N = 3 SCFTs in 4 dimensions and non-simply laced groups

In this paper we discuss various $N=3$ SCFTs in 4 dimensions and in particular those which can be obtained as a discrete gauging of an $N=4$ SYM theories with non-simply laced groups. The main goal of the project was to compute the Coulomb branch superconformal index and Higgs branch Hilbert series for the $N=3$ SCFTs that are obtained from gauging a discrete subgroup of the global symmetry group of $N=4$ Super Yang-Mills theory. The discrete subgroup contains elements of both $SU(4)$ R-symmetry group and the S-duality group of $N=4$ SYM. This computation was done for the simply laced groups (where the S-duality groups is $SL(2, \mathbb{Z})$ and Langlands dual of the the algebra $L[\mathfrak{g}]$ is simply $\mathfrak{g}$) by Bourton et al. arXiv:1804.05396, and we extended it to the non-simply laced groups. We also considered the orbifolding groups of the Coulomb branch for the cases when Coulomb branch is relatively simple; in particular, we compared them with the results of Argyres et al. arXiv:1904.10969, who classified all $N\geq 3$ moduli space orbifold geometries at rank 2 and with the results of Bonetti et al. arXiv:1810.03612, who listed all possible orbifolding groups for the freely generated Coulomb branches of $N\geq 3$ SCFTs. Finally, we have considered sporadic complex crystallographic reflection groups with rank greater than 2 and analyzed, which of them can correspond to an $N=3$ SCFT.


Introduction
In the last few years there have been many advances in the studies of superconformal field theories with extended supersymmetry in 4 dimensions. A particularly fruitful area of research was research of N = 3 SCFTs; for example, in [4] it was shown, that the relation between dimensions of Coulomb branch operators and 2a − c [5] is only true if theory doesn't possess a discrete gauge group, and in [1] it was shown, that, contrary to a long-standing belief, Coulomb branch doesn't have to be freely generated or even to be a complete intersection manifold and, in fact, is not a complete intersection manifold for a "generic" N = 3 SCFT.
Large progress has also been made in classifying superconformal field theories. In the series of papers [2,[6][7][8][9] by Argyres et al. authors have studied the Coulomb branches of N ≥ 2 SCFTs and have employed various methods to probe the relations in the holomorphic polynomial ring and the orbifolding structure of the Coulomb branch manifold. In particular, in [6] Argyres and Martone have suggested a method to refine the Coulomb limit of the superconformal index, that simplifies tracking the relations in the coordinate ring. In [2] they have classified Coulomb branches for rank-2 SCFTS with N > 2 supersymmetry. It is also possible to study Coulomb branch manifold in a "bottom-up" approach; in [3] Bonetti et al. have considered a class of N = 2 vertex operator algebras W G labeled by crystallographic complex reflection groups, that are extensions of the N = 2 super Virasoro algebra obtained by introducing additional generators; they have also found a way to recover the Macdonald limit of the superconformal index of the parent 4d theory from the corresponding vertex operator algebra, when such a theory exists. Their construction is also interesting because for every rank-r N > 2 SCFT with a freely generated Coulomb branch its complex structure can be written as C r /Γ, where Γ is a crystallographic complex reflection group acting irreducibly on C r . However, not every crystallographic complex reflection group Γ will correspond to some N = 3 SCFT. For example, none of the rank-2 crystallographic complex reflection groups G 4 , G 5 , G 8 1 can be an orbifolding group for N > 2 SCFT [2,10]. Another way to analyze the landscape of superconformal field theories is by constructing them and then computing their index. A particular example of such study can be found in a paper by Bourton et al. [1], where authors have computed Coulomb limit of the superconformal index and Higgs branch Hilbert series for various N = 3 and N = 4 SCFTs and analyzed the Coulomb branches of these theories. To construct new theories, they have considered N = 4 SCFTs with simply laced gauge groups and noticed, that these theories have an enhanced discrete global symmetry at certain values of the gauge coupling. They then refined obtained superconformal index by a fugacity for the enhanced discrete symmetry. The index of the discretely gauged daughter theory is then obtained by "integrating" over the additional fugacity, which takes values in the discrete group. The enhanced discrete symmetry is constructed from a subgroup of S-duality group 2 , so authors restrict their studies to the case of simply laced gauge groups, leaving the non-simply laced groups for the further studies.
In this paper we continued studying the landscape of N > 2 SCFTs in 4 dimensions, found more new N = 3 theories in the spirit of [1] and bridged some of the gaps between the research of Argyres et al. [2,6], Bonetti et al. [3], and Bourton et al. [1]. In order to do that, we extended the results of [1] to the non-simply laced groups. We also analyzed the geometry of the moduli space of the theories we obtained; for the cases when the Coulomb branch is freely generated the orbifolding group is a complex reflection group in agreement with [3]. The Coulomb branches of the rank-2 theories we found are in agreement with the results of [2]. Finally, we considered sporadic crystallographic complex reflection groups 3 and, using the methods of [10], checked, which of them may be an orbifolding group for N = 3 SCFTs; we compute Higgs branch Hilbert series for N = 3 SCFTs that can originate from these groups.

Coulomb branch index computation
The first part of the computation is to compute Coulomb branch limit index. It is more instructing to do these computations along the lines of [6], since the results are similar to 1 Notation for the complex crystallographic reflection groups is in agreement with with Shephard and Todd [11]. 2 It is important to notice that analysis of Bourton et al. doesn't account for the line operators, so the discrete global symmetry isn't always present and not every N = 3 theory they list actually exists [12], see appendix A for more details. 3 The non-sporadic groups were considered in [10].
the computations done as in [1], but the method from [6] gives more refined version of the index that simplifies analysis of the Coulomb branch manifold. The computation method is as follows: 1. We start with an N = 4 SYM theory with a non-simply laced gauge group (B n , C n , F 4 or G 2 ); the simply laced cases have been discussed in [1]. The non-simply laced case is more complicated than the simply laced one, so writing the S-duality group and finding the discrete symmetry group is slightly trickier. For the simply laced groups the S-duality group is simply SL(2, Z), while for the non-simply laced groups it is rather Hecke group Γ 0 (q) ≡ H 2q , where q is the square of length ratio of the short roots to the long ones; q = 2 for B n , C n , F 4 and q = 3 for G 2 . SL(2, Z) is generated by the three elements S = 0 −1 1 0 , T = ( 1 1 0 1 ), C = −1, while Γ 0 (q) is generated by C, T , A = ST q S. SL(2, Z) generators obey the relations C 2 = 1, S 2 = (ST ) 3 = C, while for Γ 0 (q) (AT ) q = C. It is important to notice that for almost every algebra (with the exception of A n ) C is a part of Weyl group and so acts on the theory trivially; only the quotient of the S-duality group by its center acts faithfully on the theory. For B n , C n it is enough to consider AT , while G 2 , F 4 require more attention. As B n , C n theories have the same Coulomb and Higgs branches, superconformal indices we consider in this paper cannot distinguish between these theories, so we will restrict ourselves to considering B n theories from now on.
2. For G 2 and F 4 similarly to [13] we should rather use a transformationS such that STS = ST q S so our S-duality group will look more like SL(2, Z), as C, T,S will obey the relations C 2 = 1,S 2 = (ST ) 2q = C. This is becauseS takes us back to the same group and if we will consider only AT transformations similarly to B n case, we will miss some of the discrete symmetries.S is not a part of the Weyl group, so it acts on the moduli space non-trivially; in [13] it was shown that for G 2 the Coulomb branch operators transform as (U 2 , U 6 )S − → (U 2 , −U 6 ), and for F 4 the rule is 3. Now we can find, for which values of the coupling τ the discrete subgroups of the S-duality group for the various theories we have considered above will leave τ unchanged. For G 2 we have three different options [13]: we can consider either 4 , for which we have two options for the symmetry group: it is realized differently comparing to the Z 2 generated byS. For B n groups we only have Z 2 × Z c 2 at τ = − 1 2 ± i 2 (generated by (AT )). As C belongs to the Weyl group and is therefore a trivial operation, only the quotient of the Hecke subgroup by its center acts faithfully on the N = 4 theory, so we drop the Z c 2 from now on. 4. The S-duality transformations transform the chiral supercharges by a phase [14]. Namely, if an element of the S-duality group transforms the SYM coupling as then the chiral supercharges transform as In particular, the transformations we have found above transform the chiral supercharges as following: Now we need to offset the action of the S-duality transformation; to do that, we will use elements of the R-symmetry group SU (4) R = SO(6) R . We can organize six real adjoint scalar fields φ I , The R-symmetry group element ρ can be represented by a simultaneous rotation in three orthogonal planes in R 6 ≃ C 3 : Then ρ rotates the complex scalars by a phase ϕ a ρ − → e iψa ϕ a , and the four chiral supercharges transform as Now we can choose ψ a in such a fashion that the combination of ρ, σ will leave Q 1 , Q 2 , Q 3 invariant; this means that for a given σ, ρ should be equal to ρ = diag(e 2iπ/n , e 2iπ/n , e −2iπ/n ), where n is the value of the denominator in the exponent at the corresponding cell of the 2.1. All in all, resulting Coulomb branch will be described by , and Z 4 for F 4 , and Z 2 for B n theories, with Γ denoting the orbifolding group of the original theory. 5. Using this knowledge, we can now compute the refined Molien series as in [6] (see equation (4.13) and derivation around it): where t i are coordinates that correspond to the Coulomb branch operators of the original theory. Applying the plethystic logarithm 5 to P J Γ k one obtains the generators of the Coulomb branch of the resulting theory, as well as the relations between them in the form where the positive coefficients count the number of generators of degree k and the negative ones count the number of relations at degree k ′ . If the Coulomb branch turns out to be not a complete intersection manifold, then F Γ (t) should be not a polynomial, but rather an infinite power series 6 .
6. We can now do a cross-check of the results we have obtained by considering the orbifolding group of the Coulomb branch directly. The orbifolding group Γ of the Coulomb branch in the original N = 4 theory is a Weyl group of a Lie algebra and, therefore, a crystallographic Coxeter group. If the Coulomb branch is freely generated, Γ ⋊ Γ k should be a crystallographic complex reflection group (see [3] Table  1 for the table of the irreducible crystallographic complex reflection groups, divided into non-Coxeter and Coxeter groups; from it one can also read the dimensions of the Coulomb branch generators). Thus the cross-check is done by computing Γ ⋊ Γ k and checking, if the resulting group is a crystallographic complex reflection group; if it is, then the degrees of its fundamental invariants should match the dimensions of the Coulomb branch operators obtained from the Molien series computations results.

Higgs branch Hilbert series computation
Next we can move on to the computation of Higgs branch Hilbert series. The algorithm for the computation is as follows 1. The Higgs branch Hilbert series can be constructed in a similar fashion. We can use the fact that N = 2 Higgs branch and N = 2 Coulomb branch are parts of the moduli space that is defined as , where µ(m) is the Möbius function. 6 This is not always true: there can be unexpected cancellations between factors in the numerator and denominator of the Molien series series (2.8) [6]. This can happen when the degree of a relation happens to be the same as that of an affine parameter in the coordinate ring, or if the degree of a syzygy happens to coincide with that of a relation, etc. As the Coulomb branch rank increases, such accidental cancellations become more likely, but, at least for the low-rank examples, one might expect that the plethystic logarithm will accurately capture the degrees and counting of generators and relations.
The complex structure of M Γ is determined by picking one left-handed supercharge in the N = 3 algebra and calling the complex scalars which are taken to left-handed Weyl spinors by the action of that supercharge the holomorphic coordinates on M Γ . The special coordinates on M Γ are not holomorphic; from every SU (3) R triplet two can be taken to be holomorphic and the third anti-holomorphic. Thus, for example, can be taken as the holomorphic coordinates (see discussion near eq. (2.7)). When we choose a N = 2 subalgebra of N = 3, we choose a minimally embedded SU (2) R ⊂ SU (3) R . Then the subspace fixed by the SU (2) R is the N = 2 Coulomb branch. If we now assume that M Γ is an orbifold, and µ τ : Γ → GL(r, C) then it can be written as with the Coulomb branch C Γ ≡ C r /µ τ (Γ) and Higgs branch Therefore, we can construct the Higgs branch Hilbert series in a fashion similar to the Coulomb branch (see [6] for more details).
2. The usual Higgs branch Hilbert series (the unrefined version) has only one fugacity that tracks scaling dimensions of the operators. Since M Γ carries a non-holomorphic U (3) R isometry, we can refine the Hilbert series as H M Γ (t, v, u 1 , u 2 ) = (2.14) The fact that the Hilbert series factorizes in three pieces is an immediate consequence of the fact that the group action on C 3r is chosen to be a direct sum of three factors ρ = µ τ ⊕ µ τ ⊕ µ τ , each of which acts independently on C r . The choice of fugacities is in agreement with the U (3) R weights of the holomorphic coordinates ([1; 0] 1 for z 1 i , [−1; 1] 1 for z 2 i , [0; 1] −1 forz 3 i ); u 1 , u 2 fugacities powers are the SU (3) R weights, t corresponds to the scaling dimension and v tracks the U (1) R charge. We can now reduce (2.15) to obtain Molien formula for Higgs and Coulomb branches: The definition of the Higgs branch Hilbert series according to [2] takes the whole series (2.15), while in [1] the authors restrict to the smaller series (2.17); this can be seen by comparing eqn. (6.14) of [1] to eqn. (4.10) of [2]. We will stick to the definition chosen in [2], since it is formulated in the N = 3 language.
3. Now let us find how N = 3 multiplets contribute to the Higgs branch Hilbert series. In order to do that, one has to find the embedding of the group Γ in GL(r, C). In our case Γ = W(g) ⋊ Z n , so we only need to find how to embed Z n properly. To check whether the Z n embedding we've chosen is correct we can plug in Γ into (2.16) and compare the results with the ones we obtained with the other method in subsection 2.1. One more sanity check is to consider terms up to t 3 in the (2.15) expansion; according to [2] the expansion for N = 3 SCFTs should go as

Gauging N = 4 G 2 SYM
The action of the generators C l of the Z l groups on the Coulomb branch coordinates is and the dimensions of the Coulomb branch operators are ∆ = 4, 6 for Γ 2 gauging of the theory, ∆ = 6, 6 for Γ 3 gauging and ∆ = 6, 12 for Γ 6 gauging. In every case Coulomb branch is freely generated. The orbifolding group for the original manifold is Weyl(g 2 ) = G(6, 6, 2) (we use Shephard and Todd notation for the complex reflection groups), and it is easy to check directly that G(6, 6, 2) ⋊ Z 2 = G(6, 3, 2), G(6, 6, 2) ⋊ Z 3 = G(6, 2, 2), G(6, 6, 2) ⋊ Z 6 = G (6, 1, 2). These groups are also present in the Table 1 of [2] 7 , so our result match theirs and fill some of the gaps in the classification of the N ≥ 3 SCFTs with rank-2 moduli spaces.
The Higgs branch Hilbert series for Z 2 gauging is for Z 3 gauging it is given by and for Z 6 gauging it is given by we can use C 2 , C 3 , C 6 as the generators of Z n ; indices for Z 3 , Z 6 gaugings differ at the t 6 order.

Gauging N = 4 F 4 SYM
The action of the generators C l of the Z l groups on the Coulomb branch coordinates is where C 2 corresponds to the discrete symmetry related toS and C ′ 2 corresponds to the discrete symmetry related to (ST ) 2 . The Coulomb branch is not freely generated in any of these three cases, for Z 4 it is not a complete intersection: , (2.25) and in the two other cases the generators obey the relations The orbifolding group for the original manifold is Weyl(f 4 ) = G 28 , and its semidirect product with Z 2 , Z 4 doesn't yield a complex reflection group. The action of the generators A l of the Z l groups on the fields φ can be chosen to be (2.28) and the Higgs branch Hilbert series is given by differ at the t 6 order.

Gauging N = 4 B n SYM
This case hasn't been analyzed in [13]. As for the theories with B n (or C n ) gauge groups S transformation takes us to another theory [12], from the S-duality side we should only consider Z 2 generated by AT . This transformation leaves the Coulomb branch invariant, so when we mix it with the R-symmetry part, we get that the action of the generator C 2 of the Γ 2 groups on the Coulomb branch operators is given by (2.34) where n corresponds to the B n gauge group. For B 2 gauge group we get that the Coulomb branch is freely generated, the dimensions of Coulomb branch operators are ∆ = 4, 4, and the orbifolding group is G(4, 2, 2), which is in agreement with [2]. For B 3 , B 4 gauge groups we found that the Coulomb branch is not freely generated, and the generators obey the relations For n ≥ 5 gauging Γ 4 yields a theory with Coulomb branch that is not a complete intersection manifold, the Molien series for e.g. B 5 is given by The action of the generators A 2 of the Z 2 groups on the fields φ can be chosen to be The Higgs branch Hilbert series for B 2 − B 5 N = 4 theories, gauged by Z 2 , at the two lowest orders is given by The difference between B 2 and B 3 indices appears at t 6 order, between B 3 and B 4 -at t 8 order and between B 4 and B 5 -at t 10 order.

N = SCFTs from complex crystallographic reflection groups
Another area of interest in N = 3 SCFT studies is theories that have freely generated Coulomb branch. Until a few years ago it was commonly believed that every N ≥ 2 theory possesses a freely generated Coulomb branch, but in [1,6] it was shown that there exist many N = 3 SCFTs that possess a non-freely generated Coulomb branch. In [10] Caorsi and Cecotti argued that for every N = 3 SCFT with a freely generated Coulomb branch the Coulomb branch is given by C r /G, where G is complex crystallographic reflection group (CCRG) 8 . However, not every CCRG can give rise to a Coulomb branch of the N = 3 SCFT; for example, a group G 8 cannot correspond to any N = 3 SCFT [10]. If a rank-k CCRG G corresponds to an N = 3 SCFT, then consistency with Dirac quantization requires that G ⊂ Sp(2k, Z) 9 . Caorsi and Cecotti argued that non-sporadic groups complex crystallographic reflection groups can always be embedded into Sp(2k, Z), and in [2] the issue of rank-2 sporadic groups has been addressed, so the groups left to analyze are G 24 , G 25 , G 26 , G 29 , G 31 , G 32 , G 33 , G 34 . In order to do that, we considered embedding of these groups into GL(k, Z[ζ]), where k is the rank of the group and ζ is primitive third root of 1 for G 25 , G 26 , G 32 , G 33 , G 34 ; ζ = i for G 29 , G 31 and ζ = √ −7 for G 24 10 . Then we computed an invariant (k × k) Hermitian form H for each of these groups. Afterwards we constructed 2k × 2k skew-symmetric form Ω from H by considering where ψ i = x i +ζy i ∈ Z[ζ] k . Ω is then obtained by clearing denominators and dividing by a non-trivial common factor for all the entries of the matrix in consideration if needed. Then the necessary and sufficient condition on whether the embedding G ֒→ GL(k, Z[ζ]) induces an embedding G ֒→ Sp(2k, Z) is simply det Ω = 1. If there indeed exists an embedding G ֒→ Sp(2k, Z), then the complex crystallographic reflection group G can correspond to an N = 3 SCFT [2]. The direct computation shows that for G = G 24 , G 25 , G 26 , G 32 , G 33 G ⊂ Sp(2k, Z). Let us list lowest terms of the Higgs branch Hilbert series expansion for G 29 , G 31 11 : We can identify that at the t 2 order the only contribution to the index comes from the N = 3 stress-tensor multiplet. The index expansion for G 31 has contributions at t 2 order and then only at t 8 .
A Existence of theories listed in Bourton et al.
In [1] Bourton et al. classify various N = 3 SCFTs obtained from N = 4 SCFTs with ADE or U (n) gauge groups. In particular, they have mixed finite cyclic subgroups of SL(2, Z) self-duality groups with the Z n ⊂ SU (R) 4 of the R-symmetry group and then gauged the resulting group; for n = 3, 4, 6 they have obtained N = 3 SCFTs, while for n = 2 they have got N = 4 SCFTs. However, there is a fine point first observed in [12] related to the fact that there might be more than one theory for a given gauge algebra g, depending on the line operators present in the theory. S, T transformations then may transform a theory with one set of line operators to a physically distinct theory with another set of line operators; an N = 4 SCFT with gauge algebra g (listed in [1]) will have a Z k ⊂ SL(2, Z) iff there is a theory which is self-dual under the corresponding S-duality transformation. Therefore, it turns out that not every N = 3 SCFT listed in [1] exists; using [12] and [6], one can find that the following N = 3 SCFTs exist: + + − + SU (4) + + + + SU (5) + − + − SO(2d), d > 1 + + + + U (d) + + + + E 6 + + − + E 7 + − + − E 8 + + + + These results are in agreement with [2], with the superficial exception of the U (2) and SO(4) Z 3 , Z 4 , Z 6 gaugings. The classification of Argyres et al. contains the Coulomb branch orbifold geometries for these theories (Table 3, entry 32 for Z 3 gauging of U (2), Table 3, entry 33 for Z 4 gauging of U (2), SO(4) and Table 4, entry 52 for Z 6 gauging of U (2) and Z 3 , Z 6 gauging of SO(4)). These entries were ruled out by Argyres et al. because the corresponding theories have two stress tensors (this can be seen from the Hilbert series analysis). However, as the mother N = 4 theory had a non-simple gauge group, getting theory with two stress tensors after gauging a discrete subgroup is the expected outcome and does not mean the geometries in question should be discarded.