.

# Li filtrations of SUSY vertex algebras

Shintarou Yanagida  Graduate School of Mathematics, Nagoya University.
Furocho, Chikusaku, Nagoya, Japan, 464-8602<EMAIL_ADDRESS>
(Date: 2021.11.10)

###### Abstract.

Any vertex algebra has a canonical decreasing filtration, called Li
filtration, whose associated graded space has a natural structure of a vertex
Poisson algebra. In this note, we introduce an analogous filtration for any
SUSY vertex algebra, which was introduced by Heluani and Kac as a superfield
formalism of a supersymmetric vertex algebra. We prove that the associated
graded superspace of our filtration has a structure of SUSY vertex Poisson
algebras. We also introduce and discuss related notions, such as Zhu’s
$C_{2}$-Poisson superalgebras, associated superschemes and singular supports,
for SUSY vertex algebras.

###### Key words and phrases:

vertex algebras, SUSY vertex algebras, Li filtration, vertex Poisson algebras,
associated schemes

The author is supported by supported by JSPS KAKENHI Grant Number 19K03399,
and also by the JSPS Bilateral Program “Elliptic algebras, vertex operators
and link invariant”.

###### Contents

  1. 0 Introduction
  2. 1 Super preliminary
  3. 2 SUSY vertex algebras
  4. 3 Li filtration of SUSY vertex algebra
  5. 4 Associated superschemes and singular supports for SUSY vertex algebras
  6. 5 Concluding comments

##  0\. Introduction

### Li filtration

A basic philosophy in the theory of vertex algebras is that they are quantized
objects. This phrase can be understood in various ways, and one of them is
given by the work of Haisheng Li [L05]. He introduced a canonical decreasing
filtration for an arbitrary vertex algebra $V$, nowadays called the _Li
filtration_ of $V$. The associated graded space has a natural structure of a
vertex Poisson algebra [FBZ04, Chap. 16], i.e., a combined structure of a
commutative vertex algebra and a vertex Lie algebra. A vertex Poisson algebra
can be considered as a Poisson object in the world of vertex algebras, and in
this sense, the existence of the Li filtration shows that we can view any
vertex algebra as a quantization of the associated graded vertex Poisson
algebra.

For explicitness, let us recall the definition of the Li filtration (see also
§ 3.2). Let $V$ be a vertex algebra, and for $a\in V$ we denote by
$a_{(n)}\in\operatorname{End}V$ the $(n)$-operator in the expansion
$Y(a,z)=\sum_{n\in\mathbb{Z}}z^{-n-1}a_{(n)}$ of the state-field
correspondence. Then the Li filtration of $V$ is a decreasing sequence of
subspaces

$V=E_{0}(V)\supset E_{1}(V)\supset\dotsb\supset E_{n}(V)\supset\dotsb$

consisting of

$\displaystyle
E_{n}(V)\coloneqq\operatorname{span}\left\\{a^{1}_{(-1-k_{1})}\dotsm
a^{r}_{(-1-k_{r})}b\,\Bigg{|}\begin{array}[]{l}r\in\mathbb{Z}_{>0},\,a^{i},b\in
V,\,k_{i}\in\mathbb{N}\\\ \text{satisfying $k_{1}+\dotsb+k_{r}\geq
n$}\end{array}\right\\}.$ (0.0.3)

Besides the philosophical reason explained above, there have been increasing
interest in and importance of the Li filtration and the associated graded
vertex Poisson algebra. In particular, there have been increasing studies with
a geometric viewpoint where one sees the spectrum of the associated graded
vertex Poisson algebra as a vertex algebra (a.k.a. chiral, or semi-infinite)
analogue of a Poisson scheme. Such geometric studies have a strong point that
the complicated structure of the original vertex algebra can be treated with
less complexity by using Poisson geometry arguments or by making appropriate
analogue of them. Among various studies, let us here only name the work of
Arakawa and Moreau [AM21], where the notion of chiral symplectic cores is
introduced and studied as a vertex Poisson analogue of symplectic leaves in
Poisson varieties.

### SUSY vertex algebras

This note is written under the influence of those Poisson-geometric studies of
vertex algebras. We focus on _SUSY vertex algebras_ introduced by Heluani and
Kac [HK07]. It is a superfield formulation of vertex algebras equipped with
supersymmetry. The detailed explanation will be given in §2, but here let us
give a brief account. There are two classes of structures, _$N_{W}=N$ SUSY
vertex algebras_ and _$N_{K}=N$ SUSY vertex algebras_, for a positive integer
$N$. Both structures are given by a linear superspace $V$, an element
$\ket{0}\in V$ and a linear operator $Y(\cdot,Z)$ on $V$ whose value for $a\in
V$ is taken in a _superfield_ of the form

$\displaystyle Y(a,Z)=\sum_{(j|J)}Z^{j|J}a_{(j|J)},\quad Z^{j|J}\coloneqq
z^{j}\zeta^{j_{1}}\dotsb\zeta^{j_{r}},\quad a_{(j|J)}\in\operatorname{End}V.$
(0.0.4)

Here $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ denotes a _supervariable_ with an even
$z$ and odd $\zeta^{i}$’s, and the index $(j|J)$ runs over integers
$j\in\mathbb{Z}$ and ordered subsets $J=\\{j_{1}<\dotsb<j_{r}\\}$ of the
ordered set $[N]\coloneqq\\{1<\dotsb<N\\}$. $V$ is called the superspace of
states, $\ket{0}$ is called the vacuum, and $Y(\cdot,Z)$ is called the state-
superfield correspondence. These should satisfy the vacuum axiom and the
locality axiom, similarly as in the even case (i.e., the case of ordinary
vertex algebras).

The difference between $N_{W}=N$ and $N_{K}=N$ cases lies in the translation
invariance. An $N_{W}=N$ SUSY vertex algebra is equipped with an even
endomorphism $T$ and odd endomorphism $S^{i}$’s for $i\in[N]$ which satisfy

$[T,Y(a,Z)]=\partial_{z}Y(a,Z),\quad[S^{i},Y(a,Z)]=\partial_{\zeta^{i}}Y(a,Z).$

Here $[\cdot,\cdot]$ denotes the supercommutator (see §1.1 for a formal
definition). On the other hand, an $N_{K}=N$ SUSY vertex algebra is equipped
with odd operations $S_{K}^{i}$’s for $i\in[N]$ satisfying

$[S_{K}^{i},Y(a,Z)]=D^{i}_{Z}Y(a,Z),\quad
D_{Z}^{i}\coloneqq\partial_{\zeta^{i}}+\zeta^{i}\partial_{z}.$

In this $N_{K}=N$ case, we automatically have
$(S_{K}^{1})^{2}=\dotsb=(S_{K}^{N})^{2}$ corresponding to
$(D_{Z}^{i})^{2}=\partial_{z}$, and setting $T\coloneqq(S_{K}^{i})^{2}$, we
have $[T,Y(a,Z)]=\partial_{z}Y(a,Z)$.

Similarly as the superfield formalism in supersymmetric quantum field
theories, a SUSY vertex algebra has a compact description of the structure.
Besides of such formal convenience, it has also much importance. Here we name
only a few studies which have geometric flavor: Heluani’s construction [H07]
of chiral algebras over super curves and superconformal curves, his another
work [H09] of SUSY structure on the chiral de Rham complex of a Calabi-Yau
manifold, and the study of super-modular property of SUSY conformal blocks for
$N_{W}=1$ case by Heluani and Van Ekeren [HV16].

### Summary of results

Now we can explain the main theme of this note. For any $N_{W}=N$ or $N_{K}=N$
SUSY vertex algebra $V$, we introduce a canonical decreasing filtration

$V=E_{0}(V)\supset E_{1}(V)\supset\dotsb\supset E_{n}(V)\supset\dotsb$

where, using the $(j|J)$-operator in (0.0.4), we set

$\displaystyle
E_{n}(V)\coloneqq\operatorname{span}\left\\{a^{1}_{(-1-k_{1}|K_{1})}\dotsm
a^{r}_{(-1-k_{r}|K_{r})}b\,\Bigg{|}\begin{array}[]{l}r\in\mathbb{Z}_{>0},\,a^{i},b\in
V,\,k_{i}\in\mathbb{N},\,K_{i}\subset[N]\\\ \text{satisfying
$k_{1}+\dotsb+k_{r}\geq n$}\end{array}\right\\}.$ (0.0.7)

We call it the _Li filtration_ of $V$ (Definitions 3.2.2 and 3.3.2).

Our definition is just neglecting the odd index $J$ and using the even case
definition (0.0.3). Despite of its simple-mindedness, we need some careful
arguments on SUSY vertex algebras to obtain the following SUSY analogue of
Li’s theorem in [L05, Theorem 2.12].

######  Theorem 1 (Theorems 3.2.6 and 3.3.6).

Let $V$ be an $N_{W}=N$ (resp. $N_{K}=N$) SUSY vertex algebra, and
$E_{n}=E_{n}(V)$ be the linear sub-superspaces of $V$ in (0.0.7). Then the
associated graded space

$\operatorname{gr}_{E}V\coloneqq{\textstyle\bigoplus}_{n\in\mathbb{N}}E_{n}/E_{n+1}$

has the following structure of an $N_{W}=N$ (resp. $N_{K}=N$) SUSY vertex
Poisson algebra (see Definitions 3.1.6 and 3.1.17). Let $a\in E_{r}$ and $b\in
E_{s}$.

  * •

The commutative multiplication $\cdot$ is

$(a+E_{r+1})\cdot(b+E_{s+1})\coloneqq a_{(-1|N)}b+E_{r+s+1}.$

  * •

The unit is $1\coloneqq\ket{0}+E_{1}$.

  * •

The even operator $\partial$ is $\partial(a+E_{r+1})\coloneqq Ta+E_{r+2}$, and
the odd operator $\delta^{i}$ for $i\in[N]$ is $\delta^{i}(a+E_{r+1})\coloneqq
S^{i}a+E_{r+2}$ (resp. the odd operator $\not{\partial}^{i}$ is
$\not{\partial}^{i}(a+E_{r+1})\coloneqq S_{K}^{i}a+E_{r+2}$).

  * •

The SUSY vertex Lie structure $Y_{-}$ is

$Y_{-}(a+E_{r+1},Z)(b+E_{s+1})\coloneqq\sum_{(j|J),\,j\geq
0}Z^{-1-j|[N]\setminus J}(a_{(j|J)}b+E_{r+s-j+1}).$

Once we have obtained the Li filtration of SUSY vertex algebras, we can
discuss various related notions which are already established in the even
case, and can expect to develop a Poisson-geometric theory of SUSY vertex
algebras. In this note, as a first step toward such a geometric theory, we
study a SUSY analogue of _associated schemes_ and _singular supports_
introduced by Arakawa [Ar12]. Our definitions and the arguments are
straightforward analogue of loc. cit. As a result, we have the following SUSY
analogue of the equivalence between the $C_{2}$-cofiniteness and the lisse
condition [Ar12, Theorem 3.3.3].

######  Theorem 2 (Theorem 4.2.9).

Let $V$ be an $N_{W}=N$ or $N_{K}=N$ SUSY vertex algebra which is finitely
strongly generated and has a lower-bounded grading (see Assumption 4.2.3 for
the detail). Then we have

$\text{$V$ is $C_{2}$-cofinite
(\lx@cref{creftype~refnum}{dfn:C2cof})}\iff\text{$V$ is lisse
(\lx@cref{creftype~refnum}{dfn:lisse})}.$

### Organization

Let us explain the organization of this note. § 1 is a preliminary part. The
first § 1.1 summarizes the super notation used throughout this note. The next
§ 1.2 is also a notational preliminary, focusing on the derivations and the
de Rham complex in the super setting. We remark that there are several
notations for the super de Rham complex in literature, and the aim is to
choose and fix one of them. The third § 1.3 is largely based on
recollections, but has some new terminologies and lemmas. There we give SUSY
analogue of the infinite jet algebras (or arc algebras), calling them the
superjet algebras and the superconformal jet algebras, corresponding to
$N_{W}=N$ and $N_{K}=N$ cases, respectively. They will be the foundation of
our study of commutative SUSY vertex algebras and SUSY vertex Poisson algebras
in the following sections.

A large part of § 2 gives recollection on the theory of SUSY vertex algebras
established in [HK07]. After introducing the superfield notation in § 2.1, we
explain the definition and basic properties of SUSY vertex algebras in § 2.2
and § 2.3, dividing the $N_{W}=N$ case and $N_{K}=N$ case, respectively. We
also give SUSY analogue of Borcherds’ iterate formula (Lemmas 2.2.10 and
2.3.13), which will be used in the proof of Theorem 1. The following § 2.4
starts our original study. There we study commutative SUSY vertex algebras,
and show that the superjet algebra (resp. the superconformal algebra) has a
natural structure of a commutative $N_{W}=N$ (resp. $N_{K}=N$) SUSY vertex
algebra.

§ 3 is the main body of this note, introducing the Li filtration of SUSY
vertex algebras and showing Theorem 1. As a preliminary, we give in § 3.1 an
exposition of SUSY vertex Poisson algebras. They are introduced in [HK07],
using the language of SUSY Lie conformal algebras. For the following analysis,
we give a restatement in terms of SUSY vertex Lie algebras. We also introduce
the level 0 SUSY vertex algebra structure on the superjet and superconformal
jet algebras (Propositions 3.1.9 and 3.1.18), which will be the foundation of
the study of associated superschemes and singular supports in § 4. The
$N_{W}=N$ case of Theorem 1 is shown in § 3.2, and the $N_{K}=N$ case is in
§ 3.3. The strategy is just mimicking Li’s argument for the even case [L05,
§2], which is based on Borcherds’ commutator formula and iterate formula.
Although the SUSY analogues of commutator formula are already given in [HK07],
we cannot find the SUSY iterate formula in literature, which is the reason for
preparing Lemmas 2.2.10 and 2.3.13.

§ 4 is an application of the previous Theorem 1, and more or less a
straightforward SUSY analogue of Arakawa’s work [Ar12], as mentioned above. In
§ 4.1, we introduce the $C_{2}$-Poisson superalgebra (Proposition 4.1.2) and
the associated superscheme (Definition 4.1.6) of an $N_{W}=N$ or $N_{K}=N$
SUSY vertex algebra, and study their basic properties (e.g. Proposition
4.1.4). One remark is that there appears Poisson superalgebras of parity
$N\bmod 2$ (Definition 3.1.7). § 4.2 gives a study of singular supports, and
shows Theorem 2.

### Notation and terminology

Here we list the notations and terminology used throughout in this paper.

  * •

For a set $S$, we denote by $\left|{S}\right|=\\#S$ its cardinality, and for
$r,s\in S$, we denote by $\delta_{r,s}$ the Kronecker delta.

  * •

We denote by $\mathbb{N}=\mathbb{Z}_{\geq 0}\coloneqq\\{0,1,2,\dotsc\\}$ the
set of non-negative integers.

  * •

For $\alpha,\beta\in\mathbb{C}$, we denote $\alpha\geq\beta$ if
$\alpha-\beta\in\mathbb{R}_{\geq 0}$.

  * •

The binomial coefficient is defined to be
$\binom{n}{m}\coloneqq\frac{1}{m!}n(n-1)\dotsb(n-m+1)$ for $m\in\mathbb{N}$
and some indeterminate or number $n$.

  * •

A ring or an algebra means a unital and associative one unless otherwise
stated. A ring action on its module is denoted by the period “.”, i.e., for a
left module $M$ over a ring $R$, we denote by $r.m$ the action of $r\in R$ on
$m\in M$.

  * •

For a linear (super)space $V$ over a field and a subset $S\subset V$, we
denote by $\operatorname{span}S$ the subspace of $V$ linearly spanned of $S$.

  * •

For a category $\mathsf{C}$, we denote $X\in\mathsf{C}$ to mean that $X$ is an
object of $\mathsf{C}$.

  * •

$\operatorname{\mathsf{Sets}}$ denotes the category of sets.

##  1\. Super preliminary

###  1.1. Super notation

We start with delivering notations for super objects. We follow [Man02, Chap.
3] and [KV11, 1.1] with slight modifications on symbols.

######  Notation 1.1.1.

Here is the first set of notations and terminologies:

  * •

We denote by
$\mathbb{Z}_{2}\coloneqq\mathbb{Z}/2\mathbb{Z}=\\{\overline{0},\overline{1}\\}$
the _parity_ or the _supergrading_.

  * •

A _super abelian group_ or a _supermodule_ means a $\mathbb{Z}_{2}$-graded
module

$M=M_{\overline{0}}\oplus M_{\overline{1}}.$

For each $\gamma\in\mathbb{Z}_{2}$, an element $v\in M_{\gamma}$ is called _of
pure parity_ , and for such $v$, we denote $p(v)\coloneqq\gamma$. We denote by
$M_{\textup{pure}}\coloneqq M_{\overline{0}}\sqcup M_{\overline{1}}$ the set
of elements of pure parity. An element $v\in M_{\overline{0}}$ is called
_even_ , and $v\in M_{\overline{1}}$ is called _odd_. If $M=M_{\overline{0}}$,
then $M$ is called _purely even_ , and if $M=M_{\overline{1}}$, then $M$ is
called _purely odd_.

  * •

A _morphism $M\to N$ of supermodules_ is a module homomorphism which preserves
the $\mathbb{Z}_{2}$-gradings. The category of supermodules is denoted by
$\mathsf{SAb}$. We denote by $\Pi\colon\mathsf{SAb}\to\mathsf{SAb}$ the parity
change functor: $(\Pi M)_{\overline{0}}=M_{\overline{1}}$ and $(\Pi
M)_{\overline{1}}=M_{\overline{0}}$. It is involutive:
$\Pi^{2}=\operatorname{id}$.

  * •

The category $\mathsf{SAb}$ of supermodules is a symmetric monoidal category
with monoidal structure $\otimes$ given by the standard graded tensor product.
The symmetry transformation is given by

$\displaystyle M\otimes N\to N\otimes M,\quad m\otimes
n\mapsto(-1)^{p(m)p(n)}n\otimes m.$ (1.1.1)

We always consider $\mathsf{SAb}$ as the category equipped with this symmetric
monoidal structure.

  * •

A _supering_ $R$ is an associative ring object in $\mathsf{SAb}$. The
_supercommutator_ in $R$ is defined by

$\displaystyle[r,s]\coloneqq rs-(-1)^{p(r)p(s)}sr$ (1.1.2)

for $r,s\in R$ of pure parity, and extended by linearity. It satisfies the
_super Jacobi rule_

$(-1)^{p(r)p(t)}[r,[s,t]]+(-1)^{p(s)p(r)}[s,[t,r]]+(-1)^{p(t)p(s)}[t,[r,s]]=0\quad(r,s,t\in
R_{\textup{pure}}).$

  * •

A _commutative superring_ is a commutative associative ring object in
$\mathsf{SAb}$, i.e., a superring whose supercommutator always vanishes. A
_morphism $R\to S$ of commutative superrings_ is a morphism in $\mathsf{SAb}$
respecting the ring structures. Given such a morphism $R\to S$ of commutative
superrings, we call $S$ a _superalgebra over $R$_ or an _$R$ -superalgebra_.
Commutative $R$-superalgebras form a category, which is denoted by
$\operatorname{\mathsf{SCom}}R$.

Hereafter we always assume:

  * •

In every commutative superring $R$, the element $2$ is invertible.

As a consequence, if $r\in R_{\overline{1}}$, then $r^{2}=\frac{1}{2}[r,r]=0$,
so all odd elements are nilpotent.

######  Notation.

Next we introduce terminologies for modules over a commutative superring $R$.

  * •

A _left $R$-supermodule $L$_ is a supermodule with a left module structure
over the underlying ring of $R$ such that the module structure $R\times L\to
L$, $r\otimes l\mapsto r.l$ is a morphism in $\mathsf{SAb}$.

A _right $R$-supermodule_ $M$ is similarly defined, and can be regarded as a
left $R$-supermodule by $r.m\coloneqq(-1)^{p(r)p(m)}m.r$ for pure elements
$r\in R$ and $m\in M$. Conversely, a left $R$-supermodule is regarded as a
right $R$-supermodule by the same correspondence. Henceforth we call a left
$R$-supermodule just by an _$R$ -supermodule_.

  * •

$R$-supermodules form a category, which is denoted by
$\operatorname{\mathsf{SMod}}R$. For $M,N\in\operatorname{\mathsf{SMod}}R$, a
_morphism_ $M\to N$ of $R$-supermodules is an $R$-linear map preserving the
parity. The category $\operatorname{\mathsf{SMod}}R$ inherits the symmetric
monoidal structure $\otimes_{R}$ from that of $\mathsf{SAb}$.

  * •

If $R=k$ is a field, then a $k$-supermodule is called a _linear
$k$-superspace_.

Let $R$ be a commutative superring. The category
$\operatorname{\mathsf{SMod}}R$ of $R$-supermodules has a natural
$\mathbb{Z}_{2}$-graded enhancement. Let $M$ and $N$ be $R$-supermodules. We
denote by $\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)$ the module
of homomorphisms $M\to N$ of the underlying ring of $R$. It is an
$R$-supermodule with

$\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)=\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)_{\overline{0}}\oplus\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)_{\overline{1}},$

where $\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)_{\overline{0}}$
and $\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)_{\overline{1}}$
denote the submodule of parity preserving and of parity changing
homomorphisms, respectively, and the $R$-action is given by $(r.f)(m)\coloneqq
r.f(m)$ for $r\in R$,
$f\in\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)$ and $m\in M$.
Note that
$\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)_{\overline{0}}=\operatorname{Hom}_{\operatorname{\mathsf{SMod}}R}(M,N)$,
the morphism set of the category $\operatorname{\mathsf{SMod}}R$.

######  Notation.

We call an element of $\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)$
an _$R$ -homomorphism_.

The $R$-supermodule $\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)$
is indeed an _internal hom_ in the symmetric monoidal category
$(\operatorname{\mathsf{SMod}}R,\otimes_{R})$. To stress this point, we denote

$\displaystyle\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(M,N):=\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)\in\operatorname{\mathsf{SMod}}R.$
(1.1.3)

The definition of internal hom claims that we have

$\operatorname{Hom}_{\operatorname{\mathsf{SMod}}R}(L,\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(M,N))\cong\operatorname{Hom}_{\operatorname{\mathsf{SMod}}R}(L\otimes_{R}M,N)$

for any $L,M,N\in\operatorname{\mathsf{SMod}}R$.

The parity change functor $\Pi$ on $\mathsf{SAb}$ naturally lifts to
$\operatorname{\mathsf{SMod}}R$. For $M\in\operatorname{\mathsf{SMod}}$, the
object $\Pi M$ is defined as in $\mathsf{SAb}$, ant the $R$-supermodule
structure is given by

$\displaystyle r.(\Pi m)\coloneqq(-1)^{p(r)}\Pi(r.m)\quad(r\in R\text{ of pure
parity},\ m\in M),$ (1.1.4)

where $\Pi m\in\Pi M$ is the element corresponding to $m\in M$. For a morphism
$f\in\operatorname{Hom}_{\operatorname{\mathsf{SMod}}R}(M,N)$, we define
$f^{\Pi}\colon\Pi M\to\Pi N$ to be the morphism which agree with $f$ as a map
of sets. Here the symbol $f^{\Pi}$ is borrowed from [Man02, Chap. 3, §1.5].
This construction gives the parity change functor
$\Pi\colon\operatorname{\mathsf{SMod}}R\to\operatorname{\mathsf{SMod}}R$.

For $f\in\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,N)$, we define
$\Pi f\in\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(M,\Pi N)$ and
$f\Pi\in\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(\Pi M,N)$ by

$\displaystyle(\Pi f)(m)\coloneqq\Pi(f(m)),\quad(f\Pi)(\Pi m)\coloneqq
f(m)\quad(m\in M).$ (1.1.5)

Then we have $f^{\Pi}=\Pi f\Pi$. More generally, we have the following
isomorphisms of $S$-modules:

$\displaystyle\begin{array}[]{ccccc}\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(\Pi
M,N)&\xleftarrow{\ \sim\
}&\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(M,N)&\xrightarrow{\
\sim\ }&\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(S,\Pi
N),\\\ f\Pi&\longmapsfrom&f&\longmapsto&\Pi f.\end{array}$ (1.1.8)

###  1.2. Super derivations and the de Rham complex

Next, we introduce derivations in super setting.

######  Definition 1.2.1 (c.f. [Man02, Chap. 3, §2.8]).

Let $R\to A$ be a morphism of commutative superrings, and $M$ be an
$A$-supermodule.

  1. (1)

An _$R$ -derivation $D:A\to M$ of parity $q\in\mathbb{Z}_{2}$_ is an
$R$-homomorphism
$D\in\operatorname{Hom}_{\operatorname{\mathsf{Mod}}R}(A,M)_{q}$ of parity $q$
(where $A,M$ are regarded as supermodules) such that $D(r)=0$ for $r\in R$,
and that the super Leibniz rule holds for $a,b\in A$ of pure parity:

$D(ab)=D(a).b+(-1)^{p(a)q}a.D(b).$

An $R$-derivation of parity $\overline{0}$ (resp. of parity $\overline{1}$) is
also called an _even $R$-derivation_ (resp. an _odd $R$-derivation_).

  2. (2)

The set of $R$-derivations $A\to M$ of parity $p$ is denoted by
$\operatorname{Der}_{R}(A,M)_{p}$, which is a module. We also define a
supermodule $\operatorname{Der}_{R}(A,M)$ by

$\operatorname{Der}_{R}(A,M)\coloneqq\operatorname{Der}_{R}(A,M)_{\overline{0}}\oplus\operatorname{Der}_{R}(A,M)_{\overline{1}}.$

Note that for an even $R$-derivation $D\colon A\to M$, the maps $\Pi D\colon
A\to\Pi M$ and $D\Pi\colon\Pi A\to M$ given in (1.1.5) are odd
$R$-derivations. Conversely, if $D$ is an odd $R$-derivation, then $\Pi D$ and
$D\Pi$ are even $R$-derivations. These are checked by using the
$R$-supermodule structure (1.1.4) of $\Pi M$.

######  Notation 1.2.2.

In the case $M=A$, we denote

$\mathfrak{X}_{A/R}^{1}=\operatorname{Der}_{R}(A)\coloneqq\operatorname{Der}_{R}(A,A),$

which is a Lie superalgebra in terms of the supercommutator (1.1.2).

The supermodule $\operatorname{Der}_{R}(A,M)$ is an $A$-supermodule, since for
$a,b,c\in A$ and $D\in\operatorname{Der}_{R}(A,M)$ of pure parity, we have
$p(a)+p(D)=p(aD)$ and

$(aD)(bc)=a\bigl{(}D(b).c+(-1)^{p(b)p(D)}b.D(c)\bigr{)}=(aD)(b).c+(-1)^{p(b)p(aD)}b.(aD)(c).$

In particular, we have a functor

$\operatorname{Der}_{R}(A,\cdot)\colon\operatorname{\mathsf{SMod}}A\longrightarrow\operatorname{\mathsf{SMod}}A.$

As in the non-super case [SP, Tag 00RM, 10.131 Differentials], this functor is
represented by the _supermodule of Kähler differentials_. In the super case,
there are two versions. For the definition, recall that a _free supermodule
over a commutative superring $R$_ is a supermodule which is a module over the
underlying algebra of $R$ with basis of pure parity. See [Man02, Chap. 3,
§1.6] for the detail.

######  Definition 1.2.3.

Let $\varphi\colon R\to A$ be a morphism of commutative superrings. Consider
the free $A$-supermodules

$E\coloneqq\bigoplus_{r\in R_{\textup{pure}}}A[r]\oplus\bigoplus_{a,b\in
S_{\textup{pure}}}A[(a,b)]\oplus\bigoplus_{f,g\in
S_{\textup{pure}}}A[(f,g)],\quad F\coloneqq\bigoplus_{a\in
S_{\textup{pure}}}A[a],$

where $R_{\textup{pure}}\coloneqq R_{\overline{0}}\sqcup R_{\overline{1}}$ and
$A_{\textup{pure}}\coloneqq A_{\overline{0}}\sqcup A_{\overline{1}}$ as sets
(see Notation 1.1.1), and the parity is defined by $p([(a,b)])\coloneqq
p(a)+p(b)$, $p([r]):=p(r)$ and $p([a])\coloneqq p(a)$. We define two morphisms

$\psi_{\textup{ev}},\psi_{\textup{od}}\colon E\longrightarrow F$

of $A$-supermodules by the following rules:

$\displaystyle\begin{split}&[r]\longmapsto[\varphi(r)],\quad[(a,b)]\longmapsto[a+b]-[a]-[b],\\\
&[(f,g)]\longmapsto\begin{cases}[fg]-f[g]-(-1)^{p(f)p(g)}g[f]&(\text{for
}\psi_{\textup{ev}})\\\ [fg]-(-1)^{p(f)}f[g]-(-1)^{p(f)p(g)}g[f]&(\text{for
}\psi_{\textup{od}})\end{cases}.\end{split}$ (1.2.1)

Then we define $A$-supermodules

$\Omega_{A/R,\textup{ev}}^{1}\coloneqq\operatorname{Cok}\psi_{\textup{ev}},\quad\Omega_{A/R,\textup{od}}^{1}\coloneqq\operatorname{Cok}\psi_{\textup{od}},$

and define maps

$d_{\textup{ev}}\colon A\longrightarrow\Omega_{A/R,\textup{ev}}^{1},\quad
d_{\textup{od}}\colon A\longrightarrow\Omega_{A/R,\textup{od}}^{1},$

to be the ones sending $a$ to the class of $[a]$. We call
$\Omega_{A/R,\textup{ev}}^{1}$ and $\Omega_{A/R,\textup{od}}^{1}$ the _$A$
-supermodules of Käler differentials of $A$ over $R$_, and call
$d_{\textup{ev}}$ and $d_{\textup{od}}$ the _universal differentials_.

The module $\Omega_{A/R,\textup{ev}}^{1}$ (resp.
$\Omega_{A/R,\textup{od}}^{1}$) is spanned by elements of the form
$a.d_{\textup{ev}}b$ (resp. $a.d_{\textup{od}}b$) with $a,b\in A$,
respectively. The rules (1.2.1) yield

$\displaystyle d_{\textup{ev}}(r)=0,$ $\displaystyle
d_{\textup{ev}}(a+b)=d_{\textup{ev}}a+d_{\textup{ev}}b,$ $\displaystyle
d_{\textup{ev}}(ab)=d_{\textup{ev}}a.b+a.d_{\textup{ev}}b$ $\displaystyle
d_{\textup{od}}(r)=0,$ $\displaystyle
d_{\textup{od}}(a+b)=d_{\textup{od}}a+d_{\textup{od}}b,$ $\displaystyle
d_{\textup{od}}(ab)=d_{\textup{od}}a.b+(-1)^{p(a)}a.d_{\textup{od}}b.$

By Definition 1.2.1, $d_{\textup{ev}}$ and $d_{\textup{od}}$ are an even and
odd $R$-derivation, respectively. We also have the natural isomorphism of
$A$-supermodules

$\displaystyle\Omega_{A/R,\textup{od}}^{1}\xrightarrow{\ \sim\
}\Pi\Omega_{A/R,\textup{ev}}^{1},\quad
a.d_{\textup{ev}}b\longmapsto\Pi(a.d_{\textup{od}}b)\quad(a,b\in A)$ (1.2.2)

Here we used the symbol $\Pi(a)$ for $a\in A$ in the same meaning as in
(1.1.4) and (1.1.5). Then, by (1.1.8), we have an isomorphism
$\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(A,\Omega_{A/R,\textup{ev}}^{1})\cong\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}R}(A,\Omega_{A/R,\textup{od}}^{1})$,
which corresponds to the identity $d_{\textup{od}}=\Pi d_{\textup{ev}}$.

The following statement is an analogue of the universality of the module of
Kähler differentials in the non-super case [SP, Tag 00RM, Lemma 10.131.3].

######  Lemma 1.2.4.

Let $R\to A$ be a morphism of commutative superrings. Then we have
isomorphisms of $A$-supermodules

$\displaystyle\begin{array}[]{ccccc}\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}A}(\Omega_{A/R,\textup{od}}^{1},M)&\xleftarrow{\
\sim\
}&\operatorname{\underline{Hom}}_{\operatorname{\mathsf{SMod}}A}(\Omega_{A/R,\textup{ev}}^{1},M)&\xrightarrow{\
\sim\ }&\operatorname{Der}_{R}(A,M),\\\
\phi\Pi&\longmapsfrom&\phi&\longmapsto&\phi\circ d_{\textup{ev}},\end{array}$
(1.2.5)

which are functorial with respect to $M\in\operatorname{\mathsf{SMod}}A$.

###### Proof.

The left isomorphism is given by (1.1.8) and (1.2.2). As for the right
isomorphism, we can easily find that the map $\phi\mapsto\phi\circ
d_{\textup{ev}}$ is an injective morphism of $A$-supermodules. Given any
$D\in\operatorname{Der}_{R}(A,M)$, we have the induced map
$[D]\colon\bigoplus_{a\in A_{\textup{pure}}}A[a]\to M$, where we used the
symbols in Definition 1.2.3. Then the condition for $D$ to be an even or odd
$R$-derivation is equivalent to the one that $[D]$ annihilates the image of
the map (1.2.1). ∎

######  Notation.

Hereafter, if no confusion may arise, we simply denote

$(\Omega_{A/R}^{1},d)\coloneqq(\Omega_{A/R,\textup{ev}}^{1},d_{\textup{ev}}).$

Let us give another description of the $A$-supermodule $\Omega_{A/R}^{1}$ as
the first order neighborhood of the diagonal
$\Delta_{A/R}\subset\operatorname{Spec}A\times_{\operatorname{Spec}R}\operatorname{Spec}A=\operatorname{Spec}A\otimes_{R}A$,
which is also an analogous statement as the non-super case.

######  Lemma 1.2.5 (c.f. [SP, Tag 00RM, Lemma 10.131.13]).

Let $R\to A$ be a morphism of commutative superrings, and
$I\coloneqq\operatorname{Ker}(A\otimes_{R}A\to A)$ be the kernel of the
multiplication map. Then the map

$\Omega_{A/R}^{1}\longrightarrow I/I^{2},\quad adb\longmapsto(a\otimes
b-ab\otimes 1\bmod I^{2})$

is an isomorphism of $A$-supermodules.

###### Proof.

The argument in the non-super case works with minor changes. ∎

Finally we explain the de Rham complex in super setting. See also [Man02,
Chap. 3, §2.5, §4.4].

######  Notation 1.2.6.

Let $R$ be a commutative superring.

  * •

For an $R$-supermodule $M$, we denote by $\operatorname{Sym}_{R}^{\bullet}M$
the _symmetric superalgebra of $M$ over $R$_. It is a commutative
$R$-superalgebra equipped with $\mathbb{N}$-grading expressed by $\bullet$.

  * •

Let $\Omega_{A/R}^{1}$ be the supermodule of Kähler differentials for a
commutative $R$-superalgebra $A$. The _de Rham complex of $A$ over $R$_ is the
complex defined to be

$\Omega_{A/R}^{\bullet}\coloneqq\bigl{(}\operatorname{Sym}_{A}^{\bullet}(\Pi\Omega_{A/R}^{1}),d_{\textup{dR}}\bigr{)},$

where the _de Rham differential_
$d_{\textup{dR}}\colon\Omega_{A/R}^{p}\to\Omega_{A/R}^{p+1}$ is induced by the
odd universal differential

$\Pi d_{\textup{ev}}=d_{\textup{od}}\colon
A\longrightarrow\Pi\Omega_{A/R,\textup{ev}}^{1}=\Omega_{A/R,\textup{od}}^{1}$

in a similar fashion as in the non-super case. If we focus on the
$A$-superalgebra structure, we denote $\Omega_{A/R}$ and call it the _de Rham
superalgebra of $A$ over $R$_.

Using the terminology of [BD04, 1.1.16], we can say that
$\operatorname{Sym}_{R}^{\bullet}M$ is an $\mathbb{N}$-graded superalgebra,
and $\Omega_{A/R}^{\bullet}$ is a dg $A$-superalgebra. Henceforth we also call
$\Omega_{A/R}^{\bullet}$ the de Rham dg superalgebra of $A$ over $R$.

$\Omega_{A/R}$ enjoys the universal property in Fact 1.2.8 below.

######  Notation 1.2.7.

Let $\Lambda_{R}[\zeta]$ be the exterior algebra of one variable $\zeta$ over
$R$. In other words, it is a commutative $R$-superalgebra consisting of the
elements of the form $r+s\zeta$ for $r,s\in R$ with $\zeta$ an odd element,
and the multiplication is given by
$(r+s\zeta)(r^{\prime}+s^{\prime}\zeta)=rr^{\prime}+(rs^{\prime}+(-1)^{(p(s)+1)p(r^{\prime})}r^{\prime}s)\zeta$.

######  Fact 1.2.8 ([KV11, Proposition 3.3]).

For a morphism $R\to A$ of commutative superrings, the de Rham superalgebra
$\Omega_{A/R}$ represents the functor
$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}(A,\cdot\otimes_{R}\Lambda_{R}[\zeta])\colon\operatorname{\mathsf{SCom}}R\to\operatorname{\mathsf{Sets}}$.
In other words, there is a functorial bijection

$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}(\Omega_{A/R},B)=\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}(A,B\otimes_{R}\Lambda_{R}[\zeta]).$

The construction $A\mapsto\Omega_{A/R}$ is functorial, so we can introduce:

######  Notation 1.2.9.

Following [KV11], we call the functor the _de Rham spectrum functor_ and
denote

$\mathcal{S}(A)\coloneqq\Omega_{A/R}.$

We can apply the functor $\mathcal{S}$ repeatedly, and the obtained
superalgebra $\mathcal{S}^{N}(A)$ for $N\in\mathbb{Z}_{>0}$ is equipped with
the de Rham differentials $d_{\textup{dR}}^{i}$ for $i=1,\dotsc,N$. These are
composable, and satisfies the anti-commutation relation
$[d_{\textup{dR}}^{i},d_{\textup{dR}}^{j}]=0$. For later reference, we
introduce:

######  Notation 1.2.10.

For an ordered set $J=\\{j_{1}<\dotsb<j_{r}\\}\subset\\{1<2<\dotsb<N\\}$, we
define $d_{\textup{dR}}^{J}\colon A\to\mathcal{S}^{N}(A)$ by
$d_{\textup{dR}}^{J}\coloneqq d_{\textup{dR}}^{j_{1}}\dotsm
d_{\textup{dR}}^{j_{r}}$.

We have an obvious multi-variable analogue of Fact 1.2.8.

######  Lemma 1.2.11.

Let $\Lambda_{R}[\zeta^{1},\dotsc,\zeta^{N}]$ be exterior algebra of $N$
variables over $R$. Then the $N$-times application of the de Rham spectrum
functor $\mathcal{S}^{N}(A)$ represents the functor
$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}(A,?\otimes_{R}\Lambda_{R}[\zeta^{1},\dotsc,\zeta^{N}])$.
In other words, we have a functorial bijection

$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}\bigl{(}\mathcal{S}^{N}(A),B\bigr{)}\xrightarrow{\
\sim\
}\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}\bigl{(}A,B\otimes_{R}\Lambda_{R}[\zeta^{1},\dotsc,\zeta^{N}]\bigr{)}.$

###  1.3. Superjet and superconformal jet algebras

We give a brief recollection on _infinite jet algebras_ , which we mean the
spectrum of infinite jet affine schemes in the sense of [EM09, §2,§3], [V07]
and [BD04, 2.3.2, 2.3.3]. We present it in super setting, following [KV11,
§4.2]. We also recall the notion of _superconformal vector fields_ or _SUSY
structure_ on super curves. In this subsection, the base field $k$ is assumed
to be of characteristic $0$.

####  1.3.1. Superjet algebras

As before, let $R$ be a commutative superring, and denote by
$\operatorname{\mathsf{SCom}}R$ the category of commutative $R$-superalgebras.
Recall that in Definition 1.2.3 we used free $R$-supermodules. In the next
Definition 1.3.1, we use the _polynomial $R$-superalgebra_ $R[x_{j}\mid j\in
J]$, where the generators $x_{j}$ are of pure parity and commutative. It can
also be called the _free commutative $R$-superalgebra_ generated by $x_{j}$’s.
We also use the set $A_{\textup{pure}}\coloneqq A_{\overline{0}}\sqcup
A_{\overline{1}}$ of elements of pure parity.

######  Definition 1.3.1 ([V07, Definition 1.3]).

Let $A\in\operatorname{\mathsf{SCom}}R$, and $f\colon R\to A$ be the structure
morphism. For $m\in\mathbb{N}\sqcup\\{\infty\\}$, define a commutative
$A$-superalgebra $\mathrm{HS}_{A/R}^{m}$ to be the quotient of the polynomial
$A$-superalgebra

$A[x^{(i)}\mid x\in A_{\textup{pure}},\,i=1,\dotsc,m]$

by the ideal $I$ generated by the following terms:

$\displaystyle f(r)^{(i)}\quad$ $\displaystyle(r\in R,\,i=1,\dotsc,m),$
$\displaystyle(x+y)^{(i)}-x^{(i)}-y^{(i)}\quad$ $\displaystyle(x,y\in
A,\,i=1,\dotsc,m),$ $\displaystyle(xy)^{(i)}-\sum_{j+k=i}x^{(j)}y^{(k)}\quad$
$\displaystyle(x,y\in A,\,i=0,\dotsc,m),$

where we denote $x^{(0)}\coloneqq x$ for $x\in A$, and if $m=\infty$ we
interpret the range “$i=1,\dotsc,m$” as $i\in\mathbb{Z}_{>0}$. We call
$\mathrm{HS}_{A/R}^{m}$ the ($A$-superalgebra of) _Hasse-Schmidt derivations_.
We also define $R$-homomorphisms $d_{i}$ ($i=0,\dotsc,m$) by

$d_{i}\colon A\longrightarrow\mathrm{HS}_{A/R}^{m},\quad
x\longmapsto(x^{(i)}\bmod I),$

and call them the _universal derivations_.

######  Remark 1.3.2.

A few remarks are in order.

  1. (1)

We have $\mathrm{HS}_{A/R}^{0}\cong A$, and
$\mathrm{HS}_{A/R}^{1}\cong\operatorname{Sym}_{A}\Omega_{A/R}^{1}$, the
symmetric $A$-superalgebra of the module $\Omega_{A/R}^{1}$ of Kähler
differentials, forgetting the $\mathbb{N}$-grading in Notation 1.2.6.

  2. (2)

$\mathrm{HS}_{A/R}^{m}$ is a commutative $R$-superalgebra by the structure
morphism $f\colon R\to A$.

  3. (3)

$d_{1}$ an even derivation of the $R$-superalgebra $A$ (Definition 1.2.1). If
the base field $k$ is of characteristic $0$, then
$d_{n}=\frac{1}{m!}d_{1}^{n}$ for $n\in\mathbb{N}$.

The Hasse-Schmidt derivations $\mathrm{HS}_{A/R}^{m}$ represents the functor
of $m$-th jets, explained in the following Fact 1.3.3.

######  Notation.

Let $R$ be a commutative superring, and $z$ be an even indeterminate.

  * •

We regard the formal power series ring $R\llbracket z\rrbracket$ as

$R\llbracket z\rrbracket=\varprojlim_{m\in\mathbb{N}}R[z]/(z^{m+1}),$

and as a complete topological superring with respect to the $z$-adic topology.

  * •

We denote by $\operatorname{\mathsf{TopSCom}}R$ the category of topological
commutative $R$-superalgebras and continuous morphisms. In particular, we have
$R\llbracket z\rrbracket\in\operatorname{\mathsf{TopSCom}}R$.

######  Fact 1.3.3 ([V07, Corollary 1.8]).

For $A\in\operatorname{\mathsf{SCom}}R$ and $m\in\mathbb{N}\cup\\{\infty\\}$,
the Hasse-Schmidt derivations $\mathrm{HS}_{A/R}^{m}$ represents the functor
$\operatorname{\mathsf{SCom}}R\to\operatorname{\mathsf{Sets}}$ given by

$\begin{cases}\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}\bigl{(}A,?\otimes_{R}R[z]/(z^{m+1})\bigr{)}&(m\in\mathbb{N})\\\
\operatorname{Hom}_{\operatorname{\mathsf{TopSCom}}R}\bigl{(}A,?\otimes_{R}R[z]/(z^{m+1})\bigr{)}&(m=\infty)\end{cases}.$

More precisely, we have a functorial bijection

$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}\bigl{(}\mathrm{HS}_{A/R}^{m},B\bigr{)}\longrightarrow\operatorname{Hom}_{\operatorname{\mathsf{SCom}}R}\bigl{(}A,B[z]/(z^{m+1})\bigr{)},\quad\phi\longmapsto\bigl{(}x\mapsto{\textstyle\sum}_{i=0}^{m}z^{i}\phi(d_{i}x)\bigr{)}$

in the case $m\in\mathbb{N}$, and a similar one given by
$\phi\mapsto\bigl{(}x\mapsto\sum_{i=0}^{\infty}z^{i}\phi(d_{i}x)\bigr{)}$ in
the case $m=\infty$.

Combining Lemma 1.2.11 and Fact 1.3.3, we have Proposition 1.3.5 below. We use
the following notations.

######  Notation 1.3.4.

Let $N\in\mathbb{Z}_{>0}$, and $k$ be a field of characteristic $0$. We denote
by $[N]$ the ordered set $\\{1<2<\dotsb<n\\}$.

  * •

for $A\in\operatorname{\mathsf{SCom}}k$ we denote

$\displaystyle A_{\infty}\coloneqq\mathrm{HS}_{A/k}^{\infty}$ (1.3.1)

and call it the (_infinite_) _jet algebra_ of $A$.

  * •

Let $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ be a set of indeterminates with $z$
even and $\zeta^{j}$ odd which are (super-) commutative in each other. We call
$Z$ an _$1|N$ -supervariable_. For $j\in\mathbb{Z}$ and an ordered subset
$J=\\{j_{1}<\dotsb<j_{r}\\}\subset[N]$, we denote

$Z^{j|J}\coloneqq z^{j}\zeta^{J}=z^{j}\zeta^{j_{1}}\dotsm\zeta^{j_{r}}.$

  * •

Let $\mathfrak{m}\coloneqq(Z)$ be the ideal of the polynomial superalgebra
$k[Z]\in\operatorname{\mathsf{SCom}}k$ generated by $z$ and $\zeta^{i}$’s. The
natural projections $k[Z]/\mathfrak{m}^{n+1}\to k[Z]/\mathfrak{m}^{n}$ form a
filtered system, and we denote

$O=O^{1|N}\coloneqq k\llbracket
Z\rrbracket=\varprojlim_{n\in\mathbb{N}}k[Z]/\mathfrak{m}^{n+1},$

regarding it as a complete topological $k$-superalgebra by $\mathfrak{m}$-adic
topology.

######  Proposition 1.3.5.

Let $N\in\mathbb{Z}_{>0}$, $k$ be a field of characteristic $0$, and
$Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ be a supervariable. For
$A\in\operatorname{\mathsf{SCom}}k$, the commutative superalgebra

$A^{O}=A^{k\llbracket Z\rrbracket}\coloneqq\mathcal{S}^{N}(A_{\infty})$

represents the functor
$\operatorname{Hom}_{\operatorname{\mathsf{TopSCom}}k}(A,\cdot\otimes_{k}O)\colon\operatorname{\mathsf{SCom}}k\to\operatorname{\mathsf{Sets}}$.
More precisely, we have a functorial bijection

$\displaystyle\operatorname{Hom}_{\operatorname{\mathsf{SCom}}k}(A^{O},B)\xrightarrow{\
\sim\
}\operatorname{Hom}_{\operatorname{\mathsf{TopSCom}}k}(A,B\otimes_{k}O),\quad\phi\longmapsto\sum_{n\in\mathbb{N},\,J\subset[N]}Z^{n|J}\phi\circ
d_{n|J},$ (1.3.2)

where $d_{n|J}\colon A\to A^{O}$ is the composition

$\displaystyle d_{n|J}\coloneqq d_{\textup{dR}}^{J}\circ d_{n}$ (1.3.3)

of the universal differentials $d_{n}\colon A\to A_{\infty}$ and the product
of the de Rham differentials $d_{\textup{dR}}^{J}$ in Notation 1.2.10. We call
$\mathcal{S}^{N}(A_{\infty})$ the (_infinite_) _$1|N$ -superjet algebra_ of
$A$.

The next Lemma 1.3.6 describes a natural basis of the superalgebra $A^{O}$. We
omit the proof.

######  Lemma 1.3.6.

In the situation of Proposition 1.3.5, we denote

$a^{[n|J]}\coloneqq d_{n|J}(a)\in A^{O}\quad(a\in
A,\,n\in\mathbb{N},\,J\subset[N]).$

Then $A^{O}$ is linearly spanned by the elements of the form

$a_{1}^{[n_{1}|J_{1}]}\dotsm a_{r}^{[n_{r}|J_{r}]}\quad(a_{i}\in
A,\,n_{i}\in\mathbb{N},\,J_{i}\subset[N]).$

Let us denote $e_{i}\coloneqq\\{i\\}\subset[N]$ for each $i\in[N]$. Then, the
operations $d_{1|0}$ and $d_{0|e_{i}}$ on $A^{O}$ are even and odd
derivations, respectively. Let us record this observation as:

######  Lemma 1.3.7.

The superalgebra $A^{O}$ has an even derivation $T$ and odd derivations
$S^{i}$ for $i\in[N]$ determined by

$T(a^{[n|J]})\coloneqq(n+1)a^{[n+1|J]},\quad
S^{i}(a^{[n|J]})\coloneqq\sigma(e_{i},J)a^{[n|J\setminus e_{i}]}.$

Here $\sigma(e_{i},J)\in\\{\pm 1,0\\}$ is determined by the following rule: If
$i\in J$, then $\sigma(e_{i},J)\coloneqq 0$. Otherwise, let $e_{i}\cup J$ be
the reordered set of the union, and determine $\sigma(e_{i},J)$ by
$\zeta^{i}\zeta^{J}=\sigma(e_{i},J)\zeta^{e_{i}\cup J}$.

For later reference, let us introduce the notion of differential
superalgebras.

######  Definition 1.3.8.

The following notions are defined over some base superring $R$, but we
suppress it.

  1. (1)

A _differential superalgebra_ is a pair $(A,D)$ of a commutative superalgebra
$A$ and a derivation on $A$. A _morphism_ $(A,D)\to(B,D^{\prime})$ of
differential superalgebras is naturally defined.

  2. (2)

A subset $I\subset A$ _generates_ $(A,D)$ if every element of $A$ can be
written as a polynomial of $D^{r}a$ with $r\in\mathbb{N}$ and $a\in A$.

We have a multi-derivation analogue of these notions: A tuple $(A,D_{i})$ of a
commutative superalgebra $A$ and derivations $D_{i}$’s is also called a
differential superalgebra. Then, by Lemmas 1.3.7 and 1.3.6, we have:

######  Lemma 1.3.9.

For any $A\in\operatorname{\mathsf{SCom}}k$, $(A^{O},T,S^{i})$ is a
differential superalgebra, and it is generated by $A$.

Now the following assertion is easily shown:

######  Lemma 1.3.10.

Let $A,B\in\operatorname{\mathsf{SCom}}k$, and assume the following
conditions.

  * •

$B$ contains $A$ as a sub-superalgebra.

  * •

$B$ has an even derivation $\partial$ and $N$ odd derivations
$\not{\partial}^{i}$ for $i\in[N]$.

  * •

The differential superalgebra $(B,\partial,\not{\partial}^{i})$ is generated
by $A$.

Then the injection $A\hookrightarrow B$ can be extended to a surjective
morphism $(A^{O},T,S^{i})\twoheadrightarrow(B,\partial,\not{\partial}^{i})$ of
differential superalgebras.

Finally, let us introduce finite superjet algebras.

######  Proposition 1.3.11.

For a supervariable $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ and $m\in\mathbb{N}$,
we denote

$O_{m}=O^{1|N}_{m}\coloneqq k[Z]/(z^{m+1}).$

Then, for $A\in\operatorname{\mathsf{SCom}}k$, the commutative superalgebra

$A^{O_{m}}=A^{k[Z]/(z^{m+1})}\coloneqq\mathcal{S}^{N}(\mathrm{HS}_{A/k}^{m})$

satisfies the universal property

$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}k}(A^{O_{m}},B)\xrightarrow{\
\sim\
}\operatorname{Hom}_{\operatorname{\mathsf{SCom}}k}(A,B\otimes_{k}O_{m}),\quad\phi\longmapsto\sum_{0\leq
n\leq m,\,J\subset[N]}Z^{n|J}\phi\circ d_{n|J},$

where $d_{n|J}$ is given by the same formula as (1.3.3). We call $A^{O_{m}}$
the _$m$ -th superjet algebra_ of $A$.

Corresponding to the truncation map $O\to O_{m}$, $z^{n}\mapsto 0$ for $n\geq
m+1$, we have a morphism of commutative superalgebras
$\mathrm{HS}_{A/k}^{m}\to\mathrm{HS}_{A/k}^{\infty}=A_{\infty}$, which induces
another morphism

$\displaystyle A^{O_{m}}\longrightarrow A^{O}.$ (1.3.4)

We call it the _projection_ of superjet algebras.

####  1.3.2. Superconformal jet algebras

Let us recollect the _super Riemann surface_ structure or the _SUSY structure_
on the superdisk $O=O^{1|N}$. We denote the topological Lie superalgebra of
$k$-derivations on $O$ (Notation 1.2.2) as

$\displaystyle\operatorname{Der}O\coloneqq\operatorname{Der}_{k}(O)=\mathfrak{X}_{O/k}^{1}.$
(1.3.5)

By [FBZ04, §6.2] and [H07, §3.1.2], the set of automorphisms

$\operatorname{Aut}O\coloneqq\operatorname{Aut}_{\operatorname{\mathsf{TopSCom}}k}(O)$

is a group superscheme over $k$, and the set of $k$-valued points is described
as

$(\operatorname{Aut}O)(k)=\Bigl{\\{}Z=(z,\zeta^{i})\mapsto\Bigl{(}\sum_{n+\left|{J}\right|\geq
1}a_{n,J}Z^{n|J},\sum_{n+\left|{J}\right|\geq
1}b^{i}_{n,J}Z^{n|J}\Bigr{)}\,|\,\left[\begin{smallmatrix}a_{1,0}&a_{0,e_{i}}\\\
b^{i}_{1,0}&b^{i}_{0,e_{i}}\end{smallmatrix}\right]\in\operatorname{GL}_{k}(1|N)\Bigr{\\}}.$

The topological Lie superalgebra of $\operatorname{Aut}O$ is denoted by

$\operatorname{Der}_{0}O\coloneqq\operatorname{Lie}(\operatorname{Aut}O),$

which is linearly compact and has a topological basis
$\\{z^{n}\partial_{z},z^{n}\partial_{\zeta^{i}}\mid n\geq
1,i\in[N]\\}\cup\\{z^{m}\zeta^{i}\partial_{z},z^{m}\zeta\partial_{\zeta^{i}}\mid
m\geq 0,i\in[N]\\}$. It is actually a Lie sub-superalgebra of
$\operatorname{Der}O$ given in (1.3.5).

######  Definition 1.3.12.

Consider the even differential 1-form on $O$

$\omega_{Z}\coloneqq
dz+\sum_{i=1}^{N}\zeta^{i}d\zeta^{i}\in(\Omega^{1}_{A/R})_{\overline{1}},$

where $d=d_{\textup{ev}}$ denotes the even universal differential (Notation
1.2.6). We define $\operatorname{Aut^{sc}}O$ to be the subgroup of
$\operatorname{Aut}O$ consisting of automorphisms preserving $\omega_{Z}$ up
to multiplication by a function. Its element is called a _superconformal
transformation_ of $\mathbb{D}$. The topological Lie superalgebra of
$\operatorname{Aut^{sc}}O$ is denoted by
$\operatorname{Der^{sc}_{0}}O\coloneqq\operatorname{Lie}\operatorname{Aut^{sc}}O$,
and its element is called a _superconformal vector field_.

Let us recall basic properties of the Lie superalgebra
$\operatorname{Der^{sc}_{0}}O$, following [H07, §3.1.3] and [HK07, Example
2.12]. For $i\in[N]$, we define an odd derivation $D_{Z}^{i}$ of $O$ by

$\displaystyle D_{Z}^{i}\coloneqq\partial_{\zeta}+\zeta\partial_{z}^{i}.$
(1.3.6)

Then $\operatorname{Der^{sc}_{0}}O$ is equal to the Lie subalgebra of
$\operatorname{Der}_{0}O$

$\operatorname{Der^{sc}_{0}}O=\\{W\in\operatorname{Der}_{0}O\mid[W,D_{Z}^{i}]=fD_{Z}^{i},\
\exists f\in O\\}.$

Also, $\operatorname{Der^{sc}_{0}}O$ consists of the derivations of the form

$D^{f}=f\partial_{z}+\frac{1}{2}(-1)^{p(f)}{\textstyle\sum}_{i=1}^{N}(D_{Z}^{i}f)D_{Z}^{i}$

with $f\in O$ of pure parity. In particular, it contains

$\displaystyle
l_{n}\coloneqq-z^{n+1}\partial_{z}-\tfrac{n+1}{2}z^{n}{\textstyle\sum}_{i=1}^{N}\zeta^{i}\partial_{\zeta^{i}}$
$\displaystyle(n\in\mathbb{N}),$ $\displaystyle
g_{r}^{j}\coloneqq-z^{r+\frac{1}{2}}(\partial_{\zeta^{i}}-\zeta^{i}\partial_{z})+\bigl{(}r+\tfrac{1}{2}\bigr{)}z^{r-\frac{1}{2}}\zeta^{j}{\textstyle\sum}_{i=1}^{N}\zeta^{i}\partial_{\zeta^{i}}$
$\displaystyle(r\in\mathbb{N}+\tfrac{1}{2},\,j\in[N]).$

In the case $N=1$, the elements $l_{n}$ ($n\in\mathbb{N}$) and $g_{r}\coloneqq
g_{r}^{1}$ ($r\in\frac{1}{2}+\mathbb{N}$) are topological generators of
$\operatorname{Der^{sc}_{0}}O$, subject to the relations

$\displaystyle[l_{m},l_{n}]=(m-n)l_{m+n},\quad[l_{n},g_{r}]=(\tfrac{n}{2}-r)g_{n+r},\quad[g_{r},g_{s}]=2l_{r+s}.$

Thus, in the case $N=1$, $\operatorname{Der^{sc}_{0}}O$ is a Lie sub-
superalgebra of the _Neveu-Schwarz algebra_ of central charge $0$ (see
(2.3.9)).

Recall that the superjet algebra $A^{O}$ enjoys a universal property:

$\operatorname{Hom}_{\operatorname{\mathsf{SCom}}k}(A^{O},k)\xrightarrow{\
\sim\ }\operatorname{Hom}_{\operatorname{\mathsf{SCom}}k}(A,O).$

Now we would like to find the odd derivations $S_{K}^{i}$ on $A^{O}$ which
corresponds to the action of
$D_{Z}^{i}=\partial_{\zeta^{i}}+\zeta^{i}\partial_{z}$ on $O$. Let
$\phi(Z)=\sum_{(n|J),\,n\geq 0}Z^{n|J}\phi_{n|J}$ be the series corresponding
to an element
$\phi\in\operatorname{Hom}_{\operatorname{\mathsf{SCom}}k}(A,O)$. Then, we
have

$(D_{Z}^{i}\phi)(Z)=\sum_{(n|J),\,n\geq 0,\,i\notin
J}Z^{n|J}\sigma(e_{i},J)\phi_{n|J\cup e_{i}}+\sum_{(n|J),\,n\geq 0,\,i\in
J}Z^{n|1}(n+1)\sigma(e_{i},J\setminus e_{i})\phi_{n+1|J\setminus e_{i}},$

where we used the same symbols as in Lemma 1.3.7. This calculation indicates
that the desired $S_{K}^{i}$ is given by

$\displaystyle
S_{K}^{i}(a^{[n|J]})\coloneqq\begin{cases}\sigma(e_{i},J)a^{[n|J\cup
e_{i}]}&(i\notin J)\\\ (n+1)\sigma(e_{i},J\setminus e_{i})a^{[n+1|J\setminus
e_{i}]}&(i\in J)\end{cases}.$ (1.3.7)

Using this action twice, we see that the square $T_{K}=(S_{K}^{i})^{2}$ is
independent of $i$, and its action is given by

$\displaystyle T_{K}(a^{[n|J]})=(n+1)a^{[n+1|J]}.$ (1.3.8)

Then, we immediately have:

######  Lemma 1.3.13.

The operators $S_{K}^{i}$’s and $T_{K}$ form a Lie superalgebra with
commutation relation

$[S_{K}^{i},S_{K}^{j}]=2\delta_{i,j}T_{K},\quad[T_{K},S_{K}^{i}]=0.$

Recalling Definition 1.3.8, we find that $(A^{O},S_{K}^{i})$ is a differential
superalgebra for any $A\in\operatorname{\mathsf{SCom}}k$. Consulting
Definition 1.3.12, we may name:

######  Definition 1.3.14.

For $A\in\operatorname{\mathsf{SCom}}k$, the differential superalgebra
$(A^{O},S_{K}^{i})$ is called the _superconformal jet algebra_ of $A$, and
denoted by $A^{O_{\textup{sc}}}$.

By Lemma 1.3.6, we have:

######  Lemma 1.3.15.

For any $A\in\operatorname{\mathsf{SCom}}k$, the superconformal jet algebra
$A^{O_{\textup{sc}}}$ is generated by $A$ as a differential superalgebra.

Then we have the following analogous of Lemma 1.3.10:

######  Lemma 1.3.16.

Let $A,B\in\operatorname{\mathsf{SCom}}k$, and assume the following
conditions.

  * •

$B$ contains $A$ as a linear sub-superalgebra.

  * •

$B$ has $N$ odd derivations $\not{\partial}^{i}$ ($i\in[N]$) which form the
Lie superalgebra in Lemma 1.3.13.

  * •

The differential superalgebra $(B,\not{\partial}^{i})$ is generated by $A$.

Then the injection $A\hookrightarrow B$ can be extended to a surjective
morphism $(A^{O},S_{K}^{i})\twoheadrightarrow(B,\not{\partial}^{i})$ of
differential superalgebras.

##  2\. SUSY vertex algebras

This section gives a recollection of SUSY vertex algebras introduced in
[HK07]. We also give a few preliminary observation on commutative SUSY vertex
algebras. We will work over a fixed field $k$ of characteristic $0$. Also we
fix a positive integer $N$, and denote the ordered set $\\{1<\dotsb<N\\}$ by
$[N]$.

###  2.1. Superfields

Here we explain the formalism of superfields used throughout the rest of this
note. The original reference is [HK07, §2.6].

Let us continue to use Notation 1.3.4. So $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$
is a $1|N$-supervariable with even $z$ and odd $\zeta^{i}$’s, and denote

$Z^{j|J}\coloneqq
z^{j}\zeta^{J}=z^{j}\zeta^{j_{1}}\zeta^{j_{2}}\dotsm\quad(j\in\mathbb{Z},\,J=\\{j_{1},j_{2},\dotsc\\}\subset[N]).$

We also use a simplified notation

$Z^{j|N}\coloneqq Z^{j|[N]}=z^{j}\zeta^{1}\dotsm\zeta^{N}.$

Below we will repeatedly use the sign notation given in [HK07, §3.1.1]. For
disjoint ordered sets $I,J\subset[N]$, we define $\sigma(I,J)\in\\{\pm 1\\}$
by

$\displaystyle\zeta^{I}\zeta^{J}=\sigma(I,J)\zeta^{I\cup J},$ (2.1.1)

where $I\cup J$ denotes the re-ordered set. If $I\cap J\neq\emptyset$, then we
define $\sigma(I,J)\coloneqq 0$. Also, for $i\in[N]$ and $J\subset[N]$, we use
the following abbreviations.

$\displaystyle e_{i}\coloneqq\\{i\\},\quad N\setminus J\coloneqq[N]\setminus
J.$ (2.1.2)

For example, we denote $\sigma(N\setminus J,e_{i})\coloneqq\sigma([N]\setminus
J,\\{i\\})$.

For a linear superspace $V$, we define $V\llbracket Z\rrbracket$ and
$V\llbracket Z^{\pm 1}\rrbracket$ to be the linear spaces of series with
coefficients in $V$:

$V\llbracket
Z\rrbracket\coloneqq\Biggl{\\{}\sum_{n\in\mathbb{N},\,J\subset[N]}Z^{n|J}v_{n|J}\
\Bigg{|}\ v_{n|J}\in V\Biggr{\\}},\quad V\llbracket Z^{\pm
1}\rrbracket\coloneqq\Biggl{\\{}\sum_{n\in\mathbb{Z},\,J\subset[N]}Z^{j|J}v_{j|J}\
\Bigg{|}\ v_{j|J}\in V\Biggr{\\}}.$

Here the summations are possibly infinite, and the index $J$ runs over ordered
subset in $[N]$. These are linear superspaces by setting the parity as
$p(Z^{j|J}v)\coloneqq\left|{J}\right|+p(v)\bmod 2$ for $v\in V$ of pure
parity. We also denote

$V(\\!(Z)\\!)\coloneqq\Biggl{\\{}\sum_{j\in\mathbb{Z},\,J\subset[N]}Z^{j|J}v_{j|J}\in
V\llbracket Z^{\pm 1}\rrbracket\ \Bigg{|}\ v_{j|J}=0\ \forall j\ll
0\Biggr{\\}}.$

Let
$\operatorname{\underline{End}}(V)\coloneqq\operatorname{\underline{Hom}}(V,V)$
be the internal end in the category of linear superspaces (see (1.1.3)). In
particular, it is a linear superspace. An _$\operatorname{\underline{End}}(V)$
-valued superfield_ of $1|N$-supervariable $Z$ is a series of the form

$\displaystyle a(Z)=\sum_{(j|J)}Z^{-1-n|N\setminus
J}a_{(j|J)}\quad(a_{(j|J)}\in\operatorname{\underline{End}}(V)),$ (2.1.3)

where the indices run over the range $j\in\mathbb{Z}$, $J\subset[N]$, and
$N\setminus J\coloneqq[N]\setminus J$, such that for each $v\in V$ we have
$a(Z)v\in V(\\!(Z)\\!)$. We denote the linear superspace of
$\operatorname{\underline{End}}(V)$-valued superfields of supervariable $Z$ by

$\operatorname{SF}(V,Z)=\operatorname{SF}(V).$

Next, we recall the locality of superfields [HK07, §3.1.2]. Let us given two
(super-)commuting $1|N$-supervariables $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ and
$W=(w,\omega_{1},\dotsc,\omega_{N})$, and let $A$ be a superalgebra. An _$A$
-valued formal distribution in two variables_ $Z$ and $W$ is a (possibly
infinite) series of the form

$a(Z,W)=\sum_{(i|I),(j|J)}Z^{i|I}W^{j|J}a_{i|I,j|J}\quad(a_{i|I,j|J}\in A)$

where the running index means $i,j\in\mathbb{Z}$ and $I,J\subset[N]$, and we
wrote $Z^{i|I}W^{j|J}\coloneqq
z^{i}\zeta^{I}w^{j}\omega^{J}=z^{i}w^{j}\zeta^{I}\omega^{J}$. We denote by
$A\llbracket Z^{\pm 1},W^{\pm 1}\rrbracket$ the linear superspace of
$A$-valued formal distributions. A formal distribution $a(Z,W)\in A\llbracket
Z^{\pm 1},W^{\pm 1}\rrbracket$ is called _local_ if there exists
$n\in\mathbb{N}$ such that

$\displaystyle(z-w)^{n}a(Z,W)=0.$ (2.1.4)

Note that the left hand side is well-defined in $A\llbracket Z^{\pm 1},W^{\pm
1}\rrbracket$.

###  2.2. $N_{W}=N$ SUSY vertex algebras

In this subsection, we cite from [HK07, §3] the notion of $N_{W}=N$ SUSY
vertex algebras. We continue to use the notation in §2.1. In particular,
$Z=(z,\zeta^{i})=(z,\zeta^{1},\dotsc,\zeta^{N})$, and $W=(w,\omega^{i})$ are
$1|N$-supervariables.

####  2.2.1. Definition and basic properties

######  Definition 2.2.1 ([HK07, Definition 3.3.1]).

An _$N_{W}=N$ SUSY vertex algebra_ (_$N_{W}=N$ SUSY VA_ for short) is a data
$(V,\ket{0},T,S^{i},Y)$ consisting of

  * •

a linear superspace $V=V_{\overline{0}}\oplus V_{\overline{1}}$, called the
_state superspace_ ,

  * •

an even vector $\ket{0}\in V_{\overline{0}}$, called the _vacuum_ ,

  * •

an even operator $T\in\operatorname{End}(V)_{\overline{0}}$,

  * •

odd operators $S^{i}\in\operatorname{End}(V)_{\overline{1}}$ for $i\in[N]$,
and

  * •

an even linear map $Y(\cdot,Z)\colon V\to\operatorname{SF}(V,Z)$, called the
_state-superfield correspondence_ ,

which satisfies the following axioms.

  * •

_Vacuum axiom_ : For any $a\in V$ and $i\in[N]$, we have

$\displaystyle Y(a,Z)\ket{0}=a+O(Z),\quad T\ket{0}=S^{i}\ket{0}=0,$ (2.2.1)

where $O(Z)$ is an element of $V\llbracket Z\rrbracket$ which vanishes at
$Z=0$ (i.e., $z=\zeta^{i}=0$).

  * •

_Translation invariance_ : For each $a\in V$, we have

$\displaystyle[T,Y(a,Z)]=\partial_{z}Y(a,Z),\quad[S,Y(a,Z)]=\partial_{\zeta^{i}}Y(a,Z)\quad(i\in[N]),$
(2.2.2)

where we denoted $\partial_{z}\coloneqq\tfrac{\partial}{\partial z}$ and
$\partial_{\zeta^{i}}\coloneqq\tfrac{\partial}{\partial\zeta^{i}}$, and used
the supercommutator (1.1.2).

  * •

_Locality axiom_ : For any $a,b\in V$, there exists an $n\in\mathbb{N}$ such
that

$\displaystyle(z-w)^{n}[Y(a,Z),Y(b,W)]=0.$ (2.2.3)

The identity is regarded as that of $(\operatorname{End}V)\llbracket Z^{\pm
1},W^{\pm 1}\rrbracket$ (see (2.1.4)).

We abbreviate $V=(V,\ket{0},T,S^{i},Y)$ if no confusion may arise.

For $a\in V$, $j\in\mathbb{Z}$ and an ordered subset $J\subset[N]$, we define
the _Fourier mode_ or the _$(j|J)$ -operator_
$a_{(j|J)}\in\operatorname{End}(V)$ by the expansion

$\displaystyle Y(a,Z)=\sum_{(j|J)}Z^{-1-j|N\setminus J}a_{(j|J)},$ (2.2.4)

where we used similar symbols as in (2.1.3). We also denote
$a_{(j|N)}\coloneqq a_{(j|[N])}$.

Using $(j|J)$-operators, the vacuum axiom (2.2.1) is equivalent to

$\displaystyle a_{(-1|N)}\ket{0}=a,\quad
a_{(n|J)}\ket{0}=0\quad(n\in\mathbb{N},\,J\subset[N]),$ (2.2.5)

and by [HK07, (3.3.1.5)], the translation invariance (2.2.2) is equivalent to

$\displaystyle[T,a_{(j|J)}]=-ja_{(j-1|J)},\quad[S^{i},a_{(j|J)}]=\begin{cases}\sigma(N\setminus
J,e_{i})a_{(j|J\setminus e_{i})}&(i\in J)\\\ 0&(i\notin J)\end{cases},$
(2.2.6)

where we used the sign $\sigma$ in (2.1.1) and the abbreviations (2.1.2).

As in the even case, the locality axiom (2.2.3) implies operator product
expansion (OPE for short). To explain it, let us cite some notation on formal
delta functions for $N_{W}=N$ SUSY VAs from [HK07, §2.3]. Let
$Z=(z,\zeta^{i})$ and $W=(w,\omega^{i})$ be two commuting
$1|N$-supervariables. We denote by

$\displaystyle i_{z,w}\colon k(\\!(Z,W)\\!)\lhook\joinrel\longrightarrow
k(\\!(Z)\\!)(\\!(W)\\!),\quad i_{w,z}\colon
k(\\!(Z,W)\\!)\lhook\joinrel\longrightarrow k(\\!(W)\\!)(\\!(Z)\\!)$ (2.2.7)

the embeddings obtained by the expansions with respect to $\frac{w}{z}$ (in
the domain $\left|{z}\right|>\left|{w}\right|$), and with respect to
$\frac{z}{w}$ (in the domain $\left|{z}\right|<\left|{w}\right|$),
respectively. For $j\in\mathbb{Z}$ and an ordered subset
$J=\\{j_{1}<\dotsb<j_{r}\\}\subset[N]$, we set

$\displaystyle(Z-W)^{j|J}\coloneqq(z-w)^{j}(\zeta-\omega)^{J}=(z-w)^{j}(\zeta^{j_{1}}-\omega^{j_{1}})\dotsm(\zeta^{j_{r}}-\omega^{j_{r}}).$
(2.2.8)

The formal $\delta$-function for $N_{W}=N$ case is given by

$\displaystyle\delta(Z,W)\coloneqq(i_{z,w}-i_{w,z})(Z-W)^{-1|N}=(i_{z,w}-i_{w,z})\frac{(\zeta-\omega)^{N}}{z-w}=(i_{z,w}-i_{w,z})\frac{(\zeta^{1}-\omega^{1})\dotsm(\zeta^{N}-\omega^{N})}{z-w}.$
(2.2.9)

For $n\in\mathbb{N}$ and $J=\\{j_{1}<\dotsb<j_{r}\\}\subset[N]$, let

$\displaystyle\partial_{W}^{n|J}\coloneqq\partial_{w}^{n}\partial_{\omega^{j_{1}}}\dotsm\partial_{\omega^{j_{r}}},\quad\partial_{W}^{(n|J)}\coloneqq\frac{(-1)^{\binom{r+1}{2}}}{j!}\partial_{W}^{j|J}.$
(2.2.10)

Then, by [HK07, §3.1.4 (4)], we have

$\displaystyle\partial_{W}^{(n|J)}\delta(Z,W)=(i_{z,w}-i_{w,z})(Z-W)^{-1-j|N\setminus
J}.$ (2.2.11)

Finally, for $f(Z)=\sum_{(j|J)}Z^{j|J}f_{j|J}\in V\llbracket Z^{\pm
1}\rrbracket$, we denote

$\displaystyle\operatorname{res}_{Z}f(Z)\coloneqq f_{-1|N}.$ (2.2.12)

Then, by [HK07, Lemma 3.1.3], any local distribution $a(Z,W)$ is decomposed in
the following finite sum.

$\displaystyle a(Z,W)=\sum_{(j|J),\,j\geq
0}\bigl{(}\partial_{W}^{(j|J)}\delta(Z,W)\bigr{)}c_{j|J}(W),\quad
c_{j|J}(W)\coloneqq\operatorname{res}_{Z}(Z-W)^{j|J}a(Z,W).$ (2.2.13)

Now, let $V$ be an $N_{W}=N$ SUSY VA, and $a,b\in V$ of pure parity. Then the
locality axiom (2.2.3) and the expansion (2.2.13) imply the following OPE
formula [HK07, Theorem 3.3.8 (5)]:

$\displaystyle[Y(a,Z),Y(b,W)]=\sum_{(j|J),\,j\geq
0}\bigl{(}\partial_{Z}^{(j|J)}\delta(Z,W)\bigr{)}Y(a_{(j|J)}b,W).$ (2.2.14)

As in the even case, an $N_{W}=N$ SUSY VA $V$ is called _strongly generated_
by a subset $\\{a^{i}\mid i\in I\\}\subset V$ if $V$ is spanned by the
elements of the form $a^{i_{1}}_{(-n_{1}|J_{1})}\dotsm
a^{i_{r}}_{(-n_{r}|J_{r})}\ket{0}$ with $r\in\mathbb{N}$,
$n_{i}\in\mathbb{Z}_{\geq 1}$ and $J_{i}\subset[N]$.

We also have the notion of a _module_ over an $N_{W}=N$ SUSY VA $V$. We omit
the detailed definition, and only give the notation: Let $M$ be a $V$-module.
For $a\in V$ and $m\in M$, we denote the $V$-action on $M$ by

$\displaystyle Y^{M}(a,Z)m=\sum_{(j|J)}Z^{-1-j|N\setminus J}a^{M}_{(j|J)}m\in
M(\\!(Z)\\!).$ (2.2.15)

####  2.2.2. Conformal $N_{W}=N$ SUSY vertex algebras

We cite from [HK07, §5] some examples of $N_{W}=N$ SUSY vertex algebras. To
explain those examples, we use the $\Lambda$-bracket introduced by [HK07],
which efficiently encodes OPE (2.2.14).

######  Notation 2.2.2 ([HK07, §3.2.1]).

Let $V$ be an $N_{W}=N$ SUSY VA.

  * •

We denote the commutative superalgebra freely generated by even $\lambda$ and
odd $\chi^{i}$ ($i\in[N]$) as

$\mathscr{L}_{W}=k[\Lambda]=k[\lambda,\chi^{1},\dotsc,\chi^{N}].$

  * •

For $m\in\mathbb{N}$ and $M=\\{m_{1}<\dotsb<m_{r}\\}\subset[N]$, we denote
$\Lambda^{m|M}\coloneqq\lambda^{m}\chi^{M}=\lambda^{m}\chi^{m_{1}}\dotsm\chi^{m_{r}}$.

  * •

Finally, for $a,b\in V$, we define
$[a_{\Lambda}b]\in\mathscr{L}_{W}\otimes_{k}V$ by

$[a_{\Lambda}b]=\sum_{(j|J),\,j\geq 0}\sigma(J,N\setminus
J)(-1)^{\binom{\left|{J}\right|+1}{2}}\frac{1}{j!}\Lambda^{j|J}a_{(j|J)}b,\quad\Lambda^{j|J}\coloneqq\lambda^{j}\chi^{j_{1}}\dotsm\chi^{j_{r}}$

We call $[a_{\Lambda}b]$ the _$\Lambda$ -bracket_.

In the rest of this § 2.2.2, we assume the base field $k=\mathbb{C}$.

######  Example 2.2.3 ([HK07, Example 5.1]).

$V(\mathscr{W}_{N})$ is an $N_{W}=N$ SUSY vertex algebra strongly generated by
an element $L$ of parity $N\bmod 2$ and elements $Q^{i}$, $i\in[N]$, of parity
$N+1\bmod 2$ whose OPEs are

$\displaystyle[L_{\Lambda}L]=(T+2\lambda)L,\quad[Q^{i}_{\Lambda}Q^{j}]=(S^{i}+\chi^{i})Q^{j}-\chi^{j}Q^{i},\quad[L_{\Lambda}Q^{i}]=(T+\lambda)Q^{i}+(-1)^{N}\chi^{i}L.$
(2.2.16)

If $N=1,2$, then it admits a central extension. For $N=1$, it is given by

$\displaystyle[L_{\Lambda}L]=(T+2\lambda)L,\quad[Q_{\Lambda}Q]=SQ+\frac{\lambda\chi}{3}C,\quad[L_{\Lambda}Q]=(T+\lambda)Q-\chi
L+\frac{\lambda^{2}}{6}C.$ (2.2.17)

For $N=2$, we have a central extension

$\displaystyle\begin{split}&[L_{\Lambda}L]=(T+2\lambda)L,\quad[L_{\Lambda}Q^{i}]=(T+\lambda)Q^{i}+\chi^{i}L\\\
&[Q^{i}_{\Lambda}Q^{j}]=S^{i}Q^{j},\quad[Q^{1}_{\Lambda}Q^{2}]=(S^{1}+\chi^{1})Q^{2}-\chi^{2}Q^{1}+\frac{\lambda}{6}{C}.\end{split}$
(2.2.18)

$V(\mathscr{W}_{N})$ is an $N_{W}=N$ SUSY analogue of Virasoro vertex algebra.
Recall that an even vertex algebra is called conformal if it has an element
which generates Virasoro vertex algebra. The following is its $N_{W}=N$ SUSY
analogue.

######  Definition 2.2.4 (First part of [HK07, Definition 5.2]).

An $N_{W}=N$ SUSY VA $(V,\ket{0},T,S^{i},Y)$ over $\mathbb{C}$ is called
_conformal_ if it has elements $\nu,\tau^{1},\dotsc,\tau^{N}$ satisfying the
following conditions.

  * •

The superfields $L(Z)\coloneqq Y(\nu,Z)$ and $Q^{i}(Z)\coloneqq Y(\tau^{i},Z)$
satisfy the OPEs (2.2.16) (or the central extension (2.2.17), (2.2.18)).

  * •

We have $\nu_{(0|0)}=2T$ and $\tau^{i}_{(0|0)}=S^{i}$.

  * •

The operator $\nu_{(1|0)}$ is semisimple and the eigenvalues $d$ are bounded
below, i.e., there is a finite subset
$\\{d_{1},\dotsc,d_{s}\\}\subset\mathbb{C}$ such that $d\in
d_{i}+\mathbb{R}_{\geq 0}$ for some $i$.

We call the elements $\nu$ and $\tau^{i}$ the _conformal elements_.

For later use, let us also cite:

######  Definition 2.2.5 (Second part of [HK07, Definition 5.2]).

An $N_{W}=N$ SUSY VA $V$ is called _strongly conformal_ if it is conformal and
the operators $\nu_{(1|0)}$ and $\sum_{i=1}^{N}\sigma(e_{i},N\setminus
e_{i})\tau^{i}_{(0|e_{i})}$ have integer eigenvalues. We call the eigenvalue
$\Delta$ of $\nu_{(1|0)}$ the _conformal weight_.

As mentioned in [HK07] and extensively studied in [H07], for a strongly
conformal $N_{W}=N$ SUSY VA $V$, we can construct a vector bundle on an
arbitrary super Riemann surface whose fiber is $V$.

####  2.2.3. Identities in $N_{W}=N$ SUSY vertex algebras

In this part, we explain basic identities valid in $N_{W}=N$ SUSY vertex
algebras, In particular, we give analogue for Borcherds’ commutator formula
(Fact 2.2.9) and iterate formula (Lemma 2.2.10), which will be used for the
discussion of Li filtration in § 3.2 and § 3.3.

Throughout this subsection, $V=(V,\ket{0},T,S^{i},Y)$ denotes an $N_{W}=N$
SUSY VA. For a supervariable $Z=(z,\zeta^{i})$, we denote

$Z\nabla\coloneqq zT+{\textstyle\sum}_{i=1}^{N}\zeta^{i}S^{i}.$

We also use the formal exponential $e^{X}=\exp
X\coloneqq\sum_{n\in\mathbb{N}}\frac{1}{n!}X^{n}$.

Using these relations and following the argument of the even (or non-super)
case in [FBZ04, 3.1.1–3.1.7], we can show the following statements.

######  Fact 2.2.6 ([HK07, Proposition 3.3.6 (1), Theorem 3.3.8 (3), (4),
Corollary 3.3.9]).

For $a\in V$ and $i\in[N]$, we have the following.

  1. (1)

$Y(a,Z)\ket{0}=e^{Z\nabla}a=e^{zT}(1+\zeta^{1}S^{1})\dotsm(1+\zeta^{N}S^{N})a$.

  2. (2)

Using the supercommutator, we have

$\displaystyle Y(Ta,Z)=\partial_{z}Y(a,Z)=[T,Y(a,Z)],\quad
Y(S^{i}a,Z)=\partial_{\zeta^{i}}Y(a,Z)=[S^{i},Y(a,Z)].$ (2.2.19)

In terms of the Fourier modes, we have

$\displaystyle(Ta)_{(j|J)}=-ja_{(j-1|J)},\quad(S^{i}a)_{(j|J)}=\sigma(e_{i},N\setminus
J)a_{(j|J\setminus e_{i})}.$ (2.2.20)

Next, we recall that operators $T$ and $S^{i}$ can be regarded as even and odd
derivations, respectively.

######  Fact 2.2.7 ([HK07, Corollary 3.3.9]).

For $a,b\in V$ of pure parity, $j\in\mathbb{Z}$, $J\subset[N]$ and $i\in[N]$,
we have

$T(a_{(j|J)}b)=(Ta)_{(j|J)}b+a_{(j|J)}(Tb),\quad
S^{i}(a_{(j|J)}b)=(-1)^{N\setminus
J}\bigl{(}(S^{i}a)_{(j|J)}b+(-1)^{p(a)}a_{(j|J)}(S^{i}b)\bigr{)}$

with $(-1)^{N\setminus J}\coloneqq(-1)^{N-\left|{J}\right|}$.

Let us also recall the skew-symmetry of the state-superfield correspondence.

######  Fact 2.2.8 ([HK07, Proposition 3.3.12]).

For $a,b\in V$ of pure parity, we have

$Y(a,Z)b=(-1)^{p(a)p(b)}e^{Z\nabla}Y(b,-Z)a.$

Now, recall Borcherds’ commutator formula in the even case [L05, (2.1)],
[FBZ04, §3.3.10, p.56]:

$\displaystyle[a_{(l)},b_{(m)}]=\sum_{j\in\mathbb{N}}\binom{l}{j}(a_{(j)}b)_{(l+m-j)}.$
(2.2.21)

We have the following $N_{W}=N$ SUSY analogue.

######  Fact 2.2.9 ([HK07, Proposition 3.3.18, (3.3.4.13)]).

For $a,b\in V$, $l\in\mathbb{Z}$ and $L\subset[N]$, we have

$[a_{(l|L)},Y(b,W)]=\sum_{(j|J),\,j\geq
0}(-1)^{\left|{J}\right|\left|{L}\right|+\left|{J}\right|N+\left|{L}\right|N}\partial_{W}^{(j|J)}W^{l|L}Y(a_{(j|J)},W)$

with $\partial_{W}^{(j|J)}$ given in (2.2.10). In terms of Fourier modes, we
have

$\displaystyle[a_{(l|L)},b_{(m|M)}]=\sum_{\begin{subarray}{c}(j|J),\\\ j\geq
0,\,J\supset L\cap
M\end{subarray}}(-1)^{\alpha}\widetilde{\sigma}\cdot\binom{l}{j}(a_{(j|J)}b)_{(l+m-j|M\cup(L\setminus
J))},$ (2.2.22)

where we denoted

$\displaystyle\alpha$
$\displaystyle\coloneqq(p(a)+N-\left|{L}\right|)(N-\left|{M}\right|)+(\left|{L}\right|-\left|{J}\right|)(n-\left|{J}\right|),$
$\displaystyle\widetilde{\sigma}$ $\displaystyle\coloneqq\sigma(J,N\setminus
J)\sigma(L,N\setminus L)\sigma(J,L\setminus J)\sigma(L\setminus J,(N\setminus
M)\setminus(L\setminus J)).$

Note that the $J=1$ part of the last formula coincides with the even case
formula (2.2.21).

Finally, let us discuss an $N_{W}=N$ SUSY analogue of Borcherds’ iterate
formula. The even case is

$\displaystyle(a_{(k)}b)_{(l)}=\sum_{j\geq
0}(-1)^{j}\binom{k}{j}(a_{(k-j)}b_{(l+j)}-(-1)^{k}b_{(k+l-j)}a_{(j)})$
(2.2.23)

for $a,b$ of an even vertex algebra and $k,l\in\mathbb{Z}$ (see [L05, (2.2)]
and [FBZ04, §3.3.10, p.56]). Here and hereafter, we denote
$a_{(k)}b_{(l)}u\coloneqq a_{(k)}(b_{(l)}u)$ for $u$ of a vertex algebra
module, and in a similar way for the super case.

######  Lemma 2.2.10.

Let $V$ be an $N_{W}=N$ SUSY VA and $M$ be a $V$-module. For $a,b\in V$ of
pure parity, $u\in M$, $k,l\in\mathbb{Z}$ and $K,L\subset[N]$, we have

$\displaystyle(a_{(k|K)}b)_{(l|L)}u=\sum_{\begin{subarray}{c}(j|J),\\\ j\geq
0,\,J\subset
K\end{subarray}}(-1)^{j+\alpha}\widetilde{\sigma}\cdot\binom{k}{j}\Bigl{(}(-1)^{(p(a)+N-\lvert{J}\rvert)\lvert{N\setminus
J^{\prime}}\rvert}a_{(k-j|J)}b_{(l+j|J^{\prime})}u$
$\displaystyle-(-1)^{k+p(a)p(b)+(p(b)+N-\lvert{J}\rvert)\lvert{N\setminus
J^{\prime}}\rvert}b_{(k+l-j|J^{\prime})}a_{(j|J)}u\Bigr{)}$

with $J^{\prime}\coloneqq L\cup(K\setminus J)$ and

$\displaystyle\alpha\coloneqq\left|{K\setminus J}\right|(1+\left|{N\setminus
J}\right|),\quad\widetilde{\sigma}\coloneqq\sigma(J,K\setminus
J)\sigma(J,N\setminus J)\sigma(L,K\setminus J)\sigma(J^{\prime},N\setminus
J^{\prime}).$

###### Proof.

Let $a,b\in V$ be elements of pure parity, and recall the OPE (2.2.14):

$[Y(a,Z),Y(b,W)]=\sum_{(j|J),\,j\geq
0}\bigl{(}\partial_{Z}^{j|J}\delta(Z,W)\bigr{)}Y(a_{(j|J)}b,W).$

Following the argument in the even case [FBZ04, §3.3.10], we multiply both
sides of this equality by some $f(Z,W)\in
k\bigl{[}Z^{1|J},W^{1|J},(Z-W)^{-1|J}\mid J\subset[N]\bigr{]}$. Then, using
(2.2.11), we have

$\displaystyle\begin{split}&\operatorname{res}_{W}\operatorname{res}_{Z}f(Z,W)a(Z)b(W)-(-1)^{p(a)p(b)}\operatorname{res}_{W}\operatorname{res}_{Z}f(Z,W)b(W)a(Z)\\\
&=\operatorname{res}_{W}\operatorname{res}_{Z-W}\sum_{(j|J),\,j\geq
0}f(Z,W)(Z-W)^{-1-j|N\setminus J}Y(a_{(j|J)}b,W).\end{split}$ (2.2.24)

To show Lemma 2.2.10, we set $f(Z,W)\coloneqq W^{l|L}(Z-W)^{k|K}$. Let us
calculate the first term in the left hand side. We have

$\displaystyle\operatorname{res}_{Z}(Z-W)^{k|K}a(Z)$
$\displaystyle=\operatorname{res}_{Z}(z-w)^{k}(\zeta-\omega)^{K}\sum_{(j|J)}z^{-1-j}\zeta^{N\setminus
J}a_{(j|J)}$
$\displaystyle=\operatorname{res}_{Z}\Bigl{[}(z-w)^{k}\cdot\sum_{M\subset
K}(-1)^{\left|{K\setminus M}\right|}\sigma(M,K\setminus
M)\zeta^{M}\omega^{K\setminus M}\cdot\sum_{(j|J)}z^{-1-j}\zeta^{N\setminus
J}a_{(j|J)}\Bigr{]}$
$\displaystyle=\operatorname{res}_{Z}\Bigl{[}\sum_{M\subset
K}\sum_{(j|J)}(-1)^{\beta}\tau\cdot(z-w)^{k}z^{-1-j}\zeta^{M\cup(N\setminus
J)}\omega^{K\setminus M}a_{(j|J)}\Bigr{]}$

with $\beta\coloneqq\left|{K\setminus M}\right|+\left|{K\setminus
M}\right|\left|{N\setminus J}\right|$ and $\tau\coloneqq\sigma(M,K\setminus
M)\sigma(M,N\setminus J)$. By definition of $\operatorname{res}_{Z}$, the
terms with $J=M$ survive, and

$\displaystyle\operatorname{res}_{Z}(Z-W)^{k|K}a(Z)=\sum_{\begin{subarray}{c}(j|J),\\\
j\geq 0,\,J\subset
K\end{subarray}}(-1)^{\beta+k-j}\tau\cdot\binom{k}{j}w^{k-j}\omega^{K\setminus
J}a_{(j|J)}.$

Then we have

$\displaystyle\operatorname{res}_{W}\operatorname{res}_{Z}W^{l|L}(Z-W)^{k|K}a(Z)b(W)$
$\displaystyle=\operatorname{res}_{W}\sum_{\begin{subarray}{c}(j|J),\\\ j\geq
0,\,J\subset
K\end{subarray}}(-1)^{\beta+k-j}\tau\cdot\binom{k}{j}W^{l|L}w^{k-j}\omega^{K\setminus
J}a_{(j|J)}\sum_{(p|P)}W^{-1-p|N\setminus P}b_{(p|P)}$
$\displaystyle=\sum_{\begin{subarray}{c}(j|J),\\\ j\geq 0,\,J\subset
K\end{subarray}}(-1)^{k-j+\beta}\upsilon\cdot\binom{k}{j}(-1)^{(p(a)+N-\lvert{J}\rvert)\lvert{N\setminus
J^{\prime}}\rvert}a_{(j|J)}b_{(l+k-j|J^{\prime})}$ (2.2.25)

with $J^{\prime}\coloneqq L\cup(K\setminus J)$ and

$\beta=\left|{K\setminus J}\right|+\left|{K\setminus
J}\right|\left|{N\setminus
J}\right|=\alpha,\quad\upsilon\coloneqq\tau\cdot\sigma(L,K\setminus
J)\sigma(J^{\prime},N\setminus J^{\prime})=\widetilde{\sigma}.$

Replacing $(j,k-j)\mapsto(k-j,j)$, we have $k-j+\alpha\mapsto j+\alpha$, and
(2.2.25) is equal to the first term in the right hand side of the statement.
By a similar calculation, we see the second terms in (2.2.24) and the
statement coincide. For the right hand side of (2.2.24), we have by (2.2.11)
that
$\operatorname{res}_{Z-W}\partial_{W}^{(j|J)}\delta(Z,W)=\delta_{j,0}\delta_{J,\emptyset}$,
which yields

$\operatorname{res}_{W}\operatorname{res}_{Z-W}\sum_{(j|J)}W^{l|L}(Z-W)^{k|K}(Z-W)^{-1-j|1-J}Y(a_{(j|J)}b,W)=(a_{(k|K)}b)_{(l|L)}.$

Hence we have the consequence. ∎

###  2.3. $N_{K}=N$ SUSY vertex algebras

In this subsection, we recall the notion of $N_{K}=N$ SUSY vertex algebras
introduced in [HK07, §4]. Throughout this subsection, we fix a positive
integer $N$.

####  2.3.1. Definition and basic properties

######  Notation 2.3.1 ([HK07, 4.1]).

For a $1|N$-supervariable $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ and $i\in[N]$, we
define odd derivations $D_{Z}^{i}$ and $\overline{D}_{Z}^{i}$ by

$D_{Z}^{i}\coloneqq\partial_{\zeta^{i}}+\zeta^{i}\partial_{z},\quad\overline{D}_{Z}^{i}\coloneqq\partial_{\zeta^{i}}-\zeta^{i}\partial_{z}.$

We have $(D_{Z}^{i})^{2}=\partial_{z}$ and
$(\overline{D}_{Z}^{i})^{2}=-\partial_{z}$.

######  Definition 2.3.2 ([HK07, Definition 4.13]).

An _$N_{K}=N$ SUSY vertex algebra_ is a data $(V,\ket{0},S_{K}^{i},Y)$
consisting of

  * •

a linear superspace $V=V_{\overline{0}}\oplus V_{\overline{1}}$, called the
_state superspace_ ,

  * •

an even element $\ket{0}\in V_{\overline{0}}$, called the _vacuum_ ,

  * •

$N$ odd operators $S_{K}^{i}\in\operatorname{End}(V)_{\overline{1}}$
($i\in[N]$),

  * •

an even linear map $Y(\cdot,Z)\colon V\to\operatorname{SF}(V,Z)$, called the
_state-superfield correspondence_ ,

which satisfies the following axioms.

  1. (1)

Vacuum axiom: $Y(a,Z)\ket{0}=a+O(Z)$ for all $a\in V$, and $S\ket{0}=0$.

  2. (2)

Translation invariance: For all $a\in V$, we have

$\displaystyle[S_{K}^{i},Y(a,Z)]=\overline{D}^{i}_{Z}Y(a,Z).$ (2.3.1)

  3. (3)

Locality axiom: For all $a,b\in V$, there is an $n\in\mathbb{N}$ such that
$(z-w)^{n}[Y(a,Z),Y(b,W)]=0$.

We abbreviate $(V,\ket{0},S_{K}^{i},Y)$ to $V$ if no confusion may arise.
Also, we often abbreviate the word “vertex algebra” to VA.

As in the $N_{W}=N$ case, we define the _Fourier mode_ or the _$(j|J)$
-operator_ $a_{(j|J)}\in\operatorname{End}(V)$ for $a\in V$, $j\in\mathbb{Z}$
and $J\subset[N]$ by the expansion (2.2.4). Then the vacuum axiom is
equivalent to the same formula as in (2.2.5), and by [HK07, (4.6.1)]111There
is a typo in the second half of [HK07, (4.6.1)]., the translation invariance
(2.3.1) is equivalent to

$\displaystyle[S_{K}^{i},a_{(j|J)}]=\begin{cases}\sigma(N\setminus
J,e_{i})a_{(j|J\setminus e_{i})}&(i\in J)\\\ -\sigma(N\setminus(J\cup
e_{i}),e_{i})ja_{(j-1|J\cup e_{i})}&(i\notin J)\end{cases}.$ (2.3.2)

We have $[S_{K}^{i},[S_{K}^{i},a_{(j|J)}]]=-ja_{(j-1|J)}$ for every $i\in[N]$,
and the super Jacobi identity implies that

$T\coloneqq\tfrac{1}{2}[S_{K}^{i},S_{K}^{i}]=(S_{K}^{i})^{2}\in\operatorname{\underline{End}}(V)_{\overline{0}}$

gives a well-defined even operator which satisfies

$\displaystyle[T,a_{(j|J)}]=-ja_{(j-1|J)}\quad(j\in\mathbb{Z},\,J\subset[N]),\quad\text{
i.e., }\ [T,Y(a,Z)]=\partial_{z}Y(a,Z).$ (2.3.3)

The operators $S_{K}^{i}$’s and $T$ satisfy the commutation relation

$[S_{K}^{i},S_{K}^{j}]=2\delta_{i,j}T,\quad[T,S_{K}^{i}]=0,$

which is the Lie superalgebra in Lemma 1.3.13.

Now let us explain the OPE formula for $N_{K}=N$ SUSY VAs, following [HK07,
§4.1, §4.2]. It involves a new delta function which is different from the
$N_{W}=N$ case (2.2.9). Let $Z=(z,\zeta^{1},\dotsc,\zeta^{N})$ and
$W=(w,\omega^{1},\dotsc,\omega^{N})$ be two commuting $1|N$-supervariables,
and let $i_{z,w}\colon k(\\!(Z,W)\\!)\hookrightarrow k(\\!(Z)\\!)(\\!(W)\\!)$
and $i_{w,z}\colon k(\\!(Z,W)\\!)\hookrightarrow k(\\!(W)\\!)(\\!(Z)\\!)$ be
the embeddings in (2.2.7). For $j\in\mathbb{Z}$ and $J\subset[N]$, we set

$\displaystyle(Z-W)_{K}^{j|J}\coloneqq\bigl{(}z-w-{\textstyle\sum}_{i=1}^{N}\zeta^{i}\omega^{i}\bigr{)}^{j}\bigl{(}\zeta-\omega\bigr{)}^{J}.$
(2.3.4)

Then the formal $\delta$-function for the $N_{K}=N$ case is defined to be

$\delta_{K}(Z,W)\coloneqq(i_{z,w}-i_{w,z})(Z-W)_{K}^{-1|N}=(i_{z,w}-i_{w,z})\frac{\zeta-\omega}{z-w-{\textstyle\sum}_{i=1}^{N}\zeta^{i}\omega^{i}}=(i_{z,w}-i_{w,z})\frac{\zeta-\omega}{z-w},$

where in the last equality we used
$(Z-W)_{K}^{-1|0}=\frac{1}{z-w-\sum_{i}\zeta^{i}\omega^{i}}=\frac{1+\sum_{i}\zeta^{i}\omega^{i}}{z-w}$.

The binomial (2.3.4) behaves with respect to the odd derivations
$D_{W}^{i}\coloneqq\partial_{\zeta^{i}}+\zeta^{i}\partial_{w}$ in the same way
as in the even case. For $n\in\mathbb{N}$ and
$J=\\{j_{1}<\dotsb<j_{r}\\}\subset[N]$, let

$\displaystyle D_{W}^{n|J}\coloneqq\partial_{w}^{n}D_{W}^{j_{1}}\dotsm
D_{W}^{j_{r}},\quad
D_{W}^{(n|J)}\coloneqq(-1)^{\binom{r+1}{2}}\tfrac{1}{n!}D_{W}^{n|J}.$ (2.3.5)

Then, by [HK07, Lemma 4.1], we have

$\displaystyle
D_{W}^{(n|J)}\delta_{K}(Z,W)=(i_{z,w}-i_{w,z})(Z-W)_{K}^{-1-j|N\setminus J}.$
(2.3.6)

Also, by [HK07, Lemma 4.4], any local distribution $a(Z,W)$ is decomposed as

$\displaystyle a(Z,W)=\sum_{(j|J),\,j\geq
0}\bigl{(}D_{W}^{(j|J)}\delta_{K}(Z,W)\bigr{)}c_{j|J}(W),\quad
c_{j|J}(W)\coloneqq\operatorname{res}_{Z}(Z-W)_{K}^{j|J}a(Z,W),$ (2.3.7)

where $\operatorname{res}_{Z}$ is defined in (2.2.12).

Now, by [HK07, Theorem 4.16 (4)], the OPE formula for an $N_{K}=N$ SUSY VA is

$\displaystyle[Y(a,Z),Y(b,W)]=\sum_{(j|J),\,j\geq
0}\bigl{(}D_{Z}^{j|J}\delta_{K}(Z,W)\bigr{)}Y(a_{(j|J)}b,W).$ (2.3.8)

We also have the notion of a _module_ $M$ over an $N_{K}=N$ SUSY VA $V$. We
will use a similar notation for the $V$-action on $M$ as in the $N_{K}=N$ case
(2.2.15).

####  2.3.2. Conformal $N_{K}=N$ SUSY vertex algebras

Here we give some examples of $N_{K}=N$ SUSY vertex algebras. We use the
following $\Lambda$-bracket to encode OPE (2.3.8), which is different from the
one for $N_{W}=N$ case (Notation 2.2.2).

######  Notation 2.3.3 ([HK07, (3.3.4.6)]).

Let $V$ be an $N_{K}=N$ SUSY VA. We define the _$\Lambda$ -bracket_
$[a_{\Lambda}b]$ for $a,b\in V$ as follows.

  * •

Let $\mathscr{L}_{K}$ be the (non-commutative) superalgebra generated by even
$\lambda$ and odd $\chi^{i}$ ($i\in[N]$) subject to the commutation relation

$[\lambda,\chi^{i}]=0,\quad[\chi^{i},\chi^{j}]=-2\lambda\delta_{i,j}.$

  * •

For $m\in\mathbb{N}$ and $M=\\{m_{1}<\dotsb<m_{r}\\}\subset[N]$, we denote
$\Lambda^{m|M}\coloneqq\lambda^{m}\chi^{M}=\lambda^{m}\chi^{m_{1}}\dotsm\chi^{m_{r}}$.

  * •

Finally, for $a,b\in V$, we define $[a\Lambda
b]\in\mathscr{L}_{K}\otimes_{k}V$ by

$[a_{\Lambda}b]\coloneqq\sum_{(m|M),\,m\geq 0}\sigma(M,N\setminus
M)(-1)^{\binom{\left|{M}\right|+1}{2}}\frac{1}{m!}\Lambda^{m|M}a_{(m|M)}b.$

Let us start with a standard example of $N_{K}=1$ SUSY VA, which comes from an
$N=1$ superconformal vertex superalgebra. In the $N_{K}=1$ case, a
supervariable is denoted by $Z=(z,\zeta)$, and a superfield is expressed as

$\displaystyle Y(a,Z)$
$\displaystyle=\sum_{j\in\mathbb{N},\,J=0,1}Z^{-1-j|1-J}a_{(j|J)}=\sum_{j\in\mathbb{N},\,J=0,1}z^{-1-j}\zeta^{1-J}a_{(j|J)}$
$\displaystyle=\sum_{j\in\mathbb{N}}z^{-1-j}a_{(j|1)}+\sum_{j\in\mathbb{N}}z^{-1-j}\zeta
a_{(j|0)}.$

Also, we denote the odd operator by $S_{K}$, and its square by $T$.

######  Example 2.3.4.

Let $V$ be an $N=1$ superconformal vertex superalgebra in the sense of [K98,
p.180]. Thus, it is a vertex superalgebra $(V,\ket{0},T,Y^{\textup{cl}})$ over
$\mathbb{C}$ together with an odd element $\tau\in V$ whose Fourier modes in
$Y^{\textup{cl}}(\tau,z)=\sum_{r\in\mathbb{Z}+\frac{1}{2}}z^{-r-\frac{3}{2}}G_{r}$
satisfy the following conditions.

  1. (1)

$G_{-\frac{1}{2}}\tau=2\nu$ with $\nu$ a Virasoro element, i.e., an even
element $\nu\in V$ whose Fourier modes in
$Y^{\textup{cl}}(\nu,z)=\sum_{n\in\mathbb{Z}}z^{-n-2}L_{n}$ satisfy
$L_{-1}=T$, $L_{0}\tau=\frac{3}{2}\tau$ and $L_{0}$ being diagonalizable.

  2. (2)

$G_{\frac{3}{2}}\tau=\frac{2}{3}c\ket{0}$ for some $c\in\mathbb{C}$.

  3. (3)

$G_{r}\tau=0$ for $k>\frac{3}{2}$.

We call $\tau$ the _Neveu-Schwarz element_ of $V$ (which is called the $N=1$
superconformal element in [K98, Definition 5.9]).

The above Fourier modes form the _Neveu-Schwarz algebra_ of central charge
$c$. It is a Lie superalgebra with central extension which is generated by odd
$G_{r}$ ($r\in\mathbb{Z}+\frac{1}{2}$) and even $L_{n}$ ($n\in\mathbb{Z}$)
subject to the following commutation relations.

$\displaystyle\begin{split}&[G_{r},G_{s}]=2L_{r+s}+\tfrac{1}{3}(r^{2}-\tfrac{1}{4})\delta_{r+s,0}c,\qquad[G_{r},L_{n}]=(r-\tfrac{n}{2})G_{n+r},\\\
&[L_{m},L_{n}]=(m-n)L_{m+n}+\delta_{m+n,0}\tfrac{m^{3}-m}{12}c.\end{split}$
(2.3.9)

Such $V$ has a structure $(\ket{0},S_{K},Y)$ of $N_{K}=1$ SUSY VA. The vacuum
$\ket{0}$ is the same one, the state-superfield correspondence $Y$ is given by

$Y(a,Z)\coloneqq Y^{\textup{cl}}(a,z)+\zeta
Y^{\textup{cl}}(G_{-\frac{1}{2}}a,z)$

for $a\in V$, and the odd operator is given by $S_{K}\coloneqq
G_{-\frac{1}{2}}=\tau_{(0|1)}$.

######  Example 2.3.5.

As a special case in Example 2.3.4, we have the _Neveu-Schwarz SUSY vertex
algebra_ whose state superspace $V$ is the linear superspace of polynomials
$\mathbb{C}[S_{K}^{n}\tau\mid n\in\mathbb{N}]$ and the vacuum is given by
$\ket{0}=1$. The state-superfield correspondence for the Neveu-Schwarz element
$\tau$ is

$Y(\tau,Z)=\sum_{r\in\mathbb{Z}+\frac{1}{2}}z^{-r-\frac{3}{2}}G_{r}+2\sum_{n\in\mathbb{Z}}z^{-n-2}\zeta
L_{n},$

and the Fourier modes

$\displaystyle\tau_{(j|1)}=G_{j-\frac{1}{2}},\quad\tau_{(j|0)}=2L_{j-1}\quad(j\in\mathbb{Z})$
(2.3.10)

satisfy the commutation relations (2.3.9). We call
$\nu\coloneqq\frac{1}{2}S_{K}\tau\in V$ the Virasoro element of $V$. Using the
$\Lambda$-bracket (Notation 2.3.3), the OPE of $Y(\tau,Z)$ is summarized as

$[\tau_{\Lambda}\tau]=(2T+\chi
S_{K}+3\lambda)\tau+\frac{c}{3}\lambda^{2}\chi\ket{0}.$

For later reference, we give some formulas.

$\displaystyle\tau=\tau_{(-1|1)}\ket{0}=G_{-\frac{3}{2}}\ket{0},\quad\nu=\tfrac{1}{2}(S_{K}\tau)_{(-1|1)}\ket{0}=\tfrac{1}{2}\tau_{(-1|0)}\ket{0}=L_{-2}\ket{0}.$
(2.3.11)

In the latter calculation, we used the recursion (2.3.12) given below.

The above Example 2.3.4 can be extended to higher $N_{K}$ cases.

######  Definition 2.3.6 (First part of [HK07, Definition 5.6]).

For $N\in\\{1,2,3,4\\}$, an $N_{K}=N$ SUSY VA $(V,\ket{0},S_{K}^{i},Y)$ is
called _conformal_ if it has an element $\tau\in V$ of parity $N\bmod 2$,
satisfying the following conditions.

  * •

Using Notation 2.3.3 of the $\Lambda$-bracket, the OPE of $Y(\tau,Z)$ is given
by

$[\tau_{\Lambda}\tau]=\bigl{(}2t+(4-N)\lambda+{\textstyle\sum}_{i=1}^{N}\chi^{i}S_{K}^{i}\bigr{)}\tau+\begin{cases}\frac{c}{3}\lambda^{3-N}\chi^{N}\ket{0}&(N\leq
3)\\\ \lambda c\ket{0}&(N=4)\end{cases}.$

  * •

Using (2.1.2), we have

$\tau_{(0|0)}=2T,\quad\tau_{(0|e_{i})}=\sigma(N\setminus
e_{i},e_{i})S_{K}^{i}.$

  * •

The operator $\tau_{(1|0)}$ is semisimple and the eigenvalues $d$ are bounded
below, i.e., there is a finite subset
$\\{d_{1},\dotsc,d_{s}\\}\subset\mathbb{C}$ such that $d\in
d_{i}+\mathbb{R}_{\geq 0}$ for some $i$.

We call $\tau$ the _conformal element_.

######  Remark.

The condition $N\leq 4$ comes from the fact that there is no central extension
of the Lie superalgebra corresponding to the $\Lambda$-bracket
$[\tau_{\Lambda}\tau]=(2t+(4-N)\lambda+{\textstyle\sum}_{i=1}^{N}\chi^{i}S_{K}^{i})\tau$.

For later use, let us also cite:

######  Definition 2.3.7 (Second part of [HK07, Definition 5.6]).

An $N_{K}=N$ SUSY VA $V$ is called _strongly conformal_ if it is conformal and
satisfies the following conditions.

  * •

$\tau_{(1|0)}$ has integer eigenvalues.

  * •

If $N=2$, then the operator $\sqrt{-1}\tau_{(0|N)}$ has integer eigenvalues.

As in the $N_{W}=N$ case (Definition 2.2.5), the strongly conformal condition
guarantees us to construct a vector bundle on an arbitrary super Riemann
surface whose fiber is $V$. See [H07] for the detail.

####  2.3.3. Identities in $N_{K}=N$ SUSY vertex algebras

We explain basic identities for $N_{K}=N$ case.. Throughout this part,
$V=(V,\ket{0},S_{K}^{i},Y)$ denotes an $N_{K}=N$ SUSY VA, and

$Z\nabla\coloneqq zT+{\textstyle\sum}_{i=1}^{N}\zeta^{i}S_{K}^{i}$

for the operators $T=(S_{K}^{i})^{2}$ and $S_{K}^{i}$ of $V$.

The following is an $N_{K}=N$ analogue of Fact 2.2.6.

######  Fact 2.3.8 ([HK07, Proposition 4.15 (1), Theorem 4.16 (3), Corollary
4.18]).

For $a\in V$, we have:

  1. (1)

$Y(a,Z)\ket{0}=e^{Z\nabla}a=e^{zT}(1+\zeta^{1}S_{K}^{1})\dotsm(1+\zeta^{N}S_{K}^{N})a$.

  2. (2)

$Y(Ta,Z)=\partial_{z}Y(a,Z)$ and
$Y(S_{K}^{i}a,Z)=D_{Z}^{i}Y(a,Z)=(\partial_{\zeta^{i}}+\zeta^{i}\partial_{z})Y(a,Z)$.
In terms of the Fourier modes, we have

$\displaystyle(Ta)_{(j|J)}=-ja_{(j-1|J)},\quad(S_{K}^{i}a)_{(j|J)}=\begin{cases}\sigma(N\setminus
J,e_{i})a_{(j|J\setminus e_{i})}&(i\in J)\\\ -j\sigma(N\setminus(J\cup
e_{i}),e_{i})a_{(j-1|J\cup e_{i})}&(i\notin J)\end{cases}.$ (2.3.12)

######  Remark 2.3.9.

Some comments are in order.

  1. (1)

As noted in [HK07, Remark 4.17] and [H09, Remark 3.3], we have

$Y(Ta,Z)=\partial_{z}Y(a,Z)=[T,Y(a,Z)],\quad
Y(S_{K}^{i}a,Z)=D_{Z}^{i}Y(a,Z)\neq\overline{D}_{Z}^{i}Y(a,Z)=[S_{K}^{i},Y(a,Z)].$

The inequality $Y(S_{K}^{i}a,Z)\neq[S_{K}^{i},Y(a,Z)]$ should be contrasted
with the $N_{W}=N$ case (2.2.19).

  2. (2)

By the equivalent form (2.2.5) of the vacuum axiom, we can recover the odd
operators $S_{K}^{i}$ from $Y$ in the way

$\displaystyle S_{K}^{i}a=(S_{K}^{i}a)_{(-1|N)}\\!\ket{0}=a_{(-1|N\setminus
e_{i})}\\!\ket{0}.$ (2.3.13)

Thus the operators $S_{K}^{i}$ can be omitted from Definition 2.3.2.
Similarly, $T$ is recovered as

$Ta=(Ta)_{(-1|N)}\\!\ket{0}=a_{(-2|N)}\\!\ket{0},$

which is a reminiscent of the even case formula $Ta=a_{(-2)}\\!\ket{0}$.

Next we recall that the odd operator $S_{K}^{i}$ can be regarded as an
derivation.

######  Fact 2.3.10 ([HK07, Corollary 4.18]).

For $a,b\in V$ of pure parity and $i\in[N]$, we have

$S_{K}^{i}(a_{(j|J)}b)=(-1)^{N\setminus
J}\bigl{(}(S_{K}^{i}a)_{(j|J)}b+(-1)^{p(a)}a_{(j|J)}(S_{K}^{i}b)\bigr{)}$

with $(-1)^{N\setminus J}\coloneqq(-1)^{N-\left|{J}\right|}$.

Let us also recall the skew-symmetry of the state-superfield correspondence.

######  Fact 2.3.11 ([HK07, Proposition 3.3.12, §4.8]).

For $a,b\in V$ of pure parity, we have

$Y(a,Z)b=(-1)^{p(a)p(b)}e^{Z\nabla}Y(b,-Z)a.$

Next, we have the following $N_{K}=N$ SUSY analogue of Borcherds’ commutator
formula
$[a_{(l)},b_{(m)}]=\sum_{j\in\mathbb{N}}\binom{l}{j}(a_{(j)}b)_{(l+m-j)}$ in
(2.2.21).

######  Fact 2.3.12 ([HK07, Proposition 3.3.18, (4.3.3), §4.10]).

For $a,b\in V$, $l\in\mathbb{Z}$ and $L\subset[N]$, we have

$[a_{(l|L)},Y(b,W)]=\sum_{(j|J),\,j\geq
0}(-1)^{\left|{J}\right|\left|{L}\right|+\left|{J}\right|N+\left|{L}\right|N}D_{W}^{(j|J)}W^{l|L}Y(a_{(j|J)},W),$

where $D_{W}^{(j|J)}$ is given in (2.3.5). In terms of Fourier modes, we have

$\displaystyle[a_{(l|L)},b_{(m|M)}]=\sum_{\begin{subarray}{c}(j|J),\,j\geq
0,\\\ M\cap(J\Delta
L)=\emptyset\end{subarray}}(-1)^{\alpha+\beta}\widetilde{\sigma}\cdot\frac{(l)^{\downarrow}_{j+\\#(J\setminus
L)}}{j!}(a_{(j|J)}b)_{(l+m-j-\\#(J\setminus L)|M\cup(J\Delta L))},$ (2.3.14)

where we denoted $(l)^{\downarrow}_{n}\coloneqq l(l-1)\dotsm(l-n+1)$ for
$n\in\mathbb{N}$, $J\Delta L\coloneqq(J\setminus L)\cup(L\setminus J)$ and

$\displaystyle\alpha$
$\displaystyle\coloneqq(p(a)+N-\left|{L}\right|)(N-\left|{M}\right|),\quad\beta\coloneqq\left|{J}\right|(N-\left|{L}\right|)+\left|{L}\right|N+\binom{\left|{J\cap
L}\right|}{2}+\binom{\left|{J}\right|+1}{2},$
$\displaystyle\widetilde{\sigma}$ $\displaystyle\coloneqq\sigma\bigl{(}J\Delta
L,N\setminus(M\cup(J\Delta L))\bigr{)}\sigma(J,N\setminus
J)\sigma(L,N\setminus L)\sigma(J\setminus L,J\cap L)\sigma(J\cap L,L\setminus
J)\sigma(J\setminus L,L\setminus J).$

In the $N_{K}=1$ case, (2.3.14) reduces to

$\displaystyle[a_{(l|0)},b_{(m|0)}]=\sum_{j\in\mathbb{N},\,J=0,1}(-1)^{\binom{J}{2}}\frac{(l)^{\downarrow}_{(j+J)}}{j!}(a_{(j|J)}b)_{(l+m-j-J|J)}\times(-1)^{p(a)+1},$
$\displaystyle[a_{(l|0)},b_{(m|1)}]=\sum_{j\in\mathbb{N},\,J=0,1}(-1)^{\binom{J}{2}}\frac{(l)^{\downarrow}_{(j+J)}}{j!}(a_{(j|J)}b)_{(l+m-j-J|1)},$
$\displaystyle[a_{(l|1)},b_{(m|0)}]=\sum_{j\in\mathbb{N},\,J=0,1}(-1)^{J+1}\binom{l}{j}(a_{(j|J)}b)_{(l+m-j|1-J)}\times(-1)^{p(a)},$
$\displaystyle[a_{(l|1)},b_{(m|1)}]=\sum_{j\in\mathbb{N},\,J=0,1}(-1)^{J+1}\binom{l}{j}(a_{(j|J)}b)_{(l+m-j|1)}$

for $a,b\in V$ of pure parity and $m,n\in\mathbb{Z}$. Note that the $J=1$ part
of $[a_{(l|1)},b_{(m|1)}]$ coincides with the even case formula (2.2.21).

Finally, let us discuss an $N_{K}=N$ SUSY analogue of Borcherds’ iterate
formula. Recall the even case formula (2.2.23): $(a_{(k)}b)_{(l)}=\sum_{j\geq
0}(-1)^{j}\binom{k}{j}(a_{(k-j)}b_{(l+j)}-(-1)^{k}b_{(k+l-j)}a_{(j)})$.

######  Lemma 2.3.13.

Let $V$ be an $N_{K}=N$ SUSY VA, and $W$ be a $V$-module. For $a,b\in V$ of
pure parity, $w\in W$, $k,l\in\mathbb{Z}$ and $K,L\subset[N]$, we have

$\displaystyle(a_{(k|K)}b)_{(l|L)}m=\sum_{(j|J),\,j\geq 0}\sum_{M\subset J\cap
K}$
$\displaystyle(-1)^{j+\alpha}\widetilde{\sigma}\cdot\binom{k}{j,\left|{J\setminus
M}\right|}$ $\displaystyle\Bigl{(}$
$\displaystyle(-1)^{(p(a)+N-\lvert{J}\rvert)\lvert{N\setminus
L^{\prime}}\rvert}a_{(k-j-\\#(J\setminus M)|J)}b_{(l+j|L^{\prime})}w$
$\displaystyle-$
$\displaystyle(-1)^{k+p(a)p(b)+(p(b)+N-\lvert{J}\rvert)\lvert{N\setminus
L^{\prime}}\rvert}b_{(l+k-j-\\#(J\setminus M)|L^{\prime})}a_{(j|J)}w\Bigr{)},$

where $\binom{k}{i,j}\coloneqq\binom{k}{i}\binom{k-i}{j}$,
$L^{\prime}\coloneqq L\cup(J\setminus M)\cup(K\setminus M)$ and

$\displaystyle\alpha\coloneqq\binom{\left|{J\setminus
M}\right|+1}{2}+\left|{J\setminus M}\right|(\left|{M}\right|+\left|{N\setminus
J}\right|)+\left|{K\setminus M}\right|(1+\left|{N\setminus
J}\right|)+\left|{N\setminus J}\right|\left|{N\setminus L^{\prime}}\right|,$
$\displaystyle\widetilde{\sigma}\coloneqq\sigma(M,K\setminus
M)\sigma(J\setminus M,M)\sigma(J,N\setminus J)\sigma(J\setminus M,K\setminus
M)\sigma(L,(J\setminus M)\cup(K\setminus M))\sigma(L^{\prime},N\setminus
L^{\prime}).$

###### Proof.

The strategy is the same as in Lemma 2.2.10. Recall the OPE for the $N_{K}=N$
case (2.3.8):

$[Y(a,Z),Y(b,W)]=\sum_{(j|J),\,j\geq
0}\bigl{(}D_{Z}^{j|J}\delta(Z,W)\bigr{)}Y(a_{(j|J)}b,W).$

For each $f\in k[Z^{1|J},W^{1|J},(Z-W)_{K}^{-1|J}\mid J\subset[N]]$, it yields
the equality

$\displaystyle\begin{split}&\operatorname{res}_{W}\operatorname{res}_{Z}f(Z,W)a(Z)b(W)-(-1)^{p(a)p(b)}\operatorname{res}_{W}\operatorname{res}_{Z}f(Z,W)b(W)a(Z)\\\
&=\operatorname{res}_{W}\operatorname{res}_{(Z-W)_{K}}\sum_{(j|J),\,j\geq
0}f(Z,W)(Z-W)^{-1-j|N\setminus J}Y(a_{(j|J)}b,W),\end{split}$ (2.3.15)

where $\operatorname{res}_{(Z-W)_{K}}$ means
$\operatorname{res}_{(z-w-\sum_{j=1}^{N}\zeta^{j}\omega^{j},\zeta^{i}-\omega^{i})}$.
We apply this equality for

$f(Z,W)\coloneqq
W^{l|L}(Z-W)_{K}^{k|K}=W^{l|L}(z-w-{\textstyle\sum}_{i=1}^{N}\zeta^{j}\omega^{j})^{k}(\zeta-\omega)^{K}.$

The first term in the left hand side of (2.3.15) is

$\displaystyle\operatorname{res}_{Z}(Z-W)_{K}^{k|K}a(Z)$
$\displaystyle=\operatorname{res}_{Z}(z-w-{\textstyle\sum}_{i=1}^{N}\zeta^{i}\omega^{i})^{k}(\zeta-\omega)^{K}\sum_{(j|J)}z^{-1-j}\zeta^{N\setminus
J}a_{(j|J)}$
$\displaystyle=\operatorname{res}_{Z}\Bigl{[}\sum_{I\subset[N]}(-1)^{\binom{\left|{I}\right|+1}{2}}\binom{k}{\left|{I}\right|}(z-w)^{k-\left|{I}\right|}\zeta^{I}\omega^{I}$
$\displaystyle\hskip 30.00005pt\cdot\sum_{M\subset K}(-1)^{\left|{K\setminus
M}\right|}\sigma(M,K\setminus M)\zeta^{M}\omega^{K\setminus
M}\cdot\sum_{(j|J)}z^{-1-j}\zeta^{N\setminus J}a_{(j|J)}\Bigr{]}$
$\displaystyle=\operatorname{res}_{Z}\Bigl{[}\sum_{I\subset[N]}\sum_{M\subset
K}\sum_{(j|J)}(-1)^{\beta}\tau\binom{k}{\left|{I}\right|}(z-w)^{k-\left|{I}\right|}z^{-1-j}\zeta^{I\cup
M\cup(N\setminus J)}\omega^{I\cup(K\setminus M)}a_{(j|J)}\Bigr{]}$

with $\beta\coloneqq\binom{\left|{I}\right|+1}{2}+\left|{K\setminus
M}\right|+\left|{I}\right|\left|{M}\right|+(\left|{I}\right|+\left|{K\setminus
M}\right|)\left|{N\setminus J}\right|$ and $\tau\coloneqq\sigma(M,K\setminus
M)\sigma(I,M)\sigma(I\cup M,N\setminus J)\sigma(I,K\setminus M)$. By
definition of $\operatorname{res}_{Z}$, the terms with $J=I\sqcup M$ survive,
and

$\displaystyle\operatorname{res}_{Z}(Z-W)_{K}^{k|K}a(Z)=\sum_{(j|J)}\sum_{M\subset
J\cap K}(-1)^{\gamma}\tau\binom{k}{\left|{J\setminus
M}\right|}\binom{k-\left|{J\setminus M}\right|}{j}w^{k-\left|{J\setminus
M}\right|-j}\omega^{(J\setminus M)\cup(K\setminus M)}a_{(j|J)},$
$\displaystyle\gamma\coloneqq\beta+k-\left|{J\setminus
M}\right|-j=k-j+\binom{\left|{J\setminus M}\right|}{2}+\left|{J\setminus
M}\right|(\left|{M}\right|+\left|{N\setminus J}\right|)+\left|{K\setminus
M}\right|(1+\left|{N\setminus J}\right|),$
$\displaystyle\tau=\sigma(M,K\setminus M)\sigma(J\setminus
M,M)\sigma(J,N\setminus J)\sigma(J\setminus M,K\setminus M).$

Then we have

$\displaystyle\operatorname{res}_{W}\operatorname{res}_{Z}W^{l|L}(Z-W)_{K}^{k|K}a(Z)b(W)$
$\displaystyle=\operatorname{res}_{W}\sum_{(j|J)}\sum_{M\subset J\cap
K}(-1)^{\gamma}\tau\binom{k}{j,\left|{J\setminus
M}\right|}W^{l|L}w^{k-j-\left|{J\setminus M}\right|}\omega^{(J\setminus
M)\cup(K\setminus M)}a_{(j|J)}\sum_{(p|P)}W^{-1-p|N\setminus P}b_{(p|P)}$
$\displaystyle=\sum_{(j|J)}\sum_{M\subset J\cap
K}(-1)^{\gamma+\left|{N\setminus J}\right|\lvert{N\setminus
L^{\prime}}\rvert}\upsilon\binom{k}{j,\left|{J\setminus
M}\right|}(-1)^{(p(a)+N-\lvert{J}\rvert)\lvert{N\setminus
L^{\prime}}\rvert}a_{(j|J)}b_{(l+k-j-\\#(J\setminus M)|L^{\prime})},$ (2.3.16)
$\displaystyle L^{\prime}\coloneqq L\cup(J\setminus M)\cup(K\setminus
M),\quad\upsilon\coloneqq\tau\cdot\sigma(L,(J\setminus M)\cup(K\setminus
M))\sigma(L^{\prime},N\setminus L^{\prime})=\widetilde{\sigma}.$

Replacing $k-j-\left|{J\setminus M}\right|$ by $j$ and vice-a-versa, we have
$\gamma+\left|{N\setminus J}\right|\left|{N\setminus L^{\prime}}\right|\mapsto
j+\alpha$, and (2.3.16) is equal to the first term in the right hand side of
the statement. By a similar calculation, we see the second term in (2.3.15)
and the statement coincide. For the right hand side of (2.3.15), we have by
(2.3.6) that
$\operatorname{res}_{(Z-W)_{K}}D_{W}^{(j|J)}\delta(Z,W)=\delta_{j,0}\delta_{J,\emptyset}$,
which yields

$\operatorname{res}_{W}\operatorname{res}_{(Z-W)_{K}}\sum_{(j|J)}W^{l|L}(Z-W)_{K}^{k|K}(Z-W)_{K}^{-1-j|1-J}Y(a_{(j|J)}b,W)=(a_{(k|K)}b)_{(l|L)}.$

Hence we have the consequence. ∎

###  2.4. Commutative SUSY vertex algebras

Hereafter, the word “SUSY vertex algebra” means an $N_{W}=N$ or $N_{K}=N$ SUSY
vertex algebra.

####  2.4.1. SUSY analogue of Borcherds’ equivalence

Recall that, in the even case, a _commutative_ vertex algebra is equivalent to
a unital commutative algebra with derivation [B86, §4], [FBZ04, §1.4]. The
following Lemmas 2.4.3 and 2.4.4 are analogue of this fact for $N_{W}=N$ and
$N_{K}=N$ SUSY vertex algebras, respectively.

######  Definition 2.4.1.

A SUSY VA $V$ is _commutative_ if we have $n=0$ for any $a,b\in V$ in the
locality axiom, i.e., $[Y(a,Z),Y(b,W)]=0$.

Similarly as in the even case, we have the following restatement:

######  Lemma 2.4.2.

A SUSY VA $V$ is commutative if and only if
$Y(a,Z)\in(\operatorname{End}V)\llbracket Z\rrbracket$ for any $a\in V$.

###### Proof.

The argument is quite similar to that for the even case [FBZ04, §1.4], but let
us write down it for completeness. Assume that $V$ is commutative. Then, for
any $a,b\in V$ of pure parity, we have

$\displaystyle
Y(a,Z)b=\left.\vphantom{1^{(}}Y(a,Z)Y(b,W)\ket{0}\right|_{W=0}=\left.(-1)^{p(a)p(b)}Y(b,W)Y(a,Z)\ket{0}\right|_{W=0}.$
(2.4.1)

The right hand side has no negative powers of $z$ by the vacuum axiom, and
hence $Y(a,Z)b\in V\llbracket Z\rrbracket$. Thus we have
$Y(a,Z)\in(\operatorname{End}V)\llbracket Z\rrbracket$.

Conversely, assume $Y(a,Z)\in(\operatorname{End}V)\llbracket Z\rrbracket$ for
any $a\in V$. Then, for any $a,b\in V$ of pure parity, both of the series
$f_{1}(Z,W)\coloneqq Y(a,Z)Y(b,W)$ and $f_{2}(Z,W)\coloneqq Y(b,W)Y(a,Z)$ are
in $(\operatorname{End}V)\llbracket Z,W\rrbracket$. Now the locality axiom
says that there exists $n\in\mathbb{N}$ such that
$(z-w)^{n}f_{1}(Z,W)=(z-w)^{n}(-1)^{p(a)p(b)}f_{2}(Z,W)$. The last equality
yields $f_{1}(Z,W)=(-1)^{p(a)p(b)}f_{2}(Z,W)$, and thus $V$ is commutative. ∎

######  Lemma 2.4.3.

Let $(V,\ket{0},T,S^{i},Y)$ be a commutative $N_{W}=N$ SUSY VA. Then $V$ is a
commutative $k$-superalgebra with multiplication $a\cdot b\coloneqq
a_{(-1|N)}b$, unit $\ket{0}$, even derivation $T$, and odd derivations $S^{i}$
(see Definition 1.2.1). Conversely, given a commutative $k$-superalgebra $V$
with multiplication $\cdot$, unit $1$, even and odd derivations $T$ and
$S^{i}$, we have a commutative $N_{W}=1$ SUSY VA $(V,1,T,S^{i},Y)$ with

$\displaystyle Y(a,Z)\coloneqq e^{Z\nabla}a=\Biggr{(}\sum_{n\geq
0}\frac{1}{n!}z^{n}T^{n}\Biggr{)}(1+\zeta^{1}S^{1})\dotsm(1+\zeta^{N}S^{N})(a\cdot?)\in(\operatorname{End}V)\llbracket
Z\rrbracket,$ (2.4.2)

where the symbol $a\cdot?$ denotes the multiplication operator.

###### Proof.

This one is also a simple analogue of the even case [FBZ04, §1.4]. First,
assume that $V$ is a commutative $N_{W}=1$ SUSY VA, and let $a,b\in V$ be of
pure parity. Then (2.4.1) yields $Y(a,Z)b=(-1)^{p(a)p(b)}b_{(-1|1)}a+O(Z,W)$,
where $O(Z,W)$ is an element vanishing at $Z=W=0$. Taking the coefficient of
$Z^{0|0}$, we have $a_{(-1|1)}b=(-1)^{p(a)p(b)}b_{(-1|1)}a$. Then, denoting
$Y_{a}:=a_{(-1|1)}$, we have $Y_{a}Y_{b}=(-1)^{p(a)p(b)}Y_{b}Y_{a}$. By the
argument in [FBZ04, §1.3.3], it shows the associativity and the commutativity
of $a\cdot b\coloneqq Y_{a}b$. The vacuum axiom yields that $\ket{0}$ is the
unit of this multiplication $\cdot$, and Fact 2.2.7 means that $T$ and $S$ are
an even and odd derivation, respectively. The converse statement is checked
directly from the formula (2.4.2), and we omit the detail. ∎

######  Lemma 2.4.4.

Let $V$ be a $k$-linear superspace. Then the structure $(V,\ket{0},S^{i},Y)$
of commutative $N_{K}=N$ SUSY VA is equivalent to the structure
$(V,\cdot,1,S^{i})$ of unital commutative $k$-superalgebra with odd
derivations $S^{i}$ ($i\in[N]$) which satisfy the commutation relation

$[S^{i},S^{j}]=2\delta_{i,j}T,\quad[T,S^{i}]=0.$

The correspondence is given by $a_{(-1|N)}b=a\cdot b$, $\ket{0}=1$ and

$\displaystyle
Y(a,Z)=e^{Z\nabla}a=e^{zT}(1+\zeta^{1}S^{1})\dotsm(1+\zeta^{N}S^{N})a.$
(2.4.3)

###### Proof.

The argument in Lemma 2.4.3 works, using Lemma 1.3.13 and Fact 2.3.10. ∎

####  2.4.2. $N_{W}=N$ SUSY VA structure of superjet algebras

Recall that in §1.3 we introduced the superjet algebra

$A^{O}=A^{k\llbracket Z\rrbracket}=\mathcal{S}^{N}(A_{\infty})$

for each $A\in\operatorname{\mathsf{SCom}}$ (Proposition 1.3.5). Here
$O=k\llbracket Z\rrbracket$ denotes the topological superalgebra
$O^{1|N}=k\llbracket z,\zeta^{1},\dotsc,\zeta^{N}\rrbracket$. We argue that
$A^{O}$ has a natural structure of $N_{W}=N$ SUSY VA (Proposition 2.4.5).

Let us recall some basic properties of $A^{O}$. It is a commutative
$k$-superalgebra equipped with the universal differentials
$d_{n|J}=d_{\textup{dR}}^{J}d_{n}$ for $n\in\mathbb{N}$ and $J\subset[N]$.
Since we are assuming $k$ is of characteristic $0$, we have
$d_{n}=\frac{1}{n!}d_{1}^{n}$ by Remark 1.3.2. Note that $d_{1}$ and
$d_{\textup{dR}}$ is an even and odd derivation, respectively. The even
derivation $d_{1}$ comes from the Hasse-Schmidt derivations
$A_{\infty}=\mathrm{HS}_{A/k}^{\infty}$, and the odd derivation
$d_{\textup{dR}}$ is the differential of the de Rham complex. Also, recall
that $A^{O}$ is generated by $a^{[n|J]}\coloneqq d_{n|J}a$ with $a\in A$,
$n\in\mathbb{N}$ and $J\subset[N]$ (see Lemma 1.3.6).

By Lemma 1.3.7, the superjet algebra $A^{O}$ has an an even derivation $T$ and
odd derivations $S^{i}$ for $i\in[N]$ with

$\displaystyle T(a^{[n|J]})\coloneqq(n+1)a^{[n+1|J]},\quad
S^{i}(a^{[n|J]})\coloneqq\sigma(e_{i},J)a^{[n|J\setminus e_{i}]}.$ (2.4.4)

Then, Lemma 2.4.3 yields:

######  Proposition 2.4.5.

For every $A\in\operatorname{\mathsf{SCom}}k$, the superjet algebra $A^{O}$
has a structure of commutative $N_{W}=N$ SUSY VA with $T$ and $S^{i}$ given by
(2.4.4).

####  2.4.3. $N_{K}=N$ SUSY VA structure of superconformal jet algebras

Recall Definition 1.3.14, which says that for any
$A\in\operatorname{\mathsf{SCom}}k$ we have the superconformal jet algebra
$A^{O_{\textup{sc}}}=(A^{O},S_{K}^{i})$. It is a unital commutative
superalgebra equipped with odd derivations

$\displaystyle
S_{K}^{i}(a^{[n|J]})\coloneqq\begin{cases}\sigma(e_{i},J)a^{[n|J\cup
e_{i}]}&(i\notin J)\\\ (n+1)\sigma(e_{i},J\setminus e_{i})a^{[n+1|J\setminus
e_{i}]}&(i\in J)\end{cases}.$ (2.4.5)

$S_{K}^{i}$’s form the Lie superalgebra in Lemma 1.3.13. Hence Lemma 2.4.4
yields:

######  Proposition 2.4.6.

For every $A\in\operatorname{\mathsf{SCom}}k$, the superjet algebra $A^{O}$
has a structure of a commutative $N_{K}=N$ SUSY VA with $S^{i}$ being the odd
derivation determined by (2.4.5).

For later reference, we give a lemma on the obtained structure on $A^{O}$.

##  3\. Li filtration of SUSY vertex algebra

In this subsection, we introduce SUSY analogue of Li’s canonical filtration
[L05]. As in the previous § 2, we fix a positive integer $N$.

###  3.1. SUSY vertex Poisson algebras

Recall that, in the even case, a vertex Poisson algebra is a data
$(V,\ket{0},T,Y_{+},Y_{-})$ combining a commutative VA structure
$(\ket{0},T,Y_{+})$ and a vertex Lie algebra structure $(V,T,Y_{-})$ with
common derivation $T$, subject to the condition that $Y_{-}$ gives Poisson-
like operations for the commutative multiplication associated $Y_{+}$. See
[FBZ04, Chap. 16] and [L05, §2] for the detail. A SUSY analogue of this notion
is introduced in [HK07], where the notion of Lie conformal algebra is used
instead of vertex Lie algebras. Below we give a definition using vertex Lie
algebras.

####  3.1.1. $N_{W}=N$ SUSY vertex Lie algebras

We begin with the introduction of $N_{W}=N$ SUSY vertex Lie algebras. Recall
that the notion of an even vertex Lie algebra is obtained by taking singular
part (called polar part in [FBZ04, 16.1.2. Definition]) of a vertex algebra.
For a linear superspace $V$ and a series $f(Z)=\sum_{(j|J)}Z^{j|J}v_{j|J}\in
V\llbracket Z^{\pm 1}\rrbracket$ of a $1|N$-supervariable
$Z=(z,\zeta^{1},\dotsc,\zeta^{N})$, we denote

$\displaystyle\operatorname{Sing}(f)\coloneqq\sum_{(j|J),\,j<0}Z^{j|J}v_{j|J}\in
Z^{-1}V[Z^{-1}]$ (3.1.1)

and call it the singular part of $f$, where
$Z^{-1}V[Z^{-1}]\coloneqq\\{\sum_{(j|J),j<0}Z^{j|J}v_{j|J}\mid v_{j|J}\in
V\\}$.

######  Definition 3.1.1.

An _$N_{W}=N$ SUSY vertex Lie algebra_is a data $(V,T,S^{i},Y_{-})$ consisting
of

  * •

a linear superspace $V$,

  * •

an even operator $T\in\operatorname{\underline{End}}(V)_{\overline{0}}$,

  * •

$N$ odd operators $S^{i}\in\operatorname{\underline{End}}(V)_{\overline{1}}$
for $i\in[N]$, and

  * •

an even linear map $Y_{-}(\cdot,Z)\colon
V\to\operatorname{Hom}(V,Z^{-1}V[Z^{-1}])$,

which satisfies the following axioms.

  1. (1)

Translation invariance: $Y_{-}(Ta,Z)=\partial_{z}Y_{-}(a,Z)$,
$Y_{-}(S^{i}a,Z)=\partial_{\zeta^{i}}Y_{-}(a,Z)$.

  2. (2)

Skew-symmetry:
$Y_{-}(a,Z)b=\operatorname{Sing}\bigl{(}(-1)^{p(a)p(b)}e^{Z\nabla}Y(b,-Z)a\bigr{)}$
with $Z\nabla\coloneqq zT+\sum_{i=1}^{N}\zeta^{i}S^{i}$.

  3. (3)

Supercommutator:
$[a_{(m|M)},Y_{-}(b,Z)]=\operatorname{Sing}\bigl{(}Y(e^{-Z\nabla}a_{(m|M)}e^{Z\nabla}b,Z)\bigr{)}$,
where $a_{(m|M)}$ denotes the Fourier mode of the expansion
$Y_{-}(a,Z)=\sum_{(m|M),m\geq 0}Z^{-m-1|N\setminus M}a_{(m|M)}$.

We often abbreviate the word “vertex Lie algebra” to VLA.

######  Lemma 3.1.2.

Given an $N_{W}=N$ SUSY VA $(V,\ket{0},T,S^{i},Y)$, we have an $N_{W}=N$ SUSY
VLA $(V,T,S^{i},Y_{-})$ by setting
$Y_{-}(a,Z)\coloneqq\operatorname{Sing}(Y(a,Z))$.

###### Proof.

The axioms in Definition 3.1.1 are consequences of the translation invariance
in Definition 2.2.1, Fact 2.2.8 and Fact 2.2.9. ∎

We also have the notion of a _module_ $M$ over an $N_{W}=N$ SUSY VLA $V$,
similar to SUSY VA modules. For $a\in V$ and $m\in M$, we denote the SUSY
vertex Lie action of $V$ on $W$ by

$\displaystyle Y^{W}_{-}(a,Z)m=\sum_{(j|J),\,j\geq 0}Z^{-1-j|N\setminus
J}a_{(j|J)}m\in M(\\!(Z)\\!).$ (3.1.2)

Our Definition 3.1.1 is equivalent to the notion of an $N_{W}=N$ SUSY Lie
conformal algebra in [HK07, Definition 3.2.2]. Using the latter, we can
describe the “Lie algebra structure” more explicitly. Let us recall Notation
2.2.2 of the $\Lambda$-bracket for $N_{W}=N$ case. In particular,
$\mathscr{L}_{W}=\mathbb{C}[\Lambda]=\mathbb{C}[\lambda,\chi^{i}]$ is the
commutative superalgebra freely generated by even $\lambda$ and odd
$\chi^{i}$’s. Note that the $\Lambda$-bracket $[a_{\Lambda}b]$ only involves
$(j|J)$-operators for $j\geq 0$, so that it can be defined also for an
$N_{W}=N$ SUSY VLA. Now, let $\mathscr{L}_{W}^{\prime}$ be another commutative
superalgebra freely generated by even $\gamma$ and odd $\eta^{i}$’s, and
define $[a_{\Gamma}b]$ similarly using
$\Gamma^{m|M}\coloneqq\gamma^{m}\eta^{M}$.

######  Fact 3.1.3 (c.f. [HK07, §3.2.1]).

Let $V$ be an $N_{W}=N$ SUSY VLA. For $a,b,c\in V$ of pure parity, we have the
following identity in $\mathscr{L}_{W}\otimes\mathscr{L}_{W}^{\prime}\otimes
V$.

$[a_{\Lambda}[b_{\Gamma}c]]=(-1)^{(p(a)+1)N}[[a_{\Lambda}b]_{\Gamma+\Lambda}c]+(-1)^{(p(a)+N)(p(b)+N)}[b_{\Gamma}[a_{\Lambda}c]].$

We can regard it as a kind of Jacobi identity. In order to understand the
signs, let us recall:

######  Definition 3.1.4 ([HK07, Definition 3.2.5]).

A _Lie superalgebra of parity $q\in\mathbb{Z}_{2}$_ is a linear superspace $V$
with a binary operation $[\cdot,\cdot]\colon V\otimes V\to V$ satisfying the
following axioms for $a,b,c\in V$ of pure parity.

  1. (1)

Skew-symmetry: $[a,b]=-(-1)^{p(a)p(b)+q}[b,a]$.

  2. (2)

Jacobi identity:
$[a,[b,c]]=-(-1)^{p(a)q+q}[[a,b],c]+(-1)^{(p(a)+q)(p(b)+q)}[b,[a,c]]$.

We call $[\cdot,\cdot]$ the _Lie bracket of parity $q$_. We also call
$(V,[\cdot,\cdot])$ an _even_ or _odd Lie superalgebra_ according to
$q=\overline{0}$ or $\overline{1}$.

######  Remark 3.1.5.

Some comments are in order.

  1. (1)

We slightly modify the terminology in [HK07], and use the phrase “of parity
$q$” instead of “degree $q$” used therein.

  2. (2)

An even Lie superalgebra is a Lie superalgebra in the standard sense.

  3. (3)

We can find in Fact 3.1.3 that “the $\Lambda=0$ part” of an $N_{W}=N$ SUSY VLA
is a Lie superalgebra of parity $N\bmod 2$. See [HK07, Lemma 3.2.7] for the
precise statement.

####  3.1.2. $N_{W}=N$ SUSY vertex Poisson algebras

Now we introduce $N_{W}=N$ SUSY vertex Poisson structure.

######  Definition 3.1.6 (c.f. [HK07, Definition 3.3.16]).

An _$N_{K}=W$ SUSY vertex Poisson algebra_ is a data
$(V,\ket{0},T,S^{i},Y_{+},Y_{-})$ consisting of

  * •

a commutative $N_{K}=N$ SUSY VA $(V,\ket{0},T,S^{i},Y_{+})$,

  * •

an $N_{W}=N$ SUSY VLA $(V,T,S^{i},Y_{-})$

such that the vertex Lie structure $Y_{-}$ is a derivation for the commutative
superalgebra structure $ab=a_{(-1|N)}b$ coming from $Y_{+}$. More precisely
speaking, for $a,b,c\in V$ of pure parity, we have

$\displaystyle
Y_{-}(a,Z)(bc)=\bigl{(}Y_{-}(a,Z)b\bigr{)}c+(-1)^{(p(a)+N)p(b)}b\bigl{(}Y_{-}(a,Z)c\bigr{)}.$
(3.1.3)

We often abbreviate the word “vertex Poisson algebra” to VPA.

Recalling Lemma 2.4.2 of commutative property, we denote the expansions of
$Y_{+}$ and $Y_{-}$ by

$\displaystyle Y_{+}(a,Z)=\sum_{(j|J),\,j<0}Z^{-1-j|1-J}a_{(j|J)},\quad
Y_{-}(a,Z)=\sum_{(j|J),\,j\geq 0}Z^{-1-j|1-J}a_{(j|J)}.$ (3.1.4)

We also have the notion of a _module_ over an $N_{W}=N$ SUSY VPA $V$, which is
a combination of the module structure over the commutative superalgebra
associated to $(V,\ket{0},T,S^{i},Y_{+})$, and the module structure over the
$N_{W}=N$ SUSY VLA $(V,T,S^{i},Y_{-})$.

As for the derivation axiom (3.1.3), let us recall:

######  Definition 3.1.7.

An _Poisson superalgebra of parity $q\in\mathbb{Z}_{2}$_ is a commutative
superalgebra $P$ together with a Lie bracket $\\{\cdot,\cdot\\}\colon P\otimes
P\to P$ of parity $q$ in the sense of Definition 3.1.4 such that

$\\{a,bc\\}=\\{a,b\\}c+(-1)^{(p(a)+q)p(b)}a\\{b,c\\}$

for $a,b,c\in P$ of pure parity. $\\{\cdot,\cdot\\}$ is called the _Poisson
bracket of parity $q$_. We also call $(V,\\{\cdot,\cdot\\})$ an _even_ or _odd
Poisson superalgebra_ according to $q=\overline{0}$ or $\overline{1}$.

######  Remark 3.1.8.

Some comments are in order.

  1. (1)

An even Poisson superalgebra is a Poisson superalgebra in the standard sense.

  2. (2)

The structure of an odd Poisson superalgebra is a part of the Gerstenhaber
algebra structure.

  3. (3)

By Definition 3.1.7 and Remark 3.1.5 (3), we see that “the $\Lambda=0$ part”
of an $N_{W}=N$ SUSY VPA is a Poisson superalgebra of parity $N\bmod 2$.

####  3.1.3. $N_{W}=N$ SUSY VPA structure on superjet algebra

Let us give a basic example of an $N_{K}=N$ SUSY VPA. Recall the
$1|N$-superjet algebra

$A^{O}=A^{k\llbracket Z\rrbracket}=\operatorname{\mathcal{S}}^{N}(A_{\infty})$

of a commutative superalgebra $A$ (Proposition 1.3.5). It has a structure of
commutative $N_{W}=N$ SUSY VA (Proposition 2.4.5). If $A$ is moreover a
Poisson superalgebra of parity $N\mod 2$ in the sense of Definition 3.1.7,
then this structure has an enhancement to vertex Poisson structure.

######  Proposition 3.1.9.

Let $(P,\\{\cdot,\cdot\\})$ be a Poisson superalgebra of parity $N\bmod 2$,
and $(P^{O},T,S^{i})$ be the commutative $N_{W}=N$ SUSY VA. Then $P^{O}$ has
an $N_{W}=N$ SUSY VPA structure such that

$\displaystyle u_{(m|M)}v=\begin{cases}\\{u,v\\}&(m=0,M=[N])\\\
0&(\text{otherwise})\end{cases}$ (3.1.5)

for $u,v\in P\subset P^{O}$, where we used the expansion (3.1.4) of $Y_{-}$.
We call is the _level $0$ SUSY VPA structure_.

######  Remark 3.1.10.

Our naming comes from the _level $0$ VPA structure_ [Ar12, Proposition 2.3.1]
of the jet algebra $P_{\infty}$ (see (1.3.1)) in the even case.

For the proof, we need the following $N_{W}=N$ SUSY analogue of [L04, Theorem
3.6, Proposition 3.10].

######  Lemma 3.1.11.

Let $A$ be a unital commutative superalgebra equipped with an even derivation
$T$ and odd derivations $S^{i}$ for $i\in[N]$. Also, let $B\subset A$ be a
linear sub-superspace generating the differential algebra $(A,T,S^{i})$ in the
sense of Definition 1.3.8.

  1. (1)

Assume that there is a morphism of linear superspaces $Y_{-}(\cdot,Z)\colon
A\to Z^{-1}(\operatorname{Der}A)\llbracket Z\rrbracket$ satisfying

    1. (i)

$Y_{-}(a,Z)a^{\prime}\in Z^{-1}A[Z^{-1}]$,

    2. (ii)

$Y_{-}(1,Z)=0$,

    3. (iii)

$Y_{-}(Ta,Z)=\partial_{z}Y_{-}(a,Z)$,
$Y_{-}(S^{i}a,Z)=\partial_{\zeta^{i}}Y_{-}(a,Z)$, and

    4. (iv)

$[T,Y_{-}(a,Z)]=\partial_{z}Y_{-}(a,Z)$,
$[S^{i},Y_{-}(a,Z)]=\partial_{\zeta^{i}}Y_{-}(a,Z)$

for $a,a^{\prime}\in A$ and $i\in[N]$. Assume moreover that for $a\in A$ and
$b\in B$ of pure parity, we have

$\displaystyle
Y_{-}(a,Z)b=\operatorname{Sing}\bigl{(}(-1)^{p(a)p(b)}e^{Z\nabla}Y_{-}(b,-Z)a\bigr{)},$
(3.1.6)

where $\operatorname{Sing}$ is given in (3.1.1), and $Z\nabla\coloneqq
zT+\sum_{i=1}^{N}\zeta^{i}S^{i}$. Then $Y_{-}$ gives an $N_{W}=N$ VPA
structure on $A$.

  2. (2)

Assume that there is a morphism of linear superspaces
$Y^{0}_{-}(\cdot,Z)\colon B\to
Z^{-1}(\operatorname{\underline{Hom}}(B,A))\llbracket Z\rrbracket$ satisfying

$\displaystyle
Y^{0}_{-}(b,Z)b^{\prime}=\operatorname{Sing}\bigl{(}(-1)^{p(b)p(b^{\prime})}e^{Z\nabla}Y_{-}(b^{\prime},-Z)b\bigr{)}\in
Z^{-1}A[Z^{-1}]$ (3.1.7)

for $b,b^{\prime}\in B$ of pure parity. Then $Y^{0}_{-}$ extends uniquely to
$Y_{-}(\cdot,Z)\colon A\to Z^{-1}(\operatorname{Der}A)\llbracket Z\rrbracket$
satisfying the conditions 1(i)–1(iv) and (3.1.6) in (1).

###### Proof.

The arguments [L04, Theorem 3.6, Proposition 3.10] works with little
modification. We omit the detail. ∎

###### Proof of Proposition 3.1.9.

Apply Lemma 3.1.11 (2) to $B=P$, $A=P^{O}$ and $Y^{0}_{-}(b,Z)\coloneqq
b_{(0|N)}$, $b_{(0|N)}b^{\prime}\coloneqq\\{b,b^{\prime}\\}$ for
$b,b^{\prime}\in P$, which obviously satisfies the condition (3.1.7). Then we
have an extension $Y_{-}$ of $Y^{0}_{-}$ on $P^{O}$, which gives the desired
VPA structure by Lemma 3.1.11 (1). ∎

######  Remark 3.1.12.

The above proof lacks explicit formulas of $a_{(m|M)}b$ for $a,b\in P^{O}$.
For a comparison with the even case [Ar12, Proposition 2.3.1] mentioned in
Remark 3.1.10, we explain how to obtain concrete formulas.

Let us consider $u_{(m|M)}a$ with $u\in P$, $a\in P^{O}$, $m\geq 0$ and
$M\subset[N]$. Note that every $a\in P^{O}$ is written as a polynomial of
$S^{L}T^{(l)}v$ with $L\subset[N]$, $l\in\mathbb{N}$ and $v\in P$, where
$S^{L}\coloneqq S^{l_{1}}S^{l_{2}}\dotsm$ for $L=\\{l_{1},l_{2},\dotsc\\}$ and
$T^{(l)}\coloneqq\frac{1}{l!}T^{l}$. Hence, by the Leibniz rule (3.1.3), it is
enough to determine $u_{(m|M)}(S^{L}T^{(l)}v)$ for $u,v\in P$ of pure parity.
We determine them by the condition 1(iv) in Lemma 3.1.11, which is equivalent
to (recall (2.2.6))

$\displaystyle[S^{i},u_{(m|M)}]=\begin{cases}\sigma(N\setminus
M,e_{m})a_{(m|M\setminus e_{i})}&(i\in M)\\\ 0&(i\notin
M)\end{cases},\quad[T,u_{(m|M)}]=-mu_{(m-1|M)}$

with the convention $u_{(-1|M)}\coloneqq 0$. These are recursion for the
desired $u_{(m|M)}(S^{L}T^{l}v)$, and using (3.1.5) as the initial condition,
we can solve them to obtain:

  * •

In case $l\geq m$, we set

$\displaystyle u_{(m|N)}(S^{L}T^{(l)}v)$
$\displaystyle\coloneqq(-1)^{\left|{L}\right|p(u)}S^{L}T^{(l-m)}\\{u,v\\},$
(3.1.8) $\displaystyle u_{(m|N\setminus L)}(S^{L}T^{(l)}v)$
$\displaystyle\coloneqq(-1)^{\left|{L}\right|p(u)+\lfloor\left|{L}\right|/2\rfloor}\bigl{(}{\textstyle\prod}_{i=1}^{\left|{L}\right|}\sigma(\\{l_{i+2},l_{i+2},\dotsc\\},e_{l_{i}})\bigr{)}T^{(l-m+\left|{I}\right|)}\\{u,v\\},$
(3.1.9)

and otherwise $u_{(m|M)}(T^{(l)}S^{L}v)\coloneqq 0$.

  * •

In case $l<m$, we set $u_{(m|M)}(T^{(l)}S^{L}v)\coloneqq 0$.

We can check directly that these formulas satisfy the condition 1(iii) in
Lemma 3.1.11. Let us also mention that the case $L=\emptyset$ in (3.1.8) is
analogous to the even case formula
$u_{(m)}(T^{l}v)=\frac{l!}{(l-m)!}T^{l-m}\\{u,v\\}$ for $l\geq m$ in [Ar12,
Proposition 2.3.1, (10)].

######  Remark 3.1.13.

Let us sketch another description of the level $0$ SUSY VPA structure,
following [Mal16, Lemma 3.1], where the level $0$ VPA structure in the even
case is treated in terms of the language of coisson algebra [BD04, §2.6].
(Although it is not shown explicitly in [Mal16] that the coisson structure
coincides with the level $0$ VPA structure in [Ar12], one can check it by
writing down the coisson product.) The following argument has some overlap
with [HK07, Remark 3.2.3], from which we borrow some symbols.

Let $\mathscr{H}_{W}=k[\partial,\delta^{1},\dotsc,\delta^{N}]$ be the
commutative superalgebra over the base field $k$ generated by an even variable
$\partial$ and odd variables $\delta^{i}$ for $i\in[N]$. We regard
$\mathscr{H}_{W}$ as a Hopf superalgebra with comultiplication
$\Delta(\partial)\coloneqq\partial\otimes 1+1\otimes\partial$,
$\Delta(\delta^{i})\coloneqq\delta^{i}\otimes 1+1\otimes\delta^{i}$ and counit
$\epsilon(\partial)=\epsilon(\delta^{i})\coloneqq 0$. On the category
$\mathsf{M}$ of right $\mathscr{H}_{W}$-modules, we can define the _$*$
-pseudo-tensor structure_ in a similar way as the even case [BD04], [Mal16,
§3.1]. A Lie algebra object $L$ in the corresponding pseudo-tensor category
$\mathsf{M}^{*}$ is called a _$\text{Lie}^{*}$ algebra_. The
$\text{Lie}^{*}$-bracket $[\cdot,\cdot]$ is an element of
$\operatorname{Hom}_{\mathscr{H}_{W}^{2}}(L\otimes_{k}L,L\otimes_{\mathscr{H}_{W}}\mathscr{H}_{W}^{2})$,
where $\mathscr{H}_{W}^{2}$ denotes the tensor product algebra
$\mathscr{H}_{W}\otimes_{k}\mathscr{H}_{W}$, and we regard it as a right
$H$-module by $\Delta$. Identifying
$L\otimes_{\mathscr{H}_{W}}\mathscr{H}_{W}^{2}$ with
$L[\partial_{1},\delta_{1}^{i}]$, where $\partial_{1}\coloneqq\partial\otimes
1$ and $\delta_{1}^{i}\coloneqq\delta^{i}\otimes 1$, we can write
$[\cdot,\cdot]$ as $[a,b]=\sum_{(j|J),j\geq
0}(a_{[j|J]}b)\partial_{1}^{j}\delta_{1}^{J}$ for some $a_{[j|J]}b\in L$. As
in the even case, such a $\text{Lie}^{*}$ algebra is equivalent to our
$N_{W}=N$ SUSY VLA, and also to an $N_{W}=N$ SUSY Lie conformal algebra in
[HK07].

The category $\mathsf{M}$ has a tensor structure $\otimes^{!}$ coming from the
comultiplication $\Delta$ on $\mathscr{H}_{W}$, and the corresponding tensor
category is denoted by $\mathsf{M}^{!}$. A commutative algebra object in
$\mathsf{M}^{!}$ is called a $\text{commutative}^{!}$ algebra. A _coisson
algebra_ $C$ is a $\text{Lie}^{*}$ algebra and a $\text{commutative}^{!}$
algebra such that the multiplication $C\otimes^{!}C\to C$ is a
$\text{Lie}^{*}$ algebra morphism.

Now we can restate Proposition 3.1.9. Let $(P,\\{\cdot,\cdot\\})$ be a Poisson
superalgebra of parity $N\bmod 2$. Note that $P^{O}\simeq
P\otimes_{k}\mathscr{H}_{W}$ with operators $T$ and $S^{i}$ corresponding to
$\partial$ and $\delta^{i}$, respectively. The statement is that
$P\otimes_{k}\mathscr{H}_{W}$ has a coisson structure whose $\text{Lie}^{*}$
bracket $\\{\cdot,\cdot\\}$ is given by
$\\{u\partial^{l}\delta^{L},v\partial^{m}\delta^{M}\\}=\\{u,v\\}\partial_{1}^{m}\delta_{1}^{L}\partial_{2}^{m}\delta_{2}^{M}$
for $u,v\in P$, $l,m\in\mathbb{N}$ and $L,M\subset[N]$, where
$\partial_{2}\coloneqq 1\otimes\partial$ and $\delta_{2}^{i}\coloneqq
1\otimes\delta^{i}$.

####  3.1.4. $N_{K}=N$ SUSY vertex Poisson algebras

Here we give an $N_{K}=N$ analogue of § 3.1.1 and § 3.1.2. We begin with:

######  Definition 3.1.14.

An _$N_{K}=N$ SUSY vertex Lie algebra_is a data $(V,S^{i},Y_{-})$ consisting
of

  * •

a linear superspace $V$,

  * •

$N$ odd operators $S^{i}\in\operatorname{\underline{End}}(V)_{\overline{1}}$
for $i\in[N]$, and

  * •

an even linear map $Y_{-}(\cdot,Z)\colon
V\to\operatorname{Hom}(V,Z^{-1}V[Z^{-1}])$,

which satisfies the following axioms.

  1. (1)

Translation invariance: $Y_{-}(S^{i}a,Z)=D^{i}_{Z}Y_{-}(a,Z)$.

  2. (2)

Skew-symmetry:
$Y_{-}(a,Z)b=\operatorname{Sing}\bigl{(}(-1)^{p(a)p(b)}e^{Z\nabla}Y(b,-Z)a\bigr{)}$
with $Z\nabla\coloneqq zT+\sum_{i=1}^{N}\zeta^{i}S^{i}$,
$T\coloneqq(S^{1})^{2}=\dotsb=(S^{N})^{2}$.

  3. (3)

Supercommutator:
$[a_{(m|M)},Y_{-}(b,Z)]=\operatorname{Sing}\bigl{(}Y(e^{-Z\nabla}a_{(m|M)}e^{Z\nabla}b,Z)\bigr{)}$,
where $a_{(m|M)}$ denotes the Fourier mode of the expansion
$Y_{-}(a,Z)=\sum_{(m|M),m\geq 0}Z^{-m-1|N\setminus M}a_{(m|M)}$.

We often abbreviate the word “vertex Lie algebra” to VLA.

As in Lemma 3.1.2, we have:

######  Lemma 3.1.15.

Given an $N_{K}=N$ SUSY VA $(V,\ket{0},S^{i},Y)$, we have an $N_{K}=N$ SUSY
VLA $(V,S^{i},Y_{-})$ by setting
$Y_{-}(a,Z)\coloneqq\operatorname{Sing}(Y(a,Z))$.

Our Definition 3.1.14 is equivalent to the notion of an $N_{K}=N$ SUSY Lie
conformal algebra in [HK07, Definition 4.10]. For later reference, we record
the Jacobi identity. Recall Notation 2.3.3 of the $\Lambda$-bracket. In
particular, $\mathscr{L}$ denotes the superalgebra generated by even $\lambda$
and odd $\chi^{i}$. Note that the $\Lambda$-bracket makes sense for an
$N_{K}=N$ SUSY VLA $V=(V,S^{i},Y_{-})$: we have

$[a_{\Lambda}b]=\sum_{(m|M),\,m\geq 0}\sigma(M,N\setminus
M)(-1)^{\binom{\left|{M}\right|+1}{2}}\frac{1}{m!}\Lambda^{m|M}a_{(m|M)}b.$

for $a,b\in V$. Also, let $\mathscr{L}^{\prime}$ be the superalgebra generated
by even $\gamma$ and odd $\eta^{i}$’s subject to the relation
$[\gamma,\eta^{i}]=0$ and $[\eta^{i},\eta^{j}]=-2\gamma\delta_{i,j}$, and
define $[a_{\Gamma}b]$ similarly using
$\Gamma^{m|M}\coloneqq\gamma^{m}\eta^{M}$.

######  Fact 3.1.16 (c.f. [HK07, Definition 4.10]).

Let $V$ be an $N_{K}=N$ SUSY VLA. For $a,b,c\in V$ of pure parity, we have the
following identity in $\mathscr{L}\otimes\mathscr{L}^{\prime}\otimes V$.

$[a_{\Lambda}[b_{\Gamma}c]]=(-1)^{(p(a)+1)N}[[a_{\Lambda}b]_{\Gamma+\Lambda}c]+(-1)^{(p(a)+N)(p(b)+N)}[b_{\Gamma}[a_{\Lambda}c]].$

######  Definition 3.1.17 (c.f. [HK07, §4.10]).

An _$N_{K}=N$ SUSY vertex Poisson algebra_ ($N_{K}=N$ SUSY VPA for short) is a
data $(V,\ket{0},S^{i},Y_{+},Y_{-})$ consisting of

  * •

a commutative $N_{K}=N$ SUSY VA $(V,\ket{0},S^{i},Y_{+})$,

  * •

an $N_{K}=N$ SUSY VLA $(V,S^{i},Y_{-})$

such that the vertex Lie structure $Y_{-}$ is a derivation for the commutative
superalgebra structure $ab=a_{(-1|N)}b$ coming from $Y_{+}$ (see (3.1.3)).

We also have the notion of a _module_ $M$ over an $N_{K}=N$ SUSY VPA $V$. The
VLA action of $V$ on $M$ is denoted as (3.1.2).

####  3.1.5. $N_{K}=N$ SUSY VPA structure on superconformal jet algebra

We have an $N_{K}=N$ analogue of § 3.1.3. Recall the $1|N$-superconformal jet
algebra

$A^{O_{\textup{sc}}}=(A^{O},S^{i})=(\Omega_{A_{\infty}},S_{K}^{i}),$

for a commutative superalgebra $A$ (Definition 1.3.14). It has a structure of
a commutative $N_{K}=N$ SUSY VA (Proposition 2.4.6), and if $A$ is moreover a
Poisson superalgebra, then this structure can be enhanced to a vertex Poisson
structure.

######  Proposition 3.1.18.

Let $(P,\\{\cdot,\cdot\\})$ be a Poisson superalgebra of parity $N\bmod 2$,
and $P^{O_{\textup{sc}}}=(P^{O},S_{K}^{i})$ be the superconformal jet algebra
of $P$. Then $P^{O_{\textup{sc}}}$ has an $N_{K}=N$ SUSY VPA structure such
that

$\displaystyle u_{(m|M)}v=\begin{cases}\\{u,v\\}&(m=0,M=[N])\\\
0&(\text{otherwise})\end{cases}$ (3.1.10)

for $u,v\in P\subset P^{O}$, where we used the expansion (3.1.4) of $Y_{-}$.
We call is the _level $0$ SUSY VPA structure_.

The proof is similar to Proposition 3.1.9, using the following lemma.

######  Lemma 3.1.19.

Let $A$ be a unital commutative superalgebra equipped with odd derivations
$S_{K}^{i}$ for $i\in[N]$ subject to the commutation relation (c.f. Lemma
1.3.13)

$[S_{K}^{i},S_{K}^{j}]=2T\delta_{i,j},\quad[S_{K}^{i},T]=0.$

Also, let $B\subset A$ be a sub-superspace generating the differential algebra
$(A,S_{K}^{i})$ (see Definition 1.3.8).

  1. (1)

Assume that there is a morphism of linear superspaces $Y_{-}(\cdot,Z)\colon
A\to Z^{-1}(\operatorname{Der}A)\llbracket Z\rrbracket$ satisfying

    1. (i)

$Y_{-}(a,Z)a^{\prime}\in Z^{-1}A[Z^{-1}]$,

    2. (ii)

$Y_{-}(1,Z)=0$,

    3. (iii)

$Y_{-}(S_{K}^{i}a,Z)=D_{Z}^{i}Y_{-}(a,Z)$,

    4. (iv)

$[S_{K}^{i},Y_{-}(a,Z)]=\overline{D}_{Z}^{i}Y_{-}(a,Z)$

for $a,a^{\prime}\in A$ and $i\in[N]$. Further assume that for $a\in A$ and
$b\in B$ of pure parity, we have

$\displaystyle
Y_{-}(a,Z)b=\operatorname{Sing}\bigl{(}(-1)^{p(a)p(b)}e^{Z\nabla}Y_{-}(b,-Z)a\bigr{)},$
(3.1.11)

where $\operatorname{Sing}$ is given in (3.1.1), and $Z\nabla\coloneqq
zT+\sum_{i=1}^{N}\zeta^{i}S_{K}^{i}$ as in Definition 3.1.14. Then $Y_{-}$
gives an $N_{K}=N$ VPA structure on $A$.

  2. (2)

Assume that there is a morphism of linear superspaces
$Y^{0}_{-}(\cdot,Z)\colon B\to
Z^{-1}(\operatorname{\underline{Hom}}(B,A))\llbracket Z\rrbracket$ satisfying

$\displaystyle
Y^{0}_{-}(b,Z)b^{\prime}=\operatorname{Sing}\bigl{(}(-1)^{p(b)p(b^{\prime})}e^{Z\nabla}Y_{-}(b^{\prime},-Z)b\bigr{)}\in
Z^{-1}A[Z^{-1}]$ (3.1.12)

for $b,b^{\prime}\in B$ of pure parity. Then $Y^{0}_{-}$ extends uniquely to
$Y_{-}(\cdot,Z)\colon A\to Z^{-1}(\operatorname{Der}A)\llbracket Z\rrbracket$
satisfying the conditions 1(i)–1(iv) and (3.1.11) in (1).

######  Remark 3.1.20.

As in the $N_{W}=N$ case (Remark 3.1.12), we have an explicit form of the VPA
structure. We will describe $u_{(m|M)}(S^{L}T^{l}v)$ for $u,v\in P$ of pure
parity. The condition 1(iv) in Lemma 3.1.19 is equivalent to (recall (2.3.2)
and (2.3.3))

$\displaystyle[S^{i},u_{(m|M)}]=\begin{cases}\sigma(N\setminus
M,e_{m})a_{(m|M\setminus e_{i})}&(i\in M)\\\ -\sigma(N\setminus(M\cup
e_{i}),e_{m})ma_{(m-1|M\cup e_{i})}&(i\notin M)\end{cases},$
$\displaystyle[T,u_{(m|M)}]=-mu_{(m-1|M)}$

with the convention $u_{(-1|M)}\coloneqq 0$. These are recursion for the
desired $u_{(m|M)}(S^{L}T^{l}v)$, and using (3.1.10) as the initial condition,
we can solve them to obtain:

  * •

In case $l\geq m$, we set

$\displaystyle u_{(m|N)}(S^{L}T^{(l)}v)$
$\displaystyle\coloneqq(-1)^{\left|{L}\right|p(u)}S^{L}T^{(l-m)}\\{u,v\\},$
$\displaystyle u_{(m|N\setminus L)}(S^{L}T^{(l)}v)$
$\displaystyle\coloneqq(-1)^{\left|{L}\right|p(u)+\lfloor\left|{L}\right|/2\rfloor}\bigl{(}{\textstyle\prod}_{i=1}^{\left|{L}\right|}\sigma(\\{l_{i+2},l_{i+2},\dotsc\\},e_{l_{i}})\bigr{)}T^{(l-m+\left|{I}\right|)}\\{u,v\\},$

and otherwise $u_{(m|M)}(T^{(l)}S^{L}v)\coloneqq 0$.

  * •

In case $l<m$, we set $u_{(m|M)}(T^{(l)}S^{L}v)\coloneqq 0$.

######  Remark 3.1.21.

We also have another description of the level $0$ SUSY VPA structure via
chiral algebra analogous to Remark 3.1.13. Instead of using the commutative
superalgebra $H=k[\partial,\not{\partial}^{i}]$, we use the non-commutative
superalgebra $\mathscr{H}=k\langle\not{\partial}_{K}^{i}\rangle$ generated by
odd variables $\not{\partial}_{K}^{i}$ subject to the relation
$[\not{\partial}_{K}^{i},\not{\partial}_{K}^{i}]=[\not{\partial}^{j},\not{\partial}_{K}^{j}]$
and $[\not{\partial}_{K}^{i},\not{\partial}_{K}^{j}]=0$ for $i\neq j$. Let us
denote
$\partial\coloneqq\frac{1}{2}[\not{\partial}_{K}^{i},\not{\partial}_{K}^{i}]$.
We regard $\mathscr{H}$ as a Hopf superalgebra, similarly as $H$. The category
$\mathsf{M}$ of right $\mathscr{H}$-modules has the $*$-pseudo-tensor
structure, and the $\text{Lie}^{*}$-bracket $[\cdot,\cdot]$ on an object
$L\in\mathsf{M}$ is an element of
$\operatorname{Hom}_{\mathscr{H}^{2}}(L\otimes_{k}L,L\otimes_{\mathscr{H}}\mathscr{H}^{2})$.
Under the identification $L\otimes_{\mathscr{H}}\mathscr{H}^{2}\cong
L\langle\not{\partial}_{1}^{i}\rangle$ with
$\not{\partial}_{1}^{i}\coloneqq\not{\partial}_{K}^{i}\otimes 1$, we can write
$[\cdot,\cdot]$ as $[a,b]=\sum_{(j|J),j\geq
0}(a_{[j|J]}b)\partial_{1}^{j}\not{\partial}_{1}^{J}$ for some $a_{[j|J]}b\in
L$. A $\text{Lie}^{*}$ algebra is equivalent to an $N_{K}=N$ SUSY VLA, and
also to an $N_{K}=N$ SUSY Lie conformal algebra in [HK07]. The equivalence is
given by $a_{[2j|J]}b=a_{(j|J)}b$.

The category $\mathsf{M}_{K}$ has a tensor structure $\otimes^{!}$ coming from
the comultiplication $\Delta$ on $\mathscr{H}$, and has a compound tensor
structure $(\otimes^{*},\otimes^{!})$. As a result, we have the notion of a
coisson algebra on $\mathsf{M}_{K}$.

Now we can restate Proposition 3.1.18. Let $(P,\\{\cdot,\cdot\\})$ be an odd
Poisson superalgebra. Note that $P^{O}\simeq P\otimes_{k}\mathscr{H}$ with the
odd operators $S^{i}$ corresponding to $\not{\partial}_{K}^{i}$. The statement
is that $P\otimes_{k}\mathscr{H}$ has a coisson structure whose
$\text{Lie}^{*}$ bracket $\\{\cdot,\cdot\\}$ is given by
$\\{u\partial^{l}\not{\partial}_{K}^{L},v\partial^{m}\not{\partial}_{K}^{M}\\}=\\{u,v\\}\partial_{1}^{l}\not{\partial}_{1}^{L}\partial_{2}^{m}\not{\partial}_{2}^{M}$
for $u,v\in P$, $l,m\in\mathbb{N}$ and $L,M\subset[N]$, where
$\not{\partial}_{2}^{i}\coloneqq 1\otimes\not{\partial}_{K}^{i}$.

###  3.2. Li filtration of $N_{W}=N$ SUSY vertex algebra

In this and the next subsections, we introduce a SUSY analogue of Li’s
canonical filtration [L05, Definition 2.7]. Let us give a brief recollection
of the even case. Let $V$ be an even vertex algebra, and $M$ be a $V$-module.
Then the Li filtration of $M$ is a decreasing sequence of subspaces

$M=E_{0}(M)\supset E_{1}(M)\supset\dotsb\supset E_{n}(M)\supset\dotsb$

which consists of

$\displaystyle
E_{n}(M)\coloneqq\operatorname{span}\left\\{a^{1}_{(-1-k_{1})}\dotsm
a^{r}_{(-1-k_{r})}m\,\Bigg{|}\begin{array}[]{l}r\in\mathbb{Z}_{>0},\,a^{i}\in
V,\,m\in M,\,k_{i}\in\mathbb{N}\\\ \text{satisfying $k_{1}+\dotsb+k_{r}\geq
n$}\end{array}\right\\}.$ (3.2.3)

Our strategy for SUSY case is to discard the odd index $J$ of
$(j|J)$-operators and to apply Li’s arguments in [L05]. Hence, some of the
arguments in loc. cit. work as they are, but some do not. The points where
Li’s argument need to be modified are the ones using Borcherds’ commutator
formula (2.2.21) and iterate formula (2.2.23).

Hereafter until the end of this subsection, $V=(V,\ket{0},T,S^{i},Y)$ denotes
an $N_{K}=N$ SUSY VA.

The construction of Li filtration in the even case [L05] starts with an
argument [L05, Proposition 2.6] on a general decreasing filtration of a vertex
algebra whose associated graded space has a natural structure of a vertex
Poisson algebra. The following statement is an $N_{W}=N$ SUSY analogue of
[L05, Proposition 2.6].

######  Proposition 3.2.1.

Let $V=E_{0}\supset E_{1}\supset\dotsb\supset E_{n}\supset\dotsb$ be a
decreasing filtration of linear sub-superspaces of $V$ such that $\ket{0}\in
E_{0}$ and

$\displaystyle a_{(j|J)}b\in E_{r+s-j-1}\quad(a\in E_{r},\,b\in
E_{s},\,j\in\mathbb{Z},\,J\subset[N]),$ (3.2.4)

where we used the convention $E_{n}\coloneqq V$ for $n\in\mathbb{Z}_{<0}$.

  1. (1)

The associated graded linear superspace

$\operatorname{gr}_{E}V\coloneqq{\textstyle\bigoplus}_{n\in\mathbb{N}}E_{n}/E_{n+1}$