aainstitutetext: Laboratory for Theoretical Fundamental Physics, EPFL, Rte de
la Sorge, CH-1015, Lausannebbinstitutetext: School of Physics and Astronomy,
Tel Aviv University, Ramat Aviv 69978, Israelccinstitutetext: Theoretical
Physics Group, The Blackett Laboratory, Imperial College London, Prince
Consort Road London, SW7 2AZ, UK

# Bootstrapping Smooth Conformal Defects in Chern-Simons-Matter Theories

Barak Gabai b Amit Sever c and De-liang Zhong

###### Abstract

The expectation value of a smooth conformal line defect in a CFT is a
conformal invariant functional of its path in space-time. For example, in
large $N$ holographic theories, these fundamental observables are dual to the
open string partition function in AdS. In this paper, we develop a bootstrap
method for studying them and apply it to conformal line defects in Chern-
Simons matter theories. In these cases, the line bootstrap is based on three
minimal assumptions – conformal invariance of the line defect, large $N$
factorization, and the spectrum of the two lowest-lying operators at the end
of the line. On the basis of these assumptions, we solve the one-dimensional
CFT on the line and systematically compute the defect expectation value in an
expansion around the straight line. We find that the conformal symmetry of a
straight defect is insufficient to fix the answer. Instead, imposing the
conformal symmetry of the defect along an arbitrary curved line leads to a
functional bootstrap constraint. The solution to this constraint is found to
be unique.

##  1 Introduction

We study line defects in conformal field theory (CFT). Such one-dimensional
operators undergo a defect Renormalization Group (RG) flow, Affleck:1991tk ;
Polchinski:2011im ; Friedan:2003yc ; Casini:2016fgb ; Cuomo:2021rkm . We focus
on the fixed points of this flow, i.e., conformal line defects. Due to the
absence of one-dimensional anomalies, such operators can be placed along an
arbitrary smooth path without breaking their conformal symmetry. In
particular, the expectation value of such an operator is a conformal invariant
functional of the path. This functional is a fundamental probe of the theory.
For example, it can serve as an order parameter for the
confinement/deconfinement transition. In holographic large-$N$ gauge theories,
the expectation value of a certain conformal defect is dual to the string
partition function in AdS.

The only known examples where such a functional is explicitly known are in 2d
YM Kazakov:1980zj , and related to it, for a certain class of BPS lines on
$S^{2}\subset{\mathbb{R}}^{4}$ in ${\cal N}=4$ SYM theory, Giombi:2009ms .
However, these examples are rather degenerate because the expectation value
depends only on a two-dimensional area that is enclosed by the loop.111Stated
differently, in these examples, the two-point function of the displacement
operator and the cusp anomalous dimension are trivial. See Drukker:1999zq ;
Polyakov:2000jg ; Polyakov:2000ti ; Semenoff:2004qr ; Giombi:2009ek ;
Toledo:2014koa ; Cooke:2017qgm  for more examples in which such functionals
were studied at different limits.

In this paper, we develop a bootstrap method to study smooth line defects. We
apply it to one of the simplest, yet nontrivial, examples of conformal
defects. These are the ones that are realized in the t’ Hooft large $N$ limit
of three-dimensional Chern-Simons (CS) theories with matter in the fundamental
representation. In these cases, we formulate the bootstrap problem in terms of
three assumptions. They consist of the conformality of the line defect, large
$N$ factorization, and the spectrum of the two lowest-lying operators at the
end of the line. We find that imposing the conformal symmetry of a flat
(straight) defect is insufficient for fixing its expectation value when the
line is not straight. On the other hand, imposing the conformal symmetry of
the defect along an arbitrary smooth path leads to new nontrivial functional
constraints. In particular, they result in a unique bootstrap solution. We use
it to solve the line CFT and compute the defect expectation value in a
systematic expansion around the straight line. Our results are in perfect
agreement with those that we have obtained in Gabai:2022vri ; Gabai:2022mya
using an all-loop resummation of the perturbative expansion. Our consideration
resembles these of conformal perturbation theory (see e.g. Cardy:1996xt ;
Zamolodchikov:1987ti ; Kutasov:1988xb ; Ranganathan:1993vj ; Gaberdiel:2008fn
; Amoretti:2017aze ; Behan:2017mwi ; Cavaglia:2022yvv  for examples of the
method and its many applications). For other methods of studying line defects
that are based on conformal symmetry, see Hartman:2022zik ; Poland:2022qrs
and references therein.

Famously, CS-matter theories possess a strong-weak bosonization duality. This
duality maps the theory coupled to bosons to another theory coupled to
fermions; see Gabai:2022mya  and the references therein. Our bootstrap axioms
are blind to which of the two dual descriptions has been used to realize the
line defects.

The paper is organized as follows. In Section 2 we set up the bootstrap
problem. We detail the axioms on which it is based and our bootstrap strategy.
As a warm-up, in Section 3, we compute the first-order variation of a mesonic
line operator. This order follows directly from our axioms and does not
require a bootstrap. We use it to set our notation and regularization scheme.
In Section 4 we bootstrap the second-order variation and use it to solve the
line CFT. All the ingridients that appear at higher-order variations are
present at the third order. In Section 5 we bootstrap it and discuss its
higher-order generalization. We end in Section 6 with a discussion.

##  2 Setup

Our main objective in this paper is to determine the expectation value of
mesonic line operators

$M[x(\cdot)]=\langle{\cal
M}[x(\cdot)]\rangle=\langle\overline{\mathcal{O}}(x_{1}){\cal
W}[x(\cdot)]\mathcal{O}(x_{0})\rangle\,,$ (2.1)

where $\cal W$ is a conformal line operator along a smooth path, and $x(s)$ is
some parameterization of the path. This path is oriented from the right
boundary operator $\mathcal{O}$ at $x_{0}\equiv x(0)$ to the conjugate left
boundary operator $\overline{\mathcal{O}}$ at $x_{1}\equiv x(1)$. Instead of
the mesonic line, we can consider the expectation value of a closed loop. As
will become clear in the following, the latter is determined from the former.

Such operators are realized in CS theory with matter in the fundamental
representation. For example, in CS theory with one fermion, ${\cal W}$ can be
a Wilson line in the fundamental representation that is stretched between
fundamental and anti-fundamental fermion fields, Gabai:2022mya .

We now detail three minimal properties of this observable, which we assume to
hold throughout this paper. They were derived in Gabai:2022mya  by an explicit
resummation of the ’t Hooft perturbative expansion.

###  2.1 Bootstrap Axioms

##### Axiom I - Conformal invariance

The first axiom concerns the conformal invariance of the line operator $\cal
W$. Under a conformal transformation, $x\to\tilde{x}$, the line operator
transforms as

${\cal W}[x(\cdot)]\quad\mapsto\quad\widetilde{\cal W}[\tilde{x}(\cdot)]={\cal
W}[\tilde{x}(\cdot)]\,.$ (2.2)

It follows from this axiom that an infinite straight line preserves an
$SL(2,{\mathbb{R}})\times U(1)$ subgroup of the three-dimensional conformal
group. Here, $SL(2,{\mathbb{R}})$ is the conformal symmetry of the line, and
the $U(1)$ factor stands for rotations in the transverse plane to the line.
Correspondingly, the operators on the line and the ones on which the line can
end are classified by their (unitary) $SL(2,{\mathbb{R}})$ representation and
their $U(1)$ transverse spin. Such operators can be divided into
$SL(2,{\mathbb{R}})$ primaries and descendants. The latter can be obtained
from the former using longitudinal derivatives. The primaries are
characterized by their dimension $\Delta$, and their transverse spin
$\mathfrak{s}$. We define the sign of the transverse spin to be correlated
with the orientation of the line such that the spin of
$x^{+}=(x^{1}+\mathrm{i}x^{2})/\sqrt{2}$ is $+1$ for a line that is oriented
in the $\hat{x}^{3}$ direction.

When the line is smooth (but not necessarily straight), we can zoom in close
to a point on the line. Locally, it is approximately straight, and we can use
the same classification of line and boundary operators as those on, and at the
ends of, a straight line. 222Namely, $\Delta$ and $\mathfrak{s}$ dictate how
the operator transforms under the $SL(2,\mathbb{R})\times U(1)$
transformations that preserve the straight line that is tangent to the line at
that point.

Operators with non-zero transverse spin are equipped with a transverse
polarization vector $n$, with ${\cal O}={\cal
O}_{\Delta,{\mathfrak{s}}}(x,n)$. A transverse rotation of this polarization
vector results in an overall phase factor of
$e^{\pm\mathrm{i}{\mathfrak{s}}\theta}$. The transverse spin of these
operators may be factional. As a result, the line expectation value acquires a
phase under a $2\pi$ rotation of one of the boundary polarization vectors in
the plane transverse to the line. In this paper, we will be concerned with the
deformation of a straight line with an arbitrary normalization, which will be
factored out. Therefore, keeping track of this $e^{\pm
2\pi\mathrm{i}{\mathfrak{s}}}$ phase is not relevant to us.333In order to have
a single-valued expectation value, one can introduce a framing vector, which
is a transverse polarization along the line. The boundary value of this vector
is the boundary polarization vector, whereas the dependence of the mesonic
line on the bulk value of this vector is topological. Having such a
topological framing vector is natural from the perturbative definition of
Wilson lines in CS theory Witten:1988hf .

##### Axiom II- Factorization

We are concerned with a “large $N$ CFT”, where correlation functions factorize
as

$\langle{\cal M}[x(\cdot),n]\,\widetilde{\cal
M}[\tilde{x}(\cdot),\tilde{n}]\rangle=\langle{\cal
M}[x(\cdot),n]\rangle\,\times\,\langle\widetilde{\cal
M}[\tilde{x}(\cdot),\tilde{n}]\rangle\,\times\,\left(1+{\mathcal{O}}(1/N)\right)\,.$
(2.3)

Another large $N$ simplification is the factorization of line operators into a
product of two boundary operators,

${\cal O}_{\text{line}}=\mathcal{O}\times\overline{\mathcal{O}}\,.$ (2.4)

In large $N$ CS-matter theory, the operators ${\cal O}_{\text{line}}$
transform in the adjoint representation, and the two boundary operators,
$\mathcal{O}$ and $\overline{\mathcal{O}}$, transform in the fundamental and
anti-fundamental, respectively.444The singlet line operators are suppressed at
large $N$. Here, the transverse polarization vectors of the right and left
boundary operators are the same.

Therefore, the properties of the line operators follow those of the left and
right boundary operators in the product (2.4). In particular, it follows from
(2.3) and (2.4) that the correlation function of operators on the line
factorizes into a product of expectation values of mesonic line operators
(2.1)

$\langle\ldots{\cal W}{\cal O}_{\text{line}}{\cal
W}\ldots\rangle=\langle\ldots{\cal
W}\mathcal{O}\rangle\times\langle\overline{\mathcal{O}}{\cal
W}\ldots\rangle\,\times\,\left(1+{\mathcal{O}}(1/N)\right)\,.$ (2.5)

Note that axioms I and II do not imply that the correlation functions of
operators on a straight line are trivial. Although their dependence on the
positions of the operators factorizes, their spectrum and (properly
normalized) overall coefficient are non-trivial. As a result, the dependence
of $M$ on the shape of the path is nontrivial. We will bootstrap them using
one more assumption about the low-lying operator spectrum.

##### Axiom III - Boundary spectrum

The spectrum of boundary operators on the left end of the line is the same as
that on the right, with the only difference being that the transverse spin is
flipped,

$(\Delta_{\overline{{\cal O}}},\mathfrak{s}_{\overline{{\cal
O}}})=(\Delta_{{{\cal O}}},-\mathfrak{s}_{{{\cal O}}})\,.$ (2.6)

This relation is inherited from the CPT symmetry of the theory, which
simultaneously reflects the transverse plane and interchanges the left and
right ends of the oriented line.

The third axiom concerns the spectrum of the low-lying $SL(2,{\mathbb{R}})$
primary boundary operators on the right end of the line. It states that the
primary operator with $\mathfrak{s}\leq{1\over 2}$ and minimal conformal
dimension is

$(\Delta_{\cal O},\mathfrak{s}_{\cal O})=(\Delta,1/2-\Delta)\,,$ (2.7)

and that the one with $\mathfrak{s}\geq{1\over 2}$ is

$(\Delta_{\cal O},\mathfrak{s}_{\cal O})=(2-\Delta,3/2-\Delta)\,,$ (2.8)

where $\Delta\in[0,1]$. We also assume that the transverse spin of these
primary operators is not degenerate. That is, all other primary operators have
a different transverse spin than those in (2.7) and (2.8). This assumption can
be traced back to having a CS-matter theory with a single flavor.

###  2.2 Discussion

One direct consequence of the axioms is that the primary operator on the line
with the lowest dimension is given by a factorized product of the right
operator (2.7) and its CPT conjugate

${\cal O}_{\text{bi-
scalar}}\equiv\mathcal{O}_{\Delta,1/2-\Delta}\times\overline{\mathcal{O}}_{\Delta,\Delta-1/2}\,,\qquad\text{with}\qquad\Delta_{\text{bi-
scalar}}=2\Delta\,,\quad\mathfrak{s}_{\text{bi-scalar}}=0\,.$ (2.9)

For $\Delta\in[0,1/2)$, it is relevant, and therefore the conformal fixed
point on the line is unstable. On the other hand, for $\Delta\in(1/2,1]$, that
scalar primary operator is irrelevant and the fixed point is stable.555In the
case where $\Delta=1/2$, the coupling in front of ${\cal O}_{\text{bi-
scalar}}$ is marginally-irrelevant and the fixed point is stable.

Chern-Simons theory with the bosonic or fermionic fundamental matter has two
different conformal line operators, one unstable and the other stable,
Gabai:2022vri ; Gabai:2022mya . By allowing $\Delta$ to take a value in
$[0,1]$, we cover both types of lines. In particular, the case where
$\Delta=1/2$ corresponds to the free bosonic description, and the cases where
$\Delta=0$ or $\Delta=1$ correspond to the free fermionic description. The
bootstrap in these cases deserves special care because, at these values,
certain relevant or irrelevant operators become marginal. In this paper, we
will avoid them by further assuming that $\Delta\neq 0,1/2,1$. These values
can still be obtained by analytic continuation.

The axiom I (2.2) states that the line is conformally invariant. Hence, the
only source in a conformal transformation of the mesonic line operators comes
from the boundary. These are taken to be $SL(2,{\mathbb{R}})$ primary
operators of dimension $\Delta_{\cal O}$ and transverse spin
$\mathfrak{s}_{\cal O}$. Correspondingly, if we transform both the path and
the boundary polarizations, the only source comes from the boundary conformal
weight,

${\widetilde{\overline{\mathcal{O}}}_{\Delta^{\prime},\mathfrak{s}^{\prime}}(\tilde{x}_{1},\tilde{n}_{1}){\cal
W}[\tilde{x}(\cdot)]\widetilde{\mathcal{O}}_{\Delta,\mathfrak{s}}(\tilde{x}_{0},\tilde{n}_{0})\over\overline{\mathcal{O}}_{\Delta^{\prime},\mathfrak{s}^{\prime}}(x_{1},n_{1}){\cal
W}[x(\cdot)]\mathcal{O}_{\Delta,\mathfrak{s}}(x_{0},n_{0})}=[\text{conformal
factor}]\,.$ (2.10)

To compute this factor, we decompose the conformal transformation at the
boundary of the line, $x\to\tilde{x}$, into translation, three-dimensional
rotation, and dilation

$\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}=\Omega(x){\Lambda^{\mu}}_{\nu}(x)\,,\qquad\text{where}\quad\det{\Lambda^{\mu}}_{\nu}=1\,.$
(2.11)

The dilatation $\Omega$ is part of the $SL(2,{\mathbb{R}})$ subgroup, under
which the boundary operator has dimension $\Delta_{\cal O}$. Hence,

$[\text{conformal
factor}]=\Omega(x_{0})^{-\Delta}\times\Omega(x_{1})^{-\Delta^{\prime}}\,.$
(2.12)

One can also rewrite this equation in a form where the polarization vector is
not transformed and the boundary spin dependence is manifest. Note that the
rotation matrix $\Lambda\in SO(3)$ is, however, not a transverse rotation.
Hence, the transverse planes before and after the conformal transformation are
not the same. The only way that the boundary polarization vector on the
transformed path would point in the same direction as that on the original
path is if it points in the direction of the intersection between these two
transverse planes. This is equivalent to starting with (2.10) and rotating
both $n$ and $\tilde{n}$, each in its transverse plane, so that they point in
the same direction. Doing so results in the phase factor

$[\text{spin factor}]=e^{\mathrm{i}{\mathfrak{s}}\theta_{0}}\times
e^{\mathrm{i}{\mathfrak{s}}^{\prime}\theta_{1}}\,,$ (2.13)

where $\theta_{0}$ and $\theta_{1}$ are the total transverse rotation angles
at the two end points. Let us focus on the right angle, $\theta_{0}$. We
denote by $e_{0}$ and $\tilde{e}_{0}^{\mu}={\Lambda^{\mu}}_{\nu}e_{0}^{\nu}$
the unit tangent vectors before and after the conformal transformation. The
deformation angle, $\theta_{0}$, is equal to the difference between two
angles. One is the angle between $\tilde{n}$ and the intersection of the two
transverse planes, $n^{\text{int}}\propto e_{0}\times\tilde{e}_{0}$, which
comes from the numerator in (2.10). From it we subtract the angle between $n$
and $n^{\text{int}}$, which comes from the denominator in (2.10). This
difference is defined in such a way that it vanishes when the conformal
transformation is trivial. All together, we have

${\widetilde{\overline{\mathcal{O}}}_{\Delta^{\prime},\mathfrak{s}^{\prime}}(\tilde{x}_{1},n_{1}^{\text{int}}){\cal
W}[\tilde{x}(\cdot)]\widetilde{\mathcal{O}}_{\Delta,\mathfrak{s}}(\tilde{x}_{0},n_{0}^{\text{int}})\over\overline{\mathcal{O}}_{\Delta^{\prime},\mathfrak{s}^{\prime}}(x_{1},n_{1}^{\text{int}}){\cal
W}[x(\cdot)]\mathcal{O}_{\Delta,\mathfrak{s}}(x_{0},n_{0}^{\text{int}})}=[\text{conformal
factor}]\times[\text{spin factor}]\,.$ (2.14)

Consider, for example, the case where the two boundary operators are primaries
with the same conformal dimension and opposite transverse spin. In this case,
the expectation value of the mesonic line operator takes the form

$\langle\overline{\mathcal{O}}_{\Delta,-\mathfrak{s}}(x_{1},n_{1}){\cal
W}[x(\cdot)]\mathcal{O}_{\Delta,\mathfrak{s}}(x_{0},n_{0})\rangle={(2n_{0}^{+}n_{1}^{-})^{\mathfrak{s}}\over|x_{0}-x_{1}|^{2\Delta}}\times
F_{\Delta}[x(\cdot)]\,,$ (2.15)

where $F$ is a conformal invariant functional of the path that we would like
to compute. Here, we have used light-cone coordinates to parameterize the
transverse planes at the two endpoints,
$\mathrm{d}^{2}s_{\perp}=2\mathrm{d}x^{+}\mathrm{d}x^{-}$.

###  2.3 Notations

We adopt the notation where the right and left boundary operators are denoted
by

${\cal O}^{(m,\ell)}\quad\text{and}\quad\overline{{\cal O}}^{(m,\ell)}\,,$
(2.16)

where, $\ell$ is the transverse spin at $\Delta=1/2$. That is, in the free
bosonic theory. Here, $m\geq 0$ is the integer $SL(2,{\mathbb{R}})$ descendant
level. The descendant operators have their dimension shifted by $m$ while
their transverse spin remains the same as that of the primary.

In this notation, the only primary operators that we assume to exist are

$\\{\mathcal{O}^{(0,0)},\mathcal{O}^{(0,1)}\\}\qquad\text{and}\qquad\\{\overline{\mathcal{O}}^{(0,0)},\overline{\mathcal{O}}^{(0,-1)}\\}\,.$
(2.17)

Nevertheless, we adopt the general notation (2.16), informed by the
retrospective understanding that the bootstrap constraints examined in this
paper are consistent only if the spectrum of boundary operators comprises two
non-degenerate, infinite towers of operators, characterized by increasing and
decreasing transverse spins, respectively

$(\Delta^{(0,\ell)},\mathfrak{s}^{(0,\ell)})=\left\\{\begin{array}[]{llcl}\left(\Delta,\frac{1}{2}-\Delta\right)&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+(|\ell|,\ell)&&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell\leq
0\,,\\\
\left(2-\Delta,\frac{3}{2}-\Delta\right)&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+\left(\ell-1,\ell-1\right)&&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell\geq
1\,,\end{array}\right.\,$ (2.18)

where $\ell\in{\mathbb{Z}}$.

The spectrum given above is invariant under the transformation of the
parameter $\Delta\to(2-\Delta)$ together with a sign flip of the transverse
spin. This symmetry transformation interchanges the two towers. We denote it
tower-swap.

We will also use the shorthand notation for the expectation value of mesonic
lines with primary boundary operators

$M^{(\ell,\ell^{\prime})}_{st}[x(\cdot)]\equiv\langle\overline{\mathcal{O}}^{(0,\ell)}(x_{s}){\cal
W}[x(\cdot)]\mathcal{O}^{(0,\ell^{\prime})}(x_{t})\rangle\,,$ (2.19)

where we have drooped the dependence of the boundary operators on their
transverse polarization vectors. In the case when the line is straight we will
also drop the dependence of $M$ on $x(\cdot)$ and express (2.19) as

$M^{(\ell,\ell^{\prime})}_{st}[x_{\text{straight}}(\cdot)]\equiv
M^{(\ell,\ell^{\prime})}_{st}\equiv\langle\\!\langle\overline{\mathcal{O}}^{(0,\ell)}(x_{s})\,\mathcal{O}^{(0,\ell^{\prime})}(x_{t})\rangle\\!\rangle\,.$
(2.20)

###  2.4 Bootstrap Strategy

Using axioms I, II and III, we can now systematically bootstrap the
expectation value of mesonic line operators along a smooth path. Our strategy
is to start from a straight mesonic line, whose only nonzero expectation
values are those that preserve the boundary spin $\ell^{\prime}=-\ell$. Their
dependence on the length of the line is fixed by symmetry to be given by

$M^{(\ell,\ell^{\prime})}_{10}=\delta_{\ell^{\prime},-\ell}\,\frac{c_{\ell}}{|x_{10}|^{2\Delta_{\ell}}}\,,$
(2.21)

where $\Delta_{\ell}=\Delta(\mathcal{O}^{(0,\ell)})$ is the dimension of
$\mathcal{O}^{(0,\ell)}$ and $c_{\ell}$ is a number that depends on our
normalization of the boundary operators. It is convenient to choose the two
boundary polarization vectors to be the same, so that the factor
$2n_{0}^{+}n_{1}^{-}=n^{2}=1$ in (2.15) and $c_{l}$ is real.

We then smoothly deform the straight line

$x_{\text{straight}}(s)\quad\rightarrow\quad x(s)=x_{\text{straight}}(s)+{\rm
v}(s)\,,$ (2.22)

and bootstrap $M[x(\cdot)]$ order by order in the smooth deformation profile
${\rm v}(s)/|x_{1}-x_{0}|$. More concretely, we express the deformed operator
in terms of operator insertions along the straight line and bootstrap their
coefficients. This allows us to explicitly compute the mesonic line
expectation value, order by order, in the deformation.

To impose conformal symmetry on the expansion, we construct a one-parameter
family of transformations, $x_{s}\to\tilde{x}_{\beta}(x_{s})$. They can be any
combination of translations, rotations, dilatation, and special conformal
transformations, that are smoothly connected to the identity at
$\beta=0$.666For example, a special conformal transformation can be
parametrized as
$\tilde{x}^{\mu}_{\beta}(x)=\frac{x^{\mu}-\beta^{2}\hat{b}^{\mu}(\hat{b}\cdot
x)}{1-2\beta(\hat{b}\cdot x)+\beta^{2}\hat{b}^{2}x^{2}}$, where $\hat{b}$ is
some unit vector. For convenience, we tune the deformation so that it deforms
the endpoints in the transverse plane only. We then expand the deformed path
$\tilde{x}_{\beta}(x_{s})=x_{s}+{\rm v}_{s}(\beta)$ in powers of $\beta$ and
reparameterize it so that at any point along the line the deformation vector
is transverse to the original straight line, ${\rm
v}_{s}(\beta)\cdot\dot{x}_{s}=0$. Because ${\rm v}_{s}(\beta)$ depends
nonlinearly on $\beta$, this expansion in powers of $\beta$ is not the same as
the expansions in powers of ${\rm v}(\beta)$. The two expansions, however,
only start to differ at the third order (see Appendix A for details).

##  3 First Order

Consider the variation of the path (2.22) to first order in ${\rm
v}/|x_{1}-x_{0}|$. It can be expressed as a linear combination of local
operators inserted on the line and at the two boundaries

$\delta M=\overline{\mathcal{O}}\delta{\cal
W}\mathcal{O}+\delta\overline{\mathcal{O}}{\cal
W}\mathcal{O}+\overline{\mathcal{O}}{\cal W}\delta\mathcal{O}\,.$ (3.1)

These boundary and line variations consist of all operators that are allowed
by symmetry.

###  3.1 Line Variation

The first order line variation can be brought to the form777In three dimension
one can also write the structure $\epsilon_{\mu\nu\rho}\dot{x}^{\mu}{\rm
v}^{\nu}\hat{\mathbb{D}}^{\rho}$. However, keeping in mind that the line is
oriented, it is not independent of ${\mathbb{D}}$ in (3.2). Explicitly, they
are related by the redefinition
${\mathbb{D}}_{\mu}=\epsilon_{\mu\nu\rho}e^{\nu}\hat{\mathbb{D}}^{\rho}$,
where $e^{\mu}=\dot{x}^{\mu}/|\dot{x}|$.

$\delta W=\int\limits_{0}^{1}\mathrm{d}s\,|\dot{x}_{s}|{\rm
v}_{s}^{\mu}\,{\cal P}[{\mathbb{D}}_{\mu}(x_{s}){\cal W}]\,,$ (3.2)

where we have used integration by parts to remove any derivatives from ${\rm
v}$. The line operator ${\mathbb{D}}$ is called the displacement operator. It
follows from (3.2) and axiom I that it is a dimension two primary operator
with transverse spin equal to one. The form of this operator is uniquely fixed
by the spectrum of lowest dimension operators, (2.7), and (2.6), to be given
by

$\displaystyle{\mathbb{D}}_{+}(x)$
$\displaystyle=\eta\left[\mathcal{O}^{(0,1)}(x,n)\times\overline{\mathcal{O}}^{(0,0)}(x,n)\right](\sqrt{2}n_{+})\,,$
(3.3) $\displaystyle{\mathbb{D}}_{-}(x)$
$\displaystyle=\eta\left[\mathcal{O}^{(0,0)}(x,n)\times\overline{\mathcal{O}}^{(0,-1)}(x,n)\right](\sqrt{2}n_{-})\,.$

Here, $n$ is an arbitrary unit transverse polarization vector, and therefore
$(\sqrt{2}n_{\pm})$ is a phase. The proportionality factor, $\eta$, depends on
our choice of normalization of the boundary operators in (2.7), (2.8), and can
be conveniently set to one. Here, we have chosen to keep it undetermined and
identical for the two transverse directions.

Using (2.21) and the form of the displacement operator (3.3) we have that

$\displaystyle\langle\\!\langle\ldots\,{\mathbb{D}}_{-}\,\mathcal{O}^{(0,\ell)}\rangle\\!\rangle$
$\displaystyle\propto\langle\\!\langle\overline{\mathcal{O}}^{(0,-1)}\,\mathcal{O}^{(0,\ell)}\rangle\\!\rangle=0\qquad\text{for}\qquad\ell\neq
1\,,$ (3.4)
$\displaystyle\langle\\!\langle\ldots\,{\mathbb{D}}_{+}\,\mathcal{O}^{(0,\ell)}\rangle\\!\rangle$
$\displaystyle\propto\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\,\mathcal{O}^{(0,\ell)}\rangle\\!\rangle\
=\ 0\qquad\text{for}\qquad\ell\neq 0\,,$ (3.5)

where the dots stand for some left operator. Similarly,

$\displaystyle\langle\\!\langle\overline{\cal
O}^{(0,\ell)}\,{\mathbb{D}}_{-}\,\ldots\rangle\\!\rangle$
$\displaystyle=0\qquad\text{for}\qquad\ell\neq 0\,,$ (3.6)
$\displaystyle\langle\\!\langle\overline{\cal
O}^{(0,\ell)}\,{\mathbb{D}}_{+}\,\ldots\rangle\\!\rangle$
$\displaystyle=0\qquad\text{for}\qquad\ell\neq-1\,.$ (3.7)

###  3.2 Boundary Variation

The boundary variation may include derivatives of ${\rm v}$,

$\delta{\cal O}={\rm v}^{\mu}\delta_{\mu}^{(0)}{\cal O}+\mathrm{d}{\rm
v}^{\mu}\delta_{\mu}^{(1)}{\cal O}+\mathrm{d}\mathrm{d}{\rm
v}^{\mu}\delta_{\mu}^{(2)}{\cal O}+\dots\,.$ (3.8)

Here, $\mathrm{d}{\rm v}=\dot{\rm v}/|\dot{x}|$, is the path derivative of the
variation vector ${\rm v}$ at the boundary. Similarly,
$\mathrm{d}\mathrm{d}{\rm v}=\ddot{\rm v}/|\dot{x}|^{2}-\dot{\rm
v}\partial_{s}|\dot{x}|/|\dot{x}|^{3}$, etc., are the higher-order
reparametrization invariant derivatives of ${\rm v}$ along the path. All terms
on the right-hand side of (3.8) must have the same conformal dimension and
spin as those of ${\cal O}$. Because $\Delta({\rm v})=-1$,
$\Delta(\mathrm{d}{\rm v})=0$, $\Delta(\mathrm{d}\mathrm{d}{\rm v})=1$, etc.,
it follows that the dimension of the boundary operators
$\delta_{\mu}^{(n)}{\cal O}$ grows with $n$. For a transverse variation, the
operators $\delta_{\mu}^{(n)}{\cal O}$ have one more unit of transverse spin
with respect to that of ${\cal O}$, while for a longitudinal variation, it
remains the same. Note that because we have already fixed the line variation
to be given by (3.2), we no longer have the freedom to shift terms between
$\delta{\cal O}$, $\delta\overline{\cal O}$, and $\delta{\cal W}$ in (3.8)
using integration by parts.

We refer to the operator $\delta_{\mu}{\cal O}\equiv\delta_{\mu}^{(0)}{\cal
O}$ as the path derivative of the boundary operator ${\cal O}$. As we go to
higher orders in the path deformation, we will have more boundary and line
operator insertions. We denote the boundary operator with higher powers of the
deformation as

${\rm v}^{\mu_{1}}{\rm v}^{\mu_{2}}\dots\rm
v^{\mu_{m}}\delta_{\mu_{1}}\delta_{\mu_{2}}\dots\delta_{\mu_{m}}{\cal O}\,.$
(3.9)

Importantly, this multi-path derivative is not a local deformation of the
path. Instead, it is the operator that stands in front of ${\rm
v}^{\mu_{1}}{\rm v}^{\mu_{2}}\dots\rm v^{\mu_{m}}$ in a smooth variation of
the path. Hence, unlike covariant derivatives, these path variations commute
by definition,

$\delta_{\mu}\delta_{\nu}{\cal O}=\delta_{\nu}\delta_{\mu}{\cal
O}\neq\delta_{\nu}\left(\delta_{\mu}{\cal O}\right)\,.$ (3.10)

Using path derivatives, the descendant boundary operators are defined as

$\mathcal{O}^{(n+1,\ell)}=\delta_{3}\mathcal{O}^{(n,\ell)}\,,\qquad\overline{\mathcal{O}}^{(n+1,\ell)}=\delta_{3}\overline{\mathcal{O}}^{(n,\ell)}$
(3.11)

where $\delta_{3}$ is the longitudinal path derivative. It is oriented in the
direction of the path, from $\mathcal{O}$ to $\overline{\mathcal{O}}$.

###  3.3 Boundary Counter-terms

The perturbative expansion of the mesonic line operator around the straight
line (2.22) is a conformal perturbation theory, parameterized by the
deformation profile. To carry it out order by order in ${\rm v}(s)$ we have to
introduce a UV cutoff and renormalize the line. We denote by $\epsilon$ the
corresponding UV cutoff, which has the dimension of length.

The resulting structure of the boundary variations is largely constrained by
symmetry. Consider, for example, the operator $\delta_{-}\mathcal{O}^{(0,1)}$.
It has transverse spin
$\mathfrak{s}=\mathfrak{s}(\mathcal{O}^{(0,1)})-1=1/2-\Delta$. According to
axiom III, the only boundary operators with this transverse spin are
$\mathcal{O}^{(0,0)}$ and its descendants $\mathcal{O}^{(n,0)}$. However,
their conformal dimension differs from that of $\delta_{-}\mathcal{O}^{(0,1)}$
by a noninteger value
$\Delta(\mathcal{O}^{(n,0)})-\Delta(\delta_{-}\mathcal{O}^{(0,1)})=n-3+2\Delta$.
Therefore, the decomposition of $\delta_{-}\mathcal{O}^{(0,1)}$ in terms of
boundary operators takes the form

$\delta_{-}\mathcal{O}^{(0,1)}=\frac{1}{\epsilon^{3-2\Delta}}\sum_{n=0}^{\infty}b_{n}\epsilon^{n}\,\mathcal{O}^{(n,0)}\,,$
(3.12)

where all coefficients, $\\{b_{n}\\}_{n=0}^{\infty}$, are scheme dependent.

The divergent terms in (3.12) are counter terms. Their coefficients are fixed
so that they cancel out divergences that result from the integration of the
displacement operator. For the case of $\delta_{-}\mathcal{O}^{(0,1)}$, this
integral takes the form

$\langle\\!\langle\ldots\,\delta{\cal
W}\,\mathcal{O}^{(0,1)}\rangle\\!\rangle=\int\limits_{0}\mathrm{d}s\,|\dot{x}|\,\langle\\!\langle\ldots\,\left[{\rm
v}_{s}^{+}{\mathbb{D}}_{+}(x_{s})+{\rm
v}_{s}^{-}{\mathbb{D}}_{-}(x_{s})\right]\,\mathcal{O}^{(0,1)}(x_{0})\rangle\\!\rangle\,,$
(3.13)

where the dots stand for some left boundary operator. Only the term with
${\mathbb{D}}_{-}$ contributes to the expectation value, see (3.4). As a
result, the integral may have a local divergence coming from the $s\to 0$
region of integration.

We choose a proper time parameterization of the straight line

$x_{\text{straight}}(s)=x_{0}+s(x_{1}-x_{0})\,,$ (3.14)

and use a point splitting regulator, $\ell>\tilde{\epsilon}=|x_{10}|\epsilon$.
With this choice, the divergence takes the form

$\displaystyle|x_{10}|\int\limits_{\tilde{\epsilon}}\mathrm{d}s\,{\rm
v}_{s}^{+}\,\langle\\!\langle\ldots\,{\mathbb{D}}_{-}(x_{s})\,\mathcal{O}^{(0,1)}(x_{0})\rangle\\!\rangle$
$\displaystyle=$
$\displaystyle\eta|x_{10}|\int\limits_{\tilde{\epsilon}}\mathrm{d}s\,{\rm
v}_{s}^{+}\,\langle\\!\langle\ldots\,\mathcal{O}^{(0,0)}(x_{s})\rangle\\!\rangle
M_{s0}^{(-1,1)}$ (3.15) $\displaystyle=$
$\displaystyle{\eta\,c_{-1}\over|x_{10}|^{3-2\Delta}}\left[{\rm
v}_{0}^{+}\langle\\!\langle\ldots\,\mathcal{O}^{(0,0)}(x_{0})\rangle\\!\rangle\int\limits_{\tilde{\epsilon}}{\mathrm{d}s\over
s^{4-2\Delta}}\right.$ $\displaystyle\qquad\qquad\ \ \left.+\left(\dot{\rm
v}_{0}^{+}\langle\\!\langle\ldots\,\mathcal{O}^{(0,0)}(x_{0})\rangle\\!\rangle+{\rm
v}_{0}^{+}|x_{10}|\langle\\!\langle\ldots\,\mathcal{O}^{(1,0)}(x_{0})\rangle\\!\rangle\right)\int\limits_{\tilde{\epsilon}}{\mathrm{d}s\over
s^{3-2\Delta}}+\ldots\right]\,.$

The ${\rm v}_{0}^{+}/\epsilon^{3-2\Delta}$ and ${\rm
v}_{0}^{+}/\epsilon^{2-2\Delta}$ divergences are canceled by setting
$b_{0}=-\eta c_{-1}/(3-2\Delta)$ and $b_{1}=-\eta c_{-1}/(2-2\Delta)$ in
(3.12). Namely, we have

$\delta_{-}\mathcal{O}^{(0,1)}=-{\eta
c_{-1}\over(3-2\Delta)\epsilon^{3-2\Delta}}\mathcal{O}^{(0,0)}-{\eta
c_{-1}\over(2-2\Delta)\epsilon^{2-2\Delta}}\mathcal{O}^{(1,0)}+O(\epsilon^{2\Delta-1})\,.$
(3.16)

Similarly, the $\dot{\rm v}_{0}^{+}/\epsilon^{2-2\Delta}$ divergences are
canceled by setting in (3.8)

$\delta_{-}^{(1)}\mathcal{O}^{(0,1)}=-{\eta
c_{-1}\over(2-2\Delta)\epsilon^{2-2\Delta}}\mathcal{O}^{(0,0)}+O(\epsilon^{2\Delta-1})\,.$
(3.17)

In this way all the divergent terms in $\delta_{-}\mathcal{O}^{(0,1)}$ are
fixed.

Similarly, for $\delta_{+}\mathcal{O}^{(0,0)}$ we have

$\delta_{+}\mathcal{O}^{(0,0)}=\frac{1}{\epsilon^{2\Delta-1}}\sum_{n=0}^{\infty}\tilde{b}_{n}\epsilon^{n}\,\mathcal{O}^{(n,1)}\,.$
(3.18)

The coefficients of the divergent terms are tuned to cancel the ones that
result from the integration of ${\mathbb{D}}_{+}$ in
$\langle\\!\langle\ldots\,\delta{\cal
W}\,\mathcal{O}^{(0,0)}\rangle\\!\rangle$.

###  3.4 The Keyhole Prescription

Consider for example the first-order variation

$\delta M^{(0,1)}=\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\delta{\cal
W}\mathcal{O}^{(0,1)}\rangle\\!\rangle+\langle\\!\langle\delta\overline{\mathcal{O}}^{(0,0)}\mathcal{O}^{(0,1)}\rangle\\!\rangle+\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\delta\mathcal{O}^{(0,1)}\rangle\\!\rangle\,,$
(3.19)

In this case, the corresponding decomposition of
$\delta\overline{\mathcal{O}}^{(0,0)}$ and $\delta\mathcal{O}^{(0,1)}$ in
terms of boundary operators (3.8) comes with fractional powers of $\epsilon$.
As discussed above, the role of these boundary operators is to remove
divergences that result from the integration of the displacement operator near
the boundary in

$\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\,\delta{\cal
W}\,\mathcal{O}^{(0,1)}\rangle\\!\rangle=|x_{10}|\int\limits_{\tilde{\epsilon}}^{1-\tilde{\epsilon}}\mathrm{d}s\,{\rm
v}_{s}^{-}\,\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\,{\mathbb{D}}_{-}(x_{s})\,\mathcal{O}^{(0,1)}\rangle\\!\rangle\,.$
(3.20)

This integral takes the form

$\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\,\delta{\cal
W}\,\mathcal{O}^{(0,1)}\rangle\\!\rangle={\eta\,c_{0}\,c_{-1}\over|x_{10}|^{3}}\int\limits_{\tilde{\epsilon}}^{1-\tilde{\epsilon}}\mathrm{d}s\,{{\rm
v}_{s}^{-}\over(1-s)^{4}}\left({1-s\over s}\right)^{2\Delta}\,.$ (3.21)

It has a branch cut of order $\Delta$ running between $s=0$ and $s=1$. The
result of regularizing this integral and canceling the divergences with the
boundary variations in (3.19) is equivalent to converting this integral into a
contour integral around the cut

$\delta
M^{(0,1)}_{10}={\mathrm{i}\eta|x_{10}|\over{2}\sin(2\pi\Delta)}\oint\limits_{[0,1]}\mathrm{d}s\,{\rm
v}_{s}^{-}M^{(0,0)}_{1s}M^{(-1,1)}_{s0}+[\text{positive powers of
}\epsilon]\,,$ (3.22)

where we have assumed that the function ${\rm v}^{-}(s)$ does not have
singularities along the segment $s\in[0,1]$. For convenience, we have also
assumed that the variation at the boundary acts transversely to the line,
${\rm v}_{0}^{3}={\rm v}_{1}^{3}=0$. Longitudinal boundary variations act
trivially by rescaling the length of the line, $|x_{01}|$ in (2.21). The
cancellation of the divergent boundary terms against the terms in (3.18) and
the conjugate of (3.12) can be understood as providing the cup or keyhole
around the end of the cut, making the integral finite.

We define the keyhole regularization prescription as888For generic $\Delta$,
all divergent terms near the boundary are power-like, therefore there is no
ambiguity in the finite part in the $\epsilon\rightarrow 0$ limit, which only
happens for a logarithmic divergence.

$\delta
M^{(0,1)}_{10}={\mathrm{i}\eta|x_{10}|\over{2}\sin(2\pi\Delta)}\oint\limits_{[0,1]}\mathrm{d}s\,{\rm
v}_{s}^{-}M^{(0,0)}_{1s}M^{(-1,1)}_{s0}\,,$ (3.23)

without additional positive powers of $\epsilon$. This prescription amounts to
a certain choice of the coefficients in the expansion of the boundary
variation. Apart from the divergent counterterms, these series consist of
boundary operators with increasing dimensions and (fractional) powers of
$\epsilon$; see, for example, (3.12). In the keyhole prescription, they are
fixed such that (3.23) holds as finite $\epsilon$.

At higher orders in the deformation of the line, these scheme-dependent terms
with a positive power of $\epsilon$ may combine with divergent ones to give
finite contributions. The total finite part, however, is scheme independent.
In Appendix B, we demonstrate this explicitly for two different schemes.

Similarly to (3.23), the keyhole prescription for $\delta M^{(-1,0)}$ reads

$\delta M^{(-1,0)}_{10}=-{\mathrm{i}\eta|x_{10}|\over
2\sin(2\pi\Delta)}\oint\limits_{[0,1]}\mathrm{d}s\,{\rm
v}_{s}^{+}M^{(-1,1)}_{1s}M^{(0,0)}_{s0}\,.$ (3.24)

In particular, for a polynomial ${\rm v}_{s}^{-}\propto s^{n}$ in (3.22), we
end up with the integral

$\frac{1}{2\mathrm{i}\sin
2\pi\Delta}\oint\limits_{[0,1]}\mathrm{d}s\frac{s^{n}}{s^{2\Delta}(1-s)^{2(2-\Delta)}}=\frac{\Gamma(2\Delta-3)\Gamma(n-2\Delta+1)}{\Gamma(n-2)}\,.$
(3.25)

Note the factor of $\Gamma(n-2)$ in the denominator. It means that $\delta
M^{(0,1)}=0$ for ${\rm v}^{-}\propto s^{0}$, ${\rm v}^{-}\propto s^{1}$, and
${\rm v}^{-}\propto s^{2}$. These correspond in turn to global translation,
rotation, and special conformal transformations, which are indeed symmetries
of the mesonic line expectation value. To be precise,
$\overline{\mathcal{O}}^{(0,0)}{\cal W}\mathcal{O}^{(0,1)}$ transforms
covariantly under conformal transformations, with non-zero weight at the
boundaries, (2.10). These weights only start to contribute at second order and
are analyzed in the next section. Hence, to the linear order, $M^{(-1,0)}$ is
invariant under these transformations. There are no further constraints
imposed by conformal symmetry at this order for mesonic line operators that
end on the four operators in (2.17).

We can now use (3.23) and (3.24) to compute the first-order variation of
$M^{(0,1)}$ and $M^{(-1,0)}$ as functional of the shape of the path. The
overall coefficient depends on our normalization of these operators. This
freedom is reflected in the factor of $\eta$ that we have chosen to keep.

##  4 Second Order

In the second order, for the first time, finite terms with undetermined
coefficients enter the line variation. We list them explicitly in Section 4.1
and then bootstrap them in Section 4.2.

###  4.1 Line Variation

The second-order variation can be divided into line and boundary contributions
as

$\displaystyle\delta^{2}{\cal M}$
$\displaystyle=\delta^{2}\bar{O}\,\mathcal{W}\,O+\delta\,\bar{O}\,\mathcal{W}\,\delta
O+\bar{O}\,\mathcal{W}\,\delta^{2}O\,$ (4.1)
$\displaystyle+\delta\bar{O}\,\delta\mathcal{W}\,O+\bar{O}\,\delta\mathcal{W}\,\delta
O+\bar{O}\,\delta^{2}\mathcal{W}\,O\,.$

Because the dynamics on the line is that of a local CFT, any term in which the
two variations act at two separated points on the line is given by two first-
order variations. New operators appear when the two variations act at the same
point, either at the boundary or in the bulk of the line. At the boundary of
the line, the second-order variation takes a similar form to the first-order
one in (3.8) and is given by

$\delta^{2}{\cal O}={\rm v}^{\mu}{\rm
v}^{\nu}\delta_{\mu}^{(0)}\delta_{\nu}^{(0)}{\cal O}+\mathrm{d}{\rm
v}^{\mu}{\rm v}^{\nu}\delta_{\mu}^{(1)}\delta_{\nu}^{(0)}{\cal
O}+\mathrm{d}{\rm v}^{\mu}\mathrm{d}{\rm
v}^{\nu}\delta_{\mu}^{(1)}\delta_{\nu}^{(1)}{\cal O}+\mathrm{d}\mathrm{d}{\rm
v}^{\mu}{\rm v}^{\nu}\delta_{\mu}^{(2)}\delta_{\nu}^{(0)}{\cal O}+\dots\,,$
(4.2)

where in our parametrization (3.14), $\mathrm{d}{\rm v}=\dot{\rm v}/|x_{10}|$,
etc, see definition below (3.8). At the bulk of the line, the second order
line variation reads

$\delta^{2}\mathcal{W}=|x_{10}|^{2}\int\limits_{0}^{1}\mathrm{d}s\,{\rm
v}_{s}^{\mu}\int\limits_{0}^{1}\mathrm{d}t\,{\rm v}_{t}^{\nu}\,{\cal
P}[{\mathbb{D}}_{\mu}(x_{s}){\mathbb{D}}_{\mu}(x_{t}){\cal
W}]+|x_{10}|\int\limits_{0}^{1}\mathrm{d}s\,{\cal
P}[\delta{\mathbb{D}}(x_{s}){\cal W}]\,.$ (4.3)

Here, using integration by parts, we have chosen the double integral to run
over two displacement operators, with no derivatives of ${\rm v}$. With this
choice, the operator $\delta{\mathbb{D}}$ takes a similar form to
$\delta^{2}{\cal O}$ in (4.2). Namely, it is a sum of operators that are
weighted by ${\rm v}^{\mu}_{s}{\rm v}^{\nu}_{s}$, and higher derivatives such
as ${\rm v}^{\mu}_{s}\mathrm{d}{\rm v}^{\nu}_{s}$, etc.

We can now repeat the analogous treatment to the first-order variation,
canceling divergences using counter-terms and keeping all terms that are
finite at $\epsilon\to 0$. The conclusion of this tedious computation is,
however, very simple. We may list only the regularized integrals and boundary
terms that are finite as $\epsilon\to 0$ and can appear in (4.1). The rest of
the terms are either divergent and are tuned to cancel out or go to zero as
the UV cutoff is removed. The coefficients in front of the finite terms are
either fixed from the first-order variation or are new, and therefore left
unfixed. In the next section, we will bootstrap them using the axioms.

In what follows, we will first focus on the case where $M=M^{(0,0)}$. As
before, we do not consider longitudinal boundary variations because they act
trivially.

####  4.1.1 Boundary Terms with One Variation

The boundary term with one variation on the right and the other on the left is
given by

$\displaystyle\langle\\!\langle\delta\overline{\mathcal{O}}^{(0,0)}\delta\mathcal{O}^{(0,0)}\rangle\\!\rangle$
$\displaystyle=\langle\\!\langle({\rm v}_{1}^{-}\delta_{-}+\mathrm{d}{\rm
v}_{1}^{-}\delta_{-}^{(1)}+\dots)\overline{\mathcal{O}}^{(0,0)}({\rm
v}_{0}^{+}\delta_{+}+\mathrm{d}{\rm
v}_{0}^{+}\delta_{+}^{(1)}+\dots)\mathcal{O}^{(0,0)}\rangle\\!\rangle$ (4.4)
$\displaystyle+\langle\\!\langle({\rm v}_{1}^{+}\delta_{+}+\mathrm{d}{\rm
v}_{1}^{+}\delta_{+}^{(1)}+\dots)\overline{\mathcal{O}}^{(0,0)}({\rm
v}_{0}^{-}\delta_{-}+\mathrm{d}{\rm
v}_{0}^{-}\delta_{-}^{(1)}+\dots)\mathcal{O}^{(0,0)}\rangle\\!\rangle\,.$

The operators in the first line have expansions of the form (3.18), with
fractional powers of $\epsilon$. The ones with negative powers of $\epsilon$
are counter-terms that cancel divergences arising from the integrals
considered below. In this order, those with positive powers of $\epsilon$
vanish as $\epsilon\to 0$. In summary, none of the terms in the first line of
(4.4) leads to a finite contribution.

The terms in the second line are operators with transverse spins
$\mathfrak{s}=-\bar{\mathfrak{s}}=(1/2-\Delta)-1$. Therefore, if they exist,
they are new operators. According to the axiom III, their dimension must be
strictly larger than that of ${\cal O}^{(0,0)}$. In particular, the operators
$\delta_{-}^{(l)}\mathcal{O}^{(0,0)}$ and
$\delta_{+}^{(l)}\overline{\mathcal{O}}^{(0,0)}$ with $l\geq 1$ come with a
positive power of $\epsilon$ and do not contribute as $\epsilon\to 0$.

We remained with the boundary operator

$\delta_{-}\mathcal{O}^{(0,0)}\equiv\epsilon^{\Delta^{(0,-1)}-\Delta^{(0,0)}-1}\Big{(}\mathcal{O}^{(0,-1)}+\sum_{n=1}^{\infty}a_{n}\epsilon^{n}\mathcal{O}^{(n,-1)}\Big{)}\,,$
(4.5)

and its CPT conjugate. The power of $\epsilon$ cannot be negative because
there is no other contribution at that transverse spin that can cancel with
it. So $\Delta^{(0,0)}<\Delta^{(0,-1)}\leq\Delta^{(0,0)}+1$. The operator
$\mathcal{O}^{(0,-1)}$ appears multiplying ${\rm v}^{-}_{0}$ in (4.4) and is
the operator of the lowest dimension with that transverse spin. Therefore, if
it exists, it is an $SL(2,{\mathbb{R}})$ primary. The operators
$\mathcal{O}^{(n,-1)}$ are the corresponding $SL(2,{\mathbb{R}})$ descendants
and the coefficients $a_{n}$ are scheme dependent. For the second line in
(4.4) to yield a finite contribute, the operator $\mathcal{O}^{(0,-1)}$ must
have conformal dimension

$\Delta^{(0,-1)}=\Delta^{(0,0)}+1=\Delta+1\,.$ (4.6)

Below we will find that, indeed, such an operator must exist. Assuming that it
does, equation (4.4) reduces to

$\langle\\!\langle\delta\overline{\mathcal{O}}^{(0,0)}\delta\mathcal{O}^{(0,0)}\rangle\\!\rangle={\rm
v}_{1}^{+}{\rm v}_{0}^{-}M_{10}^{(1,-1)}+[\text{positive powers of
}\epsilon]+[\text{negative powers of }\epsilon]\,,$ (4.7)

where

$M_{10}^{(1,-1)}\equiv\langle\\!\langle\overline{\mathcal{O}}^{(0,1)}\mathcal{O}^{(0,-1)}\rangle\\!\rangle\equiv{c_{1}\over|x_{10}|^{2\Delta+2}}\,.$
(4.8)

and $\overline{\mathcal{O}}^{(0,1)}$ is the CPT conjugate of
$\mathcal{O}^{(0,-1)}$.

####  4.1.2 Boundary Terms with Two Variations

Next, we have new boundary terms corresponding to second-order variation at
the same boundary points, one in the plus direction and the other in the minus
direction. These have the same transverse spin as that of
$\mathcal{O}^{(0,0)}$ and include all possible scalar operators
$\mathcal{O}^{(n,0)}$, weighted by the appropriate power of the cutoff.
Because the spectrum at that transverse spin is non-degenerate, no other new
operators can appear. We end up with

$\displaystyle\langle\\!\langle\bar{O}^{(0,0)}\delta^{2}O^{(0,0)}\rangle\\!\rangle=$
$\displaystyle\Big{[}{\rm v}_{0}^{+}{\rm
v}_{0}^{-}\big{(}{a_{0}\over\epsilon^{2}}+{a_{1}\over\epsilon}\delta_{3}+\gamma_{0}\delta_{3}^{2}\big{)}+\left({\rm
v}_{0}^{+}\mathrm{d}{\rm v}_{0}^{-}+{\rm v}_{0}^{-}\mathrm{d}{\rm
v}_{0}^{+}\right)\big{(}{a_{2}\over\epsilon}+\gamma_{1}\delta_{3}\big{)}$
(4.9) $\displaystyle+\gamma_{2}\left({\rm v}_{0}^{+}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{-}+{\rm v}_{0}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{+}\right)+\gamma_{3}\mathrm{d}{\rm v}_{0}^{+}\mathrm{d}{\rm
v}_{0}^{-}$ $\displaystyle+\left({\rm v}_{0}^{+}\mathrm{d}{\rm v}_{0}^{-}-{\rm
v}_{0}^{-}\mathrm{d}{\rm
v}_{0}^{+}\right)\big{(}{a_{3}\over\epsilon}+\gamma_{4}\delta_{3}\big{)}+\gamma_{5}\left({\rm
v}_{0}^{+}\mathrm{d}\mathrm{d}{\rm v}_{0}^{-}-{\rm
v}_{0}^{-}\mathrm{d}\mathrm{d}{\rm v}_{0}^{+}\right)\Big{]}M_{10}^{(0,0)}\,,$

where $\delta_{3}$ is the longitudinal path derivative (3.11). The
coefficients $a_{i}$ are counter terms that are tuned to cancel the
divergences of the integrals.

Similarly, we have

$\displaystyle\langle\\!\langle\delta^{2}\bar{O}^{(0,0)}O^{(0,0)}\rangle\\!\rangle=$
$\displaystyle\Big{[}{\rm v}_{1}^{+}{\rm
v}_{1}^{-}\big{(}{\tilde{a}_{0}\over\epsilon^{2}}-{\tilde{a}_{1}\over\epsilon}\delta_{3}+\tilde{\gamma}_{0}\bar{\delta}_{3}^{2}\big{)}+\left({\rm
v}_{1}^{+}\mathrm{d}{\rm v}_{1}^{-}+{\rm v}_{1}^{-}\mathrm{d}{\rm
v}_{1}^{+}\right)\big{(}{\tilde{a}_{2}\over\epsilon}-\tilde{\gamma}_{1}\delta_{3}\big{)}$
(4.10) $\displaystyle+\tilde{\gamma}_{2}\left({\rm
v}_{1}^{+}\mathrm{d}\mathrm{d}{\rm v}_{1}^{-}+{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{1}^{+}\right)+\tilde{\gamma}_{3}\mathrm{d}{\rm v}_{0}^{+}\mathrm{d}{\rm
v}_{0}^{-}$ $\displaystyle-\left({\rm v}_{1}^{+}\mathrm{d}{\rm v}_{1}^{-}-{\rm
v}_{1}^{-}\mathrm{d}{\rm
v}_{1}^{+}\right)\big{(}{\tilde{a}_{3}\over\epsilon}-\tilde{\gamma}_{4}\delta_{3}\big{)}-\tilde{\gamma}_{5}\left({\rm
v}_{1}^{+}\mathrm{d}\mathrm{d}{\rm v}_{1}^{-}-{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm v}_{1}^{+}\right)\Big{]}M_{10}^{(0,0)}\,,$

where the tilted coefficients are independent of the un-tilted ones.999Recall
that CPT symmetry exchanges the right and left ends of the line and the
transverse spin. Therefore, as long as we choose a regularization scheme that
is compatible with this symmetry, we are guaranteed to have
$\tilde{a}_{i}=a_{i}$ and $\tilde{\gamma}_{i}=\gamma_{i}$. In what follows, we
will not work with a symmetric scheme and these coefficients will be
different.

####  4.1.3 Single Integral

There are two types of terms with a single integral. One consists of one
boundary and one bulk variation, given by
$\langle\delta\bar{O}\delta\mathcal{W}O\rangle+\langle\bar{O}\delta\mathcal{W}\delta
O\rangle$ in (4.1). The other type of terms with a single integral have two
variations at the same point in the bulk, given by the last term in (4.3). We
now consider these contributions in turn.

The first type does not lead to a finite contribution that depends on ${\rm
v}_{s}$ at $1>s>0$. It does, however, lead to finite contributions
concentrated at the boundaries that can be absorbed in a shift of the
coefficients $\gamma_{i}$, $\tilde{\gamma}_{i}$ in (4.9), (4.10).

Explicitly, the term $\langle\bar{O}^{(0,0)}\delta\mathcal{W}\delta
O^{(0,0)}\rangle$ takes the form

$\displaystyle\langle\bar{O}^{(0,0)}\delta\mathcal{W}\delta O^{(0,0)}\rangle$
$\displaystyle=\eta\,|x_{10}|\int\mathrm{d}s\
\langle\\!\langle\bar{O}^{(0,0)}_{1}O^{(0,0)}_{s}\rangle\\!\rangle\,{\rm
v}_{s}^{-}\,\langle\\!\langle\bar{O}^{(0,-1)}_{s}\delta
O^{(0,0)}_{0}\rangle\\!\rangle$ (4.11)
$\displaystyle=\frac{\eta\,|x_{10}|}{\epsilon^{2\Delta-1}}\int\mathrm{d}s\,M_{1s}^{(0,0)}\,{\rm
v}_{s}^{-}\,\langle\\!\langle\bar{O}^{(0,-1)}_{s}\big{(}{\rm
v}_{0}^{+}\tilde{b}_{0}O^{(0,1)}_{0}+{\cal
O}(\epsilon)\big{)}\rangle\\!\rangle\,,$

where in the first line we have used the form of the displacement operator
(3.3), and in the second line we used the expansion of
$\delta_{+}\mathcal{O}^{(0,0)}_{0}$ in terms of the operators
$\mathcal{O}^{(n,1)}_{0}$, see (3.18). Similarly, we have

$\displaystyle\langle\delta\bar{O}^{(0,0)}\delta\mathcal{W}O^{(0,0)}\rangle$
$\displaystyle=\eta\,|x_{10}|\int\mathrm{d}s\
\langle\\!\langle\delta\bar{O}^{(0,0)}_{1}O^{(0,1)}_{s}\rangle\\!\rangle\,{\rm
v}_{s}^{+}\,\langle\\!\langle\bar{O}^{(0,0)}_{s}O^{(0,0)}_{0}\rangle\\!\rangle$
(4.12)
$\displaystyle=\frac{\eta\,|x_{10}|}{\epsilon^{3-2\Delta}}\int\mathrm{d}s\,\langle\\!\langle\big{(}{\rm
v}_{0}^{-}\,b_{0}\bar{O}^{(0,-1)}_{0}+{\cal
O}(\epsilon)\big{)}O^{(0,1)}_{s}\rangle\\!\rangle{\rm
v}_{s}^{+}\,M_{s0}^{(0,0)}\,,$

where we used the expansion of the boundary operator (3.12).

We see that these two integrands come with a fractional power of $\epsilon$.
The coefficients of the divergent terms, $\eta b_{0}$ and $\eta\tilde{b}_{0}$,
are tuned to cancel the divergences that come from the double integral that we
consider in the next subsection. Hence, the integrals (4.11) and (4.12) do not
lead to finite bulk contributions.

As for the first-order variation in Section 3.3, these integrals lead to
boundary terms that can be divergent or finite. Divergences are canceled by
the boundary counterterms in (4.7) and (4.9). The finite boundary terms can be
absorbed in a shift of the $\gamma_{i}$’s in (4.9).

The other type of one integral has two variations at the same point on the
line and is given by

$\displaystyle\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\delta{\mathbb{D}}(x_{s})\mathcal{O}^{(0,0)}\rangle\\!\rangle$
$\displaystyle={1\over\epsilon^{3-2\Delta}}\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\big{[}d_{1}\,{\rm
v}_{s}^{+}{\rm
v}_{s}^{-}\mathcal{O}^{(0,0)}_{s}\times\overline{\mathcal{O}}^{(0,0)}_{s}+O(\epsilon)\big{]}\mathcal{O}^{(0,0)}\rangle\\!\rangle$
(4.13) $\displaystyle+\Xi\left(\mathrm{d}{\rm
v}_{s}^{+}\mathrm{d}\mathrm{d}{\rm v}_{s}^{-}-\mathrm{d}{\rm
v}_{s}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{s}^{+}\right)\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\mathcal{O}^{(0,0)}\rangle\\!\rangle\,,$

where $d_{1}$ and $\Xi$ are numerical coefficients. Similarly to (4.12), the
integrand in the first line only has fractional powers of $\epsilon$ and,
therefore, does not contribute a finite bulk contribution. The integrand in
the second line is proportional to the identity operator on the line. It
measures the local winding of the deformation and is scheme independent. It
can be changed using integration by parts and a shift in the boundary terms.
Here we have chosen to write it in an anti-symmetric way between the plus and
minus directions.101010The sum $\mathrm{d}{\rm
v}_{s}^{+}\mathrm{d}\mathrm{d}{\rm v}_{s}^{-}+\mathrm{d}{\rm
v}_{s}^{-}\mathrm{d}\mathrm{d}{\rm v}_{s}^{+}$ is a total derivative on the
line and thus, can be absorbed into the boundary terms. Consequently, only the
anti-symmetric combination would be the genuine operator on the line.

####  4.1.4 Double Integral

The double integral takes the form

$\displaystyle\delta^{2}M_{10}^{(0,0)}\Big{|}_{\text{2-int}}^{\text{bare}}\equiv$
$\displaystyle\,\langle\bar{O}^{(0,0)}\delta^{2}\mathcal{W}\,O^{(0,0)}\rangle-|x_{10}|\int\limits\mathrm{d}s\,\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\,\delta{\mathbb{D}}(x_{s})\mathcal{O}^{(0,0)}\,\rangle\\!\rangle$
(4.14) $\displaystyle=$
$\displaystyle\,\eta^{2}\,|x_{10}|^{2}\iint\limits_{s>t}\mathrm{d}s\,\mathrm{d}t\,{\rm
v}_{s}^{-}{\rm v}_{t}^{+}\,M_{1s}^{(0,0)}M_{st}^{(-1,1)}M_{t0}^{(0,0)}\,.$

We shall now choose a regularization scheme and tune the counter-terms in the
one integral and boundary variation to arrive at a finite regularized double
integral.

For fixed $s$, the $t$-integral is the one we have considered before at first
order. We choose to regulate it using the keyhole prescription. We remain with
the $s$-integral over ${\rm v}_{s}^{-}M_{1s}^{(0,0)}\delta M_{s0}^{(-1,0)}$,
where $\delta M_{s0}^{(-1,0)}$ is the keyhole-regulated integral in (3.24).
The regions of integration near the boundaries may still lead to divergences.
Naively, close to $s=0$ we have $\delta M_{s0}^{(-1,0)}\sim{\rm
v}^{+}_{0}/s^{3}$. However, using the fact that the keyhole integral in (3.25)
vanishes for ${\rm v}_{t}^{+}\sim t^{n}=s^{n}(t/s)^{n}$ with $n=0,1,2$, we see
that $\delta M_{s0}^{(-1,0)}\sim\mathrm{d}\mathrm{d}\mathrm{d}{\rm v}^{+}_{0}$
and therefore the limit $\lim_{s\to 0}\delta M_{s0}^{(-1,0)}$ is finite. Close
to $s=1$ we have $M^{(0,0)}_{1s}\propto(1-s)^{-2\Delta}$ with
$\Delta\in(0,1)$. Therefore, for $\Delta\geq 1/2$ the integral is divergent,
and we need to subtract the divergence using boundary terms in (4.1). As
before, we choose a keyhole subtraction scheme, where we deform the integral
to go around the $s$-cut. Unlike the one variation integral (3.21), the
$s$-cut that starts at $s=1$ does not end at $s=0$. We denote the
corresponding integral as

$\delta^{2}M_{10}^{(0,0)}\Big{|}_{\text{2-int}}\equiv{\mathrm{i}\eta\,|x_{10}|\over
2\sin(2\pi\Delta)}\subset\int\limits_{(1,0}\mathrm{d}s\,{\rm
v}_{s}^{-}M_{1s}^{(0,0)}\delta M_{s0}^{(-1,0)}\,.$ (4.15)

Note that the order of integration is important – in the scheme (4.15) we
first do the $t$-integral using the keyhole prescription and only then plug
the result into the $s$-integral. Different choices of regularization of this
double integral differ by the boundary term and can be absorbed into a shift
in the $\gamma_{i}$ and $\tilde{\gamma}_{i}$ coefficients.111111We have
verified this explicitly for several different schemes. See appendix B for
details. The regularization (4.15) does not respect CPT and therefore in this
scheme $\tilde{\gamma}_{i}\neq\gamma_{i}$.

####  4.1.5 Summary

In total, the second-order variation takes the form

$\delta^{2}M_{10}^{(0,0)}=[\text{int}^{2}]+[\text{int}]+[\bar{B}B]+[B^{2}]+[\bar{B}^{2}]\
,$ (4.16)

where121212Here, the tilded coefficients are chosen such that in a PT-
symmetric scheme, they are equal to the un-tilded coefficients,
$\gamma_{i}=\tilde{\gamma}_{i}$. The PT symmetry exchanges the left and right
endpoints, exchanges the $\pm$ component, and introduces a negative sign to
all derivatives. For instance, ${\rm v}_{0}^{+}\mathrm{d}{\rm
v}_{0}^{-}\delta_{3}$ becomes ${\rm v}_{1}^{-}(-\mathrm{d}){\rm
v}_{1}^{+}(-\bar{\delta}_{3})={\rm v}_{1}^{-}\mathrm{d}{\rm
v}_{1}^{+}\bar{\delta}_{3}$.

$\displaystyle[\bar{B}B]=$ $\displaystyle{\rm v}_{1}^{+}{\rm
v}_{0}^{-}M_{10}^{(1,-1)}\,,$ (4.17) $\displaystyle[BB]=$
$\displaystyle\Big{[}\gamma_{0}{\rm v}_{0}^{+}{\rm
v}_{0}^{-}\delta_{3}^{2}+\gamma_{1}\left({\rm v}_{0}^{+}\mathrm{d}{\rm
v}_{0}^{-}+{\rm v}_{0}^{-}\mathrm{d}{\rm
v}_{0}^{+}\right)\delta_{3}+\gamma_{4}\left({\rm v}_{0}^{+}\mathrm{d}{\rm
v}_{0}^{-}-{\rm v}_{0}^{-}\mathrm{d}{\rm v}_{0}^{+}\right)\delta_{3}$
$\displaystyle+\gamma_{3}\mathrm{d}{\rm v}_{0}^{+}\mathrm{d}{\rm
v}_{0}^{-}+\gamma_{2}\left({\rm v}_{0}^{+}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{-}+{\rm v}_{0}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{+}\right)+\gamma_{5}\left({\rm v}_{0}^{+}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{-}-{\rm v}_{0}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{+}\right)\Big{]}M_{10}^{(0,0)}\,,$ $\displaystyle[\bar{B}\bar{B}]=$
$\displaystyle\Big{[}\tilde{\gamma}_{0}{\rm v}_{1}^{+}{\rm
v}_{1}^{-}\delta_{3}^{2}+\tilde{\gamma}_{1}\left({\rm v}_{1}^{+}\mathrm{d}{\rm
v}_{1}^{-}+{\rm v}_{1}^{-}\mathrm{d}{\rm
v}_{1}^{+}\right)\delta_{3}-\tilde{\gamma}_{4}\left({\rm
v}_{1}^{+}\mathrm{d}{\rm v}_{1}^{-}-{\rm v}_{1}^{-}\mathrm{d}{\rm
v}_{1}^{+}\right)\delta_{3}$ $\displaystyle+\tilde{\gamma}_{3}\mathrm{d}{\rm
v}_{1}^{+}\mathrm{d}{\rm v}_{1}^{-}+\tilde{\gamma}_{2}\left({\rm
v}_{1}^{+}\mathrm{d}\mathrm{d}{\rm v}_{1}^{-}+{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{1}^{+}\right)-\tilde{\gamma}_{5}\left({\rm
v}_{1}^{+}\mathrm{d}\mathrm{d}{\rm v}_{1}^{-}-{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm v}_{1}^{+}\right)\Big{]}M_{10}^{(0,0)}\,,$
$\displaystyle[\text{int}]=$
$\displaystyle\,\Xi\,\int\mathrm{d}s\left(\mathrm{d}{\rm
v}_{s}^{+}\mathrm{d}\mathrm{d}{\rm v}_{s}^{-}-\mathrm{d}{\rm
v}_{s}^{-}\mathrm{d}\mathrm{d}{\rm v}_{s}^{+}\right)M_{10}^{(0,0)}\,,$
$\displaystyle[\text{int}^{2}]=$
$\displaystyle-\left({\mathrm{i}\eta|x_{10}|\over{2}\sin(2\pi\Delta)}\right)^{2}\subset\int\limits_{(1,0)}\\!\\!\mathrm{d}s\,{\rm
v}_{s}^{-}M^{(0,0)}_{1s}\Big{(}\oint\limits_{[0,s]}\mathrm{d}t\,{\rm
v}_{t}^{+}M^{(-1,1)}_{st}M_{t0}^{(0,0)}\Big{)}\,.$

A change in the regularization scheme amounts to a change in the regularized
double integral, $[\text{int}^{2}]$, together with a shift in the
$\gamma_{i}$’s and $\tilde{\gamma}_{i}$’s coefficients, in such a way that the
sum (4.16) is unaffected.

####  4.1.6 The Normalization Independent Coefficients

To compute the second-order deformation of the line, we need to fix the
coefficients that appear in (4.17). These are the $\gamma_{i}$’s,
$\tilde{\gamma}_{i}$’s, $\eta$, $\Xi$, the overall normalization of
$M^{(0,0)}$ and $M^{(-1,1)}$ that we have denoted by $c_{0}$ and $c_{-1}$ in
(2.21), and $c_{1}$ in $[\bar{B}B]$. Among these, $\eta$, $c_{0}$, and
$c_{-1}$ depend on the choice of normalization of the four bottom boundary
operators in (2.17). Therefore, they are not physical by themselves. However,
the combination

$\Lambda\equiv\eta^{2}c_{0}c_{-1}\,,$ (4.18)

is independent of the normalization of the boundary operators. This
combination governs the two-point function of the displacement operator. It is
given by131313Instead, we can use the two-point function of $\mathbb{D}$ on
$\langle\\!\langle\overline{\mathcal{O}}^{(0,-1)}\mathcal{O}^{(0,1)}\rangle\\!\rangle$,
where it takes the form
${\langle\\!\langle\overline{\mathcal{O}}^{(0,-1)}{\mathbb{D}}_{+}(x_{s}){\mathbb{D}}_{-}(x_{t})\mathcal{O}^{(0,1)}\rangle\\!\rangle\over\langle\\!\langle\overline{\mathcal{O}}^{(0,-1)}\mathcal{O}^{(0,1)}\rangle\\!\rangle}=\Theta(s-t){\Lambda\over
x_{st}^{4}}\left({x_{10}x_{st}\over x_{1s}x_{t0}}\right)^{2(2-\Delta)}\,.$
(4.19) The same $\Lambda$ also governs the two-point function of the
displacement operator on the circle, where it takes the more standard form
${\langle{\mathbb{D}}_{+}(x_{0}){\mathbb{D}}_{-}(x_{\theta})\rangle_{\text{circ}}\over\langle\mathbb{I}\rangle_{\text{circ}}}={\Lambda\over[2(1-\cos\theta)]^{\Delta}}\,,$
(4.20) where the $\pm$ components are with respect to the local transverse
space. They are obtained from the straight-line ones by a conformal
transformation.

${\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}{\mathbb{D}}_{-}(x_{s}){\mathbb{D}}_{+}(x_{t})\mathcal{O}^{(0,0)}\rangle\\!\rangle\over\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\mathcal{O}^{(0,0)}\rangle\\!\rangle}=\Theta(s-t){\Lambda\over
x_{st}^{4}}\left({x_{10}x_{st}\over x_{1s}x_{t0}}\right)^{2\Delta}\,,$ (4.21)

where $\Theta$ is the step function. Here, we have divided the correlator by
the expectation value of the same straight mesonic line operator without the
insertion of displacement operators. The normalization of the displacement
operator is fixed by its physical interpretation in terms of the geometric
displacement. Hence, this ratio is manifestly independent of our choice of
normalization.

Another physical combination is the ratio

$\Sigma\equiv{c_{1}/c_{0}\over 2\Delta(2\Delta+1)}\,,$ (4.22)

where $c_{1}$ is defined in (4.8) and for convenience, we have factored out a
factor of $2\Delta(2\Delta+1)$. Provided that $\mathcal{O}^{(0,-1)}$ and
$\mathcal{O}^{(0,0)}$, and hence their normalizations, are related by the path
derivative in (4.5), this ratio is also physical. It can be thought of as the
quantum version of the boundary equation of motion, $\Box\phi=0$. To see this,
consider the operator

${\cal M}^{(0,-1)}\equiv\overline{\mathcal{O}}^{(0,0)}{\cal
W}\mathcal{O}^{(0,-1)}\,,$ (4.23)

along a straight line. It is invariant under constant translation in the plus
direction. Using (3.1), (3.2), (3.5), (3.7), and the fact that for a constant
variation vector $\mathrm{d}{\rm v}^{\mu}=\mathrm{d}\mathrm{d}{\rm v}^{\mu}=0$
in (3.8), we end with

$\langle\\!\langle\overline{\mathcal{O}}^{(0,1)}{\cal
W}\mathcal{O}^{(0,-1)}\rangle\\!\rangle+\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}{\cal
W}\,\delta_{+}\mathcal{O}^{(0,-1)}\rangle\\!\rangle=0\,.$ (4.24)

Following axiom III, the only operator with the same transverse spin and
conformal dimension as that of $\delta_{+}\mathcal{O}^{(0,-1)}$ is
$\mathcal{O}^{(2,0)}$. Hence, equation (4.24) reduces to the relation

$\delta_{+}\mathcal{O}^{(0,-1)}=-\Sigma\,\mathcal{O}^{(2,0)}\,.$ (4.25)

In the free bosonic theory, this relation becomes
$\partial_{+}\partial_{-}\phi=-{1\over 2}\partial_{3}^{2}\phi$. Similarly,
applying constant translation in the minus direction to
$\mathcal{M}^{(1,0)}\equiv\overline{\mathcal{O}}^{(1,0)}\mathcal{W}\mathcal{O}^{(0,0)}$
implies

$\delta_{-}\overline{\mathcal{O}}^{(0,1)}=-\Sigma\,\overline{\mathcal{O}}^{(2,0)}\,.$
(4.26)

In the following, we will use the axioms to bootstrap the $\gamma_{i}$’ s,
$\tilde{\gamma}_{i}$’s, $\Xi$, $\Lambda$, and $\Sigma$.

###  4.2 Bootstrap

####  4.2.1 Constraints from Conformal Symmetry of a Straight Line

Consider a general conformal transformation,
$x_{s}\to\tilde{x}_{\beta}(x_{s})$, that may shift the endpoints of the
straight line only in the transverse direction. To linear order, the resulting
deformation profile

$x_{\text{straight}}\ \rightarrow\ x_{\text{straight}}+{\rm v}_{c}\,,$ (4.27)

is a quadratic polynomial in $s$

${\rm
v}^{+}_{c}=\mathfrak{c}_{0}^{+}+\mathfrak{c}_{1}^{+}s+\mathfrak{c}_{2}^{+}s^{2}\,,\qquad{\rm
v}^{-}_{c}=\mathfrak{c}_{0}^{-}+\mathfrak{c}_{1}^{-}s+\mathfrak{c}_{2}^{-}s^{2}\,,$
(4.28)

with six independent coefficients, see Appendix A for more details.

In the case of $M^{(0,0)}$, the conformal and spin factors in (2.14) take the
form

$\displaystyle[\text{conformal
factor}]=1-2\Delta\,\delta\log|x_{10}|+\dots\,,$ (4.29)
$\displaystyle[\text{spin factor}]\qquad\
\,=1+{\mathfrak{s}}\left(\delta\theta_{0}-\delta\theta_{1}\right)+\dots\,,$

with ${\mathfrak{s}}=(1/2-\Delta)$. For the conformal transformation (4.28) we
find

$\displaystyle\delta\log|x_{10}|$ $\displaystyle=\left.({\rm
v}_{c}^{+})\right|_{0}^{1}\left.({\rm
v}_{c}^{-})\right|_{0}^{1}/|x_{10}|^{2}=(\mathfrak{c}_{1}^{-}+\mathfrak{c}_{2}^{-})(\mathfrak{c}_{1}^{+}+\mathfrak{c}_{2}^{+})/|x_{10}|^{2}\,,$
(4.30) $\displaystyle(\delta\theta_{0}-\delta\theta_{1})$
$\displaystyle={1\over 2}\left.\left({\rm v}_{c}^{+}\mathrm{d}\mathrm{d}{\rm
v}_{c}^{-}-{\rm v}_{c}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{c}^{+}\right)\right|_{0}^{1}=(\mathfrak{c}_{1}^{-}\mathfrak{c}_{2}^{+}-\mathfrak{c}_{1}^{+}\mathfrak{c}_{2}^{-})/|x_{10}|^{2}\,.$

It is therefore sufficient to keep the term linear in $\delta\log|x_{10}|$ and
$\delta\theta$ in (4.29) when working to second order in the deformation. Note
that because $\mathrm{d}\mathrm{d}\mathrm{d}{\rm v}_{c}=0$, the spin source is
the same as the spinning integral $[\text{int}]$ in (4.17), with
$\Xi={\mathfrak{s}}/2$.

In our scheme, the double integral (4.15) vanishes for a quadratic deformation
profile (3.25). Hence, the change in the expectation value (4.29) comes from
the boundary terms only. In particular, it cannot fix $\Lambda$ (4.18) because
it is independent of $\eta$, (that only appears in $[{\rm int}^{2}]$ in
(4.17)). By matching the coefficients in front of $\mathfrak{c}_{0}^{\pm}$,
$\mathfrak{c}_{1}^{\pm}$ and $\mathfrak{c}_{2}^{\pm}$ between
$[\bar{B}B]+[B^{2}]+[\bar{B}^{2}]$ and (4.29) we can reduce the set of
independent unknown coefficients to
$\\{\gamma_{0},\gamma_{2},\gamma_{4},\gamma_{5},\Sigma,\Lambda\\}$.

####  4.2.2 Constraints from Conformal Symmetry of a Curved Line

We see that the conformal symmetry of a straight line is not sufficient to fix
all the coefficients that appear in the second-order variation (4.17). Going
to higher orders in the deformation does not improve the situation because the
number of independent coefficients also grows. To determine the remaining
coefficients, we demand that the expectation value of a mesonic line operator
along an arbitrary smooth path, not necessarily straight, respects the
conformal symmetry (2.14).

Explicitly, we consider a path that we decompose into a straight line and an
arbitrary smooth deformation

$x_{a}=x_{\mathrm{straight}}+{\rm v}_{a}\,.$ (4.31)

We then apply a conformal transformation to it,

$x_{a}\ \rightarrow\ x_{ac}=x_{a}+{\rm v}_{c}=x_{\text{straight}}+{\rm
v}_{a}+{\rm v}_{c}\,.$ (4.32)

While for a finite deformation, ${\rm v}_{c}$ depends on ${\rm v}_{a}$, at
second order, they are independent. Therefore, we can simply expand
$M^{(0,0)}[x(\cdot)+{\rm v}_{a}(\cdot)+{\rm v}_{c}(\cdot)]$ to second order
using (4.16). The term of order $O({\rm v}_{c}^{2})$ was considered in the
previous section. The term of order $O({\rm v}_{a}^{2})$ is not constrained
because ${\rm v}_{a}$ is arbitrary. We then focus on the term of order $O({\rm
v}_{a}{\rm v}_{c})$ and demand that it is consistent with the conformal
transformation of the deformed path (4.31). As before, it is sufficient to
have both ${\rm v}_{a}$ and ${\rm v}_{c}$ transverse at the boundaries of the
straight line.

Because the line is not straight before the conformal transformation, the
source term, due to the boundary conformal weight (4.29), can have a support
that is linear in ${\rm v}_{c}$. For the same reason, the boundary
polarization vector $n^{\text{int}}$ in (2.14) is not a vector in the
transverse plane of the straight line. Hence, to expand the mesonic lines
around the straight line, we now have to further rotate the boundary
polarization vectors. After doing so, we can compare the expansions of the
numerator and denominator in (2.10) around the straight line. Starting with
$n$ and $\tilde{n}$ in (2.10) on either of the two sides of the line, this
results in the following sequence of rotations

$\tilde{n}\ \rightarrow\ \tilde{n}^{\text{straight}}\ \rightarrow\
n^{\text{straight}}\ \rightarrow\ n\,,$ (4.33)

where $n\rightarrow n^{\prime}$ stands for a rotation of the polarization
vector $n$ to the polarization vector $n^{\prime}$ in the plane that these two
vectors span. This sequence of rotations results in the spin factor source

$[\text{spin factor}]_{ac}=\left.{\mathfrak{s}}\left[\left({\rm
v}_{a}^{+}\mathrm{d}\mathrm{d}{\rm v}_{c}^{-}-{\rm
v}_{a}^{-}\mathrm{d}\mathrm{d}{\rm v}_{c}^{+}\right)+\left(\mathrm{d}{\rm
v}_{c}^{+}\mathrm{d}{\rm v}_{a}^{-}-\mathrm{d}{\rm v}_{c}^{-}\mathrm{d}{\rm
v}_{a}^{+}\right)/2\right]\right|_{\text{boundary}}\,.$ (4.34)

For definiteness, we take the deformed path to be a degree-four polynomial

${\rm v}_{a}(s)=\sum_{n=0}^{4}{\mathfrak{a}}_{n}^{\pm}s^{n}\,.$ (4.35)

Consequently, we require that the second order variation (4.16) agrees with
the spin and conformal factors for any value of
$\mathfrak{c}_{n=0,1,2}^{\pm}$, and $\mathfrak{a}^{\pm}_{n=0,\ldots,4}$. This
requirement, together with the constraint obtained at order ${\rm v}_{c}^{2}$,
fixes all coefficients and the normalization independent quantities to

$\displaystyle\gamma_{1}=$
$\displaystyle\gamma_{3}=-\gamma_{4}=-\tilde{\gamma}_{3}=(1/2-\Delta)/2\,,$
(4.36) $\displaystyle\gamma_{2}=$
$\displaystyle-\gamma_{5}=(\Delta-1)\gamma_{1}\,,$
$\displaystyle\tilde{\gamma}_{1}=$
$\displaystyle\tilde{\gamma}_{4}=2\tilde{\gamma}_{2}=2\tilde{\gamma}_{5}=-\gamma_{0}=1/2\,,$
$\displaystyle\tilde{\gamma}_{0}=$ $\displaystyle 0\,,$

and

$\Lambda=-\frac{1}{2\pi}(2\Delta-1)(2\Delta-2)(2\Delta-3)\sin(2\pi\Delta)\,,\quad\Xi=0\,,\quad\Sigma=1/2\,.$
(4.37)

Note that $c_{1}\propto\Sigma\neq 0$, (4.37). Hence, the operators
$\mathcal{O}^{(0,-1)}$ with dimension given in (4.6) must exist. We see that
it is required for the consistency of $M^{(0,0)}$ with conformal symmetry. If
instead of the second-order variation of $M^{(0,0)}$ we consider that of
$M^{(-1,1)}$, we conclude that the primary boundary operator
$\mathcal{O}^{(0,2)}$ must also exist, with
$\Delta^{(0,2)}=\Delta^{(0,1)}+1=3-\Delta$. The two derivations are related by
the tower-swap discussed below (2.18).

In general, we can use polynomials of arbitrarily high degree as the basis for
smooth path deformations. Every power of $s$ in this deformation leads to a
new constraint. For the consistency of the solution (4.36), (4.37), all these
constraints must be satisfied. By considering polynomials of high degree, we
have verified that this is indeed the case. At the third and higher order in
the deformation of the straight line, there are more and more coefficients to
fix. Correspondingly, in these orders, it is necessary to consider polynomial
deformations of a higher degree and, in that way, generate more and more
independent constraints.

####  4.2.3 Two Towers of Spinning Boundary Operators

One of the results of the previous section is the existence of the primary
boundary operators, $\mathcal{O}^{(0,-1)}$ and $\mathcal{O}^{(0,2)}$. Using
these operators, we can now derive the existence of two towers of primary
boundary operators with increasing transverse spin, (2.18).

Consider, for example, the second-order variation of the operator
$M^{(1,-1)}$. The main difference from the second-order variation of
$M^{(0,0)}$ is that these boundary operators do not overlap with the
displacement operator. Hence, the second-order variation of the operator
$M^{(1,-1)}$ does not contain a double integral. It does, however, contain a
single integral. The structure of this single integral is the same as in
Section 4.1.3. That is, it is proportional to the insertion of the identity on
the line or comes with a (positive) fractional power of $\epsilon$. Therefore,
the only finite terms in this case are

$\delta^{2}M_{10}^{(1,-1)}=[\text{int}]+[\bar{B}B]+[B^{2}]+[\bar{B}^{2}]\,.$
(4.38)

Using integration by parts, the single integral can be written as

$[\text{int}]=\Xi^{(1,-1)}\,\int\mathrm{d}s\left(\mathrm{d}{\rm
v}_{s}^{+}\mathrm{d}\mathrm{d}{\rm v}_{s}^{-}-\mathrm{d}{\rm
v}_{s}^{-}\mathrm{d}\mathrm{d}{\rm v}_{s}^{+}\right)M^{(1,-1)}_{10}\,.$ (4.39)

Here, we have introduced the new parameter $\Xi^{(1,-1)}$ because, in
principle, it could be different from $\Xi=0$. The boundary variation terms
represent

$[\bar{B}B]=\langle\\!\langle\delta\overline{\mathcal{O}}^{(0,1)}\delta\mathcal{O}^{(0,-1)}\rangle\\!\rangle\,,\quad[B^{2}]=\langle\\!\langle\delta^{2}\overline{\mathcal{O}}^{(0,1)}\mathcal{O}^{(0,-1)}\rangle\\!\rangle\,,\quad[\bar{B}^{2}]=\langle\\!\langle\overline{\mathcal{O}}^{(0,1)}\delta^{2}\mathcal{O}^{(0,-1)}\rangle\\!\rangle\,.$
(4.40)

They consist of all finite terms that are allowed by symmetry. First, we have

$\displaystyle\delta^{2}\overline{\mathcal{O}}^{(0,1)}$ $\displaystyle={\rm
v}_{1}^{+}{\rm
v}_{1}^{-}\bar{\beta}_{0,0}\overline{\mathcal{O}}^{(2,1)}+\mathrm{d}{\rm
v}_{1}^{+}{\rm v}_{1}^{-}\bar{\beta}_{1,0}\overline{\mathcal{O}}^{(1,1)}+{\rm
v}_{1}^{+}\mathrm{d}{\rm
v}_{1}^{-}\bar{\beta}_{0,1}\overline{\mathcal{O}}^{(1,1)}+\mathrm{d}\mathrm{d}{\rm
v}_{1}^{+}{\rm v}_{1}^{-}\bar{\beta}_{2,0}\overline{\mathcal{O}}^{(0,1)}$
(4.41) $\displaystyle+{\rm v}_{1}^{+}\mathrm{d}\mathrm{d}{\rm
v}_{1}^{-}\bar{\beta}_{0,2}\overline{\mathcal{O}}^{(0,1)}+\mathrm{d}{\rm
v}_{1}^{+}\mathrm{d}{\rm
v}_{1}^{-}\bar{\beta}_{1,1}\overline{\mathcal{O}}^{(0,1)}+\cdots\,.$

Here, we have introduced six unknown coefficients $\bar{\beta}$, whose indices
label the number of path derivatives that act on each of the variation
vectors, $\bar{\beta}_{a,b}\mathrm{d}^{a}{\rm v}_{1}^{+}\mathrm{d}^{b}{\rm
v}_{1}^{-}$, with $\mathrm{d}^{0}v=v$, $\mathrm{d}^{1}v=\mathrm{d}{\rm v}$,
etc. The expansion of $\delta^{2}\mathcal{O}^{(0,-1)}$ takes the same form,
with the replacements ${\rm v}_{1}\mapsto{\rm v}_{0}$,
$\bar{\beta}\mapsto\beta$, and $\overline{\mathcal{O}}\mapsto\mathcal{O}$.

Next, we have the boundary operator

$\delta\mathcal{O}^{(0,-1)}={\rm
v}_{0}^{-}\delta_{-}\mathcal{O}^{(0,-1)}-\Sigma{\rm
v}_{0}^{+}\mathcal{O}^{(2,0)}+\beta_{1}\mathrm{d}{\rm
v}_{0}^{+}\mathcal{O}^{(1,0)}+\beta_{2}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{+}\mathcal{O}^{(0,0)}+...$ (4.42)

where $\Sigma={1\over 2}$ was defined in (4.25) and determined in (4.37). For
the operator $\delta_{-}\mathcal{O}^{(0,-1)}$ to give a finite contribution,
it must have dimension $\Delta^{(0,-2)}=\Delta^{(0,-1)}+1=\Delta+2$. We denote
it by $\mathcal{O}^{(0,-2)}=\delta_{-}\mathcal{O}^{(0,-1)}$, if it exists.

Similarly, the expansion of $\delta_{-}\overline{\mathcal{O}}^{(0,1)}$ is
obtained from (4.42) by the replacing ${\rm v}_{1}\mapsto{\rm v}_{0}$,
$\bar{\beta}\mapsto\beta$, and $\overline{\mathcal{O}}\mapsto\mathcal{O}$.
Putting them together we get,

$\displaystyle\langle\\!\langle\delta\overline{\mathcal{O}}^{(0,1)}\delta\mathcal{O}^{(0,-1)}\rangle\\!\rangle=$
$\displaystyle\ {\rm v}_{1}^{-}{\rm v}_{0}^{+}M_{01}^{(2,-2)}$ (4.43)
$\displaystyle+\Big{(}{1\over 2}\big{[}{\beta}_{1}{\rm
v}_{1}^{-}\mathrm{d}{\rm
v}_{0}^{+}\bar{\delta}_{3}^{2}\delta_{3}-\bar{\beta}_{1}\mathrm{d}{\rm
v}_{1}^{-}{\rm v}_{0}^{+}\bar{\delta}_{3}\delta_{3}^{2}+{\beta}_{2}{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{+}\bar{\delta}_{3}^{2}+\bar{\beta}_{2}\mathrm{d}\mathrm{d}{\rm
v}_{1}^{-}{\rm v}_{0}^{+}\delta_{3}^{2}\big{]}$
$\displaystyle+\bar{\beta}_{1}{\beta}_{1}\mathrm{d}{\rm
v}_{1}^{-}\mathrm{d}{\rm
v}_{0}^{+}\bar{\delta}_{3}\delta_{3}+\bar{\beta}_{2}{\beta}_{1}\mathrm{d}\mathrm{d}{\rm
v}_{1}^{-}\mathrm{d}{\rm
v}_{0}^{+}\delta_{3}+\bar{\beta}_{1}\beta_{2}\mathrm{d}{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm v}_{0}^{+}\bar{\delta}_{3}$
$\displaystyle+\bar{\beta}_{2}\beta_{2}\mathrm{d}\mathrm{d}{\rm
v}_{1}^{-}\mathrm{d}\mathrm{d}{\rm v}_{0}^{+}+{1\over 4}{\rm v}_{1}^{-}{\rm
v}_{0}^{+}\delta_{3}^{2}\Big{)}M_{01}^{(0,0)}\,,$

where

$M_{01}^{(2,-2)}=\langle\\!\langle\overline{\mathcal{O}}^{(0,2)}\mathcal{O}^{(0,-2)}\rangle\\!\rangle=\frac{c_{2}}{|x_{10}|^{2(\Delta+2)}}\,.$
(4.44)

The primary boundary operators $\mathcal{O}^{(0,-1)}$ and
$\overline{\mathcal{O}}^{(0,1)}$ have dimension $\Delta+1$ and spins
${\mathfrak{s}}=\pm(1/2+\Delta)$. Hence, a mesonic line that ends on these
operators transforms under a conformal transformation as

${\delta^{2}\langle\overline{\mathcal{O}}^{(0,1)}{\cal
W}\mathcal{O}^{(0,-1)}\rangle\over\langle\overline{\mathcal{O}}^{(0,1)}{\cal
W}\mathcal{O}^{(0,-1)}\rangle}=-2(1+\Delta)\,\delta\log|x_{10}|+\delta[\text{spin
source}]\,,$ (4.45)

where the spin source at order ${\rm v}_{c}^{2}$ is given in (4.29) and at
order ${\rm v}_{a}{\rm v}_{c}$ in (4.34). The deformation (4.45) must
reproduce these sources. By choosing a generic polynomial ${\rm v}_{a}$ and
equating the terms of order ${\cal O}({\rm v}_{c}^{2})$ and ${\cal O}({\rm
v}_{a}{\rm v}_{c})$ on the two sides, we find a unique solution. 141414In
practice, we find that choosing ${\rm
v}_{a}^{\pm}=\mathfrak{a}_{0}^{\pm}s^{3}+\mathfrak{a}_{1}^{\pm}s^{4}+\mathfrak{a}_{2}^{\pm}s^{5}$
to be a fifth-order polynomial in $s$ determines the answer completely. The
resulting coefficients are compatible with the CPT symmetry

$\beta_{i,j}=\bar{\beta}_{j,i}\,,\qquad\beta_{i}=\bar{\beta}_{i}\,,\qquad\forall
i,j\,.$ (4.46)

They are given by

$\displaystyle c_{2}$
$\displaystyle=(\Delta+1)(2\Delta+3)c_{1}=\Delta(\Delta+1)(2\Delta+1)(2\Delta+3)c_{0}\,,$
(4.47) $\displaystyle\beta_{1,0}$
$\displaystyle=\beta_{1}=2\beta_{1,1}=-(2\Delta+1)/2\,,\qquad\Xi^{(1,-1)}=0\,,$
$\displaystyle\beta_{0,0}$
$\displaystyle=-1/2\,,\qquad\beta_{0,1}=\beta_{0,2}=0\,,\qquad\beta_{2,0}=\beta_{2}=\Delta\beta_{1}\,.$

Because in this case, there is no integral to regularize, all of these
coefficients are scheme-independent. In particular, the values of
$\beta_{0,0}=\bar{\beta}_{0,0}=-1/2$ can be interpreted as the quantum version
of the equation of motion with one extra $\partial_{+}$ derivative,
$\partial_{+}\partial_{-}(\partial_{+}\phi)=-\frac{1}{2}\partial_{3}^{2}(\partial_{+}\phi)$.

Since $c_{2}\neq 0$, the operators $\overline{\mathcal{O}}^{(0,2)}$ and
$\mathcal{O}^{(0,-2)}$, of conformal dimension $\Delta^{(0,-2)}=\Delta+2$ must
exist. We can repeat the same computation successively, starting with
$M^{(-\ell,\ell)}$ for all $\ell\leq-2$. They are related to the case of
$M^{(1,-1)}$ by substituting $\Delta\rightarrow\Delta+|\ell|$, and
$\\{c_{0},c_{1},c_{2}\\}\rightarrow\\{c_{-\ell-1},c_{-\ell},c_{1-\ell}\\}$
into the above bootstrap. In that way, we deduce the existence of infinite
towers of operators of the form $\overline{\mathcal{O}}^{(0,-\ell)}$ and
$\mathcal{O}^{(0,\ell)}$ with $\ell\leq-1$, whose spectrum is given by (2.18).
They are subject to the operator equation

$\delta_{-}\delta_{+}\overline{\mathcal{O}}^{(0,\ell)}=-\frac{1}{2}\overline{\mathcal{O}}^{(2,\ell)}\,,\quad\delta_{+}\delta_{-}\mathcal{O}^{(0,\ell)}=-\frac{1}{2}\mathcal{O}^{(2,\ell)}\,.$
(4.48)

Correspondingly, the expectation value of $M^{(\ell,\ell^{\prime})}$ is given
by (2.21), with

$c_{\ell+1}=(2\Delta+2\ell+1)(\Delta+\ell)c_{\ell}=c_{0}\prod_{j=0}^{\ell}(2\Delta+2j+1)(\Delta+j)\,,\qquad\ell\geq
0\,.$ (4.49)

Similarly, we deduce the existence of infinite towers of operators of the form
$\overline{\mathcal{O}}^{(0,-\ell)}$ and $\mathcal{O}^{(0,\ell)}$ with
$\ell\geq 1$, whose spectrum is given by (2.18). Their normalizations are
related to $c_{-1}$ as

$c_{-\ell-1}=(1+\ell-\Delta)(3+2\ell-2\Delta)\,c_{-\ell}=c_{-1}\prod_{j=1}^{\ell}(1+j-\Delta)(3+2j-2\Delta)\,,\qquad\ell\geq
1\,.$ (4.50)

Their derivation is related to that above by tower-swap, see discussion below
(2.18).

####  4.2.4 Bootstrap of Spinning Mesonic Lines

We can also apply the bootstrap to operators that have a non-zero transverse
spin and, therefore, a zero expectation value. As an example, let us examine
the operators $M^{(1,1)}_{10}$ and $M^{(0,2)}$. These operators have two units
of transverse spin, ${\mathfrak{s}}+\bar{\mathfrak{s}}=2$. Hence, to absorb
that spin and have a non-zero expectation value, we need at least two
variations in the plus direction.

##### The Operator $M^{(1,1)}_{10}$.

The only finite contributions to the second-order variation of this operator
have one bulk integration. They take the form

$\delta^{2}M^{(1,1)}_{10}=\int\limits_{0}^{1}\\!\mathrm{d}s\ \Big{[}{\rm
v}^{-}_{1}{\rm
v}^{-}_{s}\langle\\!\langle{\delta}_{-}\overline{\mathcal{O}}^{(0,1)}\mathbb{D}_{-}(x_{s})\mathcal{O}^{(0,1)}\rangle\\!\rangle+\left({\rm
v}^{-}_{s}\right)^{2}\langle\\!\langle\overline{\mathcal{O}}^{(1,0)}\mathbb{D}_{--}(x_{s})\mathcal{O}^{(0,1)}\rangle\\!\rangle\Big{]}_{\text{reg}}\,,$
(4.51)

where $\left[\dots\right]_{\text{reg}}$ stands for a regularized integral. The
boundary variation ${\delta}_{-}\overline{\mathcal{O}}^{(0,1)}$, appearing in
the second term, was defined below (4.42). The coefficients in (4.42) are
scheme-independent and were fixed in (4.47). The line operator
$\mathbb{D}_{--}$ is some linear combination of dimension $3$ and spin $2$
primary operators. Among these, only
$\mathcal{O}^{(0,-1)}\overline{\mathcal{O}}^{(0,-1)}$ can contribute to
(4.51). We denote its coefficient by $\eta^{(-1,-1)}$,

$\mathbb{D}_{--}=\eta^{(-1,-1)}\left[\mathcal{O}^{(0,-1)}\times\overline{\mathcal{O}}^{(0,-1)}\right](2n_{-}^{2})+\dots\,,$
(4.52)

where the dots stand for operators that do not overlap with either
$\mathcal{O}^{(0,1)}$ on the right end of a straight line or with
$\overline{\mathcal{O}}^{(0,1)}$ on the left end.

We proceed as before, by picking a generic enough deformation ${\rm v}_{a}$
and applying to it a conformal transformation. Because the operator ${\cal
M}^{(1,1)}$, and its first order variation $\delta{\cal M}^{(1,1)}$ have zero
expectation value, so do their conformal transformations. Consequently, at
second order, the ${\cal O}({\rm v}_{a}{\rm v}_{c})$ and ${\cal O}({\rm
v}_{c}^{2})$ terms at the right hand side of (4.51) must independently sum to
zero.

We have used the keyhole regularization scheme to evaluate the two integrals
in (4.51), which are, however, scheme independent. The vanishing of the ${\cal
O}({\rm v}_{a}{\rm v}_{c})$ contribution fixes

$\eta^{(-1,-1)}=\frac{1}{2}\,.$ (4.53)

It is worth mentioning that although the values of $\beta_{1}$ and $\beta_{2}$
were found in the preceding section, we can also fix them using the bootstrap
equation used in this section. Because these are coefficients of local
operators on the line, the result is, of course, consistent with what we have
found before. The order in which one solves the bootstrap constraints is a
matter of preference.

##### The Operator $M^{(0,2)}_{10}$.

Similarly, for this operator we have

$\displaystyle\delta^{2}M^{(0,2)}_{10}$
$\displaystyle=\int\limits_{0}^{1}\\!\\!\mathrm{d}s\ \Big{[}{\rm
v}_{s}^{-}{\rm
v}_{0}^{-}\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\mathbb{D}_{-}(x_{s})\delta_{-}\mathcal{O}^{(0,2)}\rangle\\!\rangle+({\rm
v}_{s}^{-})^{2}\langle\\!\langle\overline{\mathcal{O}}^{(0,0)}\mathbb{D}_{--}(x_{s})\mathcal{O}^{(0,2)}\rangle\\!\rangle\Big{]}_{\text{reg}}\,,$
(4.54)

Because of the different boundary spin assignment, we can now probe the
coefficient of $\mathcal{O}^{(0,0)}\overline{\mathcal{O}}^{(0,-2)}$ in
$\mathbb{D}_{--}$,

$\mathbb{D}_{--}=\eta^{(0,-2)}\left[\mathcal{O}^{(0,0)}\times\overline{\mathcal{O}}^{(0,-1)}\right](2n_{-}^{2})+\dots\,.$
(4.55)

The expansion of $\delta_{-}\mathcal{O}^{(0,2)}$ in terms boundary operators
takes the form

$\delta_{-}\mathcal{O}^{(0,2)}=-{\rm
v}_{0}^{-}\frac{1}{2}\mathcal{O}^{(2,1)}+{\beta}_{1}\mathrm{d}{\rm
v}_{0}^{-}\mathcal{O}^{(1,1)}+{\beta}_{2}\mathrm{d}\mathrm{d}{\rm
v}_{0}^{-}\mathcal{O}^{(0,1)}+...\,.$ (4.56)

We find

${\beta}_{1}=\Delta-\frac{5}{2}\,,\quad{\beta}_{2}=(2-\Delta){\beta}_{1}\,,\qquad\eta^{(0,-2)}=\frac{1}{2}\,.$
(4.57)

This result can also be obtained simply by interchanging $\Delta\to 2-\Delta$
in the deformation of $M^{(-1,-1)}$. That is due to the invariance of the
spectrum under tower-swap.

To summarize, at second order, we always find a unique bootstrap solution. It
can now be used to explicitly compute $\delta^{2}M$ for an arbitrary smooth
deformation profile.

##  5 Third Order

We can apply the bootstrap method developed in the previous sections to
higher-order deformations. At each order, we pick a regularization scheme,
list all the finite terms that can appear, and fix their coefficients by
imposing the conformal symmetry of a curved line. All the ingredients that
appear in higher orders are already present at the third order. In this
section, we carry out the method in third order explicitly and discuss the
higher-order generalizations. Our main point in doing so is to demonstrate
that the bootstrap solution is unique and that it allows one to systematically
compute the expectation value of mesonic lines to any order.

###  5.1 Line Variation

The third-order variation of any mesonic line can be divided into boundary and
bulk variations as follows

$\displaystyle\delta^{3}{\cal M}$
$\displaystyle=\delta^{3}\bar{O}\,\mathcal{W}\,O+\delta^{2}\,\bar{O}\,\mathcal{W}\,\delta
O+\bar{O}\,\mathcal{W}\,\delta^{3}O+\delta\bar{O}\,\mathcal{W}\,\delta^{2}O\,$
(5.1) $\displaystyle+\delta\bar{O}\,\delta\mathcal{W}\,\delta
O+\delta^{2}\,\bar{O}\,\delta\mathcal{W}\,O+\bar{O}\,\delta\mathcal{W}\,\delta^{2}O\,$
$\displaystyle+\bar{O}\,\delta^{2}\mathcal{W}\,\delta
O+\delta\bar{O}\,\delta^{2}\mathcal{W}\,O+\bar{O}\,\delta^{3}\mathcal{W}\,O\,.$

As before, these expressions encode many contributions. In particular, the
third-order variation at the bulk of the line, $\delta^{3}{\cal W}$, includes
the integration of three displacement operators

$[\text{int}^{3}]_{\text{bare}}=|x_{10}|^{3}\int\limits_{0}^{1}\mathrm{d}s\,{\rm
v}_{s}^{\mu}\int\limits_{0}^{s}\mathrm{d}t\,{\rm
v}_{t}^{\nu}\int\limits_{0}^{t}\mathrm{d}u\,{\rm
v}_{u}^{\rho}\,\langle\\!\langle\overline{\mathcal{O}}\,{\mathbb{D}}_{\mu}(x_{s}){\mathbb{D}}_{\mu}(x_{t}){\mathbb{D}}_{\rho}(x_{u})\mathcal{O}\rangle\\!\rangle\,.$
(5.2)

The transverse spin of a third-order deformation is either one or three.
Hence, to have a non-zero variation, we need to start with a spinning straight
mesonic line. In the following, we have chosen to present the third-order
variation of $M^{(0,1)}_{10}$.

Local, scheme-independent terms in $\delta^{3}{\cal M}$ involving one or two
variations are the same as they are at second order. For example, we have

$\displaystyle\langle\delta\bar{O}^{(0,0)}\,\delta^{2}\mathcal{W}\,O^{(0,1)}\rangle$
$\displaystyle={\rm v}_{1}^{+}\int\mathrm{d}s({\rm
v}_{s}^{-})^{2}\langle\\!\langle\overline{\mathcal{O}}^{(0,1)}\mathbb{D}_{--}(x_{s})\mathcal{O}^{(0,1)}\rangle\\!\rangle+\dots$
(5.3) $\displaystyle={\rm v}_{1}^{+}\eta^{(-1,-1)}\int\mathrm{d}s\,({\rm
v}_{s}^{-})^{2}\,M_{1s}^{(1,-1)}M_{s0}^{(-1,1)}+\dots\,,$

where $\eta^{(-1,-1)}$ was determined in (4.53). If we use the same
regularization scheme that we have used at second order, then other local
scheme-dependent terms would also be the same. For that purpose, we first
regularize the triple integral in (5.2) by converting the middle integral into
a contour integral using (3.24). We then use the keyhole prescription to
regulate the remaining right and left integrals151515To regularize the double
integral contribution to $\delta^{2}M_{10}^{(0,0)}$ we have used the scheme of
(4.15). Similarly, for the double integral in $\delta^{2}M_{10}^{(-1,1)}$ we
use
$\delta^{2}M_{10}^{(-1,1)}\Big{|}_{\text{2-int}}\equiv{\mathrm{i}\eta\,|x_{10}|\over
2\sin(2\pi\Delta)}\supset\int\limits_{1,0)}\mathrm{d}s\,{\rm v}_{s}^{-}\delta
M_{1s}^{(-1,0)}M_{s0}^{(-1,1)}\,.$ (5.4)

$[\text{int}^{3}]=\left({\mathrm{i}\eta|x_{10}|\over
2\sin(2\pi\Delta)}\right)^{2}\subset\int\limits_{(1,0}\mathrm{d}s\,{\rm
v}_{s}^{-}\supset\int\limits_{s,0)}\mathrm{d}u\,{\rm
v}_{u}^{-}\,M^{(0,0)}_{1s}\,\delta M^{(-1,0)}_{su}\,M^{(-1,1)}_{u0}\,,$ (5.5)

where the order of the last $s$ and $u$ integrals does not matter.161616This
occurs because, after the middle integral, the integrals over $s$ and $u$
factor out into two separate integrals: one depends only on $s$, and the other
only on $u$.

In this regularization scheme, which is compatible with our second-order
variation, the only new coefficients are those multiplying finite terms with
three variations at the same point. All such terms have a single integral.
They take the form

$\displaystyle\text{[int]}_{\text{mid}}=$
$\displaystyle\frac{\mathrm{i}\eta|x_{10}|}{2\sin(2\pi\Delta)}\oint\mathrm{d}s\,M_{1s}^{(0,0)}\left[\chi_{0}\,{\rm
v}^{+}_{s}({\rm v}^{-}_{s})^{2}\delta_{3}^{2}+\left(\chi_{1}\,({\rm
v}_{s}^{-})^{2}\mathrm{d}{\rm v}_{s}^{+}+\chi_{2}\,{\rm v}_{s}^{+}{\rm
v}_{s}^{-}\mathrm{d}{\rm v}_{s}^{-}\right)\delta_{3}\right.$ (5.6)
$\displaystyle+\left.\left(\chi_{3}\,{\rm v}_{s}^{-}\mathrm{d}{\rm
v}_{s}^{+}\mathrm{d}{\rm v}_{s}^{-}+\chi_{4}\,{\rm v}_{s}^{+}(\mathrm{d}{\rm
v}_{s}^{-})^{2}+\chi_{5}\,({\rm v}_{s}^{-})^{2}\mathrm{d}\mathrm{d}{\rm
v}_{s}^{+}+\chi_{6}\,{\rm v}_{s}^{-}{\rm v}_{s}^{+}\mathrm{d}\mathrm{d}{\rm
v}_{s}^{-}\right)\right]M_{s0}^{(-1,1)}\,,$

where, for convenience, we have used integration by parts to put all the
longitudinal operator derivatives on the right.171717The boundary terms that
result from these i.b.p. are counter-terms and therefore do not produce finite
contributions. We have regularized the divergences by converting the integral
into a contour integral and absorbing all factors into the $\chi$’s.

Altogether, the third-order variation takes the form

$\displaystyle\delta^{3}M^{(0,1)}_{10}=$
$\displaystyle\frac{\mathrm{i}\eta|x_{10}|}{2\sin(2\pi\Delta)}\oint\limits_{[0,1]}\mathrm{d}s\,{\rm
v}_{s}^{-}\,\Big{[}M_{1s}^{(0,0)}\left([BB]^{(-1,1)}_{s0}+\eta^{(0,-2)}{\rm
v}_{s}^{-}{\rm v}_{0}^{+}M_{s0}^{(-2,2)}\right)$ (5.7)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\
+\left([\bar{B}\bar{B}]_{1s}^{(0,0)}+\eta^{(-1,-1)}{\rm v}_{s}^{-}{\rm
v}_{1}^{+}\,M_{1s}^{(1,-1)}\right)M_{s0}^{(-1,1)}\Big{]}$
$\displaystyle+[\text{int}]_{\text{mid}}+[\text{int}^{3}]\,,$

where $[\bar{B}\bar{B}]^{(0,0)}_{1s}$ is given in (4.17) and is evaluated
along $x_{1s}$ instead of $x_{10}$. $[BB]_{s0}^{(-1,1)}$ can be obtained from
$[\bar{B}\bar{B}]^{(0,0)}_{1s}$ using tower-swap and CPT, see the discussion
below (2.18) and (5.4). In practice, $[BB]_{s0}^{(-1,1)}$ is related to
$[BB]_{s0}^{(0,0)}$ by first exchanging the $\gamma$’s with the corresponding
$\tilde{\gamma}$’s, then flipping the signs of $\gamma_{4,5}$ and
$\tilde{\gamma}_{4,5}$, and finally replacing $\Delta\to 2-\Delta$.

###  5.2 Bootstrap

To fix the coefficients $\chi_{i}$ in (5.6) we impose the consistency of (5.7)
with conformal symmetry. That is, we impose that the mesonic line transforms
covariantly under conformal transformations $x\to\tilde{x}_{\beta}(x)$. As
before, we choose the conformal transformations such that they move the
endpoints in the transverse directions. Recall that $\beta$ parameterizes a
family of conformal transformations that continuously interpolate between
$x=\tilde{x}_{0}(x)$ and $\tilde{x}_{\beta}(x)$. As opposed to the previous
orders, at third order, the expansion in the deformation of the path and the
expansion in powers of $\beta$ are not the same.

Consider first the conformal transformation of a straight line, (4.27). The
conformal deformation profile ${\rm v}_{c}$ can be expanded as

${\rm v}_{c}={\rm v}^{(1)}_{c}+{\rm v}^{(3)}_{c}+O(\beta^{4})\,,$ (5.8)

where ${\rm v}^{(n)}_{c}\propto\beta^{n}$, see Appendix A for explicit
expressions. Because $M^{(0,1)}$ has zero expectation value, the constraint
coming from the conformal variation of a straight line reads,

$\delta^{3}_{{\rm v}^{(1)}_{c}}M^{(0,1)}_{10}+\delta^{1}_{{\rm
v}^{(3)}_{c}}M^{(0,1)}_{10}=0\,,$ (5.9)

where $\delta^{1}_{{\rm v}^{(3)}_{c}}M^{(0,1)}_{10}$ is the first order
variation (3.24), evaluated with the profile ${\rm v}^{(3)}_{c}$. Similarly,
$\delta^{3}_{{\rm v}^{(1)}_{c}}M^{(0,1)}_{10}$ is the third order variation
(5.7), evaluated with the profile ${\rm v}^{(1)}_{c}$.

If instead of the straight line, we apply a conformal transformation to a
deformed line (4.31), then the conformal deformation profile admits a double
expansion, in $\beta$ and in ${\rm v}_{a}$,

${\rm v}_{c}={\rm v}^{(1,0)}_{c}+{\rm v}^{(3,0)}_{c}+{\rm v}_{c}^{(1,2)}+{\rm
v}_{c}^{(2,1)}+\dots\,,$ (5.10)

where ${\rm v}_{c}^{(n,m)}\propto\beta^{n}{\rm v}_{a}^{m}$, see Appendix A for
explicit expressions.

We now impose the conformal covariance of $M^{(0,1)}$ at orders $\beta{\rm
v}_{a}^{2}$ and $\beta^{2}{\rm v}_{a}$. The conformal and spin sources are
proportional to the mesonic line expectation value before the conformal
transformation. The latter starts at order ${\rm v}_{a}$. Hence, the
corresponding bootstrap constraint reads

$\begin{split}&\left(\delta^{3}_{{\rm v}^{(1)}_{c}+{\rm
v}_{a}}-\delta^{3}_{{\rm v}_{a}}+\delta^{1}_{{\rm v}^{(3)}_{c}+{\rm
v}_{c}^{(2,1)}+{\rm v}_{c}^{(1,2)}}\right)M^{(0,1)}_{10}\\\
=&\left([\text{conformal factor}]+[\text{spin factor}]\right)\delta^{1}_{{\rm
v}_{a}}M^{(0,1)}_{10}\,.\end{split}$ (5.11)

The terms in the bracket on the RHS all starts at order ${\cal O}(\beta^{2})$
and ${\cal O}(\beta{\rm v}_{a})$. The conformal factor reads

$[\text{conformal factor}]=-2\Delta\,\delta\log|x_{10}|\,,$ (5.12)

where the distance change is evaluated using ${\rm v}_{c}^{(1)}+{\rm v}_{a}$.
The spin factor is given by

$[\text{spin
factor}]=\frac{1}{2}\left(\mathfrak{s}S_{0}+\mathfrak{s}^{\prime}S_{1}\right)\,,$
(5.13)

where $S=S^{(1,1)}+S^{(2,0)}$ and

$\displaystyle S^{(1,1)}=$ $\displaystyle\,2(\mathrm{d}\mathrm{d}{\rm
v}_{c}^{(1)})^{-}{\rm v}^{+}_{a}-2(\mathrm{d}\mathrm{d}{\rm
v}_{c}^{(1)})^{+}{\rm v}^{-}_{a}+(\mathrm{d}{\rm
v}_{c}^{(1)})^{-}\mathrm{d}{\rm v}^{+}_{a}-(\mathrm{d}{\rm
v}_{c}^{(1)})^{+}\mathrm{d}{\rm v}^{-}_{a}\,,$ (5.14) $\displaystyle
S^{(2,0)}=$ $\displaystyle\,({\rm v}_{c}^{(1)})^{+}(\mathrm{d}\mathrm{d}{\rm
v}_{c}^{(1)})^{-}-({\rm v}_{c}^{(1)})^{-}(\mathrm{d}\mathrm{d}{\rm
v}_{c}^{(1)})^{+}\,.$

By implementing (5.11) in `Mathematica` and using a polynomial of degree four
or higher for ${\rm v}_{a}$ we obtain a unique and consistent solution for the
$\chi$’s. They are given by

$\chi_{1}=\chi_{2}=0\,,\qquad\chi_{0}=\frac{\chi_{3}}{2}=\chi_{4}=2\chi_{5}=\chi_{6}=-\frac{1}{2}\,.$
(5.15)

###  5.3 Higher Orders

The third-order analysis is sufficient to demonstrate the general structure.
We work recursively in the order of the deformation. At the $n$’th order, we
list all the finite terms that can contribute to the mesonic line expectation
value. Due to the locality of the line effective action, the coefficients of
most of these terms are determined from lower orders, provided that we keep
using the same regularization scheme. The only new terms have all the $n$
variations at the same point. This point can either be integrated along the
line or at the boundary.181818In the case of the third-order deformation of
$M^{(0,1)}$, the terms with the three variations at the same boundary point
are either irrelevant or that they are divergent counter-term. To fix these
coefficients, we impose that the deformed line, along $x_{a}$ (4.31),
transforms covariantly under conformal transformations. For that aim, we
expand $\tilde{x}_{\beta}(x_{a})$ in powers of $\beta$ and ${\rm v}_{a}$,
keeping the sum of the two powers equal to $n$. We then collect all the terms
with a fixed power $\beta^{m}{\rm v}_{a}^{n-m}$ in the variation of the
straight line. These receive contributions from all orders in the line
variation, up to order $n$. We equate the result to the expansion of the spin
source and the conformal source at order $\beta^{m}{\rm v}_{a}^{n-m}$.

By considering a polynomial deformation profile, ${\rm v}_{a}$, we generate a
sufficient set of equations to fix the solution. These equations are all
linear in the new coefficients. Generating more equations serves as a
consistency check for the solution. Once all coefficients are fixed, we can go
back and evaluate the $n$’th order line variation explicitly.

##  6 Discussion

In this paper, we have developed a bootstrap method for studying conformal
line defects. We have found that imposing the conformal symmetry of line
defects along an arbitrary smooth path leads to new nontrivial functional
constraints. Focusing on line defects in CS-matter theories at large $N$, we
have observed that these constraints lead to a unique bootstrap solution and
are therefore sufficient to fix the defect expectation value. The following is
a list of future directions that we find interesting.

The one-dimensional defect CFT that we considered is relatively simple. This
is largely due to the factorization of the correlators on the line. In
particular, we used it to compute the correlation functions between $n$
insertions of the displacement operator that arise at the $n$’th order of the
line deformation. In other theories, with more complicated line defects, such
correlators are unknown functions of the conformal cross-ratios. It would be
interesting to see if these functions can be bootstrapped in a similar way and
used to compute the expectation value of the defect. What makes us optimistic
is that the smooth line deformation profile, ${\rm v}_{a}(s)$, entering the
bootstrap constraints, is arbitrary. Therefore, it generates a functional
bootstrap equation that can be used to gradually localize the individual
deformation points and fix the $n$-point correlation of the displacement
operator.

Large $N$ CS-matter theories are expected to be holographically dual to
parity-breaking versions of Vasiliev’s higher-spin theory, Vasiliev:1992av ;
Giombi:2011kc ; Aharony:2011jz ; Aharony:2012nh ; Klebanov:2002ja ;
Sezgin:2003pt ; Chang:2012kt ; Leigh:2003gk . The massless high-spin spectrum
of this theory suggests that a unified description in terms of a tensionless,
partly topological string may exist. In particular, it is not known how to
incorporate the mesonic lines into Vasiliev’s higher-spin theory. We expect
that doing so will require a stringy description and that the mesonic lines
may even be instrumental in constructing it.

The bootstrap method that we have employed gives the defect expectation value
in an expansion around the straight line. It would be interesting to resum
this expansion and find a more explicit representation of the conformal
invariant functional of the path $F[x(\cdot)]$ in (2.15). We expect the
holographic description of the mesonic line to be such a representation.

Finally, CS-matter theories possess a higher spin symmetry that is broken only
at order $1/N$. An implication of this symmetry is the existence of protected
primary tilt operators on the line, Gabai:2022mya . It would be interesting to
fix these operators and use them to derive bootstrap constraints in the same
way that we have used the conformal symmetry. These high-spin constraints are
expected to become important at the next order in $1/N$, while at the planar
order at which we are working, they are redundant.

###### Acknowledgements.

We thank Nathan Agmon, Ofer Aharony, Nissan Itzhaki, Zohar Komargodski, Yaron
Oz, Yifan Wang, and Shimon Yankielowicz for useful discussions. AS thanks the
Simons Center for Geometry and Physics for its hospitality. AS and DlZ are
supported by the Israel Science Foundation (grant number 1197/20). During
different periods of this work, BG was partially supported by the Simons
Collaboration Grants on the Non-Perturbative Bootstrap and on Confinement and
QCD Strings and by the DOE grant DE-SC0007870. DlZ is supported in part by the
Royal Society University Research Fellowships grant URF/R1/221310
“Bootstrapping Quantum Gravity”.

##  Appendix A Conformal Transformation of a Curved Line

In this section, we study conformal transformations of arbitrary paths. We
parameterize the path by its projection on the $x^{3}$ axis,

$x_{a}(s)=({\rm v}_{a}^{1}(s),{\rm v}_{a}^{2}(s),s)\,,$ (A.1)

assuming that the transverse vector function $\vec{{\rm
v}}_{a}(s)\equiv\\{{\rm v}_{a}^{1}(s),{\rm v}_{a}^{2}(s)\\}$ is single valued
and smooth. We then apply the following conformal transformation to it

$x_{a}\quad\rightarrow\quad
x_{ac}=\mathcal{R}_{\hat{x}^{1}}(\mathfrak{c}_{1}^{2})\cdot\mathcal{R}_{\hat{x}^{2}}(\mathfrak{c}_{1}^{1})\cdot\mathcal{B}_{b}(x_{a})+\\{\mathfrak{c}_{0}^{1},\mathfrak{c}_{0}^{2},a_{3}\\}\,,$
(A.2)

where $\mathcal{R}_{\hat{n}}(\theta)$ denotes the finite rotation of angle
$\theta$ around axis $\hat{n}$. $\mathcal{B}_{b}(x_{a})$ denotes the special
conformal transformation

$\mathcal{B}_{b}(x)^{\mu}\equiv\frac{x^{\mu}-b^{\mu}x^{2}}{1-2b\cdot
x+b^{2}x^{2}}\,,\qquad{\rm with}\qquad
b=(-\mathfrak{c}_{2}^{1},-\mathfrak{c}_{2}^{2},b_{3})\,.$ (A.3)

The last term in (A.2) is a constant shift. The transverse vectors
$\vec{\mathfrak{c}}_{0}$, $\vec{\mathfrak{c}}_{1}$, and
$\vec{\mathfrak{c}}_{2}$ are a set of continuous parameters that we use to
parameterize the conformal transformation (4.28). Here, we did not include
rotations around the $\hat{x}^{3}$ axis, which leaves the straight line
invariant.

For generic values of the conformal transformation parameters, the two
endpoints transform in all three directions. Using translation invariance, we
can assume without loss of generality that the endpoint $x_{0}$ transforms
only in the transverse direction. This choice fixes the value of $a_{3}$. We
then further impose that the endpoint $x_{1}$ also transforms in the
transverse direction. This condition fixes $b_{3}$ in terms of the other
parameters. There are two solutions for $b_{3}$ and we pick the one that
connects smoothly to the identity.191919For ${\rm v}_{a}$ that does not move
the endpoints, $a_{3}=0$ and $b_{3}$ does not depend on the original path.
Otherwise, $a_{3}$ and $b_{3}$ depend on ${\rm v}_{a}$ at the endpoints.

After doing so, we are left with a conformal transformation that depends on
three transverse vectors, $\vec{\mathfrak{c}}_{1}$, $\vec{\mathfrak{c}}_{2}$
and $\vec{\mathfrak{c}}_{3}$, just as we have in equation (4.28). We assume
that these parameters are small and of the same order ${\cal O}(\beta)$, and
expand $x_{ac}$.

##### Leading Order.

To leading order, the conformal transformation just added up to the $x_{a}$
path202020To leading order there is no need to re-parameterize the deformed
path.

${x}_{ac}=x_{a}+(\vec{\mathfrak{c}}_{0}+\vec{\mathfrak{c}}_{1}s+\vec{\mathfrak{c}}_{2}s^{2},0)+O(\beta^{2})\,.$
(A.4)

##### Second Order.

There is no second-order contribution to ${x}_{ac}$ for the conformal
transformation we are considering

${x}_{ac}=x_{a}+(\vec{\mathfrak{c}}_{0}+\vec{\mathfrak{c}}_{1}s+\vec{\mathfrak{c}}_{2}s^{2},0)+O(\beta^{3})\,.$
(A.5)

##### Third Order.

In general, the conformal transformation moves the points on the line in the
transverse direction as well as along the line. As a result, at third order,
we also have to perform a reparametrization of the path, keeping the
deformation vector ${\rm v}_{c}$ transverse to the straight line.

Another subtlety that first appears in the third order is that the deformation
profile of the path due to the conformal transformation starts to depend on
${\rm v}_{a}$. It has the following decomposition

${\rm v}_{c}={\rm v}^{(1,0)}_{c}+{\rm v}^{(3,0)}_{c}+{\rm v}_{c}^{(1,2)}+{\rm
v}_{c}^{(2,1)}+\dots\,,$ (A.6)

where ${\rm v}_{c}^{(n,m)}\propto\beta^{n}{\rm v}_{a}^{m}$. The first-order
term, ${\rm v}_{c}^{(1,0)}$, is given by the second term in (A.4). The third-
order terms read

$\displaystyle\left({\rm v}^{(3,0)}_{c}\right)^{\pm}_{s}$
$\displaystyle=2{\mathfrak{c}_{2}^{\mp}}{\mathfrak{c}_{2}^{\pm}}^{2}s^{4}+2{\mathfrak{c}_{2}^{\pm}}s^{3}({\mathfrak{c}_{2}^{\mp}}{\mathfrak{c}_{1}^{\pm}}+{\mathfrak{c}_{1}^{\mp}}{\mathfrak{c}_{2}^{\pm}})$
(A.7)
$\displaystyle+\frac{1}{4}s^{2}\left(2{\mathfrak{c}_{2}^{\mp}}{\mathfrak{c}_{1}^{\pm}}^{2}+{\mathfrak{c}_{2}^{\pm}}\left(10{\mathfrak{c}_{1}^{\mp}}{\mathfrak{c}_{1}^{\pm}}-{\mathfrak{c}_{1}^{\mp}}^{2}+{\mathfrak{c}_{1}^{\pm}}^{2}\right)\right)$
$\displaystyle+\frac{1}{24}s\left(-3{\mathfrak{c}_{1}^{\mp}}^{2}{\mathfrak{c}_{1}^{\pm}}+15{\mathfrak{c}_{1}^{\mp}}{\mathfrak{c}_{1}^{\pm}}^{2}+{\mathfrak{c}_{1}^{\mp}}^{3}+3{\mathfrak{c}_{1}^{\pm}}^{3}\right)\,,$
$\displaystyle\left({\rm v}^{(1,2)}_{c}\right)^{\pm}_{s}$
$\displaystyle=s^{2}{\dot{v}^{\pm}_{0}}({\mathfrak{c}_{1}^{\mp}}{v^{\pm}_{0}}-({\mathfrak{c}_{1}^{\mp}}+2{\mathfrak{c}_{2}^{\mp}}){v^{\pm}_{1}}+{\mathfrak{c}_{1}^{\pm}}{v^{\mp}_{0}}-({\mathfrak{c}_{1}^{\pm}}+2{\mathfrak{c}_{2}^{\pm}}){v^{\mp}_{1}})$
$\displaystyle+2s\left({v^{\pm}_{s}}\left({\mathfrak{c}_{2}^{\mp}}{\dot{v}^{\pm}_{0}}-{\mathfrak{c}_{1}^{\mp}}{v^{\pm}_{0}}+({\mathfrak{c}_{1}^{\mp}}+2{\mathfrak{c}_{2}^{\mp}}){v^{\pm}_{1}}-{\mathfrak{c}_{1}^{\pm}}{v^{\mp}_{0}}+({\mathfrak{c}_{1}^{\pm}}+2{\mathfrak{c}_{2}^{\pm}}){v^{\mp}_{1}}\right)+{\mathfrak{c}_{2}^{\pm}}{v^{\mp}_{s}}{\dot{v}^{\pm}_{0}}\right)$
$\displaystyle+{\dot{v}^{\pm}_{0}}({\mathfrak{c}_{1}^{\mp}}{v^{\pm}_{s}}-{\mathfrak{c}_{1}^{\mp}}{v^{\pm}_{0}}+{\mathfrak{c}_{1}^{\pm}}{v^{\mp}_{s}}-{\mathfrak{c}_{1}^{\pm}}{v^{\mp}_{0}})-2{\mathfrak{c}_{2}^{\mp}}{v^{\pm}_{s}}^{2}\,,$
$\displaystyle\left({\rm v}^{(2,1)}_{c}\right)^{\pm}_{s}$
$\displaystyle=2{\mathfrak{c}_{2}^{\mp}}{\mathfrak{c}_{2}^{\pm}}s^{3}{\dot{v}^{\pm}_{0}}+s^{2}\left(2{\mathfrak{c}_{2}^{\pm}}^{2}{v^{\mp}_{s}}-({\mathfrak{c}_{1}^{\mp}}{\mathfrak{c}_{1}^{\pm}}+2{\mathfrak{c}_{2}^{\mp}}{\mathfrak{c}_{2}^{\pm}}){\dot{v}^{\pm}_{0}}\right)$
$\displaystyle+s\left({\mathfrak{c}_{1}^{\mp}}{\mathfrak{c}_{1}^{\pm}}{\dot{v}^{\pm}_{0}}+2({\mathfrak{c}_{1}^{\mp}}+{\mathfrak{c}_{2}^{\mp}})({\mathfrak{c}_{1}^{\pm}}+2{\mathfrak{c}_{2}^{\pm}}){v^{\pm}_{s}}-2{\mathfrak{c}_{2}^{\pm}}({\mathfrak{c}_{1}^{\mp}}{v^{\pm}_{0}}+{\mathfrak{c}_{1}^{\pm}}{v^{\mp}_{0}})+2{\mathfrak{c}_{1}^{\pm}}{\mathfrak{c}_{2}^{\pm}}{v^{\mp}_{s}}\right)$
$\displaystyle+\frac{1}{4}\left(2{\mathfrak{c}_{1}^{\pm}}({\mathfrak{c}_{1}^{\pm}}({v^{\mp}_{s}}-2{v^{\mp}_{0}})-2{\mathfrak{c}_{1}^{\mp}}{v^{\pm}_{0}})+\left(2{\mathfrak{c}_{1}^{\mp}}{\mathfrak{c}_{1}^{\pm}}-{\mathfrak{c}_{1}^{\mp}}^{2}+{\mathfrak{c}_{1}^{\pm}}^{2}\right){v^{\pm}_{s}}\right)\,.$

Here, for notational simplicity, we have omitted the index $a$ from $v_{a}$.
In this context, the lower $s$-index of $v$ indicates the position. For
example, $v_{s}^{\pm}\equiv v_{a}^{\pm}(s)$, and so on.

##  Appendix B Scheme-Independence of the Second Order Variation

The regularization scheme that we have used to regularize the double integral
in (4.15) is not unique. We can instead choose to perform the $s$ integral
first,

$\left[\iint\mathcal{P}\mathrm{d}s\,\mathrm{d}t\ldots\right]_{\text{reg
2}}={\mathrm{i}\over 2\sin(2\pi\Delta)}\
\supset\int\limits_{1,0)}\mathrm{d}t\oint\limits_{[1,t]}\mathrm{d}s\ldots\,.$
(B.1)

or to average the both orders,

$\left[\iint\mathcal{P}\mathrm{d}s\,\mathrm{d}t\ldots\right]_{\text{reg
3}}=\frac{1}{2}\times{\mathrm{i}\over
2\sin(2\pi\Delta)}\Big{[}\supset\int\limits_{1,0)}\mathrm{d}t\oint\limits_{[1,t]}\mathrm{d}s\ldots+\subset\int\limits_{(1,0}\mathrm{d}s\oint\limits_{[s,0]}\mathrm{d}t\ldots\Big{]}\,.$
(B.2)

For the first regularization scheme (B.1), we find that the values of the
invariants $\Lambda$, $\Xi$, and $\Sigma$ remain the same, see (4.37). The
values of the boundary terms, however, are swapped and can be obtained by
exchanging $\gamma_{j}\leftrightarrow\tilde{\gamma}_{j}$ in (4.36).

For the second symmetric regularization scheme (B.2) we can repeat the
bootstrap above. The values of the invariants $\Lambda$, $\Xi$, and $\Sigma$
remain the same as in (4.37). This scheme preserves the CPT symmetry, which
interchanges the left-right ends. Thus, the resulting boundary terms are
symmetric. They are given by

$\displaystyle\gamma_{0}$
$\displaystyle=\tilde{\gamma}_{0}=-\frac{1}{4}\,,\qquad$
$\displaystyle\gamma_{1}$
$\displaystyle=\tilde{\gamma}_{1}=\frac{3-2\Delta}{8}\,,$ (B.3)
$\displaystyle\gamma_{2}$
$\displaystyle=\tilde{\gamma}_{2}=\frac{\Delta(3-2\Delta)}{8},\qquad$
$\displaystyle\gamma_{3}$ $\displaystyle=\tilde{\gamma}_{3}=0,$
$\displaystyle\gamma_{4}$
$\displaystyle=\tilde{\gamma}_{4}=\frac{1+2\Delta}{8}\,,\qquad$
$\displaystyle\gamma_{5}$
$\displaystyle=\tilde{\gamma}_{5}=\frac{1}{8}(2\Delta^{2}-3\Delta-2)\,.$

From the two examples above, we can see that different regularization schemes
of the double integral result in different values for the boundary
coefficients.

## References

  * (1) I. Affleck and A. W. W. Ludwig, Universal noninteger ’ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161–164.
  * (2) J. Polchinski and J. Sully, Wilson Loop Renormalization Group Flows, JHEP 10 (2011) 059, [arXiv:1104.5077].
  * (3) D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402, [hep-th/0312197].
  * (4) H. Casini, I. Salazar Landea, and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140, [arXiv:1607.00390].
  * (5) G. Cuomo, Z. Komargodski, and A. Raviv-Moshe, Renormalization Group Flows on Line Defects, Phys. Rev. Lett. 128 (2022), no. 2 021603, [arXiv:2108.01117].
  * (6) V. A. Kazakov, WILSON LOOP AVERAGE FOR AN ARBITRARY CONTOUR IN TWO-DIMENSIONAL U(N) GAUGE THEORY, Nucl. Phys. B 179 (1981) 283–293.
  * (7) S. Giombi, V. Pestun, and R. Ricci, Notes on supersymmetric Wilson loops on a two-sphere, JHEP 07 (2010) 088, [arXiv:0905.0665].
  * (8) N. Drukker, D. J. Gross, and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006, [hep-th/9904191].
  * (9) A. M. Polyakov and V. S. Rychkov, Loop dynamics and AdS / CFT correspondence, Nucl. Phys. B 594 (2001) 272–286, [hep-th/0005173].
  * (10) A. M. Polyakov and V. S. Rychkov, Gauge field strings duality and the loop equation, Nucl. Phys. B 581 (2000) 116–134, [hep-th/0002106].
  * (11) G. W. Semenoff and D. Young, Wavy Wilson line and AdS / CFT, Int. J. Mod. Phys. A 20 (2005) 2833–2846, [hep-th/0405288].
  * (12) S. Giombi and V. Pestun, The 1/2 BPS ’t Hooft loops in N=4 SYM as instantons in 2d Yang-Mills, J. Phys. A 46 (2013) 095402, [arXiv:0909.4272].
  * (13) J. C. Toledo, Smooth Wilson loops from the continuum limit of null polygons, arXiv:1410.5896.
  * (14) M. Cooke, A. Dekel, and N. Drukker, The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines, J. Phys. A 50 (2017), no. 33 335401, [arXiv:1703.03812].
  * (15) B. Gabai, A. Sever, and D.-l. Zhong, Line Operators in Chern-Simons–Matter Theories and Bosonization in Three Dimensions, Phys. Rev. Lett. 129 (2022), no. 12 121604, [arXiv:2204.05262].
  * (16) B. Gabai, A. Sever, and D.-l. Zhong, Line operators in Chern-Simons-Matter theories and Bosonization in Three Dimensions II: Perturbative analysis and all-loop resummation, JHEP 04 (2023) 070, [arXiv:2212.02518].
  * (17) J. L. Cardy, Scaling and renormalization in statistical physics.  1996\.
  * (18) A. B. Zamolodchikov, Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory, Sov. J. Nucl. Phys. 46 (1987) 1090.
  * (19) D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153–158.
  * (20) K. Ranganathan, H. Sonoda, and B. Zwiebach, Connections on the state space over conformal field theories, Nucl. Phys. B 414 (1994) 405–460, [hep-th/9304053].
  * (21) M. R. Gaberdiel, A. Konechny, and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42 (2009) 105402, [arXiv:0811.3149].
  * (22) A. Amoretti and N. Magnoli, Conformal perturbation theory, Phys. Rev. D 96 (2017), no. 4 045016, [arXiv:1705.03502].
  * (23) C. Behan, Conformal manifolds: ODEs from OPEs, JHEP 03 (2018) 127, [arXiv:1709.03967].
  * (24) A. Cavaglià, N. Gromov, J. Julius, and M. Preti, Integrated correlators from integrability: Maldacena-Wilson line in $\mathcal{N}$ = 4 SYM, JHEP 04 (2023) 026, [arXiv:2211.03203].
  * (25) T. Hartman, D. Mazac, D. Simmons-Duffin, and A. Zhiboedov, Snowmass White Paper: The Analytic Conformal Bootstrap, in Snowmass 2021, 2, 2022.  arXiv:2202.11012.
  * (26) D. Poland and D. Simmons-Duffin, Snowmass White Paper: The Numerical Conformal Bootstrap, in Snowmass 2021, 3, 2022.  arXiv:2203.08117.
  * (27) E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351–399.
  * (28) M. A. Vasiliev, More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions, Phys. Lett. B 285 (1992) 225–234.
  * (29) S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia, and X. Yin, Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C 72 (2012) 2112, [arXiv:1110.4386].
  * (30) O. Aharony, G. Gur-Ari, and R. Yacoby, d=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories, JHEP 03 (2012) 037, [arXiv:1110.4382].
  * (31) O. Aharony, G. Gur-Ari, and R. Yacoby, Correlation Functions of Large N Chern-Simons-Matter Theories and Bosonization in Three Dimensions, JHEP 12 (2012) 028, [arXiv:1207.4593].
  * (32) I. R. Klebanov and A. M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213–219, [hep-th/0210114].
  * (33) E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP 07 (2005) 044, [hep-th/0305040].
  * (34) C.-M. Chang, S. Minwalla, T. Sharma, and X. Yin, ABJ Triality: from Higher Spin Fields to Strings, J. Phys. A 46 (2013) 214009, [arXiv:1207.4485].
  * (35) R. G. Leigh and A. C. Petkou, Holography of the N=1 higher spin theory on AdS(4), JHEP 06 (2003) 011, [hep-th/0304217].