Exploring SU(N) adjoint correlators in 3d
UUUITP-04/21
Exploring SU ( N ) adjoint correlators in 3 d Andrea Manenti a , Alessandro Vichi ba Department of Physics and Astronomy, Uppsala University,Box 516, SE-751 20 Uppsala, Sweden b Dipartimento di Fisica dell’Universit`a di Pisa and INFN,Largo Pontecorvo 3, I-56127 Pisa, Italy
Abstract
We use numerical bootstrap techniques to study correlation functions of scalarstransforming in the adjoint representation of SU ( N ) in three dimensions. Weobtain upper bounds on operator dimensions for various representations and studytheir dependence on N . We discover new families of kinks, one of which could berelated to bosonic QED . We then specialize to the cases N = 3 ,
4, which havebeen conjectured to describe a phase transition respectively in the ferromagneticcomplex projective model CP and the antiferromagnetic complex projectivemodel ACP . Lattice simulations provide strong evidence for the existence ofa second order phase transition, while an effective field theory approach does notpredict any fixed point. We identify a set of assumptions that constrain operatordimensions to small regions overlapping with the lattice predictions. a r X i v : . [ h e p - t h ] J a n ontents Q × Q OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 OPE structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Four-point structures and crossing . . . . . . . . . . . . . . . . . . . . . 9 N SU (4) 155 Focusing on SU (3) 166 Conclusions 19A Crossing equations for the adjoint and the singlet 20 A.1 Four-point structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.2 Crossing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
B Crossing equations for two adjoint scalars 24
B.1 The Q × Q (cid:48) OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24B.2 Four-point structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.3 Crossing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
C Unitary versus orthogonal group bootstrap 32D Parameters of the numerical implementation 34 Introduction
The classification of 3D Conformal Field Theories (CFTs) has seen a remarkable amountof progress in the recent years thanks to the conformal bootstrap [1, 2] (see [3, 4] fora review). This technique can be used to put rigorous bounds on operator dimensions,which often result in a precise determination of critical exponents [5–11]. The examplesstudied so far include multiple scalars [12–17], spinors [18–20], conserved currents[21, 22] and the energy momentum tensor [23]. Furthermore, several different globalsymmetry groups and representations have been considered [24–45].Four dimensional CFTs have also been the object of similar studies. For instance,the search for the conformal window in QCD-like theories has motivated the bootstrapstudy of scalars in the adjoint of SU ( N f ) [37, 38]. The bounds obtained in that caselook very smooth. The plots in our setup, on the other hand, will show many interestingand suggestive features.Our work was mainly inspired by two open problems in the condensed matterliterature. Lattice models of SU ( N ) spins have very diverse dynamics for differentvalues of N . A Monte Carlo study of a model with N = 3, defined in terms of coloredloops, has produced evidence of a new fixed point [46]. Similarly, a related lattice modelwith N = 4 displays evidences of a continuous phase transition [47]. This last exampleis particularly interesting because the Monte Carlo prediction seems to contradict anaive analysis based on the Landau-Ginzburg-Wilson theory. Our goal is to shed somelight on the nature of these phase transitions by performing a bootstrap study of fourscalars in the adjoint of SU ( N ). Since the setup can be easily generalized to arbitrary N , we also perform a systematic survey for various values of N , focusing mostly on alarge N regime. This gives us the ability to compare our results to analytic predictionsavailable in the literature, which are often obtained as a perturbative expansion in 1 /N around a known solution.We find two unrelated families of kinks at large N . A family appearing for N ≥ N − , and a family for N (cid:38) fixed points studied in [48], whereas thelatter does not seem to correspond to any known theory.In order to look more closely at the case of N = 3 we modify the setup by addinga scalar singlet operator to the correlators considered. This gives us more parametersto tune, at the expense of a more complicated system of equations. In some instances,the study of mixed correlators, together with minimal physical assumptions, has beenable to produce isolated regions in parameter space [5, 6]. We did not obtain an islandin this case, but the resulting exclusion plot is consistent with the predictions of theloop model.In the next subsections we will briefly review known results about fixed points with SU ( N ) global symmetry, both from a lattice and a field theory perspective. This willserve as a guide for interpreting the various features showing in the bounds that weobtain. In Section 2 we set up the bootstrap problem and write down the crossing See (19) and below for the definition of the [ · · · ] notation. N while in Sections 4 and 5we show the results for SU (4) and SU (3) respectively. We leave some technical detailsto the appendices. Note added:While finalizing our work we became aware of [49] which partially overlaps with ourresults at large N . We would like to thank Y.-C. He, J. Rong and N. Su for sharingtheir work with us and for coordinating the submission to the arXiv. We begin with a simple lattice model, the ( A ) CP N − , which is defined as a systemof spins z x taking values in the complex projective space CP N − , with the index x labeling the lattice site.Equivalently, we can describe the system by considering z x to take values in C N ,with the restriction ¯z x · z x = 1 and the identification z x ∼ e iθ z x for any θ ∈ [0 , π ); thelatter condition can be viewed as a U (1) gauge symmetry, since one can attach a phaseto each spin independently, i.e. locally. The Hamiltonian can be written as H CP N − = J (cid:88) (cid:104) x , y (cid:105) | ¯z x · z y | , (1)where (cid:104) x , y (cid:105) indicates that the sum runs over pairs of nearest neighbors. For negative J the system is ferromagnetic while for positive J it is antiferromagnetic.For N = 2, both the ferromagnetic and the antiferromagnetic models are equivalentto the O (3) Heisenberg model.For N = 3 the antiferromagnetic model was shown to have the same criticalexponents as the O (8) model [50] while for the ferromagnetic one there is evidenceof a fixed point in a new universality class with U (3) global symmetry [46]. The modelstudied in [46] is a system of loops with N colors in a three-dimensional lattice, butit can be shown to be equivalent to a ferromagnetic CP N − . The predicted criticalexponents of the loop model read ν loop = 0 . , γ loop = 0 . . (2)By comparison, the O (8) exponent is given by ν O (8) = 0 . N = 4, a Monte Carlo study has determined that the phase transition of theferromagnetic model is of first order [51]. On the other hand the antiferromagneticmodel exhibits a continuous phase transition with critical exponent ν = 0 . CP N − model.Its lattice formulation has the same Hamiltonian as the compact model, but the sum3s only over lattice configuration with zero monopole number. A Monte Carlo studyin [52] determined that the model for N = 25 undergoes a continuous phase transitionwith critical exponent ν = 0 . CP N − model is related to the non-compact CP N − model inthat they are characterized by the same symmetries and the suppression of monopoles,but they are distinct theories. The latter is well studied in the context of deconfinedquantum criticality. In particular, the model for N = 2 describes the phase transitionbetween the N´eel phase and the valence-bond solid (VBS) phase of two-dimensionalantiferromagnets [53, 54]. Furthermore, by non-compact CP N − model one typicallyrefers to the continuum theory. We will discuss a lattice formulation of it at the end ofthis section.The ( A ) CP N − model can also be seen as a limiting case of the Abelian Higgs model.Its Hamiltonian is defined as follows H AH = − J N (cid:88) x ,µ Re (cid:0) z x · ( λ x ,µ z x +ˆ µ ) (cid:1) + H g , (3)where H g is the gauge Hamiltonian. There are two different formulations of this model:compact and non-compact. They differ in the definitions of H g . The compact modelhas H g = − κ (cid:88) x ,µ>ν Re (cid:0) λ x ,µ λ x +ˆ µ,ν ¯ λ x +ˆ ν,µ ¯ λ x ,ν (cid:1) , (4)where the sum is over the lattice plaquettes. While in the non-compact model thefundamental degree of freedom is the gauge field λ x ,µ = e iA x ,µ and the Hamiltonian isdefined as H g = κ (cid:88) x ,µ>ν (∆ ˆ µ A x ,ν − ∆ ˆ ν A x ,µ ) , (5)where ∆ ˆ µ is the discretized derivative. The limit κ → A ) CP N − model.The continuous model, whose Lagrangian we show in Section 1.3, can be shown tohave a second order phase transition for sufficiently large N > N c , while at lower valuesof N the transition is of first order. N c was estimated by field theory techniques to be N c = 12 . .
9) [55]. A Monte Carlo study is in agreement with this estimate and putsa more stringent upper bound: 4 < N c <
10 [56].Lastly, we would like to discuss a class of lattice models for SU ( N ) antiferromagnets.These models exhibit the features of deconfined quantum criticality and are expectedto become the non-compact CP N − model in the continuum. The model is defined on abipartite square lattice with antiferromagnetic interactions J among nearest neighbors(distance 1) and ferromagnetic interactions J among next-to-nearest neighbor (distance √ SU ( N )and on odd sites we have spins in the antifundamental. Let us define P ij and Π ij as P xy = N − (cid:88) A =1 T A x · T ∗ A y , Π xy = N − (cid:88) A =1 T A x · T A y . (6)4here ( T A ) ab is a Hermitian traceless matrix (generator of SU ( N )) and x , y representthe lattice sites. P xy acts on different sublattices while Π xy acts on the same sublattice.The Hamiltonian of the model reads [57] H = − J N (cid:88) (cid:104) x , y (cid:105) P xy − J N (cid:88) (cid:104)(cid:104) x , y (cid:105)(cid:105) Π xy , (7)where (cid:104) x , y (cid:105) and (cid:104)(cid:104) x , y (cid:105)(cid:105) denote first and second neighbors respectively. This modelexhibits a deconfined quantum critical point whose critical exponents for the N´eel andVBS order parameter agree with large N estimates (which we will discuss in Section 1.3)1 + η V = 0 . N + 0 . , η N = 1 + 32 π N − . N . (8) In many cases of physical interest one can understand the critical behavior of a latticesystem also starting from a UV description in terms of a field theory of a scalar fieldwith only a few renormalizable interactions. Thanks to the properties of the RG flow,if the two UV theories belong to the same universality class, they will flow to the samefixed point in the IR.Physically this is equivalent to identifying the order parameter that describes thefluctuations near criticality and writing an effective Hamiltonian. The order parameteris chosen such that it vanishes in the disordered phase and is nonzero in the orderedphase. Thus, it is expected to be small near criticality and it make sense to consideronly the leading terms.If one is interested in describing the phase transition observed for
ACP N − , theorder parameter Q ab is a tensor of SU ( N ) transforming in the adjoint representationand odd under an additional Z symmetry. The LGW Hamiltonian reads: H = Tr[ ∂ µ Q∂ µ Q ] + r Tr[ QQ ] + u QQ ]) + v QQQQ ] . (9)The analysis of the β -functions for the couplings u and v in ε -expansion at oneloop reveals the existence of four fixed points. Two of them are well known: the freeGaussian theory ( u ∗ = v ∗ = 0) and the O ( N (cid:48) ) Wilson-Fisher fixed point ( v ∗ = 0), with N (cid:48) = N − Q . In addition thereare two fixed points, with both coupling nonzero and v <
0, that merge at N = N c andturn complex for N > N c . A Borel resummation of the five-loop ε -expansion predicts N c (cid:39) . ε and N c (cid:39) . ε [50].For N = 2 , Q = (Tr Q ) / N < N c the new fixed points can be mapped respectively to the O (3) and O (8) model. Inconclusion, the LGW analysis predicts that no fixed points exist for this model besidesthe WF ones. This is in tension with the lattice results discussed in the previous section.The LGW Hamiltonian for the ferromagnetic model is similar to the one above. Theonly difference is the addition of a cubic term w Tr[ Q ] because the order parameter doesnot have a Z symmetry anymore. On general grounds one would expect no continuous5hase transitions except the WF ones in this case as well. However, this is again indisagreement with the lattice results. In this section we will briefly review the 3 d gauge theories relevant for our bootstrapsetup. First let us focus on Abelian models. Let φ a be a scalar transforming in thefundamental of SU ( N ) and minimally coupled to a U (1) gauge field, with the followingLagrangian L AH = 14 e F µν F µν + N (cid:88) a =1 | ( ∂ µ + iA µ ) φ a | + m N (cid:88) a =1 φ a φ a + λ (cid:18) N (cid:88) a =1 φ a φ ∗ a (cid:19) . (10)This model has a fixed point for λ = m = 0 which is the (tricritical) bosonic QED anda fixed point where λ (cid:54) = 0 which is the Abelian Higgs model. In the limit where e →
0— namely when the gauge field becomes non dynamical — the latter is referred to asthe CP N − model. The name stems from its interpretation as a nonlinear sigma modelwith target space CP N − . Indeed the quartic potential gives a constraint φ a φ a = const . and the non dynamical gauge field imposes the identification φ a ∼ e iα φ a .In [48] we can find the values for various anomalous dimensions in both fixed pointsas an expansion in 1 /N . In the bosonic QED fixed point, the bilinear of φ in the adjointand in the singlet representations read, respectively∆[ | φ | ] = 1 − π N + O (cid:16) N (cid:17) , ∆[ | φ | ] = 1 + 1283 π N + O (cid:16) N (cid:17) . (11)Similar results are available for the Abelian Higgs model. However, in place of thesinglet bilinear we have the Hubbard-Stratonovich field σ , whose dimension starts with 2∆[ | φ | ] = 1 − π N + O (cid:16) N (cid:17) , ∆[ σ ] = 2 − π N + O (cid:16) N (cid:17) . (12)In the condensed matter literature | φ | is known as the N´eel order parameter andthe quantities that are typically quoted are the critical exponents, which are given as η N = 2∆[ | φ | ] − ν − = 3 − ∆[ σ ]. Also note that the above formulas appearedin [58–61] first.In [48] it was also computed the anomalous dimension of scalar quartic operators inthe representation [ N − , N − , , :∆[ | φ | N − ,N − , , ] = 2 − π N + O (cid:16) N (cid:17) , for bosonic QED , ∆[ | φ | N − ,N − , , ] = 2 − π N + O (cid:16) N (cid:17) , for Abelian Higgs . (13) They denote this representation [2 , , . . . , , , however, monopoles operatorsare neutral under the global SU ( N ) symmetry and will not appear in our setup.We decided to focus on the region in parameter space where ∆ (cid:46)
1, thereforefermionic gauge theories are typically not interesting for us as they contain heavieroperators, namely ∆[ ¯ ψψ ] ∼
2. However, the model QED-GN + considered in [48]contains a Hubbard-Stratonovich field which is in the same range of dimensions thatwe study. The Lagrangian reads L QED-GN = 14 e F µν F µν + N (cid:88) a =1 ¯ ψ a γ µ ( ∂ µ + iA µ ) ψ a + ρ N (cid:88) a =1 ¯ ψ a ψ a + µ ρ . (14)Integrating out ρ leaves us with a four-fermion interaction. The anomalous dimensionfor the lightest scalar singlet is∆[ ρ ] = 1 − π N + O (cid:16) N (cid:17) , (15)while the adjoint this time is heavier∆[( ¯ ψψ ) adj ] = 2 − π N + O (cid:16) N (cid:17) . (16)As far as we are aware, there are no such results in the context of nonabelian gaugetheories. The large N dynamics of QCD is well known and studied [65–67], but theanomalous dimensions of low-lying operators have not been computed so far. We canhowever argue, by analogy with a similar class of models studied in [68] and previouslyin [69, 61, 70], that for an SU ( M ) or SO ( M ) gauge theory with global symmetry SU ( N ), the anomalous dimensions at large N will be corrected by terms linear in M .As it was first observed in [45] and then refined in [49], the abelian or non-abeliannature of the underlining gauge theory reflects in gaps in the spectrum of scalaroperators in certain representations. Consider for instance O [ a b ][ c d ] , the smallest scalaroperator with two antisymmetric fundamental and two antisymmetric antifundamentalindexes. We will refer to this representation as [ N − ,
2] in the rest of the paper. Suchrepresentation appears in the OPE of two adjoint operators. In a Generalized FreeTheory (GFT) of an adjoint operator Q ac or in the LGW theory defined in the previoussection, the smallest scalar in the [ N − ,
2] representation is given by O [ a b ][ c d ] ∼ Q ac Q bd − Q ad Q bc . (17)Similarly, given a non-abelian gauge theory with fundamental fields φ Aa one can definethe gauge invariant scalar Q ac ∼ φ † Aa φ Ac − trace. Then, the combination in (17) is non-trivial due to the internal gauge indexes. However, if these were absent, one could notconstruct it. In the Abelian Higgs model (10) the smallest non trivial operator in the[ N − ,
2] representation that one can construct requires two extra derivatives or morefields. This reasoning is valid only in a neighbourhood of the UV description, howeverit gives us an intuition about which operators we should expect in the CFT. Hence, wedo not expect the IR fixed point of abelian gauge theories or CP N models to have lightscalars in the [ N − ,
2] representation. 7
Setup
In this section we derive the crossing equations for the four-point function of a scalaroperator Q ba transforming in the adjoint of SU ( N ). In order to keep the notationcompact we use an index-free formalism for the four-point structures. This meansthat we contract any fundamental index a with a complex polarization ¯ S a and and anyanti-fundamental index b with a complex polarization S b . Then the operator Q becomes Q ( x, S, ¯ S ) = ¯ S a S b Q ba . (18)The tracelessness properties of the operator can be encoded in the requirement S · ¯ S =0. In order to discuss the OPE it is easier to use a formalism with explicit indicesinstead. In the following we will use p i as a shorthand for the point x i together withthe polarizations that are attached to it. Q × Q OPE
We consider the single correlator of four Q ’s. The representations exchanged in theOPE Q × Q were studied in [38]. Here we rederive their results in our notation. Theexchanged representations are[ N − , ⊗ [ N − ,
1] = • + ⊕ [ N − , , − ⊕ [ N − , N − , − ⊕ [ N − , + ⊕ [ N − , N − , , + ⊕ [ N − , + ⊕ [ N − , − . (19)We adopt the notation [ λ , λ , · · · ] to denote a Young tableau with λ boxes in the firstcolumn, λ boxes in the second column and so on. For brevity we also denote the singlet(i.e. the empty tableau) as • . The subscript ± refers to the symmetry property underexchanging the two operators and, therefore, to the parity of the spin of the operatorsentering the OPE. Also note that the representations [ N − , ,
1] and [ N − , N − , (cid:104) QQQQ (cid:105) correlator we can construct the four-point structures.
Here we will fix the convention for the two-point functions of all the operators ex-changed. We also denote as T the unique two-point structure of two real symmetrictraceless operators with dimension ∆ and spin (cid:96) T = I µ ( ν · · · I µ (cid:96) ν (cid:96) ) − traces( x ) ∆ , I µν = δ µν − x ν x µ x . (20)8etting O irrep be an exchanged operator in the representation irrep we have (cid:104)O • O • (cid:105) = T , (21a) (cid:104)O [ N − , ab O [ N − , cd (cid:105) = T (cid:18) δ ad δ cb − δ ab δ cd N (cid:19) , (21b) (cid:104)O [ N − ,N − , , abcd O [ N − ,N − , , efgh (cid:105) = T (cid:18) δ a ( g δ bh ) δ e ( c δ fd ) − δ ( a (( h δ b )( c δ ( ed ) δ f ) g )) N + 2 + 2 δ a ( c δ bd ) δ e ( g δ fh ) ( N + 1)( N + 2) (cid:19) , (21c) (cid:104)O [ N − , abcd O [ N − , efgh (cid:105) = T (cid:18) δ a [ g δ bh ] δ e [ c δ fd ] − δ [ a [[ h δ b ][ c δ [ ed ] δ f ] g ]] N − δ a [ c δ bd ] δ e [ g δ fh ] ( N − N − (cid:19) , (21d) (cid:104)O [ N − , , abcd O [ N − ,N − , efgh (cid:105) = T (cid:18) δ a [ g δ bh ] δ e ( c δ fd ) + δ [ a [[ h δ b ]( c δ ( ed ) δ f ) g ]] N (cid:19) . (21e)In the above equations we use pairs of parentheses (( or [[ to avoid ambiguities whenpairs of indices being (anti)symmetrized overlap. The (anti)symmetrizations are definedas X [ ab ] = X ab − X ba and X ( ab ) = X ab + X ba . The operators in the adjoint and the singlet appear in the OPE multiplied by someKronecker δ ’s in order to achieve the correct number of indices with the right symmetryand traceleness properties. All in all the OPE reads Q ac × Q bd = (cid:88) O δ bc O +[ N − , ad + δ ad O +[ N − , bc − N (cid:0) δ ac O +[ N − , bd + δ bd O +[ N − , ac (cid:1) + (cid:88) O δ bc O − [ N − , ad − δ ad O − [ N − , bc + (cid:88) O O • (cid:16) δ ad δ bc − δ ac δ bd N (cid:17) + (cid:88) other irreps (cid:88) O O irrep abcd . (22) By taking the two-point functions of pairs of operators appearing in the OPE Q × Q wecan construct the four-point structures. We contract all the indices by the polarizations S a and ¯ S a as discussed at the beginning. We also denote as T the unique four-pointstructure of a single scalar of dimension ∆ Q T = 1( x x ) ∆ Q . (23)The four-point function can be written in terms of six real functions of the cross ratios u = x x x x and v = u | ↔ . (cid:104) Q ( p ) Q ( p ) Q ( p ) Q ( p ) (cid:105) = T (cid:88) r =1 T ( r ) f r ( u, v ) . (24)9he structures T ( r ) are chosen so that they are associated to the exchange of a singlerepresentation, as summarized in Table 1. Let us introduce the shorthand S ijkl = ( S · ¯ S i )( S · ¯ S j )( S · ¯ S k )( S · ¯ S l ) . (25)With this notation in mind we have T (1) = S (43)(21) − S + S + S + S N + 2) + S N + 1)( N + 2) ,T (2) = S − S − S − S + S − S N ,T (3) = S [43][21] − S + S + S + S N −
2) + S N − N − T (4) = S + S + S + S − S N ,T (5) = −S + S − S + S ,T (6) = S . (26)Now we can write down the conformal block expansions for each of the f ( r ) . For r = 1 , , , f r ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ r (cid:96) even (cid:0) λ r QQ O (cid:1) g ∆ ,(cid:96) ( u, v ) , (27)with = [ N − , N − , , = [ N − , = [ N − , + and = • . The otherrepresentations follow: f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , , (cid:96) odd λ [ N − , , QQ O λ [ N − ,N − , QQ O g ∆ ,(cid:96) ( u, v ) ,f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) odd (cid:0) λ [ N − , − QQ O (cid:1) g ∆ ,(cid:96) ( u, v ) . (28)Positivity holds for all sums because λ [ N − , , QQ O = (cid:0) λ [ N − ,N − , QQ O (cid:1) ∗ . The conformal blocksare normalized according to this convention g ∆ ,(cid:96) ( u, v ) ∼ u → ,v → (cid:16) u (cid:17) ∆ / P (cid:96) (cid:18) − v √ u (cid:19) , (29) P (cid:96) being a Legendre polynomial. This normalization implies g ∆ ,(cid:96) (cid:0) , (cid:1) ≥ , (cid:96) .We are now ready to write the crossing equations. First let us introduce the evenand odd combinations of the conformal blocks F ∆ ,(cid:96) = v ∆ Q g ∆ ,(cid:96) ( u, v ) − u ∆ Q g ∆ ,(cid:96) ( v, u ) ,H ∆ ,(cid:96) = v ∆ Q g ∆ ,(cid:96) ( u, v ) + u ∆ Q g ∆ ,(cid:96) ( v, u ) . (30)10 r O exchanged spin parity f [ N − , N − , ,
1] even f [ N − , ,
1] + h . c . odd f [ N − ,
2] even f [ N − , + even f [ N − , − odd f • evenTable 1: Operator contributions to the partial wave f r .The crossing equations then read V • ;0 , = (cid:88) r =1 , , , (cid:88) O ∆ ,(cid:96) ∈ r (cid:96) even (cid:0) λ r QQ O (cid:1) V r ;∆ ,(cid:96) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , , (cid:96) odd (cid:12)(cid:12) λ [ N − , , QQ O (cid:12)(cid:12) V [ N − , , ,(cid:96) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) odd (cid:0) λ [ N − , − QQ O (cid:1) V [ N − , − ;∆ ,(cid:96) , (31)where we defined the vectors V irrep ;∆ ,(cid:96) as V [ N − ,N − , , ,(cid:96) = F ∆ ,(cid:96) H ∆ ,(cid:96) , V [ N − , , ,(cid:96) = F ∆ ,(cid:96) H ∆ ,(cid:96) ,V [ N − , ,(cid:96) = F ∆ ,(cid:96) − N − N +1)( N − N +3) H ∆ ,(cid:96) N − N ( N − N − N +2) H ∆ ,(cid:96) , V [ N − , + ;∆ ,(cid:96) = F ∆ ,(cid:96) − N − N +1)( N +2) N ( N +3) H ∆ ,(cid:96)N +2 N H ∆ ,(cid:96) ,V [ N − , − ;∆ ,(cid:96) = − N +1) N +2 F ∆ ,(cid:96) N ( N − N +2) F ∆ ,(cid:96)N − N − F ∆ ,(cid:96)N ( N − ( N +2) F ∆ ,(cid:96) N +1) N +3 H ∆ ,(cid:96) − NN +2 H ∆ ,(cid:96) , V • ;∆ ,(cid:96) = ( N − N +1) N ( N +2) F ∆ ,(cid:96) ( N − N +1)2( N − N +2) F ∆ ,(cid:96) ( N − N − N ( N − F ∆ ,(cid:96)N ( N − N +1)( N − ( N +2) F ∆ ,(cid:96) − N +1) N ( N +3) H ∆ ,(cid:96) − N +1 N +2 H ∆ ,(cid:96) . (32)With the notation 0 n we mean a sequence of n zeros. If N = 3 we need to discard thethird line and the representation [ N − ,
2] is not exchanged. If N = 2 we need to discardthe second, third and fourth line and the representations [ N − ,
2] and [ N − , ,
1] arenot exchanged. 11 +++++++ ���� ���� ���� ���� ���� Δ � ������������������ Δ � Figure 1: Bound on the dimension of the first singlet scalar. These bounds have been obtainedat Λ = 19. The blue, orange, green, and red lines correspond respectively to N = 4 , , , N prediction for the O ( N −
1) model. N Here we present a systematic study for general N of bounds on the dimension of thefirst scalar operator in all representations containing even spins. Specifically we examine N = 4 , , ,
100 and occasionally other values. The bounds on the leading operatorsin the singlet representation are identical to the corresponding bounds found in the O ( N (cid:48) )-vector bootstrap, where N (cid:48) = N −
1. For what concerns other representations,any solution with O ( N (cid:48) ) symmetry is also a solution of our crossing equations but thebounds do not coincide as long as we extremize a single operator at the time. Singlets
The bound on the dimension of the first singlet scalar ∆ S shows a clear kink corre-sponding to the O ( N (cid:48) ) model under the identification φ i → Q ab . In addition there isa second set of (very mild) kinks for N = 4 , N increases. Thesebounds are shown in Figure 1. In the scalar singlet sector we do not find any newinteresting feature. Adjoint representation
As discussed in Section 2, the OPE Q × Q contains again operators in the adjointrepresentation. This offers the possibility to test the effect of a Z symmetry in the This is proven in appendix C. ��� ���� ���� ���� ���� Δ � ������������������ Δ ��� ���� ���� ���� ���� ���� ���� ���� ���� ���� Δ � ������������������ Δ ��� Figure 2: Bound on the dimension of the first adjoint scalar. We do not assume that Q is oddunder a global Z symmetry. The blue, orange, green, red, purple and brown lines correspondrespectively to N = 4 , , , , , . A kink appears for N (cid:38)
20. The bounds have beencomputed at Λ = 19. The dashed line corresponds to Generalized Free Theory (GFT).
CFT. If Q is odd under such a symmetry, then the three-point function (cid:104) QQQ (cid:105) mustvanish. When inputting gaps on the adjoint scalar sector above the external dimension∆ Q , we then have the choice to allow the presence of an isolated contribution with∆ = ∆ Q or to forbid it. This corresponds to imposing the presence of a Z symmetryunder which Q is odd or not. Interestingly, for N ≥
4, the bounds in the two casescoincide.As shown in Figure 2, a family of defined kinks arises for large values of N . Thelocation of the kink in ∆ Q moves to the left as N increases but it does not seem toconverge to the free value 1 /
2. The inspection of the extremal functional at the kinkfor the case N = 100 displays a behavior similar to the kink explored in the Ising modelin [71]: the second scalar adjoint operator decouples from the left, while the spin-2operator after the stress tensor decouples from the right.In principle one could create an island inputting gaps in these two sectors. We leavethis exercise for the future, as at present we do not have a candidate CFT for thisfamily of kinks. From Figure 2 we can speculate that this family of putative CFTsmust have a Z symmetry under which Q is odd. The small values of ∆ Q disfavorgauge theories, especially because at large- N anomalous dimensions are suppressed, seesection 1.3. We also saw in section 1.2 that a simple theory of scalars does not havefixed points for N (cid:38)
4: it would be interesting to systematically search for fixed pointsof Gross-Neveu-Yukawa like theories theories involving scalars in the adjoint (even inpresence of supersymmetry) and compare them with our results.
Representation [ N − , N − , N , moving to the right towards ∆ Q = 1 as N increases. In orderto obtain the correct scaling with N it was crucial to impose that Q is actually thelightest adjoint scalar in the theory. One can always restrict to this case with no We are grateful to authors of [49] for discussing this point. �� ��� ��� ��� ��� ��� Δ � ������������������������ Δ [ � - ��� ] ��� ��� ��� ��� ��� ��� Δ � ������������������������ Δ [ � - ��� ] Figure 3: Bounds on the dimension of the first scalar in the [ N − ,
2] representation Theblue, orange, green, red and purple lines correspond respectively to N = 4 , , , , N = 4 there are kinks at ∆ t = 0 .
54 and ∆ t = 0 .
64 respectively. For larger N the first kinkdisappears. A family of sharp kinks is visible for all N . Green, red and purple vertical dottedlines show the large- N values of ∆ Q for bosonic QED with N = 10 , , loss of generality. As shown in the right panel of Figure 3, this assumption provedvery effective, especially for large N . As discussed in Section 1.3, the lightest scalaroperator in this representation in abelian gauge theories is expected to have dimension4 + O (1 /N ), while in LGW models or non-abelian gauge theories we expect operatorswith lower dimensions. Hence, in the left panel we compare with the prediction fortricritical QED of [48]. For N = 100, where we expect to trust the large- N expansion,the kink is still a bit far away. The discrepancy gets larger for N = 20. Incidentally,the kink at N = 10 coincides exactly with the large- N prediction. We interpret this asa coincidence, since we do not expect the prediction to be reliable for this value of N .Indeed, as we will see in the next subsection, bootstrap bounds are in tension with thelarge- N prediction for N = 10. ▲▲ ▲▲ ▲▲ ▲▲▲▲ ▲▲ ▲▲ ▲▲■■ ■■ ■■■■ ■■ ■■ ■■ ��� ��� ��� ��� ��� ��� Δ � ������������������������ Δ [ � - �� � - ����� ] ▲▲■■ ���� ���� ���� ���� ���� ���� ���� Δ � ������������������������ Δ [ � - �� � - ����� ] Figure 4: Bounds on the dimension of the first scalar in the [ N − , N − , ,
1] representation.The blue, orange, green, and red lines correspond respectively to N = 4 , , , N = 4 , , , , , , , The prediction for CP N − has larger ∆ Q . In other cases bootstrap bounds have been found to disagree with large- N prediction even for largevalues of N [72]. epresentation [ N − , N − , , O ( N (cid:48) ) vector model, which is again a special solution of our crossingequations. The same kinks would also be visible in Figure 3.Despite not showing any other special feature, these bounds offer the possibilityto test the validity of the large- N expansion in the Abelian Higgs model. The leadingoperator in the representation [ N − , N − , ,
1] is made of four scalars and its dimensionhas been computed in the case of bosonic QED and in the case of Abelian Higgs model[48]. While in the former ∆ [ N − ,N − , , = 2∆ Q , in the latter che predictions are notalways in the allowed region. As expected, Figure 4 shows that for N = 4 , N prediction is noticeably wrong. On the contrary N ≥
10 is slightly inside the allowedregion. One can wonder if additional assumptions allow to get closer the predictedpoint. According to [48], for N = 10 one has ∆ S (cid:39) . S ≥ .
5, ∆ [ N − , ≥
3, ∆ adj ≥ ∆ Q andcomputed again the bound on ∆ [ N − ,N − , , . The result does not show any feature, butexcludes the large- N leading order prediction. The fact that bosonic QED seems to beat the corner is a pure coincidence, since this theory is excluded by our assumptions. Wetried several gaps, but the allowed region depends smoothly on them, with no strikingevidences of CFTs. SU (4) ���� ���� ���� ���� ���� Δ � ������������������ Δ ���� ���� ���� ���� ���� Δ � ��������������������� Δ Figure 5: Bounds on the dimension of various operators in SU (4). On the left: The blue,orange and green and red lines correspond respectively to the bond on the dimension of theleading scalar [ N − , N − , N − , ,
1] irreps. On the right: boundon the dimension of the first spin-2 operator after the stress tensor (blue) and first spin-1operator after the adjoint SU (4) current. The bounds have been computed at Λ = 19. Thelight band corresponds to the prediction of lattice simulation for ∆ Q . The darker squareshows the lattice prediction for the singlet dimension as well. Let us now focus on the case N = 4. For this specific case there is a well definedcandidate CFT to compare with. This is the ACP lattice model studied in [47].15sing lattice simulation they find a fixed point with an adjoint scalar of dimension∆ Q = 0 . ± .
02 and exactly one relevant singlet with dimension 1 . ± .
8. Our goalis to verify if the boostrap bounds show any evidence of such CFT.In Figure 5 we report the bounds for scalar and non scalar operators for the specificcase of N = 4 and compare them with the lattice determinations. The bound on T (cid:48) ,the first spin-2 operator after the stress tensor, and J (cid:48) the first spin-1 operator after theadjoint current, look promising; in addition the representation [ N − ,
2] has a milderkink in the region of interest. The situation looks remarkably similar to the case of
ARP studied in [45]. Also in this case one can check that imposing increasing gapson ∆ T (cid:48) allows to carve out islands in the (∆ Q , ∆ S ) plane that shrink around the latticeprediction, see Figure 6. Unfortunately in [45] a bootstrap study of mixed correlatorsinvolving a scalar singlet operator revealed that part of the peak in the ∆ T (cid:48) boundwas excluded. Nevertheless island overlapping with lattice prediction could be createdby imposing suitable gaps in other channels. We leave a detailed investigation of thisdirection for a future work.We conclude noticing that the scalar bounds in Figure 5 impose the presence of rela-tively small dimension operator beside the singlet, in particular ∆ [ N − ,N − , , (cid:46) .
21 inthe region of interest. It would be extremely interesting to estimate ∆ adj , ∆ [ N − ,N − , , and ∆ [ N − , in ACP on the lattice and compare with our bounds. ���� ���� ���� ���� ���� ���� Δ � ������������������������ Δ � Figure 6: Allowed regions in the plane (∆ Q , ∆ S ), with increasingly strong assumption. Theassumptions are (from light to dark): no assumptions; S is the only relevant scalar and∆ J (cid:48) ≥ T (cid:48) ≥ , . , .
7. As usual J (cid:48) denotes the next spin-1 operator after the adjointcurrent and T (cid:48) denotes the next spin-2 operator after the stress tensor. The red box is thelattice prediction for ACP . The bounds have been computed at Λ = 19. SU (3) Finally let us consider the case of N = 3. As discussed in section 1.1, there are latticeevidences of a second order phase transition, corresponding to the ferromagnetic CP model.From the group theory point of view, this case is special, since the Q × Q OPEexchanges one representation less. Let us recall the group theory decomposition in this16 ��� ���� ���� ���� ���� ���� Δ � ������������������ Δ ��� Figure 7: Bounds on the first adjoint operator for the case of SU (3), with the assumptionthat Q appears (blue) or does not appear (yellow) in its own OPE Q × Q . The dashed linecorresponds to GFT solution. The bounds have been computed at Λ = 27. The light redband corresponds to the prediction of the loop model. The vertical line corresponds to the O (8) large- N prediction. case: ⊗ = + ⊕ + ⊕ − ⊕ + ⊕ ( + ) − , (33)where in the notation of the previous section one has [ N − , → , [ N − , , → and [ N − , N − , , → . The representation [ N − ,
2] is missing.Among the bound on scalars, we only show the bound on the first adjoint scalar,with and without assuming the existence of a Z symmetry under which Q is odd. Asshown in Fugure 7 the two bounds coincide for ∆ Q (cid:46) .
54 and the differ substantially:if Q appears in the OPE Q × Q , the bound on the next scalar is much higher. Inparticular the bound displays a rounded kink in the region predicted by the loop modeldiscussed in section 1.1. This is interesting since in the ferromagnetic case the orderparameter is even under exchange of two sublattices, contrarily to the antiferromagneticcase, see discussion in [47].The bound on the singlet looks exactly as Figure 1, displaying only a sharp kinkcorresponding to the O (8) model. The loop prediction for ∆ Q and ∆ S lies well insidethe allowed region. The bound on ∆ is instead smooth and slightly above the GFTline.In Figure 8 we show instead bounds on spinning operators. The operators afterthe stress tensor and after the adjoint current have features corresponding to the O (8)model only. The other representation instead has a mild kink in correspondence of theloop model, see right panel of Figure 8.Finally we also explored bounds on the central charge C T and the OPE coefficient λ QQQ , as shown in Figure 9, which instead display a change of slope for ∆ Q ∼ . Q ∼ .
61, has a unique singlet operator inside the interval predicted by the loopmodel, ∆ S ∈ [1 . , . ��� ���� ���� ���� ���� ���� ���� Δ � ��������������������� Δ � � � Δ � � ���� ���� ���� ���� ���� ���� ���� Δ � ������������������ Δ �� + �� _ Figure 8: Bounds on the dimension of various spinning operators in SU (3). On the left: Theblue, orange and green lines correspond respectively to the bond on the dimension of: thefirst spin-2 operator in the singlet sector after the stress tensor; the first spin-1 operators inthe adjoin sector after SU (3) conserved current. On the right: bound on the dimension ofthe first spin-1 operator in the representation + . The bounds have been computed atΛ = 19. The light red band corresponds to the prediction of the loop model. The vertical linecorresponds to the O (8) large- N prediction. ���� ���� ���� ���� ���� Δ � ������������������ � � � � ���� ���� ���� ���� ���� ���� Δ � ��������������������� λ ���� Figure 9: On the left: lower bounds on the central charge C T in SU (3), normalized to thefree value for 8 real scalars. On the right: upper bounds on the OPE coefficient λ QQQ in SU (3). The bounds have been computed at Λ = 27. The light red band corresponds to theprediction of the loop model. Both bounds display a change of slope at ∆ Q (cid:39) . Although we tried many different analyses, we have been unable to create a closedregion in the (∆ Q , ∆ S ) plane. We also tried to study the mixed system of correlationfunctions involving the operator Q and the first scalar singlet S . In Figure 10, weshow a scan in the three dimensional space (∆ Q , ∆ S , ∆ Q (cid:48) ), assuming they are the onlyrelevant scalar operators in the singlet and adjoint sector (and no extra Z ). We findan allowed region consistent with the lattice prediction. In the region of interest onemust then have a second relevant adjoint scalar with dimension ∆ Q (cid:48) ∈ [1 . , . More precisely we assumed a gap in the adjoint sector ∆ Q (cid:48)(cid:48) ≥ .
5, both in the Q × Q and Q × S OPEs. igure 10: Allowed region in the space (∆ Q , ∆ Q (cid:48) , ∆ S ), the dimensions of the first two adjointscalars Q, Q (cid:48) and the first singlet scalar S , assuming that all other adjoint and singlet scalarsare irrelevant. The region has been obtained imposing crossing symmetry in the mixed systemof correlators involving Q and S . The orange shaded surface represents the upper bound on∆ Q (cid:48) from the single correlator of Q . The green region is the prediction of the loop model.The blue allowed region has been computed at Λ = 19. In this work we continue the exploration of the space of CFTs, focusing on theorieswith a scalar operator Q ba transforming in the adjoint representation of a global SU ( N )symmetry. We concentrate our attention to relatively small scalar dimensions ∆ Q (cid:46) CP N − and related models. Gauge theories with fermionic matter are indeed expected to havelarger dimension operators and their bootstrap analysis is generically harder, see [43, 73]for recent attempts.A systematic study of the correlation function (cid:104) QQQQ (cid:105) for general N revealed newfeatures and confirmed the importance of bounding non-singlet representations. Mostnotably, a family of sharp kinks appears for all N ≥ O [ a b ][ c d ] with two pairs of antisymmetric indices (we called this representation [ N − , ∼
4, in abelian gauge theories and in CP N − models, whilenon-abelian gauge theories, GFTs and LGW theories can contain smaller dimensionscalar with ∆ (cid:46)
2. It would be tempting to identify these kinks, shown in Figure 3,with the fixed points of bosonic QED , however the large- N predictions do not exactlymatch the bounds. Since these fixed points have been recently observed on the lattice[56], it would be interesting to estimate the dimension of more scalar operators, withparticular attention to the [ N − ,
2] representation.We also find a second family of kinks in the bounds on the first adjoint scalarappearing in the OPE Q × Q . The kinks appear for large values of N and they movecloser to ∆ Q = 1 / S = 1 as N increases; however, they do not seem to converge19here, as shown in Figure 2. The extremal functional spectrum reveals the presence ofa stress tensor and a rearrangement of operators on the two sides of the kink similarto the Ising model case [71]. We do not have concrete candidates for this family ofkinks: given the small values of ∆ Q it is unlikely that they correspond to a fixed pointof gauge theories. Also, since theories of only adjoint scalars with quartic interactionsdo not posses fixed points at large- N , we speculate that these kinks could be associatedto matrix models of scalars and fermions.Next, we focused on the case N = 4, in the attempt to isolate a region correspondingto the phase transition observed in ACP models by lattice simulations [47]. Thiscase is very similar to the ARP case discussed in [45]. We found several featuressupporting the existence of CFTs with critical exponents ν and η compatible with thelattice predictions. In particular, by assuming a gap on the next spin-2 operator afterthe stress tensor, one can create a closed region overlapping with the lattice results.In order to get a more reliable and gap independent result, it will be important tobootstrap mixed correlators using the techniques introduced in [10, 11].Finally, motivated by a second order phase transition observed in a loop constructionof the compact ferromagnetic CP model on the lattice [46], we studied the case of N = 3. In this case we found mild evidences of a putative CFT. Our analysis suggeststhat if such transition exists, the associated fixed point should not have a Z symmetryunder which the order parameter Q is odd: this is in agreement with the ferromagneticconstruction. Moreover, the theory would contain a second relevant adjoint scalar withdimension 1 . (cid:46) ∆ Q (cid:48) (cid:46) .
2, as we show in Figure 10. We believe a mixed correlatorstudy of
Q, Q (cid:48) , S will help settling this question.
Acknowledgments
We thank Slava Rychkov, Marten Reehorst, Andreas Stergiou, Ettore Vicari and Clau-dio Bonati for useful comments and discussions. For most of the duration of the projectAM and AV were supported by the Swiss National Science Foundation under grant no.PP00P2-163670. AM is also supported by Knut and Alice Wallenberg Foundation undergrant KAW 2016.0129 and by VR grant 2018-04438. This project has received fundingfrom the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (grant agreement no. 758903). All the numericalcomputations in this paper were run on the EPFL SCITAS cluster.
A Crossing equations for the adjoint and the singlet
In this Appendix we show the crossing equations for the mixed correlator system theadjoint and the scalar representations of SU ( N ). We will not assume a Z symmetrythat sends Q ab → − Q ab in this setup, so the nonzero correlators are (cid:104) QQQQ (cid:105) , (cid:104) QSSQ (cid:105) , (cid:104) SSQQ (cid:105) , (cid:104) QQQS (cid:105) , (cid:104) SSSS (cid:105) , (34)where Q ∈ [ N − ,
1] and S ∈ • . These equations were used to produce the plot inFigure 10. 20unction O exchanged spin parity˜ f • even h + [ N − , + even h − [ N − , − odd h [ N − , ( − (cid:96) any (cid:96) ˜ h • evenTable 2: Operator contributions to the partial waves ˜ f , h, ˜ h, h ± . A.1 Four-point structures
The four-point structures and crossing equations for (cid:104)
QQQQ (cid:105) were discussed in Sec-tion 2. Here we will list the additional structures arising in the correlators with S .Similarly as before, we define T to be the only four-point structure of four scalars, butnow with arbitrary external dimensions T = 1( x ) (∆ A +∆ B ) ( x ) (∆ C +∆ D ) (cid:18) x x (cid:19) (∆ A − ∆ B ) (cid:18) x x (cid:19) (∆ C − ∆ D ) , (35)where ∆ A,B... is either ∆ Q or ∆ S . Let us start with (cid:104) QSSQ (cid:105) and its permutation (cid:104) Q ( p ) S ( x ) S ( x ) Q ( p ) (cid:105) = T ( S · ¯ S )( S · ¯ S ) h ( u, v ) , (36a) (cid:104) S ( x ) S ( x ) Q ( p ) Q ( p ) (cid:105) = T ( S · ¯ S )( S · ¯ S ) ˜ h ( u, v ) . (36b)In h the representations exchanged are [ N − , ± according to the parity of the spinand in ˜ h only the singlet is exchanged. For the correlator with three Q ’s we have twostructures (cid:104) Q ( p ) Q ( p ) Q ( p ) S ( x ) (cid:105) = T (cid:16)(cid:0) ( S · ¯ S )( S · ¯ S )( S · ¯ S ) + ( S · ¯ S )( S · ¯ S )( S · ¯ S ) (cid:1) h + ( u, v ) + (cid:0) ( S · ¯ S )( S · ¯ S )( S · ¯ S ) − ( S · ¯ S )( S · ¯ S )( S · ¯ S ) (cid:1) h − ( u, v ) (cid:17) , (37)where the representation exchanged in h ± is the [ N − , ± . Finally the correlator (cid:104) SSSS (cid:105) has only one structure (cid:104) S ( x ) S ( x ) S ( x ) S ( x ) (cid:105) = T ˜ f ( u, v ) . (38)and exchanges a singlet. This is all summarized in Table 2. Let us now write down theexplicit conformal block expansions of ˜ f , ˜ h , h and h ± ˜ f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈• (cid:96) even ( λ • SS O ) g , ,(cid:96) ( u, v ) , (39a)˜ h ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈• (cid:96) even λ • SS O λ • QQ O g , ,(cid:96) ( u, v ) , (39b)21 ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , + (cid:96) even (cid:12)(cid:12) λ [ N − , + QS O (cid:12)(cid:12) g δ, − δ ∆ ,(cid:96) ( u, v ) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , − (cid:96) odd (cid:12)(cid:12) λ [ N − , − QS O (cid:12)(cid:12) g δ, − δ ∆ ,(cid:96) ( u, v ) , (39c) h + ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , + (cid:96) even λ [ N − , + QQ O λ [ N − , + QS O g ,δ ∆ ,(cid:96) ( u, v ) , (39d) h − ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , − (cid:96) odd λ [ N − , − QQ O λ [ N − , − QS O g ,δ ∆ ,(cid:96) ( u, v ) . (39e)For brevity we denoted δ ≡ ∆ Q − ∆ S . The conformal blocks are normalized accordingto the convention expressed in (29). That normalization implies g δ, − δ ∆ ,(cid:96) (cid:0) , (cid:1) ≥ , (cid:96), δ . A.2 Crossing equations
We now need to put together the functions defined in the previous section with thefunctions f , . . . f defined in Subsection 2.4. We also need to consider the special casewhere the operator exchanged in h or h + is Q itself or the operator exchanged in ˜ f or ˜ h is S itself. In such a case we can use the permutation symmetry of the OPE coefficients λ O O O = λ O O O to group the equations into a 3 × F ABCD ± , ∆ ,(cid:96) ( u, v ) = v ∆ C +∆ B g ∆ AB , ∆ CD ∆ ,(cid:96) ( u, v ) ± u ∆ C +∆ B g ∆ AB , ∆ CD ∆ ,(cid:96) ( v, u ) . (40)The crossing equations then read(1 1) V , (cid:18) (cid:19) = (cid:0) λ QQQ λ QSQ = λ QQS λ SSS (cid:1) V QS λ QQQ λ QSQ = λ QQS λ SSS + (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) even (cid:0) λ [ N − , + QQ O λ [ N − , + QS O (cid:1) V ,(cid:96) (cid:32) λ [ N − , + QQ O λ [ N − , + QS O (cid:33) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) odd (cid:0) λ [ N − , − QQ O λ [ N − , − QS O (cid:1) V ,(cid:96) (cid:32) λ [ N − , − QQ O λ [ N − , − QS O (cid:33) + (cid:88) O ∆ ,(cid:96) ∈ • (cid:96) even (cid:0) λ • QQ O λ • SS O (cid:1) V ,(cid:96) (cid:18) λ • QQ O λ • SS O (cid:19) + (cid:88) r =1 , (cid:88) O ∆ ,(cid:96) ∈ r (cid:96) even (cid:0) λ r QQ O (cid:1) W r ;∆ ,(cid:96) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , , (cid:96) odd (cid:0) λ [ N − , , QQ O (cid:1) W ,(cid:96) . (41)where the representations in the last line are numbered as in Table 1, namely =[ N − , N − , ,
1] and = [ N − , V r ;∆ ,(cid:96) are 2 × V QS are 3 ×
3. The W r ;∆ ,(cid:96) are instead vectors. In order to simplify the notation wewill abbreviate F ABCD ± , ∆ ,(cid:96) to f ± . The choices of the external operators are clear from thecontext. Indeed V QS ⊃ F QQQQ ± , ∆ Q , F QQQS ± , ∆ Q , F QQQS ± , ∆ Q , F QQQQ ± , ∆ S , or F QSSQ ± , ∆ Q , F SSQQ ± , ∆ S , F SSQQ ± , ∆ S , F SSSS ± , ∆ S , ,V , ∆ ,(cid:96) ⊃ (cid:32) F QQQQ ± , ∆ ,(cid:96) F QQSS ± , ∆ ,(cid:96) F QQSS ± , ∆ ,(cid:96) F SSSS ± , ∆ ,(cid:96) (cid:33) ,V r, ∆ ,(cid:96) ⊃ (cid:32) F QQQQ ± , ∆ ,(cid:96) F QQQS ± , ∆ ,(cid:96) F QQQS ± , ∆ ,(cid:96) F QSSQ ± , ∆ ,(cid:96) (cid:33) , r = 4 , ,W r, ∆ ,(cid:96) ⊃ F QQQQ ± , ∆ ,(cid:96) , r = 1 , , . (42)Furthermore we will denote a sequence of n zero matrices as n . Finally, we alsointroduce some abbreviations for the various matrices that appear: u = (cid:18) (cid:19) , l ± = (cid:18) f ± (cid:19) , s ± = (cid:18) f ± f ± (cid:19) . (43)Now we are ready to show the explicit form of the crossing vectors and matrices V ,(cid:96) = V [ N − , + ;∆ ,(cid:96) ul − l +12 s + + , V ,(cid:96) = V [ N − , − ;∆ ,(cid:96) ul − l + s − − l + , V ,(cid:96) = V • ;∆ ,(cid:96) u − s − s + + . (44)23 QS = ( N − N +1) f + N ( N +2)
00 0 0 ( N − N +1) f + N − N +2)
00 0 0 ( N − N +1) f + N − N
00 0 0 f + ( N − N ( N +1) f + ( N − ( N +2)
00 0 0 − N − N +1)( N +2) f − N ( N +3) − N +1) f − N ( N +3)
00 0 0 ( N +2) f − N − ( N +1) f − N +2
00 0 0 f − − f − − f − f + f + f + f + f + f +
00 0 0 f + , W , ∆ ,(cid:96) = (cid:32) V [ N − ,N − , , ,(cid:96) u0 (cid:33) ,W , ∆ ,(cid:96) = (cid:32) V [ N − , , ,(cid:96) u0 (cid:33) ,W , ∆ ,(cid:96) = (cid:32) V [ N − , ,(cid:96) u0 (cid:33) . (45)In the above equations V irrep ;∆ ,(cid:96) are the vectors appearing in (32). B Crossing equations for two adjoint scalars
In this appendix we show the crossing equations for the mixed correlator system oftwo different adjoint scalars. We did not use these crossing equations in the body ofthe paper, but we wish to present them anyway as they could be useful for furthergeneralizations of the present work. We use the notation and conventions set up inSection 2.
B.1 The Q × Q (cid:48) OPE
We want to study the mixed system of adjoint representations. The most generic caseconsists in the OPE Q ( p ) × Q (cid:48) ( p ) . (46)24n the above product the representations exchanged are the same as in (19) with theexception that the constraints on the spin parity are now modified. There are twomajor technical differences with the single correlator case1. The complex representation [ N − , ,
1] and its conjugate [ N − , N − , N − , ± cannot beseparated based on the parity of (cid:96) so there is a nontrivial (i.e. non diagonal)projector matrix from the space of three-point structures squared to the space offour-point structures.It will be important to study the conjugation properties of the various OPE coeffi-cients to ensure that in the crossing equations we only have positive definite contribu-tions. To do this one has to analyze the three-point functions. We will show explicitlythe correlator of the adjoint since it is the only one having two structures. For the otherrepresentations we skip the analysis and report directly the results. The two structuresfor an exchanged adjoint read (cid:104) Q ( p ) Q (cid:48) ( p ) O [ N − , ( p ) (cid:105) = T (cid:16) λ [ N − , , + QQ (cid:48) O (cid:0) ( S · ¯ S )( S · ¯ S )( S · ¯ S ) + (1 ↔ (cid:1) + iλ [ N − , , − QQ (cid:48) O (cid:0) ( S · ¯ S )( S · ¯ S )( S · ¯ S ) − (1 ↔ (cid:1)(cid:17) , (47)where T is the only three-point structure of two scalars and an operator of spin (cid:96) T = Z µ · · · Z µ (cid:96) − traces( x ) ∆ A +∆ B − ∆ O + (cid:96) ( x ) ∆ B +∆ O − ∆ A − (cid:96) ( x ) ∆ O +∆ A − ∆ B − (cid:96) , Z µ ≡ x µ x − x µ x . (48)Notice that the two tensor structures have different transformation properties underconjugation and permutation. Also notice the factor of i in the second structure. Thesymmetry and conjugations properties therefore read λ [ N − , , ± QQ (cid:48) O = (cid:0) λ [ N − , , ± QQ (cid:48) O (cid:1) ∗ , λ [ N − , , ± Q (cid:48) Q O = ± ( − (cid:96) O λ [ N − , , ± QQ (cid:48) O . (49)Now let us proceed with the remaining representations. For [ N − , N − , ,
1] we have λ [ N − ,N − , , QQ (cid:48) O = (cid:0) λ [ N − ,N − , , QQ (cid:48) O (cid:1) ∗ , λ [ N − ,N − , , Q (cid:48) Q O = ( − (cid:96) O λ [ N − ,N − , , QQ (cid:48) O . (50)Similar properties hold for the singlet and the [ N − , λ • QQ (cid:48) O = ( λ • QQ (cid:48) O ) ∗ , λ • Q (cid:48) Q O = ( − (cid:96) O λ • QQ (cid:48) O . (51) λ [ N − , QQ (cid:48) O = (cid:0) λ [ N − , QQ (cid:48) O (cid:1) ∗ , λ [ N − , Q (cid:48) Q O = ( − (cid:96) O λ [ N − , QQ (cid:48) O . (52)Finally, the [ N − , N − ,
2] is the only one that has a nontrivial conjugation property λ [ N − ,N − , QQ (cid:48) O = (cid:0) λ [ N − , , QQ (cid:48) O (cid:1) ∗ , (53a) λ [ N − ,N − , Q (cid:48) Q O = ( − (cid:96) O +1 λ [ N − ,N − , QQ (cid:48) O , (53b) λ [ N − , , Q (cid:48) Q O = ( − (cid:96) O +1 λ [ N − , , QQ (cid:48) O . (53c)25 .2 Four-point structures We will now write down the possible four-point tensor structures for any combination ofthe operators Q and Q (cid:48) . The four-point structure T is defined in (35), with ∆ A ≡ ∆ Q A .Since we assume a Z symmetry that acts trivially on Q and flips the sign of Q (cid:48) , theonly four-point functions that we have to consider have an even number of Q (cid:48) operators.Nevertheless, let us for brevity write down a general formula (with Q ≡ Q and Q ≡ Q (cid:48) ) (cid:104) Q A ( p ) Q B ( p ) Q C ( p ) Q D ( p ) (cid:105) = T (cid:88) r =1 Re T ( r ) f ABCDr ( u, v ) , (54)The structures T (1) through T (6) are even or odd under 1 ↔
2, while T (7) does nottransform in a definite way. The real part will be relevant only for T (2) , all otherstructures are taken to be real. Let us define as a shorthand S ijkl = ( S · ¯ S i )( S · ¯ S j )( S · ¯ S k )( S · ¯ S l ) . (55)The counting of all possible S ijkl is simple: due to S i · ¯ S i = 0 we have to countall permutation of four elements without 1-cycles, which results nine. However twostructures can be combined in the complex combination T (2) and one structure neverappears because we only consider correlators with an even number of Q (cid:48) . The mostconvenient basis for the T ( r ) is the one where f r gets contributions from a single OPEchannel. The choice realizing this requirement is T (1) = S (43)(21) − S + S + S + S N + 2) + S N + 1)( N + 2) ,T (2) = S − S − S − S + S − S N + S − S , ¯ T (2) = S − S − S − S + S − S N − S − S ,T (3) = S [43][21] − S + S + S + S N −
2) + S N − N − T (4) = S + S + S + S − S N ,T (5) = −S + S − S + S ,T (6) = S ,T (7) = 2 i ( S − S ) . (56)As mentioned before, the structure T (2) is complex. This is because it contains thecontributions from operators in the complex representation [ N − , , A = C and B = D then f is real andonly Re T (2) survives. A clarification for the adjoint representation is in order. Since namely T (8) = S − S . These structures are identical to those of the single correlator analysis (26) with the exceptionthat T (2) there is (cid:0) T (2) + ¯ T (2) (cid:1) here, i.e. the real part. T − (cid:104) Q A Q B Q C Q D (cid:105) ⊃ T ( r ) λ [ N − , a Q A Q B O λ [ N − , b Q C Q D O Π ab ; r G ∆ ,(cid:96) ( u, v ) , a, b = ± . (57)where Π ab ; r is a projector relating the product of three-point tensor structures to four-point tensor structures. The mixed terms a (cid:54) = b can only appear when either A (cid:54) = B or C (cid:54) = D due to the spin parity being opposite. But since we only consider the caseswith an even number of Q (cid:48) , this only occurs for (cid:104) QQ (cid:48) QQ (cid:48) (cid:105) (and (12) or (34) exchangesthereof). Thus there are only these combinationsΠ ++; r ∝ δ r , Π −− ; r ∝ δ r , and (Π + − ; r + Π − +; r ) ∝ δ r . (58)In the more general case without Z symmetry, Π ±∓ ; k would appear separately and thiswould require another structure T (8) (see footnote 8).The OPE channels associated to each partial wave f ABCDr are summarized in Table 3.Let us write down their explicit definition. For brevity we denote δ ≡ ∆ Q − ∆ Q (cid:48) . For r = 1 , , f r ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ r (cid:96) even λ r QQ O λ r Q (cid:48) Q (cid:48) O g , ,(cid:96) ( u, v ) ,f r ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ r ( − (cid:96) (cid:0) λ r QQ (cid:48) O (cid:1) g δ,δ ∆ ,(cid:96) ( u, v ) , (59)with = [ N − , N − , , = [ N − ,
2] and = • . The other representationsfollow: f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , , (cid:96) odd λ [ N − , , QQ O λ [ N − ,N − , Q (cid:48) Q (cid:48) O g , ,(cid:96) ( u, v ) ,f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , , ( − (cid:96) λ [ N − , , QQ (cid:48) O λ [ N − ,N − , QQ (cid:48) O g δ,δ ∆ ,(cid:96) ( u, v ) , (60) f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) even λ [ N − , + QQ O λ [ N − , + Q (cid:48) Q (cid:48) O g , ,(cid:96) ( u, v ) ,f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , ( − (cid:96) (cid:0) λ [ N − , + QQ (cid:48) O (cid:1) g δ,δ ∆ ,(cid:96) ( u, v ) , (61) f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) odd λ [ N − , − QQ O λ [ N − , − Q (cid:48) Q (cid:48) O g , ,(cid:96) ( u, v ) ,f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , ( − (cid:96) (cid:0) λ [ N − , − QQ (cid:48) O (cid:1) g δ,δ ∆ ,(cid:96) ( u, v ) , (62) f ( u, v ) = (cid:88) O ∆ ,(cid:96) ∈ [ N − , ( − (cid:96) λ [ N − , + QQ (cid:48) O λ [ N − , − QQ (cid:48) O g δ,δ ∆ ,(cid:96) ( u, v ) . (63)27 ABCDr (cid:104) Q A Q B O| | ¯ O Q C Q D (cid:105) If A = B If A = Cf [ N − , N − , ,
1] [ N − , N − , ,
1] only (cid:96) even f [ N − , ,
1] [ N − , N − ,
2] only (cid:96) odd f = ¯ f ¯ f [ N − , N − ,
2] [ N − , ,
1] only (cid:96) odd f = ¯ f f [ N − ,
2] [ N − ,
2] only (cid:96) even f [ N − , + [ N − , + only (cid:96) even f [ N − , − [ N − , − only (cid:96) odd f • • only (cid:96) even f [ N − , ± [ N − , ∓ not present only if A (cid:54) = B Table 3: Operator contributions to the partial wave f ABCDr .The partial waves f r are obtained by removing the ( − (cid:96) from f r , except when r = 2 , f is obtained using (53a) on the OPE coefficients. Theconformal blocks are normalized according to the convention expressed in (29). Thatnormalization implies g δ, − δ ∆ ,(cid:96) (cid:0) , (cid:1) ≥ , (cid:96), δ . B.3 Crossing equations
Using the definition of F ± , ∆ ,(cid:96) in (40), the crossing equations take the form(1 1) V , (cid:18) (cid:19) = (cid:88) r =1 , , (cid:88) O ∆ ,(cid:96) ∈ r (cid:96) even (cid:0) λ r QQ O λ r Q (cid:48) Q (cid:48) O (cid:1) V r ;∆ ,(cid:96) (cid:18) λ r QQ O λ r Q (cid:48) Q (cid:48) O (cid:19) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , , (cid:96) odd (cid:0) λ Re QQ O λ Im QQ O λ Re Q (cid:48) Q (cid:48) O λ Im Q (cid:48) Q (cid:48) O (cid:1) V ,(cid:96) λ Re QQ O λ Im QQ O λ Re Q (cid:48) Q (cid:48) O λ Im Q (cid:48) Q (cid:48) O + (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) even (cid:0) λ [ N − , + QQ O λ [ N − , + Q (cid:48) Q (cid:48) O (cid:1) V ,(cid:96) (cid:32) λ [ N − , + QQ O λ [ N − , + Q (cid:48) Q (cid:48) O (cid:33) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , (cid:96) odd (cid:0) λ [ N − , − QQ O λ [ N − , − Q (cid:48) Q (cid:48) O (cid:1) V ,(cid:96) (cid:32) λ [ N − , − QQ O λ [ N − , − Q (cid:48) Q (cid:48) O (cid:33) . (64)0 = (cid:88) r =1 , , (cid:88) O ∆ ,(cid:96) ∈ r ( − (cid:96) (cid:0) λ r QQ (cid:48) O (cid:1) W r ;∆ ,(cid:96) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , , ( − (cid:96) (cid:12)(cid:12) λ [ N − , , QQ (cid:48) O (cid:12)(cid:12) W ,(cid:96) + (cid:88) O ∆ ,(cid:96) ∈ [ N − , ( − (cid:96) (cid:0) λ [ N − , + QQ (cid:48) O λ [ N − , − QQ (cid:48) O (cid:1) W adj ;∆ ,(cid:96) (cid:32) λ [ N − , + QQ (cid:48) O λ [ N − , − QQ (cid:48) O (cid:33) . (65)28here V r (cid:54) =2;∆ ,(cid:96) and W adj ;∆ ,(cid:96) are 35 component vectors of 2 × V ,(cid:96) is a 35 component vector of 4 × W r ;∆ ,(cid:96) are ordinary 35component vectors. We also defined λ Re QQ O ≡ Re λ [ N − , , QQ O , λ Im QQ O ≡ Im λ [ N − , , QQ O . (66)In order to make the formulas more compact we abbreviate F ABCD ± , ∆ ,(cid:96) with f ± . Theindices ABCD can be deduced from context, namely the 2 × V r ;∆ ,(cid:96) ⊃ (cid:18) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) (cid:19) , W adj ;∆ ,(cid:96) ⊃ (cid:18) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) (cid:19) . (67)The 4 × V r ;∆ ,(cid:96) when split in 2 × V ,(cid:96) ⊃ (cid:18) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) F ± , ∆ ,(cid:96) (cid:19) , (68)and finally W r, ∆ ,(cid:96) has F ± , ∆ ,(cid:96) in the last 7 entries and F ± , ∆ ,(cid:96) in the remaining ones.Furthermore we will introduce some abbreviations for the various matrices that appear: u ± = (cid:18) f ±
00 0 (cid:19) , l ± = (cid:18) f ± (cid:19) , s ± = (cid:18) f ± f ± (cid:19) , a ± = (cid:18) − f ± f ± (cid:19) . (69)We use ⊗ to denote the Kronecker product. Finally we indicate with 0 n a string of n zeros and with n a string of n zero matrices. e.g. (cid:18) a bc d (cid:19) ⊗ = a b a bc d c d . ,(cid:96) = u − u + N + N +28( N +1)( N +2) s − N ( N +3)8( N +1)( N +2) s − N +38( N +1) s − N ( N +3)16( N − N +1)( N +2) s − − N +316( N +1) s − N ( N +3)8( N − N +1) s − − N + N +28( N +1)( N +2) s + − N ( N +3)8( N +1)( N +2) s + − N +38( N +1) s + − N ( N +3)16( N − N +1)( N +2) s + N +316( N +1) s + − N ( N +3)8( N − N +1) s + l − l + , V ,(cid:96) = − ⊗ u + ⊗ N − N s − ⊗ s − ⊗ N +22 N s − ⊗ N N − s − ⊗ − N − N − s − ⊗ − N − N s + ⊗ s + ⊗ − N +22 N s + ⊗ − N N − s + ⊗ N − N − s + ⊗ − ⊗ (cid:16) − (cid:17) a + ⊗ (cid:16) − (cid:17) − ⊗ l + ⊗ , (70) V ,(cid:96) = u − − ( N − N +1)( N − N +3) u + N ( N − N − N − u + N − N − s − ( N − N N − N − s − N − N +28( N − N − s −− N ( N − N − ( N − N +2) s − N − N − s − ( N − N N − ( N +1) s − − N − N − s + − ( N − N N − N − s + − N − N +28( N − N − s +( N − N N − ( N − N +2) s + − N − N − s + − N ( N − N − ( N +1) s + l − − ( N − N +1)( N − N +3) l + N ( N − N − N − l + , V ,(cid:96) = − N +1)( N − N ( N +3) u − N +2) N u + u + N − N s − N s − − N +2 N s − N − N − s − N − N s − N − N ( N − s − − N − N s + − N s + N +2 N s + − N − N − s + − N − N s + − N − N ( N − s + l − − N +1)( N − N ( N +3) l +2( N +2) N l + . (71)30 ,(cid:96) = − N +1) N +2 u − NN − u − N − N − u − N ( N − u − N +1) N +3 u + − NN +2 u + − s − − N N − s − s − NN − s − + − s + − N N − s + − s + − NN − s + − N +1) N +2 l − NN − l − N − N − l − N ( N − l − N +1) N +3 l + − NN +2 l + , V ,(cid:96) = N − N ( N +2) u − N − N − u − N − N ( N − u − N ( N − N − u − − N +1) N ( N +3) u + − N +1) N +2 u +12 s − − s − s − N N − s − N s − N − s − − s +12 s + − s + − N N − s + − N s + − N − s + N − N ( N +2) l − N − N − l − N − N ( N − l − N ( N − N − l − − N +1) N ( N +3) l + − N +1) N +2 l + . (72) W ,(cid:96) = f − f + f − f + , W ,(cid:96) = f − f + f − f + , W ,(cid:96) = f − f + f − − ( N − N +1)( N − N +3) f + N ( N − N − N − f + . (73) W adj ;∆ ,(cid:96) = u − l − s − u + l + s + − N +1) N +2 l − NN − l − N − N − l − N − (cid:16) ( N − f − N f − (cid:17) N +1) N ( N +3) (cid:16) − ( N − f + N f + (cid:17) N ( N +2) (cid:16) ( N + 2) f − − N f − (cid:17) s + , W ,(cid:96) = f − f + N − N ( N +2) f − N − N − f − N − N − N f − N ( N − N − f − − N +1) N ( N +3) f + − N +1) N +2 f + . (74)31 Unitary versus orthogonal group bootstrap
In this appendix we prove a correspondence between bounds obtained bootstrapping SU ( N ) adjoint scalars and bounds obtained bootstrapping O ( N (cid:48) ) fundamental scalars,where N (cid:48) = N −
1. In particular, we prove that the upper bounds on the dimension ofthe lightest singlet are identical. Similar theorems hold for other pairs of representationsand groups, such as fundamentals in O ( N ( N + 1) / −
1) vs. rank-2 tensors in O ( N ) [45]and bifundamentals in SU ( N ) × SU ( N ) vs. fundamentals in O (2 N ) [73]. The proofthat we will present here follows exactly the same ideas.As a first trivial observation we note that, when N (cid:48) = N −
1, the fundamentalof O ( N (cid:48) ) decomposes in the adjoint of SU ( N ). Indeed their dimensions agree. TheOPE of two O ( N (cid:48) ) fundamentals contains the singlet ( S ), the rank-2 tensor ( T ) andthe antisymmetric ( A ), while the OPE of two adjoints in SU ( N ) contains the sixrepresentations shown in (19). For brevity here we will denote them as follows[ N − , N − , ,
1] =
B , [ N − ,
2] =
C , [ N − , ,
1] =
D , [ N − , N − ,
2] = ¯
D , [ N − , ± = A ± , • = S . (75)For the above representations, the branching rules of the inclusion SU ( N ) ⊂ O ( N (cid:48) )read S → S , T → A + ⊕ B ⊕ C , A → A − ⊕ D ⊕ ¯ D . (76)Notice that this is also in agreement with the parity of the spins exchanged. Accord-ingly, any O ( N (cid:48) ) is a special SU ( N ) theory satisfying the extra constraints∆ A + = ∆ B = ∆ C , ∆ A − = ∆ D , (77)together with further constraints on OPE coefficients which we do not need for thisargument. Let us call ∆ ∗ R [ O ( N (cid:48) )] the upper bound on the lightest operator transformingin R in a O ( N (cid:48) ) theory and ∆ ∗ R [ SU ( N )] the analogous quantity for SU ( N ). Since any O ( N (cid:48) ) solution is, in particular, a SU ( N ) solution, we immediately conclude that∆ ∗ R [ SU ( N )] ≥ ∆ ∗ S [ O ( N (cid:48) )] R = S , ∆ ∗ T [ O ( N (cid:48) )] R ∈ { A + , B, C } , ∆ ∗ A [ O ( N (cid:48) )] R ∈ { A − , D } . (78)Now we would like to prove that, for the singlet, also the converse is true. To do thatwe need to reason in terms of the dual problem, i.e. in terms of positive functionalsacting on the crossing equations. The equations for O ( N (cid:48) ) can be written as (cid:88) ∆ ,(cid:96) a ( S )∆ ,(cid:96) F ∆ ,(cid:96) H ∆ ,(cid:96) + (cid:88) ∆ ,(cid:96) a ( T )∆ ,(cid:96) F ∆ ,(cid:96) (cid:0) − N (cid:48) (cid:1) F ∆ ,(cid:96) − (cid:0) N (cid:48) + 1 (cid:1) H ∆ ,(cid:96) + (cid:88) ∆ ,(cid:96) a ( A )∆ ,(cid:96) − F ∆ ,(cid:96) F ∆ ,(cid:96) − H ∆ ,(cid:96) = 0 . (79) In fact, it is the spin parity that allows us to say that A goes in A − and T goes in A + . a ( R )∆ ,(cid:96) are the OPE coefficients squared of an operator in the representation R . Wecan combine the vectors into a matrix M O ( N (cid:48) ) = F ∆ ,(cid:96) − F ∆ ,(cid:96) F ∆ ,(cid:96) (cid:0) − N (cid:48) (cid:1) F ∆ ,(cid:96) F ∆ ,(cid:96) H ∆ ,(cid:96) − (cid:0) N (cid:48) + 1 (cid:1) H ∆ ,(cid:96) − H ∆ ,(cid:96) . (80)In the dual formulation we look for functionals (cid:126)α = ( α α α ) such that the followingvector is component-wise positive (cid:0) α α α (cid:1) · M O ( N (cid:48) ) ≡ (cid:0) α S α T α A (cid:1) > . (81)If such a functional exists, then the assumptions made in the sums over ∆ are incon-sistent and the theory is excluded. Similarly, let M SU ( N ) be the matrix built out of thevectors given in (32). A positive functional (cid:126)β is such that the following inequalities aresatisfied (cid:0) β β β β β β (cid:1) · M SU ( N ) ≡ (cid:0) β B β D β C β A + β A − β S (cid:1) > . (82)And, again, the existence of a positive functional implies that the SU ( N ) theory isinconsistent. Our goal is to show that for every positive functional (cid:126)α we can constructa corresponding positive functional (cid:126)β . If we achieve this, then we obtain a set ofinequalities in the opposite direction as before, namely∆ ∗ S [ O ( N (cid:48) )] ≥ ∆ ∗ S [ SU ( N )] , ∆ ∗ T [ O ( N (cid:48) )] ≥ max R ∈{ A + ,B,C } (cid:0) ∆ ∗ R [ SU ( N )] (cid:1) , ∆ ∗ A [ O ( N (cid:48) )] ≥ max R ∈{ A − ,D } (cid:0) ∆ ∗ R [ SU ( N )] (cid:1) . (83)This, together with (78) implies ∆ ∗ S [ O ( N (cid:48) )] = ∆ ∗ S [ SU ( N )] as claimed.We are looking for a linear relation between α and β . The branching rules suggestan ansatz of the form (cid:0) β B β D β C β A + β A − β S (cid:1) = (cid:0) x α T x α A x α T x α T x α A α S (cid:1) . (84)We can solve this for β i requiring that the terms with F ∆ ,(cid:96) and H ∆ ,(cid:96) do not mix. Thereis a unique solution of the x i ’s that admits a linear map β i = T ji α j and it reads x = ( N − N ( N + 3)4 ( N −
2) ( N + 1) , x = ( N − N + 2) N − , x = ( N − N ( N + 1)( N −
2) ( N + 1) ,x = 2( N − N − N + 1)( N + 2) N ( N −
2) ( N + 1) , x = 2 NN − . (85) For completeness we also report the solution for T ji T ji = ( N − N ( N +3)4( N − N +1) N ( N +3)( N − N +1)( N − N +1) − ( N − N +2) N − N − N +2) N − ( N − N ( N +1)( N − N +1) ( N − N ( N − N − N − N +1) N − N − N +1)( N +2) N ( N − N +1) 2( N − N +2)( N − N ( N − N +1)
00 0 − N ( N +3)4( N +1)( N − − ( N − N +2) N − . x i ’s are nonnegative for N ≥
3. As a consequence of equation (84) apositive functional (cid:126)α will lead to a positive functional (cid:126)β , which is what we wanted toshow.To conclude, let us revisit the direction of the inequality that we showed at thebeginning, namely (78). Instead of presenting a purely group theoretic argument, onecould have proceeded exactly in the same way as above, via a dual formulation of theproblem. This time we are looking for a linear map from (cid:126)β to (cid:126)α . Based on the branchingrules we can propose an ansatz of the following form (cid:0) α S α T α A (cid:1) = (cid:0) β S y β A + + y β B + y β C y β A − + y β D (cid:1) . (86)As before, we find a solution with nonnegative y i ’s y = NN − , y = 2 , y = 12 , y = 1 N , y = 1 . (87)Let ˜ T be the resulting linear transformation α j = ˜ T ij β i . It is easy to check that it isjust the left inverse of the transformation shown in footnote 12, namely ˜ T · T = . D Parameters of the numerical implementation
The numerical conformal bootstrap problem was truncated according to the parametersin Table 4. The semi-definite problem was solved using sdpb [74, 75] with the choice ofparameters given in Table 5.Λ = 19 Λ = 27
Lset { , ..., } ∪ { , } { , ..., } ∪ { , , , } order
60 60 κ
14 18Table 4: Values of the various parameters appearing in the numerical bootstrapproblem. 345Parameter feasibility OPE maxIterations
500 500 maxRuntime checkpointInterval noFinalCheckpoint True FalsefindDualFeasible True FalsefindPrimalFeasible True FalsedetectDualFeasibleJump True Falseprecision
700 700 maxThreads
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