P_{q+\alpha}\Big{(}\Phi^{(1,\ldots,l)}(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{m_{1}+\ldots+m_{i}+1},z_{m_{1}+\ldots+m_{i}+1};$
$\displaystyle\qquad\ldots;v_{m_{1}+\ldots+m_{i}+k_{1}+\ldots+k_{l}},z_{m_{1}+\ldots+m_{i}+k_{1}+\ldots+k_{l}}\Big{)}\rangle.$

According to Proposition 7, as an element of
$\mathcal{W}^{(k_{1},\ldots,k_{l})}_{z_{1},\ldots,z_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}}}$

$\displaystyle\langle
w^{\prime},E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}\Big{(}v_{1},z_{1};\ldots;v_{m_{1}+\ldots+m_{l}},z_{m_{1}+\ldots+m_{l}};$
(6.4) $\displaystyle\qquad
P_{q+\alpha}\Big{(}\Phi^{(1,\ldots,l)}(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{m_{1}+\ldots+m_{i}+1},z_{m_{1}+\ldots+m_{i}+1};$
$\displaystyle\qquad\qquad\qquad\ldots;v_{m_{1}+\ldots+m_{i}+k_{1}+\ldots+k_{l}},z_{m_{1}+\ldots+m_{i}+k_{1}+\ldots+k_{l}})\Big{)}\rangle,$

is invariant under the action of $\sigma\in
S_{m_{1}+\ldots+m_{i}+k_{1}+\ldots+k_{l}}$. Thus, it possible to use this
invariance to show that (6.4) reduces to

$\displaystyle\langle
w^{\prime},E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}\Big{(}v_{k_{1}+1},z_{k_{1}+1};\ldots;v_{k_{1}+1+m_{1}},z_{k_{1}+1+m_{1}};$
$\displaystyle\qquad\ldots;v_{k_{l}+1},z_{k_{l}+1};\ldots;v_{k_{l}+1+m_{l}},z_{k_{l}+1+m_{l}};$
$\displaystyle\qquad
P_{q+\alpha}\Big{(}\Phi^{(1,\ldots,l)}(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{1},z_{1};\ldots;v_{k_{1}},z_{k_{1}};\ldots;$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
v_{k_{1}+\ldots+k_{l}},z_{k_{1}+\ldots+k_{l}})\Big{)}\Big{)}\rangle$
$\displaystyle=\langle
w^{\prime},E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}\Big{(}v_{k_{1}+1,i},x_{k_{1}+1,i};\ldots;v_{k_{l}+1+m_{l}},x_{k_{l}+1+m_{l}};$
$\displaystyle\qquad
P_{q+\alpha}\Big{(}\Phi^{(1,\ldots,l)}(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{1,i},x_{1,i};\ldots;v_{k_{i},i},x_{k_{i},i})\Big{)}\rangle.$

Similarly, for $1\leq i\leq l$

$\displaystyle\langle
w^{\prime},E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}\Big{(}v_{1},z_{1};\ldots;v_{m_{1}+\ldots+m_{l}},z_{m_{1}+\ldots+m_{l}};$
$\displaystyle\qquad
P_{q}\Big{(}Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\Phi^{(i)}(v_{m_{1}+\ldots+m_{l}+1},z_{m_{1}+\ldots+m_{l}+1};\right.$
$\displaystyle\qquad\left.\left.\ldots;v_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{i}},z_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{i}});u,\zeta_{1,i})\right),\zeta_{2,i}\right)f_{i}.\overline{u}\Big{)}\Big{)}\rangle,$

correspond to the elements of
$\mathcal{W}_{z_{1},\ldots,z_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{i}}}$. Let us
use Proposition 7 again and we arrive at

$\displaystyle\langle
w^{\prime},E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}\Big{(}v_{k_{i}+1,i},x_{k_{i}+1,i};\ldots;v_{k_{i}+m_{i}},x_{k_{i}+m_{i}};$
$\displaystyle\qquad
P_{q}\Big{(}Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\Phi^{(i)}(v_{1,i},x_{1,i};\ldots;v_{k_{i},i},x_{k_{i},i});u,\zeta_{1,i})\right),f_{i}.\overline{u}\Big{)}\Big{)}\rangle.$

Thus, the proof of Lemma is completed. ∎

Next, we prove

######  Proposition 11.

The products $\Theta\left(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l}\right.$;
$v_{1},z_{1};\ldots;v_{\theta_{l}-r},z_{\theta_{l}-r}$; $\rho_{1}$, $\ldots$,
$\rho_{l}$; $\zeta_{1,i}$, $\left.\zeta_{2,i}\right)$ (19) are regularized
transversal to $\mu_{l}-t$ vertex operators.

###### Proof.

Recall that
$\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}},z_{n_{i}+k_{i}})$,
$1\leq i\leq l$, are regularized transversal to $m_{i}-t_{i}$ vertex
operators. For the first condition of the regularized transversality: let
$l_{1,i},\ldots,l_{k_{i}-r_{i},i}\in\mathbb{Z}_{+}$ such that
$l_{1,i}+\ldots+l_{k_{i},i}=n_{i}+k_{i}-r_{i}+m_{i}-t_{i}$. For an arbitrary
$w^{\prime}_{i}\in W^{(i)}{}^{\prime}$, denote

$\displaystyle(v_{n_{i}+1},\ldots,v_{n_{i}+k_{i}},v_{n_{i}+k_{i}+1},\ldots,v_{n_{i}+k_{i}+m_{i}-t_{i}})$
$\displaystyle\qquad=(v_{n_{i}+1},\ldots,v_{n_{i}+k_{i}},v^{\prime}_{n_{i}+k_{i}+1},\ldots,v^{\prime}_{n_{i}+k_{i}+m_{i}-t_{i}}),$
$\displaystyle(z_{n_{i}+1},\ldots,z_{n_{i}+k_{i}},z_{n_{i}+k_{i}+1},\ldots,z_{n_{i}+k_{i}+m_{i}-t_{i}})$
$\displaystyle\qquad=(z_{n_{i}+1},\ldots,z_{n_{i}+k_{i}},z^{\prime}_{n_{i}+k_{i}+1},\ldots,z^{\prime}_{n_{i}+k_{i}+m_{i}-t_{i}}).$
(6.5)

Define
$\Xi_{j,i}=E^{(l_{j,i})}_{V}(v_{\varkappa_{1,i}},z_{\varkappa_{1,i}}-\varsigma_{j,i};\ldots;v_{\varkappa_{j,i}},z_{\varkappa_{j,i}}-\varsigma_{j,i};\mathbf{1}_{V})$,
where

$\displaystyle\varkappa_{1,i}=l_{1,i}+\ldots+l_{j-1,i}+1,\quad\ldots,\quad\varkappa_{j,i}=l_{1,i}+\ldots+l_{j-1,i}+l_{j},$
(6.6)

for $1\leq j\leq k_{i}-r_{i}$. Then the series

$\displaystyle\mathcal{R}^{1,k_{i}-r_{i}}_{m_{i}-t_{i}}\left(\Phi^{(i)}\right)=R\sum_{r_{1,i},\ldots,r_{k_{i}-r_{i},i}\in\mathbb{Z}}\langle
w^{\prime}_{i},\Phi^{(i)}\left(g_{i};P_{r_{1,i}}\Xi_{1,i};\varsigma_{1,i};\right.\ldots;$
(6.7)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.P_{r_{k_{i}-r_{i},i}}\Xi_{k_{i}-r_{i},i},\varsigma_{k_{i}-r_{i},i}\rangle\right)\rangle,$

is absolutely convergent when
$|z_{l_{1,i}+\ldots+l_{j-1,i}+p_{i}}-\varsigma_{j,i}|+|z_{l_{1,i}+\ldots+l_{j^{\prime}-1,i}+q}-\varsigma_{j^{\prime},i}|<|\varsigma_{j,i}-\varsigma_{j^{\prime},i}|$,
for $j$, $1\leq j^{\prime}\leq k_{i}-r_{i}$, $j\neq j^{\prime}$, and for
$1\leq p_{i}\leq l_{j,i}$ and $1\leq q_{i}\leq l_{j^{\prime},i}$. There exist
positive integers $N^{k_{i}-r_{i}}_{m_{i}-t_{i}}(v_{j,i},v_{j^{\prime},i})$,
depending only on $v_{j,i}$ and $v_{j^{\prime},i}$ for $1\leq j$,
$j^{\prime}\leq m_{i}-t_{i}$, $j\neq j^{\prime}$, such that the sum is
analytically extended to a rational function in
$(z_{1},\ldots,z_{n_{i}+k_{i}-r_{i}+m_{i}-t_{i}})$, independent of
$(\varsigma_{1,i},\ldots,\varsigma_{k_{i}-r_{i},i})$, with the only possible
poles at $x_{j,i}=x_{j^{\prime},i}$, of order less than or equal to
$N^{k_{i}-r_{i}}_{m_{i}-t_{i}}(v_{j,i},v_{j^{\prime},i})$, for $j$, $1\leq
j^{\prime}\leq k_{i}-r_{i}$, $j\neq j^{\prime}$.

Now let us consider the first condition of Definition 17 of the regularized
transversal for the product (19) of
$\Phi^{(i)}(g_{i};v_{1},z_{1};\ldots;v_{\theta_{i}},z_{\theta_{i}})$ with a
number of vertex operators. We obtain for
$\Theta\left(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{1},z_{1};\ldots;v_{\theta_{l}},z_{\theta_{l}};\rho_{1},\ldots,\rho_{l}\right)$
the following. Introduce
$l^{\prime}_{1},\ldots,l^{\prime}_{\theta_{l}-r}\in\mathbb{Z}_{+}$, such that
$l^{\prime}_{1}+\ldots+l^{\prime}_{\theta_{l}-r}=\theta_{l}-r+\mu_{l}-t$.
Define
$\Xi^{\prime}_{j^{\prime\prime}}=E^{(l^{\prime\prime}_{j^{\prime\prime}})}_{V}(v_{\varkappa^{\prime}_{1}},z_{\varkappa^{\prime}_{1}}-\varsigma^{\prime}_{j^{\prime\prime}};\ldots;v_{\varkappa^{\prime}_{j^{\prime\prime}}},z_{\varkappa^{\prime}_{j^{\prime\prime}}}-\varsigma^{\prime}_{j^{\prime\prime}};\mathbf{1}_{V})$,
where
$\varkappa^{\prime}_{1}=l^{\prime}_{1}+\ldots+l^{\prime}_{j^{\prime\prime}-1}+1$,
$\ldots$,
$\varkappa^{\prime}_{j^{\prime\prime}}=l^{\prime}_{1}+\ldots+l^{\prime}_{j^{\prime\prime}-1}+l^{\prime}_{j^{\prime\prime}}$,
for $1\leq j^{\prime\prime}\leq\theta_{l}-r$, and we take
$(\zeta^{\prime}_{1},\ldots,\zeta^{\prime}_{\theta_{k_{l}}-r})=(\zeta_{1},\ldots,\zeta_{k_{1}-r_{1}};\ldots;\zeta_{n_{l-1}+1},\ldots,\zeta_{n_{l}+k_{l}-r_{l}})$.

Then we consider

$\displaystyle\mathcal{R}^{1,\theta_{l}-r}_{\mu_{l}-r}\left(\Phi^{(1,\ldots,l)}\right)=R\sum_{r^{\prime}_{1},\ldots,r^{\prime}_{\theta_{l}-r}\in\mathbb{Z}}\widehat{\Theta}\left(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};P_{r^{\prime}_{1}}\Xi^{\prime}_{1},\varsigma^{\prime}_{1};\ldots;\right.$
(6.8)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.P_{r^{\prime}_{\theta_{l}-r}}\Xi^{\prime}_{\theta_{l}-r},\varsigma^{\prime}_{\theta_{l}-r}\right),$

and prove it’s absolutely convergence with some conditions.

The condition
$|z_{l^{\prime}_{1}+\ldots+l^{\prime}_{j-1}+p^{\prime}}$-$\varsigma^{\prime}_{j}|$
$+$ $|z_{l^{\prime}_{1}+\ldots+l^{\prime}_{j^{\prime}-1}+q^{\prime}}$\-
$\varsigma^{\prime}_{j^{\prime}}|$ $<$
$|\varsigma^{\prime}_{j}-\varsigma^{\prime}_{j^{\prime}}|$, of the absolute
convergence for (6.8) for $1\leq i^{\prime\prime}$,
$j^{\prime\prime}\leq\theta_{l}-r$, $j\neq j^{\prime}$, for $1\leq
p^{\prime}\leq l^{\prime}_{j}$, and $1\leq q^{\prime}\leq
l^{\prime}_{j^{\prime}}$, follows from the conditions (17) and (2.16). The
action of $e^{\zeta L_{W^{(i)}}(-1)}\;Y_{W^{(i)}}(.,.)$, $a=1$, $2$, in

$\displaystyle\langle
w_{i}^{\prime},e^{\zeta_{1}L_{W^{(i)}}(-1)}\;Y_{W^{(i)}}(u,-\zeta)\sum_{{r_{1},\ldots,}\atop{r_{k_{i}-r_{i}}\in\mathbb{Z}}}\Phi^{(i)}\left(g_{i};P_{r_{1,i}}\Xi_{1},\varsigma_{1};\right.P_{r_{k_{i}-r_{i}k,i}}\Xi_{k_{i}-r_{i}},\varsigma_{k_{i}-r_{i}}\rangle)\rangle,$

does not affect the absolute convergence of (6.7). Therefore,

$\displaystyle\left|\mathcal{R}^{1,\theta_{l}-r}_{\mu_{l}-t}\left(\Phi^{(1,\ldots,l)}\right)\right|$
$\displaystyle=R\left|\sum_{r^{\prime\prime}_{1},\ldots,r^{\prime\prime}_{\theta_{l}-r}\in\mathbb{Z}}\widehat{\Theta}\left(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};P_{r^{\prime}_{1}}\Xi^{\prime}_{1},\varsigma^{\prime}_{1};\ldots;P_{r^{\prime}_{\theta_{l}-r}}\Xi^{\prime}_{\theta_{l}-r},\varsigma^{\prime}_{\theta_{l}-r}\right)\right|$
$\displaystyle=\left|\sum_{u\in V_{(k)}\atop
k\in\mathbb{Z}}\widehat{\mathcal{R}}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w^{\prime}_{i},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\Big{(}\sum_{{r^{\prime}_{1},\ldots,}\atop{r^{\prime}_{\theta_{i}-r_{i}}\in\mathbb{Z}}}\Phi^{(i)}\left(g_{i};P_{r^{\prime}_{1}}\Xi^{\prime}_{1},\varsigma^{\prime}_{1};\ldots;\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\left.P_{r^{\prime}_{\theta_{i}-r_{i}}}\Xi^{\prime}_{\theta_{i}-r_{i}},\varsigma^{\prime}_{\theta_{i}-r_{i}};u,\varsigma_{1,i})\right),\varsigma_{2,i}\Big{)}\;f_{i}.\overline{u}\rangle\right|$
$\displaystyle=\left|\sum_{u\in V_{(k)}\atop
k\in\mathbb{Z}}\widehat{\mathcal{R}}\prod_{i=1}^{l}\rho_{i}^{k}\langle
w^{\prime}_{i},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\Big{(}\sum_{{r_{1,i},\ldots,}\atop{r_{k_{i}-r_{i},i}\in\mathbb{Z}}}\Phi^{(i)}\left(g_{i};P_{r_{1,i}}\Xi_{1,i},\varsigma_{1,i};\ldots;\right.\right.$
$\displaystyle\left.\left.\left.\qquad\qquad\qquad
P_{r_{k_{i}-r_{i},i}}\Xi_{k_{i}-r_{i},i},\varsigma_{k_{i}-r_{i},i};u,\varsigma_{1,i}\right),\varsigma_{2,i}\right)\;f_{i}.\overline{u}\rangle\right|$
$\displaystyle=\left|\sum_{u\in V_{(k)}\atop
k\in\mathbb{Z}}\widehat{\mathcal{R}}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w^{\prime}_{i},e^{\varsigma_{2,i}L_{W^{(i)}}(-1)}Y_{W^{(i)}}(f_{i}.\overline{u},-\varsigma_{2,i})\right.\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\left.\sum_{{r_{1,i},\ldots,}\atop{r_{k_{i}-r_{i},i}\in\mathbb{Z}}}\Phi^{(i)}\left(g_{i};P_{r_{1,i}}\Xi_{1,i},\varsigma_{1,i};\ldots;P_{r_{k,i}}\Xi_{k_{i}-r_{i},i},\varsigma_{k_{i}-r_{i},i};u,\varsigma_{1,i}\right)\;\rangle\right|\leq\left|\mathcal{R}^{1,k_{i}-r_{i}}_{m_{i}-t_{i}}\left(\Phi^{(i)}\right)\right|.$

We conclude that (6.8) is absolutely convergent. Recall that
$N^{k_{i}-r_{i}}_{m_{i}-t_{i}}(v_{i,i},v_{j,i})$ are the maximal orders of
possible poles of (6.8) at $x_{j,i}=x_{j^{\prime},i}$. From the last
expression follows that there exist positive integers
$N^{\theta_{l}-r}_{\mu_{l}-t}(v_{i^{\prime\prime},i},v_{j^{\prime\prime},i})$
for $1\leq j$, $j^{\prime}\leq k_{i}-r_{i}$, $j\neq j^{\prime}$, depending
only on $v_{i^{\prime\prime},i}$ and $v_{j^{\prime\prime},i}$ for $1\leq
i^{\prime\prime}$, $j^{\prime\prime}\leq\theta_{k_{l}}-r$,
$i^{\prime\prime}\neq j^{\prime\prime}$, such that the series (6.8) can be
analytically extended to a rational function in
$(z_{1},\ldots,z_{\theta_{l}-r})$, independent of
$(\varsigma^{\prime}_{1,i},\ldots,\varsigma^{\prime}_{\theta_{l}-r,i})$, with
extra possible poles at and $z_{j,i}=z^{\prime}_{j^{\prime},i^{\prime}}$, of
order less than or equal to
$N^{\theta_{l}-r}_{\mu_{l}-t}(v_{i^{\prime\prime},i},v_{j^{\prime\prime},i})$,
for $1\leq i^{\prime\prime}$, $j^{\prime\prime}\leq n$, $i^{\prime\prime}\neq
j^{\prime\prime}$.

Now, let us pass to the second condition of the regularized transversal for
$\Phi^{(i)}$ $(g_{i}$;
$v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{i}},x_{n_{i}+k_{i}-r_{i}})\in
C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$, and
$v_{1,i},\ldots,v_{k_{i},i}\in V$,
$(x_{1,i},\ldots,x_{k_{i}+m,i})\in\mathbb{C}$. For arbitrary
$w^{\prime}_{i}\in W^{(i)}{}^{\prime}$, the series

$\displaystyle\mathcal{R}^{2,k_{i}-r_{i}}_{m_{i}-t_{i}}\left(\Phi^{(i)}\right)=R\sum_{q_{i}\in\mathbb{C}}\langle
w^{\prime}_{i},E^{(m_{i}-t_{i})}_{W^{(i)}}\Big{(}v^{\prime}{n^{\prime}_{i}+1},z^{\prime}_{n^{\prime}_{i}+1};\ldots;v^{\prime}_{n^{\prime}_{i}+m_{i}-t_{i}},z^{\prime}_{n^{\prime}_{i}+m_{i}-t_{i}};$
$\displaystyle
P_{q_{i}}\left(\Phi^{(i)}(g_{i};v_{n^{\prime}_{i}+m_{i}-t_{i}+1},z_{n_{i}+m_{i}-t_{i}+1};\ldots;v_{n^{\prime}_{i}+m_{i}-t_{i}+k_{i}},z_{n_{i}+m_{i}-t_{i}+k_{i}})\right)\Big{)}\rangle,$
(6.9)

is absolutely convergent when $z^{\prime}_{j}\neq z^{\prime}_{j^{\prime}}$,
$j\neq j^{\prime}$, $|z^{\prime}_{j}|>|z^{\prime}_{j^{\prime}}|>0$, for $1\leq
j\leq m_{i}-t_{i}$, $m_{i}+1\leq j^{\prime}\leq k_{i}+m_{i}$, and the sum can
be analytically extended to a rational function in
$(x_{1,i},\ldots,x_{k_{i}+m_{i},i})$ with the only possible poles at
$x_{j,i}=x_{j^{\prime},i}$, of orders less than or equal to
$N^{k_{i}}_{m_{i}}(v_{i,i},v_{j,i})$, for $1\leq j$, $j^{\prime}\leq k_{i}$,
$j\neq j^{\prime}$.

In the Appendix 10 the definition (10) of the element
$E^{(\mu_{l})}_{W^{(1,\ldots,l)}}$ for
$\Phi^{(1,\ldots,l)}\in\mathcal{W}^{(1,\ldots,l)}_{z^{\prime}_{1},\ldots,z^{\prime}_{\theta_{l}-r}}$
was given. With the conditions $z_{i^{\prime\prime},i}\neq
z_{j^{\prime\prime},i},\quad i^{\prime\prime}\neq j^{\prime\prime}$, $1\leq
i\leq l$, $|z_{i^{\prime\prime},i}|>|z_{k^{\prime\prime\prime},i}|>0$, for
$i^{\prime\prime}=1,\dots,m_{1}+\ldots+m_{l}$, and
$k^{\prime\prime\prime}=m_{1}+\ldots+m_{l}+1,\dots,m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}$,
let us define

$\displaystyle\mathcal{R}^{2,k_{1}+\ldots+k_{l}-r}_{m_{1}+\ldots+m_{l}-t}\left(\Phi^{(1,\ldots,l)}\right)=R\sum_{{q_{1},\ldots,q_{l}}\in\mathbb{C}}E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}\Big{(}v_{1},z_{1};\ldots;$
$\displaystyle\qquad
v_{m_{1}+\ldots+m_{l}},z_{m_{1}+\ldots+m_{l}};P_{q_{1},\ldots,q_{l}}\Big{(}\Phi^{(1,\ldots,l)}(g_{1},\ldots,g_{l};$
$\displaystyle\qquad
v_{m_{1}+\ldots+m_{l}+1},z_{m_{1}+\ldots+m_{l}+1};\ldots;$
$\displaystyle\qquad\qquad\qquad
v_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}},z_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}};\rho_{1},\ldots,\rho_{l})\Big{)},\quad$
(6.10)

where $P_{q_{1},\ldots,q_{l}}$ stands for projections
$P_{q_{i}}:\overline{W}^{(i)}\to\mathcal{W}^{(i)}_{q_{i}}$ on the
corresponding subspaces in the tensor product $\mathcal{W}^{(1,\ldots,l)}$. In
the Appendix 10, (10) defines $E^{(m_{1}+\ldots+m_{l})}_{W^{(1,\ldots,l)}}$.
In order to get, in particular, the regularized transversal of an element
$\Phi$ with extra vertex operators, $\mathcal{R}^{2,n}_{m}(\Phi)$ (2.14) was
introduced in Subsection 2.4. We substitute the element $\Phi$ by an element
$\Phi^{(1,\ldots,l)}$ in $\Theta$. The absolute convergence of
$\mathcal{R}^{2,k_{1}+\ldots+k_{l}-r}_{m_{1}+\ldots+m_{l}-t}\left(\Phi^{(1,\ldots,l)}\right)$
defined by (6.1) with (10) provides the regularized transversal condition for
$\Phi^{(1,\ldots,l)}$ with respect to a number of extra vertex operators in
$\mathcal{W}^{(1,\ldots,l)}$. Using Lemma 8 we have

$\displaystyle\left|\mathcal{R}^{2,k_{1}+\ldots+k_{l}-r}_{m_{1}+\ldots+m_{l}-t}\left(\Phi^{(1,\ldots,l)}\right)\right|=\left|\sum_{{q_{1},\ldots,q_{l}}\in\mathbb{C}}\mathcal{R}\prod_{i=1}^{l}\langle
w^{\prime}_{i},E^{(m_{i})}_{W^{(i)}}\Big{(}v_{1},z_{1};\ldots;v_{m_{i}},z_{m_{i}};\right.$
$\displaystyle\left.P_{q_{1},\ldots,q_{l}}\Big{(}\Phi^{(i)}(g_{i};v_{m_{i}+1},z_{m_{i}+1};\ldots;v_{m_{i}+k_{i}},z_{m_{i}+k_{i}})\Big{)}\Big{)}\rangle\right|$
$\displaystyle=\left|\sum_{q_{1},\ldots,q_{l}\in\mathbb{C}}\widehat{\mathcal{R}}\prod_{i=1}^{l}\langle
w^{\prime}_{i},E^{(m_{i})}_{W^{(i)}}\Big{(}v_{1,i},x_{1,i};\ldots;v_{m_{i},i},x_{m_{i},i};\right.\qquad\qquad\qquad$
$\displaystyle\left.P_{q_{i}}\Big{(}Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\Phi^{(i)}(g_{i};v_{m_{i}+1,i},x_{m_{i}+1,i};\ldots;v_{m_{i}+k_{i},i},x_{m_{i}+k_{i},i};u,\zeta_{1,i}),\zeta_{2,i}\right)f_{i}.\overline{u}\;\Big{)}\Big{)}\rangle\right|$
$\displaystyle=\left|\sum_{q_{1},\ldots,q_{l}\in\mathbb{C}}\widehat{\mathcal{R}}\prod_{i=1}^{l}\langle
w^{\prime}_{i},E^{(m_{i})}_{W^{(i)}}\Big{(}v_{1,i},x_{1,i};\ldots;v_{m_{i},i},x_{m_{i},i};\right.$
$\displaystyle\qquad
P_{q_{i}}\Big{(}e^{\zeta_{2,i}L_{W^{(i)}}(-1)}Y_{W^{(i)}}\left(f_{i}.\overline{u},-\zeta_{2,i}\right)$
$\displaystyle\qquad\left.\Phi^{(i)}(g_{i};v_{m_{i}+1,i},x_{m_{i}+1,i};\ldots;v_{m_{i}+k_{i},i},x_{m_{i}+k_{i},i};u,\zeta_{1,i})\;\Big{)}\Big{)}\rangle\right|\leq\left|\mathcal{R}^{2,k_{i}}_{m_{i}}\left(\Phi^{(i)}\right)\right|,$

where the invariance of (19) under $\sigma\in
S_{m_{1}+\ldots+m_{l}-t+k_{1}+\ldots+k_{l}-r}$ was used. According to
Definition 17, $\mathcal{R}^{2,k_{i}}_{m_{i}}\left(\Phi^{(i)}\right)$ are
absolute convergent.

Thus, we infer that
$\mathcal{R}^{2,k_{1}+\ldots+k_{l}-r}_{m_{1}+\ldots+m_{l}-t}\left(\Phi^{(1,\ldots,l}\right)$
is absolutely convergent, and the sum (6.8) is analytically extendable to a
rational function in $(z_{1}$, $\dots$,
$z_{k_{1}+\ldots+k_{l}-r+m_{1}+\ldots+m_{l}-t})$ with the only possible poles
at $x_{j,i}=x_{j^{\prime},i}$, and at $x_{j,i}=x_{j^{\prime},i^{\prime}}$,
i.e., the only possible poles at $z_{i^{\prime\prime}}=z_{j^{\prime\prime}}$,
of orders less than or equal to
$N^{k_{1}+\ldots+k_{l}}_{m_{1}+\ldots+m_{l}}(v_{i^{\prime\prime},i},v_{j^{\prime\prime},i})$,
for $i^{\prime\prime}$, $j^{\prime\prime}=1,\ldots,k^{\prime\prime\prime}$,
$i^{\prime\prime}\neq j^{\prime\prime}$. This finishes the proof of
Proposition 11. ∎

Since we have proved that the sequence of products $\widehat{\Theta}$
$\left(f_{1}\right.$, $\ldots$, $f_{l}$; $g_{1}$, $\ldots$., $g_{l}$;
$v_{1,1}$, $x_{1,1}$; $\ldots$; $v_{k_{l},l}$, $x_{k_{l},l}$; $\rho_{1}$,
$\ldots$, $\rho_{l}$; $\zeta_{1,i}$, $\zeta_{2,i}$ $5\left.\right)$ is
regularized transversal to $\mu_{l}-t$ vertex operators (2.13) with the formal
parameters identified with the local coordinates
$c_{j,i}(p^{\prime\prime}_{j,i})$ around the points
$(p^{\prime}_{1},\ldots,p^{\prime}_{\mu_{l}-t})$ on each of the transversal
sections $U_{j,i}$, $1\leq j\leq\mu_{l}-t$, we conclude that according to
Definition 22, the sequence of products $\widehat{\Theta}$
$\left(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l}\right.$;
$v_{1,1},x_{1,1};\ldots;v_{k_{l},l},x_{k_{l},l}$; $\rho_{1},\ldots,\rho_{l}$;
$\left.\zeta_{1,i},\zeta_{2,i}\right)$ belongs to the space

$\displaystyle
C^{\theta_{l}-r}_{\mu_{l}-t}\left(V,\mathcal{W}^{(1,\ldots,l)}\right)=\bigcap_{U_{1,i}\stackrel{{\scriptstyle
h_{1,i}}}{{\hookrightarrow}}\ldots\stackrel{{\scriptstyle
h_{m_{1}+\ldots+m_{l}-1,i}}}{{\hookrightarrow}}U_{m_{1}+\ldots+m_{l}}\atop
1\leq j\leq
m_{1}+\ldots+m_{l}-t}C^{\theta_{l}-r}_{(\mu-t)}\left(V,\mathcal{W}^{(1,\ldots,l)}\right)(U_{j,i}),$

where the intersection ranges over all possible $\mu_{l}-t$-tuples of holonomy
embeddings $h_{j,i}$, $1\leq j\leq\mu_{l}-t-1$, between transversal sections
$U_{1,i},\ldots,U_{\mu_{l}-t-1,i}$ of the basis $\mathcal{U}$ for
$\mathcal{F}$. This completes the proof of Proposition 10. ∎

Since the sequence of products (18) of $\mathcal{W}^{(i)}$-spaces, $1\leq
i\leq l$, gives the tensor products of that spaces, the sequence of products
(6.3) of the corresponding
$C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$-spaces belong to the same
type of spaces. which makes further considerations much simpler.

##  7\. Properties of the multiple product sequences

Since the sequence of $(\rho_{1},\ldots,\rho_{l})$-products of elements
$\Phi^{(i)}$ $(g_{i}$; $v_{1,i},x_{1,i}$; $\ldots$; $v_{k_{i},i},x_{k_{i},i})$
$\in$ $C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$ results in an
element of
$C^{\theta_{l}-r}_{\mu_{l}-t}\left(V,\mathcal{W}^{(1,\ldots,l)},\mathcal{F}\right)$,
then the corollary below follows directly from Proposition (10) and Definition
19:

######  Corollary 1.

For the spaces $\mathcal{W}^{(i)}_{x_{1,i},\ldots,x_{k_{i},i}}$, $1\leq i\leq
l$, with the product (19) which belongs to
$\mathcal{W}^{(1,\ldots,l)}_{z_{1},\ldots,z_{\theta_{l}-r}}$, the subspace of
$\mbox{\rm
Hom}\left(V^{\otimes(\theta_{l}-r)},\mathcal{W}^{(1,\ldots,l)}_{z_{1},\ldots,z_{\theta_{l}-r}}\right)$
consisting of linear maps satisfying the $L_{V}(-1)$-derivative and the
$L_{V}(0)$-conjugation properties or regularized transversal to $\mu_{l}-t\geq
0$ vertex operators, is invariant with respect to the group
$S_{\theta_{l}-r}$. $\square$

###  7.1. Formal parameters invariance

According to Proposition 7, elements of the space
$\mathcal{W}^{(1,\ldots,l)}_{z_{1},\ldots,z_{\theta_{l}-r}}$ resulting from
the sequence of $(\rho_{1},\ldots,\rho_{l})$-products (18), (3.9) are
invariant with respect to group $\left({\rm
Aut}\;\mathcal{O}\right)^{\times(\theta_{l}-r)}_{z_{1},\ldots,z_{\theta_{l}-r}}$
of independent changes of the formal parameters. Now we prove the following

######  Corollary 2.

For $\Phi^{(i)}(g_{i};v_{1,1},x_{1,1};\ldots;v_{k_{i},i},x_{k_{i},i})\in
C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$ the sequence of products

$\displaystyle\widehat{\Theta}\left(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{1,1},x_{1,1};\ldots;v_{k_{l},l},x_{k_{l},l};\rho_{1},\ldots,\rho_{l};\zeta_{1,i},\zeta_{2,i}\right)$
(7.1)
$\displaystyle\qquad\qquad=\left(\Phi^{(i)}(g_{i};v_{1,1},x_{1,1};\ldots;v_{k_{i},i},x_{k_{i},i})\right)_{k},$

is invariant with respect to the action of the group $\left({\rm
Aut}\;\mathcal{O}\right)^{\times(\theta_{l}-r)}_{z_{1},\ldots,z_{\theta_{l}-r}}$

$\displaystyle(z_{1},\ldots,z_{\theta_{l}-r})\mapsto(\widetilde{z}_{1},\ldots,\widetilde{z}_{\theta_{l}-r})=(\varrho(z_{1}),\ldots,\varrho(z_{\theta_{l}-r})).$
(7.2)

###### Proof.

In Subsection 3.8 we proved that the sequence of multiple products (18), (3.9)
belongs to $\mathcal{W}^{(1,\ldots,l)}_{z_{1},\ldots,z_{\theta_{l}-r}}$, and
it is invariant under the action of the group $\left({\rm
Aut}\;\mathcal{O}\right)^{\times(\theta_{l}-r)}_{z_{1},\ldots,z_{\theta_{l}-r}}$.
As in to the proof of Proposition 8, it follows that the vertex operators
$\omega_{V}(v^{\prime}_{i,j},x^{\prime}_{i,j})$, $1\leq j\leq m_{i}$,
regularized transversal to $\Phi^{(i)}$ $(g_{i}$; $v_{1,1}$, $x_{1,i}$;
$\ldots$; $v_{k_{i},i}$, $x_{k_{i},i})$, are also invariant with respect to
independent changes of coordinates
$(\varrho(z_{1}),\ldots,\varrho(z_{\mu_{l}-t}))\in{\rm
Aut}\;\mathcal{O}^{\times(\mu_{l}-t)}_{z_{1},\ldots,z_{\mu_{l}-t}}$. ∎

###  7.2. The coboundary operator acting on the product for complexes

In Proposition 10 we proved that the sequence of multiple products (19) of
spaces $C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$ elements belongs to
$C^{\theta_{l}-r}_{\mu_{l}-t}$ $\left(V,\mathcal{W}^{(1,\ldots,l)}\right)$.
Thus, the product admits the action of the coboundary operators
$\delta^{\theta_{l}-r}_{\mu_{l}-t}$ and $\delta^{2-r,i}_{ex-t,i}$ defined in
(5.3) and (5.8). As we showed in Subsection 6.1, in contrast to the case of
$\mathcal{W}^{(i)}$-spaces, where the sequence of
$(\rho_{1},\ldots,\rho_{l})$-products leads to the tensor product
$\mathcal{W}^{(1,\ldots,l)}$, the products (6.3) of $C^{k_{i}}_{m_{i}}$-spaces
result in the same kind of space
$C^{k}_{m}\left(V,\mathcal{W}^{(1,\ldots,l)}\right)$ defined on
$\mathcal{W}^{(1,\ldots,l)}$.

The coboundary operators (5.3), (5.8) have a version of Leibniz law with
respect to the product (19). We will use it in Section 8 while deriving the
cohomology classes. Recall the notations $n_{i}$ of Subsection 3.2.

######  Proposition 12.

For $\Phi^{(i)}(g_{i};v_{1,1},x_{1,i};\ldots;v_{k_{i},i},x_{k_{i},i})\in
C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$, $1\leq i\leq l$, the
action of the coboundary operator $\delta_{\mu_{l}-t}^{\theta_{l}-r}$ (5.3)
(and $\delta^{2-r,i}_{ex-t,i}$ (5.8)) on the sequence of
$(\rho_{1},\ldots,\rho_{l})$-products (19), $l\geq 1$, is given by

$\displaystyle\delta_{\mu_{l}-t}^{\theta_{l}-r}\Theta(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};z_{1},v_{1};\ldots;v_{\theta_{l}-r},z_{\theta_{l}-r};\rho_{1},\ldots,\rho_{l};\zeta_{1,i},\zeta_{2,i})_{k}\qquad\qquad$
$\displaystyle=\sum\limits_{i=1}^{l}\cdot_{\rho_{1},\ldots,\rho_{l}}(-1)^{k_{i}-r_{i}}\delta^{k_{i}-r_{i}}_{m_{i}-t_{i}}\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}})_{k}.\quad$
(7.3)

###### Proof.

Due to (5.3) and (6), the action of $\delta_{\mu_{l}-t}^{\theta_{l}-r}$ on
$\Theta(f_{1},\ldots,f_{l}$; $g_{1},\ldots,g_{l}$; $z_{1},v_{1}$;$\ldots$;
$v_{\theta_{l}-r},z_{\theta_{l}-r}$; $\rho_{1},\ldots,\rho_{l}$;
$\zeta_{1,i},\zeta_{2,i})_{k}$, is given by (we assume, as before, that the
vertex operator $\omega_{V}(v_{j},z_{j}-z_{j+1})$ does not act on
$(u,\zeta_{1,i})$)

$\displaystyle\delta_{\mu_{l}-t}^{\theta_{l}-r}\Theta(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{1},z_{1};\ldots;v_{\theta_{l}-r},z_{\theta_{l}-r};\rho_{1},\ldots,\rho_{l};\zeta_{1,i},\zeta_{2,i})_{k}$
$\displaystyle=\sum_{j=1}^{\theta_{l}-r}(-1)^{j}\;\Theta(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};v_{1},z_{1};\ldots;v_{j-1},z_{j-1};$
$\displaystyle\omega_{V}(v_{j},z_{j}-z_{j+1})v_{j+1},z_{j+1};v_{j+2},z_{j+2};\ldots;v_{\theta_{l}-r},z_{\theta_{l}-r};\rho_{1},\ldots,\rho_{l};\zeta_{1,i},\zeta_{2,i})_{k}$
$\displaystyle+\Theta(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};\omega_{W^{(1)}}\left(v_{1},z_{1}\right);v_{2},z_{2};\ldots;v_{\theta_{l}-r},z_{\theta_{l}-r};\rho_{1},\ldots,\rho_{l};\zeta_{1,i},\zeta_{2,i})_{k}$
$\displaystyle+(-1)^{\theta_{l}-r+1}\Theta(f_{1},\ldots,f_{l};g_{1},\ldots,g_{l};\omega_{W^{(l)}}(v_{\theta_{l}-r+1},z_{\theta_{l}-r+1});v_{1},z_{1};\ldots;$
$\displaystyle
v_{\theta_{l}-r},z_{\theta_{l}-r};\rho_{1},\ldots,\rho_{l};\zeta_{1,i},\zeta_{2,i})_{k}.$

Recall the definition of the enumeration $n_{i}$ of $v$ and $z$-parameters
defined in Subsection 3.2. Using (18) we see that the above is equivalent to

$\displaystyle\sum_{j=1}^{\theta_{l}-r}(-1)^{j}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w^{\prime}_{i},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;\right.\qquad\qquad\qquad\qquad$
$\displaystyle\left.\;\omega_{V}(v_{j},z_{j}-z_{j+1})\;v_{j+1},z_{j+1};v_{j+2},z_{j+2};\ldots;v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i}),\zeta_{2,i}\right)\;f_{i}.\overline{u}\rangle,$
$\displaystyle+\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w_{i}^{\prime},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\left(\left(\omega_{W^{(1)}}(v_{1},z_{1})\right)^{\delta_{i,1}}\right.\right.\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\left.\left.\qquad\qquad\Phi^{(i)}(g_{i};v_{n_{i}+1+\delta_{i,1}},z_{n_{i}+1+\delta_{i,1}};\ldots;v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\right),\zeta_{2,i}\right)\;f_{i}.\overline{u}\rangle$
$\displaystyle\qquad+(-1)^{\theta_{l}-r+1}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w_{i}^{\prime},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\right.\left(\left(\omega_{W^{(l)}}(v_{n_{i+1}+1},z_{n_{i+1}+1})\right)^{\delta_{i,1}}\right.$
$\displaystyle\quad\left.\left.\Phi^{(i)}(g;v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{1}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\right),\zeta_{2,i}\right)\;f_{i}.\overline{u}\rangle.$
(7.4)

Consider the third term in (7.4)

$\displaystyle\sum\limits_{s=2}^{l}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w_{i}^{\prime},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\left(\left(\omega_{W^{(s)}}(v_{n_{i+1}+1},z_{n_{i+1}+1})\right)^{\delta_{s,i}}\right.\right.\qquad\qquad\qquad$
$\displaystyle\left.\left.\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\right),\zeta_{2,i}\right)\;f_{i}.\overline{u}\rangle\qquad$
$\displaystyle=\sum\limits_{s=2}^{l}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w_{i}^{\prime},e^{\zeta_{2,i}L_{W^{(i)}}(-1)}Y_{W^{(i)}}\left(f_{i}.\overline{u},-\zeta_{2,i}\right)\;\left(\omega_{W^{(s)}}(v_{n_{i+1}+1},z_{n_{i+1}+1})\right)^{\delta_{s,i}}$
$\displaystyle\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{1}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\rangle\qquad$
$\displaystyle=\sum\limits_{s=2}^{l}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w_{i}^{\prime},e^{\zeta_{2,i}L_{W^{(i)}}(-1)}\left(\omega_{W^{(s)}}(v_{n_{i+1}+1},z_{n_{i+1}+1})\right)^{\delta_{s,i}}Y_{W^{(i)}}\left(f_{i}.\overline{u},-\zeta_{2,i}\right)$
$\displaystyle\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\rangle.$

Due to the definition (9.5) of the intertwining operator and the locality
property (36) of vertex operators we obtain

$\displaystyle\sum\limits_{s=2}^{l}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w_{i}^{\prime},\left(\omega_{W^{(s)}}(v_{n_{i+1}+1},z_{n_{i+1}+1}+\zeta_{2,i})\right)^{\delta_{s,i}}e^{\zeta_{2,i}L_{W^{(i)}}(-1)}$
$\displaystyle\qquad
Y_{W^{(i)}}\left(f_{i}.\overline{u},-\zeta_{2,i}\right)\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\rangle.$

The insertion an arbitrary vertex algebra module $W^{(i)}$-basis
$\widetilde{w}_{i}$, and use of the definition of the intertwining operator
(9.5) results

$\displaystyle\sum\limits_{\widetilde{w}_{i}\in
W^{(i)}}\sum\limits_{s=2}^{l}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle
w^{\prime}_{i},\left(\omega_{W^{(s)}}(v_{n_{i+1}+1},z_{n_{i+1}+1}+\zeta_{2,i})\right)^{\delta_{s,i}}\widetilde{w}_{i}\rangle$
$\displaystyle\langle\widetilde{w}^{\prime}_{i},e^{\zeta_{2,i}L_{W^{(i)}}(-1)}Y_{W^{(i)}}\left(f_{i}.\overline{u},-\zeta_{2,i}\right)\Phi^{(i)}(g_{i};v_{n_{i}+1},z_{n_{i}+1};\ldots;$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad
v_{n_{i}+k_{i}-r_{i}},z_{n_{i}+k_{i}-r_{i}};u,\zeta_{1,i})\rangle$
$\displaystyle=\sum\limits_{s=2}^{l}\sum_{\widetilde{w}_{i}\in W^{(i)}\atop
k\in\mathbb{Z}}\mathcal{R}\prod_{i=1}^{l}\rho^{k}_{i}\langle\widetilde{w}^{\prime}_{i},Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\Phi^{(i)}(g_{i};v_{n_{i}},z_{n_{i}};\ldots;\right.\qquad\qquad\qquad\qquad$
$\displaystyle\left.\qquad
v_{n_{i+1}-1},z_{n_{i+1}-1};u,\zeta_{1,i}),\zeta_{2,i}\right)f_{i}.\overline{u}\rangle\langle
w^{\prime}_{i},\left(\omega_{W^{(s)}}(v_{n_{i+1}},z_{n_{i+1}}+\zeta_{2,i})\right)^{\delta_{s,i}}\widetilde{w}_{i}\rangle$
$\displaystyle=\sum\limits_{\widetilde{w}_{i}\in
W^{(i)}}\sum\limits_{s=2}^{l}\mathcal{R}\prod_{i=1}^{l-1}\rho^{k}_{i+1}\langle
w^{\prime}_{i},\left(\omega_{W^{(s)}}(v_{n_{i+1}-1},z_{n_{i+1}-1}+\zeta_{2,i}\right)^{\delta_{s,i}}\widetilde{w}_{i}\rangle$
$\displaystyle\langle
w^{\prime}_{i+1},Y^{W^{(i+1)}}_{W^{(i+1)}V^{\prime}}\left(\Phi^{(i+1)}(g_{i+1};v_{n_{i+1}},z_{n_{i+1}};\ldots;\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.v_{n_{i+2}-1},z_{n_{i+2}-1};u,\zeta_{1,i+1}),\zeta_{2,i+1}\right)f_{i+1}.\overline{u}\rangle$
$\displaystyle=\sum\limits_{s=2}^{l}\sum_{\widetilde{w}_{i}\in
W^{(i)}}\mathcal{R}\prod_{i=1}^{l-1}\rho^{k}_{i+1}\langle
w^{\prime}_{i},\left(\omega_{W^{(s)}}(v_{n_{i+1}-1},z_{n_{i+1}-1}+\zeta_{2,i}\right)^{\delta_{s,i}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
Y^{W{(i)}}_{W^{(i)}W^{(i+1)}}(\widetilde{w}_{i},\zeta)\;w_{i+1}\rangle$
$\displaystyle\langle
w^{\prime}_{i+1},Y^{W^{(i+1)}}_{W^{(i+1)}V^{\prime}}\left(\Phi^{(i+1)}(g_{i+1};v_{n_{i+1}},z_{n_{i+1}};\ldots;\right.$
$\displaystyle\qquad\qquad\qquad\left.v_{n_{i+2}-1},z_{n_{i+2}-1};u,\zeta_{1,i+1}),\zeta_{2,i+1}\right)f_{i+1}.\overline{u}\rangle.$

Now eliminate the basis $w_{i+1}$ to get

$\displaystyle=\sum\limits_{s=1}^{l}\mathcal{R}\prod_{i=1}^{l-1}\rho^{k}_{i+1}\langle
w^{\prime}_{i},e^{-L_{W^{(s-1)}}(-1)(-z_{n_{i+1}-1}-\zeta_{2,i})}e^{L_{W^{(s-1)}}(-1)(-z_{n_{i+1}-1}-\zeta_{2,i})}$
$\displaystyle\left(\omega_{W^{(s-1)}}(v_{n_{i+1}-1},z_{n_{i+1}-1}+\zeta_{2,i})\right)^{\delta_{s,i+1}}Y^{W{(i)}}_{W^{(i)}W^{(i+1)}}(\widetilde{w}_{i},\zeta)\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\;Y^{W^{(i+1)}}_{W^{(i+1)}V^{\prime}}\left(\Phi^{(i+1)}(g_{i+1};v_{n_{i+1}},z_{n_{i+1}};\ldots;v_{n_{i+2}-1},z_{n_{i+2}-1};u,\zeta_{1,i+1}),\zeta_{2,i+1}\right)f_{i+1}.\overline{u}\rangle$
$\displaystyle=\sum\limits_{s=1}^{l}\mathcal{R}\prod_{i=1}^{l-1}\rho^{k}_{i+1}\langle
w^{\prime}_{i},e^{-L_{W^{(s-1)}}(-1)(-z_{n_{i+1}-1}-\zeta_{2,i})}\left(Y^{W^{(i)}}_{W^{(i)}W^{(i)}}(\right.Y^{W{(i)}}_{W^{(i)}W^{(i+1)}}(\widetilde{w}_{i},\zeta)$
$\displaystyle
Y^{W^{(i+1)}}_{W^{(i+1)}V^{\prime}}\left(\Phi^{(i+1)}(g_{i+1};v_{n_{i+1}},z_{n_{i+1}};\ldots;v_{n_{i+2}-1},z_{n_{i+2}-1};u,\zeta_{1,i+1}),\zeta_{2,i+1}\right)f_{i+1}.\overline{u},\qquad$
$\displaystyle\left.-\zeta)\right)^{\delta_{s,i+1}}v_{n_{i+1}-1}\rangle$
$\displaystyle=\sum\limits_{s=1}^{l}\mathcal{R}\prod_{i=1}^{l-1}\rho^{k}_{i+1}\langle
w^{\prime}_{i},e^{-L_{W^{(s-1)}}(-1)(-z_{n_{i+1}-1}-\zeta_{2,i})}$
$\displaystyle\left(Y^{W^{(i)}}_{W^{(i)}W^{(i)}}(Y^{W{(i)}}_{W^{(i)}W^{(i+1)}}(\widetilde{w}_{i},\zeta)e^{L_{W^{(i+1)}}(-1)(-\zeta_{2,i+1})}Y_{W^{(i+1)}}(v_{n_{i+1}-1},\zeta)\right.$
$\displaystyle\left.\Phi^{(i+1)}(g_{i+1};v_{n_{i+1}},z_{n_{i+1}};\ldots;v_{n_{i+2}-1},z_{n_{i+2}-1};u,\zeta_{1,i+1})f_{i+1}.\overline{u},-\zeta)\right)^{\delta_{s,i+1}}\rangle$
$\displaystyle=\sum\limits_{s=1}^{l}\mathcal{R}\prod_{i=1}^{l-1}\rho^{k}_{i+1}\langle
w^{\prime}_{i+1},e^{-L_{W^{(i)}}(-1)(-z_{n_{i+1}-1}-\zeta_{2,i})}$
$\displaystyle
e^{L_{W^{(i+1)}}(-1)(-\zeta_{2,i+1})}Y_{W^{(i+1)}}(v_{n_{i+1}-1},\zeta)$
$\displaystyle\left.\Phi^{(i+1)}(g_{i+1};v_{n_{i+1}},z_{n_{i+1}};\ldots;v_{n_{i+2}-1},z_{n_{i+2}-1};u,\zeta_{1,i+1})f_{i+1}.\overline{u},-\zeta)\right)^{\delta_{s,i+1}}\rangle,$

where $\zeta=-z_{n_{i+1}-1}-\zeta_{2,i}$. Above we have made use of the
commutativity of $L_{W^{(i)}}(-1)$ and $L_{W^{(i+1)}}(-1)$, and the formula
relating the intertwining operators in the adjoint positions. Due to locality
(36) of vertex operators, and arbitrariness of $v_{k+1}\in V$ and $z_{k+1}$,
it is always possible to take
$\omega_{W^{(s-1)}}(v_{n_{i+1}-1},z_{n_{i+1}-1}+\zeta_{2,i-1}-\zeta_{2,i+1})=\omega_{W^{(s-1)}}(v_{n_{i+1}},z_{n_{i+1}})$,
for $v_{n_{i+1}}=v_{n_{i+1}-1}$,
$z_{n_{i+1}}=z_{n_{i+1}-1}+\zeta_{2,i-1}-\zeta_{2,i+1}$. We repete the same
operations with the second term of (7.4). Combining the action of
$\delta^{k_{i}}_{m_{i}}$ on $\Phi^{(i)}$, with (6) gives (12) due to (18),
(3.9). The statement of the proposition for $\delta^{2,i}_{ex,i}$ (5.8) can be
checked in the similar way. ∎

######  Remark 8.

The extra arbitrary vertex algebra element $v_{n_{i}+r_{i}-r_{i}+1}\in V$,
$1\leq i\leq l$ and the corresponding arbitrary parameter $z_{k_{i}+1,i}$ in
(5.3) comes from the action of $\delta^{k_{i}}_{m_{i}}$ on $\Phi^{(i)}\in
C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$ as a map to
$C^{k_{i}+1}_{m_{i}-1}\left(V,\mathcal{W}^{(i)}\right)$. In application to the
sequence of $(\rho_{1},\ldots,\rho_{l})$-products (19) these extra arbitrary
elements are involved in the definition of the action of
$\delta_{\mu_{l}-t}^{\theta_{l}-r}$ on
$\Phi^{(1,\ldots,l)}(g_{1},\ldots,g_{l};v_{1},z_{1};\ldots;v_{\theta_{l}-r},x_{\theta_{l}-r})$.

Note that both sides of (12) belong to the space
$C_{\mu_{l}-t-1}^{\theta_{l}-r+1}\left(V,\mathcal{W}^{(i)}\right)$. From the
coboundary operators $\delta^{k_{i}-r_{i}}_{m_{i}-t_{i}}$ in (12) we have
excluded coinciding vertex algebra elements and corresponding formal
parameters, as well as coinciding regularized transversal vertex operators.
The dependence on coinciding vertex algebra elements, parameters, and
regularized transversal to vertex operators is taken into account in the
mappings multiplying the action of the coboundary operators on $\Phi^{(i)}$.
It is easy to see following

######  Corollary 3.

The sequence of products (19) and the coboundary operators (5.3), (5.8) endow
the space $\times_{i=1}^{l}C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$
$k_{i}\geq 0$, $m_{i}\geq 0$, $l\geq 1$, with the structure of a graded
differential algebra
$\mathcal{G}\left(V,\mathcal{W}^{(1,\ldots,l)},\cdot_{\rho_{1},\ldots,\rho_{l}},\delta^{\theta_{l}-r}_{\mu_{l}-t},\mathcal{F}\right)$.
$\square$

Next, we prove the following

######  Proposition 13.

The sequence of products (19) extends the property (5.13) of the families of
chain-cochain complexes (5.15) and (9) to all sequences of products
$\cdot_{\rho_{1},\ldots,\rho_{l}}$
$C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$, $k_{i}\geq 0$, $m_{i}\geq
0$, $1\leq i\leq l$.

###### Proof.

For $\Phi^{(i)}\in C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$ we
proved in Proposition 10 that the sequence of products
$\cdot_{\rho_{1},\ldots,\rho_{l}}\left(\Phi^{(i)}\right)$ belongs to the
spaces $C^{\theta_{l}-r}_{\mu_{l}-t}\left(V,\mathcal{W}^{(i)}\right)$. Using
(12) and the chain-cochain property for $\Phi^{(i)}$ we see that

$\displaystyle\delta^{\theta_{l}-r+1}_{\mu-t-1}\circ\delta^{\theta_{l}-r}_{\mu_{l}-t}\left(\cdot_{\rho_{1},\ldots,\rho_{l}}\Phi^{(i)}\right)=0,\quad\delta^{2-r}_{ex-t}\circ\delta^{1-r}_{2-t}\left(\cdot_{\rho_{1},\ldots,\rho_{l}}\Phi^{(i)}\right)=0.$

Thus, the chain-cochain property extends to the sequence of
$(\rho_{1},\ldots,\rho_{l})$-products $\cdot_{\rho_{1},\ldots,\rho_{l}}$
$\left(C^{k_{i}}_{m_{i}}\left(V\right.\right.$,
$\left.\left.\mathcal{W}^{(i)}\right)\right)$. ∎

###  7.3. The exceptional complex

Finally, for elements of the spaces
$C^{2,i}_{ex,i}\left(V,\mathcal{W}^{(i)}\right)$ we obtain

######  Corollary 4.

The product of elements of the spaces $C^{2}_{ex}$
$\left(V,\mathcal{W}^{(ex)}\right)$ and $C^{k_{i}}_{m_{i}}$
$\left(V,\mathcal{W}^{(i)}\right)$ is given by (19),

$\displaystyle\cdot_{\rho_{1},\ldots,\rho_{l}}:\times_{i=1}^{l_{1}}C^{2,i}_{ex,i}\left(V,\mathcal{W}^{(i)}\right)\times_{j=1}^{l_{2}}C^{k_{i},i}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)\to
C^{k_{1}+\ldots+k_{l_{2}}+2l_{1}-r,i}_{m_{i}-t,i}\left(V,\mathcal{W}^{(i)}\right),$
$\cdot_{\rho_{1},\ldots,\rho_{l}}:\times_{i=1}^{l}C^{2,i}_{ex,i}\left(V,\mathcal{W}^{(i)}\right)\to
C^{4-r,i}_{0,i}\left(V,\mathcal{W}^{(i)}\right).$ (7.5)

###### Proof.

The number of formal parameters in the product (19) is
$k_{1}+\ldots+k_{l_{2}}+2l_{1}-r$. That follows from Proposition (4). Consider
the product (19) for $C^{2,i}_{ex,i}\left(V,\mathcal{W}^{(i)}\right)$ and
$C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$. As in the proof of
Proposition 10, the total number $m_{i}-t$ of vertex operators the product
$\Theta$ is regularized transversal is preserved. Thus, we have to checked
that on the right hand side of (7.5) the number of vertex operators
regularized transversal becomes $m_{i}-t$. ∎

##  8\. The multiple-product cohomology classes

In this Section proofs of the main results of this paper are provided. In
particular, we find invariant classes associated to the sequences of multiple
products for a vertex algebra cohomology for codimension one foliations.

###  8.1. The cohomology classes

In this Subsection, we introduce the cohomology classes for codimension one
foliations on complex curves associated to a grading-restricted vertex
operator algebra. The cohomology classes for a codimension one foliation [31,
15, 52] were introduced starting with an extra transversality condition on
differential forms defining a foliation, and leading to the integrability
condition. The elements of $\mathcal{E}$ in (5.3) and $\mathcal{E}_{ex}$ in
(5.10) are elements of spaces
$C^{1,i}_{\infty,i}\left(V,\mathcal{W}^{(i)}\right)$ regularized transversal
to an infinite number of vertex operators. The actions of coboundary operators
$\delta^{k_{i}}_{m_{i}}$ and $\delta^{2,i}_{ex,i}$ in (5.3) and (5.8) are
written as products similar to as differential forms in Frobenius theorem
[31]. Using the sequence of multiple products we introduce cohomology classes
of the form that are counterparts of the Godbillon class [29]. In analogy with
differential forms let us introduce

######  Definition 27.

We call a map $\Phi^{(i)}\in
C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$, closed if it represents a
closed connection
$\delta^{k_{i}}_{m_{i}}\Phi^{(i)}=G\left(\Phi^{(i)}\right)=0$. For $m_{i}\geq
1$, we call it exact if there exists $\Psi^{(i)}\in
C_{m_{i}-1}^{k_{i}+1}\left(V,\mathcal{W}^{(i)}\right)$, such that
$\Psi^{(i)}(v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{k_{i}+1},z^{\prime}_{k_{i}+1})=\delta^{k_{i}}_{m_{i}}\Phi^{(i)}(v_{1},z_{1};\ldots;v_{k_{i}},z_{k_{i}})$,
i.e., $\Psi^{(i)}$ is the form of a connection.

######  Definition 28.

For $\Phi^{(i)}\in C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$ we call
the cohomology class of mappings $\left[\Phi^{(i)}\right]$ the set of all
closed forms that differ from $\Phi^{(i)}$ by an exact mapping, i.e., for
$\Lambda^{(i)}\in C^{k_{i}-1}_{m_{i}+1}\left(V,\mathcal{W}^{(i)}\right)$,
$\left[\Phi^{(i)}\right]=\Phi^{(i)}+\delta^{k_{i}-1}_{m_{i}+1}\Lambda^{(i)}$.

In this Section we find the cohomology classes for $\Phi^{(i)}\in
C^{1}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$, $m_{i}\geq 0$, associated with
two-point connections and the first cohomology
$H^{1}_{m_{i}}\left(V,\mathcal{W}^{(i)},\mathcal{F}\right)$, as well as the
cohomology classes for $\Phi^{(i)}\in
C^{2,i}_{ex,i}\left(V,\mathcal{W}^{(i)}\right)$), associated with transversal
connections and the second cohomology
$H^{2,i}_{ex,i}\left(V,\mathcal{W}^{(i)},\mathcal{F}\right)$, of
$M/\mathcal{F}$. The cohomology classes constructed in this paper are vertex
algebra cohomology analogues of the Godbillon class [52, 29] for codimension
one foliations on complex curves.

######  Remark 9.

The existence of a affine or projective connection for codimension one
foliations on smooth manifolds leads usualy to non-vanishing corresponding
cohomology classes [1, 2].

######  Remark 10.

The cohomology classes we introduce are functionals of $v\in V$, i.e.,
$\langle w^{\prime},\Phi^{(i)}\rangle$, for $w^{\prime}_{i}\in
W^{(i)}{}^{\prime}$ varies with of $v\in V$.

###  8.2. Transversality conditions

In this Subsection we consider the general classes of cohomology invariants
which arise from the definition of the product of pairs of
$C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$-spaces. Under a natural
extra condition, the families chain-cochain complexes (5.15) and (9) allow us
to establish relations among elements of
$C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$-spaces. By analogy with
the notion of the integrability for differential forms [31], we use here the
notion of the transversality for the spaces of a complex.

######  Definition 29.

For the families chain-cochain complexes (5.15) and (9) let us require that
for chain-cochain complex spaces
$C_{m_{i_{j}}}^{k_{i_{j}}}\left(V,\mathcal{W}^{(i_{j})}\right)$, $1\leq
i_{1}<\ldots<i_{j}\leq l$, $1\leq j\leq k\leq l$ there exist subspaces
$\widetilde{C}_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)\subset
C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$, such that for
$\Phi^{(i_{j})}\in\widetilde{C}_{m_{i_{j}}}^{k_{i_{j}}}\left(V,\mathcal{W}^{(i_{j})}\right)$,
and $1\leq n\leq l$,
$\left(\ldots,\delta^{k_{i_{1}}}_{m_{i_{1}}}\Phi^{(i_{1})},\ldots,\delta^{k_{i_{k}}}_{m_{i_{k}}}\Phi^{(i_{k})},\ldots\right)=0$.
Then we call the set of subspaces
$\left\\{\widetilde{C}_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)\right\\}$
orthogonal for all spaces $C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$,
$i\neq i_{j}$ with respect to the product $\eqref{gendefgen}$. Namely,
$\delta^{k_{i_{1}}}_{m_{i_{1}}}\Phi^{(i_{1})}$, $\ldots$,
$\delta^{k_{i_{j}}}_{m_{i_{j}}}\Phi^{(i_{j})}$, are supposed to be transversal
to all other multiplicands with respect to the product (3). We call this the
generalized transversality condition for mappings of the families chain-
cochain complexes (5.15) and (9).

In particular, the simplest case of the transversality is defined for some
$1\leq i,p\leq l$ by

$\left(\ldots,\left(\delta^{k_{i}}_{m_{i}}\right)^{\delta_{i,p}}\Phi^{(i)},\ldots\right)=0.$
(8.1)

Note that in the case of differential forms considered on a smooth manifold,
the Frobenius theorem for a distribution provides the transversality condition
[31]. The fact that both sides of a differential relation belong to the same
chain-cochain complex space, applies limitations to possible combinations of
$(k_{i},m_{i})$, $1\leq i\leq j\leq l$. Below we derive the algebraic
relations occurring from the transversality condition on the families of
chain-cochain complexes (5.15) and (9). Taking into account the correspondence
(see Subsection 26) with ${\rm\check{C}}$ech-de Rham complex due to [15], we
reformulate the derivation of the product-type invariants in the vertex
algebra terms.

Recall that the Godbillon–Vey cohomology class [31] is considered on
codimension one foliations of three-dimensional smooth manifolds. In this
paper, we supply its analogue for complex curves. According to the definition
(4.1) we have $m_{i}$-tuples of one-dimesional transversal sections. In each
section we attach one vertex operator $\omega_{W^{(i)}}(u_{j},w_{j})$,
$u_{m_{i}}\in V$, $w_{m_{i}}\in U_{m_{i},i}$, $1\leq i\leq l$, $1\leq j\leq
m_{i}$. Similarly to the differential forms setup, a mapping $\Phi^{(i)}\in
C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$ defines a codimension one
foliation. As we see from (18) and (12) it satisfies the properties similar as
differential forms do.

Now, let us explain how we understand powers of an element of
$\mathcal{W}^{(i)}_{x_{1,i},\ldots,x_{k_{i},i}}$ in the multiple product
(3.9). Denote by $\Phi^{(i)}_{j_{s}}$ $=$ $\Phi^{(i)}$ $(g_{i}$; $v_{1,i}$,
$x_{1,i}$; $\ldots$; $v_{k_{i},i}$, $)$ an element of
$\mathcal{W}^{(i)}_{x_{1,i},\ldots,x_{k_{i},i}}$ placed at a position $1\leq
j_{s}\leq l$, $1\leq s\leq k$. We then have

$\displaystyle\left(\ldots,\left(\Phi^{(i)}\right)^{k},\ldots\right)=\left(\ldots,\Phi^{(i)}_{j_{1}},\ldots,\Phi^{(i)}_{j_{2}},\ldots,\Phi^{(i)}_{j_{r}},\ldots\right),\quad$
(8.2)

with $\Phi^{(i)}$ placed at some positions $(j_{1},\ldots,j_{k})$. Another
kind of transversality conditions is introduced by the following

######  Definition 30.

We call the order ${\mbox{\rm ord}}\;\Phi$ of an element $\Phi$ in a product
of the form (18) the number of appearance of $\Phi$. For two elements $\Phi$,
$\Psi$ we can also define the mutual order as ${\mbox{\rm
ord}}\;(\Phi,\Psi)=|{\mbox{\rm ord}}\;\Phi-{\mbox{\rm ord}}\;\Psi|$.

###  8.3. The commutator multiplications

In this Subsection we define further multiple products of elements of the
spaces $C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$, $1\leq i\leq l$,
suitable for the formulation of cohomology invariants. Note that when applying
the coboundary operators (5.3) (or (5.8) to corresponding $\Phi^{2,i}_{ex,i}$)
$\Phi^{(i)}(g_{i};v_{1},z_{1};\ldots;v_{n},z_{n})\in
C^{n}_{m}\left(V,\mathcal{W}^{(i)}\right)$,

$\displaystyle\delta^{k}_{m}:\Phi^{(i)}(g_{i};v_{1},z_{1};\ldots;v_{k},z_{k})\to\Phi^{(i)}(g^{\prime}_{i};v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{n+1},z^{\prime}_{n+1})\in
C^{n+1}_{m-1}\left(V,\mathcal{W}^{(i)}\right),$

one does not necessary assume that we keep the same set of vertex algebra
elements/formal parameters and vertex operators regularized transversal to for
$\delta^{n}_{m}\Phi$, i.e., the set $(g_{i};v_{1},z_{1};\ldots;v_{n},z_{n})$
may change to
$(g^{\prime}_{i};v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{n+1},z^{\prime}_{n+1})$,
though it might happen that some of them could be common with $\Phi^{(i)}$.

######  Definition 31.

For a set of indices
$(i_{1},i_{2},i_{1,2},i_{3},\ldots,i_{1,\ldots,l-1},i_{l})$ ranging in
$[1,\ldots,l]$, and corresponding complex parameters
$(\rho_{1},\rho_{2},\rho_{1,2},\ldots,\rho_{1,2,\ldots,l-1},\rho_{l})$, let us
define the additional multiple products of elements $\Phi^{(i)}$ $(g_{i}$;
$v_{n_{i}+1}$, $z_{n_{i}+1}$; $\ldots$ ; $v_{n_{i}+k_{i}-r_{i}}$,
$z_{n_{i}+k_{i}-r_{i}})$ $\in$ $C^{k_{i}}_{m_{i}}$ $\left(V\right.$,
$\left.\mathcal{W}^{(i)}\right)$, as follows (for clarity of presentation, we
omit here explicit dependence on the automorphism element, vertex algebra
elements, formal parameters, and additional $\zeta$-parameters)

$\displaystyle*_{(i_{1},i_{2},i_{1,2},i_{3},\ldots,i_{1,\ldots,l-1},i_{l})}:\times_{i=1}^{l}\mathcal{W}^{(i_{p})}_{z_{1,i_{p}},\ldots,z_{k_{p},i_{p}}}\to\mathcal{W}^{(1,\ldots,l)}_{z_{k_{1}},\ldots,z_{\theta_{l-r}}},\;$
(8.3)
$\displaystyle*_{(i_{1},i_{2},i_{1,2},i_{3},\ldots,i_{1,\ldots,l-1},i_{l})}\left(\Phi^{(i)}\right)_{1\leq
i\leq l}$
$\displaystyle\quad=\left[\left[\ldots\left[\left[\Phi^{(i_{1})},_{\cdot_{\rho_{i_{1}},\rho_{i_{2}}}}\Phi^{(i_{2})}\right],_{\cdot_{\rho_{i_{1,2}},\rho_{i_{3}}}}\Phi^{(i_{3})}\right]\ldots\right],_{\cdot_{\rho_{1,\ldots,l-1},\rho_{i_{l}}}}\Phi^{(i_{l})}\right],$

where the brackets denote the commutator with respect to the
$\cdot_{i_{p},i_{q}}$-product defined on
$\mathcal{W}^{(i_{p})}_{z_{1,i_{p}},\ldots,z_{k_{p},i_{p}}}\times\mathcal{W}^{(i_{q})}_{z_{1,i_{q}},\ldots,z_{k_{q},i_{q}}}$,
$\left[\Phi^{(i_{p})},_{i_{p},i_{q}}\Phi^{(i_{q})}\right]=\Phi^{(i_{p})}\cdot_{\rho_{i_{p}},\rho_{i_{q}}}\Phi^{(i_{q})}-\Phi^{(i_{q})}\cdot_{\rho_{i_{q}},\rho_{i_{p}}}\Phi^{(i_{p})}$,
with respect to the $\cdot_{\rho_{i_{p}},\rho_{i_{q}}}$-product (18).

We are able to use also the total $(i_{1}$, $i_{2}$, $i_{1,2}$, $\ldots$,
$i_{1,2,\ldots,i_{l-1}}$, $i_{l})$-symmetrization

${\rm
Sym}\left(*_{(i_{1},i_{2},i_{1,2},\ldots,i_{1,2,\ldots,i_{l-1}},i_{l})}\left(\Phi^{(i)}\right)_{1\leq
i\leq l}\right),$ (8.4)

of the product (8.3). The form of (8.4) is not unique of cause. We are able to
form other types of products resulting from the products (18). Nevertheless,
(8.4) is suitable for computation of cohomology invariants of foliations. Due
to the properties of the maps $\Phi^{(i)}\in
C_{m_{i}}^{k_{i}}\left(V,\mathcal{W}^{(i)}\right)$, $1\leq i\leq l$ we obtain

######  Lemma 9.

The products (8.4) belong to the space $C_{\mu_{l}-t}^{\theta_{l}-r}$
$\left(V\right.$ , $\mathcal{W}^{(1,\ldots,l)}$, $\left.\mathcal{F}\right)$.
$\square$

######  Remark 11.

For $i_{p}=i_{q}$, a self-dual bilinear pairing $\langle.,.\rangle$ for
$W^{(i_{p})}$, and $(g_{i_{p}}$; $v_{n_{i_{p}}}$, $z_{n_{i_{p}}}$;
$\ldots;v_{n_{i_{p}+1}-1},z_{n_{i_{p}+1}-1})$ $=$
$(g_{i_{q}};v_{n_{i_{q}}},z_{n_{i_{q}}};\ldots;v_{n_{i_{q}+1}-1},z_{n_{i_{q}+1}-1})$,
the product

$\displaystyle\Phi^{(i_{p})}(g_{i_{q}};v_{n_{i_{q}}},z_{n_{i_{q}}};\ldots;v_{n_{i_{q}+1}-1},z_{n_{i_{q}+1}-1})$
(8.5)
$\displaystyle\qquad*_{i_{p},i_{q}}\Phi^{(i_{p})}(g_{i_{p}};v_{n_{i_{p}}},z_{n_{i_{p}}};\ldots;v_{n_{i_{p}+1}-1},z_{n_{i_{p}+1}-1})=0.$

The product (8.3) allows to introduce cohomology invariants associated with
the condition (8.5) on $\Phi^{(i)}$. Namely, it is easy to prove the following

######  Proposition 14.

For the chain-cochain complex (5.15) elements $\Phi^{(i)}$ $\in$
$C^{k_{i}}_{m_{i}}$ $\left(V\right.$, $\left.\mathcal{W}^{(i)}\right)$
satisfying (8.5) and the transversality condition

$\delta^{k_{i_{s}}}_{m_{i_{s}}}\partial_{t}\Phi^{(i_{s})}*_{i_{s},i_{s^{\prime}}}\delta^{k_{i_{s^{\prime}}}}_{m_{i_{s^{\prime}}}}\Phi^{(i_{s^{\prime}})}=0,$
(8.6)

with $i_{s},i_{s^{\prime}}=i_{p},i_{q},i_{r}$, there exist the classes of non-
vanishing cohomology invariants of the form
$\left[\delta^{k_{i_{p}}}_{m_{i_{p}}}\Phi^{(i_{p})}*_{i_{p},i_{q}}\left(\partial_{t}\Phi^{(i_{q})}\right)^{\beta}*_{i_{p,q},i_{r}}\Phi^{(i_{r})}\right]$,
not depending on the choice of $\Phi^{(i_{s})}$. In particular, for the short
complex (9), one has $\left[\delta^{1,i_{p}}_{2,i_{p}}\Phi^{(i_{p})}\right.$
$*_{i_{p},i_{q}}$ $\left(\Phi^{(i_{q})}\right)$
$\left.*_{i_{p,q},i_{r}}\right]$,
$\left[\delta^{0,i_{p}}_{3,i_{p}}\Lambda^{(i_{p})}*_{i_{p},i_{q}}\left(\Lambda^{(i_{q})}\right)^{\beta}*_{i_{p,q},i_{r}}\Lambda^{(i_{r})}\right]$,
are invariant, i.e., they do not depend on the choices of $\Phi^{(i_{s})}$
$\in$ $C^{1,i_{s}}_{2,i_{p}}$ $\left(V,\mathcal{W}^{(i_{s})}\right)$,
$\Lambda^{(i_{s})}$ $\in$ $C^{0,i_{s}}_{3,i_{s}}$
$\left(V,\mathcal{W}^{(i_{s})}\right)$. $\square$

###  8.4. Proof of Theorem 1

Now we show that the analog of the integrability condition provides the
generalizations of the product-type invariants for codimension one foliations
on complex curves. Here we give a proof of the main statement of this paper,
Theorem 1 formulated in the Introduction.

###### Proof.

Suppose we consider products containing elements $\Phi^{(i_{s})}$,
$\Psi^{(i_{s})}$ $\in$ $C^{k_{i_{s}}}_{m_{i_{s}}}$ $\left(V\right.$,
$\left.\mathcal{W}^{(i_{s})}\right)$, with
$i_{s}=i,i^{\prime},i^{\prime\prime}$, with the mutual orders satisfying
$\mbox{\rm
ord}\left(\delta^{k_{i_{s}}}_{m_{i_{s}}}\Phi^{(i_{s})},\Psi^{(i_{s^{\prime}})}\right)<m+k-1$.
For elements $\Phi^{(i_{s})}\in
C^{k_{i_{s}}}_{m_{i_{s}}}\left(V,\mathcal{W}^{(i_{s})}\right)$, for $1\leq
i_{s}\leq n$, let us start with the foliation $\mathcal{F}$ transversality
condition [52]

$\left(\delta^{k_{i_{s}}}_{m_{i_{s}}}\partial_{t}\Phi^{(i_{s})},\delta^{k_{i_{s^{\prime}}}}_{m_{i_{s^{\prime}}}}\Phi^{(i_{s^{\prime}})}\right)=0.$
(8.7)

for any pair of $i_{s}$ and $i_{s^{\prime}}$, $1\leq i_{s},i_{s^{\prime}}\leq
n$. Then, due to associativity of the products (18), (3.9) and the definition
(8.2) of an $\mathcal{W}$-element powers it follows that

$\left(\delta^{k_{i_{s}}}_{m_{i_{s}}}\partial_{t}\Phi^{(i_{s})},\delta^{k_{i_{s^{\prime}}}}_{m_{i_{s^{\prime}}}}\left(\Phi^{(i_{s^{\prime}})}\right)^{k}\right)=0,\quad\left(\delta^{k_{i_{s}}}_{m_{i_{s}}}\partial_{t}\Phi^{(i_{s})},\left(\delta^{k_{i_{s^{\prime}}}}_{m_{i_{s^{\prime}}}}\Phi^{(i_{s^{\prime}})}\right)^{k}\right)=0.$
(8.8)

It is clear that if one of multiplicand in the product (18) is zero then the
product vanishes. Let us show that the invariant (1.1) is closed. Due to (8.7)
((8.8)

$\displaystyle\delta.\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right),$
$\displaystyle=\left((-1)^{k_{i}+1}\delta^{k_{i}+1}_{m_{i}-1}.\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)$
$\displaystyle+\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},(-1)^{k_{i^{\prime}}}\delta^{k_{i^{\prime}}}_{m_{i^{\prime}}}.\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)$
$\displaystyle+\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},(-1)^{k_{i^{\prime\prime}}}\delta^{k_{i^{\prime\prime}}}_{m_{i^{\prime\prime}}}.\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)=0,$

i.e., (1.1) is closed. Let us show non-vanishing property of (1.1). Indeed,
suppose
$\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)=0$.
Then there exists $\Gamma^{(i)}\in
C^{n}_{\mu}\left(V,\mathcal{W}^{(i)}\right)$, such that
$P^{(i)}_{(i,i,i^{\prime},i^{\prime\prime})}\delta^{k_{i}}_{m_{i}}\Phi^{(i)}=\left(\Gamma^{(i)},\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m-1},\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)$,
where $P^{(i)}_{(i,i,i^{\prime},i^{\prime\prime})}$ is the projection
$P^{(i)}_{(i,i,i^{\prime},i^{\prime\prime})}:\mathcal{W}^{(i)}\to\mathcal{W}^{(i,i,i^{\prime},i^{\prime\prime})}$.
Both sides of the last equalities should belong to the same chain-cochain
complex space. Indeed, $k_{i}+1=n+(m-1)(k_{i}+1)+\beta
k_{i^{\prime}}+kk_{i^{\prime\prime}}$, $m_{i}-1=\mu+(m-1)(m_{i}-1)+\beta
m_{i^{\prime}}+km_{i^{\prime\prime}}$. For a non-vanishing expression, $n$ or
$\mu$ should be negative. Then we obtain $(2-m)k_{i}-m+1-\beta
k_{i^{\prime}}-kk_{i^{\prime\prime}}<0$, and $(2-m)m_{i}+m-1-\beta
m_{i^{\prime}}-km_{i^{\prime\prime}}<0$. Now let us show that (1.1) is an
invariant, i.e., it does not depend on the choice of $\Phi^{(i)}\in
C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$. Substitute elements the
$\Phi^{(i)}$, $\Phi^{(i^{\prime})}$, $\Phi^{(i^{\prime\prime})}$ by elements
added by $\eta^{(i)}\in C^{k_{i}}_{m_{i}}\left(V,\mathcal{W}^{(i)}\right)$,
$\eta^{(i^{\prime})}\in
C^{k_{i^{\prime}}}_{m_{i^{\prime}}}\left(V,\mathcal{W}^{(i^{\prime})}\right)$,
$\eta^{(i^{\prime\prime})}\in
C^{k_{i^{\prime\prime}}}_{m_{i^{\prime\prime}}}\left(V,\mathcal{W}^{(i^{\prime\prime})}\right)$,
correspondingly. Since the multiple product is associative, we obtain

$\displaystyle\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}+\delta^{k_{i}}_{m_{i}}\eta^{(i)}\right)^{m},\partial_{t}\left(\Phi^{(i^{\prime})}+\eta^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}+\eta^{(i^{\prime\prime})}\right)^{k}\right)\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle=\sum\limits_{{j=0,\atop
j^{\prime}=0}}^{m,k}C^{j,j^{\prime}}_{m,k}\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{j},\left(\delta^{k_{i}}_{m_{i}}\eta^{(i)}\right)^{m-j},\partial_{t}\left(\Phi^{(i^{\prime})}+\eta^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k},\left(\eta^{(i^{\prime\prime})}\right)^{k-j^{\prime}}\right),\qquad$

where $C^{j,j^{\prime}}_{m,k}=\left({m\atop j}\right)\left({k\atop
j^{\prime}}\right)$. The expression above splits in two parts relative to
$\Phi^{(i)}$ and $\eta^{(i)}$.

$\displaystyle\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},\left(\partial_{t}\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)+\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m},\partial_{t}\left(\eta^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)\qquad\qquad\qquad$
$\displaystyle+\sum\limits_{{j=1,\atop
j^{\prime}=1}}^{m,k}C^{j,j^{\prime}}_{m,k}\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m-j},\left(\delta^{k_{i}}_{m_{i}}\eta^{(i)}\right)^{j},\partial_{t}\left(\Phi^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k-j^{\prime}},\left(\eta^{(i^{\prime\prime})}\right)^{j^{\prime}}\right)\qquad\qquad$
$\displaystyle+\sum\limits_{{j=1,\atop
j^{\prime}=1}}^{m,k}C^{j,j^{\prime}}_{m,k}\left(\left(\delta^{k_{i}}_{m_{i}}\Phi^{(i)}\right)^{m-j},\left(\delta^{k_{i}}_{m_{i}}\eta^{(i)}\right)^{j},\partial_{t}\left(\eta^{(i^{\prime})}\right)^{\beta},\left(\Phi^{(i^{\prime\prime})}\right)^{k-j^{\prime}},\left(\eta^{(i^{\prime\prime})}\right)^{j^{\prime}}\right).\qquad\qquad$

The terms except the first two vanish due to the mutual order condition of
required in the Theorem. Then one can see that the cohomology class of (1.1)
is preserved. Similarly we show that
$\left(\left(\delta^{1,i}_{2,i}\Phi^{(i)}\right)^{m}\right.$,
$\left(\partial_{t}\Phi^{(i^{\prime})}\right)$,
$\left.\left(\Phi^{(i^{\prime\prime})}\right)^{k}\right)$ and
$\left(\left(\delta^{0,i}_{3,i}\Lambda^{(i)}\right)^{m},\left(\partial_{t}\Lambda^{(i^{\prime})}\right)^{\beta},\left(\Lambda^{(i^{\prime\prime})}\right)^{k}\right)$,
are invariant, i.e., it does not depend on the choices of $\Phi^{(i_{s})}\in
C^{1,i_{s}}_{2,i_{s}}\left(V,\mathcal{W}^{(i_{s})}\right)$,
$\Lambda^{(i_{s})}\in C^{0,i_{s}}_{3,i_{s}}\left(V,\mathcal{W}^{(i)}\right)$,
with $i_{s}=i,i^{\prime},i^{\prime\prime}$, satisfying the transversality
condition (8.7) with the corresponding values of $i_{s}$, $i_{s^{\prime}}$. ∎

######  Remark 12.

In this paper we provide results concerning complex curves. They generalize to
the case of higher dimensional complex manifolds.

## Acknowledgment

The author is supported by the Institute of Mathematics, Academy of Sciences
of the Czech Republic (RVO 67985840).

##  9\. Appendix: vertex operator algebras and their modules

In this Appendix we recall basic properties of grading-restricted vertex
algebras [43] and their modules. We work over the base field $\mathbb{C}$ of
complex numbers.

###  9.1. Grading-restricted vertex algebras

######  Definition 32.

A vertex algebra $(V,Y_{V},\mathbf{1}_{V})$, [24, 47] is a $\mathbb{Z}$-graded
complex vector space $V=\coprod_{n\in\mathbb{Z}}\,V_{(n)}$, $\dim
V_{(n)}<\infty$, for each $n\in\mathbb{Z}$. It is endowed with the linear map
$Y_{V}:V\rightarrow{\rm End\;}(V)[[z,z^{-1}]]$, where $z$ is a formal
parameter, and a distinguished vector $\mathbf{1}_{V}\in V$. The evaluation of
$Y_{V}$ on $v\in V$ is called the vertex operator $Y_{V}(v)\equiv
Y_{V}(v,z)=\sum_{n\in\mathbb{Z}}v(n)z^{-n-1}$, with components
$(Y_{V}(v))_{n}=v(n)\in{\rm End\;}(V)$, where
$Y_{V}(v,z)\mathbf{1}_{V}=v+O(z)$.

######  Definition 33.

A grading-restricted vertex algebra satisfies the following conditions:

a) the grading-restriction condition: $V_{(n)}$ is finite dimensional for all
$n\in\mathbb{Z}$, and $V_{(n)}=0$ for $n\ll 0$;

b) the lower-truncation condition: for $u$, $v\in V$, $Y_{V}(u,z)v$ contains
only finitely many negative power terms, $Y_{V}(u,z)v\in V((z))$; (the space
of formal Laurent series in $z$ with coefficients in $V$);

c) the identity property: $Y_{V}(\mathbf{1}_{V},z)={\rm Id}_{V}$, the identity
operator on $V$;

d) the creation property: for $u\in V$, $Y_{V}(u,z)\mathbf{1}_{V}\in V[[z]]$,
$\lim_{z\to 0}Y_{V}(u,z)\mathbf{1}_{V}=u$;

e) the duality conditions: for $u_{1}$, $u_{2}$, $v\in V$, the dual vector
space $V_{(n)}^{*}$ to $V_{(n)}$, $v^{\prime}\in
V^{\prime}=\coprod_{n\in\mathbb{Z}}V_{(n)}^{*}$;

f) for the evaluation pairing $\langle\,.,.\rangle$, $V^{\prime}\otimes
V\to\mathbb{C}$, the series $\langle v^{\prime}$, $Y_{V}(u_{1},z_{1})$
$Y_{V}(u_{2},z_{2})v\rangle$, $|z_{1}|>|z_{2}|>0$, $\langle v^{\prime}$,
$Y_{V}(u_{2},z_{2})$ $Y_{V}(u_{1},z_{1})v\rangle$, $|z_{2}|>|z_{1}|>0$,
$\langle v^{\prime}$, $Y_{V}(Y_{V}(u_{1},z_{1}-z_{2})u_{2}$, $z_{2})v\rangle$,
$|z_{2}|>|z_{1}-z_{2}|>0$, are absolutely convergent in corresponding domains,
to a common rational function in $z_{1}$ and $z_{2}$ with the only possible
poles at $z_{1}=0=z_{2}$ and $z_{1}=z_{2}$;

g) the $L_{V}(0)$-bracket formula: let $L_{V}(0):V\to V$, be defined by
$L_{V}(0)v=nv$, $n=\mbox{\rm wt}(v)$, for $v\in V_{(n)}$. Then
$[L_{V}(0),Y_{V}(v,z)]=Y_{V}(L_{V}(0)v,z)+z\frac{d}{dz}Y_{V}(v,z)$, for $v\in
V$.

h) the $L_{V}(-1)$-derivative property: for for $v\in V$ the operator
$L_{V}(-1):V\to V$, defined by $L_{V}(-1)v=\mbox{\rm
Res}_{z}z^{-2}Y_{V}(v,z)\mathbf{1}_{V}=Y_{(-2)}(v)\mathbf{1}_{V}$, Then
$\frac{d}{dz}Y_{V}(u,z)=Y_{V}(L_{V}(-1)u,z)=[L_{V}(-1),Y_{V}(u,z)]$.

Recall [7] the following

######  Definition 34.

A grading-restricted vertex algebra $V$ is called conformal of central charge
$c\in\mathbb{C}$, if there exists a non-zero Virasoro vector $\omega\in
V_{(2)}$ such that the corresponding vertex operator
$Y_{V}(\omega,z)=\sum_{n\in\mathbb{Z}}L_{V}(n)z^{-n-2}$, is determined by
modes of Virasoro algebra $L_{V}(n):V\to V$ satisfying
$[L_{V}(m),L_{V}(n)]=(m-n)L_{V}(m+n)+\frac{c}{12}(m^{3}-m)\delta_{m+b,0}\;{\rm
Id_{V}}$.

######  Definition 35.

A quasi-conformal grading-restricted vertex algebra $V$ module $W$ vector $A$
is called primary of conformal dimension $\Delta(A)\in\mathbb{Z}_{+}$ if
$L_{W}(k)A=0$, $k>0$, $L_{W}(0)A=\Delta(A)A$.

###  9.2. Grading-restricted generalized $V$-modules

In this Subsection we describe the grading-restricted generalized $V$-module
for a grading-restricted vertex algebra $V$.

######  Definition 36.

A grading-restricted generalized $V$-module is a vector space $W$ with a
vertex operator map $Y_{W}:V\otimes W\to W[[z,z^{-1}]]$, $u\otimes w\mapsto
Y_{W}(u,w)\equiv Y_{W}(u,z)w=\sum_{n\in\mathbb{Z}}(Y_{W})_{n}(u,w)z^{-n-1}$,
and equipped with linear operators $L_{W}(0)$, $L_{W}(-1)$ on $W$ satisfying
the following conditions:

a) the grading-restriction condition: the vector space
$W=\coprod_{\alpha\in\mathbb{C}}W_{(\alpha)}$, is $\mathbb{C}$-graded, and
$W_{(\alpha)}=0$ for ${\rm Re}(\alpha)$ is sufficiently negative;

b) the lower-truncation condition: for $u\in V$, $w\in W$, $Y_{W}(u,z)w$
contains only finitely many negative power terms, i.e., $Y_{W}(u,z)w\in
W((z))$;

c) the identity property: $Y_{W}(\mathbf{1}_{V},z)={\rm Id}_{W}$, the identity
operator on $W$;

d) the duality property: for $u_{1}$, $u_{2}\in V$, $w\in W$, $W^{\prime}$ is
the dual $V$-module for $W$, $w^{\prime}\in
W^{\prime}=\coprod_{n\in\mathbb{Z}}W_{(n)}^{*}$.

e) for the bilinear pairing $\langle\,.,.\rangle$, the locality and
associativity properties require that the series

$\displaystyle\langle
w^{\prime},Y_{W}(u_{1},z_{1})Y_{W}(u_{2},z_{2})w\rangle,\;|z_{1}|>|z_{2}|>0,$
$\displaystyle\langle
w^{\prime},Y_{W}(u_{2},z_{2})Y_{W}(u_{1},z_{1})w\rangle,\;|z_{2}|>|z_{1}|>0,$
$\displaystyle\langle
w^{\prime},Y_{W}(Y_{V}(u_{1},z_{1}-z_{2})u_{2},z_{2})w\rangle,\;|z_{2}|>|z_{1}-z_{2}|>0,$
(9.1)

are absolutely convergent in $z_{1}$ and $z_{2}$ to a common rational function
on corresponding domains with the only possible poles at $z_{1}=0=z_{2}$ and
$z_{1}=z_{2}$.

f) the $L_{W}(0)$-bracket formula: for $v\in V$,

$[L_{W}(0),Y_{W}(v,z)]=Y_{W}(L_{V}(0)v,z)+z\frac{d}{dz}Y_{W}(v,z);$ (9.2)

g) $L_{W}(0)$-grading property: for $w\in W_{(\alpha)}$, there exists
$N\in\mathbb{Z}_{+}$ such that $(L_{W}(0)-\alpha)^{N}w=0$;

h) the $L_{W}(-1)$-derivative property: for $v\in V$,

$\frac{d}{dz}Y_{W}(u,z)=Y_{W}(L_{V}(-1)u,z)=[L_{W}(-1),Y_{W}(u,z)].$ (9.3)

For $z^{\prime}\in\mathbb{C}$, it follows from (9.3) that vertex operators
satisfy the translation property
$Y_{W}(u,z)=e^{-z^{\prime}L_{W}(-1)}Y_{W}(u,z+z^{\prime})e^{z^{\prime}L_{W}(-1)}$.
For $v\in V$, and $w\in W$, one defines the intertwining operator

$\displaystyle Y_{WV}^{W}:V\to W,\quad v\mapsto Y_{WV}^{W}(w,z)v,$ (9.4)
$\displaystyle Y_{WV}^{W}(w,z)v=e^{zL_{W}(-1)}Y_{W}(v,-z)w.$ (9.5)

With the grading operator $L_{W}{(0)}$, the conjugation property for
$a\in\mathbb{C}$ is

$a^{L_{W}(0)}\;Y_{W}(v,z)\;a^{-L_{W}(0)}=Y_{W}\left(a^{L_{W}(0)}v,az\right).$
(9.6)

###  9.3. Generators of Virasoro algebra and the group of automorphisms

In this Subsection we recall relations [7] between generators of Virasoro
algebra and the group of automorphisms on complex curves. An element of ${\rm
Aut}_{z}\;\mathcal{O}$ can be represented by the map
$z\mapsto\varrho=\varrho(z)$, in the form of the power series

$\varrho(z)=\sum\limits_{k\geq 1}a_{k}z^{k}.$ (9.7)

The series $\varrho(z)$ has also an exponential form

$\varrho(z)=\exp\left(\sum\limits_{k>-1}\beta_{k}\;z^{k+1}\partial_{z}\right)\left(\beta_{0}\right)^{z\partial_{z}}.z,$
(9.8)

where $\beta_{k}\in\mathbb{C}$, $k\geq 0$, is expressed via combinations of
the coefficients $a_{k}$, $k\geq 1$. In [47] one can find a representation of
Virasoro algebra modes in termrs of differential operators, i.e.,
$L_{W}(m)\mapsto-\zeta^{m+1}\partial_{\zeta}$, for $m\in\mathbb{Z}$. Expanding
(9.8) and comparing to (9.7), we arrive at a system of equations solved
recursively for all $\beta_{k}$. In [7], for $v\in V$, the following formula

$\displaystyle\left[L_{W}(n),Y_{W}(v,z)\right]=\sum_{m\geq-1}\frac{1}{(m+1)!}\left(\partial^{m+1}_{z}z^{m+1}\right)\;Y_{W}(L_{V}(m)v,z),$
(9.9)

was derived for a Virasoro generator commutation with a vertex operator. For a
vector field
$\beta(z)\partial_{z}=\sum_{n\geq-1}\beta_{n}z^{n+1}\partial_{z}$, which
belongs to local Lie algebra of ${\rm Aut}_{z}\;\mathcal{O}$, introduce the
operator $\beta=-\sum_{n\geq-1}\beta_{n}L_{W}(n)$. From the last formula it
follows

######  Lemma 10.

$\displaystyle\left[\beta,Y_{W}(v,z)\right]=\sum_{m\geq-1}\frac{1}{(m+1)!}\left(\partial^{m+1}_{z}\beta(z)\right)\;Y_{W}(L_{V}(m)v,z).$
(9.10)

In [7], the formula (9.10) is used to prove invariance of vertex operators
multiplied by conformal weight differentials.

######  Definition 37.

A grading-restricted vertex algebra $V$-module $W$ is called quasi-conformal
if it carries an action of local Lie algebra of ${\rm Aut}_{z}\;\mathcal{O}$,
and (9.10) is true for any $v\in V$; the element $L_{W}(-1)=-\partial_{z}$,
acts as the translation operator $T$, $L_{W}(0)=-z\partial_{z}$, is semi-
simple with $\mathbb{Z}$-eigenvalues; the Lie subalgebra of the positive part
of local Lie algebra of ${\rm Aut}_{z}\;\mathcal{O}^{(n)}$ acts locally
nilpotently.

###  9.4. Non-degenerate invariant bilinear pairing on $V$

In this Subsection we recall the notion of an invarian bilinear form on $V$
[24, 67, 74, 75, 76, 77]. Consider the transformation

$\gamma_{\lambda}=\left(\begin{array}[]{cc}0&\lambda\\\ -\lambda&0\\\
\end{array}\right):z\mapsto w=-\lambda^{2}/z.$ (9.11)

It is generated by $T_{\lambda}=\exp\left(\lambda
L_{V}{(-1)}\right)\;\exp\left(\lambda^{-1}L_{V}(1)\right)\;\exp\left(\lambda
L_{V}(-1)\right)$, where

$T_{\lambda}Y(u,z)T_{\lambda}^{-1}=Y\left(\exp\left(-\frac{z}{\lambda^{2}}L_{V}(1)\right)\left(-\frac{z}{\lambda}\right)^{-2L_{V}(0)}u,-\frac{\lambda^{2}}{z}\right).$

The adjoint vertex operator [47, 24] is defined by
$Y^{\dagger}(u,z)=\sum_{n\in\mathbb{Z}}u^{\dagger}(n)z^{-n-1}=T_{\lambda}Y(u,z)T_{\lambda}^{-1}$.
A bilinear pairing $\langle.,.\rangle_{\lambda}$ (see, e.g., [56, 67]) on $V$
is called invariant if for all $a$, $b$, $u\in V$, if

$\langle Y(u,z)a,b\rangle_{\lambda}=\langle
a,Y^{\dagger}(u,z)b\rangle_{\lambda},$ (9.12)

i.e., $\langle u(n)a,b\rangle_{\lambda}=\langle
a,u^{\dagger}(n)b\rangle_{\lambda}$. Therefore,

$\langle L_{V}(0)a,b\rangle_{\lambda}=\langle
a,L_{V}(0)b\rangle_{\lambda},\quad\langle a,b\rangle_{\lambda}=0,$ (9.13)

if $wt(a)\not=wt(b)$ for homogeneous $a$, $b$. In addition to that, $\langle
a,b\rangle_{\lambda}=\langle b,a\rangle_{\lambda}$. Up to normalization, if
$L_{V}(1)V_{1}=V_{0}$, the form $\langle.,.\rangle_{\lambda}$ is unique. For a
$V$-basis $\\{u^{\alpha}\\}$, the dual $V$ basis $\\{\overline{u}^{\beta}\\}$
is defined by $\langle
u^{\alpha},\overline{u}^{\beta}\rangle_{\lambda}=\delta^{\alpha\beta}$.
Similar considerations are applied to a vertex operator algebra module $W$.

##  10\. Appendix: Matrix elements for vertex algebras

In this Appendix we definitions and some properties of matrix elements for a
grading-restricted vertex algebra $V$ [43]. Let $W$ be a grading-restricted
generalized $V$-module. In this paper we consider elements
$\Phi(g;v_{1},z_{1};\ldots;v_{l},z_{l})\in\mathcal{W}$, $l\geq 0$, endowed
with an automorphism group ${\rm Aut}(V)$ elements $g$. Note that we assume
that in $\Phi(g;v_{1},z_{1};\ldots;v_{l},z_{l})$ an automorphism $g$ acts
first on elements of the corresponding module $W$. The $\overline{W}$-valued
function is given by

$\displaystyle
E^{(n)}_{W}\left(v_{1},z_{1};\ldots;v_{n},z_{n};\Phi(g;v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{l},z^{\prime}_{l})\right)$
(10.1)
$\displaystyle\qquad=E\left(\omega_{W}(v_{1},z_{1})\ldots\omega_{W}(v_{n},z_{n})\;\Phi(g;v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{l},z^{\prime}_{l})\right),$

where $\omega_{W}(dz^{\mbox{\rm wt}(v)}\otimes v,z)=Y_{W}(dz^{\mbox{\rm
wt}(v)}\otimes v,z)$, and an element $E(.)\in\overline{W}$ is given by
$\langle w^{\prime},E(g;\alpha)\rangle=R\langle w^{\prime},g.\alpha\rangle$,
$\alpha\in\overline{W}$ (here we use the notation of Subsection 3.3). Here a
group element $g$ is supposed to act both on $v^{\prime}_{j}$, $1\leq j\leq
l$, and $v_{i}$, $1\leq i\leq n$.

For a number $l$ of generalized vertex algebra $V$-modules $W^{(i)}$, denote
$\Phi^{(1,\ldots,l)}\in\mathcal{W}^{(1,\ldots,l)}_{z_{1},\ldots,z_{k_{1}+\ldots+k_{l}-r}}$.
Then we define similarly

$\displaystyle
E^{(m_{1},\ldots,m_{l})}_{W^{(1,\ldots,l)}}\left(v_{1},z_{1};\ldots;v_{m_{1}+\ldots+m_{l}},z_{m_{1}+\ldots+m_{l}};\qquad\qquad\qquad\qquad\qquad\qquad\right.$
$\displaystyle\Phi^{(1,\ldots,l)}(g_{1},\ldots,g_{l};v_{m_{1}+\ldots+m_{l}+1},z_{m_{1}+\ldots+m_{l}+1};\ldots;\qquad\qquad$
$\displaystyle\left.v_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}},z_{m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}})\right)\qquad$
$\displaystyle=\sum_{u\in
V_{(k)},\;k\in\mathbb{Z}}\;\widehat{\mathcal{R}}\prod_{i=1}^{l}\rho_{i}^{k}\langle
w^{\prime}_{i},E^{(m_{i})}_{W^{(i)}}\left(v_{1,i},x_{1,i};\ldots;v_{m_{i},i},x_{m_{i},i};\right.$
$\displaystyle\left.Y^{W^{(i)}}_{W^{(i)}V^{\prime}}\left(\Phi^{(i)}(g_{i};v_{m_{i}+1,i},x_{m_{i}+1,i};\ldots;v_{m_{i}+k_{i},i},x_{m_{i}+k_{i},i};u,\zeta_{1,i}),\zeta_{2,i}\right)f_{i}.\overline{u}\right)\rangle,\quad$
(10.2)

where $v_{j}$, $z_{j}$, $1\leq j\leq m_{1}+\ldots+m_{l}+k_{1}+\ldots+k_{l}-r$
are vertex algebra elements and formal parameters for $\Phi^{(1,\ldots,l)}$,
and $v_{i^{\prime},i}$, $x_{i^{\prime},i}$, $1\leq i^{\prime}\leq k_{i}-r_{i}$
are vertex algebra elements and formal parameters of $\Phi^{(i)}$. The form of
(10) is inspired by the regularized transversal condition for
$\Phi^{(1,\ldots,l)}$. One defines also $E^{W;(n)}_{WV^{\prime}}$ $(\Phi(g$;
$v^{\prime}_{1}$, $z^{\prime}_{1}$; $\ldots$; $v^{\prime}_{l}$,
$z^{\prime}_{l})$; $v_{1}$, $z_{1}$; $\ldots$; $v_{n}$, $z_{n})$ $=$
$E^{(n)}_{W}(v_{1},z_{1};\ldots;v_{n},z_{n};\Phi(g;v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{l},z^{\prime}_{l}))$,
which is an element of $\overline{W}_{z_{1},\ldots,z_{n}}$. In addition to
that above, we define $\left(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}\right).\Phi:V^{\otimes
m+n}\to\overline{W}_{z_{1},\ldots,z_{m+n}}$,

$\displaystyle\left(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}\right).\Phi(g;v_{1},z_{1};\ldots;v_{m+n-1},z_{m+n-1})\qquad\qquad\qquad$
$\displaystyle=E\left(\Phi\left(g;E^{(l_{1})}_{V;\mathbf{1}}(v_{1},z_{1};\ldots;v_{l_{1}},z_{l_{1}});\ldots;\right.\right.\qquad\qquad\qquad\qquad$
$\displaystyle\left.\left.E^{(l_{n})}_{V;\mathbf{1}}(v_{l_{1}+\ldots+l_{n-1}+1},z_{l_{1}+\ldots+l_{n-1}+1};\ldots;v_{l_{1}+\ldots+l_{n-1}+l_{n}},z_{l_{1}+\ldots+l_{n-1}+l_{n}})\right)\right),$
(10.3)

and $E^{(m)}_{W}.\Phi:V^{\otimes
m+n}\to\overline{W}_{z_{1},\ldots,z_{m+n-1}}$, given by

$\displaystyle E^{(m)}_{W}.\Phi(g;v_{1},z_{1};\ldots;v_{m+n},z_{m+n})$
$\displaystyle\qquad=E\left(E^{(m)}_{W}\left(v_{1},z_{1};\ldots;v_{m},z_{m};\Phi(g;v_{m+1},z_{m+1};\ldots;v_{m+n},z_{m+n})\right)\right).$

For $l_{1}=\ldots=l_{i-1}=l_{i+1}=1$, $l_{i}=m-n-1$, $1\leq i\leq n$, by
$E^{(l_{i})}_{V;\;\mathbf{1}}.\Phi$ we denote
$(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}).\Phi$, (this notation is different that of
[43]). In [43] the algebra of $E$-operators was derived:

######  Lemma 11.

$\displaystyle
E^{(1)}_{W}.E^{(1)}_{W}\left(\Phi^{(i)}\right)=E^{(2)}_{V;\mathbf{1}}(j).\left(E^{(1)}_{W}.\Phi^{(i)}\right),$
$\displaystyle\sum_{j=1}^{n}(-1)^{j}E^{(1)}_{W}.\left({\rm
Ins}_{j}\left(E^{(2)}_{V;\mathbf{1}}(j).\right)\Phi^{(i)}\right)=-\sum_{i=2}^{n+1}(-1)^{i}{\rm
Ins}_{i}\left(E^{(2)}_{V;\mathbf{1}}(j).\right)\left(E^{(1)}_{W}.\Phi^{(i)}\right),$
$\displaystyle
E^{(1)}_{W}.\left(\sigma_{n+1,1,\dots,n}\left(E_{W}^{(1)}.\Phi^{(i)}\right)\right)=\sigma_{n+2,1,\dots,n+1}\left(E^{(1)}_{W}.\left(E^{(1)}_{W}.\Phi^{(i)}\right)\right),$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i}\sum_{j=1}^{i-2}(-1)^{j}{\rm
Ins}_{i}\left(E^{(2)}_{V;\mathbf{1}}(j).\right)\left({\rm
Ins}_{j}(E^{(2)}_{V;\mathbf{1}}(j).)\Phi^{(i)}\right)$
$\displaystyle\qquad=-\sum_{i=1}^{n+1}(-1)^{i}\sum_{j=i+2}^{n+1}(-1)^{j-1}{\rm
Ins}_{i}\left(E^{(2)}_{V;\mathbf{1}}(j).\right)\left({\rm
Ins}_{j}\left(E^{(2)}_{V\mathbf{1}}.\right)\Phi^{(i)}\right),$
$\displaystyle\left(\sigma_{n+1,1,\dots,n}{\rm
Ins}_{n+1}\left(E^{(2)}_{V;\mathbf{1}}(j).\right)\left(E^{(1)}_{W}.\Phi^{(i)}\right)\right)$
$\displaystyle\qquad\qquad=\sigma_{n+2,1,\dots,n+1}\left(E^{(1)}_{W}.\sigma_{n+1,1,\dots,n}\left(E^{(1)}_{W}.\Phi^{(i)}\right)\right).\hfill\qed$

Finally, we give here a proposition proven in in [43]:

######  Proposition 15.

For a regularized transversal $\Phi:V^{\otimes
n}\to\mathcal{W}_{z_{1},\ldots,z_{n}}$ to $m$ vertex operators, one has:

a) for $p\leq m$, $\Phi$ is regularized transversal with $p$ vertex operators
and for $p$, $q\in\mathbb{Z}_{+}$ such that $p+q\leq m$ and
$l_{1},\ldots,l_{n}\in\mathbb{Z}_{+}$ such that $l_{1}+\ldots+l_{n}=p+n$,
$(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}).\Phi$ and $E^{(p)}_{W}.\Phi$ are regularized
transversal with $q$ vertex operators.

b) for $p,q\in\mathbb{Z}_{+}$ such that $p+q\leq m$,
$l_{1},\ldots,l_{n}\in\mathbb{Z}_{+}$ such that $l_{1}+\ldots+l_{n}=p+n$ and
$k_{1},\ldots,k_{p+n}\in\mathbb{Z}_{+}$ such that
$k_{1}+\ldots+k_{p+n}=q+p+n$,
$(E^{(k_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(k_{p+n})}_{V;\;\mathbf{1}}).(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}})).\Phi=(E^{(k_{1}+\ldots+k_{l_{1}})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(k_{l_{1}+\ldots+l_{n-1}+1}+\ldots+k_{p+n})}_{V;\;\mathbf{1}}).\Phi$.

c) for $p$, $q\in\mathbb{Z}_{+}$ such that $p+q\leq m$ and
$l_{1},\ldots,l_{n}\in\mathbb{Z}_{+}$ such that $l_{1}+\ldots+l_{n}=p+n$,
$E^{(q)}_{W}.((E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}).\Phi)=(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\ldots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}).(E^{(q)}_{W}.\Phi)$.

d) for $p$, $q\in\mathbb{Z}_{+}$ such that $p+q\leq m$,
$E^{(p)}_{W}.(E^{(q)}_{W}.\Phi)=E^{(p+q)}_{W}.\Phi$.

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