0}\frac{\partial^{4}}{\partial\vec{x}_{1,i}\partial\vec{x}_{2,j}\partial\vec{x}_{3,k}\partial\vec{x}_{4,l}}\int\frac{d^{4}x_{5}}{x^{2}_{15}x^{2}_{25}x^{2}_{35}x^{2}_{45}}=-\frac{8\pi^{2}}{15}\frac{I_{3}(\xi)}{\tau_{12}^{4}\tau_{34}^{4}}\Big{(}\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\Big{)}\,,$
(3.67) $\displaystyle
I_{3}(\xi)=\frac{\left(\xi^{2}-\xi+1\right)\xi^{2}}{(1-\xi)^{2}}+\frac{1}{2}\left(2\xi^{2}+\xi+2\right)\xi\log(1-\xi)+\frac{\left(2\xi^{2}-5\xi+5\right)\xi^{4}}{2(1-\xi)^{3}}\log\xi\,.$

From here we extract the four-point function of the displacement operator

$\displaystyle G_{\text{D}^{i}\text{D}^{j}\text{D}^{k}\text{D}^{l}}(\xi)$
$\displaystyle=\delta^{ij}\delta^{kl}+\delta^{ik}\delta^{jl}\xi^{4}+\delta^{il}\delta^{jk}\Big{(}\frac{\xi}{1-\xi}\Big{)}^{4}+\frac{2\varepsilon
I_{3}(\xi)}{5(N+8)}\Big{(}\delta^{ij}\delta^{kl}+\text{perms}\Big{)}+O(\varepsilon^{2})\,.$
(3.68)

In order to extract CFT data from this correlator, we decompose it in terms of
singlet $(S)$, antisymmetric $(A)$ and symmetric-traceless $(T)$ channels
under $SO(d-1)_{T}$:

$\displaystyle G_{\text{D}\text{D}\text{D}\text{D}}^{S}(\xi)$
$\displaystyle=1+\left(\frac{2}{3}+\frac{(4N+37)\varepsilon}{18(N+8)}\right)f_{4}(\xi)+\frac{5\varepsilon}{3(N+8)}\partial_{\Delta}f_{4}(\xi)+\ldots\,,$
$\displaystyle G_{\text{D}\text{D}\text{D}\text{D}}^{A}(\xi)$
$\displaystyle=2f_{5}(\xi)+\ldots\,,$ $\displaystyle
G_{\text{D}\text{D}\text{D}\text{D}}^{T}(\xi)$
$\displaystyle=\left(2+\frac{\varepsilon}{3(N+8)}\right)f_{4}(\xi)+\frac{2\varepsilon}{N+8}\partial_{\Delta}f_{4}(\xi)+\ldots\,.$
(3.69)

Notice that these operators are not necessarily the lowest-dimensional
operators in each channel. For instance, in the $(\text{D}\times\text{D})_{S}$
channel we have $\phi_{1}$ with
$\Delta_{\phi_{1}}=1+\varepsilon+O(\varepsilon^{2})$, but it does not appear
because $\lambda^{2}_{DD\phi_{1}}=O(\varepsilon^{2})$, see eq. (3.52).
Similarly, in the $(\text{D}\times\text{D})_{T}$ channel the lowest-dimension
operator is $\partial_{i}\partial_{j}\phi_{1}$ with $\Delta=3+O(\varepsilon)$,
but it is invisible in the four-point function at this order. Only in the
channel $(\text{D}\times\text{D})_{A}$ we expect the lowest-lying operators to
be $\partial_{[i}\phi_{1}\partial_{\tau}\partial_{j]}\phi_{1}$ and
$\partial_{[i}\phi_{a}\partial_{\tau}\partial_{j]}\phi_{a}$, with nearly-
degenerate dimension $\Delta=5+\ldots$. However, in order to disentangle these
operators one should do an analysis similar to the one we did for $s_{\pm}$
below eq. (3.41).

##  4 Numerical results

In this section we use numerical conformal bootstrap and the semidefinite
program solver SDPB [58] to carve out the space of line defects with
$O(2)_{F}$ global symmetry. We consider:

  1. 1.

Line defects without a tilt operator $(t)$ that preserve a $U(1)_{F}$ subgroup
of the $O(2)_{F}$ symmetry in the bulk;

  2. 2.

Line defects that break local $O(3)_{F}$ to $U(1)_{F}\simeq SO(2)_{F}$ and
therefore feature a tilt operator transforming in the vector representation of
$U(1)_{F}$.

In section 4.1 we study correlation functions involving D and $t$. Due to the
universal nature of these operators, the numerical bounds presented in section
4.1 are valid for a large class of conformal defects. In sections 4.2 and 4.3
we focus on the specific models already announced in section 3: the magnetic
line defect and the monodromy line defect. Here we use the
$\varepsilon$-expansion results of section 3 as guidance, in order to zoom in
on specific regions of the parameter space, where we expect these models to
live.

###  4.1 Bounds on universal correlators

####  4.1.1 Single-correlator with the displacement operator

We start bootstrapping correlation functions of the displacement operator D.
Recall from section 2.2.1 that D is a defect primary of scaling dimension
$\Delta_{\text{D}}=2$, transforming as a vector under $SO(2)_{T}$ and neutral
under $O(2)_{F}$. Hence, while we cannot impose the global symmetry
$O(2)_{F}$, the charge of the displacement under $SO(2)_{T}$ implies we
restrict to defects of co-dimension $q\geq 2$. In the complex notation
introduced in section 2.2.1, there are two non-equivalent orderings of the
correlation functions involving the displacement

$\langle\text{D}(\tau_{1})\text{D}(\tau_{2})\bar{\text{D}}(\tau_{3}){\bar{\text{D}}}(\tau_{4})\rangle~{},\quad\langle\text{D}(\tau_{1}){\bar{\text{D}}}(\tau_{2})\text{D}(\tau_{3}){\bar{\text{D}}}(\tau_{4})\rangle~{},$
(4.1)

and the relevant crossing equations can be obtained from eq. (2.18) upon
setting $\Delta_{\text{D}}=2$. The leading non-identity defect primaries in
the $\text{D}\times{\bar{\text{D}}}$ OPE are denoted
${(\text{D}\bar{\text{D}})}^{\pm}$ in the conventions of section 2.2.1, while
the leading primary in the $\text{D}\times\text{D}$ OPE is denoted
${\text{D}^{2}}$.

##### Gap bounds.

(a)

(b)

Figure 1: (a) Upper bounds on the dimension of the $\mathcal{S}$-parity even
scalar ${(\text{D}\bar{\text{D}})}^{+}$ as a function of the
$\mathcal{S}$-parity odd operator ${(\text{D}\bar{\text{D}})}^{-}$ and the
leading charged operator $\text{D}^{2}$. (b) Projection of the three-
dimensional allowed region in the
$(\Delta_{{(\text{D}\bar{\text{D}})}^{-}},\Delta_{{\text{D}^{2}}})$ plane for
different values of $\Delta_{{(\text{D}\bar{\text{D}})}^{+}}$. All points are
computed with $\Lambda=33,P=53$. The green and red dots correspond to the GFF
and GFB solutions respectively. The yellow dot is the extrapolation to
$\varepsilon=1$ of the $\varepsilon$-expansion predictions for the magnetic
line defect. The solid red line in (b) is the ‘agnostic’ bound for
$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}=\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$.

We start computing the upper bound on the scaling dimension of the leading
$\mathcal{S}$-parity even scalar $\Delta_{{(\text{D}\bar{\text{D}})}^{+}}$ as
we vary $\Delta_{\text{D}^{2}}$ and
$\Delta_{{{(\text{D}\bar{\text{D}})}}^{-}}$. The result is shown in the 3d
plot of figure 1(a), where the light-red shaded region represents the
‘agnostic’ bound obtained imposing the same gap on the dimension of the
$\mathcal{S}$-parity odd and $\mathcal{S}$-parity even scalars. The term
‘agnostic’ here refers to the fact that by holding
$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}=\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$
we are putting a bound on the lowest-lying singlet, whether it is parity-even
or parity-odd. In order to help visualizing the
constant-$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}$ slices, we included figure
1(b). There are three notable regions in figure 1(a):

  1. I.

This is the region with the weakest $\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$
gap assumptions, i.e. $\Delta_{{(\text{D}\bar{\text{D}})}^{-}}\simeq 0\div 3$.
In this region the upper bound is smooth: it is saturated by the ‘agnostic’
bound for both small and large values of $\Delta_{\text{D}^{2}}$, with a
cross-over behavior at around $\Delta_{\text{D}^{2}}\simeq 6$. For low values
of $\Delta_{\text{D}^{2}}$, the agnostic bound approaches the GFF value for a
real fermion $\Delta_{{(\text{D}\bar{\text{D}})}}=2\Delta_{\text{D}}+1=5$.
This is because a single real GFF satisfies the crossing equations for a
single complex scalar as long as the gap in the charged sector is
$\Delta_{\text{D}^{2}}<2\Delta_{\text{D}}=4$, see e.g. [19].

  2. II.

This is the region with intermediate values of
$\Delta_{{(\text{D}\bar{\text{D}})}^{-}}\simeq 3\div 5$. The upper bound
remains constant for $\Delta_{\text{D}^{2}}\lesssim 6$, while it drops to zero
for $\Delta_{\text{D}^{2}}\gtrsim 6$. These vertical drops can be interpreted
as an upper bound on the dimension of $\Delta_{\text{D}^{2}}$ as a function on
the gap on $\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$. As we increase the gap
on $\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$, the upper bound on
$\Delta_{\text{D}^{2}}$ becomes stronger. This region includes the GFF and GFB
solutions – see eq. (2.20)– which are indicated in the figure by green and red
dots respectively, as well as the (extrapolated) $\varepsilon$-expansion
prediction for the magnetic line defect – see eqs. (3.14) and (3.2) – which is
$(\Delta_{\text{D}^{2}},\Delta_{{(\text{D}\bar{\text{D}})}^{-}},\Delta_{{(\text{D}\bar{\text{D}})}^{+}})\simeq(3,5,1.55)$
and is marked in yellow. Interestingly, the yellow dot is close to saturating
the upper bound on $\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$, as can be seen
more clearly from figure 1(b). The $U(1)_{F}$ monodromy defect should also
live in this region, and it would be interesting to verify this by computing
e.g. the correlator
$\langle\text{D}\text{D}{\bar{\text{D}}}{\bar{\text{D}}}\rangle$ at the first
non-trivial order in the $\varepsilon$-expansion.

  3. III.

This is the region with the strongest
$\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$ gap assumptions, i.e.
$\Delta_{{(\text{D}\bar{\text{D}})}^{-}}\simeq 5\div 8$. As we enter this
region from region II, we observe that for values
$6\lesssim\Delta_{{(\text{D}\bar{\text{D}})}^{-}}\lesssim 7$, the convexity of
the bound changes and a plateau starts forming as we move towards small values
of $\Delta_{\text{D}^{2}}$. If we increase
$\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$ further, the plateau terminates
around
$(\Delta_{\text{D}^{2}},\Delta_{{(\text{D}\bar{\text{D}})}^{+}})\simeq(2.7,2.7)$.
The plateau happens to have the same height
$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}\simeq 2.7$ as the bound for large
$\Delta_{\text{D}^{2}}$ in region I. This indicates that there is a universal
upper bound $\Delta_{{(\text{D}\bar{\text{D}})}^{+}}\simeq 2.7$ when
increasing the gaps on the dimensions of all other operators.

##### Including the OPE coefficient of ${\text{D}^{2}}$.

Figure 2: Bounds on the dimension of the first singlet in the
$\text{D}\times{\bar{\text{D}}}$ OPE as a function of the gap on the dimension
$\Delta_{\text{D}^{2}}$ and the OPE coefficient
$(\lambda_{\text{D}\text{D}\text{D}^{2}})^{2}$ of the first charged operator
in the $\text{D}\times\text{D}$ OPE. The parity-even and -odd singlets are set
to have the same gap:
$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}=\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$.
The green and red dots correspond to the solutions for GFF and GFB,
respectively. $\Lambda=49,P=69$.

The light-red curve in figure 1 shows a family of solutions to crossing that
maximize the gap on the lowest-lying operator in the singlet channel. This
curve looks rather smooth in figure 1. To further investigate this family of
solutions, we repeat the gap maximization procedure but this time we keep
$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}=\Delta_{{(\text{D}\bar{\text{D}})}^{-}}$
and varying both $\Delta_{\text{D}^{2}}$ and the (squared) OPE coefficient
$(\lambda_{\text{D}\text{D}\text{D}^{2}})^{2}$. This OPE coefficient is the
most straightforward to implement, and a similar choice was made in [25, 36].
The results are shown in figure 2, whose features we now describe. The free
theory solutions are marked in the figure by green (GFF) and red (GFB) dots.
We recall that these free theory solutions are given in eqs. (2.19) and
(2.20), where it is shown that the OPE coefficients depend on the parameter
$\alpha$, which is $\alpha=-1$ for GFF and $\alpha=1$ for GFB. The red line in
the figure represents the solutions for intermediate $\alpha\in(-1,1)$; it is
well inside the allowed region and appears to be parallel to the upper bound.
The GFF solution lies outside the red line, which can be understood from the
fact that for an anti-commuting fermion $\psi_{a}$, the leading symmetric
traceless representation in the $\psi_{a}\times\phi_{b}$ OPE is
$\psi_{a}\partial_{\tau}\psi_{b}$ and has scaling dimension
$2\Delta_{\psi}+1=5$, while for a boson $\phi_{a}$ it is $\phi_{a}\phi_{b}$
and has scaling dimension $2\Delta_{\phi}=4$. A family of rising kinks appears
for values around $\Delta_{\text{D}^{2}}\simeq 5$, i.e. close to the GFF
solution. Although we have not studied the evolution of these kinks when
increasing the number of derivatives, it is plausible that they can be
explained by the vicinity of the GFF solution. For
$\Delta_{\text{D}^{2}}\gtrsim 5$, the bound on
$\Delta_{{(\text{D}\bar{\text{D}})}^{\pm}}$ quickly drops to the value
$\Delta_{{(\text{D}\bar{\text{D}})}^{\pm}}\simeq 2.7$. For
$\Delta_{{\text{D}^{2}}}\lesssim 5$, the kinks move towards smaller values of
$(\lambda_{\text{D}\text{D}\text{D}^{2}})^{2}$ until around
$\Delta_{\text{D}^{2}}\simeq 4$ and below, the upper bound becomes completely
smooth. The magnetic line defect lies well inside the allowed region, since
for this defect the first parity-even singlet has dimension
$\Delta_{{(\text{D}\bar{\text{D}})}^{+}}\simeq 1.55$.

####  4.1.2 Single-correlator with the tilt operator

Next, we consider the four-point functions of the tilt operator $t$. We recall
that $t$ is a defect primary of scaling dimension $\Delta_{t}=1$, transforming
as a vector of $O(2)_{F}$ and neutral under $SO(2)_{T}$ (see section 2.2.2).
This means that unlike the displacement bootstrap, here we are imposing the
existence of a global symmetry $O(2)_{F}$, although we cannot distinguish
between one-dimensional conformal defects with different co-dimension. In the
complex notation of section 2.2.2, the two non-equivalent orderings along the
line are

$\langle
t(\tau_{1})t(\tau_{2})\bar{t}(\tau_{3})\bar{t}(\tau_{4})\rangle\>,\quad\langle
t(\tau_{1})\bar{t}(\tau_{2})t(\tau_{3})\bar{t}(\tau_{4})\rangle\>.$ (4.2)

The leading non-identity defect primaries in the $t\times\bar{t}$ OPE are
denoted ${(t\bar{t})}^{\pm}$ in the conventions of section 2.2.1, while the
leading primary in the $t\times t$ OPE is denoted $t^{2}$. The bootstrap
equations can be obtained from eq. (2.18), setting $\Delta_{t}=1$ as external
dimensions.

##### Gap bounds.

(a)

(b)

Figure 3: Bounds on the maximal gap on the dimension of the
$\mathcal{S}$-parity even scalar ${(t\bar{t})}^{+}$ (a) or
$\mathcal{S}$-parity odd scalar ${(t\bar{t})}^{-}$ (b) vs. the
$\mathcal{S}$-parity odd (a) or even (b) scalar gap vs. the gap on the leading
charged operator $t^{2}$. $\Lambda=33,P=53$. The green and blue dots
correspond to the solutions for GFF and GFB, respectively. The yellow dot is
the 1-loop $\varepsilon$-expansion prediction for the magnetic line defect
discussed in section 3.2. The solid red line in (b) is the ‘agnostic’ bound
for $\Delta_{{(t\bar{t})}^{+}}=\Delta_{{(t\bar{t})}^{-}}$.

Here we ask the same question as for the single-correlator displacement
bootstrap, namely what is the upper bound on the scaling dimension of the
leading $\mathcal{S}$-parity even scalar $\Delta_{{(t\bar{t})}^{+}}$ as we
vary $\Delta_{t^{2}}$ and $\Delta_{{{(t\bar{t})}}^{-}}$. The results are shown
in the 3d plot of figure 3(a), where the light-red shaded region represents
the ‘agnostic’ bound obtained by imposing the same gap on the
$\mathcal{S}$-parity odd and $\mathcal{S}$-parity even scalars, and in figure
3(b). Not surprisingly, these plots show many similarities with those in
figure 1 because we used the same crossing equations, although for different
external scaling dimensions. What changes is the interpretation of the
results. There are three notable regions in 3(a):

  1. I.

This is the region with the weakest $\Delta_{{(t\bar{t})}^{-}}$ gap
assumptions, i.e. $\Delta_{{(t\bar{t})}^{-}}\simeq 0\div 1.5$. In this region
the upper bound is saturated by the ‘agnostic’ bound for both small and large
values of $\Delta_{t^{2}}$, which approaches the GFF bound for a real fermion
$\Delta_{{(t\bar{t})}}=2\Delta_{t}+1=3$ for $\Delta_{t^{2}}<2$ (cf. previous
discussion in the displacement bootstrap). At around $\Delta_{t^{2}}\simeq 3$,
the bound drops but remains smooth.

  2. II.

This is the region with intermediate values of
$\Delta_{{(t\bar{t})}^{-}}\simeq 1.5\div 3.5$. For $\Delta_{t^{2}}\lesssim 3$,
the upper bound still approaches the GFF bound at $\Delta_{{(t\bar{t})}}=3$.
For $\Delta_{t^{2}}\gtrsim 3$ the bound drops to zero, with increasingly
sharper drops until $\Delta_{{(t\bar{t})}^{-}}=3$, after which the drops
remain but become smoother. As we noted previously, these vertical drops are
due to the existing upper bound on $\Delta_{t^{2}}$, as a function of
$\Delta_{{(t\bar{t})}^{-}}$. This region also contains the GFF (green) and GFB
(blue) solutions – see eqs. (2.19) and (2.20) –, as well as the
$\varepsilon$-expansion results
$(\Delta_{t^{2}},\Delta_{{(t\bar{t})}^{-}},\Delta_{{(t\bar{t})}^{+}})\simeq(2+2/11,3,1.55)$
for the magnetic line defect (see section 3.2) which are shown as a yellow
dot.131313Here, the leading $\mathcal{S}$-parity even singlet is
${(t\bar{t})}^{+}=\phi_{1}$, with scaling dimension $\simeq 1.55$, see eq.
(2.19). The $\varepsilon$-expansion results are close to saturating the
bootstrap bound at $\Delta_{{(t\bar{t})}^{-}}\simeq 3.2$, which can be most
clearly seen in figure 3(b). It would be very interesting to know the sign of
the $O(\varepsilon^{2})$ correction to the scaling dimension of
${(t\bar{t})}^{-}$ in the magnetic line defect.

  3. III.

This is the region with the largest $\Delta_{{(t\bar{t})}^{-}}$ gap, i.e.
$\Delta_{{(t\bar{t})}^{-}}\simeq 3.5\div 5$. The convexity of the bound
changes for $4\lesssim\Delta_{{(t\bar{t})}^{-}}\lesssim 5$. The plateau that
was clearly visible in the same region in figure 1 also appears here once we
move towards small values of $\Delta_{t^{2}}$, but it is less pronounced. For
higher $\Delta_{{(t\bar{t})}^{-}}$, the plateau again terminates around
$(\Delta_{t^{2}},\Delta_{{(t\bar{t})}^{+}})\simeq(1.35,1.35)$. This is the
same height $\Delta_{{(t\bar{t})}^{+}}\simeq 1.35$ as the bound for large
$\Delta_{t^{2}}$ in region I, which can again be thought of as a universal
upper bound on $\Delta_{{(t\bar{t})}^{+}}$. These kinks are reflected in
figure 3(b) around $\Delta_{t^{2}}\sim 1.35$. The bound on
$\Delta_{{(t\bar{t})}^{-}}$ becomes infinite for
$\Delta_{{(t\bar{t})}^{+}}\lesssim 2$ and $\Delta_{t^{2}}\lesssim 1.35$, while
it is saturated by the agnostic
$\Delta_{{(t\bar{t})}^{+}}=\Delta_{{(t\bar{t})}^{-}}$ bound for
$\Delta_{t^{2}}>1.35$.

##### Including the OPE coefficient of ${t^{2}}$.

Figure 4: Bounds on the dimension of the first singlet in the $t\times\bar{t}$
OPE as a function of the gap on the dimension $\Delta_{t^{2}}$ and the OPE
coefficient $(\lambda_{ttt^{2}})^{2}$ of the first operator charged under
$O(2)_{F}$ in the $t\times t$ OPE. The gaps on the dimension of the parity-
even and -odd operators are set to be equal. The free theory solutions GFF
(green), GFB (blue), and their interpolation (blue), presented in eqs. (2.19)
and (2.20) are also shown, as is the $\varepsilon$-expansion result for the
magnetic line defect discussed in section 3.2 (yellow). $\Lambda=49,P=69$.

The $\varepsilon$-expansion results for the magnetic line defect from section
3.2 are close to saturating the bound in figure 3. However, since we do not
know the sign of the $O(\varepsilon^{2})$ correction on
$\Delta_{{(t\bar{t})}^{-}}$, we cannot predict if the point will move closer
or further away from the bound. Let us focus on the agnostic bound and impose
the same gap on $\Delta_{{(t\bar{t})}^{+}}$ and $\Delta_{{(t\bar{t})}^{-}}$.
Similarly to what we did for the displacement bootstrap, we look for bounds on
$\Delta_{{(t\bar{t})}^{\pm}}$ as a function of $(\lambda_{ttt^{2}})^{2}$ – the
(squared) OPE coefficient of the first charged operator $t^{2}$ – and
$\Delta_{t^{2}}$. The results are shown in figure 4. The free theory solutions
shown in green (GFF), and blue (GFB) – see eqs. (2.19) and (2.20) – are close
to saturating the bound. Again, the GFF solution ($\alpha=-1$) is disconnected
from the GFB solution ($\alpha=1$) and the solution for $\alpha\in(-1,1)$ (cf.
previous discussion). The $\varepsilon$-expansion result up to
$O(\varepsilon)$ for the magnetic line defect given by
$((\lambda_{ttt^{2}})^{2},\Delta_{t^{2}},\Delta_{{(t\bar{t})}})\simeq(2-\frac{4}{11},2+\frac{2}{11},1.55)$
– see eqs. (3.14), (3.17),(3.57) – and marked with the yellow dot in the
figure is below the upper bound. There are additional kinks for
$2\lesssim\Delta_{t^{2}}\lesssim 4$ around $(\lambda_{ttt^{2}})^{2}\simeq 2$,
one of which gets saturated by the GFF solution. For $\Delta_{t^{2}}\gtrsim 4$
the bound becomes horizontal and settles at $\Delta_{{(t\bar{t})}^{\pm}}\simeq
1.35$ .

####  4.1.3 Mixed-correlator with tilt and displacement

After having analyzed the single correlators of either the tilt or the
displacement operators, we consider mixed correlators that involve both at the
same time:

$\langle\text{D}(\tau_{1}){\bar{\text{D}}}(\tau_{2})\text{D}(\tau_{3}){\bar{\text{D}}}(\tau_{4})\rangle\>,\quad\langle
t(\tau_{1})\bar{t}(\tau_{2})t(\tau_{3})\bar{t}(\tau_{4})\rangle\>,\quad\langle\text{D}(\tau_{1}){\bar{\text{D}}}(\tau_{2})t(\tau_{3})\bar{t}(\tau_{4})\rangle\>.$
(4.3)

plus all other non-equivalent orderings. This is the natural system of
correlators to study the magnetic line defect for the $O(3)$ vector model,
which features a tilt operator in the vector representation of $O(2)_{F}$, as
well as a displacement in the vector of $SO(2)_{T}$. The bootstrap equations
can be found in eq. (2.22).

##### Gap bounds.

Figure 5: Bounds on the dimension of the first $\mathcal{S}$-parity even
singlet $\mathcal{O}^{+}$ in the $t\times\bar{t}$ and the
$\text{D}\times{\bar{\text{D}}}$ OPEs as a function of the scaling dimension
of ${(t\text{D})}^{\pm}$ and ${(t\bar{t})}^{-}$. The GFB solution, given in
eq. (2.25), is shown in red. The $\varepsilon$-expansion results for the
magnetic line defect are given in yellow. The light-shaded region represents
the results for the agnostic bound
$\Delta_{{(t\bar{t})}^{+}}=\Delta_{{(t\bar{t})}^{-}}$. $\Lambda=33,P=63$.

There are several operators one can study with these crossing equations. On
the one hand, we have the leading non-identity $\mathcal{S}$-parity even
primary $\mathcal{O}^{+}$ which appears both in the $t\times\bar{t}$ and the
$\text{D}\times{\bar{\text{D}}}$ OPEs. On the other hand, there are the
leading $\mathcal{S}$-parity odd primaries ${(t\bar{t})}^{-}$ and
${(\text{D}\bar{\text{D}})}^{-}$, which are in general different to each
other. Finally, there is the lowest-lying $O(2)_{F}\times SO(2)_{T}$ vector in
the $t\times\text{D}$ channel

$\displaystyle t\times\text{D}\sim{(t\text{D})}^{\pm}+\dots~{},$ (4.4)

where the superscript $\pm$ denotes the $\mathcal{S}$ parity of the operator.
In figure 5 we compute the upper bound on $\Delta_{\mathcal{O}^{+}}$, while
assuming gaps on the scaling dimensions of ${(t\bar{t})}^{-}$,
${(t\text{D})}^{\pm}$ keeping
$\Delta_{{(t\text{D})}^{+}}=\Delta_{{(t\text{D})}^{-}}$. We take all other
gaps to be very close to the unitarity bound, concretely we set them to
$\Delta>0.001$. If we are interested in the most ‘agnostic’ bound with
$\Delta_{\mathcal{O}^{+}}=\Delta_{{(t\bar{t})}^{-}}$, then the allowed region
shrinks to the light-red region of figure 5, which contains both the GFB (red
dot) and the $\varepsilon$-expansion result for the magnetic line defect. The
latter is far from saturating the upper bound and it seems hard to make
progress without further assumptions. We will come back to this issue in
section 4.3. The bounds are very uniform and show two drops, one in the
$\Delta_{{(t\bar{t})}^{-}}$ direction around $\Delta_{{(t\bar{t})}^{-}}\simeq
3$ and one in the $\Delta_{{(t\text{D})}^{\pm}}$ direction around
$\Delta_{{(t\text{D})}^{\pm}}\simeq 5$. For
$\Delta_{{(t\text{D})}^{\pm}}\gtrsim 5$ the bound on
$\Delta_{\mathcal{O}^{+}}$ becomes flat and approaches the value
$\Delta_{\mathcal{O}^{+}}\simeq 2.7$, a result we already found in the single-
correlator bootstrap of the displacement operator of section 4.1.1. In the
upper ‘cubic-shaped’ region we have that $\Delta_{\mathcal{O}^{+}}\lesssim
3.4$, which approaches the bound for a real GFF $\Delta_{\mathcal{O}^{+}}=3$.

##### Including one OPE coefficient.

In sections 4.1.1 and 4.1.2 we have seen that including one OPE coefficient
leads to interesting bounds. We repeat this strategy here and bound
$(\lambda_{t\text{D}{(t\text{D})}^{+}})^{2}$ while varying the gap
$\Delta_{{(t\text{D})}^{\pm}}$ in the agnostic region
$\Delta_{{(t\bar{t})}^{+}}=\Delta_{{(t\bar{t})}^{-}}$. It turns out that
$(\lambda_{t\text{D}{(t\text{D})}^{+}})^{2}$ is unbounded for
$\Delta_{\mathcal{O}^{+}}\lesssim 2.7$. Above this value and for
$\Delta_{\mathcal{O}^{+}}\lesssim 3.4$ there exists an upper bound which is
shown in figure 6, while for $\Delta_{\mathcal{O}^{+}}\gtrsim 3.4$ the upper
bound $(\lambda_{t\text{D}{(t\text{D})}^{+}})^{2}$ becomes negative,
consistently with the results shown in figure 5. Since we are assuming a gap
on the lowest operator $\Delta_{{(t\text{D})}^{\pm}}$, but do not make any
assumptions on the scaling dimension of the next operator in the
$t\times\text{D}$ OPE, the lower bound on
$(\lambda_{t\text{D}{(t\text{D})}^{+}})^{2}$ is at zero.

Figure 6: Bounds on $(\lambda_{t\text{D}{(t\text{D})}^{+}})^{2}$ as a function
of the scaling dimension of $\Delta_{{(t\text{D})}^{+}}$ and of the scaling
dimension of the first parity-even singlet $\Delta_{\mathcal{O}^{+}}$. The gap
on the dimension $\Delta_{{(t\text{D})}^{-}}$ is set to
$\Delta_{{(t\text{D})}^{+}}$, and all other gaps are set to 0.001. The GFB
solution given in eq. (2.25) is marked by the red dot. $\Lambda=33,P=63$.

The $\varepsilon=1$ solution of the $\varepsilon$-expansion for the magnetic
line defect is not shown, since it lies far within the bounds in the region
where the OPE coefficient $(\lambda_{t\text{D}{(t\text{D})}^{+}})^{2}$ is
unbounded.

###  4.2 Bootstrapping the monodromy line defect in the $O(2)$ model

The approach adopted so far was agnostic, in that we bounded CFT data without
committing to any particular model. In this section we pursue a complementary
approach, and combine the numerical bootstrap with the $\varepsilon$-expansion
with the goal of bootstrapping the monodromy defect in the $O(2)$ model.
Recall from section 3.1 two universal features of this monodromy defect: (i)
for generic values of the parameter $v\in[0,1)$ the flavor symmetry of the
model is $SO(2)_{F}$, which gets enhanced to $O(2)_{F}$ for $v=0,\frac{1}{2}$;
(ii) the defect spectrum contains a family of $\mathcal{S}$-parity even defect
primaries $\Psi_{s}$ with $SO(2)_{F}$ charge $v$ and transverse spin
$s\in\mathbb{Z}+v$. These two features can be combined in our numerical
bootstrap problem as follows. First, we consider a system of correlation
functions that involve the lowest-lying operator with $SO(2)_{F}$ charge $r=v$
and its partner with charge $r=1-v$. By charge conservation, the non-zero
correlation functions are

$\displaystyle\begin{split}\langle\Psi_{v}(\tau_{1})\bar{\Psi}_{v}(\tau_{2})\Psi_{v}(\tau_{3})\bar{\Psi}_{v}(\tau_{4})\rangle\>,&\quad\langle\Psi_{1-v}(\tau_{1})\bar{\Psi}_{1-v}(\tau_{2})\Psi_{1-v}(\tau_{3})\bar{\Psi}_{1-v}(\tau_{4})\rangle\>,\\\
\langle\Psi_{v}(\tau_{1})\bar{\Psi}_{1-v}&(\tau_{2})\Psi_{1-v}(\tau_{3})\bar{\Psi}_{v}(\tau_{4})\rangle\>,\end{split}$
(4.5)

which lead to the crossing equations presented in eq. (2.24). Although for
general $v$ the monodromy defect is not invariant under $O(2)_{F}$ symmetry,
but only under $SO(2)_{F}$ symmetry, we can use the same crossing equations in
both cases. The justification for this appeared in [52], but we repeat it here
for convenience. The tensor $\epsilon_{ij}$ is invariant under $SO(2)$ but not
under $O(2)$, so the antisymmetric representation is isomorphic to the singlet
representation for $SO(2)$ but not for $O(2)$. However, this does not lead to
additional relations in the crossing equations, because even when the singlet
and antisymmetric representations are isomorphic, we can distinguish them
since they contain ${\mathcal{S}}$-parity even and odd operators respectively.

(a)

(b)

Figure 7: Bounds on the dimension of the first singlet in the
$\Psi_{s}\times\bar{\Psi}_{s}$ OPE $\Delta_{{(\Psi\bar{\Psi})}^{\pm}}$ versus
the OPE coefficient of the displacement operator and the monodromy $v$. The
gaps on the dimension of the lowest-lying operators in all other channels is
set to the unitarity bound. Parity-even and odd operators are set to have the
same gaps. The gray line corresponds to the $\varepsilon$-expansion results.
$\Lambda=21,P=41$.

Second, we input the scaling dimensions of $\Psi_{s}$ using the predictions
from the $\varepsilon$-expansion in eq. (3.7), extrapolated to
$\varepsilon=1$. With this input, we put an upper bound to the dimension of
the lowest-lying operator in $\Psi_{s}\times\bar{\Psi}_{s}$, i.e.

$\displaystyle\Psi_{s}\times\bar{\Psi}_{s}={\mathds{1}}+{(\Psi\bar{\Psi})}^{\pm}+\dots,\quad
s=v,\text{~{}or~{}}v-1~{}.$ (4.6)

Note that ${(\Psi\bar{\Psi})}^{\pm}$ is a singlet under $SO(2)_{T}\times
SO(2)_{F}$, and it could be either $\mathcal{S}$-parity even or
$\mathcal{S}$-parity odd. For the $\Psi_{v}\times\bar{\Psi}_{v-1}$ OPE we
assume the lowest-lying operator is the displacement operator, which is
$\mathcal{S}$-parity even. We will further assume that there is no operator in
the $\mathcal{S}$-parity odd channel with dimension smaller than the
displacement, namely:

$\displaystyle(\Psi_{v}\times\bar{\Psi}_{v-1})^{+}=\text{D}+\dots\>,\quad(\Psi_{v}\times\bar{\Psi}_{v-1})^{-}=(\Psi_{v}\bar{\Psi}_{v-1})^{-}+\dots\>,\quad\Delta_{(\Psi_{v}\bar{\Psi}_{v-1})^{-}}\geq
2~{}.$ (4.7)

These assumptions are true in the $\varepsilon$-expansion, where one can see
at leading order that the displacement operator appears in the OPE with the
expected dimension $\Delta_{\text{D}}=2$, see appendix A for more details.

In figure 7 we plot the upper bound on $\Delta_{{(\Psi\bar{\Psi})}^{\pm}}$ as
a function of the $SO(2)_{F}$ charge $v$ and the OPE coefficient of the
displacement operator $(\lambda_{\Psi_{v}\bar{\Psi}_{1-v}\text{D}})^{2}$. Let
us stress that $v$ enters the crossing equations through the dimension of the
external operators, which we take to be the $\varepsilon$-expansion prediction
(3.7). Furthermore, since the crossing equations in eq. (4.5) are invariant
under $v\leftrightarrow 1-v$, it suffices to consider the range $0<v\leq 1/2$,
where the limiting case $v=1/2$ corresponds to the $\mathbb{Z}_{2}$ twist
defect studied in [25]. In the figure we observe a family of drops as we move
along the ($\lambda_{\Psi_{v}\bar{\Psi}_{1-v}\text{D}})^{2}$ direction, and as
we increase $v$ towards $v=1/2$ these drops become sharper and move slightly
to the right. It is promising that the $\varepsilon$-expansion results are
above and left of the free theory solutions, because this makes it possible
for the theoretical prediction to saturate the bound. However, for the results
with $\Lambda=21$ derivatives shown in figure 7(b), both the
$\varepsilon$-expansion result and the free theory solution are still somewhat
far from saturating the numerical bound.

One possibility that we explore in figure 8 is whether increasing the number
of derivatives can bring the bound closer to the analytical prediction. For
concreteness we focus on $v=1/3$,141414The choice $v=1/3$ is particularly
interesting for its connection to half-BPS defects in superconformal field
theories. For example, we expect that the monodromy defect for the Wess-Zumino
model [59] preserves supersymmetry whenever $v=1/3$. and we increase the
number of derivatives to $\Lambda=33$. We observe that the kink moves slightly
towards the left of the plot, but remains somewhat far from the analytical
prediction. Another possibility is that one needs to consider less agnostic
gap assumptions in the channels besides $\Psi_{v}\times\bar{\Psi}_{1-v}$, or
perhaps higher-order corrections from the $\varepsilon$-expansion are needed
to reconcile theory and numerics. In either case, this provides a good
motivation for a more detailed study of monodromy defects using the
$\varepsilon$-expansion, to which we hope to come back in future work.

Figure 8:  Bounds on the dimension of the first singlet in the
$\Psi_{s}\times\bar{\Psi}_{s}$ OPE $\Delta_{{(\Psi\bar{\Psi})}^{\pm}}$ versus
the OPE coefficient of the displacement operator
$(\lambda_{\Psi_{v}\bar{\Psi}_{1-v}D})^{2}$. $v=\frac{1}{3}$ and the gaps on
the dimension of the lowest-lying operators in all other channels are set to
the unitarity bound. Parity-even and odd operators are set to have the same
gaps. $\Lambda=21,P=41$ and $\Lambda=33,P=53$.

###  4.3 Bootstrapping the localized magnetic field line defect

This section studies the magnetic line defect of [5] in the bulk $O(3)$ CFT,
combining the numerical bootstrap with the $\varepsilon$-expansion results of
section 3.2. There are two features of the magnetic line defect which are
important in our analysis: (i) the model is invariant under a $O(2)_{F}\subset
O(3)$ flavor symmetry and, (ii) the defect spectrum features a tilt operator
$t$ transforming in the vector representation of $O(2)_{F}$. To exploit these
features, we consider a bootstrap problem involving the tilt $t$ and the
lowest-dimension neutral scalar $\phi_{1}$, so we consider the correlation
functions

$\displaystyle\langle\phi_{1}(\tau_{1})\phi_{1}(\tau_{2})\phi_{1}(\tau_{3})\phi_{1}(\tau_{4})\rangle~{}\>,$
$\displaystyle\quad\langle
t(\tau_{1})\bar{t}(\tau_{2})t(\tau_{3})\bar{t}(\tau_{4})\rangle~{}\>,$
$\displaystyle\langle\phi_{1}(\tau_{1})\phi_{1}(\tau_{2})$ $\displaystyle
t(\tau_{3})\bar{t}(\tau_{4})\rangle~{}\>,$ (4.8)

plus all other non-equivalent orderings. The corresponding crossing equations
are given in eq. (2.23).151515Since we are not including the displacement
operator, this system of correlation functions is agnostic about the co-
dimension of the line defect. It will enter only implicitly via our gap
assumptions which are determined by the $\varepsilon$-expansion results of
section 3.2. The inclusion of the displacement operator is more involved.
Nevertheless, it is an interesting extension and we will leave it for future
work. There are five OPE channels that enter in our discussion

$\displaystyle\begin{split}&\phi_{1}\times\phi_{1}\sim{\mathds{1}}+\phi_{1}+s_{-}+\cdots\>,\quad(t\times\bar{t})^{+}\sim{\mathds{1}}+\phi_{1}+s_{-}+\cdots\>,\quad\\\
&(t\times\bar{t})^{-}\sim A+\cdots,\quad t\times t\sim T+\dots\>,\quad
t\times\phi_{1}\sim t+V+\cdots\>,\end{split}$ (4.9)

where further details on the operators exchanged can be found in section
3.2.1. Note that the external operators are exchanged in some of the fusion
channels, which allows us to impose the extra relations
$\lambda_{\phi_{1}t\bar{t}}=\lambda_{t\bar{t}\phi_{1}}$ in the crossing
equations. Furthermore, we impose the following gaps

$\displaystyle\Delta_{s_{-}}=2.36\>,\qquad\Delta_{A}=3\>,\qquad\Delta_{T}=2.18\>,\qquad\Delta_{V}=3.18\>,$
(4.10)

where the values are the $O(\varepsilon)$ results for the scaling dimensions
from eqs. (3.15), (3.16) and (3.17). With these assumptions we bound the
scaling dimensions of $\phi_{1}$ and $t$. Although we could also fix the
scaling dimensions of $t$ and $\phi_{1}$ using the perturbative calculation
(in particular, $t$ has protected dimension $\Delta_{t}=1$), we keep them
unfixed to see how they are constrained by the numerics. This logic is
inspired by the search for the Ising model island, where (physically
motivated) gap assumptions led to constraints on the external operators
$\sigma$ and $\epsilon$ [46]. Our results are presented in figure 9, where the
allowed values in the $(\Delta_{t},\Delta_{\phi_{1}})$ plane are shown in
green.

(a)  (b)

Figure 9: Bootstrapping the $O(3)$-breaking line defect. In (a), bounds on the
scaling dimensions $\Delta_{\phi_{1}}$ and $\Delta_{t}$ for $\Lambda=21,P=41$.
The gaps are given in eq. (4.10). Allowed values of
$(\Delta_{t},\Delta_{\phi_{1}})$ are given in green. In (b), the bound on
$\Delta_{\phi_{1}}$ for $\Delta_{t}=1$ as a function of the number of
derivatives $\Lambda$, for which a fit is performed.

The lower bound for $t$ is strikingly close to 1, the numerics seems to be
rediscovering the tilt. For this value of $\Delta_{t}$, the bound goes up and
cuts out a corner. It should also be noted that there are numerical
instabilities in the region with large $\Delta_{\phi_{1}}$ outside the range
of figure 9(a). The lower bound on $\Delta_{\phi_{1}}$ is on the other hand
weaker, its value being nowhere close to the Padé extrapolation of
$\Delta_{\phi_{1}}\simeq 1.55$ (and it is even further away from the crude
$\varepsilon$-expansion result $\Delta_{\phi_{1}}=2+O(\varepsilon^{2})$). It
is nevertheless a non-trivial numerical result that such a lower bound exists
at all. Note that the magnetic line defect has no relevant operators [5], so
$\Delta_{\phi_{1}}>1$ should be expected. Our $O(\varepsilon)$ assumptions
allow for a weaker numerical bound on $\Delta_{\phi_{1}}$. For the physically
interesting value $\Delta_{t}=1$, figure 9(b) displays an extrapolation to
infinite number of derivatives of the lower bound, which converges to

$\Delta_{\phi_{1}}\gtrsim 0.76\,.$ (4.11)

Again, we cannot completely rule out the region with $\Delta_{\phi_{1}}\leq
1$. It could be that our numerics is not strong enough to rule out the
presence of relevant operators, so we should either change our assumptions or
include more external operators to the system of eq. (4.3). Another
possibility which we cannot rule out is the existence of alternative models
consistent with our assumptions but with one relevant scalar in the spectrum.
Definitely, we think this result calls for a more systematic investigation,
which nowadays can be efficiently performed with the help of the _Navigator
Function_ [60, 61] in order to search for bootstrap islands in a large
parameter space. Finally, one may wonder if our gap assumptions allow for
upper bounds on $\Delta_{\phi_{1}}$ as well. Clearly, $\Delta_{\phi_{1}}$
cannot be bigger than $\simeq 2.36$, which is when the next operator
$\Delta_{s_{-}}$ appears, see eq. (4.10). Below this threshold there seems to
be no upper bound, at least with the current number of derivatives.

##### Bounds on OPE coefficients.

Figure 10: Bounds on the OPE coefficients $\lambda_{t\bar{t}\phi_{1}}$ and
$\lambda_{\phi_{1}\phi_{1}\phi_{1}}$ as a function of the gap
$\Delta_{\phi_{1}}$ for the $O(3)$-breaking magnetic line defect. The black
dot is the prediction from the $\varepsilon$-expansion. The gaps are given in
eq. (4.10). $\Lambda=21,P=41$.

Having constrained the region $(\Delta_{\phi_{1}},\Delta_{t})$, we now set
$\Delta_{t}$ to its physical value of 1, but continue to treat
$\Delta_{\phi_{1}}$ as an external parameter. The goal is to bound the OPE
coefficients $(\lambda_{t\bar{t}\phi_{1}})^{2}$ and
$(\lambda_{\phi_{1}\phi_{1}\phi_{1}})^{2}$ as functions of
$\Delta_{\phi_{1}}$, which in turn were computed in the
$\varepsilon$-expansion in section 3.2, see eqs. (3.14) and (3.50). To this
end, we employ the strategy developed in [46], i.e. we introduce the OPE angle
$\theta$ defined as

$\tan\theta=\frac{\lambda_{\phi_{1}\phi_{1}\phi_{1}}}{\lambda_{t\bar{t}\phi_{1}}}\>,$
(4.12)

and we search for upper and lower bounds on the quantity
$(\lambda_{\phi_{1}\phi_{1}\phi_{1}})^{2}+(\lambda_{t\bar{t}\phi_{1}})^{2}$ as
a function of $\theta\in[0,\pi)$ and of $\Delta_{\phi_{1}}$. For concreteness
we restrict to the case where $0\leq\theta<\frac{\pi}{2}$ (i.e.
$\lambda_{\phi_{1}\phi_{1}\phi_{1}}\lambda_{t\bar{t}\phi_{1}}\geq 0$), which
is also compatible with the leading $\varepsilon$-expansion prediction. The
numerics is of course sensitive to the relative sign only. The results in
figure 10 show that for any value of the OPE coefficients
$\lambda_{\phi_{1}\phi_{1}\phi_{1}}$ and $\lambda_{t\bar{t}\phi_{1}}$, there
is an excluded region which is fully consistent with the lower bound from
figure 9(a). However, as it turns out, for
$\lambda_{\phi_{1}\phi_{1}\phi_{1}}/\lambda_{t\bar{t}\phi_{1}}\geq 0$, the
excluded region is larger and exists for $\Delta_{\phi_{1}}\lesssim 0.9$,
while for $\lambda_{\phi_{1}\phi_{1}\phi_{1}}/\lambda_{t\bar{t}\phi_{1}}\leq
0$, only values up to $\Delta_{\phi_{1}}\lesssim 0.7$ are excluded. For values
between $0.7\lesssim\Delta_{\phi_{1}}\lesssim 1.1$, the upper and lower bounds
on $\lambda_{t\bar{t}\phi_{1}}$ approach each other until they meet, after
which the upper bound goes to zero. This means that certain values of the OPE
coefficent $\lambda_{\phi_{1}\phi_{1}\phi_{1}}$ are ruled out by the numerical
bootstrap, even though the corresponding $\Delta_{\phi_{1}}$ values are
allowed in figure 9(a). For $\Delta_{\phi_{1}}\gtrsim 1.1$, the bound on
$\lambda_{t\bar{t}\phi_{1}}$ is always positive. There is an intriguing
feature visible starting around $\Delta_{\phi_{1}}\lesssim 0.9$ and continuing
until $\Delta_{\phi_{1}}\simeq 1.4$, where the projection onto the
$(\lambda_{\phi_{1}\phi_{1}\phi_{1}},\lambda_{t\bar{t}\phi_{1}})$ plane shows
a pronunced cusp, see figure 11(a). For higher values of $\Delta_{\phi_{1}}$
this feature disappears and the plot becomes smooth. For the quadrant where
the OPE coefficients have opposite sign this cusp is not present. In order to
better understand this feature, we have plotted two slices of the three-
dimensional plot in figure 11.

(a)  (b)

Figure 11: Bootstrapping the $O(3)$-breaking magnetic line defect. Upper
bounds on the OPE coefficient $\lambda_{\phi_{1}\phi_{1}\phi_{1}}$ as a
function of $\lambda_{t\bar{t}\phi_{1}}$. $\Lambda=21,P=41$ and
$\Lambda=33,P=63$. In (a) the external scalar has dimension
$\Delta_{\phi_{1}}=1.2$. In (b) the external scalar has dimension
$\Delta_{\phi_{1}}=1.55$. The $\varepsilon$-expansion result is shown by the
black dot.

For figure 11(a) we have chosen the value $\Delta_{\phi_{1}}=1.2$ somewhat
arbitrarily, in order to highlight a region where the cusp is clearly visible.
It appears around $\lambda_{\phi_{1}\phi_{1}\phi_{1}}\simeq 0.9$, although its
position shifts depending on the value of $\Delta_{\phi_{1}}$ and the number
of derivatives. In figure 11(b) we have chosen $\Delta_{\phi_{1}}=1.55$, which
corresponds to the best estimate of the conformal dimension of $\phi_{1}$
using a Padé approximation [5]. For this value of $\Delta_{\phi_{1}}$ the
features from figure 11(a) are gone and the bound is smooth. We have added in
figure 11(b) the prediction coming from the $\varepsilon$-expansion for
reference. We should warn the reader that this prediction is valid up to
$O(\varepsilon)$, for which $\Delta_{\phi_{1}}=2$. Our numerical plot was
obtained with a different set of assumptions, in particular for
$\Delta_{\phi_{1}}=1.55$, which explains why the dot is outside the numerical
exclusion region. One possibility is that the cusp is also present for
$\Delta_{\phi_{1}}\simeq 1.55$, but was lost due to numerical precision.
Indeed from figure 11(a) the feature becomes more pronounced when we jump from
$\Lambda=21$ to $\Lambda=33$ derivatives. It would be remarkable if the
numerics could be pushed, such that figure 11(b) starts looking more like
figure 11(a). The position of the cusp would then be a good candidate for the
magnetic line defect.

##  5 Conclusions and future directions

In this paper we have studied $O(2)$ line defects in a 3d bulk CFT with
$SO(2)$ flavor symmetry – the monodromy defect – or $O(3)$ flavor symmetry –
the magnetic line defect – using the conformal bootstrap. After writing the
corresponding crossing equations we applied the numerical machinery in order
to obtain exclusion bounds valid for generic 1d defect CFTs. We concentrated
mostly on two canonical operators: the displacement and the tilt. The
displacement operator is truly universal in that it controls the breaking of
translations due to the presence of the conformal defect. The tilt operator
can exist when a conformal defect breaks a bulk global symmetry, which is
still quite a generic phenomenon.

After obtaining general bounds that constrain the landscape of 1d theories, we
changed gears and focused on two models of interest: the monodromy defect
studied in [38, 39, 40], and the magnetic line defect described in [5]. In
order to guide the numerics we complemented our bootstrap analysis with
explicit analytic calculations in the $\varepsilon$-expansion. For the
monodromy defect, the results had already appeared in [38, 39], while for the
magnetic line defect most of the calculations presented in section 3.2 are
new.

For the monodromy defect we have kept the gap assumptions to a minimum, using
the dimension of the external operators from the $\varepsilon$-expansion
results and imposing the existence of the displacement operator. Hence the
numerical results are quite general, and they are fully consistent with the
analytical data. The results for the magnetic line defect are more
constraining. By imposing gaps coming from the $\varepsilon$-expansion, the
allowed region for certain OPE coefficients shows a series of intriguing
cusps. The location of these cusps however, does not match the best numerical
estimates of the conformal dimension of the scalar field
$\Delta_{\phi_{1}}\simeq 1.55$. Nevertheless, we consider this result an
encouraging sign. Ways in which our methods can be improved include higher
order terms in the $\varepsilon$-expansion, and higher precision in the
numerics. The hope is that with these improvements, the region where the
magnetic line defected is expected to sit also develops a cusp.

There are several avenues in which our research can be extended. A natural
question to ask is whether one could construct a monodromy defect that breaks
a bulk $O(3)$ symmetry to an $O(2)$ symmetry on the line. Such a defect would
have a tilt operator, a displacement, and fundamental fields with transverse
spin related to the charge of the monodromy. Mixed correlators with three
external operators have already been studied in [62, 63, 64] in the context of
$O(2)$ and $O(3)$ models.

Another interesting possibilty is to reproduce the $\varepsilon$-expansion
results using purely bootstrap means, without explicit Feynman diagram
calculations. This line of thinking has been particularly succesful, with
numerous analytical results obtained using only symmetry arguments and basic
structural constraints. A promising framework tailored for our setup is that
of analytic functionals, which were used in [36] to study the Ising twist line
defect with $\mathbb{Z}_{2}$ global symmetry.

Furthermore, we could extend our results to line defects with $O(N)$ flavor
symmetry. In addition to the analytical tools used in this work, we would gain
access to the large $N$ expansion, giving us perturbative results in a
different regime to compare with the numerics. At the same time, these line
defects could break a bulk $O(N+m)$ symmetry to a defect $O(N)$ symmetry,
where $N,m\in\mathbb{Z}$. In this generalization, the tilt will no longer be a
vector under $O(2)$ flavour symmetry, but will be part of another
representation.

Finally, our setup can generalized to study monodromy line defects in
superconformal models, like the monodromy defect of the Wess-Zumino model
studied in [59]. This model contains a displacement as well as a tilt operator
due to the breaking of the $R$-symmetry to a $U(1)$ symmetry on the defect,
which in turn sit inside protected multiplets of the superconformal algebra.
It would be interesting to pursue a systematic study of this setup using the
superconformal bootstrap and we leave it for the future [65].

## Acknowledgements

We would like to thank A. Antunes, N. Bobev, I. Burić, G. Cuomo, A. Kaviraj,
D. Mazáč, J. Rong, V. Schomerus, E. Trevisani and B. van Rees for discussions.
Also, we would like to thank the organizers of the conference ‘Bootstrap 2022’
for providing a place for fruitful discussions. EL would like to thank A.
Antunes and B. van Rees for their collaboration and discussions on related
projects. EL is supported by the Simons Foundation grant $\\#$488659 (Simons
Collaboration on the non-perturbative bootstrap). AG , PL and PvV acknowledge
support from the DFG through the Emmy Noether research group ‘The Conformal
Bootstrap Program’ project number 400570283, and through the German-Israeli
Project Cooperation (DIP) grant ‘Holography and the Swampland’.

##  Appendix A More data on the $O(N)$ monodromy defect in 3d

In this appendix, we give a more detailed explanation of the calculation of
the dimensions and OPE coefficients given in section 3.1 and computed in [39].

###  A.1 OPE coefficients with the displacement operator

The displacement operator appears in the OPE of two defect modes
$\Psi_{v}\times\bar{\Psi}_{v-1}\sim\text{D}+\cdots$. Hence, we are interested
in the correlators involving these defect modes, which are

$\displaystyle\begin{split}\langle\Psi_{v}(\vec{x}_{1})\bar{\Psi}_{v}(\vec{x}_{2})\Psi_{v}(\vec{x}_{3})\bar{\Psi}_{v}(\vec{x}_{4})\rangle\>,&\quad\langle\Psi_{1-v}(\vec{x}_{1})\bar{\Psi}_{1-v}(\vec{x}_{2})\Psi_{1-v}(\vec{x}_{3})\bar{\Psi}_{1-v}(\vec{x}_{4})\rangle\>,\\\
\langle\Psi_{v}(\vec{x}_{1})\bar{\Psi}_{1-v}&(\vec{x}_{2})\Psi_{1-v}(\vec{x}_{3})\bar{\Psi}_{v}(\vec{x}_{4})\rangle\>.\end{split}$
(A.1)

The results for the single-operator correlators were already computed in [25]
for a $\mathbb{Z}_{2}$ monodromy defect. For correlators with two operators
$\Psi_{1}$ and $\Psi_{2}$, the anomalous dimensions of the operators appearing
in the OPEs are computed in [39], and from their results also the OPE
coefficients can be readily derived.161616Note that our conventions differ
slightly from those used in [39]; in particular, we normalize the two-point
function as
$\langle\bar{\Psi}_{s_{1}}(\vec{x}_{1})\Psi_{s_{2}}(\vec{x}_{2})\rangle_{0}=\frac{\delta_{s_{1},s_{2}}}{\vec{x}_{12}^{2\Delta_{s_{1}}}}\>,$
(A.2) without an extra factor of $\mathcal{C}_{\Delta_{s_{1}}}$ that was
present in [39]. This does not have any impact on the anomalous dimensions,
but needs to be taken into account when computing the OPE coefficients. The
anomalous dimensions and first-order corrections to the OPE coefficients
appear in the four-point correlator as

$\displaystyle\begin{split}&\langle\Psi_{\alpha+v}(\vec{x}_{1})\bar{\Psi}_{\alpha-s+v}(\vec{x}_{2})\Psi_{\beta-s+v}(\vec{x}_{3})\bar{\Psi}_{\beta+v}(\vec{x}_{4})\rangle\in\sum_{m}\big{(}|\lambda^{(0)}_{\Psi\bar{\Psi}\bar{{\mathcal{O}}}_{\alpha}^{s,m}}|^{2}+\varepsilon|\lambda^{(1)}_{\Psi\bar{\Psi}\bar{{\mathcal{O}}}_{\alpha}^{s,m}}|^{2}\big{)}\\\
&\times\left(\delta_{\alpha\beta}+\frac{\varepsilon}{2}\Delta^{s,m}_{\alpha\beta}\partial_{m}\right)W_{d-2+s+2m,0}(\vec{x}_{i})\>,\end{split}$
(A.3)

where

$W_{\Delta,\ell}(\vec{x}_{i})=\left(\frac{\vec{x}_{24}^{2}}{\vec{x}_{14}^{2}}\right)^{\frac{1}{2}\Delta_{12}}\left(\frac{\vec{x}_{14}^{2}}{\vec{x}_{13}^{2}}\right)^{\frac{1}{2}\Delta_{34}}\frac{G_{\Delta,\ell}(u,v)}{\vec{x}_{12}^{\frac{1}{2}(\Delta_{1}+\Delta_{2})}\vec{x}_{34}^{\frac{1}{2}(\Delta_{3}+\Delta_{4})}}\>,$
(A.4)

and $G_{\Delta,\ell}$ are the conformal blocks as defined in [45]. The
variables $\alpha,\beta$ in eq. (LABEL:eq:4ptMonDef) refer to degeneracies
between operators that have the same tree-level conformal dimension and
$SO(2)_{T}$ spin. For the correlator of interest to us, we set $s=1$ and since
$\alpha,\beta=0,1,\cdots,s-1=0$ [39], we will not have to worry about said
degeneracies.

The anomalous dimensions will receive contributions from the logarithmic term
of the four-point correlator, and from the $\mathcal{O}(\varepsilon)$ term of
the dimension of the external operators. The anomalous dimension of the
external operators was given in eq. 3.7, and gives a contribution of

$\Delta^{(1);\text{disc}}=\Delta_{s_{1}}^{(1)}+\Delta_{s_{2}}^{(1)}=\frac{v(v-1)(N+2)}{2(N+8)}\left(\frac{1}{|s_{1}|}+\frac{1}{|s_{2}|}\right)$
(A.5)

to the anomalous dimension of the operators appearing in their OPEs. At tree
level, the four-point correlator is given by the free theory discussed in eq.
(2.25). At $O(\varepsilon)$, one can compute the correction to the correlator
through a contact Witten diagram with the four defect modes $\Psi_{s_{i}}$ as
external operators. These diagrams are described by the well-known D-functions
[66, 67, 68]. The first-order correction to the correlator in eq.
(LABEL:eq:4ptMonDef) is given by [39]

$\langle\Psi_{v}(\vec{x}_{1})\bar{\Psi}_{v-1}(\vec{x}_{2})\Psi_{v-1}(\vec{x}_{3})\bar{\Psi}_{v}(\vec{x}_{4})\rangle_{1}=-\pi\lambda_{*}\times
D_{\Delta_{\Psi_{v}},\Delta_{\Psi_{v-1}},\Delta_{\Psi_{v-1}},\Delta_{\Psi_{v}}}(\vec{x}_{i})\>,$
(A.6)

where $\lambda_{*}\sim O(\varepsilon)$ is given in eq. (3.6). The D-function
can be expanded in conformal blocks as [69, 70]:

$D_{\Delta_{s_{1}},\Delta_{s_{2}},\Delta_{s_{3}},\Delta_{s_{4}}}(\vec{x}_{i})=\sum_{m}P_{1}^{(12)}(m,0)W_{\Delta_{m},0}(\vec{x}_{i})+\sum_{n}P_{1}^{(34)}(n,0)W_{\Delta_{n},0}(\vec{x}_{i})\>.$
(A.7)

Note that since $\lambda_{*}\sim O(\varepsilon)$, we can evaluate the
conformal blocks at $p=d-2=2$, and the dimensions
$\Delta_{m}=\Delta_{s_{1}}+\Delta_{s_{2}}+2m$ contribute at tree level. The
coefficients $P_{1}(m,0)$ are given in [39].

When
$\Delta_{s_{1}}+\Delta_{s_{2}}-\Delta_{s_{3}}-\Delta_{s_{4}}=2k,k\in\mathbb{Z}$,
where $\Delta_{s_{i}}$ are evaluated at tree level, the coefficients have a
divergence and hence contribute to the anomalous dimension. For the correlator
we study, $\Delta_{v}+\Delta_{v-1}-\Delta_{v-1}-\Delta_{v}=0$. In this case,
the D-function can be written as [39]:

$\displaystyle\begin{split}&D_{\Delta_{s_{1}},\Delta_{s_{2}},\Delta_{s_{3}},\Delta_{s_{4}}}(\vec{x}_{i})=\sum_{m}\frac{\pi\big{(}\Pi_{i=1}^{4}(\Delta_{s_{i}})_{m}\big{)}(\Delta_{s_{1}}+\Delta_{s_{2}}+2m-1)^{2}_{-m}}{2(m!)^{2}(\Delta_{s_{1}}+\Delta_{s_{2}}+2m-1)}\Big{(}\frac{2}{\Delta_{s_{1}}+\Delta_{s_{2}}+2m-1}\\\
&+2H^{m}-2H^{m+\Delta_{s_{1}}+\Delta_{s_{2}}-2}+4H^{2m+\Delta_{s_{1}}+\Delta_{s_{2}}-2}-\sum_{i=1}^{4}H^{m+\Delta_{s_{i}}-1}-\partial_{m}\Big{)}W_{\Delta_{s_{1}}+\Delta_{s_{2}}+2m,0}(\vec{x}_{i})\>,\end{split}$
(A.8)

where $H^{m}$ are harmonic numbers. Setting $s_{1}=s_{4}=v\>,\quad
s_{2}=s_{3}=v-1$, we can extract the contribution to the anomalous dimension
from the logarithmic part:

$\Delta^{(1);\text{con}}=\frac{4}{10(2+2m)}\>,$ (A.9)

and the $O(\varepsilon)$ correction to the OPE coefficients from the non-
divergent part, for $m=0$:

$|\lambda^{(1)}_{\Psi_{v}\bar{\Psi}_{v-1}{\mathcal{O}}_{0,0}^{1,0}}|^{2}=\frac{(N+2)\left(2H^{1-v}+2H^{v}-3\right)}{2N(N+8)}\>.$
(A.10)

Combining the contribution in eq. (A.9) and the contribution in eq. (A.5), we
get

$\Delta_{0,0}^{1,m}=\frac{1}{5(1+m)}-\frac{1}{5}\>,$ (A.11)

such that indeed, for $m=0$, $\Delta_{0,0}^{1,0}=0$ and the displacement, a
protected operator, does not get any anomalous dimension.

###  A.2 The leading singlet

The anomalous dimension of the first singlet in the
$\Psi_{v}\times\bar{\Psi}_{v}$, or the $\Psi_{v-1}\times\bar{\Psi}_{v-1}$ OPE,
and its OPE coefficient, can be computed in the same way as for the
$\Psi_{v}\times\bar{\Psi}_{v-1}$ OPE. We still do not encounter degeneracies.
The correction coming from the anomalous dimensions of the external operators
is now given by

$2\Delta_{s_{i}}^{(1);\text{disc}}=\frac{v(v-1)(N+2)}{(N+8)}\frac{1}{|s_{i}|}\>,$
(A.12)

and the tree-level dimensions and OPE coefficients of the operators in the
$\Psi_{s_{i}}\times\bar{\Psi}_{s_{i}}$ OPE are given by the free theory
results described in eq. (2.19). The first-order correction to the correlator
is given by

$\langle\Psi_{s_{i}}(\vec{x}_{1})\bar{\Psi}_{s_{i}}(\vec{x}_{2})\Psi_{s_{i}}(\vec{x}_{3})\bar{\Psi}_{s_{i}}(\vec{x}_{4})\rangle_{1}=-2\pi\lambda_{*}\times
D_{\Delta_{s_{i}},\Delta_{s_{i}},\Delta_{s_{i}},\Delta_{s_{i}}}(\vec{x}_{i})\>,$
(A.13)

and we can use the same conformal block expansion of the D-functions in eq.
(A.7). Since all external dimensions are equal, the relation
$\Delta_{s_{1}}+\Delta_{s_{2}}-\Delta_{s_{3}}-\Delta_{s_{4}}=2k,k\in\mathbb{Z}$
holds. Hence, the coefficients $P_{1}(m,0)$ have a divergence and the
D-function is given by eq. (LABEL:eq:DfuncExp2). We extract the contributions
to the anomalous dimensions from the logarithmic part of eq.
(LABEL:eq:DfuncExp2) and obtain

$\Delta^{(1);\text{con}}_{v}=\frac{2(N+2)}{(N+8)(1+2v)}\>,\quad\Delta^{(1);\text{con}}_{v-1}=\frac{2(N+2)}{(N+8)(3-2v)}\>.$
(A.14)

The $O(\varepsilon)$ corrections to the OPE coefficients is now given by

$\displaystyle|\lambda^{(1)}_{\Psi_{v}\bar{\Psi}_{v}{\mathcal{O}}_{0}}|^{2}$
$\displaystyle=\frac{2(N+2)\left(2H^{v}-H^{2v}\right)}{(1+2v)N(N+8)}-\frac{2(N+2)}{(1+2v)^{2}N(N+8)}\>,$
(A.15)
$\displaystyle|\lambda^{(1)}_{\Psi_{v-1}\bar{\Psi}_{v-1}{\mathcal{O}}_{0}}|^{2}$
$\displaystyle=\frac{2(N+2)\left(2H^{1-v}-H^{2-2v}\right)}{(3-2v)N(N+8)}-\frac{2(N+2)}{(3-2v)^{2}N(N+8)}\>.$
(A.16)

Adding the results from eq. (A.14) and eq. (A.12), we obtain the following
anomalous dimensions:

$\displaystyle\Delta^{0,m}_{0,0}|_{s_{i}=v}=\frac{2(N+2)}{(N+8)(1+2v)}+\frac{(N+2)(v-1)}{(N+8)}\>,$
(A.17)
$\displaystyle\Delta^{0,m}_{0,0}|_{s_{i}=v-1}=\frac{2(N+2)}{(N+8)(3-2v)}-\frac{(N+2)v}{(N+8)}\>.$
(A.18)

All results are summarized in table 1.

OPE | $s$ | $\Delta^{(1)}$ | $\lambda^{(1)}$
---|---|---|---
$\Psi_{v}\times\bar{\Psi}_{v-1}$ | 1 | $\frac{N+2}{(N+8)(2+2m)}-\frac{(N+2)}{2(N+8)}$ | $\frac{(N+2)\left(2H^{1-v}+2H^{v}-3\right)}{2N(N+8)}$
$\Psi_{v-1}\times\bar{\Psi}_{v-1}$ | $0$ | $\frac{2(N+2)}{(N+8)(3-2v)}-\frac{(N+2)v}{(N+8)}$ | $\frac{2(N+2)\left(2H^{1-v}-H^{2-2v}\right)}{(3-2v)N(N+8)}-\frac{2(N+2)}{(3-2v)^{2}N(N+8)}$
$\Psi_{v}\times\bar{\Psi}_{v}$ | $0$ | $\frac{2(N+2)}{(N+8)(1+2v)}+\frac{(N+2)(v-1)}{(N+8)}$ | $\frac{2(N+2)\left(2H^{v}-H^{2v}\right)}{(1+2v)N(N+8)}-\frac{2(N+2)}{(1+2v)^{2}N(N+8)}$
Table 2: The $\varepsilon$-expansion results for various operators, $m=0$

We have only considered the $\Psi_{s_{1}}\times\bar{\Psi}_{s_{2}}$ channel.
One can also consider the $\Psi_{s_{1}}\times\Psi_{s_{2}}$ channel, which
contains operators of the form
${\mathcal{O}}_{\alpha}^{s,m}=\Psi_{\alpha+v}(\vec{\partial}^{2})^{m}\Psi_{s-\alpha-v}$
that have fractional spin $s=s_{1}+s_{2}=k+2v\in\mathbb{Z}+2v$. Since the
anomalous dimensions have been given in [39] and we have shown how to obtain
the OPE coefficients from their results, we will not repeat the calculation
for this channel in this work.

##  Appendix B Crossing vectors

In this appendix, we give explicit formulas for the vectors that enter the
crossing equations.

###  B.1 One complex scalar

The crossing vectors for eq. (2.18) read

$\vec{V}_{\Delta,S}^{\phi\bar{\phi}}=\begin{pmatrix}F_{-,\Delta}\\\
(-1)^{S}F_{-,\Delta}\\\
(-1)^{S}F_{+,\Delta}\end{pmatrix}\>,\qquad\vec{V}_{\Delta}^{\phi\phi}=\begin{pmatrix}0\\\
F_{-,\Delta}\\\ -F_{+,\Delta}\end{pmatrix}\>,$ (B.1)

where the shorthand notation
$F_{\pm,\Delta}=F_{\pm,\Delta}^{\phi\phi\phi\phi}(\xi)$ is understood.

###  B.2 Tilt and displacement

The crossing equations for the tilt and displacement in eq. (2.22) are written
in terms of the following crossing vectors:

$\vec{V}^{+}_{\Delta}=\begin{pmatrix}\begin{pmatrix}F^{tt,tt}_{-,\Delta}&0\\\
0&0\end{pmatrix}\\\ \begin{pmatrix}F^{tt,tt}_{-,\Delta}&0\\\
0&0\end{pmatrix}\\\\[10.00002pt] \begin{pmatrix}F^{tt,tt}_{+,\Delta}&0\\\
0&0\end{pmatrix}\\\ \begin{pmatrix}0&0\\\
0&F^{DD,DD}_{-,\Delta}\end{pmatrix}\\\ \begin{pmatrix}0&0\\\
0&F^{DD,DD}_{-,\Delta}\end{pmatrix}\\\\[10.00002pt] \begin{pmatrix}0&0\\\
0&F^{DD,DD}_{+,\Delta}\end{pmatrix}\\\ 0\\\ \begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}F^{tt,DD}_{-,\Delta}\\\\[10.00002pt]
\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}F^{tt,DD}_{+,\Delta}\\\
\end{pmatrix}\,,\quad\vec{V}^{t\bar{t},-}_{\Delta}=\begin{pmatrix}F^{tt,tt}_{-,\Delta}\\\
-F^{tt,tt}_{-,\Delta}\\\ -F^{tt,tt}_{+,\Delta}\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
0\end{pmatrix}\>,\quad\vec{V}^{D\bar{D},-}_{\Delta}=\begin{pmatrix}0\\\ 0\\\
0\\\ F^{DD,DD}_{-,\Delta}\\\ -F^{DD,DD}_{-,\Delta}\\\ -F^{DD,DD}_{+,\Delta}\\\
0\\\ 0\\\ 0\end{pmatrix}\>,$ (B.2) $\vec{V}^{tt}_{\Delta}=\begin{pmatrix}0\\\
F^{tt,tt}_{-,\Delta}\\\ -F^{tt,tt}_{+,\Delta}\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
0\end{pmatrix}\>,\quad\vec{V}^{DD}_{\Delta}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\
F^{DD,DD}_{-,\Delta}\\\ -F^{DD,DD}_{+,\Delta}\\\ 0\\\ 0\\\
0\end{pmatrix}\>,\quad\vec{V}^{tD}_{\Delta,S}=\begin{pmatrix}0\\\ 0\\\ 0\\\
0\\\ 0\\\ 0\\\ (-1)^{S}F^{tD,tD}_{-,\Delta}\\\ F^{Dt,tD}_{-,\Delta}\\\
-F^{Dt,tD}_{+,\Delta}\end{pmatrix}\>.$ (B.3)

###  B.3 One real and one complex scalar

The crossing equations for one real scalar $\phi_{1}$ and one complex scalar
$\phi_{2}$ appear in eq. (2.23), with crossing vectors

$\displaystyle\vec{V}^{+}_{\Delta}=\begin{pmatrix}\begin{pmatrix}F^{11,11}_{-,\Delta}&0\\\
0&0\end{pmatrix}\\\\[10.00002pt] \begin{pmatrix}0&0\\\
0&F^{22,22}_{-,\Delta}\end{pmatrix}\\\\[10.00002pt] \begin{pmatrix}0&0\\\
0&F^{22,22}_{-,\Delta}\end{pmatrix}\\\\[10.00002pt] \begin{pmatrix}0&0\\\
0&F^{22,22}_{+,\Delta}\end{pmatrix}\\\ 0\\\ \begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}F^{11,22}_{-,\Delta}\\\\[10.00002pt]
\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}F^{11,22}_{+,\Delta}\end{pmatrix}\>,\qquad\vec{V}^{-}_{\Delta}=\begin{pmatrix}0\\\
F^{22,22}_{-,\Delta}\\\ -F^{22,22}_{-,\Delta}\\\ -F^{22,22}_{+,\Delta}\\\ 0\\\
0\\\ 0\end{pmatrix}\>,$ (B.4)
$\displaystyle\vec{V}^{22}_{\Delta}=\begin{pmatrix}0\\\ 0\\\
F^{22,22}_{-,\Delta}\\\ -F^{22,22}_{+,\Delta}\\\ 0\\\ 0\\\
0\end{pmatrix}\>,\qquad\vec{V}^{12}_{\Delta,S}=\begin{pmatrix}0\\\ 0\\\ 0\\\
0\\\ (-1)^{S}F^{12,12}_{-,\Delta}\\\ F^{21,12}_{-,\Delta}\\\
-F^{21,12}_{+,\Delta}\end{pmatrix}\>.$ (B.5)

Here we are using
$F_{\pm,\Delta}^{ij,kl}=F_{\pm,\Delta}^{\phi_{i}\phi_{j},\phi_{k}\phi_{l}}(\xi)$,
and we do not distinguish between $\phi_{2}$ and $\bar{\phi}_{2}$ because they
have the same scaling dimension.

###  B.4 Two complex scalars

The crossing equations for two complex scalars given in eq. (2.24) are written
in terms of the following crossing vectors:

$\vec{V}_{\Delta,S}=\begin{pmatrix}\begin{pmatrix}F^{11,11}_{-,\Delta}&0\\\
0&0\end{pmatrix}\\\ \begin{pmatrix}(-1)^{S}F^{11,11}_{-,\Delta}&0\\\
0&0\end{pmatrix}\\\\[10.00002pt]
\begin{pmatrix}(-1)^{S}F^{11,11}_{+,\Delta}&0\\\ 0&0\end{pmatrix}\\\
\begin{pmatrix}0&0\\\ 0&F^{22,22}_{-,\Delta}\end{pmatrix}\\\
\begin{pmatrix}0&0\\\
0&(-1)^{S}F^{22,22}_{-,\Delta}\end{pmatrix}\\\\[10.00002pt]
\begin{pmatrix}0&0\\\ 0&(-1)^{S}F^{22,22}_{+,\Delta}\end{pmatrix}\\\ 0\\\ 0\\\
\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}F^{11,22}_{-,\Delta}\\\\[10.00002pt]
\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}F^{11,22}_{+,\Delta}\\\
\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}(-1)^{S}F^{11,22}_{-,\Delta}\\\\[10.00002pt]
\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}(-1)^{S}F^{11,22}_{+,\Delta}\\\
\end{pmatrix}\,,\quad\vec{V}^{11}_{\Delta}=\begin{pmatrix}0\\\
F^{11,11}_{-,\Delta}\\\ -F^{11,11}_{+,\Delta}\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
0\\\ 0\\\ 0\end{pmatrix}\>,\quad\vec{V}^{22}_{\Delta}=\begin{pmatrix}0\\\ 0\\\
0\\\ 0\\\ F^{22,22}_{-,\Delta}\\\ -F^{22,22}_{+,\Delta}\\\ 0\\\ 0\\\ 0\\\ 0\\\
0\\\ 0\end{pmatrix}\>,$ (B.6)
$\vec{V}^{1\bar{2}}_{\Delta,S}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
(-1)^{S}F^{12,12}_{-,\Delta}\\\ -(-1)^{S}F^{12,12}_{+,\Delta}\\\
F^{21,12}_{-,\Delta}\\\ -F^{21,12}_{+,\Delta}\\\ 0\\\
0\end{pmatrix}\>,\quad\vec{V}^{12}_{\Delta,S}=\begin{pmatrix}0\\\ 0\\\ 0\\\
0\\\ 0\\\ 0\\\ (-1)^{S}F^{12,12}_{-,\Delta}\\\ (-1)^{S}F^{12,12}_{+,\Delta}\\\
0\\\ 0\\\ F^{21,12}_{-,\Delta}\\\ -F^{21,12}_{+,\Delta}\end{pmatrix}\>.$ (B.7)

## References

  * [1] I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869–1932, [cond-mat/9512099].
  * [2] M. Billó, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri, and R. Pellegrini, Line defects in the 3d Ising model, JHEP 07 (2013) 055, [arXiv:1304.4110].
  * [3] S. Sachdev, C. Buragohain, and M. Vojta, Quantum impurity in a nearly critical two dimensional antiferromagnet, 2000.
  * [4] M. Vojta, C. Buragohain, and S. Sachdev, Quantum impurity dynamics in two-dimensional antiferromagnets and superconductors, Physical Review B 61 (Jun, 2000) 15152–15184.
  * [5] G. Cuomo, Z. Komargodski, and M. Mezei, Localized magnetic field in the O(N) model, JHEP 02 (2022) 134, [arXiv:2112.10634].
  * [6] G. Cuomo, Z. Komargodski, M. Mezei, and A. Raviv-Moshe, Spin Impurities, Wilson Lines and Semiclassics, arXiv:2202.00040.
  * [7] K. G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445–2459.
  * [8] G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1–25.
  * [9] A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005, [hep-th/0501015].
  * [10] O. J. C. Dias, G. T. Horowitz, N. Iqbal, and J. E. Santos, Vortices in holographic superfluids and superconductors as conformal defects, JHEP 04 (2014) 096, [arXiv:1311.3673].
  * [11] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [arXiv:1412.5148].
  * [12] O. Aharony, N. Seiberg, and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115, [arXiv:1305.0318].
  * [13] L. Bianchi, M. Meineri, R. C. Myers, and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076, [arXiv:1511.06713].
  * [14] L. Bianchi, S. Chapman, X. Dong, D. A. Galante, M. Meineri, and R. C. Myers, Shape dependence of holographic Rényi entropy in general dimensions, JHEP 11 (2016) 180, [arXiv:1607.07418].
  * [15] D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789–855, [arXiv:0804.2902].
  * [16] D. Gaiotto and E. Witten, S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009), no. 3 721–896, [arXiv:0807.3720].
  * [17] L. Di Pietro, D. Gaiotto, E. Lauria, and J. Wu, 3d Abelian Gauge Theories at the Boundary, JHEP 05 (2019) 091, [arXiv:1902.09567].
  * [18] L. Di Pietro, E. Lauria, and P. Niro, 3d large $N$ vector models at the boundary, SciPost Phys. 11 (2021), no. 3 050, [arXiv:2012.07733].
  * [19] M. F. Paulos, S. Rychkov, B. C. van Rees, and B. Zan, Conformal Invariance in the Long-Range Ising Model, Nucl. Phys. B 902 (2016) 246–291, [arXiv:1509.00008].
  * [20] S. Giombi and H. Khanchandani, $O(N)$ Models with Boundary Interactions and their Long Range Generalizations, arXiv:1912.08169.
  * [21] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031, [arXiv:0807.0004].
  * [22] D. Poland, S. Rychkov, and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002, [arXiv:1805.04405].
  * [23] S. M. Chester, Weizmann Lectures on the Numerical Conformal Bootstrap, arXiv:1907.05147.
  * [24] A. Bissi, A. Sinha, and X. Zhou, Selected Topics in Analytic Conformal Bootstrap: A Guided Journey, arXiv:2202.08475.
  * [25] D. Gaiotto, D. Mazac, and M. F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100, [arXiv:1310.5078].
  * [26] P. Liendo, C. Meneghelli, and V. Mitev, Bootstrapping the half-BPS line defect, JHEP 10 (2018) 077, [arXiv:1806.01862].
  * [27] A. Gimenez-Grau and P. Liendo, Bootstrapping line defects in $\mathcal{N}=2$ theories, JHEP 03 (2020) 121, [arXiv:1907.04345].
  * [28] C. Behan, Bootstrapping the long-range Ising model in three dimensions, J. Phys. A52 (2019), no. 7 075401, [arXiv:1810.07199].
  * [29] C. Behan, L. Di Pietro, E. Lauria, and B. C. Van Rees, Bootstrapping boundary-localized interactions, JHEP 12 (2020) 182, [arXiv:2009.03336].
  * [30] C. Behan, L. Di Pietro, E. Lauria, and B. C. van Rees, Bootstrapping boundary-localized interactions II. Minimal models at the boundary, JHEP 03 (2022) 146, [arXiv:2111.04747].
  * [31] N. Kobayashi and T. Nishioka, Spinning conformal defects, JHEP 09 (2018) 134, [arXiv:1805.05967].
  * [32] F. Gliozzi, P. Liendo, M. Meineri, and A. Rago, Boundary and Interface CFTs from the Conformal Bootstrap, JHEP 05 (2015) 036, [arXiv:1502.07217].
  * [33] F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP 10 (2016) 037, [arXiv:1605.04175].
  * [34] E. Lauria, P. Liendo, B. C. van Rees, and X. Zhao, Line and surface defects for the free scalar field, arXiv:2005.02413.
  * [35] C. P. Herzog and A. Shrestha, Conformal Surface Defects in Maxwell Theory are Trivial, arXiv:2202.09180.
  * [36] K. Ghosh, A. Kaviraj, and M. F. Paulos, Charging up the functional bootstrap, JHEP 10 (2021) 116, [arXiv:2107.00041].
  * [37] F. P. Toldin, F. F. Assaad, and S. Wessel, Critical behavior in the presence of an order-parameter pinning field, Physical Review B 95 (jan, 2017).
  * [38] A. Söderberg, Anomalous Dimensions in the WF O($N$) Model with a Monodromy Line Defect, JHEP 03 (2018) 058, [arXiv:1706.02414].
  * [39] S. Giombi, E. Helfenberger, Z. Ji, and H. Khanchandani, Monodromy defects from hyperbolic space, JHEP 02 (2022) 041, [arXiv:2102.11815].
  * [40] L. Bianchi, A. Chalabi, V. Procházka, B. Robinson, and J. Sisti, Monodromy defects in free field theories, JHEP 08 (2021) 013, [arXiv:2104.01220].
  * [41] M. Billò, V. Gonçalves, E. Lauria, and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091, [arXiv:1601.02883].
  * [42] A. J. Bray and M. Moore, Critical behaviour of semi-infinite systems, Journal of Physics A: Mathematical and General 10 (1977) 1927.
  * [43] G. Cuomo, M. Mezei, and A. Raviv-Moshe, Boundary conformal field theory at large charge, JHEP 10 (2021) 143, [arXiv:2108.06579].
  * [44] J. Padayasi, A. Krishnan, M. A. Metlitski, I. A. Gruzberg, and M. Meineri, The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap, arXiv:2111.03071.
  * [45] F. A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194.
  * [46] F. Kos, D. Poland, and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109, [arXiv:1406.4858].
  * [47] D. Poland, D. Simmons-Duffin, and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110, [arXiv:1109.5176].
  * [48] A. L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-Matrix, JHEP 10 (2012) 032, [arXiv:1112.4845].
  * [49] M. Lemos and P. Liendo, Bootstrapping $\mathcal{N}=2$ chiral correlators, JHEP 01 (2016) 025, [arXiv:1510.03866].
  * [50] F. Parisen Toldin, F. F. Assaad, and S. Wessel, Critical behavior in the presence of an order-parameter pinning field, prb 95 (Jan., 2017) 014401, [arXiv:1607.04270].
  * [51] A. Allais, Magnetic defect line in a critical Ising bath, arXiv:1412.3449.
  * [52] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Bootstrapping the O(N) Archipelago, JHEP 11 (2015) 106, [arXiv:1504.07997].
  * [53] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Precision islands in the Ising and $O(N)$ models, JHEP 08 (2016) 036, [arXiv:1603.04436].
  * [54] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\phi^{4}$ theories.  World Scientific Publishing Company Incorporated, 2001.
  * [55] E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071, [arXiv:1401.4361].
  * [56] E. Panzer, Feynman integrals and hyperlogarithms.  PhD thesis, Humboldt U., 2015.  arXiv:1506.07243.
  * [57] E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148–166, [arXiv:1403.3385].
  * [58] D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174, [arXiv:1502.02033].
  * [59] A. Gimenez-Grau and P. Liendo, Bootstrapping Monodromy Defects in the Wess-Zumino Model, arXiv:2108.05107.
  * [60] M. Reehorst, S. Rychkov, D. Simmons-Duffin, B. Sirois, N. Su, and B. van Rees, Navigator Function for the Conformal Bootstrap, SciPost Phys. 11 (2021) 072, [arXiv:2104.09518].
  * [61] M. Reehorst, Rigorous bounds on irrelevant operators in the 3d Ising model CFT, arXiv:2111.12093.
  * [62] M. Go and Y. Tachikawa, autoboot: A generator of bootstrap equations with global symmetry, JHEP 06 (2019) 084, [arXiv:1903.10522].
  * [63] S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Carving out OPE space and precise $O(2)$ model critical exponents, JHEP 06 (2020) 142, [arXiv:1912.03324].
  * [64] S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Bootstrapping Heisenberg magnets and their cubic instability, Phys. Rev. D 104 (2021), no. 10 105013, [arXiv:2011.14647].
  * [65] A. Gimenez-Grau, E. Lauria, P. Liendo, and P. van Vliet, Work in progress.
  * [66] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Graviton exchange and complete four point functions in the AdS / CFT correspondence, Nucl. Phys. B 562 (1999) 353–394, [hep-th/9903196].
  * [67] H. Liu and A. A. Tseytlin, On four point functions in the CFT / AdS correspondence, Phys. Rev. D 59 (1999) 086002, [hep-th/9807097].
  * [68] F. A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B599 (2001) 459–496, [hep-th/0011040].
  * [69] E. Hijano, P. Kraus, E. Perlmutter, and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146, [arXiv:1508.00501].
  * [70] C. B. Jepsen and S. Parikh, Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, JHEP 10 (2019) 268, [arXiv:1906.08405].