UUITP-30/21

Squashing and supersymmetry enhancement in three dimensions

Joseph Minahan, Usman Naseer and Charles Thull

Department of Physics and Astronomy, Uppsala University,
Box 516, SE-751 20 Uppsala, Sweden

{joseph.minahan, usman.naseer<EMAIL_ADDRESS>

We consider mass-deformed theories with $\mathcal{N}\geq 2$ supersymmetry on
round and squashed three-spheres. By embedding the supersymmetric backgrounds
in extended supergravity we show that at special values of mass deformations
the supersymmetry is enhanced on the squashed spheres. When the $3d$ partition
function can be obtained by a limit of a $4d$ index we also show that for
these special mass deformations only the states annihilated by extra
supercharges contribute to the index. By using an equivalence between
partition functions on squashed spheres and ellipsoids, we explain the
recently observed squashing independence of the partition function of mass-
deformed ABJM theory on the ellipsoid. We provide further examples of such
simplification for various $3d$ supersymmetric theories.

###### Contents

  1. 1 Introduction and summary
  2. 2 $\mathcal{N}=2$ supersymmetry and partition functions on deformed spheres
    1. 2.1 The round sphere and the squashed sphere
    2. 2.2 The ellipsoid and the squashed sphere
  3. 3 Symmetry enhancment from extended supergravity
  4. 4 $4d$ indices, $3d$ partition functions and symmetry enhancement
  5. A Partition functions of $3d$ $\mathcal{N}=2$ theories
    1. A.1 Another squashing independent point for ABJM
    2. A.2 The $N_{f}$ model
  6. B Twisted dimensional reduction of 4d $\mathcal{N}=2$ conformal supergravity backgrounds
  7. C Extra spinors in ABJM

## 1\. Introduction and summary

Localization is a powerful tool with a broad range of applications in
supersymmetric field theories [1]. In general one can put supersymmetric
theories on the round sphere $\mathbb{S}^{d}$ while preserving the same amount
of supersymmetry as the theory on flat Euclidean space $\mathbb{R}^{d}$.
Localization then gives the free energy of the theory in terms of finite
dimensional matrix model integrals which depend on various parameters of the
theory such as couplings $\vec{\tau}$ and masses $\vec{m}$. With enough
supersymmetry the matrix models can be very simple, e.g., the Gaussian matrix
model one finds for $\mathcal{N}=4$ super Yang-Mills [2].

It is still possible to localize the supersymmetric theory after squashing the
sphere, that is deforming the metric by a parameter $b$, where $b$=1
corresponds to the round sphere [3, 4, 5, 6]. The metric deformation breaks
the supersymmetry, but it can be partially restored by turning on appropriate
background fields in the supergravity multiplet à la [7]. The resulting matrix
models after localization are in general more complicated than for the round
sphere, but contain much more information. For example the derivatives of the
free energy ${\mathcal{F}}\left(b,\vec{\tau},\vec{m}\right)$ with respect to
the deformation parameter, couplings and masses give new constraints between
various integrated or partially integrated correlators of the supersymmetric
theory. Using holographic duality, these relations imply non-trivial
constraints on string theory amplitudes and can be used to rule out certain
higher-derivative terms in string theory effective actions [8, 9, 10, 11, 12,
13].

A convenient situation arises when the free energy is independent of the
squashing parameter. In this case one can obtain an infinite number of
relations between correlators without having to deal with the complicated
matrix models. This can happen in two distinct ways: (1) The free energy is
independent of the squashing without tuning any other parameters of the
theory. The free energies of this type are the $3d$ and $4d$ theories with
$\mathcal{N}=2$ supersymmetry placed on squashed spheres with $SU(2)\times
U(1)$ isometry, as studied in [3, 14]. (2) The free energy becomes independent
of the squashing when certain parameters are tuned to a special value. For
example, it was pointed out in [15, 16] that the free energy of $4d$
$\mathcal{N}=2$ theory on the squashed sphere with an adjoint hypermultiplet
of a special mass is independent of the squashing. This leads to an infinite
number of relations between (integrated) correlators of $\mathcal{N}=4$ SYM
[16], some of which were obtained earlier in [13].

A similar phenomenon was discovered by [17] for ABJM theory. The free energy
on the squashed sphere becomes squashing-independent if one deforms by a
special mass corresponding to a Cartan of the flavor symmetry $SO(2)\subset
SO(4)\times U(1)$. In this case the simplification appears miraculous as there
are complicated cancelations between different contributions to the partition
function which only become apparent after a non-trivial rewriting of the
partition function using the Cauchy determinant formula. We are thus led to
search for the underlying reason for this simplification. In this paper we
study and explain this simplification for supersymmetric theories on squashed
three-spheres. A companion paper [18] analyzes $\mathcal{N}=2^{*}$ theory on
general four-manifolds. The general strategy is to show that there is enhanced
supersymmetry at this special value of the mass.

Along the way we find various $3d$ theories whose free energy is squashing-
independent. The partition function can be assembled in terms of a classical
piece and one-loop determinants associated with the vector- and matter-
multiplets. From these ingredients we show that free energy of various
theories of an $\mathcal{N}=4$ vector-multiplet with an adjoint hypermultiplet
of mass111One can add supersymmetric mass terms by coupling the matter-
multiplet to a background $U(1)$ vector-multiplet. Here, by mass we mean the
constant value of the scalar field in this background multiplet.
$\mu=\pm{\mathrm{i}}{b+b^{-1}\over 2}$ is independent of $b$. This follows
directly from the cancellation between one-loop determinants. Similarly, we
show that the free energy of ABJM theory deformed by a mass
$\mu=\pm{\mathrm{i}}{b+b^{-1}\over 2}$ corresponding to a Cartan of the flavor
symmetry is squashing-independent. This, again, follows after a non-trivial
rewriting of the partition function that uses the Cauchy determinant
formula222The simplification in [17] occurs for
$\mu=\pm{\mathrm{i}}{b-b^{-1}\over 2}$..

To understand the squashing independence, our main tools of analysis are the
results of [19, 20] and the relations between $4d$ indices and $3d$ partition
functions. For the first kind of squashing independence, it is sufficient to
consider the transversely holomorphic foliation (THF) that determines the
partition functions of the supersymmetric theories. We show that this
structure is the same for the round sphere and the $SU(2)\times U(1)$-deformed
sphere of [3]. The $4d$ index which gives the relevant partition function
depends on a single fugacity. For the second kind of simplification, we show
that there is enhanced supersymmetry at the special mass values. For
$\mu=\pm{\mathrm{i}}{b+b^{-1}\over 2}$ there are four extra supercharges,
while for $\mu=\pm{\mathrm{i}}{b-b^{-1}\over 2}$ there are two extra
supercharges. The enhanced supersymmetry is found by embedding the $3d$
$\mathcal{N}=2$ supergravity in appropriate $3d$ extended supergravity. For
$\mu=\pm{\mathrm{i}}{b-b^{-1}\over 2}$, the THF associated with the extra
supercharge coincides with the THF for the round sphere. From this one deduces
that the partition function is independent of the squashing. Combining this
with the equivalence of partition functions between ellipsoids and squashed
spheres, one is led to the squashing independence in [13].

The rest of this paper is structured as follows: In section 2 we review
general features about supersymmetric partition functions on three manifolds
and analyze the THF on the round sphere, the $U(1)\times U(1)$ symmetric
ellipsoid and the $SU(2)\times U(1)$ symmetric squashed sphere. In section 3
we show that the supersymmetry is enhanced at the special values of the mass
parameters by considering a twisted dimensional reduction of $4d$
$\mathcal{N}=2$ supergravity which we discuss in appendix B. The corresponding
analysis for $3d\ \mathcal{N}=6$ conformal supergravity is presented in
appendix C which explains the symmetry enhancement for ABJM. In section 4 we
explain the squashing independence and the symmetry enhancement by studying
the $4d$ indices which give $3d$ partition functions upon compactification.
Appendix A contains details of the simplification of partition functions for
SYM and ABJM at special mass parameters.

## 2\. $\mathcal{N}=2$ supersymmetry and partition functions on deformed
spheres

In the literature two distinct deformations of the three-sphere have been
considered. The first deformation, which we refer to as the ellipsoid,
preserves only a $U(1)\times U(1)$ isometry. Supersymmetry on the ellipsoid as
well as localization of path integrals were discussed in [3, 21]. The second
deformation, which we refer to as the squashed sphere, preserves an
$SU(2)\times U(1)$ isometry. There are two distinct possibilities for how to
preserve some supercharges on this metric background [3, 4, 22] and they lead
to distinct results for the partition function which is equal to the round
sphere partition function in one case and the ellipsoid partition function in
the other case. The purpose of this section is to explain this using the
techniques and results from [19, 20].

On a Riemannian three manifold ${\mathcal{M}}$, the metric will break all
supersymmetries. To preserve some (rigid) supersymmetry one then has to turn
on additional background fields in the supergravity multiplet [7]. For new
minimal supergravity these fields are a gauge field $A$ for the $U(1)_{R}$
symmetry, a dual field strength $V$ for the graviphoton and a scalar field
$H$. Then every spinor $\zeta,\widetilde{\zeta}$ solving the Killing spinor
equations

$\begin{split}\nabla_{\mu}\zeta-{\mathrm{i}}\left(A_{\mu}+V_{\mu}\right)\zeta-H\gamma_{\mu}\zeta+\frac{1}{2}\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\zeta=0,\\\
\nabla_{\mu}\widetilde{\zeta}+{\mathrm{i}}\left(A_{\mu}+V_{\mu}\right)\widetilde{\zeta}-H\gamma_{\mu}\widetilde{\zeta}-\frac{1}{2}\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\widetilde{\zeta}=0\,,\end{split}$
(2.1)

corresponds to a supercharge preserved by the background.

Equivalent to the existence of at least one solution to the equations in (2.1)
is that the manifold ${\mathcal{M}}$ admits a transversely holomorphic
foliation (THF) compatible with the metric. Assuming that ${\mathcal{M}}$ is
oriented, such a THF is given by a nowhere vanishing, normalized vector field
$\xi$ subject to the constraint that the tensor
$\Phi^{\mu}{}_{\nu}\equiv\varepsilon^{\mu}{}_{\nu\rho}\xi^{\rho}$ satisfies
the integrability condition

$\Phi^{\mu}{}_{\rho}{\mathcal{L}}_{\xi}\Phi^{\rho}{}_{\nu}=0\,.$ (2.2)

The orbits of $\xi$ define leaves of a one-dimensional oriented foliation of
${\mathcal{M}}$ and $\Phi$ induces an almost complex structure on the normal
bundle of the foliation. Due to the integrability condition the almost complex
structure is constant along the orbits of $\xi$. As a consequence, one can
construct local coordinates $\left(\tau,z,\bar{z}\right)$ such that $\xi$ is
the vector

$\xi=\partial_{\tau}\,,$ (2.3)

and the metric can be written as

$ds^{2}=\left(\mathrm{d}\tau+h\left(\tau,z,\overline{z}\right)\mathrm{d}z+\overline{h}\left(\tau,z,\overline{z}\right)\mathrm{d}\overline{z}\right)^{2}+c\left(\tau,z,\overline{z}\right)^{2}\mathrm{d}z\mathrm{d}\overline{z}.$
(2.4)

For later reference we note that in such adapted coordinates the tensor $\Phi$
takes the nice form

$\Phi\indices{{}^{\mu}_{\nu}}=\begin{pmatrix}0&{\mathrm{i}}h&-{\mathrm{i}}\overline{h}\\\
0&-{\mathrm{i}}&0\\\ 0&0&{\mathrm{i}}\end{pmatrix}.$ (2.5)

The metric on the normal bundle of the foliation is hermitian, hence (2.4) is
referred to as the transversely hermitian metric (THM) compatible with the
foliation. Because of ambiguities, a THF with a compatible THM does not
correspond to a unique set of background fields. However, it is sufficient to
write down supersymmetric Lagrangians which, among other parameters, depend on
the THF, the THM and the aforementioned ambiguities.

Three-sphere and its deformations that we consider preserve at least two
supercharges of opposite R-charge, hence there exists a pair
$\zeta,\widetilde{\zeta}$ of Killing spinors. From these Killing spinors we
can construct the Killing vector

$K=\zeta\gamma^{\mu}\widetilde{\zeta}\partial_{\mu}.$ (2.6)

Assuming that $K$ generates a single isometry, we can write the normalized
vector field as $\xi=\Omega^{-1}K$, where $\Omega$ is the norm of $K$. We can
now define coordinates $\left(\psi,z,\overline{z}\right)$ where
$K=\partial_{\psi}$ and the metric takes the form

$\mathrm{d}s^{2}=\Omega\left(z,\overline{z}\right)^{2}\left(\mathrm{d}\psi+a\left(z,\overline{z}\right)\mathrm{d}z+\overline{a}\left(z,\overline{z}\right)\mathrm{d}\overline{z}\right)^{2}+c\left(z,\overline{z}\right)^{2}\mathrm{d}z\mathrm{d}\overline{z}.$
(2.7)

Crucially for our analysis, the authors in [20] showed that the deformations
of the THM couple to Q-exact terms in the Lagrangian. Hence, given a fixed THF
the free energy does not depend on the functions
$h\left(\tau,z,\overline{z}\right)$ and $c\left(\tau,z,\overline{z}\right)$ in
(2.4). Furthermore, [20] showed that the only deformations of the foliation
that affect the free energy are those in the components $\xi^{z}$ and
$\Phi\indices{{}^{z}_{\overline{z}}}$. This means in particular that we can
rescale $\xi$ without affecting the partition function. We now apply these
results to the sphere and its deformations. Infinitesimal deformations of the
THF away from the round sphere were already analyzed in [20].

### 2.1. The round sphere and the squashed sphere

In our first application we explain the squashing independence of the
partition function of supersymmetric theories on the $SU(2)\times U(1)$
deformed sphere of [3]. For this we start by discussing supersymmetry on a
round sphere of radius $r_{3}$. Using coordinates related to the Hopf
fibration the metric takes the form

$\mathrm{d}s^{2}_{\rm round}={r_{3}^{2}\over
4}\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right)^{2}+{r_{3}^{2}\over
4}\left(\mathrm{d}\theta^{2}+\sin^{2}{\theta}\mathrm{d}\phi^{2}\right).$ (2.8)

For convenience we choose the left-invariant frame

$\begin{split}e^{1}&=-{r_{3}\over
2}\left(\sin\psi\mathrm{d}\theta-\sin\theta\cos\psi\mathrm{d}\phi\right),\\\
e^{2}&={r_{3}\over
2}\left(\cos\psi\mathrm{d}\theta+\sin\theta\sin\psi\mathrm{d}\phi\right),\\\
e^{3}&=-{r_{3}\over
2}\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right),\end{split}$ (2.9)

in which all constant spinors $\zeta,\widetilde{\zeta}$ satisfy the Killing
spinor equations in (2.1) for the background fields

$A=V=0,\qquad H=-{{\mathrm{i}}\over 2r_{3}}.$ (2.10)

These spinors are invariant under the action of $SU(2)_{\rm L}\subset
SO(4)\equiv SU(2)_{\rm L}\times SU(2)_{R}$, and the preserved supergroup
$SU(2|1)\ltimes SU(2)_{\rm L}$. Out of the four supercharges preserved by the
background let us focus on the pair of Killing spinors

$\zeta=\left(1,0\right)^{\rm T},\qquad\widetilde{\zeta}=\left(0,1\right)^{\rm
T}$ (2.11)

For these the Killing vector defined in (2.6) is given by

$K=-{2\over r_{3}}\partial_{\psi}=\xi\,,$ (2.12)

where the second equality comes from the fact that $K$ is already normalized.
We can then write the round sphere metric in the form (2.4) by using the
adapted coordinates

$\tau=-{r_{3}\over 2}\psi,\qquad z=\cot{\theta\over 2}e^{{\mathrm{i}}\phi}.$
(2.13)

We then find that the functions in the transversaly hermitian metric are

$h=-r_{3}{{\mathrm{i}}\over 4z}{1-z\overline{z}\over 1+z\overline{z}},\qquad
c={r_{3}\over 1+z\overline{z}}.$ (2.14)

We now compare this to the squashed sphere. Using the same coordinates as in
the round case, deforming the metric on the sphere while preserving an
$SU(2)\times U(1)$ isometry boils down to rescaling one of the coefficients.
This leads to the metric

$\mathrm{d}s^{2}_{\rm squashed}={r_{3}^{2}\over 4}\left(b+b^{-1}\over
2\right)^{2}\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right)^{2}+{r_{3}^{2}\over
4}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right).$ (2.15)

We choose the frame to be proportional to the left-invariant frame of the
round sphere, such that $e^{3}$ picks up the prefactor $(b+b^{-1})/2$ but the
other components are unchanged. If we set the background fields to

$H=-{{\mathrm{i}}\over r_{3}}{1\over b+{b^{-1}}}\left(1-\left(b-b^{-1}\over
2\right)^{2}\right),\qquad V=-A={1\over
8}\left(b-b^{-1}\right)^{2}\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right)\,,$
(2.16)

we preserve the pair of constant Killing spinors in (2.11) and the full
supergroup is $SU(1|1)\ltimes SU(2)$. Thus, we preserve half the supercharges
as compared to the round sphere. The Killing vector then takes the form

$K=-{4\over(b+b^{-1})r_{3}}\partial_{\psi}\,,$ (2.17)

and we observe that the leaves of the foliation are preserved by the
squashing. In terms of the adapted coordinates of the round sphere this means
that $\xi^{z}$ still vanishes. Similarly one also finds that
$\Phi\indices{{}^{z}_{\overline{z}}}$ is unchanged by this squashing.
Consequently the free energy is $b$-independent.

### 2.2. The ellipsoid and the squashed sphere

For our second case we continue with the squashed sphere metric in (2.15). By
choosing a different set of background fields it is possible to preserve the
supergroup $SU(2|1)\ltimes U(1)_{\rm R}$. In this subsection we show that the
corresponding THF is equivalent to the one on the ellipsoid and hence the
partition functions on the two spaces coincide.

Let us consider the squashed sphere with the same frame as before, but now
with the background fields

$H=-{{\mathrm{i}}\over 4r_{3}}\left(b+b^{-1}\right),\qquad V=-A={1\over
8}\left(b+b^{-1}\right)\left(b-b^{-1}\right)\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right).$
(2.18)

Defining $e^{{\mathrm{i}}\Theta}=-b$ and mapping the sphere into $SU(2)$ by
setting

$g=\begin{pmatrix}\cos{\theta\over 2}e^{{{\mathrm{i}}\over
2}\left(\phi+\psi\right)}&\sin{\theta\over 2}e^{{{\mathrm{i}}\over
2}\left(\phi-\psi\right)}\\\ -\sin{\theta\over 2}e^{-{{\mathrm{i}}\over
2}\left(\phi-\psi\right)}&\cos{\theta\over 2}e^{{-{\mathrm{i}}\over
2}\left(\phi+\psi\right)}\end{pmatrix}\,,$ (2.19)

one finds that for any choice of constant spinors
$\zeta_{0},\widetilde{\zeta_{0}}$, the spinors

$\zeta=e^{{{\mathrm{i}}\over 2}\Theta\sigma_{3}}\cdot
g^{-1}\cdot\zeta_{0},\qquad\widetilde{\zeta}=e^{-{{\mathrm{i}}\over
2}\Theta\sigma_{3}}\cdot g^{-1}\cdot\widetilde{\zeta_{0}}$ (2.20)

satisfy the equation (2.1). Thus we have four independent conserved
supercharges giving us an $SU(2|1)\ltimes U(1)$ supergroup.

To study the THF we choose the pair of Killing spinors,

$\zeta_{0}=\sqrt{b+b^{-1}}\left(1,0\right)^{\rm
T},\qquad\widetilde{\zeta_{0}}=\sqrt{b+b^{-1}}\left(0,-1\right)^{\rm T}\,,$
(2.21)

which give the Killing vector

$K=\zeta\gamma^{\mu}\widetilde{\zeta}={2\over
r_{3}}\left(b-b^{-1}\right)\partial_{\psi}+{2\over
r_{3}}\left(b+b^{-1}\right)\partial_{\phi}\,.$ (2.22)

To put $K$ into the form in (2.3) we define new coordinates

$\widetilde{\psi}={r_{3}\over 2}\left(\psi-{b-b^{-1}\over
b+b^{-1}}\phi\right),\qquad\tau={r_{3}\over 2\left(b+b^{-1}\right)}\phi,$
(2.23)

such that $K=\partial_{\tau}$ and the metric takes the form

$\begin{split}\mathrm{d}s^{2}_{\rm squashed}=&\left(b+b^{-1}\over
2\right)^{2}\left(b+b^{-1}+\left(b-b^{-1}\right)\cos\theta\right)^{2}\left(\mathrm{d}\tau+{b-b^{-1}+\left(b+b^{-1}\right)\cos\theta\over\left(b+b^{-1}+\left(b-b^{-1}\right)\cos\theta\right)^{2}}\mathrm{d}\widetilde{\psi}\right)^{2}\\\
&+{r_{3}^{2}\over 4}\left(\mathrm{d}\theta^{2}+{4\over
r_{3}^{2}\left(b+b^{-1}+\left(b-b^{-1}\right)\cos\theta\right)^{2}}\left(b+b^{-1}\right)^{2}\sin^{2}\theta\mathrm{d}\widetilde{\psi}^{2}\right).\end{split}$
(2.24)

We next consider the ellipsoid. Given some smooth function $f(\vartheta)$
satisfying $f(0)=\ell_{2}$ and $f({\pi\over 2})=\ell_{1}$, the ellipsoid
metric is given by

$\mathrm{d}s^{2}_{\rm
ellipsoid}=f(\vartheta)^{2}\mathrm{d}\vartheta^{2}+\ell_{1}^{2}\cos^{2}\vartheta\mathrm{d}\phi_{1}^{2}+\ell_{2}^{2}\sin^{2}\vartheta\mathrm{d}\phi_{2}^{2}\,.$
(2.25)

As was noted in [21], the index computation of [22] explains in a
straightforward manner that the partition function on the ellipsoid does not
depend on the form of $f$, but only on its values at
$\vartheta=0,\frac{\pi}{2}$. Thus we are free to choose

$f(\vartheta)=\ell_{1}\sin^{2}\vartheta+\ell_{2}\cos^{2}\vartheta$ (2.26)

which differs from the choice in [3, 22] but will facilitate comparison with
the squashed sphere. In order to preserve supersymmetry on this metric
background, we also have to turn on the background fields

$H=\frac{{\mathrm{i}}}{f},\qquad A=-{1\over 2}\left(1-{\ell_{1}\over
f}\right)\mathrm{d}\phi_{1}+{1\over 2}\left(1-{\ell_{2}\over
f}\right)\mathrm{d}\phi_{2},\qquad V=0.$ (2.27)

The corresponding Killing spinors solving (2.1) then take the form

$\zeta=e^{{{\mathrm{i}}\over
2}\left(-\phi_{2}+\phi_{1}\right)}\begin{pmatrix}e^{{{\mathrm{i}}\over
2}\vartheta}\\\ e^{-{{\mathrm{i}}\over
2}\vartheta}\end{pmatrix},\qquad\widetilde{\zeta}=e^{-{{\mathrm{i}}\over
2}\left(-\phi_{2}+\phi_{1}\right)}\begin{pmatrix}e^{{{\mathrm{i}}\over
2}\vartheta}\\\ -e^{-{{\mathrm{i}}\over 2}\vartheta}\end{pmatrix}.$ (2.28)

This gives a $SU(1|1)\ltimes U(1)$ supergroup. In order to compare with the
squashed sphere we make the change of variables

$\psi=\phi_{1}+\phi_{2},\qquad\phi=\phi_{1}-\phi_{2},\qquad\theta=2\vartheta\,.$
(2.29)

The metric then takes the form

$\mathrm{d}s^{2}_{\rm ellipsoid}={1\over
4}f(\tfrac{\theta}{2})^{2}{\mathrm{d}\theta^{2}}+{1\over
4}\left(\ell_{1}^{2}\cos^{2}{\theta\over 2}+\ell_{2}^{2}\sin^{2}{\theta\over
2}\right)\left(\mathrm{d}\psi^{2}+\mathrm{d}\phi^{2}\right)+{1\over
2}\left(\ell_{1}^{2}\cos^{2}{\theta\over 2}-\ell_{2}^{2}\sin^{2}{\theta\over
2}\right)\mathrm{d}\psi\mathrm{d}\phi\,,$ (2.30)

and the Killing vector generated by the Killing spinors in (2.28) is

$K=2\left(\frac{1}{\ell_{1}}-\frac{1}{\ell_{2}}\right)\partial_{\psi}+2\left(\frac{1}{\ell_{1}}+\frac{1}{\ell_{2}}\right)\partial_{\phi}\,.$
(2.31)

This is the same Killing vector as we found in (2.22) for the squashed sphere
if we identify333With this identification $f={r_{3}\over
2}\left(b+b^{-1}+\left(b-b^{-1}\right)\cos\theta\right)$.  $\ell_{1}=r_{3}/b$
and $\ell_{2}=r_{3}b$. The normalized (w.r.t the respective metrics) vector
fields are proportional to each other and the leaves of the corresponding
foliations of $\mathbb{S}^{3}$ are the same.

Following the same procedure as for the squashed sphere we change to the
coordinates $\left(\widetilde{\psi},\theta,\tau\right)$ defined in (2.23). In
these coordinates $K=\partial_{\tau}$ and the metric takes the form

$\mathrm{d}s^{2}_{\rm
ellipsoid}=4\left(\mathrm{d}\tau+{\left(b+b^{-1}\right)\cos\theta-\left(b-b^{-1}\right)\over
4}\mathrm{d}\widetilde{\psi}\right)^{2}+{1\over
4}f^{2}\left(\mathrm{d}\theta^{2}+f^{-2}\left(b+b^{-1}\right)^{2}\sin^{2}\theta\mathrm{d}\widetilde{\psi}^{2}\right).$
(2.32)

Introducing the adapted coordinates with
$z=(1+\cos\theta)^{\frac{r_{3}b^{-1}}{2(b+b^{-1})}}(1-\cos\theta)^{-\frac{r_{3}b}{2(b+b^{-1})}}e^{{\mathrm{i}}\widetilde{\psi}}$
for this metric it takes the form (2.4). In the same coordinates the squashed
sphere metric (2.24) takes the form

$\mathrm{d}s^{2}=(g(z,\overline{z})\mathrm{d}\tau+h^{\prime}(z,\overline{z})\mathrm{d}{z}+\overline{h}^{\prime}(z,\overline{z})\mathrm{d}\overline{z})^{2}+c^{\prime}(z,\overline{z})^{2}\mathrm{d}z\mathrm{d}\overline{z}.$
(2.33)

From this we see that there is no consequential change in the THF when we go
from the ellipsoid to the squashed sphere and the two partition functions are
equal.

## 3\. Symmetry enhancment from extended supergravity

In this section, by embedding the $\mathcal{N}=2$ R-symmetry and the flavor
symmetry in the $\mathcal{N}=4$ R-symmetry we show that the special values of
the mass parameter in $3d$ correspond to the existence of extra
supersymmetries. The ideal framework for this computation is an off-shell
formulation of $3d$ $\mathcal{N}=4$ Poincaré supergravity but this is not
readily available. We proceed by dimensionally reducing the 4d $\mathcal{N}=2$
supergravity considered in [23, 24]. The dimensional reduction gives three
dimensional backgrounds to which $3d$ $\mathcal{N}=4$ SYM naturally couple.
Since ABJM theory cannot be obtained this way our arguments do not apply
directly. The correct setup to consider for ABJM is the $\mathcal{N}=6$
conformal supergravity [25] which we do in appendix C and show that the same
conclusions hold.

The details of the dimensional reduction are provided in appendix B and we
only state the main results here. The bosonic fields of the $3d$ backgrounds
are

$g_{\mu\nu}\qquad V_{\mu}\qquad A_{\mu}^{\rm R_{0}}\qquad A_{\mu}^{\rm
F}\qquad H\qquad\sigma.$ (3.1)

Here $V_{\mu}$ is the dual graviphoton field strength which appears in $3d$
via the reduction of the four-dimensional metric. The four-dimensional
$\mathcal{N}=2$ superconformal algebra has $U(1)_{\rm r}\times SU(2)_{\rm R}$
R-symmetry. The gauge fields and the scalars are the dimensional reduction of
the gauge field corresponding to the R-symmetry generated by the $U(1)_{\rm
r}\times U(1)_{\rm R}$ subgroup444We only allow the $SU(2)_{\rm R}$ symmetry
gauge field to take values in the Cartan subalgebra. It would be interesting
to study the generic dimensional reduction with arbitrary $SU(2)_{\rm R}$
gauge field as well as other fields in the $\mathcal{N}=2$ Weyl multiplet but
we do not attempt it here.. From the $3d$ $\mathcal{N}=2$ perspective,
$A_{\mu}^{\rm R_{0}}$ corresponds to the gauge field for the R-symmetry while
$A_{\mu}^{\rm F}$ and $\sigma$ are bosonic fields of a background $U(1)$
flavor multiplet.

Setting the variation of the gravitino to zero leads to the following
generalized Killing spinor equations (GKSE)

$\begin{split}&\left(\nabla_{\mu}-{\mathrm{i}}\left(A_{\mu}^{\rm
R_{0}}+\frac{1}{2}V_{\mu}\right)-H\gamma_{\mu}+\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\xi^{1}=0\\\
&\left(\nabla_{\mu}+{\mathrm{i}}\left(A_{\mu}^{\rm R_{0}}-A_{\mu}^{\rm
F}-\frac{1}{2}V_{\mu}\right)-\left(\sigma-H\right)\gamma_{\mu}+\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\xi^{2}=0\,.\end{split}$
(3.2)

The first equation is precisely the GKSE (2.1) for $3d$ $\mathcal{N}=2$
supersymmetric backgrounds555We scale the $V_{\mu}$ in (2.1) and then identify
$A_{\mu}=A_{\mu}^{\rm R_{0}}+{3\over 2}V_{\mu}$ . Requiring the variation of
the dilatino to vanish gives

$\begin{split}\delta\chi^{1}&=\left(D+\sigma H\right)\xi^{1}-{1\over
3}\left(\varepsilon^{\mu\nu\rho}\gamma_{\rho}\xi^{1}\partial_{\mu}\left(A_{\nu}^{\rm
R_{0}}-{3\over 2}A_{\nu}^{\rm
F}\right)+\gamma^{\mu}\xi^{1}\left(\partial_{\mu}+2{\mathrm{i}}V_{\mu}\right)\left(H-{3\over
2}\sigma\right)\right)=0,\\\ \delta\chi^{2}&=\left(D+\sigma
H\right)\xi^{2}+\frac{1}{3}\left(\varepsilon^{\mu\nu\rho}\gamma_{\rho}\xi^{2}\partial_{\mu}\left(A_{\nu}^{\rm
R_{0}}+\frac{1}{2}A_{\nu}^{\rm
F}\right)+\gamma^{\mu}\xi^{2}\left(\partial_{\mu}+2{\mathrm{i}}V_{\mu}\right)\left(H+\frac{1}{2}\sigma\right)\right)=0.\end{split}$
(3.3)

If $\xi^{2}=0$ then the constraints in (3.3) reproduce the conditions for a
$3d$ $\mathcal{N}=2$ supersymmetric theory coupled to a background $U(1)$
flavor multiplet. $\xi^{1}$ satisfies the Killing spinor equation for the
background fields

$A^{\rm R_{0}}=-\frac{1}{2}V=-{\left(b+b^{-1}\right)\left(b-b^{-1}\right)\over
16}\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right),\qquad
H={\mathrm{i}}{b+b^{-1}\over 4r_{3}}.$ (3.4)

It then follows that

$\varepsilon^{\mu\nu\rho}\partial_{\mu}A_{\nu}^{\rm
R_{0}}\gamma_{\rho}+2{\mathrm{i}}HV^{\mu}\gamma_{\mu}=0$ (3.5)

and the constraints in (3.3) simplify considerably. For constant $\sigma$ and
without imposing any further restrictions on $\xi^{1,2}$, (3.3) is satisfied
for

$D=-\sigma H,\qquad A^{\rm F}={2{\mathrm{i}}\sigma r_{3}\over b+b^{-1}}V.$
(3.6)

For $\sigma=2H$, the second GKSE is also satisfied for all $\xi^{2}$ of the
same form (2.20) as $\xi^{1}$, hence the amount of supersymmetry is doubled
for this choice of mass. For $\sigma={\mathrm{i}}{b-b^{-1}\over 2r_{3}}$ and
$\xi^{2}=\left(1,0\right)^{\rm T}$ the second GKSE becomes

$\nabla_{\mu}\xi^{2}+{\mathrm{i}}\left(b-b^{-1}\over
b+b^{-1}\right)V_{\mu}\xi^{2}+{{\mathrm{i}}\over 4}{b+b^{-1}\over
r_{3}}\xi^{2}=0,$ (3.7)

which is indeed satisfied. $\xi^{2}=\left(0,1\right)^{\rm T}$ solves the GKSE
for $\sigma=-{\mathrm{i}}{b-b^{-1}\over 2r_{3}}$. So, in this case we see that
we have two more supercharges. Note that these are exactly the supercharges
preserved in the familiar squashing discussed in subsection 2.1.

## 4\. $4d$ indices, $3d$ partition functions and symmetry enhancement

In this section we study the relation between partition functions on the
three-dimensional squashed sphere and dimensional reduction of the four-
dimensional indices [26, 27, 28, 29]. We first derive relations with the
$\mathcal{N}=1$ and $\mathcal{N}=2$ indices. We then show how various special
values of the mass parameter in the $3d$ theory correspond to limits of the
index where only states annihilated by extra supercharges contribute.

Consider a theory with $\mathcal{N}=1$ superconformal symmetry placed on
$\mathbb{S}^{1}\times\mathbb{S}^{3}$, with radii $r_{1}$ and $r_{3}$,
respectively. The $\mathcal{N}=1$ superconformal group $SU(2,2|1)$ has bosonic
generators $P_{\mu},j_{1},j_{2},j_{1}^{\pm},j_{2}^{\pm}$ for the $4d$ Poincaré
algebra, $E$ and $K_{\mu}$ for dilations and special conformal
transformations, and $\widetilde{r}$ for the $U(1)_{R}$ symmetry. Its
fermionic generators are the supercharges
$Q_{\alpha},\widetilde{Q}_{\dot{\alpha}}$ and the conformal supercharges
$S_{\alpha},\widetilde{S}_{\dot{\alpha}}$ for $\alpha,\dot{\alpha}=+,-$.

We can build an index using the supercharge $\mathcal{Q}\equiv Q_{-}$ and its
adjoint $\mathcal{Q}^{\dagger}\equiv S_{+}$. These satisfy the anti-
commutation relations

$\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}=E-2j_{1}-{3\over 2}\widetilde{r}\,,$
(4.1)

where the right hand side commutes with $\mathcal{Q}$ and
$\mathcal{Q}^{\dagger}$. Since $\mathcal{Q}$ has R-charge $\widetilde{r}=1$,
the combination $E-\frac{1}{2}\widetilde{r}$ also commutes with $\mathcal{Q}$
and $\mathcal{Q}^{\dagger}$. Hence, treating the $\mathbb{S}^{1}$ as Euclidean
time we can construct the index

$\Tr\left(-1\right)^{F}e^{-2\pi{r_{1}\over
r_{3}}(E-\frac{1}{2}\widetilde{r})}=\Tr(-1)^{F}\left(pq\right)^{\frac{1}{2}(E-{1\over
2}\widetilde{r})}\equiv\mathcal{I}(p,q)\,,$ (4.2)

where the trace is over the states in the Hilbert space on $\mathbb{S}^{3}$,
and the fugacities are defined as

$p=e^{-{2\pi bv}{r_{1}\over r_{3}}}\qquad q=e^{-{2\pi b^{-1}v}{r_{1}\over
r_{3}}}\,,$ (4.3)

with $v=2(b+b^{-1})^{-1}$,

The index $\mathcal{I}(p,q)$ is equal to the path integral

$\mathcal{I}(p,q)=\int D\phi
e^{-S_{\mathbb{S}^{1}\times\mathbb{S}^{3}}+\pi\frac{r_{1}}{r_{3}}\int_{\mathbb{S}^{3}}j^{0}_{\widetilde{r}}},$
(4.4)

where the path integration over the fields has periodic boundary conditions on
$\mathbb{S}^{1}$ for both fermions and bosons. $j^{0}_{\widetilde{r}}$ is the
$U(1)_{R}$ charge density, while under dimensional reduction we have that

$S_{\mathbb{S}^{1}\times\mathbb{S}^{3}}=2\pi r_{1}S_{\mathbb{S}^{3}}$ (4.5)

where $\mathbb{S}^{3}$ is the dimensionally reduced action on
$\mathbb{S}^{3}$. Hence, we see that the combination

$\widetilde{S}_{\mathbb{S}^{3}}=S_{\mathbb{S}^{3}}-\frac{1}{2}\int_{\mathbb{S}^{3}}j^{0}_{\widetilde{r}}$
(4.6)

is the three-dimensional action that is invariant under supersymmetry
transformations generated by $\mathcal{Q}$ and $\mathcal{Q}^{\dagger}$. Since
$Q_{+}$ and $S_{-}$ also commute with $E-\frac{1}{2}\widetilde{r}$,
$\widetilde{S}_{\mathbb{S}^{3}}$ is invariant under these supersymmetry
transformations as well. Hence, the theory on the round sphere generically
preserves four supersymmetries.

If the theory also has some flavor symmetries, labeled by $F_{i}$, then these
too can be inserted into the index since all $F_{i}$ commute with
$\mathcal{Q}$. These are put in by turning on some appropriate background
gauge fields along $\mathbb{S}^{1}$. Each $F_{i}$ comes with a fugacity
$t_{i}$, which are given by

$t_{i}=e^{-2\pi{\mathrm{i}}r_{1}m_{i}}$ (4.7)

where $m_{i}$ is the real mass parameter that appears in the dimensionally
reduced $3d$ theory. It is related to the dimensionless parameter $\mu_{i}$
that appears in the one-loop determinants by

$m_{i}=\frac{v\mu_{i}}{r_{3}}\,.$ (4.8)

The index then takes the form

$\mathcal{I}(p,q,t_{i})=\Tr(-1)^{F}\left(pq\right)^{\frac{1}{2}(E-{1\over
2}\widetilde{r})}\prod_{i}t_{i}^{F_{i}}\,.$ (4.9)

We will eventually be interested in a flavor symmetry that descends from
$\mathcal{N}=2$ superconformal symmetry in four dimensions. To this end we
extend the supercharges to $Q_{a\alpha}$, $\widetilde{Q}_{a\dot{\alpha}}$,
$S_{a\alpha}$, $\widetilde{S}_{a\dot{\alpha}}$, where $a=1,2$, and we identify
these with the $\mathcal{N}=1$ supercharges as $Q_{\alpha}=Q_{1\alpha}$,
$S_{\alpha}=S_{1\alpha}$,
$\widetilde{Q}_{\dot{\alpha}}=\widetilde{Q}_{2\dot{\alpha}}$,
$\widetilde{S}_{\dot{\alpha}}=\widetilde{S}_{2\dot{\alpha}}$. The
$\mathcal{N}=2$ R-symmetry is $SU(2)\times U(1)$ with generators R, $R^{\pm}$
and r. Comparing the anti-commutation relations, we see that the
$\mathcal{N}=1$ $U(1)_{R}$ generator is related to the $\mathcal{N}=2$
$SU(2)\times U(1)$ generators as $\widetilde{r}=\frac{2}{3}(r+2R)$. The
$\mathcal{N}=2$ supersymmetry generators that are not part of the
$\mathcal{N}=1$ generators can have flavor charges assigned to them which we
normalize to $\pm 1$. Hence, we have a $U(1)$ flavor symmetry in the
$\mathcal{N}=1$ theory, given by $f=r-R$.

Since $E-\frac{1}{2}\widetilde{r}$ does not commute with the extra
supersymmetries, the $3d$ action $\widetilde{S}_{\mathbb{S}^{3}}$ is not
invariant under their transformations. However, we can improve this by using
the flavor symmetry. In particular, if we instead consider the combination
$E-\frac{1}{2}\widetilde{r}+\frac{1}{3}f=E-R$, then this also commutes with
$\widetilde{Q}_{1\dot{\alpha}}$ and $\widetilde{S}_{1\dot{\alpha}}$. Hence
turning on a Wilson line along the $\mathbb{S}^{1}$ coupled to the appropriate
charges enhances the number of supersymmetries on $\mathbb{S}^{3}$ to eight.
The flavor symmetry also shifts the $3d$ $U(1)$ to
$r_{0}=\widetilde{r}-\frac{2}{3}f=2R$. With the shift $Q_{1\alpha}$ and
$\widetilde{Q}_{1\dot{\alpha}}$ both have $r_{0}=1$ while $S_{2\alpha}$ and
$\widetilde{S}_{2\dot{\alpha}}$ both have $r_{0}=-1$. A summary of the various
charges for the different supercharges are shown in table 1666The table is an
amended version of table 1 in [30]..

${\mathcal{Q}}$ | $SU(2)_{1}$ | $SU(2)_{2}$ | $SU(2)_{R}$ | $U(1)_{r}$ | $r_{0}$ | $f$
---|---|---|---|---|---|---
${\mathcal{Q}}_{{1}-}$ | $-\frac{1}{2}$ | $0$ | $\;\;\;\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$
${\mathcal{Q}}_{{1}+}$ | $\;\;\frac{1}{2}$ | $0$ | $\;\;\;\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$
${\mathcal{Q}}_{{2}-}$ | $-\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-1$ | $1$
${\mathcal{Q}}_{{2}+}$ | $\;\;\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-1$ | $1$
$\widetilde{\mathcal{Q}}_{{1}\dot{-}}$ | $0$ | $-\frac{1}{2}$ | $\;\;\;\frac{1}{2}$ | $-\frac{1}{2}$ | $1$ | $-1$
$\widetilde{\mathcal{Q}}_{{1}\dot{+}}$ | $0$ | $\;\;\;\frac{1}{2}$ | $\;\;\;\frac{1}{2}$ | $-\frac{1}{2}$ | $1$ | $-1$
$\widetilde{\mathcal{Q}}_{{2}\dot{-}}$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $0$
$\widetilde{\mathcal{Q}}_{{2}\dot{+}}$ | $0$ | $\;\;\;\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $0$
Table 1:  For each supercharge ${\cal Q}$ we list its quantum numbers. Here
$a=1,2$ are the $SU(2)_{R}$ indices and $\alpha=\pm$, $\dot{\alpha}=\pm$ are
the Lorentz indices. $E$ is the conformal dimension, $(j_{1},j_{2})$ the
Cartan generators of the $SU(2)_{1}\otimes SU(2)_{2}$ isometry group,
$(R\,,r)$, the Cartan generators of the $SU(2)_{R}\otimes U(1)_{r}$ R-symmetry
group, $r_{0}$ is the $3d$ R-charge and $f$ is the flavor charge.

Next consider squashing the $\mathbb{S}^{3}$ such that it preserves an
$SU(2)\times U(1)$ isometry. The charge $j_{2}$ rotates the angle $\psi$ which
parameterizes the Hopf fiber over the $\mathbb{S}^{2}$. We can then obtain the
$SU(2)\times U(1)$ squashed metric by twisted dimensional reduction, where the
rotation on the fiber as one circles the $\mathbb{S}^{1}$ is given by

$\left(x^{4},\psi\right)\sim\left(x^{4}+2\pi
r_{1},\psi+4\pi{\mathrm{i}}{b-b^{-1}\over b+b^{-1}}{r_{1}\over r_{3}}\right).$
(4.10)

The effect of the twisting is to insert the factor

$\exp\left(-4\pi{b-b^{-1}\over b+b^{-1}}{r_{1}\over
r_{3}}j_{2}\right)=\left(p\over q\right)^{j_{2}}$ (4.11)

into the index, giving us

$\mathcal{I}_{SU(2)\times U(1)}(p,q)=\Tr(-1)^{F}p^{\frac{1}{2}(E-{1\over
2}\widetilde{r}+{1\over 3}f+2j_{2})}q^{\frac{1}{2}(E-{1\over
2}\widetilde{r}+{1\over 3}f-2j_{2})}\,.$ (4.12)

This index is well-defined since $j_{2}$ commutes with $\mathcal{Q}$. It also
preserves the supersymmetry generated by $Q_{1+}$, but breaks the
$\widetilde{Q}_{1\dot{\alpha}}$ supersymmetries. Hence, four supersymmetries
are preserved in general. In fact, without breaking any further
supersymmetries we can combine this with a flavor term to give

$\mathcal{I}_{SU(2)\times U(1)}(p,q,t)=\Tr(-1)^{F}p^{\frac{1}{2}(E-{1\over
2}\widetilde{r}+{1\over 3}f+2j_{2})}q^{\frac{1}{2}(E-{1\over
2}\widetilde{r}+{1\over 3}f-2j_{2})}t^{f}$ (4.13)

where $t=e^{-2\pi{\mathrm{i}}r_{1}m}$ and $m$ has the same relation to the
dimensionless parameter $\mu$ as in (4.8). This last insertion adds a mass-
term to the $3d$ action.

We now show that for special values of $\mu$ the index simplifies. For

$\mu=-\frac{{\mathrm{i}}}{2}(b-b^{-1})\,,$ (4.14)

the index becomes

$\mathcal{I}_{SU(2)\times U(1)}(p,q)=\Tr(-1)^{F}p^{\frac{1}{2}(E-{1\over
2}\widetilde{r}+{1\over 3}f+2j_{2}+f)}q^{\frac{1}{2}(E-{1\over
2}\widetilde{r}+{1\over 3}f-2j_{2}-f)}\,.$ (4.15)

Examining the charges in table 1, we see that $\widetilde{Q}_{1\dot{+}}$
commutes with $2j_{2}+f$. Hence, with this choice of mass two extra
supersymmetries are preserved, giving six altogether. Moreover, the exponent
of $p$ equals
$\\{\widetilde{Q}_{1\dot{+}},\widetilde{Q}^{\dagger}_{1\dot{+}}\\}$ so we can
also rewrite the index in the form

$\displaystyle\mathcal{I}_{SU(2)\times
U(1)}(p,q)=\Tr(-1)^{F}q^{\frac{1}{2}(E-{1\over 2}\widetilde{r}+{1\over
3}f-2j_{2}-f)}.$ (4.16)

This makes the squashing independence explicit as all $b$ dependence can be
absorbed into a rescaling of the $\mathbb{S}^{3}$ radius $r_{3}$. For
$\mu={\mathrm{i}}{b-b^{-1}\over 2}$ supersymmetries associated with
$\widetilde{Q}_{1\dot{-}}$ are preserved. This is consistent with the analysis
in section 3.

The other point we are interested in is

$\mu={\mathrm{i}}\frac{b+b^{-1}}{2}$ (4.17)

where the index takes the form

$\displaystyle\mathcal{I}_{SU(2)\times U(1)}(p,q)$
$\displaystyle=\Tr(-1)^{F}p^{\frac{1}{2}(E-{1\over 2}\widetilde{r}+{1\over
3}f+2j_{2}-f)}q^{\frac{1}{2}(E-{1\over 2}\widetilde{r}+{1\over 3}f-2j_{2}-f)}$
(4.18)
$\displaystyle=\Tr(-1)^{F}(pq)^{\frac{1}{2}(E-r)}\left(\frac{p}{q}\right)^{j_{2}}.$
(4.19)

The exponents in this index commute with all supercharges without tildes,
giving a total of eight preserved charges. Contributions to the index then
come from multiplets satisfying $E=r$, meaning that we can simplify the index
to

$\mathcal{I}_{SU(2)\times
U(1)}(p,q)=\Tr(-1)^{F}\left(\frac{p}{q}\right)^{j_{2}}.$ (4.20)

From here we see the squashing independence of the partition function as again
$b$ can be absorbed by rescaling $r_{3}$.

In a similar fashion we can construct an index that reduces to the familiar
squashing case [3]. Taking the twisted compactification from before, we would
like the index to be of the form

$\displaystyle\Tr(-1)^{F}e^{-2\pi\frac{r_{1}}{r_{3}}(E+2iuj_{2}+\mathcal{O})}$
(4.21)

where we have defined $u=-{\mathrm{i}}\frac{b-b^{-1}}{b+b^{-1}}$. The operator
$\mathcal{O}$ is chosen such that the index will be with respect to the
supercharge ${\widetilde{\mathcal{Q}}}_{2\dot{-}}$. Given that this
supercharge commutes with $E-2j_{2}+2R+r$ and $E+j_{2}$, it is easy to see
that $\mathcal{O}=\frac{1}{3}(1-2iu)(2R+r)$. Then writing again an index in
terms of $p$ and $q$ we find

$\displaystyle\mathcal{I}_{\mathrm{HH}}$
$\displaystyle=\Tr(-1)^{F}p^{\frac{1}{2}(E+2j_{2}-\frac{2}{3}R-\frac{1}{3}r)}q^{\frac{1}{2}(E-2j_{2}+2R+r)}$
(4.22)
$\displaystyle=\Tr(-1)^{F}p^{\frac{1}{2}(E+2j_{2}-\frac{2}{3}R-\frac{1}{3}r)}.$
(4.23)

In going to the second line we used that the exponent of $q$ is equal to
$\\{{\widetilde{\mathcal{Q}}}_{2\dot{-}},{\widetilde{\mathcal{Q}}}_{2\dot{-}}^{\dagger}\\}$.
The last line shows that the dependence on $b$ of this index is trivial in the
sense that it can be absorbed into a rescaling of the $\mathbb{S}^{3}$ radius
$r_{3}$. After dimensional reduction this translates into the squashing
independence we discussed in section 2.1.

## Acknowledgements

We thank Shai Chester and Guido Festuccia for helpful comments on the
manuscript. This research is supported in part by Vetenskapsrådet under grants
#2016-03503, #2018-05572 and #2020-03339, and by the Knut and Alice Wallenberg
Foundation under grant Dnr KAW 2015.0083.

##  Appendix A Partition functions of $3d$ $\mathcal{N}=2$ theories

Here we summarize the contributions of different multiplets to the partition
function and resulting simplifications for special values of mass parameters.

The general form of the partition function is

${\cal Z}=\int\mathrm{d}^{r_{G}}\sigma e^{-\rm
S_{0}}\left(\prod_{\alpha\in\Delta_{+}}\langle\alpha,\sigma\rangle^{2}\right)Z_{\rm
1-loop}\left(b,\sigma\right),$ (A.1)

where $r_{G}$ is the rank of the gauge group, ${\rm S_{0}}$ is the action
evaluated at the localization locus. Only the Chern-Simons terms and the FI
terms contribute to ${\rm S_{0}}$. The product is over the positive roots of
the Lie algebra. $Z_{\rm 1-loop}$ is the one-loop determinant which is a
product of one-loop determinants for an $\mathcal{N}=2$ vector multiplet and
various chiral multiplets depending on the matter content of the theory. The
basic one-loop determinants are for the vector multiplet

$Z_{\mathcal{N}=2}^{\rm vec}=\prod_{\alpha\in\Delta_{+}}{\sinh\left(\pi
b\langle\sigma,\alpha\rangle\right)\sinh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\over\langle\sigma,\alpha\rangle^{2}},$
(A.2)

and for an $\mathcal{N}=2$ chiral multiplet with R-charge $q$ and mass $\mu$
in any representation of the gauge group

$Z_{\mathcal{N}=2}^{\rm ch}(q,\mu)=\prod_{\rho\in
R}s_{b}\left({{\mathrm{i}}Q\over
2}\left(1-q\right)-q\mu-\langle\sigma,\rho\rangle\right),$ (A.3)

where $s_{b}(x)$ is defined as the (regularized) infinite product

$s_{b}(x)=\prod_{m,n\geq
0}{(m+\tfrac{1}{2})b+(n+\tfrac{1}{2})b^{-1}-{\mathrm{i}}x\over(m+\tfrac{1}{2})b+(n+\tfrac{1}{2})b^{-1}+{\mathrm{i}}x}.$
(A.4)

Using these formulae it is easy to confirm that for a chiral in the adjoint
representation

$\begin{split}Z_{\mathcal{N}=2}^{\rm
ch}\left(\frac{1}{2},-{\mathrm{i}}{b+b^{-1}\over
2}\right)&=\prod_{\alpha\in\Delta_{+}}{1\over\sinh\left(\pi
b\langle\sigma,\alpha\rangle\right)\sinh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)},\\\ Z_{\mathcal{N}=2}^{\rm
ch}\left(\frac{1}{2},+{\mathrm{i}}{b+b^{-1}\over
2}\right)&=Z_{\mathcal{N}=2}^{\rm ch}\left(1,0\right)=1,\\\
Z_{\mathcal{N}=2}^{\rm ch}\left(1,\pm{\mathrm{i}}{b+b^{-1}\over
2}\right)&=\prod_{\alpha\in\Delta_{+}}\left({\sinh\left(\pi
b\langle\sigma,\alpha\rangle\right)\sinh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)}\right)^{\pm 1},\\\
Z_{\mathcal{N}=2}^{\rm ch}\left(1,\pm{\mathrm{i}}{b-b^{-1}\over
2}\right)&=\prod_{\alpha\in\Delta_{+}}\left({\sinh\pi
b\langle\sigma,\alpha\rangle\over b^{2}\sinh\pi
b^{-1}\langle\sigma,\alpha\rangle}\right)^{\pm 1}.\end{split}$ (A.5)

The $\mathcal{N}=4$ vector multiplet consists of an $\mathcal{N}=2$ vector
multiplet and an approprited adjoint $\mathcal{N}=2$ chiral multiplet with
R-charge $1$. The squashing dependence of the $\mathcal{N}=4$ vector multiplet
is trivial if we take the chiral multiplet to have mass
$\pm{\mathrm{i}}{b-b^{-1}\over 2}$. With the massless chiral multiplet the
$\mathcal{N}=4$ vector multiplet’s one-loop determinant concides with that of
the $\mathcal{N}=2$ vector multiplet.

The one-loop determinant for the $\mathcal{N}=4$ vector multiplet coincides
with that of the $\mathcal{N}=2$ vector multiplet. The $\mathcal{N}=4$
hypermultiplet consists of two $\mathcal{N}=2$ chiral multiplet with R-charge
$\frac{1}{2}$ and opposite gauge charges. At special values of masses their
one-loop determinants become

$\begin{split}Z_{\mathcal{N}=4}^{\rm
hyp}\left(\mu=\pm{\mathrm{i}}{b+b^{-1}\over
2}\right)&=\prod_{\alpha\in\Delta_{+}}{1\over\sinh\left(\pi
b\langle\sigma,\alpha\rangle\right)\sinh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)},\\\ Z_{\mathcal{N}=4}^{\rm
hyp}\left(\mu=\pm{\mathrm{i}}{b-b^{-1}\over
2}\right)&=\prod_{\alpha\in\Delta_{+}}{1\over\cosh\left(\pi
b\langle\sigma,\alpha\rangle\right)\cosh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)}.\end{split}$ (A.6)

With these ingredients it is now easy to put the various pieces together and
obtain squashing independent partition functions as well as partition
functions with very simple squashing dependence.

### A.1. Another squashing independent point for ABJM

In this section of the appendix we show squashing independence for the mass
deformed ABJM theory at a special mass given below. The partition function of
mass-deformed ABJM theory on the squashed three-sphere is given by

$\begin{split}&Z(b;m_{1},m_{2},m_{3})=e^{\frac{{\mathrm{i}}\pi}{12k}(b-b^{-1})M(M^{2}-1)}\int\frac{d^{N+M}\mu
d^{N}\nu}{N!(N+M)!}e^{{\mathrm{i}}\pi
k\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)}\\\
&\times\prod_{a>b}4\sinh\left(\pi b(\mu_{a}-\mu_{b})\right)\sinh\left(\pi
b^{-1}(\mu_{a}-\mu_{b})\right)\prod_{i>j}4\sinh\left(\pi
b(\nu_{i}-\nu_{j})\right)\sinh\left(\pi b^{-1}(\nu_{i}-\nu_{j})\right)\\\
&\times\prod_{i,a}\left\\{s_{b}\left[\frac{{\mathrm{i}}Q}{4}-\left(\mu_{a}-\nu_{i}+\frac{m_{1}+m_{2}+m_{3}}{2}\right)\right]s_{b}\left[\frac{{\mathrm{i}}Q}{4}-\left(\mu_{a}-\nu_{i}+\frac{m_{1}-m_{2}-m_{3}}{2}\right)\right]\right.\\\
&\times
s_{b}\left[\frac{{\mathrm{i}}Q}{4}-\left(-\mu_{a}+\nu_{i}+\frac{-m_{1}-m_{2}+m_{3}}{2}\right)\right]\left.s_{b}\left[\frac{{\mathrm{i}}Q}{4}-\left(-\mu_{a}+\nu_{i}+\frac{-m_{1}+m_{2}-m_{3}}{2}\right)\right]\right\\}\end{split}$
(A.7)

In the following we will show that this is squashing independent for
$m_{3}={\mathrm{i}}\frac{Q}{2}$ and $m_{1}=m_{2}$. Thus we will look at
$\mathcal{Z}=Z(b;\frac{m_{+}-m_{-}}{2},\frac{m_{+}+m_{-}}{2},{\mathrm{i}}\frac{Q}{2})$
where $m_{\pm}=m_{1}\pm m_{2}$.

With the definition eq. A.4 and the identity

$\sinh(z)=z\prod_{k\geq 1}\left(1+\frac{z^{2}}{\pi^{2}k^{2}}\right),$ (A.8)

it is easy to see that, after rescaling $(\mu,\nu)\rightarrow(\mu,\nu)/(2\pi)$
and putting some (divergent) constant factors into $\mathcal{N}_{1}$, our
partition function of interest simplifies to

$\begin{split}\mathcal{Z}&=\mathcal{N}_{1}e^{\frac{{\mathrm{i}}\pi}{12k}(b-b^{-1})M(M^{2}-1)}\\\
&\times\int d^{N+M}\mu d^{N}\nu e^{{\mathrm{i}}\pi
k\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)}\prod_{a>b}4\sinh\left(b\frac{\mu_{a}-\mu_{b}}{2}\right)\sinh\left(b^{-1}\frac{\mu_{a}-\mu_{b}}{2}\right)\\\
&\times\prod_{i>j}4\sinh\left(b\frac{\nu_{i}-\nu_{j}}{2}\right)\sinh\left(b^{-1}\frac{\nu_{i}-\nu_{j}}{2}\right)\prod_{i,a}\frac{1}{\sinh\left(b^{-1}\frac{\mu_{a}-\nu_{i}-\pi
m_{-}}{2}\right)\sinh\left(b\frac{\mu_{a}-\nu_{i}-\pi
m_{-}}{2}\right)}.\end{split}$ (A.9)

Applying the Cauchy determinant formula, this can be rewritten as

$\begin{split}\mathcal{Z}=\mathcal{N}_{1}&e^{\frac{{\mathrm{i}}\pi}{12k}(b-b^{-1})M(M^{2}-1)}\int
d^{N+M}\mu d^{N}\nu\
e^{\frac{{\mathrm{i}}k}{4\pi}\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)-\frac{M}{2}\sum_{r}(Q\mu_{r}-\pi
Qm_{-})+\frac{Q}{2}\sum_{s}Q\nu_{s}}\\\
&\times\det\left(\theta_{N,l}\frac{1}{2\sinh\frac{b^{-1}(\mu_{j}-\nu_{l}-\pi
m_{-})}{2}}+\theta_{l,N+1}e^{(M+N-l+\frac{1}{2})b^{-1}(\mu_{j}-\pi
m_{-})}\right)\\\
&\times\det\left(\theta_{N,l}\frac{1}{2\sinh\frac{b(\mu_{j}-\nu_{l}-\pi
m_{-})}{2}}+\theta_{l,N+1}e^{(M+N-l+\frac{1}{2})b(\mu_{j}-\pi
m_{-})}\right).\end{split}$ (A.10)

By expanding the determinants and reordering integration variables to get rid
of one summation, this can be simplified to

$\begin{split}\mathcal{Z}=\mathcal{N}_{2}\sum_{\sigma\in
S_{M+N}}(-1)^{\sigma}\int&d^{N+M}\mu d^{N}\nu\
e^{\frac{{\mathrm{i}}k}{4\pi}\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)-\frac{M}{2}\sum_{r}(Q\mu_{r}-\pi
Qm_{-})+\frac{Q}{2}\sum_{s}Q\nu_{s}+\frac{{\mathrm{i}}\pi}{12k}(b-b^{-1})M(M^{2}-1)}\\\
&\times{\prod_{j=N+1}^{N+M}e^{(M+N-l+\frac{1}{2})b^{-1}(\mu_{j}-\pi
m_{-})}e^{(M+N-l+\frac{1}{2})b(\mu_{\sigma}(j)-\pi
m_{-})}\over\prod_{j=1}^{N}{4\sinh\frac{b^{-1}(\mu_{j}-\nu_{j}-\pi
m_{-})}{2}\sinh\frac{b(\mu_{\sigma}(j)-\nu_{j}-\pi m_{-})}{2}}}\end{split},$
(A.11)

where we put some more irrelevant factors into the new normalization factor
$\mathcal{N}_{2}$. Fourier transforming the hyperbolic functions we get

$\begin{split}&\mathcal{Z}=\mathcal{N}_{3}\int d^{N+M}\mu d^{N}\nu
d^{N}pd^{M+N}q\
e^{\frac{{\mathrm{i}}k}{4\pi}\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)-\frac{M}{2}\sum_{r}(Q\mu_{r}-\pi
Qm_{-})+\frac{Q}{2}\sum_{s}Q\nu_{s}+\frac{{\mathrm{i}}\pi}{12k}(b-b^{-1})M(M^{2}-1)}\\\
&\times\prod_{j=1}^{N}\tanh\left(\frac{p_{j}}{2}\right)e^{\frac{ip_{j}}{2\pi
b}(\mu_{j}-\nu_{j}-\pi
m_{-})}\prod_{j=N+1}^{N+M}e^{(M+N-l+\frac{1}{2})b^{-1}(\mu_{j}-\pi
m_{-})}\prod_{s=1}^{N}\tanh\left(\frac{q_{s}}{2}\right)e^{-\frac{iq_{s}b}{2\pi}(\nu_{s}+\pi
m_{-})}\\\
&\times\prod_{s=N+1}^{N+M}\delta\left(2\pi{\mathrm{i}}(M+N-s+\frac{1}{2})+q_{s}\right)e^{-\frac{{\mathrm{i}}q_{r}bm_{-}}{2}}\times\sum_{\sigma\in
S_{N+M}}(-1)^{\sigma}\prod_{t=1}^{N+M}e^{\frac{iq_{t}b\mu_{\sigma(t)}}{2\pi}}\end{split}.$
(A.12)

Now it is straightforward to do the $\mu$ and $\nu$ integrals as they have
become Gaussian. Doing in addition some massaging of the expression, we can
write it as

$\begin{split}\mathcal{Z}=&\mathcal{N}_{4}e^{\frac{MN}{2}Q\pi
m_{-}}e^{\frac{{\mathrm{i}}\pi}{6k}M(2M^{2}+1)}\\\ &\times\sum_{\sigma\in
S_{N+M}}(-1)^{\sigma}\int
d^{N}pd^{M+N}q\prod_{j=1}^{N}\tanh\left(\frac{p_{j}}{2}\right)\tanh\left(\frac{q_{s}}{2}\right)e^{-\frac{im_{-}}{2}\left(\frac{p_{j}}{b}+q_{j}b\right)}e^{\frac{i}{2\pi
k}p_{j}(q_{\sigma(j)}-q_{j})}\\\
&\prod_{j=N+1}^{N+M}e^{\frac{1}{k}(M+N-j+\frac{1}{2})q_{\sigma(j)}}\delta\left(2\pi{\mathrm{i}}(M+N-j+\frac{1}{2})+q_{j}\right)\end{split}.$
(A.13)

Looking at this expression it is clear that the condition $m_{-}=0$ is
sufficient to guarantee squashing independence. Unfortunately we have not yet
managed to show that this is also a necessary condition.

### A.2. The $N_{f}$ model

Finally let us discuss the simplification of the partition function for the
so-called $N_{f}$ model which consists of an $\mathcal{N}=4$ vector multiplet,
an adjoint hypermultiplet and $N_{f}$ fundamental hypermultiplets with $U(N)$
gauge group [31]777We thank Shai Chester for suggesting this model to us.. The
theory has an $SU(2)\times SU(N_{f})$ flavor symmetry which can be deformed by
turning on the hypermultiplet mass parameters $m_{\rm adj}$ and $\vec{m}$ in
the Cartan of $SU(2)$ and $SU(N_{f})$ respectivley. The full one-loop
determinant depends on $\left(b,m_{\rm adj},m_{i}\right)$ where
$m_{i}=\lambda_{i}\left(m\right)$ and $\lambda_{i}$ is the $i$-th weight of
the fundamental representation of $SU(N_{f})$. We take $N_{f}$ to be even and
choose the masses to satisfy

$m_{2i-1}=-m_{2i}\qquad{\textrm{f}or}\qquad i=1,2,\cdots{N_{f}\over 2}.$
(A.14)

Taking $m_{2i}=m_{f}$ for all $i$, the one-loop determinants simplify
considerably for certain choices of $m_{\rm adj}$ and $m_{f}$. The simplified
partition functions we find are

$\begin{split}Z\left(b,\pm{\mathrm{i}}{b+b^{-1}\over
2},m_{f}=\pm{b+b^{-1}\over 2}\right)&=\prod_{\alpha\in F}\left(\sinh\left(\pi
b\langle\sigma,\alpha\rangle\right)\sinh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\right)^{-{N_{\rm f}\over 2}},\\\
Z\left(b,\pm{\mathrm{i}}{b+b^{-1}\over 2},m_{f}=\pm{b-b^{-1}\over
2}\right)&=\prod_{\alpha\in F}\left(\cosh\left(\pi
b\langle\sigma,\alpha\rangle\right)\cosh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\right)^{-{N_{\rm f}\over 2}},\\\
Z\left(b,\pm{\mathrm{i}}{b-b^{-1}\over 2},m_{f}=\pm{b+b^{-1}\over
2}\right)&={\prod_{\alpha\in\Delta_{+}}\tanh\left(\pi
b\langle\sigma,\alpha\rangle\right)\tanh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\over\prod_{\alpha\in
F}\left(\sinh\left(\pi b\langle\sigma,\alpha\rangle\right)\sinh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\right)^{{N_{\rm f}\over 2}}},\\\
Z\left(b,\pm{\mathrm{i}}{b-b^{-1}\over 2},m_{f}=\pm{b-b^{-1}\over
2}\right)&={\prod_{\alpha\in\Delta_{+}}\tanh\left(\pi
b\langle\sigma,\alpha\rangle\right)\tanh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\over\prod_{\alpha\in
F}\left(\cosh\left(\pi b\langle\sigma,\alpha\rangle\right)\cosh\left(\pi
b^{-1}\langle\sigma,\alpha\rangle\right)\right)^{{N_{\rm f}\over
2}}}.\end{split}$ (A.15)

In particular, the last choice of mass-deformations on the squashed sphere
gives a partition function which is a simple generalization of the round
sphere partition function with massless matter [32].

##  Appendix B Twisted dimensional reduction of 4d $\mathcal{N}=2$ conformal
supergravity backgrounds

In this appendix we perform a twisted dimensional reduction of four-
dimensional supersymmetric backgrounds that arise by taking the rigid limit of
the $\mathcal{N}=2$ conformal supergravity. The resulting three-dimensional
backgrounds can then be coupled naturally to three-dimensional $\mathcal{N}=4$
SYM. We use the three-dimensional backgrounds arising in this twisted
dimensional reduction to show that at special values of the chiral multiplet
masses on the squashed sphere enhanced supersymmetry is responsible for
squashing independence of the free energy.

The field content of the $4d$ $\mathcal{N}=2$ Weyl multiplet is [33]

$g_{MN}\qquad b_{M}\qquad A_{M}^{r}\qquad
V_{M}^{i}{}_{j}\qquad\psi_{M}^{i}\qquad B_{MN}\qquad\chi^{i}\qquad D_{\rm
4d}.$ (B.1)

The first five fields correspond to the gauge fields for the translations,
dilations, $U(1)_{r}\times SU(2)_{R}$ R-symmetry and the supersymmetry which
resides in the $\mathcal{N}=2$ superconformal algebra. The scalar $D_{\rm
4d}$, the two-form $B_{MN}$ and the dilatino $\chi^{i}$ are the auxliary
fields. The $\mathcal{N}=2$ SCA also contains Lorentz symmetry, conformal
supersymmetry and special conformal transformations whose gauge fields are not
independent and are determined in terms of the fields given above. We restrict
to supersymmetric backgrounds with vanishing $b_{M}$, $B_{MN}$ and

$V_{M}^{i}{}_{j}={{\mathrm{i}}\over 2}\sigma_{3}^{i}{}_{j}A_{M}^{R}.$ (B.2)

The variation of gravitino and dilatino is then given by

$\begin{split}\delta\psi_{M}^{1}&=\nabla_{M}\xi^{1}+{{\mathrm{i}}\over
2}\left(A_{M}^{r}-A_{M}^{R}\right)\xi^{1}-\gamma_{M}\eta^{1},\\\
\delta\psi_{M}^{2}&=\nabla_{M}\xi^{2}+{{\mathrm{i}}\over
2}\left(A_{M}^{r}+A_{M}^{R}\right)\xi^{2}-\gamma_{M}\eta^{2},\\\
\delta\chi^{1}&=D_{\rm 4d}\xi^{1}-{{\mathrm{i}}\over
3}\gamma^{MN}\xi^{1}\partial_{M}\left(\frac{1}{2}A_{N}^{R}+A_{N}^{r}\right),\\\
\delta\chi^{2}&=D_{\rm 4d}\xi^{2}-{{\mathrm{i}}\over
3}\gamma^{MN}\xi^{2}\partial_{M}\left(-\frac{1}{2}A_{N}^{\rm R}+A_{N}^{\rm
r}\right),\end{split}$ (B.3)

where $\eta^{1,2}$ are arbitrary spinors. Supersymmetric theories can be
placed on a manifold if the above variations vanish for a choice of background
fields and some non-trivial $\xi^{1,2}$. It is convenient to organize the
gauge fields $A^{\rm r},A^{\rm R}$ appearing above in terms of the gauge field
for the symmetry ${\rm R_{0}}$ which only rotates the fermions and the flavor
symmetry of an $\mathcal{N}=1$ subalgebra

$rA_{M}^{\rm r}+RA_{M}^{\rm R}={R_{0}\over 2}\left(A_{\mu}^{\rm
R}-A_{\mu}^{\rm r}\right)-FA_{\mu}^{\rm r}\equiv FA_{M}^{\rm
F}+R_{0}A_{M}^{{\rm R}_{0}}$ (B.4)

The spinor $\xi^{1}$ is not charged under the flavor symmetry while the spinor
$\xi^{2}$ is charged under both. The conditions for a supersymmetric
background can then be written as

$\begin{split}\nabla_{M}\xi^{1}-{\mathrm{i}}A_{M}^{\rm
R_{0}}\xi^{1}&=\gamma_{M}\eta^{1},\\\
\nabla_{M}\xi^{2}+{{\mathrm{i}}}\left(A_{M}^{\rm R_{0}}-A_{M}^{\rm
F}\right)\xi^{2}&=\gamma_{M}\eta^{2},\\\ D_{\rm 4d}\xi^{1}-{{\mathrm{i}}\over
3}\gamma^{MN}\xi^{1}\partial_{M}\left(A_{N}^{\rm R_{0}}-{3\over 2}A_{N}^{\rm
F}\right)&=0,\\\ D_{\rm 4d}\xi^{2}+{{\mathrm{i}}\over
3}\gamma^{MN}\xi^{2}\partial_{M}\left(A_{N}^{\rm R_{0}}+{1\over 2}A_{N}^{\rm
F}\right)&=0.\end{split}$ (B.5)

By dimensional reduction, we now obtain the equations for a supersymmetric
background in three dimensions. We start with the four dimensional metric

$\mathrm{d}s_{\rm
4d}^{2}=G_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}=\left(\mathrm{d}x^{4}+c_{\mu}\mathrm{d}x^{\mu}\right)^{2}+g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$
(B.6)

and frames (small latin indices denote the $3d$ frame indices)

$E^{4}=\mathrm{d}x^{4}+c_{\mu}\mathrm{d}x^{\mu},\qquad
E^{a}=e^{a}_{\mu}\mathrm{d}x^{\mu}$ (B.7)

and the inverse

$E_{4}={\partial\over\partial x^{4}},\qquad
E_{a}=e_{a}{}^{\mu}{\partial\over\partial
x^{\mu}}-e_{a}{}^{\mu}c_{\mu}{\partial\over\partial x^{4}}.$ (B.8)

The spin connections are related by

$\begin{split}\Omega_{\mu ab}&=\omega_{\mu
ab}+\frac{1}{2}c_{\mu}e_{a}^{\nu}e_{b}^{\rho}\left(\partial_{\nu}c_{\rho}-\partial_{\rho}c_{\nu}\right),\\\
\Omega_{\mu
4b}&=\frac{1}{2}e_{b}^{\nu}\left(\partial_{\mu}c_{\nu}-\partial_{\nu}c_{\mu}\right),\\\
\Omega_{\rm
4ab}&=\frac{1}{2}e^{\nu}_{b}e^{\rho}_{b}\left(\partial_{\nu}c_{\rho}-\partial_{\rho}c_{\nu}\right),\\\
\Omega_{44b}&=0.\end{split}$ (B.9)

The covariant derivatives of a spinor $\chi_{\alpha}$ which does not depend on
$x^{4}$ are related by

$\nabla_{\mu}^{\rm
4d}\chi=\nabla_{\mu}+\frac{1}{2}\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\chi+\frac{1}{2}c_{\mu}V^{\nu}\gamma_{\nu}\chi,\qquad\nabla_{4}\chi=\frac{1}{2}V^{\mu}\gamma_{\mu}\chi,$
(B.10)

where $V^{\mu}=-{{\mathrm{i}}\over
2}\varepsilon^{\mu\nu\rho}\partial_{\nu}c_{\rho}$ and our conventions for the
gamma matrices is

$\gamma^{M}_{\alpha\dot{\alpha}}=\left(\vec{\sigma},-{\mathrm{i}}\right),\qquad\widetilde{\gamma}^{M\dot{\alpha}\alpha}=\left(-\vec{\sigma},-{\mathrm{i}}\right)$
(B.11)

Let us start with a generic equation of the form

$\left(\nabla_{M}-{\mathrm{i}}A_{M}\right)\xi=\gamma_{M}\widetilde{\eta}.$
(B.12)

Using the $M=4$ component of the equation we obtain

$\widetilde{\eta}={{\mathrm{i}}\over 2}V^{\mu}\gamma_{\mu}\xi+A_{4}\xi.$
(B.13)

Using this in the $M=\mu$ component along with the gamma matrices we get

$\left(\nabla_{\mu}-{\mathrm{i}}\left(A_{\mu}+\frac{1}{2}V_{\mu}\right)-A_{4}^{\rm
4d}\gamma_{\mu}+\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\xi=0,$
(B.14)

where $A_{\mu}=A^{\rm 4d}_{\mu}-c_{\mu}A_{4}^{\rm 4d}$ is the three
dimensional one-form and $A_{4}^{\rm 4d}$ is a three dimensional scalar.
Applying this result to $\delta\psi_{M}^{1,2}=0$ we obtain

$\begin{split}&\left(\nabla_{\mu}-{\mathrm{i}}\left(A_{\mu}^{\rm
R_{0}}+\frac{1}{2}V_{\mu}\right)-H\gamma_{\mu}+\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\xi^{1}=0\\\
&\left(\nabla_{\mu}+{\mathrm{i}}\left(A_{\mu}^{\rm R_{0}}-A_{\mu}^{\rm
F}-\frac{1}{2}V_{\mu}\right)-\left(\sigma-H\right)\gamma_{\mu}+\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\xi^{2}=0,\end{split}$
(B.15)

where $H=A_{4}^{\rm R_{0}}$ and $\sigma=A_{4}^{\rm F}$. The dilatino
variations are of the form

$D_{\rm 4d}\epsilon+{\mathrm{i}}\gamma^{MN}\epsilon\partial_{M}A_{N}^{\rm
4d},$ (B.16)

which upon dimensional reduction becomes

$\left(D+\sigma
H\right)\epsilon+\varepsilon^{\mu\nu\rho}\partial_{\mu}A_{\nu}\gamma_{\rho}\epsilon+\gamma^{\mu}\epsilon\left(\partial_{\mu}+2{\mathrm{i}}V_{\mu}\right)A_{4}.$
(B.17)

Using this the two dilatino variations become

$\begin{split}\delta\chi^{1}&=\left(D+\sigma H\right)\xi^{1}-{1\over
3}\left(\varepsilon^{\mu\nu\rho}\gamma^{\rho}\xi^{1}\partial_{\mu}\left(A_{\nu}^{\rm
R_{0}}-{3\over 2}A_{\nu}^{\rm
F}\right)+\gamma^{\mu}\xi^{1}\left(\partial_{\mu}+2{\mathrm{i}}V_{\mu}\right)\left(H-{3\over
2}\sigma\right)\right)=0,\\\ \delta\chi^{2}&=\left(D+\sigma
H\right)\xi^{2}+\frac{1}{3}\left(\varepsilon^{\mu\nu\rho}\gamma^{\rho}\xi^{2}\partial_{\mu}\left(A_{\nu}^{\rm
R_{0}}\frac{1}{2}A_{\nu}^{\rm
F}\right)+\gamma^{\mu}\xi^{2}\left(\partial_{\mu}+2{\mathrm{i}}V_{\mu}\right)\left(H+\frac{1}{2}\sigma\right)\right)=0.\end{split}$
(B.18)

##  Appendix C Extra spinors in ABJM

In the main text we have shown that special values of masses correspond to
supersymmetry enhancement by embedding the $3d$ $\mathcal{N}=2$ supergravity
equations in a twisted dimensional reduction of $4d$ $\mathcal{N}=2$
supergravity. This explains the simplfication of free energy for $3d$ theories
which can be obtained by a dimensional reduction of $4d$ theory. ABJM
theories, however, cannot be obtained in such a way so the previous arguments
do not apply. In this section we consider the $3d$ $\mathcal{N}=6$ conformal
supergravity with matter coupling, whose rigid limit gives ABJM theories on
supersymmetric background. We show that our conclusions regarding the symmetry
enhancement at special values of mass hold.

The field content of the $3d$ $\mathcal{N}=6$ Weyl multiplet is

$g_{\mu\nu}\qquad A_{\mu}^{IJ}\qquad A_{\mu}\qquad E^{IJ}\qquad
D^{IJ}\qquad\psi^{I}_{\mu}\qquad\chi^{IJK},\qquad\chi^{I}$ (C.1)

where $I,J,K,\cdots=1,2,\cdots,6$. $A_{\mu}^{IJ}$ and $A_{\mu}$ are gauge
fields for the $SO(6)\times U(1)$ R-symmetry. $E^{IJ}$ and $D^{IJ}$ are scalar
fields in the ${\bf 15}$ of $SO(6)$. $\chi^{IJK}$ and $\chi^{I}$ are
dilatinos. The general conditions for supersymmetric background are obtained
by setting SUSY variations of all fermions to zero. These variations are given
in detail in eq. (2.5) of [25].

When $\mathcal{N}=6$ ABJM is coupled to this conformal supergravity then the
fields $E_{IJ}$ and $D_{IJ}$ appear in the general mass matrix for scalars and
fermions. In the $\mathcal{N}=2$ language, turning on supersymmetric
background fields for the Cartans of $SO(6)$ flavor symmetry correspond to
turning on constant $12,34$ and $56$ components of $E^{IJ}$ and $D^{IJ}$. From
the SUSY variations of fermions it is clear that in the massless case two
spinors are preserved, those corresponding to say $I=1,2$. This is because of
the fact that one need to turn on background R-fields to preserve
supersymmetry. If only background R-fields are turned on then supersymmetric
variations of various fermions in the supergravity multiplet can’t be set to
zero unless some of the spinors vanish identically. This manifestly preserves
only one-third of the supersymmetry (i.e., four supercharges).

Next we show that the supersymmetry can be enhanced to eight and six
supercharges by turning on a special mass. Without loss of generality we start
with spinors $\epsilon^{1}$ and $\epsilon^{2}$ preserved which we combine in
complex combinations $\xi^{1}$ and $\widetilde{\xi}^{1}$ which satisfy the
conformal Killing spinor equation

$\nabla_{\mu}\xi^{1}-{\mathrm{i}}A_{\mu}^{12}\xi^{1}=\gamma_{\mu}\eta^{1},\qquad\eta^{1}=-\frac{1}{2}H\xi^{1}+{\mathrm{i}}V^{\mu}\gamma_{\mu}\xi^{1},\qquad
A_{\mu}^{12}=-V_{\mu}.$ (C.2)

In addition, we want to preserve the spinors $\epsilon^{3,4}$ by turning on
scalars $E_{56}$ and $D_{56}$ and some gauge fields for the R-symmetry in the
supergravity multiplet. We again combine these spinors into complex
combinations. The relevant variations of fermions that are not trivially zero
are given by

$\begin{split}\delta\left(\psi_{\mu}^{3}+{\mathrm{i}}\psi^{4}_{\mu}\right)&=\nabla_{\mu}\xi^{2}-A_{\mu}^{34}\xi^{2}-\gamma_{\mu}\eta^{2}=0\\\
\delta\left(\lambda^{123}+{\mathrm{i}}\lambda^{124}\right)&={1\over
4\sqrt{2}}\gamma^{\mu\nu}G^{12}_{\mu\nu}\xi^{2}-{\mathrm{i}}\left(D^{56}\xi^{2}-E^{56}\eta^{2}\right),\\\
\delta\left(\lambda^{341}+{\mathrm{i}}\lambda^{342}\right)&={1\over
4\sqrt{2}}\gamma^{\mu\nu}G^{34}_{\mu\nu}\xi^{1}-{\mathrm{i}}\left(D^{56}\xi^{1}-E^{56}\eta^{1}\right).\end{split}$
(C.3)

To preserve the $\mathcal{N}=2$ supersymmetry, we see that a non-trivial
$A_{\mu}^{34}$ needs to be turned on:

$D^{56}=-\frac{1}{2}HE^{56}={{\mathrm{i}}\over
4}\left(b+b^{-1}\right)E^{56},\qquad
A_{\mu}^{34}=2{\mathrm{i}}\sqrt{2}E^{56}{r_{3}\over b+b^{-1}}V_{\mu}.$ (C.4)

The spinors $\xi^{2}$ are also preserved if we choose the flavor paramter such
that $A_{\mu}^{34}=-V_{\mu}$, this corresponds to when $E^{56}$ takes the
value $E^{56*}={{\mathrm{i}}\left(b+b^{-1}\right)\over r_{3}}{1\over
2\sqrt{2}}$. Up to an overall constant this corresponds to the second
squashing independent point for ABJM discussed below. In this case four extra
supercharges are preserved and the Killing $\xi^{2}$ is actually equal to
$\xi^{1}$.

Similarly, for the other squashing independent point, we take $\xi^{2}$ to be
an eigenspinor of $\gamma^{3}$. We then use the second equation in eq. C.3 to
determine $\eta^{2}$

$\eta^{2}=-\frac{1}{2}H\xi^{2}-{\mathrm{i}}{E^{56*}\over
E^{56}}V^{\mu}\gamma_{\mu}\xi^{2}.$ (C.5)

Using this the Killing spinor equation for $\xi^{2}$ becomes

$\nabla_{\mu}\xi^{2}-{\mathrm{i}}{E^{56}\over
E^{56*}}V_{\mu}\xi^{2}+\frac{1}{2}H\xi^{2}+{\mathrm{i}}{E^{56*}\over
E^{56}}V^{\mu}\gamma_{\mu}\xi^{2}=0.$ (C.6)

For $\xi^{2}=(1,0)$ we have $V^{\mu}\gamma_{\mu}\xi^{2}=-{b-b^{-1}\over
2r_{3}}\xi^{2}$. If $E^{56}=-{\mathrm{i}}\left(b-b^{-1}\over
2\sqrt{2}r_{3}\right)$ then the above equation becomes

$\nabla_{\mu}\xi^{2}+{\mathrm{i}}{b-b^{-1}\over
b+b^{-1}}V_{\mu}\xi^{2}+{{\mathrm{i}}\over 4}{b+b^{-1}\over r_{3}}\xi^{2}=0,$
(C.7)

which is indeed satisfied for the constant spinor.

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