Accidental Symmetries and the Conformal Bootstrap

We study an ${\cal N} = 2$ supersymmetric generalization of the three-dimensional critical $O(N)$ vector model that is described by $N+1$ chiral superfields with superpotential $W = g_1 X \sum_i Z_i^2 + g_2 X^3$. By combining the tools of the conformal bootstrap with results obtained through supersymmetric localization, we argue that this model exhibits a symmetry enhancement at the infrared superconformal fixed point due to $g_2$ flowing to zero. This example is special in that the existence of an infrared fixed point with $g_1,g_2\neq 0$, which does not exhibit symmetry enhancement, does not generally lead to any obvious unitarity violations or other inconsistencies. We do show, however, that the $F$-theorem excludes the models with $g_1,g_2\neq 0$ for $N>5$. The conformal bootstrap provides a stronger constraint and excludes such models for $N>2$. We provide evidence that the $g_2=0$ models, which have the enhanced $O(N)\times U(1)$ symmetry, come close to saturating the bootstrap bounds. We extend our analysis to fractional dimensions where we can motivate the nonexistence of the $g_1,g_2\neq 0$ models by studying them perturbatively in the $4-\epsilon$ expansion.


Introduction
The technique of F -maximization, originally proposed in [1] and further developed in [2,3], allows one to identify the superconformal U (1) R symmetry at the infrared fixed points of a wide class of three-dimensional RG trajectories that preserve N = 2 supersymmetry. Identifying this U (1) R symmetry is desirable, since the scaling dimensions of BPS operators as well as unitarity bounds satisfied by operators that belong to long multiplets are all determined by their respective R-charges [4]. Determining the R-symmetry of a superconformal field theory (SCFT) using F -maximization relies on embedding that SCFT as the infrared limit of an RG flow that starts from a free UV theory. The matter content of the UV theory and the symmetries preserved by the flow are used as inputs for the F -maximization procedure.
The R-symmetry of the infrared fixed point is then determined in terms of these inputs; the explicit procedure is described in detail in [1]. This procedure can be carried out explicitly using the supersymmetric localization results of [1,5].
In applying the above method one implicitly assumes that the symmetries of the infrared SCFT are isomorphic to those preserved by the RG flow. An important caveat of F -maximization is that it may lead to incorrect results if this assumption fails. This may happen due to extra symmetries that emerge at the IR fixed point, but are not present throughout the flow; such extra symmetries are usually referred to as accidental. Indeed, the superconformal R-symmetry at the IR fixed point could happen to be a linear combination of the various U (1) symmetries preserved by the flow and of accidental U (1) symmetries.
In such cases the F -maximization procedure has to be modified in order to take those extra symmetries into account [6][7][8]. One indication that accidental symmetries may be present is when F -maximization yields unitarity violating scaling dimensions for some chiral operators. 1 It is generally speculated that these chiral operators actually become free, resulting in the emergence of additional symmetries that were not taken into account. In some other cases, accidental symmetries can be detected by passing to a weakly coupled dual of the original theory, in which the symmetry enhancement is manifest [10]. Finally, it is possible that the RG flows where the F -theorem [2,[11][12][13] seems to be violated could also indicate the presence of additional symmetries, but there is no known systematic way for detecting them.
Our work is focused on a simple family of models whose naive IR fixed points, obtained under the assumption of no symmetry enhancement, do not always exhibit any obvious problems such as those mentioned above. Nevertheless, we will show that these naive IR fixed points are ruled out as consistent unitary SCFTs, since their CFT data violates the constraints imposed by crossing symmetry. These constraints can be efficiently implemented with the help of the numerical conformal bootstrap technique introduced in [14]. We view this result as strong evidence that accidental symmetries must be present in the IR. To our knowledge, our work represents the first example where the conformal bootstrap is used to argue for such a result. 2 The theory we examine is a certain N = 2 supersymmetric generalization of the critical O(N ) vector model. This model has a chiral super-field X and N chiral super-fields Z i interacting through a cubic superpotential where g 1 and g 2 are coupling constants. The superpotential interaction triggers a supersymmetric RG flow that preserves O(N ) × Z 3 flavor symmetry. Under the O(N ) symmetry, the Z i transform as a fundamental vector and X is a singlet. The generator of the Z 3 symmetry acts by multiplying both X and Z i by e 2πi/3 . A consequence of the O(N ) × Z 3 symmetry is that no other superpotential interactions are dynamically generated throughout the flow. 1 We refer the reader to [9] for a nice discussion and examples in the context of 4d N = 1 gauge theories. 2 The idea that the conformal bootstrap could be used to provide evidence for the emergence of accidental symmetries was already proposed in [15].
The naive guess in this case is that generically, without further tuning of parameters, the RG trajectory would end at an IR fixed point preserving only O(N ) × Z 3 flavor symmetry where both g 1 and g 2 are non-zero. 3 At this fixed point, the superconformal U (1) R charges (and hence scaling dimensions) of X and of Z i would all equal 2/3, as this is the only R-symmetry preserved by the entire flow that commutes with O(N ) × Z 3 -in this case, F -maximization is rather trivial, since there is no freedom in what the superconformal R-symmetry could be. Due to recent progress on studying supersymmetric field theories on curved manifolds, one could compute certain observables of this naive fixed point exactly. For instance, one can use supersymmetric localization to compute its S 3 free energy F = − log |Z S 3 |, or further using the results of [3], one can also calculate the O(N ) "central determined by localization, is inconsistent when N > 2. 4 The naive fixed point of (1.1) therefore does not exist when N > 2, and we expect symmetry enhancement in the infrared. A weaker result can be obtained using the F -theorem [2,[11][12][13], which can be used to argue that this naive fixed point does not exist when N > 5.
The naive fixed point of (1.1) with non-vanishing g 1 and g 2 also does not exist for N = 2.
In this case, if g 2 = 0, the theory is equivalent to the XY Z model. The X 3 coupling in (1.1) is then marginal, and using conformal perturbation theory around the XY Z model, it can be seen to flow to zero in the infrared. 5 The assumptions used in our particular bootstrap analysis do not distinguish between the XY Z model and its X 3 deformation as in (1.1), which is why we do not exclude the N = 2 case. When N = 1, we believe that generically (1.1) flows to two decoupled copies of the N = 2 super-Ising theory (the latter was studied in a bootstrap context in [18]). We will give some evidence for this proposal in Section 2.2.
Our main focus in this paper is, however, the case N > 2.
When N > 2, what we believe happens with the model (1.1) is that the IR physics is 3 It should be clear that the couplings that flow are the ones obtained after rescaling X and Z i such that their corresponding Kähler terms are canonical. For simplicity, we will not make a distinction between the superpotential and the canonical couplings in our notation. 4 Using the value of c T of this fixed point by itself does not seem to lead to an inconsistency. 5 In fact, in 3d N = 2 SCFTs, a marginal operator can be either exactly marginal or marginally irrelevant [16]. It is known that the XY Z model has one exactly marginal deformation, and that it preserves the permutation symmetry between X, Y , and Z [17]. The marginal deformation X 3 does not preserve this symmetry and must therefore be marginally irrelevant.
governed by one of two other fixed points with enhanced symmetry. Apart from the UV fixed point with g 1 = g 2 = 0, the theory (1.1) is also believed to have a fixed point where g 2 = 0 and g 1 = 0, as well as a fixed point where g 1 = 0 and g 2 = 0. The former has O(N ) × U (1) flavor symmetry, as we review in Section 2.1. The latter fixed point has U (N ) × Z 3 flavor symmetry (the Z 3 is as before and under U (N ) the Z i transform as a fundamental and X is a singlet), being the product of a free theory of N chiral multiplets and the N = 2 super-Ising model studied in [18]. Both fixed points enhance the O(N ) × Z 3 preserved by the flow to a strictly larger symmetry group.
Among the applications of the models (1.1) with N > 2 are the world volume theories of M2-branes placed at Calabi-Yau singularities. For example, the theory with N = 5 has been proposed in [19] 6 as a description of a single M2-brane placed at the tip of the conical This claim is plausible because the classical moduli space of the model (1.1) is X = 0, 5 i=1 Z 2 i = 0, which is exactly the conical Stenzel space. We will show, however, that the SO(5) symmetric theory with non-vanishing g 1 and g 2 does not exist at the quantum level. We postpone a further discussion of models that arise on M2-branes to future work.
The rest of this paper is organized as follows. In Section 2, we provide some Renormalization Group arguments for why for N > 2 one does not expect a fixed point of (1.1) with both g 1 and g 2 non-vanishing. In particular, the F -theorem rules out such models for N > 5.
In Section 3, we provide rigorous bounds on c O(N ) J when the R-charge of Z i equals 2/3 and notice that the naive fixed point of (1.1) would be inconsistent for N > 2. In this section, we actually present our bounds more generally in d space-time dimensions, with 2 < d < 4.
We leave the computation of c O(N ) J using supersymmetric localization, as well as a detailed description of the bootstrap equations to the Appendices.

Renormalization Group Arguments
In this section we present some arguments that, for N > 2, the theory (1.1) is not expected to have a fixed point with only O(N ) × Z 3 flavor symmetry. The first argument relies on a general RG flow analysis and the second on computations done in the 4 − expansion. The third argument uses the F -theorem with which we are able to exclude our putative SCFT, but only for N > 5. A more rigorous proof that this putative CFT does not exist for all N > 2 will be given in Section 3 using the conformal bootstrap.

RG Flow Analysis
Let us start by analyzing the predictions of supersymmetry for various flows in the space of couplings g 1 and g 2 of our model (1.1). Consider first deforming the free theory g 1 = g 2 = 0 only by the relevant deformation X 3 , i.e., turning on g 2 = 0, while keeping g 1 = 0 in the superpotential (1.1). The theory flows to the super-Ising SCFT plus N free fields Z i , in which the R-charges of X and Z i , are given by r Z = 1 2 and r X = 2 3 , respectively. In this case, these charges are fixed trivially by demanding the marginality of the superpotential and the fact that the Z i are free. The deformation X Z 2 i has R-charge r XZ 2 = 2 3 + 2 × 1 2 < 2, and so it is a relevant deformation of the super-Ising plus N free fields SCFT. Turning on a non-zero g 1 we therefore expect to flow to a non-trivial SCFT in the IR in which r X = r Z = 2 3 . It is easy to see that this series of RG flows is consistent with the F -theorem.
The second series of flows is slightly more involved, and starts by considering the case This superpotential preserves more flavor symmetry than (1.1): it preserves an O(N ) symmetry under which the Z i transform as a fundamental vector and X is a singlet, as well as an additional U (1) under which Z i and X have charges +1 and −2, respectively. Since (2.1) is the most general superpotential interaction preserving this O(N ) × U (1) flavor symmetry, no other superpotential interactions are generated throughout the RG flow.
The infrared limit of (2.1) is believed to be an N = 2 superconformal field theory analog of the critical O(N ) model. The scaling dimensions of the chiral operators X and Z i are related to their superconformal R-charges, which can be determined using F -maximization and supersymmetric localization as in [21]. In this case, the most general R-symmetry preserved by (2.1) is such that the R-charges of X and Z i obey with arbitrary r Z . (The superpotential has R-charge 2.) Maximizing the S 3 free energy over r Z , Ref. [21] obtained the values listed in Table 1 is normalized so that it equals 1 in a theory of N free chiral multiplets.
Coming back to the theory (1.1) (with g 1 , g 2 = 0), one may interpret it as a superpotential deformation of the IR fixed point of (2.1) by the operator X 3 . One can see, however, that if N > 2, at the fixed point of (2.1), the operator X 3 has R-charge r X 3 > 2, and so the superpotential deformation by it is irrelevant. In other words, if one starts at the interacting superconformal fixed point that has g 1 = 0 and g 2 = 0 and turns on a small non-zero value for g 2 , then g 2 flows back to zero. An N = 2 fixed point with non-zero g 1 and g 2 , if it exists at all, can therefore not be reached from the fixed point with g 2 = 0. Moreover, the arguments of [16] guarantee that if such a fixed point exists as a unitary SCFT, it is attractive in the space of couplings (g 1 , g 2 ).
To summarize, we naively expect the space of theories described by (1.1) to contain four types of fixed points. The free theory in which both g 1 and g 2 are relevant; the super-Ising model plus N free fields that preserves U (N ) × Z 3 symmetry and in which g 1 is relevant, but but we have argued that it is impossible to have an RG flow line connecting the two, in either direction. It is thus reasonable to expect that one of these two fixed points does not correspond to a unitary SCFT. In the next section we will show that it is the O(N ) × Z 3 one that does not exist.

4 − Expansion
Our second argument why for N > 2 we do not expect a fixed point with only O(N )×Z 3 flavor symmetry comes from studying the models (1.1) in the 4 − expansion. From the results of [22], we can read off the beta functions for the physical couplings in 4 − dimensions. 7 They are known up to four loops, but here we state them up to the two-loop order for brevity: 3) The equations β 1 = β 2 = 0 have the following perturbative solutions: . These redefinitions lead to two decoupled copies of the super-Ising theory whenever |g 1 | = |g 2 |, and one can indeed verify 7 The physical couplings are g phys 1 = g 1 / Z X Z 2 Z and g phys 2 = g 2 / Z 3 X , where Z X and Z Z are the wavefunction renormalization factors for X and Z, respectively, in a holomorphic scheme where the superpotential is not renormalized. Above, we use the notation g i instead of g phys i to avoid clutter. We see that there is no fixed point that has O(N ) × Z 3 global symmetry perturbatively in the 4 − expansion. It is plausible, however, that such a fixed point would appear non-perturbatively, so the 4 − expansion cannot be used to argue conclusively that such a fixed point would also be absent when = 1.

F -theorem Arguments
The F -value of the naive fixed point of (1.1), where both g 1 and g 2 are non-vanishing, can be computed using the supersymmetric localization results of [1]. It equals −(N where (z) is the function defined in [1] representing the contribution to the S 3 free energy of a chiral multiplet of R-charge 1 − z, since at the naive fixed point of (1.1) the fields X and Z i all have R-charge 2/3. Numerically, we have One could imagine deforming this fixed point by giving the X field a large superpotential mass, m X X 2 . As can be seen from solving the classical equation of motion, integrating out X then induces a quartic superpotential interaction (Z i Z i ) 2 for the Z i fields. If we then finetune the O(N )-invariant mass of the Z i fields (a smaller and smaller degree of fine-tuning is required as we take m X → ∞), the IR limit can be argued to be simply a free theory of the N fields Z i . Its F -coefficient is The RG flow we described must be possible if the fixed point with O(N )×Z 3 flavor symmetry exists. We see, however, that F O(N )×Z 3 < F N free fields when N > 5, contradicting the Ftheorem [2,[11][12][13]. This F -theorem argument thus rules out the existence of a fixed point of (1.1) with non-vanishing g 1 and g 2 for N > 5. This is a weaker bound than the one obtained using the conformal bootstrap in the next section.
The theory with g 2 = 0, (2.1), may also be deformed by the relevant operator X 2 . This flow was considered in [21] and shown to also lead to a free theory of N chiral superfields Z i . While such a flow provides a counter-example to the c T theorem, the F -theorem holds for all N [21]. This is consistent with our arguments: the enhanced symmetry theory with g 2 = 0 exists for all N .
It has been conjectured that the three-dimensional F -theorem is a special case of the Generalized F -Theorem valid in continuous dimension [23,24]. The conjecture states that Here,F U V is the generalized free energy defined in [23] corresponding to the theory (1.1) with O(N ) × Z 3 symmetry, whileF IR corresponds to the infrared fixed point obtained by deforming that theory by X 2 .
F UV >F IR , and for Wess-Zumino theories with 4 supercharges where the function F(∆) is given in eq. (5.23) of [23]. In Figure 2   R-charges of X and Z are fixed to be r X = r Z = 2/3 in any d, but the relation between the scaling dimension and R-charge is dimension dependent [18,23]: The result for c in agreement with the value listed in Table 1 for N = 2, in which case the theory (2.1) also has r X = r Z = 2/3.     Table 1. The blue triangles correspond to N = 3, 6, 9, from right to left. and is the spin. We use a similar convention for OPE coefficients.) In our normalization,  In obtaining our bounds in Figure 4 for the O(N ) current central charge, we noticed that the superconformal fixed points with enhanced O(N ) × U (1) global symmetry given in (2.1) come close to saturating these bounds, being located at certain kinks in the boundary between the allowed and disallowed regions. We hope to report on a more careful study of these SCFTs in a later publication [42].

PHY-1318681 (SG), and PHY-1314198 (IRK and RY). SSP thanks the Aspen Center for
Physics for hospitality and partial support through NSF Grant No. PHY-1066293 during the completion of this work.

A Supersymmetric Localization
(A.1) In general, for an abelian current it is convenient to define so that a free chiral multiplet with unit charge has c U (1) J = 1. From comparing (A.1) to (A.2), we see that We can now compute c U (1) J from supersymmetric localization. Let us define Y ± = Z 1 ±iZ 2 and write the superpotential (1.1) as Under U (1), Y ± have charges ±1 while all other fields are neutral, so we consider the trial R-charges where y is the trial parameter and r Z * = 2 3 at a fixed point of (1.1) with only O(N ) × Z 3 flavor symmetry. In order to study the fixed point of (2.1), one should determine r Z * by maximizing the S d free energy with respect to r Z * . Eq. (A.5) is consistent with the marginality of the superpotential (A.4) and the coefficients y are precisely equal to the U (1) charge.
The S d free energy is whereF is the quantity defined in [23] that interpolates between the familiar Free Energies defined in integer dimensions. Here, we used ∆ = (d − 1)r/2 as the relation between the scaling dimension and R-charge for a chiral operator. For a single chiral super-field, F can be compactly defined by its derivative: Regardless of r Z * , F is maximized when y = 0. The coefficient c can be computed according to the results of [3] as where the constant of proportionality is fixed by requiring that c  Table 2 for numerical results in various d.
If we wish to compute c for the fixed point of (2.1) with O(N )×U (1) flavor symmetry, we should first maximize F with respect to r Z * . In d = 3, this maximization was performed in [21] and the result is also listed in Table 1. In Table 1 and work with Z Ii , (I = 1, 2 being a fundamental SO(2) R index) instead of Z i andZ i . We will examine the four-point function: which includes all orderings of two Z's and twoZ's at once.
The operators appearing in the Z Ii ×Z Jj OPE can be classified according to their transformation properties under SO(2) R × O(N ). We have singlets or rank-two traceless symmetric tensors of SO(2) R (corresponding to operators that have zero or ±2r Z R-charge, respectively) denoted by S and T , as well as singlets, rank-two traceless symmetric tensors, or rank-two anti-symmetric tenors of O(N ) denoted by s, t, and a, respectively. Due to Bose symmetry, the operators of the type T s and T t must have even spin, those of the type T a should have odd spin, and there are no spin restrictions on the other operators.
Forgetting about supersymmetry and performing the s-channel OPE in (B.1), one can write the four-point function as a sum over conformal blocks G ∆, for identical scalar operators [43]. We have 8 and F (1) In the sum (B.2), we sum only over conformal primaries O.
Supersymmetry relates some of the OPE coefficients in (B.2) to one another. As explained in [18], in the T channel, there is only one operator per superconformal multiplet contributing to (B.2). In the S channel, there are generically four operators contributing that have related OPE coefficients. Their contributions can be grouped into a superconformal block. As in [18], let us define We also defineG ∆, to be the same expression as (B.5) with the middle two terms multiplied by (−1). Then, combining the contributions coming from the same superconformal multiplet amounts to the replacements where the V ∆, are given by Here, we defined (B.10) The operator spectrum is further constrained due to the N = 2 supersymmetry [18,45]. Specifically, by generalizing the reasoning in [18] for N = 2 SCFTs to include O(N ) symmetry, we find the constraints listed in Table 3. . This quantity is related to the OPE coefficient λ Sa,d−1,1 with which the O(N ) current, j µ ij , appears in the Z i ×Z j OPE. Since j µ ij is a superconformal descendant, its OPE coefficient λ Sa,d−1,1 is related to the OPE coefficient  Table 3. The numerical results presented in the main text were generated as follows. We used a Mathematica script to generate the conformal blocks G ∆, (u, v) in arbitrary dimension, using the recursion formula for scalar conformal blocks [46]. We implemented the semidefinite programming required by the numerical bootstrap using SDPB [47], for which we used the parameters specified in the first column of Table 1 in the SDPB manual [47]. The convergence of our results was tested by varying the maximum number of derivatives, Λ, of the functionals α-specifically, we notice that the bound for Λ = 19 differ from those with Λ = 21 by 10 −3 in the value of c O(N ) J . Furthermore, as we increase the search space in α, the space of allowed theories can only become smaller, which implies that once a theory is excluded by the numerical bootstrap at a given value of Λ, it is rigorously excluded from the space of all mathematically consistent unitary CFTs.