key: cord-0277991-suownp5b
authors: He, Yin-Chen; Rong, Junchen; Su, Ning
title: A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions
date: 2021-01-18
journal: nan
DOI: 10.21468/scipostphys.11.6.111
sha: 5f4b68d4a3e15426dcb8039223eef8eab779f3ae
doc_id: 277991
cord_uid: suownp5b

We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions $d>2$. In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our recipe is based on the notion of emph{decoupling operator}, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of $2d$ Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling 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 of motion, which has interesting consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the scalar QED which has a $U(1)$ gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its kinks and islands in both $d=3$ and $d=2+epsilon$ dimensions.

Coupling gapless particles with gauge fields is one of the few known ways to obtain an interacting conformal field theory in dimensions d > 2. These gauge theory type of CFTs have interesting applications in both the high energy [1] [2] [3] and condensed matter physics. In condensed matter system, such CFTs describe phase transitions or gapless phases beyond conventional Landau's symmetry breaking paradigm [4] [5] [6] [7] [8] [9] [10] [11] [12] , and they have interesting properties such as fractionalization and long-range entanglement. Understanding such CFTs may pave the way towards several long-standing problems in condensed matter, including critical quantum spin liquids [6] [7] [8] and plateau transitions of fractional quantum Hall states [9] [10] [11] [12] .

Compared to the Wilson-Fisher (WF) CFTs, these gauge theory CFTs are poorly understood. Recently, conformal bootstrap [13] became a powerful technique to study CFTs in dimensions higher than 2d [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] (see a review [25] ). The numerical bootstrap obtained critical exponents of 3d Ising [14] and O(2) WF [23] with the world record precision, and importantly, has solved the long-standing inconsistency between experiments and Monte-Carlo simulations of O(2) WF [23] as well as the cubic instability of O(3) WF [24] . However, so far the gauge theory CFTs resist to be tackled by bootstrap [26] [27] [28] [29] . The main challenge is built in the fundamental philosophy of bootstrap, namely characterizing a theory without relying on a specific Lagrangian. More concretely, in a bootstrap study one typically inputs the global symmetry of the theory, and utilizes the consistency of crossing equations to constrain or to compute the scaling dimensions of operators in certain representations of the global symmetry. For a Wilson-Fisher type of CFT, it is believed that one can uniquely define it by 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 could have similar or even identical global symmetries. Their lowest lying operators would sit in the same representation and have similar scaling dimensions. Therefore, it is unclear how to detect the gauge group in a typical bootstrap calculation. [30] . There were efforts to bootstrap the 4pt of either fermion bilinears [26, 28] or monopole operators [27] , but no unambiguous signature of gauge theories is found.

In d = 2 dimensions it is pretty common that distinct CFTs have the same global symmetry. However, numerical bootstrap successfully detects some of these CFTs, including the 2d Ising CFT out of minimal models [31] and the SU (2) 1 Wess-Zumino-Witten (WZW)

CFT out of SU (2) k WZW theories [32, 33] . It is found that the Ising CFT (SU (2) 1 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 to the existence of a family of CFTs sharing the same global symmetry and similar operator spectrum. Compared to their cousins, the Ising CFT and SU (2) 1 WZW are special because they have null operators at low levels. These null operators will lead to some non-analyticity in the numerical bound, resulting in a kink [15, 33, 34] .

The examples in 2d suggest that, the existence a family of cousins with the same global symmetries is not an obstacle of bootstrapping a CFT, it could instead guide us to find the right condition (i.e. null operator condition) to bootstrap the CFT of interest. Theoretically, the existence of the null operator at a certain level can serve as a defining feature of 2d minimal model [35] . One cannot help to wonder if a similar physics also exists for higher dimensional CFTs, and can it be further utilized in the bootstrap study? We provide a positive answer to this by exploring gauge theories and in particular, their relations with the 2d 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 2d WZW CFTs in higher dimensions. Similar to the Kac-Moody algebra, the structure of decoupling operators of gauge theories are sensitive to the representations of global symmetry. Moreover, the color number 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 the equation of motion (EOM) can lead to the phenomenon of operator missing in the CFT spectrum [15, 16, 36, 37] . Theoretically, it was understood that as the consequence of the EOM of φ 4 theory, i.e. φ = gφ 3 , φ 3 becomes a descendent of φ. In other words, the operator φ 3 becomes missing in the primary operator spectrum of WF CFTs. This structure can further serve as an algebraic definition of WF CFTs in 4 − dimensions [36] .

Numerically, one can also impose the condition of φ 3 being missing by adding a large gap above φ in the O(N ) vector channel, this is indeed how one obtains the famous bootstrap island of WF CFTs [14, 16] . We will push this idea further by exploring the consequence of EOMs on high level missing operators. Such higher level missing operators are actually rather straightforward to visualize. For example, it is natural to expect that φ( φ − gφ 3 ) is missing as well. We will elaborate more on this and its consequence in the main text.

To be concrete, we will discuss the idea of decoupling operators and their bootstrap application in the context of a prototypical gauge theory, namely the scalar QED. It is described by N f flavor critical bosons coupled to a U (1) gauge field,

The global symmetry of the scalar QED is P SU

. One fundamental (gauge invariant) operators of this theory are the boson bilinearφ i φ j − δ i j /Nφ k φ k andφ i φ i , which are in the SU (N f ) adjoint and singlet representation, respectively. This theory is dual to the CP N f −1 model, i.e., a non-linear sigma model (NLσM) on the target space (1) . Within the NLσM formulation, one can access the scalar QED fixed point using the 2 + expansion [38] [39] [40] . It is worth noting that in d = 3 dimensions, there is one extra emergent symmetry called U (1) top , which corresponds to the flux conservation of the gauge field. There will be a new type of primary operators, called monopoles [41] , that are charged under U (1) top . In this paper, we will not study monopole operators.

For a large enough N f , the scalar QED in 2 < d < 4 dimensions will flow into an interacting CFT as one tunes the mass of bosons to a critical value. In a given dimension d, there exists a critical N * f (d) below which the scalar QED fixed point will disappear by colliding with the tri-critical QED fixed point (see definition below) [42] [43] [44] [45] . In other words, only if N f > N * f (d) the scalar QED will be a real CFT 2 . It is believed that N * f (d) monotonically increases with d, but its precise form is unknown. From 2 + and 4 − expansion, it is found that N * f (d → 2) → 0 [38] [39] [40] and N * f (d → 4) ≈ 183 [42] . It remains an open problem regarding the value of N * f in d = 3 dimensions [48] [49] [50] . The N f = 2 scalar QED in d = 3 dimensions is one of the dual descriptions of the widely studied deconfined phase transition in condensed matter literature [4, 5] . There are extensive studies to discuss whether it is truly a CFT in the deep infrared [43, [51] [52] [53] [54] .

The paper is organized as follows. In Sec. II we will introduce the notation of decoupling operators. In particular, In Sec. II A we will show that the null operators of 2d WZW CFTs 2 We shall note it is an exception for the N f = 1 scalar QED in 3d as it is dual to the O(2) WF [46, 47] .

can be interpreted as decoupling operators of gauge theories. We then discuss decoupling operators of bosonic gauge theories in Sec. II B. In Sec. III we discuss the consequence of EOMs on the CFT spectrum. In Sec. IV we will present our numerical results of the scalar QED. In particular, by imposing the information of decoupling operators we show the scalar QED in 3 dimensions (Sec. IV B) and 2 + dimensions (Sec. IV C) can be solved using conformal bootstrap : we have obtained kinks and islands of scalar QED. We will conclude in Sec. V, and will provide more numerical data in the appendix.

Note added: Upon the completion of this work we became aware of an independent work [55] that overlaps with ours.

In this section, we will define what we mean by the decoupling operator and discuss several concrete examples in 2d CFTs and higher dimensional gauge theories.

The decoupling operator of a CFT of interest A can be defined by embedding A into a 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 as one continuously tunes to the CFT A. A textbook example is the 2d minimal model M q,q−1 for which one can promote the integer valued q to be real valued, which then interpolates all the minimal models M q,q−1 . This is more than a conceptual interpolation, indeed we can explicitly write down a number of crossing symmetric correlation functions that continuously depend on q 3 . As one continuously tunes q, there are operators decoupled from the spectrum at integer valued q. These decoupling operators are indeed null operators for a specific theory M q,q−1 [34] . Similarly, for the 2d WZW CFTs, one can promote the integer valued WZW level k to be real valued, and ask how are operators decoupled as one continuously varies k (see Sec. II A for more details). Different from the example of the 2d minimal model, the decoupling (null) operators are lying in representations that strongly depend 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 [56] .

Although null operators of 2D CFTs can be defined as decoupling operators, null operators certainly have deeper implications in the algebra of CFTs, e.g. they can act as differential operators that annihilate correlation functions of primary operators. The decoupling operators, on the other hand, may or may not have such fundamental applications in the operator algebra of higher dimensional CFTs. It will be interesting to understand the similarity and difference between 2d null operators and higher dimensional decoupling operators 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 a decoupling operator in the context of the SU (N ) k WZW CFT. Let us start with a simple case, i.e. SU (2) k WZW theory. It has a global symmetry SO(4) ∼ = SU (2) L ×SU (2) R /Z 2 , and its Kac-Moody primary operators |j, j are in the SO(4) representations (SU (2) L , SU (2) R ) = (j, j) with j = 0, 1/2, 1, · · · , k/2 4 . So |1, 1 is a Kac-Moody primary of SU (2) k≥2 WZW CFT, while it becomes null in the SU (2) 1 WZW CFT. Now we create an interpolation between all the SU (2) k WZW CFTs by promoting the integer valued k to be real valued. More precisely, the four-point correlation function (4pt) of any primary operator of the SU (2) k WZW CFTs is an analytical function of k, so there is no obstacle to promote k to be real valued. For our purpose it is enough to consider the 4pt of the Kac-Moody primary |1/2, 1/2 , which is a SO(4) vector and we will call it φ i :

Here N = 4 and G S [z,z], G T [z,z], G A [z,z] corresponds to the 4pt's in the channels of the 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 [56] . We are primarily concerned with the Kac-Moody primary |1, 1 , which is in the channel of rank-2 symmetric traceless tensor. The 4pt corresponding in this channel is,

with

Decomposing this 4pt into the global conformal blocks, one obtains the low lying spectrum to be ∆ = 4 k+2 , 2, · · · . The first operator (denoted as t) is nothing but the Kac-Moody primary |1, 1 , while the second operator is a global primary obtained by applying Kac-Moody current operator to the vacuum, i.e. J L J R |0, 0 . We can also work out the OPE square λ 2 φφt of |1, 1 ,

The above formula is positive definite for k > 1, and it vanishes precisely at k = 1. In other words the Kac-Moody primary |1, 1 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 |1, 1 of SU (2) 1 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 operator of gauge theories, without relying on any precise knowledge of the correlation function or operator spectrum of the WZW CFTs. The key is to recognize a dual description for the SU (N ) k WZW CFTs, namely a gauge theory with N flavors of 2-component Dirac fermions interacting with a U (k) gauge field. For the case of k = 1, this duality can be proved exactly as the U (1) gauge theory is integrable [57] . For a general level-k WZW CFT, there are reasonable evidences suggesting that they are dual to a QCD 2 theory (e.g. see [58] and references therein), although the QCD 2 is not integrable anymore. This operator exists for arbitrary k, and its scaling dimension is

In the gauge theory, this operator is nothing but 2-fermion operators, schematically written asψ c l 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 of 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 operatorψ c l ψ r,c is the SU (N ) bi-fundamental and its scaling dimension is ∆ = 1 + O(1/N ) in the N k limit. We have matched the lowest primary operators 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 asψ c 1 l 1ψ gauge fields. We will discuss it in the following subsection. It is also worth mentioning that, in higher dimensions (e.g. 3d) one can also make a straightforward connection between fermionic gauge theories and WZW CFTs [59, 60] . The details are a bit off the theme of the current paper, we will elaborate more in the Appendix.

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

We can simply start by classifying gauge invariant operators (constructed by bosonic field) in these gauge theories. We denote boson operators as φ f,c andφ 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. 

and adjoint, respectively. Their large−N f scaling dimensions are ∆ = 2 + O(1/N f ) and

Things become interesting as one moves to m = 2. Let us ask what is the lowest

where both the upper and lower indices are antisymmetric. To construct an operator in this representation, one needs at least 4 bosons, 

while for N c = 1 (e.g. scalar QED) the spectrum is ∆ = 2(d − 2) + 2 + O(1/N f ), · · · . In other words,

[f 3 ,f 4 ] channel, the lowest operator of U (N c > 1) gauge theories is decoupling 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

This structure of decoupling operators is almost identical to the null operator structure of 2d WZW CFTs, and the color number N c plays the role of WZW level k. Similar structures can also be found in theories with other gauge groups 6 .

The notion of decoupling operator was formulated by identifying a family of CFTs with the identical global symmetry. In the numerical bootstrap, one can impose gap conditions based on the structure of the decoupling operator to isolate the theory of interest from their cousins. On the practical side, depending on the scheme of bootstrap, one may also need to consider other theories that are consistent with crossing equations being bootstrapped.

For example, we will be bootstrapping the 4pt of SU (N f ) adjoint boson bilinears, so besides the U (N c ) scalar gauge theory we also need to consider other theories that contain such operator:

1. Tri-critical QED: It corresponds to the UV fixed point of the scalar QED. It can also be described by Eq. (6), but different from the scalar QED, hitting the tri-critical QED fixed point requires the fine tuning of two singlet operators, i.e., |φ| 2 and (|φ| 2 ) 2 .

The relation between the tri-critical QED and scalar QED is similar to the relation between the Gaussian and WF CFT. 4. Chern-Simons (CS) gauge theories: In 3d one can add a quantized CS term to the U (1) gauge field at any integer level N , N/4π µνρ a µ ∂ ν a ρ , leading to a family of parity breaking CFTs [63] . Similarly, one can also consider QCD theories with finite CS terms.

5. Generalized free field (GFF) theory: it is worth noting that there could be different

The last four theories do not have the identical symmetry as the scalar QED, but bootstrapping SU (N f ) adjoint will not be able to tell the difference 8 .

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. Some consequences of EOMs have already been discussed [36, 37] and been used in the bootstrap analysis [16, 64] . Here we push the idea further, in specific we will discuss 1) the consequence of the EOM of gauge field; 2) high level spectrum due to the EOM. These results will help us to distinguish the scalar QED from its other cousins, particularly the tri-critical QED,

O(2N f ) * , and GFF-A. 7 We adopt the terminology in condensed matter literatures [62] . 8 One can also consider more complicated gauge theories, e.g., critical bosons coupled to a product gauge

with G i to be Lie groups such as U , SU . The decoupling operator can be used to exclude theories that contain non-Abelian subgroups (gauge group), i.e. ∃N i c > 1. So the remaining theory one needs to consider has a gauge field U (1) m , which happens to be equivalent to the scalar QED.

One can easily analyze the consequence of EOM on the operator spectrum in the perturbative regime, including the large N f limit, 2 + limit and 4 − limit. Here we will consider the large N f limit. It is known that in the large N f limit, the Lagrangian of the theory can be written as [44] ,

Here σ is a Hubbard-Stratonovich auxiliary field, and the terms σ 2 and F 2 µν are dropped as they are irrelevant. They are three EOMs (of φ, σ and A µ respectively),

The first two are similar to the EOMs of the WF CFTs, with the difference that the conventional derivative ∂ µ is replaced by the covariant derivative D µ = ∂ µ − iA µ . For the brevity of notation we will also write D µ D µ = . The last one is unique for gauge theories 9 .

All the operators of the scalar QED can be built using φ i , σ and A µ . Except for monopole operators in 3d, all other operators' scaling dimensions are simply the summation of its constituents' scaling dimensions (∆ φ i , ∆ σ , ∆ Aµ ) = (d/2 − 1, 2, 1), up to 1/N f corrections.

It is important to note that any operators proportional to the EOM (e.g.φ i φ j |φ| 2 and (φ k ← → D ν φ k )φ i φ j ) shall be removed from the operator spectrum 10 . This would then distinguish the scalar QED from its cousins. Table I 10 This was known in the context of large-N f WF CFTs, and was also discussed in the large-N f QED theory [65] . 11 For a parity preserving theory (e.g. scalar QED), only parity even operators can appear in the OPE of two parity even scalars (e.g. SU (N f ) adjoint boson bilinear operators). 

and we skip the concrete forms of operators in the last row as it is not illuminating to write them down explicitly. also comment on Chern-Simons theories. The operator spectrum of Chern-Simons theories is similar to the scalar QED, however it does not have parity symmetry. This could be used to distinguish the scalar QED from Chern-Simons theories as we will elaborate later.

It is worth emphasizing that even though we analyze EOMs caused missing operators in the perturbative regime, these results are qualitatively correct in the non-pertrubtative regime. Therefore, it is not only necessary but also safe to input these information in a bootstrap study.

In this section we will switch gear to numerical results. We will study the scalar QED in 2 < d ≤ 4 dimensions and will start with the single correlator of SU (N f ) adjoint operators, For the bootstrap equations, one can check Ref. [26] .

We will denote the low lying scalar operators in the singlet channel as s, s , · · · ; scalar operators in the adjoint channel as a, a · · · ; l = 1 operators in the adjoint channel as J µ , J µ , · · · . Besides the single correlator of a, we will also present some results of mix correlators of a and s. We note that a appears in the OPE of a × a, so we impose this condition in all the numerics, for example we require that all the scalars in the adjoint channel should be no smaller than ∆ a . Physically this gap condition does not introduce any assumption 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 spectrum of the scalar QED. In 3d, the large-N f calculation [44, 66] gives

From 2 + expansion [38] [39] [40] , one has It is also worth noting the tri-critical QED in 3d has [44] ,

Other results of spectrum will be discussed below when needed.

As we discussed in Sec. II B, the lowest operator in the AĀ channel of non-Abelian gauge theories becomes decoupled in the abelian gauge theories (e.g. scalar QED, tri-critical QED,

O(2N * f )), so it is natural to bound AĀ channel gap ∆ AĀ to see if this operator decoupling can be detected. Concretely, for abelian gauge theories (and GFF-A) we have

while for non-Abelian gauge theories (and GFF-B) we have 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) 1 WZW CFT.

Given that the WZW CFTs' null operators can be viewed as gauge theories' decoupling operators, it is very tempting to conjecture that the AĀ kinks here correspond to the scalar QED. A careful analysis from both the numerical and theoretical perspective suggests that the AĀ kink is unfortunately not the scalar QED. Although the 1/N f correction of ∆ AĀ is unknown, we can compare ∆ a of the kinks with the large-N f results Eq. (11). In Fig. 1 we also plot large-N f ∆ a of SU (100) and SU (1000) scalar QED, which shows considerably large discrepancies to the kinks. Take a closer look at the data, the SU (100) kink sits around ∆ a ≈ 0.953, while the large N f results gives ∆ a ≈ 0.984. The discrepancy between these two numbers is around 3/N f . Similarly, this is also the case for SU (1000), which has ∆ a ≈ 0.995 and ∆ a ≈ 0.998 for the kink and large N f , respectively. This large discrepancy does not seem to be caused by a numerical convergence issue, as the differences of ∆ a between Λ = 19, 27, 35 are small. Theoretically, it is indeed easy to convince oneself that the AĀ kink cannot be the scalar QED. That is because the tri-critical QED also has ∆ AĀ = 2(d−2)+2+O(1/N f ), and its ∆ a (Eq. (15)) is smaller than that of the scalar QED (Eq. (11)). As a side note, in theory the AĀ kink could be the tri-critical QED, but numerically it does not seem be so as their large-N f ∆ a 's also have more than 2/N f discrepancy from the numerical kink.

We have also studied AĀ bound in other dimensions (see Fig. 2 ). We find that the AĀ Therefore, the single correlator can capture the essential physics of the AĀ decoupling from non-Abelian gauge theories to Abelian gauge theories. However, the AĀ kink does not correspond to any known CFT. This result inspires us that, instead of bounding ∆ AĀ we can impose a gap in the AĀ channel to exclude all the non-Abelian gauge theories. We will pursue this in the remaining part of this paper. Fig. 3 and the physical gaps of different theories. Most physical gaps have already been analyzed above in Table I . For Chern-Simons theories, we have J µ = a · ε µνρ F νρ , whose scaling dimension is ∆ = 3 + O(1/N f ). In the scalar QED such operator also exists, but it is a parity odd operator, hence will not appear in the a × a OPE.

-critical QED GFF-A O(2N * f ) QCD Chern-Simons ∆ AĀ 3 4 + O( 1 N f ) 4 + O( 1 N f ) 4 + O( 1 N f ) 4 + O( 1 N f ) 2 + O( 1 N f ) 4 + O( 1 N f ) ∆ J µ 3.1 4 + O( 1 N f ) 3 + O( 1 N f ) 3 + O( 1 N f ) 3 + O( 1 N f ) 4 + O( 1 N f ) 3 + O( 1 N f ) ∆ SS 3.1 4 + O( 1 N f ) 3 + O( 1 N f ) 3 + O( 1 N f ) 4 + O( 1 N f ) 4 + O( 1 N f ) 4 + O( 1 N f )

Interestingly, by imposing gaps ∆ AĀ ≥ 3, ∆ J µ ≥ 3.1, and ∆ SS ≥ 3.1 in the operator spectrum, we are able to obtain bootstrap islands of scalar QED in d = 3 dimensions by scanning the ∆ a -∆ SS space, as shown in Fig. 3 . Fig. 3 ) [16] , but three operators mix could drastically shrink the island [24] . It will be very interesting to mix a with SS to see how small the scalar QED island will shrink to, and to see if a scalar QED island will also exist for small N f using an accessible Λ.

The appearance of scalar QED islands strongly advocates our proposed recipes for bootstrapping gauge theories. We also remark that, a basic requirement to isolate a CFT of interest into an island is to impose a set of gaps that exclude other crossing symmetric theo- [16] . Secondly, the mixed correlator has stronger constraining power and better numerical convergence.

The above discussed scalar QED islands in the ∆ a -∆ SS space also exist in 2+ dimensions.

Moreover, in 2 + limit the numerical convergence becomes much faster, and we are able to obtain bootstrap island for small N f (e.g. N f = 4) with a small Λ. We will not repeat such discussions here. It turns out illuminating to study ∆ s bound in 2 + dimensions, as will be detailed in this section.

We first add a mild gap ∆ AĀ ≥ 2(d − 2) + 1 in the AĀ channel that excludes all the non-Abelian gauge theories (as well as the GFF-B). Having excluded all the non-Abelian gauge theories, the remaining cousins of the scalar QED that are consistent with the crossing symmetry are the tri-critical QED, GFF-A and O(2N f ) * . As we discussed in Sec. III the difference between the scalar QED and tri-critical QED/GFF is that, the former contains φ 4 interactions, while the latter does not. This difference 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 by bounding the O(N ) singlet [16] . So one may expect that the scalar QED would appear as a kink if one bounds the SU (N f ) singlet ∆ s . Fig. 4 shows the numerical bounds of ∆ s of N f = 4 12 , which has a kink that is close to the 2 + expansion results of scalar QED 13 . We further impose a gap in the second low lying singlet ∆ s ≥ 3, 3.5 and scan the feasible region of ∆ s . The ∆ s gap carves out a large region, leaving a sharp tip where the scalar QED sits in. This phenomenon is similar to that of Ising CFT, for which imposing further constraints will carve the feasible region into a small island [14] . Below we will show that the feasible region of scalar QED also shrinks to an island in the ∆ a -∆ s space with proper conditions imposed.

It is good to pause here to elaborate a bit more on the philosophy of imposing gap conditions in bootstrap calculations. As we have explained, in many cases, in particular for gauge theories, it is necessary to impose gaps in order to exclude other theories that are also consistent with crossing equations. On the other hand, in bootstrap calculations it is common that imposing gaps will carve out feasible regions, possibly leaving a kink on the numerical bounds. Sometimes, the kink is floating, namely it is moving as the gap changes (see appendix for more details). Such floating kink does not unambiguously correspond to an isolated CFT. On the practical side, it is hard to extract useful information about the physical theory from a floating kink unless one already has the knowledge of precise values of the gaps. In contrast, the kink in Fig. 4 is stable, namely it does not move as long as the gap (∆ AĀ ) is in a finite window. We have explicitly checked that the kink and numerical bounds are almost identical for different values of gap, i.e. ∆ AĀ ≥ 2(d − 2) + 1 and ∆ AĀ ≥ 2(d − 2) + 1.5. On the other hand, if one removes the ∆ AĀ gap, ∆ s bound gets modified significantly (the black curve in Fig. 4(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 type theory. These results justify our decoupling operator based recipes for bootstrapping gauge 12 The results of different N f 's are rather similar, so we just choose N f = 4 as a representative one. 13 The discrepancy is of order O( 3 ) and O( 2 ) for ∆ a and ∆ s , respectively. theories, in specific the ∆ AĀ 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 corresponds to the ∆ AĀ jump shown in Fig. 1-2 . It is noticeable that ∆ s is pretty small in this region, supporting again that the ∆ AĀ 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. the leftmost kink) of the numerical bounds.

To get an island of the scalar QED, we need to find conditions to exclude all its cousins. Table II , we impose the following mild gaps in the operator spectrum,

and we successfully isolate the scalar QED into a small island (in the ∆ a − ∆ s space) with the single correlator in d = 2.01 dimensions, as shown in Fig. 5(a) . The first three gaps have very clear physical meanings, they serve to exclude non-Abelian gauge theories, tri-critical QED/GFF, and O(2N * f ). The last gap ∆ SS is rather mysterious, we do not have a clear idea what theory does it exclude. Removing any of these four gaps, the scalar QED will not be isolated to an island any more. Somewhat surprisingly, by increasing the dimensions slightly, say d = 2.1, the single correlator can not isolate an island any more. The mixed correlator can still yield an island with a high Λ = 35 14 and more aggressive (but still 14 Λ = 27 does not produce an island. physical) gap conditions (Fig. 5(b) ), i.e., ∆ AĀ ≥ 2(d − 2) + 1, ∆ s ≥ 3.5, ∆ J µ ≥ d + 0.5,

The appearance of scalar QED kinks and islands in d = 2 + dimension again advocate our proposed recipes for bootstrapping critical gauge theories. These nice results, however, do not sustain to d = 3 dimensions. More detailed numerical observations and discussions can be found in Appendix B.

We have introduced the notion of decoupling operators of critical gauge theories in dimensions d > 2. The decoupling operator is the higher dimensional reminiscent of null operators of 2d WZW CFTs, and it can efficiently detect the rank of the gauge group. Based on the information of decoupling operators, one can then impose gap conditions in bootstrap calculations to isolate gauge theories of interest from other theories. As an illustrative example, we study a prototypical critical gauge theory, i.e., the scalar QED. We firstly identified the concrete decoupling operators of the scalar QED, and then showed how to use them in a bootstrap study.

In both the 3d large-N f limit and the d = 2 + limit, we have successfully obtained kinks as well as islands of the scalar QED, by imposing mild gap conditions inspired by the physics of decoupling operators and EOMs. We shall remark that, even though these two limits can be accessed using perturbative expansions, our bootstrap calculations do not rely on any of these perturbative results. The gap conditions we imposed are very mild that are likely to hold for any N f in 3d. The success of bootstrap calculations, however, does not sustain to the most interesting case, i.e., small N f in 3d. The failure for small N f in 3d might be due to the poor numerical convergence. It is possible that the mixed correlator bootstrap between a and SS will improve the numerical convergence significantly and solve the long-standing problem regarding the properties of small N f scalar QED in 3d. We will leave this for the future study.

One interesting question is what does the AĀ 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-2 symmetric 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 traceless tensor [61] . We believe these kinks may have similar physical mechanisms. They could either be 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, given that they are close to gauge theories in the parameter space. Understanding them may help to eventually solve the gauge theories in 3d.

We have showed how to use the decoupling operator in the AĀ channel to bootstrap U (1) gauge theories. In a similar fashion, one can bootstrap a non-Abelian gauge theory with a specific gauge group U (N c = m) by using the decoupling operators in the antisym- we will argue that the Grassmannian WZW models have simple UV completions, i.e., Dirac fermions coupled to a gauge field.

The UV completion of the 3d leve-k U (2N )/(U (N ) × U (N )) WZW model is the QCD 3 -Gross-Neveu model,

Here α µ is a SU (k) gauge field, ψ i Dirac fermions are in the SU (k) fundamental presentation. In the SSB phase, the Dirac fermions are gapped, integrating out of them will generate a level-k WZW term [70] . The level k (instead of 1) comes from the color multiplicity of Dirac fermions due to the SU (k) gauge field. Therefore, we have proved that the SSB fixed point of the QCD 3 -Gross-Neveu model and the level-k U (2N )/(U (N ) × U (N )) WZW are dual to each other.

Given that phase diagrams of two models match and the SSB phase of two models are dual, it is natural to conjecture that the QCD 3 -Gross-Neveu model is the UV completion of There is an interesting sanity check for this duality. The Grassmannian U (2N )/(U (N ) × U (N )) has a nontrivial π 2 = Z leading to Skyrmion operators. The Skyrmion is either a boson or fermion depending on the evenness and oddness of k [60] . The Skyrmion can be identified as the baryon operator of the SU (k) gauge theory, whose statistics also depends on k. In this appendix we will provide more detailed numerical data, and most of the data will focus on 2 + dimensions.

Firstly, let us briefly comment on floating kinks and stable kinks. As we have explained in the main text, the floating kink means the kink is moving as the imposed gap changes, while the stable kink means that the kink is not moving as long as the imposed gap lies in a finite window. Fig. 6 shows a concrete comparison between floating kinks and stable kinks. The floating kinks in Fig. 6 (a) clearly show dependence on the values of ∆ SS gap. In contrast, the stable kinks in Fig. 6(b) show little dependence on the value of the gap. 

To have a more intuitive idea about the magic of EOMs, we have investigated how the bound of ∆ J µ evolves with ∆ a . As shown in Fig. 7 , the scalar QED sits at a sharp spike, which is well separated from O(2N f ) * . This is the consequence of EOM of gauge field, as discussed in Table I . The sharp spike also explains why the gap of ∆ J µ helps to isolate the scalar QED into an island. Another noteworthy observation is that convergence quickly becomes difficult as the dimension d increases slightly. In d = 2.01 dimensions ( Fig. 7 Fig. 7(b) ). This also explains why the single correlator does not produce an island in d = 2.1 dimensions for N f = 4. We also want to remark that the convergence becomes easier for a larger N f , e.g. ∆ J µ still has a spike in d = 2.3 dimensions for SU (100) with Λ = 19. This also agrees that in 3d we are able to obtain islands in the ∆ S − ∆ SS space for large N f (i.e. Fig. 3 ).

Finally, let us investigate how the scalar QED kinks evolve as we approach d = 3 dimen- sions. In a given dimension there exists a critical N * f (d) below which the scalar QED will lose its conformality. It remains an open question about the precise value of N * f in d = 3 dimensions. To avoid the unnecessary complexity, we choose a large N f = 100 to monitor how the scalar QED kink evolves as the dimension increases. (Fig. 8(a) ), similar to N f = 4 in Fig. 4 the numerical bound has a sharp kink that is close to the 2 + result (∆ a , ∆ s ) = (0.0998, 1.9998) of the scalar QED. As d increases, the scalar QED kink becomes weak in d = 2.4 ( Fig. 8(b) ), and finally becomes invisible in d = 2.7 (Fig. 8(c) ) and d = 3 dimensions (Fig. 8(d) ).

It is unclear that why the scalar QED kink disappears for d's close to 3 17 . One possible 17 The scalar QED kink being disappearing shall not be ascribed to the physics of fixed point annihilation as N f = 100 shall be large enough the the scalar QED being conformal in d = 3 dimensions. explanation is that the numerical convergence becomes harder as d increases, which can be clearly seen by comparing the numerical bounds of Λ = 19 and Λ = 27 in Fig. 8(b)-(d) . It is also worth noting that, in d = 3 dimensions, the numerical bound of ∆ s is much larger than the value (∆ ≈ 2) of the scalar QED. However, based on our numerical data there is no 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 improved significantly by imposing a mild gap ∆ SS ≥ ∆ a 18 , as shown in Fig. 9 (b). In contrast, in d = 2.1 dimensions ( Fig. 9 [73] we find that on the boundary of feasible region one roughly has ∆ SS ≈ 2∆ a , i.e., a relation expected for the scalar QED. Also recall that in Fig. 5 , to get the scalar QED island (in the ∆ a − ∆ s space) in 2 + dimensions it is necessary to impose this mysterious gap ∆ SS ≥ ∆ a .

These observations suggest that this gap excludes some crossing symmetric solutions for the 18 We note that this gap can be further relaxed, but we have not examined it carefully to find the most optimal gap condition.

bootstrap equations, but we are not able to identify any candidate theory. Nevertheless, in d = 3 dimensions with this extra gap imposed the scalar QED kink still does not show up 19 , and the numerical bounds of ∆ s are still higher than that of the scalar QED. It is possible that one needs to exclude other theories by imposing extra gap conditions in order to spot the scalar QED kink in d = 3 dimensions. We leave this for future exploration.

Electric -magnetic duality in supersymmetric nonAbelian gauge theories

The Large N limit of superconformal field theories and supergravity

Conformal technicolor

Deconfined quantum critical points

Quantum criticality beyond the landau-ginzburg-wilson paradigm

Algebraic spin liquid as the mother of many competing orders

Properties of an algebraic spin liquid on the kagome lattice

Unifying description of competing orders in two-dimensional quantum magnets

Scaling theory of the fractional quantum hall effect

The leftmost kink corresponds to the AĀ kink, which shall not be the scalar QED as we explained earlier

Global phase diagram in the quantum hall effect

Mott transition in an anyon gas

Emergent Multi-Flavor QED 3 at the Plateau Transition between Fractional Chern Insulators: Applications to Graphene Heterostructures

Bounding scalar operator dimensions in 4D CFT

Bootstrapping Mixed Correlators in the 3D Ising Model

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

Bootstrapping the O(N) Archipelago

Precision Islands in the Ising and O(N ) Models

The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT

Bootstrapping minimal N = 1 superconformal field theory in three dimensions

Bootstrapping the Minimal 3D SCFT

Bootstrapping 3D fermions

Bootstrapping 3D fermions with global symmetries

Carving out OPE space and precise O(2) model critical exponents

Bootstrapping Heisenberg Magnets and their Cubic Instability

The conformal bootstrap: Theory, numerical techniques, and applications

Bootstrap experiments on higher dimensional cfts

Towards bootstrapping qed3

Solving qed 3 with conformal bootstrap

Searching for gauge theories with the conformal bootstrap

Monopole Taxonomy in Three-Dimensional Conformal Field Theories

Universal Constraints on Conformal Operator Dimensions

Applied Conformal Bootstrap

Non-Wilson-Fisher kinks of O(N) numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs

Unitary subsector of generalized minimal models

Applied Conformal Field Theory

The -expansion from conformal field theory

Anomalous dimensions in CFT with weakly broken higher spin symmetry

Phase transitions in nonlinear abelian higgs models

Three-loop ß-functions of non-linear σ models on symmetric spaces

Renormalization group functions of cpn-1 non-linear σ-model and ncomponent scalar qed model

Action of hedgehog instantons in the disordered phase of the (2+ 1)-dimensional cpn-1 model

First-order phase transitions in superconductors and smectic-a liquid crystals

Deconfined quantum criticality, scaling violations, and classical loop models

Easy-plane QED 3 's in the large N f limit

Walking, weak first-order transitions, and complex CFTs

Phase transition in a lattice model of superconductivity

Mandelstam-'t hooft duality in abelian lattice models

Continuous quantum phase transition between an antiferromagnet and a valence-bond solid in two dimensions: Evidence for logarithmic corrections to scaling

Lattice model for the SU(n) néel to valence-bond solid quantum phase transition at large n

Lattice Abelian-Higgs model with noncompact gauge fields

Evidence for deconfined quantum criticality in a two-dimensional heisenberg model with four-spin interactions

Scaling in the fan of an unconventional quantum critical point

Deconfined criticality: Generic first-order transition in the su(2) symmetry case

Emergent so(5) symmetry at the Néel to valence-bond-solid transition

Exploring SU (N ) adjoint correlators in 3d

Conformal field theory

Gauge invariance and mass. ii

Two-dimensional Quantum Field Theory, examples and applications

Stiefel liquids: possible non-Lagrangian quantum criticality from intertwined orders

A symmetry breaking scenario for QCD 3

Bootstrapping traceless symmetric O(N ) scalars

Handbook of Magnetism and Advanced Magnetic

Transitions between the quantum hall states and insulators induced by periodic potentials

Bootstrapping the N = 1 wess-zumino models in three dimensions

Anomalous dimensions of scalar operators in qed3

Quantum criticality of u(1) gauge theories with fermionic and bosonic matter in two spatial dimensions

3D CFT Archipelago from Single Correlator Bootstrap

A Semidefinite Program Solver for the Conformal Bootstrap

Theta-terms in nonlinear sigma-models

Renormalization of the nonlinear σ model in 2+ dimensionsapplication to the heisenberg ferromagnets

Anomalous dimensions of composite operators near two dimensions for ferromagnets with o(n) symmetry

Bootstrapping conformal field theories with the extremal functional method

YCH would like to thank Chong Wang and Liujun Zou for the stimulating discussions and collaborations on 3d WZW models, and Zheng Zhou for the discussions on the large−N f

In this appendix, we will discuss some examples that show direct connections between WZW CFTs and 3d gauge theories. The physics discussed here is not new, it is the recollection of the results in Ref. [59, 60] .Despite of the pure algebraic definition, 2d WZW CFTs also have a Lagrangian formulation, namely a non-linear sigma model (NLσM) on a (Lie) group manifold G (SU (N ), U Sp(2N ), etc.) supplemented with a level k WZW term [56] ,g is a matrix field valued in a unitary presentation of the Lie group. The first term is the ordinary kinetic term of NLσM, the second term is the WZW term defined in the 3dimensional extended space. k is quantized and corresponds to the homotopy class π 3 (G) = Z. One shall also have π 2 (G) = 0 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 3d a non-trivial WZW term requires the target space G to satisfy π 4 (G) = Z and π 3 (G) = 0. There are several target spaces, including Grassmannian and Stiefel manifold (e.g. (4)), satisfying this requirement. One important difference in 3d is that the NLσM is non-renormalizable, making it hard to analyze 15 . Nevertheless, it was argued that [59] 16 , there are three fixed points as the coupling strength a 2 increases from 0:

1. An attractive fixed point of spontaneous symmetry breaking (SSB) phase at a 2 = 0.The ground state manifold is the target space of NLσM.

3. An attractive conformal fixed point preserving all the symmetries.The last attractive conformal fixed point is the 3d version of the 2d WZW CFT, while the first two fixed points merge into the Gaussian fixed point in 2d.Ref. [59] studied such 3d WZW models on the Stiefel manifold, here we discuss a simpler situation-the 3d Grassmannian U (2N )/(U (N ) × U (N )) WZW models [60] . In particular, 15 A theory being non-renormalizable does not necessarily mean it is non-sensible. For the context of NLσM, we know that it can describe the WF CFTs although it is non-renormalizable in d > 2 dimensions. 16 Ref. [59] studied Stiefel manifold, but it should be readily generalized to other manifold.