# Derivation of all structure constants for boundary Liouville CFT

Morris Ang  ,  Guillaume Remy  ,  Xin Sun  and  Tunan Zhu

###### Abstract.

We prove that the probabilistic definition of the most general boundary three-
point and bulk-boundary structure constants in Liouville conformal field
theory (LCFT) agree respectively with the formula proposed by Ponsot-Techsner
(2002) and by Hosomichi (2001). These formulas also respectively describe the
fusion kernel and modular kernel of the Virasoro conformal blocks, which are
important functions in various contexts of mathematical physics. As an
intermediate step, we obtain the formula for the boundary reflection
coefficient of LCFT proposed by Fateev-Zamolodchikov-Zamolodchikov (2000). Our
proof relies on the boundary Belavin-Polyakov-Zamolodchikov differential
equation recently proved by the first named author, and inputs from the
coupling theory of Liouville quantum gravity (LQG) and Schramm Loewner
evolution. Our results supply all the structure constants needed to perform
the conformal bootstrap for boundary LCFT. They also yield exact descriptions
for the joint law of the area and boundary lengths of basic LQG surfaces,
including quantum triangles and two-pointed quantum disks.

##  1\. Introduction

Liouville conformal field theory (LCFT) describes the law of the conformal
factor of random surfaces in Liouville quantum gravity. Introduced by Polyakov
[Pol81] in theoretical physics, LCFT was recently made rigorous in probability
theory, first in the case of the Riemann sphere in [DKRV16], and then in the
case of a simply connected domain with boundary in [HRV18]; see also [DRV16,
Rem18, GRV19] for the case of other topologies. Following the framework of
[BPZ84], the correlation functions of LCFT can be solved by the conformal
bootstrap program. In the probabilistic framework, this was recently carried
out on surfaces without boundary [KRV20], [GKRV20, GKRV21]. The initial input
of the conformal bootstrap are the _structure constants_. For surfaces without
boundary, it is the three-point correlation function on the sphere. It has an
exact expression called the DOZZ formula which was proposed in [DO94, ZZ96]
and proved in [KRV20].

For LCFT on surfaces with boundary, the structure constants are the
correlation functions on the disk with three points on the boundary, or one
point in the bulk and one on the boundary. The theory involves both the bulk
and the boundary Liouville potentials (see the Liouville action (1.3)). When
the bulk Liouville potential is absent, the structure constants were obtained
by Remy and Zhu [RZ22]. When there is one bulk insertion and no boundary ones,
the structure constant was obtained by Ang, Remy, and Sun in [ARS23],
confirming an earlier proposal of Fateev, Zamolodchikov and Zamolodchikov
(FZZ) in [FZZ00]. The conformal bootstrap is also applicable in the boundary
case and B. Wu [Wu22] recently proved a bootstrap formula for the annulus with
one insertion at each boundary.

In this paper we obtain the exact formula for the boundary three-point
structure constant proposed by Ponsot and Techsner [PT02] and for the bulk-
boundary structure constant proposed by Hosomichi [Hos01], in the most general
case where both the boundary and bulk Liouville potentials are present. As an
intermediate step, we obtain the formula for the boundary two-point function
of LCFT — also known as the boundary reflection coefficient — which was
proposed in [FZZ00]. This completes the derivation of all structure constants
required for the conformal bootstrap of boundary LCFT. We rely on a novel
Belavin-Polyakov-Zamolodchikov (BPZ) differential equation for LCFT on the
disk established in [Ang23] and inputs from the coupling theory of Liouville
quantum gravity and Schramm Loewner evolution. See Section 1.2 for a summary
of our method.

The Liouville structure constants have intrinsic interest beyond the conformal
bootstrap. The boundary three-point structure constant and the bulk-boundary
structure constant agree modulo prefactors with the fusion kernel and the
modular kernel of the Virasoro conformal blocks [PT02], an important special
function with various interpretations in mathematical physics. Moreover, the
boundary reflection coefficient gives the joint area and boundary length
distribution of an important family of Liouville quantum gravity surfaces
called two-pointed quantum disks [DMS21, AHS23a]; the boundary three-point
structure constant gives the corresponding information for a more general
family of Liouville quantum gravity surfaces called quantum triangles [ASY22,
AHS23b].

###  1.1. Main results

We start by recalling the probabilistic construction of LCFT on the disk from
[HRV18], which is adapted from the construction on the Riemann sphere
performed in [DKRV16]. By conformal invariance we will use the upper half
plane ${\mathbb{H}}$ as the base domain. Our presentation is brief with more
details provided in Section 2. Fix the global constants

(1.1) $\gamma\in(0,2)\quad\textrm{ and }\quad
Q=\frac{\gamma}{2}+\frac{2}{\gamma}.$

In physics LCFT is defined using a formal path integral. Fix $N$ points
$z_{i}\in\mathbb{H}$ with associated weights $\alpha_{i}\in\mathbb{R}$ and
similarly $M$ points $s_{j}\in\mathbb{R}$ with associated weights
$\beta_{j}\in\mathbb{R}$. The correlation function associated to these points
is given by the formal integral

(1.2)
$\displaystyle\left\langle\prod_{i=1}^{N}e^{\alpha_{i}\phi(z_{i})}\prod_{j=1}^{M}e^{\frac{\beta_{i}}{2}\phi(s_{i})}\right\rangle=\int_{\phi:\mathbb{H}\mapsto\mathbb{R}}D\phi\prod_{i=1}^{N}e^{\alpha_{i}\phi(z_{i})}\prod_{j=1}^{M}e^{\frac{\beta_{j}}{2}\phi(s_{j})}e^{-S_{L}(\phi)},$

where $S_{L}$ is the Liouville action given by:

(1.3) $\displaystyle
S_{L}(\phi)=\frac{1}{4\pi}\int_{\mathbb{H}}\left(|\partial^{\hat{g}}\phi|^{2}+QR_{\hat{g}}\phi+4\pi\mu
e^{\gamma\phi}\right)d\lambda_{\hat{g}}+\frac{1}{2\pi}\int_{\mathbb{R}}\left(QK_{\hat{g}}\phi+2\pi\mu_{B}e^{\frac{\gamma}{2}\phi}\right)d\lambda_{\partial\hat{g}}.$

Here $\hat{g}$ is a background metric with $R_{\hat{g}}$ and
$\lambda_{\hat{g}}$ representing the curvature and volume in the bulk,
respectively, and $(K_{\hat{g}},\lambda_{\partial\hat{g}})$ being their
boundary counterparts. The terms involving $\mu$ and $\mu_{B}$ are the bulk
and boundary Liouville potentials, respectively. Here $\mu>0$ and $\mu_{B}$ is
a complex valued function on $\mathbb{R}$ which is piecewise constant in
between boundary insertions. We assume the $s_{j}$’s are chosen in
counterclockwise order on $\mathbb{R}$ and denote by $\mu_{j}$ the value of
$\mu_{B}$ on the interval $(s_{j-1},s_{j})$, with convention $s_{0}=-\infty$,
$s_{M+1}=\infty$. We always assume $\Re(\mu_{j})\geq 0$. Furthermore we
frequently work under the following conditions on weights
$\alpha_{i},\beta_{j}$ known as the _Seiberg bound_ :

(1.4)
$\sum_{i=1}^{N}\alpha_{i}+\sum_{j=1}^{M}\frac{\beta_{j}}{2}>Q,\quad\alpha_{i}<Q,\quad\textrm{and}\quad\beta_{j}<Q.$

We now give a rigorous probabilistic meaning to (1.2) for $N=0$, $M=3$ and for
$N=1$, $M=1$. Let $P_{\mathbb{H}}$ be the probability measure corresponding to
the free-boundary Gaussian free field on ${\mathbb{H}}$ normalized to have
average zero on the semi-circle $\partial\mathbb{D}\cap{\mathbb{H}}$. Let the
infinite measure $\mathrm{LF}_{\mathbb{H}}(d\phi)$ be the law of
$\phi(z)=h(z)-2Q\log|z|_{+}+\mathbf{c}$, where $|z|_{+}:=\max(|z|,1)$ and
$(h,\mathbf{c})$ is sampled according to $P_{\mathbb{H}}\times[e^{-Qc}\,dc]$.
We call the field $\phi$ sampled from $\mathrm{LF}_{{\mathbb{H}}}$ a Liouville
field on ${\mathbb{H}}$. This definition of $\mathrm{LF}_{\mathbb{H}}$
corresponds to choosing the background metric in (1.3) to be
$\hat{g}(z)=|z|_{+}^{-4}$. We define the bulk and boundary Gaussian
multiplicative chaos (GMC) measures of $\phi$ as the limits (see e.g. [Ber17,
RV14]):

${\mathcal{A}}_{\phi}=\lim_{{\varepsilon}\to
0}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma\phi_{\epsilon}(z)}d^{2}z\textrm{ on
}{\mathbb{H}},\qquad\text{and}\qquad{\mathcal{L}}_{\phi}=\lim_{{\varepsilon}\to
0}\epsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}\phi_{\epsilon}(z)}dz\textrm{
on }{\mathbb{R}}.$

Given three points $s_{1},s_{2},s_{3}$ lying counterclockwise on
${\mathbb{R}}$, let $L_{j}$ be the ${\mathcal{L}}_{\phi}$-length of the
counterclock arc on $\partial{\mathbb{H}}$ from $s_{j-1}$ to $s_{j}$, namely
$L_{j}={\mathcal{L}}_{\phi}(s_{j-1},s_{j})$, where we identify $s_{0}$ as
$s_{3}$. Fix $\mu>0$ and $\mu_{1},\mu_{2},\mu_{3}\geq 0$. For
$\beta_{1},\beta_{2},\beta_{3}\in{\mathbb{R}}$ satisfying the bounds of (1.4),
the _boundary three-point correlation function_ of LCFT on ${\mathbb{H}}$ with
bulk cosmological constant $\mu>0$ and boundary cosmological constants
$(\mu_{j})_{1\leq j\leq 3}$ is defined by:

(1.5)
$\left\langle\prod_{j=1}^{3}e^{\frac{\beta_{j}}{2}\phi(s_{j})}\right\rangle=\int\prod_{j=1}^{3}e^{\frac{\beta_{j}}{2}\phi(s_{j})}\cdot
e^{-\mu{\mathcal{A}}_{\phi}({\mathbb{H}})-\sum_{j=1}^{3}\mu_{j}L_{j}}\mathrm{LF}_{\mathbb{H}}(d\phi).$

Similarly, consider a point $z\in\mathbb{H}$ and a point $s\in\mathbb{R}$. Let
the weights $\alpha,\beta\in\mathbb{R}$ obey (1.4), and let $\mu,\mu_{B}>0$.
We define the _bulk-boundary correlation function_ as:

(1.6) $\left\langle
e^{\alpha\phi(z)}e^{\frac{\beta}{2}\phi(s)}\right\rangle=\int
e^{\alpha\phi(z)}e^{\frac{\beta}{2}\phi(s)}\cdot
e^{-\mu{\mathcal{A}}_{\phi}({\mathbb{H}})-\mu_{B}{\mathcal{L}}_{\phi}(\mathbb{R})}\mathrm{LF}_{\mathbb{H}}(d\phi).$

Here although $\phi$ is only a generalized function, the factors
$\prod_{j=1}^{3}e^{\frac{\beta_{j}}{2}\phi(s_{j})}$ and
$e^{\alpha\phi(z)}e^{\frac{\beta}{2}\phi(s)}$ in the integrands can be made
sense of by regularization and Girsanov’s theorem. Moreover, the Seiberg
bounds (1.4) ensure that both integrations in (1.5) and (1.6) are finite. We
will review this construction in full detail in Section 2.2.

Due to conformal symmetry, the 3-point boundary correlation function has the
following form

(1.7)
$\left\langle\prod_{j=1}^{3}e^{\frac{\beta_{j}}{2}\phi(s_{j})}\right\rangle=\frac{H\mu^{\frac{2Q-\beta_{1}-\beta_{2}-\beta_{3}}{2\gamma}}}{|s_{1}-s_{2}|^{\Delta_{\beta_{1}}+\Delta_{\beta_{2}}-\Delta_{\beta_{3}}}|s_{1}-s_{3}|^{\Delta_{\beta_{1}}+\Delta_{\beta_{3}}-\Delta_{\beta_{2}}}|s_{2}-s_{3}|^{\Delta_{\beta_{2}}+\Delta_{\beta_{3}}-\Delta_{\beta_{1}}}},$

while the bulk-boundary correlation has the form:

(1.8) $\left\langle
e^{\alpha\phi(z)}e^{\frac{\beta}{2}\phi(s)}\right\rangle=\frac{G\mu^{\frac{2Q-2\alpha-\beta}{2\gamma}}}{|z-\overline{z}|^{2\Delta_{\alpha}-\Delta_{\beta}}|z-s|^{2\Delta_{\beta}}}.$

Here the conformal dimensions are given by:

$\Delta_{\alpha}=\frac{\alpha}{2}(Q-\frac{\alpha}{2}),\quad\textrm{and}\quad\Delta_{\beta}=\frac{\beta}{2}(Q-\frac{\beta}{2}).$

The function $H$ only depends on $\beta_{j}$ and $\mu_{j}/\sqrt{\mu}$ for
$1\leq j\leq 3$, while $G$ only depends on $\alpha,\beta$ and
$\mu_{B}/\sqrt{\mu}$. We call $H$ the _boundary three-point structure
constant_ and $G$ the _bulk-boundary structure constant_ for LCFT. For
simplicity, in the rest of the paper we will set the bulk cosmological
constant to be $\mu=1$ and write $H$ as
$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}$ and $G$ as
$G_{\mu_{B}}(\alpha,\beta)$.

The conjectural formulas in physics for $H$ and $G$ requires a crucial change
of variable:

(1.9)
$\mu_{B}(\sigma):=\sqrt{\frac{1}{\sin(\pi\frac{\gamma^{2}}{4})}}\cos\left(\pi\gamma(\sigma-\frac{Q}{2})\right).$

For $\Re(\mu_{B}(\sigma))>0$, we choose
$\Re(\sigma)\in(-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2})$.
Ponsot-Teschner [PT02] proposed a remarkable formula for $H$ under the
reparametrization:

(1.10) $H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}:=H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{B}(\sigma_{1}),\mu_{B}(\sigma_{2}),\mu_{B}(\sigma_{3}))}.$

The Ponsot-Teschner formula is expressed in terms of the Barnes’ double Gamma
function $\Gamma_{\frac{\gamma}{2}}(x)$ and the double sine function
$S_{\frac{\gamma}{2}}(x)=\frac{\Gamma_{\frac{\gamma}{2}}(x)}{\Gamma_{\frac{\gamma}{2}}(Q-x)}$,
which are prevalent in LCFT. Both functions admit a meromorphic extension on
$\mathbb{C}$ with an explicit pole structure; see Section A.2 for more
details. Given a function $f$ that will be taken to be
$\Gamma_{\frac{\gamma}{2}}$ or $S_{\frac{\gamma}{2}}$ below, we will use the
shorthand notation:

$f(a\pm b):=f(a-b)f(a+b)\quad\textrm{and}\quad f(a\pm b\pm
c):=f(a-b-c)f(a-b+c)f(a+b-c)f(a+b+c).$

The Ponsot-Teschner function is given by:111Our formula contains a $2\pi$
factor that was not present in [PT02]. We believe that it was an error as it
was claimed on [PT02, Page 8] that the residue of $H_{\mathrm{PT}}$ at
$\beta_{1}=2Q-\beta_{2}-\beta_{3}$ is 1, which is only true if the $2\pi$
factor is included. Moreover, this factor is consistent with the checks
performed in [RZ22] and with the $2\pi$ factor in $G_{\mathrm{Hos}}$.

(1.11)
$\displaystyle{H_{\mathrm{PT}}}\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=2\pi\left(\frac{\pi(\frac{\gamma}{2})^{2-\frac{\gamma^{2}}{2}}\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}\right)^{\frac{2Q-(\beta_{1}+\beta_{2}+\beta_{3})}{2\gamma}}$
$\displaystyle\times\frac{\Gamma_{\frac{\gamma}{2}}(Q-\frac{\beta_{2}}{2}\pm\frac{Q-\beta_{1}}{2}\pm\frac{Q-\beta_{3}}{2})}{S_{\frac{\gamma}{2}}(\frac{\beta_{3}}{2}-\sigma_{1}+\frac{Q}{2}\pm(\frac{Q}{2}-\sigma_{3}))S_{\frac{\gamma}{2}}(\frac{\beta_{1}}{2}+\sigma_{1}-\frac{Q}{2}\pm(\frac{Q}{2}-\sigma_{2}))\Gamma_{\frac{\gamma}{2}}(Q)\prod_{j=1}^{3}\Gamma_{\frac{\gamma}{2}}(Q-\beta_{j})}$
$\displaystyle\times\int_{\mathcal{C}}\frac{S_{\frac{\gamma}{2}}(\frac{Q-\beta_{2}}{2}+\sigma_{3}\pm(\frac{Q}{2}-\sigma_{2})+r)S_{\frac{\gamma}{2}}(\frac{Q}{2}\pm\frac{Q-\beta_{3}}{2}+\sigma_{3}-\sigma_{1}+r)}{S_{\frac{\gamma}{2}}(\frac{3Q}{2}\pm\frac{Q-\beta_{1}}{2}-\frac{\beta_{2}}{2}+\sigma_{3}-\sigma_{1}+r)S_{\frac{\gamma}{2}}(2\sigma_{3}+r)S_{\frac{\gamma}{2}}(Q+r)}\frac{dr}{\mathbf{i}}.$

Here $\mathcal{C}$ is a properly chosen contour from $-\mathbf{i}\infty$ to
$\mathbf{i}\infty$ such that the integral is meromorphic - namely a ratio of
two holomorphic functions - in all of the six variables in the entire complex
plane. We provide more details on the function ${H_{\mathrm{PT}}}$ and the
contour $\mathcal{C}$ in Section 2.1 below. The first main result of our paper
is the derivation of Ponsot-Teschner formula.

######  Theorem 1.1.

Let $\gamma\in(0,2)$, $\sum_{j=1}^{3}\beta_{j}>2Q$, $\beta_{j}<Q$, and
$\mu_{j}=\mu_{B}(\sigma_{j})>0$ for $j=1,2,3$. Then

(1.12) $\displaystyle H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}={H_{\mathrm{PT}}}\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}.$

Similarly the formula for the function $G$ will be expressed in terms of
$\alpha,\beta,\sigma$ where $\sigma$ is related to $\mu_{B}$ still by the
relation (1.9). We thus write
$G(\alpha,\beta,\sigma):=G_{\mu_{B}}(\alpha,\beta)$. The exact formula
$G_{\mathrm{Hos}}$ proposed for $G$ by Hosomichi [Hos01] is then given
by:222For $G_{\mathrm{Hos}}$ we have an extra
$2^{\Delta_{\beta}-2\Delta_{\alpha}}$ compared with [Hos01] which comes from a
different convention for the renormalization of the bulk insertion. This
factor is consistent with the FZZ formula of our previous work [ARS23].

$\displaystyle G_{\mathrm{Hos}}(\alpha,\beta,\sigma)=$ $\displaystyle 2\pi
2^{\Delta_{\beta}-2\Delta_{\alpha}}\left(\frac{\pi(\frac{\gamma}{2})^{2-\frac{\gamma^{2}}{2}}\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}\right)^{\frac{1}{\gamma}(Q-\alpha-\frac{\beta}{2})}\frac{\Gamma_{\frac{\gamma}{2}}(2Q-\frac{\beta}{2}-\alpha)\Gamma_{\frac{\gamma}{2}}(\alpha-\frac{\beta}{2})\Gamma_{\frac{\gamma}{2}}(Q-\frac{\beta}{2})^{3}}{\Gamma_{\frac{\gamma}{2}}(Q-\alpha)\Gamma_{\frac{\gamma}{2}}(Q-\beta)\Gamma_{\frac{\gamma}{2}}(\alpha)\Gamma_{\frac{\gamma}{2}}(Q)\Gamma_{\frac{\gamma}{2}}(\frac{\beta}{2})}$
$\displaystyle\times\int_{\mathcal{C}}\frac{d\sigma_{2}}{\mathbf{i}}e^{2\mathbf{i}\pi(Q-2\sigma)\sigma_{2}}\frac{S_{\frac{\gamma}{2}}(\frac{1}{2}(\alpha+\frac{\beta}{2}-Q)\pm\sigma_{2})}{S_{\frac{\gamma}{2}}(\frac{1}{2}(\alpha-\frac{\beta}{2}+Q)\pm\sigma_{2})}.$

The contour integral in this expression for $G_{\mathrm{Hos}}$ goes from
$-\mathbf{i}\infty$ to $\mathbf{i}\infty$ and is converging at
$\pm\mathbf{i}\infty$ if and only if the parameters $\alpha,\beta,\sigma$ obey
the conditions $\frac{\Re(\beta)}{2}<Q$,
$\Re(\sigma)\in\left[\frac{\Re(\beta)}{4},Q-\frac{\Re(\beta)}{4}\right]$. See
also Section 2.1 for more details on this function. Our second main result is
thus:

######  Theorem 1.2.

Let $\gamma\in(0,2)$, $\alpha+\frac{\beta}{2}>Q$, $\alpha<Q$, $\beta<Q$, and
$\mu_{B}=\mu_{B}(\sigma)>0$. Then:

(1.13) $\displaystyle
G(\alpha,\beta,\sigma)=G_{\mathrm{Hos}}(\alpha,\beta,\sigma).$

Our last main result concerns the reflection coefficient for boundary LCFT.
Formally speaking, this quantity is defined as
$R_{\mu_{1},\mu_{2}}(\beta):=|s_{1}-s_{2}|^{2\Delta_{\beta}}\left\langle
e^{\frac{\beta}{2}\phi(s_{1})}e^{\frac{\beta}{2}\phi(s_{2})}\right\rangle$ for
$\beta\in{\mathbb{R}}$, which by conformal symmetry does not depend on
$s_{1},s_{2}\in{\mathbb{R}}$. The name reflection coefficient comes from the
fact the values of $H$ at $\beta_{1}$ and at the reflected $2Q-\beta_{1}$ are
related by a factor of $R$; see (3.35). Although this is in the same spirit as
(1.7), there are multiple subtleties in the rigorous definition of
$R_{\mu_{1},\mu_{2}}(\beta)$. First, instead of integrating
$e^{\frac{\beta}{2}\phi(s_{1})}e^{\frac{\beta}{2}\phi(s_{2})}\mathrm{LF}_{\mathbb{H}}(d\phi)$
following (1.5), we need to integrate against
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$, which is the law of the two-
pointed quantum disk with $\beta$-insertion. This is because
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$ can be viewed as
$e^{\frac{\beta}{2}\phi(s_{1})}e^{\frac{\beta}{2}\phi(s_{2})}\mathrm{LF}_{\mathbb{H}}(d\phi)$
modulo the redundant conformal symmetries of ${\mathbb{H}}$ fixing
$s_{1},s_{2}$. The measure ${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$ was
first introduced in [DMS21] and implicitly used in [RZ22]. It describes the
law of a quantum surface with two boundary marked points. We recall its
precise definition in Section 2.4.

Another subtlety in defining $R_{\mu_{1},\mu_{2}}(\beta)$ is that we cannot
directly integrate $e^{-A-\mu_{1}L_{1}-\mu L_{2}}$ over
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$ as suggested by (1.5), where $A$
and $L_{1},L_{2}$ are the area and the two boundary lengths of a sample from
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$, respectively. In fact, there is
no $\beta$ such that this integration is finite. The same issue arises in
[ARS23] where the LCFT correlation function on the disk with one bulk
insertion is considered. In both cases, one needs to truncate the function
$e^{-x}$ near $x=0$, after which the integral is finite for some range of
$\beta$. Concretely, for $\mu_{1}\geq 0,\mu_{2}\geq 0$, we define

(1.14)
$R_{\mu_{1},\mu_{2}}(\beta):=\frac{2(Q-\beta)}{\gamma}\int(e^{-A-\mu_{1}L_{1}-\mu_{2}L_{2}}-1)\,\mathrm{d}{{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)\quad\textrm{for
}\beta\in(\frac{2}{\gamma},Q).$

Then $R_{\mu_{1},\mu_{2}}(\beta)$ is indeed finite. We will give the full
detail of the integral in (1.14) in Section 2.4, including its finiteness.
Here we still use the convention that bulk cosmological constant $\mu=1$ as in
the definition of
$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}$. The
prefactor $\frac{2(Q-\beta)}{\gamma}$ is to match with [RZ22].

In the seminal work [FZZ00], Fateev, Zamolodchikov and Zamolodchikov proposed
a formula for the boundary reflection coefficient under the same
reparametrization $\mu_{B}(\sigma)$ as in (1.10):

(1.15)
$R(\beta,\sigma_{1},\sigma_{2}):=R_{\mu_{B}(\sigma_{1}),\mu_{B}(\sigma_{2})}(\beta).$

Their formula is:

(1.16) $\displaystyle
R_{\mathrm{FZZ}}(\beta,\sigma_{1},\sigma_{2})=\left(\frac{\pi(\frac{\gamma}{2})^{2-\frac{\gamma^{2}}{2}}\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}\right)^{\frac{Q-\beta}{\gamma}}\frac{\Gamma_{\frac{\gamma}{2}}(\beta-Q)S_{\frac{\gamma}{2}}(Q\pm(Q-\sigma_{1}-\sigma_{2})-\frac{\beta}{2})}{\Gamma_{\frac{\gamma}{2}}(Q-\beta)S_{\frac{\gamma}{2}}(\frac{\beta}{2}\pm(\sigma_{2}-\sigma_{1}))}.$

Our last main result is the confirmation of their proposal.

######  Theorem 1.3.

Let $\gamma\in(0,2)$, $\beta\in(\frac{2}{\gamma},Q)$, and
$\mu_{j}=\mu_{B}(\sigma_{j})>0$ for $j=1,2$. Then

(1.17) $\displaystyle
R(\beta,\sigma_{1},\sigma_{2})=R_{\mathrm{FZZ}}(\beta,\sigma_{1},\sigma_{2}).$

Figure 1 summarizes all the structure constants discussed above of the
boundary LCFT, including for completeness the bulk one-point function
$U_{\mu_{B}}(\alpha)$ which was proven in [ARS23] to be equal to the FZZ
formula.

Figure 1. Structure constants of boundary Liouville CFT.

###  1.2. Overview of the proof

Our proof strategy can be summarized as follows.

  * Step 1

Take as input the BPZ equation stated in Theorem 3.4 coming from [Ang23].

  * Step 2

In the case of one degenerate and three generic boundary points this equation
reduces to a hypergeometric equation. Using the solution theory for such
equations and the asymptotic analysis of Gaussian multiplicative chaos, one
obtains a set of functional equations for the probabilistically defined $H$
and $R$ called _shift equations_. This is the content of Section 3.

  * Step 3

Prove that under certain conditions the solution to these shift equations is
unique. Since both $R,R_{\mathrm{FZZ}}$ and $H,{H_{\mathrm{PT}}}$ are
respectively solutions to these shift equations and they obey the conditions
for uniqueness, this implies Theorems 1.1 and 1.3.

  * Step 4

Using a relation between $G$ and $R$ recently proved by [Wu23], we deduce
Theorem 1.2 from Theorem 1.3. This step and the previous one are performed in
Section 4.

There are several novelties in our proof to overcome new difficulties, as we
highlight below:

  * (1).

In the previous works on boundary theory [Rem20, RZ22], a handy fact is that
the Liouville correlation functions can be written as a moment of Gaussian
multiplicative chaos times an explicit prefactor. Under this form the
probabilistic definition can be extended meromorphically to a range where the
shift equations make sense. This is no longer the case in our paper. We have
to introduce proper truncations in the spirit of (1.14) to extend the
probabilistic definitions of $H$ and $R$, and handle analytic issues coming
with this complication. This is performed in Section 2. Along the same lines
the fact that the Liouville correlations contain both area and boundary GMC
measures complicates the proof of several results, including the limit giving
$R$ from $H$ (see Section 5.1), and the proof of the operator product
expansions (see Appendix C).

  * (2).

At several places in the proofs we use a blending of the techniques of the
exact solvability of LCFT and the LQG/SLE coupling, namely conformal welding
and mating-of-trees theory. We now summarize the main places where the latter
theory enters the picture. Firstly, the BPZ equation from [Ang23] was proved
using the mating-of-trees theorem. Secondly, in the derivation of the shift
equations for $R$, we need the exact value of
$R(\gamma,\sigma_{1},\sigma_{2})$ coming from the description of the quantum
disk in the mating-of-trees theory [DMS21, AG21]. This is in contrast to
[RZ22] where the bulk potential is absent and the corresponding input was
supplied by [RZ20]. Thirdly, the relation between $R$ and $G$ in our Step 4
relies on a conformal welding result of [Wu23], and the FZZ formula whose
proof in [ARS23] again uses conformal welding and mating-of-trees.

  * (3).

The fact that we are working with the complicate formulas ${H_{\mathrm{PT}}}$
and $G_{\mathrm{Hos}}$ introduces several technicalities. The shift equations
satisfied by ${H_{\mathrm{PT}}}$ consist of three term relations, which is
significantly more complicated than the two term relations obeyed by
$R_{\mathrm{FZZ}}$ or used in [KRV20]. The uniqueness argument for such
relations is quite involved and is carried out in Section 4.2. Similarly,
obtaining the formula $G_{\mathrm{Hos}}$ requires an involved computation and
uses non-trivial identities of integrals of the functions
$\Gamma_{\frac{\gamma}{2}},S_{\frac{\gamma}{2}}$; see Section 4.3. Finally, we
need a number of technical computations on
${H_{\mathrm{PT}}},G_{\mathrm{Hos}}$ collected in Appendix A.

###  1.3. Connection with the fusion and modular kernels of Virasoro conformal
blocks

Conformal blocks are fundamental analytic functions that appear in the
conformal bootstrap equations in CFT (see (1.20)). There are two key linear
transformations describing the symmetries of the space of conformal blocks.
The two kernels appearing in these two transformations are called the fusion
and modular kernels respectively. For rational CFT, the existence and
properties of such kernels are well known in physics thanks to the work of
Verlinde [Ver88], and Moore-Seiberg [MS89]. In mathematics they are understood
in the context of modular tensor category; see e.g. [BK01]. For generic
Virasoro conformal blocks with central charge $c>25$, the existence and
explicit expressions of the two kernels were conjectured in [PT99, PT01] and
[Tes03], respectively; see also [Tes01, CEZ23]. In a work in preparation, the
second and the third named authors of this paper will prove these conjectures
jointly with P. Ghosal and Y. Sun.

As argued by Ponsot-Teschner [PT02], the fusion kernel is related to the
boundary three-point Liouville correlation. By matching exact formulas, we
found that the modular kernel is related to the bulk-boundary Liouville
correlation in a similar manner. Let us now explicitly write these relations.
Below we will use the notation $\alpha_{i}^{\prime}$ for the alpha parameters
appearing in the fusion and modular kernel to clearly distinguish them from
the alpha and beta parameters of the current paper. First we give the relation
for the fusion kernel, under the parameter identification

$\displaystyle\beta_{1}=\alpha_{2}^{\prime},\>\>\beta_{2}=Q-\mathbf{i}P^{\prime},\>\>\beta_{3}=\alpha_{3}^{\prime},\>\>2\sigma_{1}=Q+\mathbf{i}P,\>\>2\sigma_{2}=\alpha_{1}^{\prime},\>\>2\sigma_{3}=\alpha_{4}^{\prime},$

we have

(1.18)
$\displaystyle\mathcal{M}_{\alpha^{\prime}_{1},\alpha^{\prime}_{2},\alpha^{\prime}_{3},\alpha^{\prime}_{4}}^{\mathsf{fusion}}(P,P^{\prime})=\frac{1}{2\pi}\left(\frac{\pi(\frac{\gamma}{2})^{2-\frac{\gamma^{2}}{2}}\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}\right)^{\frac{\beta_{1}+\beta_{2}+\beta_{3}-2Q}{2\gamma}}\frac{\Gamma_{\frac{\gamma}{2}}(Q)\Gamma_{\frac{\gamma}{2}}(Q-\beta_{1})\Gamma_{\frac{\gamma}{2}}(Q-\beta_{3})}{\Gamma_{\frac{\gamma}{2}}(\beta_{2}-Q)}$
$\displaystyle\times\frac{\Gamma_{\frac{\gamma}{2}}(Q\pm(2\sigma_{1}-Q))\Gamma_{\frac{\gamma}{2}}(\frac{\beta_{2}}{2}\pm\frac{Q-2\sigma_{2}}{2}\pm\frac{Q-2\sigma_{3}}{2})S_{\frac{\gamma}{2}}(\frac{Q+\beta_{2}-\beta_{3}}{2}\pm\frac{Q-\beta_{1}}{2})}{\Gamma_{\frac{\gamma}{2}}(Q-\frac{\beta_{1}}{2}\pm\frac{Q-2\sigma_{1}}{2}\pm\frac{Q-2\sigma_{2}}{2})\Gamma_{\frac{\gamma}{2}}(Q-\frac{\beta_{3}}{2}\pm\frac{Q-2\sigma_{1}}{2}\pm\frac{Q-2\sigma_{3}}{2})}H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}.$

Next we give the relation for the modular kernel. Under the parameter
identification

$\displaystyle\alpha=Q+\mathbf{i}P^{\prime},\quad\beta=\alpha^{\prime},\quad
2\sigma=Q+\mathbf{i}P,$

we have

(1.19)
$\displaystyle\mathcal{M}_{\alpha^{\prime}}^{\mathsf{modular}}(P,P^{\prime})$
$\displaystyle=-\frac{\pi}{2}2^{2\Delta_{\alpha}-\Delta_{\beta}}\left(\frac{\gamma}{2}\right)^{(\frac{2}{\gamma}-\frac{\gamma}{2})(Q-\alpha)+1}\left(\frac{\pi(\frac{\gamma}{2})^{2-\frac{\gamma^{2}}{2}}\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}\right)^{\frac{1}{\gamma}(\alpha+\frac{\beta}{2}-Q)}$
$\displaystyle\times\frac{\Gamma_{\frac{\gamma}{2}}(Q\pm(2\sigma-Q))\Gamma_{\frac{\gamma}{2}}(Q-\beta)\Gamma_{\frac{\gamma}{2}}(Q)}{\Gamma(\frac{2}{\gamma}(\alpha-Q))\Gamma(\frac{\gamma\alpha}{4}-\frac{\gamma^{2}}{4})\Gamma_{\frac{\gamma}{2}}(Q-\frac{\beta}{2}\pm(2\sigma-Q))\Gamma_{\frac{\gamma}{2}}(Q-\frac{\beta}{2})^{2}}G(\alpha,\beta,\sigma).$

The relation (1.18) between the Virasoro fusion kernel and boundary Liouville
three-point function was inspired by a similar relation in the context of
Virasoro minimal models [Run99]. We expect that the relation (1.19) we
discovered between the modular kernel and the bulk-boundary correlation
function has a similar interpretation, but this remains to be understood.

###  1.4. Outlook and perspective

####  1.4.1. Conformal bootstrap for boundary LCFT

Conformal bootstrap is a program for computing correlation functions of a CFT
on a Riemann surface with marked points, based on the decomposition of the
marked surface into simpler pieces. The data in this program are the spectrum
of its Hamiltonian and the structure constants. For LCFT on surfaces without
boundary, the bootstrap program was completely carried out in [KRV20, GKRV20,
GKRV21], where the structure constant is the 3-point spherical correlation
function given by the DOZZ formula [KRV20].

For the boundary LCFT, the structure constants needed for conformal bootstrap
are precisely the boundary three-point and bulk-boundary correlation functions
whose expressions we have determined in Theorems 1.1 and 1.2. The FZZ formula
will also appear as a special case of the bulk-boundary correlation with the
boundary weight $\beta$ set to zero. A first case of the conformal bootstrap
for a surface with boundary has recently been proved in [Wu22] on the annulus
with one insertion at each boundary. This bootstrap formula involves the bulk-
boundary correlation $G$. The general conformal bootstrap for boundary LCFT,
which is the counterpart of [GKRV21], is a work in progress by the authors of
[GKRV21, Wu22]. As an example involving the boundary three-point function $H$,
we state the bootstrap equation for the boundary four-point function:

(1.20)
$\displaystyle\left\langle\prod_{j=1}^{4}e^{\frac{\beta_{j}}{2}\phi(s_{j})}\right\rangle=\frac{1}{4\pi}s^{(\frac{Q^{2}}{4}-\Delta_{\beta_{1}}-\Delta_{\beta_{2}})}\int_{\mathbb{R}}H^{(\beta_{1},\beta_{2},Q+\mathbf{i}P)}_{(\mu_{1},\mu_{2},\mu_{3})}H^{(\beta_{3},\beta_{4},Q-\mathbf{i}P)}_{(\mu_{3},\mu_{4},\mu_{1})}s^{\frac{P^{2}}{4}}\mathcal{F}_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(s,P)dP.$

Here the marked points $s_{j}$ are at the locations $(0,s,1,\infty)$. The
correlation function depends on four cosmological constants
$(\mu_{j})_{j=1,\dots,4}$, and the function
$\mathcal{F}_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(s,P)$ is the conformal
block.

####  1.4.2. Implications for GMC measures

In the spirit of the discussions in [ARS23, Section 1.3] and [RZ22, Sections
1.3 and 1.4], we give here some consequences of the exact formulas we have
derived for the GMC measures. First of all if one sets all boundary
cosmological constants to $0$, the probabilistic expressions for $H$ and $G$
will reduce to a moment of the Gaussian multiplicative chaos area measure of
$\mathbb{H}$. Thus our results give exact formulas for these 2d GMC measures.

The function $R$ can also be used to give the tail expansion of Gaussian
multiplicative chaos. In particular, we expect that when
$\beta\in(\frac{\gamma}{2},Q)$

$\displaystyle\mathbb{P}\left(\int_{\mathbb{H}\cap\mathbb{D}}\frac{1}{|x|^{\gamma\beta}}e^{\gamma
h(x)}d^{2}x>u^{2}\right)\underset{u\rightarrow\infty}{\sim}-\frac{1}{\Gamma(1-\frac{2(Q-\beta)}{2})}\frac{R_{\mu_{1}=0,\mu_{2}=0}(\beta)}{u^{\frac{2}{\gamma}(Q-\beta)}}.$

Furthermore, the joint law of the area and boundary length distribution under
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$, which is encoded by
$R_{\mu_{1},\mu_{2}}(\beta)$, should describe the asymptotic behavior of

$\displaystyle\mathbb{P}\left(\int_{\mathbb{H}\cap\mathbb{D}}\frac{1}{|x|^{\gamma\beta}}e^{\gamma
h(x)}d^{2}x>u^{2},\int_{-1}^{1}\frac{1}{|r|^{\frac{\gamma\beta}{2}}}e^{\frac{\gamma}{2}h(r)}dr>u\right).$

See [RV19, Cer22] on the role of reflection coefficients in the tail
expansions of GMC measures.

####  1.4.3. Interaction between the integrability of LCFT and the mating-of-
trees theory.

We see from (1.14) that the boundary reflection coefficient $R$ encodes the
joint law of the length and area of the two-pointed quantum disk sampled from
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$. With Yu, the first and third
named authors considered in [ASY22] the LQG surfaces with three boundary
marked points sampled from
$\prod_{j=1}^{3}e^{\frac{\beta_{j}}{2}\phi(s_{j})}\mathrm{LF}_{\mathbb{H}}(d\phi)$,
which they called the quantum triangles. The boundary structure constant $H$
encodes the joint area and length distribution of these surfaces. As shown in
[AHS23a, AHS23b, ARS23, ASY22, Wu23, NQSZ23], quantum disks and triangles
behave nicely under conformal welding and facilitate a mutually beneficial
interaction between the integrability of LCFT and of the mating-of-trees
theory. Results of this paper are not only outcomes of this interaction, but
also enrich it in several ways. For example, in a work in preparation with Yu,
the first and third named authors will derive an exact formula for a quantity
in mating-of-trees which is shown in [BHSY23] to encode the expected
proportion of inversions in a skew Brownian permuton recently introduced in
[Bor23]. Our result on $H$ plays a crucial role there. Moreover, conformal
welding results give various relations between the fusion kernel and modular
kernel of Virasoro conformal blocks. We plan to investigate the possible
connection between these relations and those which are well-known in the
modular tensor category context [MS89].

###  1.5. Outline of the paper

The rest of our paper is organized as follows. In Section 2 we give precise
definitions of the functions ${H_{\mathrm{PT}}}$ and $G_{\mathrm{Hos}}$, of
the probabilistic expressions for $H$, $G$, and $R$, and prove the required
analytic properties of these functions. In Section 3 we prove the shift
equations for $H$ and $R$. In Section 4 we complete the proof of our main
theorems using these shift equations. In Section 5 we supply the proofs of two
facts on $R$ that were used in Section 3. In the appendix we provide several
technical ingredients needed in the main text.

Acknowledgements. The authors are grateful for the hospitality of the
Institute for Advanced Study (IAS) at Princeton, where the final stage of the
project took place. We would like to very warmly thank Lorenz Eberhardt who
helped us fix the uniqueness argument for the boundary three-point function by
suggesting to write the functional equations in matrix form. We also thank
Baojun Wu and Da Wu for helpful discussions. M.A. was partially supported by
NSF grant DMS-1712862, and by the Simons Foundation as a Junior Fellow at the
Simons Society of Fellows. G.R. was partially supported by an NSF mathematical
sciences postdoctoral research fellowship, NSF Grant DMS-1902804, and by a
member fellowship from IAS. X.S. was partially supported by the NSF grant
DMS-2027986, the NSF Career award 2046514, a start-up grant from the
University of Pennsylvania, and by a member fellowship from IAS.

##  2\. Definitions and meromorphicity of ${H_{\mathrm{PT}}}$,
$G_{\mathrm{Hos}}$, $H$, $G$, and $R$

In this section we start by giving a full definition of the functions
${H_{\mathrm{PT}}}$ and $G_{\mathrm{Hos}}$ and the probabilistic definition of
$H$ and $G$ under the Seiberg bounds (1.4). We then given an extended
definition of $H$ and the definition of $R$ using a truncation procedure. We
also state and prove analyticity results for these functions. Throughout this
section and the next we will frequently refer to a function of several complex
variables as being meromorphic on some domain. By this terminology we simply
mean that it can be expressed as the ratio of two holomorphic functions of
several variables on that domain.

###  2.1. The Ponsot-Teschner and Hosimichi functions

We start by giving full details on the definition of ${H_{\mathrm{PT}}}$ given
by equation (1.11) and some of its properties. We first clarify how the
contour $\mathcal{C}$ is chosen. For a function $f$, recall our notation
$f(a\pm b)=f(a+b)f(a-b)$. Consider the integral

(2.1)
$\mathcal{J}_{\mathrm{PT}}:=\int_{\mathcal{C}}\frac{S_{\frac{\gamma}{2}}(\frac{Q-\beta_{2}}{2}+\sigma_{3}\pm(\frac{Q}{2}-\sigma_{2})+r)S_{\frac{\gamma}{2}}(\frac{Q}{2}\pm\frac{Q-\beta_{3}}{2}+\sigma_{3}-\sigma_{1}+r)}{S_{\frac{\gamma}{2}}(\frac{3Q}{2}\pm\frac{Q-\beta_{1}}{2}-\frac{\beta_{2}}{2}+\sigma_{3}-\sigma_{1}+r)S_{\frac{\gamma}{2}}(2\sigma_{3}+r)S_{\frac{\gamma}{2}}(Q+r)}\frac{dr}{\mathbf{i}},$

so that ${H_{\mathrm{PT}}}$ and $\mathcal{J}_{\mathrm{PT}}$ are then related
by an explicit prefactor containing the functions $\Gamma_{\frac{\gamma}{2}}$
and $S_{\frac{\gamma}{2}}$ as in (1.11). To describe how to choose the contour
of integration $\mathcal{C}$, we must look at the locations of the poles in
$r$ of the integrand. Recall that the function $S_{\frac{\gamma}{2}}$ has
poles at $x=-n\frac{\gamma}{2}-m\frac{2}{\gamma}$ and zeros at
$x=Q+n\frac{\gamma}{2}+m\frac{2}{\gamma}$ for any $n,m\in\mathbb{N}$ (here and
throughout the paper $\mathbb{N}$ contains $0$). We will group the poles of
the integrand into two groups, the ones coming from the poles of the numerator
and the ones coming from the zeros of the denominator. More precisely consider
the set of poles in the lattices extending to the left

$\displaystyle
P^{-}_{\mathrm{PT}}=\left\\{\xi-\frac{n\gamma}{2}-\frac{2m}{\gamma}\Bigg{|}n,m\in\mathbb{N},\xi\in\\{\frac{\beta_{2}}{2}-\sigma_{2}-\sigma_{3},\frac{\beta_{2}}{2}-Q+\sigma_{2}-\sigma_{3},-\frac{\beta_{3}}{2}-\sigma_{3}+\sigma_{1},\frac{\beta_{3}}{2}-Q-\sigma_{3}+\sigma_{1}\\}\right\\},$

and the set of poles in the lattices extending to the right:

$\displaystyle
P^{+}_{\mathrm{PT}}=\left\\{\xi+\frac{n\gamma}{2}+\frac{2m}{\gamma}\bigg{|}n,m\in\mathbb{N},\xi\in\\{-\frac{\beta_{1}}{2}+\frac{\beta_{2}}{2}-\sigma_{3}+\sigma_{1},-Q+\frac{\beta_{1}}{2}+\frac{\beta_{2}}{2}-\sigma_{3}+\sigma_{1},-2\sigma_{3}+Q,0\\}\right\\}.$

Assume the parameters $\beta_{i},\sigma_{i}$ are chosen so that
$P^{-}_{\mathrm{PT}}\cap P^{+}_{\mathrm{PT}}=\emptyset$. Then the contour
$\mathcal{C}$ in $\mathcal{J}_{\mathrm{PT}}$ goes from $-\mathbf{i}\infty$ to
$\mathbf{i}\infty$ passing to the right to all the poles in
$P^{-}_{\mathrm{PT}}$ and to the left of all the poles in
$P^{+}_{\mathrm{PT}}$. In the region of large or very negative imaginary part,
namely away from the region containing the poles, the contour $\mathcal{C}$ is
chosen simply to be a vertical line on the axis $\mathbf{i}\mathbb{R}$. Figure
2 below provides an illustration of the contour $\mathcal{C}$ and the lattices
of poles $P^{-}_{\mathrm{PT}}$ and $P^{+}_{\mathrm{PT}}$. The following lemma
verifies that this integral is converging at $\pm\mathbf{i}\infty$.

Figure 2. The contour of integration $\mathcal{C}$ in the function
${H_{\mathrm{PT}}}$.

######  Lemma 2.1.

The integral in (2.1) is absolutely converging near both $\mathbf{i}\infty$
and $-\mathbf{i}\infty$.

###### Proof.

We need to determine the asymptotic in $r$ of the integrand of the contour
integral as $r$ goes to $\pm\mathbf{i}\infty$. For this we simply need to use
the asymptotic of $S_{\frac{\gamma}{2}}$ given by equation (A.13). Using this
fact, one obtains that the integrand as $r\rightarrow+\mathbf{i}\infty$ is
equivalent to $c_{1}e^{3\mathbf{i}\pi Qr}$ and as
$r\rightarrow-\mathbf{i}\infty$ to $c_{2}e^{-3\mathbf{i}\pi Qr}$ for some
constants $c_{1},c_{2}\in\mathbb{C}$ independent of $r$. Since $Q>0$, the
integral is absolutely convergent. ∎

When the parameters $\beta_{i},\sigma_{i}$ are such that
$P^{-}_{\mathrm{PT}}\cap P^{+}_{\mathrm{PT}}\neq\emptyset$, the function
$\mathcal{J}_{\mathrm{PT}}$ have a pole which can be determined by a residue
computation; see the following Lemma 2.2.

######  Lemma 2.2.

The poles of the function $\mathcal{J}_{\mathrm{PT}}$ occur when the set
$\\{n\frac{\gamma}{2}+m\frac{2}{\gamma}:n,m\in\mathbb{N}\\}$ contains any of
the following

$\begin{array}[]{llll}\frac{\beta_{1}}{2}-\sigma_{1}-\sigma_{2},&\quad
Q-\frac{\beta_{1}}{2}-\sigma_{1}-\sigma_{2},&\quad\frac{\beta_{2}}{2}-\sigma_{2}+\sigma_{3}-Q,&\quad\frac{\beta_{2}}{2}-\sigma_{2}-\sigma_{3},\\\\[5.69046pt]
-Q+\sigma_{2}+\frac{\beta_{1}}{2}-\sigma_{1},&\quad\sigma_{2}-\sigma_{1}-\frac{\beta_{1}}{2},&\quad-2Q+\frac{\beta_{2}}{2}+\sigma_{3}+\sigma_{2},&\quad-Q+\frac{\beta_{2}}{2}-\sigma_{3}+\sigma_{2},\\\\[5.69046pt]
\frac{\beta_{1}}{2}-\frac{\beta_{2}}{2}-\frac{\beta_{3}}{2},&\quad
Q-\frac{\beta_{1}}{2}-\frac{\beta_{2}}{2}-\frac{\beta_{3}}{2},&\quad-Q-\frac{\beta_{3}}{2}+\sigma_{3}+\sigma_{1},&\quad-\frac{\beta_{3}}{2}-\sigma_{3}+\sigma_{1},\\\\[5.69046pt]
-Q+\frac{\beta_{1}}{2}-\frac{\beta_{2}}{2}+\frac{\beta_{3}}{2},&\quad\frac{\beta_{3}}{2}-\frac{\beta_{1}}{2}-\frac{\beta_{2}}{2},&\quad-2Q+\frac{\beta_{3}}{2}+\sigma_{3}+\sigma_{1},&\quad-Q+\frac{\beta_{3}}{2}-\sigma_{3}+\sigma_{1}.\end{array}$

###### Proof.

The proof follows exactly the same steps as the one of [RZ22, Lemma 5.11],
except that in our case we have four $S_{\frac{\gamma}{2}}$ functions in the
numerator and denominator of the integrand of the contour integral of
$\mathcal{J}_{\mathrm{PT}}$ (instead of three in the case of [RZ22, Lemma
5.11]). We nonetheless recall below the steps of the proof and explain how to
derive the list of poles.

As long as the parameters $\beta_{i},\sigma_{i}$ are chosen such that
$P^{-}_{\mathrm{PT}}\cap P^{+}_{\mathrm{PT}}=\emptyset$, it is always possible
to choose the contour $\mathcal{C}$ so that it goes from $-\mathbf{i}\infty$
to $\mathbf{i}\infty$, passing to the right of the poles of
$P^{-}_{\mathrm{PT}}$ and to the left of the poles of $P^{+}_{\mathrm{PT}}$.
On the other hand, it is impossible to choose such a contour when
$P^{-}_{\mathrm{PT}}\cap P^{+}_{\mathrm{PT}}\neq\emptyset$, since at least one
pole from $P^{-}_{\mathrm{PT}}$ collapses with a pole from
$P^{+}_{\mathrm{PT}}$, This situation will lead to a pole for the function
$\mathcal{J}_{\mathrm{PT}}$. To properly identify this pole, one can start by
deforming the contour $\mathcal{C}$ to cross one of the two collapsing poles
before they collapse and pick up a residue contribution; see the proof of
Lemma A.2 in Appendix A.3 for an instance of such deformation of contour. This
residue term viewed as a function of $\sigma_{i},\beta_{i}$ have a pole at the
value of parameters $\sigma_{i},\beta_{i}$ that caused one pole of
$P^{+}_{\mathrm{PT}}$ to collapse with a pole of $P^{-}_{\mathrm{PT}}$.
Therefore the poles of $\mathcal{J}_{\mathrm{PT}}$ are at the values of
$\beta_{i},\sigma_{i}$ where $P^{-}_{\mathrm{PT}}\cap
P^{+}_{\mathrm{PT}}\neq\emptyset$ which is precisely the list given in the
lemma. ∎

From here it is immediate that ${H_{\mathrm{PT}}}$ is a meromorphic function
on $\mathbb{C}^{6}$.

######  Lemma 2.3.

The function ${H_{\mathrm{PT}}}$ is meromorphic on $\mathbb{C}^{6}$, namely a
ratio of two holomorphic functions on $\mathbb{C}^{6}$ of the six parameters
$\beta_{1},\beta_{2},\beta_{3},\sigma_{1},\sigma_{2},\sigma_{3}$.

###### Proof.

In Lemma 2.2 we have established that $\mathcal{J}_{\mathrm{PT}}$ is
meromorphic on $\mathbb{C}^{6}$ with prescribed poles. Since
${H_{\mathrm{PT}}}$ is equal to $\mathcal{J}_{\mathrm{PT}}$ times an explicit
prefactor containing $\Gamma_{\frac{\gamma}{2}}$ and $S_{\frac{\gamma}{2}}$
functions which are meromorphic, this implies the claim. ∎

We now repeat a similar analysis for the contour integral

$\displaystyle\mathcal{J}_{\mathrm{Hos}}:=\int_{\mathcal{C}}\frac{d\sigma_{2}}{\mathbf{i}}e^{2\mathbf{i}\pi(Q-2\sigma)\sigma_{2}}\frac{S_{\frac{\gamma}{2}}(\frac{1}{2}(\alpha+\frac{\beta}{2}-Q)-\sigma_{2})S_{\frac{\gamma}{2}}(\frac{1}{2}(\alpha+\frac{\beta}{2}-Q)+\sigma_{2})}{S_{\frac{\gamma}{2}}(\frac{1}{2}(\alpha-\frac{\beta}{2}+Q)+\sigma_{2})S_{\frac{\gamma}{2}}(\frac{1}{2}(\alpha-\frac{\beta}{2}+Q)-\sigma_{2})}.$

defining the function $G_{\mathrm{Hos}}$. There is one notable difference,
which is that there will be a condition on the parameters for the contour
integral to converge at $\pm\mathbf{i}\infty$. In fact, using again the
asymptotic of $S_{\frac{\gamma}{2}}$ in (A.13), the contour integral is
converging at $\pm\mathbf{i}\infty$ if and only if:

$\displaystyle\frac{\Re(\beta)}{2}<Q,\quad\textrm{and}\quad\Re(\sigma)\in\left[\frac{\Re(\beta)}{4},Q-\frac{\Re(\beta)}{4}\right].$

Despite this condition it is still possible to analytically continue the
function $\mathcal{J}_{\mathrm{Hos}}$ to a meromorphic function of its three
parameters $\alpha,\beta,\sigma$ on $\mathbb{C}^{3}$. We do not pursue this
here, see [Ebe23] for more details on this fact. For the choice of the contour
of integration, similarly as for $\mathcal{J}_{\mathrm{PT}}$, it has to pass
to the left of the lattice $P_{\mathrm{Hos}}^{+}$ of poles extending to the
right, and to the right of the lattice $P_{\mathrm{Hos}}^{-}$ of poles
extending to the left, where

(2.2) $\displaystyle P_{\mathrm{Hos}}^{-}$
$\displaystyle=\left\\{\xi-\frac{n\gamma}{2}-\frac{2m}{\gamma}\quad\Bigg{|}\quad
n,m\in\mathbb{N},\quad\xi\in\\{\frac{1}{2}(Q-\alpha-\frac{\beta}{2}),\frac{1}{2}(\alpha-\frac{\beta}{2}-Q)\\}\right\\},$
(2.3) $\displaystyle P_{\mathrm{Hos}}^{+}$
$\displaystyle=\left\\{\xi+\frac{n\gamma}{2}+\frac{2m}{\gamma}\quad\bigg{|}\quad
n,m\in\mathbb{N},\quad\xi\in\\{\frac{1}{2}(\alpha+\frac{\beta}{2}-Q),\frac{1}{2}(Q-\alpha+\frac{\beta}{2})\\}\right\\}.$

###  2.2. Definition of $H$ and $G$ under the Seiberg bound

Let $h$ be the free boundary Gaussian free field on the upper half plane
${\mathbb{H}}=\\{z\in{\mathbb{C}}:\Im(z)>0\\}$ with covariance kernel

(2.4)
$\mathbb{E}[h(x)h(y)]=G_{\mathbb{H}}(x,y):=\log\frac{1}{|x-y||x-\bar{y}|}+2\log|x|_{+}+2\log|y|_{+},$

where $|x|_{+}:=\max(|x|,1)$ and in the sense that
$\mathbb{E}[(h,f)(h,g)]=\iint f(x)\mathbb{E}[h(x)h(y)]g(y)dxdy$, for smooth
test functions $f$ and $g$. Let $P_{\mathbb{H}}$ be the law of $h$, so that
$P_{\mathbb{H}}$ is a probability measure on the negatively indexed Sobolev
space $H^{-1}({\mathbb{H}})$ ([She07, Dub09]). The particular covariance
kernel corresponds to requiring the field to have average 0 on the unit
circle.

Following [AHS23b], we introduce the Liouville field on ${\mathbb{H}}$,
possibly with boundary insertions.

######  Definition 2.4 (Liouville field).

Let $(h,\mathbf{c})$ be sampled from $P_{\mathbb{H}}\times[e^{-Qc}dc]$ and set
$\phi=h(z)-2Q\log|z|_{+}+\mathbf{c}$. We write $\mathrm{LF}_{{\mathbb{H}}}$ as
the law of $\phi$, and call a sample from $\mathrm{LF}_{{\mathbb{H}}}$ a
_Liouville field on ${\mathbb{H}}$_.

######  Definition 2.5 (Liouville field with insertions).

Let $(\beta_{j},s_{j})\in{\mathbb{R}}\times\partial{\mathbb{H}}$ for
$j=1,\dots,M$, where $M\geq 1$ and the $s_{j}$ are pairwise distinct. Sample
$(h,\mathbf{c})$ from
$C_{{\mathbb{H}}}^{(\beta_{j},s_{j})_{j}}P_{\mathbb{H}}\times[e^{(\frac{1}{2}\sum_{j}\beta_{j}-Q)c}dc]$
where

$C_{{\mathbb{H}}}^{(\beta_{j},s_{j})_{j}}=\prod_{j=1}^{M}|s_{j}|_{+}^{-\beta_{j}(Q-\frac{\beta_{j}}{2})}\prod_{1\leq
j<k\leq M}e^{\frac{\beta_{j}\beta_{k}}{4}G_{\mathbb{H}}(s_{j},s_{k})}.$

Let
$\phi(z)=h(z)-2Q\log|z|_{+}+\sum_{j=1}^{M}\frac{\beta_{j}}{2}G_{\mathbb{H}}(z,s_{j})+\mathbf{c}$.
We write $\mathrm{LF}_{\mathbb{H}}^{(\beta_{j},s_{j})_{j}}$ for the law of
$\phi$ and call a sample from
$\mathrm{LF}_{\mathbb{H}}^{(\beta_{j},s_{j})_{j}}$ the _Liouville field on
${\mathbb{H}}$ with insertions $(\beta_{j},s_{j})_{1\leq j\leq M}$_.

We can also define Liouville fields with an insertion at $\infty$. We will
need the case
$\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}$,
which can be defined by
$\lim_{s\to\infty}|s|^{2\Delta_{3}}\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},s)}$
with $\Delta_{3}=\frac{\beta_{3}}{2}(Q-\frac{\beta_{3}}{2})$. Here we give a
more explicit definition which is equivalent to the limiting procedure; see
[AHS23b, Lemma 2.9].

######  Definition 2.6.

Fix $\beta_{1},\beta_{2},\beta_{3}\in{\mathbb{R}}$. Set
$s_{1}=0,s_{2}=1,s_{3}=\infty$ and $G_{\mathbb{H}}(z,\infty)=2\log|z|_{+}$.
Sample $(h,\mathbf{c})$ from
$P_{\mathbb{H}}\times[e^{(\frac{1}{2}\sum_{j}\beta_{j}-Q)c}dc]$. We write
$\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}$
for the law of $\phi$ where
$\phi(z)=h(z)-2Q\log|z|_{+}+\sum_{j=1}^{3}\frac{\beta_{j}}{2}G_{\mathbb{H}}(z,s_{j})+\mathbf{c}$.

Given a sample $h$ from $P_{\mathbb{H}}$, the associated quantum area and
length measure are defined by

(2.5) ${\mathcal{A}}_{h}=\lim_{{\varepsilon}\to
0}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
h_{\epsilon}(z)}d^{2}z,\quad\text{and}\quad{\mathcal{L}}_{h}=\lim_{{\varepsilon}\to
0}\epsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}h_{\epsilon}(z)}dz,$

where the limits hold in probability in the weak topology of measures. The
existence of limits is well-known from Gaussian multiplicative chaos (GMC);
see e.g. [Ber17, RV14].

######  Remark 2.7.

Note that it is also possible to define the GMC measures by renormalizing by
the variance. The difference between the two conventions is given by:

$\displaystyle\mathcal{A}_{h}=\frac{|z|_{+}^{2\gamma^{2}}}{|z-\bar{z}|^{\frac{\gamma^{2}}{2}}}\lim_{{\varepsilon}\to
0}e^{\gamma
h_{\epsilon}(z)-\frac{\gamma^{2}}{2}\mathbb{E}[h_{\epsilon}(z)^{2}]}d^{2}z,\quad\text{and}\quad\mathcal{L}_{h}=|z|_{+}^{\frac{\gamma^{2}}{2}}\lim_{{\varepsilon}\to
0}e^{\frac{\gamma}{2}h_{\epsilon}(z)-\frac{\gamma^{2}}{8}\mathbb{E}[h_{\epsilon}(z)^{2}]}dz.$

Given a sample $\phi$ from $\mathrm{LF}_{\mathbb{H}}$ or
$\mathrm{LF}_{\mathbb{H}}^{(\beta_{j},s_{j})_{j}}$, we similarly define
${\mathcal{A}}_{\phi}$ and ${\mathcal{L}}_{\phi}$ as in (2.5) with $h$
replaced by $\phi$. This allows us to rigorously define the function $H$ in
(1.7) for a certain range of parameters.

######  Definition 2.8.

Let $\mu_{i}\geq 0$ for $i=1,2,3$. Suppose
$\beta_{1},\beta_{2},\beta_{3}\in{\mathbb{R}}$ satisfy the Seiberg bound:

(2.6) $\sum_{i=1}^{3}\beta_{i}>2Q\quad\textrm{and}\quad\beta_{i}<Q.$

Then the boundary three-point structure constant is defined by

(2.7) $H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}:=\int
e^{-{\mathcal{A}}_{\phi}({\mathbb{H}})-\mu_{1}{\mathcal{L}}_{\phi}(-\infty,0)-\mu_{2}{\mathcal{L}}_{\phi}(0,1)-\mu_{3}{\mathcal{L}}_{\phi}(1,+\infty)}\;\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}(\mathrm{d}\phi).$

In a similar way we can give the probabilistic definition of $G$. We start
with the definition:

######  Definition 2.9.

Fix $\alpha,\beta\in{\mathbb{R}}$. Sample $(h,\mathbf{c})$ from
$2^{-\frac{\alpha^{2}}{2}}P_{\mathbb{H}}\times[e^{(\alpha+\frac{\beta}{2}-Q)c}dc]$.
We write $\mathrm{LF}_{\mathbb{H}}^{(\alpha,\mathbf{i}),(\beta,0)}$ for the
law of $\phi$ where $\phi(z)=h(z)-2Q\log|z|_{+}+\alpha
G_{\mathbb{H}}(z,\mathbf{i})+\frac{\beta}{2}G_{\mathbb{H}}(z,0)+\mathbf{c}$.

We can now give the precise definition of the structure constant
$G_{\mu_{B}}(\alpha,\beta)$.

######  Definition 2.10.

Let $\mu_{B}\geq 0$. Suppose $\alpha,\beta\in{\mathbb{R}}$ satisfy the Seiberg
bound:

(2.8) $\alpha+\frac{\beta}{2}>Q,\quad\alpha<Q,\quad\textrm{and}\quad\beta<Q.$

Then the bulk-boundary structure constant is defined by:

(2.9) $G_{\mu_{B}}(\alpha,\beta):=\int
e^{-{\mathcal{A}}_{\phi}({\mathbb{H}})-\mu_{B}{\mathcal{L}}_{\phi}(\mathbb{R})}\;\mathrm{LF}_{\mathbb{H}}^{(\alpha,\mathbf{i}),(\beta,0)}(\mathrm{d}\phi).$

###  2.3. Meromorphic extension of $H$

To make sense of shift equations for $H$ we meromorphically extend the range
of its definition via an integration by parts, based on the following lemma.

######  Lemma 2.11.

In the range of parameters as in Definition 2.8, set
$s=\frac{1}{2}\sum\beta_{i}-Q$ and

(2.10)
$\hat{H}^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}:=\int(\gamma(s+\frac{\gamma}{2})A+\frac{\gamma^{2}}{2}A(\sum_{i}\mu_{i}L_{i})+\frac{\gamma^{2}}{4}(\sum_{i}\mu_{i}L_{i})^{2})e^{-A-\sum_{i}\mu_{i}L_{i}}\,\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}(\mathrm{d}\phi).$

where
$A={\mathcal{A}}_{\phi}({\mathbb{H}}),L_{1}={\mathcal{L}}_{\phi}(-\infty,0),L_{2}={\mathcal{L}}_{\phi}(0,1)$
and $L_{3}={\mathcal{L}}_{\phi}(1,\infty)$. Then

$s(s+\frac{\gamma}{2})H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}=\hat{H}^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}.$

###### Proof.

Let $h$ be a Gaussian free field and
$\widetilde{h}:=h-2Q\log|\cdot|_{+}+\sum\frac{\beta_{i}}{2}G_{\mathbb{H}}(\cdot,s_{i})$,
where $(s_{1},s_{2},s_{3})=(0,1,\infty)$. Write
$\widetilde{A}={\mathcal{A}}_{\widetilde{h}}({\mathbb{H}})$ and
$\widetilde{L}_{1}={\mathcal{L}}_{\widetilde{h}}(-\infty,0),\widetilde{L}_{2}={\mathcal{L}}_{\widetilde{h}}(0,1),\widetilde{L}_{3}={\mathcal{L}}_{\widetilde{h}}(1,+\infty)$.
Since $A=e^{\gamma c}\widetilde{A}$ and $L_{i}=e^{\gamma
c/2}\widetilde{L}_{i}$, by Definition 2.8, we have

$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2}\mu_{3})}=\mathbb{E}\left[\int
e^{sc}\cdot e^{-e^{\gamma c}\widetilde{A}-e^{\gamma
c/2}\sum_{i=1}^{3}\mu_{i}\widetilde{L}_{i}}\,dc\right].$

For $s>0$, $a>0$ and $\ell\in{\mathbb{C}}$ with $\Re\ell>0$, applying
integration by parts twice we have

$\displaystyle\int e^{sc}\cdot e^{-e^{\gamma c}a-e^{\gamma
c/2}\ell}\,dc=-\int(\frac{1}{s}e^{sc})((-\gamma e^{\gamma
c}a-\frac{\gamma}{2}e^{\gamma c/2}\ell)e^{-e^{\gamma c}a-e^{\gamma
c/2}\ell})\,dc$ $\displaystyle=\frac{\gamma}{s}\int e^{sc}(e^{\gamma
c}a)e^{-e^{\gamma c}a-e^{\gamma c/2}\ell}\,dc+\frac{\gamma}{2s}\int
e^{sc}(e^{\gamma c/2}\ell)e^{-e^{\gamma c}a-e^{\gamma c/2}\ell}\,dc$
$\displaystyle=\frac{\gamma}{s}\int e^{sc}(e^{\gamma c}a)e^{-e^{\gamma
c}a-e^{\gamma
c/2}\ell}\,dc+\frac{\gamma}{2s}\frac{\gamma}{s+\frac{\gamma}{2}}\int
e^{sc}(e^{\gamma c}a)(e^{\gamma c/2}\ell)e^{-e^{\gamma c}a-e^{\gamma
c/2}\ell}\,dc$
$\displaystyle+\frac{\gamma}{2s}\frac{\frac{\gamma}{2}}{s+\frac{\gamma}{2}}\int
e^{sc}(e^{\gamma c/2}\ell)^{2}e^{-e^{\gamma c}a-e^{\gamma c/2}\ell}\,dc.$

Now setting $s=\frac{1}{2}\sum\beta_{i}-Q$, $a=\widetilde{A}$ and
$\ell=\sum_{i}\mu_{i}\widetilde{L}_{i}$ we get the desired equality:

$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2}\mu_{3})}=\int(\frac{\gamma}{s}A+\frac{\gamma}{2s}\frac{\gamma}{s+\frac{\gamma}{2}}A(\sum_{i}\mu_{i}L_{i})+\frac{\gamma}{2s}\frac{\frac{\gamma}{2}}{s+\frac{\gamma}{2}}(\sum_{i}\mu_{i}L_{i})^{2})e^{-A-\sum_{i}\mu_{i}L_{i}}\,\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}(\mathrm{d}\phi).$

∎

The function $\hat{H}$ has a better analytic property as shown in the
following proposition.

######  Proposition 2.12.

Set
$V=\\{(\beta_{1},\beta_{2},\beta_{3}):Q-\frac{1}{2}\sum\beta_{i}<\gamma\wedge\frac{2}{\gamma}\wedge\min_{i}(Q-\beta_{i})\textrm{
and }\beta_{i}<Q\textrm{ for each }i\\}$. When
$(\beta_{1},\beta_{2},\beta_{3})\in V$ and $\Re\mu_{i}\geq 0$ for $i=1,2,3$,
the integral in (2.10) defining $\hat{H}$ is absolutely convergent. Moreover,
for each $(\beta_{1},\beta_{2},\beta_{3})\in V$, the function $\hat{H}$ is
holomorphic on $\\{(\mu_{1},\mu_{2},\mu_{3}):\Re\mu_{i}>0\\}$ and continuous
on $\\{(\mu_{1},\mu_{2},\mu_{3}):\Re\mu_{i}\geq 0\\}$. Finally, for each
$(\mu_{1},\mu_{2},\mu_{3})$ satisfying $\Re\mu_{i}>0$, the function $\hat{H}$
can be analytically extended on a complex neighborhood in ${\mathbb{C}}^{3}$of
$V$.

We defer the proof of Proposition 2.12 to Section 2.5 and proceed to extend
the definition of $H$.

######  Definition 2.13.

Given $(\beta_{1},\beta_{2},\beta_{3})\in V$ in Proposition 2.12 and
$\Re\mu_{i}\geq 0$ for $i=1,2,3$, define

(2.11)
$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}:=\frac{1}{s(s+\frac{\gamma}{2})}\hat{H}^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}.$

Recall that $H$ from Theorem 1.1 is expressed in the $\sigma$ variable via a
change of variable. We now transfer Proposition 2.12 in terms of the $\sigma$
variable.

######  Proposition 2.14.

For $V,s$, and
$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}$ as in
Definition 2.13, we define

(2.12) $\displaystyle H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}:=H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{B}(\sigma_{1}),\mu_{B}(\sigma_{2}),\mu_{B}(\sigma_{3}))}\quad\textrm{where
}\mu_{B}(\sigma)=\sqrt{\frac{1}{\sin(\pi\frac{\gamma^{2}}{4})}}\cos\left(\pi\gamma(\sigma-\frac{Q}{2})\right).$

Let
${\mathcal{B}}=(-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2})\times\mathbb{R}$
and
$\overline{{\mathcal{B}}}=[-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2}]\times\mathbb{R}$.
Then for each $(\beta_{1},\beta_{2},\beta_{3})\in V$, the function
$(\sigma_{1},\sigma_{2},\sigma_{3})\mapsto H$ is holomorphic on
$\mathcal{B}^{3}$ and continuous on $\overline{\mathcal{B}}^{3}$. Moreover,
for each $(\sigma_{1},\sigma_{2},\sigma_{3})\in\mathcal{B}^{3}$, the function
$(\beta_{1},\beta_{2},\beta_{3})\mapsto s(s+\frac{\gamma}{2})H$ is analytic on
a complex neighborhood in ${\mathbb{C}}^{3}$of $V$.

###### Proof.

The function $\mu_{B}(\sigma)$ is a holomorphic bijection between
$\\{\mu\in\mathbb{C}:\Re\mu>0\\}$ and ${\mathcal{B}}^{3}$. Moreover, $\mu_{B}$
is continuous on $\overline{{\mathcal{B}}}^{3}$. Therefore Proposition 2.12
yields Proposition 2.14. ∎

###  2.4. Probabilistic definition of $R$ via truncations

We first recall the definition of two pointed quantum disk
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$ used in the definition of $R$.
Following [AHS23b, Section 2], we describe it using the horizontal strip
${\mathcal{S}}={\mathbb{R}}\times(0,\pi)$. Let
$h_{\mathcal{S}}(z)=h_{\mathbb{H}}(e^{z})$, where $h_{\mathbb{H}}$ is sampled
from $P_{\mathbb{H}}$. We call $h_{\mathcal{S}}$ a free-boundary GFF on
${\mathcal{S}}$. The field $h_{\mathcal{S}}$ can be written as
$h_{\mathcal{S}}=h^{\mathrm{c}}+h^{\ell}$, where $h^{\mathrm{c}}$ is constant
on each vertical line, and $h^{\ell}$ has mean zero on all such lines [DMS21,
Section 4.1.6]. We call $h^{\ell}$ the _lateral component_ of the free-
boundary GFF on ${\mathcal{S}}$.

######  Definition 2.15.

Fix $\beta<Q$. Let

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

where $(B_{s})_{s\geq 0}$ is a standard Brownian motion conditioned on
$B_{2s}-(Q-\beta)s<0$ for all $s>0$,333Here we condition on a zero probability
event. This can be made sense of via a limiting procedure. and
$(\widetilde{B}_{s})_{s\geq 0}$ is an independent copy of $(B_{s})_{s\geq 0}$.
Let $h^{1}(z)=Y_{\Re z}$ for each $z\in{\mathcal{S}}$. Let
$h^{2}_{\mathcal{S}}$ be independent of $h^{1}$ and have the law the lateral
component of the free-boundary GFF on ${\mathcal{S}}$. Let
$\psi=h^{1}+h^{2}_{\mathcal{S}}$. Let $\mathbf{c}$ be a real number sampled
from $\frac{\gamma}{2}e^{(\beta-Q)c}dc$ independent of $\psi$ and
$\phi=\psi+\mathbf{c}$. Let ${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$ be
the infinite measure describing the law of $\phi$.

For $\phi$ in Definition 2.15, define
${\mathcal{A}}_{\phi}=\lim_{{\varepsilon}\to
0}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma\phi_{\epsilon}(z)}d^{2}z$ on
${\mathcal{S}}$ and ${\mathcal{L}}_{\phi}=\lim_{{\varepsilon}\to
0}\epsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}\phi_{\epsilon}(z)}dz$ on
$\partial{\mathcal{S}}$. Write $A={\mathcal{A}}_{\phi}({\mathcal{S}})$ as the
total ${\mathcal{A}}_{\phi}$-area, and write $L_{1}$, $L_{2}$ as the
${\mathcal{L}}_{\phi}$-length of top and bottom boundary arc of
${\mathcal{S}}$, respectively. It appears that $\int e^{-A-\mu_{1}L_{1}-\mu
L_{2}}\,\mathrm{d}{{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)=\infty$. For
$\beta\in(\frac{2}{\gamma},Q)$, the correct definition is via truncation and
we have the following counterpart of Lemma 2.11.

######  Lemma 2.16.

Let $\beta\in(\frac{2}{\gamma},Q)$ and write $A,L_{1},L_{2}$ for the quantum
area and boundary arc lengths of a sample from
${{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta)$ as above. Then the following
integral defining $R_{\mu_{1},\mu_{2}}(\beta)$ is finite:

(2.13)
$R_{\mu_{1},\mu_{2}}(\beta):=\frac{2(Q-\beta)}{\gamma}\int(e^{-A-\mu_{1}L_{1}-\mu_{2}L_{2}}-1)\,\mathrm{d}{{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta).$

Furthermore set $s=\beta-Q$ and

(2.14)
$\hat{R}_{\mu_{1},\mu_{2}}(\beta):=\frac{-2s}{\gamma}\int(\gamma(s+\frac{\gamma}{2})A+\frac{\gamma^{2}}{2}A(\sum_{i}\mu_{i}L_{i})+\frac{\gamma^{2}}{4}(\sum_{i}\mu_{i}L_{i})^{2})e^{-A-\sum_{i}\mu_{i}L_{i}}\,d{{\mathcal{M}}^{\textrm{disk}}_{2}}(\beta).$

Then
$s(s+\frac{\gamma}{2})R_{\mu_{1},\mu_{2}}(\beta)=\hat{R}_{\mu_{1},\mu_{2}}(\beta)$.

###### Proof.

Recall the field $\psi$ used in Definition 2.15 and write
$\widetilde{A}=A_{\psi}({\mathcal{S}})$ and
$L_{1}={\mathcal{L}}_{\psi}({\mathbb{R}}\times\\{\pi\\}),L_{2}={\mathcal{L}}_{\psi}({\mathbb{R}})$.
By definition we have

$R_{\mu_{1},\mu_{2}}(\beta)=-\frac{2s}{\gamma}\mathbb{E}[\int\frac{\gamma}{2}e^{sc}\cdot(e^{-e^{\gamma
c}\widetilde{A}-e^{-\gamma
c/2}\sum_{i=1}^{2}\mu_{i}\widetilde{L}_{i}}-1)\,dc].$

We have $s\in(-\frac{\gamma}{2},0)$. For $a>0$ and $\ell\in{\mathbb{C}}$ with
$\Re\ell>0$, applying integration by parts twice as in the proof of Lemma
2.11, the expression $\int e^{sc}\cdot(e^{-e^{\gamma c}a-e^{\gamma
c/2}\ell}-1)\,dc$ equals

$\displaystyle\frac{\gamma}{s}\int e^{sc}(e^{\gamma c}a)e^{-e^{\gamma
c}a-e^{\gamma
c/2}\ell}\,dc+\frac{\gamma}{2s}\frac{\gamma}{s+\frac{\gamma}{2}}\int
e^{sc}(e^{\gamma c}a)(e^{\gamma c/2}\ell)e^{-e^{\gamma c}a-e^{\gamma
c/2}\ell}\,dc$
$\displaystyle+\frac{\gamma}{2s}\frac{\frac{\gamma}{2}}{s+\frac{\gamma}{2}}\int
e^{sc}(e^{\gamma c/2}\ell)^{2}e^{-e^{\gamma c}a-e^{\gamma c/2}\ell}\,dc.$

Applying this with $a=\widetilde{A}$ and
$\ell=\sum_{i=1}^{2}\mu_{i}\widetilde{L}_{i}$ gives the desired result.

∎

The following proposition extends the definition of $R$ in the same way as for
$H$ in Proposition 2.12. We defer its proof to Appendix B as it is similar to
that of Proposition 2.12.

######  Proposition 2.17.

For $\beta\in((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$ and
$\Re\mu_{1},\Re\mu_{2}\geq 0$, the integral in (2.14) defining $\hat{R}$ is
absolutely convergent. Moreover, for each
$\beta\in((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$, the
function $(\mu_{1},\mu_{2})\mapsto\hat{R}$ is holomorphic on
$\\{\Re\mu_{1}>0,\Re\mu_{2}>0\\}$ and continuous on $\\{\Re\mu_{1}\geq
0,\Re\mu_{2}\geq 0\\}$. Finally, for $\Re\mu_{1}>0,\Re\mu_{2}>0$, the function
$\beta\mapsto\hat{R}$ can be analytically extended on a complex neighborhood
of $((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$.

######  Definition 2.18.

For $\beta\in((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$ and
$\Re\mu_{1},\Re\mu_{2}\geq 0$, writing $s=\beta-Q$, we define

(2.15)
$R_{\mu_{1},\mu_{2}}(\beta):=\frac{1}{s(s+\frac{\gamma}{2})}\hat{R}_{\mu_{1},\mu_{2}}(\beta).$

We have the following counterpart of Proposition 2.14 on the change of
variable.

######  Proposition 2.19.

Recall $\mu_{B}(\sigma)$ and
${\mathcal{B}}=(-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2})\times\mathbb{R}$
in Proposition 2.14. Let
$R(\beta,\sigma_{1},\sigma_{2}):=R_{\mu_{B}(\sigma_{1}),\mu_{B}(\sigma_{2})}(\beta)$.
For each
$\beta\in((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$, the
function $(\sigma_{1},\sigma_{2})\mapsto R$ is holomorphic on
$\mathcal{B}^{2}$ and continuous on $\overline{\mathcal{B}}^{2}$. For each
$(\sigma_{1},\sigma_{2})\in\mathcal{B}^{2}$, the function $\beta\mapsto
s(s+\frac{\gamma}{2})R$ is analytic on a complex neighborhood of
$((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$.

###### Proof.

This follows from Proposition 2.17 similarly as in the proof of Proposition
2.14. ∎

######  Remark 2.20.

The meromorphic extension of $H$ can also be done by truncations in the spirit
of (2.13). Indeed, suppose $\mu_{1},\mu_{2},\mu_{3}\in{\mathbb{C}}$ satisfy
$\Re\mu_{i}\geq 0$ for $i=1,2,3$, and $(\beta_{1},\beta_{2},\beta_{3})$
satisfy
$0<Q-\frac{1}{2}\sum\beta_{i}<\frac{\gamma}{2}\wedge\min_{i}(Q-\beta_{i})$,
then

(2.16)
$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}=\int\left(e^{-{\mathcal{A}}_{\phi}({\mathbb{H}})-\sum_{i=1}^{3}\mu_{i}L_{i}}-1\right)\;\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}(\mathrm{d}\phi)$

If instead $(\beta_{1},\beta_{2},\beta_{3})$ satisfy
$\frac{\gamma}{2}<Q-\frac{1}{2}\sum\beta_{i}<\gamma\wedge\frac{2}{\gamma}\wedge\min_{i}(Q-\beta_{i})$,
then

(2.17)
$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}=\int\left(e^{-{\mathcal{A}}_{\phi}({\mathbb{H}})-\sum_{i=1}^{3}\mu_{i}L_{i}}-1+\sum_{i=1}^{3}\mu_{i}L_{i}\right)\;\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}(\mathrm{d}\phi).$

Both (2.16) and (2.17) follow from integration by parts. One can further
extend the range of $H$ via further truncations of $e^{-x}$. We leave these
details to the interested readers as we will not need them in this paper.
Similarly, we record without a detailed proof that when
$\beta\in((Q-\gamma)\vee\frac{\gamma}{2},\frac{2}{\gamma})$,

$R_{\mu_{1},\mu_{2}}(\beta)=\frac{2(Q-\beta)}{\gamma}\int\left(e^{-A-\sum_{i=1}^{2}\mu_{i}L_{i}}-1+\sum_{i=1}^{2}\mu_{i}L_{i}\right)\;d{\mathcal{M}}_{2}^{\mathrm{disk}}(\beta).$

###  2.5. Analytic continuation of $H$: proof of Proposition 2.12

We first prove Proposition 2.21 which will immediately imply Proposition 2.12.
Our argument follows the same strategy as in [KRV20] for the sphere case, with
the additional difficulty that the correlation functions are no longer GMC
moments due to the presence of both bulk and boundary Liouville potentials.

######  Proposition 2.21.

Recall the domain $V$ from Definition 2.11 and Proposition 2.12. Fix $\mu_{i}$
with $\Re\mu_{i}>0$ for $i=1,2,3$. Then for each of

$\displaystyle(\beta_{1},\beta_{2},\beta_{3})$ $\displaystyle\mapsto\int
Ae^{-A-\sum\mu_{i}L_{i}}\,\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0)(\beta_{2},1),(\beta_{3},\infty)}(d\phi),$
$\displaystyle(\beta_{1},\beta_{2},\beta_{3})$ $\displaystyle\mapsto\int
A(\sum\mu_{i}L_{i})e^{-A-\sum\mu_{i}L_{i}}\,\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0)(\beta_{2},1),(\beta_{3},\infty)}(d\phi),$
$\displaystyle(\beta_{1},\beta_{2},\beta_{3})$
$\displaystyle\mapsto\int(\sum\mu_{i}L_{i})^{2}e^{-A-\sum\mu_{i}L_{i}}\,\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0)(\beta_{2},1),(\beta_{3},\infty)}(d\phi),$

the integral converges when $(\beta_{1},\beta_{2},\beta_{3})\in V$, and the
function extends analytically to a neighborhood of $V$ in ${\mathbb{C}}^{3}$.
Here, we write $A={\mathcal{A}}_{\phi}({\mathbb{H}})$,
$L_{1}={\mathcal{L}}_{\phi}(-\infty,0)$, $L_{2}={\mathcal{L}}_{\phi}(0,1)$ and
$L_{3}={\mathcal{L}}_{\phi}(1,\infty)$.

We only prove the result for the second function in detail as the other two
can be proved similarly with the same inputs. To simplify notation, we will
assume that $\mu_{1}=\mu_{2}=\mu_{3}=1$ but the same argument works for the
general case. By the change of coordinates in [AHS23b, Lemma 2.20, Proposition
2.16], modulo an explicit prefactor the second function becomes

$f(\beta_{1},\beta_{2},\beta_{3})=\int{\mathcal{A}}_{\phi}({\mathbb{H}}){\mathcal{L}}_{\phi}({\mathbb{R}})e^{-{\mathcal{A}}_{\phi}({\mathbb{H}})-{\mathcal{L}}_{\phi}({\mathbb{R}})}\,\mathrm{LF}_{\mathbb{H}}^{(\beta_{j},p_{j})_{j}}(d\phi)\quad\textrm{with
}(p_{1},p_{2},p_{3})=(0,2,-2).$

Therefore, it suffices to prove that $f$ extends analytically to a
neighborhood of $V$ in ${\mathbb{C}}^{3}$. To show this, we will approximate
$f$ using truncations of ${\mathbb{H}}$ away from the insertions.

######  Lemma 2.22.

For $r\geq 1$, let
${\mathbb{H}}_{r}:={\mathbb{H}}\backslash\bigcup_{i}B_{e^{-r}}(p_{i})$ and
${\mathbb{R}}_{r}:={\mathbb{R}}\backslash\bigcup_{i}B_{e^{-r}}(p_{i})$. For a
Gaussian free field $h$ sampled from $P_{\mathbb{H}}$, write
$A_{r}:={\mathcal{A}}_{h-2Q\log|\cdot|_{+}}({\mathbb{H}}_{r})$ and
$L_{r}:={\mathcal{L}}_{h-2Q\log|\cdot|_{+}}({\mathbb{R}}_{r})$. For
$x\in{\mathbb{R}}$, write $h_{r}(x)$ for the average of $h$ on $\partial
B_{e^{-r}}(x)\cap{\mathbb{H}}$. For $r\geq 1$, let

$f_{r}(\beta_{1},\beta_{2},\beta_{3}):=\int_{\mathbb{R}}dc\,e^{(\frac{1}{2}\sum\beta_{i}-Q)c}\mathbb{E}\left[\prod_{i}e^{\frac{\beta_{i}}{2}h_{r}(p_{i})-\frac{\beta_{i}^{2}}{4}r}(e^{\gamma
c}A_{r})(e^{\frac{\gamma}{2}c}L_{r})\exp(-e^{\gamma
c}A_{r}-e^{\frac{\gamma}{2}c}L_{r})\right]$

Then each point in $V$ has a neighborhood in ${\mathbb{C}}^{3}$ on which
$f_{r}$ converges uniformly as $r\to\infty$. Moreover,
$\lim_{r\to\infty}f_{r}(\beta_{1},\beta_{2},\beta_{3})=f(\beta_{1},\beta_{2},\beta_{3})$
for all $(\beta_{1},\beta_{2},\beta_{3})\in V$.

Proposition 2.21 follows immediately from Lemma 2.22.

###### Proof of Proposition 2.21.

Since $f_{r}$ is analytic, by Lemma 2.22, there is a neighborhood of $V$ on
which $\lim_{r\to\infty}f_{r}$ is holomorphic. Moreover, this limit agrees
with $f$ on $V$, hence is the desired analytic continuation of $f$. ∎

###### Proof of Lemma 2.22.

If we condition on $h|_{{\mathbb{H}}_{r}}$, then
$(h_{r+1}(p_{i})-h_{r}(p_{i}))_{i=1,2,3}$ is a triple of conditionally
independent Gaussians. Therefore

$f_{r}(\beta_{1},\beta_{2},\beta_{3})=\int_{\mathbb{R}}dc\,e^{(\frac{1}{2}\sum\beta_{i}-Q)c}\mathbb{E}\left[\prod_{i=1}^{3}e^{\frac{\beta_{i}}{2}h_{r+1}(p_{i})-\frac{\beta_{i}^{2}}{4}(r+1)}(e^{\gamma
c}A_{r})(e^{\frac{\gamma}{2}c}L_{r})\exp(-e^{\gamma
c}A_{r}-e^{\frac{\gamma}{2}c}L_{r})\right].$

Write $g(a,\ell):=ae^{-a}\ell e^{-\ell}$. We have
$f_{r+1}(\beta_{1},\beta_{2},\beta_{3})-f_{r}(\beta_{1},\beta_{2},\beta_{3})$
equals

$\displaystyle\int_{\mathbb{R}}dc\,e^{(\frac{1}{2}\sum\beta_{i}-Q)c}\mathbb{E}\left[\prod_{i}e^{\frac{\beta_{i}}{2}h_{r+1}(p_{i})-\frac{\beta_{i}^{2}}{4}(r+1)}(g(e^{\gamma
c}A_{r+1},e^{\frac{\gamma}{2}c}L_{r+1})-g(e^{\gamma
c}A_{r},e^{\frac{\gamma}{2}c}L_{r}))\right].$

Fix a point in $V$ and let $O$ be a neighborhood of the point. We will require
$O$ to be sufficiently small as explained below. For
$(\beta_{1},\beta_{2},\beta_{3})\in O$ write $\beta_{j}=u_{j}+v_{j}\mathbf{i}$
where $u_{j},v_{j}\in{\mathbb{R}}$. Set
$\tilde{s}:=\frac{1}{2}\sum_{i}u_{i}-Q$. By the definition of $V$, we can
choose $O$ small enough such that $\tilde{s}>-\gamma$. Write
$\widetilde{A}_{r}:={\mathcal{A}}_{\widetilde{h}}({\mathbb{H}}_{r})$ and
$\widetilde{L}_{r}:={\mathcal{L}}_{\widetilde{h}}({\mathbb{R}}_{r})$ where
$\widetilde{h}=h+\sum\frac{u_{i}}{2}G_{\mathbb{H}}(\cdot,p_{i})-2Q\log|\cdot|_{+}$.
By Girsanov’s theorem,

$\displaystyle|f_{r+1}(\beta_{1},\beta_{2},\beta_{3})-f_{r}(\beta_{1},\beta_{2},\beta_{3})|$
$\displaystyle\leq\int_{\mathbb{R}}dc\,e^{\tilde{s}c}\mathbb{E}\left[\left|\prod_{i}e^{\frac{\beta_{i}}{2}h_{r+1}(p_{i})-\frac{\beta_{i}^{2}}{4}(r+1)}(g(e^{\gamma
c}A_{r+1},e^{\frac{\gamma}{2}c}L_{r+1})-g(e^{\gamma
c}A_{r},e^{\frac{\gamma}{2}c}L_{r}))\right|\right]$ (2.18) $\displaystyle\leq
C_{0}e^{\frac{r+1}{4}\sum
v_{i}^{2}}\int_{\mathbb{R}}dc\,e^{\tilde{s}c}\mathbb{E}\left[|g(e^{\gamma
c}\widetilde{A}_{r+1},e^{\frac{\gamma}{2}c}\widetilde{L}_{r+1})-g(e^{\gamma
c}\widetilde{A}_{r},e^{\frac{\gamma}{2}c}\widetilde{L}_{r})|\right],$

where
$C_{0}=\sup_{O}\mathbb{E}[\prod_{i}e^{\frac{u_{i}}{2}h_{r+1}(p_{i})-\frac{u_{i}^{2}}{4}(r+1)}]$
is a constant not depending on $r$, which is finite if $O$ is sufficiently
small. Now, using the triangle inequality, and the fact that $\ell\mapsto\ell
e^{-\ell}$ is bounded and Lipschitz, we have for any $a_{r+1}>a_{r}>0$ and
$\ell_{r+1}>\ell_{r}>0$ that

$|g(a_{r+1},\ell_{r+1})-g(a_{r},\ell_{r})|\leq
C_{1}((\ell_{r+1}-\ell_{r})a_{r+1}e^{-a_{r+1}}+(a_{r+1}-a_{r})e^{-a_{r+1}}+a_{r}(e^{-a_{r}}-e^{-a_{r+1}})),$

where $C_{1}$ is a universal constant. This gives three terms to bound. Now
using the identity $\int e^{tc}e^{-e^{\gamma
c}a}\,\mathrm{d}c=\frac{1}{\gamma}\Gamma(\frac{t}{\gamma})a^{-\frac{t}{\gamma}}$
for $t,\ell>0$, we have

$\displaystyle\int_{\mathbb{R}}dc\,e^{(\tilde{s}+\gamma+\frac{\gamma}{2})c}\mathbb{E}[(\widetilde{L}_{r+1}-\widetilde{L}_{r})\widetilde{A}_{r+1}e^{-e^{\gamma
c}\widetilde{A}_{r+1}}]$
$\displaystyle=\frac{1}{\gamma}\Gamma(\frac{\tilde{s}}{\gamma}+\frac{3}{2})\mathbb{E}[(\widetilde{L}_{r+1}-\widetilde{L}_{r})\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-\frac{1}{2}}],$
$\displaystyle\int_{\mathbb{R}}dc\,e^{(\tilde{s}+\gamma)c}\mathbb{E}[(\widetilde{A}_{r+1}-\widetilde{A}_{r})e^{-e^{\gamma
c}\widetilde{A}_{r+1}}]$
$\displaystyle=\frac{1}{\gamma}\Gamma(\frac{\tilde{s}}{\gamma}+1)\mathbb{E}[(\widetilde{A}_{r+1}-\widetilde{A}_{r})\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1}],$
$\displaystyle\int_{\mathbb{R}}dc\,e^{(\tilde{s}+\gamma)c}\mathbb{E}[\widetilde{A}_{r}(e^{-e^{\gamma
c}\widetilde{A}_{r}}-e^{-e^{\gamma c}\widetilde{A}_{r+1}})]$
$\displaystyle=\frac{1}{\gamma}\Gamma(\frac{\tilde{s}}{\gamma}+1)\mathbb{E}[\widetilde{A}_{r}(\widetilde{A}_{r}^{-\frac{\tilde{s}}{\gamma}-1}-\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1})].$

Now Lemma 2.22 follows from Lemma 2.23, which we state right below.

######  Lemma 2.23.

For each point in $V$ there is a neighborhood $U\subset{\mathbb{R}}^{3}$ and a
constant $C>0$ such that the following holds. For $(u_{1},u_{2},u_{3})\in U$
and $r\geq 1$, with $\tilde{s}$, $\widetilde{A}_{r}$ and $\widetilde{L}_{r}$
defined above,

$\mathbb{E}[(\widetilde{A}_{r+1}-\widetilde{A}_{r})\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1}],\quad\mathbb{E}[\widetilde{A}_{r}(\widetilde{A}_{r}^{-\frac{\tilde{s}}{\gamma}-1}-\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1})],\quad\mathbb{E}[(\widetilde{L}_{r+1}-\widetilde{L}_{r})\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-\frac{1}{2}}]$

are all bounded by $Ce^{-r/C}$. Moreover, write
$\widetilde{A}_{\infty}:={\mathcal{A}}_{\widetilde{h}}({\mathbb{H}})$ and
$\widetilde{L}_{\infty}:={\mathcal{L}}_{\widetilde{h}}({\mathbb{R}})$. Then
the same holds with $\widetilde{A}_{\infty},\widetilde{L}_{\infty}$ in place
of $\widetilde{A}_{r+1},\widetilde{L}_{r+1}$.

By Lemma 2.23, we can choose $O$ small enough and $C$ large enough so that
$|f_{r+1}(\beta_{1},\beta_{2},\beta_{3})-f_{r}(\beta_{1},\beta_{2},\beta_{3})|<Ce^{(\frac{1}{4}\sum
v_{i}^{2}-C^{-1})r}$ uniformly in $O$. By possibly shrinking $O$ further so
that $\sum v_{i}^{2}-1/C<-1/2C$ for all $(\beta_{1},\beta_{2},\beta_{3})\in
O$, we see that the $f_{r}$ uniformly converge in $O$ as desired.

It remains to prove
$\lim_{r\to\infty}f_{r}(\beta_{1},\beta_{2},\beta_{3})=f(\beta_{1},\beta_{2},\beta_{3})$
for $(\beta_{1},\beta_{2},\beta_{3})\in V$. In this case since
$\beta_{1},\beta_{2},\beta_{3}$ are real we have

$|f(\beta_{1},\beta_{2},\beta_{3})-f_{r}(\beta_{1},\beta_{2},\beta_{3})|\leq
C_{0}\int_{\mathbb{R}}dc\,e^{(\frac{1}{2}\sum\beta_{i}-Q)c}\mathbb{E}[|g(e^{\gamma
c}\widetilde{A}_{\infty},e^{\frac{\gamma}{2}c}\widetilde{L}_{\infty})-g(e^{\gamma
c}\widetilde{A}_{r},e^{\frac{\gamma}{2}c}\widetilde{L}_{r})|].$

Now by the same argument as before with the
$\widetilde{A}_{\infty},\widetilde{L}_{\infty}$ case of Lemma 2.23 being
applied, we get
$\lim_{r\to\infty}f_{r}(\beta_{1},\beta_{2},\beta_{3})=f(\beta_{1},\beta_{2},\beta_{3})$.
∎

###### Proof of Lemma 2.23.

Fix a choice of $(\tilde{\beta}_{1},\tilde{\beta}_{2},\tilde{\beta}_{3})\in
V$; we will find a suitably small neighborhood $U$ of this point. We will only
explain the $A_{r+1},L_{r+1}$ case in detail since the $A_{\infty},L_{\infty}$
follows from the same argument. Moreover, we will focus on a fixed
$(u_{1},u_{2},u_{3})\in U$, but all inputs in the argument vary continuously
in $(u_{1},u_{2},u_{3})$ which gives the desired uniform bounds in $U$. We now
prove the three bounds one by one. We write $x\lesssim y$ to denote that
$x\leq ky$ for some constant $k$ not depending on $r$ (but which may depend on
$U$).

First bound. For ${\varepsilon}>0$, we have
$\mathbb{E}[\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1}(\widetilde{A}_{r+1}-\widetilde{A}_{r})]\leq{\varepsilon}\mathbb{E}[\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1}]+\mathbb{E}[1_{\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}}\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}}]$.
By Hölder inequality,
$\mathbb{E}[1_{\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}}\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}}]\leq\mathbb{E}[\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}p}]^{1/p}\mathbb{P}[\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}]^{1/q}$
for $p,q>1$ satisfying $\frac{1}{p}+\frac{1}{q}=1$. By the definition of $V$,
we can choose $U$ and $p$ small enough such that both
$-\frac{\tilde{s}}{\gamma}-1$ and $-\frac{\tilde{s}}{\gamma}p$ lie in
$(-\infty,\frac{2}{\gamma^{2}}\wedge\frac{1}{\gamma}\min(Q-\tilde{\beta}_{i}))$.
By [HRV18, Corollary 6.11], for
$\lambda<\frac{2}{\gamma^{2}}\wedge\frac{1}{\gamma}\min(Q-\tilde{\beta}_{i})$
we have
$\mathbb{E}[\widetilde{A}_{r+1}^{\lambda}]<\mathbb{E}[\widetilde{A}_{1}^{\lambda}]+\mathbb{E}[\widetilde{A}_{\infty}^{\lambda}]<\infty$.
Therefore
$\mathbb{E}[\widetilde{A}_{r+1}^{-\frac{\tilde{s}}{\gamma}-1}(\widetilde{A}_{r+1}-\widetilde{A}_{r})]\lesssim{\varepsilon}+\mathbb{P}[\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}]^{1/q}$.
Now, for $m>0$ small enough so that $Q-\max_{i}\tilde{\beta}_{i}-m\gamma>0$,
by the multifractal spectrum of LQG (e.g. [BP, Theorem 3.23] with minor
adaptation for the boundary setting) we have

(2.19)
$\mathbb{P}[\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}]\leq{\varepsilon}^{-m}\mathbb{E}[(\widetilde{A}_{r+1}-\widetilde{A}_{r})^{m}]\lesssim{\varepsilon}^{-m}e^{-m\gamma(Q-\max_{i}\tilde{\beta}_{i}-m\gamma)r}$

where the term $e^{m\frac{\gamma}{2}r\max_{i}\tilde{\beta}_{i}}$ comes from
the log singularities added to the field. Choosing ${\varepsilon}=e^{-r/C}$
for large enough $C$ yields
$\mathbb{E}[\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-1}(\widetilde{A}_{r+1}-\widetilde{A}_{r})]\leq
Ce^{-r/C}$.

Second bound. Choose $U$ small enough so that $-\frac{\tilde{s}}{\gamma}-1<0$.
For $a_{r+1}>a_{r}>0$ we have
$a_{r}(a_{r}^{-\frac{s}{\gamma}-1}-a_{r+1}^{-\frac{s}{\gamma}-1})\lesssim(a_{r+1}-a_{r})a_{r}^{-\frac{s}{\gamma}-1}$.
As in the proof of the first equality, for ${\varepsilon}>0$ we have

$\mathbb{E}[\widetilde{A}_{r}(\widetilde{A}_{r}^{-\frac{s}{\gamma}-1}-\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-1})]\lesssim{\varepsilon}\mathbb{E}[\widetilde{A}_{r}^{-\frac{s}{\gamma}-1}]+\mathbb{E}[1_{\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}}\widetilde{A}_{r}^{-\frac{s}{\gamma}}]\lesssim{\varepsilon}+\mathbb{E}[\widetilde{A}_{r}^{-\frac{s}{\gamma}p}]^{1/p}\mathbb{P}[\widetilde{A}_{r+1}-\widetilde{A}_{r}>{\varepsilon}]^{1/q}.$

Now the same argument gives
$\mathbb{E}[\widetilde{A}_{r}(\widetilde{A}_{r}^{-\frac{s}{\gamma}-1}-\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-1})]\leq
Ce^{-r/C}$ for large enough $C$.

Third bound. Arguing as in the first case, we have

$\mathbb{E}[(\widetilde{L}_{r+1}-\widetilde{L}_{r})\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-\frac{1}{2}}]\lesssim{\varepsilon}+\mathbb{E}[(\widetilde{L}_{r+1}\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-\frac{1}{2}})^{p}]^{1/p}\mathbb{P}[\widetilde{L}_{r+1}-\widetilde{L}_{r}>{\varepsilon}]^{1/q}.$

We now show
$\mathbb{E}[(\widetilde{L}_{r+1}\widetilde{A}^{-\frac{s}{\gamma}-\frac{1}{2}})^{p}]$
is uniformly bounded in $r$, from which the same argument gives
$\mathbb{E}[(\widetilde{L}_{r+1}-\widetilde{L}_{r})\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-\frac{1}{2}}]\leq
Ce^{-r/C}$ for large enough $C$.

Since
$\frac{1}{2}+(-\frac{s}{\gamma}-\frac{1}{2})=-\frac{s}{\gamma}<\frac{2}{\gamma^{2}}\wedge\min_{i}\frac{1}{\gamma}(Q-\beta_{i})$
we can choose $\lambda_{1},\lambda_{2}$ satisfying
$\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}=1$ such that
$\frac{1}{2}\lambda_{1}$ and $(-\frac{s}{\gamma}-\frac{1}{2})\lambda_{2}$ are
each bounded above by
$\frac{2}{\gamma^{2}}\wedge\frac{1}{\gamma}\min_{i}(Q-\beta_{i})$. Hölder’s
inequality gives
$\mathbb{E}[(\widetilde{L}_{r+1}\widetilde{A}^{-\frac{s}{\gamma}-\frac{1}{2}})^{p}]\leq\mathbb{E}[\widetilde{L}_{r+1}^{\lambda_{1}p}]^{1/\lambda_{1}}\mathbb{E}[(\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-\frac{1}{2}})^{p\lambda_{2}}]^{1/\lambda_{2}}$.
Choose $p>1$ sufficiently small and recall our moment bound for
$\widetilde{A}_{r+1}$ in the first case. By [HRV18, Corollary 3.10] the
analogous bound
$\mathbb{E}[\widetilde{L}_{r+1}^{\lambda}]<\mathbb{E}[\widetilde{L}_{1}^{\lambda}]+\mathbb{E}[\widetilde{L}_{\infty}^{\lambda}]<\infty$
holds for
$\lambda<\frac{4}{\gamma^{2}}\wedge\frac{2}{\gamma}\min(Q-\beta_{i})$.
Therefore
$\mathbb{E}[(\widetilde{L}_{r+1}\widetilde{A}_{r+1}^{-\frac{s}{\gamma}-\frac{1}{2}})^{p}]$
is uniformly bounded as needed. ∎

###### Proof of Proposition 2.12.

The analyticity in $\mu$ follows from Morera’s theorem. The meromorphicity in
$\beta$ follows from Proposition 2.21, where we proved the desired properties
for each term in $\hat{H}$. ∎

##  3\. Shift equations for $H$ and $R$

Our goal is to derive a set of shift equations (Theorems 3.1 and 3.2) that
will completely specify $H$ and $R$. We will always assume
$\gamma\neq\sqrt{2}$ — this is a technical limitation inherited from Theorem
3.4 from [Ang23]. In the next section we will use these shift equations to
first derive the exact formulas for $H$ and $R$ when $\gamma^{2}$ is
irrational, and then use continuity in $\gamma$ to remove this restriction. In
particular the assumption $\gamma\neq\sqrt{2}$ can be removed from Theorems
3.1 and 3.2 once these exact formulas are proved. For concision we will not
explicitly state the $\gamma\neq\sqrt{2}$ assumption in the lemmas and
propositions of this section.

To state the shift equations we introduce the following two notations:

(3.1)
$q=\frac{2Q-\beta_{1}-\beta_{2}-\beta_{3}+\chi}{\gamma}\quad\textrm{and}\quad
g_{\chi}(\sigma)=\left(\sin(\frac{\pi\gamma^{2}}{4})\right)^{-\chi/\gamma}\cos\left(2\pi\chi(\sigma-\frac{Q}{2})\right).$

Recalling (1.9), note that for $\chi=\frac{\gamma}{2}$ one has
$g_{\frac{\gamma}{2}}(\sigma)=\mu_{B}(\sigma)$. We will again refer to a
function of several complex variables as being meromorphic on a domain if it
can be expressed as the ratio of two holomorphic functions of several
variables on the domain.

######  Theorem 3.1 (Shift equations for $H$).

Assume $\gamma\neq\sqrt{2}$, let $\chi=\frac{\gamma}{2}$ or $\frac{2}{\gamma}$
and fix
$\sigma_{3}\in[-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2}-\frac{\gamma}{4}]\times\mathbb{R}$.
The function $H$ can be jointly meromorphically extended to a complex
neighborhood of $\mathbb{R}^{3}$ in $(\beta_{1},\beta_{2},\beta_{3})$ and to
$\mathbb{C}^{2}$ in $(\sigma_{1},\sigma_{2})$. Under this extension, it obeys

(3.2) $\displaystyle H\begin{pmatrix}\beta_{1},\beta_{2}-\chi,\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=\frac{\Gamma(\chi(\beta_{1}-\chi))\Gamma(1-\chi\beta_{2}+\chi^{2})}{\Gamma(\chi(\beta_{1}-\chi+q\frac{\gamma}{2}))\Gamma(1-\chi\beta_{2}+\chi^{2}-q\frac{\gamma\chi}{2})}H\begin{pmatrix}\beta_{1}-\chi,\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\chi}{2},\sigma_{3}\end{pmatrix}$
$\displaystyle+\frac{\chi^{2}\pi^{\frac{2\chi}{\gamma}}}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2\chi}{\gamma}}}\frac{\Gamma(1-\chi\beta_{1})\Gamma(1-\chi\beta_{2}+\chi^{2})\left(g_{\chi}(\sigma_{1})-g_{\chi}(\sigma_{2}+\frac{\beta_{1}}{2})\right)}{\sin(\pi\chi(\chi-\beta_{1}))\Gamma(1+\frac{q\gamma\chi}{2})\Gamma(2-\chi(\beta_{1}+\beta_{2}-2\chi+q\frac{\gamma}{2}))}H\begin{pmatrix}\beta_{1}+\chi,\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\chi}{2},\sigma_{3}\end{pmatrix},$

and

(3.3)
$\displaystyle\frac{\chi^{2}\pi^{\frac{2\chi}{\gamma}-1}}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2\chi}{\gamma}}}\Gamma(1-\chi\beta_{2})\left(g_{\chi}(\sigma_{3})-g_{\chi}(\sigma_{2}+\frac{\beta_{2}}{2})\right)H\begin{pmatrix}\beta_{1},\beta_{2}+\chi,\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\chi}{2},\sigma_{3}\end{pmatrix}$
$\displaystyle=\frac{\Gamma(\chi(\beta_{1}-\chi))}{\Gamma(-q\frac{\gamma\chi}{2})\Gamma(-1+\chi(\beta_{1}+\beta_{2}-2\chi+q\frac{\gamma}{2}))}H\begin{pmatrix}\beta_{1}-\chi,\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}$
$\displaystyle+\frac{\chi^{2}\pi^{\frac{2\chi}{\gamma}}}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2\chi}{\gamma}}}\frac{\left(g_{\chi}(\sigma_{1})-g_{\chi}(\sigma_{2}-\frac{\beta_{1}}{2}+\frac{\chi}{2})\right)\Gamma(1-\chi\beta_{1})}{\sin(\pi\chi(\chi-\beta_{1}))\Gamma(1-\chi(\beta_{1}-\chi+q\frac{\gamma}{2}))\Gamma(\chi\beta_{2}-\chi^{2}+q\frac{\gamma\chi}{2})}H\begin{pmatrix}\beta_{1}+\chi,\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}.$

######  Theorem 3.2 (Shift equations for $R$).

Assume $\gamma\neq\sqrt{2}$ and let $\chi=\frac{\gamma}{2}$ or
$\frac{2}{\gamma}$. The function $R(\beta,\sigma_{1},\sigma_{2})$ can be
jointly meromorphically extended to a complex neighborhood of $\mathbb{R}$ in
$\beta$ and to $\mathbb{C}^{2}$ in $(\sigma_{1},\sigma_{2})$. Under this
extension, it obeys

(3.4)
$\displaystyle\frac{R(\beta,\sigma_{1},\sigma_{2})}{R(\beta+\chi,\sigma_{1}-\frac{\chi}{2},\sigma_{2})}$
$\displaystyle=c_{\chi}(\gamma)\Gamma(-1+\chi\beta-\chi^{2})\Gamma(1-\chi\beta)\left(g_{\chi}(\sigma_{2})-g_{\chi}(\sigma_{1}-\frac{\beta}{2})\right),$
(3.5)
$\displaystyle\frac{R(\beta,\sigma_{1},\sigma_{2})}{R(\beta+\chi,\sigma_{1}+\frac{\chi}{2},\sigma_{2})}$
$\displaystyle=c_{\chi}(\gamma)\Gamma(-1+\chi\beta-\chi^{2})\Gamma(1-\chi\beta)\left(g_{\chi}(\sigma_{2})-g_{\chi}(\sigma_{1}+\frac{\beta}{2})\right),$

where $c_{\frac{\gamma}{2}}(\gamma)=-\frac{1}{\Gamma(-\frac{\gamma^{2}}{4})}$
and
$c_{\frac{2}{\gamma}}(\gamma)=\frac{4}{\gamma^{2}}\pi^{\frac{4}{\gamma^{2}}-1}\Gamma(1-\frac{\gamma^{2}}{4})^{-\frac{4}{\gamma^{2}}}$.

The proofs of Theorems 3.1 and 3.2 are based on the Belavin-Polyakov-
Zamolodchikov (BPZ) equations and the operator product expansions (OPE), a
strategy first used in [KRV20] and carried out for boundary LCFT in the
absence of bulk Liouville potential in [RZ22]. In our case, the high level
idea is to consider two special cases of four-point correlation functions as
deformations of the $H$ function which satisfy the BPZ equations proved in
[Ang23]. They reduce to hypergeometric equations after a change of variable.
The OPE is the asymptotic expansion by which we relate the coefficients of the
solution space of hypergeometric equations to $H$. The shift equations on $H$
are then simply the connection formulas on the solution space. The shift
equations on $R$ are obtained by taking a suitable limit of the equations on
$H$.

Although the same shift equations as in Theorems 3.1 and 3.2 were derived in
[RZ22] except that the function $g_{\chi}$ is replaced by
$e^{2\chi\pi\mathbf{i}(\sigma-\frac{Q}{2})}$, there are two additional
difficulties that we must overcome. The first is that in [RZ22] $H$ and $R$
(denoted by $\overline{H}$ and $\overline{R}$ in [RZ22]) can reduce to moments
of GMC up to a prefactor. This is not possible in our case. We are forced to
work with the more involved Definitions 2.13 and 2.18 which introduces extra
constraints on the parameters. The second is that [RZ22] used the value of
$R(\beta,\sigma_{1},\sigma_{2})$ at $\mu_{1}=0$ as an input, which was known
from [RZ20]. For us this is not available. Instead we use the mating-of-trees
theory to get the needed explicit formula for
$R(\gamma,\sigma_{1},\sigma_{2})$; see Lemma 3.18. Finally, we have to first
prove Theorem 3.2 based on some partial results towards Theorem 3.1, and then
completely prove Theorem 3.1.

The rest of this section is organized as follows. In Section 3.1 we recall the
BPZ equation from [Ang23]. In Sections 3.2 and 3.3 we establish respectively
Theorems 3.1 and 3.2 in the case $\chi=\frac{\gamma}{2}$ and in a limited
range of parameters. In Section 3.4 we prove the reflection principle for $H$
and $R$ and show they can be meromorphically extended to the full range of
parameters claimed in Theorems 3.1 and 3.2. In Section 3.5 we prove Theorem
3.2 in the case $\chi=\frac{2}{\gamma}$, which is the most technically
involved and novel part of this section. Finally in Section 3.6 we prove
Theorem 3.1 for $\chi=\frac{2}{\gamma}$. We provide in Appendix A the
necessary background on hypergeometric equations and several facts on the
Ponsot-Teschner formula ${H_{\mathrm{PT}}}$. We defer the proofs of two facts
on the function $R$ to Section 5 and the proofs of the OPE lemmas to Appendix
C.

###  3.1. The BPZ equation for a deformation of $H$

Before stating the BPZ differential equation, let us give a generic definition
of the boundary four-point function, presented after having applied the
Girsanov theorem to all the insertions. This definition was first proposed in
[HRV18], modulo the fact that [HRV18] does not consider the varying boundary
cosmological constant.

######  Definition 3.3.

(Boundary four-point function) Consider four points $s_{1}<s_{2}<s_{3}<s_{4}$
in $\mathbb{R}$ with associated weights
$\beta_{1},\beta_{2},\beta_{3},\beta_{4}\in\mathbb{R}$ satisfying the
conditions $\beta_{i}<Q$ and $\sum_{i}\beta_{i}>2Q$. Consider also
$\mu_{1},\mu_{2},\mu_{3},\mu_{4}$ satisfying the condition $\Re(\mu_{i})\geq
0$. Then define

$\displaystyle\langle\prod_{j=1}^{4}B_{\beta_{j}}^{\sigma_{j},\sigma_{j+1}}(s_{j})\rangle=\prod_{j<j^{\prime}}|s_{j}-s_{j^{\prime}}|^{-\frac{\beta_{j}\beta_{j^{\prime}}}{2}}\int_{\mathbb{R}}dc\,e^{-\frac{\gamma\widetilde{q}c}{2}}\mathbb{E}\Bigg{[}\exp\Bigg{(}-e^{\gamma
c}\int_{\mathbb{H}}\frac{\hat{g}(x)^{\frac{\gamma^{2}}{4}(\widetilde{q}+1)}}{\prod_{j=1}^{4}|x-s_{j}|^{\gamma\beta_{j}}}e^{\gamma
h(x)}d^{2}x$ $\displaystyle-e^{\frac{\gamma
c}{2}}\int_{\mathbb{R}}\frac{\hat{g}(r)^{\frac{\gamma^{2}\widetilde{q}}{8}}}{\prod_{j=1}^{4}|r-s_{j}|^{\frac{\gamma\beta_{j}}{2}}}e^{\frac{\gamma}{2}h(r)}d\mu_{\partial}(r)\Bigg{)}\Bigg{]},$

where
$\widetilde{q}=\frac{2Q-\beta_{1}-\beta_{2}-\beta_{3}-\beta_{4}}{\gamma}$,
$\hat{g}(x)=|x|_{+}^{-4}$ and where the boundary measure is given by:

(3.6)
$d\mu_{\partial}(r)/dr=\sum_{j=1}^{3}\mu_{j+1}\mathbf{1}_{s_{j}<r<s_{j+1}}+\mu_{1}\mathbf{1}_{r\notin(s_{1},s_{4})}.$

We now state parameter ranges for which the four-point function with the
degenerate insertion will be well-defined. Recall
${\mathcal{B}}=(-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2})\times\mathbb{R}$
and
$\overline{{\mathcal{B}}}=[-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2}]\times\mathbb{R}$.
For $\chi=\frac{\gamma}{2}$ or $\frac{2}{\gamma}$, consider the following
ranges of parameters

(3.7)
$\beta_{i}\in(-\infty,Q),\>\>\beta_{1}+\beta_{2}+\beta_{3}>2Q+\chi,\quad\sigma_{1}-\frac{\chi}{2}\in\overline{{\mathcal{B}}},\>\>\sigma_{2}-\frac{\chi}{2}\in\overline{{\mathcal{B}}},\>\>\sigma_{1}\in\overline{{\mathcal{B}}},\>\>\sigma_{2}\in\overline{{\mathcal{B}}},\>\>\sigma_{3}\in\overline{{\mathcal{B}}},$

and:

(3.8)
$\beta_{i}\in(-\infty,Q),\>\>\beta_{1}+\beta_{2}+\beta_{3}>2Q+\chi,\quad\sigma_{1}+\frac{\chi}{2}\in\overline{{\mathcal{B}}},\>\>\sigma_{2}+\frac{\chi}{2}\in\overline{{\mathcal{B}}},\>\>\sigma_{1}\in\overline{{\mathcal{B}}},\>\>\sigma_{2}\in\overline{{\mathcal{B}}},\>\>\sigma_{3}\in\overline{{\mathcal{B}}}.$

Note that each set of constraints specifies a nonempty parameter range.
Indeed, for (3.7), if all three $\beta_{i}$ are close to $Q$ then
$\beta_{1}+\beta_{2}+\beta_{3}>2Q+\chi$ is true. For the $\sigma_{i}$, the
band with its boundary included $\overline{{\mathcal{B}}}$ has width
$\frac{1}{\gamma}$. Therefore it is possible to have both
$\sigma_{1}-\frac{\chi}{2}$ and $\sigma_{1}$ in $\overline{{\mathcal{B}}}$,
but in the case $\chi=\frac{2}{\gamma}$ this means that
$\sigma_{1}-\frac{\chi}{2}$ and $\sigma_{1}$ are exactly on the boundary of
$\overline{{\mathcal{B}}}$. The same is true for $\sigma_{2}$. Then the
parameter range (3.8) works the same. Under these constraints we have the
following BPZ equation obtained in [Ang23].

######  Theorem 3.4 ([Ang23]).

Suppose $\gamma\neq\sqrt{2}$. Consider parameters
$(\beta_{i},\sigma_{i})_{i=1,2,3}$ obeying (3.7) or (3.8). Consider disjoint
points $s,s_{1},s_{2},s_{3}\in\mathbb{R}$ with associated weights
$-\chi,\beta_{1},\beta_{2},\beta_{3}$ where $\chi=\frac{\gamma}{2}$ or
$\frac{2}{\gamma}$. Define the following differential operator:

(3.9)
$\mathcal{D}:=\frac{1}{\chi^{2}}\partial_{ss}+\sum_{j=1}^{3}\frac{1}{s-s_{j}}\partial_{s_{j}}+\sum_{j=1}^{3}\frac{\Delta_{\beta_{j}}}{(s-s_{j})^{2}}.$

Then for $s_{1}<s<s_{2}<s_{3}$ we have

(3.10) $\displaystyle\mathcal{D}\left\langle
B_{\beta_{1}}^{\sigma_{1}\pm\frac{\chi}{2},\sigma_{2}\pm\frac{\chi}{2}}(s_{1})B_{-\chi}^{\sigma_{2}\pm\frac{\chi}{2},\sigma_{2}}(s)B_{\beta_{2}}^{\sigma_{2},\sigma_{3}}(s_{2})B_{\beta_{3}}^{\sigma_{3},\sigma_{1}\pm\frac{\chi}{2}}(s_{3})\right\rangle=0,$

and for $s<s_{1}<s_{2}<s_{3}$:

(3.11) $\displaystyle\mathcal{D}\left\langle
B_{-\chi}^{\sigma_{1}\pm\frac{\chi}{2},\sigma_{1}}(s)B_{\beta_{1}}^{\sigma_{1},\sigma_{2}}(s_{1})B_{\beta_{2}}^{\sigma_{2},\sigma_{3}}(s_{2})B_{\beta_{3}}^{\sigma_{3},\sigma_{1}\pm\frac{\chi}{2}}(s_{3})\right\rangle=0.$

Above when $\pm\frac{\chi}{2}$ equals $-\frac{\chi}{2}$ the parameters must
obey (3.7) and when it equals $\frac{\chi}{2}$ must obey (3.8).

We point out that very recently Cerclé [Cer24] gave a new proof of Theorem 3.4
without using mating of trees. Moreover, Baverez and Wu [BW23] suggested a
proof strategy; see [BW23, Theorem 1.3] and the discussion afterwards.

We now use conformal invariance to perform a change of variable that will
place the boundary spectator insertions $s_{1},s_{2},s_{3}$ at $0,1,\infty$
and the degenerate insertion at $s$ will be mapped to the cross ratio $t$ of
$s_{1},s_{2},s_{3},s$. The concrete conformal map we use for this change of
variable is thus:

(3.12)
$\displaystyle\psi(u)=\frac{(u-s_{1})(s_{2}-s_{3})}{(u-s_{3})(s_{2}-s_{1})}.$

By setting $t=\psi(s)$ we have $t\in(0,1)$ when $s_{1}<s<s_{2}$ and
$t\in(-\infty,0)$ when $s<s_{1}$. Now under the constraint of (3.7), consider
the functional

(3.13) $\displaystyle H_{\chi}(t)=\int_{\mathbb{R}}dc\,e^{-\frac{\gamma
qc}{2}}\mathbb{E}\Bigg{[}\exp\Bigg{(}-e^{\gamma c}$
$\displaystyle\int_{\mathbb{H}}\frac{|x-t|^{\gamma\chi}\hat{g}(x)^{\frac{\gamma^{2}}{4}(q+1)}}{|x|^{\gamma\beta_{1}}|x-1|^{\gamma\beta_{2}}}e^{\gamma
h(x)}d^{2}x$ $\displaystyle-e^{\frac{\gamma
c}{2}}\int_{\mathbb{R}}\frac{|r-t|^{\frac{\gamma\chi}{2}}\hat{g}(r)^{\frac{\gamma^{2}q}{8}}}{|r|^{\frac{\gamma\beta_{1}}{2}}|r-1|^{\frac{\gamma\beta_{2}}{2}}}e^{\frac{\gamma}{2}h(r)}d\mu_{\partial}^{t}(r)\Bigg{)}\Bigg{]},$

where the boundary measure $d\mu_{\partial}^{t}(r)$ is defined by:

$\displaystyle
d\mu_{\partial}^{t}(r)/dr=\left\\{\begin{array}[]{ll}g_{\frac{\gamma}{2}}(\sigma_{1}-\frac{\chi}{2})\mathbf{1}_{r<0}+g_{\frac{\gamma}{2}}(\sigma_{2}-\frac{\chi}{2})\mathbf{1}_{0<r<t}+g_{\frac{\gamma}{2}}(\sigma_{2})\mathbf{1}_{t<r<1}+g_{\frac{\gamma}{2}}(\sigma_{3})\mathbf{1}_{r>1},&\mbox{for
}t\in(0,1),\\\
g_{\frac{\gamma}{2}}(\sigma_{1}-\frac{\chi}{2})\mathbf{1}_{r<t}+g_{\frac{\gamma}{2}}(\sigma_{1})\mathbf{1}_{t<r<0}+g_{\frac{\gamma}{2}}(\sigma_{2})\mathbf{1}_{0<r<1}+g_{\frac{\gamma}{2}}(\sigma_{3})\mathbf{1}_{r>1},&\mbox{for
}t<0.\end{array}\right.$

Since the change of $\sigma$ on the left and right or $t$ can be either
$+\frac{\chi}{2}$ or $-\frac{\chi}{2}$, we similarly consider the function
$\tilde{H}_{\chi}(t)$ with the other choice under the constraints of (3.8),
which is given by the same expression as $H_{\chi}(t)$ except one needs to
replace $d\mu_{\partial}^{t}(r)$ by the following measure

$\displaystyle
d\tilde{\mu}_{\partial}^{t}(r)/dr=\left\\{\begin{array}[]{ll}g_{\frac{\gamma}{2}}(\sigma_{1}+\frac{\chi}{2})\mathbf{1}_{r<0}+g_{\frac{\gamma}{2}}(\sigma_{2}+\frac{\chi}{2})\mathbf{1}_{0<r<t}+g_{\frac{\gamma}{2}}(\sigma_{2})\mathbf{1}_{t<r<1}+g_{\frac{\gamma}{2}}(\sigma_{3})\mathbf{1}_{r>1},&\mbox{for
}t\in(0,1),\\\
g_{\frac{\gamma}{2}}(\sigma_{1}+\frac{\chi}{2})\mathbf{1}_{r<t}+g_{\frac{\gamma}{2}}(\sigma_{1})\mathbf{1}_{t<r<0}+g_{\frac{\gamma}{2}}(\sigma_{2})\mathbf{1}_{0<r<1}+g_{\frac{\gamma}{2}}(\sigma_{3})\mathbf{1}_{r>1},&\mbox{for
}t<0.\end{array}\right.$

We could also of course define in a similar manner $H_{\chi}(t)$,
$\tilde{H}_{\chi}(t)$ for $t\in(1,\infty)$, but this will not be needed. By a
direct change of variable applied to the differential operator of Theorem 3.4,
$H_{\chi}(t)$ and $\tilde{H}_{\chi}(t)$ obey the hypergeometric equation. We
state this in the next proposition.

######  Proposition 3.5.

Under the parameter constraint of (3.7), the function $H_{\chi}$ obeys

(3.14)
$t(1-t)\partial_{t}^{2}H_{\chi}+(C-(A+B+1)t)\partial_{t}H_{\chi}-ABH_{\chi}=0\quad\textrm{for
}t\in(-\infty,0)\cup(0,1),$

with $A,B,C$ given by:

(3.15) $A=-q\frac{\gamma\chi}{2},\quad
B=-1+\chi(\beta_{1}+\beta_{2}-2\chi+q\frac{\gamma}{2}),\quad
C=\chi(\beta_{1}-\chi).$

The exact same result holds for $\tilde{H}_{\chi}$ under the parameter
constraints (3.8).

The following analyticity result for $H_{\frac{\gamma}{2}}$ and
$\tilde{H}_{\frac{\gamma}{2}}$ will be used in the proof of Lemma 3.11 below.

######  Lemma 3.6.

Set $\chi=\frac{\gamma}{2}$. Fix the $\sigma_{i}$ in the parameter range (3.7)
but with the boundary of the band being excluded, namely
$\sigma_{1}-\frac{\gamma}{4}\in{\mathcal{B}}$,
$\sigma_{2}-\frac{\gamma}{4}\in{\mathcal{B}}$,
$\sigma_{1},\sigma_{2},\sigma_{3}\in{\mathcal{B}}$. Then the function
$H_{\chi}$ is meromorphic in all three $\beta_{1},\beta_{2},\beta_{3}$ in a
complex neighborhood of the subdomain of $\mathbb{R}^{3}$ given by the
constraints
$\beta_{i}\in(-\infty,Q),\>\>\beta_{1}+\beta_{2}+\beta_{3}>2Q+\chi$. The same
claim holds for $\tilde{H}_{\chi}$, except this time the $\sigma_{i}$ need to
obey $\sigma_{1}+\frac{\gamma}{4}\in{\mathcal{B}}$,
$\sigma_{2}+\frac{\gamma}{4}\in{\mathcal{B}}$,
$\sigma_{1},\sigma_{2},\sigma_{3}\in{\mathcal{B}}$.

###### Proof.

This result follows from a direct adaptation of the proof of the claim of
analyticity in the $\beta_{i}$ for $H$ of Proposition 2.12. The fact that we
have an extra insertion on the boundary poses no additional difficulty. Here
we are also in the range of parameters where the Seiberg bounds are satisfied
and thus no truncation procedure is required. ∎

Before further analysis, we recall here the range of parameters where the
functions $H$ and $R$ have been defined in Section 2. For $H$ the range on the
$\beta_{i}$ and $\sigma_{i}$ is given in Definition 2.13:

(3.16)
$\left\\{(\beta_{1},\beta_{2},\beta_{3},\sigma_{1},\sigma_{2},\sigma_{3})\>:\>\beta_{i}<Q,\>\>Q-\frac{1}{2}\sum\beta_{i}<\gamma\wedge\frac{2}{\gamma}\wedge\min_{i}(Q-\beta_{i}),\>\>\sigma_{i}\in\overline{{\mathcal{B}}}\right\\}.$

For $R$ the range of $\beta$ and $\sigma_{i}$ is given in Definition 2.18:

(3.17)
$\left\\{(\beta,\sigma_{1},\sigma_{2})\>:\>\frac{\gamma}{2}\vee(\frac{2}{\gamma}-\frac{\gamma}{2})<\beta<Q,\>\>\sigma_{i}\in\overline{{\mathcal{B}}}\right\\}.$

We have kept the $\beta_{i}$ to be real parameters above but by the results of
Section 2 we have shown analytic continuation of $H$ and $R$ in the
$\beta_{i}$ to a complex neighborhood of the real interval where they are
defined.

###  3.2. The $\frac{\gamma}{2}$-shift equations for $H$

Using the functions $H_{\frac{\gamma}{2}}(t)$ and
$\tilde{H}_{\frac{\gamma}{2}}(t)$, we prove the shift equations on $H$ in
Theorem 3.1 in the case when $\chi=\frac{\gamma}{2}$ and the parameter range
is chosen such that the $\beta_{i}$ and $\sigma_{i}$ parameters of each $H$ in
the shift equations belong to the domain (3.16).

######  Lemma 3.7.

Set $\chi=\frac{\gamma}{2}$. The shift equation (3.2) holds in the parameter
range

(3.18) $\beta_{1}<\frac{2}{\gamma},\quad\beta_{2},\beta_{3}<Q,\quad
Q-\frac{1}{2}(\beta_{1}+\beta_{2}+\beta_{3}-\frac{\gamma}{2})<\gamma\wedge\frac{2}{\gamma}\wedge\min_{i}(Q-\beta_{i}),\quad\sigma_{1},\sigma_{2},\sigma_{3}\in{\mathcal{B}},\quad\sigma_{2}+\frac{\gamma}{4}\in{\mathcal{B}},$

and the shift equation (3.3) holds in parameter range:

(3.19) $\beta_{1},\beta_{2}<\frac{2}{\gamma},\quad\beta_{3}<Q,\quad
Q-\frac{1}{2}(\beta_{1}+\beta_{2}+\beta_{3}-\frac{\gamma}{2})<\gamma\wedge\frac{2}{\gamma}\wedge\min_{i}(Q-\beta_{i}),\quad\sigma_{1},\sigma_{2},\sigma_{3}\in{\mathcal{B}},\quad\sigma_{2}+\frac{\gamma}{4}\in{\mathcal{B}}.$

###### Proof.

Here we are always assuming $\chi=\frac{\gamma}{2}$. We first take a look at
the parameter ranges given by (3.18) and (3.19). For $\chi=\frac{\gamma}{2}$,
the shift equation (3.2) contains the following three $H$ functions:

$\displaystyle
H\begin{pmatrix}\beta_{1},\beta_{2}-\frac{\gamma}{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix},\quad
H\begin{pmatrix}\beta_{1}-\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}\end{pmatrix},\quad
H\begin{pmatrix}\beta_{1}+\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}\end{pmatrix}.$

The $\beta_{i}$ and $\sigma_{i}$ parameters of each $H$ must be in the range
(3.16). This gives (3.18). Similarly, the second shift equations (3.3)
contains the $H$ functions

$\displaystyle
H\begin{pmatrix}\beta_{1},\beta_{2}+\frac{\gamma}{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}\end{pmatrix},\quad
H\begin{pmatrix}\beta_{1}-\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix},\quad
H\begin{pmatrix}\beta_{1}+\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix},$

and the $\beta_{i}$ and $\sigma_{i}$ then need to obey the constraint (3.19).
It is easy to check that these parameter constraints are non-empty. The
condition of (3.18) is weaker than (3.19), and (3.19) can be satisfied if all
the $\beta_{i}$ parameters are chosen in the interval
$(\frac{2}{\gamma}-\epsilon,\frac{2}{\gamma})$ for a small $\epsilon>0$.

Let us now derive the first shift equation (3.2). We assume
$\beta_{i},\sigma_{i}$ obey (3.7) in order for $H_{\frac{\gamma}{2}}$ to be
well-defined, plus the following extra constraint on $\beta_{1}$:

(3.20) $\frac{\gamma}{2}<\beta_{1}<\frac{2}{\gamma}.$

By Proposition 3.5, the function $t\mapsto H_{\frac{\gamma}{2}}(t)$ obeys the
hypergeometric equation for $t\in(0,1)$. Using the basis of solutions of the
hypergeometric equation recalled in Section A.1, we can write the following
solutions around $t=0$ and $t=1$, under the assumption that neither $C$,
$C-A-B$, or $A-B$ are integers.444The values excluded here are recovered by a
simple continuity argument in $\gamma$. For $t\in(0,1)$:

(3.21) $\displaystyle H_{\frac{\gamma}{2}}(t)$
$\displaystyle=C_{1}F(A,B,C,t)+C_{2}^{+}t^{1-C}F(1+A-C,1+B-C,2-C,t)$
$\displaystyle=B_{1}F(A,B,1+A+B-C,1-t)+B_{2}^{-}(1-t)^{C-A-B}F(C-A,C-B,1+C-A-B,1-t).$

Here, $C_{1},C_{2}^{+},B_{1},B_{2}^{-}$ are the real constants that
parametrize the solution space around $t=0$ and $t=1$. We now express
$C_{1},C_{2}^{+},B_{1}$ using $H\begin{pmatrix}\cdot,\cdot,\cdot\\\
\cdot,\cdot,\cdot\end{pmatrix}$ and then relate them.

First, by sending $t\to 0$ and inspecting the definition (3.13) of
$H_{\chi}(t)$, we have:

(3.22) $C_{1}=\lim_{t\to
0}H_{\chi}(t)=H\begin{pmatrix}\beta_{1}-\chi,\beta_{2},\beta_{3}\\\
\sigma_{1}-\frac{\chi}{2},\sigma_{2},\sigma_{3}\end{pmatrix}.$

Next to find $C_{2}^{+}$ we go at higher order in the $t\rightarrow 0_{+}$
limit. For this we use the asymptotic expansion of
$H_{\frac{\gamma}{2}}(t)-H_{\frac{\gamma}{2}}(0)$ given by the operator
product expansion (OPE) from Lemma C.1. Under the parameter constraints (3.7)
and (3.20), Lemma C.1 gives

(3.23) $\displaystyle
H_{\frac{\gamma}{2}}(t)-H_{\frac{\gamma}{2}}(0)=C_{2}^{+}t^{1-C}+o(t^{1-C}),$

where

(3.24) $\displaystyle C_{2}^{+}=$
$\displaystyle-\frac{\Gamma(-1+\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2})}{\Gamma(-\frac{\gamma^{2}}{4})}\left(g_{\frac{\gamma}{2}}(\sigma_{1}-\frac{\gamma}{4})-g_{\frac{\gamma}{2}}(\sigma_{2}+\frac{\beta_{1}}{2}-\frac{\gamma}{4})\right)H\begin{pmatrix}\beta_{1}+\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1}-\frac{\gamma}{4},\sigma_{2},\sigma_{3}\end{pmatrix}.$

To align with (3.2), we can write (3.24) as

(3.25)
$C_{2}^{+}=\chi^{2}\pi^{\frac{2\chi}{\gamma}-1}\frac{\Gamma(-1+\chi\beta_{1}-\chi^{2})\Gamma(1-\chi\beta_{1})}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2\chi}{\gamma}}}\left(g_{\chi}(\sigma_{1}-\frac{\chi}{2})-g_{\chi}(\sigma_{2}+\frac{\beta_{1}}{2}-\frac{\chi}{2})\right)H\begin{pmatrix}\beta_{1}+\chi,\beta_{2},\beta_{3}\\\
\sigma_{1}-\frac{\chi}{2},\sigma_{2},\sigma_{3}\end{pmatrix}.$

Similarly as for (3.22), by sending $t\to 1$ we get:

(3.26) $\displaystyle B_{1}=\lim_{t\to
1}H_{\chi}(t)=H\begin{pmatrix}\beta_{1},\beta_{2}-\chi,\beta_{3}\\\
\sigma_{1}-\frac{\chi}{2},\sigma_{2}-\frac{\chi}{2},\sigma_{3}\end{pmatrix}.$

By the connection formula (A.7), with $\chi=\frac{\gamma}{2}$ we have

(3.27) $\displaystyle B_{1}$
$\displaystyle=\frac{\Gamma(\chi(\beta_{1}-\chi))\Gamma(1-\chi\beta_{2}+\chi^{2})}{\Gamma(\chi(\beta_{1}-\chi+q\frac{\gamma}{2})\Gamma(1-\chi\beta_{2}+\chi^{2}-q\frac{\gamma\chi}{2})}C_{1}+\frac{\Gamma(2-\chi\beta_{1}+\chi^{2})\Gamma(1-\chi\beta_{2}+\chi^{2})}{\Gamma(1+\frac{q\gamma\chi}{2})\Gamma(2-\chi(\beta_{1}+\beta_{2}-2\chi+q\frac{\gamma}{2}))}C_{2}^{+},$

This implies the shift equation (3.2) for $\chi=\frac{\gamma}{2}$ in the range
of parameters constraint by (3.7) and (3.20), after performing furthermore the
replacement $\sigma_{1}\rightarrow\sigma_{1}+\frac{\gamma}{4}$ and
$\sigma_{2}\rightarrow\sigma_{2}+\frac{\gamma}{4}$ (which also shifts the
domain where $\sigma_{1},\sigma_{2}$ belong). To lift these constraints we
then invoke the analyticity of $H$ as a function of its parameters given by
Proposition 2.14. We have thus shown that (3.2) holds for
$\chi=\frac{\gamma}{2}$ in the parameter range given by (3.18).

Now we repeat these steps with $\tilde{H}_{\frac{\gamma}{2}}$ to obtain the
shift equation with the opposite phase. For $t\in(0,1)$ we have

(3.28) $\displaystyle\tilde{H}_{\frac{\gamma}{2}}(t)$
$\displaystyle=\tilde{C}_{1}F(A,B,C,t)+\tilde{C}_{2}^{+}t^{1-C}F(1+A-C,1+B-C,2-C,t)$
$\displaystyle=\tilde{B}_{1}F(A,B,1+A+B-C,1-t)+\tilde{B}_{2}^{-}(1-t)^{C-A-B}F(C-A,C-B,1+C-A-B,1-t).$

Similarly as for $C_{1},C_{2}^{+},B_{1}$, we get the following expressions for
$\tilde{C}_{1},\tilde{C}_{2}^{+},\tilde{B}_{2}^{-}$ in terms of $H$:

(3.29) $\tilde{C}_{1}=\lim_{t\to
0}\tilde{H}_{\chi}(t)=H\begin{pmatrix}\beta_{1}-\chi,\beta_{2},\beta_{3}\\\
\sigma_{1}+\frac{\chi}{2},\sigma_{2},\sigma_{3}\end{pmatrix}.$ (3.30)
$\tilde{C}_{2}^{+}=\chi^{2}\pi^{\frac{2\chi}{\gamma}-1}\frac{\Gamma(-1+\chi\beta_{1}-\chi^{2})\Gamma(1-\chi\beta_{1})}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2\chi}{\gamma}}}\left(g_{\chi}(\sigma_{1}+\frac{\chi}{2})-g_{\chi}(\sigma_{2}-\frac{\beta_{1}}{2}+\frac{\chi}{2})\right)H\begin{pmatrix}\beta_{1}+\chi,\beta_{2},\beta_{3}\\\
\sigma_{1}+\frac{\chi}{2},\sigma_{2},\sigma_{3}\end{pmatrix}.$ (3.31)
$\tilde{B}_{2}^{-}=\chi^{2}\pi^{\frac{2\chi}{\gamma}-1}\frac{\Gamma(-1+\chi\beta_{2}-\chi^{2})\Gamma(1-\chi\beta_{2})}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2\chi}{\gamma}}}\left(g_{\chi}(\sigma_{3})-g_{\chi}(\sigma_{2}+\frac{\beta_{2}}{2})\right)H\begin{pmatrix}\beta_{1},\beta_{2}+\chi,\beta_{3}\\\
\sigma_{1}+\frac{\chi}{2},\sigma_{2}+\frac{\chi}{2},\sigma_{3}\end{pmatrix}.$

Again by the connection formula (A.7), with $\chi=\frac{\gamma}{2}$ we have

(3.32)
$\displaystyle\tilde{B}_{2}^{-}=\frac{\Gamma(\chi(\beta_{1}-\chi))\Gamma(-1+\chi\beta_{2}-\chi^{2})}{\Gamma(-q\frac{\gamma\chi}{2})\Gamma(-1+\chi(\beta_{1}+\beta_{2}-2\chi+q\frac{\gamma}{2}))}\tilde{C}_{1}+\frac{\Gamma(2-\chi\beta_{1}+\chi^{2}))\Gamma(-1+\chi\beta_{2}-\chi^{2})}{\Gamma(1-\chi(\beta_{1}-\chi+q\frac{\gamma}{2})\Gamma(\chi\beta_{2}-\chi^{2}+q\frac{\gamma\chi}{2})}\tilde{C}_{2}^{+}.$

Therefore (3.3) holds for $\chi=\frac{\gamma}{2}$ in the parameter range given
by (3.19) as before. ∎

######  Remark 3.8.

The two shift equations we derive using respectively $H_{\chi}$ and
$\tilde{H}_{\chi}$ do not have exactly the same shape. The reason is that in
the proof above we used the connection formula between the coefficients
$C_{1},C_{2}^{+},B_{1}$ to derive (3.2), and used the connection formula
between $\tilde{C}_{1},\tilde{C}_{2}^{+},\tilde{B}_{2}^{-}$ to derive (3.3).
One may wonder why we did not use the connection formula between
$\tilde{C}_{1},\tilde{C}_{2}^{+},\tilde{B}_{1}$ in the case of
$\tilde{H}_{\chi}$ as well. The main reason is that with our choice it is
possible to perform a linear combination of (3.2) and (3.3) to obtain a shift
equation of $H$ that only shifts the parameter $\beta_{1}$. This will be
crucial to establish a uniqueness statement for the solutions to (3.2) and
(3.3), see the proof of Theorem 1.1 in Section 4.2.

###  3.3. The $\frac{\gamma}{2}$-shift equations for $R$

The next step is to derive the $\frac{\gamma}{2}$-shift equation for the
reflection coefficient $R$. The key idea is to take a suitable limit of the
$\frac{\gamma}{2}$-shift equations for $H$ that we have established in Lemma
3.7 to make $R$ appear from $H$. For this purpose we give the following result
expressing $R$ as a limit of $H$.

######  Lemma 3.9.

Suppose the $\beta_{i}$ are in the range given by (3.16) and the $\sigma_{i}$
in ${\mathcal{B}}$. Suppose $Q>\beta_{1}>\beta_{2}\vee\frac{\gamma}{2}$,
$\beta_{2}>0$ and $\beta_{1}-\beta_{2}<\beta_{3}<Q$. Then the following limit
holds:

$\displaystyle\lim_{\beta_{3}\downarrow\beta_{1}-\beta_{2}}(\beta_{2}+\beta_{3}-\beta_{1})H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=2R(\beta_{1},\sigma_{1},\sigma_{2}).$

We defer the proof of Lemma 3.9 to Section 5.1 and proceed to prove the
$\frac{\gamma}{2}$-shift equation for $R$.

######  Lemma 3.10.

Consider
$\beta_{1}\in(\frac{\gamma}{2}\vee(\frac{2}{\gamma}-\frac{\gamma}{2}),\frac{2}{\gamma})$
and $\sigma_{1},\sigma_{2}\in\mathbb{C}$ such that
$\sigma_{1},\sigma_{2},\sigma_{2}+\frac{\gamma}{4}$ all belong to
${\mathcal{B}}$. Then $R(\beta_{1},\sigma_{1},\sigma_{2})$ obeys

(3.33) $\displaystyle
R(\beta_{1},\sigma_{1},\sigma_{2})=-\frac{\Gamma(-1+\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2})}{\Gamma(-\frac{\gamma^{2}}{4})}\left(g_{\frac{\gamma}{2}}(\sigma_{1})-g_{\frac{\gamma}{2}}(\sigma_{2}+\frac{\beta_{1}}{2})\right)R(\beta_{1}+\frac{\gamma}{2},\sigma_{1},\sigma_{2}+\frac{\gamma}{4}).$

Similarly for
$\beta_{1}\in(0\vee(\frac{2}{\gamma}-\gamma),\frac{2}{\gamma}-\frac{\gamma}{2})$
and the same constraint on $\sigma_{1},\sigma_{2}$ as previously,

(3.34) $\displaystyle
R(\beta_{1}+\frac{\gamma}{2},\sigma_{1},\sigma_{2}+\frac{\gamma}{4})=-\frac{\Gamma(-1+\frac{\gamma\beta_{1}}{2})\Gamma(1-\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})}{\Gamma(-\frac{\gamma^{2}}{4})}\left(g_{\frac{\gamma}{2}}(\sigma_{1})-g_{\frac{\gamma}{2}}(\sigma_{2}-\frac{\beta_{1}}{2})\right)R(\beta_{1}+\gamma,\sigma_{1},\sigma_{2}).$

###### Proof.

We derive the first shift equation (3.33) by taking a limit of (3.2)
established in Lemma 3.7. Fix a
$\beta_{1}\in(\frac{\gamma}{2}\vee(\frac{2}{\gamma}-\frac{\gamma}{2}),\frac{2}{\gamma})$.
Consider two parameters $\epsilon,\eta>0$ chosen small enough and set
$\beta_{2}=\beta_{1}-\epsilon$,
$\beta_{3}=\beta_{1}-\beta_{2}+\frac{\gamma}{2}+\eta=\frac{\gamma}{2}+\epsilon+\eta$.
For this parameter choice the condition (3.18) required in Lemma 3.7 is
satisfied. By Lemma 3.9 we get:

$\displaystyle\lim_{\beta_{3}\downarrow\beta_{1}-\beta_{2}+\frac{\gamma}{2}}(\beta_{2}+\beta_{3}-\beta_{1}-\frac{\gamma}{2})H\begin{pmatrix}\beta_{1},\beta_{2}-\frac{\gamma}{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=2R(\beta_{1},\sigma_{1},\sigma_{2}),$
$\displaystyle\lim_{\beta_{3}\downarrow\beta_{1}-\beta_{2}+\frac{\gamma}{2}}(\beta_{2}+\beta_{3}-\beta_{1}-\frac{\gamma}{2})\frac{\Gamma(\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{2}}{2}+\frac{\gamma^{2}}{4})}{\Gamma(\frac{\gamma\beta_{1}}{2}+(q-1)\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma}{2}\beta_{2}-(q-1)\frac{\gamma^{2}}{4}))}H\begin{pmatrix}\beta_{1}-\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}\end{pmatrix}=0,$
$\displaystyle\lim_{\beta_{3}\downarrow\beta_{1}-\beta_{2}+\frac{\gamma}{2}}(\beta_{2}+\beta_{3}-\beta_{1}-\frac{\gamma}{2})\Bigg{[}\frac{\Gamma(2+\frac{\gamma^{2}}{4}-\frac{\gamma\beta_{1}}{2})\Gamma(1-\frac{\gamma\beta_{2}}{2}+\frac{\gamma^{2}}{4})\Gamma(-1+\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2})}{\Gamma(1+\frac{\gamma^{2}}{4})\Gamma(2-\frac{\gamma}{2}(\beta_{1}+\beta_{2})-(q-2)\frac{\gamma^{2}}{4}))\Gamma(-\frac{\gamma^{2}}{4})}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\left(g_{\frac{\gamma}{2}}(\sigma_{1})-g_{\frac{\gamma}{2}}(\sigma_{2}+\frac{\beta_{1}}{2})\right)H\begin{pmatrix}\beta_{1}+\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}\end{pmatrix}\Bigg{]}$
$\displaystyle=2\frac{\Gamma(-1+\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2})}{\Gamma(-\frac{\gamma^{2}}{4})}\left(g_{\frac{\gamma}{2}}(\sigma_{1})-g_{\frac{\gamma}{2}}(\sigma_{2}+\frac{\beta_{1}}{2})\right)R(\beta_{1}+\frac{\gamma}{2},\sigma_{1},\sigma_{2}+\frac{\gamma}{4}).$

In the above three limits the second one is actually trivial, only the first
and third involve using Lemma 3.9. Putting these limits together using (3.2)
leads to (3.33). By using the alternative function
$\tilde{H}_{\frac{\gamma}{2}}(t)$ along the same lines we obtain the desired
relation (3.34) between
$R(\beta_{1}+\frac{\gamma}{2},\sigma_{1},\sigma_{2}+\frac{\gamma}{4})$ and
$R(\beta_{1}+\gamma,\sigma_{1},\sigma_{2})$. ∎

###  3.4. Analytic continuation for $H$ and $R$

In this subsection we will analytically extend the functions $H$ and $R$ to a
larger domain of parameters than the one we are currently working with. For
this purpose, we must first derive the so-called reflection principle for $H$
and $R$ which will be obtain by performing the operator product expansion
(OPE) with reflection in the case $\chi=\frac{\gamma}{2}$. We rely extensively
on Lemma C.2 giving the Taylor expansion using the reflection coefficient.
Finally we also extend the validity of Lemma 3.9 to a larger range of
parameters.

######  Lemma 3.11 (Reflection principle for $H$).

Consider parameters $\sigma_{1},\sigma_{2},\sigma_{3}$,
$\beta_{1},\beta_{2},\beta_{3}$ satisfying (3.16) and (3.17) so that $H$ and
$R$ are well-defined. One can meromorphically extend $\beta_{1}\mapsto H$
beyond the point $\beta_{1}=Q$ by the following relation:

(3.35) $\displaystyle H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=R(\beta_{1},\sigma_{1},\sigma_{2})H\begin{pmatrix}2Q-\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}.$

The quantity $H\begin{pmatrix}2Q-\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}$ is thus well-defined as long as
the parameters of $H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}$ and
$R(\beta_{1},\sigma_{1},\sigma_{2})$ respectively obey the constraints of
(3.16) and (3.17). Similarly, for $\sigma_{1},\sigma_{2}\in{\mathcal{B}}$, we
can analytically extend $\beta_{1}\mapsto R(\beta_{1},\sigma_{1},\sigma_{2})$
to the range
$((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q+(\frac{2}{\gamma}\wedge\gamma))$
thanks to the relation:

(3.36) $\displaystyle
R(\beta_{1},\sigma_{1},\sigma_{2})R(2Q-\beta_{1},\sigma_{1},\sigma_{2})=1.$

###### Proof.

Throughout the proof we keep the same notations as used in the proof of Lemma
3.7 for the solution space of the hypergeometric equation satisfied by
$H_{\frac{\gamma}{2}}(t)$ for $t\in(0,1)$. We assume the parameters
$\beta_{i}$, $\sigma_{i}$ obey the condition (3.7) in order for
$H_{\frac{\gamma}{2}}(t)$ to be well-defined. We also add the condition
$\beta_{1}\in(Q-\beta_{0},Q)$ so that we can apply the result of Lemma C.2 to
get:

(3.37)
$C_{2}^{+}=R(\beta_{1},\sigma_{1}-\frac{\gamma}{4},\sigma_{2}-\frac{\gamma}{4})H\begin{pmatrix}2Q-\beta_{1}-\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1}-\frac{\gamma}{4},\sigma_{2},\sigma_{3}\end{pmatrix}.$

The key argument is to observe that since by Lemma 3.6 $\beta_{1}\mapsto
H_{\frac{\gamma}{2}}(t)$ is complex analytic in $\beta_{1}$ so is the
coefficient $C_{2}^{+}$. By using this combined with the analyticity of $R$
and $H$, we can extend the range of validity of equation (3.37) from
$\beta_{1}\in(Q-\beta_{0},Q)$ to $\beta_{1}\in(\frac{2}{\gamma},Q)$, still
under the constraint of (3.7). Now equation (3.24) derived in the the proof of
Lemma 3.7 gives us an alternative expression for $C_{2}^{+}$, which is valid
for $\beta_{1}\in(\frac{\gamma}{2},\frac{2}{\gamma})$. The analyticity of
$\beta_{1}\mapsto C_{2}^{+}$ in a complex neighborhood of $\frac{2}{\gamma}$
then implies that one can “glue” together the two expressions for $C_{2}^{+}$.
More precisely, by setting equal the right hand sides of (3.24) and (3.37) and
after performing the parameter replacement
$\sigma_{i}\rightarrow\sigma_{i}+\frac{\gamma}{4}$ for $i=1,2,3$, one obtains
the equality

(3.38) $\displaystyle
R(\beta_{1},\sigma_{1},\sigma_{2})H\begin{pmatrix}2Q-\beta_{1}-\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}+\frac{\gamma}{4}\end{pmatrix}$
$\displaystyle=\frac{\Gamma(-1+\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2})}{\Gamma(-\frac{\gamma^{2}}{4})}\left(g_{\frac{\gamma}{2}}(\sigma_{1})-g_{\frac{\gamma}{2}}(\sigma_{2}+\frac{\beta_{1}}{2})\right)H\begin{pmatrix}\beta_{1}+\frac{\gamma}{2},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2}+\frac{\gamma}{4},\sigma_{3}+\frac{\gamma}{4}\end{pmatrix},$

which provides the desired analytic continuation of $H$. To land on the form
of the reflection equation given in the lemma one needs to replace $\beta_{1}$
by $\beta_{1}-\frac{\gamma}{2}$. This transforms
$R(\beta_{1},\sigma_{1},\sigma_{2})$ into
$R(\beta_{1}-\frac{\gamma}{2},\sigma_{1},\sigma_{2})$ which we can shift back
to $R(\beta_{1},\sigma_{1},\sigma_{2}+\frac{\gamma}{4})$ using the shift
equation (3.33). Lastly we perform the parameter replacement
$\sigma_{2}+\frac{\gamma}{4}$ to $\sigma_{2}$ and
$\sigma_{3}+\frac{\gamma}{4}$ to $\sigma_{3}$. This implies the reflection
principle (3.35) for $H$. The relation (3.36) for $R$ follows as an immediate
consequence. ∎

We now use the shift equations we have derived to analytically continue $H$
and $R$ both in the parameters $\beta_{i}$ and $\sigma_{i}$. The analytic
continuations will be defined in a larger range of parameters than (3.16) and
(3.17) required for the GMC expressions to be well-defined. The proofs follow
closely the ones of [RZ22]. We start with the case of $R$ which is very
straightforward.

######  Lemma 3.12.

(Analytic continuation of $R(\beta_{1},\sigma_{1},\sigma_{2})$) For all
$\sigma_{1},\sigma_{2}\in{\mathcal{B}}$, the meromorphic function
$\beta_{1}\mapsto R(\beta_{1},\sigma_{1},\sigma_{2})$ originally defined on
the interval $((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$
extends to a meromorphic function defined in a complex neighborhood of
$\mathbb{R}$ and satisfying the shift equation:

(3.39)
$\displaystyle\frac{R(\beta_{1},\sigma_{1},\sigma_{2})}{R(\beta_{1}+\gamma,\sigma_{1},\sigma_{2})}=(\frac{\gamma}{2})^{4}\frac{\Gamma(-1+\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma\beta_{1}}{2})\Gamma(-1+\frac{\gamma\beta_{1}}{2})}{\sin(\pi\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma^{2}}{4})^{2}}$
$\displaystyle\times
4\sin(\frac{\gamma\pi}{2}(\frac{\beta}{2}-\sigma_{1}-\sigma_{2}+Q))\sin(\frac{\gamma\pi}{2}(\frac{\beta}{2}+\sigma_{1}+\sigma_{2}-Q))\sin(\frac{\gamma\pi}{2}(\frac{\beta}{2}+\sigma_{2}-\sigma_{1}))\sin(\frac{\gamma\pi}{2}(\frac{\beta}{2}+\sigma_{1}-\sigma_{2})).$

Furthermore, for a fixed $\beta_{1}$ in the above complex neighborhood of
$\mathbb{R}$, the function $R(\beta_{1},\sigma_{1},\sigma_{2})$ extends to a
meromorphic function of $(\sigma_{1},\sigma_{2})$ on $\mathbb{C}^{2}$.

###### Proof.

This proof is performed in [RZ22] but we sketch it below as it is quite short.
Fix first the parameters $\sigma_{1},\sigma_{2}$ in ${\mathcal{B}}$. Thanks to
Proposition 2.17, $\beta_{1}\mapsto R(\beta_{1},\sigma_{1},\sigma_{2})$ is
meromorphic in a complex neighborhood of the real interval
$((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q)$, and thanks to
the reflection principle given by Lemma 3.11, it is meromorphic in a complex
neighborhood of
$((\frac{2}{\gamma}-\frac{\gamma}{2})\vee\frac{\gamma}{2},Q+(\frac{2}{\gamma}\wedge\gamma))$.
Note that the length of this interval is strictly greater than $\gamma$.

Combining the two shift equations of Lemma 3.10, we obtain the shift equation
(3.39) relating $R(\beta_{1}+\gamma,\sigma_{1},\sigma_{2})$ and
$R(\beta_{1},\sigma_{1},\sigma_{2})$, which does not shift the
$\sigma_{1},\sigma_{2}$ parameters. Since $R(\beta_{1},\sigma_{1},\sigma_{2})$
is defined in a complex neighborhood of a real interval of length strictly
bigger than $\gamma$, the shift equation (3.39) can be used to meromorphically
extend $\beta_{1}\mapsto R(\beta_{1},\sigma_{1},\sigma_{2})$ to a complex
neighborhood of the whole real line.

Now for any $\beta_{1}$ fixed in this complex neighborhood of $\mathbb{R}$, we
perform the analytic continuation to $\mathbb{C}^{2}$ in the variables
$\sigma_{1},\sigma_{2}$. For this one can simply apply either of the shift
equation of Lemma 3.10. This is possible since the band ${\mathcal{B}}$ has
width strictly greater than $\frac{\gamma}{4}$. Hence the result. ∎

######  Lemma 3.13.

(Analytic continuation of $H$) Fix
$\sigma_{3}\in[-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2}-\frac{\gamma}{4}]\times\mathbb{R}$
and fix $\sigma_{1},\sigma_{2}\in{\mathcal{B}}$. Then the function
$(\beta_{1},\beta_{2},\beta_{3})\mapsto H$ originally defined in the parameter
range given by (3.16) extends to a meromorphic function of the three variables
in a small complex neighborhood of $\mathbb{R}^{3}$. Now fix
$\beta_{1},\beta_{2},\beta_{3}$ in this complex neighborhood of
$\mathbb{R}^{3}$, keeping $\sigma_{3}$ still fixed in
$[-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2}-\frac{\gamma}{4}]\times\mathbb{R}$.
The function $(\sigma_{1},\sigma_{2})\mapsto H$ then extends to a meromorphic
function of $\mathbb{C}^{2}$.

###### Proof.

This proof also follows exactly the proof of [RZ22, Lemma 3.6], with one
notable difference highlighted below that prevents us from performing an
analytic continuation in $\sigma_{3}$. The main idea of [RZ22] is to use the
shift equations on $H$ to successively extend the domain where $H$ is defined
to larger and larger regions until it has been extended to the region claimed
in the lemma. Keeping the notations of [RZ22], we choose as the starting
domain the region given by:

(3.40)
$\mathcal{E}_{1}:=\left\\{\beta_{i}\in(\frac{2}{\gamma}-\delta,Q)\times[-\nu,\nu],\>\sigma_{i}\in(-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2})\times\mathbb{R}\right\\}.$

By $\beta_{i}\in(\frac{2}{\gamma}-\delta,Q)\times[-\nu,\nu]$ we mean
$\Re(\beta_{i})\in(\frac{2}{\gamma}-\delta,Q)$ and
$\Im(\beta_{i})\in[-\nu,\nu]$, the same convention is used for the domain of
$\sigma_{i}$. We have introduced $\delta,\nu>0$ chosen small enough so that
(3.16) holds for $\beta_{i}\in(\frac{2}{\gamma}-\delta,Q)$ and one can apply
Proposition 2.12 to show analyticity of $H$ in all of its variables on
$\mathcal{E}_{1}$. The condition on $\delta$ is
$\delta<\frac{1}{\gamma}-\frac{\gamma}{4}$. By applying the shift equations on
$H$ as performed in the proof of [RZ22, Lemma 3.6], one can meromorphically
extend $H$ to the region:

(3.41)
$\mathcal{E}_{7}:=\left\\{\beta_{i}\in\mathbb{R}\times[-\nu,\nu],\>(\sigma_{1},\sigma_{2})\in\mathbb{C}^{2},\>\sigma_{3}\in(-\frac{1}{2\gamma}+\frac{Q}{2},\frac{1}{2\gamma}+\frac{Q}{2}-\frac{\gamma}{4})\times\mathbb{R}\right\\}.$

Note here that we have performed a cyclic permutation with respect to the
indices $1,2,3$ of the parameters in comparison to the domain
$\mathcal{E}_{7}$ written in [RZ22]. This thus gives the desired domain of
analytic continuation of $H$ claimed in the statement of the lemma.

Contrarily to what is done in [RZ22], we are not able to lift the constraint
on $\sigma_{3}$. Indeed, in the case $\mu=0$ when $H$ has an expression
reducing to a moment of a boundary GMC measure, if one adds a global constant
to each of the $\sigma_{i}$ the function $H$ is simply changed by a global
phase. In our case this property does not hold which is why we are not able to
perform the extension in the variable $\sigma_{3}$. Notice though that once
Theorem 1.1 is established, the exact formula ${H_{\mathrm{PT}}}$ implies that
$H$ extends meromorphically in all its parameters to $\mathbb{C}^{6}$. ∎

We finish this subsection by extending Lemma 3.9 to a larger range of
parameters that will be required for the next subsection. This is a novel
difficulty that was not present in [RZ22] because in our case we had to
perform a truncation procedure to define $H$ and $R$ while in [RZ22] they
simply reduce to moments of GMC.

######  Lemma 3.14.

Suppose $\beta_{1},\beta_{2},\beta_{3}$ satisfy
$Q-\frac{1}{2}\sum_{i}\beta_{i}<\frac{2}{\gamma}\wedge\min_{i}(Q-\beta_{i})$,
$Q>\beta_{1}>\beta_{2}\vee\frac{\gamma}{2}$ and
$\beta_{1}-\beta_{2}<\beta_{3}<Q$. Let also
$\sigma_{1}\in\overline{{\mathcal{B}}}$ and
$\sigma_{2},\sigma_{3}\in{\mathcal{B}}$. Then the following limit holds

$\displaystyle\lim_{\beta_{3}\rightarrow\beta_{1}-\beta_{2}}(\beta_{2}+\beta_{3}-\beta_{1})H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=2R(\beta_{1},\sigma_{1},\sigma_{2}),$

where the function $R$ should be understood by the analytic continuation given
in Lemma 3.12.

###### Proof.

By the meromorphicity of $H$ function from Lemma 3.13 and the residue theorem,
we have

$\displaystyle\lim_{\beta_{3}\rightarrow\beta_{1}-\beta_{2}}(\beta_{2}+\beta_{3}-\beta_{1})H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}=\frac{1}{2\pi\mathbf{i}}\int_{\mathcal{C}_{\epsilon}(\beta_{1}-\beta_{2})}H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}d\beta_{3}$

where above $\mathcal{C}_{\epsilon}(\beta_{1}-\beta_{2})$ is a circle of
radius $\epsilon$ around $\beta_{1}-\beta_{2}$. By Lemma 3.9, the equality

$\displaystyle\frac{1}{2\pi\mathbf{i}}\int_{\mathcal{C}_{\epsilon}(\beta_{1}-\beta_{2})}H\begin{pmatrix}\beta_{1},\beta_{2},\beta_{3}\\\
\sigma_{1},\sigma_{2},\sigma_{3}\end{pmatrix}d\beta_{3}=2R(\beta_{1},\sigma_{1},\sigma_{2})$

holds in the parameter range given by Lemma 3.9. By the analytic property of
$H$ and $R$, this equality extends to the larger parameter range in Lemma
3.14. This concludes the proof. ∎

###  3.5. The $\frac{2}{\gamma}$-shift equations for $R$

We now derive the $\frac{2}{\gamma}$-shift equations on
$R(\beta_{1},\sigma_{1},\sigma_{2})$.

######  Lemma 3.15.

For all $\sigma_{1},\sigma_{2}\in\mathbb{C}$, the meromorphic function