[j]_{\Delta}^{R}\,$, where $\,j\,$ and
$\,R\,$ are the Lorentz and R-symmetry Dynkin labels of the highest weight
state (HWS) in the multiplet and $\,\Delta\,$ is its conformal dimension, the
spectrum contains five unprotected long multiplets

$L\bar{L}[0]^{0}_{\Delta_{1}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[0]^{0}_{\tilde{\Delta}_{\pm}}\ ,$ (3.7)

with conformal dimensions given by

$\begin{array}[]{rcl c
rcll}\Delta_{1}&=&\tfrac{1}{2}+\tfrac{1}{2}\sqrt{\tfrac{17+33\varphi^{2}}{1+\varphi^{2}}}&\hskip
14.22636pt,&\hskip 14.22636pt\Delta_{\pm}&=&\tfrac{1}{2}+\tfrac{2+(\varphi\pm
1)\varphi}{\sqrt{2(1+\varphi^{2})}}&,\\\\[11.38109pt]
\tilde{\Delta}_{-}&=&\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1+8\varphi^{2}}&\hskip
14.22636pt,&\hskip
14.22636pt\tilde{\Delta}_{+}&=&\tfrac{1}{2}+\tfrac{1}{2}\sqrt{\tfrac{17+\varphi^{2}}{1+\varphi^{2}}}&.\end{array}$
(3.8)

In addition there are one short and two semi-short protected multiplets with
integer conformal dimension $\,\Delta=2\,$, namely,

$A_{1}\bar{A}_{1}[1]_{2}^{0}\hskip 14.22636pt,\hskip
14.22636ptL\bar{B}_{1}[0]_{2}^{2}\hskip 14.22636pt,\hskip
14.22636ptB_{1}\bar{L}[0]_{2}^{-2}\ ,$ (3.9)

where $\,A_{1}\bar{A}_{1}[1]^{0}_{2}\,$ is the stress-energy tensor multiplet
of the $\,\mathcal{N}=2\,$ CFT3’s.

The multiplets in (3.7)-(3.9) describe a $\,\mathbb{Z}_{2}$-invariant subset
of the spectrometry performed in [1] within the context of the
$\,\textrm{G}_{\textrm{max}}=[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$
maximal supergravity. The two semi-short multiplets
$\,L\bar{B}_{1}[0]_{2}^{2}\,$ and $\,B_{1}\bar{L}[0]_{2}^{-2}\,$ in (3.9)
contain the two real marginal operators investigated in [1]. They are
$\,\mathbb{Z}_{2}$-even enabling us to capture them also within the context of
half-maximal supergravity. The scalar modulus $\,\varphi\,$ in (3.10) is dual
to one such marginal operators. The other marginal operator is dual to a
different modulus $\,\chi\,$ that will be studied in detail in Section 4. Last
but not least, the spectrum in (3.7)-(3.9) does not contain a
$\,\mathfrak{u}(1)_{\textrm{F}}\,$ flavour current multiplet
$\,A_{2}\bar{A}_{2}[0]_{1}^{0}\,$ present in [1]. This multiplet is
$\,\mathbb{Z}_{2}$-odd and therefore projected away when truncating from
maximal to half-maximal supergravity. This fact has some consequences we touch
upon in the conclusions.

###  3.3 A $\,(\varphi\,;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$ AdS4
solutions

Let us now consider the effect of turning on the deformation parameter
$\,\tilde{\varphi}\,$, i.e. $\,\tilde{\varphi}\neq\pm 1\,$ in the embedding
tensor (2.21). As already anticipated, turning on this parameter produces new
$\,\mathcal{N}=2\,$ supersymmetric AdS4 solutions which can still be found
analytically.

At generic values of the deformation parameter $\,\tilde{\varphi}\,$, the
locus of the $\,\mathcal{N}=2\,$ AdS4 solutions in (3.5) changes to

$z_{1}=-\bar{z}_{3}=i\,c\,\sqrt{\dfrac{1+\varphi^{2}}{1+\tilde{\varphi}^{2}}}\hskip
5.69054pt,\hskip 5.69054ptz_{2}=i\,c\hskip 5.69054pt,\hskip
5.69054ptz_{4}=-\bar{z}_{6}=\dfrac{-\varphi+i}{\sqrt{1+\varphi^{2}}}\hskip
5.69054pt,\hskip
5.69054ptz_{5}=z_{7}=\dfrac{\mp\tilde{\varphi}+i}{\sqrt{1+\tilde{\varphi}^{2}}}\
,$ (3.10)

with the $\,\mp\,$ sign in (3.10) being correlated with the $\,\pm\,$ sign in
(2.21). Notice that having $\,\textrm{Im}z_{1,2,3}>0\,$ still requires
$\,c>0\,$, and also the reflection symmetry
$\,\tilde{\varphi}\rightarrow-\tilde{\varphi}\,$. The vacuum energy turns out
to be independent of the embedding tensor deformation $\,\tilde{\varphi}\,$
and, therefore, still given by

$V_{0}=-3\,g^{2}\,c^{-1}\ .$ (3.11)

Taking the limit $\,\tilde{\varphi}\rightarrow\pm\infty\,$ becomes
pathological as $\,\textrm{Im}z_{1,3,5,7}=0\,$ hinting at some
decompactification regime. This decompactification regime resonates well with
the fact that taking $\,\tilde{\varphi}\rightarrow\pm\infty\,$ changes the
gauge group to a new one being nilpotent (see discussion below (2.25)-(2.26)).

###  3.4 $\mathfrak{osp}(2|4)\,$ superconformal multiplets

The full half-maximal supergravity spectrum at this
$\,(\varphi\,;\tilde{\varphi})$-family of AdS4 solutions can be arranged into
multiplets of the $\,\mathfrak{osp}(2|4)\,$ superconformal symmetry of the
dual $\,\mathcal{N}=2\,$ CFT3’s. The spectrum contains five unprotected long
multiplets

$L\bar{L}[0]^{0}_{\Delta_{1}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[0]^{0}_{\tilde{\Delta}_{\pm}}\ ,$ (3.12)

with conformal dimensions given by

$\begin{array}[]{lcl}\Delta_{1}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{9+25\tilde{\varphi}^{2}+\varphi^{2}\left(17+49\tilde{\varphi}^{2}\right)}{\left(1+\varphi^{2}\right)\left(1+\tilde{\varphi}^{2}\right)}}\
,\\\\[17.07164pt]
\Delta_{\pm}&=&1+\frac{1}{2}\sqrt{\frac{4\varphi^{4}+\varphi^{2}\left(13\tilde{\varphi}^{2}+9\right)+4\tilde{\varphi}^{4}+9\tilde{\varphi}^{2}+5}{\left(\varphi^{2}+1\right)\left(\tilde{\varphi}^{2}+1\right)}-\frac{4\left(\varphi^{2}+\tilde{\varphi}^{2}+1\right)\pm
8\varphi^{3}\tilde{\varphi}\pm
4\varphi\tilde{\varphi}\left(2\tilde{\varphi}^{2}+2-\sqrt{\left(\varphi^{2}+1\right)\left(\tilde{\varphi}^{2}+1\right)}\right)}{\sqrt{\left(\varphi^{2}+1\right)\left(\tilde{\varphi}^{2}+1\right)}}}\
,\\\\[17.07164pt]
\tilde{\Delta}_{\pm}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{5\left(1+\tilde{\varphi}^{2}\right)+8\left(\varphi^{4}+\tilde{\varphi}^{4}\right)+\varphi^{2}\left(\tilde{\varphi}^{2}+9\right)\pm
4\sqrt{4\varphi^{8}+8\varphi^{6}-4\varphi^{4}\left(\tilde{\varphi}^{4}+3\tilde{\varphi}^{2}-1\right)-4\varphi^{2}\tilde{\varphi}^{2}\left(3+\tilde{\varphi}^{2}\right)+\left(1+\tilde{\varphi}^{2}+2\tilde{\varphi}^{4}\right)^{2}}}{\left(1+\varphi^{2}\right)\left(1+\tilde{\varphi}^{2}\right)}}\
.\end{array}$ (3.13)

In addition, there are one short and two semi-short protected multiplets with
integer conformal dimension $\,\Delta=2\,$. These are the same as in (3.9),
namely,

$A_{1}\bar{A}_{1}[1]^{0}_{2}\hskip 14.22636pt,\hskip
14.22636ptL\bar{B}_{1}[0]^{2}_{2}\hskip 14.22636pt,\hskip
14.22636ptB_{1}\bar{L}[0]^{-2}_{2}\ ,$ (3.14)

where $\,A_{1}\bar{A}_{1}[1]^{0}_{2}\,$ is the stress-energy tensor multiplet
of the dual $\,\mathcal{N}=2\,$ CFT3’s.

###  3.5 Special loci

The $\,(\varphi\,;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$ AdS4
solutions in (3.10) generically preserves a $\,\textrm{U}(1)_{\textrm{R}}\,$
symmetry. The latter can be seen from the normalised gravitino masses which
are given by

$\begin{array}[]{lll}mL&=&1\,\,\,\,(\times 2)\hskip 14.22636pt,\hskip
14.22636pt\dfrac{1+\varphi^{2}+\tilde{\varphi}^{2}\pm\varphi\,\tilde{\varphi}}{\sqrt{\left(1+\varphi^{2}\right)\left(1+\tilde{\varphi}^{2}\right)}}\
.\end{array}$ (3.15)

As a result, the marginal deformation $\,\varphi\,$ turns out to be compatible
with the embedding tensor deformation $\,\tilde{\varphi}\,$. Moreover, they
both enter the AdS4 solutions (3.10) and the normalised gravitino masses
(3.15) in a very symmetric fashion.

A detailed inspection of the normalised gravitino masses in (3.15) singles out
four special cases to be further investigated:

$\begin{array}[]{lllll}i)&\varphi=\tilde{\varphi}=0&\Rightarrow&mL=1\,\,\,\,(\times
4)&,\\\\[5.69054pt] ii)&\varphi=\pm\tilde{\varphi}\neq
0&\Rightarrow&mL=1\,\,\,\,(\times
3)\,\,\,\,,\,\,\,\,3-\dfrac{2}{1+\tilde{\varphi}^{2}}&,\\\\[11.38109pt]
iii)&\varphi=0&\Rightarrow&mL=1\,\,\,\,(\times
2)\,\,\,\,,\,\,\,\,\sqrt{1+\tilde{\varphi}^{2}}\,\,\,\,(\times
2)&,\\\\[8.53581pt] iv)&\tilde{\varphi}=0&\Rightarrow&mL=1\,\,\,\,(\times
2)\,\,\,\,,\,\,\,\,\sqrt{1+\varphi^{2}}\,\,\,\,(\times 2)&.\end{array}$ (3.16)

Note that the case $\,i)\,$ sits at the intersection of the one-dimensional
slicings $\,ii)\,$, $\,iii)\,$ and $\,iv)\,$. A diagram of the
$\,(\varphi\,;\tilde{\varphi})$-family of AdS4 solutions in (3.10) is shown in
Figure 1. In the figure, and in the rest of the work, we have denoted by
$\,\mathcal{N}\,\&\,\textrm{G}_{0}\,$ the number $\,\mathcal{N}\,$ of four-
dimensional supersymmetries and the residual symmetry group
$\,\textrm{G}_{0}\,$ of a given AdS4 solution.

####  3.5.1 $\mathcal{N}=3\,$ line of supersymmetry enhancement

The two involutions $\,\varphi=\pm\tilde{\varphi}\,$ respectively yield
$\,\Delta_{\mp}=\frac{3}{2}\,$ so that the corresponding long multiplet in
(3.12) hits the unitary bound. We will set $\,\varphi=\tilde{\varphi}\,$ (red
dashed line in Figure 1) without loss of generality by virtue of the
reflection symmetry $\,\tilde{\varphi}\rightarrow-\tilde{\varphi}\,$ of (3.10)
and (3.13).

The conformal dimensions in (3.13) reduce in this case to131313One has that
$\,\tilde{\Delta}_{-}=2\,$ and
$\,\tilde{\Delta}_{+}=3-\tfrac{2}{1+\tilde{\varphi}^{2}}\,$ for
$\,|\tilde{\varphi}|\geq 1\,$ whereas $\,\tilde{\Delta}_{+}=2\,$ and
$\,\tilde{\Delta}_{-}=3-\tfrac{2}{1+\tilde{\varphi}^{2}}\,$ for
$\,|\tilde{\varphi}|\leq 1\,$.

$\Delta_{1}=4-\tfrac{2}{1+\tilde{\varphi}^{2}}\hskip 11.38109pt,\hskip
11.38109pt\Delta_{-}=\tfrac{3}{2}\hskip 11.38109pt,\hskip
11.38109pt\Delta_{+}=\tfrac{7}{2}-\tfrac{2}{1+\tilde{\varphi}^{2}}\hskip
11.38109pt,\hskip 11.38109pt\tilde{\Delta}_{-}=2\hskip 11.38109pt,\hskip
11.38109pt\tilde{\Delta}_{+}=3-\tfrac{2}{1+\tilde{\varphi}^{2}}\ .$ (3.17)

As a result, the long multiplet $\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{-}}\,$
hits the unitarity bound and splits into one short and two semi-short
multiplets. The multiplets in (3.12) then reduce to

$\begin{array}[]{rcl}L\bar{L}[0]^{0}_{\Delta_{1}}&\to&L\bar{L}[0]^{0}_{4-\frac{2}{1+\tilde{\varphi}^{2}}}\\\\[17.07164pt]
L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{-}}\,\,\oplus\,\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{+}}&\to&\left[A_{1}\bar{A_{1}}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,\,\oplus\,\,A_{2}\bar{L}[0]^{-1}_{2}\,\,\oplus\,\,L\bar{A_{2}}[0]^{1}_{2}\right]\,\,\oplus\,\,L\bar{L}[\tfrac{1}{2}]^{0}_{\frac{7}{2}-\frac{2}{1+\tilde{\varphi}^{2}}}\\\\[17.07164pt]
L\bar{L}[0]_{\tilde{\Delta}_{-}}^{0}\,\,\oplus\,\,L\bar{L}[0]_{\tilde{\Delta}_{+}}^{0}&\to&L\bar{L}[0]_{2}^{0}\,\,\oplus\,\,L\bar{L}[0]_{3-\frac{2}{1+\tilde{\varphi}^{2}}}^{0}\end{array}$
(3.18)

The enhancement to $\,\mathcal{N}=3\,$ supersymmetry originates from the short
multiplet $\,A_{1}\bar{A_{1}}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,$ in (3.18)
which contains a massless gravitino.

[1][25][32][40]

Figure 1: Web of $\,\mathcal{N}=2\,$ CFT3’s dual to the
$\,(\varphi\,;\tilde{\varphi})$-family of AdS4 solutions in (3.10). CFT3’s at
$\,{\tilde{\varphi}=1}\,$ (dotted blue line) are dual to AdS4 solutions of the
$\,[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$ maximal
supergravity and have been studied and characterised in [1]. The green and
blue/red circles have a type IIB dual: the S-fold backgrounds in [32] and
[25], respectively. The $\mathcal{N}=4\,$ CFT3 sitting at
$\,\varphi=\tilde{\varphi}=0\,$ (red star) is dual to an exotic AdS4 solution
originally presented in [40] and classified as non-geometric therein. A
generic CFT3 in the diagram features
$\,\mathcal{N}=2\,\&\,\textrm{U}(1)_{\textrm{R}}\,$ symmetry.

Alternatively, the mass spectrum in (3.14) and (3.18) can be arranged into
unitary superconformal multiplets of the $\,\mathfrak{osp}(3|4)\,$ algebra.
Following closely the notation141414Our convention for the Lorentz and
R-symmetry Dynkin labels differs from the one in [54]:
$\,j=\frac{1}{2}j_{\tiny{\cite[cite]{[\@@bibref{}{Cordova:2016emh}{}{}]}}}\,$
and
$\,R=\frac{1}{2}R_{\tiny{\cite[cite]{[\@@bibref{}{Cordova:2016emh}{}{}]}}}\,$.
of [54], we find a set of multiplets

$L[0]_{3-\frac{2}{1+\tilde{\varphi}^{2}}}^{0}\hskip 14.22636pt,\hskip
14.22636ptB_{1}[0]_{2}^{2}\hskip 14.22636pt,\hskip
14.22636ptA_{1}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\ ,$ (3.19)

with $\,A_{1}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,$ corresponding to the stress-
energy multiplet of the dual $\,\mathcal{N}=3\,$ CFT3. The unprotected long
multiplet in (3.19) is simply a rearrangement of the
$\,\tilde{\varphi}$-dependent long multiplets in (3.18).

####  3.5.2 $\mathcal{N}=4\,$ points of supersymmetry enhancement

There are two isolated points in the space of conformal dimensions (3.17) at
which $\,\Delta_{1}\,$ and $\,\tilde{\Delta}_{\pm}\,$ corresponding to a
$\,[j]=[0]\,$ HWS are integer valued whereas $\,\Delta_{\pm}\,$ corresponding
to a $\,[j]=[\frac{1}{2}]\,$ HWS are half-integer valued.151515There is also
the limit $\,\tilde{\varphi}\rightarrow\pm\infty\,$ for which the gauging
changes drastically pointing at a decompactification regime (see discussion
below (2.25)-(2.26)). Studying this limit goes beyond the scope of this work.
Let us look at these two special points in more detail.

#####  $\bullet$ Point $\,\varphi=\tilde{\varphi}=1\,$:

At this point (red/blue circle in Figure 1), the conformal dimensions in
(3.17) simplify to

$\Delta_{1}=3\hskip 14.22636pt,\hskip 14.22636pt\Delta_{-}=\tfrac{3}{2}\hskip
14.22636pt,\hskip 14.22636pt\Delta_{+}=\tfrac{5}{2}\hskip 14.22636pt,\hskip
14.22636pt\tilde{\Delta}_{\pm}=2\ .$ (3.20)

Therefore, there are no further long multiplets in (3.18) hitting the
unitarity bound. Instead, they simply reduce to

$\begin{array}[]{rcl}L\bar{L}[0]^{0}_{\Delta_{1}}&\to&L\bar{L}[0]^{0}_{3}\\\\[11.38109pt]
L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{-}}\,\,\oplus\,\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{+}}&\to&\left[A_{1}\bar{A_{1}}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,\,\oplus\,\,A_{2}\bar{L}[0]^{-1}_{2}\,\,\oplus\,\,L\bar{A_{2}}[0]^{1}_{2}\right]\,\,\oplus\,\,L\bar{L}[\tfrac{1}{2}]^{0}_{\frac{5}{2}}\\\\[17.07164pt]
L\bar{L}[0]_{\tilde{\Delta}_{-}}^{0}\,\,\oplus\,\,L\bar{L}[0]_{\tilde{\Delta}_{+}}^{0}&\to&L\bar{L}[0]_{2}^{0}\,\,\,\,(\times
2)\end{array}$ (3.21)

The multiplets in (3.14) and (3.21) precisely describe the
$\mathbb{Z}_{2}$-even sector of the
$\,{\mathcal{N}=4\,\&\,\textrm{SO}(4)}_{\textrm{R}}\,$ S-fold of [25]. In
other words, the field content of half-maximal supergravity only captures an
$\,\mathcal{N}=3\,$ subsector of the $\,\mathcal{N}=4\,$ S-fold CFT3 of [25].

In $\,\mathfrak{osp}(3|4)\,$ language, the unprotected long multiplet in
(3.19) does not split and the spectrum (3.19) simply reduces to

$L[0]_{2}^{0}\hskip 14.22636pt,\hskip 14.22636ptB_{1}[0]_{2}^{2}\hskip
14.22636pt,\hskip 14.22636ptA_{1}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\ .$ (3.22)

#####  $\bullet$ Point $\,\varphi=\tilde{\varphi}=0\,$:

At this special point (red star in Figure 1) the conformal dimensions in
(3.17) reduce to

$\Delta_{1}=2\hskip 14.22636pt,\hskip 14.22636pt\Delta_{\pm}=\frac{3}{2}\hskip
14.22636pt,\hskip 14.22636pt\tilde{\Delta}_{-}=2\hskip 14.22636pt,\hskip
14.22636pt\tilde{\Delta}_{+}=1\ .$ (3.23)

This implies that the three long multiplets
$\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\,$ and
$\,L\bar{L}[0]^{0}_{\tilde{\Delta}_{+}}\,$ hit the unitarity bound and split,
each of them producing one short and two semi-short multiplets. More
concretely, the multiplets in (3.18) decompose as

$\begin{array}[]{rcl}L\bar{L}[0]^{0}_{\Delta_{1}}&\to&L\bar{L}[0]^{0}_{2}\\\\[11.38109pt]
L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}&\to&\left[A_{1}\bar{A_{1}}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,\,\oplus\,\,A_{2}\bar{L}[0]^{-1}_{2}\,\,\oplus\,\,L\bar{A_{2}}[0]^{1}_{2}\right]\,\,\,\,(\times
2)\\\\[17.07164pt]
L\bar{L}[0]_{\tilde{\Delta}_{-}}^{0}\,\,\oplus\,\,L\bar{L}[0]_{\tilde{\Delta}_{+}}^{0}&\to&L\bar{L}[0]_{2}^{0}\,\,\oplus\,\,\Big{[}A_{2}\bar{A_{2}}[0]^{0}_{1}\,\,\oplus\,\,B_{1}\bar{L}[0]^{-2}_{2}\,\,\oplus\,\,L\bar{B_{1}}[0]^{2}_{2}\Big{]}\end{array}$
(3.24)

There is this time an enhancement to $\,\mathcal{N}=4\,$ supersymmetry
originating from the two short multiplets
$\,A_{1}\bar{A_{1}}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,$ in (3.24) each one
containing a massless gravitino. In addition, there is the shortening
associated with $\,\tilde{\Delta}_{+}=1\,$ which provides an additional
massless vector multiplet.

In $\,\mathfrak{osp}(3|4)\,$ language, the unprotected long multiplet in
(3.19) hits the unitarity bound and splits into two short multiplets as
$\,L[0]^{0}_{1}\,\rightarrow\,A_{2}[0]^{0}_{1}\,\oplus B_{1}[0]^{2}_{2}\,$.
The spectrum in (3.19) then reduces to

$A_{2}[0]^{0}_{1}\hskip 14.22636pt,\hskip
14.22636ptB_{1}[0]^{2}_{2}\,\,\,\,(\times 2)\hskip 14.22636pt,\hskip
14.22636ptA_{1}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\ ,$ (3.25)

as a consequence of the supersymmetry enhancement to $\,\mathcal{N}=4\,$ in
the exotic CFT3.

####  3.5.3 $\textrm{U}(1)_{\textrm{F}}\,$ flavour symmetry enhancement

The identification $\,\varphi=0\,$ (vertical axis in Figure 1) gives rise to a
$\,\textrm{U}(1)_{\textrm{F}}\,$ flavour symmetry enhancement in the
corresponding CFT3’s. At this value it occurs that $\,\tilde{\Delta}_{-}=1\,$,
the long multiplet $\,L\bar{L}[0]_{\tilde{\Delta}_{-}}^{0}\,$ in (3.12) hits
the unitarity bound and splits again into one short and two semi-short
multiplets. The multiplets in (3.12) reduce to

$\begin{array}[]{rcl}L\bar{L}[0]^{0}_{\Delta_{1}}&\to&L\bar{L}[0]^{0}_{\tfrac{1}{2}+\tfrac{1}{2}\sqrt{25-\frac{16}{1+\tilde{\varphi}^{2}}}}\\\\[17.07164pt]
L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}&\to&L\bar{L}[\tfrac{1}{2}]^{0}_{1+\frac{1}{2}\sqrt{5+4\tilde{\varphi}^{2}-4\sqrt{\tilde{\varphi}^{2}+1}}}\,\,\,\,(\times
2)\\\\[17.07164pt]
L\bar{L}[0]_{\tilde{\Delta}_{-}}^{0}\,\,\oplus\,\,L\bar{L}[0]_{\tilde{\Delta}_{+}}^{0}&\to&\Big{[}A_{2}\bar{A}_{2}[0]^{0}_{1}\,\,\oplus\,\,L\bar{B}_{1}[0]_{2}^{2}\,\,\oplus\,\,B_{1}\bar{L}[0]^{-2}_{2}\Big{]}\,\,\oplus\,\,L\bar{L}[0]^{0}_{\tfrac{1}{2}+\tfrac{1}{2}\sqrt{9+\frac{16\,\tilde{\varphi}^{4}}{1+\tilde{\varphi}^{2}}}}\end{array}$
(3.26)

where $\,A_{2}\bar{A}_{2}[0]^{0}_{1}\,$ is a massless vector multiplet
reflecting the $\,\textrm{U}(1)_{\textrm{F}}\,$ flavour symmetry enhancement
in the dual $\,\mathcal{N}=2\,$ CFT3’s.

Note that the two degenerated long multiplets
$\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\,$ in (3.26) hit the unitarity
bound at the special point $\,\tilde{\varphi}=0\,$. At this point, each of
them splits as

$\begin{array}[]{llllllll}L\bar{L}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}&\rightarrow&A_{1}\bar{A}_{1}[1]^{0}_{\frac{3}{2}}&\oplus&A_{2}\bar{L}[0]^{-1}_{2}&\oplus&L\bar{A}_{2}[0]^{1}_{2}\end{array}$
(3.27)

recovering the exotic $\,\mathcal{N}=4\,$ CFT3.

###  3.6 General axion deformations of the exotic $\mathcal{N}=4\,$ CFT3

Setting $\,\varphi=\tilde{\varphi}=0\,$ in (3.10) reproduces the exotic
$\,\mathcal{N}=4\,$ AdS4 solution with $\,\textrm{SO}(4)_{\textrm{R}}\,$
symmetry originally reported in [40] (red star in Figure 1). Let us recall
that it is located at the (rescaled by $c$) origin of the scalar geometry,
namely,

$z_{1,2,3}=i\,c\,\hskip 28.45274pt,\hskip 28.45274ptz_{4,5,6,7}=i\ ,$ (3.28)

and preserves the compact part of the gauging, namely,

$\textrm{SO}(4)_{\textrm{R}}\,\sim\,\textrm{SO}(3)_{1}\times\textrm{SO}(3)_{2}\,\subset\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\
.$ (3.29)

Three new scalar moduli can be turned on at this AdS4 solution which are
parameterised by three axions $\,\chi_{1,2,3}\in\mathbb{R}\,$ of the axion-
like type investigated for the S-fold solutions in [35, 36, 41, 34].
Activating $\,\chi_{1,2,3}\,$ changes the location of the AdS4 solution (3.28)
to

$z_{1,2,3}=c\,\,(-\chi_{1,2,3}+i)\hskip 28.45274pt,\hskip
28.45274ptz_{4,5,6,7}=i\ ,$ (3.30)

keeping the vacuum energy at the value $\,V_{0}=-3\,g^{2}c^{-1}$. Therefore,
the moduli fields $\,\chi_{i}\,$ ($i=1,2,3$) are naturally identified with
marginal deformations of the $\,\mathcal{N}=4\,$ exotic CFT3.

The explicit computation of the normalised gravitino masses at the AdS4
solution (3.30) yields

$mL\,\,=\,\,\sqrt{1+\omega_{1}^{2}}\hskip 11.38109pt,\hskip
11.38109pt\sqrt{1+\omega_{2}^{2}}\hskip 11.38109pt,\hskip
11.38109pt\sqrt{1+\omega_{3}^{2}}\hskip 11.38109pt,\hskip
11.38109pt\sqrt{1+\omega_{4}^{2}}\ ,$ (3.31)

with

$\chi_{i}=\omega_{j}+\omega_{k}\hskip 14.22636pt(i\neq j\neq k)\hskip
28.45274pt\textrm{ and }\hskip
28.45274pt\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}=0\ .$ (3.32)

It proves very convenient to introduce a set
$\,\omega_{A}=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}\,$ of
deformation parameters subject to the constraint
$\,\sum_{A=1}^{4}\omega_{A}=0$. The twelve normalised vector masses can then
be very symmetrically written as

$m^{2}L^{2}=1+\omega_{A}^{2}+\omega_{B}^{2}\pm\sqrt{1+(\omega_{A}+\omega_{B})^{2}+4\,\omega_{A}^{2}\,\omega_{B}^{2}}\hskip
22.76219pt\textrm{ with }\hskip 22.76219ptA<B\ ,$ (3.33)

whereas the normalised scalar masses are given by

$\begin{array}[]{rcl}m^{2}L^{2}&=&-1+\omega_{A}^{2}+\omega_{B}^{2}\pm\sqrt{1+(\omega_{A}+\omega_{B})^{2}+4\,\omega_{A}^{2}\,\omega_{B}^{2}}\hskip
22.76219pt\textrm{ with }\hskip 22.76219ptA<B\ ,\\\\[8.53581pt]
&&4\,\,\,,\,\,\,-2\,\,\,(\times 2)\,\,\,,\,\,\,0\,\,\,(\times
15)\,\,\,,\,\,\,\lambda_{1,\ldots,8}\,\,\,(\times 1)\ .\end{array}$ (3.34)

We cannot provide a closed form for the eight normalised scalar masses
$\,\lambda_{1,\ldots,8}\,$. They correspond to the eigenvalues of the
$\,\chi_{1,2,3}$-dependent matrix

$\small{\left(\begin{array}[]{cccccccc}\hat{\chi}^{2}-2&0&2\chi_{2}\chi_{3}&2\chi_{1}\chi_{3}&2\chi_{1}\chi_{2}&0&0&0\\\
0&\hat{\chi}^{2}-2&2\chi_{1}&2\chi_{2}&2\chi_{3}&2\chi_{2}^{2}-\hat{\chi}^{2}&2\chi_{3}^{2}-\hat{\chi}^{2}&2\chi_{1}^{2}-\hat{\chi}^{2}\\\
2\chi_{2}\chi_{3}&2\chi_{1}&\hat{\chi}^{2}&2\chi_{1}\chi_{2}&2\chi_{1}\chi_{3}&-2\chi_{1}&-2\chi_{1}&0\\\
2\chi_{1}\chi_{3}&2\chi_{2}&2\chi_{1}\chi_{2}&\hat{\chi}^{2}&2\chi_{2}\chi_{3}&0&-2\chi_{2}&-2\chi_{2}\\\
2\chi_{1}\chi_{2}&2\chi_{3}&2\chi_{1}\chi_{3}&2\chi_{2}\chi_{3}&\hat{\chi}^{2}&-2\chi_{3}&0&-2\chi_{3}\\\
0&2\chi_{2}^{2}-\hat{\chi}^{2}&-2\chi_{1}&0&-2\chi_{3}&\hat{\chi}^{2}&2-\hat{\chi}^{2}+2\chi_{1}^{2}&2-\hat{\chi}^{2}+2\chi_{3}^{2}\\\
0&2\chi_{3}^{2}-\hat{\chi}^{2}&-2\chi_{1}&-2\chi_{2}&0&2-\hat{\chi}^{2}+2\chi_{1}^{2}&\hat{\chi}^{2}&2-\hat{\chi}^{2}+2\chi_{2}^{2}\\\
0&2\chi_{1}^{2}-\hat{\chi}^{2}&0&-2\chi_{2}&-2\chi_{3}&2-\hat{\chi}^{2}+2\chi_{3}^{2}&2-\hat{\chi}^{2}+2\chi_{2}^{2}&\hat{\chi}^{2}\\\
\end{array}\right),}$ (3.35)

with $\,\hat{\chi}^{2}\equiv\chi_{1}^{2}+\chi_{2}^{2}+\chi_{3}^{2}\,$. A
numerical scan of the mass spectrum in (3.34) as a function of
$\,\omega_{A}\,$ shows that there exist regions in parameter space for which
non-supersymmetric solutions become perturbatively unstable, e.g. whenever
$\,\omega_{1}=\omega_{2}=-\omega_{3}=-\omega_{4}>0.3052\,$.

It follows from (3.33) that the number of massless vectors is given by the
number $\,n_{p}\,$ of pairs $\,\omega_{A}=\omega_{B}\,$ with $\,A<B\,$. This
yields the following classification of CFT3 duals in terms of the four
parameters $\,\omega_{A}\,$:

  * $\circ$

All four $\,\omega$’s are zero: $\mathcal{N}=4$ with
$\,\textrm{SO}(4)_{\textrm{R}}\,$ symmetry

  * $\circ$

Three $\,\omega$’s are zero: same case as before by virtue of
$\,\sum\omega_{A}=0$

  * $\circ$

Exactly two $\,\omega$’s are zero: $\,\mathcal{N}=2\,$ with
$\,\textrm{SO}(2)_{\textrm{R}}\,$ symmetry

  * $\circ$

Exactly one $\,\omega\,$ is zero: $\,\mathcal{N}=1\,$ with

    * –

$\textrm{SO}(2)_{\textrm{F}}\,$ flavour symmetry if two of the remaining
$\,\omega$’s are equal

    * –

no flavour symmetry otherwise

  * $\circ$

All $\,\omega$’s are non-zero: $\mathcal{N}=0\,$ with

    * –

$\textrm{SO}(2)_{\textrm{F}}\times\textrm{SO}(2)_{\textrm{F}}\,$ if there are
two pairs of $\omega$’s of equal value, one pair with the opposite sign of the
other

    * –

$\textrm{SO}(3)_{\textrm{F}}\,$ if three $\,\omega$’s are identified

    * –

$\textrm{SO}(2)_{\textrm{F}}$ if only two $\,\omega$’s are identified

    * –

no flavour symmetry otherwise

In summary, the number of supersymmetries (both of the AdS4 solution and the
dual CFT3) matches the number of parameters $\,\omega_{A}=0$, and the AdS4
solution features an orthogonal symmetry group of dimension $\,n_{p}$. Amongst
the $\,n_{p}\,$ pairs, each pair $\,\omega_{A}=\omega_{B}=0\,$ adds one
generator to the orthogonal R-symmetry group of the dual CFT3 whereas each
pair $\,\omega_{A}=\omega_{B}\neq 0\,$ adds one generator to an orthogonal
flavour symmetry group in the CFT3. Finally, observe that there is a case
compatible with a single axion, let us denote it $\,\chi$, that preserves
$\,\mathcal{N}=2\,$ and $\,\textrm{SO}(2)_{\textrm{R}}\,$ symmetry within
half-maximal supergravity. The consequences of turning on this axion
$\,\chi\,$ will be investigated in detail in Section 4.

####  3.6.1 Relation to axion-like deformations of S-folds?

Axion-like deformations (generically also denoted by $\,\chi$’s) of S-fold
solutions of type IIB supergravity have been geometrically characterised in
[36, 41] (see [34] for a light review). In these solutions, they cause a
breaking of symmetries by global effects in the ten dimensional geometry [35,
36]. More concretely, axion-like deformations break the symmetry group of an
S-fold down to its Cartan subgroup. The number of such deformations was shown
to match the dimension of the Cartan subalgebra of the symmetry group of the
S-fold solution before turning on any deformation.

The way the $\,\chi$’s enter the AdS4 solution in (3.30) is identical to the
known examples of axion-like deformations of S-folds (cf. eq.$(2.18)$ in
[32]). However one must be cautious about giving the same interpretation to
all the $\,\chi$’s in (3.30). The reason is twofold:

  * $i)\,$

We find three different axion deformations $\,\chi_{1,2,3}\,$ for the exotic
$\,\mathcal{N}=4\,$ AdS4 solution with $\,\textrm{SO}(4)_{\textrm{R}}\,$
symmetry and not two, as one would naively expect from the number of Cartan
generators of $\,\textrm{SO}(4)_{\textrm{R}}\,$.

  * $ii)\,$

Generic values of $\,\chi_{1,2,3}\,$ totally break the
$\,\textrm{SO}(4)_{\textrm{R}}\,$ symmetry of the undeformed solution: the
Cartan subgroup of $\,\textrm{SO}(4)_{\textrm{R}}\,$ is generically not
preserved. As a result, the pattern of symmetry breaking is very different
from the one induced by the axion-like deformations in the S-fold solutions
[36, 41].

As a result, while the $\,\mathcal{N}=0\,$ case with
$\,\textrm{SO}(2)_{\textrm{F}}\times\textrm{SO}(2)_{\textrm{F}}\,$ symmetry
preserves the Cartan subgroup of $\,\textrm{SO}(4)_{\textrm{R}}\,$ and,
therefore, stands a chance of having a geometrical interpretation alike the
axion-like deformations of S-folds, the other cases appear (at first glance)
to be different as the Cartan subgroup of $\,\textrm{SO}(4)_{\textrm{R}}\,$ is
not preserved. Of special interest will be the case of the single axion
$\,\chi\,$ mentioned above, which we move to discuss in the next section. This
axion preserves $\,\mathcal{N}=2\,$ supersymmetry, an
$\,\textrm{SO}(2)_{\textrm{R}}\subset\textrm{SO}(4)_{\textrm{R}}\,$ symmetry
within half-maximal supergravity, and combines with the modulus $\,\varphi\,$
to generalise the conformal manifold of $\,\mathcal{N}=2\,$ CFT3’s in (1.3) to
arbitrary values of the parameter $\,\tilde{\varphi}$.

##  4 $\textrm{U}(1)_{\textrm{R}}$-invariant sector

In this section we construct a particular $\,\textrm{U}(1)_{\textrm{R}}\,$
invariant sector of half-maximal supergravity that suffices to capture the
modulus $\,\chi\,$ dual to the second marginal deformation spanning with
$\,\varphi\,$ the $\,\mathcal{N}=2\,$ conformal manifold in (1.3).

###  4.1 The $\,\mathcal{N}=2\,$ three-vector and two-hyper model

To our knowledge, there is no explicit construction of the
$\,\textrm{U}(1)_{\textrm{R}}\,$ invariant sector of half-maximal supergravity
of relevance for this work. So we will present it in some detail. In order to
construct it, let us first introduce the set of $\,\textrm{SO}(6,6)\,$
generators

$\left[t_{MN}\right]_{P}{}^{Q}=2\,\eta_{P[M}\,\delta^{Q}_{N]}\ ,$ (4.1)

where $\,M=1\ldots,6,\bar{1},\ldots\bar{6}\,$ is a fundamental
$\,\textrm{SO}(6,6)\,$ index in the light-cone basis. We choose the specific
$\,\textrm{U}(1)_{\textrm{R}}\,$ generator to be

$t_{\textrm{U}(1)_{\textrm{R}}}=\left(t_{5\bar{1}}-t_{1\bar{5}}\right)-\left(t_{6\bar{2}}-t_{2\bar{6}}\right)\
,$ (4.2)

which is embedded in the duality group of half-maximal supergravity as

$\textrm{SL}(2)\times\textrm{SO}(6,6)\supset\textrm{SL}(2)\times\textrm{SO}(2,2)\times\textrm{SO}(4,4)\supset\textrm{SL}(2)\times\textrm{SO}(2,2)\times\textrm{SU}(2,2)\times\textrm{U}(1)_{\textrm{R}}\
.$ (4.3)

From the commutant of $\,\textrm{U}(1)_{\textrm{R}}\,$ in the embedding chain
(4.3), the scalar manifold invariant under $\,\textrm{U}(1)_{\textrm{R}}\,$ is
identified with

$\mathcal{M}^{{}^{\textrm{U}(1)_{\textrm{R}}}}_{\textrm{scal}}=\underbrace{\frac{\textrm{SL}(2)}{\textrm{SO}(2)}\times\frac{\textrm{SO}(2,2)}{\textrm{SO}(2)\times\textrm{SO}(2)}}_{\left[\dfrac{\textrm{SL}(2)}{\textrm{SO}(2)}\right]^{3}}\times\frac{\textrm{SU}(2,2)}{\textrm{S}\left(\textrm{U}(2)\times\textrm{U}(2)\right)}\
.$ (4.4)

This $\,\textrm{U}(1)_{\textrm{R}}$-invariant sector of half-maximal
supergravity can be described as an $\,\mathcal{N}=2\,$ gauged supergravity
coupled to three vector multiplets and two hyper-multiplets. Within this
$\,\mathcal{N}=2\,$ sector, the gauge group is

$\textrm{G}_{\mathcal{N}=2}=\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)_{\gamma}\times\mathbb{R}_{a}\times\mathbb{R}_{\epsilon}\subset\textrm{ISO}(3)\times\textrm{ISO}(3)\
,$ (4.5)

with all the fields being inert under the $\,\textrm{U}(1)_{\textrm{R}}\,$
factor. The three complex scalars in the vector multiplets span the
$\,\left[\textrm{SL}(2)/\textrm{SO}(2)\right]^{3}\,$ factor of the scalar
geometry (4.4). They are identified with $\,z_{7}\,$ and $\,(z_{2},z_{5})\,$
in the $\,\mathbb{Z}_{2}^{2}$-invariant sector of Section 3. The scalar matrix
$\,M_{\alpha\beta}\,$ spanned by $\,z_{7}\,$ was given in (2.11). The part of
the $\,M_{MN}\,$ matrix in (2.12) spanned by $\,(z_{2},z_{5})\,$ was
constructed in terms of the $\,2\times 2\,$ blocks $\,G_{2}\,$ and $\,B_{2}\,$
in (3.2). Alternatively, it can directly be constructed from the coset
representative

$\mathcal{V}_{\textrm{SO}(2,\,2)}=e^{-\chi_{2}\,t_{4\bar{3}}+\chi_{5}\,t_{43}}\,\,e^{-\frac{1}{2}\,\left[\,\varphi_{2}\,(t_{4\bar{4}}-t_{3\bar{3}})+\varphi_{5}(t_{4\bar{4}}+t_{3\bar{3}})\,\right]}\
,$ (4.6)

such that $\,z_{2,5}=-\chi_{2,5}+ie^{-\varphi_{2,5}}$. There is also the part
of the scalar matrix $\,M_{MN}\,$ that depends on the scalars in the
quaternionic Kähler (QK) space
$\,\left[\textrm{SU}(2,2)/\textrm{S}(\textrm{U}(2)\times\textrm{U}(2))\right]\sim\left[\textrm{SO}(2,4)/(\textrm{SO}(2)\times\textrm{SO}(4))\right]$.
Following the coset construction of [55], we will first introduce the
generators

$\begin{array}[]{rclcrcl}H_{1}&=&t_{1\bar{1}}+t_{5\bar{5}}+t_{6\bar{6}}+t_{2\bar{2}}&\hskip
14.22636pt,&\hskip
14.22636ptH_{2}&=&t_{1\bar{1}}+t_{5\bar{5}}-t_{6\bar{6}}-t_{2\bar{2}}\\\\[5.69054pt]
{E_{2}}^{3}&=&-t_{15}&\hskip 14.22636pt,&\hskip
14.22636ptV^{23}&=&t_{26}\\\\[5.69054pt]
U_{1}{}^{3}&=&-t_{12}+t_{56}+t_{16}-t_{25}&\hskip 14.22636pt,&\hskip
14.22636ptU_{1}{}^{2}&=&t_{2\bar{1}}+t_{2\bar{5}}+t_{6\bar{1}}-t_{6\bar{5}}\\\\[5.69054pt]
U_{2}{}^{3}&=&t_{12}-t_{56}+t_{16}-t_{25}&\hskip 14.22636pt,&\hskip
14.22636ptU_{2}{}^{2}&=&t_{2\bar{1}}-t_{2\bar{5}}-t_{6\bar{1}}-t_{6\bar{5}}\end{array}$
(4.7)

and construct the coset representative as

$\mathcal{V}_{\textrm{SO}(2,4)}=e^{\frac{1}{2\sqrt{2}}\,U}\,\,\,e^{a\,V^{23}}\,\,\,e^{h\,E_{2}{}^{3}}\,\,\,e^{-\frac{1}{4}\,\left[\,(\phi_{2}+\phi_{1})H_{1}+(\phi_{2}-\phi_{1})H_{2}\,\right]}\
,$ (4.8)

with

$U=-(\tilde{\zeta}_{0}-\tilde{\zeta}_{1})\,U_{1}{}^{2}-(\zeta^{0}+\zeta^{1})\,U_{2}{}^{2}-(\zeta^{0}-\zeta^{1})\,U_{1}{}^{3}+(\tilde{\zeta}_{0}+\tilde{\zeta}_{1})\,U_{2}{}^{3}\
.$ (4.9)

The scalar matrix $\,M_{MN}\in\textrm{SO}(6,6)\,$ is then obtained as
$\,M=\mathcal{V}\,\mathcal{V}^{t}\,$ using the factorised coset representative

$\mathcal{V}=\mathcal{V}_{\textrm{SO}(2,2)}\,\,\mathcal{V}_{\textrm{SO}(2,4)}\
.$ (4.10)

In order to complete the characterisation of the $\,2+12\,$ scalars in the
$\,\textrm{U}(1)_{\textrm{R}}$-invariant sector, we will now look at the
metric on the scalar manifold (4.4) which can be extracted from the kinetic
terms in (2.13). An explicit computation gives

$\begin{array}[]{rcl}\mathcal{L}_{\textrm{kin}}&=&-\tfrac{1}{4}\left(d\varphi_{7}^{2}+e^{2\varphi_{7}}\,d\chi_{7}^{2}\right)-\tfrac{1}{4}\left(d\varphi_{2}^{2}+e^{2\varphi_{2}}\,d\chi_{2}^{2}\right)-\tfrac{1}{4}\left(d\varphi_{5}^{2}+e^{2\varphi_{5}}\,d\chi_{5}^{2}\right)\\\\[8.53581pt]
&-&\tfrac{1}{4}\left(D\phi_{2}^{2}+e^{2\phi_{2}}\,Dh^{2}\right)\\\\[8.53581pt]
&-&\frac{1}{4}\left[d\phi_{1}^{2}+e^{2\phi_{1}}\Big{(}Da+\tfrac{1}{2}\big{(}\zeta^{0}D\tilde{\zeta}_{0}+\zeta^{1}D\tilde{\zeta}_{1}-\tilde{\zeta}_{0}D\zeta^{0}-\tilde{\zeta}_{1}D\zeta^{1}\big{)}\Big{)}^{2}\right]\\\\[11.38109pt]
&-&\tfrac{1}{8}\left[e^{\phi_{1}-\phi_{2}}\big{(}D\zeta^{0}+D\zeta^{1}\big{)}^{2}+e^{\phi_{1}-\phi_{2}}\big{(}D\tilde{\zeta}_{0}-D\tilde{\zeta}_{1}\big{)}^{2}\right.\\\\[8.53581pt]
&&\left.\,\,\,\,\,\,+\,e^{\phi_{1}+\phi_{2}}\Big{(}D\zeta^{0}-D\zeta^{1}+h\,(D\tilde{\zeta}_{0}-D\tilde{\zeta}_{1})\Big{)}^{2}\right.\\\\[8.53581pt]
&&\left.\,\,\,\,\,\,+\,e^{\phi_{1}+\phi_{2}}\Big{(}D\tilde{\zeta}_{0}+D\tilde{\zeta}_{1}-h\,(D\zeta^{0}+D\zeta^{1})\Big{)}^{2}\right]\
,\end{array}$ (4.11)

where we have introduced the kinetic term notation
$\,dX\,dY\equiv\partial_{\mu}X\,\partial^{\mu}Y\,$ and $\,DX\,DY\equiv
D_{\mu}X\,D^{\mu}Y\,$ for two generic scalars $\,X\,$ and $\,Y$. The covariant
derivatives in (4.11) include a gauge connection for the gauge group in (4.5).
They can be straightforwardly computed from (2.14) and take the form

$D\phi_{2}=d\phi_{2}-4g\,A^{{}^{(\epsilon)}}\,h\hskip 11.38109pt,\hskip
11.38109ptDh=dh-2g\,A^{{}^{(\epsilon)}}\,\left(e^{-2\phi_{2}}-h^{2}\right)\hskip
11.38109pt,\hskip 11.38109ptDa=da-2g\,A^{{}^{(a)}}\ ,$ (4.12)

together with

$\begin{array}[]{ccc}D\zeta^{0}=d\zeta^{0}-g\,A^{{}^{(\epsilon)}}\,(\tilde{\zeta}_{0}+\tilde{\zeta}_{1})-2g\,A^{{}^{(\gamma)}}\,\tilde{\zeta}_{0}&,&D\zeta^{1}=d\zeta^{1}-g\,A^{{}^{(\epsilon)}}\,(\tilde{\zeta}_{0}+\tilde{\zeta}_{1})+2g\,A^{{}^{(\gamma)}}\,\tilde{\zeta}_{1}\
,\\\\[8.53581pt]
D\tilde{\zeta}_{0}=d\tilde{\zeta}_{0}-g\,A^{{}^{(\epsilon)}}\,(\zeta^{1}-\zeta^{0})+2g\,A^{{}^{(\gamma)}}\,\zeta^{0}&,&D\tilde{\zeta}_{1}=d\tilde{\zeta}_{1}-g\,A^{{}^{(\epsilon)}}\,(\zeta^{0}-\zeta^{1})-2g\,A^{{}^{(\gamma)}}\,\zeta^{1}\
,\end{array}$ (4.13)

in terms of three linear combinations of vectors $\,A_{\mu}{}^{\alpha M}\,$
given by

$\begin{array}[]{rcll}A^{{}^{(\epsilon)}}_{\mu}&\equiv&\sqrt{2}\,c\,\frac{1-\tilde{\varphi}^{2}}{1+\tilde{\varphi}^{2}}\,A_{\mu}{}^{+3}+\frac{\sqrt{2}}{\sqrt{1+\tilde{\varphi}^{2}}}\left(A_{\mu}{}^{+\bar{3}}+c\,\tilde{\varphi}\,A_{\mu}{}^{-3}\right)&,\\\\[11.38109pt]
A^{{}^{(a)}}_{\mu}&\equiv&\sqrt{2}\,c\,\frac{1-\tilde{\varphi}^{2}}{1+\tilde{\varphi}^{2}}\,A_{\mu}{}^{-\bar{4}}+\frac{\sqrt{2}}{\sqrt{1+\tilde{\varphi}^{2}}}\left(A_{\mu}{}^{-4}+c\,\tilde{\varphi}\,A_{\mu}{}^{+\bar{4}}\right)&,\\\\[11.38109pt]
A^{{}^{(\gamma)}}_{\mu}&\equiv&\frac{\sqrt{2}}{\sqrt{1+\tilde{\varphi}^{2}}}\left(A_{\mu}{}^{+3}+A_{\mu}{}^{-\bar{4}}\right)\
.\par\end{array}$ (4.14)

The vectors $\,A^{{}^{(\epsilon)}}_{\mu}\,$, $\,A^{{}^{(a)}}_{\mu}\,$ and
$\,A^{{}^{(\gamma)}}_{\mu}\,$ in (4.14) respectively gauge the factors
$\,\mathbb{R}_{\epsilon}\,$ and $\,\mathbb{R}_{a}\,$ and
$\,\textrm{U}(1)_{\gamma}\,$ in (4.5). There is an additional vector field
$\,A^{{}^{(\textrm{R})}}_{\mu}\,$ associated with the
$\,\textrm{U}(1)_{\textrm{R}}\,$ generator in (4.2) under which all the
scalars in this sector of the theory are invariant.

###  4.2 Warming up: the $(\varphi,\chi)$-family of $\,\mathcal{N}=2\,$ AdS4
solutions of [1]

Let us start by recovering the two-parameter $(\varphi,\chi)$-family of
$\,\mathcal{N}=2\,$ AdS4 solutions of the
$\,[\textrm{SO}(1,1)\times\textrm{SO}(6)]\ltimes\mathbb{R}^{12}\,$ maximal
supergravity put forward in [1]. As explained in Section 2.3, we must first of
all set $\,\tilde{\varphi}^{2}=1\,$ in order to make contact with the maximal
theory. Then, the two-parameter $(\varphi,\chi)$-family of $\,\mathcal{N}=2\,$
AdS4 solutions of [1] is recovered within the
$\textrm{U}(1)_{\textrm{R}}$-invariant sector of half-maximal supergravity as
follows. The SK scalars are fixed to

$z_{5}=z_{7}=\dfrac{\mp 1+i}{\sqrt{2}}\hskip 22.76219pt\textrm{ and }\hskip
22.76219ptz_{2}=i\,c\ ,$ (4.15)

where, as before, the $\,\mp\,$ sign in (4.15) is correlated with the
$\,\pm\,$ sign in (2.21). The QK scalars are fixed to

$\begin{array}[]{c}h+i\,e^{-\phi_{2}}=i\,\dfrac{\sqrt{2}}{c\,(1+\varphi^{2})}\hskip
14.22636pt,\hskip 14.22636pte^{-\phi_{1}}=\dfrac{c}{\sqrt{2}}\
,\\\\[17.07164pt] a=0\hskip 14.22636pt,\hskip
14.22636pt\zeta^{0}=\zeta^{1}=\dfrac{c\,\chi}{\sqrt{2}}\hskip
14.22636pt,\hskip
14.22636pt\tilde{\zeta}_{0}=\tilde{\zeta}_{1}=\dfrac{\varphi}{\sqrt{1+\varphi^{2}}}\
.\end{array}$ (4.16)

This $\,\mathcal{N}=2\,$ family of AdS4 solutions comes along with a vacuum
energy

$V_{0}=-3g^{2}c^{-1}\ ,$ (4.17)

for any value of the moduli fields $\,(\varphi,\chi)\,$ thus identifying them
with marginal deformations in the dual CFT3’s. It is worth emphasising that
(4.15)-(4.16) provides an explicit realisation of the $(\varphi,\chi)$-family
of $\,\mathcal{N}=2\,$ AdS4 solutions of [1] in a specific supergravity
model.161616This two-parameter $(\varphi,\chi)$-family of solutions was
constructed in [1] by applying a $\,\chi$-dependent $\,\textrm{E}_{7(7)}\,$
duality transformation to the one-parameter $\varphi$-family in (3.5).

#### Marginal deformation and $\mathfrak{osp}(2|4)\,$ superconformal
multiplets

The explicit computation of the normalised mass spectrum recovers the half-
maximal ($\mathbb{Z}_{2}$-invariant) subset of multiplets within the maximal
content of [1]. This consists of five unprotected long multiplets

$L\bar{L}[0]^{0}_{\Delta_{1}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[0]^{0}_{\tilde{\Delta}_{\pm}}\ ,$ (4.18)

with conformal dimensions171717The conformal dimensions
$\,(\Delta_{1},\Delta_{+},\Delta_{-},\tilde{\Delta}_{+},\tilde{\Delta}_{-})\,$
map into $\,(\beta_{3},\beta_{5},\beta_{4},\beta_{1},\beta_{2})\,$ in
eq.$(4.5)$ of [1] upon the identification
$\,\chi=\sqrt{1+\tilde{\varphi}^{2}}\,\chi_{\tiny{\cite[cite]{[\@@bibref{}{Bobev:2021yya}{}{}]}}}$
with $\,\tilde{\varphi}=1\,$.

$\begin{array}[]{cll}\Delta_{1}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{17+33\varphi^{2}}{1+\varphi^{2}}}\
,\\\\[17.07164pt]
\Delta_{\pm}&=&\frac{1}{2}+\frac{\sqrt{\left(2+\varphi^{2}\right)^{2}+\chi^{2}}\,\pm\,\varphi}{\sqrt{2\left(1+\varphi^{2}\right)}}\
,\\\\[17.07164pt]
\tilde{\Delta}_{\pm}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{9+4\varphi^{4}+5\varphi^{2}+4\chi^{2}\pm
4(\varphi^{4}+\varphi^{2}+\chi^{2}-2)}{\varphi^{2}+1}}\ .\end{array}$ (4.19)

There are also the short and semi-short protected multiplets given in (3.9),
namely,

$A_{1}\bar{A}_{1}[1]_{2}^{0}\hskip 14.22636pt,\hskip
14.22636ptL\bar{B}_{1}[0]_{2}^{2}\hskip 14.22636pt,\hskip
14.22636ptB_{1}\bar{L}[0]_{2}^{-2}\ ,$ (4.20)

where $\,A_{1}\bar{A}_{1}[1]^{0}_{2}\,$ is identified with the stress-energy
tensor multiplet of the $\,\mathcal{N}=2\,$ CFT3’s.

###  4.3 A $\,(\varphi,\chi\,;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$
AdS4 solutions

The two-parameter family of $\,\mathcal{N}=2\,$ AdS4 solutions in
(4.15)-(4.16) can be generalised to arbitrary values of the deformation
parameter $\,\tilde{\varphi}\,$. The SK scalars are given by

$z_{5}=z_{7}=\dfrac{\mp\tilde{\varphi}+i}{\sqrt{1+\tilde{\varphi}^{2}}}\hskip
22.76219pt\textrm{ and }\hskip 22.76219ptz_{2}=i\,c\ ,$ (4.21)

whereas the QK scalars take the form

$\begin{array}[]{c}h+i\,e^{-\phi_{2}}=i\,\dfrac{\sqrt{1+\tilde{\varphi}^{2}}}{c\,(1+\varphi^{2})}\hskip
14.22636pt,\hskip
14.22636pte^{-\phi_{1}}=\dfrac{c}{\sqrt{1+\tilde{\varphi}^{2}}}\
,\\\\[17.07164pt] a=0\hskip 14.22636pt,\hskip
14.22636pt\zeta^{0}=\zeta^{1}=\dfrac{c\,\chi}{\sqrt{1+\tilde{\varphi}^{2}}}\hskip
14.22636pt,\hskip
14.22636pt\tilde{\zeta}_{0}=\tilde{\zeta}_{1}=\dfrac{\varphi}{\sqrt{1+\varphi^{2}}}\
.\end{array}$ (4.22)

The vacuum energy at this $\,\mathcal{N}=2\,$ family of AdS4 solutions is
still given by

$V_{0}=-3g^{2}c^{-1}\ ,$ (4.23)

for any value of the deformation parameter $\,\tilde{\varphi}\,$ as well as of
the moduli fields $\,(\varphi,\chi)\,$ dual to marginal deformations. It is
worth highlighting that, at any value of $\,\tilde{\varphi}$, turning on
$\,(\varphi,\chi)\,$ activates the hypermultiplet scalars
$\,(\zeta^{0},\zeta^{1};\tilde{\zeta}_{0},\tilde{\zeta}_{1})\,$ spanning the
Heisenberg fiber of the QK geometry. Activating these scalars automatically
breaks the compact $\,\textrm{U}(1)_{\gamma}\,$ factor of the gauge group
(4.5), as it can be seen from (4.13).

###  4.4 $\mathfrak{osp}(2|4)\,$ superconformal multiplets

The half-maximal supergravity spectrum at generic $\,\tilde{\varphi}\,$ of the
$\,\mathcal{N}=2\,$ $(\varphi,\chi)$-family of AdS4 solutions can be arranged
into multiplets of the $\,\mathfrak{osp}(2|4)\,$ superconformal symmetry of
the would-be dual $\,\mathcal{N}=2\,$ CFT3’s. The spectrum contains five
unprotected long multiplets

$L\bar{L}[0]^{0}_{\Delta_{1}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[0]^{0}_{\tilde{\Delta}_{\pm}}\ ,$ (4.24)

with conformal dimensions given by

$\begin{array}[]{cll}\Delta_{1}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{9+25\tilde{\varphi}^{2}+\varphi^{2}\left(17+49\tilde{\varphi}^{2}\right)}{\left(1+\varphi^{2}\right)\left(1+\tilde{\varphi}^{2}\right)}}\
,\\\\[17.07164pt]
\Delta_{\pm}&=&\frac{1}{2}+\frac{\sqrt{\left(1+\varphi^{2}+\tilde{\varphi}^{2}\right)^{2}+\chi^{2}}\,\pm\,\varphi\tilde{\varphi}}{\sqrt{\left(1+\varphi^{2}\right)\left(1+\tilde{\varphi}^{2}\right)}}\
,\\\\[17.07164pt]
\tilde{\Delta}_{\pm}&=&\scriptsize{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{5+8\varphi^{4}+\varphi^{2}\left(\tilde{\varphi}^{2}+9\right)+8\tilde{\varphi}^{4}+5\tilde{\varphi}^{2}+8\chi^{2}\pm
4\sqrt{\Theta}}{\left(\varphi^{2}+1\right)\left(\tilde{\varphi}^{2}+1\right)}}}\
,\end{array}$ (4.25)

where

$\begin{array}[]{lll}\Theta&=&4\varphi^{8}+8\varphi^{6}-4\varphi^{4}\left(\tilde{\varphi}^{4}+3\tilde{\varphi}^{2}-1-2\chi^{2}\right)-4\varphi^{2}\left(\tilde{\varphi}^{4}+3\tilde{\varphi}^{2}-2\chi^{2}\right)\\\\[5.69054pt]
&&\left(\tilde{\varphi}^{2}\left(1+2\tilde{\varphi}^{2}\right)+1-2\chi^{2}\right)^{2}+4\left(\tilde{\varphi}^{2}-1\right)^{2}\chi^{2}.\end{array}$
(4.26)

In addition, there are one short and two semi-short protected multiplets with
integer conformal dimension $\,\Delta=2\,$. These are the same as in (3.9),
namely,

$A_{1}\bar{A}_{1}[1]^{0}_{2}\hskip 14.22636pt,\hskip
14.22636ptL\bar{B}_{1}[0]^{2}_{2}\hskip 14.22636pt,\hskip
14.22636ptB_{1}\bar{L}[0]^{-2}_{2}\ ,$ (4.27)

where $\,A_{1}\bar{A}_{1}[1]^{0}_{2}\,$ is the stress-energy tensor multiplet
of the dual $\,\mathcal{N}=2\,$ CFT3’s. The two moduli fields $\,\varphi\,$
and $\,\chi\,$ in (4.22) belong to the semi-short multiplets
$\,L\bar{B}_{1}[0]^{2}_{2}\,$ and $\,B_{1}\bar{L}[0]^{-2}_{2}$.

###  4.5 Special loci

The computation of the four gravitino masses in the half-maximal theory yields

$m\,L=1\,\,\,\,(\times 2)\hskip 14.22636pt,\hskip
14.22636pt\frac{\sqrt{(1+\varphi^{2}+\tilde{\varphi}^{2})^{2}+\chi^{2}}\pm\varphi\,\tilde{\varphi}}{\sqrt{\left(1+\varphi^{2}\right)\left(1+\tilde{\varphi}^{2}\right)}}\
,$ (4.28)

which consistently reduces to (3.15) when setting $\,\chi=0\,$. Notice that,
even when turning on $\,\chi\neq 0\,$, the marginal deformation $\,\varphi\,$
and the embedding tensor parameter $\,\tilde{\varphi}\,$ continue entering the
gravitino masses (4.28) in a symmetric fashion. A quick inspection of (4.28)
shows that supersymmetry enhancement to $\,\mathcal{N}>2\,$ is no longer
possible whenever $\,\chi\neq 0$. Still we will look at two special cases. The
first case is $\,\tilde{\varphi}=0\,$ which accounts for the
$\,\mathcal{N}=2\,$ marginal deformations of the $\,\mathcal{N}=4\,$ exotic
CFT3. The second one is $\,\varphi=0\,$ which describes the effect of the
modulus $\,\chi\,$ in a genuine half-maximal supergravity at generic
$\,\tilde{\varphi}$.

####  4.5.1 $(\varphi,\chi)$-deformations of the $\,\mathcal{N}=4\,$ exotic
CFT3

Setting $\,\tilde{\varphi}=0\,$ in the general expressions of the previous
section one is left with the $\,\mathcal{N}=2\,$ $\,(\varphi,\chi)$ marginal
deformations of the exotic $\,\mathcal{N}=4\,$ CFT3. More concretely, we find
in this case five unprotected long multiplets

$L\bar{L}[0]^{0}_{\Delta_{1}}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[\tfrac{1}{2}]^{0}_{\Delta}\,\,\,\,(\times 2)\hskip
14.22636pt,\hskip 14.22636ptL\bar{L}[0]^{0}_{\tilde{\Delta}_{\pm}}\ ,$ (4.29)

with conformal dimensions

$\begin{array}[]{cll}\Delta_{1}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{9+17\varphi^{2}}{1+\varphi^{2}}}\
,\\\\[17.07164pt]
\Delta&=&\frac{1}{2}+\frac{\sqrt{\left(1+\varphi^{2}\right)^{2}+\chi^{2}}}{\sqrt{1+\varphi^{2}}}\
,\\\\[17.07164pt]
\tilde{\Delta}_{\pm}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{5+8\chi^{2}+8\varphi^{4}+9\varphi^{2}\pm
4\sqrt{1+4\left(\varphi^{4}+\varphi^{2}+\chi^{2}\right)^{2}}}{1+\varphi^{2}}}\
.\end{array}$ (4.30)

Notice that, unlike for the
$\,\mathcal{N}=4\,\&\,\textrm{SO}(4)_{\textrm{R}}\,$ S-fold in (4.18), the
multiplets $\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta_{\pm}}\,$ get degenerated in
this case. In addition, there are also the short and semi-short protected
multiplets given in (4.27). The long multiplets
$\,L\bar{L}[\tfrac{1}{2}]^{0}_{\Delta}\,$ and
$\,L\bar{L}[0]^{0}_{\tilde{\Delta}_{-}}\,$ hit the unitarity bound at the
special value $\,\varphi=\chi=0\,$ and split as

$\begin{array}[]{lcl}L\bar{L}[\tfrac{1}{2}]^{0}_{\frac{3}{2}}&\rightarrow&A_{1}\bar{A}_{1}[\tfrac{1}{2}]_{\frac{3}{2}}^{0}\,\,\oplus\,\,A_{2}\bar{L}[0]_{2}^{-1}\,\,\oplus\,\,L\bar{A}_{2}[0]_{2}^{1}\\\\[11.38109pt]
L\bar{L}[0]^{0}_{1}&\rightarrow&A_{2}\bar{A}_{2}[0]_{1}^{0}\,\,\oplus\,\,B_{1}\bar{L}[0]_{2}^{-2}\,\,\oplus\,\,L\bar{B}_{1}[0]_{2}^{2}\end{array}$
(4.31)

recovering the undeformed exotic $\,\mathcal{N}=4\,$ CFT3.

####  4.5.2 $(\chi\,;\tilde{\varphi})$-family of $\,\mathcal{N}=2\,$ AdS4
solutions

In Section 3.3 we characterised the $\,(\varphi\,;\tilde{\varphi})$-family of
$\,\mathcal{N}=2\,$ CFT3’s at $\,\chi=0\,$. Let us now take the complementary
case $\,\varphi=0\,$ and characterise the $\,(\chi\,;\tilde{\varphi})$-family
of $\,\mathcal{N}=2\,$ CFT3’s. Since the axion-like deformations are by now
well understood geometrically for the S-fold backgrounds at
$\,\tilde{\varphi}=1\,$ [36], it would be interesting to investigate whether a
higher-dimensional geometric interpretation at arbitrary values of
$\,\tilde{\varphi}\,$ is still be possible.

Setting $\,\varphi=0\,$, the half-maximal spectrum of dual operators contains
the five unprotected long multiplets

$L\bar{L}[0]_{\Delta_{1}}^{0}\hskip 14.22636pt,\hskip
14.22636ptL\bar{L}[\tfrac{1}{2}]_{\Delta}^{0}\,\,\,\,(\times 2)\hskip
14.22636pt,\hskip 14.22636ptL\bar{L}[0]_{\tilde{\Delta}_{\pm}}^{0}\ ,$ (4.32)

this time with conformal dimensions given by

$\begin{array}[]{ccl}\Delta_{1}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{9+25\tilde{\varphi}^{2}}{1+\tilde{\varphi}^{2}}}\
,\\\\[11.38109pt]
\Delta&=&\frac{1}{2}+\frac{\sqrt{\left(1+\tilde{\varphi}^{2}\right)^{2}+\chi^{2}}}{\sqrt{1+\tilde{\varphi}^{2}}}\
,\\\\[11.38109pt]
\tilde{\Delta}_{\pm}&=&\frac{1}{2}+\frac{1}{2}\sqrt{\frac{8\tilde{\varphi}^{4}+5\tilde{\varphi}^{2}+8\chi^{2}+5\pm
4\sqrt{\Theta_{\varphi\to 0}}}{1+\tilde{\varphi}^{2}}}\ ,\end{array}$ (4.33)

with

$\Theta_{\varphi\to
0}=\left(\tilde{\varphi}^{2}\left(1+2\tilde{\varphi}^{2}\right)+1-2\chi^{2}\right)^{2}+4\left(\tilde{\varphi}^{2}-1\right)^{2}\chi^{2}.$
(4.34)

There are also the short and semi-short protected multiplets given in (4.27).
As a check of consistency, the $\,\chi$-family of $\,\mathcal{N}=2\,$ S-folds
in [32] is recovered at $\,\tilde{\varphi}=1$. We also recover the results of
Section 3.6 upon setting $\,\tilde{\varphi}=0\,$ together with
$\chi_{1}=-\chi_{3}=\chi\,$ and $\,\chi_{2}=0\,$ (up to a
$\,\textrm{U}(1)_{\gamma}\,$ transformation (see eq.(4.36)) of angle
$\,\gamma=-\frac{\pi}{4}$).

###  4.6 On the conformal manifold of $\,\mathcal{N}=2\,$ CFT3’s

Given the supergravity model in Section 4.1, which includes vector fields and
gaugings of scalar isometries, the holographic Zamolodchikov metric on a
conformal manifold of CFT3’s cannot be obtained simply by direct substitution
of the AdS4 solution (4.21)-(4.22) into the scalar kinetic terms (4.11). The
reason being that a solution like (4.21)-(4.22) can be brought to a different,
but physically equivalent, form upon a gauge transformation.

The infinitesimal
$\,\textrm{U}(1)_{\gamma}\times\mathbb{R}_{a}\times\mathbb{R}_{\epsilon}\,$
gauge transformations entering the covariant derivatives in (4.13) – recall
that all the fields within this sector are invariant under
$\,\textrm{U}(1)_{\textrm{R}}$ – can be integrated to finite transformations.
To describe such finite transformations we will introduce three complex fields

$z=h+i\,e^{-\phi_{2}}\hskip 14.22636pt,\hskip
14.22636pt\psi_{0}=\zeta_{0}+i\,\tilde{\zeta}_{0}\hskip 14.22636pt,\hskip
14.22636pt\psi_{1}=\tilde{\zeta}_{1}+i\,\zeta_{1}\ ,$ (4.35)

in terms of which the compact $\,\textrm{U}(1)_{\gamma}\,$ transformation acts
as

$\psi_{0}\rightarrow e^{i\,\gamma}\,\psi_{0}\hskip 28.45274pt\text{ and
}\hskip 28.45274pt\psi_{1}\rightarrow e^{i\,\gamma}\,\psi_{1}\ ,$ (4.36)

the non-compact $\,\mathbb{R}_{a}\,$ acts as a shift

$a\rightarrow a+c_{a}\ ,$ (4.37)

and $\,\mathbb{R}_{\epsilon}\,$ acts as a fractional linear transformation

$z\rightarrow\frac{z}{\epsilon\,z+1}\hskip 28.45274pt\text{and}\hskip
28.45274pt\begin{pmatrix}\psi_{0}\\\
\psi_{1}\end{pmatrix}\rightarrow\begin{pmatrix}1-i\,\frac{\epsilon}{2}&\frac{\epsilon}{2}\\\
\frac{\epsilon}{2}&1+i\,\frac{\epsilon}{2}\end{pmatrix}\,\begin{pmatrix}\psi_{0}\\\
\psi_{1}\end{pmatrix}\ .$ (4.38)

As a result, the gauge-fixed solution in (4.21)-(4.22) can be gauge-released
by acting on it with (4.36)-(4.38). This action introduces three additional
(yet unphysical) parameters $\,(\gamma,c_{a},\epsilon)\,$ in the solution
(4.21)-(4.22), and the naive pull-back of the metric (4.11) on different
gauge-fixed solutions will depend on the choice of gauge. For example,
choosing $\,c_{a}=f(\varphi,\chi)\,$ in (4.37) and performing the pull-back of
the metric (4.11), one encounters different Zamolodchikov metrics for
different choices of the function $\,f(\varphi,\chi)\,$. Of course, the catch
is that we are considering solutions which are gauge-equivalent, and therefore
physically equivalent, as different. In order to perform the gauge-fixing
properly, one should not pick up a subspace within the gauge-released space of
solutions and perform the naive pull-back of the metric onto it. Instead, one
must study the quotient of the gauge-released space of solutions by the gauge
group. This quotient space is the one being dual to a conformal manifold with
a uniquely defined Zamolodchikov metric.

For the sake of concreteness, let us particularise to our specific gauge-
released or ambient scalar geometry as described by the kinetic terms in
(4.11). We first need to identify three independent one-forms on the scalar
geometry which are to be declared as “pure gauge” or unphysical, and then
quotient the scalar geometry by them. How to identify such three one-forms in
field space is a physical question. And the answer to that question comes from
the vector sector which, despite being set to zero at the supergravity
solution, still provides equations of motion that must hold. In short, we must
quotient the geometry by the one-form currents $\,J^{{}^{(\gamma)}}$,
$\,J^{{}^{(a)}}$ and $\,J^{{}^{(\epsilon)}}$ acting as sources for the vector
fields that have been set to zero at the supergravity solution. This implies
that we should first put those one-forms to zero in the kinetic terms (4.11)
before reading off the Zamolodchikov metric by performing the pull-back of the
ambient metric on any gauge-fixed subspace of solutions.

Let us illustrate the procedure described above by looking at the gauge-fixing
of the $\,\mathbb{R}_{a}\,$ shift symmetry in (4.37) spanned by the vector
field $\,A^{{}^{(a)}}$. The scalar $\,a\,$ plays the role of a Stückelberg
field for the massive vector $\,A^{{}^{(a)}}$ – recall that
$\,Da=da+2\,g\,A^{{}^{(a)}}$ in (4.12) – and the associated current computed
from (4.11) reads

$J^{{}^{(a)}}\equiv\,g\,e^{2\phi_{1}}*\left[Da+\tfrac{1}{2}\big{(}\zeta^{0}D\tilde{\zeta}_{0}+\zeta^{1}D\tilde{\zeta}_{1}-\tilde{\zeta}_{0}D\zeta^{0}-\tilde{\zeta}_{1}D\zeta^{1}\big{)}\right]\
.$ (4.39)

Quotienting the scalar geometry by this (field space) one-form implies that
(4.39) must be set to zero identically, i.e. $J^{{}^{(a)}}=0\,$, when
evaluating any quantity at a gauge-fixed solution like (4.21)-(4.22). In
particular, its contribution to the third line in the kinetic terms (4.11)
must be removed before reading off the Zamolodchikov metric from it.
Proceeding similarly with the contributions coming from the remaining one-form
currents $\,J^{{}^{(\epsilon)}}\,$ and $\,J^{{}^{(\gamma)}}\,$ acting as
sources for $\,A^{{}^{(\epsilon)}}\,$ and $\,A^{{}^{(\gamma)}}$, the resulting
Zamolodchikov metric becomes independent of the deformation parameter
$\,\tilde{\varphi}\,$ and reads

$ds_{\textrm{CM}}^{2}=\dfrac{1+2\varphi^{2}}{2(1+\varphi^{2})^{2}}\big{(}d\varphi^{2}+(1+\varphi^{2})\,d\chi^{2}\big{)}\
.$ (4.40)

This metric in the conformal manifold of $\,\mathcal{N}=2\,$ CFT3’s at generic
$\,\tilde{\varphi}\,$ agrees with that of [1] upon the identification
$\,\chi=\sqrt{1+\tilde{\varphi}^{2}}\,\chi_{\scriptsize{\cite[cite]{[\@@bibref{}{Bobev:2021yya}{}{}]}}}$
with $\,\tilde{\varphi}=1\,$.

The moduli $\,(\varphi,\chi)\,$ belong to the hypermultiplet sector in the
AdS4 solution (4.21)-(4.22). Recalling that, for AdS4 solutions preserving
$\,\mathcal{N}=2\,$, the hypermultiplet moduli space must be a Kähler
submanifold of the quaternionic Kähler geometry [56], the conformal manifold
in (4.40) must be Kähler. The Kählericity of the metric (4.40) can be checked
as follows. Let us first introduce a set of so-called isothermal coordinates
for which the metric is conformal to the Euclidean metric. These are given by
$\,x=\chi\,$ and $\,y=\textrm{arcsinh}\varphi\,$ so that the Zamolodchikov
metric in (4.40) is brought to the form

$ds_{\textrm{CM}}^{2}=\tfrac{1}{2}\,\Omega(y)^{2}\left(dx^{2}+dy^{2}\right)\hskip
22.76219pt\textrm{ with }\hskip 22.76219pt\Omega(y)^{2}\equiv 1+\tanh^{2}y\ .$
(4.41)

Introducing the complex coordinate $\,z=x+iy\,$ one arrives at

$ds_{\textrm{CM}}^{2}=g_{z\bar{z}}\,dz\,d\bar{z}\hskip 22.76219pt\textrm{ with
}\hskip
22.76219ptg_{z\bar{z}}=\tfrac{1}{2}\,\left(1+\tanh^{2}\left[\tfrac{-i(z-\bar{z})}{2}\right]\right)\
,$ (4.42)

where $\,g_{z\bar{z}}=\frac{\partial^{2}K}{\partial z\partial\bar{z}}\,$ can
be derived from the real Kähler potential

$K(z,\bar{z})=|z|^{2}-\log\left[\cosh^{2}\left(\tfrac{-i(z-\bar{z})}{2}\right)\right]\
.$ (4.43)

#### More on gauge-fixing and Zamolodchikov metric

The scalar manifold $\,\mathcal{M}_{\text{scal}}\,$ in (2.5) of half-maximal
supergravity is endowed with a canonical Riemannian metric $\,g$, prior to any
gauge fixing, and a left action of a gauge group G , e.g.,
$\,\textrm{G}=\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ in our case. In
general, the action of G on $\,\mathcal{M}_{\text{scal}}\,$ is not free –
there are fixed loci under the action of the compact part of G – and thus the
quotient space $\,\textrm{G}\backslash\mathcal{M}_{\text{scal}}\,$ is not a
manifold. In supergravity, the action of G on $\,\mathcal{M}_{\text{scal}}\,$
is well-behaved and we can chop the fixed loci out of
$\,\mathcal{M}_{\text{scal}}\,$ to define a new manifold
$\,\tilde{M}_{\text{scal}}\,$ on which G has a free action.

In order to establish a connection with the $\,\mathcal{N}=2\,$ supergravity
model of Section 4.1, we will focus on the
$\,\textrm{U}(1)_{\textrm{R}}$-invariant sector of half-maximal supergravity
for which
$\,\textrm{G}\rightarrow{\textrm{G}_{\mathcal{N}=2}=\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)_{\gamma}\times\mathbb{R}_{a}\times\mathbb{R}_{\epsilon}}\,$
and $\,\mathcal{M}_{\text{scal}}\,$ in (2.5) reduces to the one in (4.4). From
(4.13), the $\,\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)_{\gamma}\,$
compact part of $\,\textrm{G}_{\mathcal{N}=2}\,$ leaves invariant the scalar
locus defined by the condition
$\,\tilde{\zeta}_{0}=\tilde{\zeta}_{1}=\zeta^{0}=\zeta^{1}=0\,$.181818In the
gauge-fixed supergravity solution of (4.21)-(4.22), this locus corresponds to
setting to zero the marginal deformations, i.e., $\,\varphi=\chi=0\,$.
Starting from the gauge-released solution extending the gauged-fixed one in
(4.21)-(4.22) with three parameters $\,(\gamma,c_{a},\epsilon)\,$ (see
discussion below (4.38)) and removing the fixed locus under
$\,\textrm{U}(1)_{\gamma}$, we can finally define a manifold of supergravity
solutions $\,\mathcal{S}\subset\tilde{\mathcal{M}}_{\text{scal}}\,$ on which
$\,\textrm{G}_{\mathcal{N}=2}\,$ acts freely. This gives a structure of
principal bundle

$\pi:\mathcal{S}\rightarrow\textrm{G}_{\mathcal{N}=2}\backslash\mathcal{S}\ .$
(4.44)

The quotient space $\,\textrm{G}_{\mathcal{N}=2}\backslash\mathcal{S}\,$ is
the object dual to the $\,\mathcal{N}=2\,$ conformal manifold of CFT3’s, and
it is on this quotient space that we must define metric $\,g_{\textrm{CM}}\,$
dual to the Zamolodchikov metric in (4.40).

We can always decompose the tangent space of $\,\mathcal{S}\,$ as
$\,T\mathcal{S}=V\mathcal{S}\oplus H\mathcal{S}\,$ where
$\,V\mathcal{S}=\text{ker}\,\pi_{*}\,$ and
$\,H\mathcal{S}=[V\mathcal{S}]^{\perp}\,$ is the orthogonal complement of
$\,V\mathcal{S}\,$ with respect to the metric $\,g$. The tangent space
$\,T\mathcal{S}\,$ should then be understood as the space of all small
deformations: $\,H\mathcal{S}\,$ corresponding to physical deformations and
$\,V\mathcal{S}\,$ being the space of unphysical deformations. The latter are
exactly the infinitesimal gauge transformations, and a projector
$\,\text{Pr}_{H\mathcal{S}}\,$ onto $\,H\mathcal{S}\,$ can be defined that
projects them away. Finally, for any
$\,x\in\textrm{G}_{\mathcal{N}=2}\backslash\mathcal{S}\,$ there is a
$\,p\in\mathcal{S}\,$ such that $\,\pi(p)=x$. Then, for any pair of vectors
$\,(v,\,v^{\prime})\,$ in
$\,T(\textrm{G}_{\mathcal{N}=2}\backslash\mathcal{S})\,$ we can choose vectors
$\,(w,\,w^{\prime})\,$ in $\,T\mathcal{S}\,$ such that $\,\pi_{*}w=v\,$ and
$\,\pi_{*}w^{\prime}=v^{\prime}$. In this manner the Zamolodchikov metric can
be defined as the map

$g_{\textrm{CM}}:T(\textrm{G}_{\mathcal{N}=2}\backslash\mathcal{S})^{\otimes
2}\rightarrow\mathbb{R}:v\otimes v^{\prime}\rightarrow
g_{\textrm{CM}}(v,\,v^{\prime})=g\left(\,\text{Pr}_{H\mathcal{S}}w,\,\text{Pr}_{H\mathcal{S}}w^{\prime}\,\right)\,.$
(4.45)

This is a well defined metric on
$\,\textrm{G}_{\mathcal{N}=2}\backslash\mathcal{S}\,$ and it is equivalent to
the prescription we have given in our example. Importantly, it does not depend
on the choice of gauge fixing nor the invariant subsector used to find the
supergravity solution within the full theory.

##  5 Final remarks

In the present paper we have initiated a holographic study of new CFT3’s with
$\,\mathcal{N}=2,3,4\,$ supersymmetry using an effective four-dimensional
gauged supergravity approach. The rich structure of multi-parametric families
of supersymmetric AdS4 solutions we have just started to identify in the half-
maximal gauged supergravities with
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings raises some
immediate questions.

Perhaps the most obvious question is whether or not the
$\,\tilde{\varphi}$-family of
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings of half-maximal
supergravity we have presented in Sections 2.3 and 2.4 (and, more ambitiously,
its generalisation in Appendix A) describes consistent truncations of ten- or
eleven-dimensional supergravity down to four dimensions. In this respect,
since turning on the embedding parameter $\,\tilde{\varphi}\,$ (i.e.
$\,\tilde{\varphi}\neq\pm 1\,$ in our parameterisation) only affects the
embedding of the non-compact translational generators in the gauge algebra
(see e.g. (2.25)-(2.26)), the family of $\,\mathcal{N}=2\,$ AdS4 solutions we
have found may stand a chance of being upliftable to new (possibly only
locally geometric) type II or M-theory backgrounds. However, it could still
happen – as for the $\,\omega\,$ deformation of the $\textrm{SO}(8)$-gauged
supergravity [57] – that only very specific values of $\,\tilde{\varphi}\,$
enjoy a higher-dimensional interpretation, the natural ones being
$\,\tilde{\varphi}=\pm 1\,\textrm{ and }\,0\,$. The case
$\,\tilde{\varphi}=\pm 1\,$ is by now known to uplift to type IIB S-fold
backgrounds. Examples are the $\,\mathcal{N}=4\,$ S-fold of [25] and its
marginal deformations [32, 1, 41]. The case $\,\tilde{\varphi}=0\,$ remains to
be understood. But it would certainly be disappointing if a supergravity
solution like the exotic AdS4 vacuum of [40] with such a (conjectured but)
highly symmetric $\,\mathcal{N}=4\,$ CFT3 dual – together with its marginal
deformations presented in Sections 3.6 and 4.5.1 – ended up being in the
Swampland.

Let us further comment on the exactly marginal deformation of the exotic
$\,\mathcal{N}=4\,$ CFT3 dual to the modulus $\,\chi\,$ in (4.22) preserving
$\,\mathcal{N}=2\,$ supersymmetry. As discussed in Section 3.6.1, turning on
$\,\chi\,$ breaks the original $\,\textrm{SO(4)}_{\textrm{R}}\,$ symmetry down
to a $\,\textrm{U}(1)_{\textrm{R}}\,$ factor within the Cartan subgroup
$\,\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)_{\textrm{F}}\subset\textrm{SO}(4)_{\textrm{R}}$.
As a result, the whole Cartan subgroup is not preserved and a geometric
interpretation of the modulus $\,\chi\,$ along the lines of the axion-like
deformations of S-folds seems a priori unplausible within the context of half-
maximal supergravity. However, the situation is more subtle: when setting
$\,\tilde{\varphi}=\pm 1\,$, the embedding of this solution into maximal
supergravity provides an additional flavour current multiplet
$\,A_{2}\bar{A}_{2}[0]_{1}^{0}\,$ that accounts for the
$\,\textrm{U}(1)_{\textrm{F}}\,$ symmetry [1]. In this case, the
$\,\textrm{U}(1)_{\textrm{F}}\subset\textrm{SO}(4)_{\textrm{R}}\,$ is actually
not broken but projected out by the $\,\mathbb{Z}_{2}\,$ symmetry truncating
maximal to half-maximal supergravity. Therefore, if a ten- or eleven-
dimensional uplift exists for the AdS4 solution (4.21)-(4.22) at
$\,\tilde{\varphi}=0\,$ (or more generically at $\,\tilde{\varphi}\neq\pm
1\,$), it is still possible that some symmetries have been truncated away in
the half-maximal effective description. Needless to say, an uplift (if any) to
ten or eleven dimensions is required in order to settle this question.

The effective four-dimensional gauged supergravity approach adopted in this
work provides us with some guidance in order to guess what the potential
higher-dimensional realisation of the AdS4 solutions could be. Remarkably,
amongst the extra quadratic constraints in (2.8), only those in the
$\,(\mathbf{1},\mathbf{462^{\prime}})\,$ irrep of
$\,\textrm{SL}(2)\times\textrm{SO}(6,6)\,$ are violated in the AdS4 solutions
with $\,\tilde{\varphi}\neq\pm 1$.191919The extra quadratic constraints in
(2.8) living in the $\,(\mathbf{3},\mathbf{1})\,$ irrep are not violated in
our half-maximal supergravity models. These constraints appeared as the four-
dimensional incarnation of the $\,\textrm{SL}(2)\,$ extension [58] of the so-
called section constraint in double field theory [59]. Assuming (as for the
S-folds) a type IIB origin, and since in a type IIB duality frame the
$\,\textrm{SL}(2)\,$ factor of the duality group is identified with S-duality,
one possibility is the presence of $\,\textrm{SL}(2)$-singlet branes (or bound
states) in the corresponding ten-dimensional backgrounds. It could be
interesting to look at possible uplifts incorporating such branes in a smeared
limit. For example, try to add smeared D3-branes to the S-fold setups
previously investigated in the literature. This could shed some light on how
to incorporate sources in S-fold backgrounds. Another possibility is that the
breaking of supersymmetries in the four-dimensional supergravity Lagrangian
(from maximally to half-maximally supersymmetric) is not related to the
inclusion of sources but, instead, it stems from geometry. This would be more
in the spirit of the half-maximal consistent truncations of [60] and, perhaps,
some generalised frame could be constructed for these half-maximal
supergravities along the lines of [58] in order to systematically uplift any
four-dimensional solution.

Another interesting line to explore is the possible relation between the AdS4
solutions of the $\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings
of half-maximal supergravity and various classes of type IIB backgrounds of
the form $\,\textrm{AdS}_{4}\times\textrm{M}_{6}\,$ that have been constructed
directly in ten dimensions using different techniques: pure spinor formalism,
G-structures, non-abelian T-duality, … (see [61, 62, 63, 64] for an incomplete
list). Identifying the field theory duals of these ten-dimensional solutions
is a laborious and generically non-systematic task: one first makes an
educated guess for the field theory duals and then runs as many holographic
tests as possible. Thinking along these lines, it would be very interesting to
establish whether or not the general eight-parameter family of
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings of half-maximal
supergravity we have presented in Appendix A describes classes of consistent
truncations of type IIB supergravity on
$\,\textrm{M}_{6}=\textrm{S}^{2}\times\textrm{S}^{2}\times\Sigma\,$ with
$\,\Sigma\,$ being a Riemann surface. If such a connection exists and is well
established via generalised geometry or extended field theory techniques, then
exploiting the four-dimensional effective description would provide a way to
characterise the CFT3’s dual to such type IIB solutions (presumably related to
IR fixed points of Gaiotto–Hanany–Witten-like brane constructions) without
having to work out their ten-dimensional uplift explicitly. For example, as we
have done in this work, the conformal dimensions of the low lying operators in
the dual CFT3’s could be extracted directly using four-dimensional data,
namely, from the mass spectrum of the supergravity fields in the half-maximal
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gauged supergravity. Also
the new techniques for Kaluza-Klein (KK) spectrometry put forward in [65]
could be applied to the type IIB backgrounds of the form
$\,\textrm{AdS}_{4}\times\textrm{S}^{2}\times\textrm{S}^{2}\times\Sigma\,$
(see [35, 37, 33, 66] for a study of the spectrum of KK modes around the type
IIB S-folds at $\,\tilde{\varphi}=\pm 1\,$) upon suitable adjustment of the
techniques to the context of half-maximal supergravity.

Finally, the analysis performed in Appendix A shows that the results in the
main text can be straightforwardly generalised to include different embedding
parameters $\,\tilde{\varphi}_{1,2}\,$ and $\,c_{1,2}$, as well as independent
gauge couplings $\,g_{1,2}$, for each of the $\,\textrm{ISO}(3)_{1,2}\,$
factors in the gauge group. In particular, having two independent gauge
couplings $\,g_{1,2}\,$ permits to collapse or flatten-out one
$\,\textrm{S}^{2}\,$ while keeping the other $\,\textrm{S}^{2}\,$ at finite
size. This suggests an a priori much larger structure of supersymmetric AdS4
solutions with new potentially interesting CFT3 duals. Also going beyond the
$\,\mathbb{Z}_{2}^{2}\,$ and $\,\textrm{U}(1)_{\textrm{R}}\,$ invariant
sectors investigated in this work could accommodate new families of AdS4
solutions with additional flat directions dual to new marginal deformations in
the dual CFT3’s. These are all open questions and aspects we plan to continue
exploring in the future.

## Acknowledgements

We are grateful to Antón Faedo, Emanuel Malek, Carlos Núñez and Daniel Waldram
for interesting conversations. The work of AG and MCh-B is partially supported
by the Ministry of Science and Universities through the Spanish grant
MCIU-22-PID2021-123021NB-I00 and by FICYT through the Asturian grant SV-
PA-21-AYUD/2021/52177. CS is supported by IISN-Belgium (convention 4.4503.15)
and is a Research Fellow of the F.R.S.-FNRS (Belgium).

##  Appendix A $\textrm{ISO}(3)\times\textrm{ISO}(3)$ gaugings of half-maximal
supergravity

In this appendix we analyse the set of possible embeddings of an
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gauging of half-maximal
supergravity of the form

$\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,\,\subset\,\,\textrm{SL}(2)\times\textrm{SO}(3,3)_{1}\times\textrm{SO}(3,3)_{2}\,\,\subset\,\,\textrm{SL}(2)\times\textrm{SO}(6,6)\
.$ (A.1)

We have attached labels $\,{}_{1}\,$ and $\,{}_{2}\,$ to keep track of each
repeated factor. Following the notation of [40], and building upon the results
of [48], a gauging of this type is totally encoded in a set of eight embedding
matrices [67]. Since $\,\textrm{SO}(3,3)_{1}\sim\textrm{SL}(4)_{1}\,$, the
most general embedding202020This is so up to equivalent solutions of the
quadratic constraints in (2.7). of the first $\,\textrm{ISO}(3)_{1}\,$ factor
is encoded in four $\,4\times 4\,$ matrices given by

$Q_{+}^{(1)}=\left(\begin{array}[]{cc}-a_{0}^{\prime}&0\\\
0&0\times\mathbb{I}_{3}\end{array}\right)\hskip 22.76219pt,\hskip
22.76219pt\bar{Q}^{(1)}_{+}=\left(\begin{array}[]{cc}0&0\\\
0&\tilde{c}_{1}^{\prime}\times\mathbb{I}_{3}\end{array}\right)\ ,$ (A.2)
$Q_{-}^{(1)}=\left(\begin{array}[]{cc}-b_{0}^{\prime}&0\\\
0&0\times\mathbb{I}_{3}\end{array}\right)\hskip 22.76219pt\,,\hskip
22.76219pt\bar{Q}^{(1)}_{-}=\left(\begin{array}[]{cc}0&0\\\
0&\tilde{d}_{1}^{\prime}\times\mathbb{I}_{3}\end{array}\right)\ ,$ (A.3)

with $\,\tilde{c}_{1}^{\prime}\neq 0\,$. Equivalently, the first
$\,\textrm{ISO}(3)_{1}\,$ factor is specified by four embedding tensor
components of the form

$f_{+\bar{a}bc}=\tilde{c}_{1}^{\prime}\,\epsilon_{\bar{a}bc}\hskip
14.22636pt,\hskip 14.22636ptf_{-abc}=-b_{0}^{\prime}\,\epsilon_{abc}\hskip
14.22636pt,\hskip 14.22636ptf_{+abc}=-a_{0}^{\prime}\,\epsilon_{abc}\hskip
14.22636pt,\hskip
14.22636ptf_{-\bar{a}bc}=\tilde{d}_{1}^{\prime}\,\epsilon_{\bar{a}bc}\ .$
(A.4)

Analogously, the second $\,\textrm{ISO}(3)_{2}\,$ factor is encoded in another
set of four $\,4\times 4\,$ matrices of the form

$Q_{+}^{(2)}=\left(\begin{array}[]{cc}0&0\\\
0&\tilde{c}_{2}\times\mathbb{I}_{3}\end{array}\right)\hskip 22.76219pt,\hskip
22.76219pt\bar{Q}^{(2)}_{+}=\left(\begin{array}[]{cc}a_{3}&0\\\
0&0\times\mathbb{I}_{3}\end{array}\right)\ ,$ (A.5)
$Q_{-}^{(2)}=\left(\begin{array}[]{cc}0&0\\\
0&\tilde{d}_{2}\times\mathbb{I}_{3}\end{array}\right)\hskip 22.76219pt,\hskip
22.76219pt\bar{Q}^{(2)}_{-}=\left(\begin{array}[]{cc}b_{3}&0\\\
0&0\times\mathbb{I}_{3}\end{array}\right)\ ,$ (A.6)

with $\,\tilde{d}_{2}\neq 0\,$. The components of the embedding tensor for the
second $\,\textrm{ISO}(3)_{2}\,$ factor are then given by

$f_{-i\bar{j}\bar{k}}=\tilde{d}_{2}\,\epsilon_{i\bar{j}\bar{k}}\hskip
14.22636pt,\hskip
14.22636ptf_{+\bar{i}\bar{j}\bar{k}}=a_{3}\,\epsilon_{\bar{i}\bar{j}\bar{k}}\hskip
14.22636pt,\hskip
14.22636ptf_{-\bar{i}\bar{j}\bar{k}}=b_{3}\,\epsilon_{\bar{i}\bar{j}\bar{k}}\hskip
14.22636pt,\hskip
14.22636ptf_{+i\bar{j}\bar{k}}=\tilde{c}_{2}\,\epsilon_{i\bar{j}\bar{k}}\ .$
(A.7)

Together, (A.4) and (A.7) account for all the components of the embedding
tensor $\,f_{\alpha MNP}\,$ that are activated in the class (A.1) of
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings of half-maximal
supergravity we investigate in this work.

###  A.1 Quadratic constraints and algebra structure

The embedding tensor components in (A.4) and (A.7) automatically satisfy the
quadratic constraints of half-maximal supergravity. However, the computation
of the additional constraints in (2.8) for this multi-parameteric family of
$\,\textrm{ISO}(3)_{1}\times\textrm{ISO}(3)_{2}\,$ gaugings yields

$f_{\alpha MNP}\,f_{\beta}{}^{MNP}=0\hskip 17.07164pt\textrm{ and }\hskip
17.07164pt\left.\epsilon^{\alpha\beta}\,f_{\alpha[MNP}\,f_{\beta
QRS]}\right|_{\textrm{SD}}=0\,\,\Leftrightarrow\,\,\left\\{\begin{array}[]{c}b_{0}^{\prime}\,\tilde{c}_{2}-a_{0}^{\prime}\,\tilde{d}_{2}=0\\\\[5.69054pt]
b_{3}\,\tilde{c}_{1}^{\prime}-a_{3}\,\tilde{d}_{1}^{\prime}=0\end{array}\right.\
.$ (A.8)

The antisymmetry of the commutators $\,[\,T_{\alpha M}\,,\,T_{\beta
N}\,]=f_{\alpha MN}{}^{P}\,T_{\beta P}\,$ for this general class of embeddings
imposes a set of linear relations between the generators of the form

$(\tilde{c}_{1}^{\prime})^{2}\,T_{-a}=\tilde{d}_{1}^{\prime}\,\tilde{c}_{1}^{\prime}\,T_{+a}+(a_{0}^{\prime}\,\tilde{d}_{1}^{\prime}-b_{0}^{\prime}\,\tilde{c}_{1}^{\prime})\,T_{+\bar{a}}\hskip
22.76219pt,\hskip
22.76219pt\tilde{c}_{1}^{\prime}\,T_{-\bar{a}}=\tilde{d}_{1}^{\prime}\,T_{+\bar{a}}\
,$ (A.9)

and

$(\tilde{d}_{2})^{2}\,T_{+\bar{i}}=\tilde{d}_{2}\,\tilde{c}_{2}\,T_{-\bar{i}}+(a_{3}\,\tilde{d}_{2}-b_{3}\,\tilde{c}_{2})\,T_{-i}\hskip
22.76219pt,\hskip 22.76219pt\tilde{d}_{2}\,T_{+i}=\tilde{c}_{2}\,T_{-i}\ .$
(A.10)

Choosing the independent generators to be $\,(T_{+a},T_{+\bar{a}})\,$ and
$\,(T_{-i},T_{-\bar{i}})\,$, one finds a set of non-trivial commutation
relations of the form

$\begin{array}[]{llll}\left[\,T_{+a},T_{+b}\,\right]&=&\tilde{c}_{1}^{\prime}\,\epsilon_{ab}{}^{c}\,T_{+c}-a_{0}^{\prime}\,\epsilon_{ab}{}^{\bar{c}}\,T_{+\bar{c}}\
,\\\\[11.38109pt]
\left[\,T_{+a},T_{+\bar{b}}\,\right]&=&\tilde{c}_{1}^{\prime}\,\epsilon_{a\bar{b}}{}^{\bar{c}}\,T_{+\bar{c}}\
,\\\\[11.38109pt] \left[\,T_{+\bar{a}},T_{+\bar{b}}\,\right]&=&0\
,\end{array}$ (A.11)

for the first ISO(3)1 factor in the gauge group and, similarly,

$\begin{array}[]{llll}\left[\,T_{-\bar{i}},T_{-\bar{j}}\,\right]&=&\,\tilde{d}_{2}\,\epsilon_{\bar{i}\bar{j}}{}^{\bar{k}}\,T_{-\bar{k}}+b_{3}\,\epsilon_{\bar{i}\bar{j}}{}^{k}\,T_{-k}\
,\\\\[11.38109pt]
\left[\,T_{-\bar{i}},T_{-j}\,\right]&=&\tilde{d}_{2}\,\epsilon_{\bar{i}j}{}^{k}\,T_{-k}\
,\\\\[11.38109pt] \left[\,T_{-i},T_{-j}\,\right]&=&0\ ,\end{array}$ (A.12)

for the second ISO(3)2 factor. Note that, for each of the
$\,\textrm{ISO}(3)_{1,2}\,$ factors, there are two parameters entering the
commutation relations in (A.11) and (A.12) and two additional parameters
specifying the linear combinations of generators in (A.9) and (A.10).

###  A.2 $\mathcal{N}=1\,$ superpotentials

It is also interesting to investigate the dynamics of the seven moduli fields
$\,z_{I}\,$, $\,I=1,\ldots,7\,$, in the $\,\mathbb{Z}_{2}^{2}$-invariant
sector of half-maximal supergravity coupled to six vector multiplets. This
sector is described by the $\,\mathcal{N}=1\,$ supergravity multiplet coupled
to seven chiral superfields with Kähler potential

$K=-\sum_{I=1}^{7}\log[-i(z_{I}-\bar{z}_{I})]\ ,$ (A.13)

and a superpotential given by

$\begin{array}[]{lll}W&=&\Big{[}-a_{3}+a_{0}^{\prime}\,z_{4}z_{5}z_{6}-\tilde{c}_{2}\,(z_{1}z_{4}+z_{2}z_{5}+z_{3}z_{6})+\tilde{c}_{1}^{\prime}\,(z_{1}z_{5}z_{6}+z_{2}z_{4}z_{6}+z_{3}z_{4}z_{5})\Big{]}\\\\[17.07164pt]
&+&\Big{[}\phantom{-}b_{3}-b_{0}^{\prime}\,z_{4}z_{5}z_{6}+\tilde{d}_{2}\,(z_{1}z_{4}+z_{2}z_{5}+z_{3}z_{6})-\tilde{d}_{1}^{\prime}\,(z_{1}z_{5}z_{6}+z_{2}z_{4}z_{6}+z_{3}z_{4}z_{5})\Big{]}\,z_{7}\
.\end{array}$ (A.14)

Note that the eight gauging parameters in (A.4) and (A.7) enter the
superpotential of the model.

In this work we have made a simple choice of gauging parameters. More
concretely, we have chosen the same embedding for the two
$\,\textrm{ISO(3)}_{1,2}\,$ factors in the gauging. This choice drastically
simplifies the analysis of supersymmetric vacua. These vacua satisfy the set
of supersymmetric (or F-flatness) conditions

$F_{I}\equiv D_{I}W=\partial_{I}W+(\partial_{I}K)W=0\ ,$ (A.15)

with $\,I=1,\ldots 7\,$.

###  A.3 Back to our model

The specific model discussed in Section 2.3 corresponds to a simple fixing of
the gauging parameters in (A.4) of the form
$\,\tilde{d}_{1}^{\prime}=\tilde{c}_{2}=0\,$ and

$\begin{array}[]{rcrcrc}\tilde{c}_{1}^{\prime}=\frac{2\,\sqrt{2}\,g}{\sqrt{1+\tilde{\varphi}^{2}}}&,&-b_{0}^{\prime}=\pm
2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}}{\sqrt{1+\tilde{\varphi}^{2}}}&,&-a_{0}^{\prime}=-2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}^{2}-1}{\tilde{\varphi}^{2}+1}&,\\\\[17.07164pt]
\tilde{d}_{2}=\frac{2\,\sqrt{2}\,g}{\sqrt{1+\tilde{\varphi}^{2}}}&,&a_{3}=\pm
2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}}{\sqrt{1+\tilde{\varphi}^{2}}}&,&b_{3}=-2\sqrt{2}\,g\,c\,\frac{\tilde{\varphi}^{2}-1}{\tilde{\varphi}^{2}+1}&.\end{array}$
(A.16)

Plugging (A.16) into (A.14) yields a superpotential

$\begin{array}[]{lll}W&=&\dfrac{2\,\sqrt{2}\,g}{\sqrt{1+\tilde{\varphi}^{2}}}\,\Big{[}z_{1}z_{5}z_{6}+z_{2}z_{4}z_{6}+z_{3}z_{4}z_{5}+(z_{1}z_{4}+z_{2}z_{5}+z_{3}z_{6})\,z_{7}\Big{]}\\\\[11.38109pt]
&-&\dfrac{2\,\sqrt{2}\,g}{\sqrt{1+\tilde{\varphi}^{2}}}\,c\,\Big{[}\pm\tilde{\varphi}\,(1-z_{4}z_{5}z_{6}z_{7})+\dfrac{1-\tilde{\varphi}^{2}}{\sqrt{1+\tilde{\varphi}^{2}}}(z_{4}z_{5}z_{6}-z_{7})\Big{]}\
.\end{array}$ (A.17)

We have verified that the AdS4 solutions in (3.10) solve the F-flatness
equations in (A.15) constructed from (A.17).

Lastly, as a further check of consistency, setting $\,\tilde{\varphi}^{2}=1\,$
recovers the maximal theory. Namely, the superpotential in (A.17) reduces to

$W=2\,g\,\Big{[}z_{1}z_{5}z_{6}+z_{2}z_{4}z_{6}+z_{3}z_{4}z_{5}+(z_{1}z_{4}+z_{2}z_{5}+z_{3}z_{6})\,z_{7}\Big{]}\pm
2\,g\,c\,\left(1-z_{4}z_{5}z_{6}z_{7}\right)\ ,$ (A.18)

in agreement with the result of [32].

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