aainstitutetext: Science Division, New York University Abu Dhabi, Saadiyat
Island, Abu Dhabi, United Arab Emiratesbbinstitutetext: Courant Institute of
Mathematical Sciences, New York University, 251 Mercer Street, New York, NY
10012, USAccinstitutetext: Department of Mathematics, Vrije Universiteit
Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, The
Netherlandsddinstitutetext: Center for Cosmology and Particle Physics, New
York University, 726 Broadway, New York, NY 10003, USA

# The Brownian loop soup stress-energy tensor

Federico Camia a Valentino F. Foit a Alberto Gandolfi d Matthew Kleban
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Brownian loop soup (BLS) is a conformally invariant statistical ensemble
of random loops in two dimensions characterized by an intensity $\lambda>0$.
Recently, we constructed families of operators in the BLS and showed that they
transform as conformal primary operators. In this paper we provide an explicit
expression for the BLS stress-energy tensor and compute its operator product
expansion with other operators. Our results are consistent with the conformal
Ward identities and our previous result that the central charge is
$c=2\lambda$. In the case of domains with boundary we identify a boundary
operator that has properties consistent with the boundary stress-energy
tensor. We show that this operator generates local deformations of the
boundary and that it is related to a boundary operator that induces a Brownian
excursion starting or ending at its insertion point.

##  1 Introduction

The Brownian loop soup (BLS) lawler2004brownian  is an ideal gas of Brownian
loops in two dimensions with a distribution chosen so that it is invariant
under local conformal transformations. The distribution depends on a single
parameter, the intensity $\lambda>0$. The BLS is implicit in the work of
Symanzik osti_4117149  on Euclidean quantum field theory, more precisely, in
the representation of the partition function of Euclidean fields in terms of
random paths that are locally statistically equivalent to Brownian motion.
This representation can be made precise for the Gaussian free field, in which
case the random paths are independent of each other and can be generated as a
Poisson process. Viewed this way, the BLS consists of $2\lambda$ Gaussian free
fields (a generally non-integer number).

The BLS is closely related not only to Brownian motion and the Gaussian free
field but also to the Schramm-Loewner Evolution (SLE) and Conformal Loop
Ensembles (CLEs). It provides an interesting and useful link between Brownian
motion, field theory, and statistical mechanics. Inspired and partly motivated
by these connections, in Camia_2016 ; Camia_2020 ; camia2021scalar  we
introduced and studied operators that compute properties of the BLS,
discovering new families of conformal primary fields and analyzing the
operator content of the emerging conformal field theory.

In this paper, we derive the bulk stress-energy tensor of the theory as well
as the boundary stress-energy tensor in the upper half-plane. We also analyze
the relation of the boundary stress-energy tensor to domain perturbations and
boundary condition changing operators.

###  1.1 Preliminary definitions

The Brownian Loop Soup (BLS) is an ideal gas of planar loops. If $A$ is a set
of loops, the partition function of the BLS restricted to loops from $A$ can
be written as

$\displaystyle{Z_{A}=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}\left(\mu^{\text{loop}}(A)\right)^{n}},$
(1)

where $\lambda>0$ is a constant and $\mu^{\text{loop}}$ is a measure on planar
loops called _Brownian loop measure_ and defined as

$\displaystyle\mu^{\text{loop}}\equiv\int_{\mathbb{C}}\int_{0}^{\infty}\frac{1}{2\pi
t^{2}}\,\mu^{br}_{z,t}\,dt\,d{\bf A}(z),$ (2)

where $\bf A$ denotes area and $\mu^{br}_{z,t}$ is the complex Brownian bridge
measure with starting point $z$ and duration $t$.111We note that the Brownian
loop measure should be interpreted as a measure on “unrooted” loops, that is,
loops without a specified starting point. Unrooted loops are equivalence
classes of rooted loops. The interested reader is referred to
lawler2004brownian  for more details. $Z_{A}$ can be thought of as the grand
canonical partition function of a system of loops with fugacity $\lambda$, and
the BLS can be shown to be conformally invariant and to have central charge
$c=2\lambda$, see lawler2004brownian ; Camia_2016 .

As in camia2021scalar , in this paper we will only be concerned with the of
outer boundaries of Brownian loops. More precisely, given a planar loop
$\gamma$ in $\mathbb{C}$, its outer boundary $\ell=\ell(\gamma)$ is the
boundary of the unique infinite component of ${\mathbb{C}}\setminus\gamma$.
Note that, for any planar loop $\gamma$, $\ell(\gamma)$ is always a simple
closed curve, i.e., a closed loop without self-intersections. Hence, in this
paper, we will work with collections ${\mathcal{L}}$ of simple loops $\ell$
which are the outer boundaries of the loops from a BLS and for us, with a
slight abuse of terminology, a BLS will be a collection of simple loops.

Given a simple loop $\ell$, let $\bar{\ell}$ denote its _interior_ , i.e., the
unique, bounded, simply connected component of $\mathbb{C}\setminus\ell$. In
other words, a point $z$ belongs to $\bar{\ell}$ if $\ell$ disconnects $z$
from infinity, in which case we write $z\in\bar{\ell}$. In Camia_2016 , the
authors studied the correlation functions of the _layering operator_ 222In
this paper we use the terms _field_ and _operator_ interchangeably.

$\displaystyle
V_{\beta}(z)=\exp\Big{(}{i\beta\sum_{\ell:z\in\bar{\ell}}\sigma_{\ell}}(z)\Big{)},$
(3)

where $\sigma_{\ell}$ are independent, symmetric, $(\pm 1)$-valued Boolean
variables associated to the loops. One difficulty arises immediately due to
the scale invariance of the BLS, which implies that the sum at the exponent is
infinite with probability one. This difficulty can be overcome by imposing a
short-distance cutoff $\delta>0$ on the diameter of loops333An additional
infrared cutoff or a “charge neutrality” or “charge conservation” condition
may be necessary in some circumstances — we refer the interested reader to
Camia_2016  for more details. (essentially removing from the loop soup all
loops with diameter smaller than $\delta$), which produces a cutoff version
$V^{\delta}_{\beta}$ of $V_{\beta}$. The cutoff $\delta$ can be removed by
rescaling $V^{\delta}_{\beta}$ by $\delta^{-2\Delta(\beta)}$, with
$\Delta(\beta)=\frac{\lambda}{10}(1-\cos\beta)$, and sending $\delta\to 0$.
When $\delta\to 0$, the $n$-point correlation functions of
$\delta^{-2\Delta(\beta)}V^{\delta}_{\beta}$ converge to conformally covariant
quantities Camia_2016 , showing that the limiting field is a scalar conformal
primary field with real and positive conformal dimensions varying continuously
as periodic functions of $\beta$, namely as

$\displaystyle\Delta(\beta)=\bar{\Delta}(\beta)=\frac{\lambda}{10}(1-\cos\beta).$
(4)

This limiting field is further studied in Camia_2020 , where its canonically
normalized version is denoted by $\mathcal{O}_{\beta}$.444By _canonically
normalized_ we mean that
$\Braket{\mathcal{O}_{\beta}(z)\mathcal{O}_{-\beta}(z^{\prime})}_{\mathbb{C}}=|z-z^{\prime}|^{-4\Delta(\beta)}$.

For a domain $D$ with a boundary $\partial D$ and a point $\zeta\in\partial
D$, the _boundary field_ $\mathcal{B}^{\delta}_{\varepsilon}(\zeta)$ studied
in this paper counts the number of loops $\ell$ with diameter at least
$\delta$ passing within a short-distance $\varepsilon$ of $\zeta$. This is a
boundary version of the _edge field_ studied in camia2021scalar  and discussed
in Section 1.2 below. As in camia2021scalar , the cutoffs
$\delta,\varepsilon>0$ are necessary to keep
$\mathcal{B}^{\delta}_{\varepsilon}(\zeta)$ from being infinite or identically
zero. We show that they can be removed (i.e. sent to zero) by placing the
field $\mathcal{B}^{\delta}_{\varepsilon}$ inside $n$-point correlation
functions with $n\geq 2$ and renormalizing it by an appropriate power of
$\varepsilon$.

###  1.2 The edge counting operator

For the reader’s convenience, in this section we provide a brief introduction
to the _edge counting operator_ introduced in camia2021scalar . Some of the
properties of the edge counting operator will be essential to the analysis
carried out in this paper.

The edge counting operator $\mathcal{E}(z)$, for $z$ in the _interior_ of a
domain $D$, is defined in camia2021scalar  as the limit

$\displaystyle\begin{split}{\mathcal{E}}(z):=\frac{\hat{c}}{\sqrt{\lambda}}\lim_{\varepsilon\to
0}\varepsilon^{-2/3}E_{\varepsilon}(z),\end{split}$ (5)

with

$\displaystyle\begin{split}E_{\varepsilon}(z):=&\lim_{\delta\to
0}\left(N^{\delta}_{\varepsilon}(z)-\Braket{N^{\delta}_{\varepsilon}(z)}_{D}\right)\\\
=&\lim_{\delta\to
0}\left(N^{\delta}_{\varepsilon}(z)-\lambda\mu^{\text{loop}}_{D}\left(\text{diam}(\ell)>\delta,\ell\cap
B_{\varepsilon}(z)\neq\emptyset\right)\right).\end{split}$ (6)

Here $N^{\delta}_{\varepsilon}(z)$ counts the number of loops $\gamma$ with
diameter555The diameter of a loop is the largest distance between any two
point on the loop. $\text{diam}(\gamma)=\text{diam}(\ell)$ larger than
$\delta$ and whose “edge” $\ell$ (the outer boundary) comes
$\varepsilon-$close to $z$. The angle brackets $\langle\cdot\rangle_{D}$
denote expectation with respect to the Brownian loop soup in $D$ (of fixed
intensity $\lambda$), $\mu^{\text{loop}}_{D}$ is the Brownian loop measure
$\mu^{\text{loop}}$ restricted to $D$, i.e. the unique (up to a multiplicative
constant) conformally invariant measure on simple planar loops
2005math…..11605W , and $B_{\varepsilon}(z)$ is the disk of radius
$\varepsilon$ centered at $z$.

The subtraction of the mean in (6) is needed because the mean is divergent in
the limit $\delta\to 0$, due to the scale invariance of the BLS. With this,
the field $E_{\varepsilon}$ is well defined in the sense of correlation
functions. More precisely, it is shown in camia2021scalar  that, for any
collection of points $z_{1},\ldots,z_{n}\in D$ at distance greater than
$2\varepsilon$ from each other, with $n\geq 2$, the following limit exists:

$\displaystyle\langle E_{\varepsilon}(z_{1})\ldots
E_{\varepsilon}(z_{n})\rangle_{D}:=\lim_{\delta\to 0}\langle
E^{\delta}_{\varepsilon}(z_{1})\ldots
E^{\delta}_{\varepsilon}(z_{n})\rangle_{D}.$ (7)

The exponent $2/3$ used in (5) is a consequence of the fact that the
$\mu^{\text{loop}}_{D}$-measure of “macroscopic” loops coming to distance
$\varepsilon$ of $z$ in the interior of $D$ tends to zero as
$\varepsilon^{2/3}$ when $\varepsilon\to 0$. This can be understood using a
deep connection 2005math…..11605W  between the Brownian loop measure
$\mu^{\text{loop}}_{D}$ and a conformally invariant model of non-simple loops,
called CLE6, which emerges from the scaling limit of critical two-dimensional
percolation Camia_2006 . Using this connection, the exponent $2/3$ is directly
related to the probability of a three-arm event in the bulk in two-dimensional
percolation PhysRevLett.83.1359 ; Smirnov2001CRITICALEF .

The field $\mathcal{E}$ is only formally defined by (5) because it cannot be
evaluated pointwise, as is customary in (Euclidean) quantum field theory.
However, it is shown in camia2021scalar  that $\mathcal{E}$ has well-defined
$n$-point functions and behaves like a conformal primary field with dimensions
$(1/3,1/3)$. The constant $\hat{c}$ and the multiplicative factor
$\frac{\hat{c}}{\sqrt{\lambda}}$ are chosen so that the field is canonically
normalized, that is,

$\displaystyle\Braket{{\mathcal{E}}(z_{1}){\mathcal{E}}(z_{2})}_{\mathbb{C}}=|z_{1}-z_{2}|^{-4/3}.$
(8)

More generally, the $n$-point functions of the edge operator can be expressed
as

$\displaystyle\langle{\mathcal{E}}(z_{1})\ldots{\mathcal{E}}(z_{n})\rangle_{D}=\frac{\hat{c}^{n}}{\lambda^{n/2}}\sum_{(S_{1},\ldots,S_{r})\in\mathcal{S}}\lambda^{r}\alpha^{S_{1}}_{D}\ldots\alpha^{S_{r}}_{D},$
(9)

where $\mathcal{S}=\mathcal{S}(z_{1},\ldots,z_{n})$ denotes the set of all
partitions of $\\{z_{1},\ldots,z_{n}\\}$ such that each element $S_{l}$ of
$(S_{1},\ldots,S_{r})\in\mathcal{S}$ has cardinality $|S_{l}|\geq 2$ and, for
any subset $S_{l}=\\{z_{j_{1}},\ldots,z_{j_{k}}\\}$ of
$\\{z_{1},\ldots,z_{n}\\}$,

$\displaystyle\alpha^{S_{l}}_{D}\equiv\alpha_{D}^{z_{j_{1}},\ldots,z_{j_{k}}}:=\lim_{\varepsilon\to
0}\varepsilon^{-2k/3}\mu_{D}^{\text{loop}}\big{(}\ell\cap
B_{\varepsilon}(z_{j_{m}})\neq\emptyset,m=1,\ldots,k\big{)}.$ (10)

It is shown in camia2021scalar  that, if $D\subseteq{\mathbb{C}}$ is either
the complex plane $\mathbb{C}$ or the upper-half plane $\mathbb{H}$ or any
domain conformally equivalent to $\mathbb{H}$, for any collection of distinct
points $z_{1},\ldots,z_{k}\in D$ with $k\geq 2$, the above limit exists and
has the following property. If $D^{\prime}$ is a domain conformally equivalent
to $D$ and $f:D\to D^{\prime}$ is a conformal map, then

$\alpha_{D^{\prime}}^{f(z_{1}),\ldots,f(z_{k})}=\left(\prod_{j=1}^{k}|f^{\prime}(z_{k})|^{-2/3}\right)\alpha_{D}^{z_{1},\ldots,z_{k}}.$
(11)

This, combined with (9), implies that

$\Braket{\mathcal{E}(f(z_{1}))\ldots\mathcal{E}(f(z_{n}))}_{D^{\prime}}=\left(\prod_{j=1}^{k}|f^{\prime}(z_{k})|^{-2/3}\right)\Braket{\mathcal{E}(z_{1})\ldots\mathcal{E}(z_{n})}_{D}.$
(12)

###  1.3 Summary of the main results and structure of the paper

  1. Section 2.

We provide integral expressions of the bulk stress-energy tensor $T$ that can
be used to compute certain correlation functions involving $T$. We verify that
the Ward identities involving these correlation functions are satisfied,
confirming the validity of our expressions for $T$.

  2. Section 3.

We study a boundary version of the edge counting operator $\mathcal{E}$ (5),
namely

$\displaystyle\begin{split}{\mathcal{B}}(\zeta):=\frac{1}{\sqrt{\lambda}}\lim_{\varepsilon\to
0}\varepsilon^{-2}E_{\varepsilon}(\zeta),\end{split}$ (13)

where $\zeta\in\partial D$. We show that, when $D$ is the upper half-plane
$\mathbb{H}$ and $x_{1},x_{2}\in\mathbb{R}$,

$\displaystyle\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}=|x_{1}-x_{2}|^{-4}.$
(14)

We identify a new operator, formally

$\displaystyle\begin{split}{\mathcal{T}}(\zeta):=\sqrt{\lambda}{\mathcal{B}}(\zeta)=\lim_{\varepsilon\to
0}\varepsilon^{-2}E_{\varepsilon}(\zeta),\end{split}$ (15)

whose behavior is consistent with the role of boundary stress-energy tensor.
We derive part of the operator product expansion of
$\mathcal{T}\times\mathcal{T}$ (66) and check the Ward identity
(LABEL:eq:Ward).

  3. Section 4.

We link the operator $\mathcal{T}$ to local deformations in the boundary of
the domain and to the insertion of a pair of operators that generate a
Brownian excursion in the domain (with the insertion points as starting and
ending points of the excursion) and appear to behave like _boundary condition
changing operators_.

##  2 The bulk stress-energy tensor

In this section we show how to express the bulk stress-energy tensor $T$ in
the BLS in terms of the edge counting operator $\mathcal{E}$ and in terms of
the vertex layering operator $\mathcal{O}$. From these expressions, we compute
two examples of correlation functions of primary fields with $T$ and show that
they satisfy the conformal Ward identities, confirming the validity of our
expressions for $T$. The strategy of this section is inspired by Doyon_2006 ;
Gamsa_2006 . In particular, the techniques of Gamsa_2006 , where the authors
derive the central charge of the $O(n\to 0)$ model, have been used in
camia2021scalar  to determine the central charge of the BLS.

###  2.1 Identification

The holomorphic and anti-holomorphic components of the stress-energy tensor
can be understood as the generators of conformal transformations. The
holomorphic component of the stress-energy tensor in a two-dimensional CFT can
generally be understood as the level-2 descendant of the identity operator

$\displaystyle(L_{-2}\mathbb{1})(z)\equiv T(z).$ (16)

The anti-holomorphic component $\bar{T}$ is equivalently given by
$\bar{L}_{-2}$. Their conformal dimensions are $(2,0)$ and $(0,2)$ for the
holomorphic and anti-holomorphic component, respectively.

(a) Non-zero $C_{\mathcal{E}\mathcal{E}}^{(p,p^{\prime})}$

(b) Non-zero $C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{(p,p^{\prime})}$

Figure 1: The non-zero three-point function coefficients are shown. Rows and
columns label $(p,p^{\prime})$ for primary operators with dimensions
$(p/3,p^{\prime}/3)$. Left: between two edge-counting operators. Right:
between two vertex operators. The operators giving contributions to $T$,
corresponding to $\mathbb{1}$ (left) and $\mathbb{1},J,W$ (right) are marked
in red (hatched).

Our strategy is to identify the stress-energy tensor starting from the
operator product expansion (OPE) of two primary operators $\mathcal{A}_{1}$
and $\mathcal{A}_{2}$ (see Section 6.6.3 of DiFrancesco:639405 )

$\displaystyle\begin{split}\mathcal{A}_{1}(z+\epsilon)\times\mathcal{A}_{2}(z)=\sum_{\mathcal{P}}\sum_{\\{k,\bar{k}\\}}&C^{\mathcal{P}}_{12}\beta^{\mathcal{P}\\{k\\}}_{12}\bar{\beta}^{\mathcal{P}\\{\bar{k}\\}}_{12}\epsilon^{\Delta_{\mathcal{P}}-\Delta_{1}-\Delta_{2}+K}\bar{\epsilon}^{\bar{\Delta}_{\mathcal{P}}-\bar{\Delta}_{1}-\bar{\Delta}_{2}+\bar{K}}\\\
&\cdot L_{-k_{1}}\ldots
L_{-k_{N}}\bar{L}_{-\bar{k}_{1}}\ldots\bar{L}_{-\bar{k}_{\bar{N}}}\mathcal{P}(z),\end{split}$
(17)

where the outer sum runs over all primary operators $\mathcal{P}$, and the
inner sum is over all collections of indices (the descendant levels)
$k_{i},\bar{k}_{i}$ with $K=\sum_{k_{i}\in\\{k\\}}k_{i}$ and
$\bar{K}=\sum_{\bar{k}_{i}\in\\{\bar{k}\\}}\bar{k}_{i}$.
$\beta_{12}^{\mathcal{P}\\{k\\}},\bar{\beta}_{12}^{\mathcal{P}\\{\bar{k}\\}}$
are numerical coefficients fully determined by the Virasoro algebra (they
depend on the central charge and the conformal dimensions of the operators
involved), and $C_{12}^{\mathcal{P}}$ are three-point function coefficients of
the theory. If we choose $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ so that (17)
contains the identity operator then the stress tensor will appear as a
descendent. In particular, the identity is contained in the OPE

$\displaystyle\begin{split}\mathcal{E}(z+\epsilon)\times\mathcal{E}(z)&=|\epsilon|^{-4/3}\left(\mathbb{1}+\epsilon^{2}\frac{2\Delta_{\mathcal{E}}}{c}T(z)+\ldots\right)\\\
&=|\epsilon|^{-4/3}\left(\mathbb{1}+\epsilon^{2}\frac{1/3}{\lambda}T(z)+\ldots\right),\end{split}$
(18)

where $\Delta_{\mathcal{E}}=1/3$ is the conformal dimension of $\mathcal{E}$
and $c=2\lambda$ is the central charge, and where higher descendants, as well
as other conformal blocks and the anti-holomorphic components, have been
omitted. The term containing $T(z)$ appears at order
$O(\epsilon^{-2/3+2}\bar{\epsilon}^{-2/3})$ in (18), and no other term appears
at the same order. (The anti-holomorphic component $\bar{T}$ would appear at
order $O(\epsilon^{-2/3}\bar{\epsilon}^{-2/3+2})$.) This can be deduced from
the analysis carried out in Camia_2020 , as shown in Figure 1(a). Thus one can
identify

$\displaystyle T(z)=\frac{\lambda}{1/3}\lim_{\rho\to
0}\rho^{-2/3}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\mathcal{E}(z+\rho
e^{i\theta})\mathcal{E}(z),$ (19)

since the integral in (19) singles out the term of the expansion in the right
hand side of (18) that corresponds to the stress-energy tensor. As shown in
Figure 2, the loops that contribute to the integral in (19) are those that
come close to $z$ and that intersect the circle of radius $\rho$ centered at
$z$. This provides a geometric interpretation for the stress-energy tensor.
$\bar{T}$ is obtained by replacing $e^{-2i\theta}$ with $e^{2i\theta}$.

Figure 2: Every loop whose outer boundary comes close to the point $z$ and
intersects the dashed circle somewhere contributes to $T(z)$ in (19). The red
points represent the insertion points of the edge-counting operators
$\mathcal{E}$ at $z$ and $z^{\prime}=z+\rho e^{i\theta}$.

As a consequence of conformal invariance, correlation functions involving the
stress-energy tensor in the bulk obey the conformal Ward identities

$\displaystyle\Braket{T(z)\prod_{j=1}^{N}\mathcal{A}_{j}(z_{j})}_{\mathbb{C}}=\sum_{j=1}^{N}\left(\frac{\Delta_{j}}{(z-z_{j})^{2}}+\frac{1}{z-z_{j}}\partial_{z_{j}}\right)\Braket{\prod_{j=1}^{N}\mathcal{A}_{j}(z_{j})}_{\mathbb{C}},$
(20)

where $\mathcal{A}_{j}$ denotes a generic primary operator with conformal
dimension $\Delta_{j}$. In the next section, we test (19) using (20).

###  2.2 Correlation functions

We now use the full-plane mixed four-point function
$\Braket{\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2}){\mathcal{E}}(z_{3}){\mathcal{E}}(z_{4})}_{\mathbb{C}}$
to compute
$\braket{T(z)\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}}$
and $\braket{T(z)\mathcal{E}(z_{1})\mathcal{E}(z_{2})}_{\mathbb{C}}$, which
can then be compared to the Ward identities (20). (See Section 1.1 for the
definition of $\mathcal{O}_{\beta}(z)$.) The four-point function was computed
in camia2021scalar  and can be written as

$\displaystyle\begin{split}&\Braket{\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2}){\mathcal{E}}(z_{3}){\mathcal{E}}(z_{4})}_{\mathbb{C}}\\\
&=\Braket{\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}}\Big{[}\hat{\alpha}_{\mathbb{C}}^{z_{3},z_{4}}-(1-\cos\beta)\hat{\alpha}^{z_{3},z_{4}}_{z_{1}|z_{2};\mathbb{C}}+\lambda(1-\cos\beta)^{2}\hat{\alpha}^{z_{3}}_{z_{1}|z_{2};\mathbb{C}}\hat{\alpha}^{z_{4}}_{z_{1}|z_{2};\mathbb{C}}\Big{]},\end{split}$
(21)

where, letting $z_{jk}=z_{j}-z_{k}$,

$\displaystyle\begin{split}\Braket{\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}}&=|z_{12}|^{-4\Delta(\beta)}\\\
\Braket{\mathcal{E}(z_{3})\mathcal{E}(z_{4})}_{\mathbb{C}}=\hat{\alpha}^{z_{3},z_{4}}_{\mathbb{C}}&=|z_{34}|^{-4/3}\\\
\hat{\alpha}^{z_{3}}_{z_{1}|z_{2};{\mathbb{C}}}&=\frac{2^{7/6}\pi}{3^{1/4}\sqrt{5}\,\Gamma(1/6)\Gamma(4/3)}\left|\frac{z_{12}}{z_{13}z_{23}}\right|^{2/3}\\\
\hat{\alpha}^{z_{3},z_{4}}_{z_{1}|z_{2};\mathbb{C}}&=\frac{|z_{34}|^{-4/3}-Z_{\text{twist}}}{2},\end{split}$
(22)

and

$\displaystyle\begin{split}&Z_{\text{twist}}=Z_{\text{twist}}(z_{1},z_{2};z_{3},z_{4})\\\
&=\left|\frac{z_{13}z_{24}}{z_{34}^{2}z_{23}z_{14}}\right|^{2/3}\left[\left|{}_{2}F_{1}\left(-\frac{2}{3},\frac{1}{3};\frac{2}{3},x\right)\right|^{2}-\frac{4\Gamma\left(\frac{2}{3}\right)^{6}}{\Gamma\left(\frac{4}{3}\right)^{2}\Gamma\left(\frac{1}{3}\right)^{4}}|x|^{2/3}\left|{}_{2}F_{1}\left(-\frac{1}{3},\frac{2}{3};\frac{4}{3},x\right)\right|^{2}\right]\end{split}$
(23)

corresponds to Eq. (52) of Simmons_2009  with
$x=\frac{z_{12}z_{34}}{z_{13}z_{24}}$.

We can then apply (19) to (LABEL:eq:OOEE) using (22) by expanding
$\hat{\alpha}^{z+\rho e^{i\theta}}_{z_{1}|z_{2};\mathbb{C}}$ in powers of
$\rho$. Observe that the terms independent of $\theta$ (e.g.,
$\hat{\alpha}_{\mathbb{C}}^{z,z+\rho e^{i\theta}}=\rho^{-3/4}$) do not
contribute because of the integral $\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}$,
while the terms containing powers of $\rho$ greater than $2/3$ give zero in
the limit $\rho\to 0$. This gives

$\displaystyle\begin{split}&\Braket{T(z)\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}}\\\
&=\frac{\lambda}{1/3}\lim_{\rho\to
0}\rho^{-2/3}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\Braket{\mathcal{E}(z+\rho
e^{i\theta})\mathcal{E}(z)\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}}\\\
&=\frac{3\lambda(1-\cos\beta)}{2}|z_{12}|^{-4\Delta(\beta)}\lim_{\rho\to
0}\rho^{-2/3}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}Z_{\text{twist}}(z_{1},z_{2};z,z+\rho
e^{i\theta})\,.\end{split}$ (24)

Using the fact that $\rho^{-2/3}Z_{\text{twist}}(z_{1},z_{2};z,z+\rho
e^{i\theta})$ is analytic and expanding it in powers of $\rho$, performing the
integral, and sending $\rho\to 0$, we obtain

$\displaystyle\begin{split}\Braket{T(z)\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}}=\Delta(\beta)\,\frac{(z_{1}-z_{2})^{2-2\Delta(\beta)}(\bar{z}_{1}-\bar{z}_{2})^{-2\Delta(\beta)}}{(z-z_{1})^{2}(z-z_{2})^{2}}\,,\end{split}$
(25)

which corresponds to the Ward identity (20) for
$N=2,\mathcal{A}_{1,2}=\mathcal{O}_{\pm\beta}$, and
$\Delta_{j}=\Delta(\beta)$.

In a very similar manner, one can obtain
$\Braket{T(z)\mathcal{E}(z_{3})\mathcal{E}(z_{4})}_{\mathbb{C}}$ using the
fact that the identity block is also contained in the OPE of
$\mathcal{O}_{\beta}(z+\epsilon)\times\mathcal{O}_{-\beta}(z)$. The term
containing $T(z)$ appears at order
$O(\epsilon^{-2\Delta(b)+2}\bar{\epsilon}^{-2\Delta(b)})$ (the anti-
holomorphic component $\bar{T}$ appears at order
$O(\epsilon^{-2\Delta(\beta)}\bar{\epsilon}^{-2\Delta(\beta)+2})$), but in
this case there are other contributions to the OPE at the same order. The
relevant operators are conformal primaries with dimensions $(1,0)$ and
$(2,0)$, and we denote them $J$ and $W$, respectively. They are shown in
Figure 1(b), which is obtained using results from Camia_2020 . More precisely,
in the case of two layering vertex operators, (17) gives

$\displaystyle\begin{split}&\mathcal{O}_{\beta}(z+\epsilon)\times\mathcal{O}_{-\beta}(z)\\\
&=|\epsilon|^{-4\Delta(\beta)}\epsilon^{2}\left(\beta_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{\mathbb{1}\\{2\\}}(L_{-2}\mathbb{1})(z)+\beta_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J\\{1\\}}C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J}(L_{-1}J)(z)++C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{W}W(z)\right)+\ldots\\\
&=|\epsilon|^{-4\Delta(\beta)}\epsilon^{2}\left(\frac{2\Delta(\beta)}{c}T(z)+\frac{1}{2}C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J}\partial_{z}J(z)+C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{W}W(z)\right)+\ldots,\end{split}$
(26)

where higher descendants, other conformal blocks, as well as the anti-
holomorphic components, have been omitted. The coefficients
$\beta_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{\mathbb{1}\\{2\\}}$ and
$\beta_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J\\{1\\}}$ are determined by
the Virasoro algebra and are given by $2\Delta(\beta)/c$ and $1/2$,
respectively DiFrancesco:639405 ; the central charge of the BLS is
$c=2\lambda$, $(L_{-1}J)(z)=\partial_{z}J(z)$, and
$C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{\mathbb{1}}=1$ because of the
(canonical) normalization of $\mathcal{O}_{\beta}$. Finally, the three-point
function coefficients $C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J}$ and
$C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{W}$ can be computed666In
Camia_2020  we found that
$C_{\mathcal{O}_{\beta_{1}}\mathcal{O}_{\beta_{2}}}^{J}=0$ for generic
$\beta_{1},\beta_{2}$. In the special case $\beta_{1}+\beta_{2}=0\mod 2\pi$
the coefficient is given in (27). as explained in Camia_2020 , which gives

$\displaystyle\begin{split}\left(C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J}\right)^{2}&=\frac{\lambda}{10}\sin^{2}\beta\\\
\left(C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{W}\right)^{2}&=\frac{1}{2}\left(C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J}\right)^{4}.\end{split}$
(27)

Based on these considerations, one can write

$\displaystyle\begin{split}T(z)&=\frac{\lambda}{\Delta(\beta)}\bigg{[}\lim_{\rho\to
0}\rho^{4\Delta(\beta)-2}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\mathcal{O}_{\beta}(z+\rho
e^{i\theta})\mathcal{O}_{-\beta}(z)\\\
&\qquad-\frac{1}{2}C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{J}\partial_{z}J(z)-C_{\mathcal{O}_{\beta}\mathcal{O}_{-\beta}}^{W}W(z)\bigg{]}.\end{split}$
(28)

Applying (28), (LABEL:eq:OOEE) and (22), and using the fact that
$\braket{J\mathcal{E}\mathcal{E}}_{\mathbb{C}}=\braket{W\mathcal{E}\mathcal{E}}_{\mathbb{C}}=0$,
as can be seen from Figure 1(a), we obtain

$\displaystyle\begin{split}&\Braket{T(z){\mathcal{E}}(z_{3}){\mathcal{E}}(z_{4})}_{\mathbb{C}}\\\
&\quad=\frac{\lambda}{\Delta(\beta)}\lim_{\rho\to
0}\rho^{4\Delta(\beta)-2}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\Braket{\mathcal{O}_{\beta}(z+\rho
e^{i\theta})\mathcal{O}_{-\beta}(z){\mathcal{E}}(z_{3}){\mathcal{E}}(z_{4})}_{\mathbb{C}}\\\
&\quad=-\frac{\lambda(1-\cos\beta)}{\Delta(\beta)}\lim_{\rho\to
0}\rho^{-2}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\,\hat{\alpha}^{z_{3},z_{4}}_{z|z+\rho
e^{i\theta};\mathbb{C}}\\\
&\quad=\frac{\lambda(1-\cos\beta)}{2\Delta(\beta)}\lim_{\rho\to
0}\rho^{-2}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\,Z_{\text{twist}}(z,z+\rho
e^{i\theta};z_{3},z_{4})\\\
&\quad=\frac{1}{3}\frac{(z_{3}-z_{4})^{4/3}}{(z-z_{3})^{2}(z-z_{4})^{2}(\bar{z}_{3}-\bar{z}_{4})^{2/3}}\,.\end{split}$
(29)

This is the correct form of the conformal Ward identity in the bulk, i.e.
(20), with $N=2,\mathcal{A}_{j}=\mathcal{E}$ and $\Delta_{j}=1/3$.

We point out that using (28) in

$\displaystyle\Braket{\mathcal{O}_{\tilde{\beta}}(z+\rho
e^{i\theta})\mathcal{O}_{-\tilde{\beta}}(z)\mathcal{O}_{\beta}(z_{1})\mathcal{O}_{-\beta}(z_{2})}_{\mathbb{C}},$
(30)

which has been derived in Camia_2020 , and performing a similar analysis that
takes into account the contributions from $J$ and $W$ from (LABEL:OPEapplied),
leads again to (25), as expected.

To conclude this section, we compute the bulk stress-energy tensor two-point
function, which is fixed by conformal invariance to be
$\Braket{T(z)T(z^{\prime})}_{\mathbb{C}}=c/2/(z-z^{\prime})^{4}$. We apply
(18) again to (LABEL:eq:calculation) by setting $z_{3}=z^{\prime}+\rho
e^{i\theta},\bar{z}_{3}=\bar{z}^{\prime}+\rho
e^{-i\theta},z_{4}=z^{\prime},\bar{z}_{4}=\bar{z}^{\prime}$ in
(LABEL:eq:calculation) and obtain

$\displaystyle\begin{split}&\Braket{T(z)T(z^{\prime})}_{\mathbb{C}}\\\
&=\frac{\lambda}{1/3}\lim_{\rho\to
0}\rho^{-2/3}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta~{}e^{-2i\theta}\frac{1}{3}\frac{(\rho
e^{i\theta})^{4/3}}{(z-z^{\prime}-\rho e^{i\theta})^{2}(z-z^{\prime})^{2}(\rho
e^{-i\theta})^{2/3}}\\\ &=\frac{\lambda}{(z-z^{\prime})^{4}}.\end{split}$ (31)

This is indeed the correct form of the two-point function of stress-energy
tensors in a theory with central charge $c=2\lambda$.

##  3 The boundary stress-energy tensor

In this section we introduce a boundary version of the edge operator discussed
in Section 1.2 and study some of its properties. In particular, we show that
it is essentially the boundary stress-energy tensor (up to a factor of
$\sqrt{\lambda}$).

Figure 3: A Brownian loop and its outer boundary in the upper-half plane. Such
a loop would contribute to the two-point function of boundary edge counting
operators inserted at $x_{1}$ and $x_{2}$ because the loop comes within
distance $\varepsilon$ of both points.

###  3.1 The boundary edge counting operator and its correlation functions

The same analysis briefly discussed in the Section 1.2 can be carried out for
a point $\zeta$ on the boundary $\partial D$ of a domain $D$, where $D$ is
either a finite domain with a sufficiently smooth boundary or the upper half
plane $\mathbb{H}$. In this case, using the connection with critical
percolation and CLE6, the relevant probability is that of a three-arm event at
the boundary, which gives an exponent $2$ instead of $2/3$. This is consistent
with the well-known fact that in a renormalizable quantum field theory
operators at the boundary require a different renormalization from those in
the bulk CARDY2006333  (producing, in general, a different set of conformal
dimensions) and leads to the following (formal) definition of _boundary edge
counting operator_

$\displaystyle\begin{split}{\mathcal{B}}(\zeta):=\frac{\hat{c}_{b}}{\sqrt{\lambda}}\lim_{\varepsilon\to
0}\varepsilon^{-2}E_{\varepsilon}(\zeta),\end{split}$ (32)

where $\hat{c}_{b}$ is chosen so that ${\mathcal{B}}$ is canonically
normalized when $D=\mathbb{H}$, that is, for points $x_{1},x_{2}$ on the real
line,

$\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}=|x_{1}-x_{2}|^{-4}.$
(33)

At the end of Section 3.2, we will see that $\hat{c}_{b}=1$.

Definition (32) is convenient because it uses the edge counting operator
analyzed in camia2021scalar , but we could have equivalently defined
$\mathcal{B}$ by counting the loops crossing an infinitesimal slit attached to
the boundary of the domain. This alternative definition, reminiscent of the
definitions of the operators discussed in FRIEDRICH2002947 ; Doyon_2006 ,
would not change our conclusions.

The existence of the $n$-point functions of the field ${\mathcal{B}}$ follows
from the same arguments used in camia2021scalar . First of all, we note that
the proof that the limit in (7) exists extends trivially to the case in which
(some of) the points $z_{j}$ are moved to the boundary of the domain (see
Lemma A.1 of camia2021scalar ). This means that

$\displaystyle\Braket{E_{\varepsilon}(x_{1})\ldots
E_{\varepsilon}(x_{n})}_{\mathbb{H}}=\lim_{\delta\to
0}\Braket{E^{\delta}_{\varepsilon}(x_{1})\ldots
E^{\delta}_{\varepsilon}(x_{n})}_{\mathbb{H}}$ (34)

exists for all $x_{1},\ldots,x_{n}\in\mathbb{R}$ with $n\geq 2$, which allows
us to state the following result.

######  Proposition 3.1.

For any collection of distinct points $x_{1},\ldots,x_{n}$ on the real line,
with $n\geq 2$, the following limit exists:

$\displaystyle\begin{split}\Braket{\mathcal{B}(x_{1})\ldots\mathcal{B}(x_{n})}_{\mathbb{H}}\quad:=\frac{\hat{c}_{b}^{n}}{\lambda^{n/2}}\lim_{\varepsilon\to
0}\varepsilon^{-2n}\Braket{E_{\varepsilon}(x_{1})\ldots
E_{\varepsilon}(x_{n})}_{\mathbb{H}}.\end{split}$ (35)

Moreover

$\Braket{\mathcal{B}(x_{1})\ldots\mathcal{B}(x_{n})}_{\mathbb{H}}=\frac{\hat{c}_{b}^{n}}{\lambda^{n/2}}\sum_{(S_{1},\ldots,S_{r})\in\mathcal{S}}\lambda^{r}\beta^{S_{1}}\ldots\beta^{S_{r}},$
(36)

where $\mathcal{S}=\mathcal{S}(x_{1},\ldots,x_{n})$ denotes the set of all
partitions of $\\{x_{1},\ldots,x_{n}\\}$ such that each element $S_{l}$ of
$(S_{1},\ldots,S_{r})\in\mathcal{S}$ has cardinality $|S_{l}|\geq 2$ and, for
any subset $S_{l}=\\{x_{j_{1}},\ldots,x_{j_{k}}\\}$ of
$\\{x_{1},\ldots,x_{n}\\}$,

$\displaystyle\beta^{S_{l}}\equiv\beta^{x_{j_{1}},\ldots,x_{j_{k}}}:=\lim_{\varepsilon\to
0}\varepsilon^{-2k}\mu^{\text{loop}}_{\mathbb{H}}\left(\ell\cap
B_{\varepsilon}(x_{j_{m}}),\,m=1,\ldots,k\right).$ (37)

Proof. The proposition is analogous to Theorem 2.3 of camia2021scalar  and its
proof follows directly from the arguments used in the proofs of Lemmas 2.1 and
2.2 of camia2021scalar . ∎

The two-point function
$\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}$ can be
calculated exactly, as follows. For any $s>0$, the scale invariance of the BLS
implies that

$\displaystyle\begin{split}\Braket{{\mathcal{B}}(sx_{1}){\mathcal{B}}(sx_{2})}_{\mathbb{H}}&=\frac{\hat{c}_{b}^{2}}{\lambda}\lim_{\varepsilon\to
0}\varepsilon^{-4}\Braket{E_{\varepsilon}(sx_{1})E_{\varepsilon}(sx_{2})}_{\mathbb{H}}\\\
&=s^{-4}\,\frac{\hat{c}_{b}^{2}}{\lambda}\lim_{\varepsilon\to
0}\left(\frac{\varepsilon}{s}\right)^{-4}\Braket{E_{\varepsilon/s}(x_{1})E_{\varepsilon/s}(x_{2})}_{\mathbb{H}}\\\
&=s^{-4}\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}.\end{split}$
(38)

Since $\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}$ can
only depend on $|x_{1}-x_{2}|$, this means that

$\displaystyle\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}\sim|x_{1}-x_{2}|^{-4},$
(39)

and we can choose $\hat{c}_{b}$ so that the proportionality constant is $1$.
(As mentioned above, it will turn out that $\hat{c}_{b}=1$.)

###  3.2 Identification of the boundary stress-energy tensor

We now present the main results of this section, which allow us to identify
$\sqrt{\lambda}\mathcal{B}$ with the boundary stress-energy tensor of the
theory. To this end, we recall the formal definition of _layering vertex
operator_ in the upper-half plane Camia_2020

$\displaystyle\tilde{\mathcal{O}}_{\beta}(z):=\lim_{\delta\to
0}(\hat{c}_{\mathbb{H}}\delta)^{-2\Delta(\beta)}\exp{\Big{(}i\beta\sum_{\stackrel{{\scriptstyle\ell\in\mathcal{L}^{\delta}}}{{z\in\bar{\ell}}}}\sigma_{\ell}\Big{)}},$
(40)

where $\Delta(\beta)=\bar{\Delta}(\beta)=\frac{\lambda}{10}(1-\cos\beta)$,
$\sigma_{\ell}$ is a symmetric $(\pm 1)$-valued random variable associated to
loop $\ell$, and $\mathcal{L}^{\delta}$ is a BLS from which all loops of
diameter smaller than $\delta$ have been removed, $z\in\bar{\ell}$ means that
$z$ is in the interior of the domain bounded by $\ell$ (it is disconnected
from infinity by $\ell$); $\hat{c}_{\mathbb{H}}$ is chosen so that

$\Braket{\tilde{\mathcal{O}}_{\beta}(z)}_{\mathbb{H}}=|z-\bar{z}|^{-2\Delta(\beta)}.$
(41)

Equation (41) is obtained in Camia_2020 , while the existence and conformal
covariance properties of more general $n$-point functions are proved in
Camia_2016 . Essentially, when a layering operator is inserted at $z$, each
loop $\ell$ such that $z\in\bar{\ell}$ picks up a phase
$e^{i\beta\sigma_{\ell}}$. We note that the layering vertex operators are
scalar operators.

######  Proposition 3.2.

For any choice of points $z_{1},z_{2}\in\mathbb{H}$, we have that

$\displaystyle\begin{split}&\Braket{\mathcal{B}(0)\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}=-\hat{c}_{b}\sqrt{\lambda}(1-\cos\beta)\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&\quad\cdot\left\\{\frac{2}{5}\left[\left(\operatorname{Im}\left(\frac{1}{z_{1}}\right)\right)^{2}+\left(\operatorname{Im}\left(\frac{1}{z_{2}}\right)\right)^{2}\right]-\frac{4}{5}\operatorname{Im}\left(\frac{1}{z_{1}}\right)\operatorname{Im}\left(\frac{1}{z_{2}}\right)G(\sigma(z_{1},z_{2}))\right\\},\end{split}$
(42)

where $\sigma(z_{1},z_{2})=\frac{|z_{1}-z_{2}|^{2}}{|z_{1}-\bar{z}_{2}|^{2}}$,

$\displaystyle
G(\sigma)=1-\sigma\,{}_{2}F_{1}\left(1,\frac{4}{3};\frac{5}{3};1-\sigma\right),$
(43)

and ${}_{2}F_{1}$ is the hypergeometric function.

Proof. Using (3.6) and (3.7) from camia2021scalar  applied to the upper half-
plane and with $z_{3}=0$ and the exponent $2/3$ replaced by $2$, we obtain an
analog of (3.8) therein, namely

$\displaystyle\begin{split}&\Braket{\tilde{\mathcal{B}(0)\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}=-\lambda(1-\cos\beta)\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&\qquad\cdot\frac{\hat{c}_{b}}{\sqrt{\lambda}}\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu^{\text{loop}}_{\mathbb{H}}\big{(}\ell\cap
B_{\varepsilon}(0)\neq\emptyset,\ell\text{ separates
}z_{1},z_{2}\big{)}.\end{split}$ (44)

In order to evaluate the limit in (LABEL:eq:OOB-proof), we use the fact that

$\displaystyle\begin{split}\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu^{\text{loop}}_{\mathbb{H}}\big{(}\ell\cap
B_{\varepsilon}(0)\neq\emptyset,\ell\text{ separates
}z_{1},z_{2}\big{)}=\mu^{\text{bub}}_{\mathbb{H}}(\ell\text{ separates
}z_{1},z_{2}),\end{split}$ (45)

where $\mu^{\text{bub}}_{\mathbb{H}}$ is the measure on Brownian bubbles in
$\mathbb{H}$, pinned at the origin, defined in lawler2004brownian , and the
equality follows from Theorem 1 of lawler2004brownian .

As explained in Remark 7.1 of Lawler_2003  and in beliaev2013some ,
$\mu^{\text{bub}}_{\mathbb{H}}$ is closely related to the SLE8/3 excursion
measure $\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}$; more precisely
$\mu^{\text{bub}}_{\mathbb{H}}=\frac{8}{5}\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}$.
Therefore we have

$\displaystyle\begin{split}&\Braket{\mathcal{B}(0)\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&=-\sqrt{\lambda}\hat{c}_{b}(1-\cos\beta)\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\frac{8}{5}\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}(\ell\text{
separates }z_{1},z_{2})\\\
&=-\sqrt{\lambda}\hat{c}_{b}(1-\cos\beta)\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&\qquad\cdot\frac{8}{5}\left(\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}(z_{1}\in\bar{\ell})+\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}(z_{2}\in\bar{\ell})-2\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}(z_{1},z_{2}\in\bar{\ell})\right).\end{split}$
(46)

Since $\mu^{\text{SLE}_{8/3}}$ is obtained by multiplying the law of an SLE8/3
$\varepsilon$-bubble (i.e., an SLE8/3 excursion in $\mathbb{H}$ starting at
the origin and ending at $\varepsilon$) by $\varepsilon^{-2}$ and passing to
the limit, the proof is concluded by using (3.2) and (3.3) from Proposition 1
of beliaev2013some  to evaluate the three terms in the last line of the last
equation. ∎

The importance of Proposition 3.2 lies in the fact that it allows us to
identify the boundary stress-energy tensor of our CFT. The ingredients for
this identification are

  * •

the Ward identity at the presence of a boundary (see Eq. (24) of CARDY2006333
)

$\displaystyle\begin{split}&\Braket{\mathcal{T}(x)\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&=\sum_{j=1}^{2}\left(\frac{\Delta(\beta)}{(x-z_{j})^{2}}+\frac{1}{x-z_{j}}\partial_{z_{j}}+\frac{\Delta(\beta)}{(x-\bar{z}_{j})^{2}}+\frac{1}{x-\bar{z}_{j}}\partial_{\bar{z}_{j}}\right)\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}},\end{split}$
(47)

where $\mathcal{T}$ denotes the boundary stress-energy tensor with
$x\in\mathbb{R}$,

  * •

the expression (see Eq. (2.2) of Camia_2020 )

$\displaystyle\begin{split}&\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}=|z_{1}-z_{2}|^{-4\Delta(\beta)}|z_{1}-\bar{z}_{2}|^{4\Delta(\beta)}|z_{1}-\bar{z}_{1}|^{-2\Delta(\beta)}|z_{2}-\bar{z}_{2}|^{-2\Delta(\beta)}\\\
&\qquad\cdot\exp\left[-2\Delta(\beta)(1-\sigma(z_{1},z_{2})){}_{3}F_{2}\big{(}1,1,4/3;2,5/3;1-\sigma(z_{1},z_{2})\big{)}\right],\end{split}$
(48)

where ${}_{3}F_{2}$ is the hypergeometric function.777We note that the
expression (LABEL:eq:OO), which was derived in Camia_2020  using results from
han2017brownian ; beliaev2013some , is rigorous.

Plugging (LABEL:eq:OO) into the right hand side of (LABEL:eq:Ward), sending
$x\to 0$, and comparing with (LABEL:eq:OOB) shows that the result equals
$\frac{\sqrt{\lambda}}{\hat{c}_{b}}\Braket{\mathcal{B}(0)\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}$.
By translation invariance, this means that, on the real line, we can make the
identification $\mathcal{T}=\frac{\sqrt{\lambda}}{\hat{c}_{b}}\mathcal{B}$.
According to (33), for any two points $x_{1},x_{2}\in\mathbb{R}$,
$\mathcal{T}$ would then have the two-point function

$\displaystyle\Braket{{\mathcal{T}}(x_{1}){\mathcal{T}}(x_{2})}_{\mathbb{H}}=\frac{\lambda}{\hat{c}_{b}^{2}\,|x_{1}-x_{2}|^{4}},$
(49)

but since the stress-energy tensor $\mathcal{T}$ satisfying (LABEL:eq:Ward) is
assumed to be normalized in such a way that

$\displaystyle\Braket{{\mathcal{T}}(z_{1}){\mathcal{T}}(z_{2})}_{\mathbb{H}}=\frac{c/2}{|z_{1}-z_{2}|^{4}},$
(50)

where $c$ denotes the central charge (see, e.g., Eq. (8) of CARDY2006333 ),
from the relation $c=2\lambda$ valid for the BLS with intensity $\lambda$, we
conclude that $\hat{c}_{b}=1$ and determine the boundary stress-energy tensor
to be

$\displaystyle\mathcal{T}=\sqrt{\lambda}\mathcal{B}.$ (51)

In other words, the discussion above leads to the following result.

######  Proposition 3.3.

For any choice of distinct points $z_{1},z_{2}\in\mathbb{H}$ and
$x\in\mathbb{R}$, $\mathcal{T}(x)=\sqrt{\lambda}\mathcal{B}(x)$ satisfies the
Ward identity (LABEL:eq:Ward).

###  3.3 Operator product expansion

In this section we discuss the OPE of $\mathcal{T}\times\mathcal{T}$. In order
to do that, we first express the mixed four-point function
$\Braket{\mathcal{B}(x_{1})\mathcal{B}(x_{2})\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}$
in terms of $\mu^{\text{loop}}_{\mathbb{H}}$-weights. For that purpose, it is
useful to introduce some additional notation. We define

$\displaystyle\begin{split}\beta^{x}_{z}:=\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu^{\text{loop}}_{\mathbb{H}}\big{(}\ell\cap
B_{\varepsilon}(x)\neq\emptyset,z\in\bar{\ell}\big{)}\end{split}$ (52)

and

$\displaystyle\beta^{x_{1},x_{2}}_{z_{1}|z_{2}}=\beta^{x_{1},x_{2}}_{z_{2}|z_{1}}:=\lim_{\varepsilon\to
0}\varepsilon^{-4}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{j})\neq\emptyset\text{ for }j=1,2;\ell\text{ separates
}z_{1},z_{2}),$ (53)

where the existence of both limits follows from the proof of Lemma 2.2 of
camia2021scalar .

######  Proposition 3.4.

For any choice of points $z_{1},z_{2}\in\mathbb{H}$ and
$x_{1},x_{2}\in\mathbb{R}$, with the notation of Proposition 3.2, we have that

$\displaystyle\begin{split}&\Braket{\mathcal{B}(x_{1})\mathcal{B}(x_{2})\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\left[\beta^{x_{1},x_{2}}-(1-\cos\beta)\beta^{x_{1},x_{2}}_{z_{1}|z_{2}}+\lambda(1-\cos\beta)^{2}\beta^{x_{1}}_{z_{1}|z_{2}}\beta^{x_{2}}_{z_{1}|z_{2}}\right]\\\
&=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\left[\Braket{\mathcal{B}(x_{1})\mathcal{B}(x_{2})}_{\mathbb{H}}-(1-\cos\beta)\beta^{x_{1},x_{2}}_{z_{1}|z_{2}}+\lambda(1-\cos\beta)^{2}\beta^{x_{1}}_{z_{1}|z_{2}}\beta^{x_{2}}_{z_{1}|z_{2}}\right].\end{split}$
(54)

Proof. The result follows from carrying out the analysis leading to (5.6) in
camia2021scalar  to the upper half-plane, with the exponents $2/3$ replaced by
$2$. The last line of (LABEL:eq:OOBB) follows from Proposition 3.1, which
allows us to make the identification
$\Braket{\mathcal{B}(x_{1})\mathcal{B}(x_{2})}_{\mathbb{H}}=\beta^{x_{1},x_{2}}$.
∎

We now discuss (LABEL:eq:OOBB) in some detail. First of all, we note that,
arguing as in the proof of Proposition 3.2 and using again Proposition 1 of
beliaev2013some , one can find an explicit expression for the last term
between square brackets of the last line of (LABEL:eq:OOBB), namely using

$\displaystyle\begin{split}\beta^{x}_{z_{1}|z_{2}}&=\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu^{\text{loop}}_{\mathbb{H}}\big{(}\ell\cap
B_{\varepsilon}(x)\neq\emptyset,\ell\text{ separates
}z_{1},z_{2}\big{)}=\mu^{\text{bub}}_{\mathbb{H},x}\big{(}\ell\text{ separates
}z_{1},z_{2}\big{)}\\\
&=\frac{8}{5}\left(\mu^{\text{SLE}_{8/3}}_{\mathbb{H},x}(z_{1}\in\bar{\ell})+\mu^{\text{SLE}_{8/3}}_{\mathbb{H},x}(z_{2}\in\bar{\ell})-2\mu^{\text{SLE}_{8/3}}_{\mathbb{H},x}(z_{1},z_{2}\in\bar{\ell})\right)\\\
&=\frac{2}{5}\frac{(\operatorname{Im}z_{1})^{2}}{\left(\text{dist}(x,z_{1})\right)^{4}}+\frac{2}{5}\frac{(\operatorname{Im}z_{2})^{2}}{\left(\text{dist}(x,z_{2})\right)^{4}}-\frac{4}{5}\frac{\operatorname{Im}z_{1}}{\left(\text{dist}(x,z_{1})\right)^{2}}\frac{\operatorname{Im}z_{2}}{\left(\text{dist}(x,z_{2})\right)^{2}}G(\sigma(z_{1},z_{2})),\end{split}$
(55)

where $\mu^{\text{bub}}_{\mathbb{H},x}$ is the measure of Brownian bubbles in
$\mathbb{H}$ pinned at $x$ and $\mu^{\text{SLE}_{8/3}}_{\mathbb{H},x}$ is the
measure of SLE8/3 excursions in $\mathbb{H}$ from $x$ (in other words, they
are the measures $\mu^{\text{SLE}_{\text{bub}}}_{\mathbb{H}}$ and
$\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}$, respectively, after a translation of
$\overline{\mathbb{H}}$ that moves the origin to $x$).

Next, we note that the first two terms between square brackets in the last
line of (LABEL:eq:OOBB) diverge as $|x_{1}-x_{2}|\to 0$. This is clear for
$\Braket{{\mathcal{B}}(x_{1}){\mathcal{B}}(x_{2})}_{\mathbb{H}}=|x_{1}-x_{2}|^{-4}$,
so we analyze the remaining term. We need to understand the behavior of
$\beta^{x_{1},x_{2}}_{z_{1}|z_{2}}$ as $|x_{1}-x_{2}|\to 0$. For any $s>0$,
the scale invariance of the BLS implies that

$\displaystyle\begin{split}\beta^{sx_{1},sx_{2}}_{z_{1}|z_{2}}&=\lim_{\varepsilon\to
0}\varepsilon^{-4}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(sx_{j})\neq\emptyset\text{ for }j=1,2;\ell\text{ separates
}z_{1},z_{2})\\\ &=s^{-4}\lim_{\varepsilon\to
0}(\varepsilon/s)^{-4}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon/s}(x_{j})\neq\emptyset\text{ for }j=1,2;\ell\text{ separates
}z_{1}/s,z_{2}/s)\\\
&=s^{-4}\beta^{x_{1},x_{2}}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}.\end{split}$
(56)

In particular,
$\beta^{0,s}_{z_{1}|z_{2}}=s^{-4}\beta^{0,1}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}$.

For $s$ very small, $z_{1}/s$ and $z_{2}/s$ are very far from $0$ and $1$, and
one can expect that
$\beta^{0,1}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}\sim\beta^{0}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}$.
This can be justified using the relation between $\mu^{\text{loop}}$ and
critical percolation (see the proof of Lemma 2.2 in the appendix of
camia2021scalar ). Essentially, $\mu_{\mathbb{H}}^{\text{loop}}$ of the set of
loops getting $\varepsilon$-close to 0 and 1 and separating $z_{1}/s$ from
$z_{2}/s$ can be expressed in terms of the probability that a critical
percolation cluster contained in the upper half-plane gets close to $0$ and
$1$ and “swallows” $z_{1}/s$ but not $z_{2}/s$ or vice versa. Because of
properties of percolation (namely the fact that a percolation configuration
can be obtained by a local exploration process and the fact that two such
processes are independent when they happen in spatially separated regions),
the probability of such an event in a percolation model with a small mesh size
$a$ is well approximated by the product of four terms: (i) the probability
that the semi-annulus of inner radius $\varepsilon$ and outer radius $1/2$
centered at 0 contains a three-arm event (i.e., contains a percolation cluster
crossing the semi-annulus without touching the real line), (ii) the
probability that the semi-disk of inner radius $\varepsilon$ and outer radius
$1/2$ centered at 1 contains a three-arm event, (iii) the probability that the
clusters crossing the semi-annuli centered at 0 and 1 are connected, (iv) the
probability that the cluster crossing the semi-annulus centered at $0$
“swallows” $z_{1}/s$ but not $z_{2}/s$ or vice versa. In the scaling limit,
$a\to 0$, the first two terms are equal and scale like $\varepsilon^{2}$:
(i)$=$(ii) $\sim\varepsilon^{2}$. Term (iii) is bounded away from $0$ and $1$
as $a\to 0$ by the well-known (and celebrated) Russo-Seymour-Welsh theorem, so
it can be neglected. Since $\beta^{0,1}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}$ is
obtained by multiplying $\mu^{\text{loop}}_{\mathbb{H}}$ by
$\varepsilon^{-4}$, we can use one factor of $\varepsilon^{-2}$ to compensate
for (ii). If we now combine (i) and (iv) and multiply them by the remaining
factor of $\varepsilon^{-2}$, as $\varepsilon\to 0$, we obtain
$\beta^{0}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}$. Putting everything together and
using the last line of (55), we obtain
$\beta^{0,1}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}\sim\beta^{0}_{\frac{z_{1}}{s}|\frac{z_{2}}{s}}\sim
s^{2}$, which suggests that $\beta^{0,s}_{z_{1}|z_{2}}\sim s^{-2}$ as $s\to
0$, and more generally,

$\displaystyle\beta^{x_{1},x_{2}}_{z_{1}|z_{2}}\sim|x_{1}-x_{2}|^{-2}\text{ as
}|x_{1}-x_{2}|\to 0.$ (57)

Combined with (LABEL:eq:OOBB), these observations lead to the conclusion that,

$\displaystyle\begin{split}\Braket{\mathcal{B}(x_{1})\mathcal{B}(x_{2})\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}|x_{1}-x_{2}|^{-4}\left[1+O\left(|x_{1}-x_{2}|^{2}\right)\right],\end{split}$
(58)

which implies that the OPE of $\mathcal{B}\times\mathcal{B}$ starts with

$\displaystyle\begin{split}\mathcal{B}(x)\times\mathcal{B}(x^{\prime})=\frac{\mathbb{1}}{|x-x^{\prime}|^{4}}+\ldots,\end{split}$
(59)

where $\mathbb{1}$ denotes the identity operator.

The identification $\mathcal{T}=\sqrt{\lambda}\,\mathcal{B}$, made in the
previous section, gives the OPE

$\displaystyle\begin{split}\mathcal{T}(x)\times\mathcal{T}(x^{\prime})=\frac{\lambda}{|x-x^{\prime}|^{4}}\,\mathbb{1}+\ldots,\end{split}$
(60)

which is consistent with $\mathcal{T}$ being the stress-energy tensor of a CFT
with central charge $c=2\lambda$.

In order to guess the next term in the OPE of $\mathcal{T}\times\mathcal{T}$,
note that

$\displaystyle\begin{split}\beta^{x_{1},x_{2}}_{z_{1}|z_{2}}&=\lim_{\varepsilon\to
0}\varepsilon^{-4}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{j})\neq\emptyset\text{ for }j=1,2;\ell\text{ separates
}z_{1},z_{2})\\\ &=\lim_{\varepsilon\to
0}\varepsilon^{-4}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates }z_{1},z_{2})\\\
&\quad+\lim_{\varepsilon\to
0}\varepsilon^{-4}\Big{[}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{j})\neq\emptyset\text{ for }j=1,2;\ell\text{ separates
}z_{1},z_{2})\\\ &\qquad-\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates
}z_{1},z_{2})\Big{]}\\\ &=\lim_{\varepsilon\to
0}\varepsilon^{-4}\Big{[}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates }z_{1},z_{2})\\\
&\qquad-\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset,\ell\cap
B_{\varepsilon}(x_{2})=\emptyset;\ell\text{ separates }z_{1},z_{2})\Big{]}\\\
&=\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates }z_{1},z_{2})\\\
&\qquad\cdot\varepsilon^{-2}\Big{[}1-\frac{\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset,\ell\cap
B_{\varepsilon}(x_{2})=\emptyset;\ell\text{ separates
}z_{1},z_{2})}{\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates
}z_{1},z_{2})}\Big{]}.\end{split}$ (61)

Observing that

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates
}z_{1},z_{2})=\mu_{\mathbb{H},x_{1}}^{\text{bub}}(\ell\text{ separates
}z_{1},z_{2})$ (62)

is finite, (57) suggests that, as $\varepsilon\to 0$,

$\displaystyle\begin{split}\varepsilon^{-2}\left[1-\frac{\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset,\ell\cap
B_{\varepsilon}(x_{2})=\emptyset;\ell\text{ separates
}z_{1},z_{2})}{\mu_{\mathbb{H}}^{\text{loop}}(\ell\cap
B_{\varepsilon}(x_{1})\neq\emptyset;\ell\text{ separates
}z_{1},z_{2})}\right]\end{split}$ (63)

converges to a function $f(x_{1},x_{2})$ such that $\lim_{|x_{1}-x_{2}|\to
0}|x_{1}-x_{2}|^{2}f(x_{1},x_{2})=\text{const}$. If we make this assumption,
we are led to the conclusion that

$\displaystyle\begin{split}&\Braket{\mathcal{B}(x_{1})\mathcal{B}(x_{2})\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}=\frac{\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}}{|x_{1}-x_{2}|^{4}}\\\
&\quad+\frac{\text{const}}{\sqrt{\lambda}}\frac{\Braket{\mathcal{B}(x_{1})\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}}{|x_{1}-x_{2}|^{2}}+o(|x_{1}-x_{2}|^{-2}),\end{split}$
(64)

which implies the OPEs

$\displaystyle\begin{split}\mathcal{B}(x)\times\mathcal{B}(x^{\prime})=\frac{1}{|x-x^{\prime}|^{4}}\,\mathbb{1}+\frac{\text{const}}{\sqrt{\lambda}}\frac{1}{|x-x^{\prime}|^{2}}\,\mathcal{B}(x)+\ldots\end{split}$
(65)

and

$\displaystyle\begin{split}\mathcal{T}(x)\times\mathcal{T}(x^{\prime})=\frac{\lambda}{|x-x^{\prime}|^{4}}\,\mathbb{1}+\frac{\text{const}}{|x-x^{\prime}|^{2}}\,\mathcal{T}(x)+\ldots.\end{split}$
(66)

The general form of the OPE of the stress energy tensor with itself (see,
e.g., Eq. (8) of CARDY2006333 ) is consistent with (66) and the fact that the
central charge of the theory is $2\lambda$, and implies that $\text{const}=2$.

##  4 Domain perturbations and boundary condition changing operators

In the first part of this section (Section 4.1) we discuss the relation
between local boundary deformations and the operator $\mathcal{T}$. In the
second part (Section 4.2), we introduce boundary operators that generate
Brownian excursions and show that they behave like _boundary condition
changing operators_ and how they are related to $\mathcal{T}$.

###  4.1 Domain perturbations

Roughly speaking, inserting the stress-energy tensor at $z$ corresponds to
applying a conformal transformation that preserves $\infty$ and has a simple
pole at $z$. Let $\mathbb{H}_{\varepsilon}$ denote the upper half-plane from
which a disk of radius $\varepsilon$ center at the origin has been removed,
i.e. $\mathbb{H}_{\varepsilon}:=\mathbb{H}\setminus B_{\varepsilon}(0)$. The
map

$\displaystyle f_{\varepsilon}(z)=z+\frac{\varepsilon^{2}}{z}$ (67)

preserves $\infty$, has a simple pole at $0$, and transforms
$\mathbb{H}_{\varepsilon}$ to $\mathbb{H}$ confrormally. (Consequently, it
maps $\mathbb{C}\setminus B_{\varepsilon}(0)$ conformally to $\mathbb{C}$.)
Therefore, in the limit $\varepsilon\to 0$, mapping $\mathbb{H}_{\varepsilon}$
to $\mathbb{H}$ confrormally, which is equivalent to applying
$f_{\varepsilon}$ to $\mathbb{H}_{\varepsilon}$, should in some sense be also
equivalent to inserting the boundary stress energy tensor at $0$ in the upper
half-plane.

Following ideas from camia_2016_book  (see Proposition 2.6) and
10.1214/18-EJP200  (see Proposition 4.3), we will show that $\mathcal{T}$ can
be used to express the effect of domain perturbations on the two-point
function of layering vertex operators, corroborating its identification with
the boundary stress-energy tensor. For concreteness, we will work in the upper
half-plane.

It follows from the definition (40) of $\tilde{\mathcal{O}}_{\beta}$ and Eq.
(4.3) of Camia_2016  that

$\displaystyle\begin{split}\frac{\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}_{\varepsilon}}}{\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}}=\exp{\left(\lambda(1-\cos\beta)\mu^{\text{loop}}_{\mathbb{H}}(\ell\cap
B_{\varepsilon}(0)\neq\emptyset,\ell\text{ separates
}z_{1},z_{2})\right)},\end{split}$ (68)

which we can also write as

$\displaystyle\begin{split}&\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}_{\varepsilon}}-\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&=\lambda(1-\cos\beta)\mu^{\text{loop}}_{\mathbb{H}}(\ell\cap
B_{\varepsilon}(0)\neq\emptyset,\ell\text{ separates
}z_{1},z_{2})\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\\\
&\qquad+O\left(\left(\mu^{\text{loop}}_{\mathbb{H}}(\ell\cap
B_{\varepsilon}(0)\neq\emptyset,\ell\text{ separates
}z_{1},z_{2})\right)^{2}\right).\end{split}$ (69)

This allows us to define a “derivative” with respect to domain perturbations:

$\displaystyle\begin{split}&\partial_{D}\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{D}\Big{|}_{D=\mathbb{H}}\\\
&:=\lim_{\varepsilon\to
0}\frac{\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}-\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}_{\varepsilon}}}{\varepsilon^{2}}\\\
&=\lambda(1-\cos\beta)\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}_{\varepsilon}}\lim_{\varepsilon\to
0}\varepsilon^{-2}\mu^{\text{loop}}_{\mathbb{H}}(\ell\cap
B_{\varepsilon}(0)\neq\emptyset,\ell\text{ separates }z_{1},z_{2})\\\
&=-\Braket{\mathcal{T}(0)\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}},\end{split}$
(70)

where the last equality follows from (LABEL:eq:OOB-proof) and shows the
appearance of the operator $\mathcal{T}$ inserted at 0. From this expression,
using (LABEL:eq:OOB) one can obtain an explicit formula for the derivative
above.

###  4.2 Boundary condition changing operators

Working for concreteness in the upper half-plane $\mathbb{H}$, in this section
we consider a Brownian loop soup in $\mathbb{H}$ with an additional Brownian
excursion $\gamma_{0,x}$ in $\mathbb{H}$ from the origin $0$ to a point $x$ on
the real line, independent of the loop soup. We think of $\gamma_{0,x}$ as
being generated by the insertion of a pair of boundary changing operators at
$0$ and $x$, and we think of the concatenation of $\gamma_{0,x}$ with the
interval $[0,x]$ as forming a loop added to the loop soup, with outer boundary
b. The boundary curve b is assigned an independent random sign
$\sigma_{\text{b}}=\pm 1$ with equal probability. In this model, expectations
will be denoted by $\Braket{\Phi(0)}{\cdot}{\Phi(x)}_{\mathbb{H}}$, where
$\Phi(y)$ denotes a boundary operator inserted at $y$. We will also use
$\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}$ to denote the probability
distribution of a Brownian excursion888A Brownian excursion in $\mathbb{H}$
from $0$ to $x$ is a Brownian path starting at $0$ and ending at $x$
conditioned to stay in $\mathbb{H}$. Defining such an object requires some
care because it implies conditioning on en event of zero probability. However,
this can be done and $\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}$ is a well
defined probability measure. in $\mathbb{H}$ from $0$ to $x$ and
$\mathbb{E}^{\text{B}}_{\mathbb{H};0,x}$ to denote expectation with respect to
$\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}$, while $\mathbf{I}(\cdot)$ will
denote the indicator function. With this notation, using (40), we have

$\displaystyle\begin{split}&\Braket{\Phi(0)\left
missing}{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}{\Phi(x)}_{\mathbb{H}}=\lim_{\delta\to
0}(\hat{c}_{\mathbb{H}}\delta)^{-4\Delta(\beta)}\Braket{\exp{\Big{[}i\beta\Big{(}\sum_{\stackrel{{\scriptstyle\ell\in\mathcal{L}^{\delta}}}{{z_{1}\in\bar{\ell}}}}\sigma_{\ell}-\sum_{\stackrel{{\scriptstyle\ell\in\mathcal{L}^{\delta}}}{{z_{2}\in\bar{\ell}}}}\sigma_{\ell}\Big{)}\Big{]}}}_{\mathbb{H}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\sum_{\sigma_{\text{b}}=\pm
1}\frac{1}{2}\mathbb{E}^{\text{bub}}_{\mathbb{H};0,x}\left[e^{i\beta\sigma_{\text{b}}\mathbf{I}(z_{1}\in\bar{\text{b}})}e^{-i\beta\sigma_{\text{b}}\mathbf{I}(z_{2}\in\bar{\text{b}})}\right]\\\
&=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{\beta}(z_{2})}_{\mathbb{H}}\left(\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}(\text{b
does not separate
}z_{1},z_{2})+\cos\beta\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}(\text{b
separates }z_{1},z_{2})\right)\\\
&=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{\beta}(z_{2})}_{\mathbb{H}}\left[1-(1-\cos\beta)\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}(\text{b
separates }z_{1},z_{2})\right].\end{split}$ (71)

Now let $\Gamma_{0,x}$ denote an SLE8/3 in the upper half-plane from $0$ to
$x$, with distribution $\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}$. The
equivalence
$\mu^{\text{bub}}_{\mathbb{H}}=\frac{8}{5}\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}$
and the relation between $\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}$ and
$\mu^{\text{bub}}_{\mathbb{H}}$ on the one hand and between
$\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}$ and
$\frac{8}{5}\mu^{\text{SLE}_{8/3}}_{\mathbb{H}}$, on the other, imply that

$\displaystyle\mathbb{P}^{\text{B}}_{\mathbb{H};0,x}(\text{b separates
}z_{1},z_{2})=\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(\Gamma_{0,x}\text{
separates }z_{1},z_{2}),$ (72)

which gives

$\displaystyle\begin{split}&\Braket{\Phi(0)\left
missing}{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}{\Phi(x)}_{\mathbb{H}}\\\
&=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\left[1-(1-\cos\beta)\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(\Gamma_{0,x}\text{
separates }z_{1},z_{2})\right].\end{split}$ (73)

This leads to two interesting conclusions. On the one hand, using (3.2) and
(3.3) from Proposition 1 of beliaev2013some , and letting $\bar{\Gamma}_{0,x}$
denote the interior of the loop formed by the concatenation of $\Gamma_{0,x}$
with $[0,x]$, we have

$\displaystyle\begin{split}&\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(\Gamma_{0,x}\text{
separates }z_{1},z_{2})\\\
&=\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(z_{1}\in{\bar{\Gamma}_{0,x}})+\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(z_{2}\in{\bar{\Gamma}_{0,x}})-2\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(z_{1},z_{2}\in{\bar{\Gamma}_{0,x}})\\\
&=\frac{5}{8}\left\\{\frac{2}{5}\left[\left(\operatorname{Im}\left(\frac{1}{z_{1}}\right)\right)^{2}+\left(\operatorname{Im}\left(\frac{1}{z_{2}}\right)\right)^{2}\right]-\frac{4}{5}\operatorname{Im}\left(\frac{1}{z_{1}}\right)\operatorname{Im}\left(\frac{1}{z_{2}}\right)G(\sigma(z_{1},z_{2}))\right\\}x^{2}+o(x^{2}),\end{split}$
(74)

where $G$ and $\sigma$ are the same as in Proposition 3.2. Plugging this
expression into (LABEL:eq:OO_bc-ops_bis) and using Proposition 3.2
(remembering that $\hat{c}_{b}=1$ and $\mathcal{T}=\sqrt{\lambda}\mathcal{B}$)
gives

$\displaystyle\begin{split}&\Braket{\Phi(0)\left
missing}{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}{\Phi(x)}_{\mathbb{H}}\\\
&=\Braket{\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}+\frac{5}{8}\Braket{\mathcal{T}(0)\tilde{\mathcal{O}}_{\beta}(z_{1})\tilde{\mathcal{O}}_{-\beta}(z_{2})}_{\mathbb{H}}\frac{x^{2}}{\lambda}+o(x^{2}),\end{split}$
(75)

showing the appearance of the boundary stress-energy tensor.

On the other hand, it follows from the analysis in Section VI of Simmons_2009
that999The analysis in Section VI of Simmons_2009  is carried out in a strip
but, as the authors point out at the end of the section, the quantities
involved are conformally invariant, so the results apply to the upper half-
plane as well.

$\displaystyle\begin{split}1-2\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(\Gamma_{0,x}\text{
separates
}z_{1},z_{2})=\frac{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)\phi_{2,1}(z_{1})\phi_{2,1}(z_{2})}^{O(0)}_{\mathbb{H}}}{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)}^{O(0)}_{\mathbb{H}}},\end{split}$
(76)

where the expectations in the right hand side are computed in the $O(n\to 0)$
model in the upper half-plane and $\phi_{1,2}(x)$ is a _boundary condition
changing operator_ applied at $x\in\mathbb{R}$ and $\phi_{2,1}(z)$ is a _twist
operator_ applied at $z\in\mathbb{H}$. Combined with (LABEL:eq:OO_bc-ops_bis),
(76) gives, for $\beta=\pi$,

$\displaystyle\begin{split}&\Braket{\Phi(0)\left
missing}{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}{\Phi(x)}_{\mathbb{H}}\\\
&=\Braket{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}_{\mathbb{H}}\left[1-2\mathbb{P}^{\text{SLE}_{8/3}}_{\mathbb{H};0,x}(\tilde{\gamma}_{0,x}\text{
separates }z_{1},z_{2})\right]\\\
&=\Braket{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}_{\mathbb{H}}\frac{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)\phi_{2,1}(z_{1})\phi_{2,1}(z_{2})}^{O(0)}_{\mathbb{H}}}{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)}^{O(0)}_{\mathbb{H}}},\end{split}$
(77)

which leads to an interesting equivalence between quantities in the BLS
conformal field theory and in the $O(0)$ model, namely

$\displaystyle\begin{split}&\frac{\Braket{\Phi(0)\left
missing}{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}{\Phi(x)}_{\mathbb{H}}}{\Braket{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}_{\mathbb{H}}}=\frac{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)\phi_{2,1}(z_{1})\phi_{2,1}(z_{2})}^{O(0)}_{\mathbb{H}}}{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)}^{O(0)}_{\mathbb{H}}}.\end{split}$
(78)

Note that the equivalence is valid for all $\lambda\geq 0$.

If we now let $\lambda\to 0$, all loops from the loop soup are suppressed and

$\displaystyle\begin{split}\lim_{\lambda\to
0}\Braket{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}_{\mathbb{H}}=1,\end{split}$
(79)

and we get

$\displaystyle\begin{split}\Braket{\Phi(0)\left
missing}{\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}{\Phi(x)}_{\mathbb{H}}^{\lambda=0}=\frac{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)\phi_{2,1}(z_{1})\phi_{2,1}(z_{2})}^{O(0)}_{\mathbb{H}}}{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)}^{O(0)}_{\mathbb{H}}}.\end{split}$
(80)

When $\beta=\pi$, the operators
$\tilde{\mathcal{O}}_{\beta}=\tilde{\mathcal{O}}_{\pi}$ act exactly like the
twist operators $\phi_{2,1}$. This suggests that we can think of the boundary
operator $\Phi$ that inserts a Brownian excursion as a boundary condition
changing operator $\phi_{1,2}$ inserting a chordal SLE8/3. More precisely, we
make the identification

$\displaystyle\begin{split}\Braket{\Phi(0)\left
missing}{\,\cdot\,}{\Phi(x)}^{\lambda=0}_{\mathbb{H}}=\frac{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)\,\cdot\,}^{\lambda=0}_{\mathbb{H}}}{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)}^{\lambda=0}_{\mathbb{H}}},\end{split}$
(81)

where $\phi_{1,2}$ is interpreted as a primary operator with scaling dimension
$\Delta=5/8$ (corresponding to the “1-leg operator” in the upper half plane
with $\Delta(\kappa)=\frac{6-\kappa}{2\kappa}$ for $\kappa=8/3$). Inserting
(81) into (LABEL:eq:PhiOOPhi) leads to

$\displaystyle\begin{split}&\frac{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}^{\lambda=0}_{\mathbb{H}}}{\Braket{\phi_{1,2}(0)\phi_{1,2}(x)}^{\lambda=0}_{\mathbb{H}}}\\\
&\quad=1+\frac{5}{8}x^{2}\lim_{\lambda\to
0}\frac{1}{\lambda}\Braket{\mathcal{T}(0)\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}_{\mathbb{H}}+o(x^{2}),\end{split}$
(82)

which corresponds to the OPE

$\displaystyle\begin{split}\phi_{1,2}(0)\times\phi_{1,2}(x)=\lvert
x\rvert^{-5/4}\left(\mathbb{1}+\frac{5}{8}\lim_{\lambda\to
0}\frac{\mathcal{T}(0)}{\lambda}x^{2}+o(x^{2})\right),\end{split}$ (83)

showing how the boundary stress-energy tensor emerges in the limit in which
two boundary condition changing operators are brought together.

We note that
$\frac{1}{\lambda}\Braket{\mathcal{T}(0)\tilde{\mathcal{O}}_{\pi}(z_{1})\tilde{\mathcal{O}}_{\pi}(z_{2})}_{\mathbb{H}}$
is independent of $\lambda$, so that the limit in (LABEL:eq:phiphiOO) is not
necessary. It is tempting to conjecture that one can remove the limit also
from (83), obtaining an OPE for $\phi_{1,2}\times\phi_{1,2}$ valid for all
values of $\lambda>0$. The resulting expression

$\displaystyle\begin{split}\phi_{1,2}(0)\times\phi_{1,2}(x)&=\lvert
x\rvert^{-5/4}\left(\mathbb{1}+\frac{5}{8}\frac{\mathcal{T}(0)}{\lambda}x^{2}+o(x^{2})\right)\\\
&=\lvert
x\rvert^{-5/4}\left(\mathbb{1}+\frac{2\Delta}{c}\mathcal{T}(0)x^{2}+o(x^{2})\right)\end{split}$
(84)

with $\Delta=5/8$ and $c=2\lambda$ is consistent with an OPE for primary
operators with scaling dimension $5/8$, with structure constant
$C^{\mathcal{T}}_{\phi_{1,2},\phi_{1,2}}=1$ for all $\lambda$, see (17).

It was argued in Camia_2020  that the $n\to 0$ limit of the $O(n)$ model,
which describes single self-avoiding loops, is closely related to the the BLS
in the $\lambda\to 0$ limit. We saw additional evidence for this
identification in this section, where we identified the boundary changing
operator $\Phi$ with the $O(n)$ model operator $\phi_{1,2}$ and presented
evidence that this identification holds for finite $\lambda$.

It is curious to note that a similar identification can be made for
$\mathcal{T}$. Its scaling dimension, $\Delta=2$, coincides with that of a
“2-leg” boundary operator in the $O(0)$ model, namely
$\Delta(\kappa)=(8-\kappa)/\kappa$ for $\kappa=8/3$. Geometrically,
$\mathcal{T}$ counts the number of loops coming close to its insertion point
on the boundary (see Figure 3). This geometric interpretation coincides with
that of the 2-leg operator, which acts as a sink or source of two self-
avoiding random walks at its insertion point that avoid each other and the
boundary of the domain. Moreover, although the ensemble of loops of the BLS
contains an infinite number of loops, if one considers the set of points $x$
on the real line such that the semi-disk of radius $\varepsilon$ centered at
$x$ intersects a “macroscopic” loop, its fractal dimension goes to $0$ as
$\varepsilon\to 0$, regardless of the value of $\lambda$. This allows us to
conjecture that $\mathcal{T}$ can be identified with the 2-leg operator for
all $\lambda$. Additionally, it gives an explanation for why the scaling
dimension of $\mathcal{B}$ (and $\mathcal{T}$) is independent of $\lambda$.

The discussion in this section can in principle be extended to operators other
than $\tilde{\mathcal{O}}_{\beta}$ and to the case in which multiple pairs of
boundary condition changing operators are inserted at the boundary.

##  5 Conclusions and future perspectives

For conformal field theories associated to the BLS, we have provided integral
expressions of the bulk stress-energy tensor $T$ based on the OPE of
$\mathcal{E}\times\mathcal{E}$ and of
$\mathcal{O}_{\beta}\times\mathcal{O}_{-\beta}$. These expressions can be used
to compute certain correlation functions involving $T$ itself, and we have
done this for two specific examples, verifying that the corresponding Ward
identities are satisfied. It would be interesting to extend this analysis to
$J$, the dimension $(1,0)$ current that also occurs in these OPEs (see Figure
1, position (3,0)).

Furthermore, we have identified a new operator $\mathcal{T}$ whose properties
are consistent with those of a boundary stress-energy tensor. The operator
$\mathcal{T}$ is essentially a boundary version of the edge counting operator
introduced in camia2021scalar , and is reminiscent of the boundary stress-
energy tensor discussed in FRIEDRICH2002947 ; Doyon_2006 . A full verification
that $\mathcal{T}$ is the boundary stress-energy tensor, for example by
verifying all Ward identities involving $\mathcal{T}$, is an interesting open
problem.

We have shown that the insertion of $\mathcal{T}$ is linked to local
deformations of the boundary of the domain. This is in the spirit of the
discussion at the end of Section 6 of 2005math…..11605W , where it is argued
that the measure on simple loops induced by $\mu^{\text{loop}}$ is well suited
to the study of local deformations of the complex structure of Riemann
surfaces. It would indeed be interesting to study the CFTs arising from the
BLS on Riemann surfaces, in particular in the case of the torus.

The operator $\mathcal{T}$ can also be linked to the insertion of a pair of
boundary operators that generate a Brownian excursion between the insertion
points. These operators appear to behave like boundary condition changing
operators with scaling dimension $\Delta=5/8$, consistent with
$\Delta(\kappa)=\frac{6-\kappa}{2\kappa}$ for $\kappa=8/3$. The appearance of
the value $\kappa=8/3$ is not surprising since the outer boundary of a
Brownian loop is locally distributed like an SLE8/3 curve. It would also be
interesting to further explore the properties of these putative boundary
condition changing operators.

###### Acknowledgements.

The work of M.K. is partially supported by the NSF through the grant
PHY-1820814.

## References

  * [1] Gregory F Lawler and Wendelin Werner.  The Brownian loop soup.  Probability theory and related fields, 128(4):565–588, 2004.
  * [2] K Symanzik.  Euclidean quantum field theory.  Rend. Scu. Int. Fis. Enrico Fermi 45: 152-226, 1 1969.
  * [3] Federico Camia, Alberto Gandolfi, and Matthew Kleban.  Conformal correlation functions in the Brownian loop soup.  Nuclear Physics B, 902:483–507, Jan 2016.
  * [4] Federico Camia, Valentino F. Foit, Alberto Gandolfi, and Matthew Kleban.  Exact correlation functions in the Brownian Loop Soup.  Journal of High Energy Physics, 2020(7), Jul 2020.
  * [5] Federico Camia, Valentino F. Foit, Alberto Gandolfi, and Matthew Kleban.  Scalar conformal primary fields in the Brownian loop soup, 2021.
  * [6] Wendelin Werner.  The conformally invariant measure on self-avoiding loops.  Journal of the American Mathematical Society, 21(1):137–169, 2008\.
  * [7] Federico Camia and Charles M. Newman.  Two-Dimensional Critical Percolation: The Full Scaling Limit.  Communications in Mathematical Physics, 268(1):1–38, Sep 2006\.
  * [8] Michael Aizenman, Bertrand Duplantier, and Amnon Aharony.  Path-Crossing Exponents and the External Perimeter in 2D Percolation.  Phys. Rev. Lett., 83:1359–1362, Aug 1999.
  * [9] Stanislav Smirnov and Wendelin Werner.  Critical exponents for two-dimensional percolation.  Mathematical Research Letters, 8:729–744, 2001.
  * [10] B. Doyon, V. Riva, and J. Cardy.  Identification of the Stress-Energy Tensor Through Conformal Restriction in SLE and Related processes.  Communications in Mathematical Physics, 268(3):687–716, Sep 2006\.
  * [11] Adam Gamsa and John Cardy.  Correlation functions of twist operators applied to single self-avoiding loops.  Journal of Physics A: Mathematical and General, 39(41):12983–13003, Sep 2006.
  * [12] Philippe Di Francesco, Pierre Mathieu, and David Sénéchal.  Conformal Field Theory.  Graduate texts in contemporary physics. Springer, New York, NY, 1997.
  * [13] Jacob J H Simmons and John Cardy.  Twist operator correlation functions in O(n) loop models.  Journal of Physics A: Mathematical and Theoretical, 42(23):235001, May 2009.
  * [14] J. Cardy.  Boundary Conformal Field Theory.  In Jean-Pierre Françoise, Gregory L. Naber, and Tsou Sheung Tsun, editors, Encyclopedia of Mathematical Physics, pages 333–340. Academic Press, Oxford, 2006.
  * [15] Roland Friedrich and Wendelin Werner.  Conformal fields, restriction properties, degenerate representations and SLE.  Comptes Rendus Mathematique, 335(11):947–952, 2002.
  * [16] Gregory Lawler, Oded Schramm, and Wendelin Werner.  Conformal restriction: The chordal case.  Journal of the American Mathematical Society, 16(4):917–955, Jun 2003.
  * [17] Dmitry Beliaev and Fredrik Johansson Viklund.  Some remarks on SLE bubbles and Schramm’s two-point observable.  Communications in Mathematical Physics, 320(2):379–394, 2013.
  * [18] Yong Han, Yuefei Wang, and Michel Zinsmeister.  On the Brownian Loop Measure.  Journal of Statistical Physics, 175(5):987–1005, March 2019.
  * [19] Federico Camia.  Scaling Limits, Brownian Loops, and Conformal Fields.  In Advances in Disordered Systems, Random Processes and Some Applications, page 205–269. Cambridge University Press, 2016.
  * [20] Tim van de Brug, Federico Camia, and Marcin Lis.  Spin systems from loop soups.  Electronic Journal of Probability, 23(none):1 – 17, 2018.