0}\left(\left\langle\bm{\mathrm{W}}_{\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}+\frac{1}{8}\sum_{l=1}^{N}\left(\frac{w(\alpha_{l})}{(z_{0}-z_{l})^{3}}+\frac{\bm{\mathcal{W}_{-1,\varepsilon}}(z_{l},\alpha_{l})}{(z_{0}-z_{l})^{2}}+\frac{\bm{\mathcal{W}_{-2,\varepsilon}}(z_{l},\alpha_{l})}{z_{0}-z_{l}}\right)\left\langle\prod_{k=1}^{N}V_{\alpha_{k},\varepsilon}(z_{k})\right\rangle_{\delta}\right)=0.$

To finish up with the proof of our main result, it remains to show that the
limiting quantities

$\bm{\mathcal{W}}_{-i}(z_{l},\alpha_{l})\left\langle\bm{\mathrm{V}}\right\rangle=\lim\limits_{\delta\rightarrow
0}\lim\limits_{\varepsilon\rightarrow
0}\bm{\mathcal{W}}_{-i,\varepsilon}(z_{l},\alpha_{l})\left\langle\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}$

do exist for $i=1,2$. Convergence of
$\bm{\mathcal{W}}_{-1}(z_{l},\alpha_{l})\left\langle\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}$
follows from the very same argument that allows to prove the differentiability
of the correlation functions. Namely quantities of the form

$\int_{\mathbb{C}}\frac{1}{(z_{l}-x)_{\varepsilon}}\left\langle V_{\gamma
e_{i},\varepsilon}(x)\bm{\mathrm{V}}_{\varepsilon}\right\rangle d^{2}x$

admit a limit when $\varepsilon\rightarrow 0$. As for
$\bm{\mathcal{W}}_{-2,\varepsilon}(z_{l},\alpha_{l})\left\langle\bm{\mathrm{V}_{\varepsilon}}\right\rangle_{\delta}$,
we note that

$q\left(B(e_{i},\alpha)-B(\alpha,e_{i})\right)+2C(\alpha,\gamma
e_{i}-\alpha,e_{i})=\frac{4}{\gamma}\left\langle
h_{2},e_{i}\right\rangle\omega_{3-i}(\alpha)\left(1+\frac{1}{2}\left\langle\alpha,\gamma
e_{i}\right\rangle\right).$

As a consequence symmetrization identities allow to rewrite the integrals in
$\bm{\mathcal{W}}_{-2,\varepsilon}(z_{k},\alpha_{k})\left\langle\bm{\mathrm{V}_{\varepsilon}}\right\rangle$
as

$\displaystyle-\mu\gamma\sum_{i=1}^{2}\int_{\mathbb{C}}\sum_{l\neq
k}\left(\frac{C^{\sigma}(\alpha_{k},\alpha_{l},e_{i})}{(z_{k}-z_{l})_{\varepsilon}(z_{k}-x)_{\varepsilon}}-\frac{\left\langle
h_{2},e_{i}\right\rangle\omega_{3-i}(\alpha_{k})\left\langle\alpha_{l},e_{i}\right\rangle}{(z_{l}-x)_{\varepsilon}(z_{k}-x)_{\varepsilon}}\right)\left\langle
V_{\gamma
e_{1},\varepsilon}(x)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x$
$\displaystyle+(\mu\gamma)^{2}\int_{\mathbb{C}^{2}}\frac{\left\langle
e_{1},\alpha_{k}\right\rangle}{(z_{k}-x_{1})_{\varepsilon}(x_{1}-x_{2})_{\varepsilon}}+\frac{\left\langle
e_{2},\alpha_{k}\right\rangle}{(z_{k}-x_{2})_{\varepsilon}(x_{1}-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{1},\varepsilon}(x_{1})V_{\gamma
e_{2},\varepsilon}(x_{2})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}.$

We can proceed in the same way as in the proof that the correlation functions
are $C^{2}$ (see Subsection 6.3) to show that the limit does exist. This wraps
up the proof of our main result.

###  4.4. Global Ward identities

The Ward identity (1.22) that we have just proved shows that the probabilistic
model thus defined is indeed consistent with the expectations of the physics
literature and may be understood as a manifestation of the higher-spin
symmetry enjoyed by the $\mathfrak{sl}_{3}$ Toda theory. In addition to this
identity there is a second building block which is fundamental in the study of
Toda theories: the existence of global Ward identities (1.27) which are
manifestations of the (global) higher-spin symmetry of the theory. In order to
prove that the equations (1.27) hold we will prove that the higher-spin
current is “holomorphic at infinity”, a fact which is axiomatic in the physics
literature and by which is meant that

$\bm{\mathrm{W}}(z_{0})\sim\frac{1}{z_{0}^{6}}$

as $z_{0}\rightarrow\infty$. To do so we rely on the fact that
$\bm{\mathrm{W}}$ behaves like a covariant tensor of order three in the
following sense.

######  Proposition 4.1.

Under the Seiberg bounds (2.25) and for any Möbius transform $\psi$ of the
plane

(4.1)
$\left\langle\bm{\mathrm{W}}(z_{0})\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle=\psi^{\prime}(z_{0})^{3}\prod_{l=1}^{N}\left\lvert\psi^{\prime}(z_{l})\right\rvert^{2\Delta_{\alpha_{l}}}\left\langle\bm{\mathrm{W}}(\psi(z_{0}))\prod_{k=1}^{N}V_{\alpha_{k}}(\psi(z_{k}))\right\rangle.$

In particular when $\psi:z\mapsto\frac{1}{z}$ the asymptotic behaviour of
$\bm{\mathrm{W}}(z_{0})$ as $z_{0}\rightarrow\infty$ is given by

(4.2)
$\left\langle\bm{\mathrm{W}}(z_{0})\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle\sim-\frac{1}{z_{0}^{6}}\prod_{l=1}^{N}\left\lvert
z_{l}\right\rvert^{-4\Delta_{\alpha_{l}}}\left\langle\bm{\mathrm{W}}(0)\prod_{k=1}^{N}V_{\alpha_{k}}(\frac{1}{z_{k}})\right\rangle.$

Assume for now that this statement holds. Then we see that, when
$z_{0}\rightarrow\infty$ the left-hand side in the Ward identity (1.22) is
therefore asymptotic to $\frac{1}{z_{0}^{6}}$. On the other hand the leading
term in the right-hand side of Equation (1.22) is given (up to the prefactor
$-\frac{1}{8}$) by

$\frac{1}{z_{0}}\sum_{l=1}^{N}\bm{\mathcal{W}_{-2}}(z_{l},\alpha_{l})\left\langle\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle.$

Therefore in order for these two asymptotics to be consistent we need to
assume that that the $n=0$ global Ward identity (1.27) holds:

$\sum_{l=1}^{N}\bm{\mathcal{W}_{-2}}(z_{l},\alpha_{l})\left\langle\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle=0.$

We may proceed in the same way for the other terms that appear in the
asymptotic of the right-hand side of Equation (1.27). To do so let us wite the
right-hand side of Equation (1.22) as

$-\frac{1}{8}\sum_{l=1}^{N}\left(\frac{1}{z_{0}^{3}}\frac{w(\alpha_{l})}{(1-\frac{z_{l}}{z_{0}})^{3}}+\frac{1}{z_{0}^{2}}\frac{\bm{\mathcal{W}_{-1}^{(l)}}}{(1-\frac{z_{l}}{z_{0}})^{2}}+\frac{1}{z_{0}}\frac{\bm{\mathcal{W}_{-2}^{(l)}}}{1-\frac{z_{l}}{z_{0}}}\right)\left\langle\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle$

and expand in powers of $\frac{1}{z_{0}}$ the terms of the form
$\frac{1}{(1-\frac{z_{l}}{z_{0}})^{p}}$, $p=1,2,3$. Then the asymptotic as
$z_{0}\rightarrow\infty$ can be expanded as negative powers of $z_{0}$, and
corresponding coefficients in this expansion are given by

$\frac{1}{z_{0}^{p}}\sum_{l=1}^{N}\left(z_{l}^{p-1}\bm{\mathcal{W}_{-2}}(z_{l},\alpha_{l})+(p-1)z_{l}^{p-2}\bm{\mathcal{W}_{-1}}(z_{l},\alpha_{l})+\frac{(p-1)(p-2)}{2}z_{l}^{n-2}w(\alpha_{l})\right)\left\langle\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle.$

Therefore to be consistent with the asymptotic prescribed by Equation (4.2) we
need to have, as soon as $0\;\leqslant\;n\;\leqslant\;4$,

$\sum_{l=1}^{N}\left(z_{l}^{n}\bm{\mathcal{W}_{-2}}(z_{l},\alpha_{l})+nz_{l}^{n-1}\bm{\mathcal{W}_{-1}}(z_{l},\alpha_{l})+\frac{n(n-1)}{2}z_{l}^{n-3}w(\alpha_{l})\right)\left\langle\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle=0.$

In other words the global Ward identities (1.27) hold.

######  Remark 4.2.

With a little bit of extra work it is possible to show that the right-hand
side in the asymptotic of Equation (4.2) is given by

$-\frac{1}{8}\sum_{l=1}^{N}\left(z_{l}^{5}\bm{\mathcal{W}_{-2}}(z_{l},\alpha_{l})+5z_{l}^{4}\bm{\mathcal{W}_{-1}}(z_{l},\alpha_{l})+10z_{l}^{3}w(\alpha_{l})\right)\left\langle\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle$

so we may not learn anything new from the exact value of the leading term in
the asymptotic of the higher-spin current.

###### Proof of Proposition 4.1.

Let us come back to the regularized correlation functions
$\left\langle\bm{\mathrm{W}}_{\varepsilon}(z_{0})\prod_{k=1}^{N}V_{\alpha_{k},\varepsilon}(z_{k})\right\rangle$
defined thanks to the mollified field
$\varphi_{\varepsilon}=\bm{\mathrm{X}}_{\varepsilon}+\frac{Q}{2}\ln\hat{g}_{\varepsilon}+\bm{c}$.
Since the latter is smooth, its derivatives and the ones of
$\varphi_{\varepsilon}\circ\psi$ are well-defined. They are given by

$\displaystyle\partial\left(\varphi_{\varepsilon}\circ\psi\right)$
$\displaystyle=\psi^{\prime}\partial\varphi_{\varepsilon}\circ\psi$
$\displaystyle\partial^{2}\left(\varphi_{\varepsilon}\circ\psi\right)$
$\displaystyle=\psi^{\prime\prime}\partial\varphi_{\varepsilon}\circ\psi+(\psi^{\prime})^{2}\partial^{2}\varphi_{\varepsilon}\circ\psi$
$\displaystyle\partial^{3}\left(\varphi_{\varepsilon}\circ\psi\right)$
$\displaystyle=\psi^{\prime\prime\prime}\partial\varphi_{\varepsilon}\circ\psi+3\psi^{\prime\prime}\psi^{\prime}\partial^{2}\varphi_{\varepsilon}\circ\psi+(\psi^{\prime})^{3}\partial^{3}\varphi_{\varepsilon}\circ\psi.$

We can proceed in the same way for the map
$\ln\left\lvert\psi^{\prime}\right\rvert$ and yields

$\displaystyle\partial\left(\ln\left\lvert\psi^{\prime}\right\rvert\right)=\frac{\psi^{\prime\prime}}{2\psi^{\prime}}\quad\text{and}\quad\partial^{2}\left(\ln\left\lvert\psi^{\prime}\right\rvert\right)=\frac{\psi^{\prime\prime\prime}\psi^{\prime}-(\psi^{\prime\prime})^{2}}{2(\psi^{\prime})^{2}}\cdot$

We can now apply the tensor $\bm{\mathrm{W}}$ to the field
$\varphi_{\varepsilon}\circ\psi+Q\ln\left\lvert\psi^{\prime}\right\rvert$
instead of $\varphi_{\varepsilon}$, yielding

$\displaystyle\bm{\mathrm{W}}(\varphi_{\varepsilon}\circ\psi+Q\ln\left\lvert\psi^{\prime}\right\rvert)=-\frac{q^{2}}{8}\left\langle
h_{2},(\psi^{\prime})^{3}\partial^{3}\varphi_{\varepsilon}+3\psi^{\prime\prime}\psi^{\prime}\partial^{2}\varphi_{\varepsilon}+\psi^{\prime\prime\prime}\partial\varphi_{\varepsilon}\right\rangle\circ\psi$
$\displaystyle+\frac{q}{4}\Big{(}\psi^{\prime}\left\langle
h_{2}-h_{1},\psi^{\prime\prime}\partial\varphi_{\varepsilon}+(\psi^{\prime})^{2}\partial^{2}\varphi_{\varepsilon}\right\rangle\left\langle
h_{1},\partial\varphi_{\varepsilon}\right\rangle+\psi^{\prime}\left\langle
h_{3}-h_{2},\psi^{\prime\prime}\partial\varphi_{\varepsilon}+(\psi^{\prime})^{2}\partial^{2}\varphi_{\varepsilon}\right\rangle\left\langle
h_{3},\partial\varphi_{\varepsilon}\right\rangle$
$\displaystyle+q\frac{\psi^{\prime\prime}}{2\psi^{\prime}}\left\langle
3h_{2},\psi^{\prime\prime}\partial\varphi_{\varepsilon}+(\psi^{\prime})^{2}\partial^{2}\varphi_{\varepsilon}\right\rangle+q\frac{\psi^{\prime\prime\prime}\psi^{\prime}-(\psi^{\prime\prime})^{2}}{2\psi^{\prime}}\left\langle
h_{2},\partial\varphi_{\varepsilon}\right\rangle\Big{)}\circ\psi$
$\displaystyle+(\psi^{\prime})^{3}\left\langle
h_{1},\partial\varphi_{\varepsilon}\right\rangle\left\langle
h_{2},\partial\varphi_{\varepsilon}\right\rangle\left\langle
h_{3},\partial\varphi_{\varepsilon}\right\rangle\circ\psi+\frac{q}{2}\psi^{\prime}\psi^{\prime\prime}\left\langle
h_{2},\partial\varphi_{\varepsilon}\right\rangle\left\langle
h_{3}-h_{1},\partial\varphi_{\varepsilon}\right\rangle\circ\psi-q^{2}\frac{(\psi^{\prime\prime})^{2}}{4\psi^{\prime}}\left\langle
h_{2},\partial\varphi_{\varepsilon}\right\rangle\circ\psi.$

Therefore we end up with the equality

$\bm{\mathrm{W}}(\varphi_{\varepsilon}\circ\psi+Q\ln\left\lvert\psi^{\prime}\right\rvert)=(\psi^{\prime})^{3}\bm{\mathrm{W}}(\varphi_{\varepsilon})\circ\psi.$

On the other hand we know from [8, Equation (3.8)] that, provided that the
limit exists,

(4.3) $\lim\limits_{\varepsilon\rightarrow 0}\left\langle
F(\varphi_{\varepsilon})\prod_{k=1}^{N}V_{\alpha_{k},\varepsilon}(z_{k})\right\rangle=\prod_{k=1}^{N}\left\lvert\psi^{\prime}(z_{k})\right\rvert^{2\Delta_{\alpha_{k}}}\lim\limits_{\varepsilon\rightarrow
0}\left\langle
F(\varphi_{\varepsilon}\circ\psi+Q\ln\left\lvert\psi^{\prime}\right\rvert)\prod_{k=1}^{N}V_{\alpha_{k},\varepsilon}(\psi(z_{k}))\right\rangle$

for any continuous map $F$ on
$\mathrm{H}^{-1}(\mathbb{C}\to\mathfrak{h}_{3}^{*},\hat{g})$. As a consequence
with $F=\bm{\mathrm{W}}$, in the $\varepsilon\rightarrow 0$ limit we will be
left with

$\left\langle\bm{\mathrm{W}}(z_{0})\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})\right\rangle=\psi^{\prime}(z_{0})^{3}\prod_{k=1}^{N}\left\lvert\psi^{\prime}(z_{k})\right\rvert^{2\Delta_{\alpha_{k}}}\left\langle\bm{\mathrm{W}}(\psi(z_{0}))\prod_{k=1}^{N}V_{\alpha_{k}}(\psi(z_{k}))\right\rangle.$

This is the result we were looking for. ∎

##  5\. Degenerate fields and a BPZ-type equation

In this section we shed light on another manifestation, beyond the Ward
identities (1.22) and (1.27), of the higher-spin symmetry enjoyed by the
$\mathfrak{sl}_{3}$ Toda theory. This is achieved by showing existence of
degenerate fields —that correspond to singular vectors in $W$-algebras— which
in turn implies a BPZ-type differential equation on certain correlation
functions. To do so we rely on the explicit form of the $W$-descendent fields
shown in the present document. To the best of our knowledge this approach is
entirely new and differs from the standard method based on the study of
singular vectors of $W$-algebras. For more details on this method see _e.g._
Parts 2 and 3 in [17]. Throughout this section we will denote by
$\bm{\mathrm{V}}=\prod_{k=1}^{N}V_{\alpha_{k}}(z_{k})$ a product of Vertex
Operators and by $\bm{\mathrm{V}}_{\varepsilon}$ its regularized counterpart.

###  5.1. A first application to degenerate fields at level one

To start with let us investigate the existence of (partially) degenerate
fields at the level one. These correspond to Vertex Operators $V_{\alpha}$ for
which the weight $\alpha$ takes a special value, which allows to write down
the quantity $\bm{\mathcal{W}_{-1}}$ in terms of derivatives of the
correlation functions.

As explained throughout this document, the first $W$-descendent can be
expressed in terms of the Toda field via

$\bm{\mathcal{W}_{-1}^{(1)}}\left\langle\bm{\mathrm{V}}\right\rangle=\lim\limits_{\varepsilon\rightarrow
0}\left\langle\bm{\mathrm{W}_{-1,\varepsilon}}(z_{1},\alpha_{1})\bm{\mathrm{V}}_{\varepsilon}\right\rangle$

where we have set, as before,

$\bm{\mathrm{W}_{-1,\varepsilon}}(z,\alpha)=-qB(\alpha,\partial\varphi_{\varepsilon}(z))-2C(\alpha,\alpha,\partial\varphi_{\varepsilon}(z)).$

Our goal here is to express $\bm{\mathcal{W}_{-1}}$ in terms of derivatives of
the field, and to do so we may introduce the first Virasoro descendent of
$V_{\alpha_{1}}$ by setting

$\bm{\mathcal{L}_{-1}^{(1)}}\left\langle\bm{\mathrm{V}}\right\rangle\coloneqq\partial_{z_{1}}\left\langle\bm{\mathrm{V}}\right\rangle.$

Then we use Equation (2.22) to rewrite the derivative of the correlation
functions (we omit the $\delta$-dependence of the correlation functions since
$z_{1}$ does not appear in the correlation functions we consider) as

$\partial_{z_{1}}\left\langle\bm{\mathrm{V}}_{\varepsilon}\right\rangle=\Big{\langle}\left\langle\alpha_{1},\partial\varphi_{\varepsilon}(z_{1})\right\rangle\bm{\mathrm{V}}_{\varepsilon}\Big{\rangle}.$

Hence the Virasoro descendent
$\bm{\mathcal{L}_{-1}^{(1)}}\left\langle\bm{\mathrm{V}}\right\rangle$ can be
represented as the limit

$\lim\limits_{\varepsilon\rightarrow
0}\Big{\langle}\left\langle\alpha_{1},\partial\varphi_{\varepsilon}(z_{1})\right\rangle\bm{\mathrm{V}}_{\varepsilon}\Big{\rangle}.$

As a consequence we can express the first $W$-descendent in terms of the first
Virasoro descendent when $\bm{\mathrm{W}_{-1,\varepsilon}}(z,\alpha)$ and
$\left\langle\alpha,\partial\varphi_{\varepsilon}(z)\right\rangle$ are
proportional. Therefore we will say that the primary field is degenerate at
the level one when there exists a real $\kappa$ for which

$\bm{\mathrm{W}_{-1,\varepsilon}}(z,\alpha)=\kappa\left\langle\alpha,\partial\varphi_{\varepsilon}(z)\right\rangle.$

Put differently we need to find $\alpha$ such that for all
$u\in\mathfrak{h}^{*}$,

$-qB(\alpha,u)-2C(\alpha,\alpha,u)=\kappa\left\langle\alpha,u\right\rangle.$

Taking respectively $u=e_{1}$ and $u=e_{2}$ and setting
$\alpha\coloneqq\alpha^{1}\omega_{1}+\alpha^{2}\omega_{2}$ we see that we must
have

$\displaystyle\kappa\alpha^{1}=q\alpha^{1}-2\alpha^{1}\frac{2\alpha^{2}+\alpha^{1}}{3}\quad\text{and}\quad\kappa\alpha^{2}=-q\alpha^{2}+2\alpha^{2}\frac{2\alpha^{1}+\alpha^{2}}{3}$

which implies that either $\alpha^{1}\alpha^{2}=0$ or
$q=\alpha^{1}+\alpha^{2}$.

######  Proposition 5.1.

Degenerate fields at level one are given by the $V_{\alpha}$ with $\alpha$ of
the form $\chi\omega_{1}$ or $\chi\omega_{2}$ for $\chi<q$, or
$\lambda\omega_{1}+(q-\lambda)\omega_{2}$ for some $0<\lambda<q$. In that case
we have the relation

(5.1) $\bm{\mathcal{W}_{-1}^{(1)}}\left\langle
V_{\alpha}(z_{1})\prod_{k=2}^{N}V_{\alpha_{k}}(z_{k})\right\rangle=\frac{3w(\alpha)}{2\Delta_{\alpha}}\partial_{z_{1}}\left\langle
V_{\alpha}(z_{1})\prod_{k=2}^{N}V_{\alpha_{k}}(z_{k})\right\rangle,$

valid as soon as the $\bm{z}$ are distinct and the $\bm{\alpha}$ satisfies the
Seiberg bounds (2.25).

###  5.2. Degenerate fields at the levels two and three

We may proceed in the same way for other descendents at the levels two and
three. Vertex Operators that satisfy similar requirements as above but for
descendents at the second or third level will be called _degenerate fields at
the second or third level_. These are also sometimes referred to as _fully
degenerate fields_ by contrast with degenerate fields at the first level only.
In order to keep notations simple we will omit the Wick product convention in
this subsection: for instance quantities such as
$\left\langle\partial\varphi(z),\partial\varphi(z)\right\rangle$ are to be
understood as
$:\left\langle\partial\varphi(z),\partial\varphi(z)\right\rangle:$ . Similarly
we no longer use the $\theta$-regularization of the correlation functions in
order not to overload notations.

We start by considering the Virasoro descendents of order two. We define them
by the expressions

(5.2)
$\displaystyle\bm{\mathrm{L}_{-(1,1)}}(z,\alpha)\coloneqq\left\langle\alpha,\partial^{2}\varphi(z)\right\rangle+\left\langle\alpha,\partial\varphi(z)\right\rangle^{2},$
(5.3) $\displaystyle\bm{\mathrm{L}_{-2}}(z,\alpha)\coloneqq\left\langle
Q+\alpha,\partial^{2}\varphi(z)\right\rangle-\left\langle\partial\varphi(z),\partial\varphi(z)\right\rangle.$

By doing so it is immediate to see that

$\displaystyle\left\langle\bm{\mathrm{L}_{-(1,1)}}(z_{1},\alpha_{1})\bm{\mathrm{V}}\right\rangle$
$\displaystyle=\partial_{z_{1}}^{2}\left\langle\bm{\mathrm{V}}\right\rangle$
$\displaystyle=\left(\bm{\mathcal{L}_{-1}^{(1)}}\right)^{2}\left\langle\bm{\mathrm{V}}\right\rangle.$

The effect of the $L_{-2}$ descendent on the correlation functions is less
obvious; however conformal invariance allows to claim the following statement.

######  Lemma 5.2.

Set
$\bm{\mathcal{L}_{-2}^{(1)}}\left\langle\bm{\mathrm{V}}\right\rangle\coloneqq\lim\limits_{\varepsilon\rightarrow
0}\left\langle\bm{\mathrm{L}_{-2,\varepsilon}}(z,\alpha)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle$.
Then

(5.4) $\bm{\mathcal{L}_{-2}^{(1)}}\left\langle
V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle=\sum_{k=2}^{N}\left(\frac{\partial_{z_{k}}}{z-z_{k}}+\frac{\Delta_{\alpha_{k}}}{(z-z_{k})^{2}}\right)\left\langle
V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle.$

###### Proof.

At the regularized level, the right-hand side divided by the non-zero quantity
$\left\langle V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle$ is given by

$\displaystyle=\sum_{k\neq
l}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{2(z-z_{k})(z_{l}-z_{k})_{\varepsilon}}+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\alpha\right\rangle}{2(z-z_{k})(z-z_{k})_{\varepsilon}}+\frac{\Delta_{\alpha_{k}}}{(z-z_{k})^{2}}\quad+\rm
I$ $\displaystyle=\sum_{k\neq
l}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{4(z_{l}-z_{k})}\left(\frac{1}{z-z_{k}}-\frac{1}{z-z_{l}}\right)+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},Q+\alpha\right\rangle}{2(z-z_{k})^{2}}-\frac{\left\langle\alpha_{k},\alpha_{k}\right\rangle}{4(z-z_{k})^{2}}\quad+o_{\varepsilon}(1)\quad+\rm
I$ $\displaystyle=-\sum_{k\neq
l}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{4(z-z_{k})(z-z_{l})}+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},Q+\alpha\right\rangle}{2(z-z_{k})^{2}}-\frac{\left\langle\alpha_{k},\alpha_{k}\right\rangle}{4(z-z_{k})^{2}}\quad+o_{\varepsilon}(1)\quad+\rm
I$
$\displaystyle=-\sum_{k,l=1}^{N}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{4(z-z_{k})(z-z_{l})}+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},Q+\alpha\right\rangle}{2(z-z_{k})^{2}}\quad+o_{\varepsilon}(1)\quad+\rm
I,$

where $\left\langle V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle\rm I$ is equal
to:

$\frac{\mu\gamma}{2}\sum_{i=1}^{2}\int_{\mathbb{C}}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},e_{i}\right\rangle}{2(z-z_{k})(z_{k}-x)_{\varepsilon}}\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x.$

Using the same reasoning as in Section 4 involving symmetrisation and Stokes’
formula, the latter is equal (up to terms that vanish in the
$\varepsilon\rightarrow 0$ limit) to

$\displaystyle\frac{\mu\gamma}{2}\sum_{i=1}^{2}\int_{\mathbb{C}}\left(\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},e_{i}\right\rangle}{2(z-z_{k})(z-x)_{\varepsilon}}-\frac{\left\langle\alpha,e_{i}\right\rangle+\frac{2}{\gamma}}{(z-x)^{2}_{\varepsilon}}\right)\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x$
$\displaystyle-\frac{(\mu\gamma)^{2}}{4}\sum_{i,j=1}^{2}\int_{\mathbb{C}^{2}}\frac{\left\langle
e_{i},e_{j}\right\rangle}{(z-x_{1})_{\varepsilon}(z-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}.$

This expression coincides with the left-hand side in the
$\varepsilon\rightarrow 0$ limit by applying Gaussian integration by parts in
the same spirit as in Section 4, concluding the proof:

$\displaystyle\left\langle\bm{\mathrm{L}_{-2,\varepsilon}}(z,\alpha)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle$
$\displaystyle=\sum_{k=1}^{N}\frac{\left\langle
Q+\alpha,\alpha_{k}\right\rangle}{2(z-z_{k})^{2}_{\varepsilon}}\left\langle
V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle-\frac{\mu\gamma}{2}\sum_{i=1}^{2}\int_{\mathbb{C}}\frac{\left\langle
Q+\alpha,e_{i}\right\rangle}{(z-x)^{2}_{\varepsilon}}\left\langle V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle
d^{2}x$ $\displaystyle-$
$\displaystyle\sum_{k,l=1}^{N}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{4(z-z_{k})_{\varepsilon}(z-z_{l})_{\varepsilon}}\left\langle
V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle$
$\displaystyle+\frac{\mu\gamma}{4}$
$\displaystyle\sum_{i=1}^{2}\int_{\mathbb{C}}\left(\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},e_{i}\right\rangle}{(z-z_{k})_{\varepsilon}(z-x)_{\varepsilon}}+\frac{\gamma\left\langle
e_{i},e_{i}\right\rangle}{(z-x)_{\varepsilon}(z-x)_{\varepsilon}}\right)\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle
d^{2}x$ $\displaystyle-$
$\displaystyle\frac{(\mu\gamma)^{2}}{4}\sum_{i,j=1}^{2}\int_{\mathbb{C}^{2}}\frac{\left\langle
e_{i},e_{j}\right\rangle}{(z-x_{1})_{\varepsilon}(z-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle
d^{2}x_{1}d^{2}x_{2}.$

∎

Thanks to the expression of the Virasoro descendents at the level two, we can
readily search for degenerate fields at the level two. As before, we think of
them as Vertex Operators for which the $W$-descendent at the level two,
$\bm{\mathrm{W}_{-2}}(z,\alpha)$, can be expressed as a linear combination of
Virasoro descendents at the level two. Put differently, these are $\alpha$
such that there exist $\kappa_{1},\kappa_{2}$ real numbers with

$\kappa_{1}\bm{\mathrm{L}_{-(1,1)}}+\kappa_{2}\bm{\mathrm{L}_{-2}}=\bm{\mathrm{W}_{-2}}.$

The latter implies that $\alpha=\alpha_{1}\omega_{1}+\alpha_{2}\omega_{2}$ is
a solution of the following set of equations

$\displaystyle\left\\{\begin{array}[]{cc}\kappa_{1}\alpha_{1}^{2}-2\kappa_{2}&=2\frac{2\alpha_{2}+\alpha_{1}}{3}\\\
\kappa_{1}\alpha_{2}^{2}-2\kappa_{2}&=-2\frac{2\alpha_{1}+\alpha_{2}}{3}\\\
\kappa_{1}\alpha_{1}\alpha_{2}+\kappa_{2}&=2\frac{\alpha_{1}-\alpha_{2}}{3}\\\
\kappa_{1}\alpha_{1}+\kappa_{2}(q+\alpha_{1})&=-(q+\alpha_{1})\frac{2\alpha_{2}+\alpha_{1}}{3}\\\
\kappa_{1}\alpha_{2}+\kappa_{2}(q+\alpha_{2})&=(q-\alpha_{2})\frac{2\alpha_{1}+\alpha_{2}}{3}.\end{array}\right.$

Explicit computations show the following.

######  Proposition 5.3.

Degenerate fields at the level two are given by the Vertex Operators
$V_{\alpha}$ with $\alpha$ of the form $-\chi\omega_{i}$ where $\chi$ is
either $\frac{2}{\gamma}$ or $\gamma$ and $i\in\\{1,2\\}$. In that case

(5.5)
$\begin{split}\bm{\mathrm{W}_{-2}}(\cdot,\alpha)&=-\frac{4}{\chi}\bm{\mathrm{L}_{-(1,1)}}(\cdot,\alpha)-\frac{4\chi}{3}\bm{\mathrm{L}_{-2}}(\cdot,\alpha),\text{
hence}\\\ \bm{\mathcal{W}_{-2}^{(1)}}\left\langle
V_{\alpha}(z_{1})\prod_{k=2}^{N}V_{\alpha_{k}}(z_{k})\right\rangle&=-\left(\frac{4}{\chi}\left(\bm{\mathcal{L}_{-1}^{(1)}}\right)^{2}+\frac{4\chi}{3}\bm{\mathcal{L}_{-2}^{(1)}}\right)\left\langle
V_{\alpha}(z_{1})\prod_{k=2}^{N}V_{\alpha_{k}}(z_{k})\right\rangle.\end{split}$

A similar reasoning remains true when we turn to descendents at the level
three; nevertheless calculations are slightly more involved. Like before, we
shall first introduce the three Virasoro descendents at the third level by
setting

(5.6)
$\begin{split}\bm{\mathrm{L}_{-(1,1,1)}}(z,\alpha)\coloneqq&\left\langle\alpha,\partial^{3}\varphi(z)\right\rangle+3\left\langle\alpha,\partial\varphi(z)\right\rangle\left\langle\alpha,\partial^{2}\varphi(z)\right\rangle+\left\langle\alpha,\partial\varphi(z)\right\rangle^{3}\\\
\bm{\mathrm{L}_{-(1,2)}}(z,\alpha)\coloneqq&\left\langle
Q+\alpha,\partial^{3}\varphi(z)\right\rangle+\left\langle
Q+\alpha,\partial^{2}\varphi(z)\right\rangle\left\langle\alpha,\partial\varphi(z)\right\rangle\\\
&-2\left\langle\partial^{2}\varphi(z),\partial\varphi(z)\right\rangle-\left\langle\partial\varphi(z),\partial\varphi(z)\right\rangle\left\langle\alpha,\partial\varphi(z)\right\rangle\cdot\end{split}$

These are such that (provided that the objects exist)

(5.7)
$\begin{split}\left\langle\bm{\mathrm{L}_{-(1,1,1)}}(z_{1},\alpha_{1})\bm{\mathrm{V}}\right\rangle&=\left(\bm{\mathcal{L}_{-1}^{(1)}}\right)^{3}\left\langle\bm{\mathrm{V}}\right\rangle\\\
\left\langle\bm{\mathrm{L}_{-(1,2)}}(z_{1},\alpha_{1})\bm{\mathrm{V}}\right\rangle&=\bm{\mathcal{L}_{-1}^{(1)}}\bm{\mathcal{L}_{-2}^{(1)}}\left\langle\bm{\mathrm{V}}\right\rangle\cdot\end{split}$

We also introduce

(5.8) $\bm{\mathrm{L}_{-3}}(z,\alpha)\coloneqq\left\langle
Q+\frac{\alpha}{2},\partial^{3}\varphi(z)\right\rangle-2\left\langle\partial^{2}\varphi(z),\partial\varphi(z)\right\rangle$

which is has been defined in order to satisfy the below equation:

(5.9)
$\left\langle\bm{\mathrm{L}_{-3}}(z_{1},\alpha_{1})\bm{\mathrm{V}}\right\rangle=\sum_{k=2}^{N}\left(\frac{\partial_{z_{k}}}{(z_{1}-z_{k})^{2}}+\frac{2\Delta_{\alpha_{k}}}{(z_{1}-z_{k})^{3}}\right)\left\langle\bm{\mathrm{V}}\right\rangle.$

To see that we may proceed like in the proof of Lemma 5.2 and the very same
arguments still apply. Nonetheless to motivate this claim simply note that

$\displaystyle\sum_{k=1}^{N}\left(\frac{\partial_{z_{k}}}{(z-z_{k})^{2}}+\frac{2\Delta_{\alpha_{k}}}{(z-z_{k})^{3}}\right)\left\langle
V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle$ $\displaystyle=\sum_{k\neq
l}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{2(z-z_{k})^{2}(z_{l}-z_{k})}+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\alpha\right\rangle}{2(z-z_{k})^{3}}+\frac{2\Delta_{\alpha_{k}}}{(z-z_{k})^{3}}\quad+\quad\text{integral
terms}$ $\displaystyle=\sum_{k\neq
l}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{4(z_{l}-z_{k})}\left(\frac{1}{(z-z_{k})^{2}}-\frac{1}{(z-z_{l})^{2}}\right)+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},Q+\frac{\alpha}{2}\right\rangle}{(z-z_{k})^{3}}-\frac{\left\langle\alpha_{k},\alpha_{k}\right\rangle}{2(z-z_{k})^{3}}\quad+\quad\text{integral
terms}$ $\displaystyle=-\sum_{k\neq
l}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{2(z-z_{k})^{2}(z-z_{l})}+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},Q+\frac{\alpha}{2}\right\rangle}{(z-z_{k})^{3}}-\frac{\left\langle\alpha_{k},\alpha_{k}\right\rangle}{2(z-z_{k})^{3}}\quad+\quad\text{integral
terms}$
$\displaystyle=-\sum_{k,l=1}^{N}\frac{\left\langle\alpha_{k},\alpha_{l}\right\rangle}{2(z-z_{k})^{2}(z-z_{l})}+\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},Q+\frac{\alpha}{2}\right\rangle}{(z-z_{k})^{3}}\quad+\quad\text{integral
terms}$
$\displaystyle=\Big{\langle}-2\left\langle\partial^{2}\varphi(z),\partial\varphi(z)\right\rangle
V_{\alpha}(z)\bm{\mathrm{V}}\Big{\rangle}+\Big{\langle}\langle
Q+\frac{\alpha}{2},\partial^{3}\varphi(z)\rangle
V_{\alpha}(z)\bm{\mathrm{V}}\Big{\rangle}.$

We now turn to the $W$-descendents at the third level. With a reasoning
similar to the one of Lemma 5.2 we end up with:

######  Lemma 5.4.

Let us set

(5.10) $\begin{split}\bm{\mathrm{W}_{-3}}(z,\alpha)&\coloneqq
q^{2}\left\langle
h_{2},\partial^{3}\varphi(z)\right\rangle+\frac{q}{2}\left(2B(\partial^{3}\varphi(z),\alpha)-B(\alpha,\partial^{3}\varphi(z))\right)-C(\partial^{3}\varphi(z),\alpha,\alpha)\\\
&-2qB(\partial^{2}\varphi(z),\partial\varphi(z))+4C(\partial^{2}\varphi(z),\partial\varphi(z),\alpha)+4C(\partial\varphi(z),\partial^{2}\varphi(z),\alpha)\\\
&-8\left\langle h_{1},\partial\varphi(z)\right\rangle\left\langle
h_{2},\partial\varphi(z)\right\rangle\left\langle
h_{3},\partial\varphi(z)\right\rangle.\end{split}$

Then

(5.11) $\lim\limits_{\varepsilon\rightarrow
0}\left\langle\bm{\mathrm{W}_{-3,\varepsilon}}(z_{1},\alpha_{1})\bm{\mathrm{V}}_{\varepsilon}\right\rangle=\sum_{k=2}^{N}\left(\frac{\bm{\mathcal{W}_{-2}^{(k)}}}{z_{1}-z_{k}}+\frac{\bm{\mathcal{W}_{-1}^{(k)}}}{(z_{1}-z_{k})^{2}}+\frac{w(\alpha_{k})}{(z_{1}-z_{k})^{3}}\right)\left\langle\bm{\mathrm{V}}\right\rangle.$

###### Proof.

Computations parallel the ones made in Section 4. The right-hand side in
(5.11), when divided by $\left\langle
V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle$, can be expanded as

$\displaystyle\sum_{k,l,p\text{
distinct}}\frac{C(\alpha_{k},\alpha_{l},\alpha_{p})}{(z-z_{k})(z_{k}-z_{l})_{\varepsilon}(z_{k}-z_{p})_{\varepsilon}}+\sum_{k\neq
l}\frac{C^{\sigma}(\alpha,\alpha_{k},\alpha_{l})}{(z-z_{k})(z_{k}-z)_{\varepsilon}(z_{k}-z_{l})_{\varepsilon}}$
$\displaystyle+\sum_{k\neq
l}\frac{q\left(B(\alpha_{l},\alpha_{k})-B(\alpha_{k},\alpha_{l})\right)+2C(\alpha_{k},\alpha_{l}-\alpha_{k},\alpha_{l})}{2(z-z_{k})(z_{k}-z_{l})_{\varepsilon}^{2}}+\frac{qB(\alpha_{k},\alpha_{l})+2C(\alpha_{k},\alpha_{k},\alpha_{l})}{2(z-z_{k})^{2}(z_{k}-z_{l})_{\varepsilon}}$
$\displaystyle+\sum_{k=1}^{N}\frac{q\left(B(\alpha,\alpha_{k})-B(\alpha_{k},\alpha)\right)+2C(\alpha_{k},\alpha-\alpha_{k},\alpha)}{2(z-z_{k})(z-z_{k})_{\varepsilon}^{2}}+\frac{qB(\alpha_{k},\alpha)+2C(\alpha_{k},\alpha_{k},\alpha)}{2(z-z_{k})^{2}(z_{k}-z)_{\varepsilon}}+\frac{w(\alpha_{k})}{(z-z_{k})^{3}}$
$\displaystyle+\text{integral terms}.$

The first quantity can be easily dealt with by recursive application of
symmetrisation identities:

$\displaystyle\sum_{k,l,p\text{
distinct}}\frac{C(\alpha_{k},\alpha_{l},\alpha_{p})}{(z-z_{k})(z_{k}-z_{l})(z_{k}-z_{p})}=\frac{1}{3}\sum_{k,l,p\text{
distinct}}\frac{C(\alpha_{k},\alpha_{l},\alpha_{p})}{(z-z_{k})(z-z_{l})(z-z_{p})}$
$\displaystyle=\sum_{k,l,p}\frac{\left\langle\alpha_{k},h_{1}\right\rangle\left\langle\alpha_{l},h_{2}\right\rangle\left\langle\alpha_{p},h_{3}\right\rangle}{(z-z_{k})(z-z_{l})(z-z_{p})}-\sum_{k,l}\frac{C(\alpha_{k},\alpha_{k},\alpha_{l})}{(z-z_{k})^{2}(z-z_{l})}-\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},h_{1}\right\rangle\left\langle\alpha_{k},h_{2}\right\rangle\left\langle\alpha_{k},h_{3}\right\rangle}{(z-z_{k})^{3}}.$

Similarly we can write that

$\displaystyle\sum_{k\neq
l}\frac{qB(\alpha_{k},\alpha_{l})+2C(\alpha_{k},\alpha_{k},\alpha_{l})}{2(z-z_{k})^{2}(z_{k}-z_{l})}+\frac{q\left(B(\alpha_{l},\alpha_{k})-B(\alpha_{k},\alpha_{l})\right)+2C(\alpha_{k},\alpha_{l}-\alpha_{k},\alpha_{l})}{2(z-z_{k})(z_{k}-z_{l})^{2}}$
$\displaystyle=\sum_{k\neq
l}\frac{qB(\alpha_{k},\alpha_{l})+2C(\alpha_{k},\alpha_{k},\alpha_{l})}{2(z-z_{k})^{2}(z_{k}-z_{l})}+\frac{q\left(B(\alpha_{l},\alpha_{k})-B(\alpha_{k},\alpha_{l})\right)+2C(\alpha_{k},\alpha_{l}-\alpha_{k},\alpha_{l})}{4(z-z_{k})(z-z_{l})(z_{k}-z_{l})}$
$\displaystyle=\sum_{k\neq
l}\frac{qB(\alpha_{k},\alpha_{l})+2C(\alpha_{k},\alpha_{k},\alpha_{l})}{2(z-z_{k})^{2}(z-z_{l})}$
$\displaystyle=\sum_{k,l}\frac{qB(\alpha_{k},\alpha_{l})+2C(\alpha_{k},\alpha_{k},\alpha_{l})}{2(z-z_{k})^{2}(z-z_{l})}-\sum_{k=1}^{N}\frac{qB(\alpha_{k},\alpha_{k})+2C(\alpha_{k},\alpha_{k},\alpha_{k})}{2(z-z_{k})^{3}}\cdot$

Therefore the renormalized right-hand side in (5.11) is actually equal to

(5.12)
$\begin{split}&\sum_{k,l,p}\frac{\left\langle\alpha_{k},h_{1}\right\rangle\left\langle\alpha_{l},h_{2}\right\rangle\left\langle\alpha_{p},h_{3}\right\rangle}{(z-z_{k})(z-z_{l})(z-z_{p})}+\sum_{k,l}\frac{qB(\alpha_{k},\alpha_{l})-2C^{\sigma}(\alpha,\alpha_{k},\alpha_{l})}{2(z-z_{k})^{2}(z-z_{l})}\\\
&+\sum_{k=1}^{N}\frac{-2q^{2}\left\langle\alpha_{k},h_{2}\right\rangle+q\left(B(\alpha,\alpha_{k})-2B(\alpha_{k},\alpha)\right)+2C(\alpha_{k},\alpha,\alpha)}{2(z-z_{k})^{3}}+o_{\varepsilon}(1)\hskip
5.69046pt+\text{integral terms}.\end{split}$

It remains to treat the integral terms. This is done with a treatment similar
to the one we have used in Section 4, the only difference being that there is
an extra Vertex Operator within the correlation function. Nevertheless along
the same lines we get (up to a factor $\left\langle
V_{\alpha,\varepsilon}(z)V_{\varepsilon}\right\rangle$ and a
$o_{\varepsilon}(1)$ quantity) a sum of the terms

$\displaystyle-\mu\gamma\sum_{i=1}^{2}\int_{\mathbb{C}}\left(\frac{-\frac{2q}{\gamma}\left\langle
e_{i},h_{2}\right\rangle}{(z-x)^{3}_{\varepsilon}}+\sum_{k=1}^{N}\left(\frac{qB(\alpha_{k},e_{i})+2C(\alpha_{k},\alpha_{k},e_{i})}{2(z-z_{k})^{2}(z-x)_{\varepsilon}}+\frac{qB(e_{i},\alpha_{k})+2\gamma
C(e_{i},e_{i},\alpha_{k})}{2(z-z_{k})(z-x)^{2}_{\varepsilon}}\right)\right.$
$\displaystyle\left.\hskip
142.26378pt+\sum_{k,l=1}^{N}\frac{C(\alpha_{k},\alpha_{l},e_{i})}{(z-z_{k})(z-z_{l})(z-x)_{\varepsilon}}\right)\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x$
$\displaystyle+(\mu\gamma)^{2}\sum_{i,j=1}^{2}\int_{\mathbb{C}^{2}}\left(\sum_{k=1}^{N}\frac{C(e_{i},e_{j},\alpha_{k})}{(z-z_{k})(z-x_{1})_{\varepsilon}(z-x_{2})_{\varepsilon}}+\frac{qB(e_{i},e_{j})+\gamma
C(e_{i},e_{i},e_{j})}{2(z-x_{1})_{\varepsilon}^{2}(z-x_{2})_{\varepsilon}}\right)$
$\displaystyle\hskip 227.62204pt\left\langle V_{\gamma
e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}$
$\displaystyle-(\mu\gamma)^{3}\sum_{i,j,f=1}^{2}\int_{\mathbb{C}^{3}}\frac{\left\langle
e_{i},h_{1}\right\rangle\left\langle e_{j},h_{2}\right\rangle\left\langle
e_{f},h_{3}\right\rangle}{(z-x_{1})_{\varepsilon}(z-x_{2})_{\varepsilon}(z-x_{3})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\gamma
e_{f},\varepsilon}(x_{3})V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle
d^{2}x_{1}d^{2}x_{2}d^{2}x_{3}$

coming from the reasoning developed in Section 4, and additional ones given by

$\displaystyle-\mu\gamma\sum_{i=1}^{2}\left\langle
h_{2},e_{i}\right\rangle\int_{\mathbb{C}}\left\langle V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\left(\frac{\left\langle
2Q+(\frac{2}{\gamma}-3\gamma)\omega_{3-i},\alpha\right\rangle+2\left\langle
e_{i},\alpha\right\rangle\left\langle\omega_{3-i},\alpha\right\rangle}{2(z-x)^{3}_{\varepsilon}}\right.$
$\displaystyle\hskip
56.9055pt\left.+\sum_{k=1}^{N}\frac{\left\langle\alpha,e_{i}\right\rangle\omega_{3-i}(\alpha_{k})+\left\langle\alpha_{k},e_{i}\right\rangle\omega_{3-i}(\alpha)}{(z-z_{k})^{2}(z-x)_{\varepsilon}}+\frac{\left\langle\alpha,e_{i}\right\rangle\omega_{3-i}(\alpha_{k})+\left\langle\alpha_{k},e_{i}\right\rangle\omega_{3-i}(\alpha)}{(z-z_{k})(z-x)^{2}_{\varepsilon}}\right)d^{2}x$
$\displaystyle-(\mu\gamma)^{2}\sum_{i,j=1}^{2}\int_{\mathbb{C}^{2}}\frac{C^{\sigma}(\alpha,e_{i},e_{j})}{(z-x_{1})_{\varepsilon}^{2}(z-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}$

that account for the extra Vertex operator in the correlation function and
which are obtained by recursive integration by parts. Therefore using Gaussian
integration by parts as well as the explicit expression of $B$ and $C$ we see
that the expression (5.12) coincides with
$\left\langle\bm{\mathrm{W}_{-3,\varepsilon}}(z,\alpha)V_{\alpha,\varepsilon}(z)\bm{\mathrm{V}}_{\varepsilon}\right\rangle$,
up to a term that vanishes in the $\varepsilon\rightarrow 0$ limit. ∎

We are now in position to address the question of finding degenerate fields at
the level three.

######  Proposition 5.5.

Degenerate fields at the levels two and three are given by the $V_{\alpha}$
with $\alpha$ of the form $-\chi\omega_{1}$ or $-\chi\omega_{2}$ with
$\chi\in\\{\gamma,\frac{2}{\gamma}\\}$. In that case both relations

(5.13)
$\bm{\mathrm{W}_{-2}}=-\frac{4}{\chi}\bm{\mathrm{L}_{-(1,1)}}-\frac{4\chi}{3}\bm{\mathrm{L}_{-2}}\quad\text{and}$
(5.14)
$\bm{\mathrm{W}_{-3}}=-\left(\frac{\chi}{3}+\frac{2}{\chi}\right)\bm{\mathrm{L}_{-3}}+\frac{4}{\chi}\bm{\mathrm{L}_{-(1,2)}}+\frac{8}{\chi^{3}}\bm{\mathrm{L}_{-(1,1,1)}},$

are valid as soon as the $\bm{z}$ are distinct and that $\bm{\alpha}$
satisfies the Seiberg bounds (2.25).

As a consequence correlation functions of the form $\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle$ are solutions of the
following BPZ identity:

(5.15)
$\left[\frac{8}{\chi^{3}}\partial_{z}^{3}+\frac{4}{\chi}\partial_{z}\sum_{k=1}^{N}\left(\frac{\partial_{z_{k}}}{z-z_{k}}+\frac{\Delta_{\alpha_{k}}}{(z-z_{k})^{2}}\right)+\left(\frac{\chi}{3}+\frac{2}{\chi}\right)\sum_{k=1}^{N}\left(\frac{\partial_{z_{k}}}{(z-z_{k})^{2}}+\frac{2\Delta_{\alpha_{k}}}{(z-z_{k})^{3}}\right)\right.\\\
\left.-\sum_{k=1}^{N}\left(\frac{\bm{\mathcal{W}_{-2}^{(k)}}}{z-z_{k}}+\frac{\bm{\mathcal{W}_{-1}^{(k)}}}{(z-z_{k})^{2}}+\frac{w(\alpha_{k})}{(z-z_{k})^{3}}\right)\right]\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle=0.$

###### Proof.

Equation (5.15) simply corresponds to inserting the equality in Equation
(5.14) within a correlation function and using Equations (5.7), (5.9) and
(5.11). Proving Equation (5.14) follows from calculations very similar to
those we have done for degenerate fields at the levels one and two. ∎

###  5.3. Implications on a four-point correlation function

The expression of the BPZ-type equation (5.15) is not always tractable when it
comes to deriving exact expressions for the correlation functions. However
when considering a small number of Vertex Operators some cancellations occur,
allowing to express the differential equation in several variables (5.15) as a
differential equation in only one variable. Indeed let us consider a four-
point correlation function with one point at infinity $\left\langle
V_{\alpha}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle$,
defined by the limit

(5.16) $\left\langle
V_{\alpha}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle\coloneqq\lim\limits_{z^{\prime}\rightarrow\infty}\left\lvert
z^{\prime}\right\rvert^{4\Delta_{\alpha_{\infty}}}\left\langle
V_{\alpha}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(z^{\prime})\right\rangle.$

This limit is non-zero and admits the following representation:

$\left\langle
V_{\alpha}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle=\left\lvert
z\right\rvert^{-\left\langle\alpha_{0},\alpha\right\rangle}\left\lvert
z-1\right\rvert^{-\left\langle\alpha_{1},\alpha\right\rangle}\mathcal{H}(z)$

where $\mathcal{H}(z)$ is equal to, with
$\bm{\alpha}=(\alpha,\alpha_{0},\alpha_{1},\alpha_{\infty})$:

$\left(\prod_{i=1}^{2}\frac{\Gamma(s_{i})\mu_{i}^{-s_{i}}}{\gamma}\right)\mathds{E}\left[\prod_{i=1}^{2}\left(\int_{\mathbb{C}}\frac{\hat{g}(y_{i})^{-\frac{\gamma}{4}\sum_{k=1}^{4}\left\langle\alpha_{k},e_{i}\right\rangle}}{\prod_{k=1}^{3}\left\lvert
z_{k}-y_{i}\right\rvert^{\gamma\left\langle\alpha_{k},e_{i}\right\rangle}}M^{\gamma
e_{i},\hat{g}}(d^{2}y_{i})\right)^{-s_{i}}\right].$

Then by using the global Ward identities we can express the different
quantities $\bm{\mathcal{W}_{-i}^{(k)}}$ for $k=0,1,\infty$ and $i=1,2$ in
terms of $\bm{\mathcal{W}_{-i}}$, $i=1,2$, and $\bm{\mathcal{W}_{-1}^{(1)}}$.
Heuristically this is due to the fact that there are six such quantities
linked by five linearly independent constraints. Inverting this system of
equations yields the following:

(5.17) $\begin{split}\bm{\mathcal{W}_{-3}}\left\langle
V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle=\left[-\left(\frac{2}{z}+\frac{1}{z-1}\right)\bm{\mathcal{W}_{-2}}-\left(\frac{1}{z^{2}}+\frac{2}{z(z-1)}\right)\bm{\mathcal{W}_{-1}}+\frac{\bm{\mathcal{W}_{-1}^{(1)}}}{(z(z-1))^{2}}\right.&\\\
\left.-\frac{w_{\alpha}+w_{\infty}}{z^{2}(z-1)}-\frac{w_{0}}{z^{3}(z-1)}+\frac{w_{1}}{z(z-1)^{2}}\left(\frac{1}{z}+\frac{1}{z-1}\right)\right]\left\langle
V_{\alpha}(z)\bm{\mathrm{V}}\right\rangle&.\end{split}$

In the special case where the Vertex Operator $V_{\alpha}$ is fully
degenerate, we can use Propositions 5.1 and 5.5 to rewrite the above using
differential operators:

(5.18)
$\begin{split}&\left[\frac{8}{\chi^{3}}\partial_{z}^{3}+\frac{4}{\chi}\partial_{z}\bm{\mathcal{L}_{-2}}-\left(\frac{\chi}{3}+\frac{2}{\chi}\right)\bm{\mathcal{L}_{-3}}\right]\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle\\\
=&\left[\left(\frac{2}{z}+\frac{1}{z-1}\right)\left(\frac{4}{\chi}\partial_{z}^{2}+\frac{4\chi}{3}\bm{\mathcal{L}_{-2}}\right)-\left(\frac{1}{z^{2}}+\frac{2}{z(z-1)}\right)\left(\frac{5\chi}{3}+\frac{2}{\chi}\right)\partial_{z}+\frac{\bm{\mathcal{W}_{-1}^{(1)}}}{(z(z-1))^{2}}\right.\\\
&\left.-\frac{w+w_{\infty}}{z^{2}(z-1)}-\frac{w_{0}}{z^{3}(z-1)}+\frac{w_{1}}{z(z-1)^{2}}\left(\frac{1}{z}+\frac{1}{z-1}\right)\right]\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle.\end{split}$

We proceed in the same way for the Virasoro descendents. Using the three
global Ward identities given by conformal covariance of the correlation
function of Vertex Operators (see [8, Theorem 3.1]) we end up with

(5.19) $\begin{split}\bm{\mathcal{L}_{-3}}\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle&=\left[\frac{3z^{2}-3z+1}{(z(z-1))^{2}}\partial_{z}+\frac{\Delta-\Delta_{\infty}}{z(z-1)}\left(\frac{1}{z}+\frac{1}{z-1}\right)\right.\\\
&\left.+\frac{\Delta_{0}}{z^{2}(z-1)}\left(\frac{2}{z}+\frac{1}{z-1}\right)-\frac{\Delta_{1}}{z(z-1)^{2}}\left(\frac{1}{z}+\frac{2}{z-1}\right)\right]\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle\\\
\bm{\mathcal{L}_{-2}}\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle&=\left[\frac{2z-1}{z(1-z)}\partial_{z}+\frac{\Delta_{\infty}-\Delta}{z(z-1)}-\frac{\Delta_{0}}{z^{2}(z-1)}+\frac{\Delta_{1}}{z(z-1)^{2}}\right]\left\langle
V_{-\chi\omega_{1}}(z)\bm{\mathrm{V}}\right\rangle.\end{split}$

Combining the two last equations shows that applying the following
differential operator to a four-point correlation function with a degenerate
field $\left\langle
V_{-\chi\omega_{1}}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle$

(5.20)
$\begin{split}&\frac{8z^{2}(z-1)}{\chi^{3}}\partial_{z}^{3}+\frac{4}{\chi}z(3-5z)\partial^{2}_{z}\\\
&+\Big{[}\frac{2}{\chi}\left(\frac{4z^{2}-5z+2}{z-1}+2z(\Delta_{\infty}-\Delta)-2\Delta_{0}+2\Delta_{1}\frac{z}{z-1}\right)+\chi\frac{12z^{2}-15z+4}{z-1}\Big{]}\partial_{z}\\\
&+\frac{2}{\chi}\left((1-2z)\frac{\Delta_{\infty}-\Delta}{z-1}+(3z-2)\frac{\Delta_{0}}{z(z-1)}-(3z-1)\frac{\Delta_{1}}{(z-1)^{2}}\right)\\\
&+\frac{\chi}{3}\left((7-10z)\frac{\Delta_{\infty}-\Delta}{z-1}+(9z-6)\frac{\Delta_{0}}{z(z-1)}+(7-9z)\frac{\Delta_{1}}{(z-1)^{2}}\right)\\\
&+w+w(\infty)+\frac{w(0)}{z}-\frac{w(1)}{(z-1)^{2}}\left(2z-1\right)\end{split}$

will yield the quantity $\frac{\bm{\mathcal{W}_{-1}^{(1)}}}{z-1}\left\langle
V_{-\chi\omega_{1}}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle$.

If we further assume that the Vertex Operator $V_{\alpha_{1}}(1)$ is semi-
degenerate, _i.e._ that $\alpha_{1}=\kappa\omega_{2}$ for some real $\kappa$,
then this last quantity may be expressed as a derivative in $z$ of the
correlation function:

$\displaystyle\bm{\mathcal{W}_{-1}}(\alpha_{1},1)\left\langle
V_{-\chi\omega_{1}}(z)V_{\alpha_{0}}(0)V_{\alpha_{1}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle=(q-2\left\langle
h_{1},\alpha_{1}\right\rangle)\partial_{z_{1}}\Big{|}_{z_{1}=1}\left\langle
V_{-\chi\omega_{1}}(z)V_{\alpha_{0}}(0)V_{\kappa\omega_{2}}(z_{1})V_{\alpha_{\infty}}(\infty)\right\rangle$
$\displaystyle=(q-2\left\langle
h_{1},\alpha_{1}\right\rangle)\left(z\partial_{z}+\sum_{k=1}^{4}\Delta_{k}-2\Delta_{\infty}\right)\left\langle
V_{-\chi\omega_{1}}(z)V_{\alpha_{0}}(0)V_{\kappa\omega_{2}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle.$

The first equality follows from Proposition 5.1 while the second one is a
consequence of the conformal covariance of the correlation functions. Then
some (lengthy) algebraic manipulations show that we can write

$\left\langle
V_{-\chi\omega_{1}}(z)V_{\alpha_{0}}(0)V_{\kappa\omega_{2}}(1)V_{\alpha_{\infty}}(\infty)\right\rangle=\left\lvert
z\right\rvert^{\chi\left\langle h_{1},\alpha_{0}\right\rangle}\left\lvert
z-1\right\rvert^{\frac{\chi\kappa}{3}}\mathcal{H}(z)$

with $\mathcal{H}$ solution of the hypergeometric differential equation of
order three

(5.21)
$\begin{split}&\Big{[}z\left(A_{1}+z\partial_{z}\right)\left(A_{2}+z\partial_{z}\right)\left(A_{3}+z\partial_{z}\right)-\left(B_{1}-1+z\partial_{z}\right)\left(B_{2}-1+z\partial_{z}\right)z\partial_{z}\Big{]}\mathcal{H}=0.\end{split}$

In the above equation we have set

(5.22)
$\begin{split}A_{i}&:=\frac{\chi}{2}\left\langle\alpha_{0}+\kappa\omega_{2}-\chi\omega_{1}-Q,h_{1}\right\rangle+\frac{\chi}{2}\left\langle\alpha_{\infty}-Q,h_{i}\right\rangle\\\
B_{i}&:=1+\frac{\chi}{2}\left\langle\alpha_{0}-Q,h_{1}-h_{i+1}\right\rangle.\end{split}$

The fact that $\mathcal{H}$ is a (at least distributional) solution of
Equation (5.21) allows to claim that $\mathcal{H}$ is actually a real analytic
function via a standard elliptic regularity argument. Indeed, we can apply the
differential operator $\partial_{\bar{z}}^{3}$ to Equation (5.21). By doing so
we see that $\mathcal{H}$, viewed as a function of two real variables, is a
solution of a partial differential equation $\mathrm{P}\mathcal{H}=0$ with
analytic coefficients on $\mathbb{R}^{2}\setminus\\{(0,0);(1,0)\\}$, and whose
term of highest degree is given by $z^{3}(z-1)\Delta^{3}$ ($z=x+iy$) where
$\Delta$ is the standard Laplace operator $\Delta
f(x,y)=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial
y^{2}}$. In particular $\mathcal{H}$ is seen to be a solution of
$\mathrm{P}\mathcal{H}=0$ where $\mathrm{P}$ is an analytic hypoelliptic
operator on $\mathbb{R}^{2}\setminus\\{(0,0);(1,0)\\}$. This implies that
$\mathcal{H}$ is real analytic on $\mathbb{C}\setminus\\{0,1\\}$.

###  5.4. Explicit expression for the descendent fields: some perspectives

One of the contributions of the present document is to provide an explicit
expression for the descendent fields. In the above subsections we have shed
light on some applications of this property to the study of degenerate fields
at the first, second and third level, and its implications on a BPZ-type
differential equation. Let us now stress some future directions of work
allowed by this new framework.

####  5.4.1. Liouville theory

Using the same reasoning as above it is possible to write down an explicit
expression of the descendent fields in Liouville theory. Namely if
$\lambda=(\lambda_{1},\cdots,\lambda_{r})$ is a finite, non-increasing
sequence of positive integers (a _Young diagram_) we can write

(5.23)
$\bm{\mathrm{L}_{-\lambda_{1}}}\cdots\bm{\mathrm{L}_{-\lambda_{r}}}V_{\alpha}(z)=\bm{\mathrm{L}_{-\lambda}}(z,\alpha)V_{\alpha}(z),$

where $\bm{\mathrm{L}_{-\lambda}}(z,\alpha)$ admits an expansion of the form

(5.24)
$\bm{\mathrm{L}_{-\lambda}}(z,\alpha)=\sum_{\left\lvert\mu\right\rvert=\left\lvert\lambda\right\rvert}a_{\mu|\lambda}(\alpha)\partial^{\mu}\varphi.$

Here the sum ranges over Young diagrams with
$\left\lvert\mu\right\rvert\coloneqq\sum_{i\;\geqslant\;1}\mu_{i}=\left\lvert\lambda\right\rvert$
and we have denoted

$\partial^{\mu}\varphi\coloneqq\partial^{\mu_{1}}\varphi\times\cdots\times\partial^{\mu_{r}}\varphi.$

Coefficients $a_{\mu|\lambda}(\alpha)$ can be determined via a simple
recursive procedure. Determining these coefficients could provide an
algorithmic way of deriving BPZ-type equations for general singular vectors
$V_{\left\langle r,s\right\rangle}\coloneqq
V_{-(r-1)\frac{\gamma}{2}-(s-1)\frac{2}{\gamma}}$. We stress that this method,
new to the best of our knowledge, can be implemented within the probabilistic
framework of Liouville theory, in contrast with the usual one based on
irreducible representations of Verma modules [19].

####  5.4.2. Toda theories

The reasoning presented in the context of Liouville theory extends to the
study of Toda theories. Namely it is possible to write down $W$-descendents in
a form similar to that of Equations (5.23) and (5.24). This could be the
starting point to the derivation of BPZ-type equations for Toda theories.
Again one of the key advantages of the formulation presented is that it could
be implemented within our probabilistic framework and provide a general
procedure for rigorously proving that BPZ-type equations hold true for Toda
theories. In particular this representation should allow us to recover some of
the results from [7], [2] and [36].

##  6\. Fusion estimates and Auxiliary computations

In this last section we provide some auxiliary results to prove the regularity
of the correlation functions.

###  6.1. Proof of Lemma 3.1

The first one corresponds to the fact that the definition we have provided for
regularizing the map $x\mapsto\frac{1}{x^{p}}$ gives the expected result in
the $\varepsilon\rightarrow 0$ limit.

###### Proof of Lemma 3.1.

First assume that we are given $0<\varepsilon<\frac{\left\lvert
x\right\rvert}{4R}$. If $\left\lvert
x+\varepsilon(z_{1}-z_{2})\right\rvert\;\leqslant\;\frac{\left\lvert
x\right\rvert}{2}$, then necessarily either $\left\lvert
z_{1}\right\rvert\;\geqslant\;R$ or $\left\lvert
z_{2}\right\rvert\;\geqslant\;R$, and this implies that
$\eta(z_{1})\eta(z_{2})$ vanishes since $\eta$ is compactly supported in the
domain $B(0,R)$. Therefore we can apply integration by parts and the change of
variables $z_{i}\leftrightarrow\varepsilon z_{i}$ to get

$\frac{1}{(x)_{\varepsilon}^{p}}=\int_{\left\lvert
x+\varepsilon(z_{1}-z_{2})\right\rvert>\frac{\left\lvert
x\right\rvert}{2}}\frac{1}{(x+\varepsilon(z_{1}-z_{2}))^{p}}\eta(z_{1})\eta(z_{2})d^{2}z_{1}d^{2}z_{2}.$

As a consequence

$\frac{x^{p}}{(x)_{\varepsilon}^{p}}-1=\int_{\left\lvert
x+\varepsilon(z_{1}-z_{2})\right\rvert>\frac{\left\lvert
x\right\rvert}{2}}\frac{x^{p}-(x+\varepsilon(z_{1}-z_{2}))^{p}}{(x+\varepsilon(z_{1}-z_{2}))^{p}}\eta(z_{1})\eta(z_{2})d^{2}z_{1}d^{2}z_{2}.$

We also know that, since we have assumed that $0<\varepsilon<\frac{\left\lvert
x\right\rvert}{4R}$, we have the bound
$\left\lvert\frac{\varepsilon(z_{1}-z_{2})}{x}\right\rvert<1$ on the domain
where $\eta(z_{1})\eta(z_{2})$ does not vanish. As a consequence we can expand
the integrand as a power series in the variable
$\frac{\varepsilon(z_{1}-z_{2})}{x}$. Since $\eta$ is compactly supported, the
integral of this power series is absolutely convergent. Therefore, we conclude
by noticing that the first two terms in the expansion vanish in the limit (the
first one is identically zero; for the second order term we use the
$z_{1}\leftrightarrow z_{2}$ symmetry), so that we can factorize by
$\left(\frac{\varepsilon}{x}\right)^{2}$.

Now if we assume that $\left\lvert x\right\rvert\;\leqslant\;4R\varepsilon$
then

$\frac{x^{p}}{(x)_{\varepsilon}^{p}}=\int_{\left\lvert
x+\varepsilon(z_{1}-z_{2})\right\rvert>\frac{\left\lvert
x\right\rvert}{2}}\frac{(\frac{x}{\varepsilon})^{p}}{(\frac{x}{\varepsilon}+(z_{1}-z_{2}))^{p}}\eta(z_{1})\eta(z_{2})d^{2}z_{1}d^{2}z_{2}.$

Since $\left\lvert x\right\rvert\;\leqslant\;4R\varepsilon$, the integrand can
be bounded by the quantity $3^{-p}$, uniformly on $\varepsilon>0$. This allows
to conclude our proof of Lemma 3.1. ∎

###  6.2. Technical estimates

For future convenience we rewrite the correlation functions as

$\left\langle\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}=\left(\prod_{i=1}^{2}\frac{\Gamma(s_{i})\mu_{i}^{-s_{i}}}{\gamma}\right)\prod_{j<k}\left\lvert
z_{j}-z_{k}\right\rvert_{\varepsilon}^{-\langle\alpha_{j},\alpha_{k}\rangle}H_{\varepsilon,\delta}(\bm{z},\bm{\alpha})$

where $s_{i}$ is the one in Equation (2.23) and

(6.1)
$H_{\varepsilon,\delta}(\bm{z},\bm{\alpha})\coloneqq\mathds{E}\left[\prod_{i=1}^{2}\left(\int_{\mathbb{C}}\theta_{\delta}(z_{0}-y_{i})\frac{\hat{g}_{\varepsilon}(y_{i})^{-\frac{\gamma}{4}\sum_{k=1}^{N}\left\langle\alpha_{k},e_{i}\right\rangle}}{\prod_{k=1}^{N}\left\lvert
z_{k}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle\alpha_{k},e_{i}\right\rangle}}M^{\gamma
e_{i},\hat{g}}_{\varepsilon}(d^{2}y_{i})\right)^{-s_{i}}\right].$

We also introduce for finite complex vectors
$\bm{x}_{1}\coloneqq\left(x_{1}^{(1)},\cdots,x_{1}^{(r_{1})}\right),\bm{x}_{2}\coloneqq\left(x_{2}^{(1)},\cdots,x_{2}^{(r_{2})}\right)$
the notation
$H^{(\bm{x}_{1},\bm{x}_{2})}_{\varepsilon,\delta}(\bm{z},\bm{\alpha})$, which
is defined in a way similar to $H_{\varepsilon,\delta}(\bm{z},\bm{\alpha})$
but for the correlation functions where extra Vertex Operators with directions
$\gamma e_{1}$ or $\gamma e_{2}$ are inserted:

$\left\langle\prod_{i=1}^{2}V_{\gamma
e_{i},\varepsilon}(\bm{x}_{i})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\coloneqq\left\langle\prod_{k=1}^{r_{1}}V_{\gamma
e_{1},\varepsilon}\left(x_{1}^{(k)}\right)\prod_{j=1}^{r_{2}}V_{\gamma
e_{2},\varepsilon}\left(x_{2}^{(j)}\right)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}.$

As before, this correlation is the product of a constant, a GFF prefactor and
an expectation $H_{\varepsilon,\delta}(\bm{z},\bm{\alpha})$. The exact
expression for $H_{\varepsilon,\delta}(\bm{z},\bm{\alpha})$ is lengthy to
write down: one should take Equation (6.1) and appropriately modify indices.

####  6.2.1. Bounds on the Toda correlation functions: proof of Lemma 3.2

Let us start by observing that with the $\varepsilon$-regularization, no
singularities come from the prefactor. Moreover, viewed as a function of
$x_{i}^{(l)}$, the quantity $\left\lvert
x_{i}^{(l)}-z_{k}\right\rvert_{\varepsilon}^{-\gamma\left\langle
e_{i},\alpha_{k}\right\rangle}$ can be bounded by some constant times
$\left\lvert 1+x_{i}^{(l)}\right\rvert_{\varepsilon}^{-\gamma\left\langle
e_{i},\alpha_{k}\right\rangle}$. As a consequence this prefactor together with
the constant part can be bounded by

(6.2) $C_{\varepsilon}\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{-\gamma\left\langle\sum\limits_{k}\alpha_{k},e_{1}\right\rangle}\prod_{m=1}^{r_{2}}\left\lvert
1+|x_{2}^{(m)}|\right\rvert_{\varepsilon}^{-\gamma\left\langle\sum\limits_{k}\alpha_{k},e_{2}\right\rangle}\prod_{(i,l)\neq(j,m)}\left\lvert
x_{i}^{(l)}-x_{j}^{(m)}\right\rvert_{\varepsilon}^{-\frac{\gamma^{2}}{2}A_{ij}}.$

For item (1), we need to study the expectation part
$H^{(\bm{x}_{1},\bm{x}_{2})}_{\varepsilon,\delta}$ for large
$\bm{x}_{1},\bm{x}_{2}$. Inside the integrals in the expression
$H^{(\bm{x}_{1},\bm{x}_{2})}_{\varepsilon,\delta}$, we identify singularities
of the form with $i=1,2$:

$\frac{1}{\prod_{k=1}^{N}\left\lvert
z_{k}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle\alpha_{k},e_{i}\right\rangle}}\frac{1}{\prod_{l=1}^{r_{1}}\left\lvert
x_{1}^{(l)}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle
e_{1},e_{i}\right\rangle}}\frac{1}{\prod_{m=1}^{r_{2}}\left\lvert
x_{2}^{(m)}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle
e_{2},e_{i}\right\rangle}}\cdot$

We can factorize out these deterministic factors on $\bm{x}_{1},\bm{x}_{2}$
“at infinity” from the expectation. Then what is left in the expectation is
approximately the part corresponding to $H_{\varepsilon,\delta}$. More
precisely, write

$\frac{1}{\prod_{l=1}^{r_{1}}\left\lvert
x_{1}^{(l)}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle
e_{1},e_{i}\right\rangle}}=\frac{\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{\gamma\left\langle
e_{1},e_{i}\right\rangle}}{\prod_{l=1}^{r_{1}}\left\lvert
x_{1}^{(l)}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle
e_{1},e_{i}\right\rangle}}\cdot\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{-\gamma\left\langle
e_{1},e_{i}\right\rangle}$

and similarly for $\bm{x}_{2}$. Remark that the first factor above goes to $1$
as $x^{(l)}\to\infty$. It follows that, for some positive constant $C$,

$\begin{split}H^{(\bm{x}_{1},\bm{x}_{2})}_{\varepsilon,\delta}&\;\leqslant\;C\prod_{i=1}^{2}\left(\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{-\gamma\left\langle
e_{1},e_{i}\right\rangle}\prod_{m=1}^{r_{2}}\left\lvert
1+|x_{2}^{(m)}|\right\rvert_{\varepsilon}^{-\gamma\left\langle
e_{2},e_{i}\right\rangle}\right)^{-s_{i}}\\\
&\;\leqslant\;C\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{\gamma\left\langle\sum\limits_{k}\alpha_{k}-2Q+\gamma
r_{1}e_{1}+\gamma r_{2}e_{2},e_{1}\right\rangle}\prod_{m=1}^{r_{2}}\left\lvert
1+|x_{2}^{(m)}|\right\rvert_{\varepsilon}^{\gamma\left\langle\sum\limits_{k}\alpha_{k}-2Q+\gamma
r_{1}e_{1}+\gamma r_{2}e_{2},e_{2}\right\rangle}.\end{split}$

Combining with the bound (6.2) for the prefactor yields:

$\begin{split}&\left\langle\prod_{i=1}^{r_{1}}V_{\gamma
e_{1},\varepsilon}\left(x_{1}^{(i)}\right)\prod_{j=1}^{r_{2}}V_{\gamma
e_{2},\varepsilon}\left(x_{2}^{(j)}\right)\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\\\
&\;\leqslant\;C_{\varepsilon}\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{\gamma\left\langle-2Q+\gamma
r_{1}e_{1}+\gamma r_{2}e_{2},e_{1}\right\rangle}\prod_{m=1}^{r_{2}}\left\lvert
1+|x_{2}^{(m)}|\right\rvert_{\varepsilon}^{\gamma\left\langle-2Q+\gamma
r_{1}e_{1}+\gamma
r_{2}e_{2},e_{2}\right\rangle}\prod_{(i,l)\neq(j,m)}\left\lvert
x_{i}^{(l)}-x_{j}^{(m)}\right\rvert_{\varepsilon}^{-\frac{\gamma^{2}}{2}A_{ij}}\\\
&=C_{\varepsilon}\prod_{l=1}^{r_{1}}\left\lvert
1+|x_{1}^{(l)}|\right\rvert_{\varepsilon}^{-4}\prod_{m=1}^{r_{2}}\left\lvert
1+|x_{2}^{(m)}|\right\rvert_{\varepsilon}^{-4}\prod_{(i,l)\neq(j,m)}\left(\frac{\left\lvert
1+x_{i}^{(l)}\right\rvert\left\lvert 1+x_{j}^{(m)}\right\rvert}{\left\lvert
x_{i}^{(l)}-x_{j}^{(m)}\right\rvert_{\varepsilon}}\right)^{\frac{\gamma^{2}}{2}A_{ij}}.\end{split}$

The last term on the right-hand side being bounded this finishes the proof of
item (1).

For item (2), the same reasoning remains valid and therefore the same bounds
are still true when looking at the behaviour near infinity of the correlation
functions when we take the $\varepsilon,\delta\to 0$ limit of the correlation
functions.

For item (3), we investigate the behaviour when $\varepsilon$ goes to zero of
the fusion of two insertions, that is when $\left\lvert
z_{1}-z_{2}\right\rvert\rightarrow 0$ but with all other insertions staying at
fixed positive distance at least $\rho$. Suppose that for both $k=1,2$,
$\left\langle\alpha_{1}+\alpha_{2}-Q,e_{k}\right\rangle<0$. In this case,
$H_{\varepsilon,\delta}$ remains bounded when $z_{1}$ and $z_{2}$ merge (since
in that case the expectation remains well-defined according to [8, Lemma 4.1])
and therefore the behaviour of the correlation functions is governed by the
prefactor, of which the singularity is of order $\left\lvert
z_{1}-z_{2}\right\rvert^{-\left\langle\alpha_{1},\alpha_{2}\right\rangle}$.

The analysis is a bit more subtle when
$\left\langle\alpha_{1}+\alpha_{2},e_{2}\right\rangle\;\geqslant\;\gamma+\frac{2}{\gamma}$
but
$\left\langle\alpha_{1}+\alpha_{2},e_{1}\right\rangle<\gamma+\frac{2}{\gamma}$.
However we can adapt the result in the Liouville case by a simple argument.
Indeed, by Hölder’s inequality we have that for positive $p_{1},p_{2}$ with
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=1$,

$H_{\varepsilon,\delta}\;\leqslant\;\prod_{i=1}^{2}\mathds{E}\left[\left(\int_{\mathbb{C}}\theta_{\delta}(z_{0}-y_{i})\frac{\hat{g}_{\varepsilon}(y_{i})^{-\frac{\gamma}{4}\sum_{k=1}^{N}\left\langle\alpha_{k},e_{i}\right\rangle}}{\prod_{k=1}^{N}\left\lvert
z_{k}-y_{i}\right\rvert_{\varepsilon}^{\gamma\left\langle\alpha_{k},e_{i}\right\rangle}}M^{\gamma
e_{i},\hat{g}}_{\varepsilon}(d^{2}y_{i})\right)^{-p_{i}s_{i}}\right]^{\frac{1}{p_{i}}}.$

We can further ignore the $\theta_{\delta}$ term since $z_{0}$ is away from
the singularities in the integrand. Since
$\left\langle\alpha_{1}+\alpha_{2},e_{1}\right\rangle<\gamma+\frac{2}{\gamma}$
and $s_{1}>0$, the expectation corresponding to the index $i=1$ remains
bounded for any $p_{1}>1$ (see again [8, Lemma 4.1]). The behaviour of the
expectation with index $i=2$ when $z_{1}$ and $z_{2}$ merge is reduced to the
Liouville case (the so-called freezing estimate, see [27, Lemma 6.5]): for any
$\eta>0$,

$\mathds{E}\left[\left(\int_{\mathbb{C}}\frac{\hat{g}_{\varepsilon}(y_{2})^{-\frac{\gamma}{4}\sum_{k=1}^{N}\left\langle\alpha_{k},e_{2}\right\rangle}}{\prod_{k=1}^{N}\left\lvert
z_{k}-y_{2}\right\rvert_{\varepsilon}^{\gamma\left\langle\alpha_{k},e_{2}\right\rangle}}M^{\gamma
e_{2},\hat{g}}_{\varepsilon}(d^{2}y_{2})\right)^{-p_{2}s_{2}}\right]\;\leqslant\;C_{\varepsilon}\left\lvert
z_{1}-z_{2}\right\rvert^{\frac{\left\langle\alpha_{1}+\alpha_{2}-Q,e_{2}\right\rangle^{2}}{2}-\eta}.$

Choosing $p_{2}$ close enough to $1$, we see that the two-point fusion
estimate is governed by

(6.3) $\left\lvert
z_{1}-z_{2}\right\rvert^{-\left\langle\alpha_{1},\alpha_{2}\right\rangle}\left\lvert
z_{1}-z_{2}\right\rvert^{-\eta^{\prime}+\frac{1}{2}\left\langle\alpha_{1}+\alpha_{2}-Q,e_{2}\right\rangle^{2}}.$

Item (4) is a direct consequence of the previous bound.

For item (5): the bound of item (3) ensures local integrability near $z_{1}$
of $\left\langle V_{\gamma e_{1}}(x)\bm{\mathrm{V}}\right\rangle$ in
$L^{p}(\mathbb{C})$ for $1\;\leqslant\;p<\frac{4}{4-\gamma^{2}}$ if we have
$\frac{2}{\gamma}-\gamma\;\leqslant\;\left\langle\alpha_{0},e_{1}\right\rangle<\frac{2}{\gamma}+\gamma$,
for
$1\;\leqslant\;p<\frac{2}{\gamma\left\langle\alpha_{0},e_{1}\right\rangle}$ if
we assume that
$0<\left\langle\alpha_{0},e_{1}\right\rangle<\frac{2}{\gamma}-\gamma$, and in
$L^{\infty}(\mathbb{C})$ if
$\left\langle\alpha_{0},e_{1}\right\rangle\;\leqslant\;0$. ∎

####  6.2.2. A generalized fusion estimate

We now provide a generalized fusion estimate for which we consider the
situation where finitely many points merge within a correlation functions. In
our applications, we only need the special case where the weights of the
merging Vertex Operators are of the form $\gamma e_{i}$, for $i=1,2$, and
where points merge pairwise. This will be the setup that we investigate in
this section.

We assume that $\bm{z}$ are distinct and that $\bm{\alpha}$ are such that the
Seiberg bounds (2.25) hold. Without loss of generality we assume that
$\rho\coloneqq\min\limits_{1\;\leqslant\;k\;\leqslant\;N}\left\lvert
z_{k}\right\rvert$ is positive. We are interested in estimating the following
correlation functions with $2p$ extra points:

$\left\langle\prod_{l=1}^{p}V_{\gamma e_{\sigma(l)}}(x_{l})V_{\gamma
e_{\tau(l)}}(y_{l})\bm{\mathrm{V}}\right\rangle$

where $\sigma(l),\tau(l)\in\\{1,2\\}$ so the extra weights are either $\gamma
e_{1}$ or $\gamma e_{2}$.

To get a reasonable estimate, we need to first reorder the points
$(\bm{x},\bm{y})$ in such a way that the collision can only happen pairwise,
i.e. between each pair $x_{i}$ and $y_{i}$. In short, we restrict the
locations on the points: we assume that $x_{l}$, $y_{l}$ belong to sets
$A_{l}$, $B_{l}$ in such a way that any two of these sets are disjoint, except
for $A_{l}$ and $B_{l}$ with the same indices.

More precisely, we introduce domains $\bm{A}\coloneqq A_{1}\times\cdots\times
A_{p}$ and $\bm{B}\coloneqq B_{1}\times\cdots\times B_{p}$ such that:

  * •

For any $1\;\leqslant\;l\;\leqslant\;p$, $A_{l}$ and $B_{l}$ are either annuli
or balls, and are contained in $B(0,\frac{\rho}{2})$. This is to avoid
collisions between $(\bm{x},\bm{y})$ and the given $\bm{z}$;

  * •

There exists some positive distance $d>0$, for which the
$(A_{l})_{1\;\leqslant\;l\;\leqslant\;p}$ are at distance at least $d$ one to
the other (and similarly for the $(B_{l})_{1\;\leqslant\;l\;\leqslant\;p}$);

  * •

For any $1\;\leqslant\;l,m\;\leqslant\;p$, the distance between $A_{l}$ and
$B_{m}$ is zero if and only if $l=m$.

######  Lemma 6.1.

There exist two positive constants $C$ and $\zeta$ such that, for any
$(\bm{x},\bm{y})\in\bm{A}\times\bm{B}$:

(6.4) $\left\langle\prod_{l=1}^{p}V_{\gamma
e_{\sigma(l)},\varepsilon}(x_{l})V_{\gamma
e_{\tau(l)},\varepsilon}(y_{l})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\;\leqslant\;C\prod_{l=1}^{p}\left\lvert
x_{l}-y_{l}\right\rvert^{-2+\zeta}$

uniformly in $\varepsilon$ and $\delta$. In particular the integral

(6.5)
$\int_{\bm{A}}\int_{\bm{B}}\frac{1}{\prod_{l=1}^{p}(x_{l}-y_{l})}\left\langle\prod_{i=1}^{2}V_{\gamma
e_{i}}(\bm{x}_{i})V_{\gamma e_{i}}(\bm{y}_{i})\bm{\mathrm{V}}\right\rangle
d^{2}\bm{x}d^{2}\bm{y}$

is absolutely convergent.

###### Proof of Lemma 6.1.

The proof of this claim parallels the one in the Liouville case [32, Lemma
3.1]. The study is slightly more involved in the present case because of the
presence of an additional GFF in the construction of the correlation
functions, but the basic idea remains the same.

We separate as before between the prefactor and the expectation term, see
Equation (6.1). Since the domains are disjoint except for $A_{l}$ and $B_{m}$
with $l=m$, the prefactor part is bounded by

$C\prod_{l=1}^{p}\left\lvert x_{l}-y_{l}\right\rvert^{-\gamma^{2}\left\langle
e_{\sigma(l)},e_{\tau(l)}\right\rangle}$

where $C$ is some positive constant.

For the expectation term the analysis is slightly more subtle, but one can
simplify the problem by noticing that when $\sigma(l)\neq\tau(l)$, there is no
singularity in the integral (since the sign in the power is the opposite one).
As a consequence, and without loss of generality, we may assume
$\sigma(l)=\tau(l)=1$ for $l\in E_{1}\coloneqq\\{1,\dots,r\\}$ and
$\sigma(l)=\tau(l)=2$ for $l\in E_{2}\coloneqq\\{r+1,\dots,p\\}$. Therefore
the integrand can be bounded by some constant times

$\mathds{E}\left[\prod_{i=1,2}\left(\int_{\mathbb{C}}\frac{F_{i}(w_{i})}{\prod_{l\in
E_{i}}\left\lvert
x_{l}-w_{i}\right\rvert_{\varepsilon}^{2\gamma^{2}}\left\lvert
y_{l}-w_{i}\right\rvert_{\varepsilon}^{2\gamma^{2}}}M^{\gamma
e_{i},\hat{g}}_{\varepsilon}(d^{2}w_{i})\right)^{-s_{i}}\right]$

where $F_{1}$ and $F_{2}$ are smooth in a neighbourhood of the $x_{l},y_{l}$.
We are now to distinguish between two cases:

  * •

If $\gamma<\sqrt{\frac{2}{3}}$, then when $x_{l}$ and $y_{l}$ merge the
singularity remains integrable since in that case
$4\gamma^{2}<\gamma\left\langle Q,e_{i}\right\rangle$ so the expectation still
makes sense according to [8, Lemma 4.1]. Therefore if
$\gamma<\sqrt{\frac{2}{3}}$, the fusion estimate is governed by the prefactor,
which scales as $|x_{l}-y_{l}|^{-2\gamma^{2}}$ with
$-2\gamma^{2}>-\frac{4}{3}>-2$.

  * •

If $\sqrt{\frac{2}{3}}\;\leqslant\;\gamma<\sqrt{2}$, we claim that, for any
positive $\zeta$, it is bounded by

$\prod_{l=1}^{p}\left\lvert
x_{l}-y_{l}\right\rvert^{\frac{(3\gamma-\frac{2}{\gamma})^{2}}{2}-\zeta}.$

For $\sqrt{\frac{2}{3}}\;\leqslant\;\gamma<\sqrt{2}$, one checks that
$-2\gamma^{2}+\frac{(3\gamma-\frac{2}{\gamma})^{2}}{2}>-2$.

We now prove the last claim following [32, Lemma 3.1]. Using that the
exponents $s_{i}$ are positive, we see that we can bound the expectation by
(some constant times)

$\mathds{E}\left[\left(\sum_{l=1}^{r}\int_{B(x_{l},\frac{d}{4})}\frac{M^{\gamma
e_{1},\hat{g}}_{\varepsilon}(d^{2}w_{1})}{\left\lvert
x_{l}-w_{1}\right\rvert_{\varepsilon}^{2\gamma^{2}}\left\lvert
y_{l}-w_{1}\right\rvert_{\varepsilon}^{2\gamma^{2}}}\right)^{-s_{1}}\left(\sum_{l=r+1}^{p}\int_{B(x_{l},\frac{d}{4})}\frac{M^{\gamma
e_{2},\hat{g}}_{\varepsilon}(d^{2}w_{2})}{\left\lvert
x_{l}-w_{2}\right\rvert_{\varepsilon}^{2\gamma^{2}}\left\lvert
y_{l}-w_{2}\right\rvert_{\varepsilon}^{2\gamma^{2}}}\right)^{-s_{2}}\right].$

We can suppose that the GFFs in different balls $B(x_{l},\frac{d}{4})$ are
independent: this is a classical manipulation of Kahane’s convexity
inequality, since the covariance of different GFFs in different balls is
uniformly bounded from below and above by a global constant (see the proof of
[8, Lemma 4.1]). Furthermore, we use the elementary inequality, for positive
$a$’s and $s>0$,

$\left(\sum\limits_{l=1}^{r}a_{l}\right)^{-s}\;\leqslant\;\prod\limits_{l=1}^{r}a_{l}^{-s/r}$

to reduce the problem to the following estimate: for $s>0$ and any $\zeta>0$
there exists a positive constant $C$ such that

$\mathds{E}\left[\left(\int_{B(x,\frac{d}{4})}\frac{M^{\gamma
e_{1},\hat{g}}_{\varepsilon}(d^{2}w)}{\left\lvert
x-w\right\rvert_{\varepsilon}^{2\gamma^{2}}\left\lvert
y-w\right\rvert_{\varepsilon}^{2\gamma^{2}}}\right)^{-s}\right]\;\leqslant\;C\left\lvert
x-y\right\rvert^{\frac{(3\gamma-\frac{2}{\gamma})^{2}}{2}-\zeta}.$

Since for $\gamma^{2}\;\geqslant\;\frac{2}{3}$,
$4\gamma\;\geqslant\;q=\gamma+\frac{2}{\gamma}$, the classical freezing
estimate [27, Lemma 6.5] yields the bound
$|x-y|^{\frac{(4\gamma-q)^{2}}{2}-\zeta}$ for any $\zeta>0$, which is the
above.

For the second item of Lemma 6.1 we use the first estimate to see that all one
needs to prove is that for positive $\zeta$ the quantity

$\int_{\bm{A}}\int_{\bm{B}}\frac{1}{\prod_{l=1}^{p}\left\lvert
x_{l}-y_{l}\right\rvert^{3-\zeta}}d^{2}\bm{x}d^{2}\bm{y}$

is finite. Since the domains are disjoint we see that it is enough to check
that the two-fold integral

$\int_{B(0,\frac{\rho}{6})}\int_{A(0,\frac{\rho}{6},\frac{\rho}{3})}\frac{1}{\left\lvert
x-y\right\rvert^{3-\zeta}}d^{2}xd^{2}y$

converges. This can be easily seen by using polar coordinates. ∎

###  6.3. Second order differentiability for the correlation functions
(Proposition 3.6)

Recall the expression for the first derivative of the regularized correlation
functions (we keep the same notations and performed an integration by parts
with Stokes’ formula):

$\begin{split}&\partial_{z_{0}}\left\langle
V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}={}-\frac{1}{2}\sum_{k=1}^{N}\frac{\left\langle\alpha_{0},\alpha_{k}\right\rangle}{(z_{0}-z_{k})_{\varepsilon}}\left\langle
V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\\\
&+\mu\sum_{i=1}^{2}\int_{B(z_{0},r)^{c}}\frac{\gamma\left\langle\alpha_{0},e_{i}\right\rangle}{2}\frac{\theta_{\delta}(z_{0}-x)}{(z_{0}-x)_{\varepsilon}}\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x\\\
&+\mu\sum_{i=1}^{2}\oint_{\partial
B(z_{0},r)}\frac{\theta_{\delta}(z_{0}-\xi)}{(z_{0}-\xi)_{\varepsilon}}\left\langle
V_{\gamma
e_{i},\varepsilon}(\xi)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\frac{\sqrt{-1}d\bar{\xi}}{2}\\\
&-\mu\sum_{i=1}^{2}\int_{A}\sum_{k=1}^{N}\frac{\gamma\left\langle\alpha_{k},e_{i}\right\rangle}{2}\frac{\theta_{\delta}(z_{0}-x)}{(z_{k}-x)_{\varepsilon}}\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x\\\
&-\sum_{i,j=1}^{2}\frac{(\mu\gamma)^{2}\left\langle
e_{i},e_{j}\right\rangle}{2}\int_{A}\int_{B(z_{0},r)^{c}}\frac{\theta_{\delta}(z_{0}-x_{1})\theta_{\delta}(z_{0}-x_{2})}{(x_{1}-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}.\end{split}$

where $A=A(z_{0},\delta/2,r)$ is the annulus of radii $\delta/2$ and $r$
centered at $z_{0}$. We have already seen that the correlation functions are
$C^{1}$ with respect to $z_{0}$. It follows that the first three terms above
are $C^{1}$ in $z_{0}$, since the integrand remains smooth as
$\varepsilon,\delta\to 0$ (we stay away from the singular point $z_{0}$).

To prove that the correlation functions are $C^{2}$, we differentiate the two
last terms and prove that the $\varepsilon,\delta\to 0$ limit is finite.
Differentiating the penultimate line with respect to $z_{0}$ yields

$\begin{split}&+\mu\sum_{i=1}^{2}\int_{A}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle}{2(z_{k}-x)_{\varepsilon}}\left(\sum_{l=1}^{N}\frac{\theta_{\delta}(z_{0}-x)\left\langle\alpha_{0},\alpha_{l}\right\rangle}{2(z_{0}-z_{k})_{\varepsilon}}+\frac{\left\langle\alpha_{0},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x)}{2(z_{0}-x)_{\varepsilon}}\right)\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x\\\
&-\mu\sum_{i=1}^{2}\int_{A}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle}{2(z_{k}-x)_{\varepsilon}}\partial_{z_{0}}\theta_{\delta}(z_{0}-x)\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x\\\
&-\mu^{2}\sum_{i,j=1}^{2}\int_{A}\int_{\mathbb{C}}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x_{1})}{2(z_{k}-x_{1})_{\varepsilon}}\frac{\left\langle\alpha_{0},\gamma
e_{j}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(z_{0}-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}\\\
&+\mu^{2}\sum_{i,j=1}^{2}\int_{A}\int_{\mathbb{C}}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x_{1})}{2(z_{k}-x_{1})_{\varepsilon}}\partial_{z_{0}}\theta_{\delta}(z_{0}-x_{2})\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}.\end{split}$

For the terms involving derivatives of $\theta_{\delta}$, we again integrate
them by parts:

$\displaystyle\mu\sum_{i=1}^{2}\oint_{\partial
B(z_{0},r)}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle}{2(z_{k}-\xi)_{\varepsilon}}\theta_{\delta}(z_{0}-\xi)\left\langle
V_{\gamma
e_{i},\varepsilon}(\xi)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}\frac{\sqrt{-1}d\bar{\xi}}{2}+\hskip
156.49014pt$
$\displaystyle\mu\sum_{i=1}^{2}\int_{A}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle}{2(z_{k}-x)_{\varepsilon}}\left(\sum_{l=1}^{N}\frac{\theta_{\delta}(z_{0}-x)\left\langle\alpha_{0},\alpha_{l}\right\rangle}{2(z_{0}-z_{k})_{\varepsilon}}+\frac{\left\langle\alpha_{l},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x)}{2(x-z_{l})_{\varepsilon}}+\frac{\theta_{\delta}(z_{0}-x)}{2(x-z_{k})_{\varepsilon}}\right)\left\langle
V_{\gamma
e_{i},\varepsilon}(x)V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}$
$\displaystyle d^{2}x$
$\displaystyle-\mu^{2}\sum_{i,j=1}^{2}\int_{A}\int_{B(z_{0},r)^{c}}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x_{1})}{2(z_{k}-x_{1})_{\varepsilon}}\times\hskip
256.0748pt$ $\displaystyle\left(\frac{\left\langle\alpha_{0},\gamma
e_{j}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(z_{0}-x_{2})_{\varepsilon}}+\frac{\left\langle\gamma
e_{i},\gamma
e_{j}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(x_{1}-x_{2})_{\varepsilon}}-\partial_{z_{0}}\theta_{\delta}(z_{0}-x_{2})\right)\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}$
$\displaystyle d^{2}x_{2}$
$\displaystyle-\mu^{2}\sum_{i,j=1}^{2}\int_{A^{2}}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x_{1})}{2(z_{k}-x_{1})_{\varepsilon}}\times\hskip
284.52756pt$ $\displaystyle\left(\frac{\left\langle\alpha_{0},\gamma
e_{j}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(z_{0}-x_{2})_{\varepsilon}}+\frac{\left\langle\gamma
e_{i},\gamma
e_{j}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(x_{1}-x_{2})_{\varepsilon}}-\partial_{z_{0}}\theta_{\delta}(z_{0}-x_{2})\right)\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}$
$\displaystyle d^{2}x_{2}.$

All but the two last lines remain bounded as $\varepsilon,\delta\rightarrow 0$
thanks to Lemmas 3.2 and 6.1.

Using again integration by parts for the last line we get regular terms plus:

$\displaystyle-\mu^{2}\sum_{i,j=1}^{2}\int_{A^{2}}\sum_{k,l=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x_{1})}{2(z_{k}-x_{1})_{\varepsilon}}$
$\displaystyle\frac{\left\langle\alpha_{k},\gamma
e_{j}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(z_{k}-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}$
$\displaystyle+\mu^{3}\sum_{i,j,l=1}^{2}\int_{A^{2}}\int_{\mathbb{C}}\sum_{k=1}^{N}\frac{\left\langle\alpha_{k},\gamma
e_{i}\right\rangle\theta_{\delta}(z_{0}-x_{1})}{2(z_{k}-x_{1})_{\varepsilon}}$
$\displaystyle\frac{\left\langle\gamma e_{j},\gamma
e_{l}\right\rangle\theta_{\delta}(z_{0}-x_{2})}{2(x_{3}-x_{2})_{\varepsilon}}$
$\displaystyle\left\langle V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\gamma
e_{l},\varepsilon}(x_{3})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}d^{2}x_{3}.$

Using the symmetries in the $x_{2},x_{3}$ variables, the last integral
vanishes on $A^{3}$. This means that this term remains bounded when
$\varepsilon,\delta\rightarrow 0$ by using Lemma 6.1.

To finish up with the proof that the correlation functions are $C^{2}$ it
remains to take care of the derivative of the term

$-\sum_{i,j=1}^{2}\frac{(\mu\gamma)^{2}\left\langle
e_{i},e_{j}\right\rangle}{2}\int_{A}\int_{B(z_{0},r)^{c}}\frac{\theta_{\delta}(z_{0}-x_{1})\theta_{\delta}(z_{0}-x_{2})}{(x_{1}-x_{2})_{\varepsilon}}\left\langle
V_{\gamma e_{i},\varepsilon}(x_{1})V_{\gamma
e_{j},\varepsilon}(x_{2})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}.$

We proceed in the same way by using integration by parts to get rid of the
terms involving derivatives of $\theta_{\delta}$. We may follow the same lines
as in the previous computation, apart from the fact that the new integration
domains we will consider either contain or don’t the previous integration
variables. We won’t write down the details here since they are quite lengthy
and not informative, nonetheless we see that in the end it is enough to show
that the integrals (here $\rho$ is some positive number)

$\int_{A^{2}}\int_{B(z_{0},r)^{c}}\frac{\mathds{1}_{\left\lvert
x_{1}-x_{2}\right\rvert>\rho}}{(x_{1}-x_{3})(x_{2}-x_{3})}\left\langle\prod\limits_{k=1}^{3}V_{\gamma
e_{\star},\varepsilon}(x_{k})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}d^{2}x_{3}$

and

$\int_{A^{2}}\int_{(B(z_{0},r)^{c})^{2}}\frac{\mathds{1}_{\left\lvert
x_{1}-x_{2}\right\rvert>\rho}\mathds{1}_{\left\lvert
x_{3}-x_{4}\right\rvert>\rho}}{(x_{1}-x_{3})(x_{2}-x_{4})}\left\langle\prod\limits_{k=1}^{4}V_{\gamma
e_{\star},\varepsilon}(x_{k})V_{\alpha_{0},\varepsilon}(z_{0})\bm{\mathrm{V}}_{\varepsilon}\right\rangle_{\delta}d^{2}x_{1}d^{2}x_{2}d^{2}x_{3}d^{2}x_{4}$

are absolutely convergent, which follows from Lemma 6.1. Therefore the
$\frac{\partial^{2}}{\partial z_{0}^{2}}$ derivative of the correlation
functions are well-defined. Treating the mixed derivatives
$\frac{\partial^{2}}{\partial z_{0}\partial\bar{z}_{0}}$ leads to the same
conclusion. Hence existence of both the $\frac{\partial^{2}}{\partial
z_{0}^{2}}$ and $\frac{\partial^{2}}{\partial z_{0}\partial\bar{z}_{0}}$
derivatives: this implies that the correlation functions are $C^{2}$.

######  Remark 6.2.

The same reasoning shows that the correlation functions are smooth, in
conjunction with Lemma 6.1. This is in agreement with the Liouville case as
shown in [32].

### Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated
or analysed.

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