# Liouville conformal field theory and the quantum zipper

Morris Ang

###### Abstract

Sheffield showed that conformally welding a $\gamma$-Liouville quantum gravity
(LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with
parameter $\kappa=\gamma^{2}$ as the interface, and Duplantier-Miller-
Sheffield proved similar stories for $\kappa=\frac{16}{\gamma^{2}}$ for
$\gamma$-LQG surfaces with boundaries decorated by looptrees of disks or by
continuum random trees. We study these dynamics for LQG surfaces coming from
Liouville conformal field theory (LCFT). At stopping times depending only on
the curve, we give an explicit description of the surface and curve in terms
of LCFT and SLE. This has applications to both LCFT and SLE. We prove the
boundary BPZ equation for LCFT, which is crucial to solving boundary LCFT.
With Yu we will prove the reversibility of whole-plane
$\operatorname{SLE}_{\kappa}$ for $\kappa>8$ via a novel radial mating-of-
trees, and show the space of LCFT surfaces is closed under conformal welding.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Reverse $\operatorname{SLE}_{\kappa,\rho}$ and the GFF partition function
  4. 4 Sheffield’s coupling and the $\kappa\leq 4$ LCFT zipper
  5. 5 The $\kappa>8$ LCFT zipper
  6. 6 The $\kappa\in(4,8)$ LCFT zipper
  7. 7 The boundary BPZ equation for Liouville conformal field theory

##  1 Introduction

Polyakov introduced a canonical one-parameter family of random surfaces called
_Liouville quantum gravity (LQG)_ to make sense of summation over surfaces
[Pol81]. The _mating-of-trees_ framework studies LQG through its coupling with
random curves called _Schramm-Loewner evolution (SLE)_. Let $\kappa>0$ and
$\gamma=\min(\sqrt{\kappa},\frac{4}{\sqrt{\kappa}})$.
$\operatorname{SLE}_{\kappa}$ is a simple curve when $\kappa\leq 4$, self-
intersecting when $\kappa\in(4,8)$, and space-filling when $\kappa\geq 8$.
When $\kappa\in(0,4]$ there is an infinite-volume $\gamma$-LQG surface which,
when decorated by an independent $\operatorname{SLE}_{\kappa}$ curve, is
invariant in law under the operation of conformally welding the two boundary
arcs according to their random length measures; this is called the _quantum
zipper_ [She16, HP21, KMS22]. Similar stories hold for other ranges of
$\kappa$ when the boundary of the LQG surface is modified to have non-trivial
topology [DMS21].

Starting with these stationary quantum zippers, the mating-of-trees approach
develops a theory of conformal welding of special LQG surfaces, culminating in
landmark results such as the equivalence of the Brownian map and LQG [MS20]
and the convergence of random planar maps to LQG [GMS21, HS23]. A recent
program [AHS22, ARS22, AS21, ASY22] extends the conformal welding theory to a
larger class of LQG surfaces which arise from _Liouville conformal field
theory_ (LCFT). In these conformal weldings, whole boundary arcs are glued at
once, in contrast to the quantum zipper where the gluing is incremental.

We study the quantum zipper dynamics of [She16, DMS21] applied to random
surfaces arising from LCFT, applying a different zipping mechanism for each
parameter range of $\kappa$. For $\kappa\in(0,4]$, we conformally weld the
left and right boundaries of the LQG surface. For $\kappa\in(4,8)$, we add a
Poisonnian collection of looptrees of LQG disks to the boundary, then mate the
forested boundaries. For $\kappa>8$, we attach a pair of correlated continuum
random trees to the boundary arcs of the LQG surface, then mate the continuum
random trees. This gives an LQG surface with an interface curve, see figure
below. In all three regimes, when the process is run until a stopping time
depending only on the curve, we give an explicit description of the joint law
of the field and curve. Roughly speaking, we show the curve is described by
reverse $\operatorname{SLE}_{\kappa}$ and the field is described by LCFT; see
Theorems 1.1, 1.2, 1.5 and 1.7 for details. The $\kappa=8$ requires further
arguments and is not treated in this work.

We then give an application of the LCFT quantum zipper. Belavin, Polyakov and
Zamolodchikov (BPZ) proposed differential equations for conformal field
theories [BPZ84], which were rigorously proved for LCFT in [KRV19] and used in
the landmark computation of the LCFT three-point function [KRV20]. There are
substantial conceptual and technical difficulties in adapting the argument of
[KRV19] to boundary LCFT. We instead prove the boundary LCFT BPZ equations via
SLE martingales from the quantum zipper. These equations require a non-trivial
coupling of cosmological constants (1.3) conjectured from special cases
[FZZ00]. To the best of our knowledge, our argument gives the first conceptual
explanation of (1.3), even at the physics level of rigor.

Our results have consequences for both LCFT and SLE. In [ARSZ] the BPZ
equations will be used to prove that the boundary three-point function and the
bulk-boundary function of LCFT agree with formulas proposed by Ponsot-Teschner
[PT02] and Hosomichi [Hos01] respectively; this is the boundary analog of
[KRV20] and provides the initial data for the boundary LCFT conformal
bootstrap. With Pu Yu we will establish a radial mating-of-trees via the LCFT
quantum zipper, and use it to prove the reversibility of whole-plane
$\operatorname{SLE}_{\kappa}$ for $\kappa>8$. Whole-plane reversibility was
shown for $\kappa\in(0,4]$ and $\kappa\in(4,8)$ by [Zha15] and [MS17]
respectively, and our result will resolve the remaining case. This answers two
conjectures of [VW20]. Finally, with Pu Yu we will prove that the conformal
welding of LCFT surfaces of arbitrary genus gives a curve-decorated LCFT
surface, extending the program of [AHS22, ARS22, AS21, ASY22].

###  1.1 Liouville quantum gravity and Schramm-Loewner evolution

We first briefly recall some preliminaries; a detailed introduction is given
in Section 2. The free boundary Gaussian free field (GFF) on a simply-
connected domain $D\subset\mathbbm{C}$ is the Gaussian process on $D$ whose
covariance kernel is the Green function; it can be understood as a random
generalized function $h$ [She07]. For $\gamma\in(0,2]$ and $h$ a variant of
the GFF, the $\gamma$-LQG area measure $\mathcal{A}_{h}$ on $D$ and boundary
measure $\mathcal{L}_{h}$ on $\partial D$ are heuristically defined by
$\mathcal{A}_{h}(dz)=e^{\gamma h(z)}dz$ and
$\mathcal{L}_{h}(dx)=e^{\frac{\gamma}{2}h(x)}dx$. These definitions are made
rigorous via regularization and renormalization [DS11, DRSV14].

Suppose $g:D\to\widetilde{D}$ is a conformal map and $h$ a generalized
function on $D$. We define the generalized function $g\bullet_{\gamma}h$ on
$\widetilde{D}$ by

$g\bullet_{\gamma}h:=h\circ g^{-1}+Q\log|(g^{-1})^{\prime}|,\qquad
Q=\frac{\gamma}{2}+\frac{2}{\gamma}.$

A _quantum surface_ is an equivalence class of pairs $(D,h)$ where
$(D,h)\sim_{\gamma}(\widetilde{D},\widetilde{h})$ if there exists a conformal
map $g:D\to\widetilde{D}$ such that $\widetilde{h}=g\bullet_{\gamma}h$
[She16]. The pair $(D,h)$ is called an embedding of the quantum surface. The
$\gamma$-LQG area and boundary measure are intrinsic to the quantum surface:
If $h$ is a variant of the GFF on $D$, then writing $g_{*}$ for the
pushforward under $g$, we have
$g_{*}\mathcal{A}_{h}=\mathcal{A}_{\widetilde{h}}$ and
$g_{*}\mathcal{L}_{h}=\mathcal{L}_{\widetilde{h}}$ where
$\widetilde{h}=g\bullet_{\gamma}h$.

Schramm-Loewner evolution (SLE) [Sch00] is a canonical random planar curve
which describes the scaling limits of many critical 2D statistical physics
models, e.g. [Smi01, LSW04, SS09, CDCH+14]. The parameter $\kappa>0$ describes
the “roughness” of the $\operatorname{SLE}_{\kappa}$ curve: the curve is
simple when $\kappa\in(0,4]$, self-intersecting (but not self-crossing) when
$\kappa\in(4,8)$, and space filling when $\kappa\geq 8$. We will work with the
_reverse_ variant of $\operatorname{SLE}_{\kappa}$ where the curve grows from
its base.

Liouville conformal field theory (LCFT) is the quantum field theory arising
from the Liouville action introduced by the physicist Polyakov in his work on
quantum gravity and string theory [Pol81]. LCFT was rigorously constructed on
the sphere in [DKRV16] by making sense of the path integral for the Liouville
action, and has since been extended to other surfaces [HRV18, Rem18, DRV16,
GRV19]. In a series of recent breakthroughs, the _correlation functions_ of
LCFT on closed surfaces were rigorously computed, by first solving for the
three-point correlation function [KRV20], and then implementing the conformal
bootstrap program to recursively obtain all higher order correlation functions
[GKRV22, GKRV21].

In this paper we focus on LCFT on the disk, parametrized by the upper half-
plane $\mathbbm{H}$. There is an infinite measure
${{\mathrm{LF}}}_{\mathbbm{H}}$ on the space of generalized functions on
$\mathbbm{H}$ obtained by an additive perturbation of the GFF, which we call
the _Liouville field_ ; see Definition 2.1. For $\delta>0$ and finitely many
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ we can make
sense of the measure
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j}),(\delta,\infty)}=\prod_{j}e^{\alpha_{j}\phi(z_{j})}e^{\delta\phi(\infty)}{{\mathrm{LF}}}_{\mathbbm{H}}(d\phi)$
via regularization and renormalization. This is the Liouville field with
_insertions_ of size $\alpha_{i}$ at $z_{i}$ and an insertion of size $\delta$
at $\infty$. The correlation functions of LCFT (sometimes written as
$\langle\prod e^{\alpha_{j}\phi_{j}(z_{j})}e^{\delta\phi(\infty)}\rangle$) are
defined as functionals of
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j}),(\delta,\infty)}$, see for
instance (1.5).

###  1.2 The $\kappa\leq 4$ LCFT quantum zipper

Let $\kappa\in(0,4]$ and $\gamma=\sqrt{\kappa}$. See Figure 1 for a summary of
the LCFT quantum zipper in this range.

Let $n\geq 0$, let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ such that
$z_{1},\dots,z_{n}$ are distinct, and let $\delta\in\mathbbm{R}$. Sample a
field
$\phi_{0}\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}$.
Let $s>0$ satisfy
$\mathcal{L}_{\phi_{0}}(-\infty,0),\mathcal{L}_{\phi_{0}}(0,\infty)>s$. For
each $u\in(0,s]$ let $p_{u}\in(0,\infty)$ and $q_{u}\in(-\infty,0)$ be the
points such that
$\mathcal{L}_{\phi_{0}}([0,p_{u}])=\mathcal{L}_{\phi_{0}}([q_{u},0])=u$. We
want to glue the boundary arcs $[q_{s},0]$ and $[0,p_{s}]$ of $\mathbbm{H}$
together, identifying $q_{u}$ with $p_{u}$ for $u\in(0,s]$. Almost surely
there is a simple curve $\hat{\eta}_{s}:[0,s]\to\overline{\mathbbm{H}}$ such
that $\hat{\eta}_{s}\cap\mathbbm{R}=\hat{\eta}_{s}(s)$ and a conformal map
$\hat{g}_{s}:\mathbbm{H}\to\mathbbm{H}\backslash\hat{\eta}_{s}$ fixing
$\infty$ such that $\hat{g}_{s}(p_{u})=\hat{g}_{s}(q_{u})=\hat{\eta}_{s}(u)$
for all $u\leq s$; this is called a _conformal welding_. The pair
$(\hat{\eta}_{s},\hat{g}_{s})$ is unique modulo conformal automorphisms of
$\mathbbm{H}$, so specifying the _hydrodynamic normalization_
$\lim_{z\to\infty}\hat{g}_{s}(z)-z=0$ uniquely defines
$(\hat{\eta}_{s},\hat{g}_{s})$. The existence and uniqueness of
$(\hat{\eta}_{s},\hat{g}_{s})$ was shown in [She16] for $\gamma<2$; for
$\gamma=2$ existence was established by [HP21] and uniqueness by [KMS22] (see
also [MMQ21]).

Thus, for
$\phi_{0}\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}$,
we can define a process $(\hat{\eta}_{s},\hat{g}_{s})$ for
$s<\min(\mathcal{L}_{\phi_{0}}(-\infty,0),\mathcal{L}_{\phi_{0}}(0,\infty))$.
The _half-plane capacity_ of $\hat{\eta}_{s}$ is
$\mathrm{hcap}(\hat{\eta}_{s}):=\lim_{z\to\infty}z(\hat{g}_{s}(z)-z)$. We
reparametrize time to get a process $(\eta_{t},g_{t})$ such that
$\mathrm{hcap}(\eta_{t})=2t$. Define $\phi_{t}=g_{t}\bullet_{\gamma}\phi_{0}$
and $W_{t}=\eta_{t}\cap\mathbbm{R}$. See Figure 1.

We first give a description of the law of the field and curve when the process
is run until a stopping time before any marked points are zipped into the
curve.

Figure 1:  Let $\kappa\leq 4$ and $\gamma=\sqrt{\kappa}$. Left: Liouville
field $\phi_{0}$ with insertions at $z_{1}\in\mathbbm{H}$ and
$z_{2}\in\mathbbm{R}$. Middle: We conformally weld the left and right boundary
arcs of $\phi_{0}$ according to quantum length to get
$(\phi_{t},\eta_{t})_{t\geq 0}$. Time is parametrized by half-plane capacity
of the curve. For a stopping time $\tau$ for
$\mathcal{F}_{t}=\sigma(\eta_{t})$ such that $g_{\tau}(z_{2})$ has not yet hit
the curve, Theorem 1.1 describes the law of $(\phi_{\tau},\eta_{\tau})$ in
terms of the Liouville field and reverse SLE. Right: At the time $T$ that
$g_{T}(z_{2})$ hits $W_{T}$, if the quantum length measure near $W_{T}$ is
finite, we can continue the conformal welding process to “zip the marked point
into the bulk”, e.g. until the depicted time $\tau^{\prime}$. Theorem 1.2
describes the law of $(\phi_{\tau^{\prime}},\eta_{\tau^{\prime}})$.

######  Theorem 1.1.

In the setting immediately above, assume $z_{1},\dots,z_{n}\neq 0$ and let
$\tau$ be a stopping time with respect to the filtration
$\mathcal{F}_{t}=\sigma(\eta_{t})$ such that a.s.
$g_{\tau}(z_{j})\not\in\eta_{\tau}$ for all $j$. Let
$I=\\{i\>:\>z_{i}\in\mathbbm{H}\\}$ and $B=\\{b\>:\>z_{b}\in\mathbbm{R}\\}$.
Then the law of $(\phi_{\tau},\eta_{\tau})$ is

$\prod_{i\in I}|g_{\tau}^{\prime}(z_{i})|^{2\Delta_{\alpha_{i}}}\prod_{b\in
B}|g_{\tau}^{\prime}(z_{b})|^{\Delta_{2\alpha_{b}}}{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},W_{\tau}),(\alpha_{j},g_{\tau}(z_{j}))_{j},(\delta,\infty)}(d\phi)\operatorname{rSLE}^{\tau}_{\kappa}(d\eta),$

where $\Delta_{\alpha}:=\frac{\alpha}{2}(Q-\frac{\alpha}{2})$ and
$\operatorname{rSLE}_{\kappa}^{\tau}$ is the law of reverse
$\operatorname{SLE}_{\kappa}$ run until the stopping time $\tau$.

Informally, zipping up a Liouville field until a stopping time $\tau$ that
depends only on the zipping interface, the curve $\eta_{\tau}$ is described by
reverse $\operatorname{SLE}_{\kappa}$, and given $\eta_{\tau}$ the resulting
field is described by a Liouville field with insertions at locations
determined by $\eta_{\tau}$.

In Theorem 1.1 the condition that $g_{\tau}(z_{j})\not\in\eta_{\tau}$ for all
$j$ is necessary, since otherwise the law of the curve would be singular with
respect to reverse $\operatorname{SLE}_{\kappa}$. In Theorem 1.2, we allow
boundary marked points to be zipped into the bulk, by using an
$\operatorname{SLE}_{\kappa}$ variant called _reverse
$\operatorname{SLE}_{\kappa}$ with force points_ (see Section 2.3).
Regardless, the zipping procedure can only be run until the _continuation
threshold_ , defined as the first time $t\leq\infty$ that any neighborhood of
$W_{t}$ in $\mathbbm{R}$ has infinite quantum length, i.e.
$\mathcal{L}_{\phi_{t}}((W_{t}-\varepsilon,W_{t}+\varepsilon))=\infty$ for all
$\varepsilon>0$. Once the continuation threshold is hit, there is no canonical
way to continue the conformal welding.

For finitely many $(a_{j},p_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$
such that the points $p_{j}$ are distinct, we define

$\mathcal{Z}((a_{j},p_{j})_{j})=\prod_{i\in
I}(2\operatorname{Im}p_{i})^{-a_{i}^{2}/2}\prod_{j<k}e^{a_{j}a_{k}G(p_{j},p_{k})},\quad
I=\\{i:p_{i}\in\mathbbm{H}\\},$ (1.1)

where $G(p,q)=-\log|p-q|-\log|p-\overline{q}|$. If the $p_{j}$ are not
distinct, we combine all pairs $(a,p)$ with the same $p$ by summing their
$a$’s to get a collection $(a^{\prime}_{j},p^{\prime}_{j})$ with
$p^{\prime}_{j}$ distinct, and define
$\mathcal{Z}((a_{j},p_{j})_{j}):=\mathcal{Z}((a^{\prime}_{j},p^{\prime}_{j})_{j})$.

######  Theorem 1.2.

In the setting above Theorem 1.1, let $\tau$ be a stopping time with respect
to the filtration $\mathcal{F}_{t}=\sigma(\eta_{t})$ which is not beyond the
continuation threshold. Then the law of $(\phi_{\tau},\eta_{\tau})$ is

$\frac{\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j})}{\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},W_{\tau}),(\alpha_{j},g_{\tau}(z_{j}))_{j})}{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},W_{\tau}),(\alpha_{j},g_{\tau}(z_{j}))_{j},(\delta,\infty)}(d\phi)\operatorname{rSLE}^{\tau}_{\kappa,\rho}(d\eta)$
(1.2)

where $\operatorname{rSLE}_{\kappa,\rho}^{\tau}$ denotes the law of reverse
$\operatorname{SLE}_{\kappa,\rho}$ with a force point at $z_{j}$ of weight
$\rho_{j}=2\sqrt{\kappa}\alpha_{j}$ for each $j$, run until the stopping time
$\tau$.

We emphasize that while we study the same Liouville field as e.g. [AHS22,
ARS22, AS21, ASY22], for boundary insertions our notation differs by a factor
of 2, see Remark 2.4. The present choice of notation simplifies the statement
of Theorem 1.2 since boundary insertions zipped into the bulk maintain the
same value of $\alpha$.

######  Remark 1.3.

Let
$R_{t}:=\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j})/\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},W_{t}),(\alpha_{j},g_{t}(z_{j}))_{j})$
be the prefactor in (1.2) at time $t$. If $\tau$ is a time such that
$g_{\tau}(z_{j})=W_{\tau}$ for some $j$, then
$\lim_{t\uparrow\tau}R_{t}\in\\{0,\infty\\}$ whereas $R_{\tau}\in(0,\infty)$.
This apparent discontinuity is only cosmetic: the definition of
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},W_{t}),(\alpha_{j},g_{t}(z_{j}))_{j},(\delta,\infty)}$
includes a factor of
$C_{\kappa}^{(-\frac{1}{\sqrt{\kappa}},W_{t}),(\alpha_{j},g_{t}(z_{j}))_{j},(\delta,\infty)}$
(defined in (2.2)), and
$C_{\kappa}^{(-\frac{1}{\sqrt{\kappa}},W_{t}),(\alpha_{j},g_{t}(z_{j}))_{j},(\delta,\infty)}R_{t}$
is continuous in $t$.

###  1.3 The $\kappa\in(4,8)$ LCFT quantum zipper

We now need to work with _beaded quantum surfaces_ which can have nontrivial
topology. Suppose $(D,h)$ are such that $D\subset\mathbbm{C}$ is a closed set
such that each connected component of the interior of $D$ and its prime-end
boundary is homeomorphic to the closed disk, and $h$ is a generalized function
defined only on the interior of $D$. We say that
$(D,h)\sim_{\gamma}(\widetilde{D},\widetilde{h})$ if there is a homeomorphism
$g:D\to\widetilde{D}$ which is conformal on each component of the interior of
$D$ and which satisfies $\widetilde{h}=g\bullet_{\gamma}h$. A _beaded quantum
surface_ is an equivalence class of pairs $(D,h)$ under $\sim_{\gamma}$.

Let $\kappa\in(4,8)$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. In this regime
there is a natural beaded quantum surface called a _rooted generalized quantum
disk_. It is obtained by sampling an excursion of a $\frac{\kappa}{4}$-stable
Lévy process with no negative jumps, constructing a looptree whose nodes
correspond to the jumps of the excursion, and for each node sampling an
independent quantum disk with boundary length given by the jump size. This
gives a beaded quantum surface with a marked boundary point, whose law is an
infinite measure (since the Lévy excursion measure is infinite). Its boundary
has a _generalized quantum length measure_ coming from the time
parametrization of the Lévy excursion.

A _forested line_ is the beaded quantum surface obtained by attaching a
Poissonian collection of marked generalized quantum disks to either the top or
bottom of the half-line $(0,\infty)$. We equip the side without attached disks
with a quantum length measure given by Lebesgue measure on $(0,\infty)$,
whereas the other side inherits the generalized boundary length measure from
the generalized quantum disks. The law of the forested line is a probability
measure. For details on these constructions see [MSW21, Section 2.2] or
[DMS21, Section 1.4.2].

A quantum wedge is a canonical scale-invariant $\gamma$-LQG surface with a
_weight_ parameter $W>0$. When $W\geq\frac{\gamma^{2}}{2}$ the quantum wedge
is called _thick_ and has the half-plane topology, and when
$W<\frac{\gamma^{2}}{2}$ the quantum wedge is called _thin_ and is the
concatenation of a chain of countably many “beads”. See Section 2.4 for
definitions.

Proposition 1.4 below states that there is a way to mate a pair of independent
forested lines according to quantum forested length to obtain a beaded curve-
decorated quantum surface.

######  Proposition 1.4 ([DMS21, Theorem 1.15]).

Sample a weight $(2-\frac{\gamma^{2}}{2})$ quantum wedge and let $\eta$ be a
concatenation of independent
$\operatorname{SLE}_{\kappa}(\frac{\kappa}{2}-4;\frac{\kappa}{2}-4)$ curves in
each bead. Then $\eta$ divides the quantum wedge into two independent forested
lines, whose forested boundaries are identified according to generalized
quantum length. Moreover the curve-decorated quantum wedge is measurable with
respect to the pair of forested lines.

We now explain a quantum zipper for the Liouville field with forested
boundary, see Figure 2. Let $\mathrm{FL}_{\kappa}$ be the law of the pair of
forested lines in Proposition 1.4 (so $\mathrm{FL}_{\kappa}$ is a probability
measure). Let $n\geq 0$, let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ such that
$z_{1},\dots,z_{n}$ are distinct, and let $\delta\in\mathbbm{R}$. Sample
$(\phi_{0},(F_{L},F_{R}))\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}\times\mathrm{FL}_{\kappa}$
and glue $F_{L}$ (resp. $F_{R}$) to the boundary of $(\mathbbm{H},\phi_{0})$
according to quantum length, starting at $0$ and gluing to the left (resp.
right)111If $\mathcal{L}_{\phi_{0}}((-\infty,0))<\infty$ we only glue the
initial segment of $F_{L}$ to $(-\infty,0)$, and cut and discard the remainder
of the forested line, and proceed likewise for the right boundary.. This gives
a beaded quantum surface. For $s>0$ such that the generalized quantum lengths
to the left and right of $0$ are at least $s$, let $p_{s}$ and $q_{s}$ be the
forested boundary points to the left and right of $0$ having generalized
quantum length $s$ from $0$. We mate the forested lines until we have
identified $p_{s}$ and $q_{s}$, obtaining a curve-decorated beaded quantum
surface (with the curve being the interface). We embed the curve-containing
connected component via the hydrodynamic normalization, to get
$(\mathbbm{H},\hat{\phi}_{s},\hat{\eta}_{s})$. That is, if $\hat{g}_{s}$ is
the conformal map from $\mathbbm{H}$ to the unbounded connected component of
$\mathbbm{H}\backslash\hat{\eta}_{s}$ satisfying
$\lim_{z\to\infty}\hat{g}_{s}(z)-z=0$, then
$\hat{g}_{s}^{-1}\bullet_{\gamma}\hat{\phi}_{s}=\phi_{0}$.

We reparametrize the process $(\hat{\phi}_{s},\hat{\eta}_{s},\hat{g}_{s})$
according to half-plane capacity to get $(\phi_{t},\eta_{t},g_{t})$ such that
$\mathrm{hcap}(\eta_{t})=2t$. Let $W_{t}$ be the endpoint of $\eta_{t}$ lying
in $\mathbbm{R}$. The _continuation threshold_ is the first time $t\leq\infty$
that any neighborhood of $W_{t}$ has infinite generalized quantum length.

Figure 2:  Let $\kappa\in(4,8)$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. Left: We
sample a Liouville field $\phi_{0}$ and attach independent forested lines to
the left and right boundary arcs. Marked points are not depicted. Right:
Mating the forested boundary arcs corresponds to “zipping up the quantum
zipper”. The curve $\eta_{t}$ is the interface in $\mathbbm{H}$ between the
purple and green forests. The conformal map $g_{t}$ sends the pink region on
the left to that on the right and satisfies $\lim_{z\to\infty}g_{t}(z)-z=0$.

######  Theorem 1.5.

For $\kappa\in(4,8)$ and $\gamma=\frac{4}{\sqrt{\kappa}}$, the conclusions of
Theorems 1.1 and 1.2 hold for the process $(\phi_{t},\eta_{t})$ constructed
immediately above.

###  1.4 The $\kappa>8$ LCFT quantum zipper

In this section, we describe a mating of correlated continuum random trees.
Let $(X_{t})_{t\geq 0}$ and $(Y_{t})_{t\geq 0}$ be correlated Brownian motions
with $\operatorname{Var}(X_{t})=\operatorname{Var}(Y_{t})=\mathbbm{a}^{2}t$
and
$\operatorname{Cov}(X_{t},Y_{t})=-\mathbbm{a}^{2}\cos(\frac{4\pi}{\kappa})t$
where $\mathbbm{a}^{2}=2/\sin(\frac{4\pi}{\kappa})$. We can construct a
_continuum random tree_ from $(X_{t})_{t\geq 0}$ by plotting the graph of
$t\mapsto X_{t}$, letting $S\subset\mathbbm{R}^{2}$ be the set of points lying
on or below the graph, and identifying points in $S$ which lie on a horizontal
chord which stays below the graph; see Figure 3. In the same way we may
construct a continuum random tree from $(Y_{t})_{t\geq 0}$, so
$(X_{t},Y_{t})_{t\geq 0}$ describes a pair of correlated continuum random
trees. We write $\mathrm{CRT}_{\kappa}$ for the law of $(X_{t},Y_{t})_{t\geq
0}$.

Figure 3:  Left: Plot the graphs of the correlated Brownian motions $X_{t}$
and $Y_{t}$ (orange and green) and draw horizontal segments below the graphs
(black). Right: Identifying points on the same horizontal segment gives a pair
of correlated continuum random trees.

Recall that when $\kappa\geq 8$, the $\operatorname{SLE}_{\kappa}$ curve is
space-filling. In this regime, the seminal mating-of-trees theorem of
Duplantier, Miller and Sheffield can be stated as follows. See Figure 4.

######  Proposition 1.6.

Let $\kappa\geq 8$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. Let
$(\mathbbm{H},\phi,0,\infty,\eta)$ be an embedding of a weight
$(2-\frac{\gamma^{2}}{2})$ quantum wedge decorated by an independent
$\operatorname{SLE}_{\kappa}$ curve $\eta$. Parametrize $\eta$ by quantum
area, so $A_{\phi}(\eta([0,t]))=t$. On the counterclockwise (resp. clockwise)
boundary arc of $\eta([0,t])$ from $0$ to $\eta(t)$, let $X_{t}^{-}$ and
$X_{t}^{+}$ (resp. $Y_{t}^{-}$ and $Y_{t}^{+}$) be the quantum lengths of the
boundary segments in $\mathbbm{R}$ and $\mathbbm{H}$ respectively. Then the
law of $(X_{t},Y_{t}):=(X_{t}^{+}-X_{t}^{-},Y_{t}^{+}-Y_{t}^{-})$ is
$\mathrm{CRT}_{\kappa}$. Moreover, the curve-decorated quantum surface
$(\mathbbm{H},\phi,0,\infty,\eta)/{\sim_{\gamma}}$ is measurable with respect
to $(X_{t},Y_{t})_{t\geq 0}$.

The measurability statement and the fact that $(X_{t},Y_{t})$ evolve as
correlated Brownian motion was shown in [DMS21], the correlation
$\cos(\frac{\pi\gamma^{2}}{4})$ was obtained in [GHMS17], and the value of
$\mathbbm{a}^{2}$ was identified in [ARS22]. (For the $\kappa\in(4,8)$ mating-
of-trees theorem of [DMS21], see Proposition 7.6.)

Figure 4:  Left: A pair of correlated continuum random trees. Right: There is
a way of mating the pair of CRTs to get a weight $(2-\frac{\gamma^{2}}{2})$
quantum wedge decorated by an independent (space-filling)
$\operatorname{SLE}_{\kappa}$ curve. Middle: Running the mating procedure
until the quantum area is $t$, we write
$X_{t}^{-},X_{t}^{+},Y_{t}^{-},Y_{t}^{+}$ for the quantum lengths of the red,
orange, blue and green boundary arcs respectively, and set
$(X_{t},Y_{t})=(X_{t}^{+}-X_{t}^{-},Y_{t}^{+}-Y_{t}^{-})$.

We can now state the quantum zipper for the Liouville field with correlated
CRTs glued to the boundary, see Figure 5. Suppose $\kappa\geq 8$ and
$\gamma=\frac{4}{\sqrt{\kappa}}$. Sample $(\phi_{0},(X_{t},Y_{t})_{t\geq
0})\sim{{\mathrm{LF}}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}\times\mathrm{CRT}_{\kappa}$
and let $X_{s}^{-}=-\inf_{u\leq s}X_{u}$ and $Y_{s}^{-}=-\inf_{u\leq s}Y_{u}$.
For $s>0$ such that $L_{\phi_{0}}((-\infty,0))>X_{s}^{-}$ and
$L_{\phi_{0}}((0,\infty))>Y_{s}^{-}$, let $a_{s}\in(-\infty,0)$ and
$b_{s}\in(0,\infty)$ satisfy $L_{\phi_{0}}((a_{s},0))=X_{s}^{-}$ and
$L_{\phi_{0}}((0,b_{s}))=Y_{s}^{-}$. Mate the pair of continuum random trees
for $s$ units of quantum area to get a quantum surface with boundary arcs of
lengths $X_{s}^{-},X_{s}^{+},Y_{s}^{-},Y_{s}^{+}$, and conformally weld the
boundary arcs of lengths $X_{s}^{-}$ and $Y_{s}^{-}$ to the boundary arcs
$(a_{s},0)$ and $(0,b_{s})$ of $(\mathbbm{H},\phi_{0})$ respectively. Embed
the resulting curve-decorated quantum surface via the hydrodynamic
normalization to get $(\mathbbm{H},\hat{\phi}_{s},\hat{\eta}_{s})$. That is,
if $\hat{g}_{s}$ is the conformal map from $\mathbbm{H}$ to the unbounded
connected component of $\mathbbm{H}\backslash\hat{\eta}_{s}$ satisfying
$\lim_{z\to\infty}\hat{g}_{s}(z)-z=0$, then
$\hat{g}_{s}^{-1}\bullet_{\gamma}\hat{\phi}_{s}=\phi_{0}$.

We reparametrize the process $(\hat{\phi}_{s},\hat{\eta}_{s},\hat{g}_{s})$
according to half-plane capacity to get $(\phi_{t},\eta_{t},g_{t})$ such that
$\mathrm{hcap}(\eta_{t})=2t$. Let $W_{t}$ be the endpoint of $\eta_{t}$ lying
in $\mathbbm{R}$. The continuation threshold is the first time $t$ that any
neighborhood of $W_{t}$ in $\mathbbm{R}$ has infinite quantum length.

Figure 5:  Let $\kappa\geq 8$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. Left: We
sample a Liouville field $\phi_{0}$ and attach correlated continuum random
trees to the left and right boundaries. Marked points are not depicted. Right:
Mating the trees corresponds to “zipping up the quantum zipper”. The curve
$\eta_{t}$ is the interface in $\mathbbm{H}$ between the orange and green
trees. The conformal map $g_{t}$ sends the pink region on the left to that on
the right.

######  Theorem 1.7.

For $\kappa>8$ and $\gamma=\frac{4}{\sqrt{\kappa}}$, the conclusions of
Theorems 1.1 and 1.2 hold for the process $(\phi_{t},\eta_{t})$ constructed
immediately above.

######  Remark 1.8.

Theorem 1.7 should also hold for $\kappa=8$ but our argument does not apply to
this critical value, see Remark 5.11. We leave the $\kappa=8$ LCFT quantum
zipper as an open problem.

######  Remark 1.9.

Suppose $\kappa\in(4,8)$ and $\gamma=\frac{4}{\sqrt{\kappa}}$, and sample
$(\phi_{0},(X_{t},Y_{t})_{t\geq 0})$ from
${{\mathrm{LF}}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}\times\mathrm{CRT}_{\kappa}$.
For $s>0$, mating the continuum random trees for $s$ units of quantum area
gives a beaded quantum surface, which a.s. produces a non-simply-connected
quantum surface when conformally welded with $(\mathbbm{H},\phi_{0})$. At the
exceptional times $s$ that $X_{s}^{+}=Y_{s}^{+}=0$, however, the conformal
welding yields a simply connected quantum surface. Restricting to these times
and replacing the space-filling interfaces with non-space-filling curves
measurable with respect to them, the result is the process described by
Theorem 1.5. See Section 7.3 for more details.

###  1.5 BPZ equation for Liouville conformal field theory

We now give an application of the quantum zipper to LCFT, by proving a novel
Belavin-Polyakov-Zamolodchikov (BPZ) equation for correlation functions with a
degenerate boundary insertion.

In this section we distinguish bulk and boundary insertions with different
letters. Let $m,n\geq 0$. Let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\mathbbm{H}$ for $j\leq m$ and assume
the $z_{j}$ are distinct. Let
$-\infty=x_{0}<x_{1}<\dots<x_{n}<x_{n+1}=+\infty$ be boundary points, let
$\beta_{1},\dots,\beta_{n}\in\mathbbm{R}$, and let $\delta\in\mathbbm{R}$. Let
$\beta_{*}\in\\{-\frac{\gamma}{2},-\frac{2}{\gamma}\\}$.

For $k=0,\dots,n$ let $I_{k}=(x_{k},x_{k+1})$. Fix an index
$k_{*}\in\\{0,\dots,n\\}$ and let $w\in I_{k_{*}}$. Let $I_{L}=(x_{k_{*}},w)$
and $I_{R}=(w,x_{k_{*}+1})$. See Figure 6. For $k\neq k_{*}$ let
$\mu_{k}\in\mathbbm{C}$ satisfy $\operatorname{Re}\mu_{k}\geq 0$, and assume
$\mu_{L},\mu_{R}\in\mathbbm{C}$ satisfy
$\operatorname{Re}\mu_{L},\operatorname{Re}\mu_{R}\geq 0$ and are defined in
terms of $\sigma_{L},\sigma_{R}\in\mathbbm{C}$ as follows:

$\mu_{L}=g(\sigma_{L}),\,\,\mu_{R}=g(\sigma_{R})\quad\text{ where
}g(\sigma)=\frac{\cos(\pi\gamma(\sigma-\frac{Q}{2}))}{\sqrt{\sin(\pi\gamma^{2}/4)}}\text{
and }\sigma_{L}-\sigma_{R}=\pm\frac{\beta_{*}}{2}.$ (1.3)

Figure 6:  Marked boundary points on $\partial\mathbbm{H}$ and the intervals
between them. The bulk insertions are $(\alpha_{j},z_{j})_{j}$ (not depicted),
the boundary insertions are
$(\frac{\beta_{*}}{2},w),(\frac{\delta}{2},\infty)$ and
$(\frac{\beta_{k}}{2},x_{k})$ for $1\leq k\leq n$. Each boundary interval
$I_{\bullet}$ has an associated boundary cosmological constant
$\mu_{\bullet}$.

Suppose the _Seiberg bounds_ hold:

$\sum_{j}\alpha_{j}+\sum_{k}\frac{\beta_{k}}{2}+\frac{\delta}{2}+\frac{\beta_{*}}{2}>Q,\qquad\alpha_{j},\beta_{k}<Q\text{
for all }j,k,\quad\delta<Q.$ (1.4)

We define the LCFT correlation function
$F_{\beta_{*}}(w,(z_{j})_{j},(x_{k})_{k})$ to be

${{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{\beta_{*}}{2},w),(\alpha_{j},z_{j})_{j},(\frac{\beta_{k}}{2},x_{k}),(\frac{\delta}{2},\infty)}[\exp(-\mathcal{A}_{\phi}(\mathbbm{H})-\sum_{\begin{subarray}{c}0\leq
k\leq n\\\ k\neq
k_{*}\end{subarray}}\mu_{k}\mathcal{L}_{\phi}(I_{k})-\mu_{L}\mathcal{L}_{\phi}(I_{L})-\mu_{R}\mathcal{L}_{\phi}(I_{R}))].$
(1.5)

Because the Seiberg bounds hold, the integral (1.5) converges absolutely
[HRV18, Theorem 3.1].

######  Theorem 1.10.

For $(\gamma,\beta_{*})\neq(\sqrt{2},-\frac{\gamma}{2})$, the correlation
functions $F_{\beta_{*}}$ are smooth and satisfy the BPZ equation

$\mathopen{}\mathclose{{}\left(\frac{1}{\beta_{*}^{2}}\partial_{ww}+\sum_{j}(\frac{1}{w-z_{j}}\partial_{z_{j}}+\frac{1}{w-\overline{z}_{j}}\partial_{\overline{z}_{j}})+\sum_{k}\frac{1}{w-x_{k}}\partial_{x_{k}}+\sum_{j}\operatorname{Re}\frac{2\Delta_{\alpha_{j}}}{(w-z_{j})^{2}}+\sum_{k}\frac{\Delta_{\beta_{k}}}{(w-x_{k})^{2}}}\right)F_{\beta_{*}}=0.$

If we assume the $\kappa=8$ quantum zipper not proved in this paper, the proof
of Theorem 1.10 carries through to the remaining case
$(\gamma,\beta_{*})=(\sqrt{2},-\frac{\gamma}{2})$.

Here is a proof sketch. Consider the quantum zipper with
$\kappa=\frac{4}{\beta_{*}^{2}}$, and let $A_{t},L_{t},R_{t}$ be the quantum
area and left and right quantum boundary lengths at time $t$. For
$\beta_{*}=-\frac{2}{\gamma}$ and $\kappa\leq 4$, the coupling of $\sigma_{L}$
and $\sigma_{R}$ gives $\mu_{L}+\mu_{R}=0$, so
$\mu_{L}L_{t}+\mu_{R}R_{t}=\mu_{L}(L_{t}-R_{t})$. This quantity is invariant
under the conformal welding zipper process, giving rise to an
$\operatorname{SLE}_{\kappa}$ martingale from which the BPZ equation is
immediate. For $\beta_{*}=-\frac{\gamma}{2}$ and $\kappa>4$, although
$e^{-A_{t}-\mu_{L}L_{t}-\mu_{R}R_{t}}$ is not invariant, it evolves as a
martingale (Lemma 7.3); the proof uses the mating-of-trees Brownian motion and
requires the coupling (1.3). This again gives an $\operatorname{SLE}_{\kappa}$
martingale and thus the BPZ equation. These arguments only yield that
$F_{\beta_{*}}$ is a weak solution, but a hypoellipticity argument following
[Dub15, PW19] then implies that $F_{\beta_{*}}$ is smooth, completing the
proof.

In their pioneering work, Belavin, Polyakov and Zamolodchikov used
representation theoretic methods to derive BPZ equations for the sphere
[BPZ84]. These were recently mathematically proved for LCFT [KRV19] using a
subtle argument involving cancellations of not absolutely convergent
integrals. See also [Rem20, RZ20, RZ22] for BPZ equations on the disk with
bulk cosmological constant zero, that is, in (1.5) the term
$-\mathcal{A}_{\phi}(\mathbbm{H})$ is removed.

When the bulk cosmological constant is nonzero, the BPZ equations do not hold
unless there is a coupling of the cosmological constants $\mu_{L}$ and
$\mu_{R}$ via (1.3). This was proposed by [FZZ00] after examining special
cases. As far as we know, there was no prior conceptual explanation of this
constraint, even in the physics literature. Our martingale argument explains
(1.3).

######  Remark 1.11.

The statement of Theorem 1.10 was chosen for simplicity. Our argument is quite
robust, and many of the conditions can be loosened.

  * •

We can choose $\beta_{k}\geq Q$ if the boundary cosmological constants for the
intervals adjacent to $x_{k}$ are zero. We can choose $\delta\geq Q$ if
$\mu_{0}=\mu_{n}=0$.

  * •

The condition that
$\operatorname{Re}\mu_{k},\operatorname{Re}\mu_{L},\operatorname{Re}\mu_{R}\geq
0$ can be relaxed so long as (1.5) converges absolutely.

  * •

The bound
$\sum_{j}\alpha_{j}+\sum_{k}\frac{\beta_{k}}{2}+\frac{\delta}{2}+\frac{\beta_{*}}{2}>Q$
is needed to ensure convergence of (1.5). Sometimes the correlation function
can be defined even when (1.5) is nonconvergent, by introducing _truncations_
where the exponential in (1.5) is replaced by $e^{z}-1$ or $e^{z}-1-z$ for
instance. See e.g. [AHS22, Proposition 3.4], [ARS22, (1.7)] and [Cer22,
Theorem B].

We note that the smoothness result of Theorem 1.10 is itself rather
nontrivial. [KRV19] proved the bulk correlation functions are $C^{2}$ using
mollifications of the GFF, and this was extended to $C^{\infty}$ by [Oik19].
It is not clear to us whether these arguments can be adapted for boundary
LCFT.

###  1.6 Outlook

We state here a few applications of the LCFT quantum zipper, and mention some
future directions and open questions.

####  1.6.1 Integrability of boundary LCFT

The basic objects of conformal field theories are their correlation functions,
and solving a conformal field theory means obtaining exact formulae for them.
For the case of LCFT on surfaces without boundary, this was carried out in a
series of landmark works that proved the three-point structure constant equals
the DOZZ formula proposed in physics [KRV20], then made rigorous the conformal
bootstrap program of physicists to recursively solve for all correlation
functions on all surfaces [GKRV22, GKRV21].

A similar program is currently being carried out for LCFT on surfaces with
boundary. With Remy and Sun we computed the one-point bulk structure constant
using mating-of-trees [ARS22]. Taking the BPZ equation Theorem 1.10 as input,
in future work with Remy, Sun and Zhu [ARSZ] we compute the boundary three-
point structure constant and the bulk-boundary structure constant, making
rigorous the formulas of Ponsot-Teschner [PT02] and Hosomichi [Hos01]
respectively. Our work provides all the initial data needed to compute all
correlation functions on surfaces with boundary. The boundary conformal
bootstrap program has been initiated, see [Wu22].

####  1.6.2 Reversibility of whole plane $\operatorname{SLE}_{\kappa}$ when
$\kappa>8$

For $\kappa\in(0,8]$, chordal SLE is reversible in the following sense. Let
$f:\mathbbm{H}\to\mathbbm{H}$ be a conformal automorphism with $f(0)=\infty$
and $f(\infty)=0$. If $\eta$ is $\operatorname{SLE}_{\kappa}$ in $\mathbbm{H}$
from $0$ to $\infty$, then the time-reversal of $f\circ\eta$ has the same law
as $\eta$ up to time-reparametrization [Zha08, MS16]. However, reversibility
fails for $\kappa>8$ chordal $\operatorname{SLE}_{\kappa}$ [MS17].

_Whole-plane SLE_ is a variant of SLE in $\mathbbm{C}$ that starts at $\infty$
and targets $0$. Its reversibility was established by [Zha15] for $\kappa\leq
4$ and [MS17] for $\kappa\in(4,8]$. For $\kappa>8$ the reversibility of whole-
plane $\operatorname{SLE}_{\kappa}$ was conjectured in [VW20] via
reversibility of the $\kappa\to\infty$ large deviations rate function, which
they obtained from a field-foliation coupling they interpret as describing a
“$\kappa\to\infty$ radial mating-of-trees”. Inspired by [VW20], in future work
with Pu Yu we establish a radial mating-of-trees using the LCFT quantum
zipper, then exploit mating-of-trees reversibility and LCFT reversibility to
show whole-plane $\operatorname{SLE}_{\kappa}$ reversibility for $\kappa>8$.

####  1.6.3 A general theory of conformal welding in LCFT

It is known in many cases that conformally welding quantum surfaces described
by LCFT produces a quantum surface also described by LCFT. This was first
demonstrated for quantum wedges [She16], and many more cases followed [DMS21,
HP21, MSW22, MSW21, AHS23, AHS22, ARS22, AS21, ASY22]. In all cases the
welding interfaces are either chords or loops, and the surfaces have genus 0.
In forthcoming work with Pu Yu, we will use the LCFT quantum zipper to obtain
radial conformal weldings and subsequently develop a systematic framework that
produces conformal weldings of quantum surfaces of arbitrary genus.

####  1.6.4 Other BPZ equations in random conformal geometry

Our argument uses the mating-of-trees framework to prove the boundary BPZ
equation for LCFT on the disk. The BPZ equation for LCFT on the sphere has
already been shown [KRV19], but can something similar to our proof of Theorem
1.10 give an alternative proof?

The _conformal loop ensemble (CLE)_ [She09, SW12] is a canonical conformally
invariant collection of loops that locally look like SLE [She09], and arises
as the scaling limit of the collection of interfaces of statistical physics
models. It is expected that CLE is described by a CFT – indeed a suitably-
defined CLE three-point function agrees with the _generalized minimal model_
CFT structure constant [AS21] – so CLE multipoint functions should satisfy the
BPZ equations. In the present work we obtained the boundary LCFT BPZ equation
from a BPZ equation for SLE (in the sense that using Itô calculus on the
martingale of Lemma 3.1 gives a second-order differential equation resembling
BPZ), using mating-of-trees. One might hope to turn this argument around:
could a BPZ equation for CLE be obtained from the BPZ equation for LCFT
[KRV19] using mating-of-trees?

Organization of the paper. In Section 2 we recall some preliminaries about
LQG, LCFT, SLE and the mating-of-trees framework. In Section 3 we explain that
reverse $\operatorname{SLE}_{\kappa,\rho}$ is reverse
$\operatorname{SLE}_{\kappa}$ weighted by a GFF partition function. In Section
4 we adapt a Neumann GFF quantum zipper of [DMS21] to obtain the $\kappa\leq
4$ LCFT quantum zipper. In Section 5 we prove the $\kappa>8$ LCFT quantum
zipper, and in Section 6 the $\kappa\in(4,8)$ LCFT quantum zipper. Finally, we
establish the LCFT boundary BPZ equations in Section 7.

Acknowledgements. We thank Guillaume Remy, Xin Sun and Tunan Zhu for earlier
discussions on alternative approaches towards proving the boundary LCFT BPZ
equations. We thank Yoshiki Fukusumi, Xin Sun, Yilin Wang and Pu Yu for
helpful discussions. The author was supported by the Simons Foundation as a
Junior Fellow at the Simons Society of Fellows.

##  2 Preliminaries

###  2.1 The Gaussian free field and Liouville quantum gravity

Let $m$ be the uniform probability measure on the unit half-circle
$\\{z\>:\>z\in\mathbbm{H},|z|=1\\}$. The Dirichlet inner product is defined by
$\langle f,g\rangle_{\nabla}=(2\pi)^{-1}\int_{\mathbbm{H}}\nabla f\cdot\nabla
g$. Consider the collection of smooth functions $f$ with $\langle
f,f\rangle_{\nabla}<\infty$ and $\int f(z)m(dz)=0$, and let $H$ be its Hilbert
space closure with respect to $\langle\cdot,\cdot\rangle_{\nabla}$. Let
$(f_{n})$ be an orthonormal basis of $H$ and let $(\alpha_{n})$ be a
collection of independent standard Gaussians. The summation

$h=\sum_{n}\alpha_{n}f_{n}$

a.s. converges in the space of distributions. Then $h$ is the _Gaussian free
field on $\mathbbm{H}$ normalized so $\int h(z)m(dz)=0$_ [DMS21, Section
4.1.4].

Write $|z|_{+}:=\max(|z|,1)$. For $z,w\in\overline{\mathbbm{H}}$ we define

$\begin{gathered}G_{\mathbbm{H}}(z,w)=-\log|z-w|-\log|z-\overline{w}|+2\log|z|_{+}+2\log|w|_{+},\\\
G_{\mathbbm{H}}(z,\infty)=\lim_{w\to\infty}G_{\mathbbm{H}}(z,w)=2\log|z|_{+}.\end{gathered}$
(2.1)

The GFF $h$ is the centered Gaussian field with covariance structure formally
given by $\mathbbm{E}[h(z)h(w)]=G_{\mathbbm{H}}(z,w)$. This is formal because
$h$ is a distribution and so does not admit pointwise values, but for smooth
compactly supported functions $f,g$ on $\mathbbm{H}$ we have
$\mathbbm{E}[(h,f)(h,g)]=\iint G_{\mathbbm{H}}(z,w)f(z)g(w)\,dz\,dw$.

Suppose $\phi=h+g$ where $g$ is a (random) function on $\mathbbm{H}$ which is
continuous at all but finitely many points. Let $\phi_{\varepsilon}(z)$ denote
the average of $\phi$ on $\partial B_{\varepsilon}(z)\cap\mathbbm{H}$. For
$\gamma\in(0,2)$, the $\gamma$-LQG area measure $\mathcal{A}_{\phi}$ on
$\mathbbm{H}$ can be defined by the almost sure weak limit
$\mathcal{A}_{\phi}(dz)=\lim_{\varepsilon\to
0}\varepsilon^{\gamma^{2}/2}e^{\gamma\phi_{\varepsilon}(z)}dz$ [DS11].
Similarly, the $\gamma$-LQG boundary length measure $\mathcal{L}_{\phi}$ on
$\mathbbm{R}$ can be defined by $\mathcal{L}_{\phi}(dx):=\lim_{\varepsilon\to
0}\varepsilon^{\gamma^{2}/4}e^{\frac{\gamma}{2}\phi_{\varepsilon}(x)}dx$. For
the critical parameter $\gamma=2$ a correction is needed to make the measure
nonzero; we set $\mathcal{A}_{\phi}(dz)=\lim_{\varepsilon\to
0}(\log(1/\varepsilon)-\phi_{\varepsilon}(z))\varepsilon^{2}e^{2\phi_{\varepsilon}(z)}dz$
and $\mathcal{L}_{\phi}(dz)=\lim_{\varepsilon\to
0}(\log(1/\varepsilon)-\frac{1}{2}\phi_{\varepsilon}(x))\varepsilon
e^{\phi_{\varepsilon}(x)}dx$ [DRSV14]. See for instance [Lac22, Pow20] for
more details.

###  2.2 The Liouville field

Let $\gamma\in(0,2]$ be the LQG parameter, and
$Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Write $|z|_{+}:=\max(|z|,1)$.

######  Definition 2.1.

Let $(h,\mathbf{c})$ be sampled from $P_{\mathbbm{H}}\times[e^{-Qc}\,dc]$ and
let $\phi(z)=h(z)-2Q\log|z|_{+}+\mathbf{c}$. We call $\phi$ the _Liouville
field on $\mathbbm{H}$_ and denote its law by ${{\mathrm{LF}}}_{\mathbbm{H}}$.

Define

$C_{\gamma}^{(\alpha,z)}=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ll}|z|_{+}^{-2\alpha(Q-\alpha)}(2\operatorname{Im}z)^{-\alpha^{2}/2}&\mbox{if
}z\in\mathbbm{H}\\\ |z|_{+}^{-2\alpha(Q-\alpha)}&\mbox{if
}z\in\mathbbm{R}\end{array}}\right..$

For $\delta\in\mathbbm{R}$, $n\geq 0$ and
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ for $j\leq n$
such that the $z_{j}$ are distinct, let

$C_{\gamma}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}=\prod_{j}C_{\gamma}^{(\alpha_{j},z_{j})}e^{\alpha_{j}\delta
G_{\mathbbm{H}}(z_{j},\infty)}\times\prod_{1\leq j<k\leq
n}e^{\alpha_{j}\alpha_{k}G_{\mathbbm{H}}(z_{j},z_{k})}.$ (2.2)

More generally, if the $(\alpha_{j},z_{j})$ are such that the $z_{j}$ are not
distinct, we combine all pairs $(\alpha,z)$ with the same $z$ by summing their
$\alpha$’s to get a collection $(\alpha^{\prime}_{j},z_{j}^{\prime})$ where
the $z_{j}^{\prime}$ are distinct, and define
$C_{\gamma}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}=C_{\gamma}^{(\alpha_{j}^{\prime},z_{j}^{\prime})_{j},(\delta,\infty)}$.

######  Definition 2.2.

For $\delta\in\mathbbm{R}$, $n\geq 0$ and
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$, let
$(h,\mathbf{c})$ be sampled from
$C^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}_{\gamma}P_{\mathbbm{H}}\times[e^{(\sum_{j}\alpha_{j}+\delta-Q)c}\,dc]$,
and set

$\phi(z)=h(z)+\sum_{j}\alpha_{j}G_{\mathbbm{H}}(z,z_{j})+(\delta-Q)G_{\mathbbm{H}}(z,\infty)+\mathbf{c}.$

We call $\phi$ the _Liouville field with insertions $(\alpha_{j},z_{j})_{j}$,
$(\delta,\infty)$_ and we write
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}$ for
its law.

Note that
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}$
depends implicitly on the parameter $\gamma$. When $\delta=0$ there is no
insertion at $\infty$, so we write $C_{\gamma}^{(\alpha_{j},z_{j})_{j}}$ and
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j}}$ rather than
$C_{\gamma}^{(\alpha_{j},z_{j})_{j},(0,\infty)}$ and
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(0,\infty)}$.

######  Remark 2.3.

When $\sum_{j}\alpha_{j}+\delta=Q$, Definition 2.2 has the following
simplification. Sample $(h,\mathbf{c})\sim
C^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}_{\gamma}P_{\mathbbm{H}}\times dc$,
and let

$\phi(z)=h(z)+\sum_{j}\alpha_{j}G(z,z_{j})+\mathbf{c}$ (2.3)

where $G(z,w)=-\log|z-w|-\log|z-\overline{w}|$. Then the law of $\phi$ is
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}$.
Moreover, we have the simplification
$C^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}_{\gamma}=\mathcal{Z}((\alpha_{j},z_{j})_{j})$
with $\mathcal{Z}$ defined in and below (1.1).

######  Remark 2.4.

Let $m,n\geq 0$, let $(\alpha_{j},z_{j})\in\mathbbm{R}\times\mathbbm{H}$ for
$j\leq m$, let $(\beta_{k},x_{k})\in\mathbbm{R}\times\mathbbm{R}$ for $k\leq
n$, and let $\beta\in\mathbbm{R}$. The measure that we call
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\frac{1}{2}\beta_{k},x_{k})_{k},(\frac{1}{2}\beta,\infty)}$
is instead called
“$\mathrm{LF}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\beta_{k},x_{k})_{k},(\beta,\infty)}$”
in [AHS22, ARS22]; there, the boundary insertions are described by their log-
singularities (near $x_{k}$ the field blows up as $-\beta_{k}\log|\cdot-
x_{k}|$), whereas our notation instead uses the Green function coefficient
(near $x_{k}$ the field blows up like
$\frac{\beta_{k}}{2}G_{\mathbbm{H}}(\cdot,x_{k})$). Our notation is more
convenient for this paper since the Green function coefficient is invariant
when a boundary point is zipped into the bulk.

Finally, we will need the following rooted measure statement: sampling a point
from the quantum area measure of a Liouville field is equivalent to adding a
$\gamma$-insertion to the field.

######  Proposition 2.5.

Suppose $\gamma\in(0,2)$. Let $n\geq 0$ and
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ for $j\leq n$,
and let $\delta\in\mathbbm{R}$. Then

$\mathcal{A}_{\phi}(dz)\,{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}(d\phi)={{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\gamma,z),(\delta,\infty)}(d\phi)\,dz.$

Formally, Proposition 2.5 holds because
$\mathcal{A}_{\phi}(dz)=\lim_{\varepsilon\to
0}\varepsilon^{\gamma^{2}/2}e^{\gamma\phi_{\varepsilon}(z)}dz$ and
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\gamma,z),(\delta,\infty)}(d\phi)=\lim_{\varepsilon\to
0}\varepsilon^{\gamma^{2}/2}e^{\gamma\phi_{\varepsilon}(z)}{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},z_{j})_{j},(\delta,\infty)}(d\phi)$,
where $\phi_{\varepsilon}(z)$ is the average of $\phi$ on $\partial
B_{\varepsilon}(z)\cap\mathbbm{H}$. A rigorous proof of the analogous
statement for $\mathbbm{C}$ is given in [AHS22, Lemma 2.31]; the proof of
Proposition 2.5 is identical so we omit it.

###  2.3 Reverse Schramm-Loewner evolution

In this section we recall some basic properties of SLE, see [Kem17] for a more
detailed introduction. Let $\kappa>0$. Let
$\rho_{1},\dots,\rho_{n}\in\mathbbm{R}$ and let
$z_{1},\dots,z_{n}\in\overline{\mathbbm{H}}$ be distinct points. With
$(B_{t})_{t\geq 0}$ a standard Brownian motion, the driving function
$(W_{t})_{t\geq 0}$ for reverse $\operatorname{SLE}_{\kappa,\rho}$ is defined
by the stochastic differential equations

$\begin{gathered}W_{0}=0,\quad
dW_{t}=\sum_{j=1}^{n}\operatorname{Re}\mathopen{}\mathclose{{}\left(\frac{-\rho_{j}}{Z^{j}_{t}-W_{t}}}\right)\,dt+\sqrt{\kappa}\,dB_{t},\\\
Z^{j}_{0}=z_{j},\quad dZ^{j}_{t}=-\frac{2}{Z^{j}_{t}-W_{t}}\,dt\quad\text{for
}j=1,\dots,n.\end{gathered}$ (2.4)

For each $z\in\mathbbm{H}$ and $s\geq 0$, let $(g_{s,t}(z))_{t\geq s}$ be the
solution to $g_{s,s}(z)=z$,
$\frac{d}{dt}g_{s,t}(z)=-\frac{2}{g_{s,t}(z)-W_{t}}$. This defines a family of
conformal maps $g_{s,t}$; we also write $g_{t}:=g_{0,t}$. For each $t>0$ we
can define a curve $\eta_{t}:[0,t]\to\overline{\mathbbm{H}}$ by
$\eta_{t}(u):=\lim_{z\to W_{t-u}}g_{t-u,t}(z)$. We call the family of curves
$(\eta_{t})_{t\geq 0}$ _reverse $\operatorname{SLE}_{\kappa}$ with force
points at $z_{j}$ of weight $\rho_{j}$_. Note that $g_{t}$ is the unique
conformal map from $\mathbbm{H}$ to the unbounded connected component of
$\mathbbm{H}\backslash\eta_{t}$ such that $\lim_{z\to\infty}g_{t}(z)-z=0$, and
that reverse SLE is parametrized by half-plane capacity in the sense that
$\mathrm{hcap}(\eta_{t}):=\lim_{z\to\infty}z(g_{t}(z)-z)$ equals $2t$. The
curve $\eta_{t}$ is simple when $\kappa\leq 4$, self-intersecting (but not
self-crossing) when $\kappa\in(4,8)$, and space-filling when $\kappa\geq 8$.

Figure 7:  Reverse $\operatorname{SLE}_{\kappa}$ for $\kappa\leq 4$; similar
diagrams with different topologies can be drawn for $\kappa\in(4.8)$ and
$\kappa\geq 8$. The curve grows from its base. As depicted,
$\eta_{t}(0)=W_{t}$ and $\eta_{t}(t)=\lim_{z\to W_{0}}g_{t}(z)$.

In the case of no force points ($n=0$), this process is well-defined for all
time. For $n\geq 0$, by absolute continuity with respect to the $n=0$ case,
the SDE (2.4) can be run until the time $\tau$ that $g_{t}(z_{j})=W_{t}$ for
some $j$. Let $J_{\tau}=\\{j\>:\>g_{\tau}(z_{j})=W_{\tau}\\}$. If $\sum_{j\in
J_{\tau}}\rho_{j}<\frac{\kappa}{2}+4$, then there is a unique way of
continuing the process such that $g_{t}(z_{j})\in\mathbbm{H}$ for all $t>\tau$
[DMS21, Proposition 3.8]. Applying this continuation procedure to each time a
force point hits $W_{t}$, we see that reverse
$\operatorname{SLE}_{\kappa,\rho}$ is well-defined until the _continuation
threshold_ : the first time $\tau\leq\infty$ that the sum of the weights of
force points hitting $W_{\tau}$ is at least $\frac{\kappa}{2}+4$.

###  2.4 Quantum wedges and quantum cells

In this section we define the _quantum wedges_ introduced in [She16, DMS21],
and the _quantum cell_ which arises from partially mating correlated continuum
random trees.

We first define quantum wedges. For symmetry reasons it is easier to work in
the strip $\mathcal{S}:=\mathbbm{R}\times(0,\pi)$ than in $\mathbbm{H}$. Let
$m$ be the uniform probability measure on the segment $\\{0\\}\times(0,\pi)$.
As in Section 2.1, consider the space of smooth functions $f$ on $\mathcal{S}$
with $\langle f,f\rangle_{\nabla}$ and $\int f(z)m(dz)=0$, and let $H$ be the
Hilbert space closure with respect to $\langle\cdot,\cdot\rangle_{\nabla}$. As
before, let $h=\sum_{j}\alpha_{j}f_{j}$ where the $\alpha_{j}$ are i.i.d.
standard Gaussians and $\\{f_{j}\\}$ is an orthonormal basis for $H$. We call
$h$ the _GFF on $\mathcal{S}$ normalized so $\int h(z)m(dz)=0$_, and denote
its law by $P_{\mathcal{S}}$.

We can decompose $H=H_{\mathrm{av}}\oplus H_{\mathrm{lat}}$ where
$H_{\mathrm{av}}$ (resp. $H_{\mathrm{lat}}$) is the subspace of functions
which are constant (resp. have mean zero) on $\\{t\\}\times(0,\pi)$ for all
$t\in\mathbbm{R}$; this gives a decomposition
$h=h_{\mathrm{av}}+h_{\mathrm{lat}}$ of $h$ into independent components.

We now define thick quantum wedges. These are $\gamma$-LQG surfaces with the
half-plane topology, and have a _weight_ parameter
$W\geq\frac{\gamma^{2}}{2}$.

######  Definition 2.6 (Thick quantum wedge).

For $W\geq\frac{\gamma^{2}}{2}$, let
$\alpha=\frac{1}{2}(Q+\frac{\gamma}{2}-\frac{W}{\gamma})$. Let

$Y_{t}=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ll}B_{2t}+(Q-2\alpha)t&\mbox{if
}t\geq 0\\\ \widetilde{B}_{-2t}+(Q-2\alpha)t&\mbox{if }t<0\end{array}}\right.$

where $(B_{s})_{s\geq 0}$ is standard Brownian motion, and independently
$(\widetilde{B}_{s})_{s\geq 0}$ is standard Brownian motion conditioned on
$\widetilde{B}_{2s}-(Q-2\alpha)s<0$ for all $s>0$. Let
$\phi_{\mathrm{av}}(z)=Y_{\operatorname{Re}z}$, let $\phi_{\mathrm{lat}}$ be
the projection of an independent GFF onto $H_{\mathrm{lat}}$, and let
$\phi_{0}=\phi_{\mathrm{av}}+\phi_{\mathrm{lat}}$. We call the quantum surface
$(\mathcal{S},\phi_{0},-\infty,+\infty)/{\sim_{\gamma}}$ a _weight $W$ quantum
wedge_ and write $\mathcal{M}^{\mathrm{wed}}(W)$ for its law.

Thin quantum wedges are an ordered collection of two-pointed quantum surfaces.

######  Definition 2.7 (Thin quantum wedge).

For $W\in(0,\frac{\gamma^{2}}{2})$, sample a Bessel process of dimension
$\delta=1+\frac{2W}{\gamma^{2}}$. It decomposes as a countable ordered
collection of Bessel excursions; for each excursion $e$ we construct a two-
pointed quantum surface
$\mathcal{B}_{e}=(\mathcal{S},\phi^{e},-\infty,\infty)/{\sim_{\gamma}}$ as
follows. Let $(Y_{t}^{e})_{t\in\mathbbm{R}}$ be any time-reparametrization of
$\frac{2}{\gamma}\log e$ which has quadratic variation $2dt$. Let
$\phi_{\mathrm{av}}^{e}(z)=Y_{\operatorname{Re}z}^{e}$, let
$\phi^{e}_{\mathrm{lat}}$ be the projection of a GFF (independent of
everything else) onto $H_{\mathrm{lat}}$, and set
$\phi^{e}=\phi_{\mathrm{av}}^{e}+\phi_{\mathrm{lat}}^{e}$. Then the _weight
$W$ quantum wedge_ is the ordered collection $(\mathcal{B}_{e})$.

Now, to avoid topological complications, assume $\gamma\in(0,\sqrt{2}]$ and
$\kappa=\frac{16}{\gamma^{2}}$. Let $(\mathbbm{H},h,0,\infty)$ be an embedding
of a sample from $\mathcal{M}^{\mathrm{wed}}(2-\frac{\gamma^{2}}{2})$. Let
$\eta$ be an independent $\operatorname{SLE}_{\kappa}$ from $0$ to $\infty$;
parametrize $\eta$ by quantum area so $\mathcal{A}_{h}(\eta([0,a]))=a$ for all
$a>0$. Let $D_{a}=\eta([0,a])$ and let $p=\eta(0)$ and $q=\eta(a)$. Let
$x_{L}$ (resp. $x_{R}$) be the last point on the left (resp. right) boundary
arc of $\mathbbm{H}$ hit by $\eta$ before time $a$.

######  Definition 2.8.

We call the $\operatorname{SLE}_{\kappa}$-decorated quantum surface
$\mathcal{C}_{a}=(D_{a},h,\eta|_{[0,a]},p,q,x_{L},x_{R})/{\sim_{\gamma}}$ an
_area $a$ quantum cell_. We denote its law by $P_{a}$.

Recall that there is a way of mating correlated continuum random trees to
obtain LQG decorated by SLE, see Proposition 1.6. By its definition, the area
$a$ quantum cell is the decorated quantum surface arising from mating a pair
of correlated Brownian motions $(X_{t},Y_{t})_{t\geq
0}\sim\mathrm{CRT}_{\kappa}$ and stopping when the quantum area is $a$.

######  Remark 2.9.

When $(\gamma,\kappa)\neq(\sqrt{2},8)$, the quantum cell $\mathcal{C}_{a}$ is
measurable with respect to $(D_{a},h,\eta|_{[0,a]})/{\sim_{\gamma}}$, that is,
the four marked points can be recovered from
$(D_{a},h,\eta|_{[0,a]})/{\sim_{\gamma}}$. First, we can recover two of the
marked points $p=\eta(0)$ and $q=\eta(a)$. Let $\psi:D_{a}\to\mathbbm{H}$ be a
conformal map with $\psi(p)=0$ and $\psi(q)=\infty$. From the imaginary
geometry construction of space-filling $\operatorname{SLE}_{\kappa}$ [MS17],
modulo time-parametrization the law of $\psi\circ\eta$ given $\psi(x_{L})$ and
$\psi(x_{R})$ is $\operatorname{SLE}_{\kappa,\rho}$ with force points of size
$\kappa/2-4$ at $\psi(x_{L})$ and $\psi(x_{R})$. Since $\kappa/2-4\neq 0$ the
points $\psi(x_{L})$ and $\psi(x_{R})$ are measurable with respect to
$\psi\circ\eta$, so we can recover $x_{L}$ and $x_{R}$ also. In this paper, we
will not consider the $(\gamma,\kappa)=(\sqrt{2},8)$ case, so for notational
simplicity we will frequently omit the marked points.

Since Brownian motion is reversible, it is unsurprising that the area $a$
quantum cell is reversible:

######  Lemma 2.10.

$\mathcal{C}_{a}$ is symmetric in law in the following sense. If
$\widetilde{\eta}$ is the time-reversal of $\eta|_{[0,a]}$, then
$(D_{a},h,\eta|_{[0,a]},p,q,x_{L},x_{R})/{\sim_{\gamma}}$ and
$(D_{a},h,\widetilde{\eta},q,p,x_{R},x_{L})/{\sim_{\gamma}}$ have the same
law.

###### Proof.

[DMS21, Theorem 1.9] states that if a _$\gamma$ -quantum cone
$(\mathbbm{C},h,0,\infty)/{\sim_{\gamma}}$_ is decorated by an independent
$\operatorname{SLE}_{\kappa}$ curve $\eta$ from $\infty$ to $\infty$, then
parametrizing $\eta:\mathbbm{R}\to\mathbbm{C}$ by quantum area and fixing
$\eta(0)=0$, the quantum surface
$(\eta((0,\infty)),h,0,\infty)/{\sim_{\gamma}}$ has law
$\mathcal{M}^{\mathrm{wed}}(2-\frac{\gamma^{2}}{2})$. Thus
$(\eta([0,a]),h,\eta|_{[0,a]})/{\sim_{\gamma}}$ has law $P_{a}$. Let
$z=\eta(a)$, then [DMS21, Theorem 1.9] states that
$(\mathbbm{C},h(\cdot+z),\eta(\cdot+a)-z)/{\sim_{\gamma}}$ also has the law of
a $\gamma$-quantum cone decorated by an independent
$\operatorname{SLE}_{\kappa}$ from $\infty$ to $\infty$, and by the
reversibility of this $\operatorname{SLE}_{\kappa}$ curve, the decorated
quantum surface $(\mathbbm{C},h(\cdot+z),\eta(a-\cdot)-z)/{\sim_{\gamma}}$
also has this law. Thus, $(\eta([0,a]),h,\tilde{\eta})$ also has law $P_{a}$,
where $\tilde{\eta}:[0,a]\to\mathbbm{C}$ is given by
$\tilde{\eta}(t)=\eta(a-t)$. ∎

##  3 Reverse $\operatorname{SLE}_{\kappa,\rho}$ and the GFF partition
function

We make no claim of originality for the material in this section, and present
it here for completeness. See the end of this section for references.

Let $n\geq 0$ and let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ for $j\leq n$,
where the $z_{j}$ are distinct. Let $w\in\mathbbm{R}$. Let $\kappa>0$,
$\gamma=\min(\sqrt{\kappa},\frac{4}{\sqrt{\kappa}})$. The GFF partition
function $\mathcal{Z}$ defined in (1.1), times a factor arising from
uniformizing in $\mathbbm{H}$, is a martingale observable for reverse SLE:

######  Lemma 3.1.

Let $\kappa>0$. Let $n\geq 0$, and let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ for $j\leq n$.
Let $I=\\{j:z_{j}\in\mathbbm{H}\\}$ and $B=\\{j:z_{j}\in\mathbbm{R}\\}$.
Sample reverse $\operatorname{SLE}_{\kappa}$ with no force points (as in
(2.4)) and define

$M_{t}=\prod_{i\in I}|g_{t}^{\prime}(z_{i})|^{2\Delta_{\alpha_{i}}}\prod_{b\in
B}|g_{t}^{\prime}(z_{b})|^{\Delta_{2\alpha_{b}}}\mathcal{Z}\big{(}(-\frac{1}{\sqrt{\kappa}},W_{t}),(\alpha_{j},g_{t}(z_{j}))_{j}\big{)}$

where $\Delta_{\alpha}:=\frac{\alpha}{2}(Q-\frac{\alpha}{2})$ and
$Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Let $\tau$ be a stopping time such that
almost surely $g_{t}(z_{j})\neq W_{t}$ for all $t\leq\tau$. Then $M_{t}$ is a
martingale up until time $\tau$.

###### Proof.

To lighten notation, we assume there are only bulk insertions (i.e.
$z_{j}\in\mathbbm{H}$ for all $j$); the general case follows from the same
arguments. We write $Z_{j}=g_{t}(z_{j})$ and write $W=W_{t}$, leaving the
$t$-dependence implicit to simplify notation. Our goal is to compute $d\log
M_{t}$. We have

$\displaystyle\log M_{t}=$
$\displaystyle\sum_{j}2\Delta_{\alpha_{j}}\log|g_{t}^{\prime}(z_{j})|-\frac{\alpha_{j}^{2}}{2}\log(2\operatorname{Im}Z_{j})+\frac{2\alpha_{j}}{\sqrt{\kappa}}\log|Z_{j}-W|+\sum_{1\leq
j<k\leq m}\alpha_{j}\alpha_{k}G(Z_{j},Z_{k}).$ (3.1)

By the definition of the reverse Loewner flow and $Z_{j}=g_{t}(z_{j})$, we
have

$dW=\sqrt{\kappa}dB_{t},\enspace d\langle W\rangle=\kappa dt,\enspace
dZ_{j}=-\frac{2dt}{Z_{j}-W},\enspace
d\overline{Z}_{j}=-\frac{2dt}{\overline{Z}_{j}-W},\enspace
d\log|g_{t}^{\prime}(z_{j})|=\operatorname{Re}\frac{2dt}{(Z_{j}-W)^{2}}.$

The last identity holds since $d(g_{t}(z))=-\frac{2}{g_{t}(z)-W}dt$ implies
$d(g_{t}^{\prime}(z))=-\frac{2}{(g_{t}(z)-W)^{2}}g_{t}^{\prime}(z)dt$. Using
these we have
$d\log(2\operatorname{Im}Z_{j})=d\log(Z_{j}-\overline{Z}_{j})=\frac{1}{Z_{j}-\overline{Z}_{j}}dZ_{j}+\frac{1}{\overline{Z}_{j}-Z_{j}}d\overline{Z}_{j}=\frac{1}{|Z_{j}-W|^{2}}dt$
and, since $\log|Z_{j}-W|=\operatorname{Re}\log(Z_{j}-W)$, we have
$d\log|Z_{j}-W|=\operatorname{Re}\frac{\sqrt{\kappa}}{W-Z_{j}}dB_{t}-\operatorname{Re}\frac{\sqrt{\kappa}Q}{(W-Z_{j})^{2}}dt$.
Thus

$\displaystyle
d\mathopen{}\mathclose{{}\left(2\Delta_{\alpha_{j}}\log|g_{t}^{\prime}(z_{j})|-\frac{\alpha_{j}^{2}}{2}\log(2\operatorname{Im}Z_{j})+\frac{2\alpha_{j}}{\sqrt{\kappa}}\log|Z_{j}-W|}\right)$
(3.2)
$\displaystyle=\operatorname{Re}\frac{2\alpha_{j}}{W-Z_{j}}dB_{t}-(\operatorname{Re}\frac{1}{(Z_{j}-W)^{2}}+\frac{1}{|Z_{j}-W|^{2}})\alpha_{j}^{2}dt=\operatorname{Re}\frac{2\alpha_{j}}{W-Z_{j}}dB_{t}-2\alpha_{j}^{2}\mathopen{}\mathclose{{}\left(\operatorname{Re}\frac{1}{Z_{j}-W}}\right)^{2}dt.$

In the last equality, we used the identity
$\operatorname{Re}(z^{2})+|z|^{2}=2(\operatorname{Re}z)^{2}$. Next, since
$G(Z_{j},z)=-\operatorname{Re}(\log(Z_{j}-z)+\log(Z_{j}-\overline{z}))$, we
have
$dG(Z_{j},z)=\operatorname{Re}\mathopen{}\mathclose{{}\left((\frac{1}{Z_{j}-z}+\frac{1}{Z_{j}-\overline{z}})\frac{2}{Z_{j}-W}}\right)dt$,
so

$\displaystyle d$ $\displaystyle
G(Z_{j},Z_{k})=\operatorname{Re}\mathopen{}\mathclose{{}\left((\frac{1}{Z_{j}-Z_{k}}+\frac{1}{Z_{j}-\overline{Z}_{k}})\frac{2}{Z_{j}-W}+(\frac{1}{Z_{k}-Z_{j}}+\frac{1}{Z_{k}-\overline{Z}_{j}})\frac{2}{Z_{k}-W}}\right)dt$
$\displaystyle=\operatorname{Re}\mathopen{}\mathclose{{}\left(-\frac{2}{(Z_{j}-W)(Z_{k}-W)}-\frac{2}{(Z_{j}-W)(\overline{Z}_{k}-W)}}\right)dt=-4\operatorname{Re}\mathopen{}\mathclose{{}\left(\frac{1}{Z_{k}-W}}\right)\operatorname{Re}\mathopen{}\mathclose{{}\left(\frac{1}{Z_{j}-W}}\right)dt.$

Combining this with (3.2), we can compute $d\log M_{t}$ from (3.1):

$d\log
M_{t}=(\sum_{j}\operatorname{Re}\frac{2\alpha_{j}}{W-Z_{j}})dB_{t}-\frac{1}{2}(\sum_{j}\operatorname{Re}\frac{2\alpha_{j}}{W-Z_{j}})^{2}dt.$

Thus, $M_{t}$ is a martingale, completing the proof for the case of only bulk
insertions. The general case is essentially the same: we get the same
expression for $d\log M_{t}$ so $M_{t}$ is a martingale. ∎

######  Proposition 3.2.

In the setting of Lemma 3.1, the law of reverse $\operatorname{SLE}_{\kappa}$
run until $\tau$ and weighted by $\frac{M_{\tau}}{M_{0}}$ is precisely that of
$\operatorname{SLE}_{\kappa,\rho}$ run until the stopping time $\tau$, where
at each $z_{j}$ there is a force point of weight
$\rho_{j}=2\sqrt{\kappa}\alpha_{j}$.

###### Proof.

We use the notation
${\mathcal{Z}_{\kappa}^{\alpha}}(w,(z_{j})_{j}):=\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},w),(\alpha_{j},z_{j})_{j})$.
By Girsanov’s theorem, under the reweighted law, the driving function
satisfies the SDE

$dW_{t}=\sqrt{\kappa}\,dB_{t}+\kappa\partial_{w}\log{\mathcal{Z}_{\kappa}^{\alpha}}(W_{t},(g_{t}(z_{j}))_{j})dt.$

We have
$\partial_{w}\log{\mathcal{Z}_{\kappa}^{\alpha}}(w,(z_{j})_{j})=\sum_{j}-\frac{2}{\sqrt{\kappa}}\alpha_{j}\partial_{w}\operatorname{Re}\log(z_{j}-w)=\sum_{j}\operatorname{Re}\mathopen{}\mathclose{{}\left(\frac{-2\alpha_{j}/\sqrt{\kappa}}{z_{j}-w}}\right)$,
so the SDE agrees with (2.4) as desired. ∎

The observation that reverse SLE with force points is intimately related to
the Coulomb gas partition function is not original. See for instance [Fuk17,
Ale19, KK20, KM21, Kos21]. A similar perspective arose earlier in the setting
of forward SLE. See e.g. [BB03, BBK05, Kyt06].

##  4 Sheffield’s coupling and the $\kappa\leq 4$ LCFT zipper

In this section we prove Theorems 1.1 and 1.2. These results are fairly
straightforward consequences of Sheffield’s coupling of LQG and SLE.

A version of the following was first proved in [She16], and the full version
where force points may be zipped into the bulk was proved in [DMS21]. Recall
$G(z,w)=-\log|z-w|-\log|z-\overline{w}|$.

######  Proposition 4.1 ([DMS21, Theorem 5.1]).

Let $\kappa>0$, $\gamma=\min(\sqrt{\kappa},\frac{4}{\sqrt{\kappa}})$, let
$n\geq 0$ and $(\rho_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ for
$j\leq n$. Suppose that $(W_{t})_{t\geq 0}$ is the driving function for
reverse $\operatorname{SLE}_{\kappa,\rho}$ with force points at $z_{j}$ of
size $\rho_{j}$, and let $g_{t}$ be the Loewner map. For each $t\geq 0$ let

${\mathfrak{h}}_{t}(z)=-\frac{1}{\sqrt{\kappa}}G(z,W_{t})+\frac{1}{2\sqrt{\kappa}}\sum_{j=1}^{n}\rho_{j}G(g_{t}(z),g_{t}(z_{j}))+Q\log|g_{t}^{\prime}(z)|.$

Let $h$ be an independent Neumann GFF modulo additive constant on
$\mathbbm{H}$. Suppose $\tau$ is an a.s. finite stopping time such that $\tau$
occurs before or at the continuation threshold for $W_{t}$. Then, as
distributions modulo additive constant,

${\mathfrak{h}}_{0}+h\stackrel{{\scriptstyle
d}}{{=}}{\mathfrak{h}}_{\tau}+h\circ g_{\tau}.$ (4.1)

Note that if there is a force point at $0$ with weight
$\rho\geq\frac{\kappa}{2}+4$ then the reverse $\operatorname{SLE}_{\kappa}$
immediately hits the continuation threshold, i.e. $\tau=0$.

As we see next, adding a constant chosen from Lebesgue measure to the field
essentially gives Theorem 1.2 when
$\delta=Q-\sum_{j}\alpha_{j}+\frac{1}{\sqrt{\kappa}}$.

######  Proposition 4.2.

Let $\kappa>0$ and $\gamma=\min(\sqrt{\kappa},\frac{4}{\sqrt{\kappa}})$. Let
$n\geq 0$ and let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ for $j\leq n$.
Let $\delta=Q-\sum_{j}\alpha_{j}+\frac{1}{\sqrt{\kappa}}$. Let $(\eta_{t})$ be
reverse $\operatorname{SLE}_{\kappa,\rho}$ with force points at $z_{j}$ of
size $\rho_{j}=2\sqrt{\kappa}\alpha_{j}$, run until a stopping time $\tau$
which a.s. occurs before or at the continuation threshold. Let $g_{\tau}$ be
the conformal map from $\mathbbm{H}$ to the unbounded connected component of
$\mathbbm{H}\backslash\eta_{\tau}$ such that
$\lim_{z\to\infty}g_{\tau}(z)-z=0$. Then for any nonnegative measurable
function $F$,

$\frac{\mathrm{LF}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}[F(\phi)]}{\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j})}=\mathbbm{E}\mathopen{}\mathclose{{}\left[\frac{\mathrm{LF}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},W_{\tau}),(\alpha_{j},g_{\tau}(z_{j}))_{j},(\delta,\infty)}[F(g_{\tau}^{-1}\bullet_{\gamma}\phi)]}{\mathcal{Z}((-\frac{1}{\sqrt{\kappa}},W_{\tau}),(\alpha_{j},g_{\tau}(z_{j}))_{j})}}\right].$
(4.2)

###### Proof.

We will use the Liouville field description from Remark 2.3. Write
$\mathfrak{f}_{\tau}(z)=-\frac{1}{\sqrt{\kappa}}G(z,W_{\tau})+\sum_{j}\alpha_{j}G(z,g_{\tau}(z_{j}))$.
Writing $\mathbbm{E}$ to denote expecation with respect to $\eta_{\tau}$ and
independently $h\sim P_{\mathbbm{H}}$, the right hand side of (4.2) equals

$\mathbbm{E}[\int_{\mathbbm{R}}F(g_{\tau}^{-1}\bullet_{\gamma}(h+\mathfrak{f}_{\tau}+c))\,dc]=\mathbbm{E}[\int_{\mathbbm{R}}F(h\circ
g_{\tau}+\mathfrak{h}_{\tau}+c)\,dc].$

By Proposition 4.1 this equals
$\mathbbm{E}[\int_{\mathbbm{R}}F(h+\mathfrak{h}_{0}+c)\,dc]$, which is the
left hand side of (4.2). ∎

To remove the constraint that
$\delta=Q-\sum_{j}\alpha_{j}+\frac{1}{\sqrt{\kappa}}$, in Proposition 4.5 we
will weight by the average of the field on very large semicircles to change
the value of $\delta$, see Lemma 4.4. As an intermediate step we need the
following collection of identities. Recall $G_{\mathbbm{H}}$ from (2.1) and
$G(z,w)=-\log|z-w|-\log|z-\overline{w}|$.

######  Lemma 4.3.

Suppose $K\subset\overline{\mathbbm{H}}$ is compact, $\mathbbm{H}\backslash K$
is simply connected and there is a conformal map
$g:\mathbbm{H}\to\mathbbm{H}\backslash K$ such that
$\lim_{|z|\to\infty}g(z)-z=0$. Let $\varepsilon>0$ and let
$\theta_{\varepsilon,\infty}$ be the uniform probability measure on
$\\{z\in\mathbbm{H}:|z|=1/\varepsilon\\}$. Suppose
$g(-1/\varepsilon),g(1/\varepsilon)\in\mathbbm{R}$ and $u\in
g(B_{1/\varepsilon}(0)\cap\mathbbm{H})$. Then $\int
G(u,v)(g_{*}\theta_{\varepsilon,\infty})(dv)=2\log\varepsilon$. Moreover,
$\int\log g^{\prime}(v)\theta_{\varepsilon,\infty}(dv)=0$. Finally, if
$|g(z)|>1$ whenever $|z|=1/\varepsilon$, then $\int
G_{\mathbbm{H}}(u,v)(g_{*}\theta_{\varepsilon,\infty})(dv)=G_{\mathbbm{H}}(u,\infty)$
and $\int
G_{\mathbbm{H}}(\infty,v)(g_{*}\theta_{\varepsilon,\infty})(dv)=-2\log\varepsilon$.

###### Proof.

Extend $g$ by Schwarz reflection to a map with image
$\mathbbm{C}\backslash(K\cup\overline{K})$. Define the map
$f:B_{1/\varepsilon}(0)\backslash\\{0\\}\to\mathbbm{C}$ by
$f(v)=(u-g(\frac{1}{v}))^{-2}v^{-2}$; this is a holomorphic map which extends
continuously to $f(0)=1$, hence the extended map
$f:B_{1/\varepsilon}(0)\to\mathbbm{C}$ is holomorphic. Since $\log|f|$ is
harmonic we have $(\log|f|,\theta_{\varepsilon,0})=\log|f(0)|=0$, and
rephrasing this gives
$\int-2\log|u-g(\frac{1}{v})|\theta_{\varepsilon,0}(dv)=2\log\varepsilon$. Now
the change of variables $v^{\prime}=\frac{1}{v}$ gives the assertion. The
second assertion is proved similarly. The third assertion follows from
$G_{\mathbbm{H}}(u,v)=G(u,v)-G(0,v)+G_{\mathbbm{H}}(u,\infty)$ and
$G_{\mathbbm{H}}(\infty,v)=-G(0,v)$ for $|v|>1$. ∎

Weighting the Liouville field by the field average near $\infty$ changes the
value of $\delta$:

######  Lemma 4.4.

Let $\gamma\in(0,2]$. In the setting of Lemma 4.3, suppose
$(\alpha_{j},w_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ and
$\delta\in\mathbbm{R}$. Let
$\mathbbm{H}_{\varepsilon}=B_{1/\varepsilon}(0)\cap\mathbbm{H}$. Then for any
nonnegative function $F$ such that $F(\phi)$ depends only on
$\phi|_{g(\mathbbm{H}_{\varepsilon})}$,

${{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},w_{j})_{j},(\delta,\infty)}[\varepsilon^{(\delta^{\prime}-\delta)(\delta^{\prime}+\delta-2Q)}e^{(\delta^{\prime}-\delta)(g^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon,\infty})}F(\phi)]={{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},w_{j})_{j},(\delta^{\prime},\infty)}[F(\phi)].$

###### Proof.

Write
$\theta_{\varepsilon}=(\delta^{\prime}-\delta)\theta_{\varepsilon,\infty}$.
Let $Z_{\varepsilon}=\mathbbm{E}[e^{(h,g_{*}\theta_{\varepsilon})}]$ where
$h\sim P_{\mathbbm{H}}$, then Lemma 4.3 gives
$Z_{\varepsilon}=\varepsilon^{-(\delta^{\prime}-\delta)^{2}}$. With
$\mathfrak{f}(z):=\sum_{j}\alpha_{j}G_{\mathbbm{H}}(z,w_{j})+(\delta-Q)G_{\mathbbm{H}}(z,\infty)$,
Lemma 4.3 also gives

$\displaystyle(\mathfrak{f},g_{*}\theta_{\varepsilon})$
$\displaystyle=(\delta^{\prime}-\delta)\sum_{j}2\alpha_{j}\log|w_{j}|_{+}-2(\delta^{\prime}-\delta)(\delta-Q)\log\varepsilon$
$\displaystyle=\log\frac{C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta^{\prime},\infty)}}{C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta,\infty)}}-2(\delta^{\prime}-\delta)(\delta-Q)\log\varepsilon.$

Writing $\phi=h+\mathfrak{f}+c$, Lemma 4.3 gives
$(g^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon})=(\phi,g_{*}\theta_{\varepsilon})+Q(\log|g^{\prime}|,\theta_{\varepsilon})=(\phi,g_{*}\theta_{\varepsilon})$,
so

$e^{(g^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon})}=\varepsilon^{-2(\delta^{\prime}-\delta)(\delta-Q)}e^{(\delta^{\prime}-\delta)c}\frac{C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta^{\prime},\infty)}}{C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta,\infty)}}e^{(h,g_{*}\theta_{\varepsilon})}.$

Thus, writing
$\widetilde{F}(\phi)=\varepsilon^{(\delta^{\prime}-\delta)(\delta^{\prime}+\delta-2Q)}e^{(\delta^{\prime}-\delta)(g^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon,\infty})}F(\phi)$,
we have

$\displaystyle{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha_{j},w_{j})_{j},(\delta,\infty)}[\widetilde{F}(\phi)]$
$\displaystyle=C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta,\infty)}\int_{-\infty}^{\infty}e^{(\sum_{j}\alpha_{j}+\delta-Q)c}\mathbbm{E}[\widetilde{F}(h+\mathfrak{f}+c)]\,dc$
$\displaystyle=C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta^{\prime},\infty)}\int_{-\infty}^{\infty}e^{(\sum_{j}\alpha_{j}+\delta^{\prime}-Q)c}\mathbbm{E}[\frac{e^{(h,\theta_{\varepsilon})}}{Z_{\varepsilon}}F(h+\mathfrak{f}+c)]\,dc$
$\displaystyle=C_{\gamma}^{(\alpha_{j},w_{j})_{j},(\delta^{\prime},\infty)}\int_{-\infty}^{\infty}e^{(\sum_{j}\alpha_{j}+\delta^{\prime}-Q)c}\mathbbm{E}[F(h+\mathfrak{f}+\int
G_{\mathbbm{H}}(\cdot,v)g_{*}\theta_{\varepsilon}(dv)+c)]\,dc.$

where in the last line we use Girsanov’s theorem. Lemma 4.3 gives for any
$u\in g(\mathbbm{H}_{\varepsilon})$ that $\int
G(u,v)g_{*}\theta_{\varepsilon}(dv)=(\delta^{\prime}-\delta)G_{\mathbbm{H}}(u,\infty)$,
so since $F$ only depends on the field on $g(\mathbbm{H}_{\varepsilon})$, the
last line equals the right hand side of the desired identity. ∎

The following is a quick consequence of Lemma 4.4.

######  Proposition 4.5.

Suppose any of Theorems 1.1, 1.2, 1.5 or 1.7 holds for insertions
$(\alpha_{j},z_{j})_{j}$ and $(\delta,\infty)$. Then the same theorem holds
for insertions $(\alpha_{j},z_{j})$ and $(\delta^{\prime},\infty)$ where
$\delta^{\prime}$ is arbitrary.

###### Proof.

We discuss the cases of Theorems 1.1 and 1.2 ($\kappa\leq 4$) in detail; the
$\kappa>4$ results are obtained identically. Proposition 3.2 implies that
Theorem 1.1 is equivalent to Theorem 1.2 when $\tau$ a.s. satisfies
$g_{\tau}(z_{j})\not\in\eta_{\tau}$ for all $j$. Thus it suffices to discuss
only Theorem 1.2.

Assume Theorem 1.2 for insertions $(\alpha_{j},z_{j})_{j}$ and
$(\delta,\infty)$: starting with a field $\phi_{0}$ sampled from
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}$,
zipping up gives $(\phi_{\tau},\eta_{\tau})$ whose law is given by (1.2).

Let $\varepsilon>0$ and
$\mathbbm{H}_{\varepsilon}=B_{1/\varepsilon}(0)\cap\mathbbm{H}$. Assume that
$\tau$ satisfies
$g_{\tau}(-1/\varepsilon),g_{\tau}(1/\varepsilon)\in\mathbbm{R}$ and
$|g_{\tau}(z)|>1$ for $z\in\mathbbm{H}_{\varepsilon}$. Weight by
$\varepsilon^{(\delta^{\prime}-\delta)(\delta^{\prime}+\delta-2Q)}e^{(\delta^{\prime}-\delta)(\phi_{0},\theta_{\varepsilon,\infty})}$.
By Lemma 4.4 with $K=\emptyset$, the weighted law of
$\phi_{0}|_{\mathbbm{H}_{\varepsilon}}$ agrees with that of
$\phi|_{\mathbbm{H}_{\varepsilon}}$ where
$\phi\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta^{\prime},\infty)}$.
By Lemma 4.4 with $K$ the hull of $\eta_{\tau}$, the weighted law of
$(\phi_{\tau}|_{g_{\tau}(\mathbbm{H}_{\varepsilon})},\eta_{\tau})$ agrees with
that of $(\phi|_{g_{\tau}(\mathbbm{H}_{\varepsilon})},\eta)$ where
$(\phi,\tau)$ has law given by (1.2) with $\delta$ replaced by
$\delta^{\prime}$. Our assumptions on $\tau$ imply that the zipping up
procedure until time $\tau$ only depends on
$\phi_{0}|_{\mathbbm{H}_{\varepsilon}}$. Thus sending $\varepsilon\to 0$ gives
Theorem 1.2 for insertions $(\alpha_{j},z_{j})_{j},(\delta^{\prime},\infty)$.
∎

Combining the above, we now prove the $\kappa\leq 4$ theorems.

###### Proof of Theorem 1.2.

Suppose $-\frac{1}{\sqrt{\kappa}}+\sum_{j}\alpha_{j}+\delta=Q$. Proposition
4.2 implies that if we sample $(\phi,\eta)$ from (1.2) and let
$g:\mathbbm{H}\to\mathbbm{H}\backslash\eta$ be the conformal map such that
$\lim_{z\to\infty}g(z)-z=0$, then the law of
$\phi_{0}=g^{-1}\bullet_{\gamma}\phi$ is
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}$.
Moreover, $(\phi,\eta)$ is obtained from $\phi_{0}$ by conformal welding
according to quantum length; this is proved for $\kappa<4$ in [She16] and for
$\kappa=4$ in [HP21, KMS22]. Thus Theorem 1.2 holds when
$-\frac{1}{\sqrt{\kappa}}+\sum_{j}\alpha_{j}+\delta=Q$. Proposition 4.5 then
removes this constraint. ∎

###### Proof of Theorem 1.1.

This is immediate from Theorem 1.2 and Proposition 3.2. ∎

##  5 The $\kappa>8$ LCFT zipper

In this section we prove Theorem 1.7. There are two key steps. First, we need
to produce the quantum cell in the setting of LCFT. In Section 5.1 we prove
that the uniform embedding of the quantum wedge is a Liouville field, and in
Section 5.2 we use this to transfer a statement about cutting a quantum cell
from a quantum wedge to one about cutting a quantum cell from a Liouville
field. Second, we must show the quantum cell arises in the time-evolution
described by Sheffield’s coupling (see Proposition 4.2). This is accomplished
for a special case in Section 5.3 by a technical limiting argument. In Section
5.4 we bootstrap the special case to the full result.

###  5.1 Uniform embedding of quantum wedge

We show that when a quantum wedge is embedded in the upper half-plane
uniformly at random, the resulting field is a Liouville field.

######  Proposition 5.1.

Let $W>\frac{\gamma^{2}}{2}$ and
$\alpha=\frac{1}{2}(Q+\frac{\gamma}{2}-\frac{W}{\gamma})$. Sample
$(\mathcal{W},T)\sim\mathcal{M}^{\mathrm{wed}}(W)\times dT$ and let
$(\mathbbm{H},\phi_{0},0,\infty)$ be an embedding of $\mathcal{W}$ chosen in a
way not depending on $T$. Let $f_{T}(z)=e^{T}z$. Then the law of
$\phi=f_{T}\bullet_{\gamma}\phi_{0}$ is
$\frac{1}{Q-2\alpha}{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha,0),(Q-\alpha,\infty)}$.

Proposition 5.1 is analogous to [AHS22, Theorem 2.13] which proves a similar
statement for quantum disks. The proof hinges on the following Brownian motion
identity.

######  Lemma 5.2.

Let $\alpha<\frac{Q}{2}$. The random processes $X_{1},X_{2}$ defined below
have the same law:

  * •

Let $P$ be the law of $Y_{t}$ from Definition 2.6. Sample $(Y,T)\sim P\times
dT$ and let $X_{1}(t)=Y_{t-T}$.

  * •

Let $P^{\prime}$ be the law of standard two-sided Brownian motion
$(B_{t})_{t\in\mathbbm{R}}$ (with $B_{0}=0$). Sample
$((B_{t})_{t\in\mathbbm{R}},\mathbf{c})$ from
$(Q-2\alpha)^{-1}P^{\prime}\times dc$, and let
$X_{2}(t)=B_{2t}+(Q-2\alpha)t+\mathbf{c}$.

###### Proof.

We claim that for any $y\in\mathbbm{R}$ we have
$X_{1}+y\stackrel{{\scriptstyle d}}{{=}}X_{1}$. Indeed, the process $Y_{t}$ is
a variance 2 Brownian motion with drift $(Q-2\alpha)$ started at $-\infty$ and
run until it reaches $+\infty$, so setting $\widetilde{Y}_{t}=Y_{t}+y$ and
$\widetilde{\tau}=\inf\\{t:\widetilde{Y}_{t}\geq 0\\}$, we have
$(\widetilde{Y}_{t+\widetilde{\tau}})_{t\in\mathbbm{R}}\stackrel{{\scriptstyle
d}}{{=}}(Y_{t})_{t\in\mathbbm{R}}$. Now the translation invariance of Lebesgue
measure $dT$ yields $X_{1}+y\stackrel{{\scriptstyle d}}{{=}}X_{1}$. We will
refer to this fact as _translation symmetry_.

We first show that the law of $X_{1}(0)$ is $(Q-2\alpha)^{-1}dx$. By Fubini’s
theorm, the measure of $\\{X_{1}(0)\in(0,b)\\}$ is
$\mathbbm{E}[\int_{\mathbbm{R}}1_{Y_{t}\in(0,b)}\,dt]$. This expectation is
$((Q-2\alpha)^{-1}+o(1))b$ as $b\to\infty$; indeed the drift term of
$Y_{t}=B_{2t}+(Q-2\alpha)t$ dominates the Gaussian fluctuations of $B_{2t}$
for large $t$. Translation symmetry implies the law of $X_{1}(0)$ is $c\,dx$
for some constant $c$, and comparing with the above gives $c=1/(Q-2\alpha)$.

Now, let $I$ be a finite interval. We show that conditioned on
$X_{1}|_{(-\infty,0]}$ and $X_{1}(0)\in I$, the process $X_{1}|_{[0,\infty)}$
agrees in law with $B_{2t}+(Q-2\alpha)t+X_{1}(0)$ where $(B_{t})_{t\geq 0}$ is
standard Brownian motion. By translation symmetry we may assume
$I\subset(0,\infty)$, then we may restrict to the event $\\{T>0\\}$ since
$Y_{t}<0$ for $t<0$. By the Markov property of Brownian motion, given
$(Y_{t})_{(-\infty,T]}$ and $Y_{T}\in I$, the conditional law of
$(Y_{t})_{[T,\infty)}$ is $B_{2t}+(Q-2\alpha)t+Y_{T}$; rephrasing in terms of
$X_{t}$ gives the desired statement.

Finally, a similar argument gives that conditioned on $X_{1}|_{[0,\infty)}$
and $X_{1}(0)\in I$, the process $X_{1}|_{(-\infty,0]}$ agrees in law with
$B_{-2t}+(Q-2\alpha)t+X_{1}(0)$. Since $X_{1}(0)\stackrel{{\scriptstyle
d}}{{=}}X_{2}(0)$, and the conditional law of $X_{1}$ given $X_{1}(0)=x$
agrees with the conditional law of $X_{2}$ given $X_{2}(0)=x$ for all $x$, we
conclude $X_{1}\stackrel{{\scriptstyle d}}{{=}}X_{2}$. ∎

###### Proof of Proposition 5.1.

Let $\log:\mathbbm{H}\to\mathcal{S}$ be the confomal map sending
$(0,1,\infty)$ to $(-\infty,0,+\infty)$, and let
$\phi^{\mathcal{S}}_{0}=\log\bullet_{\gamma}\phi_{0}$. By definition there is
a random $x\in\mathbbm{R}$ such that
$(\phi^{\mathcal{S}}_{0},T)\stackrel{{\scriptstyle
d}}{{=}}(\phi_{\mathrm{av}}(\cdot+x)+\phi_{\mathrm{lat}}(\cdot+x),T^{\prime})$
where the law of $T^{\prime}$ is Lebesgure measure on $\mathbbm{R}$ and
$(\phi_{\mathrm{av}},\phi_{\mathrm{lat}})$ are independently sampled as in
Definition 2.6. By the translation invariance of Lebesgue measure on
$\mathbbm{R}$, and the translation invariance in law of $\phi_{\mathrm{lat}}$,
Lemma 5.2 implies
$\phi_{\mathrm{av}}(\cdot+x-T^{\prime})+\phi_{\mathrm{lat}}(\cdot+x-T^{\prime})\stackrel{{\scriptstyle
d}}{{=}}X_{2}(\operatorname{Re}\cdot)+h_{\mathrm{lat}}$ where $X_{2}$ is as in
Lemma 5.2 and $h_{\mathrm{lat}}$ is the projection of an independent GFF on
$\mathcal{S}$ to $H_{\mathrm{lat}}(\mathcal{S})$. Since the projection of a
GFF to $H_{\mathrm{av}}(\mathcal{S})$ has the law of
$(B_{2t})_{t\in\mathbbm{R}}$ where $B_{t}$ is standard two-sided Brownian
motion, we conclude that $\log\bullet_{\gamma}(f_{T}\bullet_{\gamma}\phi_{0})$
agrees in law with $h+(Q-2\alpha)\operatorname{Re}(\cdot)+\mathbf{c}$ where
$(h,\mathbf{c})\sim\frac{1}{Q-2\alpha}P_{\mathcal{S}}\times dc$. But by
definition
$\log^{-1}\bullet_{\gamma}(h+(Q-2\alpha)\operatorname{Re}(\cdot)+\mathbf{c})$
has law
$\frac{1}{Q-2\alpha}{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha,0),(Q-\alpha,\infty)}$,
concluding the proof. ∎

###  5.2 Cutting a quantum cell from the Liouville field

The goal of this section is to prove the following. We write
$\operatorname{SLE}_{\kappa}$ for the law of forward
$\operatorname{SLE}_{\kappa}$ in $(\mathbbm{H},0,\infty)$. Recall that $P_{a}$
is the law of the area $a$ quantum cell.

######  Proposition 5.3.

Suppose $\kappa>8$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. Sample a triple
$(\psi,z,\eta)$ from the measure

$\mathcal{A}_{\psi}(dz)\,{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\psi)\times\operatorname{SLE}_{\kappa}(d\eta).$
(5.1)

Parametrize $\eta$ by quantum area and let $A$ be the time $\eta$ first hits
$z$. Let $g:\mathbbm{H}\to\mathbbm{H}\backslash\eta([0,A])$ be the unique
conformal map with $g(0)=z$ and $g(w)=w+O(1)$ as $w\to\infty$, and let
$\phi=\psi\circ g+Q\log|g^{\prime}|$. Let
$\mathcal{C}=(\eta([0,A]),h,\eta|_{[0,A]})/{\sim_{\gamma}}$. Then the law of
$(\phi,\mathcal{C},A)$ is

${{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi)P_{A}(d\mathcal{C})1_{A>0}dA.$
(5.2)

To that end, we prove a similar statement for the quantum wedge (Lemma 5.4),
then transfer to the Liouville field using uniform embedding (Proposition
5.1).

For the next lemma, let $P(d\psi_{0})$ be any law on fields such that the law
of $(\mathbbm{H},\psi_{0},0,\infty)/{\sim_{\gamma}}$ is
$\mathcal{M}^{\mathrm{wed}}(2-\frac{\gamma^{2}}{2})$. For instance $P$ could
be the law of the field of Definition 2.6 after parametrizing in $\mathbbm{H}$
via $z\mapsto e^{z}$.

######  Lemma 5.4.

Suppose $\kappa>8$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. Sample
$(z_{0},\psi_{0},\eta_{0})\sim\mathcal{A}_{\psi_{0}}(dz_{0})\,P(d\psi_{0})\operatorname{SLE}_{\kappa}(d\eta_{0})$.
Parametrize $\eta_{0}$ by quantum area and let $A$ be the time it hits
$z_{0}$. Then the marginal law of $A$ is the Lebesgue measure
$\mathrm{Leb}_{(0,\infty)}$, and the conditional law of the pair of quantum
surfaces
$(\mathbbm{H}\backslash\eta_{0}([0,A]),\psi_{0},z_{0},\infty)/{\sim_{\gamma}}$
and $(\eta_{0}([0,A]),\psi_{0},\eta_{0}|_{[0,A]})/{\sim_{\gamma}}$ given $A$
is $\mathcal{M}^{\mathrm{wed}}(2-\frac{\gamma^{2}}{2})\times P_{A}$.

###### Proof.

We construct the same random objects $(z_{0},\psi_{0},\eta_{0})$ of the lemma
statement from a different approach. Sample
$(A,\psi_{0},\eta_{0})\sim\mathrm{Leb}_{(0,\infty)}(dA)\,P(d\psi_{0})\operatorname{SLE}_{\kappa}(d\eta_{0})$.
Let $z_{0}=\eta(A)$. Since
$\mathcal{A}_{\psi_{0}}=\eta_{*}\mathrm{Leb}_{(0,\infty)}$, the law of
$(z_{0},\psi_{0},\eta_{0})$ is
$\mathcal{A}_{\psi_{0}}(dz_{0})P(d\psi_{0})\operatorname{SLE}_{\kappa}(d\eta_{0})$.
By our construction the marginal law of $A$ is $\mathrm{Leb}_{(0,\infty)}$.

Condition on $A$. The conditional law of $(\psi_{0},\eta_{0})$ is
$P(d\psi_{0})\operatorname{SLE}_{\kappa}(d\eta_{0})$. By definition, the law
of $\mathcal{C}_{A}:=(\eta([0,A]),\psi_{0},\eta_{0}|_{[0,A]})/{\sim_{\gamma}}$
is $P_{A}$. Finally, by the last two claims of [DMS21, Theorem 1.9],
$(\eta_{0}([A,\infty)),\psi_{0},\eta(A),\infty)/{\sim_{\gamma}}$ has law
$\mathcal{M}^{\mathrm{wed}}(2-\frac{\gamma^{2}}{2})$ and is independent of
$\mathcal{C}_{A}$.

∎

We are now ready to prove Proposition 5.3.

###### Proof of Proposition 5.3.

Sample
$(z_{0},\psi_{0},\eta_{0},T)\sim\mathcal{A}_{\psi_{0}}(dz_{0})\,P(d\psi_{0})\operatorname{SLE}_{\kappa}(d\eta_{0})\,dT$.
Let $f_{T}(z)=e^{T}z$, and let
$(z,\psi,\eta)=(f_{T}(z_{0}),f_{T}\bullet_{\gamma}\psi_{0},f_{T}\circ\eta_{0})$.
By Proposition 5.1 and the scale-invariance of $\operatorname{SLE}_{\kappa}$
the law of $(z,\psi,\eta)$ is (5.1) times $(Q-\frac{3\gamma}{2})^{-1}$.

Let $A$ be the time that $\eta_{0}$ hits $z_{0}$. Let
$g_{0}:\mathbbm{H}\to\mathbbm{H}\backslash\eta_{0}([0,A])$ be the conformal
map with $g_{0}(0)=z_{0}$ and $g_{0}(w)=w+O(1)$ as $w\to\infty$, and let
$\phi_{0}=\psi_{0}\circ g_{0}+Q\log|g_{0}^{\prime}|$. Let
$g:\mathbbm{H}\to\mathbbm{H}\backslash\eta([0,A])$ be the conformal map with
$g(0)=z$ and $\lim_{w\to\infty}g(w)-w=0$, and let $\phi=\psi\circ
g+Q\log|g_{T}^{\prime}|$. Let
$\mathcal{C}=(\eta([0,A]),\psi,\eta|_{[0,A]})/{\sim_{\gamma}}=(\eta_{0}([0,A]),\psi_{0},\eta_{0}|_{[0,A]})/{\sim_{\gamma}}$.

By Lemma 5.4 the law of
$((\mathbbm{H},\phi_{0},0,\infty)/{\sim_{\gamma}},\mathcal{C},A,T)$ is
$\mathcal{M}^{\mathrm{wed}}(2-\frac{\gamma^{2}}{2})\,P_{A}(d\mathcal{C})\,1_{A>0}dA\,dT$.
It is easy to check that $f_{T}\circ g_{0}=g\circ f_{T}$, so
$f_{T}\bullet_{\gamma}\phi_{0}=\phi$. Thus by Proposition 5.1 the law of
$(\phi,\mathcal{C},A)$ is (5.2) times $(Q-\frac{3\gamma}{2})^{-1}$. ∎

###  5.3 The $n=0$ case of Theorem 1.7

In this section, we prove the following.

######  Proposition 5.5.

Theorem 1.7 holds for the special case where $n=0$.

In Lemma 5.6 we construct a process $(\phi_{t},\eta_{t})_{t\geq 0}$ on field-
curve pairs where the evolution is given by Sheffield’s coupling. In
Proposition 5.7 we run this process until the time $\tau$ that the quantum
area has increased by a Lebesgue-typical amount, and identify the law of
$(\phi_{\tau},\eta_{\tau})$. In Proposition 5.9 we use Propositions 5.3 and
5.7 to show that $(\phi_{\tau},\eta_{\tau})$ comes from conformally welding
$\phi_{0}$ with a quantum cell. Proposition 5.5 then follows quickly.

We write $\operatorname{SLE}_{\kappa}^{t}$ to denote the law of forward
$\operatorname{SLE}_{\kappa}$ in $\mathbbm{H}$ from $0$ to $\infty$ run for
time $t$ (so its half-plane capacity is $2t$), and write
$\operatorname{rSLE}_{\kappa}^{t}$ for reverse SLE in $(\mathbbm{H},0,\infty)$
run for time $t$. Let $\mathfrak{F}$ be the space of distributions on
$\mathbbm{H}$ and let $\mathfrak{C}$ be the space of bounded parametrized
curves in $\overline{\mathbbm{H}}$ equipped with the metric
$d(\eta_{1},\eta_{2})=|T_{1}-T_{2}|+\sup_{t\leq T_{1}\wedge
T_{2}}|\eta_{1}(t)-\eta_{2}(t)|$ where $T_{i}$ is the duration of $\eta_{i}$.

######  Lemma 5.6.

For any $\kappa>0$ and $\gamma=\min(\sqrt{\kappa},\frac{4}{\sqrt{\kappa}})$,
there is an infinite measure $M$ on
$C([0,\infty),\mathfrak{F}\times\mathfrak{C})$ such that, for
$(\phi_{t},\eta_{t})_{t\geq 0}$ sampled from $M$,

  1. i)

$\eta_{t}$ satisfies $\mathrm{hcap}(\eta_{t})=2t$ and is parametrized by half-
plane capacity (i.e. $\mathrm{hcap}(\eta_{t}([0,s]))=2s$);

  2. ii)

for $\tau$ an a.s. finite stopping time for the filtration
$\mathcal{F}_{t}=\sigma(\eta_{t})$, the law of $(\phi_{\tau},\eta_{\tau})$ is
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},\eta_{\tau}(0)),(Q-\frac{3\gamma}{4},\infty)}\operatorname{rSLE}_{\kappa}^{\tau}$;

  3. iii)

for $t_{1}<t_{2}$, let
$g_{t_{1},t_{2}}:\mathbbm{H}\to\mathbbm{H}\backslash\eta_{t_{2}}([0,t_{2}-t_{1}])$
be the conformal map with
$g_{t_{1},t_{2}}(\eta_{t_{1}}(0))=\eta_{t_{2}}(t_{2}-t_{1})$ and
$\lim_{z\to\infty}g_{t_{1},t_{2}}(z)-z=0$. Then
$\phi_{t_{1}}=g_{t_{1},t_{2}}^{-1}\bullet_{\gamma}\phi_{t_{2}}$.

###### Proof.

Fix $T$. Sample
$(\phi_{T},\eta_{T})\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},\eta_{\tau}(0)),(Q-\frac{3\gamma}{4},\infty)}\operatorname{rSLE}_{\kappa}^{T}$,
for $0\leq t<T$ let $\eta_{t}$ be the reverse SLE curve at time $t$, for
$t_{1}<t_{2}<T$ define
$g_{t_{1},t_{2}}:\mathbbm{H}\to\mathbbm{H}\backslash\eta_{t_{2}}([0,t_{2}-t_{1}])$
as above, and let $\phi_{t}=g_{t,T}^{-1}\bullet_{\gamma}\phi_{T}$. We claim
that $M_{T}$ satisfies the desired conditions up to time $T$. i) is immediate
by definition. For ii), we want to determine the law of the field and curve
when $(\phi_{T},\eta_{T})$ is “unzipped” for $(T-\tau)$ units of time; this
follows from the strong Markov property of reverse
$\operatorname{SLE}_{\kappa}$, and Proposition 4.2 with
$(n,\delta)=(0,Q-\frac{3\gamma}{4})$ and with its stopping time equal to
$T-\tau$. Next, by ii) the law of $(\phi_{t_{2}},\eta_{t_{2}})$ is
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},\eta_{t_{2}}(0)),(Q-\frac{3\gamma}{4},\infty)}\operatorname{rSLE}_{\kappa}^{t_{2}}$,
so applying Proposition 4.2 with $n=0$, $\delta=Q-\frac{3\gamma}{4}$, and
stopping time $t_{2}-t_{1}$ gives iii). Finally, continuity of
$(\phi_{t})_{t\geq 0}$ is immediate from iii), and continuity of reverse
$\operatorname{SLE}$ is well known.

The measure $M_{T}$ satisfies the desired properties up to time $T$. Moreover
for $T^{\prime}>T$, if we sample $(\phi_{t},\eta_{t})_{t\leq T^{\prime}}\sim
M_{T^{\prime}}$, then by ii) the law of $(\phi_{t},\eta_{t})_{t\leq T}$ is
$M_{T}$. Thus the Kolmogorov extension theorem gives the existence of $M$. ∎

For $(\phi_{t},\eta_{t})_{t\geq 0}\sim M$, for each $t$ let
$W_{t}:=\eta_{t}(0)\in\mathbbm{R}$. We point out that the field-curve pair
$(\phi_{t}(\cdot+W_{t}),\eta_{t}-W_{t})$ is the translation of
$(\phi_{t},\eta_{t})$ sending $\eta_{t}(0)$ to $0$.

######  Proposition 5.7.

Let $\kappa>8$ and $\gamma=\frac{4}{\sqrt{\kappa}}$. Sample
$(\\{(\phi_{t},\eta_{t})\\}_{t\geq 0},A)$ from $M\times 1_{A>0}\,dA$. Let
$\tau>0$ be the time $t$ that $\mathcal{A}_{\phi_{t}}(\eta_{t}([0,t]))=A$.
Then the law of $(\phi_{\tau}(\cdot+W_{\tau}),\eta_{\tau}-W_{\tau},\tau)$ is
$C\cdot{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}\operatorname{SLE}_{\kappa}^{t}1_{t>0}dt$
for some constant $C\in(0,\infty)$.

At high level, the proof of Proposition 5.7 goes as follows, see Figure 8.
First, if we instead sample $(\\{(\phi_{t},\eta_{t})\\}_{t\geq
0},A,T)\sim\delta^{-1}1_{T\in[\tau,\tau+\delta]}M\times 1_{A>0}\,dA\times dT$
then the marginal law of $(\phi_{\tau},\eta_{\tau},\tau)$ is the same as in
Proposition 5.7. Secondly, by the definition of $\tau$ the field $\phi_{T}$
has a singularity $\gamma G(\cdot,z)$ at $z=\eta_{T}(T-\tau)$ since $z$ is
“quantum typical” (Proposition 2.5). As $\delta\to 0$ we have $T-\tau\to 0$ so
$z\to W_{T}$, so in the limit the field $\phi_{T}$ has the singularity $\gamma
G(\cdot,W_{T})-\frac{\gamma}{4}G(\cdot,W_{T})$ at $W_{T}$. Finally, since
$T\to\tau$ as $\delta\to 0$, we have
$(\phi_{T}(\cdot+W_{T}),\eta_{T}-W_{T},T)\to(\phi_{\tau}(\cdot+W_{\tau}),\eta_{\tau}-W_{\tau},\tau)$.
The primary complication is in taking limits of infinite measures; below we
truncate on events to work with finite measures. We state the finiteness of
one such event as Lemma 5.8.

###### Proof.

To simplify notation, let
$(\phi^{1},\eta^{1}):=(\phi_{\tau}(\cdot+W_{\tau}),\eta_{\tau}-W_{\tau})$. Let
$\rho$ be the uniform probability measure on $B_{1/2}(i)$ (the exact choice of
$\rho$ is unimportant). Let $N>0$ and define

$E_{N}:=\\{\tau,|(\phi^{1},\rho)|,|(f^{-1}\bullet_{\gamma}\phi^{1},\rho)|<N\\}$
(5.3)

where $f:\mathbbm{H}\to\mathbbm{H}\backslash\eta^{1}([0,\tau])$ is the
conformal map with $f(0)=\eta^{1}(\tau)$ and $f(z)=z+O(1)$ as $z\to\infty$.
Let $P_{N}$ denote the conditional law of $(\phi^{1},\eta^{1},\tau)$ given
$E_{N}$; this is well defined because the measure of $E_{N}$ is finite (Lemma
5.8). Likewise let $\widetilde{P}_{N}$ denote the conditional law of
$(\phi,\eta,t)\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}\operatorname{SLE}_{\kappa}^{t}1_{t>0}dt$
given $\\{t,|(\phi,\rho)|,|(g^{-1}\bullet_{\gamma}\phi)|<N\\}$ with
$g:\mathbbm{H}\to\mathbbm{H}\backslash\eta([0,t])$ the conformal map with
$g(0)=\eta(t)$ and $g(z)=z+O(1)$ as $z\to\infty$. We will show that
$P_{N}=\widetilde{P}_{N}$; sending $N\to\infty$ concludes the proof.

Sample $(\\{(\phi_{t},\eta_{t})\\}_{t\geq 0},A,T)$ from $1_{T>\tau}M\times
1_{A>0}\,dA\times 1_{T>0}dT$ where as before $\tau$ is the time $t$ that
$\mathcal{A}_{\phi_{t}}(\eta_{t}([0,t]))=A$. As before let
$(\phi^{1},\eta^{1}):=(\phi_{\tau}(\cdot+W_{\tau}),\eta_{\tau}-W_{\tau})$. Let
$F_{\delta}=\\{T-\tau<\delta\\}$. Let
$(\phi^{2},\eta^{2})=(\phi_{T}(\cdot+W_{T}),\eta_{T}-W_{T})$ and let
$G_{\varepsilon}=\\{\eta^{2}([0,T-\tau])\subset B_{\varepsilon}(0)\\}$. See
Figure 8.

Claim 1: Law of $(\phi^{1},\eta^{1},\tau)$ given $E_{N}\cap F_{\delta}\cap
G_{\varepsilon}$ converges to $P_{N}$ as $\delta\to 0$ then $\varepsilon\to
0$.
Let $X=T-\tau$, then the conditional law of $(\phi^{1},\eta^{1},\tau,x)$ given
$E_{N}\cap F_{\delta}$ is $P_{N}\times\delta^{-1}1_{x\in[0,\delta]}\,dx$. As
$\delta\to 0$, we have $X\to 0$ in probability, so by continuity of the
Loewner chain the diameter of $\eta^{2}([0,T-\tau])$ shrinks to 0 in
probability. Hence the conditional probability of $G_{\varepsilon}$ given
$E_{N}\cap F_{\delta}$ is $1-o_{\delta}(1)$. Thus, as $\delta\to 0$ then
$\varepsilon\to 0$, the conditional law of $(\phi^{1},\eta^{1})$ given
$E_{N}\cap F_{\delta}\cap G_{\varepsilon}$ converges to $P_{N}$ in total
variation.

Claim 2: Law of $(\phi^{2},\eta^{2},T)$ given $E_{N}\cap F_{\delta}\cap
G_{\varepsilon}$ converges to $\widetilde{P}_{N}$ as $\delta\to 0$ then
$\varepsilon\to 0$.
By the second property of $M$ in Lemma 5.6 applied to time $T$, and the fact
that for fixed $t$ the curves $\operatorname{SLE}_{\kappa}^{t}$ and centered
$\operatorname{rSLE}_{\kappa}^{t}$ have the same law, the unconditioned law of
$(\phi^{2},\eta^{2},A,T)$ is

$1_{A\in(0,\mathcal{A}_{\phi^{2}}(\eta^{2}([0,T]))}dA\,{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{2})\operatorname{SLE}_{\kappa}^{T}(d\eta^{2})1_{T>0}dT.$

Let $z=\eta^{2}(T-\tau)$. The conditional law of $z$ given
$(\phi^{2},\eta^{2},T)$ is the probability measure proportional to
$\mathcal{A}_{\phi^{2}}|_{\eta^{2}([0,T])}$. Thus, using Proposition 2.5, the
unconditioned law of $(\phi^{2},\eta^{2},z,T)$ is

$\displaystyle
1_{z\in\eta^{2}([0,T])}\mathcal{A}_{\phi^{2}}(dz){{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{2})\operatorname{SLE}_{\kappa}^{T}(d\eta^{2})1_{T>0}dT$
$\displaystyle={{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(\gamma,z),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{2})1_{z\in\eta^{2}([0,T])}dz\operatorname{SLE}_{\kappa}^{T}(d\eta^{2})1_{T>0}dT.$
(5.4)

Now we express the event $E_{N}\cap F_{\delta}\cap G_{\varepsilon}$ in terms
of $(\phi^{2},\eta^{2},z,T)$. Let $\tau_{z}$ be the time $\eta^{2}$ hits $z$,
let $f_{1}:\mathbbm{H}\to\mathbbm{H}\backslash\eta^{2}([0,\tau_{z}])$ (resp.
$f_{0}:\mathbbm{H}\to\mathbbm{H}\backslash\eta^{2}([0,T])$) be the conformal
map sending $0$ to $\eta^{2}(\tau_{z})$ (resp. $\eta^{2}(T)$) and with
asymptotic behavior $w\mapsto w+O(1)$ as $w\to\infty$. Then

$E_{N}=\\{T-\tau_{z},|(f_{1}^{-1}\bullet_{\gamma}\phi^{2},\rho)|,|(f_{2}^{-1}\bullet_{\gamma}\phi^{2},\rho)|<N\\},$

$F_{\delta}=\\{\tau_{z}<\delta\\}$ and
$G_{\varepsilon}=\\{\eta^{2}([0,\tau_{z}])\subset B_{\varepsilon}(0)\\}$.

We now have the description (5.4) of the unconditioned law of
$(\phi^{2},\eta^{2},z,T)$, and the previous paragraph’s description of
$E_{N},F_{\delta},G_{\varepsilon}$. Since $z\to 0$ by the definition of
$G_{\varepsilon}$ and since $\lim_{z\to
0}G_{\mathbbm{H}}(\cdot,z)=G_{\mathbbm{H}}(\cdot,0)$ in $\mathfrak{F}$, the
law of $(\phi^{2},\eta^{2},T)$ conditioned on $E_{N}\cap F_{\delta}\cap
G_{\varepsilon}$ converges to $\widetilde{P}_{N}$ as $\delta\to 0$ then
$\varepsilon\to 0$.

Conclusion. By the definition of $G_{\varepsilon}$ the conformal map $f_{1}$
converges to the identity map as $\delta\to 0$, so combining Claims 1 and 2
gives $P_{N}=\widetilde{P}_{N}$. ∎

Figure 8:  Figure for Proposition 5.7. Sample
$(\\{(\phi_{t},\eta_{t})\\}_{t\geq 0},A,T)$ from $1_{T>\tau}M\times
1_{A>0}\,dA\times 1_{T>0}dT$ and let $\tau$ be the time the curve (grey) has
quantum area $A$. Roughly speaking, $E_{N}$ is the event that $\tau$ and
certain field observables of $\phi_{0}$ and $\phi^{1}$ are not too large,
$F_{\delta}=\\{T-\tau<\delta\\}$, and $G_{\varepsilon}$ is the event that the
dark grey region is small in a Euclidean sense. As $\delta\to 0$ then
$\varepsilon\to 0$, $(\phi^{2},\eta^{2})$ approximates $(\phi^{1},\eta^{1})$.
We use this to show $P_{N}=\widetilde{P}_{N}$.

######  Lemma 5.8.

In the setting of Proposition 5.7, with
$(\phi^{1},\eta^{1}):=(\phi_{\tau}(\cdot+W_{\tau}),\eta_{\tau}-W_{\tau})$ and
$E_{N}$ defined in (5.3), the $(M\times 1_{A>0}\,dA)$-measure of $E_{N}$ is
finite.

###### Proof.

The event $\\{\tau<N\\}$ agrees with
$\\{\mathcal{A}_{\phi_{N}}(\eta_{N}([0,N]))<A\\}$. Thus, the $(M\times
1_{A>0}\,dA)$-measure of $E_{N}$ is
$M[\mathcal{A}_{\phi_{N}}(\eta_{N}([0,N]))1_{\widetilde{E}_{N}}]$, where
$\widetilde{E}_{N}=\\{|(\phi_{\tau},\rho)|,|(\phi_{0},\rho)|<N\\}$. By the
second property of $M$ from Lemma 5.6, and the fact that for fixed $t$ the
curves $\operatorname{SLE}_{\kappa}^{t}$ and centered
$\mathrm{rSLE}_{\kappa}^{t}$ have the same law, the law of
$(\phi_{N}(\cdot+W_{N}),\eta_{N}-W_{N})$ is
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}\operatorname{SLE}_{\kappa}^{N}$.
Thus it suffices to show that

$({{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}\times\operatorname{SLE}_{\kappa}^{N})[\mathcal{A}_{\phi}(\eta([0,N]))1_{|(f\bullet_{\gamma}\phi,\rho)|<N}]<\infty,$
(5.5)

where $f:\mathbbm{H}\backslash\eta([0,N])\to\mathbbm{H}$ is the conformal map
sending $f(\eta(N))=0$ and with $f(z)=z+O(1)$ as $z\to\infty$.

We first claim that for some $C>0$,

$\mathbbm{P}\mathopen{}\mathclose{{}\left[\eta([0,N])\not\subset
R_{x}}\right]<2e^{-x^{2}/C^{2}}\quad\text{ for
}\eta\sim\operatorname{SLE}_{\kappa}^{N}\text{ and }x>0,\text{ where
}R_{x}:=[-x,x]\times[0,2\sqrt{N}].$ (5.6)

Indeed, the driving function $W_{t}$ of $\operatorname{SLE}_{\kappa}$ is a
multiple of Brownian motion so $\sup_{0<t<N}|W_{t}|$ is sub-Gaussian, and for
compact $K\subset\overline{\mathbbm{H}}$ such that $\mathbbm{H}\backslash K$
is simply connected, $\mathrm{hcap}(K)=2N$ implies
$K\subset\mathbbm{R}\times[0,2\sqrt{N}]$.

Next, fix $K$ (we will later set $K=\eta$) and let $f:\mathbbm{H}\backslash
K\to\mathbbm{H}$ be a conformal map fixing $\infty$. Then

$\displaystyle{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}[\mathcal{A}_{\phi}(R_{x})1_{|(f\bullet_{\gamma}\phi,\rho)|<N}]=\int_{\mathbbm{R}}e^{-\gamma
c}\mathbbm{E}[\mathcal{A}_{\widetilde{h}+c}(R_{x})1_{|(f\bullet_{\gamma}(\widetilde{h}+c),\rho)|<N)}]\,dc$
$\displaystyle=\int_{\mathbbm{R}}\mathbbm{E}[\mathcal{A}_{\widetilde{h}}(R_{x})1_{|(f\bullet_{\gamma}\widetilde{h},\rho)+c|<N)}]\,dc=2N\mathbbm{E}[\mathcal{A}_{\widetilde{h}}(R_{x})],$

where the expectation is taken over $h\sim P_{\mathbbm{H}}$ and we write
$\widetilde{h}=h-\frac{\gamma}{4}G_{\mathbbm{H}}(\cdot,0)+(Q-\frac{3\gamma}{4})G_{\mathbbm{H}}(\cdot,\infty)$,
we use $-\frac{\gamma}{4}+(Q-\frac{3\gamma}{4})-Q=-\gamma$ in the first
equality and $\mathcal{A}_{\widetilde{h}+c}=e^{\gamma
c}\mathcal{A}_{\widetilde{h}}$ in the second equality, and we interchange
$\mathbbm{E}$ and $\int$ in the third equality. Since for some $C^{\prime}$ we
have $(\widetilde{h}-h)\leq C^{\prime}\log x$ on $R_{x}$,

$\mathbbm{E}[\mathcal{A}_{\widetilde{h}}(R_{x})]\leq
x^{C^{\prime}}\mathbbm{E}[\mathcal{A}_{h}(R_{x})]=x^{C^{\prime}}\int_{R_{x}}(2\operatorname{Im}z)^{-\frac{\gamma^{2}}{2}}|z|_{+}^{2\gamma^{2}}dz\lesssim
x^{C^{\prime}+2\gamma^{2}+1}.$

where we use here the formula for $G_{\mathbbm{H}}$ from (2.1). Thus,
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}[\mathcal{A}_{\phi}(R_{x})1_{|(f\bullet_{\gamma}\phi,\rho)|<N}]$
is bounded above by a polynomial in $x$; combining with (5.6) gives the
desired (5.5).

∎

######  Proposition 5.9.

In the setting of Proposition 5.7, let
$\mathcal{C}=(\eta_{\tau}([0,\tau]),\phi_{\tau},\eta_{\tau})/{\sim_{\gamma}}$.
Then the law of $(\phi_{0},\mathcal{C},A)$ is

${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}\,P_{a}\,1_{a>0}da.$

###### Proof.

Sample $(\\{(\phi_{t},\eta_{t})\\}_{t\geq 0},A_{1},A_{2})$ from $M\times
1_{A_{1},A_{2}>0}dA_{1}dA_{2}$. Let $\tau^{1}$ (resp. $\tau^{2}$) be the time
$t$ that $\mathcal{A}_{\phi_{t}}(\eta_{t}([0,t]))$ equals $A_{1}$ (resp.
$A_{1}+A_{2}$). Let
$\mathcal{C}_{2}=(\eta_{\tau^{2}}([0,\tau^{2}-\tau^{1}]),\phi_{\tau^{2}},\eta_{\tau^{2}}|_{[0,\tau^{2}-\tau^{1}]})/{\sim_{\gamma}}$.
We will show that the law of $(\phi_{0},\mathcal{C}_{2},A_{1},A_{2})$ is

${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{0})P_{A_{2}}(d\mathcal{C}_{2})1_{A_{1},A_{2}>0}dA_{1}dA_{2}.$
(5.7)

To simplify notation, let
$(\phi^{j},\eta^{j})=(\phi_{j}(\cdot+W_{\tau_{j}}),\eta_{j}-W_{\tau_{j}})$ for
$j=1,2$ and let $\eta^{12}=\eta^{2}|_{[0,\tau^{2}-\tau^{1}]}$. Let
$z=\eta^{2}(\tau^{2}-\tau^{1})$. See Figure 9.

Figure 9:  Figure for Proposition 5.9. The pair $(\phi^{j},\eta^{j})$
corresponds to $(\phi_{\tau_{j}},\eta_{\tau_{j}})$ translated so that the
curve starts from $0$ rather than $W_{\tau_{j}}$. The curve $\eta^{12}$ is the
initial segment of $\eta^{2}$.

Let $S=A_{1}+A_{2}$, then by a change of variables the law of
$(\\{(\phi_{t},\eta_{t})\\}_{t\geq 0},A_{1},S)$ is
$M\,1_{A_{1}\in[0,S]}dA_{1}\,1_{S>0}dS$. By Proposition 5.7 the law of
$(A_{1},\phi^{2},\eta^{2},\tau^{2})$ is

$C\cdot
1_{A_{1}\in[0,S]}dA_{1}\,{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{2})\operatorname{SLE}_{\kappa}^{\tau^{2}}(d\eta^{2})1_{\tau^{2}>0}d\tau^{2},\quad
S:=\mathcal{A}_{\phi^{2}}(\eta^{2}([0,\tau^{2}])).$

Since $z$ is the point such that $\eta^{2}$ covers $S-A_{1}$ units of quantum
area before hitting $z$, we conclude the law of
$(z,\phi^{2},\eta^{2},\tau^{2})$ is

$C\cdot
1_{z\in\eta^{2}([0,\tau^{2}])}\mathcal{A}_{\phi^{2}}(dz){{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{2})\operatorname{SLE}_{\kappa}^{\tau^{2}}(d\eta^{2})1_{\tau^{2}>0}d\tau^{2}.$

By Lemma 5.10 stated below, the law of
$(\phi^{2},\eta^{1},\eta^{12},z,\tau^{1})$ is

$C\cdot\operatorname{SLE}_{\kappa}^{z}(d\eta^{12})\,\mathcal{A}_{\phi^{2}}(dz)\,{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{2})\operatorname{SLE}_{\kappa}^{\tau^{1}}(d\eta^{1})\,1_{\tau^{1}>0}d\tau^{1}$

where $\operatorname{SLE}_{\kappa}^{z}$ denotes forward
$\operatorname{SLE}_{\kappa}$ run until the time it hits $z$. By Proposition
5.3, the law of $(\phi^{1},\eta^{1},\tau^{1},\mathcal{C}_{2},A_{2})$ is

$C\cdot{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{1})\operatorname{SLE}_{\kappa}^{\tau^{1}}(d\eta^{1})1_{\tau^{1}>0}d\tau^{1}P_{A_{2}}(d\mathcal{C}_{2})1_{A_{2}>0}dA_{2}.$

By Proposition 5.7 applied to the measure
$C\cdot{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi^{1})\operatorname{SLE}_{\kappa}^{\tau^{1}}(d\eta^{1})1_{\tau^{1}>0}d\tau^{1}$,
we conclude that the law of $(\phi_{0},\mathcal{C}_{2},A_{1},A_{2})$ is (5.7).

As a consequence of (5.7), if we sample $(\\{(\phi_{t},\eta_{t})\\}_{t\geq
0},A_{1},A_{2})$ from
$M\times\mathrm{Unif}_{[0,\varepsilon]}(dA_{1})1_{A_{2}>0}dA_{2}$ where
$\mathrm{Unif}_{[0,\varepsilon]}$ is the uniform probability measure on
$[0,\varepsilon]$, then the law of $(\phi_{0},\mathcal{C}_{2},A_{1},A_{2})$ is
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}(d\phi_{0})P_{A_{2}}(d\mathcal{C}_{2})\mathrm{Unif}_{[0,\varepsilon]}(dA_{1})1_{A_{2}>0}dA_{2}$.
Sending $\varepsilon\to 0$ yields the desired result. ∎

######  Lemma 5.10.

Let $z\in\mathbbm{H}$ and consider a pair $(\eta,T)\sim
1_{E}\operatorname{SLE}_{\kappa}^{t}1_{t>0}dt$ where $E$ is the event that $z$
lies in the range of the curve. Let $\tau_{z}$ be the time $\eta$ hits $z$,
let $\eta^{12}=\eta|_{[0,\tau_{z}]}$, let $T^{1}=T-\tau_{z}$ and let
$\eta^{1}=f^{-1}\circ\eta(\cdot+\tau_{z})|_{[0,T^{1}]}$ where
$f:\mathbbm{H}\to\mathbbm{H}\backslash\eta([0,\tau_{z}])$ is the conformal map
with $f(0)=z$ and $f(w)=w+O(1)$ as $w\to\infty$. Then the law of
$(\eta^{12},\eta^{1},T^{1})$ is
$\operatorname{SLE}_{\kappa}^{z}\operatorname{SLE}_{\kappa}^{t^{1}}1_{t^{1}>0}dt^{1}$,
where $\operatorname{SLE}_{\kappa}^{z}$ is the law of
$\operatorname{SLE}_{\kappa}$ run until it hits $z$.

###### Proof.

From the change of variables $T^{1}=T-\tau_{z}$, the law of $T^{1}$ is
$1_{t^{1}>0}dt^{1}$. Conditioned on $T^{1}$ the conditional law of $\eta^{12}$
is $\operatorname{SLE}_{\kappa}^{z}$, and by the domain Markov property,
conditioned on $T^{1}$ and $\eta^{12}$ the conditional law of $\eta^{1}$ is
$\operatorname{SLE}_{\kappa}^{T^{1}}$. ∎

###### Proof of Proposition 5.5.

By Proposition 1.6 and Definition 2.8, the area $a$ quantum cell is the
quantum surface obtained by mating $(X_{t},Y_{t})_{t\geq
0}\sim\mathrm{CRT}_{\kappa}$ for quantum area $a$.

Recall $M$ from Lemma 5.6. Proposition 5.9 implies that the time-evolution of
$M$ for quantum area $a$ arises from conformally welding an area $a$ quantum
cell to $\phi_{0}$, and so corresponds to mating trees sampled from
$\mathrm{CRT}_{\kappa}$ for quantum area $a$. Sending $a\to\infty$, we
conclude that $M$ is the process of Theorem 1.7 when $n=0$ and
$\delta=Q-\frac{3\gamma}{4}$. The first property of $M$ in Lemma 5.6 is the
desired description of $(\phi_{\tau},\eta_{\tau})$, so we obtain Theorem 1.7
for $n=0,\delta=Q-\frac{3\gamma}{4}$. Finally, we extend to arbitrary
$\delta\in\mathbbm{R}$ by Proposition 4.5. ∎

######  Remark 5.11.

The main difficulty in adapting our proof of Proposition 5.5 to the case
$\gamma=\sqrt{2},\kappa=8$ is in proving the analog of Proposition 5.7, where
${{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{3\gamma}{4},0),(Q-\frac{3\gamma}{4},\infty)}={{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{Q}{2},0),(\frac{Q}{2},\infty)}$
has to be replaced by $\lim_{\beta\uparrow
Q}\frac{1}{Q-\beta}{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{\beta}{2},0),(\frac{Q}{2},\infty)}$.
This limit is constructed in [ASY22, Section 2.5]. Our present argument does
not apply since the $(M\times 1_{A>0}\,dA)$-measure of $E_{N}$ is infinite,
that is, Lemma 5.8 fails for $\kappa=8$. One could try using a finite event
such as $E_{N}\cap\\{A<N\\}$, or instead directly tackle the $\kappa=8$
variant of Proposition 5.5 by taking a $\kappa\to 8$ limit.

###  5.4 Proof of Theorem 1.7

In this section, we prove Theorem 1.7. The first step is to extend Proposition
5.5 to the setting where insertions are allowed but the process stops before
any insertions hit the curve:

######  Proposition 5.12.

Theorem 1.7 holds for any stopping time $\tau$ such that a.s.
$g_{\tau}(z_{j})\not\in\eta_{\tau}$ for all $j$.

Given this, we can complete the proof of Theorem 1.7.

###### Proof of Theorem 1.7.

Proposition 4.2 gives us a process
$(\widetilde{\phi}_{t},\widetilde{\eta}_{t})_{0\leq t\leq\tau}$ such that
$\widetilde{\phi}_{0}$ has law
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},0),(\alpha_{j},z_{j})_{j},(\delta,\infty)}$
and $(\widetilde{\phi}_{\tau},\widetilde{\eta}_{\tau})$ has law (1.2), where
$\tau$ is a stopping time for $\mathcal{F}_{t}=\sigma(\widetilde{\eta}_{t})$.
Let $S=\\{t:g_{t}(z_{j})=W_{t}\text{ for some }j\\}$. By Proposition 5.12, if
$\sigma,\sigma^{\prime}$ are stopping times for $\mathcal{F}_{t}$ such that
$[\sigma,\sigma^{\prime}]\cap S=\emptyset$, then on $[\sigma,\sigma^{\prime}]$
this process can alternatively be described by mating continuum random trees
independent of $\mathcal{F}_{\sigma}$ until the stopping time
$\sigma^{\prime}$. Since $S$ is at most finite, by continuity we conclude that
the _whole_ process $(\widetilde{\phi}_{t},\widetilde{\eta}_{t})_{0\leq
t\leq\tau}$ arises from the mating-of-trees procedure described in Theorem
1.7. This completes the proof. ∎

The proof of Proposition 5.12 is identical to that of Proposition 4.5, but
with different calculations which we detail below. We will show how to weight
the Liouville field to introduce insertions in Lemma 5.14, then apply this to
the special case of Theorem 1.7 to obtain the more general statement desired.

For Lemma 5.14 we need Lemma 4.3 and the following. Recall
$G(z,w)=-\log|z-w|-\log|z-\overline{w}|$.

######  Lemma 5.13.

Suppose $K\subset\overline{\mathbbm{H}}$ is compact, $\mathbbm{H}\backslash K$
is simply connected and there is a conformal map
$g:\mathbbm{H}\to\mathbbm{H}\backslash K$ such that
$\lim_{|z|\to\infty}g(z)-z=0$. Let $\varepsilon>0$. Then for
$z_{0}\in\mathbbm{H}$ such that $\operatorname{Im}z_{0}>\varepsilon$ and
$u\not\in g(B_{\varepsilon}(z_{0}))$ we have $\int
G(u,v)(g_{*}\theta_{\varepsilon,z_{0}})(dv)=G(u,g(z_{0}))$, where
$\theta_{\varepsilon,z_{0}}$ is the uniform probability measure on $\partial
B_{\varepsilon}(z_{0})$. For $x_{0}\in\partial\mathbbm{H}$ such that
$g([x_{0}-\varepsilon,x_{0}+\varepsilon])\subset\mathbbm{R}$ and $u\not\in
g(B_{\varepsilon}(x_{0})\cap\mathbbm{H})$ we have $\int
G(u,v)(g_{*}\theta_{\varepsilon,x_{0}})(dv)=G(u,g(x_{0}))$, where
$\theta_{\varepsilon,x_{0}}$ is the uniform probability measure on $\partial
B_{\varepsilon}(x_{0})\cap\mathbbm{H}$.

###### Proof.

For the first assertion, the function $G(u,g(\cdot))$ is harmonic on
$\overline{B_{\varepsilon}(z_{0})}$ because $G(u,\cdot)$ is harmonic and $g$
is conformal. Thus, by the mean value property of harmonic functions, $\int
G(u,v)(g_{*}\theta_{\varepsilon,z_{0}})(dv)=\int
G(u,g(v))\theta_{\varepsilon,z_{0}}(dv)=G(u,g(z_{0}))$. The second assertion
is proved similarly. ∎

Now we add insertions to the Liouville field by weighting. We place
constraints on the insertions so that they sum to $Q$; this allows us to work
with $G(\cdot,\cdot)$ rather than the more complicated
$G_{\mathbbm{H}}(\cdot,\cdot)$. Recall that $\theta_{\varepsilon,\infty}$ is
the uniform probability measure on $\partial
B_{1/\varepsilon}(0)\cap\mathbbm{H}$.

######  Lemma 5.14.

In the setting of Lemma 5.13, let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\overline{\mathbbm{H}}$ and
$\beta=-\sum_{j}\alpha_{j}$. Let $I=\\{i:z_{i}\in\mathbbm{H}\\}$ and
$B=\\{b:z_{b}\in\mathbbm{R}\\}$. Suppose $\varepsilon>0$ satisfies
$|z_{j}-z_{k}|>2\varepsilon$ for $j\neq k$,
$\operatorname{Im}z_{i}>\varepsilon$ for $i\in I$,
$g((z_{b}-\varepsilon,z_{b}+\varepsilon))\subset\mathbbm{R}$ for $b\in B$, and
$g(-1/\varepsilon),g(1/\varepsilon)\in\mathbbm{R}$. Let
$\mathbbm{H}_{\varepsilon}^{g}=\mathbbm{H}\backslash(g(\mathbbm{H}\backslash
B_{1/\varepsilon}(0))\cup\bigcup_{j}g(B_{\varepsilon}(z_{j})))$ and
$\theta_{\varepsilon}:=\sum_{j}\alpha_{j}\theta_{\varepsilon,z_{j}}+\beta\theta_{\varepsilon,\infty}$.
For any $(\alpha,w)\in\mathbbm{R}\times\mathbbm{H}_{\varepsilon}^{g}$ and
nonnegative function $F(\phi)$ depending only on
$\phi|_{\mathbbm{H}_{\varepsilon}^{g}}$,

$\displaystyle{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha,w),(Q-\alpha,\infty)}[\varepsilon^{-\beta^{2}+2\alpha\beta}\prod_{i\in
I}\varepsilon^{\alpha_{i}^{2}/2}\prod_{b\in
B}\varepsilon^{\alpha_{b}^{2}}\times
e^{(g^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon})}F(\phi)]$
$\displaystyle=\prod_{i\in
I}|g_{\tau}^{\prime}(z_{i})|^{2\Delta_{\alpha_{i}}}\prod_{b\in
B}|g_{\tau}^{\prime}(z_{b})|^{\Delta_{2\alpha_{b}}}{{\mathrm{LF}}}_{\mathbbm{H}}^{(\alpha,w),(\alpha_{j},g(z_{j}))_{j},(\beta+Q-\alpha,\infty)}[F(\phi)].$

###### Proof.

The proof of this is identical to that of Lemma 4.4; the only difference is in
the computations of
$Z_{\varepsilon}:=\mathbbm{E}[e^{(h,g_{*}\theta_{\varepsilon})}]$ and
$e^{(g_{t}^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon})}$. To shorten
notation write $\theta_{\varepsilon,j}=\theta_{\varepsilon,z_{j}}$.

To compute $Z_{\varepsilon}$ we first need
$\operatorname{Var}((h,g_{*}\theta_{\varepsilon}))$. Since $\int
1\,g_{*}\theta_{\varepsilon}(dv)=0$, the law of
$(h,g_{*}\theta_{\varepsilon})$ when $h\sim P_{\mathbbm{H}}$ agrees with that
when $h$ is instead considered as a distribution modulo additive constant.
Thus, $\operatorname{Var}((h,g_{*}\theta_{\varepsilon}))=\iint
G(u,v)g_{*}\theta_{\varepsilon}(du)g_{*}\theta_{\varepsilon}(dv)$ where
$G(u,v)=-\log|u-v|-\log|u-\overline{v}|$.

For $i\in I$ we have

$\displaystyle\iint
G(u,v)g_{*}\theta_{\varepsilon,i}(du)g_{*}\theta_{\varepsilon,i}(dv)=\int
G(g(z_{i}),g(v))\theta_{\varepsilon,i}(dv)=-\log|g^{\prime}(z_{i})|+\log(2\operatorname{Im}(g(z_{i})))+\log\varepsilon,$

where in the first equality we use Lemma 5.13 and in the second equality we
use the fact that the function $f:B_{\varepsilon}(z_{i})\to\mathbbm{R}$
defined by $f(w)=-\log|g_{t}(w)-g_{t}(z_{i})|+\log|w-z_{i}|$ for $w\neq z_{i}$
and $f(z_{i})=-\log|g_{t}^{\prime}(z_{i})|$ is harmonic. Similarly, $\iint
G(u,v)g_{*}\theta_{\varepsilon,b}(du)g_{*}\theta_{\varepsilon,b}(dv)=-2\log|g_{\tau}^{\prime}(z_{b})|+2\log\varepsilon$,
and proceeding similarly we can compute all cross-terms. Summing gives the
value of $\operatorname{Var}((h,g_{*}\theta_{\varepsilon}))$, and finally

$Z_{\varepsilon}=e^{\frac{1}{2}{\operatorname{Var}}((h,g_{*}\theta_{\varepsilon}))}=\varepsilon^{-\beta^{2}+\sum_{i}\alpha_{i}^{2}/2+\sum_{b}\alpha_{b}^{2}}\prod_{i\in
I}|g^{\prime}(z_{i})|^{-\alpha_{i}^{2}/2}\prod_{b\in
B}|g^{\prime}(z_{b})|^{-\alpha_{b}^{2}}C_{\gamma}^{(\alpha_{j},g(z_{j}))_{j},(\beta,\infty)}.$

Next, by Lemmas 4.3 and 5.13 we have
$(G(\cdot,w),g_{*}\theta_{\varepsilon})=-2\beta\log\varepsilon+\sum_{j}\alpha_{j}G(z_{j},w)$,
so with $\phi=h+\alpha G(\cdot,w)+c$, we have

$e^{(g^{-1}\bullet_{\gamma}\phi,\theta_{\varepsilon})}=\varepsilon^{2\alpha\beta}\frac{C_{\gamma}^{(\alpha,w),(\alpha_{j},g(z_{j}))_{j},(\delta,\infty)}}{C_{\gamma}^{(\alpha_{j},g(z_{j}))_{j},(\beta,\infty)}}e^{(h,g_{*}\theta_{\varepsilon})}.$

The rest of the argument is identical to that of Lemma 4.4. ∎

###### Proof of Proposition 5.12.

By Proposition 5.5, Theorem 1.7 holds for $n=0,\delta=Q+\frac{\gamma}{4}$.
Applying the argument of Proposition 4.5 with Lemma 5.14 as input, we obtain
the $n\geq 0$ case when the insertions $(\alpha_{j},z_{j})_{j}$ and
$(\delta,\infty)$ satisfy
$-\frac{1}{\sqrt{\kappa}}+\sum_{j}\alpha_{j}+\delta=Q$. We were able to use
Lemma 5.14 because of the condition $g_{\tau}(z_{j})\not\in\eta_{\tau}$. A
final application of Proposition 4.5 removes this constraint. ∎

##  6 The $\kappa\in(4,8)$ LCFT zipper

In this section we prove Theorem 1.5.

[DMS21, Theorem 6.9] says that when a particular GFF variant is cut by a
forward $\operatorname{SLE}_{\kappa}$ with $\kappa\in(4,8)$, the connected
components in the complement of the curve are a pair of independent forested
lines. In fact, their argument (together with the addition of a constant
$\mathbf{c}\sim dc$) proves the following stronger result.

######  Proposition 6.1.

Theorem 1.5 holds for $n=0$ and $\delta=Q+\frac{\gamma}{4}$.

###### Proof.

The setup of [DMS21, Theorem 6.9] starts with a field
$h_{0}=h+\frac{\gamma}{2}\log|\cdot|$ where $h\sim P_{\mathbbm{H}}$, and uses
Proposition 4.1 to get a field and curve $h_{t},\eta_{t}$ where the law of
$\eta_{t}$ is reverse $\operatorname{SLE}_{\kappa}$ run for time $t$ and the
conditional law of $h_{t}$ given $\eta_{t}$ is
$\widetilde{h}+\frac{\gamma}{2}\log|\cdot-\eta_{t}(0)|$ where $\widetilde{h}$
is a Neumann GFF with a normalization that depends on $\eta_{t}$.

[DMS21, Theorem 6.9] states that the bounded connected components in
$\mathbbm{H}\backslash\eta_{t}$ are a countable collection of quantum surfaces
arising from a Poisson point process of LQG disks, and by the definition of
forested line in [DMS21] these quantum surfaces come from a pair of
independent forested lines. In fact, their argument not only proves
independence of these quantum surfaces, but also implicitly proves their
independence from $h_{0}$; this gives the independence of $h_{0}$ and the
forested lines. The fact that $(h_{t},\eta_{t})$ is obtained by mating the
forested lines as in Proposition 1.4 follows from local absolute continuity
with respect to that setting.

Finally, if we add a constant $\mathbf{c}$ chosen from Lebesgue measure on
$\mathbbm{R}$, both $h_{0}$ and $h_{t}(\cdot+\eta_{t}(0))$ have the marginal
law of the Liouville field
${{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{\gamma}{4},0),(Q+\frac{\gamma}{4},\infty)}$.
Since forested lines are scale-invariant in law (i.e. are invariant in law
under adding a constant to the field), this completes the proof of Theorem 1.5
when $n=0,\delta=Q+\frac{\gamma}{4}$ and $\tau=t$ is a deterministic time. The
result where $\tau\in\\{2^{-m}k:k,m\in\mathbbm{Z}\\}$ a.s. is then immediate,
and a limit gives the result for general $\tau$ (this is the same argument
proving the strong Markov property of Brownian motion from the Markov
property). ∎

###### Proof of Theorem 1.5.

Starting with Proposition 6.1, Proposition 4.5 gives Theorem 1.5 for $n=0$.
Then the arguments of Sections 5.4 yield the full Theorem 1.5. ∎

##  7 The boundary BPZ equation for Liouville conformal field theory

In this section we use the LCFT quantum zipper to prove the boundary BPZ
equations stated in Theorem 1.10. Let $m,n\geq 0$. Let
$(\alpha_{j},z_{j})\in\mathbbm{R}\times\mathbbm{H}$ for $j\leq m$ and assume
$z_{1},\dots,z_{m}$ are distinct. Let
$-\infty=x_{0}<x_{1}<\dots<x_{n}<x_{n+1}=+\infty$, let
$\beta_{1},\dots,\beta_{n}\in\mathbbm{R}$ and let $\delta\in\mathbbm{R}$. Let
$\beta_{*}\in\\{-\frac{\gamma}{2},-\frac{2}{\gamma}\\}$ and recall the LCFT
correlation function $F_{\beta_{*}}(w,(z_{j})_{j},(x_{k})_{k})$ defined in
(1.5).

In Section 7.1 we prove the BPZ equation holds for
$\beta_{*}=-\frac{2}{\gamma}$, in Section 7.2 we handle the case
$\beta_{*}=-\frac{\gamma}{2}$ and $\gamma\in(0,\sqrt{2})$ and in Section 7.3
we settle the case $\beta_{*}=-\frac{\gamma}{2}$ and $\gamma\in(\sqrt{2},2)$
case. These sections use the coupling of LCFT with
$\operatorname{SLE}_{\kappa}$ for $\kappa\in(0,4],(8,\infty),(4,8)$
respectively.

###### Proof of Theorem 1.10.

For $\gamma\in(0,2]$ the $\beta_{*}=-\frac{2}{\gamma}$ BPZ equation is shown
in Lemma 7.2. For $\gamma\in(0,\sqrt{2})$ and $\beta_{*}=-\frac{\gamma}{2}$ it
is shown in Lemma 7.5, and for $\gamma\in(\sqrt{2},2)$ and
$\beta_{*}=-\frac{\gamma}{2}$ see Lemma 7.8. Finally, when $\gamma=2$ we have
$-\frac{\gamma}{2}=-\frac{2}{\gamma}$ so the $\beta_{*}=-\frac{\gamma}{2}$ BPZ
equation has already been settled by the first case. ∎

###  7.1 Case: $\beta_{*}=-\frac{2}{\gamma}$

Let $\kappa\leq 4$ and $\gamma=\sqrt{\kappa}$. Consider reverse
$\operatorname{SLE}_{\kappa}$ where the driving function $W_{t}$ is given by
$W_{0}=w$ and $dW_{t}=\sqrt{\kappa}dB_{t}$. Let $(\eta_{t})_{t\geq 0}$ be the
family of curves and $g_{t}$ the corresponding Loewner maps, and let
$T\leq\infty$ be the first time $t$ that $g_{t}(x_{k})\in\eta_{t}$ for some
$k$. For $t<T$ define

$M_{t}=\prod_{j}|g_{t}^{\prime}(z_{j})|^{2\Delta_{\alpha_{j}}}\prod_{k}|g_{t}^{\prime}(x_{k})|^{\Delta_{\beta_{k}}}F_{-\frac{2}{\gamma}}(W_{t},(g_{t}(z_{j}))_{j},(g_{t}(x_{k}))_{k}).$
(7.1)

######  Lemma 7.1.

$M_{t}$ is a local martingale.

###### Proof.

Let $\overline{F}_{-\frac{2}{\gamma}}(w,(z_{j})_{j},(x_{k})_{k})<\infty$ be
defined via (1.5) except with the integrand replaced by its absolute value,
and let
$\overline{M}_{t}=\prod_{j}|g_{t}^{\prime}(z_{j})|^{2\Delta_{\alpha_{j}}}\prod_{k}|g_{t}^{\prime}(x_{k})|^{\Delta_{\beta_{k}}}\overline{F}_{-\frac{2}{\gamma}}(w,(z_{j})_{j},(x_{k})_{k})$,
so $\overline{M}_{t}\geq|M_{t}|$. For $N>0$ let
$T_{N}=T\wedge\inf\\{t:\overline{M}_{t}\geq N\\}$. Since
$t\mapsto\overline{M}_{t}$ is continuous on $[0,T)$ we have
$\lim_{N\to\infty}T_{N}=T$ almost surely. Thus we are done once we show
$M_{t}$ stopped at $T_{N}$ is a martingale.

It suffices to show that for stopping times $\tau_{1}\leq\tau_{2}\leq T_{N}$
we have $\mathbbm{E}[M_{\tau_{2}}\mid\eta_{\tau_{1}}]=M_{\tau_{1}}$. Instead,
to simplify notation, we show that if $\tau\leq T_{N}$ is a stopping time then
$\mathbbm{E}[M_{\tau}]=M_{0}$; the proof of the desired claim is identical.

Sample
$\phi_{0}\sim{{\mathrm{LF}}}_{\mathbbm{H}}^{(\frac{\beta_{*}}{2},w),(\alpha_{j},z_{j})_{j},(\frac{\beta_{k}}{2},x_{k})_{k},(\frac{\delta}{2},\infty)}$,
and define $(\phi_{t},\eta_{t})$ by conformally welding the boundary arcs to
the left and right of $w$ as in Theorem 1.1. Let
$A_{t}=\mathcal{A}_{\phi_{t}}(\mathbbm{H})$, let
$L_{k,t}=\mathcal{L}_{\phi_{t}}(g_{t}(I_{k}))$ for $k\neq k_{*}$, and let
$L_{t}=\mathcal{L}_{\phi_{t}}((g_{t}(x_{k_{*}}),W_{t}))$ and
$R_{t}=\mathcal{L}_{\phi_{t}}((W_{t},g_{t}(x_{k_{*}+1}))$. Since the conformal
welding does not affect the quantum area nor quantum lengths of boundary
segments not adjacent to $w$, the processes $A_{t}$ and $L_{k,t}$ are
constant. Moreover, the conformal welding identifies segments of equal quantum
length, so $L_{t}-R_{t}=L_{0}-R_{0}$ for all $t$. Thus,
$G_{t}=\exp(-A_{t}-\sum_{k\neq k_{*}}\mu_{k}L_{k,t}-\mu_{L}(L_{t}-R_{t}))$ is
constant as $t$ varies. Consequently, writing $\mathbbm{E}$ to denote
expectation with respect to $\operatorname{rSLE}_{\kappa}^{\tau}$ and using
Theorem 1.1, we have

$\displaystyle M_{0}$
$\displaystyle={{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},w),(\alpha_{j},z_{j})_{j},(\frac{\beta_{k}}{2},x_{k})_{k},(\frac{\delta}{2},\infty)}[G_{0}]$
(7.2)
$\displaystyle=\mathbbm{E}[\prod_{j}|g_{\tau}^{\prime}(z_{j})|^{2\Delta_{\alpha_{j}}}\prod_{k}|g_{\tau}^{\prime}(x_{k})|^{\Delta_{\beta_{k}}}{{\mathrm{LF}}}_{\mathbbm{H}}^{(-\frac{1}{\sqrt{\kappa}},W_{\tau}),(\alpha_{j},g_{\tau}(z_{j}))_{j},(\frac{\beta_{k}}{2},g_{\tau}(x_{k}))_{k},(\frac{\delta}{2},\infty)}[G_{\tau}]]=\mathbbm{E}[M_{\tau}].$

Here, the integrals converge absolutely by the definition of $T_{N}$. This
completes the proof. ∎

If we assume that $F_{-\frac{2}{\gamma}}$ is smooth, then setting the drift
term of $dM_{t}$ to zero immediately gives Theorem 1.10 for
$\beta_{*}=-\frac{2}{\gamma}$. Since we do not have smoothness a priori, we
only know $F_{-\frac{2}{\gamma}}$ is a weak solution to the BPZ equation. A
hypoellipticity argument originally due to [Dub15, Lemma 5] shows that weak
solutions are also strong solutions; we instead follow [PW19, Lemma 4.4,
Proposition 2.6] which is closer to our setting.

######  Lemma 7.2.

Theorem 1.10 holds for $\beta_{*}=-\frac{2}{\gamma}$.

###### Proof.

Consider the diffusion
$X_{t}=(W_{t},(g_{t}(z_{j}))_{j},(g_{t}(x_{k}))_{k},(|g_{t}^{\prime}(z_{j})|)_{j},(|g_{t}^{\prime}(x_{k})|)_{k})$
where $W_{t}$ and $g_{t}$ are defined above (7.1). For each $t$ we have
$X_{t}\in\mathbbm{R}\times\mathbbm{H}^{m}\times\mathbbm{R}^{n}\times\mathbbm{R}^{m}\times\mathbbm{R}^{n}$.
We denote the coordinates of an element of
$\mathbbm{R}\times\mathbbm{H}^{m}\times\mathbbm{R}^{n}\times\mathbbm{R}^{m}\times\mathbbm{R}^{n}$
by
$(\mathbf{w},(\mathbf{z}_{j})_{j},(\mathbf{x}_{k})_{k},(\mathbf{a}_{j})_{j},(\mathbf{b}_{k})_{k})$;
with this notation, since $dg_{t}(z)=-\frac{2}{g_{t}(z)-W_{t}}dt$ and
$d|g_{t}^{\prime}(z)|=\operatorname{Re}\frac{2|g_{t}^{\prime}(z)|}{(g_{t}(z)-W_{t})^{2}}dt$,
the infinitesimal generator $A$ of $X_{t}$ is

$A=\frac{\kappa}{2}\partial_{\mathbf{w}}^{2}-\sum_{j}(\frac{2}{\mathbf{z}_{j}-\mathbf{w}}\partial_{\mathbf{z}_{j}}+\frac{2}{\overline{\mathbf{z}}_{j}-\mathbf{w}}\partial_{\overline{\mathbf{z}}_{j}})-\sum_{k}\frac{2}{\mathbf{x}_{k}-\mathbf{w}}\partial_{\mathbf{x}_{k}}+\sum_{j}\operatorname{Re}\frac{2\mathbf{a}_{j}}{(\mathbf{z}_{j}-\mathbf{w})^{2}}\partial_{\mathbf{a}_{j}}+\sum_{j}\frac{2\mathbf{b}_{k}}{(\mathbf{x}_{k}-\mathbf{w})^{2}}\partial_{\mathbf{b}_{k}}.$

Let
$F(\mathbf{w},(\mathbf{z}_{j})_{j},(\mathbf{x}_{k})_{k},(\mathbf{a}_{j})_{j},\mathbf{b}_{k})_{k})=\prod_{j}\mathbf{a}_{j}^{2\Delta_{\alpha_{j}}}\prod_{k}\mathbf{b}_{k}^{\Delta_{\beta_{k}}}F_{-\frac{2}{\gamma}}(\mathbf{w},(\mathbf{z}_{j})_{j},(\mathbf{x}_{k})_{k})$.
Since $M_{t}$ is a local martingale (Lemma 7.1), $F$ is a weak solution to
$AF=0$. The product rule then yields

$0=A\Big{(}\prod_{j}\mathbf{a}_{j}^{2\Delta_{\alpha_{j}}}\prod_{k}\mathbf{b}_{k}^{\Delta_{\beta_{k}}}F_{-\frac{2}{\gamma}}(\mathbf{w},(\mathbf{z}_{j})_{j},(\mathbf{x}_{k})_{k})\Big{)}=\prod_{j}\mathbf{a}_{j}^{2\Delta_{\alpha_{j}}}\prod_{k}\mathbf{b}_{k}^{\Delta_{\beta_{k}}}\mathcal{D}F_{-\frac{2}{\gamma}}(\mathbf{w},(\mathbf{z}_{j})_{j},(\mathbf{x}_{k})_{k})$

where $\mathcal{D}$ is the differential operator on
$\mathbbm{R}\times\mathbbm{H}^{m}\times\mathbbm{R}^{n}$ given by

$\mathcal{D}:=\frac{\kappa}{2}\partial_{\mathbf{w}}^{2}-\sum_{j}(\frac{2}{\mathbf{z}_{j}-\mathbf{w}}\partial_{\mathbf{z}_{j}}+\frac{2}{\overline{\mathbf{z}}_{j}-\mathbf{w}}\partial_{\overline{\mathbf{z}}_{j}})-\sum_{k}\frac{2}{\mathbf{x}_{k}-\mathbf{w}}\partial_{\mathbf{x}_{k}}+\sum_{j}\operatorname{Re}\frac{4\Delta_{\alpha_{j}}}{(\mathbf{z}_{j}-\mathbf{w})^{2}}+\sum_{j}\frac{2\Delta_{\beta_{k}}}{(\mathbf{x}_{k}-\mathbf{w})^{2}}.$

We conclude that $\mathcal{D}F_{-\frac{2}{\gamma}}=0$ in the distributional
sense.

The operator $\mathcal{D}$ is called _hypoelliptic_ if the weak solutions of