On 4d rank-one \( \mathcal{N}=3 \) superconformal field theories

Journal of High Energy Physics, Sep 2016

We study the properties of 4d \( \mathcal{N}=3 \) superconformal field theories whose rank is one, i.e. those that reduce to a single vector multiplet on their moduli space of vacua. We find that the moduli space can only be of the form ℂ3/ℤ ℓ for ℓ=1, 2, 3, 4, 6, and that the supersymmetry automatically enhances to \( \mathcal{N}=4 \) for ℓ=1, 2. In addition, we determine the central charges a and c in terms of ℓ, and construct the associated 2d chiral algebras, which turn out to be exotic \( \mathcal{N}=2 \) supersymmetric W-algebras.

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On 4d rank-one \( \mathcal{N}=3 \) superconformal field theories

Received: April Published for SISSA by Springer 0 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 University of Tokyo , Bunkyo-ku, Tokyo 113-0033 , Japan 2 Kyoto University , Kyoto 606-8502 , Japan 3 Department of Physics, Faculty of Science 4 Yukawa Institute for Theoretical Physics 5 Open Access , c The Authors We study the properties of 4d N = 3 superconformal eld theories whose rank is one, i.e. those that reduce to a single vector multiplet on their moduli space of vacua. We nd that the moduli space can only be of the form C3=Z` for `=1; 2; 3; 4; 6, and that the supersymmetry automatically enhances to N = 4 for `=1; 2. In addition, we determine the central charges a and c in terms of `, and construct the associated 2d chiral algebras, which turn out to be exotic N = 2 supersymmetric W-algebras. gauge theory ArXiv ePrint: 1602.01503 Conformal and W Symmetry; Extended Supersymmetry; Supersymmetric - 3 superconformal eld theories 1 Introduction and summary Basic properties Allowed forms of the moduli space Higgs branch operators Coulomb branch operators 3.2 Identifying the 2d N =2 super Virasoro multiplet 3.3 2d operators corresponding to Higgs branch operators Construction of the associated 2d chiral algebras ` = 1 ` = 2 ` = 3 ` = 4; 5; 6 A The 4d N = 3 superconformal algebras Detailed computations Introduction and summary several years on the supersymmetric dynamics tell us, however, that there are many `nonLagrangian' theories, i.e. strongly-coupled eld theories which do not have obvious Lagrangian descriptions. erties were rst discussed in a paper by Aharony and Evtikhiev [1] from early December orientifolds in F-theory.1 2015. Later in the same month, Garc a-Etxebarria and Regalado made a striking discovpurely eld-theoretical manner. We mainly restrict attention to rank-1 theories, where the rank is de ned as the dimension of the Coulomb branch of the theory considered as an N = 2 theory. We will nd ` = 1; 2; 3; 4; 6, In addition we construct the 2d chiral algebras associated in the sense of [5] to these rank-1 N = 3 theories. We will nd the following: conformal algebra. The Jacobi identities of these operators close only for a nite number of central charges, including c2d = 1) as predicted from the construction of [5]. Furthermore, the null relation correctly encodes the structure of the moduli space of supersymmetric vacua at this value of the central charge. Further studies of these chiral algebras will uncover the spectrum of BPS local operators All the ndings in this note are consistent with, but do not prove, the existence of we compute in this note do not distinguish them. The rest of the note is organized as follows: in section 2, we study basic properties of of the associated 2d chiral algebra. In section 4, we use the results obtained so far to although no concrete models were identi ed there. The authors thank T. Nishioka for bringing this referIIA and type IIB setups. name given by Kodaira; (u) is the scaling dimension of the Coulomb branch operator u; the complexi ed coupling at the generic points on the Coulomb branch; and g is the SL(2; Z) monodromy around the origin. On the row for , ! is a third root of unity, and arb. means that Note added: when this paper is completed, the authors learned from P. Argyres, M. Lotito, Y. Lu and M. Martone that they have an upcoming paper [13] which has a small overlap with but is largely complementary to this paper. The authors thank them for sharing the draft in advance. Basic properties Allowed forms of the moduli space be a one-dimensional scale-invariant Seiberg-Witten geometry. Its classi cation is wellknown: one just needs to go through Kodaira's list of singularities of elliptic brations and keep only the ones where the modulus of the elliptic ber is constant. The resulting list is reproduced in table 1. In particular, the scaling dimension of the Coulomb branch operator u is xed to be one of the eight possible values listed there. = z` and J = z+z satisfying branch is then a hyperkahler cone of quaternionic dimension one. Such a one-dimensional isometry. This restricts to be of the form Z`. Let (z+; z ) be the coordinates of C2 before therefore relates the operator W and u, as we will see this in more detail in section 3.1. has dimension ` and U(1) charge `, and J is the moment map of the U(1) (Q1; Q2; Q3; Q4) 7! ( 1=2Q1; 1=2Q2; 1=2Q3; 3=2Q4): The action of the duality transformation by g on the supercharges can be found e.g. in section 2.2 of [15]: (Q1; Q2; Q3; Q4) 7! 1=2(Q1; Q2; Q3; Q4): Combined, we see that the action on the four supercharges is given by This means that the integer ` should also be a number allowed as (u). We conclude that where ` is one from the list above. Note that away from the origin, the moduli space is smooth. As such, the theory is locally plet, which determines its action of the four supercharges as (Q1; Q2; Q3; Q4) 7! (Q1; Q2; Q3; sponding moment map operators of dimension two. This in turn implies that the avor Finally, let us determine the central charges a and c of these theories labeled by `. c = 1 X(2 (ui) where the sum runs over the independent generators of the Coulomb branch operators. This relation was originally conjectured in [16] and a derivation that applies to a large 2The analysis of the supercharges here is completely the same as the one given in Garc a-Etxebarria and Regalado [2] done in F-theory. The point here is that it can be phrased in a completely eld-theoretical manner. used in [17] is satis ed by strongly-coupled theories we are discussing here, but the authors think it is quite plausible.3 Assuming the validity of the general formula, we then have gravity-gravity anomaly, which is proportional to a c, is conserved. On the Higgs branch the theory is just N = 4, and therefore a c = 0. From the known value of 2a c above, we conclude that c = a = c = As mentioned above, the derivation here is not completely watertight, but we give a rather non-trivial consistency check in the rest of the paper. realizations of these theories. vacuum moduli space is simply C 3 without any singularity, and therefore we can safely conclude that the only such theory is a theory of a single free U(1) vector multiplet. For algebra su(2). The gauge group can either be SU(2) or SO(3), depending on which we have two subtly di erent theories.4 started from the F-theory setup of the form R1;3 T 2 where the last T 2 describes the axiodilaton of the Type IIB theory, took the quotient (C3 T 2)=Zk, and probed this as we saw above. There is a caveat however: we cannot directly identify the integer k governing the F-theory background and the integer ` governing the moduli space of the superconformal SL(2; Z) duality of the type IIB. One is the O3 plane and the other is the SL(2; Z) 3It is known that this relation fails in gauge theories where part of the gauge symmetry is disconnected both have 2a latter. In this note, when we speak about the moduli space of vacua, we declare that we do not impose the invariance under the disconnected part of the gauge group, or whatever that concept corresponds to in non-Lagrangian theories. The author expects that this relation holds under this condition. as discussed in Footnote 3. super Yang-Mills theory with gauge algebra so(2), whereas the latter gives that with gauge algebra su(2). As discussed in Footnote 3, we declare that when we discuss the moduli space we do not gauge by the disconnected part of the gauge group, and then the former F-theory. Depending on the version, we will have a di erent discrete quotient C3=Z` ! C3=Zk where the left hand side is the moduli space of the superconformal theory and the right hand side is the F-theory background. We do not yet know which version of the Zk quotient gives which divisor ` of k. understood and requires further study, and the details will be reported elsewhere [18]. We would like to point out that, even assuming this, the F-theory construction gives a realization; we do not yet know whether there are multiple subtly di erent versions of the theory for each ` = 3; 4; 6 either. mass deformations through the construction of the Seiberg-Witten curves and di erentials is currently under investigation by Argyres, Lotito, Lu and Martone [25, 26]. The properties describing the IV might similarly correspond to some of the entries in the same table. We immediately notice, however, that there are no entries corresponding to the III and II singularities marked there as having u(1) avor symmetry. This does not yet preclude the existence of acting on the mass parameters were considered as coming from the Weyl symmetry. In particular, in their construction, those marked as having su(2) avor symmetry can be interpreted as having Z2 n U(1) symmetry. This point clearly needs further study.6 follow the conventions of [27]. 6The authors thank P. Argyres, M. Lotito, Y. Lu and M. Martone for instructive discussions on this point, and for sharing their upcoming paper [13]. )commutation relations in our notation are summarized in appendix A. In particular, the fermionic generators are Q ; _ = _ and I = 1; 2; 3. This are respectively given by + = The Rij for i; j = 1; 2 and R33 generate an su(2)R we take u(1)F to be generated by The quantum numbers of the supercharges are listed in table 2 together with the eigenvalues of the following linear combinations of charges: u(1)F subalgebra of u(3). Here _ + +_ = where E is the scaling dimension and j1; j2 are the so(4) spins such that M+ j2. We will use the above two linear combinations of charges to discuss, in 1 ; (Q1 )yg = E 2 fQ~2 _ ; (Q~2 _ )yg = E The anti-commutation relations (A.3) imply various unitarity bounds on operators. In particular, the presence of the third set of supercharges implies the following unitar 3 ; (Q3 )yg = E 2 fQ~3 _ ; (Q~3 _ )yg = E This particularly means that any scalar operator should have E 1 ; (Q1 )yg = E 2 fQ~2 _ ; (Q~2 _ )yg = E N = 3 unitarity bounds (3.4) reduce to Moreover, Higgs branch operators are annihilated by all of SI other SI ; S~I _ on Higgs branch operators breaks one of the unitarity bounds in (3.5). = (QI )y and S~I _ = (Q~I _ )y of the Higgs branch chiral ring, W +; W (`; `); (`; `) and (2; 0), the W In particular, W + is annihilated by Q annihilated by Q 1 ; Q~2 _ ; Q~3 _ (and their conjugates). 3 (and their conjugates), while W Coulomb branch operators Let us next consider Coulomb branch operators, which are de ned as scalar local operators unitarity bounds Higgs branch operators 1 ; (Q1 )y and Q~2 _ ; (Q~2 _ )y for called Higgs branch operators. They are de ned as local operators annihilated by all of and _ = _ . Since they saturate the follow2 fQ~1 _ ; (Q~1 _ )yg = E 2 fQ~2 _ ; (Q~2 _ )yg = E 2j2 + 2R + r and therefore have E = we see that Coulomb branch operators saturate the rst unitarity bound in (3.4), and therefore are annihilated not only by Q~1 _ ; Q~2 _ (and their conjugates) but also by Q (and its conjugate).8 From the unitarity bounds (3.4) and (3.7), we also see that they are N = 3 superconformal primaries. is determined by the fact that u can be regarded as a Higgs branch operator with respect to another set of N = 2 supercharges, say Q 2 ; Q~2 _ . With this new choice of N = 2 symmetry, Q is annihilated by Q 1 and Q~1 _ are regarded as the \third" set of supercharges. Since u 3 ; Q~2 _ and their conjugates, it is indeed regarded as a Higgs branch anti-chiral part of the \third" set of supercharges, Q~1 _ . This implies that u is mapped to (and vice versa) by exchanging (Q1 ; Q~1 _ ) and (Q3 ; Q~3 _ ). Since this exchanging is a part of the U(3)R symmetry of the theory, we see that the scaling dimension of u is given (u) = ) = `.9 a Coulomb branch operator to a Higgs branch operator saturating the second unitarity In this sub-section, we show that the 2d chiral algebra [T ] corresponding in the sense First of all, let us recall that Schur operators are de ned as local operators with j1 + j2 = E j2 = The unitarity implies that they are operators annihilated by Q Any local operator which is not a Schur operator has 1 > 0 or 2 > 0. It was shown in [5] algebra. In particular, every 4d Schur operator O maps to a 2d local operator [O] with 2d chiral operator product expansions (OPEs) determined by 4d OPEs. The 2d chiral algebra always contains a Virasoro subalgebra with the identi cation L0 = E discussed in [28]. 9Exchanging (Q2 ; Q~2 _ ) and (Q3 ; Q~3 _ ) maps the conjugate of u to W + and vice versa. In other words, 10This also follows from the fact that R 2 R 3 = r 2 3 11This was also noticed by O. Aharony, M. Evtikhiev and R. Yacoby (unpublished). F2 has only integer eigenvalues as R11 R22 = 2R. following bosonic Schur operators: F = 0.12 The corresponding 2d operator is the 2d stress tensor. was denoted by J in section 2. The corresponding 2d operator is an a ne U(1) current. Other bosonic Schur operators will be discussed in the next sub-section. table 2, we see that Q + and Q~3+_ have 1 = 3 and Q~3+_ on the above bosonic ones. For example, Q3+J 11 and Q~3+_ J 11 are two fermionic Schur operators, which are non-vanishing due to the unitarity bounds (3.4).14 Moreover, section 3.1. Then, as shown in appendix B, the corresponding 2d operators 2 = 0 and therefore act on the space of are Virasoro primaries. From (3.9), we see that their holomorphic dimension is 32 . Moreover, G and G respectively have charge +1 and 1 under J since Q + and Q~3+_ have U(1)F 3 Let us next consider fQ3+; Q~3+_ gJ 11 and [Q3+; Q~3+_ ]J 11. While the former is a conmoreover is a conformal primary. According to [5, 27, 30], the only such Schur operator is the highest weight component of the SU(2)R current, J+11+_ , in the stress tensor multiplet. Assuming that the theory T contains only one stress tensor multiplet,15 we conclude that J+11+_ = 12Here, we follow the convention of [5]. Namely, J+11+_ is the highest weight su(2)R so(4) component of the SU(2)R current J _ 13On the other hand, Q3 and Q~3 _ have either 1 > 0 or 2 > 0, and therefore their actions cannot create any Schur operator. They map any local operator to a non-Schur operator or zero. 14In the language of [27], these operators are respectively in the D 12 (0;0) and the D 12 (0;0) multiplets. 15If T is the union of two or more mutually non-interacting SCFTs, it contains more than one stress tensor two or more separate N = 2 SCFTs. superconformal multiplet as the stress tensor.16 This means that the corresponding 2d chiral operators J; G; G and T are also in a 2d super multiplet. It is a standard fact that in 2d, the energy momentum tensor T , a U(1) current J , and two fermionic dimension 3/2 currents G; G of U(1) charge c2d = and L0 = E R in [5],17 and our identi cation (3.12) means G 1 = G 1 = G 1 = G 1 = 2 S3+; J0 = F : It is then straightforward to show that, under these identi cations, L0; L 1; J0 and G 1 ; G 1 generate a subalgebra of su(2; 2j3) which acts as sl(2j1) on the space of Schur 2 2 2d operators corresponding to Higgs branch operators In addition to the above Schur operators, the Higgs branch operators are all Schur operators. We here show the following two statements: 1. For any Higgs branch operator O, [O] is a superprimary operator. 2. For any Higgs branch operator O with E = F , [O] is a(n) (anti-)chiral superpri In the next section, we will use the second statement to identify the 2d chiral algebras Let us rst show the rst statement. Suppose that O is a Higgs branch operator. Since O is a Hall-Littlewood operator in the language of [5, 30], [O] is a Virasoro primary in two dimensions (as shown in section 3.2.4 of [5] and reviewed in appendix B). Therefore we only need to show that [O] is annihilated by Gn+ 12 for n superconformal primary as shown in section 3.1, it is annihilated by (Q3+)y; (Q~3+_ )y. This 0. Since O is an N =3 means that [O] is annihilated by G 1 and G 1 . Therefore, for all n 2 2 Gn+ 12 = Gn+ 12 = 16Further actions of Q3+ or Q~3 +_ on these operators do not create any new Schur operators up to their conformal descendants. 17The extra factor of 21 comes from our di erent normalization of P _ and K _ . also annihilate [O]. Finally, 3 L2G 1 [O] ; 3 L2G 1 [O] ; Hence, [O] is a superprimary in two dimensions. are vanishing because G 1 [O] and G 1 [O] are Virasoro primaries (see appendix B). 2 2 annihilated by Q 3 (or Q~3 _ ). This particularly means that the corresponding 2d operator, [O], is annihilated by G 1 (or G 1 ). Thus, we see that any Higgs branch operator with 2 2 E = F (or E = F ) maps to an anti-chiral (or chiral) superprimary in two dimensions. Construction of the associated 2d chiral algebras Based on the properties we uncovered in the previous section, here we proceed to the congenerators of the 2d chiral algebra, as was shown in [5]. In our setup, the Higgs branch operators in 4d are generated by W+, W and J , whose dimensions are `, `, 2 and the U(1) charges are `, `, 0 respectively, with one relation As shown in section 3.2, [J ] is the bottom component of the super energy momentum the following shorthand notations for them: J := [J ]; W := [W+]; W := [W ]: In the cases studied previously in the literature e.g. [5{7], it was often the case that the entire 2d chiral algebras were generated by taking repeated operator product expansions of the Higgs branch operators. We use this empirical feature as a working hypothesis and will nd out that it leads to a consistent answer. As it is important, let us record here our bosonic chiral primary W and a bosonic antichiral primary W , both of dimension `=2. sistent with the 4d central charge c4d = (2` 1)=4 derived in (2.8) with the standard mapping c2d = 12c4d. Furthermore, we see that the construction automatically leads to a null relation of the form W W / J 3 + (composite operators constructed from J and (super)derivatives); reproducing the Higgs branch relation. reproduced for c2d = 21, and for ` = 6, the allowed c2d are 15 and 18, with the Higgs branch relation reproduced for c2d = Before proceeding, we note that the 2d chiral algebras satisfying the assumption above This choice was more natural for a 2d unitary algebra, since the spin of W and W is for ` = 4 and ` = 6 with bosonic W were constructed in [33], with the allowed central charges as listed above. The null relation leading to the Higgs branch relation was not expansion was developed and described.18 dinate Z consists of the bosonic coordinate z and the fermionic coordinates and . We mostly follow the convention of Krivonos and Thielemans [34], where the Mathematica We de ne the superderivatives to be Then a chiral super eld W and an antichiral W satisfy D = @ D = @ DW = 0; DW = 0 respectively. The operator product expansions can be usefully done using covariant com Z12 = z1 12 = 1 12 = 1 Then the energy momentum super eld J (Z) has the operator product expansion J (Z1)J (Z2) c=3 + 12 12J and a superprimary O with dimension expansion with J given by and U(1) charge F has the operator product J (Z1)O(Z2) 18Note that they called the operators satisfying DW antichiral primary, but we call such operators chiral primary and vice versa. They also had a typo in their super OPE of the superconformal algebra in their (7), where c=4 should be c=3. Here, in the equations (4.7)and (4.8) and below, the operators on the right hand sides of In our convention the (anti)chiral primaries are those with = F=2 ( Note that our 2d algebra is not unitary, and therefore, = F=2 does not immediately imply that the antichiral derivative to vanish. Rather, we use the fact that W and W come from 4d Higgs branch operators W+ and W (anti)chiral primaries. The normal ordered product of two operators O1 and O2 is de ned as the constant term, i.e. the term without any power of 12, 12 or Z12 in the operator product expansion of O1 and O2. Note that this does not always agree with the normal ordered product of two operators de ned as the constant part of the operator product expansion of the bottom components on the non-superspace parametrized only by z. The normal ordered product of more than two operators are de ned by recursively taking the operator product expansions from the right, i.e. O1O2O3 = (O1(O2(O3 of an N = 3 theory to conclude that they are Our computational strategy is quite simple. We rst require the operator product expansions of J with itself (4.7), and that W , W have the operator product expansions with respect to J given by (4.8) where = `=2 and F = `, and that W (Z1)W (Z2) W (Z1)W (Z2) The only operator product expansion that needs to be worked out is that of W and W . Our assumption implies that only J and composite operators constructed out of it appear in the singular part of this operator product expansion. Demanding that W (Z1)W (Z2) to be annihilated by D1 and D2, we nd that it has the form W (Z1)W (Z2) d=1 Z12d 2 Z12 where Od is an operator of dimension d constructed out of J and its (super)derivatives. normalization of W and W . Demanding the closure of the Jacobi identity among J , W and W then xes all other Od. Note that this is just the standard fact that when we x the normalization of a primary (this time, the identity operator) in an operator product expansion, the contribution of all the descendants are automatically xed. The explicit expressions for Od are given in [33]. The only nontrivial procedure is to check the closure of the Jacobi identity among W , W and W ; the analysis of the Jacobi identity for the triple W , W and W is similar, thanks to the discrete symmetry exchanging W and W . The computations can be performed easily and quickly using SOPEN2defs, the Mathematica package written by Krivonos and Thielemans [34]. On a 2012 notebook computer, the computation time was dominated by the time needed to type expressions into a notebook. The entire computation of Jacobi identities etc. took at most a few minutes. The Mathematica notebook detailing the computations below is available as ancillary les on the arXiv page for this paper. ` = 1 1=z and two neutral fermions , of dimension 1 with agrees with the expected formula c2d = 3(2` 1). In fact this case was already essentially discussed in [5]. ` = 2 As such, the operator product expansions close for arbitrary value of c2d. Explicitly, we need to choose O0 = c=3 and O1 = J . It is still instructive to see when there can be null relations representing the Higgs branch relation W W / J 2. In the language of the 2d chiral algebra, this should correspond to a null relation of the form (a1J 2 + a2J 0 + a3[D; D]J ) = 0: It turns out, however, that only the rst choice makes the left hand side of (4.11) to be an 9 = 3(2` 1 2 [D; D]J is given by the Sugawara stress tensor associated with the SU(2) current formed by W , J and W . This is consistent with the discussion in section 5.3.2 of [5]. Before 9 leads to new null operators X = D@W J DW + 2(DJ )W; X = D@W + J DW ` = 3 general values of c2d, since the failure of the Jacobi identity contains terms proportional to 15. Note that 12c4d = With this value of the central charge, the W W operator product expansion (4.10) is O0 = ; O1 = J; O2 = The failure of the Jacobi identity for W , W , W now contains only terms proportional to X = D@W J DW + 3(DJ )W and DX = the Jacobi identity for W , W and W closes. nd that the Jacobi identity for W , W and W closes after demanding that the composite operator X = D@W + J DW One further nds that the operator product of X and W is regular, while that of X and W contains operators whose scaling dimensions are larger than that of X . Similar statements hold X . This guarantees that X and X are the operators with lowest dimension among the null states to be removed. Another null state is obtained by taking the operator product expansion of W with X , whose coe cient of 12=Z122 is proportional to Y = 36W W (J 3 + 9(@J )J + 6J [D; D]J + 6DJ DJ + 6[D; D]@J + 7@2J ): This operator is null, and correctly represents the 4d Higgs branch relation W W / J 3. ` = 4; 5; 6 failure of the Jacobi identity for W , W , W contains terms proportional to the identity operator times (c 9 and 12, we nd that the Jacobi identity can be satis ed by imposing a null relation. But we nd that the null relation is only consistent with the expected Higgs branch relation when c2d = 21 = is c = 15 and 18. Again, the null relation is compatible with the Higgs branch 33 = relation only for c2d = null operators are X = D@W J DW + `(DJ )W; X = D@W + J DW see (4.13), (4.15), (4.16). consistent for c2d = 27 = 1), and the null relation are generated by the same X and X given in (4.18). A descendant by W of X generates a new null relation of the form W W / J 5 + (operators involving (super)derivatives). Note that the existence of the theory with ` = 5 in four dimensions. The analysis so far suggests that there is a series of super W-algebras generated by of dimension `=2 with c2d = 1), with the basic null elds as given in (4.18). The operator product expansion of W and W has the form (4.10). A descendant of the null eld seems to automatically give the relation of the form `2W W = where the coe cients are guessed from the examples so far. Note that we normalized W two dimensions, to see whether such a series of 2d chiral algebras indeed exists. Acknowledgments The authors thank S. Nawata for helping locate the literature on W-algebras, and T. Nishioka for bringing the paper [3] to the authors' attention. The authors also thank P. Argyres, M. Lotito, Y. Lu, M. Martone for instructive discussions on their papers [13, 25, 26]. T.N. would like to thank M. Buican and Y.T. would like to thank O. Aharony for illuminating discussions in related projects. T.N. is the Yukawa Research Fellow supported by Yukawa Memorial Foundation, and the work of Y.T. is partially supported in part by JSPS Grant-in-Aid for Scienti c Research No. 25870159, and by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. The 4d N = 3 superconformal algebras RI J ; RK L = JK RI L The fermionic generators of su(2; 2j3) are Q and _ = _ . Their R-charges can be read o from I ; Q~I _ and SI ; S~I _ for I = 1; 2; 3; [RI J ; SK ] = 1 I K 1 I [RI J ; Q~K _ ] = [RI J ; S~K _ ] = JK S~I _ 4 J S 1 I ~K _ : The anti-commutation relations among Q I ; Q~I _ ; SI ; S~I _ are given by I ; SJ g = 2 JI ~I _ ; Q~J _ g =2 JI __ H 4 JI M~ _ _ + 4 __ RI J ; I ; Q~J _ g = 2 JI P _ ; ~I _ ; SJ g = 2 JI K _ : Here H is the Hamiltonian whose eigenvalue is the scaling dimension, and M generators of so(4) subalgebra of su(2; 2). They satisfy I ] = I ] = [M~ _ _ ; Q~I _ ] = On the other hand, P _ and K [H; SI ] = 2 SI ; [H; S~I _ ] = ; SI ] = [M~ _ _ ; S~I _ ] = _ have the following commutation relations with the su I ] = 2 P _ ; SI ] = [K _ Q~I _ ] = 2 __ SI ; P _ ; S~I _ ] = The hermiticity is given by (QI )y = SI ; )y = M (Q~I _ )y = S~I _ ; )y = M~ (RI J )y = RJ I ; (H)y = H ; (P _ )y = K Detailed computations We here show that G [Q~3+_ J 11] are Virasoro primaries. To that end, we rst recall the argument of sub-section 3.2.4 of [5], where the authors proved that any Hall-Littlewood (HL) operators map to Virasoro primaries in two dimensions. Here, HL operators are de ned as local operators annihilated by Q 1 ; Q~2 _ and their conjugates, and therefore are Schur operators. The unitarity bounds in (3.5) imply that HL operators have E = 2R r; j1 = of Schur operators, and that O1 is a HL operator. We also use a short-hand notation [Oi] for the corresponding 2d operators. Then the OPE of O^1 and the 2d stress tensor T (z) is of the form where hi is the eigenvalue of L0 for O^i. From equation (3.6) of [5], hi is given by T (z) O^1(0) = hi = R(i) + j1(i) + j2(i); where R(i) and (j1(i); j2(i)) are the SU(2)R charge and the spins of Oi. Since any Schur operator satisfy r = j2 j1, this is equivalent to hi = R(i) + jr(i)j + 2 min (j1(i); j2(i)) ; h1 = R(1) + jr(1)j: Therefore (B.1) is rewritten as under U(1)R. T (z)O^1(0) = R(i). The U(1)R charge dependence drops out because T (z) is neutral Recall here that the 2d stress tensor T (z) is given by a linear combination of the 4d SU(2)R current J++_ [5]. Since the SU(2)R current is an SU(2)R triplet, T (z) is a linear 1 or 0, namely R(i) = depending on i. Moreover, from (3.8), we see that Schur operators have j1 = j2 = Virasoro primary. is replaced by whose right-hand sides are positive semi-de nite because of the unitarity bounds (3.5). Therefore the worst possible singularity in (B.5) is of order three. On the other hand, since any Hall-Littlewood operator is a conformal primary, it is annihilated by L1.19 Therefore the singularity of order three in (B.5) vanishes. This means that O^1(z) is a T (z)O^1(0) = However, the worst possible singularity is still of order three since, as discussed in [5], any 2d OPE corresponding to a 4d OPE should be single-valued. Moreover, since Q3+J 11 and Q~3+_ J 11 are conformal primaries, the corresponding 2d operator O^1 is annihilated by L1. Therefore the worst singularity in the above OPE is of order two. Thus, we see that [Q~3+_ JF11] are Virasoro primaries. Note here that exactly the same argument tells us that, for any Higgs branch operator O, it follows that [O]; [Q3+O] and [Q~3+_ O] are Virasoro primaries.20 19Recall that L1 is identi ed with K +_+. See equation (2.19) of [5]. as the 2d stress tensor. 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Takahiro Nishinaka, Yuji Tachikawa. On 4d rank-one \( \mathcal{N}=3 \) superconformal field theories, Journal of High Energy Physics, 2016, 116, DOI: 10.1007/JHEP09(2016)116