# A roadmap for bootstrapping critical gauge theories: decoupling operators of
conformal field theories in $d>2$ dimensions

Yin-Chen He<EMAIL_ADDRESS>Perimeter Institute for
Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada  Junchen Rong
<EMAIL_ADDRESS>DESY Hamburg, Theory Group, Notkestraße 85, D-22607
Hamburg, Germany  Ning Su<EMAIL_ADDRESS>Department of Physics,
University of Pisa, I-56127 Pisa, Italy

###### Abstract

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 _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.

###### Contents

  1. I Introduction
  2. II Decoupling operators in gauge theories
    1. II.1 Null operator as a decoupling operator: the $SU(N)_{k}$ WZW CFT
    2. II.2 Decoupling operator of bosonic gauge theories
  3. III Consequence of the equation of motion
  4. IV Numerical results
    1. IV.1 Kinks of the $A\bar{A}$ bound
    2. IV.2 Scalar QED islands in $3$ dimensions
    3. IV.3 Scalar QED kinks and islands in $2+\epsilon$ dimensions
  5. V Conclusion and outlook
  6. A 3d WZW models and gauge theories
  7. B More numerical data

##  I Introduction

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
Seiberg (1995); Maldacena (1999); Luty and Okui (2006) and condensed matter
physics. In condensed matter system, such CFTs describe phase transitions or
gapless phases beyond conventional Landau’s symmetry breaking paradigm Senthil
_et al._ (2004a, b); Hermele  _et al._ (2005, 2008); Song  _et al._ (2019);
Jain  _et al._ (1990); Kivelson  _et al._ (1992); Chen  _et al._ (1993); Lee
_et al._ (2018), 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 Hermele  _et al._ (2005, 2008); Song
_et al._ (2019) and plateau transitions of fractional quantum Hall states Jain
_et al._ (1990); Kivelson  _et al._ (1992); Chen  _et al._ (1993); Lee  _et
al._ (2018).

Compared to the Wilson-Fisher (WF) CFTs, these gauge theory CFTs are poorly
understood. Recently, conformal bootstrap Rattazzi  _et al._ (2008) became a
powerful technique to study CFTs in dimensions higher than $2d$ Kos  _et al._
(2014); El-Showk  _et al._ (2014); Kos  _et al._ (2015, 2016); Simmons-Duffin
(2017); Rong and Su (2018); Atanasov  _et al._ (2018); Iliesiu  _et al._
(2016, 2018); Chester  _et al._ (2019, 2020) (see a review Poland  _et al._
(2019)). The numerical bootstrap obtained critical exponents of $3d$ Ising Kos
_et al._ (2014) and $O(2)$ WF Chester  _et al._ (2019) with the world record
precision, and importantly, has solved the long-standing inconsistency between
experiments and Monte-Carlo simulations of $O(2)$ WF Chester  _et al._ (2019)
as well as the cubic instability of $O(3)$ WF Chester  _et al._ (2020).
However, so far the gauge theory CFTs resist to be tackled by bootstrap
Nakayama (2018); Chester and Pufu (2016a); Li (2018); Li and Poland (2021).
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.

As a concrete example, one can consider a family of theories described by
$N_{f}$ flavors of two-component Dirac fermions coupled to a $U(N_{c})$ gauge
field in $d=3$ dimensions. For a 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 be a
number of distinct CFTs that correspond to different $N_{c}$’s. These CFTs
have identical global symmetries, i.e. $(SU(N_{f})\times
U(1)_{top})/\mathbf{Z}_{N_{f}}$111For the precise global symmetry one may need
to further quotient out certain discrete symmetries. Such global part of the
global symmetry is not important for our discussion.. The $SU(N_{f})$
corresponds to the flavor symmetries of Dirac fermions, while the $U(1)_{top}$
symmetry corresponds to the $U(1)$ gauge flux conservation of $U(N_{c})$ gauge
group. The most important low lying (scalar) operators are 1) fermion
bilinears which are $SU(N_{f})$ adjoint but neutral under $U(1)_{top}$, its
scaling dimension is $\Delta=2+O(1/N_{f})$; 2) $2\pi$ monopole operators which
are charged under $U(1)_{top}$ and also carries a non-trivial representation
of $SU(N_{f})$ (which is independent of $N_{c}$), its scaling dimension is
$\Delta=0.265N_{f}-0.0383-0.516(N_{c}-1)+O(1/N_{f})$ Dyer  _et al._ (2013).
There were efforts to bootstrap the 4pt of either fermion bilinears Nakayama
(2018); Li (2018) or monopole operators Chester and Pufu (2016a), 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 Rychkov and
Vichi (2009) and the $SU(2)_{1}$ Wess-Zumino-Witten (WZW) CFT out of
$SU(2)_{k}$ WZW theories Ohtsuki (2016); He  _et al._ (2021). 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 El-Showk  _et al._ (2014); Behan (2018); He  _et al._
(2021).

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 Ginsparg (1988). 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 Kos  _et al._ (2015); El-Showk  _et al._ (2014); Rychkov and
Tan (2015); Giombi and Kirilin (2016). Theoretically, it was understood that
as the consequence of the EOM of $\phi^{4}$ theory, i.e.
$\square\phi=g\phi^{3}$, $\phi^{3}$ becomes a descendent of $\phi$. In other
words, the operator $\phi^{3}$ becomes missing in the primary operator
spectrum of WF CFTs. This structure can further serve as an algebraic
definition of WF CFTs in $4-\epsilon$ dimensions Rychkov and Tan (2015).
Numerically, one can also impose the condition of $\phi^{3}$ being missing by
adding a large gap above $\phi$ in the $O(N)$ vector channel, this is indeed
how one obtains the famous bootstrap island of WF CFTs Kos  _et al._ (2015,
2014). 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
$\phi(\square\phi-g\phi^{3})$ 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,

$\mathcal{L}=\sum_{i=1}^{N_{f}}|(\partial_{\mu}-iA_{\mu})\phi_{i}|^{2}+m^{2}|\phi|^{2}+\frac{g}{4}|\phi|^{4}+\frac{1}{4e^{2}}F_{\mu\nu}^{2}.$
(1)

The global symmetry of the scalar QED is
$PSU(N_{f})=\frac{SU(N_{f})}{Z_{N_{f}}}$. One fundamental (gauge invariant)
operators of this theory are the boson bilinear
$\bar{\phi}^{i}\phi_{j}-\delta^{i}_{j}/N\bar{\phi}^{k}\phi_{k}$ and
$\bar{\phi}^{i}\phi_{i}$, 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 (NL$\sigma$M) on the target space
$\textrm{CP}^{N_{f}-1}\cong\frac{U(N_{f})}{U(N_{f}-1)\times U(1)}$. Within the
NL$\sigma$M formulation, one can access the scalar QED fixed point using the
$2+\epsilon$ expansion Lawrie and Athrone (1983); Hikami (1981, 1979). 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 Murthy
and Sachdev (1990), 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) Halperin  _et al._ (1974); Nahum  _et
al._ (2015a); Benvenuti and Khachatryan (2019); Gorbenko  _et al._ (2018). In
other words, only if $N_{f}>N_{f}^{*}(d)$ the scalar QED will be a real CFT
222We shall note it is an exception for the $N_{f}=1$ scalar QED in $3d$ as it
is dual to the $O(2)$ WF Dasgupta and Halperin (1981); Peskin (1978).. It is
believed that $N_{f}^{*}(d)$ monotonically increases with $d$, but its precise
form is unknown. From $2+\epsilon$ and $4-\epsilon$ expansion, it is found
that $N_{f}^{*}(d\rightarrow 2)\rightarrow 0$ Lawrie and Athrone (1983);
Hikami (1981, 1979) and $N_{f}^{*}(d\rightarrow 4)\approx 183$ Halperin  _et
al._ (1974). It remains an open problem regarding the value of $N_{f}^{*}$ in
$d=3$ dimensions Sandvik (2010); Kaul and Sandvik (2012); Bonati  _et al._
(2020). 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 Senthil  _et al._ (2004a, b). There are extensive studies to
discuss whether it is truly a CFT in the deep infrared Sandvik (2007); Melko
and Kaul (2008); Kuklov  _et al._ (2008); Nahum  _et al._ (2015b, a).

The paper is organized as follows. In Sec. II we will introduce the notation
of decoupling operators. In particular, In Sec. II.1 we will show that the
null operators of $2d$ WZW CFTs can be interpreted as decoupling operators of
gauge theories. We then discuss decoupling operators of bosonic gauge theories
in Sec. II.2. 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.2) and $2+\epsilon$ dimensions (Sec.
IV.3) 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 Manenti and Vichi (2021) that overlaps with ours.

##  II Decoupling operators in gauge theories

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 $\mathcal{A}$ can be defined by
embedding $\mathcal{A}$ 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 $\mathcal{A}$. A textbook example is the
$2d$ minimal model $\mathcal{M}_{q,q-1}$ for which one can promote the integer
valued $q$ to be real valued, which then interpolates all the minimal models
$\mathcal{M}_{q,q-1}$. This is more than a conceptual interpolation, indeed we
can explicitly write down a number of crossing symmetric correlation functions
that continuously depend on $q$ 333In the end, a fully consistent solution of
all crossing symmetric correlation functions would only admit discrete
(integer valued) $q$.. 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 $\mathcal{M}_{q,q-1}$ Behan
(2018). 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.1 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 Di Francesco  _et al._
(2012).

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.

###  II.1 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 $SO(4)\cong SU(2)_{L}\times SU(2)_{R}/Z_{2}$, and its Kac-Moody
primary operators $|j,j\rangle$ are in the $SO(4)$ representations
$(SU(2)_{L},SU(2)_{R})=(j,j)$ with $j=0,1/2,1,\cdots,k/2$ 444In this notation,
$(1/2,1/2)$ corresponds to the $SO(4)$ vector, $(1,1)$ corresponds to the
rank-2 symmetric traceless tensor, $(1,0)\oplus(0,1)$ corresponds to the
rank-2 anti-symmetric tensor.. So $|1,1\rangle$ is a Kac-Moody primary of
$SU(2)_{k\geq 2}$ 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 $|1/2,1/2\rangle$, which is a $SO(4)$ vector and we will
call it $\phi_{i}$:

$\displaystyle\langle\phi_{i}(x_{1})\phi_{j}(x_{2})\phi_{k}(x_{3})\phi_{l}(x_{4})\rangle=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}\left[\frac{\delta_{ij}\delta_{kl}}{N}G^{S}[z,\bar{z}]\right.$
$\displaystyle+(\frac{1}{2}\delta_{il}\delta_{jk}+\frac{1}{2}\delta_{ik}\delta_{jl}-\frac{1}{N}\delta_{ij}\delta_{kl})G^{T}[z,\bar{z}]$
$\displaystyle\left.+(\frac{1}{2}\delta_{il}\delta_{jk}-\frac{1}{2}\delta_{ik}\delta_{jl})G^{A}[z,\bar{z}]\right].$
(2)

Here $N=4$ and $G^{S}[z,\bar{z}]$, $G^{T}[z,\bar{z}]$, $G^{A}[z,\bar{z}]$
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 Di Francesco  _et al._
(2012). We are primarily concerned with the Kac-Moody primary $|1,1\rangle$,
which is in the channel of rank-2 symmetric traceless tensor. The 4pt
corresponding in this channel is,

$\displaystyle\frac{G^{T}[z,\bar{z}]}{((1-z)(1-\bar{z}))^{\frac{1}{2k+4}}}=\frac{2z\bar{z}}{k^{2}}A[z,\bar{z}]+2(z\bar{z})^{\frac{2}{k+2}}\left(\frac{\Gamma^{2}(\frac{1}{k+2})\Gamma^{2}(\frac{3}{k+2})}{\Gamma^{4}(\frac{2}{k+2})}-\frac{4\Gamma^{2}(\frac{-2}{k+2})\Gamma^{2}(\frac{3}{k+2})}{\Gamma^{2}(\frac{2}{k+2})\Gamma^{2}(\frac{-1}{k+2})}\right)B[z,\bar{z}],$
(3)

with

$\displaystyle A[z,\bar{z}]$
$\displaystyle={}_{2}F_{1}(\frac{k+1}{k+2},\frac{k+3}{k+2},\frac{2k+2}{k+2},z)\,{}_{2}F_{1}(\frac{k+1}{k+2},\frac{k+3}{k+2},\frac{2k+2}{k+2},\bar{z}),$
(4) $\displaystyle B[z,\bar{z}]$
$\displaystyle={}_{2}F_{1}(\frac{1}{k+2},\frac{3}{k+2},\frac{2}{k+2},z)\,{}_{2}F_{1}(\frac{1}{k+2},\frac{3}{k+2},\frac{2}{k+2},\bar{z}).$

Decomposing this 4pt into the global conformal blocks, one obtains the low
lying spectrum to be $\Delta=\frac{4}{k+2},2,\cdots$. The first operator
(denoted as $t$) is nothing but the Kac-Moody primary $|1,1\rangle$, while the
second operator is a global primary obtained by applying Kac-Moody current
operator to the vacuum, i.e. $J_{L}J_{R}|0,0\rangle$. We can also work out the
OPE square $\lambda_{\phi\phi t}^{2}$ of $|1,1\rangle$,

$\lambda^{2}_{\phi\phi
t}=\frac{\Gamma^{2}(\frac{1}{k+2})\Gamma^{2}(\frac{3}{k+2})}{2\Gamma^{4}(\frac{2}{k+2})}-\frac{2\Gamma^{2}(\frac{-2}{k+2})\Gamma^{2}(\frac{3}{k+2})}{\Gamma^{2}(\frac{2}{k+2})\Gamma^{2}(\frac{-1}{k+2})}.$
(5)

The above formula is positive definite for $k>1$, and it vanishes precisely at
$k=1$. In other words the Kac-Moody primary $|1,1\rangle$ 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 $|1,1\rangle$ 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 Schwinger (1962).
For a general level-$k$ WZW CFT, there are reasonable evidences suggesting
that they are dual to a QCD2 theory (e.g. see Abdalla (1997) and references
therein), although the QCD2 is not integrable anymore.

The global symmetry of both the $SU(N)_{k}$ WZW and the $U(k)$ gauge theory is
$SU(N)_{L}\times SU(N)_{R}$, so we can consider the (global) primary operator
spectrum of these theories in various representations of $SU(N)_{L}\times
SU(N)_{R}$. 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
$\Delta=\frac{N^{2}-1}{N(N+k)}$, which is $\Delta\approx 1+O(1/N)$ in the
limit of $N\gg k$. In the gauge theory, this operator is nothing but 2-fermion
operators, schematically written as $\bar{\psi}^{c}_{l}\psi_{r,c}$. We use a
convention that the right (left) moving fermion $\psi$ ($\bar{\psi}$) 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 operator $\bar{\psi}^{c}_{l}\psi_{r,c}$ is
the $SU(N)$ bi-fundamental and its scaling dimension is $\Delta=1+O(1/N)$ in
the $N\gg k$ 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 as
$\bar{\psi}^{c_{1}}_{l_{1}}\bar{\psi}^{c_{2}}_{l_{2}}\psi_{r_{3},c_{3}}\psi_{r_{4},c_{4}}$.
The two left (right) moving fermions shall be totally antisymmetric, so we
shall have either the flavor indices or the color indices anti-symmetric, and
meanwhile keep the other indices symmetric. We need to further contract the
color indices of left and right moving fermions to get a gauge invariant
operator. For $k=1$, however, anti-symmetrizing color indices is not an
option, leaving the only possibility to be anti-symmetrizing the flavor
indices. Therefore, for $k>1$ there are two different 4-fermion operators
(that are Lorentz scalar) which are in the $SU(N)_{L}\times SU(N)_{R}$
representations $A_{L}A_{R}$ and $T_{L}T_{R}$. 555Here $T_{L}T_{R}$
($A_{L}A_{R}$) refers to rank-2 symmetric (anti-symmetric) tensor of
$SU(N)_{L}$ and $SU(N)_{R}$. 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, whose scaling dimensions are
$\Delta=\frac{2(N-2)(N+1)}{N(N+k)}$ and $\Delta=\frac{2(N-1)(N+2)}{N(N+k)}$.
In the limit of $N\gg k$, these two scaling dimensions are $\Delta=2+O(1/N)$
matching what we expect for 4-fermion operators. On the other hand, when $k=1$
there is only one 4-fermions operator in the channel $A_{L}A_{R}$, as the
other operator in the channel $T_{L}T_{R}$ becomes a decoupling operator due
to the low rank of the gauge group. This again nicely matches the physics of
$SU(N)_{k}$ WZW CFTs, namely at $k=1$ the Kac-Moody primary in the
$T_{L}T_{R}$ channel becomes null (the Kac-Moody primary in the $A_{L}A_{R}$
channel is still intact). It is straightforward to generalize 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
understood as decoupling operators in the context of 2d gauge theories. From
the gauge theory side, we can also generalize the analysis to higher
dimensions. A complexity is that fermion is in the spinor representation of
$SO(d)$ Lorentz rotation, which has a strong dependence on the spacetime
dimension $d$. It turns out that it is easiest to discuss the idea of
decoupling operators in the context of bosonic gauge theories, namely critical
bosons coupled to 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 Zou  _et al._ (2021); Komargodski and Seiberg (2018).
The details are a bit off the theme of the current paper, we will elaborate
more in the Appendix.

###  II.2 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 $USp(2N_{c})$ is rather
straightforward.

We can simply start by classifying gauge invariant operators (constructed by
bosonic field) in these gauge theories. We denote boson operators as
$\phi_{f,c}$ and $\bar{\phi}^{f,c}$, which are $SU(N_{f})$ ($U(N_{c})$)
fundamental and anti-fundamental, respectively. $f=1,\cdots,N_{f}$ and
$c=1,\cdots,N_{c}$ correspond to the flavor and color index. To keep the
$U(1)\subset U(N_{c})$ gauge invariance, we shall only consider operators like
$\bar{\phi}^{f_{1},c_{1}}\cdots\bar{\phi}^{f_{m},c_{m}}\phi_{f_{m+1},c_{m+1}}\cdots\phi_{f_{2m},c_{2m}}$.
Among these operators, one should further choose $SU(N_{c})$ gauge invariant
ones. Let us start with $m=1$, i.e. boson bilinears
$\bar{\phi}^{f_{1},c_{1}}{\phi}_{f_{2},c_{2}}$. Apparently, to keep
$SU(N_{c})$ invariance there are only two operators,
$\bar{\phi}^{f_{1},c_{1}}{\phi}_{f_{1},c_{1}}$ and
$\bar{\phi}^{f_{1},c_{1}}{\phi}_{f_{2},c_{1}}-\delta^{f_{1}}_{f_{2}}/N_{f}\bar{\phi}^{f,c_{1}}{\phi}_{f,c_{1}}$,
which are the $SU(N_{f})$ singlet and adjoint, respectively. Their
large$-N_{f}$ scaling dimensions are $\Delta=2+O(1/N_{f})$ and
$\Delta=d-2+O(1/N_{f})$ for $N_{c}\ll N_{f}$.

Things become interesting as one moves to $m=2$. Let us ask what is the lowest
operator in the representation $A^{[f_{1},f_{2}]}_{[f_{3},f_{4}]}$, where both
the upper and lower indices are antisymmetric. To construct an operator in
this representation, one needs at least 4 bosons,
$\bar{\phi}^{f_{1},c_{1}}\bar{\phi}^{f_{2},c_{2}}{\phi}_{f_{3},c_{3}}{\phi}_{f_{4},c_{4}}$.
If $N_{c}\geq 2$, one can simultaneously antisymmetrize the flavor indices
(i.e. $[f_{1},f_{2}]$, $[f_{3},f_{4}]$) and the color indices (i.e.
$[c_{1},c_{2}]$, $[c_{3},c_{4}]$) of
$\bar{\phi}^{f_{1},c_{1}}\bar{\phi}^{f_{2},c_{2}}$ and
${\phi}_{f_{3},c_{3}}{\phi}_{f_{4},c_{4}}$, 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
$\Delta=2(d-2)+O(1/N_{f})$. When $N_{c}=1$, in contrast, antisymmetrizing the
color indices of two identical bosons will vanish. So the lowest operator in
the required representation shall involve two covariant derivatives,
schematically written as
$(\bar{\phi}^{f_{1}}D_{\mu}\bar{\phi}^{f_{2}})({\phi}_{f_{3}}D_{\mu}{\phi}_{f_{4}})$.
Its scaling dimension is $\Delta=2(d-2)+2+O(1/N_{f})$. Therefore, in the
$A^{[f_{1},f_{2}]}_{[f_{3},f_{4}]}$ channel the QCD gauge theories ($N_{c}>1$)
have the spectrum $\Delta=2(d-2)+O(1/N_{f}),2(d-2)+2+O(1/N_{f}),\cdots$, while
for $N_{c}=1$ (e.g. scalar QED) the spectrum is
$\Delta=2(d-2)+2+O(1/N_{f}),\cdots$. In other words,

  * •

_In the $SU(N_{f})$ $A^{[f_{1},f_{2}]}_{[f_{3},f_{4}]}$ 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 $SU(N_{f})$ anti-symmetric representation
$A^{[i_{1},\cdots,i_{m}]}_{[j_{1},\cdots,j_{m}]}$ of $N_{c}>m-1$ is decoupling
at $N_{c}\leq m-1$._

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 666An independent work Reehorst  _et al._ (2020) has a
similar analysis for $SO(2)$ gauge theory in the context of $SO(N)$ invariant
CFTs..

##  III Consequence of the equation of motion

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. 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., $|\phi|^{2}$ and $(|\phi|^{2})^{2}$. The relation between the
tri-critical QED and scalar QED is similar to the relation between the
Gaussian and WF CFT.

  2. 2.

$SU(N_{c})$ QCD: $N_{f}$ flavor of critcal bosons coupled to a $SU(N_{c}>1)$
gauge field.

  3. 3.

$O(2N_{f})^{*}$ 777We adopt the terminology in condensed matter literatures
Sachdev (2007).: This theory is nothing but replacing the $U(1)$ gauge field
of scalar QED in Eq. (6) with a discrete gauge field (e.g. say $Z_{N}$). It is
almost identical to the $O(2N_{f})$ WF except only gauge invariant operators
are physically allowed in $O(2N_{f})^{*}$. Equivalently, one can also consider
branching $O(2N_{f})$ into $SU(N_{f})\times U(1)$, and only consider the
$U(1)$ neutral sector. In this branching, the $O(2N_{f})$ symemtric rank-2
traceless tensor becomes the $SU(N_{f})$ adjoint.

  4. 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 $N$,
$N/4\pi\epsilon_{\mu\nu\rho}a_{\mu}\partial_{\nu}a_{\rho}$, leading to a
family of parity breaking CFTs Wen and Wu (1993). Similarly, one can also
consider QCD theories with finite CS terms.

  5. 5.

Generalized free field (GFF) theory: it is worth noting that there could be
different GFFs. One type of GFF (dubbed GFF-A) is made of the $SU(N_{f})$
fundamental $\phi_{i}$, meaning that the $SU(N_{f})$ adjoint is constructed by
$\phi^{i}\phi_{j}$. The other GFF (dubbed GFF-B) is directly made of
$SU(N_{f})$ adjoint $A^{i}_{j}$. One difference between these two GFFs is, the
OPE $(\phi^{i}\phi_{j})\times(\phi^{k}\phi_{l})$ in GFF-A contains
$\phi^{i}\phi_{j}$, while $A^{i}_{j}\times A^{k}_{l}$ in GFF-B does not
contain $A^{i}_{j}$.

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
888One can also consider more complicated gauge theories, e.g., critical
bosons coupled to a product gauge field $G_{1}(N_{c}^{1})\times
G_{2}(N_{c}^{2})\cdots$, 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. $\exists N_{c}^{i}>1$. So the remaining theory
one needs to consider has a gauge field $U(1)^{m}$, which happens to be
equivalent to the scalar QED..

The decoupling operator we identified in Sec. II.2 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 Rychkov
and Tan (2015); Giombi and Kirilin (2016) and been used in the bootstrap
analysis Kos  _et al._ (2015); Rong and Su (2019) . 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.

One can easily analyze the consequence of EOM on the operator spectrum in the
perturbative regime, including the large $N_{f}$ limit, $2+\epsilon$ limit and
$4-\epsilon$ 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 Benvenuti and Khachatryan (2019),

$\mathcal{L}=\sum_{i=1}^{N_{f}}|(\partial_{\mu}-iA_{\mu})\phi_{i}|^{2}+\sigma|\phi|^{2},$
(6)

Here $\sigma$ is a Hubbard-Stratonovich auxiliary field, and the terms
$\sigma^{2}$ and $F_{\mu\nu}^{2}$ are dropped as they are irrelevant. They are
three EOMs (of $\phi$, $\sigma$ and $A_{\mu}$ respectively),

$\displaystyle D_{\mu}D_{\mu}\phi_{i}$ $\displaystyle=$
$\displaystyle\sigma\phi_{i},$ (7) $\displaystyle\bar{\phi}^{i}\phi_{i}$
$\displaystyle=$ $\displaystyle|\phi|^{2}=0,$ (8)
$\displaystyle\bar{\phi}^{i}\overleftrightarrow{D}_{\nu}\phi_{i}$
$\displaystyle=$ $\displaystyle 0.$ (9)

The first two are similar to the EOMs of the WF CFTs, with the difference that
the conventional derivative $\partial_{\mu}$ is replaced by the covariant
derivative $D_{\mu}=\partial_{\mu}-iA_{\mu}$. For the brevity of notation we
will also write $D_{\mu}D_{\mu}=\square$. The last one is unique for gauge
theories 999Here $\bar{\phi}^{i}\overleftrightarrow{D}_{\nu}\phi_{i}$ stands
for
$\bar{\phi}^{i}[(\partial_{\mu}-iA_{\mu})\phi_{i}]-[(\partial_{\mu}+iA_{\mu})\bar{\phi}_{i}]\phi_{i}$..

| Level | GFF-A | Scalar QED | tri-critical QED | $O(2N_{f})^{*}$
---|---|---|---|---|---
Singlet $l=0$ | $\Delta=1+O(\frac{1}{N_{f}})$ | $|\phi|^{2}$ | None | $|\phi|^{2}$ | None
$\Delta=2+O(\frac{1}{N_{f}})$ | $(|\phi|^{2})^{2}$ | $\sigma$ | $(|\phi|^{2})^{2}$ | $\sigma$
Adjoint $l=0$ | $\Delta=1+O(\frac{1}{N_{f}})$ | $\bar{\phi}^{i}\phi_{j}$ | $\bar{\phi}^{i}\phi_{j}$ | $\bar{\phi}^{i}\phi_{j}$ | $\bar{\phi}^{i}\phi_{j}$
$\Delta=2+O(\frac{1}{N_{f}})$ | $\bar{\phi}^{i}\phi_{j}|\phi|^{2}$ | None | $\bar{\phi}^{i}\phi_{j}|\phi|^{2}$ | None
$\Delta=3+O(\frac{1}{N_{f}})$ |  $\bar{\phi}^{i}\phi_{j}(|\phi|^{2})^{2}$, $\bar{\phi}^{i}\square\phi_{j}$ | $\bar{\phi}^{i}\phi_{j}\sigma$ | $\bar{\phi}^{i}\phi_{j}(|\phi|^{2})^{2}$ | $\bar{\phi}^{i}\phi_{j}\sigma$
Singlet $l=1$ | $\Delta=2$ | $\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{i}$ | None | None | $\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{i}$
Adjoint $l=1$ | $\Delta=2$ | $\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{j}$ | $\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{j}$ | $\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{j}$ | $\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{j}$
$\Delta=3+O(\frac{1}{N_{f}})$ |  $O_{1}$, $O_{2}$, $O_{3}$ | None |  $O_{1}$, $O_{2}$ | $O_{3}$
$\Delta=4+O(\frac{1}{N_{f}})$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Table 1: List of low lying (parity even) primary operators of various theories
in $3d$ and the large $N_{f}$ limit. For notational brevity we omit terms like
$-1/N_{f}\delta^{i}_{j}|\phi|^{2}$ for the operators in the adjoint
representation. In the table we have
$O_{1}=(\bar{\phi}^{i}D_{\mu}\phi_{j})|\phi|^{2}$,
$O_{2}=(\bar{\phi}^{i}\phi_{j})\partial_{\mu}|\phi|^{2}$,
$O_{3}=(\bar{\phi}^{i}\phi_{j})(\bar{\phi}^{k}D_{\mu}\phi_{k})$, and we skip
the concrete forms of operators in the last row as it is not illuminating to
write them down explicitly.

All the operators of the scalar QED can be built using $\phi_{i}$, $\sigma$
and $A_{\mu}$. Except for monopole operators in 3d, all other operators’
scaling dimensions are simply the summation of its constituents’ scaling
dimensions $(\Delta_{\phi_{i}},\Delta_{\sigma},\Delta_{A_{\mu}})=(d/2-1,2,1)$,
up to $1/N_{f}$ corrections. It is important to note that any operators
proportional to the EOM (e.g. $\bar{\phi}^{i}\phi_{j}|\phi|^{2}$ and
$(\bar{\phi}^{k}\overleftrightarrow{D}_{\nu}\phi_{k})\bar{\phi}^{i}\phi_{j}$)
shall be removed from the operator spectrum 101010This was known in the
context of large-$N_{f}$ WF CFTs, and was also discussed in the large-$N_{f}$
QED theory Chester and Pufu (2016b).. This would then distinguish the scalar
QED from its cousins. Table 1 listed the low lying (parity even 111111For 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).) primary operators of the scalar QED and its cousins,
including the GFF-A, tric-critical QED, $O(2N_{f})^{*}$ in the large-$N_{f}$
limit in 3d. Comparing the scalar QED with its cousins, one can find that the
former 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 no operator in the scalar QED at the level
$\Delta=1+O(1/N_{f})$, as the operator $\bar{\phi}^{i}\phi_{i}$ should be
deleted due to the EOM of $\sigma$, $\bar{\phi}^{i}\phi_{i}=0$. Similarly,
from the EOM of $A_{\mu}$, we know that in the scalar QED any operator
proportional to the gauge current
$J_{\mu}^{g}=\bar{\phi}^{i}\overleftrightarrow{D}_{\mu}\phi_{i}$ should be
absent. This would then distinguish the scalar QED from the $O(2N_{f})^{*}$.
Let us 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.

##  IV Numerical results

In this section we will switch gear to numerical results. We will study the
scalar QED in $2<d\leq 4$ dimensions and will start with the single correlator
of $SU(N_{f})$ adjoint operators,
$a=\bar{\phi}^{i}\phi_{j}-\delta^{i}_{j}/N_{f}|\phi|^{2}$. The OPE $a\times a$
is,

$a\times
a=S^{+}+Adj^{+}+A\bar{A}^{+}+S\bar{S}^{+}+Adj^{-}+S\bar{A}^{-}+A\bar{S}^{-}.$
(10)

Here $S$ and $Adj$ refer $SU(N_{f})$ singlet and adjoint. $A\bar{A}$,
$S\bar{S}$, $S\bar{A}$, and $A\bar{S}$ are rank-4 tensors with two upper and
two lower indices. The naming convention of these representations is rather
simple, for example $A\bar{A}$ means that both the upper and lower indices are
anti-symmetric, while $S\bar{A}$ means that the lower indices are symmetric
and the upper indices are antisymmetric. The upper script $\pm$ means the
intermediate channel has even or odd spins. For the bootstrap equations, one
can check Ref. Nakayama (2018).

We will denote the low lying scalar operators in the singlet channel as
$s,s^{\prime},\cdots$; scalar operators in the adjoint channel as
$a,a^{\prime}\cdots$; $l=1$ operators in the adjoint channel as
$J_{\mu},J^{\prime}_{\mu},\cdots$. 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 $a\times a$, 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 $\Delta_{a}$. 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 $\Lambda=27$ (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 Kaul
and Sachdev (2008); Benvenuti and Khachatryan (2019) gives

$\displaystyle\Delta_{a}$ $\displaystyle=$ $\displaystyle
1-\frac{48}{3\pi^{2}N_{f}}+O(1/N_{f}^{2}),$ (11) $\displaystyle\Delta_{s}$
$\displaystyle=$ $\displaystyle 2-\frac{144}{3\pi^{2}N_{f}}+O(1/N_{f}^{2}).$
(12)

From $2+\epsilon$ expansion Lawrie and Athrone (1983); Hikami (1981, 1979),
one has

$\displaystyle\Delta_{a}$ $\displaystyle=$
$\displaystyle\epsilon-\frac{2}{N_{f}}\epsilon^{2}+O(\epsilon^{3}),$ (13)
$\displaystyle\Delta_{s}$ $\displaystyle=$ $\displaystyle
2-\frac{2}{N_{f}}\epsilon^{2}+O(\epsilon^{3}).$ (14)

It is also worth noting the tri-critical QED in $3d$ has Benvenuti and
Khachatryan (2019),

$\displaystyle\Delta_{a}$ $\displaystyle=$ $\displaystyle
1-\frac{64}{3\pi^{2}N_{f}}+O(1/N_{f}^{2}),$ (15) $\displaystyle\Delta_{s}$
$\displaystyle=$ $\displaystyle 1+\frac{128}{3\pi^{2}N_{f}}+O(1/N_{f}^{2}).$
(16)

Other results of spectrum will be discussed below when needed.

###  IV.1 Kinks of the $A\bar{A}$ bound

As we discussed in Sec. II.2, the lowest operator in the $A\bar{A}$ 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 $A\bar{A}$ channel gap $\Delta_{A\bar{A}}$ to see if this operator
decoupling can be detected. Concretely, for abelian gauge theories (and GFF-A)
we have

$\Delta_{A\bar{A}}=2(d-2)+2+O(1/N_{f}),$ (17)

while for non-Abelian gauge theories (and GFF-B) we have

$\Delta_{A\bar{A}}=2(d-2)+O(1/N_{f}).$ (18)

Figure 1: The numerical bounds of the lowest operator in the $A\bar{A}$
channel for $SU(10)$, $SU(100)$, and $SU(1000)$ CFTs in $d=3$. The dashed line
corresponds to the large$-N_{f}$ results of $\Delta_{a}$ for the $N_{f}=100$
and $N_{f}=1000$ scalar QED. The orange circle corresponds to
$(\Delta_{a},\Delta_{A\bar{A}})=(d-2,2(d-2)+2)$.

Fig. 1 shows the numerical bounds of $\Delta_{A\bar{A}}$ in $3d$. The
numerical bounds show clear kinks for different $N_{f}$’s, and the kink
evolves into a vertical jump from $(\Delta_{a},\Delta_{A\bar{A}})=(1,2)$ to
$(\Delta_{a},\Delta_{A\bar{A}})=(1,4)$ as $N_{f}\rightarrow\infty$. The
appearance of the $A\bar{A}$ kink can be ascribed to the decoupling operator
theorem of Abelian gauge theories we discussed above. In particular, in the
large-$N_{f}$ limit the Abelian gauge theories are living in the space after
the jump, while the 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 He  _et al._ (2021). 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 $A\bar{A}$ kinks here correspond to the
scalar QED. A careful analysis from both the numerical and theoretical
perspective suggests that the $A\bar{A}$ kink is unfortunately not the scalar
QED. Although the $1/N_{f}$ correction of $\Delta_{A\bar{A}}$ is unknown, we
can compare $\Delta_{a}$ of the kinks with the large-$N_{f}$ results Eq. (11).
In Fig. 1 we also plot large-$N_{f}$ $\Delta_{a}$ 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 $\Delta_{a}\approx
0.953$, while the large $N_{f}$ results gives $\Delta_{a}\approx 0.984$. The
discrepancy between these two numbers is around $3/N_{f}$. Similarly, this is
also the case for $SU(1000)$, which has $\Delta_{a}\approx 0.995$ and
$\Delta_{a}\approx 0.998$ 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 $\Delta_{a}$ between $\Lambda=19,27,35$ are small.
Theoretically, it is indeed easy to convince oneself that the $A\bar{A}$ kink
cannot be the scalar QED. That is because the tri-critical QED also has
$\Delta_{A\bar{A}}=2(d-2)+2+O(1/N_{f})$, and its $\Delta_{a}$ (Eq. (15)) is
smaller than that of the scalar QED (Eq. (11)). As a side note, in theory the
$A\bar{A}$ kink could be the tri-critical QED, but numerically it does not
seem be so as their large-$N_{f}$ $\Delta_{a}$’s also have more than $2/N_{f}$
discrepancy from the numerical kink.

Figure 2: The numerical bounds of the lowest operator in the $A\bar{A}$
channel for $SU(N_{f})$ CFTs in $d=2.1$ (left) and $d=4$ dimensions (right).
The dashed line in the left corresponds to the $2+\epsilon$ results of
$\Delta_{a}$ for $N_{f}=4$ and $N_{f}=100$ scalar QED. The orange circle
corresponds to $(\Delta_{a},\Delta_{A\bar{A}})=(d-2,2(d-2)+2)$.

We have also studied $A\bar{A}$ bound in other dimensions (see Fig. 2). We
find that the $A\bar{A}$ kinks still exist in $2<d\leq 4$ dimensions, and the
kink approaches $(\Delta_{a},\Delta_{A\bar{A}})=(d-2,2(d-2)+2)$ as
$N_{f}\rightarrow\infty$. It is worth noting that, in 4d for $N_{f}\neq\infty$
the $A\bar{A}$ kink does not sit at $(\Delta_{a},\Delta_{A\bar{A}})=(2,6)$.
This again suggests this kink should not be identified as the scalar QED or
tri-critical QED, as both of them shall flow to the Gaussian fixed point in
$4d$. In $d=2.1$ dimensions, the $A\bar{A}$ kinks again have large deviations
to the $2+\epsilon$ results of the scalar QED.

Therefore, the single correlator can capture the essential physics of the
$A\bar{A}$ decoupling from non-Abelian gauge theories to Abelian gauge
theories. However, the $A\bar{A}$ kink does not correspond to any known CFT.
This result inspires us that, instead of bounding $\Delta_{A\bar{A}}$ we can
impose a gap in the $A\bar{A}$ channel to exclude all the non-Abelian gauge
theories. We will pursue this in the remaining part of this paper.

###  IV.2 Scalar QED islands in $3$ dimensions

| Gap imposed | Scalar QED | Tri-critical QED | GFF-A | $O(2N_{f}^{*})$ | QCD | Chern-Simons
---|---|---|---|---|---|---|---
$\Delta_{A\bar{A}}$ | $3$ | $4+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$ | $2+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$
$\Delta_{J^{\prime}_{\mu}}$ | $3.1$ | $4+O(\frac{1}{N_{f}})$ | $3+O(\frac{1}{N_{f}})$ | $3+O(\frac{1}{N_{f}})$ | $3+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$ | $3+O(\frac{1}{N_{f}})$
$\Delta_{S\bar{S}^{\prime}}$ | $3.1$ | $4+O(\frac{1}{N_{f}})$ | $3+O(\frac{1}{N_{f}})$ | $3+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$ | $4+O(\frac{1}{N_{f}})$
Table 2: The imposed spectrum gaps for the scalar QED island in Fig. 3 and the
physical gaps of different theories. Most physical gaps have already been
analyzed above in Table 1. For Chern-Simons theories, we have
$J_{\mu}^{\prime}=a\cdot\varepsilon_{\mu\nu\rho}F_{\nu\rho}$, whose scaling
dimension is $\Delta=3+O(1/N_{f})$. In the scalar QED such operator also
exists, but it is a parity odd operator, hence will not appear in the $a\times
a$ OPE. Figure 3: Scalar QED islands in $3d$: (a) $N_{f}=1000$, (b)
$N_{f}=100$. The allowed regions (shaded) are obtained by imposing gaps in
operator spectrum summarized in Table 2. The green circles are the
large$-N_{f}$ results of scalar QED, with
$\Delta_{S\bar{S}}=2-\frac{48}{3\pi^{2}N_{f}}$Benvenuti and Khachatryan (2019)
and $\Delta_{a}$ in Eq. (11).

Interestingly, by imposing gaps $\Delta_{A\bar{A}}\geq 3$,
$\Delta_{J^{\prime}_{\mu}}\geq 3.1$, and $\Delta_{S\bar{S}^{\prime}}\geq 3.1$
in the operator spectrum, we are able to obtain bootstrap islands of scalar
QED in $d=3$ dimensions by scanning the $\Delta_{a}$-$\Delta_{S\bar{S}}$
space, as shown in Fig. 3. These three gaps are pretty mild compared to the
real gaps of scalar QED, and they have very clear physical meanings: 1)
$\Delta_{A\bar{A}}\geq 3$ serves to exclude non-Abelian gauge theories (i.e.
QCD); 2) $\Delta_{J^{\prime}_{\mu}}\geq 3.1$ serves to exclude $O(2N_{f})^{*}$
and Chern-Simons theories; 3) $\Delta_{S\bar{S}}\geq 3.1$ serves to exclude
tri-critical QED and GFF-A. Table 2 gives a summary of imposed gaps and
physical gaps of various theories.

The numerics seems to converge better for large $N_{f}$. For instance, for
$N_{f}=1000$ we can get an island with $\Lambda=19$, while for $N_{f}=100$
$\Lambda=19$ does not yield an island. Moreover, for $N_{f}=50$ we are not
able to obtain an island up to $\Lambda=35$. It could be that for small
$N_{f}$ the scalar QED island still exist for large $\Lambda$, which, however,
is beyond our computational power. Up to $\Lambda=35$, the size of the island
is proportional to $1/N_{f}$. It is unknown if the islands will further shrink
as $\Lambda$ increases. It is worth remarking that in $O(N)$ WF bootstrap, the
island of $O(3)$ WF from two operators mix also shrinks rather slowly with
$\Lambda$ (in a similar rate as Fig. 3) Kos  _et al._ (2015), but three
operators mix could drastically shrink the island Chester  _et al._ (2020). It
will be very interesting to mix $a$ with $S\bar{S}$ 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 $\Lambda$.

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 theories. From this perspective, the single
correlator bootstrap could also do the job of isloating a CFT as long as it
can access a set of such gaps, and it has also been demonstrated for $O(N)$ WF
Li and Su (2017). The mixed correlator bootstrap, on the other hand, is
certainly more powerful than the single correlator bootstrap for several
reasons. Firstly, the mixed correlator can access 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
operator in the $O(N)$ vector channel, i.e., $\phi|\phi|^{2}$ Kos  _et al._
(2015). Secondly, the mixed correlator has stronger constraining power and
better numerical convergence.

###  IV.3 Scalar QED kinks and islands in $2+\epsilon$ dimensions

Figure 4: The numerical bounds of $SU(4)$ singlet $\Delta_{s}$ of $SU(4)$ CFTs
in $d=2.1$ dimension. The allowed regions (shaded) are obtained with the gaps
$\Delta_{A\bar{A}}\geq 2(d-2)+1$ and $\Delta_{s^{\prime}}$ itemized in the
figure. The green circle corresponds to scaling dimensions of the scalar QED
from epsilon expansion $(\Delta_{a},\Delta_{s})\approx(0.095,1.995)$. The
right panel is the zoomed in plot of the left panel. In the left panel, we
also plot the numerical bounds (i.e. black curve) that does not have any gap
imposed.

The above discussed scalar QED islands in the $\Delta_{a}$-$\Delta_{S\bar{S}}$
space also exist in $2+\epsilon$ dimensions. Moreover, in $2+\epsilon$ 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 $\Lambda$. We
will not repeat such discussions here. It turns out illuminating to study
$\Delta_{s}$ bound in $2+\epsilon$ dimensions, as will be detailed in this
section.

We first add a mild gap $\Delta_{A\bar{A}}\geq 2(d-2)+1$ in the $A\bar{A}$
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 $\phi^{4}$ 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 Kos  _et al._ (2015). So one
may expect that the scalar QED would appear as a kink if one bounds the
$SU(N_{f})$ singlet $\Delta_{s}$.

Fig. 4 shows the numerical bounds of $\Delta_{s}$ of $N_{f}=4$ 121212The
results of different $N_{f}$’s are rather similar, so we just choose $N_{f}=4$
as a representative one., which has a kink that is close to the $2+\epsilon$
expansion results of scalar QED 131313The discrepancy is of order
$O(\epsilon^{3})$ and $O(\epsilon^{2})$ for $\Delta_{a}$ and $\Delta_{s}$,
respectively.. We further impose a gap in the second low lying singlet
$\Delta_{s^{\prime}}\geq 3,3.5$ and scan the feasible region of $\Delta_{s}$.
The $\Delta_{s^{\prime}}$ 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 Kos  _et al._ (2014). Below we will show that the feasible region
of scalar QED also shrinks to an island in the $\Delta_{a}$-$\Delta_{s}$ 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
($\Delta_{A\bar{A}}$) 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. $\Delta_{A\bar{A}}\geq 2(d-2)+1$ and $\Delta_{A\bar{A}}\geq
2(d-2)+1.5$. On the other hand, if one removes the $\Delta_{A\bar{A}}$ gap,
$\Delta_{s}$ 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 $\Delta_{a}$) which is likely to be a WF type theory. These results
justify our decoupling operator based recipes for bootstrapping gauge
theories, in specific the $\Delta_{A\bar{A}}$ 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 corresponds to the $\Delta_{A\bar{A}}$ jump shown in Fig. 1-2. It is
noticeable that $\Delta_{s}$ is pretty small in this region, supporting again
that the $\Delta_{A\bar{A}}$ 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.
the leftmost kink) of the numerical bounds.

Figure 5: The islands of the scalar QED with $N_{f}=4$. The green circles mark
$2+\epsilon$ results of the scalar QED. (a) $d=2.01$ dimensions: the feasible
regions are obtained from the single correlator. (b) $d=2.1$ dimensions: The
feasible region is obtained from the $a$, $s$ mixed correlator.

To get an island of the scalar QED, we need to find conditions to exclude all
its cousins. Similar to Table 2, we impose the following mild gaps in the
operator spectrum,

$\Delta_{A\bar{A}}\geq 2d-3,\Delta_{s^{\prime}}\geq
3,\Delta_{J_{\mu}^{\prime}}\geq 2d-2.8,\Delta_{S\bar{S}}\geq\Delta_{a},$ (19)

and we successfully isolate the scalar QED into a small island (in the
$\Delta_{a}-\Delta_{s}$ 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 $\Delta_{S\bar{S}}$ 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 $\Lambda=35$
141414$\Lambda=27$ does not produce an island. and more aggressive (but still
physical) gap conditions (Fig. 5(b)), i.e., $\Delta_{A\bar{A}}\geq 2(d-2)+1$,
$\Delta_{s^{\prime}}\geq 3.5$, $\Delta_{J_{\mu}^{\prime}}\geq d+0.5$,
$\Delta_{S\bar{S}}\geq\Delta_{a}$, $\Delta_{a^{\prime}}\geq\Delta_{a}+1.5$.

The appearance of scalar QED kinks and islands in $d=2+\epsilon$ 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.

##  V Conclusion and outlook

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 $d=2+\epsilon$ 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 $S\bar{S}$ 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 $A\bar{A}$ 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-2 symmetric tracless tensor of the $O(N)$
theories (this kink was dubbed non-WF kink) He  _et al._ (2021). Also a
similar kink was recently observed in bootstrapping $O(N)$ rank-2 symmetric
traceless tensor Reehorst  _et al._ (2020). 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 $A\bar{A}$ 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 antisymmetric representations
$A^{[f_{1},\cdots,f_{n+1}]}_{[f_{n+2},\cdots,f_{2n+2}]}$ of $SU(N_{f})$ with
$n\leq m$. For example, in the channel
$A^{[f_{1},\cdots,f_{m+1}]}_{[f_{m+2},\cdots,f_{2m+2}]}$ the lowest operator
of different gauge theories will have distinct scaling dimensions: 1) the
$U(N_{c}>m)$ gauge theories have $\Delta=(m+1)(d-2)+O(1/N_{f})$; 2) the
$U(N_{c}=m)$ gauge theories have $\Delta=(m+1)(d-2)+2+O(1/N_{f})$; 3) the
$U(N_{c}<m)$ gauge theories have $\Delta\geq(m+1)(d-2)+4+O(1/N_{f})$. We also
remark that as a concrete example we analyzed decoupling operators of theories
with a $U(N_{c})$ gauge field coupled to bosons. It is straightforward to
generalize to other gauge groups (e.g. $SU(N_{c})$, $SO(N_{c})$,
$USp(2N_{c})$) as well as fermions coupled to gauge fields. It will be
interesting to try our decoupling operator based recipes to tackle other gauge
theories. In particular, exciting progress might be made by using advanced
bootstrap techniques such as mixing spinning operators Erramilli  _et al._
(2020).

On the phenomenological level the decoupling operators of gauge theories share
several similarities with the null operators of $2d$ WZW CFTs. As detailed in
Sec. II.1 the null operators of $SU(N)_{k}$ WZW CFTs can be even considered as
decoupling operators of $2d$ $U(k)$ gauge theories. In the context of $2d$
CFTs the null operator has important applications, e.g. they can act as
differential operators that annihilate correlation functions. It is an open
question whether a similar application also exists for the decoupling
operators of gauge theories in a general dimension. The progress might be made
by looking for an exact interpolation between gauge theories with different
gauge groups, which is similar to the interpolation between WZW CFTs with
different WZW levels.

## Acknowledgements

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$-N_{f}$ equation of motion, which benefit current
work. We thank Slava Rychkov for his critical reading of our manuscript and
for his various suggestions. Research at Perimeter Institute is supported in
part by the Government of Canada through the Department of Innovation, Science
and Industry Canada and by the Province of Ontario through the Ministry of
Colleges and Universities. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement no. 758903). The work of J.R. is
supported by the DFG through the Emmy Noether research group The Conformal
Bootstrap Program project number 400570283. The numerics is solved using SDPB
program Simmons-Duffin (2015) and simpleboot
(https://gitlab.com/bootstrapcollaboration/simpleboot). The computations in
this paper were run on the Symmetry cluster of Perimeter institute, and on the
EPFL SCITAS cluster 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 would like to thank the hospitality of Institute of
Physics Chinese Academy of Sciences and the Center for Advanced Study,
Tsinghua University while part of the work was finished.

##  Appendix A 3d WZW models and gauge theories

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. Komargodski and Seiberg (2018);
Zou  _et al._ (2021).

Despite of the pure algebraic definition, 2d WZW CFTs also have a Lagrangian
formulation, namely a non-linear sigma model (NL$\sigma$M) on a (Lie) group
manifold $G$ ($SU(N)$, $USp(2N)$, etc.) supplemented with a level $k$ WZW term
Di Francesco  _et al._ (2012),

$\mathcal{L}=\frac{1}{4a^{2}}\int
d^{2}x\,\textrm{Tr}(\partial^{\mu}g^{-1})(\partial_{\mu}g)+k\cdot\frac{i}{24\pi}\epsilon_{\mu\nu\rho}\int_{B}d^{3}x\textrm{Tr}((\hat{g}^{-1}\partial^{\mu}\hat{g})(\hat{g}^{-1}\partial^{\nu}\hat{g})(\hat{g}^{-1}\partial^{\rho}\hat{g})).$
(20)

$g$ is a matrix field valued in a unitary presentation of the Lie group. The
first term is the ordinary kinetic term of NL$\sigma$M, the second term is the
WZW term defined in the 3-dimensional extended space. $k$ is quantized and
corresponds to the homotopy class $\pi_{3}(G)=\mathbf{Z}$. One shall also have
$\pi_{2}(G)=\mathbf{0}$ in order for the WZW term to be well defined. The
Lagrangian 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
$\pi_{4}(G)=\mathbf{Z}$ and $\pi_{3}(G)=\mathbf{0}$. There are several target
spaces, including Grassmannian and Stiefel manifold (e.g. $SO(N)/SO(4)$),
satisfying this requirement. One important difference in $3d$ is that the
NL$\sigma$M is non-renormalizable, making it hard to analyze 151515A theory
being non-renormalizable does not necessarily mean it is non-sensible. For the
context of NL$\sigma$M, we know that it can describe the WF CFTs although it
is non-renormalizable in $d>2$ dimensions.. Nevertheless, it was argued that
Zou  _et al._ (2021) 161616Ref. Zou  _et al._ (2021) studied Stiefel manifold,
but it should be readily generalized to other manifold., there are three fixed
points as the coupling strength $a^{2}$ increases from $0$:

  1. 1.

An attractive fixed point of spontaneous symmetry breaking (SSB) phase at
$a^{2}=0$. The ground state manifold is the target space of NL$\sigma$M.

  2. 2.

A repulsive fixed point of order-disorder phase transitions.

  3. 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. Zou  _et al._ (2021) studied such $3d$ WZW models on the Stiefel
manifold, here we discuss a simpler situation–the $3d$ Grassmannian
$U(2N)/(U(N)\times U(N))$ WZW models Komargodski and Seiberg (2018). In
particular, 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-$k$ $U(2N)/(U(N)\times U(N))$ WZW model is
the QCD3-Gross-Neveu model,

$\mathcal{L}=\sum_{i=1}^{2N}\bar{\psi}^{i}\gamma_{\mu}(\partial_{\mu}-i\alpha_{\mu})\psi_{i}+\lambda\phi^{i}_{j}\left(\bar{\psi}_{j}\psi_{i}-\frac{1}{2N}\delta^{j}_{i}\bar{\psi}\psi\right)+\textrm{Tr}((\partial\phi)^{2})+m\textrm{Tr}(\phi^{2})+u_{1}\textrm{Tr}(\phi^{4})+u_{2}(\textrm{Tr}(\phi^{2}))^{2}.$
(21)

Here $\alpha_{\mu}$ is a $SU(k)$ gauge field, $\psi_{i}$ Dirac fermions are in
the $SU(k)$ fundamental presentation. $\phi^{i}_{j}$ is a bosonic field in the
$SU(2N)$ adjoint representation, and it is coupled to the adjoint mass term of
the Dirac fermions.

The QCD3-Gross-Neveu model model has three fixed points, corresponding to a
SSB phase with ground state manifold $U(2N)/(U(N)\times U(N))$, QCD3-Gross-
Neveu CFT, and QCD3 CFT. The QCD3-Gross-Neveu CFT fixed point is unstable, and
will flow to either the SSB or the QCD3 CFT depending on the sign of
$m\textrm{Tr}(\phi^{2})$. This phase diagram coincides with that of
$U(2N)/(U(N)\times U(N))$ WZW models. In the SSB phase of the QCD3-Gross-Neveu
model, one can define a NL$\sigma$M model on the target space
$U(2N)/(U(N)\times U(N))$. In the SSB phase, the Dirac fermions are gapped,
integrating out of them will generate a level-$k$ WZW term Abanov and Wiegmann
(2000). 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 QCD3-Gross-Neveu model and the level-$k$
$U(2N)/(U(N)\times U(N))$ 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 QCD3-Gross-Neveu model is the
UV completion of 3d WZW model on $U(2N)/(U(N)\times U(N))$ manifold. In
particular,

  * •

_The IR conformal fixed point of the $3d$ level$-k$ $U(2N)/(U(N)\times U(N))$
WZW model is dual to the QCD3 CFTs with $N_{f}=2N$ Dirac fermions coupled to a
$SU(k)$ gauge field._

There is an interesting sanity check for this duality. The Grassmannian
$U(2N)/(U(N)\times U(N))$ has a nontrivial $\pi_{2}=\mathbf{Z}$ leading to
Skyrmion operators. The Skyrmion is either a boson or fermion depending on the
evenness and oddness of $k$ Komargodski and Seiberg (2018). The Skyrmion can
be identified as the baryon operator of the $SU(k)$ gauge theory, whose
statistics also depends on $k$.

Similarly, one can derive,

  * •

_The IR conformal fixed point of the $3d$ level$-k$ $SO(2N)/(SO(N)\times
SO(N))$ WZW model is dual to the QCD3 CFTs with $N_{f}=2N$ Dirac fermions
coupled to a $SO(k)$ gauge field._

  * •

_The IR conformal fixed point of the $3d$ level$-k$ $USp(4N)/(USp(2N)\times
USp(2N))$ WZW model is dual to the QCD3 CFTs with $N_{f}=2N$ Dirac fermions
coupled to a $USp(2k)$ gauge field._

##  Appendix B More numerical data

Figure 6: Floating kinks versus stable kinks in $d=2.1$ dimensions with a
global symmetry $SU(4)$. (a) Example of floating kinks. (b) Example of stable
kinks. The feasible regions calculated with the gap conditions
$\Delta_{A\bar{A}}\geq 1.2$ and $\Delta_{A\bar{A}}\geq 1.7$ are almost
identical to each other.

In this appendix we will provide more detailed numerical data, and most of the
data will focus on $2+\epsilon$ 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 $\Delta_{S\bar{S}}$ gap. In
contrast, the stable kinks in Fig. 6(b) show little dependence on the value of
the gap.

Figure 7: The numerical bounds of $\Delta_{J_{\mu}^{\prime}}$ of $SU(4)$ CFTs
with gap $\Delta_{A\bar{A}}\geq 2(d-2)+1$, $\Delta_{s}\geq 1$,
$\Delta_{S\bar{S}}\geq\Delta_{a}$ in (a) $d=2.01$ dimensions, (b) $d=2.07$
dimensions. The shaded regions are allowed regions. The green circles mark the
scalar QED and the green stars mark $O(2N_{f})^{*}$, up to $O(\epsilon^{3})$
and $O(\epsilon)$ corrections for $\Delta_{a}$ and
$\Delta_{J_{\mu}^{\prime}}$, respectively 181818In $O(2N_{f})^{*}$, $a$ (i.e.
$SU(N_{f})$ adjoint) corresponds to the rank-2 symmetric traceless tensor of
the $O(2N_{f})$ WF CFT. Its scaling dimension from $2+\epsilon$ expansion is
$\Delta_{a}=\frac{2N_{f}\epsilon}{2N_{f}-2}-\frac{2N_{f}\epsilon^{2}}{(-2+2N_{f})^{2}}+O(\epsilon^{3})$
Brézin and Zinn-Justin (1976); Brézin  _et al._ (1976)..  Figure 8: The
numerical bounds of $SU(N_{f})$ singlet $\Delta_{s}$ of $SU(100)$ CFTs in
$d=2.1$ (a), $d=2.4$ (b), $d=2.7$ (c), and $d=3$ (d) dimensions. The data is
obtained with a gap condition $\Delta_{A\bar{A}}\geq 2(d-2)+1$ imposed, and
the allowed regions do not change under a tighter $\Delta_{A\bar{A}}$ gap,
e.g. $\Delta_{A\bar{A}}\geq 2(d-2)+1.5$. The green circles mark the scalar
QED: (a) it corresponds to the $2+\epsilon$ expansion results
$(\Delta_{a},\Delta_{s})\approx(0.0998,1.9998)$; (d) it corresponds to the
large-$N_{f}$ results $(\Delta_{a},\Delta_{s})\approx(0.984,1.951)$.

To have a more intuitive idea about the magic of EOMs, we have investigated
how the bound of $\Delta_{J_{\mu}^{\prime}}$ evolves with $\Delta_{a}$. As
shown in Fig. 18, 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 1. The sharp spike also explains why the gap of
$\Delta_{J^{\prime}_{\mu}}$ 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. 18(a))
$\Delta_{J_{\mu}^{\prime}}$ has a sharper spike for a small $\Lambda=11$, and
a larger $\Lambda=19$ does not improve the bound significantly. In contrast,
in $d=2.07$ dimensions $\Lambda=11$ does not produce a spike at all, while the
spike shows up weakly for $\Lambda=19$ and becomes sharper for $\Lambda=27$.
Moving to a higher dimension (e.g. $d=2.1$) the spike does not show up even
for $\Lambda=27$ (the feasible region looks similar to that of $d=2.07$ with
$\Lambda=11$ in Fig. 18(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.
$\Delta_{J_{\mu}^{\prime}}$ still has a spike in $d=2.3$ dimensions for
$SU(100)$ with $\Lambda=19$. This also agrees that in $3d$ we are able to
obtain islands in the $\Delta_{S}-\Delta_{S\bar{S}}$ space for large $N_{f}$
(i.e. Fig. 3).

Finally, let us investigate how the scalar QED kinks evolve as we approach
$d=3$ dimensions. 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 shows the numerical bounds of $\Delta_{s}$ in $d=2.1,2.4,2.7,3$
dimensions. In every plot there is a sharp vertical kink on the leftmost side
of the feasible region. This kink is the $A\bar{A}$ kink discussed in Sec.
IV.1, and does not correspond to the scalar QED. In $d=2.1$ dimensions (Fig.
8(a)), similar to $N_{f}=4$ in Fig. 4 the numerical bound has a sharp kink
that is close to the $2+\epsilon$ result
$(\Delta_{a},\Delta_{s})=(0.0998,1.9998)$ 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 $3$
191919The 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.. One possible explanation
is that the numerical convergence becomes harder as $d$ increases, which can
be clearly seen by comparing the numerical bounds of $\Lambda=19$ and
$\Lambda=27$ in Fig. 8(b)-(d). It is also worth noting that, in $d=3$
dimensions, the numerical bound of $\Delta_{s}$ is much larger than the value
($\Delta\approx 2$) 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 $\Lambda\rightarrow\infty$.

Figure 9: The numerical bounds of $SU(N_{f})$ singlet $\Delta_{s}$ of
$SU(100)$ CFTs in $d=2.1$ (a), and $d=3$ (b) dimensions. The green circles
mark the scalar QED: (a) it corresponds to the $2+\epsilon$ expansion results
$(\Delta_{a},\Delta_{s})\approx(0.0998,1.9998)$; (d) it corresponds to the
large-$N_{f}$ results $(\Delta_{a},\Delta_{s})\approx(0.984,1.951)$. The light
orange and orange feasible regions are obtained with the gap condition i)
$\Delta_{A\bar{A}}\geq 2(d-2)+1$, ii) $\Delta_{A\bar{A}}\geq 2(d-2)+1$ and
$\Delta_{S\bar{S}}\geq\Delta_{a}$. The feasible regions do not change under
tighter (but still physical) conditions, e.g. $\Delta_{A\bar{A}}\geq
2(d-2)+1.5$ and $\Delta_{S\bar{S}}\geq 1.5\Delta_{a}$.

A curious observation is that, in $d=3$ dimensions the numerical bounds are
improved significantly by imposing a mild gap
$\Delta_{S\bar{S}}\geq\Delta_{a}$ 202020We note that this gap can be further
relaxed, but we have not examined it carefully to find the most optimal gap
condition., as shown in Fig. 9(b). In contrast, in $d=2.1$ dimensions (Fig.
9(a)) by imposing $\Delta_{S\bar{S}}\geq\Delta_{a}$ the numerical bounds are
only improved a little, and the position of the kink does not move. On the
other hand, the numerical bounds (for both $d=2.1$ and $d=3$) are not further
improved under a tighter gap condition, e.g. $\Delta_{S\bar{S}}\geq
1.5\Delta_{a}$. From the Extremal Functional Method (EFM) El-Showk and Paulos
(2013) we find that on the boundary of feasible region one roughly has
$\Delta_{S\bar{S}}\approx 2\Delta_{a}$, i.e., a relation expected for the
scalar QED. Also recall that in Fig. 5, to get the scalar QED island (in the
$\Delta_{a}-\Delta_{s}$ space) in $2+\epsilon$ dimensions it is necessary to
impose this mysterious gap $\Delta_{S\bar{S}}\geq\Delta_{a}$. These
observations suggest 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 with this extra gap imposed the
scalar QED kink still does not show up 212121The leftmost kink corresponds to
the $A\bar{A}$ kink, which shall not be the scalar QED as we explained
earlier., and the numerical bounds of $\Delta_{s}$ 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.

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