# Irreducibility of Virasoro representations in Liouville CFT

Guillaume Baverez  Humboldt-Universität zu Berlin, Institut für Mathematik,
Rudower Chaussee 25, 12489 Berlin, Deutschland<EMAIL_ADDRESS>and  Baojun Wu  Beijing International Center for Mathematical Research (BICMR)
<EMAIL_ADDRESS>
###### Abstract.

In the context of Liouville conformal field theory, we construct the highest-
weight representations of the Virasoro algebra at the degenerate values of the
conformal weight (Kac table). We show that these modules are irreducible,
giving a complete characterisation of the algebraic structure of Liouville
CFT. It also implies that all singular vectors vanish, which is one of the
main asumptions usually made in theoretical physics. Our proof uses inputs
from both probability theory and algebra, and gives new probabilistic content
to the Kac table.

Combining the information that singular vectors vanish with the main geometric
properties of conformal blocks, we deduce that conformal blocks involving
degenerate primary fields satisfy null-vector equations. These equations take
the form of PDEs on the Teichmüller space of the underlying surface and
generalise previous works in several directions.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation and background
    2. 1.2 Main results
    3. 1.3 Proof outline
    4. 1.4 Outlook
  2. 2 Background
    1. 2.1 Gaussian fields and Liouville Hilbert space
    2. 2.2 Semigroups
    3. 2.3 Feigin-Fuchs modules
  3. 3 Singular modules
    1. 3.1 Notations
    2. 3.2 Preliminaries
    3. 3.3 Meromorphic continuation
    4. 3.4 Study on the Kac table
    5. 3.5 Descendant states vs. singular integrals
    6. 3.6 Conclusion
  4. 4 BPZ equations
    1. 4.1 Teichmüller spaces and the semigroup of annuli
    2. 4.2 Coordinates and deformations
    3. 4.3 Line bundles over $\mathcal{T}_{\Sigma,m+1}$
    4. 4.4 Conformal blocks and proof of Theorem 1.3
  5. 5 Differential operators on $\mathcal{T}_{\Sigma,m+1}$
    1. 5.1 The Riemann sphere
    2. 5.2 Elliptic curves
    3. 5.3 Hyperbolic surfaces
  6. A Martingales of the Ornstein-Uhlenbeck process
  7. B The Virasoro algebra

##  1\. Introduction

###  1.1. Motivation and background

Ever since the groundbreaking work of Belavin, Polyakov and Zamolodchikov
[BPZ84], two-dimensional conformal field theory (CFT) has been rooted in the
representation theory of the Virasoro algebra. The main idea is that conformal
symmetry – as an infinite-dimensional Lie group – is so constraining that it
should be possible to classify all CFTs from algebraic principles and
consistency conditions. This has been extremely successful in physics for the
following reason: given a model of statistical mechanics at criticality
assumed to possess a conformally invariant scaling limit, it should be
possible to identify its corresponding CFT from a few basic properties of the
model. On the other hand, it raises some problems from the mathematical point
of view: indeed, it is usually very difficult to prove that a given CFT is
fully consistent since the CFT axioms are typically over-constraining.

The main objects in the algebraic structure of CFT are the highest-weight
modules for the Virasoro algebra, which are well-understood by standard
representation theory. These modules depend on two parameters: the _central
charge_ $c_{\mathrm{L}}\in\mathbb{C}$ (which is usually fixed) and the
_conformal weight_ $\alpha\in\mathbb{C}$. The corresponding Verma module is
irreducible except for a discrete set of values of $\alpha$ known as the _Kac
table_ (see Appendix B). A typical axiom in CFT is that all representations in
the spectrum are irreducible so that certain linear relations hold for
descendant fields in a degenerate module (i.e. a module with a conformal
weight in the Kac table). This information is used to show that a correlation
function involving a degenerate field solves a certain PDE. These PDEs are now
collectively known as the _null-vector equations_ or _BPZ equations_
(following the founding work [BPZ84]). They have been extensively used in the
physics literature to solve the minimal models and the Liouville CFT (see
[DFMS97, Rib14] for reviews).

One way to mathematically answer the question of consistency is to construct
an explicit representation of the algebraic structure of a given CFT. For the
Liouville CFT, this program has been initiated in a series of works by David,
Guillarmou, Kupiainen, Rhodes & Vargas [DKRV16, DRV16, GRV19], where the path
integral has been constructed on surfaces of arbitrary topology. A subset of
all BPZ equations has been established in this framework, and they were key
inputs into the proof of the “DOZZ formula” [KRV19, KRV20], which is the exact
formula for the three-point correlation function on the Riemann sphere.
Following suit, these equations were used – together with the mating-of-trees
theory [DMS14] – to prove similar results in the framework of boundary LCFT
[RZ20, ARS21, ARSZ23]. Recently, some important steps have been made towards a
full understanding of the algebraic structure of this CFT. The spectrum was
computed in [GKRV23], and the Virasoro representation was constructed
probabilistically as a family of unbounded operators on the Hilbert space
[BGK+23b]. The authors also defined the highest-weight modules for all values
of the conformal weight _except_ the Kac table. One of the contributions of
this work is to extend this result to the Kac table and show that these
modules are irreducible. Let us stress that the irreducibility is taken as an
axiom in physics, while it is a theorem in our given representation. As a
consequence, we deduce null-vector equations for conformal blocks on surfaces
of arbitrary topologies, and for all values of the Kac table. This generalises
previous results in three different directions [KRV19, Zhu20, GRSS20].

###  1.2. Main results

Here and in the sequel, $\gamma$ and $\mu$ are parameters satisfying

$\gamma\in(0,2);\qquad\mu>0.$

We will sometimes allow $\mu$ to take complex values, provided
$\mathrm{Re}(\mu)>0$. We also set,

$Q:=\frac{\gamma}{2}+\frac{2}{\gamma}>2;\qquad c_{\mathrm{L}}:=1+6Q^{2}>25.$

Given $r,s\in\mathbb{N}^{*}=\mathbb{Z}_{>0}$, we set

$\alpha_{r,s}:=(1-r)\frac{\gamma}{2}+(1-s)\frac{2}{\gamma},$

and for all $\alpha\in\mathbb{C}$:

$\Delta_{\alpha}:=\frac{\alpha}{2}\left(Q-\frac{\alpha}{2}\right).$

Observe that $\Delta_{\alpha}=\Delta_{2Q-\alpha}$.

####  1.2.1. The Poisson operator

Let $\mathcal{F}=\mathbb{C}[(\varphi_{n})_{n=1}^{\infty}]$ be the space of
polynomials in countably many complex variables. A sequence
$\mathbf{k}=(k_{n})_{n=1}^{\infty}$ of non-negative integers with only
finitely many non-zero terms will be called an _(integer) partition_ : it
represents a partition of the integer
$|\mathbf{k}|:=\sum_{n=1}^{\infty}nk_{n}$, also called the _level_ of
$\mathbf{k}$. The _length_ of $\mathbf{k}$ is
$\ell(\mathbf{k}):=\sum_{n=1}^{\infty}k_{n}$. The set of all partitions (resp.
partitions of level $N$) is denoted $\mathcal{T}$ (resp. $\mathcal{T}_{N}$),
and we set $p(N):=\\#\mathcal{T}_{N}$. By convention, $p(0)=1$ and the unique
element of $\mathcal{T}_{0}$ is called the _empty partition_.

Given $\mathbf{k}\in\mathcal{T}$, we set
$\varphi^{\mathbf{k}}:=\prod_{n=1}^{\infty}\varphi_{n}^{k_{n}}\in\mathcal{F}$
and we say that $\varphi^{\mathbf{k}}$ is at level $|\mathbf{k}|$. We let
$\delta_{n}=(0,...,0,1,0...)$ be the partition of $n$ coding the monomial
$\varphi_{n}$, so that the monomial $\varphi^{\mathbf{k}}$ is coded by
$\sum_{n=1}^{\infty}k_{n}\delta_{n}$. We have a grading of $\mathcal{F}$ by
the level

$\mathcal{F}=\oplus_{N\in\mathbb{N}}\mathcal{F}_{N},$

with $\mathcal{F}_{N}$ spanned by the monomials at level $N$.

One can define an infinite measure
$\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$ on the Sobolev space
$H^{-1}(\mathbb{S}^{1})$, and the _Liouville Hilbert space_ is
$\mathcal{H}=L^{2}(\mathrm{d}c\otimes\mathbb{P}_{S^{1}})$. Here,
$\mathbb{P}_{S^{1}}$ is a Gaussian measure on the space of fields with
vanishing mean, and $\mathrm{d}c$ is a Lebesgue measure on $\mathbb{R}$ (see
Section 2.1). Functions (or distributions) with vanishing mean on the circle
are written in Fourier modes
$\varphi(e^{i\theta})=\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\varphi_{n}e^{ni\theta}$,
with $\varphi_{-n}=\bar{\varphi}_{n}$. This allows us to identify
$\mathcal{F}$ as a linear subspace of $L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$.

Acting on $\mathcal{H}$ are two positive, essentially self-adjoint operators
$\mathbf{H}^{0}$, $\mathbf{H}$, called respectively the _free field and
Liouville Hamiltonians_. The Liouville Hamiltonian is of the form
$\mathbf{H}=\mathbf{H}^{0}+\mu e^{\gamma c}V(\varphi)$, where $V(\varphi)$ is
a certain potential. The quadratic form associated with $\mathbf{H}$ is
denoted $\mathcal{Q}$, and its domain $\mathcal{D}(\mathcal{Q})$. The
Liouville Hamiltonian defines a bounded operator
$\mathbf{H}:\mathcal{D}(\mathcal{Q})\to\mathcal{D}^{\prime}(\mathcal{Q})$,
where $\mathcal{D}^{\prime}(\mathcal{Q})$ is the topological dual of
$\mathcal{D}(\mathcal{Q})$. This operator and its quadratic form were
extensively studied in [GKRV23, Section 5]. Both operators $\mathbf{H}^{0}$,
$\mathbf{H}$ generate contraction semi-groups
$(e^{-t\mathbf{H}^{0}})_{t\;\geqslant\;0}$,
$(e^{-t\mathbf{H}})_{t\;\geqslant\;0}$ or _heat flows_. The operator
$\mathbf{H}^{0}$ is easy to diagonalise [GKRV23, Section 4], and the function
$e^{(\alpha-Q)c}\varphi^{\mathbf{k}}$ is a generalized eigenstate of
$\mathbf{H}^{0}$ with eigenvalue $2\Delta_{\alpha}+|\mathbf{k}|$, for each
$\alpha\in\mathbb{C}$ and partition $\mathbf{k}$.

One of the main inputs of [GKRV23] is to diagonalise $\mathbf{H}$. It is shown
in Sections 6-7 that the long-time asymptotic of the heat flow of $\mathbf{H}$
can produce generalized eigenstates. More precisely, for all
$\chi\in\mathcal{F}_{|\mathbf{k}|}$ and $\alpha$ in a neighborhood of
$-\infty$ (depending on the level of $\chi$, see [GKRV23, Proposition 7.2]),
we have the following limit in a weighted space $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$:

$\mathcal{P}_{\alpha,|\mathbf{k}|}(\chi)=\underset{t\to\infty}{\lim}\,e^{t(2\Delta_{\alpha}+|\mathbf{k}|)}e^{-t\mathbf{H}}\left(\chi
e^{(\alpha-Q)c}\right),$

and the map $\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is analytic in this
region. The region provided in [GKRV23, Proposition 7.2] gets smaller as the
level of the polynomial increases. For each $\alpha$,

$\mathcal{P}_{\alpha}=\oplus_{N\in\mathbb{N}}\mathcal{P}_{\alpha,N}:\mathcal{F}\to
e^{-\beta c}\mathcal{D}(\mathcal{Q})$

is a linear map and is called the _Poisson operator_ , where
$\mathcal{P}_{\alpha,N}$ is the restriction of $\mathcal{P}_{\alpha}$ to the
(finite-dimensional) space $\mathcal{F}_{N}$.

It is easy to deduce from [GKRV23, BGK+23b] that the Poisson operator extends
meromorphically to all $\alpha\in\mathbb{C}$ with values in $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$such that $\beta>Q-\mathrm{Re}(\alpha)$. Our first
result is to discard the possibility of poles.

######  Theorem 1.1.

For each $\chi\in\mathcal{F}$, the map
$\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is analytic on the whole
$\alpha$-plane, i.e. it has no poles.

This theorem is proved in Section 3, and an overview of the main steps of the
proof is given in Section 1.3.

####  1.2.2. Free field modules and Liouville modules

For each $\alpha\in\mathbb{C}$, the _Sugawara construction_ is a
representation $(\mathbf{L}_{n}^{0,\alpha})_{n\in\mathbb{Z}}$ of the Virasoro
algebra on $\mathcal{F}$. Given a partition
$\nu=(\nu_{1},...,\nu_{\ell})\in\mathcal{T}$ of length $\ell$ (see Remark 1),
we write
$\mathbf{L}_{-\nu}=\mathbf{L}_{-\nu_{\ell}}^{0,\alpha}...\mathbf{L}_{-\nu_{2}}^{0,\alpha}\mathbf{L}_{-\nu_{1}}^{0,\alpha}$.
The _Free Field module_ $\mathcal{V}_{\alpha}^{0}$ is the highest-weight
Virasoro module generated by the Virasoro descendants of the vacuum vector,
namely

$\mathcal{V}_{\alpha}^{0}:=\mathrm{span}\,\\{\mathbf{L}_{-\nu}^{0,\alpha}\mathds{1}|\,\nu\in\mathcal{T}\\},$

where $\mathds{1}$ denotes the constant function 1. Here, the span is
algebraic, meaning we take finite linear combinations. This module has central
charge $c_{\mathrm{L}}$ and highest-weight $\Delta_{\alpha}$ (note that
$\Delta_{2Q-\alpha}=\Delta_{\alpha}$). The free field module can be identified
with the image of the canonical map from Verma module to the _Feigin-Fuchs
module_. Roughly speaking, the Feigin-Fuchs module is the Fock space, but we
consider it as a Virasoro module, for explicit definition, see Section 2.3.
The classification of Feigin-Fuchs modules is known [Fre92] and reviewed in
Section 2.3. Briefly, for $\mathrm{Re}(\alpha)<Q$, the module
$\mathcal{V}_{2Q-\alpha}^{0}$ is always Verma (hence
$\mathcal{V}_{2Q-\alpha}^{0}\simeq\mathcal{F}$ linearly), while
$\mathcal{V}_{\alpha}^{0}$ can be the quotient of the Verma by a non-trivial
submodule. Thus, there is a canonical projection
$\Phi_{\alpha}^{0}:\mathcal{V}_{2Q-\alpha}^{0}\to\mathcal{V}_{\alpha}^{0}$,
well-defined and analytic in the region $\mathrm{Re}(\alpha)<Q$. We then
define
$\Phi_{\alpha}:=\mathcal{P}_{\alpha}\circ\Phi_{\alpha}^{0}:\mathcal{F}\to
e^{-\beta c}\mathcal{D}(\mathcal{Q})$, and set the _Liouville module_

$\mathcal{V}_{\alpha}:=\mathrm{ran}\,\Phi_{\alpha}\simeq\mathcal{F}/\ker(\Phi_{\alpha}).$
(1.1)

If we don’t assume Theorem 1.1, this expression is only well-defined on the
domain of analyticity of $\mathcal{P}_{\alpha}$. However, $\Phi_{\alpha}$ has
an explicit expression in terms of generalised eigenstates of the Liouville
Hamiltonian, and this expression was proved to extend analytically to the
whole region $\\{\mathrm{Re}(\alpha)\;\leqslant\;Q\\}$. Moreover, the
reflection relation $\Phi_{2Q-\alpha}=R(\alpha)\Phi_{\alpha}$ defines an
analytic extension to the whole $\alpha$-plane [GKRV23, BGK+23b], where
$R(\alpha)$ is the _reflection coefficient_ [KRV20]. Due to the analyticity of
$\Phi_{\alpha}$, we get that $\mathcal{P}_{\alpha}$ extends meromorphically to
the region $\\{\mathrm{Re}(\alpha)<Q\\}$, and its poles may only occur on the
zeros of $\Phi_{\alpha}^{0}$. By standard representation theory, these zeros
are contained in the _Kac table_ :

  * •

$kac=\\{\alpha\mid\Delta_{\alpha}=\Delta_{\alpha_{r,s}}\text{ for some
}r,s\in\mathbb{N}^{*}\\}$

  * •

$kac^{-}=\\{\alpha\mid\alpha=\alpha_{r,s}\text{ for some
}r,s\in\mathbb{N}^{*}\\}$

  * •

$kac^{+}=\\{\alpha\mid\alpha=2Q-\alpha_{r,s}\text{ for some
}r,s\in\mathbb{N}^{*}\\}$,

where we recall $\alpha_{r,s}=(1-r)\frac{\gamma}{2}+(1-s)\frac{2}{\gamma}$. It
is easy to observe that $kac=kac^{+}\sqcup kac^{-}$. Moreover, we have
$kac^{-}\subset(-\infty,0]$, while $kac^{+}=2Q-kac^{-}\subset[2Q,\infty)$.
Thus, $\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is meromorphic on $\mathbb{C}$
with possible poles contained in $kac^{-}$, for all $\chi\in\mathcal{F}$.

The next result is that $\mathcal{V}_{\alpha}$ is the highest-weight Virasoro
module, and we give a complete classification. For reader’s convenience, we
gather the notation of all modules we need in this paper.

Notations of modules

  * •

Verma module $M(c_{\mathrm{L}},\Delta_{\alpha})$, we refer to Appendix B for
terminology and notations;

  * •

Irreducible (quotient) module $V(c_{\mathrm{L}},\Delta_{\alpha})$, we refer to
Appendix B for terminology and notations;

  * •

Feigin-Fuchs module $\mathcal{F}(c_{\mathrm{L}},\alpha)$ is the Fock space
$\mathcal{F}$ as a Virasoro module, usually written as $\mathcal{F}$ if there
is no ambiguity;

  * •

Free Field module
$\mathcal{V}_{\alpha}^{0}:=\mathrm{span}\,\\{\mathbf{L}_{-\nu}^{0,\alpha}\mathds{1}|\,\nu\in\mathcal{T}\\}$,

  * •

Liouville module
$\mathcal{V}_{\alpha}:=\mathrm{ran}\,\Phi_{\alpha}\simeq\mathcal{F}/\ker(\Phi_{\alpha})$.

We will prove this theorem in Section 3.6, simultaneously with Theorem 1.1.

######  Theorem 1.2.

For $\alpha\in\mathbb{C}$. The space $\mathcal{V}_{\alpha}$ defined in (1.1)
is the highest-weight representation of the Virasoro algebra. Moreover, it has
the following structure:

  * •

For $\alpha\notin kac$, $\mathcal{V}_{\alpha}$ is isomorphic to the Verma
module $M(c_{\mathrm{L}},\Delta_{\alpha})$ and it is irreducible.

  * •

For $\alpha\in kac^{-}$, $\mathcal{V}_{\alpha}$ is isomorphic to the
irreducible quotient $V(c_{\mathrm{L}},\Delta_{\alpha})$ of the Verma module
$M(c_{\mathrm{L}},\Delta_{\alpha})$ by the maximal proper submodule.

  * •

For $\alpha\in kac^{+}$, $\mathcal{V}_{\alpha}$ is isomorphic to
$V(c_{\mathrm{L}},\Delta_{\alpha})$ when
$\alpha\notin(Q+\frac{2}{\gamma}\mathbb{N}^{*})\cup(Q+\frac{\gamma}{2}\mathbb{N}^{*})$
and $\mathcal{V}_{\alpha}=0$ when
$\alpha\notin(Q+\frac{2}{\gamma}\mathbb{N}^{*})\cup(Q+\frac{\gamma}{2}\mathbb{N}^{*})$.

Only the second item is new. Indeed, the first item was done in [BGK+23b],
while the third corresponds to the reflection formula stated above. The
reflection coefficient has zeros on
$\frac{\gamma}{2}\mathbb{N}^{*}\cup\frac{2}{\gamma}\mathbb{N}^{*}$, which is
why these values are excluded. The Virasoro representation
$(\mathbf{L}_{n})_{n\in\mathbb{Z}}$ on $\mathcal{V}_{\alpha}$ is the one
constructed probabilistically in [BGK+23b], and the Poisson operator has the
property of intertwining the free field representation
$(\mathbf{L}_{n}^{0,\alpha})$ with the Liouville one.

The relevance of this theorem is two-fold. On the fundamental level, it gives
a complete classification of the algebraic structure of Liouville CFT. On the
practical level, we get non-trivial relations between Liouville eigenstates
for $\alpha$ in the Kac table, which are precisely the relations that are
exploited through the BPZ equations. In a highest-weight representation, a
_singular vector_ is one which is both a descendant and a highest-weight
vector. In particular, any non-zero singular vector generates a non-trivial
proper submodule. Theorem 1.2 implies that for each
$\alpha_{r,s}=Q-\frac{\gamma}{2}r-\frac{2}{\gamma}s$,
$\mathcal{V}_{\alpha_{r,s}}$ does not have any non-zero singular vector. On
the other hand, the corresponding Verma module does have singular vectors, so
the canonical projection maps singular vectors to zero. With the notation from
the previous section, this means that there exists $\chi\in\mathcal{F}_{rs}$
such that $\Phi_{\alpha_{r,s}}(\chi)=0$. Thus, there is a non-trivial linear
combination

$\mathbf{S}_{\alpha_{r,s}}=\sum_{|\nu|=rs}\sigma_{\nu}^{r,s}\mathbf{L}_{-\nu}$
(1.2)

such that

$\mathbf{S}_{\alpha_{r,s}}\Psi_{\alpha_{r,s}}=0.$

Moreover, the coefficients $(\sigma_{\nu}^{r,s})_{|\nu|=rs}$ are universal in
the sense that they only depend on $c_{\mathrm{L}}$ and $\alpha_{r,s}$. At
level two, standard representation theory gives
$(\alpha^{2}\mathbf{L}_{-2}+\mathbf{L}_{-1}^{2})\Psi_{\alpha}=0$, for
$\alpha\in\\{\alpha_{1,2},\alpha_{2,1}\\}$.

We stress that it would be quite difficult to prove such relations by analytic
means, even for the level 2 above. In principle, one can find explicit
formulae for the descendants (see the proof of Theorem 1.1), but the
combinatorics soon become intractable. Instead, Theorem 1.2 gives us all the
relations at once for all $r,s\in\mathbb{N}^{*}$.

####  1.2.3. BPZ equations

The vanishing of singular vectors implies that conformal blocks evaluated on
primary fields at the degenerate weights satisfy certain PDEs on Teichmüller
space.

Let $\Sigma$ be a compact surface of genus $\mathbf{g}$, $m\in\mathbb{N}$, and
$\mathcal{T}_{\Sigma,m+1,\uparrow}^{\epsilon}$ be the Teichmüller space of
$\Sigma$ defined in Section 4.1. A point in this Teichmüller space is a
Riemann surface $(\Sigma,J)$ with $m+1$ distinct marked points
$\boldsymbol{p}=(p_{0},...,p_{m})\in\Sigma^{m+1}$, non-zero tangent vectors
$(\xi_{0},...,\xi_{m})\in\prod_{j=0}^{m}T_{p_{j}}\Sigma$ at the marked points,
and conformal embeddings of discs
$\boldsymbol{\psi}=(\psi_{0},...,\psi_{m}):e^{\epsilon}\mathbb{D}\to\Sigma$
such that $\psi_{j}(0)=p_{j}$,
$(\mathrm{d}\psi_{j})_{0}(\partial_{z})=\xi_{j}$ (with $z$ the standard
coordinate on $\mathbb{D}$), and the images $\psi_{j}(\bar{\mathbb{D}})$ are
pairwise disjoint. Note that $z_{j}:=\psi_{j}^{-1}$ defines a local coordinate
at $p_{j}$ vanishing there. Upon cutting out the discs, we can think of a
point in $\mathcal{T}_{\Sigma,m+1,\uparrow}^{\epsilon}$ as the gluing of a
Riemann surface with $m+1$ boundary components and $m+1$ discs.

In the constructions of [GKRV21, BGK+23a], conformal blocks depend a priori on
a choice of decomposition of the surface into complex pairs of pants, and on
the choice of disc embeddings/local coordinate at the marked points. The
dependence is mild, and can be encoded into certain line bundles over
$\mathcal{T}_{\Sigma,m+1}$. Namely, one can define holomorphic line bundles
$\mathscr{L}^{c_{\mathrm{L}}}$ and $\mathscr{L}^{\boldsymbol{\Delta}}$ over
$\mathcal{T}_{\Sigma,m+1}$ with connections $\nabla^{c_{\mathrm{L}}}$ and
$\nabla^{\boldsymbol{\Delta}}$ respectively. These definitions are detailed in
Section 4.3.

As recalled in Section 4.2, an element
$\mathbf{v}\in\mathrm{Vect}^{\epsilon}(\mathbb{S}^{1})$ generates a
deformation in $\mathcal{T}_{\Sigma,m+1}$, i.e. it represents a tangent vector
there. Using the embedding $\psi_{0}$, $\mathbf{v}$ is viewed as a holomorphic
vector field in a neighbourhood of the loop $\psi_{0}(\mathbb{S}^{1})$
surrounding the marked point $p_{0}$ (we always view the deformation from
$p_{0}$). There are two possible cases in which the induced deformation is
trivial: if either $\mathbf{v}$ extends holomorphically to $\mathbb{D}$, or
$(\psi_{0})_{*}\mathbf{v}$ extends meromorphically to the whole surface, with
only possible poles at the punctures. Hence, the tangent spaces to
$\mathcal{T}_{\Sigma,m+1}$ are equivalence classes of vector fields in
$\mathbf{v}\in\mathrm{Vect}^{\epsilon}(\mathbb{S}^{1})$, denoted
$[\mathbf{v}]$. Moreover, we have $m$ extra vector fields $\partial_{z_{j}}$
on $\mathcal{T}_{\Sigma,m+1}$ generating the motion of the marked points. Note
that $\partial_{z_{j}}$ fits in the above framework since it corresponds to
the constant vector field at the marked point $p_{j}$. This description of the
deformation space of Teichmüller space is called “Virasoro uniformisation” and
is reviewed in Section 4.2.2.

Given $\mathbf{v}\in\mathrm{Vect}^{\epsilon}(\mathbb{S}^{1})$, we can then
define a differential operators on $\mathcal{T}_{\Sigma,m+1}$ as the
connection operator in the direction $[\mathbf{v}]$:

$\mathcal{L}_{\mathbf{v}}:=\nabla_{[\mathbf{v}]}.$

In particular, we set $\mathcal{L}_{n}:=\mathcal{L}_{z^{n+1}\partial_{z}}$.
Since $[z^{n+1}\partial_{z}]=0$ for $n\;\geqslant\;0$, only the operators with
$n\;\leqslant\;-1$ are non-trivial. Moreover, we have

$\mathcal{L}_{-1}^{(j)}:=\nabla_{\partial_{z_{j}}}=\partial_{z_{j}}.$

Concretely, if we fix a holomorphic frame
$(\partial_{\tau_{k}})_{1\;\leqslant\;k\;\leqslant\;3\mathbf{g}-3}$ on
$\mathcal{T}_{\Sigma}$ obtained (say) from plumbing coordinates (see Section
4.2.1), there are some holomorphic coefficients
$(\varpi_{k},\varpi_{-1,j},\varpi_{0,j})_{1\;\leqslant\;k\;\leqslant\;3\mathbf{g}-3,\,0\;\leqslant\;j\;\leqslant\;m}$
on $\mathcal{T}_{\Sigma,m+1}$ such that

$\mathcal{L}_{\mathbf{v}}=\sum_{k=1}^{3\mathbf{g}-3}\varpi_{k}\partial_{\tau_{k}}+\sum_{j=0}^{m}\left(\varpi_{-1,j}\partial_{z_{j}}+\Delta_{j}\varpi_{0,j}\right).$
(1.3)

These coefficients encode the change of basis between Virasoro uniformisation
and more standard coordinates on Teichmüller space. The computation is
explicit in genus $\mathbf{g}\;\leqslant\;1$ (See Section 5). Finally, for a
partition $\nu=(\nu_{1},...,\nu_{\ell})\in\mathcal{T}$, we define

$\mathcal{L}_{-\nu}:=\mathcal{L}_{-\nu_{\ell}}...\mathcal{L}_{-\nu_{2}}\mathcal{L}_{-\nu_{1}},$

which is a differential operator of order $\ell$ on
$\mathcal{T}_{\Sigma,m+1}$.

From Theorem 1.3, we have the existence of a non-trivial relation
$\mathbf{S}_{\alpha_{r,s}}\Psi_{\alpha_{r,s}}=0$ at level $rs$, for all
$r,s\in\mathbb{N}^{*}$, where the operator $\mathbf{S}_{\alpha_{r,s}}$ has
been defined in (1.2). We can associate to this operator a differential
operator

$\mathcal{S}_{\alpha_{r,s}}=\sum_{|\nu|=rs}\sigma_{\nu}^{r,s}\mathcal{L}_{-\nu}$

of order $rs$ on Teichmüller space. We stress that this operator depends on
the topological type of the surface (genus and number of marked points), while
the operator $\mathbf{S}_{\alpha_{r,s}}$ is universal.

In the probabilistic framework, conformal blocks have been introduced in
[GKRV23, GKRV21] and studied intrinsically in [BGK+23a]. Here, we will only be
needing their main geometric properties, which are recalled in Definition 4.1.
Briefly, a conformal block is a holomorphic section of $\mathscr{L}$ with
values in $\mathcal{D}^{\prime}(\mathcal{Q})$ satisfying additional
properties. In the next statement, the _Seiberg bounds_ are the conditions
$\mathrm{Re}(\alpha_{j})<Q$ for all $j\in\\{0,1,...,m\\}$, and
$\sum_{j=0}^{m}\mathrm{Re}(\alpha_{j})-Q(2-2\mathbf{g})>0$.

######  Theorem 1.3.

Let $m\in\mathbb{N}$,
$\boldsymbol{\alpha}=(\alpha_{0},...,\alpha_{m})\in\mathbb{C}^{m+1}$
satisfying the Seiberg bounds, and suppose that $\alpha_{0}=\alpha_{r,s}$ for
some $r,s\in\mathbb{N}^{*}$. For any conformal block
$\mathcal{B}\in\Gamma(\mathscr{L}\otimes\mathcal{D}^{\prime}(\mathcal{Q})^{m+1})$
in the sense of Section 4.4, we have

$\mathcal{S}_{\alpha_{r,s}}\mathcal{B}\left(\otimes_{j=0}^{m}\Psi_{\alpha_{j}}\right)=0.$

This result follows from the two main properties characterizing conformal
blocks: the conformal Ward identities and the fact that they are horizontal
with respect to the connection defined by the Virasoro algebra.

As mentioned above, the expression of $\mathbf{S}_{\alpha_{r,s}}$ is purely
representation theoretical, while the operator $\mathcal{S}_{\alpha_{r,s}}$
depends on the topological type of the surface. In Section 5, we explain how
the Virasoro algebra is represented as differential operators on Teichmüller
space, i.e. we give an expression for the operators $\mathcal{L}_{n}$. In a
nutshell, we compute the change of basis between the “Virasoro uniformisation”
and another set of holomorphic coordinates on Teichmüller space (e.g. the
plumbing coordinates). Perhaps the most interesting case is genus one, where
we have a non-trivial moduli space, and the computations are explicit using
elliptic functions. This shows how to use Theorem 1.3 to deduce the explicit
PDEs satisfied by conformal blocks. Similar conclusions appear elsewhere
[EO87], but our method seems new.

###  1.3. Proof outline

The proofs of Theorems 1.1 and 1.2 use a combination of probabilistic and
algebraic arguments which we summarise here. First, we stress that the two
results are intimately related. Indeed, from the existence of a Virasoro
representation on the Liouville side [BGK+23b], it is fairly easy to see that
$\ker\Phi_{\alpha_{r,s}}$ is a Virasoro submodule for all
$r,s\in\mathbb{N}^{*}$, i.e. $\mathcal{V}_{\alpha_{r,s}}$ is a highest-weight
representation. This immediately implies
$\ker\Phi_{\alpha_{r,s}}\subset\ker\Phi_{\alpha_{r,s}}^{0}$ since the latter
is the maximal proper submodule ([Fre92] and Section 3.6). The converse
inclusion is equivalent to Theorem 1.1.

Now, we explain the main steps of the proof of Theorem 1.1.

  1. 1.

Using Gaussian integration by parts/Girsanov’s theorem, we exhibit a
probabilistic formula for the descendant states, valid for $\alpha$ in a
neighbourhood of $-\infty$ (Lemma 3.11). This formula is essentially a linear
combination of integrals of the form $\mathcal{I}_{r,\boldsymbol{s}}(\alpha)$
as defined in (3.5). These are disc amplitudes with an $\alpha$-insertion at 0
and several $\gamma$-insertions integrated over the unit disc. This reduces
the question to the analytic continuation of these integrals, and their study
on the Kac table.

  2. 2.

The integrals have singularities at zero, which is why they converge
absolutely only for small enough $\alpha$. Using the fusion estimate of
Proposition 3.2, we get a sharp bound for the region of absolute convergence
and analyticity in $\alpha$ of these integrals.

  3. 3.

We find two induction formulae (3.12),(3.14), expressing a singular integral
in terms of simpler ones. This allows us to study the pole structure of the
singular integrals recursively. As a byproduct, the induction formulae
highlight the special role played by the Kac table.

  4. 4.

From the first three inputs, we deduce that for each $r,s\in\mathbb{N}^{*}$,
the Poisson operator is analytic at $\alpha_{r,s}$ and levels
$\;\leqslant\;rs$, _provided_ that $\gamma^{2}\not\in\mathbb{Q}$ (Corollary
3.6).

  5. 5.

Using the Virasoro representation constructed in [BGK+23b], the analyticity
propagates to all levels (Proposition 3.7). This implies that the Poisson
operator $\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is analytic for _all_
$\alpha$ and _all_ $\chi\in\mathcal{F}$, _provided_
$\gamma^{2}\not\in\mathbb{Q}$.

  6. 6.

We exploit a continuity property in $\gamma$ (Proposition 3.9) to extend the
result from $\gamma^{2}\not\in\mathbb{Q}$ to arbitrary $\gamma\in(0,2)$. This
is done in Section 3.6.

###  1.4. Outlook

The proof of Theorem 1.2 relies on the analyticity of the Poisson operator for
each state in the module. In a follow-up work, we study some cases in which
the Poisson operator has poles on the Kac table. These are known as the (level
2) _higher-equations of motions_ , both in the bulk and the boundary case.

Let
$\mathbf{S}^{0,\alpha}:=\alpha^{2}\mathbf{L}_{-2}^{0,\alpha}+(\mathbf{L}_{-1}^{0,\alpha})^{2}$
and define $\widetilde{\mathbf{S}}^{0,\alpha}$ similarly with the
antiholomorphic representation [GKRV23]. For $rs=2$, we show that
$\underset{\alpha\to\alpha_{r,s}}{\lim}\,\mathcal{P}_{\alpha}(\mathbf{S}^{0,\alpha}\mathbf{S}^{0,\alpha}\mathds{1})=B_{r,s}\Psi_{\alpha_{-r,s}}^{\prime}$
for some constant $B_{r,s}$, where $\Psi_{\alpha_{-r,s}}$ is the highest-
weight state of weight
$\alpha_{-r,s}=(1+r)\frac{\gamma}{2}+(1-s)\frac{2}{\gamma}$ and the prime
means derivative in $\alpha$. This equation has been conjectured in [Zam04],
and we prove it by computing the residue of the Poisson operator at
$\alpha_{r,s}$.

The states of boundary LCFT are associated to the half-disc
$\mathbb{D}\cap\mathbb{H}$. In addition to the bulk GMC measure $M_{\gamma}$,
there is a boundary measure
$\mathrm{d}L_{\gamma}(x)=e^{\frac{\gamma}{2}X-\frac{\gamma^{2}}{8}\mathbb{E}[X^{2}]}\mathrm{d}x$,
which are coupled via the “boundary cosmological constants”
$\mu_{\mathrm{L}}$, $\mu_{\mathrm{R}}$ (one for the left interval $(-1,0)$ and
the other for the right interval $(0,1)$). We can define a semigroup and a
Poisson operator as in the bulk case. For $rs=2$, we show that this Poisson
operator has a pole at $\alpha_{r,s}$, which is a scalar multiple of the
boundary primary state $\Psi_{\alpha_{-r,s}}$. Moreover, this scalar is
polynomial in the cosmological constants. In the case $(r,s)=(1,2)$, this
polynomial is $\mu_{\mathrm{L}}+\mu_{\mathrm{R}}$ (up to constant). In the
case $(r,s)=(2,1)$, this polynomial is
$\mu\sin(\pi\frac{\gamma^{2}}{4})-\mu_{\mathrm{L}}^{2}+2\mu_{\mathrm{L}}\mu_{\mathrm{R}}\cos(\pi\frac{\gamma^{2}}{4})-\mu_{\mathrm{R}}^{2}$.
The zero set of these polynomials define a region where the singular vector
vanishes, i.e. where the BPZ equations hold. This had been proposed in physics
[FZZ00, BB10] and the BPZ equation has been proved in [Ang23] for correlation
functions on the disc. We will prove the boundary HEMs at level 2 in an
upcoming work.

### Acknowledgements

G.B. thanks Joona Oikarinen for discussions on the smoothness of Liouville
correlation functions. We are also grateful to Colin Guillarmou and Rémi
Rhodes helpful discussions.

##  2\. Background

###  2.1. Gaussian fields and Liouville Hilbert space

Let $\mathbb{P}_{\mathbb{S}^{1}}$ be the law of the centred, Gaussian,
$\log$-correlated field on the circle, i.e. samples of
$\mathbb{P}_{\mathbb{S}^{1}}$ are Gaussian fields $\varphi$ on the circle with
covariance

$\mathbb{E}[\varphi(e^{i\theta})\varphi(e^{i\theta^{\prime}})]=\log\frac{1}{|e^{i\theta}-e^{i\theta^{\prime}}|},\qquad\forall\theta,\theta^{\prime}\in\mathbb{R}.$

With $\mathbb{P}_{\mathbb{S}^{1}}$-probability 1, $\varphi\in
H^{-s}(\mathbb{S}^{1})$ for all $s>0$. In Fourier modes, one has

$\varphi(e^{i\theta})=\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\varphi_{n}e^{ni\theta},$

where $\varphi_{-n}=\bar{\varphi}_{n}$, and $(\varphi_{n})_{n\;\geqslant\;1}$
are independent complex Gaussians $\mathcal{N}_{\mathbb{C}}(0,\frac{1}{2n})$.
The harmonic extension $P\varphi\in\mathcal{C}^{\infty}(\mathbb{D})$ is

$P\varphi(z)=\sum_{n=1}^{\infty}\varphi_{n}z^{n}+\varphi_{-n}\bar{z}^{n},\qquad\forall
z\in\mathbb{D}.$

Its covariance structure is

$\mathbb{E}[P\varphi(z)P\varphi(w)]=\log\frac{1}{|1-z\bar{w}|}=:G_{\partial}(z,w).$
(2.1)

In particular, we have $\mathbb{E}[P\varphi(z)^{2}]=\log\frac{1}{1-|z|^{2}}$.
The _Liouville Hilbert space_ is

$\mathcal{H}:=L^{2}(\mathrm{d}c\otimes\mathbb{P}_{S^{1}}),$

where $\mathrm{d}c$ is Lebesgue measure on $\mathbb{R}$. Samples of the
(infinite) measure $\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$ are written
$c+\varphi$. We define a dense subspace $\mathcal{C}$ of
$L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$ as follows. A functional
$F\in\mathcal{C}^{0}(H^{-1}(\mathbb{S}^{1}))$ belongs to $\mathcal{C}$ if
there exists $N\in\mathbb{N}$ and $f\in\mathcal{C}^{\infty}(\mathbb{C}^{N})$
such that $F(\varphi)=f(\varphi_{1},...,\varphi_{N})$ for all $\varphi\in
H^{-1}(\mathbb{S}^{1})$, and $f$ and all its derivatives have at most
exponential growth at $\infty$. Let $\mathcal{C}^{\prime}$ be the topological
dual of $\mathcal{C}$. We will refer to $\mathcal{C}$ (resp.
$\mathcal{C}^{\prime}$) as the space of _test functions_ (resp.
_distributions_).

We will frequently introduce some extra randomness in the form of an
independent Dirichlet free field in $\mathbb{D}$. This is a centred Gaussian
field $X_{\mathbb{D}}$ with covariance structure

$\mathbb{E}[X_{\mathbb{D}}(z)X_{\mathbb{D}}(w)]=\log\left|\frac{1-z\bar{w}}{z-w}\right|=:G_{\mathbb{D}}(z,w).$
(2.2)

We will always denote

$X:=X_{\mathbb{D}}+P\varphi,$

which is a $\log$-correlated Gaussian field in $\mathbb{D}$:

$\mathbb{E}[X(z)X(w)]=\log\frac{1}{|z-w|}=:G(z,w)$ (2.3)

Let $\rho\in\mathcal{C}^{\infty}(\mathbb{C})$ be a radially symmetric bump
function, with total integral $1$. For $\epsilon>0$, we write
$\rho_{\epsilon}(z):=\epsilon^{-2}\rho(\frac{z}{\epsilon})$, and
$X_{\epsilon}:=X\star\rho_{\epsilon}$. The _Gaussian multiplicative chaos_
(GMC) measure of $X$ with parameter $\gamma\in(0,2)$ is the following weak
limit of measures

$\mathrm{d}M_{\gamma}(z):=\underset{\epsilon\to
0}{\lim}\,\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}(z)}|\mathrm{d}z|^{2},$

where the convergence holds in probability. This limit exists, is non-trivial,
and is independent of the choice of regularisation [Kah85, Ber17, RV14].

###  2.2. Semigroups

One can define two strongly continuous, contraction semigroups on
$\mathcal{H}$, with (unbounded) generators $\mathbf{H}^{0}$, $\mathbf{H}$
[GKRV23]. These generators are positive, essentially self-adjoint operators.
Explictly, the _free field semigroup_
$e^{-t\mathbf{H}^{0}}\in\mathcal{L}(\mathcal{H})$ has the expression

$e^{-t\mathbf{H}^{0}}F(c+\varphi):=e^{-\frac{Q^{2}}{2}t}\mathbb{E}_{\varphi}\left[F(c+B_{t}+\varphi_{t})\right],\qquad\forall
F\in\mathcal{H},$

where $B_{t}=\int_{0}^{2\pi}X(e^{-t+i\theta})\frac{\mathrm{d}\theta}{2\pi}$,
and $\varphi_{t}(e^{i\theta})=X(e^{-t+i\theta})-B_{t}$. The notation
$\mathbb{E}_{\varphi}$ means conditionally on $\varphi$ (i.e. we integrate
over the Dirichlet free field $X_{\mathbb{D}}$). It is known that
$(B_{t})_{t\;\geqslant\;0}$ evolves as a standard Brownian motion. On the
other hand, the Fourier modes of $\varphi_{t}$ evolve as independent Ornstein-
Uhlenbeck processes. In fact, $\mathbb{P}_{\mathbb{S}^{1}}$ is the invariant
measure for the dynamics generated by $\mathbf{H}^{0}$.

The _Liouville semigroup_ $e^{-t\mathbf{H}}\in\mathcal{L}(\mathcal{H})$ has
the exprssion

$e^{-t\mathbf{H}}F(c+\varphi)=e^{-\frac{Q^{2}}{2}t}\mathbb{E}_{\varphi}\left[F(c+B_{t}+\varphi_{t})e^{-\mu
e^{\gamma c}\int_{\mathbb{A}_{t}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma
Q}}}\right],\qquad\forall F\in\mathcal{H},$ (2.4)

where

$\mathbb{A}_{t}:=\\{z\in\mathbb{C}|\,e^{-t}<|z|<1\\}.$

The two generators are related by

$\mathbf{H}=\mathbf{H}^{0}+\mu e^{\gamma c}V(\varphi),$

where the potential $V$ is only defined in $\mathcal{C}^{\prime}$, namely

$\mathbb{E}[V(\varphi)F(\varphi)]=\int_{0}^{2\pi}\mathbb{E}[F(\varphi-\gamma\log|e^{i\theta}-\cdot|)]\mathrm{d}\theta,$

for all test functions $F$. By the Girsanov transform, one formally has
$V(\varphi)=\int_{0}^{2\pi}e^{\gamma\varphi(e^{i\theta})-\frac{\gamma^{2}}{2}\mathbb{E}[\varphi(e^{i\theta})^{2}]}\mathrm{d}\theta$,
and (2.4) can be understood as a Feynman-Kac formula [GKRV23].

The generator $\mathbf{H}$ is called the _Liouville Hamiltonian_. It has an
associated quadratic form $\mathcal{Q}$ with domain
$\mathcal{D}(\mathcal{Q})$, and $\mathbf{H}$ defines a bounded operator
$\mathbf{H}:\mathcal{D}(\mathcal{Q})\to\mathcal{D}^{\prime}(\mathcal{Q})$,
where $\mathcal{D}^{\prime}(\mathcal{Q})$ is the continuous dual of
$\mathcal{D}(\mathcal{Q})$ [GKRV23]. More generally, $\mathbf{H}$ extends to a
bounded operator on weighted spaces, i.e. $\mathbf{H}:e^{-\beta
c}\mathcal{D}(\mathcal{Q})\to e^{-\beta c}\mathcal{D}^{\prime}(\mathcal{Q})$
is continuous for all $\beta>0$.

###  2.3. Feigin-Fuchs modules

Let $\alpha\in\mathbb{C}$ arbitrary. On $L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$,
we have a representation of the Heisenberg algebra
$(\mathbf{A}_{n})_{n\in\mathbb{Z}}$ given for $n>0$ by

$\mathbf{A}_{n}^{\alpha}=\frac{i}{2}\partial_{n};\qquad\mathbf{A}_{-n}^{\alpha}=\frac{i}{2}(\partial_{-n}-2n\varphi_{n});\qquad\mathbf{A}_{0}^{\alpha}=\frac{i}{2}\alpha.$

These operators satisfy the commutation relations
$[\mathbf{A}_{n}^{\alpha},\mathbf{A}_{m}^{\alpha}]=\frac{n}{2}\partial_{n,m}$
on $\mathcal{C}$. For $n\neq 0$, we have the hermiticity relations
$(\mathbf{A}_{n}^{\alpha})^{*}=\mathbf{A}_{-n}^{\alpha}$ on
$L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$. Given a partition $\mathbf{k}$, we set
$\mathbf{A}_{-\mathbf{k}}:=\prod_{n=1}^{\infty}(\mathbf{A}_{-n}^{\alpha})^{k_{n}}$.

We call the constant function $\mathds{1}\in
L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$ the _vacuum vector_. The Heisenberg
representation gives the space
$\mathcal{F}=\mathbb{C}[(\varphi_{n})_{n\;\geqslant\;1}]$ a structure of
highest-weight Heisenberg module, i.e.

$\mathcal{F}=\mathrm{span}\,\\{\pi_{\mathbf{k}}:=\mathbf{A}_{-\mathbf{k}}\mathds{1}|\,\mathbf{k}\in\mathcal{T}\\}.$

The level gives a grading

$\mathcal{F}=\oplus_{N\in\mathbb{N}}\mathcal{F}_{N},$

and we have $\dim\mathcal{F}_{N}=p(N)$.

The _Sugawara construction_ is a representation
$(\mathbf{L}_{n}^{0,\alpha})_{n\in\mathbb{Z}}$ of the Virasoro algebra on
$L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$, where $\mathbf{L}_{n}^{0,\alpha}$ is the
following quadratic expression in the Heisenberg algebra

$\displaystyle\mathbf{L}_{n}^{0,\alpha}:=i(\alpha-(n+1)Q)\mathbf{A}_{n}+\sum_{m\neq\\{0,n\\}}\mathbf{A}_{n-m}\mathbf{A}_{m},\qquad
n\neq 0;$
$\displaystyle\mathbf{L}_{0}^{0,\alpha}:=\Delta_{\alpha}+2\sum_{m=1}^{\infty}\mathbf{A}_{-m}\mathbf{A}_{m}.$

On $\mathcal{C}$, these operator satisfy the commutation relations

$[\mathbf{L}_{n}^{0,\alpha},\mathbf{L}_{m}^{0,\alpha}]=(n-m)\mathbf{L}_{n+m}^{0,\alpha}+\frac{c_{\mathrm{L}}}{12}(n^{3}-n)\delta_{n,m}.$

We also have the hermiticity relations
$(\mathbf{L}_{n}^{0,\alpha})^{*}=\mathbf{L}_{-n}^{0,2Q-\bar{\alpha}}$ on
$L^{2}(\mathbb{P}_{\mathbb{S}^{1}})$.

When we consider $\mathcal{F}$ as a moulde of Virasoro algebra
$\mathbf{L}_{n}^{0,\alpha}$, we call it _Feigin-Fuchs module_.

Given a partition $\nu=(\nu_{1},...,\nu_{\ell})$, we set
$\mathbf{L}_{-\nu}^{0,\alpha}=\mathbf{L}_{-\nu_{\ell}}^{0,\alpha}...\mathbf{L}_{-\nu_{1}}^{0,\alpha}$.
The descendant state

$\mathcal{Q}_{\alpha,\nu}:=\mathbf{L}_{-\nu}^{0,\alpha}\mathds{1}\in\mathcal{F}$

is a polynomial of level $|\nu|$.

Let $\mathcal{V}_{\alpha}^{0}$ be the Virasoro highest-weight representation
obtained by acting with the Virasoro operators on the vacuum vector, i.e.

$\mathcal{V}_{\alpha}^{0}:=\mathrm{span}\\{\mathcal{Q}_{\alpha,\nu}|\nu\in\mathcal{T}\\}\subset\mathcal{F},$

where the span is algebraic. We also define
$\mathcal{V}_{\alpha}^{0,N}:=\mathcal{V}_{\alpha}^{0}\cap\mathcal{F}_{N}$. The
module $\mathcal{V}_{\alpha}^{0}$ has central charge $c_{\mathrm{L}}=1+6Q^{2}$
and highest-weight $\Delta_{\alpha}=\frac{\alpha}{2}(Q-\frac{\alpha}{2})$.
Recall that $\Delta_{\alpha}=\Delta_{2Q-\alpha}$. For $\alpha\not\in kac$, we
have $\mathcal{V}_{\alpha}^{0}\simeq\mathcal{F}$ linearly, and
$\mathcal{V}_{\alpha}^{0}$ is isomorphic to the Verma module.

From now on, we assume $\mathrm{Re}(\alpha)<Q$, so that
$2Q-\mathrm{Re}(\alpha)>Q$. By [Fre92, Theorem 1], the module
$\mathcal{V}_{2Q-\alpha}^{0}$ is Verma, namely for each level
$N\in\mathbb{N}$, the polynomials $(\mathcal{Q}_{2Q-\alpha,\nu})_{|\nu|=N}$
are linearly independent and form a basis of the space $\mathcal{F}_{N}$.
Hence, we have a canonical linear map
$\Phi_{\alpha}^{0}:\mathcal{V}_{2Q-\alpha}^{0}\to\mathcal{V}_{\alpha}^{0}$
such that for all partitions $\nu$,

$\Phi_{\alpha}^{0}(\mathbf{L}_{-\nu}^{0,2Q-\alpha}\mathds{1})=\mathbf{L}_{-\nu}^{0,\alpha}\mathds{1},$
(2.5)

i.e.
$\Phi_{\alpha}^{0}(\mathcal{Q}_{2Q-\alpha,\nu})=\mathcal{Q}_{\alpha,\nu}$. The
map $\Phi_{\alpha}^{0}$ is the canonical linear surjection from the Verma
module $\mathcal{V}_{2Q-\alpha}^{0}\simeq\mathcal{F}$ to the highest-weight
module $\mathcal{V}_{\alpha}^{0}$. It preserves the level and we denote by
$\Phi_{\alpha}^{0,N}:=\Phi_{\alpha}^{0}|_{\mathcal{F}_{N}}$ its restriction to
level $N$. Moreover, $\ker(\Phi_{\alpha}^{0})$ is a submodule of
$\mathcal{V}_{2Q-\alpha}^{0}$ and we have a canonical isomorphism of highest-
weight Virasoro representations

$\mathcal{V}_{\alpha}^{0}=\mathrm{ran}(\Phi_{\alpha}^{0})\simeq\mathcal{V}_{2Q-\alpha}^{0}/\ker(\Phi_{\alpha}^{0}).$

Again by [Fre92], the determinant of $\Phi_{\alpha}^{0,N}$ is given by the
celebrated Kac formula:

$\det(\Phi_{\alpha}^{0,N})=\prod_{1\;\leqslant\;rs\;\leqslant\;N}(\Delta_{\alpha}-\Delta_{\alpha_{r,s}})^{p(N-rs)}.$

This implies that $\ker(\Phi_{\alpha}^{0})=\\{0\\}$ if and only if $\alpha$
does not belong to the Kac table. In particular,
$\ker(\Phi_{\alpha_{r,s}}^{0,rs})$ is non-trivial.

In fact, we claim the stronger statement that $\mathcal{V}_{\alpha_{r,s}}^{0}$
is irreducible. Indeed, it is a fact that any non-trivial proper submodule of
$\mathcal{V}_{\alpha_{r,s}}^{0}$ is a highest-weight representation [KRR13],
so it is generated by a highest-weight vector at positive level (such vectors
are called _singular_). Thus, to prove the claim, it suffices to show that all
singular vectors vanish. Let $\chi\in\mathcal{V}_{\alpha_{r,s}}^{0}$ be such a
vector. By definition, we have $\mathbf{L}_{\nu}^{0,\alpha_{r,s}}\chi=0$ for
all $\nu\in\mathcal{T}$. Thus, by the hermiticity relations,
$\langle\chi,\mathcal{Q}_{2Q-\alpha_{r,s},\nu}\rangle_{L^{2}(\mathbb{P}_{\mathbb{S}^{1}})}=0$
for all $\nu\in\mathcal{T}$, i.e.
$\chi\in(\mathcal{V}_{2Q-\alpha_{r,s}}^{0})^{\perp}$. But we have seen that
$\mathcal{V}_{2Q-\alpha_{r,s}}^{0}=\mathcal{F}$, so $\chi=0$.

Let us summarise our findings.

######  Lemma 2.1.

Let $c_{\mathrm{L}}>25$ and $\alpha\in\mathbb{C}$.

  * •

If $\alpha\not\in kac^{-}$, $\mathcal{V}_{\alpha}^{0}$ is isomorphic to the
Verma module $M(c_{\mathrm{L}},\Delta_{\alpha})$.

  * •

If $\alpha\in kac^{-}$, $\mathcal{V}_{\alpha}^{0}$ is isomorphic to the
irreducible quotient of the Verma module $\mathcal{V}_{2Q-\alpha}^{0}$ by the
maximal proper submodule $\ker(\Phi_{\alpha}^{0})$.

Finally, we show that the map $\alpha\mapsto\Phi_{\alpha}^{0,N}$ is analytic
in the region $\mathrm{Re}(\alpha)<Q$ for all levels $N\in\mathbb{N}$. Since
the Sugawara expression is polynomial in $\alpha$, there exist coefficients
$m_{\nu,\mathbf{k}}\in\mathbb{C}[\alpha]$ such that

$\mathcal{Q}_{\alpha,\nu}=\sum_{|\mathbf{k}|=|\nu|}m_{\nu,\mathbf{k}}(\alpha)\pi_{\mathbf{k}}.$

The family of matrices $\alpha\mapsto
m^{N}(2Q-\alpha)=(m_{\nu,\mathbf{k}}(2Q-\alpha))_{|\nu|=|\mathbf{k}|=N}$ is
analytic and invertible in the region $\mathrm{Re}(\alpha)<Q$. Since
$\mathrm{GL}_{p(N)}(\mathbb{C})$ is an analytic Lie group, we deduce that
$M^{N}(2Q-\alpha):=m^{N}(2Q-\alpha)^{-1}$ exists and is analytic in the the
same region. We have just shown the existence of analytic coefficients
$M_{\mathbf{k},\nu}(2Q-\alpha)$ in the region $\alpha<Q$ such that for all
integer partitions $\mathbf{k}:$

$\Phi_{\alpha}^{0}(\pi_{\mathbf{k}})=\sum_{|\nu|=|\mathbf{k}|}M_{\mathbf{k},\nu}(2Q-\alpha)\mathcal{Q}_{\alpha,\nu}.$

We stress that the last formula does not extend analytically to the region
$\mathrm{Re}(\alpha)>Q$, precisely because some matrices $m^{N}(\alpha)$ are
not invertible on the Kac table, so their inverse $M^{N}(\alpha)$ will have a
pole there.

######  Example 1.

Frenkel’s result implies that $\mathcal{V}_{2Q-{\alpha_{r,s}}}^{0}$ is Verma
but $\mathcal{V}_{\alpha_{r,s}}^{0}$ is not [Fre92]. Thus, the singular vector
in $\mathcal{V}_{2Q-\alpha_{r,s}}^{0}$ does not vanish. For the benefit of the
reader, it may be useful to illustrate this fact on an explicit example at
level 2. By a straightforward computation, one finds for all
$\alpha\in\mathbb{C}$,

$\displaystyle\mathbf{L}_{-2}^{0,\alpha}\mathds{1}=2(Q+\alpha)\varphi_{2}-\varphi_{1}^{2};\qquad(\mathbf{L}_{-1}^{0,\alpha})^{2}\mathds{1}=2\alpha\varphi_{2}+\alpha^{2}\varphi_{1}^{2}.$

To check when these two polynomials are colinear, we look for linear
combinations cancelling $\varphi_{1}^{2}$, leading to

$(\alpha^{2}\mathbf{L}_{-2}^{0,\alpha}+(\mathbf{L}_{-1}^{0,\alpha})^{2})\mathds{1}=2\alpha(\alpha^{2}+\alpha
Q+1)\varphi_{2}=2\alpha(\alpha-\alpha_{1,2})(\alpha-\alpha_{2,1})\varphi_{2}.$

This polynomial vanishes for the degenerate weights
$\alpha_{1,2},\alpha_{2,1}<Q$, but not for the dual weights
$2Q-\alpha_{1,2},2Q-\alpha_{2,1}$. The root $\alpha=0$ is artificial and
corresponds to the level 1 computation: we have
$\mathbf{L}_{-1}^{0,\alpha}=\alpha\varphi_{1}$, which vanishes for
$\alpha=\alpha_{1,1}=0$ (but not for the dual weight $2Q-\alpha_{1,1}=2Q$).

##  3\. Singular modules

In this section, we prove Theorems 1.1 and 1.2, following the outline of
Section 1.3. We introduce and study the singular integrals in Sections 3.2,
3.3 and 3.4. Then, we establish the link between singular integrals and
descendants states in Section 3.5. Finally, Section 3.6 puts the pieces
together to conclude the proofs. We mention that the objects of this section
bear some similarities with the ones encoutered in [Oik19], but the treatment
we give them is rather different.

###  3.1. Notations

Recall $\mathbb{A}_{t}:=\\{e^{-t}<|z|<1\\}\subset\mathbb{C}$. For
$r\in\mathbb{N}^{*}$, we set

$\displaystyle\Delta_{t}^{r}:=\\{\boldsymbol{w}\in\mathbb{A}_{t}^{r}|\,\forall
1\;\leqslant\;k<l\;\leqslant\;r,\,|w_{k}-w_{l}|>e^{-t}\\}$
$\displaystyle\Delta_{t}^{r-1}(w):=\\{\boldsymbol{w}\in\mathbb{A}_{t}^{r-1}|\,(\boldsymbol{w},w)\in\Delta_{t}^{r}\\}.$

and for each $\boldsymbol{w}\in\Delta_{t}^{r}$,

$D_{t}(\boldsymbol{w}):=\cup_{j=1}^{r}D(w_{j},e^{-t}),$

where $D(w,\epsilon)$ is the (open) disc centred at $w$ and of radius
$\epsilon$ . Note that $\Delta_{t}^{r}$ is invariant under the action of the
$r^{\text{th}}$ symmetric group permuting the variables.

In this section, we represent an integer partition by a pair
$(r,\boldsymbol{s})$ where $r\in\mathbb{N}^{*}$ is the _length_ , and
$\boldsymbol{s}$ is a non-increasing sequence of positive integers
$\boldsymbol{s}=(s_{1},...,s_{r})$ of length $r$. Given
$r,s\in\mathbb{N}^{*}$, we use the shorthand notation $(r,(s))$ for the
“rectangular” partition $(r,(s,s,...,s))$ of length $r$. The level of the
partition is $|\boldsymbol{s}|=\sum_{j=1}^{r}s_{j}$, and we define
$\mathcal{T}_{N}$ (resp. $\mathcal{T}$) to be the set of integer partitions of
level $N$ (resp. all partitions). We set

$r^{*}:=\max\\{j|\,s_{j}=s_{1}\\}.$

The $j^{\text{th}}$ _elementary move_
$\tau_{j}:\mathcal{T}_{N}\to\mathcal{T}_{N}$ is defined as follows (see Figure
1). For all $(r,\boldsymbol{s})\in\mathcal{T}_{N}$ and $j\;\leqslant\;r$,

  * •

If $s_{j}=1$, $\tau_{j}(r,\boldsymbol{s})=(r,\boldsymbol{s})$.

  * •

If $s_{j}\;\geqslant\;2$, let $j^{*}:=\max\\{l|\,s_{l}=s_{j}\\}$.

    * –

If $s_{j}\in\\{s_{r},s_{r}+1\\}$, then $\tau_{j}(r,\boldsymbol{s})$ is the
partition $(r+1,\widetilde{\boldsymbol{s}})$, with
$\widetilde{s}_{j^{*}}=s_{j^{*}}-1$, $\widetilde{s}_{r+1}=1$, and
$\widetilde{s}_{l}=s_{l}$ for $l\not\in\\{j^{*},r+1\\}$.

    * –

If $s_{j}\;\geqslant\;s_{r}+2$, let
$k:=\min\\{l|\,s_{l}\;\leqslant\;s_{j}-2\\}$. Then
$\tau_{j}(r,\boldsymbol{s})$ is the partition $(r,\widetilde{\boldsymbol{s}})$
with $\widetilde{s}_{j^{*}}=s_{j}-1$, $\widetilde{s}_{k}=s_{k}+1$, and
$\widetilde{s}_{l}=s_{l}$ for $l\not\in\\{k,j^{*}\\}$.

Informally, an elementary move consists in taking a box from the
$j^{\text{th}}$ column and move it to the first available column to the right,
in such a way that we still get a partition.

Figure 1.  From left to right, applying the first elementary move back-to-back
in $\mathcal{T}_{4}$. To a partition $(r,\boldsymbol{s})$, we associate the
singularity $\prod_{j=1}^{r}w_{j}^{-s_{j}}$, where the $w_{j}$ are variables
in the unit disc.

We define a partial order $\preceq$ on $\mathcal{T}_{N}$ by declaring
$(\widetilde{r},\widetilde{\boldsymbol{s}})\preceq(r,\boldsymbol{s})$ if
$(\widetilde{r},\widetilde{\boldsymbol{s}})$ can be obtained from
$(r,\boldsymbol{s})$ by a (possibly empty) sequence of elementary moves. One
can check that the unique minimal (resp. maximal) partition in
$\mathcal{T}_{N}$ is $(N,(1))$ (resp. $(1,(N))$). The associated strict order
is denoted by $\prec$. The _distance_ of a partition $(r,\boldsymbol{s})$ to
$(N,(1))$ is the minimal number of elementary moves required to reach
$(N,(1))$ from $(r,\boldsymbol{s})$. It is always a finite number. The
distance can be used to prove statements by induction on
$(\mathcal{T}_{N},\preceq)$: the initialisation is to prove that the statement
is true for the minimal partition, while the hereditity is to prove that the
statement is true for partitions at distance $d+1$ if it is true for
partitions at distance $\;\leqslant\;d$.

######  Remark 1.

In this work, integer partitions (a.k.a. Young diagrammes) appear in three
different forms and we use three different parametrisations:

  * •

We code elements of $\mathcal{F}$ using integer sequences
$\mathbf{k}=(k_{1},...,k_{n},...)$ with finitely many non-zero terms. The
length of the partition is $\ell(\mathbf{k})=\sum_{n\in\mathbb{N}^{*}}k_{n}$,
and its level is $|\mathbf{k}|=\sum_{n\in\mathbb{N}^{*}}nk_{n}$. The
corresponding polynomial is
$\varphi^{\mathbf{k}}=\prod_{n\in\mathbb{N}^{*}}\varphi_{n}^{k_{n}}$.

  * •

We code Virasoro descendants using finite, non-increasing integer sequences
$\nu=(\nu_{1},...,\nu_{\ell})$. The length is $\ell(\nu)=\ell$ and the level
is $|\nu|=\sum_{j=1}^{\ell}\nu_{j}$. The corresponding descendant is
$\mathbf{L}_{-\nu}=\mathbf{L}_{-\nu_{\ell}}...\mathbf{L}_{-\nu_{1}}$.

  * •

We code singular integrals (see below) using pairs $(r,\boldsymbol{s})$ as
above. The length is $\ell(r,\boldsymbol{s})=r$ and the level is
$|\boldsymbol{s}|=\sum_{j=1}^{r}s_{j}$. The corresponding singularity is
$\prod_{j=1}^{r}w_{j}^{-s_{j}}$.

These different parametrisations correspond to different representations of
the descendant states. In all cases, we write $\mathcal{T}_{N}$ the set of
partitions of level $N$, and $p(N)=\\#\mathcal{T}_{N}$. By convention,
$p(0)=1$ and the unique element of $\mathcal{T}_{0}$ is called the _empty
partition_.

###  3.2. Preliminaries

For $\alpha<Q$ and $t>0$, we define a function
$\Psi_{\alpha}^{t}:\Delta_{t}^{r}\to e^{-\beta c}\mathcal{D}(\mathcal{Q})$
such that for $\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$-a.e.
$(c,\varphi)$,

$\displaystyle\Psi_{\alpha}^{t}(\boldsymbol{w})$
$\displaystyle:=\underset{\epsilon\to
0}{\lim}\,e^{(\alpha+r\gamma-Q)c}\mathbb{E}_{\varphi}\left[\epsilon^{\frac{\alpha^{2}}{2}}e^{\alpha
X_{\epsilon}(0)}\prod_{j=1}^{r}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}(w_{j})}e^{-\mu e^{\gamma c}\int_{\mathbb{A}_{t}\setminus
D_{t}(\boldsymbol{w})}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}(z)}|\mathrm{d}z|^{2}}\right]$ (3.1)
$\displaystyle=e^{(\alpha+r\gamma-Q)c}\prod_{j=1}^{r}|w_{j}|^{-\gamma\alpha}e^{\gamma
P\varphi(w_{j})-\frac{\gamma^{2}}{2}G_{\partial}(w_{j},w_{j})}\prod_{1\;\leqslant\;k<l\;\leqslant\;r}e^{\gamma^{2}G_{\mathbb{D}}(w_{k},w_{l})}$
$\displaystyle\qquad\times\mathbb{E}_{\varphi}\left[e^{-\mu e^{\gamma
c}\int_{\mathbb{A}_{t}\setminus
D_{t}(\boldsymbol{w})}\prod_{j=1}^{r}e^{\gamma^{2}G_{\mathbb{D}}(z,w_{j})}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right].$

The last expression follows from the Girsanov transform.

For each $t>0$ and $\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$-a.e.,
$\Psi_{\alpha}^{t}$ is invariant under permutation of its variables. Moreover,
$\Psi_{\alpha}^{t}(\boldsymbol{w})\in e^{-\beta c}\mathcal{D}(\mathcal{Q})$
for all $\beta>(Q-\alpha-r\gamma)_{+}$, and
$\Psi_{\alpha}^{t}(\boldsymbol{w})$ converges as $t\to\infty$ in $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$ and
$\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$-a.e., to the limit

$\Psi_{\alpha}(\boldsymbol{w})=e^{(\alpha+r\gamma-Q)c}\prod_{j=1}^{r}|w_{j}|^{-\gamma\alpha}e^{\gamma
P\varphi(w_{j})-\frac{\gamma^{2}}{2}G_{\partial}(w_{j},w_{j})}\prod_{1\;\leqslant\;k<l\;\leqslant\;r}e^{\gamma^{2}G_{\mathbb{D}}(w_{k},w_{l})}\mathbb{E}_{\varphi}\left[e^{-\mu
e^{\gamma
c}\int_{\mathbb{D}}\prod_{j=1}^{r}e^{\gamma^{2}G_{\mathbb{D}}(z,w_{j})}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right].$
(3.2)

From [GKRV23, appendix C.1], we know that
$\alpha\mapsto\Psi_{\alpha}(\boldsymbol{w})$ admits an analytic continuation
in a complex neighborhood of $(-\infty,Q)$. The states
$\Psi_{\alpha}(\boldsymbol{w})$ and integrals
$\mathcal{I}_{r,\boldsymbol{s}}(\alpha)$ are always viewed as elements of a
weighted space $e^{-\beta c}\mathcal{D}(\mathcal{Q})$ for some arbitrary real
number $\beta>(Q-\alpha-r\gamma)_{+}$.

######  Remark 2.

Given a function $f\in L^{1}(\Delta_{t}^{r})$, the extension of $f$ by zero in
$\mathbb{D}^{r}\setminus\bar{\Delta}_{t}^{r}$ gives a continuous embedding
$L^{1}(\Delta_{t}^{r})\hookrightarrow L^{1}(\mathbb{D}^{r})$, so we will
freely identify functions in $L^{1}(\Delta_{t}^{r})$ with functions in
$L^{1}(\mathbb{D}^{r})$.

The key identity to our results is the following derivative formula for
$\Psi_{\alpha}(\cdot)$. This formula has been written in various forms
elsewhere [KRV19, Oik19, GKRV23], and its proof is a simple consequence of
Gaussian integration by parts.

######  Lemma 3.1.

For all $F\in\mathcal{C}$, the map
$\boldsymbol{w}\mapsto\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]$ is
smooth , and we have

$\displaystyle\partial_{w_{1}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]$
$\displaystyle=\left(\alpha\gamma\partial_{w_{1}}G(w_{1},0)+\gamma^{2}\sum_{j=2}^{r}\partial_{w_{1}}G(w_{1},w_{j})\right)\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]$
(3.3) $\displaystyle\quad-\mu\gamma^{2}\int_{\mathbb{A}_{t}\setminus
D_{t}(\boldsymbol{w})}\partial_{w_{1}}G(w_{1},w_{r+1})\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w},w_{r+1})F]|\mathrm{d}w_{r+1}|^{2}$
$\displaystyle\quad+\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})\nabla
F(\partial_{w_{1}}G_{\partial}(w_{1},\cdot\,))]$
$\displaystyle\quad-\mu\int_{\partial
D(w_{1},e^{-t})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w},w_{r+1})F]\mathrm{d}\bar{w}_{r+1}.$

By the last term, we mean the following: for all $w_{1}\in\mathbb{D}$ and
$e^{i\theta}\in\mathbb{S}^{1}$ we have
$\partial_{w_{1}}G_{\partial}(w_{1},e^{i\theta})=\frac{1}{2}\frac{e^{-i\theta}}{1-e^{-i\theta}w_{1}}=\frac{1}{2}\sum_{n=0}^{\infty}e^{-i(n+1)\theta}w_{1}^{n}$,
and

$\nabla
F(\partial_{w_{1}}G_{\partial}(w_{1},\cdot\,))=\frac{1}{2}\sum_{n=1}^{\infty}w_{1}^{n-1}\partial_{-n}F.$
(3.4)

For $F\in\mathcal{C}$, the sum is actually finite (since $F$ depends only on
finitely many modes) and $\nabla
F(\partial_{w_{1}}G_{\partial}(w_{1},\cdot\,))\in\mathcal{C}$.

###### Proof.

The first two terms come from Gaussian integration by parts with respect to
$X$, and the third term comes from
$\mathbb{E}[\partial_{w_{1}}P\varphi(w_{1})F(\varphi)]$. And the last term
comes from $w_{1}$ derivatives of $\int_{\mathbb{A}_{t}\setminus
D_{t}(\boldsymbol{w})}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}(z)}|\mathrm{d}z|^{2}$. ∎

For a partition $(r,\boldsymbol{s})$, we introduce

$\mathcal{I}_{r,\boldsymbol{s}}^{t}(\alpha):=\int_{\Delta_{t}^{r}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{\boldsymbol{w}^{\boldsymbol{s}}}\in
e^{-\beta c}\mathcal{D}(\mathcal{Q}).$ (3.5)

Our goal is to investigate the existence and analyticity (in $\alpha$) of a
limit $\mathcal{I}_{r,\boldsymbol{s}}$ as $t\to\infty$. The next proposition
is a criterion for the absolute integrability of $\Psi_{\alpha}^{t}(\cdot)$ on
$\mathbb{D}^{r}$, which refines [GKRV23, Proposition 7.2]. It gives a bound
for arbitrary partitions $(r,\boldsymbol{s})\in\mathcal{T}$, but we will
actually only need it for the rectangular partition $(r,(s,...,s))$.

######  Proposition 3.2.

Let $(r,\boldsymbol{s})\in\mathcal{T}$. There exists a complex neighbourhood
of $(-\infty,\alpha_{r,|\boldsymbol{s}|/r})$ such that,
$\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$-a.e., we have

$\underset{t\to\infty}{\lim}\,\int_{\mathbb{A}_{t}^{r}}\Psi_{\alpha}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2\boldsymbol{s}}}<\infty.$

As a consequence, the map
$\alpha\mapsto\mathcal{I}_{r,\boldsymbol{s}}(\alpha)\in e^{-\beta
c}\mathcal{D}(\mathcal{Q})$ is analytic in a neighbourhood of
$(-\infty,Q-(\frac{\gamma}{2}r+\frac{2}{\gamma}\frac{|\boldsymbol{s}|}{2r}))$.
We denote this neighborhood by $\mathcal{W}_{r,\boldsymbol{s}}$. We call
$\mathcal{W}_{r,\boldsymbol{s}}$ the locally uniform convergence region.

For the empty partition, we will denote the corresponding neighborhood by
$\mathcal{W}_{0}$. This neighborhood contains the half-line $(-\infty,Q)$ by
[GKRV23, Appendix C.1].

###### Proof.

Note that the limit exists
$\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}$-a.e. by monotonicity, and it
suffices to study the convergence along the sequence of annuli
$\mathbb{A}_{n}$ for integer $n$. For simplicity, we assume
$s_{1}...=s_{r}=s$, and we will indicate the changes for the general case at
the end of the proof.

Recall the expression for $\Psi_{\alpha}(\boldsymbol{w})$ in the last line of
(3.2). Integrating over $\mathbb{P}_{\mathbb{S}^{1}}$, we have

$\mathbb{E}[\Psi_{\alpha}(\boldsymbol{w})]=e^{(\alpha+r\gamma-Q)c}\prod_{j=1}^{r}|w_{j}|^{-\gamma\alpha}\prod_{1\;\leqslant\;k<l\;\leqslant\;r}|w_{k}-w_{l}|^{-\gamma^{2}}\mathbb{E}\left[e^{-\mu
e^{\gamma
c}\int_{\mathbb{D}}\prod_{j=1}^{r}|z-w_{j}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]$
(3.6)

Moreover, for all $t>0$, the function $F:\mu\mapsto
e^{(\alpha-Q)c}\mathbb{E}\left[e^{-\mu e^{\gamma
c}\int_{\mathbb{A}_{t}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]$
is smooth in $(0,\infty)$ (since the GMC is a positive random variable), and
we have

$\displaystyle\partial_{\mu}^{r}F(\mu)$
$\displaystyle=e^{(\alpha+r\gamma-Q)c}\int_{\mathbb{A}_{t}^{r}}\prod_{j=1}^{r}|w_{j}|^{-\gamma\alpha}\prod_{1\;\leqslant\;k<l\;\leqslant\;r}|w_{k}-w_{l}|^{-\gamma^{2}}\mathbb{E}\left[e^{-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}\prod_{j=1}^{r}|z-w_{j}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]|\mathrm{d}\boldsymbol{w}|^{2}$
$\displaystyle\;\geqslant\;e^{-2trs}\int_{\mathbb{A}_{t}^{r}}\mathbb{E}[\Psi_{\alpha}(\boldsymbol{w})]\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}.$

Thus, $\boldsymbol{w}\mapsto\Psi_{\alpha}(\boldsymbol{w})$ is integrable on
$\mathbb{A}_{t}^{r}$ for each $t>0$.

Now, we prove the result by induction on $r$. For $r=1$, we have
$\mathbb{E}[\Psi_{\alpha}(w)]\;\leqslant\;C|w|^{-\gamma\alpha}$ as $w\to 0$,
hence
$\int_{\mathbb{D}}\mathbb{E}[\Psi_{\alpha}(w)]\frac{|\mathrm{d}w|^{2}}{|w|^{2s}}<\infty$
for $\alpha<\alpha_{1,s}$. In particular,
$\int_{\mathbb{D}}\Psi_{\alpha}(w)\frac{|\mathrm{d}w|^{2}}{|w|^{2s}}$ is
finite almost surely in this case.

Let $r\;\geqslant\;2$. Suppose the result holds for all $r^{\prime}<r$, and
let us prove it at rank $r$. Given $m<n\in\mathbb{N}$, we use the notation
$\mathbb{A}_{m,n}:=\\{e^{-n}<|w|<e^{-m}\\}$. For all $n\in\mathbb{N}^{*}$, we
write

$\int_{\mathbb{A}^{r}_{0,n+1}}\Psi_{\alpha}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}=\sum_{r^{\prime}=0}^{r-1}\binom{r}{r^{\prime}}\int_{\mathbb{A}_{0,1}^{r-r^{\prime}}}\int_{\mathbb{A}_{1,n+1}^{r^{\prime}}}\Psi_{\alpha}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}+\int_{\mathbb{A}_{1,n+1}^{r}}\Psi_{\alpha}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}.$
(3.7)

By the induction hypothesis for $r^{\prime}$ (since
$\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$ is bounded on
$\mathbb{A}_{0,1}^{r-r^{\prime}}$), all the terms in the sum converge
$(\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})$-a.e. as $n\to\infty$. To
treat the last term in the RHS, we use the scaling $w_{j}\mapsto ew_{j}$ and
(3.6) to get

$\displaystyle\int_{\mathbb{A}_{1,n+1}^{r}}\mathbb{E}[\Psi_{\alpha}(\boldsymbol{w})]\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$
$\displaystyle\quad=e^{r\gamma\alpha+\frac{1}{2}\gamma^{2}r(r-1)+2r(s-1)}\int_{\mathbb{A}_{0,n}^{r}}\prod_{j=1}^{r}|w_{j}|^{-\gamma\alpha}\prod_{1\;\leqslant\;k<l\;\leqslant\;r}|w_{k}-w_{l}|^{\gamma^{2}}$
(3.8) $\displaystyle\qquad\times
e^{(\alpha+r\gamma-Q)c}\mathbb{E}\left[e^{-\mu e^{\gamma
c}\int_{\mathbb{D}}\prod_{j=1}^{r}|z-\frac{w_{j}}{e}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}.$

Note that the constant prefactor equals $e^{r\gamma(\alpha-\alpha_{r,s})}$.
Next, we integrate the last line over the zero mode. We fix
$\beta\;\geqslant\;0$ such that $\beta=0$ if $\alpha+r\gamma-Q>0$, and
$\beta>Q-\alpha-r\gamma$ otherwise. Writing
$\sigma:=\frac{1}{\gamma}(\beta+\alpha+r\gamma-Q)>0$, we have

$\displaystyle\int_{\mathbb{R}}e^{(\beta+\alpha+r\gamma-Q)c}\mathbb{E}\left[e^{-\mu
e^{\gamma
c}\int_{\mathbb{D}}\prod_{j=1}^{r}|z-\frac{w_{j}}{e}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]\mathrm{d}c$
$\displaystyle\qquad=\frac{1}{\gamma}\mu^{-\sigma}\Gamma(\sigma)\mathbb{E}\left[\left(\int_{\mathbb{D}}\prod_{j=1}^{r}|z-\frac{w_{j}}{e}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}\right)^{-\sigma}\right]$
$\displaystyle\qquad\;\leqslant\;\frac{1}{\gamma}\mu^{-\sigma}\Gamma(\sigma)\mathbb{E}\left[\left(\int_{e^{-1}\mathbb{D}}\prod_{j=1}^{r}|z-\frac{w_{j}}{e}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}\right)^{-\sigma}\right]$
$\displaystyle\qquad=\frac{1}{\gamma}\mu^{-\sigma}\Gamma(\sigma)e^{\frac{1}{2}\gamma^{2}\sigma^{2}}e^{(-r\gamma^{2}-\gamma\alpha+\gamma
Q)\sigma}\mathbb{E}\left[\left(\int_{\mathbb{D}}\prod_{j=1}^{r}|z-w_{j}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}\right)^{-\sigma}\right]$
$\displaystyle\qquad=e^{-\frac{1}{2}((\alpha+r\gamma-Q)^{2}-\beta^{2})}\int_{\mathbb{R}}e^{(\alpha+\beta+r\gamma-Q)c}\mathbb{E}\left[e^{-\mu
e^{\gamma
c}\int_{\mathbb{D}}\prod_{j=1}^{r}|z-w_{j}|^{-\gamma^{2}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]\mathrm{d}c,$

where we used multifractal scaling in the second-to-last line. Plugging this
into (3.8), we get

$\int_{\mathbb{A}_{1,n+1}^{r}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}\;\leqslant\;e^{r\gamma(\alpha-\alpha_{r,s})}e^{-\frac{1}{2}((\alpha+r\gamma-Q)^{2}-\beta^{2})}\int_{\mathbb{A}_{0,n}^{r}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}.$
(3.9)

Recall that for each $r^{\prime}\in\\{0,...,r-1\\}$, the term
$\int_{\mathbb{A}_{0,1}^{r-r^{\prime}}\times\mathbb{A}_{1,n}^{r^{\prime}}}\Psi_{\alpha}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$
converges almost everywhere as $n\to\infty$. By monotone convergence, we
deduce that
$\int_{\mathbb{A}_{0,1}^{r-r^{\prime}}\times\mathbb{A}_{1,n}^{r^{\prime}}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$
converges to a finite limit, denoted $\int_{A_{0,1}^{r-r^{\prime}}\times
e^{-1}\mathbb{D}^{r^{\prime}}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{S^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$.
Going back to (3.7), we have

$\displaystyle\int_{\mathbb{A}_{0,n+1}^{r}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$
$\displaystyle\;\leqslant\;\sum_{r^{\prime}=0}^{r-1}\binom{r}{r^{\prime}}\int_{\mathbb{A}_{0,1}^{r-r^{\prime}}}\int_{e^{-1}\mathbb{D}^{r^{\prime}}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$
$\displaystyle\quad+e^{r\gamma(\alpha-\alpha_{r,s})}e^{-\frac{1}{2}((\alpha+r\gamma-Q)^{2}-\beta^{2})}\int_{\mathbb{A}_{0,n}^{r}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{L^{1}(e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}})}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}.$

In the case $\alpha+r\gamma-Q>0$, we have $\beta=0$ so
$e^{r\gamma(\alpha-\alpha_{r,s})}e^{-\frac{1}{2}(\alpha+r\gamma-Q)^{2}}<1$ for
$\alpha<\alpha_{r,s}$. In the other case, we can choose $\beta$ arbitrarily
close to $Q-r\gamma-\alpha$ to make this constant less than 1 as well. Thus,
$\underset{n\to\infty}{\lim}\int_{\mathbb{A}_{0,n}^{r}}\left\|\Psi_{\alpha}(\boldsymbol{w})\right\|_{e^{\beta
c}\mathrm{d}c\otimes\mathbb{P}_{\mathbb{S}^{1}}}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}<\infty$,
which also implies
$\underset{n\to\infty}{\lim}\,\int_{\mathbb{A}_{0,n}^{r}}\Psi_{\alpha}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}<\infty$
almost everywhere.

Finally, we explain the changes in the case of arbitrary $\boldsymbol{s}$.
First, the binomial coefficient in the RHS of (3.7) must be replaced by a sum
over all subsets of $\\{1,...,r\\}$ of cardinality $r^{\prime}$. Second, the
scaling constant $e^{r\gamma(\alpha-\alpha_{r,s})}$ in (3.8) is replaced by
$e^{r\gamma(\alpha-\alpha_{r,|\boldsymbol{s}|/r})}$. The rest of the argument
is unchanged.

With this convergence in hand, we can apply dominated convergence to get that
$\mathcal{I}_{r,\boldsymbol{s}}^{t}\underset{t\to\infty}{\to}\mathcal{I}_{r,\boldsymbol{s}}$
locally uniformly in $\mathcal{W}_{r,\boldsymbol{s}}$. Since
$\mathcal{I}_{r,\boldsymbol{s}}^{t}$ is analytic for each $t>0$, its local
uniform limit is also analytic. ∎

The next proposition extends the previous one by saying that adding some
insertion points with no singularities do not change the integrability
properties of the integrals.

######  Proposition 3.3.

Let $(r,\boldsymbol{s})\in\mathcal{T}$, and $\widetilde{r}\in\mathbb{N}^{*}$.
For all $\alpha\in\mathcal{W}_{r,\boldsymbol{s}}$, the following limit exists
in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$:

$\underset{t\to\infty}{\lim}\,\int_{\mathbb{A}_{t}^{r+\widetilde{r}}}\Psi_{\alpha}(\boldsymbol{w},\widetilde{\boldsymbol{w}})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2\boldsymbol{s}}}|\mathrm{d}\widetilde{\boldsymbol{w}}|^{2}.$

As a consequence, the map
$\alpha\mapsto\underset{t\to\infty}{\lim}\,\int_{\mathbb{A}_{t}^{r+\widetilde{r}}}\Psi_{\alpha}(\boldsymbol{w},\widetilde{\boldsymbol{w}})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{\boldsymbol{w}^{\boldsymbol{s}}}|\mathrm{d}\widetilde{\boldsymbol{w}}|^{2}$
is well-defined and analytic on $\mathcal{W}_{r,\boldsymbol{s}}$, with values
in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$.

###### Proof.

In this proof, we write explicitly the dependence on $\mu$ and
$\widetilde{r}$, i.e. we write
$\mathcal{I}_{r,(s)}^{\mu,\widetilde{r}}(\alpha)$, and similarly
$\Psi_{\alpha}^{\mu,\widetilde{r}}$.

The function $\mu\mapsto\Psi_{\alpha}^{\mu,0}(\boldsymbol{w})$ is analytic in
the region $\mathrm{Re}(\mu)>0$, with values in $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$. On the other hand, we have

$\partial_{\mu}^{\widetilde{r}}\Psi_{\alpha}^{\mu,0}(\boldsymbol{w})=(-1)^{\widetilde{r}}\int_{\mathbb{D}^{\widetilde{r}}}\Psi_{\alpha}^{\mu,\widetilde{r}}(\boldsymbol{w},\widetilde{w}_{1},...,\widetilde{w}_{r})\prod_{j=1}^{\widetilde{r}}|\mathrm{d}\widetilde{w}_{j}|^{2},$
(3.10)

so the RHS lies in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$ since the LHS does.
Note that (3.10) also holds for the regularised function
$\Psi_{\alpha}^{\mu,0,t}(\boldsymbol{w})$. By uniform absolute convergence for
$\alpha\in\mathcal{W}_{r,s}$ and Cauchy’s integral formula on a small (but
fixed) loop around $\mu$, we then get for all $\mathrm{Re}(\mu)>0$ and
$\alpha\in\mathcal{W}_{r,s}$:

$\displaystyle\int_{\mathbb{D}^{r+\widetilde{r}}}\Psi_{\alpha}^{\mu,\widetilde{r}}(\boldsymbol{w},\widetilde{\boldsymbol{w}})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}|\mathrm{d}\widetilde{\boldsymbol{w}}|^{2}$
$\displaystyle=(-1)^{\widetilde{r}}\partial_{\mu}^{\widetilde{r}}\int_{\mathbb{D}^{r}}\int_{\mathbb{D}^{r}}\Psi_{\alpha}^{\mu,0}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}$
$\displaystyle=\frac{(-1)^{\widetilde{r}}\widetilde{r}!}{2i\pi}\oint\int_{\mathbb{D}^{r}}\Psi_{\alpha}^{\widetilde{\mu},0}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{|\boldsymbol{w}|^{2s}}\frac{\mathrm{d}\widetilde{\mu}}{(\widetilde{\mu}-\mu)^{\widetilde{r}+1}}.$

By the previous proposition, the last line is well-defined and analytic with
values in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$ for all
$\alpha\in\mathcal{W}_{r,s}$, and it is the limit of its $t$-regularisation.

∎

######  Remark 3.

In the sequel, we will say that an integral is _of type $(r,\boldsymbol{s})$_
if it is of the form

$\int_{\mathbb{D}^{r+\widetilde{r}}}\Psi_{\alpha}(\boldsymbol{w},\widetilde{\boldsymbol{w}})f(\boldsymbol{w},\widetilde{\boldsymbol{w}})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{\boldsymbol{w}^{\boldsymbol{s}}}|\mathrm{d}\widetilde{\boldsymbol{w}}|^{2},$
(3.11)

where $f$ is a smooth function on $\bar{\mathbb{D}}^{r+\widetilde{r}}$
converging smoothly on $\bar{\mathbb{D}}^{r+\widetilde{r}}$. By the previous
proposition, integrals of type $(r,\boldsymbol{s})$ converge absolutely in
$\mathcal{W}_{r,\boldsymbol{s}}$ and are analytic there.

We can also replace $\Psi_{\alpha}$ by its regularised version
$\Psi_{\alpha}^{t}$, and $f$ by a family $f_{t}$ of smooth functions in
$\bar{\mathbb{D}}^{r}$ converging smoothly to $f$ as $t\to\infty$. In this
case, the regularised integral is also said to be of type
$(r,\boldsymbol{s})$, and it converges to (3.11) as $t\to\infty$, for
$\alpha\in\mathcal{W}_{r,\boldsymbol{s}}$.

Our goal in the remainder of this section is to show that
$\mathcal{I}_{r,\boldsymbol{s}}$ extends analytically to a complex
neighborhood of $(-\infty,Q)$. We will do so in two steps. In Section 3.3, we
prove that $\mathcal{I}_{r,\boldsymbol{s}}$ has a meromorphic continuation
with possible poles on the Kac table. In Section 3.4, we exclude the
possibility that $\mathcal{I}_{r,\boldsymbol{s}}$ has a pole, i.e. we show
that $\mathcal{I}_{r,\boldsymbol{s}}$ extends analytically on the Kac table.

###  3.3. Meromorphic continuation

In this section, we prove that $\mathcal{I}_{r,\boldsymbol{s}}$ extends
meromorphically to the region $\mathcal{W}_{0}$, with possible poles contained
in the Kac table. We proceed in two steps. In Proposition 3.4, we prove that
the extension holds for the minimal partitions $(N,(1))$,
$N\in\mathbb{N}^{*}$. In Proposition 3.5, we prove that the extension holds
for all partitions in $\mathcal{T}_{N}$, for each $N\in\mathbb{N}^{*}$. Both
results are proved by induction.

Recall the region $\mathcal{W}_{r,\boldsymbol{s}}\subset\mathbb{C}$ introduced
in Proposition 3.2. For simplicity, we will write
$\mathcal{W}_{r}:=\mathcal{W}_{r,(1)}$. We have
$\mathcal{W}_{r+1}\subset\mathcal{W}_{r}$ for each $r\in\mathbb{N}$, and
$\mathcal{W}_{0}$ contains the half-line $(-\infty,Q)$.

######  Proposition 3.4.

Let $r\in\mathbb{N}$. The function
$\mathcal{I}_{r,(1)}:\mathcal{W}_{r}\to\mathcal{C}^{\prime}$ extends
meromorphically to $\mathcal{W}_{0}$. Moreover, the set of its poles is
contained in
$\\{\alpha_{r^{\prime},1}|\,1\;\leqslant\;r^{\prime}\;\leqslant\;r\\}$.

###### Proof.

We will prove that for each $r\in\mathbb{N}^{*}$, $\mathcal{I}_{r,(1)}$ can be
expressed as a sum of integrals involving only $r-1$ singularities of order 1
(i.e. integrals of the form $\mathcal{I}_{r-1,(1)}$). The result will then
follow by an induction on $r$.

Integrating formula (3.3) for $\partial_{w_{r}}\Psi_{\alpha}^{t}$ on
$\Delta^{r}_{t}$ with respect to the measure
$\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}|\mathrm{d}w_{r}|^{2}$,
we get (after an $\epsilon$-regularisation, which is harmless due to the fact
that we are on $\Delta^{r}_{t}$)

$\displaystyle\int_{\Delta^{r}_{t}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\left(\alpha\gamma\partial_{w_{r}}G(w_{r},0)+\gamma^{2}\sum_{j=1}^{r-1}\partial_{w_{r}}G(w_{r},w_{j})\right)\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}|\mathrm{d}w_{r}|^{2}$
(3.12)
$\displaystyle\quad=-\int_{\Delta_{t}^{r}}\mathbb{E}\left[\Psi_{\alpha}^{t}(\boldsymbol{w})\nabla
F(\partial_{w_{1}}G_{\partial}(w_{1},\cdot\,))\right]\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}|\mathrm{d}w_{r}|^{2}$
$\displaystyle\qquad+\mu\gamma^{2}\int_{\Delta_{t}^{r+1}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w},w_{r+1})F]\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}\partial_{w_{r}}G(w_{r},w_{r+1})|\mathrm{d}w_{r}|^{2}|\mathrm{d}w_{r+1}|^{2}$
$\displaystyle\qquad+\int_{\Delta_{t}^{r-1}}\int_{\partial
D_{t}(w_{1},...,w_{r-1})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\mathrm{d}\bar{w}_{r}\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}$
$\displaystyle\qquad+\mu\int_{\Delta^{r}_{t}}\int_{\partial
D(w_{r},e^{-t})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w},w_{r+1})F]\mathrm{d}\bar{w}_{r+1}\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}|\mathrm{d}w_{r}|^{2}$
$\displaystyle\quad=:\mathcal{I}_{1}^{t}+\mathcal{I}_{2}^{t}+\mathcal{I}_{3}^{t}+\mathcal{I}_{4}^{t},$

where $\mathcal{I}_{1}^{t}$ denotes the first line and so on.

Let us focus on the singularities coming from the LHS. By permutation symmetry
of $\Psi_{\alpha}^{t}$ and antisymmetry of
$(w_{r},w_{j})\mapsto\frac{1}{w_{r}-w_{j}}$, we have for all
$1\;\leqslant\;j\;\leqslant\;r-1$,

$\displaystyle\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{1}{w_{r}-w_{j}}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}|\mathrm{d}w_{r}|^{2}$
$\displaystyle=\frac{1}{2}\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{1}{w_{r}-w_{j}}\left(\frac{1}{w_{j}}-\frac{1}{w_{r}}\right)|\mathrm{d}w_{j}|^{2}|\mathrm{d}w_{r}|^{2}$
$\displaystyle=\frac{1}{2}\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}\frac{|\mathrm{d}w_{r}|^{2}}{w_{r}}.$

Therefore, since $\partial_{w_{r}}G_{\mathbb{D}}(w_{r},0)=-\frac{1}{2w_{r}}$
and $\partial_{w_{r}}G(w_{r},w_{1})=-\frac{1}{2}\frac{1}{w_{r}-w_{1}}$, we see
that the LHS is a multiple of $\mathcal{I}_{r,(1)}^{t}$. Namely, (3.12) can be
rewritten as:

$-\frac{\gamma}{2}(\alpha-\alpha_{r,1})\mathcal{I}_{r,(1)}^{t}=\mathcal{I}_{1}^{t}+\mathcal{I}_{2}^{t}+\mathcal{I}_{3}^{t}+\mathcal{I}_{4}^{t}.$
(3.13)

Let us now study the RHS of (3.12). Recalling (3.4), we see that
$\mathcal{I}^{t}_{1}$ is treated by the induction hypothesis at rank $r-1$.
Using the antisymmetry of $(w_{r},w_{r+1})\mapsto\frac{1}{w_{r}-w_{r+1}}$, we
see that $\mathcal{I}_{2}^{t}=0$. Finally, we focus on $\mathcal{I}_{3}^{t}$.
We have $\partial
D_{t}(w_{1},...,w_{r-1})=\mathbb{S}^{1}\cup\mathcal{C}_{t}(w_{1},...,w_{r-1})$,
where $\mathcal{C}_{t}(w_{1},...,w_{r-1})$ is a union of small circles centred
at 0, $w_{1},...,w_{r-1}$. We need to control the contribution of
$\mathcal{C}_{t}$. By passing to the $\epsilon$-regularisation, we see that
all the other bulk terms in (3.12) are uniformly integrable in the region
$\mathcal{W}_{r}$. This is then also true for the contribution of
$\mathcal{C}_{t}$, namely for all $\alpha\in\mathcal{W}_{r}$:

$\displaystyle\underset{t\to\infty}{\lim}\,\int_{\Delta_{t}^{r-1}}\int_{\mathcal{C}_{t}(w_{1},...,w_{r-1})}\Psi_{\alpha}^{t}(\boldsymbol{w})\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}\mathrm{d}\bar{w}_{r}=\underset{\epsilon\to
0}{\lim}\,\underset{t\to\infty}{\lim}\,\int_{\Delta_{t}^{r-1}}\int_{\mathcal{C}_{t}(w_{1},...,w_{r-1})}\Psi_{\alpha}^{t,\epsilon}(\boldsymbol{w})\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}\mathrm{d}\bar{w}_{r}=0.$

We will write $\widetilde{\mathcal{I}}_{3}^{t}$ for the contribution of
$\mathbb{S}^{1}$ in $\mathcal{I}_{3}^{t}$, namely

$\displaystyle\widetilde{\mathcal{I}}_{3}^{t}$
$\displaystyle:=\int_{\mathbb{S}^{1}}\int_{\Delta_{t}^{r-1}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\prod_{j=1}^{r-1}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}}\mathrm{d}\bar{w}_{r}$

By the Girsanov transform, for $w_{r}\in\mathbb{S}^{1}$, the change of measure
$e^{\gamma\varphi(w_{r})-\frac{\gamma^{2}}{2}G_{\partial}(w_{r},w_{r})}$
amounts to shifting $P\varphi$ by the (deterministic) function $w\mapsto\gamma
G_{\partial}(w_{r},w)$, which is smooth in $\mathbb{D}$. Therefore, this shift
does not change anything about the convergence of $\Psi_{\alpha}^{t}$ in
$\mathcal{C}^{\prime}$. Thus, $\widetilde{\mathcal{I}}_{3}^{t}$ falls in the
case of the induction hypothesis at rank $r-1$, i.e. converges
$\mathcal{I}_{3}^{t}(\alpha)$ converges to a limit $\mathcal{I}_{3}(\alpha)$
for $\alpha\in\mathcal{W}_{r-1}$, and $\mathcal{I}_{3}$ is analytic in
$\mathcal{W}_{r-1}$. Similarly, we know $\mathcal{I}_{4}^{t}$ converges to $0$
for $\alpha\in\mathcal{W}_{r+1}$.

Therefore, for all $\alpha\in\mathcal{W}_{r+1}\subset\mathcal{W}_{r-1}$,
(3.13) has a limit as $t\to\infty$, which is given by

$-\frac{\gamma}{2}(\alpha-\alpha_{r,1})\mathcal{I}_{r,(1)}(\alpha)=\mathcal{I}_{1}(\alpha)+\mathcal{I}_{3}(\alpha).$

Moreover, we have shown that the RHS has an analytic continuation to the
region $\mathcal{W}_{r-1,1}$ and LHS is analytic in $\mathcal{W}_{r}$ by
proposition 3.2. This defines a meromorphic extension of $\mathcal{I}_{r,(1)}$
in the region $\mathcal{W}_{r}$, with a unique possible pole at
$\alpha=\alpha_{r,1}$.∎

######  Proposition 3.5.

Let $(r,\boldsymbol{s})\in\mathcal{T}$. The function
$\mathcal{I}_{r,\boldsymbol{s}}:\mathcal{W}_{r,\boldsymbol{s}}\to\mathcal{C}^{\prime}$
extends meromorphically to $\mathcal{W}_{0}$. Moreover, the set of its poles
is contained in the Kac table.

###### Proof.

We will prove the result by induction on
$(\mathcal{T}_{|\boldsymbol{s}|},\prec)$. The initialisation holds by the
previous proposition since $(|\boldsymbol{s}|,(1))$ is the smallest partition
in $\mathcal{T}_{|\boldsymbol{s}|}$. For the heredity, we will show that
$\mathcal{I}_{r,\boldsymbol{s}}$ can be expressed as a sum of integrals of
strictly lower type.

By an integration by parts, we get for all
$(w_{2},...,w_{r})\in\Delta_{t}^{r-1}$

$\displaystyle(s_{1}-1)\int_{D_{t}(w_{2},...,w_{r})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}}}$
$\displaystyle=\frac{i}{2}\int_{\partial
D_{t}(w_{2},...,w_{r})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}$
$\displaystyle\quad+\int_{D_{t}(w_{2},...,w_{r})}\partial_{w_{1}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\frac{\mathrm{d}\bar{w}_{1}}{w_{1}^{s_{1}-1}}$

Combining with (3.3) integrated with respect to
$\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\prod_{j=2}^{r}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}^{s_{j}}}$,
we get

$\displaystyle\int_{\Delta_{t}^{r}}\left(\frac{s_{1}-1}{w_{1}}+\alpha\gamma\partial_{w_{1}}G(w_{1},0)+\gamma^{2}\sum_{j=2}^{r^{*}}\partial_{w_{1}}G(w_{1},w_{j})\right)\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\prod_{k=2}^{r}\frac{|\mathrm{d}w_{k}|^{2}}{w_{k}^{s_{k}}}$
(3.14)
$\displaystyle\quad=-\gamma^{2}\sum_{j=r^{*}+1}^{r}\int_{\Delta_{t}^{r}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\partial_{w_{1}}G(w_{1},w_{j})\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\prod_{k=2}^{r}\frac{|\mathrm{d}w_{k}|^{2}}{w_{k}^{s_{k}}}$
$\displaystyle\qquad+\int_{\Delta_{t}^{r}}\mathbb{E}\left[\Psi_{\alpha}^{t}(\boldsymbol{w})\nabla
F(\partial_{w_{1}}G_{\partial}(w_{1},\cdot\,))\right]\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\prod_{k=2}^{r}\frac{|\mathrm{d}w_{k}|^{2}}{w_{k}^{s_{k}}}$
$\displaystyle\qquad-\mu\gamma^{2}\int_{\Delta_{t}^{r+1}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\partial_{w_{1}}G(w_{1},w_{r+1})\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\prod_{k=2}^{r}\frac{|\mathrm{d}w_{k}|^{2}}{w_{k}^{s_{k}}}|\mathrm{d}w_{r+1}|^{2}$
$\displaystyle\qquad-\frac{i}{2}\int_{\partial
D_{t}(w_{2},...,w_{r})}\int_{\Delta_{t}^{r-1}(w_{1})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\prod_{k=2}^{r}\frac{|\mathrm{d}w_{k}|^{2}}{w_{k}^{s_{k}}}\frac{\mathrm{d}\bar{w}_{1}}{w_{1}^{s_{1}-1}}$
$\displaystyle\qquad-\mu\int_{\Delta^{r}_{t}}\int_{\partial
D(w_{1},e^{-t})}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w},w_{r+1})F]\mathrm{d}\bar{w}_{r+1}\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\prod_{j=2}^{r}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}^{s_{j}}}$
$\displaystyle\quad=:\mathcal{J}_{1}^{t}+\mathcal{J}_{2}^{t}+\mathcal{J}_{3}^{t}+\mathcal{J}_{4}^{t}+\mathcal{J}_{5}^{t}.$

Let us first focus on the LHS. As in the proof of Proposition 3.4, we can
symmetrise the terms involving
$\partial_{w_{1}}G(w_{1},w_{j})=-\frac{1}{2}\frac{1}{w_{1}-w_{j}}$ for
$1\;\leqslant\;j\;\leqslant\;r^{*}$ to see that the LHS is a multiple of
$\mathcal{I}_{r,\boldsymbol{s}}^{t}$. Namely, (3.14) can be written as

$-\frac{\gamma}{2}(\alpha-\alpha_{r^{*},s_{1}})\mathcal{I}^{t}_{r,\boldsymbol{s}}=\mathcal{J}_{1}^{t}+\mathcal{J}_{2}^{t}+\mathcal{J}_{3}^{t}+\mathcal{J}_{4}^{t}+\mathcal{J}_{5}^{t}.$
(3.15)

Next, we study the integrals appearing in the RHS of (3.14), starting with
$\mathcal{J}_{1}^{t}$. For $r^{*}+1\;\leqslant\;j\;\leqslant\;r$, we
symmetrise again
$\partial_{w_{1}}G(w_{1},w_{j})=-\frac{1}{2}\frac{1}{w_{1}-w_{j}}$ to get
(using $s_{1}-1\;\geqslant\;s_{j}$

$\displaystyle\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{1}{w_{1}-w_{j}}\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-1}}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}^{s_{j}}}$
$\displaystyle=\frac{1}{2}\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{1}{w_{1}-w_{j}}\left(\frac{1}{w_{1}^{s_{1}-1}w_{j}^{s_{j}}}-\frac{1}{w_{1}^{s_{j}}w_{j}^{s_{1}-1}}\right)|\mathrm{d}w_{1}|^{2}|\mathrm{d}w_{j}|^{2}$
$\displaystyle=\frac{1}{2}\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{1}{w_{1}^{s_{1}-1}w_{j}^{s_{1}-1}}\frac{w_{j}^{s_{1}-s_{j}-1}-w_{1}^{s_{1}-s_{j}-1}}{w_{1}-w_{j}}|\mathrm{d}w_{1}|^{2}|\mathrm{d}w_{j}|^{2}$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{s_{1}-s_{j}-1}\int_{\Delta_{t}^{2}}\Psi_{\alpha}^{t}(\boldsymbol{w})\frac{|\mathrm{d}w_{1}|^{2}}{w_{1}^{s_{1}-l}}\frac{|\mathrm{d}w_{j}|^{2}}{w_{j}^{s_{j}+l}}.$

For each $1\;\leqslant\;l\;\leqslant\;s_{1}-s_{j}-1$, the permutation
$(\widetilde{r},\widetilde{\boldsymbol{s}})$ obtained from
$(r,\boldsymbol{s})$ by decreasing (resp. increasing) $s_{1}$ (resp. $s_{j}$)
by $l$ is strictly smaller than $(r,\boldsymbol{s})$ (with respect to
$\prec$). Therefore, $\mathcal{J}_{1}^{t}$ is absolutely convergent is a
region $\mathcal{W}_{r^{\prime},\boldsymbol{s}^{\prime}}$ for some
$(r^{\prime},\boldsymbol{s}^{\prime})\prec(r,\boldsymbol{s})$, and the limit
$\mathcal{J}_{1}$ is analytic in this region. The same treatment and
conclusion can be given to the integral $\mathcal{J}_{3}^{t}$, and we denote
its limit by $\mathcal{J}_{3}$. Recalling (3.4), the same statement is true
for $\mathcal{J}_{2}^{t}$, and we denote its limit by $\mathcal{J}_{2}$.
Finally, we focus on $\mathcal{J}_{4}^{t}$. Writing $\partial
D_{t}(w_{2},...,w_{r})=\mathbb{S}^{1}\cup\mathcal{C}_{t}(w_{2},...,w_{r})$ as
in the proof of Proposition 3.4 and arguing similarly, we get that the
contribution of $\mathcal{C}_{t}$ tends to zero as $t\to\infty$, for all
$\alpha\in\mathcal{W}_{r,\boldsymbol{s}}$. As in the proof of Proposition 3.4,
we then define

$\displaystyle\widetilde{\mathcal{J}}_{4}^{t}$
$\displaystyle:=-\int_{\mathbb{S}^{1}}\int_{\Delta^{r-1}_{t}}\mathbb{E}[\Psi_{\alpha}^{t}(\boldsymbol{w})F]\prod_{k=2}^{r}\frac{|\mathrm{d}w_{k}|^{2}}{w_{k}^{s_{k}}}\frac{\mathrm{d}\bar{w}_{1}}{w_{1}^{s_{1}-1}}$

and $\widetilde{\mathcal{J}}_{4}^{t}$ also converges absolutely in a strictly
larger region $\mathcal{W}$ than $\mathcal{W}_{r,\boldsymbol{s}}$. For
$\mathcal{J}_{5}(\alpha)$, we can show it goes to $0$ when
$\alpha\in\mathcal{W}_{r+1,\boldsymbol{s}}$. Thus, (3.15) has a limit as
$t\to\infty$, which satisfies for all
$\alpha\in\mathcal{W}_{r+1,\boldsymbol{s}}$

$-\frac{\gamma}{2}(\alpha-\alpha_{r^{*},s_{1}})\mathcal{I}_{r,\boldsymbol{s}}(\alpha)=\mathcal{J}_{1}(\alpha)+\mathcal{J}_{2}(\alpha)+\mathcal{J}_{4}(\alpha).$
(3.16)

The RHS is a sum of integrals indexed by strictly lower partitions than
$(r,\boldsymbol{s})$. By the induction hypothesis, it extends meromorphically
to $\mathcal{W}_{0}$, with possible poles on the Kac table. Therefore, the RHS
of (3.16) defines a meromorphic extension of $\mathcal{I}_{r,\boldsymbol{s}}$
to $\mathcal{W}_{0}$. The set of its poles is the set of poles of the RHS,
possibly augmented with $\alpha_{r^{*},s_{1}}$. ∎

###  3.4. Study on the Kac table

We have shown that $\mathcal{I}_{r,\boldsymbol{s}}$ extends meromorphically to
a complex neighborhood of $\\{\alpha<Q\\}$, with possible poles on the Kac
table. In this section, we prove some special cases of Theorem 1.1. Later in
Section 3.6, we will combine these results with representation theoretic
information to deduce Theorems 1.1 & 1.2 in all generality.

In the course of our analysis, we will need to treat the case
$\gamma^{2}\not\in\mathbb{Q}$ first. The extension to all $\gamma\in(0,2)$
will be handled by a continuity argument, which is contained in Proposition
3.9. Whenever necessary, we will make the $\gamma$-dependence explicit by
writing
$\mathcal{I}_{r,\boldsymbol{s}}^{\gamma}(\alpha)=\mathcal{I}_{r,\boldsymbol{s}}(\alpha)$.

We will denote by $\mathcal{P}_{r,\boldsymbol{s}}\subset kac^{-}$the set of
poles of $\mathcal{I}_{r,\boldsymbol{s}}$. It follows straightforwardly from
the induction formula that for all $(r,\boldsymbol{s})\in\mathcal{T}_{N}$, we
have

$\mathcal{P}_{r,\boldsymbol{s}}\subset\left\\{\alpha_{r^{*},s_{1}}\right\\}\bigcup\left(\underset{(\widetilde{r},\widetilde{\boldsymbol{s}})\prec(r,\boldsymbol{s})}{\bigcup}\mathcal{P}_{\widetilde{r},\widetilde{\boldsymbol{s}}}\right),$

######  Proposition 3.6.

Suppose $\gamma^{2}\not\in\mathbb{Q}$, and let $r,s\in\mathbb{N}^{*}$. For all
partitions $(\widetilde{r},\widetilde{\boldsymbol{s}})\in\mathcal{T}_{rs}$,
the map
$\alpha\mapsto\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha)$
is regular at $\alpha_{r,s}$.

###### Proof.

First, since $\alpha_{r,s}<Q-(\frac{\gamma}{2}r+\frac{2}{\gamma}\frac{s}{2})$,
Proposition 3.2 gives that the integral $\mathcal{I}_{r,(s)}$ is regular at
$\alpha_{r,s}$. Moreover, the partition $(r,(s))$ is the only one at level
$rs$ having $r^{*}=r$ and $s_{1}=s$.

Let $(\widetilde{r},\widetilde{\boldsymbol{s}})\in\mathcal{T}_{rs}$ arbitrary.
By the induction formulae (3.12), (3.14), we can express
$(\alpha-\alpha_{\widetilde{r}^{*},\widetilde{s}_{1}})\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}$
in terms of integrals of lower type. We can iterate the formula until we reach
either an integral of type $(r,(s))$ or of empty type, which are both analytic
at $\alpha=\alpha_{r,s}$ by virtue of Lemma 3.3 and Proposition 3.2 (note that
$\alpha_{r,s}<Q-(\frac{\gamma}{2}r+\frac{2}{\gamma}\frac{s}{2})$). Since the
numbers
$\\{\alpha_{r^{\prime},s^{\prime}}|\,r^{\prime},s^{\prime}\in\mathbb{N}^{*}\\}$
are pairwise distinct, the possible poles of $\mathcal{I}_{r,\boldsymbol{s}}$
obtained by this procedure are all distinct from $\alpha_{r,s}$. ∎

######  Corollary 3.7.

Suppose $\gamma^{2}\not\in\mathbb{Q}$, and let $r,s\in\mathbb{N}^{*}$. For all
partitions $(\widetilde{r},\widetilde{\boldsymbol{s}})\in\mathcal{T}_{rs}$ and
all $\nu\in\mathcal{T}$, the map
$\alpha\mapsto\mathbf{L}_{-\nu}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha)\in
e^{-\beta c}\mathcal{D}(\mathcal{Q})$ is regular at $\alpha=\alpha_{r,s}$.

###### Proof.

We prove this by induction on the length of $\nu$.

The case $\nu=\eset$ is the content of Corollary 3.6.

Suppose the result holds for all partitions of length $\ell$, for some
$\ell\in\mathbb{N}$. Let $\nu_{1}\;\geqslant\;...\;\geqslant\;\nu_{\ell}$ be a
partition of length $\ell$. Let $n\;\geqslant\;\nu_{1}$, and let
$\widetilde{\nu}=(n,\nu_{1},...,\nu_{\ell})$. By the induction hypothesis, the
family
$\mathbf{L}_{-\nu}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha)\in
e^{-\beta c}\mathcal{D}(\mathcal{Q})$ is analytic in a neighbourhood
$\mathcal{U}$ of $\alpha=\alpha_{r,s}$. Since $\mathbf{L}_{-n}:e^{-\beta
c}\mathcal{D}(\mathcal{Q})\to e^{-\beta c}\mathcal{D}^{\prime}(\mathcal{Q})$
is bounded, we get that
$\mathbf{L}_{-\widetilde{\nu}}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}:\mathcal{U}\to
e^{-\beta c}\mathcal{D}^{\prime}(\mathcal{Q})$. Moreover, we know that the
family
$\mathbf{L}_{-\widetilde{\nu}}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}:\mathcal{U}\setminus\\{\alpha_{r,s}\\}\to
e^{-\beta c}\mathcal{D}(\mathcal{Q})$ is analytic. By Cauchy’s integral
formula, we then have the following equality in $e^{-\beta
c}\mathcal{D}^{\prime}(\mathcal{Q})$:

$\mathbf{L}_{-\widetilde{\nu}}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha_{r,s})=\frac{1}{2i\pi}\oint\mathbf{L}_{-\widetilde{\nu}}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha)\frac{\mathrm{d}\alpha}{\alpha-\alpha_{r,s}},$

where the integral is over a loop around $\alpha_{r,s}$ contained in
$\mathcal{U}$. This shows that
$\mathbf{L}_{-\widetilde{\nu}}\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha_{r,s})$
is actually well-defined in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$, and the
map
$\alpha\mapsto\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}(\alpha)$
is regular at $\alpha_{r,s}$ with values in $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$. ∎

The final results of this section provides the continuity property that will
allow us to extend the results from $\gamma^{2}\not\in\mathbb{Q}$ to arbitrary
$\gamma\in(0,2)$. Essentially, we will prove that the singular integrals
$\mathcal{I}_{r,\boldsymbol{s}}$ are continuous with respect to
$\gamma\in(0,2)$, with values in $\mathcal{C}^{\prime}$.

######  Lemma 3.8.

Let $F$ be a Fréchet space, and $(f_{n})_{n\;\geqslant\;0}$ be a bounded
sequence in $\mathcal{C}^{1}([0,1];F)$ such that $f_{n}$ converges pointwise
to some function $f:[0,1]\to\mathcal{C}$. Then, $f$ is continuous on $[0,1]$.

###### Proof.

Suppose first that $(F,\left\|\cdot\right\|_{F})$ is Banach. Define the
constant
$C:=\sup_{n\;\geqslant\;0}\left\|f_{n}\right\|_{\mathcal{C}^{1}([0,1];F)}$,
which is finite by hypothesis. For all $n\;\geqslant\;0$, and all
$t_{0},t\in[0,1]$, we have
$\left\|f_{n}(t)-f_{n}(t_{0})\right\|_{F}\;\leqslant\;\int_{t_{0}}^{t}\left\|f_{n}^{\prime}(s)\right\|_{F}\mathrm{d}s\;\leqslant\;C|t-t_{0}|$.
Taking $n\to\infty$ gives

$\left\|f(t)-f(t_{0})\right\|_{F}=\underset{n\to\infty}{\lim}\,\left\|f_{n}(t)-f_{n}(t_{0})\right\|_{F}\;\leqslant\;C|t-t_{0}|,$

showing that $f$ is Lipschitz. In the general case, the topology of $F$ is
generated by a countable family of seminorms, and we can use the previous
estimate for each one of these seminorms. ∎

######  Proposition 3.9.

For all partitions $(r,\boldsymbol{s})\in\mathcal{T}$, the function
$(\gamma,\alpha)\mapsto\mathcal{I}_{r,\boldsymbol{s}}^{\gamma}(\alpha)\in\mathcal{C}^{\prime}$
is continuous on the set $\\{(\gamma,\alpha)|\,\gamma\in(0,2)\text{ and
}\alpha\in\mathcal{W}_{r,\boldsymbol{s}}\\}$.

Note that the region $\mathcal{W}_{r,\boldsymbol{s}}$ depends on $\gamma$, but
this dependence is continuous, so we will keep it implicit in order to ease
notations.

###### Proof.

For $\alpha\in\mathcal{W}_{r,\boldsymbol{s}}$, $\gamma_{1},...,\gamma_{r}$ in
a neighbourhood of $\gamma$, and $\epsilon>0$, let us consider

$\displaystyle
F^{\epsilon}_{\gamma}(\boldsymbol{w}):=e^{(\alpha+\sum_{j=1}^{r}\gamma_{j}-Q)c}\mathbb{E}_{\varphi}\left[\epsilon^{\frac{\alpha^{2}}{2}}e^{\alpha
X_{\epsilon}(0)}\prod_{j=1}^{r}\epsilon^{\frac{\gamma_{j}^{2}}{2}}e^{\gamma_{j}X_{\epsilon}(w_{j})}e^{-\mu
e^{\gamma c}\int_{\mathbb{D}}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}}|\mathrm{d}z|^{2}}\right];$
$\displaystyle\mathcal{J}_{r,\boldsymbol{s}}^{\boldsymbol{\gamma},\epsilon}(\gamma):=\int_{\mathbb{D}^{r}}F_{\gamma}^{\epsilon}(\boldsymbol{w})\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{\boldsymbol{w}^{\boldsymbol{s}}},$

where we write $\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{r})$.

For each $\epsilon>0$,
$\mathcal{J}_{r,\boldsymbol{s}}^{\boldsymbol{\gamma},\epsilon}$ is
differentiable in $\gamma$, and we have

$\displaystyle\frac{\partial\mathcal{J}_{r,\boldsymbol{s}}^{\boldsymbol{\gamma},\epsilon}}{\partial\gamma}$
$\displaystyle=-\mu\int_{\mathbb{D}}\mathbb{E}_{\varphi}\left[\left(c+X_{\epsilon}(w)+\gamma\log\epsilon\right)\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}(w)}\epsilon^{\frac{\alpha^{2}}{2}}e^{\alpha
X_{\epsilon}(0)}\prod_{j=1}^{r}\epsilon^{\frac{\gamma_{j}^{2}}{2}}e^{\gamma_{j}X_{\epsilon}(w_{j})}e^{-\mu
e^{\gamma c}\int_{\mathbb{D}}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
X_{\epsilon}}|\mathrm{d}z|^{2}}\right]|\mathrm{d}w|^{2}\frac{|\mathrm{d}\boldsymbol{w}|^{2}}{\boldsymbol{w}^{\boldsymbol{s}}}$
$\displaystyle=-\mu\frac{\partial}{\partial\gamma_{r+1}}_{|\gamma_{r+1}=\gamma}\mathcal{J}_{r+1,(\boldsymbol{s},0)}^{(\boldsymbol{\gamma},\gamma_{r+1}),\epsilon}$

Now, the family of functions
$\gamma_{r+1}\mapsto\mathcal{J}_{r+1,(\boldsymbol{s},0)}^{(\boldsymbol{\gamma},\gamma_{r+1}),\epsilon}$
is analytic and uniformly bounded. Moreover, it converges to the analytic
limit
$\mathcal{J}_{r+1,(\boldsymbol{s},0)}^{(\boldsymbol{\gamma},\gamma_{r+1})}$ as
$\epsilon\to 0$, for $\gamma_{r+1}$ is a neighbourhood of $\gamma$. By the
previous lemma, we deduce that
$\gamma\mapsto\mathcal{I}_{r,\boldsymbol{s}}^{\gamma}\in\mathcal{C}^{\prime}$
is continuous on $\mathcal{W}_{r,\boldsymbol{s}}$. ∎

######  Corollary 3.10.

For all partitions $(r,\boldsymbol{s})\in\mathcal{T}$, the function
$(\gamma,\alpha)\mapsto\mathcal{I}_{r,\boldsymbol{s}}^{\gamma}(\alpha)\in\mathcal{C}^{\prime}$
is continuous on the set

$\\{(\gamma,\alpha)|\,\gamma\in(0,2)\text{ and
}\alpha\in\mathcal{W}_{0}\\}\setminus\cup_{r^{\prime},s^{\prime}\in\mathbb{N}^{*}}\left\\{(\gamma,(1-r^{\prime})\frac{\gamma}{2}+(1-s^{\prime})\frac{2}{\gamma})|\,\gamma\in(0,2)\right\\}.$
(3.17)

For each $\gamma\in(0,2)$, the function
$\alpha\mapsto\mathcal{I}^{\gamma}_{r,\boldsymbol{s}}(\alpha)\in\mathcal{C}^{\prime}$
is meromorphic in $\mathcal{W}_{0}$, with possible poles on the Kac table.

###### Proof.

By the induction formulae (3.12), (3.14), we can express
$(\alpha-\alpha_{\widetilde{r}^{*},\widetilde{s}_{1}})\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}$
in terms of integrals of lower type, with a larger domain of continuity in
$(\gamma,\alpha)$ as given by Proposition 3.9. Iterating the formula, we find
a polynomial $P(\alpha)=\prod_{k}(\alpha-\alpha_{r_{k},s_{k}})$ such that
$P(\alpha)\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}^{\gamma}(\alpha)$
is continuous on $\\{(\gamma,\alpha)|\gamma\in(0,2)\text{ and
}\alpha\in\mathcal{W}_{0}\\}$. Thus,
$\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}$ is continuous on the
set (3.17), and the function
$\alpha\mapsto\mathcal{I}_{\widetilde{r},\widetilde{\boldsymbol{s}}}^{\gamma}(\alpha)$
is meromorphic in $\mathcal{W}_{0}$ for each $\gamma\in(0,2)$, with possible
poles in the Kac table. ∎

###  3.5. Descendant states vs. singular integrals

The goal of this section is to express the descendant states using the
singular integrals studied in the previous subsection.

In the statement of the next lemma,
$\boldsymbol{\varepsilon}=(\varepsilon_{n})_{n\in\mathbb{N}^{*}}$ is a
sequence of complex numbers, and we use the multi-index notation. The
differential operator $\frac{\partial}{\partial\varepsilon_{n}}$ denotes the
complex differential in $\varepsilon_{n}$.

######  Lemma 3.11.

Let $F\in\mathcal{C}$ and $\mathbf{k}\in\mathcal{T}$. For all $t>0$, we have

$\displaystyle
e^{t(2\Delta_{\alpha}+|\mathbf{k}|)}\mathbb{E}\left[e^{-t\mathbf{H}}\left(\varphi^{\mathbf{k}}e^{(\alpha-Q)c}\right)F\right]$
$\displaystyle\qquad=e^{(\alpha-Q)c}\frac{\partial^{\ell(\mathbf{k})}}{\partial\boldsymbol{\varepsilon}^{\mathbf{k}}}_{|\boldsymbol{\varepsilon}=0}\mathbb{E}\left[\exp\left(-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}\prod_{n\in\mathbb{N}^{*}}e^{\gamma\mathrm{Re}\left(\frac{\varepsilon_{n}}{n}z^{-n}\right)}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}\right)F\left(\varphi+\sum_{n\in\mathbb{N}^{*}}\frac{1}{n}\mathrm{Re}(\varepsilon
e^{-ni\theta})\right)\right]$

###### Proof.

Recall that we write $X=X_{\mathbb{D}}+P\varphi$ the GFF in the disc, where
$X_{\mathbb{D}}$ is a Dirichlet GFF and $P\varphi$ is the harmonic extension
of the boundary field, which is written in Fourier modes as

$P\varphi(z)=2\mathrm{Re}\left(\sum_{n=1}^{\infty}\varphi_{n}z^{n}\right).$

Let us define
$\varphi_{n}(t):=\frac{1}{2\pi}\int_{0}^{2\pi}X(e^{-t+i\theta})e^{-ni\theta}\mathrm{d}\theta$,
as well as $X_{n}(t):=\mathrm{Re}(\varphi_{n}(t))$,
$Y_{n}(t):=\mathrm{Im}(\varphi_{n}(t))$. From [GKRV23, Section 4], $X_{n}$ and
$Y_{n}$ are independent OU processes with parameters
$(\kappa,\sigma)=(n,\frac{1}{\sqrt{2}})$ (notation from Appendix A).

For notational simplicity, we only treat the case
$\varphi^{\mathbf{k}}=\varphi_{n}^{k}$ for some $k\in\mathbb{N}^{*}$. The
general case is then a straightforward consequence of the independence of the
processes $(X_{n}(t),Y_{n}(t))_{n\in\mathbb{N}^{*}}$. For each
$\varepsilon\in\mathbb{C}$, we consider the martingale

$\displaystyle\mathcal{E}_{\varepsilon}(t)$
$\displaystyle:=e^{e^{nt}\left(2\mathrm{Re}(\varepsilon)X_{n}(t)-\frac{\mathrm{Re}(\varepsilon)^{2}}{n}\sinh(nt)\right)}e^{e^{nt}\left(-2\mathrm{Im}(\varepsilon)Y_{n}(t)-\frac{\mathrm{Im}(\varepsilon)^{2}}{n}\sinh(nt)\right)}$
$\displaystyle=e^{e^{nt}\left(2\mathrm{Re}(\varepsilon\varphi_{n}(t))-\frac{|\varepsilon|^{2}}{n}\sinh(nt)\right)}$

Note that $\mathcal{E}_{\varepsilon}(t)$ is real analytic in $\varepsilon$,
and

$\mathcal{E}_{\varepsilon}(t)=e^{e^{nt}\varepsilon\varphi_{n}(t)}(1+\bar{\varepsilon}O(1))=(1+\bar{\varepsilon}O(1))\sum_{k=0}^{\infty}\frac{\varepsilon^{k}}{k!}e^{nkt}\varphi_{n}(t)^{k},$

so that

$e^{nkt}\varphi_{n}(t)^{k}=\frac{\partial^{k}}{\partial\varepsilon^{k}}_{|\varepsilon=0}\mathcal{E}_{\varepsilon}(t).$

Next, we evaluate the effect of reweighting the measure by the martingale
$\mathcal{E}_{\varepsilon}(t)$. Since $\mathrm{Re}(\varphi_{n}(t))$ and
$\mathrm{Im}(\varphi_{n}(t))$ are independent, Appendix A tells us that the
reweighting amounts to the shifts

$\mathrm{Re}(\varphi_{n}(t))\mapsto\mathrm{Re}(\varphi_{n}(t))+\frac{\mathrm{Re}(\varepsilon)}{n}\sinh(nt);\qquad\mathrm{Im}(\varphi_{n}(t))\mapsto\mathrm{Im}(\varphi_{n}(t))-\frac{\mathrm{Im}(\varepsilon)}{n}\sinh(nt),$

or $\varphi_{n}(t)\mapsto\varphi_{n}(t)+\frac{\bar{\varepsilon}}{n}\sinh(nt)$.
Writing $z=e^{-t+i\theta}\in\mathbb{D}$, the corresponding shift of the field
is

$\displaystyle X(z)\mapsto
X(e^{-t+i\theta})+\frac{2}{n}\sinh(nt)\mathrm{Re}(\varepsilon
e^{-ni\theta})=X(z)+\frac{1}{n}\mathrm{Re}(\varepsilon(z^{-n}-\bar{z}^{n})).$

By Girsanov’s theorem, we then get

$\displaystyle
e^{(\alpha-Q)c}\mathbb{E}_{\varphi}\left[\mathcal{E}_{\varepsilon}(t)e^{-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]$
$\displaystyle=e^{2\mathrm{Re}(\varepsilon\varphi_{n})}e^{(\alpha-Q)c}\mathbb{E}_{\varphi}\left[\exp\left(-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}e^{\frac{\gamma}{n}\mathrm{Re}\left(\varepsilon(z^{-n}-\bar{z}^{n})\right)}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}\right)\right].$
(3.18)

Now, we take the expectation over $\varphi$. Again by Girsanov’s theorem,
reweighting $\mathbb{P}_{\mathbb{S}^{1}}$ by
$e^{2\mathrm{Re}(\varepsilon\varphi_{n})-\frac{|\varepsilon|^{2}}{2n}}$
amounts to the shift
$\varphi_{n}\mapsto\varphi_{n}+\frac{\bar{\varepsilon}}{2n}$. The
corresponding shift of the harmonic extension is $P\varphi(z)\mapsto
P\varphi(z)+\frac{1}{n}\mathrm{Re}(\varepsilon\bar{z}^{n})$. Thus, integrating
(3.18) over $\mathbb{P}_{\mathbb{S}^{1}}$ gives

$\displaystyle
e^{(\alpha-Q)c}\mathbb{E}\left[\mathcal{E}_{\varepsilon}(t)e^{-\mu e^{\gamma
c}\int_{\mathbb{A}_{t}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}F(\varphi)\right]$
(3.19)
$\displaystyle\qquad=e^{\frac{|\varepsilon|^{2}}{2n}}e^{(\alpha-Q)c}\mathbb{E}\left[\exp\left(-\mu
e^{\gamma c}\int_{\mathbb{A}_{t}}e^{\frac{\gamma}{n}\mathrm{Re}(\varepsilon
z^{-n})}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}\right)F\left(\varphi+\frac{1}{n}\mathrm{Re}(\varepsilon
e^{-ni\theta})\right)\right]$

Finally, we differentiate $k$ times at $\varepsilon=0$. The RHS coincides with
the RHS of the statement of the lemma. As for the LHS, we have

$\displaystyle
e^{(\alpha-Q)c}\frac{\partial^{k}}{\partial\varepsilon^{k}}\mathbb{E}\left[\mathcal{E}_{\varepsilon}(t)e^{-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}F(\varphi)\right]$
$\displaystyle=e^{nkt}e^{(\alpha-Q)c}\mathbb{E}\left[\varphi_{n}(t)^{k}e^{-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}\frac{\mathrm{d}M_{\gamma}(z)}{|z|^{\gamma\alpha}}}\right]$
$\displaystyle=e^{t(2\Delta_{\alpha}+nk)}e^{-t\mathbf{H}}\left(\varphi_{n}^{k}e^{(\alpha-Q)c}\right).$

As mentioned at the beginning of the proof, the case of general $\mathbf{k}$
follows by reweighting by the product of martingales
$\mathcal{E}_{\varepsilon_{n}}(t)=e^{e^{nt}(2\mathrm{Re}(\varepsilon_{n}\varphi_{n}(t))-\frac{|\varepsilon|^{2}}{n}\sinh(nt))}$
and using independence.

∎

The previous lemma will allow us to express descendant states as singular
integrals. Let us list two important consequences of this representation.

######  Lemma 3.12.

Suppose $\gamma^{2}\not\in\mathbb{Q}$. For all $r,s\in\mathbb{N}^{*}$ and
$\mathbf{k}\in\mathcal{T}_{rs}$, the map
$\alpha\mapsto\mathcal{P}_{\alpha}(\varphi^{\mathbf{k}})\in e^{-\beta
c}\mathcal{D}(\mathcal{Q})$ is analytic at $\alpha_{r,s}$.

###### Proof.

Let $F\in\mathcal{C}$. We will use Lemma 3.11 to evaluate
$\mathbb{E}[\mathcal{P}_{\alpha}(\mathcal{Q})F]$ in terms of singular
integrals. By the Leibniz rule, the differential operator
$\frac{\partial^{\ell(\mathbf{k})}}{\partial\boldsymbol{\varepsilon}^{\mathbf{k}}}$
hits alternatively
$F(\varphi+\sum_{n\in\mathbb{N}^{*}}\frac{1}{n}\mathrm{Re}(\varepsilon
e^{-ni\theta}))$ and the potential. We have
$\partial_{\varepsilon}F(\varphi+\frac{1}{n}\mathrm{Re}(\varepsilon
e^{-ni\theta}))=\frac{1}{2n}\partial_{-n}F(\varphi+\frac{1}{n}\mathrm{Re}(\varepsilon
e^{-ni\theta}))\in\mathcal{C}$. Taking higher order derivatives, we see that
all the terms produced by differentiating $F$ are in $\mathcal{C}$.

Next, we evaluate the derivative of the potential, starting at order 1:

$\frac{\partial}{\partial\varepsilon_{n}}e^{-\mu e^{\gamma
c}\int_{\mathbb{A}_{t}}e^{\frac{\gamma}{n}\mathrm{Re}(\varepsilon_{n}z^{-n})}\mathrm{d}M_{\gamma}(z)}=-\mu\frac{\gamma}{2n}e^{\gamma
c}\int_{\mathbb{A}_{t}}e^{\gamma
X(w)-\frac{\gamma^{2}}{2}\mathbb{E}[X(w)^{2}]}e^{\frac{\gamma}{n}\mathrm{Re}(\varepsilon_{n}w^{-n})}e^{-\mu
e^{\gamma
c}\int_{\mathbb{A}_{t}}e^{\frac{\gamma}{n}\mathrm{Re}(\varepsilon_{n}z^{-n})}\mathrm{d}M_{\gamma}(z)}\frac{|\mathrm{d}w|^{2}}{w^{n}}.$

When we iterate this formula and evaluate it at $\boldsymbol{\varepsilon}=0$,
we see that

$\frac{\partial^{\ell(\mathbf{k})}}{\partial\boldsymbol{\varepsilon}^{\mathbf{k}}}_{|\boldsymbol{\varepsilon}=0}\mathbb{E}_{\varphi}\left[e^{\alpha
X(0)-\frac{\alpha^{2}}{2}\mathbb{E}[X(0)^{2}]}e^{-\mu e^{\gamma
c}\int_{\mathbb{A}_{t}}\prod_{n\in\mathbb{N}^{*}}e^{\frac{\gamma}{n}\mathrm{Re}(\varepsilon_{n}z^{-n})}\mathrm{d}M_{\gamma}(z)}\right]\in\mathcal{C}^{\prime}\otimes\mathbb{C}_{rs}[\mu],$

i.e. it is a polynomial of degree $rs$ in $\mu$ with coefficients in
$\mathcal{C}^{\prime}$. Moreover, these coefficients are linear combinations
of singular integrals at level $rs$ studied in the previous subsection. By
Proposition 3.2, all these singular integrals are analytic at $\alpha_{r,s}$,
so we deduce that $\alpha\mapsto\mathcal{P}_{\alpha}(\varphi^{\mathbf{k}})$ is
analytic at $\alpha_{r,s}$.

This shows that $\alpha\mapsto\mathcal{P}_{\alpha}(\varphi^{\mathbf{k}})$ is
analytic at $\alpha_{r,s}$, with values in $\mathcal{C}^{\prime}$. By Cauchy’s
formula, we have the equality in $\mathcal{C}^{\prime}$

$\mathcal{P}_{\alpha_{r,s}}(\varphi^{\mathbf{k}})=\frac{1}{2i\pi}\oint\mathcal{P}_{\alpha}(\varphi^{\mathbf{k}})\frac{\mathrm{d}\alpha}{\alpha-\alpha_{r,s}},$

where the integral is over a small loop around $\alpha_{r,s}$. On the other
hand, we know that $\alpha\mapsto\mathcal{P}_{\alpha}(\varphi^{\mathbf{k}})$
takes values in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$ away from the Kac
table, so the previous display shows that
$\mathcal{P}_{\alpha_{r,s}}(\varphi^{\mathbf{k}})\in e^{-\beta
c}\mathcal{D}(\mathcal{Q})$. The same equation shows that
$\alpha\mapsto\mathcal{P}_{\alpha}(\varphi^{\mathbf{k}})$ is analytic around
$\alpha_{r,s}$, with values in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$. ∎

Recall the Virasoro representation $(\mathbf{L}_{n})_{n\in\mathbb{Z}}$
introduced in [BGK+23b, Theorem 1.1]: for each $n\in\mathbb{Z}$, these
operators can be defined as bounded operators $\mathbf{L}_{n}:e^{-\beta
c}\mathcal{D}(\mathcal{Q})\to e^{-\beta
c}\mathcal{D}^{\prime}(\mathcal{Q})$111The theorem is stated for
$n\;\geqslant\;0$, but it extends easily to
$\mathbf{L}_{-n}=\mathbf{L}_{n}^{*}$ since
$|\langle\mathbf{L}_{-n}F,G\rangle_{\mathcal{H}}|=|\langle
F,\mathbf{L}_{n}G\rangle_{\mathcal{H}}|\;\leqslant\;C\left\|F\right\|_{\mathcal{D}(\mathcal{Q})}\left\|G\right\|_{\mathcal{D}(\mathcal{Q})}$
for all $F,G\in\mathcal{C}$. The same applies to weighted spaces..

######  Lemma 3.13.

Let $r,s\in\mathbb{N}^{*}$ and $\chi\in\mathcal{F}$ such that
$\alpha\mapsto\mathcal{P}_{\alpha}(\chi)\in e^{-\beta
c}\mathcal{D}(\mathcal{Q})$ is analytic at $\alpha_{r,s}$. For all
$\nu\in\mathcal{T}$, the map
$\alpha\mapsto\mathbf{L}_{-\nu}\mathcal{P}_{\alpha}(\chi)\in e^{-\beta
c}\mathcal{D}(\mathcal{Q})$ is analytic at $\alpha_{r,s}$.

###### Proof.

We work by induction on the length $\ell=\ell(\nu)$. The case $\ell=0$ is true
by hypothesis. Suppose the result holds at rank $\ell\;\geqslant\;0$, and let
us prove it at rank $\ell+1$. Let $(\nu,\nu_{\ell+1})$ be a partition of
length $\ell+1$ (with $\nu$ a partition of length $\ell$). Since
$\mathbf{L}_{-\nu_{\ell+1}}:e^{-\beta c}\mathcal{D}(\mathcal{Q})\to e^{-\beta
c}\mathcal{D}^{\prime}(\mathcal{Q})$ is continuous, the map
$\alpha\mapsto\mathbf{L}_{-\nu_{\ell+1}}(\mathbf{L}_{-\nu}\mathcal{P}_{\alpha}(\chi))$
is analytic at $\alpha_{r,s}$, with values in $e^{-\beta
c}\mathcal{D}^{\prime}(\mathcal{Q})$. To show it’s indeed valued in $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$, we write Cauchy’s formula

$\mathcal{P}_{\alpha_{r,s}}(\chi)=\frac{1}{2i\pi}\oint\mathcal{P}_{\alpha}(\chi)\frac{\mathrm{d}\alpha}{\alpha-\alpha_{r,s}},$

where the contour is a small loop around $\alpha_{r,s}$ which doesn’t
intersect the Kac table. Now, for $\alpha$ away from the Kac table,
$\mathcal{P}_{\alpha}(\chi)$ is a linear combination of Virasoro descendants
$(\Psi_{\alpha,\nu})_{|\nu|=rs}$, which is uniformly bounded in $e^{-\beta
c}\mathcal{D}(\mathcal{Q})$. Indeed, this follows from the construction of the
module $\mathcal{V}_{\alpha}$ from [BGK+23b]. Hence,
$\mathcal{V}_{\alpha_{r,s}}(\chi)\in e^{-\beta c}\mathcal{D}(\mathcal{Q})$,
and $\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is analytic in a neighbourhood
of $\alpha_{r,s}$, with values in $e^{-\beta c}\mathcal{D}(\mathcal{Q})$. ∎

###  3.6. Conclusion

From the previous subsections, we have a family of Poisson operators:

$\left\\{\begin{aligned} &\mathcal{W}_{0}\times\mathcal{F}&&\to e^{-\beta
c}\mathcal{D}(\mathcal{Q});\\\
&(\alpha,\chi)&&\mapsto\mathcal{P}_{\alpha}(\chi)\end{aligned}\right.$

For each $\chi\in\mathcal{F}$, the map
$\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is meromorphic in $\mathcal{W}_{0}$
with possible poles in the Kac table $kac^{-}$. For each fixed
$\alpha\in\mathcal{W}_{0}\setminus kac^{-}$,
$\mathcal{P}_{\alpha}:\mathcal{F}\to e^{-\beta c}\mathcal{D}(\mathcal{Q})$ is
a linear map. The explicit expression of $\mathcal{P}_{\alpha}(\chi)$ is a
linear combination of the singular integrals studied in Section 3.

From the explicit expression
$\Phi_{\alpha}(\mathbf{L}_{-\nu}^{0,2Q-\alpha}\mathds{1})=\Psi_{\alpha,\nu}$,
the composition
$\Phi_{\alpha}=\mathcal{P}_{\alpha}\circ\Phi_{\alpha}^{0}:\mathcal{F}\to
e^{-\beta c}\mathcal{D}(\mathcal{Q})$ is well-defined and analytic in
$\mathcal{W}_{0}$. In particular, the poles of $\mathcal{P}$ may only occur at
the zeros of $\Phi_{\alpha}^{0}$ (which is consistent with the fact that
$\Phi_{\alpha}^{0}$ may vanish only on the Kac table). We then define
$\mathcal{V}_{\alpha}\subset e^{-\beta c}\mathcal{D}(\mathcal{Q})$ by

$\mathcal{V}_{\alpha}:=\mathrm{ran}(\Phi_{\alpha})\simeq\mathcal{F}/\ker(\Phi_{\alpha}).$

Moreover, the explicit expression of $\Phi_{\alpha}$ gives

$\mathcal{V}_{\alpha}=\mathrm{span}\,\\{\Psi_{\alpha,\nu}=\mathbf{L}_{-\nu}\Psi_{\alpha}|\,\nu\in\mathcal{T}\\}.$

More precisely, for $\alpha\not\in kac$, [BGK+23b] showed that
$(\mathcal{V}_{\alpha},(\mathbf{L}_{n})_{n\in\mathbb{Z}})$ is a highest-weight
representation, and the Poissson operator intertwines the free field and
Liouville representation, i.e.
$\Phi_{\alpha}\circ\mathbf{L}_{n}^{0,2Q-\alpha}=\mathbf{L}_{n}\circ\Phi_{\alpha}$
(here, we slightly abuse notations by identifying $\mathbf{L}_{n}$ with its
restriction to $\mathcal{V}_{\alpha}$). The Virasoro relations extend to
$\mathcal{V}_{\alpha_{r,s}}$ by continuity, so that
$\mathcal{V}_{\alpha_{r,s}}$ is also a highest-weight representation,
generated by the highest-weight vector $\Psi_{\alpha_{r,s}}$. In particular,
$\ker\Phi_{\alpha_{r,s}}$ is a submodule of
$(\mathcal{V}_{2Q-\alpha_{r,s}}^{0},(\mathbf{L}_{n}^{0,2Q-\alpha_{r,s}})_{n\in\mathbb{Z}})$.
It remains to show that it is isomorophic to $\mathcal{V}_{\alpha_{r,s}}^{0}$,
i.e. $\ker\Phi_{\alpha_{r,s}}=\ker\Phi_{\alpha_{r,s}}^{0}$. Since
$\mathcal{V}_{\alpha_{r,s}}^{0}$ is the quotient of the Verma module by the
maximal proper submodule, we have immediately
$\ker\Phi_{\alpha_{r,s}}\subset\ker\Phi_{\alpha_{r,s}}^{0}$. The next
proposition gives the converse inclusion and ends the proofs of Theorems 1.1
and 1.2.

######  Proposition 3.14.

For all $r,s\in\mathbb{N}^{*}$, we have
$\ker\Phi_{\alpha_{r,s}}^{0}\subset\ker\Phi_{\alpha_{r,s}}$. Thus,
$\alpha\mapsto\mathcal{P}_{\alpha}(\chi)$ is analytic at $\alpha_{r,s}$ for
all $\chi\in\mathcal{F}$.

###### Proof.

_Case $\gamma^{2}\not\in\mathbb{Q}$._

In this case, $\ker\Phi_{\alpha_{r,s}}^{0}$ is the Virasoro-module (for the
representation $(\mathbf{L}_{n}^{0,2Q-\alpha_{r,s}})_{n\in\mathbb{Z}}$)
generated by the level $rs$-singular vector. Moreover, $\Phi_{\alpha}^{0}$ has
only simple zeros. Let
$\chi_{r,s}\in\mathcal{V}_{2Q-\alpha_{r,s}}\simeq\mathcal{F}$ be the
level-$rs$ singular vector. Then,
$\frac{1}{\alpha-\alpha_{r,s}}\Phi_{\alpha}^{0}(\chi)\to\chi_{r,s}^{\prime}$
for some $\chi_{r,s}^{\prime}\in\mathcal{F}_{rs}$. From Lemma 3.6,
$\alpha\mapsto\mathcal{P}_{\alpha}(\chi^{\prime})$ is analytic at
$\alpha_{r,s}$, so that
$\frac{1}{\alpha-\alpha_{r,s}}\Phi_{\alpha}(\chi)\to\mathcal{P}_{\alpha_{r,s}}(\chi_{r,s}^{\prime})$.
In particular, $\Phi_{\alpha_{r,s}}(\chi_{r,s})=0$.

It remains to study the descendant states. By the intertwining property and
Proposition 3.7, we have

$\displaystyle\frac{1}{\alpha-\alpha_{r,s}}\Phi_{\alpha_{r,s}}(\mathbf{L}_{-\nu}^{0,2Q-\alpha_{r,s}}\chi_{r,s})=\frac{1}{\alpha-\alpha_{r,s}}\mathbf{L}_{-\nu}\mathcal{P}_{\alpha}(\Phi_{\alpha}^{0}(\chi_{r,s}))\underset{\alpha\to\alpha_{r,s}}{\to}\,\mathbf{L}_{-\nu}\mathcal{P}_{\alpha_{r,s}}(\chi_{r,s}^{\prime}).$

In particular,
$\Phi_{\alpha_{r,s}}(\mathbf{L}_{-\nu}^{0,2Q-\alpha_{r,s}}\chi_{r,s})=0$.

_General case._

Let $(\gamma_{n})\in(0,2)^{\mathbb{N}}$ such that
$\gamma^{2}_{n}\not\in\mathbb{Q}$ and $\gamma_{n}\to\gamma$ as $n\to\infty$.
Let $\chi\in\mathcal{F}$. By Corollary 3.10, we have for all
$p\in\mathbb{N}^{*}$,

$\oint\mathcal{P}_{\alpha}^{\gamma}(\chi)(\alpha-\alpha_{r,s})^{p-1}\mathrm{d}\alpha=\underset{n\to\infty}{\lim}\,\oint\mathcal{P}_{\alpha}^{\gamma_{n}}(\chi)(\alpha-\alpha_{r,s})^{p-1}\mathrm{d}\alpha=0,$

where the contour is a small loop around $\alpha_{r,s}$ which is uniformly
bounded away from the Kac table (this contour exists since the Kac table is
discrete and depends continuously on $\gamma$). This proves that the polar
part of $\mathcal{P}_{\alpha_{r,s}}^{\gamma}(\chi)$ vanishes at
$\alpha_{r,s}$, i.e. $\mathcal{P}_{\alpha}^{\gamma}(\chi)$ is regular at
$\alpha_{r,s}$. ∎

######  Remark 4.

For $\gamma^{2}\not\in\mathbb{Q}$, the previous proof identifies
$\ker\Phi_{\alpha_{r,s}}$ with the module generated by
$\Phi_{\alpha_{r,s}}(\chi_{r,s}^{\prime})=\mathbf{S}_{\alpha_{r,s}}\Psi_{\alpha_{r,s}}^{\prime}\in
e^{-\beta c}\mathcal{D}(\mathcal{Q})$, where we denote
$\Psi_{\alpha}^{\prime}=\partial_{\alpha}\Psi_{\alpha}$, and
$\mathbf{S}_{\alpha_{r,s}}$ is the combination of Virasoro generators at level
$rs$ creating the singular vector. This vector is of highest-weight
$\Delta_{\alpha_{r,s}}+rs$ for the representation
$(\mathbf{L}_{n})_{n\in\mathbb{Z}}$, but is a primary of weight
$\Delta_{\alpha_{r,s}}$ for the antiholomorphic representation
$(\widetilde{\mathbf{L}}_{n})_{n\in\mathbb{Z}}$. In particular,
$\widetilde{\mathbf{S}}_{\alpha_{r,s}}\mathbf{S}_{\alpha_{r,s}}\Psi_{\alpha_{r,s}}^{\prime}$
is a singular vector for the antiholomorphic representation. It turns out that
the Poisson operator $\alpha\mapsto\mathcal{P}_{\alpha}(\chi_{r,s}^{\prime})$
has a pole at $\alpha_{r,s}$, which is the phenomenon underlying the higher
equations of motions [Zam04]. This will be studied in subsequent work.

##  4\. BPZ equations

In this section, we prove Theorem 1.3 using Theorem 1.2. Recall that Theorem
1.3 states that conformal blocks involving degenerate primary fields satisfy
certain PDEs on Teichmüller space. This is due to the linear relations holding
in the degenerate representations of Theorem 1.2. To deduce the PDEs from the
fact that singular vectors vanish, we need to express the action of Virasoro
generators as differential operators on Teichmüller space. Analytically, these
PDEs can be interpreted as the fact that conformal blocks exponentiate the
Virasoro algebra. The geometric formulation of this property is that blocks
are horizontal sections of a vector bundle with connection form given by the
stress-energy tensor.

Most of the material of this section is not new, and is a summary of the setup
and results of [BGK+23a]. In Sections 4.1,4.2 and 4.3, we give some background
on Teichmüller theory and define some line bundles over Teichmüller space. In
Section 4.4, we define conformal blocks as
$\mathcal{D}^{\prime}(\mathcal{Q})$-valued sections of a line-bundle over
Teichmüller space, satisfying certain geometric properties. The proof of
Theorem 1.3 is an easy consequence of this definition.

###  4.1. Teichmüller spaces and the semigroup of annuli

####  4.1.1. Teichmüller spaces

Let $\Sigma$ be a compact surface of genus $\mathbf{g}$ and $m\in\mathbb{N}$
such that $2\mathbf{g}-2+m+1>0$ (i.e. $m\;\geqslant\;3$ if $\mathbf{g}=0$). We
denote by $\mathcal{T}_{\Sigma,m+1}$ the set of equivalence classes of tuples
$(\Sigma,J,\boldsymbol{p})$ where $J\in\Gamma(\mathrm{End}(T\Sigma))$ is an
(integrable) almost complex structure on $\Sigma$, and
$\boldsymbol{p}=(p_{0},...,p_{m})\in\Sigma^{m}$ are pairwise distinct marked
points. The equivalence relation is as follows: two tuples
$(\Sigma,J,\boldsymbol{p})$ and
$(\Sigma,\widetilde{J},\widetilde{\boldsymbol{p}})$ are equivalent if there
exists a diffeomorphism $\phi\in\mathrm{Diff}(\Sigma)$ isotopic to the
identity such that $\phi^{*}J=J$ and $\phi^{-1}(\widetilde{p}_{j})=p_{j}$ for
all $j=1,...,m$. The set $\mathcal{T}_{\Sigma,m+1}$ is called the _Teichmüller
space_ of $\Sigma$ with $m+1$ marked points, and it is a complex analytic
manifold of dimension $3\mathbf{g}-3+m+1$. We always require at least one
marked point, since ultimately we want to insert a degenerate primary field at
some marked point.

Next, we define $\mathcal{T}_{\Sigma,m+1,\uparrow}$ to be the set of
equivalence classes of tuples $(\Sigma,J,\boldsymbol{p},\boldsymbol{\xi})$,
where $(\Sigma,J,\boldsymbol{p})$ is as above, and
$\boldsymbol{\xi}=(\xi_{0},...,\xi_{m})\in\prod_{j=0}^{m}(T_{p_{j}}\Sigma\setminus\\{0\\})$
are non-zero tangent vectors at the marked points222For a complex analytic
manifold $\mathcal{T}$, we will always denote by $T\mathcal{T}$ the
_holomorphic_ tangent bundle, namely we drop the customary
$(1,0)$-superscript.. The equivalence relation is as follows: two tuples
$(\Sigma,J,\boldsymbol{p})$ and
$(\Sigma,\widetilde{J},\widetilde{\boldsymbol{p}})$ are equivalent if
$(\Sigma,J,\boldsymbol{p})$ is equivalent to
$(\Sigma,\widetilde{J},\widetilde{\boldsymbol{p}}))$ in the previous sense,
and the corresponding diffeomorphism satisfies
$\mathrm{d}\phi_{p_{j}}(\xi_{j})=\widetilde{\xi_{j}}$ for all $j=0,...,m$. It
is known that $\mathcal{T}_{\Sigma,m+1,\uparrow}$ is a complex analytic
manifold of dimension $3\mathbf{g}-3+2m+2$ and is a principal
$\mathbb{C}^{m+1}$-bundle over $\mathcal{T}_{\Sigma,m+1}$ [ABO18, Theorem 2.2,
Proposition 2.3]. Let

$\pi_{\Sigma,m+1,\uparrow}:\mathcal{T}_{\Sigma,m+1,\uparrow}\to\mathcal{T}_{\Sigma,m}$

be the corresponding projection. For each $j=0,...,m$, the Dehn twist along a
simple closed curve homotopic to $p_{j}$ acts on
$\mathcal{T}_{\Sigma,m+1,\uparrow}$. The quotient of
$\mathcal{T}_{\Sigma,m+1,\uparrow}$ by the action generated by these Dehn
twists is a $(\mathbb{C}^{*})^{m+1}$-bundle over $\mathcal{T}_{\Sigma,m+1}$.
Concretely, the fibre over $(\Sigma,J,\boldsymbol{p})$ is (isomorphic to) the
universal cover of
$\prod_{j=0}^{m}(T_{p_{j}}\Sigma\setminus\\{0\\})\simeq(\mathbb{C}^{*})^{m+1}$.
We denote by $\xi_{j}\cdot\tau_{j}$ the action of $\tau_{j}\in\mathbb{C}$ in
the fibre $\pi_{\Sigma,m+1,\uparrow}^{-1}\\{\Sigma,J,\boldsymbol{p}\\}$. A
full Dehn twist corresponds to $\tau_{j}=1$.

Now, we want to attach local coordinates at the marked points. For
$\epsilon>0$, let $\mathcal{T}_{\Sigma,m+1,\uparrow}^{\epsilon}$ be the set of
equivalence classes of tuples
$(\Sigma,J,\boldsymbol{p},\boldsymbol{\xi},\boldsymbol{\psi})$, where
$(\Sigma,J,\boldsymbol{p},\boldsymbol{\xi})$ is as above, and
$\boldsymbol{\psi}=(\psi_{0},...,\psi_{m})$ are conformal embeddings
$\psi_{j}:e^{\epsilon}\mathbb{D}\to(\Sigma,J)$ such that no two images
overlap, $\psi_{j}(0)=z_{j}$ and
$(\mathrm{d}\psi_{j})_{0}(\partial_{z})=\xi_{j}$ for all $j=0,...,m$, where
$z$ is the standard coordinate on $\mathbb{C}$. Two such tuples
$(\Sigma,J,\boldsymbol{p},\boldsymbol{\xi},\boldsymbol{\psi})$,
$(\Sigma,\widetilde{J},\widetilde{\boldsymbol{p}},\widetilde{\boldsymbol{\xi}},\widetilde{\boldsymbol{\psi}})$
are equivalent if $(\Sigma,J,\boldsymbol{p},\boldsymbol{\xi})$ is equivalent
to
$(\Sigma,\widetilde{J},\widetilde{\boldsymbol{p}},\widetilde{\boldsymbol{\xi}})$
in the $\mathcal{T}_{\Sigma,m+1,\uparrow}$-sense, and the corresponding
diffeomorphism $\phi$ is such that $\phi\circ\psi_{j}=\widetilde{\psi}_{j}$
for all $j=0,...,m$. The extra data $\psi_{j}$ gives a local coordinate
$z_{j}:=\psi_{j}^{-1}$ around $p_{j}$ and vanishing there. The space
$\mathcal{T}_{\Sigma,m+1,\uparrow}^{\epsilon}$ is an infinite dimensional
complex analytic Fréchet manifold; it is fibred over
$\mathcal{T}_{\Sigma,m+1,\uparrow}$, and we denote by

$\pi_{\Sigma,m+1,\uparrow}^{\epsilon}:\mathcal{T}_{\Sigma,m+1,\uparrow}^{\epsilon}\to\mathcal{T}_{\Sigma,m+1,\uparrow}$

the corresponding (holomorphic) projection. Finally, set

$\pi_{\Sigma,m+1}^{\epsilon}:=\pi_{\Sigma,m+1,\uparrow}\circ\pi_{\Sigma,m+1,\uparrow}^{\epsilon}:\mathcal{T}_{\Sigma,m+1,\uparrow}^{\epsilon}\to\mathcal{T}_{\Sigma,m+1}.$

Notice that for
$\boldsymbol{\psi}_{1},\boldsymbol{\psi}_{2}\in(\pi_{\Sigma,m+1,\uparrow}^{\epsilon})^{-1}\\{\Sigma,J,\boldsymbol{p},\boldsymbol{\xi}\\}$,
the conformal maps
$\boldsymbol{\psi}_{2}^{-1}\circ\boldsymbol{\psi}_{1}:e^{\epsilon}\mathbb{D}\to\mathbb{C}$
fix zero and have unit conformal radius, namely
$(\psi_{2,j}^{-1}\circ\psi_{1,j})^{\prime}(0)=1$ for all $j=1,...,m$. The
space of such conformal maps is simply connected: given $\psi$ with
$\psi(0)=0$, $\psi^{\prime}(0)=1$, the path
$\psi_{t}(z):=\frac{1}{t}\psi(tz)$, $t\in[0,1]$, contracts $\psi$ to the
identity. Finally, we note the following fact: if $\mathbf{v}_{j}$ is a
holomorphic vector field in the neighbourhood of $p_{j}$, then its power
series expansion at order 1 is independent of the choice of conformal
embedding $\psi_{j}$.

####  4.1.2. The semigroup of annuli

When $\Sigma$ is a topological annulus (a sphere minus two discs), we get a
related Teichmüller space called the _semigroup of annuli_ , which we denote
by $\mathcal{S}^{\epsilon}$. Any element of $\mathcal{S}^{\epsilon}$ is
represented uniquely by a pair $(f_{\mathrm{in}},f_{\mathrm{out}})$, where
$f_{\mathrm{in}}$ (resp. $f_{\mathrm{out}}$) is conformal on
$e^{\epsilon}\mathbb{D}$ (resp. $e^{-\epsilon}\mathbb{D}^{*}$, with
$\mathbb{D}^{*}=\widehat{\mathbb{C}}\setminus\bar{\mathbb{D}})$),
$f_{\mathrm{in}}(0)=0$, $f_{\mathrm{out}}(\infty)=\infty$,
$f_{\mathrm{in}}^{\prime}(0)=1$, and the two maps have non-overlapping images.
The annulus is identified with the region
$A=\widehat{\mathbb{C}}\setminus(f_{\mathrm{in}}(\bar{\mathbb{D}})\cup
f_{\mathrm{out}}(\bar{\mathbb{D}^{*}}))$, and the boundaries of $A$ are
parametrised by
$f_{\mathrm{in}}|_{\mathbb{S}^{1}},f_{\mathrm{out}}|_{\mathbb{S}^{1}}$
respectively. Following [BGK+23a], this representative is called the _model
form_. We refer to $f_{\mathrm{in}}(\mathbb{S}^{1})$ (resp.
$f_{\mathrm{out}}(\mathbb{S}^{1})$) as the incoming (resp. outgoing) boundary
component. The semigroup structure is as follows (see Figure 2): given
$A_{1},A_{2}\in\mathcal{S}^{\epsilon}$, the annulus $A_{2}\\#A_{1}$ is
obtained by gluing the incoming boundary of $A_{2}$ to the outgoing boundary
of $A_{1}$ according to the boundary parametrisations. This operation is
complex analytic in the Fréchet topology on $\mathcal{S}^{\epsilon}$.

Figure 2. The gluing of two annuli $A_{2},A_{1}$ with model forms
$(f_{\mathrm{in},2},f_{\mathrm{out},2})$ and
$(f_{\mathrm{in},1},f_{\mathrm{out},1})$. There are conformal embeddings
$\psi_{2},\psi_{1}$ defined on $A_{2},A_{1}$ respectively such that the model
form $(f_{\mathrm{in}},f_{\mathrm{out}})$ of the gluing $A_{2}\\#A_{1}$
satisfies the following equalities on $\mathbb{S}^{1}$:
$f_{\mathrm{out}}=\psi_{2}\circ f_{\mathrm{out},2}$,
$f_{\mathrm{in}}=\psi_{1}\circ f_{\mathrm{in},1}$, $\psi_{2}\circ
f_{\mathrm{in},2}=\psi_{1}\circ f_{\mathrm{out},1}$. The dashed curve is the
gluing interface $\psi_{2}\circ
f_{\mathrm{in},2}(\mathbb{S}^{1})=\psi_{1}\circ
f_{\mathrm{out},1}(\mathbb{S}^{1})$.

The fundamental group of $\mathcal{S}^{\epsilon}$ is isomorphic to
$\mathbb{Z}$ and is generated by the Dehn twist along the non-contractible
cycle. The universal cover $\widetilde{\mathcal{S}}^{\epsilon}$ is described
by triples $(f_{\mathrm{in}},f_{\mathrm{out}},\tau)$ where $\tau\in\mathbb{R}$
and $(f_{\mathrm{in}},f_{\mathrm{out}})\in\mathcal{S}^{\epsilon}$ with the
extra requirement that $f_{\mathrm{out}}^{\prime}(\infty)\in\mathbb{R}_{+}$.
The canonical projection
$\widetilde{\mathcal{S}}^{\epsilon}\to\mathcal{S}^{\epsilon}$ is
$(f_{\mathrm{in}},f_{\mathrm{out}},\tau)\mapsto(f_{\mathrm{in}},f_{\mathrm{out}}(e^{2i\pi\tau}\cdot\,))$.

Let $\mathcal{S}^{\epsilon}_{+}$ be the sub-semigroup of
$\mathcal{S}^{\epsilon}$ consisting of annuli $A\in\mathcal{S}^{\epsilon}$
with model form $(f_{\mathrm{in}},f_{\mathrm{out}})$ such that
$f_{\mathrm{out}}$ is a Möbius transformation of the sphere. We can identify
$\mathcal{S}^{\epsilon}_{+}$ with the space of conformal maps preserving the
unit disc:

$\mathcal{S}^{\epsilon}_{+}\simeq\\{f\text{ conformal in
}e^{\epsilon}\mathbb{D},\,f(0)=0,\,f(\bar{\mathbb{D}})\subset\mathbb{D}\\}.$

The semigroup structure in $\mathcal{S}^{\epsilon}_{+}$ is simply the
composition of maps. The fundamental group is also isomorphic to $\mathbb{Z}$,
and the universal cover $\widetilde{\mathcal{S}}^{\epsilon}$ is described by
pairs $\widetilde{f}=(f,\tau)$ where $\tau\in\mathbb{R}$ and
$f\in\mathcal{S}^{\epsilon}$ satisfies $f^{\prime}(0)\in\mathbb{R}_{+}$. The
canonical projection is $(f,\tau)\mapsto f(e^{2i\pi\tau}\cdot\,)$. The
semigroup structure in $\widetilde{\mathcal{S}}_{+}^{\epsilon}$ is
$(f_{2},\tau_{2})\circ(f_{1},\tau_{1})=(f_{2}\circ f_{1},\tau_{2}+\tau_{1})$.
For $\tau\in\mathbb{H}$, we define the annulus
$\mathbb{A}_{\tau}\in\widetilde{\mathcal{S}}^{\epsilon}_{+}$ by the pair
$(e^{-2\pi\mathrm{Im}\tau}\mathrm{id}_{\mathbb{D}},\mathrm{Re}\,\tau)$, and we
call it the _straight annulus_ of modulus $\tau$.

There is an action of $\widetilde{\mathcal{S}}^{\epsilon}_{+}$ on
$\mathcal{T}^{\epsilon}_{\Sigma,1,\uparrow}$ by:

$(\Sigma,J,p,\xi,\psi)\cdot(f,\tau):=\left(\Sigma,J,p,\xi\cdot(\tau+\frac{1}{2i\pi}\log
f^{\prime}(0)),\psi\circ f\right).$

This action preserves the fibres of the projection $\pi^{\epsilon}_{\Sigma,1}$
(i.e. it only changes the tangent vector and local coordinate). Moreover, the
action is transitive in the fibres in the following sense. For all
$(\xi_{1},\psi_{1}),(\xi_{2},\psi_{2})\in(\pi_{\Sigma,m}^{\epsilon})^{-1}\\{(\Sigma,J,p)\\}$,
there exists $(f_{1},\tau_{1})$,
$(f_{2},\tau_{2})\in\widetilde{\mathcal{S}}_{+}^{\epsilon}$ such that
$(\Sigma,J,p,\xi_{1},\psi_{1})\cdot(f_{1},\tau_{1})=(\Sigma,J,p,\xi_{2},\psi_{2})\cdot(f_{2},\tau_{2})$.
Indeed, pick any analytic disc $\mathcal{D}\subset\Sigma$ such that
$\bar{\mathcal{D}}\subset\psi_{1}(\mathbb{D})\cap\psi_{2}(\mathbb{D})$ and
pick $f_{1}$ (resp. $f_{2}$) such that $\psi_{1}\circ f_{1}$ (resp.
$\psi_{2}\circ f_{2}$) is a conformal map from $\mathbb{D}$ to $\mathcal{D}$,
and choose $\tau_{1},\tau_{2}\in\mathbb{R}$ such that the arguments match. All