notesntList of Notes

MI-HET-822

Singular limits in STU supergravity

Gabriel Larios1 , Christopher N. Pope1,2 and Haoyu Zhang1

1 Mitchell Institute for Fundamental Physics and Astronomy,
Texas A&M University, College Station, TX, 77843, USA

2 DAMTP, Centre for Mathematical Sciences, Cambridge University,
Wilberforce Road, Cambridge CB3 OWA, UK

Abstract

> We analyse the STU sectors of the four-dimensional maximal gauged
> supergravities with gauge groups ${\rm SO(8)}$,
> $\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ and
> $[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$, and construct
> new domain-wall black-hole solutions in $D=4$. The consistent Kaluza-Klein
> embedding of these theories is obtained using the techniques of Exceptional
> Field Theory combined with the 4$d$ tensor hierarchies, and their respective
> uplifts into $D=11$ and type IIB supergravities are connected through
> singular limits that relate the different gaugings.

###### Contents

  1. 1 Introduction
  2. 2 Gauged STU Supergravity
    1. 2.1 Tensor and duality hierarchies
  3. 3 E7(7) Exceptional Field Theory
    1. 3.1 M-theory section
    2. 3.2 Type IIB section
    3. 3.3 Generalised Scherk-Schwarz Ansätze
  4. 4 Supergravity Embeddings
    1. 4.1 SO(8) gauging on $S^{7}$
    2. 4.2 $[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$ gauging on $S^{5}\times S^{1}$
  5. 5 Black Hole and Domain Wall Solutions
    1. 5.1 AdS-BH limits
    2. 5.2 DW-Ansätze
  6. 6 Discussion
  7. A STU Truncation
  8. B A $D=9$ Detour
    1. B.1 Reduction from $D=11$ to $D=9$
    2. B.2 Type IIB reduction to $9$ dimensions

##  1 Introduction

On certain backgrounds, string theory admits a consistent truncation to a
subset of modes in its Kaluza-Klein (KK) spectrum whose complete non-linear
dynamics is captured by a gauged supergravity in lower dimensions. Notable
examples include the reduction of eleven-dimensional supergravity on ${\rm
AdS}_{4}\times S^{7}$ [1, 2, 3, 4] and on ${\rm AdS}_{7}\times S^{4}$ [5, 6,
7], and the reduction of type IIB supergravity on ${\rm AdS}_{5}\times S^{5}$
[8, 9, 10].

Such consistent truncations are in general difficult to construct, but when
they exist they constitute a fundamental tool for obtaining solutions in ten
and eleven dimensions, as the consistency of the truncation guarantees that
every solution of the lower-dimensional gauged supergravity can be embedded
into a configuration that solves the equations of motion of the parent theory.
This approach has been particularly fruitful in holographic set-ups, where the
gauged supergravity techniques have made possible the construction of hundreds
of different AdS solutions – see [11, 12, 13, 14, 15] for recent surveys –
that are dual to different conformal field theories (CFTs) and can be used,
among other things, as a playground to check Swampland conjectures, such as
the AdS conjecture [16] or the CFT distance conjecture [17]. Furthermore, the
relative simplicity of the lower-dimensional theories has allowed the analysis
of other holographically relevant phenomena, such as black holes [18, 19]
describing finite temperature states in the CFT, or domain walls realising CFT
interfaces [20, 21] and RG-flows between different CFTs [22, 23].
Additionally, the existence of these consistent truncations also facilitates
even some intrinsically higher-dimensional computations such as the spectrum
masses of the KK modes [24].

Recently, reformulations of string theory based on its duality symmetries [25,
26] have played a pivotal rôle in our understanding of consistent truncations
[27, 28]. In fact, at present there are theorems [29] that guarantee the
existence of these reductions for wide classes of theories and backgrounds in
terms of suitable factorisations of duality-covariant fields. However,
explicit KK Ansätze for the standard higher-dimensional metric and fluxes
keeping the entire dependence on the lower-dimensional fields and their
derivatives are often not known due to the intricate nature of the dictionary
relating the original supergravities to the duality-covariant counterparts. A
convenient way to describe the embedding of these non-trivial profiles makes
use of the tensor hierarchy, which is a supplement to the $p$-forms in the
ungauged supergravity that are introduced so as to guarantee that the gauged
theory is formally covariant under the original global symmetry group [30, 31,
32, 33]. The redundancies so introduced can be eliminated at the level of the
field strengths, and an explicit KK Ansatz in terms of the original fields and
their derivatives can thus be obtained [34, 35, 36]. These extra forms allow
us to trade some complicated dualisations with respect to the internal metric
for much simpler dualisations with respect to the four-dimensional metric.

It has been observed [37, 15] that there exists a non-trivial network of
gauged supergravitites in four dimensions that are connected by singular
limits of their moduli. When the original gauged supergravity admits an uplift
to string theory, one can follow this limit also in higher dimensions, and
thereby connect one consistent truncation to another. This technique can also
be employed to construct new solutions from known ones [38]. In this paper, we
employ these relations to describe the truncation of type IIB supergravity on
${\rm AdS}_{4}\times S^{5}\times S^{1}$. There are in fact several
inequivalent truncations on this geometry, and in this work we focus on the
ones in which the resulting gaugings are
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ [39] and
$[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$ [40, 41]. The
latter can be understood as a modification of the former where an extra vector
is dyonically coupled to the matter fields. Even though the consistency of
these truncations is well know [42], complete KK Ansätze are unavailable. For
simplicity, we focus here on the truncation of these four-dimensional theories
down to the STU sector so that only six scalars and four vectors can have a
non-trivial profile. The STU model obtained from the
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ theory can be obtained as a singular
limit of the STU model corresponding to the SO(8) gauging of [19], and we show
that the $S^{5}\times S^{1}$ background in type IIB can be obtained as a
singular limit of eleven-dimensional supergravity on the $S^{7}$ followed by a
circle reduction and T-duality. The explicit Ansatz embedding both theories in
type IIB is presented in equations (4.32)–(4.37).

The rest of the paper is structured as follows. In the next section we
introduce the 4$d$ models which we are going to analyse, and discuss some of
their properties and their reformulation in terms of the tensor hierarchy.
Section 3 briefly reviews E7(7)-Exceptional Field Theory (ExFT) [43] and the
description of consistent truncations down to maximal gauged supergravities in
the language of generalised Scherk-Schwarz reductions [28], which will be
subsequently used to describe the uplift of the STU models into M-theory on
$S^{7}$ and type IIB supergravity on $S^{5}\times S^{1}$, and the consistency
of these embedding is explicitly checked in a simpler sub-truncation. New
solutions in these gaugings are then constructed in section 5 and we conclude
by discussing some possible future directions and include further technical
details in two appendices.

##  2 Gauged STU Supergravity

We consider $D=4$ theories that arise as a truncation of $\mathcal{N}=8$
gauged supergravity [44, 45] by requiring invariance under the maximal torus
of the relevant gauge group – see appendix A for the expression of the
generators of these Cartan subalgebras in terms of the generators of E7(7).
The bosonic field content of these theories consists of the metric, four
vectors and six (pseudo)scalars corresponding to the scalar manifold

$\left(\frac{\textrm{SL}(2,\mathbb{R})}{\textrm{SO}(2)}\right)^{3}\subset\frac{\rm
E_{7(7)}}{\rm SU(8)}\,,$ (2.1)

parametrised by $u_{i}=\chi_{i}-ie^{-\varphi_{i}}$, with $i=1,2,3$. The
bosonic sector of the Lagrangians of these STU supergravities can then be
written as

$\mathcal{L}=(R-V)\,\textrm{vol}_{4}+\mathcal{L}_{\rm NLSM}+\mathcal{L}_{\rm
vec}\,.$ (2.2)

The scalar kinetic terms read

$\mathcal{L}_{\rm
NLSM}=\tfrac{1}{2}\sum_{i}\big{[}d\varphi_{i}\wedge*d\varphi_{i}+e^{2\varphi_{i}}d\chi_{i}\wedge*d\chi_{i}\big{]}\,,$
(2.3)

and the vector kinetic terms are given by

$\mathcal{L}_{\rm vec}=\tfrac{1}{2}\mathcal{I}_{{\rm ab}}F^{\rm
a}\wedge*F^{\rm b}+\tfrac{1}{2}\mathcal{R}_{{\rm ab}}F^{\rm a}\wedge F^{\rm
b}\,,$ (2.4)

with ${\rm a}=1,2,3,4$. These non-minimal couplings can be extracted from the
symmetric coset representative of the maximal theory in (A.9) via the block
decomposition in (A.12) after the identifications in (A.11). The result can be
given as the period matrix  $\mathcal{N}_{{\rm ab}}=\mathcal{R}_{{\rm
ab}}+i\mathcal{I}_{{\rm ab}}$ 111This period matrix recovers the kinetic and
Chern-Simons terms in [46] under $\chi_{i}^{\rm there}=-\chi_{i}^{\rm here}$
and the following redefinition of the vector fields:  $F_{1}^{\rm
here}=F_{4}^{\rm there}\,,\qquad F_{2}^{\rm here}=F_{3}^{\rm there}\,,\qquad
F_{3}^{\rm here}=-F_{1}^{\rm there}\,,\qquad F_{4}^{\rm here}=F_{2}^{\rm
there}\,.$ (2.5) In section 4.1, this same relabelling applies to the
coordinates $\mu^{\rm a}$, $\phi^{\rm a}$, and similarly to $W_{\rm a}$ and
$Z_{\rm a}$.

$\mathcal{N}_{{\rm
ab}}=\frac{i}{W}\begin{pmatrix}-Y^{2}_{1}\,Y^{2}_{2}\,\tilde{Y}^{2}_{3}&q_{1}\,Y^{2}_{1}&q_{3}\,\tilde{Y}^{2}_{3}&q_{2}\,Y^{2}_{2}\\\
q_{1}\,Y^{2}_{1}&-Y^{2}_{1}\,\tilde{Y}^{2}_{2}\,Y^{2}_{3}&q_{2}\,\tilde{Y}^{2}_{2}&q_{3}\,Y^{2}_{3}\\\
q_{3}\,\tilde{Y}^{2}_{3}&q_{2}\,\tilde{Y}^{2}_{2}&-\tilde{Y}^{2}_{1}\,\tilde{Y}^{2}_{2}\,\tilde{Y}^{2}_{3}&q_{1}\,\tilde{Y}^{2}_{1}\\\
q_{2}\,Y^{2}_{2}&q_{3}\,Y^{2}_{3}&q_{1}\,\tilde{Y}^{2}_{1}&-\tilde{Y}^{2}_{1}\,Y^{2}_{2}\,Y^{2}_{3}\\\
\end{pmatrix}\,,$ (2.6)

in terms of the shorthands [46]

$\tilde{Y}^{2}_{i}=e^{-\varphi_{i}}+e^{\varphi_{i}}\,\chi_{i}^{2}\,,\qquad\qquad
Y^{2}_{i}=e^{\varphi_{i}}\,,\qquad\qquad b_{i}=e^{\varphi_{i}}\chi_{i}\,,$
(2.7)

and

$W=P_{0}-i\,\tilde{P}_{0}\,,\qquad\qquad
q_{i}=i\,b_{i}+b_{j}\,b_{k}\,,\quad{\rm with}\quad i\neq j\neq k\,,$ (2.8)

with

$P_{0}=1+b_{1}^{2}+b_{2}^{2}+b_{3}^{2}\,,\qquad\text{and}\qquad\tilde{P}_{0}=2b_{1}b_{2}b_{3}\,.$
(2.9)

For the potential, we consider the parent $\mathcal{N}=8$ supergravity to have
one of the following gauge groups: $\textrm{SO}(8)$ [47], its CSO contraction
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ [48, 39] or the dyonic CSO
$[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$ [40, 41]
gaugings, which all admit a higher-dimensional interpretation. These gaugings
can be described by embedding tensors with non-vanishing components in the
$\bm{36^{\prime}}\oplus\bm{36}$ of $\textrm{SL}(8,\mathbb{R})$ as in (A.1),
with the components $\theta_{AB}$ and $\xi^{AB}$ given by

$\displaystyle\theta$ $\displaystyle=g\,{\rm diag}(1,1,1,1,1,1,x,x)\,,$ (2.10)
$\displaystyle\xi$ $\displaystyle=m\,{\rm
diag}(0,0,0,0,0,0,\tilde{x},\tilde{x})\,,$

where $g$ (resp. $m$) is the electric (resp. magnetic) coupling constant. By
construction, these are valid gaugings satisfying the linear and quadratic
constraints for the embedding tensor [44, 42] whenever $x\tilde{x}=0$. The
three inequivalent choices are

$\displaystyle\theta_{{{\scriptscriptstyle(8)}}}$ $\displaystyle=g\,{\rm
diag}(1,1,1,1,1,1,1,1)\,,\qquad\xi_{{{\scriptscriptstyle(8)}}}$
$\displaystyle=0\,,$ (2.11) $\displaystyle\theta_{{{\scriptscriptstyle(6)}}}$
$\displaystyle=g\,{\rm
diag}(1,1,1,1,1,1,0,0)\,,\qquad\xi_{{{\scriptscriptstyle(6)}}}$
$\displaystyle=0\,,$ $\displaystyle\theta_{{\scriptscriptstyle(6c)}}$
$\displaystyle=g\,{\rm
diag}(1,1,1,1,1,1,0,0)\,,\qquad\xi_{{\scriptscriptstyle(6c)}}$
$\displaystyle=m\,{\rm diag}(0,0,0,0,0,0,1,1)\,,$

with the labels respectively denoting the three different gauge groups above,
with $c=m/g\neq 0$ in the latter case.

In the STU truncation, these embedding tensors induce Fayet-Iliopoulos
gaugings, whose potentials read

$\displaystyle V_{{{\scriptscriptstyle(8)}}}$
$\displaystyle=-4g^{2}\,\sum_{i}\big{(}\tilde{Y}^{2}_{i}+Y^{2}_{i}\big{)}\,,$
(2.12a) $\displaystyle V_{{{\scriptscriptstyle(6)}}}$
$\displaystyle=V_{{\scriptscriptstyle(}6c)}=-4g^{2}\,\big{(}\tilde{Y}^{2}_{1}+Y^{2}_{2}+Y^{2}_{3}\big{)}\,.$
(2.12b)

The potential (2.12a) only admits one critical point. It sits at the scalar
origin and corresponds to the SO(8) maximally supersymmetric solution. In
turn, the potentials (2.12b) do not possess any extremum in this sector.

Even though the $V_{{\scriptscriptstyle(6c)}}$ potential is blind to the value
of the magnetic coupling $m$, the fermion couplings in this theory do depend
on it. A similar situation has been previously encountered [49] in the STU
truncation of the dyonically-gauged ${\rm SO}(8)$ supergravity. In fact, the
truncated theory is not supersymmetric for non-vanishing $m$. On the other
hand, the electric cases are gauged $\mathcal{N}=2$ supegravities coupled to
three vector multiplets. To see this, observe that theories with
$\mathcal{N}=2$ supersymmetries can necessarily be recovered in the canonical
perspective of [50, 51] in terms of special Kähler and quaternionic
structures. For the scalar manifold (2.1), the special holomorphic section can
be taken as

$\Omega^{M}(z)=\\{1,u_{1},u_{2},u_{3},-u_{1}u_{2}u_{3},u_{2}u_{3},u_{1}u_{3},u_{1}u_{2}\\}\,,$
(2.13)

in terms of the special holomorphic coordinates in (2.1). This section
describes the geometry of the scalar manifold and encodes its Kähler potential
as

$\mathcal{K}=-{\rm
log}\big{[}i\,\bar{\Omega}^{M}\mathbb{C}_{MN}{\Omega}^{N}\big{]}=-{\rm
log}\big{[}-i(z_{1}-\bar{z}_{1})(z_{2}-\bar{z}_{2})(z_{3}-\bar{z}_{3})\big{]}\,,$
(2.14)

with $\mathbb{C}_{MN}$ the symplectic form on ${\rm Sp(8,\mathbb{R})}$. Purely
Fayet-Iliopoulos gaugings have a potential [50]

$V=-4g^{2}\Big{(}g^{i\bar{i}}\partial_{i}V^{M}\partial_{\bar{i}}\bar{V}^{N}-3V^{M}\bar{V}^{N}\Big{)}\vartheta_{M}\vartheta_{N}\,,$
(2.15)

with $V^{M}=e^{-\mathcal{K}/2}\Omega^{M}(z)$ a section of the special
U(1)-bundle, $g_{i\bar{i}}=\partial_{i}\partial_{\bar{i}}\mathcal{K}$ the
hermitean metric associated to the Kähler potential, and $\vartheta_{M}$ the
embedding tensor describing how the ${\rm U(1)}$ gauge group sits into the
${\rm SU}(2)$ R-symmetry group. The potential obtained from the truncation of
the SO(8) gauging is given in this language by

$\vartheta^{{{\scriptscriptstyle(8)}}}_{M}=g\,\big{(}-1,0,0,0,0,1,1,1\big{)}\,,$
(2.16)

whereas the embedding tensor corresponding to $V_{{{\scriptscriptstyle(6)}}}$
is given by

$\vartheta^{{\scriptscriptstyle(6)}}_{M}=g\,\big{(}-1,0,0,0,0,0,1,1\big{)}\,.$
(2.17)

For the electric theories, the $\mathcal{N}=2$ supersymmetry variations and
fermionic mass-like terms in the the Lagrangian associated to these embedding
tensors agree with the truncation of the $\mathcal{N}=8$ fermion shifts
associated to (2.11). However, for the dyonic gauging in (2.11), the fermion
shifts carry dependences on $m$ which can not be recovered in the
$\mathcal{N}=2$ language.

The potentials in (2.12b) can be obtained from $V_{{{\scriptscriptstyle(8)}}}$
by means of a singular scaling. For that, the scalars must be redefined as

$\displaystyle\varphi_{1}$ $\displaystyle\mapsto\varphi_{1}-k\,,\qquad$
$\displaystyle\varphi_{2,3}\mapsto\varphi_{2,3}+k\,,$ (2.18)
$\displaystyle\chi_{1}$ $\displaystyle\mapsto e^{k}\chi_{1}\,,\qquad$
$\displaystyle\chi_{2,3}\mapsto e^{-k}\chi_{2,3}\,,$

the gauge coupling as $g\mapsto e^{-k/2}g$, and the gauge fields must also be
scaled as

$A_{1,2,3}\mapsto e^{k/2}A_{1,2,3}\,,\qquad A_{4}\mapsto e^{-3k/2}A_{4}\,,$
(2.19)

whilst the metric remains invariant. The singular limit $k\to\infty$ on (2.2)
after these redefinitions maps the Lagrangian
$\mathcal{L}_{{\scriptscriptstyle(8)}}$ into
$\mathcal{L}_{{\scriptscriptstyle(6)}}$, also including the fermion couplings.

###  2.1 Tensor and duality hierarchies

The equations of motion of gauged $D=4$, $\mathcal{N}=8$ supergravity can be
written in a formally covariant E7(7)–covariant formulation [52, 31, 30] if
one introduces a set of redundant fields in the so-called tensor hierarchy.
For generic gaugings of $D=4$, $\mathcal{N}=8$ supergravity, one requires [31,
30] one-forms in the $\bm{56}$ representation of E7(7), two-forms in the
$\bm{133}$ and three-forms in the $\bm{912}$, together with a set of four-
forms that will not play a rôle in the following. These redundancies can be
eliminated at the level of the field strengths through a chain of dualities
that relate them to the original fields and their derivatives [32, 33], so
that combinations of the equations of motion for the original fields are
recovered from the Bianchi identities for the forms in the tensor hierarchy.
In the following, we show how these reformulations apply to the STU
supergravities of interest.

The equations of motion stemming from (2.2) for the one-forms and scalars are

$\displaystyle d\big{[}\mathcal{I}_{\rm ab}*F^{\rm b}+\mathcal{R}_{\rm
ab}F^{\rm b}\big{]}$ $\displaystyle=0\,,$ (2.20) $\displaystyle
d*d\varphi_{i}-e^{2\varphi_{i}}d\chi_{i}\wedge*d\chi_{i}-\tfrac{1}{2}\partial_{\varphi_{i}}\mathcal{I}_{{\rm
ab}}F^{\rm a}\wedge*F^{\rm
b}-\tfrac{1}{2}\partial_{\varphi_{i}}\mathcal{R}_{\rm ab}F^{\rm a}\wedge
F^{\rm b}+\partial_{\varphi_{i}}V\textrm{vol}_{4}$ $\displaystyle=0\,,$
$\displaystyle
d(e^{2\varphi_{i}}*d\chi_{i})-\tfrac{1}{2}\partial_{\chi_{i}}\mathcal{I}_{\rm
ab}F^{\rm a}\wedge*F^{\rm b}-\tfrac{1}{2}\partial_{\chi_{i}}\mathcal{R}_{\rm
ab}F^{\rm a}\wedge F^{\rm b}+\partial_{\chi_{i}}V\textrm{vol}_{4}$
$\displaystyle=0\,,$

which can be interpreted as Bianchi identities for two- and three-form field
strengths, respectively. For gaugings inside ${\rm SL}(8,\mathbb{R})$, as the
ones considered in this work, the $p$-forms in the E7(7) tensor hierarchy are
conveniently decomposed into one-forms $A^{AB}$, $\tilde{A}_{AB}$, two-forms
$B_{A}{}^{B}$, $B_{ABCD}$, and three-forms $C^{AB}$, $\tilde{C}_{AB}$,
$C_{A}{}^{BCD}$, $\tilde{C}^{A}{}_{BCD}$ in the
$\bm{28}\oplus\bm{28^{\prime}}$, $\bm{63}\oplus\bm{70}$ and
$\bm{36}\oplus\bm{36^{\prime}}\oplus\bm{420}\oplus\bm{420^{\prime}}$ of ${\rm
SL}(8,\mathbb{R})$. Their associated field strengths are given by

$\displaystyle H_{{\scriptscriptstyle(2)}}^{AB}$
$\displaystyle=dA^{AB}+\theta_{CD}A^{C[A}\wedge
A^{B]D}-\xi^{[A|C}\tilde{A}_{CD}\wedge A^{D|B]}+2\xi^{C[A}B_{C}{}^{B]}\,,$
$\displaystyle H_{{{\scriptscriptstyle(2)}}AB}$
$\displaystyle=d\tilde{A}_{AB}-\xi^{CD}\tilde{A}_{C[A}\wedge\tilde{A}_{B]D}+\theta_{[A|C}A^{CD}\wedge\tilde{A}_{D|B]}-2B_{[A}{}^{C}\theta_{B]C}\,,$
$\displaystyle H_{{{\scriptscriptstyle(3)}}A}{}^{B}$
$\displaystyle=DB_{A}{}^{B}+\tfrac{1}{2}\tilde{A}_{AC}\wedge
dA^{CB}+\tfrac{1}{2}A^{BC}\wedge d\tilde{A}_{CA}$
$\displaystyle\quad+\tfrac{1}{2}\theta_{DE}\tilde{A}_{AC}\wedge A^{CD}\wedge
A^{EB}+\tfrac{1}{6}\theta_{AC}A^{CD}\wedge\tilde{A}_{DE}\wedge A^{EB}$
$\displaystyle\quad-\tfrac{1}{2}\xi^{CD}\tilde{A}_{AC}\wedge\tilde{A}_{DE}\wedge
A^{EB}-\tfrac{1}{6}\xi^{EB}\tilde{A}_{AC}\wedge A^{CD}\wedge\tilde{A}_{DE}$
$\displaystyle\qquad+2\,\theta_{AC}C^{CB}-2\,\tilde{C}_{AC}\xi^{CB}-\tfrac{1}{8}\delta_{A}{}^{B}({\rm
Trace})\,,$ $\displaystyle H_{{{\scriptscriptstyle(3)}}ABCD}$
$\displaystyle=DB_{ABCD}+\tfrac{1}{4}\tilde{A}_{[AB}\wedge
d\tilde{A}_{CD]}-\tfrac{1}{96}\epsilon_{ABCDEFGH}A^{EF}\wedge dA^{GH}$
$\displaystyle\quad+\tfrac{1}{6}\Big{(}\theta_{[A|E}A^{EF}-\tilde{A}_{[A|E}\xi^{EF}\Big{)}\wedge\tilde{A}_{F|B}\wedge\tilde{A}_{CD]}$
$\displaystyle\quad+\tfrac{1}{144}\epsilon_{ABCDEFGH}A^{EF}\wedge\Big{(}\xi^{GI}\tilde{A}_{IJ}-A^{GI}\theta_{IJ}\Big{)}\wedge
A^{JH}$
$\displaystyle\quad+\tfrac{1}{3}\theta_{E[A}\tilde{C}^{E}{}_{BCD]}-\tfrac{1}{12}\epsilon_{ABCDEFGH}\xi^{HI}C_{I}{}^{EFG}\,,$
(2.21)

and similarly for the four-forms $H_{{\scriptscriptstyle(4)}}^{AB}$,
$\tilde{H}_{{{\scriptscriptstyle(4)}}AB}$,
$H_{{{\scriptscriptstyle(4)}}A}{}^{BCD}$,
$\tilde{H}_{{\scriptscriptstyle(4)}}^{A}{}_{BCD}$. Here the covariant
derivatives are given by the ${\rm SL}(8,\mathbb{R})$ decomposition of

$D=d+\Theta_{M}{}^{\alpha}A^{M}\wedge(t_{\alpha})_{({\rm R})}\,,$ (2.22)

with $(t_{\alpha})_{({\rm R})}$ the E7(7) generators in the appropriate
representation. For instance,

$\displaystyle DB_{A}{}^{B}$
$\displaystyle=dB_{A}{}^{B}+\theta_{AC}A^{CD}\wedge
B_{D}{}^{B}+\theta_{CD}A^{BC}\wedge B_{A}{}^{D}$ (2.23)
$\displaystyle\qquad\qquad+\xi^{CD}\tilde{A}_{CA}\wedge
B_{D}{}^{B}+\xi^{BD}\tilde{A}_{CD}\wedge B_{A}{}^{C}\,,$ $\displaystyle
DB_{ABCD}$
$\displaystyle=dB_{ABCD}+4B_{[ABC|E}\wedge\Big{(}\xi^{EF}\tilde{A}_{F|D]}-A^{EF}\theta_{F|D]}\Big{)}\,.$

These field strengths obey the Bianchi identities

$\displaystyle DH_{{\scriptscriptstyle(2)}}^{AB}$
$\displaystyle=2\xi^{C[A}H_{{{\scriptscriptstyle(3)}}C}{}^{B]}\,,\qquad\qquad
DH_{{{\scriptscriptstyle(2)}}AB}=-2H_{{{\scriptscriptstyle(3)}}[A}{}^{C}\theta_{B]C}\,,$
$\displaystyle DH_{{{\scriptscriptstyle(3)}}A}{}^{B}$
$\displaystyle=\tilde{H}_{{{\scriptscriptstyle(2)}}AC}\wedge
H_{{\scriptscriptstyle(2)}}^{CB}+2\theta_{AC}H_{{\scriptscriptstyle(4)}}^{CB}-2\tilde{H}_{{{\scriptscriptstyle(4)}}AC}\xi^{CB}-\tfrac{1}{8}\delta_{A}{}^{B}({\rm
Trace})\,,$ $\displaystyle DH_{{{\scriptscriptstyle(3)}}ABCD}$
$\displaystyle=\tfrac{1}{4}\tilde{H}_{{{\scriptscriptstyle(2)}}[AB}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}CD]}-\tfrac{1}{96}\epsilon_{ABCDEFGH}H_{{{\scriptscriptstyle(2)}}}^{EF}\wedge
H_{{{\scriptscriptstyle(2)}}}^{GH}$
$\displaystyle\qquad+\tfrac{1}{3}\theta_{E[A}\tilde{H}_{{\scriptscriptstyle(4)}}^{E}{}_{BCD]}-\tfrac{1}{12}\epsilon_{ABCDEFGH}\xi^{HI}H_{{{\scriptscriptstyle(4)}}I}{}^{EFG}\,,$
(2.24)

which, together with the E7(7) duality relations [32, 33]

$\displaystyle H_{{{\scriptscriptstyle(2)}}AB}$
$\displaystyle=\tfrac{1}{2}\mathcal{R}_{AB\,CD}\,H_{{{\scriptscriptstyle(2)}}}{}^{CD}+\tfrac{1}{2}\mathcal{I}_{AB\,CD}\,*H_{{{\scriptscriptstyle(2)}}}{}^{CD}\,,$
(2.25) $\displaystyle H_{{{\scriptscriptstyle(3)}}\alpha}$
$\displaystyle=-\tfrac{1}{12}(t_{\alpha})_{M}{}^{P}\,M_{NP}*DM^{MN}\,,$
$\displaystyle H_{{{\scriptscriptstyle(4)}}\alpha}{}^{M}$
$\displaystyle=-\tfrac{1}{84}\Big{[}(t_{\alpha})_{P}{}^{R}\,X_{NQ}{}^{S}\,M^{MN}\Big{(}M^{PQ}M_{RS}+7\delta^{P}_{S}\delta^{Q}_{R}\Big{)}\Big{]}\Big{|}_{\bm{912}}\textrm{vol}_{4}\,,$

recover the equations of motion for the vectors and scalars. Further details
on the E7(7) generators $(t_{\alpha})_{M}{}^{P}$, structure constants
$X_{MN}{}^{P}$ and scalar representative $M_{MN}$ can be found in appendix A.

Demanding invariance under the $\mathcal{H}$ algebra in (A.4) allows us to
consistently truncate this field content to

4+4 one-forms | : | $A^{\rm a}\,,\ \tilde{A}_{\rm a}\,,$
---|---|---
3+4+6 two-forms | : | $B_{p}\,,\ B^{\prime}_{\rm a}\,,\ B_{\rm ab}\,,$
4+4+12+12 three-forms | : |  $C_{\rm a}\,,$ $\tilde{C}_{\rm a}\,,$ $C_{\rm ab}\,,$ $\tilde{C}_{\rm ab}\,,$
(2.26)

with ${\rm a}=1,2,3,4\,,$ and $p=1,2,3$. The forms $B_{\rm ab}$, $C_{\rm ab}$,
$\tilde{C}_{\rm ab}$ are off-diagonal, and $B_{\rm ab}$ are also symmetric
under exchange of indices. See (A.11)–(A.15) for the relation between the
preserved $p$-forms in (2.26) and the ${\rm SL}(8,\mathbb{R})$ objects. For
convenience, we also introduce an extra two-form $B_{4}$, constrained as

$B_{1}+B_{2}+B_{3}+B_{4}=0\,,$ (2.27)

and extend the index as ${\rm a}=(p,4)$. If we restrict our attention to the
class of gaugings in (2.10), we can further truncate consistently the field
content to

4+4 one-forms | : | $A^{\rm a}\,,\ \tilde{A}_{\rm a}\,,$
---|---|---
3+6 two-forms | : | $B_{p}\,,\ B_{\rm ab}\,,$
4+4+3+6 three-forms | : |  $C_{\rm a}\,,$ $\tilde{C}_{\rm a}\,,$ $C_{p4}\,,$ $\tilde{C}_{p4}\,,$ $\tilde{C}_{12}=\tilde{C}_{43}\,,$ $\tilde{C}_{23}=\tilde{C}_{41}\,,$ $\tilde{C}_{31}=\tilde{C}_{42}\,,$
(2.28)

with $\tilde{C}_{pq}=\tilde{C}_{qp}$. The field strengths for these gauge
potentials can be obtained by implementing these truncations at the level of
the ${\rm SL}(8,\mathbb{R})$ field strengths in (2.1). The resulting
expressions read

$\displaystyle H_{{{\scriptscriptstyle(2)}}}^{\rm a}$ $\displaystyle=dA^{\rm
a}\,,\qquad\qquad\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}=d\tilde{A}_{\rm
a}\,,$ (2.29) $\displaystyle H_{{{\scriptscriptstyle(3)}}{\rm a}}$
$\displaystyle=dB_{\rm a}+2\,\theta_{\rm a}\,C_{\rm a}-2\,\xi_{\rm
a}\,\tilde{C}_{\rm a}-\tfrac{1}{2}A^{\rm a}\wedge d\tilde{A}_{\rm
a}-{\tfrac{1}{2}}\tilde{A}_{\rm a}\wedge dA^{\rm a}$
$\displaystyle\qquad\qquad-\tfrac{1}{4}\sum_{\rm b}\Big{[}2\,\theta_{\rm
b}\,C_{\rm b}-2\,\xi_{\rm b}\,\tilde{C}_{\rm b}-{\tfrac{1}{2}}A^{\rm b}\wedge
d\tilde{A}_{\rm b}-{\tfrac{1}{2}}\tilde{A}_{\rm b}\wedge dA^{\rm b}\Big{]}\,,$
$\displaystyle H_{{{\scriptscriptstyle(3)}}{\rm ab}}$ $\displaystyle=dB_{\rm
ab}+\tfrac{1}{12}\tilde{A}_{\rm(a}\wedge d\tilde{A}_{\rm
b)}+\tfrac{1}{6}\big{(}\theta_{\rm a}\tilde{C}_{\rm ab}+\theta_{\rm
b}\tilde{C}_{\rm ba}\big{)}$ $\displaystyle\qquad\qquad+\tfrac{1}{24}\sum_{\rm
cd}S_{\rm abcd}\big{[}-A^{\rm c}\wedge dA^{\rm d}+4\,\xi^{\rm c}C_{\rm
cd}\big{]}\,,$ $\displaystyle H_{{{\scriptscriptstyle(4)}}{\rm a}}$
$\displaystyle=dC_{\rm
a}\,,\qquad\qquad\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm a}}=d\tilde{C}_{\rm
a}\,,$

and similarly for $H_{{{\scriptscriptstyle(4)}}{\rm ab}}$ and
$\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm ab}}$. In (2.29) there are no sums
unless explicitly indicated, and the embedding tensor components $\theta_{\rm
a}$ and $\xi_{\rm a}$ are related to $\theta_{AB}$ and $\xi^{AB}$ in (2.11)
following the same pattern as the three-forms in (A.14). We have also
introduced the totally symmetric tensor

$S_{\rm abcd}=S_{\rm(abcd)}\,,\qquad S_{1234}=1\,.$ (2.30)

From their definition in terms of potentials, it is easy to check that the
field strengths satisfy the Bianchi identities

$\displaystyle dH_{{{\scriptscriptstyle(2)}}}^{\rm a}$
$\displaystyle=d\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm
a}}=dH_{{{\scriptscriptstyle(4)}}{\rm
a}}=d\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm
a}}=dH_{{{\scriptscriptstyle(4)}}{\rm
ab}}=d\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm ab}}=0\,,$ (2.31)
$\displaystyle dH_{{{\scriptscriptstyle(3)}}{\rm a}}$
$\displaystyle=2\,\theta_{\rm a}H_{{{\scriptscriptstyle(4)}}{\rm
a}}-2\,\xi_{\rm a}\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm
a}}-H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}-\tfrac{1}{4}({\rm
sum})\,,$ $\displaystyle dH_{{{\scriptscriptstyle(3)}}{\rm ab}}$
$\displaystyle=\tfrac{1}{12}\tilde{H}_{\rm a}\wedge\tilde{H}_{\rm
b}+\tfrac{1}{6}\big{(}\theta_{\rm a}\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm
ab}}+\theta_{\rm b}\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm ba}}\big{)}$
$\displaystyle\qquad\qquad+\tfrac{1}{24}\sum_{\rm cd}S_{\rm
abcd}\big{[}-H^{\rm c}\wedge H^{\rm d}+4\,\xi^{\rm
c}H_{{{\scriptscriptstyle(4)}}{\rm cd}}\big{]}\,.$

Implementing the restrictions in (A.11)–(A.15) on (2.25), we can express the
dual field strengths in terms of the original STU fields and their
derivatives. The magnetic two-form field strengths are then given by

$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}1}$
$\displaystyle=\frac{1}{|W|^{2}}\Big{[}c_{1}(\tilde{P}_{0}-P_{0}*)H_{{\scriptscriptstyle(2)}}^{1}+Y^{2}_{1}(a_{1}*-\tilde{a}_{1})H_{{\scriptscriptstyle(2)}}^{2}+\tilde{Y}^{2}_{3}(a_{3}*-\tilde{a}_{3})H_{{\scriptscriptstyle(2)}}^{3}+Y^{2}_{2}(a_{2}*-\tilde{a}_{2})H_{{\scriptscriptstyle(2)}}^{4}\Big{]}\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}2}$
$\displaystyle=\frac{1}{|W|^{2}}\Big{[}Y^{2}_{1}(a_{1}*-\tilde{a}_{1})H_{{\scriptscriptstyle(2)}}^{1}+c_{2}(\tilde{P}_{0}-P_{0}*)H_{{\scriptscriptstyle(2)}}^{2}+\tilde{Y}^{2}_{2}(a_{2}*-\tilde{a}_{2})H_{{\scriptscriptstyle(2)}}^{3}+Y^{2}_{3}(a_{3}*-\tilde{a}_{3})H_{{\scriptscriptstyle(2)}}^{4}\Big{]}\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}3}$
$\displaystyle=\frac{1}{|W|^{2}}\Big{[}\tilde{Y}^{2}_{3}(a_{3}*-\tilde{a}_{3})H_{{\scriptscriptstyle(2)}}^{1}+\tilde{Y}^{2}_{2}(a_{2}*-\tilde{a}_{2})H_{{\scriptscriptstyle(2)}}^{2}+c_{3}(\tilde{P}_{0}-P_{0}*)H_{{\scriptscriptstyle(2)}}^{3}+\tilde{Y}^{2}_{1}(a_{1}*-\tilde{a}_{1})H_{{\scriptscriptstyle(2)}}^{4}\Big{]}\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}4}$
$\displaystyle=\frac{1}{|W|^{2}}\Big{[}Y^{2}_{2}(a_{2}*-\tilde{a}_{2})H_{{\scriptscriptstyle(2)}}^{1}+Y^{2}_{3}(a_{3}*-\tilde{a}_{3})H_{{\scriptscriptstyle(2)}}^{2}+\tilde{Y}^{2}_{1}(a_{1}*-\tilde{a}_{1})H_{{\scriptscriptstyle(2)}}^{3}+c_{4}(\tilde{P}_{0}-P_{0}*)H_{{\scriptscriptstyle(2)}}^{4}\Big{]}\,,$
(2.32)

with $W$ in (2.8) and

$\displaystyle a_{i}$
$\displaystyle=b_{j}b_{k}\big{[}P_{0}-2\,b_{i}^{2}\big{]}\,,\qquad\quad\tilde{a}_{i}=b_{i}\,P_{0}+\frac{\tilde{P}_{0}^{2}}{2b_{i}}\,,\qquad\text{with}\quad
i\neq j\neq k\,,$ (2.33) $\displaystyle c_{1}$
$\displaystyle=Y_{1}^{2}Y_{2}^{2}\tilde{Y}_{3}^{2}\,,\qquad
c_{2}=Y_{1}^{2}\tilde{Y}_{2}^{2}Y_{3}^{2}\,,\qquad
c_{3}=\tilde{Y}_{1}^{2}\tilde{Y}_{2}^{2}\tilde{Y}_{3}^{2}\,,\qquad
c_{4}=\tilde{Y}_{1}^{2}Y_{2}^{2}Y_{3}^{2}\,.$

Similarly, the three-form field strengths are

$\displaystyle
H_{{{\scriptscriptstyle(3)}}1}=\frac{1}{2}*(-d\varphi_{1}-d\varphi_{2}+d\varphi_{3}+e^{2\varphi_{1}}\chi_{1}d\chi_{1}+e^{2\varphi_{2}}\chi_{2}d\chi_{2}-e^{2\varphi_{3}}\chi_{3}d\chi_{3})\,,$
(2.34) $\displaystyle
H_{{{\scriptscriptstyle(3)}}2}=\frac{1}{2}*(-d\varphi_{1}+d\varphi_{2}-d\varphi_{3}+e^{2\varphi_{1}}\chi_{1}d\chi_{1}-e^{2\varphi_{2}}\chi_{2}d\chi_{2}+e^{2\varphi_{3}}\chi_{3}d\chi_{3})\,,$
$\displaystyle
H_{{{\scriptscriptstyle(3)}}3}=\frac{1}{2}*(d\varphi_{1}+d\varphi_{2}+d\varphi_{3}-e^{2\varphi_{1}}\chi_{1}d\chi_{1}-e^{2\varphi_{2}}\chi_{2}d\chi_{2}-e^{2\varphi_{3}}\chi_{3}d\chi_{3})\,,$
$\displaystyle
H_{{{\scriptscriptstyle(3)}}4}=\frac{1}{2}*(d\varphi_{1}-d\varphi_{2}-d\varphi_{3}-e^{2\varphi_{1}}\chi_{1}d\chi_{1}+e^{2\varphi_{2}}\chi_{2}d\chi_{2}-e^{2\varphi_{3}}\chi_{3}d\chi_{3})\,.$

for the ones in the $\bm{63}$ of ${\rm SL}(8,\mathbb{R})$, and

$\begin{gathered}H_{{{\scriptscriptstyle(3)}}12}=\frac{1}{12}*e^{2\varphi_{1}}d\chi_{1}\,,\qquad\quad
H_{{{\scriptscriptstyle(3)}}14}=\frac{1}{12}*e^{2\varphi_{2}}d\chi_{2}\,,\qquad\quad
H_{{{\scriptscriptstyle(3)}}24}=\frac{1}{12}*e^{2\varphi_{3}}d\chi_{3}\,,\\\\[5.0pt]
H_{{{\scriptscriptstyle(3)}}34}=\frac{1}{12}*\big{[}2\chi_{1}d\varphi_{1}+(1-e^{-2\varphi_{1}}\chi_{1}^{2})d\chi_{1}\big{]}\,,\\\\[5.0pt]
H_{{{\scriptscriptstyle(3)}}23}=\frac{1}{12}*\big{[}2\chi_{2}d\varphi_{2}+(1-e^{-2\varphi_{2}}\chi_{2}^{2})d\chi_{2}\big{]}\,,\\\\[5.0pt]
H_{{{\scriptscriptstyle(3)}}13}=\frac{1}{12}*\big{[}2\chi_{3}d\varphi_{3}+(1-e^{-2\varphi_{3}}\chi_{3}^{2})d\chi_{3}\big{]}\,,\\\\[-14.0pt]
\end{gathered}$ (2.35)

for the ones in the $\bm{70}$. The expressions for the four-form field
strengths in terms of the scalars depend on the choice of embedding tensor.
Employing (2.25) for the class of gaugings in (2.10), we obtain

$\displaystyle H_{{{\scriptscriptstyle(4)}}1}$
$\displaystyle=2g\big{[}\tilde{Y}_{1}^{2}+x\,\tilde{Y}_{2}^{2}+Y_{3}^{2}\big{]}\textrm{vol}_{4}\,,\qquad\quad\tilde{H}_{{{\scriptscriptstyle(4)}}1}=2m\,\tilde{x}\,Y_{2}^{2}\,\textrm{vol}_{4}\,,$
(2.36) $\displaystyle H_{{{\scriptscriptstyle(4)}}2}$
$\displaystyle=2g\big{[}\tilde{Y}_{1}^{2}+Y_{2}^{2}+x\,\tilde{Y}_{3}^{2}\big{]}\textrm{vol}_{4}\,,\qquad\quad\tilde{H}_{{{\scriptscriptstyle(4)}}2}=2m\,\tilde{x}\,Y_{3}^{2}\,\textrm{vol}_{4}\,,$
$\displaystyle H_{{{\scriptscriptstyle(4)}}3}$
$\displaystyle=2g\big{[}x\,Y_{1}^{2}+Y_{2}^{2}+Y_{3}^{2}\big{]}\textrm{vol}_{4}\,,\qquad\quad\tilde{H}_{{{\scriptscriptstyle(4)}}3}=2m\,\tilde{x}\,\tilde{Y}_{1}^{2}\,\textrm{vol}_{4}\,,$
$\displaystyle H_{{{\scriptscriptstyle(4)}}4}$
$\displaystyle=2g\big{[}Y_{1}^{2}+\tilde{Y}_{2}^{2}+\tilde{Y}_{3}^{2}\big{]}\textrm{vol}_{4}\,,\hskip
40.68723pt\tilde{H}_{{{\scriptscriptstyle(4)}}4}=0\,,$

and

$\displaystyle
H_{{{\scriptscriptstyle(4)}}34}=2m\tilde{x}\chi_{1}e^{\varphi_{1}}\textrm{vol}_{4}\,,\qquad\quad$
$\displaystyle
H_{{{\scriptscriptstyle(4)}}14}=2m\tilde{x}\chi_{2}e^{\varphi_{2}}\textrm{vol}_{4}\,,\qquad\quad$
$\displaystyle
H_{{{\scriptscriptstyle(4)}}24}=2m\tilde{x}\chi_{3}e^{\varphi_{3}}\textrm{vol}_{4}\,,$
(2.37)
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}12}=-2g\chi_{1}e^{\varphi_{1}}\textrm{vol}_{4}\,,\qquad\quad$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}23}=-2g\chi_{2}e^{\varphi_{2}}\textrm{vol}_{4}\,,\qquad\quad$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}31}=-2g\chi_{3}e^{\varphi_{3}}\textrm{vol}_{4}\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}34}=-2gx\chi_{1}e^{\varphi_{1}}\textrm{vol}_{4}\,,\qquad\quad$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}14}=-2gx\chi_{2}e^{\varphi_{2}}\textrm{vol}_{4}\,,\qquad\quad$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}24}=-2gx\chi_{3}e^{\varphi_{3}}\textrm{vol}_{4}\,.$

The forms in (2.36) can be used to reproduce the potentials in (2.12) as

$\sum_{\rm a}\big{[}\theta_{\rm a}H_{{{\scriptscriptstyle(4)}}\rm a}+\xi^{\rm
a}\tilde{H}_{{{\scriptscriptstyle(4)}}\rm a}\big{]}=-V\textrm{vol}_{4}\,.$
(2.38)

As noted before, inserting the duality relations (2.1) into the Bianchi
identities (2.31), we obtain the scalar-Maxwell equations in (2.20) stemming
from the Lagrangian (2.2). Similarly, the Bianchi identities for the three-
form field strengths are identically verified if the equations of motion for
the scalars are satisfied. The derivatives of the scalars that appear in
equations of motion for the dilatons are retrieved from

$\displaystyle
H_{{{\scriptscriptstyle(3)}}1}+H_{{{\scriptscriptstyle(3)}}2}=-2\,G_{uv}\,k^{u}[h_{1}]*d\Phi^{v}\,,$
$\displaystyle\qquad\quad
H_{{{\scriptscriptstyle(3)}}1}+H_{{{\scriptscriptstyle(3)}}4}=-2\,G_{uv}\,k^{u}[h_{2}]*d\Phi^{v}\,,$
(2.39) $\displaystyle
H_{{{\scriptscriptstyle(3)}}2}+H_{{{\scriptscriptstyle(3)}}4}$
$\displaystyle=-2\,G_{uv}\,k^{u}[h_{3}]*d\Phi^{v}\,,$

with $\Phi^{u}$ a collective name for the six scalars
$(\varphi_{i},\,\chi_{i})$ and $G_{uv}$ the non-linear sigma-model metric in
(2.3). The Killing vectors $k[h_{i}]$ are defined in (A.10). Similarly, the
three-forms in (2.35) are related to $k[e_{i}]$ and $k[f_{i}]$ in (A.10) and
encode the derivatives of the axions in their equations of motion. The
derivatives of the scalar potential in (2.20) are accounted for by the four-
form field strengths as

$\displaystyle\theta_{1}H_{{{\scriptscriptstyle(4)}}1}+\theta_{2}H_{{{\scriptscriptstyle(4)}}2}-\xi_{1}\tilde{H}_{{{\scriptscriptstyle(4)}}1}-\xi_{2}\tilde{H}_{{{\scriptscriptstyle(4)}}2}-\tfrac{1}{2}\sum_{\rm
a}\big{(}\theta_{a}H_{{{\scriptscriptstyle(4)}}{\rm
a}}-\xi_{a}\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm a}}\big{)}=$
$\displaystyle\,\tfrac{1}{4}k^{u}[h_{1}]\partial_{u}V\,\textrm{vol}_{4}\,,$
(2.40)
$\displaystyle\theta_{1}H_{{{\scriptscriptstyle(4)}}1}+\theta_{4}H_{{{\scriptscriptstyle(4)}}4}-\xi_{1}\tilde{H}_{{{\scriptscriptstyle(4)}}1}-\xi_{4}\tilde{H}_{{{\scriptscriptstyle(4)}}4}-\tfrac{1}{2}\sum_{\rm
a}\big{(}\theta_{a}H_{{{\scriptscriptstyle(4)}}{\rm
a}}-\xi_{a}\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm a}}\big{)}=$
$\displaystyle\,\tfrac{1}{4}k^{u}[h_{2}]\partial_{u}V\,\textrm{vol}_{4}\,,$
$\displaystyle\theta_{2}H_{{{\scriptscriptstyle(4)}}2}+\theta_{4}H_{{{\scriptscriptstyle(4)}}4}-\xi_{2}\tilde{H}_{{{\scriptscriptstyle(4)}}2}-\xi_{4}\tilde{H}_{{{\scriptscriptstyle(4)}}4}-\tfrac{1}{2}\sum_{\rm
a}\big{(}\theta_{a}H_{{{\scriptscriptstyle(4)}}{\rm
a}}-\xi_{a}\tilde{H}_{{{\scriptscriptstyle(4)}}{\rm a}}\big{)}=$
$\displaystyle\,\tfrac{1}{4}k^{u}[h_{3}]\partial_{u}V\,\textrm{vol}_{4}\,,$
$\displaystyle
g\tilde{H}_{{{\scriptscriptstyle(4)}}12}=\tfrac{1}{28}k^{u}[e_{1}]\partial_{u}V\textrm{vol}_{4}\,,\quad
g\tilde{H}_{{{\scriptscriptstyle(4)}}23}=\tfrac{1}{28}k^{u}[e_{2}]\partial_{u}V\textrm{vol}_{4}\,,\quad
g\tilde{H}_{{{\scriptscriptstyle(4)}}31}$
$\displaystyle=\tfrac{1}{28}k^{u}[e_{3}]\partial_{u}V\textrm{vol}_{4}\,,$

and, finally, the non-minimal couplings to the vector fields follow from

$\displaystyle
H_{{\scriptscriptstyle(2)}}^{1}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}1}+H_{{\scriptscriptstyle(2)}}^{2}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}2}-\tfrac{1}{2}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}$
$\displaystyle=\tfrac{1}{4}k^{u}[h_{1}]\partial_{u}\Big{(}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}\Big{)}\,,$ (2.41)
$\displaystyle
H_{{\scriptscriptstyle(2)}}^{1}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}1}+H_{{\scriptscriptstyle(2)}}^{4}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}4}-\tfrac{1}{2}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}$
$\displaystyle=\tfrac{1}{4}k^{u}[h_{2}]\partial_{u}\Big{(}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}\Big{)}\,,$ $\displaystyle
H_{{\scriptscriptstyle(2)}}^{2}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}2}+H_{{\scriptscriptstyle(2)}}^{4}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}4}-\tfrac{1}{2}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}$
$\displaystyle=\tfrac{1}{4}k^{u}[h_{3}]\partial_{u}\Big{(}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}\Big{)}\,,$

in the equation for the dilations, and

$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}1}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}2}-H_{{\scriptscriptstyle(2)}}^{3}\wedge
H_{{\scriptscriptstyle(2)}}^{4}$
$\displaystyle=\tfrac{1}{2}k^{u}[e_{1}]\partial_{u}\Big{(}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}\Big{)}\,,$ (2.42)
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}1}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}4}-H_{{\scriptscriptstyle(2)}}^{2}\wedge
H_{{\scriptscriptstyle(2)}}^{3}$
$\displaystyle=\tfrac{1}{2}k^{u}[e_{2}]\partial_{u}\Big{(}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}\Big{)}\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}2}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}4}-H_{{\scriptscriptstyle(2)}}^{1}\wedge
H_{{\scriptscriptstyle(2)}}^{3}$
$\displaystyle=\tfrac{1}{2}k^{u}[e_{3}]\partial_{u}\Big{(}\sum_{\rm
a}H_{{{\scriptscriptstyle(2)}}}^{\rm
a}\wedge\tilde{H}_{{{\scriptscriptstyle(2)}}{\rm a}}\Big{)}\,,$

in the one for the axions.

Turning now our attention to the singular limit relating the different
gaugings, under the scaling (2.18) the three-form field strengths in (2.34)
stay invariant, the magnetic two-form field strengths have opposite scaling to
the electric forms in (2.19),

$\tilde{H}_{{{\scriptscriptstyle(2)}}1,2,3}\mapsto
e^{-k/2}\tilde{H}_{{{\scriptscriptstyle(2)}}1,2,3}\,,\qquad\tilde{H}_{{{\scriptscriptstyle(2)}}4}\mapsto
e^{3k/2}\tilde{H}_{{{\scriptscriptstyle(2)}}4}\,,$ (2.43)

and the electric four-form field strengths reduce to

$H_{{{\scriptscriptstyle(4)}}\rm a}\mapsto
e^{k}\bar{H}_{{{\scriptscriptstyle(4)}}\rm a}\,,$ (2.44)

with $\bar{H}_{{{\scriptscriptstyle(4)}}\rm a}$ the four-forms given by (2.36)
with $x=0$ and $\tilde{x}=0$. The scaling of the other four-forms, which will
not be relevant in the following, can be computed in the same way from (2.36)
and (2.37).

In the sequel, we will embed the previous $D=4$ gauged supergravities into
type IIB and $D=11$ supergravity using a duality-covariant reformulation of
the latter higher-dimensional theories known as Exceptional Field Theory, and
encoding part of the dependence on the $4d$ fields in terms of a subset of the
forms in the tensor hierarchy.

##  3 E7(7) Exceptional Field Theory

The bosonic field content of ExFT [43] is given by

$\\{\bm{e}_{\mu}{}^{a}\,,\ \bm{M}_{MN}\,,\ \bm{A}_{\mu}{}^{M}\,,\
\bm{B}_{\mu\nu}{}_{\alpha}\,,\ \bm{B}_{\mu\nu}{}_{M}\\}\,,$ (3.1)

with all fields depending on both “external”, $x^{\mu}$, $\mu=0,\dots,3$, and
extended “internal” coordinates, $Y^{M}$, $M=1,\dots,56$. This dependence and
the fields themselves are restricted by the section constraints [43]

$(t_{\alpha})^{MN}Q_{M}\otimes Q_{N}=0\,,\qquad\Omega^{MN}Q_{M}\otimes
Q_{N}=0\,,$ (3.2)

with $Q_{M}\in\\{\partial_{M}\,,\ \bm{B}_{\mu\nu}{}_{M}\\}$ and the
derivatives acting on any combination of fields or gauge parameters in the
theory. Here, $(t_{\alpha})_{M}{}^{N}$ are the algebra generators and indices
are raised and lowered with the invariant symplectic form $\Omega_{MN}$ as

$\Omega^{MN}=\begin{pmatrix}0&\mathbbm{1}_{28}\\\
-\mathbbm{1}_{28}&0\end{pmatrix}\,,\qquad V^{M}=\Omega^{MN}V_{N}\,,\qquad
V_{M}=V^{N}\Omega_{NM}\,.$ (3.3)

These constraints are needed for the generalised Lie derivative to close into
a local E7(7) gauge algebra. Variations under the latter are given by [43]

$\delta_{\Lambda}V^{M}\equiv\mathcal{L}_{\Lambda}V^{M}=\Lambda^{N}\partial_{N}V^{M}-12\,\mathbb{P}^{M}{}_{N}{}^{K}{}_{L}\,\partial_{K}\Lambda^{L}\,V^{N}+\lambda(V)\,\partial_{N}\Lambda^{N}\,V^{M}\,,$
(3.4)

where
$\mathbb{P}^{M}{}_{N}{}^{K}{}_{L}=(t_{\alpha})_{N}{}^{M}\,(t^{\alpha})_{L}{}^{K}$
is the projector onto the adjoint representation and $\lambda(V)$ is the
weight associated to the generalised vector $V^{M}$. To solve the constraints
(3.2), E7(7) can be reduced down to ${\rm GL}(7,\mathbb{R})$ (M-theory
section) or ${\rm GL}(6,\mathbb{R})\times{\rm SL}(2,\mathbb{R})$ (Type IIB
section). After this reduction, the variation $\delta_{\Lambda}$ encodes the
behaviour of the different fields under both “internal” diffeomorphisms and
gauge transformations.

###  3.1 M-theory section

For $D=11$ supergravity, we use conventions in which our fields are subject to
the action

$S=\int
d^{11}x\sqrt{|\hat{g}_{11}|}\Big{[}\hat{R}_{11}-\tfrac{1}{2}|\hat{F}_{{\scriptscriptstyle(4)}}|^{2}\Big{]}-\tfrac{1}{6}\int\hat{A}_{{\scriptscriptstyle(3)}}\wedge\hat{F}_{{\scriptscriptstyle(4)}}\wedge\hat{F}_{{\scriptscriptstyle(4)}}\,,$
(3.5)

with $\hat{F}_{{\scriptscriptstyle(4)}}=d\hat{A}_{{\scriptscriptstyle(3)}}$
and contraction of indices with weight one denoted by $|\cdot|^{2}$, i.e.

$|\hat{F}_{{\scriptscriptstyle
p}}|^{2}=\tfrac{1}{p!}F_{\hat{\mu}_{1}\dots\hat{\mu}_{p}}\,F^{\hat{\mu}_{1}\dots\hat{\mu}_{p}}\,.$
(3.6)

Under the structure group relevant for a seven-dimensional internal space, the
extended ExFT coordinates decompose as222The coordinate index in $y^{i}$
should not be confused with the index labelling the different factors in
(2.1).

E7(7) | $\supset$ | ${\rm SL}(8,\mathbb{R})$ | $\supset$ | ${\rm GL}(7,\mathbb{R})$
---|---|---|---|---
$\bm{56}$ | $\to$ | $\bm{28}\oplus\bm{28^{\prime}}$ | $\to$ |  $\bm{7}_{+3}\oplus\bm{21^{\prime}}_{+1}\oplus\bm{21}_{-1}\oplus\bm{7^{\prime}}_{-3}$ ,
$\\{Y^{M}\\}$ | $\to$ | $\\{Y^{AB},\,Y_{AB}\\}$ | $\to$ |  $\\{y^{i}\,,\ y_{ij}\,,\ y^{ij}\,,\ y_{i}\\}$ ,
(3.7)

with $y^{i}=Y^{i8}$, etc. The section constraint (3.2) can be solved by
imposing

$\partial^{ij}=\partial_{ij}=\partial^{i}=0\,,\qquad{\rm
and}\qquad\bm{B}_{\mu\nu}{}^{ij}=\bm{B}_{\mu\nu}{}_{ij}=\bm{B}_{\mu\nu}{}^{i}=0\,,$
(3.8)

keeping only the $\bm{B}_{\mu\nu}{}_{i}$ components of
$\bm{B}_{\mu\nu}{}_{M}$. Similarly, the objects in the adjoint representation
of E7(7) break according to

E7(7) | $\supset$ | ${\rm SL}(8,\mathbb{R})$ | $\supset$ | ${\rm GL}(7,\mathbb{R})$
---|---|---|---|---
$\bm{133}$ | $\to$ | $\bm{63}\oplus\bm{70}$ | $\to$ |  $\bm{7^{\prime}}_{+4}\oplus\bm{35}_{+2}\oplus(\bm{48}\oplus\bm{1})_{0}\oplus\bm{35^{\prime}}_{-2}\oplus\bm{7}_{-4}$ ,
$\\{t_{\alpha}\\}$ | $\to$ | $\\{t_{A}{}^{B},\,t_{ABCD}\\}$ | $\to$ |  $\\{t_{i}\,,\ t^{ijk}\,,\ t_{0}\,,\ t_{i}{}^{j}\,,\ t_{ijk}\,,\ t^{i}\\}$ .
(3.9)

To make contact with M-theory, one needs to split the $D=11$ structure group
${\rm GL}(11,\mathbb{R})\supset{\rm GL}(4,\mathbb{R})\times{\rm
GL}(7,\mathbb{R})$. Then, the metric and three-form of $D=11$ supergravity
give rise to the following fields:

$\\{g_{\mu\nu}\,,\ A_{\mu}{}^{i}\,,\ \phi_{ij}\,;\ C_{ijk}\,,\ C_{ij\rho}\,,\
C_{i\nu\rho}\,,\ C_{ijklmn}\\}\,,$ (3.10)

where all of the fields depend on both $x^{\mu}$ and $y^{i}$. The forms
$C_{ijk}\,,\ C_{ij\rho}\,,\ C_{i\nu\rho}$ are related to the components of the
eleven dimensional $\hat{A}_{{\scriptscriptstyle(3)}}$ through the usual
Kaluza-Klein decomposition with flattening and unflattening of indices. The
$C_{ijklmn}$ components are dual to the external legs $C_{\mu\nu\rho}$ through

$\displaystyle F_{{{\scriptscriptstyle(7)}}}$
$\displaystyle=*_{11}dC^{\rm{\scriptscriptstyle(ext)}}_{{\scriptscriptstyle(3)}}$
(3.11)
$\displaystyle=dC_{{{\scriptscriptstyle(6)}}}-\frac{1}{2}C_{{{\scriptscriptstyle(3)}}}\wedge
dC_{{{\scriptscriptstyle(3)}}}\,,$

with

$C^{\rm{\scriptscriptstyle(ext)}}_{{{\scriptscriptstyle(3)}}}=\tfrac{1}{3!}C_{\mu\nu\rho}dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\rho}\,,\qquad
C_{{{\scriptscriptstyle(3)}}}=\tfrac{1}{3!}C_{ijk}dy^{i}\wedge dy^{j}\wedge
dy^{k}\,,$ (3.12)

as required by the infinitesimal transformations

$\delta
C_{{{\scriptscriptstyle(3)}}}=L_{\xi}C_{{{\scriptscriptstyle(3)}}}+d\Lambda_{{{\scriptscriptstyle(2)}}}\,,\qquad\delta
C_{{{\scriptscriptstyle(6)}}}=L_{\xi}C_{{{\scriptscriptstyle(6)}}}+d\Lambda_{{{\scriptscriptstyle(5)}}}-\frac{1}{2}d\Lambda_{{{\scriptscriptstyle(2)}}}\wedge
C_{{{\scriptscriptstyle(3)}}}\,.$ (3.13)

These fields are then related to the ExFT fields branched according to (3.7)
and (3.9). For the tensor-like degrees of freedom, the dictionary reads

$\bm{e}_{\mu}{}^{a}=\phi^{1/4}e_{\mu}{}^{a}\,,\qquad\bm{A}_{\mu}{}^{i}=A_{\mu}{}^{i}\,,\qquad\bm{A}_{\mu}{}_{ij}=C_{\mu
ij}\,,$ (3.14)

and $C_{\mu\nu i}$ are related to $\bm{B}_{\mu\nu\,\alpha}$ and
$\bm{B}_{\mu\nu\,M}$. For the $D=4$ scalars,

$\bm{M}^{ij}=\phi^{-1/2}\phi^{ij}\,,\quad\bm{M}^{i}{}_{jk}=3\phi^{-1/2}\phi^{il}C_{ljk}\,,\quad\bm{M}^{i}{\,}^{jk}=-\frac{3}{20}\phi^{-1/2}\phi^{i[j}\varepsilon^{k]l_{1}l_{2}l_{3}l_{4}l_{5}l_{6}}C_{l_{1}l_{2}l_{3}l_{4}l_{5}l_{6}}$
(3.15)

for the components of the $\bm{M}^{MN}$ coset representative of E7(7)/SU(8).
Here and throughout, $\phi={\rm det}\,\phi_{ij}$. Conversely, the internal
components of the three- and six-form can be given as

$C_{ijk}=\frac{1}{3}\phi^{1/2}\phi_{l[i}\bm{M}^{l}{}_{jk]}\,,\qquad
C_{l_{1}l_{2}l_{3}l_{4}l_{5}l_{6}}=-\frac{20}{3}\phi^{1/2}\phi_{ij}\bm{M}^{i\;jk}\varepsilon_{kl_{1}l_{2}l_{3}l_{4}l_{5}l_{6}}\,.$
(3.16)

In the following, we will show that in the context of consistent truncations,
one can circumvent the dualisation in (3.11) by working in terms of the
$p$-forms of the four-dimensional tensor hierarchy, which also cleanly account
for the information corresponding to the $4d$ vectors $A_{\mu}^{i}$ and
$C_{\mu ij}$, and two-forms $C_{\mu\nu i}$.

###  3.2 Type IIB section

Similarly to the $D=11$ case, the extended ExFT coordinates decompose under
${\rm GL}(6,\mathbb{R})\times\textrm{SL}(2,\mathbb{R})$ following

E7(7) | $\supset$ | ${\rm GL}(6,\mathbb{R})\times\textrm{SL}(2,\mathbb{R})$
---|---|---
$\bm{56}$ | $\to$ |  $(\bm{6},\bm{1})_{+2}\oplus(\bm{6}^{\prime},\bm{2})_{+1}\oplus(\bm{20},\bm{1})_{0}\oplus(\bm{6},\bm{2})_{-1}\oplus(\bm{6}^{\prime},\bm{1})_{-2}$ ,
$\\{Y^{M}\\}$ | $\to$ |  $\\{y^{i}\,,\ y_{ia}\,,\ y^{ijk}\,,\ y^{ia}\,,\ y_{i}\\}$ ,
(3.17)

now with $i=1,\dots,6$ and $a=1,2$. The section constraint (3.2) can be solved
by requiring that all fields and parameters only depend on $y^{i}$ and that
the only non-zero component of the constrained two-form is
$\bm{B}_{\mu\nu}{}_{i}$. In turn, the objects in the adjoint representation
adhere to

E7(7) | $\supset$ | ${\rm GL}(6,\mathbb{R})\times\textrm{SL}(2,\mathbb{R})$
---|---|---
$\bm{133}$ | $\to$ | $(\bm{1},\bm{2})_{3}\oplus(\bm{15^{\prime}},\bm{1})_{2}\oplus(\bm{15},\bm{2})_{1}\oplus(\bm{35}+\bm{1},\bm{1})_{0}\hbox{\hskip 42.67912pt}$
|  |  $\hbox{\hskip 42.67912pt}\oplus(\bm{1},\bm{3})_{0}\oplus(\bm{15^{\prime}},\bm{2})_{-1}\oplus(\bm{15},\bm{1})_{-2}\oplus(\bm{1},\bm{2})_{-3}$ ,
$\\{t_{\alpha}\\}$ | $\to$ |  $\\{t_{a}\,,\ t_{ij}\,,\ t^{ij\,a}\,,\ t_{i}{}^{j}\,,\ t_{0}\,,\ t_{a}{}^{b}\,,\ t_{ij\,a}\,,\ t^{ij}\,,\ \tilde{t}_{a}\\}$ .
(3.18)

Contact with type IIB supergravity is now achieved after splitting the ten-
dimensional structure group ${\rm GL}(10,\mathbb{R})$ into ${\rm
GL}(4,\mathbb{R})\times{\rm GL}(6,\mathbb{R})$ so that the bosonic fields read

$\\{g_{\mu\nu}\,,\ \phi_{ij}\,,\ \Phi\,,\ C_{0}\,,\ C_{\mu\nu}{}^{a}\,,\
C_{\mu i}{}^{a}\,,\ C_{ij}{}^{a}\,,\ C_{\mu\nu\rho\sigma}\,,\ C_{\mu\nu\rho
i}\,,\ C_{\mu\nu ij}\,,\ C_{\mu ijk}\,,\ C_{ijkl}\\}\,,$ (3.19)

again with all of them depending both on $x^{\mu}$ and $y^{i}$ and taking into
account the flattening and unflattening of indices with the Kaluza-Klein
vector. Here, we use conventions in which the type IIB pseudoaction is given
by

$S=\int
d^{10}x\sqrt{|\hat{g}_{10}|}\Big{[}\hat{R}_{10}-\tfrac{1}{2}(\partial\hat{\Phi})^{2}-\tfrac{1}{2}e^{2\hat{\Phi}}|\hat{F}_{{\scriptscriptstyle(1)}}|^{2}-\tfrac{1}{2}e^{-\hat{\Phi}}|\hat{H}_{{\scriptscriptstyle(3)}}|^{2}-\tfrac{1}{2}e^{\hat{\Phi}}|\hat{F}_{{\scriptscriptstyle(3)}}|^{2}-\tfrac{1}{4}|\hat{F}_{{\scriptscriptstyle(5)}}|^{2}\Big{]}+\int\mathcal{L}_{\rm
top}\,,$ (3.20)

with $|\hat{F}_{{\scriptscriptstyle(p)}}|^{2}$ and
$|\hat{H}_{{{\scriptscriptstyle(3)}}}|^{2}$ defined in (3.6), and a
topological term $\mathcal{L}_{\rm
top}=\hat{C}_{{\scriptscriptstyle(4)}}\wedge\hat{H}_{{\scriptscriptstyle(3)}}\wedge\hat{F}_{{\scriptscriptstyle(3)}}$,
and the field strengths given by

$\hat{H}_{{\scriptscriptstyle(3)}}=d\hat{B}_{{\scriptscriptstyle(2)}}\,,\quad\hat{F}_{{\scriptscriptstyle(1)}}=d\hat{C}_{{\scriptscriptstyle(0)}}\,,\quad\hat{F}_{{\scriptscriptstyle(3)}}=d\hat{C}_{{\scriptscriptstyle(2)}}-\hat{C}_{{\scriptscriptstyle(0)}}\,\hat{H}_{{\scriptscriptstyle(3)}}\,,\quad\hat{F}_{{\scriptscriptstyle(5)}}=d\hat{C}_{{\scriptscriptstyle(4)}}+\frac{1}{2}\epsilon_{ab}\,\hat{C}_{{\scriptscriptstyle(2)}}^{a}\wedge
d\hat{C}_{{\scriptscriptstyle(2)}}^{b}\,,$ (3.21)

with
$\hat{C}_{{\scriptscriptstyle(2)}}^{a}=(\hat{B}_{{\scriptscriptstyle(2)}},\hat{C}_{{\scriptscriptstyle(2)}})$.
In addition to the equations of motion obtained from (3.20), the self-duality
of $\hat{F}_{{\scriptscriptstyle(5)}}$ needs also to be imposed. The relation
to the ExFT fields branched according to (3.17) and (3.18) then is

$\displaystyle\bm{M}^{ij}$
$\displaystyle=\phi^{-1/2}\phi^{ij}\,,\qquad\qquad\bm{M}^{i}{}_{ja}=\phi^{-1/2}\phi^{ik}\epsilon_{ab}C_{kj}{}^{b}\,,$
(3.22) $\displaystyle\bm{M}^{i}{}_{jkl}$
$\displaystyle=\phi^{-1/2}\phi^{im}\big{(}24C_{mjkl}+3\epsilon_{ab}C_{m[j}{}^{a}C_{kl]}{}^{b}\big{)}\,,$
$\displaystyle\bm{M}_{ia}{}_{jb}$
$\displaystyle=6\,\phi^{-1/2}\phi_{ij}\big{(}\mathfrak{m}_{ab}-\epsilon_{ac}\epsilon_{bd}C_{kl}{}^{c}C^{kl}{}^{d}\big{)}\,,$

for the $D=4$ scalars, with the IIB axiodilation encoded in the
$\textrm{SL}(2,\mathbb{R})$ matrix as

$\mathfrak{m}=e^{\Phi}\begin{pmatrix}e^{-2\Phi}+C_{0}^{2}&-C_{0}\\\
-C_{0}&1\end{pmatrix}\,.$ (3.23)

Similarly, we have $\bm{e}_{\mu}{}^{a}=\phi^{1/4}e_{\mu}{}^{a}$ and
$\bm{A}_{\mu}{}^{i}=A_{\mu}{}^{i}$ for the tensor-like components of the
metric, and the other $p$-form contributions we will phrase in terms of
objects in the tensor hierarchy when we turn to consistent truncations in the
sequel.

To make contact with the $S^{5}\times S^{1}$ configurations, we need to
further decompose under ${\rm GL}(5,\mathbb{R})\times{\rm
GL}(1,\mathbb{R})\times\textrm{SL}(2,\mathbb{R})$ via $i=(\textrm{i},6)$ with
$\textrm{i}=1,\dots,5$. This subgroup is common to both
$\textrm{SL}(8,\mathbb{R})$ and ${\rm
GL}(6,\mathbb{R})\times\textrm{SL}(2,\mathbb{R})$, and therefore its
representations can be given in terms of $\textrm{SL}(8,\mathbb{R})$ pairs. In
our conventions,

$\displaystyle V^{i}=(V^{\textrm{i}6},\ V_{78})\,,\qquad V^{ijk}$
$\displaystyle=(\epsilon^{ijk\textrm{l}\textrm{m}6}V_{\textrm{l}\textrm{m}},\
V^{\textrm{i}\textrm{j}}\delta^{k}_{6})\,,\qquad V_{i}=(V_{\textrm{i}6},\
V^{78})\,,$ (3.24) $\displaystyle V_{ia}=$ $\displaystyle(V_{\textrm{i}a},\
V^{6a})\,,\qquad V^{ia}=(V^{\textrm{i}a},\ V_{6a})\,,$

and for the coordinates we further introduce
$y^{\textrm{i}}=\hat{y}^{\textrm{i}}$ and $y^{6}=\tilde{y}$.

###  3.3 Generalised Scherk-Schwarz Ansätze

The ExFT fields (3.1) can be parametrised in terms of $D=4$ $\mathcal{N}=8$
supergravity fields via a Scherck-Schwarz Ansatz [28]333In the following, we
add bars to the E7(7) and ${\rm SL}(8,\mathbb{R})$ indices in the previous
section to distinguish flat and curved counterparts.

$\displaystyle\bm{g}_{\mu\nu}(x,Y)$
$\displaystyle=\rho^{-2}(Y)\,g_{\mu\nu}(x)\,,$ (3.25)
$\displaystyle\bm{M}_{MN}(x,Y)$
$\displaystyle=U_{M}{}^{\bar{M}}(Y)U_{N}{}^{\bar{N}}(Y)M_{\bar{M}\bar{N}}(x)\,,$
$\displaystyle\bm{A}_{\mu}{}^{M}(x,Y)$
$\displaystyle=\rho^{-1}(Y)(U^{-1})^{M}{}_{\bar{M}}(Y)A_{\mu}{}^{\bar{M}}(x)\,,$
$\displaystyle\bm{B}_{\mu\nu}{}_{\alpha}(x,Y)$
$\displaystyle=\rho^{-2}(Y)U_{\alpha}{}^{\bar{\alpha}}(Y)B_{\mu\nu}{}_{\bar{\alpha}}(x)\,,$
$\displaystyle\bm{B}_{\mu\nu}{}_{M}(x,Y)$
$\displaystyle=-2\rho^{-2}(Y)(U^{-1})^{P}{}_{\bar{S}}(Y)\partial_{M}U_{P}{}^{\bar{R}}(Y)(t^{\bar{\alpha}})_{\bar{R}}{}^{\bar{S}}B_{\mu\nu}{}_{\bar{\alpha}}(x)\,.$

This Ansatz provides a consistent truncation of the ExFT equations of motion
down to $D=4$ maximal supergravity provided that the twist matrix and scaling
factor define a generalised frame
$\mathcal{U}_{\bar{M}}{}^{M}=\rho^{-1}(U^{-1})^{M}{}_{\bar{M}}$ such that

$\mathcal{L}_{\mathcal{U}_{\bar{M}}}\mathcal{U}_{\bar{N}}=X_{\bar{M}\bar{N}}{}^{\bar{P}}\,\mathcal{U}_{\bar{P}}\,,$
(3.26)

with $\mathcal{L}$ given in (3.4) and $X_{\bar{M}\bar{N}}{}^{\bar{P}}$ a set
of constants to be identified with the embedding tensor of the lower-
dimensional supergravity, specified in our cases by (2.11) through
$X_{\bar{M}\bar{N}}{}^{\bar{P}}=\Theta_{\bar{M}}{}^{\bar{\alpha}}(t_{\bar{\alpha}})_{\bar{N}}{}^{\bar{P}}$,
with $\Theta_{\bar{M}}{}^{\bar{\alpha}}$ given in (A.1) in terms of the
components in (2.10).

For the gaugings under consideration, the twist matrix can be written as

$\displaystyle U_{M}{}^{\bar{M}}(y)$
$\displaystyle=\left(\begin{array}[]{c;{2pt/2pt}c}U_{AB}{}^{\bar{A}\bar{B}}(y)&0\\\
\hdashline[2pt/2pt]0&(U^{-1})^{AB}{}_{\bar{A}\bar{B}}(y)\\\
\end{array}\right)\,,$ (3.29)

with

$U_{AB}{}^{\bar{A}\bar{B}}(y)=2\,U_{[A}{}^{\bar{A}}(y)U_{B]}{}^{\bar{B}}(y)\,.$
(3.30)

The $S^{7}$ reduction [3] can be described via (3.25) [28] in terms of the
scaling function

$\rho=(1-g^{2}|y|^{2})^{1/4}\,.$ (3.31)

The components of the SL$(8,\mathbb{R})$ matrix $U_{A}{}^{\bar{A}}(y)$ in
(3.30) under the splitting (3.7) for local and global indices are

$\displaystyle U_{i}{}^{\bar{j}}$
$\displaystyle=(1-g^{2}|y|^{2})^{-1/8}\big{(}\delta_{i}^{j}+g^{2}y^{i}\,y^{j}\,K_{1}(y)\big{)}\,,$
(3.32) $\displaystyle U_{i}{}^{\bar{8}}$
$\displaystyle=g\,(1-g^{2}|y|^{2})^{3/8}\,y^{i}\,,$ $\displaystyle
U_{8}{}^{\bar{j}}$ $\displaystyle=g\,(1-g^{2}|y|^{2})^{3/8}\,y^{j}K_{1}(y)\,,$
$\displaystyle U_{8}{}^{\bar{8}}$ $\displaystyle=(1-g^{2}|y|^{2})^{7/8}\,,$

with

$K_{1}(y)=-{}_{2}F_{1}\big{(}1,3,\tfrac{1}{2},1-g^{2}|y|^{2}\big{)}\,,$ (3.33)

and $|y|^{2}=\delta_{ij}y^{i}\,y^{j}$. In terms of these coordinates, the
round metric obtained through (3.15) by setting the scalars to zero reads

$ds^{2}(S^{7}_{\rm
round})=\delta_{ij}dy^{i}\,dy^{j}+\frac{g^{2}(\delta_{ij}y^{i}\,dy^{j})^{2}}{1-g^{2}|y|^{2}}\,.$
(3.34)

The uplift of the dyonic CSO gaugings of [40] can also be described via (3.25)
with block diagonal twist matrix (3.29) with (3.30) [42]. For the
$[\textrm{SO}(6)\times{\rm SO(2)}]\ltimes\mathbb{R}^{12}$ gauging, the scaling
function reads

$\rho=\hat{\rho}(\hat{y})\tilde{\rho}(\tilde{y})\equiv(1-g^{2}|\hat{y}|^{2})^{1/4}(1-m^{2}\tilde{y}^{2})^{1/4}\,,$
(3.35)

and the components of the SL$(8,\mathbb{R})$ matrix $U_{A}{}^{\bar{A}}(y)$
under the splitting (3.24) for local and global indices are

$\displaystyle U_{\textrm{i}}{}^{\bar{\textrm{j}}}$
$\displaystyle=\hat{\rho}^{-1/2}\tilde{\rho}^{1/2}\big{(}\delta_{\textrm{i}}^{\textrm{j}}+g^{2}\,\hat{y}^{\textrm{i}}\,\hat{y}^{\textrm{j}}\,K_{2}(\hat{y})\big{)}\,,\qquad\quad$
$\displaystyle U_{7}{}^{\bar{7}}=\hat{\rho}^{-1/2}\tilde{\rho}^{1/2}\,,$
(3.36) $\displaystyle U_{\textrm{i}}{}^{\bar{6}}$
$\displaystyle=g\,\hat{\rho}^{3/2}\tilde{\rho}^{1/2}\,\hat{y}^{\textrm{i}}\,,$
$\displaystyle
U_{7}{}^{\bar{8}}=-m\,\,\hat{\rho}^{-1/2}\tilde{\rho}^{-3/2}\,\tilde{y}\,,$
$\displaystyle U_{6}{}^{\bar{\textrm{j}}}$
$\displaystyle=g\,\hat{\rho}^{3/2}\tilde{\rho}^{1/2}\,\hat{y}^{\textrm{j}}\,K_{2}(\hat{y})\,,$
$\displaystyle
U_{8}{}^{\bar{7}}=m\,\hat{\rho}^{-1/2}\tilde{\rho}^{-3/2}\,\tilde{y}\,,$
$\displaystyle U_{6}{}^{\bar{6}}$
$\displaystyle=\hat{\rho}^{7/2}\tilde{\rho}^{1/2}\,,$ $\displaystyle
U_{8}{}^{\bar{8}}=\hat{\rho}^{-1/2}\tilde{\rho}^{1/2}\,,$

with
$|\hat{y}|^{2}=\delta_{\textrm{i}\textrm{j}}\,\hat{y}^{\textrm{i}}\,\hat{y}^{\textrm{j}}$
and

$K_{2}(\hat{y})=-{}_{2}F_{1}\big{(}1,2,\tfrac{1}{2},1-g^{2}|\hat{y}|^{2}\big{)}\,.$
(3.37)

The generalised frames (3.31)-(3.33) and (3.35)-(3.37) can be checked to
satisfy (3.26) with the embedding tensors corresponding to (2.11) via (A.1).

##  4 Supergravity Embeddings

###  4.1 SO(8) gauging on $S^{7}$

In the following, it is convenient to introduce coordinates $(\mu^{\rm a},\
\phi^{\rm a})$, with $\rm a=1,2,3,4$, as444These coordinates are related to
the ones in [46] through the redefinitions in footnote 1.

$\displaystyle y^{1}=g^{-1}\mu_{1}\cos\,\phi_{1}\,,\qquad$ $\displaystyle
y^{3}=g^{-1}\mu_{2}\cos\,\phi_{2}\,,\qquad$ $\displaystyle
y^{5}=g^{-1}\mu_{3}\cos\,\phi_{3}\,,\qquad$ $\displaystyle
y^{7}=g^{-1}\mu_{4}\cos\,\phi_{4}\,,$ (4.1) $\displaystyle
y^{2}=g^{-1}\mu_{1}\sin\,\phi_{1}\,,$ $\displaystyle
y^{4}=g^{-1}\mu_{2}\sin\,\phi_{2}\,,$ $\displaystyle
y^{6}=g^{-1}\mu_{3}\sin\,\phi_{3}\,,$

constrained by

$\displaystyle\sum_{\rm a=1}^{4}\mu_{\rm a}^{2}=1\,.$ (4.2)

In terms of these coordinates, the metric with (3.25) and (3.32) becomes [46]

$\displaystyle ds_{11}^{2}=$
$\displaystyle\,\Xi_{1}^{1/3}ds_{4}^{2}+g^{-2}\Xi_{1}^{-2/3}\Big{[}\sum_{\rm
a}Z_{a}(d\mu_{\rm a}^{2}+\mu_{\rm a}^{2}D\phi_{\rm a}^{2})$ (4.3)
$\displaystyle-2b_{2}b_{3}(\mu_{3}^{2}\mu_{4}^{2}D\phi_{3}D\phi_{4}+\mu_{1}^{2}\mu_{2}^{2}D\phi_{1}D\phi_{2})$
$\displaystyle-2b_{1}b_{3}(\mu_{2}^{2}\mu_{3}^{2}D\phi_{2}D\phi_{3}+\mu_{1}^{2}\mu_{4}^{2}D\phi_{1}D\phi_{4})$
$\displaystyle-2b_{1}b_{2}(\mu_{1}^{2}\mu_{3}^{2}D\phi_{1}D\phi_{3}+\mu_{2}^{2}\mu_{4}^{2}D\phi_{2}D\phi_{4})$
$\displaystyle+b_{1}^{2}(\mu_{1}d\mu_{1}+\mu_{2}d\mu_{2})^{2}+b_{2}^{2}(\mu_{2}d\mu_{2}+\mu_{3}d\mu_{3})^{2}+b_{3}^{2}(\mu_{1}d\mu_{1}+\mu_{3}d\mu_{3})^{2}\Big{]}\,,$

where

$\displaystyle\Xi_{1}=$
$\displaystyle\,\tilde{Y}^{2}_{1}\tilde{Y}^{2}_{2}Y^{2}_{3}\,\mu_{1}^{4}+\tilde{Y}^{2}_{1}Y^{2}_{2}\tilde{Y}^{2}_{3}\,\mu_{2}^{4}+Y^{2}_{1}Y^{2}_{2}Y^{2}_{3}\,\mu_{3}^{4}+Y^{2}_{1}\tilde{Y}^{2}_{2}\tilde{Y}^{2}_{3}\,\mu_{4}^{4}$
(4.4)
$\displaystyle+(Y^{2}_{2}\tilde{Y}^{2}_{2}+Y^{2}_{3}\tilde{Y}^{2}_{3})\big{(}Y^{2}_{1}\mu_{3}^{2}\mu_{4}^{2}+\tilde{Y}^{2}_{1}\mu_{1}^{2}\mu_{2}^{2}\big{)}$
$\displaystyle+(Y^{2}_{1}\tilde{Y}^{2}_{1}+Y^{2}_{3}\tilde{Y}^{2}_{3})\big{(}Y^{2}_{2}\mu_{2}^{2}\mu_{3}^{2}+\tilde{Y}^{2}_{2}\mu_{1}^{2}\mu_{4}^{2}\big{)}$
$\displaystyle+(Y^{2}_{1}\tilde{Y}^{2}_{1}+Y^{2}_{2}\tilde{Y}^{2}_{2})\big{(}Y^{2}_{3}\mu_{1}^{2}\mu_{3}^{2}+\tilde{Y}^{2}_{3}\mu_{2}^{2}\mu_{4}^{2}\big{)}\,,$

and

$Z_{\rm a}=\mu^{2}_{\rm a}+W_{\rm a}\,,$ (4.5)

with

$\displaystyle
W_{1}=Y^{2}_{2}\tilde{Y}^{2}_{3}\mu_{2}^{2}+Y^{2}_{1}Y^{2}_{2}\mu_{3}^{2}+Y^{2}_{1}\tilde{Y}^{2}_{3}\mu_{4}^{2}\,,\qquad
W_{2}=\tilde{Y}^{2}_{2}Y^{2}_{3}\mu_{1}^{2}+Y^{2}_{1}Y^{2}_{3}\mu_{3}^{2}+Y^{2}_{1}\tilde{Y}^{2}_{2}\mu_{4}^{2}\,,$
(4.6) $\displaystyle
W_{3}=\tilde{Y}^{2}_{1}\tilde{Y}^{2}_{2}\mu_{1}^{2}+\tilde{Y}^{2}_{1}\tilde{Y}^{2}_{3}\mu_{2}^{2}+\tilde{Y}^{2}_{2}\tilde{Y}^{2}_{3}\mu_{4}^{2}\,,\qquad
W_{4}=\tilde{Y}^{2}_{1}Y^{2}_{3}\mu_{1}^{2}+\tilde{Y}^{2}_{1}Y^{2}_{2}\mu_{2}^{2}+Y^{2}_{2}Y^{2}_{3}\mu_{3}^{2}\,.$

For this gauging, the covariant derivatives on the angles denote the fibering
with the four vectors

$D\phi_{\rm a}=d\phi_{\rm a}-gA^{\rm a}\,.$ (4.7)

with $A^{\rm a}$ given in terms of SL$(8,\mathbb{R})$ objects in (A.11). The
three-form potential can be written in terms of the 4$d$ potentials in (2.28)
suitably coupled to the $S^{7}$ coordinates, and an internal contribution as
dictated by (3.15). The link between the tensor hierarchy fields and the
sphere coordinates is in fact fixed by their respective ${\rm
SL}(8,\mathbb{R})$ structure, and gauge invariance demands that the different
terms combine into the field strengths in (2.29) when acted upon by the
exterior derivative. Notably, only a subset of the forms in (2.28), dubbed
“restricted tensor hierarchy” in [33, 34], enters the KK Ansatz. For the STU
truncation, the eleven-dimensional three-form can be decomposed as simply

$\hat{A}_{{\scriptscriptstyle(3)}}=\sum_{\rm a=1}^{4}\Big{[}-C_{\rm
a}\,\mu_{\rm a}^{2}+\tfrac{1}{2g}\big{[}B_{\rm a}+\tfrac{1}{2}A^{\rm
a}\wedge\tilde{A}_{\rm a}\big{]}\wedge d(\mu^{2}_{\rm
a})-\tfrac{1}{2g^{2}}\tilde{A}_{\rm a}\wedge d(\mu_{\rm a}^{2})\wedge
D\phi^{\rm a}\Big{]}+C_{{\scriptscriptstyle(3)}}\,,$ (4.8)

with the overall scaling fixed by the equations of motion. This result agrees
with the truncation of [34] through (A.11)–(A.15). The expression for the
internal three-form can also be obtained from (3.15) with (3.25) and (3.32),
and reads

$C_{{\scriptscriptstyle(3)}}=\frac{1}{2}\sum_{\rm abc}C_{\rm a,bc}d\mu^{\rm
a}\wedge D\phi^{\rm b}\wedge D\phi^{\rm c}\,,$ (4.9)

with components $C_{\rm a,bc}=C_{\rm a,[bc]}$ given by

$\displaystyle C_{\rm a,12}\,d\mu^{\rm a}$
$\displaystyle=\frac{-b_{1}}{2\,g^{3}\,\Xi_{1}}\Big{[}\mu_{1}^{2}\,W_{2}\,d(\mu_{2}^{2})-\mu_{2}^{2}\,W_{1}\,d(\mu_{1}^{2})+\mu_{1}^{2}\mu_{2}^{2}\big{(}Y_{2}^{2}\tilde{Y}_{2}^{2}\,d\alpha_{23}-Y_{3}^{2}\tilde{Y}_{3}^{2}\,d\alpha_{13}\big{)}\Big{]}\,,$
(4.10) $\displaystyle C_{\rm a,34}\,d\mu^{\rm a}$
$\displaystyle=\frac{-b_{1}}{2\,g^{3}\,\Xi_{1}}\Big{[}\mu_{3}^{2}\,W_{4}\,d(\mu_{4}^{2})-\mu_{4}^{2}\,W_{3}\,d(\mu_{3}^{2})-\mu_{3}^{2}\mu_{4}^{2}\big{(}Y_{2}^{2}\tilde{Y}_{2}^{2}\,d\alpha_{23}+Y_{3}^{2}\tilde{Y}_{3}^{2}\,d\alpha_{13}\big{)}\Big{]}\,,$
$\displaystyle C_{\rm a,23}\,d\mu^{\rm a}$
$\displaystyle=\frac{-b_{2}}{2\,g^{3}\,\Xi_{1}}\Big{[}\mu_{2}^{2}\,W_{3}\,d(\mu_{3}^{2})-\mu_{3}^{2}\,W_{2}\,d(\mu_{2}^{2})+\mu_{2}^{2}\mu_{3}^{2}\big{(}Y_{3}^{2}\tilde{Y}_{3}^{2}\,d\alpha_{13}-Y_{1}^{2}\tilde{Y}_{1}^{2}\,d\alpha_{12}\big{)}\Big{]}\,,$
$\displaystyle C_{\rm a,14}\,d\mu^{\rm a}$
$\displaystyle=\frac{-b_{2}}{2\,g^{3}\,\Xi_{1}}\Big{[}\mu_{1}^{2}\,W_{4}\,d(\mu_{4}^{2})-\mu_{4}^{2}\,W_{1}\,d(\mu_{1}^{2})-\mu_{1}^{2}\mu_{4}^{2}\big{(}Y_{1}^{2}\tilde{Y}_{1}^{2}\,d\alpha_{12}+Y_{3}^{2}\tilde{Y}_{3}^{2}\,d\alpha_{13}\big{)}\Big{]}\,,$
$\displaystyle C_{\rm a,13}\,d\mu^{\rm a}$
$\displaystyle=\frac{-b_{3}}{2\,g^{3}\,\Xi_{1}}\Big{[}\mu_{1}^{2}\,W_{3}\,d(\mu_{3}^{2})-\mu_{3}^{2}\,W_{1}\,d(\mu_{1}^{2})+\mu_{1}^{2}\mu_{3}^{2}\big{(}Y_{2}^{2}\tilde{Y}_{2}^{2}\,d\alpha_{23}-Y_{1}^{2}\tilde{Y}_{1}^{2}\,d\alpha_{12}\big{)}\Big{]}\,,$
$\displaystyle C_{\rm a,24}\,d\mu^{\rm a}$
$\displaystyle=\frac{-b_{3}}{2\,g^{3}\,\Xi_{1}}\Big{[}\mu_{2}^{2}\,W_{4}\,d(\mu_{4}^{2})-\mu_{4}^{2}\,W_{2}\,d(\mu_{2}^{2})-\mu_{2}^{2}\mu_{4}^{2}\big{(}Y_{1}^{2}\tilde{Y}_{1}^{2}\,d\alpha_{12}+Y_{2}^{2}\tilde{Y}_{2}^{2}\,d\alpha_{23}\big{)}\Big{]}\,,$

and shorthands

$\alpha_{12}=\mu_{1}^{2}+\mu_{2}^{2}\,,\qquad\alpha_{23}=\mu_{2}^{2}+\mu_{3}^{2}\,,\qquad\alpha_{34}=\mu_{3}^{2}+\mu_{4}^{2}\,,\qquad\rm{etc.}$
(4.11)

This result matches (4.19) of [46] upon making the identifications in footnote
1. The eleven-dimensional four-form is then produced by the exterior
derivative of (4.8), which by using (2.29) can be given as

$\displaystyle\hat{F}_{{\scriptscriptstyle(4)}}=$ $\displaystyle\sum_{\rm
a=1}^{4}\Big{[}-H_{{{\scriptscriptstyle(4)}}\rm a}\,\mu_{\rm
a}^{2}+\tfrac{1}{2g}H_{{{\scriptscriptstyle(3)}}\rm a}\wedge d(\mu^{2}_{\rm
a})+\tfrac{1}{2g^{2}}\tilde{H}_{{{\scriptscriptstyle(2)}}\rm a}\wedge
d(\mu_{\rm a}^{2})\wedge D\phi^{\rm a}\Big{]}$ (4.12)
$\displaystyle+g\sum_{\rm abc}C_{\rm a,bc}d\mu^{\rm a}\wedge D\phi^{\rm
b}\wedge H_{{\scriptscriptstyle(2)}}^{\rm c}+\frac{1}{2}\sum_{\rm
abcd}\partial_{d}C_{\rm a,bc}d\mu^{\rm d}\wedge d\mu^{\rm a}\wedge D\phi^{\rm
b}\wedge D\phi^{\rm c}$ $\displaystyle\qquad+\frac{1}{2}\sum_{\rm
abc}\partial_{\mu}C_{\rm a,bc}dx^{\mu}\wedge d\mu^{\rm a}\wedge D\phi^{\rm
b}\wedge D\phi^{\rm c}\,,$

and thus making use of only the original fields appearing in the
$\mathcal{N}=8$ action and their derivatives through the duality relations
(2.1)–(2.36), with the last equation particularised for this gauging to

$\displaystyle H_{{{\scriptscriptstyle(4)}}1}$
$\displaystyle=2g(\tilde{Y}_{1}^{2}+\tilde{Y}_{2}^{2}+Y_{3}^{2})\textrm{vol}_{4}\,,\qquad\quad
H_{{{\scriptscriptstyle(4)}}2}=2g(\tilde{Y}_{1}^{2}+Y_{2}^{2}+\tilde{Y}_{3}^{2})\textrm{vol}_{4}\,,$
(4.13) $\displaystyle H_{{{\scriptscriptstyle(4)}}3}$
$\displaystyle=2g(Y_{1}^{2}+Y_{2}^{2}+Y_{3}^{2})\textrm{vol}_{4}\,,\qquad\quad
H_{{{\scriptscriptstyle(4)}}4}=2g(Y_{1}^{2}+\tilde{Y}_{2}^{2}+\tilde{Y}_{3}^{2})\textrm{vol}_{4}\,.$

#### Singular limit

Scaling the fields and couplings as in (2.18) and (2.19), the configuration
remains finite up to a trombone scaling if we also transform the internal
coordinates as

$\mu_{4}\mapsto e^{-k}\mu_{4}\,,$ (4.14)

with the coordinates $(\mu_{\rm\bar{a}},\ \phi_{\rm\bar{a}},\ \phi_{4})$
invariant. Doing so, the warping (4.4) factorises into

$\Xi_{1}\mapsto\bar{\Xi}_{1}=e^{k}\,H\,\Xi_{2}\,,$ (4.15)

with

$H=\tilde{Y}_{2}^{2}\mu_{1}^{2}+\tilde{Y}_{3}^{2}\mu_{2}^{2}+Y_{1}^{2}\mu_{3}^{2}\,,\qquad\qquad\Xi_{2}=W_{4}\,.$
(4.16)

Accordingly, the metric becomes

$\displaystyle ds_{11}^{2}\mapsto d\bar{s}_{11}^{2}=$
$\displaystyle\,e^{k/3}\Big{\\{}(H\,\Xi_{2})^{1/3}ds_{4}^{2}+g^{-2}\,(H\,\Xi_{2})^{-2/3}\Big{[}\,\Xi_{2}(d\mu_{4}^{2}+\mu_{4}^{2}d\phi_{4}^{2})$
$\displaystyle\qquad+H\big{(}Y_{2}^{2}(d\mu_{1}^{2}+\mu_{1}^{2}D\phi_{1}^{2})+Y_{3}^{2}(d\mu_{2}^{2}+\mu_{2}^{2}D\phi_{2}^{2})+\tilde{Y}_{1}^{2}(d\mu_{3}^{2}+\mu_{3}^{2}D\phi_{3}^{2})\big{)}$
$\displaystyle\qquad\quad-\big{(}b_{2}\mu_{1}^{2}D\phi_{1}+b_{3}\mu_{2}^{2}D\phi_{2}+b_{1}\mu_{3}^{2}D\phi_{3}\big{)}^{2}\Big{]}\Big{\\}}\,,$
(4.17)

and the three-form components (4.10) transform as

$C_{a,bc}d\mu^{a}\mapsto e^{k/2}\bar{C}_{a,bc}d\mu^{a}\,,$ (4.18)

with

$\displaystyle\bar{C}_{\rm a,12}\,d\mu^{\rm
a}=\frac{b_{1}\,\mu_{1}\mu_{2}}{g^{3}\,\Xi_{2}}\big{(}Y_{2}^{2}\mu_{2}d\mu_{1}$
$\displaystyle-Y_{3}^{2}\mu_{1}d\mu_{2}\big{)}\,,\qquad\bar{C}_{\rm
a,23}\,d\mu^{\rm
a}=\frac{b_{2}\,\mu_{2}\mu_{3}}{g^{3}\,\Xi_{2}}\big{(}Y_{3}^{2}\mu_{3}d\mu_{2}-\tilde{Y}_{1}^{2}\mu_{2}d\mu_{3}\big{)}\,,$
(4.19) $\displaystyle\bar{C}_{\rm a,13}\,d\mu^{\rm
a}=\frac{b_{3}\,\mu_{1}\mu_{3}}{g^{3}\,\Xi_{2}}\big{(}Y_{2}^{2}\mu_{3}d\mu_{1}-\tilde{Y}_{1}^{2}\mu_{1}d\mu_{3}\big{)}\,,$

and

$\displaystyle\bar{C}_{\rm a,41}\,d\mu^{\rm
a}=\frac{b_{2}}{2g^{3}H}\mu_{1}^{2}\,d(\mu_{4}^{2})\,,$
$\displaystyle\qquad\qquad\bar{C}_{\rm a,42}\,d\mu^{\rm
a}=\frac{b_{3}}{2g^{3}H}\mu_{2}^{2}\,d(\mu_{4}^{2})\,,$ (4.20)
$\displaystyle\bar{C}_{\rm a,43}\,d\mu^{\rm a}$
$\displaystyle=\frac{b_{1}}{2g^{3}H}\mu_{3}^{2}\,d(\mu_{4}^{2})\,.$

Therefore, the four-form (4.12) becomes

$\displaystyle\hat{F}_{{\scriptscriptstyle(4)}}\mapsto\bar{\hat{F}}_{{\scriptscriptstyle(4)}}$
$\displaystyle=e^{k/2}\Big{\\{}\sum_{\rm\bar{a}=1}^{3}\Big{[}-\bar{H}_{{{\scriptscriptstyle(4)}}\rm\bar{a}}\,\mu_{\rm\bar{a}}^{2}+\tfrac{1}{2g}H_{{{\scriptscriptstyle(3)}}\rm\bar{a}}\wedge
d(\mu^{2}_{\rm\bar{a}})+\tfrac{1}{2g^{2}}\tilde{H}_{{{\scriptscriptstyle(2)}}\rm\bar{a}}\wedge
d(\mu_{\rm\bar{a}}^{2})\wedge D\phi^{\rm\bar{a}}\Big{]}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\sum_{\rm\bar{a}\bar{b}\bar{c}}d\Big{[}\bar{C}_{\rm\bar{a},\bar{b}\bar{c}}d\mu^{\rm\bar{a}}\wedge
D\phi^{\rm\bar{b}}\wedge D\phi^{\rm\bar{c}}\Big{]}$ (4.21)
$\displaystyle\qquad\qquad+\tfrac{1}{2g^{2}}\tilde{H}_{{{\scriptscriptstyle(2)}}4}\wedge
d(\mu_{4}^{2})\wedge d\phi^{4}-\sum_{\rm\bar{a}}d\Big{[}\bar{C}_{\rm
4,\bar{a}4}D\phi^{\rm\bar{a}}\Big{]}\wedge d\mu^{4}\wedge
d\phi^{4}\Big{\\}}\,,$

with the $\bar{H}_{{{\scriptscriptstyle(4)}}\rm\bar{a}}$ in (2.44) reading

$\bar{H}_{1}=2g(\tilde{Y}_{1}^{2}+Y_{3}^{2})\textrm{vol}_{4}\,,\qquad\bar{H}_{2}=2g(\tilde{Y}_{1}^{2}+Y_{2}^{2})\textrm{vol}_{4}\,,\qquad\bar{H}_{3}=2g(Y_{2}^{2}+Y_{3}^{2})\textrm{vol}_{4}\,.$
(4.22)

#### IIA reduction and dualisation to IIB

The metric and four-form in (4.1) and (4.1) formally describe a warped
compactification on $S^{5}\times\mathbb{R}^{2}$, with the $\mathbb{R}^{2}$
factor parameterised by $(\mu_{4},\,\phi_{4})$ and the sphere by
$(\mu_{\rm\bar{a}},\,\phi_{\rm\bar{a}})$ satisfying

$\displaystyle\sum_{\rm\bar{a}=1}^{3}\mu_{\rm\bar{a}}^{2}=1\,,$ (4.23)

which follows from (4.2) after taking the $k\to\infty$ limit on (4.14).
Introducing coordinates $z_{1,2}$ as

$z_{1}=g^{-1}\mu_{4}\cos\phi_{4}\,,\qquad\quad
z_{2}=g^{-1}\mu_{4}\sin\phi_{4}\,,$ (4.24)

we can promote the $\mathbb{R}^{2}$ factor into a two-torus by imposing
$z_{1,2}\sim z_{1,2}+2\pi R_{1,2}$. The eleven-dimensional configuration,
(4.1) and (4.1), can then be interpreted as a type IIA supergravity solution
upon reducing on one of the circles, say $z_{2}$. The resulting IIA geometry
has an $S^{1}$ factor that allows one to perform a T-duality transformation
and in this way obtain a solution of type IIB supergravity.

To ease our notation, we introduce the following shorthands

$\displaystyle X_{{\scriptscriptstyle(4)}}$
$\displaystyle=\sum_{\rm\bar{a}=1}^{3}\Big{[}-\bar{H}_{{{\scriptscriptstyle(4)}}\rm\bar{a}}\,\mu_{\rm\bar{a}}^{2}+\tfrac{1}{2g}H_{{{\scriptscriptstyle(3)}}\rm\bar{a}}\wedge
d(\mu^{2}_{\rm\bar{a}})+\tfrac{1}{2g^{2}}\tilde{H}_{{{\scriptscriptstyle(2)}}\rm\bar{a}}\wedge
d(\mu_{\rm\bar{a}}^{2})\wedge D\phi^{\rm\bar{a}}\Big{]}$
$\displaystyle\qquad\quad+\frac{1}{2}\sum_{\rm\bar{a}\bar{b}\bar{c}}d\Big{[}\bar{C}_{\rm\bar{a},\bar{b}\bar{c}}d\mu^{\rm\bar{a}}\wedge
D\phi^{\rm\bar{b}}\wedge D\phi^{\rm\bar{c}}\Big{]}\,,$ $\displaystyle
X_{{\scriptscriptstyle(2)}}$
$\displaystyle=\tilde{H}_{{{\scriptscriptstyle(2)}}4}+d\Big{[}\frac{1}{gH}\big{(}b_{2}\,\mu_{1}^{2}D\phi_{1}+b_{3}\,\mu_{2}^{2}D\phi_{2}+b_{1}\,\mu_{3}^{2}D\phi_{3}\big{)}\Big{]}\,,$
(4.25)

such that the eleven-dimensional four-form can be written as

$\bar{\hat{F}}_{{\scriptscriptstyle(4)}}=X_{{\scriptscriptstyle(4)}}+X_{{\scriptscriptstyle(2)}}\wedge
dz_{1}\wedge dz_{2}\,.$ (4.26)

Both $X_{{\scriptscriptstyle(2)}}$ and $X_{{\scriptscriptstyle(4)}}$ are
exact, and a representative potential for $X_{{\scriptscriptstyle(2)}}$ can be
read off from (4.25) to be

$\mathcal{A}=\tilde{A}_{4}+\frac{1}{gH}\Big{[}b_{2}\,\mu_{1}^{2}D\phi_{1}+b_{3}\,\mu_{2}^{2}D\phi_{2}+b_{1}\,\mu_{3}^{2}D\phi_{3}\Big{]}\,.$
(4.27)

Following the conventions stated in appendix B, the type IIA configuration
resulting from reduction of (4.1) and (4.26) on $z_{2}$ is

$\displaystyle e^{\phi_{\rm IIA}}$
$\displaystyle=\frac{\Xi_{2}^{1/4}}{\sqrt{H}}\,,\quad
H_{{{\scriptscriptstyle(3)}}\rm IIA}=X_{{\scriptscriptstyle(2)}}\wedge
dz_{1}\,,\quad F_{{{\scriptscriptstyle(2)}}\rm IIA}=0\,,\quad
F_{{{\scriptscriptstyle(4)}}\rm IIA}=X_{{\scriptscriptstyle(4)}}\,,$
$\displaystyle ds_{\rm IIA}^{2}$
$\displaystyle=H^{1/4}\Xi_{2}^{3/8}ds^{2}_{4}+g^{-2}\,H^{-3/4}\Xi_{2}^{-5/8}\Big{[}g^{2}\,\Xi_{2}\,dz_{1}^{2}-(b_{2}\mu_{1}^{2}D\phi_{1}+b_{3}\mu_{2}^{2}D\phi_{2}+b_{1}\mu_{3}^{2}D\phi_{3})^{2}$
$\displaystyle\quad\quad+H\big{(}Y_{2}^{2}(d\mu_{1}^{2}+\mu_{1}^{2}D\phi_{1}^{2})+Y_{3}^{2}(d\mu_{2}^{2}+\mu_{2}^{2}D\phi_{2}^{2})+\tilde{Y}_{1}^{2}(d\mu_{3}^{2}+\mu_{3}^{2}D\phi_{3}^{2})\big{)}\Big{]}\,,$
(4.28)

and the type IIB solution obtained by T-dualising along the $z_{1}$ direction
employing the relations (B.8)-(B.9) reads

$\displaystyle\phi_{\rm IIB}=\chi=0\,,$ $\displaystyle ds_{\rm
IIB}^{2}=\Xi_{2}^{1/2}ds^{2}_{4}+g^{-2}\,\Xi_{2}^{-\frac{1}{2}}\big{[}Y_{2}^{2}(d\mu_{1}^{2}+\mu_{1}^{2}D\phi_{1}^{2})+Y_{3}^{2}(d\mu_{2}^{2}+\mu_{2}^{2}D\phi_{2}^{2})+\tilde{Y}_{1}^{2}(d\mu_{3}^{2}+\mu_{3}^{2}D\phi_{3}^{2})$
$\displaystyle\quad\quad\quad-H^{-1}(b_{2}\mu_{1}^{2}D\phi_{1}+b_{3}\mu_{2}^{2}D\phi_{2}+b_{1}\mu_{3}^{2}D\phi_{3})^{2}+g^{2}H(dz_{1}+\mathcal{A})^{2}\big{]}\,,$
$\displaystyle H_{{{\scriptscriptstyle(3)}}\rm
IIB}=F_{{{\scriptscriptstyle(3)}}\rm IIB}=0\,,$ $\displaystyle
F_{{{\scriptscriptstyle(5)}}\rm
IIB}=(1+*_{10})\big{[}X_{{\scriptscriptstyle(4)}}\wedge\big{(}dz_{1}+\mathcal{A}\big{)}\big{]}\,.$
(4.29)

###  4.2 $[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$ gauging
on $S^{5}\times S^{1}$

The type IIB configurations that uplift from $D=4$ gauged supergravity with
gauging specified by
$(\theta_{{\scriptscriptstyle(6c)}},\xi_{{\scriptscriptstyle(6c)}})$ in (2.11)
can also be obtained from (3.25), by employing the twist in (3.35)–(3.37) and
the ExFT dictionary. For the STU sector, this configuration is related to the
singular limit of the $S^{7}$ configurations in M-theory in the last section.
To accomplish this, we use the coordinates
$(\mu_{\rm\bar{a}},\,\phi_{\rm\bar{a}})$ above related to $y^{i}$ as

$\displaystyle\hat{y}^{1}=g^{-1}\mu_{1}\cos\,\phi_{1}\,,\qquad$
$\displaystyle\hat{y}^{3}=g^{-1}\mu_{2}\cos\,\phi_{2}\,,\qquad$
$\displaystyle\hat{y}^{5}=g^{-1}\mu_{3}\cos\,\phi_{3}\,,$ (4.30)
$\displaystyle\hat{y}^{2}=g^{-1}\mu_{1}\sin\,\phi_{1}\,,$
$\displaystyle\hat{y}^{4}=g^{-1}\mu_{2}\sin\,\phi_{2}\,,$

and constrained by (4.23). A coordinate $z$ can also be introduced such that

$\tilde{y}=m^{-1}\sin mz\,,$ (4.31)

which reduces to $\tilde{y}=z$ in the vanishing $m$ limit. The metric obtained
from (3.22) reads

$\displaystyle d\hat{s}_{10}^{2}=$
$\displaystyle\,\Xi_{2}^{1/2}ds_{4}^{2}+g^{-2}\Xi_{2}^{-1/2}\Big{[}Y^{2}_{2}(d\mu_{1}^{2}+\mu_{1}^{2}D\phi_{1}^{2})+Y^{2}_{3}(d\mu_{2}^{2}+\mu_{2}^{2}D\phi_{2}^{2})+\tilde{Y}^{2}_{1}(d\mu_{3}^{2}+\mu_{3}^{2}D\phi_{3}^{2})$
(4.32)
$\displaystyle\qquad\qquad\qquad\qquad\enspace-H^{-1}\big{(}b_{2}\mu_{1}^{2}D\phi_{1}+b_{3}\mu_{2}^{2}D\phi_{2}+b_{1}\mu_{3}^{2}D\phi_{3}\big{)}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+H\Big{(}g\,dz+g\tilde{A}_{4}+H^{-1}\big{(}b_{2}\mu_{1}^{2}D\phi_{1}+b_{3}\mu_{2}^{2}D\phi_{2}+b_{1}\mu_{3}^{2}D\phi_{3}\big{)}\Big{)}^{2}\Big{]}\,,$

where $\Xi_{2}$ and $H$ are given in (4.16) and, for this gauging, the
covariant derivatives on the angles denote the fibering with the four vectors

$D\phi_{\rm\bar{a}}=d\phi_{\rm\bar{a}}-gA^{\rm\bar{a}}\,,$ (4.33)

with $A^{\rm\bar{a}}=\\{A^{12}\,,\ A^{34}\,,\ A^{56}\\}$ in terms of
SL$(6,\mathbb{R})$ indices. Note that for these flat indices, ${\rm
SL}(6,\mathbb{R})\subset{\rm SL}(8,\mathbb{R})$. In this sector, the
axiodilaton vanishes

$\Phi=0\,,\qquad\quad C_{0}=0\,,$ (4.34)

as well as the two-form potentials

$C_{{{\scriptscriptstyle(2)}}}{}^{a}=0\,.$ (4.35)

Finally, the four-form gauge potential can also be derived from the ExFT
dictionary. The purely internal contribution can be obtained from (3.22) to be

$\displaystyle C_{{{\scriptscriptstyle(4)}}}$
$\displaystyle=-g^{-4}\mu_{1}\mu_{2}(1+K_{2})\frac{\cot\phi_{3}}{\mu_{3}^{2}}d\mu_{1}\wedge
d\mu_{2}\wedge d\phi_{1}\wedge d\phi_{2}$ (4.36)
$\displaystyle\quad-g^{-4}\mu_{1}\mu_{2}\Big{[}K_{2}(\mu_{2}d\mu_{1}-\mu_{1}d\mu_{2})+\Xi_{2}^{-1}\tilde{Y}_{1}^{2}(Y_{2}^{2}\mu_{2}d\mu_{1}-Y_{3}^{2}\mu_{1}d\mu_{2})\Big{]}\wedge
d\phi_{1}\wedge d\phi_{2}\wedge d\phi_{3}$
$\displaystyle\qquad+\tfrac{1}{2g^{3}}\Xi_{2}^{-1}\Big{[}b_{1}\big{(}Y_{2}^{2}\mu_{2}^{2}d(\mu_{1}^{2})-Y_{3}^{2}\mu_{1}^{2}d(\mu_{2}^{2})\big{)}\wedge
d\phi_{1}\wedge d\phi_{2}$
$\displaystyle\qquad\qquad\qquad+b_{2}\big{(}Y_{2}^{2}\mu_{3}^{2}d(\mu_{1}^{2})-\tilde{Y}_{1}^{2}\mu_{1}^{2}d(\mu_{3}^{2})\big{)}\wedge
d\phi_{1}\wedge d\phi_{3}$
$\displaystyle\qquad\qquad\qquad\quad+b_{3}\big{(}Y_{3}^{2}\mu_{3}^{2}d(\mu_{2}^{2})-\tilde{Y}_{1}^{2}\mu_{2}^{2}d(\mu_{3}^{2})\big{)}\wedge
d\phi_{2}\wedge d\phi_{3}\Big{]}\wedge dz\,.$

We recognise that, stripped of $dz$, the contributions in the last three lines
in (4.36) precisely match the coefficients
$\bar{C}_{\rm\bar{a},\bar{b}\bar{c}}$ in (4.19), and the derivatives of the
first two lines are dual to the contributions in (4.19) given by the forms in
the tensor hierarchy. Therefore, the associated self-dual five-form field
strength is given by

$\displaystyle\hat{F}_{{{\scriptscriptstyle(5)}}}$
$\displaystyle=(1+*_{10})\Big{\\{}\Big{[}\sum_{\rm\bar{a}=1}^{3}\Big{(}-H_{{{\scriptscriptstyle(4)}}\rm\bar{a}}\,\mu_{\rm\bar{a}}^{2}+\tfrac{1}{2g}H_{{{\scriptscriptstyle(3)}}\rm\bar{a}}\wedge
d(\mu^{2}_{\rm\bar{a}})+\tfrac{1}{2g^{2}}\tilde{H}_{{{\scriptscriptstyle(2)}}\rm\bar{a}}\wedge
d(\mu_{\rm\bar{a}}^{2})\wedge D\phi^{\rm\bar{a}}\Big{)}$ (4.37)
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\sum_{\rm\bar{a}\bar{b}\bar{c}}d\Big{(}\bar{C}_{\rm\bar{a},\bar{b}\bar{c}}d\mu^{\rm\bar{a}}\wedge
D\phi^{\rm\bar{b}}\wedge D\phi^{\rm\bar{c}}\Big{)}\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\wedge\Big{[}dz+\tilde{A}_{4}+\frac{1}{gH}\Big{(}b_{2}\,\mu_{1}^{2}D\phi_{1}+b_{3}\,\mu_{2}^{2}D\phi_{2}+b_{1}\,\mu_{3}^{2}D\phi_{3}\Big{)}\Big{]}\Big{\\}}\,,$

with $\bar{C}_{\rm\bar{a},\bar{b}\bar{c}}$ given in (4.19), and the field
strengths in the tensor hierarchy given by (2.1), (2.34) and (2.36) with $x=0$
and $\tilde{x}=1$, which coincide with the ones in (4.22). The uplift formulae
(4.32)–(4.37) precisely match the type IIB configuration obtained in (4.1) if
one identifies the angles on $S^{1}$.

Even though the gauge coupling $m$ does not enter into the Kaluza-Klein
Ansätze for the type IIB bosonic fields, it mediates the relation between ten-
and four-dimensional spinors, as expected from the different fermion couplings
observed in four dimensions.

To perform a thorough check of the Ansatz in (4.32)–(4.37), we will restrict
our attention to a simpler truncation that identifies the fields as

$\varphi_{1}=-\varphi_{2}=-\varphi_{3}=\tfrac{1}{\sqrt{3}}\varphi\,,\qquad\quad\chi_{1,2,3}=\tfrac{1}{\sqrt{3}}\chi\,,\qquad\quad
A^{1,2,3}_{{\scriptscriptstyle(1)}}=\tfrac{1}{\sqrt{3}}A_{{\scriptscriptstyle(1)}}\,,$
(4.38)

and leaves $A^{4}_{{\scriptscriptstyle(1)}}$ unfixed. For the
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ gauging, this theory can be obtained as
a circle reduction of the $\mathcal{N}=4$ ${\rm SU(2)\times U(1)}$ gauged
theory in $D=5$ [53, 54] truncated so that the five-dimensional scalar
vanishes and the gauge group is reduced as ${\rm U(1)}\subset{\rm U(1)\times
U(1)}\subset{\rm SU(2)\times U(1)}$. After the circle reduction, one can
identify $A^{4}_{{\scriptscriptstyle(1)}}$ with the dual of the KK vector. In
the following, we only consider configurations with $\chi=0$, which per (2.31)
cannot be dyonically charged so as to guarantee
$H_{{\scriptscriptstyle(2)}}^{\rm a}\wedge H_{{\scriptscriptstyle(2)}}^{\rm
b}=0$. The Lagrangian (2.2) for the $\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ and
$[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$ gaugings then
becomes

$e^{-1}\mathcal{L}=R-\tfrac{1}{2}(\partial\varphi)^{2}-\tfrac{1}{4}e^{a\varphi}H^{2}-\tfrac{1}{4}e^{-\varphi/a}(H^{4})^{2}+12g^{2}e^{-a\varphi}\,.$
(4.39)

with $H_{{{\scriptscriptstyle(2)}}}=dA_{{\scriptscriptstyle(1)}}$. To make
contact with the STU models in (2.2), the dilaton coupling needs to be set as
$a=1/\sqrt{3}$, but we find it convenient to keep it unspecified at the four-
dimensional level.

For $a=1/\sqrt{3}$, we can embed any solution of the $4d$ theory in ten
dimensions. From (4.32), the metric reads

$\displaystyle d\hat{s}_{10}^{2}=$
$\displaystyle\,e^{\frac{-\varphi}{\sqrt{3}}}ds_{4}^{2}+g^{-2}\Big{[}d\mu_{1}^{2}+d\mu_{2}^{2}+d\mu_{3}^{2}+\mu_{1}^{2}D\phi_{1}^{2}+\mu_{2}^{2}D\phi_{2}^{2}+\mu_{3}^{2}D\phi_{3}^{2}\Big{]}+e^{\frac{2\varphi}{\sqrt{3}}}\big{(}dz+\tilde{A}_{4}\big{)}^{2}\,,$
(4.40)

and the tensor hierarchy fields in (4.42) allow us to write the five-form as

$\displaystyle\hat{F}_{{{\scriptscriptstyle(5)}}}$
$\displaystyle=(1+*_{10})\Big{\\{}\Big{[}-4g\,e^{-\frac{\varphi}{\sqrt{3}}}\,\textrm{vol}_{4}-\tfrac{1}{2\sqrt{3}g^{2}}e^{\frac{\varphi}{\sqrt{3}}}\,*H_{{\scriptscriptstyle(2)}}\wedge\sum_{\rm\bar{a}=1}^{3}\big{(}d(\mu_{\rm\bar{a}}^{2})\wedge
D\phi^{\rm\bar{a}}\big{)}\Big{]}\wedge\big{(}dz+\tilde{A}_{4}\big{)}\Big{\\}}$
(4.41)
$\displaystyle=\Big{(}-4g\,e^{-\frac{\varphi}{\sqrt{3}}}\,\textrm{vol}_{4}-\tfrac{1}{2\sqrt{3}g^{2}}e^{\frac{\varphi}{\sqrt{3}}}\,*H_{{\scriptscriptstyle(2)}}\wedge\sum_{\rm\bar{a}=1}^{3}\big{(}d(\mu_{\rm\bar{a}}^{2})\wedge
D\phi^{\rm\bar{a}}\big{)}\Big{)}\wedge\big{(}dz+\tilde{A}_{4}\big{)}$
$\displaystyle\qquad-4g^{-4}\,\mu_{1}\mu_{2}d\mu_{1}\wedge d\mu_{2}\wedge
D\phi^{1}\wedge D\phi^{2}\wedge D\phi^{3}$
$\displaystyle\qquad\quad+\tfrac{1}{\sqrt{3}g^{3}}\,H_{{\scriptscriptstyle(2)}}\wedge\Big{[}\mu_{1}\mu_{2}(\mu_{1}d\mu_{2}-\mu_{2}d\mu_{1})\wedge
D\phi^{1}\wedge D\phi^{2}$
$\displaystyle\qquad\qquad+\mu_{2}\mu_{3}(\mu_{2}d\mu_{3}-\mu_{3}d\mu_{2})\wedge
D\phi^{2}\wedge
D\phi^{3}+\mu_{1}\mu_{3}(\mu_{1}d\mu_{3}-\mu_{3}d\mu_{1})\wedge
D\phi^{1}\wedge D\phi^{3}\Big{]}\,,$

using the duality relations (2.1)–(2.36), that reduce to

$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}\,1,2,3}$
$\displaystyle=-\tfrac{1}{\sqrt{3}}\,e^{\frac{\varphi}{\sqrt{3}}}\,*H_{{\scriptscriptstyle(2)}}\,,\qquad\quad\;\tilde{H}_{{{\scriptscriptstyle(2)}}\,4}=-e^{-\sqrt{3}\varphi}\,*H_{{{\scriptscriptstyle(2)}}}^{4}\,,$
(4.42) $\displaystyle H_{{{\scriptscriptstyle(3)}}\,1,2,3}$
$\displaystyle=-\tfrac{1}{2\sqrt{3}}*d\varphi\,,\qquad\qquad\quad\,\;H_{{{\scriptscriptstyle(3)}}\,4}=\tfrac{\sqrt{3}}{2}*d\varphi\,,$
$\displaystyle H_{{{\scriptscriptstyle(4)}}\,1,2,3}$
$\displaystyle=4g\,e^{-\frac{\varphi}{\sqrt{3}}}\,\textrm{vol}_{4}\,,\qquad\quad\enspace\quad\,H_{{{\scriptscriptstyle(4)}}\,4}=6g\,e^{\frac{\varphi}{\sqrt{3}}}\,\textrm{vol}_{4}\,,$

upon using the truncation (4.38) with $\chi=0$. From (3.20), the Bianchi
identity for the five-form and Einstein equations are the only equations to be
checked in ten dimensions. Since the axiodilation and two-forms in (4.34) and
(4.35) are zero, the equation of motion for the type IIB five-form amounts to
demanding that $F_{{{\scriptscriptstyle(5)}}}$ be closed. In (4.41), it is
straightforward to see that this is in turn an immediate consequence of the
four-dimensional Bianchi identities and equations of motion from (4.39).

For vanishing axiodilaton and two-forms, Einstein equations in 10$d$ in turn
reduce to

$\hat{G}=\hat{T}=-\frac{1}{480}\Big{[}\hat{F}_{\hat{\mu}\hat{\rho}_{1}\hat{\rho}_{2}\hat{\rho}_{3}\hat{\rho}_{4}}\hat{F}_{\hat{\nu}}{}^{\hat{\rho}_{1}\hat{\rho}_{2}\hat{\rho}_{3}\hat{\rho}_{4}}-\frac{1}{10}\hat{g}_{\hat{\mu}\hat{\nu}}\hat{F}_{\hat{\rho}_{1}\hat{\rho}_{2}\hat{\rho}_{3}\hat{\rho}_{4}\hat{\rho}_{5}}\hat{F}^{\hat{\rho}_{1}\hat{\rho}_{2}\hat{\rho}_{3}\hat{\rho}_{4}\hat{\rho}_{5}}\Big{]}v^{\hat{\mu}}v^{\hat{\nu}}\,,$
(4.43)

with $\hat{G}=\hat{G}_{\hat{\mu}\hat{\nu}}v^{\hat{\mu}}v^{\hat{\nu}}$ the
Einstein tensor for $\hat{g}_{10}$. We find it convenient to expand our
tensors in the one-form basis

$v^{\hat{\mu}}=\big{\\{}dx^{\mu},\ v^{m}\big{\\}}=\Big{\\{}dx^{\mu},\
d\alpha,\ d\beta,\ D\phi^{1},\ D\phi^{2},\ D\phi^{3},\ Dz\Big{\\}}\,,$ (4.44)

where $dx^{\mu}$ is a coordinate basis for the four-dimensional spacetime and
the angles on the sphere are given by

$\mu_{1}=\sin\alpha\cos\beta\,,\qquad\quad\mu_{2}=\sin\alpha\sin\beta\,,\qquad\quad\mu_{3}=\cos\alpha\,.$
(4.45)

In this basis, the Einstein tensor reads

$\displaystyle\hat{G}_{\mu\nu}$
$\displaystyle=G_{\mu\nu}-\tfrac{1}{6}e^{\frac{\varphi}{\sqrt{3}}}\Big{(}H_{\mu\rho}H_{\nu}{}^{\rho}-\tfrac{1}{4}g_{\mu\nu}H_{\rho\sigma}H^{\rho\sigma}\Big{)}-\tfrac{1}{2}e^{\sqrt{3}\varphi}\Big{(}\tilde{H}_{4\mu\rho}\tilde{H}_{4\nu}{}^{\rho}-\tfrac{1}{4}g_{\mu\nu}\tilde{H}_{4\rho\sigma}\tilde{H}_{4}^{\rho\sigma}\Big{)}$
$\displaystyle\qquad-\tfrac{1}{2}\big{(}\partial_{\mu}\varphi\partial_{\nu}\varphi-\tfrac{1}{2}g_{\mu\nu}\partial_{\rho}\varphi\partial^{\rho}\varphi\big{)}-10g^{2}e^{-\frac{\varphi}{\sqrt{3}}}g_{\mu\nu}\,,$
(4.46a) $\displaystyle\hat{G}_{\mu m}v^{m}$
$\displaystyle=\tfrac{1}{\sqrt{3}g}\nabla^{\nu}\big{(}e^{\frac{\varphi}{\sqrt{3}}}H_{\nu\mu}\big{)}\big{(}\mu_{1}^{2}D\phi_{1}+\mu_{2}^{2}D\phi_{2}+\mu_{3}^{2}D\phi_{3}\big{)}-\nabla^{\nu}\big{(}e^{\sqrt{3}\varphi}\tilde{H}_{4\nu\mu}\big{)}Dz\,,$
(4.46b) $\displaystyle\hat{G}_{mn}v^{m}v^{n}$
$\displaystyle=\tfrac{1}{2}e^{\frac{\varphi}{\sqrt{3}}}\Big{[}g^{\mu\nu}G_{\mu\nu}+\tfrac{1}{12}e^{\frac{\varphi}{\sqrt{3}}}H_{\mu\nu}H^{\mu\nu}+\tfrac{1}{4}e^{\sqrt{3}\varphi}\tilde{H}_{4\mu\nu}\tilde{H}_{4}^{\mu\nu}-\tfrac{1}{\sqrt{3}}\square\varphi$
$\displaystyle\qquad\quad+\tfrac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-12g^{2}e^{-\frac{\varphi}{\sqrt{3}}}\Big{]}(d\alpha^{2}+\sin^{2}\\!\alpha
d\beta^{2}+\mu_{1}^{2}D\phi_{1}^{2}+\mu_{2}^{2}D\phi_{2}^{2}+\mu_{3}^{2}D\phi_{3}^{2})$
$\displaystyle\quad+\tfrac{1}{2}e^{\sqrt{3}\varphi}\Big{[}g^{\mu\nu}G_{\mu\nu}+\tfrac{1}{12}e^{\frac{\varphi}{\sqrt{3}}}H_{\mu\nu}H^{\mu\nu}+\tfrac{3}{4}e^{\sqrt{3}\varphi}\tilde{H}_{4\mu\nu}\tilde{H}_{4}^{\mu\nu}-\sqrt{3}\,\square\varphi$
$\displaystyle\qquad\quad+\tfrac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-20g^{2}e^{-\frac{\varphi}{\sqrt{3}}}\Big{]}Dz^{2}$
$\displaystyle\quad+\tfrac{1}{12g^{2}}e^{\frac{2\varphi}{\sqrt{3}}}H_{\mu\nu}H^{\mu\nu}\big{(}\mu_{1}^{2}D\phi_{1}+\mu_{2}^{2}D\phi_{2}+\mu_{3}^{2}D\phi_{3}\big{)}^{2}$
$\displaystyle\qquad+\tfrac{1}{2\sqrt{3}}e^{\frac{4\varphi}{\sqrt{3}}}H_{\mu\nu}\tilde{H}_{4}^{\mu\nu}\big{(}\mu_{1}^{2}D\phi_{1}+\mu_{2}^{2}D\phi_{2}+\mu_{3}^{2}D\phi_{3}\big{)}Dz\,,$
(4.46c)

and the stress-energy tensor has components

$\displaystyle\hat{T}_{\mu\nu}$
$\displaystyle=-\tfrac{1}{3}e^{-\frac{\varphi}{\sqrt{3}}}\Big{[}12g^{2}g_{\mu\nu}-e^{\frac{2\varphi}{\sqrt{3}}}\Big{(}H_{\mu\rho}H_{\nu}{}^{\rho}-\tfrac{1}{4}g_{\mu\nu}H_{\rho\sigma}H^{\rho\sigma}\Big{)}\Big{]}\,,$
(4.47a) $\displaystyle\hat{T}_{\mu m}v^{m}$ $\displaystyle=0\,,$ (4.47b)
$\displaystyle\hat{T}_{mn}v^{m}v^{n}$
$\displaystyle=4g^{2}(d\alpha^{2}+\sin^{2}\\!\alpha
d\beta^{2}+\mu_{1}^{2}D\phi_{1}^{2}+\mu_{2}^{2}D\phi_{2}^{2}+\mu_{3}^{2}D\phi_{3}^{2})$
$\displaystyle\quad-\tfrac{1}{12}e^{\frac{2\varphi}{\sqrt{3}}}\Big{(}48g^{2}+e^{\frac{2\varphi}{\sqrt{3}}}H_{\mu\nu}H^{\mu\nu}\Big{)}Dz^{2}$
$\displaystyle\quad+\tfrac{1}{12g^{2}}e^{\frac{2\varphi}{\sqrt{3}}}H_{\mu\nu}H^{\mu\nu}\big{(}\mu_{1}^{2}D\phi_{1}+\mu_{2}^{2}D\phi_{2}+\mu_{3}^{2}D\phi_{3}\big{)}^{2}$
$\displaystyle\qquad+\tfrac{1}{6g}e^{\sqrt{3}\varphi}H_{\mu\nu}(*H)^{\mu\nu}\big{(}\mu_{1}^{2}D\phi_{1}+\mu_{2}^{2}D\phi_{2}+\mu_{3}^{2}D\phi_{3}\big{)}Dz\,.$
(4.47c)

It is immediate to verify that the external components of (4.43) are satisfied
on the four-dimensional Einstein equations, the mixed components amount to the
Maxwell equation for $H$ and Bianchi identity for $H^{4}$, and the internal
components of (4.43) are a combination of the trace of the Einstein equations
and the equations of motion for the scalars.

##  5 Black Hole and Domain Wall Solutions

In the previous section, we have described how to embed any solution of the
four-dimensional gauged STU models presented in section 2 into M-theory or
type IIB supergravity. We now switch gears to present new black-hole solutions
in these theories, first by considering singular limits of previously known
solutions and later by directly solving the equations of motion for a suitable
Ansatz.

###  5.1 AdS-BH limits

Setting the axions to zero, the potential (2.12a) for the STU truncation of
the ${\rm SO(8)}$ gauging reads

$V=-8g^{2}(\cosh\varphi_{1}+\cosh\varphi_{2}+\cosh\varphi_{3})\,.$ (5.1)

This theory admits four-charge AdS-black hole solutions [19],

$\displaystyle ds_{4}^{2}$
$\displaystyle=-(H_{1}H_{2}H_{3}H_{4})^{-1/2}f\,dt^{2}+(H_{1}H_{2}H_{3}H_{4})^{1/2}\big{(}f^{-1}dr^{2}+r^{2}d\Omega_{2,\kappa}^{2}\big{)}\,,$
(5.2) $\displaystyle A^{i}$
$\displaystyle=H_{i}^{-1}\frac{\sqrt{\sigma_{\kappa}+\kappa\,\sinh^{2}(\sqrt{\sigma_{\kappa}}\beta_{i})}}{\sinh(\sqrt{\sigma_{\kappa}}\beta_{i})}dt\,,$
$\displaystyle e^{2\varphi_{1}}$
$\displaystyle=\frac{H_{1}H_{2}}{H_{3}H_{4}}\,,\qquad
e^{2\varphi_{2}}=\frac{H_{1}H_{3}}{H_{2}H_{4}}\,,\qquad
e^{2\varphi_{3}}=\frac{H_{1}H_{4}}{H_{2}H_{3}}\,,$

with

$f=\kappa-\frac{\mu}{r}+4g^{2}r^{2}\,H_{1}H_{2}H_{3}H_{4}\qquad\quad\text{and}\qquad\quad
H_{i}=1+\frac{\mu}{r}\Big{(}\frac{1}{\sqrt{\sigma_{\kappa}}}\sinh(\sqrt{\sigma_{\kappa}}\beta_{i})\Big{)}^{2}\,.$
(5.3)

Here, $d\Omega^{2}_{2,\kappa}$ denotes the line element for the unit radius
metric on the sphere, torus and hyperboloid for $\kappa=1,0,-1$, respectively
given by

$d\Omega^{2}_{2,\kappa}=\begin{cases}d\theta^{2}+\sin^{2}\\!\theta\,d\phi^{2},&\kappa=+1\,,\\\
d\theta^{2}+d\phi^{2},&\kappa=0\,,\\\
d\theta^{2}+\sinh^{2}\\!\theta\,d\phi^{2},&\kappa=-1\,,\end{cases}$ (5.4)

and we also define

$\sigma_{\kappa}=\lim_{\epsilon\to
0^{+}}\text{sign}(\kappa+\epsilon)=\begin{cases}+1,&\kappa=+1,0\,,\\\
-1,&\kappa=-1\,.\end{cases}$ (5.5)

These black holes have charges555Our $\beta_{i}$ parameters are always taken
to be real. These configurations agree with those in [19] if one identifies
$\beta_{i}^{\text{there}}=i\,\beta_{i}^{\text{here}}$ in the hyperbolic case.

$q_{i}=\frac{\pi\mu}{\sqrt{\sigma_{\kappa}}}\sinh(\sqrt{\sigma_{\kappa}}\beta_{i})\,\sqrt{\sigma_{\kappa}+\kappa\,\sinh^{2}(\sqrt{\sigma_{\kappa}}\beta_{i})}=\begin{cases}\tfrac{1}{2}\pi\mu\sinh
2\beta_{i}\,,&\kappa=+1\,,\\\ \pi\mu\sinh\beta_{i}\,,&\kappa=0\,,\\\
\tfrac{1}{2}\pi\mu\sin 2\beta_{i}\,,&\kappa=-1\,.\end{cases}$ (5.6)

Domain-wall limits of these solutions can be constructed by considering
rescalings of the metric and vectors and shifts of the dilatons so that the
equations of motion remain invariant up to a change in the scalar potential
resulting from the loss of some of the terms in (5.1) after taking a singular
limit [38]. Taking

$\varphi_{i}\mapsto\varphi_{i}+\lambda\,,\qquad\quad\text{with}\qquad\lambda=\log(\tilde{g}/g)\,,$
(5.7)

the potential (5.1) becomes

$V=-4\tilde{g}^{2}(e^{\varphi_{1}}+e^{\varphi_{2}}+e^{\varphi_{3}})-4g^{2}(e^{-\varphi_{1}}+e^{-\varphi_{2}}+e^{-\varphi_{3}})\,,$
(5.8)

and the equations of motion are solved by (5.2) with now

$\displaystyle
H_{1}=\Big{(}\frac{g}{\tilde{g}}\Big{)}^{2}\Big{[}1+\frac{\mu}{r}\Big{(}\frac{1}{\sqrt{\sigma_{\kappa}}}\sinh(\sqrt{\sigma_{\kappa}}\beta_{1})\Big{)}^{2}\Big{]}\,,$
(5.9) $\displaystyle H_{i\neq
1}=1+\frac{\mu}{r}\Big{(}\frac{1}{\sqrt{\sigma_{\kappa}}}\sinh(\sqrt{\sigma_{\kappa}}\beta_{i\neq
1})\Big{)}^{2}\,,$

and

$f=\kappa-\frac{\mu}{r}+4\tilde{g}^{2}r^{2}\,H_{1}H_{2}H_{3}H_{4}\,.$ (5.10)

In this case, it is possible to take the $g\to 0$ limit of the potential (5.8)
while keeping non-trivial solutions provided that
$g\sinh(\sqrt{\sigma_{\kappa}}\beta_{i})$ can be kept constant by sending
$\beta_{i}$ to infinity. Note that this cannot be achieved for the hyperbolic
horizon ($\kappa=-1$) case above, and therefore only domain walls with
spherical and toroidal horizons can be constructed with this method.

We note that the potential (5.8) after the $g\to 0$ limit corresponds to the
STU sector of a maximal supergravity with
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ gauging which is related by duality to
the one discussed above. In particular, instead of by the embedding tensor
$\Theta_{{\scriptscriptstyle(6)}}$ in (2.11), it is described by

$\displaystyle\theta_{AB}=0\,,\qquad\xi^{AB}=0\,,\qquad\zeta^{A}{}_{BCD}=0\,,$
(5.11) $\displaystyle\tilde{\zeta}_{4}{}^{123}=$
$\displaystyle-\tilde{\zeta}_{3}{}^{124}=\tilde{\zeta}_{2}{}^{134}=-\tilde{\zeta}_{1}{}^{234}=\tilde{\zeta}_{5}{}^{678}=-\tilde{\zeta}_{6}{}^{578}=\tilde{g}\,,$

with only non-trivial components in the $\bm{420}$ representation. Using the
same scalings as the ones studied in section 4.1, this gauged supergravity can
be proved to uplift consistently into an $S^{5}\times T^{2}$ configuration of
$D=11$ supergravity.

The potential (2.12b) can be similarly obtained by taking

$\varphi_{1}\mapsto\varphi_{1}-\lambda\,,\qquad\varphi_{2,3}\mapsto\varphi_{2,3}+\lambda\,,\qquad\text{with}\qquad\lambda=\log(\tilde{g}/g)\,,$
(5.12)

so that (5.1) becomes

$V=-4\tilde{g}^{2}(e^{-\varphi_{1}}+e^{\varphi_{2}}+e^{\varphi_{3}})-4g^{2}(e^{\varphi_{1}}+e^{-\varphi_{2}}+e^{-\varphi_{3}})\,.$
(5.13)

However, in this case the rescaled solution is (5.2) with

$\displaystyle
H_{2}=\Big{(}\frac{g}{\tilde{g}}\Big{)}^{-1}\Big{[}1+\frac{\mu}{r}\Big{(}\frac{1}{\sqrt{\sigma_{\kappa}}}\sinh(\sqrt{\sigma_{\kappa}}\beta_{2})\Big{)}^{2}\Big{]}\qquad\text{and}$
(5.14) $\displaystyle H_{i\neq
2}=\Big{(}\frac{g}{\tilde{g}}\Big{)}\Big{[}1+\frac{\mu}{r}\Big{(}\frac{1}{\sqrt{\sigma_{\kappa}}}\sinh(\sqrt{\sigma_{\kappa}}\beta_{i\neq
2})\Big{)}^{2}\Big{]}\,,$

so that the $g\to 0$ limit is ill-defined.

###  5.2 DW-Ansätze

The fact that domain-wall solutions for the
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ and
$[\textrm{SO}(6)\times\textrm{SO}(2)]\ltimes\mathbb{R}^{12}$ theories cannot
be constructed through singular limits of the AdS-black holes in [19] does not
mean that such solutions do not exist. To find them, we consider the simpler
truncation in (4.38) and the Ansatz

$\displaystyle
ds_{4}^{2}=-e^{2u}dt^{2}+e^{-2u}dr^{2}+L^{2}e^{2A}d\Omega^{2}_{2,\kappa}\,,\qquad
A_{{\scriptscriptstyle(1)}}=v\,dt\,,\qquad
A_{{{\scriptscriptstyle(1)}}}^{4}=w\,dt\,,$ (5.15)

with $\varphi,u,v,w$ and $A$ being functions on $r$ only, and $L$ a constant
radius. The equations of motion stemming from (4.39) reduce to

$\displaystyle\varphi^{\prime\prime}+2(u^{\prime}+A^{\prime})\varphi^{\prime}+\tfrac{1}{2}a\,e^{a\varphi-2u}(v^{\prime})^{2}-\tfrac{1}{2a}\,e^{-\varphi/a-2u}(w^{\prime})^{2}-12a\,g^{2}e^{-a\varphi-2u}=0\,,$
(5.16) $\displaystyle(e^{2A+a\varphi}v^{\prime})^{\prime}=0\,,$
$\displaystyle(e^{2A-\varphi/a}w^{\prime})^{\prime}=0\,,$ $\displaystyle
A^{\prime\prime}+A^{\prime}+\tfrac{1}{4}(\varphi^{\prime})^{2}=0\,,$
$\displaystyle
A^{\prime\prime}+2A^{\prime}(u^{\prime}+A^{\prime})+\tfrac{1}{4}e^{a\varphi-2u}(v^{\prime})^{2}+\tfrac{1}{4}e^{-\varphi/a-2u}(w^{\prime})^{2}-12g^{2}e^{-a\varphi-2u}-\tfrac{\kappa}{L^{2}}\,e^{-2A-2u}=0\,,$
$\displaystyle
u^{\prime\prime}+2u^{\prime}(u^{\prime}+A^{\prime})-\tfrac{1}{4}e^{a\varphi-2u}(v^{\prime})^{2}-\tfrac{1}{4}e^{-\varphi/a-2u}(w^{\prime})^{2}-12g^{2}e^{-a\varphi-2u}=0\,,$

with primes denoting derivatives with respect to $r$. For any $a$ in the range
$-\sqrt{3}<a<\sqrt{3}$, this system of equations admits a solution in the
$\kappa=0$ case, which reads

$\displaystyle
ds_{4}^{2}=-f\,dt^{2}+f^{-1}dr^{2}+L^{2}\Big{(}\frac{r}{r_{0}}\Big{)}^{2/(1+a^{2})}d\Omega^{2}_{2,0}\,,$
(5.17) $\displaystyle
A_{{\scriptscriptstyle(1)}}=\frac{\lambda}{r}\,dt\,,\qquad\quad
A_{{\scriptscriptstyle(1)}}^{4}=0\,,$ $\displaystyle
e^{\varphi}=\Big{(}\frac{r}{r_{0}}\Big{)}^{2a/(1+a^{2})}\,,$

with

$f=\frac{1+a^{2}}{4}\left[\frac{24g^{2}r_{0}^{2}(1+a^{2})}{3-a^{2}}\Big{(}\frac{r}{r_{0}}\Big{)}^{2/(1+a^{2})}+\frac{\lambda^{2}}{r_{0}^{2}}\Big{(}\frac{r_{0}}{r}\Big{)}^{2/(1+a^{2})}\right]-\frac{\mu}{r_{0}}\Big{(}\frac{r_{0}}{r}\Big{)}^{(1-a^{2})/(1+a^{2})}\,,$
(5.18)

for $r_{0}$ a length parameter and $\lambda$ and $\mu$ constants encoding the
charge and mass of the solution. This class of solutions recovers (A.6) of
[38] for $a=1$, while for $a=1/\sqrt{3}$ it reduces to

$\displaystyle
ds_{4}^{2}=-\sqrt{\frac{r}{r_{0}}}\left[4g^{2}r_{0}r+\frac{\lambda^{2}}{3r^{2}}-\frac{\mu}{r}\right]\,dt^{2}+\sqrt{\frac{r_{0}}{r}}\left[4g^{2}r_{0}r+\frac{\lambda^{2}}{3r^{2}}-\frac{\mu}{r}\right]^{-1}dr^{2}+L^{2}\Big{(}\frac{r}{r_{0}}\Big{)}^{3/2}d\Omega^{2}_{2,0}\,,$
$\displaystyle A_{{\scriptscriptstyle(1)}}=\frac{\lambda}{r}\,dt\,,\qquad\quad
A_{{\scriptscriptstyle(1)}}^{4}=0\,,$ $\displaystyle
e^{\varphi}=\Big{(}\frac{r}{r_{0}}\Big{)}^{\sqrt{3}/2}\,,$ (5.19)

which describes a charged black hole with domain-wall asymptotics. At $r=0$,
this solution has a curvature singularity for all possible values of the
charge and mass. When

$\mu^{3}>3g^{2}\lambda^{4}r_{0}\,,$ (5.20)

the singularity is covered by two horizons sitting at

$r_{\pm}=\sqrt{\frac{\mu}{3g^{2}r_{0}}}\cos\left[\frac{1}{3}\arccos\left(-g\lambda^{2}\sqrt{\frac{3r_{0}}{\mu^{3}}}\right)-\frac{\pi}{3}\pm\frac{\pi}{3}\right]$
(5.21)

that coalesce when the bound (5.20) is saturated. When
$\mu^{3}<3g^{2}\lambda^{4}r_{0}$, the solution displays a naked singularity.

For this choice of dilaton coupling, we can embed the solution in ten
dimensions using (4.40)–(4.41) with the tensor hierarchy fields in (4.42),
which become

$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}\,1,2,3}$
$\displaystyle=-\frac{\lambda L^{2}}{\sqrt{3}r_{0}^{2}}d\theta\wedge
d\phi\,,\qquad\quad\tilde{H}_{{{\scriptscriptstyle(2)}}\,4}=0\,,$ (5.22)
$\displaystyle H_{{{\scriptscriptstyle(3)}}\,1,2,3}$
$\displaystyle=-\frac{1}{3}H_{{{\scriptscriptstyle(3)}}\,4}=\frac{L^{2}}{4r}\Big{(}\frac{r}{r_{0}}\Big{)}^{2}\left[4g^{2}r_{0}r+\frac{\lambda^{2}}{3r^{2}}-\frac{\mu}{r}\right]dt\wedge
d\theta\wedge d\phi\,,$ $\displaystyle H_{{{\scriptscriptstyle(4)}}\,1,2,3}$
$\displaystyle=4gL^{2}\,\Big{(}\frac{r}{r_{0}}\Big{)}\,dt\wedge dr\wedge
d\theta\wedge
d\phi\,,\qquad\,H_{{{\scriptscriptstyle(4)}}\,4}=6gL^{2}\,\Big{(}\frac{r}{r_{0}}\Big{)}^{2}\,dt\wedge
dr\wedge d\theta\wedge d\phi\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}\,1,2,3}$
$\displaystyle=2mL^{2}\,\Big{(}\frac{r}{r_{0}}\Big{)}\,dt\wedge dr\wedge
d\theta\wedge d\phi\,,\quad\enspace\
\tilde{H}_{{{\scriptscriptstyle(4)}}\,4}=0\,.$

Other solutions to the equations (5.16) can be found for specific values of
the dilaton coupling $a$. If we commit to the choice $a=1/\sqrt{3}$
corresponding to the STU model, a class of solutions with spherical horizon is
given by

$\displaystyle
ds_{4}^{2}=-f\,dt^{2}+f^{-1}dr^{2}+\lambda^{2}\Big{(}\frac{r}{r_{0}}\Big{)}^{3/2}d\Omega^{2}_{2,1}\,,\qquad\text{with}\quad
f=\sqrt{\frac{r}{r_{0}}}\left[4g^{2}r_{0}r+\frac{r_{0}^{2}}{\lambda^{2}}-\frac{\mu}{r}\right]\,,$
(5.23) $\displaystyle A_{{\scriptscriptstyle(1)}}=0\,,\qquad\qquad
A^{4}_{{\scriptscriptstyle(1)}}=\frac{r}{\lambda}\,dt\,,\qquad\qquad
e^{\varphi}=\Big{(}\frac{r}{r_{0}}\Big{)}^{\sqrt{3}/2}\,.$

Note that this solution only exists for non-vanishing electric charge, and for
$\mu>0$ it has a single horizon at radius

$r_{H}=\frac{r_{0}}{8g^{2}\lambda^{2}}\left[\sqrt{1+\frac{16g^{2}\lambda^{4}\mu}{r_{0}^{3}}}-1\right]\,.$
(5.24)

For this solution, the type IIB uplift is given by (4.40)–(4.41) with tensor
hierarchy fields

$\displaystyle\tilde{H}_{{{\scriptscriptstyle(2)}}\,1,2,3}$
$\displaystyle=0\,,\qquad\quad\tilde{H}_{{{\scriptscriptstyle(2)}}\,4}=\lambda\sin\theta\,d\theta\wedge
d\phi\,,$ (5.25) $\displaystyle H_{{{\scriptscriptstyle(3)}}\,1,2,3}$
$\displaystyle=-\frac{1}{3}H_{{{\scriptscriptstyle(3)}}\,4}=\frac{\lambda^{2}}{4r}\Big{(}\frac{r}{r_{0}}\Big{)}^{2}\left[4g^{2}r_{0}r+\frac{r_{0}^{2}}{\lambda^{2}}-\frac{\mu}{r}\right]\sin\theta\,dt\wedge
d\theta\wedge d\phi\,,$ $\displaystyle H_{{{\scriptscriptstyle(4)}}\,1,2,3}$
$\displaystyle=4g\lambda^{2}\,\Big{(}\frac{r}{r_{0}}\Big{)}\,\sin\theta\,dt\wedge
dr\wedge d\theta\wedge
d\phi\,,\qquad\,H_{{{\scriptscriptstyle(4)}}\,4}=6g\lambda^{2}\,\Big{(}\frac{r}{r_{0}}\Big{)}^{2}\,\sin\theta\,dt\wedge
dr\wedge d\theta\wedge d\phi\,,$
$\displaystyle\tilde{H}_{{{\scriptscriptstyle(4)}}\,1,2,3}$
$\displaystyle=2m\lambda^{2}\,\Big{(}\frac{r}{r_{0}}\Big{)}\,\sin\theta\,dt\wedge
dr\wedge d\theta\wedge d\phi\,,\quad\enspace\
\tilde{H}_{{{\scriptscriptstyle(4)}}\,4}=0\,.$

##  6 Discussion

In this paper, we have shown that singular limits of gauged supergravities can
offer insights not only into how to construct new solutions in the resulting
gauged supergravity, but also on its consistent uplift to higher dimensions if
the resulting gauged supergravity can itself be obtained as a consistent
truncation. We have exemplified this idea relating the STU sector of the
electrically gauged SO(8) supergravity to the STU sector of the CSO theory
with the $\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ gauging, and we have used the
known consistent uplift of the former theory into M-theory on $S^{7}$ to
construct an embedding of the latter into type IIB supergravity on
$S^{5}\times S^{1}$. It would be interesting to investigate if singular limits
such as the ones we studied can be used to relate gauged supergravities with
larger field contents.

To describe the uplifts of these gauged supergravities, we have employed
techniques that exploit the formal duality-covariance of the embedding tensor
formulation of the gauging. In particular, apart from making use of the
generalised Scherk-Schwarz factorisation of the higher-dimensional fields
expressed in the language of Exceptional Field Theory, we have recast all the
contributions of $p$-form fields in $D=4$ in terms of the four-dimensional
tensor hierarchy (suitably restricted to the STU sector). This has provided a
simple way of circumventing some complicated dualisations in ExFT involving
the internal metric. To illustrate the power of this technique, we have
explicitly checked that the higher-dimensional equations of motion follow from
the four-dimensional ones, and constructed new families of charged black hole
solutions in $4d$. These black holes involve non-trivial scalar profiles, and
their asymptotics do not approach an AdS solution, but a domain wall.

A close cousin of the dyonic CSO theory we have considered comes equipped with
an $[\textrm{SO}(6)\times\textrm{SO}(1,1)]\ltimes\mathbb{R}^{12}$ gauge group.
This theory has been shown to uplift into type IIB supergravity on
$S^{5}\times S^{1}$ with a non-geometric patching of the circle [42]. This
theory possesses a very rich structure of AdS vacua [55], both supersymmetric
and non-supersymmetric, including continuous families realising a holographic
conformal manifold [56, 57, 58, 59]. It will be interesting to extend our
results to describe more explicitly the uplift of this gauging, and to
construct in this way ten-dimensional solutions arising from non-trivial
profiles of the fields in the consistent truncation. These new solutions may
play an important rôle in understanding the holography of the ${\rm T}[{\rm
U}(N)]$ theories and the $J$-folds of $\mathcal{N}=4$ SYM that are conjectured
to be dual to this solution [60].

## Acknowledgements

We would like to thank Camille Eloy, Henning Samtleben and Oscar Varela for
interesting discussions and comments. HZ would like to thank the Harvard
Physics Department and CCPP at New York University for kind hospitality during
the later stages of this project. GL is supported by endowment funds from the
Mitchell Family Foundation. CNP is supported in part by DOE grant DE-
SC0010813.

##  Appendix A STU Truncation

In its $\textrm{SL}(8,\mathbb{R})$ basis, the generators of E7(7) can be split
as $t_{\alpha}=\\{t_{A}{}^{B},\ t_{[ABCD]}\\}$, with the index $A=1,\dots,8$
labelling the fundamental of $\textrm{SL}(8,\mathbb{R})$ and $t_{A}{}^{A}=0$.
The gaugings considered are described by an embedding tensor
$\Theta_{M}{}^{\alpha}$ with non-trivial components

$\Theta_{AB}{}^{C}{}_{D}=2\delta_{[A}{}^{C}\theta_{B]D}\,,\qquad\quad\Theta^{ABC}{}_{D}=2\delta_{D}{}^{[A}\xi^{B]C}\,,$
(A.1)

where $\theta_{AB}$ and $\xi^{AB}$ respectively belong to the
$\bm{36^{\prime}}$ and $\bm{36}$ of $\textrm{SL}(8,\mathbb{R})$. This
embedding tensor determines the covariant derivatives as

$\displaystyle D$ $\displaystyle=d+A^{M}\Theta_{M}{}^{\alpha}t_{\alpha}$ (A.2)
$\displaystyle=d+A^{AB}\,\theta_{BC}\,t_{A}{}^{C}+\tilde{A}_{AB}\,\xi^{BC}\,t_{C}{}^{A}\,.$

and appears in the potential and mass matrices through the generators
$X_{MN}{}^{P}=\Theta_{M}{}^{\alpha}(t_{\alpha})_{N}{}^{P}$.

For these gaugings, the gauge group $G$ is embedded in ${\rm E}_{7(7)}$ as

$G\subset\textrm{SL}(8,\mathbb{R})\subset{\rm E}_{7(7)}\,.$ (A.3)

The $\mathcal{H}={\rm SO}(2)^{4}$ Cartan subalgebra of the gaugings discussed
in the main text can be taken to be666For the
$\textrm{SO}(6)\ltimes\mathbb{R}^{12}$ gauging, $T_{78}$ strictly speaking
corresponds to a global symmetry commuting with the gauge group, following the
$m\rightarrow 0$ limit.

$T_{12}=t_{1}{}^{2}-t_{2}{}^{1}\,,\qquad
T_{34}=t_{3}{}^{4}-t_{4}{}^{3}\,,\qquad
T_{56}=t_{5}{}^{6}-t_{6}{}^{5}\,,\qquad T_{78}=t_{7}{}^{8}-t_{8}{}^{7}\,,$
(A.4)

and its commutant within ${\rm E}_{7(7)}$ is

${\rm Comm}_{\mathcal{H}}\,{\rm
E}_{7(7)}=\textrm{SL}(2,\mathbb{R})_{1}\times\textrm{SL}(2,\mathbb{R})_{2}\times\textrm{SL}(2,\mathbb{R})_{3}\,.$
(A.5)

Each factor is generated by $\\{h_{i},e_{i},f_{i}\\}$, which are given in
terms of ${\rm E}_{7(7)}$ generators as

$\displaystyle h_{1}$
$\displaystyle=\tfrac{1}{4}\big{(}t_{1}{}^{1}+t_{2}{}^{2}+t_{3}{}^{3}+t_{4}{}^{4}-t_{5}{}^{5}-t_{6}{}^{6}-t_{7}{}^{7}-t_{8}{}^{8}\big{)}\,,$
(A.6) $\displaystyle\qquad e_{1}=12\,t_{1234}\,,\qquad\qquad
f_{1}=12\,t_{5678}\,,$ $\displaystyle h_{2}$
$\displaystyle=\tfrac{1}{4}\big{(}t_{1}{}^{1}+t_{2}{}^{2}-t_{3}{}^{3}-t_{4}{}^{4}-t_{5}{}^{5}-t_{6}{}^{6}+t_{7}{}^{7}+t_{8}{}^{8}\big{)}\,,$
$\displaystyle\qquad e_{2}=12\,t_{1278}\,,\qquad\qquad f_{2}=12\,t_{3456}\,,$
$\displaystyle h_{3}$
$\displaystyle=\tfrac{1}{4}\big{(}-t_{1}{}^{1}-t_{2}{}^{2}+t_{3}{}^{3}+t_{4}{}^{4}-t_{5}{}^{5}-t_{6}{}^{6}+t_{7}{}^{7}+t_{8}{}^{8}\big{)}\,,$
$\displaystyle\qquad e_{3}=12\,t_{3478}\,,\qquad\qquad f_{3}=12\,t_{1256}\,,$

with non-vanishing brackets

$[h_{i},\,e_{j}]=e_{i}\,\delta_{ij}\,,\qquad[h_{i},\,f_{j}]=-f_{i}\,\delta_{ij}\,,\qquad[e_{i},\,f_{j}]=2h_{i}\,\delta_{ij}\,.$
(A.7)

The representative of the scalar manifold (2.1) is taken to be

$\mathcal{V}=e^{\chi_{1}e_{1}+\chi_{2}e_{2}+\chi_{3}e_{3}}\,e^{-\varphi_{1}h_{1}-\varphi_{2}h_{2}-\varphi_{3}h_{3}}\,,$
(A.8)

which leads to (2.2) through the symmetric matrix
$M_{MN}=(\mathcal{V}\mathcal{V}^{T})_{MN}$ and its inverse as

$\mathcal{L}_{\rm NLSM}=-\tfrac{1}{48}dM_{MN}\wedge*dM^{MN}\,.$ (A.9)

The symmetry group (A.5) is then realised through the Killing vectors

$k[h_{i}]=2\partial_{\varphi_{i}}-2\chi_{i}\partial_{\chi_{i}}\,,\qquad\quad
k[e_{i}]=\partial_{\chi_{i}}\,,\qquad\quad
k[f_{i}]=2\chi_{i}\partial_{\varphi_{i}}+(e^{-2\varphi_{i}}-\chi_{i}^{2})\partial_{\chi_{i}}\,,$
(A.10)

which close into (A.7) under the Lie bracket.

There are four electric vectors $A^{\rm a}$ and four magnetic counterparts
$\tilde{A}_{\rm a}$ in the $\mathcal{H}$-invariant truncation of the maximal
theory. In terms of the original $\bm{28}\oplus\bm{28}^{\prime}$ of the
$\mathcal{N}=8$ theory, they are given by

$A^{1}=A^{12}\,,$ | $A^{2}=A^{34}\,,$ | $A^{3}=A^{56}\,,$ | $A^{4}=A^{78}\,,$
---|---|---|---
$\tilde{A}_{1}=\tilde{A}_{12}\,,$ | $\tilde{A}_{2}=\tilde{A}_{34}\,,$ | $\tilde{A}_{3}=\tilde{A}_{56}\,,$ | $\tilde{A}_{4}=\tilde{A}_{78}\,.$
(A.11)

These identifications allow one to extract the non-minimal gauge couplings
from $M$ through the block decomposition

$M=\begin{pmatrix}-\mathcal{I}-\mathcal{R}\mathcal{I}^{-1}\mathcal{R}&\mathcal{R}\mathcal{I}^{-1}\\\
\mathcal{I}^{-1}\mathcal{R}&-\mathcal{I}^{-1}\end{pmatrix}\,,$ (A.12)

and the index restriction $A^{M}=(A^{[AB]},\tilde{A}_{[AB]})\to(A^{\rm
a},\tilde{A}_{\rm a})$ in (A.11).

Similarly, the preserved two-forms in (2.26) are related to the ${\rm
SL}(8,\mathbb{R})$ objects through

$\displaystyle B_{1}=B_{1}{}^{1}=B_{2}{}^{2}\,,\hskip
27.0ptB_{2}=B_{3}{}^{3}=B_{4}{}^{4}\,,\hskip
27.0ptB_{3}=B_{5}{}^{5}=B_{6}{}^{6}\,,\hskip
27.0ptB_{4}=B_{7}{}^{7}=B_{8}{}^{8}\,,$ $\displaystyle
B^{\prime}_{1}=B_{1}{}^{2}=-B_{2}{}^{1}\,,\hskip
18.0ptB^{\prime}_{2}=B_{3}{}^{4}=-B_{4}{}^{3}\,,\hskip
18.0ptB^{\prime}_{3}=B_{5}{}^{6}=-B_{6}{}^{5}\,,\hskip
18.0ptB^{\prime}_{4}=B_{7}{}^{8}=-B_{8}{}^{7}\,,$ $\displaystyle
B_{12}=B_{1234}\,,\hskip 27.0ptB_{13}=B_{1256}\,,\hskip
27.0ptB_{14}=B_{1278}\,,$ $\displaystyle B_{23}=B_{3456}\,,\hskip
27.0ptB_{24}=B_{3478}\,,\hskip 27.0ptB_{34}=B_{5678}\,,$ (A.13)

and the three-forms as

$C_{1}=C^{11}=C^{22}\,,$ | $C_{2}=C^{33}=C^{44}\,,$ | $C_{3}=C^{55}=C^{66}\,,$ | $C_{4}=C^{77}=C^{88}\,,$
---|---|---|---
$\tilde{C}_{1}=\tilde{C}_{11}=\tilde{C}_{22}\,,$ | $\tilde{C}_{2}=\tilde{C}_{33}=\tilde{C}_{44}\,,$ | $\tilde{C}_{3}=\tilde{C}_{55}=\tilde{C}_{66}\,,$ | $\tilde{C}_{4}=\tilde{C}_{77}=\tilde{C}_{88}\,,$
(A.14)

for those in the $\bm{36}\oplus\bm{36^{\prime}}$ of ${\rm SL}(8,\mathbb{R})$,
and and those in the $\bm{420}\oplus\bm{420^{\prime}}$ as

$\begin{gathered}C_{12}=C_{1}{}^{234}=-C_{2}{}^{134}\,,\hskip
27.0ptC_{13}=C_{1}{}^{256}=-C_{2}{}^{145}\,,\hskip
27.0ptC_{14}=C_{1}{}^{278}=-C_{2}{}^{178}\,,\\\\[5.0pt]
C_{21}=C_{3}{}^{412}=-C_{4}{}^{312}\,,\hskip
27.0ptC_{23}=C_{3}{}^{456}=-C_{4}{}^{356}\,,\hskip
27.0ptC_{24}=C_{3}{}^{478}=-C_{4}{}^{378}\,,\\\\[5.0pt]
\tilde{C}_{12}=\tilde{C}^{1}{}_{234}=-\tilde{C}^{2}{}_{134}\,,\hskip
27.0pt\tilde{C}_{13}=\tilde{C}^{1}{}_{256}=-\tilde{C}^{2}{}_{145}\,,\hskip
27.0pt\tilde{C}_{14}=\tilde{C}^{1}{}_{278}=-\tilde{C}^{2}{}_{178}\,,\\\\[5.0pt]
\tilde{C}_{21}=\tilde{C}^{3}{}_{412}=-\tilde{C}^{4}{}_{312}\,,\hskip
27.0pt\tilde{C}_{23}=\tilde{C}^{3}{}_{456}=-\tilde{C}^{4}{}_{356}\,,\hskip
27.0pt\tilde{C}_{24}=\tilde{C}^{3}{}_{478}=-\tilde{C}^{4}{}_{378}\,,\\\\[5.0pt]
etc\,.\\\\[-12.0pt] \end{gathered}$ (A.15)

##  Appendix B A $D=9$ Detour

###  B.1 Reduction from $D=11$ to $D=9$

The relation between the eleven-dimensional and type IIA fields in Einstein
frame reads

$\displaystyle d\hat{s}_{11}^{2}$
$\displaystyle=e^{-\frac{1}{6}\phi_{A}}ds_{\rm
IIA}^{2}+e^{\frac{4}{3}\phi_{A}}(dz_{2}+\mathcal{A}^{\prime})^{2}\,,$ (B.1)
$\displaystyle\hat{F}_{{{\scriptscriptstyle(4)}}}$
$\displaystyle=F_{{{\scriptscriptstyle(4)}}\rm
IIA}+H_{{{\scriptscriptstyle(3)}}\rm
IIA}\wedge(dz_{2}+\mathcal{A}^{\prime})\,.$

Further reducing type IIA on another circle, we get maximal supergravity in
$D=9$. The reduction Ansatz is

$\displaystyle ds_{\rm IIA}^{2}$
$\displaystyle=e^{-\frac{1}{2\sqrt{7}}\varphi_{A}}ds_{9}^{2}+e^{\frac{\sqrt{7}}{2}\varphi_{A}}\big{(}dz_{1}+\mathcal{A}^{\rm
KK}\big{)}^{2}\,,$ (B.2) $\displaystyle H_{{{\scriptscriptstyle(3)}}\rm IIA}$
$\displaystyle=H^{A}_{{\scriptscriptstyle(3)}}+H^{A}_{{\scriptscriptstyle(2)}}\wedge\big{(}dz_{1}+\mathcal{A}^{\rm
KK}\big{)}\,,$
$\displaystyle\mathcal{F}^{\prime}_{{{\scriptscriptstyle(2)}}\rm IIA}$
$\displaystyle=\mathcal{F}^{A}_{{{\scriptscriptstyle(2)}}}+\mathcal{F}^{A}_{{{\scriptscriptstyle(1)}}}\wedge\big{(}dz_{1}+\mathcal{A}^{\rm
KK}\big{)}\,,$ $\displaystyle F_{{{\scriptscriptstyle(4)}}\rm IIA}$
$\displaystyle=F^{A}_{{\scriptscriptstyle(4)}}+F^{A}_{{\scriptscriptstyle(3)}}\wedge\big{(}dz_{1}+\mathcal{A}^{\rm
KK}\big{)}\,,$

with $\mathcal{F}^{\prime}_{{{\scriptscriptstyle(2)}}\rm
IIA}=d\mathcal{A}^{\prime}$ and the $D=9$ field strengths given by

$\displaystyle\mathcal{F}_{{\scriptscriptstyle(1)}}^{A}=dA_{{\scriptscriptstyle(0)}}^{\rm
RR}\,,$
$\displaystyle\mathcal{F}_{{\scriptscriptstyle(2)}}^{A}=dA_{{\scriptscriptstyle(1)}}^{\rm
RR}-dA_{{\scriptscriptstyle(0)}}^{\rm
RR}\wedge\mathcal{A}_{{\scriptscriptstyle(1)}}^{\rm KK}\,,\qquad\quad
H_{{\scriptscriptstyle(2)}}^{A}=dA_{{\scriptscriptstyle(1)}}^{\rm NS}\,,$
$\displaystyle
H_{{\scriptscriptstyle(3)}}^{A}=dA_{{\scriptscriptstyle(2)}}^{\rm
NS}-dA_{{\scriptscriptstyle(1)}}^{\rm
NS}\wedge\mathcal{A}_{{\scriptscriptstyle(1)}}^{\rm KK}\,,\qquad\quad\
F_{{\scriptscriptstyle(3)}}^{A}=dA_{{\scriptscriptstyle(2)}}^{\rm
RR}+A_{{\scriptscriptstyle(1)}}^{\rm NS}\wedge
dA_{{\scriptscriptstyle(1)}}^{\rm RR}-A_{{\scriptscriptstyle(0)}}^{\rm
RR}\,dA_{{\scriptscriptstyle(2)}}^{\rm NS}\,,$ $\displaystyle
F_{{\scriptscriptstyle(4)}}^{A}=dA_{{\scriptscriptstyle(3)}}^{\rm
RR}+A_{{\scriptscriptstyle(2)}}^{\rm NS}\wedge
dA_{{\scriptscriptstyle(1)}}^{\rm RR}-(dA_{{\scriptscriptstyle(2)}}^{\rm
RR}-A_{{\scriptscriptstyle(0)}}^{\rm RR}\,dA_{{\scriptscriptstyle(2)}}^{\rm
NS}-A_{{\scriptscriptstyle(1)}}^{\rm NS}\wedge
dA_{{\scriptscriptstyle(1)}}^{\rm
RR})\wedge\mathcal{A}_{{\scriptscriptstyle(1)}}^{\rm KK}\,.$ (B.3)

Comparing with (4.1) and (4.1), we find

$\displaystyle e^{\phi_{\rm
IIA}}=\frac{\Xi_{2}^{1/4}}{\sqrt{H}}\,,\quad\enspace
e^{-\tfrac{2\sqrt{7}}{3}\varphi_{A}}=\frac{H}{\sqrt{\Xi_{2}}}\,,\quad\enspace
A_{{\scriptscriptstyle(1)}}^{\rm NS}=\mathcal{A}\,,$ $\displaystyle
A_{{\scriptscriptstyle(0)}}^{\rm RR}=\mathcal{A}^{\rm KK}=A^{\rm
RR}_{{{\scriptscriptstyle(1)}}}=A^{\rm
NS}_{{{\scriptscriptstyle(2)}}}=A_{{\scriptscriptstyle(2)}}^{\rm
RR}=0\,,\quad\enspace dA_{{\scriptscriptstyle(3)}}^{\rm
RR}=X_{{\scriptscriptstyle(4)}}\,,$ (B.4)

with $\mathcal{A}$ in (4.27), and

$\displaystyle ds_{9}^{2}$
$\displaystyle=H^{-6/7}\Xi_{2}^{-4/7}\Big{[}H\Xi_{2}\,ds_{4}^{2}-(\mu_{1}^{2}b_{2}D\phi_{1}+\mu_{2}^{2}b_{3}D\phi_{2}-\mu_{3}^{2}b_{1}D\phi_{3})^{2}$
(B.5)
$\displaystyle\qquad+H\big{(}Y_{2}^{2}(d\mu_{1}^{2}+\mu_{1}^{2}D\phi_{1}^{2})+Y_{3}^{2}(d\mu_{2}^{2}+\mu_{2}^{2}D\phi_{2}^{2})+\tilde{Y}_{1}^{2}(d\mu_{3}^{2}+\mu_{3}^{2}D\phi_{3}^{2})\big{)}\Big{]}\,.$

###  B.2 Type IIB reduction to $9$ dimensions

The same nine-dimensional theory can be obtained starting from type IIB
supergravity. In this case, the Kaluza-Klein Ansatz reads

$\displaystyle ds_{\rm IIB}^{2}$
$\displaystyle=e^{-\frac{1}{2\sqrt{7}}\varphi_{B}}ds_{9}^{2}+e^{\frac{\sqrt{7}}{2}\varphi_{B}}(dz_{1}+\mathcal{B}^{\rm
KK})^{2}\,,$ (B.6) $\displaystyle F_{{{\scriptscriptstyle(5)}}\rm IIB}$
$\displaystyle=(1+*_{10})[F^{\rm B}_{{\scriptscriptstyle(4)}}\wedge
dz_{1}]\,,$ $\displaystyle F_{{{\scriptscriptstyle(3)}}\rm IIB}$
$\displaystyle=F^{\rm B}_{{\scriptscriptstyle(3)}}+F^{\rm
B}_{{\scriptscriptstyle(2)}}\wedge(dz_{1}+\mathcal{B}^{\rm KK})\,,$
$\displaystyle H_{{{\scriptscriptstyle(3)}}\rm IIB}$ $\displaystyle=H^{\rm
B}_{{\scriptscriptstyle(3)}}+H^{\rm
B}_{{\scriptscriptstyle(2)}}\wedge(dz_{1}+\mathcal{B}^{\rm KK})\,,$

and the field strengths can be given as [61]

$\displaystyle H^{\rm B}_{{\scriptscriptstyle(2)}}=dB^{\rm
NS}_{{\scriptscriptstyle(1)}}\,,\qquad F^{\rm
B}_{{\scriptscriptstyle(2)}}=dB^{\rm RR}_{{\scriptscriptstyle(1)}}-B^{\rm
NS}_{{\scriptscriptstyle(1)}}\wedge d\chi\,,$ (B.7) $\displaystyle H^{\rm
B}_{{\scriptscriptstyle(3)}}=dB^{\rm NS}_{{\scriptscriptstyle(2)}}-dB^{\rm
NS}_{{\scriptscriptstyle(1)}}\wedge\mathcal{B}^{\rm
KK}_{{\scriptscriptstyle(1)}}\,,$ $\displaystyle F^{\rm
B}_{{\scriptscriptstyle(3)}}=dB^{\rm RR}_{{\scriptscriptstyle(2)}}+B^{\rm
NS}_{{\scriptscriptstyle(2)}}\wedge d\chi-(dB^{\rm
RR}_{{\scriptscriptstyle(1)}}-B^{\rm NS}_{{\scriptscriptstyle(1)}}\wedge
d\chi)\wedge\mathcal{B}^{\rm KK}_{{\scriptscriptstyle(1)}}\,,$ $\displaystyle
F^{\rm B}_{{\scriptscriptstyle(4)}}=dB^{\rm
RR}_{{\scriptscriptstyle(3)}}+B^{\rm RR}_{{\scriptscriptstyle(1)}}\wedge
dB^{\rm NS}_{{\scriptscriptstyle(2)}}-B^{\rm
NS}_{{\scriptscriptstyle(1)}}\wedge dB^{\rm
RR}_{{\scriptscriptstyle(2)}}-B^{\rm NS}_{{\scriptscriptstyle(1)}}\wedge
B^{\rm NS}_{{\scriptscriptstyle(2)}}\wedge d\chi\,.$

In this context, T-duality between the two type II theories stems from the
uniqueness of maximal $D=9$ supergravity. The relation between IIA and IIB
fields can thus be obtained by relating the nine-dimensional scalars, metric
and potentials in (B.1) and (B.7). Using the rules in [61] (disregarding the
doubled fields there) on (B.1), we find that the 9$d$ metrics are equal, the
dilatons are given by

$e^{\phi_{B}}=1\,,\qquad
e^{\varphi_{B}}=H^{2/\sqrt{7}}\,\Xi_{2}^{-1/\sqrt{7}}\,,$ (B.8)

and the only non-vanishing $p$-forms are

$\displaystyle B_{{\scriptscriptstyle(1)}}^{\rm KK}=\mathcal{A}\,,\qquad
B_{{\scriptscriptstyle(3)}}^{\rm RR}=A_{{\scriptscriptstyle(3)}}^{\rm RR}\,.$
(B.9)

The total type IIB configuration is given in (4.1) in the main text.

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