A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in d>2 dimensions
AA roadmap for bootstrapping gauge theories: decouplingoperators of conformal field theories in d > dimensions Yin-Chen He, ∗ Junchen Rong, † and Ning Su ‡ Perimeter Institute for Theoretical Physics,Waterloo, Ontario N2L 2Y5, Canada DESY Hamburg, Theory Group, Notkestraße 85, D-22607 Hamburg, Germany Department of Physics, University of Pisa, I-56127 Pisa, Italy
Abstract
We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gaugetheories in dimensions d >
2. In particular, we provide a simple and workable answer to thequestion of how to detect the gauge group in the bootstrap calculation. Our recipe is based onthe notion of decoupling operator , which has a simple (gauge) group theoretical origin, and isreminiscent of the null operator of 2 d Wess-Zumino-Witten CFTs in higher dimensions. Using thedecoupling operator we can efficiently detect the rank (i.e. color number) of gauge groups, e.g.,by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation ofmotion, which has interesting consequences in the CFT spectrum as well. As an application ofour recipes, we study a prototypical gauge theory, namely the scalar QED which has a U (1) gaugefield interacting with critical bosons. In d = 2 + (cid:15) dimensions we successfully solve it by obtaininga kink as well as an island of the scalar QED. Further attempt towards the 3 d scalar QED is alsodiscussed. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] J a n ONTENTS
I. Introduction 2II. Decoupling operators in gauge theories 6A. Null operator as a decoupling operator: the SU ( N ) k WZW CFT 7B. Decoupling operator of bosonic gauge theories 10III. Consequence of the equation of motion 11IV. Numerical results 15A. Kinks of the A ¯ A bound 16B. Scalar QED kinks in 2 + (cid:15) dimensions 18C. Scalar QED islands in 2 + (cid:15) dimensions 20D. Towards the 3 d scalar QED 23V. Conclusion and outlook 26Acknowledgments 27A. 3d WZW models and gauge theories 29B. Floating kinks v.s. stable kinks 31References 32 I. INTRODUCTION
Coupling gapless particles with gauge fields is one of the few known ways to obtainan interacting conformal field theory in dimensions d >
2. These gauge theory type ofCFTs have interesting applications in both the high energy [1–3] and condensed matterphysics. In condensed matter system, such CFTs describe phase transitions or gaplessphases beyond conventional Landau’s symmetry breaking paradigm [4–12], and they haveinteresting properties such as fractionalization and long-range entanglement. Understandingsuch CFTs may pave the way towards several long-standing problems in condensed matter,2ncluding critical quantum spin liquids [6–8] and plateau transitions of fractional quantumHall states [9–12].Compared to the Wilson-Fisher (WF) CFTs, these gauge theory CFTs are poorly under-stood. Recently, conformal bootstrap [13] becomes a powerful technique to study CFTs indimensions higher than 2 d [14–24] (see a review [25]). The numerical bootstrap obtainedcritical exponents of 3 d Ising [14] and O (2) WF [23] with the world record precision, and im-portantly, has solved the long-standing inconsistency between experiments and Monte-Carlosimulations of O (2) WF [23] as well as the cubic instability of O (3) WF [24]. However, so farthe gauge theory CFTs resist to be tackled by bootstrap [26–29]. The main challenge is builtin the fundamental philosophy of bootstrap, namely characterizing a theory without rely-ing on a specific Lagrangian. More concretely, in a bootstrap study one typically inputs theglobal symmetry of the theory, and utilizes the consistency of crossing equations to constrainor to compute the scaling dimensions of operators in certain representations of the globalsymmetry. For a Wilson-Fisher type of CFT, it is believed that one can uniquely define itby specifying its global symmetry as well as the representation of the order parameter, i.e.,the lowest lying operators. In contrast, gauge theories with distinct gauge groups typicallyhave similar or even identical global symmetries. Their lowest lying operators usually sit inthe same representation and have similar scaling dimensions.As a concrete example, one can consider a family of theories described by N f flavors oftwo-component Dirac fermions coupled to a U ( N c ) gauge field in d = 3 dimensions. Fora given color number N c , most theories in the infrared (IR) will flow into CFTs when N f is larger than a critical value N ∗ f . In other words, for a large enough N f there will bea number of distinct CFTs that correspond to different N c ’s. These CFTs have identicalglobal symmetries, i.e. ( SU ( N f ) × U (1) top ) / Z N f . The SU ( N f ) corresponds to the flavorsymmetries of Dirac fermions, while the U (1) top symmetry corresponds to the U (1) gaugeflux conservation of U ( N c ) gauge group. The most important low lying (scalar) operatorsare 1) fermion bilinears which are SU ( N f ) adjoint but neutral under U (1) top , its scalingdimension is ∆ = 2 + O (1 /N f ); 2) 2 π monopole operators which are charged under U (1) top and also carries a non-trivial representation of SU ( N f ) (which is independent of N c ), itsscaling dimension is ∆ = 0 . N f − . − . N c −
1) + O (1 /N f ) [30]. There were For the precise global symmetry one may need to further quotient out certain discrete symmetries. Suchglobal part of the global symmetry is not important for our discussion. d = 2 dimensions it is pretty common that distinct CFTs have the same globalsymmetry. However, numerical bootstrap successfully detects some of these CFTs, includingthe 2 d Ising CFT out of minimal models [31] and the SU (2) Wess-Zumino-Witten (WZW)CFT out of SU (2) k WZW theories [32, 33]. It is found that the Ising CFT ( SU (2) WZW)sits at the kink of numerical bounds, while its cousins in the minimal model ( SU (2) k WZW)saturate the numerical bounds on the right (left) hand side of the kink. More interestingly,the phenomenon that these CFTs appear at kinks of bootstrap bound is closely related tothe existence of a family of CFTs sharing the same global symmetry and similar operatorspectrum. Compared to their cousins, the Ising CFT and SU (2) WZW are special becausethey have null operators at low levels. These null operators will lead to some non-analyticityin the numerical bound, resulting in a kink [15, 33, 34].The examples in 2 d suggests that, the existence a family of cousins with the same globalsymmetries is not an obstacle of bootstrapping a CFT, it could instead guide us to find theright condition (i.e. null operator condition) to bootstrap the interested CFT. Theoretically,the existence of the null operator at a certain level can serve as a defining feature of 2dminimal model [35]. One cannot help to wonder if a similar physics also exists for higherdimensional CFTs, and can it be further utilized in the bootstrap study? We provide apositive answer to this by exploring gauge theories and in particular, their relations withthe 2 d WZW CFTs. We will show that in gauge theories there exists a family of operators,we dub decoupling operators, that are reminiscent of null operators of 2 d WZW CFTs inhigher dimensions. Similar to the Kac-Moody algebra, the structure of decoupling operatorsof gauge theories are sensitive to the representations of global symmetry. Moreover, the colornumber N c of the gauge group plays the role of the WZW level (i.e. k ) in WZW CFTs.We also explore another related observation in higher dimensional CFTs, namely theequation of motion (EOM) can lead to the phenomenon of operator missing (more preciselyconformal multiplet recombination [36]) in the CFT spectrum [15, 16]. Theoretically, it wasunderstood that as the consequence of the EOM of φ theory, i.e. (cid:3) φ = gφ , φ becomes adescendent of φ . In other words, the operator φ becomes missing in the primary operatorspectrum of WF CFTs. This structure can further serve as an algebraic definition of WFCFTs in 4 − (cid:15) dimensions [36]. Numerically, one can also impose the condition of φ being4issing by adding a large gap above φ in the O ( N ) vector channel, this is indeed how oneobtains the famous bootstrap island of WF CFTs [14, 16]. We will push this idea furtherby arguing that the EOMs also lead to missing operators at high levels. Such higher levelmissing operators are actually rather straightforward to visualize. For example, it is naturalto expect that φ ( (cid:3) φ − gφ ) is missing as well. We will elaborate more on this and itsconsequence in the main text.To be concrete, we will discuss the idea of decoupling operators and their bootstrapapplication in the context of a simple gauge theory, namely the scalar QED. It is describedby N f flavor critical bosons coupled to a U (1) gauge field, L = N f (cid:88) i =1 | ( ∂ µ − iA µ ) φ i | + m | φ | + g | φ | + 14 e F µν . (1)The global symmetry of the scalar QED is P SU ( N f ) = SU ( N f ) Z Nf . The most fundamental(gauge invariant) operators of this theory are the boson bilinear ¯ φ i φ j − δ ij /N ¯ φ k φ k and ¯ φ i φ i ,which are in the SU ( N f ) adjoint and singlet representation, respectively. This theory isalso called CP N f − model, as it can be defined as a non-linear sigma model (NL σ M) on thetarget space CP N f − ∼ = U ( N f ) U ( N f − × U (1) . Within the NL σ M formulation, one can access thescalar QED fixed point using the 2 + (cid:15) expansion [37–39]. It is worth noting that in d = 3dimensions, there is one extra emergent symmetry called U (1) top , which corresponds to theflux conservation of the gauge field. There will be a new type of primary operators, calledmonopoles [40], that are charged under U (1) top . In this paper, we will not study monopoleoperators.For a large enough N f , the scalar QED in 2 < d < d , there exists a critical N ∗ f ( d ) below which the scalar QED fixed point will disappearby colliding with the tri-critical QED fixed point (see definition below) [41–44]. In otherwords, only if N f > N ∗ f ( d ) the scalar QED will be a real CFT . It is believed that N ∗ f ( d )monotonically increases with d , but its precise form is unknown. From 2 + (cid:15) and 4 − (cid:15) expansion, it is found that N ∗ f ( d → → N ∗ f ( d → ≈
183 [41]. It remains anopen problem regarding the value of N ∗ f in d = 3 dimensions [47–49]. The N f = 2 scalar QEDin d = 3 dimensions is one of the dual descriptions of the widely studied deconfined phase We shall note it is an exception for the N f = 1 scalar QED in 3 d as it is dual to the O (2) WF [45, 46]. d WZW CFTscan be interpreted as decoupling operators of gauge theories. We then discuss decouplingoperators of bosonic gauge theories in Sec. II B. In Sec. III we discuss the consequence ofEOMs on the CFT spectrum. In Sec. IV we will present our numerical results of the scalarQED. In particular, by imposing the information of decoupling operators we successfullysolving the scalar QED in 2 + (cid:15) dimensions: we have obtained the scalar QED kink inSec. IV B and the scalar QED island in Sec. IV C. We further approach d = 3 dimensionsfrom d = 2 + (cid:15) dimensions in Sec. IV D. We find that the success in d = 2 + (cid:15) does notsustain to d = 3 dimensions, namely the scalar QED kink disappears in d = 3 dimensions.Possible reasons have been analyzed. We will conclude in Sec. V, and the appendix A willserve to discuss some concrete connections between 3 d WZW models and gauge theories.
Note added:
Upon the completion of this work we became aware of an independentwork [54] that overlap with ours.
II. DECOUPLING OPERATORS IN GAUGE THEORIES
In this section, we will define what we mean by the decoupling operator and discussseveral concrete examples in 2d CFTs and higher dimensional gauge theories.The decoupling operator of an interested CFT A can be defined by embedding A intoa family of CFTs that share the same global symmetry and similar operator spectrum.Then one can construct a possible continuous interpolation between these different theories,and define decoupling operators as operators that decouple from the theories’ spectrum oralgebra as one continuously tunes to the CFT A . A textbook example is the 2 d minimalmodel M q,q − for which one can promote the integer valued q to be real valued, which theninterpolates all the minimal models M q,q − . This is more than a conceptional interpolation,indeed we can explicitly write down a number of crossing symmetric correlation functionsthat continuously depend on q . As one continuously tunes q , there are operators decoupled In the end, a fully consistent solution of all crossing symmetric correlation functions would only admitdiscrete (integer valued) q . q . These decoupling operators are indeed null operatorsfor a specific theory M q,q − [34]. Similarly, for the 2 d WZW CFTs, one can promote theinteger valued WZW level k to be real valued, and ask how are operators decoupled as onecontinuously varies k (see Sec. II A for more details). Different from the example of the 2 d minimal model, the decoupling (null) operators are lying in representations that stronglydepend on k ’s: for the SU ( N ) k = l WZW CFT, all the Kac-Moody primaries in the rank- m symmetric tensor representation with m > l are becoming null operators [55].Although null operators of 2D CFTs can be defined as decoupling operators, the nulloperators certainly have deeper implications in the algebra of CFTs, e.g. they can actas differential operators that annihilate correlation functions of primary operators. Thedecoupling operators, on the other hand, may or may not have such fundamental applicationsin the operator algebra of higher dimensional CFTs. It will be interesting to understandthe similarity and difference between 2d null operators and higher dimensional decouplingoperators in the future. A. Null operator as a decoupling operator: the SU ( N ) k WZW CFT
In this section, we will elaborate more on how to view the null operator of 2d CFTs as adecoupling operator in the context of the SU ( N ) k WZW CFT. Let us start with a simplecase, i.e. SU (2) k WZW theory. It has a global symmetry SO (4) ∼ = SU (2) L × SU (2) R /Z , andits Kac-Moody primary operators | j, j (cid:105) are in the SO (4) representations ( SU (2) L , SU (2) R ) =( j, j ) with j = 0 , / , , · · · , k/ . So | , (cid:105) is a Kac-Moody primary of SU (2) k ≥ WZWCFT, while it becomes null in the SU (2) WZW CFT.Now we create an interpolation between all SU (2) k WZW CFTs by promoting the integervalued k to be real valued. More precisely, the four-point correlation function (4pt) of anyprimary operator of the SU (2) k WZW CFTs is an analytical function of k , so there is noobstacle to promote k to be real valued. For our purpose it is enough to consider the 4pt of In this notation, (1 / , /
2) corresponds to the SO (4) vector, (1 ,
1) corresponds to the rank-2 symmetrictraceless tensor, (1 , ⊕ (0 ,
1) corresponds to the rank-2 anti-symmetric tensor. | / , / (cid:105) , which is a SO (4) vector and we will call it φ i : (cid:104) φ i ( x ) φ j ( x ) φ k ( x ) φ l ( x ) (cid:105) = 1 x φ x φ (cid:20) N δ ij δ kl G S [ z, ¯ z ]+( 12 δ il δ jk + 12 δ ik δ jl − N δ ij δ kl ) G T [ z, ¯ z ]+( 12 δ il δ jk − δ ik δ jl ) G A [ z, ¯ z ] (cid:21) . (2)Here N = 4 and G S [ z, ¯ z ], G T [ z, ¯ z ], G A [ z, ¯ z ] corresponds to the 4pt’s in the channels ofthe SO (4) singlet, rank-2 symmetric traceless tensor, and rank-2 anti-symmetry tensor.The precise form of these 4pt’s can be found in textbooks such as [55]. We are primarilyconcerned with the Kac-Moody primary | , (cid:105) , which is in the channel of rank-2 symmetrictraceless tensor. The 4pt corresponding to this channel is, G T [ z, ¯ z ]((1 − z )(1 − ¯ z )) k +4 = 2 z ¯ zk A [ z, ¯ z ] + 2( z ¯ z ) k +2 (cid:32) Γ ( k +2 )Γ ( k +2 )Γ ( k +2 ) − ( − k +2 )Γ ( k +2 )Γ ( k +2 )Γ ( − k +2 ) (cid:33) B [ z, ¯ z ] , (3)with A [ z, ¯ z ] = F ( k + 1 k + 2 , k + 3 k + 2 , k + 2 k + 2 , z ) F ( k + 1 k + 2 , k + 3 k + 2 , k + 2 k + 2 , ¯ z ) ,B [ z, ¯ z ] = F ( 1 k + 2 , k + 2 , k + 2 , z ) F ( 1 k + 2 , k + 2 , k + 2 , ¯ z ) . (4)Decomposing this 4pt into the global conformal block, one obtains the low lying spectrumto be ∆ = k +2 , , · · · . The first operator (denoted as t ) is nothing but the Kac-Moodyprimary | , (cid:105) , while the second operator is a global primary obtained by applying Kac-Moody current operator to the vacuum, i.e. J L J R | , (cid:105) . We can also work out the OPEsquare λ φφt , λ φφt = Γ ( k +2 )Γ ( k +2 )2Γ ( k +2 ) − ( − k +2 )Γ ( k +2 )Γ ( k +2 )Γ ( − k +2 ) . (5)The above formula is positive definite for k >
1, and it vanishes precisely at k = 1. Inother words the Kac-Moody primary | , (cid:105) gets decoupled from operator spectrum at k = 1.Therefore, in this natural interpolation of SU (2) k WZW CFTs we can view the null operator | , (cid:105) of SU (2) WZW as a decoupling operator.The above discussion can be easily generalized to the SU ( N ) k WZW CFTs. Interestingly,in the large- N limit we can directly relate the Kac-Moody null operator to the decoupling op-erator of gauge theories, without relying on any precise knowledge of the correlation functionor operator spectrum of the WZW CFTs. The key is to recognize a gauge theory description8or the SU ( N ) k WZW CFTs, namely a gauge theory with N flavors of 2-component Diracfermions interacting with a U ( k ) gauge field. For the case of k = 1, this duality can beproved exactly as the U (1) gauge theory is integrable [56]. For a general level- k WZW CFT,there are reasonable evidences suggesting that they are dual to a QCD theory (e.g. see [57]and references therein), although the QCD is not integrable anymore.The global symmetry of both the SU ( N ) k WZW and the gauge theory is SU ( N ) L × SU ( N ) R , so we can consider the (global) primary operator spectrum of these two theories indifferent representations of SU ( N ) L × SU ( N ) R . Let us warm up with the lowest weight Kac-Moody primary (except for the vacuum), i.e., a bi-fundamental of SU ( N ) L and SU ( N ) R .This operator exists for arbitrary k , and its scaling dimension is ∆ = N − N ( N + k ) , which is∆ ≈ O (1 /N ) in the limit of N (cid:29) k . In the gauge theory, this operator is nothing but2-fermion operators, schematically written as ¯ ψ cl ψ r,c . We use a convention that the right(left) moving fermion ψ ( ¯ ψ ) is the fundamental (anti-fundamental) of U ( k ) gauge field, and c is the index for its SU ( k ) subgroup. The index l and r refer to the index of SU ( N ) L and SU ( N ) R . So this 2-fermion operator ¯ ψ cl ψ r,c is the SU ( N ) bi-fundamental and its scalingdimension is ∆ = 1 + O (1 /N ) in the N (cid:29) k limit. We have matched the lowest primaryoperators of SU ( N ) k WZW with the 2-fermion operators of U ( k ) gauge theories.Let us now move to the 4-fermion operators (that are Lorentz scalar) of gauge theories.Such operator can be schematically written as ¯ ψ c l ¯ ψ c l ψ r ,c ψ r ,c . The two left (right) movingfermions shall be totally antisymmetric, so we shall have either the flavor indices or thecolor indices anti-symmetric, and meanwhile keep the other indices symmetric. We need tofurther contract the color indices of left and right moving fermions to get a gauge invariantoperator. For k = 1, however, anti-symmetrizing color indices is not an option, leaving theonly possibility to be anti-symmetrizing the flavor indices. Therefore, for k > SU ( N ) L × SU ( N ) R representations A L A R and T L T R . What are these operators in the SU ( N ) k WZW CFTs?There are nothing but the Kac-Moody primaries in the A L A R and T L T R channel, whosescaling dimensions are ∆ = N − N +1) N ( N + k ) and ∆ = N − N +2) N ( N + k ) . In the limit of N (cid:29) k ,these two scaling dimensions are ∆ = 2 + O (1 /N ) matching what we expect for 4-fermionoperators. On the other hand, when k = 1 there is only one 4-fermions operator in thechannel A L A R , as the other operator in the channel T L T R becomes a decoupling operator Here T L T R ( A L A R ) refers to rank-2 symmetric (anti-symmetric) tensor of SU ( N ) L and SU ( N ) R . SU ( N ) k WZW CFTs, namely at k = 1 the Kac-Moody primary in the T L T R channel becomesnull (the Kac-Moody primary in the A L A R channel is still intact). It is straightforward togeneralize to other Kac-Moody null operators for higher k ’s, as well as to other WZW CFTs.Therefore, on the phenomenological level null operators of 2d WZW CFT can be under-stood as decoupling operators in the context of 2d gauge theories. From the gauge theoryside, we can also generalize the analysis to higher dimensions. A complexity is that thefermion is in the spinor representation of SO ( d ) Lorentz rotation, which has a strong de-pendence on the spacetime dimension d . It turns out that it is easiest to discuss the idea ofdecoupling operators in the context of bosonic gauge theories, namely critical bosons coupledto gauge fields. We will discuss it in the following subsection. It is also worth mentioningthat, in higher dimensions (e.g. 3 d ) one can also make a straightforward connection betweenfermionic gauge theories and WZW CFTs [58, 59]. The details are a bit off the theme ofthe current paper, we will elaborate more in the Appendix. B. Decoupling operator of bosonic gauge theories
In this subsection, we will discuss the decoupling operators of bosonic gauge theories,namely critical bosons coupled to gauge fields. We will explain the idea in the context of U ( N c ) gauge theories, and the generalization to other gauge groups SU ( N c ), SO ( N c ), and U Sp (2 N c ) is rather straightforward.We can simply start by classifying gauge invariant operators (constructed by bosonicfield) in these gauge theories. We denote boson operators as φ f,c and ¯ φ f,c , which are SU ( N f )( U ( N c )) fundamental and anti-fundamental, respectively. f = 1 , · · · , N f and c = 1 , · · · , N c correspond to the flavor and color index. To keep the U (1) ⊂ U ( N c ) gauge invariance, weshall only consider operators like ¯ φ f ,c · · · ¯ φ f m ,c m φ f m +1 ,c m +1 · · · φ f m ,c m . Among these opera-tors, one should further choose SU ( N c ) gauge invariant ones. Let us start with m = 1,i.e. boson bilinears ¯ φ f ,c φ f ,c . Apparently, to keep SU ( N c ) invariance there are onlytwo operators, ¯ φ f ,c φ f ,c and ¯ φ f ,c φ f ,c − δ f f /N f ¯ φ f,c φ f,c , which are the SU ( N f ) singletand adjoint, respectively. Their large − N f scaling dimensions are ∆ = 2 + O (1 /N f ) and∆ = d − O (1 /N f ) for N c (cid:28) N f .Things become interesting as one moves to m = 2. Let us ask what is the lowest10perator in the representation A [ f ,f ][ f ,f ] , where both the upper and lower indices are anti-symmetric. To construct an operator in this representation, one needs at least 4 bosons,¯ φ f ,c ¯ φ f ,c φ f ,c φ f ,c . If N c ≥
2, one can simultaneously antisymmetrize the flavor in-dices (i.e. [ f , f ], [ f , f ]) and the color indices (i.e. [ c , c ], [ c , c ]) of ¯ φ f ,c ¯ φ f ,c and φ f ,c φ f ,c , and then contract their color indices to get a SU ( N c ) gauge invariant operator.This will then give an operator in the required representation, with a scaling dimension∆ = 2( d −
2) + O (1 /N f ). When N c = 1, in contrast, antisymmetrizing the color indices oftwo identical bosons will vanish. So the lowest operator in the required representation shallinvolve two covariant derivatives, schematically written as ( ¯ φ f D µ ¯ φ f )( φ f D µ φ f ). Its scalingdimension is ∆ = 2( d −
2) + 2 + O (1 /N f ). Therefore, in the A [ f ,f ][ f ,f ] channel the QCD gaugetheories ( N c >
1) have the spectrum ∆ = 2( d −
2) + O (1 /N f ) , d −
2) + 2 + O (1 /N f ) , · · · ,while for N c = 1 (e.g. scalar QED) the spectrum is ∆ = 2( d −
2) + 2 + O (1 /N f ) , · · · . Inother words, • In the SU ( N f ) A [ f ,f ][ f ,f ] channel, the lowest operator of U ( N c > gauge theories isdecoupling at N c = 1 . One can easily generalize above discussions to arbitrary N c , • In the interpolation between U ( N c ) gauge theories, the lowest lying operator in the SU ( N f ) anti-symmetric representation A [ i , ··· ,i m ][ j , ··· ,j m ] of N c > m − is decoupling at N c ≤ m − . This structure of decoupling operators is almost identical to the null operator structure of2 d WZW CFTs, and the color number N c plays the role of WZW level k . Similar structurescan also be found in theories with other gauge groups . III. CONSEQUENCE OF THE EQUATION OF MOTION
The notion of decoupling operator was formulated by identifying a family of CFTs withthe identical global symmetry. In the numerical bootstrap, one can impose gap conditionsbased on the structure of the decoupling operator to isolate the interested theory from their An independent work [60] has a similar analysis for SO (2) gauge theory in the context of SO ( N ) invariantCFTs. SU ( N f ) adjoint boson bilinears, so besidesthe U ( N c ) scalar gauge theory we also need to consider other theories that contain suchoperator:1. Tri-critical QED: It corresponds to the UV fixed point of the scalar QED. It can alsobe described by Eq. (1), but different from the scalar QED, hitting the tri-criticalQED fixed point requires the fine tuning of two singlet operators, i.e., φ and φ . Therelation between the tri-critical QED and scalar QED is similar to the relation betweenthe Gaussian and WF CFT.2. SU ( N c ) QCD: N f flavor of critcal bosons coupled to a SU ( N c >
1) gauge field.3. O (2 N f ) ∗ : This theory is nothing but replacing the U (1) gauge field of scalar QEDin Eq. (1) with a discrete gauge field (e.g. say Z N ). It is almost identical to the O (2 N f ) WF except only gauge invariant operators are physically allowed in O (2 N f ) ∗ .Equivalently, one can also consider branching O (2 N f ) into SU ( N f ) × U (1), and onlyconsider the U (1) neutral sector. In this branching, the O (2 N f ) symemtric rank-2traceless tensor becomes the SU ( N f ) adjoint.4. Chern-Simons (CS) gauge theories: In 3 d one can add a quantized CS term to the U (1) gauge field at any integer level N , N/ π(cid:15) µνρ a µ ∂ ν a ρ , leading to a family of paritybreaking CFTs [62]. Similarly, one can also consider QCD theories with finite CSterms.5. Generalized free field (GFF) theory: it is worth noting that there could be differentGFFs. One type of GFF (dubbed GFF-A) is made of the SU ( N f ) fundamental φ i ,meaning that the SU ( N f ) adjoint is constructed by φ i φ j . The other GFF (dubbedGFF-B) is directly made of SU ( N f ) adjoint A ij . One difference between these twoGFFs is, the OPE ( φ i φ j ) × ( φ k φ l ) in GFF-A contains φ i φ j , while A ij × A kl in GFF-Bdoes not contain A ij . We adopt the terminology in condensed matter literatures [61]. SU ( N f ) adjoint will not be able to tell the difference.The decoupling operator we identified in Sec. II B can be used to exclude SU ( N c >
1) gauge theories and GFF-B, while for other theories we need to rely on EOMs. Someconsequences of EOMs have already been discussed [36] and been used in the bootstrapanalysis [16, 63] . Here we push the idea further, in specific we will discuss 1) the consequenceof the EOM of gauge field; 2) high level spectrum due to the EOM. These results will helpus to distinguish the scalar QED from its other cousins, particularly the tri-critical QED, O (2 N f ) ∗ , and GFF-A.The scalar QED has two EOMs, D µ D µ φ i = gφ i ¯ φ j φ j = gφ i | φ | , (6) e ¯ φ i D ν φ i = ∂ µ F µν . (7)The first one is similar to the EOM of the WF CFTs, with the difference that the conventionalderivative ∂ µ is replaced by the covariant derivative D µ = ∂ µ − ia µ . For the brevity of notationwe will also write D µ D µ = (cid:3) . The second one is unique for gauge theories.These EOMs are obtained from the UV Langrangian, so one should not view them asprecise equations in the IR. The essential physics is that, due to EOMs certain seeminglyindependent operators becomes dependent, e.g. φ i and φ i | φ | . In the UV their relation isEq. (6), and under RG the precise form of the two operators as well as the coupling constant( g ) will be running, but the fact that these two operators are dependent is unchanged underRG. One can make this statement more precise, for example, in the framework of 4 − (cid:15) expansion [36]. In the extreme limit, N f = ∞ or (cid:15) = 0, the scaling operators of the theoryare simply the composites of UV operators in the Lagrangian (i.e. φ , A µ ). At a finite N f or (cid:15) , these UV operators will be dressed under RG. For the notation brevity, we will stilldenote scaling operators as composites of UV operators in the following discussion.The consequence of operators being dependent is that the IR CFT has missing operatorscompared to GFF or other CFTs that do not have the same EOM. Let us firstly spell outthe consequence of EOMs: • For a theory with the EOM A = B , any operator involving A − B , e.g. O · ( A − B ) ,becomes missing. In other words, when classifying operators, O · A and O · B shouldbe regarded as the same operator, for an arbitrary operator O . evel GFF-A Scalar QED tri-critical QED O (2 N f ) ∗ Singlet l = 0 ∆ = 2 + O ( (cid:15) ) | φ | = s | φ | = s | φ | = s | φ | = s ∆ = 4 + O ( (cid:15) ) ¯ φ i (cid:3) φ i , | φ | | φ | ¯ φ i (cid:3) φ i , | φ | | φ | Adjoint l = 0 ∆ = 2 + O ( (cid:15) ) ¯ φ i φ j = a ¯ φ i φ j = a ¯ φ i φ j = a ¯ φ i φ j = a ∆ = 4 + O ( (cid:15) ) ¯ φ i (cid:3) φ j , ¯ φ i φ j | φ | ¯ φ i φ j | φ | ¯ φ i (cid:3) φ j , ¯ φ i φ j | φ | ¯ φ i φ j | φ | Singlet l = 1 ∆ = 3 − (cid:15) ¯ φ i D µ φ i = J gµ None None ¯ φ i D µ φ j = J gµ Adjoint l = 1 ∆ = 3 − (cid:15) ¯ φ i D µ φ j = J fµ ¯ φ i D µ φ j = J fµ ¯ φ i D µ φ j = J fµ ¯ φ i D µ φ j = J fµ ∆ = 5 + O ( (cid:15) ) J fµ · s , a · ∂ µ s , a · J gµ J fµ · s , a · ∂ µ s , a · ∂ ν F µν J fµ · s , a · ∂ µ s , a · ∂ ν F µν J fµ · s , a · ∂ µ s , a · J gµ TABLE I. List of low lying primary operators in d = 4 − (cid:15) dimensions. For notational brevity weomit terms like − /N f δ ij | φ | for the operators in the adjoint representation. Table I lists low lying primary operators of the scalar QED and its cousins, including theGFF-A, tric-critical QED, O (2 N f ) ∗ in the 4 − (cid:15) limit. Comparing the GFF-A with thescalar QED, one can find that the latter has several operators missing in specific channels.These missing operators are the consequences of EOMs. For example, in the channel of SU ( N f ) singlet l = 0, there is only one primary operator | φ | in the scalar QED at thelevel ∆ = 4 + O ( (cid:15) ), as the operator ¯ φ i (cid:3) φ j becomes redundant due to the EOM in Eq. (6).Similarly, from the second EOM in Eq. (7), we know that in the scalar QED the gauge current J gµ = ¯ φ i D µ φ i becomes the descendent of F µν . Correspondingly, when counting the primaryoperator one shall skip all the operators that inolve J gµ and only count those involving ∂ ν F µν .Table I only lists the first few low lying channels, it is straightforward (although tedious)to generalize to higher levels. The operators at the same level have the same leading orderscaling dimension, but O ( (cid:15) ) corrections can lead to significant differences for (cid:15) ∼
1. Forexample, in 3 d (i.e. (cid:15) = 1) the scalar QED has ∆ s = 2 − / ( π N f ) + O (1 /N f ), while thetri-critical QED has ∆ s = 1 + 128 / (3 π N f ) + O (1 /N f ) [43]. More generally, for arbitrarydimensions in the large − N f limit, we have ∆ s = 2 + O (1 /N f ) for the scalar QED, and∆ s = d − O (1 /N f ) for the tri-critical QED. We will discuss more about scaling dimensions We will not discuss the Chern-Simons theory as it only exists in 3 d . But we note that the missing operatorsfrom EOMs we discussed here can still be used to exclude Chern-Simons theory in 3 d , as we will elaboratelater in the paper.
14f various operators in the following section in the context of numerical study.
IV. NUMERICAL RESULTS
In this section we will switch gear to numerical results. We will study the scalar QED in2 < d ≤ SU ( N f ) adjoint operators, a = ¯ φ i φ j − δ ij /N f | φ | . The OPE a × a is, a × a = Singlet + + Adjoint + + A ¯ A + + S ¯ S + + Adjoint − + S ¯ A − + A ¯ S − . (8)Here A ¯ A , S ¯ S , S ¯ A , and A ¯ S are rank-4 tensors with two upper and two lower indices. Thenaming convention of these representations is rather simple, for example A ¯ A means thatboth the upper and lower indices are anti-symmetric, while S ¯ A means that the lower indicesare symmetric and the upper indices are antisymmetric. The upper script ± means theintermediate channel has even or odd spins. For the bootstrap equations, one can checkRef. [26].We will denote the low lying scalar operators in the singlet channel as s, s (cid:48) , · · · ; scalaroperators in the adjoint channel as a, a (cid:48) · · · ; l = 1 operators in the adjoint channel as J µ , J (cid:48) µ , · · · . Besides the single correlator of a , we will also present some results of mixcorrelators of a and s . We note that a appears in the OPE of a × a , so we impose thiscondition in all the numerics, for example we require that all the scalars in the adjointchannel should be no smaller than ∆ a . Physically this gap condition does not introduce anyassumption to the CFT spectrum, but it does modify the numerical bounds significantly.Most results are calculated with Λ = 27 (the number of derivatives included in the numerics)unless stated otherwise.Before going to details, we will summarize some known results about the low lying spec-trum of the scalar QED. In 3 d , the large- N f calculation [43, 64] gives∆ a = 1 − π N f + O (1 /N f ) , (9)∆ s = 2 − π N f + O (1 /N f ) . (10)From 2 + (cid:15) expansion [37–39], one has∆ a = (cid:15) − N f (cid:15) + O ( (cid:15) ) , (11)∆ s = 2 − N f (cid:15) + O ( (cid:15) ) . (12)15 .70 0.75 0.80 0.85 0.90 0.95 1.00 1.051.52.02.53.03.54.04.5 ∆ A ¯ A ∆ a SU (10) SU (100) SU (1000) FIG. 1. The numerical bounds of the lowest operator in the A ¯ A channel for SU (10), SU (100), and SU (1000) CFTs in 3 d . The dashed line corresponds to the large − N f results of ∆ a for N f = 100and N f = 1000 scalar QED. The orange circle corresponds to (∆ a , ∆ A ¯ A ) = ( d − , d −
2) + 2).
It is also worth noting the tri-critical QED in 3 d has [43],∆ a = 1 − π N f + O (1 /N f ) , (13)∆ s = 1 + 1283 π N f + O (1 /N f ) . (14)Other results of spectrum will be discussed below when needed. A. Kinks of the A ¯ A bound As we discussed in Sec. II B, the lowest operator in the A ¯ A channel of non-Abelian gaugetheories becomes missing in the abelian gauge theories (e.g. scalar QED, tri-critical QED, O (2 N ∗ f )), so it is natural to bound A ¯ A channel gap ∆ A ¯ A to see if this operator decouplingcan be detected. Concretely, for abelian gauge theories (and GFF-A) we have∆ A ¯ A = 2( d −
2) + 2 + O (1 /N f ) , (15)while for non-Abelian gauge theories (and GFF-B) we have∆ A ¯ A = 2( d −
2) + O (1 /N f ) . (16)Fig. 1 shows the numerical bounds of ∆ A ¯ A in 3 d . The numerical bounds show clear kinksfor different N f ’s, and the kink evolves into a vertical jump from (∆ a , ∆ A ¯ A ) = (1 ,
2) to16∆ a , ∆ A ¯ A ) = (1 ,
4) as N f → ∞ . The appearance of the A ¯ A kink can be ascribed to thedecoupling operator theorem of Abelian gauge theories we discussed above. In particular, inthe large- N f limit the Abelian gauge theories are living in the space after the jump, whilethe non-Abelian gauge theories may live in the space before the jump.We note that this family of kinks is very similar to the non-WF kinks of O ( N ) theories [33].In particular, in 2d the O (4) non-WF kink exactly corresponds to the SU (2) WZW CFT.Given that the WZW CFTs’ null operators can be viewed as gauge theories’ decouplingoperators, it is very tempting to conjecture that the A ¯ A kinks here correspond to the scalarQED. A careful analysis from both the numerical and theoretical perspective suggests thatthe A ¯ A kink is unfortunately not the scalar QED. Although the 1 /N f correction of ∆ A ¯ A isunknown, we can compare ∆ a of the kinks with the large- N f results Eq. (9). In Fig. 1 wealso plot large- N f ∆ a of SU (100) and SU (1000) scalar QED, which shows considerably largediscrepancies to the kinks. Take a closer look at the data, the SU (100) kink sits around∆ a ≈ . N f results gives ∆ a ≈ . /N f . Similarly, this is also the case for SU (1000), which has ∆ a ≈ . a ≈ .
998 for the kink and large N f , respectively. This large discrepancy does not seemto be caused by a numerical convergence issue, as the differences of ∆ a between Λ = 19 , , A ¯ A kink cannot bethe scalar QED. That is because the tri-critical QED also has ∆ A ¯ A = 2( d −
2) + 2 + O (1 /N f ),and its ∆ a (Eq. (13)) is smaller than that of the scalar QED (Eq. (9)). As a side note, intheory the A ¯ A kink could be the tri-critical QED, but numerically it does not seem be soas their large- N f ∆ a ’s also have more than 2 /N f discrepancy from the numerical kink.We have also studied A ¯ A bound in other dimensions (see Fig. 2). We find that the A ¯ A kinks still exist in 2 < d ≤ a , ∆ A ¯ A ) = ( d − , d −
2) + 2) as N f → ∞ . It is worth noting that, in 4d for N f (cid:54) = ∞ the A ¯ A kink does not sit at(∆ a , ∆ A ¯ A ) = (2 , d . In d = 2 . A ¯ A kinks again have large deviations to the 2 + (cid:15) results of the scalar QED.Therefore, the single correlator can capture the essential physics of the A ¯ A decouplingfrom non-Abelian gauge theories to Abelian gauge theories. However, the A ¯ A kink does notcorrespond to any known CFT. This result inspires us that, instead of bounding ∆ A ¯ A wecan impose a gap in the A ¯ A channel to exclude all the non-Abelian gauge theories. We will17 .05 0.06 0.07 0.08 0.09 0.100.00.51.01.52.02.53.0 ∆ a ∆ A ¯ A ∆ a ∆ A ¯ A SU (4) SU (100) SU (100) SU (1000) FIG. 2. The numerical bounds of the lowest operator in the A ¯ A channel for SU ( N f ) CFTs in d = 2 . d = 4 dimensions (right). The dashed line in the left corresponds to the2 + (cid:15) results of ∆ a for N f = 4 and N f = 100 scalar QED. The orange circle corresponds to(∆ a , ∆ A ¯ A ) = ( d − , d −
2) + 2). pursue this in the remaining part of this paper.
B. Scalar QED kinks in (cid:15) dimensions
It turns out that it is easiest to bootstrap scalar QED in 2 + (cid:15) dimensions. For all resultspresented below, we add a gap ∆ A ¯ A ≥ d − A ¯ A channel unless stated otherwise.This gap excludes all the non-Abelian gauge theories (as well as the GFF-B) and is far belowthe physical gap of the Abelian gauge theories. Also numerically we have checked that thenumerical bounds are almost identical if a tighter gap (e.g. ∆ A ¯ A ≥ d −
2) + 1 . O (2 N f ) ∗ . As we discussed in Sec. III the difference between the scalar QED and tri-criticalQED/GFF is that, the former contains φ interactions, while the latter does not. Thisdifference is similar to the difference between the WF CFT and GFF/Gaussian. For the O ( N ) WF CFT, it is well known that one can detect it as a kink that is above the GFF bybounding the O ( N ) singlet [16]. So one may expect that the scalar QED would appear asa kink if one bounds the SU ( N f ) singlet ∆ s .Fig. 3 shows the numerical bounds of ∆ s of N f = 4 , which has a kink that is close to The results of different N f ’s are rather similar, so we just choose N f = 4 as a representative one. .05 0.06 0.07 0.08 0.09 0.10 0.110.51.01.52.0 (a) ∆ a ∆ s ∆ a ∆ s ∆ s ≥ ∆ s ∆ s ≥ s ≥ . s ≥ ∆ s ∆ s ≥ s ≥ . FIG. 3. The numerical bounds of SU (4) singlet ∆ s of SU (4) CFTs in d = 2 . A ¯ A ≥ d −
2) + 1 gap and the ∆ s (cid:48) gap itemizedin the figure. The green circle corresponds to scaling dimensions of the scalar QED from epsilonexpansion (∆ a , ∆ s ) ≈ (0 . , . the 2 + (cid:15) expansion results of scalar QED . We further impose a gap in the second lowlying singlet ∆ s (cid:48) ≥ , . s . The ∆ s (cid:48) gap carves out a largeregion, leaving a sharp tip where the scalar QED sits in. This phenomenon is similar to thatof Ising CFT, for which imposing further constraints will carve the feasible region into asmall island [14]. In the following subsection we will show that the feasible region of scalarQED also shrinks to an island with proper condition imposed.It is good to pause here to elaborate a bit more on the philosophy of imposing gapconditions in the bootstrap calculation. As we have explained, in many cases, in particularfor gauge theories, it is necessary to impose gaps in order to exclude other theories that arealso consistent with crossing equations. On the other hand, in bootstrap calculation it iscommon that imposing gaps will carve out feasible regions, possibly leaving a kink on thenumerical bounds. Sometimes, the kink is floating, namely it is moving as the gap changes The discrepancy is of order O ( (cid:15) ) and O ( (cid:15) ) for ∆ a and ∆ s , respectively. A ¯ A ) is in a finite window. We have explicitly checked that the kink andnumerical bounds are almost identical for different values of gap, i.e. ∆ A ¯ A ≥ d −
2) + 1and ∆ A ¯ A ≥ d −
2) + 1 .
5. On the other hand, if one removes the ∆ A ¯ A gap, ∆ s boundgets modified significantly (the black curve in Fig. 3(a)): The scalar QED kink disappears,but there is one kink close to the unitary bound (of ∆ a ) which is likely to be a WF typetheory. These results justify our decoupling operator based recipes for bootstrapping gaugetheories, in specific the ∆ A ¯ A gap is serving to exclude all the non-Abelian gauge theories.We also remark that there is a vertical kink on the leftmost feasible region. It correspondsto ∆ A ¯ A jump (Fig. 1-2) reported in the previous subsection. It is noticeable that ∆ s is prettysmall in this region, supporting again that the ∆ A ¯ A kink (jump) cannot be the scalar QED.It will be interesting to know if the tri-critical QED lives in any special region (e.g. theleftmost kink) of the numerical bounds. C. Scalar QED islands in (cid:15) dimensions
To get an island of the scalar QED, we need to find conditions to exclude all its cousins.As shown above, the gap ∆ A ¯ A ≥ d −
2) + 1 can exclude all the non-Abelian gauge theoriesand GFF-B; ∆ s (cid:48) ≥ O (2 N ∗ f ) which lives on the right hand side of scalar QED. Comparing these two theoriesin Table I, one can find that due to the EOM Eq. (7) the scalar QED has missing operatorsrelated to the gauge current J gµ = ¯ φ i D µ φ i . This will be the key for us to exclude O (2 N f ) ∗ .We now conjecture several expressions for scaling dimensions of various operators bynaively summing up the scaling dimensions of its constituents. It in principle can be verifiedby the large- N f or (cid:15) computation. For the O (2 N ∗ f ), the second lowest operator ( J (cid:48) µ ) in theadjoint, l = 1 channel is J gµ · a (in either 2 + (cid:15) limit or large- N f limit),∆ J (cid:48) µ = ∆ J gµ · a = d − a + O ( (cid:15) ) · O (1 /N f ) = 2 d − O ( (cid:15) ) · O (1 /N f ) . (17)20 .00990 0.00995 0.01000 0.01005 0.010101.99941.99961.99982.00002.0002 ∆ a ∆ s (a) (b) Λ = 19Λ = 27 Λ = 35 ∆ a ∆ s FIG. 4. The islands of the scalar QED with N f = 4. The green circles mark 2 + (cid:15) results of thescalar QED. (a) d = 2 .
01 dimensions: the feasible regions are obtained from the single correlator.(b) d = 2 . a , s mixed correlator. In contrast, for the scalar QED we have∆ J fµ · s = d + 1 + O ( (cid:15) ) · O (1 /N f ) , ∆ a · ∂ µ s = d + 1 + O ( (cid:15) ) · O (1 /N f ) , ∆ a · ∂ ν F µν = d + 1 + O ( (cid:15) ) · O (1 /N f ) , (18)Therefore, if we impose a gap for ∆ J (cid:48) µ in d < O (2 N f ) ∗ . We also remark that this gap can also exclude Chern-Simons theory in 3 d . InChern-Simons theory J (cid:48) µ = a · ε µνρ F νρ , whose scaling dimension is ∆ = 3 + O (1 /N f ). In thescalar QED such operator also exists, but it is a parity odd operator, hence will not appearin the a × a OPE.In d = 2 .
01 dimensions (Fig. 4(a)), we successfully isolate the scalar QED into a smallisland with the single correlator by imposing the gap conditions as shown in Table II. Aswe have detailed above, the first three gaps have very clear physical meanings, they serve toexclude non-Abelian gauge theories, tri-critical QED/GFF, and O (2 N ∗ f ). The last gap ∆ S ¯ S is rather mysterious, we do not have a clear idea what theory does it exclude. Removing anyof these four gaps, the scalar QED will not be isolated to an island any more. Somewhatsurprisingly, by increasing the dimensions slightly, say d = 2 .
1, the single correlator cannot isolate an island any more. The mixed correlator can still yield an island with a high21 ap imposed Scalar QED Tri-critical QED GFF-A O (2 N ∗ f ) QCD∆ A ¯ A d −
2) + 1 2( d −
2) + 2 2( d −
2) + 2 2( d −
2) + 2 2( d −
2) + 2 2( d − s (cid:48) d −
2) 2( d −
2) 4 4∆ J (cid:48) µ d − . d + 1 2 d − d − d − d + 1∆ S ¯ S ∆ a a a a a a TABLE II. The imposed gaps for the scalar QED island in Fig. 4(a) and the physical gaps of differ-ent theories. We note that for these physical gaps, the values are up to O ( (cid:15) ) O (1 /N f ) corrections,where (cid:15) = d − Λ = 35 and more aggressive (but still physical) gap conditions (Fig. 4(b)), i.e., ∆ A ¯ A ≥ d −
2) + 1, ∆ s (cid:48) ≥ .
5, ∆ J (cid:48) µ ≥ d + 0 .
5, ∆ S ¯ S ≥ ∆ a , ∆ a (cid:48) ≥ ∆ a + 1 . O ( N ) WF intoan island [65]. The mixed correlator bootstrap, on the other hand, is certainly more powerfulthan the single correlator bootstrap for several reasons. Firstly, the mixed correlator canaccess new decoupling/missing operators that are absent in the single correlator bootstrap.For instance, the mixed correlator of O ( N ) vector and singlet can detect the missing operatorin the O ( N ) vector channel, i.e., φ | φ | [16]. Secondly, the mixed correlator has strongerconstraining power and better numerical convergence, as illustrated in our numerical resultsof Fig. 4(b).To have a more intuitive idea about the magic of EOMs, we further investigate how thebound of ∆ J (cid:48) µ evolves with ∆ a . As shown in Fig. 5, the scalar QED sits at a sharp spike, whichis well separated from O (2 N f ) ∗ . This agrees with our conjectures of ∆ J (cid:48) µ in Eq. (17) andEq. (18), that is a consequence of the EOM in Eq. (7). The sharp spike also explains why thegap of ∆ J (cid:48) µ helps to isolate the scalar QED into an island. Another noteworthy observation isthat convergence quickly becomes difficult as the dimension d increases slightly. In d = 2 . Λ = 27 does not produce an island. .010 0.011 0.012 0.013 0.0140.00.51.01.52.0 ∆ a ∆ J µ − ∆ J µ ∆ a ∆ J µ − ∆ J µ (a) (b) Λ = 11 Λ = 19 Λ = 11 Λ = 19 Λ = 27
FIG. 5. The numerical bounds of ∆ J (cid:48) µ of SU (4) CFTs with gap ∆ A ¯ A ≥ d −
2) + 1, ∆ s ≥ S ¯ S ≥ ∆ a . The shaded regions are allowed regions. The green circles mark the scalar QED andthe green stars mark O (2 N f ) ∗ , up to O ( (cid:15) ) and O ( (cid:15) ) corrections for ∆ a and ∆ J (cid:48) µ , respectively a .(a) d = 2 .
01 dimensions, (b) d = 2 .
07 dimensions. a In O (2 N f ) ∗ , a (i.e. SU ( N f ) adjoint) corresponds to the rank-2 symmetric traceless tensor of the O (2 N f )WF CFT. Its scaling dimension from 2 + (cid:15) expansion is ∆ a = N f (cid:15) N f − − N f (cid:15) ( − N f ) + O ( (cid:15) ) [66, 67]. dimensions (Fig. 5(a)) ∆ J (cid:48) µ has a sharper spike for a small Λ = 11, and a larger Λ = 19 doesnot improve the bound significantly. In contrast, in d = 2 .
07 dimensions Λ = 11 does notproduce a spike at all, while the spike shows up weakly for Λ = 19 and becomes sharper forΛ = 27. Moving to a higher dimension (e.g. d = 2 .
1) the spike does not show up even forΛ = 27 (the feasible region looks similar to that of d = 2 .
07 with Λ = 11 in Fig. 5(b)). Thisalso explains why the single correlator does not produce an island in d = 2 . N f , e.g. ∆ J (cid:48) µ stillhas a spike in d = 2 . SU (100) with Λ = 19. D. Towards the d scalar QED Finally, we move towards d = 3 dimensions. In a given dimension there exists a critical N ∗ f ( d ) below which the scalar QED becomes gapped. It remains an open questions about23 .097 0.098 0.099 0.100 0.101 0.1020.51.01.52.0 ∆ a ∆ s Λ = 27 Λ = 27Λ = 19 d = 2 . d = 2 . Λ = 27Λ = 19Λ = 27Λ = 19 ∆ a ∆ s d = 2 . d = 3∆ a ∆ s ∆ a ∆ s (a) (b)(c) (d) FIG. 6. The numerical bounds of SU ( N f ) singlet ∆ s of SU (100) CFTs in d = 2 . d = 2 . d = 2 . d = 3 (d) dimensions. The data is obtained with a gap condition ∆ A ¯ A ≥ d − A ¯ A gap, e.g. ∆ A ¯ A ≥ d − . (cid:15) expansion results (∆ a , ∆ s ) ≈ (0 . , . N f results (∆ a , ∆ s ) ≈ (0 . , . the precise value of N ∗ f in d = 3 dimensions. To avoid the unnecessary complexity, we choosea large N f = 100 to monitor how the scalar QED kink evolves as the dimension increases.Fig. 6 shows the numerical bounds of ∆ s in d = 2 . , . , . , A ¯ A kink discussed in Sec. IV A, and does not correspond to the scalar QED. In d = 2 . N f = 4 in Fig. 3 the numerical bound has a sharp kinkthat is close to the 2 + (cid:15) result (∆ a , ∆ s ) = (0 . , . d increases,the scalar QED kink becomes weak in d = 2 . d = 2 . d = 3 dimensions (Fig. 6(d)).It is unclear that why the scalar QED kink disappears for d ’s close to 3 . One possible The scalar QED kink being disappearing shall not be ascribed to the physics of fixed point annihilation .097 0.098 0.099 0.100 0.101 0.1020.51.01.52.0 ∆ s ∆ a (a) (b) 0.94 0.96 0.98 1.00 1.02 1.041.52.02.53.03.54.0 ∆ s ∆ a ∆ S ¯ S ≥ d/ − S ¯ S ≥ ∆ a ∆ S ¯ S ≥ d/ − S ¯ S ≥ ∆ a FIG. 7. The numerical bounds of SU ( N f ) singlet ∆ s of SU (100) CFTs in d = 2 . d = 3 (b) dimensions. The green circles mark the scalar QED: (a) it corresponds to the 2 + (cid:15) expansion results (∆ a , ∆ s ) ≈ (0 . , . N f results (∆ a , ∆ s ) ≈ (0 . , . A ¯ A ≥ d −
2) + 1, ii) ∆ A ¯ A ≥ d −
2) + 1 and ∆ S ¯ S ≥ ∆ a . The feasible regions do not changeunder tighter (but still physical) conditions, e.g. ∆ A ¯ A ≥ d −
2) + 1 . S ¯ S ≥ . a . explanation is that the numerical convergence becomes harder as d increases, which can beclearly seen by comparing the numerical bounds of Λ = 19 and Λ = 27 in Fig. 6(b)-(d). Itis also worth noting that, in d = 3 dimensions, the numerical bound of ∆ s is much largerthan the value (∆ ≈
2) of the scalar QED. However, based on our numerical data there isno indication that the scalar QED kink will show up in d = 3 dimensions as Λ → ∞ .A curious observation is that, in d = 3 dimensions the numerical bounds are improvedsignificantly by imposing a mild gap ∆ S ¯ S ≥ ∆ a , as shown in Fig. 7(b). In contrast,in d = 2 . S ¯ S ≥ ∆ a the numerical bounds are onlyimproved a little, and the position of the kink does not move. On the other hand, thenumerical bounds (for both d = 2 . d = 3) are not further improved under a tightergap condition, e.g. ∆ S ¯ S ≥ . a . From the Extremal Functional Method (EFM) [68] wefind that on the boundary of feasible region one roughly has ∆ S ¯ S ≈ a , i.e., a relation as N f = 100 shall be large enough the the scalar QED being conformal in d = 3 dimensions. We note that this gap can be further relaxed, but we have not examined it carefully to find the mostoptimal gap condition. (cid:15) dimensions it is necessary to impose this mysterious gap ∆ S ¯ S ≥ ∆ a . These observationssuggest that this gap excludes some crossing symmetric solutions for the bootstrap equations,but we are not able to identify any candidate theory. Nevertheless, in d = 3 dimensions withthis extra gap imposed the scalar QED kink still does not show up , and the numericalbounds of ∆ s are still higher than that of the scalar QED. It is possible that one needs toexclude other theories by imposing extra gap conditions in order to spot the scalar QEDkink in d = 3 dimensions. We leave this for future exploration. V. CONCLUSION AND OUTLOOK
We have introduced the notion of decoupling operators of gauge theories in dimensions d >
2. The decoupling operator is the higher dimensional reminiscent of null operators of2d WZW CFTs, and it can efficiently detect the rank of the gauge group. Based on theinformation of decoupling operators, one can then impose gap conditions in the bootstrapcalculation to isolate interested gauge theories from other theories. As an illustrative ex-ample, we study a prototypical gauge theory, i.e., the scalar QED. We firstly identified theconcrete decoupling operators of the scalar QED, and then showed how to use them in abootstrap study. In 2 + (cid:15) dimensions, we have successfully solved it with the numericalbootstrap, namely we have obtained a kink as well as an island of the scalar QED, by im-posing mild gap conditions inspired by the physics of decoupling operators and EOMs. Thesuccess, however, does not sustain to the most interesting case, i.e., d = 3 dimensions. Thefailure in d = 3 dimensions might be due to the poor numerical convergence or some keymissing ingredients (i.e. gap conditions). We will leave this for the future study.An interesting question is what does the A ¯ A kink in Fig. 1 and Fig. 2 correspond to?This family of kinks shares a lot of similarities as the vertical jump in the bound of rank-2symmetric tracless tensor of the O ( N ) theories (this kink was dubbed non-WF kink) [33].Also a similar kink was recently observed in bootstrapping O ( N ) rank-2 symmetric tracelesstensor [60]. We believe these kinks may have similar physical mechanisms. They could eitherbe unknown CFTs or artifacts of numerical bootstrap. Even if they are numerical artifacts,the crossing symmetric solution at the kink may have certain relations to gauge theories, The leftmost kink corresponds to the A ¯ A kink, which shall not be the scalar QED as we explained earlier. d .In this paper we showed how to use the decoupling operator in the A ¯ A channel to boot-strap U (1) gauge theories. In a similar fashion, one can bootstrap a non-Abelian gaugetheory with a specific gauge group U ( N c = m ) by using the decoupling operators in theantisymmetric representations A [ f , ··· ,f n +1 ][ f n +2 , ··· ,f n +2 ] of SU ( N f ) with n ≤ m . For example, in thechannel A [ f , ··· ,f m +1 ][ f m +2 , ··· ,f m +2 ] the lowest operator of different gauge theories will have distinct scal-ing dimensions: 1) the U ( N c > m ) gauge theories have ∆ = ( m + 1)( d −
2) + O (1 /N f ); 2) the U ( N c = m ) gauge theories have ∆ = ( m + 1)( d −
2) + 2 + O (1 /N f ); 3) the U ( N c < m ) gaugetheories have ∆ ≥ ( m + 1)( d −
2) + 4 + O (1 /N f ). We also remark that as a concrete examplewe analyzed decoupling operators of theories with a U ( N c ) gauge field coupled to bosons. Itis straightforward to generalize to other gauge groups (e.g. SU ( N c ), SO ( N c ), U Sp (2 N c )) aswell as fermions coupled to gauge fields. It will be interesting to try our decoupling operatorbased recipes to tackle other gauge theories. In particular, exciting progress might be madeby using advanced bootstrap techniques such as mixing spinning operators [69].On the phenomenological level the decoupling operators of gauge theories share severalsimilarities with the null operators of 2 d WZW CFTs. As detailed in Sec. II A the nulloperators of SU ( N ) k WZW CFTs can be even considered as decoupling operators of 2 d U ( k )gauge theories. In the context of 2 d CFTs the null operator has important applications, e.g.they can act as differential operators that annihilate correlation functions. It is an openquestion whether a similar application also exists for the decoupling operators of gaugetheories in a general dimension. The progress might be made by looking for an exactinterpolation between gauge theories with different gauge groups, which is similar to theinterpolation between WZW CFTs with different WZW levels.
ACKNOWLEDGMENTS
YCH would like to thank Chong Wang and Liujun Zou for the stimulating discussions andcollaborations on 3 d WZW models, which benefit current work. We thank Slava Rychkov forhis critical reading of our manuscript and for his various suggestions. Research at Perime-ter Institute is supported in part by the Government of Canada through the Departmentof Innovation, Science and Industry Canada and by the Province of Ontario through the27inistry of Colleges and Universities. This project has received funding from the Euro-pean Research Council (ERC) under the European Union’s Horizon 2020 research and in-novation programme (grant agreement no. 758903). The work of J.R. is supported bythe DFG through the Emmy Noether research group The Conformal Bootstrap Programproject number 400570283. The numerics is solved using SDPB program [70] and sim-pleboot (https://gitlab.com/bootstrapcollaboration/simpleboot). The computations in thispaper were run on the Symmetry cluster of Perimeter institute, and on the EPFL SCITAScluster funded by the Swiss National Science Foundation under grant no. PP00P2-163670.NS would like to thank his parents for support during the COVID-19 pandemic. NS wouldlike to thank the hospitality of Institute of Physics Chinese Academy of Sciences and theCenter for Advanced Study, Tsinghua University while part of the work was finished.28 ppendix A: 3d WZW models and gauge theories
In this appendix, we will discuss some examples that show direct connections betweenWZW CFTs and 3d gauge theories. The physics discussed here is not new, it is the recol-lection of the results in Ref. [58, 59].Despite of the pure algebraic definition, 2d WZW CFTs also have a Lagrangian formu-lation, namely a non-linear sigma model (NL σ M) on a (Lie) group manifold G ( SU ( N ), U Sp (2 N ), etc.) supplemented with a level k WZW term [55], L = 14 a (cid:90) d x Tr( ∂ µ g − )( ∂ µ g ) + k · i π (cid:15) µνρ (cid:90) B d x Tr((ˆ g − ∂ µ ˆ g )(ˆ g − ∂ ν ˆ g )(ˆ g − ∂ ρ ˆ g )) . (A1) g is a matrix field valued in a unitary presentation of the Lie group. The first term isthe ordinary kinetic term of NL σ M, the second term is the WZW term defined in the 3-dimensional extended space. k is quantized and corresponds to the homotopy class π ( G ) = Z . One shall also have π ( G ) = in order for the WZW term to be well defined. TheLagrangian has a conformal fixed point (i.e. WZW CFT) at a finite coupling strength.It is straightforward to generalize the WZW Lagrangian to a higher dimension. In 3da non-trivial WZW term requires the target space G to satisfy π ( G ) = Z and π ( G ) = . There are several target spaces, including Grassmannian and Stiefel manifold (e.g. SO ( N ) /SO (4)), satisfying this requirement. One important difference in 3 d is that theNL σ M is non-renormalizable, making it hard to analyze . Nevertheless, it was arguedthat [58] , there are three fixed points as the coupling strength a increases from 0:1. An attractive fixed point of spontaneous symmetry breaking (SSB) phase at a = 0.The ground state manifold is the target space of NL σ M.2. A repulsive fixed point of order-disorder phase transitions.3. An attractive conformal fixed point preserving all the symmetries.The last attractive conformal fixed point is the 3 d version of the 2 d WZW CFT, while thefirst two fixed points merge into the Gaussian fixed point in 2 d .Ref. [58] studied such 3 d WZW models on the Stiefel manifold, here we discuss a simplersituation–the 3 d Grassmannian U (2 N ) / ( U ( N ) × U ( N )) WZW models [59]. In particular, A theory being non-renormalizable does not necessarily mean it is non-sensible. For the context of NL σ M,we know that it can describe the WF CFTs although it is non-renormalizable in d > Ref. [58] studied Stiefel manifold, but it should be readily generalized to other manifold.
29e will argue that the Grassmannian WZW models have simple UV completions, i.e., Diracfermions coupled to a gauge field.The UV completion of the 3 d leve- k U (2 N ) / ( U ( N ) × U ( N )) WZW model is the QCD -Gross-Neveu model, L = N (cid:88) i =1 ¯ ψ i γ µ ( ∂ µ − iα µ ) ψ i + λφ ij (cid:18) ¯ ψ j ψ i − N δ ji ¯ ψψ (cid:19) + m Tr( φ )+ u Tr( φ )+ u (Tr( φ )) . (A2)Here α µ is a SU ( k ) gauge field, ψ i Dirac fermions are in the SU ( k ) fundamental presentation. φ ij is a bosonic field in the SU (2 N ) adjoint representation, and it is coupled to the adjointmass term of the Dirac fermions.The QCD -Gross-Neveu model model has three fixed points, corresponding to a SSBphase with ground state manifold U (2 N ) / ( U ( N ) × U ( N )), QCD -Gross-Neveu CFT, andQCD CFT. The QCD -Gross-Neveu CFT fixed point is unstable, and will flow to eitherthe SSB or the QCD CFT depending on the sign of m Tr( φ ). This phase diagram coincideswith that of U (2 N ) / ( U ( N ) × U ( N )) WZW models. In the SSB phase of the QCD -Gross-Neveu model, one can define a NL σ M model on the target space U (2 N ) / ( U ( N ) × U ( N )).In the SSB phase, the Dirac fermions are gapped, integrating out of them will generate alevel- k WZW term [71]. The level k (instead of 1) comes from the color multiplicity of Diracfermions due to the SU ( k ) gauge field. Therefore, we have proved that the SSB fixed pointof the QCD -Gross-Neveu model and the level- k U (2 N ) / ( U ( N ) × U ( N )) WZW are dual toeach other.Given that phase diagrams of two models match and the SSB phase of two models aredual, it is natural to conjecture that the QCD -Gross-Neveu model is the UV completion of3d WZW model on U (2 N ) / ( U ( N ) × U ( N )) manifold. In particular, • The IR conformal fixed point of the d level − k U (2 N ) / ( U ( N ) × U ( N )) WZW modelis dual to the QCD CFTs with N f = 2 N Dirac fermions coupled to a SU ( k ) gaugefield. There is an interesting sanity check for this duality. The Grassmannian U (2 N ) / ( U ( N ) × U ( N )) has a nontrivial π = Z leading to Skyrmion operators. The Skyrmion is either aboson or fermion depending on the evenness and oddness of k [59]. The Skyrmion can beidentified as the baryon operator of the SU ( k ) gauge theory, whose statistics also dependson k . 30imilarly, one can derive, • The IR conformal fixed point of the d level − k SO (2 N ) / ( SO ( N ) × SO ( N )) WZWmodel is dual to the QCD CFTs with N f = 2 N Dirac fermions coupled to a SO ( k ) gauge field. • The IR conformal fixed point of the d level − k U Sp (4 N ) / ( U Sp (2 N ) × U Sp (2 N )) WZW model is dual to the QCD CFTs with N f = 2 N Dirac fermions coupled to a
U Sp (2 k ) gauge field. Appendix B: Floating kinks v.s. stable kinks ∆ a ∆ s ∆ S ¯ S ≥ .
05 ∆ S ¯ S ≥ .
15 ∆ S ¯ S ≥ .
17 ∆ S ¯ S ≥ . A ¯ A ≥ . A ¯ A ≥ . A ¯ A ≥ . ∆ a ∆ s (b)(a) FIG. 8. Floating kinks versus stable kinks in d = 2 . SU (4).(a) Example of floating kinks. (b) Example of stable kinks. The feasible regions calculated withthe gap conditions ∆ A ¯ A ≥ . A ¯ A ≥ . In this appendix, we compare numerical results of floating kinks and stable kinks. As wehave explained in the main text, the floating kink means the kink is moving as the imposedgap changes, while the stable kink means that the kink is not moving as long as the imposedgap lies in a finite window. Fig. 8 shows a concrete comparison between floating kinks and31table kinks. The floating kinks in Fig. 8(a) clearly show dependence on the values of ∆ S ¯ S gap. In contrast, the stable kinks in Fig. 8(b) show little dependence on the value of thegap. [1] N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theories,” Nucl.Phys. B , 129–146 (1995), arXiv:hep-th/9411149.[2] Juan Martin Maldacena, “The Large N limit of superconformal field theories and supergrav-ity,” Int. J. Theor. Phys. , 1113–1133 (1999), arXiv:hep-th/9711200.[3] Markus A. Luty and Takemichi Okui, “Conformal technicolor,” JHEP , 070 (2006),arXiv:hep-ph/0409274.[4] T. Senthil, Ashvin Vishwanath, Leon Balents, Subir Sachdev, and Matthew P. A. Fisher,“Deconfined quantum critical points,” Science , 1490 (2004).[5] T. Senthil, Leon Balents, Subir Sachdev, Ashvin Vishwanath, and Matthew P. A. Fisher,“Quantum criticality beyond the landau-ginzburg-wilson paradigm,” Phys. Rev. B , 144407(2004).[6] Michael Hermele, T. Senthil, and Matthew P. A. Fisher, “Algebraic spin liquid as the motherof many competing orders,” Phys. Rev. B , 104404 (2005), arXiv:cond-mat/0502215 [cond-mat.str-el].[7] Michael Hermele, Ying Ran, Patrick A. Lee, and Xiao-Gang Wen, “Properties of an algebraicspin liquid on the kagome lattice,” Phys. Rev. B , 224413 (2008), arXiv:0803.1150 [cond-mat.str-el].[8] Xue-Yang Song, Chong Wang, Ashvin Vishwanath, and Yin-Chen He, “Unifying descriptionof competing orders in two-dimensional quantum magnets,” Nature Communications , 4254(2019), arXiv:1811.11186 [cond-mat.str-el].[9] J. K. Jain, S. A. Kivelson, and Nandini Trivedi, “Scaling theory of the fractional quantumhall effect,” Phys. Rev. Lett. , 1993 (1990).[10] Steven Kivelson, Dung-Hai Lee, and Shou-Cheng Zhang, “Global phase diagram in the quan-tum hall effect,” Phys. Rev. B , 2223 (1992).[11] Wei Chen, Matthew P. A. Fisher, and Yong-Shi Wu, “Mott transition in an anyon gas,” Phys.Rev. B , 13749 (1993).
12] Jong Yeon Lee, Chong Wang, Michael P. Zaletel, Ashvin Vishwanath, and Yin-Chen He,“Emergent Multi-Flavor QED at the Plateau Transition between Fractional Chern Insu-lators: Applications to Graphene Heterostructures,” Physical Review X , 031015 (2018),arXiv:1802.09538 [cond-mat.str-el].[13] Riccardo Rattazzi, Vyacheslav S. Rychkov, Erik Tonni, and Alessandro Vichi, “Boundingscalar operator dimensions in 4D CFT,” Journal of High Energy Physics , 031 (2008),arXiv:0807.0004 [hep-th].[14] Filip Kos, David Poland, and David Simmons-Duffin, “Bootstrapping Mixed Correlators inthe 3D Ising Model,” JHEP , 109 (2014), arXiv:1406.4858 [hep-th].[15] Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David Simmons-Duffin,and Alessandro Vichi, “Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents,” J. Stat. Phys. , 869 (2014), arXiv:1403.4545[hep-th].[16] Filip Kos, David Poland, David Simmons-Duffin, and Alessandro Vichi, “Bootstrapping theO(N) Archipelago,” JHEP , 106 (2015), arXiv:1504.07997 [hep-th].[17] Filip Kos, David Poland, David Simmons-Duffin, and Alessandro Vichi, “Precision Islands inthe Ising and O ( N ) Models,” JHEP , 036 (2016), arXiv:1603.04436 [hep-th].[18] David Simmons-Duffin, “The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT,”JHEP , 086 (2017), arXiv:1612.08471 [hep-th].[19] Junchen Rong and Ning Su, “Bootstrapping minimal N = 1 superconformal field theory inthree dimensions,” (2018), arXiv:1807.04434 [hep-th].[20] Alexander Atanasov, Aaron Hillman, and David Poland, “Bootstrapping the Minimal 3DSCFT,” JHEP , 140 (2018), arXiv:1807.05702 [hep-th].[21] Luca Iliesiu, Filip Kos, David Poland, Silviu S. Pufu, David Simmons-Duffin, and RanYacoby, “Bootstrapping 3D fermions,” Journal of High Energy Physics , 120 (2016),arXiv:1508.00012 [hep-th].[22] Luca Iliesiu, Filip Kos, David Poland, Silviu S. Pufu, and David Simmons-Duffin, “Boot-strapping 3D fermions with global symmetries,” Journal of High Energy Physics , 36(2018), arXiv:1705.03484 [hep-th].[23] Shai M. Chester, Walter Landry, Junyu Liu, David Poland, David Simmons-Duffin, Ning Su,and Alessandro Vichi, “Carving out OPE space and precise O (2) model critical exponents,” , 015002 (2019),arXiv:1805.04405 [hep-th].[26] Yu Nakayama, “Bootstrap experiments on higher dimensional cfts,” International Journal ofModern Physics A , 1850036 (2018).[27] Shai M. Chester and Silviu S. Pufu, “Towards bootstrapping qed3,” Journal of High EnergyPhysics (2016), 10.1007/jhep08(2016)019.[28] Zhijin Li, “Solving qed with conformal bootstrap,” (2018), arXiv:1812.09281 [hep-th].[29] Zhijin Li and David Poland, “Searching for gauge theories with the conformal bootstrap,”(2020), arXiv:2005.01721 [hep-th].[30] Ethan Dyer, M´ark Mezei, and Silviu S. Pufu, “Monopole Taxonomy in Three-DimensionalConformal Field Theories,” arXiv e-prints , arXiv:1309.1160 (2013), arXiv:1309.1160 [hep-th].[31] Vyacheslav S. Rychkov and Alessandro Vichi, “Universal Constraints on Conformal OperatorDimensions,” Phys. Rev. D , 045006 (2009), arXiv:0905.2211 [hep-th].[32] Tomoki Ohtsuki, Applied Conformal Bootstrap , Ph.D. thesis, University of Tokyo (2016).[33] Yin-Chen He, Junchen Rong, and Ning Su, “Non-wilson-fisher kinks of o ( n ) numericalbootstrap: from the deconfined phase transition to a putative new family of cfts,” (2020),arXiv:2005.04250 [hep-th].[34] Connor Behan, “Unitary subsector of generalized minimal models,” Phys. Rev. D , 094020(2018).[35] Paul Ginsparg, “Applied Conformal Field Theory,” arXiv e-prints , hep-th/9108028 (1988),arXiv:hep-th/9108028 [hep-th].[36] Slava Rychkov and Zhong Ming Tan, “The (cid:15) -expansion from conformal field theory,” Journalof Physics A Mathematical General , 29FT01 (2015), arXiv:1505.00963 [hep-th].[37] ID Lawrie and C Athrone, “Phase transitions in nonlinear abelian higgs models,” Journal ofPhysics A: Mathematical and General , L587 (1983).[38] S Hikami, “Three-loop ß-functions of non-linear σ models on symmetric spaces,” Physics etters B , 208 (1981).[39] Shinobu Hikami, “Renormalization group functions of cpn-1 non-linear σ -model and n-component scalar qed model,” Progress of Theoretical Physics , 226 (1979).[40] Ganpathy Murthy and Subir Sachdev, “Action of hedgehog instantons in the disordered phaseof the (2+ 1)-dimensional cpn- 1 model,” Nucl. Phys. B , 557 (1990).[41] B. I. Halperin, T. C. Lubensky, and Shang-keng Ma, “First-order phase transitions in super-conductors and smectic- a liquid crystals,” Phys. Rev. Lett. , 292 (1974).[42] Adam Nahum, J.T. Chalker, P. Serna, M. Ortuno, and A. M. Somoza, “Deconfined quan-tum criticality, scaling violations, and classical loop models,” Physical Review X (2015),10.1103/physrevx.5.041048.[43] Sergio Benvenuti and Hrachya Khachatryan, “Easy-plane QED ’s in the large N f limit,”Journal of High Energy Physics , 214 (2019), arXiv:1902.05767 [hep-th].[44] Victor Gorbenko, Slava Rychkov, and Bernardo Zan, “Walking, weak first-order transitions,and complex CFTs,” Journal of High Energy Physics , 108 (2018), arXiv:1807.11512[hep-th].[45] C. Dasgupta and B. I. Halperin, “Phase transition in a lattice model of superconductivity,”Phys. Rev. Lett. , 1556 (1981).[46] Michael E Peskin, “Mandelstam-’t hooft duality in abelian lattice models,” Annals of Physics , 122 (1978).[47] Anders W. Sandvik, “Continuous quantum phase transition between an antiferromagnet and avalence-bond solid in two dimensions: Evidence for logarithmic corrections to scaling,” Phys.Rev. Lett. , 177201 (2010).[48] Ribhu K. Kaul and Anders W. Sandvik, “Lattice model for the SU( n ) n´eel to valence-bondsolid quantum phase transition at large n ,” Phys. Rev. Lett. , 137201 (2012).[49] Claudio Bonati, Andrea Pelissetto, and Ettore Vicari, “Lattice Abelian-Higgs model withnoncompact gauge fields,” arXiv e-prints , arXiv:2010.06311 (2020), arXiv:2010.06311 [cond-mat.stat-mech].[50] Anders W. Sandvik, “Evidence for deconfined quantum criticality in a two-dimensional heisen-berg model with four-spin interactions,” Physical Review Letters (2007), 10.1103/phys-revlett.98.227202.[51] Roger G. Melko and Ribhu K. Kaul, “Scaling in the fan of an unconventional quantum critical oint,” Physical Review Letters (2008), 10.1103/physrevlett.100.017203.[52] A. B. Kuklov, M. Matsumoto, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, “Deconfinedcriticality: Generic first-order transition in the su(2) symmetry case,” Phys. Rev. Lett. ,050405 (2008).[53] Adam Nahum, P. Serna, J.T. Chalker, M. Ortuno, and A.M. Somoza, “Emergent so(5)symmetry at the N´eel to valence-bond-solid transition,” Physical Review Letters (2015),10.1103/physrevlett.115.267203.[54] Andrea Manenti and Alessandro Vichi, “Exploring SU ( N ) adjoint correlators in 3 d ,” to ap-pear .[55] Philippe Di Francesco, Pierre Mathieu, and David S´en´echal, Conformal field theory (SpringerScience & Business Media, 2012).[56] Julian Schwinger, “Gauge invariance and mass. ii,” Phys. Rev. , 2425 (1962).[57] E. Abdalla, “Two-dimensional Quantum Field Theory, examples and applications,” arXive-prints , hep-th/9704192 (1997), arXiv:hep-th/9704192 [hep-th].[58] Liujun Zou, Yin-Chen He, and Chong Wang, “Stiefel liquids: possible non-Lagrangian quan-tum criticalities from intertwined orders,” to appear .[59] Zohar Komargodski and Nathan Seiberg, “A symmetry breaking scenario for QCD ,” Journalof High Energy Physics , 109 (2018), arXiv:1706.08755 [hep-th].[60] Marten Reehorst, Maria Refinetti, and Alessandro Vichi, “Bootstrapping traceless symmetric O ( N ) scalars,” arXiv e-prints , arXiv:2012.08533 (2020), arXiv:2012.08533 [hep-th].[61] Subir Sachdev, “Quantum phase transitions,” Handbook of Magnetism and Advanced Mag-netic Materials (2007).[62] Xiao-Gang Wen and Yong-Shi Wu, “Transitions between the quantum hall states and insula-tors induced by periodic potentials,” Phys. Rev. Lett. , 1501 (1993).[63] Junchen Rong and Ning Su, “Bootstrapping the N = 1 wess-zumino models in three dimen-sions,” (2019), arXiv:1910.08578 [hep-th].[64] Ribhu K. Kaul and Subir Sachdev, “Quantum criticality of u(1) gauge theories with fermionicand bosonic matter in two spatial dimensions,” Phys. Rev. B , 155105 (2008).[65] Zhijin Li and Ning Su, “3D CFT Archipelago from Single Correlator Bootstrap,” arXiv e-prints , arXiv:1706.06960 (2017), arXiv:1706.06960 [hep-th].[66] E. Br´ezin and J. Zinn-Justin, “Renormalization of the nonlinear σ model in 2+ (cid:15) dimensions— pplication to the heisenberg ferromagnets,” Phys. Rev. Lett. , 691 (1976).[67] E. Br´ezin, J. Zinn-Justin, and J. C. Le Guillou, “Anomalous dimensions of composite op-erators near two dimensions for ferromagnets with o ( n ) symmetry,” Phys. Rev. B , 4976(1976).[68] Sheer El-Showk and Miguel F. Paulos, “Bootstrapping conformal field theories withthe extremal functional method,” Physical Review Letters (2013), 10.1103/phys-revlett.111.241601.[69] Rajeev S. Erramilli, Luca V. Iliesiu, Petr Kravchuk, Walter Landry, David Poland, andDavid Simmons-Duffin, “blocks 3d: Software for general 3d conformal blocks,” arXiv e-prints, arXiv:2011.01959 (2020), arXiv:2011.01959 [hep-th].[70] David Simmons-Duffin, “A Semidefinite Program Solver for the Conformal Bootstrap,” JHEP , 174 (2015), arXiv:1502.02033 [hep-th].[71] A. G. Abanov and P. B. Wiegmann, “Theta-terms in nonlinear sigma-models,” Nuclear PhysicsB , 685–698 (2000), arXiv:hep-th/9911025 [hep-th]., 685–698 (2000), arXiv:hep-th/9911025 [hep-th].