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

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.


Introduction
There had been no known 3D 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 C 4 /Z k . Similarly, no 4D N = 3 SCFT had been known until the authors of [2] discovered the first example using D3-branes on the F-theory singularity (C 3 × T 2 )/Z k for k = 3, 4 or 6. 1 As discussed in [2], the duality between M-theory and F-theory suggests that the compactification of these 4D 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 N = 3 SCFTs and ABJM theories. 1 A general discussion on 4D N = 3 SCFTs was first given in [3]. More detailed analysis on explicit One 4D quantity whose behavior under the compactification is well-understood for generic 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 × 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 [5][6][7]. This leads us to comparing the superconformal index of the 4D 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 N = 3 SCFTs is rather non-trivial, unlike for generic 4D N = 2 SCFTs. On the one hand, the bosonic (zero-form) global symmetry of the ABJM theory is so(6) R × u(1) b , where u(1) b is a flavor symmetry commuting with the N = 6 R-symmetry. On the other hand, the bosonic (zeroform) global symmetry of 4D N = 3 SCFTs is u(3) R R-symmetry. 2 Naively, one might think that the u(3) R in four dimensions is mapped to a u(3) sub-algebra of so(6) R under the compactification. However, this naive map would lead to various contradictions as discussed below.
First, the naive map u(3) R ֒→ so(6) R contradicts with a general relation between supersymmetry enhancement of the 4D index and squashing independence of the 3D partition function [8] 3 . 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 [9] for the ABJM theory when a special value of an (imaginary) mass parameter is turned on. This was then interpreted in [8] as a supersymmetry enhancement on S 3 b , or on S 1 × 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 [8]. As we will show in Sec. 3, the naive map u(3) R ֒→ 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 u(3) R ֒→ so(6) R can be seen in the moduli 2 Here, we normalize the center u(1) of u(3) R so that the 4D chiral supercharges Q I α are in the fundamental representation of u(3) R . Note also that a 4D N = 3 SCFT cannot have a continuous N = 3 flavor symmetry without having extra supersymmetries [3]. space of vacua. Let us consider the case of rank-one N = 3 SCFTs [10]. The moduli space is then C 3 /Z k for k = 3, 4 or 6, where the Z k -action is such that C 3 ∋ (z 1 , z 2 , z 3 ) → (ωz 1 , ω −1 z 2 , ωz 3 ) for ω ∈ Z k . The 3D reduction of this 4D N = 3 SCFT is believed to be the ABJM theory with gauge group U(1) k × U(1) −k , whose moduli space is C 4 /Z k with the Z k -action such that C 4 ∋ (z 1 , z 2 , z 3 , z 4 ) → (ωz 1 , ω −1 z 2 , ωz 3 , ω −1 z 4 ). The emergence of the extra direction (parameterized by z 4 ) is a general phenomenon for the compactification of 4D N ≥ 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 u(3) R symmetry while z 4 is neutral. 4 From the 3D viewpoint, however, the so(6) R R-symmetry non-trivially acts on all z 1 , z 2 , z 3 and z 4 . 5 Therefore, if the naive inclusion u(3) R ֒→ so(6) R is correct, there must be a u (3) sub-algebra of 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 u(3) R symmetry in four dimensions is mixed with the u(1) b symmetry in three dimensions under the S 1 compactification. Indeed, we show that there is a unique u(3) sub-algebra of so(6) R × u(1) b that, when identified as the 3D reduction of the 4D u(3) R symmetry, resolves all the above contradictions. Our result would be important when comparing the superconformal index of the 4D N = 3 theories and the sphere partition function of their 3D reductions.
Indeed, given the above mixing between u(3) R and u(1) b , one can now establish a clear correspondence between the supersymmetry enhancement of the 4D N = 3 SCFT on S 1 ×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 N = 3 cousin there. In Sec. 3, we consider the S 1 -compactification of the 4D N = 3 SCFT and discuss the Rsymmetry mixing, focusing on the case that the 3D theory is the ABJM theory. In Sec. 4, 4 We 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. 5 To be more specific, (z 1 , z 2 , z 3 ) forms a fundamental representation of u(3) R , while (z 1 , z 2 , z 3 , z 4 ) forms a fundamental representation of su(4) ≃ so(6) R . Here, the action of u(3) R is different from that of u(3) R only in the action of its center; the former is obtained by rescaling the charge of u(1) c ⊂ u(3) R by 1/2. Indeed, our normalization of the charge of u(1) c ⊂ u(3) R is such that the 4D chiral supercharges Q I α are in its fundamental representation, while the action of the center of u(3) R is such that (z 1 , 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 N = 3 vector multiplet (which is equivalent to an 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. we discuss a divergence that arises in the reduction of 4D N = 3 superconformal index.
We finally conclude in Sec. 5. In appendix A, we discuss in some more detail how the 4D and 3D symmetries are related under the S 1 -compactification. Here, we briefly review the relation between limits of 4D superconformal index and the 3D squashing independence [8].
The superconformal index of a 4D N = 2 SCFT is defined by where the trace is taken over the space of local operators, (j 1 , j 2 ) are 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 N = 2 superconformal algebra reviewed in appendix A.1, R and r are written as Since the exponents of p, q, t and a i commute with Q 2− and S 2− (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 → q, is known to provide a variety of application. 6 One important feature of the Schur limit is that the resulting index (called the "Schur index") turns out to be independent of p [8,13]. The reason for this is the following. First note that, when t = q, the index is written as There are now four supercharges, Q 1 − , Q 2− , S 1 − and S 2− , 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, vanishes for all such operators. As a result, the Schur index has no p-dependence and is expressed as The absence of p-dependence is clearly a consequence of the supersymmetry enhancement triggered by t → q [8,13].
Let us now consider the compactification of the 4D N = 2 SCFT on S 1 . In the deep infrared, we end up with a 3D 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) [5][6][7]14]. In this limit, we set and take the limit β → 0 with b, m and M i kept fixed. 7 Then the 4D index reduces to the partition function Z S 3 b of the 3D 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. 6 For instance, it gives rise to a general connection between 4D N = 2 SCFTs and non-unitary vertex algebras [11]. It also simplifies the TQFT expression for the index of class S theories [12]. 7 The special combination t/ √ pq appears here since it is a flavor fugacity from the viewpoint of the N = 1 supersymmetry involving the preserved supercharge Q 2− . From the 3D viewpoint, the Schur limit t → q corresponds to constraining the 3D mass parameter m as 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 [8]. Indeed, the index is now independent of p, and therefore the b-dependence is only in q = e −βb −1 . We see that this b-dependence can be removed by where rescaling M i is necessary for keeping a i = e −iβM i unchanged. Hence, the Schur limit t → q in four dimensions generally leads to squashing independence in three dimensions [8].
It was also found in [8] that the 3D squashing independence also occurs at In four dimensions, this corresponds to the condition t = pq, under which the 4D index is expressed as This index counts operators annihilated by Q 1+ , Q 2− , S 1+ and S 2− , and for such operators vanishes. Therefore, the above expression for the index reduces to 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. 8

4D N = 3 SCFTs and 3D 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 N = 3 SCFTs and 3D ABJM theories.
We will focus on 4D N = 3 SCFTs without N = 4 supersymmetry, although its generalization to 4D N = 4 SCFTs is straightforward and will be discussed in [17]. The bosonic global symmetry of 4D (genuine) N = 3 SCFT is just the u(3) R symmetry; there is no "N = 3 flavor symmetry." This u(3) R contains the N = 2 R-symmetry, su(2) R × u(1) r , and the N = 2 flavor symmetry, u(1) f . Therefore, the most general expression for the superconformal index is written as where we define the u(1) f charge by as in [10]. For generic values of the fugacities, the above index preserves only Q 2− and S 2− .
Since the discussions in the previous sub-section are also applied to N = 3 SCFTs, we see that there are two special values of the fugacity t that lead to supersymmetry enhancement: (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 N = 2 supersymmetry with a flavor charge f , no such supersymmetry enhancement occurs since f commutes with all N = 2 supercharges.
However, when the theory has an 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 Q 3 − (and its conjugate S 3 − ) by setting In this case, the expression for the index reduces to We see from Table 1 that it indeed preserves Q 3 − and S 3 − in addition to Q 2− and S 2− .
Note that one can rewrite (2.16) as Since the index is contributed only from operators annihilated by Q 2− , Q 3 − , S 2− and S 3 − , the expression (2.17) can be reduced to 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 The condition (2.15) is then translated as Under this condition, the index depends only on q = e −βb −1 and , and therefore the b-dependence can be removed by the replacement

R-symmetry mixing under the S 1 -compactification
We here demonstrate that the above general discussion on squashing independence of 3D N ≥ 4 theories puts a strong constraint on the relation between the R-symmetries of a 4D 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 u(3) R to so(6) R × u(1) b by the compactification, we need to understand which u (3) sub-algebra of the latter corresponds to the 3D reduction of the former. While a naive expectation might be that u(3) R simply descends to a sub-algebra of so(6) R , that would contradicts with the general discussions in the previous section.

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 Q s α for s = 1, · · · , 6 so that we can distinguish them from the 4D supercharges Q I α and Q Iα . We arrange them so that Q I α and Q Iα descend to We also denote by R 4D I and R 3D I a basis of u(3) R and so(6) R , respectively. We take them so that Q I α and Q Iα respectively have charge +1 and −1 under R 4D I but are neutral under R 4D J( =I) , and similarly that Q 2I−1 . Note that, if the 4D R-symmetry u(3) R was mapped into so(6) R by the compactification, we would have However, since the center u(1) c ⊂ u(3) R can be mixed with u(1) b in the ABJM theory, the most general identification is written as where J 3D U (1) b is the charge of the u(1) b flavor symmetry in three dimensions, and ξ is a real number. 9 When ξ = 0, there is a mixing between u(1) c and u(1) b . Indeed, we will show below that ξ does not vanish.
Let us now write the superconformal index (2.12) of the 4D theory in terms of R 3D i and J 3D U (1) b . From Table 1, we see that This implies that the index (2.12) can be expressed as Using the identification (3.2), one can rewrite this in terms of the 3D R-charges and J 3D (3.5) 9 As discussed below, we will normalize J 3D U(1) b so that the 3D twisted chiral multiplet with R 3D 1 = R 3D 3 = 1/2 has charge J 3D U(1) b = 1/2.
Since this index preserves Q 2− , one can regard it as an N = 1 index associated with (Q 2 α , Q 2α ). In this case, t/ √ pq and a pq/t are regarded as flavor fugacities from the N = 1 viewpoint. In three dimensions, these fugacities reduce to two mass parameters associated with a 3D N = 2 flavor symmetry. For the ABJM theory, the N = 2 flavor symmetry is the product of so(4) ⊂ so(6) R and 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 so(2) 2 ⊂ so(4) ⊂ so(6) R , and by m 1 that associated with u(1) b , as in [9]. The expression (3.5) implies that these masses are related to the 4D fugacities as 10 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 ξ, which characterizes the mixing between u(1) c and u(1) b . Below, we will uniquely fix the value of ξ so that it is consistent with the general discussion on squashing independence in the previous section.

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 10 Note here that R 3D 2 is identified as the superconformal R-charge of the 3D N = 2 superconformal algebra associated with Q 3 α ± iQ 4 α . Therefore, (1/ √ pq) R 3D 2 in (3.5) corresponds to the background gauge field for the R-symmetry that reduces to the 3D N = 2 superconformal R-symmetry under the compactification. Turning on such a background gauge field is necessary for preserving the supersymmetry on S 1 × S 3 [18].
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 Here we will show that this general constraint uniquely fixes the value of ξ 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 [19]: where s b (z) is the double-sine function Recall here that m 1 is the mass parameter associated with u(1) b , and m 2 and m 3 are those associated with so(2) 2 ⊂ so(4). The above normalizations of m 1 , m 2 and m 3 imply that a 3D (twisted) chiral multiplet has R 3D 1 = R 3D 3 = J 3D U (1) b = 1/2. It was shown in [9] that when m 2 = i b−b −1 2 , the above matrix model expression (3.10) reduces to a function only of where m ± ≡ m 3 ± m 1 . This means that, when (3.8) is imposed, (3.13) Note that, for generic values of ξ (including ξ = 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 ξ = −1 . (3.14) Indeed, only for this value of ξ, the b-dependence of Z S 3 b can be removed by (3.9). Therefore we conclude that the correct value of ξ is ξ = −1. Note that this implies a non-trivial mixing between the 4D u(3) R symmetry and the u(1) b flavor symmetry of the ABJM theory!

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 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 [8] that the b-dependence of Z S 3 b disappears in the case that m 2 = −i(b + b −1 )/2 and m + = 0. 11 This is a strong evidence for our identification ξ = −1.
Another limit we consider is (2.20), corresponding to a = √ 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 To see this is indeed the case, first note that our identification (3.7) is now expressed as (3.18) 11 To be precise, the condition discussed in [8] is It was shown in [9] 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 ξ = −1, we see that Z S 3 b is a function of the single combination of variables b (2m 2 + ib) . (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 ξ = −1.

Divergence and flat directions
Having identified the correct R-symmetry mixing ξ = −1, we see that the mass parameters m 1 , m 2 and m 3 of the ABJM theory are constrained by when it is obtained as the S 1 -compactification of a 4D N = 3 SCFT. Here, M is related to the fugacity a for the U(1) f symmetry by a = e −iβM , while m is related to the fugacity 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 N i,j=1 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 N = 4 super Yang-Mills (SYM) theories [20], 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 × S 3 up to a prefactor. For generic 4D N = 2 SCFTs, the small S 1 limit of the S 1 × S 3 partition function Z S 1 ×S 3 behaves as [6,21] log 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 N = 4 super Yang-Mills theories, the above formula is modified as [20] log 5) where N is the complex dimension of the (N = 2) Coulomb branch. Note that the first term in (4.4) drops out since N = 4 superconformal symmetry implies a = c. The divergent term N log(2π/β) in (4.5) is interpreted to mean that Z S 3 obtained in the small S 1 limit of Z S 1 ×S 3 has a power-law divergence as [20] where Λ 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. 12 Note that this divergence is purely three-dimensional; Z S 1 ×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 Λ is introduced. The other directions on the Coulomb branch are lifted by the background gauge fields and masses. 13 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 12 Here, the 3D "Coulomb branch" is a sub-space of the 3D moduli space on which the action of su(2) R is trivial, where su(2) L × su(2) R is an 3D N = 4 R-symmetry. 13 Note that the FI-parameter of the ABJM theory is already converted into a mass parameter via N = 6 supersymmetry. Therefore, the Coulomb branch can be lifted by changing the values of the mass parameters. dimension of the 4D Coulomb branch. This suggest that, even for 4D N = 3 SCFTs, the β → 0 limit of the S 1 × S 3 partition function behaves as (4.5). 14 The above discussion implies that our identification ξ = −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 N = 3 SCFTs by setting N = 1. The 4D moduli space is then C 3 /Z k for k = 3, 4 or 6, and the 3D moduli space is C 4 /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 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 ×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, 15 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.
14 Note that any 4D N = 3 SCFT has a = c [3] and therefore the first term in the RHS of (4.4) vanishes for all 4D N ≥ 3 SCFTs. 15 It is known that every Coulomb branch operator in any 4D N ≥ 2 SCFT is neutral under N = 2 flavor symmetries [15]. See also [16] for its higher spin generalization.

Conclusions and Discussions
In this paper, we have studied the S 1 -compactification of 4D 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 u(1) of u(3) R in four dimensions can mix with an abelian flavor symmetry of the 3D 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 N = 3 cousin) of the 4D index correctly reproduces the squashing independence of Z S 3 b found in [9]. Our result implies that the recent discussions in [8] 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 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 u(3) R implies that the 3D limit of 4D N = 3 index leads to a divergence. A similar divergence appears in the case of 4D N = 4 SYM theories. This seems to suggest that the small S 1 limit of the superconformal index of 4D N ≥ 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 N = 2 Argyres-Douglas (AD) SCFTs [7]. When a generic AD theory is compactified on S 1 , there is a mixing between the 4D u(1) r symmetry and 3D topological u(1) global symmetry. This is because the 4D N = 2 R-symmetry, su(2) R × u(1) r , needs to enhance to the 3D N = 4 R-symmetry, so(4) R ≃ su(2) R ×su(2) C , under the compactification; without the mixing between u(1) r and the topological u(1), such an enhancement is prohibited by the presence of Coulomb branch operators of fractional 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 N = 3 R-symmetry, u(3) R , and the 3D u(1) b symmetry. Since u(1) b can be regarded as a topological 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 N = 3 SCFTs have no Coulomb branch operators of fractional u(1) r charges. The R-symmetry mixing in our case just reflects the N = 3 superconformal symmetry.
In general, our result gives a necessary condition for a 3D N = 6 SCFT to have its 4D uplift. Indeed, whenever a 3D N = 6 SCFT can be realized by compactifying a 4D 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 N = 3 cousins a = √ t/p. 16 Therefore, if the space of mass parameters of a 3D N = 6 SCFT has no such special sub-spaces, one can conclude that there exists no 4D N = 3 SCFT whose S 1 -compactification gives rise to that 3D N = 6 SCFT.
There are clearly many future directions, some of which we list below: • While we have focused on 4D N = 3 SCFTs whose 3D reduction is the ABJM theory, there are many other N = 6 Chern-Simons matter theories [22]. 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 N = 6 SCFT contains an N = 6 flavor current [23] (See also Sec. 5.4.6 of [24]). In the case of ABJM theories, this is the u(1) b current. Based on our discussion, it is natural to expect that there is a mixing between the 4D u(3) R current and this 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 N = 3 SCFT. Several limits of the superconformal index of 4D N = 3 SCFTs are computed in [25], 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 N = 6 theory. For this purpose, one needs to take into account the R-symmetry mixing that we discussed in this paper.
[R rs , Q αt ] = i(δ rt Q αs − δ st Q αr ) , The hermiticity is expressed as Let us consider the S 1 -compactification of a 4D N = 3 SCFT, which leads to a 3D N = 6 SCFT in the deep infrared. We here discuss how the 4D N = 3 supersymmetry is related to the 3D 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 Q I α , Q Iα , R I J , M α β , Mαα and To that end, we first identify the 3D supercharges as and the first relation in (A.28). 18 We now turn to the R-symmetry. Recall that the 4D R-symmetry is u(3) R while the 3D R-symmetry is so(6) R . Along the RG-flow from four dimensions to three dimensions, the manifest global symmetry is only u(3) R , which is in the deep infrared identified with a sub-algebra of the product of so(6) R and the N = 6 flavor symmetry. In the main text, we argue that the center u(1) of u(3) R is mixed with the N = 6 flavor symmetry. This is generally expressed (up to automorphims) as where J 3D flavor is an N = 6 flavor symmetry that mixes with the center of u(3) R and ξ is a real parameter. When the 3D theory is the ABJM theory, J 3D flavor is the u(1) b charge. 19 and ξ = −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, su(3) ⊂ u(3) R , is embedded in the 3D global symmetry. Indeed, under the S 1 -compactification, the 4D R-charges R I J for I = J are identified as the following linear combinations of the 3D R-charges R rs : 20 R  18 In the reduction to three dimensions, we set P αβ − P βα = 0 since we only keep the zero modes in the compactified direction. 19 In comparison to Eq. (3.2), we identify J 3D flavor = 1 4 J 3D U(1) b . 20 One can summarize these six relations in R I J = 1 2 (R 2J 2I−1 + R 2I 2J−1 − iR 2J−1 2I−1 − iR 2J 2I ). Note that R rs = −R sr .
It is straightforward to see that the above identifications are consistent with the 3D and 4D commutation relations for the R-charges and supercharges.