Kazama-Suzuki coset construction and its inverse

TL;DR

This paper generalizes the inverse Kazama-Suzuki coset construction to arbitrary Lie algebra types, establishing categorical equivalences and proving regularity and braided monoidal structures for associated vertex operator superalgebras.

Contribution

It extends the inverse coset construction beyond type A₁ and proves regularity and monoidal category structure for the resulting superalgebras.

Findings

Categorical equivalence between module categories established.

Regularity of the coset superalgebra proven.

Braided monoidal category structure shown for modules.

Abstract

We study the representation theory of the Kazama-Suzuki coset vertex operator superalgebra associated with the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type $A_{1}$, B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin introduced another coset construction, which is "inverse" of the Kazama-Suzuki coset construction. In this paper we generalize the latter coset construction to arbitrary type and establish a categorical equivalence between the categories of certain modules over an affine vertex operator algebra and the corresponding Kazama-Suzuki coset vertex operator superalgebra. Moreover, when the affine vertex operator algebra is regular, we prove that the corresponding Kazama-Suzuki coset vertex operator superalgebra is also regular and the category of its ordinary modules carries a braided monoidal category structure by the theory of vertex tensor categories.

Equations

$$(\text{Affine VOAs})\to (\mathcal{N}=2\text{VOSAs}),$$

$$(\mathcal{N}=2\text{VOSAs})\to (\text{Affine VOAs}).$$

$$V^{k+h^{\vee }}(\mathfrak{h})\to V_{\mathsf{af}}\otimes V^{+},$$

$$V_{\sf sc}:={\sf Com}\big{(}\mathcal{H}^{+},V_{\sf af}\otimes V^{+}\big{)}.$$

$$c_{\mathsf{sc}}:=\frac{k\mathsf{dim}\mathfrak{g}}{k+h^{\vee }}+\frac{1}{2}\mathsf{dim}{\mathfrak{h}}^{\perp }-\mathsf{dim}\mathfrak{h}$$

$$f_{\sf FST}\colon V_{\sf af}\to{\sf Com}\big{(}\mathcal{H}^{-},V_{\sf sc}\otimes V^{-}\big{)}$$

$${\Omega }^{+}(\lambda ):{\mathcal{C}}_{[\lambda ]}(V_{\mathsf{af}})\to {\mathcal{C}}_{[{\lambda }_{\mathsf{sc}}]}(V_{\mathsf{sc}}),{\Omega }^{-}({\lambda }_{\mathsf{sc}}):{\mathcal{C}}_{[{\lambda }_{\mathsf{sc}}]}(V_{\mathsf{sc}})\to {\mathcal{C}}_{[\lambda ]}(V_{\mathsf{af}})$$

$$\mathcal{C}_{k}(\mathfrak{g}):={\sf Com}\big{(}\mathcal{H}^{+},L_{k}(\mathfrak{g})\otimes V^{+}\big{)}.$$

$$\mathcal{E}^{+}:={\sf Com}\Big{(}{\sf Com}\big{(}\mathcal{H}^{+},L_{k}(\mathfrak{g})\otimes V^{+}\big{)},L_{k}(\mathfrak{g})\otimes V^{+}\Big{)}$$

$$\sqrt{k+h^{\vee }}{Q}_l,$$

$$\mathcal{E}^{-}:={\sf Com}\Big{(}{\sf Com}\big{(}\mathcal{H}^{-},\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}\big{)},\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}\Big{)}$$

$$\sqrt{-(k+h^{\vee })}{Q}_l\oplus {\mathbb{Z}}^{\frac{1}{2}\mathsf{dim}{\mathfrak{h}}^{\perp }-\mathsf{dim}\mathfrak{h}}.$$

$$L^{+}:=\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}\alpha^{+},\ \ \langle\alpha^{+},\beta^{+}\big{\rangle}:=\delta_{\alpha,\beta}\ (\alpha,\beta\in\Delta^{+})$$

$$H_{\alpha }:=\frac{(\alpha ,\alpha )}{2}{\alpha }^{\vee }\in \mathfrak{h},$$

$$H_{\alpha }^{+}$$

$${\alpha }_{\mathsf{f}}:=\sum _{\beta \in {\Delta }^{+}}(\alpha ,\beta ){\beta }^{+}\in L^{+}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

$$⟨{\alpha }_{\mathsf{f}},{\beta }_{\mathsf{f}}⟩=\sum _{\gamma \in {\Delta }^{+}}(\alpha ,\gamma )(\gamma ,\beta )=h^{\vee }(\alpha ,\beta )$$

$$V_{\sf sc}:={\sf Com}\bigl{(}\mathcal{H}^{+}\!,V_{\sf af}\otimes V^{+}\bigr{)}$$

$$\mathcal{C}_{k}(\mathfrak{g}):={\sf Com}\bigl{(}\mathcal{H}^{+}\!,L_{k}(\mathfrak{g})\otimes V^{+}\bigr{)}$$

$$\mathcal{C}^{k}(\mathfrak{g}):={\sf Com}\bigl{(}\mathcal{H}^{+}\!,V^{k}(\mathfrak{g})\otimes V^{+}\bigr{)}.$$

$${\iota }_{+}:V_{\mathsf{sc}}\otimes {\mathcal{H}}^{+}\to V_{\mathsf{af}}\otimes V^{+};A\otimes B\mapsto A_{-1}B$$

$$\left\{J_{\alpha }:=\frac{1}{k}{H}_{\alpha ,-1}{1}_{\mathsf{af}}\otimes 1_{V^{+}}+1_{\mathsf{af}}\otimes {\alpha }_{-1}^{+}{1}_{V^{+}}\in (V_{\mathsf{af}}\otimes V^{+}{)}^{\bar{0}}|\alpha \in {\Delta }^{+}\right\}$$

$$\displaystyle J_{\alpha}(z)J_{\beta}(w)\sim\frac{g_{\alpha,\beta}}{(z-w)^{2}}:=\frac{\frac{(\alpha,\beta)}{k}+\delta_{\alpha,\beta}}{(z-w)^{2}},\$$

$$\sum _{\alpha \in {\Delta }^{+}}(\lambda ,\alpha )\alpha =h^{\vee }\lambda$$

$$(g_{\alpha ,\beta }^{\ast }{)}_{\alpha ,\beta \in {\Delta }^{+}}:={\left(-\frac{(\alpha ,\beta )}{k+h^{\vee }}+{\delta }_{\alpha ,\beta }\right)}_{\alpha ,\beta \in {\Delta }^{+}}$$

$$J_{\alpha }^{\ast }:=\sum _{\beta \in {\Delta }^{+}}{g}_{\alpha ,\beta }^{\ast }{J}_{\beta }=\frac{1}{k+h^{\vee }}{H}_{\alpha }^{+}+1_{\mathsf{af}}\otimes {\alpha }_{-1}^{+}{1}_{V^{+}}\in {\mathcal{H}}_{\mathsf{sc}}$$

$$J_{\alpha }^{\ast }(z)J_{\beta }(w)\sim \frac{{\delta }_{\alpha ,\beta }}{(z-w{)}^2},J_{\alpha }^{\ast }(z)J_{\beta }^{\ast }(w)\sim \frac{g_{\alpha ,\beta }^{\ast }}{(z-w{)}^2}$$

$$P_{\mathsf{sc}}^{\vee }:=\underset{\alpha \in {\Delta }^{+}}{⨁}\mathbb{Z}{J}_{\alpha }^{\ast },{\mathfrak{h}}_{\mathsf{sc}}:=\underset{\alpha \in {\Delta }^{+}}{⨁}\mathbb{C}{J}_{\alpha }^{\ast },$$

$$V_{\mathsf{sc}}=\underset{\gamma \in Q_{\mathsf{sc}}}{⨁}(V_{\mathsf{sc}}{)}_{\gamma }$$

$$(V_{\mathsf{sc}}{)}_{\gamma }:=\{A\in V_{\mathsf{sc}}|H_{0}A=\gamma (H)A\text{ for any }H\in {\mathfrak{h}}_{\mathsf{sc}}\}.$$

Full text

1. 1 Introduction
2. 2 Fermionization and Defermionization
3. 3 Main Theorem
4. 4 Proof of Main Theorem
5. 5 Regular cases
6. A Vertex superalgebras and modules
7. B Twisted sector

1. Introduction

Representation theory of vertex operator superalgebras plays a fundamental role in the mathematical study of the corresponding $2$-dimensional conformal field theories, see e.g. [FZ92], [NT05], [Hua08]. When the symmetry of the Virasoro algebra on a given vertex operator superalgebra is extended to that of $\mathbb{Z}/2\mathbb{Z}$-graded generalizations of the Virasoro algebra, known as superconformal algebras (e.g. [KvdL89]), the corresponding theory is naturally expected to provide an example of superconformal field theories. Though such generalizations are originally motivated by purely physical applications (e.g. [NS71], [Ram71]), the superconformal algebra symmetry turns out to be non-trivially related with several areas of mathematics, see e.g. [KRW03], [BZHS08], [EOT11], [Wit12], [KW16].

In this paper, we study the representation theory of a certain specific family of vertex operator superalgebras with the $\mathcal{N}=2$ superconformal algebra symmetry, which is given by the Kazama–Suzuki coset construction [KS89] associated with the pair of a complex simple Lie algebra $\mathfrak{g}$ and its Cartan subalgebra $\mathfrak{h}$. Our main tool to study such vertex operator superalgebras is a generalization of the Feigin–Semikhatov–Tipunin coset construction introduced in [FST98] for $\mathfrak{g}=\mathfrak{sl}_{2}$. Roughly speaking, these two constructions are Heisenberg coset constructions (see [CKLR18] for the general theory and many examples), which are “mutually inverse” to each other:

1. (1)

the Kazama–Suzuki coset construction (“fermionization”)

[TABLE] 2. (2)

the Feigin–Semikhatov–Tipunin coset construction (“defermionization”)

[TABLE]

One of the most important features of the above bidirectional constructions is that they relate the representation theory of $\mathbb{Z}/2\mathbb{Z}$-graded objects to that of purely even111 We say that a $\mathbb{Z}/2\mathbb{Z}$-graded vector space $V=V^{\bar{0}}\oplus V^{\bar{1}}$ is purely even if $V^{\bar{1}}=\{0\}$. Throughout this paper, we regard all the vertex algebras as purely even vertex superalgebras. objects. In fact, we establish a categorical equivalence between $\mathbb{C}$-linear additive categories of certain modules over affine vertex operator algebras and the corresponding $\mathcal{N}=2$ superconformal vertex operator superalgebras.

1.1. Main result

In order to explain the main result, we first give a brief review about the Kazama–Suzuki coset construction for the pair $(\mathfrak{g},\mathfrak{h})$ as above.

Let $k\in\mathbb{C}\setminus\{-h^{\vee}\}$, where $h^{\vee}$ is the dual Coxeter number of $\mathfrak{g}$. We denote by $V^{k}(\mathfrak{g})$ the universal affine vertex operator algebra (see Lemma A.1) associated with the invariant bilinear form $kB$, where $B$ is the normalized symmetric invariant bilinear form on $\mathfrak{g}$. We simply write $V_{\sf af}$ for $V^{k}(\mathfrak{g})$ or its simple quotient $L_{k}(\mathfrak{g})$.

As is described in [HT91, §2], the adjoint $\mathfrak{h}$-actions on $\mathfrak{g}$ and on the orthogonal complement $\mathfrak{h}^{\perp}$ of $\mathfrak{h}$ with respect to $B$ give rise to an injective vertex superalgebra homomorphism (see §2.1 for details)

[TABLE]

where $V^{k+h^{\vee}}(\mathfrak{h})$ is the Heisenberg vertex operator algebra associated with the bilinear form $(k+h^{\vee})B|_{\mathfrak{h}\times\mathfrak{h}}$ and $V^{+}$ is the Clifford vertex operator superalgebra associated with $\mathfrak{h}^{\perp}$ and $B|_{\mathfrak{h}^{\perp}\times\mathfrak{h}^{\perp}}$. In this paper, we write $\mathcal{H}^{+}$ for the image of (1.1) and denote the corresponding commutant (usually referred to as coset) vertex superalgebra by

[TABLE]

As a special case of [HT91, Theorem 2.5] (see also [KS89, §3]), one can see that $V_{\sf sc}$ has the $\mathcal{N}=2$ superconformal algebra symmetry:

Theorem 1.1** ([HT91]).**

The coset vertex superalgebra $V_{\sf sc}$ carries a structure of an $\mathcal{N}=2$ superconformal vertex operator superalgebra222 It is clear that both $\mathcal{C}^{k}(\mathfrak{g})$ and $\mathcal{C}_{k}(\mathfrak{g})$ are of CFT type, that is, every $L_{0}$-eigenvalue is non-negative and the $L_{0}$-eigenspace of eigenvalue [math] is $1$-dimensional. of central charge

[TABLE]

in the sense of [HM02, Definition 1.1] (see also [Ada99, Definition 1.1]).

Next we move on to the Feigin–Semikhatov–Tipunin coset construction. In [FST98], B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin introduced a coset construction of the affine Lie algebra $\widehat{\mathfrak{sl}}_{2}$ from the $\mathcal{N}=2$ superconformal algebra and the lattice vertex superalgebra associated with the negative-definite lattice $\sqrt{-1}\mathbb{Z}$. After a while, D. Adamović reformulated their construction purely in terms of vertex superalgebras in [Ada99, §4]. The key result in this paper is an appropriate generalization of their construction to arbitrary type as follows.

Proposition 1.2** (Proposition 2.11).**

Assume that $k\in\mathbb{C}\setminus\{0,-h^{\vee}\}$. Let $V^{-}$ be the lattice vertex superalgebra associated with the negative-definite integral lattice $\sqrt{-1}\mathbb{Z}^{{\sf dim}\mathfrak{h}}$. Then, for a certain Heisenberg vertex subalgebra $\mathcal{H}^{-}$ of $V_{\sf sc}\otimes V^{-}$ (see Lemma 2.8 for the definition), there exists an explicit isomorphism

[TABLE]

of purely even vertex operator superalgebras.

For each coset $[\lambda]\in\mathfrak{h}^{*}\!/Q$, where $Q$ is the root lattice of $\mathfrak{g}$, we introduce certain $\mathbb{C}$-linear full subcategories $\mathscr{C}_{[\lambda]}(V_{\sf af})$ and $\mathscr{C}_{[\lambda_{\sf sc}]}(V_{\sf sc})$ of weak $V_{\sf af}$-modules and weak $V_{\sf sc}$-modules, respectively (see Definition 3.1 and Proposition 3.5). Now we can state our main result as follows:

Theorem 1.3** (Theorem 3.9).**

Let $k\in\mathbb{C}\setminus\{0,-h^{\vee}\}$ and $\lambda\in\mathfrak{h}^{*}$. Then the following $\mathbb{C}$-linear functors

[TABLE]

defined in §3.2 are mutually quasi-inverse to each other. Moreover, these functors satisfy a certain equivariance property with respect to the spectral flow automorphisms (see §4.3 for details).

We should mention that the above result for $\mathfrak{g}=\mathfrak{sl}_{2}$ is essentially the same as [Sat16, Theorem 4.4, 7.7, and Corollary 6.3] and the original idea in this case goes back to [FST98, Theorem IV. 10]. It is worth noting that, when $\mathfrak{g}\neq\mathfrak{sl}_{2}$, the same proof as [Sat16, Theorem 4.4] does not work due to the existence of a non-trivial lattice vertex subsuperalgebra of $V_{\sf sc}$ (see §3.3). See Remark 4.7 for details.

1.2. Regular cases

It is well-known that the simple affine vertex operator algebra $L_{k}(\mathfrak{g})$ is regular if $k$ is a positive integer (see [DLM97, Theorem 3.7]). In this case, we obtain further information about the simple vertex operator superalgebra

[TABLE]

We first determine the bicommutant of the Heisenberg vertex subalgebra $\mathcal{H}^{+}$ and $\mathcal{H}^{-}$ in $L_{k}(\mathfrak{g})\otimes V^{+}$ and $\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}$, respectively.

Proposition 1.4** (Proposition 5.2 and 5.6).**

Assume that $k$ is a positive integer.

1. (1)

The bicommutant vertex superalgebra

[TABLE]

is purely even and isomorphic to the lattice vertex algebra associated with

[TABLE]

where $Q_{l}$ is the even integral sublattice of the root lattice $Q$ spanned by long roots (cf. [DW11, Proposition 4.11]). 2. (2)

The bicommutant vertex superalgebra

[TABLE]

is isomorphic to the lattice vertex superalgebra associated with

[TABLE]

Next, by using [Miy15, Corollary 2] and [CKLR18, Theorem 4.12] (cf. [CM16, Theorem 5.24]), we prove the following.

Theorem 1.5** (Theorem 5.7).**

When $k$ is a positive integer, both $\mathcal{C}_{k}(\mathfrak{g})$ and its even part $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$ are regular.

At last, we obtain the following theorem as a special case of [CKM17, Theorem 3.65], which is based on the theory of vertex tensor categories developed by Y.-Z. Huang and J. Lepowsky (see [HL95a], [HL95b], [HL95c], [Hua95], [Hua05], and references therein).

Theorem 1.6** (Theorem 5.12).**

When $k$ is a positive integer, the semisimple $\mathbb{C}$-linear abelian category of $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-modules carries a braided monoidal category structure333 Note that the symmetric center (see [Müg03, Definition 2.9]) of this braided monoidal category contains at least two non-isomorphic simple objects, the monoidal unit $\mathcal{C}_{k}(\mathfrak{g})$ and its $\mathbb{Z}/2\mathbb{Z}$-parity reversed object. In particular, it is not modular in the sense of [Müg03, Definition 3.1]. induced by [Hua05, Theorem 3.7] and [CKM17, Theorem 3.65].

We should note that Theorem 1.5 (resp. Theorem 1.6) for $\mathfrak{g}=\mathfrak{sl}_{2}$ is proved by D. Adamović in [Ada01, Theorem 8.1] (resp. by Y.-Z. Huang and A. Milas in [HM02, Theorem 4.8]).

1.3. Structure of the paper

We organize this paper as follows. In §2 we define the coset vertex operator superalgebra $V_{\sf sc}$ and its “Cartan subalgebra” $\mathfrak{h}_{\sf sc}$ in an explicit way. Our main tool, a generalization of the Feigin–Semikhatov–Tipunin coset construction, is introduced in §2.3. In §3, after introducing an appropriate category of modules and two families of functors $\Omega^{+}$ and $\Omega^{-}$, we give the precise statement of our main result. The proof of the main theorem is given in §4. At last, in §5, we discuss the case that the level $k$ is a positive integer.

**Acknowledgments: ** The author would like to express his gratitude to Shun-Jen Cheng, Ching Hung Lam, Masahiko Miyamoto, Hiromichi Yamada, and Hiroshi Yamauchi for helpful discussions and valuable advice. He also would like to express his appreciation to Tomoyuki Arakawa, Kenichiro Tanabe, Shintarou Yanagida, and many others for constructive comments.

2. Fermionization and Defermionization

2.1. Kazama–Suzuki coset construction

In this subsection, we review the vertex superalgebraic formulation of the Kazama-Suzuki coset constructions. See [KS89] and [HT91] for details.

Let $\mathfrak{g}$ be a simple complex Lie algebra with a fixed Borel subalgebra $\mathfrak{b}$. Let $\mathfrak{h}$ be the corresponding Cartan subalgebra and $\Delta$ (resp. $\Delta^{+},\Pi$) the set of (resp. positive, simple) roots associated with $(\mathfrak{g},\mathfrak{b})$. We write $B=B(?,?)\colon\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$ for the normalized symmetric invariant bilinear form on $\mathfrak{g}$.

Define a positive-definite integral lattice $L^{+}$ of rank $N:=|\Delta^{+}|$ by

[TABLE]

and take a bimultiplicative map $\epsilon^{+}\colon L^{+}\times L^{+}\to\{\pm 1\}$ in the same way as $\epsilon\colon\mathbb{Z}^{N}\times\mathbb{Z}^{N}\to\{\pm 1\}$ in [Kac98, Example 5.5a]444 Our discussion below does not depend on the choice of the lattice isomorphism $L^{+}\simeq\mathbb{Z}^{N}$.. Then we write $V^{+}$ for the lattice vertex superalgebra associated with $(L^{+},\epsilon^{+})$, which is isomorphic to the $N$-times tensor product of the charged free fermions. Unless otherwise specified, we follow the standard notation of lattice vertex superalgebras used in [Kac98].

Let $h^{\vee}$ be the dual Coxeter number of $\mathfrak{g}$ and $k\in\mathbb{C}\setminus\{-h^{\vee}\}$. We denote by $V_{\sf af}$ the universal affine vertex operator algebra $V^{k}(\mathfrak{g})$ of level $k$ or its simple quotient vertex operator algebra $L_{k}(\mathfrak{g})$. Based on the setting of the Kazama–Suzuki coset construction, we consider a Heisenberg vertex subalgebra of $V_{\sf af}\otimes V^{+}$ as follows.

Lemma 2.1**.**

For $\alpha\in\Delta$, we set

[TABLE]

where $(?,?)\colon\mathfrak{h}^{*}\times\mathfrak{h}^{*}\to\mathbb{C}$ is the normalized bilinear form induced by the restriction of $B$ to $\mathfrak{h}\times\mathfrak{h}$ and $\alpha^{\vee}\in\mathfrak{h}$ is the coroot of $\alpha$. We also set

[TABLE]

where $\mathbf{1}_{\sf af}$ is the vacuum vector of $V_{\sf af}$ and

[TABLE]

Then the vertex subalgebra $\mathcal{H}^{+}$ of $V_{\sf af}\otimes V^{+}$ generated by the set $\{H^{+}_{\alpha}\,|\,\alpha\in\Pi\}$ is the Heisenberg vertex algebra of rank $\ell:={\sf dim}\,\mathfrak{h}$.

Proof.

Since we have

[TABLE]

for $\alpha,\beta\in\Pi$, there exists a unique vertex algebra isomorphism $V^{k+h^{\vee}}(\mathfrak{h})\to\mathcal{H}^{+}$ such that $H_{\alpha,-1}\mathbf{1}_{\sf af}\mapsto H^{+}_{\alpha}$ for $\alpha\in\Pi$. ∎

Let $\omega^{\sf af}$, $\omega^{\sf f}$, and $\omega^{\sf b}$ be the standard conformal vectors of central charge $c_{\sf af}:=\frac{k\,{\sf dim}\,\mathfrak{g}}{k+h^{\vee}}$, $N$, and $\ell$ of $V_{\sf af}$, $V^{+}$, and $\mathcal{H}^{+}$, respectively (see e.g. [Kac98] for details).

Lemma 2.2**.**

The coset vertex superalgebra

[TABLE]

together with $\omega^{\sf sc}:=\omega^{\sf af}\otimes\mathbf{1}_{V^{+}}+\mathbf{1}_{\sf af}\otimes\omega^{\sf f}-\omega^{\sf b}$ forms a vertex operator superalgebra of central charge $c_{\sf sc}:=c_{\sf af}+N-\ell.$

Proof.

It follows from the general theory of coset vertex superalgebras (see e.g. [FZ92, Theorem 5.1]). ∎

Proposition 2.3**.**

The coset vertex operator superalgebra

[TABLE]

is a unique simple quotient vertex operator superalgebra of

[TABLE]

Proof.

Let $I$ be the maximal proper ideal of $V^{k}(\mathfrak{g})$. Since $V^{k}(\mathfrak{g})\otimes V^{+}$ and $I\otimes V^{+}$ are completely reducible as weak $\mathcal{H}^{+}$-modules by the same argument as in [FLM89, §1.7], it follows that $\mathcal{C}_{k}(\mathfrak{g})$ is a quotient of $\mathcal{C}^{k}(\mathfrak{g})$. The simplicity and uniqueness follow from the same argument of [CKLR18, Proposition 3.2] (see also [ACKL17, Lemma 2.1]) and the fact that $\mathcal{C}^{k}(\mathfrak{g})$ is of CFT type, respectively. ∎

Lemma 2.4**.**

The following homomorphism

[TABLE]

of vertex operator superalgebras is injective.

Proof.

Since $\mathcal{H}^{+}$ is simple and $V_{\sf af}\otimes V^{+}$ is completely reducible as a weak $\mathcal{H}^{+}$-module, the injectivity of $\iota_{+}$ follows. ∎

2.2. Cartan subalgebra

In this subsection, we introduce a Heisenberg vertex subalgebra of $V_{\sf sc}$. From now on, we always assume that $k\in\mathbb{C}\setminus\{0,-h^{\vee}\}$.

Lemma 2.5**.**

The following set

[TABLE]

generates a vertex subalgebra $\mathcal{H}_{\sf sc}$ of $V_{\sf sc}$, which is isomorphic to the Heisenberg vertex algebra of rank $N$.

Proof.

Since it is easy to verify that each $J_{\alpha}$ lies in $V_{\sf sc}$, we only need to check the non-degeneracy. By a direct computation, we have

[TABLE]

for any $\alpha,\beta\in\Delta^{+}$. By using the formula

[TABLE]

for $\lambda\in\mathfrak{h}^{*}$, we can verify that

[TABLE]

is the inverse matrix of $(g_{\alpha,\beta})_{\alpha,\beta\in\Delta^{+}}.$ This completes the proof. ∎

By the inverse matrix in the above proof, we obtain the following.

Corollary 2.6**.**

We set

[TABLE]

for $\alpha\in\Delta^{+}$. Then we have

[TABLE]

for any $\alpha,\beta\in\Delta^{+}$.

Lemma 2.7**.**

We set

[TABLE]

and Q_{\sf sc}:=\big{\{}\lambda\in(\mathfrak{h}_{\sf sc})^{*}\,\big{|}\,\lambda(H)\in\mathbb{Z}\text{ for any }H\in P^{\vee}_{\sf sc}\big{\}}. Then $V_{\sf sc}$ decomposes into a direct sum

[TABLE]

of $\mathfrak{h}_{\sf sc}$-weight subspaces, where

[TABLE]

Proof.

It follows from the fact that the operator $J_{\alpha,0}^{*}$ coincides with $\mathbf{1}_{\sf af}\otimes\,\alpha_{0}^{+}$ on $V_{\sf sc}$ for any $\alpha\in\Delta^{+}$. ∎

2.3. Feigin–Semikhatov–Tipunin coset construction

In [FST98], B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin studied a coset construction of $V^{k}(\mathfrak{sl}_{2})$ in the tensor product of $\mathcal{C}^{k}(\mathfrak{sl}_{2})$ and the lattice vertex superalgebra associated with $\sqrt{-1}\mathbb{Z}$ (see also [Ada99, §5]). In this subsection, we generalize their coset construction to arbitrary type.

We define a negative-definite integral lattice $L^{-}$ of rank $\ell$ by

[TABLE]

and a bimultiplicative map $\epsilon^{-}\colon L^{-}\times L^{-}\to\{\pm 1\}$ by

[TABLE]

for $\alpha,\beta\in\Pi$. We write $V^{-}$ for the lattice vertex superalgebra associated with $(L^{-},\epsilon^{-})$555 One can easily verify that $\epsilon^{-}$ satisfies the condition

$\epsilon^{-}(\zeta,\zeta^{\prime})\epsilon^{-}(\zeta^{\prime},\zeta)=(-1)^{\langle\zeta,\zeta^{\prime}\rangle+\langle\zeta,\zeta\rangle\langle\zeta^{\prime},\zeta^{\prime}\rangle}$

for any $\zeta,\zeta^{\prime}\in L^{-}$ in [Kac98, Theorem 5.5 (b)].. Since the lattice $L^{-}$ is negative-definite, the vertex superalgebra $V^{-}$ has a natural conformal structure of central charge $\ell$ and the set of the $L_{0}$-eigenvalues is not bounded below.

Lemma 2.8**.**

For each $\alpha\in\Delta^{+}$, we set

[TABLE]

where $\mathbf{1}_{\sf sc}$ is the vacuum vector of $V_{\sf sc}$. Then the vertex subalgebra $\mathcal{H}^{-}$ of $V_{\sf sc}\otimes V^{-}$ generated by $\{H^{-}_{\alpha}\,|\,\alpha\in\Delta^{+}\}$ is the Heisenberg vertex algebra of rank $N$.

Proof.

By Corollary 2.6, we have

[TABLE]

for $\alpha,\beta\in\Delta^{+}$. We define an $N\times N$ matrix $(G^{*}_{\alpha,\beta})_{\alpha,\beta\in\Delta^{+}}$ by

[TABLE]

where $(c_{\alpha,\beta})_{\alpha,\beta\in\Pi}$ is the inverse of the symmetrized Cartan matrix \big{(}(\alpha,\beta)\big{)}_{\alpha,\beta\in\Pi} and $\alpha^{*}\in\mathfrak{h}$ is the fundamental coweight defined by $\beta(\alpha^{*})=\delta_{\alpha,\beta}$ for $\beta\in\Pi$. Then we can verify that $(G^{*}_{\alpha,\beta})_{\alpha,\beta\in\Delta^{+}}$ is the inverse matrix of $(G_{\alpha,\beta})_{\alpha,\beta\in\Delta^{+}}$ and thus the latter is non-singular. ∎

Let $Q$ be the root lattice666 The root lattice $Q$ with the normalized bilinear form $(?,?)$ is integral (resp. even) if and only if $\mathfrak{g}$ is not of type $G_{2}$ (resp. is of type $A,D,E$). of $\mathfrak{g}$. We define $\mathbb{Z}$-linear maps $f^{\pm}_{\sf af}\colon Q\to L^{\pm}$ by

[TABLE]

for $\gamma\in Q$. Then we obtain the next lemma as a generalization of the “anti-Kazama–Suzuki mapping” for $\mathfrak{g}=\mathfrak{sl}_{2}$ in [FST98, Lemma III.5] and [Ada99, Lemma 5.1] to arbitrary type.

Lemma 2.9**.**

There exists a unique vertex superalgebra homomorphism

[TABLE]

such that

[TABLE]

for any $\alpha\in\Delta$ and $\beta\in\Pi$, where $X_{\alpha}\in\mathfrak{g}$ is a root vector associated with the root $\alpha$ and normalized by $[X_{\alpha},X_{-\alpha}]=\alpha^{\vee}$.

Proof.

First we prove the above assignment induces a vertex superalgebra homomorphism from $V^{k}(\mathfrak{g})$ to $V^{k}(\mathfrak{g})\otimes V^{+}\otimes V^{-}$. It suffices to show that the above elements in the right-hand side obey the same OPE as in the left-hand side. By the general theory of lattice vertex superalgebras, we have

[TABLE]

for elements $\xi,\xi^{\prime}$ of a general non-degenerate integral lattice, where $\epsilon$ is an appropriate $2$-cocycle of the lattice (see e.g. [Kac98, §5.5]). In particular, since we have $\langle f^{+}_{\sf af}(\gamma),f^{+}_{\sf af}(\gamma^{\prime})\rangle+\langle f^{-}_{\sf af}(\gamma),f^{-}_{\sf af}(\gamma^{\prime})\rangle=0$ for any $\gamma,\gamma^{\prime}\in Q$, we obtain

[TABLE]

Then, by (2.3), (2.4), and some computations, we obtain

[TABLE]

for any $\alpha,\beta\in\Delta$. In addition, we also obtain

[TABLE]

Thus we get a homomorphism from $V^{k}(\mathfrak{g})$ to $V^{k}(\mathfrak{g})\otimes V^{+}\otimes V^{-}$.

Next we need to prove that the image of the homomorphism is actually contained in the commutant of $\mathcal{H}^{-}$ in $\mathcal{C}^{k}(\mathfrak{g})\otimes V^{-}$. It suffices to check that $(H_{\alpha}^{+}(z)\otimes\operatorname{\sf id}_{V^{-}})\widetilde{X}_{\beta}(w)\sim 0$ and $H_{\alpha}^{-}(z)\widetilde{X}_{\beta}(w)\sim 0$ for any $\alpha\in\Delta^{+}$ and $\beta\in\Delta$. Since they are verified by straightforward computations, we omit the detail. ∎

Corollary 2.10**.**

The homomorphism (2.2) gives rise to conformal vertex superalgebra777 Note that $V^{-}$ is not a vertex operator superalgebra. homomorphisms $\iota_{-}\colon V^{k}(\mathfrak{g})\otimes\mathcal{H}^{-}\to\mathcal{C}^{k}(\mathfrak{g})\otimes V^{-}$ and $\iota_{\sf af}\colon V^{k}(\mathfrak{g})\otimes\mathcal{H}^{+}\otimes\mathcal{H}^{-}\to V^{k}(\mathfrak{g})\otimes V^{+}\otimes V^{-}.$

As a matter of fact, the following stronger statement holds.

Proposition 2.11**.**

The homomorphism (2.2) is an isomorphism. In particular, it descends to their simple quotients and gives rise to conformal vertex superalgebra homomorphisms $\iota_{-}\colon L_{k}(\mathfrak{g})\otimes\mathcal{H}^{-}\to\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}$ and $\iota_{\sf af}\colon L_{k}(\mathfrak{g})\otimes\mathcal{H}^{+}\otimes\mathcal{H}^{-}\to L_{k}(\mathfrak{g})\otimes V^{+}\otimes V^{-}.$

The proof of Proposition 2.11 is postponed to Corollary 4.5.

3. Main Theorem

In this section, by using the notation defined in the previous section, we give the precise statement of our main result (Theorem 3.9). Recall that the pair $(V_{\sf af},V_{\sf sc})$ stands for \big{(}V^{k}(\mathfrak{g}),\mathcal{C}^{k}(\mathfrak{g})\big{)} or \big{(}L_{k}(\mathfrak{g}),\mathcal{C}_{k}(\mathfrak{g})\big{)}. Throughout this section, we assume that $k\in\mathbb{C}\setminus\{0,-h^{\vee}\}$ and the letter $V$ always stands for $V_{\sf af}$ or $V_{\sf sc}$.

3.1. Setting of the category of modules

For $V=V_{\sf af}$ (resp. $V_{\sf sc}$), we set

[TABLE]

where $P^{\vee}$ is the coweight lattice in $\mathfrak{h}$.

Definition 3.1**.**

Let $\lambda\in\mathfrak{t}_{V}^{*}$ and set

[TABLE]

Then we define $\mathscr{C}_{[\lambda]}(V)$ to be a full subcategory of $V\text{-\sf gMod}$ (see §A.3 for the definition) whose object $M=M^{\bar{0}}\oplus M^{\bar{1}}$ satisfies the following conditions:

1. (1)

For each $\bar{i}\in\mathbb{Z}/2\mathbb{Z}$, the $\mathbb{Z}/2\mathbb{Z}$-homogeneous subspace $M^{\bar{i}}$ decomposes into a direct sum

[TABLE]

of $\mathfrak{t}_{V}$-weight spaces, where

[TABLE] 2. (2)

For each $\bar{i}\in\mathbb{Z}/2\mathbb{Z}$ and $\mu\in[\lambda]$, the $\mathfrak{t}_{V}$-weight space $M_{\mu}^{\bar{i}}$ further decomposes into a direct sum

[TABLE]

of finite-dimensional subspaces, where

[TABLE]

In addition, for any $h\in\mathbb{C}$, we have $M_{\mu}^{\bar{i}}(h-r)=\{0\}$ if $r\gg 0$.

We write $M_{\mu}$ and $M_{\mu}(h)$ for $M_{\mu}^{\bar{0}}\oplus M_{\mu}^{\bar{1}}$ and $M_{\mu}^{\bar{0}}(h)\oplus M_{\mu}^{\bar{1}}(h)$, respectively.

Definition 3.2**.**

For an object $M$ of $\mathscr{C}_{[\lambda]}(V)$, we define the string function of $M$ through $\mu\in[\lambda]$ by

[TABLE]

and the formal character of $M$ by

[TABLE]

3.2. Definition of the functors $\Omega^{+}$ and $\Omega^{-}$

In this subsection, we assume Proposition 2.11 whenever we consider the case of $V=\mathcal{C}_{k}(\mathfrak{g})$. Note that the other cases are proved independently of Proposition 2.11.

We set

[TABLE]

From now on, we fix the following linear isomorphisms

[TABLE]

and denote the induced linear isomorphisms by

[TABLE]

Definition 3.3**.**

Let $\lambda,\mu\in\mathfrak{t}_{V}^{*}$ and $M$ be an object of $\mathscr{C}_{[\lambda]}(V)$.

1. (1)

When $V=V_{\sf af}$, we define a $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf sc}$-module by

[TABLE] 2. (2)

When $V=V_{\sf sc}$, we define a $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf af}$-module888 When $V=\mathcal{C}_{k}(\mathfrak{g})$, we use Proposition 2.11. by

[TABLE]

For a morphism $f\colon M^{1}\to M^{2}$ of $\mathscr{C}_{[\lambda]}(V)$, we define a linear map

[TABLE]

by the restriction of $f\otimes{\sf id}_{V^{\pm}}$ to the subspace $\Omega^{\pm}(\mu)(M^{1})$ of $M^{1}\otimes V^{\pm}$.

We denote the purely even Heisenberg Fock $\mathcal{H}^{\pm}$-module of highest weight $\mu^{\pm}\in(\mathfrak{t}^{\pm})^{*}$ by $\mathcal{H}^{\pm}_{\mu^{\pm}}$ and its even highest weight vector by $\ket{\mu^{\pm}}$.

Lemma 3.4**.**

For $V=V_{\sf af}$, there exists a unique isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf sc}\otimes\mathcal{H}^{+}$-modules such that $v\otimes\ket{\nu^{+}(\mu)}\mapsto v$ for any $v\in\Omega^{+}(\mu)(M)$. Similarly, for $V=V_{\sf sc}$, there exists a unique isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf af}\otimes\mathcal{H}^{-}$-modules such that $v\otimes\ket{\nu^{-}(\mu)}\mapsto v$ for any $v\in\Omega^{-}(\mu)(M)$.

Proof.

Since $\iota_{\pm}^{*}(M\otimes V^{\pm})$ is completely reducible as a $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{H}^{\pm}$-module by the same argument as in [FLM89, §1.7] and $\Omega^{\pm}(\mu)(M)=\{0\}$ holds for any $\mu\in\mathfrak{t}_{V}^{*}\setminus[\lambda]$, we get the above isomorphisms. ∎

Proposition 3.5**.**

We have the following.

1. (1)

Let $\lambda\in\mathfrak{h}^{*}$ and $\mu\in[\lambda]$. Then the assignment $\Omega^{+}(\mu)$ gives rise to a $\mathbb{C}$-linear functor

[TABLE]

where $\mu_{\sf sc}\in(\mathfrak{h}_{\sf sc})^{*}$ is defined by

[TABLE]

for $\alpha\in\Delta^{+}$. 2. (2)

Let $\lambda\in(\mathfrak{h}_{\sf sc})^{*}$ and $\mu\in[\lambda]$. Then the assignment $\Omega^{-}(\mu)$ gives rise to a $\mathbb{C}$-linear functor

[TABLE]

where $\mu_{\sf af}\in\mathfrak{h}^{*}$ is defined by

[TABLE]

for any $\alpha\in\Pi$.

Proof.

Since the proof of (1) and that of (2) are simliar, we only verify (1) here. Let $M$ be an object of $\mathscr{C}_{[\lambda]}(V_{\sf af})$. As the functoriality is obvious, it suffices to show that $\Omega^{+}(\mu)(M)$ is an object of $\mathscr{C}_{[\mu_{\sf sc}]}(V_{\sf sc})$. By the definition of $\Omega^{+}(\mu)(M)$, the operator $H_{\alpha,0}^{+}$ acts on $\Omega^{+}(\mu)(M)$ as scalar $\nu^{+}(\mu)(H_{\alpha}^{+})=\mu(H_{\alpha})$. Therefore $J_{\alpha,0}$ acts on $\Omega^{+}(\mu)(M)$ as scalar $\mu_{\sf sc}(J_{\alpha})$ modulo $\mathbb{Z}$ and every $\mathfrak{h}_{\sf sc}$-weight of $\Omega^{+}(\mu)(M)$ lies in the coset $[\mu_{\sf sc}]$. The other conditions of $\mathscr{C}_{[\mu_{\sf sc}]}(V_{\sf sc})$ are easily verified by the corresponding conditions of $\mathscr{C}_{[\lambda]}(V_{\sf af})$. This completes the proof. ∎

The next lemma is obvious by the definition.

Lemma 3.6**.**

For any $\lambda\in\mathfrak{h}^{*}$, we have $(\lambda_{\sf sc})_{\sf af}=\lambda$.

3.3. The sublattice $K$

We define a $\mathbb{Z}$-linear map $g_{\sf af}^{+}\colon L^{+}\to Q$ by

[TABLE]

for $\xi\in L^{+}$ and regard its kernel $K$ as a positive-definite sublattice of $L^{+}$. Since we have $g_{\sf af}^{+}\circ f^{+}_{\sf af}={\sf id}_{Q}$, it is easy to verify that the set

[TABLE]

forms a $\mathbb{Z}$-basis of $K$.

Lemma 3.7**.**

Let $V_{K}$ be the lattice vertex superalgebra associated with the lattice $K$. We identify $V_{K}$ with the vertex subsuperalgebra of $V_{\sf af}\otimes V^{+}$ generated by $\{\mathbf{1}_{\sf af}\otimes\mathrm{e}^{\xi}\,|\,\xi\in K\}$. Then $V_{K}$ is contained in $V_{\sf sc}$. In addition, we have

[TABLE]

for any $\alpha\in\Delta^{+}$.

Proof.

Since $\langle\alpha_{\sf f},\xi(\beta)\rangle=0$ holds for any $\alpha\in\Pi$ and $\beta\in\Delta^{+}\setminus\Pi$, the set of generators is contained in $V_{\sf sc}$. The equality (3.4) follows from

[TABLE]

∎

With the help of the vertex subsuperalgebra $V_{K}$ of $V_{\sf sc}$, we obtain the ‘converse’ of Lemma 3.6 as follows.

Proposition 3.8**.**

Let $\lambda\in(\mathfrak{h}_{\sf sc})^{*}$. Assume that $\mathscr{C}_{[\lambda]}(V_{\sf sc})$ contains at least one non-zero object. Then $[(\mu_{\sf af})_{\sf sc}]=[\lambda]$ holds for any $\mu\in[\lambda]$.

Proof.

Take $\mu\in[\lambda]$. When $\mathfrak{g}=\mathfrak{sl}_{2}$, we have $(\mu_{\sf af})_{\sf sc}=\mu$ and $[(\mu_{\sf af})_{\sf sc}]=[\mu]=[\lambda]$. Now we assume that $\mathfrak{g}\neq\mathfrak{sl}_{2}$. It suffices to show that $\mu(J_{\alpha})-(\mu_{\sf af})_{\sf sc}(J_{\alpha})$ lies in $\mathbb{Z}$ for any $\alpha\in\Delta^{+}$. Since $K$ is non-trivial and $V_{K}$ is regular (see [DLM97, Theorem 3.16]), any non-zero object of $\mathscr{C}_{[\lambda]}(V_{\sf sc})$ decomposes into a non-empty direct sum of simple ordinary $V_{K}$-modules. Therefore we have \mu\big{(}\widetilde{\xi}(\alpha)\big{)}\in\mathbb{Z} for any $\alpha\in\Delta^{+}$. On the other hand, by the definition of $(\mu_{\sf af})_{\sf sc}$ and (3.4), we also have (\mu_{\sf af})_{\sf sc}\big{(}\widetilde{\xi}(\alpha)\big{)}=0. Then, by using (3.4) and $\mu(J_{\beta})=(\mu_{\sf af})_{\sf sc}(J_{\beta})$ for any $\beta\in\Pi$, we obtain $\mu(J_{\alpha})-(\mu_{\sf af})_{\sf sc}(J_{\alpha})\in\mathbb{Z}$ for any $\alpha\in\Delta^{+}$. We thus complete the proof. ∎

3.4. Main Theorem

Our main result is as follows:

Theorem 3.9**.**

Let $k\in\mathbb{C}\setminus\{0,-h^{\vee}\}$.

1. (1)

For any $[\lambda]\in\mathfrak{h}^{*}/Q$ and $\mu\in[\lambda]$, the following $\mathbb{C}$-linear functors

[TABLE]

are mutually quasi-inverse to each other. 2. (2)

For any $\lambda\in\mathfrak{h}^{*}$ and $\gamma\in Q$, there exist an isomorphism

[TABLE]

of categories and an isomorphism

[TABLE]

of functors from $\mathscr{C}_{[\lambda]}(V_{\sf af})$ to $\mathscr{C}_{[(\lambda-\gamma)_{\sf sc}]}(V_{\sf sc})$. 3. (3)

For any $\lambda\in(\mathfrak{h}_{\sf sc})^{*}$ and $\gamma\in Q_{\sf sc}$, there exist an isomorphism

[TABLE]

of categories and an isomorphism

[TABLE]

of functors from $\mathscr{C}_{[\lambda]}(V_{\sf sc})$ to $\mathscr{C}_{[(\lambda+\gamma)_{\sf af}]}(V_{\sf af})$.

Remark 3.10**.**

Following the notation and terminology of [BE17], we give a comment on supercategory structures. Since the category $V\text{-\sf gMod}$ is a $\mathbb{C}$-linear supercategory (see Lemma A.2 for the definition), so is the full subcategory $\mathscr{C}_{[\lambda]}(V)$. It is clear by the definition that $\Omega^{+}(\mu)$ and $\Omega^{-}(\mu_{\sf sc})$ are superfunctors in the sense of [BE17, Definition 1.1]. In addition, by the proof in the next section (see Remark 4.4 and 4.12), the underlying categories $\underline{\mathscr{C}_{[\lambda]}(V_{\sf af})}$ and $\underline{\mathscr{C}_{[\mu_{\sf sc}]}(V_{\sf sc})}$ in the sense of [BE17, Definition 1.1] are also categorical equivalent to each other.

4. Proof of Main Theorem

In this subsection, we fix $\lambda\in\mathfrak{h}^{*}$ and $\mu\in[\lambda]$.

4.1. Faithfulness of $\Omega^{+}(\mu)$

In this subsection, without using Proposition 2.11, we prove that

[TABLE]

is naturally isomorphic to the identity functor. As corollaries, we obtain the proof of Proposition 2.11 and the faithfulness of $\Omega^{+}(\mu)$.

4.1.1. Character formula

Similarly to (3.3), we define $\mathbb{Z}$-linear maps $g_{\sf af}^{-}\colon L^{-}\to Q$ and $g_{\sf sc}^{\pm}\colon L^{\pm}\to Q_{\mathcal{C}^{k}(\mathfrak{g})}$ by

[TABLE]

for $\xi\in L^{+}$ and $\zeta\in L^{-}$.

Proposition 4.1**.**

Let $M$ be an object of \mathscr{C}_{[\lambda]}\big{(}V^{k}(\mathfrak{g})\big{)}. Then we have

[TABLE]

and

[TABLE]

where $\Delta_{\mu}^{+}:=\frac{1}{k+h^{\vee}}\frac{(\mu,\mu)}{2}$ is the lowest $L_{0}^{\sf b}$-eigenvalue of $\mathcal{H}^{+}_{\nu^{+}(\mu)}$.

Proof.

We first prove (4.1). One can verify that the restriction of the isomorphism (3.1) for $V=V^{k}(\mathfrak{g})$ gives an even linear isomorphism

[TABLE]

where V^{+}_{\xi}:=\big{\{}v\in V^{+}\,\big{|}\,\xi^{\prime}_{0}v=\langle\xi,\xi^{\prime}\rangle v\text{ for any }\xi^{\prime}\in L^{+}\big{\}}. It follows by direct calculation that $M_{\mu+g_{\sf af}^{+}(\xi)}\otimes V^{+}_{\xi}$ is the $\mathfrak{t}_{\mathcal{C}^{k}(\mathfrak{g})}$-eigenspace of eigenvalue $\mu_{\sf sc}+g_{\sf sc}^{+}(\xi)$, the $(L^{\sf af}_{0}\otimes\operatorname{\sf id}+\operatorname{\sf id}\otimes L^{\sf f}_{0})$-graded dimension of $M_{\mu+g_{\sf af}^{+}(\xi)}\otimes V^{+}_{\xi}$ is given by $s^{\mu+g_{\sf af}^{+}(\xi)}_{M}(q)q^{\frac{1}{2}\langle\xi,\xi\rangle}\eta(q)^{-N}$, and the $L^{\sf b}_{0}$-graded dimension of $\mathcal{H}^{+}_{\nu^{+}(\mu)}$ is given by $q^{\Delta_{\mu}^{+}}\eta(q)^{-\ell}$. By combining them, we obtain the required formula (4.1).

Next we verify (4.2). For an arbitrary object $\mathcal{M}$ in \mathscr{C}_{[\mu_{\sf sc}]}\big{(}\mathcal{C}^{k}(\mathfrak{g})\big{)}, in the same way as above, we can verify that the restriction of the isomorphism (3.2) for $V=\mathcal{C}^{k}(\mathfrak{g})$ gives

[TABLE]

and

[TABLE]

where V^{-}_{\zeta}:=\big{\{}v\in V^{-}\,\big{|}\,\zeta^{\prime}_{0}v=\langle\zeta,\zeta^{\prime}\rangle v\text{ for any }\zeta^{\prime}\in L^{-}\big{\}}. On the other hand, by some computation, we obtain

[TABLE]

for any $\gamma\in Q$, where

[TABLE]

Then, by using (4.3) for $\mathcal{M}=\Omega^{+}(\mu)(M)$ together with (4.1) and (4.4), we can derive the character formula (4.2). ∎

4.1.2. Twisted embedding

Let $M$ be an object of \mathscr{C}_{[\lambda]}\big{(}V^{k}(\mathfrak{g})\big{)}. We define an even linear operator $\mathcal{H}$ on $M\otimes V^{+}\otimes V^{-}$ by

[TABLE]

Then, by the condition (2) of Definition 3.1, the formal sum ${\sf exp}(\mathcal{H})$ defines an even linear automorphism on $M\otimes V^{+}\otimes V^{-}$.

Lemma 4.2**.**

The assignment

[TABLE]

for $v\in M_{\mu+\gamma}$ and $\gamma\in Q$ uniquely extends to an injective morphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})\otimes\mathcal{H}^{+}\otimes\mathcal{H}^{-}$-modules.

Proof.

First we prove that the assignment (4.5) defines a $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})$-module homomorphism from $M\otimes\ket{\nu^{+}(\mu)}\otimes\ket{\nu^{-}(\mu_{\sf sc})}$ to $\iota_{\sf af}^{*}(M\otimes V^{+}\otimes V^{-}).$ It suffices to prove that

[TABLE]

for any $\alpha\in\Pi\sqcup(-\Pi)$, $X_{\alpha}\in\mathfrak{g}_{\alpha}$, and $m\in\mathbb{Z}$. By direct calculation, we obtain

[TABLE]

for any $\alpha,\beta\in\Pi$ and $p,q,r\in\mathbb{Z}$. By using them, for $\alpha\in\Pi$, we have

[TABLE]

where

[TABLE]

It is clear that $\Phi$ lies in $V^{k}(\mathfrak{g})\otimes V^{+}_{f^{+}_{\sf af}(\gamma+\alpha)}\otimes V^{-}_{f^{-}_{\sf af}(\gamma+\alpha)}$. By straightforward computations, we have $(\operatorname{\sf id}\otimes(\alpha^{\prime})^{+}_{n}\otimes\operatorname{\sf id})\Phi=(\operatorname{\sf id}\otimes\operatorname{\sf id}\otimes(\alpha^{\prime\prime})^{-}_{n})\Phi=0$ for any $\alpha^{\prime}\in\Delta^{+}$, $\alpha^{\prime\prime}\in\Pi$, and $n\in\mathbb{Z}_{>0}$. Therefore, by the uniquness of singular vector in the Heisenberg Fock module $V^{+}_{f^{+}_{\sf af}(\gamma+\alpha)}\otimes V^{-}_{f^{-}_{\sf af}(\gamma+\alpha)}$, there exists $\Phi_{\sf af}\in V^{k}(\mathfrak{g})$ such that $\Phi=\Phi_{\sf af}\otimes e^{f^{+}_{\sf af}(\gamma+\alpha)}\otimes e^{f^{-}_{\sf af}(\gamma+\alpha)}.$ Then, by the explicit form of $\Psi$, we conclude that $\Phi_{\sf af}=X_{\alpha,m}v$. This proves the formula (4.7) for $\alpha\in\Pi$. Since the proof for $\alpha\in-\Pi$ is the same, we omit it.

Next we prove that the assignment (4.5) uniquely extends to (4.6). By a direct computation, we obtain

[TABLE]

for $\alpha\in\Pi$ and $\beta\in\Delta^{+}$. Hence (4.5) uniquely extends to (4.6).

At last, as the injectivity of (4.5) follows from the bijectivity of the operator ${\sf exp}(\mathcal{H})$ on $M\otimes V^{+}\otimes V^{-}$, the induced homomorphism (4.6) is also injective. ∎

4.1.3. Natural isomorphism

Proposition 4.3**.**

Let $M$ be an object of \mathscr{C}_{[\lambda]}\big{(}V^{k}(\mathfrak{g})\big{)}. Then the injective homomorphism (4.6) gives rise to an isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})$-modules, which is natural in $M$.

Proof.

First we prove that $\mathscr{F}^{\mu}_{M}$ is a $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})$-module isomorphism. Since $\mathscr{F}^{\mu}_{M}$ is a well-defined injective morphism of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})$-modules by Lemma 4.2, it suffices to show that $\mathscr{F}^{\mu}_{M}$ is surjective. The surjectivity of $\mathscr{F}^{\mu}_{M}$ follows from its injectivity and the character formula (4.2).

Next we prove that $\mathscr{F}^{\mu}_{M}$ is natural in $M$. Let $f\colon M^{1}\to M^{2}$ be a morphism of \mathscr{C}_{[\lambda]}\big{(}V^{k}(\mathfrak{g})\big{)}. By using the explicit form (4.5) of $\mathscr{F}^{\mu}_{M}$, we have

[TABLE]

for any $v\in M^{1}$. We thus complete the proof. ∎

Remark 4.4**.**

It is clear by the proof that the above natural isomorphism is an even supernatural transformation in the sense of [BE17, Definition 1.1].

Now we give the proof of Proposition 2.11.

Corollary 4.5**.**

Proposition 2.11 holds.

Proof.

Since $f_{\sf FST}=\mathscr{F}^{0}_{V^{k}(\mathfrak{g})}$, it follows from Proposition 4.3. ∎

By using Proposition 2.11, we obtain the following.

Proposition 4.6**.**

The composed functor $\Omega^{-}(\mu_{\sf sc})\circ\Omega^{+}(\mu)\colon\mathscr{C}_{[\lambda]}(V_{\sf af})\to\mathscr{C}_{[\lambda]}(V_{\sf af})$ is naturally isomorphic to the identity functor of $\mathscr{C}_{[\lambda]}(V_{\sf af})$. In particular, the functor $\Omega^{+}(\mu)\colon\mathscr{C}_{[\lambda]}(V_{\sf af})\to\mathscr{C}_{[\mu_{\sf sc}]}(V_{\sf sc})$ is faithful.

Proof.

By Proposition 2.11, we can verify that Proposition 4.3 remains true if we replace $V^{k}(\mathfrak{g})$ by $L_{k}(\mathfrak{g})$. Therefore (4.8) gives a desired natural isomorphism. ∎

Remark 4.7**.**

When $\mathfrak{g}=\mathfrak{sl}_{2}$, one can describe an explicit natural isomorphism from the identity functor of $\mathscr{C}_{\mu_{\sf sc}}(V_{\sf sc})$ to $\Omega^{+}(\mu)\circ\Omega^{-}(\mu_{\sf sc})$ in the same way as [Sat16, §5.3]. On the other hand, when $\mathfrak{g}\neq\mathfrak{sl}_{2}$, there exist certain “inner automorphisms” of $V_{\sf sc}$ induced by the lattice $K$ (see Lemma 4.16) and the same argument as [Sat16, §5.3] is not applicable.

4.2. Essentially surjectivity and fullness of $\Omega^{+}(\mu)$

In this subsection, we prove that $\Omega^{+}(\mu)\colon\mathscr{C}_{[\lambda]}(V_{\sf af})\to\mathscr{C}_{[\mu_{\sf sc}]}(V_{\sf sc})$ is essentially surjective and full.

We first recall the theory of Li’s $\Delta$-operators developed in [Li97]. Let $(\mathcal{V}=\mathcal{V}^{\bar{0}}\oplus\mathcal{V}^{\bar{1}},Y,\mathbf{1},\omega)$ be a conformal vertex superalgebra. We use the following notation introduced in [Li97, (3.4)]:

[TABLE]

for $h\in\mathcal{V}^{\bar{0}}$.

Lemma 4.8** ([Li97]).**

Let $(\mathcal{M},Y_{\mathcal{M}})$ be a $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{V}$-module. We assume that there exist $h\in\mathcal{V}^{\bar{0}}$ and $c\in\mathbb{C}\setminus\{0\}$ satisfying the following conditions:

1. (1)

$L_{n}h=\delta_{n,0}h$ holds for any $n\geq 0$, 2. (2)

$h_{n}h=\delta_{n,1}c\mathbf{1}$ holds for any $n\geq 0$, 3. (3)

the operator $h_{0}$ on $\mathcal{V}$ is diagonalizable and has only integral eigenvalues, 4. (4)

$E^{+}(-h,-z)v\in\mathcal{V}[z,z^{-1}]$ holds for any $v\in\mathcal{V}$.

Then Li’s $\Delta$-operator

[TABLE]

lies in the group $G^{0}(\mathcal{V})$ introduced in [Li97, §2]. In particular, we can twist the $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{V}$-module structure $Y_{\mathcal{M}}$ on $\mathcal{M}$ to obtain a new $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{V}$-module $\mathcal{M}^{h}:=(\mathcal{M},Y_{\mathcal{M}^{h}})$ defined by

[TABLE]

Proof.

See [Li97, Proposition 2.1 and 3.2] for the proof. ∎

We now focus on the case of $\mathcal{V}=\mathcal{C}^{k}(\mathfrak{g})\otimes\mathcal{H}^{+}$.

Lemma 4.9**.**

Let $\mathcal{M}$ be an object of \mathscr{C}_{[\mu_{\sf sc}]}\big{(}\mathcal{C}^{k}(\mathfrak{g})\big{)}. We consider the direct sum

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{C}^{k}(\mathfrak{g})\otimes\mathcal{H}^{+}$-modules, where $\boldsymbol{\Pi}$ is the $\mathbb{Z}/2\mathbb{Z}$-parity reversing functor and

[TABLE]

Then the $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{C}^{k}(\mathfrak{g})\otimes\mathcal{H}^{+}$-module structure on $\widetilde{\mathcal{M}}$ can be extended to a $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})\otimes V^{+}$-module structure on $\widetilde{\mathcal{M}}$.

Proof.

First we extend the $\mathbb{Z}/2\mathbb{Z}$-graded action of $V_{K}$ on $\widetilde{\mathcal{M}}$ to that of $V^{+}$. Since we have \iota_{+}\big{(}h(\gamma)\big{)}=\mathbf{1}_{\sf af}\otimes f^{+}_{\sf af}(\gamma), the action of the Heisenberg vertex algebra generated by \big{\{}h(\alpha)\,\big{|}\,\alpha\in\Pi\big{\}} on $\widetilde{\mathcal{M}}$ can be extended to that of $V_{f^{+}_{\sf af}(Q)}$ along with the $2$-cocycle $\epsilon^{+}$ (cf. [LL12, Theorem 6.5.18]). More precisely, we define a family of $\mathbb{Z}/2\mathbb{Z}$-homogeneous linear automorphisms $\{\mathbf{S}^{h(\gamma)}\,|\,\gamma\in Q\}$ on $\widetilde{\mathcal{M}}$ by

[TABLE]

and then the following $\mathbb{Z}/2\mathbb{Z}$-homogeneous mutually local fields999 The $\mathbb{Z}/2\mathbb{Z}$-parity of $\mathrm{e}^{f^{+}_{\sf af}(\gamma)}(z)$ is given by $\langle f^{+}_{\sf af}(\gamma),f^{+}_{\sf af}(\gamma)\rangle\ {\sf mod}\ 2\mathbb{Z}$.

[TABLE]

on $\widetilde{\mathcal{M}}$ generate the $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{f^{+}_{\sf af}(Q)}$-module structure. On the other hand, by direct calculation, we have

[TABLE]

for any $\gamma\in Q$. By the general theory of regular lattice vertex superalgebra, there exists a family of $\mathbb{Z}/2\mathbb{Z}$-homogeneous linear automorphisms $\{\mathbf{S}^{\xi}\,|\,\xi\in K\}$ of $\mathcal{M}$ such that

[TABLE]

for any $\xi\in K$. Then, by using the explicit description (4.10) and (4.11), one can verify that the following $\mathbb{Z}/2\mathbb{Z}$-homogeneous fields

[TABLE]

are mutually local and generate the $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{+}$-module structure on $\widetilde{\mathcal{M}}$ (cf. [Li96, Theorem 3.2.10]).

Next, we further extend the above $V^{+}$-action on $\widetilde{\mathcal{M}}$ to a $V^{k}(\mathfrak{g})\otimes V^{+}$-action which is compatible with the original $\mathcal{C}^{k}(\mathfrak{g})\otimes\mathcal{H}^{+}$-action101010 Since $\mathcal{C}^{k}(\mathfrak{g})$ and $V^{+}\simeq\mathbf{1}_{\sf af}\otimes V^{+}$ generate the whole $V^{k}(\mathfrak{g})\otimes V^{+}$, such an extension is unique if it exists.. By using the residue product of mutually local fields (see e.g. [Kac98]), we define the following mutually local even fields:

[TABLE]

for $\alpha\in\Delta$, where $\Psi_{\alpha}:=X_{\alpha,-1}\mathbf{1}_{\sf af}\otimes\mathrm{e}^{f^{+}_{\sf af}(\alpha)}\otimes\mathbf{1}^{+}$. Then, by using the Borcherds identity for mutually local fields, we can verify that the set of mutually local fields \big{\{}X_{\alpha}(z),\,H_{\alpha}(z)\,\big{|}\,\alpha\in\Delta\big{\}} generates a $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})$-module structure on $\widetilde{\mathcal{M}}$, which commutes with the $V^{+}$-action on $\widetilde{\mathcal{M}}$. ∎

Since $L^{+}\simeq\mathbb{Z}^{N}$ is a positive-definite unimodular lattice, the canonical pairing

[TABLE]

gives an isomorphism of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})\otimes V^{+}$-modules.

Lemma 4.10**.**

Let $\mathcal{M}$ and $\widetilde{\mathcal{M}}$ be as in Lemma 4.9. Then the $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})$-module ${\sf Hom}_{V^{+}\text{-\sf Mod}}(V^{+},\widetilde{\mathcal{M}})$ is an object of \mathscr{C}_{[\lambda]}\big{(}V^{k}(\mathfrak{g})\big{)}.

Proof.

Since $\mathcal{M}$ is an object of \mathscr{C}_{[\mu_{\sf sc}]}\big{(}\mathcal{C}^{k}(\mathfrak{g})\big{)}, by the same computation as in Proposition 4.1, we can verify that every $\mathfrak{h}$-weight of ${\sf Hom}_{V^{+}\text{-\sf Mod}}(V^{+},\widetilde{\mathcal{M}})$ lies in $[\lambda]$ and each string function of ${\sf Hom}_{V^{+}\text{-\sf Mod}}(V^{+},\widetilde{\mathcal{M}})$ has the lowest exponent in $q$. Thus ${\sf Hom}_{V^{+}\text{-\sf Mod}}(V^{+},\widetilde{\mathcal{M}})$ is an object of \mathscr{C}_{[\lambda]}\big{(}V^{k}(\mathfrak{g})\big{)}. ∎

Proposition 4.11**.**

The functor $\Omega^{+}(\mu)\colon\mathscr{C}_{[\lambda]}(V_{\sf af})\to\mathscr{C}_{[\mu_{\sf sc}]}(V_{\sf sc})$ is essentially surjective and full.

Proof.

By Proposition 4.6, it suffices to prove the case of (V_{\sf af},V_{\sf sc})=\big{(}V^{k}(\mathfrak{g}),\mathcal{C}^{k}(\mathfrak{g})\big{)}.

We first prove that $\Omega^{+}(\mu)$ is essentially surjective. It suffices to show that $\mathcal{M}$ is isomorphic to $\Omega^{+}(\mu)(M)$ for $M:={\sf Hom}_{V^{+}\text{-\sf Mod}}(V^{+},\widetilde{\mathcal{M}})$ as a $\mathbb{Z}/2\mathbb{Z}$-graded $\mathcal{C}^{k}(\mathfrak{g})$-module. By using Lemma 4.8, we obtain a $\mathbb{Z}/2\mathbb{Z}$-graded $\mathcal{H}^{+}$-module isomorphism

[TABLE]

for $\alpha\in\Pi$. Hence we can rewrite the $\mathbb{Z}/2\mathbb{Z}$-graded decomposition (4.9) as

[TABLE]

where $h_{\sf sc}[\gamma]:=\sum_{\alpha\in\Pi}\gamma(\alpha^{*})J_{\alpha}^{*}$. Therefore, by (3.1) and Schur’s lemma, we conclude that $\mathcal{M}\simeq\Omega^{+}(\mu)(M).$

Next we prove that $\Omega^{+}(\mu)$ is full. Let $\mathcal{M}^{i}$ be an object of \mathscr{C}_{[\mu_{\sf sc}]}\big{(}\mathcal{C}^{k}(\mathfrak{g})\big{)} and $\widetilde{\mathcal{M}}^{i}$ the extended $\mathbb{Z}/2\mathbb{Z}$-graded weak $V^{k}(\mathfrak{g})\otimes V^{+}$-module as in Lemma 4.9 for $i\in\{1,2\}$. Then an arbitrary morphism $f\colon\mathcal{M}^{1}\to\mathcal{M}^{2}$ of weak $\mathcal{C}^{k}(\mathfrak{g})$-modules extends to a morphism of weak $V^{k}(\mathfrak{g})\otimes V^{+}$-modules defined by

[TABLE]

where $f(\gamma):=f\otimes\operatorname{\sf id}_{\mathcal{H}^{+}_{\nu^{+}(\mu)}}\colon\widetilde{\mathcal{M}}^{1}(\gamma)\to\widetilde{\mathcal{M}}^{2}(\gamma)$. Then, by the $\mathbb{Z}/2\mathbb{Z}$-graded linear isomorphisms

[TABLE]

we conclude that $f=\Omega^{+}(\mu)(\widetilde{f})$. This completes the proof. ∎

Remark 4.12**.**

By the proof, the superfunctor $\Omega^{+}(\mu)$ is also evenly dense in the sense of [BE17, Definition 1.1].

4.3. Spectral flow equivariance

In this subsection, we prove (2) and (3) in Theorem 3.9.

4.3.1. Equivariance of $\Omega^{+}$

Lemma 4.13**.**

Let $\lambda\in\mathfrak{h}^{*}$ and $\gamma\in Q$. For an object $\mathcal{M}$ and a morphism $f\colon\mathcal{M}^{1}\to\mathcal{M}^{2}$ of $\mathscr{C}_{[\lambda_{\sf sc}]}(V_{\sf sc})$, we set ${\sf U}^{\gamma}_{\sf sc}(\mathcal{M}):=\boldsymbol{\Pi}^{\langle f^{+}_{\sf af}(\gamma),f^{+}_{\sf af}(\gamma)\rangle}\mathcal{M}^{h_{\sf sc}[\gamma]}$ and ${\sf U}^{\gamma}_{\sf sc}(f):=f\colon{\sf U}^{\gamma}_{\sf sc}(\mathcal{M}^{1})\to{\sf U}^{\gamma}_{\sf sc}(\mathcal{M}^{2}),$ respectively. Then the assignment ${\sf U}^{\gamma}_{\sf sc}$ defines a $\mathbb{C}$-linear functor

[TABLE]

which is a categorical isomorphism with the inverse ${\sf U}^{-\gamma}_{\sf sc}$.

Proof.

By the two isomorphisms (3.1) and (4.12) of $\mathbb{Z}/2\mathbb{Z}$-graded $V_{\sf sc}\otimes\mathcal{H}^{+}$-modules, the weak $V_{\sf sc}$-module ${\sf U}^{\gamma}_{\sf sc}(\mathcal{M})$ turns out to lie in $\mathscr{C}_{[(\lambda-\gamma)_{\sf sc}]}(V_{\sf sc})$. The rest of the proof is straightforward and we omit it. ∎

Remark 4.14**.**

It is known that ${\sf U}^{\gamma}_{\sf sc}$ induces the spectral flow automorphism of the $\mathcal{N}=2$ superconformal algebra. See [HT91, Proposition 4.2] for details.

Theorem 4.15**.**

For any $\lambda\in\mathfrak{h}^{*}$ and $\gamma\in Q$, we have an isomorphism

[TABLE]

of functors from $\mathscr{C}_{[\lambda]}(V_{\sf af})$ to $\mathscr{C}_{[(\lambda-\gamma)_{\sf sc}]}(V_{\sf sc})$.

Proof.

By using (3.1) and (4.12), we can verify that the natural isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf af}\otimes V^{+}$-modules for an object $M$ of $\mathscr{C}_{[\lambda]}(V_{\sf af})$ gives rise to a natural isomorphism $\Omega^{+}(\lambda-\gamma)(M)\xrightarrow{\simeq}{\sf U}^{\gamma}_{\sf sc}\circ\Omega^{+}(\lambda)(M)$ of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf sc}$-modules. ∎

4.3.2. Equivariance of $\Omega^{-}$

Lemma 4.16**.**

Let $\lambda\in(\mathfrak{h}_{\sf sc})^{*}$ and $\xi\in K$. For an object $\mathcal{M}$ and a morphism $f\colon\mathcal{M}^{1}\to\mathcal{M}^{2}$ of $\mathscr{C}_{[\lambda]}(V_{\sf sc})$, we set ${\sf U}^{\xi}_{\sf sc}(\mathcal{M}):=\boldsymbol{\Pi}^{\langle\xi,\xi\rangle}\mathcal{M}^{\xi}$ and ${\sf U}^{\xi}_{\sf sc}(f):=f\colon{\sf U}^{\xi}_{\sf sc}(\mathcal{M}^{1})\to{\sf U}^{\xi}_{\sf sc}(\mathcal{M}^{2})$, respectively. Then the assignment ${\sf U}^{\xi}_{\sf sc}$ defines a $\mathbb{C}$-linear functor

[TABLE]

which is naturally isomorphic to the identity functor of $\mathscr{C}_{[\lambda]}(V_{\sf sc})$.

Proof.

By Proposition 3.8, we may assume that $[\lambda]=[(\lambda_{\sf af})_{\sf sc}]$. For an object $M$ of $\mathscr{C}_{[\lambda_{\sf af}]}(V_{\sf af})$, in the same way as the proof of Theorem 4.15, the natural isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf af}\otimes V^{+}$-modules gives rise to a natural isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf sc}$-modules. Then, by Theorem 3.9, we get the conclusion. ∎

Theorem 4.17**.**

Let $\lambda\in(\mathfrak{h}_{\sf sc})^{*}$ and $\gamma\in Q_{\sf sc}$.

1. (1)

For an object $M$ and a morphism $f\colon M^{1}\to M^{2}$ of $\mathscr{C}_{[\lambda_{\sf af}]}(V_{\sf af})$, we set ${\sf U}^{\gamma}_{\sf af}(M):=\boldsymbol{\Pi}^{\langle f^{+}_{\sf sc}(\gamma),f^{+}_{\sf sc}(\gamma)\rangle}M^{h_{\sf af}[\gamma]}$ and ${\sf U}^{\gamma}_{\sf af}(f):=f\colon{\sf U}^{\gamma}_{\sf af}(M^{1})\to{\sf U}^{\gamma}_{\sf af}(M^{2})$, respectively, where

[TABLE]

Then the assignment ${\sf U}^{\gamma}_{\sf af}$ defines a $\mathbb{C}$-linear functor

[TABLE]

which is a categorical isomorphism with the inverse ${\sf U}^{-\gamma}_{\sf af}$. 2. (2)

We have an isomorphism

[TABLE]

of functors from $\mathscr{C}_{[\lambda]}(V_{\sf sc})$ to $\mathscr{C}_{[(\lambda+\gamma)_{\sf af}]}(V_{\sf af})$.

Proof.

To prove (1) and (2), it suffices to construct a natural isomorphism from $\Omega^{-}(\lambda+\gamma)$ to ${\sf U}^{\gamma}_{\sf af}\circ\Omega^{-}(\lambda)$. For each $\alpha\in\Delta^{+}$, we set

[TABLE]

By using the formula $\alpha^{*}=\sum_{\beta\in\Pi}c_{\alpha,\beta}H_{\beta}$ and tedious but straightforward calculation, we obtain

[TABLE]

where

[TABLE]

Since we have a $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{H}^{-}$-module isomorphism

[TABLE]

in the same way as Theorem 4.15, we can verify that the natural isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf sc}\otimes V^{-}$-modules for an object $\mathcal{M}$ of $\mathscr{C}_{[\lambda]}(V_{\sf sc})$ gives rise to a natural isomorphism $\Omega^{-}(\lambda+\gamma)(\mathcal{M})\simeq{\sf U}^{\gamma}_{\sf af}\circ\Omega^{-}(\lambda)(\mathcal{M})$ of $\mathbb{Z}/2\mathbb{Z}$-graded weak $V_{\sf af}$-modules. ∎

5. Regular cases

Throughout this section, we assume that $k$ is a positive integer. In this section, we give additional information about the representation theory of $\mathcal{C}_{k}(\mathfrak{g})$.

5.1. Bicommutant of $\mathcal{H}^{+}$

Let $Q_{l}$ be the even integral sublattice of the root lattice $Q$ spanned by long roots. By the normalization of $(?,?)$ on $\mathfrak{h}^{*}$, we have

[TABLE]

We first recall that C. Dong and Q. Wang give a proof of the following fact (see also [DLY09, §4] for the case of $\mathfrak{g}=\mathfrak{sl}_{2}$).

Proposition 5.1** ([DW11, Proposition 4.11]).**

The vertex operator algebra extension

[TABLE]

of the Heisenberg vertex subalgebra $\mathcal{H}$ generated by $\mathfrak{h}$ in $L_{k}(\mathfrak{g})$ is isomorphic to the lattice vertex operator algebra associated with $\sqrt{k}Q_{l}$.

The next proposition is proved in a similar way as [DW11, Proposition 4.11] and we omit the detail (see [CL19, §8.2] in the case of $\mathfrak{g}=\mathfrak{sl}_{2}$).

Proposition 5.2**.**

The vertex operator superalgebra extension

[TABLE]

of the Heisenberg vertex subalgebra $\mathcal{H}^{+}$ in $L_{k}(\mathfrak{g})\otimes V^{+}$ is purely even and isomorphic to the lattice vertex operator algebra associated with $\sqrt{k+h^{\vee}}Q_{l}$. Moreover, we have

[TABLE]

Proof.

Note that the even part $(V^{+})^{\bar{0}}$ is the lattice vertex operator algebra associated with the even sublattice

[TABLE]

of $L^{+}$. Then we can apply a straightforward modification of the proof of [DW11, Proposition 4.11]. ∎

5.2. Bicommutant of $\mathcal{H}^{-}$

By (3.2) and Theorem 4.17, we have an isomorphism

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded weak $L_{k}(\mathfrak{g})\otimes\mathcal{H}^{-}$-modules. We first study the structure of {\sf U}^{\gamma}_{\sf af}\big{(}L_{k}(\mathfrak{g})\big{)} by the following lemma.

Lemma 5.3**.**

Let $\theta\in P^{\vee}$. Then L_{k}(\mathfrak{g})^{\theta}:=\big{(}L_{k}(\mathfrak{g})\big{)}^{\theta_{-1}\mathbf{1}_{\sf af}} is isomorphic to $L_{k}(\mathfrak{g})$ as a weak $L_{k}(\mathfrak{g})$-module if and only if $\theta\in Q^{\vee}$.

Proof.

Define a Lie algebra automorphism ${\sf t}_{\theta}$ of $\widehat{\mathfrak{g}}=\widehat{\mathfrak{g}}_{B}$ by

[TABLE]

for $\alpha\in\Delta$, $n\in\mathbb{Z}$, and $H\in\mathfrak{h}$. Then, by a direct computation, we obtain

[TABLE]

for any $X\in\mathfrak{g}$. By the same argument in [HT91, (4-1)], the automorphism ${\sf t}_{\theta}$ is inner if $\theta\in Q^{\vee}$. On the other hand, since ${\sf t}_{\theta}$ for $\theta\in P^{\vee}\setminus Q^{\vee}$ corresponds to a non-trivial Dynkin diagram automorphism and the induced $P^{\vee}/Q^{\vee}$-action on the set of special indices is simply transitive (see e.g. [Wak01, §1.2] for details), we thus conclude that the highest weight of $L_{k}(\mathfrak{g})^{\theta}$ coincides with that of $L_{k}(\mathfrak{g})$ if and only if $\theta\in Q^{\vee}$. ∎

Remark 5.4**.**

Following [HT91, Proposition 4.1], we call the automorphism $\mathfrak{t}_{\theta}$ the spectral flow automorphism of $\widehat{\mathfrak{g}}$ associated with $\theta\in P^{\vee}$.

We use the next lemma to determine the vertex superalgebra structure of the bicommutant of $\mathcal{H}^{-}$.

Lemma 5.5** (cf. [CKLR18, Theorem 4.1]).**

Let $(\mathcal{V}=\mathcal{V}^{\bar{0}}\oplus\mathcal{V}^{\bar{1}},Y,\mathbf{1},\omega)$ be a simple conformal vertex superalgebra and \big{(}L,\langle?,?\rangle\big{)} a non-degenerate integral lattice. Assume that there exists an injective $\mathbb{Z}$-linear map $i\colon L\to\mathcal{V}^{\bar{0}}$ such that

1. (1)

$L_{n}i(h)=\delta_{n,0}i(h)$ for any $h\in L$ and $n\geq 0$, 2. (2)

$i(h)_{n}i(h^{\prime})=\delta_{n,1}\langle h,h^{\prime}\rangle\mathbf{1}$ for any $n\geq 0$, 3. (3)

we have an isomorphism

[TABLE]

of (not necessarily $\mathbb{Z}/2\mathbb{Z}$-graded) weak $\mathcal{H}_{L}$-modules, where $\mathcal{H}_{L}$ is the Heisenberg vertex subalgebra generated by $i(L)$ in $\mathcal{V}$ and $(\mathcal{H}_{L})^{i(h)}$ is the weak $\mathcal{H}_{L}$-module defined in Lemma 4.8.

Then $\mathcal{V}$ is isomorphic to the lattice vertex superalgebra associated with the lattice $L$ as a vertex superalgebra.

Proof.

It follows from a straightforward generalization of [LX95, Theorem 3.14] (see [Xu98, Theorem 6.3.1] for details). ∎

Now we prove the following proposition.

Proposition 5.6**.**

The conformal vertex supralgebra extension

[TABLE]

of the Heisenberg vertex subalgebra $\mathcal{H}^{-}$ in $\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}$ is isomorphic to the lattice vertex superalgebra associated with $\sqrt{-(k+h^{\vee})}Q_{l}\oplus\mathbb{Z}^{N-\ell}.$

Proof.

In order to apply Lemma 5.5, we verify the assumptions in Lemma 5.5.

We first prove that $\mathcal{E}^{-}$ is simple. By Proposition 2.11 and the regularity of $L_{k}(\mathfrak{g})$, the vertex superalgebra $\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}$ decomposes into a direct sum of simple ordinary {\sf Com}\big{(}\mathcal{H}^{-},\mathcal{C}_{k}(\mathfrak{g})\otimes V^{-}\big{)}-modules. Since $V^{-}$ is strongly $L^{-}$-graded in the sense of [HLZ07, Definition 2.23], we can define the contragredient $V^{-}$-module in the sense of [HLZ07, Definition 2.35]. Then, we can apply the same argument in [ACKL17, Lemma 2.1] to our case and conclude that $\mathcal{E}^{-}$ is simple.

Next we verify the conditions (1), (2), and (3) in Lemma 5.5. Define a $\mathbb{Z}$-linear map $i\colon\sqrt{-(k+h^{\vee})}Q_{l}\oplus\mathbb{Z}^{N-\ell}\to(\mathcal{E}^{-})^{\bar{0}}$ by

[TABLE]

for $\gamma\in Q_{l}$ and $\beta\in\Delta^{+}\setminus\Pi$, where $\{\varepsilon_{\beta}\,|\,\beta\in\Delta^{+}\setminus\Pi\}$ is an orthonormal $\mathbb{Z}$-basis of $\mathbb{Z}^{N-\ell}$. Then the condition (1) is clear and the condition (2) follows from (2.1) and direct computations. At last, we verify the condition (3). It suffices to prove that, for $\gamma\in Q_{\sf sc}$, the $\mathbb{Z}/2\mathbb{Z}$-graded weak $L_{k}(\mathfrak{g})$-module {\sf U}^{\gamma}_{\sf af}\big{(}L_{k}(\mathfrak{g})\big{)} is isomorphic to $L_{k}(\mathfrak{g})$ if and only if $h^{\gamma}$ lies in the image of the map $i$. This is verified by Lemma 5.3 and some computations. We thus complete the proof. ∎

5.3. Regularity and Unitarity

In this subsection, we prove that $\mathcal{C}_{k}(\mathfrak{g})$ is regular and has a unitary structure.

We first prove the regularity. By Proposition 5.2, we obtain the following coset realization

[TABLE]

Then we obtain the following.

Theorem 5.7**.**

Both $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$ and $\mathcal{C}_{k}(\mathfrak{g})$ are regular.

Proof.

We first prove the regularity of $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$. By [ABD04, Theorem 4.5], it suffices to prove that $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$ is rational and $C_{2}$-cofinite. Then, by (5.1) and Proposition 5.2, the $C_{2}$-cofiniteness follows from [Miy15, Corollary 2] and the rationality follows from [CKLR18, Theorem 4.12].

Next we prove the regularity of $\mathcal{C}_{k}(\mathfrak{g})$. By a straightforward generalization of [ABD04, Theorem 4.5], it suffices to prove that $\mathcal{C}_{k}(\mathfrak{g})$ is rational and $C_{2}$-cofinite. The rationality of $\mathcal{C}_{k}(\mathfrak{g})$ follows from [DH12, Proposition 3.6]. The $C_{2}$-cofiniteness of $\mathcal{C}_{k}(\mathfrak{g})$ follows from [Miy04, Lemma 2.4] and the fact that $\mathcal{C}_{k}(\mathfrak{g})$ is finitely generated over the regular vertex operator algebra $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$. ∎

Corollary 5.8**.**

We have

[TABLE]

where P_{\sf sc}^{k}:=\big{\{}\mu_{\sf sc}\in(\mathfrak{h}_{\sf sc})^{*}\,\big{|}\,\mu\in P\big{\}}.

Proof.

The right hand side is a full subcategory of the left hand side by the definition. Since the left hand side is a semisimple abelian category with finitely many (up to isomorphism) simple objects, it suffices to prove that every simple object in the left hand side lies in the right hand side. Let $\mathcal{M}$ be a simple $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-module. Since $\mathcal{M}$ decomposes into a direct sum of finite-dimensional $L_{0}$-eigenspaces, the abelian Lie algebra $\mathfrak{h}_{\sf sc}$ acts diagonally on $\mathcal{M}$. Therfore, by Proposition 3.8, Theorem 3.9, and [FZ92, Theorem 3.1.3], there exists $\mu\in P$ such that $\mathcal{M}$ is a simple object of \mathscr{C}_{[\mu_{\sf sc}]}\big{(}\mathcal{C}_{k}(\mathfrak{g})\big{)}. ∎

We next prove the unitarity.

Lemma 5.9**.**

There exists an anti-linear involution $\phi$ of $\mathcal{C}_{k}(\mathfrak{g})$ such that \big{(}\mathcal{C}_{k}(\mathfrak{g}),\phi\big{)} forms a unitary vertex operator superalgebra in the sense of [AL17, §2.1].

Proof.

By [DL14, Theorem 4.7] and [AL17, Proposition 2.4 and Theorem 2.9], both $L_{k}(\mathfrak{g})\otimes V^{+}$ and $\mathcal{E}^{+}$ have a unitary structure. Then, by [DL14, Corollary 2.8], the unitary structure of $L_{k}(\mathfrak{g})\otimes V^{+}$ inherits to $\mathcal{C}_{k}(\mathfrak{g})$. ∎

We close this subsection with the following result.

Proposition 5.10**.**

Every simple $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-module appears as a direct summand of the tensor product of some simple $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $L_{k}(\mathfrak{g})$-module and the lattice vertex operator superalgebra $V^{+}$. In particular, every simple $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-module is unitarizable.

Proof.

Let $\mathcal{M}$ be a simple $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-module. By Corollary 5.8, we may assume that $\mathcal{M}$ is a simple object of \mathscr{C}_{[\mu_{\sf sc}]}\big{(}\mathcal{C}_{k}(\mathfrak{g})\big{)} for some $\mu\in P$. Then, by Theorem 3.9, we obtain $\mathcal{M}\simeq\Omega^{+}(\mu)\circ\Omega^{-}(\mu_{\sf sc})(\mathcal{M})$ as a direct summand of $\Omega^{-}(\mu_{\sf sc})(\mathcal{M})\otimes V^{+}$. Therefore the unitarizability of $\mathcal{M}$ follows from [DL14, Theorem 4.8] and the same discussion as the proof of [DL14, Corollary 2.8]. ∎

5.4. Braided monoidal structure

In this subsection, owing to the recent developments in the general theory of vertex superalgebra extensions by [HKL15], [CKL19], and [CKM17], we prove that the category of $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-modules carries a braided monoidal category structure.

As a consequence of the theory of vertex tensor categories developed by Y.-Z. Huang and J. Lepowsky (see [HL95a], [HL95b], [HL95c], [Hua95], [Hua05], and references therein), we obtain the following.

Proposition 5.11**.**

The semisimple $\mathbb{C}$-linear abelian category of purely even $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$-modules has a braided monoidal category structure induced by [Hua05, Theorem 3.7] (see also [HL94, Theorem 4.2 and 4.4]).

Proof.

Since $\mathcal{C}_{k}(\mathfrak{g})^{\bar{0}}$ is regular by Theorem 5.7, all the conditions in [Hua05, Theorem 3.5] are satisfied. Therefore it follows from [Hua05, Theorem 3.7]. ∎

We now arrive at the last result in this paper.

Theorem 5.12**.**

The semisimple $\mathbb{C}$-linear abelian category

[TABLE]

of $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{C}_{k}(\mathfrak{g})$-modules has a braided monoidal category structure induced by [CKM17, Theorem 3.65].

Proof.

By [CKL19, Theorem 3.14], the category $\mathscr{C}$ is equivalent to the underlying category of the supercategory {\sf Rep}^{0}\big{(}\mathcal{C}_{k}(\mathfrak{g})\big{)} defined in [CKL19, Definition 2.16] (see also [KJO02, Definition 1.8] for details). Since these categories carry a natural braided monoidal category structure by [CKM17, Theorem 3.65] (see also [CKM17, Remark 3.56 and 3.67]), we complete the proof. ∎

Remark 5.13**.**

When $\mathfrak{g}=\mathfrak{sl}_{2}$, Theorem 5.12 is proved by Y.-Z. Huang and A. Milas in [HM02, Theorem 4.8]. Note that Proposition 2.11 for $\mathfrak{g}=\mathfrak{sl}_{2}$ (see also [Ada99, Theorem 5.1]) plays a crucial role in their proof of the convergence and extension property for products of intertwining operators. See [HM02, §3] for details.

Example 5.14**.**

Assume that $\mathfrak{g}$ is simply-laced and $k=1$. By Proposition 5.2 and the isomoprhism $L_{1}(\mathfrak{g})\simeq V_{Q}$ known as the Frenkel–Kac construction, we have \mathcal{C}_{1}(\mathfrak{g})\simeq{\sf Com}\big{(}V_{\sqrt{1+h^{\vee}}Q},V_{Q\oplus\mathbb{Z}^{N}}\big{)}. Then this coset vertex operator superalgebra turns out to be isomorphic to the lattice vertex superalgebra associated with

[TABLE]

By a direct computation, we can verify that its dual quotient group is given by

[TABLE]

Appendix A Vertex superalgebras and modules

A.1. Universal affine VOAs

Throughout this subsection, $\mathfrak{g}$ stands for a finite-dimensional reductive Lie algebra over $\mathbb{C}$. Let $B\colon\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$ be a symmetric invariant bilinear form and $\widehat{\mathfrak{g}}_{B}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}K$ the corresponding affinization of $\mathfrak{g}$, i.e. the commutation relations are given by

[TABLE]

for $X,Y\in\mathfrak{g}$ and $n,m\in\mathbb{Z}$, where $X_{n}$ stands for $X\otimes t^{n}\in\widehat{\mathfrak{g}}_{B}$. The next lemma is well-known.

Lemma A.1**.**

Let $\mathbb{C}_{B}$ be a $1$-dimensional representation of $\widehat{\mathfrak{g}}_{\geq 0}:=\mathfrak{g}\otimes\mathbb{C}[t]\oplus\mathbb{C}K$ defined by $\mathfrak{g}\otimes\mathbb{C}[t].1=\{0\}$ and $K.1=1$. Then there exists a unique vertex algebra structure on the induced module

[TABLE]

which is strongly generated by

[TABLE]

for $X\in\mathfrak{g}$, which is called the universal affine vertex algebra associated with $(\mathfrak{g},B)$.

When $\mathfrak{g}$ is an $n$-dimensional abelian Lie algebra and $B$ is non-degenerate, we call $V(\mathfrak{g},B)$ the Heisenberg vertex algebra of rank $n$. It is well-known that Heisenberg vertex algebras of fixed rank are simple and isomorphic to each other.

A.2. Coset vertex superlagebras

Let $\mathcal{V}$ be a vertex superalgebra and $\mathcal{W}$ its vertex subsuperalgebra. Then the $\mathbb{Z}/2\mathbb{Z}$-graded subspace

[TABLE]

gives a vertex subsuperalgebra of $\mathcal{V}$, called the coset vertex superalgebra with respect to the pair $(\mathcal{W},\mathcal{V})$. By the general theory of vertex superalgebras (see e.g. [Kac98]), the even linear map

[TABLE]

gives a vertex superalgebra homomorphism.

A.3. Weak modules

Let $(\mathcal{V}=\mathcal{V}^{\bar{0}}\oplus\mathcal{V}^{\bar{1}},Y,\mathbf{1})$ be a vertex superalgebra. Let $M=M^{\bar{0}}\oplus M^{\bar{1}}$ be a $\mathbb{Z}/2\mathbb{Z}$-graded vector space and

[TABLE]

a not necessarily $\mathbb{Z}/2\mathbb{Z}$-homogeneous (resp. even) linear map. We write

[TABLE]

for $A\in\mathcal{V}$. Then the above pair $(M,Y_{M})$ is called a (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) weak $\mathcal{V}$-module if the following conditions hold for any $A\in\mathcal{V}^{\bar{a}}$, $B\in\mathcal{V}^{\bar{b}}$, and $v\in M$:

**Field axiom: **

$A^{M}_{n}v=0$ holds for $n\gg 0$,

**Vacuum axiom: **

${\bf 1}^{M}_{n}v=\delta_{n,-1}v$ holds for any $n\in\mathbb{Z}$,

**The Borcherds identity: **

[TABLE]

holds for any $p,q,r\in\mathbb{Z}$.

Let $(M^{i},Y_{M^{i}})$ be (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) weak $\mathcal{V}$-modules for $i\in\{1,2\}$. A not necessarily $\mathbb{Z}/2\mathbb{Z}$-homogeneous (resp. even) linear map $f\colon M^{1}\rightarrow M^{2}$ is a morphism of (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) weak $V$-modules if

[TABLE]

for any $A\in\mathcal{V}$ and $n\in\mathbb{Z}$.

Let $\mathcal{V}\text{-\sf Mod}$ be the $\mathbb{C}$-linear abelian category of weak $\mathcal{V}$-modules and $\mathcal{V}\text{-\sf gMod}$ the full subcategory111111 We note that the $\mathbb{C}$-linear additive category $\mathcal{V}\text{-\sf gMod}$ is not abelian in general. of $\mathcal{V}\text{-\sf Mod}$ whose objects are $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{V}$-modules.

Lemma A.2**.**

The categories $\mathcal{V}\text{-\sf Mod}$ and $\mathcal{V}\text{-\sf gMod}$ are $\mathbb{C}$-linear supercategories, that is, each space of morphisms is a $\mathbb{Z}/2\mathbb{Z}$-graded vector space over $\mathbb{C}$ and composition of morphisms induces an even $\mathbb{C}$-linear map. In addition, the underlying category $\underline{\mathcal{V}\text{-\sf gMod}}$ of the supercategory $\mathcal{V}\text{-\sf gMod}$ in the sense of [BE17, Definition 1.1] coincides with the $\mathbb{C}$-linear abelian category of $\mathbb{Z}/2\mathbb{Z}$-graded weak $\mathcal{V}$-modules.

A.4. Ordinary modules

In this subsection, we assume that $(\mathcal{V}=\mathcal{V}^{\bar{0}}\oplus\mathcal{V}^{\bar{1}},Y,\mathbf{1})$ is a vertex operator superalgebra with respect to a conformal vector $\omega\in\mathcal{V}^{\bar{0}}$. Then a (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) weak $\mathcal{V}$-module $(M,Y_{M})$ is called a (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) ordinary $\mathcal{V}$-module if $M$ decomposes into a direct sum

[TABLE]

of (resp. $\mathbb{Z}/2\mathbb{Z}$-homogeneous) finite-dimensional subspaces $M_{h}:=\{v\in M\,|\,L_{0}v:=\omega^{M}_{1}v=hv\}$ such that $M_{h+n}=\{0\}$ for any $h\in\mathbb{C}$ and $n\ll 0$.

Similarly to the previous subsection, we write $\mathcal{V}$-mod (resp. $\mathcal{V}\text{-\sf gmod}$) for the full subsupercategory of $\mathcal{V}$-Mod whose objects are (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) ordinary $\mathcal{V}$-modules. Then the underlying category $\underline{\mathcal{V}\text{-\sf gmod}}$ of the $\mathbb{C}$-linear supercategory $\mathcal{V}\text{-\sf gmod}$ is the $\mathbb{C}$-linear abelian category of $\mathbb{Z}/2\mathbb{Z}$-graded ordinary $\mathcal{V}$-modules.

Appendix B Twisted sector

In this section, we give a remark on the twisted sector of $V_{\sf sc}$ for $k\in\mathbb{C}\setminus\{0,-h^{\vee}\}$.

We set

[TABLE]

Then the parity involution $\sigma\colon V_{\sf sc}\to V_{\sf sc}$, which is defined by $\sigma|_{V_{\sf sc}^{\bar{i}}}=(-1)^{i}\operatorname{\sf id}_{V_{\sf sc}^{\bar{i}}}$ for $i\in\{0,1\}$, satisfies that $\sigma={\sf exp}(2\pi\sqrt{-1}J^{*}_{0})$.

Lemma B.1**.**

The category of (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) $\mathbb{Z}$-twisted positive energy $V_{\sf sc}$-modules121212 Every $\mathbb{Z}$-twisted positive energy $V_{\sf sc}$-module is a weak $V_{\sf sc}$-module by the definition. See [DSK06, Example 2.14] for details. is isomorphic to that of (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) $\frac{1}{2}\mathbb{Z}$-twisted positive energy $V_{\sf sc}$-modules. Here we refer the reader to [DSK06, Definition 2.21] for the definition of $\Gamma$-twisted positive energy modules, where $\Gamma=\mathbb{Z}$ or $\frac{1}{2}\mathbb{Z}$.

Proof.

For a (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) $\mathbb{Z}$-twisted positive energy $V_{\sf sc}$-module $(\mathcal{M},Y_{\mathcal{M}})$, we define

[TABLE]

by Y_{\mathcal{M}}^{\sigma}(v,z):=Y_{\mathcal{M}}\big{(}\Delta(-J^{*},z)v,z\big{)} for $v\in V_{\sf sc}$. Then $\mathcal{M}^{\sigma}:=(\mathcal{M},Y_{\mathcal{M}}^{\sigma})$ is a (resp. $\mathbb{Z}/2\mathbb{Z}$-graded) $\frac{1}{2}\mathbb{Z}$-twisted positive energy $V_{\sf sc}$-module and the assignment $\mathcal{M}\mapsto\mathcal{M}^{\sigma}$ gives rise to a desired categorical isomorphism. ∎

Remark B.2**.**

By [HT91, Theorem 2.5], every $\mathbb{Z}$-twisted (resp. $\frac{1}{2}\mathbb{Z}$-twisted) weak $V_{\sf sc}$-module admits an action of the $\mathcal{N}=2$ superconformal algebra in the Neveu–Schwarz sector (resp. in the Ramond sector) of central charge $c_{\sf sc}$.