Seeking SUSY fixed points in the $4-\epsilon$ expansion

We use the $4-\epsilon$ expansion to search for fixed points corresponding to $2+1$ dimensional $\mathcal{N}$=1 Wess-Zumino models of $N_{\Phi}$ scalar superfields interacting through a cubic superpotential. In the $N_{\Phi}=3$ case, we classify all SUSY fixed points that are perturbatively unitary. In the $N_{\Phi}=4$ and $N_{\Phi}=5$ cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For $N_{\Phi}=4$ we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For $N_{\Phi}=5$, we go through all Lie subgroups of O(5) and then use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with $N_{\Phi}=\frac{\rm N(N-1)}{2}-1$ and O(N)/Z2 symmetry, that exists for arbitrary integer N$\geq3$.


INTRODUCTION
In recent years, 2 + 1 dimensional superconformal field theories (SCFTs) with minimal (N = 1) supersymmetry have received significant attention [1][2][3][4][5][6][7][8][9][10]. Perhaps the simplest model with this symmetry is the so-called N = 1 super-Ising model, which was proposed in [1] to describe a quantum critical point at the boundary of a 3 + 1 dimensional topological superconductor. Because the super-Ising model preserves time reversal symmetry, its spectrum contains only one relevant operator which is time-reversal invariant. This means that a generic renormalization group flow (not necessarily supersymmetric) can reach the fixed point by tuning a single parameter. This property is called emergent supersymmetry and might be realizable in experiment.
Apart from the super-Ising, a whole zoo of theories have been identified and they are sometimes related by dualities [4][5][6][7]. In particular, a certain super QED was shown to be dual to an N = 1 Wess-Zumino model [5,6]. This duality has many proprieties that resemble the self duality of nonsupersymmetric QED coupled to two complex scalars, which describes the famous de-confinement quantum critical point [11,12].
The study of 2 + 1 dimensional N = 1 SCFT has also been boosted by technical improvements on standard techniques. For example, it was observed in [3] that if one sets the number of Dirac fermions to N f = 1 4 in four dimensions, after analytic continuation to 4 − dimensions, one ends up with a single three-dimensional Majorana fermion when = 1. By studying the Gross-Neveu-Yukawa model with N f = 1 4 Dirac fermions and one real scalar, one gets a perturbative fixed-point which is in good agreement with the N = 1 super-Ising model in 2 + 1 dimensions. This technique can be easily generalized to more general 2 + 1 dimensional N = 1 Wess-Zumino models with superpotential 1 In other words, one can study the corresponding Gross-Neveu-Yukawa model 2) with N f = N Φ 4 . Parallel to these analytical developments, progress has also been made using the highly successful numerical bootstrap program [13,14]. Recent works that studied 2 + 1 dimensional N = 1 SCFTs using numerical techniques include [2,[8][9][10]. In particular, when applied to the N = 1 super-Ising model, the scaling dimension of the superfield Φ can be determined to very high precision [8].
In this work we would like to find an organizational scheme for three-dimensional models with N = 1 supersymmetry. Our analysis will be perturbative and rely in the standard -expansion. The logic being that a one-loop analysis should give us at least a qualitative understanding of potential fixed points. Once we know a fixed point exists, and have a basic idea of its spectrum, we can use more powerful techniques like the modern numerical bootstrap in order to solve it to high precision. This work is then similar to [15][16][17], where scalar models in several dimensions were studied using a similar method.
Our approach is then the following, we use the 4− expansion to search for N = 1 SCFTs with a small number of scalar superfields interacting through the superpotential (1.1). We systematically study all the cases when the number scalar superfields N Φ is less or equal than 5. In Section 3, we start with the N Φ = 3 case, which will also include all the possible solutions for N Φ = 1, 2. For N Φ = 3 the superpotential has 10 independent couplings h ijk , by solving the one-loop beta functions we list in Table I all the possible fixed points with N Φ ≤ 3 that are perturbatively unitary. We then increase the number of scalar to N Φ = 4 in Section 4 and to N Φ = 5 in Section 5. We focus on the SUSY fixed points where the scalar superfields form a single irreducible representation of the symmetry group. We call these fixed points "irreducible fixed points". This means the coupling constants h ijk = a g a d a ijk are a linear superposition of the degree three invariant tensors of the symmetry group. The invariant tensors are traceless and also satisfy This is essentially the trace condition used in [18], and has important physical consequences. It implies there is only one boson mass operator φ i φ i that is invariant under the symmetry group of the fixed points. For the most stable fixed point of the beta function, this operator is also the only relevant operator that preserves both time-reversal symmetry and the flavor symmetry of the SCFT. Similar to the super-Ising model, non-supersymmetric RG flows will reach the supersymmetric fixed point as long as one tunes the coupling of the boson mass term to zero. In other words, these systems have emergent supersymmetry. The results of the classification are listed in Table II. In the N Φ = 4 case, it turns out that there is only one group that gives us a fixed point where the four superfields are fully interacting. This result is based on the analysis of all the subgroups of O(4) in [19]. In the N Φ = 5 case, we focus on Lie subgroups and finite subgroups of O(5) with orders less or equal to 800. In Section 6, we observe that some fixed points for low values of N Φ can be considered the first entry in a family of fixed points that can be generalized to higher values of N Φ . For example, the O(3)/Z2 SCFT presented in Table II can be generalized to a new family of SCFTs preserving the O(N)/Z2 symmetry. This family complements the three families (the super Potts models, the SU(N) invariant SCFTs and the F4-family of SCFTs) of Wess-Zumino models already studied in [10].   The one-loop beta function of the N = 1 SCFT Wess-Zumino model with N Φ scalar superfields coupled through the superpotential (1.1) is given by [3], and the anomalous dimension matrix (whose eigenvalues give us the anomalous dimension) reads Like the scalar theory in 6 − dimensions, this is a gradient flow [15,16,20]. In other words, Along the RG flow to the IR, A decreases and at the fixed points, Here γ i are the eigen-values of the anomalous dimension matrix.

N Φ = 3 REDUCIBLE FIXED POINTS
The strategy we employ in this section is identical to the one described in [16] to study scalar φ 3 theory in 6 − dimensions. Notice that the coupling constant h ijk is a fully symmetric tensor and can be characterized as Here λ ijk is symmetry and traceless. We parameterize the potential as and all other v's andλ's vanishing. The S3 invariant fixed point (S3 super Potts model) of two scalars plus a decoupled scalar superfield is located at and all other v's andλ's vanishing. The Lagrangian of the S 3 super Potts model can in fact be re-written as an N =2 supersymmetric Lagrangian with a single complex chiral superfield [8]. This explains the anomalous dimension γ Φ = 1 6 . The Z2 fixed point with two coupled scalars plus one decoupled scalar superfield is located at and all other v's andλ's vanishing. The S4 invariant super Potts model is located at and all other v's andλ's vanishing. The S3 invariant fixed point with all three scalars fully coupled is located at A summary of the previous analysis is presented in Table I. We can compare our results with the perturbative fixed points of scalar φ 3 theory in 6 − dimension, which were studied in [16]. Our entries of Table I turn out to be in one-to-one correspondence with the fixed points of scalar φ 3 theory in 6 − dimensions. In 6 − dimensions, only the O(2) fixed point and the K4 fixed point are perturbatively unitary. In our case, it seems that supersymmetry improves the situation a lot, and all the fixed points become perturbatively unitary.

N Φ = 4 IRREDUCIBLE FIXED POINTS
We now discuss the case when the scalar superfield transforms as an irreducible representation of the symmetry group. To construct the superpotential (1.1), the symmetry group needs to preserve a rank-3 fully symmetric invariant tensor d a ijk . The tensors d a ijk are necessarily traceless, such that the representation is reducible. Another important condition is that the invariant tensors satisfy 2 This is analogous to the trace condition used in [18]. We can now normalize d a ijk to satisfy If the symmetry group preserves a single d ijk , the beta function and the anomalous dimension become Here T 3 is defined through assuming d ijk satisfies the normalization (4.3). Using the fact that 2 Otherwise one can build projectors of the form which projects the vector representation into invariant subspace. Here e1 is the first eigenvalue of dimndjmn.

From the beta function (4.4), we know that the one-loop fixed point exists if and only if
The irreducible subgroups of O(4) were studied extensively in [19]. It was discovered that only five of the groups have a four dimensional irreducible real representation that preserves a degree three polynomial 3 . The result is recalled in Table III. The five polynomials are The corresponding fully symmetric invariant tensor can be easily constructed from these polyno- Of these five polynomials only two of them are independent. We can pick these two independent polynomials to be I 3 and I 1 . The polynomial I 4 is related to I 3 by an O(3) rotation. The group that preserves I 4 is Z5 Z4, which is, in fact, a subgroup of S5 (which preserves I 3 ). The two groups have the same degree three polynomial but different degree four polynomials. This means that to break the symmetry from S5 to Z5 Z4, it is necessary to include quartic terms in the superpotential (1.1). Since we include only cubic terms (the quartic terms are not only irrelevant, but also break time-reversal symmetry), the symmetry of our Lagrangian will be S5. Also, I 5 is related to I 1 by an O(3) rotation. Including either I 1 or I 5 in the superpotential (1.1) will preserve the symmetry (S3×S3) Z2. The subgroup (Z3×Z3) Z2 preserves two polynomials I 1 and I 2 . However, any combination of the two polynomial aI 1 + bI 2 can be brought back to the form cI 1 by an O(3) rotation (the rotation parameter depends on a and b). This means a superpotential of the form W = g 1 I 1 + g 2 I 2 in fact preserves the symmetry (S3×S3) Z2 at any of point of the (g 1 , g 2 )-plane. This of course is also true for the SCFT fixed point.

The fixed points and anomalous dimensions
We can now use the explicit form of the polynomial I 3 to study the S5 invariant fixed point. Plug I 3 into (4.10), re-scale it to satisfy the normalization (4.3), and then plug it in (4.5), we get Since T 3 > − 3 4 , the fixed point exists, and from (4.4) we know This model belongs to the family of N =1 Potts models studied in [10].
Similarly, we can also use the polynomial I 1 to study the (S3×S3) Z2 invariant fixed point, we get Since T 3 > − 3 4 , the fixed point exists, and we get (4.14) Consider a superpotential W = gI 1 , we know that So that the model is simply two decoupled copies of the S3 super Potts model.  4 The invariant tensor of such a representation can be constructed using the δ ij of O(3), we will postpone the details of the construction to Section 6 6.2, where we discuss the generalization of this fixed point to a series SCFTs with the scalar superfields transforming in the T irrep of O(N). We note down here the T 3 constant, , the one-loop fixed point exists, the anomalous dimension is and the corresponding A function reads We now use the GAP system [21] for computational discrete algebra and the Small Groups library [22] to search for finite groups with 5 dimensional faithful irreducible representations, similar to the study done in [23]. Since all representations of finite groups are unitary representations, these finite groups are subgroups of U(5). There are two theorems that are especially useful in seeking faithful irreducible representations [23].
• Theorem 1 Suppose a finite group G has a p dimensional irreducible representation, then Ord(G)/p is an integer.
• Theorem 2 Suppose a finite group G has a p dimensional faithful irreducible representation, then this irreducible representation has a single character p in the character table.
The proof of Theorem 1 can be found in [24] Page 288. The proof of Theorem 2 was given in [23] Appendix A.1. The SmallGroup library allows one to specify a finite group using two integers [p, q], the first integer indicates the order of the group, while the second integer enumerates all finite groups with order p. We first select finite groups that are not Abelian. For example, the following command l := Filtered( AllSmallGroups(60) ,x -> IsAbelian(x)=false);; (5.6) gives all non-Abelian groups with order 60. The GAP system also allows us to calculate the character table easily Display(CharacterTable(SmallGroup(60,5)));. (5.7) To select 5 dimensional irreps, we can use psi := Filtered( Irr( CharacterTable( SmallGroup(60,5) ) ), x -> Degree( x ) = 5);;.
The symbol psi is now a list which contains all the characters of the 5 dimensional irreps: The characters contain a single 5, so according to Theorem 2 this is a faithful irrep. It is sometimes useful to check the structure description of the finite group StructureDescription(SmallGroup(60,5));. (5.10) From the output, we learn that SmallGroup(60,5) is the alternating group of five elements, or "A5". Since our goal is to search for degree 3 invariant polynomials, we can also use the command MolienSeries(psi [1]); (5.11) to obtain the Molien Series of the corresponding irrep. The Molien series is a generating function which counts the number of invariant polynomials of a certain degree. Here |G| is the order of the group G, ρ(g) is a representation of G. Doing a series expansion of M (z), the coefficient of z n counts the number of invariant polynomials of degree n. The Molien series of the five-dimensional irrep of A5 is So that it has one degree two invariant polynomial, two degree three polynomials and two degree four polynomials. There are more than 10 7 finite groups of order less or equal than 800. Among them, we found only 109 groups that have at least one 5 dimensional faithful irrep. They are listed in Table V. Among these 109 finite groups, it turns out 13 of them are also subgroups of O(5). That is, the finite groups preserve a bilinear form, or in other words, their Molien series takes the form (5.14) These groups are marked with an "*" in Table V. Among the 13 finite subgroups of O(5), only four of them have irreducible representations that preserve at least one degree three invariant polynomial. The results are listed in Table IV. For a fixed N Φ , bigger groups tend to preserve less invariant polynomials, which leads us to conjecture that this list is complete. 5

The group A5
The SmallGroup([60,5]) is the alternating group of five elements. As is explicit from the Molien series, it has a five dimensional irrep, which has a degree two invariant polynomial and two degree ID

Group Irrep Molien Series
Molien Series (expansion) [60, 5] A5 5 [120, 34] S5 5 a z 12 +z 10 +z 9 +z 8 +z 6 +1 to calculate a matrix representation of the generators of the group (5.16) Here (1,2,3,4,5) and (1,2,3) denote the cyclic permutations that generate the alternating group. We can now calculate the invariant tensor using these matrices, for example, we can use the Mathematica function "NullSpace[ ]". Notice that the generators from "IrreducibleRepresentationsDixon" are not necessarily orthogonal. The bi-linear form that is preserved by g 1 and g 2 is not proportional to the identity matrix. To fix this, we perform the Cholesky Decomposition of Ω = L T L. Then the new representation of the generators in terms of the orthogonal matrices is In this basis, we found the two degree three invariant polynomials to be and (5.20)

The group S5
The permutation group of five elements has two five dimensional irreps denoted by the Young diagram and . For simplicity, we will denote them as 5 a and 5 b . The group A5 can be embedded in S5 through the standard way. That is, S5 contains all the permutations of five elements, while A5 contains even permutation that permutes the five elements. Under this embedding, both 5 a and 5 d branch into the 5 dimensional irrep of A5.
5 a → 5, One can similarly work out the invariant polynomials by using the generators from "IrreducibleRep-resentationsDixon". It turns out that the invariant polynomial of 5 a is related to K 1 by an O(5) rotation, while the invariant polynomial of 5 b is related to K 2 by an O(5) rotation.

The group A6 and S6
The group S6 contains all permutations of six elements. There are two 5 dimensional irreps that preserve a degree-three polynomial. One of them is the standard irreps of permutation groups "5 std ". We will denote the other irrep as "5 ". As is clear from Table IV, the two irreps in fact have the same Molien series. This is only possible if the two irreps have exactly the same invariant polynomials. Physically, this means a potential V (φ) with φ transforms in the 5 std irrep can be also interpreted as a potential with φ transforms in the 5 irrep.
The symmetric group S5 can be embedded in S6 in the standard way. While S6 contains all the permutation that permutes six elements, S5 contains all the permutation that permutes five of the six elements. Under this embedding An invariant polynomial of the parent group S6 should also be an invariant polynomial of the subgroup S5. We, therefore, know the degree three invariant polynomial of 5 is simply K 1 . Since 5 std and 5 have the same Molien series, the invariant polynomial of 5 std should also be K 1 .
From Table IV, we know the Molien series of the group A6 and S6 are the same up to z 4 terms. Physically, this means we can not break the symmetry from S6 to A5 until we introduce Φ 5 terms in the (super-)potential. Since we consider only cubic terms of the superpotential, we should consider only the group S6.

The beta functions and fixed points
We can now consider the beta function of the N =1 Lagrangian with superpotential W = g 1 K 1 + g 2 K 2 , (5.23) with K 1 and K 2 defined in (5.19) and (5.20). The beta function is . (5.24) and the corresponding anomalous dimension is We find the following four fixed points:

SN× SN bi-standard fixed points
We discuss here the irreducible fixed points first. The decoupled S3 Potts model studied in Section 4 4.1 can also be understoodd as an SCFT with the four scalar superfields Φ i transforming in the bi-standard representation of S3× S3. This SCFT can be easily generalized to a higher number of scalar superfields, if we take the scalars to transform in the (r 1 , . . . , r n ) representation of G 1 × . . . G n . The constant T 3 defined in (4.5) is simply the product of the T 3 's of the individual G's, Interestingly, if we take G 1 = G 1 = . . . G n = G, and if n is an even number, The one-loop N =1 SUSY fixed point is guaranteed to exist. Take the group to be SN× SN, we have then Also the A-function is The projector to the T irrep is Here we use the birdtrack notation [25]. Solid lines mean the Kronecker δ ij , the unfilled box means symmetrization, and the filled box means anti-symmetrization. The invariant tensor for the T irrep of O(N) is simply, = . (6.6) We rescale the invariant tensor to satisfy the normalization (4.3), which can now be represented as = . (6.7) 15 After contracting the Kronecker delta's, we get Notice T 3 ≥ −3/14 for N ≥ 3, which satisfies the condition (4.8). So that the fixed points exist at one loop. The anomalous dimension is then . (6.10) The A-function is

Generalization of A5
The group A5 is very interesting. As far as we know, this is the first example of a group that has an irrep that satisfies the trace condition (preserves a single degree two polynomial) and preserves more than one degree three polynomial. To generalize this to the higher-component case we use the birdtrack technique [25]. To be more specific, we search for subgroups of SN groups that have an irrep with the Molien series 1 + z 2 + 2z 3 + 2z 4 + O z 5 . (6.14) Again we decompose two vector irreps of O(n) as S, A and T. If a subgroup of O(n) preserves a rank three symmetric traceless tensor, the T irrep of O(n) gets further decomposed into two irreps: The invariant tensor of the standard irrep of SN satisfies a special condition. That is, the A irrep of O(n) does not decompose. In term of birdtracks, it implies the following identity: is the dimension of the standard irrep of SN. The coefficient is fixed by contracting the top two legs. From the above relation we can show that the invariant tensor of SN satisfies with We now want to generalize the group A5 to a group with higher n. According to the Molien series, we first introduce another invariant tensor The Molien series also tells us that the irrep has only two degree four invariant polynomials. This means the two invariant tensors satisfy Contracting two legs with δ ij , we get The proof is given in Appendix A. Contracting two legs of (6.20) with , we get = P (6.27) with P = A 2 − 1 (n − 2) 2(n + 2) + A 2 T 3 . (6.28) Contracting two legs of (6.21) with , we obtain 2B 3(n + 2) Since the two tensor are independent, we get According to (6.25), this leads to A = − (n − 1)(n + 10) 6(n − 2)(n + 1) . (6.31) Using (6.20) and (6.21), after a long calculation, we get = 5n 2 − 71n + 86 12(n − 2)(n + 1) + 66 − 15n −8n 2 + 8n + 16 . (6.32) From this we know how the anti-symmetric subspace can be decomposed using the projectors: P 1 = (n + 1)(5n − 22) (n + 10)(5n − 13) + 12 −n 2 + n + 2 (n + 10)(5n − 13) , (6.33) and P 2 = 54(n − 2) (n + 10)(5n − 13) + 12(n − 2)(n + 1) (n + 10)(5n − 13) . (6.34) They satisfy P 1 P 1 = P 1 , and P 2 P 2 = P 2 . (6.35) The dimension of the corresponding irrep is Here we use to denote the Clebsch-Gordan coefficients (T i ) ab of P 2 × P 2 → n. This diagram is a sum of squares, and is therefore non-negative. This puts a constraint on the values that n can take. Notice that n is a positive integer, so that 3 ≤ n ≤ 40 . The n = 5 solution corresponds to the group A5 we have studied, while the n = 26 solution might correspond to a subgroup of S27. Let us assume that such a group exists and calculate the beta function and anomalous dimensions. Take the superpotential to be using the birdtrack relations (6.24), (6.26) and (6.27), we get , (6.42) and the corresponding anomalous dimension is  3 19 ) has γ Φ / = 7 18 and A/ 2 = − 1 9 2548π 2 . Notice the birdtrack rules (6.20) and (6.21) are very similar to the rules that lead to the classification of the F4-family of Lie groups, see [25] equation (19.16). Other birdtrack conditions lead to the classification of E6, E7 and E8 family of Lie Groups. Deligne conjectured that there exist categories interpolating these exceptional groups [26,27]. See [28] for how to use the Deligne category to make sense of O(N) invariant quantum field theories at non-integer N. It is tempting to conjecture that (6.20) and (6.21) also lead to a new Deligne category. We leave this for future work.

The N =1 Potts models
The N =1 Potts models were already studied in [10]. We recall here the procedure to construct the invariant tensor d ijk according to [29]. In [10], the d ijk were constructed explicitly and used to calculated γ Φ to two loops in . The result was shown to be consistent with the result from the non-perturbative bootstrap. Take The overall constant is chosen so that the invariant tensor satisfies the normalization (4.3). From the above definition of d ijk , we get and (6.47) 21

Generalization of reducible fixed points
Take the superpotential to be The β functions for the three coupling turn out to be The anomalous dimension is given by The constant T 3 is defined in (4.5). Take d ijk to be the invariant tensor of the standard irrep of SN (6.45). At The anomalous dimension is given by The A-function is In three dimensions, the large N limit of the fixed points is dual to N =1 higher-spin theory on AdS 4 through AdS/CFT correspondence [30,31].
The fixed point is located at .

DISCUSSION
A natural generalization of the N =1 analysis of this paper is to study N =2 SCFTs. Consider the following N =2 superpotential The β function is given by and the anomalous dimension matrix reads The scalar superfield Φ is complex so that the coupling h ijk transform in the fully symmetric representation of U(N). By studying subgroups of U(N) and analyzing their degree three invariant polynomials we can study the corresponding N =2 superconformal fixed points. Among the 109 finite subgroups of U(5) with 5 dimensional faithful irreducible representations, we found a single group whose Molien Series takes the form The group has a SmallGroup id [180,19] and structure description GL (2,4). This is the general linear group of degree two over the field of four elements. Notice that the corresponding irrep has no degree two invariant polynomials, it is in fact a complex irrep. This irrep has two degree three polynomials. Using these two polynomials to construct an N =2 superpotential as in (7.1), and analyzing the beta function, we get a full conformal manifold. The group GL (2,4) is in fact the direct product of A5 and cyclic group of order three Z3. The invariant polynomials of GL(2,4) are simply K 1 and K 2 as defined in (5.19) and (5.20 It might be interesting to study this conformal manifold in more detail. A similar study of a conformal manifold containing the XYZ model was performed in [32]. Notice the degree three invariant polynomials studied in this paper can be used to construct a Landau theory with cubic terms. The Landau criterion says that a second-order phase transition is possible only if the irrep of the order parameter has no degree three invariant polynomials. This however is known to fail in two dimensions. For example, the 3 state Potts model is known to go through a second order phase transition at the critical temperature. The corresponding Landau theory with a cubic term has an infra-red conformal fixed point. It will be interesting to understand whether the Landau theories analogous to the SCFTs studied in this paper lead to unitary conformal field theories in two dimensions. The problem perhaps can be studied using conformal bootstrap techniques. Due to the special form of the superpotential (1.1), we focus here on finite subgroups of O(4) and O(5) which preserve degree three invariant polynomials. It will also be interesting to study degree four polynomials, which can then be used to construct λφ 4 theories in 4 − dimensions. The early works in the 80's focused on λφ 4 theories where the scalars form a single irreducible representation of the symmetry group. In [19,33], all the subgroups of O(4) and the corresponding λφ 4 theories were studied extensively. The work of [34][35][36][37] studied the λφ 4 theories with N=6 and N=8 scalars forming irreducible representations of the 230 crystallographic space groups (see also [38]). Recently, there was also revived interest on fixed points of reducible λφ 4 theories where the scalars form more than one irrep of the symmetry groups [15,17,39,40]. Some of these perturbative fixed points were shown to survive in three dimensions using the numerical bootstrap [41][42][43][44][45][46][47]. A full classification of the irreducible λφ 4 theories with N scalars will need the classification of subgroups of O(N), this can be difficult when N is large. The method used in Section 5 5.2 based on the GAP system can be easily generalized to study λφ 4 theories. It will be interesting to see whether we can discover new fixed points in the 4 − expansion.
As mentioned in the introduction, the numerical bootstrap method has proven to be very powerful in studying 2 + 1 dimensional N =1 SCFTs. In particular, the work of [10] studied three infinite families of Wess-Zumino models. It will be interesting to apply the numerical bootstrap to the new O(N)/Z2 family discovered in this work. As usual, due to the non-perturbative nature of the numerical bootstrap, we expect rigorous numerical results that can then be compared to other methods such as the -expansion. We leave this interesting analysis for future work.