# The quantum group dual
of the first-row subcategory
for the generic Virasoro VOA

Shinji Koshida, Kalle Kytölä  Department of Physics, Faculty of Science and
Engineering, Chuo University, Kasuga, Bunkyo, Tokyo 112-8551, Japan
<EMAIL_ADDRESS>Department of Mathematics and Systems Analysis,
Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland
<EMAIL_ADDRESS>
###### Abstract.

In several examples it has been observed that a module category of a vertex
operator algebra (VOA) is equivalent to a category of representations of some
quantum group. The present article is concerned with developing such a duality
in the case of the Virasoro VOA at generic central charge; arguably the most
rudimentary of all VOAs, yet structurally complicated. We do not address the
category of all modules of the generic Virasoro VOA, but we consider the
infinitely many modules from the first row of the Kac table. Building on an
explicit quantum group method of Coulomb gas integrals, we give a new proof of
the fusion rules, we prove the analyticity of compositions of intertwining
operators, and we show that the conformal blocks are fully determined by the
quantum group method. Crucially, we prove the associativity of the
intertwining operators among the first-row modules, and find that the
associativity is governed by the $6j$-symbols of the quantum group. Our
results constitute a concrete duality between a VOA and a quantum group, and
they will serve as the key tools to establish the equivalence of the first-row
subcategory of modules of the generic Virasoro VOA and the category of
(type-1) finite-dimensional representations of
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$.

###### Contents

  1. 1 Introduction
  2. 2 Background on the quantum group method
    1. 2.1 The quantum group
    2. 2.2 The correspondence with functions
    3. 2.3 Series expansions of the functions
  3. 3 Construction of conformal block vectors for the quantum group
    1. 3.1 The quantum $6j$-symbols
    2. 3.2 The construction of the conformal block vectors
  4. 4 Generic Virasoro VOA
    1. 4.1 Some notational conventions
    2. 4.2 Generic Virasoro vertex operator algebra
    3. 4.3 Modules and intertwining operators for VOAs
    4. 4.4 Modules and intertwining operators for the generic Virasoro VOA
  5. 5 Compositions of intertwining operators
    1. 5.1 Fusion rules and normalized intertwining operators
    2. 5.2 Space of formal series
    3. 5.3 System of differential equations
    4. 5.4 Functions from the quantum group method
    5. 5.5 Some applications
  6. 6 Associativity of intertwining operators
    1. 6.1 The setup and statement of associativity
    2. 6.2 The special case with highest weight vectors
    3. 6.3 Reduction to the initial terms
  7. A Proofs of analyticity and Frobenius series
    1. A.1 Elementary power series estimates
    2. A.2 Power series for the functions from the quantum group method
  8. B Construction of the intertwining operators for Verma modules
    1. B.1 Construction of a linear map
    2. B.2 Some properties of the linear map
    3. B.3 Translation property
    4. B.4 Jacobi identity

##  1\. Introduction

#### Conformal field theories, vertex operator algebras, and quantum groups

Two-dimensional conformal field theories (CFT) are an outstanding example of
extremely fruitful interaction of physics and mathematics [DFMS97, Gaw99,
Hua12, Nah00]. Their physical applications include string theory [GSW87] and
critical phenomena in planar statistical physics [Mus10], and they are among
the best understood examples of quantum field theories. In mathematics, ideas
from CFT have been instrumental to the Monster simple group [FLM89], tensor
categories [BKJ01], subfactors [Kaw15], moduli spaces [FBZ04], the geometric
Langlands program [Fre07], and conformally invariant random geometry [BB04],
among others.

As with quantum field theories in general, the mathematical axiomatization and
construction of CFTs are vast challenges, but CFTs possess remarkable
structure that has enabled a highly successful algebraic axiomatization based
on vertex operator algebras (VOA) [Kac97, LL04, Hua12]. A VOA is essentially
the chiral symmetry algebra of a CFT as envisioned in the seminal work of
Belavin, Polyakov, and Zamolodchikov [BPZ84a, BPZ84b]. The symmetry algebra
always contains the Virasoro algebra responsible for the conformal symmetry
itself. Virasoro vertex operator algebras are thus fundamental in that they
incorporate only and exactly the minimal amount of symmetry that any CFT
possesses. At special choices of the central charge $c$, a key parameter of
CFTs, there may actually exist two different Virasoro VOAs: the universal
Virasoro VOA [LL04] (maximally large) and its irreducible quotient, the
minimal Virasoro VOA [Wan93]. At generic values of $c$, however, the universal
Virasoro VOA itself is irreducible, and we call it a generic Virasoro VOA. The
topic of this article is such generic Virasoro VOAs.

Generic Virasoro VOAs have been studied very little in comparison with many
other VOAs. The main reason is that they fail most structural properties that
have enabled significant progress. In particular, they are far from rational
VOAs [FZ92, Zhu96], which possess a semisimple category of modules with
finitely many simple objects. Generic Virasoro VOAs admit first of all
infinitely many simple modules, and many more indecomposable but not
irreducible ones. We are lacking even the description of the general
indecomposable modules, let alone the category of such modules equipped with
the desired structures of a tensor product and braiding. In a notable recent
progress [CJH+21], the category of $C_{1}$-cofinite modules of the generic
Virasoro VOA was studied, and the tensor product constructed in it.

An intriguing aspect of conformal field theories, and of the corresponding
VOAs, is a hidden quantum group symmetry. In a number of prominent examples, a
representation category of a suitable quantum group has been found to agree
with a module category of a VOA — often together with the tensor products and
braiding in the categories. The case of (VOAs based on) Wess–Zumino–Witten
(WZW) CFTs at various levels have been treated in [Dri89, KL94, McR16], and
the corresponding quantum group was a $q$-deformation of the finite
dimensional Lie algebra of the corresponding WZW theory. Another well studied
example is the triplet $W$-algebra of logarithmic CFT [FGST06b, FGST06a, NT11,
KS11, TW13, GN21, CLR21], whose representation category is equivalent to that
of the restricted quantum group of $\mathfrak{sl}_{2}$. Though not as much as
a categorical equivalence, a certain structure related to a quantum group has
been also observed in the context of Liouville CFT [TV14].

#### First row subcategory of modules for the generic Virasoro VOA

The category of all modules of the generic Virasoro VOA being hopelessly
complicated, we focus here on a subcategory we call the first row subcategory.
It is the semisimple category whose infinitely many simple modules are the
irreducible Virasoro highest weight modules “in the first row of the Kac
table”, i.e., with highest weights $h=h_{1,s}$, $s\in\mathbb{Z}_{>0}$, when
$h_{r,s}$, $r,s\in\mathbb{Z}_{>0}$, denote the usual Kac labeled highest
weights [Kac79]. These correspond to a certain infinite set of (chiral)
primary fields in a CFT, which has been found to be relevant in particular to
questions in conformally invariant random geometry — the two simplest of these
primary fields after the identity, with Kac labeled conformal weights
$h_{1,2}$ and $h_{1,3}$, correspond to SLE-type curves’ starting points
[BB03b, Dub07] and boundary visit points [BB03a, JJK16, Dub15], respectively.
For such SLE applications, the generic Virasoro VOA corresponds to generic
values of the key parameter $\kappa>0$ of SLEs [Wer04, RS05, Law08], and is
completely natural.

By contrast to the general Kac labeled highest weights $h_{r,s}$,
$r,s\in\mathbb{Z}_{>0}$, at the first row highest weights $h_{1,s}$,
$s\in\mathbb{Z}_{>0}$ one has truly explicit expressions of singular vectors
in the Virasoro Verma module, by the Benoit–St. Aubin (BSA) formula [BSA88].
Correspondingly the BPZ partial differential equations [BPZ84a] for the
correlation functions of these primary fields are explicit. This is a feature
that facilitates the analysis of the first row subcategory, but resorting to
the explicit partial differential equations does not in principle seem
essential.

Our analysis of this first row subcategory of the generic Virasoro VOA is
based on a quantum group method of [KP20], which is a concrete and practical
version of the hidden quantum group symmetry [MR89, PS90, RRRA91, FW91, SV91,
Var95, GRAS96] developed with applications [KP16, JJK16] to random geometry in
mind. The corresponding quantum group is $\mathcal{U}_{q}(\mathfrak{sl}_{2})$,
a $q$-deformation of $\mathfrak{sl}_{2}$, at a deformation parameter $q$ which
is not a root-of-unity. Correspondingly the category of (type-1) finite-
dimensional representations of this quantum group is semisimple, with
infinitely many irreducible representations, and it is equipped with tensor
products and braiding [Lus93, BKJ01].

We consider it noteworthy that our VOA without extended symmetries of Lie
group type (present, e.g., in WZW models) has a quantum group counterpart in
this way, and that our VOA, which is irrational with very complicated
representation theory, corresponds to a quantum group with extremely well-
behaved category of finite-dimensional representations (albeit with infinitely
many irreducibles). The case more often seen before has been rational VOAs
with good module categories, and complicated root-of-unity quantum groups
whose representation categories are “semisimplified” for certain purposes.

In general terms, our main results are that the first row subcategory of
modules of the generic Virasoro VOA is stable under fusion, and detailed
calculations of the fusions with the quantum group method.

#### Methods

After reviewing the characterization and construction of intertwining
operators among the first row modules, we show that the intertwining operators
and their arbitrary compositions are the correlation functions obtained with
the quantum group method. A priori, the compositions of intertwining operators
of a VOA are formal series, but this shows that they are actual analytic
functions given by explicit integral formulas. In particular one obtains
convergence of the series, and straightforward methods of analytic
continuation.

Showing associativity of tensor products of modules of VOAs is generally a
very difficult task, and one of the main obstacles to constructing the
appropriate tensor category of modules of a VOA [HL92, HL94, HL95a, HL95b,
HL95c, Hua95, Hua05, HLZ14, HLZ10a, HLZ10b, HLZ10c, HLZ10d, HLZ10e, HLZ11a,
HLZ11b], see also [HKJL15, Section 2] for a review. The difficulties lie
partly in the fact that the formal series are not even supposed to correspond
to single-valued functions, so nontrivial branch choices are inevitable, and
yet the convergence and analyticity of the formal series is far from obvious.
The explicit analytic expressions from the quantum group method enable our
proof of associativity. It is also the explicit expressions that show the
equivalence of the tensor categories of the finite-dimensional representations
of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ and of the first-row modules of the
generic Virasoro VOA. The operator product expansion (OPE) coefficients of the
corresponding primary fields, in particular, are explicit, and involve the
quantum $6j$-symbols.

The braiding in the tensor category of VOA modules makes the multivaluedness
of the functions even more manifest. We will postpone the construction of the
braiding in the first row subcategory to a subsequent article, but the key to
it is similarly the explicit analytical expressions that are amenable to
analytic continuation.

#### Novelty and advantages of the approach

Our results provide a very satisfactory VOA to quantum group duality for the
fundamentally important generic Virasoro VOA, especially when combined with
the follow-up work establishing the equivalence of the tensor categories of
the first row modules of the VOA and of (type-1) finite-dimensional
representations of the quantum group $\mathcal{U}_{q}(\mathfrak{sl}_{2})$. The
formulation is practical, and in particular allows us to perform VOA
calculations with very straightforward linear algebra in finite-dimensional
representations of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$.

Conversely the method sheds light onto the quantum group method of [KP20].
Notably, VOA techniques can be used to systematize the calculation of general
series expansion coefficients of the correlation functions obtained from the
method. Moreover, the result gives a rather satisfactory characterization of
the space of solutions to BSA PDEs obtained from the quantum group method: it
says that the obtained solutions are exactly the linear span of the conformal
blocks, which can be described combinatorially, and in this sense all
solutions relevant to CFTs are included. By contrast, direct analytical
description of the solution space is complicated already in the particular
case involving only second order BSA PDEs [FK15].

Underlying the method and the results is the key observation that the
intertwining operators in the first row subcategory for the generic Virasoro
VOA are described by explicit analytic functions, not just formal series.

#### Related works

The recent article [CJH+21] also treats the question about the tensor products
in a subcategory of modules for the generic universal Virasoro VOA. The
category of all $C_{1}$-cofinite modules considered there is more general than
the first row category considered in the present article. Our approach is thus
less general, but it is fully explicit, and additionally shows the intimate
relationship with the quantum group.

Very recently, in [GN21], a ribbon tensor equivalence was established between
a module category of the Virasoro VOA at a central charge lying in a specific
series and a module category of the quantum $\mathrm{SL}_{2}$ at a root of
unity. That article employs results about tensor categories directly, and
specifically the fact that the tensor categories in question have a
distinguished generator. The first row subcategory of the generic Virasoro VOA
is also generated (as a tensor category) by a single module. With the methods
of [GN21] one could therefore expect to obtain a complementary viewpoint to
the relationship of the $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ and our first row
subcategory, which would be directly category theoretical but not as explicit
about the correlation functions as our approach.

The setting of the article [TV14] involves the same VOA and the same quantum
group as the present work, and the authors also observed a duality between
modules of the two. The modules of both are, however, different from what we
consider here. In other words, the results of us and [TV14] thus pertain to
different CFTs despite the fact that the VOA and the quantum group are the
same.

The quantum group method of [KP20] relies on tensor products of finite-
dimensional representations of the quantum group
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$ at generic values of the deformation
parameter $q$ when the representation theory is semisimple. Questions about it
often reduce to the commutant of the quantum group on the tensor product
representation, via a general $q$-Schur-Weyl duality. This approach to
calculations with the quantum group method has been developed in particular by
Flores and Peltola, in a series of articles [FP18b, FP18a, FP20, FP21]. The
commutant is a generalization of Temperley-Lieb algebas, and Flores and
Peltola have developed specific representation theoretic tools that are
suitable for explicit calculations in the quantum group method. Some of the
calculations in the present article, especially related to the $6j$ symbols,
are closely parallel to such a $q$-Schur-Weyl duality approach. The results we
need are, however, sufficiently concrete and tractable directly, so we do not
need to introduce the commutant algebra and its presentation by generators and
relations.

Ideas of reconstructing intertwining operators for the Virasoro VOA from
integral formulas appeared already in the seminal article [Fel89], and more
recently [KKP19] contains a conjecture of a special case of the precise
relationship between the quantum group method and the generic Virasoro VOA
that we establish here.

As a future perspective, it would be desirable to develop the method to a more
systematic one and facilitate generalizations in particular to non-semisimple,
root of unity cases. For that purpose, we view the theories of twisted
(co)homologies [AK11, TW14] and Nichols algebras [Len21] as particularly
promising.

### Acknowledgments

SK is supported by the Grant-in-Aid for JSPS Fellows (No. 19J01279).

KK wishes to thank Eveliina Peltola and Steven Flores for discussions and for
sharing their relevant work with us, Steven Flores and David Radnell for
contributions to our study group, and Yi-Zhi Huang for accepting the
invitation to give a minicourse in Helsinki and for sharing valuable insights.

KK also wishes to thank the organizers of the conference “The Mathematical
Foundations of Conformal Field Theory and Related Topics” for hospitality.
Numerous discussions at the conference, particularly with Ingo Runkel,
Shashank Kanade, Jinwei Yang, and Robert McRae, were very helpful for this
work.

##  2\. Background on the quantum group method

In this section we review the method of [KP20], by which one construct
functions of relevance to conformal field theories from vectors in tensor
product representations of a quantum group. In Section 3 we select specific
vectors in such representations, which will correspond to the conformal blocks
that are crucial to all of our main results: the construction (Section 4) of
the intertwining operators among the first-row modules of the generic Virasoro
VOA, their compositions (Section 5), and associativity (Section 6).

The textbook [Kas95] uses conventions similar to ours about the quantum group
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$. Our specific choices and notations are
identical to [KP20].

###  2.1. The quantum group

The quantum group $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ is the Hopf algebra
defined as follows. Fix a non-zero complex number
$q\in\mathbb{C}\setminus\left\\{0\right\\}$.

#### Definition of the quantum group

As an algebra, $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ is generated by elements

$\displaystyle E,F,K,K^{-1},$

subject to relations

$\displaystyle KK^{-1}=\;$ $\displaystyle 1=K^{-1}K,$ $\displaystyle KE=\;$
$\displaystyle q^{2}\,EK,$ $\displaystyle EF-FE=\;$
$\displaystyle\frac{1}{q-q^{-1}}\big{(}K-K^{-1}\big{)}$ $\displaystyle KF=\;$
$\displaystyle q^{-2}\,FK.$

The Hopf algebra structure on $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ is uniquely
determined by the coproduct, an algebra homomorphism
${\Delta}\colon\mathcal{U}_{q}(\mathfrak{sl}_{2})\to\mathcal{U}_{q}(\mathfrak{sl}_{2})\otimes\mathcal{U}_{q}(\mathfrak{sl}_{2})$,
whose values on the generators are

$\displaystyle{\Delta}(K)=K\otimes K,\qquad{\Delta}(E)=E\otimes K+1\otimes
E,\qquad{\Delta}(F)=F\otimes 1+K^{-1}\otimes F.$

#### Representations of the quantum group

We consider the representations of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ which
continuously $q$-deform the finite-dimensional representations of
$\mathfrak{sl}_{2}$. These are specified by a highest weight
$\lambda\in\mathbb{N}$. The $(\lambda+1)$-dimensional representation
$\mathsf{M}_{\lambda}$ of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ has a basis
$(u_{j}^{(\lambda)})_{j=0}^{\lambda}$ in which the generator $K$ acts
diagonally

$\displaystyle K.u_{j}^{(\lambda)}=\;$ $\displaystyle
q^{\lambda-2j}\,u_{j}^{(\lambda)},\qquad\text{for $j=0,\dots,\lambda$,}$

and the generators $E$ and $F$ act as raising and lowering operators

$\displaystyle E.u_{j}^{(\lambda)}=\;$ $\displaystyle\begin{cases}0&\text{ if
$j=0$}\\\ \llbracket
j\rrbracket\,\llbracket\lambda+1-j\rrbracket\,u_{j-1}^{(\lambda)}&\text{ if
$0<j\leq\lambda$,}\end{cases}$ $\displaystyle F.u_{j}^{(\lambda)}=\;$
$\displaystyle\begin{cases}u_{j+1}^{(\lambda)}\phantom{\llbracket
j\rrbracket\,\llbracket\lambda+1-j\rrbracket\,}&\text{ if $0\leq
j<\lambda$}\\\ 0&\text{ if $j=\lambda$,}\end{cases}$

where we used the $q$-integers defined by

$\displaystyle\llbracket n\rrbracket=\frac{q^{n}-q^{-n}}{q-q^{-1}}.$

The representations $\mathsf{M}_{\lambda}$, $\lambda\in\mathbb{N}$, are
irreducible if $q$ is not a root of unity, as we will assume throughout the
present article.

#### Tensor product representations

Using the coproduct
${\Delta}\colon\mathcal{U}_{q}(\mathfrak{sl}_{2})\to\mathcal{U}_{q}(\mathfrak{sl}_{2})\otimes\mathcal{U}_{q}(\mathfrak{sl}_{2})$,
we can equip the tensor product
$\mathsf{V}^{\prime}\otimes\mathsf{V}^{\prime\prime}$ of any two
representations $\mathsf{V}^{\prime},\mathsf{V}^{\prime\prime}$ of
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$ with the structure of a representation.
Coassociativity of ${\Delta}$ ensures that we can unambiguoulsy define triple
tensor products such as
$\mathsf{V}^{\prime}\otimes\mathsf{V}^{\prime\prime}\otimes\mathsf{V}^{\prime\prime\prime}$,
as well as further iterated tensor products. One should note, however, that
due to the lack of cocommutativity of ${\Delta}$, we can not canonically
identify $\mathsf{V}^{\prime}\otimes\mathsf{V}^{\prime\prime}$ with
$\mathsf{V}^{\prime\prime}\otimes\mathsf{V}^{\prime}$. In the category of
modules that we will consider, such identifications can be done by braiding,
but the choice of braiding direction must be specified.

The rest of the section assumes that $q$ is not a root of unity,

$\displaystyle q^{n}\neq 1\qquad\text{ for all
}n\in\mathbb{Z}\setminus\left\\{0\right\\}.$

Then the tensor products of the representations $\mathsf{M}_{\lambda}$,
$\lambda\in\mathbb{N}$, are completely reducible, and the Clebsch-Gordan
decomposition is the same as for $\mathfrak{sl}_{2}$

(2.1)
$\displaystyle\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}\;\cong\;\bigoplus_{\ell=0}^{\min(\mu,\lambda)}\mathsf{M}_{\mu+\lambda-2\ell},$

see, e.g., [KP20, Lemma 2.4]. Let us therefore define the selection rule set
associated to $\mu,\lambda\in\mathbb{N}$ as the set of those $\sigma$ such
that a copy of $\mathsf{M}_{\sigma}$ is contained in
$\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$, i.e.,

$\displaystyle\mathrm{Sel}(\mu,\lambda):=\left\\{\sigma\in\mathbb{N}\;\Big{|}\;\sigma+\mu+\lambda\equiv
0\;(\mathrm{mod}\;2),\;|\mu-\lambda|\leq\sigma\leq\mu+\lambda\right\\}.$

Note the symmetries

$\displaystyle\mathrm{Sel}(\mu,\lambda)=\mathrm{Sel}(\lambda,\mu)\qquad\text{
and }$
$\displaystyle\sigma\in\mathrm{Sel}(\mu,\lambda)\quad\Longleftrightarrow\quad\mu\in\mathrm{Sel}(\sigma,\lambda)\quad\Longleftrightarrow\quad\lambda\in\mathrm{Sel}(\mu,\sigma).$

The most convenient formulation of the Clebsch-Gordan rule for our purposes is
in terms of the following embedding.

######  Lemma 2.1.

Let $\mu,\lambda\in\mathbb{N}$. Then we have

$\displaystyle\mathrm{dim}\Big{(}\mathrm{Hom}_{\mathcal{U}_{q}(\mathfrak{sl}_{2})}(\mathsf{M}_{\sigma},\;\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu})\Big{)}=\begin{cases}1&\text{
if }\sigma\in\mathrm{Sel}(\mu,\lambda)\\\ 0&\text{ otherwise}.\end{cases}$

In the case $\sigma\in\mathrm{Sel}(\mu,\lambda)$, any
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$-module map
$\mathsf{M}_{\sigma}\to\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$ is
proportional to the embedding

(2.2)
$\displaystyle{\iota}_{\;{\sigma}}^{{\lambda},{\mu}}\;\colon\;\mathsf{M}_{\sigma}\,\hookrightarrow\;\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$

which is uniquely determined by

$\displaystyle{\iota}_{\;{\sigma}}^{{\lambda},{\mu}}\big{(}u_{0}^{(\sigma)}\big{)}=\;$
$\displaystyle\sum_{i,j=0}^{\frac{\mu+\lambda-\sigma}{2}}\delta_{i+j,\frac{\mu+\lambda-\sigma}{2}}\,(-1)^{j}\,\frac{\llbracket\mu-j\rrbracket!\llbracket\lambda-i\rrbracket!}{\llbracket\mu\rrbracket!\llbracket\lambda\rrbracket!\llbracket
i\rrbracket!\llbracket
j\rrbracket!}\frac{q^{j(\mu+1-j)}}{(q-q^{-1})^{\frac{\mu+\lambda-\sigma}{2}}}\;(u_{i}^{(\lambda)}\otimes
u_{j}^{(\mu)}).$

###### Proof.

This follows directly from, e.g., [KP20, Lemma 2.4]. ∎

Since for $\sigma\in\mathrm{Sel}(\mu,\lambda)$ the multiplicity of the
irreducible representation $\mathsf{M}_{\sigma}$ in the tensor product
$\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$ is one, there exist a unique
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$-module map

(2.3)
$\displaystyle{\hat{\pi}}_{{\lambda},{\mu}}^{\;{\sigma}}\colon\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}\to\mathsf{M}_{\sigma}\qquad\text{
such that
}\qquad{\hat{\pi}}_{{\lambda},{\mu}}^{\;{\sigma}}\circ{\iota}_{\;{\sigma}}^{{\lambda},{\mu}}=\mathrm{id}_{\mathsf{M}_{\sigma}}.$

The projection

$\displaystyle{\pi}_{{\lambda},{\mu}}^{\;{\sigma}}\colon\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}\to\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu},$

from $\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$ to its unique
subrepresentation isomorphic to $\mathsf{M}_{\sigma}$ then agrees with the
composition of ${\hat{\pi}}_{{\lambda},{\mu}}^{\;{\sigma}}$ with the embedding
${\iota}_{\;{\sigma}}^{{\lambda},{\mu}}$,

(2.4)
$\displaystyle{\pi}_{{\lambda},{\mu}}^{\;{\sigma}}={\iota}_{\;{\sigma}}^{{\lambda},{\mu}}\circ{\hat{\pi}}_{{\lambda},{\mu}}^{\;{\sigma}}.$

###  2.2. The correspondence with functions

Throughout this section, we parametrize $q$ by $\kappa$ via

$\displaystyle q=q(\kappa)=\;e^{\mathbf{i}4\pi/\kappa},$

and assume that $\kappa\in(0,\infty)\setminus\mathbb{Q}$. Then $q$ has unit
modulus, but is not a root of unity.

For $\lambda\in\mathbb{N}$, we write

(2.5) $\displaystyle
h(\lambda):=\frac{\lambda\,\big{(}2(\lambda+2)-\kappa\big{)}}{2\kappa}$

for the conformal weight of a module in the “first row of the Kac table” (see
Section 4).

For $N\in\mathbb{N}$, let us denote by

$\displaystyle\mathfrak{X}_{N}:=\left\\{(x_{1},\ldots
x_{N})\in\mathbb{R}^{N}\;\big{|}\;x_{1}<\cdots<x_{N}\right\\}$

the chamber of $N$ ordered real variables. We also fix parameters
$\lambda_{1},\ldots,\lambda_{N}\in\mathbb{N}$, and sometimes refer to all of
them collectively as

$\underline{{\lambda}}=(\lambda_{1},\ldots,\lambda_{N}).$

We are interested in functions $F\colon\mathfrak{X}_{N}\to\mathbb{C}$ that
satisfy certain properties motivated by conformal field theory: $N$ linear
partial differential equations, asymptotics as two variables approach each
other, as well as translation invariance, homogeneity, and sometimes
covariance under more general Möbius transformations.

Specifically, for each $j\in\left\\{1,\ldots,N\right\\}$, we define a
Benoit–Saint-Aubin partial differential operator

(2.6)
$\displaystyle\mathscr{D}^{(j)}:=\sum_{k=1}^{\lambda_{j}+1}\sum_{\begin{subarray}{c}p_{1},\ldots,p_{k}\geq
1\\\
p_{1}+\cdots+p_{k}=\lambda_{j}+1\end{subarray}}\frac{(-4/\kappa)^{1+\lambda_{j}-k}\,\lambda_{j}!^{2}}{\prod_{u=1}^{k-1}(\sum_{i=1}^{u}p_{i})(\sum_{i=u+1}^{k}p_{i})}\;\mathscr{L}^{(j)}_{-p_{1}}\cdots\mathscr{L}^{(j)}_{-p_{k}},$

of order $\lambda_{j}+1$, where

(2.7) $\displaystyle\mathscr{L}^{(j)}_{n}:=-\sum_{\begin{subarray}{c}1\leq
i\leq N\\\ i\neq
j\end{subarray}}\bigg{(}(x_{i}-x_{j})^{1+n}\frac{\partial}{\partial{x_{i}}}+(1+n)h(\lambda_{i})\,(x_{i}-x_{j})^{n}\bigg{)},\qquad\text{for
}n\in\mathbb{Z}.$

The parameters $\lambda_{1},\ldots,\lambda_{N}\in\mathbb{N}$ as well as the
number of variables $N$ are implicit in this notation. Note that with a fixed
$j$, the operators $\mathscr{L}^{(j)}_{n}$ satisfy the Witt algebra
commutation relations

$\displaystyle[\mathscr{L}^{(j)}_{m},\mathscr{L}^{(j)}_{n}]=(m-n)\mathscr{L}^{(j)}_{m+n},\qquad\text{
for }m,n\in\mathbb{Z}.$

The correspondence involves the representation of the quantum group
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$ constructed as the tensor product of the
irreducible representations
$\mathsf{M}_{\lambda_{1}},\ldots,\mathsf{M}_{\lambda_{N}}$. The tensorands are
ordered from left to right in the reverse order of the index, and we use the
shorthand notation

(2.8)
$\displaystyle\bigotimes_{i=1}^{N}\mathsf{M}_{\lambda_{i}}\;=\;\mathsf{M}_{\lambda_{N}}\otimes\cdots\otimes\mathsf{M}_{\lambda_{1}}$

for this ordering convention. Similarly we denote, e.g.,

$\displaystyle\bigotimes_{i<j}\mathsf{M}_{\lambda_{i}}\,=\;$
$\displaystyle\mathsf{M}_{\lambda_{j-1}}\otimes\cdots\otimes\mathsf{M}_{\lambda_{1}},$
$\displaystyle\bigotimes_{i>j}\mathsf{M}_{\lambda_{i}}\,=\;$
$\displaystyle\mathsf{M}_{\lambda_{N}}\otimes\cdots\otimes\mathsf{M}_{\lambda_{j+1}}.$

For the projection ${\pi}_{{\lambda_{j+1}},{\lambda_{j}}}^{\;{\tau}}$ of (2.4)
applied in the two consecutive tensorands with indices $j,j+1$, we use the
notation

$\displaystyle{\pi}_{\\{{j},{j+1}\\}}^{\;\;{\tau}}\,\colon\;$
$\displaystyle\bigotimes_{i=1}^{N}\mathsf{M}_{\lambda_{i}}\to\bigotimes_{i=1}^{N}\mathsf{M}_{\lambda_{i}}$
$\displaystyle{\pi}_{\\{{j},{j+1}\\}}^{\;\;{\tau}}\,=\;$
$\displaystyle\Big{(}\big{(}\bigotimes_{i>j+1}\mathrm{id}_{\mathsf{M}_{\lambda_{i}}}\big{)}\otimes{\pi}_{{\lambda_{j+1}},{\lambda_{j}}}^{\;{\;\;\tau}}\otimes\big{(}\bigotimes_{i<j}\mathrm{id}_{\mathsf{M}_{\lambda_{i}}}\big{)}\Big{)},$

and when the projection
${\hat{\pi}}_{{\lambda_{j+1}},{\lambda_{j}}}^{\;{\sigma}}$ is applied instead
(thus reducing the number of tensorands by one), we use the notation

$\displaystyle{\hat{\pi}}_{\\{{j},{j+1}\\}}^{\;\;{\tau}}\,\colon\;$
$\displaystyle\bigotimes_{j=1}^{N}\mathsf{M}_{\lambda_{j}}\to\Big{(}\big{(}\bigotimes_{i>j+1}\mathsf{M}_{\lambda_{i}}\big{)}\otimes\mathsf{M}_{\tau}\otimes\big{(}\bigotimes_{i<j}\mathsf{M}_{\lambda_{i}}\big{)}\Big{)}$
$\displaystyle{\hat{\pi}}_{\\{{j},{j+1}\\}}^{\;\;{\tau}}\,=\;$
$\displaystyle\Big{(}\big{(}\bigotimes_{i>j+1}\mathrm{id}_{\mathsf{M}_{\lambda_{i}}}\big{)}\otimes{\hat{\pi}}_{{\lambda_{j+1}},{\lambda_{j}}}^{\;{\;\;\tau}}\otimes\big{(}\bigotimes_{i<j}\mathrm{id}_{\mathsf{M}_{\lambda_{i}}}\big{)}\Big{)}.$

Within the tensor product (2.8), we are primarily concerned with the subspace

(2.9)
$\displaystyle\mathcal{H}_{\underline{{\lambda}}}:=\left\\{v\in\bigotimes_{j=1}^{N}\mathsf{M}_{\lambda_{j}}\;\bigg{|}\;E.v=0\right\\}$

consisting of highest weight vectors.

######  Theorem 2.2 ([KP20]).

There is a family of linear mappings
$\mathcal{F}\colon\mathcal{H}_{\underline{{\lambda}}}\to\mathcal{C}^{\infty}(\mathfrak{X}_{N})$
indexed by $\underline{{\lambda}}\in\bigsqcup_{N\in\mathbb{N}}\mathbb{N}^{N}$,
normalized so that for $N=1$ and any $\lambda_{1}\in\mathbb{N}$ we have
$\mathcal{F}[u_{0}^{(\lambda_{1})}](x_{1})\equiv 1$, and with the following
properties:

  * (PDE)

For any $u\in\mathcal{H}_{\underline{{\lambda}}}$ the function
$F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ satisfies

$\displaystyle\mathscr{D}^{(j)}F(x_{1},\ldots,x_{N})\,=\,0$ for all
$j=1,\ldots,N$ and $(x_{1},\ldots,x_{N})\in\mathfrak{X}_{N}$
$\displaystyle\,.$

  * (COV)

For any $u\in\mathcal{H}_{\underline{{\lambda}}}$ the function
$F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ is translation
invariant,

(2.10) $\displaystyle F(x_{1}+t,\ldots,x_{N}+t)\,=\,F(x_{1},\ldots,x_{N})$ for
all $(x_{1},\ldots,x_{N})\in\mathfrak{X}_{N}$ and $t\in\mathbb{R}$ .

If, moreover, $u$ is a Cartan eigenvector,
$u\in\mathcal{H}_{\underline{{\lambda}}}\cap\mathrm{Ker}(K-q^{\sigma})$, then
the function $F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ is
translation invariant and homogeneous,

(2.11) $\displaystyle
F(sx_{1}+t,\ldots,sx_{N}+t)\,=\,s^{h(\sigma)-\sum_{i=1}^{N}h(\lambda_{i})}\;F(x_{1},\ldots,x_{N})$
for all $(x_{1},\ldots,x_{N})\in\mathfrak{X}_{N}$ and $t\in\mathbb{R}$, $s>0$
.

Finally, if $u$ lies in a trivial subrepresentation,
$u\in\mathcal{H}_{\underline{{\lambda}}}\cap\mathrm{Ker}(K-1)$, then the
function $F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ is fully
Möbius-covariant in the sense that for any
$(x_{1},\ldots,x_{N})\in\mathfrak{X}_{N}$ and any
$\mathfrak{M}(z)=\frac{az+b}{cz+d}$ such that
$\mathfrak{M}(x_{1})<\cdots<\mathfrak{M}(x_{N})$, we have

(2.12) $\displaystyle
F(\mathfrak{M}(x_{1}),\ldots,\mathfrak{M}(x_{N}))\,=\,\prod_{i=1}^{N}\mathfrak{M}^{\prime}(x_{i})^{-h(\lambda_{i})}\;F(x_{1},\ldots,x_{N}).$

  * (ASY)

If $u\in\mathcal{H}_{\underline{{\lambda}}}$ lies in the subrepresentation
corresponding to the irreducible $\mathsf{M}_{\tau}$ in the tensor product of
the $j$:th and $j+1$:th factors, i.e., if
$u={\pi}_{\\{{j},{j+1}\\}}^{\;\;{\tau}}(u)$, then the function
$F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ has the expansion

$\displaystyle F(x_{1},\ldots,x_{j},x_{j+1},\ldots,x_{N})$ $\displaystyle=\;$
$\displaystyle
B\,(x_{j+1}-x_{j})^{{\Delta}}\,\Big{(}\hat{F}\big{(}x_{1},\ldots,\xi,\ldots,x_{N}\big{)}+\mathfrak{o}(1)\Big{)}\qquad\text{
as $x_{j},x_{j+1}\to\xi$},$

where $\hat{F}=\mathcal{F}[\hat{u}]$ with

$\displaystyle\hat{u}=\;$
$\displaystyle{\hat{\pi}}_{\\{{j},{j+1}\\}}^{\;\;{\tau}}(u)\;\in\;\Big{(}\big{(}\bigotimes_{i>j+1}\mathsf{M}_{\lambda_{i}}\big{)}\otimes\mathsf{M}_{\tau}\otimes\big{(}\bigotimes_{i<j}\mathsf{M}_{\lambda_{i}}\big{)}\Big{)},$

and
$B=B_{{\lambda_{j+1}},{\lambda_{j}}}^{\;{\;\;\tau}}=\frac{1}{((\lambda_{j}+\lambda_{j+1}-\tau)/2)!}\prod_{p=1}^{(\lambda_{j}+\lambda_{j+1}-\tau)/2}\frac{\Gamma(1+\frac{4}{\kappa}p)\,\Gamma(1-\frac{4}{\kappa}(1+\lambda_{j}-p))\,\Gamma(1-\frac{4}{\kappa}(1+\lambda_{j+1}-p))}{\Gamma(1+\frac{4}{\kappa})\,\Gamma(2-\frac{2}{\kappa}(4-2p+\lambda_{j}+\lambda_{j+1}+\tau))}$
and
${\Delta}={\Delta}_{{\lambda_{j+1}},{\lambda_{j}}}^{\;{\;\;\tau}}=h(\tau)-h(\lambda_{j})-h(\lambda_{j+1})$.

###  2.3. Series expansions of the functions

The method of Theorem 2.2 in fact yields not only smooth functions, but
analytic functions which have Frobenius series expansions on the codimension
one boundaries of the chamber $\mathfrak{X}_{N}$. These Frobenius series will
be important in Sections 5 and 6. We start with a general definitions about
the assumptions we use on parametrized power series, and then state the series
expansion results. The proofs are left to Appendix A.

#### Controlled parametrized power series

We will need to expand the functions of Section 2.2 as power series
recursively one variable at a time. Therefore, we will treat one of the
variables as the variable of the power series, and the other variables as
parameters. In order to be able to perform the natural operations on the power
series, we need the following type of control of the power series coefficients
locally uniformly over the the parameters.

######  Definition 2.3.

Let $\Omega\subset\mathbb{R}^{m}$ be an open set and
$c_{k}\colon\Omega\to\mathbb{C}$ smooth functions for ${k\in\mathbb{N}}$. For
$R>0$, we say that $(c_{k})_{k\in\mathbb{N}}$ are locally uniformly
$R$-controlled power series coefficients if for every compact $K\subset\Omega$
and every multi-index $\alpha\in\mathbb{N}^{m}$ we have

$\displaystyle\limsup_{k\to\infty}\big{(}\sup_{y\in
K}|\partial^{\alpha}c_{k}(y)|\big{)}^{1/k}\leq\frac{1}{R}.$

As simpler terminology, in the above situation we may just say that the power
series

$\displaystyle\sum_{k=0}^{\infty}c_{k}(y)\,z^{k}$

parametrized by $y\in\Omega$ is locally uniformly $R$-controlled. Note that by
the Cauchy-Hadamard formula for the radius of convergence, this implies in
particular that for any $y\in\Omega$ the radius of convergence of the power
series itself and its coefficientwise derivatives with respect to the
parameters $y$ have radius of convergence at least $R$.

#### Analyticity and Frobenius series statements

The analyticity statement in a single variable for the functions from Theorem
2.2 is the following.

######  Lemma 2.4.

Let $F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ be the function
associated to any $u\in\mathcal{H}_{\underline{{\lambda}}}$, and let
$(x_{1},\ldots,x_{N})\in\mathfrak{X}_{N}$, and let
$j\in\left\\{1,\ldots,N\right\\}$. Then we have a power series expansion

$\displaystyle
F(x_{1},\ldots,x_{j-1},z_{j},x_{j+1},\ldots,x_{N})\;=\;\sum_{k=0}^{\infty}c_{k}(x_{1},\ldots,x_{j-1},x_{j+1},\ldots,x_{N})\;\big{(}z_{j}-x_{j}\big{)}^{k}$

in the $j$:th variable. For fixed $x_{j}\in\mathbb{R}$ and $R>0$, viewing the
other variables $(x_{i})_{i\neq j}$ as parameters, on the subset
$\Omega\subset\mathbb{R}^{N-1}$ defined by the conditions $x_{1}<\cdots<x_{N}$
and $\min_{i\neq j}|x_{i}-x_{j}|>R$, the power series is locally uniformly
$R$-controlled.

The proof is elementary, but it is instructive as a preparation for the
consideration of the Frobenius series, so we give it in Appendix A.

The Frobenius series statement that we will use is the following. Variants of
this formulation with obvious modifications to the statement and proof could
be done as well.

######  Lemma 2.5.

Let $j\in\left\\{2,\ldots,N\right\\}$. Suppose that
$\tau\in\mathrm{Sel}(\lambda_{j-1},\lambda_{j})$ and that
$u\in\mathcal{H}_{\underline{{\lambda}}}$ is such that
$u={\pi}_{\\{{j-1},{j}\\}}^{\;\;{\tau}}(u)$. The function
$F=\mathcal{F}[u]\colon\mathfrak{X}_{N}\to\mathbb{C}$ associated to $u$ has a
Frobenius series expansion in the variable $z=x_{j}-x_{j-1}$

$\displaystyle
F\big{(}x_{1},\ldots,x_{j-1},(x_{j-1}+z),x_{j+1},\ldots,x_{N}\big{)}\;=\;z^{{\Delta}}\,\sum_{k=0}^{\infty}c_{k}(x_{1},\ldots,x_{j-1},x_{j+1},\ldots,x_{N})\;z^{k}$

where the indicial exponent is
${\Delta}=h(\tau)-h(\lambda_{j})-h(\lambda_{j-1})$. For fixed $R>0$, viewing
the other variables $(x_{i})_{i\neq j}$ as parameters, on the subset
$\Omega\subset\mathbb{R}^{N-1}$ defined by the conditions $x_{1}<\cdots<x_{N}$
and $\min_{i\neq j,j-1}|x_{i}-x_{j-1}|>R$, the power series part of this
Frobenius series is locally uniformly $R$-controlled, and for $0<z<R$ the
Frobenius series converges to the function $F$ on the left hand side.

######  Remark 2.6.

The leading coefficient $c_{0}$ of the Frobenius series in Lemma 2.5 is
related to the value of the function associated to
$\hat{u}={\hat{\pi}}_{\\{{j-1},{j}\\}}^{\;\;{\tau}}(u)$ as follows

$\displaystyle
c_{0}(x_{1},\ldots,x_{j-1},x_{j+1},\ldots,x_{N})\;=\;B\times\mathcal{F}[\hat{u}](x_{1},\ldots,x_{j-1},x_{j+1},\ldots,x_{N}),$

where $B=B_{{\lambda_{j}},{\lambda_{j-1}}}^{\;{\;\;\tau}}$ is as in Theorem
2.2. This can be seen from the (ASY) part of Theorem 2.2 (or more directly
from the proofs).

The proof of Lemma 2.5 is based on an elaboration of ideas from [KP20], in
particular those leading to part (ASY) of Theorem 2.2. Since this result will
be crucially relied on in the present article, we outline the proof in
Appendix A.

##  3\. Construction of conformal block vectors for the quantum group

In this section we construct specific vectors in tensor product
representations of the quantum group $\mathcal{U}_{q}(\mathfrak{sl}_{2})$,
which will correspond to our basis of “conformal blocks”. More precisely, in
the subsequent sections, compositions of intertwining operators in the first
row category of modules for the generic Virasoro VOA will be obtained from
these vectors via the quantum group method of Section 2.

###  3.1. The quantum $6j$-symbols

The embeddings
${\iota}_{\;{\sigma}}^{{\lambda},{\mu}}\;\colon\;\mathsf{M}_{\sigma}\,\hookrightarrow\;\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$
in Lemma 2.1 can be combined in different ways, and the relationships between
the choices are given by the quantum $6j$-symbols.

######  Lemma 3.1.

Let $\lambda_{1},\lambda_{2},\lambda_{3}\in\mathbb{N}$. Then for any
$\sigma\in\mathbb{N}$, the space

$\displaystyle\mathrm{Hom}_{\mathcal{U}_{q}(\mathfrak{sl}_{2})}\big{(}\mathsf{M}_{\sigma},\;\mathsf{M}_{\lambda_{3}}\otimes\mathsf{M}_{\lambda_{2}}\otimes\mathsf{M}_{\lambda_{1}}\big{)}$

has one basis consisting of

$\displaystyle\big{(}{\iota}_{\;{\nu}}^{{\lambda_{3}},{\lambda_{2}}}\otimes\mathrm{id}_{\mathsf{M}_{\lambda_{1}}}\big{)}\circ{\iota}_{\;{\sigma}}^{{\nu},{\lambda_{1}}},\qquad\nu\in\mathrm{Sel}(\sigma,\lambda_{1})\cap\mathrm{Sel}(\lambda_{3},\lambda_{2}),$

and another basis consisting of

$\displaystyle\big{(}\mathrm{id}_{\mathsf{M}_{\lambda_{3}}}\otimes{\iota}_{\;{\kappa}}^{{\lambda_{2}},{\lambda_{1}}}\big{)}\circ{\iota}_{\;{\sigma}}^{{\lambda_{3}},{\kappa}},\qquad\kappa\in\mathrm{Sel}(\sigma,\lambda_{3})\cap\mathrm{Sel}(\lambda_{2},\lambda_{1}).$

###### Proof.

Clearly the given maps are $\mathcal{U}_{q}(\mathfrak{sl}_{2})$-module maps
$\mathsf{M}_{\sigma}\to\mathsf{M}_{\lambda_{3}}\otimes\mathsf{M}_{\lambda_{2}}\otimes\mathsf{M}_{\lambda_{1}}$.
It follows straightforwardly from coassociativity and Clebsch-Gordan
decompositions that the two collections both span and are linearly
independent. ∎

The expansions of the elements of the second basis of Lemma 3.1 with respect
to the first basis are denoted as in the following:

(3.3)
$\displaystyle\big{(}\mathrm{id}_{\mathsf{M}_{\lambda_{3}}}\otimes{\iota}_{\;{\kappa}}^{{\lambda_{2}},{\lambda_{1}}}\big{)}\circ{\iota}_{\;{\sigma}}^{{\lambda_{3}},{\kappa}}=\;$
$\displaystyle\sum_{\nu}\Big{\\{}\begin{array}[]{ccc}{\lambda_{3}}&{\lambda_{2}}&{\nu}\\\
{\lambda_{1}}&{\sigma}&{\kappa}\end{array}\Big{\\}}\big{(}{\iota}_{\;{\nu}}^{{\lambda_{3}},{\lambda_{2}}}\otimes\mathrm{id}_{\mathsf{M}_{\lambda_{1}}}\big{)}\circ{\iota}_{\;{\sigma}}^{{\nu},{\lambda_{1}}}.$
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$}}}}}}}}}$

The coefficients
$\Big{\\{}\begin{array}[]{ccc}{\lambda_{3}}&{\lambda_{2}}&{\nu}\\\
{\lambda_{1}}&{\sigma}&{\kappa}\end{array}\Big{\\}}$ in these expansions are
called the quantum $6j$ symbols.

###  3.2. The construction of the conformal block vectors

Fix

$\displaystyle\underline{{\lambda}}=(\lambda_{0},\lambda_{1},\ldots,\lambda_{N},\lambda_{\infty})\in\mathbb{N}^{N+2}$

throughout. Note that compared to Section 2 we now have two additional labels,
$\lambda_{0}$ and $\lambda_{\infty}$. They will later be seen to have the
interpretations of labels of primary fields at the origin and at infinity.

######  Definition 3.2.

A sequence

$\displaystyle\underline{{\varsigma}}=({\varsigma}_{0},{\varsigma}_{1},\ldots,{\varsigma}_{N-1},{\varsigma}_{N})\in\mathbb{N}^{N+1}$

is said to be $\underline{{\lambda}}$-admissible, if we have

$\displaystyle{\varsigma}_{0}=\lambda_{0},\qquad{\varsigma}_{N}=\lambda_{\infty},$
and
$\displaystyle\lambda_{j}\in\mathrm{Sel}({\varsigma}_{j},{\varsigma}_{j-1})$
for all $j=1,\ldots,N$

The following picture should serve as a visual guide to what is going on at
the level of representations of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ (and in
fact at the level of modules of the generic Virasoro VOA in later sections);
the $\underline{{\lambda}}$-admissibility of the sequence
$\underline{{\varsigma}}$ exactly ensures that the selection rules are
satisfied at every vertex of this picture:

We seek to associate a conformal block to such an admissible sequence, and for
that purpose we will first associate to it a suitable vector

$\displaystyle
u_{\underline{{\varsigma}}}\;\in\;\Big{(}\bigotimes_{j=1}^{N}\mathsf{M}_{\lambda_{j}}\Big{)}\otimes\mathsf{M}_{\lambda_{0}}.$

Our convention is that tensor products are formed in the order with the
indices increasing from right to left, i.e., the space above is

$\mathsf{M}_{\lambda_{N}}\otimes\bigg{(}\mathsf{M}_{\lambda_{N-1}}\otimes\Big{(}\cdots\otimes\big{(}\mathsf{M}_{\lambda_{2}}\otimes(\mathsf{M}_{\lambda_{1}}\otimes\mathsf{M}_{\lambda_{0}})\big{)}\cdots\Big{)}\bigg{)}.$

Above we have placed the parentheses to illustrate the idea according to which
the vector $u_{\underline{{\varsigma}}}$ is chosen. The construction of the
vector is done with the composition

(3.6)

of embeddings

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Namely, we take $u_{\underline{{\varsigma}}}$ to be the image of the highest
weight vector $u_{0}^{(\lambda_{\infty})}\in\mathsf{M}_{\lambda_{\infty}}$
under this composition of embeddings,

(3.7) $\displaystyle u_{\underline{{\varsigma}}}\;:=\;\;$
$\displaystyle\bigg{(}\Big{(}\big{(}\bigotimes_{j=2}^{N}\mathrm{id}_{\mathsf{M}_{\lambda_{j}}}\big{)}\otimes{\iota}_{\;{{\varsigma}_{1}}}^{{\lambda_{1}},{{\varsigma}_{0}}}\Big{)}\circ\cdots\circ\Big{(}\mathrm{id}_{\mathsf{M}_{\lambda_{N}}}\otimes{\iota}_{\;{{\varsigma}_{N-1}}}^{{\lambda_{N-1}},{{\varsigma}_{N-2}}}\Big{)}\circ{\iota}_{\;{\lambda_{\infty}}}^{{\lambda_{N}},{{\varsigma}_{N-1}}}\bigg{)}(u_{0}^{(\lambda_{\infty})}).$

######  Lemma 3.3.

The vector $u_{\underline{{\varsigma}}}$ in (3.7) satisfies

$\displaystyle E.u_{\underline{{\varsigma}}}=0\qquad\text{ and }\qquad
K.u_{\underline{{\varsigma}}}=q^{\lambda_{\infty}}\,u_{\underline{{\varsigma}}}.$

###### Proof.

These properties are satisfied by the vector
$u_{0}^{(\lambda_{\infty})}\in\mathsf{M}_{\lambda_{\infty}}$, and the mapping
applied on this vector in (3.7) is a
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$-module map. ∎

More formally, the reason for the choice of $u_{\underline{{\varsigma}}}$ is
the following projection conditions. We use partial compositions of the
following sequence

of projections

######  Proposition 3.4.

For each $j=1,\dots,N$, denote by
${P}^{(j)}(u_{\underline{{\varsigma}}})\in\big{(}\bigotimes_{i\geq
j}\mathsf{M}_{\lambda_{i}}\big{)}\otimes\mathsf{M}_{{\varsigma}_{j-1}}$ the
image of $u_{\underline{{\varsigma}}}$ under the composition of the first
$j-1$ projections above. The following conditions hold for the vector
$u_{\underline{{\varsigma}}}$ in (3.7):

$\displaystyle{P}^{(j)}(u_{\underline{{\varsigma}}})=\;$
$\displaystyle\Big{(}\big{(}\bigotimes_{i>j}\mathrm{id}_{\mathsf{M}_{\lambda_{i}}}\big{)}\otimes{\pi}_{{\lambda_{j}},{{\varsigma}_{j-1}}}^{\;{{\varsigma}_{j}}}\Big{)}\big{(}{P}^{(j)}(u_{\underline{{\varsigma}}})\big{)}$

for each $j=1,\ldots,N$. Moreover, $u_{\underline{{\varsigma}}}$ is up to a
multiplicative constant the unique vector in
$\big{(}\bigotimes_{j=1}^{N}\mathsf{M}_{\lambda_{j}}\big{)}\otimes\mathsf{M}_{\lambda_{0}}$
for which the above conditions hold.

###### Proof.

The conditions for vector $u_{\underline{{\varsigma}}}$ follow directly from
its construction, using the relationships
${\pi}_{{\lambda},{\mu}}^{\;{\sigma}}\circ{\iota}_{\;{\sigma}}^{{\lambda},{\mu}}={\iota}_{\;{\sigma}}^{{\lambda},{\mu}}$
and
${\hat{\pi}}_{{\lambda},{\mu}}^{\;{\sigma}}\circ{\iota}_{\;{\sigma}}^{{\lambda},{\mu}}=\mathrm{id}_{\mathsf{M}_{\sigma}}$
between the projections and embeddings.

Uniqueness (up to multiplicative constants) can be shown by an induction over
$N$, using the multiplicity-free branching rule (2.1). ∎

##  4\. Generic Virasoro VOA

This section introduces the main algebraic structure of the present work, the
generic Virasoro vertex operator algebra. We also define its modules and
intertwining operators between modules.

From the point of view of physics, vertex operator algebras serve as the
chiral algebras of conformal field theories, and the case of the Virasoro VOA
is appropriate for the case with conformal symmetry alone. The modules of a
VOA correspond to the (conformal families of) fields in the CFT. Intertwining
operators are the building blocks of the correlation functions of these
fields.

This section is organized as follows. In Section 4.1 we introduce notation and
fix conventions about formal series. In Section 4.2 we introduce the Virasoro
algebra and its highest weight representations, as well as the VOAs based on
them. Section 4.3 contains the general definition of modules and intertwining
operators of VOAs, and Section 4.4 concentrates on the specific case of the
first row modules of the generic Virasoro VOA. The specific result about the
fusion rules, in particular, is given in Section 4.4. Much of the topic of
this section can be found in textbooks. To the extent possible, in our
presentation we follow [LL04] in Sections 4.1–4.2, and [Xu98, Li99] in
Sections 4.1–4.3. The more specific fusion rule statement of Section 4.4 has
been obtained through a different method in [FZ12].

###  4.1. Some notational conventions

Let us first fix some notational conventions.

#### General conventions

When a statement depends on a real number $m$ (integer, natural number, …), we
use the quantifier “for $m\gg 0$” (resp. “for $m\ll 0$”) to mean that the
statement holds for all sufficiently large $m$ (resp. sufficiently small $m$),
i.e., that there exists some $m_{0}$ such that the statement holds for all
$m>m_{0}$ (resp. for all $m<m_{0}$).

#### Formal series

We will have to consider various types of formal series: polynomials, Laurent
polynomials, formal power series, formal Laurent series, and formal series of
yet more general types. The formal series are formal sum expressions with
terms which are a coefficient times a power of a formal variable. The
coefficients are always taken to lie in some complex vector space, and
consequently also the spaces of formal series are naturally vector spaces with
addition and scalar multiplication defined coefficientwise.

Let $V$ be a vector space, and let $\mathfrak{z}$ be a formal variable.

The space of formal power series with coefficients in $V$ is

(4.1) $\displaystyle
V[[\mathfrak{z}]]=\left\\{\sum_{n\in\mathbb{N}}v_{n}\,\mathfrak{z}^{n}\;\bigg{|}\;v_{n}\in
V\text{ for all }n\in\mathbb{N}\right\\},$

and the space of polynomials is the subspace $V[\mathfrak{z}]\subset
V[[\mathfrak{z}]]$ consisting of those formal power series
$\sum_{n\in\mathbb{N}}v_{n}\,\mathfrak{z}^{n}$ which only have finitely many
non-zero coefficients, i.e., $v_{n}=0$ for all $n\gg 0$.

Similarly the space of formal Laurent series with coefficients in $V$ is

(4.2) $\displaystyle V[[\mathfrak{z}^{\pm
1}]]=\left\\{\sum_{m\in\mathbb{Z}}v_{m}\,\mathfrak{z}^{m}\;\bigg{|}\;v_{m}\in
V\text{ for all }m\in\mathbb{Z}\right\\},$

and space of Laurent polynomials is the subspace $V[\mathfrak{z}^{\pm
1}]\subset V[[\mathfrak{z}^{\pm 1}]]$ consisting of those formal Laurent
series $\sum_{m\in\mathbb{Z}}v_{m}\,\mathfrak{z}^{m}$ which only have finitely
many non-zero coefficients, i.e., $v_{m}=0$ for $|m|\gg 0$. The residue of a
formal Laurent series is defined as

$\displaystyle\mathrm{Res}_{\mathfrak{z}}\Big{(}\sum_{m\in\mathbb{Z}}v_{m}\,\mathfrak{z}^{m}\Big{)}=v_{-1}.$

The space of general formal series with coefficients in $V$ is

(4.3) $\displaystyle
V\\{\mathfrak{z}\\}=\left\\{\sum_{\alpha\in\mathbb{C}}v_{\alpha}\,\mathfrak{z}^{\alpha}\;\bigg{|}\;v_{\alpha}\in
V\text{ for all }\alpha\in\mathbb{C}\right\\}.$

Elements of any of the above are typically denoted by e.g.
$f(\mathfrak{z})=\sum_{i}v_{i}\,\mathfrak{z}^{i}$, to explicitly indicate the
formal variable $\mathfrak{z}$, and to emphasize the analogue with functions.

Series with several formal variables are defined by considering series in one
variable with coefficients in a vector space of formal series of other
variables, and natural identifications are made without comment: we set, e.g.,
$V[[\mathfrak{z},\mathfrak{w}]]=\big{(}V[[\mathfrak{z}]]\big{)}[[\mathfrak{w}]]=\big{(}V[[\mathfrak{w}]]\big{)}[[\mathfrak{z}]]$.

#### Binomial expansion convention

We follow the commonly used binomial expansion convention according to which
the power of a binomial in formal variables is always expanded in non-negative
integer powers of the second variable: for example if
$\mathfrak{z},\mathfrak{w}$ are two formal variables and $\beta\in\mathbb{C}$,
we interpret

(4.4) $\displaystyle(\mathfrak{z}+\mathfrak{w})^{\beta}=\;$
$\displaystyle\sum_{n=0}^{\infty}{\beta\choose
n}\,\mathfrak{z}^{\beta-n}\,\mathfrak{w}^{n}\;\in\;\mathbb{C}\\{\mathfrak{z}\\}[[\mathfrak{w}]],$
$\displaystyle\text{ where }\qquad{\beta\choose n}:=\;$
$\displaystyle\frac{1}{n!}\,\prod_{j=0}^{n-1}(\beta-j).$

The convention does require some caution: for instance for
$n\in\mathbb{Z}_{>0}$ the two series

$\displaystyle(\mathfrak{z}-\mathfrak{w})^{-n}\neq(-\mathfrak{w}+\mathfrak{z})^{-n}$

are not equal, but are different Laurent series expansions of the same
rational function function. The series on the left is in
$\mathbb{C}[[\mathfrak{z}^{\pm 1},\mathfrak{w}]]$ and is convergent in the
region $|\mathfrak{z}|>|\mathfrak{w}|$, while the series on the right is in
$\mathbb{C}[[\mathfrak{z},\mathfrak{w}^{\pm 1}]]$ and is convergent in the
region $|\mathfrak{w}|>|\mathfrak{z}|$. In the case of non-integer $\beta$,
note that non-integer powers are placed on only one of the formal variables,
leading to different branch choice issues when specializing the formal
variables to actual complex values.

#### The formal delta function

The formal delta-function in the formal variable $\mathfrak{z}$ is the formal
Laurent series

(4.5)
$\displaystyle\delta(\mathfrak{z})=\sum_{m\in\mathbb{Z}}\mathfrak{z}^{m}\,\in\,\mathbb{C}[[\mathfrak{z}^{\pm
1}]].$

If $\mathfrak{z},\mathfrak{w},\mathfrak{u}$ are formal variables, with the
binomial expansion convention we interpret

$\displaystyle\delta\Big{(}\frac{\mathfrak{z}-\mathfrak{w}}{\mathfrak{u}}\Big{)}=\sum_{m\in\mathbb{Z}}\sum_{n=0}^{\infty}{m\choose
n}(-1)^{n}\,\mathfrak{u}^{-m}\,\mathfrak{z}^{m-n}\,\mathfrak{w}^{n}\;\in\,\mathbb{C}[[\mathfrak{u}^{\pm
1},\mathfrak{z}^{\pm 1},\mathfrak{w}]].$

#### Multiplication of formal series

In the case when $V=A$ is an associative algebra, e.g., $V=\mathbb{C}$ or
$V=\mathrm{End}(W)$ for some vector space $W$, multiplication of formal series
of particular types may be meaningful: e.g. the product
$a(\mathfrak{z})\,b(\mathfrak{z})$ of two formal power series
$a(\mathfrak{z}),b(\mathfrak{z})\in A[[\mathfrak{z}]]$ is well-defined in
$A[[\mathfrak{z}]]$ (there are finitely many contributions to the coefficient
of any $\mathfrak{z}^{n}$), and the product $f(\mathfrak{z})\,g(\mathfrak{z})$
of a general series $f(\mathfrak{z})\in A\\{\mathfrak{z}\\}$ with a Laurent
polynomial $g(\mathfrak{z})\in A[\mathfrak{z}^{\pm 1}]$ is well-defined in
$A\\{\mathfrak{z}\\}$ (there are finitely many contributions to the
coefficient of any $\mathfrak{z}^{\alpha}$).

Likewise, suitable formal series with complex coefficients can be multiplied
with suitable series with coefficients in complex vector spaces: e.g., the
multiplication of a Laurent polynomial
$r(\mathfrak{z})\in\mathbb{C}[\mathfrak{z}^{\pm 1}]$ with a formal power
series $h(\mathfrak{z})\in V[[\mathfrak{z}]]$ is well-defined in
$V[[\mathfrak{z}^{\pm 1}]]$ (there are finitely many contributions to the
coefficient of any $\mathfrak{z}^{m}$).

Where well-defined, we use any such products without explicit comment in what
follows.

###  4.2. Generic Virasoro vertex operator algebra

#### Virasoro algebra

The Virasoro algebra is the complex Lie algebra

$\displaystyle\mathfrak{vir}=\;\bigoplus_{n\in{\mathbb{Z}}}\mathbb{C}L_{n}\;\oplus\;\mathbb{C}C$

with the Lie bracket determined by

$\displaystyle[L_{m},L_{n}]=\;$
$\displaystyle(m-n)\,L_{m+n}+\frac{m^{3}-m}{12}\,\delta_{m+n,0}\,C\qquad\text{
for $m,n\in{\mathbb{Z}}$,}$ $\displaystyle[C,\mathfrak{vir}]=\;$
$\displaystyle 0.$

To describe the relevant representations of $\mathfrak{vir}$, we introduce the
Lie subalgebras

$\displaystyle\mathfrak{vir}_{>0}:=\;$
$\displaystyle\bigoplus_{n>0}\mathbb{C}L_{n},$
$\displaystyle\mathfrak{vir}_{<0}:=\;$
$\displaystyle\bigoplus_{n<0}\mathbb{C}L_{n},$
$\displaystyle\mathfrak{vir}_{0}:=\;$
$\displaystyle\mathbb{C}L_{0}\oplus\mathbb{C}C,$
$\displaystyle\mathfrak{vir}_{\geq 0}:=\;$
$\displaystyle\mathfrak{vir}_{0}\oplus\mathfrak{vir}_{>0}.$

The universal enveloping algebra of $\mathfrak{vir}$ is denoted by
$\mathcal{U}(\mathfrak{vir})$.

#### Verma module

For $c,h\in\mathbb{C}$, the Verma module $M(c,h)$ of central charge $c$ and
conformal weight $h$ is defined as the quotient of the universal enveloping
algebra by the left ideal generated by the elements $C-c1$, $L_{0}-h1$, and
$L_{n}$ for $n>0$, i.e.

(4.6) $\displaystyle
M(c,h):=\mathcal{U}(\mathfrak{vir})\,\Big{/}\,\Big{(}\mathcal{U}(\mathfrak{vir})(C-c1)+\mathcal{U}(\mathfrak{vir})(L_{0}-h1)+\mathcal{U}(\mathfrak{vir})\,\mathfrak{vir}_{>0}\Big{)}.$

By construction, the vector $\bar{w}_{c,h}:=[1]\in M(c,h)$ satisfies

$\displaystyle C\bar{w}_{c,h}=\;$ $\displaystyle c\,\bar{w}_{c,h},$
$\displaystyle L_{0}\bar{w}_{c,h}=\;$ $\displaystyle h\,\bar{w}_{c,h},$
$\displaystyle L_{n}\,\bar{w}_{c,h}=\;$ $\displaystyle 0\quad\text{ for
$n>0$},$

and it is a cyclic vector, i.e., it generates the whole representation,
$\mathcal{U}(\mathfrak{vir})\bar{w}_{c,h}=M(c,h)$.

Owing to the Poincaré–Birkhoff–Witt (PBW) theorem, as a vector space the Verma
module is isomorphic to $\mathcal{U}(\mathfrak{vir}_{<0})$; a PBW basis for
$M(c,h)$ consists of vectors

(4.7) $\displaystyle L_{-n_{k}}\cdots L_{-n_{1}}\,\bar{w}_{c,h},$

where $k\in\mathbb{N}$ and $0<n_{1}\leq n_{2}\leq\cdots\leq n_{k}$.

The central element $C$ acts as the scalar $c$ on $M(c,h)$. The element
$L_{0}$ is diagonalizable and has eigenvalues $h+d$, $d\in\mathbb{N}$, and we
use this to define a grading of the Verma module

$\displaystyle
M(c,h)=\bigoplus_{d=0}^{\infty}\mathrm{Ker}\Big{(}L_{0}-(h+d)\,\mathrm{id}_{M(c,h)}\Big{)}.$

The homogeneous subspaces in this grading are finite-dimensional, since the
basis elements (4.7) are eigenvectors of $L_{0}$, with eigenvalues $h+d$,
where $d=n_{1}+\cdots+n_{k}$.

The Verma module has a filtration associated with the PBW basis, which we will
use extensively. For each $p\in\mathbb{Z}_{\geq 0}$, we define the subspace

(4.8)
$\displaystyle\mathscr{F}^{p}M(c,h):=\mathrm{span}\left\\{L_{-n_{k}}\cdots
L_{-n_{1}}\,\bar{w}_{c,h}\;\Big{|}\;0<n_{1}\leq\cdots\leq n_{k},\ k\leq
p\right\\}\;\subset\;M(c,h)$

spanned by basis vectors (4.7) with “PBW word-length” at most $p$. The PBW
filtration is the increasing sequence of subspaces

$\mathbb{C}\bar{w}_{c,h}=\mathscr{F}^{0}M(c,h)\,\subset\,\cdots\,\subset\,\mathscr{F}^{p}M(c,h)\,\subset\,\mathscr{F}^{p+1}M(c,h)\,\subset\,\cdots\,\subset\,M(c,h),$

which clearly has the property that
$\bigcup_{p\in\mathbb{N}}\mathscr{F}^{p}M(c,h)=M(c,h)$. Note furthermore that
each subspace $\mathscr{F}^{p}M(c,h)$ is itself a representation of the Lie
subalgebra $\mathfrak{vir}_{\geq 0}\subset\mathfrak{vir}$.

#### Highest weight modules

If $\chi$ is a non-zero vector in a representation $\mathcal{W}$ of
$\mathfrak{vir}$, which for some $\eta\in\mathbb{C}$ and $c\in\mathbb{C}$
satisfies

$\displaystyle L_{0}\chi=\eta\chi,\qquad C\chi=c\chi,\qquad\text{ and }\qquad
L_{n}\chi=0\;\text{ for all $n>0$,}$

then we call $\chi\in\mathcal{W}$ a singular vector.

If a singular vector $\chi\in\mathcal{W}$ generates the whole representation,

$\displaystyle\mathcal{U}(\mathfrak{vir})\chi=\mathcal{W},$

then it is called a highest weight vector, the representation $\mathcal{W}$ is
called a highest weight representation, and the $L_{0}$-eigenvalue $\eta$ and
the $C$-eigenvalue $c$ of $\chi$ are called its conformal weight and central
charge, respectively. As an example, the Verma module $M(c,h)$ is a highest
weight representation and $\bar{w}_{c,h}$ its highest weight vector. Note that
a highest weight vector in a given representation is necessarily unique up to
non-zero scalar multiples.

By construction the Verma module $M(c,h)$ has the universal property that for
any highest weight representation $\mathcal{W}$ with the same central charge
$c$ and highest weight $h$, there exists a surjective
$\mathcal{U}(\mathfrak{vir})$-module map

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathcal{W}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

As a consequence, any highest weight representation is isomorphic to a
quotient of a Verma module by a proper subrepresentation (possibly zero). In
particular, a highest weight representation $\mathcal{W}$ with highest weight
$h$ also admits a $\mathbb{N}$-grading by eigenvalues of
$L_{0}-h\,\mathrm{id}_{\mathcal{W}}$, and the homogeneous subspaces
$\mathrm{Ker}\big{(}L_{0}-(h+d)\,\mathrm{id}_{\mathcal{W}}\big{)}$ are finite-
dimensional. The PBW filtration of a Verma module is also inherited to its
quotient: letting $\mathscr{F}^{p}\mathcal{W}$ denote the image of
$\mathscr{F}^{p}M(c,h)$ under the above surjection, we obtain an increasing
filtration

$\displaystyle\mathbb{C}\chi=\mathscr{F}^{0}\mathcal{W}\,\subset\,\mathscr{F}^{1}\mathcal{W}\,\subset\,\cdots\,\subset\,\mathscr{F}^{p}\mathcal{W}\,\subset\,\mathscr{F}^{p+1}\mathcal{W}\,\subset\,\cdots\,\subset\,\mathcal{W}.$

Since the surjection is a $\mathcal{U}(\mathfrak{vir})$-module map, each
$\mathscr{F}^{p}\mathcal{W}$ is a representation of $\mathfrak{vir}_{\geq 0}$.

The unique irreducible highest weight representation with given $c,h$ is the
quotient of the Verma module $M(c,h)$ by its maximal proper subrepresentation.
A basic fact is that all subrepresentations of Verma modules are generated by
singular vectors, and that at most two singular vectors are needed to generate
a given subrepresentation [FF90], see also [IK11, Chapter 6].

The easiest example of a non-trivial submodule of a Verma module appears when
${h=0}$: the vector $L_{-1}\bar{w}_{c,0}$ is a singular vector in the Verma
module $M(c,0)$ and thus generates a proper subrepresentation
$\mathcal{U}(\mathfrak{vir})L_{-1}\bar{w}_{c,0}\subset M(c,0)$. This easy case
will be relevant for the universal Virasoro vertex operator algebra, but we
first give other examples that are the building blocks of the module category
that we study.

#### The Kac table and its first row

The highest weights $h$ for which the Verma module $M(c,h)$ is not irreducible
form what is called the Kac table: these highest weights $h_{r,s}$ are indexed
by two positive integers $r,s$.

Let us parametrize central charges $c$ by $\kappa$ via111This parametrization
of the central charges $c\leq 1$ by $\kappa>0$ arises naturally in the context
Schramm-Loewner Evolutions (SLE). It is a $2$-to-$1$ parametrization (except
at $c=1$ and $\kappa=4$): the values $\kappa$ and
$\kappa^{\prime}=\frac{16}{\kappa}$ give rise to the same $c$. In these two
cases with the same central charge $c$ we will introduce different categories
of modules for the same vertex operator algebra. What we call “the first row
of the Kac table” for $\kappa^{\prime}$ becomes instead “the first column of
the Kac table” for $\kappa$ — notice that
$h_{r,s}(\kappa^{\prime})=h_{s,r}(\kappa)$ in the formula below. Thus with
this $2$-to-$1$ parametrization also the latter category becomes covered even
if we explicitly only treat the former.

(4.9) $\displaystyle
c=c(\kappa):=1-\frac{3(\kappa-4)^{2}}{2\kappa}=13-6\Big{(}\frac{\kappa}{4}+\frac{4}{\kappa}\Big{)}$

Then an explicit formula for the Kac table highest weights is

$h_{r,s}(\kappa):=\frac{16(s^{2}-1)+8\kappa(1-rs)+\kappa^{2}(r^{2}-1)}{16\kappa},\qquad\text{
for $r,s\in\mathbb{Z}_{>0}$.}$

We will be interested in highest weight modules with conformal weights in the
first row of the Kac table, i.e., with $r=1$. These conformal weights are
exactly the ones in (2.5); we have

(4.10) $\displaystyle
h(\lambda)=h_{1,1+\lambda}(\kappa)=\frac{\lambda(2(\lambda+2)-\kappa)}{2\kappa},\qquad\text{
for }\lambda\in\mathbb{N}.$

######  Lemma 4.1 ([FF90]).

Assume $\kappa\not\in\mathbb{Q}$ and $\lambda\in\mathbb{N}$, and let $c$ and
$h(\lambda)$ be as in (4.9) and (4.10). Then the maximal proper
subrepresentation of the Verma module $M(c,h(\lambda))$ is isomorphic to the
Verma module $M(c,\,h(\lambda)+1+\lambda)$. In particular, the irreducible
highest weight representation with highest weight $h(\lambda)$ is the quotient

(4.11) $\displaystyle
Q_{\lambda}:=M(c,h(\lambda))\,\big{/}\,M(c,\,h(\lambda)+1+\lambda).$

###### Proof.

See, e.g., [FF90] or [IK11, Section 5.3.1]. With the above assumptions, the
Verma module $M(c,h(\lambda))$ falls in “class $I$” following the terminology
of Iohara and Koga [IK11]. ∎

The singular vector in $M(c,h(\lambda))$ that generates the maximal proper
subrepresentation $M(c,\,h(\lambda)+1+\lambda)\subset M(c,h(\lambda))$ is
given by the explicit formula of [BSA88],

$\displaystyle S_{\lambda}\,\bar{w}_{c,h(\lambda)}\,\in\;$ $\displaystyle
M(c,h(\lambda)),\qquad\text{where}$ (4.12) $\displaystyle S_{\lambda}\,:=\;$
$\displaystyle\sum_{k=1}^{\lambda+1}\sum_{\begin{subarray}{c}p_{1},\dots,p_{k}\geq
1\\\
p_{1}+\cdots+p_{k}=\lambda+1\end{subarray}}\frac{(-4/\kappa)^{\lambda+1-k}\;(\lambda!)^{2}}{\prod_{u=1}^{k-1}(\sum_{i=1}^{u}p_{i})(\sum_{i=u+1}^{k}p_{i})}L_{-p_{1}}\cdots
L_{-p_{k}}\;\in\;\mathcal{U}(\mathfrak{vir}).$

#### Virasoro vertex operator algebras

A vertex operator algebra is an $\mathbb{N}$-graded vector space

$\displaystyle V=\bigoplus_{d\in\mathbb{N}}V_{(d)}$

equipped with two distinguished non-zero vectors,

$\displaystyle\text{the vacuum vector }\,\mathsf{1}\in V_{(0)}\qquad\text{ and
}\qquad\text{the conformal vector }\,\omega\in V_{(2)},$

as well as a vertex operator map

$\displaystyle Y(\cdot,\bm{\zeta})\colon V\to\mathrm{End}(V)[[\bm{\zeta}^{\pm
1}]],$

all subject to axioms which are explicitly given in, e.g., [LL04].

Let $c\in\mathbb{C}$ be given. The vector $L_{-1}\bar{w}_{c,0}$ is a singular
vector in the Verma module $M(c,0)$ of highest weight $h=0$. Consider the
highest weight representation obtained as the quotient of the Verma module by
the proper subrepresentation
$\mathcal{U}(\mathfrak{vir})L_{-1}\bar{w}_{c,0}\subset M(c,0)$ generated by
this singular vector,

$V_{c}:=M(c,0)\,\big{/}\,\mathcal{U}(\mathfrak{vir})L_{-1}\bar{w}_{c,0}.$

It is known [LL04, Theorem 6.1.5] that the vector space $V_{c}$ can be
equipped with the structure of a vertex operator algebra (VOA), uniquely fixed
by the following. The vacuum vector is $\mathsf{1}=[\bar{w}_{c,0}]$, the
conformal vector is $\omega=L_{-2}\mathsf{1}=[L_{-2}\bar{w}_{c,0}]$, and the
Laurent modes of the vertex operator

$\displaystyle
Y(\omega,\bm{\zeta})=\sum_{n\in\mathbb{Z}}\bm{\zeta}^{-2-n}L_{n}\;\in\,\mathrm{End}(V_{c})[[\bm{\zeta},\bm{\zeta}^{-1}]]$

corresponding to the conformal vector are the Virasoro generators $L_{n}$,
$n\in\mathbb{Z}$ (acting as endomorphisms of $V_{c}$). This vertex operator
algebra $V_{c}$ is known as the universal Virasoro vertex operator algebra
with central charce $c$.

The irreducible highest weight representation of the Virasoro algebra with
highest weight $h=0$ can be also obtained as the quotient of $V_{c}$ by its
maximal proper submodule. A simple vertex operator algebra can be formed as
the corresponding quotient of the universal Virasoro VOA $V_{c}$ [LL04,
Theorem 6.1.5], and we refer to this as the simple Virasoro vertex operator
algebra with central charce $c$. For generic central charges, $c=c(\kappa)$
with $\kappa\notin\mathbb{Q}$, the representation $V_{c}$ is in fact already
irreducible by itself (case $\lambda=0$ of Lemma 4.1), so the simple Virasoro
VOA coincides with the universal Virasoro VOA $V_{c}$. In this article we
focus on the generic case: specifically we assume that $c=c(\kappa)$ as in
(4.9) with

(4.13) $\displaystyle\kappa\in(0,+\infty)\setminus\mathbb{Q}.$

For clarity, we then call $V_{c}$ the generic Virasoro vertex operator
algebra, although it is also both the universal Virasoro VOA and the simple
Virasoro VOA. As a warning we already point out that, despite being a simple
VOA, the generic Virasoro VOA $V_{c}$ fails many key properties assumed in
most VOA theory: $V_{c}$ is not $C_{2}$-cofinite, it has infinitely many
simple modules, it has modules which are not completely reducible, etc.

It is instructive to contrast the generic case we consider with what happens
for rational central charges $c$ — which are relevant, e.g., for minimal
models of CFT. For typical rational central charges the representation $V_{c}$
is not irreducible, and correspondingly the universal Virasoro VOA and the
simple Virasoro VOA are different. The simple Virasoro VOA has been studied
extensively in these cases, and is a model case of a well-behaved VOA: it is
in particular $C_{2}$-cofinite, has finitely many simple modules, has
semisimple category of modules closed under fusion, satisfies Verlinde’s
formula, etc. The failure of such good properties for the generic Virasoro VOA
is probably one of the main reasons the generic Virasoro VOA has not been
extensively studied yet.

###  4.3. Modules and intertwining operators for VOAs

We introduce the notions of modules and intertwining operators for a general
vertex operator algebra $V$.

#### Modules for vertex operator algebras

A module for a vertex operator algebra $V$ is a vector space $W$ equipped with
a linear map

$\displaystyle Y_{W}(\cdot,\bm{\zeta})\colon
V\to\mathrm{End}(W)[[\bm{\zeta}^{\pm 1}]]$

subject to the following conditions. For $v\in V$, let us denote by
$v^{W}_{(n)}\in\mathrm{End}(W)$ the coefficient of $\bm{\zeta}^{-1-n}$ in the
formal series $Y_{W}(v,\bm{\zeta})$, so that

(4.14) $\displaystyle
Y_{W}(v,\bm{\zeta})=\sum_{m\in\mathbb{Z}}\bm{\zeta}^{-1-m}\,v^{W}_{(m)}.$

For the coefficients of the series associated to the conformal vector
$\omega\in V$, we use the notation $L^{W}_{n}:=\omega^{W}_{(1+n)}$,
$n\in\mathbb{Z}$ so that
$Y_{W}(\omega,\bm{\zeta})=\sum_{n\in\mathbb{Z}}\bm{\zeta}^{-2-n}L^{W}_{n}$.
When the module $W$ is sufficiently clear from the context, we even omit the
superscript and denote simply $L_{n}=L^{W}_{n}$. The required conditions then
read:

  * •

We have $Y_{W}(\mathsf{1},\bm{\zeta})=\mathrm{id}_{W}$.

  * •

For any $v_{1},v_{2}\in V$, the following Jacobi identity holds:

(4.15)
$\displaystyle\bm{\zeta}_{0}^{-1}\,\delta\Big{(}\frac{\bm{\zeta}_{1}-\bm{\zeta}_{2}}{\bm{\zeta}_{0}}\Big{)}\,Y_{W}(v_{1},\bm{\zeta}_{1})\,Y_{W}(v_{2},\bm{\zeta}_{2})-\bm{\zeta}_{0}^{-1}\,\delta\Big{(}\frac{\bm{\zeta}_{2}-\bm{\zeta}_{1}}{-\bm{\zeta}_{0}}\Big{)}\,Y_{W}(v_{2},\bm{\zeta}_{2})\,Y_{W}(v_{1},\bm{\zeta}_{1})$
$\displaystyle=\;$
$\displaystyle\bm{\zeta}_{2}^{-1}\,\delta\Big{(}\frac{\bm{\zeta}_{1}-\bm{\zeta}_{0}}{\bm{\zeta}_{2}}\Big{)}\,Y_{W}\big{(}Y(v_{1},\bm{\zeta}_{0})v_{2},\,\bm{\zeta}_{2}\big{)}.$

  * •

The operator $L^{W}_{0}\in\mathrm{End}(W)$ is diagonalizable, its eigenspaces
$W_{(\eta)}:=\mathrm{Ker}\big{(}L^{W}_{0}-\eta\,\mathrm{id}_{W}\big{)}$ are
finite-dimensional, and $W_{(\eta)}=\left\\{0\right\\}$ when
$\mathfrak{Re}(\eta)\ll 0$.

A module $W$ thus has an $L^{W}_{0}$-eigenspace decomposition

(4.16) $\displaystyle W=\bigoplus_{\eta\in\mathbb{C}}W_{(\eta)}.$

The coefficients of the module vertex operator (4.14) respect this
decomposition in the following sense.

######  Lemma 4.2.

For any homogeneous element $v\in V_{(d)}$ of the vertex operator algebra and
any $m\in\mathbb{Z}$ and $\eta\in\mathbb{C}$ we have

$\displaystyle v^{W}_{(m)}\,W_{(\eta)}\subset W_{(\eta-m+d-1)}.$

###### Proof.

This is a standard result; it follows from

$\displaystyle[L_{0},v^{W}_{(m)}]=(-m+d-1)v^{W}_{(m)},\quad m\in\mathbb{Z},$

which in turn follows from applying the Jacobi identity (4.15) in the case
that $v_{1}=\omega$ and $v_{2}=v$. (Similar calculations will be done in some
more detail in a more general case in Lemma 4.6 and Corollary 4.7, for
example.) ∎

#### Contragredient modules

Suppose that $W$ is a module for a vertex operator algebra $V$. Considering
the duals $W_{(\eta)}^{*}=\mathrm{Hom}(W_{(\eta)},\mathbb{C})$ of the finite-
dimensional $L^{W}_{0}$-eigenspaces $W_{(\eta)}$ in the decomposition (4.16),
the restricted dual (also called graded dual) is the space

(4.17) $\displaystyle W^{\prime}=\bigoplus_{\eta\in\mathbb{C}}W_{(\eta)}^{*}.$

It can be equipped with the structure of a $V$-module [FHL93, Xu98] so that
the module vertex operator $Y_{W^{\prime}}(\cdot,\bm{\zeta})$ on $W^{\prime}$
is defined by the formula

$\displaystyle\Braket{Y_{W^{\prime}}(v,\bm{\zeta})w^{\prime},w}=\Braket{w^{\prime},Y_{W}(e^{\bm{\zeta}L_{1}}(-\bm{\zeta}^{-2})^{L_{0}}v,\bm{\zeta}^{-1})w},$
$\displaystyle\text{ for }v\in V,\ w^{\prime}\in W^{\prime},\ w\in W.$

This $V$-module $W^{\prime}$ is called the contragredient module of $W$.
Double contragredients are isomorphic to the original modules,
$W^{\prime\prime}\cong W$.

######  Lemma 4.3.

We have $L_{n}^{W^{\prime}}=(L_{-n}^{W})^{\top}$ for any $n\in\mathbb{Z}$,
i.e., for any $w^{\prime}\in W^{\prime}$ and $w\in W$ we have

$\displaystyle\langle{L_{n}^{W^{\prime}}\,w^{\prime},w}\rangle=\langle{w^{\prime},L_{-n}^{W}\,w}\rangle.$

###### Proof.

Taking $v=\omega$ and using
$e^{\bm{\zeta}L_{1}}(-\bm{\zeta}^{-2})^{L_{0}}\omega=\bm{\zeta}^{-4}\,\omega$,
this is just the equality of the coefficients of $\bm{\zeta}^{-2-n}$ in the
defining formula above. ∎

#### Intertwining operators

As an analogy to that the tensor product of modules over a Hopf algebra is
governed by the coproduct structure, the fusion product of modules of a VOA
$V$ is governed by the notion of intertwining operators.

######  Definition 4.4.

Let $W_{1},W_{0},W_{\infty}$ be three modules for a VOA $V$, with respective
module vertex operators
$Y_{W_{1}}(\cdot,\bm{\zeta}),Y_{W_{0}}(\cdot,\bm{\zeta}),Y_{W_{\infty}}(\cdot,\bm{\zeta})$.
An intertwining operator of type $\binom{W_{\infty}}{W_{1}\;W_{0}}$ is a
linear map

$\displaystyle\mathcal{Y}(\cdot,\bm{x})\colon
W_{1}\to\mathrm{Hom}(W_{0},W_{\infty})\\{\bm{x}\\}$

satisfying the Jacobi identity

(4.18)
$\displaystyle\bm{\xi}^{-1}\,\delta\left(\frac{\bm{\zeta}-\bm{x}}{\bm{\xi}}\right)\,Y_{W_{\infty}}(v,\bm{\zeta})\,\mathcal{Y}(w,\bm{x})-\bm{\xi}^{-1}\,\delta\left(\frac{\bm{x}-\bm{\zeta}}{-\bm{\xi}}\right)\,\mathcal{Y}(w,\bm{x})\,Y_{W_{0}}(v,\bm{\zeta})$
$\displaystyle=\;$
$\displaystyle\bm{x}^{-1}\,\delta\left(\frac{\bm{\zeta}-\bm{\xi}}{\bm{x}}\right)\mathcal{Y}\Big{(}Y_{W_{1}}(v,\bm{\xi})w,\,\bm{x}\Big{)}$

and the translation property

(4.19)
$\mathcal{Y}\big{(}L_{-1}^{W_{1}}w,\bm{x}\big{)}=\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\mathcal{Y}(w,\bm{x})$

for all $v\in V$ and $w\in W_{1}$.

The space of intertwining operators of type $\binom{W_{\infty}}{W_{1}\;W_{0}}$
is a vector space, which we will denote by

$\displaystyle\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}.$

It is known that spaces of intertwining operators admit the following
symmetry.

######  Proposition 4.5 ([HL95b]).

Let $W_{0}$, $W_{1}$, and $W_{\infty}$ be modules of a VOA $V$. Then, we have
linear isomorphisms

$\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}\;\simeq\;\mathcal{I}\binom{W_{\infty}}{W_{0}\;W_{1}}\;\simeq\;\mathcal{I}\binom{W_{0}^{\prime}}{W_{1}\;W_{\infty}^{\prime}}.$

The next lemma is useful for calculations with intertwining operators, and it
is in fact equivalent to the property (4.18).

######  Lemma 4.6.

Let $\mathcal{Y}(\cdot,\bm{x})$ be an intertwining operator of type
$\binom{W_{\infty}}{W_{1}\;W_{0}}$. Then, for any $p,q\in{\mathbb{Z}}$, $v\in
V$, $w\in W_{1}$, we have

$\displaystyle\mathrm{Res}_{\bm{\zeta}}\Big{(}Y_{W_{\infty}}(v,\bm{\zeta})\,\mathcal{Y}(w,\bm{x})\;\bm{\zeta}^{p}\,(\bm{\zeta}-\bm{x})^{q}\Big{)}-\mathrm{Res}_{\bm{\zeta}}\Big{(}\mathcal{Y}(w,\bm{x})\,Y_{W_{0}}(v,\bm{\zeta})\;\bm{\zeta}^{p}\,(-\bm{x}+\bm{\zeta})^{q}\Big{)}$
$\displaystyle=\;$
$\displaystyle\mathrm{Res}_{\bm{\zeta}_{0}}\Big{(}\mathcal{Y}\big{(}Y_{W_{1}}(v,\bm{\zeta}_{0})w,\bm{x}\big{)}\;(\bm{x}+\bm{\zeta}_{0})^{p}\,\bm{\zeta}_{0}^{q}\Big{)}.$

###### Proof.

Take the terms in (4.18) proportional to $\bm{\xi}^{-q-1}$ to obtain

$\displaystyle
Y_{W_{\infty}}(v,\bm{\zeta})\mathcal{Y}(w,\bm{x})(\bm{\zeta}-\bm{x})^{q}-\mathcal{Y}(w,\bm{x})Y_{W_{0}}(v,\bm{\zeta})(-\bm{x}+\bm{\zeta})^{q}$
$\displaystyle=\;$
$\displaystyle\mathrm{Res}_{\bm{\xi}}\bigg{(}\bm{\xi}^{q}\bm{x}^{-1}\,\delta\Big{(}\frac{\bm{\zeta}-\bm{\xi}}{\bm{x}}\Big{)}\,\mathcal{Y}\big{(}Y_{W_{1}}(v,\bm{\xi})w,\bm{x}\big{)}\bigg{)}$
$\displaystyle=\;$
$\displaystyle\sum_{m\in{\mathbb{Z}}}\sum_{k=0}^{\infty}(-1)^{k}\binom{m}{k}\bm{\zeta}^{m-k}\bm{x}^{-m-1}\mathcal{Y}(v_{(q+k)}w,\bm{x}).$

We further multiply $\bm{\zeta}^{p}$ and take the residue with respect to
$\bm{\zeta}$ to obtain

$\displaystyle\mathrm{Res}_{\bm{\zeta}}\Big{(}Y_{W_{\infty}}(v,\bm{\zeta})\,\mathcal{Y}(w,\bm{x})\;\bm{\zeta}^{p}(\bm{\zeta}-\bm{x})^{q}\Big{)}-\mathrm{Res}_{\bm{\zeta}}\Big{(}\mathcal{Y}(w,\bm{x})\,Y_{W_{0}}(v,\bm{\zeta})\;\bm{\zeta}^{p}(\bm{\zeta}-\bm{x})^{q}\Big{)}$
$\displaystyle=\;$
$\displaystyle\sum_{k=0}^{\infty}\binom{p}{k}\bm{x}^{p-k}\mathcal{Y}(v_{(q+k)}w,\bm{x})$
$\displaystyle=\;$
$\displaystyle\mathrm{Res}_{\bm{\zeta}_{0}}\Big{(}\mathcal{Y}\big{(}Y_{W_{1}}(v,\bm{\zeta}_{0})w,\,\bm{x}\big{)}\;(\bm{x}+\bm{\zeta}_{0})^{p}\bm{\zeta}_{0}^{q}\Big{)}.$

This proves the assertion. ∎

From the above formulas, we get the following equations for intertwining
operators, which give the basis of many recursive constructions with respect
to the PBW filtration (4.8) that we will use.

######  Corollary 4.7.

Let $\mathcal{Y}(\cdot,\bm{x})$ be an intertwining operator of type
$\binom{W_{\infty}}{W_{1}\;W_{0}}$.

  1. (1)

For any $w\in W_{1}$, $n\in{\mathbb{Z}}$, we have

$\displaystyle\mathcal{Y}(L_{n}^{W_{1}}\,w,\,\bm{x})$ $\displaystyle=$
$\displaystyle\;\mathrm{Res}_{\bm{\zeta}}\Big{(}Y_{W_{\infty}}(\omega,\bm{\zeta})\,\mathcal{Y}(w,\bm{x})\,(\bm{\zeta}-\bm{x})^{n+1}-\mathcal{Y}(w,\bm{x})\,Y_{W_{0}}(\omega,\bm{\zeta})\,(-\bm{x}+\bm{\zeta})^{n+1}\Big{)}$
$\displaystyle=\;$
$\displaystyle\sum_{k=0}^{\infty}\binom{n+1}{k}\,(-\bm{x})^{k}\,L_{n-k}^{W_{\infty}}\,\mathcal{Y}(w,\bm{x})-\sum_{k=0}^{\infty}\binom{n+1}{k}\,(-\bm{x})^{n-k+1}\,\mathcal{Y}(w,\bm{x})\,L_{k-1}^{W_{0}}.$

  2. (2)

For any $w\in W_{1}$, $n\in{\mathbb{Z}}$, we have

$\displaystyle
L_{n}^{W_{\infty}}\,\mathcal{Y}(w,\bm{x})-\mathcal{Y}(w,\bm{x})\,L_{n}^{W_{0}}=\;$
$\displaystyle\sum_{k=0}^{\infty}\binom{n+1}{k}\,\bm{x}^{n-k+1}\,\mathcal{Y}(L_{k-1}^{W_{1}}\,w,\bm{x}).$

  3. (3)

For any $w\in W_{1}$, we have

$\displaystyle
L_{-1}^{W_{\infty}}\,\mathcal{Y}(w,\bm{x})-\mathcal{Y}(w,\bm{x})\,L_{-1}^{W_{0}}=\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\mathcal{Y}(w,\bm{x}).$

######  Remark 4.8.

From here on, we will not use as careful notation as above to indicate in
which modules the different $L_{n}$’s act. We will instead abuse the notation
slightly and write

$\displaystyle
L_{n}^{W_{\infty}}\,\mathcal{Y}(w,\bm{x})-\mathcal{Y}(w,\bm{x})\,L_{n}^{W_{0}}=\;$
$\displaystyle\big{[}L_{n},\mathcal{Y}(w,\bm{x})\big{]}$

in, e.g., parts (2) and (3) above.

###### Proof of Corollary 4.7.

Recall that $L_{n}=\omega_{(n+1)}$, $n\in{\mathbb{Z}}$. By setting $v=\omega$,
$p=0$, $q=n+1$, we see (1). By setting $v=\omega$, $p=n+1$, $q=0$, we see (2).
In particular the latter gives

$\displaystyle\big{[}L_{-1},\mathcal{Y}(w,\bm{x})\big{]}=\;$
$\displaystyle\mathcal{Y}(L_{-1}w,\bm{x}),$

and the right hand side here equals
$\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\mathcal{Y}(w,\bm{x})$ by the
translation property (4.19) of intertwining operators. ∎

###  4.4. Modules and intertwining operators for the generic Virasoro VOA

We now discuss modules and intertwining operators focusing on the case of the
generic Virasoro VOA $V_{c}$, and describe the subcategory of modules that is
the topic of this article. We continue to parametrize the central charge
$c\leq 1$ by $\kappa>0$ as in (4.9). Throughout we make the genericity
assumption that $\kappa\notin\mathbb{Q}$.

#### Modules

Suppose that $\mathcal{W}$ is a representation of the Virasoro algebra where
$C$ acts as $c\,\mathrm{id}_{\mathcal{W}}$, and $L_{0}$ acts diagonalizably
with finite-dimensional eigenspaces and eigenvalues with real part bounded
from below. Then $\mathcal{W}$ has a unique structure of a module for the
universal Virasoro VOA $V_{c}$ such that

$\displaystyle
Y_{\mathcal{W}}(\omega,\bm{\zeta})=\sum_{n\in\mathbb{Z}}\bm{\zeta}^{-n-2}L_{n},$

i.e., $\omega^{\mathcal{W}}_{(m)}=L_{m-1}\in\mathrm{End}(\mathcal{W})$, see,
e.g., [LL04, Theorem 6.1.7].

In particular any of the Verma modules $M_{\lambda}=M(c,h(\lambda))$,
$\lambda\in\mathbb{N}$, and their irreducible quotient representations

$Q_{\lambda}=M_{\lambda}\,\big{/}\,M(c,\,h(\lambda)+1+\lambda),\qquad\lambda\in\mathbb{N}$

in Lemma 4.1 becomes a module for the generic Virasoro VOA $V_{c}$. We call
these $Q_{\lambda}$, $\lambda\in\mathbb{N}$ first row modules, and we denote
the module vertex operators in them simply by

$\displaystyle Y_{\lambda}(\cdot,\bm{\zeta})\colon
V_{c}\to\mathrm{End}(Q_{\lambda})[[\bm{\zeta}^{\pm 1}]].$

Irreducible highest weight representations are self-dual in the following
sense.

######  Lemma 4.9.

Let $\mathcal{W}$ be an irreducible highest weight representation of Virasoro
algebra, viewed as a module for the universal Virasoro VOA $V_{c}$. Then the
contragradient module $\mathcal{W}^{\prime}$ is isomorphic to $\mathcal{W}$.

###### Proof.

Let $\bar{w}\in\mathcal{W}_{(h)}$ be a highest weight vector in $\mathcal{W}$,
and choose $\bar{w}^{\prime}\in\mathcal{W}_{(h)}^{*}$ such that
$\langle\bar{w}^{\prime},\bar{w}\rangle=1$. From Lemma 4.3 we see that
$\bar{w}^{\prime}\in\mathcal{W}^{\prime}$ is a singular vector with highest
weight $h$ in $\mathcal{W}^{\prime}$. Then, $\mathcal{W}^{\prime}$ contains
the subrepresentation
$\mathcal{U}(\mathfrak{vir})\bar{w}^{\prime}\subset\mathcal{W}^{\prime}$,
which is a highest weight representation with the same highest weight $h$.
Since $\mathcal{W}$ is irreducible, there is a surjective module homomorphism
$\mathcal{U}(\mathfrak{vir})\bar{w}^{\prime}\twoheadrightarrow\mathcal{W}$,
which in particular implies that
$\mathrm{dim}((\mathcal{U}(\mathfrak{vir})\bar{w}^{\prime})_{(\eta)})\geq\mathrm{dim}(\mathcal{W}_{(\eta)})$
for all $\eta\in\mathbb{C}$. On the other hand, by construction, we have
$\mathrm{dim}(\mathcal{W}^{\prime}_{(\eta)})=\mathrm{dim}(\mathcal{W}_{(\eta)})$,
$\eta\in\mathbb{C}$. Therefore we get that

$\displaystyle\mathrm{dim}(\mathcal{W}^{\prime}_{(\eta)})=\mathrm{dim}(\mathcal{W}_{(\eta)})\leq\mathrm{dim}((\mathcal{U}(\mathfrak{vir})\bar{w}^{\prime})_{(\eta)})\leq\mathrm{dim}(\mathcal{W}^{\prime}_{(\eta)}),\qquad\text{
for }\eta\in\mathbb{C},$

and we can conclude that
$\mathcal{U}(\mathfrak{vir})\bar{w}^{\prime}=\mathcal{W}^{\prime}\simeq\mathcal{W}$.
∎

######  Corollary 4.10.

The first-row modules are self-dual, $Q_{\lambda}^{\prime}\cong Q_{\lambda}$
for any $\lambda\in\mathbb{N}$.

#### Intertwining operators among highest weight modules

Let us now suppose that $W_{1}$, $W_{0}$, $W_{\infty}$ are three modules for
the universal Virasoro VOA $V_{c}$, and that each of $W_{1}$, $W_{0}$, and
$W_{\infty}^{\prime}$ are highest weight modules (note also that by Lemma 4.9,
the contragredient $W_{\infty}^{\prime}$ is a highest weight module for
example if $W_{\infty}$ itself is an irreducible highest weight module). Let
$\bar{w}_{1}$, $\bar{w}_{0}$, $\bar{w}_{\infty}^{\prime}$ be highest weight
vectors in $W_{1}$, $W_{0}$, $W_{\infty}^{\prime}$, respectively, and denote
the corresponding highest weights by $h_{1},h_{0},h_{\infty}$.

For an intertwining operator operator
$\mathcal{Y}\in\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}$, define

$\displaystyle\mathrm{Init}[\mathcal{Y}](\bm{x}):=\braket{\bar{w}_{\infty}^{\prime},\;\mathcal{Y}(\bar{w}_{1},\bm{x})\,\bar{w}_{0}}$

and call this the initial term of the intertwining operator $\mathcal{Y}$. The
initial term defines a linear map

$\mathrm{Init}\colon\;\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}\to\,{\mathbb{C}}\\{\bm{x}\\}.$

We make a few simple general observations in this setup.

######  Lemma 4.11.

The initial term of any
$\mathcal{Y}\in\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}$ is of the form

$\displaystyle\mathrm{Init}[\mathcal{Y}](\bm{x})=A\,\bm{x}^{h_{\infty}-h_{1}-h_{0}},\qquad\text{
for some $A\in\mathbb{C}$.}$

###### Proof.

Using Corollary 4.7(2) for $n=0$, Lemma 4.3, and the eigenvector equations
$L_{0}\bar{w}_{1}=h_{1}\,\bar{w}_{1}$, $L_{0}\bar{w}_{0}=h_{0}\,\bar{w}_{0}$,
$L_{0}\bar{w}_{\infty}^{\prime}=h_{\infty}\,\bar{w}_{\infty}^{\prime}$, we
calculate

$\displaystyle h_{\infty}\;\mathrm{Init}[\mathcal{Y}](\bm{x})=\;$
$\displaystyle\braket{L_{0}\bar{w}_{\infty}^{\prime},\;\mathcal{Y}(\bar{w}_{1},\bm{x})\,\bar{w}_{0}}$
$\displaystyle=\;$
$\displaystyle\braket{\bar{w}_{\infty}^{\prime},\;L_{0}\,\mathcal{Y}(\bar{w}_{1},\bm{x})\,\bar{w}_{0}}$
$\displaystyle=\;$
$\displaystyle\braket{\bar{w}_{\infty}^{\prime},\;\big{(}\bm{x}\,\mathcal{Y}(L_{-1}\bar{w}_{1},\bm{x})+\mathcal{Y}(L_{0}\bar{w}_{1},\bm{x})\big{)}\,\bar{w}_{0}}$
$\displaystyle+\braket{\bar{w}_{\infty}^{\prime},\;\mathcal{Y}(\bar{w}_{1},\bm{x})\,L_{0}\bar{w}_{0}}$
$\displaystyle=\;$
$\displaystyle\Big{(}\bm{x}\,\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}+h_{1}+h_{0}\Big{)}\;\mathrm{Init}[\mathcal{Y}](\bm{x}).$

Thus the initial term satisfies
$\bm{x}\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\mathrm{Init}[\mathcal{Y}](\bm{x})=(h_{\infty}-h_{1}-h_{0})\,\mathrm{Init}[\mathcal{Y}](\bm{x})$.
The solution space of this differential equation is one dimensional and
spanned by $\bm{x}^{h_{\infty}-h_{1}-h_{0}}$, so the assertion follows. ∎

The following straightforward proposition contains a key idea, that of an
induction based on PBW-filtrations, so we do it in detail here. In later
proofs (including our main results), we will then not always write out
explicitly all of the cases, as the ideas are similar.

######  Proposition 4.12.

An intertwining operator
$\mathcal{Y}\in\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}$ is uniquely
determined by its initial term $\mathrm{Init}[\mathcal{Y}](\bm{x})$. In
particular, by Lemma 4.11 we thus have

$\displaystyle\mathrm{dim}\;\mathcal{I}\binom{W_{\infty}}{W_{1}\;W_{0}}\;\leq\;1,$

i.e., if a non-zero intertwining operators exists, it is unique up to a
multiplicative constant.

###### Proof.

Suppose that $\mathrm{Init}[\mathcal{Y}](\bm{x})=0$. We will prove that then
$\braket{{w}_{\infty}^{\prime},\;\mathcal{Y}({w}_{1},\bm{x})\,{w}_{0}}=0$ for
all ${w}_{\infty}^{\prime}\in W_{\infty}^{\prime}$, ${w}_{1}\in W_{1}$,
${w}_{0}\in W_{0}$. It will follow that $\mathcal{Y}=0$. From this we can
conclude that the initial term indeed determines the interwining operator.

The proof of the above is done by induction with respect to the total PBW word
length for the PBW filtrations of the three highest weight representations
$W_{1}$, $W_{0}$, $W_{\infty}^{\prime}$. So assume that
$\braket{{w}_{\infty}^{\prime},\;\mathcal{Y}({w}_{1},\bm{x})\,{w}_{0}}=0$
whenever ${w}_{\infty}^{\prime}\in\mathscr{F}^{p_{1}}W_{\infty}^{\prime}$,
${w}_{1}\in\mathscr{F}^{p_{2}}W_{1}$, ${w}_{0}\in\mathscr{F}^{p_{3}}W_{0}$ are
such that the total word lengths satisfy $p_{1}+p_{2}+p_{3}\leq p$. Now let
$n>0$, and consider any such ${w}_{\infty}^{\prime},{w}_{1},{w}_{0}$. From
Corollary 4.7(2) we get that

$\displaystyle\braket{L_{-n}{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}=\;$
$\displaystyle\braket{{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x})L_{n}{w}_{0}}+\bm{x}^{n+1}\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\braket{{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}$
$\displaystyle+\sum_{k=1}^{\infty}\binom{n+1}{k}\bm{x}^{n-k+1}\,\braket{{w}_{\infty}^{\prime},\mathcal{Y}(L_{k-1}{w}_{1},\bm{x}){w}_{0}}$
$\displaystyle=\;$ $\displaystyle 0,$

where each term of the second expression vanished by the induction hypothesis
(note that also $L_{n}{w}_{0}\in\mathscr{F}^{p_{3}-1}W_{0}$ and
$L_{k-1}{w}_{1}\in\mathscr{F}^{p_{2}}W_{1}$ above). Similarly we get

$\displaystyle\braket{{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x})L_{-n}{w}_{0}}=\;$
$\displaystyle\braket{L_{n}{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}-\bm{x}^{-n+1}\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\;\braket{{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}$
$\displaystyle-\sum_{k=1}^{\infty}\binom{-n+1}{k}\bm{x}^{-n-k+1}\;\braket{{w}_{\infty}^{\prime},\mathcal{Y}(L_{k-1}{w}_{1},\bm{x}){w}_{0}}$
$\displaystyle=\;$ $\displaystyle 0.$

Finally, using Corollary 4.7(1–3), we similarly get

$\displaystyle\braket{{w}_{\infty}^{\prime},\mathcal{Y}(L_{-n}{w}_{1},\bm{x}){w}_{0}}=\;$
$\displaystyle\sum_{k=0}^{\infty}\binom{-n+1}{k}(-\bm{x})^{k}\;\braket{L_{n+k}{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}$
$\displaystyle-\sum_{k=1}^{\infty}\binom{-n+1}{k}(-\bm{x})^{-n-k+1}\;\braket{{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x})L_{k-1}{w}_{0}}$
$\displaystyle-(-\bm{x})^{-n+1}\,\braket{L_{1}{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}$
$\displaystyle+(-\bm{x})^{-n+1}\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\,\braket{{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x}){w}_{0}}$
$\displaystyle=\;$ $\displaystyle 0.$

These three cases complete the induction step, by establishing that
$\braket{{w}_{\infty}^{\prime},\;\mathcal{Y}({w}_{1},\bm{x})\,{w}_{0}}$
vanishes also whenever
${w}_{\infty}^{\prime}\in\mathscr{F}^{p_{1}}W_{\infty}^{\prime}$,
${w}_{1}\in\mathscr{F}^{p_{2}}W_{1}$, ${w}_{0}\in\mathscr{F}^{p_{3}}W_{0}$ are
such that the total word lengths satisfy $p_{1}+p_{2}+p_{3}\leq p+1$. ∎

#### Intertwining operators among first row modules

To find all intertwining operators among the first row modules $Q_{\lambda}$,
in view of Proposition 4.12 it suffices to determine when non-zero
intertwining operators can exist. We call the conditions for the existence
selection rules.222These selection rules turn out to take exactly the same
form as the selection rules which determine when an irreducible representation
of the quantum group $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ appears in the
tensor product of two others.

The singular vectors (4.12) lead to necessary conditions. Fix
$\lambda\in\mathbb{N}$, and introduce the corresponding polynomial of
$h_{\infty},h_{0}$

(4.20) $\displaystyle P_{\lambda}(h_{0},h_{\infty})$ $\displaystyle:=\;$
$\displaystyle\sum_{k=1}^{\lambda+1}\sum_{\begin{subarray}{c}p_{1},\dots,p_{k}\geq
1\\\
p_{1}+\cdots+p_{k}=\lambda+1\end{subarray}}\frac{(-4/\kappa)^{\lambda+1-k}\;(\lambda!)^{2}}{\prod_{u=1}^{k-1}(\sum_{i=1}^{u}p_{i})(\sum_{i=u+1}^{k}p_{i})}\prod_{j=1}^{k}(-1)^{p_{j}}\Big{(}\sum_{i>j}p_{i}+h(\lambda)-h_{\infty}+p_{j}h_{0}\Big{)}.$

######  Lemma 4.13.

Suppose that $W_{0}$, $W_{\infty}$ are modules for $V_{c}$ such that $W_{0}$
and $W_{\infty}^{\prime}$ are highest weight modules with highest weights
$h_{0}$ and $h_{\infty}$, respectively. For each $\lambda\in\mathbb{N}$, the
linear map

$\mathcal{I}\binom{W_{\infty}}{Q_{\lambda}\;W_{0}}\to\mathcal{I}\binom{W_{\infty}}{M_{\lambda}\;W_{0}};\qquad\mathcal{Y}(\cdot,\bm{x})\mapsto\mathcal{Y}(\pi_{\lambda}(\cdot),\bm{x}),$

where $\pi_{\lambda}:M_{\lambda}\to Q_{\lambda}$ is the canonical projection,
is an embedding of the space of intertwining operators. Furthermore, this
embedding is an isomorphism if and only if the highest weights satisfy the
polynomial equation

$\displaystyle P_{\lambda}(h_{0},h_{\infty})=0.$

###### Proof.

It is clear that for any
$\mathcal{Y}\in\mathcal{I}\binom{W_{\infty}}{Q_{\lambda}\;W_{0}}$, the formula
$\mathcal{Y}(\pi_{\lambda}(\cdot),\bm{x})$ indeed gives an intertwining
operator of type $\binom{W_{\infty}}{M_{\lambda}\;W_{0}}$, so the map is well-
defined.

For its injectivity, observe the following. The initial terms satisfy
$\mathrm{Init}[\mathcal{Y}(\cdot,\bm{x})]=\mathrm{Init}[\mathcal{Y}(\pi_{\lambda}(\cdot),\bm{x})]$.
Now if $\mathcal{Y}(\pi_{\lambda}(\cdot),\bm{x})=0$, we have
$\mathrm{Init}[\mathcal{Y}(\cdot,\bm{x})]=0$, which by Proposition 4.12
implies that also $\mathcal{Y}(\cdot,\bm{x})=0$. Injectivity follows.

The embedding is an isomorphism if and only if all intertwining operators
$\mathcal{Y}\in\mathcal{I}\binom{W_{\infty}}{M_{\lambda}\;W_{0}}$ factor
through the irreducible quotient in the sense that we have
$\mathcal{Y}(w,\bm{x})=0$ for all $w$ in the maximal proper submodule of
$M_{\lambda}$. As in Proposition 4.12, we see that the factorization is
equivalent to the single condition
$\braket{\bar{w}_{\infty}^{\prime},\,\mathcal{Y}(S_{\lambda}\bar{w}_{c,h(\lambda)},\,\bm{x})\,\bar{w}_{0}}=0$,
where $S_{\lambda}\bar{w}_{c,h(\lambda)}$ is the singular vector defined in
(4.12); just note that the singular vector generates the maximal proper
submodule of $M_{\lambda}$, which is isomorphic to
$M(c,h(\lambda)+\lambda+1)$.

Let $\bar{w}_{0}$ and $\bar{w}_{\infty}^{\prime}$ be highest weight vectors in
$W_{0}$ and $W_{\infty}^{\prime}$, respectively, and let $\bar{w}_{\lambda}$
be a highest weight vector in $M_{\lambda}$. Denote the initial term of
$\mathcal{Y}$ by

$\displaystyle
f(\bm{x})=\mathrm{Init}[\mathcal{Y}](\bm{x})=\braket{\bar{w}_{\infty}^{\prime},\,\mathcal{Y}(\bar{w}_{\lambda},\bm{x})\,\bar{w}_{0}}.$

By Lemma 4.11, we must have

$\displaystyle f(\bm{x})=A\,\bm{x}^{\Delta},\qquad\text{
where}\qquad\Delta=h_{\infty}-h_{0}-h(\lambda).$

Again by a calculation using the formulas of Corollary 4.7 (in fact the
special case ${w}_{\infty}^{\prime}=\bar{w}_{\infty}^{\prime}$ and
${w}_{0}=\bar{w}_{0}$ of the last calculation in the proof of Proposition
4.12), we get for any $n>0$

$\displaystyle\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}(L_{-n}{w}_{1},\bm{x})\bar{w}_{0}}$
$\displaystyle=\;$
$\displaystyle(-\bm{x})^{-n+1}\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\,\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x})\bar{w}_{0}}-(1-n)h_{0}(-\bm{x})^{-n}\;\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x})\bar{w}_{0}}$
$\displaystyle=\;$
$\displaystyle\mathscr{L}_{-n}\,\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}({w}_{1},\bm{x})\bar{w}_{0}},$

where we introduced the differential operator

$\displaystyle\mathscr{L}_{-n}=\;$
$\displaystyle(-\bm{x})^{-n}\Big{(}(n-1)h_{0}-\bm{x}\frac{\mathrm{d}}{\mathrm{d}{\bm{x}}}\Big{)}.$

Recursively used, this allows to reduce the following expression to a
differential operator acting on the initial term

(4.21)
$\displaystyle\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}(L_{-n_{k}}\cdots
L_{-n_{1}}\bar{w}_{\lambda},\bm{x})\bar{w}_{0}}=\;$
$\displaystyle\mathscr{L}_{-n_{k}}\cdots\mathscr{L}_{-n_{1}}\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}(\bar{w}_{0},\bm{x})L_{k-1}\bar{w}_{0}}$
$\displaystyle=\;$
$\displaystyle\mathscr{L}_{-n_{k}}\cdots\mathscr{L}_{-n_{1}}\;f(\bm{x}).$

With induction on $k$, starting from $f(\bm{x})=A\,\bm{x}^{\Delta}$ and using
the explicit differential operators $\mathscr{L}_{-n}$, we find

$\displaystyle\mathscr{L}_{-n_{k}}\cdots\mathscr{L}_{-n_{1}}\;f(\bm{x})=\;$
$\displaystyle
A\,\bm{x}^{\Delta-\sum_{j}n_{j}}\,\prod_{j=1}^{k}(-1)^{p_{j}}\Big{(}\sum_{i<j}n_{i}-\Delta+(n_{j}-1)h_{0}\Big{)}$

From the formula (4.12) for the singular vector
$S_{\lambda}\bar{w}_{c,h(\lambda)}$, using (4.21), we get

$\displaystyle\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}(S_{\lambda}\bar{w}_{1},\bm{x})\bar{w}_{0}}=\;$
$\displaystyle\sum_{k=1}^{\lambda+1}\sum_{\begin{subarray}{c}p_{1},\dots,p_{k}\geq
1\\\
p_{1}+\cdots+p_{k}=\lambda+1\end{subarray}}\frac{(-4/\kappa)^{\lambda+1-k}\;(\lambda!)^{2}}{\prod_{u=1}^{k-1}(\sum_{i=1}^{u}p_{i})(\sum_{i=u+1}^{k}p_{i})}\mathscr{L}_{-p_{1}}\cdots\mathscr{L}_{-p_{k}}\;f(\bm{x}).$

The vanishing
$\braket{\bar{w}_{\infty}^{\prime},\mathcal{Y}(S_{\lambda}\bar{w}_{1},\bm{x})\bar{w}_{0}}=0$
therefore amounts to a differential equation for the initial term
$f(\bm{x})=\mathrm{Init}[\mathcal{Y}](\bm{x})$. With the explicit formula
$f(\bm{x})=A\,\bm{x}^{\Delta}$, this differential equation simplifies to

$\displaystyle 0=\;$ $\displaystyle
A\,\bm{x}^{\Delta-1-\lambda}\;P_{\lambda}(h_{0},h_{\infty})$

The intertwining operator $\mathcal{Y}$ is non-zero only if $A\neq 0$, and in
this case the desired factorization is equivalent to
$P_{\lambda}(h_{0},h_{\infty})=0$. ∎

For fixed $\lambda\in\mathbb{N}$ and $h_{0}\in\mathbb{C}$, the equation
$P_{\lambda}(h_{0},h_{\infty})$ is a degree $\lambda+1$ polynomial equation
for $h_{\infty}$. It is possible to find the roots by a direct calculation
[FZ12]. With the quantum group method, the following easier argument works as
well.

######  Proposition 4.14.

Let $\lambda\in\mathbb{N}$ and $\mu\in\mathbb{Q}$. Then we have the
factorization

$\displaystyle
P_{\lambda}\big{(}h(\mu),h_{\infty}\big{)}=\prod_{\ell=0}^{\lambda}\big{(}h_{\infty}-h(\lambda+\mu-2\ell)\big{)}.$

###### Proof.

Recall from (4.10) that $h(\mu)$ is a quadratic polynomial in $\mu$,
specifically
$h(\mu)=\frac{1}{\kappa}\mu^{2}+\big{(}\frac{2}{\kappa}-\frac{1}{2}\big{)}\mu$.
Note also that

$\displaystyle h(\rho+m)-h(\rho)=\;$
$\displaystyle\frac{m}{\kappa}\Big{(}m+2\rho+\big{(}2-\frac{\kappa}{2}\big{)}\Big{)}.$

Recalling that $\kappa\notin\mathbb{Q}$, we see in particular that if
$\rho\in\mathbb{Q}$, then all $h(\rho+m)$, $m\in\mathbb{Z}$, are distinct.

First fix $\lambda\in\mathbb{N}$ and
$\ell\in\left\\{0,1,\ldots,\lambda\right\\}$. Now if $\mu\in\mathbb{N}$ is
such that $\mu\geq\lambda$, then we have
$\sigma=\lambda+\mu-2\ell\in\mathrm{Sel}(\lambda,\mu)$, so we may consider the
non-zero vector
$u={\iota}_{\;{\sigma}}^{{\lambda},{\mu}}\big{(}u_{0}^{(\sigma)}\big{)}\in\mathsf{M}_{\lambda}\otimes\mathsf{M}_{\mu}$.
The associated function

$\displaystyle F=\mathcal{F}[u]\colon\mathfrak{X}_{2}\to\mathbb{C}$

satisfies the BSA PDEs $\mathscr{D}^{(j)}F(x_{0},x_{1})=0$, for $j=0,1$, where
$\mathscr{D}^{(j)}$ is given by (2.6). From the translation and scaling
covariance and asymptotics given in Theorem 2.2 it follows that in this case
the function must be simply

$\displaystyle F(x_{0},x_{1})=B\,(x_{1}-x_{0})^{\Delta},$

where $\Delta=h(\lambda+\mu-2\ell)-h(\lambda)-h(\mu)$ and
$B=B_{{\lambda+\mu-2\ell},{\lambda}}^{\;{\;\;\mu}}\neq 0$.

But the BSA PDE for a function of the above form reads

$\displaystyle 0=\;$ $\displaystyle
B\,(x_{1}-x_{0})^{\Delta-\lambda-1}\;\sum_{k=1}^{\lambda+1}\sum_{\begin{subarray}{c}p_{1},\ldots,p_{k}\geq
1\\\
p_{1}+\cdots+p_{k}=\lambda+1\end{subarray}}\frac{(-4/\kappa)^{1+\lambda-k}\,\lambda!^{2}}{\prod_{u=1}^{k-1}(\sum_{i=1}^{u}p_{i})(\sum_{i=u+1}^{k}p_{i})}$
$\displaystyle\qquad\qquad\qquad\qquad\prod_{j=1}^{k}(-1)^{p_{j}}\Big{(}-\Delta+\sum_{i>j}p_{i}+(p_{j}-1)h(\mu)\Big{)}$
$\displaystyle=\;$ $\displaystyle
B\,(x_{1}-x_{0})^{\Delta-\lambda-1}\,P_{\lambda}\big{(}h(\mu),h(\lambda+\mu-2\ell)\big{)}.$

We conclude that $P_{\lambda}\big{(}h(\mu),h(\lambda+\mu-2\ell)\big{)}=0$. Now
observe that both $\mu\mapsto h(\mu)$ and $\mu\mapsto h(\lambda+\mu-2\ell)$
are quadratic polynomials in $\mu$, so also $\mu\mapsto
P_{\lambda}\big{(}h(\mu),h(\lambda+\mu-2\ell)\big{)}$ is a polynomial in
$\mu$. We have just shown that this polynomial vanishes for all integers
$\mu\geq\lambda$, so it must be identically zero:

$\displaystyle
P_{\lambda}\big{(}h(\mu),h(\lambda+\mu-2\ell)\big{)}=0\qquad\text{ for all
$\lambda\in\mathbb{N}$, $\ell\in\left\\{0,1,\ldots,\lambda\right\\}$,
$\mu\in\mathbb{C}$.}$

Now let $\lambda\in\mathbb{N}$ and $\mu\in\mathbb{Q}$ be fixed. Consider the
polynomial $h_{\infty}\mapsto P_{\lambda}\big{(}h(\mu),h_{\infty}\big{)}$ of
degree $\lambda+1$. From the above we see that
$h_{\infty}=h(\lambda+\mu-2\ell)$ are roots of this polynomial, when
$\ell=0,1,\ldots,\lambda$. Since $\kappa\not\in\mathbb{Q}$, these $\lambda+1$
roots are distinct, and so we must have

$\displaystyle
P_{\lambda}\big{(}h(\mu),h_{\infty}\big{)}=\mathrm{const.}\times\prod_{\ell=0}^{\lambda}\big{(}h_{\infty}-h(\lambda+\mu-2\ell)\big{)}.$

From the defining formula (4.20) one sees that the leading term of this
polynomial of $h_{\infty}$ is $(h_{\infty})^{1+\lambda}$, so the constant of
proportionality above is in fact $1$. ∎

The following theorem states that there is always a nontrivial intertwining
operator among an arbitrary triple of Verma modules. The theorem can be seen
as a particular case of a more general one in [Li99]. For self-containedness,
we give the proof for the case of our interest in Appendix B.

######  Theorem 4.15.

For $h_{1},h_{0},h_{\infty}\in\mathbb{C}$,

$\dim\mathcal{I}{\binom{M(c,h_{\infty})^{\prime}}{M(c,h_{1})\;M(c,h_{0})}}=1.$

The fusion rules among first row modules of the generic Virasoro VOA now
follow from Lemma 4.13, Proposition 4.14 and Theorem 4.15.

######  Corollary 4.16.

Let $\lambda,\mu,\nu\in\mathbb{N}$. Then,

$\displaystyle\mathrm{dim}\mathcal{I}\binom{Q_{\nu}}{Q_{\lambda}\;Q_{\mu}}=\begin{cases}1&\nu\in\mathrm{Sel}(\lambda,\mu),\\\
0&\mbox{otherwise}.\end{cases}$

###### Proof.

Recall that the first row modules are self-dual (Corollary 4.10). We consider
the following sequence of embeddings

$\displaystyle\mathcal{I}\binom{Q_{\nu}}{Q_{\lambda}\;Q_{\mu}}\to\mathcal{I}\binom{Q_{\nu}}{M_{\lambda}\;Q_{\mu}}\simeq\mathcal{I}\binom{Q_{\nu}}{Q_{\mu}\;M_{\lambda}}\to\mathcal{I}\binom{Q_{\nu}}{M_{\mu}\;M_{\lambda}}\simeq\mathcal{I}\binom{M_{\lambda}^{\prime}}{Q_{\nu}\;M_{\mu}}$
$\displaystyle\to\mathcal{I}\binom{M_{\lambda}^{\prime}}{M_{\nu}\;M_{\mu}}\simeq\mathcal{I}\binom{M_{\nu}^{\prime}}{M_{\lambda}\;M_{\mu}},$

where we used the symmetry in Proposition 4.5. Then, due to Lemma 4.13, the
composed embedding is an isomorphism if and only if the highest weights
satisfy the conditions

$\displaystyle
P_{\lambda}(h(\mu),h(\nu))=P_{\nu}(h(\lambda),h(\mu))=P_{\mu}(h(\nu),h(\lambda))=0.$

It is an easy manipulation of the factorization in Proposition 4.14 to see
that these conditions333 It can also be seen that two of these conditions
imply the other one.  are equivalent to $\nu\in\mathrm{Sel}(\lambda,\mu)$. ∎

##  5\. Compositions of intertwining operators

Intertwining operators, defined in the previous section, give $3$-point
correlation functions of conformal field theories (or alternatively, in a
geometric interpretation, amplitudes in a pair-of-pants Riemann surface with
three parametrized boundary components [Hua12]). Importantly, they serve as
the building blocks of more general correlation functions. Namely, for a
multipoint correlation function (or an amplitude on a Riemann surfaces with
more than three parametrized boundary components), one forms an appropriate
composition of intertwining operators.

An intertwining operator is a formal series, with coefficients that are linear
operators between modules of the VOA. Compositions of intertwining operators
are then a priori formal series in several formal variables, also with
coefficients that are (composed) linear operators between modules. To get
actual correlation functions, however, one considers suitable matrix elements
of the formal series of linear operators, and crucially, one must then address
the convergence of the corresponding series.

In this section, we first study properties of the compositions of the
intertwining operators between modules of the first row subcategory for the
generic Virasoro VOA as formal power series. Then we establish analoguous
properties for the functions obtained by the quantum group method —
specifically the ones corresponding to the conformal block vectors of Section
3. The main result of this section (Theorem 5.9) is that the formal series of
the composition of intertwining operators coincide with suitable power series
expansions of the actual conformal block functions, and that these series are
convergent in the appropriate domains.

More precisely, the section is organized as follows. In Section 5.1 we fix our
normalization of the intertwining operators, and in Section 5.2 we introduce
the specific spaces of formal series that will be used. Section 5.3 contains
the proofs of two key properties; that (the highest weight matrix elements of)
compositions of the intertwining operators satisfy a system of partial
differential equations, and that the formal series solutions to this PDE
system are unique (up to scalar multiples). In Section 5.4 we detail a
recursive series expansion procedure for the quantum group functions, and use
this and the partial differential equations to prove the main result (Theorem
5.9) that the formal series of the composition of intertwining operators
converge to the functions corresponding to the conformal block vectors. In
Section 5.5 we comment on some first applications of the result.

###  5.1. Fusion rules and normalized intertwining operators

A comparison of Corollary 4.16 and Lemma 2.1 directly gives

$\displaystyle\dim\mathcal{I}\binom{Q_{\lambda_{\infty}}}{Q_{\lambda_{1}}\;Q_{\lambda_{0}}}=\;$
$\displaystyle\dim\mathrm{Hom}_{\mathcal{U}_{q}(\mathfrak{sl}_{2})}\big{(}\mathsf{M}_{\lambda_{\infty}},\;\mathsf{M}_{\lambda_{1}}\otimes\mathsf{M}_{\lambda_{0}}\big{)}$
$\displaystyle=\;$ $\displaystyle\begin{cases}1&\text{ if
}\lambda_{\infty}\in\mathrm{Sel}(\lambda_{1},\lambda_{0})\\\ 0&\text{
otherwise.}\end{cases}$

######  Definition 5.1.

When $\lambda_{\infty}\in\mathrm{Sel}(\lambda_{1},\lambda_{0})$ so that
nonzero intertwining operators exist, we denote by

$\displaystyle\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}\in\mathcal{I}\binom{Q_{\lambda_{\infty}}}{Q_{\lambda_{0}}\;Q_{\lambda_{1}}}$

the unique intertwining operator normalized so that

(5.1)
$\displaystyle\mathrm{Init}[\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}](\bm{x})=\;B\,\bm{x}^{\Delta},$

where

$\displaystyle B=B_{{\lambda_{1}},{\lambda_{0}}}^{\;{\;\lambda_{\infty}}}\neq
0\qquad\text{ and
}\qquad\Delta={\Delta}_{{\lambda_{1}},{\lambda_{0}}}^{\;{\;\lambda_{\infty}}}=h(\lambda_{\infty})-h(\lambda_{1})-h(\lambda_{0})$

are as in Theorem 2.2.

The above normalization of the intertwining operator
$\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}$ is chosen so
that that the “matrix element”
$\Braket{{\bar{w}}_{\lambda_{\infty}},\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}({\bar{w}}_{\lambda_{1}},\bm{x})\,{\bar{w}}_{\lambda_{0}}}=\mathrm{Init}[\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}](\bm{x})=B\,\bm{x}^{\Delta}$
between highest weight vectors formally coincides with the function
$x\mapsto\mathcal{F}[u_{\underline{{\varsigma}}}](0,x)$ constructed with the
quantum group method from the conformal block vector
$u_{\underline{{\varsigma}}}\in\mathsf{M}_{\lambda_{1}}\otimes\mathsf{M}_{\lambda_{0}}$
corresponding to
${\underline{{\varsigma}}=(\lambda_{0},\lambda_{1},\lambda_{\infty})}$ as in
Section 3.

###  5.2. Space of formal series

For a sequence
$\underline{{\lambda}}=(\lambda_{0},\lambda_{1}\dots,\lambda_{N},\lambda_{\infty})\in\mathbb{N}^{N+2}$,
and a $\underline{{\lambda}}$-admissible sequence
$\underline{{\varsigma}}=({\varsigma}_{0},{\varsigma}_{1},\ldots,{\varsigma}_{N-1},{\varsigma}_{N})$,
we consider the composition of intertwining operators

$\displaystyle\mathcal{Y}_{\lambda_{N}\,{\varsigma}_{N-1}}^{\;{\varsigma}_{N}}(w_{N},{\bm{x}_{N}})\mathcal{Y}_{\lambda_{N-1}\,{\varsigma}_{N-2}}^{\;{\varsigma}_{N-1}}(w_{N-1},{\bm{x}_{N-1}})\cdots\mathcal{Y}_{\lambda_{2}\,{\varsigma}_{1}}^{\;{\varsigma}_{2}}(w_{2},{\bm{x}_{2}})\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}(w_{1},{\bm{x}_{1}}),$

which is a priori a formal series in $\bm{x}_{1},\ldots,\bm{x}_{N}$ with
coefficients that are linear maps ${Q_{{\varsigma}_{0}}\to
Q_{{\varsigma}_{N}}}$. We consider in particular the “matrix element” between
highest weight states

(5.2) $\displaystyle
C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(\bm{x}_{1},\dots,\bm{x}_{N}):=\Braket{{\bar{w}}_{{\varsigma}_{N}}^{\prime},\,\mathcal{Y}_{\lambda_{N}\,{\varsigma}_{N-1}}^{\;{\varsigma}_{N}}({\bar{w}}_{\lambda_{N}},\bm{x}_{N})\cdots\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}({\bar{w}}_{\lambda_{1}},\bm{x}_{1})\,{\bar{w}}_{\lambda_{0}}}.$

As in the proof of Proposition 4.12, the general matrix elements

$\displaystyle\Braket{w_{\infty}^{\prime},\,\mathcal{Y}_{\lambda_{N}\,{\varsigma}_{N-1}}^{\;{\varsigma}_{N}}(w_{N},\bm{x}_{N})\cdots\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}(w_{1},\bm{x}_{1})\,w_{0}}$

with $w_{i}\in Q_{\lambda_{i}}$, $i=0,1,\dots,N$, and $w_{\infty}^{\prime}\in
Q_{\lambda_{\infty}}^{\prime}$, can be determined by a recursion using the
Jacobi identity (4.18) (in the specific form of Corollary 4.7) from this
matrix element (5.2); this particular matrix element serves a role analogous
to the initial term of an intertwining operator. A priori, (5.2) is a formal
series in $\bm{x}_{1},\ldots,\bm{x}_{n}$ with complex coefficients, i.e., an
element of $\mathbb{C}\\{\bm{x}_{1},\ldots,\bm{x}_{n}\\}$. But in fact the
series has a more particular structure: it is essentially a formal power
series (with non-negative powers!) in the ratios of the variables.

######  Lemma 5.2.

The series (5.2) belongs to the space

(5.3)
$\displaystyle\mathbb{C}[[\bm{x}_{N-1}/\bm{x}_{N}]][[\bm{x}_{N-2}/\bm{x}_{N-1}]]\cdots[[\bm{x}_{1}/\bm{x}_{2}]]\bm{x}_{N}^{\Delta_{N}}\cdots\bm{x}_{1}^{\Delta_{1}},$

with $\Delta_{i}=h({\varsigma}_{i})-h(\lambda_{i})-h({\varsigma}_{i-1})$ for
$i=1,\dots,N$.

We make a preliminary to prove Lemma 5.2. For each $\lambda\in\mathbb{N}$ and
$n\in\mathbb{N}$, we set $Q_{\lambda}(n):=(Q_{\lambda})_{(h(\lambda)+n)}$ as
the eigenspace of $L_{0}$ corresponding to the eigenvalue $h(\lambda)+n$.
Suppose that a triple $(\lambda_{\infty},\lambda_{1},\lambda_{0})$ satisfies
the selection rule $\lambda_{\infty}\in\mathrm{Sel}(\lambda_{1},\lambda_{0})$,
and take the intertwining operator
$\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}$ of type
$\binom{Q_{\lambda_{\infty}}}{Q_{\lambda_{1}}\;Q_{\lambda_{0}}}$. For $w\in
Q_{\lambda_{1}}(k)$, $k\in\mathbb{N}$, using Corollary 4.7(2) for $L_{0}$ like
in Lemma 4.2, we find that
$\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}(w,\bm{x})$ takes
the form

$\mathcal{Y}_{\lambda_{1}\,\lambda_{0}}^{\;\lambda_{\infty}}(w,\bm{x})=\sum_{m\in\mathbb{Z}}w_{(m)}\bm{x}^{\Delta-m-1}\qquad\text{
with }\Delta=h(\lambda_{\infty})-h(\lambda_{1})-h(\lambda_{0}),$

where $w_{(m)}Q_{\lambda_{0}}(n)\subset Q_{\lambda_{\infty}}(k-m+n-1)$ for
$m\in\mathbb{Z}$, $n\in\mathbb{N}$.

We prove a slightly more general result than Lemma 5.2:

######  Lemma 5.3.

Let $w\in Q_{{\varsigma}_{0}}(n)$, $n\in\mathbb{N}$. Then, the formal series
defined by

$\displaystyle
C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(w;\bm{x}_{1},\dots,\bm{x}_{N}):=\Braket{{\bar{w}}_{{\varsigma}_{N}}^{\prime},\mathcal{Y}_{\lambda_{N}\,{\varsigma}_{N-1}}^{\;{\varsigma}_{N}}({\bar{w}}_{\lambda_{N}},\bm{x}_{N})\cdots\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}({\bar{w}}_{\lambda_{1}},\bm{x}_{1})w}$

lies in the space

$\displaystyle\mathbb{C}[[\bm{x}_{N-1}/\bm{x}_{N}]][[\bm{x}_{N-2}/\bm{x}_{N-1}]]\cdots[[\bm{x}_{1}/\bm{x}_{2}]]\,\bm{x}_{N}^{\Delta_{N}}\cdots\bm{x}_{2}^{\Delta_{2}}\bm{x}_{1}^{\Delta_{1}-n}.$

###### Proof.

We prove this by induction on $N$. When $N=1$, we have

$\displaystyle\braket{{\bar{w}}_{{\varsigma}_{1}}^{\prime},\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}({\bar{w}}_{\lambda_{1}},\bm{x}_{1})w}$
$\displaystyle\;=\sum_{m\in\mathbb{Z}}\braket{{\bar{w}}_{{\varsigma}_{1}}^{\prime},({\bar{w}}_{\lambda_{1}})_{(m)}w}\bm{x}_{1}^{\Delta_{1}-m-1}.$

Here, since $({\bar{w}}_{\lambda_{1}})_{(m)}w\subset
Q_{{\varsigma}_{1}}(-m+n-1)$, the matrix element
$\braket{{\bar{w}}_{{\varsigma}_{1}}^{\prime},({\bar{w}}_{\lambda_{1}})_{(m)}w}$
vanishes unless $m=n-1$. Therefore, we can see that

$\displaystyle\braket{{\bar{w}}_{{\varsigma}_{1}}^{\prime},\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}({\bar{w}}_{\lambda_{1}},\bm{x}_{1})w}$
$\displaystyle=\braket{{\bar{w}}_{{\varsigma}_{1}}^{\prime},({\bar{w}}_{\lambda_{1}})_{(n-1)}w}\bm{x}_{1}^{\Delta_{1}-n},$

which is the $N=1$ case.

Then assume that the assertion holds for $N\leq k-1$ with some $k>1$. Then, we
have

$\displaystyle
C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(w;\bm{x}_{1},\dots,\bm{x}_{k})$
$\displaystyle\;=\sum_{m\in\mathbb{Z}}\Braket{{\bar{w}}_{{\varsigma}_{k}}^{\prime},\mathcal{Y}_{\lambda_{k}\,{\varsigma}_{k-1}}^{\;{\varsigma}_{k}}({\bar{w}}_{\lambda_{k}},\bm{x}_{k})\cdots\mathcal{Y}_{\lambda_{2}\,{\varsigma}_{1}}^{\;{\varsigma}_{2}}({\bar{w}}_{\lambda_{2}},\bm{x}_{2})({\bar{w}}_{\lambda_{1}})_{(m)}w}\bm{x}_{1}^{\Delta_{1}-m-1}.$

Again, since $({\bar{w}}_{\lambda_{1}})_{(m)}w\in
Q_{{\varsigma}_{1}}(-m+n-1)$, the matrix element vanishes unless $m\leq n-1$.
In other words, we have the expansion

$\displaystyle
C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(w;\bm{x}_{1},\dots,\bm{x}_{k})$
$\displaystyle=\sum_{\ell=0}^{\infty}\Braket{{\bar{w}}_{{\varsigma}_{k}}^{\prime},\mathcal{Y}_{\lambda_{k}\,{\varsigma}_{k-1}}^{\;{\varsigma}_{k}}({\bar{w}}_{\lambda_{k}},\bm{x}_{k})\cdots\mathcal{Y}_{\lambda_{2}\,{\varsigma}_{1}}^{\;{\varsigma}_{2}}({\bar{w}}_{\lambda_{2}},\bm{x}_{2})({\bar{w}}_{\lambda_{1}})_{(n-\ell-1)}w}\bm{x}_{1}^{\Delta_{1}-n+\ell}$
$\displaystyle=\sum_{\ell=0}^{\infty}C^{\underline{{\lambda}}^{\prime}}_{\underline{{\varsigma}}^{\prime}}\left(({\bar{w}}_{\lambda_{1}})_{(n-\ell-1)}w;\bm{x}_{2},\dots,\bm{x}_{k}\right)\bm{x}_{1}^{\Delta_{1}-n+\ell},$

where we set
$\underline{{\lambda}}^{\prime}=({\varsigma}_{1},\lambda_{2},\ldots,\lambda_{k},\lambda_{\infty})$
and
$\underline{{\varsigma}}^{\prime}=({\varsigma}_{1},\dots,{\varsigma}_{k})$. By
the induction hypothesis, each coefficient
$C^{\underline{{\lambda}}^{\prime}}_{\underline{{\varsigma}}^{\prime}}\left(({\bar{w}}_{\lambda_{1}})_{(n-\ell-1)}w;\bm{x}_{2},\dots,\bm{x}_{k}\right)$
lies in

$\displaystyle\mathbb{C}[[\bm{x}_{k-1}/\bm{x}_{k}]]\cdots[[\bm{x}_{2}/\bm{x}_{3}]]\,\bm{x}_{2}^{\Delta_{2}-\ell}.$

Consequently, we have the desired result at $N=k$. ∎

###  5.3. System of differential equations

Due to the quotienting out of the singular vector in the first row module
$Q_{\lambda}=M_{\lambda}\,\big{/}\,\mathcal{U}(\mathfrak{vir})S_{\lambda}\bar{w}_{c,h(\lambda)}$,
the matrix element
$C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(\bm{x}_{1},\dots,\bm{x}_{N})$
satisfies a certain system of differential equations. Given a sequence
$\underline{{\lambda}}=(\lambda_{0},\dots,\lambda_{N},\lambda_{\infty})\in\mathbb{N}^{N+2}$,
we introduce the differential operators

(5.4) $\displaystyle\widetilde{\mathscr{L}}_{n}^{(j)}:=$
$\displaystyle(-x_{j})^{n}\left(-x_{j}\sum_{1\leq i\leq
N}\frac{{\partial}}{{\partial}x_{i}}-(1+n)h(\lambda_{0})\right)$
$\displaystyle-\sum_{\begin{subarray}{c}1\leq i\leq N\\\ i\neq
j\end{subarray}}(x_{i}-x_{j})^{n}\left((x_{i}-x_{j})\frac{{\partial}}{{\partial}x_{i}}+(1+n)h(\lambda_{i})\right)$

for $j=1,\dots,N$ and $n\in\mathbb{Z}$, and

(5.5)
$\displaystyle\widetilde{\mathscr{D}}^{(j)}=\sum_{k=1}^{\lambda_{j}+1}\sum_{\begin{subarray}{c}p_{1},\dots,p_{k}\geq
1\\\
p_{1}+\cdots+p_{k}=\lambda_{j}+1\end{subarray}}\frac{(-4/\kappa)^{1+\lambda_{j}-k}\lambda_{j}!}{\prod_{u=1}^{k-1}(\sum_{i=1}^{u}p_{i})(\sum_{i=u+1}^{k}p_{i})}\widetilde{\mathscr{L}}_{-p_{1}}^{(j)}\cdots\widetilde{\mathscr{L}}_{-p_{k}}^{(j)}$

for $j=1,\dots,N$. These differential operators
$\widetilde{\mathscr{L}}_{n}^{(j)}$ and $\widetilde{\mathscr{D}}^{(j)}$
essentially correspond to how the differential operators
$\mathscr{L}^{(j)}_{n}$ and $\mathscr{D}^{(j)}$ of (2.7) and (2.6) act on
translation invariant functions; see Lemma 5.7 for a precise statement. Notice
that the actions of $\widetilde{\mathscr{L}}_{n}^{(j)}$ and
$\widetilde{\mathscr{D}}^{(j)}$ on the space

$\displaystyle\mathbb{C}[\bm{x}_{N}]\,[[\bm{x}_{N-1}/\bm{x}_{N}]][[\bm{x}_{N-2}/\bm{x}_{N-1}]]\cdots[[\bm{x}_{1}/\bm{x}_{2}]]\,[\bm{x}_{1}^{-1}]\;\bm{x}_{N}^{\Delta_{N}}\cdots\bm{x}_{1}^{\Delta_{1}}$

of formal series are canonically determined in such a way that

$\displaystyle x_{j}\mapsto\bm{x}_{j}\qquad\text{ and
}\qquad\frac{{\partial}}{{\partial}x_{i}}\mapsto\frac{{\partial}}{{\partial}\bm{x}_{i}}\quad\text{
for }i=1,\dots,N,$

and the factors $(x_{i}-x_{j})^{n}$ are expanded as the following formal power
series

$\displaystyle(x_{i}-x_{j})^{n}\;\mapsto\;\begin{cases}(-\bm{x}_{j})^{n}\,\sum_{k=0}^{\infty}\binom{n}{k}(-1)^{k}\,(\bm{x}_{i}/\bm{x}_{j})^{k}&\text{
for }i<j\\\
\;\bm{x}_{i}^{n}\;\sum_{k=0}^{\infty}\binom{n}{k}(-1)^{k}\,(\bm{x}_{j}/\bm{x}_{i})^{k}&\text{
for }i>j,\end{cases}$

i.e., $(x_{i}-x_{j})^{n}\mapsto(\bm{x}_{i}-\bm{x}_{j})^{n}$ for $i>j$ but
$(x_{i}-x_{j})^{n}\mapsto(-\bm{x}_{j}+\bm{x}_{i})^{n}$ for $i<j$.

The following standard lemma will be instrumental for us. We include the
details here to concretely demonstrate the source of the specific power series
expansions above.

######  Lemma 5.4.

Let
$\underline{{\lambda}}=(\lambda_{0},\dots,\lambda_{N},\lambda_{\infty})\in\mathbb{N}^{N+2}$
be a sequence and
$\underline{{\varsigma}}=({\varsigma}_{0},\dots,{\varsigma}_{N})$ be
$\underline{{\lambda}}$-admissible. Then, the matrix element
$C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(\bm{x}_{1},\dots,\bm{x}_{N})$
solves the following system of differential equations

$\widetilde{\mathscr{D}}^{(j)}C^{\underline{{\lambda}}}_{\underline{{\varsigma}}}(\bm{x}_{1},\dots,\bm{x}_{N})=0\qquad\text{
for }j=1,\dots,N.$

###### Proof.

Fix $j\in\left\\{1,\ldots,N\right\\}$. For $w\in Q_{\lambda_{j}}$, consider

$\displaystyle
X(w;\bm{x}_{1},\dots,\bm{x}_{N}):=\Braket{{\bar{w}}_{{\varsigma}_{N}}^{\prime},\mathcal{Y}_{\lambda_{N}\,{\varsigma}_{N-1}}^{\;{\varsigma}_{N}}({\bar{w}}_{\lambda_{N}},\bm{x}_{N})\cdots\mathcal{Y}_{\lambda_{j}\,{\varsigma}_{j-1}}^{\;{\varsigma}_{j}}(w,\bm{x}_{j})\cdots\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}({\bar{w}}_{\lambda_{1}},\bm{x}_{1}){\bar{w}}_{\lambda_{0}}}.$

Then we can use Corollary 4.7(1) to calculate
$X(L_{-p}w;\bm{x}_{1},\dots,\bm{x}_{N})$, for $p>0$,

$\displaystyle X(L_{-p}w;\bm{x}_{1},\dots,\bm{x}_{N})$ $\displaystyle=\;$
$\displaystyle\phantom{+}\sum_{k=0}^{\infty}\binom{1-p}{k}(-\bm{x}_{j})^{k}$
$\displaystyle\qquad\qquad\Braket{{\bar{w}}_{{\varsigma}_{N}}^{\prime},\mathcal{Y}_{\lambda_{N}\,{\varsigma}_{N-1}}^{\;{\varsigma}_{N}}({\bar{w}}_{\lambda_{N}},\bm{x}_{N})\cdots
L_{-p-k}\,\mathcal{Y}_{\lambda_{j}\,{\varsigma}_{j-1}}^{\;{\varsigma}_{j}}(w,\bm{x}_{j})\cdots\mathcal{Y}_{\lambda_{1}\,{\varsigma}_{0}}^{\;{\varsigma}_{1}}({\bar{w}}_{\lambda_{1}},\bm{x}_{1}){\bar{w}}_{\lambda_{0}}}$
$\displaystyle-\sum_{k=0}^{\infty}\binom{1-p}{k}(-\bm{x}_{j})^{1-p-k}$