aainstitutetext: Department of Mathematical Sciences, Durham University,
Durham, DH1 3LE, United Kingdombbinstitutetext: Institut de Physique
Théorique, Université Paris Saclay, CNRS, CEA, F-91191, Gif-sur-Yvette,
Franceccinstitutetext: Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians Universität, Theresienstraße 37, 80333 München,
Germanyddinstitutetext: Department of Physics and Astronomy, Johns Hopkins
University, 3400 North Charles Street, Baltimore, MD 21218, USA

# Consistent truncations from the geometry of sphere bundles

Federico Bonetti b Ruben Minasian c Valentí Vall Camell d and Peter Weck
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we present a unified perspective on sphere consistent
truncations based on the classical geometric properties of sphere bundles. The
backbone of our approach is the global angular form for the sphere. A
universal formula for the Kaluza-Klein ansatz of the flux threading the
$n$-sphere captures the full nonabelian isometry group $SO(n+1)$ and scalar
deformations associated to the coset $SL(n+1,\mathbb{R})/SO(n+1)$. In all
cases, the scalars enter the ansatz in a shift by an exact form. We find that
the latter can be completely fixed by imposing mild conditions, motivated by
supersymmetry, on the scalar potential arising from dimensional reduction of
the higher dimensional theory. We comment on the role of the global angular
form in the derivation of the topological couplings of the lower-dimensional
theory, and on how this perspective could provide inroads into the study of
consistent truncations with less supersymmetry.

LMU-ASC 54/22

##  1 Introduction

Consistency of truncations, _i.e._ ensuring that any solution of a lower-
dimensional theory obtained via Kaluza-Klein (KK) reduction automatically
satisfies the higher-dimensional equations of motion, is of obvious practical
interest. The spectrum of the lower-dimensional theory is determined by
linearized KK analysis. However, nonlinear modifications away from the
infinitesimal neighbourhood of the ground state are required in order to
capture the interactions in the reduced theory and ensure that all higher-
dimensional equations of motion are satisfied. The problem has received much
recent (and not so-recent) attention, and many instances of consistent
truncation have been elaborated in the literature. The conditions for a given
truncation to be consistent are relatively straightforward when the reduction
is on a group manifold, or a quotient theoreof: only the modes invariant under
the group action should be kept Scherk:1979zr . However, the conditions for
consistent truncation for more general internal manifolds are significantly
less well understood.

This paper is an attempt to present a unified and synthetic approach to the
construction of consistent truncations, based on the geometric properties of
the fibre bundle employed in the reduction. We shall discuss here only the
cases of sphere reductions (collected in Table 1), where our formulae can be
compared with known uplift results Pilch:1984xy ; Nastase:1999cb ;
Nastase:1999kf ; deWit:1983vq ; deWit:1984nz ; deWit:1985iy ; deWit:1986mz ;
deWit:1986oxb ; Nicolai:2011cy ; deWit:2013ija ; Hohm:2013pua ;
Godazgar:2013dma ; Hohm:2014qga ; Godazgar:2015qia ; Varela:2015ywx ;
Kim:1985ez ; Cvetic:1999xp ; Lu:1999bw ; Cvetic:1999xx ; Cvetic:2000eb ;
Khavaev:1998fb ; Cvetic:2000nc ; Pilch:2000ue ; Cassani:2010uw ; Liu:2010sa ;
Gauntlett:2010vu ; Skenderis:2010vz ; Baguet:2015sma ; Guarino:2015jca ;
Guarino:2015qaa ; Guarino:2015vca ; Cvetic:2000dm . Our analysis retains the
gauge fields for the full nonabelian isometry group of the sphere as well as
scalar fields encoding its deformation. The justification and content of our
approach can be summarized in three main points:

  1. 1.

The popular approach, mostly based on variants of exceptional field theory
(see _e.g._ Lee:2014mla ; Hohm:2014qga ; Baguet:2015sma ; Ciceri:2016dmd ;
Cassani:2016ncu ; Berman:2020tqn ; Malek:2020yue ), solves the problem of
combining the lower-dimensional fields into their higher-dimensional
progenitors using clever applications of representations of duality
symmetries. In contrast, the starting point of our analysis is a classical
geometric object—the global angular form bott1982differential  $e_{n}$ on an
$n$-sphere bundle over the $AdS$ space in question. The naive space of
deformations on an $n$-sphere is given by $SL(n+1,\mathbb{R})/SO(n+1)$, and
the only group-theoretic fact required is that the denominator and the
numerator of this coset embed maximally into the duality group and its maximal
compact subgroup, respectively.111In case of IIB reductions, one needs to
incorporate $SL(2)$ and $U(1)$ factors, respectively. However, they can and
will be ignored here. In other words, our approach offers an alternative and
in some ways simpler starting point for constructing a given consistent
truncation ansatz.

When restricted to each fibre, $e_{n}$ is the generator of the top cohomology
of the fibre, _i.e._ the volume form. The pull-back of its exterior derivative
gives the Euler class of the sphere bundle (notably it vanishes for even $n$).
More details can be found in section 2.1. Crucially, for any given sphere
consistent truncation, the flux threading the internal sphere can be
reconstructed from this object—or rather, as we shall see, from an appropriate
incarnation of it, which we denote $e_{n}^{\prime}$.

  2. 2.

The embedding of $e_{n}^{\prime}$ into the consistent truncation ansatz (see
section 2) allows us to see several important universal features.

First, the flux through the undeformed $n$-sphere bundle is always of the form

$\hat{\mathcal{F}}_{n}|_{\mbox{\tiny no defs}}=e_{n}^{\prime}.$ (1)

This can be thought of as a common starting point for all the sphere
consistent truncations. Away from the vacuum solution, the sphere is deformed
and the solution needs to accommodate the associated scalar modes, which are
encoded in a symmetric unimodular matrix, $T$. The inclusion of these scalars
simply amounts to shifting $e_{n}^{\prime}$ by an exact form. On a case-by-
case basis there can also be contributions from extra fields.

While an exact shift to $e_{n}^{\prime}$ would in principle still allow the
addition of an infinite number of new terms, we demonstrate that the ambiguity
can be fixed by considering the contributions of the kinetic terms for
$\hat{\mathcal{F}}_{n}$ to the scalar potential. General supersymmetry
arguments tell us that the scalar potential is quadratic in $T$ (see section 3
for details). The part of the integral of $|\hat{\mathcal{F}}_{n}|^{2}$ on the
$n$-sphere that yields no-derivative terms is not manageable, and in general
is not polynomial in $T$. However this is not the only contribution to the
scalar potential. The other comes from the reduction of the higher dimensional
Hilbert-Einstein term. This contribution is even less pleasant, but it admits
a choice of parameters from the deformed higher-dimensional metric that
collects all non-polynomial (non-quadratic) contributions into the integral of
a perfect square, while still leaving a free parameter. This contribution can
be cancelled against the contribution from the flux kinetic term, provided the
final free parameter is set to zero. Remarkably, by choosing the values for
these free parameters as described above, we land precisely on the exact shift
to $e_{n}^{\prime}$ and on the values for $n$ and $D$ (the dimension $D$ of
the full theory) which are required for consistent truncation.

As an aside, these statements can be compared with those in Lee:2014mla  based
on the generalised parallelizability of spheres. There trivial $SO(n+1)$
connections were considered. In that case, $e_{n}^{\prime}={\rm vol}_{n}$, and
the shift to (1) is indeed exact, while the relevant parameters are fixed
based on the requirement of off-shell supersymmetry, to the same values found
here. Note that arguments based on the scalar potential of the lower-
dimensional theory have also appeared in Nastase:2000tu .

  3. 3.

Equation (1) is already sufficient for obtaining lower-dimensional topological
Chern-Simons couplings.222This is true _e.g._ for $AdS_{7}$ and $AdS_{5}$
reductions, as in Freed:1998tg ; Harvey:1998bx . In $AdS_{4}$ the Chern-Simons
couplings involve scalar fields, and require extra care. Hence one way of
describing our approach would be by saying that it provides a consistent
completion of the reduction in the topological sector. The latter in turn
functions as a backbone for the full consistent Kaluza-Klein ansatz. This
perspective could offer a fruitful counterpart to approaches rooted in
exceptional duality groups, whose details are in general dependent on the
dimensions and the amount of supersymmetry.

For many $AdS_{5}$ theories with less supersymmetry, where the lower-
dimensional symmetry based approaches are less powerful, the Chern-Simons
couplings can and have been obtained by suitable modifications of
$e_{n}^{\prime}$ Bah:2019rgq ; Bah:2020jas ; Bah:2020uev . In other words, in
all these cases a new starting point similar to (1) is already available.
While there are a number of technical challenges in completing to the full
solution from this starting point, better understanding the general
construction of $e_{n}^{\prime}$ with scalar deformations and its embedding
into the consistent truncation ansatz are worth exploring.

##  2 Global angular forms and sphere truncations

Higher dimensional theory | Sphere |  Flux $\hat{\mathcal{F}}$ | Dilaton | Gauge group | Extra fields
---|---|---|---|---|---
bosonic sector of $D=11$ sugra | $S^{4}$ | $\hat{G}_{4}$ | no | $SO(5)$ | 5 three-forms
bosonic sector of $D=11$ sugra | $S^{7}$ | $\hat{*}\hat{G}_{4}$ | no | $SO(8)$ | 35 pseudoscalars
$SL(2,\mathbb{R})$ singlet sector of $D=10$ type IIB sugra | $S^{5}$ | $\hat{F}_{5}$ | no | $SO(6)$ | none
bosonic sector of massive $D=10$ type IIA sugra | $S^{6}$ | $\hat{*}\hat{F}_{4}$ | yes | $ISO(7)$ |  see sec. 2.3.4
(2) with $p=2$ in $D$ dim.  | $S^{2}$ | $\hat{F}_{2}$ | yes | $SO(3)$ | none
(2) with $p=3$ in $D$ dim.  | $S^{3}$ | $\hat{F}_{3}$ | yes | $SO(4)$ | 1 two-form
(2) with $p=3$ in $D$ dim.  | $S^{D-3}$ | $\hat{*}F_{3}$ | yes | $SO(D-2)$ | none
Table 1: Consistent truncations on spheres considered in this section. For
each case, the higher-dimensional theory is indicated, as well as the
dimensionality of the sphere used in the truncation. The quantity
$\hat{\mathcal{F}}$ denotes the $D$-dimensional flux that threads the sphere;
in each case we indicate which field strength (or Hodge dual thereof) in the
$D$-dimensional theory is identified with $\hat{\mathcal{F}}$. The column
‘Dilaton’ indicates whether the kinetic term for $\hat{\mathcal{F}}$ in the
$D$-dimensional theory comes with a dilaton prefactor. We also indicate the
gauge group of the lower-dimensional gauged model. Finally, the column ‘Extra
fields’ collects the bosonic fields that have to be added for consistency, in
addition to the gauge fields of $SO(n+1)$ and the scalars parametrizing the
coset $SL(n+1,\mathbb{R})/SO(n+1)$.

In this section we consider the sphere consistent truncations listed in Table
1. Notice that for 11d supergravity on $S^{4}$ and $S^{7}$, for type IIB
supergravity on $S^{5}$, and for massive type IIA supergravity on $S^{6}$ the
consistent truncation applies to the full content of the model, including
fermions. For simplicity, throughout this work we restrict to the bosonic
sectors (and, for type IIB, to the $SL(2,\mathbb{R})$ singlet sector
consisting of the Einstein frame metric and the self-dual five-form flux). We
use $D$ to denote the spacetime dimension of the higher-dimensional model, $d$
for the dimension of the lower-dimensional, and $n$ for the dimension of the
sphere (so that clearly $D=d+n$). We use a hat for fields in $D$ dimensions.
The $D$-dimensional theory contains, among other fields, an $n$-form field
strength that threads the $S^{n}$, which we denote $\hat{\mathcal{F}}_{n}$.

For the fourth, fifth, and sixth entry in Table 1 the starting $D$-dimensional
theory is a bosonic model with action

$S_{(D)}=\int\bigg{[}\hat{R}\hat{*}1-\frac{1}{2}d\hat{\phi}\wedge\hat{*}d\hat{\phi}-\frac{1}{2}e^{-a\hat{\phi}}\hat{F}_{p}\wedge\hat{*}\hat{F}_{p}\bigg{]}\
,$ (2)

where $\hat{R}\hat{*}1$ is the standard Einstein-Hilbert term, $\hat{\phi}$ is
a real scalar (dilaton), and $\hat{F}_{p}$ is the closed field strength of a
$(p-1)$-form gauge potential. The positive constant $a$ is given by

$a^{2}=4-\frac{2(p-1)(D-p-1)}{D-2}=4-\frac{2(n-1)(D-n-1)}{D-2}\ ,$ (3)

which is the value required for the consistent truncation to be possible
Cvetic:2000dm . In the second step, we have observed that the dimensionality
$n$ of the sphere equals $p$ if the flux threading the sphere is
$\hat{F}_{p}$, and $D-p$ if it is its Hodge dual $\hat{*}\hat{F}_{p}$. In both
cases we get the same value of $a^{2}$ as a function of $D$, $n$.

In each case, the modes retained in the truncation include the following
bosonic fields in the $d$-dimensional supergravity theory:

  * •

metric;

  * •

gauge fields associated to the $SO(n+1)$ isometry of the round sphere $S^{n}$;

  * •

real scalars parametrizing the coset space $SL(n+1,\mathbb{R})/SO(n+1)$;

  * •

a real dilaton, if there is a ‘yes’ in the pertinent column of Table 1.

Crucially, consistency of the truncation might require that we keep additional
modes in $d$ dimensions, as reported in the last column of Table 1.

Our main objects of interest are the uplift formulae for the $D$-dimensional
metric, dilaton, and $n$-form flux $\hat{\mathcal{F}}_{n}$ that threads the
sphere. In particular, we focus on the parts of the uplift formulae that only
contain the $SO(n+1)$ gauge fields and the $SL(n+1,\mathbb{R})/SO(n+1)$
scalars. We observe a simple regular pattern, described in detail below, which
extends the analysis of Nastase:2000tu ; Lee:2014mla  by considering
simultaneously the $SO(n+1)$ gauge fields and the $SL(n+1,\mathbb{R})/SO(n+1)$
scalars.

The detailed forms of the uplift formulae depend on the specific case
considered; we describe each case in turn in the following subsections. The
general structure of the uplift formulae, however, can be presented uniformly
for all cases. The $D$-dimensional metric, dilaton, and $n$-form flux
$\hat{\mathcal{F}}_{n}$ take the form

$\displaystyle d\hat{s}^{2}_{D}$
$\displaystyle=Y^{c_{1}}(yTy)^{\frac{n-1}{D-2}}\bigg{[}ds^{2}_{d}+g^{-2}Y^{-\frac{2}{n+1}}\frac{1}{yTy}T^{-1}_{IJ}Dy^{I}Dy^{J}\bigg{]}+\text{(extra)}\
,$ (4) $\displaystyle e^{\frac{2s}{a}\hat{\phi}}$
$\displaystyle=Y^{c_{2}}(yTy)^{-1}\times\text{(extra)}\ ,$ (5)
$\displaystyle\hat{\mathcal{F}}_{n}$
$\displaystyle=e_{n}^{\prime}+\frac{1}{\mathcal{V}_{n}n!}d\bigg{[}\frac{n}{n-1}\frac{1}{yTy}\epsilon
y(Ty)(Dy)^{n-1}\bigg{]}+\text{(extra)}\ ,$ (6) $\displaystyle e^{\prime}_{n}$
$\displaystyle=\frac{1}{\mathcal{V}_{n}n!}\epsilon
y(Dy)^{n}+\frac{1}{\mathcal{V}_{n}n!}\sum_{j=1}^{j_{\rm
max}}\frac{g^{j}}{j!}(n/2)_{j}\epsilon yF^{j}(Dy)^{n-2j}\ .$ (7)

Our notation is as follows. We use $ds^{2}_{d}$ for the metric in $d$
dimensions, while $Y$ is a positive scalar that encodes the $d$-dimensional
dilaton, if present (if absent, the above formulae are understood with
$Y\equiv 1$). The positive constant $a$ was defined in (3) while the constants
$c_{1}$, $c_{2}$ are given by

$c_{1}=\frac{2n}{(D-2)(n+1)}\ ,\qquad
c_{2}=\frac{1}{a^{2}}\frac{4(D-n-2)(n-1)}{(D-2)(n+1)}\ .$ (8)

The quantity $s$ is a sign, determined by how the $D$-dimensional dilaton
enters the kinetic term for the flux that threads the sphere, $S_{(D)}\sim\int
e^{-sa\hat{\phi}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}+\dots$.
The symbol $\mathcal{V}_{n}$ stands for the volume of the round sphere $S^{n}$
of radius 1,

$\mathcal{V}_{n}=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma(\frac{n+1}{2})}\ ,$ (9)

while $(n/2)_{j}$ is the descending Pochhammer symbol, given by $(n/2)_{0}:=1$
and

$(n/2)_{j}=\frac{n}{2}\left(\frac{n}{2}-1\right)\left(\frac{n}{2}-2\right)\dots\left(\frac{n}{2}-j+1\right)\
.$ (10)

The indices $I$, $J$, $K=1,\dots,n+1$ are vector indices of $SO(n+1)$. Unless
otherwise stated, they are raised and lowered with the $SO(n+1)$ invariant
tensor $\delta_{IJ}$ and its inverse $\delta^{IJ}$. The quantities $y^{I}$ are
constrained coordinates on the sphere $S^{n}$. They are defined via the
standard embedding of the round unit $S^{n}$ in flat $\mathbb{R}^{n+1}$ in
Cartesian coordinates, and therefore satisfy

$\delta_{IJ}y^{I}y^{J}=1\ .$ (11)

The 1-forms $Dy^{I}$ are defined as

$Dy^{I}=dy^{I}+gA^{IJ}y_{J}\ ,$ (12)

where the 1-forms $A^{IJ}=A^{[IJ]}$ are the external $d$-dimensional gauge
fields of $SO(n+1)$, and $g$ is a constant parameter. The associated field
strength reads

$F^{IJ}=dA^{IJ}+gA^{IK}\wedge A_{K}{}^{J}\ .$ (13)

The quantities $T_{IJ}$ are the entries of a symmetric, positive definite,
unimodular matrix,

$T_{IJ}=T_{JI}\ ,\qquad\det T_{IJ}=1\ .$ (14)

The matrix $T$ depends only on the external $d$-dimensional spacetime
coordinates and parametrizes the $n(n+3)/2$ real scalars of the coset
$SL(n+1,\mathbb{R})/SO(n+1)$.

In (4)-(7) we have introduced some shorthand notation used throughout this
work. First of all, we write

$yTy=y_{I}T^{IJ}y_{J}\ ,\qquad(Ty)^{I}=T^{IJ}y_{J}\ .$ (15)

Secondly, we have suppressed $I$ indices and wedge products of differential
forms in expressions involving the epsilon tensor of $SO(n+1)$. For example,

$\displaystyle\epsilon y(Ty)(Dy)^{n-1}$ $\displaystyle=\epsilon_{IJK_{1}\dots
K_{n-1}}y^{I}(Ty)^{J}Dy^{K_{1}}\wedge\dots\wedge Dy^{K_{n-1}}\ ,$ (16)
$\displaystyle\epsilon yF^{2}(Dy)^{n-4}$
$\displaystyle=\epsilon_{LJ_{1}K_{1}J_{2}K_{2}I_{1}\dots
I_{n-4}}y^{L}F^{J_{1}K_{1}}\wedge F^{J_{2}K_{2}}\wedge
Dy^{I_{1}}\wedge\dots\wedge Dy^{I_{n-4}}\ .$ (17)

As mentioned above, the quantity $\hat{\mathcal{F}}_{n}$ is the flux threading
the $S^{n}$. We normalize $\hat{\mathcal{F}}_{n}$ so that its flux through
$S^{n}$ is one,

$\int_{S^{n}}\hat{\mathcal{F}}_{n}=1\ .$ (18)

The physical flux is in general an integer multiple of
$\hat{\mathcal{F}}_{n}$. We comment on the structure of $e_{n}^{\prime}$ in
greater detail below. Interestingly, $e^{\prime}_{n}$ does not contain the
external scalars $T_{IJ}$. The latter enter $\hat{\mathcal{F}}_{n}$ only via
the total derivative term in (6). This is an essential feature of the
presentation (6) of the uplift formula for the $n$-form flux. It has been
derived in Lee:2014mla  in the case in which the gauge fields of $SO(n+1)$ are
set to zero.333 The relative factor between $\epsilon y(Dy)^{n}$ and
$\frac{1}{yTy}\epsilon y(Ty)(Dy)^{n-1}$ in (6), (7) is $\frac{n}{n-1}$,
correcting a typo in equation (2.50) of Lee:2014mla  (the factor $2(d-2)!$ in
the denominator of $A^{\prime}$ should be $(d-1)!$).  Furthermore, we note
that the $d$-dimensional dilaton $Y$ does not enter the uplift formula for
$\hat{\mathcal{F}}_{n}$.

As pointed out above, retaining only the metric, the gauge fields $A^{IJ}$,
the scalars $T^{IJ}$, and the dilaton $Y$ (if there is a ‘yes’ entry in Table
1) in the lower-dimensional model might not be consistent, and extra fields
might be needed (see last column of Table 1). When extra fields are present,
they generically enter the uplift formulae for $d\hat{s}^{2}_{D}$,
$\hat{\phi}$, and/or $\hat{\mathcal{F}}_{n}$. We have used the notation
(extra) in (4)-(6) as a reminder of this caveat.

The quantity $e_{n}^{\prime}$ is closely related to the canonical global
angular form for an $S^{n}$ bundle. Therefore, before proceeding, we briefly
review the salient aspects of global angular forms.

###  2.1 Brief review of global angular forms

We follow Harvey:1998bx , see also the textbook bott1982differential . Let $E$
be a real oriented vector bundle of rank $n+1$ over a base manifold $B$.
Suppose $E$ is equipped with a connection and a metric. Let $S(E)$ denote the
associated unit-sphere bundle: if $p\in B$ and we choose Cartesian coordinates
$y^{I}$, $I=1,\dots,n+1$ on the $\mathbb{R}^{n+1}$ fiber of $E$ at $p$, the
sphere fiber of $S(E)$ at $p$ is described by (11). We use $E_{0}$ to denote
the complement of the zero-section in $E$. One can prove that there exists a
globally defined $n$-form on $E_{0}$, which we denote $e_{n}$444In the
mathematics literature the global angular form is usually normalized in such a
way that it integrates to 2 on the sphere fibers, so that $e_{n}^{\rm
here}=\tfrac{1}{2}e_{n}^{\rm maths}$. For $n$ even, the closed form
$e_{n}^{\rm maths}$ represents an _integral_ cohomology class. , with the
following properties:

  * •

The form $e_{n}$ restricted to the fibers of $S(E)$ reduces to the standard
volume form on $S^{n}$, normalized to integrate to $1$,

$\int_{S^{n}}e_{n}=1\ .$ (19)

  * •

The exterior derivative of $e_{n}$ is given by

$de_{n}=\left\\{\begin{array}[]{ll}0&\text{if $n$ is even,}\\\
-\pi^{*}\chi_{n+1}(E)&\text{if $n$ is odd.}\end{array}\right.$ (20)

Here $\pi^{*}$ is the pullback by the projection $\pi:E_{0}\rightarrow B$ onto
the base space. The $(n+1)$-form $\chi_{n+1}(E)$ is a polynomial in the
curvature of the bundle $E$, given explicitly below in (23), that represents
the Euler characteristic of $E$ as a cohomology class in
$H^{n+1}(B;\mathbb{Z})$.

We can exhibit an explicit local expression for $e_{n}$ in terms of the
constrained coordinates $y^{I}$ and the components $F^{IJ}$ of the field
strength of the $SO(n+1)$ connection on $E$ Harvey:1998bx ,

$e_{n}=\frac{1}{\mathcal{V}_{n}n!}\sum_{j=0}^{\lfloor
n/2\rfloor}\frac{g^{j}}{j!}(n/2)_{j}\epsilon yF^{j}(Dy)^{n-2j}\ .$ (21)

The unit sphere volume $\mathcal{V}_{n}$ and the Pochhammer symbols
$(n/2)_{j}$ have been introduced in (9), (10).

The term with $j=0$ in (21) of schematic form $\epsilon y(Dy)^{n}$ describes
the volume form on the round $S^{n}$, normalized to integrate to 1. The
derivative of the $j=0$ term generates a term with one $F$ factor,
schematically $\epsilon yF(Dy)^{n-1}$. The relative coefficient between the
$j=0$ and $j=1$ term in (21) is engineered in such a way that this term linear
in $F$ cancels in $de_{n}$, leaving behind only one term with two $F$’s. This
can be verfied with the help of some Schouten identities and of the Bianchi
identities

$0=DF^{IJ}:=dF^{IJ}+gA^{IK}\wedge F_{K}{}^{J}+gA^{JK}\wedge F^{I}{}_{K}\
,\quad DDy^{I}=gF^{IJ}y_{J}\ .$ (22)

The relative coefficient between the $j=1$ and $j=2$ terms in (21) is
similarly engineered to guarantee the cancellation of the terms with two $F$’s
in $de_{n}$, leaving behind only a term with three $F$’s. For even $n$ this
pattern of cancellations in $de_{n}$ continues until we reach $de_{n}=0$. For
odd $n$, the pattern of cancellations proceeds until we get a $de_{n}$ that is
purely horizontal, _i.e._ without any $Dy$ factors,

$\displaystyle\text{$n=2m-1$:}\;\;\;\;de_{2m-1}$
$\displaystyle=-\pi^{*}\chi_{2m}(E)\ ,$ (23) $\displaystyle\chi_{2m}(E)$
$\displaystyle=\frac{g^{m}}{2^{m}m!(2\pi)^{m}}\epsilon_{I_{1}J_{1}\dots
I_{m}J_{m}}F^{I_{1}J_{1}}\wedge\dots\wedge F^{I_{m}J_{m}}={\rm
Pf}\left(\frac{gF}{2\pi}\right)\ .$

The quantity $e_{n}^{\prime}$ that enters the uplift formula (6) is
constructed out of the canonical global angular form $e_{n}$. More precisely,
depending on the specific case under examination from Table 1, we go from
$e_{n}$ to $e_{n}^{\prime}$ by truncating the expansion of $e_{n}$ in powers
of the field strenghts $F^{IJ}$. In other words, $e_{n}^{\prime}$ is obtained
by truncating the sum over $j$ in (21). The details of the truncation are
reported below for each case.

###  2.2 Interpretation of the uplift formula for $\hat{\mathcal{F}}_{n}$

Let us comment on the form (6) of the uplift formula for the flux
$\hat{\mathcal{F}}_{n}$ threading the sphere.

If the gauge fields of $SO(n+1)$ are set to zero, and the scalar matrix
$T_{IJ}$ is set to the identity, the expression for $\hat{\mathcal{F}}_{n}$
clearly reduces to the volume form on the round $S^{n}$. This is a closed form
that represents the generator of the cohomology group
$H^{n}(S^{n};\mathbb{Z})$. The physical flux is given by an integer times this
generator, by virtue of flux quantization.

Let us imagine to turn on the scalar fields $T_{IJ}$, keeping the gauge fields
zero for the moment. Formula (6) states that, under the continuous deformation
parametrized by $T_{IJ}$, $\hat{\mathcal{F}}_{n}$ is shifted by an exact piece
of the form $d[(yTy)^{-1}\epsilon y(Ty)(dy)^{n-1}]$. It follows that the de
Rham cohomology class of $\hat{\mathcal{F}}_{n}$ is unmodified. This is to be
expected, since a continuous deformation cannot change the integral value of
the flux threading the sphere. The detailed form of the exact deformation of
$\hat{\mathcal{F}}_{n}$ when the scalars $T_{IJ}$ are turned on (but the gauge
fields are zero) was derived in Lee:2014mla  using the notion of generalized
parallelizability for spheres. In section 3 below we offer a different
argument (similar to the considerations of Nastase:2000tu ) that determines
both the functional form and the numerical prefactor of the exact deformation.

We may alternatively start from the volume form on the round $S^{n}$ and turn
on the $SO(n+1)$ gauge fields, keeping $T_{IJ}$ fixed to be the identity
matrix. In this case the total $D$-dimensional spacetime should be regarded as
a sphere fibration. The ‘naked’ volume form $\epsilon y(dy)^{n}$ is no longer
a well-defined $n$-form in spacetime. Rather, it must be promoted by means of
the replacement $dy^{I}\rightarrow Dy^{I}$. The resulting $n$-form $\epsilon
y(Dy)^{n}$ is indeed globally defined, yet it fails to be closed: its
derivative takes the form $\epsilon F(Dy)^{n-1}$. The terms in the sum (7)
with $j=1,2,\dots,j_{\rm max}$ can be interpreted as corrective terms: as
discussed in the previous section around (22), adding the term with $j=1$
ensures that the non-closure of $e^{\prime}_{n}$ is of the form $\epsilon
F^{2}(Dy)^{n-3}$; further adding the term with $j=2$ yields a non-closure of
the form $\epsilon F^{3}(Dy)^{n-5}$; and so on.

Finally, let us comment on the case in which both the gauge fields of
$SO(n+1)$ and the scalars $T_{IJ}$ are turned on. The uplift formula (6),
based on all the explicit examples we have studied, exhibits a particularly
simple structure. Indeed, the dependence on the scalar fields $T_{IJ}$ is
entirely confined inside a total derivative, even after the gauge fields are
activated. Furthermore, the $(n-1)$-form inside the total derivative is
$(yTy)^{-1}\epsilon y(Ty)(Dy)^{n-1}$. This quantity is obtained from the exact
deformation when the gauge fields as zero by means of the minimal replacement
$dy^{I}\rightarrow Dy^{I}$ inside the total derivative. A priori, additional
terms could have been added inside the total derivative after turning on the
gauge fields, such as terms proportional to $\epsilon y(Ty)F(Dy)^{n-3}$. In
all examples we have studied, however, we observe that such terms are not
generated. In section 4, in the case of $D=11$ supergravity on $S^{4}$, we
offer a different perspective on this fact, based on a formula of Bott and
Cattaneo BottCattaneo .

###  2.3 Detailed uplift formulae

Let us now examine each case in Table 1 in turn.

####  2.3.1 $D=11$ supergravity on $S^{4}$

11d supergravity can be consistently truncated on $S^{4}$ Pilch:1984xy ;
Nastase:1999cb ; Nastase:1999kf  to 7d maximal $SO(5)$ gauged supergravity
Pernici:1984xx . The flux that threads the sphere is clearly the closed
$G_{4}$ flux of 11d supergravity,

$\hat{\mathcal{F}}_{4}\propto\hat{G}_{4}\ .$ (24)

Recall that, by definition, $\hat{\mathcal{F}}_{4}$ is rescaled in such a way
as to integrate to 1 on $S^{4}$. In this case there is no dilaton in the
higher-dimensional theory, and thus no dilaton among the modes that are kept
in seven dimensions. The bosonic content of 7d maximal $SO(5)$ gauged
supergravity consists of the metric, the $SO(5)$ gauge fields, 20 real scalars
parametrizing the coset $SL(5,\mathbb{R})/SO(5)$, and five 3-forms
$c^{I}_{3}$, transforming in the vector representation of $SO(5)$. The latter
are the ‘extra’ fields reported in the last column of Table 1.

The complete uplift formulae for this consistent truncation are given in
Nastase:1999kf , see also Cvetic:2000ah . By applying some Schouten
identities, they can be recast in the following form,

$\displaystyle d\hat{s}^{2}_{11}$
$\displaystyle=(yTy)^{1/3}\bigg{[}ds^{2}_{7}+g^{-2}\frac{1}{yTy}T^{-1}_{IJ}Dy^{I}Dy^{J}\bigg{]}\
,$ (25) $\displaystyle\hat{\mathcal{F}}_{4}$
$\displaystyle=e_{4}+\frac{1}{\mathcal{V}_{4}4!}d\bigg{[}\frac{4}{3}\frac{1}{yTy}\epsilon
y(Ty)(Dy)^{3}\bigg{]}+d(y_{I}c^{I}_{3})\ ,$ (26) $\displaystyle e_{4}$
$\displaystyle=\frac{1}{\mathcal{V}_{4}4!}\bigg{[}\epsilon
y(Dy)^{4}+2g\epsilon yF(Dy)^{2}+g^{2}\epsilon yF^{2}\bigg{]}\ .$ (27)

We observe that in this case the quantity $e_{4}^{\prime}$ that enters the
uplift formula for $\hat{\mathcal{F}}_{4}$ is exactly identified with the
canonical global angular form $e_{4}$. In other words, the sum over $j$ in
(21) is not truncated. Moreover, let us point out that the extra three-forms
$c_{3}^{I}$ do not enter the uplift formula for the 11d metric, and enter the
uplift formula for the 11d four-form flux in a very simple way, via a total
derivative.

####  2.3.2 $D=11$ supergravity on $S^{7}$

11d supergravity can be consistently truncated on $S^{7}$ deWit:1983vq ;
deWit:1984nz ; deWit:1985iy ; deWit:1986mz ; deWit:1986oxb ; Nicolai:2011cy ;
deWit:2013ija ; Hohm:2013pua ; Godazgar:2013dma ; Hohm:2014qga ;
Godazgar:2015qia ; Varela:2015ywx  to 4d maximal $SO(8)$ gauged supergravity
deWit:1981sst ; deWit:1982bul . The flux that threads the sphere is now

$\hat{\mathcal{F}}_{7}\propto\hat{*}\hat{G}_{4}\ .$ (28)

The bosonic content of 4d maximal $SO(8)$ gauged supergravity consists of the
metric, the $SO(8)$ gauge fields, and 70 real scalars parametrizing the coset
$E_{7(7)}/(SU(8)/\mathbb{Z}_{2})$. Out of these 70 scalars, 35 have positive
intrinsic parity (proper scalars) and 35 have negative intrinsic parity
(pseudoscalars). The 35 proper scalars parametrize the coset
$SL(8,\mathbb{R})/SO(8)$. From this point of view, the remaining 35
pseudoscalars are regarded as ‘extra’ fields in the terminology of Table 1.

The complete uplift formulae for the 11d metric and four-form flux for this
consistent truncation are given in Varela:2015ywx , building on deWit:1983vq ;
deWit:1984nz ; deWit:1985iy ; deWit:1986mz ; deWit:1986oxb ; Nicolai:2011cy ;
deWit:2013ija ; Hohm:2013pua ; Godazgar:2013dma ; Hohm:2014qga ;
Godazgar:2015qia . In order to verify the general formulae (4), (6), the task
at hand is: (i) turn off the 35 pseudoscalars, so that only the
$SL(8,\mathbb{R})/SO(8)$ scalars remain; (ii) compute $\hat{G}_{7}$ by taking
the Hodge dual of the $\hat{G}_{4}$ flux given in Varela:2015ywx . Some steps
of these computations are reported in appendix A.3. The result reads

$\displaystyle d\hat{s}^{2}_{11}$
$\displaystyle=(yTy)^{2/3}\bigg{[}ds^{2}_{4}+g^{-2}\frac{1}{yTy}T^{-1}_{IJ}Dy^{I}Dy^{J}\bigg{]}+\text{(extra)}\
,$ (29) $\displaystyle\hat{\mathcal{F}}_{7}$
$\displaystyle=e_{7}^{\prime}+\frac{1}{\mathcal{V}_{7}7!}d\bigg{[}\frac{7}{6}\frac{1}{yTy}\epsilon
y(Ty)(Dy)^{6}\bigg{]}+\text{(extra)}\ ,$ (30) $\displaystyle e_{7}^{\prime}$
$\displaystyle=\frac{1}{\mathcal{V}_{7}7!}\bigg{[}\epsilon
y(Dy)^{7}+\frac{7}{2}g\epsilon yF(Dy)^{5}\bigg{]}\ .$ (31)

We notice that the 7-form $e_{7}^{\prime}$ that enters the uplift formula is
now distinct from the canonical global angular form $e_{7}$ given by (21). In
the canonical $e_{7}$, we encounter terms with up to three $F$’s,
schematically $e_{7}\sim\epsilon y(Dy)^{7}+\epsilon yF(Dy)^{5}+\epsilon
yF^{2}(Dy)^{3}+\epsilon yF^{3}Dy$. In this case the base of the $S^{7}$
fibration is 4d spacetime, and thus the $F^{3}$ term vanishes for dimensional
reasons. The $F^{2}$ term in $e_{7}$, however, survives on a 4d base space. It
constitutes the difference between the canonical $e_{7}$ and its truncated
counterpart $e_{7}^{\prime}$.

It is worth emphasizing that in this consistent truncation the extra fields
(the 35 pseudo scalars) enter both the uplift formula for the metric and
seven-flux in non-trivial ways. While the uplift formulae retaining all 70
scalars are known explicitly Varela:2015ywx , we were not able to recast them
in a simple form in the same spirit as in (6). We leave this problem for
future research.

####  2.3.3 $D=10$ type IIB supergravity on $S^{5}$

10d type IIB supergravity can be consistently truncated on $S^{5}$ Kim:1985ez
; Cvetic:1999xp ; Lu:1999bw ; Cvetic:1999xx ; Cvetic:2000eb ; Khavaev:1998fb ;
Cvetic:2000nc ; Pilch:2000ue ; Cassani:2010uw ; Liu:2010sa ; Gauntlett:2010vu
; Skenderis:2010vz ; Baguet:2015sma  to 5d maximal $SO(6)$ gauged supergravity
Gunaydin:1984qu ; Pernici:1985ju ; Gunaydin:1985cu . The flux that threads the
sphere is the self-dual Ramond-Ramond five-form flux $\hat{F}_{5}$. To discuss
the uplift formula for $\hat{F}_{5}$, we find it convenient to introduce a
five-form $\hat{\mathcal{F}}_{5}$ that satisfies

$\hat{\mathcal{F}}_{5}+\hat{*}\hat{\mathcal{F}}_{5}\propto\hat{F}_{5}\ .$ (32)

The bosonic content of 5d maximal $SO(6)$ gauged supergravity consists of the
metric, the $SO(6)$ gauge fields, 42 real scalars parametrizing the coset
$E_{6(6)}/USp(8)$, and a collection of twelve real two-form potentials
$B_{2}^{I\alpha}$, where $I=1,\dots,6$ is a fundamental index of
$SL(6,\mathbb{R})$ and $\alpha=1,2$ is a fundamental index of
$SL(2,\mathbb{R})$. Recall that $SL(6,\mathbb{R})\times SL(2,\mathbb{R})$ is a
maximal subgroup of $E_{6(6)}$. The $SL(2,\mathbb{R})$ factor is identified
with the global $SL(2,\mathbb{R})$ symmetry of classical type IIB supergravity
in ten dimensions.

The complete uplift formulae for this consistent truncation are given in
Baguet:2015sma . In this work, however, we restrict our attention to a subset
of the bosonic fields of the 5d gauged supergravity, namely those that are
inert under $SL(2,\mathbb{R})$. In ten dimensions, the bosonic fields that are
singlets of $SL(2,\mathbb{R})$ are the Einstein frame metric and the five-form
flux. They form a 10d consistent bosonic subsector of the full type IIB
supergravity. This 10d bosonic theory can be consistently truncated on $S^{5}$
to the bosonic theory obtained from 5d maximal $SO(6)$ supergravity by
discarding the fermions, the two-forms $B_{2}^{I\alpha}$, and carving out a
scalar submanifold $SL(6,\mathbb{R})/SO(6)$ out of the full $E_{6(6)}/USp(8)$
scalar manifold. The uplift formulae for this truncation are given in
Cvetic:2000nc  and can be rearranged to take the form555 The quantity we
denote $\hat{\mathcal{F}}_{5}$ is denoted $\hat{*}\hat{G}_{(5)}$ in
Cvetic:2000nc , where it is given in equation (4).

$\displaystyle d\hat{s}^{2}_{10}$
$\displaystyle=(yTy)^{1/2}\bigg{[}ds^{2}_{5}+g^{-2}\frac{1}{yTy}T^{-1}_{IJ}Dy^{I}Dy^{J}\bigg{]}\
,$ (33) $\displaystyle\hat{\mathcal{F}}_{5}$
$\displaystyle=e_{5}^{\prime}+\frac{1}{\mathcal{V}_{5}5!}d\bigg{[}\frac{5}{4}\epsilon
y(Ty)(Dy)^{4}\bigg{]}\ ,$ (34) $\displaystyle e_{5}^{\prime}$
$\displaystyle=\frac{1}{\mathcal{V}_{5}5!}\bigg{[}\epsilon
y(Dy)^{5}+\frac{5}{2}g\epsilon yF(Dy)^{3}\bigg{]}\ .$ (35)

We notice that the 5-form $e_{5}^{\prime}$ that enters the uplift formula is
again distinct from the canonical global angular form $e_{5}$ given by (21).
In the canonical $e_{5}$, we encounter terms with up to two $F$’s,
schematically $e_{5}\sim\epsilon y(Dy)^{5}+\epsilon yF(Dy)^{3}+\epsilon
yF^{2}Dy$. The terms with two $F$’s is not identically zero on external 5d
spacetime, yielding indeed a non-zero difference between $e_{5}$ and
$e_{5}^{\prime}$.

####  2.3.4 Massive $D=10$ type IIA supergravity on $S^{6}$

Massive type IIA supergravity can be consistently truncated on $S^{6}$ to
maximal dyonically gauged $ISO(7)$ supergravity Guarino:2015jca ;
Guarino:2015qaa ; Guarino:2015vca . The bosonic fields of this 4d theory
consists of the metric, electric gauge fields for the gauge group
$ISO(7)=SO(7)\ltimes\mathbb{R}^{7}$ of dimension $21+7$, 70 real scalars
parametrizing the coset $E_{7(7)}/(SU(8)/\mathbb{Z}_{2})$, as well as 7
magnetic gauge fields and 7 2-form potentials. The presence of magnetic gauge
fields and 2-form potentials is due to a magnetic gauging of the
$\mathbb{R}^{7}$ subgroup of $ISO(7)$ Guarino:2015qaa ; deWit:2007kvg .

The full uplift formulae for this consistent truncation are given in
Guarino:2015vca . We want to study the terms in these formulae that contain
the 4d metric, the electric gauge fields associated to the $SO(7)$ subgroup of
$ISO(7)$, as well as a subset of the scalar fields, consisting of the scalars
$T_{IJ}$ in the coset $SL(7,\mathbb{R})/SO(7)$, together with one extra real
scalar $Y$. All other bosonic fields of the truncation as regarded as ‘extra’
in the terminology of Table 1. In analogy to the consistent truncations based
on the bosonic action (2), discussed below in section 2.3.5, we may think of
$Y$ as a 4d dilaton. The precise embedding of $T_{IJ}$ and $Y$ into
$E_{7(7)}/(SU(8)/\mathbb{Z}_{2})$ is described in appendix A.4.

Let us focus on the uplift formulae for the 10d Einstein frame metric, the 10d
dilaton, and the 10d flux that threads the $S^{6}$. The latter is identified
with

$\hat{\mathcal{F}}_{6}\propto e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{4}\ ,$
(36)

where $\hat{F}_{4}$ is the Ramond-Ramond 4-form field strength. Notice the
appearance of a dilaton prefactor: it is due to the fact that the 10d
equations of motion take the form
$d(e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{4})=\dots$. The proportionality
constant in (36) is determined by our choice of normalization for
$\hat{\mathcal{F}}_{6}$ (which integrates to 1 on $S^{6}$).

The task at hand is to start from the uplift formulae of Guarino:2015vca , in
the simplified setting in which we set to zero the 4d fields we are not
keeping track of. After obtaining the expression for $\hat{F}_{4}$, we can
compute $e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{4}$. Some details of this
derivation are reported in appendix A.4. We find the following uplift
formulae,

$\displaystyle d\hat{s}^{2}_{10}$
$\displaystyle=Y^{\frac{3}{14}}(yTy)^{\frac{5}{8}}\bigg{[}ds^{2}_{4}+g^{-2}\frac{Y^{-\frac{2}{7}}}{yTy}T^{-1}_{IJ}Dy^{I}Dy^{J}\bigg{]}+\text{(extra)}\
,$ (37) $\displaystyle e^{4\hat{\phi}}$
$\displaystyle=Y^{\frac{20}{7}}(yTy)^{-1}\times\text{(extra)}\ ,$ (38)
$\displaystyle\hat{\mathcal{F}}_{6}$
$\displaystyle=e_{6}^{\prime}+\frac{1}{\mathcal{V}_{6}6!}d\bigg{[}\frac{6}{5}\epsilon
y(Ty)(Dy)^{5}\bigg{]}+\text{(extra)}\ ,$ (39) $\displaystyle e_{6}^{\prime}$
$\displaystyle=\frac{1}{\mathcal{V}_{6}6!}\bigg{[}\epsilon
y(Dy)^{6}+3g\epsilon yF(Dy)^{4}\bigg{]}\ .$ (40)

Interestingly, all $Y$ and $dY$ terms eventually drop out from the expression
for $\hat{\mathcal{F}}_{6}$, which only contains the $SO(7)$ gauge fields and
the $SL(7,\mathbb{R})/SO(7)$ scalars. The quantity $e_{6}^{\prime}$ is a
truncation of the canonical global angular form $e_{6}$ as given in (21).
Indeed, on a 4d base space the canonical $e_{6}$ contains three terms,
schematically $e_{6}\sim\epsilon y(Dy)^{6}+\epsilon yF(Dy)^{4}+\epsilon
yF^{2}Dy^{2}$, while $e_{6}^{\prime}$ lacks the $F^{2}$ term.

The powers of $yTy$ and $Y$ in (37), (38) are in agreement with the general
expressions (4), (5). The sign $s$ is $+1$. This is due to the fact that we
are adopting the conventions of Guarino:2015vca  for the action of massive
$D=10$ type IIA supergravity—see (A.1) therein. In particular, the kinetic
term for the Ramond-Ramond 4-form flux is of the form
$e^{+\frac{1}{2}\hat{\phi}}\hat{F}_{4}\wedge\hat{*}\hat{F}_{4}$. In terms of
the dual flux $\hat{\mathcal{F}}_{6}\propto
e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{4}$, this term reads schematically
$e^{-\frac{1}{2}\hat{\phi}}\hat{\mathcal{F}}_{6}\wedge\hat{*}\hat{\mathcal{F}}_{6}$,
from which we infer $s=+1$.

####  2.3.5 Consistent truncations based on (2)

These three cases are discussed in Cvetic:2000dm , where the full uplift
formulae are given. The uplift formulae for the $D$-dimensional metric and
dilaton take the form666The matrix $T$ in our expressions is always
unimodular. It corresponds to the matrix denoted $\tilde{T}$ in Cvetic:2000dm
.

$\displaystyle
d\hat{s}^{2}_{D}=Y^{c_{1}}(yTy)^{\frac{n-1}{D-2}}\bigg{[}ds^{2}_{d}+g^{-2}Y^{-\frac{2}{n+1}}\frac{1}{yTy}T^{-1}_{IJ}Dy^{I}Dy^{J}\bigg{]}\
,\quad e^{\frac{2s}{a}\hat{\phi}}=Y^{c_{2}}(yTy)^{-1}\ .$ (41)

The constants $c_{1}$, $c_{2}$ are as in (8). The sign $s$ is $+1$ for the
reductions on $S^{2}$, $S^{3}$ and $-1$ for the reduction on $S^{D-3}$. This
can be seen from (2). For the reductions on $S^{2}$, $S^{3}$ the flux that
threads the sphere is the ‘electric’ flux $\hat{F}_{p}$, whose kinetic term is
of the form $e^{-a\hat{\phi}}\hat{F}_{p}\wedge\hat{*}\hat{F}_{p}$. For the
reduction on $S^{D-3}$ the flux threading the sphere is the ‘magnetic’ flux
$e^{-a\hat{\phi}}\hat{*}\hat{F}_{3}$. Written in terms of
$e^{-a\hat{\phi}}\hat{*}\hat{F}_{3}$, the kinetic term for $\hat{F}_{p}$ in
(2) takes the form
$e^{+a\hat{\phi}}(e^{-a\hat{\phi}}\hat{*}\hat{F}_{3})\wedge\hat{*}(e^{-a\hat{\phi}}\hat{*}\hat{F}_{3})$,
from which we observe the anticipated flip in sign in the exponent of the
dilaton prefactor. We adopt the same normalization for $Y$ as in Cvetic:2000dm
. The kinetic term for $Y$ in the lower-dimensional action takes the form

$S_{(d)}\supset\int-\frac{1}{2}c_{5}Y^{-2}dY\wedge*dY\ ,\qquad
c_{5}=\frac{4(D-n-2)}{a^{2}(D-2)(n+1)}\ .$ (42)

The canonically normalized lower-dimensional dilaton $\phi$ is related to $Y$
by $Y=e^{\frac{1}{\sqrt{c_{5}}}\phi}$.

Let us now discuss the uplift formulae for the flux threading the sphere, in
each of the three cases. For the reduction on $S^{2}$,
$\hat{\mathcal{F}}_{2}\propto\hat{F}_{2}$ with

$\displaystyle\hat{\mathcal{F}}_{2}$
$\displaystyle=e_{2}+\frac{1}{\mathcal{V}_{2}2!}d\bigg{[}2\frac{1}{yTy}\epsilon_{IJK}y^{I}(Ty)^{J}Dy^{K}\bigg{]}\
,$ (43) $\displaystyle e_{2}$
$\displaystyle=\frac{1}{\mathcal{V}_{2}2!}\epsilon_{IJK}y^{I}(Dy^{J}\wedge
Dy^{K}+gF^{JK})\ .$ (44)

The notation $\mathcal{V}_{n}$ was introduced in (9). This result is given in
equation (55) of Cvetic:2000dm  in terms of the quantity denoted $T^{ij}$
there, which corresponds to $Y^{1/3}T_{IJ}$ in our notation. Remarkably, all
occurrences of $Y$ and its derivatives drop away from the expression for
$\hat{\mathcal{F}}_{2}$. We also notice that $e_{2}$ is the canonical global
angular form (21), satisfying $de_{2}=0$.

For the reduction on $S^{3}$ we find similarly
$\hat{\mathcal{F}}_{3}\propto\hat{F}_{3}$ with

$\displaystyle\hat{\mathcal{F}}_{3}$
$\displaystyle=e_{3}+\frac{1}{\mathcal{V}_{3}3!}d\bigg{[}\frac{3}{2}\frac{1}{yTy}\epsilon_{IJKL}y^{I}(Ty)^{J}Dy^{K}\wedge
Dy^{L}\bigg{]}+f_{3}\ ,$ (45) $\displaystyle e_{3}$
$\displaystyle=\frac{1}{\mathcal{V}_{3}3!}\epsilon_{IJKL}y^{I}\left(Dy^{J}\wedge
Dy^{J}\wedge Dy^{K}+\frac{3}{2}gF^{JK}\wedge Dy^{L}\right)\ .$ (46)

The quantity $f_{3}$ is the field strength of a 2-form potential in the
$d=D-3$ dimensional theory. This 2-form potential is an extra field needed for
the consistency of the truncation, see Table 1. Up to normalization, $f_{3}$
is the same as the field $F_{(3)}$ in the notation of Cvetic:2000dm . We
observe once more that all factors of $Y$ and $dY$ drop from the uplift
formula for the flux threading the sphere. The quantity $e_{3}$ coincides in
this case with the canonical (21). Indeed, the derivative of $e_{3}$ is purely
horizontal,

$de_{3}=-\frac{g^{2}}{8(2\pi)^{2}}\epsilon_{IJKL}F^{IJ}\wedge F^{KL}\ .$ (47)

Recall that the $D$-dimensional flux $\hat{\mathcal{F}}_{3}$ is closed. Its
closure holds because the extra 2-form potential with field strength $f_{3}$
has a non-trivial Bianchi identity involving the $SO(4)$ gauge fields
Cvetic:2000dm , with $df_{3}$ exhibiting the same structure as the RHS of
(47). In fact, in computing $d\hat{\mathcal{F}}_{3}$ we verify an exact
cancellation between $de_{3}$ and $df_{3}$.

We finally turn to the reduction on $S^{D-3}$. The flux threading the sphere
is in this case given by

$\hat{\mathcal{F}}_{D-3}\propto e^{-a\hat{\phi}}\hat{*}\hat{F}_{3}\ .$ (48)

We have defined the dual flux $\hat{\mathcal{F}}_{D-3}$ by including a
suitable power of the $D$-dimensional dilaton. This is motivated by the fact
that the EOM for $\hat{F}_{3}$ derived from the action (2) reads

$d(e^{-a\hat{\phi}}\hat{*}\hat{F}_{3})=0\ .$ (49)

Thus, in $D$ dimensions $\hat{F}_{3}$ is closed off-shell and
$\hat{\mathcal{F}}_{D-3}$ is closed on-shell.

The uplift formula for $\hat{\mathcal{F}}_{D-3}$ is given in (47) of
Cvetic:2000dm , which can be written as

$\displaystyle\hat{\mathcal{F}}_{D-3}$
$\displaystyle=e^{\prime}_{D-3}+\frac{1}{\mathcal{V}_{D-3}(D-3)!}d\bigg{[}\frac{D-3}{D-4}\frac{1}{yTy}\epsilon
y(Ty)(Dy)^{D-4}\bigg{]}\ ,$ (50) $\displaystyle e^{\prime}_{D-3}$
$\displaystyle=\frac{1}{\mathcal{V}_{D-3}(D-3)!}\bigg{[}\epsilon
y(Dy)^{D-3}+\frac{D-3}{2}\epsilon yF(Dy)^{D-5}\bigg{]}\ .$ (51)

We are adopting a compact notation analogous to that in (16). In analogy with
the other cases discussed in this section, we verify that all occurrences of
$Y$ and $dY$ drop out from the uplift formula for $\hat{\mathcal{F}}_{D-3}$.
In this case, we find generically a truncated version $e^{\prime}_{D-3}$ of
the canonical global angular form $e_{D-3}$, because the sum over $j$ in (21)
is truncated after the first term. (For $D\leq 6$ the sum over $j$ in the
canonical global angular form $e_{D-3}$ stops anyway at 1; thus for $D\leq 6$
we actually have $e^{\prime}_{D-3}=e_{D-3}$.) The fact that we truncate the
sum over $j$ at $j=1$ implies that, for general $D$, $de^{\prime}_{D-3}$
consists of one term quadratic in $F$, of the schematic form $\epsilon
F^{2}(Dy)^{D-6}$. The lower-dimensional theory, however, lives in
$d=D-(D-3)=3$ dimensions, implying that $de^{\prime}_{D-3}=0$ for dimensional
reasons. We this conclude that the form of the uplift formula (50) is such
that $d\hat{\mathcal{F}}_{D-3}=0$ holds identically, without use of the 3d
equations of motion. The closure of $\hat{F}_{3}$, on the other hand, holds
after using them.

##  3 Reduction of the action and exact flux deformation

In this section we give an argument that explains the form of the exact piece
in the uplift formula (6) for the flux that threads the $n$-sphere. The
argument is based on the derivation of the scalar potential of the lower-
dimensional model from the $D$-dimensional action integrated on $S^{n}$ (see
also Nastase:2000tu ).

###  3.1 Models without dilaton

Let us first consider the cases in which, in the $D$-dimensional theory, the
kinetic term for the flux threading the sphere has no dilaton prefactor. The
relevant terms in the $D$-dimensional action are the Einstein-Hilbert term and
the kinetic term for $\hat{\mathcal{F}}_{n}$,

$S_{(D)}=\int\bigg{[}\hat{R}\hat{*}1-\frac{1}{2g^{2}_{\mathcal{F}}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}\bigg{]}\
.$ (52)

Recall that, in our conventions, $\hat{\mathcal{F}}_{n}$ is normalized to
integrate to $1$ on $S^{n}$. As a result, its kinetic term is not necessarily
canonically normalized in $D$-dimensions, but rather comes with a coupling
constant $g^{-2}_{\mathcal{F}}$. The following argument does not depend on the
value of $g^{-2}_{\mathcal{F}}$. We also notice that we have suppressed the
overall $D$-dimensional Newton’s constant from the action (52).

Our goal is to integrate (52) on $S^{n}$ to derive couplings in the
$d$-dimensional effective action. More precisely, we seek to determine the
potential for the scalars $T_{IJ}$. To this end, we may ignore the terms with
the gauge fields of $SO(n+1)$, and further consider the simpler case in which
the profile for the $T_{IJ}$ scalars in $d$ dimensions is constant,

$A_{IJ}=0\ ,\qquad dT_{IJ}=0\ .$ (53)

To perform the dimensional reduction of the Einstein-Hilbert term in (52) we
need an ansatz for the $D$-dimensional metric, capturing the contributions of
the $T_{IJ}$ scalars. We use the ansatz

$d\hat{s}^{2}_{D}=(yTy)^{b_{1}}\Big{[}ds^{2}_{d}+g^{-2}(yTy)^{b_{2}}T^{-1}_{IJ}dy^{I}dy^{J}\Big{]}\
,$ (54)

where $b_{1}$, $b_{2}$ are constant parameters. In due course, they will be
fixed to their expected values $b_{1}=(n-1)/(D-2)$, $b_{2}=-1$, cfr. (4). The
$D$-dimensional Ricci scalar and volume element of the metric (54) can now be
evaluated by means of a straightforward computation. We keep track of terms
that contribute to the $d$-dimensional scalar potential and to the
$d$-dimensional Einstein-Hilbert term. Further details of the derivation are
reported in appendix B.1. The result reads

$\displaystyle\int_{S^{n}}\hat{R}\hat{*}1$ $\displaystyle\supset
g^{-n}\sqrt{-g_{d}}R[g_{d}]\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{\frac{1}{2}b_{1}(D-2)+\frac{1}{2}b_{2}n+\frac{1}{2}}$
$\displaystyle+g^{2-n}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{\frac{1}{2}b_{1}(D-2)+\frac{1}{2}b_{2}(n-2)+\frac{1}{2}}\mathcal{G}\
.$ (55)

Here $\xi^{m}$ are local coordinates on $S^{n}$, $g_{d}$, $R[g_{d}]$ are the
determinant and Ricci scalar of the $d$-dimensional metric, and
$\accentset{\circ}{g}$ is the determinant of the metric on the round unit
sphere. We have introduced the shorthand notation

$\displaystyle\mathcal{G}$ $\displaystyle=\frac{({\rm
Tr}\,T)^{2}}{yTy}-\frac{{\rm Tr}\,T^{2}}{yTy}-\frac{2\,({\rm
Tr}\,T)\,(yT^{2}y)}{(yTy)^{2}}+\frac{2\,yT^{3}y}{(yTy)^{2}}+\mathcal{K}\bigg{[}\frac{(yT^{2}y)^{2}}{(yTy)^{3}}-\frac{(yT^{3}y)}{(yTy)^{2}}\bigg{]}\
,$ (56)

where the constant $\mathcal{K}$ is given in terms of $D$, $n$, $b_{1}$,
$b_{2}$ by

$\mathcal{K}=-2b_{2}b_{1}(D-2)(n-1)-b_{1}^{2}(D-2)(D-1)-b_{2}^{2}(n-2)(n-1)\
.$ (57)

In the quantities $yT^{2}y$ and $yT^{3}y$ we are suppressing $SO(n+1)$ indices
as in (15).

To perform the integration of the $\hat{\mathcal{F}}_{n}$ kinetic term in (52)
over the $n$-sphere, we need an ansatz capturing the dependence of
$\hat{\mathcal{F}}_{n}$ on the scalars $T_{IJ}$. Since we are working under
the simplifying assumptions (53), $\hat{\mathcal{F}}_{n}$ must be proportional
to the volume form of the round metric on $S^{n}$, where the proportionality
factor may depend on $y^{I}$ and $T_{IJ}$,

$\hat{\mathcal{F}}_{n}=\frac{1}{n!}f(y,T)\epsilon_{I_{0}I_{1}\dots
I_{n}}y^{I_{0}}dy^{I_{1}}\wedge\dots\wedge dy^{I_{n}}\ .$ (58)

The reduction is reported in appendix B.1. The result reads

$\displaystyle\int_{S^{n}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}$
$\displaystyle=g^{n}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{\frac{b_{1}D}{2}-b_{1}n-\frac{b_{2}n}{2}-\frac{1}{2}}f(y,T)^{2}\
.$ (59)

In the $d$-dimensional action the Einstein-Hilbert term should not be
multiplied by a non-trivial function of $T_{IJ}$. This leads us to set the
exponent of $yTy$ in the first line of (3.1) to zero,

$\frac{1}{2}b_{1}(D-2)+\frac{1}{2}b_{2}n+\frac{1}{2}=0\ .$ (60)

For the integrand in the second line of (3.1) we then find the quantity
$(yTy)^{-b_{2}}\mathcal{G}$. The matrix $T_{IJ}$ is unimodular, but it is
convenient to regard the quantities $yTy$ and $\mathcal{G}$ as functions of an
arbitrary non-singular symmetric matrix. In this sense, we can formally
consider their behavior under a constant rescaling $T_{IJ}\rightarrow\lambda
T_{IJ}$. We see that $yTy$ and $\mathcal{G}$ are homogeneous of degree 1 in
$T$, implying that the combination $(yTy)^{-b_{2}}\mathcal{G}$ has degree
$1-b_{2}$. We know, however, that the scalar potential of the $d$-dimensional
theory should be a quadratic function of $T$. This is known to be the case for
all examples in Table 1 with a ‘no’ in the dilaton column. These
considerations lead us to set $1-b_{2}=2$. In combination with (60) this fixes
$b_{1}$ and $b_{2}$ to their expected values,

$b_{1}=\frac{n-1}{D-2}\ ,\qquad b_{2}=-1\ .$ (61)

The overall integrand in the second line of (3.1) then becomes
$(yTy)\mathcal{G}$.

To proceed, we observe that we are free to add to $(yTy)\mathcal{G}$ any total
divergence on $S^{n}$, without affecting the value of the integral. Exploiting
this freedom, (3.1) can be brought to the form (see appendix B.1 for details)

$\displaystyle\int_{S^{n}}\hat{R}\hat{*}1\supset
g^{-n}\mathcal{V}_{n}\sqrt{-g_{d}}R[g_{d}]+g^{2-n}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}\,\mathcal{I}\
,$ $\displaystyle\mathcal{I}=2\bigg{[}\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm
Tr}\,T\bigg{]}^{2}-\bigg{(}1+\frac{k}{2}\bigg{)}\,{\rm
Tr}\,(T^{2})+\bigg{(}\frac{1}{2}+\frac{k}{4}-\frac{k^{2}}{32}\bigg{)}\,({\rm
Tr}\,T)^{2}\ ,$ (62)

where we have introduced the shorthand notation

$k:=\frac{(n-1)(D-n-1)}{D-2}-2\ .$ (63)

Note that inside the integrand $\mathcal{I}$ all dependence on $y^{I}$ is now
collected into a perfect square. Moreover, using (61), (63) we can rewrite
(59) in the form

$\displaystyle\int_{S^{n}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}$
$\displaystyle=g^{n}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{2+k}f(y,T)^{2}\
.$ (64)

Upon integrating $\mathcal{I}$ against the volume form
$\sqrt{\accentset{\circ}{g}}$, we obtain a function of $T_{IJ}$ which
represents the contribution of the $D$-dimensional Einstein-Hilbert term to
the $d$-dimensional scalar potential. The resulting function of $T_{IJ}$ is
manifestly homogeneous of degree 2 under $T_{IJ}\rightarrow\lambda T_{IJ}$. It
is not, however, a quadratic function. In fact, it is not a rational function
of the entries of $T_{IJ}$. This is due to the contribution from the square
$[\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm Tr}\,T]^{2}$ in the integrand. This is
point is clarified in appendix B.3. Non-quadratic contributions to the scalar
potential must drop out when the Einstein-Hilbert contribution is added to
(64). Comparison of (62), (64) immediately suggests a simple and natural
mechanism to achieve this goal: the term
$[\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm Tr}\,T]^{2}$ in $\mathcal{I}$ is
cancelled against $(yTy)^{2+k}f(y,T)^{2}$ in (64) at the level of the
_integrand_. The remaining terms in $\mathcal{I}$ are a $y$-independent
quadratic function of $T$.

For the rest of this section, the aforementioned cancellation between
$[\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm Tr}\,T]^{2}$ and
$(yTy)^{2+k}f(y,T)^{2}$ is our working assumption. Therefore, we set

$f(y,T)=2g_{\mathcal{F}}g^{1-n}(yTy)^{-\frac{k}{2}}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{4-k}{8}\frac{{\rm
Tr}\,T}{yTy}\bigg{]}\ .$ (65)

A necessary condition for the identification (65) to hold is that the integral
of the RHS on $S^{n}$ with measure $\sqrt{\accentset{\circ}{g}}$ be
independent of $T$. Indeed, the integral of the LHS must be independent of
$T$, because turning on $T$ must deform $\hat{\mathcal{F}}_{n}$ by an exact
piece (the quantized flux through $S^{n}$ cannot be modified by a continuous
deformation). In appendix B.2 we prove that a necessary condition for the
integral of the RHS of (65) to be independent of $T$ is

$k=0\ .$ (66)

Crucially, this condition is also sufficient to ensure that the integral of
$f(T,y)$ is independent of $T$. This follows from the observation that

$\epsilon y(dy)^{n}+d\bigg{[}\frac{n}{n-1}\frac{1}{yTy}\epsilon
y(Ty)(dy)^{n-1}\bigg{]}=-\frac{2}{n-1}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{1}{2}\frac{{\rm
Tr}\,T}{yTy}\bigg{]}\epsilon y(dy)^{n}\ .$ (67)

The integral of the LHS over $S^{n}$ is manifestly independent of $T$ by
virtue of Stokes’ theorem.

To summarize, our assumptions are: (1) turning on $T$ deforms the flux by an
exact piece; (2) the perfect square in the Einstein-Hilbert and the
contribution from the kinetic term for $\hat{\mathcal{F}}_{n}$ cancel against
each other at the level of the $S^{n}$ integrand. Under these assumptions, we
determine uniquely the form of $f(y,T)$ and therefore of
$\hat{\mathcal{F}}_{n}$ via (58), up to the overall normalization. The latter
is readily fixed by requiring $\int_{S^{n}}\hat{\mathcal{F}}_{n}=1$. The
result is

$\hat{\mathcal{F}}_{n}=\frac{1}{\mathcal{V}_{n}n!}\epsilon
y(dy)^{n}+\frac{1}{\mathcal{V}_{n}n!}d\bigg{[}\frac{n}{n-1}\frac{1}{yTy}\epsilon
y(Ty)(dy)^{n-1}\bigg{]}\ ,$ (68)

in perfect agreement with (6) and Lee:2014mla . Both the functional form of
the exact deformation of the flux, as well as its numerical coefficient
relative to the undeformed flux, are determined777More precisely, the
$(n-1)$-form inside the total derivative is completely fixed up to shifts by
closed $(n-1)$-forms, which clearly would not modify the expression for
$\hat{\mathcal{F}}_{n}$. .

Remarkably, in the process of our derivation we have found the condition
$k=0$, which is a constraint on the possible values of $n$, $D$. In fact, this
condition selects precisely the values for $n$, $D$ for which a consistent
truncation is possible with the field content we are considering. Setting (63)
equal to zero, one finds that the only integer solutions with $n<D$ are given
by

$(n,D)=(1,2),(4,11),(5,10),(7,11)\ .$ (69)

While the first of these would describe reduction on a circle from 2d to 1d,
the remaining three correspond to familiar consistent truncations. This is a
striking result. Based on the requirement that the $d$-dimensional scalar
potential be quadratic in the scalars $T_{IJ}$ and that the flux be deformed
by an exact piece, we are able to back out the possible dimensions of the
sphere and the lower-dimensional model. Similar conclusions were drawn in
Nastase:2000tu  based on slightly different arguments.888Compared to
Nastase:2000tu , our argument is based on a cancellation at the level of the
_integrand_ on $S^{n}$. Thus, we do not have to evaluate explicitly $S^{n}$
integrals of rational functions of $y$, $T$, which in general yield non-
rational functions of the entries of $T$, see the example (164) in appendix
B.1.

###  3.2 Models with dilaton

The analysis of the previous section can be repeated for the cases in which
the kinetic term for $\hat{\mathcal{F}}_{n}$ comes with a non-trivial dilaton
prefactor. The relevant terms in the $D$-dimensional action are now

$S_{(D)}=\int\bigg{[}\hat{R}\hat{*}1-\frac{1}{2g^{2}_{\mathcal{F}}}e^{-sa\hat{\phi}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}-\frac{1}{2}d\hat{\phi}\wedge\hat{*}d\hat{\phi}\bigg{]}\
.$ (70)

where $a$ is a positive constant and $s$ is a sign. Below we will see how the
value (3) for $a$ is selected. Notice that the $D$-dimensional dilaton
$\hat{\phi}$ has a canonically normalized kinetic term. Our goal is to
determine the scalar potential of the $d$-dimensional theory. As a result, we
may work under the simplifying assumptions

$A^{IJ}=0\ ,\qquad dT_{IJ}=0\ ,\qquad dY=0\ ,$ (71)

where $Y$ encodes the $d$-dimensional dilaton.

The metric ansatz must be modified, to take into account $Y$. We adopt the
following modification of (54),

$d\hat{s}^{2}_{D}=(yTy)^{b_{1}}Y^{b^{\prime}_{1}}\Big{[}ds^{2}_{d}+g^{-2}(yTy)^{b_{2}}Y^{b^{\prime}_{2}}T^{-1}_{IJ}dy^{I}dy^{J}\Big{]}\
,$ (72)

where $b_{1}$, $b_{2}$, $b^{\prime}_{1}$, $b^{\prime}_{2}$ are constant
parameters, to be fixed momentarily. The reduction of the $D$-dimensional
Einstein-Hilbert term can be performed analogously to the previous section.
The result reads

$\displaystyle\int_{S^{n}}\hat{R}\hat{*}1$ $\displaystyle\supset
g^{-n}Y^{\frac{b_{1}^{\prime}D}{2}+\frac{b_{2}^{\prime}n}{2}-b_{1}^{\prime}}\sqrt{-g_{d}}R[g_{d}]\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{\frac{1}{2}b_{1}(D-2)+\frac{1}{2}b_{2}n+\frac{1}{2}}$
$\displaystyle+g^{2-n}Y^{\frac{b_{1}^{\prime}D}{2}+\frac{b_{2}^{\prime}n}{2}-b_{1}^{\prime}-b_{2}^{\prime}}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{\frac{1}{2}b_{1}(D-2)+\frac{1}{2}b_{2}(n-2)+\frac{1}{2}}\mathcal{G}\
,$ (73)

with the same $\mathcal{G}$ as in (56).

The reduction of the kinetic term for $\hat{\mathcal{F}}_{n}$ is also
analogous to the previous case. We adopt the same ansatz for the flux as (58),
except that now we allow $f$ to be a function of $Y$, too,

$\hat{\mathcal{F}}_{n}=\frac{1}{n!}f(y,T,Y)\epsilon_{I_{0}I_{1}\dots
I_{n}}y^{I_{0}}dy^{I_{1}}\wedge\dots\wedge dy^{I_{n}}\ .$ (74)

We also use the following ansatz for the $D$-dimensional dilaton,

$e^{s\hat{\phi}}=(yTy)^{b_{3}}Y^{b_{3}^{\prime}}\ ,$ (75)

where $b_{3}$, $b_{3}^{\prime}$ are real constant parameters. Reducing the
kinetic term for $\hat{\mathcal{F}}_{n}$ yields

$\displaystyle\int_{S^{n}}e^{-sa\hat{\phi}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}=$
(76)
$\displaystyle\quad=g^{n}Y^{\frac{b_{1}^{\prime}D}{2}-b_{1}^{\prime}n-\frac{b_{2}^{\prime}n}{2}-ab_{3}^{\prime}}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{\frac{b_{1}D}{2}-b_{1}n-\frac{b_{2}n}{2}-\frac{1}{2}-ab_{3}}f(y,T,Y)^{2}\
.$

Finally, we have to take into account the contributions to the $d$-dimensional
scalar potential originating from the kinetic term for $\hat{\phi}$. We have

$\displaystyle-\frac{1}{2}\int_{S^{n}}d\hat{\phi}\wedge\hat{*}d\hat{\phi}$
$\displaystyle=2g^{2-n}b_{3}^{2}Y^{\frac{b_{1}^{\prime}D}{2}-b_{1}^{\prime}+\frac{b_{2}^{\prime}n}{2}-b_{2}^{\prime}}\times$
(77)
$\displaystyle\times\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}\,(yTy)^{\frac{b_{1}D}{2}-b_{1}+\frac{b_{2}n}{2}-b_{2}+\frac{1}{2}}\,\,\bigg{[}\frac{(yT^{2}y)^{2}}{(yTy)^{3}}-\frac{(yT^{3}y)}{(yTy)^{2}}\bigg{]}\
.$

In order to get a canonical Einstein-Hilbert term in $d$ dimensions from
(3.2), we set

$\frac{1}{2}b_{1}(D-2)+\frac{1}{2}b_{2}n+\frac{1}{2}=0\
,\qquad\frac{b_{1}^{\prime}D}{2}+\frac{b_{2}^{\prime}n}{2}-b_{1}^{\prime}=0\
.$ (78)

Exactly as in the previous section, we can examine how the second line of
(3.2) scales with $T$, impose an overall scaling of degree $2$, and thus
obtain another linear relation between $b_{1}$ and $b_{2}$. The net result is
to fix these parameters as before, see (61). Based for instance on the results
of Cvetic:2000dm , we know that the scalar potential, written as a function of
unimodular $T$ and $Y$, is also homogeneous in rescalings of $Y$, with degree
$2/(n+1)$. We may alternatively regard this condition as a way of (partially)
fixing ambiguities related to redefinitions of $Y$. If we demand that the
second line of (3.2) be homogeneous of degree $2/(n+1)$ in $Y$ we get a second
linear relation in $b_{1}^{\prime}$, $b_{2}^{\prime}$, which combined with
(78) gives us

$b^{\prime}_{1}=\frac{2n}{(D-2)(n+1)}\ ,\qquad b^{\prime}_{2}=-\frac{2}{n+1}\
.$ (79)

These values guarantee that (72) matches with our previous expression (4).

Having fixed $b_{1}$, $b_{2}$ according to (61), we can mimick the same steps
as in the previous section. By adding a suitable total divergence on $S^{n}$,
we can eliminate the $yT^{3}y$ terms from the sum of (3.2) and (77), obtaining
the simpler form

$\displaystyle\int_{S^{n}}\bigg{[}\hat{R}\hat{*}1-\frac{1}{2}d\hat{\phi}\wedge\hat{*}d\hat{\phi}\bigg{]}=g^{-n}\mathcal{V}_{n}\sqrt{-g_{d}}R[g_{d}]+g^{2-n}Y^{\frac{2}{n+1}}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}\,\tilde{\mathcal{I}}\
,$
$\displaystyle\tilde{\mathcal{I}}=2\bigg{[}\frac{yT^{2}y}{yTy}-\frac{4-\tilde{k}}{8}{\rm
Tr}\,T\bigg{]}^{2}-\bigg{(}1+\frac{\tilde{k}}{2}\bigg{)}\,{\rm
Tr}\,(T^{2})+\bigg{(}\frac{1}{2}+\frac{\tilde{k}}{4}-\frac{\tilde{k}^{2}}{32}\bigg{)}\,({\rm
Tr}\,T)^{2}\ ,$ (80)

where we have introduced

$\tilde{k}=\frac{(n-1)(D-n-1)}{D-2}-2+2b_{3}^{2}\ .$ (81)

Remarkably, the functional form of the integrand in (3.2) is exactly the same
as in (62), up to the replacement $k\rightarrow\tilde{k}$.

Making use of (61), (79), and (81), the contribution (76) from the flux
kinetic term takes the form

$\displaystyle-\frac{1}{2g_{\mathcal{F}}^{2}}\int_{S^{n}}e^{-sa\hat{\phi}}\hat{\mathcal{F}}_{n}\wedge\hat{*}\hat{\mathcal{F}}_{n}=$
$\displaystyle\qquad=-\frac{1}{2g_{\mathcal{F}}^{2}}g^{n}Y^{\frac{2n(D-n-1)}{(D-2)(n+1)}-ab_{3}^{\prime}}\sqrt{-g_{d}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{2+\tilde{k}-ab_{3}-2b_{3}^{2}}f(y,T,Y)^{2}\
.$ (82)

Once again, a natural simple mechanism suggests itself to eliminate non-
rational functions of $T$ originating from the square in $\tilde{\mathcal{I}}$
in (3.2): a cancellation against $f^{2}$ at the level of the integrand. We
thus proceed assuming

$f(y,T,Y)=2g_{\mathcal{F}}g^{1-n}Y^{\frac{ab_{3}^{\prime}}{2}+\frac{D-Dn+n^{2}+n-2}{(D-2)(n+1)}}(yTy)^{-\frac{\tilde{k}}{2}+\frac{1}{2}ab_{3}+b_{3}^{2}}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{4-\tilde{k}}{8}\frac{{\rm
Tr}\,T}{yTy}\bigg{]}\ .$ (83)

As in the previous section, we assume that $f(y,T,Y)$ originates from an exact
deformation of the round volume form on $S^{n}$. A necessary condition for the
identification (83) to be possible is therefore that the $S^{n}$ integral of
the RHS with measure $\sqrt{\accentset{\circ}{g}}$ be independent of $T$ and
$Y$. Independence on $Y$ is easily established by tuning the value of
$b_{3}^{\prime}$,

$b_{3}^{\prime}=\frac{2(n-1)(D-n-2)}{a(D-2)(n+1)}\ .$ (84)

This value guarantees that (75) agrees with our previous expressions (5), (8).
In appendix B.2 we study when the integral of (83) becomes independent of $T$,
as we vary the parameters $\tilde{k}$, $\tfrac{1}{2}ab_{3}+b_{3}^{2}$. We
identify two possibilities.

The first possibility is to set

$\tilde{k}=0\ ,\qquad\frac{1}{2}ab_{3}+b_{3}^{2}=0\ .$ (85)

For these values of $\tilde{k}$, $\tfrac{1}{2}ab_{3}+b_{3}^{2}$ the RHS of
(83) becomes the same as in the case without dilaton (65) with $k=0$. We
already know from the discussion around (67) that the integral is independent
of $T$, for any symmetric matrix $T$, unimodular or not.

As observed above, in the cases without a dilaton the condition $k=0$ fixes
the allowed values of $D$, $n$. In the present context, we can solve the
conditions (85) in terms of the parameters $a$ and $b_{3}$. We find

$a=-2b_{3}\ ,\qquad a^{2}=4-\frac{2(n-1)(D-n-1)}{D-2}\ .$ (86)

This is in agreement with (3): we have thus found a different argument that
singles out the value of $a^{2}$ for which the truncation is possible
Cvetic:2000dm . Moreover, the relation $b_{3}=-a/2$ ensure the compatibility
of (75) with our previous formula (5).

Let us now turn to the second possible choice for $\tilde{k}$,
$\tfrac{1}{2}ab_{3}+b_{3}^{2}$ that renders the integral of (83) independent
of $T$,

$\tilde{k}=\frac{4(n+1)}{n+3}\
,\qquad\frac{1}{2}ab_{3}+b_{3}^{2}=-\frac{(n-1)(n+1)}{2(n+3)}\ .$ (87)

This case is qualitatively different from (85): as discussed in appendix B.2,
if we choose (87) the integral of (83) becomes a constant times $(\det
T)^{-\frac{1}{2}}$, hence a constant for unimodular $T$, but not for a generic
symmetric $T$. The values (87) are not compatible with the uplift formulae
recorded earlier and checked against the literature. The interpretation of
(87) seems more elusive and is left for future investigation.

##  4 Bott-Cattaneo formula and $S^{4}$ truncation

In the previous section we have studied a simplified setting in which the
$SO(n+1)$ gauge fields are set to zero, and the $T$ scalars to a constant. We
have demonstrated that $\hat{\mathcal{F}}_{n}$ must contain an exact term
given by the derivative of the $(n-1)$-form $(yTy)^{-1}\epsilon
y(Ty)(dy)^{n-1}$, with the appropriate numerical prefactor. Let us now suppose
we turn on the $SO(n+1)$ gauge fields, and we relax the assumption that $T$ is
constant. Gauge invariance requires the replacement $dy\rightarrow Dy$,
leading to establish that the exact deformation of $\hat{\mathcal{F}}_{n}$
must contain the term $(yTy)^{-1}\epsilon y(Ty)(Dy)^{n-1}$. A priori,
additional terms might be generated inside the exact deformation, proportional
to $F$ and/or $DT$. The explicit analysis of the uplift formulae in all
examples collected in table 1, however, shows that this does not happen. While
we cannot furnish a first-principle proof of this fact, we can point out an
interesting connection with some results of Bott and Cattaneo BottCattaneo
regarding the fiber integrals of global angular forms for even-dimensional
spheres.

##### Reminder on the Bott-Cattaneo formula.

Recall from section 2.1 that the canonical global angular form $e_{n}$ for an
$n$-sphere with $n$ even is a closed form in the total space of an $S^{n}$
fibration over a base space $B$. In this setting, we can consider the $k$th
power of $e_{n}$ and fiber-integrate it along the $S^{n}$ directions to obtain
a closed $n(k-1)$-form on the base space $B$. The result of this operation is
the content of the Bott-Cattaneo formula BottCattaneo ,

$\int_{S^{2m}}(e_{2m})^{2s+2}=0\
,\qquad\int_{S^{2m}}(e_{2m})^{2s+1}=2^{-2s}(p_{m})^{s}\ ,\qquad s=0,1,2,\dots$
(88)

Here $n=2m$ is the dimension of the sphere, and $p_{m}$ denotes the $m$th
Pontryagin _form_ constructed with the curvature $F^{IJ}$ of the $SO(n+1)$
gauge fields. This is a polynomial in $F^{IJ}$ of degree $m$, a closed
$4m$-form on the base space $B$ with integral periods. Its cohomology class
corresponds to an element of $H^{2m}(B;\mathbb{Z})$.

##### Application to the reduction of $D=11$ supergravity on $S^{4}$.

In this case, the flux $\hat{\mathcal{F}}_{4}$ that threads the $S^{4}$ is the
$\hat{G}_{4}$ flux of 11d supergravity. The 11d action contains a Chern-Simons
term $\hat{C}_{3}\hat{G}_{4}\hat{G}_{4}$. We are interested in analyzing the
7d couplings that are generated by fiber-integrating this Chern-Simons term
along $S^{4}$. Following Freed:1998tg ; Harvey:1998bx , this is most easily
performed by regarding 11d spacetime as the boundary of an auxiliary 12d
space, which is an $S^{4}$ fibration over an 8d base $B_{8}$. The physical 7d
spacetime is the boundary of $B_{8}$. We can consider the formal 12-form
$\hat{G}_{4}^{3}$, do the fiber-integration to get an 8-form on $B_{8}$, and
consider its restriction to the 7d boundary of $B_{8}$.

We know that $\hat{G}_{4}\propto\hat{\mathcal{F}}_{4}$ takes the form

$\hat{\mathcal{F}}_{4}=e_{4}+d\omega_{3}\ ,$ (89)

where $\omega_{3}$ is a globally-defined 3-form. We thus can write

$\int_{S^{4}}\hat{\mathcal{F}}_{4}^{3}=\int_{S^{4}}\Big{(}e_{4}^{3}+d\omega_{11}\Big{)}=\int_{S^{4}}e_{4}^{3}+d\int_{S^{4}}\omega_{11}=\tfrac{1}{4}p_{2}+d\int_{S^{4}}\omega_{11}\
.$ (90)

In the previous expressions we have defined the 11-form

$\omega_{11}=3e_{4}^{2}\wedge\omega_{3}+3e_{4}\wedge\omega_{3}\wedge
d\omega_{3}+\omega_{3}\wedge(d\omega_{3})^{2}\ ,$ (91)

and we have used the fact that fiber-integration and exterior derivative
commute. In the final step of (90) we have applied the Bott-Cattaneo formula
(88).

The RHS of (90) is an 8-form that encodes topological couplings in the 7d
action. The term with $p_{2}$ in the 8d bulk corresponds to a Chern-Simons
form on the 7d boundary. Such coupling is indeed found in the action of 7d
maximal $SO(5)$ gauged supergravity Pernici:1984xx .

The term $d\int_{S^{4}}\omega_{11}$ in (90) is an exact deviation from the
Bott-Cattaneo formula. At the level of cohomology classes, the Bott-Cattaneo
relation persists, but it is no longer valid at the level of differential
forms. The term $d\int_{S^{4}}\omega_{11}$ in the 8d bulk can generate
additional topological couplings $\int_{S^{4}}\omega_{11}$ on the 7d boundary.

We may now observe the following. We know that the consistent truncation
selects the following form for $\omega_{3}$,

$\omega_{3}=\frac{1}{\mathcal{V}_{4}4!}\frac{4}{3}\frac{1}{yTy}\epsilon
y(Ty)(Dy)^{3}\ .$ (92)

Plugging this $\omega_{3}$ into (91), we verify that the 7-form
$\int_{S^{4}}\omega_{11}$ vanishes identically.999In fact, this is true as
soon as $\omega_{3}$ is proportional to $Dy^{I}\wedge Dy^{J}\wedge Dy^{K}$,
since it is easy to verify that, in this case, no term in $\omega_{11}$ has
four factors $Dy$, which is necessary to yield a non-zero result upon fiber-
integrating over $S^{4}$. Thus, the Bott-Cattaneo relation persist at the
level of differential forms, and no additional topological couplings are
generated in the 7d action.

This remarkable property is generically lost if we consider a more general
$\omega_{3}$. For example, we can imagine adding a term with one field
strength $F$, of the form

$\omega_{3}=\frac{1}{\mathcal{V}_{4}4!}\bigg{[}\frac{4}{3}\frac{1}{yTy}\epsilon
y(Ty)(Dy)^{3}+\xi_{2}\frac{1}{yTy}\epsilon y(Ty)FDy\bigg{]}\ ,$ (93)

where $\xi_{2}$ is a constant coefficient. Notice that the new term goes to
zero if we set $T=\mathbb{I}$. We can repeat the computation of
$\int_{S^{4}}\omega_{11}$ with the new $\omega_{3}$ in (93). The result is
non-zero, of the schematic form

$\int_{S^{4}}\omega_{11}\sim\mathcal{S}(T)_{I_{1}\dots
I_{8}}DT^{I_{1}I_{2}}\wedge F^{I_{3}I_{4}}\wedge F^{I_{5}I_{6}}\wedge
F^{I_{7}I_{8}}\ .$ (94)

Here $\mathcal{S}(T)_{I_{1}\dots I_{8}}$ stands for a tensor with 8 free
$SO(5)$ indices, constructed with $T$. A non-zero $\int_{S^{4}}\omega_{11}$
implies that the Bott-Cattaneo formula is no longer valid at the level of
forms.

To summarize, the consistent truncation selects an exact deformation with
precisely three $Dy$ legs. Such a deformation is non-generic, in that it
ensures that the Bott-Cattaneo relation persists at the level of differential
forms. A deviation from a term with three $Dy$’s generically induces
topological terms in the 7d action of the form (94).101010Such terms have four
derivatives. This is more than the three derivatives in the supergravity
action built by the Noether procedure from the Einstein-Hilbert term. (The
term with three derivatives is the Chern-Simons term $d^{-1}p_{2}$.) In a
maximal supergravity theory, the first higher-derivative corrections are
expected to emerge at eight derivatives. These naive considerations suggest
that terms such as (94) might be incompatible with supersymmetry.

##  5 Outlook

In this work we have made progress towards a unified approach to consistent
truncations on spheres, centered around the classical geometric notion of
global angular form. The latter is a mathematical object defined for an
$n$-sphere bundle over a base space. In applications to consistent
truncations, the base space is the spacetime of the lower-dimensional theory,
and the sphere fiber is the internal space used in the compactification.

Our main results are equations (4)-(6). They describe universal features of
the consistent Kaluza-Klein ansätze that are common to all the cases listed in
Table 1. In particular, our formulae capture the contributions of the
$SO(n+1)$ gauge fields, the $SL(n+1,\mathbb{R})/SO(n+1)$ scalars describing
the naive deformation space of $S^{n}$, and of the dilaton (when present). The
ansatz (6) for the flux threading the sphere takes a particularly compact
form, with the $SL(n+1,\mathbb{R})/SO(n+1)$ scalars entering only via an exact
shift. We have also shown how this exact shift is determined by computing the
scalar potential of the lower-dimensional model by dimensional reduction, and
imposing mild constraints on its functional dependence on the scalars.

An important problem that deserves further investigation is the analysis of
the ‘extra fields’ that are required for consistency of the truncation in some
cases (see last column of Table 1). In particular, it is desirable to identify
a way of detecting the necessity of extra fields based on the uplift formulae
(4)-(6), without relying on the structure of supersymmetry multiplets in the
lower-dimensional supergravity theory, and without performing a full explicit
check that the lower-dimensional equations of motion imply the higher-
dimensional ones. Progress in this direction would be particularly useful in
light of applications to consistent truncations with less-than-maximal
supersymmetry. In this context, matter supermultiplets can appear in the
lower-dimensional model, making it more challenging to identify which modes
should be retained for consistency of the reduction. It would also be
beneficial to try to uncover general patterns in the way in which extra
fields, when present, modify the uplift formulae (4)-(6), with the aim to
learn lessons to be applied to more general setups with less supersymmetry.

In our approach, it is convenient to regard the construction of the consistent
Kaluza-Klein ansatz for the flux as a two-step procedure. The first step is
the identification of the flux with the relevant (truncated) global angular
form $e^{\prime}_{n}$, see (1). At this stage we capture the $SO(n+1)$ gauge
fields related to the isometry of the sphere. As observed in the introduction,
this stage is sufficient for the determination of the Chern-Simons terms in
the lower-dimensional supergravity (for the $AdS_{7}$ and $AdS_{5}$ cases).
The second step is the analysis of the effect of turning on scalar
deformations.

The analog of the starting point (1) is available for more general setups, in
which the internal space is not necessarily a sphere and the $d$-dimensional
theory has less than maximal supersymmetry. In fact, we may consider replacing
$S^{n}$ with a compact internal space $M_{n}$, which is then fibered over a
base space $B_{d}$, $M_{n}\hookrightarrow X_{d+n}\rightarrow B_{d}$. If
$M_{n}$ has a continuous isometry group $G$, we turn on $G$-connections on the
bundle $X_{d+n}$. They correspond to gauge fields with gauge group $G$ in the
$d$-dimensional theory formulated on $B_{d}$. In this context, an important
task is the identification of globally defined forms on the total space
$X_{d+n}$ which reduce to a given closed form on $M_{n}$ if restricted to the
fiber directions.111111Mathematically, this is related to promoting an element
of $H^{\bullet}(M_{n};\mathbb{R})$ to a $G$-equivariant cohomology class in
$H^{\bullet}_{G}(M_{6};\mathbb{R})$. This process can be obstructed WU1993381
. The physical interpretation of such obstructions involves the Stückelberg
mechanism for $p$-form gauge fields PhysRevD.89.086007  (see also Bah:2021mzw
; Bah:2021hei  for an analysis of this phenomenon in 4d QFTs engineered with
M5-branes).  In Bah:2019rgq ; Bah:2020jas ; Bah:2020uev  this problem has been
studied systematically to compute anomalies in brane engineering and
holography (see also Hosseini:2020vgl ). The methods and results of
Bah:2019rgq ; Bah:2020jas ; Bah:2020uev  could also provide starting points
for the construction of consistent Kaluza-Klein ansätze.121212For instance,
$M_{6}$ could be an $S^{4}$ bundle over a Riemann surface. See
MatthewCheung:2019ehr ; Cassani:2020cod  for consistent truncations with such
an internal space.

###### Acknowledgements.

We thank C. Hull and D. Waldram for useful discussions. RM is supported in
part by ERC Grant 787320-QBH Structure and by ERC Grant
772408-Stringlandscape. The work of VVC was supported by the Alexander von
Humboldt Foundation via a Feodor Lynen fellowship. The work of FB is supported
by the Simons Collaboration Grant on Categorical Symmetries. PW is supported
in part by NSF grant PHY-2112699.

##  Appendix A Analysis of uplift formulae

###  A.1 Conventions for round spheres

The unit round $S^{n}$ is defined by the equation

$\delta_{IJ}y^{I}y^{J}=1$ (95)

in $\mathbb{R}^{n+1}$ with Cartesian coordinates $y^{I}$, $I=1,\dots,n+1$.
Throughout this appendix we raise/lower $I$ indices with $\delta$. We use
$\xi^{m}$, $m=1,\dots,n$ for local coordinates on $S^{n}$, with
$\partial_{m}=\partial/\partial\xi^{m}$. The round metric on $S^{n}$ reads

$\accentset{\circ}{g}_{mn}=\delta_{IJ}\partial_{m}y^{I}\partial_{n}y^{J}\ .$
(96)

It admits the Killing vectors

$\mathcal{K}_{IJ}^{m}=\accentset{\circ}{g}^{mn}(y_{I}\partial_{n}y_{J}-y_{J}\partial_{n}y_{I})\
,$ (97)

where $\accentset{\circ}{g}^{mn}$ is the inverse of
$\accentset{\circ}{g}_{mn}$. A useful identity is

$\accentset{\circ}{g}^{mn}\partial_{m}y^{I}\partial_{n}y^{J}=\delta^{IJ}-y^{I}y^{J}\
.$ (98)

The quantities $y^{I}$, regarded as a set of $n+1$ scalar functions on
$S^{n}$, satisfy

$\accentset{\circ}{\nabla}_{m}\accentset{\circ}{\nabla}_{n}y^{I}=-y^{I}\accentset{\circ}{g}_{mn}\
,$ (99)

where $\accentset{\circ}{\nabla}_{m}$ denotes the Levi-Civita connection
associated to $\accentset{\circ}{g}_{mn}$. As a consequence of (99), the
quantities $y^{I}$ are eigenfunctions of the Laplacian constructed with the
round metric,

$\accentset{\circ}{g}^{mn}\accentset{\circ}{\nabla}_{m}\accentset{\circ}{\nabla}_{n}y^{I}=-ny^{I}\
.$ (100)

###  A.2 Formulae for Hodge stars

Let us consider a $D$-dimensional metric of the form

$d\hat{s}^{2}_{D}=e^{2A}ds^{2}_{d}+g_{mn}D\xi^{m}D\xi^{n}\ ,$ (101)

where $\xi^{m}$, are local coordinates on the $n$-sphere with constrained
coordinates $y^{I}$. The 1-forms $D\xi^{m}$ are related to the 1-forms
$Dy^{I}$ in (12) by

$Dy^{I}=\partial_{m}y^{I}D\xi^{m}\ ,\qquad
D\xi^{m}=d\xi^{m}-\tfrac{1}{2}gA^{IJ}\mathcal{K}_{IJ}^{m}\ ,$ (102)

with $\mathcal{K}^{m}_{IJ}$ as in (97). The relation (98) is useful in
verifying the compatibility between (102) and (12). We allow the quantities
$e^{2A}$ and for $g_{mn}$ in (101) to depend both on the external coordinates
and on the coordinates on $S^{n}$.

Suppose $\alpha$ is a $q$-form with legs along the external spacetime
$ds^{2}_{d}$ only. The following identity holds,

$\displaystyle\hat{*}(\alpha\wedge Dy^{I_{1}}\wedge\dots\wedge
Dy^{I_{p}})=(-)^{p(d-q)}e^{(d-2q)A}\frac{\sqrt{g}}{\sqrt{\accentset{\circ}{g}}}(*\alpha)\wedge$
(103)
$\displaystyle\qquad\qquad\wedge\frac{1}{(n-p)!}\epsilon_{J_{0}J_{1}\dots
J_{p}K_{1}\dots K_{n-p}}y^{J_{0}}Q^{I_{1}J_{1}}\dots
Q^{I_{p}J_{p}}Dy^{K_{1}}\wedge\dots\wedge Dy^{K_{n-p}}\ .$

The symbol $\hat{*}$ on the LHS denotes the Hodge star with respect to the
metric (101), while $*$ on the RHS is the Hodge star with respect to the
external metric $ds^{2}_{d}$. The quantities $g$, $\accentset{\circ}{g}$ are
the determinants of $g_{mn}$, $\accentset{\circ}{g}_{mn}$, respectively. The
symmetric tensor $Q^{IJ}$ is defined as

$Q^{IJ}=g^{mn}\partial_{m}y^{I}\partial_{n}y^{J}\ ,$ (104)

where $g^{mn}$ is the inverse of $g_{mn}$ in (101).

The formula (103) takes a simpler form if specialized to a metric $g_{mn}$ of
the form

$g_{mn}=e^{2A^{\prime}}T^{-1}_{IJ}\partial_{m}y^{I}\partial_{n}y^{J}\ ,$ (105)

where $A^{\prime}$ is an arbitrary function of external and internal
coordinates and $T_{IJ}$ is a symmetric unimodular matrix, independent on the
coordinates on $S^{n}$. In this case, the inverse $g^{mn}$ of $g_{mn}$ can be
written in closed form in terms of the Killing vectors (97),

$g^{mn}=e^{-2A^{\prime}}\frac{1}{2(yTy)}T^{IJ}T^{KL}\mathcal{K}^{m}_{IK}\mathcal{K}^{m}_{JL}\
,$ (106)

as it may be verified using (98). Plugging this expression into the $Q$ tensor
(104) and using again (98) we obtain

$Q^{IJ}=e^{-2A^{\prime}}\bigg{[}T^{IJ}-\frac{(Ty)^{I}(Ty)^{J}}{yTy}\bigg{]}\
.$ (107)

The determinant of $g_{mn}$ in (105) can also be written in closed form (see
_e.g._ Lee:2014mla ),

$\det g_{mn}=e^{2A^{\prime}n}(yTy)\det\accentset{\circ}{g}_{mn}\
,\qquad\frac{\sqrt{g}}{\sqrt{\accentset{\circ}{g}}}=e^{A^{\prime}n}(yTy)^{1/2}\
,$ (108)

where we recalled that $T$ is unimodular.

Let us close this section by sketching the derivation of (103). We can write
the metric (101) in the form

$d\hat{s}^{2}_{D}=e^{2A}ds^{2}_{d}+\delta_{\bar{m}\bar{n}}e^{\bar{m}}e^{\bar{n}}\
,\qquad e^{\bar{m}}=e^{\bar{m}}{}_{p}D\xi^{p}\ ,$ (109)

where a bar denotes a flat index of $S^{n}$ and the vielbein components
$e^{\bar{m}}{}_{p}$ satisfy
$\delta_{\bar{m}\bar{n}}e^{\bar{m}}{}_{p}e^{\bar{n}}{}_{q}=g_{pq}$. Note that
$e^{\bar{m}}{}_{p}$ depends in general on both internal and external
coordinates. By using the inverse $e_{\bar{m}}{}^{p}$ of $e^{\bar{m}}{}_{p}$
we can write

$\alpha\wedge Dy^{I_{1}}\wedge\dots
Dy^{I_{p}}=\partial_{m_{1}}y^{I_{1}}\dots\partial_{m_{p}}y^{I_{p}}e_{\bar{n}_{1}}{}^{m_{1}}\dots
e_{\bar{n}_{p}}{}^{m_{p}}\alpha\wedge e^{\bar{n}_{1}}\wedge\dots\wedge
e^{\bar{n}_{p}}\ .$ (110)

The Hodge star of $\alpha\wedge e^{\bar{n}_{1}}\wedge\dots\wedge
e^{\bar{n}_{p}}$ splits into the wedge product of $*\alpha$ and
$*_{g}(e^{\bar{n}_{1}}\wedge\dots\wedge e^{\bar{n}_{p}})$, where $*_{g}$ is
the Hodge star with respect to $g_{mn}$. In the splitting, we generate a
suitable power of the warp factor $A$, and the sign factor $(-)^{p(d-q)}$ from
a reordering of indices.

We may now evaluate the quantity

$\displaystyle\partial_{m_{1}}y^{I_{1}}\dots\partial_{m_{p}}y^{I_{p}}e_{\bar{n}_{1}}{}^{m_{1}}\dots
e_{\bar{n}_{p}}{}^{m_{p}}*_{g}(e^{\bar{n}_{1}}\wedge\dots\wedge
e^{\bar{n}_{p}})=$
$\displaystyle=\frac{1}{(n-p)!}\partial_{m_{1}}y^{I_{1}}\dots\partial_{m_{p}}y^{I_{p}}e_{\bar{n}_{1}}{}^{m_{1}}\dots
e_{\bar{n}_{p}}{}^{m_{p}}\epsilon^{\bar{n}_{1}\dots\bar{n}\bar{r}_{1}\dots\bar{r}_{n-p}}e_{\bar{r}_{1}}\wedge\dots\wedge
e_{\bar{r}_{n-p}}\ .$ (111)

Here the epsilon symbol with flat indices takes values in the set $\\{0,\pm
1\\}$. Making use of $e^{\bar{m}}=e^{\bar{m}}{}_{p}D\xi^{p}$ we can also
recast the same quantity in a way that does not make explicit reference to the
vielbein,

$\displaystyle\partial_{m_{1}}y^{I_{1}}\dots\partial_{m_{p}}y^{I_{p}}e_{\bar{n}_{1}}{}^{m_{1}}\dots
e_{\bar{n}_{p}}{}^{m_{p}}*_{g}(e^{\bar{n}_{1}}\wedge\dots\wedge
e^{\bar{n}_{p}})=$
$\displaystyle=\frac{1}{(n-p)!}\partial_{m_{1}}y^{I_{1}}\dots\partial_{m_{p}}y^{I_{p}}g^{m_{1}s_{1}}\dots
g^{m_{p}s_{p}}\epsilon_{s_{1}\dots s_{p}r_{1}\dots
r_{n-p}}D\xi^{r_{1}}\wedge\dots\wedge D\xi^{r_{n-p}}\ .$ (112)

Now the epsilon tensor with lower curved indices stands for the volume form of
the metric $g_{mn}$ and therefore takes values in the set
$\\{0,\pm\sqrt{g}\\}$.

The final step is to relate the epsilon tensor of the metric $g_{mn}$ to that
of the round metric $\accentset{\circ}{g}_{mn}$, and write the latter in terms
of the contrained coordinates $y^{I}$,

$\epsilon_{m_{1}\dots
m_{n}}=\frac{\sqrt{g}}{\sqrt{\accentset{\circ}{g}}}\accentset{\circ}{\epsilon}_{m_{1}\dots
m_{n}}=\frac{\sqrt{g}}{\sqrt{\accentset{\circ}{g}}}\epsilon_{I_{0}I_{1}\dots
I_{n}}y^{I_{0}}\partial_{m_{1}}y^{I_{1}}\dots\partial_{m_{n}}y^{I_{n}}\ .$
(113)

Plugging this into (112) and making use of $\partial_{m}y^{I}D\xi^{m}=Dy^{I}$
and the definition (104) of the tensor $Q^{IJ}$, we arrive at the desired
formula (103).

###  A.3 $D=11$ supergravity on $S^{7}$

The complete uplift formulae for this consistent truncation are given in
Varela:2015ywx , including all scalars from the coset
$E_{7(7)}/(SU(8)/\mathbb{Z}_{2})$. The latter enter the uplift formulae via
the symmetric matrices $\mathcal{M}_{MN}$, $\mathcal{I}_{\Lambda\Sigma}$,
$\mathcal{R}_{\Lambda\Sigma}$. Here $M,N=1,\dots,56$ are indices of the
$\mathbf{56}$ of $E_{7(7)}$, $\Lambda,\Sigma=1,\dots,28$ are indices of the
$\mathbf{28}$ of $SL(8,\mathbb{R})$. A $\Lambda$ index is equivalent to pair
of antisymmetrized fundamental indices $[IJ]$ of $SL(8,\mathbb{R})$. In terms
of $SL(8,\mathbb{R})\subset E_{7(7)}$, a lower $M$ index splits into a lower
$\Lambda$ index, and an upper $\Lambda$ index. The matrix $\mathcal{M}_{MN}$
can be written in block form as

$\mathcal{M}_{MN}=\begin{pmatrix}\mathcal{M}_{\Lambda\Sigma}&\mathcal{M}_{\Lambda}{}^{\Sigma}\\\
\mathcal{M}^{\Lambda}{}_{\Sigma}&\mathcal{M}^{\Lambda\Sigma}\end{pmatrix}\ .$
(114)

The four blocks of $\mathcal{M}_{MN}$ and the matrices
$\mathcal{I}_{\Lambda\Sigma}$, $\mathcal{R}_{\Lambda\Sigma}$ are related by
(see _e.g._ Guarino:2015qaa )

$\mathcal{M}_{\Lambda\Sigma}=-(\mathcal{I}+\mathcal{R}\mathcal{I}^{-1}\mathcal{R})_{\Lambda\Sigma}\
,\quad\mathcal{M}_{\Lambda}{}^{\Sigma}=\mathcal{M}^{\Sigma}{}_{\Lambda}=(\mathcal{R}\mathcal{I}^{-1})_{\Lambda}{}^{\Sigma}\
,\quad\mathcal{M}^{\Lambda\Sigma}=-(\mathcal{I}^{-1})^{\Lambda\Sigma}\ .$
(115)

On the RHS of the previous relations, $\Lambda$ indices are contracted
according to matrix multiplication.131313When we identify a $\Lambda$ index
with an antisymmetric pair $[IJ]$, we insert a factor $1/2$ for each
contracted pair. For instance,
$(\mathcal{R}\mathcal{I}^{-1})_{[I_{1}I_{2}]}{}^{[J_{1}J_{2}]}=\tfrac{1}{2}\mathcal{R}_{[I_{1}I_{2}][K_{1}K_{2}]}(\mathcal{I}^{-1})^{[K_{1}K_{2}][J_{1}J_{2}]}$
and
$(\mathcal{I}\mathcal{I}^{-1})_{[I_{1}I_{2}]}{}^{[J_{1}J_{2}]}=\tfrac{1}{2}\mathcal{I}_{[I_{1}I_{2}][K_{1}K_{2}]}(\mathcal{I}^{-1})^{[K_{1}K_{2}][J_{1}J_{2}]}=\mathbb{I}_{[I_{1}I_{2}]}{}^{[J_{1}J_{2}]}:=2\delta_{[I_{1}}^{J_{1}}\delta_{I_{2}]}^{J_{2}}$.
The matrix $\mathcal{I}_{\Lambda\Sigma}$ is invertible and negative definite.
If we set to zero the 35 pseudoscalars, and only retain the 35 proper scalars,
we have

$\mathcal{R}_{\Lambda\Sigma}=0\
,\qquad\mathcal{I}_{\Lambda\Sigma}=\mathcal{I}_{[IJ][KL]}=-T^{-1}_{IK}T^{-1}_{JL}+T^{-1}_{IL}T^{-1}_{JK}\
.$ (116)

The quantity $T^{-1}_{IJ}$ is the inverse of the symmetric unimodular matrix
parametrizing the 35 proper scalars of $SL(8,\mathbb{R})/SO(8)$.

The specialization (116) implies considerable simplifications in the general
uplift formulae of Varela:2015ywx . The expressions for the internal metric
and the warp factor are given in (24) and (26) of Varela:2015ywx  and can be
unpacked with the help of the expressions (C.3) in Guarino:2015qaa  for the
generators of $SL(8,\mathbb{R})\subset E_{7(7)}$. The result is

$d\hat{s}^{2}_{11}=(yTy)^{2/3}ds^{2}_{4}+g^{-2}(yTy)^{-1/3}T^{-1}_{IJ}Dy^{I}Dy^{J}\
,$ (117)

with $(yTy)$ and $Dy^{I}$ as in (4), (12).

The uplift formula for $\hat{G}_{4}$ is (27) in Varela:2015ywx  (where the
4-form flux is denoted $\hat{F}_{(4)}$). Thanks to (116), the term $dA$ drops
away. The quantities $\mathcal{H}_{(4)}^{IJ}$, $\mathcal{H}_{(3)I}{}^{J}$,
$\tilde{\mathcal{H}}_{(2)IJ}$ in (27) of Varela:2015ywx  can be evaluated
using (16) in Varela:2015ywx . The components of the $X$ tensor that appears
there can be found for instance in (C.9), (C.10) of Guarino:2015qaa  (for the
$SO(8)$ gauging, plug $\theta_{AB}=\delta_{AB}$ and $\xi^{AB}=0$ in those
equations). We arrive at the following expression for the 4-form flux (in our
notation),

$\displaystyle\hat{G}_{4}$ $\displaystyle=-\frac{1}{2g^{2}}(*F)^{JL}\wedge
Dy^{I}\wedge
Dy^{K}T^{-1}_{IJ}T^{-1}_{KL}-\frac{1}{g}T^{JK}y_{K}Dy^{I}\wedge*DT^{-1}_{IJ}$
$\displaystyle+g{\rm vol}_{4}\Big{[}({\rm Tr}T)(yTy)-2(yTTy)\Big{]}\ .$ (118)

Here $yTTy=y_{I}T^{IJ}T_{JK}y^{K}$ and ${\rm vol}_{4}$ is the volume form of
the external metric $ds^{2}_{4}$. We may now compute the Hodge star of
$\hat{G}_{4}$ with the help of the identities collected in section A.2 and
verify that the resulting $\hat{G}_{7}$ takes the form (30) given in the main
text.

###  A.4 Massive $D=10$ type IIA supergravity on $S^{6}$

In this subsection, we use $A$, $B=1,\dots,8$ for fundamental indices of
$SL(8,\mathbb{R})$, reserving $I$, $J=1,\dots,7$ to fundamental indices of
$SL(7,\mathbb{R})$ or $SO(7)$. The full set of scalar fields in four
dimensions is again parametrized in terms of the matrices $\mathcal{M}_{MN}$,
$\mathcal{I}_{\Lambda\Sigma}$, $\mathcal{R}_{\Lambda\Sigma}$ as above. Now, we
identify $\Lambda$ indices with antisymmetrized pairs $[AB]$. In a first step,
we freeze to zero all pseudoscalar modes, so that

$\mathcal{R}_{\Lambda\Sigma}=0\
,\qquad\mathcal{I}_{\Lambda\Sigma}=\mathcal{I}_{[AB][CD]}=-\mathcal{T}^{-1}_{AC}\mathcal{T}^{-1}_{BD}+\mathcal{T}^{-1}_{AD}\mathcal{T}^{-1}_{BC}\
.$ (119)

The quantity $\mathcal{T}$ is a symmetric unimodular $8\times 8$ matrix.
Notice that (119) is nothing but (116), written in a slightly different
notation that is better suited to discuss the problem at hand.

Next, let us perform the index splitting $A\rightarrow(I,8)$. We restrict the
matrix $\mathcal{T}^{AB}$ to take the following block-diagonal form,

$\mathcal{T}^{AB}=\begin{pmatrix}\mathcal{T}^{IJ}&\mathcal{T}^{I8}\\\
\mathcal{T}^{8J}&\mathcal{T}^{88}\end{pmatrix}=\begin{pmatrix}Y^{1/7}T^{IJ}&0\\\
0&Y^{-1}\end{pmatrix}\ .$ (120)

Here $Y$ is a positive real scalar and $T^{IJ}$ is the symmetric unimodular
matrix that parametrizes $SL(7,\mathbb{R})/SO(7)$.

The uplift formulae for the metric and the dilaton can be extracted from
(3.14), (3.18), (3.23) in Guarino:2015vca . Since we only keep a subset of the
scalars, the quantities denoted $A_{m}$, $B_{mn}$, $A_{mnp}$ there are all
zero. One may then readily reproduce the formulae (37), (39) given in the main
text.

Next, we turn to the evaluation of the Ramond-Ramond 4-form field strength. It
is denoted $\hat{F}_{(4)}$ in Guarino:2015vca  and it is given by (A.4)
therein in terms of the $p$-form potentials of type IIA. Using (3.12) in
Guarino:2015vca , keeping in mind that $A_{m}=B_{mn}=A_{mnp}=0$ for us, we
confirm that $\hat{F}_{(4)}$ takes the same form as in (3.27) in
Guarino:2015vca , with the terms implicit in the ellipses being zero in our
simplified setting. This step can be checked making use of the relations
(2.7), (2.8), (2.9) in Guarino:2015qaa . The next task is then to evaluate the
quantities $\mathcal{H}_{(4)}^{IJ}$, $\mathcal{H}_{(3)I}{}^{J}$,
$\tilde{\mathcal{H}}_{(2)IJ}$ in (3.27) of Guarino:2015vca . This can be done
using (2.23), (2.21), (2.19) in Guarino:2015qaa . The latter can be unpacked
using (C.3), (C.9) in the same reference, keeping in mind that for the dyonic
gauging at hand the correct values of $\theta_{AB}$, $\xi^{AB}$ are given in
(C.15) of Guarino:2015qaa . We obtain

$\displaystyle\mathcal{H}_{(3)I}{}^{J}$
$\displaystyle=-T^{-1}_{IK}*DT^{KJ}+\frac{1}{7}\delta_{I}{}^{J}T^{-1}_{KL}*DT^{LK}\
,$ (121) $\displaystyle\tilde{\mathcal{H}}_{(2)IJ}$
$\displaystyle=-Y^{-\frac{2}{7}}T^{-1}_{IK}T^{-1}_{JL}*\mathcal{H}_{(2)}^{KL}\
,$ (122) $\displaystyle\mathcal{H}_{(4)}^{IJ}$
$\displaystyle=Y^{\frac{2}{7}}\Big{[}T^{IJ}{\rm Tr}\,T-2(TT)^{IJ}\Big{]}\ .$
(123)

Here $\mathcal{H}_{(2)}^{KL}$ is the notation of Guarino:2015qaa  for the
field strengths $F^{KL}$. Interestingly, all $Y$ and $dY$ factors drop away
from $\mathcal{H}_{(3)I}{}^{J}$.

Once $\hat{F}_{(4)}$ is computed, we may turn to
$e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{(4)}$, which may be evaluated with
the help of the identities of section A.2. Taking the Hodge star with respect
to the metric (37) generates additional powers of $Y$. These conspire with the
powers of $Y$ in (121)-(123) to ensure that $\hat{*}\hat{F}_{(4)}$ has a
single overall power of $Y$, which is precisely cancelled by the prefactor
$e^{\frac{1}{2}\hat{\phi}}$. We thus confirm that
$e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{(4)}$ does not contain $Y$. Finally,
we verify that $e^{\frac{1}{2}\hat{\phi}}\hat{*}\hat{F}_{(4)}$ takes the form
specified in (39), (40), up to an overall constant normalization factor.

##  Appendix B Scalar potential from reduction of the $D$-dimensional action

###  B.1 Reduction of the action: cases without dilaton

##### Reduction of the Einstein-Hilbert term.

We find it convenient to write the metric ansatz (54) in the form

$d\hat{s}^{2}_{D}=(yTy)^{b_{1}}d\bar{s}^{2}_{D}\ ,\qquad
d\bar{s}^{2}_{D}=ds^{2}_{d}+d\tilde{s}^{2}_{n}\ ,\qquad
d\tilde{s}^{2}_{n}=g^{-2}(yTy)^{b_{2}}\gamma_{mn}d\xi^{m}d\xi^{n}\ .$ (124)

Here $\xi^{m}$ are local coordinates on $S^{n}$, and we have introduced the
notation

$\gamma_{mn}=T^{-1}_{IJ}\partial_{m}y^{I}\partial_{n}y^{J}\ .$ (125)

Geometrically, $\gamma_{mn}$ describes an ellipsoid, whose principal axes are
determined by the eigenvalues of the matrix $T_{IJ}$.

Our first task is the evaluation of the Ricci scalar of the metric
$d\hat{s}^{2}_{D}$. In a first step, we express the Ricci scalar of
$d\hat{s}^{2}_{D}$ in terms of that of $d\bar{s}^{2}_{D}$, making use of the
fact that these two $D$-dimensional line elements are related by a Weyl
rescaling by the factor $(yTy)^{b_{1}}$. Let $\hat{\mu}$, $\hat{\nu}$ be
curved indices in $D$ dimensions. The following formula is useful,

$\displaystyle\text{if}\;\;\hat{g}_{\hat{\mu}\hat{\nu}}$
$\displaystyle=e^{2\varphi}\bar{g}_{\hat{\mu}\hat{\nu}}\ ,$ $\displaystyle
R[\hat{g}_{\hat{\mu}\hat{\nu}}]$
$\displaystyle=e^{-2\varphi}\,\Big{[}R[\bar{g}_{\hat{\mu}\hat{\nu}}]-2(D-1)\,\bar{g}^{\hat{\mu}\hat{\nu}}\,\overline{\nabla}_{\hat{\mu}}\overline{\nabla}_{\hat{\nu}}\varphi-(D-1)(D-2)\,\bar{g}^{\hat{\mu}\hat{\nu}}\,\overline{\nabla}_{\hat{\mu}}\varphi\,\overline{\nabla}_{\hat{\nu}}\varphi\Big{]}\
.$ (126)

The symbol $\overline{\nabla}_{\hat{\mu}}$ denotes the Levi-Civita connection
of the $D$-dimensional metric $d\bar{s}^{2}_{D}$. We apply (B.1) with
$e^{2\varphi}=(yTy)^{b_{1}}$.

Next, we observe that the metric $d\bar{s}^{2}_{D}$ is a direct product
between the $d$-dimensional metric $ds^{2}_{d}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ and
the $n$-dimensional metric
$d\tilde{s}^{2}_{n}=\tilde{g}_{mn}d\xi^{m}d\xi^{n}$. (We have introduced
$\mu$, $\nu$, which are curved indices in $d$ dimensions.) This holds because
we are working under the simplifying assumptions (53) of no gauge fields and
constant $T$. As a result, we have the simple formula

$R[\bar{g}_{\hat{\mu}\hat{\nu}}]=R[g_{\mu\nu}]+R[\tilde{g}_{mn}]\ .$ (127)

Furthermore, the $D$-dimensional Levi-Civita connection
$\overline{\nabla}_{\hat{\mu}}$ also splits in a trivial way,

$\overline{\nabla}_{\hat{\mu}}=(\overline{\nabla}_{\mu},\overline{\nabla}_{m})=(\nabla_{\mu},\tilde{\nabla}_{m})\
.$ (128)

Here $\nabla_{\mu}$ is the Levi-Civita connection of $g_{\mu\nu}$ and
$\tilde{\nabla}_{m}$ is that of $\tilde{g}_{mn}$. The conformal factor
$e^{2\varphi}=(yTy)^{b_{1}}$ in (B.1) depends on the $S^{n}$ coordinates only
(because we are assuming $T$ is constant). As a result, we can write

$\displaystyle R[\hat{g}_{\hat{\mu}\hat{\nu}}]$
$\displaystyle=e^{-2\varphi}\,\Big{[}R[g_{\mu\nu}]+R[\tilde{g}_{mn}]-2(D-1)\,\tilde{g}^{mn}\,\tilde{\nabla}_{m}\tilde{\nabla}_{n}\varphi-(D-1)(D-2)\,\tilde{g}^{mn}\,\tilde{\nabla}_{m}\varphi\,\tilde{\nabla}_{n}\varphi\Big{]}\
.$ (129)

We may now proceed by writing quantities associated to $\tilde{g}_{mn}$ in
terms of quantities associated to $\gamma_{mn}$. The two metrics are related
as $\tilde{g}_{mn}=e^{2\varphi^{\prime}}\gamma_{mn}$ with
$e^{2\varphi^{\prime}}=g^{-2}(yTy)^{b_{2}}$. For the Ricci scalar
$R[\tilde{g}_{mn}]$ we can use a formula completely analogous to (B.1). We
also have immediately $\tilde{\nabla}_{m}\varphi=\partial_{m}\varphi$. The
Laplacian term $\tilde{g}^{mn}\,\tilde{\nabla}_{m}\tilde{\nabla}_{n}\varphi$
can be addressed with the help of the identity

$\displaystyle\text{if}\;\;\tilde{g}_{mn}=e^{2\varphi^{\prime}}\gamma_{mn}\
,\;\tilde{g}^{mn}\,\tilde{\nabla}_{m}\tilde{\nabla}_{n}\varphi=e^{-2\varphi^{\prime}}\,\Big{[}\gamma^{mn}\nabla^{(\gamma)}_{m}\nabla^{(\gamma)}_{n}\varphi+(n-2)\gamma^{mn}\partial_{m}\varphi\partial_{n}\varphi^{\prime}\Big{]}\
.$ (130)

We have introduced the notation $\nabla^{(\gamma)}_{m}$ for the Levi-Civita
connection of $\gamma_{mn}$.

Retracing all the steps outlined above, we arrive at the following formula for
the Ricci scalar of the $D$-dimensional metric $d\hat{s}^{2}_{D}$,

$\displaystyle R[\hat{g}_{\hat{\mu}\hat{\nu}}]$
$\displaystyle=(yTy)^{-b_{1}}\,R[g_{\mu\nu}]+g^{2}\,(yTy)^{-b_{1}-b_{2}}\,\bigg{[}R[\gamma_{mn}]+\mathcal{K}_{\rm(L)}\,\gamma^{mn}\,\nabla^{(\gamma)}_{m}\nabla^{(\gamma)}_{n}\log(yTy)$
$\displaystyle\qquad+\mathcal{K}_{\rm(G)}\,\gamma^{mn}\,\partial_{m}\log(yTy)\partial_{n}\log(yTy)\bigg{]}\
.$ (131)

We have introduced the constants

$\displaystyle\mathcal{K}_{\rm(L)}$ $\displaystyle=b_{1}-b_{1}D+b_{2}-b_{2}n\
,$ $\displaystyle\mathcal{K}_{\rm(G)}$
$\displaystyle=-\frac{b_{1}^{2}}{4}(D-1)(D-2)-\frac{b_{2}^{2}}{4}(n-1)(n-2)-\frac{b_{1}b_{2}}{2}\,(D-1)(n-2)\
.$ (132)

The volume form of the $D$-dimensional metric $d\hat{s}^{2}_{D}$ is written in
terms of the $d$-dimensional metric $g_{\mu\nu}$ and the $n$-dimensional
metric $\gamma_{mn}$ as

$\sqrt{-\hat{g}_{D}}=g^{-n}\,(yTy)^{b_{1}D/2+b_{2}n/2}\,\sqrt{-g_{d}}\,\sqrt{\gamma}\
.$ (133)

The $D$-dimensional Einstein-Hilbert Lagrangian
$\sqrt{-\hat{g}_{D}}R[\hat{g}_{\hat{\mu}\hat{\nu}}]$ is readily written by
combining (B.1) and (133).

We are free to add to $\sqrt{-\hat{g}_{D}}R[\hat{g}_{\hat{\mu}\hat{\nu}}]$ any
total divergence on $S^{n}$, of the form
$\sqrt{\gamma}\nabla^{(\gamma)}_{m}(\dots)^{m}$, because we are interested in
the integral of the $D$-dimensional action over $S^{n}$. If we add a term of
the form

$\sqrt{\gamma}\nabla^{(\gamma)}_{m}\Big{(}\sqrt{-g_{d}}\gamma^{mn}\nabla^{(\gamma)}_{n}(yTy)^{\frac{b_{1}D}{2}-b_{1}+\frac{b_{2}n}{2}-b_{2}}\Big{)}$
(134)

with the appropriate constant prefactor, we eliminate the Laplacian term
$\gamma^{mn}\,\nabla^{(\gamma)}_{m}\nabla^{(\gamma)}_{n}\log(yTy)$. We thus
get the simpler expression

$\displaystyle\sqrt{-\hat{g}_{D}}R[\hat{g}_{\hat{\mu}\hat{\nu}}]=g^{-n}(yTy)^{-b_{1}+b_{1}D/2+b_{2}n/2}\sqrt{-g_{d}}\,\sqrt{\gamma}R[g_{\mu\nu}]$
(135)
$\displaystyle+g^{-n+2}\,(yTy)^{-b_{1}-b_{2}+b_{1}D/2+b_{2}n/2}\sqrt{-g_{d}}\,\sqrt{\gamma}\bigg{[}R[\gamma_{mn}]+\mathcal{K}^{\prime}_{\rm(G)}\,\gamma^{mn}\,\partial_{m}\log(yTy)\partial_{n}\log(yTy)\bigg{]}\
,$

where the new constant $\mathcal{K}^{\prime}_{\rm(G)}$ is given by

$\mathcal{K}^{\prime}_{\rm(G)}=\mathcal{K}_{\rm(G)}-\mathcal{K}_{\rm(L)}\left(\frac{b_{1}D}{2}-b_{1}+\frac{b_{2}n}{2}-b_{2}\right)\
.$ (136)

In a final step, we rewrite the quantities expressed in terms of the ellipsoid
metric $\gamma_{mn}$ in terms of the round metric $\accentset{\circ}{g}_{mn}$
defined in (96). The following identities are useful,

$\displaystyle\sqrt{\gamma}=(yTy)^{1/2}\,\sqrt{\accentset{\circ}{g}}\
,\qquad\gamma^{mn}\,\partial_{m}(yTy)\partial_{n}(yTy)=4\,(yT^{3}y)-4\,\frac{(yT^{2}y)^{2}}{yTy}\
.$ (137)

We also need the expression of the Ricci scalar of the ellipsoid metric
Nastase:2000tu ,

$R[\gamma_{mn}]=\frac{({\rm Tr}\,T)^{2}}{yTy}-\frac{{\rm
Tr}\,T^{2}}{yTy}-\frac{2\,({\rm
Tr}\,T)\,(yT^{2}y)}{(yTy)^{2}}+\frac{2\,yT^{3}y}{(yTy)^{2}}\ .$ (138)

Combining all the above ingredients, we finally arrive at the result quoted in
(3.1), (56).

As explained in the main text, the values of the parameters $b_{1}$, $b_{2}$
are fixed according to (61). After that, we find it convenient to add another
total divergence on $S^{n}$. More precisely, we add a constant multiple of

$\displaystyle\sqrt{\gamma}\nabla^{(\gamma)}_{m}\Big{(}\sqrt{-g_{d}}\gamma^{mn}\nabla^{(\gamma)}_{n}(yTy)^{1/2}\Big{)}=$
$\displaystyle=2\sqrt{-g_{d}}\sqrt{\accentset{\circ}{g}}\bigg{[}\frac{(yT^{2}y)^{2}}{(yTy)^{2}}-\frac{(yT^{3}y)}{yTy}+\frac{1}{2}\,{\rm
Tr}(T^{2})-\frac{({\rm Tr}\,T)\,(yT^{2}y)}{2\,(yTy)}\bigg{]}\ .$ (139)

Adding this total divergence allows us to cancel the terms with $yT^{3}y$ and
complete a perfect square, thus arriving at the result (62) quoted in the main
text.

##### Reduction of kinetic term for the flux.

The kinetic term for $\hat{\mathcal{F}}_{n}$ is proportional to the quantity

$\sqrt{-\hat{g}_{D}}|\hat{\mathcal{F}}_{n}|^{2}=\frac{1}{n!}\sqrt{-\hat{g}_{D}}\hat{\mathcal{F}}_{\hat{\mu}_{1}\dots\hat{\mu}_{n}}\hat{\mathcal{F}}_{\hat{\nu}_{1}\dots\hat{\nu}_{n}}\hat{g}^{\hat{\mu}_{1}\hat{\nu}_{1}}\dots\hat{g}^{\hat{\mu}_{n}\hat{\nu}_{n}}\
.$ (140)

The ansatz (58) for $\hat{\mathcal{F}}_{n}$ implies that the only non-zero
components of $\hat{\mathcal{F}}_{n}$ are

$\hat{\mathcal{F}}_{p_{1}\dots
p_{n}}=f(y,T)\accentset{\circ}{\epsilon}_{p_{1}\dots p_{n}},$ (141)

where $\accentset{\circ}{\epsilon}$ is the volume form of the round metric
$\accentset{\circ}{g}_{mn}$ on $S^{n}$. In a first step, we can rewrite (140)
in terms of the $D$-dimensional metric $d\bar{s}^{2}_{D}$, generating an
overall $yTy$ prefactor,

$\sqrt{-\hat{g}_{D}}|\hat{\mathcal{F}}_{n}|^{2}=\frac{1}{n!}(yTy)^{b_{1}D/2-b_{1}n}\sqrt{-\hat{g}_{D}}\hat{\mathcal{F}}_{\hat{\mu}_{1}\dots\hat{\mu}_{n}}\hat{\mathcal{F}}_{\hat{\nu}_{1}\dots\hat{\nu}_{n}}\hat{g}^{\hat{\mu}_{1}\hat{\nu}_{1}}\dots\hat{g}^{\hat{\mu}_{n}\hat{\nu}_{n}}\
.$ (142)

Next, we use the fact that $d\bar{s}^{2}_{D}$ is a direct product metric,
$d\bar{s}^{2}_{D}=ds^{2}_{d}+d\tilde{s}^{2}_{n}$, and that
$\hat{\mathcal{F}}_{n}$ has only internal legs,

$\displaystyle\sqrt{-\hat{g}_{D}}|\hat{\mathcal{F}}_{n}|^{2}=\frac{1}{n!}(yTy)^{b_{1}D/2-b_{1}n}\sqrt{-g_{d}}\sqrt{\tilde{g}}\hat{\mathcal{F}}_{p_{1}\dots
p_{n}}\hat{\mathcal{F}}_{q_{1}\dots
q_{n}}\tilde{g}^{p_{1}q_{1}}\dots\tilde{g}^{p_{n}q_{n}}\ .$ (143)

We proceed by trading the metric $\tilde{g}_{mn}$ with the ellipsoid metric
$\gamma_{mn}$ and using (141),

$\displaystyle\sqrt{-\hat{g}_{D}}|\hat{\mathcal{F}}_{n}|^{2}=\frac{1}{n!}g^{n}(yTy)^{b_{1}D/2-b_{1}n-b_{2}n/2}\sqrt{-g_{d}}\sqrt{\gamma}f^{2}\accentset{\circ}{\epsilon}_{p_{1}\dots
p_{n}}\accentset{\circ}{\epsilon}_{q_{1}\dots
q_{n}}\gamma^{p_{1}q_{1}}\dots\gamma^{p_{n}q_{n}}\ .$ (144)

To compute the contraction of two _round_ volume forms
$\accentset{\circ}{\epsilon}$ with the _ellipsoid_ inverse metric, we use
$\det\gamma_{mn}=(yTy)\det\accentset{\circ}{g}_{mn}$ (valid for unimodular
$T$) to write

$\accentset{\circ}{\epsilon}_{p_{1}\dots
p_{n}}\accentset{\circ}{\epsilon}_{q_{1}\dots
q_{n}}\gamma^{p_{1}q_{1}}\dots\gamma^{p_{n}q_{n}}=(yTy)^{-1}{\epsilon}^{(\gamma)}_{p_{1}\dots
p_{n}}{\epsilon}^{(\gamma)}_{q_{1}\dots
q_{n}}\gamma^{p_{1}q_{1}}\dots\gamma^{p_{n}q_{n}}=n!(yTy)^{-1}\ ,$ (145)

where ${\epsilon}^{(\gamma)}$ is the volume form of the ellipsoid metric
$\gamma_{mn}$. We thus reproduce the result (64) quoted in the main text.

###  B.2 Integrals on $S^{n}$ and independence on $T$

Our goal is to determine under which conditions the following quantity has an
integral over $S^{n}$ that is independent of $T$ (provided $\det T=1$),

$h(T,y)=(yTy)^{-k/2+u}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{4-k}{8}\frac{{\rm
Tr}\,T}{yTy}\bigg{]}\ .$ (146)

We have included two independent constant parameters $k$, $u$, in order to be
able to apply the present argument to the case with a dilaton.

We consider a $T$ matrix of the form $T_{IJ}=\exp(t)_{IJ}$, with ${\rm
Tr}\,t=0$, so that $\det T=1$. We can expand $h(e^{t},y)$ in a power series
around $t=0$. The integral over $S^{n}$ can be performed order by order with
the help of identities

$\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}y^{I_{1}}\dots
y^{I_{2p}}=\frac{(2p-1)!!\,\delta^{(I_{1}I_{2}}\delta^{I_{3}I_{4}}\dots\delta^{I_{2p-1}I_{2p})}}{(n+1)(n+3)\dots(n+2p-1)}\
,\qquad p=1,2,3,\dots$ (147)

For simplicity, let us first freeze $u=0$, keeping $k$ as only free parameter.
This is enough for applications to the cases without dilaton. We find

$\displaystyle\frac{1}{\mathcal{V}_{n}}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}h(e^{t},y)=\frac{1}{8}((k-4)n+k+4)$
(148)
$\displaystyle+\frac{k\left(k^{2}(n+1)-kn(n+2)+7k+4(n-3)(n+1)\right)}{32(n+1)(n+3)}{\rm
Tr}(t^{2})$
$\displaystyle-\frac{k(k-n-1)\left(2k^{2}(n+1)+k(19-(n-2)n)+4(n-7)(n+1)\right)}{96(n+1)(n+3)(n+5)}{\rm
Tr}(t^{3})+\mathcal{O}(t^{4})\ .$

We see that the only way to ensure that the result is independent of $t$,
without imposing any restriction on $t$ (except tracelessness), is to select

$k=0\ .$ (149)

We know that (146) with $(k,u)=(0,0)$ is related to a total derivative, thanks
to the identity (67), repeated here for convenience,

$\epsilon y(dy)^{n}+d\bigg{[}\frac{n}{n-1}\frac{1}{yTy}\epsilon
y(Ty)(dy)^{n-1}\bigg{]}=-\frac{2}{n-1}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{1}{2}\frac{{\rm
Tr}\,T}{yTy}\bigg{]}\epsilon y(dy)^{n}\ .$ (150)

This relation demonstrates that

$(k,u)=(0,0)\quad\Rightarrow\quad\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}h(y,T)=-\frac{n-1}{2}\mathcal{V}_{n}\
,$ (151)

which is indeed non-zero for $n\geq 2$ and independent of $T$.

Let us now repeat the analysis keeping both $k$ and $u$ as independent
parameters, as required for applications to the cases with a dilaton. We have
studied the integral of (146) up to fourth order in $t$. By collecting the
coefficients of the ${\rm Tr}(t^{2})$, ${\rm Tr}(t^{3})$, ${\rm Tr}(t^{4})$,
and $[{\rm Tr}(t^{2})]^{2}$ structures, and setting them to zero, we obtain a
set of relations among $k$, $u$, $n$, which admit the following solutions,

$\displaystyle(k,u)=(0,0)\
,\quad(k,u)={\scriptstyle(\frac{4(n-1)}{n+1},-\frac{(n-3)(n-1)}{2(n+1)})}\
,\quad(k,u)={\scriptstyle(\frac{4(n+1)}{n+3},-\frac{(n-1)(n+1)}{2(n+3)})}\ .$
(152)

These three options are the only possible values of $(k,u)$ for which the
integral of (146) has a chance of being independent of $T$, for unimodular
$T$. We now study each case and prove that the integral has indeed this
property.

We have already encountered the case $(k,u)=0$, and we have already
demonstrated in (151) that for these values of $k$, $u$ the integral of $h$ is
indeed independent of $T$.

For the case
$(k,u)={\scriptstyle(\frac{4(n-1)}{n+1},-\frac{(n-3)(n-1)}{2(n+1)})}$ we use
the relation

$d\bigg{[}(yTy)^{-\frac{n+1}{2}}\epsilon
y(Ty)(dy)^{n-1}\bigg{]}=-\frac{n+1}{n}(yTy)^{1-\frac{n+1}{2}}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{1}{n+1}\frac{{\rm
Tr}\,T}{yTy}\bigg{]}\epsilon y(dy)^{n}\ .$ (153)

We conclude that

$(k,u)={\scriptstyle(\frac{4(n-1)}{n+1},-\frac{(n-3)(n-1)}{2(n+1)})}\quad\Rightarrow\quad\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}h(y,T)\equiv
0\ .$ (154)

While indeed independent of $T$, a flux that is identically zero is not
acceptable, because it does not reproduce the non-zero flux of the case
$T=\mathbb{I}$.

The third option in (152) requires a more careful analysis. We start from

$\displaystyle\sqrt{\accentset{\circ}{g}}\accentset{\circ}{\nabla}^{m}\accentset{\circ}{\nabla}_{m}(yTy)^{\lambda}$
$\displaystyle=\sqrt{\overline{g}}\,\Big{[}2\,\lambda\,(yTy)^{\lambda-1}\,{\rm
Tr}\,T-2\,\lambda\,(n+1)\,(yTy)^{\lambda}$
$\displaystyle+4\,\lambda\,(\lambda-1)\,(yTy)^{\lambda-2}\,(yT^{2}y)-4\,\lambda\,(\lambda-1)\,(yTy)^{\lambda}\Big{]}\
,$ (155)

where $\accentset{\circ}{\nabla}^{m}\accentset{\circ}{\nabla}_{m}$ is the
scalar Laplace operator constructed with the round metric on $S^{n}$ and
$\lambda$ is a real constant. If we specialize to $\lambda=-(n+1)/2$, we
obtain

$\sqrt{\accentset{\circ}{g}}\accentset{\circ}{\nabla}^{m}\accentset{\circ}{\nabla}_{m}(yTy)^{\lambda}=(n+1)(n+3)\sqrt{\accentset{\circ}{g}}(yTy)^{-\frac{n+1}{2}}\bigg{[}\frac{yT^{2}y}{(yTy)^{2}}-\frac{1}{n+3}\frac{{\rm
Tr}\,T}{yTy}-\frac{2}{n+3}\bigg{]}\ .$ (156)

Since the LHS is a total divergence, we conclude that

$(k,u)={\scriptstyle(\frac{4(n+1)}{n+3},-\frac{(n-1)(n+1)}{2(n+3)})}\quad\Rightarrow\quad\int_{S^{n}}\sqrt{\accentset{\circ}{g}}h(y,T)=\frac{2}{n+3}\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{-\frac{n+1}{2}}\
.$ (157)

The integral on the RHS can be evaluated as follows. We start from the
standard Gaussian integral

$\int_{\mathbb{R}^{n+1}}d^{n+1}xe^{-\frac{1}{2}xTx}=(2\pi)^{\frac{n+1}{2}}(\det
T)^{-\frac{1}{2}}\ ,$ (158)

where $x^{I}$ are Cartesian coordinates on $\mathbb{R}^{n+1}$. We introduce
polar coordinates $x^{I}=ry^{I}$, with $y^{I}y_{I}=1$. We can then write

$\displaystyle\int_{\mathbb{R}^{n+1}}d^{n+1}xe^{-\frac{1}{2}xTx}$
$\displaystyle=\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}\int_{0}^{\infty}drr^{n}e^{-\frac{1}{2}r^{2}(yTy)}$
$\displaystyle=\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}2^{\frac{n-1}{2}}(yTy)^{-\frac{n+1}{2}}\Gamma(\tfrac{n+1}{2})\
.$ (159)

Comparing (158) and (159) and recalling (9), we find

$\int_{S^{n}}d^{n}\xi\sqrt{\accentset{\circ}{g}}(yTy)^{-\frac{n+1}{2}}=\mathcal{V}_{n}(\det
T)^{-\frac{1}{2}}\ .$ (160)

Using (157) we conclude

$(k,u)={\scriptstyle(\frac{4(n+1)}{n+3},-\frac{(n-1)(n+1)}{2(n+3)})}\quad\Rightarrow\quad\int_{S^{n}}\sqrt{\accentset{\circ}{g}}h(y,T)=\frac{2}{n+3}\mathcal{V}_{n}(\det
T)^{-\frac{1}{2}}\ .$ (161)

The result is not identically zero and depends on $T$. The dependence on $T$,
however, drops out after imposing the unimodularity constraint.

To summarize: the integral of $h$ in (146) is independent of $T$ for
unimodular $T$ if and only if one of the following three cases is realized:

  * •

$(k,u)=(0,0)$: the integral of $h$ is non-zero and independent of $T$ without
using the condition $\det T=1$;

  * •

$(k,u)={\scriptstyle(\frac{4(n-1)}{n+1},-\frac{(n-3)(n-1)}{2(n+1)})}$: the
integral of $h$ is identically zero for any symmetric $T$ without using the
condition $\det T=1$;

  * •

$(k,u)={\scriptstyle(\frac{4(n+1)}{n+3},-\frac{(n-1)(n+1)}{2(n+3)})}$: the
integral of $T$ is a non-zero constant times $(\det T)^{-\frac{1}{2}}$, and
thus independent of $T$ using $\det T=1$.

###  B.3 Features of non-trivial integrals over $S^{n}$

The reduction of the Einstein-Hilbert term yields the integrand $\mathcal{I}$
in (62). The terms with a non-trivial $y$ dependence are collected in the
perfect square $[\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm Tr}\,T]^{2}$. This is a
rational function of $y$, $T$, homogeneous of degree 2 under a formal
rescaling $T_{IJ}\rightarrow\lambda T_{IJ}$. (In the present discussion, we
relax the unimodularity constraint on $T$ and study integrals of expressions
such as $\frac{yT^{2}y}{yTy}$ as functions of an arbitrary symmetric $T$.) If
we integrate $[\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm Tr}\,T]^{2}$ over $S^{n}$
with measure $\sqrt{\accentset{\circ}{g}}$, we get a a function of $T_{IJ}$
which is manifestly homogeneous of degree 2. It is not, however, a quadratic
function, nor a rational function of the entries of $T$.

This can be verified explicitly in the example $n=2$, choosing for simplicity
a diagonal $T$ of the form

$T_{IJ}={\rm diag}(t,t,s)\ ,$ (162)

where $t$, $s$ are positive parameters. The constrained coordinates $y^{I}$
may be parametrized in terms of an interval coordinate $-1\leq\mu\leq 1$ and
an angle $\phi$ with period $2\pi$,

$y^{1}=\sqrt{1-\mu^{2}}\cos\phi\ ,\qquad y^{2}=\sqrt{1-\mu^{2}}\sin\phi\
,\qquad y^{3}=\mu\ .$ (163)

The round metric on $S^{2}$ in these coordinates reads
$d\accentset{\circ}{s}^{2}=\frac{d\mu^{2}}{1-\mu^{2}}+(1-\mu^{2})d\phi^{2}$
and therefore $\sqrt{\accentset{\circ}{g}}=1$. We also check that the quantity
$[\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm Tr}\,T]^{2}$ is independent of $\phi$,
due to the fact that the first two diagonal entries of $T$ are equal. We are
thus left to compute an integration in the variable $\mu$ only. Considering
for definiteness the case $s>t$, we find141414Our findings in this explicit
example are not compatible with some of the identities quoted in
Nastase:2000tu .

$\frac{1}{2\pi}\int_{S^{2}}\sqrt{\accentset{\circ}{g}}\bigg{[}\frac{yT^{2}y}{yTy}-\frac{4-k}{8}{\rm
Tr}\,T\bigg{]}^{2}=st+\frac{((k+4)s+2kt)^{2}}{32}-\frac{s((k+2)s+2kt)\arctan\sqrt{\frac{s}{t}-1}}{2\sqrt{\frac{s}{t}-1}}\
.$ (164)

The RHS is homogeneous of degree 2 under a simultaneous rescaling
$(t,s)\rightarrow(\lambda t,\lambda s)$, but it is not a rational function of
$t$, $s$.

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