H_{b}^{0,1}(S^{3})^{\otimes n-1}$. For $a_{0}\in H_{b}^{0,0}(S^{3})$ and
$\bar{a}_{i}\epsilon\in H_{b}^{0,1}(S^{3})$, we have

$\\{a_{0},\bar{a}_{1}\epsilon,\dots,\bar{a}_{n-1}\epsilon\\}_{\bar{\pi},n}=\sum_{\sigma\in
S_{n-1}}p\\{\dots
h\\{h\\{a_{0},\bar{a}_{\sigma(1)}\epsilon\\},\bar{a}_{\sigma(2)}\epsilon\\},\dots,\bar{a}_{\sigma(n-1)}\epsilon\\}.$
(6.28)

We define the following constant

$\displaystyle(\pi_{n})^{p,q;r,s}_{u_{1},v_{1},\dots,u_{n-1},v_{n-1}}:=$
(6.29)
$\displaystyle\prod_{i}\frac{(u_{i}+v_{i}+1)!}{u_{i}!v_{i}!}\times\operatorname{Tr}_{S^{3}}(w_{1}^{p}w_{2}^{q}p\\{\dots
h\\{h\\{w_{1}^{r}w_{2}^{s},\bar{w}_{1}^{u_{1}}\bar{w}_{2}^{v_{1}}\epsilon\\},\bar{w}_{1}^{u_{2}}\bar{w}_{2}^{v_{2}}\epsilon\\},\dots,\bar{w}_{1}^{u_{n-1}}\bar{w}_{2}^{v_{n-1}}\epsilon\\}).$

We have that $(\pi_{n})^{p,q;r,s}_{u_{1},v_{1},\dots,u_{n-1},v_{n-1}}$ is non
zero only when the following condition is hold

$u_{1}+\dots+u_{n-1}=p+r-2n+3,\quad v_{1}+\dots+v_{n-1}=q+s-2n+3.$ (6.30)

Then the $n$-th bracket $\\{...\\}_{\bar{\pi},n}$ on the subspace
$H_{b}^{0,0}(S^{3})\otimes H_{b}^{0,1}(S^{3})^{\otimes n-1}$ can be expressed
as follows

$\displaystyle\prod_{i}\frac{(u_{i}+v_{i}+1)!}{u_{i}!v_{i}!}\\{w_{1}^{r}w_{2}^{s},\bar{w}_{1}^{u_{1}}\bar{w}_{2}^{v_{1}}\epsilon,\dots,\bar{w}_{1}^{u_{n-1}}\bar{w}_{2}^{v_{n-1}}\epsilon\\}_{\bar{\pi},n}=$
(6.31) $\displaystyle\left(\sum_{\sigma\in
S_{n-1}}(\pi_{n})^{p,q;r,s}_{u_{\sigma(1)},v_{\sigma(1)},\dots,u_{\sigma(n-1)},v_{\sigma(n-1)}}\right)\frac{(p+q+1)}{p!q!}\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon,$

with $p$ and $q$ satisfying 6.30.

This expression of the higher brackets also leads us to an expansion of the
effective interaction of the KK theory of Poisson BF theory

$\displaystyle\sum_{n\geq 3}\sum_{p,q,r,s\geq
0}\sum_{\begin{subarray}{c}u_{1}+\dots u_{n-1}=p+r-2n+3\\\ v_{1}+\dots
v_{n-1}=q+s-2n+3\end{subarray}}\left(\sum_{\sigma\in
S_{n-1}}(\pi_{n})^{p,q;r,s}_{u_{\sigma(1)},v_{\sigma(1)},\dots,u_{\sigma(n-1)},v_{\sigma(n-1)}}\right)$
(6.32)
$\displaystyle\left(\frac{1}{2(n-2)!}\int\mathbf{H}[p,q]\mathbf{H}[r,s]\mathbf{w}[u_{1},v_{1}]\tilde{\mathbf{w}}[u_{2},v_{2}]\dots\tilde{\mathbf{w}}[u_{n-1},v_{n-1}]\right.$
$\displaystyle\left.+\frac{1}{(n-1)!}\int\tilde{\mathbf{H}}[p,q]\mathbf{H}[r,s]\tilde{\mathbf{w}}[u_{1},v_{1}]\dots\tilde{\mathbf{w}}[u_{n-1},v_{n-1}]\right).$

We provide an explicit expression for the constant
$(\pi_{3})^{p,q;r,s}_{u_{1},v_{1},u_{2},v_{2}}$ in D.2.

##  7 Kodaira Spencer gravity

###  7.1 A brief review

##### algebraic structure of Polyvector fields

The closed-string sector of the topological B-model can be described by a
quantum field theory on space time called the Kodaira-Spencer theory of
gravity [39]. This theory is generalized in [40] by turning on gravitational
descendants and is called BCOV theory. To introduce this theory, we first
introduce the space of polyvector fields on a Calabi-Yau three-fold $X$. Let

$\mathrm{PV}^{j,i}(X)=\Omega^{0,i}(X,\wedge^{j}TX)$ (7.1)

be the bundle of $(0,i)$ forms with coefficients in the $j$-th exterior power
of the holomorphic tangent bundle of $X$. We call them poly-vector fields on
$X$. The anti-holomorphic Dolbeault differential $\bar{\partial}$ on forms
extended to polyvector fields and gives us a differential

$\bar{\partial}:\mathrm{PV}^{j,i}(X)\to\mathrm{PV}^{j,i+1}(X).$ (7.2)

Wedge product together with the differential $\bar{\partial}$ makes
$\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)$
a differential graded commutative algebra.

Next we can identify $\mathrm{PV}^{j,i}(X)$ with $\Omega^{3-j,i}$ via
contracting poly-vector fields with the holomorphic volume form $\Omega_{X}$
on $X$.

$\displaystyle\mathrm{PV}^{j,i}(X)$ $\displaystyle\cong\Omega^{3-j,i},$ (7.3)
$\displaystyle\alpha$ $\displaystyle\mapsto\alpha\vee\Omega_{X}.$

The differential $\partial$ on forms
$\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}$
defines an operator on
$\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)$
via the above isomorphism

$\partial:\mathrm{PV}^{j,i}(X)\to\mathrm{PV}^{j-1,i}(X).$ (7.4)

The $\partial$ operator on polyvector fields is not a derivation with respect
to the wedge product. The failure of it being a derivation is measured by the
following bracket

$\\{\alpha,\beta\\}:=\partial(\alpha\wedge\beta)-(\partial\alpha)\wedge\beta-(-1)^{|\alpha|}\alpha\wedge(\partial\beta),$
(7.5)

which coincides with the Schouten-Nijenhuis bracket on polyvector fields (up
to a sign). The fundamental algebraic structures of polyvector fields on
Calabi-Yau geometry can be summarized by saying that the tuple
$\\{\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X),\bar{\partial},\wedge,\partial,\\{-,-\\}\\}$
defines a differential graded Gerstenhaber-Batalin-Vilkovisky algebra. The BV
theory of Kodaira-Spencer gravity, or BCOV theory, is built on this algebraic
structure.

In a local coordinate, the algebra of polyvector fields can be written as

$\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{C}^{3})=C^{\infty}(\mathbb{C}^{3})[\eta^{i},d\bar{z}_{i}].$
(7.6)

The differential $\partial$ can be given by the expressions

$\partial=\sum_{i=1}^{3}\frac{\partial}{\partial\eta^{i}}\frac{\partial}{\partial
z_{i}}.$ (7.7)

The bracket can be given by the expression

$\\{\alpha,\beta\\}=\sum_{i=1}^{3}\frac{\partial\alpha}{\partial\eta^{i}}\frac{\partial\beta}{\partial
z_{i}}+(-1)^{|\alpha|}\frac{\partial\alpha}{\partial
z_{i}}\frac{\partial\beta}{\partial\eta^{i}}.$ (7.8)

##### Kodaira-Spencer theory

In the original formulation, Kodaira-Spencer theory has BV fields described by
divergence free polyvector fields

$\mu\in\ker\partial\subset\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)[2].$
(7.9)

The Lagrangian takes the form

$\frac{1}{2}\int_{X}(\partial^{-1}\mu)\wedge\bar{\partial}\mu+\frac{1}{6}\int_{X}\mu\wedge\mu\wedge\mu.$
(7.10)

The equation of motion can be obtained by varying the action functional, which
is

$\bar{\partial}\mu+\frac{1}{2}\\{\mu,\mu\\}=0.$ (7.11)

One can see that the BV formalism of Kodaira-Spencer theory is encoded in the
dg Lie algebra $(\ker\partial,\bar{\partial},\\{-,-\\})$, together with an
ill-defined pairing $\int_{X}\partial^{-1}\otimes\mathrm{Id}$. Though the
kinetic term for Kodaira-Spencer theory is ill-defined, this theory has a
well-defined propagator which allows one to analyze the theory in perturbation
theory.

There is a consistent truncation of Kodaira-Spencer theory by simply dropping
the fields in $\mathrm{PV}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}$ and
$\mathrm{PV}^{2,\mathbin{\vbox{\hbox{{$\bullet$}}}}}$. We are left with fields
in
$\ker\partial\subset\mathrm{PV}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)[2]$
and action functional of the same form as 7.10. This theory is called the
$(1,0)$ Kodaira-Spencer theory, or type I Kodaira Spencer gravity because its
relationship to the ordinary B-model is similar to the relationship of
physical type I string theory to the IIB string theory [41]. It is a pleasant
feature shown in [7] that the full Kodaira-Spencer theory is equivalent to an
Abelian Chern-Simons theory valued in $\Pi\mathbb{C}^{2}$ coupled with the
$(1,0)$ Kodaira-Spencer theory. Therefore, for the analysis of KK reduction of
Kodaira-Spencer theory, we only need to focus on the fields in
$\mathrm{PV}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)$.

In [40], a different formulation of Kodaira-Spencer theory, called BCOV
theory, is proposed, where the constraint $\partial\mu=0$ is imposed
homologically. This theory produces a better behaved BV theory and admits
rigorous perturbation quantization [40]. However, in this paper, we still
proceed with the old formulation of Kodaira-Spencer gravity for simplicity.

###  7.2 Some useful definitions

The space of Polyvector fields on $\mathbb{C}^{3}\backslash\mathbb{C}$ is
given by

$\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{C}^{3}\backslash\mathbb{C})=\mathbb{C}^{\infty}(\mathbb{C}^{3}\backslash\mathbb{C})[\partial_{z},\partial_{z_{1}},\partial_{z_{2}},d\bar{z},d\bar{z}_{1},d\bar{z}_{2}].$
(7.12)

We extend the isomorphism 3.1 to an isomorphism

$K:\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{C}^{3}\backslash\mathbb{C})\to\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{C}\times\mathbb{R}_{>0})\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})[\partial_{z},\eta_{1},\eta_{2}],$
(7.13)

where we simply maps $\partial_{z_{i}}$ to $\eta_{i}$. Using this isomorphism,
the BV algebra structure on polyvector fields induces a BV algebra structure
on
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})[\partial_{z},\eta_{1},\eta_{2}]$.
For example, the BV operator is defined by $K\circ\partial\circ K^{-1}$.

By identifying the space of polyvector fields
$\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}$
with
$\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{Dol}$
using the volume form, we can further identify the above complex with
$\Omega^{(\mathbin{\vbox{\hbox{{$\bullet$}}}}),\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{3d}(\mathbb{C}\times\mathbb{R}_{>0})\otimes\Omega_{b}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$.
However, it will be more convenient to use the space
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{C}\times\mathbb{R}_{>0})\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})[\partial_{z},\eta_{1},\eta_{2}]$
when we deal with the open-closed coupling in later sections.

We are interested in the $SU(2)$ decomposition of the above complex. For this
purpose, we construct the following two $SU(2)$ invariant vector fields

$\displaystyle\eta$ $\displaystyle=\bar{w}_{2}\eta_{1}-\bar{w}_{1}\eta_{2},$
(7.14) $\displaystyle\hat{\eta}$ $\displaystyle=w_{1}\eta_{1}+w_{2}\eta_{2}.$

Note that $\eta$ is the basis of the bundle $T^{\mathbb{C}}S^{3}\cap
T^{(1,0)}\mathbb{C}^{2}$ where $\mathbb{C}^{2}$ is the ambient space of
$S^{3}$. The vector fields $\partial_{z}$, $\eta$ and $\hat{\eta}$ generate
the whole polyvector fields because

$\displaystyle\eta_{1}$ $\displaystyle=w_{2}\eta+\bar{w}_{1}\hat{\eta},$
(7.15) $\displaystyle\eta_{2}$
$\displaystyle=-w_{1}\eta-\bar{w}_{2}\hat{\eta},$

and

$\eta\wedge\hat{\eta}=\eta_{1}\wedge\eta_{2}.$ (7.16)

The vector fields $\eta$ and $\hat{\eta}$ act on the complex
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$
via the Schouten bracket. These actions are important because they determine
the interactions of the Kodaira-Spencer theory and its coupling with the
holomorphic Chern-Simons theory.

First, we analyze the action of the two vector fields $\eta$ and $\hat{\eta}$
on $\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$. This can be
obtained from the $SU(2)$ decomposition of
$\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$. Since both vector
fields are $SU(2)$ invariant, on each irreducible subspace they act as either
zero or scalar multiples of the identity onto an irreducible subspace of the
same $SU(2)$ representation. For $\eta$, we have

$\eta:\begin{array}[]{l}\mathcal{H}_{0,\bar{j}}\to 0\\\
\mathcal{H}_{j,\bar{j}}\overset{\simeq}{\to}\mathcal{H}_{j-\frac{1}{2},\bar{j}+\frac{1}{2}}\end{array}.$
(7.17)

The action of $\eta$ on
$\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$ is completely
characterized by its action on the highest weight vector of each space
$\mathcal{H}_{j,\bar{j}}$

$\eta(w_{1}^{2j}\bar{w}_{2}^{2\bar{j}})=2jw_{1}^{2j-1}\bar{w}_{2}^{2\bar{j}+1}.$
(7.18)

Using this vector $\eta$ we can define another action on
$\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$ by
$\partial(-\wedge\eta)$. We can check that $\partial(-\wedge\eta)=\eta(-)$.

We see that the operator $\eta$ has a very similar behavior as the homotopy
operator $h$ defined in Section 4.1 except that $\eta$ is of degree $0$. It is
useful to define a degree zero operator
$\tilde{h}:\oplus\mathcal{H}_{j,\bar{j}}\to\oplus\mathcal{H}_{j,\bar{j}}$ as
follows

$\tilde{h}(Y)=h(Y\epsilon),\quad\text{for any
}Y\in\oplus\mathcal{H}_{j,\bar{j}}.$ (7.19)

Then the operator $\eta$ and $\tilde{h}$ are related as follows

$\eta|_{\mathcal{H}_{j,\bar{j}}}=(2j)(2\bar{j}+1)\tilde{h}|_{\mathcal{H}_{j,\bar{j}}}.$
(7.20)

This relation will be useful later. We can use it to reduce computation in
Kodaira-Spencer theory into computation in holomorphic Chern-Simons theory.

Now we consider the other $SU(2)$ invariant vector $\hat{\eta}$. The action of
$\hat{\eta}$ on $\Omega_{b}^{0,0}(S^{3})$ is given by

$\hat{\eta}:\begin{array}[]{l}\mathcal{H}_{j,j}\to 0\\\
\mathcal{H}_{j,\bar{j}}\overset{\simeq}{\to}\mathcal{H}_{j,\bar{j}}\quad\text{
for }j\neq\bar{j}\end{array}.$ (7.21)

It act on the highest weight vector of $\mathcal{H}_{j,\bar{j}}$ by the
constant
$\hat{\eta}(w_{1}^{2j}\bar{w}_{2}^{2\bar{j}})=(2j-2\bar{j})w_{1}^{2j}\bar{w}_{2}^{2\bar{j}}$.
Therefore we have

$\hat{\eta}|_{\mathcal{H}_{j,\bar{j}}}=(2j-2\bar{j})\mathrm{Id}_{\mathcal{H}_{j,\bar{j}}}.$
(7.22)

We can also consider the operation defined by $\partial(-\wedge\hat{\eta})$.
Unlike $\eta$, the operations $\partial(-\wedge\hat{\eta})\neq\hat{\eta}(-)$.
We find that

$\displaystyle\partial(-\wedge\hat{\eta})|_{\mathcal{H}_{j,\bar{j}}}=(2j-2\bar{j}+2)\mathrm{Id}_{\mathcal{H}_{j,\bar{j}}},$
(7.23)
$\displaystyle\partial(-\wedge\hat{\eta})|_{\mathcal{H}_{j,\bar{j}}\epsilon}=(2j-2\bar{j})\mathrm{Id}_{\mathcal{H}_{j,\bar{j}}}.$

In fact, the action of $\hat{\eta}$ on
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}$ is also nontrivial. For
example, $\hat{\eta}(t)$ is computed by
$(z_{1}\partial_{z_{1}}+z_{2}\partial_{z_{2}})(r)=\frac{1}{2}r$. We have

$\hat{\eta}(f(t))=\frac{1}{2}t\partial_{t}f(t).$ (7.24)

This means that $\hat{\eta}$ will have a nontrivial action on the topological
$t$ direction. Terms like this will be annoying in our later formulation of
the KK theory. However, we can proceed as we deal with the Poisson BF theory.
We first pass to the $\hat{d}$ cohomology without introducing any higher
product. By passing to the $\hat{d}$ cohomology, the action of $\eta$ and
$\hat{\eta}$ on $\mathcal{O}_{\mathbb{C}}$ is trivial. As a consequence, we
can forget the action of $\eta$ and $\hat{\eta}$ on
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}$ and only consider their
action on $\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$. This
greatly simplifies our later analysis of the KK theory.

###  7.3 KK reduction

It was shown in [7] that the $PV^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}$ and
$PV^{2,\mathbin{\vbox{\hbox{{$\bullet$}}}}}$ part of Kodaira-Spencer fields
can be reformulated as a (super) abelian holomorphic Chern-Simons theory. So
it suffices to focus on the $PV^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}$
fields. Kodaira-Spencer theory is more subtle than our previous examples
because fields are subject to the constraint $\partial\bm{\mu}=0$. This
equation cut out a subspace of
$\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})\partial_{z}\oplus\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})\eta\oplus\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})\hat{\eta}$
that forms the tower of KK fields. One can proceed as in our previous
examples. The Dolbeault differential and the Schouten bracket of polyvector
fields defined a $L_{\infty}$ structure on the cohomology of the subspace
subject to the $\partial$ constraint. However, this approach will end up with
cumbersome notation. To simplify the discussion, we directly write down the KK
fields that solve the $\partial$ constraint. We find the following fields

$\displaystyle\bm{\beta}=$ $\displaystyle\sum_{p+q\geq
1}\frac{(p+q+1)!}{p!q!}(\bm{\beta}[p,q]\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon\partial_{z}+\frac{1}{p+q}\partial_{z}\bm{\beta}[p,q]\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon\hat{\eta}),$
(7.25) $\displaystyle\bm{\alpha}=$
$\displaystyle(p+q)\bm{\alpha}[p,q]w_{1}^{p}w_{2}^{q}\eta-\sum_{p+q\geq
1}\bm{\alpha}[p,q]\eta(w_{1}^{p}w_{2}^{q})\hat{\eta}.$

Note that the KK tower start from the modes $p+q=1$.

We can check that both of the fields are in the $\bar{\partial}_{b}$
cohomology and the kernel of $\partial$. The corresponding action functional
can be obtained from the Kodaira-Spencer action functional as follows

$\int\operatorname{Tr}_{S^{3}}\left((\partial^{-1}\bm{\beta})\hat{d}\bm{\alpha}+\bm{\beta}\bm{\alpha}\bm{\alpha}\right).$
(7.26)

With the help of the decomposition into KK towers, we can explicit compute the
$\partial^{-1}$ operation on the fields:

$\displaystyle\partial^{-1}\bm{\beta}=$
$\displaystyle\frac{(p+q+1)!}{p!q!}(\frac{1}{p+q}\bm{\beta}[p,q]\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon\partial_{z}\wedge\hat{\eta}),$
(7.27) $\displaystyle\partial^{-1}\bm{\alpha}=$
$\displaystyle\bm{\alpha}[p,q]w_{1}^{p}w_{2}^{q}\eta\wedge\hat{\eta}.$

This feature displays the major advantage of working with the KK theory. The
KK theory utilizes the $SU(2)$ action on fields that allows us to solve the
$\partial^{-1}$ operation, which is annoying in the analysis of the original
$6d$ theory. We can expand the action functional explicitly in terms of the KK
modes as follows

$\sum\int\bm{\beta}[p,q]\hat{d}\bm{\alpha}[p,q]+\sum(ps-
qr)\int\bm{\alpha}[p,q]\bm{\alpha}[r,s]\bm{\beta}[p+r-1,q+s-1].$ (7.28)

The coefficient in front of the quadratic term is computed by

$\frac{(p+q+1)!}{p!q!}\operatorname{Tr}_{S^{3}}(\bar{w}_{1}^{p}\bar{w}_{2}^{q}w_{1}^{p}w_{2}^{q})=1.$
(7.29)

The coefficient in front of the cubic term is computed by

$\displaystyle\frac{(p+q+s+r-1)!}{(p+r-1)!(s+m-1)!}\operatorname{Tr}_{S^{3}}(\eta(w_{1}^{p}w_{2}^{q})\hat{\eta}(w_{1}^{r}w_{2}^{s})\bar{w}_{1}^{p+r-1}\bar{w}_{2}^{q+s-1})-(p\leftrightarrow
rq\leftrightarrow s)$ (7.30) $\displaystyle=(ps-qr).$

We see that the action functional takes the same form as the $3d$ holomorphic
topological theory. As we will see later, this feature allows us to easily
compute the boundary chiral algebra.

The coefficient in front of the cubic term is the same as the structure
constant of the Lie algebra $\mathrm{Ham}(\mathbb{C}^{2})$. Therefore, this
part of Kodaira-Spencer theory is the same as the holomorphic twist of $3d$
$\mathcal{N}=2$ theory associated to a vector multiplet valued in
$\mathrm{Ham}(\mathbb{C}^{2})$. Of course, higher structures exist and should
give rise to higher order interaction terms. We briefly discuss them in the
next section.

Similarly, we have the following fields

$\displaystyle\tilde{\bm{\beta}}=$ $\displaystyle\sum_{p,q\geq
0}\frac{(p+q+1)!}{p!q!}\tilde{\bm{\beta}}[p,q]\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon\eta,$
(7.31) $\displaystyle\tilde{\bm{\alpha}}=$ $\displaystyle\sum_{p,q\geq
0}\partial_{z}\tilde{\bm{\alpha}}[p,q]w_{1}^{p}w_{2}^{q}\hat{\eta}-(p+q+2)\tilde{\bm{\alpha}}[p,q]w_{1}^{p}w_{2}^{q}\partial_{z}.$

This KK tower starts from the modes $p,q=0$. The corresponding action
functional is given by

$\int(\partial^{-1}\tilde{\bm{\beta}})\hat{d}\tilde{\bm{\alpha}}+\tilde{\bm{\beta}}\tilde{\bm{\alpha}}\tilde{\bm{\alpha}}.$
(7.32)

We can compute the $\partial^{-1}$ operation on the fields:

$\displaystyle\partial^{-1}\tilde{\bm{\beta}}=$
$\displaystyle\frac{(p+q+1)!}{p!q!}(\frac{1}{p+q+2}\tilde{\bm{\beta}}[p,q]\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon\eta\wedge\hat{\eta}),$
(7.33) $\displaystyle\partial^{-1}\tilde{\bm{\alpha}}=$
$\displaystyle\tilde{\bm{\alpha}}[p,q]w_{1}^{p}w_{2}^{q}\partial_{z}\wedge\hat{\eta}.$

This allows us to expand the action functional explicitly as follows

$\displaystyle\sum\int\tilde{\bm{\beta}}[p,q]\hat{d}\tilde{\bm{\alpha}}[p,q]+(p+q+2)\int\bm{\beta}[p+r,q+s]\bm{\alpha}[p,q]\partial_{z}\bm{\alpha}[r,s]$
(7.34)
$\displaystyle-(r+s+2)\int\bm{\beta}[p+r,q+s]\partial_{z}\bm{\alpha}[p,q]\bm{\alpha}[r,s].$

Due to the $\partial_{z}$ that is present in the action functional, this
theory does not come from any $3d$ $\mathcal{N}=2$ theory. However, it still
shares many similar features with the holomorphic twisted theory and can be
analyzed in a similar fashion.

There are also coupling between the two part of Kodaira-Spencer fields, given
as follows

$\int\operatorname{Tr}_{S^{3}}\bm{\beta}\bm{\alpha}\tilde{\bm{\alpha}}+\tilde{\bm{\beta}}\bm{\alpha}\tilde{\bm{\alpha}}.$
(7.35)

We emphasize that the subspace of $PV^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}$
cut out by the constraint $\partial\bm{\mu}=0$ does not have a canonical
choice of basis. In other words, the KK theory of Kodaria-Spencer gravity can
be formulated by different representatives of KK fields that have different
action functional. The different choices of basis should be related by a
recombination of the KK modes. We made our above choice to make the boundary
chiral algebra compatible with the celestial chiral algebra studied in [13].

###  7.4 Higher order interaction

Like our previous examples, higher order interactions for the KK theory of
Kodaira-Spencer gravity exist and can be obtained from homotopy transfer.
However, we need to be careful here because the homotopy operator in this case
is not the homotopy operator $h$ defined in Section 4.1 anymore. The reason
for this is that the CR differential takes a slightly different form on
$\mathrm{PV}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},\mathbin{\vbox{\hbox{{$\bullet$}}}}}$
under the basis $\eta,\hat{\eta}$. We can compute the following

$\bar{\partial}_{b}(\partial_{z})=\bar{\partial}_{b}(\hat{\eta})=0,\;\;\bar{\partial}_{b}(\eta)=\epsilon\hat{\eta}.$
(7.36)

This implies that the CR differential $\bar{\partial}_{b}$ act on
$\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})\eta$ as follows

$\displaystyle\bar{\partial}_{b}:\mathcal{H}_{j,\bar{j}}\eta$
$\displaystyle\to\begin{array}[]{l
l}\mathcal{H}_{j,\bar{j}}\epsilon\hat{\eta}&\bar{j}=0,\\\
\mathcal{H}_{j+\frac{1}{2},\bar{j}-\frac{1}{2}}\epsilon\eta\oplus\mathcal{H}_{j,\bar{j}}\epsilon\hat{\eta}&\bar{j}\geq\frac{1}{2},\end{array}$
(7.37) $\displaystyle Y\eta$
$\displaystyle\mapsto\bar{\partial}_{b}(Y)\eta+Y\epsilon\hat{\eta}.$

Accordingly, we get a new homotopy operator $\mathds{h}$. Since
$\bar{\partial}_{b}$ act on $\Omega_{b}^{0,0}(S^{3})\partial_{z}$ and
$\Omega_{b}^{0,0}(S^{3})\hat{\eta}$ in the same way as it act on
$\Omega_{b}^{0,0}(S^{3})$, the new homotopy operator $\mathds{h}$ acts on
$\Omega_{b}^{0,1}(S^{3})\partial_{z}$ and $\Omega_{b}^{0,1}(S^{3})\hat{\eta}$
in the same way as $h$ act on them. The only difference is the action of
$\mathds{h}$ on $\Omega_{b}^{0,1}(S^{3})\eta$, given as follows:

$\displaystyle\mathds{h}:\mathcal{H}_{j,\bar{j}}\epsilon\eta$
$\displaystyle\to\begin{array}[]{l l}0&j=0,\\\
\mathcal{H}_{j-\frac{1}{2},\bar{j}+\frac{1}{2}}\epsilon\eta&j=\frac{1}{2},\\\
\mathcal{H}_{j-\frac{1}{2},\bar{j}+\frac{1}{2}}\epsilon\eta\oplus\mathcal{H}_{j-1,\bar{j}+1}\epsilon\hat{\eta}&j\geq
1,\end{array}$ (7.38) $\displaystyle Y\epsilon\eta$
$\displaystyle\mapsto\tilde{h}(Y)\eta-\tilde{h}^{2}(Y)\hat{\eta}.$

Then higher order interaction of the KK theory is computed by terms like

$\int\partial^{-1}\bm{\beta}\\{\bm{\alpha},\mathds{h}\\{\bm{\alpha},\bm{\beta}\\}\\}+\dots$
(7.39)

We leave it to future work to study these terms in detail.

###  7.5 Coupling to holomorphic Chern-Simons theory

BCOV theory can be coupled to the holomorphic Chern-Simons theory. The coupled
theory has a unique quantization [17] and gives us a concrete realization of
open-closed string field theory. In this paper, we are interested in the
leading order coupling between the Kodaria spencer fields and the holomorphic
Chern-Simons theory. In particular, the coupling between an elements
$\bm{\mu}=\bm{\mu}^{i}\partial_{z_{i}}\in\mathrm{PV}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)\cap\ker\partial$
and the holomorphic Chern-Simons is given by

$\frac{1}{2}\int\Omega_{X}\operatorname{Tr}(\boldsymbol{\EuScript{A}}\wedge\bm{\mu}^{i}(\partial_{z_{i}}\boldsymbol{\EuScript{A}})).$
(7.40)

We are interested in the coupling after the KK reduction. If we substitute
$\bm{\mu}$ with $\bm{\alpha}$ and $\bm{\beta}$ in the above coupling, we get
the following terms

$\displaystyle\frac{1}{2}\int\operatorname{Tr}(\bm{\beta}[p+r,q+s]\mathbf{A}[p,q]\partial_{z}\mathbf{A}[r,s])+\frac{1}{2}\frac{r+s}{p+q+r+s}\int\operatorname{Tr}(\partial_{z}\bm{\beta}[p+r,q+s]\mathbf{A}[p,q]\mathbf{A}[r,s])$
(7.41) $\displaystyle+\frac{1}{2}(ps-
qr)\int\operatorname{Tr}(\bm{\alpha}[r,s]\mathbf{B}[p+r-1,q+s-1]\mathbf{A}[p,q])$

If we substitute $\bm{\mu}$ with $\tilde{\bm{\alpha}}$ and
$\tilde{\bm{\beta}}$, we get the following terms

$\displaystyle-\frac{1}{2}\frac{(ps-
qr)}{p+q+r+s}\int\operatorname{Tr}(\tilde{\bm{\beta}}[p+r-1,q+s-1]\mathbf{A}[p,q]\mathbf{A}[r,s])$
(7.42)
$\displaystyle\frac{1}{2}(p+q)\int\operatorname{Tr}(\partial_{z}\tilde{\bm{\alpha}}[r,s]\mathbf{B}[p+r,q+s]\mathbf{A}[p,q])-\frac{1}{2}(r+s+2)\int\operatorname{Tr}(\tilde{\bm{\alpha}}[r,s]\mathbf{B}[p+r,q+s]\partial_{z}\mathbf{A}[p,q]).$

Just like our previous examples, there are higher order interactions coming
from homotopy transfer. 666There are also higher order couplings between BCOV
theory and holomorphic Chern-Simons theory. They should also appear in the KK
theory but should be distinguished from the higher order interaction discussed
in this section that comes from the homotopy transfer of leading order
coupling. They can be computed by the same method of summing over trees as
before, replacing the $2$-brackets by the action of Kodaira-Spencer fields on
the Chern-Simons form. The simplest one is the following

$I_{OC}=\int\mathbf{B}[\mathbf{A},h(\bm{\beta}\mathbf{A})].$ (7.43)

Using the expansion 7.25 of the Kodaira-Spencer fields, we have

$\displaystyle I=$ $\displaystyle\sum_{\begin{subarray}{c}u+\tilde{u}=p+r-1\\\
v+\tilde{v}=q+s-1\end{subarray}}(m_{3})^{p,q;r,s}_{u,v}\int\operatorname{Tr}\mathbf{B}[\tilde{u},\tilde{v}]\left[\mathbf{A}[p,q],\left(\bm{\beta}[u,v]\partial_{z}\mathbf{A}[r,s]+\partial_{z}\bm{\beta}[u,v]\frac{r+s}{u+v}\mathbf{A}[r,s]\right)\right]$
(7.44)
$\displaystyle+(m_{3})^{r,s;p,q}_{u,v}\int\operatorname{Tr}\mathbf{B}[\tilde{u},\tilde{v}]\left[\left(\partial_{z}\mathbf{A}[p,q]\bm{\beta}[u,v]+\frac{p+q}{u+v}\mathbf{A}[p,q]\partial_{z}\bm{\beta}[u,v]\right),\mathbf{A}[r,s]\right].$

There are also coupling of the form
$\int\mathbf{B}[\mathbf{A},h(\bm{\alpha}\mathbf{B})]+\dots$. Finding their
explicit expansion requires us to compute expression of the form
$pM(\eta(w_{1}^{r}w_{2}^{s}),\hat{\eta}hM(w_{1}^{p}w_{2}^{q},\bar{w}_{1}^{u}\bar{w}_{2}^{v}\epsilon))$.
We leave them to future work.

###  7.6 Coupling to holomorphic BF theory

Kodaria-Spencer gravity can also be coupled to the holomorphic BF theory. The
coupling takes the same form as the coupling with holomorphic Chern-Simons
theory

$\frac{1}{2}\int\Omega_{X}\operatorname{Tr}(\boldsymbol{\EuScript{A}}\wedge\bm{\mu}^{i}(\partial_{z_{i}}\boldsymbol{\EuScript{A}})).$
(7.45)

The only difference is that $\boldsymbol{\EuScript{A}}$ is the field of
holomorphic BF theory now. According to 5.2, after KK reduction, the field
$\boldsymbol{\EuScript{A}}$ decompose into KK fields $\mathbf{A}[p,q]$ and
$\tilde{\mathbf{B}}[p,q]$. Since the coupling takes the same form as in the
last section, it is straightforward to write down the first-order coupling of
the KK fields. We have

$\displaystyle\frac{1}{2}\int\operatorname{Tr}(\bm{\beta}[p+r,q+s]\mathbf{A}[p,q]\partial_{z}\mathbf{A}[r,s])+\frac{1}{2}\frac{r+s}{p+q+r+s}\int\operatorname{Tr}(\partial_{z}\bm{\beta}[p+r,q+s]\mathbf{A}[p,q]\mathbf{A}[r,s])$
(7.46) $\displaystyle+\frac{1}{2}(ps-
qr)\int\operatorname{Tr}(\bm{\alpha}[r,s]\tilde{\mathbf{B}}[p+r-1,q+s-1]\mathbf{A}[p,q]).$

Similarly, we have the coupling with the
$\tilde{\bm{\alpha}},\tilde{\bm{\beta}}$ fields

$\displaystyle-\frac{1}{2}\frac{(ps-
qr)}{p+q+r+s}\int\operatorname{Tr}(\tilde{\bm{\beta}}[p+r-1,q+s-1]\mathbf{A}[r,s]\mathbf{A}[p,q])$
(7.47)
$\displaystyle\frac{1}{2}(p+q)\int\operatorname{Tr}(\partial_{z}\tilde{\bm{\alpha}}[r,s]\tilde{\mathbf{B}}[p+r,q+s]\mathbf{A}[p,q])-\frac{1}{2}(r+s+2)\int\operatorname{Tr}(\tilde{\bm{\alpha}}[r,s]\tilde{\mathbf{B}}[p+r,q+s]\partial_{z}\mathbf{A}[p,q]).$

Higher order couplings also exists and we leave them to future work.

##  8 Theories in deformed geometry

###  8.1 The deformed geoemtry

In the twisted holography setup, we place a stack of $N$ topological B-branes
wrapping the complex line $\mathbb{C}=\\{z_{i}=0\\}$ in $\mathbb{C}^{3}$. This
brane configuration sources non-trivial flux and induces gravitational
backreaction. The backreaction of such a stack of branes is analyzed in [7].
We briefly review the results here.

Solving the equation of motion of Kodaira-Spencer gravity with a localized
source on the brane gives us the following Beltrami differential

$\beta_{0}=\frac{2N}{8\pi^{2}}\frac{\bar{z}_{1}d\bar{z}_{2}-\bar{z}_{2}d\bar{z}_{1}}{(|z_{1}|^{2}+|z_{2}|^{2})^{2}}\partial_{z}.$
(8.1)

This Beltrami differential defines a new complex structure on
$\mathbb{C}^{3}\backslash\mathbb{C}$. Holomorphic functions of this new
complex structure satisfy

$(\bar{\partial}+\beta)F=0,\;$ (8.2)

where $\beta$ act on $F$ as a vector field. Solving this equation, we obtained
the new holomorphic coordinates, which are $z_{1},z_{2}$ and

$\displaystyle u_{1}$
$\displaystyle=z_{1}z-\frac{N\bar{z}_{2}}{|z_{1}|^{2}+|z_{2}|^{2}},$ (8.3)
$\displaystyle u_{2}$
$\displaystyle=z_{2}z+\frac{N\bar{z}_{1}}{|z_{1}|^{2}+|z_{2}|^{2}}.$

These coordinates are not independent. They satisfy

$u_{1}z_{1}-u_{1}z_{2}=N.$ (8.4)

This is the equation defining the deformed conifold $X_{N}$ as a submanifold
of $\mathbb{C}^{4}$. If we collect the coordinates into a single matrix

$g=\begin{pmatrix}z_{1}&z_{2}\\\ u_{1}&u_{2}.\end{pmatrix}$ (8.5)

Then the defining equation of the deformed conifold becomes

$\det g=N.$ (8.6)

Therefore the deformed geometry is also equivalent to $SL_{2}(\mathbb{C})$.

###  8.2 Holomorphic Chern-Simons in deformed geometry, I

In this section, we analyze the KK theory of holomorphic Chern-Simons in the
deformed geometry. One way to understand the effect of the new geometry on the
holomorphic Chern-Simons theory is through the open-closed coupling. Note that
new complex structure of $SL_{2}(\mathbb{C})$ is completely encoded in the
Beltrami differential
$\beta_{0}=\frac{2N}{8\pi^{2}}\frac{\bar{z}_{1}d\bar{z}_{2}-\bar{z}_{2}d\bar{z}_{1}}{(|z_{1}|^{2}+|z_{2}|^{2})^{2}}\partial_{z}$,
which can be regarded as a background value of the dynamical closed string
fields. The interaction of complex structure moduli with the open string field
is described by the coupling introduced in Section 7.5

$\int\Omega_{X}\operatorname{Tr}(\frac{1}{2}\boldsymbol{\EuScript{A}}\wedge\bm{\mu}^{i}(\partial_{z_{i}}\boldsymbol{\EuScript{A}})).$
(8.7)

Therefore, working in the new geometry is equivalent to turning on a
background Beltrami differential $\beta_{0}$ in the above open-closed
coupling.

Via the coordinate transform 3.1 we rewrite the Beltrami differential
$\beta_{0}$ as $N\epsilon\partial_{z}$, where we absorbed a constant factor
which can be restored from a rescaling. We see that by turning on this
background Beltrami differential, the open-closed coupling term gives us the
following:

$N\int\operatorname{Tr}(\bm{\mathcal{A}}\wedge\epsilon\partial_{z}\bm{\mathcal{A}}),$
(8.8)

where
$\bm{\mathcal{A}}\in\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\Omega_{b}^{0,0}(S^{3})\otimes\mathfrak{g}$
is the KK fields of holomorphic Chern-Simons theory before we pass to the
$\bar{\partial}_{b}$ cohomology. This term describes the deformed action
functional for holomorphic Chern-Simons theory in the $SL_{2}(\mathbb{C})$
geometry.

We would like to understand the corresponding deformation for the KK theory
after we pass to the $\bar{\partial}_{b}$ cohomology. A naive answer is that
we simply restrict the above term to the fields $\mathbf{A}$. This will
introduce a term
$N\int\operatorname{Tr}(\mathbf{A}[0,0]\epsilon\partial_{z}\mathbf{A}[0,0])$,
which is a Chern-Simons deformation to the zero KK modes. This is compatible
with the observation in [7]. However, by integrating out massive fields, there
are higher order corrections to this zero mode answer.

We can see this by analyzing the complex of KK fields. Note that adding the
deformation 8.8 is equivalent to add the differential $N\epsilon\partial_{z}$
to the complex
$(\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}+\bar{\partial}_{b})$.
Therefore, the new theory on deformed geometry is described by the following
complex

$(\Omega_{3d}\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}+\bar{\partial}_{b}+N\epsilon\partial_{z}).$
(8.9)

Due to the presence of the term $N\epsilon\partial_{z}$, new differential will
be generated after we take the $\bar{\partial}_{b}$ cohomology of the above
complex. It will be a laborious work to follow the standard procedure of
spectral sequence to obtain the right answer. Here, we use the homological
perturbation lemma (we provide more details of homological perturbation lemma
in C.4), which gives us the correct differential on
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes
H_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$ to make it quasi-
isomorphic to the above complex 8.9. We have

$(\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes
H_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}+D),$ (8.10)

where the differential $D$ is obtained by homological perturbation lemma,
given by the following formula

$D=(1-N\epsilon\partial_{z}h)^{-1}\epsilon\partial_{z}=N\epsilon\partial_{z}+N^{2}\epsilon\partial_{z}h\epsilon\partial_{z}+\dots$
(8.11)

Using the degree $0$ operator $\tilde{h}$ defined in 7.19, the above
expression can be simplified to

$D=\sum_{k\geq 1}N^{k}\epsilon\tilde{h}^{k-1}\partial_{z}^{k}.$ (8.12)

Given this new differential, we find the following deformed action functional

$I^{(1)}_{O}=\int\frac{1}{2}\operatorname{Tr}(\mathbf{A}D\mathbf{A})=\sum_{k\geq
1}I^{(1,k)}_{O},$ (8.13)

where

$I^{(1,k)}_{O}=\frac{1}{2}N^{k}\int\operatorname{Tr}(\mathbf{A}\epsilon\tilde{h}^{k-1}\partial_{z}^{k}\mathbf{A}).$
(8.14)

We can simplify the above expression by analyzing the operator
$\tilde{h}^{k}$. Note that the action of $\tilde{h}^{k}$ takes the following
form

$\tilde{h}^{k}:\begin{array}[]{l}\mathcal{H}_{j,\bar{j}}\to 0,\\\
\mathcal{H}_{j,\bar{j}}\overset{\simeq}{\to}\mathcal{H}_{j-\frac{k}{2},\bar{j}+\frac{k}{2}},\end{array}\begin{array}[]{l}\text{
for }j<\frac{k}{2},\\\ \text{ for }j\geq\frac{k}{2}.\end{array}$ (8.15)

When non-zero, $\tilde{h}^{k}$ acts as the following constant

$\tilde{h}^{k}|_{\mathcal{H}_{j,\bar{j}}}=\sqrt{\frac{(2j-k)!(2\bar{j})!}{(2j)!(2\bar{j}+k)!}}.$
(8.16)

We are interested in the action of $\tilde{h}^{k}$ on $\mathcal{H}_{k,0}$. It
is easy to find that

$\tilde{h}^{k}(e^{(k)}_{i})=\frac{1}{k!}\bar{e}^{(k)}_{i}.$ (8.17)

Transform this result into the unnormalized basis $\\{w_{1}^{p}w_{2}^{q}\\}$,
we obtain

$I^{(1,k)}_{O}=\sum_{p+q=k-1}N^{k}\frac{(-1)^{p}}{(k-1)!}\frac{p!q!}{k!}\frac{1}{2}\int\operatorname{Tr}(\mathbf{A}[p,q]dz\partial_{z}^{k}\mathbf{A}[q,p]).$
(8.18)

These interaction terms describe part of the deformation of the KK theory of
holomorphic Chern-Simons in the deformed geometry.

###  8.3 An alternative approach

In this section, we briefly comment on performing the KK reduction directly on
the deformed conifold. The deformed conifold has a $SU(2)_{R}$ symmetry by
matrix multiplication from the right. The quotient

$SL_{2}(\mathbb{C})\to SL_{2}(\mathbb{C})/SU(2)_{R}$ (8.19)

is precisely the hyperbolic space – the Euclidean version of $AdS_{3}$.

The quotient space is parameterized by coordinates $\rho=gg^{\dagger}$. We
have

$\rho=\begin{pmatrix}r^{2}&\bar{z}r^{2}\\\
zr^{2}&\frac{N^{2}}{16\pi^{4}}\frac{1}{r^{2}}+|z|^{2}r^{2}\end{pmatrix},$
(8.20)

where $r=(z_{1}\bar{z}_{1}+z_{2}\bar{z}_{2})^{\frac{1}{2}}$. Therefore, we can
also use the coordinates $(z,\bar{z},r)$ for the quotient.

KK reduction of field theory on $SL_{2}(\mathbb{C})$ amounts to analyzing the
Dolbeault complex

$(\Omega^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(SL_{2}(\mathbb{C})),\bar{\partial}).$
(8.21)

Recall that $SL_{2}(\mathbb{C})$ is obtained as a deformation of
$\mathbb{C}^{3}\backslash\mathbb{C}$. The spaces of functions (not necessarily
holomorphic) on them and the decomposition of the Dolbeault complexes under
the $SU(2)_{R}$ action are the same. The only difference is their complex
structure, encoded in the Dolbeault differential $\bar{\partial}$. Note that
the new complex structure of $SL_{2}(\mathbb{C})$ can be obtained by adding
the Beltrami differential 8.1 to the Dolbeault differential $\bar{\partial}$
of $\mathbb{C}^{3}\backslash\mathbb{C}$. Therefore, the Dolbeault complex on
$SL_{2}(\mathbb{C})$ is equivalent to the complex obtained by adding the
Beltrami differential 8.1 to the Dolbeault complex 3.10 of
$\mathbb{C}^{3}\backslash\mathbb{C}$. This is precisely the complex 8.9 we
find in the last section.

In summary, by performing KK reduction directly on $SL_{2}(\mathbb{C})$, we
will end up with the same answer as in the last section using open-closed
coupling. This is expected given the self consistency of the open-closed
string theory.

###  8.4 Holomorphic Chern-Simons in deformed geometry, II

In the last section, we discussed how the quadratic terms of the KK theory are
modified in the deformed geometry. These terms correspond to a deformed
differential of the complex of KK fields. In fact, all the products and higher
products constructed in Section 4.4 that build the action functional of KK
theory are deformed in the new geometry. For example, we commented on the
higher order open-closed coupling of the KK theory in Section 7.5. By letting
$\bm{\beta}[0,0]=N\epsilon\partial_{z}$ in the formula 7.44, we find a
deformed action of the form $\int BA\partial_{z}A$.

To correctly obtain all deformation, we need to use all the data in the
special deformation retract (SDR). Recall from Section 4.1 that the SDR we
used to build the product and higher product on CR cohomology is given by the
following

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to32.9pt{\pgfpicture\makeatletter\hbox{\hskip 28.7693pt\lower-14.45697pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.50003}{0.86601}{-0.86601}{0.50003}{-0.09999pt}{-5.2945pt}\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{3d}\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}+\bar{\partial}_{b})\overset{p}{\underset{i}{\rightleftarrows}}(\Omega_{3d}\otimes
H_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}).$ (8.22)

Using homological perturbation theory, we obtain the new SDR after adding
$N\epsilon\partial_{z}$ to the differential

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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{3d}\otimes\Omega_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}+\bar{\partial}_{b}+N\epsilon\partial_{z})\overset{p^{\prime}}{\underset{i^{\prime}}{\rightleftarrows}}(\Omega_{3d}\otimes
H_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3}),\hat{d}+D).$ (8.23)

The differential $D$ is given in the previous section. The other operations
$h^{\prime},p^{\prime},i^{\prime}$ are given by

$\displaystyle h^{\prime}$ $\displaystyle=\sum_{k\geq
0}(N^{k}\tilde{h}^{k}\partial_{z}^{k})h,$ (8.24) $\displaystyle p^{\prime}$
$\displaystyle=p(\sum_{k\geq 0}N^{k}\tilde{h}^{k}\partial_{z}^{k}),$
$\displaystyle i^{\prime}$ $\displaystyle=\sum_{k\geq
0}(N^{k}\tilde{h}^{k}\partial_{z}^{k})i.$

The new $A_{\infty}$ structure on $\Omega_{3d}\otimes
H_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$ is given by the homotopy
transfer theorem. They can be computed by the same method as in Section 4.3,
but now with the new maps $i^{\prime},p^{\prime},h^{\prime}$.

For instance, the deformed product $m_{2}^{\prime}$ is defined by
$m_{2}^{\prime}(-,-)=p^{\prime}\circ M(i^{\prime}(-),i^{\prime}(-))$. For the
purpose of obtaining the corresponding deformed action, it suffices to compute
it on $(\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{3d}\otimes
H^{0,0}_{CR}(S^{3}))^{\otimes 2}$. In this case, $m_{2}^{\prime}$ is
simplified to

$m_{2}^{\prime}(f_{1}Y_{1},f_{2}Y_{2})=\sum_{k_{1},k_{2}\geq
0}N^{k_{1}+k_{2}}(\partial_{z}^{k_{1}}f_{1})(\partial_{z}^{k_{2}}f_{2})pM(\tilde{h}^{k_{1}}Y_{1},\tilde{h}^{k_{2}}Y_{2}),\quad
f_{i}\in\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{3d},Y_{i}\in
H^{0,0}_{CR}(S^{3}).$ (8.25)

We denote the map
$m_{2}^{(k_{1},k_{2})}(Y_{1},Y_{2})=pM(\tilde{h}^{k_{1}}Y_{1},\tilde{h}^{k_{2}}Y_{2})$,
then

$m_{2}^{\prime}(f_{1}Y_{1},f_{2}Y_{2})=\sum_{k_{1},k_{2}\geq
0}N^{k_{1}+k_{2}}(\partial_{z}^{k_{1}}f_{1})(\partial_{z}^{k_{2}}f_{2})m_{2}^{(k_{1},k_{2})}(Y_{1},Y_{2}).$
(8.26)

We have that $m_{2}^{(0,0)}=m_{2}$ by definition. We illustrate the
computation of $m_{2}^{(1,0)}$ and $m_{2}^{(0,1)}$. To compute
$m_{2}^{(1,0)}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})$, we first compute
$\tilde{h}(w_{1}^{p}w_{2}^{q})$. Using the definition 7.19 of $\tilde{h}$ and
expand the harmonic polynomial into polynomials of $w_{i},\bar{w}_{i}$, we
find

$\tilde{h}(w_{1}^{p}w_{2}^{q})=\frac{1}{p+q}(pw_{1}^{p-1}w_{2}^{q}\bar{w}_{2}-qw_{1}^{p}w_{2}^{q-1}\bar{w}_{1}).$
(8.27)

Therefore,

$m_{2}^{(1,0)}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})=\frac{ps-
qr}{(p+q)(p+q+r+s)}w_{1}^{p+r-1}w_{2}^{q+s-1}.$ (8.28)

Similarly, $m_{2}^{(0,1)}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})=-\frac{ps-
qr}{(r+s)(p+q+r+s)}w_{1}^{p+r-1}w_{2}^{q+s-1}$. Using these deformed products,
we obtain the first-order deformed action functional

$\displaystyle I^{(2,1)}_{O}=$ $\displaystyle N\sum_{p,q,r,s}\frac{ps-
qr}{(p+q+r+s)}\left(\frac{1}{p+q}\int\operatorname{Tr}\mathbf{B}[p+r-1,q+s-1][\partial_{z}\mathbf{A}[p,q],\mathbf{A}[r,s]]\right.$
(8.29)
$\displaystyle-\left.\frac{1}{r+s}\int\operatorname{Tr}\mathbf{B}[p+r-1,q+s-1][\mathbf{A}[p,q],\partial_{z}\mathbf{A}[r,s]]\right).$

In order to compute all other $m_{2}^{(k_{1},k_{2})}$, one can use our formula
4.29 for the product of two arbitrary $S^{3}$ harmonics. Here, we provide a
different method. Note that $m_{2}^{(k_{1},k_{2})}$ is compatible with the
$SU(2)$ action and send $\mathcal{H}_{j_{1},0}\otimes\mathcal{H}_{j_{2},0}$ to
$\mathcal{H}_{j_{1}+j_{2}-k_{1}-k_{2},0}$. Therefore the corresponding matrix
elements must be proportional to the Clebsch-Gordan coefficients in the
orthonormal basis. To determine the constant of proportionality it suffices to
compute one non-zero value of $m_{2}^{(k_{1},k_{2})}$. First, using 8.16 we
find that
$\tilde{h}^{k}(e^{(j)}_{m})=\sqrt{\frac{(2j-k)!}{(2j)!k!}}e^{(j-\frac{k}{2},\frac{k}{2})}_{m}$.
The easiest case of a non-zero product is the product between the highest
weight vector $e^{(j-\frac{k}{2},\frac{k}{2})}_{j}$ and the lowest weight
vector. We have

$\displaystyle
m_{2}^{(k_{1},k_{2})}(e^{(j_{1})}_{j_{1}},e^{(j_{2})}_{-j_{2}})=\sqrt{\frac{(2j_{1}-k_{1})!(2j_{2}-k_{2})!}{k_{1}!k_{2}!(2j_{1})!(2j_{2})!}}m_{2}(e^{(j_{1}-\frac{k_{1}}{2},\frac{k_{1}}{2})}_{j_{1}},e^{(j_{2}-\frac{k_{2}}{2},\frac{k_{2}}{2})}_{-j_{2}})$
(8.30)
$\displaystyle=(-1)^{k_{2}}\frac{\sqrt{(2j_{1}+1)(2j_{2}+1)}(2j_{1}-k_{1})!(2j_{2}-k_{2})!}{k_{1}!k_{2}!(2j_{1}+2j_{2}-k_{1}-k_{2}+1)!}\sqrt{\frac{(2j_{1}+2j_{2}-2k_{1}-2k_{2}+1)!}{(2j_{1}-k_{1}-k_{2})!(2j_{2}-k_{1}-k_{2})!}}e^{(j_{1}+j_{2}-k_{1}-k_{2})}_{j_{1}-j_{2}}.$

This implies

$\displaystyle
m_{2}^{(k_{1},k_{2})}(e^{(j_{1})}_{m_{1}},e^{(j_{2})}_{m_{2}})=$
$\displaystyle(-1)^{k_{2}}\sqrt{\frac{(2j_{1}+1)(2j_{2}+1)(2j_{1}+2j_{2}-2k_{1}-2k_{2})!}{(2j_{1}+2j_{2}-k_{1}-k_{2}+1)!(2j_{1})!(2j_{2})!(2j_{1}-k_{1}-k_{2})!(2j_{2}-k_{1}-k_{2})!}}$
(8.31)
$\displaystyle\times\frac{(2j_{1}-k_{1})!(2j_{2}-k_{2})!}{k_{1}!k_{2}!}C^{j_{1},j_{2},j_{1}+j_{2}-k_{1}-k_{2}}_{m_{1},m_{2},m_{1}+m_{2}}e^{(j_{1}+j_{2}-k_{1}-k_{2})}_{m_{1}+m_{2}}.$

Transforming this result into the unnormalized basis
$\\{w_{1}^{p}w_{2}^{q}\\}$ (see formula (2.18) in [42]), we have

$m_{2}^{(k_{1},k_{2})}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})=\frac{(-1)^{k_{2}}R_{k}(p,q,r,s)w_{1}^{p+r-k}w_{2}^{q+s-k}}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k+1]_{k}},\quad
k=k_{1}+k_{2},$ (8.32)

where we used the descending Pochhammer symbol $[a]_{n}=\frac{a!}{(a-n)!}$.
The constant $R_{k}(p,q,r,s)$ 777This constant also appears in the definition
of $W_{\infty}$ algebra [42], and we are following the notation of [25] is
defined by

$R_{k}(p,q,r,s):=\sum_{i}(-1)^{i}\binom{k}{i}[p]_{k-i}[q]_{i}[r]_{i}[s]_{k-i}.$
(8.33)

Using this deformed product, we can write down the deformed cubic action of
all orders:

$\displaystyle I^{(2,k)}_{O}=$ $\displaystyle
N^{k}\sum_{p,q,r,s}\sum_{k_{1}+k_{2}=k}\frac{(-1)^{k_{2}}R_{k}(p,q,r,s)}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k+1]_{k}}$
(8.34)
$\displaystyle\times\int\operatorname{Tr}\mathbf{B}[p+r-k,q+s-k][\partial_{z}^{k_{1}}\mathbf{A}[p,q],\partial_{z}^{k_{2}}\mathbf{A}[r,s]].$

We can check that by taking $k=1$ in the above formula, we reproduce the first
order deformed action $I^{(2,1)}_{O}$ in 8.29.

As we have mentioned, the whole $A_{\infty}$ algebra structure on
$\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes
H_{b}^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(S^{3})$ is deformed in the new
geometry. They can be computed by the same method presented in this paper.
Understanding the deformed $A_{\infty}$ structure allows us to have a complete
expansion of the action functional in the deformed $SL_{2}(\mathbb{C})$
geometry. This can, in principle, give us the full tree level boundary chiral
algebra.

###  8.5 Kodaira-Spencer gravity in deformed geometry

Analogous to the holomorphic Chern-Simons theory, we can also understand
Kodaira-Spencer gravity in deformed geometry as a deformation of the theory on
flat space. There are two differences in the analysis. Firstly, we add the
following differential to the complex:

$\\{N\epsilon\partial_{z},-\\}.$ (8.35)

Secondly, we use the homotopy operator $\mathds{h}$ defined in 7.38 in the
homological perturbation. After passing to the CR cohomology, the deformed
differential is given by

$\displaystyle\mathds{D}$
$\displaystyle=(1-\\{N\epsilon\partial_{z},-\\}\circ\mathds{h})^{-1}\circ\\{N\epsilon\partial_{z},-\\}$
(8.36) $\displaystyle=\sum_{k\geq 1}\mathds{D}^{(k)},$

where

$\mathds{D}^{(k)}=(\\{N\epsilon\partial_{z},-\\}\circ\mathds{h})^{k-1}\circ\\{N\epsilon\partial_{z},-\\}.$
(8.37)

This deformed differential corresponds to the following deformed quadratic
action

$\int\partial^{-1}\bm{\alpha}\mathds{D}\bm{\alpha}.$ (8.38)

To compute this deformed action, we need to compute the action of $\mathds{D}$
on the fields. We have

$\displaystyle\mathds{D}^{(k)}(f(z)Y(w_{i},\bar{w}_{i})\partial_{z})$
$\displaystyle=N^{k}\partial_{z}^{k}(f)\tilde{h}^{k-1}(Y)\epsilon\partial_{z},$
(8.39) $\displaystyle\mathds{D}^{(k)}(f(z)Y(w_{i},\bar{w}_{i})\hat{\eta})$
$\displaystyle=N^{k}\partial_{z}^{k}(f)\tilde{h}^{k-1}(Y)\epsilon\hat{\eta}-2kN^{k}\partial_{z}^{k-1}(f)\tilde{h}^{k-1}(Y)\epsilon\partial_{z},$
$\displaystyle\mathds{D}^{(k)}(f(z)Y(w_{i},\bar{w}_{i})\eta)$
$\displaystyle=N^{k}\partial_{z}^{k}(f)\tilde{h}^{k-1}(Y)\epsilon\eta-(k-1)N^{k}\partial^{k}_{z}(f)\tilde{h}^{k}(Y)\epsilon\hat{\eta}$
$\displaystyle+k(k-1)N^{k}\partial_{z}^{k-1}(f)\tilde{h}^{k}(Y)\partial_{z}.$

Thus we can compute

$\displaystyle\mathds{D}^{(k)}(\bm{\alpha})=\sum_{p,q}N^{k}(p+q)$
$\displaystyle\left(k\partial^{k}_{z}\bm{\alpha}[p,q]\tilde{h}^{k}(w_{1}^{p}w_{2}^{q})\epsilon\hat{\eta}-\partial^{k}_{z}\bm{\alpha}[p,q]\tilde{h}^{k-1}(w_{1}^{p}w_{2}^{q})\epsilon\eta\right.$
(8.40)
$\displaystyle\left.-k(k+1)\partial^{k-1}_{z}\bm{\alpha}[p,q]\tilde{h}^{k}(w_{1}^{p}w_{2}^{q})\epsilon\partial_{z}\right).$

Using the above formula, we can expand the deformed action
$\int\operatorname{Tr}\partial^{-1}\bm{\alpha}\mathds{D}\bm{\alpha}$. The
constant coefficient in front of the term
$\int\bm{\alpha}[q,p]dz\partial_{z}^{k-1}\bm{\alpha}[p,q]$ is computed by
$(p+q)k(k+1)\operatorname{Tr}_{S^{3}}(w_{1}^{q}w_{2}^{p}\tilde{h}^{k}(w_{1}^{p}w_{2}^{q}))$.
In fact, this is already computed in 8.17,8.18. Therefore, we have the
following deformed quadratic term

$I^{(1,k)}_{C}=\sum_{p+q=k}N^{k}(-1)^{q}\frac{p!q!}{((k-1)!)^{2}}\frac{1}{2}\int\bm{\alpha}[q,p]dz\partial_{z}^{k-1}\bm{\alpha}[p,q].$
(8.41)

Using the same method as we analyze the deformed cubic action for holomorphic
Chern-Simons, we obtain the following deformed cubic action for Kodaira-
Spencer gravity

$I^{(2,k)}_{C}=\sum_{k_{1}+k_{2}=k}\int\operatorname{Tr}\bm{\beta}(\mathds{h}\circ\mathds{D}^{(k_{1})}(\bm{\alpha}))(\mathds{h}\circ\mathds{D}^{(k_{2})}(\bm{\alpha})),$
(8.42)

where we assumed that $\mathds{h}\circ\mathds{D}^{(k)}=\mathrm{Id}$ for $k=0$.
Using the expansion 8.40 of $\mathds{D}^{(k)}(\bm{\alpha})$, we have

$\displaystyle\mathds{h}\circ\mathds{D}^{(k)}(\bm{\alpha})=\sum_{p,q}N^{k}(p+q)$
$\displaystyle\left((k+1)\partial^{k}_{z}\bm{\alpha}[p,q]\tilde{h}^{k+1}(w_{1}^{p}w_{2}^{q})\hat{\eta}-\partial^{k}_{z}\bm{\alpha}[p,q]\tilde{h}^{k}(w_{1}^{p}w_{2}^{q})\eta\right.$
(8.43)
$\displaystyle\left.-k(k+1)\partial^{k-1}_{z}\bm{\alpha}[p,q]\tilde{h}^{k+1}(w_{1}^{p}w_{2}^{q})\partial_{z}\right).$

We can expand the deformed cubic action using the above formula. The constant
coefficient in front of the term
$\int\bm{\beta}[p+r-k-1,q+s-k-1]\bm{\alpha}[p,q]\bm{\alpha}[r,s]$ can be
computed using the deformed product
$m_{2}^{(k_{1}+1,k_{2})}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})$ and
$m_{2}^{(k_{1},k_{2}+1)}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})$ obtained in
the last section. We have the following

$\displaystyle I^{(2,k)}_{C}=$
$\displaystyle\sum_{k_{1}+k_{2}=k}\frac{(-1)^{k_{2}}(p+q)(r+s)R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q+r+s-k]_{k+1}}\left(\frac{1}{[p+q]_{k_{1}+1}[r+s]_{k_{2}}}+\frac{1}{[p+q]_{k_{1}}[r+s]_{k_{2}+1}}\right)$
(8.44)
$\displaystyle\times\int\bm{\beta}[p+r-k-1,q+s-k-1]\partial_{z}^{k_{1}}\bm{\alpha}[p,q]\partial_{z}^{k_{2}}\bm{\alpha}[r,s].$

###  8.6 Open-closed coupling in deformed geometry

The open-closed coupling terms studied in Section 7.5 are also deformed in the
$SL_{2}(\mathbb{C})$ background. To understand the deformation, we first
understand the flat space coupling as coming from a bracket of the complex
$\ker\partial\cap\mathrm{PV}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)\oplus\Omega^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)$

$\displaystyle\\{-,-\\}:\mathrm{PV}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)\otimes\Omega^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)$
$\displaystyle\to\Omega^{0,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X),$ (8.45)
$\displaystyle\bm{\mu}^{i}\partial_{z_{i}}\otimes\bm{\gamma}$
$\displaystyle\mapsto\bm{\mu}^{i}\partial_{z_{i}}(\bm{\gamma}).$

######  Remark 8.1.

Only the leading order coupling of the
$\mathrm{PV}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}(X)$ part of Kodaira-
Spencer fields and the holomorphic Chern-Simons can be understood as a
$2$-bracket. Other parts of the coupling correspond to $L_{\infty}$ maps on
the space of fields [17].

Given this bracket that encodes (part of) the open-closed coupling, we can use
the same techniques as before to compute the deformed action. For example, we
have the following deformed coupling terms with fields
$\mathbf{B},\mathbf{A},\bm{\alpha}$

$I_{OC}^{(2,k)}[\mathbf{B},\mathbf{A},\bm{\alpha}]=\sum_{k_{1}+k_{2}=k}\int\operatorname{Tr}\mathbf{B}\left(\mathds{h}\circ\mathds{D}^{k_{1}}(\bm{\alpha})(N^{k_{2}}\tilde{h}^{k_{2}}\partial_{z}^{k_{2}}\mathbf{A})\right).$
(8.46)

Using the expansion 8.43 for
$\mathds{h}\circ\mathds{D}^{(k_{1})}(\bm{\alpha})$ and the deformed product
$m_{2}^{(k_{1}+1,k_{2})}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})$ and
$m_{2}^{(k_{1},k_{2}+1)}(w_{1}^{p}w_{2}^{q},w_{1}^{r}w_{2}^{s})$ obtained in
the previous section, we can easily expand the above deformed action as
follows

$\displaystyle I_{OC}^{(2,k)}[\mathbf{B},\mathbf{A},\bm{\alpha}]=$
$\displaystyle\sum_{k_{1}+k_{2}=k}\left(\frac{r+s-2k_{2}}{p+q-k_{1}}+1\right)\frac{(p+q)(-1)^{k_{2}}R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k]_{k+1}}$
(8.47)
$\displaystyle\times\int\operatorname{Tr}\mathbf{B}[p+r-k-1,q+s-k-1]\partial_{z}^{k_{1}}\bm{\alpha}[p,q]\partial_{z}^{k_{2}}\mathbf{A}[r,s]$
$\displaystyle+\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\ k_{1}\geq 1,k_{2}\leq
k-1\end{subarray}}\frac{(p+q)(-1)^{k_{2}}R_{k+1}(p,q,r,s)}{(k_{1}-1)!k_{2}![p+q]_{k_{1}+1}[r+s]_{k_{2}}[p+q+r+s-k]_{k+1}}$
$\displaystyle\times\int\operatorname{Tr}\mathbf{B}[p+r-k-1,q+s-k-1]\partial_{z}^{k_{1}-1}\bm{\alpha}[p,q]\partial_{z}^{k_{2}+1}\mathbf{A}[r,s].$

By a change of summation variable: $k_{1}\rightarrow k_{1}+1,k_{2}\rightarrow
k_{2}-1$, the second term in the above formula can be rewritten as

$\frac{-k_{2}(r+s-k_{2}+1)}{(p+q-k_{1})(p+q-k_{1}-1)}\frac{(p+q)(-1)^{k_{2}}R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k]_{k+1}}\int\operatorname{Tr}\mathbf{B}[...]\partial_{z}^{k_{1}}\bm{\alpha}[p,q]\partial_{z}^{k_{2}}\mathbf{A}[r,s].$
(8.48)

Since the coefficient in the above formula vanishes for $k_{2}=0$, the
summation range can be taken to be all $k_{1}+k_{2}=k$. Thus we can rewrite
the deformed coupling as follows

$\displaystyle I_{OC}^{(2,k)}[\mathbf{B},\mathbf{A},\bm{\alpha}]=$
$\displaystyle\sum_{k_{1}+k_{2}=k}\frac{(p+q-k-1)}{(p+q-1)}\frac{(p+q)(-1)^{k_{2}}R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q-2]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k-1]_{k}}$
(8.49)
$\displaystyle\times\int\operatorname{Tr}\mathbf{B}[p+r-k-1,q+s-k-1]\partial_{z}^{k_{1}}\bm{\alpha}[p,q]\partial_{z}^{k_{2}}\mathbf{A}[r,s].$

Similarly, we can write down the deformed coupling between the fields
$\bm{\beta},\mathbf{A}$

$\displaystyle I_{OC}^{(2,k)}[\mathbf{A},\mathbf{A},\bm{\beta}]=$
$\displaystyle\sum_{k_{1}+k_{2}=k}\frac{(-1)^{k_{2}}R_{k}(p,q,r,s)}{k_{!}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k+1]_{k}}$
(8.50)
$\displaystyle\times\left(\int\bm{\beta}[p+r-k,q+s-k]\partial_{z}^{k_{1}}\mathbf{A}[p,q]\partial_{z}^{k_{2}+1}\mathbf{A}[r,s]\right.$
$\displaystyle\left.+\frac{r+s-2k_{2}}{p+q+r+s-2k}\int\partial_{z}\bm{\beta}[p+r-k,q+s-k]\partial_{z}^{k_{1}}\mathbf{A}[p,q]\partial_{z}^{k_{2}}\mathbf{A}[r,s]\right).$

There are also coupling between the fields $\mathbf{A},\mathbf{B}$ and the
other parts of the Kodaira-Spencer fields. We leave them to future analysis.

###  8.7 Moyal-Weyl product and higher spin

In our previous analysis, we observe that the constant $R_{k}(p,q,r,s)$ 8.33
plays an important role in the construction of the cubic action for various KK
theories in the deformed geometry. In this section, we briefly recall its
connection with higher spin theory.

First, we recall the definition of the Moyal-Weyl product, which is a non-
commutative deformation of the commutative product of functions on
$\mathbb{R}^{2n}$ given by a constant Poisson tensor. In our case, we focus on
the Moyal-Weyl deformation of $\mathbb{C}[w_{1},w_{2}]$ equipped with the
Poisson tensor $\epsilon_{ij}\partial_{i}\partial_{j}$. The Moyal-Weyl product
$\star$ is given by

$f\star
g:=m(e^{\frac{\hbar}{2}\epsilon_{ij}\partial_{i}\otimes\partial_{j}}f\otimes
g),\quad f,g\in\mathbb{C}[w_{1},w_{2}],$ (8.51)

where $m$ is the commutative product map $m(a\otimes b)=ab$, and the
exponential is treated as a power series. Expanding the exponential to the
first order, one can find

$(w_{1}^{p}w_{2}^{q})\star(w_{1}^{r}w_{2}^{s})=w_{1}^{p+r}w_{2}^{q+s}+\frac{\hbar}{2}(ps-
qr)w_{1}^{p+r-1}w_{2}^{q+s-1}+\mathcal{O}(\hbar^{2}).$ (8.52)

It is straightforward to expand the exponential to all orders, and we find
that the Moyal-Weyl product is given by

$(w_{1}^{l}w_{2}^{m})\star(w_{1}^{r}w_{2}^{s})=\sum_{k}(\frac{\hbar}{2})^{k}\frac{1}{k!}R_{k}(p,q,r,s)w_{1}^{p+r-k}w_{2}^{q+s-k}.$
(8.53)

In fact, The Moyal-Weyl product and the constant $R_{k}(p,q,r,s)$ appear in
many higher spin related contexts. The global symmetry algebra of higher spin
theory- the higher spin algebra $\mathfrak{hs}$ can be defined using the
Moyal-Weyl algebra. Vasiliev’s formulation of higher spin equation is
constructed based on the Moyal-Weyl algebra, which controls its first-order
interaction. The constant $R_{k}(p,q,r,s)$ also appears in the definition of
$\mathcal{W}_{\infty}$ algebra, which is the chiral algebra of higher spin
current [43].

Indeed, our KK theory obtained by compactification is naturally a higher spin
theory in three dimension. When we perform the compactification from
$SL_{2}(\mathbb{C})$, the KK theory can be understood as a higher spin theory
on $SL_{2}(\mathbb{C})/SU(2)$, which is precisely the Euclidean $AdS_{3}$.
Therefore, it is not a coincidence that the same constant $R_{k}(p,q,r,s)$
appears in the cubic interactions of the KK theory.

The KK theory we obtained contains an infinite series of effective
interactions beyond cubic interactions. This is the same for the Vasiliev’s
theory of higher spin. Though Vasiliev’s higher spin theory does not have an
action functional, the $L_{\infty}$ algebra controlling the equation of motion
is nontrivial and contains an infinite tower of higher brackets. The
$2$-bracket is, of course, defined by the Moyal bracket. The $3$-bracket is
first constructed in [44] and is observed in [45] to correspond to the
Feigin–Felder–Shoikhet cocycle [46] of Weyl algebra. This $3$-bracket can also
be represented as a specific correlator in a topological quantum mechanics of
a free particle on the circle [47]. It will be an interesting problem to see
if the quadratic vertex of our KK theory given by a $3$-bracket on the fields
can have a similar interpretation.

##  9 Twisted holography from boundary chiral algebra

In this section, we revisit a model of twisted holography proposed in [7].
This model captures a protected subsector of the canonical example of
$AdS_{5}/CFT_{4}$ duality [1, 2]. The “gravity” side of the duality is the
string field theory of $B$-model topological string on a $SL_{2}(\mathbb{C})$
background. The “gauge theory” side of the duality involves a large $N$ chiral
algebra of gauged $\beta\gamma$ systems. In this paper, we present nontrivial
results in matching the chiral algebras from both sides of the duality.

###  9.1 Chiral algebra on topological D-branes

In this section, we introduce the large $N$ chiral algebra studied in [7].
This is the world volume theory on a stack of $N$ topological B-branes
wrapping $\mathbb{C}\subset\mathbb{C}^{3}$ with the inclusion of $K|K$ space-
filling branes. This chiral algebra can also be regarded as coming from the
$4d$ $\mathcal{N}=2$ superconformal field theory through the procedure of
[48]. In one word, this is the ”twisted” version of the gauge theory side of
the AdS/CFT correspondence.

The chiral algebra involves of the following system

  * •

A $bc$ system $(b(z),c(z))$ valued in $\mathfrak{gl}_{N}$:

$b(z)c(w)\sim\frac{1}{z-w}.$ (9.1)

  * •

a $\beta\gamma$ system (or sometime called symplectic boson)
$(Z^{1}(z),Z^{2}(z))$ valued in the adjoint $\mathfrak{gl}_{N}$:

$Z^{1}(z)Z^{2}(w)\sim\frac{1}{z-w}.$ (9.2)

  * •

a $\beta\gamma$ system $(I(z),J(z))$ valued in the fundamental representation
$\mathbb{C}^{K|K}\otimes\mathbb{C}^{N}$:

$I(z)J(w)\sim\frac{1}{z-w}$ (9.3)

.

The BRST operator takes the form

$Q_{BRST}=\oint\operatorname{Tr}cZ^{1}Z^{2}+\operatorname{Tr}IcJ+\frac{1}{2}\operatorname{Tr}bc^{2}.$
(9.4)

In the large $N$ limit, the BRST reduction is exactly computed in [7]. The
BRST cohomology comprises polynomials of the following single-trace operators.

$\displaystyle A^{(n)}$
$\displaystyle=\operatorname{Tr}(Z^{(i_{1}}Z^{i_{1}}\dots Z^{i_{n})}),$ (9.5)
$\displaystyle B^{(n)}$
$\displaystyle=\operatorname{Tr}(bZ^{(i_{1}}Z^{i_{1}}\dots Z^{i_{n})}),$
$\displaystyle C^{(n)}$ $\displaystyle=\operatorname{Tr}(\partial
cZ^{(i_{1}}Z^{i_{1}}\dots Z^{i_{n})}),$ $\displaystyle D^{(n)}$
$\displaystyle=\epsilon_{ij}\operatorname{Tr}(\partial
Z^{i}Z^{(j}Z^{i_{1}}\dots Z^{i_{n})})+\dots,$ $\displaystyle
E^{(n)}_{\mathfrak{t}}$
$\displaystyle=\operatorname{Tr}(\mathfrak{t}IZ^{(i_{1}}Z^{i_{1}}\dots
Z^{i_{n})}J).$

In the twisted holography setup, the $A,B,C,D$ towers of operators correspond
to the boundary operators of the Kodaira-Spencer theory, and the $E$ tower of
operators correspond to the boundary operators of the holomorphic Chern-
Simons. Following our form of the twisted holography conjecture, we will see
that this large $N$ chiral algebra can be matched with the boundary chiral
algebra of the corresponding KK theory with a proper choice of boundary
condition.

###  9.2 Boundary algebras from holomorphic Chern-Simons

The “gravity” side of the duality is given by the Kodaira-Spencer gravity
coupled with holomorphic Chern-Simons theory in the deformed geometry
$SL_{2}(\mathbb{C})$. First, we focus on the holomorphic Chern-Simons part.

Recall that our $3d$ fields have mode expansion

$\displaystyle\mathbf{A}(t,z,\bar{z})$ $\displaystyle=\sum_{p,q\geq
0}\mathbf{A}[p,q](t,z,\bar{z})w_{1}^{p}w_{2}^{q},$ (9.6)
$\displaystyle\mathbf{B}(t,z,\bar{z})$ $\displaystyle=\sum_{p,q\geq
0}\frac{(p+q+1)!}{p!q!}\mathbf{B}[p,q](t,z,\bar{z})\bar{w}_{1}^{p}\bar{w}_{2}^{q}\epsilon.$

They are related to the original $6d$ fields as follows

$\boldsymbol{\EuScript{A}}=\sum_{p,q\geq
0}\mathbf{A}[p,q](r,z,\bar{z})z_{1}^{p}z_{2}^{q}+\frac{(p+q+1)!}{p!q!}\mathbf{B}[p,q](r,z,\bar{z})\frac{\bar{z}_{1}^{p}\bar{z}_{2}^{q}}{r^{2(p+q)}}\frac{\epsilon_{ij}\bar{z}_{i}d\bar{z}_{j}}{r^{4}}.$
(9.7)

where $r=(z_{1}\bar{z}_{1}+z_{2}\bar{z}_{2})^{\frac{1}{2}}$. We can see that
the natural boundary condition we get by requiring the original $6d$ fields to
extend to $r=0$ is to impose the following condition

$\mathbf{B}[p,q]|_{t=0}=0\text{ for all }p,q.$ (9.8)

This is the Neumann boundary condition for the $3d$ KK theory. The
corresponding boundary algebra should reproduce the chiral algebra of the
original $6d$ theory restricted along $\mathbb{C}$. We are more interested in
the transverse boundary condition. This is the Dirichlet boundary condition
given by

$\mathbf{A}[p,q]|_{t=0}=0\text{ for all }p,q$ (9.9)

According to the argument we presented in the introduction, the boundary
algebra for this boundary condition should coincide with the universal defect
algebra on $\mathbb{C}$. Then one of the prediction of twisted holography
implies that this boundary chiral algebra is the same as the large $N$ chiral
algebra on B-brane, generated by the $E$ towers of fields.

######  Remark 9.1.

The above statement is actually not precise. Because the OPE of open string
operators will eventually involve closed string operators, and the holomorphic
Chern-Simons theory alone suffers from anomaly which requires the introduction
of Kodaira-Spencer theory to cancel. A precise matching should involve the
inclusion of Kodaira-Spencer gravity and their coupling. We provide some
examples of matching the coupling terms and the corresponding B-brane OPE in
9.7.

For the Dirichlet boundary condition, the (perturbative) boundary local
operators are generated by the lowest component $B$ of $\mathbf{B}$ and its
derivatives:

$\partial_{z}^{n}B[p,q](z).$ (9.10)

It is easy to see that there is a one-to-one correspondence of boundary local
operators and the large $N$ single trace operators from the $E$ tower
operators. Let

$E^{(p,q)}=IZ^{(i_{1}}Z^{i_{2}}\dots Z^{i_{n})}J,\quad\text{where
}\\{i_{1},\dots
i_{n}\\}=\\{\overbrace{1,\dots,1}^{p},\overbrace{2,\dots,2}^{q}\\}.$ (9.11)

Then we expect the following correspondence

$\partial_{z}^{n}B[p,q](z)\Leftrightarrow\partial_{z}^{n}E^{(p,q)}(z).$ (9.12)

In the next section, we provide some nontrivial results showing that OPE’s of
the corresponding operators are also the same.

In fact, our $3d$ KK theories are holomorphic topological cousins of the $3d$
Chern-Simons theory. In this formulation, the twisted holography duality are
reminiscent of the familiar holography duality between the Chern-Simons theory
and the WZW model [49].

###  9.3 Matching the OPE

In this section, we compare the OPE of the boundary chiral algebra and the
large $N$ chiral algebra. On the one side, the OPE for the chiral algebra on
B-brane is computed by wick contraction of the matrix fields. On the other
side, the boundary OPE is computed by Feynman diagrams through bulk
interaction vertices. We will show that different wick contraction patterns
have gravity interpretations as different bulk interaction vertices.

It is convenient to organize the $E$ tower of fields into a generating
function

$E^{(n)}(\lambda;z):=I(z)Z(\lambda;z)^{n}J(z),$ (9.13)

where

$Z(\lambda;z):=Z^{1}(z)\lambda_{1}+Z^{2}(z)\lambda_{2}.$ (9.14)

In this notation, the OPE between the $Z$ fields can be written as

$Z(\lambda;z)Z(\lambda^{\prime};0)\sim\frac{[\lambda\lambda^{\prime}]}{z},$
(9.15)

where
$[\lambda\lambda^{\prime}]=\lambda_{1}\lambda^{\prime}_{2}-\lambda_{2}\lambda^{\prime}_{1}$.
The generating field $E^{(n)}(x;z)$ is related to the symmetrized operator
$E^{(p,q)}(z)$ as follows

$E^{(n)}(\lambda;z)=\sum_{p+q=n}\binom{p+q}{p}E^{(p,q)}(z)\lambda_{1}^{p}\lambda_{2}^{q}.$
(9.16)

####  9.3.1 Quadratic vertex and two-point function

In this section, we analyze the tree-level boundary OPE from the quadratic
vertices of the KK theory. First, we briefly review how to perform such kind
of computations in the $3d$ holomorphic topological theories. We refer to [19]
for a detailed analysis.

The kinetic term $\int\operatorname{Tr}\mathbf{B}\hat{d}\mathbf{A}$ of the
$3d$ theory gives us a propagator connecting the superfields $\mathbf{B}$ and
$\mathbf{A}$

$\mathbf{B}$$\mathbf{A}$

In the presence of boundary, the corresponding propagator is obtained using
the method of reflection [19]. Omitting the color factor, we have

$P_{\partial}(z,t;z^{\prime},t^{\prime})=\frac{1}{2}(P(z,t;z^{\prime},t^{\prime})-P(z,t;z^{\prime},-t^{\prime})),$
(9.17)

where $P(z,t;z^{\prime},t^{\prime})$ is the pure bulk propagator

$P(z,t;z^{\prime},t^{\prime})=\frac{1}{8\pi
i}\frac{(\bar{z}-\bar{z}^{\prime})d(t-t^{\prime})-\frac{1}{2}(t-t^{\prime})d(\bar{z}-\bar{z}^{\prime})}{(|z-z^{\prime}|^{2}+(t-t^{\prime})^{2})^{\frac{3}{2}}},$
(9.18)

The action functional of the KK theory of holomorphic Chern-Simons on
$SL_{2}(\mathbb{C})$ consists of the flat space action functional and
deformation terms coming from the deformed geometry. For the quadratic
vertices, the flat space part
$\int\operatorname{Tr}\mathbf{B}[p,q]\hat{d}\mathbf{A}[p,q]$ is the kinetic
term which gives us the propagator connecting the fields $\mathbf{B}[p,q]$ and
$\mathbf{A}[p,q]$. The remaining part coming from $SL_{2}(\mathbb{C})$
deformation is analyzed in 8.2. We have

$I^{(1,k)}_{O}=\sum_{p+q=k-1}N^{k}\frac{(-1)^{p}}{(k-1)!}\frac{p!q!}{k!}\frac{1}{2}\int\operatorname{Tr}(\mathbf{A}[p,q]dz\partial_{z}^{k}\mathbf{A}[q,p]).$
(9.19)

We have the following tree-level Feynman diagram associated with the vertex
$I^{(1,k)}$

$I^{(1,p+q+1)_{O}}$$\mathbf{B}[p,q]$$\mathbf{B}[q,p]$

The corresponding Feynman integral is computed by

$\int_{z^{\prime}\in\mathbb{C},s\geq
0}\frac{1}{(|z-z^{\prime}|^{2}+s^{2})^{\frac{3}{2}}}\partial_{z^{\prime}}^{p+q+1}\frac{1}{(|z^{\prime}|^{2}+s^{2})^{\frac{3}{2}}}\bar{z}sdsdz^{\prime}d\bar{z}^{\prime}.$
(9.20)

The evaluation of this Feynman integral is given in Appendix F and we find

$\frac{(p+q+1)!}{z^{p+q+2}}.$ (9.21)

Therefore, the corresponding OPE is given by

$B[p,q](z)B[q,p](0)\sim(-1)^{q}N^{p+q+1}\frac{p!q!}{(p+q)!}\frac{1}{z^{p+q+2}},$
(9.22)

where we dropped the Lie algebra indices to simplify the notation. It is easy
to restore them by adding the Killing form into the above formula.

On the B-brane side, this OPE should be matched with the full contraction of
the $E$ tower operators, which is also the two-point function

$\langle
E^{(n)}(\lambda,z)E^{(n)}(\lambda^{\prime},0)\rangle=\frac{N^{n+1}[\lambda\lambda^{\prime}]^{n}}{z^{n+2}}.$
(9.23)

Expanding $E^{(n)}(\lambda,z)$ into the symmetrized operator $E^{(p,q)}$, we
find

$\langle
E^{(p,q)}(z)E^{(q,p)}(0)\rangle=(-1)^{q}N^{p+q+1}\frac{p!q!}{(p+q)!}\frac{1}{z^{p+q+2}}.$
(9.24)

This gives the same result as in the boundary chiral algebra. In fact, one
should understand this matching of two-point functions as fixing the
normalization of the operators on the two sides. The real non-trivial matching
start in the next section, when we analyze OPE from higher order vertices.

####  9.3.2 Cubic vertex and three-point function

In this section, we analyze the (deformed) cubic vertices and the
corresponding chiral OPE. We proceed from the easiest cases to the most
general case. As in the last section, we omit the Lie algebra indices. They
can be easily restored by adding the structure constant $f_{ab}^{c}$ to our
formula.

##### flat space action

First, we look at the flat space cubic vertex $I^{(2,0)}_{O}$.

$I^{(2,0)}_{O}=\int\operatorname{Tr}\mathbf{B}[p+r,q+s][\mathbf{A}[p,q],\mathbf{A}[r,s]].$
(9.25)

This vertex gives us the following tree-level Feynman diagram

$I^{(2,0)}_{O}$$B[p,q]$$B[r,s]$$B[p+r,q+s]$

The corresponding Feynman integral is the same as the integral in the last
section but without $\partial_{z^{\prime}}$ derivatives, which gives us the
following OPE

$B[p,q](z)B[r,s](0)\sim\frac{1}{z}B[p+r,q+s](0).$ (9.26)

On the B-brane side, this should be matched with the OPE of a single $I-J$
contraction. For example,

$\wick{(IZ^{i_{1}}\dots\c{J})(z)(\c{I}Z^{j_{1}}\dots
J)(0)}=\frac{1}{z}IZ^{i_{1}}\dots Z^{j_{1}}\dots J.$ (9.27)

######  Remark 9.2.

We also have the contraction $\wick{(\c{I}Z^{i_{1}}\dots
J)(IZ^{j_{1}}\dots\c{J})}$. However, this term gives us the same combinatorial
factor and pole and should be attributed to the Lie algebra factor that we
omitted.

For the symmetrized operators, we have

$\text{contraction $I-J$ :
}\;E^{(p,q)}(z)E^{(r,s)}(0)=\frac{1}{z}E^{(p+r,q+s)}(0)+\dots.$ (9.28)

This OPE can also be written as

$E^{(n)}(\lambda;z)E^{(n^{\prime})}(\lambda^{\prime};0)\sim\frac{1}{z}\frac{n^{\prime}!}{(n+n^{\prime})!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n}E^{(n+n^{\prime})}(\lambda^{\prime};0)+\dots$
(9.29)

where
$\lambda\cdot\partial_{\lambda^{\prime}}=\lambda_{1}\partial_{\lambda_{1}^{\prime}}+\lambda_{2}\partial_{\lambda_{2}^{\prime}}$.

We emphasize that there are secretly other terms in this OPE. In fact, after
the $I,J$ contraction of the symmetrized operators, we obtain
$IZ^{(i_{1}}\dots Z^{i_{n})}Z^{(j_{1}}\dots Z^{j_{n})}J$. This is, in general,
not the symmetrized operator $IZ^{(i_{1}}\dots Z^{i_{n}}Z^{j_{1}}\dots
Z^{j_{n})}J$. However, we can always manipulate the final expression into a
sum of the symmetrized operator $IZ^{(i_{1}}\dots Z^{i_{n}}Z^{j_{1}}\dots
Z^{j_{n})}J$ and other operators in the BRST representative.

For example, an $I,J$ contraction of $IZ^{1}J$ and $IZ^{2}J$ is
$IZ^{1}Z^{2}J$, which is not symmetrized. Using the BRST relation, we find
that $I[Z^{1},Z^{2}]J$ is cohomologous to $IJIJ$. Therefore the term
$IZ^{1}Z^{2}J$ is cohomologous to $IZ^{(1}Z^{2)}J+\frac{1}{2}IJIJ$. The
remaining terms like $IJIJ$ are also important, and we will come back to them
in the next section.

##### first order deformed action

We can also consider the deformed cubic vertices. As a simple example, we
consider the first order deformation $I_{O}^{(2,1)}$ in Section 8.4. Up to a
total derivative, this interaction vertex is given by

$\displaystyle I_{O}^{(2,1)}=$ $\displaystyle N\sum\frac{ps-
qr}{(p+q)(r+s)}\int\operatorname{Tr}\mathbf{B}[p+r-1,q+s-1][\partial_{z}\mathbf{A}[p,q],\mathbf{A}[r,s]]$
(9.30) $\displaystyle+$ $\displaystyle N\sum\frac{ps-
qr}{(p+q+r+s)}\frac{1}{r+s}\int\operatorname{Tr}\partial_{z}\mathbf{B}[p+r-1,q+s-1][\mathbf{A}[p,q],\mathbf{A}[r,s]].$

The corresponding tree-level diagram gives us the following OPE

$\displaystyle B[p,q](z)B[r,s](0)\sim$ $\displaystyle\frac{N}{z^{2}}\frac{ps-
qr}{(p+q)(r+s)}B[p+r-1,q+s-1](0)$ (9.31) $\displaystyle+\frac{N}{z}\frac{ps-
qr}{(p+q+r+s)}\frac{1}{r+s}\partial_{z}B[p+r-1,q+s-1](0).$

This should be matched with the OPE of a single $I-J$ contraction together
with a $Z-Z$ contraction next to the $I-J$ contraction. We have

$\wick{(I(z)Z(\lambda;z)\dots\c{2}Z(\lambda;z)\c{1}J(z))(\c{1}I(0)\c{2}Z(\lambda^{\prime},0)\dots
J(0))}=\frac{N[\lambda\lambda^{\prime}]}{z^{2}}I(z)Z(\lambda,z)^{n-1}Z(\lambda^{\prime},0)^{n^{\prime}-1}J(0).$
(9.32)

Using BRST relation to symmetrize the operators and drop operators in other
BRST representatives, we get

$\displaystyle E^{(n)}(\lambda;z)E^{(n^{\prime})}(\lambda^{\prime};0)$
$\displaystyle\sim\frac{N[\lambda\lambda^{\prime}]}{z^{2}}\frac{(n^{\prime}-1)!}{(n+n^{\prime}-2)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-1}E^{(n+n^{\prime}-2)}(\lambda^{\prime};0)$
(9.33)
$\displaystyle+\frac{N[\lambda\lambda^{\prime}]}{z}\frac{n}{n+n^{\prime}}\frac{(n^{\prime}-1)!}{(n+n^{\prime}-2)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-1}\partial_{z}E^{(n+n^{\prime}-2)}(\lambda^{\prime};0)+\dots$

We can rewrite the above OPE using the symmetrized operator $E^{(p,q)}(z)$.
The coefficient for the $\frac{1}{z^{2}}E$ term can be computed by

$\frac{\binom{p+q-1}{q}\binom{r+s-1}{r}-\binom{p+q-1}{p}\binom{r+s-1}{s}}{\binom{p+q}{p}\binom{r+s}{s}}=\frac{ps-
qr}{(p+q)(r+s)}.$ (9.34)

The coefficient for the $\frac{1}{z}\partial_{z}E$ term have an extra
$\frac{n}{n+n^{\prime}}=\frac{p+q}{p+q+r+s}$ factor. Therefore, the
corresponding OPE is given by

$\displaystyle E^{(p,q)}(z)E^{(r,s)}(0)\sim$
$\displaystyle\frac{N}{z^{2}}\frac{ps-qr}{(p+q)(r+s)}E^{(p+r-1,q+s-1)}(0)$
(9.35) $\displaystyle+\frac{N}{z}\frac{ps-
qr}{(p+q+r+s)}\frac{1}{r+s}\partial_{z}E^{(p+r-1,q+s-1)}(0).$

##### all orders deformed action

Computation of the OPE corresponding to the higher order deformed action is
similar. Recall from Section 8.4 that the $k-$th order deformed cubic action
$I^{(2,k)}_{O}$ takes the following form

$\displaystyle I^{(2,k)}_{O}=$ $\displaystyle
N^{k}\sum_{p,q,r,s}\sum_{k_{1}+k_{2}=k}\frac{(-1)^{k_{2}}R_{k}(p,q,r,s)}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k+1]_{k}}$
(9.36)
$\displaystyle\times\int\operatorname{Tr}\mathbf{B}[p+r-k,q+s-k][\partial_{z}^{k_{1}}\mathbf{A}[p,q],\partial_{z}^{k_{2}}\mathbf{A}[r,s]].$

In this case, we have $\partial_{z}$ derivatives of total order $k$. The
Feynman integral contribute a constant $k!$ factor and a pole
$\frac{1}{z^{k+1}}$. Therefore, the constant coefficient of the leading term
$\frac{N^{k}}{z^{k+1}}B$ is given by

$k!\sum_{k_{1}+k_{2}=k}\frac{R_{k}(p,q,r,s)}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k+1]_{k}}=\frac{R_{k}(p,q,r,s)}{[p+q]_{k}[r+s]_{k}},$
(9.37)

where we used the formula E.10 derived in the Appendix. This give the OPE

$B[p,q](z)B[r,s](0)\sim\frac{N^{k}}{z^{k+1}}\frac{R_{k}(p,q,r,s)}{[p+q]_{k}[r+s]_{k}}B[p+r-k,q+s-k](0)+\dots$
(9.38)

To compute other terms with derivatives of $B$, we move the $\partial_{z}$
derivatives from one $\mathbf{A}$ field to the other $\mathbf{A}$ field and
the $\mathbf{B}$ field in the above cubic action. For example,
$\mathbf{B}[\partial_{z}^{k_{1}}\mathbf{A},\partial_{z}^{k_{2}}\mathbf{A}]$
can be rewritten as
$\sum_{l}(-1)^{k_{2}}\binom{k_{2}}{l}\partial_{z}^{l}\mathbf{B}[\partial_{z}^{k_{1}+k_{2}-l}\mathbf{A},\mathbf{A}]$
up to a total $\partial_{z}$ derivatives. We see that the term with
$\partial_{z}^{l}B$ has pole $\frac{1}{z^{k-l+1}}$. The corresponding
coefficient is computed by

$\displaystyle(k-l)!\sum_{k_{1}+k_{2}=k}\binom{k_{2}}{l}\frac{R_{k}(p,q,r,s)}{k_{1}!k_{2}![p+q]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k+1]_{k}}$
(9.39)
$\displaystyle=\frac{R_{k}(p,q,r,s)}{l![p+q]_{k-l}[r+s]_{k}[p+q+r+s-2k+l+1]_{l}},$

where we used the formula E.12. The full OPE corresponding to the deformed
cubic action $I^{(2,k)}_{O}$ is thus given by

$B[p,q](z)B[r,s](0)\sim\sum_{l=0}^{k}\frac{N^{k}}{z^{k-l+1}}\frac{R_{k}(p,q,r,s)\times\partial_{z}^{l}B[p+r-k,q+s-k](0)}{l![p+q]_{k-l}[r+s]_{k}[p+q+r+s-2k+l+1]_{l}}.$
(9.40)

On the B brane side, this OPE should correspond to a single $I-J$ contraction
together with $k$ adjacent $Z-Z$ contractions. Such contraction produce a
$N^{k}$ factor and a pole $\frac{1}{z^{k+1}}$. We have

$\wick{(I(z)...\c{3}Z(\lambda;z)...\c{2}Z(\lambda;z)\c{1}J(z))(\c{1}I(0)\c{2}Z(\lambda^{\prime};0)...\c{3}Z(\lambda^{\prime};0)...J(0))}=\frac{N^{k}}{z^{k+1}}I(z)Z(\lambda;z)^{n-k}Z(\lambda^{\prime};0)^{n^{\prime}-k}J(0).$
(9.41)

By symmetrizing the $Z$ operators, we can rewrite this OPE as follows

$E^{(n)}(\lambda;z)E^{(n^{\prime})}(\lambda^{\prime};0)\sim\frac{N^{k}[\lambda\lambda^{\prime}]^{k}}{z^{k+1}}\frac{(n^{\prime}-k)!}{(n+n^{\prime}-2k)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-k}E^{(n+n^{\prime}-2k)}(\lambda^{\prime};0)\dots,$
(9.42)

where we omitted terms with derivatives of $E$. Expanding the operators into
$E^{(p,q)}$, we can compute the constant coefficient by

$\sum_{i=0}^{k}\frac{(-1)^{i}\binom{k}{i}\binom{p+q-k}{q-i}\binom{r+s-k}{r-i}}{\binom{p+q}{p}\binom{r+s}{r}}=\frac{R_{k}(p,q,r,s)}{[p+q]_{k}[r+s]_{k}},$
(9.43)

which follows from the definition 8.33 of the constant $R_{k}(p,q,r,s)$. This
matches the leading term of 9.40. To obtain the coefficients of the derivative
terms, we use the planar three point function

$\langle
E^{(n)}(\lambda,z)E^{(n^{\prime})}(\lambda^{\prime},z^{\prime})E^{(n^{\prime\prime})}(\lambda^{\prime\prime},z^{\prime\prime})\rangle=\frac{N^{\frac{n+n^{\prime}+n^{\prime\prime}}{2}+1}[\lambda\lambda^{\prime}]^{\frac{n+n^{\prime}-n^{\prime\prime}}{2}}[\lambda^{\prime}\lambda^{\prime\prime}]^{\frac{n^{\prime}+n^{\prime\prime}-n}{2}}[\lambda\lambda^{\prime\prime}]^{\frac{n+n^{\prime\prime}-n^{\prime}}{2}}}{(z-z^{\prime})^{\frac{n+n^{\prime}-n^{\prime\prime}}{2}+1}(z^{\prime}-z^{\prime\prime})^{\frac{n^{\prime}+n^{\prime\prime}-n^{\prime}}{2}+1}(z-z^{\prime\prime})^{\frac{n+n^{\prime\prime}-n^{\prime}}{2}+1}},$
(9.44)

which is computed via a direct computation of wick contraction. Expanding this
three point function we find that

$\displaystyle\langle
E^{(n)}(\lambda,z)E^{(n^{\prime})}(\lambda^{\prime},z^{\prime})E^{(n^{\prime\prime})}(\lambda^{\prime\prime},z^{\prime\prime})\rangle$
(9.45) $\displaystyle=$
$\displaystyle\frac{N^{\frac{n+n^{\prime}+n^{\prime\prime}}{2}+1}[\lambda\lambda^{\prime}]^{\frac{n+n^{\prime}-n^{\prime\prime}}{2}}[\lambda^{\prime}\lambda^{\prime\prime}]^{\frac{n^{\prime}+n^{\prime\prime}-n}{2}}[\lambda\lambda^{\prime\prime}]^{\frac{n+n^{\prime\prime}-n^{\prime}}{2}}}{(z-z^{\prime})^{\frac{n+n^{\prime}-n^{\prime\prime}}{2}+1}}\sum_{l\geq
0}\frac{(\frac{n+n^{\prime\prime}-n^{\prime}}{2}+l)!}{(\frac{n+n^{\prime\prime}-n^{\prime}}{2})!l!}(-1)^{l}\left(\frac{z-z^{\prime}}{z^{\prime}-z^{\prime\prime}}\right)^{l}\frac{1}{(z^{\prime}-z^{\prime\prime})^{n^{\prime\prime}+2}}$
$\displaystyle=$ $\displaystyle\sum_{l\geq
0}\frac{N^{\frac{n+n^{\prime}+n^{\prime\prime}}{2}+1}[\lambda\lambda^{\prime}]^{\frac{n+n^{\prime}-n^{\prime\prime}}{2}}}{(z-z^{\prime})^{\frac{n+n^{\prime}-n^{\prime\prime}}{2}-l+1}}\frac{[\frac{n+n^{\prime\prime}-n^{\prime}}{2}+l]_{l}(\frac{n^{\prime}+n^{\prime\prime}-n}{2})!}{[n^{\prime\prime}+l+1]_{l}n^{\prime\prime}!}(\lambda\cdot\partial_{\lambda^{\prime}})^{\frac{n+n^{\prime\prime}-n^{\prime}}{2}}[\lambda^{\prime}\lambda^{\prime\prime}]^{n^{\prime\prime}}\frac{\partial_{z^{\prime}}^{l}}{l!}\frac{1}{(z^{\prime}-z^{\prime\prime})^{n^{\prime\prime}+2}}$

Using the two-point function, we can extract from the above expression the
following OPE

$E^{(n)}(\lambda;z)E^{(n^{\prime})}(\lambda^{\prime};0)\sim\sum_{l=0}^{k}\frac{N^{k}[\lambda\lambda^{\prime}]^{k}}{z^{k-l+1}}\frac{[n-k+l]_{l}(n^{\prime}-k)!}{[n+n^{\prime}-2k+l+1]_{l}(n+n^{\prime}-2k)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-k}\frac{\partial_{z}^{l}}{l!}E^{(n+n^{\prime}-2k)}(\lambda^{\prime};0).$
(9.46)

We can check that we get exactly the OPE of the boundary chiral algebra 9.40.

####  9.3.3 Quartic vertex

The quartic vertex is constructed using the higher product $m_{3}$ that we
studied in Section 4.3. We have the following Feynman diagram.

$I^{(3,0)}$$B[p,q]$$B[r,s]$$B[u_{1},v_{1}]$$B[u_{2},v_{2}]$

The corresponding OPE is given by

$B_{a}[p,q](z)B_{b}[r,s](0)\sim\sum_{\begin{subarray}{c}u_{1}+u_{2}=p+r-1\\\
v_{1}+v_{2}=q+s-1\end{subarray}}\sum_{c,d,e,f}\frac{K^{fc}}{z}\left((m_{3})^{p,q;r,s}_{u_{1},v_{1}}f_{bf}^{e}f_{ae}^{d}-(m_{3})^{r,s;p,q}_{u_{1},v_{1}}f_{af}^{e}f_{be}^{d}\right)B_{c}[u_{1},v_{1}]B_{d}[u_{2},v_{2}](0).$
(9.47)

where we used the shorthand
$(m_{3})^{p,q;r,s}_{u,v}:=(m_{3})^{p,q;r,s}_{u,v;p+r-1-u,q+s-1-v}$. We
included the Lie algebra factor here since it becomes important in the
analysis. Though the above formula is for general Lie algebra, in the
holographic setting we need to use the Lie algebra $\mathfrak{gl}(K|K)$ or
$\mathfrak{gl}(K)$. Strictly speaking, the bulk anomaly of the open-closed
coupled theory cancels only for the super Lie algebra $\mathfrak{gl}(K|K)$.
However, the OPE matching at this level also works for $\mathfrak{gl}(K)$. For
simplicity, we use $\mathfrak{gl}(K)$ here. The result can be easily
generalized to $\mathfrak{gl}(K|K)$ by taking care of the $\pm$ sign. Note
that $\mathfrak{gl}(K)$ has a canonical basis given by the elementary matrices
$\\{E_{a_{1}a_{2}}\\}_{1\leq a_{1},a_{2}\leq K}$. Therefore we replace the
indices $a$ by $a_{1}a_{2}$ in the above formula. The Killing form
$K^{a_{1}a_{2};b_{1}b_{2}}$ is given by

$K^{a_{1}a_{2};b_{1}b_{2}}=\delta_{a_{1}b_{2}}\delta_{a_{2}b_{1}}.$ (9.48)

The structure constant can be extracted from the commutation relation

$[E_{a_{1}a_{2}},E_{b_{1}b_{2}}]=\delta_{a_{2}b_{1}}E_{a_{1}b_{2}}-\delta_{a_{1}b_{2}}E_{b_{1}a_{2}}.$
(9.49)

Then the formula 9.47 can be expanded as

$\displaystyle B_{a_{1}a_{2}}[p,q](z)$ $\displaystyle
B_{b_{1}b_{2}}[r,s](0)\sim\frac{1}{z}\sum_{\begin{subarray}{c}u_{1}+u_{2}=p+r-1\\\
v_{1}+v_{2}=q+s-1\end{subarray}}((m_{3})^{r,s;p,q}_{u_{1},v_{1}}-(m_{3})^{p,q;r,s}_{u_{1},v_{1}})B_{a_{1}b_{2}}[u_{1},v_{1}]B_{b_{1}a_{2}}[u_{2},v_{2}](0)$
(9.50)
$\displaystyle+\sum_{c}\left((m_{3})^{p,q;r,s}_{u_{1},v_{1}}\delta_{a_{1}b_{2}}B_{b_{1}c}[u_{1},v_{1}]B_{ca_{2}}[u_{2},v_{2}]-(m_{3})^{r,s;p,q}_{u_{1},v_{1}}\delta_{a_{2}b_{1}}B_{a_{1}c}[u_{1},v_{1}]B_{cb_{2}}[u_{2},v_{2}]\right).$

Naively, we expect that the corresponding wick contraction on the B-brane side
is given by a single $Z-Z$ contraction

$\wick{(I...\c{Z}^{i_{k}}...J)(z)(I...\c{Z}^{j_{l}}...J)(0)}=\frac{1}{z}(IZ^{j_{1}}...Z^{j_{l-1}}Z^{i_{k}+1}...J)(0)(IZ^{i_{1}}...Z^{i_{k-1}}Z^{j_{l+1}}Z^{j_{1}}...J)(0).$
(9.51)

However, as we have mentioned in the last section, after a single $I-J$
contraction, we need to manipulate the final expression into BRST
representative. In this process, we can trade the commutator $[Z^{1},Z^{1}]$
with $\sum J_{a}I_{a}$. Therefore, OPE of the form $(I\dots J)(z)(I\dots
J)(0)\sim\frac{1}{z}(I\dots J)(I\dots J)$ can also be generated in the single
$I-J$ wick contraction studied in the last section.

It is a tedious work to analyze the combinatorial factor of the wick
contractions in the general situation. To simplify the discussion, we study
some special cases.

The simplest example we can look at is the OPE of $I_{a_{1}}Z^{1}J_{a_{2}}$
and $I_{b_{1}}Z^{2}J_{b_{2}}$. A single $Z-Z$ contraction gives us

$\wick{(I_{a_{1}}\c{Z}^{1}J_{a_{2}})(z)(I_{b_{1}}\c{Z}^{2}J_{b_{2}})(0)}\sim\frac{1}{z}(I_{a_{1}}J_{b_{2}})(I_{b_{1}}J_{a_{2}})(0).$
(9.52)

A single $I-J$ contraction gives us

$\displaystyle\wick{(I_{a_{1}}Z^{1}\c{J}_{a_{2}})(z)(\c{I}_{b_{1}}Z^{2}J_{b_{2}})(0)}$
$\displaystyle+\wick{(\c{I}_{a_{1}}Z^{1}J_{a_{2}})(z)(I_{b_{1}}Z^{2}\c{J}_{b_{2}})(0)}$
(9.53)
$\displaystyle\sim-\frac{1}{2z}\sum_{c}(\delta_{a_{2}b_{1}}(I_{a_{1}}J_{c})(I_{c}J_{b_{2}})(0)+\delta_{a_{1}b_{2}}(I_{b_{1}}J_{c})(I_{c}J_{a_{2}})),$

where we omitted the operator $IZ^{(1}Z^{2)}J$ that analyzed in the last
section.

By computing $m_{3}$ using 4.39, we find that
$(m_{3})^{1,0;0,1}_{0,0}=-\frac{1}{2}$ and
$(m_{3})^{0,1;1,0}_{0,0}=\frac{1}{2}$ . Therefore we have the following OPE

$\displaystyle B_{a_{1}a_{2}}[1,0](z)B_{b_{1}b_{2}}[0,1](0)$
$\displaystyle\sim\frac{1}{z}\Bigl{(}B_{a_{1}b_{2}}[0,0]B_{b_{1}a_{2}}[0,0](0)$
(9.54)
$\displaystyle-\frac{1}{2}\sum_{c}\delta_{a_{2}b_{1}}B_{a_{1}c}[0,0]B_{cb_{2}}[0,0](0)-\delta_{a_{1}b_{2}}B_{b_{1}c}[0,0]B_{ca_{2}}[0,0](0)\Bigr{)}.$

This matches exactly the corresponding OPE on B-brane.

As a more nontrivial example, we use the formula 4.39 to compute the following
value of $m_{3}$

$\displaystyle(m_{3})^{1,0;r,s}_{u,v}$
$\displaystyle=-\frac{r!s!(u+v+1)!(r+s-u-v-1)!}{(r+s+1)!u!v!(r-u)!(s-v-1)!},$
(9.55) $\displaystyle(m_{3})^{r,s;1,0}_{u,v}$
$\displaystyle=\frac{r!s!(u+v)!(r+s-u-v)!}{(r+s+1)!u!v!(r-u)!(s-v-1)!}.$

Therefore, we have the following OPE

$\displaystyle B_{a_{1}a_{2}}[1,0](z)B_{b_{1}b_{2}}[r,s](0)$ (9.56)
$\displaystyle\sim$
$\displaystyle\frac{1}{z}\sum_{u,v}\binom{r+s}{r}^{-1}\binom{u+v}{u}\binom{r+s-u-v-1}{r-u}\Bigl{(}B_{a_{1}b_{2}}[u,v]B_{b_{1}a_{2}}[r-u,s-v-1](0)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{c}\frac{r+s-u-v}{r+s+1}\delta_{a_{2}b_{1}}B_{a_{1}c}[u,v]B_{cb_{2}}[r-u,s-v-1](0)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{c}\frac{u+v}{r+s+1}\delta_{a_{1}b_{2}}B_{b_{1}c}[u,v]B_{ca_{2}}[r-u,s-v-1](0)\Bigr{)}.$

On the B-brane side, we can compute a single $Z-Z$ contraction as follows

$E_{a_{1}a_{2}}^{(1)}(\lambda;z)E_{b_{1}b_{2}}^{(n^{\prime})}(\lambda^{\prime},0)\sim\sum_{n^{\prime\prime}}\frac{[\lambda\lambda^{\prime}]}{z}E^{(n^{\prime\prime})}_{a_{1}b_{2}}(\lambda^{\prime},0)E^{(n^{\prime}-n^{\prime\prime}-1)}_{b_{1}a_{2}}(\lambda^{\prime},0).$
(9.57)

Expanding the above formula with
$E_{b_{1}b_{2}}^{(n^{\prime})}(\lambda^{\prime},0)=\sum_{r+s=n^{\prime}}\binom{r+s}{r}E_{b_{1}b_{2}}^{(r,s)}(0)\lambda_{1}^{\prime
r}\lambda_{2}^{\prime s}$, we find

$E_{a_{1}a_{2}}^{(1,0)}(z)B_{b_{1}b_{2}}^{(r,s)}(0)\sim\frac{1}{z}\sum_{u,v}\binom{r+s}{r}^{-1}\binom{u+v}{u}\binom{r+s-u-v-1}{r-u}E_{a_{1}b_{2}}^{(u,v)}B_{b_{1}a_{2}}^{(r-u,s-v-1)}(0).$
(9.58)

This matches with the first term of the boundary OPE of 9.56. To match the
remaining terms, we consider the single $I-J$ contraction. As in previous
discussion, we use the BRST relation and find that
$I\dots[Z(\lambda),Z(\lambda^{\prime})]\dots J$ is cohomologous to
$[\lambda\lambda^{\prime}]I\dots JI\dots J$. Therefore, after dropping the
symmetrized operator that is already studied in the last section, we have the
following OPE

$\displaystyle
E_{a_{1}a_{2}}^{(1)}(\lambda;z)E_{b_{1}b_{2}}^{(n^{\prime})}(\lambda^{\prime},0)$
$\displaystyle\sim-\frac{[\lambda\lambda^{\prime}]}{z}\sum_{n^{\prime\prime}}\sum_{c}\left(\frac{n^{\prime}-n^{\prime\prime}}{n^{\prime}+1}\delta_{a_{2}b_{1}}E^{(n^{\prime\prime})}_{a_{1}c}(\lambda^{\prime},0)E^{(n^{\prime}-n^{\prime\prime}-1)}_{cb_{2}}(\lambda^{\prime},0)\right.$
(9.59)
$\displaystyle\left.+\frac{n^{\prime\prime}+1}{n^{\prime}+1}\delta_{a_{1}b_{2}}E^{(n^{\prime\prime})}_{b_{1}c}(\lambda^{\prime},0)E^{(n^{\prime}-n^{\prime\prime}-1)}_{ca_{2}}(\lambda^{\prime},0)\right).$

Expanding this formula, we find

$\displaystyle
E_{a_{1}a_{2}}^{(1,0)}(z)B_{b_{1}b_{2}}^{(r,s)}(0)\sim-\frac{1}{z}\sum_{u,v}\sum_{c}\binom{r+s}{r}^{-1}\binom{u+v}{u}\binom{r+s-u-v-1}{r-u}$
(9.60)
$\displaystyle\times\left(\frac{r+s-u-v}{r+s+1}\delta_{a_{2}b_{1}}E_{a_{1}c}^{(u,v)}B_{cb_{2}}^{(r-u,s-v-1)}(0)+\frac{u+v+1}{r+s+1}\delta_{a_{1}b_{2}}E_{b_{1}c}^{(u,v)}B_{ca_{2}}^{(r-u,s-v-1)}(0)\right).$

This gives the remaining terms of the boundary OPE of 9.56.

As we have discussed, the deformed geometry will introduce deformation to the
higher operations and gives us deformed interaction vertices. We expect that
the deformed quartic action $I^{(3,k)}_{O}$ corresponds to the wick
contraction of $k+1$ adjacent $Z-Z$ fields, and also the wick contraction of a
single $I-J$ together with $k$ adjacent $Z-Z$ fields. It will be interesting
to explore these OPE.

###  9.4 Comments on higher order interaction

In this section we briefly analyze OPE coming from higher order vertices.
First, we study the quintic interaction vertices. By looking at the Lie
algebra factor, we see that this interaction term produce the following terms
in the OPE of $B_{a_{1}a_{2}}(z)B_{b_{1}b_{2}}(0)$

$\displaystyle\sum_{c,c^{\prime}}\delta_{a_{2}b_{1}}B_{a_{1}c}B_{cc^{\prime}}B_{c^{\prime}b_{2}}$
$\displaystyle,\quad\sum_{c,c^{\prime}}\delta_{a_{1}b_{2}}B_{b_{1}c}B_{cc^{\prime}}B_{c^{\prime}a_{2}},$
(9.61) $\displaystyle\sum_{c}B_{a_{1}c}B_{cb_{2}}B_{b_{1}a_{2}}$
$\displaystyle,\quad\sum_{c}B_{a_{1}b_{2}}B_{b_{1}c}B_{ca_{2}}.$

Naively, wick contraction of two $IZ\dots J$ operators cannot produce
operators with three or more pairs of $I,J$ fields. However, as we have
analyzed in the last section, we trade the commutator $[Z^{1},Z^{2}]$ with
$\sum_{c}J_{c}I_{c}$ in this process of symmetrize the adjoint fields. This
will gives OPE that produce three or more pairs of $I,J$ fields as desired.
For example, a single $Z-Z$ contraction of $I_{a_{1}}\dots J_{a_{2}}$ and
$I_{b_{1}}\dots J_{b_{2}}$ gives $I_{a_{1}}\dots J_{b_{2}}I_{b_{1}}\dots
J_{a_{2}}$. Symmetrizing this operator gives the following two possibilities
with three pairs of $I,J$ fields.

$\sum_{c}(I_{a_{1}}\dots J_{c})(I_{c}\dots J_{b_{2}})(I_{b_{1}}\dots
J_{a_{2}}),\quad\sum_{c}(I_{a_{1}}\dots J_{b_{2}})(I_{b_{1}}\dots
J_{c})(I_{c}\dots J_{a_{2}}).$ (9.62)

They match the operators in the second line of 9.61. We can also consider a
single $I-J$ contraction. This will gives us the operators
$\delta_{a_{2}b_{1}}I_{a_{1}}\dots J_{b_{2}}$ and
$\delta_{a_{1}b_{2}}I_{b_{1}}\dots J_{a_{2}}$. To symmetrize this operator we
need to use $[Z^{1},Z^{2}]\sim\sum_{c}J_{c}I_{c}$ more than once in general.
This gives the following two possibilities with three pairs of $I,J$ fields.

$\sum_{c,c^{\prime}}\delta_{a_{2}b_{1}}(I_{a_{1}}\dots J_{c})(I_{c}\dots
J_{c^{\prime}})(I_{c^{\prime}}\dots
J_{b_{2}}),\quad\sum_{c,c^{\prime}}\delta_{a_{1}b_{2}}(I_{b_{1}}\dots
J_{c})(I_{c}\dots J_{c^{\prime}})(I_{c^{\prime}}\dots J_{a_{2}}).$ (9.63)

They match the operators in the first line of 9.61. So far, we have only
considered the matching of Lie algebra indices in the OPE. It will be a
nontrivial and interesting prediction of twisted holography that the
$A_{\infty}$ structure on the Cauchy-Riemann cohomology can reproduce the
combinatorial factor coming from matrix contraction and symmetrization of the
operators.

More generally, we expect that the full tree-level OPE for the KK theory on
deformed geometry takes the following schematic form

$\displaystyle B_{a_{1}a_{2}}[p,q](z)B_{b_{1}b_{2}}[r,s](0)\sim\sum_{n\geq
1}\sum_{k\geq 0}\sum_{\begin{subarray}{c}u_{1}+\dots u_{n}=p+r-k-n+1\\\
v_{1}+\dots
v_{n}=q+s-k-n+1\end{subarray}}\frac{N^{k}}{z^{k+1}}\times(\text{some
constants})\times$ (9.64)
$\displaystyle\sum_{c_{1},\dots,c_{n-1}}\delta_{a_{2}b_{1}}B_{a_{1}c_{1}}[u_{1},v_{1}]B_{c_{1}c_{2}}[u_{2},v_{2}]\dots
B_{c_{n-1}b_{2}}[u_{n},v_{n}]+\delta_{a_{1}b_{2}}B_{b_{1}c_{1}}[u_{1},v_{1}]B_{c_{1}c_{2}}[u_{2},v_{2}]\dots
B_{c_{n-1}a_{2}}[u_{n},v_{n}]$
$\displaystyle\sum_{c_{1},\dots,c_{n-2}}\sum_{l=1}^{n-1}B_{a_{1}c_{1}}[u_{1},v_{1}]B_{c_{1}c_{2}}[u_{2},v_{2}]\dots
B_{c_{l-1}b_{2}}[u_{l},v_{l}]B_{b_{1}c_{l}}[u_{l+1},v_{l+1}]B_{c_{1}c_{l+1}}[u_{l+2},v_{l+2}]\dots
B_{c_{n-2}a_{2}}[u_{n},v_{n}]$ $\displaystyle+\text{derivatives}.$

Under the normalization $B[p,q]\to N^{-\frac{p+q+2}{2}}B[p,q]$, we find that
the above OPE’s are all of order $\frac{1}{N}$. Dropping the $\frac{1}{N^{2}}$
and higher terms in the full OPE provides us with an interesting limit of the
holography chiral algebra beyond the planar limit. In fact, this limit gives
us a Poisson vertex algebra. Matching this Poisson vertex algebra structure is
a very nontrivial but also approachable test of the twisted holography.

###  9.5 Chiral algebra from abelian Chern-Simons

As explained in [7], the $\mathrm{PV}^{2,0}$ and $\mathrm{PV}^{0,2}$ part of
Kodaira-Spencer theory can be reformulated as a super abelian Chern-Simons
theory valued in $\Pi\mathbb{C}^{2}$ 888This notation simply means the abelian
Lie super algebra of $2$-dimension in degree $1$.. This abelian Lie algebra
has bilinear pairing given by the symplectic form on $\mathbb{C}^{2}$.
Therefore, we can directly apply our previous results about the holomorphic
Chern-Simons theory. After the KK reduction, we have the following field
content

$\displaystyle\mathbf{A}^{\prime}$
$\displaystyle\in\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\Pi\mathbb{C}^{2}[w_{1},w_{2}][1],$
(9.65) $\displaystyle\mathbf{B}^{\prime}$
$\displaystyle\in\Omega_{3d}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},(1)}\otimes\Pi\mathbb{C}^{2}[\bar{w}_{,}\bar{w}_{2}].$

Since $\Pi\mathbb{C}^{2}$ is an abelian algebra, all brackets and higher
brackets $l_{k}$ vanish for this theory. The only nontrivial terms are the
quadratic action and its deformation discussed in 8.2. We immediately obtain
the action functional for this theory

$\sum_{p,q}\int\epsilon_{ab}\mathbf{B}^{\prime}_{a}[p,q]\hat{d}\mathbf{A}^{\prime}_{b}[p,q]+\frac{(-1)^{q}N^{p+q+1}}{(p+q+1)!}\frac{p!q!}{(p+q)!}\int\epsilon_{ab}\mathbf{A}^{\prime}_{a}[p,q]\partial_{z}^{p+q+1}\mathbf{A}^{\prime}_{b}[q,p].$
(9.66)

Taking the Dirichlet boundary condition, the boundary chiral algebra is
generated by

$\partial_{z}^{k}B^{\prime}_{a}[p,q],\;\;a=1,2.$ (9.67)

As we have discussed, the quadratic term gives the following OPE

$B^{\prime}_{1}[p,q](z)B^{\prime}_{2}[q,p](0)\sim(-1)^{q}N^{p+q+1}\frac{p!q!}{(p+q)!}\frac{1}{z^{p+q+2}}.$
(9.68)

On the B brane side, this part of boundary algebra should be matched with the
$B^{(n)}$ and $C^{(n)}$ towers of fields.

$\displaystyle\operatorname{Tr}(bZ^{(i_{1}}Z^{i_{1}}\dots Z^{i_{n})}),$ (9.69)
$\displaystyle\operatorname{Tr}(\partial cZ^{(i_{1}}Z^{i_{1}}\dots
Z^{i_{n})}).$

The corresponding OPE is given by the full wick contraction as follows

$\wick{\operatorname{Tr}(\c{4}b\c{3}Z^{i_{1}}\c{2}Z^{i_{2}}\dots\c{1}Z^{i_{n}})(z)\operatorname{Tr}(\c{4}\partial
c\c{1}Z^{\bar{i}_{n}}\dots\c{2}Z^{\bar{i}_{2}}\c{3}Z^{\bar{i}_{1}})(0)}=(-1)^{\\#\text{
of }2\text{ in }\\{i_{n}\\}}\frac{N^{n+1}}{z^{n+2}}$ (9.70)

where $\bar{i}=2$ if $i=1$ and $\bar{i}=1$ if $i=2$. Taking into account of
the combinatorial factor of symmetrized trace, we have

$\wick{\operatorname{Tr}(\c{4}b\c{3}Z^{(i_{1}}\dots\c{2}Z^{i_{n})})(z)\operatorname{Tr}(\c{4}\partial
c\c{2}Z^{(\bar{i}_{n}}\dots\c{3}Z^{\bar{i}_{1})}))(0)}=(-1)^{q}N^{p+q+1}\frac{p!q!}{(p+q)!}\frac{1}{z^{p+q+2}},$
(9.71)

where $p$ is the number of $1$ in $\\{i_{n}\\}$ and $q$ is the number of $2$
in $\\{i_{n}\\}$. This is the same as the corresponding boundary OPE.

Since there is no other interaction vertex in the abelian theory itself, we
don’t have any other OPE in this sector of fields. This is compatible with the
Large N chiral algebra. In fact, any contractions of
$\operatorname{Tr}(bZ^{i_{1}}\dots)$ and $\operatorname{Tr}(\partial c\dots
Z^{j_{1}})$ will not produce operators in the same towers of fields. The
nontrivial terms in the OPE thus only involve other towers of fields. On the
gravity side, these OPE come from BCOV interactions and open-closed couplings.

###  9.6 Chiral algebra from Kodaira-Spencer gravity

In this section, we analyze the boundary chiral algebra for the
$(\bm{\alpha},\bm{\beta})$ part of Kodaira Spencer fields defined in Section
7.3. Similar to the analysis of holomorphic Chern-Simons, we choose the
boundary condition such that

$\bm{\alpha}|_{t=0}=0.$ (9.72)

The corresponding boundary chiral algebra is generated by the boundary fields
$\beta[p,q](z)$ and all their derivatives $\partial_{z}^{n}\beta[p,q]$.

On the B-brane side, the corresponding single trace operator is given by the
$A$ towers of fields. Let

$A^{(p,q)}=\operatorname{Tr}Z^{(i_{1}}Z^{i_{2}}\dots
Z^{i_{n})},\quad\text{where }\\{i_{1},\dots
i_{n}\\}=\\{\overbrace{1,\dots,1}^{p},\overbrace{2,\dots,2}^{q}\\}.$ (9.73)

We show in this section that OPE’s of $A^{(p,q)}$ operators are matched with
that of the boundary $\beta[p,q]$ operators. It is convenient to define the
generating fields as follows

$A^{(n)}(\lambda;z):=\operatorname{Tr}Z(\lambda;z)^{n}.$ (9.74)

They are related to $A^{(p,q)}$ by

$A^{(n)}(\lambda;z)=\sum_{p+q=n}\binom{p+q}{l}A^{(p,q)}(z)\lambda_{1}^{l}\lambda_{2}^{m}.$
(9.75)

Now we consider the boundary chiral algebra for the $\beta$ fields. First, we
consider the quadratic action. The Feynman diagram and the corresponding
Feynman integral are the same as in Section 9.3.1. We can immediately write
down the corresponding OPE using the deformed quadratic action $I^{(1,k)}_{C}$
8.18. We have

$\beta[p,q](z)\beta[q,p](0)\sim(-1)^{q}\frac{p!q!}{(p+q-1)!}\frac{N^{p+q}}{z^{p+q}}.$
(9.76)

On the B-brane side, this OPE corresponds to a full contraction, which is also
the two-point correlation function of two $A$ tower operators. Using the
generating fields, we have

$\langle
A^{(n)}(\lambda;z)A^{(n)}(\lambda^{\prime};0)\rangle=nN^{n}\frac{[\lambda\lambda^{\prime}]^{n}}{z^{n}}.$
(9.77)

Expanding $A^{(n)}(\lambda;z)$ into $A^{(p,q)}(z)$ we get exactly the OPE
9.76.

As a next step, we consider the cubic action in flat space obtained in Section
7.3. We find the following OPE

$\beta[p,q](z)\beta[r,s](0)\sim(ps-qr)\beta[p+r-1,q+s-1](0).$ (9.78)

Similarly, we can proceed to consider deformed cubic action $I_{C}^{(2,k)}$
obtained in 8.44.

$I^{(2,k)}_{C}$$\beta[p,q]$$\beta[r,s]$$\beta[p+r-k-1,q+s-k-1]$

The leading order (without any $z$ derivatives) coefficient is computed as
follows

$\displaystyle\sum_{k_{1}+k_{2}=k}\frac{k!(p+q)(r+s)R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q+r+s-k]_{k+1}}\left(\frac{1}{[p+q]_{k_{1}+1}[r+s]_{k_{2}}}+\frac{1}{[p+q]_{k_{1}}[r+s]_{k_{2}+1}}\right)$
(9.79)
$\displaystyle=\sum_{k_{1}+k_{2}=k}\frac{k!R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q+r+s-k]_{k+1}}\left(\frac{r+s}{[p+q-1]_{k_{1}}[r+s]_{k_{2}}}+\frac{p+q}{[p+q]_{k_{1}}[r+s-1]_{k_{2}}}\right)$
$\displaystyle=\frac{R_{k+1}(p,q,r,s)}{[p+q-1]_{k}[r+s-1]_{k}}.$

Therefore, we have the following OPE

$\beta[p,q](z)\beta[r,s](0)\sim\frac{N^{k}}{z^{k+1}}\frac{R_{k+1}(p,q,r,s)}{[p+q-1]_{k}[r+s-1]_{k}}\beta[p+r-k-1,q+s-k-1](0)+\dots$
(9.80)

Note that by letting $k=0$ in the above formula, th result is the same as the
flat space OPE 9.78.

The corresponding B-brane OPE is given by $k+1$ adjacent contraction of $Z$
fields. For example, we have the following contraction between
$A^{(n)}(\lambda;z)$ and $A^{(n^{\prime})}(\lambda^{\prime};0)$

$\displaystyle\wick{\operatorname{Tr}(Z(\lambda;z)\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}\c{3}Z(\lambda;z)\c{2}\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}\c{1}Z(\lambda;z))\operatorname{Tr}(\c{1}Z(\lambda^{\prime};0)\c{2}\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}\c{3}Z(\lambda^{\prime};0)\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}Z(\lambda^{\prime};0))}$
(9.81) $\displaystyle=$ $\displaystyle
N^{k}\frac{[\lambda\lambda^{\prime}]^{k+1}}{z^{k+1}}\operatorname{Tr}Z(\lambda;z)^{n-k-1}Z(\lambda^{\prime};0)^{n^{\prime}-k-1}.$

There are $nn^{\prime}$ such contractions in total. As in the previous
section, we use the BRST relation to symmetrize the operator and discard
operators in different BRST representatives. Omitting terms with derivatives
of $A$ field, we have

$A^{(n)}(\lambda;z)A^{(n^{\prime})}(\lambda^{\prime};0)\sim
nn^{\prime}\frac{N^{k}[\lambda\lambda^{\prime}]^{k+1}}{z^{k+1}}\frac{(n^{\prime}-k-1)!}{(n+n^{\prime}-k-2)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-k-1}A^{(n+n^{\prime}-k-2)}(\lambda^{\prime};0)+\dots$
(9.82)

Expanding $A^{(n)}(\lambda;z)$ into $A^{(p,q)}(z)$ in the above formula, we
get exactly the OPE 9.80.

We leave it to future work to analyze the terms with derivatives.

We can also look at the coupling between holomorphic Chern-Simons theory and
the Kodaira-Spencer gravity studied in Section 7.5. When we take the Lie
algebra to be the abelian Lie algebra $\Pi\mathbb{C}^{2}$ as in the last
section, this coupling actually reflects the self-interaction of Kodaira-
Spencer gravity. For example, from the coupling 7.41 we obtain the following

$\frac{1}{2}\int\bm{\beta}[p+r,q+s]\mathbf{A}_{1}[p,r]\partial_{z}\mathbf{A}_{2}[r,s]+\dots$
(9.83)

This gives us the following boundary OPE

$B_{1}^{\prime}[p,q](z)B_{2}^{\prime}[r,s]\sim\frac{1}{z^{2}}\beta[p+r,q+s]$
(9.84)

On the B brane side, this OPE is given by a contraction of $\partial c$ and
$b$ fields

$\wick{\operatorname{Tr}(\c{b}Z^{(i_{1}}\dots
Z^{i_{n})})(z)\operatorname{Tr}(\c{\pa}cZ^{(i^{\prime}_{1}}\dots
Z^{i^{\prime}_{n^{\prime}})}))(0)}\sim\frac{1}{z^{2}}\operatorname{Tr}Z^{(i_{1}}Z^{i_{n}}\dots
Z^{i^{\prime}_{1}}Z^{i^{\prime}_{n^{\prime}})}.$ (9.85)

We will study OPE of this type in more detail in the next section.

###  9.7 Chiral from open-closed coupling

In this section, we analyze the OPE coming from the coupling between
holomorphic Chern-Simons and the Kodaira-Spencer gravity.

##### closed + open $\rightarrow$ open

First, we look at the flat space coupling between $\mathbf{B},\mathbf{A}$ and
$\bm{\alpha}$ fields 7.41. This coupling gives us the following OPE

$\beta[p,q](z)B_{a}[r,s](0)\sim\frac{ps-qr}{z}B_{a}[p+r-1,q+s-1](0).$ (9.86)

Similarly, we can consider the deformed coupling
$I_{OC}^{(2,k)}[\mathbf{B},\mathbf{A},\bm{\alpha}]$ in 8.49.

$I^{(2,k)}_{OC}$$\beta[p,q]$$B_{a}[r,s]$$B_{a}[p+r-k-1,q+s-k-1]$

The coefficient for the leading order term (without $z$ derivatives term) is
computed by

$\displaystyle\sum_{k_{1}+k_{2}=k}\frac{(p+q-k-1)}{(p+q-1)}\frac{k!R_{k+1}(p,q,r,s)}{k_{1}!k_{2}![p+q-2]_{k_{1}}[r+s]_{k_{2}}[p+q+r+s-k-1]_{k}}$
(9.87) $\displaystyle=\frac{R_{k+1}(p,q,r,s)}{[p+q-1]_{k}[r+s]_{k}}.$

Therefore we have the following OPE

$\beta[p,q](z)B_{a}[r,s](0)\sim\frac{N^{k}}{z^{k+1}}\frac{R_{k+1}(p,q,r,s)}{[p+q-1]_{k}[r+s]_{k}}B_{a}[p+r-k-1,q+s-k-1]+\dots\;,$
(9.88)

where we omitted terms with derivatives of $B$.

On the B-brane side, this correspond to the wick contraction of $k+1$ adjacent
$Z$ fields between $A^{(n)}(\lambda;z)$ and
$E^{(n^{\prime})}(\lambda^{\prime},0)$ operators. For example,

$\displaystyle\wick{\operatorname{Tr}(Z(\lambda;z)\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}\c{3}Z(\lambda;z)\c{2}\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}\c{1}Z(\lambda;z))I(0)\c{1}Z(\lambda;0)\c{2}\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}\c{3}Z(\lambda;0)\makebox[10.00002pt][c]{$\cdot\hss\cdot\hss\cdot$}Z(\lambda;0)J(0)}$
(9.89) $\displaystyle=$ $\displaystyle
N^{k}\frac{[\lambda\lambda^{\prime}]^{k+1}}{z^{k+1}}I(0)Z(\lambda;z)^{n-k-1}Z(\lambda^{\prime};0)^{n^{\prime}-k-1}J(0).$

There are $n(n^{\prime}-k)$ such contractions. As in the previous section, we
use the BRST relation to symmetrize the operator and discard operators in
different BRST representatives. Omitting terms with derivatives of $E$ field,
we have

$A^{(n)}(\lambda;z)E^{(n^{\prime})}_{a_{1}a_{2}}(\lambda^{\prime};0)\sim
n(n^{\prime}-k)\frac{N^{k}[\lambda\lambda^{\prime}]^{k+1}}{z^{k+1}}\frac{(n^{\prime}-k-1)!}{(n+n^{\prime}-k-2)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-k-1}E^{(n+n^{\prime}-2k-2)}_{a_{1}a_{2}}(\lambda^{\prime};0)+\dots$
(9.90)

Expanding the generating fields into component, we find that this OPE is
exactly the boundary OPE 9.88

##### open + open $\rightarrow$ closed

We can also consider the coupling between $\mathbf{A},\mathbf{A}$ and
$\bm{\beta}$ fields. The flat space coupling gives us the following OPE

$B_{a}[p,q](z)B_{b}[r,s](0)\sim-\frac{K_{ab}}{z^{2}}\beta[p+r,q+s](0)+\dots$
(9.91)

The deformed coupling $I_{OC}^{(2,k)}[\mathbf{A},\mathbf{A},\bm{\beta}]$
8.50gives us the following OPE

$B_{a}[p,q](z)B_{b}[r,s](0)\sim-\frac{N^{k}K_{ab}}{z^{k+2}}\frac{(k+1)R_{k}(p,q,r,s)}{[p+q]_{k}[r+s]_{k}}\beta[p+r-k,q+s-k](0)+\dots$
(9.92)

where we omitted terms with derivatives of $\beta$.

On the B-brane side, this OPE corresponds to the wick contraction of two $I-J$
fields and $k$ pairs of $Z$ fields. The contracted $Z$ fields must be
alongside one of the $I$ or $J$ fields. As a illustration, we list all
possible contractions for $2$ pair of $Z$ fields

$\wick{(\c{4}I\dots\c{3}Z\c{2}Z\c{1}J)(z)(\c{1}I\c{2}Z\c{3}Z\dots\c{4}J)(0)}\quad\wick{(\c{4}I\c{3}Z\dots\c{2}Z\c{1}J)(z)(\c{1}I\c{2}Z\dots\c{3}Z\c{4}J)(0)}\quad\wick{(\c{4}I\c{3}Z\c{2}Z\dots\c{1}J)(z)(\c{1}I\dots\c{2}Z\c{3}Z\c{4}J)(0)}$
(9.93)

For contraction of $k$ pairs of $Z$ fields, there are $k+1$ numbers of such
contractions in total. This gives us the following

$E^{(n)}_{a_{1}a_{2}}(\lambda;z)E^{(n^{\prime})}_{b_{1}b_{2}}(\lambda^{\prime};0)\sim-\frac{N^{k}[\lambda\lambda^{\prime}]^{k}}{z^{k+2}}\delta_{a_{1}b_{2}}\delta_{a_{2}b_{1}}(k+1)\frac{(n^{\prime}-k)!}{(n+n^{\prime}-k)!}(\lambda\cdot\partial_{\lambda^{\prime}})^{n-k}A^{(n+n^{\prime}-2k)}(\lambda^{\prime};0)+\dots$
(9.94)

Expanding this OPE we get exactly 9.92.

Finally, we briefly analyze boundary OPE corresponding to the quartic open-
closed coupling.

$I^{(3,0)}_{OC}$$B_{a}[p,q]$$B_{b}[r,s]$$\beta[u,v]$$B_{c}[p+r-u-1,q+s-v-1]$

The coupling 7.44 lead us the following boundary OPE

$B_{a}[p,q](z)B_{b}[r,s](0)\sim\frac{f_{ab}^{c}}{z^{2}}\sum_{u,v}\left((m_{3})^{p,q;r,s}_{u,v}-(m_{3})^{r,s;p,q}_{u,v}\right)\beta[u,v]B_{c}[p+r-u-1,q+s-v-1](0)+\dots$
(9.95)

In the special case $p=1,q=0$, this is computed in 9.55 and gives us the
following

$\displaystyle B_{a}[1,0](z)B_{b}[r,s](0)\sim$
$\displaystyle\frac{f_{ab}^{c}}{z^{2}}\sum_{u,v}\binom{r+s}{r}^{-1}\binom{u+v}{u}\binom{r+s-u-v-1}{r-u}$
(9.96) $\displaystyle\times\beta[u,v]B_{c}[p+r-u-1,q+s-v-1](0)\dots$

The corresponding wick contraction on B-brane is given by a single $I-J$
contraction, together with a single $Z-Z$ contraction that at least one $Z$ is
not adjacent to the contracted $I-J$. This gives us the following

$E^{(1)}_{a_{1}a_{2}}(\lambda,z)E^{(n^{\prime})}_{b_{1}b_{2}}(\lambda^{\prime},z)\sim\sum_{n^{\prime\prime}\geq
1}\frac{[\lambda\lambda^{\prime}]}{z^{2}}A^{(n^{\prime\prime})}(\lambda^{\prime},0)(\delta_{a_{2}b_{1}}E^{(n^{\prime}-n^{\prime\prime}-1)}_{a_{1}b_{2}}(\lambda^{\prime},0)-\delta_{a_{1}b_{2}}E^{(n^{\prime}-n^{\prime\prime}-1)}_{a_{2}b_{1}}(\lambda^{\prime},0))+\dots$
(9.97)

Expanding the generating fields into component, we get the same constant
coefficient as the expansion of 9.57. We can check that this OPE match
precisely the boundary OPE 9.96.

###  9.8 Comments on loops and non-planar contributions

In this section, we briefly analyze a simple loop diagram that contributes to
the OPE. We consider the following simplest loop diagrams.

$A^{a}[2,0]$$\partial_{z}A^{b}[u-1,v+1]$$\beta[u,v]$$A^{b}[0,2]$$B_{b}[u-1,v+1]$$\alpha[u,v]$$B_{a}[2,0]$$B_{b}[0,2]$$B_{c}[0,0]$

By a dimensional analysis [19], one can show that the Feynman integral gives
us a pole of order $3$. The Lie algebra factor gives us $f_{ab}^{c}$.
Therefore the corresponding OPE takes the following form

$B_{a}[2,0](z)B_{b}[0,2](0)\sim\frac{f_{ab}^{c}}{z^{3}}B_{c}[0,0](0).$ (9.98)

We will see in Section 10.3 that the summation over the internal KK modes in
the above diagram actually diverges. We assume that a proper regularization
can be employed to make sense of the summation.

There is a natural candidate for the corresponding OPE on the B brane side. We
consider the following non-planar wick contraction

$\wick{(I_{a_{1}}\c{3}Z^{1}\c{2}Z^{1}\c{1}J_{a_{2}})(z)(\c{1}I_{b_{1}}\c{3}Z^{2}\c{2}Z^{2}J_{b_{2}})(0)}=\frac{1}{z^{3}}\delta_{a_{2}b_{1}}I_{a_{1}}J_{b_{2}}(0).$
(9.99)

If we contract instead $I_{a_{1}}-J_{b_{2}}$, we get the term
$\frac{1}{z^{3}}\delta_{a_{1}b_{2}}I_{a_{2}}J_{b_{1}}(0)$. Together with the
above term, they reproduce exactly the boundary 1-loop OPE.

As we have discussed in Section 9.4, the tree-level OPE actually defined a
limit of the full chiral algebra and corresponds to the structure of a Poisson
vertex algebra (or coisson algebra). Computing loop Feynman diagrams in the
holomorphic topological theory can be understood mathematically as solving the
deformation quantization problem of coisson algebra outlined in [50].

$\framebox[1.1pt]{Coisson algebra}\overset{\text{"deformation
quantization"}}{\Longrightarrow}\framebox[1.1pt]{Full chiral
algebra}\overset{?}{\leftrightsquigarrow}\framebox[1.1pt]{Full B-brane chiral
algebra}$

Suppose we can find a consistent regularization to deal with the divergence in
summing the KK modes, then it will be interesting to match loops contribution
to the boundary OPE with non-planar wick contractions of the B-brane chiral
algebra.

##  10 Celestial holography from boundary chiral algebra

###  10.1 Celestial holography and twistor correspondence

In this section, we review the various relationships between celestial
holography and twistor correspondence following [13].

Recall that the twistor space $\mathbb{PT}$ of flat Euclidean spacetime can be
identified with the total space of the rank $2$ bundle

$\mathcal{O}(1)\oplus\mathcal{O}(1)\to\mathbb{CP}^{1}.$ (10.1)

More invariantly, let $S_{+}$,$S_{-}$ denote the two spinor representation of
$\mathrm{Spin}(4)=SU(2)\times SU(2)$. Then we can identify $\mathbb{PT}$ with
the total space of the bundle $\mathcal{O}(1)\otimes
S_{-}\to\mathbb{P}(S_{+})$. The action of $\mathrm{Spin}(4)$ on $\mathbb{PT}$
is thus evident.

There is a non-holomorphic fibration

$\pi:\mathbb{PT}\to\mathbb{R}^{4}$ (10.2)

called twistor fibration. It has the property that for any
$u\in\mathbb{R}^{4}$, $\pi^{-1}(u)\cong\mathbb{CP}_{u}^{1}$ and is called a
twistor line. The twistor fibration can be explicitly written in a local
coordinate. Let $z$ be a coordinates on $\mathbb{CP}^{1}$ and $v_{i}$, $i=1,2$
be the coordinates on the $\mathcal{O}(1)$ fiber. Then a point
$(u_{1},\dots,u_{4})\in\mathbb{R}^{4}$ correspond to the twistor line
$\mathbb{CP}_{u}^{1}$ defined by

$\displaystyle v_{1}=x_{1}+ix_{2}+z(x_{3}-ix_{4}),$ (10.3) $\displaystyle
v_{2}=x_{3}+ix_{4}-z(x_{1}-ix_{2}),$

which is also called the incidence relations.

The original Penrose-Ward correspondence [51] states that there is a natural
bijection between vector bundle on $\mathbb{R}^{4}$ equipped with self-dual
connection and holomorphic vector bundles on $\mathbb{PT}$.

A more refined version of Penrose-Ward correspondence (see [52, 53]) states
that the self-dual Yang-Mills theory on $\mathbb{R}^{4}$ is equivalent to the
holomorphic BF theory on $\mathbb{PT}$.

There are many variations of Penrose-Ward correspondence for holomorphic
theory on twistor theory and four dimensional quantum field theories. We list
some of them:

  1. 1.

Holomorphic Chern-Simons theory on twistor space and four dimensional WZW
model [54, 55, 56]

  2. 2.

Holomorphic Chern-Simons theory on super twistor space $\mathbb{PT}^{3|4}$ and
four dimensional self-dual $\mathcal{N}=4$ SYM [20].

  3. 3.

Holomorphic Poisson BF theory on twistor space and self-dual Einstein gravity
[57, 58].

The relationships between celestial holography and twistor correspondence
start by looking at the algebra of the twistor theory
$\mathrm{Obs}_{6d}|_{\pi^{-1}(0)}$ restricted to the twistor line
$\pi^{-1}(0)\cong\mathbb{CP}^{1}$. This is a dg chiral algebra, and its Koszul
dual chiral algebra $(\mathrm{Obs}_{6d}|_{\pi^{-1}(0)})^{!}$ plays the
fundamental role in this story. As is explained in [13], we have the following
dictionary between the chiral algebra $(\mathrm{Obs}_{6d}|_{\pi^{-1}(0)})^{!}$
and the four dimensional theory

  1. 1.

The generators of the chiral algebra are in bijection with single-particle
conformal primary states of the four-dimensional theory.

  2. 2.

Conformal blocks of the chiral algebra are in bijection with local operators
in the four-dimensional theory.

  3. 3.

Scattering amplitudes of the $4d$ theory in the presence of a chosen local
operator at a fixed position are equal to correlation functions of the chiral
algebra defined using the corresponding conformal block.

In this section, we apply our results of the KK theory to compute the Koszul
dual chiral algebra $(\mathrm{Obs}_{6d}|_{\pi^{-1}(0)})^{!}$ in holomorphic BF
theory and holomorphic Poisson BF theory. We work in the open patch
$\mathbb{C}^{3}\subset\mathbb{PT}$ of twistor space. Note that the twistor
line $\pi^{-1}(0)$ is exactly the boundary $\mathbb{CP}^{1}$ at $r=0$ in our
previous study. Therefore the chiral algebra
$(\mathrm{Obs}_{6d}|_{\pi^{-1}(0)})^{!}$ can be computed by the boundary
chiral algebra of the KK theory with transversal boundary condition. This is
the same boundary condition we analyzed in the twisted holography chiral
algebra.

###  10.2 Chiral algebra for holomorphic BF theory

In this section, we study the (tree-level) boundary OPE for the holomorphic BF
theory.

The $3d$ KK theory of holomorphic BF theory have fields
$\mathbf{A},\tilde{\mathbf{A}},\mathbf{B},\tilde{\mathbf{B}}$. The original
$6d$ fields are related to the KK fields as follows

$\displaystyle\boldsymbol{\EuScript{A}}$ $\displaystyle=\sum_{p,q\geq
0}\mathbf{A}[p,q]z_{1}^{p}z_{2}^{q}+\frac{(p+q+1)!}{p!q!}\tilde{\mathbf{B}}[p,q]\frac{\bar{z}_{1}^{p}\bar{z}_{2}^{q}}{r^{2(p+q)}}\frac{\epsilon_{ij}\bar{z}_{i}d\bar{z}_{j}}{r^{4}},$
(10.4) $\displaystyle\boldsymbol{\EuScript{B}}$ $\displaystyle=\sum_{p,q\geq
0}\tilde{\mathbf{A}}[p,q]z_{1}^{p}z_{2}^{q}+\frac{(p+q+1)!}{p!q!}\mathbf{B}[p,q]\frac{\bar{z}_{1}^{p}\bar{z}_{2}^{q}}{r^{2(p+q)}}\frac{\epsilon_{ij}\bar{z}_{i}d\bar{z}_{j}}{r^{4}}.$

We choose the boundary condition such that $\mathbf{A},\tilde{\mathbf{A}}=0$
at the boundary. The boundary chiral algebra is generated by fields
$B,\tilde{B}$ and their derivatives. In the celestial holography context, the
$6d$ geometry is not deformed. The flat space action functional for the KK
theory has all the information we need to compute the boundary chiral algebra.
Given the interactions 5.12, the tree-level Feynman diagrams computing the OPE
of two boundary $B$ fields read as follows

$B$$B$$B$$B$$B$$B$$\tilde{B}$$B$$B$$B$$\tilde{B}$$\tilde{B}$$\cdots$

The corresponding boundary OPE takes the following schematic form

$B(z)B(0)\sim\frac{1}{z}\left(B(0)+(m_{3})B\tilde{B}(0)+(m_{4})B\tilde{B}\tilde{B}(0)\dots\right).$
(10.5)

More precisely, the first term

$B_{a}(z)B_{b}(0)\sim\frac{f^{c}_{ab}}{z}B_{c}(0)$ (10.6)

is the Kac-Moody chiral OPE. It is shown in [13] that the corresponding
correlation function reproduces the Parke-Taylor formula. The second term is
given by the quadratic interaction, a similar term also appears in the twisted
holography chiral algebra in Section 9.3.3. More explicitly, this term can be
written as follows

$B_{a}[p,q](z)B_{b}[r,s](0)\sim\sum_{\begin{subarray}{c}u_{1}+u_{2}=p+r-1\\\
v_{1}+v_{2}=q+s-1\end{subarray}}\sum_{c,d,e,f}\frac{K^{fc}}{2z}\left((m_{3})^{p,q;r,s}_{u_{1},v_{1}}f_{bf}^{e}f_{ae}^{d}-(m_{3})^{r,s;p,q}_{u_{1},v_{1}}f_{af}^{e}f_{be}^{d}\right)B_{c}[u_{1},v_{1}]\tilde{B}_{d}[u_{2},v_{2}](0),$
(10.7)

where we used the shorthand
$(m_{3})^{p,q;r,s}_{u,v}:=(m_{3})^{p,q;r,s}_{u,v;p+r-1-u,q+s-1-v}$, which can
be computed by the formula 4.39 and 4.45. A special case of this formula is
obtained in [13].

Using the full action functional of the KK theory 4.5 (though we write the
action for holomorphic Chern-Simons, it is easy to generalize it to
holomorphic BF theory), one can explicitly write down the remaining OPE.

$\displaystyle B_{a}[p,q](z)B_{b}[r,s](0)\sim\sum_{n\geq
2}\frac{1}{(n-2)!}\sum_{\begin{subarray}{c}u_{1}+\dots u_{n-1}=p+r-n+2\\\
v_{1}+\dots
v_{n-1}=q+s-n+2\end{subarray}}\sum_{c_{1},\dots,c_{n-1}}\sum_{\sigma\in
S_{n-1}}(m_{n})^{p,q;r,s}_{u_{\sigma(1)},v_{\sigma(1)};\dots;u_{\sigma(n-1)},v_{\sigma(n-1)}}$
(10.8) $\displaystyle\times
K(t_{a},[...[[t_{b},t^{c_{\sigma(1)}}],t^{c_{\sigma(2)}}],...,t^{c_{\sigma(n-1)}}])B_{c_{1}}[u_{1},v_{1}]\tilde{B}_{c_{2}}[u_{2},v_{2}]\cdots\tilde{B}_{c_{n-1}}[u_{n-1},v_{n-1}].$

We recollect our results from Section 4.4 that compute the constant
$(m_{n})^{p,q;r,s}_{u_{1},v_{1};\dots;u_{n-1},v_{n-1}}$. First, we change our
variable as follows

$\begin{cases}&p=j_{1}+m_{1}\\\
&q=j_{1}-m_{1}\end{cases},\quad\begin{cases}&r=j_{2}+m_{2}\\\
&s=j_{2}-m_{2}\end{cases},\quad\begin{cases}&u_{i}=\bar{j}_{i}-\bar{m}_{i}\\\
&v_{i}=\bar{j}_{i}+\bar{m}_{i}\end{cases}.$ (10.9)

Note that the constraint $u_{1}+\dots u_{n-1}=p+r-n+2$, $v_{1}+\dots
v_{n-1}=q+s-n+2$ is equivalent to

$\bar{j}_{1}+\bar{j}_{2}+\dots+\bar{j}_{n-1}=j_{1}+j_{2}-n+2,\quad\bar{m}_{1}+\bar{m}_{2}+\dots\bar{m}_{n}=-(m_{1}+m_{2}).$
(10.10)

Then we have

$\displaystyle(m_{n})^{p,q;r,s}_{u_{1},v_{1};\dots;u_{n-1},v_{n-1}}=\sqrt{\frac{(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{1}+m_{2})!}{(2j_{1}+1)!(2j_{2}+1)!}}\left(\prod_{l=1}^{n-1}\sqrt{\frac{(2\bar{j}_{l}+1)!}{(\bar{j}_{l}+\bar{m}_{l})!(\bar{j}_{l}-\bar{m}_{l})!}}\right)$
(10.11) $\displaystyle\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-2}\\\
i_{l-1}\leq
i_{l}\leq\min\\{2j_{2}-l,2\bar{J}_{l}+l-1\\}\end{subarray}}\sqrt{2j_{2}-n+3}\prod_{l=2}^{n-1}\sqrt{\frac{(2j_{2}-l+3)(2\bar{j}_{l}+1)(2J_{l-1}-2i_{l-1}+1)}{(2j_{2}-l+2-i_{l-1})(2\bar{J}_{l-1}+l-1-i_{l-1})}}$
$\displaystyle
C^{j_{2},\bar{j}_{1};j_{2}+\bar{j}_{1}-i_{1}}_{m_{2},\bar{m}_{1};m_{2}+\bar{m}_{1}}\left.\left(\prod_{l=2}^{n-1}\begin{Bmatrix}\bar{J}_{l-1}+\frac{l-1}{2}&j_{2}-\frac{l-1}{2}&J_{l-1}-i_{l-1}\\\
J_{l}-i_{l}&\bar{j}_{l}&\bar{J}_{l}+\frac{l-1}{2}\end{Bmatrix}C^{J_{l-1}-i_{l-1},\bar{j}_{l},J_{l}-i_{l}}_{m_{2}+\bar{M}_{l-1},m_{l},m_{2}+\bar{M}_{l}}\right)\right|_{i_{n-1}=2j_{2}-n+2}.$

where

$\displaystyle\bar{J}_{l}=\bar{j}_{1}+\dots+\bar{j}_{l},$ (10.12)
$\displaystyle J_{l}=j_{2}+\bar{J}_{l},$
$\displaystyle\bar{M}_{l}=\bar{m}_{1}+\dots+\bar{m}_{l}.$

Though our current computation of the boundary OPE is completely at the tree-
level, the interaction vertices we used are effective interactions that
already contain Feynman diagrams in themselves. These tree-level results
should reproduce loop level computations in the Koszul dual picture. Indeed,
the $BB\sim\frac{1}{z}B\tilde{B}$ OPE given above is from a one loop
computation in [13]. More generally, the $BB\sim B\tilde{B}^{n}$ OPE should
correspond to a $n$ loop computation in the Koszul dual picture and also a $n$
loop result in the form factor computation (However they are not the full loop
results).

Similarly, The tree-level Feynman diagrams for a boundary $B$ field and a
$\tilde{B}$ field consist of the following

$B$$\tilde{B}$$\tilde{B}$$B$$\tilde{B}$$\tilde{B}$$\tilde{B}$$B$$\tilde{B}$$\tilde{B}$$\tilde{B}$$\tilde{B}$$\cdots$

The corresponding boundary OPE is given by

$\displaystyle
B_{a}[p,q](z)\tilde{B}_{b}[r,s](0)\sim\frac{f^{c}_{ab}}{z}\tilde{B}_{c}[p+r,q+s](0)$
(10.13) $\displaystyle+$ $\displaystyle\sum_{n\geq
3}\frac{1}{(n-1)!}\sum_{\begin{subarray}{c}u_{1}+\dots u_{n-1}=p+r-n+2\\\
v_{1}+\dots
v_{n-1}=q+s-n+2\end{subarray}}\sum_{c_{1},\dots,c_{n-1}}\sum_{\sigma\in
S_{n-1}}(m_{n})^{p,q;r,s}_{u_{\sigma(1)},v_{\sigma(1)};\dots;u_{\sigma(n-1)},v_{\sigma(n-1)}}$
$\displaystyle
K(t_{a},[...[[t_{b},t^{c_{\sigma(1)}}],t^{c_{\sigma(2)}}],...,t^{c_{\sigma(n-1)}}])\tilde{B}_{c_{1}}[u_{1},v_{1}]\tilde{B}_{c_{2}}[u_{2},v_{2}]\cdots\tilde{B}_{c_{n-1}}[u_{n-1},v_{n-1}].$

As is explained in [13], holomorphic BF theory alone suffers from an anomaly
on twistor space. As a result, we might eventually get into trouble with the
boundary chiral algebra at the full quantum level. For holomorphic BF theory,
the anomaly can be canceled by coupling with the free limit of Kodaira-Spencer
theory via a Green-Schwarz mechanism [41]. In four dimension, the Kodaira-
Spencer field introduces an axion field coupled with the self-dual gauge
theory. For the boundary chiral algebra, the Kodaira Spencer fields introduce
two towers of boundary generators. We can easily write their OPE via the
coupling of KK theory we obtained in Section 7.6

$\displaystyle B_{a}[p,q](z)\beta[r,s](0)$ $\displaystyle\sim\frac{ps-
qr}{z}\tilde{B}_{a}[p+r-1,q+s-1],$ (10.14) $\displaystyle
B_{a}[p,q](z)\tilde{\beta}[r,s](0)$
$\displaystyle\sim-\frac{(p+q+r+s+2)}{z^{2}}\tilde{B}_{a}[p+r,q+s]+\frac{r+s+2}{z}\partial_{z}\tilde{B}_{a}[p+r,q+s],$
$\displaystyle B_{a}[p,q]B_{b}[r,s]$
$\displaystyle\sim-\frac{K_{ab}}{z^{2}}\beta[p+r,q+s]+\frac{r+s}{p+r+q+s}\frac{K_{ab}}{z}\partial_{z}\beta[p+r,q+s]$
$\displaystyle-\frac{ps-
qr}{p+q+r+s}\frac{K_{ab}}{z}\tilde{\beta}[p+r-1,q+s-1].$

Under a rescaling
$\beta[r,s]\to(r+s)\beta[r,s],\tilde{\beta}[r,s]\to(r+s+2)\tilde{\beta}[r,s]$,
we reproduce the OPE in [13] computed using Koszul duality 999In [13], the
authors used a different notation for the fields, which is related to ours by
$B\to J,\tilde{B}\to\tilde{J},\beta\to E,\tilde{\beta}\to F$.

It won’t be hard to generalize our method to compute all the coupling of the
KK theory of Kodaira-Spencer gravity and holomorphic BF theory. This should
lead to a complete understanding of the ”tree-level” correction to the above
OPE. We leave this problem to future work.

###  10.3 Comments on loop corrections

So far, we have only considered tree-level boundary OPE. Loop corrections to
the boundary OPE are also important. In this section, we briefly analyze some
loop effects. We will see that they does not behave so well in the KK theory.
Divergence happens when we sum over the infinite KK tower and appropriate
regularization is needed.

As an example, we consider the following loop diagrams.

$A^{a}[p,q]$$B_{d}[u,v]$$A^{e}[r+u,s+v]$$A^{b}[r,s]$$A^{d}[u,v]$$B_{e}[r+u,s+v]$$B_{a}[p,q]$$B_{b}[r,s]$$\tilde{B}_{c}[p+r-1,q+s-1]$$B_{a}[p,q]$$B_{b}[r,s]$$\tilde{B}_{c}[p+r-1,q+s-1]$

The corresponding Feynman integral is actually difficult to evaluate and we
won’t compute it in this paper. Nevertheless, simply by a dimensional analysis
[19], one can show that it gives us a pole of order $2$. So we expect to have
OPE of the following form

$B(z)B(0)\sim\frac{1}{z^{2}}\tilde{B}(0).$ (10.15)

The corresponding color factor and the KK modes factor can be evaluated by
contracting the cubic and quartic interaction vertices. The first diagram
gives us the following

$\left(\sum_{u,v}\sum_{d,e,f,g}(m_{3})^{p,q;r+u,s+v}_{u,v}K^{df}f_{ef}^{g}f_{ag}^{c}+(m_{3})^{r+u,s+v;p,q}_{u,v}K^{df}f_{af}^{g}f_{eg}^{c}\right)f_{bd}^{e}.$
(10.16)

The first term gives the Lie algebra constant $2h^{\vee}f_{ab}^{c}$, where
$h^{\vee}$ is the dual coxter number. To compute the constant factor from the
KK modes, it will be easier to label the KK modes using quantum mechanics
notation. We change our notation by the following

$\begin{cases}&p=j_{1}+m_{1}\\\
&q=j_{1}-m_{1}\end{cases},\quad\begin{cases}&r=j_{2}+m_{2}\\\
&s=j_{2}-m_{2}\end{cases},\quad\begin{cases}&u=\bar{j}-\bar{m}\\\
&v=\bar{j}+\bar{m}\end{cases}.$ (10.17)

Recall that we denoted $N(j,m)=\sqrt{\frac{(j-m)!(j+m)!}{(2j+1)!}}$. Then we
have

$\displaystyle\sum_{u,v}(m_{3})^{p,q;r+u,s+v}_{u,v}=\sum_{\bar{j},\bar{m}}\sum_{i}\frac{(-1)^{\bar{j}-\bar{m}}N(j_{1},m_{1})N(j_{2}+\bar{j},m_{2}-\bar{m})}{N(\bar{j},\bar{m})N(j_{1}+j_{2}-1,m_{1}+m_{2})}\begin{Bmatrix}j_{2}+\bar{j}-\frac{1}{2}&\bar{j}+\frac{1}{2}&j_{2}+2\bar{j}-i\\\
j_{1}+j_{2}-1&j_{1}&j_{1}+j_{2}+\bar{j}-\frac{1}{2}\end{Bmatrix}$ (10.18)
$\displaystyle(-1)^{2j_{1}+2j_{2}-i+1}\sqrt{\frac{(2j_{1}+1)(2j_{2}+2\bar{j})(2j_{2}+2\bar{j}+1)(2j_{2}+4\bar{j}-2i+1)}{(2j_{2}+2\bar{j}-i)(2\bar{j}-i+1)}}C^{j_{2}+\bar{j},\bar{j};j_{2}+2\bar{j}-i}_{m_{2}-\bar{m},\bar{m};m_{2}}C^{j_{2}+2\bar{j}-i,j_{1};j_{1}+j_{2}-1}_{m_{2},m_{1};m_{1}+m_{2}}.$

Note that
$\frac{N(j_{2}+\bar{j},m_{2}-\bar{m})}{N(j_{2},m_{2})N(\bar{j},\bar{m})}=(-1)^{\bar{j}+\bar{m}}\sqrt{2\bar{j}+1}C^{j_{2}+\bar{j},\bar{j};j_{2}}_{m_{2}-\bar{m},\bar{m};m_{2}}$.
Moreover,
$\sum_{\bar{m}}C^{j_{2}+\bar{j},\bar{j};j_{2}}_{m_{2}-\bar{m},\bar{m};m_{2}}C^{j_{2}+\bar{j},\bar{j};j_{2}+2\bar{j}-i}_{m_{2}-\bar{m},\bar{m};m_{2}}=\delta_{j_{2},j_{2}+2\bar{j}-i}$.
Therefore, summation over $i$ in the above formula can be simplified to
letting $i=2\bar{j}$. We also have

$\begin{Bmatrix}j_{2}+\bar{j}-\frac{1}{2}&\bar{j}+\frac{1}{2}&j_{2}\\\
j_{1}+j_{2}-1&j_{1}&j_{1}+j_{2}+\bar{j}-\frac{1}{2}\end{Bmatrix}=\frac{(-1)^{2j_{1}+2j_{2}+2\bar{j}-1}\sqrt{(2j_{1})(2\bar{j}+1)}}{\sqrt{(2j_{1}+2j_{2})(2j_{1}+2j_{2}-1)(2j_{2}+2\bar{j})(2j_{2}+2\bar{j}+1)}}.$
(10.19)

Consequently,

$\displaystyle\sum_{u,v}(m_{3})^{p,q;r+u,s+v}_{u,v}$ (10.20)
$\displaystyle=\sum_{\bar{j}\in\frac{1}{2}\mathbb{Z}_{\geq
0}}(2\bar{j}+1)\frac{N(j_{1},m_{1})N(j_{2},m_{2})}{N(j_{1}+j_{2}-1,m_{1}+m_{2})}\sqrt{\frac{(2j_{1})(2j_{1}+1)(2j_{2}+1)}{(2j_{2})(2j_{1}+2j_{2})(2j_{1}+2j_{2}-1)}}C^{j_{2},j_{1};j_{1}+j_{2}-1}_{m_{2},m_{1};m_{1}+m_{2}}$
$\displaystyle=\left(\sum_{n\geq 0}(n+1)\right)\frac{(ps-
qr)}{(r+s)(p+q+r+s)}.$

Unfortunately, the summation $\sum_{n\geq 0}(n+1)$ does not converge.
Nevertheless, it still gives us the dependence on the $p,q,r,s$ indices.

The second term in 10.16 gives a Lie algebra factor $h^{\vee}f_{ab}^{c}$. The
KK modes part is computed by

$\displaystyle\sum_{u,v}(m_{3})^{r+u,s+v;p,q}_{u,v}=\sum_{\bar{j},\bar{m}}\sum_{i}\frac{(-1)^{2j_{1}+2j_{2}+2\bar{j}-i+1}N(j_{1},m_{1})N(j_{2},m_{2})}{N(j_{1}+j_{2}-1,m_{1}+m_{2})}\begin{Bmatrix}j_{1}-\frac{1}{2}&\bar{j}+\frac{1}{2}&j_{1}+\bar{j}-i\\\
j_{1}+j_{2}-1&j_{2}+\bar{j}&j_{1}+j_{2}+\bar{j}-\frac{1}{2}\end{Bmatrix}$
(10.21)