Irregular vertex algebras

TL;DR

This paper introduces irregular vertex algebras, extending fundamental properties like associativity and operator product expansions, and provides elementary examples to illustrate their structure.

Contribution

It defines irregular vertex algebras and proves key properties, expanding the theoretical framework of vertex operator algebras.

Findings

Irregular vertex algebras are formally introduced.

Key properties like associativity are established for irregular cases.

Elementary examples of irregular vertex algebras are provided.

Abstract

We introduce the notion of irregular vertex (operator) algebras. The irregular versions of fundamental properties, such as Goddard uniqueness theorem, associativity and operator product expansions are formulated and proved. We also give some elementary examples of irregular vertex operator algebras.

Equations

$$Y_{\mathcal{M}}:V⟶{\mathrm{End}}_{{\mathcal{D}}_S}(\mathcal{M})[[z^{\pm 1}]],A\mapsto Y_{\mathcal{M}}(A,z)=\sum _{n\in \mathbb{Z}}{A}_{(n)}^{\mathcal{M}}{z}^{-n-1}$$

$$Y:V⟶\mathrm{End}(V)[[z^{\pm 1}]],A\mapsto Y(A,z)=\sum _{n\in \mathbb{Z}}{A}_{(n)}{z}^{-n-1}$$

$$(z-w{)}^N[Y(A,z),Y(B,w)]=0$$

$$Y(A,z)Y(B,w)=\sum _{n=0}^N\frac{Y(C_n,w)}{(z-w{)}^{n+1}}+\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}Y(A,z)Y(B,w)\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}$$

$${\mathcal{A}}_{\lambda }(z)=\sum _{n\in \mathbb{Z}}{\mathcal{A}}_{\lambda ,n}{z}^{-n-1}\in \mathrm{Hom}({\mathcal{U}}_{\mu }^{\circ },{\overline{\mathcal{U}}}_{\lambda +\mu }^{\circ })[[z^{\pm 1}]]$$

$${\mathcal{A}}_{\lambda }(z){\mathcal{B}}_{\mu }\in e^{\mathfrak{f}(z;\lambda ,\mu )}{\mathcal{U}}_{\lambda +\mu }^{\circ }((z))$$

$$Y:{\mathcal{U}}_{\lambda }^{\circ }⟶\mathrm{Hom}\left({\mathcal{U}}_{\mu }^{\circ },{\overline{\mathcal{U}}}_{\lambda +\mu }^{\circ }\right)[[z^{\pm 1}]],$$

$$(z-w{)}^N\left(e_{|z|>|w|}^{-\mathfrak{f}(z-w;\lambda ,\mu )}Y({\mathcal{A}}_{\lambda },z)Y({\mathcal{B}}_{\mu },w)-e_{|w|>|z|}^{-\mathfrak{f}(z-w;\lambda ,\mu )}Y({\mathcal{B}}_{\mu },w)Y({\mathcal{A}}_{\lambda },z)\right)=0$$

$$Y({\mathcal{A}}_{\lambda },z)Y({\mathcal{B}}_{\mu },w)=e^{\mathfrak{f}(z-w;\lambda ,\nu )}\left(\sum _{n=0}^N\frac{Y({\mathcal{C}}_{\lambda +\mu ,n},w)}{(z-w{)}^{n+1}}+\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}Y({\mathcal{A}}_{\lambda },z)Y({\mathcal{B}}_{\mu },w)\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}\right)$$

$$[{\partial }_{\lambda },Y({\mathcal{A}}_{\lambda },z)]=Y({\partial }_{\lambda }{\mathcal{A}}_{\lambda },z)$$

$$\mathrm{Heis}=\underset{n\in \mathbb{Z}}{⨁}\mathbb{C}{a}_n\oplus \mathbb{C}1$$

$$Y(a_{-n_1}{a}_{-n_2}\cdots a_{-n_k}|0⟩,z):=\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}{\partial }_z^{(n_1-1)}a(z){\partial }_z^{(n_2-1)}a(z)\cdots {\partial }_z^{(n_k-1)}a(z)\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}$$

$$a(z)a(w)=\frac{1}{(z-w{)}^2}+\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}a(z)a(w)\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}.$$

$$Y(|\lambda ⟩,z)=\sum _{k=0}^{\infty }\frac{Y(a_{-1}^k|0⟩,z)}{k!}=\sum _{k=0}^{\infty }\frac{\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}a(z{)}^k\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}}{k!}=\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}{e}^{\lambda a(z)}\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}$$

$$Y(|\lambda ⟩,z)|\mu ⟩=\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}{e}^{\lambda a(z)}\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}{e}^{\mu a_{-1}}|0⟩=e^{\lambda \mu /z^2}(|\lambda +\mu ⟩+o(z))$$

$$Y(a_{-n_1}\cdots a_{-n_k}|\lambda ⟩,z)=\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}{\partial }_z^{(n_1-1)}a(z)\cdots {\partial }_z^{(n_k-1)}a(z)e^{\lambda a(z)}\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}.$$

$$[{\partial }_{\lambda },Y(|\lambda ⟩,z)]=Y({\partial }_{\lambda }|\lambda ⟩,z)=Y(a_{-1}|\lambda ⟩,z)=\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}a(z)Y(|\lambda ⟩,z)\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}.$$

$$Y(|\lambda ⟩,z)Y(a_{-1}|\mu ⟩,w)=e_{|z|>|w|}^{\lambda \mu /(z-w{)}^2}(\frac{\lambda Y(|\lambda +\mu ⟩,w)}{(z-w{)}^2}$$

$$+\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}Y(|\lambda ⟩,z)Y(a_{-1}|\mu ⟩,w)\text{raise}2.5pt\underset{{}^{\circ }}{\overset{{}_{\circ }}{}}).$$

$$Y(|\lambda ⟩,z)a_{-1}|\mu ⟩=e^{\lambda \mu /z^2}\left(\frac{\lambda |\lambda +\mu ⟩}{z^2}+\frac{\lambda a_{-2}|\lambda +\mu ⟩}{z}+o(1)\right).$$

$$M[[z^{\pm 1}]{]}_{\ell }:=\prod _{n\in \mathbb{Z}}{M}_{\ell +n}{z}^n.$$

$${\mathrm{Hom}}_R(M,N)=\underset{n\in \mathbb{Z}}{⨁}{\mathrm{Hom}}_R(M,N{)}_n$$

$$\sigma :S\times S⟶S.$$

$$[t^{k+1}{\partial }_t,t^{\ell +1}{\partial }_t]=(\ell -k)t^{k+\ell +1}{\partial }_t$$

$${\rho }_S:{\mathrm{Der}}_0(\mathbb{C}[[t]])⟶{\Theta }_S,t^{k+1}{\partial }_t\mapsto D_k$$

$$[{\rho }_{S^2}(t^{k+1}{\partial }_t),{\sigma }^{\ast }f]={\sigma }^{\ast }[D_k,f]$$

$$D_k:=\sum _{j=1}^{r-k}j{\lambda }_{j+k}\frac{\partial }{\partial {\lambda }_j}(k=0,\dots ,r-1)$$

$$Y:V⟶\mathrm{End}(V)[[z^{\pm 1}]],A\mapsto Y(A,z)=\sum _{n\in \mathbb{Z}}{A}_{(n)}{z}^{-n-1}$$

$$(z-w{)}^N[Y(A,z),Y(B,w)]=0$$

$$Y_{\mathcal{M}}:V⟶{\mathrm{End}}_{{\mathcal{D}}_S}(\mathcal{M})[[z^{\pm 1}]],A\mapsto Y_{\mathcal{M}}(A,z)=\sum _{n\in \mathbb{Z}}{A}_{(n)}^{\mathcal{M}}{z}^{-n-1},$$

$$Y_{\mathcal{M}}(A,z)Y_{\mathcal{M}}(B,w)\mathcal{C}\in \mathcal{M}((z))((w))$$

Full text

1. 1 Introduction
2. 2 Coherent state modules
3. 3 Irregular vertex operator algebras
4. 4 Associativity and operator product expansions
5. 5 Irregular Heisenberg vertex operator algebras
6. 6 Irregular Virasoro vertex operator algebras

1. Introduction

The vertex algebras, the definition of which was introduced by Borcherds [Bor86] and the foundation of the theory of which was developed by Frenkel-Lepowsky-Meurman [FLM88], may be seen as a mathematical language of the two-dimensional conformal field theory initiated by Belavin-Polyakov-Zamolodchikov [BPZ84].

Recently, several people [Gai13, GT12, JNS08, NS10, Nag15, Nag18] study irregular singularities in conformal field theory. They are mainly motivated by Alday-Gaiotto-Tachikawa (AGT) correspondence [AGT10] and their applications. We would like to note that the notion of coherent states plays a fundamental role in these studies.

In the present paper, we shall initiate an attempt to give a mathematical language of irregular singularities in conformal field theory by introducing the notions of coherent state modules and irregular vertex $($operator$)$ algebras. The main result of this paper is to formulate and prove the irregular versions of fundamental properties of irregular vertex algebras. We also give some elementary examples of coherent state modules and irregular vertex algebras.

In this introduction, we shall explain these notions and examples. We will explain the notion of coherent state module in Section 1.1, irregular vertex algebras and their fundamental properties in Section 1.2, and the examples of irregular vertex algebras in Section 1.3.

1.1. Coherent states and irregular singularities in conformal field theory

In conformal field theory on a Riemann sphere, Belavin-Polyakov-Zamolodchikov [BPZ84] and Knizhnik-Zamolodchikov [KZ84] found that the chiral correlation functions (conformal blocks) of vertex operators corresponding to highest weight vectors in minimal models and in Wess-Zumino-Witten models respectively, satisfy certain systems of differential equations with regular singularities. These equations are known as the BPZ equation and the KZ equation respectively.

Mathematically, these vertex operators can be interpreted as intertwining operators [FHL93] associated with highest weight vectors of modules over vertex algebras. The conformal block is then given by the composition of intertwining operators [Hua03].

In [AGT10], Alday-Gaiotto-Tachikawa found the relationship between Virasoro conformal blocks and Nekrasov partition functions of $\mathcal{N}=2$ superconformal gauge theories. The relation is now known as the AGT correspondence. In order to generalize the AGT correspondence to Nekrasov partition functions of asymptotically free gauge theories, Gaiotto [Gai13] introduced irregular conformal blocks as the counter parts in CFT side.

In his construction of the irregular conformal block, a coherent state, which is a simultaneous eigenvector of some positive modes of the Virasoro algebra, plays a central role. He found that the coherent state corresponds to a state creating an irregular singularity of the stress-energy tensor (irregular state), while a highest weight vector corresponds to a state creating a regular singularity (regular state).

Correlation functions of vertex operators corresponding to irregular states are called irregular conformal blocks. More general studies of such irregular states for the Virasoro algebra were given in [BMT12, GT12]. In particular, Gaiotto-Teschner [GT12] constructed an irregular state as a certain collision limit (confluence) of regular states and characterized this state as an element of a $\mathcal{D}$-module. Irregular states for the affine Lie algebras and the $W_{3}$-algebra were studied in [GLP] and [KMST13].

We note that in the side of mathematics, the idea that non-highest weight states of the affine Lie algebra create irregular singularities of the KZ equation was already appeared in [FFTL10] and [JNS08] before [Gai13].

In Section 2, based on these various studies of coherent states and irregular singularities in conformal field theory, we introduce the notion of a coherent state module over a vertex algebra (Definition 2.4). Let $V$ be a vertex algebra and $S$ be a positively graded vector space over $\mathbb{C}$. Denote by $\mathcal{D}_{S}$ the ring of differential operators on $S$. The coherent state $V$-module $\mathcal{M}$ on $S$ is a $\mathcal{D}_{S}$-module with $\mathcal{D}_{S}$-linear vertex operators

[TABLE]

and a distinguished vector $|{\rm coh}\rangle\in\mathcal{M}$, called a coherent state, together with some axioms. We call $S$ the space of internal parameters. The definition is motivated by the description of coherent states of the Virasoro algebra in [GT12] as follows: in the process of the confluence of $(r+1)$ regular states, the resulting irregular state obtains $r$ new parameters and the action of positive modes of the Virasoro algebra is given by differential operators of these $r$ parameters. We call them internal parameters of the coherent state.

Conformal structures, the action of stress-energy tensors, on vertex algebras and their modules give coordinate change rules of vertex operators. They play important roles in coordinate-free approach for various concepts and theory on higher genus Riemann surfaces (see [FBZ04]). In Section 2.3, we give the definition of a conformal structure on a coherent state module. The appearance of internal parameters makes the definition a little more complicated than the usual modules since we also need to consider coordinate changes of internal parameters.

In the subsequent, we will study irregular conformal blocks as the dual space of coinvariants associated to conformal coherent state modules. As an application, the confluent KZ equation [JNS08] is described as an integrable connection on the irregular conformal block associated with coherent state modules over the vertex algebra $V_{k}(\mathfrak{sl}_{2})$. In the future work, we will also discuss the relationship between $3$ points irregular conformal blocks of coherent state modules and the irregular type intertwining operators (see also Section 1.2).

1.2. Irregular vertex algebras

The definition of a vertex algebras was introduced by Borcherds [Bor86] and a vertex operator algebra (vertex algebra with a conformal structure) was defined by Frenkel-Lepowsky-Meurman [FLM88]. Nowadays, various equivalent definitions of vertex algebras are known. We shall adopt the axioms in [FKRW95] (see also [FBZ04, Kac98]) known as Goddard’s axioms [God89] since our definition of irregular vertex algebra is a generalization of them.

The main point is the definition of vertex operators. For a vertex algebra $V$ (see Definition 2.3), vertex operators are given by the state-field correspondence

[TABLE]

and they become fields, namely $Y(A,z)B\in V((z))$ for all $A,B\in V$. They also satisfy the locality axiom

[TABLE]

for sufficiently large $N$. As a consequence of the axioms, we have the operator product expansion (OPE)

[TABLE]

where $C_{n}\in V$ are some states and {\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}Y(A,z)Y(B,w){\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}} is the normally ordered product, which is smooth along with $z=w$. Thus in usual vertex algebras, all singularities are poles and this is the reason why correlation functions of usual vertex operators only have regular singularities.

Our irregular vertex algebras are constructed on a particular coherent state modules, called envelopes of vertex algebras. A coherent state module $\mathcal{U}$ on a space of internal parameter $S$ is called an envelope of a vertex algebra $V$ if it contains $V$ in the fiber $\mathcal{U}_{0}$ on the origin $0\in S$ and satisfies some compatibility conditions (see Definition 3.11). To consider irregular vertex operators for irregular states in $\mathcal{U}$, we also need to consider the singular locus $H\subset S$ since the composition of two irregular vertex operators may have singularities not only on $z=w$ but also on some divisor $H\subset S$. Therefore we need to introduce a $\mathcal{D}_{S}$-module $\mathcal{U}^{\circ}$ satisfies $\mathcal{U}\subset\mathcal{U}^{\circ}\subset\mathcal{U}(*H)$ with some good properties where $\mathcal{U}(*H)$ is a localization of $\mathcal{U}$ along $H$ (see Definition 3.1).

We can generalize the notion of field to have exponential type essential singularities. Denote by $\mathcal{U}_{\mu}$ the fiber of $\mathcal{U}$ on $\mu\in S$. An irregular field with an irregularity ${\mathfrak{f}}(z;\lambda,\mu)$ and an internal parameter $\lambda\in S$ is a $\mathrm{Hom}(\,\mathcal{U}_{\mu}^{\circ},\overline{\mathcal{U}}_{\lambda+\mu}^{\circ}\,)$-valued formal power series

[TABLE]

satisfies the condition

[TABLE]

for $\mathcal{B}_{\mu}\in\mathcal{U}_{\mu}^{\circ}$ where ${\mathfrak{f}}(z;\lambda,\mu)=\sum_{k=1}^{2r}c_{l}(\lambda,\mu)z^{-k}$ is a polynomial of $z^{-1}$ with coefficients $c_{k}(\lambda,\mu)\in\mathcal{O}_{S^{2}}$ and $\overline{\mathcal{U}}_{\lambda+\mu}^{\circ}$ is a certain completion of $\mathcal{U}_{\lambda+\mu}^{\circ}$. Thus the irregular field $\mathcal{A}_{\lambda}(z)$ has an exponential type essential singularity at $z=0$, but after dividing the factor $e^{{\mathfrak{f}}(z;\lambda,\mu)}$ it becomes a usual field. We also note that an irregular field with an internal parameter $\lambda$ shifts internal parameters of states by $\lambda$. Since the product of $e^{{\mathfrak{f}}(z;\lambda,\mu)}$ and an element in $\mathcal{U}^{\circ}_{\lambda+\mu}(\!(z)\!)$ has infinite sums in $\mathcal{U}^{\circ}_{\lambda+\mu}$, we consider the completion $\overline{\mathcal{U}}_{\lambda+\mu}^{\circ}$ and regard $\mathcal{A}_{\lambda}(z)$ as an element in $\mathrm{Hom}(\,\mathcal{U}_{\mu}^{\circ},\overline{\mathcal{U}}_{\lambda+\mu}^{\circ}\,)[\![z^{\pm 1}]\!]$. This condition for irregular fields are first considered in [Nag15] as a part of characterization of irregular vertex operators for the Virasoro algebra. In the definition of irregular vertex algebras, we assume that irregular vertex operators given by the state-field correspondence

[TABLE]

become irregular fields with a fixed irregularity ${\mathfrak{f}}(z;\lambda,\mu)$.

We also require that these irregular fields satisfy the irregular locality axiom

[TABLE]

for $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$, $\mathcal{B}_{\mu}\in\mathcal{U}_{\mu}^{\circ}$ and sufficiently large $N$. Here, $e^{-{\mathfrak{f}}(z-w;\lambda,\mu)}_{|z|>|w|}$ and $e^{-{\mathfrak{f}}(z-w;\lambda,\mu)}_{|w|>|z|}$ denote the expansions of $e^{{\mathfrak{f}}(z-w;\lambda,\mu)}$ to their respective domains (see Section 3.4 for more detail). As a consequence, we can show the following:

Theorem 1.1** (Theorem 4.10. OPE for irregular vertex operators).**

[TABLE]

where $\mathcal{C}_{\lambda+\mu,n}\in\mathcal{U}_{\lambda+\mu}^{\circ}$ are some states with an internal parameter $\lambda+\mu$.

We need to note that the definition of normally ordered product for irregular fields is a littele complicated (see Definition 4.7).

Another new feature of irregular vertex operators is that the state-field correspondence is a $\mathcal{D}$-module homomorphism, which implies that it satisfies

[TABLE]

for a vector field $\partial_{\lambda}$ along the direction of the parameter $\lambda$ (See Remark 3.13 for more precise). This yields additional integrable differential equations on the space of internal parameters $S$ for correlation functions.

The irregular vertex operator given in the present paper is a prototype of a more general object, the irregular type intertwining operator among general conformal coherent state modules. (See [FHL93],[Hua05, Hua03] for regular type intertwining operators.) Actually, Nagoya [Nag15] considered the irregular type intertwining operator among two coherent state modules and one highest weight module. In the future work, we will give the definition of the irregular type intertwining operator.

1.3. Examples of irregular vertex algebras

We will give two classes of elementary examples of irregular vertex algebras in Section 5 and Section 6. We see basic ideas for the easiest case.

In Section 5, we define irregular version of Heisenberg vertex algebra ${\mathcal{F}}^{(r)}$. The construction is based on the ideas of [NS10]. In the following, we consider the case $r=1$. Recall the Heisenberg Lie algebra

[TABLE]

with the relations $[a_{n},a_{m}]=n\,\delta_{m+n,0}\mathbf{1}$ and $[a_{n},\mathbf{1}]=0$. Let $|0\rangle$ be the vacuum characterized by $a_{n}|0\rangle=0$ for $n\geq 0$ and consider the Fock space ${\mathcal{F}}$ generated by $|0\rangle$ over $\mathrm{Heis}$. Then, the Fock space ${\mathcal{F}}=\bigoplus\mathbb{C}a_{-n_{1}}\cdots a_{-n_{k}}|0\rangle$ has the vertex algebra structure by the state-field correspondence

[TABLE]

where $a(z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}$ is the bosonic current and $\partial_{z}^{(n)}=(n!)^{-1}\partial_{z}^{n}$. Bosonic currents $a(z)$ and $a(w)$ have the OPE

[TABLE]

Now we consider the coherent state $|\lambda\rangle$ characterized by $a_{1}|\lambda\rangle=\lambda|\lambda\rangle$ and $a_{n}|\lambda\rangle=0$ for $n=0$ and $n\geq 2$. We can realize this coherent state as $|\lambda\rangle=e^{\lambda a_{-1}}|0\rangle$ in a certain completion of ${{\mathcal{F}}}\otimes\mathbb{C}[\lambda]$ with $\deg\lambda=-1$. We set $S_{\lambda}={\mathrm{Spec}\ \!}\mathbb{C}[\lambda]$ and assume it the space of internal parameters. Consider a family of Fock spaces on $S_{\lambda}$ generated by $|\lambda\rangle$ over $\mathrm{Heis}$ and denote it by ${\mathcal{F}}_{\lambda}^{(1)}$. Then ${\mathcal{F}}_{\lambda}^{(1)}=\bigoplus\mathbb{C}[\lambda]a_{-n_{1}}\cdots a_{-n_{k}}|\lambda\rangle$. The space ${\mathcal{F}}_{\lambda}^{(1)}$ is called the envelope of the Heisenberg vertex algebra ${\mathcal{F}}$. Let us consider the vertex operators corresponding to states in ${\mathcal{F}}_{\lambda}^{(1)}$. For the coherent state $|\lambda\rangle$, since $|\lambda\rangle=\sum_{k=0}^{\infty}\frac{a_{-1}^{k}|0\rangle}{k!}$, it is natural to assume

[TABLE]

where {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}e^{\lambda a(z)}{{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}=e^{\lambda a_{+}(z)}e^{\lambda a_{-}(z)}. We can easily check that $Y(|\lambda\rangle,z)|0\rangle|_{z=0}=|\lambda\rangle$.

The vertex operator $Y(|\lambda\rangle,z)$ is not a field in the usual sense but an irregular filed with the irregularity ${\mathfrak{f}}(z;\lambda,\mu)=\lambda\mu/z^{2}$ since the direct computation by using the Baker-Campbell-Hausdorff formula gives

[TABLE]

where $o(z)$ is positive powers of $z$. So we call $Y(|\lambda\rangle,z)$ the irregular vertex operator. For a general state $a_{-n_{1}}\cdots a_{-n_{k}}|\lambda\rangle$, we can define irregular vertex operators as

[TABLE]

The property that the vertex operation $Y$ is a $\mathcal{D}$-module homomorphism implies

[TABLE]

Actually, Y(|\lambda\rangle,z)={{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}e^{\lambda a(z)}{{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} is the solution of the differential equation [\partial_{\lambda},Y(|\lambda\rangle,z)]={{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}a(z)Y(|\lambda\rangle,z){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} under the initial condition $Y(|0\rangle,z)=\mathrm{id}$.

Thus, we can define irregular vertex operators $Y\colon{{\mathcal{F}}}^{(1)}_{\lambda}\to\mathrm{Hom}({{\mathcal{F}}}^{(1)}_{\mu},\overline{{{\mathcal{F}}}}_{\lambda+\mu}^{(1)})[\![z^{\pm 1}]\!]$. Finally, we see an example of the OPE for irregular vertex operators. For $Y(|\lambda\rangle,z)$ and $Y(a_{-1}|\mu\rangle,z)$, we have

[TABLE]

It follows from the computation

[TABLE]

In Section 6, we will define the irregular version $\mathrm{Vir}_{c}^{(r)}$ of the Virasoro vertex algebra $\mathrm{Vir}_{c}$ for $r\in\mathbb{Z}_{>0}$. The space of internal parameter of $\mathrm{Vir}_{c}^{(r)}$ will be the same as that of ${{\mathcal{F}}}^{(r)}$. The construction will be given via the free field realization of Virasoro vertex algebra to the Heisenberg vertex algebra. The main difference is that $\mathrm{Vir}_{c}^{(r)}$ have singularity in the space of internal parameters. In the case $r=1$ the singular locus will be $\lambda=0$. This kind of singularity also appears in the definition of irregular vertex operators for Virasoro Verma modules in [Nag15].

1.4. Notations

Throughout this paper, the term “grading” refers to the $\mathbb{Z}$-grading. For a graded module $M=\bigoplus_{k\in\mathbb{Z}}M_{k}$ over a graded $\mathbb{C}$-algebra $R=\bigoplus_{k\in\mathbb{Z}}R_{k}$, the symbol $M[\![z^{\pm 1}]\!]=\bigoplus_{\ell}M[\![z^{\pm 1}]\!]_{\ell}$ denote the graded $R$-module whose degree $\ell$ part $M[\![z^{\pm 1}]\!]_{\ell}$ is

[TABLE]

In other words, we set $\deg z=-1$ and only consider finite sums of homogeneous power series in this paper. We define modules $M[\![z]\!]$ (resp. $M(\!(z)\!)$) of positive formal power series (resp. formal Laurent series) with coefficients in $M$ in a similar way.

For two graded $R$-modules $M,N$ and an integer $n$, $\mathrm{Hom}_{R}(M,N)_{n}$ denotes the $\mathbb{C}$-vector space of $R$-linear morphisms of degree $n$. We then put

[TABLE]

and regard it as a graded $R$-module. We also set $\mathrm{End}_{R}(M)\coloneqq\mathrm{Hom}_{R}(M,M)$.

The derivation with respect to a (local, formal,…) coordinate $x$ is denoted by $\partial_{x}$ or $\frac{\partial}{\partial x}$. We also set $\partial_{x}^{(n)}\coloneqq(n!)^{-1}\partial_{x}^{n}$ for $n\in\mathbb{Z}_{\geq 0}$.

For an affine scheme $X={\mathrm{Spec}\ \!}A$, we identify the structure sheaf $\mathcal{O}_{X}$ with the ring $A$ of global sections as an abuse of the notation. Sheaves of modules over $\mathcal{O}_{X}$ are also identified with the $A$-modules of their global sections. In this paper, we only consider the case $X={\mathrm{Spec}\ \!}\mathbb{C}[x_{1},\dots,x_{n}]$ for some $n$. Then we denote $\mathbb{C}[x_{1},\dots,x_{n}]$ by $\mathcal{O}_{X}$. We also set $\Theta_{X}\coloneqq\bigoplus_{j=1}^{n}\mathcal{O}_{X}\partial_{x_{j}}$ and $\mathcal{D}_{X}=\mathbb{C}\langle x_{i},\partial_{x_{j}}\mid i,j=1,\dots,n\rangle$ with the relation $[\partial_{x_{i}},x_{j}]=\delta_{i,j}$.

Acknowledgement

The first author would like to thank to Hajime Nagoya for teaching basic ideas of his various works on irregular conformal blocks. The second author would like to thank Takuro Mochizuki and Jeng-Daw Yu for their encouragement. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. The first author is supported by JSPS KAKENHI Grant Number 16K17588 and 16H06337. The second author is supported by JSPS KAKENHI Grant Number JP18H05829.

2. Coherent state modules

In this section, we introduce the notion of coherent state modules.

2.1. Space of internal parameters

Let $\mathbb{C}[\lambda_{j}]_{j\in J}$ be the graded ring of polynomials with variables $\lambda_{j}$ indexed by a finite set $J$. We assume that the degree $d_{j}\coloneqq\deg\lambda_{j}$ are all negative. We call the spectrum $S\coloneqq{\mathrm{Spec}\ \!}\mathbb{C}[\lambda_{j}]_{j\in J}$ a space of internal parameters. We set $\mathcal{O}_{S}\coloneqq\mathbb{C}[\lambda_{j}]_{j\in J}$ to simplify the notation (See Section 1.4). We consider $S$ as an additive algebraic group in a natural way. The addition morphism is denoted by

[TABLE]

The grading defines a $\mathbb{C}^{*}$-action on $S$ naturally. The addition $\sigma$ is $\mathbb{C}^{*}$-equivariant.

Let $\mathcal{D}_{S}=\mathbb{C}\langle\lambda_{j},\partial_{\lambda_{j}}\rangle_{j\in J}$ denote the ring of differential operators on $\mathcal{O}_{S}$. This is a graded algebra such that $\deg\partial_{\lambda_{j}}=-d_{j}$. We also set $\Theta_{S}\coloneqq\bigoplus_{j\in J}\mathcal{O}_{S}\partial_{\lambda_{j}}\subset\mathcal{D}_{S}$, which is naturally equipped with the structure of a graded Lie algebra. Let $\mathrm{Der}(\mathbb{C}[\![t]\!])$ denote the Lie algebra $\mathbb{C}[\![t]\!]\partial_{t}$ with the usual Lie bracket, i.e.

[TABLE]

The grading is given by $\deg t=-1$, $\deg\partial_{t}=1$. Let $\mathrm{Der}_{0}(\mathbb{C}[\![t]\!])$ denote the Lie subalgebra $t\mathbb{C}[\![t]\!]\partial_{t}$ of $\mathrm{Der}(\mathbb{C}[\![t]\!])$.

Definition 2.1**.**

A $\mathrm{Der}_{0}(\mathbb{C}[\![t]\!])$-structure on $S$ is a grade preserving Lie algebra homomorphism

[TABLE]

such that $[D_{0},f]=\ell f,\text{ and}$

[TABLE]

for $f\in\mathcal{O}_{S,-\ell}$, $k\in\mathbb{Z}_{\geq}0$, where $\rho_{S^{2}}\colon\mathrm{Der}_{0}(\mathbb{C}[\![t]\!])\to\Theta_{S^{2}}$ denotes the diagonal action induced from $\rho_{S}$.

Example 2.2** (see [Ike18, Lemma 4.5]).**

Fix $r\in\mathbb{Z}_{>0}$. Let $S\coloneqq{\mathrm{Spec}\ \!}\mathbb{C}[\lambda_{j}]_{j=1}^{r}$ be a space of internal parameter with $\deg\lambda_{j}=-j$. Set

[TABLE]

and $D_{k}=0$ for $k\geq r$. The morphism $\rho_{S}\colon{\mathrm{Der}}_{0}(\mathbb{C}[\![t]\!])\longrightarrow\Theta_{S}$, $t^{k+1}\partial_{t}\mapsto D_{k}$ defines the $\mathrm{Der}_{0}(\mathbb{C}[\![t]\!])$-structure on $S$.

2.2. Coherent state modules

We firstly recall the definition of vertex algebras. In this paper, we only consider $\mathbb{Z}$-graded vertex algebras:

Definition 2.3**.**

A (graded) vertex algebra is a tuple $V=(V,|0\rangle,T,Y)$ of a graded vector space $V=\bigoplus_{n\in\mathbb{Z}}V_{n}$, a non-zero vector $|0\rangle\in V_{0}$, a degree one endomorphism $T$ on $V$, and a grade-preserving homomorphism

[TABLE]

with the following properties:

• •

(vacuum axiom) $Y(|0\rangle,z)=\mathrm{id}_{V}$ and $Y(A,z)|0\rangle\in A+zV[\![z]\!]$ for any $A\in V$.

• •

(translation axiom) $[T,Y(A,z)]=\partial_{z}Y(A,z)$ for any $A\in V$ and $T|0\rangle=0$.

• •

(field axiom) $Y(A,z)B\in V(\!(z)\!)$ for any $A,B\in V$.

• •

(locality axiom) For any $A,B\in V$, there exists a positive integer $N$ such that

[TABLE]

in $\mathrm{End}(V)[\![z^{\pm 1},w^{\pm 1}]\!]$. Here, $w$ denotes a copy of $z$ with $\deg w=-1$.

We then consider the following family of modules over $V$.

Definition 2.4** (Coherent state modules).**

Let $V$ be a vertex algebra. Let $S$ be a space of internal parameters (Section 2.1). A coherent state $V$-module on $S$ is a triple $(\mathcal{M},Y_{\mathcal{M}},|{\rm coh}\rangle)$ of a $\mathbb{C}$-graded $\mathcal{D}_{S}$-module $\mathcal{M}$, a degree zero $\mathbb{C}$-linear map

[TABLE]

and a homogeneous global section $|{\rm coh}\rangle$ of $\mathcal{M}$ (called a coherent state of $\mathcal{M}$) such that the following properties hold:

1. (1)

$Y_{\mathcal{M}}(|0\rangle,z)=\mathrm{id}_{\mathcal{M}}$. 2. (2)

$Y_{\mathcal{M}}(A,z)\mathcal{B}$ is in $\mathcal{M}(\!(z)\!)$ for any $A\in V$ and $\mathcal{B}\in\mathcal{M}$. 3. (3)

For any $A,B\in V$ and $\mathcal{C}\in\mathcal{M}$, the three elements

[TABLE]

are the expansions of the same element in $\mathcal{M}[\![z,w]\!][z^{-1},w^{-1},(z-w)^{-1}]$ to their respective domains. 4. (4)

$\mathcal{M}$ is generated by $|{\rm coh}\rangle$ over $V$ and $\mathcal{D}_{S}$, i.e. if a $\mathcal{D}_{S}$-submodule $\mathcal{M}^{\prime}\subset\mathcal{M}$ contains $|{\rm coh}\rangle$ and is closed under operations $A_{(n)}^{\mathcal{M}}$ for any $A\in V$ and $n\in\mathbb{Z}$, then we have $\mathcal{M}^{\prime}=\mathcal{M}$. 5. (5)

Let $\mathcal{M}_{\mathcal{O}}$ denote the smallest graded $\mathcal{O}_{S}$-submodule of $\mathcal{M}$ such that $|{\rm coh}\rangle$ is contained in $\mathcal{M}_{\mathcal{O}}$ and closed under operations $A_{(n)}^{\mathcal{M}}$ for any $A\in V$ and $n\in\mathbb{Z}$. Then, the support of the quotient module $\mathcal{M}/\mathcal{M}_{\mathcal{O}}$ is either an empty set or co-dimension one subvariety in $S$.

Remark 2.5**.**

Let $\widetilde{U}(V)$ be the complete topological associative algebra associated to $V$ (see [FBZ04, Definition 4.3.1]). By the same argument as [FBZ04, Theorem 5.1.6], the condition $(1)$, $(2)$, and $(3)$ is equivalent to the condition that $\mathcal{M}$ is a smooth $\widetilde{U}(V)$-module. By definition, the action of $\widetilde{U}(V)$ is compatible with the action of $\mathcal{D}_{S}$, and hence we may consider $\mathcal{M}$ as a module over $\widetilde{U}(V)\otimes_{\mathbb{C}}\mathcal{D}_{S}$. The condition $(4)$ is then equivalent to the condition $\mathcal{M}=(\widetilde{U}(V)\otimes_{\mathbb{C}}\mathcal{D}_{S})|{\rm coh}\rangle$. The module $\mathcal{M}_{\mathcal{O}}$ in the condition $(5)$ is expressed as $\mathcal{M}_{\mathcal{O}}=(\widetilde{U}(V)\otimes_{\mathbb{C}}\mathcal{O}_{S})|{\rm coh}\rangle$.

Definition 2.6** (Singular locus of coherent state modules).**

Let $\mathcal{M}$ be a coherent state $V$-module. Let $\mathcal{M}_{\mathcal{O}}$ be the $\mathcal{O}_{S}$-submodule defined in (5) of Definition 2.4. The singular locus $H$ of the coherent state module $\mathcal{M}$ is the support of $\mathcal{M}/\mathcal{M}_{\mathcal{O}}$:

[TABLE]

which is by assumption a $\mathbb{C}^{*}$-invariant Zariski closed subvariety in $S$. The coherent state module $\mathcal{M}$ is called non-singular along $S$ if $H$ is empty.

2.3. Conformal coherent state modules

Recall that the Virasoro algebra is a Lie algebra $\mathrm{Vir}=(\bigoplus_{n\in\mathbb{Z}}\mathbb{C}L_{n})\oplus\mathbb{C}C$ whose Lie bracket is defined as follows:

[TABLE]

Fix a complex number $c\in\mathbb{C}$. Let $\mathbb{C}_{c}$ be an one dimensional representation of the Lie subalgebra $\mathrm{Vir}_{\geqslant-1}\coloneqq\bigoplus_{n\geq-1}\mathbb{C}L_{n}\oplus\mathbb{C}C$ defined by $L_{n}1=0$ for $n\geq-1$ and $C\cdot 1=c1$. Take a induced representation

[TABLE]

where $U(\mathrm{Vir})$ (resp. $U(\mathrm{Vir}_{\geqslant-1})$) denotes the universal enveloping algebra of $\mathrm{Vir}$ (resp. $\mathrm{Vir}_{\geqslant-1}$), and $v_{c}$ denotes the image of $1\otimes 1$. Define the $\mathbb{Z}$-gradation on $\mathrm{Vir}_{c}$ by the formulas $\deg L_{-n}=n$ and $\deg v_{c}=0$. The power series

[TABLE]

defines a field on $\mathrm{Vir}_{c}$.

A Virasoro vertex algebra with central charge $c$ is a vertex algebra

[TABLE]

where $Y(\cdot,z):\mathrm{Vir}_{c}\to\mathrm{End}(\mathrm{Vir}_{c})[\![z^{\pm 1}]\!]$ is defined as follows:

[TABLE]

where {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\cdot{{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} denotes the normally ordered product.

A conformal structure of central charge $c\in\mathbb{C}$ on a vertex algebra $V$ is a degree two non-zero vector $\omega\in V_{2}$ called a conformal vector such that the Fourier coefficients $L_{n}^{V}$ of the corresponding vertex operator

[TABLE]

satisfy $L_{-1}^{V}=T$ and $L_{2}^{V}\omega=\frac{c}{2}|0\rangle$. A vertex algebra with a conformal structure is called a vertex operator algebra, or a conformal vertex algebra. The following properties of vertex operator algebra is well known (see [FBZ04, Lemma 3.4.5]):

• •

$\{L_{n}^{V}\}_{n}$ satisfies the relation (2.2) replacing $C$ by $c\cdot\mathrm{id}_{V}$.

• •

$L_{0}^{V}=n\cdot\mathrm{id}_{V_{n}}$ on $V_{n}$ for every $n\in\mathbb{Z}$.

• •

There exists a unique morphism $\mathrm{Vir}_{c}\to V$ of vertex algebras such that $v_{c}\mapsto|0\rangle$, $L_{-2}v_{c}\mapsto\omega$.

Definition 2.7**.**

Let $\mathcal{M}=(\mathcal{M},Y_{\mathcal{M}},|{\rm coh}\rangle)$ be a coherent state $V$-module over $S$. Assume that $V$ has a conformal vector $\omega$, and $S$ has a $\mathrm{Der}_{0}(\mathbb{C}[\![t]\!])$-structure

[TABLE]

(see Definition 2.1). Let $L^{\mathcal{M}}_{n}$ be the Fourier coefficients of the action

[TABLE]

Then, $\mathcal{M}$ is called conformal if there exist differential operators $\mathcal{L}_{k}=h_{k}+D_{k}\in\mathcal{O}_{S}\oplus\Theta_{S}$ such that the following properties hold:

1. (1)

$L_{k}^{\mathcal{M}}|{\rm coh}\rangle=\mathcal{L}_{k}|{\rm coh}\rangle$ for $k\in\mathbb{Z}_{\geq 0}$. 2. (2)

The map

[TABLE]

is a Lie algebra homomorphism. 3. (3)

$L_{0}^{\mathcal{M}}-D_{0}$ is the grading operator on $\mathcal{M}$ (i.e. $(L_{0}^{\mathcal{M}}-D_{0})m=\deg(m)m$ for any homogeneous section $m\in\mathcal{M}$) and $L_{k}^{\mathcal{M}}-\mathcal{L}_{k}$ is locally finite for $k>0$ (i.e. for any $m\in\mathcal{M}$, there exists $N>0$ such that $(L_{k}^{\mathcal{M}}-\mathcal{L}_{k})^{N}m=0$).

2.4. An example of conformal coherent state module

We shall give an example of conformal coherent state module, following the idea of Gaiotto-Teschner [GT12, Section 2.1.2]. Take the space of internal parameter $S$ as in Example 2.2 for fixed $r>0$. Fix complex numbers $\rho$ and $\lambda_{0}$. Set $h_{k}(\lambda)\coloneqq\frac{1}{2}\sum_{j=0}^{k}\lambda_{j}\lambda_{k-j}-\rho(k+1)\lambda_{k}$ for $k=0,\dots,2r$, where we put $\lambda_{\ell}=0$ for $\ell>r$. Put $h\coloneqq h_{0}=2^{-1}\lambda_{0}(\lambda_{0}-2\rho)$. Put $\mathcal{L}_{k}\coloneqq h_{k}+D_{k}$ for $k=0,\dots,2r$ and $\mathcal{L}_{m}\coloneqq 0$ for $m>2r$.

Lemma 2.8**.**

For $k,\ell\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Let $\mathrm{Vir}_{\geqslant 0}$ denote the Lie subalgebra $\bigoplus_{n\geq 0}\mathbb{C}L_{n}\oplus\mathbb{C}C$ of $\mathrm{Vir}$. By Lemma 2.8,

[TABLE]

defines left $U(\mathrm{Vir}_{\geqslant 0})$-module structure on $\mathcal{D}_{S}$, where $U(\mathrm{Vir}_{\geqslant 0})$ denotes the universal enveloping algebra of $\mathrm{Vir}_{\geqslant 0}$.

Definition 2.9**.**

We set

[TABLE]

Set $|{\rm coh}\rangle\coloneqq 1\otimes 1\in\mathcal{M}_{c,h}^{(r)}$. We define the $\mathbb{C}$-grading of $\mathcal{M}_{c,h}^{(r)}$ so that $\deg|{\rm coh}\rangle=h$.

Proposition 2.10**.**

The pair $(\mathcal{M}_{c,h}^{(r)},|{\rm coh}\rangle)$ naturally equips with the structure of conformal coherent state $\mathrm{Vir}_{c}$-module.

Proof.

By construction, $\mathcal{M}_{c,h}^{(r)}$ is a $\mathrm{Vir}$-module with the property that for any section $s\in\mathcal{M}_{c,h}^{(r)}$ there exists $N>0$ such that $[L_{n},s]=0$ $(n>N)$. It follows from this fact that $\mathcal{M}_{c,h}^{(r)}$ is a smooth $\widetilde{U}(\mathrm{Vir}_{c})$-module, which implies the conditions (1), (2), (3) in Definition 2.4 (see Remark 2.5 and [FBZ04, Section 5.1.8]).

Again by the construction, we have the expression

[TABLE]

This expression implies $(4)$. Since $\partial_{\lambda_{r}}$ is not in $(\mathcal{M}_{c,h}^{(r)})_{\mathcal{O}}$, the quotient $\mathcal{M}_{c,h}^{(r)}/(\mathcal{M}_{c,h}^{(r)})_{\mathcal{O}}$ is non-empty. The (proof of) Lemma 4.3 in [Ike18] implies that the singularity of $\mathcal{M}_{c,h}^{(r)}$ is $H=\{\lambda_{r}=0\}$. Hence we obtain $(5)$. The conformality is trivial by construction. ∎

Remark 2.11**.**

At each point $\lambda^{o}=(\lambda_{1}^{o},\dots,\lambda_{r}^{o})$ with $\lambda_{r}^{o}\neq 0$, the fiber $\mathcal{M}_{c,h}^{(r)}|_{\lambda^{o}}$ of $\mathcal{M}_{c,h}^{(r)}$ have PBW basis

[TABLE]

where $\mathcal{P}^{r}$ denote the set of non-decreasing finite sequence

[TABLE]

of integers and $L_{\bm{n}}=L_{n_{1}}\cdots L_{n_{\ell}}$ (we also set $L_{\emptyset}=1$). Hence $\mathcal{M}_{c,h}^{(r)}|_{\lambda^{0}}$ is a universal Whittaker module of type $h_{r}(\lambda^{o}),\dots,h_{2r}(\lambda^{o})$ in the sense of [FJK12, Definition 2.2]. In particular, $\mathcal{M}_{c,h}^{(r)}|_{\lambda^{o}}$ is a simple module ([FJK12, Corollary 2.2], see also [LGZ11, Theorem 7] and [NS10, Remark 2.2]).

3. Irregular vertex operator algebras

In this section, we introduce the notion of irregular vertex algebras.

3.1. Notations on pullbacks

We shall fix the notation for pull backs of $\mathcal{D}$-modules over the space of internal parameters. Let $S={\mathrm{Spec}\ \!}(\mathbb{C}[\lambda_{j}]_{j\in J})$ be a space of internal parameters. To specify coordinate functions on $S$, we use the notation $S_{\lambda}={\mathrm{Spec}\ \!}(\mathbb{C}[\lambda_{j}]_{j\in J})$, $\mathcal{O}_{S_{\lambda}}\coloneqq\mathbb{C}[\lambda_{j}]_{j\in J}$, and $\mathcal{D}_{S_{\lambda}}=\mathbb{C}\langle\lambda_{j},\partial_{\lambda_{j}}\rangle_{j\in J}$ instead of $S$, $\mathcal{O}_{S}$, and $\mathcal{D}_{S}$. In the rest of this paper, we often use the direct product of two or three copies of $S$. To distinguish each component together with specified coordinate functions in the direct product, we write

[TABLE]

instead of $S^{3}={\mathrm{Spec}\ \!}(\mathbb{C}[\lambda_{j}]_{j\in J}^{\otimes 3})$. We also use the notation $S_{\lambda,\mu}^{2}\coloneqq S_{\lambda}\times S_{\mu}$ and $S_{\lambda,\mu,\nu}^{3}\coloneqq S_{\lambda}\times S_{\mu}\times S_{\nu}$.

Let $\sigma_{\lambda+\mu}\colon S^{2}_{\lambda,\mu}\to S_{\xi}$ be the addition. For a $\mathcal{D}_{S_{\xi}}$-module $\mathcal{M}$, we denote the pull back of $\mathcal{M}$ w.r.t. $\sigma_{\lambda+\mu}$ by $\mathcal{M}_{\lambda+\mu}\coloneqq\sigma_{\lambda+\mu}^{*}\mathcal{M}$ . Here we can explicitly write as

[TABLE]

where the tensor product is given through the morphism

[TABLE]

The action of $\mathcal{D}_{S^{2}_{\lambda,\mu}}=\mathbb{C}\langle\lambda_{j},\partial_{\lambda_{j}},\mu_{j},\partial_{\mu_{j}}\rangle_{j\in J}$ on $\mathcal{M}_{\lambda+\mu}$ is given by

[TABLE]

Since $\mathcal{D}_{S_{\lambda}}$ and $\mathcal{D}_{S_{\mu}}$ are subalgebras of $\mathcal{D}_{S^{2}_{\lambda,\mu}}$, the module $\mathcal{M}_{\lambda+\mu}$ naturally has both $\mathcal{D}_{S_{\lambda}}$-module and $\mathcal{D}_{S_{\mu}}$-module structures. Similarly, we can define $\mathcal{D}_{S^{3}_{\lambda,\mu,\nu}}$-module $\mathcal{M}_{\lambda+\mu+\nu}$ by using $\sigma_{\lambda+\mu+\nu}\colon S^{3}_{\lambda,\mu,\nu}\to S_{\xi}$.

3.2. Filtered small lattices and exponential twists

Let $S$ be a space of internal parameters. Let $V$ be a vertex algebra. Let $\mathcal{M}$ be a coherent state $V$-module over $S$. As discussed in Section 2.2, $\mathcal{M}$ has the singular divisor $H=\mathrm{Supp}(\mathcal{M}/\mathcal{M}_{\mathcal{O}})$.

We shall consider the localization of $\mathcal{M}$ along $H$. In other words, we consider $\mathcal{M}(*H)\coloneqq\mathcal{M}\otimes\mathcal{O}_{S}(*H)$, where $\mathcal{O}_{S}(*H)$ denotes the ring of rational functions on $S$ with poles in $H$. By definition, we have $\mathcal{M}(*H)=\mathcal{M}_{\mathcal{O}}\otimes\mathcal{O}_{S}(*H)$.

Definition 3.1**.**

A filtered small lattice of $\mathcal{M}(*H)$ is a pair $(\mathcal{M}^{\circ},F^{\bullet})$ of a $\mathcal{D}_{S}$-submodule $\mathcal{M}^{\circ}\subset\mathcal{M}(*H)$ and a decreasing filtration $F^{\bullet}(\mathcal{M}^{\circ})$ of $\mathcal{O}_{S}$-submodules indexed by $\mathbb{Z}$ with the following properties:

• (L)

We have $\mathcal{M}\subset\mathcal{M}^{\circ}\subset\mathcal{M}(*H)$ and hence $\mathcal{M}^{\circ}(*H)=\mathcal{M}(*H)$.

• (F1)

There exists an integer $N$ such that we have $F^{N}(\mathcal{M}^{\circ})=\mathcal{M}^{\circ}$.

• (F2)

We have $\bigcap_{m\in\mathbb{Z}}F^{m}(\mathcal{M}^{\circ})=0$.

• (F3)

For $k,\ell\in\mathbb{Z}$ and $m\in\mathbb{C}$, we have $\mathcal{D}_{S,k}\cdot F^{\ell}(\mathcal{M}^{\circ}_{m})\subset F^{\ell-k}(\mathcal{M}^{\circ}_{m+k})$.

• (F4)

For any $A\in V$, $n\in\mathbb{Z}$, $A_{(n)}^{\mathcal{M}}F^{k}(\mathcal{M}^{\circ})\subset F^{k}(\mathcal{M}^{\circ})$.

Let $\overline{\mathcal{M}}^{\circ}$ denote the completion of $U^{\circ}$ with respect to $F^{\bullet}(U^{\circ})$ in the category of $\mathbb{Z}$-graded $\mathcal{O}_{S}$-modules. We have a natural isomorphism $\overline{\mathcal{M}}^{\circ}_{k}\simeq\prod_{\ell\in\mathbb{Z}}\mathrm{Gr}_{F}^{\ell}(\mathcal{M}^{\circ}_{k})$, where $\mathrm{Gr}_{F}^{\ell}(\mathcal{M}^{\circ}_{k})\coloneqq F^{\ell}(\mathcal{M}_{k}^{\circ})/F^{\ell+1}(\mathcal{M}_{k}^{\circ})$. By condition (F3), $\overline{\mathcal{M}}^{\circ}$ naturally equips with the structure of $\mathcal{D}_{S}$-module. By condition (F4), $A_{(n)}^{\mathcal{M}}$ acts on $\mathcal{M}^{\circ}$ and $\overline{\mathcal{M}}^{\circ}$. However, we do not assume that $\mathcal{M}^{\circ}$ (or, $\overline{\mathcal{M}}^{\circ}$) is a coherent state $V$-module.

A coherent state module $\mathcal{M}$ is called small if it admits a filtered small lattice. In this section, we assume that $\mathcal{M}$ is small and fix a filtered small lattice $(\mathcal{M}^{\circ},F^{\bullet})$. We note that $\overline{\mathcal{M}}_{\lambda+\mu}^{\circ}$ is the completion of $\mathcal{M}^{\circ}_{\lambda+\mu}$ with respect to the filtration $F^{\bullet}(\mathcal{M}^{\circ}_{\lambda+\mu})\coloneqq\sigma_{\lambda+\mu}^{*}F^{\bullet}(\mathcal{M}^{\circ})$.

We shall define the product

[TABLE]

as follows: Take a power series

[TABLE]

with $f_{n}(\lambda,\mu)\in\mathcal{O}_{S^{2}_{\lambda,\mu},-n}$, and the homogeneous series

[TABLE]

with $p\in\mathbb{Z},d=\deg u(z;\lambda,\mu)\in\mathbb{C}$, $u_{m}(\lambda,\mu)\in\mathcal{M}^{\circ}_{\lambda+\mu,m+d}$.

Take the biggest integer $N$ such that $F^{N}\mathcal{M}^{\circ}=\mathcal{M}^{\circ}$ (such $N$ exists by (F1) and (F2) in Definition 3.1). Then, by the condition (F3), $f_{n}(\lambda;\mu)u_{m}(\lambda,\mu)$ is in $F^{N+n}(\mathcal{M}^{\circ}_{\lambda+\mu,m-n})$. It follows that the infinite sum

[TABLE]

converges in $\overline{\mathcal{M}}_{\lambda+\mu}^{\circ}$. Hence, we can define (3.1) by

[TABLE]

Here, we note that $\mathcal{M}^{\circ}_{\lambda+\mu}(\!(z)\!)=\bigoplus_{d\in\mathbb{C}}\mathcal{M}^{\circ}_{\lambda+\mu}(\!(z)\!)_{d}$ in our notation (see Section 1.4).

Let $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]_{0}$ denote the ring of degree zero sections of $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]$. Let $\mathcal{O}_{S^{2}}[\![z^{-1}]\!]^{\times}_{0}$ denote the abelian group of degree zero invertible elements in $\mathcal{O}_{S^{2}}[\![z^{-1}]\!]$. Then the map

[TABLE]

defines a morphism of abelian groups.

Definition 3.2**.**

For ${\mathfrak{f}}(z;\lambda,\mu)\in z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]_{0}$, let $e^{{\mathfrak{f}}(z;\lambda,\mu)}\mathcal{M}^{\circ}_{\lambda+\mu}(\!(z)\!)$ denote the image of

[TABLE]

by the morphism (3.1) (see also (3.2)).

3.3. Irregular fields and their compositions

Definition 3.3** (Irregular fields).**

Let $\mathcal{M}_{\mu}$ be a coherent state module on $S_{\mu}$ with a filtered small lattice $(\mathcal{M}_{\mu}^{\circ},F^{\bullet})$. An irregular field with an irregularity ${\mathfrak{f}}(z;\lambda,\mu)\in z^{-1}\mathcal{O}_{S^{2}_{\lambda,\mu}}[z^{-1}]_{0}$ on $\mathcal{M}_{\mu}$ (with respect to $(\mathcal{M}_{\mu}^{\circ},F^{\bullet})$) is an element $\mathcal{A}_{\lambda}(z)$ in

[TABLE]

with the following property: for any $\mathcal{B}_{\mu}\in\mathcal{M}_{\mu}^{\circ}$, we have

[TABLE]

Remark 3.4**.**

We also call an element

[TABLE]

irregular field with an irregularity ${\mathfrak{f}}\left(z;\lambda+\mu,\nu\right)$ if we have

[TABLE]

for any $\mathcal{C}_{\nu}\in\mathcal{M}^{\circ}_{\nu}$. We can also generalize the notion of irregular fields in a similar way.

Similarly to Section 3.2, we can naturally define the product

[TABLE]

Then, we define $e^{{\mathfrak{f}}(z;\lambda,\mu+\nu)+{\mathfrak{f}}(w;\mu,\nu)}\mathcal{M}^{\circ}(\!(z)\!)(\!(w)\!)$, $e^{{\mathfrak{f}}(z;\lambda,\nu)+{\mathfrak{f}}(w;\mu,\nu)}\mathcal{M}^{\circ}(\!(z)\!)(\!(w)\!)$, and so on in a way similar to Definition 3.2.

Definition 3.5**.**

For irregular fields $\mathcal{A}_{\lambda}(z)$ and $\mathcal{B}_{\mu}(w)$ with irregularity ${\mathfrak{f}}$, we define the composition $\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(w)$, which is an element in

[TABLE]

as follows: Since $\mathcal{A}_{\lambda}(z)$ and $\mathcal{B}_{\mu}(w)$ are irregular fields, we have expansions

[TABLE]

where $\mathcal{A}^{\prime}_{\lambda}(\eta)_{(m)}$ $(m\in\mathbb{Z})$ and $\mathcal{B}^{\prime}_{\mu}(\nu)_{(n)}$ $(n\in\mathbb{Z})$ are in $\mathrm{Hom}_{\mathcal{O}_{S_{\eta}}}(\mathcal{M}^{\circ}_{\eta},\mathcal{M}^{\circ}_{\lambda+\eta})$ and $\mathrm{Hom}_{\mathcal{O}_{S_{\nu}}}(\mathcal{M}^{\circ}_{\nu},\mathcal{M}^{\circ}_{\mu+\nu})$, respectively. We then define

[TABLE]

We can easily check that the composition $\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(w)$ is independent of the parameters $\nu_{i}$. For each $\mathcal{C}_{\nu}\in\mathcal{M}^{\circ}_{\nu}$, we have

[TABLE]

3.4. Exponentially twisted Lie bracket

Lemma 3.6**.**

Let $e^{-{\mathfrak{f}}(z-w)}_{|z|>|w|}$ denote the expansion of $e^{-{\mathfrak{f}}(z-w)}\in\mathcal{O}_{S^{2}}[\![(z-w)^{-1}]\!]$ in $\mathcal{O}_{S^{2}}[\![z^{-1},w]\!]$. Then, we have $e^{-{\mathfrak{f}}(z-w)}_{|z|>|w|}e^{{\mathfrak{f}}(z)}\in\mathcal{O}_{S^{2}}[z^{-1}][\![w]\!]$.

Proof.

A priori, $e^{-{\mathfrak{f}}(z-w)}_{|z|>|w|}e^{{\mathfrak{f}}(z)}$ is in $\mathcal{O}_{S^{2}}[\![z^{-1},w]\!]$. Consider the Taylor expansion

[TABLE]

where each coefficient $c_{k}(z)$ is the restriction of

[TABLE]

to $w=0$. The left hand side of (3.4) is $1$ when when we restrict it to $w=0$. Hence we have $c_{0}(z)=1$. For general $k\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Hence we obtain that $c_{k}(z)\in\mathcal{O}_{S^{2}}[z^{-1}]$ by the induction on $k$. ∎

Definition 3.7**.**

For an element ${\mathfrak{f}}(z;\lambda,\mu)$ of $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]_{0}$, consider the following properties:

• •

(skew symmetry) ${\mathfrak{f}}(z;\lambda,\mu)={\mathfrak{f}}(-z;\mu,\lambda)$.

• •

(additivity) ${\mathfrak{f}}(z;\lambda+\mu,\nu)={\mathfrak{f}}(z;\lambda,\nu)+{\mathfrak{f}}(z;\mu,\nu)$.

The set of sections of $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]_{0}$ with these properties is denoted by $\mathrm{Irr}(S)$.

Corollary 3.8**.**

For two irregular fields $\mathcal{A}_{\lambda}(z)$ and $\mathcal{B}_{\mu}(w)$ with an irregularity ${\mathfrak{f}}$ in $\mathrm{Irr}(S)$ and an element $\mathcal{C}_{\nu}\in\mathcal{M}_{\nu}$, we have

[TABLE]

Definition 3.9** (Exponentially twisted Lie bracket).**

For two irregular fields $\mathcal{A}_{\lambda}(z)$, and $\mathcal{B}_{\mu}(w)$ with an irregularity ${\mathfrak{f}}\in\mathrm{Irr}(S)$, the exponentially twisted Lie bracket

[TABLE]

is defined as

[TABLE]

The two irregular fields $\mathcal{A}_{\lambda}(z)$ and $\mathcal{B}_{\mu}(w)$ are called mutually ${\mathfrak{f}}$-local if there exists an integer $N$ such that

[TABLE]

Lemma 3.10**.**

For mutually ${\mathfrak{f}}$-local irregular fields $\mathcal{A}_{\lambda}(z)$ and $\mathcal{B}_{\mu}(w)$, and for any $\mathcal{C}_{\nu}\in\mathcal{M}^{\circ}_{\nu}$, the two elements

[TABLE]

are the expansions of the same element in

[TABLE]

to their respective domains.

3.5. Envelopes of vertex algebras

Let $V$ be a vertex algebra and $S$ be a space of internal parameters. Let $(\mathcal{U},Y_{\mathcal{U}},|{\rm coh}\rangle)$ be a $\mathbb{Z}$-graded coherent state $V$-module over $S$ with $\deg|{\rm coh}\rangle=0$. We consider the morphism

[TABLE]

where $A^{\mathcal{U}}_{(-1)}$ is defined in (2.1).

Assume moreover that we have a filtered small lattice $(\mathcal{U}^{\circ},F^{\bullet})$ of $\mathcal{U}$. Consider $\mathcal{U}_{\mathcal{O}}$ as submodules of $\mathcal{U}^{\circ}$. Let $\mathcal{U}_{\mathcal{O}}|^{\circ}_{0}$ denote the fiber of $\mathcal{U}_{\mathcal{O}}$ at the origin as a submodule of $\mathcal{U}^{\circ}$. In other words, we put

[TABLE]

where $\mathfrak{m}_{S,0}$ denote the maximal ideal of ${\mathcal{O}}_{S}$ corresponding to the origin of $S$. We note that $\mathcal{U}_{\mathcal{O}}\cap\mathfrak{m}_{S,0}\mathcal{U}^{\circ}$ is a $V$-submodule of $\mathcal{U}_{\mathcal{O}}$. Hence $\mathcal{U}_{\mathcal{O}}|_{0}^{\circ}$ is equipped with the structure of $V$-module.

Definition 3.11**.**

A coherent state $V$-module $(\mathcal{U},Y_{\mathcal{U}},|{\rm coh}\rangle)$ on $S$ together with the filtered small lattice $(\mathcal{U}^{\circ},F^{\bullet})$ is called an envelope of $V$ if the morphism $\Psi$ is injective, and the composition

[TABLE]

of $\Psi$ and the quotient map is an isomorphism of $V$-modules.

For an envelope $\mathcal{U}$ of $V$, we identify $\mathcal{U}_{\mathcal{O}}|_{0}^{\circ}$ and $V$ via $\overline{\Psi}$.

3.6. Definition of irregular vertex algebras

For an envelope

[TABLE]

of $V$ on $S_{\lambda}$ (resp. $S_{\mu}$), write $\mathcal{U}_{\lambda}\coloneqq\mathcal{U}$ and $|\lambda\rangle\coloneqq|{\rm coh}\rangle$ (resp. $\mathcal{U}_{\mu}\coloneqq\mathcal{U}$ and $|\mu\rangle\coloneqq|{\rm coh}\rangle$). We also use the notation $\Psi_{\lambda}(v)$ instead of $\Psi(v)$ for $v\in V$ and so on. Set $|\lambda+\mu\rangle\coloneqq\sigma_{\lambda+\mu}^{*}|{\rm coh}\rangle\in\mathcal{U}_{\lambda+\mu}$.

We note that an irregular field $\mathcal{A}_{\lambda}(z)$ on $\mathcal{U}$ with irregularity ${\mathfrak{f}}\in\mathrm{Irr}(S)$ defines an element in $\mathrm{Hom}(V,{\mathcal{U}}_{\lambda}^{\circ})[\![z^{\pm 1}]\!]$ in the following way: For a vector $v\in V$ consider $\mathcal{A}_{\lambda}(z)\Psi_{\mu}(v)\in e^{{\mathfrak{f}}(z;\lambda,\mu)}\mathcal{U}_{\lambda+\mu}^{\circ}(\!(z)\!)$, and take the restriction to $\mu=0$. Since ${\mathfrak{f}}(z;\lambda,\mu)$ is a degree zero element in $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]$ and satisfies the skew symmetry, we have ${\mathfrak{f}}(z;\lambda,0)=0$. Hence we have $\mathcal{A}_{\lambda}(z)\Psi_{\mu}(v)|_{\mu=0}\in\mathcal{U}_{\lambda}^{\circ}(\!(z)\!)$, which is denoted by $\mathcal{A}_{\lambda}(z)v$ for short. In particular, we can define $\mathcal{A}_{\lambda}(z)|0\rangle$.

For an endomorphism $T^{\mathcal{U}}\in\mathrm{End}(\mathcal{U}^{\circ})$ and an irregular field $\mathcal{A}_{\lambda}(z)$, we define the Lie bracket $[T^{\mathcal{U}},\mathcal{A}_{\lambda}(z)]$ by

[TABLE]

Consider $\mathrm{Hom}_{\mathcal{D}_{S_{\mu}}}\left(\mathcal{U}^{\circ}_{\mu},\overline{\mathcal{U}}^{\circ}_{\lambda+\mu}\right)[\![z^{\pm 1}]\!]$ as a $\mathcal{D}_{S_{\lambda}}$-module by the $\mathcal{D}_{S_{\lambda}}$-module structure on $\overline{\mathcal{U}}^{\circ}_{\lambda+\mu}$ and $[\partial_{\lambda_{j}},z^{n}]=0$ for $n\in\mathbb{Z}$. We shall define the notion of irregular vertex algebra as follows:

Definition 3.12** (Irregular vertex algebra).**

Let $V$ be a vertex algebra. Let $S$ be a space of internal parameters. An irregular vertex algebra for $V$ on $S$ is a tuple $(\mathcal{U},{Y},{\mathfrak{f}},T^{\mathcal{U}})$ of an envelope $\mathcal{U}=(\mathcal{U},Y_{\mathcal{U}},|{\rm coh}\rangle,(\mathcal{U}^{\circ},F^{\bullet}))$ of $V$ on $S$, a grade preserving $\mathcal{D}_{S_{\lambda}}$-module morphism

[TABLE]

an element ${\mathfrak{f}}\in\mathrm{Irr}(S)$, and an endomorphism $T^{\mathcal{U}}\in\mathrm{End}_{\mathcal{D}_{S}}(\mathcal{U}^{\circ})_{1}$ with the following properties:

• •

(irregular field axiom) For every $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$, the series ${Y}(\mathcal{A}_{\lambda},z)$ is an irregular field with the irregularity ${\mathfrak{f}}(z;\lambda,\mu)$.

• •

(irregular locality axiom) For any $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$, $\mathcal{B}_{\mu}\in\mathcal{U}_{\mu}^{\circ}$, two irregular fields ${Y}(\mathcal{A}_{\lambda},z)$ and ${Y}(\mathcal{B}_{\mu},w)$ are mutually ${\mathfrak{f}}$-local.

• •

(vacuum axiom) For any $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$, we have ${Y}(\mathcal{A}_{\lambda},z)|0\rangle\in\mathcal{U}_{\lambda}^{\circ}[\![z]\!]$, and ${Y}(\mathcal{A}_{\lambda},z)|0\rangle|_{z=0}=\mathcal{A}_{\lambda}$.

• •

(coherent state axiom) We have ${Y}(|\lambda\rangle,z)|\mu\rangle\in e^{{\mathfrak{f}}(z;\lambda,\mu)}\mathcal{U}_{\lambda+\mu}^{\circ}[\![z]\!]$, and

[TABLE]

• •

(translation axiom) We have $[T^{\mathcal{U}},{Y}(\mathcal{A}_{\lambda},z)]=\partial_{z}{Y}(\mathcal{A}_{\lambda},z)$ for any $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$.

• •

(compatibility condition) For any ${A}\in V$, $\mathcal{B}_{\mu}\in\mathcal{U}_{\mu}$, the restriction of ${Y}(\Psi_{\lambda}(A),z)\mathcal{B}_{\mu}$ to $\lambda=0$ is $Y_{\mathcal{U}}(A,z)\mathcal{B}_{\mu}$. We also have

[TABLE]

where $*|_{0}^{\circ}$ denotes the restriction as a section of $\mathcal{U}^{\circ}$.

Remark 3.13**.**

The endomorphism $T^{\mathcal{U}}$ will be denoted by $T$ if it is not confusing. The condition that $Y(\cdot,z)$ is a morphism of $\mathcal{D}_{S_{\lambda}}$-modules and takes values in $\mathrm{Hom}_{\mathcal{D}_{S_{\mu}}}\left(\mathcal{U}^{\circ}_{\mu},\overline{\mathcal{U}}^{\circ}_{\lambda+\mu}\right)[\![z^{\pm 1}]\!]$ implies that for $\mathcal{A}_{\lambda}\in\mathcal{U}^{\circ}$,

[TABLE]

for any $j\in J$. Thus, Fourier coefficients $\mathcal{A}_{\lambda,(n)}$ of the irregular vertex operator $Y(\mathcal{A}_{\lambda},z)=\sum_{n\in\mathbb{Z}}\mathcal{A}_{\lambda,(n)}z^{-n-1}$ are independent of the parameters $\mu_{j}$. A irregular vertex algebra whose coherent state module is non-singular is called a non-singular irregular vertex algebra.

Examples of irregular vertex algebras will be given in Section 5 and Section 6.

3.7. Conformal structures

We shall define the notion of conformal structure on irregular vertex algebras i.e. irregular vertex operator algebras. Let $V$ be a vertex operator algebra, i.e. a vertex algebra $V$ together with the conformal vector $\omega$ (see Section 2.3). Let $S$ be a space of internal parameters with a conformal structure

[TABLE]

Define vector fields $D_{j}^{\lambda}\in\Theta_{S_{\lambda}}$ (resp. $D_{j}^{\mu}\in\Theta_{S_{\mu}}$) for $j=0,\dots,r-1$ as the images of $-t^{j+1}\partial_{t}$ via the above map $\rho_{S_{\lambda}}$ (resp. $\rho_{S_{\mu}}$). We consider the action of $D_{j}^{\lambda}$ and $D_{j}^{\mu}$ on $\mathcal{O}_{S^{2}_{\lambda,\mu}}$. Let ${\mathfrak{f}}(z;\lambda,\mu)$ be an irregularity on $S$. Since ${\mathfrak{f}}(z;\lambda,\mu)$ is degree zero, i.e. ${\mathfrak{f}}(z;\lambda,\mu)\in z^{-1}\mathcal{O}_{S^{2}_{\lambda,\mu}}[z^{-1}]_{0}$, we have

[TABLE]

Definition 3.14**.**

An irregularity ${\mathfrak{f}}(z;\lambda,\mu)$ is called conformal if it satisfies the differential equations

[TABLE]

for any non-negative integer $j$. An irregular vertex algebra $\mathcal{U}$ is called an irregular vertex operator algebra if $\mathcal{U}$ and ${\mathfrak{f}}$ are conformal and $T^{\mathcal{U}}=L_{-1}^{\mathcal{U}}$.

4. Associativity and operator product expansions

In this section, we shall prove the three fundamental properties of irregular vertex algebras: Goddard uniqueness theorem, associativity, and operator product expansions. The proofs are almost parallel to the classical ones under suitable formulations.

4.1. Goddard Uniqueness theorem

Let $(\mathcal{U},(\mathcal{U}^{\circ},F^{\bullet}),{Y},{\mathfrak{f}},{T})$ be an irregular vertex algebra for a vertex algebra $V$ on a space $S$ of internal parameters. The following is an analog of Goddard Uniqueness theorem:

Theorem 4.1**.**

Let $\mathcal{A}_{\lambda}(z)$ be an irregular field on $\mathcal{U}$ with the irregularity ${\mathfrak{f}}$. If

1. (1)

for any $b_{\mu}\in\mathcal{U}_{\mu}^{\circ}$, irregular fields $\mathcal{A}_{\lambda}(z)$ and ${Y}(b_{\mu},w)$ are mutually ${\mathfrak{f}}$-local, 2. (2)

for an element $a_{\lambda}\in\mathcal{U}^{\circ}_{\lambda}$, we have $\mathcal{A}_{\lambda}(z)|0\rangle={Y}(a_{\lambda},z)|0\rangle$,

then we have $\mathcal{A}_{\lambda}(z)={Y}(a_{\lambda},z)$.

Proof.

Let $b_{\mu}$ be an element in $\mathcal{U}_{\mu}^{\circ}$. By the assumptions and the irregular locality axiom (Definition 3.12 (2)), there exists a positive integer $N$ such that

[TABLE]

Therefore, we obtain

[TABLE]

Since we can restrict both sides to $w=0$, we get

[TABLE]

This implies the theorem. ∎

Lemma 4.2**.**

For any $\mathcal{A}\in\mathcal{U}^{\circ}$, we have ${Y}(\mathcal{A},z)|0\rangle=e^{zT}\mathcal{A}$.

Proof.

Take the expansion ${Y}(\mathcal{A},z)=\sum_{n\in\mathbb{Z}}\mathcal{A}_{(n)}z^{-n-1}$, $\mathcal{A}_{(n)}\in\mathrm{Hom}(\mathcal{U}_{\mu}^{\circ},\overline{\mathcal{U}}_{\lambda+\mu}^{\circ})$. By the vacuum axiom in Definition 3.12, we have $\mathcal{A}_{(n)}|0\rangle=0$ for $n\geq 0$, and $\mathcal{A}_{(-1)}|0\rangle=\mathcal{A}$. By the translation axiom, we have $\partial_{z}{Y}(\mathcal{A},z)=T\mathcal{A}(z)|0\rangle.$ Hence we obtain $n\mathcal{A}_{(-n-1)}|0\rangle=T\mathcal{A}_{(-n)}$. Therefore, we obtain $\mathcal{A}_{(-n-1)}|0\rangle=(n!)^{-1}T^{n}\mathcal{A}$. This implies the lemma. ∎

Corollary 4.3**.**

Assume that an irregular field $\mathcal{A}_{\lambda}(z)\in\mathrm{Hom}(\mathcal{U}_{\mu}^{\circ},\overline{\mathcal{U}}_{\lambda+\mu}^{\circ})[\![z^{\pm 1}]\!]$ and ${Y}(\mathcal{B}_{\mu},w)$ are mutually ${\mathfrak{f}}$-local for any $\mathcal{B}_{\mu}\in\mathcal{U}^{\circ}_{\mu}$,

[TABLE]

for some $a_{\lambda}\in\mathcal{U}^{\circ}_{\lambda}$, and $\partial_{z}\mathcal{A}_{\lambda}(z)|0\rangle=T\mathcal{A}_{\lambda}(z)|0\rangle$. Then we obtain $\mathcal{A}_{\lambda}(z)={Y}(a_{\lambda},z)$.

Lemma 4.4**.**

For any $\mathcal{A}_{\lambda}\in\mathcal{U}^{\circ}_{\lambda}$, we have

[TABLE]

in $\mathrm{Hom}(\mathcal{U}_{\mu}^{\circ},\overline{\mathcal{U}}_{\lambda+\mu}^{\circ})[\![z^{\pm 1}]\!]$, where $(z+w)^{-1}$ is expanded in $\mathbb{C}(\!(z)\!)(\!(w)\!)$.

Proposition 4.5** (skew symmetry).**

For $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$, $\mathcal{B}_{\mu}\in\mathcal{U}_{\mu}^{\circ}$, we have

[TABLE]

in $e^{{\mathfrak{f}}(z;\lambda,\mu)}\mathcal{U}_{\lambda+\mu}^{\circ}(\!(z)\!)$.

Proof.

For sufficiently large $N\in\mathbb{Z}$, we have

[TABLE]

If we take $N$ sufficiently large, any term in these equalities are in $\mathcal{U}_{\lambda+\mu}^{\circ}[\![z,w]\!]$. Hence we can restrict them to $w=0$ and obtain

[TABLE]

This implies the proposition. ∎

4.2. Associativity

The following theorem is a generalization of the associativity to irregular vertex algebras:

Theorem 4.6**.**

For any $\mathcal{A}_{\lambda}\in\mathcal{U}^{\circ}_{\lambda}$, $\mathcal{B}_{\mu}\in\mathcal{U}^{\circ}_{\mu}$, and $\mathcal{C}_{\nu}\in\mathcal{U}^{\circ}_{\nu}$, the three elements

[TABLE]

are the expansions of the same element in

[TABLE]

to their respective domains.

Take an expansion

[TABLE]

where $\mathcal{A}^{\prime}_{\lambda}(\mu)_{(n)}\in\mathrm{Hom}(\mathcal{U}^{\circ}_{\mu},\mathcal{U}^{\circ}_{\lambda+\mu})$ for each $n\in\mathbb{Z}$. Since $e^{-{\mathfrak{f}}(z-w;\lambda,\mu)}{Y}(\mathcal{A}_{\lambda},z-w)\mathcal{B}_{\mu}$ is in $\mathcal{U}^{\circ}_{\lambda+\mu}(\!(z-w)\!)$, we have the expansion

[TABLE]

for sufficiently large $N$.

Then, the composition $e^{-{\mathfrak{f}}(z;\lambda,\nu)}_{|w|>|z-w|}e^{-{\mathfrak{f}}(z-w;\lambda,\mu)}{Y}({Y}(\mathcal{A}_{\lambda},z-w)\mathcal{B}_{\mu},w)\mathcal{C}_{\nu}$ is defined as

[TABLE]

Here, note that we have $e^{-{\mathfrak{f}}(z;\lambda,\nu)}_{|w|>|z-w|}e^{{\mathfrak{f}}(w;\lambda,\nu)}\in\mathcal{O}_{S^{2}_{\lambda,\nu}}[w^{-1}][\![z-w]\!]$ by Lemma 3.6. Hence $e^{-{\mathfrak{f}}(z;\lambda,\nu)}_{|w|>|z-w|}{Y}(\mathcal{A}^{\prime}_{\lambda}(\mu)_{(n)}\mathcal{B}_{\mu},w)\mathcal{C}_{\nu}$ is in $e^{{\mathfrak{f}}(w;\mu,\nu)}\mathcal{U}^{\circ}_{\lambda+\mu+\nu}(\!(w)\!)[\![z-w]\!]$.

Proof of Theorem 4.6.

By the skew symmetry (Proposition 4.5), we have

[TABLE]

Since $e^{-{\mathfrak{f}}(z-w;\lambda,\mu)}_{|z|>|w|}{Y}(\mathcal{A}_{\lambda},z)e^{wT}{Y}(\mathcal{C}_{\nu},-w)\mathcal{B}_{\mu}$ is in $e^{{\mathfrak{f}}(z;\lambda,\nu)+{\mathfrak{f}}(w;\mu,\nu)}\mathcal{U}^{\circ}_{\lambda+\mu+\nu}(\!(z)\!)(\!(w)\!)$, the last equality makes sense. Then, again by Lemma 4.4, this equals to

[TABLE]

Therefore, the two elements

[TABLE]

are the expansions of the same element in

[TABLE]

to the modules $e^{{\mathfrak{f}}(w;\mu,\nu)}\mathcal{U}^{\circ}_{\lambda+\mu+\nu}(\!(z)\!)(\!(w)\!)$ and $e^{{\mathfrak{f}}(w;\mu,\nu)}\mathcal{U}^{\circ}_{\lambda+\mu+\nu}(\!(z-w)\!)(\!(w)\!)$ respectively.

By the skew symmetry, we have

[TABLE]

Hence, we obtain

[TABLE]

in $\overline{\mathcal{U}}_{\lambda+\mu+\nu}[\![w^{\pm 1},(z-w)^{\pm 1}]\!]$. By Lemma 3.10, (4.1) and (4.2) are the expansion of the same element. This proves the theorem. ∎

4.3. Normally ordered product and operator product expansion

We shall define the normally ordered product for irregular fields:

Definition 4.7** (normally ordered product).**

For an irregular field $\mathcal{A}_{\lambda}(z)$ with an expansion $\mathcal{A}_{\lambda}(z)=e^{{\mathfrak{f}}(z;\lambda,\nu)}\sum_{n\in\mathbb{Z}}\mathcal{A}^{\prime}_{\lambda}(\nu)_{(n)}z^{-n-1}$, set

[TABLE]

and

[TABLE]

Let $\mathcal{B}_{\mu}(w)$ be an irregular field with irregularity ${\mathfrak{f}}(w;\mu,\nu)$ and set $\mathcal{B}_{\mu}^{\prime}(w;\nu)\coloneqq e^{-{\mathfrak{f}}(w;\mu,\nu)}\mathcal{B}_{\mu}(w)$. The *normally ordered product * {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(w){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} of $\mathcal{A}_{\lambda}(z)$ and $\mathcal{B}_{\mu}(w)$ is defined by

[TABLE]

The restriction of {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(w){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} to $z=w$ is well defined and is denoted by {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(z){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}. Note that {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(z){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} is again an irregular field with irregularity ${\mathfrak{f}}(z;\lambda+\mu,\nu)$. Actually, we can check that {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(z){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} does not depend on the parameters $\nu_{i}$ by using the presentation in Lemma 4.8 below.

The following two lemmas can be proved by the same way as the classical case:

Lemma 4.8**.**

The restriction {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(w)\mathcal{B}_{\mu}(w){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} equals to $e^{{\mathfrak{f}}(w;\lambda+\mu,\nu)}$ times

[TABLE]

where $\delta(z-w)_{-}\coloneqq\sum_{n=0}^{\infty}w^{n}/z^{n+1}$ and $\delta(z-w)_{+}\coloneqq\sum_{n>0}z^{n-1}/w^{n}$.

Lemma 4.9** (Dong’s lemma).**

Let $\mathcal{A}_{\lambda}(z),\mathcal{B}_{\mu}(w),\mathcal{C}_{\nu}(u)$ be irregular fields with an irregularity ${\mathfrak{f}}$. Assume that each two of three fields are mutually ${\mathfrak{f}}$-local. Then, the normally ordered product {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\mathcal{A}_{\lambda}(z)\mathcal{B}_{\mu}(z){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} and $\mathcal{C}_{\nu}(w)$ are mutually ${\mathfrak{f}}$-local.

Theorem 4.10** (Operator product expansion).**

For any $\mathcal{A}_{\lambda}\in\mathcal{U}_{\lambda}^{\circ}$ and $\mathcal{B}_{\mu}\in\mathcal{U}^{\circ}_{\mu}$, there is an equality

[TABLE]

where ${Y}(\mathcal{A}_{\lambda},z)=e^{{\mathfrak{f}}(z;\lambda,\mu)}\sum_{n\in\mathbb{Z}}\mathcal{A}^{\prime}_{\lambda}(\mu)_{(n)}z^{-n-1}$ and both sides are expanded in the domain $|z|>|w|$.

Proof.

By the associativity, it remains to show that

[TABLE]

for every non-negative integer $n$. By the direct computation, we have

[TABLE]

The irregular field {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}\left(\partial_{w}^{(n)}{Y}(\mathcal{A}_{\lambda},w)\right){Y}(\mathcal{B}_{\mu},w){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}} and ${Y}(\mathcal{C}_{\nu},z)$ are mutually ${\mathfrak{f}}$-local for every $\mathcal{C}_{\nu}$ by the Dong’s lemma (Lemma 4.9). Then, by the Goddard uniqueness theorem (Theorem 4.1), we obtain (4.3). ∎

As an easy consequence, we obtain the following:

Corollary 4.11**.**

The composition $e^{-{\mathfrak{f}}(z-w;\lambda,\mu)}_{|z|>|w|}{Y}(|\lambda\rangle,z){Y}(|\mu\rangle,w)$ can be restricted to $z=w$. Moreover, the restriction equals to {{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}{Y}(|\lambda\rangle,z){Y}(|\mu\rangle,z){{\raise 2.5pt\hbox{\mathop{\hphantom{\cdot}}\limits^{{}{\circ}}{{}^{\circ}}}}}={Y}(|\lambda+\mu\rangle,z).

5. Irregular Heisenberg vertex operator algebras

In this section, following the ideas of [NS10], we shall define the irregular vertex algebras for the Heisenberg vertex algebra.

5.1. Heisenberg vertex algebra

Let us briefly recall the definition of Heisenberg vertex algebra to fix the notation. Let ${{\mathcal{F}}}$ denote the $\mathbb{Z}_{\geq 0}$-graded vector space of graded polynomial ring $\mathbb{C}[x_{n}]_{n>0}$ of variables $x_{n}$ with $\deg x_{n}=n\in\mathbb{Z}_{>0}$. Let $|0\rangle\in{{\mathcal{F}}}$ denote the unit of $\mathbb{C}[x_{n}]_{n>0}$.

Define an endomorphism $T$ as the derivation on $\mathbb{C}[x_{n}]_{n>0}$ with $Tx_{n}=nx_{n+1}$. Fix a non-zero complex number $\kappa$. Let $a_{-n}$ (resp. $a_{n}$) denote the multiplication of $x_{n}$, (resp. the derivation $2\kappa n{\partial_{x_{n}}}$) for $n>0$. Set $a_{0}\coloneqq 0\in\mathrm{End}({{\mathcal{F}}})$. The power series $a(z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}$ defines an field on ${{\mathcal{F}}}$.

Define $Y(\cdot,z)\colon{{\mathcal{F}}}\to\mathrm{End}({{\mathcal{F}}})[\![z^{\pm 1}]\!]$ by

[TABLE]

for $n_{1},\dots,n_{k}\in\mathbb{Z}_{>0}$, $k>0$. We also set $Y(|0\rangle,z)=\mathrm{id}$. Then, the tuple

[TABLE]

is known to be a vertex algebra called Heisenberg vertex algebra.

5.2. Coherent state module

Fix a positive integer $r$. Let $S\coloneqq{\mathrm{Spec}\ \!}(\mathbb{C}[\lambda_{j}]_{j=1}^{r})$ be the space of internal parameters with $\deg\lambda_{j}=-j$.

Consider the completion $\overline{{{\mathcal{F}}}}\coloneqq\prod_{n\geq 0}{{\mathcal{F}}}_{n}$, where ${{\mathcal{F}}}_{n}$ is the degree $n$-part of ${{\mathcal{F}}}$. The tensor product $\overline{{{\mathcal{F}}}}_{S}\coloneqq\overline{{{\mathcal{F}}}}\otimes_{\mathbb{C}}\mathcal{O}_{S}$ is a $\mathbb{Z}$-graded $\mathcal{O}_{S}$-module whose degree $n$-part is given by

[TABLE]

where ${{\mathcal{F}}}_{n+j}=0$ for $n+j<0$. It is also considered as a $\mathcal{D}_{S}$-module in an obvious way. We also set

[TABLE]

for $m\in\mathbb{Z}$ and ${\mathrm{End}}({{{\mathcal{F}}}})_{S}\coloneqq\bigoplus_{m}{\mathrm{End}}({{{\mathcal{F}}}})_{S,m}$.

Lemma 5.1**.**

Every element in ${\mathrm{End}}({{{\mathcal{F}}}})_{S}$ defines an endomorphism on $\overline{{{\mathcal{F}}}}_{S}$.

Proof.

Fix $n,m\in\mathbb{Z}$. We have an natural morphism from $\overline{{{\mathcal{F}}}}_{S,n}\otimes\mathrm{End}({{\mathcal{F}}})_{S,m}$ to $\overline{{{\mathcal{F}}}}_{S,n+m}$ as follows:

[TABLE]

This gives the conclusion. ∎

Set

[TABLE]

Corollary 5.2**.**

The exponential

[TABLE]

defines an automorphism on $\overline{{{\mathcal{F}}}}_{S}$.

Definition 5.3**.**

We set ${{\mathcal{F}}}^{(r)}\coloneqq\Phi_{\lambda}^{(r)}({{\mathcal{F}}}\otimes\mathcal{O}_{S})\subset\overline{{{\mathcal{F}}}}_{S}$, and $|{\rm coh}\rangle\coloneqq\Phi_{\lambda}^{(r)}(|0\rangle\otimes 1)$.

We shall show that the pair $({{\mathcal{F}}}^{(r)},|{\rm coh}\rangle)$ is equipped with the structure of non-singular coherent state ${{\mathcal{F}}}_{\kappa}$-module over $S$.

Lemma 5.4**.**

We have $\left[2\kappa n\partial_{\lambda_{n}},\Phi_{\lambda}^{(r)}\right]=a_{-n}\Phi_{\lambda}^{(r)}$ for $n=1,\dots,r$.

By this lemma, we obtain the following.

Corollary 5.5**.**

${{\mathcal{F}}}^{(r)}$* is a $\mathcal{D}_{S}$-submodule of $\overline{{{\mathcal{F}}}}_{S}$.*

Lemma 5.6**.**

As endomorphisms on $\overline{{{\mathcal{F}}}}_{S}$, we have the commutation relation

[TABLE]

Proof.

By definition, we have

[TABLE]

Hence, we obtain

[TABLE]

This implies the lemma. ∎

Corollary 5.7**.**

We have

[TABLE]

as an $\mathcal{O}_{S}$-module.

It also follows from Lemma 5.6 that $a_{n}$ $(n\in\mathbb{Z})$ acts on ${{\mathcal{F}}}^{(r)}$. We can define

[TABLE]

in a way similar to (5.1).

Proposition 5.8**.**

The tuple ${{\mathcal{F}}}^{(r)}_{\kappa}\coloneqq\left({{\mathcal{F}}}^{(r)},Y_{{{\mathcal{F}}}^{(r)}}(\cdot,z),|{\rm coh}\rangle\right)$ is a non-singular coherent state ${{\mathcal{F}}}$-module, which is an envelope of ${{\mathcal{F}}}_{\kappa}$.

5.3. Irregular vertex algebra structure

Lemma 5.9**.**

Let $F^{\bullet}({{\mathcal{F}}}^{(r)})$ be the decreasing filtration on ${{\mathcal{F}}}^{(r)}$ defined by

[TABLE]

Then, $({{\mathcal{F}}}^{(r)},F^{\bullet})$ is a filtered small lattice of ${{\mathcal{F}}}^{(r)}$ i.e. satisfies the conditions in Definition 3.1.

Proof.

The condition (L) requires nothing since $H$ is empty. Since $F^{0}({{\mathcal{F}}}^{(r)})$ equals to ${{\mathcal{F}}}^{(r)}$, condition (F1) holds. The conditions (F2), (F4) (resp. (F3), (F5)) are the corollaries of Lemma 5.6 (resp. Lemma 5.4). ∎

Lemma 5.10**.**

We have an isomorphism

[TABLE]

Proof.

We have

[TABLE]

This proves the lemma. ∎

Define $\overline{{{\mathcal{F}}}}_{S^{2}}$ and $\mathrm{End}({{\mathcal{F}}})_{S^{2}}$ by replacing $S$ with $S^{2}$ in the definition of $\overline{{{\mathcal{F}}}}_{S}$ and $\mathrm{End}({{\mathcal{F}}})_{S}$, respectively. We can replace $S$ with $S^{2}$ in Lemma 5.1 and Lemma 5.10. We shall consider the following extension of $Y(\cdot,z)$:

Definition 5.11**.**

Define the morphism of $\mathcal{D}_{S^{2}}$-modules

[TABLE]

by $\overline{Y}\left(\sum_{k=0}^{\infty}A_{n+k}\otimes P_{-k}(\lambda,\mu),z\right)\coloneqq\sum_{k=0}^{\infty}Y(A_{k+n},z)\otimes P_{-k}(\lambda,\mu)$ for each homogeneous element $\sum_{k=0}^{\infty}A_{n+k}\otimes P_{-k}(\lambda,\mu)\in{{\mathcal{F}}}_{S^{2},n}$.

We identify $\overline{{{\mathcal{F}}}}_{S_{\lambda}}$ with a subspace of $\overline{{{\mathcal{F}}}}_{S^{2}_{\lambda,\mu}}$ along with the $S_{\lambda}$-axis. Then $\mathcal{D}_{S_{\lambda}}$-module structures on $\overline{{{\mathcal{F}}}}_{S_{\lambda}}$ and $\overline{{{\mathcal{F}}}}_{S^{2}_{\lambda,\mu}}$ are compatible. We also regards the coherent state module ${\mathcal{F}}_{\lambda}^{(r)}\subset\overline{{{\mathcal{F}}}}_{S_{\lambda}}$ defined in Definition 5.3 as a subspace of $\overline{{{\mathcal{F}}}}_{S^{2}_{\lambda,\mu}}$ through the above identification. We do the same thing above for $\mu$.

We note that the pull back ${{\mathcal{F}}}^{(r)}_{\lambda+\mu}\coloneqq\sigma_{\lambda+\mu}^{*}{{\mathcal{F}}}^{(r)}$ (see Section 3.1) is a $\mathcal{D}_{S^{2}_{\lambda,\mu}}$-submodule of $\overline{{{\mathcal{F}}}}_{S^{2}_{\lambda,\mu}}$ by definition. The completion $\overline{{{\mathcal{F}}}}^{(r)}_{\lambda+\mu}$ of ${{\mathcal{F}}}^{(r)}_{\lambda+\mu}$ is naturally identified with $\overline{{{\mathcal{F}}}}_{S^{2}_{\lambda,\mu}}$. By Lemma 5.1, we can restrict $\overline{Y}(\cdot,z)$ to ${{\mathcal{F}}}^{(r)}_{\lambda}$:

[TABLE]

and ${Y}(\cdot,z)$ is a $\mathcal{D}_{S_{\lambda}}$-module morphism by definition.

Set

[TABLE]

Lemma 5.12**.**

There are equalities

[TABLE]

for $k,n_{1},\dots,n_{k}\in\mathbb{Z}_{>0}$.

Proof.

By definition, we have

[TABLE]

and hence

[TABLE]

This proves the first equality. The second equality can also be proved similarly. ∎

Note that we have $\varphi_{\lambda}^{(r)}(z)_{+}|_{z=0}=\varphi_{\lambda}^{(r)}$ and $\varphi_{\lambda}^{(r)}(z)_{+}|0\rangle=0$.

Corollary 5.13**.**

For any $\mathcal{A}_{\lambda}\in{{\mathcal{F}}}^{(r)}_{\lambda}$, we have ${Y}(\mathcal{A}_{\lambda},z)|0\rangle\in{{\mathcal{F}}}^{(r)}_{\lambda}[\![z]\!]$, and

[TABLE]

Lemma 5.14**.**

Put

[TABLE]

We have $e^{-{\mathfrak{f}}_{\kappa}(z;\lambda,\mu)}{Y}(|\lambda\rangle,z)|\mu\rangle\in{{\mathcal{F}}}^{(r)}_{\lambda+\mu}[\![z]\!]$, and $e^{-{\mathfrak{f}}_{\kappa}(z;\lambda,\mu)}{Y}(|\lambda\rangle,z)|\mu\rangle|_{z=0}=|\lambda+\mu\rangle$.

Moreover, for any $\mathcal{A}_{\lambda}\in{{\mathcal{F}}}_{\lambda}^{(r)}$, ${Y}(\mathcal{A}_{\lambda},z)$ is an irregular field on ${{\mathcal{F}}}^{(r)}_{\mu}$ with the irregularity ${\mathfrak{f}}_{\kappa}(z;\lambda,\mu)$.

Proof.

Since we have $[\partial_{z}^{(p-1)}a(z)_{+},a_{-q}]=0$ and

[TABLE]

for $p,q\in\mathbb{Z}_{>0}$, we have $[\varphi_{\lambda}^{(r)}(z)_{+},\varphi_{\mu}^{(r)}]=0$ and

[TABLE]

Using this equality, we obtain that

[TABLE]

We obtain the first statement by Corollary 5.13.

The latter statement can be proved by using Corollary 5.7, Lemma 5.12 and (5.4). ∎

Lemma 5.15**.**

There is an equality

[TABLE]

Proof.

We have $[\partial_{z}^{(p-1)}a(z)_{+},\partial_{w}^{(q-1)}a(w)_{+}]=[\partial_{z}^{p-1}a(z)_{-},\partial_{w}^{(q-1)}a(w)_{-}]=0$,

[TABLE]

where $(z-w)^{-p-q}|_{|z|>|w|}$ denotes the expansion in positive powers of $w/z$. Hence we obtain

[TABLE]

By the Baker-Campbell-Hausdorff formula, we obtain the lemma. ∎

Since $[T,\Phi_{\lambda}^{(r)}]=\sum_{j=1}^{r}\lambda_{j}a_{-j-1}\Phi_{\lambda}^{(r)}$, $T$ naturally acts on ${{\mathcal{F}}}^{(r)}$.

Theorem 5.16**.**

The tuple $\left({{\mathcal{F}}}_{\kappa}^{(r)},({{\mathcal{F}}}^{(r)},F^{\bullet}),{Y}(\cdot,z),T,{\mathfrak{f}}_{\kappa}\right)$ is an irregular vertex algebra for ${{\mathcal{F}}}_{\kappa}$.

Proof.

Let us check the axioms in Definition 3.12. The translation axiom and the compatibility condition are trivial by the construction. The vacuum axiom is proved in Corollary 5.13. The irregular field axiom and coherent state axiom are proved in Lemma 5.14. It remains to prove the irregular locality axiom.

By Lemma 5.15, ${Y}(|\lambda\rangle,z)$ and ${Y}(|\mu\rangle,w)$ are mutually ${\mathfrak{f}}_{\kappa}$-local. Then, by the compatibility, we can apply the Dong’s lemma (Lemma 4.9), to obtain the ${\mathfrak{f}}_{\kappa}$-locality in general. ∎

5.4. Conformal structures

In this subsection, we shall show that ${{\mathcal{F}}}^{(r)}$ is an irregular vertex operator algebra if $\kappa=1/2$. Recall that the space $S={\mathrm{Spec}\ \!}\mathbb{C}[\lambda_{j}]_{j=1}^{r}$ is equipped with the $\mathrm{Der}_{0}(\mathbb{C}[\![t]\!])$-structure as explained in Example 2.2. We firstly prove the following:

Lemma 5.17**.**

The irregularity ${\mathfrak{f}}_{\kappa}(z;\lambda,\mu)$ is conformal (see Definition 3.14).

Proof.

We have

[TABLE]

Since $\partial_{z}^{(m+1)}z^{j+1}=\binom{j+1}{m+1}z^{j-m}$ for $m\geq-1$, we have

[TABLE]

Hence the coefficient of $\lambda_{u}\mu_{v}/z^{w}$ ($u,v,w>0$) with $u+v=j+w$ in (3.6) is given by

[TABLE]

Hence we need to show

[TABLE]

The left hand side of this equation is the coefficients of $x^{u}$ in $(1+x)^{-(w-u)}$, and the right hand side is that of $x^{u}$ in $(1+x)^{-(v+1)}(1+x)^{j+1}=(1+x)^{j-v}$. Hence we obtain the lemma. ∎

Let the complex number $\kappa$ be $1/2$. Take a complex number $\rho$, and put $c=1-12\rho^{2}$. It is known that $\omega_{\rho}\coloneqq\frac{1}{2}a_{-1}^{2}+\rho a_{-2}$ is a conformal vector of the Heisenberg vertex algebra ${{\mathcal{F}}}$. Let $h_{k}$ $(k=0,\dots,2r)$ (and hence $\mathcal{L}_{k}$) as in Section 2.4 with $\lambda_{0}=0$. For the simplicity of the notation, we denote

[TABLE]

Lemma 5.18**.**

For $s\geq 0$, we have

[TABLE]

Proof.

By Lemma 5.6, we have

[TABLE]

On the other hand, by Lemma 5.4 we have

[TABLE]

This proves the lemma. ∎

Corollary 5.19**.**

For $k\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Proposition 5.20**.**

The irregular Heisenberg vertex algebra ${{\mathcal{F}}}^{(r)}$ for $\kappa=1/2$ is an irregular vertex operator algebra.

Proof.

Lemma 2.8 shows that $\rho_{{{\mathcal{F}}}^{(r)}};t^{k+1}\partial_{t}\mapsto-(L_{k}-\mathcal{L}_{k})$ is a Lie algebra homomorphism.

For $v_{k+d}\in{{\mathcal{F}}}_{k+d}$ and $f_{k}(\lambda)\in\mathcal{O}_{S,-k}$, by Lemma 5.18, we have

[TABLE]

This proves that $L_{0}-\mathcal{L}_{0}$ acts as the grading operator on ${{\mathcal{F}}}^{(r)}$.

By Lemma 5.18, for any $v_{\lambda}\in{{\mathcal{F}}}\otimes\mathcal{O}_{S}$, we have

[TABLE]

Since

[TABLE]

is locally nilpotent on ${\mathcal{F}}\otimes{\mathcal{O}}_{S}$, we can deduce that $L_{s}-\mathcal{L}_{s}$ is locally nilpotent. ∎

6. Irregular Virasoro vertex operator algebras

We shall give a definition of irregular Virasoro vertex algebra via the free field realization.

6.1. Saturated vertex subalgebras

Let $(\mathcal{V},{Y},{\mathfrak{f}})$ be a non-singular irregular vertex operator algebra for a vertex operator algebra $V$ on $S$. In particular, a filtration $F^{\bullet}(\mathcal{V})$ with the properties (F1)-(F4) is fixed. Let $W\subset V$ be a vertex operator subalgebra of $V$. Let $\mathcal{U}\subset\mathcal{V}$ be the smallest $\mathcal{D}_{S}$-submodule which contains $|{\rm coh}\rangle\in\mathcal{V}$ and is closed under the operation $A_{(n)}^{\mathcal{V}}$ for every $A\in W$ and every $n\in\mathbb{Z}$.

Definition 6.1**.**

The vertex operator subalgebra $W\subset V$ is saturated with respect to $\mathcal{V}$ if $\mathcal{U}$ is a coherent state $W$-module with singularity $H$ and $\mathcal{U}(*H)=\mathcal{V}(*H)$.

If $W$ is saturated, set $\mathcal{U}^{\circ}\coloneqq\mathcal{V}\subset\mathcal{U}(*H)$ and $F^{\bullet}(\mathcal{U}^{\circ})\coloneqq F^{\bullet}(\mathcal{V})$. Then we have the following lemma:

Lemma 6.2**.**

\big{(}\mathcal{U},(\mathcal{U}^{\circ},F^{\bullet}),{Y},{\mathfrak{f}}\big{)}* is an irregular vertex operator algebra for $W$.*

Proof.

The only non-trivial point is that $(\mathcal{U},Y_{\mathcal{U}},(\mathcal{U}^{\circ},F^{\bullet}))$ is an envelope of $W$. In other words, we need to show that the morphism

[TABLE]

defined in Definition 3.11, is an isomorphism. Since $\mathcal{V}$ is non-singular, we have an isomorphism

[TABLE]

Since $\overline{\Psi}_{W}$ is the restriction of $\overline{\Psi}_{V}$, it is injective.

It remains to prove that $\overline{\Psi}_{W}$ is surjective. Since $\overline{\Psi}_{V}$ is an isomorphism of $V$-modules, we have

[TABLE]

for $A\in V$ and $n\geq 0$. By the construction, a section of $\mathcal{U}_{\mathcal{O}}$ can be expressed as an ${\mathcal{O}}_{S}$-linear combination of the sections of the form

[TABLE]

for some $A_{1},\dots,A_{k}\in W$, and $n_{1}\leq\cdots\leq n_{k}\in\mathbb{Z}$. If $n_{k}\leq-1$, then (6.2) is the image of

[TABLE]

by ${\Psi}_{W}\colon W\to\mathcal{U}_{\mathcal{O}},\quad A\mapsto A_{(-1)}|{\rm coh}\rangle$. If $n_{k}\geq 0$, then by (6.1), the class of (6.2) in $\mathcal{U}_{\mathcal{O}}|^{\circ}_{0}$ is zero. Hence we obtain the lemma. ∎

6.2. Irregular Virasoro vertex algebra via free field realization

Recall that $\mathrm{Vir}_{c}$ denotes the Virasoro vertex algebra (Section 2.3). Let ${{\mathcal{F}}}$ be the Heisenberg vertex algebra with $\kappa\coloneqq 1/2$. The irregular Heisenberg algebra ${{\mathcal{F}}}^{(r)}$ is also considered in the case $\kappa=1/2$.

Consider the morphism $\mathrm{Vir}_{c}\to{{\mathcal{F}}}$ given by the conformal vector $\omega=\frac{1}{2}a_{-1}^{2}+\rho a_{-2}$ for a complex number $\rho$ and $c=1-12\rho^{2}$. We assume that $c$ is generic so that we have $\mathrm{Vir}_{c}\subset{\mathcal{F}}$.

Definition 6.3**.**

Let $\mathrm{Vir}_{c}^{(r)}$ denote the smallest $\mathcal{D}_{S}$-submodule of ${{\mathcal{F}}}^{(r)}$ which is closed under all operations of the form $A_{(n)}^{{{\mathcal{F}}}^{(r)}}$ for $A\in\mathrm{Vir}_{c}$ and $n\in\mathbb{Z}$.

Let $\mathcal{M}_{c,0}^{(r)}$ denote the coherent state module defined in Section 2.4 with $\lambda_{0}=0$. The relation between $\mathrm{Vir}_{c}^{(r)}$ and $\mathcal{M}_{c,0}^{(r)}$ is given by the following proposition:

Proposition 6.4**.**

We have a unique morphism

[TABLE]

of $\mathcal{D}_{S}\otimes_{\mathbb{C}}U(\mathrm{Vir})$-modules such that the coherent state of $\mathcal{M}^{(r)}_{c,0}$ maps to that of ${{\mathcal{F}}}^{(r)}$. The morphism $(\ref{VIR})$ is injective and the image of $(\ref{VIR})$ coincides with $\mathrm{Vir}_{c}^{(r)}$.

Proof.

By Corollary 5.5, there is a unique morphism $\mathcal{D}_{S}\otimes_{\mathbb{C}}U(\mathrm{Vir})\longrightarrow{{\mathcal{F}}}^{(r)}$ which sends $1\otimes 1$ to $|\lambda\rangle$. By (5.5), the above morphism uniquely induces the morphism (6.3). On the one hand, by Remark 2.11, the morphism (6.3) is injective at each point $\lambda^{o}=(\lambda_{1}^{o},\dots,\lambda_{r}^{o})$ with $\lambda_{r}^{o}\neq 0$. On the other hand, since $\mathcal{M}^{(r)}_{c,0}$ is a free ${\mathcal{O}}_{S}$-module (see (2.4)), the kernel of (6.3) should be torsion free. Hence we have that the kernel is the zero module, which means that (6.3) is injective. The coincidence of the image with $\mathrm{Vir}_{c}^{(r)}$ follows from the minimality in the definition of $\mathrm{Vir}_{c}^{(r)}$. ∎

Corollary 6.5**.**

$\mathrm{Vir}_{c}^{(r)}$* is a coherent state $\mathrm{Vir}_{c}$-module with singularity $\{\lambda_{r}=0\}$.*

The following theorem is the main theorem of this section.

Theorem 6.6**.**

The Virasoro vertex algebra $\mathrm{Vir}_{c}$ is saturated in the Heisenberg vertex algebra ${{\mathcal{F}}}$ with respect to the irregular vertex algebra ${{\mathcal{F}}}^{(r)}$.

Proof.

We have already checked that $\mathrm{Vir}_{c}^{(r)}$ is coherent state $\mathrm{Vir}_{c}$-module with singular divisor $H=\{\lambda_{r}=0\}$. Hence it remains to show that $\mathrm{Vir}_{c}^{(r)}(*H)={{\mathcal{F}}}^{(r)}(*H)$. We shall prove this by showing the following proposition inductively on $n$:

• $(\bm{P}_{n})$

Every section $a_{-n_{1}}\cdots a_{-n_{k}}|\lambda\rangle$ with $n_{j}>0$ $(1\leq j\leq k)$ and $\sum_{j=1}^{k}n_{k}\leq n$ is in $\mathrm{Vir}_{c}^{(r)}(*H)$.

The proposition $(\bm{P}_{0})$ is trivial since $|\lambda\rangle\in\mathrm{Vir}_{c}^{(r)}$. Assume that $(\bm{P}_{n-1})$ holds. Let $a_{-n_{1}}\cdots a_{-n_{k}}|\lambda\rangle$ be an arbitrary section with $\sum_{j=1}^{k}n_{j}=n$. If $0<n_{1}\leq r$, then

[TABLE]

If $n_{1}>r$, consider the action of $L_{-n_{1}+r}$ on $a_{-n_{2}}\cdots a_{-n_{k}}|\lambda\rangle$:

[TABLE]

Each term other than $\lambda_{r}a_{-n_{1}}a_{-n_{2}}\cdots a_{-n_{k}}|\lambda\rangle$ is in $\mathrm{Vir}_{c}^{(r)}(*H)$ by $(\bm{P}_{n-1})$. We also have $L_{-n_{1}+r}a_{-n_{2}}\cdots a_{-n_{k}}|\lambda\rangle\in\mathrm{Vir}_{c}^{(r)}(*H)$. Hence we obtain that the element $a_{-n_{1}}\cdots a_{-n_{k}}|\lambda\rangle$ is in $\mathrm{Vir}_{c}^{(r)}(*H)$. This proves $(\bm{P}_{n})$ and hence the theorem. ∎

By the Lemma 6.2, we can define the irregular Virasoro vertex algebra:

Definition 6.7**.**

We set $\mathcal{V}ir^{(r)}_{c}\coloneqq{{\mathcal{F}}}^{(r)}$, which is considered as a filtered small lattice of $\mathrm{Vir}_{c}^{(r)}(*H)$. The irregular vertex operator algebra

[TABLE]

is called an irregular Virasoro vertex algebra.

Remark 6.8**.**

The quotient

[TABLE]

is isomorphic to the usual Verma module for the Virasoro algebra, while the quotient

[TABLE]

is isomorphic to $\mathrm{Vir}_{c}$ via $\overline{\Psi}_{\mathrm{Vir}_{c}}$.

Remark 6.9**.**

By the Theorem 6.6, at least theoretically, we can describe the vertex operators $Y(\mathcal{A}_{\lambda},z)$ for $\mathcal{A}_{\lambda}\in\mathcal{V}ir_{c}^{(r)}$ only in terms of the Virasoro algebra and the coherent states after the localization, although the computation is very complicated in practice. For example, in the case $r=1$ and $\rho=0$, we have

[TABLE]

where we put $\lambda=\lambda_{1}$ and $\mu=\mu_{1}$.

Concluding remarks

The key point of our construction in Section 5 was the Baker-Campbell-Hausdorff formula used in Lemma 5.15. We expect that our way of constructing an irregular vertex algebra used in Section 5 can be easily generalized to the vertex algebras generated by finitely many free fields in the sense of [Kac98, Definition] although we have only treated the case where the vertex algebra is generated by a free field, for simplicity (and to give a canonical conformal structure).

Then, we also expect that Lemma 6.2 in Section 6 (or its generalization) would be useful in the construction of irregular versions of Kac-Moody vertex algebras and $\mathcal{W}$-algebras. In other words, we expect that we may define irregular versions of $V_{k}(\mathfrak{g})$ and $\mathcal{W}(\mathfrak{g})$ via their respective free field realizations. We shall refer [Nag15, Nag18] and [GLP, KMST13] as studies on irregular conformal blocks for $V_{k}(\mathfrak{sl}_{2})$ and $\mathcal{W}_{3}$-algebra, respectively. The details of these expectations would be given in the subsequent studies.