$\tilde{\tau}_{12}\left(\tilde{\mathscr{U}}\right)=\mathbb{H}\setminus\\{SL_{2}(\mathbb{Z})(i)\\}.$

Lemma proved.

∎

Here, we make the following deformation of the modular discriminant. Composing
the $t^{3}$ deformations (178) and (186) with the last equation of (176), we
obtain

$\begin{split}Q=\frac{\Delta(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})}{\tilde{y}^{6}},\end{split}$
(188)

where

$\begin{split}\Delta(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})&=\frac{\Delta\left(\tilde{\tau}=\tilde{\tau}_{12}+\sum_{n=1}^{\infty}\frac{\tilde{\tau}_{12,n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right)}{\left(1+\sum_{n=1}^{\infty}\frac{\tilde{r}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right)^{6}},\\\
&=\Delta(\tilde{\tau}_{12})+\sum_{n=1}^{\infty}\Delta_{n}(\tilde{\tau}_{12})\left(Q^{\frac{1}{3}}t^{3}\right)^{n}.\end{split}$
(189)

######  Lemma 4.5.

Let the power series $w_{1},w_{2},w_{3}$ be defined in (160). Then, the
function

$\tilde{\tau}_{1}\tilde{\tau}_{2}:=\frac{\tilde{w}_{1}(t^{1}-\lambda,Q,t^{3})}{\tilde{w}_{3}(t^{1}-\lambda,Q,t^{3})}$

satisfies the following relation

$t^{3}=\tilde{w}_{2}^{2}-4\tilde{w}_{1}\tilde{w}_{3}.$ (190)

Moreover, consider

$\left(\tilde{\tau}_{1}-\tilde{\tau}_{2}\right)^{2}:=\frac{t^{3}}{\tilde{w}_{3}^{2}}.$

Hence, the following relation holds true

$\tilde{\tau}_{1}\tilde{\tau}_{2}=\tilde{\tau}_{12}^{2}-\frac{1}{4}\frac{Q^{\frac{1}{3}}t^{3}}{\Delta^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})}.$

###### Proof.

The relation (190) holds true due to Lemma 4.2. Furthermore,

$\begin{split}\tilde{\tau}_{1}\tilde{\tau}_{2}&=\tilde{\tau}_{12}^{2}-\frac{\left(\tilde{\tau}_{1}-\tilde{\tau}_{2}\right)^{2}}{4},\\\
&=\tilde{\tau}_{12}^{2}-\frac{\left(t^{3}\right)^{2}}{4\tilde{w}_{3}^{2}},\\\
&=\tilde{\tau}_{12}^{2}-\frac{1}{4}\frac{Q^{\frac{1}{3}}t^{3}}{\Delta^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})}.\\\
\end{split}$

∎

Since, the Extended Gauss-Manin connection (10) has the same monodromy group
of the Gauss-Manin connection (8), the periods
$\tilde{\tau}_{1},\tilde{\tau}_{2},\tilde{y}$ have the following
transformation law due to (165)

$\begin{split}A(\tilde{\tau}_{1},\tilde{\tau}_{2},\tilde{y})&=\left(\frac{a\tilde{\tau}_{1}+b}{c\tilde{\tau}_{1}+d},\frac{a\tilde{\tau}_{2}+b}{c\tilde{\tau}_{2}+d},\tilde{y}\left(c\tilde{\tau}_{1}+d\right)\left(c\tilde{\tau}_{1}+d\right)\right),\\\
B(\tilde{\tau}_{1},\tilde{\tau}_{2},\tilde{y})&=(\tilde{\tau}_{1},\tilde{\tau}_{2},-\tilde{y}),\\\
\end{split}$ (191)

where ${\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}}\in SL_{2}(\mathbb{Z}).$
Using the change of coordinates (164), we have that $\tilde{\tau}_{12}$ has
the following transformation law under (191):

$\begin{split}&\text{If}\quad(\tilde{\tau}_{1},\tilde{\tau}_{2})\mapsto(\tilde{\tau}_{1}+1,\tilde{\tau}_{2}+1),\implies\tilde{\tau}_{12}\mapsto\tilde{\tau}_{12}+1,\\\
&\text{If}\quad(\tilde{\tau}_{1},\tilde{\tau}_{2})\mapsto(\frac{-1}{\tilde{\tau}_{1}},\frac{-1}{\tilde{\tau}_{2}}),\implies\tilde{\tau}_{12}\mapsto\frac{-\tilde{\tau}_{12}}{\tilde{\tau}_{1}\tilde{\tau}_{2}}.\\\
\end{split}$ (192)

Alternatively, due to Lemma 4.5 the group action (192) can be written as

$\begin{split}&\text{If}\quad(\tilde{\tau}_{1},\tilde{\tau}_{2})\mapsto(\tilde{\tau}_{1}+1,\tilde{\tau}_{2}+1),\implies\tilde{\tau}_{12}\mapsto\tilde{\tau}_{12}+1,\\\
&\text{If}\quad(\tilde{\tau}_{1},\tilde{\tau}_{2})\mapsto(\frac{-1}{\tilde{\tau}_{1}},\frac{-1}{\tilde{\tau}_{2}}),\implies\tilde{\tau}_{12}\mapsto\frac{-\tilde{\tau}_{12}}{\left(\tilde{\tau}_{12}^{2}-\frac{1}{4}\frac{Q^{\frac{1}{3}}t^{3}}{\Delta^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})}\right)}.\\\
\end{split}$ (193)

Note that for $t^{3}$ small enough the group action (193) gives rise to an
automorphism of $\mathbb{H}$. On another hand, the $A_{1}\times
PSL_{2}(\mathbb{Z})$ action in the periods $\tilde{\tau},\tilde{r},t^{3}$ is
the following

$\begin{split}A(\tilde{\tau},\tilde{r},t^{3})&=\left(\frac{a\tilde{\tau}+b}{c\tilde{\tau}+d},\tilde{r}\left(c\tilde{\tau}+d\right)^{2},t^{3}\right),\\\
B(\tilde{\tau},\tilde{r},t^{3})&=(\tilde{\tau},-\tilde{r},t^{3}).\\\
\end{split}$ (194)

where ${\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}}\in SL_{2}(\mathbb{Z}).$
Therefore, the map (178) has the following transformation law

$\begin{split}&\tilde{\tau}\left(\tilde{\tau}_{12}+1,Q^{\frac{1}{3}}t^{3}\right)=\tilde{\tau}\left(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}\right)+1,\\\
&\tilde{\tau}\left(\frac{-\tilde{\tau}_{12}}{\left(\tilde{\tau}_{12}^{2}-\frac{1}{4}\frac{Q^{\frac{1}{3}}t^{3}}{\Delta^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})}\right)},Q^{\frac{1}{3}}t^{3}\right)=\frac{-1}{\tilde{\tau}\left(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}\right)},\\\
\end{split}$ (195)

At this stage, we can prove the main theorem of this section

######  Theorem 4.6.

  1. 1.

Let the function

$\tilde{\tau}_{12}:\mathbb{H}\setminus{SL_{2}(\mathbb{Z})(i)}\mapsto\mathbb{H}\setminus{SL_{2}(\mathbb{Z})(i)}$

given by

$\tilde{\tau}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})=\tilde{\tau}_{12}+\sum_{n=1}^{\infty}\frac{\tilde{\tau}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}$

be defined in (178). Then, the Landau Ginzburg superpotential of Big Quantum
Cohomology of $\mathbb{CP}^{2}$ is a family of functions

$\lambda(\tilde{\tau}_{12},t^{1},Q,t^{3}):\mathbb{H}\mapsto\mathbb{C}$

with holomorphic dependence in the parameter space

$\Big{\\{}(t^{1},Q^{\frac{1}{3}},Q^{\frac{1}{3}}t^{3})\in\mathbb{C}\times\mathbb{C}^{*}\times\mathbb{C}:|Q^{\frac{1}{3}}t^{3}|<\left(\frac{1}{a}\right)^{\frac{1}{3}}\Big{\\}}$

and given by

$\lambda(\tilde{\tau}_{12},t^{1},Q,t^{3})=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})$
(196)

where

$\begin{split}J^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})&:=J^{\frac{1}{3}}\left(\tilde{\tau}_{12}+\sum_{n=1}^{\infty}\frac{\tilde{\tau}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right),\\\
&=J^{\frac{1}{3}}(\tilde{\tau}_{12})+\sum_{n=1}^{\infty}J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})\left(Q^{\frac{1}{3}}t^{3}\right)^{n},\end{split}$
(197)

and

$J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12}):=\frac{1}{n!}\sum_{k=0}^{n}k!\frac{\partial^{k}J^{\frac{1}{3}}}{\partial\tilde{\tau}_{12}}(\tilde{\tau}_{12})B_{n,k}\left(1!\frac{\tilde{\tau}_{1}(\tilde{\tau}_{12})}{\Delta^{\frac{1}{3}}(\tilde{\tau}_{12})},2!\frac{\tilde{\tau}_{2}(\tilde{\tau}_{12})}{\Delta^{\frac{2}{3}}(\tilde{\tau}_{12})},..,n!\frac{\tilde{\tau}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\right).$

The functions $B_{n,k}(x_{1},..,x_{n})$ are the Partial Bell polynomials
defined in (264).

The coefficients $J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})$ belong to the
following ring

$J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})\in\Delta^{\frac{-n}{3}}\mathbb{C}[E_{2},E_{4},E_{6}].$

In addition, the correspondent Abelian differential $\phi$ is given by

$\phi=-\frac{2^{\frac{5}{2}}}{2\pi}\frac{\Delta^{\frac{1}{6}}\left(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}\right)}{Q^{\frac{1}{6}}}d\tilde{\tau}_{12}.,$
(198)

where $\Delta\left(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}\right)$ is defined
in (189).

  2. 2.

Denote by $\Gamma^{(3)}_{Q^{\frac{1}{3}}t^{3}}$ the image of the group
$\Gamma^{(3)}$ under the group homomorphism (193). Then the Landau Ginzburg
superpotential of Big Quantum Cohomology of $\mathbb{CP}^{2}$ (196) is
$\Gamma^{(3)}_{Q^{\frac{1}{3}}t^{3}}$\- invariant.

###### Proof.

We obtain the Landau-Ginzburg superpotential of $QH^{*}\mathbb{(}{CP}^{2})$
for $t^{3}\neq 0$ by applying the Dubrovin construction of Landau Ginzburg
superpotential, see Theorem 2.4. More specifically, in order to obtain the LG
superpotential, we invert (175) in $\lambda$, which is done in practice by
composing (178) with (135) :

Using Faa di Bruno formula (265)

$\begin{split}\lambda(\tilde{\tau}_{12},t^{1},Q,t^{3})&=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}\left(\tilde{\tau}_{12}+\sum_{n=1}^{\infty}\frac{\tilde{\tau}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right),\\\
&=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}\left(\tilde{\tau}_{12}+\sum_{n=1}^{\infty}n!\frac{\tilde{\tau}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\frac{\left(Q^{\frac{1}{3}}t^{3}\right)^{n}}{n!}\right),\\\
&=t^{1}+3Q^{\frac{1}{3}}\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}k!\frac{\partial^{k}J^{\frac{1}{3}}}{\partial\tilde{\tau}_{12}}B_{n,k}\left(1!\frac{\tilde{\tau}_{1}}{\Delta^{\frac{1}{3}}},2!\frac{\tilde{\tau}_{2}}{\Delta^{\frac{2}{3}}},..,n!\frac{\tilde{\tau}_{n}}{\Delta^{\frac{n}{3}}}\right)\right)\frac{\left(Q^{\frac{1}{3}}t^{3}\right)^{n}}{n!},\\\
&=t^{1}+3Q^{\frac{1}{3}}\sum_{n=0}^{\infty}J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})\left(Q^{\frac{1}{3}}t^{3}\right)^{n},\\\
&=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}).\\\
\end{split}$ (199)

The coefficients $J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})$ are polynomial in
$\frac{\tau_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}$,
which belongs to
$\Delta^{\frac{-n}{3}}\mathbb{C}[E_{2},E_{4},E_{6},E_{6}^{-1}]$ due to
corollary 4.2.1 and lemma 4.4. But the Taylor expansion of any Landau-Ginzburg
superpotential near the discriminant is of the form

$\lambda=u_{i}+\lambda^{\prime\prime}(\tilde{\tau}_{12}^{0})\frac{\left(\tilde{\tau}_{12}-\tilde{\tau}_{12}^{0}\right)^{2}}{2}+O\left(\tilde{\tau}_{12}-\tilde{\tau}_{12}^{0}\right)^{3},$

which is holomorphic. Hence, we can extend the domain of Landau-Ginzburg
superpotential from $\mathbb{H}\setminus\\{E_{6}((\tilde{\tau}_{12})=0\\}$ to
$\mathbb{H}$, which also implies that the coefficients
$J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})$ are not rational in $E_{6}$.
Moreover, the Landau-Ginzburg superpotential is holomorphic with respect its
parameters $(t^{1},Q^{\frac{1}{3}},Q^{\frac{1}{3}}t^{3})$ in the space

$\Big{\\{}(t^{1},Q^{\frac{1}{3}},Q^{\frac{1}{3}}t^{3})\in\mathbb{C}\times\mathbb{C}^{*}\times\mathbb{C}:|Q^{\frac{1}{3}}t^{3}|<\left(\frac{1}{a}\right)^{\frac{1}{3}}\Big{\\}}$

because, we can extend the parameter space over the discriminant locus as it
was discussed in the remark 4.2.

Here, we conclude that the pair

$(\lambda=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}),\phi=d\tilde{w}_{2})$

serve as a Landau Ginzburg superpotential for Big Quantum Cohomology of
$\mathbb{CP}^{2}$, where $\tilde{w}_{2}$ is the pullback of
$\tilde{w}_{2}=w_{2}(t^{1}-\lambda,Q,t^{3})$, via the map (196).

The domain of (199) is a family of $\mathbb{H}$ parametrised by

$(\tau_{12},y,Q^{\frac{1}{3}}t^{3})\in\left(\mathbb{H}\setminus\\{E_{6}(\tau_{12})=0\\}\right)\times\mathbb{C}^{*}\times
D\left(0,\left(\frac{1}{a}\right)^{\frac{1}{3}}\right)$

because of the parametrised LG superpotential (199) and the change of
coordinates (174). Furthermore, this family of $\mathbb{H}$ is biholomorphic
to the following family of manifolds

$\begin{split}\mathbb{H}_{(\tau_{12},y,Q^{\frac{1}{3}}t^{3})}=\bigg{\\{}&(\tilde{\tau}_{12},\tilde{y},\tau_{12},y,Q^{\frac{1}{3}}t^{3})\in(\mathbb{H}\times\mathbb{C}^{*})^{2}\times
D\left(0,\left(\frac{1}{a}\right)^{\frac{1}{3}}\right):\\\
&\tilde{y}=\frac{2^{\frac{3}{2}}}{(2\pi)^{2}}\left(\frac{\Delta(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})}{Q}\right)^{\frac{1}{6}},E_{6}(\tau_{12})\neq
0\bigg{\\}}\end{split}$ (200)

Let the space $\Omega_{w_{2},y}$ be defined by

$\Omega_{w_{2},y}=\\{(w_{2},y)\in\mathbb{C}^{*}\times\mathbb{C}^{*}:\Im\left(\frac{-w_{2}}{2y}\right)>0\\}.$

Consider the following isomorphism
$S:\mathbb{H}\times\mathbb{C}^{*}\times\mathbb{H}\times\mathbb{C}^{*}\mapsto\Omega_{\tilde{w}_{2},\tilde{y}}\times\Omega_{w_{2},y}$

$S(\tilde{\tau}_{12},\tilde{y},\tau_{12},y)=(\tilde{w}_{2},\tilde{y},w_{2},y),\quad\text{where}\quad\tilde{w}_{2}=-2\tilde{y}\tilde{\tau}_{12},\quad
w_{2}=-2y\tau_{12}.$

Hence, the section
$d\tilde{w}_{2}\in\Gamma\left(T^{*}\left(\Omega_{\tilde{w}_{2},\tilde{y}}\times\Omega_{w_{2},y}\right)\times
D\left(0,\left(\frac{1}{a}\right)^{\frac{1}{3}}\right)\right)$ projected to
the submanifold $\mathbb{H}_{(\tau_{12},y,Q^{\frac{1}{3}}t^{3})}$ defined in
(200) is the section

$d\tilde{w}_{2}\in\Gamma\left(T^{*}\left(\mathbb{H}_{(\tau_{12},y,Q^{\frac{1}{3}}t^{3})}\right)\right)$

and is given by

$\begin{split}d\tilde{w}_{2}&=-2\tilde{y}d\tilde{\tau}_{12},\\\
&=-\frac{2^{\frac{5}{2}}}{2\pi}\frac{\Delta^{\frac{1}{6}}\left(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}\right)}{Q^{\frac{1}{6}}}d\tilde{\tau}_{12}.\end{split}$

Furthermore, by definition the function
$J^{\frac{1}{3}}\left(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3}\right)$ has the
following form

$\begin{split}J^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})&=J^{\frac{1}{3}}\left(\tilde{\tau}=\tilde{\tau}_{12}+\sum_{n=1}^{\infty}\frac{\tilde{\tau}_{n}(\tilde{\tau}_{12})}{\Delta^{\frac{n}{3}}(\tilde{\tau}_{12})}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right),\\\
&=J^{\frac{1}{3}}(\tilde{\tau}(\tilde{\tau}_{12})).\end{split}$ (201)

Here, it is convenient to suppress the dependence of $Q^{\frac{1}{3}}t^{3}$ in
$\tilde{\tau}(\tilde{\tau}_{12})$ to avoid a heavy notation. Then, by the
construction of the group homomorphism (193), the action of
$\gamma\in\Gamma^{(3)}_{Q^{\frac{1}{3}}t^{3}}$ in the domain of (201) is given
by

$\begin{split}J^{\frac{1}{3}}(\gamma\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})&=J^{\frac{1}{3}}(\tilde{\tau}(\gamma\tilde{\tau}_{12})),\\\
&=J^{\frac{1}{3}}\left(\frac{a\tilde{\tau}\left(\tilde{\tau}_{12}\right)+b}{c\tilde{\tau}\left(\tilde{\tau}_{12}\right)+d}\right),\quad{\begin{pmatrix}a&b\\\
c&d\\\ \end{pmatrix}}\in\Gamma^{(3)},\\\
&=J^{\frac{1}{3}}(\tilde{\tau}(\tilde{\tau}_{12})).\end{split}$ (202)

In the last line of (202), we have used that the function
$J^{\frac{1}{3}}(\tilde{\tau})$ is $\Gamma^{(3)}$-invariant and the matrix
${\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}}\in\Gamma^{(3)}$ in the second
line is the inverse of $\gamma\in\Gamma^{(3)}_{Q^{\frac{1}{3}}t^{3}}$ under
the group homomorphism (193). Theorem proved.

∎

######  Remark 4.4.

Due to the remark 3.2, the zeros of the Landau Ginzburg superpotential (196)
is achieved when $\tilde{\tau}_{12}=\tau_{12}$. Therefore, the coefficients
$J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})$ can be also obtained by the relation

$\begin{split}t^{1}&=3Q^{\frac{1}{3}}J^{\frac{1}{3}}(\tau_{12},Q^{\frac{1}{3}}t^{3}).\\\
\end{split}$ (203)

More specifically, given the coefficients $t^{1}_{n}$ and $Q_{n}$ in (174), we
can derive $J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})$ by using the Taylor
expansions of $t^{1}$ and $Q$ in $\left(\tau_{1}-\tau_{2}\right)^{2}$ and
substituting in (203).

###  4.2 From canonical to flat coordinates

The objective of this section is to provide a geometric interpretation of the
$t^{3}$ deformation by considering the analytic continuation of solutions from
the flat coordinate system of the Dubrovin connection. To achieve this, we
begin with a brief overview of the theory of n-dimensional semisimple
Dubrovin-Frobenius manifolds, we refer to [21] and [26] for this part.

Let’s consider flat sections of the Dubrovin connection constrained by the
first condition of (1). We can then select the holomorphic part of (35) as the
fundamental matrix solution:

$H(z,t)=(\eta^{\beta\gamma}\partial_{\gamma}\tilde{t}_{\alpha})=\sum_{p=0}^{\infty}H_{p}(t)z^{p},$
(204)

if we prefer to work in Saito flat coordinates $(t^{1},t^{2},\dots,t^{n})$, or
alternatively, we can choose the holomorphic part of (36):

$\Psi(z,t)=(Y_{i\alpha})=\sum_{p=0}^{\infty}\Psi_{p}(u)z^{p},$ (205)

if we prefer the canonical coordinates $(u_{1},u_{2},\dots,u_{n})$. The
coefficients of the matrices are related by the following:

$\Psi_{i\alpha}=\psi_{i\beta}\eta^{\beta\gamma}\partial_{\gamma}\tilde{t}_{\alpha}.$
(206)

The flat coordinate system of the first condition of (1) deforms the Saito
flat coordinates $(t^{1},t^{2},\dots,t^{n})$ as follows:

$\tilde{t}_{\alpha}(z,t)=t_{\alpha}+\sum_{p=1}^{\infty}H_{\alpha,p}(t)z^{p},\quad
t_{\alpha}=\eta_{\alpha\beta}t^{\beta}.$ (207)

Here, $H_{\alpha,p}$ is determined by the following recursion:

$\begin{split}H_{\alpha,0}&=t_{\alpha}=\eta_{\alpha\beta}t^{\beta},\\\
\partial_{\gamma}\partial_{\beta}H_{\alpha,p+1}&=c^{\epsilon}_{\gamma\beta}\partial_{\epsilon}H_{\alpha,p},\quad
p=0,1,2,\dots\end{split}$ (208)

In this setting, based on exercise 2.7 in [21], the following identity holds
true:

$t_{\alpha}=\langle\nabla H_{\alpha,0},\nabla
H_{1,1}\rangle=\eta^{\mu\lambda}\partial_{\mu}H_{\alpha,0}\partial_{\lambda}H_{1,1}.$
(209)

Thus, substituting the equation (207) into (206) and the change of basis

$\frac{\partial}{\partial
t^{\alpha}}=\frac{\psi_{i\alpha}}{\psi_{i1}}\frac{\partial}{\partial u_{i}},$
(210)

we obtain

$\Psi_{i\alpha,p}=\frac{\partial_{i}H_{\alpha,p}}{\psi_{i1}},\quad\text{where}\quad\Psi_{p}=\left(\Psi_{i\alpha,p}\right).$
(211)

Therefore, by substituting (211) and (210) into (209), we get

$t_{\alpha}(u)=\sum_{i=1}^{n}\Psi_{i\alpha,0}(u)\Psi_{i1,1}(u),\quad\alpha=1,2,\dots,n.$
(212)

The functions (212) are multivalued functions within the domain:

$\\{(u_{1},u_{2},\dots,u_{n})\in\mathbb{C}^{n}:u_{i}\neq u_{j},\text{ if
}i\neq j\\}=\mathbb{C}^{n}\setminus\\{u_{i}=u_{j}\\},$

since they are derived from solutions of the Dubrovin flat coordinate system
within the semisimple locus. To express (212) more clearly, consider the
compatibility condition of the Dubrovin connection flat coordinate system in
canonical coordinates (18), (19), given by:

$\frac{\partial^{2}Y}{\partial z\partial u_{i}}=\frac{\partial^{2}Y}{\partial
u_{i}\partial z},$

which specifically appears as:

$\begin{split}\frac{\partial V}{\partial u_{i}}&=[V_{i},V],\\\
[U,V_{i}]&=[E_{i},V].\end{split}$ (213)

Furthermore, due to the following fact:

$[U,V_{k}]=[E_{k},V]\implies\left(V_{k}\right)_{ij}=\frac{\delta_{ki}-\delta_{kj}}{u_{i}-u_{j}}V_{ij},$

the system (213) can be expressed as:

$\begin{split}\frac{\partial V}{\partial u_{i}}&=[V_{i},V],\\\
\frac{\partial\Psi}{\partial u_{i}}&=V_{i}\Psi.\end{split}$ (214)

######  Remark 4.5.

The first equation (214) can be presented as a time-dependent Hamiltonian
system:

$\frac{\partial V}{\partial u_{i}}=\\{V,H_{i}\\},\quad i=1,2,\dots,n,$

with a quadratic Hamiltonian:

$H_{i}(V;u)=\frac{1}{2}\sum_{i\neq j}\frac{V_{ij}^{2}}{u_{i}-u_{j}},\quad
i=1,2,\dots,n,$ (215)

employing the Poisson bracket:

$\\{V_{ij},V_{kl}\\}=V_{il}\delta_{jk}-V_{jl}\delta_{ik}+V_{jk}\delta_{il}-V_{ik}\delta_{jl}.$
(216)

The Hamiltonians $H_{i}(V;u)$ (215) pairwise commute with respect to the
Poisson bracket (216), i.e., $\\{H_{i},H_{j}\\}=0$ for any $i,j$. Hence, the
one-form:

$\sum_{i=1}^{n}H_{i}(V;u)du_{i}$

forms a closed form for any solution of (214). This implies the local
existence of a function $\tau_{I}(u)$ such that:

$\frac{\partial\ln\tau_{I}(u)}{\partial u_{i}}=H_{i}(V;u),\quad
i=1,2,\dots,n.$ (217)

The function $\tau_{I}$ defined in (217) is referred to as the isomonodromic
$\tau$ function of the system (214). In the context of semisimple Quantum
Cohomology, $\tau_{I}$ satisfies a significant identity. Specifically, if
$F^{1}$ represents the generating function of genus $1$ Gromov Witten
invariants, then:

$F^{1}=\log\frac{\tau_{I}}{J^{\frac{1}{24}}},\quad\text{where}\quad
J=\det\left(\frac{\partial t^{\alpha}}{\partial
u_{i}}\right)=\psi_{11}\dots\psi_{n1}.$

Here, our focus centers on the particular case of $3D$ Dubrovin Frobenius
manifolds. Therefore, we will use the results obtained by Guzzetti in section
6.1 of [26]. Consider the metric $\eta$ and matrix $V(u_{1},u_{2},u_{3})$:

$\begin{split}\eta&=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix},\quad
V(u)=\begin{pmatrix}0&-\Omega_{3}&\Omega_{2}\\\ \Omega_{3}&0&-\Omega_{1}\\\
-\Omega_{2}&\Omega_{1}&0\end{pmatrix}.\end{split}$

Recall that the columns of matrix $\Psi$ are the eigenvectors of matrix $V$.
Moreover, it satisfies:

$\Psi^{T}\Psi=\eta.$

Therefore, we can write $\Psi$ in the following form:

$\Psi(u)=\begin{pmatrix}\frac{E_{11}}{f}&E_{12}&E_{13}f\\\
\frac{E_{21}}{f}&E_{22}&E_{23}f\\\
\frac{E_{31}}{f}&E_{32}&E_{33}f\end{pmatrix},$

where

$\begin{split}E_{11}&=\frac{\Omega_{1}\Omega_{2}-\mu\Omega_{3}}{2\mu^{2}},\quad
E_{12}=\frac{\Omega_{1}}{i\mu},\quad
E_{13}=-\frac{\Omega_{1}\Omega_{2}+\mu\Omega_{3}}{\Omega_{1}^{2}+\Omega_{3}^{2}},\\\
E_{21}&=-\frac{\Omega_{1}^{2}+\Omega_{3}^{2}}{2\mu^{2}},\quad
E_{22}=\frac{\Omega_{2}}{i\mu},\quad E_{23}=1,\\\
E_{31}&=\frac{\Omega_{2}\Omega_{3}+\mu\Omega_{1}}{2\mu^{2}},\quad
E_{32}=\frac{\Omega_{3}}{i\mu},\quad
E_{33}=-\frac{\Omega_{2}\Omega_{3}-\mu\Omega_{3}}{\Omega_{1}^{2}+\Omega_{3}^{2}},\end{split}$
(218)

and $f(u)$ is determined by:

$\frac{\partial\Psi}{\partial u_{i}}=V_{i}\Psi.$

Here, it is convenient to make the following global change of coordinates:

$F:\mathbb{C}^{3}\setminus\\{u_{i}=u_{j}\\}\mapsto\mathbb{C}\times\mathbb{C}^{*}\times\left(\mathbb{C}\setminus\\{0,1\\}\right),\quad
F(u_{1},u_{2},u_{3})=(v,H,x)$

where

$v=u_{1}+u_{2}+u_{3},\quad H=u_{2}-u_{1},\quad
x=\frac{u_{3}-u_{1}}{u_{2}-u_{1}}.$ (219)

From the standard theory of Dubrovin Frobenius manifolds [19], [21], [26], the
action of the unit vector field $e$ and Euler vector field $E$ in the matrix
$V$ are given by:

$\sum_{i=1}^{3}\partial_{i}V=0,\quad\sum_{i=1}^{3}u_{i}\partial_{i}V=0.$ (220)

Hence, considering the change of coordinates (219) and the symmetries of the
matrix $V$ in (220), we have the following:

$V(u_{1},u_{2},u_{3})=V\left(\frac{u_{3}-u_{1}}{u_{2}-u_{1}}\right)=V(x).$

Consequently, the system (213) reduces to:

$\begin{split}\frac{d\Omega_{1}}{dx}&=\frac{\Omega_{2}\Omega_{3}}{x},\\\
\frac{d\Omega_{2}}{dx}&=\frac{\Omega_{1}\Omega_{3}}{1-x},\\\
\frac{d\Omega_{3}}{dx}&=\frac{\Omega_{1}\Omega_{2}}{x(x-1)}.\end{split}$ (221)

In the specific case of $QH^{*}(\mathbb{CP}^{2})$, Guzzetti derived the Saito
flat coordinates $(t^{1},t^{2},t^{3})$ as functions of $(v,H,x)$ in Section
6.2.2 of [26], specifically in his equations (6.7), (6.8), and (6.9):

$\begin{split}t^{1}(u)&=u_{1}+a(x)H,\\\
t^{2}(u)&=3\ln(H)+3\int^{x}\frac{d\zeta}{\zeta+\frac{E_{21}E_{22}}{E_{31}E_{32}}},\\\
t^{3}(u)&=-9\frac{c(x)}{b(x)^{2}H},\end{split}$ (222)

where

$\begin{split}a(x)&=E_{21}E_{23}+xE_{31}E_{33},\\\
b(x)&=E_{22}E_{21}+xE_{32}E_{31},\\\ c(x)&=E_{21}^{2}+xE_{31}^{2}.\\\
\end{split}$ (223)

In Appendix E of [19], Dubrovin proved that any semisimple 3D Dubrovin
Frobenius manifold is associated with a special solution of $PVI$. More
concretely, consider the twisted Gauss Manin connection

$\begin{split}\left(U-\lambda
I\right)\partial_{\beta}\chi+C_{\beta}\mu\chi&=0,\\\ \left(U-\lambda
I\right)\partial_{\lambda}\chi-\mu\chi&=0.\\\ \end{split}$

In canonical coordinates, the corresponding $3\times 3$ system of differential
equations is given by

$\begin{split}\frac{dX}{d\lambda}&=-\mu\left[\frac{A_{1}}{\lambda-
u_{1}}+\frac{A_{2}}{\lambda-u_{2}}+\frac{A_{3}}{\lambda-u_{3}}\right]X,\\\
\frac{\partial X}{\partial u_{i}}&=\mu\frac{A_{i}}{\lambda-u_{i}}X,\\\
\end{split}$ (224)

where

$A_{i}=\begin{pmatrix}\psi_{i1}\psi_{i3}&0&-\psi_{i3}^{2}\\\
\psi_{i1}\psi_{i2}&0&-\psi_{i2}\psi_{i3}\\\
\psi_{i1}^{2}&0&-\psi_{i1}\psi_{i3}\\\
\end{pmatrix},\mu=\mu\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\\\
\end{pmatrix},X=\begin{pmatrix}X_{1}\\\ X_{2}\\\ X_{3}\\\ \end{pmatrix},$

$\frac{\partial u_{i}}{\partial t^{\alpha}}=\frac{\psi_{i\alpha}}{\psi_{i1}}$

The twisted Gauss-Manin connection (224) can be split into a $2\times 2$
system of differential equations due to the fact that the matrix
$\mu=diag(\mu,0,-\mu)$ has a $0$ eigenvalue. Indeed,

$\begin{split}\frac{dX}{d\lambda}&=-\mu\left[\frac{A_{1}}{\lambda-
u_{1}}+\frac{A_{2}}{\lambda-u_{2}}+\frac{A_{3}}{\lambda-u_{3}}\right]X,\\\
\frac{\partial X}{\partial u_{i}}&=\mu\frac{A_{i}}{\lambda-u_{i}}X,\\\
\end{split}$ (225)

where

$A_{i}=\begin{pmatrix}\psi_{i1}\psi_{i3}&-\psi_{i3}^{3}\\\
\psi_{i1}^{2}&-\psi_{i1}\psi_{i3}\\\
\end{pmatrix},\mu=\mu\begin{pmatrix}1&0\\\ 0&-1\\\
\end{pmatrix},X=\begin{pmatrix}X_{1}\\\ X_{3}\\\ \end{pmatrix},$

$\frac{\partial u_{i}}{\partial t^{\alpha}}=\frac{\psi_{i\alpha}}{\psi_{i1}}.$

Using the new coordinates $q,p$ given by the relations

$\begin{split}0=&\left[\sum_{i=1}^{3}\frac{A_{i}}{q-u_{i}}\right]_{12},\\\
p=&\left[\sum_{i=1}^{3}\frac{A_{i}}{q-u_{i}}\right]_{11},\end{split}$

the compatibility conditions of the twisted Gauss Manin connection (225) are
given by

$\begin{split}\frac{\partial q}{\partial
u_{i}}&=\frac{P(q)}{P^{\prime}(u_{i})}\left[2p+\frac{1}{q-u_{i}}\right],\\\
\frac{\partial p}{\partial
u_{i}}&=\frac{P^{\prime}(q)p^{2}+\left(2q+u_{i}-\sum_{i=1}^{3}u_{i}\right)p+\mu(1-\mu)}{P^{\prime}(u_{i})},\\\
\end{split}$

where

$\begin{split}P(\lambda)=\left(\lambda-u_{1}\right)\left(\lambda-
u_{2}\right)\left(\lambda-u_{3}\right)\end{split}$

Eliminating $p$ from the system, we obtain a second-order differential
equation for the function $q=q(u_{1},u_{2},u_{3})$. Using a change of
coordinates

$\begin{split}x&=\frac{u_{3}-u_{1}}{u_{2}-u_{1}},\\\
y&=\frac{q-u_{1}}{u_{2}-u_{1}},\\\ \end{split}$ (226)

we have that $y=y(x)$ solves a particular one-parameter family of Painlevé
$VI$ equation, denoted by $PVI_{\mu}$,

$\begin{split}y^{\prime\prime}=&\frac{1}{2}\left[\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-x}\right]\left(y^{\prime}\right)^{2}-\left[\frac{1}{x}+\frac{1}{x-1}+\frac{1}{y-x}\right]y^{\prime}\\\
&+\frac{1}{2}\frac{y(y-1)(y-x)}{x^{2}(x-1)^{2}}\left[(2\mu-1)^{2}+\frac{x(x-1)}{(y-x)^{2}}\right].\end{split}$
(227)

The parameter $\mu$ coincides with the $\mu$ of the monodromy data
$(\mu,R,S,C)$ of some 3D Dubrovin Frobenius manifold.

######  Remark 4.6.

A differential equation is said to possess the Painlevé property if the
problematic singularities of its corresponding solutions, such as essential
singularities and branch points, are always at the coefficients of the
differential equation. Additionally, the movable singularities—those dependent
on the integration constants—are poles. In particular, the Painlevé VI (PVI)
equation has the Painlevé property. Consequently, the potential locations of
essential singularities, branch points, and so on, are confined to $0$, $1$,
and $\infty$, while the positions of the poles depend on the integration
constants. In this context, the locus of movable poles of $PVI_{\mu}$, known
as the Malgrange divisor $MD_{\mu}$, coincides with the zeros and poles
divisor associated with the isomonodromic $\tau$ function defined in (217)
projected onto $\mathbb{C}\setminus\\{0,1\\}$. The functions $a(x)$, $b(x)$,
$c(x)$ in (223) are rational functions of $x,y(x),y^{\prime}(x)$, see section
6.1 of [26]. Due to the Painlevé property, the Saito flat coordinates
$t^{1},t^{2},t^{3}$ (222) are meromorphic functions in
$\mathbb{C}\times\mathbb{C}^{*}\times\left(\mathbb{C}\setminus\\{0,1\\}\right)$.

From here, we return to the case of $QH^{*}(\mathbb{CP}^{2})$. Note that the
hypersurface $t^{3}=0$ in canonical coordinates corresponds to the
hypersurface

$\\{(v,H,e^{\frac{-\pi
i}{3}})\in\mathbb{C}\times\mathbb{C}^{*}\times\mathbb{C}\setminus\\{0,1\\}\\}.$

Indeed, substituting the small canonical coordinates (144) in (219), we obtain
the desired result.

Hence, the Big Quantum Cohomology is an expansion around $x=e^{-\frac{\pi
i}{3}}$. Following section 6.5 of [26], define $s:=x-e^{\frac{-\pi i}{3}}$ and
expand $\Omega_{1},\Omega_{2},\Omega_{3}$ in (221) around $e^{\frac{\pi
i}{3}}$:

$\begin{split}\Omega_{1}(x)&=-\frac{i\sqrt{3}}{2}-\left(\frac{1}{6}+\frac{i\sqrt{3}}{6}\right)s+\frac{i\sqrt{3}}{9}s^{2}+O(s^{3}),\\\
\Omega_{2}(x)&=\frac{i\sqrt{3}}{2}+\left(\frac{1}{6}-\frac{i\sqrt{3}}{6}\right)s-\frac{i\sqrt{3}}{9}s^{2}+O(s^{3}),\\\
\Omega_{3}(x)&=\frac{i\sqrt{3}}{2}-\frac{1}{3}s+\frac{2i\sqrt{3}}{9}s^{2}+O(s^{3}).\end{split}$
(228)

Furthermore, expanding $y(x)$:

$\begin{split}y(x)&=\frac{1}{2}-\frac{i\sqrt{3}}{6}+\frac{1}{3}s-\frac{i\sqrt{3}}{3}s^{2}+O(s^{3}).\end{split}$

Substituting (228) in (218) ,(223) and (222), the Saito flat coordinates have
the following expansion:

$\begin{split}t^{1}(u)&=u_{1}+\left(\frac{1}{2}-\frac{i\sqrt{3}}{6}s+O(s^{2})\right)H,\\\
Q(u)&=\frac{i\sqrt{3}}{143}\left(1+i\sqrt{3}s+O(s^{2})\right)H^{3},\\\
t^{3}(u)&=\left(-9s+O(s^{2})\right)H^{-1}.\end{split}$

Then, the expansion of coordinates $Q^{\frac{1}{3}}t^{3}$ is given by

$\begin{split}Q^{\frac{1}{3}}t^{3}(x)&=\sum_{n=1}^{\infty}Q_{n}\left(x-e^{-\frac{\pi
i}{3}}\right)^{n}\\\
&=\left(\frac{3}{2}-\frac{i\sqrt{3}}{2}\right)\left(x-e^{\frac{-\pi
i}{3}}\right)-\left(\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)\left(x-e^{\frac{-\pi
i}{3}}\right)^{2}+O\left(\left(x-e^{\frac{-\pi
i}{3}}\right)^{3}\right)\end{split}$ (229)

is a holomorphic function in some neighbourhood $U(e^{\frac{-\pi i}{3}})$ of
$e^{\frac{-\pi i}{3}}\in\mathbb{C}\setminus\\{0,1\\}$.

At this stage, we can state the main result of this section

######  Lemma 4.7.

The composition of the modular lambda function (150) restricted to a suitable
neighborhood of $e^{\frac{2\pi i}{3}}$ with the map (229) is a local
biholomorphism given by the following power series

$Q^{\frac{1}{3}}t^{3}(z)=\sum_{n=0}^{\infty}\tilde{Q}_{n}\left(z-e^{\frac{2\pi
i}{3}}\right)^{n},$ (230)

where

$\tilde{Q}_{n}=\frac{\sum_{k=0}^{n}k!Q_{k}B_{n,k}(x_{1},x_{2},..,x_{n})}{n!},$

$x_{m}$ are the coefficients of the modular lambda function (150) around
$e^{\frac{2\pi i}{3}}$, i.e.,

$x(z)=\sum_{m=0}^{\infty}x_{m}\frac{\left(z-e^{\frac{2\pi
i}{3}}\right)^{m}}{m!},$

and $B_{n,k}$ are the Partial Bell polynomials given by the formula (264).

###### Proof.

The map defined by (229) is a local biholomorphism

$Q^{\frac{1}{3}}t^{3}(x):U(e^{\frac{-\pi
i}{3}})\subset\mathbb{C}\setminus\\{0,1\\}\mapsto
Q^{\frac{1}{3}}t^{3}\left(U\left(e^{\frac{-\pi i}{3}}\right)\right)\subset
D\left(0,\left(\frac{1}{a}\right)^{\frac{1}{3}}\right).$

Indeed, the point $e^{\frac{-\pi i}{3}}\in\mathbb{C}\setminus\\{0,1\\}$ lies
within the semisimple locus and serves as a holomorphic point for the
functions given by (228). Since the Jacobian of the transition function
between canonical and flat Saito coordinates

$J=\det\left(\frac{\partial t^{\alpha}}{\partial
u_{i}}\right)=\prod_{i=1}^{3}\psi_{i1}$

does not vanish in the semisimple locus, the map (229) is a local
biholomorphism.

Recall that the preimage of $e^{\frac{-\pi i}{3}}$ under the modular lambda
function (150) is $e^{\frac{2\pi i}{3}}$ mod $\Gamma(2)$. Moreover, the
modular lambda function (150) is a local biholomorphism. In particular, the
modular lambda function (150) is a local biholomorphism around $e^{\frac{2\pi
i}{3}}$. Therefore, composing the modular lambda function (150) restricted to
a suitable neighborhood $U(e^{\frac{2\pi i}{3}})$ of $e^{\frac{2\pi
i}{3}}\in\mathbb{H}$ with (229), we obtain the desire local biholomorphism

$Q^{\frac{1}{3}}t^{3}(x(z)):U(e^{\frac{2\pi i}{3}})\subset\mathbb{H}\mapsto
U(e^{\frac{-\pi i}{3}})\subset\mathbb{C}\setminus\\{0,1\\}\mapsto
Q^{\frac{1}{3}}t^{3}\left(x\left(U\left(e^{\frac{2\pi
i}{3}}\right)\right)\right)\subset
D\left(0,\left(\frac{1}{a}\right)^{\frac{1}{3}}\right).$ (231)

Furthermore, we use (229), (150) and using the Faa di Bruno formula (265) to
compute the explicit power series composition:

$\begin{split}Q^{\frac{1}{3}}t^{3}(x(z))&=\sum_{n=0}^{\infty}Q_{n}\left(x(z)-e^{\frac{-\pi
i}{3}}\right)^{n},\\\
&=\sum_{n=0}^{\infty}n!Q_{n}\frac{\left(x(z)-e^{\frac{-\pi
i}{3}}\right)^{n}}{n!},\\\
&=\sum_{n=0}^{\infty}n!\frac{Q_{n}}{n!}\left(\sum_{m=0}^{\infty}x_{m}\frac{\left(z-e^{\frac{2\pi
i}{3}}\right)^{m}}{m!}\right)^{n},\\\
&=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}k!Q_{k}B_{n,k}(x_{1},x_{2},..,x_{n})\right)\frac{\left(z-e^{\frac{2\pi
i}{3}}\right)^{n}}{n!},\\\
&=\sum_{n=0}^{\infty}\tilde{Q}_{n}\left(z-e^{\frac{2\pi i}{3}}\right)^{n}.\\\
\end{split}$

Lemma proved.

∎

######  Remark 4.7.

The coefficients of (230) can be explicitly computed to any desired order
using the Guzzetti algorithm (229) or by employing the alternative algorithm
derived in Appendix 5.3.

###  4.3 Canonical coordinates for Big Quantum Cohomology of $\mathbb{CP}^{2}$

In this section, the goal is to establish a formula expressing the canonical
coordinates of Quantum Cohomology for $\mathbb{CP}^{2}$ in terms of the Saito
flat coordinates.

The canonical coordinates are the roots (15). In particular, $g^{\alpha\beta}$
and $\eta^{\alpha\beta}$ for Big Quantum Cohomology are given by

$\begin{split}&g^{\alpha\beta}=E^{\epsilon}c^{\alpha\beta}_{\epsilon}\\\
&=\begin{pmatrix}\frac{3}{(t^{3})^{3}}[9\Phi^{\prime\prime}(X)-9\Phi^{\prime}(X)+2\Phi(X)]&\frac{2}{(t^{3})^{2}}[3\Phi^{\prime\prime}(X)-\Phi^{\prime}(X)]&t^{1}\\\
\frac{2}{(t^{3})^{2}}[3\Phi^{\prime\prime}(X)-\Phi^{\prime}(X)]&t^{1}+\frac{\Phi^{\prime\prime}(X)}{t^{3}}&3\\\
t^{1}&3&-t^{3}\end{pmatrix},\end{split}$ (232)
$\begin{split}&\eta^{\alpha\beta}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\end{split}$ (233)

where $\Phi(X)$ is defined by (23).

In Example 3.10.34 of [22], Dubrovin and Zhang made the following ansartz

$u_{i}=t^{1}+\frac{9+\Phi^{\prime\prime}(X)-z_{i}}{t^{3}}$ (234)

Substituting (232), (233) and (234) in (15), we obtain that
$z_{1},z_{2},z_{3}$ are the roots of the following cubic equation

$z^{3}-s_{1}z^{2}+s_{2}z-s_{3}=0$ (235)

where

$\begin{split}s_{1}&=27+2\Phi^{\prime\prime},\\\
s_{2}&=243+6\Phi-15\Phi^{\prime}+27\Phi^{\prime\prime}+\left(\Phi^{\prime\prime}\right)^{2}\\\
s_{3}&=\left(27+2\Phi^{\prime}-3\Phi^{\prime\prime}\right)^{2}\\\ \end{split}$
(236)

Summarising, we have the following lemma

######  Lemma 4.8.

The canonical coordinates of Quantum Cohomology $u_{1},u_{2},u_{3}$ as
function of the Saito flat coordinates $t^{1},t^{2},t^{3}$ are given by

$u_{k}=t^{1}+\frac{1}{t^{3}}\sum_{n=1}^{\infty}A_{n}^{k}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}.$
(237)

where the coefficients $A_{k}$ are given explicitly by

$\begin{split}&\tilde{A}_{3k}=\frac{k^{2}N_{k}}{(3k-1)!},\\\
&\sum_{n_{2}=2}^{3n}3\cos\left(\frac{2\pi(n_{2}-1)}{3}\right)\tilde{A}_{3n-n_{2}+1}\tilde{A}_{n_{2}-1}=\left(6-15n-9n^{2}\right)\frac{N_{n}}{(3n-1)!},\\\
&\sum_{n_{2}=1}^{3n-2}\sum_{n_{3}=1}^{3n-n_{2}-1}3\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right)\tilde{A}_{3n-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}=\left(54-243n\right)\frac{N_{n}}{(3n-1)!}+\delta_{n}\\\
\end{split}$ (238)

where

$A_{n}^{k}=\tilde{A}_{n}\left(e^{\frac{2\pi i}{3}}\right)^{nk}$ (239)
$\delta_{n}=\begin{cases}0&\text{if }n=1\\\
\tilde{\delta}_{n},&\text{otherwise}\end{cases}$
$\begin{split}\tilde{\delta}_{n}=&\sum_{n_{2}=2}^{n}\frac{\left(6(n_{2}-1)-3(n-n_{2}+1)(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-4(n-n_{2}+1)(n_{2}-1)\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-9(n_{1}-n_{2}+1)^{2}(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}.\\\
\end{split}$ (240)

###### Proof.

See appendix 5.2.

∎

Moreover, the transition function matrix $\psi_{i\alpha}$ can be obtained by
finding the eigenvectors related to (15) as done in Example 3.10.34 of [22].
More precisely, the transition function matrix $\psi_{i\alpha}$ is the matrix
such that the columns are eigenvectors of

$g^{*}\eta$

which is given by

$\begin{split}\psi_{i1}&=h_{i},\\\
\psi_{i2}&=\frac{h_{i}\left(-27-2\Phi^{\prime}+3\Phi^{\prime\prime}+3z_{i}\right)}{z_{i}t^{3}},\\\
\psi_{i3}&=\frac{h_{i}\left(81+6\Phi^{\prime}-9\Phi^{\prime\prime}-18z_{i}+z_{i}^{2}-z_{i}^{2}\Phi^{\prime\prime}\right)}{z_{i}(t^{3})^{2}},\\\
\end{split}$ (241)

where $h_{i}$ have the form

$\begin{split}h_{1}=\frac{t^{3}\sqrt{z_{1}}}{\sqrt{\left(z_{1}-z_{2}\right)}\sqrt{\left(z_{1}-z_{3}\right)}},\quad
h_{2}=\frac{t^{3}\sqrt{z_{2}}}{\sqrt{\left(z_{2}-z_{1}\right)}\sqrt{\left(z_{2}-z_{3}\right)}},\quad
h_{3}=\frac{t^{3}\sqrt{z_{3}}}{\sqrt{\left(z_{3}-z_{1}\right)}\sqrt{\left(z_{3}-z_{2}\right)}}.\end{split}$
(242)

###  4.4 Geometric Isomonodromic deformation

The aim of this section is to give a geometric interpretation for the
isomonodromic deformation of the Hauptmoduln $\gamma_{2}(\tilde{\tau})$ (102).
For this purpose, we derive a $t^{3}$ deformation of the equianharmonic
elliptic curve

$\begin{split}y^{2}&=4(x-u_{1})(x-u_{2})(x-u_{3}),\\\ &=4(x-t^{1})^{3}-27Q,\\\
\end{split}$

######  Lemma 4.9.

Consider the canonical coordinates $(u_{1},u_{2},u_{3})$ of the Quantum
Cohomology of $\mathbb{CP}^{2}$, obtained from (234) and (236), along with the
cubic equation:

$y^{2}=4\left(\lambda-u_{1}\right)\left(\lambda-u_{2}\right)\left(\lambda-
u_{3}\right).$ (243)

Then, the pair $(\lambda,y)$ satisfying (243) can be expressed as follows:

$\begin{split}\lambda&=t^{1}+\frac{\Phi^{\prime\prime}(Q(t^{3})^{3})}{3t^{3}}-4^{\frac{1}{3}}3Q^{\frac{1}{3}}\wp(v,z),\\\
y&=i3^{\frac{3}{2}}2Q^{\frac{1}{2}}\wp^{\prime}(v,z).\\\ \end{split}$ (244)

###### Proof.

Substituting (234) in (243), we obtain

$\begin{split}\tilde{y}^{2}&=4\left(\tilde{\lambda}-z_{1}\right)\left(\tilde{\lambda}-z_{2}\right)\left(\tilde{\lambda}-z_{3}\right),\\\
=&4\left(\tilde{\lambda}^{3}-s_{1}\tilde{\lambda}^{2}+s_{2}\tilde{\lambda}-s_{3}\right)\end{split}$
(245)

where $s_{1},s_{2},s_{3}$ are given by (236) and

$\begin{split}y&=\frac{i\tilde{y}}{\left(t^{3}\right)^{\frac{3}{2}}},\\\
\lambda&=t^{1}+\frac{9+\Phi^{\prime\prime}-\tilde{\lambda}}{t^{3}}.\end{split}$
(246)

To express (245) in the depressed form of the cubic equation, we make the
following substitution:

$\begin{split}\tilde{\lambda}&=\hat{\lambda}+\frac{s_{1}}{3},\\\
\hat{y}&=\tilde{y},\\\ z_{i}&=\hat{e}_{i}+\frac{s_{1}}{3},\end{split}$ (247)

in (245), then we obtain

$\begin{split}\hat{y}^{2}&=4\left(\hat{\lambda}-\hat{e}_{1}\right)\left(\hat{\lambda}-\hat{e}_{2}\right)\left(\hat{\lambda}-\hat{e}_{3}\right),\\\
&=4\hat{\lambda}^{3}-g_{2}\hat{\lambda}-g_{3},\end{split}$ (248)

where

$\begin{split}\hat{g}_{2}&=-4\left(\hat{e}_{1}\hat{e}_{2}+\hat{e}_{1}\hat{e}_{3}+\hat{e}_{2}\hat{e}_{3}\right),\\\
\hat{g}_{3}&=4\hat{e}_{1}\hat{e}_{2}\hat{e}_{3}.\end{split}$

Moreover, consider the following rescaling

$\begin{split}\hat{\lambda}=\left(4^{\frac{1}{3}}3Q^{\frac{1}{3}}t^{3}\right)\lambda^{r}\quad,\hat{e}_{i}=\left(4^{\frac{1}{3}}3Q^{\frac{1}{3}}t^{3}\right)e_{i},\quad\hat{y}=\left(4^{\frac{1}{3}}3Q^{\frac{1}{3}}t^{3}\right)^{\frac{3}{2}}y^{r}.\\\
\end{split}$ (249)

Hence, substituting (249) in (248):

$\begin{split}(y^{r})^{2}&=4\left(\lambda^{r}-e_{1}\right)\left(\lambda^{r}-e_{2}\right)\left(\lambda^{r}-e_{3}\right),\\\
&=4(\lambda^{r})^{3}-g_{2}\lambda^{r}-g_{3},\end{split}$ (250)

where

$\begin{split}g_{2}&=-4\left(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3}\right),\\\
g_{3}&=4e_{1}e_{2}e_{3}.\end{split}$

Due to the uniformization of elliptic curves, we can parametrize the cubic
equation (250) using the Weierstrass $\wp$ function and its derivatives. In
other words, we express (250) in the form:

$\begin{split}\left(\wp^{\prime}(v,z)\right)^{2}&=4\wp(v,z)^{3}-g_{2}(z)\wp(v,z)-g_{3}(z)\end{split}$
(251)

Substituting (251), (249), (247) in (246) we obtain,

$\begin{split}\lambda&=t^{1}+\frac{\Phi^{\prime\prime}(Q(t^{3})^{3})}{3t^{3}}-4^{\frac{1}{3}}3Q^{\frac{1}{3}}\wp(v,z),\\\
y&=i3^{\frac{3}{2}}2Q^{\frac{1}{2}}\wp^{\prime}(v,z).\\\ \end{split}$

Lemma proved.

∎

Our objective is to extend the family of elliptic curves (244) to a LG
superpotential for $QH^{*}(\mathbb{CP}^{2})$. To begin, we examine the
following auxiliary lemma.

######  Lemma 4.10.

The Landau Ginzburg superpotential of Quantum Cohomology of $\mathbb{CP}^{2}$
is a family of functions

$\lambda(\tilde{\tau}_{12},t^{1},Q^{\frac{1}{3}},z):\mathbb{H}\mapsto\mathbb{C}$

parametrised by

$(t^{1},Q^{\frac{1}{3}},z)\in\\{(t^{1},Q^{\frac{1}{3}},z)\in\mathbb{C}\times\mathbb{C}^{*}\times
U(e^{\frac{2\pi
i}{3}})\\}\subset\mathbb{C}\times\mathbb{C}^{*}\times\mathbb{H}$

and given by

$\lambda(\tilde{\tau}_{12},t^{1},Q,z)=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}(\tilde{\tau}_{12},z)$
(252)

where $J^{\frac{1}{3}}(\tilde{\tau}_{12},z)$ is the composition of the
function $J^{\frac{1}{3}}(\tilde{\tau}_{12},Q^{\frac{1}{3}}t^{3})$ defined in
(197) and $Q^{\frac{1}{3}}t^{3}(z)$ defined in (231). i.e.

$\begin{split}J^{\frac{1}{3}}(\tilde{\tau}_{12},z)&:=\sum_{n=0}^{\infty}J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})\left(Q^{\frac{1}{3}}t^{3}(z)\right)^{n},\\\
&=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}k!J_{k}^{\frac{1}{3}}(\tilde{\tau}_{12})B_{n,k}\left(1!\tilde{Q}_{1},..,m!\tilde{Q}_{m}\right)\right)\frac{\left(z-e^{\frac{2\pi
i}{3}}\right)^{n}}{n!},\end{split}$ (253)

where $B_{n,k}$ are the Bell polynomials defined in (264). In addition, the
correspondent Abelian differential $\phi$ is given by

$\phi=\frac{\Delta^{\frac{1}{6}}(\tilde{\tau}_{12},z)}{Q^{\frac{1}{6}}}d\tilde{\tau}_{12},$

where $\Delta(\tilde{\tau}_{12},z)$ is the composition of (231) with (189).

###### Proof.

Doing the change of coordinates (231) in the Landau Ginzburg superpotential
(196) and in Abelian differential (198) we get the desired result. More
explicitly, substituting (231) in (197), we obtain

$\begin{split}\lambda(\tilde{\tau}_{12},t^{1},Q,t^{3})&=t^{1}+3Q^{\frac{1}{3}}\sum_{n=0}^{\infty}J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})\left(Q^{\frac{1}{3}}t^{3}\right)^{n},\\\
&=t^{1}+3Q^{\frac{1}{3}}\sum_{n=0}^{\infty}n!J_{n}^{\frac{1}{3}}(\tilde{\tau}_{12})\frac{\left(\sum_{k=0}^{\infty}\tilde{Q}_{k}\left(z-e^{\frac{2\pi
i}{3}}\right)^{k}\right)^{n}}{n!},\\\
&=t^{1}+3Q^{\frac{1}{3}}\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}k!J_{k}^{\frac{1}{3}}(\tilde{\tau}_{12})B_{n,k}\left(1!\tilde{Q}_{1},..,m!\tilde{Q}_{m}\right)\right)\frac{\left(z-e^{\frac{2\pi
i}{3}}\right)^{n}}{n!},\\\
&=t^{1}+3Q^{\frac{1}{3}}J^{\frac{1}{3}}(\tilde{\tau}_{12},z).\\\ \end{split}$
(254)

∎

Our next goal is to generalise the Cohn identity (152) for $z\in\mathbb{H}$
near $e^{\frac{2\pi i}{3}}$.

######  Theorem 4.11.

The Landau Ginzburg superpotential of Big Quantum Cohomology of
$\mathbb{CP}^{2}$ is a family of functions

$\lambda(\tilde{\tau}_{12},v_{0},\omega,z):\mathbb{H}\mapsto\mathbb{C}$

parametrised by

$\\{(v_{0},\omega,z)\in\mathbb{C}\times\mathbb{C}^{*}\times\mathbb{H}:v_{0}\neq\mathbb{Z}\oplus
z\mathbb{Z},z\in U(e^{\frac{2\pi i}{3}})\\}$

and given by

$\lambda(v(\tilde{\tau}_{12},z),v_{0},\omega,z)=\frac{\wp(v_{0},z)}{\left(2\omega\right)^{2}}-\frac{\wp(v(\tilde{\tau}_{12},z),z)}{\left(2\omega\right)^{2}},$

where for $z\in\mathbb{H}$ close enough to $e^{\frac{2\pi i}{3}}$,
$v(\tilde{\tau}_{12},z)$ is the universal covering of
$\mathbb{C}\setminus\\{\mathbb{Z}\oplus z\mathbb{Z}\\}$. In addition, the
correspondent Abelian differential $\phi$ is given by

$\phi=\left[\frac{\Delta^{\frac{1}{6}}(\tilde{\tau}_{12},z)}{\frac{\partial
v(\tilde{\tau}_{12},z)}{\partial\tilde{\tau}_{12}}}\right]2\omega
dv(\tilde{\tau}_{12},z),$

where $\Delta(\tilde{\tau}_{12},z)$ is the composition of (231) with (189).

###### Proof.

Recall that the critical values of the Uniformization (244) are the canonical
coordinates (237) by construction. The Abelian differential which gives the
correct Landau Ginzburg superpotential is $\phi=d\tilde{w}_{2}$. But, in order
to implement the Abelian differential in the spectral curve (243), we need to
construct, for a fixed $z\in\mathbb{H}$, a change of coordinates

$v_{z}:\mathbb{H}\mapsto\mathbb{C}\setminus{\mathbb{Z}\oplus
z\mathbb{Z}},\quad\tilde{\tau}_{12}\mapsto
v_{z}(\tilde{\tau}_{12}):=v(\tilde{\tau}_{12},z)$ (255)

such that

$\begin{split}\lambda(\tilde{\tau}_{12},t^{1},Q,z)&=t^{1}+Q^{\frac{1}{3}}\frac{\Phi^{\prime\prime}\left(Q(t^{3})^{3}\left(z\right)\right)}{3Q^{\frac{1}{3}}t^{3}(z)}-Q^{\frac{1}{3}}\wp(v(\tilde{\tau}_{12},z),z).\\\
\end{split}$ (256)

Then, the change of coordinates (255) is concretely constructed by comparing
(256) with (254). More explicitly, by the relation below

$\begin{split}J^{\frac{1}{3}}(\tilde{\tau}_{12},z)&=\frac{\Phi^{\prime\prime}\left(Q(t^{3})^{3}\left(z\right)\right)}{9Q^{\frac{1}{3}}t^{3}(z)}-4^{\frac{1}{3}}\wp(v(\tilde{\tau}_{12},z),z)\\\
\end{split}$ (257)

The identity (257) can be understood as generalised Cohn identity. Indeed, the
map (255) is defined by a composition of the deformed $J^{\frac{1}{3}}$
function (254) restricted to $\mathbb{H}\setminus{SL_{2}(\mathbb{Z})(i)}$ and
the inverse of the Weierstrass $\wp$ function restricted to
$\mathbb{C}\setminus\\{e_{1}(z),e_{2}(z),e_{3}(z)\\}$, for fixed $z$ close
enough to $e^{\frac{2\pi i}{3}}$. More precisely, consider the functions

$\begin{split}&J_{z}^{\frac{1}{3}}:\mathbb{H}\setminus{SL_{2}(\mathbb{Z})(i)}\mapsto\mathbb{C}\setminus\\{e_{1}^{*}(z),e^{*}_{2}(z),e^{*}_{3}(z)\\},\quad\tilde{\tau}_{12}\mapsto
J^{\frac{1}{3}}(\tilde{\tau}_{12},z),\\\
&\wp:\left(\mathbb{C}\setminus{\mathbb{Z}\oplus
z\mathbb{Z}}\right)\setminus\big{\\{}\frac{\mathbb{Z}}{2}\oplus
z\frac{\mathbb{Z}}{2}\big{\\}}\mapsto\mathbb{C}\setminus\\{e_{1}(z),e_{2}(z),e_{3}(z)\\},\quad
v\mapsto\wp(v,z).\end{split}$

where

$e_{i}^{*}(z)=\frac{\Phi^{\prime\prime}\left(Q(t^{3})^{3}\left(z\right)\right)}{9Q^{\frac{1}{3}}t^{3}(z)}-4^{\frac{1}{3}}e_{i}(z).$

Then, for small enough $z$, the composition

$\begin{split}v(\tilde{\tau}_{12},z)=\left(-4^{\frac{1}{3}}\wp+\frac{\Phi^{\prime\prime}}{9Q^{\frac{1}{3}}t^{3}}\right)^{-1}\left(J_{z}^{\frac{1}{3}}\left(\tilde{\tau}_{12}\right)\right):\mathbb{H}\setminus\\{E_{6}=0\\}\mapsto\left(\mathbb{C}\setminus{\mathbb{Z}\oplus
z\mathbb{Z}}\right)\setminus\big{\\{}\frac{\mathbb{Z}}{2}\oplus
z\frac{\mathbb{Z}}{2}\big{\\}}\end{split}$ (258)

is a holomorphic surjective function, because it is a composition of
holomorphic surjective functions. Moreover, for $z$ close enough to
$e^{\frac{2\pi i}{3}}$, the derivative of the function (258)

$v^{\prime}(\tilde{\tau}_{12},z)=v^{\prime}(\tilde{\tau}_{12})+\sum_{n=1}^{\infty}v_{n}(\tau_{12})\left(z-e^{\frac{2\pi
i}{3}}\right)^{n}$

is close to its leading term

$v^{\prime}(\tilde{\tau}_{12})=\Delta^{\frac{1}{6}}\left(\tilde{\tau}_{12}\right),$

which is a non-vanishing function on $\mathbb{H}$. Therefore, we can extend
(258) to a local biholomorphism on $\mathbb{H}$

$\begin{split}v(\tilde{\tau}_{12},z):\mathbb{H}\mapsto\
\mathbb{C}\setminus{\left(\mathbb{Z}\oplus z\mathbb{Z}\right)}.\end{split}$
(259)

The function (259) is a $\pi_{1}(\mathbb{C}\setminus{(\mathbb{Z}\oplus
z\mathbb{Z}}))$-invariant function. Indeed, let $\gamma\in\Gamma^{(3)}$ and
$\tilde{\tau}_{12}(\tilde{\tau})$ the function defined in (177), then consider
the group homomorphism defined by

$\gamma_{z}\tilde{\tau}_{12}(\tilde{\tau}):=\tilde{\tau}_{12}(\gamma\tilde{\tau}).$
(260)

Since (177) is an isomonodromic deformation equation, the following property
holds

$\begin{split}J^{\frac{1}{3}}(\tilde{\tau}_{12}(\tilde{\tau}),z)&=J^{\frac{1}{3}}(\tilde{\tau}).\end{split}$

In addition,

$\begin{split}J^{\frac{1}{3}}(\gamma_{z}\tilde{\tau}_{12}(\tilde{\tau}),z)&=J^{\frac{1}{3}}(\tilde{\tau}_{12}(\gamma\tilde{\tau}),z),\\\
&=J^{\frac{1}{3}}(\gamma\tilde{\tau}),\\\
&=J^{\frac{1}{3}}(\tilde{\tau}).\end{split}$ (261)

The group homomorphism (260) induces another group homomorphism

$\bar{\gamma}_{z}v\left(\tilde{\tau}_{12}(\tilde{\tau}),z\right):=v\left(\tilde{\tau}_{12}(\gamma\tilde{\tau}),z\right).$
(262)

Denoting the image of (260) by $\Gamma^{(3)}_{z}$ and $Im\Psi_{z}$,
$Ker\Psi_{z}$ the image and kernel of (262) respectively, we have that

$\mathbb{H}/\Gamma^{(3)}_{z}=\left(\mathbb{H}/Ker\Psi_{z}\right)/Im\Psi_{z}=\mathbb{C}\setminus{\left(\mathbb{Z}\oplus
z\mathbb{Z}\right)}/Im\Psi_{z},$

$\mathbb{H}/Ker\Psi_{z}=\mathbb{C}\setminus{\left(\mathbb{Z}\oplus
z\mathbb{Z}\right)}=\mathbb{H}/\pi_{1}(\mathbb{C}\setminus{\left(\mathbb{Z}\oplus
z\mathbb{Z}\right)}).$

Hence, the map (259) is a local biholomorphism, which is
$\pi_{1}(\mathbb{C}\setminus{(\mathbb{Z}\oplus z\mathbb{Z}}))$-invariant.
Then, the map (259) can be identified with the quotient map

$\pi:\mathbb{H}\mapsto\mathbb{H}/\pi_{1}(\mathbb{C}\setminus{\mathbb{Z}\oplus
z\mathbb{Z}}),$

which is a covering map, since $\pi_{1}(\mathbb{C}\setminus{\mathbb{Z}\oplus
z\mathbb{Z}})$ acts properly discontinuously on $\mathbb{H}$.

Therefore, (255) is a family of universal covering of
$\mathbb{C}\setminus{\mathbb{Z}\oplus z\mathbb{Z}}$, which is an isomonodromic
deformation of the universal covering of the Equianharmonic Lattice.

Moreover, the correspondent Abelian differential $\phi=d\tilde{w}_{2}$ in
these coordinates is given by

$\begin{split}\phi&=d\tilde{w}_{2},\\\ &=\tilde{y}d\tilde{\tau}_{12},\\\
&=\frac{\Delta^{\frac{1}{6}}(\tilde{\tau}_{12},z)}{Q^{\frac{1}{6}}}d\tilde{\tau}_{12},\\\
&=\left[\frac{\Delta^{\frac{1}{6}}(\tilde{\tau}_{12},z)}{\frac{\partial
v(\tilde{\tau}_{12},z)}{\partial\tilde{\tau}_{12}}}\right]2\omega\frac{\partial
v(\tilde{\tau}_{12},z)}{\partial\tilde{\tau}_{12}}d\tilde{\tau}_{12},\\\
&=\left[\frac{\Delta^{\frac{1}{6}}(\tilde{\tau}_{12},z)}{\frac{\partial
v(\tilde{\tau}_{12},z)}{\partial\tilde{\tau}_{12}}}\right]2\omega
dv(\tilde{\tau}_{12},z).\end{split}$

Theorem proved.

∎

######  Remark 4.8.

The second generalised Cohn identity is obtained by substituting (257) in
(251) as follows

$\begin{split}&\left(\wp^{\prime}(v(\tilde{\tau}_{12},z),z)\right)^{2}=\\\
&-J(\tilde{\tau}_{12},z)-\frac{\Phi^{\prime\prime}}{4^{\frac{1}{3}}Q^{\frac{1}{3}}t^{3}}J^{\frac{2}{3}}(\tilde{\tau}_{12},z)-\left(\frac{\left(\Phi^{\prime\prime}\right)^{2}}{4^{\frac{2}{3}}Q^{\frac{2}{3}}(t^{3})^{2}}+g_{2}(z)\right)J^{\frac{1}{3}}(\tilde{\tau}_{12},z)-\frac{\left(\Phi^{\prime\prime}\right)^{3}}{4*27Q(t^{3})^{3}}-g_{3}(z).\end{split}$
(263)

######  Remark 4.9.

The title of this subsection was inspired by Doran’s paper [17], which
utilizes the moduli space of elliptic curves over $\mathbb{CP}^{1}$ to
construct algebraic-geometric solutions of Painlevé VI. In the context of this
manuscript, we also have a Painlevé VI associated with isomonodromic
deformation induced by the Gauss-Manin connection of $QH^{*}(\mathbb{CP}^{2})$
as mentioned in Section 4.2. It would be interesting to compare both
constructions in a more general setting, since geometric isomonodromic
deformation coming from Gromov Witten potential could give a good source of
interesting moduli space of elliptic curves over $\mathbb{CP}^{1}$.

##  5 Appendix

###  5.1 Composition of Power Series and Faa di Bruno formula

In this subsection, we state the necessary definitions and results on
composition of power series, referencing Chapter 11 of [10].

######  Definition 5.1.

[10] The polynomials
$B_{n}(x_{1},x_{2},..,x_{n}),B_{n,k}(x_{1},x_{2},..,x_{n})$ in the variables
$x_{1},x_{2},..,x_{n}$, defined by the sum

$\begin{split}B_{n}&=\sum_{k_{1}+2k_{2}+3k_{3}+..+nk_{n}=n}\frac{n!}{k_{1}!(1!)k_{2}!(2!)^{k_{2}}..k_{n}!(n!)^{k_{n}}}x_{1}^{k_{1}}x_{2}^{k_{2}}...x_{n}^{k_{n}}\\\
B_{n,k}&=\sum_{\begin{subarray}{c}k_{1}+..+k_{n}=k\\\
k_{1}+2k_{2}+3k_{3}+..+nk_{n}=n\end{subarray}}\frac{n!}{k_{1}!(1!)k_{2}!(2!)^{k_{2}}..k_{n}!(n!)^{k_{n}}}x_{1}^{k_{1}}x_{2}^{k_{2}}...x_{n}^{k_{n}}\\\
\end{split}$ (264)

are called exponential Bell partition polynomial and partial Bell partition
polynomial respectively.

######  Theorem 5.1.

[10][Faa di Bruno formula] Let $f(u)$ and $g(t)$ be two functions of real
variables for which all the derivatives

$g_{r}=\left[\frac{d^{r}g(t)}{dt^{r}}\right]_{t=a},\quad r=0,1...,\quad
f_{k}=\left[\frac{d^{k}f(u)}{du^{k}}\right]_{u=(g(a)},\quad k=0,1...$

exist. Then the derivatives of the composite function $h(t)=f(g(t))$,

$h_{n}=\left[\frac{d^{n}h(u)}{dt^{n}}\right]_{t=a},\quad n=0,1...$

are given by

$\begin{split}h_{n}&=\sum_{k=0}^{n}f_{k}B_{n,k}(g_{1},g_{2},..,g_{n})\\\
&=B_{n}(fg_{1},fg_{2},..,fg_{n}).\end{split}$ (265)

######  Definition 5.2.

[10] The polynomial $C_{n,s}=C_{n,s}(x_{1},x_{2},..,x_{n})$ in the variables
$x_{1},x_{2},..,x_{n}$, defined for a real (or complex) number s by the sum

$\begin{split}C_{n,s}&=\sum_{k_{1}+2k_{2}+3k_{3}+..+nk_{n}=n}\frac{n!}{k_{1}!(1!)k_{2}!(2!)^{k_{2}}..k_{n}!(n!)^{k_{n}}}{{s}\choose{k}}x_{1}^{k_{1}}x_{2}^{k_{2}}...x_{n}^{k_{n}}\\\
\end{split}$

is called potential partition polynomial, where

${{s}\choose{k}}=\frac{s(s-1)(s-2)..(s-k+1))}{k!}.$

######  Theorem 5.2.

[10]

The generating function of the potential partition polynomials
$C_{n,s}(x_{1},..x_{n})$, $n=0,1,..$, for fixed k, is given by

$C_{s}(t)=\sum_{n=0}^{\infty}C_{n,s}(x_{1},..x_{n})\frac{t^{n}}{n!}=\left[1+\left(g(t)-x_{0}\right)\right]^{s},$
(266)

where

$g(t)=\sum_{r=0}^{\infty}x_{r}\frac{t^{r}}{r!}.$

###  5.2 Canonical coordinates as functions of Gromov Witten invariants

######  Lemma 5.3.

Let be $\zeta=e^{\frac{2\pi i}{3}}$ and

$\begin{split}b_{k_{1},k_{2}}&:=\zeta^{k_{1}}+\zeta^{k_{1}+2k_{2}}+\zeta^{2k_{1}},\\\
c_{n_{1},n_{2},n_{3}}&=\zeta^{n_{1}+2n_{2}},\\\
\tilde{c}_{n_{1},n_{2},n_{3}}&:=c_{n_{1},n_{2},n_{3}}+c_{n_{1},n_{3},n_{2}}+c_{n_{3},n_{2},n_{1}}\\\
&+c_{n_{3},n_{1},n_{2}}+c_{n_{2},n_{1},n_{3}}+c_{n_{2},n_{3},n_{1}}.\end{split}$
(267)

for $k,k_{1},k_{2}k_{3}\in\mathbb{Z}$. Then the following identities holds

$\begin{split}\zeta^{k}+\zeta^{2k}+\zeta^{3k}&=1+2\cos\left(\frac{2\pi
k}{3}\right),\\\ b_{k,k}&=1+2\cos\left(\frac{2\pi k}{3}\right),\\\
c_{k,k,k}&=1,\end{split}$ (268)
$b_{k_{1}-k_{2},k_{2}}+b_{k_{2},k_{1}-k_{2}}=\begin{cases}6\cos\left(\frac{2\pi
k_{2}}{3}\right)&\text{if }k_{1}\in 3\mathbb{Z}\\\
0,&\text{otherwise}\end{cases}$
$\tilde{c}_{k_{1}-k_{2}-k_{3},k_{2},k_{3}}=\begin{cases}6\cos\left(\frac{2\pi(k_{2}+2k_{3})}{3}\right)&\text{if
}k_{1}\in 3\mathbb{Z}\\\ 0,&\text{otherwise}\end{cases}$

###### Proof.

$\begin{split}\zeta^{k}+\zeta^{2k}+\zeta^{3k}&=\zeta^{k}+\zeta^{2k}+1\\\
&=2\left(\frac{e^{\frac{2\pi ik}{3}}+e^{\frac{-2\pi ik}{3}}}{2}\right)+1\\\
&=1+2\cos\left(\frac{2\pi k}{3}\right).\\\ \end{split}$ (269)

Define $b_{k_{1},k_{2}}$

$\begin{split}b_{k_{1},k_{2}}:=\zeta^{k_{1}}+\zeta^{k_{1}+2k_{2}}+\zeta^{2k_{1}}.\end{split}$
(270)

Then,

$\begin{split}b_{k,k}&=\zeta^{k}+\zeta^{3k}+\zeta^{2k}\\\
&=1+2\cos\left(\frac{2\pi k}{3}\right).\\\ \end{split}$ (271)

Moreover,

$\begin{split}b_{k_{1}-k_{2},k_{2}}&=\zeta^{k_{1}-k_{2}}+\zeta^{k_{1}+k_{2}}+\zeta^{2(k_{1}-k_{2})}\\\
b_{k_{2},k_{1}-k_{2}}&=\zeta^{k_{2}}+\zeta^{2k_{1}-k_{2}}+\zeta^{2k_{2}}\\\
\end{split}$ (272)

  * •

If $k_{1}\in 3\mathbb{Z}$

$\begin{split}b_{k_{1}-k_{2},k_{2}}+b_{k_{2},k_{1}-k_{2}}&=\zeta^{-k_{2}}+\zeta^{k_{2}}+\zeta^{-2k_{2}}+\zeta^{k_{2}}+\zeta^{-k_{2}}+\zeta^{2k_{2}}\\\
&=\zeta^{2k_{2}}+\zeta^{k_{2}}+\zeta^{k_{2}}+\zeta^{k_{2}}+\zeta^{2k_{2}}+\zeta^{2k_{2}}\\\
&=3\left(\zeta^{2k_{2}}+\zeta^{k_{2}}\right)\\\ &=6\cos\left(\frac{2\pi
k_{2}}{3}\right).\end{split}$ (273)

  * •

If $k_{1}\in 1+3\mathbb{Z}$

$\begin{split}b_{k_{1}-k_{2},k_{2}}+b_{k_{2},k_{1}-k_{2}}&=\zeta^{1-k_{2}}+\zeta^{k_{2}+1}+\zeta^{2-2k_{2}}+\zeta^{k_{2}}+\zeta^{2-k_{2}}+\zeta^{2k_{2}}\\\
&=\zeta^{1+2k_{2}}+\zeta^{k_{2}+1}+\zeta^{k_{2}+2}+\zeta^{k_{2}}+\zeta^{2(k_{2}+1)}+\zeta^{2k_{2}}\\\
&=2\cos\left(\frac{2\pi(1-k_{2})}{3}\right)+2\cos\left(\frac{2\pi
k_{2}}{3}\right)+2\cos\left(\frac{2\pi(1+k_{2})}{3}\right)\\\ &=0.\end{split}$
(274)

  * •

If $k_{1}\in 2+3\mathbb{Z}$

$\begin{split}b_{k_{1}-k_{2},k_{2}}+b_{k_{2},k_{1}-k_{2}}&=\zeta^{2-k_{2}}+\zeta^{k_{2}+2}+\zeta^{1-2k_{2}}+\zeta^{k_{2}}+\zeta^{1-k_{2}}+\zeta^{2k_{2}}\\\
&=\zeta^{2-2k_{2}}+\zeta^{k_{2}+2}+\zeta^{2(2-k_{2})}+\zeta^{k_{2}}+\zeta^{2(k_{2}+2)}+\zeta^{2k_{2}}\\\
&=2\cos\left(\frac{2\pi(2-k_{2})}{3}\right)+2\cos\left(\frac{2\pi
k_{2}}{3}\right)+2\cos\left(\frac{2\pi(2+k_{2})}{3}\right)\\\ &=0.\end{split}$
(275)

Let be the tensor defined by

$\begin{split}c_{n_{1},n_{2},n_{3}}=\zeta^{n_{1}+2n_{2}}\end{split}$ (276)

It is straightforward the following

$\begin{split}c_{n,n,n}&=\zeta^{3n}\\\ &=1.\end{split}$ (277)

Moreover, setting

$\begin{split}\tilde{c}_{n_{1},n_{2},n_{3}}:=&c_{n_{1},n_{2},n_{3}}+c_{n_{1},n_{3},n_{2}}+c_{n_{3},n_{2},n_{1}}\\\
&+c_{n_{3},n_{1},n_{2}}+c_{n_{2},n_{1},n_{3}}+c_{n_{2},n_{3},n_{1}}\end{split}$
(278)

we have

$\begin{split}\tilde{c}_{n_{1},n_{2},n_{3}}:=&\zeta^{n_{1}+2n_{2}}+\zeta^{n_{1}+2n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{n_{3}+2n_{1}}+\zeta^{n_{2}+2n_{1}}+\zeta^{n_{2}+2n_{3}}.\end{split}$
(279)

In particular,

$\begin{split}\tilde{c}_{n_{1}-n_{2}-n_{3},n_{2},n_{3}}=&\zeta^{n_{1}+n_{2}-n_{3}}+\zeta^{n_{1}-n_{2}+n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{-n_{3}+2n_{1}-2n_{2}}+\zeta^{-n_{2}+2n_{1}-2n_{3}}+\zeta^{n_{2}+2n_{3}}.\end{split}$
(280)

  * •

If $n_{1}\in 3\mathbb{Z}$

$\begin{split}\tilde{c}_{n_{1}-n_{2}-n_{3},n_{2},n_{3}}&=\zeta^{n_{2}-n_{3}}+\zeta^{-n_{2}+n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{-n_{3}-2n_{2}}+\zeta^{-n_{2}-2n_{3}}+\zeta^{n_{2}+2n_{3}}\\\
&=\zeta^{n_{2}+2n_{3}}+\zeta^{2n_{2}+n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{2n_{3}+n_{2}}+\zeta^{2n_{2}+n_{3}}+\zeta^{n_{2}+2n_{3}}\\\
&=3\left(\zeta^{n_{2}+2n_{3}}+\zeta^{2n_{2}+n_{3}}\right)\\\
&=3\left(\zeta^{n_{2}+2n_{3}}+\zeta^{-n_{2}-2n_{3}}\right)\\\
&=6\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right).\\\ \end{split}$
(281)

  * •

If $n_{1}\in 1+3\mathbb{Z}$

$\begin{split}\tilde{c}_{n_{1}-n_{2}-n_{3},n_{2},n_{3}}&=\zeta^{1+n_{2}-n_{3}}+\zeta^{1-n_{2}+n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{-n_{3}-2n_{2}+2}+\zeta^{-n_{2}-2n_{3}+2}+\zeta^{n_{2}+2n_{3}}\\\
&=\zeta^{1+n_{2}-n_{3}}+\zeta^{2+2n_{2}-2n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{-n_{3}-2n_{2}+2}+\zeta^{-n_{2}-2n_{3}+1}+\zeta^{n_{2}+2n_{3}}\\\
&=2\cos\left(\frac{2\pi(1+n_{2}-n_{3})}{3}\right)+2\cos\left(\frac{2\pi(2n_{2}+n_{3})}{3}\right)\\\
&+2\cos\left(\frac{2\pi(1-n_{2}+n_{3})}{3}\right)\\\ &=0.\end{split}$ (282)

  * •

If $n_{1}\in 2+3\mathbb{Z}$

$\begin{split}\tilde{c}_{n_{1}-n_{2}-n_{3},n_{2},n_{3}}&=\zeta^{2+n_{2}-n_{3}}+\zeta^{2-n_{2}+n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{-n_{3}-2n_{2}+1}+\zeta^{-n_{2}-2n_{3}+1}+\zeta^{n_{2}+2n_{3}}\\\
&=\zeta^{2+n_{2}-n_{3}}+\zeta^{1+2n_{2}-2n_{3}}+\zeta^{n_{3}+2n_{2}}\\\
&+\zeta^{-n_{3}-2n_{2}+1}+\zeta^{-n_{2}-2n_{3}+2}+\zeta^{n_{2}+2n_{3}}\\\
&=2\cos\left(\frac{2\pi(2+n_{2}-n_{3})}{3}\right)+2\cos\left(\frac{2\pi(2n_{2}+n_{3})}{3}\right)\\\
&+2\cos\left(\frac{2\pi(2-n_{2}+n_{3})}{3}\right)\\\ &=0.\end{split}$ (283)

∎

######  Lemma 5.4.

Let be $u_{1},u_{2},u_{3}$ the canonical coordinates of Quantum Cohomology as
function of the Saito flat coordinates $t^{1},t^{2},t^{3}$. i.e.

$u_{k}=t^{1}+\frac{1}{t^{3}}\sum_{n=1}^{\infty}A_{n}^{k}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}.$
(284)

Then, the coefficients $A_{k}$ are given explicitly by

$\begin{split}&\tilde{A}_{3n}=\frac{n^{2}N_{n}}{(3n-1)!},\\\
&\sum_{n_{2}=2}^{3n}3\cos\left(\frac{2\pi(n_{2}-1)}{3}\right)\tilde{A}_{3n-n_{2}+1}\tilde{A}_{n_{2}-1}=\left(6-15n-9n^{2}\right)\frac{N_{n}}{(3n-1)!},\\\
&\sum_{n_{2}=1}^{3n-2}\sum_{n_{3}=1}^{3n-n_{2}-1}3\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right)\tilde{A}_{3n-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}=\left(54-243n+243n^{2}\right)\frac{N_{n}}{(3n-1)!}+\delta_{n}\\\
\end{split}$ (285)

where

$A_{n}^{k}=\tilde{A}_{n}\left(e^{\frac{2\pi i}{3}}\right)^{nk}$ (286)
$\delta_{n}=\begin{cases}0&\text{if }n=1\\\
\tilde{\delta}_{n},&\text{otherwise}\end{cases}$
$\begin{split}\tilde{\delta}_{n}=&\sum_{n_{2}=2}^{n}\frac{\left(6(n_{2}-1)-3(n-n_{2}+1)(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-4(n-n_{2}+1)(n_{2}-1)\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-9(n_{1}-n_{2}+1)^{2}(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}.\\\
\end{split}$ (287)

###### Proof.

The canonical coordinates $u_{1},u_{2},u_{3}$ can be written as

$u_{i}=t^{1}+\frac{9+\Phi^{\prime\prime}-z_{i}}{t^{3}}$ (288)

where $z_{i}$ is are the roots of

$\begin{split}(z-z_{1})(z-z_{2})(z-z_{3})&=\\\
z^{3}-s_{1}z^{2}+s_{2}z-s_{3}&=0\end{split}$ (289)

where

$\begin{split}s_{1}&=z_{1}+z_{2}+z_{3}=27+2\Phi^{\prime\prime},\\\
s_{2}&=z_{1}z_{2}+z_{2}z_{3}+z_{1}z_{3}=243+6\Phi-15\Phi^{\prime}+27\Phi^{\prime\prime}+\left(\Phi^{\prime\prime}\right)^{2}\\\
s_{3}&=z_{1}z_{2}z_{3}=\left(27+2\Phi^{\prime}-3\Phi^{\prime\prime}\right)^{2}\\\
\end{split}$ (290)

Define

$f_{k}:=\frac{1}{t^{3}}\sum_{n=1}^{\infty}A_{n}^{k}\left(Q^{\frac{1}{3}}t^{3}\right)^{n},$
(291)

substituting (288) in (311)

$z_{i}=9+\Phi^{\prime\prime}-f_{i}.$ (292)

Substituting (292) in (290), we obtain

$\begin{split}f_{1}+f_{2}+f_{3}&=\Phi^{\prime\prime}\\\
f_{1}f_{2}+f_{2}f_{3}+f_{1}f_{3}&=6\Phi-15\Phi^{\prime}-9\Phi^{\prime\prime}\\\
f_{1}f_{2}f_{3}&=54\Phi-243\Phi^{\prime}+243\Phi^{\prime\prime}+6\Phi\Phi^{\prime\prime}-4{\Phi^{\prime}}^{2}-3\Phi^{\prime}\Phi^{\prime\prime}-9{\Phi^{\prime\prime}}^{2}.\\\
\end{split}$ (293)

Using (21) in the first two equation of the right-hand side of (295), we have

$\begin{split}\Phi^{\prime\prime}&=\sum_{n=1}^{\infty}n^{2}\frac{N_{n}}{(3n-1)!}\left(Q(t^{3}\right)^{3})^{n}\\\
6\Phi-15\Phi^{\prime}-9\Phi^{\prime\prime}&=\sum_{n=1}^{\infty}\left(6-15n-9n^{2}\right)\frac{N_{n}}{(3n-1)!}\left(Q(t^{3}\right)^{3})^{n}\\\
\end{split}$ (294)

The third equation of the right-hand side of (295) is bit more involved

$\begin{split}&54\Phi-243\Phi^{\prime}+243\Phi^{\prime\prime}+6\Phi\Phi^{\prime\prime}-3\Phi^{\prime}\Phi^{\prime\prime}-4\left(\Phi^{\prime}\right)^{2}-9\left(\Phi^{\prime\prime}\right)^{2}=\\\
&=\sum_{n=1}^{\infty}\left(54-243n+243n^{2}\right)\frac{N_{n}}{(3n-1)!}\left(Q(t^{3}\right)^{3})^{n}\\\
&+\sum_{n_{1}=1}^{\infty}\sum_{n_{2}=1}^{\infty}\left(6n_{2}-3n_{1}n_{2}^{2}-4n_{1}n_{2}-9n_{1}^{2}n_{2}^{2}\right)\frac{N_{n_{1}}}{(3n_{1}-1)!}\frac{N_{n_{2}}}{(3n_{2}-1)!}\left(Q(t^{3}\right)^{3})^{n_{1}+n_{2}}\end{split}$
(295)

Using the following double infinite sum identity

$\sum_{k_{1}=1}^{\infty}\sum_{k_{2}=1}^{\infty}C_{k_{1},k_{2}}=\sum_{k_{1}=2}^{\infty}\left(\sum_{k_{2}=2}^{k_{1}}C_{k_{1}-k_{2}+1,k_{2}-1}\right)$
(296)

in the equation (295)

$\begin{split}&554\Phi-243\Phi^{\prime}+243\Phi^{\prime\prime}+6\Phi\Phi^{\prime\prime}-4{\Phi^{\prime}}^{2}-3\Phi^{\prime}\Phi^{\prime\prime}-9{\Phi^{\prime\prime}}^{2}=\\\
&=\sum_{n=1}^{\infty}\left(54-243n+243n^{2}\right)\frac{N_{n}}{(3n-1)!}\left(Q(t^{3}\right)^{3})^{n}\\\
&+\sum_{n_{1}=2}^{\infty}\left(\sum_{n_{2}=2}^{n_{1}}\frac{\left(6(n_{2}-1)-3(n_{1}-n_{2}+1)(n_{2}-1)^{2}\right)}{(3n_{1}-3n_{2}+2)!(3n_{2}-4)!}N_{n_{1}-n_{2}+1}N_{n_{2}-1}\right)\left(Q(t^{3}\right)^{3})^{n_{1}}\\\
&+\sum_{n_{1}=2}^{\infty}\left(\sum_{n_{2}=2}^{n_{1}}\frac{\left(-4(n_{1}-n_{2}+1)(n_{2}-1)\right)}{(3n_{1}-3n_{2}+2)!(3n_{2}-4)!}N_{n_{1}-n_{2}+1}N_{n_{2}-1}\right)\left(Q(t^{3}\right)^{3})^{n_{1}}\\\
&+\sum_{n_{1}=2}^{\infty}\left(\sum_{n_{2}=2}^{n_{1}}\frac{\left(-9(n_{1}-n_{2}+1)^{2}(n_{2}-1)^{2}\right)}{(3n_{1}-3n_{2}+2)!(3n_{2}-4)!}N_{n_{1}-n_{2}+1}N_{n_{2}-1}\right)\left(Q(t^{3}\right)^{3})^{n_{1}}\\\
\end{split}$ (297)

On another hand, setting

$A_{n}^{k}=\tilde{A}_{n}\left(e^{\frac{2\pi i}{3}}\right)^{nk}$ (298)

and using the first equation of (268), we have the first equation of the left
hand side of (295) can be written as

$\begin{split}f_{1}+f_{2}+f_{3}&=\sum_{n=1}^{\infty}\tilde{A}_{n}\left(\left(e^{\frac{2\pi
i}{3}}\right)^{n}+\left(e^{\frac{2\pi i}{3}}\right)^{2n}+\left(e^{\frac{2\pi
i}{3}}\right)^{3n}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\\\
&=\sum_{n=1}^{\infty}\tilde{A}_{n}\left(1+\cos(\frac{2\pi
n}{3})\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\\\
&=\sum_{n=1}^{\infty}3\tilde{A}_{3n}\left(Q(t^{3}\right)^{3})^{n}\\\
\end{split}$ (299)

Comparing equation (299) with (294) and (295), we have

$\tilde{A}_{3n}=\frac{n^{2}N_{n}}{3(3n-1)!}.$ (300)

The second equation of the left hand side of (295) can be written as

$\begin{split}f_{1}f_{2}+f_{2}f_{3}+f_{1}f_{3}&=\sum_{n_{1}=1}^{\infty}\sum_{n_{2}=1}^{\infty}\tilde{A}_{n_{1}}\tilde{A}_{n_{2}}b_{n_{1},n_{2}}\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}+n_{2}}\\\
&=\sum_{n_{1}=2}^{\infty}\left(\sum_{n_{2}=2}^{n_{1}}\tilde{A}_{n_{1}-n_{2}+1}\tilde{A}_{n_{2}-1}b_{n_{1}-n_{2}+1,n_{2}-1}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}}\\\
\end{split}$ (301)

where

$\begin{split}b_{n_{1},n_{2}}&=\left(e^{\frac{2\pi
i}{3}}\right)^{n_{1}+2n_{2}}+\left(e^{\frac{2\pi
i}{3}}\right)^{n_{1}}+\left(e^{\frac{2\pi i}{3}}\right)^{2n_{1}}\\\
\end{split}$ (302)

using lemma 5.3 in the equation (301)

$\begin{split}&\sum_{n_{1}=2}^{\infty}\left(\sum_{n_{2}=2}^{n_{1}}\tilde{A}_{n_{1}-n_{2}+1}\tilde{A}_{n_{2}-1}b_{n_{1}-n_{2}+1,n_{2}-1}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}}=\\\
&=\sum_{n_{1}=1}^{\infty}\left(\sum_{n_{2}=2}^{3n_{1}}3\cos\left(\frac{2\pi(n_{2}-1)}{3}\right)\tilde{A}_{3n_{1}-n_{2}+1}\tilde{A}_{n_{2}-1}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{3n_{1}}\\\
&=\sum_{n_{1}=1}^{\infty}\left(\sum_{n_{2}=2}^{3n_{1}}3\cos\left(\frac{2\pi(n_{2}-1)}{3}\right)\tilde{A}_{3n_{1}-n_{2}+1}\tilde{A}_{n_{2}-1}\right)\left(Q(t^{3})^{3}\right)^{n_{1}}\\\
\end{split}$ (303)

Comparing (303) with (294)

$\begin{split}\sum_{n_{2}=2}^{3n}3\cos\left(\frac{2\pi(n_{2}-1)}{3}\right)\tilde{A}_{3n-n_{2}+1}\tilde{A}_{n_{2}-1}=\left(6-15n-9n^{2}\right)\frac{N_{n}}{(3n-1)!}.\\\
\end{split}$ (304)

Using the triple infinite sum identity was used in

$\sum_{k_{1}=1}^{\infty}\sum_{k_{2}=1}^{\infty}\sum_{k_{3}=1}^{\infty}C_{k_{1},k_{2},k_{3}}=\sum_{k_{1}=3}^{\infty}\left(\sum_{k_{2}=1}^{k_{1}-2}\sum_{k_{3}=1}^{k_{1}-k_{2}-1}C_{k_{1}-k_{2}-k_{3},k_{2},k_{3}}\right).$
(305)

The third equation of the left hand side of (295) can be written as

$\begin{split}f_{1}f_{2}f_{3}&=\sum_{n_{1}=1}^{\infty}\sum_{n_{2}=1}^{\infty}\sum_{n_{3}=1}^{\infty}\tilde{A}_{n_{1}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}c_{n_{1},n_{2},n_{3}}\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}+n_{2}+n_{3}}\\\
&=\sum_{n_{1}=3}^{\infty}\left(\sum_{n_{2}=1}^{n_{1}-2}\sum_{n_{3}=1}^{n_{1}-n_{2}-1}\tilde{A}_{n_{1}-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}c_{n_{1}-n_{2}-n_{3},n_{2},n_{3}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}}\\\
\end{split}$ (306)

where

$\begin{split}c_{n_{1},n_{2},n_{3}}=\left(e^{\frac{2\pi
i}{3}}\right)^{n_{1}+2n_{2}+3n_{3}}.\end{split}$ (307)

Then

$\begin{split}&\sum_{n_{1}=3}^{\infty}\left(\sum_{n_{2}=1}^{n_{1}-2}\sum_{n_{3}=1}^{n_{1}-n_{2}-1}\tilde{A}_{n_{1}-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}c_{n_{1}-n_{2}-n_{3},n_{2},n_{3}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}}=\\\
&=\sum_{n_{1}=3}^{\infty}\left(\sum_{n_{2}=1}^{n_{1}-2}\sum_{n_{3}=1}^{n_{1}-n_{2}-1}\tilde{A}_{n_{1}-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}\left(e^{\frac{2\pi
i}{3}}\right)^{n_{1}+n_{2}+2n_{3}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n_{1}}\\\
&=\sum_{n_{1}=1}^{\infty}\left(\sum_{n_{2}=1}^{3n_{1}-2}\sum_{n_{3}=1}^{3n_{1}-n_{2}-1}3\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right)\tilde{A}_{3n_{1}-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{3n_{1}}\\\
&=\sum_{n_{1}=1}^{\infty}\left(\sum_{n_{2}=1}^{3n_{1}-2}\sum_{n_{3}=1}^{3n_{1}-n_{2}-1}3\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right)\tilde{A}_{3n_{1}-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}\right)\left(Q(t^{3})^{3}\right)^{n_{1}}\\\
\end{split}$ (308)

Comparing (308) with (297)

$\begin{split}\sum_{n_{2}=1}^{3n-2}\sum_{n_{3}=1}^{3n-n_{2}-1}3\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right)\tilde{A}_{3n-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}=\left(54-243n+243n^{2}\right)\frac{N_{n}}{(3n-1)!}+\delta_{n}\\\
\end{split}$ (309)

where

$\delta_{n}=\begin{cases}0&\text{if }n=1\\\
\tilde{\delta}_{n},&\text{otherwise}\end{cases}$
$\begin{split}\tilde{\delta}_{n}=&\sum_{n_{2}=2}^{n}\frac{\left(6(n_{2}-1)-3(n-n_{2}+1)(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-4(n-n_{2}+1)(n_{2}-1)\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-9(n_{1}-n_{2}+1)^{2}(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}.\\\
\end{split}$ (310)

∎

######  Corollary 5.4.1.

Let be $u_{1},u_{2},u_{3}$ the canonical coordinates of Quantum Cohomology as
function of the Saito flat coordinates $t^{1},t^{2},t^{3}$. i.e.

$u_{k}=t^{1}+\frac{1}{t^{3}}\sum_{n=1}^{\infty}A_{n}^{k}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}.$
(311)

Then, the coefficients $A_{k}$ are given recursively by

$\begin{split}\tilde{A}_{3n}&=\frac{n^{2}N_{n}}{(3n-1)!},\\\
\tilde{A}_{3n-1}&=\frac{1}{3\tilde{A}_{1}}\left[\left(6-15n-9n^{2}\right)\frac{N_{n}}{(3n-1)!}-\sum_{n_{2}=3}^{3n-1}3\cos\left(\frac{2\pi(n_{2}-1)}{3}\right)\tilde{A}_{3n-n_{2}+1}\tilde{A}_{n_{2}-1}\right],\\\
\tilde{A}_{3n-2}&=\frac{1}{9\tilde{A}_{1}^{2}}\left[\left(54-243n+243n^{2}\right)\frac{N_{n}}{(3n-1)!}+\delta_{n}-\sum_{n_{3}=2}^{3n-2}3\cos\left(\frac{2\pi\left(1+2n_{3}\right)}{3}\right)\tilde{A}_{3n-1-n_{3}}\tilde{A}_{1}\tilde{A}_{n_{3}}\right]\\\
&-\frac{1}{9\tilde{A}_{1}^{2}}\left[\sum_{n_{2}=2}^{3n-3}\sum_{n_{3}=1}^{3n-n_{2}-1}3\cos\left(\frac{2\pi\left(n_{2}+2n_{3}\right)}{3}\right)\tilde{A}_{3n-n_{2}-n_{3}}\tilde{A}_{n_{2}}\tilde{A}_{n_{3}}\right],\end{split}$
(312)

where

$A_{n}^{k}=\tilde{A}_{n}\left(e^{\frac{2\pi i}{3}}\right)^{nk}$ (313)
$\delta_{n}=\begin{cases}0&\text{if }n=1\\\
\tilde{\delta}_{n},&\text{otherwise}\end{cases}$
$\begin{split}\tilde{\delta}_{n}=&\sum_{n_{2}=2}^{n}\frac{\left(6(n_{2}-1)-3(n-n_{2}+1)(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-4(n-n_{2}+1)(n_{2}-1)\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}\\\
&+\sum_{n_{2}=2}^{n}\frac{\left(-9(n_{1}-n_{2}+1)^{2}(n_{2}-1)^{2}\right)}{(3n-3n_{2}+2)!(3n_{2}-4)!}N_{n-n_{2}+1}N_{n_{2}-1}.\\\
\end{split}$ (314)

###  5.3 Coefficients of the cross ratio function

######  Lemma 5.5.

Let the cross ratio function be defined by the formula

$\begin{split}f(Q^{\frac{1}{3}}t^{3})&=\frac{u_{3}-u_{1}}{u_{2}-u_{1}}.\\\
&=\sum_{n=0}^{\infty}f_{n}\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\end{split}$
(315)

Then, the Taylor series of (315) is given by

$\begin{split}f(Q^{\frac{1}{3}}t^{3})&=\sum_{n=0}^{\infty}\sum_{m=0}^{n}\left(\frac{A_{n+1-m}^{3}-A_{n+1-m}^{1}}{A_{1}^{2}-A_{1}^{1}}\right)C_{m,-1}(y_{1},y_{2},..,y_{m})\left(Q^{\frac{1}{3}}t^{3}\right)^{n},\\\
\end{split}$ (316)

where

$y_{n}=\frac{A_{n+1}^{2}-A_{n+1}^{1}}{A_{1}^{2}-A_{1}^{1}},$

and $A_{n}$ is defined in (237).

###### Proof.

The coefficients $f_{n}$ (315) is obtained by substituting (237) in (315) and
using theorem 5.2. Indeed,

$\begin{split}\frac{u_{3}-u_{1}}{u_{2}-u_{1}}&=\frac{\sum_{n=1}^{\infty}\left(A_{n}^{3}-A_{n}^{1}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}}{\sum_{n=1}^{\infty}\left(A_{n}^{2}-A_{n}^{1}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}}\\\
&=\left(\sum_{n=0}^{\infty}\left(\frac{A_{n+1}^{3}-A_{n+1}^{1}}{A_{1}^{2}-A_{1}^{1}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right)\left(1+\sum_{n=1}^{\infty}\left(\frac{A_{n+1}^{2}-A_{n+1}^{1}}{A_{1}^{2}-A_{1}^{1}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right)^{-1}\\\
&=\left(\sum_{n=0}^{\infty}\left(\frac{A_{n+1}^{3}-A_{n+1}^{1}}{A_{1}^{2}-A_{1}^{1}}\right)\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right)\left(\sum_{n=0}^{\infty}C_{n,-1}(y_{1},y_{2},..,y_{n})\left(Q^{\frac{1}{3}}t^{3}\right)^{n}\right)\\\
&=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\left(\frac{A_{n+1}^{3}-A_{n+1}^{1}}{A_{1}^{2}-A_{1}^{1}}\right)C_{m,-1}(y_{1},y_{2},..,y_{m})\left(Q^{\frac{1}{3}}t^{3}\right)^{n+m}\\\
&=\sum_{n=0}^{\infty}\sum_{m=0}^{n}\left(\frac{A_{n+1-m}^{3}-A_{n+1-m}^{1}}{A_{1}^{2}-A_{1}^{1}}\right)C_{m,-1}(y_{1},y_{2},..,y_{m})\left(Q^{\frac{1}{3}}t^{3}\right)^{n},\\\
\end{split}$ (317)

where

$y_{n}=\frac{A_{n+1}^{2}-A_{n+1}^{1}}{A_{1}^{2}-A_{1}^{1}},$

and the following Double sum identity was used

$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_{n,m}=\sum_{n=0}^{\infty}\sum_{m=0}^{n}C_{n-m,m}.$
(318)

Lemma proved. ∎

###  5.4 Coefficients of Modular lambda function

Recall the the Eisenstein series $E_{2}(\tau),E_{4}(\tau),E_{6}(\tau)$ at the
point $\tau=e^{\frac{2\pi i}{3}}$ have the following values

$E_{2}(e^{\frac{2\pi i}{3}})=\frac{2\sqrt{3}}{\pi},\quad E_{4}(e^{\frac{2\pi
i}{3}})=0,\quad E_{6}(e^{\frac{2\pi
i}{3}})=\frac{\left(\Gamma(\frac{1}{3})\right)^{18}}{4^{6}}.$ (319)

Moreover, there exist the following relation between $x,x^{\prime}$ and
$\Delta$

$\begin{split}\frac{2^{8}}{3^{3}}\left(2\pi\right)^{-12}\Delta(\tau)=\frac{(x^{\prime})^{6}}{x^{4}(x-1)^{4}}\end{split}$
(320)

The special values of $x$ at $\tau=e^{\frac{2\pi i}{3}}$ is

$x(e^{\frac{2\pi i}{3}})=e^{\frac{\pi i}{3}}$ (321)

Substituting (321), (319) in (320), we compute $x^{\prime}(e^{\frac{2\pi
i}{3}})$. Computing

$\begin{split}\frac{2^{8}}{3^{3}}\left(2\pi\right)^{-12}\frac{d^{n}\log\Delta(\tau)}{d\tau^{n}}=\frac{d^{n}}{d\tau^{n}}\left(\frac{(x^{\prime})^{6}}{x^{4}(x-1)^{4}}\right)\end{split}$
(322)

we obtain in the left hand side (322) a polynomial expression in terms of
$E_{2},E_{4},E_{6}$ and in the right hand side a rational expression of
$\frac{d^{n}x(\tau)}{d\tau^{n}}$. Then, we can derive
$\frac{d^{n}x(\tau)}{d\tau^{n}}$ for any order $n$ by using the recursive
relation (322).

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