#  $S^{1}$ Reduction of 4D $\mathcal{N}=3$ SCFTs and
Squashing Independence of ABJM Theories

Tomoki Nakanishi1,2♢ and Takahiro Nishinaka1,2,3♣

###### Abstract

We study the compactification of 4D $\mathcal{N}=3$ superconformal field
theories (SCFTs) on $S^{1}$, focusing on the relation between the 4D
superconformal index and 3D partition function on the squashed sphere
$S^{3}_{b}$. Since the center $\mathfrak{u}(1)$ of the $\mathfrak{u}(3)$
R-symmetry of the 4D theory can mix with an $\mathcal{N}=6$ abelian flavor
symmetry in three dimensions, the precise 4D/3D relation for the global
symmetry is not obvious. Focusing on the case in which the 3D theory is the
ABJM theory we demonstrate that the above R-symmetry mixing can be precisely
identified by considering the Schur limit (and/or its $\mathcal{N}=3$ cousin)
of the 4D index. As a result, we generalize to the ABJM theories recent
discussions on the connection between supersymmetry enhancement of the 4D
index and squashing independence of the $S^{3}_{b}$ partition function.

Contents

toc

##  1 Introduction

There had been no known 3D $\mathcal{N}=6$ superconformal field theories
(SCFTs) until the authors of [1] discovered a special series of Chern-Simons
matter theories with this amount of supersymmetry. These theories are now
called ABJM theories, and are identified as the low energy description of
M2-branes on $\mathbb{C}^{4}/\mathbb{Z}_{k}$.

Similarly, no 4D $\mathcal{N}=3$ SCFT had been known until the authors of [2]
discovered the first example using D3-branes on the F-theory singularity
$(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$ for $k=3,4$ or $6$.111A general
discussion on 4D $\mathcal{N}=3$ SCFTs was first given in [3]. More detailed
analysis on explicit constructions of 4D $\mathcal{N}=3$ SCFTs was given in
[4, 5]. As discussed in [2], the duality between M-theory and F-theory
suggests that the compactification of these 4D $\mathcal{N}=3$ theories on
$S^{1}$ gives rise to ABJM theories (for specific values of the Chern-Simons
level). Then it would be desirable to find a clear correspondence between
physical quantities of 4D $\mathcal{N}=3$ SCFTs and ABJM theories.

One 4D quantity whose behavior under the compactification is well-understood
for generic $\mathcal{N}=2$ SCFTs is the superconformal index. Since the index
can be thought of as the (normalized) partition function of the 4D theory on
$S^{1}\times S^{3}_{b}$ with background gauge fields turned on, it is reduced
to the $S^{3}_{b}$ partition function of the 3D theory by the $S^{1}$
reduction. This relation between the 4D index and $S^{3}_{b}$ partition
function has been carefully studied and confirmed in various examples [6, 7,
8, 9]. This leads us to comparing the superconformal index of the 4D
$\mathcal{N}=3$ SCFTs and the $S^{3}_{b}$ partition function of the ABJM
theories.

However, it turns out that the 3D reduction of the R-symmetry of 4D
$\mathcal{N}=3$ SCFTs is rather non-trivial, unlike for generic 4D
$\mathcal{N}=2$ SCFTs. On the one hand, the bosonic (zero-form) global
symmetry of the ABJM theory is
$\mathfrak{so}(6)_{R}\times\mathfrak{u}(1)_{b}$, where $\mathfrak{u}(1)_{b}$
is a flavor symmetry commuting with the $\mathcal{N}=6$ R-symmetry. On the
other hand, the bosonic (zero-form) global symmetry of 4D $\mathcal{N}=3$
SCFTs is $\mathfrak{u}(3)_{R}$ R-symmetry.222Here, we normalize the center
$\mathfrak{u}(1)$ of $\mathfrak{u}(3)_{R}$ so that the 4D chiral supercharges
$Q^{I}_{\alpha}$ are in the fundamental representation of
$\mathfrak{u}(3)_{R}$. Note also that a 4D $\mathcal{N}=3$ SCFT cannot have a
continuous $\mathcal{N}=3$ flavor symmetry without having extra
supersymmetries [3].  Naively, one might think that the $\mathfrak{u}(3)_{R}$
in four dimensions is mapped to a $\mathfrak{u}(3)$ sub-algebra of
$\mathfrak{so}(6)_{R}$ under the compactification. However, this naive map
would lead to various contradictions as discussed below.

First, the naive map $\mathfrak{u}(3)_{R}\hookrightarrow\mathfrak{so}(6)_{R}$
contradicts with a general relation between supersymmetry enhancement of the
4D index and squashing independence of the 3D partition function [10]333Here,
we mean by “supersymmetry enhancement” that the number of preserved
supercharges on $S^{1}\times S^{3}_{b}$ is increased by taking a special limit
of background gauge fields. . Here, by squashing independence, we mean the
property that the $b$-dependence of the $S^{3}_{b}$ partition function
$Z_{S^{3}_{b}}$ can be absorbed by rescaling mass parameters. Such a
phenomenon was recently observed in [11] for the ABJM theory when a special
value of an (imaginary) mass parameter is turned on. This was then interpreted
in [10] as a supersymmetry enhancement on $S^{3}_{b}$, or on $S^{1}\times
S^{3}_{b}$ in the case when the theory has a 4D uplift. In the latter case,
the special value of the 3D mass parameter corresponds to a limit of a
fugacity in the 4D superconformal index. In particular, the Schur limit of the
4D index (i.e., Schur index) generally leads to squashing independence of
$Z_{S^{3}_{b}}$ under the compactification [10]. As we will show in Sec. 3,
the naive map $\mathfrak{u}(3)_{R}\hookrightarrow\mathfrak{so}(6)_{R}$
discussed above, however, contradicts with this general relation between the
4D Schur limit and 3D squashing independence.

Another contradiction of the naive map
$\mathfrak{u}(3)_{R}\hookrightarrow\mathfrak{so}(6)_{R}$ can be seen in the
moduli space of vacua. Let us consider the case of rank-one $\mathcal{N}=3$
SCFTs [12]. The moduli space is then $\mathbb{C}^{3}/\mathbb{Z}_{k}$ for
$k=3,4$ or $6$, where the $\mathbb{Z}_{k}$-action is such that
$\mathbb{C}^{3}\ni(z_{1},z_{2},z_{3})\to(\omega
z_{1},\,\omega^{-1}z_{2},\,\omega z_{3})$ for $\omega\in\mathbb{Z}_{k}$. The
3D reduction of this 4D $\mathcal{N}=3$ SCFT is believed to be the ABJM theory
with gauge group $U(1)_{k}\times U(1)_{-k}$, whose moduli space is
$\mathbb{C}^{4}/\mathbb{Z}_{k}$ with the $\mathbb{Z}_{k}$-action such that
$\mathbb{C}^{4}\ni(z_{1},z_{2},z_{3},z_{4})\to(\omega
z_{1},\,\omega^{-1}z_{2},\,\omega z_{3},\,\omega^{-1}z_{4})$. The emergence of
the extra direction (parameterized by $z_{4}$) is a general phenomenon for the
compactification of 4D $\mathcal{N}\geq 2$ theories; the dimension of the
Coulomb branch is doubled by the compactification. Here, $z_{1},z_{2}$ and
$z_{3}$ are charged under the 4D $\mathfrak{u}(3)_{R}$ symmetry while $z_{4}$
is neutral.444We here assume there is no non-trivial quantum corrections to
the moduli space under the $S^{1}$ compactification, which we believe is a
natural assumption with this large amount of supersymmetry. From the 3D
viewpoint, however, the $\mathfrak{so}(6)_{R}$ R-symmetry non-trivially acts
on all $z_{1},z_{2},z_{3}$ and $z_{4}$.555To be more specific,
$(z_{1},\overline{z}_{2},z_{3})$ forms a fundamental representation of
$\widetilde{\mathfrak{u}(3)}_{R}$, while
$(z_{1},\overline{z}_{2},z_{3},\overline{z}_{4})$ forms a fundamental
representation of $\mathfrak{su}(4)\simeq\mathfrak{so}(6)_{R}$. Here, the
action of $\widetilde{\mathfrak{u}(3)}_{R}$ is different from that of
$\mathfrak{u}(3)_{R}$ only in the action of its center; the former is obtained
by rescaling the charge of $\mathfrak{u}(1)_{c}\subset\mathfrak{u}(3)_{R}$ by
$1/2$. Indeed, our normalization of the charge of
$\mathfrak{u}(1)_{c}\subset\mathfrak{u}(3)_{R}$ is such that the 4D chiral
supercharges $\mathcal{Q}^{I}_{\alpha}$ are in its fundamental representation,
while the action of the center of $\widetilde{\mathfrak{u}(3)}_{R}$ is such
that $(z_{1},\overline{z}_{2},z_{3})$ are in its fundamental representation.
Since $z_{1},z_{2}$ and $z_{3}$ are identified as the VEVs of the three chiral
multiplets in an $\mathcal{N}=3$ vector multiplet (which is equivalent to an
$\mathcal{N}=4$ vector multiplet for CPT reasons), we see that these
normalizations differ by a factor two. The authors thank Y. Tachikawa for
pointing out this fact.  Therefore, if the naive inclusion
$\mathfrak{u}(3)_{R}\hookrightarrow\mathfrak{so}(6)_{R}$ is correct, there
must be a $\mathfrak{u}(3)$ sub-algebra of $\mathfrak{so}(6)_{R}$ that keeps
$z_{4}$ invariant. It turns out, however, that there is no such sub-algebra.

In this paper, we show that the above two contradictions are resolved when the
$\mathfrak{u}(3)_{R}$ symmetry in four dimensions is mixed with the
$\mathfrak{u}(1)_{b}$ symmetry in three dimensions under the $S^{1}$
compactification. Indeed, we show that there is a unique $\mathfrak{u}(3)$
sub-algebra of $\mathfrak{so}(6)_{R}\times\mathfrak{u}(1)_{b}$ that, when
identified as the 3D reduction of the 4D $\mathfrak{u}(3)_{R}$ symmetry,
resolves all the above contradictions. Our result would be important when
comparing the superconformal index of the 4D $\mathcal{N}=3$ theories and the
sphere partition function of their 3D reductions. Indeed, given the above
mixing between $\mathfrak{u}(3)_{R}$ and $\mathfrak{u}(1)_{b}$, one can now
establish a clear correspondence between the supersymmetry enhancement of the
4D $\mathcal{N}=3$ SCFT on $S^{1}\times S^{3}_{b}$ and the squashing
independence of the ABJM theories on $S^{3}_{b}$.

The organization of this paper is as follows. In Sec. 2, we give a brief
review of the relation between the supersymmetry enhancement of the 4D index
and the squashing independence of the 3D partition function. We also discuss
its $\mathcal{N}=3$ cousin there. In Sec. 3, we consider the
$S^{1}$-compactification of the 4D $\mathcal{N}=3$ SCFT and discuss the
R-symmetry mixing, focusing on the case that the 3D theory is the ABJM theory.
In Sec. 4, we discuss a divergence that arises in the reduction of 4D
$\mathcal{N}=3$ superconformal index. We finally conclude in Sec. 5. In
appendix Appendix A. 4D $\mathcal{N}=3$ and 3D $\mathcal{N}=6$ superconformal
algebras, we discuss in some more detail how the 4D and 3D symmetries are
related under the $S^{1}$-compactification.

##  2 Squashing independence of 3D theories arising from 4D $\mathcal{N}\geq
2$ SCFTs

| $j_{1}$ | $j_{2}$ | $R$ | $r$ | $f$ | $j_{2}+j_{1}-r$ | $j_{2}-j_{1}-R-2r-\frac{3}{2}f$
---|---|---|---|---|---|---|---
$\mathcal{Q}^{1}{}_{+}$ | $+\frac{1}{2}$ | $0$ | $+\frac{1}{2}$ | $+\frac{1}{2}$ | $0$ | $0$ | $-2$
$\mathcal{Q}^{1}{}_{-}$ | $-\frac{1}{2}$ | $0$ | $+\frac{1}{2}$ | $+\frac{1}{2}$ | $0$ | $-1$ | $-1$
$\mathcal{Q}^{2}{}_{+}$ | $+\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $+\frac{1}{2}$ | $0$ | $0$ | $-1$
$\mathcal{Q}^{2}{}_{-}$ | $-\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $+\frac{1}{2}$ | $0$ | $-1$ | $0$
$\widetilde{\mathcal{Q}}_{1\dot{+}}$ | $0$ | $+\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $0$ | $1$ | $2$
$\widetilde{\mathcal{Q}}_{1\dot{-}}$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $0$ | $0$ | $1$
$\widetilde{\mathcal{Q}}_{2\dot{+}}$ | $0$ | $+\frac{1}{2}$ | $+\frac{1}{2}$ | $-\frac{1}{2}$ | $0$ | $1$ | $1$
$\widetilde{\mathcal{Q}}_{2\dot{-}}$ | $0$ | $-\frac{1}{2}$ | $+\frac{1}{2}$ | $-\frac{1}{2}$ | $0$ | $0$ | $0$
$\mathcal{Q}^{3}{}_{+}$ | $+\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $+1$ | $1$ | $-1$
$\mathcal{Q}^{3}{}_{-}$ | $-\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $+1$ | $0$ | $0$
$\widetilde{\mathcal{Q}}_{3\dot{+}}$ | $0$ | $+\frac{1}{2}$ | $0$ | $+\frac{1}{2}$ | $-1$ | $0$ | $1$
$\widetilde{\mathcal{Q}}_{3\dot{-}}$ | $0$ | $-\frac{1}{2}$ | $0$ | $+\frac{1}{2}$ | $-1$ | $-1$ | $0$
Table 1: Quantum numbers of supercharges

###  2.1 4D $\mathcal{N}=2$ SCFTs and 3D $\mathcal{N}=4$ theories

Here, we briefly review the relation between limits of 4D superconformal index
and the 3D squashing independence [10].

The superconformal index of a 4D $\mathcal{N}=2$ SCFT is defined by

$\displaystyle
I=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}-r}q^{j_{2}+j_{1}-r}t^{r+R}\prod_{i}a_{i}^{f_{i}}~{},$
(2.1)

where the trace is taken over the space of local operators, $(j_{1},j_{2})$
are $\mathfrak{so}(4)$ spins, $R,r$ are respectively the $SU(2)_{R}$ and
$U(1)_{r}$ charges, and $f_{i}$ are the flavor charges. In terms of the
generators of the 4D $\mathcal{N}=2$ superconformal algebra reviewed in
appendix A.1, $R$ and $r$ are written as

$\displaystyle
R\equiv\frac{1}{2}\left(\mathcal{R}^{1}{}_{1}-\mathcal{R}^{2}{}_{2}\right)~{},\qquad
r\equiv\mathcal{R}^{1}{}_{1}+\mathcal{R}^{2}{}_{2}~{}.$ (2.2)

Since the exponents of $p,q,t$ and $a_{i}$ commute with
$\widetilde{\mathcal{Q}}_{2\dot{-}}$ and $\widetilde{\mathcal{S}}^{2\dot{-}}$
(Table 1), the index receives contributions only from operators annihilated by
these supercharges. Therefore we say that the above index preserves two
supercharges.

The superconformal index has several interesting limits of parameters. In
particular, the Schur limit, $t\to q$, is known to provide a variety of
application.666For instance, it gives rise to a general connection between 4D
$\mathcal{N}=2$ SCFTs and non-unitary vertex algebras [13]. It also simplifies
the TQFT expression for the index of class $\mathcal{S}$ theories [14]. One
important feature of the Schur limit is that the resulting index (called the
“Schur index”) turns out to be independent of $p$ [15, 10]. The reason for
this is the following. First note that, when $t=q$, the index is written as

$\displaystyle
I=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}-r}q^{j_{2}+j_{1}+R}\prod_{i}a_{i}^{f_{i}}~{}.$
(2.3)

There are now four supercharges,
$\mathcal{Q}^{1}{}_{-},\,\widetilde{\mathcal{Q}}_{2\dot{-}},\,\mathcal{S}_{1}{}^{-}$
and $\widetilde{\mathcal{S}}^{2\dot{-}}$, commuting with the exponents of
$p,q$ and $a_{i}$. Therefore, the Schur index preserves four supercharges, and
receives contributions only from operators annihilated by them. We see that
the exponent of $p$,

$\displaystyle
j_{2}-j_{1}-r=\left\\{\mathcal{Q}^{1}{}_{-},\mathcal{S}_{1}{}^{-}\right\\}-\left\\{\widetilde{\mathcal{Q}}_{2\dot{-}},\widetilde{\mathcal{S}}^{2\dot{-}}\right\\}~{},$
(2.4)

vanishes for all such operators. As a result, the Schur index has no
$p$-dependence and is expressed as

$\displaystyle I\big{|}_{t\to
q}=\text{Tr}(-1)^{F}q^{j_{2}+j_{1}+R}\prod_{i}a_{i}^{f_{i}}~{}.$ (2.5)

The absence of $p$-dependence is clearly a consequence of the supersymmetry
enhancement triggered by $t\to q$ [15, 10].

Let us now consider the compactification of the 4D $\mathcal{N}=2$ SCFT on
$S^{1}$. In the deep infrared, we end up with a 3D $\mathcal{N}=4$ SCFT. Its
$S^{3}_{b}$ partition function can be obtained as the small $S^{1}$ limit of
the 4D superconformal index (2.1) [6, 7, 8, 9, 16]. In this limit, we set

$\displaystyle p=e^{-\beta b}~{},\qquad q=e^{-\beta
b^{-1}}~{},\qquad\frac{t}{\sqrt{pq}}=e^{-i\beta m}~{},\qquad a_{i}=e^{-i\beta
M_{i}}~{},$ (2.6)

and take the limit $\beta\to 0$ with $b,m$ and $M_{i}$ kept fixed.777The
special combination $t/\sqrt{pq}$ appears here since it is a flavor fugacity
from the viewpoint of the $\mathcal{N}=1$ supersymmetry involving the
preserved supercharge $\widetilde{Q}_{2\dot{-}}$. Then the 4D index reduces to
the partition function $Z_{S^{3}_{b}}$ of the 3D $\mathcal{N}=4$ theory on
$S^{3}_{b}$ (up to a prefactor). The parameters $m$ and $M_{i}$ are identified
as mass deformation parameters of the 3D theory.

From the 3D viewpoint, the Schur limit $t\to q$ corresponds to constraining
the 3D mass parameter $m$ as

$\displaystyle m=i\frac{b-b^{-1}}{2}~{}.$ (2.7)

With this value of $m$, the supersymmetry preserved on $S^{3}_{b}$ is
enhanced. As a result, the $b$-dependence of $Z_{S^{3}_{b}}$ can be removed by
rescaling the remaining mass parameters $M_{i}$ [10]. Indeed, the index is now
independent of $p$, and therefore the $b$-dependence is only in $q=e^{-\beta
b^{-1}}$. We see that this $b$-dependence can be removed by

$\displaystyle\beta\to\beta b~{},\qquad M_{i}\to b^{-1}M_{i}~{},$ (2.8)

where rescaling $M_{i}$ is necessary for keeping $a_{i}=e^{-i\beta M_{i}}$
unchanged. Hence, the Schur limit $t\to q$ in four dimensions generally leads
to squashing independence in three dimensions [10].

It was also found in [10] that the 3D squashing independence also occurs at

$\displaystyle m=-i\frac{b+b^{-1}}{2}~{}.$ (2.9)

In four dimensions, this corresponds to the condition $t=pq$, under which the
4D index is expressed as

$\displaystyle
I\big{|}_{t=pq}=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}+R}q^{j_{2}+j_{1}+R}\prod_{i}a_{i}^{f_{i}}~{}.$
(2.10)

This index counts operators annihilated by
$\widetilde{\mathcal{Q}}_{1\dot{+}},\,\widetilde{\mathcal{Q}}_{2\dot{-}},\,\widetilde{\mathcal{S}}^{1\dot{+}}$
and $\widetilde{\mathcal{S}}^{2\dot{-}}$ , and for such operators
$j_{2}+R=\frac{1}{2}\\{\widetilde{\mathcal{Q}}_{1\dot{+}},\,\widetilde{\mathcal{S}}^{1\dot{+}}\\}-\frac{1}{2}\\{\widetilde{\mathcal{Q}}_{2\dot{-}},\,\widetilde{\mathcal{S}}^{2\dot{-}}\\}$
vanishes. Therefore, the above expression for the index reduces to

$\displaystyle
I\big{|}_{t=pq}=\text{Tr}(-1)^{F}\left(\frac{q}{p}\right)^{j_{1}}\prod_{i}a_{i}^{f_{i}}~{}.$
(2.11)

Since the index involves only one combination of the superconformal
fugacities, $q/p$, its 3D reduction leads to $Z_{S^{3}_{b}}$ whose
$b$-dependence can again be removed by rescaling mass parameters. Note,
however, that the index (2.11) is always divergent when the 4D theory has a
freely generated Coulomb branch, since all the Coulomb branch operators
contribute to it with $j_{1}=R=f_{i}=0$.888By “Coulomb branch operators,” we
mean local operators annihilated by all anti-chiral $\mathcal{N}=2$
supercharges. For the fact that $f_{i}=0$ for all Coulomb branch operators,
see [17, 18].

###  2.2 4D $\mathcal{N}=3$ SCFTs and 3D $\mathcal{N}=6$ theories

When the 4D theory has an extra supersymmetry, one can generalize the above
discussion. Such a generalization will be useful when we discuss in the next
section the relation between 4D $\mathcal{N}=3$ SCFTs and 3D ABJM theories.

We will focus on 4D $\mathcal{N}=3$ SCFTs without $\mathcal{N}=4$
supersymmetry, although its generalization to 4D $\mathcal{N}=4$ SCFTs is
straightforward and will be discussed in [19]. The bosonic global symmetry of
4D (genuine) $\mathcal{N}=3$ SCFT is just the $\mathfrak{u}(3)_{R}$ symmetry;
there is no “$\mathcal{N}=3$ flavor symmetry.” This $\mathfrak{u}(3)_{R}$
contains the $\mathcal{N}=2$ R-symmetry,
$\mathfrak{su}(2)_{R}\times\mathfrak{u}(1)_{r}$, and the $\mathcal{N}=2$
flavor symmetry, $\mathfrak{u}(1)_{f}$. Therefore, the most general expression
for the superconformal index is written as

$\displaystyle
I=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}-r}q^{j_{2}+j_{1}-r}t^{r+R}a^{f}~{},$ (2.12)

where we define the $\mathfrak{u}(1)_{f}$ charge by

$\displaystyle
f\equiv\mathcal{R}^{1}{}_{1}+\mathcal{R}^{2}{}_{2}+2\mathcal{R}^{3}{}_{3}~{},$
(2.13)

as in [12]. For generic values of the fugacities, the above index preserves
only $\widetilde{\mathcal{Q}}_{2\dot{-}}$ and
$\widetilde{\mathcal{S}}^{2\dot{-}}$. Since the discussions in the previous
sub-section are also applied to $\mathcal{N}=3$ SCFTs, we see that there are
two special values of the fugacity $t$ that lead to supersymmetry enhancement:

$\displaystyle t=q~{},\qquad t=pq~{}.$ (2.14)

In three dimensions, they correspond to two special values of a mass parameter
that give rise to squashing independence.

We now ask whether a similar supersymmetry enhancement occurs at a special
value of the fugacity $a$. If the theory has only $\mathcal{N}=2$
supersymmetry with a flavor charge $f$, no such supersymmetry enhancement
occurs since $f$ commutes with all $\mathcal{N}=2$ supercharges. However, when
the theory has an $\mathcal{N}=3$ supersymmetry and $f$ is defined by (2.13),
there exists a supercharge that does not commute with $f$. Such a supercharge
can always be preserved by setting $a$ to a special value.

For instance, one can preserve $\mathcal{Q}^{3}{}_{-}$ (and its conjugate
$\mathcal{S}_{3}{}^{-}$) by setting

$\displaystyle a=\frac{\sqrt{t}}{p}~{}.$ (2.15)

In this case, the expression for the index reduces to

$\displaystyle
I\big{|}_{a=\frac{\sqrt{t}}{p}}=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}-r-f}q^{j_{2}+j_{1}-r}t^{r+R+\frac{f}{2}}~{}.$
(2.16)

We see from Table 1 that it indeed preserves $\mathcal{Q}^{3}{}_{-}$ and
$\mathcal{S}_{3}{}^{-}$ in addition to $\widetilde{\mathcal{Q}}_{2\dot{-}}$
and $\widetilde{\mathcal{S}}^{2\dot{-}}$. Note that one can rewrite (2.16) as

$\displaystyle
I\big{|}_{a=\frac{\sqrt{t}}{p}}=\text{Tr}(-1)^{F}\left(pt\right)^{\frac{1}{2}\left(-\delta_{1}+\delta_{2}\right)}q^{j_{2}+j_{1}-r}\left(\frac{p}{t}\right)^{\frac{1}{2}(j_{2}-j_{1}-R-2r-\frac{3}{2}f)}$
(2.17)

where
$\delta_{1}\equiv\\{\widetilde{\mathcal{Q}}_{2\dot{-}},\,\widetilde{\mathcal{S}}^{2\dot{-}}\\}\frac{1}{2}E-j_{2}-R+\frac{1}{2}r$
and
$\delta_{2}\equiv\\{\mathcal{Q}^{3}{}_{-},\,\mathcal{S}_{3}{}^{-}\\}=\frac{1}{2}E-j_{1}-\frac{1}{2}f+\frac{1}{2}r$.
Since the index is contributed only from operators annihilated by
$\widetilde{\mathcal{Q}}_{2\dot{-}},\,\mathcal{Q}^{3}{}_{-},\,\widetilde{\mathcal{S}}^{2\dot{-}}$
and $\mathcal{S}_{3}{}^{-}$, the expression (2.17) can be reduced to

$\displaystyle
I\big{|}_{a=\frac{\sqrt{t}}{p}}=\text{Tr}(-1)^{F}q^{j_{2}+j_{1}-r}\left(\frac{p}{t}\right)^{\frac{1}{2}\left(j_{2}-j_{1}-R-2r-\frac{3}{2}f\right)}~{}.$
(2.18)

The fact that the $(pt)$-dependence drops out in (2.18) implies a new
condition for the squashing independence in three dimensions. To see this,
first let us write things in terms of the 3D parameters using

$\displaystyle p=e^{-\beta b}~{},\qquad q=e^{-\beta
b^{-1}}~{},\qquad\frac{t}{\sqrt{pq}}=e^{-i\beta m}~{},\qquad a=e^{-i\beta
M}~{}.$ (2.19)

The condition (2.15) is then translated as

$\displaystyle M=\frac{m}{2}+i\frac{3b-b^{-1}}{4}~{}.$ (2.20)

Under this condition, the index depends only on $q=e^{-\beta b^{-1}}$ and
$p/t=e^{i\beta m-\beta\frac{b-b^{-1}}{2}}$, and therefore the $b$-dependence
can be removed by the replacement

$\displaystyle\beta\to\beta b~{},\qquad m\to b^{-1}m-i\frac{b-b^{-1}}{2}~{}.$
(2.21)

This implies that, when (2.20) is imposed, the $b$-dependence of the
$S^{3}_{b}$ partition function can be removed by $m\to
b^{-1}m-i\frac{b-b^{-1}}{2}$.

##  3 R-symmetry mixing under the $S^{1}$-compactification

We here demonstrate that the above general discussion on squashing
independence of 3D $\mathcal{N}\geq 4$ theories puts a strong constraint on
the relation between the R-symmetries of a 4D $\mathcal{N}=3$ SCFT and its 3D
reduction. For simplicity, we focus on 4D SCFTs whose 3D reduction is the ABJM
theory. Since the bosonic (zero-form) global symmetry is enhanced from
$\mathfrak{u}(3)_{R}$ to $\mathfrak{so}(6)_{R}\times\mathfrak{u}(1)_{b}$ by
the compactification, we need to understand which $\mathfrak{u}(3)$ sub-
algebra of the latter corresponds to the 3D reduction of the former. While a
naive expectation might be that $\mathfrak{u}(3)_{R}$ simply descends to a
sub-algebra of $\mathfrak{so}(6)_{R}$, that would contradicts with the general
discussions in the previous section.

###  3.1 4D fugacities and 3D masses

To see the above statement, let us first make a concrete connection between
the R-charges in four and three dimensions.

We denote the 3D supercharges by $\mathsf{Q}^{s}_{\alpha}$ for $s=1,\cdots,6$
so that we can distinguish them from the 4D supercharges
$\mathcal{Q}^{I}{}_{\alpha}$ and $\widetilde{\mathcal{Q}}_{I\dot{\alpha}}$. We
arrange them so that $\mathcal{Q}^{I}{}_{\alpha}$ and
$\widetilde{\mathcal{Q}}_{I\dot{\alpha}}$ descend to
$\frac{1}{\sqrt{2}}(\mathsf{Q}^{2I-1}{}_{\alpha}\pm
i\mathsf{Q}^{2I}{}_{\alpha})$ in three dimensions. We also denote by
$R^{\text{4D}}_{I}$ and $R^{\text{3D}}_{I}$ a basis of $\mathfrak{u}(3)_{R}$
and $\mathfrak{so}(6)_{R}$, respectively. We take them so that
$\mathcal{Q}^{I}{}_{\alpha}$ and $\widetilde{\mathcal{Q}}_{I\dot{\alpha}}$
respectively have charge $+1$ and $-1$ under $R^{\text{4D}}_{I}$ but are
neutral under $R^{\text{4D}}_{J(\neq I)}$, and similarly that
$\mathsf{Q}^{2I-1}{}_{\alpha}\pm i\mathsf{Q}^{2I}{}_{\alpha}$ have charge $\pm
1$ under $R^{\text{3D}}_{I}$ but are neutral under $R^{\text{3D}}_{J(\neq
I)}$. Note that, if the 4D R-symmetry $\mathfrak{u}(3)_{R}$ was mapped into
$\mathfrak{so}(6)_{R}$ by the compactification, we would have

$\displaystyle R_{I}^{\text{4D}}=R_{I}^{\text{3D}}~{}.$ (3.1)

However, since the center $\mathfrak{u}(1)_{c}\subset\mathfrak{u}(3)_{R}$ can
be mixed with $\mathfrak{u}(1)_{b}$ in the ABJM theory, the most general
identification is written as

$\displaystyle R^{\text{4D}}_{I}=R^{\text{3D}}_{I}+\xi
J^{\text{3D}}_{U(1)_{b}}~{},$ (3.2)

where $J^{\text{3D}}_{U(1)_{b}}$ is the charge of the $\mathfrak{u}(1)_{b}$
flavor symmetry in three dimensions, and $\xi$ is a real number.999As
discussed below, we will normalize $J^{\text{3D}}_{U(1)_{b}}$ so that the 3D
twisted chiral multiplet with $R^{\text{3D}}_{1}=R^{\text{3D}}_{3}=1/2$ has
charge $J^{\text{3D}}_{U(1)_{b}}=1/2$. When $\xi\neq 0$, there is a mixing
between $\mathfrak{u}(1)_{c}$ and $\mathfrak{u}(1)_{b}$. Indeed, we will show
below that $\xi$ does not vanish.

Let us now write the superconformal index (2.12) of the 4D theory in terms of
$R_{i}^{\text{3D}}$ and $J^{\text{3D}}_{U(1)_{b}}$. From Table 1, we see that

$\displaystyle R^{\text{4D}}_{1}=R+r+\frac{f}{2}~{},\qquad
R^{\text{4D}}_{2}=-R+r+\frac{f}{2}~{},\qquad R^{\text{4D}}_{3}=f~{}.$ (3.3)

This implies that the index (2.12) can be expressed as

$\displaystyle
I=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}}q^{j_{2}+j_{1}}\left(\frac{t}{\sqrt{pq}}\right)^{R^{\text{4D}}_{1}}\left(\frac{1}{\sqrt{pq}}\right)^{R^{\text{4D}}_{2}}\left(a\sqrt{\frac{pq}{t}}\right)^{R^{\text{4D}}_{3}}~{}.$
(3.4)

Using the identification (3.2), one can rewrite this in terms of the 3D
R-charges and $J_{U(1)_{b}}^{\text{3D}}$ as

$\displaystyle
I=\text{Tr}(-1)^{F}p^{j_{2}-j_{1}}q^{j_{2}+j_{1}}\left(\frac{t}{\sqrt{pq}}\right)^{R^{\text{3D}}_{1}}\left(\frac{1}{\sqrt{pq}}\right)^{R^{\text{3D}}_{2}}\left(a\sqrt{\frac{pq}{t}}\right)^{R^{\text{3D}}_{3}}\left(a\sqrt{\frac{t}{pq}}\right)^{\xi
J^{\text{3D}}_{U(1)_{b}}}~{}.$ (3.5)

Since this index preserves $\widetilde{Q}_{2\dot{-}}$, one can regard it as an
$\mathcal{N}=1$ index associated with
$(Q^{2}{}_{\alpha},\,\widetilde{Q}_{2\dot{\alpha}})$. In this case,
$t/\sqrt{pq}$ and $a\sqrt{pq/t}$ are regarded as flavor fugacities from the
$\mathcal{N}=1$ viewpoint. In three dimensions, these fugacities reduce to two
mass parameters associated with a 3D $\mathcal{N}=2$ flavor symmetry. For the
ABJM theory, the $\mathcal{N}=2$ flavor symmetry is the product of
$\mathfrak{so}(4)\subset\mathfrak{so}(6)_{R}$ and $\mathfrak{u}(1)_{b}$. While
this product is of rank three, only two independent mass parameters arise in
the reduction of the 4D index, since the remaining one is associated with an
accidental symmetry in three dimensions.

To be more specific, let us denote by $m_{2}$ and $m_{3}$ the mass parameters
associated with
$\mathfrak{so}(2)^{2}\subset\mathfrak{so}(4)\subset\mathfrak{so}(6)_{R}$, and
by $m_{1}$ that associated with $\mathfrak{u}(1)_{b}$, as in [11]. The
expression (3.5) implies that these masses are related to the 4D fugacities
as101010Note here that $R_{2}^{\text{3D}}$ is identified as the superconformal
R-charge of the 3D $\mathcal{N}=2$ superconformal algebra associated with
$\mathsf{Q}^{3}_{\alpha}\pm i\mathsf{Q}^{4}_{\alpha}$. Therefore,
$(1/\sqrt{pq})^{R^{\text{3D}}_{2}}$ in (3.5) corresponds to the background
gauge field for the R-symmetry that reduces to the 3D $\mathcal{N}=2$
superconformal R-symmetry under the compactification. Turning on such a
background gauge field is necessary for preserving the supersymmetry on
$S^{1}\times S^{3}$ [20].

$\displaystyle\frac{t}{\sqrt{pq}}=e^{-i\beta m_{2}}~{},\qquad
a\sqrt{\frac{pq}{t}}=e^{-i\beta
m_{3}}~{},\qquad\left(a\sqrt{\frac{t}{pq}}\right)^{\xi}=e^{-i\beta m_{1}}~{}.$
(3.6)

Since $p$ and $q$ are related to the squashing parameter by $p=e^{-\beta b}$
and $q=e^{-\beta b^{-1}}$ , for fixed $b$ only two of $m_{1},m_{2}$ and
$m_{3}$ are independent. In terms of $m$ and $M$ defined in (2.19), the mass
parameters $m_{i}$ are expressed as

$\displaystyle
m_{1}=\xi\left(M+\frac{m}{2}+i\frac{b+b^{-1}}{4}\right)~{},\qquad
m_{2}=m~{},\qquad m_{3}=M-\frac{m}{2}-i\frac{b+b^{-1}}{4}$ (3.7)

We stress that the three mass parameters of the ABJM theory are constrained
here as above since one linear combination of them is associated with an
accidental symmetry in three dimensions and does not have a 4D counterpart.
One important question is then which two linear combinations have 4D
counterparts. The answer to this question depends on the value of $\xi$, which
characterizes the mixing between $\mathfrak{u}(1)_{c}$ and
$\mathfrak{u}(1)_{b}$. Below, we will uniquely fix the value of $\xi$ so that
it is consistent with the general discussion on squashing independence in the
previous section.

###  3.2 R-symmetry mixing consistent with the squashing independence

Let us consider the Schur limit of the 4D index by imposing (2.7). Under
(3.7), this condition is equivalent to

$\displaystyle m_{1}=\xi\left(M+i\frac{b}{2}\right)~{},\qquad
m_{2}=i\frac{b-b^{-1}}{2}~{},\qquad m_{3}=M-i\frac{b}{2}$ (3.8)

Recall that the Schur limit of the 4D index generally leads to $Z_{S^{3}_{b}}$
whose $b$-dependence can be removed by rescaling the remaining mass parameter
$M$ as

$\displaystyle M\to b^{-1}M~{}.$ (3.9)

Here we will show that this general constraint uniquely fixes the value of
$\xi$ in (3.8).

To that end, let us first look at $Z_{S^{3}_{b}}$ of the ABJM theory. The
supersymmetric localization leads to the following formula for $Z_{S^{3}_{b}}$
[21, 22, 23]:

$\displaystyle Z_{S^{3}_{b}}$
$\displaystyle\sim\int\frac{\prod_{i=1}^{N}d\mu_{i}d\nu_{i}}{(N!)^{2}}e^{i\pi
k\left(\sum_{i=1}^{N}(\nu_{i}^{2}-\mu_{i}^{2})\right)}\prod_{i<j}\sinh\left(\pi
b(\mu_{i}-\mu_{j})\right)\sinh\left(\pi b^{-1}(\mu_{i}-\mu_{j})\right)$
$\displaystyle\qquad\times\prod_{i<j}\sinh\left(\pi
b(\nu_{i}-\nu_{j})\right)\sinh\left(\pi b^{-1}(\nu_{i}-\nu_{j})\right)$
$\displaystyle\qquad\times\prod_{i,j=1}^{N}\Bigg{[}s_{b}\left(\frac{iQ}{4}-\left(\mu_{i}-\nu_{j}+\frac{m_{1}+m_{2}+m_{3}}{2}\right)\right)s_{b}\left(\frac{iQ}{4}-\left(\mu_{i}-\nu_{j}+\frac{m_{1}-m_{2}-m_{3}}{2}\right)\right)$
$\displaystyle\qquad\times
s_{b}\left(\frac{iQ}{4}-\left(-\mu_{i}+\nu_{j}+\frac{-m_{1}-m_{2}+m_{3}}{2}\right)\right)s_{b}\left(\frac{iQ}{4}-\left(-\mu_{i}+\nu_{j}+\frac{-m_{1}+m_{2}-m_{3}}{2}\right)\right)\Bigg{]}~{},$
(3.10)

where $s_{b}(z)$ is the double-sine function

$\displaystyle
s_{b}(z)\equiv\prod_{k,\ell=0}^{\infty}\frac{\left(k+\frac{1}{2}\right)b+\left(\ell+\frac{1}{2}\right)b^{-1}-iz}{\left(k+\frac{1}{2}\right)b+\left(\ell+\frac{1}{2}\right)b^{-1}+iz}~{}.$
(3.11)

Recall here that $m_{1}$ is the mass parameter associated with
$\mathfrak{u}(1)_{b}$, and $m_{2}$ and $m_{3}$ are those associated with
$\mathfrak{so}(2)^{2}\subset\mathfrak{so}(4)$. The above normalizations of
$m_{1},m_{2}$ and $m_{3}$ imply that a 3D (twisted) chiral multiplet has
$R_{1}^{\text{3D}}=R_{3}^{\text{3D}}=J_{U(1)_{b}}^{\text{3D}}=1/2$.

It was shown in [11] that when $m_{2}=i\frac{b-b^{-1}}{2}$, the above matrix
model expression (3.10) reduces to a function only of

$\displaystyle b^{-1}m_{+}~{},\qquad bm_{-}~{},$ (3.12)

where $m_{\pm}\equiv m_{3}\pm m_{1}$. This means that, when (3.8) is imposed,
$Z_{S^{3}_{b}}$ depends only on

$\displaystyle b^{-1}\left((\xi+1)M+(\xi-1)i\frac{b}{2}\right)~{},\qquad
b\left((\xi-1)M+(\xi+1)i\frac{b}{2}\right)~{}.$ (3.13)

Note that, for generic values of $\xi$ (including $\xi=0$), the $b$-dependence
of $Z_{S^{3}_{b}}$ cannot be removed by the rescale (3.9), which is
inconsistent with the general discussion on the squashing independence. For
the $S^{3}_{b}$ partition function of the ABJM theory to have the right
squashing independence, one must have

$\displaystyle\xi=-1~{}.$ (3.14)

Indeed, only for this value of $\xi$, the $b$-dependence of $Z_{S^{3}_{b}}$
can be removed by (3.9). Therefore we conclude that the correct value of $\xi$
is $\xi=-1$. Note that this implies a non-trivial mixing between the 4D
$\mathfrak{u}(3)_{R}$ symmetry and the $\mathfrak{u}(1)_{b}$ flavor symmetry
of the ABJM theory!

###  3.3 Consistency check with the other limits

In the previous section, we have seen that there are two more special values
of the mass parameters, (2.9) and (2.20), that give rise to squashing
independence in three dimensions. We here check if (3.14) is consistent with
it.

Let us first consider (2.9), corresponding to $t=pq$ in four dimensions. In
this case, our identification (3.7) of mass parameters reduces to

$\displaystyle m_{1}=\xi M~{},\qquad m_{2}=-i\frac{b+b^{-1}}{2}~{},\qquad
m_{3}=M~{}.$ (3.15)

When $\xi=-1$, this implies

$\displaystyle m_{+}\equiv m_{3}+m_{1}=0~{}.$ (3.16)

We now ask whether the squashing independence occurs when the above conditions
on the mass parameters are imposed. Indeed, it was shown in appendix A.1 of
[10] that the $b$-dependence of $Z_{S^{3}_{b}}$ disappears in the case that
$m_{2}=-i(b+b^{-1})/2$ and $m_{+}=0$.111111To be precise, the condition
discussed in [10] is $m_{3}=i(b+b^{-1})/2$ and $m_{1}-m_{2}=0$ . Since
$Z_{S^{3}_{b}}$ is invariant under $m_{2}\leftrightarrow m_{3}$, this is
equivalent to $m_{2}=i(b+b^{-1})/2$ and $m_{1}-m_{3}=0$. Using the invariance
of $Z_{S^{3}_{b}}$ under $m_{2}\to-m_{2}$ and $m_{3}\to-m_{3}$ , this leads to
the squashing independence at (3.16). This is a strong evidence for our
identification $\xi=-1$.

Another limit we consider is (2.20), corresponding to $a=\sqrt{t}/p$ in four
dimensions. In this case, we expect that the $b$-dependence of $Z^{3}_{b}$ can
be removed by replacing $m_{2}$ as

$\displaystyle m_{2}\to b^{-1}m_{2}-i\frac{b-b^{-1}}{2}~{}.$ (3.17)

To see this is indeed the case, first note that our identification (3.7) is
now expressed as

$\displaystyle m_{1}=\xi\left(m_{2}+ib\right)~{},\qquad
m_{3}=i\frac{b-b^{-1}}{2}~{}.$ (3.18)

It was shown in [11] that, when $m_{3}=i(b-b^{-1})/2$, the partition function
$Z_{S^{3}_{b}}$ depends only on $b^{-1}(m_{2}+m_{1})$ and $b(m_{2}-m_{1})$.
Therefore, in the case of $\xi=-1$, we see that $Z_{S^{3}_{b}}$ is a function
of the single combination of variables

$\displaystyle b\left(2m_{2}+ib\right)~{}.$ (3.19)

It is now straightforward to show that the $b$-dependence of $Z_{S^{3}_{b}}$
is removed by (3.17) as expected. This is another strong evidence for our
identification $\xi=-1$.

##  4 Divergence and flat directions

Having identified the correct R-symmetry mixing $\xi=-1$, we see that the mass
parameters $m_{1},m_{2}$ and $m_{3}$ of the ABJM theory are constrained by

$\displaystyle m_{1}=-\left(M+\frac{m}{2}+i\frac{b+b^{-1}}{4}\right)~{},\qquad
m_{2}=m~{},\qquad m_{3}=M-\frac{m}{2}-i\frac{b+b^{-1}}{4}~{},$ (4.1)

when it is obtained as the $S^{1}$-compactification of a 4D $\mathcal{N}=3$
SCFT. Here, $M$ is related to the fugacity $a$ for the $U(1)_{f}$ symmetry by
$a=e^{-i\beta M}$, while $m$ is related to the fugacity $t$ by $t=e^{-i\beta
m-(b+b^{-1})/2}$. The constraint $m_{1}+m_{2}+m_{3}=-i(b+b^{-1})/2$ is
interpreted to mean that the 3D global symmetry corresponding to the mass
parameter $(m_{1}+m_{2}+m_{3})$ is accidental in three dimensions and has no
counterpart in four dimensions.

One important fact is that the $S^{3}_{b}$ partition function of the ABJM
theory is always divergent when (4.1) is imposed. Indeed, the expression
(3.10) for $Z_{S^{3}_{b}}$ contains the factor

$\displaystyle\prod_{i,j=1}^{N}s_{b}\left(\frac{iQ}{4}-\left(\mu_{i}-\nu_{j}+\frac{m_{1}+m_{2}+m_{3}}{2}\right)\right)=\prod_{i,j=1}^{N}s_{b}\left(\frac{iQ}{2}-\mu_{i}+\nu_{j}\right)~{},$
(4.2)

which is divergent whenever $\mu_{i}=\nu_{j}$. The divergence in
$Z_{S^{3}_{b}}$ is schematically

$\displaystyle\left(s_{b}\left(\frac{iQ}{2}\right)\right)^{N}=\left(\prod_{k,\ell=0}^{\infty}\frac{(k+1)b+(\ell+1)b^{-1}}{kb+\ell
b^{-1}}\right)^{N}~{}.$ (4.3)

Note that this divergence occurs even for the most general values of the mass
parameters, $m$ and $M$, that have 4D counterparts.

A similar divergence appears in the small $S^{1}$ limit of the superconformal
index of $\mathcal{N}=4$ super Yang-Mills (SYM) theories [24], as we will
review below in this paragraph. Recall that the superconformal index is
equivalent to the partition function of the theory on $S^{1}\times S^{3}$ up
to a prefactor. For generic 4D $\mathcal{N}=2$ SCFTs, the small $S^{1}$ limit
of the $S^{1}\times S^{3}$ partition function $Z_{S^{1}\times S^{3}}$ behaves
as [8, 25, 26, 27, 28, 29, 30]

$\displaystyle\log Z_{S^{1}\times S^{3}}\sim\frac{8\pi^{2}}{\beta}(a-c)+\log
Z_{S^{3}}+\mathcal{O}(\beta)~{},$ (4.4)

where $a$ and $c$ are two conformal anomalies, and $Z_{S^{3}}$ is the $S^{3}$
partition function of the 3D reduction of the 4D theory. However, for
$\mathcal{N}=4$ super Yang-Mills theories, the above formula is modified as
[24]

$\displaystyle\log Z_{S^{1}\times S^{3}}\sim
N\log\frac{2\pi}{\beta}+\mathcal{O}(\beta^{0})~{},$ (4.5)

where $N$ is the complex dimension of the ($\mathcal{N}=2$) Coulomb branch.
Note that the first term in (4.4) drops out since $\mathcal{N}=4$
superconformal symmetry implies $a=c$. The divergent term $N\log(2\pi/\beta)$
in (4.5) is interpreted to mean that $Z_{S^{3}}$ obtained in the small $S^{1}$
limit of $Z_{S^{1}\times S^{3}}$ has a power-law divergence as [24]

$\displaystyle Z_{S^{3}}\sim\Lambda^{N}~{},$ (4.6)

where $\Lambda$ is the cutoff for the vacuum expectation value (VEV) of the
Coulomb branch operators. The expression (4.6) implies that there are $N$ flat
directions in the 3D Coulomb branch on $S^{3}$ when the 3D theory is obtained
by an RG-flow from four dimensions.121212Here, the 3D “Coulomb branch” is a
sub-space of the 3D moduli space on which the action of $\mathfrak{su}(2)_{R}$
is trivial, where $\mathfrak{su}(2)_{L}\times\mathfrak{su}(2)_{R}$ is an 3D
$\mathcal{N}=4$ R-symmetry. Note that this divergence is purely three-
dimensional; $Z_{S^{1}\times S^{3}}$ is not divergent when the radius of
$S^{1}$ is non-vanishing.

We now argue that our divergence in (4.3) is precisely of the form (4.6).
Indeed, each double-sine function in (4.2) arises from the path integral of a
massless chiral multiplet whose VEV parameterizes a sub-space of the Coulomb
branch. Therefore, the zeros in the denominator of (4.3) imply that the 3D
Coulomb branch has $N$ flat directions, leading to the behavior (4.6) when the
cutoff $\Lambda$ is introduced. The other directions on the Coulomb branch are
lifted by the background gauge fields and masses.131313Note that the FI-
parameter of the ABJM theory is already converted into a mass parameter via
$\mathcal{N}=6$ supersymmetry. Therefore, the Coulomb branch can be lifted by
changing the values of the mass parameters.

Note that the above $N$ flat directions disappear when the constraint
$m_{1}+m_{2}+m_{3}=-i(b+b^{-1})/2$ is relaxed. In other words, this constraint
on the mass parameters is such that the $N$ directions on the Coulomb branch
is unlifted. Here, $N$ is precisely the complex dimension of the 4D Coulomb
branch. This suggest that, even for 4D $\mathcal{N}=3$ SCFTs, the $\beta\to 0$
limit of the $S^{1}\times S^{3}$ partition function behaves as
(4.5).141414Note that any 4D $\mathcal{N}=3$ SCFT has $a=c$ [3] and therefore
the first term in the RHS of (4.4) vanishes for all 4D $\mathcal{N}\geq 3$
SCFTs.

The above discussion implies that our identification $\xi=-1$ for the
R-symmetry mixing also resolves the second contradiction discussed in Sec. 1.
To see this, let us focus on the 4D rank-one $\mathcal{N}=3$ SCFTs by setting
$N=1$. The 4D moduli space is then $\mathbb{C}^{3}/\mathbb{Z}_{k}$ for $k=3,4$
or $6$, and the 3D moduli space is $\mathbb{C}^{4}/\mathbb{Z}_{k}$. We denote
their local coordinates by $(z_{1},z_{2},z_{3})$ and
$(z_{1},z_{2},z_{3},z_{4})$, respectively. The extra coordinate $z_{4}$ of the
3D moduli space corresponds to the VEV of the scalar field in the massless
chiral multiplet discussed above. The fact that the $z_{4}$-direction is flat
(even with the most general mass parameters arising from four dimensions
turned on) implies that, under our identification of the R-charges (3.2) with
$\xi=-1$, the coordinate $z_{4}$ is neutral under $\mathfrak{u}(3)_{R}$ as
expected. This resolves the second contradiction. Note that this neutrality of
$z_{4}$ is possible only when there is a non-trivial mixing between the
$\mathfrak{u}(1)_{c}$ and $\mathfrak{u}(1)_{b}$.

Finally, we comment that there is an extra divergence in $Z_{S^{3}_{b}}$ when
we set $m_{2}=m=-i(b+b^{-1})/2$. Indeed, in this case, we have $m_{1}+m_{3}=0$
and therefore

$\displaystyle\prod_{i,j=1}^{N}s_{b}\left(\frac{iQ}{4}-\left(-\mu_{i}+\nu_{j}+\frac{-m_{1}+m_{2}-m_{3}}{2}\right)\right)$
(4.7)

has exactly the same divergence as (4.2). The reason for this extra divergence
is that, when $m=-i(b+b^{-1})/2$, the 4D partition function $Z_{S^{1}\times
S^{3}}$ is already divergent even before taking the small $S^{1}$ limit. To
see this, recall that $m=-i(b+b^{-1})/2$ corresponds to $t=pq$ in the 4D
index, as discussed in Sec. 2.1. Since a 4D Coulomb branch operator of
$U(1)_{r}$ charge $r$ contributes $(t/pq)^{r}$ to the index,151515It is known
that every Coulomb branch operator in any 4D $\mathcal{N}\geq 2$ SCFT is
neutral under $\mathcal{N}=2$ flavor symmetries [17]. See also [18] for its
higher spin generalization. the condition $t=pq$ sets all the index
contributions from an infinite number of Coulomb branch operators to $1$,
leading to a divergence. This reflects the fact that the 4D Coulomb branch is
not lifted when $t=pq$. This is in contrast to the case of generic values of
$m$, in which case the 4D index is finite but the 3D partition function
$Z_{S^{3}}$ is divergent.

##  5 Conclusions and Discussions

In this paper, we have studied the $S^{1}$-compactification of 4D
$\mathcal{N}=3$ SCFTs, focusing on the relation between the 4D superconformal
index and 3D partition function $Z_{S^{3}_{b}}$. In particular, we have argued
that the center $\mathfrak{u}(1)$ of $\mathfrak{u}(3)_{R}$ in four dimensions
can mix with an abelian flavor symmetry of the 3D $\mathcal{N}=6$ theory
obtained by the compactification. In the case that the 3D theory is the ABJM
theory, we have shown that such an R-symmetry mixing does occur and is
uniquely fixed so that the Schur limit (and/or its $\mathcal{N}=3$ cousin) of
the 4D index correctly reproduces the squashing independence of
$Z_{S^{3}_{b}}$ found in [11]. Our result implies that the recent discussions
in [10] on the connection between supersymmetry enhancement of the 4D index
and squashing independence of $Z_{S^{3}_{b}}$ can also be applied to the ABJM
theories.

The R-symmetry mixing we have found is also consistent with the expectation
that the 4D $\mathfrak{u}(3)_{R}$ trivially acts on the sub-space of the 3D
Coulomb branch that does not have a 4D counterpart (and therefore purely
three-dimensional). This trivial action of $\mathfrak{u}(3)_{R}$ implies that
the 3D limit of 4D $\mathcal{N}=3$ index leads to a divergence. A similar
divergence appears in the case of 4D $\mathcal{N}=4$ SYM theories. This seems
to suggest that the small $S^{1}$ limit of the superconformal index of 4D
$\mathcal{N}\geq 3$ SCFTs always behaves as (4.5).

We note that the R-symmetry mixing that we have discussed in this paper is
similar (but different) to the one for the compactification of $\mathcal{N}=2$
Argyres-Douglas (AD) SCFTs [9]. When a generic AD theory is compactified on
$S^{1}$, there is a mixing between the 4D $\mathfrak{u}(1)_{r}$ symmetry and
3D topological $\mathfrak{u}(1)$ global symmetry. This is because the 4D
$\mathcal{N}=2$ R-symmetry, $\mathfrak{su}(2)_{R}\times\mathfrak{u}(1)_{r}$,
needs to enhance to the 3D $\mathcal{N}=4$ R-symmetry,
$\mathfrak{so}(4)_{R}\simeq\mathfrak{su}(2)_{R}\times\mathfrak{su}(2)_{C}$,
under the compactification; without the mixing between $\mathfrak{u}(1)_{r}$
and the topological $\mathfrak{u}(1)$, such an enhancement is prohibited by
the presence of Coulomb branch operators of fractional $\mathfrak{u}(1)_{r}$
charges. The R-symmetry mixing we have discussed in this paper is, in
contrast, the mixing between the center of the 4D $\mathcal{N}=3$ R-symmetry,
$\mathfrak{u}(3)_{R}$, and the 3D $\mathfrak{u}(1)_{b}$ symmetry. Since
$\mathfrak{u}(1)_{b}$ can be regarded as a topological $\mathfrak{u}(1)$
symmetry of the Chern-Simons matter theory, our R-symmetry mixing shares some
characteristics with the one for the AD theories. A big difference is,
however, 4D $\mathcal{N}=3$ SCFTs have no Coulomb branch operators of
fractional $\mathfrak{u}(1)_{r}$ charges. The R-symmetry mixing in our case
just reflects the $\mathcal{N}=3$ superconformal symmetry.

In general, our result gives a necessary condition for a 3D $\mathcal{N}=6$
SCFT to have its 4D uplift. Indeed, whenever a 3D $\mathcal{N}=6$ SCFT can be
realized by compactifying a 4D $\mathcal{N}=3$ SCFT, there must be special
sub-spaces in the space of 3D mass parameters in which $Z_{S^{3}}$ is
squashing independent. These sub-spaces correspond to the 4D Schur limit $t=q$
and its $\mathcal{N}=3$ cousins $a=\sqrt{t}/p$.161616The 3D partition function
$Z_{S^{3}}$ must also be squashing independent at $t=pq$, when the 3D theory
has a 4D uplift. Therefore, if the space of mass parameters of a 3D
$\mathcal{N}=6$ SCFT has no such special sub-spaces, one can conclude that
there exists no 4D $\mathcal{N}=3$ SCFT whose $S^{1}$-compactification gives
rise to that 3D $\mathcal{N}=6$ SCFT.

There are clearly many future directions, some of which we list below:

  * •

While we have focused on 4D $\mathcal{N}=3$ SCFTs whose 3D reduction is the
ABJM theory, there are many other $\mathcal{N}=6$ Chern-Simons matter theories
[31, 32]. It would be interesting to identify the R-symmetry mixing for these
other cases. In particular, the generalization to the ABJ theory must be
straightforward.

  * •

It is known that the stress-tensor multiplet of every 3D $\mathcal{N}=6$ SCFT
contains an $\mathcal{N}=6$ flavor current [33] (See also Sec. 5.4.6 of [34]).
In the case of ABJM theories, this is the $\mathfrak{u}(1)_{b}$ current. Based
on our discussion, it is natural to expect that there is a mixing between the
4D $\mathfrak{u}(3)_{R}$ current and this $\mathcal{N}=6$ flavor current. It
would be interesting to see if there is a universal formula for this
R-symmetry mixing.

  * •

It is desirable to obtain a closed form expression for the superconformal
index (or at least its Schur limit) of a 4D $\mathcal{N}=3$ SCFT. Several
limits of the superconformal index of 4D $\mathcal{N}=3$ SCFTs are computed in
[35, 36, 37, 38, 39], but a closed form expression for the index with a non-
trivial flavor fugacity is not known. Once we find such an expression, we
should be able to study its small $S^{1}$ limit. Then it would be interesting
to see if this small $S^{1}$ limit is identical (up to a prefactor) to
$Z_{S^{3}_{b}}$ of the 3D $\mathcal{N}=6$ theory. For this purpose, one needs
to take into account the R-symmetry mixing that we discussed in this paper.

Acknowledgments

We are grateful to M. Buican, Y. Hatsuda, Y. Nakayama, Y. Tachikawa and Y.
Yoshida for helpful discussions. T. Nakanishi’s research is partially
supported by JST, the establishment of university fellowships towards the
creation of science technology innovation, Grant Number JPMJFS2138. T.
Nishinaka’s research is partially supported by JSPS KAKENHI Grant Numbers
JP18K13547 and JP21H04993. This work was also partially supported by Osaka
Central Advanced Mathematical Institute: MEXT Joint Usage/Research Center on
Mathematics and Theoretical Physics JP- MXP0619217849.

## Appendix A. 4D $\mathcal{N}=3$ and 3D $\mathcal{N}=6$ superconformal
algebras

In this section, we discuss in some more detail how the 4D and 3D symmetries
are related under the $S^{1}$-compactification.

###  A.1 4D $\mathcal{N}=3$ superconformal algebra

In this subsection, we review the 4D $\mathcal{N}=3$ superconformal algebra
[40]. We follow the convention of Appendix A of [13] unless otherwise stated.
We use $\sigma^{\mu}=(I_{2},\sigma^{1},\sigma^{3},\sigma^{2})$ and
$\overline{\sigma}^{\mu}=(-I_{2},\sigma^{1},\sigma^{3},\sigma^{2})$ with
$I_{2}$ being the $2\times 2$ identity matrix.171717Note that this means
$\overline{\sigma}^{\mu\dot{\alpha}\alpha}=\epsilon^{\alpha\beta}\sigma^{\mu}_{\beta\dot{\beta}}\epsilon^{\dot{\beta}\dot{\alpha}}$.

The 4D $\mathcal{N}=3$ superconformal algebra (for Minkowski signature) is
$\mathfrak{su}(2,2|3)$, which contains
$\mathfrak{su}(2,2)\oplus\mathfrak{u}(3)_{R}$ as the bosonic sub-algebra. We
take a basis of $\mathfrak{su}(2,2|3)$ as

$\displaystyle\overbrace{\mathcal{M}_{\alpha}\\!^{\beta}~{},\quad\widetilde{\mathcal{M}}^{\dot{\alpha}}\\!_{\dot{\beta}}~{},\quad\mathcal{P}_{\alpha\dot{\beta}}~{},\quad\mathcal{K}^{\dot{\alpha}\beta}~{},\quad\mathcal{H}}^{\mathfrak{su}(2,2)}~{},\quad\overbrace{\mathcal{R}^{I}\\!_{J}~{}}^{\mathfrak{u}(3)_{R}},\quad\mathcal{Q}^{I}\\!_{\alpha}~{},\quad\widetilde{\mathcal{Q}}_{I\dot{\alpha}}~{},\quad\mathcal{S}_{I}\\!^{\alpha}~{},\quad\widetilde{\mathcal{S}}^{I\dot{\alpha}}~{},$
(A.1)

where $I,J=1,2,3$, and the first six are bosonic while the last four are
fermionic. The non-vanishing commutation relations for the bosonic charges are
written as

$\displaystyle[\mathcal{M}_{\alpha}{}^{\beta},\,\mathcal{M}_{\gamma}{}^{\delta}]=\delta_{\gamma}^{\beta}\mathcal{M}_{\alpha}{}^{\delta}-\delta_{\alpha}^{\delta}\mathcal{M}_{\gamma}{}^{\beta}~{}$
$\displaystyle,\qquad[\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\beta}},\,\widetilde{\mathcal{M}}^{\dot{\gamma}}{}_{\dot{\delta}}]=\delta^{\dot{\alpha}}_{\dot{\delta}}\widetilde{\mathcal{M}}^{\dot{\gamma}}{}_{\dot{\beta}}-\delta^{\dot{\gamma}}_{\dot{\beta}}\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\delta}}~{},$
(A.2)
$\displaystyle[\mathcal{M}_{\alpha}{}^{\beta},\,\mathcal{P}_{\gamma\dot{\gamma}}]=\mathcal{P}_{\alpha\dot{\gamma}}\delta^{\beta}_{\gamma}-\frac{1}{2}\delta^{\beta}_{\alpha}\mathcal{P}_{\gamma\dot{\gamma}}~{}$
$\displaystyle,\qquad[\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\beta}},\,\mathcal{P}_{\gamma\dot{\gamma}}]=\delta^{\dot{\alpha}}_{\dot{\gamma}}\mathcal{P}_{\gamma\dot{\beta}}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{P}_{\gamma\dot{\gamma}}~{},$
(A.3)
$\displaystyle[\mathcal{M}_{\alpha}{}^{\beta},\,\mathcal{K}^{\dot{\gamma}\gamma}]=-\mathcal{K}^{\dot{\gamma}\beta}\delta^{\gamma}_{\alpha}+\frac{1}{2}\delta^{\beta}_{\alpha}\mathcal{K}^{\dot{\gamma}\gamma}~{}$
$\displaystyle,\qquad[\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\beta}},\,\mathcal{K}^{\dot{\gamma}\gamma}]=-\mathcal{K}^{\dot{\alpha}\gamma}\delta^{\dot{\gamma}}_{\dot{\beta}}+\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{K}^{\dot{\gamma}\gamma}~{},$
(A.4)
$\displaystyle[\mathcal{K}^{\dot{\alpha}\alpha},\,\mathcal{P}_{\beta\dot{\beta}}]$
$\displaystyle=\frac{1}{4}(\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{M}_{\beta}{}^{\alpha}+\delta^{\alpha}_{\beta}\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\beta}}+\delta^{\alpha}_{\beta}\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{H})~{},$
(A.5)
$\displaystyle[\mathcal{H},\,\mathcal{P}_{\alpha\dot{\alpha}}]=\mathcal{P}_{\alpha\dot{\alpha}}~{}$
$\displaystyle,\qquad[\mathcal{H},\,\mathcal{K}^{\dot{\alpha}\alpha}]=-\mathcal{K}^{\dot{\alpha}\alpha}~{},$
(A.6)
$\displaystyle\qquad\quad[\mathcal{R}^{I}\\!_{J},\mathcal{R}^{K}\\!_{L}]=$
$\displaystyle\delta^{K}_{J}\mathcal{R}^{I}\\!_{L}-\delta^{I}_{L}\mathcal{R}^{K}\\!_{J}~{}.$
(A.7)

Our notation for the R-charges is such that

$\displaystyle[\mathcal{R}^{I}\\!_{J},\mathcal{Q}^{K}_{\alpha}]=\delta^{K}_{J}\mathcal{Q}^{I}\\!_{\alpha}-\frac{1}{4}\delta^{I}_{J}\mathcal{Q}^{K}\\!_{\alpha}~{},$
$\displaystyle\quad[\mathcal{R}^{I}\\!_{J},\widetilde{\mathcal{Q}}_{K\dot{\alpha}}]=-\delta^{I}_{K}\widetilde{\mathcal{Q}}_{J\dot{\alpha}}+\frac{1}{4}\delta^{I}_{J}\widetilde{\mathcal{Q}}_{K\dot{\alpha}}~{},$
(A.8)
$\displaystyle[\mathcal{R}^{I}\\!_{J},\widetilde{\mathcal{S}}^{K\dot{\alpha}}]=\delta^{K}_{J}\widetilde{\mathcal{S}}^{I\dot{\alpha}}-\frac{1}{4}\delta^{I}_{J}\widetilde{\mathcal{S}}^{K\dot{\alpha}}~{},$
$\displaystyle\quad[\mathcal{R}^{I}\\!_{J},\mathcal{S}_{K}\\!^{\alpha}]=-\delta^{I}_{K}\mathcal{S}_{J}\\!^{\alpha}+\frac{1}{4}\delta^{I}_{J}\mathcal{S}_{K}\\!^{\alpha}~{}.$
(A.9)

The non-trivial anti-commutation relations for the supercharges are written as

$\displaystyle\\{\mathcal{Q}^{I}_{\alpha},\widetilde{\mathcal{Q}}_{J\dot{\alpha}}\\}=2\delta^{I}_{J}\mathcal{P}_{\alpha\dot{\alpha}}~{}$
$\displaystyle,\quad\\{\mathcal{Q}^{I}_{\alpha},\mathcal{S}_{J}^{\beta}\\}=\frac{1}{2}\delta^{I}_{J}\delta^{\beta}_{\alpha}\mathcal{H}+\delta^{I}_{J}\mathcal{M}_{\alpha}\\!^{\beta}-\delta^{\beta}_{\alpha}\mathcal{R}^{I}\\!_{J}~{},$
(A.10)
$\displaystyle\\{\widetilde{\mathcal{S}}^{I\dot{\alpha}},\mathcal{S}_{J}^{\alpha}\\}=2\delta^{I}_{J}\mathcal{K}^{\dot{\alpha}\alpha}~{}$
$\displaystyle,\quad\\{\widetilde{\mathcal{S}}^{I\dot{\alpha}},\widetilde{\mathcal{Q}}_{J\dot{\beta}}\\}=\frac{1}{2}\delta^{I}_{J}\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{H}+\delta^{I}_{J}\widetilde{\mathcal{M}}^{\dot{\alpha}}\\!_{\dot{\beta}}+\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{R}^{I}\\!_{J}~{}.$
(A.11)

Here, our normalization of $\mathcal{P}_{\alpha\dot{\alpha}}$ and
$\mathcal{K}^{\dot{\alpha}\alpha}$ are different from that in [13] by a factor
two. The remaining non-vanishing commutation relations are as follows:

$\displaystyle[\mathcal{H},\mathcal{Q}^{I}\\!_{\alpha}]=\frac{1}{2}\mathcal{Q}^{I}\\!_{\alpha}~{},\quad[\mathcal{H},\widetilde{\mathcal{Q}}_{I\dot{\alpha}}]=\frac{1}{2}\widetilde{\mathcal{Q}}_{I\dot{\alpha}}~{}$
$\displaystyle,\quad[\mathcal{H},\mathcal{S}_{I}\\!^{\alpha}]=-\frac{1}{2}\mathcal{S}_{I}\\!^{\alpha}~{},\quad[\mathcal{H},\widetilde{\mathcal{S}}^{I\dot{\alpha}}]=-\frac{1}{2}\widetilde{\mathcal{S}}^{I\dot{\alpha}}~{},$
(A.12)
$\displaystyle[\mathcal{M}_{\alpha}\\!^{\beta},\mathcal{Q}^{I}\\!_{\gamma}]=\delta^{\beta}_{\gamma}\mathcal{Q}^{I}\\!_{\alpha}-\frac{1}{2}\delta^{\beta}_{\alpha}\mathcal{Q}^{I}\\!_{\gamma}~{}$
$\displaystyle,\quad[\mathcal{M}_{\alpha}\\!^{\beta},\mathcal{S}_{I}\\!^{\gamma}]=-\delta^{\gamma}_{\alpha}\mathcal{S}_{I}\\!^{\beta}+\frac{1}{2}\delta^{\beta}_{\alpha}\mathcal{S}_{I}\\!^{\gamma}~{},$
(A.13)
$\displaystyle[\widetilde{\mathcal{M}}^{\dot{\alpha}}\\!_{\dot{\beta}},\widetilde{\mathcal{Q}}_{I\dot{\gamma}}]=\delta^{\dot{\alpha}}_{\dot{\gamma}}\widetilde{\mathcal{Q}}_{I\dot{\beta}}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\widetilde{\mathcal{Q}}^{I}\\!_{\gamma}~{}$
$\displaystyle,\quad[\widetilde{\mathcal{M}}^{\dot{\alpha}}\\!_{\dot{\beta}},\widetilde{\mathcal{S}}^{I\dot{\gamma}}]=-\delta^{\dot{\gamma}}_{\dot{\beta}}\widetilde{\mathcal{S}}^{I\dot{\alpha}}+\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\widetilde{\mathcal{S}}^{I\dot{\gamma}}~{},$
(A.14)
$\displaystyle[\mathcal{K}^{\dot{\alpha}\alpha},\mathcal{Q}^{I}\\!_{\beta}]=\frac{1}{2}\delta^{\alpha}_{\beta}\widetilde{\mathcal{S}}^{I\dot{\alpha}}~{}$
$\displaystyle,\quad[\mathcal{K}^{\dot{\alpha}\alpha},\widetilde{\mathcal{Q}}_{I\dot{\beta}}]=\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{S}_{I}\\!^{\alpha}~{},$
(A.15)
$\displaystyle[\mathcal{P}_{\alpha\dot{\alpha}},\mathcal{S}_{I}^{\beta}]=-\frac{1}{2}\delta^{\beta}_{\alpha}\widetilde{\mathcal{Q}}_{I\dot{\alpha}}~{}$
$\displaystyle,\quad[\mathcal{P}_{\alpha\dot{\alpha}},\widetilde{\mathcal{S}}^{I\dot{\beta}}]=-\frac{1}{2}\delta^{\dot{\beta}}_{\dot{\alpha}}\mathcal{Q}^{I}\\!_{\alpha}~{}.$
(A.16)

The hermiticity is such that

$\displaystyle\mathcal{H}^{\dagger}$
$\displaystyle=\mathcal{H}~{},\qquad(\mathcal{P}_{\alpha\dot{\beta}})^{\dagger}=\mathcal{K}^{\dot{\beta}\alpha}~{},\qquad(\mathcal{M}_{\alpha}{}^{\beta})^{\dagger}=\mathcal{M}_{\beta}{}^{\alpha}~{},\qquad(\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\beta}})^{\dagger}=\widetilde{\mathcal{M}}^{\dot{\beta}}{}_{\dot{\alpha}}~{},$
(A.17) $\displaystyle(\mathcal{Q}^{I}_{\alpha})^{\dagger}$
$\displaystyle=\mathcal{S}_{I}^{\alpha}~{},\qquad(\widetilde{\mathcal{Q}}_{I\dot{\alpha}})^{\dagger}=\widetilde{\mathcal{S}}^{I\dot{\alpha}}~{},\qquad(\mathcal{R}^{I}{}_{J})^{\dagger}=\mathcal{R}^{J}{}_{I}~{}.$
(A.18)

###  A.2 3D $\mathcal{N}=6$ superconformal algebra

We here review the 3D $\mathcal{N}=6$ superconformal algebra, following
Appendix B of [41]. We use the 3D gamma matrices
$\gamma^{i}_{\alpha\beta}=(I_{2},\,\sigma^{1},\,\sigma^{3})_{\alpha\beta}$ and
$\overline{\gamma}^{i\alpha\beta}=(-I_{2},\,\sigma^{1},\,\sigma^{3})$.

The 3D $\mathcal{N}=6$ superconformal algebra is $\mathfrak{osp}(6|4)$ whose
basis we take as

$\displaystyle\overbrace{M_{\alpha}{}^{\beta}~{},\quad
P_{\alpha\beta}~{},\quad K^{\alpha\beta}~{},\quad
D}^{\mathfrak{so}(3,2)}~{},\quad\overbrace{R_{rs}~{}}^{\mathfrak{so}(6)_{R}},\quad
Q_{\alpha r}~{},\quad S^{\alpha}{}_{r}~{},$ (A.19)

where $P_{\alpha\beta}$ and $K^{\alpha\beta}$ are symmetric under
$\alpha\leftrightarrow\beta$ while $R_{st}$ is anti-symmetric under
$s\leftrightarrow t$. They satisfy the following (anti-)commutation relations:

$\displaystyle[M_{\alpha}{}^{\beta},\,M_{\gamma}{}^{\delta}]$
$\displaystyle=-\delta_{\alpha}^{\delta}M_{\gamma}{}^{\beta}+\delta^{\beta}_{\gamma}M_{\alpha}{}^{\delta}~{},$
(A.20) $\displaystyle[M_{\alpha}{}^{\beta},\,P_{\gamma\delta}]$
$\displaystyle=\delta^{\beta}_{\gamma}P_{\alpha\delta}+\delta^{\beta}_{\delta}P_{\alpha\gamma}-\delta^{\beta}_{\alpha}P_{\gamma\delta}~{},$
(A.21) $\displaystyle[M_{\alpha}{}^{\beta},\,K^{\gamma\delta}]$
$\displaystyle=-\delta^{\gamma}_{\alpha}K^{\beta\delta}-\delta^{\delta}_{\alpha}K^{\beta\gamma}+\delta^{\beta}_{\alpha}K^{\gamma\delta}~{},$
(A.22) $\displaystyle[K^{\alpha\beta},\,P_{\gamma\delta}]$
$\displaystyle=4\delta^{(\alpha}_{(\gamma}M_{\delta)}{}^{\beta)}+4\delta^{(\alpha}_{(\gamma}\delta^{\beta)}_{\delta)}D~{},$
(A.23) $\displaystyle[D,\,P_{\alpha\beta}]=P_{\alpha\beta}~{}$
$\displaystyle,\qquad[D,\,K^{\alpha\beta}]=-K^{\alpha\beta}~{},$ (A.24)
$\displaystyle[R_{rs},\,R_{tu}]$
$\displaystyle=i\left(\delta_{rt}R_{su}-\delta_{st}R_{ru}-\delta_{ru}R_{st}+\delta_{su}R_{rt}\right)~{},$
(A.25) $\displaystyle[R_{rs},\,Q_{\alpha t}]$
$\displaystyle=i(\delta_{rt}Q_{\alpha s}-\delta_{st}Q_{\alpha r})~{},$ (A.26)
$\displaystyle[R_{rs},\,S^{\alpha}{}_{t}]$
$\displaystyle=i\left(\delta_{rt}S^{\alpha}{}_{s}-\delta_{st}S^{\alpha}{}_{r}\right)$
(A.27) $\displaystyle\\{Q_{\alpha r},\,Q_{\beta
s}\\}=2\delta_{rs}P_{\alpha\beta}~{}$
$\displaystyle,\qquad\\{S^{\alpha}{}_{r},\,S^{\beta}{}_{s}\\}=-2\delta_{rs}K^{\alpha\beta}~{},$
(A.28) $\displaystyle\\{Q_{\alpha r},\,S^{\beta}{}_{s}\\}$
$\displaystyle=2i\left(\delta_{rs}\delta^{\beta}_{\alpha}D+\delta_{rs}M_{\alpha}{}^{\beta}-i\delta^{\beta}_{\alpha}R_{rs}\right)$
(A.29) $\displaystyle[D,Q_{\alpha r}]=\frac{1}{2}Q_{\alpha r}~{}$
$\displaystyle,\qquad[D,S^{\alpha}{}_{r}]=-\frac{1}{2}S^{\alpha}{}_{r}~{},$
(A.30) $\displaystyle[M_{\alpha}{}^{\beta},\,Q_{\gamma r}]$
$\displaystyle=\delta^{\beta}_{\gamma}Q_{\alpha
r}-\frac{1}{2}\delta^{\alpha}_{\beta}Q_{\gamma r}~{},$ (A.31)
$\displaystyle[M_{\alpha}{}^{\beta},\,S^{\gamma}{}_{r}]$
$\displaystyle=-\delta^{\gamma}_{\alpha}S^{\beta}{}_{r}+\frac{1}{2}\delta^{\beta}_{\alpha}S^{\gamma}{}_{r}~{},$
(A.32) $\displaystyle[K^{\alpha\beta},\,Q_{\gamma
r}]=-2i\delta^{(\alpha}_{\gamma}S^{\beta)}{}_{r}~{}$
$\displaystyle,\qquad[P_{\alpha\beta},\,S^{\gamma}{}_{r}]=-2i\delta^{\gamma}_{(\alpha}Q_{\beta)r}$
(A.33)

The hermiticity is expressed as

$\displaystyle
D^{\dagger}=D~{},\qquad(P_{\alpha\beta})^{\dagger}=K^{\alpha\beta}~{},\qquad(M_{\alpha}{}^{\beta})^{\dagger}=M_{\beta}{}^{\alpha}~{},\qquad(Q_{\alpha
r})^{\dagger}=-iS^{\alpha}{}_{r}~{},\qquad(R_{rs})^{\dagger}=R_{rs}~{}.$
(A.34)

###  A.3 Reducting 4D $\mathcal{N}=3$ to 3D $\mathcal{N}=6$

Let us consider the $S^{1}$-compactification of a 4D $\mathcal{N}=3$ SCFT,
which leads to a 3D $\mathcal{N}=6$ SCFT in the deep infrared. We here discuss
how the 4D $\mathcal{N}=3$ supersymmetry is related to the 3D $\mathcal{N}=6$
supersymmetry.

Note that the $S^{1}$-compactification explicitly breaks the conformal
symmetry. We therefore focus on the super Poincaré sub-algebra generated by
$\mathcal{Q}_{\alpha}^{I},\,\widetilde{\mathcal{Q}}_{I\dot{\alpha}},\,\mathcal{R}^{I}{}_{J},\,\mathcal{M}_{\alpha}{}^{\beta},\,\widetilde{\mathcal{M}}^{\dot{\alpha}}{}_{\dot{\alpha}}$
and $\mathcal{P}_{\alpha\dot{\beta}}$. The relevant (anti-)commutation
relations are Eqs. (A.2), (A.3), (A.7), (A.8) and the left relations of
(A.10), (A.13) and (A.14) . Below, we will identify how these relations are
mapped to the (anti-)commutation relations for the 3D $\mathcal{N}=6$ super
Poincaré algebra.

To that end, we first identify the 3D supercharges as

$\displaystyle Q_{\alpha
r}=\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2}}\left(\mathcal{Q}^{I}_{\alpha}+\widetilde{\mathcal{Q}}_{I\alpha}\right)\quad\text{when}\;\;r=2I-1\;\;\text{
for }\;\;I\in\\{1,2,3\\}\\\
\frac{i}{\sqrt{2}}\left(\mathcal{Q}^{I}_{\alpha}-\widetilde{\mathcal{Q}}_{I\alpha}\right)\quad\text{when}\;\;r=2I\;\;\text{
for }\;\;I\in\\{1,2,3\\}\\\ \end{array}\right.~{}.$ (A.37)

Note that the 3D Lorentz symmetry
$\mathfrak{so}(1,2)\simeq\mathfrak{sl}(2,\mathbb{R})$ is the diagonal sub-
algebra of the 4D Lorentz symmetry
$\mathfrak{so}(1,3)\simeq\mathfrak{sl}(2,\mathbb{C})\simeq\mathfrak{sl}(2,\mathbb{R})^{2}$,
and therefore we identify $\dot{\alpha}$ and $\alpha$ under the
compactification. This means that the 3D Lorentz generator is identified as

$\displaystyle
M_{\alpha}{}^{\beta}=\mathcal{M}_{\alpha}{}^{\beta}+\widetilde{\mathcal{M}}^{\beta}{}_{\alpha}~{}.$
(A.38)

The three-dimensional translations are

$\displaystyle
P_{\alpha\beta}=\frac{1}{2}\left(\mathcal{P}_{\alpha\beta}+\mathcal{P}_{\beta\alpha}\right)~{},$
(A.39)

which removes the “fourth” component of the momentum. One can check that these
identifications correctly reproduce the 3D (anti-)commutation relations
(A.20), (A.21), (A.31) and the first relation in (A.28).181818In the reduction
to three dimensions, we set
$\mathcal{P}_{\alpha\beta}-\mathcal{P}_{\beta\alpha}=0$ since we only keep the
zero modes in the compactified direction.

We now turn to the R-symmetry. Recall that the 4D R-symmetry is
$\mathfrak{u}(3)_{R}$ while the 3D R-symmetry is $\mathfrak{so}(6)_{R}$. Along
the RG-flow from four dimensions to three dimensions, the manifest global
symmetry is only $\mathfrak{u}(3)_{R}$, which is in the deep infrared
identified with a sub-algebra of the product of $\mathfrak{so}(6)_{R}$ and the
$\mathcal{N}=6$ flavor symmetry. In the main text, we argue that the center
$\mathfrak{u}(1)$ of $\mathfrak{u}(3)_{R}$ is mixed with the $\mathcal{N}=6$
flavor symmetry. This is generally expressed (up to automorphims) as

$\displaystyle\mathcal{R}^{1}{}_{1}$
$\displaystyle=-\frac{3}{4}R_{12}+\frac{1}{4}R_{34}+\frac{1}{4}R_{56}+\frac{\xi}{4}J_{\text{flavor}}^{\text{3D}}~{},$
(A.40) $\displaystyle\mathcal{R}^{2}{}_{2}$
$\displaystyle=+\frac{1}{4}R_{12}-\frac{3}{4}R_{34}+\frac{1}{4}R_{56}+\frac{\xi}{4}J^{\text{3D}}_{\text{flavor}}~{},$
(A.41) $\displaystyle\mathcal{R}^{3}{}_{3}$
$\displaystyle=+\frac{1}{4}R_{12}+\frac{1}{4}R_{34}-\frac{3}{4}R_{56}+\frac{\xi}{4}J^{\text{3D}}_{\text{flavor}}~{},$
(A.42)

where $J^{\text{3D}}_{\text{flavor}}$ is an $\mathcal{N}=6$ flavor symmetry
that mixes with the center of $\mathfrak{u}(3)_{R}$ and $\xi$ is a real
parameter. When the 3D theory is the ABJM theory,
$J^{\text{3D}}_{\text{flavor}}$ is the $\mathfrak{u}(1)_{b}$ charge.191919In
comparison to Eq. (3.2), we identify
$J^{\text{3D}}_{\text{flavor}}=\frac{1}{4}J_{U(1)_{b}}^{\text{3D}}$.and
$\xi=-1$ for the normalization that we used in the main text.

One can also identify how the non-abelian part of the 4D R-symmetry,
$\mathfrak{su}(3)\subset\mathfrak{u}(3)_{R}$, is embedded in the 3D global
symmetry. Indeed, under the $S^{1}$-compactification, the 4D R-charges
$\mathcal{R}^{I}{}_{J}$ for $I\neq J$ are identified as the following linear
combinations of the 3D R-charges $R_{rs}$:202020One can summarize these six
relations in
$\mathcal{R}^{I}{}_{J}=\frac{1}{2}\left(R_{2J\,2I-1}+R_{2I\,2J-1}-iR_{2J-1\,2I-1}-iR_{2J\,2I}\right)$.
Note that $R_{rs}=-R_{sr}$.

$\displaystyle\mathcal{R}^{1}{}_{2}$
$\displaystyle=\frac{1}{2}(-R_{14}+R_{23}+iR_{13}+iR_{24})~{},\qquad\mathcal{R}^{2}{}_{1}=\frac{1}{2}(-R_{14}+R_{23}-iR_{13}-iR_{24})~{},$
(A.43) $\displaystyle\mathcal{R}^{2}{}_{3}$
$\displaystyle=\frac{1}{2}(-R_{36}+R_{45}+iR_{35}+iR_{46})~{},\qquad\mathcal{R}^{3}{}_{2}=\frac{1}{2}(-R_{36}+R_{45}-iR_{35}-iR_{46})~{},$
(A.44) $\displaystyle\mathcal{R}^{3}{}_{1}$
$\displaystyle=\frac{1}{2}(-R_{16}+R_{25}-iR_{15}-iR_{26})~{},\qquad\mathcal{R}^{1}{}_{3}=\frac{1}{2}(-R_{16}+R_{25}+iR_{15}+iR_{26})~{}.$
(A.45)

It is straightforward to see that the above identifications are consistent
with the 3D and 4D commutation relations for the R-charges and supercharges.

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P2H4 | Ag |  CH4:PH3 |  5 keV e- | 130 | 0.017 | 19 | 2897 | 282 | – | – | 2.2E+18 | 6180 | Turner et al. (2016)
CH2OCHCN | HOPG |  C2H3CN | O | 145 | 0.017 | 17 | 18968 | 328 | – | – | 1.9E+19 | 7220 | Kimber et al. (2018)
CH2OCHCN | HOPG |  C2H3CN | O | 177 | 0.017 | 16 | 25582 | 401 | – | – | 3.8E+19 | 8980 | Kimber et al. (2018)
CH3COCN | HOPG |  C2H3CN | O | 145 | 0.017 | 17 | 22310 | 328 | – | – | 2.2E+19 | 7250 | Kimber et al. (2018)
CH3COCN | HOPG |  C2H3CN | O | 177 | 0.017 | 16 | 30089 | 401 | – | – | 4.4E+19 | 9000 | Kimber et al. (2018)
CHOCH2CN | HOPG |  C2H3CN | O | 145 | 0.017 | 17 | 18968 | 328 | – | – | 1.9E+19 | 7220 | Kimber et al. (2018)
CHOCH2CN | HOPG |  C2H3CN | O | 177 | 0.017 | 16 | 25582 | 401 | – | – | 3.8E+19 | 8980 | Kimber et al. (2018)
H3SiC(H)CO | Rh |  SiH4:CO |  5 keV e- | 130 | 0.008 | 18 | 9491 | 307 | – | – | 7.9E+18 | 6440 | Tarczay et al. (2017)
(CH3)2NCHO | HOPG |  (CH3)2NCHO | No | 186 | 1.000 | 15 | 38565 | 445 | 1.0E+14 | 6370 | 6.7E+19 | 8780 | Ulbricht et al. (2006)
CH3C(O)OCH3 | Ni(111) |  CH3C(O)OCH3 | No | 165 | 0.500 | 16 | 29623 | 401 | – | – | 4.1E+19 | 7810 | Zahidi et al. (1994)
CH3CH2OCHO | HOPG |  CH3CH2OCHO | No | 120 | 0.500 | 19 | 18004 | 291 | 4.2E+20 | 5200 | 1.3E+19 | 5500 | Salter et al. (2019)
CH3CH2OCHO | ASW |  CH3CH2OCHO | No | 134 | 0.500 | 18 | 21245 | 325 | 1.0E+16 | 5830 | 1.9E+19 | 6210 | Salter et al. (2019)
CH3CH2OCHO |  CH3CH2OH |  CH3CH2OCHO | No | 116 | 0.500 | 19 | 17111 | 282 | – | – | 1.2E+19 | 5300 | Salter et al. (2019)
CH3CH2OCHO | Ni(111) |  CH3CH2OCHO | No | 136 | 0.500 | 17 | 21722 | 330 | – | – | 2.0E+19 | 6310 | Zahidi et al. (1994)
HCOCOOH | Ag |  H2O:CO |  5 keV e- | 210 | 0.017 | 14 | 35060 | 510 | – | – | 7.8E+19 | 10840 | Eckhardt et al. (2019)
[CH3NH${}_{3}^{+}$][NCO-] | Au |  HNCO:CH4 | UV | 214 | 0.083 | 14 | 0 | 520 | – | – | 6.8E+17 | 9690 | Ligterink et al. (2018a)
HC5N | ASW |  HC5N | No | 150 | 0.012 | 16 | 0 | 369 | 1.0E+13 | 10820 | – | – | Coupeaud et al. (2008)
NH2CH2COOH | Au |  CO:CH3NH2:O2 | H | 245 | 0.083 | 13 | 55385 | 603 | – | – | 1.7E+20 | 12480 | Ioppolo et al. (2021)
NH2CH2COOH |  Al2O3 |  NH2CH2COOH | No | 310 | 1.000 | 11 | 78829 | 763 | – | – | 3.9E+20 | 15350 | Tzvetkov et al. (2004b)
NH2CH2COOH | ASW / LDA |  NH2CH2COOH | No | 310 | 1.000 | 11 | 78829 | 763 | – | – | 3.9E+20 | 15350 | Tzvetkov et al. (2004a)
CH3CH2CH2SH | Au(111) |  CH3CH2CH2SH | No | 205 | 0.240 | 14 | 43957 | 511 | – | – | 9.6E+19 | 10070 | Rzeźnicka et al. (2005)
CH3OCOOH | Ag |  CO2:CH3OH |  5 keV e- | 195 | 0.008 | 14 | 33355 | 486 | – | – | 6.6E+19 | 10150 | Marks et al. (2023)
HOCH2COOH | Mo(110) |  CH3OH |  20 eV e- | 235 | 0.017 | 13 | 51211 | 586 | – | – | 1.5E+20 | 12300 | Boamah et al. (2014)
HOCH2COOH | Ag |  CO2:CH3OH |  5 keV e- | 247 | 0.008 | 13 | 55184 | 616 | – | – | 1.7E+20 | 13160 | Marks et al. (2023)
NCCCCN | ASW | NCCCCN | No | 159 | 0.013 | 16 | 0 | 396 | 1.0E+13 | 5050 | – | – | Guennoun et al. (2005)
NCCCCN | Au | NCCCCN | No | – | 0.013 | 0 | 0 | – | 1.0E+13 | 6130 | – | – | Guennoun et al. (2005)
[NH${}_{4}^{+}$][CH3COO-] | Au |  CH3COOH:NH3 | No | 199 | 0.200 | 14 | 0 | 503 | 3.0E+20 | 9980 | 1.3E+19 | 9410 | Kruczkiewicz et al. (2021)
C6H6 | Rh |  C2H2 |  5 keV e- | 140 | 0.008 | 17 | 4872 | 358 | – | – | 5.1E+18 | 6880 | Abplanalp et al. (2019)
C6H6 | ASW |  C6H6 | No | 141 | 0.050 | 17 | 4924 | 361 | 1.0E+13 | 4690 | 5.2E+18 | 6680 | Bahr & Kempter (2007)
C6H6 | HOPG |  C6H6 | No | 148 | 0.500 | 16 | 5296 | 379 | 1.0E+13 | 5470 | 6.2E+18 | 6710 | Salter et al. (2018)
C6H6 | Graphene |  C6H6 | No | 160 | 1.000 | 16 | 5953 | 409 | 1.0E+17 | 6560 | 8.1E+18 | 7200 | Smith & Kay (2018)
C6H6 | Cu(111) |  C6H6 | No | 155 | 4.000 | 16 | 5676 | 397 | – | – | 7.3E+18 | 6730 | Xi et al. (1994)
C6H6 | Cu(111) |  C6H6 | No | 195 | 4.000 | 14 | 8009 | 499 | – | – | 1.6E+19 | 8670 | Xi et al. (1994)
C6H6 | Cu(111) |  C6H6 | No | 213 | 4.000 | 14 | 9143 | 545 | – | – | 2.2E+19 | 9560 | Xi et al. (1994)
C6H6 | Cu(111) |  C6H6 | No | 225 | 4.000 | 13 | 9927 | 576 | – | – | 2.7E+19 | 10150 | Xi et al. (1994)
C6H6 | (A) silica |  C6H6 | No | – | 0.100 | 0 | 0 | – | 1.0E+13 | 6250 | – | – | Thrower et al. (2009)
C6H6 | (A) silica |  C6H6 | No | – | 0.100 | 0 | 0 | – | 1.0E+13 | 5890 | – | – | Thrower et al. (2009)
C6H6 | (A) silica |  C6H6 | No | – | 0.100 | 0 | 0 | – | 1.0E+13 | 5290 | – | – | Thrower et al. (2009)
C6H6 | (A) silica |  C6H6 | No | – | 0.100 | 0 | 0 | – | 1.0E+13 | 4990 | – | – | Thrower et al. (2009)
C6H6 | (A) silica |  C6H6 | No | – | 0.100 | 0 | 0 | – | 1.0E+13 | 4930 | – | – | Thrower et al. (2009)
C6H6 | (A) silica |  C6H6 | No | – | 0.100 | 0 | 0 | – | 1.0E+13 | 4750 | – | – | Thrower et al. (2009)
C6H6 | HOPG |  C6H6 | No | 151 | 0.750 | 16 | 5458 | 386 | 9.0E+15 | 5770 | 6.6E+18 | 6800 | Ulbricht et al. (2006)
H2PCOOH | Ag |  PH3:CO2 |  5 keV e- | 192 | 0.017 | 14 | 32421 | 491 | – | – | 6.4E+19 | 9850 | Zhu et al. (2018a)
[NH${}_{4}^{+}$][NH2COO-] | Au |  CO2:NH3 | Thermal | 240 | 0.083 | 13 | 104054 | 614 | 1.0E+13 | 8300 | 6.9E+19 | 12000 | Noble et al. (2014)
[NH${}_{4}^{+}$][NH2COO-] | (A) carbon |  CO2:NH3 | Thermal | 240 | 0.033 | 13 | 104054 | 614 | – | – | 6.9E+19 | 12220 | Potapov et al. (2019)
[NH${}_{4}^{+}$][NH2COO-] |  MgFeSiO4 |  CO2:NH3 | Thermal | 240 | 0.033 | 13 | 104054 | 614 | – | – | 6.9E+19 | 12220 | Potapov et al. (2019)
[NH${}_{4}^{+}$][NH2COO-] | KBr |  CO2:NH3 | Thermal | 240 | 0.033 | 13 | 104054 | 614 | – | – | 6.9E+19 | 12220 | Potapov et al. (2019)
CH3P2H3 | Ag |  CH4:PH3 |  5 keV e- | 135 | 0.017 | 17 | 14034 | 354 | – | – | 1.4E+19 | 6670 | Turner et al. (2016)
PH2CH2PH2 | Ag |  CH4:PH3 |  5 keV e- | 165 | 0.017 | 15 | 18920 | 433 | – | – | 2.8E+19 | 8310 | Turner et al. (2016)
C4H4S | Au(111) |  C4H4S | No | 186 | 3.000 | 14 | 36209 | 513 | 1.0E+13 | 5540 | 7.2E+19 | 8590 | Liu et al. (2002)
C4H4S | Au(111) |  C4H4S | No | 142 | 3.000 | 16 | 24153 | 391 | – | – | 2.8E+19 | 6390 | Liu et al. (2002)
c-C6H12 | Graphene |  C6H6 | No | 153 | 1.000 | 15 | 46526 | 422 | 1.7E+16 | 6440 | 6.3E+19 | 7190 | Smith & Kay (2018)
C6H14 |  Al2O3(0001) |  C6H14 | No | – | – | 0 | 0 | – | 5.4E+09 | 5230 | – | – | Slayton et al. (1995)
C6H14 | MgO(100) |  C6H14 | No | 144 | 0.600 | 16 | 70101 | 406 | 1.0E+16 | 5580 | 8.5E+19 | 6870 | Tait et al. (2005a)
CH3COCOCH3 | Ag |  CH3CHO |  5 keV e- | 236 | 0.008 | 12 | 90763 | 666 | – | – | 3.0E+20 | 12690 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag |  CH3COCOCH3 | No | 135 | 0.008 | 16 | 39268 | 381 | – | – | 4.2E+19 | 6920 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag |  CH3COCOCH3 | No | 158 | 0.008 | 15 | 49719 | 446 | – | – | 7.3E+19 | 8210 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag | P:A |  5 keV e- | 160 | 0.008 | 15 | 50666 | 451 | – | – | 7.6E+19 | 8320 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag | P:A |  5 keV e- | 206 | 0.008 | 13 | 74018 | 581 | – | – | 1.8E+20 | 10950 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag | P:A |  5 keV e- | 236 | 0.008 | 12 | 90763 | 666 | – | – | 3.0E+20 | 12690 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag |  H2O:CH3CHO |  5 keV e- | 160 | 0.008 | 15 | 50666 | 451 | – | – | 7.6E+19 | 8320 | Kleimeier et al. (2020b)
CH3COCOCH3 | Ag |  H2O:CH3CHO |  5 keV e- | 170 | 0.008 | 14 | 55490 | 480 | – | – | 9.4E+19 | 8890 | Kleimeier et al. (2020b)
CH2C(OH)COOH | Ag |  CH3CHO:CO2 |  5 keV e- | 200 | 0.008 | 13 | 61496 | 577 | – | – | 1.5E+20 | 10580 | Kleimeier et al. (2020a)
CH3COCOOH | Ag |  CH3CHO:CO2 |  5 keV e- | 235 | 0.008 | 12 | 81777 | 679 | – | – | 2.7E+20 | 12610 | Kleimeier et al. (2020a)
CH3COCOOH | Ag |  CH3COCOOH | No | 185 | 0.008 | 14 | 57120 | 534 | – | – | 1.2E+20 | 9730 | Kleimeier et al. (2020a)
HOCH2CH(OH)CHO | Au |  HOCH2CHO:CO | H | 215 | 0.333 | 13 | 93892 | 635 | – | – | 2.7E+20 | 10720 | Fedoseev et al. (2017)
HOCH2CH(OH)CHO | Cu |  H2O:H2CO | UV | 225 | 0.033 | 12 | 100518 | 664 | – | – | 3.1E+20 | 11780 | Layssac et al. (2020)
HOCH2CH(OH)CHO | Pt(111) | G | No | 230 | 4.000 | 12 | 103887 | 679 | – | – | 3.4E+20 | 10970 | McManus et al. (2014)
HOCH2CH(OH)CHO | Pt(111) | G | No | 300 | 4.000 | 11 | 154758 | 886 | – | – | 8.6E+20 | 14660 | McManus et al. (2014)
CH3C6H5 | HOPG |  CH3C6H5 | No | 149 | 0.500 | 15 | 20673 | 450 | 1.0E+13 | 5650 | 2.9E+19 | 6980 | Salter et al. (2018)
CH3C6H5 | HOPG |  CH3C6H5 | No | 179 | 1.000 | 14 | 27221 | 540 | 1.0E+19 | 8180 | 5.5E+19 | 8410 | Ulbricht et al. (2006)
HOCH(CH2OH)2 | Mo(110) |  CH3OH |  20 eV e- | 280 | 0.017 | 11 | 158901 | 845 | – | – | 7.8E+20 | 15180 | Boamah et al. (2014)
HOCH(CH2OH)2 | Au |  HOCH2CHO:CO | H | 240 | 0.333 | 12 | 126098 | 724 | – | – | 4.6E+20 | 12120 | Fedoseev et al. (2017)
HOCH(CH2OH)2 | Ag |  CH3OH |  5 keV e- | 238 | 0.008 | 12 | 124525 | 718 | – | – | 4.4E+20 | 12890 | Kaiser & Maksyutenko (2015)
HOCH(CH2OH)2 | Cu |  H2O:H2CO | UV | 250 | 0.033 | 12 | 134060 | 755 | – | – | 5.3E+20 | 13250 | Layssac et al. (2020)
HOCH(CH2OH)2 | Ag |  CH3OH |  5 keV e- | 234 | 0.008 | 12 | 121399 | 706 | – | – | 4.2E+20 | 12660 | Maity et al. (2014b)
HOCH(CH2OH)2 | Mo(110) |  CH3OH |  1k eV e- | 280 | 0.017 | 11 | 158901 | 845 | – | – | 7.8E+20 | 15180 | Sullivan et al. (2016)
CH3P(O)(OH)2 | Ag |  H2O:CH4:PH3 |  5 keV e- | 250 | 0.017 | 11 | 62113 | 787 | – | – | 2.5E+20 | 13240 | Turner et al. (2019)
S3 | ? |  H2S:CH3OH:NH3:H2O |  10 keV e- | 282 | 0.008 | 11 | 38093 | 888 | – | – | 2.0E+20 | 15100 | Mahjoub et al. (2017)
S3 | ? |  H2S | UV | 275 | 0.023 | 11 | 73366 | 866 | – | – | 3.6E+20 | 14600 | Jimenez-Escobar & Muñoz Caro (2011)
P3H5 | Ag |  CH4:PH3 |  5 keV e- | 162 | 0.017 | 14 | 25261 | 521 | – | – | 4.4E+19 | 8230 | Turner et al. (2016)
C6H5CCH | Rh |  C2H2 |  5 keV e- | 170 | 0.008 | 13 | 55058 | 569 | – | – | 1.1E+20 | 8910 | Abplanalp et al. (2019)
c-C6H5CN | Au(100) |  C6H5CN | No | 285 | 4.000 | 10 | 119327 | 963 | 1.0E+14 | 9020 | 6.8E+20 | 13850 | Solomun et al. (1989)
c-C6H5CN | Cu |  C6H6:CH3CN |  2 keV e- | 183 | 0.033 | 13 | 61397 | 618 | – | – | 1.4E+20 | 9400 | Maksyutenko et al. (2022)
C6H5CHCH2 | Rh |  C2H2 |  5 keV e- | 163 | 0.008 | 13 | 111250 | 556 | – | – | 2.1E+20 | 8640 | Abplanalp et al. (2019)
C6H5CHCH2 | Ag(111) |  C8H8 | No | 265 | 2.000 | 11 | 230615 | 904 | 1.0E+13 | 7940 | 1.2E+21 | 13180 | Wei et al. (2004)
C6H5CHCH2 | Ag(111) |  C8H8 | No | 211 | 2.000 | 12 | 163848 | 720 | 1.0E+13 | 5650 | 5.2E+20 | 10280 | Wei et al. (2004)
C6H5CHCH2 | Ag(100) |  C8H8 | No | 330 | 5.000 | 9 | 320471 | 1126 | 2.0E+11 | 8540 | 2.5E+21 | 16440 | Williams et al. (2003)
(CH3)2C6H4 | HOPG |  (CH3)2C6H4 | No | 155 | 0.500 | 14 | 54464 | 539 | 3.1E+17 | 6130 | 9.5E+19 | 7450 | Salter et al. (2018)
(CH3)2C6H4 | HOPG |  (CH3)2C6H4 | No | 153 | 0.500 | 14 | 109806 | 532 | 1.0E+16 | 5990 | 1.9E+20 | 7460 | Salter et al. (2018)
C6H5CHO | ASW |  C6H5CHO | No | 180 | 0.017 | 13 | 126806 | 626 | – | 4810 | 3.0E+20 | 9500 | Lasne et al. (2012)
CH3CH2C6H5 | HOPG |  CH3CH2C6H5 | No | 205 | 1.000 | 12 | 173146 | 713 | 5.0E+19 | 9500 | 5.3E+20 | 10130 | Ulbricht et al. (2006)
C4H5N3O | Au | Cytosine | No | 433 | 0.250 | 8 | 1212828 | 1577 | 1.0E+13 | 14670 | 1.7E+22 | 23820 | Demers et al. (2002)
C4H5N3O | Au(100) | Cytosine | No | 428 | 0.250 | 8 | 1191882 | 1559 | – | 15400 | 1.7E+22 | 23530 | Demers et al. (2002)
C8H18 |  Al2O3(0001) |  C8H18 | No | – | – | 0 | 0 | – | 1.6E+12 | 7350 | – | – | Slayton et al. (1995)
C8H18 | MgO(100) |  C8H18 | No | 175 | 0.600 | 12 | 250305 | 655 | 7.9E+17 | 7560 | 6.0E+20 | 8730 | Tait et al. (2005a)
C5H6N2O2 | Au | Thymine | No | 373 | 0.250 | 8 | 519385 | 1542 | 1.0E+13 | 12510 | 6.2E+21 | 20090 | Demers et al. (2002)
C5H6N2O2 | Au(100) | Thymine | No | 378 | 0.250 | 8 | 529864 | 1563 | – | 13350 | 6.5E+21 | 20380 | Demers et al. (2002)
C10H8 | Rh |  C10H8 |  5 keV e- | 185 | 0.008 | 11 | 114642 | 777 | – | – | 3.4E+20 | 9930 | Abplanalp et al. (2019)
C10H8 | HOPG |  C10H8 | No | 235 | 1.000 | 10 | 164130 | 987 | 6.0E+16 | 9260 | 7.9E+20 | 11740 | Ulbricht et al. (2006)
S4 | ? |  H2S:CH3OH:NH3:H2O |  10 keV e- | 282 | 0.008 | 9 | 118863 | 1184 | – | – | 8.3E+20 | 15500 | Mahjoub et al. (2017)
C6H15N3 | Au |  HCOOH:CH3CHNH | Thermal | 230 | 0.067 | 10 | 363127 | 973 | – | – | 1.7E+21 | 12280 | Vinogradoff et al. (2012)
C5H5N5 | Au(100) | Adenine | No | 448 | 0.250 | 7 | 841125 | 1984 | – | 15760 | 1.6E+22 | 24620 | Demers et al. (2002)
C5H5N5 | Au | Adenine binding site 1 | No | 363 | 0.250 | 8 | 613484 | 1608 | 1.0E+13 | 12150 | 7.5E+21 | 19600 | Östblom et al. (2005)
C5H5N5 | Au |  Adenine binding site 2 | No | 443 | 0.250 | 7 | 827083 | 1962 | 1.0E+13 | 14910 | 1.5E+22 | 24320 | Östblom et al. (2005)
C5H5N5 | Au |  Adenine binding site 3 | No | 483 | 0.250 | 7 | 941595 | 2139 | 1.0E+13 | 16360 | 2.0E+22 | 26710 | Östblom et al. (2005)
C10H22 | MgO(100) |  C10H22 | No | 204 | 0.600 | 10 | 679726 | 950 | 1.3E+19 | 9370 | 2.7E+21 | 10520 | Tait et al. (2005a)
CH3I | ASW |  CH3I | No | 115 | 3.000 | 14 | 721 | 536 | – | – | 9.3E+17 | 4760 | DeSimone et al. (2013)
CH3I | (P) ASW |  CH3I | No | 115 | 3.000 | 14 | 721 | 536 | – | – | 9.3E+17 | 4760 | DeSimone et al. (2013)
C5H5N5O | Au(100) | Guanine | No | 488 | 0.250 | 6 | 1451100 | 2418 | – | 17560 | 3.6E+22 | 27260 | Demers et al. (2002)
C5H5N5O | Au | Guanine | No | 453 | 0.250 | 7 | 1297821 | 2244 | 1.0E+13 | 15280 | 2.7E+22 | 25160 | Östblom et al. (2005)
[C6H15N3H+][HCOO-] | Au |  HCOOH:CH3CHNH | Thermal | 270 | 0.067 | 8 | 0 | 1550 | – | – | 1.7E+21 | 14460 | Vinogradoff et al. (2012)
[C6H15N3H+][HCOO-] | Au |  HCOOH:C6H15N3 | Thermal | 260 | 0.067 | 8 | 0 | 1493 | – | – | 1.7E+21 | 13910 | Vinogradoff et al. (2012)
C14H10 | Rh |  C2H2 |  5 keV e- | 232 | 0.008 | 9 | 491510 | 1355 | – | – | 3.2E+21 | 13020 | Abplanalp et al. (2019)
C24H12 | HOPG |  C24H12 | No | 390 | 2.000 | 5 | 1572181 | 3839 | 2.0E+16 | 15280 | 4.9E+22 | 21010 | Ulbricht et al. (2006)
C32H14 | HOPG |  C32H14 | No | 490 | 2.000 | 4 | 15999447 | 6399 | 5.0E+21 | 27660 | 1.0E+24 | 28010 | Ulbricht et al. (2006)
C60 | HOPG |  C60 | No | 573 | 2.000 | 3 | 25544230 | 13536 | 2.0E+13 | 19600 | 4.1E+24 | 33630 | Ulbricht et al. (2006)
Table 4: continued. 888aPOM = Polyoxymethylene, AAT = Acetaldehyde Ammonia
Trimer, AATF salt = Acetaldehyde Ammonia Trimer Formate salt. b(P) = porous,
(C) = crystalline cP:C:E = CH3CH2CHO:CO:CH3CH3, P:A = CH3COCOCH3:CH3CHO, G =
HOCH2CH(OH)CHO dH = Hydrogenation, O = Oxygenation

##  Appendix B Elemental ratios

Based on the desorption parameters found in this work and assumed abundances,
the ice elemental composition is determined throughout a protoplanetary disk,
see Sect. 4.4. The radial variations in elemental ratios are presented in Fig.
12. Figure 13 presents the [Phosphorus]/[Carbon] ratio plotted against
temperature.

Figure 12: Ice elemental composition as [X]/[Carbon] for Hydrogen, Nitrogen,
Oxygen, and Sulphur as a function of protoplanetary disk radius. Peak
desorption radii for selected species are indicated in the plot (vertical
dashed lines). Elemental compositions of Soluble Organic Matter (SOM) obtained
from the Murchison meteorite (Schmitt-Kopplin et al. 2010) and averaged
Insoluble Organic Matter (IOM) (Alexander et al. 2017) are indicated
(horizontal dashed lines). The desorption profiles are simulated with a first-
order Polanyi-Wigner equation, variable surface coverages, and a heating rate
of 1 K century-1. An average protoplanetary disk temperature profile of
$T(r)=200\times(r/1AU)^{0.62}$ (Andrews & Williams 2007) is used to determine
the radii. Figure 13: Ice elemental composition [Phosphorus]/[Carbon] as a
function of temperature. The desorption profiles are simulated with a first-
order Polanyi-Wigner equation, variable surface coverages, and a heating rate
of 1 K century-1.