# Cohomology of Jacobi forms

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences
Zitna 25, Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We define and compute a cohomology of the space of Jacobi forms based on
precise analogues of Zhu reduction formulas derived in [Zhu, BKT, MTZ]. A
counterpart of the Bott-Segal theorem for the reduction cohomology of Jacobi
forms on the torus is proven. It is shown that the reduction cohomology for
Jacobi forms is given by the cohomology of $n$-point connections over a
deformed vertex algebra bundle defined on the torus. The reduction cohomology
for Jacobi forms for a vertex algebra is determined in terms of the space of
analytical continuations of solutions to Knizhnik-Zamolodchikov equations.

###### Key words and phrases:

Cohomology; Jacobi forms; Vertex algebras; Knizhnik-Zamolodchikov equations

##  1\. Introduction

The natural problem of computation of continuous cohomologies for non-
commutative structures on manifolds has proven to be a subject of great
geometrical interest [BS, Fei, Fuks, Wag]. For Riemann surfaces, and even for
higher dimension complex manifolds, the classical cohomology of holomorphic
vector fields is often trivial [Kaw, Wag]. In [Fei] Feigin has obtained
various results concerning (co)-homology of cosimplicial objects associated to
holomorphic vector fields $Lie(M)$. Vertex algebra [BZF, FHL, K] theory of
automorphic forms [Fo] goes back to celebrated Moonshine problem [MT]. Most of
$n$-point characteristic functions [FS, FHL, KZ, MT, Zhu] for vertex algebras
deliver examples of modular forms with respect to appropriate groups attached
to geometry of corresponding underlying manifolds. $n$-point functions are
subject to action of differential operators with specific analytical behavior
[BKT, GK, GN, Ob]. In this paper we develop ideas and previous results on
cohomology of Jacobi forms originating from algebraic and geometrical
procedures in conformal field theory [FS, TUY]. This paper aims at developing
algebraic, differential geometry, and topological methods for the
investigation of cohomology theories of Jacobi forms generated by vertex
algebras, with applications in algebraic topology, number theory and
mathematical physics.

In most cases of lower genera Riemann surfaces, there exist algebraic formulas
relating $n$-point functions with $n-1$-point functions in a linear way for
fixed genus $g$ [Zhu, MT, MTZ]. The reduction cohomology is defined via
reduction formulas [Zhu, BKT] relating $n$-point characteristic functions with
$(n-1)$-functions. Our new algebraic and geometrical approach for computation
of reduction (co)homology involves vertex algebras and applications of
techniques [Huang, Y] used in conformal field theory. Computation of moduli
forms reduction cohomology is useful in further studies of constructions in
algebraic topology, analytical and geometrical structure of spaces of modular
forms originating from the description of vertex algebras by means of
characteristic functions on manifolds. The main aim of the reduction
cohomology is to describe non-commutative structures in terms of commutative
ones. In contrast to more geometrical methods in classical cohomology for Lie
algebras [Fuks], the reduction cohomology pays more attention to the
differential, analytical, and automorphic structure of chain complex elements
constructed by means of characteristic functions for non-commutative elements
of vertex algebras with complex parameters. Computational methods involving
reduction formulas proved their effectiveness in conformal field theory [KMI,
KMII, MT, MT1, MTZ, TZ, DLM, Miy]. Though the Zhu reduction formulas were
obtained for odrinary $n$-point functions of vertex operators, it also works
for multi-parametric automorphisms inserted into traces written for the torus
case. Then coefficients in the reduction formulas are expressed in terms of
quasi-modular forms. Since quasi-modular forms are holomorphic on the complex
upper half-plane $\mathbb{H}$, then it follows that $n$-point Jacobi functions
are also holomorphic. The plan of this paper is the following. We define the
reduction cohomology, chain condition, and co-boundary operator for complexes
of Jacobi forms. Specific examples of coboundary operators are provided
subject to various conditions on vertex algebra elements. A statement relating
$n$-th reduction cohomology with analytic extensions of solutions to a
counterpart of Knizhnik–Zamolodchikov equation is proven, and its geometrical
meaning is found. In appendixes we recall the notions of quasi-modular forms,
reduction formulas for Jacobi functions, and vertex algebras. Quasi-Jacobi
forms have found applications in vertex algebra theory in [HE], for
characteristic functions of topological $N=2$ vertex algebras, Gromov-Witten
potentials [Kaw], computation of elliptic genera [Lib]related to Jacobi zero-
point functions, Landau-Ginzburg orbifolds [KYY].

##  2\. Chain complex for vertex algebra $n$-point functions

In this section we will give definition of a chain complex associated to the
space of Jacobi forms [BKT] defined by vertex algebras.

###  2.1. Spaces of $n$-point Jacobi functions via vertex operators

Let us fix a vertex algebra $V$. We denote by ${\bf
v}_{n}=(v_{1},\ldots,v_{n})\in V^{\otimes n}$ a tuple of vertex algebra
elements (see Appendix 3.4 for definition of a vertex algebra). Mark $n$
points ${\bf p}_{n}=(p_{1},\ldots,p_{n})$ on the torus ${\mathcal{T}}$. Denote
by ${\bf z}_{n}=(z_{1},\ldots,z_{n})$ local coordinates around ${\bf
p}_{n}\in{\mathcal{T}}$. Let us introduce the notation: ${\bf
x}_{n}=\left({\bf v}_{n},{\bf z}_{n}\right)$.

In [MTZ] we considered the orbifold Jacobi $n$-point functions associated with
a vertex operator superalgebra [K] (see Appendix 3.4), with an automorphism
inserted in traces. Let $\sigma\in\mathrm{Aut}(V)$ denote the parity
automorphism

$\sigma a=(-1)^{p(a)}a.$ (2.1)

Let $g\in\mathrm{Aut}(V)$ denote any other automorphism which commutes with
$\sigma$. Let $W$ be a $V$-module. Assume that $W$ is stable under both
$\sigma$ and $g$, i.e., $\sigma$ and $g$ act on $W$.

######  Definition 1.

The $n$-point Jacobi function on $W$ for ${\bf x}_{n}\in V^{\otimes
n}\times\mathbb{C}^{n}$, and $g\in\mathrm{Aut}(V)$ is defined by

$\displaystyle\mathcal{Z}_{W}^{J}({\bf x};g,\tau)=\mathrm{STr}_{W}\left({\bf
Y}_{W}\left({\bf q}^{L_{V}(0)}{\bf v},{\bf
q}\right)_{n}\;g\;q^{L_{V}(0)-c/24}\right),$ (2.2)

$q=\exp(2\pi i\tau)$, $q_{i}=\exp(z_{i})$, $1\leq i\leq n$.

Here $\mathrm{STr}_{W}$ denotes the supertrace defined by

$\mathrm{STr}_{W}(X)=Tr_{W}(\sigma
X)=Tr_{W_{\bar{0}}}(X)-Tr_{W_{\bar{1}}}(X).$ (2.3)

The orbifold Jacobi zero-point function for general $g$ is then

$\mathcal{Z}_{W}(g,\tau)=\mathrm{STr}_{W}\left(g\;q^{L_{V}(0)-c/24}\right).$
(2.4)

In particular, when $V$ is a vertex operator algebra with Virasoro vector
$\omega$ of central charge $c_{V}$. Consider an element $J\in V_{1}$ such that
$J(0)$ acts semisimply on $V$. For ${\bf v}_{n}\in V^{\otimes n}$, on
${\mathcal{T}}$, and a weak $V$-module $W$ [MT], the Jacobi $n\geq 0$-point
function is

$\displaystyle\mathcal{Z}_{W}^{J}\left({\bf x}_{n};B\right)={\rm
Tr}_{W}\left({\bf Y}\left(e^{{\bf z}\;L_{V}(0)}{\bf v},e^{\bf
z}\right)_{n}\zeta^{J(0)}q^{L(0)}\right),$ (2.5)

where $B$ denotes parameters of $\mathcal{Z}_{W}^{J}$ including $z$ and
$\tau$, and

$\zeta=q_{z}=e^{2\pi iz}.$

The Jacobi one-point function, for $v\in V$, is given by

$\displaystyle\mathcal{Z}_{W}^{J}\left(x_{1};B\right)={\rm
Tr}_{W}\left(o_{\,0}(v_{1})\;\zeta^{J(0)}\;q^{L(0)}\right),$

which does not depend on $z_{1}$. Here
$o_{\,0}(v_{1})=v_{1}(\textup{wt}\;v_{1}-1)$ (see Appendix 3.4), with
$z\in\mathbb{C}$, and $\tau$ being the modular parameter of ${\mathcal{T}}$.

######  Definition 2.

For a $V$-module $W$, we consider the spaces of $n$-point Jacobi forms

$C^{n}(W)=\left\\{\mathcal{Z}_{W}^{J}\left({\bf x}_{n},;B\right),n\geq
0,\right\\}.$

The coboundary operator $\delta^{n}({\bf x}_{n+1})$ on $C^{n}(W)$-space is
defined according to the reduction formulas (see section 3 and Appendix 6)
[BKT, MTZ] for $V$-module $W$ Jacobi forms.

######  Definition 3.

For $n\geq 0$, and any $x_{n+1}\in V\times\mathbb{C}$, define

$\displaystyle\delta^{n}({\bf x}_{n+1}):C^{n}(W)$ $\displaystyle{\rightarrow}$
$\displaystyle C^{n+1}(W),$ (2.6)

with non-commutative operators $T_{j}(v[m].)$, $j\geq 0$, is given by the
reduction formulas (2.7)

$\displaystyle\delta^{n}\left({\bf
x}_{n+1}\right)\mathcal{Z}_{W}^{J}\left({\bf x}_{n};B\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0\atop m\geq 0}^{n}f_{k,m}({\bf
x}_{n+1};B)\;T_{k}(v_{n+1}[m].)\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right),$
(2.7)

The operators $T_{k}(v[m].)$ are insertion operators of vertex algebra modes
$v[m].$, $m\geq 0$, into $\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ at
the $k$-th entry:

$\displaystyle T_{k}(v[m].)\;\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$
$\displaystyle=$
$\displaystyle\mathcal{Z}^{J}_{W}\left(T_{k}\left(v[m]\right).{\bf
x}_{n};B\right),$

where we use the notation

$(\Gamma.)_{k}\;{\bf
x}_{n}=\left(x_{1},\ldots,\Gamma.x_{k},\ldots,x_{n}\right),$

for an operator $\Gamma$ acting on $k$-th entry.

######  Remark 1.

The reductions formulas have an interpretation in terms of torsors [BZF]
(Chapter 6). In such formulation ${\bf x}_{n}$ is a torsor with respect to the
group of transformation of the space $V^{\otimes\;n}\times\mathbb{C}^{n}$. In
particular, from (2.7) we see that $T_{k}\left(u[m].\right)$-operators act on
$V^{\otimes\;n}$-entries of ${\bf x}_{n}$, while $f_{k,m}({\bf
x}_{n+1};B)$-functions act on ${\bf z}_{n}$ of $\mathcal{Z}^{J}_{W}({\bf
x}_{n};B)$ as a complex function.

For $n\geq 0$, let us denote by ${\mathfrak{V}}_{n}$ the subsets of all ${\bf
x}_{n}\in V^{\otimes n}\times\mathbb{C}^{n}$, such that the chain condition

$\delta^{n+1}({\bf x}_{n+1})\;\delta^{n}({\bf
x}_{n})\;\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)=0,$ (2.8)

for the coboundary operators (2.7) for complexes $C^{n}(W)$ is satisfied.
Explicitly, the chain condition (2.8) leads to an infinite $n\geq 0$ set of
equations involving functions $f_{k,m}\left({\bf x}_{n+1};B\right)$ and
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$:

$\displaystyle\left(\sum\limits_{k^{\prime},k=0\atop m^{\prime},m\geq
0}^{n+1,n}f_{k^{\prime},m^{\prime}}\left({\bf
x}_{n+1};B\right)f_{k,m}\left({\bf
x}_{n};B\right)T_{k^{\prime}}(v_{n+2}[m^{\prime}].)\;T_{k}(v_{n+1}[m].)\right)\mathcal{Z}^{J}_{W}\left({\bf
x}_{n};B\right)=0.$

######  Remark 2.

(2.1) contain finite series and narrows the space of compatible $n$-point
functions. It follows from considerations of [BKT], the subspaces of
$C^{n}(W)$, $n\geq 0$, of $n$-point Jacobi forms such that the condition (2.1)
is fulfilled for reduction cohomology complexes are non-empty. Indeed, the
condition (2.1) represents an infinite $n\geq 0$ set of functional-
differential equations (with finite number of summands) on converging complex
functions $\mathcal{Z}^{J}_{W}({\bf x}_{n};B)$ defined for $n$ local complex
variables on ${\mathcal{T}}$ with functional coefficients $f_{k,m}\left({\bf
x}_{n+1};B\right)$ (in our examples in subsection 3.1–3.4, these are
generalizations of elliptic functions) on ${\mathcal{T}}$. Note that all
vertex algebra elements of ${\bf v}_{n}\in V^{\otimes n}$, as non-commutative
parameters are not present in final form of functional-differential equations
since they incorporated into either matrix elements, traces, etc. According to
the theory of such equations [FK, Gu], each equation in the infinite set of
(2.1) always have a solution in domains they are defined. Thus, there always
exist solutions of (2.1) defining $\mathcal{Z}^{J}_{W}\in C^{n}(W)$, and they
are not empty.

######  Definition 4.

The spaces with conditions (2.1) constitute a semi-infinite chain complex

$0\longrightarrow
C^{0}\stackrel{{\scriptstyle\delta^{0}(x_{1})}}{{\longrightarrow}}C^{1}\stackrel{{\scriptstyle\delta^{1}({\bf
x}_{2})}}{{\longrightarrow}}\ldots\;\;\stackrel{{\scriptstyle\delta^{n-2}({\bf
x}_{n-1})}}{{\longrightarrow}}C^{n-1}\stackrel{{\scriptstyle\delta^{n-1}({\bf
x}_{n})}}{{\longrightarrow}}C^{n}\stackrel{{\scriptstyle\delta^{n}({\bf
x}_{n+1})}}{{\longrightarrow}}\ldots.$ (2.10)

For $n\geq 1$, we call corresponding cohomology

$H^{n}_{J}(W)={\rm Ker}\;\delta^{n}({\bf x}_{n+1})/{\rm Im}\;\delta^{n-1}({\bf
x}_{n}),$ (2.11)

the $n$-th reduction cohomology of a vertex algebra $V$-module $W$ on
${\mathcal{T}}$.

##  3\. Reduction formulas and examples of coboundary operators for Jacobi
$n$-point functions

###  3.1. The main formula for coboundary operator

In this subsection, using Propositions 5 and 6 of [BKT] (see Appendix 6), we
introduce the definition of a coboundary operator associated to the most
general (up to certain assumptions) reduction formulas available for Jacobi
forms. Recall the definition of square bracket vertex operators from Appendix
6.4. Summing (6.12) over $l$ multiplied by $z^{l-1}_{n+1}$,
$Z_{W}^{J}\left(v_{n+1}[-l].x_{1},{\bf x}_{2,n};B\right)$. and using
associativity of vertex operators we formulate the following definition of the
coboundary operator

######  Definition 5.

Let $v_{n+1}\in V$ such that

$v_{n+1}[l].v_{k}=0,$

for $l\geq 1$, $1\leq k\leq n$, and such that

$J(0)v_{n+1}=\alpha v_{n+1}.$

with $\alpha\in\mathbb{C}$. Then the coboundary operator is given by (2.7)
with the summation over $l\in\mathbb{Z}$, i.e.,

$\displaystyle\delta^{n}\left({\bf
x}_{n+1}\right)\mathcal{Z}_{W}^{J}\left({\bf
x}_{n};z,\tau\right)=\sum\limits_{l\in\mathbb{Z}\atop{m\geq 0,\atop
k=0}}^{n}f_{k,m}({\bf
x}_{n+1};B)\;T_{k}(v_{n+1}[m].)\;\mathcal{Z}^{J}_{W}\left({\bf
x}_{n};z,\tau\right),$ (3.1) $\displaystyle f_{0}({\bf
x}_{n+1};B)\;T_{0}(v[m].)=\sum\limits_{l\in\mathbb{Z}}(-1)^{l+1}\;\delta_{\alpha
z,{\mathbb{Z}\tau}+\mathbb{Z}}\;\frac{\lambda^{l-1}}{(l-1)!}\;z_{n+1}^{l-1}\;T_{0}(o_{\lambda}(v_{n+1})),$
$\displaystyle f_{k,m}({\bf
x}_{n+1};B)=\sum\limits_{l\in\mathbb{Z}}(-1)^{m+1}\binom{m+l-1}{m}\;z_{n+1}^{l-1}\;F_{k,m}({\bf
x}_{n+1};l,\alpha z,\tau),$ (3.2)

where

$\displaystyle F_{k,m}({\bf x}_{n+1};l,\alpha
z,\tau)=\delta_{0,m}\;T^{1-\delta_{\alpha
z,{\mathbb{Z}\tau}+\mathbb{Z}}}.\widetilde{E}_{m+l,\lambda}\left((1-\delta_{\alpha
z,{\mathbb{Z}\tau}+\mathbb{Z}})\;\alpha z,\tau\right)$
$\displaystyle\qquad+T^{1-\delta_{\alpha
z,{\mathbb{Z}\tau}+\mathbb{Z}}}.\widetilde{P}_{m+l,\left(1-\delta_{\alpha
z,{\mathbb{Z}\tau}+\mathbb{Z}}\right)\lambda}\left(\frac{z_{1}-z_{k}}{2\pi
i},(1-\delta_{\alpha z,{\mathbb{Z}\tau}+\mathbb{Z}})\;\alpha z,\tau\right),$

with tilde applying operator $T$, i.e.,

$\displaystyle T.{E}_{m+l,\lambda}$ $\displaystyle=$
$\displaystyle\widetilde{E}_{m+l,\lambda},$ $\displaystyle
T.{P}_{m+l,\lambda}$ $\displaystyle=$
$\displaystyle\widetilde{P}_{m+l,\lambda},$

and $\widetilde{E}_{m+k,\lambda}(\alpha z,\tau)$,
$\widetilde{P}_{m+l,\lambda}(z^{\prime},\alpha z,\tau)$ given by (5.6) and
(5.5) correspondingly.

###  3.2. The simplest coboundary operator

For certain further restriction on $v_{n+1}$, we are able to define the
simplest version of coboundary operator for the reduction cohomology. Using
propositions 3 and 4 proven in [BKT] (see Appendix 6), we obtain

######  Definition 6.

For $v_{n+1}$, with $J(0)v_{n+1}=\alpha v_{n+1}$, $\alpha\in\mathbb{C}$, the
coboundary operator is defined by (2.7) with

$\displaystyle f_{0}({\bf x}_{n+1};\alpha
z,\tau)\;T_{0}(v_{n+1}[m])=\delta_{\alpha
z,\lambda\tau+\mu\in{\mathbb{Z}\tau+\mathbb{Z}}}\;e^{-{z_{n+1}}\lambda}\;T_{0}(o_{\lambda}(v_{n+1})),$
$\displaystyle f_{k,m}({\bf z}_{n+1};\lambda,k,\alpha
z,\tau)=T^{1-\delta_{\alpha
z,\lambda\tau+\mu\in{\mathbb{Z}\tau+\mathbb{Z}}}}{P}_{m+1,\lambda}\left(\frac{z_{n+1}-z_{k}}{2\pi
i},(1-\delta_{\alpha z,\lambda\tau+\mu\in{\mathbb{Z}\tau+\mathbb{Z}}})\;\alpha
z,\tau\right),$ (3.3)

with $\widetilde{P}_{m+1,\lambda}\left({\bf z}_{n+1},\alpha z,\tau\right)$
defined in (5.5).

###  3.3. Coboundary operator for a shifted Virasoro vector

Suppose that $J(0)a=\alpha a$ for $\alpha\not\in\mathbb{Z}\setminus\\{0\\}$,
and define a $V$-automorphism $g\in\mathrm{Aut}(V)$ by

$\displaystyle g=e^{2\pi i\frac{\mu}{\alpha}J(0)},$

for $\mu\in\mathbb{Z}$ for which $ga=a$. Then Corollary 2 follows from
Proposition 6 of [MTZ] which states that

$\sum_{k=1}^{n}\mbox{\rm Tr}_{W}\left(T_{k}(a[0].){\bf Y}\left(e^{{\bf
z}L_{V}(0)}v,e^{\bf z}\right)_{n}g\;q^{L(0)}\right)=0.$ (3.4)

For

$J(0)v_{k}=\alpha_{k}v_{k},$

for $k=1,\dots,n$, For the case of shifted Virasoro vector [MTZ] (see Appendix
3.4) we find In a similar fashion, we can relate Proposition 4 to Theorem 2 of
[MTZ] for the above shifted Virasoro grading $L_{h}(0)$ and with $g=e^{2\pi
i\frac{\mu}{\alpha}J(0)}$. Using Theorem 2 of [MTZ] we give the following

######  Definition 7.

The shifted coboundary operator for the shifted Jacobi form

$\displaystyle\mathcal{Z}_{W}^{J}({\bf x}_{n+1};h,\mu,\alpha,z,\tau)=\mbox{\rm
Tr}_{W}\left({\bf Y}\left(e^{\bf z\;L_{h}(0)}{\bf v},e^{\bf
x}\right)_{n+1}\;g\;q^{L_{h}(0)}\right),$ (3.5)

is given by (2.7) with

$\displaystyle f_{0}({\bf
x}_{n+1};B)\;T_{0}(v_{n+1}[m].)=T_{0}(o_{h}(v_{n+1})),$ (3.6) $\displaystyle
f_{k,m}({\bf x}_{n+1};B)=P_{m+1}\left(\frac{z_{n+1}-z_{k}}{2\pi
i},\tau\right)$ (3.7)

and $T_{k}(v_{n+1}[m]_{h}.)$, where

$o_{h}(v_{n+1})=v_{n+1}(\textup{wt}_{h}(v_{n+1})-1)=v_{n+1}(\textup{wt}(v_{n+1})-1+\mu)=o_{\mu}(v_{n+1}).$
(3.8)

###  3.4. Vertex operator superalgebra case

For the case of orbifold Jacobi $n$-point functions, we have the following.
Let $v_{n+1}$ be homogeneous of weight $\textup{wt}(v_{n+1})\in\mathbb{R}$ and
define $\phi\in U(1)$ by

$\phi=\exp(2\pi i\;\textup{wt}(v_{n+1})).$ (3.9)

We also take $v_{n+1}$ to be an eigenfunction under $g$ with

$gv_{n+1}=\theta^{-1}v_{n+1}$ (3.10)

for some $\theta\in U(1)$ so that

$g^{-1}v_{n+1}(k)g=\theta v_{n+1}(k).$ (3.11)

Then we have

######  Definition 8.

Let $v,\theta$ and $\phi$ be as as above. Then the coboundary operator is
defined by

$\displaystyle f_{0}({\bf
x}_{n+1};B)\;T_{0}(v_{n+1}[m])=\delta_{\theta,1}\delta_{\phi,1}T_{0}(o_{0}(v_{n+1})),$
$\displaystyle f_{k,m}({\bf x}_{n+1};B)=p(v_{n+1},{\bf
v}_{k-1})\;P_{m+1}\left[\begin{array}[]{c}\theta\\\
\phi\end{array}\right](z_{n+1}-z_{k},\tau)$ (3.14)

where the deformed Weierstrass functions are defined in (5.9)) (see Appendix
5.2).

Note that, as it was shown in [MTZ], the orbifold Jacobi function case is
related to the shifted Virasoro vector case above.

##  4\. Cohomology

In this section we compute the reduction cohomology defined by (2.10)–(2.11).

###  4.1. The $n$-th cohomology and analytic extensions of solutions to
Knizhnik-Zamolodchikov equations

The main result of this paper is the following.

######  Proposition 1.

Under assumptions of subsections 3.1–3.4, the $n$-th reduction cohomology of
the space of Jacobi forms for $V$-module $W$ is given by space of analytical
continuations of solutions $\mathcal{Z}_{W}^{J}\left({\bf x}_{n};B\right)$ to
the Knizhnik-Zamolodchikov equation

$\sum_{k=0}^{n}\sum_{m\geq 0}f_{k,m}\left({\bf
x}_{n};B\right)T_{k}(v_{n+1}[m]_{\beta}.)\;\mathcal{Z}_{M}^{J}\left({\bf
x}_{n};B\right)=0,$ (4.1)

with $x_{i}\notin{\mathfrak{V}}_{i}$, $\beta=h$ for a shifted Virasoro element
and zero otherwise, for $1\leq i\leq n$. These are given by the spaces of
quasi-modular forms in terms of series of deformed Weierstrass functions,
defined in Appendix 5.2, recursively generated by reduction formulas (2.7).

  * Proof:

The $n$-th reduction cohomology is defined by the subspace of $C^{n}(W)$ of
functions $\mathcal{Z}^{J}\left({\bf x}_{n};B\right)$ satisfying (4.1), modulo
the subspace of $C^{n}(W)$ $n$-point functions $\mathcal{Z}^{J}_{W}\left({\bf
x}^{\prime}_{n};B\right)$ resulting from:

$\displaystyle\mathcal{Z}^{J}_{W}\left({\bf x}^{\prime}_{n};B\right)$
$\displaystyle=$ $\displaystyle\left(\sum\limits_{k=1}^{n-1}\sum\limits_{m\geq
0}f_{k,m}\left({\bf
x}_{n};B\right)\;T^{(g)}_{k}(v^{\prime}_{n}[m]_{\beta})\right)\;\mathcal{Z}^{J}_{W}\left({\bf
x}^{\prime}_{n-1};B\right).$

Subject to other fixed parameters, $n$-point functions are completely
determined by all choices ${\bf x}_{n}\in V^{\otimes n}\times\mathbb{C}^{n}$
which does not belong to $\mathfrak{V}$. Thus, the reduction cohomology can be
treated as depending on set of ${\bf x}_{n}$ only with appropriate action of
endomorphisms generated by $x_{n+1}$. Consider a non-vanishing solution
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ to (4.1) for some ${\bf
x}_{n}$. Let us use the reduction formulas (2.7) recursively for each $x_{i}$,
$1\leq i\leq n$ of ${\bf x}_{n}$ in order to express
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ in terms of the partition
function $\mathcal{Z}^{J}_{W}\left(B\right)$, i.e., we obtain

$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)={\mathcal{D}}({\bf
x}_{n};B)\;\mathcal{Z}^{J}_{W}\left(B\right),$ (4.3)

as in [MT, MTZ, TZ]. Thus, $x_{i}\notin{\mathfrak{V}}_{i}$ for $1\leq i\leq
n$, i.e., at each stage of the recursion procedure reproducing (4.3),
otherwise $\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ is zero. Therefore,
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B,\right)$ is explicitly known and is
repsented as a series of auxiliary functions ${\mathcal{D}}({\bf
x}_{n};B\tau)$ depending on $V$. Consider now $\mathcal{Z}^{J}_{W}\left({\bf
x}^{\prime}_{n};B\right)$ given by (Proof:). It is either vanishes when
$v_{n-i}\in{\mathfrak{V}}_{n-i}$, $2\leq i\leq n$, or given by (4.3) with
${\bf x}^{\prime}_{n}$ arguments.

The way the reduction relations (2.7) were derived in [Y] is exactly the same
as for the vertex algebra derivation [KZ, TK] for the Knizhnik-Zamolodchikov
equations. Namely, one considers a double integration of
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ along small circles around two
auxiliary variables with the action of appropriate reproduction kernels
inserted. Then, these procedure leads to recursion formulas relating
$\mathcal{Z}^{J}_{W}({\bf x}_{n+1};B)$ and $\mathcal{Z}^{J}_{W}({\bf
x}_{n};B)$ with functional coefficients depending on the nature of the vertex
algebra $V$. Thus, (4.1) can be seen as a version of the Knizhnik-
Zamolodchikov equation. In [Y, MT, MTZ, BKT] formulas for $n$-point functions
in various specific examples of $V$ and configuration of Riemann surfaces were
explicitly obtained.

In terms of $x_{n+1}$, by using (6.13), one transfers in (4.1) the action of
$v_{n+1}$-modes into an analytical continuation of
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ multi-valued holomorphic
functions to domains ${\mathcal{T}}_{n}\subset{\mathcal{T}}$ with $z_{i}\neq
z_{j}$ for $i\neq j$. Namely, in (4.1), the operators
$T_{k}(v_{n+1}[m]_{\beta}.)$ act by certain modes $v_{n+1}[m].$ of a vertex
algebra element $v_{n+1}$ on ${\bf v}_{n}\in V^{\otimes n}$. Using vertex
algebra associativity we express the action of of operators
$T_{k}(v_{n+1}[m].)$ in terms of modes $v_{n+1}[m]$ inside vertex operators in
actions of $V$-modes on the whole vertex operator at expense of a shift of
their formal parameters ${\bf z}_{n}$ by $z_{n+1}$, i.e., ${\bf
z}^{\prime}_{n}={\bf z}_{n}+z_{n+1}$. Note that under such associativity
transformations $v$-part of ${\bf x}_{n}$, i.e., ${\bf v}_{n}$ do not change.
Thus, the $n$-th reduction cohomology of a $V$-module $W$ is given by the
space of analytical continuations of $n$-point functions
$\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right)$ with ${\bf
x}_{n-1}\notin{\mathfrak{V}}_{n-1}$ that are solutions to the Knizhnik-
Zamolodchikov equations (4.1). The above analytic extensions for the Knizhnik-
Zamolodchikov equations generated by $x_{n+1}$ and with coefficients provided
by functions $f_{k,m}\left({\bf x}_{n+1};B\right)$ on ${\mathcal{T}}$. ∎

One can make connection with the first cohomology of grading-restricted vertex
algebras in terms of derivations, and to the second cohomology in terms of
square-zero extensions of $V$ by $W$ [Huang]. In certain cases of coboundary
operators, we are able to compute the $n$-th cohomology even more explicitly
by using reduction formulas in terms of generalized elliptic functions In
particular, for orbifold $n$-point Jacobi functions associated to a vertex
operator superalgebra described in Appendix 3.4, we obtain from [MTZ]

######  Corollary 1.

For ${\bf v}_{n}\notin{\mathfrak{V}}_{n}$, the $n$-th cohomology is given by
the space of determinants of $n\times n$-matrices containing deformed elliptic
functions depending on $z_{i}-z_{j}$, $1\leq i,j\leq n$, for all possible
combinations of ${\bf v}_{n}$-modes.

###  4.2. Geometrical meaning of reduction formulas and conditions (2.1)

In this section we show that the Jacobi forms reduction formulas (2.7) appear
as of multipoint connections on a vector bundle over ${\mathcal{T}}$
generalizing ordinary holomorphic connections on complex curves [BZF]. Let us
recall the notion of a multipoint connection which will be useful for further
identifying reduction cohomology in this subsection. Motivated by the
definition of a holomorphic connection for a vertex algebra bundle (cf.
Section 6, [BZF] and [Gu]) over a smooth complex curve, we introduce the
definition of the multiple point connection over ${\mathcal{T}}$.

######  Definition 9.

Let $\mathcal{V}$ be a holomorphic vector bundle over ${\mathcal{T}}$, and
$\mathcal{T}_{0}\subset{\mathcal{T}}$ be its subdomain. Denote by
${\mathcal{S}\mathcal{V}}$ the space of sections of $\mathcal{V}$. A multi-
point connection $\mathcal{G}$ on $\mathcal{V}$ is a $\mathbb{C}$-multi-linear
map

$\mathcal{G}:{\mathcal{T}}^{\times n}\times V^{\otimes n}\to\mathbb{C},$

such that for any holomorphic function $f$, and two sections $\phi(p)$ and
$\psi(p^{\prime})$ of $\mathcal{V}$ at points $p$ and $p^{\prime}$ on
$\mathcal{T}_{0}\subset{\mathcal{T}}$ correspondingly, we have

$\sum\limits_{q,q^{\prime}\in\mathcal{T}_{0}\subset{\mathcal{T}}}\mathcal{G}\left(f(\psi(q)).\phi(q^{\prime})\right)=f(\psi(p^{\prime}))\;\mathcal{G}\left(\phi(p)\right)+f(\phi(p))\;\mathcal{G}\left(\psi(p^{\prime})\right),$
(4.4)

where the summation on left hand side is performed over locuses of points $q$,
$q^{\prime}$ on $\mathcal{T}_{0}$. We denote by ${\mathcal{C}on}^{n}$ the
space of $n$-point connections defined over ${\mathcal{T}}$.

Geometrically, for a vector bundle $\mathcal{V}$ defined over ${\mathcal{T}}$,
a multi-point connection (4.4) relates two sections $\phi$ and $\psi$ at
points $p$ and $p^{\prime}$ with a number of sections on
$\mathcal{T}_{0}\subset{\mathcal{T}}$.

######  Definition 10.

We call

$G(\phi,\psi)=f(\phi(p))\;\mathcal{G}\left(\psi(p^{\prime})\right)+f(\psi(p^{\prime}))\;\mathcal{G}\left(\phi(p)\right)-\sum\limits_{q,q^{\prime}\in\mathcal{T}_{0}\subset\mathcal{T}}\mathcal{G}\left(f(\psi(q^{\prime})).\phi(q)\right),$
(4.5)

the form of a $n$-point connection $\mathcal{G}$. The space of $n$-point
connection forms will be denoted by $G^{n}$.

Here we prove the following

######  Lemma 1.

Jacobi $n$-point forms (2.2) generated by reduction formulas (2.7) are
$n$-point connections on the space of automorphisms $g$ deformed sections of
the vertex algebra bundle $\mathcal{V}$ associated to $V$. For $n\geq 0$, the
$n$-th reduction cohomology of Jacobi forms is given by

$H^{n}_{J}(W)=H^{n}_{J}(\mathcal{S}\mathcal{V}_{g})={\mathcal{C}on}^{n}/G^{n-1},$
(4.6)

is isomorphic to the cohomology of the space of deformed
$\mathcal{V}$-sections.

######  Remark 3.

Proposition 1 is a deformed section vertex algebra version of the main
proposition of [BS, Wag], i.e., the Bott–Segal theorem for Riemann surfaces.

  * Proof:

In [BZF] (Chapter 6, subsection 6.5.3) the vertex operator bundle
$\mathcal{V}$ was explicitly constructed. It is easy to see that $n$-point
connections are holomorphic connection on the bundle $\mathcal{V}$ with the
following identifications. For non-vanishing $f(\phi(p))$ let us set

$\displaystyle\mathcal{G}=\mathcal{Z}^{J}_{W}\left({\bf x}_{n};B\right),$
$\displaystyle\psi(p^{\prime})=\left({\bf x}_{n+1}\right),$
$\displaystyle\phi(p)=\left({\bf x_{n}}\right),$
$\displaystyle\mathcal{G}\left(f(\psi(q)).\phi(q^{\prime})\right)=T_{k}(v[m]_{\beta}.)\;\mathcal{Z}^{J}_{W}\left({\bf
x}_{n};B\right),$
$\displaystyle-\frac{f(\psi(p^{\prime}))}{f(\phi(p))}\;\mathcal{G}\left(\phi(p)\right)=f_{0}\left({\bf
x}_{n+1};B\right)\;T_{0}(o_{\lambda}(v_{n+1}))\;\mathcal{Z}^{J}_{W}\left({\bf
x}_{n};B\right),$ $\displaystyle
f^{-1}(\phi(p))\sum\limits_{{q}_{n},{q^{\prime}}_{n}\atop\in{\mathcal{T}}_{0}\subset{\mathcal{T}}}\mathcal{G}\left(f(\psi(q)).\phi(q^{\prime})\right)=\sum\limits_{k=1\atop
m\geq 0}^{n}f_{k,m}\left({\bf
x}_{n+1};B\right)T_{k}(v[m]_{\beta}.)\;\mathcal{Z}^{J}_{W}\left({\bf
x}_{n};B\right).$

Thus, the formula (Proof:) gives (2.7). Recall [BZF] the construction of the
vertex algebra bundle $\mathcal{V}$. Here we use a Virasoro vector shifred
version of it. According to Proposition 6.5.4 of [BZF], one canonically (i.e.,
coordinate independently) associates ${\rm End}\;\mathcal{V}$-valued sections
$\mathcal{Y}_{p}$ of the $g$-twisted bundle $\mathcal{V}^{*}$ (the bundle dual
to $\mathcal{V}$). The intrinsic, i.e., coordinate independent, vertex algebra
operators are defined by [BZF]

$\langle u,\left({\mathcal{Y}}_{\bf p}^{*}(i({\bf
v}_{n}))\right)_{n}\;g\;v\rangle=\langle u,{\bf Y}({\bf x}_{n})v\rangle.$

to matrix elements of a number of vertex operators on appropriate punctured
disks around points with local coordinates ${\bf z}_{n}$ on ${\mathcal{T}}$.
The spaces of such $\mathcal{V}$-sections for each $n$ of is described by
identifications (Proof:). Taking into account the construction of Section 6
(subsection 6.6.1, in particular, construction 6.6.4, and Proposition 6.6.7)
of [BZF], we see that $n$-point functions are connections on the space of
sections of $\mathcal{V}$, and the reduction cohomology (2.11) is represented
by (4.6). ∎

The geometrical meaning of (2.1) consists in the following. Since in (2.7)
operators act on vertex algebra elements only, we can interpret it as a
relation on modes of $V$ with functional coefficients. In particular, all
operators $T$ change vertex algebra elements by action either of
$o(v)=v_{{\textup{wt}}v-1}$, or positive modes of $v[m].$, $m\geq 0$. Recall
that for $n$-point Jacobi forms are quasi-modular forms. Moreover, the
reduction formulas (2.7) can be used to prove modular invariance for higher
$n$ Jacobipoint functions. Due to automorphic properties of $n$-point
functions, (2.1) can be also interpreted as relations among modular forms. It
also defines a complex variety in ${\bf z}_{n}\in\mathbb{C}^{n}$ with non-
commutative parameters ${\bf v}_{n}\in V^{\otimes n}$. As most identities
(e.g., trisecant identity [Fay, Mu] and triple product identity [K, MTZ]) for
$n$-point functions (2.1) has its algebraic-geometrical meaning. The condition
(2.1) relates finite series of vertex algebra correlations functions on
${\mathcal{T}}$ with elliptic functions [Zhu, MT, MTZ]. Since $n$-point Jacobi
forms are quasi-modular forms, we treat (2.1) as a source of new identities on
such forms.

## Acknowledgments

The author would like to thank H. V. Lê and A. Lytchak for related
discussions.

##  5\. Appendix: Quasi-Jacobi forms

In this appendix we recall definitions and properties of Jacobi and quasi-
Jacobi forms [BKT]. First, we provide the definition of ordinary Jacobi forms
[EZ]. Let $\mathbb{H}$ be the upper-half plane.

######  Definition 11.

Let $k$, $m\in\mathbb{N}_{0}$, and $\chi$ be a rational character for a one
dimensional representation of the Jacobi group
$\textup{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$. A holomorphic Jacobi form of
weight $k$ and index $m$ on $\textup{SL}_{2}(\mathbb{Z})$ with rational
multiplier $\chi$ is a holomorphic function

$\phi:\mathbb{C}\times\mathbb{H}\to\mathbb{C},$

and

$\gamma.\tau=\frac{a\tau+b}{c\tau+d},$

which satisfies the following conditions. Let
$\gamma\in\textup{SL}_{2}(\mathbb{Z})$,

$\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

Then, for $(\lambda,\mu)\in\mathbb{Z}\times\mathbb{Z}$,

$\phi\Big{|}_{k,m}\left(\gamma,(\lambda,\mu)\right)=\chi\left(\gamma,(\lambda,\mu)\right)\phi,$
(5.1)

where for a function

$\phi:\mathbb{C}\times\mathbb{H}\to\mathbb{C},$
$\phi\Big{|}_{k,m}\left(\gamma,(\lambda,\mu)\right)(z,\tau)\\\
=(c\tau+d)^{-k}e\left(-\frac{cm(z+\lambda\tau+\mu)^{2}}{c\tau+d}+m\left(\lambda^{2}\tau+2\lambda
z\right)\right)\phi\left(\frac{z+\lambda\tau+\mu}{c\tau+d},\gamma.\tau\right).$

with $e(w)=e^{2\pi iw}$. For a multiplier $\chi$,

$\chi\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\chi\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix},(0,0)\right),\quad\chi(\lambda,\mu)=\chi\left(\begin{pmatrix}1&0\\\
0&1\end{pmatrix},(\lambda,\mu)\right),$

and $N_{1},N_{2}\in\mathbb{N}$ uniquely defined by

$\chi\left(\begin{matrix}1&1\\\ 0&1\end{matrix}\right)=e^{2\pi
i\frac{a_{1}}{N_{1}}},\quad\chi(0,1)=e^{2\pi i\frac{a_{2}}{N_{2}}},$

where for $a_{j}\in\mathbb{N}$, $\gcd(a_{j},N_{j})=1$. The function $\phi$ has
a Fourier expansion of the form with $q=e(\tau)$, $\zeta=e(z)$,

$\displaystyle\phi\left(z,\tau\right)=\sum_{n\in\mathbb{N}_{0}+\rho_{1}}\sum_{r\in\mathbb{Z}+\rho_{2}\atop{r^{2}\leq
4nm}}c(n,r)q^{n}\zeta^{r},$ (5.2)

where $\rho_{j}=\frac{a_{j}}{N_{j}}\pmod{\mathbb{Z}}$ with $0\leq\rho_{j}<1$.

We next consider quasi-Jacobi forms as introduced in [Lib].

######  Definition 12.

An almost meromorphic Jacobi form of weight $k$, index $0$, and depth $(s,t)$
is a meromorphic function in
$\mathbb{C}\\{q,\zeta\\}[z^{-1},\frac{z_{2}}{\tau_{2}},\frac{1}{\tau_{2}}]$
with $z=z_{1}+iz_{2},\ \tau=\tau_{1}+i\tau_{2}$) satisfying (5.1), and which
has degree at most $s$, $t$ in $\frac{z_{2}}{\tau_{2}}$, $\frac{1}{\tau_{2}}$,
respectively.

######  Definition 13.

A quasi-Jacobi form of weight $k$, index $0$, and depth $(s,t)$ is defined by
the constant term of an almost meromorphic Jacobi form of index $0$ considered
as a polynomial in $\frac{z_{2}}{\tau_{2}}$, $\frac{1}{\tau_{2}}$.

###  5.1. Modular and elliptic functions

For a variable $x$, set

$D_{x}=\frac{1}{2\pi i}\frac{\partial}{\partial x},$

and $q_{x}=e^{2\pi ix}$. Define for

$m\in\mathbb{N}=\\{\ell\in\mathbb{Z}:\ell>0\\},$

the elliptic Weierstrass functions

$\displaystyle P_{1}(w,\tau)=$
$\displaystyle-\sum_{n\in\mathbb{Z}\backslash\\{0\\}}\frac{q_{w}^{n}}{1-q^{n}}-\frac{1}{2},$
(5.3) $\displaystyle P_{m+1}(w,\tau)=$
$\displaystyle\frac{\left(-1\right)^{m}}{m!}D_{w}^{m}\left(P_{1}(w,\tau)\right)=\frac{(-1)^{m+1}}{m!}\sum_{n\in\mathbb{Z}\backslash\\{0\\}}\frac{n^{m}q_{w}^{n}}{1-q^{n}}.$

Next, we have

######  Definition 14.

The modular Eisenstein series $E_{k}(\tau)$, defined by $E_{k}=0$ for $k$ for
odd and $k\geq 2$ even

$\displaystyle E_{k}(\tau)$
$\displaystyle=-\frac{B_{k}}{k!}+\frac{2}{(k-1)!}\sum\limits_{n\geq
1}\frac{n^{k-1}q^{n}}{1-q^{n}},$

where $B_{k}$ is the $k$-th Bernoulli number defined by

$(e^{z}-1)^{-1}=\displaystyle{\sum\limits_{k\geq 0}\frac{B_{k}}{k!}z^{k-1}}.$

It is convenient to define $E_{0}=-1$. $E_{k}$ is a modular form for $k>2$ and
a quasi-modular form for $k=2$. Therefore,

$\displaystyle
E_{k}(\gamma\tau)=(c\tau+d)^{k}E_{k}(\tau)-\delta_{k,2}\frac{c(c\tau+d)}{2\pi
i}.$

######  Definition 15.

For $w$, $z\in\mathbb{C}$, and $\tau\in\mathbb{H}$ let us define

$\displaystyle\widetilde{P}_{1}(w,z,\tau)=-\sum_{n\in\mathbb{Z}}\frac{q_{w}^{n}}{1-q_{z}q^{n}}.$

We also have

######  Definition 16.

$\widetilde{P}_{m+1}(w,z,\tau)=\frac{(-1)^{m}}{m!}D_{w}^{m}\left(\widetilde{P}_{1}(w,z,\tau)\right)=\frac{(-1)^{m+1}}{m!}\sum_{n\in\mathbb{Z}}\frac{n^{m}q_{w}^{n}}{1-q_{z}q^{n}}.$
(5.4)

It is thus useful to give

######  Definition 17.

For $m\in\mathbb{N}_{0}$, let

$\displaystyle P_{m+1,\lambda}\left(w,\tau\right)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{m+1}}{m!}\sum_{n\in\mathbb{Z}\backslash\\{-\lambda\\}}\frac{n^{m}q_{w}^{n}}{1-q^{n+\lambda}}.$
(5.5)

On notes that

$P_{1,\lambda}\left(w,\tau\right)=q_{w}^{-\lambda}(P_{1}(w,\tau)+1/2),$

with

$\displaystyle P_{m+1,\lambda}\left(w,\tau\right)$
$\displaystyle=\frac{(-1)^{m}}{m!}D_{w}^{m}\left(P_{1,\lambda}\left(w,\tau\right)\right).$

We also consider the expansion

$\displaystyle P_{1,\lambda}(w,\tau)=\frac{1}{2\pi iw}-\sum_{k\geq
1}E_{k,\lambda}(\tau)(2\pi iw)^{k-1},$

where we find [Zag]

$\displaystyle E_{k,\lambda}(\tau)$
$\displaystyle=\sum_{j=0}^{k}\frac{\lambda^{j}}{j!}E_{k-j}(\tau).$ (5.6)

######  Definition 18.

We define another generating set $\widetilde{E}_{k}(z,\tau)$ for $k\geq 1$
together with $E_{2}(\tau)$ given by [Ob]

$\displaystyle\widetilde{P}_{1}(w,z,\tau)=\frac{1}{2\pi iw}-\sum_{k\geq
1}\widetilde{E}_{k}(z,\tau)(2\pi iw)^{k-1},$ (5.7)

where we find that for $k\geq 1$,

$\displaystyle\widetilde{E}_{k}(z,\tau)=$
$\displaystyle-\delta_{k,1}\frac{q_{z}}{q_{z}-1}-\dfrac{B_{k}}{k!}+\frac{1}{(k-1)!}\sum_{m,n\geq
1}\left(n^{k-1}q_{z}^{m}+(-1)^{k}n^{k-1}q_{z}^{-m}\right)q^{mn},$ (5.8)

and $\widetilde{E}_{0}(z,\tau)=-1$.

###  5.2. Deformed elliptic functions

In this subsection we recall the definition of deformed elliptic functions
[DLM, MTZ]. Let $(\theta,\phi)\in U(1)\times U(1)$ denote a pair of modulus
one complex parameters with $\phi=\exp(2\pi i\lambda)$ for $0\leq\lambda<1$.

######  Definition 19.

For $z\in\mathbb{C}$, $\tau\in\mathbb{H}$ we define deformed Weierstrass
functions for $k\geq 1$ as

$P_{k}\left[\begin{array}[]{c}\theta\\\
\phi\end{array}\right](z,\tau)=\frac{(-1)^{k}}{(k-1)!}\sum\limits_{n\in\mathbb{Z}+\lambda}^{\prime}\frac{n^{k-1}q_{z}^{n}}{1-\theta^{-1}q^{n}},$
(5.9)

for $q=q_{2\pi i\tau}$ where $\sum\limits^{\prime}$ means we omit $n=0$ if
$(\theta,\phi)=(1,1)$.

The functions (5.9) converge absolutely and uniformly on compact subsets of
the domain $\left|q\right|<\left|q_{z}\right|<1$ [DLM]. For $k\geq 1$,

$P_{k}\left[{\begin{array}[]{c}\theta\\\
\phi\end{array}}\right](z,\tau)=\frac{(-1)^{k-1}}{(k-1)!}\frac{d^{k-1}}{dz^{k-1}}P_{1}\left[{\begin{array}[]{c}\theta\\\
\phi\end{array}}\right](z,\tau).$ (5.10)

##  6\. Appendix: Reduction formulas for Jacobi $n$-point functions

In this subsection we recall the reduction formulas derived in [MTZ, BKT].

###  6.1. Vertex operator superalgebra case

Recall [MTZ], the following

######  Proposition 2.

Suppose that $v_{n+1}\in V$ is homogeneous of integer weight
$\textup{wt}(v_{n+1})\in\mathbb{Z}$. Then we have

$\sum\limits_{k=1}^{n}p(v_{n+1},{\bf
v}_{k-1})\;\mathcal{Z}^{J}_{w}\left((v[0].)_{k}{\bf v}_{n},B\right)=0,$ (6.1)

with $p(v_{n+1},{\bf v}_{k-1})$ given by

$p(A,B_{1}\ldots B_{r-1})=\left\\{\begin{array}[]{cc}1\text{ }&\text{for
}r=1\\\ (-1)^{p(A)[p(B_{1})+...+p(B_{r-1})]}&\text{ for
}r>1\end{array}\right..$ (6.2)

Let $v_{n+1}$ be homogeneous of weight $\textup{wt}(v_{n+1})\in\mathbb{R}$ and
define $\phi\in U(1)$ by

$\phi=\exp(2\pi i\;\textup{wt}(v_{n+1})).$ (6.3)

We also take $v_{n+1}$ to be an eigenfunction under $g$ with

$gv_{n+1}=\theta^{-1}v_{n+1}$ (6.4)

for some $\theta\in U(1)$ so that

$g^{-1}v_{n+1}(k)g=\theta v_{n+1}(k).$ (6.5)

Then we obtain the following generalization of Zhu’s Proposition 4.3.2 [Zhu]
for the $n$-point function:

######  Theorem 1.

Let $v_{n+1},\theta$ and $\phi$ be as as above. Then for any ${\bf v}_{n}\in
V^{\otimes n}$ we have

$\displaystyle\mathcal{Z}_{W}^{J}\left({\bf
x_{n+1}};B\right)=\delta_{\theta,1}\delta_{\phi,1}\mathrm{STr}_{W}\left(o(v_{n+1})\;{\bf
Y}_{W}\left({\bf q}^{L_{V}(0)}{\bf v},{\bf
q}\right)_{n}\;g\;q^{L(0)-c/24}\right)$ $\displaystyle+\sum\limits_{k=1\atop
m\geq 0}^{n}p(v_{n+1},{\bf v}_{k-1})\;P_{m+1}\left[\begin{array}[]{c}\theta\\\
\phi\end{array}\right](z_{n+1}-z_{k},\tau)\mathcal{Z}_{W}^{J}((v_{n+1}[m])_{k}.{\bf
v}_{n};B).$ (6.8)

The deformed Weierstrass function is defined in (5.9)).

###  6.2. The first reduction formula

Suppose that $v_{n+1}\in V$ with

$L(0)v_{n+1}=\textup{wt}(v_{n+1})v_{n+1},$ $J(0)v_{n+1}=\alpha v_{n+1},$

for $\alpha\in\mathbb{C}$. The simplest case of reduction formulas for modes

$v_{n+1}(\textup{wt}(v_{n+1})-1+\beta)=o_{\beta}(v_{n+1}),$

by $\beta\in\mathbb{Z}$, is given in [BKT]:

######  Lemma 2.

For all $k\in\mathbb{C}$, we have

$\displaystyle\left(1-\zeta^{-\alpha}q^{\beta}\right)\;T_{0}(o_{\beta}(v_{n+1}))\;\mathcal{Z}^{J}_{W}({\bf
x}_{n},B)$ $\displaystyle\qquad=\sum\limits_{k=1}^{n}\sum_{m\geq
0}\mathcal{Z}_{M}^{J}\left(\left(e^{z_{k}\beta}\frac{\beta^{m}}{m!}v_{n+1}[m].\right)_{k}{\bf
x}_{n};B\right).$

Lemma 2 implies the following corollary.

######  Corollary 2.

Let $J(0)v_{n+1}=\alpha v_{n+1}$. If $\alpha
z=\lambda\tau+\mu\in{\mathbb{Z}\tau}+\mathbb{Z}$, then

$\sum_{k=1}^{n}\sum_{m\geq
0}\mathcal{Z}_{M}^{J}\left(\left(e^{z_{k}\lambda}\frac{\lambda^{m}}{m!}v_{n+1}[m].\right)_{k}{\bf
v}_{n};B\right)=0.$ (6.9)

We now provide the following reduction formula for formal Jacobi $n$-point
functions [BKT]. For eigenstates $v_{n+1}$ with respect to $J(0)$ we obtain:

######  Proposition 3.

Let ${\bf x}_{n+1}\in V^{\otimes(n+1)}\times\mathbb{C}^{n+1}$, with
$J(0)v_{n+1}=\alpha v_{n+1}$, $\alpha\in\mathbb{C}$. For $\alpha
z\notin{\mathbb{Z}\tau}+\mathbb{Z}$, we have

$\displaystyle\mathcal{Z}_{M}^{J}\left({\bf
x}_{n+1};B\right)=\sum_{k=1}^{n}\sum_{m\geq
0}\widetilde{P}_{m+1}\left(\frac{z_{n+1}-z_{k}}{2\pi i},\alpha
z,\tau\right)\mathcal{Z}_{W}^{J}\left((v_{n+1}[m].)_{k}\;{\bf
x}_{n};B\right).$ (6.10)

######  Proposition 4.

Let ${\bf x}_{n+1}\in V^{\otimes(n+1)}\times\mathbb{C}^{n+1}$, with
$J(0)v_{n+1}=\alpha v_{n+1}$. For $\alpha
z=\lambda\tau+\mu\in{\mathbb{Z}\tau+\mathbb{Z}}$, we have

$\displaystyle\mathcal{Z}_{W}^{J}\left({\bf x}_{n+1};B\right)$
$\displaystyle\quad=e^{-z_{n+1}\lambda}\mbox{\rm
Tr}_{W}\left(v_{n+1}(\textup{wt}(v_{n+1})-1+\lambda)\;{\bf Y}\left(e^{{\bf
z}L(0)}{\bf v},e^{\bf x}\right)_{n}\zeta^{J(0)}q^{L(0)}\right)$
$\displaystyle\quad\quad+\sum_{k=1}^{n}\sum_{m\geq
0}P_{m+1,\lambda}\left(\frac{z_{n+1}-z_{k}}{2\pi
i},\tau\right)\mathcal{Z}_{M}^{J}\left((v_{n+1}[m].)_{k}{\bf
x}_{n});B\right),$ (6.11)

with $P_{m+1,\lambda}\left(w,\tau\right)$ defined in (5.5).

Next we provide the reduction formula for Jacobi $n$-point functions.

######  Proposition 5.

Let ${\bf x}_{n+1}\in V^{\otimes(n+1)}\times\mathbb{C}^{n+1}$, with
$J(0)v_{n+1}=\alpha v_{n+1}$. For $l\geq 1$ and $\alpha
z\notin{\mathbb{Z}\tau}+\mathbb{Z}$, we have

$\displaystyle\mathcal{Z}_{W}^{J}\left(v_{n+1}[-l].x_{1},{\bf
x}_{2,n});B\right)$ $\displaystyle\quad=\sum_{m\geq
0}(-1)^{m+1}\binom{m+l-1}{m}\widetilde{G}_{m+l}(\alpha
z,\tau)\mathcal{Z}_{W}^{J}\left(v_{n+1}[m].x_{1}),{\bf x}_{2,n};B\right)$
$\displaystyle\quad\quad+\sum_{k=2}^{n}\sum_{m\geq
0}(-1)^{l+1}\binom{m+l-1}{m}\widetilde{P}_{m+l}\left(\frac{z_{1}-z_{k}}{2\pi
i},\alpha z,\tau\right)\mathcal{Z}_{W}^{J}\left(v_{n+1}[m].{\bf
x}_{n};B\right).$ (6.12)

Propositions 4 and 5 imply the next result [BKT]:

######  Proposition 6.

Let ${\bf x}_{n+1}\in V^{\otimes(n+1)}\times\mathbb{C}^{n+1}$, with
$J(0)v_{n+1}=\alpha v_{n+1}$. For $l\geq 1$ and $\alpha
z=\lambda\tau+\mu\in{\mathbb{Z}\tau}+\mathbb{Z}$, we have

$\displaystyle\mathcal{Z}_{W}^{J}\left(v_{n+1}[-l].x_{1},{\bf
x}_{2,n});B\right)$
$\displaystyle\quad=(-1)^{l+1}\frac{\lambda^{l-1}}{(l-1)!}{\rm
Tr}_{W}\left(v_{n+1}(\lambda+\textup{wt}(v_{n+1})-1){\bf Y}\left(e^{{\bf
x}L_{V}(0)}{\bf v},e^{\bf x}\right)_{n}\zeta^{J(0)}q^{L(0)}\right)$
$\displaystyle\quad\quad+\sum\limits_{m\geq
0}(-1)^{m+1}\binom{m+l-1}{m}{E}_{m+l,\lambda}(\tau)\;\mathcal{Z}_{W}^{J}\left(v_{n+1}[m].x_{1},{\bf
x}_{2,n};B\right)$ $\displaystyle\quad\quad+\sum\limits_{k=2}^{n}\sum_{m\geq
0}(-1)^{l+1}\binom{m+l-1}{m}P_{m+l,\lambda}\left(\frac{x_{1}-x_{k}}{2\pi
i},\tau\right)\;\mathcal{Z}_{W}^{J}\left(v_{n+1}[m].{\bf x}_{n};B\right),$

for ${E}_{k,\lambda}$ given in (5.6).

######  Remark 4.

In the case $\alpha=0$ we have that $\lambda=\mu=0$ and Propositions 4 and 6
imply the standard results of [Zhu] or [MTZ] with
$a(\lambda+\textup{wt}(a)-1)=o(a)$.

###  6.3. Vertex operator (super)algebras

In this subsection we recall the notion of vertex operator (super)algebras [B,
FHL, FLM, K, MN]. Let $V$ be a superspace, i.e., a complex vector space

$V=V_{\bar{0}}\oplus V_{\bar{1}}=\oplus_{\alpha}V_{\alpha},$

with index label $\alpha$ in $\mathbb{Z}/2\mathbb{Z}$ so that each $a\in V$
has a parity $p(a)\in\mathbb{Z}/2\mathbb{Z}$.

An $\mathbb{C}$-graded vertex operator superalgebra is defined by
$(V,Y,\mathbf{1}_{V},\omega)$ where $V$ is a superspace with a
$\mathbb{C}$-grading where

$V=\oplus_{r\geq r_{0}}V_{r},$

for some $r_{0}$ and with parity decomposition

$V_{r}=V_{\bar{0},r}\oplus V_{\bar{1},r}.$

$\mathbf{1}_{V}\in V_{\bar{0},0}$ is the vacuum vector and $\omega\in
V_{\bar{0},2}$ the conformal vector with properties described below. The
vertex operator $Y$ is a linear map

$Y:V\rightarrow(\mathrm{End}V)[[z,z^{-1}]],$

for formal variable $z$, so that for any vector $x=(a,v)\in
V\times\mathbb{C}$,

$Y(x)=\sum_{n\in\mathbb{Z}}a(n)z^{-n-1}.$ (6.13)

The component operators (modes) $a(n)\in\mathrm{End}V$ are such that

$a(n)\mathbf{1}_{V}=\delta_{n,-1}a,$

for $n\geq-1$ and

$a(n)V_{\alpha}\subset V_{\alpha+p(a)},$ (6.14)

for $a$ of parity $p(a)$.

The vertex operators satisfy the locality property for all
$x_{i}=(v_{i},z_{i})\in V\times\mathbb{C}$, $i=1,2$,

$(z_{1}-z_{2})^{N}[Y(x_{1}),Y(x_{2})]=0,$ (6.15)

for $N\gg 0$, where the commutator is defined in the graded sense, i.e.,

$[Y(x_{1}),Y(x_{2})]=Y(x_{1})Y(x_{2})-(1)^{p(v_{1})p(v_{2})}Y(x_{2})Y(x_{1}).$

The vertex operator for the vacuum is

$Y(\mathbf{1}_{V},z)={\rm Id}_{V},$

whereas that for $\omega$ is

$Y(\omega,z)=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2},$ (6.16)

where $L(n)$ forms a Virasoro algebra for central charge $c$

$[L_{V}(m),L_{V}(n)]=(m-n)L_{V}(m+n)+\frac{c}{12}(m^{3}-m)\delta_{m,-n}{\rm
Id}_{V}.$ (6.17)

$L_{V}(-1)$ satisfies the translation property

$Y\left(L_{V}(-1)x\right)=\frac{d}{dz}Y(x).$ (6.18)

$L_{V}(0)$ describes the $\mathbb{C}$-grading with

$L_{V}(0)a=\textup{wt}(a)a,$

for weight $\textup{wt}(a)\in\mathbb{C}$ and

$V_{r}=\\{a\in V|\textup{wt}(a)=r\\}.$ (6.19)

We quote the standard commutator property of vertex operator superalgebra,
e.g., [K, FHL, MN], for $x_{1}=(a,z_{1})$, $x=(b,z_{2})$

$[a(m),Y(x)]=\sum\nolimits_{j\geq 0}\binom{m}{j}Y(a(j).x)z_{1}^{m-j}.$ (6.20)

Taking $a=\omega$ this implies for $b$ of weight $\textup{wt}(b)$ that

$[L_{V}(0),b(n)]=(\textup{wt}(b)-n-1)b(n),$ (6.21)

so that

$b(n)V_{r}\subset V_{r+\textup{wt}(b)-n-1}.$ (6.22)

In particular, we define for $a$ of weight $\textup{wt}(a)$ the zero mode

$o_{\lambda}(a)=\left\\{\begin{array}[]{c}a(\textup{wt}(a)-1+\lambda),\text{ \
for }\textup{wt}(a)\in\mathbb{Z}\\\ 0\text{, \ otherwise,}\end{array}\right.$
(6.23)

which is then extended by linearity to all $a\in V$.

###  6.4. Square bracket formalism

Define the square bracket operators for $V$ by

$\displaystyle
Y[x]=Y\left(e^{z\;L(0)}v,e^{z}-1\right)=\sum_{n\in\mathbb{Z}}v[n]z^{-n-1}.$
(6.24)

For $v$ of weight $\textup{wt}(v)$ and $k\in\mathbb{Z}$ (see [Zhu, Lemma
4.3.1]), we have

$\displaystyle\sum_{j\geq 0}\binom{k+\textup{wt}(v)-1}{j}v(j)=\sum_{m\geq
0}\frac{k^{m}}{m!}v[m].$ (6.25)

The square bracket operators form an isomorphic vertex operator algebra with
Virasoro vector

$\displaystyle\widetilde{\omega}=\omega-\frac{c}{24}\mathbbm{1}_{V}.$

Let us now introduce [DMs] the shifted Virasoro vector

$\omega_{h}=\omega+h(-2)\mathbbm{1}_{V},$

where

$h=-\frac{\lambda}{\alpha}J,$

for $\lambda\in\mathbb{Z}$. Then the shifted grading operator is

$\displaystyle L_{h}(0)=L(0)-h(0)=L(0)+\frac{\lambda}{\alpha}J(0).$

Denote the square bracket vertex operator for the shifted Virasoro vector by

$Y[x]_{h}=Y\left(e^{z\;L_{h}(0)}v,e^{z}-1\right)=\sum_{n\in\mathbb{Z}}v[n]_{h}\;z^{-n-1},$

Therefore,

$Y[a,z]_{h}=e^{z\lambda}Y[a,z],$

or equivalently,

$\displaystyle a[n]_{h}=\sum_{m\geq 0}\frac{\lambda^{m}}{m!}a[n+m].$ (6.26)

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# Multi-color continuous-variable quantum entanglement in dissipative Kerr
solitons

Ming Li  Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  Yan-Lei Zhang  Key Laboratory of
Quantum Information, CAS, University of Science and Technology of China, Hefei
230026, China  CAS Center For Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
P. R. China.  Xin-Biao Xu  Key Laboratory of Quantum Information, CAS,
University of Science and Technology of China, Hefei 230026, China  CAS Center
For Excellence in Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, P. R. China.  Chun-Hua
Dong  Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  Hong X. Tang  Department of
Electrical Engineering, Yale University, New Haven, CT 06511, USA  Guang-Can
Guo  Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  Chang-Ling Zou<EMAIL_ADDRESS>Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.

###### Abstract

In a traveling wave microresonator, the cascaded four-wave mixing between
optical modes allows the generation of frequency combs, including the
intriguing dissipative Kerr solitons (DKS). Here, we theoretically investigate
the quantum fluctuations of the comb and reveal the quantum feature of the
soliton. It is demonstrated that the fluctuations of Kerr frequency comb lines
are correlated, leading to multi-color continuous-variable entanglement. In
particular, in the DKS state, the coherent comb lines stimulate photon-pair
generation and also coherent photon conversion between all optical modes, and
exhibit all-to-all connection of quantum entanglement. The broadband multi-
color entanglement is not only universal, but also is robust against practical
imperfections, such as extra optical loss or extraordinary frequency shift of
a few modes. Our work reveals the prominent quantum nature of DKSs, which is
of fundamental interest in quantum optics and also holds potential for quantum
network and distributed quantum sensing applications.

_Introduction.-_ Over the past decades, nonlinear optics technologies become
the backbone of modern optics, enabling frequency conversion, optical
parametric oscillation, ultrafast modulation, and non-classical quantum
sources. Especially, various nonlinear optics mechanisms enabled the
generation of the optical frequency comb, which has attracted lots of research
interests due to its revolutionary applications in precision spectroscopy,
astronomy detection, optical clock, and optical communication (fortier201920,
; foltynowicz2011optical, ; picque2019frequency, ; Papp:14, ;
obrzud2019microphotonic, ; marin2017microresonator, ; menicucci2008one, ).
Recently, by harnessing the enhanced nonlinear optical effect in a
microresonator, the Kerr frequency comb, in particular, the dissipative Kerr
soliton (DKS) as phase-locked frequency comb, has been extensively studied
(kippenberg2018dissipative, ). Such DKS have been realized on photonic chips
with various $\chi^{(3)}$ materials, showing advances in compact size,
scalability, low power consumption, great spectral range and large repetition
rate (gaeta2019photonic, ; Wan:20, ; Bruch2021, ; bai2020brillouin, ).

Therefore, the DKS is appealing for photonic quantum information science. From
one aspect, the DKS produces an array of stable and locked coherent laser
sources with equally spaced frequencies, which provides a coherent laser
source for driving nonlinear light-matter interaction as well as stable
frequency reference of local oscillators for heterodyne measurements via
frequency multiplexing. By taking advantage of the coherence over a large
frequency band, the quantum key distribution utilizing DKS has been
experimentally studied (wang2020quantum, ), which promises commercialized
high-speed quantum communication. From another aspect, the Kerr nonlinear
interaction is inherently a quantum parametric process that describes the
annihilation of two photons and simultaneous generation of signal-idler photon
pairs, which imply quantum correlations among comb lines. When working below
the threshold, the DKS devices have been exploited to generate both discrete-
and continuous-variable (CV) quantum entangled states (reimer2016generation, ;
Cui2020, ; zhu2020chip, ; PhysRevResearch.2.023138, ), holding the potential
for one-way quantum computing and high-dimensional entanglement between
different colors that could be distributed over the quantum network
(kues2019quantum, ). For the case above the threshold, although previous
studies of $\chi^{(2)}$ optical combs indicate the potential applications of
comb in the multipartite cluster-state generation for quantum computing
(menicucci2008one, ; PhysRevLett.107.030505, ; PhysRevLett.108.083601, ), the
quantum nature of DKS have not been studied excepting one pioneer work by
Chembo (chembo2016quantum, ), who pointed out the mode-pairs locating
symmetrically in two sides of the pump mode exhibit significant squeezing.

In this Letter, we theoretically investigated the CV entanglement between comb
lines in a microcavity and compare the quantum features of comb in different
states, i.e. the state below the threshold, primary comb, and DKS. We showed
that the DKS could stimulate pair-generation and coherent conversion
interactions between optical modes and thus produces a complex network links
all optical modes. Evaluated by the entanglement logarithm negativity, it is
demonstrated that a large number of comb lines with different colors are
entangled together when the cavity is prepared at the soliton state. In
particular, a group of modes show all-to-all quantum entanglement, indicating
that arbitrary two modes over a large bandwidth could be entangled. The all-
to-all entanglement persists even some modes in this complex network are
eliminated. This multi-color quantum-entangled state (coelho2009three, ) holds
great potential in application in multi-user quantum communication, quantum
teleportation network (van2000multipartite, ; furusawa2007quantum, ) and
quantum-enhanced measurements (PhysRevLett.111.093603, ; simon2017quantum, ;
guo2020distributed, ; xia2020demonstration, ).

_Model and principle.-_ Figure 1(a) illustrates an optical microring cavity
sided-coupled to a waveguide for generating dissipative Kerr solitons
(kippenberg2018dissipative, ). By injecting a pump laser into a resonance of
the cavity, the intracavity optical field builds up and the photons are
converted between optical modes via cascaded four-wave mixing (FWM) due to the
Kerr nonlinearity, leading to a broad comb spectrum at the output. For
investigating the DKS in the microcavity, we consider a group of $2N+1$ modes
belongs to the same mode family, with the spatial distributions described by
$\Psi\left(\overrightarrow{r},\theta\right)=\phi\left(\overrightarrow{r}\right)e^{im\theta}$
(strekalov2016nonlinear, ). Here, $m\in\mathbf{m}$ denotes the mode index,
which corresponds to the angular momentum of the resonant modes and
$\mathbf{m}=\left\\{-N,-N-1,...,-1,0,1,...,N-1,N\right\\}$. Due to the
dispersion, the resonant frequencies could be expanded with respect to the
index as $\omega_{m}=\omega_{0}+mD_{1}+\frac{m^{2}}{2!}D_{2}+....$
(chembo2013spatiotemporal, ). The Hamiltonian describing this multimode system
reads (guo2018efficient, )

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=-N}^{N}\delta_{i}a_{i}^{\dagger}a_{i}+g_{0}\sum_{ijkl}\delta\left(i+j-k-l\right)a_{i}^{\dagger}a_{j}^{\dagger}a_{k}a_{l}$
(1)

in the rotating frame of
$\sum_{j=-N}^{N}\left(\omega_{p}+jD_{1}\right)a_{j}^{\dagger}a_{j}$. Here,
$\delta_{j}=\omega_{0}-\omega_{p}+j^{2}D_{2}/2$ is the mode detuning by
neglecting higher order dispersion terms, $a_{i}^{\dagger}$($a_{i}$) is the
photon creation (annihilation) operator, $g_{0}$ is the vaccum coupling
strength of the Kerr nonlinearity, $\delta(\cdot)$ is the Kronecker delta
function and reflects the phase-matching condition $i+j=k+l$ according to the
angular momentum conservation (guo2018efficient, ; strekalov2016nonlinear, ).
The nonlinear interaction terms describe the simultaneous annihilation and
creation of photon-pairs in mode-pair $\left(i,j\right)$ and
$\left(k,l\right)$, with the total photon numbers are conserved. By taking all
permutations of indices $\left\\{i,j,k,l\right\\}$ in the summation, the
Hamiltonian is Hermitian and involves all FWM terms, including self-phase
modulation, cross-phase modulation, degenerate and non-degenerate FWMs.

Figure 1: (a) Schematic of dissipative Kerr soliton generation in an optical
microring cavity with a continuous-wave laser drive. (b) Four-wave mixing
processes classified by the conserved total angular momentum $\xi$, with
photon-pairs generated or annihilated simultaneously in mode-pairs
$\left(i,j\right)$ with $\xi=i+j$. The paired modes are labeled by the same
color and orientation, and the index $\xi$ can be taken from $-2N$ to $2N$.
Here we only show $\xi=-1,0,4$ as an illustration. Blue arrow: coherent photon
conversion between modes $m=-4,0$ stimulated by $m=-1,3$, with $\xi=-1$.

In Eq. (1), the number of FWM terms grows rapidly with the total mode number
as $\sim\left(2N+1\right)^{3}$. Since the total angular momentum [$\xi=i+j$ of
mode-pair $\left(i,j\right)$] conserves, we use the $\xi$ to classify
different FWM terms, as all the mode-pairs of $\xi$ could interact with each
other. As shown by Fig. 1(b), each horizontal line of $\xi$ represents the
group of signal-idler mode-pairs which are labeled by arrows with the same
color and orientation. The dashed orange arrow represents the degenerate case
$i=j$. It is anticipated that the photon-pair generation for mode-pair
$\left(i,j\right)$ would be stimulated by all the coherent light fields
belongs to the same $\xi$, which might lead to the bipartite CV entanglement
by the effective interaction
$\left(a_{i}^{\dagger}a_{j}^{\dagger}+a_{i}a_{j}\right)$ for all pairs in
$\xi$. Meanwhile, a single mode also connects to multiple $\xi$. As an
example, for mode $m=-3$ in Fig. 1(b), it could be entangled with mode $m=3$
with $\xi=0$, and $m=2$ with $\xi=-1$, as well as $m=7$ with $\xi=4$. In
addition, for the phase-matching $i+j=k+l$, there are also coherent photon
conversion processes between mode $\left(i,l\right)$ simulated by mode
$\left(j,k\right)$ and vice verse, which also distributes the quantum
correlations between different modes. Consequently, the FWM in the
microresonator results in a complex network, with optical modes serving as
nodes, and the links corresponding to photon-pair generation and coherent
photon conversion.

The complex network of FWM in a microcavity (Fig. 1(b)) implies complicated
dynamics of optical fields, which can be numerically evaluated by solving the
quantum dynamics of bosonic modes according to the Hamiltonian [Eq. (1)].
Instead of solving the intractable many-body nonlinear equations via full
quantum theory, we adopt the mean-field treatment of the strongly pumped
system, i.e. the mean and fluctuation of optical field in each mode are solved
separately, due to the weak nonlinearity $g_{0}/\kappa\ll 1$ in practice
(PhysRevApplied.13.034030, ). The operator of the cavity mode $a_{i}$ is
approximated by the sum of a classical field described by a _amplitude_
$\alpha_{i}$ and a _fluctuation_ described by a bosonic operator $\delta
a_{i}$. According to the Heisenberg equation and discarding the high-order
terms of the fluctuation operators, the dynamics of the classical fields and
their fluctuations follow

$\displaystyle\frac{d}{dt}\alpha_{i}$ $\displaystyle=$
$\displaystyle\beta_{i}\alpha_{i}-ig_{0}\sum_{jkl}\alpha_{j}^{*}\alpha_{k}\alpha_{l}+\varepsilon_{p}\delta\left(i\right),$
(2) $\displaystyle\frac{d}{dt}\delta a_{i}$ $\displaystyle=$
$\displaystyle\beta_{i}\delta a_{i}+\sqrt{2\kappa_{a}}a_{i}^{in}$ (3)
$\displaystyle-ig_{0}\sum_{jkl}\left(\alpha_{k}\alpha_{l}\delta
a_{j}^{\dagger}+\alpha_{j}^{*}\alpha_{l}\delta
a_{k}+\alpha_{j}^{*}\alpha_{k}\delta a_{l}\right),$

respectively. Here, the summation takes over all possible permutation
$\left(j,k,l\right)$ with $i=k+l-j$, $\beta_{i}=-i\delta_{i}-\kappa_{i}$,
$\kappa_{j}$ is the total amplitude decay rate of the $j$-th mode,
$\varepsilon_{p}$ is the pump strength, $a_{i}^{in}$ is the input noise on
mode $i$ and fulfills $\langle
a_{i}^{in}(t)(a_{i}^{in})^{\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime})$.
From Eq. (3), the last three terms represent the classical-field-stimulated
photon-pair generation in modes $\left(i,j\right)$, and also the coherent
photon conversion between modes $\left(i,k\right)$ and $\left(i,l\right)$.
Rather than merely produce down-conversion photons by drive lasers
(reimer2016generation, ), these terms imply very rich quantum dynamics in the
complex network, with the quantum correlations directly generated through
photon-pair generation and also indirectly generated through coherently re-
distributing the generated photons among the modes.

Figure 2: Quantum entanglement in different comb states. (a)-(d) Optical
spectra of below-threshold comb state (a), primary comb (b), two-soliton state
(c), and single-soliton state (d). (e)-(h) The corresponding entanglement
measure $E_{\mathcal{N}}$ between all mode-pairs. The $E_{\mathcal{N}}>0$ is
painted with colors whereas $E_{\mathcal{N}}=0$ is painted white.

The classical fields have been extensively studied in previous works
(hansson2014numerical, ; guo2018efficient, ). With the amplitudes of classical
fields $\alpha_{i}$ obtained, the CV quantum correlation
(braunstein2005quantum, ) between the comb lines can be evaluated by solving
the dynamics of fluctuations [Eq. (3)]. It is convenient to represent the
fluctuations by the “amplitude” and “phase” field quadratures $\delta
X_{i}=\left(\delta a^{\dagger}+\delta a\right)/\sqrt{2}$, $\delta
Y_{i}=i\left(\delta a^{\dagger}-\delta a\right)/\sqrt{2}$. The dynamics of the
quadratures $\overrightarrow{Q}=\left\\{\delta X_{-N},\delta Y_{-N},\delta
X_{-N+1},\delta Y_{-N+1},...,\delta X_{N},\delta Y_{N}\right\\}^{T}$ follow
$\frac{d}{dt}\overrightarrow{Q}=\mathbf{M}\cdot\overrightarrow{Q}+\overrightarrow{n}(t),$where
$\overrightarrow{n}^{T}=\left\\{\sqrt{2\kappa_{-N}}X_{-N}^{in},\sqrt{2\kappa_{-N}}Y_{-N}^{in},...\right\\}$
is the input noise, and $\mathbf{M}$ is a $(4N+2)\times(4N+2)$ matrix derived
from Eq. (3). Then, the correlation matrix $\mathbf{V}$ for all modes can be
solved following the deviation in Ref. (vitali2007optomechanical, ) and we
obtain

$\displaystyle\mathbf{M}\mathbf{V}+\mathbf{V}\mathbf{M}^{T}$ $\displaystyle=$
$\displaystyle-\mathbf{D},$ (4)

where the element of the correlation matrix $V_{ij}=\langle
Q_{i}Q_{j}+Q_{j}Q_{i}\rangle/2$, the noise term $D_{ij}=\langle
n_{i}n_{j}+n_{j}n_{i}\rangle/2$ can be derived from $\langle
X_{i}^{in}(t)X_{j}^{in}(t^{\prime})\rangle=\delta(i-j)\delta(t-t^{\prime})$
and $\langle X_{i}^{in}(t)Y_{j}^{in}(t^{\prime})\rangle=0$. The bipartite CV
entanglement between comb lines could be evaluated by the logarithmic
negativity

$\displaystyle E_{\mathcal{N}}^{mn}$ $\displaystyle=$
$\displaystyle\mathrm{max}[0,-\ln\sqrt{2}\eta],$ (5)

where $\eta=\sqrt{\Theta-\sqrt{\Theta^{2}-4\mathrm{det}V}}$,
$\Theta=\mathrm{det}A+\mathrm{det}B-2\mathrm{det}C$, with $A$, $B$ and $C$ are
the elements of $V^{mn}=\\{\\{A,C\\},\\{C^{T},B\\}\\}$, which is a sub-matrix
of $V$ representing the bipartite correlation matrix between $\left\\{\delta
X_{m},\delta Y_{m},\delta X_{n},\delta Y_{n}\right\\}$. $E_{\mathcal{N}}$ is a
measure of the CV entanglement (PhysRevA.70.022318, ; PhysRevA.65.032314, ;
vitali2007optomechanical, ), and the mode-pair is entangled only if
$E_{\mathcal{N}}^{mn}>0$. By calculating all combinations of the modes, one
obtains the entanglement matrix $\mathbf{E}_{\mathcal{N}}$ of the cavity
field.

_Multi-color entanglement.-_ Figure 2 depicts the typical results of Kerr
combs in a microring resonator at different states (PhysRevA.89.063814, ;
guo2018efficient, ), with the upper and bottom rows show the classical
intracavity fields and the entanglement $E_{\mathcal{N}}$ between mode-pairs.
Here, we consider a monochromatic field driving on the $0$-th mode to
initially excite the mode-pairs with $\xi=0$, and the system parameters are
chosen from a typical AlN microring in the experimental work
(guo2018efficient, ).

(i) Below the threshold [Fig. 2(a) and (e)]. For weak pump power, the
parametric gain on mode-pair of $\xi=0$ can not compensate the dissipation in
these modes, thus the system stays below the optical parametric oscillation
threshold and generates thermal photon-pairs by the spontaneous parametric
down-conversion (SPDC) (reimer2016generation, ). From the comb spectrum of
classical field [Fig. 2(a)], only the $0$-th mode is sufficiently excited, and
the intracavity fields in all the other modes are negligible. Figure 2(e)
shows the matrix $\mathbf{E}_{\mathcal{N}}$, which only has positive value at
its diagonal elements, indicating that only the photon-pair generation between
mode-pair $\left(+l,-l\right)$ for $\xi=0$ are initiated. Due to the
dispersion of the cavity resonance, modes with indices away from $0$
experience poorer phase-matching condition and thus the $E_{\mathcal{N}}$
decays with $l$.

(ii) Primary comb [Figs. 2(b) and (f)]. As the pump power increases above the
OPO threshold and the cavity field can be prepared into a stable Turing
pattern (Fig. 2(b)). In this state, several equally-spaced comb lines are
efficiently excited, with the space approximately be $m\times$FSR determined
by the dissipation rate $\kappa$ and the dispersion $D_{2}$
(PhysRevA.89.063814, ). Even though only $\xi=0$ is initially excited, the
generated comb lines can stimulate the cascaded FWM and produce entanglement
for mode-pairs belongs to other $\xi$. Consequently,
$\mathbf{E}_{\mathcal{N}}$ in Fig. 2(f) shows non-zero values at elements on
several diagonal lines, with each diagonal line corresponding to $\xi=2nm$
with $n\in\mathbb{Z}$. In the primary comb, the power of the comb lines
decreases with the mode index, the degree of entanglement decreases from the
main-diagonal line ($\xi=0$) to the high-order diagonal ($\xi=2nm\neq 0$).

(iii) Soliton state [Figs. 2(c), (d), (g) and (h)]. With appropriate laser
power and frequency detuning, the intracavity field can be driven to the
soliton states, which are ultrashort pulses circulating inside the cavity
(herr2014temporal, ; kippenberg2018dissipative, ). As shown by the spectra in
Fig.2(c)-(d), when tens of modes are efficiently excited in both the two and
single soliton states, the envelopes show a profile of
$\mathrm{sech}^{2}$-function. The strong comb line with index number $l$
generated by the the pump mode can further drive the mode-pairs with $\xi=2l$,
leading to the positive $E_{\mathcal{N}}$ on the corresponding diagonals of
the entanglement matrix in Fig. 2(g) and 2(h). Due to the intensity
distribution of the spectra, the entanglement matrix
$\mathbf{E}_{\mathcal{N}}$ of the two-soliton state has positive values only
on diagonal lines corresponding to even mode index ($m/2\in\mathbb{Z})$, while
all diagonals near the center have positive values for the single-soliton
state. Since multi-soliton state has higher energy than the single-soliton
state (herr2014temporal, ), the FWMs are excited more efficiently, resulting
in a higher $E_{\mathcal{N}}$. For the case in Fig. 2, the two-soliton state
has a maximum $E_{\mathcal{N}}$ of 0.259, in comparison with 0.117 for the
single-soliton state.

_All-to-all entanglement.-_ For single-soliton state, modes are excited
efficiently so that the multi-color entanglement is distinct from other cases.
In particular, we can find a group of modes [mode index $|i|<12$ in Fig. 2(h)]
where all-pair of modes are entangled. The all-to-all entanglement indicates a
fully connected complex network. This phenomena manifests the distinct
physical mechanisms of entanglement generation in the soliton state: two-mode
and single-mode ($l$ [$\propto\left(a_{l}^{2}+a_{l}^{\dagger 2}\right)$] due
to the comb lines in $\xi=2l$) squeezing generation, and the linear conversion
between two modes $\left(i,j\right)$ stimulated by comb lines
$\left(\xi-i,\xi-j\right)$
[$\propto\left(\alpha_{\xi-i}^{*}a_{i}^{\dagger}\alpha_{\xi-j}a_{j}+h.c.\right)$]
for all $\xi$, compared to the SPDC in conventional studies [Fig. 2(a)]. Since
the modes have relatively uniform intensities around the $0$-th mode, i.e.
$\left|\alpha_{j}\right|$ decays with $j$ as indicated by the
$\mathrm{sech}^{2}$-function, the soliton state has higher degree of
entanglement at the center compared with modes away from the pump in Figs.
2(g) and (h).

According to Eq. 3, the entanglement is stimulated by classical fields and
thus the all-to-all entanglement region could be further spread by flatten the
comb spectrum. Since the spectrum bandwidth of DKS could be efficiently
controlled by the dispersion $D_{2}$ (herr2014temporal, ), the multi-color
entanglement for various $D_{2}$ is investigated. Figures 3(a)-(b) show the
the entanglement matrix $\mathbf{E}_{\mathcal{N}}$ for $D_{2}/\kappa=1.645$
and $D_{2}/\kappa=0.329$, respectively. The $D_{2}/\kappa=0.329$ case supports
an all-connected group involving more modes than that of the
$D_{2}/\kappa=1.645$ case but has less degree of entanglement, as marked by
the black square. Defining the edge length of the square as the size of the
fully connected group, the size of the quantum state decreases with the
dispersion $D_{2}$, as summarized in Fig. 3(c). All-to-all entanglement among
more than $50$ colors is predicted for $D_{2}/\kappa=0.329$. Thus, by
designing a microring with smaller dispersion and larger cavity size,
potentially hundreds of modes can be all-to-all entangled. By distributing the
photons to different users via wavelength multiplexing, this all-to-all
entangled state is promising to build a multi-party quantum teleportation
network (yonezawa2004demonstration, ).

Figure 3: All-to-all quantum entanglement via dispersion engineering. (a)-(b)
The matrices $\mathbf{E}_{\mathcal{N}}$ of single-soliton state with
$D_{2}/\kappa=1.645$ (a) and $D_{2}/\kappa=0.329$ (b), respectively. (c) The
relationship between the size of the group and the cavity dispersion. The
frequency and strength of the pump field are fixed.

_Entanglement and comb percolation.-_ For a complex network, it is always
curious about how the system performs if some nodes are removed. The network
with defects corresponds to the optical cavities in practice, where some modes
show extraordinary low quality factor or large frequency shifting, due to the
perturbation of environments or avoid-mode crossing induced by other mode
families. The absorption and shifting of the resonance can significantly
suppress the intensities in these modes and modify the soliton spectrum.
Therefore, we further investigate the multi-color entanglement of the soliton
state in a defective mode family. The defects are simulated by magnifying the
dissipation rate of certain modes to $1000\>\kappa$ to eliminate the mode
density of state and thus suppresses the corresponding comb lines. As shown by
the numerical results in Figs. 4(a)-(b), the single-soliton state can still
exist even if $5$ modes are eliminated, and the spectra remain a envelope of
$\mathrm{sech}^{2}$-function except the lines of the lossy modes being
suppressed. Due to the nature of the complex network, each mode can
participate many different frequency mixing processes with different $\xi$,
thus the network is still fully connected even in the absence of a few nodes,
and showing a percolation of the comb generation in a microcavity. From the
entanglement matrix of the soliton state, as shown in Figs. 4(c) and (d), only
the elements that are associated with the very lossy modes are affected,
whereas the entanglement between the rest of modes are almost unaffected. The
percolation of the entanglement demonstrates the robustness of the multi-color
entanglement of DKS against loss or other distortion of the mode density of
states in practical experiments. Therefore, such a robust, self-organizing,
entangled quantum source holds great potential for future applications in non-
ideal environments.

Figure 4: Multi-color entanglement in a defective FWM network. (a)-(b) The
soliton spectrum with the dissipation rate of $1$ (a) and $5$ (b) consecutive
modes are magnified by 1000 times. (c)-(d) The entanglement matrices of the
corresponding soliton states.

_Conclusion.-___ The multi-color CV entanglement in the DKS comb is
investigated. In the microresonator, a complex network of optical modes are
formed via the Kerr nonlinearity, with the optical fluctuations of different
modes are connected through-pair-generation and conversion interactions. It is
demonstrated that the modes in the complex network show all-to-all
entanglement, and also exhibits robustness against the defects of the network,
which is induced by extra optical losses or other experimental imperfections.
Combing with other nonlinear optical effects, the multi-color entanglement of
DKS could be further extended to other frequency bands, such as mid-infrared
and ultraviolet wavelengths, by $\chi^{(2)}$-nonlinearity (guo2018efficient, ;
Bruch2021, ) or Raman scattering (jung2014green, ; karpov2016raman, ;
xue2017second, ). By multiplexing the color of DKS to other degrees of freedom
of photons, such as the polarization, transverse waveguide mode, and time-bin,
high-dimension quantum cluster state could be generated
(larsen2019deterministic, ; asavanant2019generation, ). Our results reveal the
interesting dynamics of quantum entanglement generation associated with DKS,
and also indicate the great potential of DKS in quantum information science,
such as high-dimensional CV quantum sources, multiparty quantum teleportation
network (pirandola2015advances, ), and distributed quantum metrology
(guo2020distributed, ; zhang2020distributed, ).

###### Acknowledgements.

This work was funded by the National Key Research and Development Program
(Grant No. 2016YFA0301300), the National Natural Science Foundation of China
(Grant No.11874342, 11934012, 11904316, 11704370, and 11922411), Anhui
Initiative in Quantum Information Technologies (Grant No. AHY130200), and
Anhui Provincial Natural Science Foundation (Grant No. 2008085QA34) and the
China Postdoctoral Science Foundation (Grant No. 2019M662153). ML and CLZ was
also supported by the Fundamental Research Funds for the Central Universities,
and the State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, China. This work was partially
carried out at the USTC Center for Micro and Nanoscale Research and
Fabrication.

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