Let $\ell$ be admissible at $u^{(0)}$. An $\ell$-cell of
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ is any connected component of the open
dense subset of points $u\in\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ such that
$u_{1},...,u_{n}$ are pairwise distinct and $\ell$ is admissible at $u$.

######  Proposition 4.1 ([CDG19]).

An $\ell$-cell is homeomorphic to a ball.

$\delta_{i}$$\delta_{j}$$\lambda_{i}$$\lambda_{j}$$\lambda_{i}$$\delta_{i}$$\delta_{j}$$u_{a}$$u_{b}$$\overline{B}(\lambda_{i};\epsilon_{0})$$\overline{B}(\lambda_{j};\epsilon_{0})$$\lambda_{j}$$u_{a}$$u_{b}$$\overline{B}(\lambda_{j};\epsilon_{0})$$\overline{B}(\lambda_{i};\epsilon_{0})$
Figure 2. Points $\lambda_{i}$, $\lambda_{j}$ and $u_{a}$, $u_{b}$ are
represented on the same complex plane. The thick line has slope $\pi/2-\phi$.
As $u$ varies, for values of $\epsilon_{0}$ sufficiently small (left figure)
the Stokes rays $R_{ab}$ and $R_{ba}$ associated with $u_{a}$ in the disk
$\overline{B}(\lambda_{i};\epsilon_{0})$ and $u_{b}$ in the disk of
$\overline{B}(\lambda_{j};\epsilon_{0})$ do not cross the line $\ell$. If the
disks have radius exceeding $\min_{1\leq i\leq s}\delta_{i}$ as in (4.2) (see
right figure) then the Stokes rays $R_{ab}$, $R_{ba}$ cross the line $\ell$.

We notice that, if $u(p)$ is in a $\ell$-cell then $p$ lies in an
$\ell$-chamber. Thus, if $\mathcal{D}$ is an open subset whose closure is
contained in a cell of $\mathcal{U}_{\epsilon_{0}}(u^{(0)})$, according to
Theorems 2.9, point (3), the system

$\frac{dY}{dz}=\left(U+\frac{V(u)}{z}\right)Y,$ (4.4)

for $u\in\mathcal{D}$ admits two fundamental solutions $Y^{(0)}_{\rm
right/left}(z,u)$ uniquely determined by the canonical asymptotic
representation $Y^{(0)}_{\rm right/left}(z,u)\sim Y_{\rm formal}(z,u)$ as in
(2.19) valid in the sectors $\Pi_{\rm left/right}(\phi)$ respectively. It
follows from the proof of Theorem 2.8 that $Y_{\rm formal}(z,u)$ is
$u$-holomorphic in
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})\backslash\Delta_{\Omega}$. By Remark 2.8
actually the asymptotic representation is valid in wider sectors
$\mathcal{S}_{\rm left/right}(u)$, defined as the sectors which contain
$\Pi_{\rm left/right}(\phi)$ and extends up to the nearest Stokes rays. By
Theorem 2.12 the above system with $u\in\mathcal{D}$ is isomonodromic, so that
the Stokes matrix $S$ defined in formula (2.24) is constant.

Let us now turn our attention to the coalescence point $u^{(0)}$. From the
results of [CDG19] – and more generally in [BJL79b] \- it follows that there
are a unique formal solution at $u^{(0)}$,

$\mathring{Y}_{\rm
formal}(z)=\left(\mathbbm{1}+\sum_{k=1}^{\infty}\frac{\mathring{G}_{k}}{z^{k}}\right)e^{zU},$

and unique actual solutions $\mathring{Y}^{(0)}_{\rm left}(z)$ and
$\mathring{Y}^{(0)}_{\rm right}(z)$, with asymptotic representation given by
$\mathring{Y}_{\rm formal}(z)$ in $\Pi_{\rm left/right}$, and in wider sectors
$\mathcal{S}_{\rm left}(u^{(0)})$ and $\mathcal{S}_{\rm right}(u^{(0)})$
respectively. The Stokes matrices of $\mathring{Y}^{(0)}_{\rm right}(z)$ and
$\mathring{Y}^{(0)}_{\rm left}(z)$ are defined by

$\mathring{Y}^{(0)}_{\rm left}(z)=\mathring{Y}^{(0)}_{\rm
right}(z)\mathring{S},\quad\quad\mathring{Y}^{(0)}_{\rm left}(e^{2\pi
i}z)=\mathring{Y}^{(0)}_{\rm
right}(z)\mathring{S}_{-},\quad\quad\mathring{S}_{-}=\mathring{S}^{T}.$

A priori, the following problems could emerge.

  1. (1)

The asymptotic representations

$Y^{(0)}_{\rm left/right}(z,u)\sim Y_{\rm formal}(z,u),\quad\text{for
}|z|\to\infty\text{ and }z\in\bigcap_{u\in\mathcal{D}}\mathcal{S}_{\rm
left/right}(u)\supsetneq\Pi_{\rm left/right}(\phi)$

do no longer hold for $u$ outside the cell containing $\mathcal{D}$.

  2. (2)

The coefficients $G_{k}(u)$’s of (2.19) may be divergent at $\Delta_{\Omega}$.

  3. (3)

The locus $\Delta_{\Omega}$ is expected to be a locus of singularities for the
solutions $Y_{\rm formal}(z,u)$ in (2.19) and $Y^{(0)}_{\rm left/right}(z,u)$.
$Y_{\rm formal}(z,u)$

  4. (4)

The Stokes matrix $S$ may differ from $\mathring{S}$.

We notice that the system (4.4) at $u^{(0)}$ also has a fundamental solution
in Levelt form at $z=0$,

$\mathring{Y}_{0}(z)=\Psi(u^{(0)})(I+\mathcal{O}(z))z^{\mu}z^{\mathring{R}},$
(4.5)

with a certain exponent $\mathring{R}$. Hence, a central connection matrix
$\mathring{C}$ is defined by

$\mathring{Y}^{(0)}_{\rm right}(z)=\mathring{Y}_{0}(z)\mathring{C}$

We recall that $\Psi(u)$ is holomorphic in the whole
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$, so that $V_{ij}(u)$ vanishes along
$\Delta_{\Omega}$ whenever $u_{i}=u_{j}$ (see Lemma 2.5). These are sufficient
conditions to apply the main theorem of [CDG19], adapted and particularised to
the case of Frobenius manifolds, which becomes the following:

######  Theorem 4.1.

Let $M$ be a semisimple Frobenius manifold,
$p_{0}\in\mathcal{B}_{M}\setminus\mathcal{K}_{M}$ and $\Omega\subseteq
M_{ss}=M\setminus\mathcal{K}_{M}$ an open connected neighborhood of $p_{0}$
with the property of Remark 2.4 on which a holomorphic branch for canonical
coordinates $u\colon\Omega\to\mathbb{C}^{n}$ and $\Psi\colon\Omega\to
GL_{n}(\mathbb{C})$ has been fixed. Let $\epsilon_{0}$ be a real positive
number as above, and consider the corresponding neighborhood
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ of $u^{(0)}=u(p_{0})$. Then

  1. (1)

The coefficients $G_{k}(u)$, $k\geq 1$, in (2.19) are holomorphic over
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$,

$G_{k}(u^{(0)})=\mathring{G}_{k}\quad\text{ and }\quad Y_{\rm
formal}(z,u^{(0)})=\mathring{Y}_{\rm formal}(z).$

  2. (2)

$Y^{(0)}_{\rm left}(z,u)$, $Y^{(0)}_{\rm right}(z,u)$, can be $u$-analytically
continued as single-valued holomorphic functions141414Hence, they are
holomoprhic on $\mathcal{R}\times\mathcal{U}_{\epsilon_{0}}(u^{(0)})$. on
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$. Moreover

$Y^{(0)}_{\rm left/right}(z,u^{(0)})=\mathring{Y}^{(0)}_{\rm left/right}(z).$

  3. (3)

For any solution $\mathring{Y}_{0}(z)$ as in (4.5) there exists a fundamental
solution $Y_{0}(z,u)$ in Levelt form (2.23) such that

$Y_{0}(z,u^{(0)})=\mathring{Y}_{0}(z),\quad\quad R=\mathring{R}.$

  4. (4)

For any positive $\epsilon_{1}<\epsilon_{0}$, the asymptotic relations

$Y^{(0)}_{\rm
left/right}(z,u)\sim\left(\mathbbm{1}+\sum_{k=1}^{\infty}\frac{G_{k}(u)}{z^{k}}\right)e^{zU},\quad
z\to\infty\hbox{ in }\Pi_{\rm left/right}(\phi),$ (4.6)

hold uniformly in $u\in\mathcal{U}_{\epsilon_{1}}(u^{(0)})$. In particular
they hold also at points of
$\Delta_{\Omega}\cap\mathcal{U}_{\epsilon_{1}}(u^{(0)})$ and at $u^{(0)}$.

  5. (5)

For any $u\in\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ consider the sectors
$\widehat{\mathcal{S}}_{\rm right}(u)$ and $\widehat{\mathcal{S}}_{\rm
left}(u)$ which contain the sectors $\Pi_{\rm right}(\phi)$ and $\Pi_{\rm
left}(\phi)$ respectively, and extend up to the nearest Stokes rays in the set
$\mathfrak{R}(u)$ defined above. Let

$\widehat{\mathcal{S}}_{\rm
left/right}=\bigcap_{u~{}\in~{}\mathcal{U}_{\epsilon_{0}}(u^{(0)})}\widehat{\mathcal{S}}_{\rm
left/right}(u).$

Observe that for sufficiently small $\varepsilon>0$ the sectors

$\Pi^{\varepsilon}_{\rm{right}}(\phi):=\left\\{z\in\mathcal{R}\colon\phi-\pi-\varepsilon<\arg\
z<\phi+\varepsilon\right\\},$
$\Pi^{\varepsilon}_{\rm{left}}(\phi):=\left\\{z\in\mathcal{R}\colon\phi-\varepsilon<\arg\
z<\phi+\pi+\varepsilon\right\\},$

are strictly contained in $\widehat{\mathcal{S}}_{\rm right}$ and
$\widehat{\mathcal{S}}_{\rm left}$ respectively. Then, the asymptotic
relations (4.6) actually hold in the sectors $\widehat{\mathcal{S}}_{\rm
left/right}$.

  6. (6)

The monodromy data $\mu$, $R$, $C$, $S$ of system (4.4), defined and constant
in an open subset $\mathcal{D}$ of a cell of
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$, are actually defined and constant at
any $u\in\mathcal{U}_{\epsilon_{1}}(u^{(0)})$, namely the system is
isomonodromic in $\mathcal{U}_{\epsilon_{1}}(u^{(0)})$. They coincide with the
data $\mu$, $\mathring{R}$, $\mathring{C}$, $\mathring{S}$ associated to the
fundamental solutions $\mathring{Y}_{\rm left/right}(z)$ and
$\mathring{Y}_{0}(z)$ of system (4.4) at $u^{(0)}$. The entries of
$S=(S_{ij})_{i,j=1}^{n}$ satisfy the vanishing condition (1.14), namely

$S_{ij}=S_{ji}=0\quad\text{ for all $i\neq j$ such that }\quad
u_{i}^{(0)}=u_{j}^{(0)}.$ (4.7)

This Theorem allows us to obtain the monodromy data $\mu$, $R$, $C$, $S$ in a
neighbourhood of a coalescence point just by computing them _at_ the
coalescence point, namely just by computing $\mu$, $\mathring{R}$,
$\mathring{C}$, $\mathring{S}$. Its importance has been explained in the
Introduction and will be illustrated in subsequent sections.

######  Remark 4.1.

Suppose that $S$ is upper triangular. By formula (4.7) it follows that in any
$\ell$-cell of $\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ the order of the
canonical coordinates is triangular, according to Definition 3.2, and at most
in one cell the order is lexicographical (Definition 3.1). Moreover, if
$\lambda_{1}$, …, $\lambda_{s}$ are in lexicographical order, then the order
of canonical coordinates is lexicographical in exactly one cell.

###  4.1. Reconstruction of monodromy data of the whole manifold

The monodromy data of the Frobenius manifold can be obtained from those
computed in Theorem 4.1 around $u^{(0)}$. Without loss of generality, let us
suppose that the ordering (4) is such that $\lambda_{1},\dots,\lambda_{s}$ are
in $\ell$\- lexicographical order. Then, the matrix $S$ computed at the
coalescence point $u^{(0)}$ is upper triangular. Therefore, by Theorem 4.1,
the matrix is constant and upper triangular in the whole polydisc
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$. In particular, it is upper triangular
in every cell of $\mathcal{U}_{\epsilon_{1}}(u^{(0)})$. This means that
$u_{1},...,u_{n}$ are in triangular order (Definition 3.2) in each such cell,
and in particular they are in lexicographical order in only one of these cells
(Definition 3.1, Remark 4.1). Note that any permutation of canonical
coordinates preserving the sets $I_{1},\dots,I_{s}$ of (4.3) maintains the
upper triangular structure of $S$, namely the triangular order of
$u_{1},...,u_{n}$ in each cell of $\mathcal{U}_{\epsilon_{o}}(u^{(0)})$. The
permutation changes the cell where the order is lexicographical. Now, each
cell of the polydisc $\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ is contained in a
chamber of the manifold (identifying coordinates with points of the manifold,
which is possible because of the holomorphy of canonical coordinates near
_semisimple_ coalescent points). Let us start from the cell of
$\mathcal{U}_{\epsilon_{0}}(u^{(0)})$ where $u_{1},...,u_{n}$ are in
lexicographical order. The monodromy data of Theorem 4.1 in this cell are the
constant data of the chamber containing the cell (Theorems 2.4 and 2.12).
Since in this chamber $u_{1},...,u_{n}$ are in lexicographical order (and
distinct!), we can apply the action of the braid group to $S$ and $C$, as
dictated by formulae (3.3), (3.5). In this way, the monodromy data for any
other chamber of the manifold are obtained, as explained in Section 3.

##  5\. First detailed example of application of Theorem 4.1: the $A_{3}$
Frobenius manifold. Stokes phenomenon for Pearcey-type oscillating integrals
from Hankel functions

With the example of $A_{3}$ Frobenius manifold below, we show how Theorem 4.1
allows the computation of monodromy data in an elementary way, by means of
Hankel special functions. Moreover, we apply the results of section 3,
especially showing how the braid group can be used to reconstruct the data for
the whole manifold, starting from a coalescence point. The reader not
interested in a general introduction to Frobenius manifolds associated with
singularity theory may skip Sections 5.1 and 5.2 and go directly to Section
5.3.

###  5.1. Singularity Theory and Frobenius Manifolds

Let $f$ be a quasi-homogeneous polynomial on $\mathbb{C}^{m}$ with an isolated
simple singularity at $0\in\mathbb{C}^{m}$. According to V.I. Arnol’d [Arn72]
simple singularities are classified by simply-laced Dynkin diagrams $A_{n}\
(\text{with }n\geq 1),D_{n}\ (\text{with }n\geq 4),E_{6},E_{7},E_{8}$.
Denoting by $(x_{1}\dots,x_{m})$ the coordinates in $\mathbb{C}^{m}$ (for
singularities of type $A_{n}$ we consider $m=1$), the classification of simple
singularities is summarized in Table 1. Let $\mu$ be the Milnor number of $f$
(note that $\mu=n$ for $A_{n}$, $D_{n}$ and $E_{n}$), and

$f(x,a):=f(x)+\sum_{i=1}^{\mu}a_{i}\phi_{i}(x),$

be a miniversal unfolding of $f$, where $a$ varies in a ball
$B\subseteq\mathbb{C}^{\mu}$, and $(\phi_{1}(x),\dots,\phi_{\mu}(x))$ is a
basis of the Milnor ring. Using K. Saito’s theory of _primitive forms_
[Sai83], a flat metric and a Frobenius manifold structure can be defined on
the base space $B$ [BV92]. See also [Sai93]. For any fixed $a\in B$, let the
critical points be $x_{i}(a)=(x^{(1)}_{i},\dots,x^{(m)}_{i})$,
$i=1,\dots,\mu$, defined by the condition $\partial_{x_{\alpha}}f(x_{i},a)=0$
for any $\alpha=1,\dots,m$, with critical values $u_{i}(a):=f(x_{i}(a),a)$.
The open ball $B$ can be stratified as follows:

  1. (1)

the stratum of _generic points_ , i.e. points where both critical points
$x^{(i)}$’s and critical values $u_{i}$’s are distinct;

  2. (2)

the _Maxwell stratum_ , which is the closure of the set of points with
distinct critical points $x^{(i)}$’s but some coalescing critical values
$u_{i}$’s;

  3. (3)

the _caustic_ , where some critical points coalesce.

The union of the Maxwell stratum and the caustic is called _function
bifurcation diagram_ $\Xi$ of the singularity (see [Arn93] and [AGZV88]).

The complement of the caustic consists exclusively of semisimple points of the
Frobenius manifold, and the critical values $u_{i}(a):=f(x_{i}(a),a)$ are the
canonical coordinates.

In this section we want to show how one can reconstruct local information near
semisimple points in the Maxwell stratum, by invoking Theorem 4.1. We will
focus on the simplest example of $A_{3}$.

######  Remark 5.1.

For simplicity of exposition, here we focus only on the case of Frobenius
structures associated with simple singularities. The existence of a primitive
forms for arbitrary singularities was proved by M. Saito in [Sai89].

| Singularity | Versal Deformation
---|---|---
$A_{n}$ | $f(x)=x^{n+1}$ | $f(x,a)=x^{n+1}+a_{n-1}x^{n-1}+\dots+a_{1}x+a_{0}$
$D_{n}$ | $f(x)=x_{1}^{n-1}+x_{1}x_{2}^{2}$ | $f(x,a)=x_{1}^{n-1}+x_{1}x_{2}^{2}+a_{n-1}x_{1}^{n-2}+\dots+a_{1}+a_{0}x_{2}$
$E_{6}$ | $f(x)=x_{1}^{4}+x_{2}^{3}$ | $f(x,a)=x_{1}^{4}+x_{2}^{3}+a_{6}x_{1}^{2}x_{2}+a_{5}x_{1}x_{2}+a_{4}x^{2}_{1}+a_{3}x_{2}+a_{2}x_{1}+a_{1}$
$E_{7}$ | $f(x)=x_{1}^{3}x_{2}+x_{2}^{3}$ | $f(x,a)=x^{3}_{1}x_{2}+x^{3}_{2}+a_{1}+a_{2}x_{2}+a_{3}x_{1}x_{2}+a_{4}x_{1}x_{2}+a_{5}x_{1}+a_{6}x^{2}_{1}+a_{7}x_{2}$
$E_{8}$ | $f(x)=x^{5}_{1}+x^{3}_{2}$ | $f(x,a)=x^{5}_{1}+x^{3}_{2}+a_{8}x^{3}_{1}x_{2}+a_{7}x^{2}_{1}x_{2}+a_{6}x^{3}_{1}+a_{5}x_{1}x_{2}+a_{4}x^{2}_{1}+a_{3}x_{2}+a_{2}x_{1}+a_{1}$
Table 1. Arnol’d’s classification of simple singularities, and their
corresponding miniversal deformations.

###  5.2. Frobenius structure of type $A_{n}$

( [DVV91], [Dub96], [Dub99b],[Dub99a])

Let us consider the affine space $M\cong\mathbb{C}^{n}$ of all polynomials

$f(x,a)=x^{n+1}+a_{n-1}x^{n-1}+\dots+a_{1}x+a_{0},$

where $(a_{0},\dots,a_{n-1})\in M$ are used as coordinates. We call
_bifurcation diagram_ $\Xi$ of the singularity $A_{n}$ the set of polynomials
in $M$ with some coalescing critical values. The bifurcation diagram $\Xi$ is
an algebraic subvariety in $M$ which consists of two irreducible components
(the derivative w.r.t. the variable $x$ will be denoted by
$(\cdot)^{\prime}$):

  * •

the caustic $\mathcal{K}$, which is the set of polynomials with degenerate
critical points (i.e. solutions of the system of equations
$f^{\prime}(x,a)=f^{\prime\prime}(x,a)=0$)151515The equation of the caustic is
$\Delta(f^{\prime})=0$, where
$\Delta(f^{\prime}):=\operatorname{Res}(f^{\prime},f^{\prime\prime})$ is the
discriminant of the polynomial $f^{\prime}(x,a)$. The reader can consult the
monograph [GKZ94], Chapter 12.;

  * •

The Maxwell stratum $\mathcal{M}$, defined as the closure of the set of
polynomials with some coalescing critical values but different critical
points.

For more information about the topology and geometry of (the complement of)
these strata, the reader can consult the paper [Nek93], and the monograph
[Vas92]. There is a naturally defined covering map $\rho\colon\widetilde{M}\to
M$ of degree $n!$, whose fiber over a point $f(x,a)$ consists of total
orderings of its critical points. On $\widetilde{M}$, $x_{1},\dots,x_{n}$ are
well defined functions such that

$f^{\prime}(x,\rho(w))=(n+1)\prod_{i=1}^{n}(x-x_{i}(w)),\quad
w\in\widetilde{M}.$

The caustic $\mathcal{K}$ is the ramification locus of the covering $\rho$.
For any simply connected open subset $U\subseteq M\setminus\mathcal{K}$, we
can choose a connected component $W$ of $\rho^{-1}(U)$. The restriction of the
functions $x_{1}$, …, $x_{n}$ on $W$ defines single-valued functions of $a\in
U$, which are local branches of $x_{1}$, …, $x_{n}$. For further details see
[Man99].

We define on $M$ the following structures:

  1. (1)

a free sheaf of rank $n$ of $\mathcal{O}_{M}$-algebras: this is the sheaf of
Jacobi-Milnor algebras

$\frac{\mathcal{O}_{M}[x]}{f^{\prime}(x,a)\cdot\mathcal{O}_{M}[x]}.$

For fixed $a\in M$, the fiber of this sheaf is the algebra
$\mathbb{C}[x]/\langle f^{\prime}(x,a)\rangle$. We also define an
$\mathcal{O}_{M}$-linear _Kodaira-Spencer_ isomorphism
$\kappa\colon\mathscr{T}_{M}\to\mathcal{O}_{M}[x]/\langle
f^{\prime}(x,a)\rangle$ which associates to a vector field $\xi$ the class
$\mathfrak{L}_{\xi}(f)=\xi(f)\mod f^{\prime}$. In particular, for any
$\alpha=0,\dots,n-1$ the class $\partial_{a_{i}}f$ is associated with the
vector field $\partial_{a_{i}}$. In this way we introduce a product $\circ$ of
vector fields defined by

$\xi\circ\zeta:=\kappa^{-1}\left(\xi(f)\cdot\zeta(f)\mod f^{\prime}\right).$

The product $\circ$ is associative, commutative and with unit
$\partial_{a_{0}}$. We call Euler vector field the distinguished vector field
$E$ corresponding to the class $f\mod f^{\prime}$ under the Kodaira-Spencer
map $\kappa$. An elementary computation shows that

$E=\sum_{i=0}^{n-1}\frac{n+1-i}{n+1}a_{i}\frac{\partial}{\partial
a_{i}},\quad\quad\mathfrak{L}_{E}(\circ)=\circ.$

  2. (2)

A _symmetric bilinear form_ $\eta$, defined at a fixed point $a\in M$ as the
Grothendieck residue

$\eta_{a}(\xi,\zeta):=\frac{1}{2\pi
i}\int_{\Gamma_{a}}\frac{\xi(f)(u,a)\cdot\zeta(f)(u,a)}{f^{\prime}(u,a)}du,$
(5.1)

where $\Gamma_{a}$ is a circle, positively oriented, bounding a disc
containing all the roots of $f^{\prime}(u,a)$. It is a nontrivial fact that
the bilinear form $\eta$ is non-degenerate (for a proof, see [AGZV88]) and
_flat_ (explicit flat coordinates can be found in [SYS80]: notice that the
natural coordinates $a_{i}$’s are not flat). Notice that

$\mathfrak{L}_{E}\eta=\frac{n+3}{n+1}\eta.$

######  Theorem 5.1.

The manifold $M$, endowed with the tensors $(\eta,\circ,\partial_{a_{0}},E)$,
is a Frobenius manifold of charge $\frac{n-1}{n+1}$. The caustic
$\mathcal{K}_{M}$, defined as in Definition 2.11, coincides with the caustic
$\mathcal{K}$ of the singularity $A_{n}$ defined above. By analytic
continuation, the semisimple Frobenius structures extends on the unramified
covering space $\rho^{-1}(M\backslash\mathcal{K})\subseteq\widetilde{M}$.
Critical values define a system of canonical coordinates.

The reader can find detailed proofs in [Dub96], [Dub99b], [Man99], [Sab08]. If
$a$ is a given point of $M\setminus\mathcal{K}$, i.e. such that $f(x,a)$ has
$n$ distinct Morse critical points ${x}_{1},\dots,{x}_{n}$, then the elements

$\pi_{i}(a):=\kappa^{-1}\left(\frac{f^{\prime}(x,a)}{f^{\prime\prime}({x}_{i},a)(x-x_{i})}\right)\quad\text{for
}i=1,\dots,n$

are idempotents of $(T_{a}M,\circ_{a})$. This follows from the equality
$f^{\prime}(x,a)=(n+1)\prod_{i=1}^{n}(x-x_{i})$. Consider now the critical
points $x_{1}(a),\dots,x_{n}(a)$ locally well defined as functions of $a$
varying in a simply connected open set away from the caustic. The critical
values $u_{i}(a):=f(x_{i}(a),a)$ for $i=1,\dots,n$ can also be considered as
functions on the same set. Since $\det\left(\frac{\partial u_{i}}{\partial
a_{j}}\right)$ is the Vandermonde determinant of $x_{i}(a)$’s, the functions
$u_{i}$’s define a system of local coordinates on $M$. In order to see that
$\pi_{i}\equiv\frac{\partial}{\partial u_{i}}$, it is sufficient to prove that
$\kappa(\partial{u_{i}})(x_{j})=\delta_{ij}$, i.e. $\frac{\partial f}{\partial
u_{i}}(x_{i})=\delta_{ij}$. This follows from the equalities

$\frac{\partial f(x_{i}(a),a)}{\partial
a_{j}}=(x_{i}(a))^{j},\quad\frac{\partial}{\partial
a_{j}}=\sum_{i}(x_{i}(a))^{j}\frac{\partial}{\partial u_{i}}.$

###  5.3. The case of $A_{3}$: reduction of the system for deformed flat
coordinates

We consider the space $M$ of polynomials

$f(x;a)=x^{4}+a_{2}x^{2}+a_{1}x+a_{0},$

where $a_{0},a_{1},a_{2}\in\mathbb{C}$ are “natural” coordinates on $M$. The
Residue Theorem implies that the metric $\eta$, defined on $M$ as in (5.1),
can be expressed as

$\eta_{a}(\xi,\zeta)=-\operatorname{res}_{u=\infty}\frac{\xi(f)(u,a)\cdot\zeta(f)(u,a)}{f^{\prime}(u,a)}du,$

and consequently

$\eta_{a}(\partial_{i},\partial_{j})=\operatorname{res}_{v=0}\frac{v^{1-i-j}}{4+2a_{2}v^{2}+a_{1}v^{3}}dv,$

where $\partial_{i}=\frac{\partial}{\partial
a_{i}},\partial_{j}=\frac{\partial}{\partial a_{j}}$. So we find that

$\eta_{a}=\begin{pmatrix}0&0&\frac{1}{4}\\\ 0&\frac{1}{4}&0\\\
\frac{1}{4}&0&-\frac{a_{2}}{8}\end{pmatrix}.$

Note that $a_{0},a_{1},a_{2}$ are not flat coordinates for $\eta$. The
commutative and associative product defined on each tangent space $T_{a}M$,
using the Kodaira-Spencer map, is given by the structure constants at a
generic point $a\in M$:

$\partial_{0}\circ\partial_{i}=\partial_{i}\quad\text{for all }i,$
$\partial_{1}\circ\partial_{1}=\partial_{2},\quad\quad\partial_{1}\circ\partial_{2}=-\frac{1}{2}a_{2}\partial_{1}-\frac{1}{4}a_{1}\partial_{0},\quad\quad\partial_{2}\circ\partial_{2}=-\frac{1}{2}a_{2}\partial_{2}-\frac{1}{4}a_{1}\partial_{1}.$

The Euler vector field is

$E:=\sum_{i=0}^{2}\frac{4-i}{4}a_{i}\partial_{i}=a_{0}\partial_{0}+\frac{3}{4}a_{1}\partial_{1}+\frac{1}{2}a_{2}\partial_{2}.$

With such a structure $M$ is a Frobenius manifold. The (1,1)-tensor
$\mathcal{U}$ of multiplication by $E$ is:

$\mathcal{U}(a)=\left(\begin{array}[]{ccc}a_{0}&-\frac{1}{8}a_{1}a_{2}&-\frac{3}{16}a_{1}^{2}\\\
\frac{3a_{1}}{4}&a_{0}-\frac{a_{2}^{2}}{4}&-\frac{1}{2}a_{1}a_{2}\\\
\frac{a_{2}}{2}&\frac{3}{4}a_{1}&a_{0}-\frac{a_{2}^{2}}{4}\\\
\end{array}\right)$

Up to a multiplicative constant, the discriminant of the characteristic
polynomial of $\mathcal{U}$ is equal to

$a_{1}^{2}\left(8a_{2}^{3}+27a_{1}^{2}\right)^{3}$

and so the bifurcation set of the Frobenius manifold is the locus

$\mathcal{B}=\left\\{a_{1}=0\right\\}\cup\left\\{8a_{2}^{3}+27a_{1}^{2}=0\right\\}.$

The irreducible component $\left\\{a_{1}=0\right\\}$ is the Maxwell stratum,
whereas the irreducible component $\left\\{8a_{2}^{3}+27a_{1}^{2}=0\right\\}$
is the caustic.

Let us focus on the set $\left\\{a_{1}=0\right\\}$, and let us look for
semisimple points on it. It is enough to consider the multiplication by the
vector field $\lambda\partial_{1}+\mu\partial_{2}$
($\lambda,\mu\in\mathbb{C}$), and show that it has distinct eigenvalues. This
is a (1,1)-tensor with components at points $(a_{0},a_{1},a_{2})$ equal to

$\left(\begin{array}[]{ccc}0&-\frac{\mu}{4}a_{1}&-\frac{\lambda}{2}a_{1}\\\
\lambda&-\frac{\mu}{2}a_{2}&-\frac{\lambda}{2}a_{2}-\frac{\mu}{4}a_{1}\\\
\mu&\lambda&-\frac{\mu}{2}a_{2}\\\ \end{array}\right)$

whose characteristic polynomial, at points $(a_{0},0,a_{2})$, has discriminant

$-\frac{1}{8}\lambda^{2}a_{2}^{3}\left(2\lambda^{2}+\mu^{2}a_{2}\right)^{2}.$

So, the points $(a_{0},0,a_{2})$ with $a_{2}\neq 0$ are semisimple points of
the bifurcation set, namely they belong to the Maxwell stratum. In view of
Theorem 5.1, they are semisimple coalescence points of Definition 1.1. We
would like to study deeper the behavior of the Frobenius structure near points
$(a_{0},a_{1},a_{2})=(0,0,h)$ of the Maxwell stratum, with fixed $a_{0}=0$ and
with $h\in\mathbb{C}^{*}$.

######  Remark 5.2.

The points $(a_{0},0,0)$, instead, are not semisimple because we have
evidently $\partial_{2}^{2}=0$ on them.

Let us introduce flat coordinates $t_{1},t_{2},t_{3}$ defined by

$\left\\{\begin{array}[]{@{}l@{}}a_{0}=t_{1}+\frac{1}{8}t^{2}_{3},\\\ \\\
a_{1}=t_{2},\\\ \\\ a_{2}=t_{3}\\\ \end{array}\right.\quad
J=\left(\frac{\partial a_{i}}{\partial
t_{j}}\right)_{i,j}=\begin{pmatrix}1&0&\frac{1}{4}t_{3}\\\ 0&1&0\\\
0&0&1\end{pmatrix}.$

In flat coordinates we have:

$\eta=\left(\begin{array}[]{ccc}0&0&\frac{1}{4}\\\ 0&\frac{1}{4}&0\\\
\frac{1}{4}&0&0\\\
\end{array}\right),\quad\mathcal{U}(t_{1},t_{2},t_{3})=\left(\begin{array}[]{ccc}t_{1}&\frac{-5}{16}t_{2}t_{3}&-\frac{3}{16}t_{2}^{2}+\frac{1}{32}t_{3}^{3}\\\
\frac{3t_{2}}{4}&t_{1}-\frac{t_{3}^{2}}{8}&\frac{-5}{16}t_{2}t_{3}\\\
\frac{t_{3}}{2}&\frac{3t_{2}}{4}&t_{1}\\\
\end{array}\right),\quad\mu=\begin{pmatrix}-\frac{1}{4}&&\\\ &0&\\\
&&\frac{1}{4}\end{pmatrix}.$

Thus, the second system in (2.1)

$\partial_{z}\xi=\left(\mathcal{U}^{T}-\frac{1}{z}\mu\right)\xi$

reads

$\left\\{\begin{array}[]{@{}l@{}}\partial_{z}\xi_{1}=\frac{3}{4}\xi_{2}t_{2}+\frac{1}{2}\xi_{3}t_{3}+\xi_{1}\left(t_{1}+\frac{1}{4z}\right),\\\
\\\
\partial_{z}\xi_{2}=-\frac{5}{16}\xi_{1}t_{2}t_{3}+\xi_{2}\left(t_{1}-\frac{t_{3}^{2}}{8}\right)+\frac{3}{4}\xi_{3}t_{2},\\\
\\\
\partial_{z}\xi_{3}=\xi_{1}\left(-\frac{3}{16}t_{2}^{2}+\frac{1}{32}t_{3}^{3}\right)-\frac{5}{16}\xi_{2}t_{2}t_{3}+\xi_{3}\left(t_{1}-\frac{1}{4z}\right).\end{array}\right.$
(5.2)

We know that, if $(t_{1},t_{2},t_{3})$ is a semisimple point of the Frobenius
manifold then the monodromy data are well defined, and that these are
invariant under (small) deformations of $t_{1},t_{2},t_{3}$ by Theorem 2.12
and Theorem 4.1. The bifurcation set is now

$\left\\{t_{2}=0\right\\}\cup\left\\{8t_{3}^{3}+27t_{2}^{2}=0\right\\}.$

Now, if we fix $a_{0}=0$, the tensor $\mathcal{U}$ at $(0,a_{1},h)$, i.e.
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},t_{2},h\right)$, is

$\mathcal{U}\left(-\frac{1}{8}h^{2},t_{2},h\right)=\left(\begin{array}[]{ccc}-\frac{h^{2}}{8}&-\frac{5h}{16}t_{2}&\frac{1}{32}\left(h^{3}-6t_{2}^{2}\right)\\\
\frac{3t_{2}}{4}&-\frac{h^{2}}{4}&-\frac{5h}{16}t_{2}\\\
\frac{h}{2}&\frac{3t_{2}}{4}&-\frac{h^{2}}{8}\\\ \end{array}\right).$ (5.3)

The bifurcation locus is reached for $a_{1}=t_{2}=0$. At these points, i.e.

$(t_{1},t_{2},t_{3})=\Bigl{(}-\frac{1}{8}h^{2},0,h\Bigr{)},$

we have

$\mathcal{U}\Bigl{(}-\frac{1}{8}h^{2},0,h\Bigr{)}=\left(\begin{array}[]{ccc}-\frac{h^{2}}{8}&0&\frac{h^{3}}{32}\\\
0&-\frac{h^{2}}{4}&0\\\ \frac{h}{2}&0&-\frac{h^{2}}{8}\\\ \end{array}\right).$

Define the function

$X(a):=\left[-9a_{1}+\sqrt{3}(27a_{1}^{2}+8a_{2}^{3})^{\frac{1}{2}}\right]^{\frac{1}{3}},$

which has branch points along the caustic
$\mathcal{K}=\left\\{a_{2}=0\right\\}\cup\\{27a_{1}^{2}+8a_{2}^{3}=0\\}$. Fix
a branch of $X$ on a simply connected domain in $M\setminus\mathcal{K}$, that
we also denote by $X(a)$. The critical points $x_{1},x_{2},x_{3}$ of $f(x,a)$
are equal to

$x_{i}(a):=\frac{\vartheta_{i}\cdot a_{2}}{2\sqrt{3}\cdot
X(a)}-\frac{\vartheta_{i}\cdot X(a)}{2\cdot 3^{2/3}},$

where

$\vartheta_{1}:=-1,\quad\vartheta_{2}:=\frac{1-i\sqrt{3}}{2},\quad\vartheta_{3}:=\frac{1+i\sqrt{3}}{2}$

are the cubic roots of $(-1)$. Of course, different choices of branches of $X$
correspond to permutations of the $x_{i}$’s. After some computations, we find
the following expression for $\Psi$:

$\Psi(t)=\left.\left(\begin{array}[]{ccc}\frac{\sqrt{6x_{1}^{2}+a_{2}}}{2\sqrt{2}\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)}&-\frac{\left(x_{2}+x_{3}\right)\sqrt{6x_{1}^{2}+a_{2}}}{2\sqrt{2}\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)}&-\frac{\sqrt{6x_{1}^{2}+a_{2}}\left(a_{2}-4x_{2}x_{3}\right)}{8\sqrt{2}\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)}\\\
\\\
\frac{\sqrt{6x_{2}^{2}+a_{2}}}{2\sqrt{2}\left(x_{1}-x_{2}\right)\left(x_{3}-x_{2}\right)}&\frac{\left(x_{1}+x_{3}\right)\sqrt{6x_{2}^{2}+a_{2}}}{2\sqrt{2}\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)}&\frac{\sqrt{6x_{2}^{2}+a_{2}}\left(a_{2}-4x_{1}x_{3}\right)}{8\sqrt{2}\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)}\\\
\\\
\frac{\sqrt{6x_{3}^{2}+a_{2}}}{2\sqrt{2}\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)}&\frac{\left(x_{1}+x_{2}\right)\sqrt{6x_{3}^{2}+a_{2}}}{2\sqrt{2}\left(x_{1}-x_{3}\right)\left(x_{3}-x_{2}\right)}&\frac{\left(a_{2}-4x_{1}x_{2}\right)\sqrt{6x_{3}^{2}+a_{2}}}{8\sqrt{2}\left(x_{1}-x_{3}\right)\left(x_{3}-x_{2}\right)}\\\
\end{array}\right)\right|_{a=a(t)},$

where

$a_{0}=t_{1}+\frac{1}{8}t_{3}^{2},\quad a_{1}=t_{2},\quad a_{2}=t_{3}.$

The canonical coordinates are $u_{i}(t)=f(x_{i}(a(t)),a(t))$. Near the point
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$, i.e. for small
$t_{2}$, we find:

$\displaystyle\begin{split}u_{1}(t_{2};h)=&-\frac{t_{2}^{2}}{4h}+\frac{t_{2}^{4}}{16h^{4}}-\frac{t_{2}^{6}}{16h^{7}}+\frac{3t_{2}^{8}}{32h^{10}}+O\left(|t_{2}|^{10}\right),\\\
u_{2}(t_{2};h)=&-\frac{h^{2}}{4}+\frac{i\sqrt{h}t_{2}}{\sqrt{2}}+\frac{t_{2}^{2}}{8h}+\frac{it_{2}^{3}}{16\sqrt{2}h^{5/2}}-\frac{t_{2}^{4}}{32h^{4}}-\frac{21it_{2}^{5}}{512\sqrt{2}h^{11/2}}\\\
&+\frac{t_{2}^{6}}{32h^{7}}+\frac{429it_{2}^{7}}{8192\sqrt{2}h^{17/2}}-\frac{3t_{2}^{8}}{64h^{10}}-\frac{46189it_{2}^{9}}{524288\sqrt{2}h^{23/2}}+O\left(|t_{2}|^{10}\right),\\\
u_{3}(t_{2};h)=&-\frac{h^{2}}{4}-\frac{i\sqrt{h}t_{2}}{\sqrt{2}}+\frac{t_{2}^{2}}{8h}-\frac{it_{2}^{3}}{16\sqrt{2}h^{5/2}}-\frac{t_{2}^{4}}{32h^{4}}+\frac{21it_{2}^{5}}{512\sqrt{2}h^{11/2}}\\\
&+\frac{t_{2}^{6}}{32h^{7}}-\frac{429it_{2}^{7}}{8192\sqrt{2}h^{17/2}}-\frac{3t_{2}^{8}}{64h^{10}}+\frac{46189it_{2}^{9}}{524288\sqrt{2}h^{23/2}}+O\left(|t_{2}|^{10}\right),\end{split}$
(5.4) $\displaystyle\Psi(t_{2})=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}\sqrt{h}}&0&\frac{\sqrt{h}}{4\sqrt{2}}\\\
\frac{i}{2\sqrt{h}}&-\frac{1}{2\sqrt{2}}&-\frac{1}{8}\left(i\sqrt{h}\right)\\\
\frac{i}{2\sqrt{h}}&\frac{1}{2\sqrt{2}}&-\frac{1}{8}\left(i\sqrt{h}\right)\\\
\end{array}\right)+t_{2}\left(\begin{array}[]{ccc}0&-\frac{1}{2\sqrt{2}h^{3/2}}&0\\\
-\frac{3}{8\sqrt{2}h^{2}}&-\frac{i}{16h^{3/2}}&-\frac{5}{32\sqrt{2}h}\\\
\frac{3}{8\sqrt{2}h^{2}}&-\frac{i}{16h^{3/2}}&\frac{5}{32\sqrt{2}h}\\\
\end{array}\right)$
$\displaystyle+t_{2}^{2}\left(\begin{array}[]{ccc}-\frac{3}{4\sqrt{2}h^{7/2}}&0&\frac{1}{16\sqrt{2}h^{5/2}}\\\
-\frac{39i}{128h^{7/2}}&\frac{15}{128\sqrt{2}h^{3}}&-\frac{41i}{512h^{5/2}}\\\
-\frac{39i}{128h^{7/2}}&-\frac{15}{128\sqrt{2}h^{3}}&-\frac{41i}{512h^{5/2}}\\\
\end{array}\right)+t_{2}^{3}\left(\begin{array}[]{ccc}0&\frac{5}{8\sqrt{2}h^{9/2}}&0\\\
\frac{303}{512\sqrt{2}h^{5}}&\frac{125i}{1024h^{9/2}}&\frac{265}{2048\sqrt{2}h^{4}}\\\
-\frac{303}{512\sqrt{2}h^{5}}&\frac{125i}{1024h^{9/2}}&-\frac{265}{2048\sqrt{2}h^{4}}\\\
\end{array}\right)+O(t_{2}^{4}).$

Hence, at points $(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$,
canonical coordinates $u_{i}(0;h)$ are

$(u_{1},u_{2},u_{3})=\left(0,-\frac{h^{2}}{2},-\frac{h^{2}}{2}\right)$

and the system (5.2) reduces to

$\left\\{\begin{array}[]{@{}l@{}}\partial_{z}\xi_{1}=\left(-\frac{h^{2}}{8}+\frac{1}{4z}\right)\xi_{1}+\frac{h}{2}\xi_{3},\\\
\\\ \partial_{z}\xi_{2}=-\frac{h^{2}}{4}\xi_{2},\\\ \\\
\partial_{z}\xi_{3}=\frac{h^{3}}{32}\xi_{1}-\left(\frac{h^{2}}{8}+\frac{1}{4z}\right)\xi_{3}.\end{array}\right.$
(5.5)

The second equation yields

$\xi_{2}(z)=c\cdot e^{-\frac{h^{2}}{4}z},\quad c\in\mathbb{C}.$

From the first equation we find that

$\xi_{3}=\frac{2}{h}\left(\partial_{z}\xi_{1}+\frac{h^{2}}{8}\xi_{1}-\frac{1}{4z}\xi_{1}\right),$
(5.6)

and so from the third equation we obtain

$\frac{2}{h}\xi_{1}^{\prime\prime}(z)+\frac{h}{2}\xi_{1}^{\prime}(z)+\frac{3}{8z^{2}h}\xi_{1}=0.$

Making the ansatz

$\xi_{1}=z^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}\Lambda(z),$

the equation for $\Lambda$ becomes the following Bessel equation:

$64z^{2}\Lambda^{\prime\prime}(z)+64z\Lambda^{\prime}(z)-(4+z^{2}h^{4})\Lambda(z)=0.$
(5.7)

Therefore, $\xi_{1}$ is of the form

$\xi_{1}=z^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}\left(c_{1}H^{(1)}_{\frac{1}{4}}\left(\frac{ih^{2}}{8}z\right)+c_{2}H^{(2)}_{\frac{1}{4}}\left(\frac{ih^{2}}{8}z\right)\right),\quad
c_{1},c_{2}\in\mathbb{C}$

where $H^{(1)}_{\nu}(z),H^{(2)}_{\nu}(z)$ stand for the Hankel functions of
the first and second kind of parameter $\nu=1/4$. Notice that if $\Lambda(z)$
is a solution of equation (5.7), then also $\Lambda(e^{\pm i\pi}z)$ is a
solution.

###  5.4. Computation of Stokes and Central Connection matrices

In order to compute the Stokes matrix, let us fix the line $\ell$ coinciding
with the real axis. Such a line is admissible for all points
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$ with

$|\operatorname{Re}h|\neq|\operatorname{Im}h|,\quad h\in\mathbb{C}^{*}.$

Indeed, the Stokes rays for
$(u_{1},u_{2},u_{3})=(0,-\frac{1}{4}h^{2},-\frac{1}{4}h^{2})$ are

$z=\pm i\rho\overline{h}^{2}\quad\quad\Longrightarrow\quad\quad\arg
z=\frac{\pi}{2}-2\arg h\text{ (mod $\pi$)}.$

Thus, admissibility corresponds to $\frac{1}{2}\pi-2\arg h\neq k\pi$,
$k\in\mathbb{Z}$. Let us compute the Stokes matrix in the case

$-\frac{\pi}{4}<\arg h<\frac{\pi}{4}.$

The asymptotic expansion for fundamental solutions $\Xi_{\rm left},\Xi_{\rm
right}$ of the system (5.5), is

$\displaystyle\eta\Psi^{-1}\left(\mathbbm{1}+O\left(\frac{1}{z}\right)\right)e^{zU}$
$\displaystyle=\Psi^{T}\left(\mathbbm{1}+O\left(\frac{1}{z}\right)\right)e^{zU}$
$\displaystyle=\left(\mathbbm{1}+O\left(\frac{1}{z}\right)\right)\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}\sqrt{h}}&\frac{i}{2\sqrt{h}}e^{-\frac{1}{4}h^{2}z}&\frac{i}{2\sqrt{h}}e^{-\frac{1}{4}h^{2}z}\\\
0&-\frac{1}{2\sqrt{2}}e^{-\frac{1}{4}h^{2}z}&\frac{1}{2\sqrt{2}}e^{-\frac{1}{4}h^{2}z}\\\
\frac{\sqrt{h}}{4\sqrt{2}}&-\frac{i}{8}e^{-\frac{1}{4}h^{2}z}\sqrt{h}&-\frac{i}{8}e^{-\frac{1}{4}h^{2}z}\sqrt{h}\\\
\end{array}\right),$

being

$U:=\Psi\mathcal{U}\Psi^{-1}=\operatorname{diag}(u_{1},u_{2},u_{3})=\operatorname{diag}\left(0,-\frac{h^{2}}{4},-\frac{h^{2}}{4}\right).$

For the admissible line $\ell$ and for the above labelling of canonical
coordinates the Stokes matrix must be of the form prescribed by Theorem 2.10:

$S=\begin{pmatrix}1&0&0\\\ \alpha&1&0\\\ \beta&0&1\end{pmatrix}$ (5.8)

for some constants $\alpha,\beta\in\mathbb{C}$ to be determined. This means
that the last two columns of $\Xi_{\rm left}$ must be the analytic
continuation of $\Xi_{\rm right}$.

######  Lemma 5.1.

The following asymptotic expansions hold:

  * •

if $m\in\mathbb{Z}$, then

$H_{\frac{1}{4}}^{(1)}\left(e^{im\pi}\frac{ih^{2}}{8}z\right)\sim\sqrt{\frac{2}{\pi}}\left(e^{im\pi}\frac{ih^{2}}{8}z\right)^{-\frac{1}{2}}e^{-\frac{3i\pi}{8}}\exp\left(-e^{im\pi}\frac{h^{2}}{8}z\right)$

in the sector

$-\frac{3}{2}\pi-m\pi-\arg(h^{2})<\arg z<\frac{3}{2}\pi-m\pi-\arg(h^{2});$

  * •

if $m\in\mathbb{Z}$, then

$H_{\frac{1}{4}}^{(2)}\left(e^{im\pi}\frac{ih^{2}}{8}z\right)\sim\sqrt{\frac{2}{\pi}}\left(e^{im\pi}\frac{ih^{2}}{8}z\right)^{-\frac{1}{2}}e^{\frac{3i\pi}{8}}\exp\left(e^{im\pi}\frac{h^{2}}{8}z\right)$

in the sector

$-\frac{5}{2}\pi-m\pi-\arg(h^{2})<\arg z<\frac{\pi}{2}-m\pi-\arg(h^{2}).$

###### Proof.

These formulae easily follow from the following well-known asymptotic
expansion of Hankel functions (see [Wat44]):

$H^{(1)}_{\nu}(z)\sim\sqrt{\frac{2}{\pi
z}}\exp\left(i\left(z-\frac{\nu}{2}\pi-\frac{\pi}{4}\right)\right),\quad\quad-\pi+\delta\leq\arg
z\leq 2\pi-\delta,$

$\delta$ being any positive acute angle. Analogously,

$H^{(2)}_{\nu}(z)\sim\sqrt{\frac{2}{\pi
z}}\exp\left(-i\left(z-\frac{\nu}{2}\pi-\frac{\pi}{4}\right)\right),\quad\quad-2\pi+\delta\leq\arg
z\leq\pi-\delta.$

∎

Using Lemma 5.1, we obtain

$\Xi_{\rm
left}(z)=\begin{pmatrix}\xi^{L}_{(1),1}&\xi^{L}_{(2),1}&\xi^{L}_{(3),1}\\\
&&\\\
0&-\frac{e^{-\frac{1}{4}h^{2}z}}{2\sqrt{2}}&\frac{e^{-\frac{1}{4}h^{2}z}}{2\sqrt{2}}\\\
&&\\\ *&*&*\end{pmatrix},\quad\Xi_{\rm
right}(z)=\begin{pmatrix}\xi^{R}_{(1),1}&\xi^{R}_{(2),1}&\xi^{R}_{(3),1}\\\
&&\\\
0&-\frac{e^{-\frac{1}{4}h^{2}z}}{2\sqrt{2}}&\frac{e^{-\frac{1}{4}h^{2}z}}{2\sqrt{2}}\\\
&&\\\ *&*&*\end{pmatrix}$ (5.9)

where

$\xi^{L}_{(2),1}(z)=\xi^{L}_{(3),1}(z)=\xi^{R}_{(2),1}(z)=\xi^{R}_{(3),1}(z)=\frac{i\sqrt{\pi}}{8}h^{\frac{1}{2}}e^{i\frac{5}{8}\pi}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}H^{(1)}_{\frac{1}{4}}\left(\frac{ih^{2}}{8}z\right),$

with the required asymptotic expansion in the following sector containing both
$\Pi_{\rm left}$ and $\Pi_{\rm right}$

$\left\\{z\in\mathcal{R}\colon-\frac{3}{2}\pi-\arg(h^{2})<\arg
z<\frac{3}{2}\pi-\arg(h^{2})\right\\},$

and

$\displaystyle\xi^{L}_{(1),1}(z)$
$\displaystyle=\frac{\sqrt{\pi}}{4\sqrt{2}}h^{\frac{1}{2}}e^{i\frac{\pi}{8}}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}H^{(1)}_{\frac{1}{4}}\left(e^{-i\pi}\frac{ih^{2}}{8}z\right),$
$\displaystyle\xi^{R}_{(1),1}(z)$
$\displaystyle=\frac{\sqrt{\pi}}{4\sqrt{2}}h^{\frac{1}{2}}e^{-i\frac{\pi}{8}}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}H^{(2)}_{\frac{1}{4}}\left(\frac{ih^{2}}{8}z\right),$

with the required expansion respectively in the sectors

$\left\\{z\in\mathcal{R}\colon-\frac{\pi}{2}-\arg(h^{2})<\arg
z<\frac{5}{2}\pi-\arg(h^{2})\right\\}\supseteq\Pi_{\rm left},$
$\left\\{z\in\mathcal{R}\colon-\frac{5}{2}\pi-\arg(h^{2})<\arg
z<\frac{\pi}{2}-\arg(h^{2})\right\\}\supseteq\Pi_{\rm right}.$

The entries of $\Xi_{\rm left},\Xi_{\rm right}$ denoted by $*$ are
reconstructed from the first rows, by applying equation (5.6).

From the second rows of $\Xi_{\rm left},\Xi_{\rm right}$ we can immediately
say that the entries $\alpha,\beta$ of (5.8) must be equal. Specializing the
following well-known connection formula for Hankel special functions

$\sin(\nu\pi)H^{(1)}_{\nu}(ze^{m\pi
i})=-\sin((m-1)\nu\pi)H^{(1)}_{\nu}(z)-e^{-\nu\pi
i}\sin(m\nu\pi)H^{(2)}_{\nu}(z),\quad m\in\mathbb{Z},$ (5.10)

to the case $m=-1,\nu=\frac{1}{4}$, we easily obtain

$\xi^{L}_{(1),1}(z)=\xi^{R}_{(1),1}(z)-\xi^{R}_{(2),1}(z)-\xi^{R}_{(3),1}(z)$

which means that $\alpha=\beta=-1$. So, we have obtained that, at points
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$ with

$|\operatorname{Re}h|>|\operatorname{Im}h|,\quad-\frac{\pi}{4}<\arg
h<\frac{\pi}{4}$

(and consequently in their neighborhood, by Theorem 4.1) the Stokes matrix is

$S=\begin{pmatrix}1&0&0\\\ -1&1&0\\\ -1&0&1\end{pmatrix}.$ (5.11)

In order to compute the central connection matrix, we observe that the $A_{3}$
Frobenius manifold structure is _non-resonant_ , i.e. the components of the
tensor $\mu$ are such that $\mu_{\alpha}-\mu_{\beta}\notin\mathbb{Z}$ for
$\alpha\neq\beta$. This implies that the $(\eta,\mu)$-parabolic orthogonal
group is trivial, and that the fundamental system of (5.5) near the origin
$z=0$ can be uniquely chosen in such a way that

$\Xi_{0}(z)=(\eta+O(z))z^{\mu}.$ (5.12)

Now, let us recall the following Mellin–Barnes integral representations of
Hankel functions (see [Wat44])

$H^{(1)}_{\nu}(z)=-\frac{\cos(\nu\pi)}{\pi^{\frac{5}{2}}}e^{i(z-\pi\nu)}(2z)^{\nu}\int_{-\infty
i}^{\infty
i}\Gamma(s)\Gamma(s-2\nu)\Gamma\left(\nu+\frac{1}{2}-s\right)(-2iz)^{-s}ds,$
$H^{(2)}_{\nu}(z)=\frac{\cos(\nu\pi)}{\pi^{\frac{5}{2}}}e^{-i(z-\pi\nu)}(2z)^{\nu}\int_{-\infty
i}^{\infty
i}\Gamma(s)\Gamma(s-2\nu)\Gamma\left(\nu+\frac{1}{2}-s\right)(2iz)^{-s}ds,$

which are valid for

  * •

$2\nu\notin 2\mathbb{Z}+1$,

  * •

respectively in the sectors $|\arg(\mp iz)|<\frac{3}{2}$,

  * •

and where the integration path separates the poles of
$\Gamma(s)\Gamma(s-2\nu)$ from those of
$\Gamma\left(\nu+\frac{1}{2}-s\right)$.

Specializing these integral forms to $\nu=\frac{1}{4}$, and deforming the
integration path so that it reduces to positively oriented circles around the
poles

$s\in\frac{1}{2}-\frac{1}{2}\mathbb{N},$

we immediately obtain the following expansion of the solution
$\xi^{(1)}_{1,R},\xi^{(2)}_{1,R},\xi^{(3)}_{1,R}$ for the points
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right),\text{ with
}-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$, valid for small values of $|z|$:

######  Lemma 5.2.

At the points $(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right),\text{
with }-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$ the following expansion holds:

$\displaystyle\xi^{R}_{(1),1}(z)=$
$\displaystyle\frac{(1+i)\Gamma\left(\frac{5}{4}\right)}{\sqrt{\pi}}z^{\frac{1}{4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right)h\Gamma\left(\frac{3}{4}\right)}{\sqrt{\pi}}z^{3/4}$
$\displaystyle-\frac{\left(\frac{1}{32}+\frac{i}{32}\right)h^{2}\Gamma\left(\frac{1}{4}\right)}{\sqrt{\pi}}z^{5/4}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right)h^{3}\Gamma\left(\frac{3}{4}\right)}{\sqrt{\pi}}z^{7/4}+O\left(|z|^{9/4}\right),$

$\displaystyle\xi^{R}_{(2),1}(z)=\xi^{R}_{(3),1}(z)=$
$\displaystyle\frac{i\Gamma\left(\frac{5}{4}\right)}{\sqrt{\pi}}z^{\frac{1}{4}}-\frac{4ih\Gamma\left(\frac{11}{4}\right)}{21\sqrt{\pi}}z^{3/4}-\frac{ih^{2}\Gamma\left(\frac{5}{4}\right)}{8\sqrt{\pi}}z^{5/4}+\frac{ih^{3}\Gamma\left(\frac{11}{4}\right)}{42\sqrt{\pi}}z^{7/4}+O\left(|z|^{9/4}\right).$

Moreover, using equation (5.6) we find that

$\xi^{R}_{(1),3}(z)=\frac{\left(\frac{1}{4}-\frac{i}{4}\right)\Gamma\left(\frac{3}{4}\right)}{\sqrt{\pi}}z^{-\frac{1}{4}}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right)h^{2}\Gamma\left(\frac{3}{4}\right)}{\sqrt{\pi}}z^{3/4}+O\left(|z|^{5/4}\right),$
$\xi^{R}_{(2),3}(z)=\xi^{R}_{(3),3}(z)=-\frac{4i\Gamma\left(\frac{11}{4}\right)}{21\sqrt{\pi}}z^{-\frac{1}{4}}+\frac{ih^{2}\Gamma\left(\frac{11}{4}\right)}{42\sqrt{\pi}}z^{3/4}+O\left(|z|^{5/4}\right).$

###### Proof.

These expansions are the first terms of the expressions

$\displaystyle\xi^{R}_{(1),1}(z)=$
$\displaystyle\frac{\sqrt{\pi}}{4\sqrt{2}}h^{\frac{1}{2}}e^{-i\frac{\pi}{8}}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}\cdot$
$\displaystyle\cdot\left(\frac{e^{-i\left(-\frac{\pi}{4}+\frac{ih^{2}z}{8}\right)}(ih^{2}z)^{\frac{1}{4}}}{2\pi^{5/2}}\cdot
2\pi
i\sum_{n=0}^{\infty}\underset{s=\frac{1}{2}-\frac{n}{2}}{\operatorname{res}}\left(\Gamma(s)\Gamma(s-2\nu)\Gamma\left(\nu+\frac{1}{2}-s\right)\left(e^{i\pi}\frac{h^{2}z}{4}\right)^{-s}\right)\right),$

and

$\displaystyle\xi^{R}_{(2),1}(z)=$
$\displaystyle\xi^{R}_{(3),1}(z)=\frac{i\sqrt{\pi}}{8}h^{\frac{1}{2}}e^{i\frac{5}{8}\pi}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}\cdot$
$\displaystyle\cdot\left(-\frac{e^{i\left(-\frac{\pi}{4}+\frac{1}{8}ih^{2}z\right)}(ih^{2}z)^{\frac{1}{4}}}{2\pi^{5/2}}\cdot
2\pi
i\sum_{n=0}^{\infty}\underset{s=\frac{1}{2}-\frac{n}{2}}{\operatorname{res}}\left(\Gamma(s)\Gamma(s-2\nu)\Gamma\left(\nu+\frac{1}{2}-s\right)\left(-e^{i\pi}\frac{h^{2}z}{4}\right)^{-s}\right)\right).$

∎

By a direct comparison between these expansion of solution $\Xi_{\rm
right}(z)$ of (5.9) and the dominant term of (5.12), namely

$\left(\begin{array}[]{ccc}0&0&\frac{z^{\frac{1}{4}}}{4}\\\ 0&\frac{1}{4}&0\\\
\frac{z^{-\frac{1}{4}}}{4}&0&0\\\ \end{array}\right),$

we obtain the central connection matrix

$C=\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}(1-i)\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)\\\
\\\ 0&-\sqrt{2\pi}&\sqrt{2\pi}\\\ \\\
(1+i)\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)\\\
\end{array}\right).$

Notice that such a matrix satisfies all the constraints of Theorem 2.11.

We can put the Stokes matrix in triangular form using two different
permutations of the canonical coordinates $(0,-h^{2}/4,-h^{2}/4)$, namely

  * •

re-labelling $(u_{1},u_{2},u_{3})\mapsto(u_{2},u_{3},u_{1})$, corresponding to
the permutation matrix

$P=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\end{pmatrix},$

  * •

or re-labelling $(u_{1},u_{2},u_{3})\mapsto(u_{3},u_{2},u_{1})$, corresponding
to the permutation matrix

$P=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix}.$

In both cases these are the lexicographical orders of two different
$\ell$-cells which divide any sufficiently small neighborhood of the point
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$, with
$|\operatorname{Re}h|>|\operatorname{Im}h|$ and $-\frac{\pi}{4}<\arg
h<\frac{\pi}{4}$, in which Theorem 4.1 applies. Using both permutations, the
Stokes matrix becomes

$S_{\text{lex}}=PSP^{-1}=\begin{pmatrix}1&0&-1\\\ 0&1&-1\\\
0&0&1\end{pmatrix},$ (5.13)

which can be thought as in the lexicographical form in one of the
$\ell$-cells. The central connection matrix, instead, has the following
lexicographical forms in the two $\ell$-cells:

$C_{\text{lex}}=\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}-i\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)&(1-i)\Gamma\left(\frac{3}{4}\right)\\\
\\\ \mp\sqrt{2\pi}&\pm\sqrt{2\pi}&0\\\ \\\
i\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)&(1+i)\Gamma\left(\frac{1}{4}\right)\\\
\end{array}\right),$ (5.14)

where we take the first sign if the lexicographical order is the relabeling
$(u_{1},u_{2},u_{3})\mapsto(u_{2},u_{3},u_{1})$, the second if it is the re-
labeling $(u_{1},u_{2},u_{3})\mapsto(u_{3},u_{2},u_{1})$.

###  5.5. A “tour” in the Maxwell stratum: reconstruction of neighboring
monodromy data

From the data (5.13) and (5.14), by an action of the braid group, we can
compute $S$ and $C$ in the neighborhood of all other points
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$ with
$|\operatorname{Re}h|\neq|\operatorname{Im}h|$. As an example, let us
determine the Stokes matrix for points

$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right),\quad\text{with
}\frac{\pi}{4}<\arg h<\frac{3}{4}\pi.$

Starting from a point in the region $-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$ and
moving counter-clockwise towards the region $\frac{\pi}{4}<\arg
h<\frac{3}{4}\pi$, the two coalescing canonical coordinates
$u_{2}=u_{3}=-\frac{1}{2}h^{2}$ move in the $u_{i}$’s-plane counter-clockwise
w.r.t. $u_{1}=0$. For example, in Figure 3 we move along a curve $h\mapsto
he^{i\frac{\pi}{2}}$, starting in $-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$. At
$\arg h=\frac{\pi}{4}$, the Stokes rays
$R_{12}=\\{z=-i\rho\overline{h}^{2},~{}\rho>0\\}$ and
$R_{21}=\\{z=i\rho\overline{h}^{2},~{}\rho>0\\}$ cross the real line $\ell$,
and a braid must act on the monodromy data.
In order to determine the braid and the transformed monodromy data, we proceed
according to the prescription of Section 4.1, as follows.

Figure 3. The triple $(u_{1},u_{2},u_{2})$ is represented by three points
$u_{1}$, $u_{2}$, $u_{3}$ in $\mathbb{C}$. We move along $h\mapsto
he^{i\frac{\pi}{2}}$, starting from $-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$. The
two dashed regions in the left and right figures correspond respectively to
$-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$ and $\frac{\pi}{4}<\arg
h<\frac{3\pi}{4}$.

(1) We split the coalescing canonical coordinates, for example by considering
the point

$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},\varepsilon
e^{i\varphi},h\right),\quad\text{with }-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$
(5.15)

for chosen $\varphi$ and $\epsilon$, being $\varepsilon$ small (so that
$\varepsilon^{2}\ll\varepsilon$). The corresponding canonical coordinates

$\displaystyle u_{1}$ $\displaystyle=O(\varepsilon^{2}),$ (5.16)
$\displaystyle u_{2}$
$\displaystyle=-\frac{h^{2}}{4}+\varepsilon|h|^{\frac{1}{2}}\exp\left[i\left(\frac{\arg
h}{2}+\varphi+\frac{\pi}{2}\right)\right]+O(\varepsilon^{2}),$ (5.17)
$\displaystyle u_{3}$
$\displaystyle=-\frac{h^{2}}{4}+\varepsilon|h|^{\frac{1}{2}}\exp\left[i\left(\frac{\arg
h}{2}+\varphi-\frac{\pi}{2}\right)\right]+O(\varepsilon^{2}),$ (5.18)

give a point $(u_{1},u_{2},u_{3})$ which lies in one of the two cells
(Definition 4.1) which divide a polydisc centred at
$(u_{1},u_{2},u_{3})=(0,-\frac{1}{2}h^{2},-\frac{1}{2}h^{2})$. The Stokes rays
are

$R_{12}=\\{z=-i\rho\overline{h}^{2}+O(\varepsilon),~{}\rho>0\\},\quad\quad
R_{13}=\\{z=-i\rho\overline{h}^{2}+O(\varepsilon),~{}\rho>0\\},$
$R_{23}=\left\\{z=\rho\exp\left[-i\left(\frac{\arg
h}{2}+\varphi+\pi\right)\right]+O(\varepsilon^{2}),~{}\rho>0\right\\},$ (5.19)

and opposite ones $R_{21}$, $R_{31}$, $R_{32}$. Notice that in order for the
real line $\ell$ to remain admissible, we choose $\varphi\neq
k\pi-\frac{1}{2}\arg h$, $k\in\mathbb{Z}$, $-\frac{\pi}{4}<\arg
h<\frac{\pi}{4}$. The position of $R_{23}$ w.r.t. the real line $\ell$ is
determined by the sign of $\cos\left(\frac{\arg
h}{2}+\varphi+\frac{\pi}{2}\right)$. As long as $\varphi$ varies in such a way
that $\operatorname{sgn}\cos\left(\frac{\arg
h}{2}+\varphi+\frac{\pi}{2}\right)$ does not change, then $R_{23}$ does not
cross $\ell$. See Figure 5. This means that $(u_{1},u_{2},u_{3})$ remains
inside the same cell, i.e. the point corresponding to coordinates (5.15)
remains inside an $\ell$-chamber, where the Isomonodromy Theorem 2.12 applies.

(2) The Stokes matrix must be put in triangular form $S_{\text{lex}}$ (5.13).
In particular,

  * •

if $\cos\left(\frac{\arg h}{2}+\varphi+\frac{\pi}{2}\right)<0$, then $R_{23}$
is on the _left_ of $\ell$, and the lexicographical order is given by the
permutation
$(u_{1},u_{2},u_{3})\mapsto(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=(u_{2},u_{3},u_{1})$;

  * •

if $\cos\left(\frac{\arg h}{2}+\varphi+\frac{\pi}{2}\right)>0$, then $R_{23}$
is on the _right_ of $\ell$, and the lexicographical order is given by the
permutation
$(u_{1},u_{2},u_{3})\mapsto(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=(u_{3},u_{2},u_{1})$.

We choose the cell where the triangular order coincides with the
lexicographical order. The passage to the other $\ell$-cell is obtained by a
counter-clockwise rotation of $u_{1}^{\prime}$ w.r.t. $u_{2}^{\prime}$, which
corresponds to the action of the elementary braid $\beta_{12}$. Its action
(3.3) is a permutation matrix, since $(S_{\text{lex}})_{12}=0$; it is a
trivial action on $S_{\text{lex}}$, but not on $C_{\text{lex}}$, as (5.14)
shows.

(3) We move along a curve $h\mapsto he^{i\frac{\pi}{2}}$ in the $h$-plane from
a point (5.15) up to a point

$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},\varepsilon
e^{i\varphi^{\prime}},h\right),\quad\text{with }\frac{\pi}{4}<\arg
h<\frac{3}{4}\pi,$

for some $\varphi^{\prime}\neq k\pi-\frac{1}{2}\arg h$, $k\in\mathbb{Z}$,
$\frac{\pi}{4}<\arg h<\frac{3\pi}{4}$. The transformation in Figure 3, due to
the splitting, can substituted by the sequence of transformations in Figure 4,
each step corresponding to an elementary braid. Each elementary braid
corresponds to a Stokes ray crossing clock-wise the real line $\ell$ as $h$
varies along the curve $h\mapsto he^{i\frac{\pi}{2}}$.161616 Notice that the
ray $R_{23}$ rotates slower than $R_{12},R_{13}$: namely, the angular velocity
of $R_{23}$ is approximately (i.e. modulo negligible corrections in powers of
$\varepsilon$) equal to $\frac{1}{4}$ the one of $R_{12},R_{13}$.

Figure 4. The transition in Figure 3 by splitting and elementary steps. After
the splitting, we obtain a point $(u_{1},u_{2},u_{3})$, as in (5.16)-(5.18),
lying in an $\ell$-cell of the polydisc centred at
$(u_{1},u_{2},u_{3})=(0,-\frac{1}{2}h^{2},-\frac{1}{2}h^{2})$ of the left part
of Figure 3. The transformation of Figure 3 is obtained by successive steps
following the arrows. The final step is the right part of Figure 3. The first
elementary braid is $\beta_{12}$ (because $u_{1}^{\prime}=u_{2}$,
$u_{2}^{\prime}=u_{3}$ in the the upper left figure). The second is
$\beta_{23}$ (after relabelling in lexicographical order,
$u_{2}^{\prime}=u_{2}$ and $u_{3}^{\prime}=u_{1}$ in the upper right figure).
The third is $\beta_{12}$.

The total braid is then factored into the product of the elementary braids as
in Figure 6, namely

$\beta_{12}\beta_{23}\beta_{12},\quad\text{or}\quad\beta_{12}\beta_{23}\beta_{12}\beta_{23}.$

Applying formulae (3.3),(3.5), we obtain

$S_{\text{lex}}^{\beta_{12}\beta_{23}\beta_{12}}=S_{\text{lex}}^{\beta_{12}\beta_{23}\beta_{12}\beta_{23}}=\begin{pmatrix}1&1&1\\\
0&1&0\\\ 0&0&1\end{pmatrix}.$ (5.20)

These are the monodromy data in the two $\ell$-cells of a polydisc centred at
the point

$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right),\quad\text{with
}\frac{\pi}{4}<\arg h<\frac{3}{4}\pi.$

The braid $\beta_{23}$ is responsible for the passage from one cell to the
other. Its action
$A^{\beta_{23}}(S_{\text{lex}}^{\beta_{12}\beta_{23}\beta_{12}})$ is a
permutation matrix, since
$(S_{\text{lex}}^{\beta_{12}\beta_{23}\beta_{12}})_{23}=0$, which explains the
equality in (5.20). By the action (3.5), the central connection matrix (5.14),
instead, assumes the following two forms (differing for a permutation of the
second and third column)

$C_{\text{lex}}^{\beta_{12}\beta_{23}\beta_{12}}=\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}(1+i)\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)\\\
\\\ 0&\pm\sqrt{2\pi}&\mp\sqrt{2\pi}\\\ \\\
(1-i)\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)\\\
\end{array}\right),$
$C_{\text{lex}}^{\beta_{12}\beta_{23}\beta_{12}\beta_{23}}=\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}(1+i)\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)\\\
\\\ 0&\mp\sqrt{2\pi}&\pm\sqrt{2\pi}\\\ \\\
(1-i)\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)\\\
\end{array}\right).$

$u_{2}$$A$$B$$C$$D$$E$$F$$A$$B$$C$$D$$E$$F$$\ell$ Figure 5. In the left
picture we represent relative positions of $u_{3}$ w.r.t $u_{2}$ such that the
real line $\ell$ is admissible. On the right, we represent the corresponding
positions of the Stokes ray $R_{23}$. Notice that if we let vary $u_{3}$, by a
deformation of the parameter $\varphi$, starting from $A$, going through $B$
up to $C$, the corresponding Stokes ray does not cross the line $\ell$, and no
braids act. If we continue the deformation of $\varphi$ from $C$ to $D$, an
elementary braid acts on the monodromy data. $u_{2}=u_{3}$$u_{1}$One wayfor
splittinganother wayfor
splitting$u_{1}$$u_{2}$$u_{3}$$u_{2}$$u_{3}$$u_{1}$$u_{1}$$u_{1}$$u_{2}$$u_{2}$$u_{3}$$u_{3}$$\beta_{12}$$\beta_{12}\beta_{23}\beta_{12}$$\beta_{23}$$\beta_{23}$$\beta_{12}$$\beta_{12}\beta_{23}\beta_{12}$$\beta_{12}$111122223333
Figure 6. In the picture we represent $u_{1}$, $u_{2}$, $u_{3}$ as points in
$\mathbb{C}$. On the left we describe all the braids necessary to pass from a
neighborhood of $(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$ with
$-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$ to one with $\frac{\pi}{4}<\arg
h<\frac{3}{4}\pi$. Different columns of this diagram correspond to different
$\ell$-cells of the same neighborhood. The passage from such one cell to the
other is through an action of an elementary braid ($\beta_{12}$ or
$\beta_{23}$) acting as a permutation matrix. In the picture on the right, we
show the decomposition of the global transformation in elementary ones.

In Table 2 we show the monodromy data for other values of $\arg h$, with the
corresponding braid. In Figure 6 we represent the braid corresponding to the
passage from $-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$ to $\frac{5}{4}\pi<\arg
h<\frac{7}{4}\pi$.

$\beta_{12}$$\beta_{23}\beta_{12}$$\beta_{23}$$\beta_{12}\beta_{23}$$\beta_{12}$$\beta_{23}\beta_{12}$$\beta_{23}$$u_{b}$$u_{a}$$u_{1}$$2$$1$$3$
Figure 7. Using the diagram representation of the braid group as mapping class
group of the punctured disk, we draw the braids acting along a curve $h\mapsto
e^{\frac{3\pi i}{2}}h$, starting from the chambers close to
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$ with
$-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$, and reaching the ones with
$\frac{5}{4}\pi<\arg h<\frac{7}{4}\pi$. The braids in red describe mutations
of the split pair $u_{2},u_{3}$: their action on the monodromy data is a
permutation matrix. In the central disk, the blue numbers refer to the
lexicographical order w.r.t. the real axis $\ell$ (i.e. from the left to the
right). The braids are the same for both cases $(a,b)=(2,3)$ and $(3,2)$.

######  Remark 5.3.

The reader can re-obtain this result by direct computation observing that, for
points

$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right),\text{ with
}\frac{\pi}{4}<\arg h<\frac{3}{4}\pi,$

the left and right solutions of (5.5) defining the Stokes matrix171717Notice
that for the points with $\frac{\pi}{4}<\arg h<\frac{3}{4}\pi$ the original
labelling of canonical coordinates
$(u_{1},u_{2},u_{3})=\left(0,-\frac{h^{2}}{4},-\frac{h^{2}}{4}\right)$ already
put the Stokes matrix in upper triangular form. are of the form (5.9) with:

$\xi^{L}_{(1),1}=\xi^{R}_{(1),1}=\frac{\sqrt{\pi}}{4\sqrt{2}}h^{\frac{1}{2}}e^{i\frac{\pi}{8}}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}H^{(1)}_{\frac{1}{4}}\left(e^{-i\pi}\frac{ih^{2}}{8}z\right),$
$\xi^{L}_{(2),1}(z)=\xi^{L}_{(3),1}(z)=\frac{i\sqrt{\pi}}{8}h^{\frac{1}{2}}e^{i\frac{3}{8}\pi}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}H^{(2)}_{\frac{1}{4}}\left(e^{-3i\pi}\frac{ih^{2}}{8}z\right),$
$\xi^{R}_{(2),1}(z)=\xi^{R}_{(3),1}(z)=\frac{i\sqrt{\pi}}{8}h^{\frac{1}{2}}e^{i\frac{5}{8}\pi}z^{\frac{1}{2}}e^{-\frac{zh^{2}}{8}}H^{(1)}_{\frac{1}{4}}\left(\frac{ih^{2}}{8}z\right),$

having the expected asymptotic expansions in suitable sectors containing
$\Pi_{{\rm left}}$ and/or $\Pi_{\rm right}$ by Lemma 5.1. Thus, by some
manipulation of formulae (5.10) and

$\sin(\nu\pi)H^{(2)}_{\nu}(ze^{m\pi i})=e^{\nu\pi
i}\sin(m\nu\pi)H^{(1)}_{\nu}(z)+\sin((m+1)\nu\pi)H^{(2)}_{\nu}(z),\quad
m\in\mathbb{Z},$

one sees that

$\xi^{L}_{(2),1}(z)=\xi^{R}_{(1),1}(z)+\xi^{R}_{(2),1}(z),\quad\xi^{L}_{(3),1}(z)=\xi^{R}_{(1),1}(z)+\xi^{R}_{(3),1}(z),$

which are equivalent to (5.20). For the computation of the central connection
matrix, one can use analogous Puiseux series expansions of the solution
$\Xi_{\rm right}(z)$, obtained from the integral representation of Hankel
functions given above.

| $S_{\text{lex}}$ | $C_{\text{lex}}$ | Braid
---|---|---|---
$-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$ | $\begin{pmatrix}1&0&-1\\\ 0&1&-1\\\ 0&0&1\end{pmatrix}$ | $\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}-i\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)&(1-i)\Gamma\left(\frac{3}{4}\right)\\\ \\\ \mp\sqrt{2\pi}&\pm\sqrt{2\pi}&0\\\ \\\ i\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)&(1+i)\Gamma\left(\frac{1}{4}\right)\\\ \end{array}\right)$ | $\color[rgb]{1,0,0}{\beta_{12}}$
$\frac{\pi}{4}<\arg h<\frac{3}{4}\pi$ | $\begin{pmatrix}1&1&1\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}(1+i)\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)&-i\Gamma\left(\frac{3}{4}\right)\\\ \\\ 0&\pm\sqrt{2\pi}&\mp\sqrt{2\pi}\\\ \\\ (1-i)\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)&i\Gamma\left(\frac{1}{4}\right)\\\ \end{array}\right)$ | $\beta_{12}\beta_{23}\beta_{12}\color[rgb]{1,0,0}{\beta_{23}}$
$\frac{3}{4}\pi<\arg h<\frac{5}{4}\pi$ | $\left(\begin{array}[]{ccc}1&0&-1\\\ 0&1&-1\\\ 0&0&1\\\ \end{array}\right)$ | $\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}\Gamma\left(\frac{3}{4}\right)&\Gamma\left(\frac{3}{4}\right)&(1+i)\Gamma\left(\frac{3}{4}\right)\\\ \\\ \mp\sqrt{2\pi}&\pm\sqrt{2\pi}&0\\\ \\\ \Gamma\left(\frac{1}{4}\right)&\Gamma\left(\frac{1}{4}\right)&(1-i)\Gamma\left(\frac{1}{4}\right)\\\ \end{array}\right)$ | $(\beta_{12}\beta_{23})^{3}\color[rgb]{1,0,0}{\beta_{12}}$
$\frac{5}{4}\pi<\arg h<\frac{7}{4}\pi$ | $\left(\begin{array}[]{ccc}1&1&1\\\ 0&1&0\\\ 0&0&1\\\ \end{array}\right)$ | $\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}(-1+i)\Gamma\left(\frac{3}{4}\right)&\Gamma\left(\frac{3}{4}\right)&\Gamma\left(\frac{3}{4}\right)\\\ \\\ 0&\pm\sqrt{2\pi}&\mp\sqrt{2\pi}\\\ \\\ (-1-i)\Gamma\left(\frac{1}{4}\right)&\Gamma\left(\frac{1}{4}\right)&\Gamma\left(\frac{1}{4}\right)\\\ \end{array}\right)$ | $(\beta_{12}\beta_{23})^{3}\beta_{12}\beta_{23}\beta_{12}\color[rgb]{1,0,0}{\beta_{23}}$
$\frac{7}{4}\pi<\arg h<\frac{9}{4}\pi$ | $\left(\begin{array}[]{ccc}1&0&-1\\\ 0&1&-1\\\ 0&0&1\\\ \end{array}\right)$ | $\frac{1}{\pi^{\frac{1}{2}}}\left(\begin{array}[]{ccc}i\Gamma\left(\frac{3}{4}\right)&i\Gamma\left(\frac{3}{4}\right)&(-1+i)\Gamma\left(\frac{3}{4}\right)\\\ \\\ \mp\sqrt{2\pi}&\pm\sqrt{2\pi}&0\\\ \\\ -i\Gamma\left(\frac{1}{4}\right)&-i\Gamma\left(\frac{1}{4}\right)&(-1-i)\Gamma\left(\frac{1}{4}\right)\\\ \end{array}\right)$ | $(\beta_{12}\beta_{23})^{6}\color[rgb]{1,0,0}{\beta_{12}}$
Table 2. For different values of $\arg h$, we tabulate the monodromy data
$(S_{\text{lex}},C_{\text{lex}})$, in lexicographical order, in the two
$\ell$-cells which divide a sufficiently small neighborhood of the point
$(t_{1},t_{2},t_{3})=\left(-\frac{1}{8}h^{2},0,h\right)$. The difference of
the data in the two $\ell$-cells (just a permutation of two columns in the
central connection matrix) is obtained by applying the braid written in red:
if it is not applied the sign to be read is the first one, the second one
otherwise. Notice that the central element $(\beta_{12}\beta_{23})^{3}$ acts
trivially on the Stokes matrices, and by a left multiplication by
$M_{0}^{-1}=\operatorname{diag}(i,1,-i)$ on the central connection matrix.

###  5.6. Monodromy data as computed outside the Maxwell stratum

In this section, we compute the Stokes matrix $S$ at non-coalescence points in
a neighbourhood of a coalescence one, by means of oscillatory integrals. We
show that $S$ coincides with that obtained at the coalescence point in the
previous section. Moreover, we explicitly show that the fundamental matrices
converge to those computed at the coalescence point, exactly as prescribed by
our Theorem 4.1.

The system (5.2) admits solutions given in terms of oscillating integrals,

$\displaystyle\xi_{1}(z,t)$
$\displaystyle=z^{\frac{1}{2}}\int_{\gamma}\exp\left\\{z\cdot
f(x,t)\right\\}dx,$ (5.21) $\displaystyle\xi_{2}(z,t)$
$\displaystyle=z^{\frac{1}{2}}\int_{\gamma}x\exp\left\\{z\cdot
f(x,t)\right\\}dx,$ (5.22) $\displaystyle\xi_{3}(z,t)$
$\displaystyle=z^{\frac{1}{2}}\int_{\gamma}\left(x^{2}+\frac{1}{4}t_{3}\right)\exp\left\\{z\cdot
f(x,t)\right\\}dx,$ (5.23)

where $f(x,t)=x^{4}+t_{3}x^{2}+t_{2}x+t_{1}+\frac{1}{8}t_{3}^{2}$. Here
$\gamma$ is any cycle along which $\operatorname{Re}(z\cdot f(x,t))\to-\infty$
for $|x|\to+\infty$, i.e. a relative cycle in
$H_{1}(\mathbb{C},\mathbb{C}_{T,z,t})$, with

$\mathbb{C}_{T,z,t}:=\left\\{x\in\mathbb{C}\colon\operatorname{Re}(zf(x,t))<-T\right\\},\quad\text{with
}T\text{ very large positive number. }$

First, we show that the Stokes matrix at points in $\ell$-chambers near the
coalescence point $(t_{1},t_{2},t_{3})=(-\frac{1}{8}h^{2},0,h)$ coincide with
the one previously computed, in accordance with Theorem 4.1. In what follows
we will focus on the $\ell$-chamber made of points
$(t_{1},t_{2},t_{3})=(-\frac{1}{8}h^{2},\varepsilon e^{i\phi},h)$, where
$-\frac{\pi}{4}<\arg h<\frac{\pi}{4}$, and $\varepsilon,\phi$ are small
positive numbers. For points in this $\ell$-chamber, the Stokes rays are
disposed as described in Figure 9.

$R_{21}$$R_{31}$$R_{32}$$R_{12}$$R_{13}$$R_{23}$ Figure 8. Disposition of the
Stokes rays for a point in the chosen $\ell$-chamber.

$\frac{\pi}{8}$$\mathcal{I}_{3}$$\mathcal{I}_{1}$$\mathcal{I}_{2}$ Figure 9.
Integration contours $\mathcal{I}_{i}$ which define the functions
$\mathfrak{I}_{i}$’s.

Notice that in order to compute the Stokes matrix at a semisimple point with
distinct canonical coordinates it suffices to know the first rows of
$\Xi_{\text{left/right}}$. Assuming that $z\in\mathbb{R}_{+}$, we define the
following three functions obtained from the integrals (5.21) with integration
cycles $\mathcal{I}_{i}$ as in Figure 9:

$\mathfrak{I}_{i}(z,t_{2}):=\int_{\mathcal{I}_{i}}\exp\left(z(x^{4}+hx^{2}+t_{2}x)\right)dx,\quad
i=1,2,3.$ (5.24)

For the specified integration cycles, the integrals
$\mathfrak{I}_{i}(z,t_{2})$ are convergent in the half-plane $|\arg
z|<\frac{\pi}{2}$. A continuous deformation of a path ${\mathcal{I}_{i}}$,
which maintains its asymptotic directions in the shaded sectors, yields a
convergent integral and defines the analytic continuation of
$\mathfrak{I}_{i}(z,t_{2})$ on the whole sector $|\arg z|<\pi$. If we vary $z$
(excluding $z=0$), the shaded regions continuously rotate clockwise or
counterclockwise. In order to obtain the analytic continuation of the
functions $\mathfrak{I}_{i}(z,t_{2})$ to the whole universal cover
$\mathcal{R}$, we can simply rotate the integration contours
$\mathcal{I}_{i}$. This procedure also makes it clear that the functions
$\mathfrak{I}_{i}$ have monodromy of order $4$: indeed as $\arg z$ increases
or decreases by $2\pi$, the shaded regions are cyclically permuted.

In order to obtain information about the asymptotic expansions of the
functions $\mathfrak{I}_{i}$, we associate to any critical point $x_{i}$ a
relative cycle $\mathcal{L}_{i}$, called _Lefschetz thimble_ , defined as the
set of points of $\mathbb{C}$ which can be reached along the downward
geodesic-flow

$\frac{dx}{d\tau}=-\bar{z}\frac{\partial\bar{f}}{\partial\bar{x}},\quad\frac{d\bar{x}}{d\tau}=-z\frac{\partial
f}{\partial{x}}$ (5.25)

starting at the critical point $x_{i}$ for $\tau\to-\infty$. Morse and Picard-
Lefschetz Theory guarantees that the cycles $\mathcal{L}_{i}$ are smooth one-
dimensional submanifolds of $\mathbb{C}$, _piecewise smoothly_ dependent on
the parameters $z,t$, and they represent a basis for the relative homology
groups $H_{1}(\mathbb{C},\mathbb{C}_{T,z,t})$. Moreover, the Lefschetz
thimbles are steepest descent paths: namely, $\operatorname{Im}(zf(x,t))$ is
constant on each connected component of
$\mathcal{L}_{i}\setminus\left\\{x_{i}\right\\}$ and
$\operatorname{Re}(zf(x,t))$ is strictly decreasing along the flow (5.25).
Thus, after choosing an orientation, the paths of integration defining the
functions $\mathfrak{I}_{i}$ can be expressed as integer combinations of the
thimbles $\mathcal{L}_{i}$ for any value of $z$:

$\mathcal{I}_{i}=n_{1}\mathcal{L}_{1}+n_{2}\mathcal{L}_{2}+n_{3}\mathcal{L}_{3},\quad
n_{i}\in\mathbb{Z}.$ (5.26)

If we let $z$ vary, the Lefschetz thimbles change. When $z$ crosses a Stokes
ray, Lefschetz thimbles jump discontinuously, as shown in Figure 10. In
particular, for $z$ on a Stokes ray there exists a flow line of (5.25)
connecting two critical points $x_{i}$’s.

$x_{1}$$x_{2}$$x_{3}$$x_{1}$$x_{3}$$x_{2}$$x_{1}$$x_{3}$$x_{2}$

Figure 10. Discontinuous change of a Lefschetz thimbles. As $z$ varies in
$\mathcal{R}$, we pass from the configuration on the left to the one on the
right. The middle configuration is realized when $z$ is on a Stokes ray: in
this case there is a downward geodesic-flow line connecting two critical
points $x_{1}$ and $x_{3}$.

This discontinuous change of the thimbles implies a discontinuous change of
the integer coefficients $n_{i}$ in (5.26), and a discontinuous change of the
leading term of the asymptotic expansions of the functions
$\mathfrak{I}_{i}$’s. Using the notations introduced in Figure 11, in each
configuration the following identities hold:

$(A)\colon\left\\{\begin{array}[]{@{}l@{}}\mathcal{I}_{1}=\mathcal{L}_{1},\\\
\mathcal{I}_{2}=\mathcal{L}_{2},\\\
\mathcal{I}_{3}=\mathcal{L}_{3},\end{array}\right.\quad\quad(B)\colon\left\\{\begin{array}[]{@{}l@{}}\mathcal{I}_{1}=\mathcal{L}_{1}+\mathcal{L}_{2},\\\
\mathcal{I}_{2}=\mathcal{L}_{2},\\\ \mathcal{I}_{3}=\mathcal{L}_{3},\\\
\end{array}\right.\quad\quad(C)\colon\left\\{\begin{array}[]{@{}l@{}}\mathcal{I}_{1}=\mathcal{L}_{1}+\mathcal{L}_{2},\\\
\mathcal{I}_{2}=\mathcal{L}_{2},\\\
\mathcal{I}_{3}=-\mathcal{L}_{1}+\mathcal{L}_{3},\\\ \end{array}\right.$
$(D)\colon\left\\{\begin{array}[]{@{}l@{}}\mathcal{I}_{1}=\mathcal{L}_{1}-\mathcal{L}_{3},\\\
\mathcal{I}_{2}=\mathcal{L}_{2},\\\ \mathcal{I}_{3}=\mathcal{L}_{3},\\\
\end{array}\right.\quad\quad(E)\colon\left\\{\begin{array}[]{@{}l@{}}\mathcal{I}_{1}=\mathcal{L}_{1}-\mathcal{L}_{3},\\\
\mathcal{I}_{2}=\mathcal{L}_{1}+\mathcal{L}_{2},\\\
\mathcal{I}_{3}=\mathcal{L}_{3}.\end{array}\right.$

$\mathcal{I}_{1}$$\mathcal{I}_{1}$$\mathcal{I}_{1}$$\mathcal{I}_{1}$$\mathcal{I}_{1}$$\mathcal{I}_{2}$$\mathcal{I}_{2}$$\mathcal{I}_{2}$$\mathcal{I}_{2}$$\mathcal{I}_{2}$$\mathcal{I}_{3}$$\mathcal{I}_{3}$$\mathcal{I}_{3}$$\mathcal{I}_{3}$$\mathcal{L}_{2}$$\mathcal{L}_{2}$$\mathcal{L}_{2}$$\mathcal{L}_{3}$$\mathcal{L}_{3}$$\mathcal{L}_{3}$$\mathcal{L}_{1}$$\mathcal{L}_{1}$$\mathcal{L}_{1}$$\mathcal{L}_{2}$$\mathcal{L}_{2}$$\mathcal{L}_{3}$$\mathcal{L}_{3}$$\mathcal{I}_{3}$$\mathcal{L}_{1}$$\mathcal{L}_{1}$$(A)$$(B)$$(C)$$(E)$$(D)$

Figure 11. In this figure it is shown how the Lefschetz thimbles
$\mathcal{L}_{i}$’s (continuous lines), and the integrations contours
$\mathcal{I}_{i}$’s (dotted lines) change by analytic continuation with
respect to the variable $z$. The configuration $(A)$ corresponds to the case
$\arg z=0$. Increasing $\arg z$ the configuration $(B)$ and $(C)$ are reached
after crossing the Stokes rays $R_{31}$, and $R_{21}$ respectively. Decreasing
$\arg z$, we obtain the configurations $(D)$ and $(E)$ after crossing the rays
$R_{12}$ and $R_{13}$ respectively. Note that when $z$ crosses the Stokes rays
$R_{32}$ and $R_{23}$ no Lefschetz thimble changes, coherently with the
detailed analysis done in [CDG19].

By a straightforward application of the Laplace method we find that, at least
for sufficiently small positive values of $\arg z$, the following asymptotic
expansions hold

$\mathfrak{I}_{i}(z,t_{2})=\pi^{\frac{1}{2}}iz^{-\frac{1}{2}}(6x_{i}^{2}+h)^{-\frac{1}{2}}e^{zu_{i}}\left(1+O\left(\frac{1}{z}\right)\right).$

Since the deformations of the thimbles $\mathcal{I}_{2},\mathcal{I}_{3}$
happen for values of $z$ for which the exponent $e^{zu_{1}}$ is subdominant,
we immediately conclude that the functions

$\xi^{L}_{(2),1}(z,t_{2})=\xi^{R}_{(2),1}(z,t_{2})=\pm
i\pi^{-\frac{1}{2}}z^{\frac{1}{2}}\frac{6x_{2}^{2}+h}{2\sqrt{2}(x_{1}-x_{2})(x_{3}-x_{2})}\mathfrak{I}_{2}(z,t_{2}),$
(5.27) $\xi^{L}_{(3),1}(z,t_{2})=\xi^{R}_{(3),1}(z,t_{2})=\pm
i\pi^{-\frac{1}{2}}z^{\frac{1}{2}}\frac{6x_{3}^{2}+h}{2\sqrt{2}(x_{1}-x_{3})(x_{2}-x_{3})}\mathfrak{I}_{3}(z,t_{2}),$
(5.28)

have asymptotic expansions

$\Psi_{21}e^{zu_{2}}\left(1+O\left(\frac{1}{z}\right)\right),\quad\Psi_{31}e^{zu_{3}}\left(1+O\left(\frac{1}{z}\right)\right),$

respectively, both in $\Pi_{\rm left}$ and $\Pi_{\rm right}$. Thus, we can
immediately say that the Stokes matrix computed at a point
$(t_{1},t_{2},t_{3})=(-\frac{1}{8}h^{2},\varepsilon e^{i\phi},h)$ is of the
form

$S=\begin{pmatrix}1&0&0\\\ *&1&0\\\ *&0&1\end{pmatrix}.$

Note that the arbitrariness of the orientations of the Lefschetz thimbles can
be incorporated in the choice of the entries of the $\Psi$ matrix, and hence
it will affect the monodromy data by the action of the group
$(\mathbb{Z}/2\mathbb{Z})^{3}$.

After a careful analysis of the deformations of the Lefschetz thimbles, one
finds that the solutions $\xi_{(1),1}^{L}(z,t_{2})$,
$\xi_{(1),1}^{R}(z,t_{2})$ are respectively given by

$\displaystyle\xi_{(1),1}^{R}(z,t_{2})$ $\displaystyle=\pm
i\Psi_{11}\pi^{-\frac{1}{2}}z^{\frac{1}{2}}(6x_{1}^{2}+h)^{\frac{1}{2}}\left(\mathfrak{I}_{1}(z,t_{2})+\mathfrak{I}_{3}(z,t_{2})\right),$
(5.29) $\displaystyle\xi_{(1),1}^{L}(z,t_{2})$ $\displaystyle=\pm
i\Psi_{11}\pi^{-\frac{1}{2}}z^{\frac{1}{2}}(6x_{1}^{2}+h)^{\frac{1}{2}}\left(\mathfrak{I}_{1}(z,t_{2})-\mathfrak{I}_{2}(z,t_{2})\right),$
(5.30)

having the asymptotic expansion

$\Psi_{11}e^{zu_{1}}\left(1+O\left(\frac{1}{z}\right)\right)$

in $\Pi_{\rm right}$ and $\Pi_{\rm left}$ respectively. This immediately
allows one to compute the remaining entries of the Stokes matrix

$\displaystyle S_{21}$
$\displaystyle=\frac{\Psi_{11}(6x_{1}^{2}+h)^{\frac{1}{2}}}{\Psi_{21}(6x_{2}^{2}+h)^{\frac{1}{2}}}=\pm\frac{(6x_{1}^{2}+h)(x_{3}-x_{2})}{(x_{1}-x_{3})(6x_{2}^{2}+h)}\equiv\pm
1,$ $\displaystyle S_{31}$
$\displaystyle=\frac{\Psi_{11}(6x_{1}^{2}+h)^{\frac{1}{2}}}{\Psi_{31}(6x_{3}^{2}+h)^{\frac{1}{2}}}=\pm\frac{(6x_{1}^{2}+h)(x_{2}-x_{3})}{(x_{1}-x_{2})(6x_{3}^{2}+h)}\equiv\pm
1.$

This result is independent on the point
$(t_{1},t_{2},t_{3})=(-\frac{1}{8}h^{2},\varepsilon e^{i\phi},h)$ of the
chosen $\ell$-chamber. It coincides with the Stokes matrix obtained at the
coalescence point $(t_{1},t_{2},t_{3})=(-\frac{1}{8}h^{2},0,h)$, in complete
accordance with our Theorem 4.1.

######  Remark 5.4.

It is interesting to note that the isomonodromy condition in this context is
equivalent to the condition

$\frac{f^{\prime\prime}(x_{1})}{f^{\prime\prime}(x_{2})}=-\frac{x_{1}-x_{3}}{x_{2}-x_{3}},$

a relation that the reader can easily show to be valid for any polynomial
$f(x)$ of fourth degree with three non-degenerate critical points
$x_{1},x_{2},x_{3}$.

Our Theorem 4.1 also states that, as $t_{2}\to 0$ the solutions (5.27),
(5.28), (5.29), (5.30) must converge to the ones computed in the previous
section at the coalescence point. We show this explicitly below. In order to
do this, it suffices to set $t_{2}=0$ in the integral (5.24). With the change
of variable $x=2^{-\frac{1}{4}}z^{-\frac{1}{4}}s^{\frac{1}{2}}$ , we obtain

$\displaystyle\mathfrak{I}_{2}(z,0)=\mathfrak{I}_{3}(z,0)$
$\displaystyle=2^{-\frac{5}{4}}z^{-\frac{1}{4}}\int_{L}\exp\left\\{\frac{s^{2}}{2}+\left(\frac{hz^{\frac{1}{2}}}{\sqrt{2}}\right)s\right\\}ds$
$\displaystyle=2^{-\frac{5}{4}}z^{-\frac{1}{4}}(2\pi)^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}D_{-\frac{1}{2}}\left(\frac{hz^{\frac{1}{2}}}{\sqrt{2}}\right)$
$\displaystyle=2^{-\frac{3}{2}}e^{-\frac{h^{2}z}{8}}h^{\frac{1}{2}}K_{\frac{1}{4}}\left(\frac{h^{2}z}{8}\right)$
$\displaystyle=\pi i\cdot
2^{-\frac{5}{2}}h^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}e^{\frac{\pi
i}{8}}H^{(1)}_{\frac{1}{4}}\left(\frac{ih^{2}z}{8}\right).$

Here $D_{\nu}(z)$ is the Weber parabolic cylinder function of order $\nu$,
with integral representation ([AS70], page 688)

$D_{-\frac{1}{2}}(z)=\pm\frac{e^{\frac{1}{2}z^{2}}}{(2\pi)^{\frac{1}{2}}}\int_{L}s^{-\frac{1}{2}}\exp\left(\frac{s^{2}}{2}+zs\right)ds,\quad\text{where}\left\\{\begin{array}[]{@{}l@{}}(+)\text{
if }-\frac{3\pi}{2}+2k\pi<\arg s<-\frac{\pi}{2}+2k\pi,\\\ (-)\text{ if
}\frac{\pi}{2}+2k\pi<\arg s<\frac{3\pi}{2}+2k\pi,\end{array}\right.$

the integration contour $L$ being the one represented in Figure 13, together
with the identities

$\quad
D_{-\frac{1}{2}}(z)=\left(\frac{z}{2\pi}\right)^{\frac{1}{2}}K_{\frac{1}{4}}\left(\frac{1}{4}z^{2}\right),\quad
K_{\nu}\left(z\right)=\left\\{\begin{array}[]{@{}l@{}}\frac{\pi
i}{2}e^{\frac{\nu\pi i}{2}}H^{(1)}_{\nu}\left(ze^{\frac{\pi i}{2}}\right),\\\
\\\ -\frac{\pi i}{2}e^{-\frac{\nu\pi i}{2}}H^{(2)}_{\nu}\left(ze^{-\frac{\pi
i}{2}}\right).\end{array}\right.$

$L$

Figure 12. Integration contour $L$ used in the integral representation of the
Weber parabolic cylinder functions.

$x_{2}$$x_{1}$$x_{3}$$\mathcal{I}_{1}^{1}$$\mathcal{I}_{1}^{2}$ Figure 13. For
$t_{2}=0$, we can decompose the integration cycle $\mathcal{I}_{1}$ into two
pieces, $\mathcal{I}_{1}^{1},\mathcal{I}_{1}^{2}$ used to define the functions
$\mathfrak{I}_{1}^{1}$ and $\mathfrak{I}_{1}^{2}$. The continuous lines
represent the Lefschetz thimbles through the critical points $x_{i}$’s.

It follows that

$\displaystyle\xi^{L}_{(2),1}(z,0)=\xi^{R}_{(2),1}(z,0)$ $\displaystyle=\pm
i\pi^{-\frac{1}{2}}z^{\frac{1}{2}}\frac{6x_{2}^{2}+h}{2\sqrt{2}(x_{1}-x_{2})(x_{3}-x_{2})}\mathfrak{I}_{2}(z,0)$
$\displaystyle=\pm\frac{i\sqrt{\pi}}{8}h^{\frac{1}{2}}e^{\frac{5i\pi}{8}}z^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}H^{(1)}_{\frac{1}{4}}\left(\frac{ih^{2}z}{8}\right),$

which coincides (up to an irrelevant sign) with the solution computed in the
previous section at the coalescence point. The computations for
$\xi_{(3),1}^{L}(z,0)=\xi_{(3),1}^{R}(z,0)$ are identical.

The computations for $\xi_{(1),1}^{R}$ and $\xi_{(1),1}^{L}$ are a bit more
laborious. First of all let us observe that the integral

$g(z):=\int_{0}^{\infty}\exp\left(-\frac{t^{2}}{2}-zt\right)t^{-\frac{1}{2}}dt$

is convergent for all $z\in\mathbb{C}$, defining an entire function181818This
is in accordance with the expression of $g$ in terms of the modified Bessel
function $K$, which gives  $g(e^{\pm\pi
i}z)=2^{-\frac{1}{2}}e^{\frac{z^{2}}{4}}e^{\pm\frac{\pi
i}{2}}z^{\frac{1}{2}}K_{\frac{1}{4}}\left(e^{\pm 2\pi
i}\frac{z^{2}}{4}\right).$ From the symmetry $K_{\frac{1}{4}}(e^{4\pi
i}z)=-K_{\frac{1}{4}}(z)$ we deduce that $g(e^{-\pi i}z)=g(e^{\pi i}z)$..
Moreover we have

$g(z)=\sqrt{\pi}e^{\frac{z^{2}}{4}}D_{-\frac{1}{2}}(z)=2^{-\frac{1}{2}}e^{\frac{z^{2}}{4}}z^{\frac{1}{2}}K_{\frac{1}{4}}\left(\frac{z^{2}}{4}\right).$

With a change of variable $t=e^{-i\theta}\tau$ that rotates the half line
$\mathbb{R}_{+}$ by $\theta$, we find the following identity

$g(z)=e^{-i\frac{\theta}{2}}\int_{e^{i\theta}\mathbb{R}_{+}}\exp\left(-e^{-2i\theta}\frac{\tau^{2}}{2}-e^{-i\theta}z\tau\right)\tau^{-\frac{1}{2}}d\tau.$
(5.31)

For $t_{2}=0$ the integral $\mathfrak{I}_{1}(z,0)$ splits into two pieces:

$\mathfrak{I}_{1}(z,0)=\mathfrak{I}_{1}^{1}(z)+\mathfrak{I}_{1}^{2}(z),\quad\mathfrak{I}_{1}^{i}(z):=\int_{\mathcal{I}_{1}^{i}}\exp(z(x^{4}+hx^{2}))dx,\quad
i=1,2$

where the paths $\mathcal{I}_{1}^{i}$ are as in Figure 13. Setting
$x=2^{-\frac{1}{4}}z^{-\frac{1}{4}}s^{\frac{1}{2}}$, the image of the paths
$\mathcal{I}_{1}^{i}$ are in two different sheets of the Riemann surface with
local coordinate $s$. Keeping track of this, and of the orientations of the
modified paths, using formula (5.31) for $\theta=\frac{3\pi i}{2},\frac{5\pi
i}{2}$ and a small deformation of the paths of integration, we find that

$\displaystyle\mathfrak{I}_{1}^{1}(z)$
$\displaystyle=2^{-\frac{5}{4}}z^{-\frac{1}{4}}\left(-\int_{e^{\frac{3\pi
i}{2}}\mathbb{R}_{+}}\exp\left\\{\frac{s^{2}}{2}+\frac{hz^{\frac{1}{2}}}{\sqrt{2}}s\right\\}s^{-\frac{1}{2}}ds\right)$
$\displaystyle=-2^{-\frac{5}{4}}z^{-\frac{1}{4}}e^{\frac{3\pi
i}{4}}g\left(e^{\frac{\pi i}{2}}\frac{hz^{\frac{1}{2}}}{\sqrt{2}}\right)$
$\displaystyle=-2^{-\frac{5}{4}}z^{-\frac{1}{4}}e^{\frac{3\pi i}{4}}\cdot
2^{-\frac{1}{2}}e^{-\frac{h^{2}z}{8}}\left(e^{\frac{\pi
i}{2}}\frac{hz^{\frac{1}{2}}}{\sqrt{2}}\right)^{\frac{1}{2}}K_{\frac{1}{4}}\left(e^{\pi
i}\frac{h^{2}z}{8}\right)=\frac{1}{4}e^{-\frac{h^{2}z}{8}}h^{\frac{1}{2}}K_{\frac{1}{4}}\left(e^{\pi
i}\frac{h^{2}z}{8}\right),$

and

$\displaystyle\mathfrak{I}_{1}^{2}(z)$
$\displaystyle=2^{-\frac{5}{4}}z^{-\frac{1}{4}}\left(\int_{e^{\frac{5\pi
i}{2}}\mathbb{R}_{+}}\exp\left\\{\frac{s^{2}}{2}+\frac{hz^{\frac{1}{2}}}{\sqrt{2}}s\right\\}s^{-\frac{1}{2}}ds\right)$
$\displaystyle=2^{-\frac{5}{4}}z^{-\frac{1}{4}}e^{\frac{5\pi
i}{4}}g\left(e^{-\frac{\pi i}{2}}\frac{hz^{\frac{1}{2}}}{\sqrt{2}}\right)$
$\displaystyle=2^{-\frac{5}{4}}z^{-\frac{1}{4}}e^{\frac{5\pi i}{4}}\cdot
2^{-\frac{1}{2}}e^{-\frac{h^{2}z}{8}}\left(e^{-\frac{\pi
i}{2}}\frac{hz^{\frac{1}{2}}}{\sqrt{2}}\right)^{\frac{1}{2}}K_{\frac{1}{4}}\left(e^{-\pi
i}\frac{h^{2}z}{8}\right)=-\frac{1}{4}e^{-\frac{h^{2}z}{8}}h^{\frac{1}{2}}K_{\frac{1}{4}}\left(e^{-\pi
i}\frac{h^{2}z}{8}\right).$

Thus, in the limit $t_{2}=0$ we find that

$\displaystyle\xi_{(1),1}^{R}(z,0)$ $\displaystyle=\pm
i\Psi_{11}\pi^{-\frac{1}{2}}z^{\frac{1}{2}}(6x_{1}^{2}+h)^{\frac{1}{2}}\left(\mathfrak{I}_{1}^{1}(z)+\mathfrak{I}_{1}^{2}(z)+\mathfrak{I}_{3}(z,0)\right)$
$\displaystyle=\pm
i2^{-\frac{5}{2}}\pi^{-\frac{1}{2}}z^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}h^{\frac{1}{2}}\left\\{K_{\frac{1}{4}}\left(e^{i\pi}\frac{h^{2}z}{8}\right)-K_{\frac{1}{4}}\left(e^{-i\pi}\frac{h^{2}z}{8}\right)+2^{\frac{1}{2}}K_{\frac{1}{4}}\left(\frac{h^{2}z}{8}\right)\right\\}$
$\displaystyle=\pm\pi^{\frac{1}{2}}z^{\frac{1}{2}}2^{-\frac{7}{2}}e^{-\frac{h^{2}z}{8}}h^{\frac{1}{2}}e^{-\frac{\pi
i}{8}}\left\\{H^{(2)}_{\frac{1}{4}}\left(e^{\frac{i\pi}{2}}\frac{h^{2}z}{8}\right)+e^{\frac{\pi
i}{4}}H^{(1)}_{\frac{1}{4}}\left(e^{-\frac{i\pi}{2}}\frac{h^{2}z}{8}\right)+2^{\frac{1}{2}}H^{(2)}_{\frac{1}{4}}\left(e^{-\frac{i\pi}{2}}\frac{h^{2}z}{8}\right)\right\\}$
$\displaystyle=\pm\pi^{\frac{1}{2}}2^{-\frac{3}{2}}z^{\frac{1}{2}}e^{-\frac{h^{2}z}{8}}h^{\frac{1}{2}}e^{-\frac{\pi
i}{8}}H^{(2)}_{\frac{1}{4}}\left(e^{\frac{i\pi}{2}}\frac{h^{2}z}{8}\right),$

which is exactly (modulo irrelevant signs) the solution at the coalescence
point as computed in the previous section. We leave as an exercise for the
reader to show that all other solutions $\xi_{(i),j}^{R/L}(z)$ converge to the
ones computed at the coalescence point.

##  6\. Second example of application of Theorem 4.1: Quantum cohomology of
the Grassmannian $\mathbb{G}_{2}(\mathbb{C}^{4})$ and $\Gamma$-conjecture

In this section we prove Theorem 6.2, which is Theorem 1.4 of the
Introduction. This is one of the main results of the paper. We also prove
Proposition 6.1, which we believe is important.

The problem is to compute the monodromy data for the Frobenius manifold known
as Quantum cohomology of the Grassmannian $\mathbb{G}_{2}(\mathbb{C}^{4})$.
This manifold has a locus of coalescent semisimple points, known as small
quantum cohomology. Hence, the computation is completely justified by our
Theorem 4.1. Our explicit computation, which yields Theorem 6.2 (Theorem 1.4),
seems to be missing from the literature. The result allows us to clarify and
verify, by analytic methods and in completely explicit way, the so called
$\Gamma$-conjecture (as formulated by the second author in [Dub98], then
refined191919The detailed comparison between the explicit computations of the
monodromy data for complex Grassmannians and the $\Gamma$-classes proposed in
[GGI16], will be the content of a forthcoming paper [CDG18]. in [Dub13]) in
the case the quantum cohomology of the Grassmannian
$\mathbb{G}_{2}(\mathbb{C}^{4})$.

###  6.1. Notations in Gromov-Witten Theory

Let $X$ be a smooth complex projective variety with odd-vanishing cohomology

$H^{2k+1}(X;\mathbb{C})=0,\quad k\geq 0.$

Let us fix a homogeneous basis $(T_{1},T_{2},\dots,T_{N})$ of
$H^{\bullet}(X;\mathbb{C})=\bigoplus_{k}H^{2k}(X;\mathbb{C})$ such that

  * •

$T_{1}=1$ is the unity of the cohomology ring;

  * •

$\deg T_{\alpha}=:2q_{\alpha}$;

  * •

$T_{2},\dots,T_{r}$ span $H^{2}(X;\mathbb{C})$.

We will denote by $\eta\colon H^{\bullet}(X;\mathbb{C})\times
H^{\bullet}(X;\mathbb{C})\to\mathbb{C}$ the Poincaré metric

$\eta(\xi,\zeta):=\int_{X}\xi\cup\zeta,\quad\quad\quad\eta_{\alpha\beta}:=\int_{X}T_{\alpha}\cup
T_{\beta}.$

We also introduce the _Novikov ring_
$\Lambda:=\mathbb{C}[\\![Q_{2},\dots,Q_{r}]\\!]$, and the symbol

$Q^{\beta}:=Q_{2}^{\int_{\beta}T_{2}}\cdot\dots\cdot
Q_{r}^{\int_{\beta}T_{r}}.$

Let $X_{g,n,\beta}$ be the moduli space of stable maps with target $X$, of
genus $g$, with $n$ distinct marked points and of degree $\beta\in
H_{2}(Z;\mathbb{Z})$. We will denote by

$\langle\tau_{d_{1}}\gamma_{1},\dots,\tau_{d_{n}}\gamma_{n}\rangle^{X}_{g,n,\beta}:=\int_{[X_{g,n,\beta}]^{\text{vir}}}\bigwedge_{i=1}^{n}\operatorname{ev}^{*}_{i}(\gamma_{i})\cup\psi_{i}^{d_{i}}$

the value of the _Gromov-Witten invariant_ (with _gravitational descendants_ ,
if some of the $d_{i}$’s is nonzero), where

  * •

$\gamma_{1},\dots,\gamma_{n}\in H^{\bullet}(X;\mathbb{C})$,

  * •

$(d_{1},\dots,d_{n})\in\mathbb{N}^{n}$,

  * •

$\psi_{1},\dots,\psi_{n}\in H^{2}(X_{g,n,\beta};\mathbb{Q})$ are the universal
cotangent line classes,

  * •

$\operatorname{ev}_{i}\colon X_{g,n,\beta}\to X$ is the evaluation map at the
$i$-th marked point,

  * •

$[X_{g,n,\beta}]^{\text{vir}}$ stands for the _virtual fundamental class_.
Recall that degree of the virtual cycle is equal to the virtual dimension
(over $\mathbb{R}$)

$\operatorname{vir\
dim}_{\mathbb{R}}X_{g,n,\beta}=2(1-g)\dim_{\mathbb{C}}X-2\int_{\beta}\omega_{X}+2(3g-3+n).$

It is convenient to collect Gromov-Witten invariants with descendants as
coefficients of a generating function, called _genus g gravitational Gromov-
Witten potential_ , or simply _genus g Free Energy_

$\mathcal{F}_{g}^{X}(\gamma):=\sum_{n=0}^{\infty}\sum_{\beta\in\operatorname{Eff}(X)}\frac{q^{\beta}}{n!}\langle\underbrace{\gamma,\dots,\gamma}_{n\text{
times}}\rangle_{g,n,\beta}^{X},$

the set $\operatorname{Eff}(X)\subseteq H_{2}(X;\mathbb{Z})$ being the set of
effective classes of $X$. Introducing (infinitely many) coordinates
$\mathbb{t}:=(t^{\alpha,p})_{\alpha,p}$

$\gamma=\sum_{\alpha,p}t^{\alpha,p}\tau_{p}T_{\alpha},$

the free energy $\mathcal{F}^{X}_{g}\in\Lambda[\\![{\mathbb{t}}]\\!]$ can be
seen as a function on the _large phase-space_ , and restricting the free
energy to the _small phase space_ (naturally identified with
$H^{\bullet}(X;\mathbb{C})$),

$F^{X}_{g}(t^{1,0},\dots,t^{N,0}):=\mathcal{F}^{X}_{g}(\mathbb{t})|_{t^{\alpha,p}=0,\
p>0},$

one obtains the generating function of the Gromov-Witten invariants of genus
$g$. It will also be convenient to introduce the _genus g correlation
functions_ defined by the derivatives

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{d_{1}}T_{\alpha_{1}},\dots,\tau_{d_{n}}T_{\alpha_{n}}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{g}:=\frac{\partial}{\partial
t^{\alpha_{1},d_{1}}}\dots\frac{\partial}{\partial
t^{\alpha_{n},d_{n}}}\mathcal{F}^{X}_{g}.$

Let $t^{\alpha}:=t^{\alpha,0}$. By the Divisor axiom, the genus $0$ Gromov-
Witten potential $F^{X}_{0}(t)$ can be seen as an element of the ring
$\mathbb{C}[\\![t^{1},Q_{2}e^{t^{2}},\dots,Q_{r}e^{t^{r}},t^{r+1},\dots,t^{N}]\\!]$:
in what follows we will be interested in the cases in which $F^{X}_{0}$ is the
series expansion of an analytic function, i.e.

$F^{X}_{0}\in\mathbb{C}\left\\{t^{1},Q_{2}e^{t^{2}},\dots,Q_{r}e^{t^{r}},t^{r+1},\dots,t^{N}\right\\}.$

Without loss of generality, we can put $q_{1}=q_{2}=\dots=q_{r}=1$, and
$F^{X}_{0}(t)$ defines an analytic function in an open neighborhood
$\Omega\subseteq H^{\bullet}(X;\mathbb{C})$ of the point

$t^{i}=0,\quad
i=1,r+1,\dots,N;\quad\quad\quad\operatorname{Re}t^{i}\to-\infty,\quad
i=2,3,\dots,r.$ (6.1)

The function $F^{X}_{0}$ is a solution of WDVV equations ([KM94], [Man99],
[Tia94], [Voi96]), and thus it defines an analytic Frobenius manifold
structure on $\Omega$. Using the canonical identifications of tangent spaces
$T_{p}\Omega\cong H^{\bullet}(X;\mathbb{C})\colon\partial_{t^{\alpha}}\mapsto
T_{\alpha}$, the unit vector field is $e=\partial_{t^{1}}\equiv 1$, and the
Euler vector field is

$E:=c_{1}(X)+\sum_{\alpha=1}^{N}\left(1-\frac{1}{2}\deg
T_{\alpha}\right)t^{\alpha}T_{\alpha}.$

The resulting Frobenius structure is called _quantum cohomology of_ $X$,
denoted $QH^{\bullet}(X)$. Notice that at the classical limit point (6.1) the
algebra structure on the tangent spaces coincides with the classical
cohomological algebra structure. Notice that, because of the Divisor axiom,
the Frobenius structure is $2\pi i$-periodic in the 2-nd cohomology
directions: the structure can be considered as defined on an open region of
the quotient $H^{\bullet}(X;\mathbb{C})/2\pi iH^{2}(X;\mathbb{Z})$.

Below, we focus on the small quantum cohomology of
$\mathbb{G}_{2}(\mathbb{C}^{4})$, namely the restriction to the locus
$H^{2}(\mathbb{G}_{2}(\mathbb{C}^{4});\mathbb{C})$, with coordinates
$(0,t^{2},0,...,0)$.

###  6.2. Small Quantum Cohomology of $\mathbb{G}_{2}(\mathbb{C}^{4})$

####  6.2.1. Generalities and proof of its Semisimplicity

For simplicity, let us use the notation
$\mathbb{G}:=\mathbb{G}_{2}(\mathbb{C}^{4})$. From the general theory of
Schubert Calculus formulated in the previous chapter, it is known that
$H^{\bullet}(\mathbb{G};\mathbb{C})$ is a complex vector space of dimension 6,
and a basis is given by _Schubert classes_ :

$\sigma_{0}:=1,\ \sigma_{1},\ \sigma_{2},\ \sigma_{1,1},\ \sigma_{2,1},\
\sigma_{2,2}$

where $\sigma_{\lambda}$ is a generator of
$H^{2|\lambda|}(\mathbb{G};\mathbb{C})$. By posing

$v_{1}:=\sigma_{0},\ v_{2}:=\sigma_{1},\ v_{3}:=\sigma_{2},\
v_{4}:=\sigma_{1,1},\ v_{5}:=\sigma_{2,1},\ v_{6}:=\sigma_{2,2},$

we will denote by $t^{i}$ the coordinates with respect to $v_{i}$. The
coordinates in the small quantum cohomology are

$t=(0,t^{2},0,...,0).$

By Pieri–Bertram and Giambelli formulas one finds that the matrix of the
Poincaré pairing

$\eta(\alpha,\beta):=\int_{\mathbb{G}}\alpha\wedge\beta$

with respect to the above basis is given by

$\eta=\begin{pmatrix}0&0&0&0&0&c\\\ 0&0&0&0&c&0\\\ 0&0&c&0&0&0\\\
0&0&0&c&0&0\\\ 0&c&0&0&0&0\\\ c&0&0&0&0&0\end{pmatrix},\quad
c:=\int_{\mathbb{G}}\sigma_{2,2}.$

Using Pieri–Bertram formula we deduce that the matrix of the operator of
multiplication by $\lambda\sigma_{1}+\mu\sigma_{1,1}$ is

$\begin{pmatrix}0&0&\mu q&0&\lambda q&0\\\ \lambda&0&0&0&\mu q&\lambda q\\\
0&\lambda&0&0&0&\mu q\\\ \mu&\lambda&0&0&0&0\\\ 0&\mu&\lambda&\lambda&0&0\\\
0&0&0&\mu&\lambda&0\end{pmatrix},\quad\quad\quad q:=e^{t^{2}}.$ (6.2)

The discriminant of the characteristic polynomial of this matrix is

$16777216~{}\lambda^{4}\mu^{2}q^{8}(\lambda^{4}+\mu^{4}q)^{6}$

and so, if $\lambda\neq 0,\mu\neq 0$ and $\lambda^{4}+q\mu^{4}\neq 0$, its
eigenvalues are pairwise distinct. This is a sufficient condition to state
that the quantum cohomology of $\mathbb{G}$ is semisimple.
Notice that the value at the point $p$ of coordinates $(0,t^{2},0,\dots,0)$ of
the Euler field of quantum cohomology $QH^{\bullet}(\mathbb{G})$ is 202020We
identify $T_{p}H^{\bullet}(\mathbb{G})$ with $H^{\bullet}(\mathbb{G})$ in the
canonical way. given by the first Chern class $c_{1}(\mathbb{G})=4\sigma_{1}$:

$E|_{p}=4\frac{\partial}{\partial t^{2}}\equiv 4\sigma_{1}.$

The matrix $\mathcal{U}$ of multiplication by $E$ at the point $p$ is given by
posing $\lambda=4$, $\mu=0$ in (6.2):

$\mathcal{U}(0,t^{2},0,\dots,0)\equiv
4~{}\mathcal{C}_{2}(0,t^{2},0,\dots,0)=\begin{pmatrix}0&0&0&0&4q&0\\\
4&0&0&0&0&4q\\\ 0&4&0&0&0&0\\\ 0&4&0&0&0&0\\\ 0&0&4&4&0&0\\\
0&0&0&0&4&0\end{pmatrix}.$

The characteristic polynomial is $p(z)=z^{6}-1024qz^{2}$, so that 0 is an
eigenvalue with multiplicity 2. Therefore, the semisimple points with
coordinates $(0,t^{2},0,\dots,0)$ are _semisimple coalescence points in the
bifurcation set_.

####  6.2.2. Idempotents at the points $(0,t^{2},0,\dots,0)$

The multiplication by $\sigma_{1}+\sigma_{1,1}$ has pairwise distinct
eigenvalues, at least at points for which $t^{2}\neq i\pi(2k+1)$. Putting
$\lambda=\mu=1$ in (6.2), we deduce that the characteristic polynomial of this
operator is

$p(z)=(q+z^{2})(-4q+q^{2}-8qz-2qz^{2}+z^{4}).$

So the six eigenvalues are

$\displaystyle iq^{\frac{1}{2}},\quad-iq^{\frac{1}{2}},$
$\displaystyle\varepsilon_{1}:=-i\sqrt{2}q^{\frac{1}{4}}-q^{\frac{1}{2}},\quad\varepsilon_{2}:=i\sqrt{2}q^{\frac{1}{4}}-q^{\frac{1}{2}},\quad\varepsilon_{3}:=-\sqrt{2}q^{\frac{1}{4}}+q^{\frac{1}{2}},\quad\varepsilon_{4}:=\sqrt{2}q^{\frac{1}{4}}+q^{\frac{1}{2}},$

and the corresponding eigenvectors are

$\displaystyle\pi_{1}:=$ $\displaystyle-q-
iq^{\frac{1}{2}}\sigma_{2}+iq^{\frac{1}{2}}\sigma_{1,1}+\sigma_{2,2},\quad\quad\pi_{2}:=-q+iq^{\frac{1}{2}}\sigma_{2}-iq^{\frac{1}{2}}\sigma_{1,1}+\sigma_{2,2},$
$\displaystyle\pi_{2+i}:=$
$\displaystyle(q^{2}+q\varepsilon_{i}^{2})+(-q^{2}+2q\varepsilon_{i}+q\varepsilon_{i}^{2})\sigma_{1}+(2q+2q\varepsilon_{i})\sigma_{2}+(2q+2q\varepsilon_{i})\sigma_{1,1}$
$\displaystyle+(-2q-q\varepsilon_{i}+\varepsilon_{i}^{3})\sigma_{2,1}+(q+\varepsilon_{i}^{2})\sigma_{2,2}.$

Then,

$\pi_{i}\cdot\pi_{j}=0\ \ \text{if }i\neq
j,\quad\pi_{i}^{2}=\lambda_{i}\pi_{i}\ \ \text{where }\lambda_{i}>0;$

as a consequence, these vectors are orthogonal since, for $i\neq j$,
$\eta(\pi_{i},\pi_{j})=\eta(\pi_{i}\cdot\pi_{j},1)=\eta(0,1)=0$. Introducing
the normalized eigenvectors

$f_{i}:=\frac{\pi_{i}}{\eta(\pi_{i},\pi_{i})^{\frac{1}{2}}}$

we obtain an orthonormal frame of normalized idempotent vectors, for any
choice of the sign of the square roots.
Let us now introduce a matrix $\Psi=(\psi_{ij})$ such that

$\frac{\partial}{\partial
t_{\alpha}}=\sum_{i}\psi_{i\alpha}f_{i},\quad\quad\alpha=1,2,...,n.$

Note that necessarily we have

$\Psi^{T}\Psi=\eta,\quad\psi_{i1}=\frac{\eta(\pi_{i},1)}{\eta(\pi_{i},\pi_{i})^{\frac{1}{2}}}.$

After some computations, we obtain

$\Psi=\frac{c^{\frac{1}{2}}}{2}\begin{pmatrix}-iq^{-\frac{1}{2}}&0&-1&1&0&iq^{\frac{1}{2}}\\\
-iq^{-\frac{1}{2}}&0&1&-1&0&iq^{\frac{1}{2}}\\\
\frac{1}{\sqrt{2}q^{\frac{1}{2}}}&-\frac{i}{q^{\frac{1}{4}}}&-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&iq^{\frac{1}{4}}&\frac{q^{\frac{1}{2}}}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}q^{\frac{1}{2}}}&\frac{i}{q^{\frac{1}{4}}}&-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&-iq^{\frac{1}{4}}&\frac{q^{\frac{1}{2}}}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}q^{\frac{1}{2}}}&-\frac{1}{q^{\frac{1}{4}}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&-q^{\frac{1}{4}}&\frac{q^{\frac{1}{2}}}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}q^{\frac{1}{2}}}&\frac{1}{q^{\frac{1}{4}}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&q^{\frac{1}{4}}&\frac{q^{\frac{1}{2}}}{\sqrt{2}}\\\
\end{pmatrix}.$

This matrix diagonalizes $\mathcal{U}$ as follows

$\displaystyle U:=\Psi\mathcal{U}\Psi^{-1}$
$\displaystyle=(\Psi^{T})^{-1}\widehat{\mathcal{U}}\Psi^{T}=$
$\displaystyle=\begin{pmatrix}u_{1}&&&&&\\\ &u_{2}&&&&\\\ &&u_{3}&&&\\\
&&&u_{4}&&\\\ &&&&u_{5}&\\\ &&&&&u_{6}\end{pmatrix}$
$\displaystyle=4\sqrt{2}q^{\frac{1}{4}}\left(\begin{array}[]{cccccc}0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&-i&0&0&0\\\ 0&0&0&i&0&0\\\ 0&0&0&0&-1&0\\\ 0&0&0&0&0&1\\\
\end{array}\right)$

The eigenvalues $u_{i}$ stand for $u_{i}(0,t^{2},..,0)$. Note that

$u_{i}(0,t^{2},0,...,0)=q^{\frac{1}{4}}u_{i}(0,0,...,0)=e^{\frac{t^{2}}{4}}u_{i}(0,0,...,0).$
(6.3)

####  6.2.3. Differential system expressing the Flatness of the Deformed
Connection

The matrix $\mu$ is given by

$\mu=\text{diag}\Bigl{(}-2,-1,0,0,1,2\Bigr{)},\quad\quad\hbox{ with
eigenvalues }\mu_{\alpha}=\frac{\deg(\partial/\partial_{\alpha})-4}{2},\quad
1\leq\alpha\leq 6.$ (6.4)

Consider the system (2.1), rewritten as follows:

$\displaystyle\partial_{z}\xi=(\widehat{\mathcal{U}}-\frac{1}{z}\mu)~{}\xi$
(6.5) $\displaystyle\partial_{2}\xi=z~{}\widehat{\mathcal{C}_{2}}~{}\xi$ (6.6)

where $\xi$ is a column vector, whose components are
$\xi_{i}=\partial_{i}\tilde{t}(t,z)$ (derivatives of a deformed flat
coordinate), and

$\widehat{\mathcal{U}}:=\eta~{}\mathcal{U}~{}\eta^{-1}=\begin{pmatrix}0&4&0&0&0&0\\\
0&0&4&4&0&0\\\ 0&0&0&0&4&0\\\ 0&0&0&0&4&0\\\ 4q&0&0&0&0&4\\\
0&4q&0&0&0&0\end{pmatrix},\quad\quad\widehat{\mathcal{C}_{2}}\equiv\frac{1}{4}~{}\widehat{\mathcal{U}}.$

Introducing a new function $\phi$ defined by

$\phi(t,z):=\frac{\xi_{1}(t,z)}{z^{2}}$

the first equation of the system becomes a single linear differential equation

$z^{4}\partial_{z}^{5}\phi+10z^{3}\partial_{z}^{4}\phi+25z^{2}\partial_{z}^{3}\phi+15z\partial_{z}^{2}\phi+(1-1024qz^{4})\partial_{z}\phi-2048qz^{3}\phi=0$
(6.7)

and the solution can be reconstructed from

$\displaystyle\xi_{1}$ $\displaystyle=z^{2}\phi$ $\displaystyle\xi_{2}$
$\displaystyle=\frac{1}{4}z^{2}\partial_{z}\phi$ $\displaystyle\xi_{3}$
$\displaystyle=\frac{1}{32}(z\partial_{z}\phi+z^{2}\partial_{z}^{2}\phi)+h$
$\displaystyle\xi_{4}$
$\displaystyle=\frac{1}{32}(z\partial_{z}\phi+z^{2}\partial_{z}^{2}\phi)-h$
(6.8) $\displaystyle\xi_{5}$
$\displaystyle=\frac{1}{128}(\partial_{z}\phi+3z\partial_{z}^{2}\phi+z^{2}\partial_{z}^{3}\phi)$
$\displaystyle\xi_{6}$
$\displaystyle=\frac{1}{512}(-512qz^{2}\phi+\frac{1}{z}\partial_{z}\phi+7\partial_{z}^{2}\phi+6z\partial_{z}^{3}\phi+z^{2}\partial_{z}^{4}\phi)$

with a constant $h$.

######  Remark 6.1.

The third and the fourth equalities (6.2.3) follow from the fact that

$\xi_{3}+\xi_{4}=\frac{1}{16}(z\partial_{z}\phi+z^{2}\partial_{z}^{2}\phi),\quad\partial_{z}(\xi_{3}-\xi_{4})=0,\quad\partial_{2}(\xi_{3}-\xi_{4})=0$

so that $\xi_{3}-\xi_{4}=2h$ is a constant.

From system (6.6) it follows that

$\partial_{2}\phi=\frac{z}{4}\partial_{z}\phi$

which implies the following functional form:

$\phi(t_{2},z)=\Phi\left(zq^{\frac{1}{4}}\right).$

As a consequence, our problem (6.7) reduces to the solution of a single scalar
ordinary differential equation for a function $\Phi(w)$, $w=zq^{\frac{1}{4}}$:

$w^{4}\Phi^{(5)}+10w^{3}\Phi^{(4)}+25w^{2}\Phi^{(3)}+15w\Phi^{\prime\prime}+(1-1024w^{4})\Phi^{\prime}-2048w^{3}\Phi=0.$

Multiplying by $w\in\mathbb{C}^{*}$, we can rewrite this equation in a more
compact form:

$\Theta^{5}\Phi-1024w^{4}\Theta\Phi-2048w^{4}\Phi=0$ (6.9)

where $\Theta$ is the Euler’s differential operator $w\frac{d}{dw}$. Moreover,
defining $\varpi:=4w^{4}$, and writing $\Phi(w)=\tilde{\Phi}(\varpi)$, we can
rewrite the equation in the form

$\Theta^{5}_{\varpi}\tilde{\Phi}-\varpi\Theta_{\varpi}\tilde{\Phi}-\frac{1}{2}\varpi\tilde{\Phi}=0\quad\quad\quad\Theta_{\varpi}:=\varpi\frac{d}{d\varpi}=\frac{1}{4}\Theta_{w}.$

####  6.2.4. Expected Asymptotic Expansions

Let $\Xi$ be a fundamental matrix solution of system (6.5), and let $Y$ be
defined by

$\Xi=\eta\Psi^{-1}Y.$ (6.10)

Then, $Y$ is a fundamental solution of system (2.17). The asymptotic theory
for such $Y$’s has been explained in Section 2.3, and Theorem 2.9 applies. To
the formal solution (2.18) it corresponds a formal matrix solution

$\Xi_{\rm formal}=\eta\Psi^{-1}G(z)^{-1}e^{zU}.$

To the fundamental solutions $Y_{\rm left/right}$, there correspond solutions
$\Xi_{\rm left/right}$. For fixed $t^{2}$, then $\Xi_{\rm
left/right}(t^{2},z)\equiv\Xi_{\rm left/right}(e^{\frac{t^{2}}{4}}z)$ has the
following asymptotic expansion for $z\to\infty$

$\Xi_{\rm
left/right}(t^{2},z)=\eta\Psi^{-1}\left(1+O\left(\frac{1}{z}\right)\right)e^{zU}=$
$=\frac{c^{\frac{1}{2}}}{2}\left(I+O\left(\frac{1}{z}\right)\right)\left(\begin{array}[]{cccccc}-\frac{ie^{zu_{1}}}{q^{\frac{1}{2}}}&-\frac{ie^{zu_{2}}}{q^{\frac{1}{2}}}&\frac{e^{zu_{3}}}{\sqrt{2}q^{\frac{1}{2}}}&\frac{e^{zu_{4}}}{\sqrt{2}q^{\frac{1}{2}}}&\frac{e^{zu_{5}}}{\sqrt{2}q^{\frac{1}{2}}}&\frac{e^{zu_{6}}}{\sqrt{2}q^{\frac{1}{2}}}\\\
0&0&-\frac{ie^{zu_{3}}}{q^{\frac{1}{4}}}&\frac{ie^{zu_{4}}}{q^{\frac{1}{4}}}&-\frac{e^{zu_{5}}}{q^{\frac{1}{4}}}&\frac{e^{zu_{6}}}{q^{\frac{1}{4}}}\\\
-e^{zu_{1}}&e^{zu_{2}}&-\frac{e^{zu_{3}}}{\sqrt{2}}&-\frac{e^{zu_{4}}}{\sqrt{2}}&\frac{e^{zu_{5}}}{\sqrt{2}}&\frac{e^{zu_{6}}}{\sqrt{2}}\\\
e^{zu_{1}}&-e^{zu_{2}}&-\frac{e^{zu_{3}}}{\sqrt{2}}&-\frac{e^{zu_{4}}}{\sqrt{2}}&\frac{e^{zu_{5}}}{\sqrt{2}}&\frac{e^{zu_{6}}}{\sqrt{2}}\\\
0&0&ie^{zu_{3}}q^{\frac{1}{4}}&-ie^{zu_{4}}q^{\frac{1}{4}}&-e^{zu_{5}}q^{\frac{1}{4}}&e^{zu_{6}}q^{\frac{1}{4}}\\\
ie^{zu_{1}}q^{\frac{1}{2}}&ie^{zu_{2}}q^{\frac{1}{2}}&\frac{e^{zu_{3}}q^{\frac{1}{2}}}{\sqrt{2}}&\frac{e^{zu_{4}}q^{\frac{1}{2}}}{\sqrt{2}}&\frac{e^{zu_{5}}q^{\frac{1}{2}}}{\sqrt{2}}&\frac{e^{zu_{6}}q^{\frac{1}{2}}}{\sqrt{2}}\\\
\end{array}\right).$ (6.11)

The first row of the above matrix gives the asymptotics of $\phi(z,t^{2})$.
The correct value of $h$ in (6.2.3) must be determined in order to match with
the asymptotics of the third and fourth rows of (6.11). We find

$h=-\frac{c^{\frac{1}{2}}}{2},\quad\text{for the first column,}$ (6.12)
$h=\frac{c^{\frac{1}{2}}}{2},\quad\text{for the second column,}$ (6.13)

$h=0,\quad\text{for the remaining columns.}$ (6.14)

Indeed, for $\phi$ corresponding to the first two columns we have respectively

$\phi(z,t^{2})=-\frac{c^{\frac{1}{2}}}{2}\frac{ie^{zu_{1}}}{z^{2}q^{\frac{1}{2}}}\left(1+O\left(\frac{1}{z}\right)\right)\quad\text{or}\quad\phi(z,t^{2})=-\frac{c^{\frac{1}{2}}}{2}\frac{ie^{zu_{2}}}{z^{2}q^{\frac{1}{2}}}\left(1+O\left(\frac{1}{z}\right)\right).$

Since $u_{1}=u_{2}=0$, the above expressions become

$\phi(z,t^{2})=-\frac{c^{\frac{1}{2}}}{2}\frac{i}{z^{2}q^{\frac{1}{2}}}\left(1+O\left(\frac{1}{z}\right)\right).$

Then

$\frac{1}{32}(z\partial_{z}\phi+z^{2}\partial^{2}_{z}\phi)=O\left(\frac{1}{z}\right).$

Comparing with the matrix elements $(3,1)$, $(4,1)$ and $(3,2)$, $(4,2)$ of
(6.11) respectively, we obtain (6.12) and (6.13). For the remaining columns we
proceed in the same way and find (6.14).

###  6.3. Solutions of the Differential Equation

####  6.3.1. Solutions of $\Theta^{5}\Phi-1024w^{4}\Theta\Phi-2048w^{4}\Phi=0$
and their asymptotics

The equation

$\Theta^{5}_{\varpi}\tilde{\Phi}-\varpi\Theta_{\varpi}\tilde{\Phi}-\frac{1}{2}\varpi\tilde{\Phi}=0,$

belongs to the class of generalized hypergeometric differential equations (see
[Luk69], [AS70], [PK01], [OLBC10] and references therein). By applying the
Mellin transform $\mathfrak{M}$ to it, we obtain the finite difference
equation

$s^{5}\tilde{\tau}(s)=\left(s+\frac{1}{2}\right)\tilde{\tau}(s+1),\quad\quad\tilde{\tau}(s):=\mathfrak{M}(\tilde{\Phi})(s):=\int_{0}^{\infty}\tilde{\Phi}(t)t^{s-1}dt,$
(6.15)

whose solutions are of the form

$\tilde{\tau}(s)=\frac{\Gamma(s)^{5}}{\Gamma\left(s+\frac{1}{2}\right)}\psi(s),\quad\psi(s)=\psi(s+1).$

Hence, we expect that solutions of (6.9) are of the form

$\Phi(w)=\frac{1}{2\pi
i}\int_{\Lambda}\frac{\Gamma(s)^{5}}{\Gamma\left(s+\frac{1}{2}\right)}\psi(s)4^{-s}w^{-4s}ds,$

for suitable chosen paths of integration $\Lambda$. Actually, the lemma below
holds:

######  Lemma 6.1.

The following functions are solutions of the generalized hypergeometric
equation (6.9):

  * •

the function

$\Phi_{1}(w):=\frac{1}{2\pi
i}\int_{\Lambda_{1}}\frac{\Gamma(s)^{5}}{\Gamma\left(s+\frac{1}{2}\right)}4^{-s}w^{-4s}ds,$

defined for $-\frac{\pi}{2}<\arg w<\frac{\pi}{2}$, and where $\Lambda_{1}$ is
any line in the complex plane from the point $c-i\infty$ to $c+i\infty$ for
any $0<c$;

  * •

the function

$\Phi_{2}(w):=\frac{1}{2\pi
i}\int_{\Lambda_{2}}\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{i\pi
s}4^{-s}w^{-4s}ds,$

defined for $-\frac{\pi}{2}<\arg w<\pi$, and where $\Lambda_{2}$ is any line
in the complex plane from the point $c-i\infty$ to $c+i\infty$ for any
$0<c<\frac{1}{2}$.

Before giving the proof of this Lemma, we recall the following well-known
useful results ( see e.g. [WW96], [OLBC10], [Luk69])

######  Theorem 6.1 (Stirling).

The following estimate holds

$\log\Gamma(s)=\left(s-\frac{1}{2}\right)\log
s-s+\frac{1}{2}\log(2\pi)+O\left(\frac{1}{|s|}\right)$

for $s\to\infty$ and $|\arg s|<\pi$, and where $\log$ stands for the principal
branch of the complex logarithm.

######  Corollary 6.1.

For $|t|\to+\infty$ we have

$|\Gamma(\sigma+it)|=\sqrt{2\pi}|t|^{\sigma-\frac{1}{2}}e^{-\frac{\pi}{2}|t|}\left(1+O\left(\frac{1}{|t|}\right)\right),$

uniformly on any strip of the complex plane
$\sigma_{1}\leq\sigma\leq\sigma_{2}$.

###### Proof of the Lemma 6.1.

First of all let us prove that the functions $\Phi_{1}$, $\Phi_{2}$ are well
defined on the above regions. We denote by $\mathcal{I}_{1}$ and
$\mathcal{I}_{2}$ the integrands in $\Phi_{1}$ and $\Phi_{2}$ respectively,
and $s=c+it$. By Corollary 6.1 we have that

$|\mathcal{I}_{1}|\sim(2\pi)^{2}|t|^{5(c-\frac{1}{2})}e^{-\frac{5\pi}{2}|t|}|t|^{-c}e^{\frac{\pi}{2}|t|}e^{-c\log
4}e^{-4c\log|w|+4t\arg w}.$

The dominant part is

$e^{-\frac{5\pi}{2}|t|}e^{\frac{\pi}{2}|t|}e^{4t\arg w}.$

In order to have $|\mathcal{I}_{1}|\to 0$ for $t\to+\infty$ we must impose

$-\frac{5\pi}{2}+\frac{\pi}{2}+4\arg w<0,\quad\text{i.e. }\arg
w<\frac{\pi}{2};$

analogously, for $t\to-\infty$ we have to impose

$\frac{5\pi}{2}-\frac{\pi}{2}+4\arg w>0,\quad\text{i.e. }\arg
w>-\frac{\pi}{2}.$

Analogously, from Corollary 6.1 we deduce that

$|\mathcal{I}_{2}|\sim(2\pi)^{3}|t|^{5(c-\frac{1}{2})}e^{-\frac{5\pi}{2}|t|}|-t|^{-c}e^{-\frac{\pi}{2}|-t|}e^{-\pi
t}e^{-c\log 4}e^{-4c\log|w|+4t\arg w},$

and now the dominant part is

$e^{-\frac{5\pi}{2}|t|}e^{-\frac{\pi}{2}|-t|}e^{-\pi t}e^{4t\arg w}.$

In order to have $|\mathcal{I}_{2}|\to 0$ for $t\to\pm\infty$, we find

$-\frac{5\pi}{2}-\frac{\pi}{2}-\pi+4\arg w<0,\quad\text{i.e. }\arg w<\pi,$
$\frac{5\pi}{2}+\frac{\pi}{2}-\pi+4\arg w>0,\quad\text{i.e. }\arg
w<-\frac{\pi}{2}.$

Let us now prove that $\Phi_{1}$ and $\Phi_{2}$ are solutions of equation
(6.9). We have that

$\displaystyle\Theta^{5}\Phi_{1}(w)$ $\displaystyle=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{1}}-s^{5}\frac{\Gamma(s)^{5}}{\Gamma\left(s+\frac{1}{2}\right)}4^{-s}w^{-4s}ds$
$\displaystyle=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{1}}-\left(s+\frac{1}{2}\right)\frac{\Gamma(s+1)^{5}}{\Gamma\left(s+\frac{3}{2}\right)}4^{-s}w^{-4s}ds$

because $z\Gamma(z)=\Gamma(z+1)$. Changing variable $t:=s+1$, and consequently
shifting the line of integration $\Lambda_{1}$ to $\Lambda_{1}+1$, we have

$\displaystyle\Theta^{5}\Phi_{1}(w)$ $\displaystyle=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{1}+1}-\left(t-\frac{1}{2}\right)\frac{\Gamma(t)^{5}}{\Gamma\left(t+\frac{1}{2}\right)}4^{-t}\cdot
4w^{-4(t-1)}dt$ $\displaystyle=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{1}+1}\left(-4t\right)\frac{\Gamma(t)^{5}}{\Gamma\left(t+\frac{1}{2}\right)}4^{-t}w^{-4(t-1)}dt+\frac{2\cdot
4^{5}}{2\pi
i}\int_{\Lambda_{1}+1}\frac{\Gamma(t)^{5}}{\Gamma\left(t+\frac{1}{2}\right)}4^{-t}w^{-4(t-1)}dt.$

Note that in the region between $\Lambda_{1}$ and $\Lambda_{1}+1$ the two last
integrands have no poles; so $\int_{\Lambda_{1}+1}=\int_{\Lambda_{1}}$ by
Cauchy Theorem. This shows that

$\Theta^{5}\Phi_{1}=4^{5}w^{4}\Theta\Phi_{1}+2\cdot 4^{5}w^{4}\Phi_{1}.$

Analogously we have

$\displaystyle\Theta^{5}\Phi_{2}$ $\displaystyle=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{2}}-s^{5}\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{i\pi
s}4^{-s}w^{-4s}ds$ $\displaystyle=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{2}}-\left(s+\frac{1}{2}\right)\Gamma(s+1)^{5}\Gamma\left(-\frac{1}{2}-s\right)e^{i\pi(s+1)}4^{-s}w^{-4s}ds$

where the second identity follows from $z\Gamma(z)=\Gamma(z+1)$. Note that the
integrand function is holomorphic at $s=-\frac{1}{2}$: indeed we have

$\lim_{s\to{-\frac{1}{2}}}\left(s+\frac{1}{2}\right)\Gamma\left(-\frac{1}{2}-s\right)=-1.$

So in the strip of the complex plane $-1<\text{Re }s<\frac{1}{2}$ there are no
poles, and by Cauchy Theorem, we can change path of integration by shifting
$\Lambda_{2}$ to $\Lambda_{2}-1$:

$\Theta^{5}\Phi_{2}=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{2}-1}-s^{5}\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{i\pi
s}4^{-s}w^{-4s}ds.$

Posing now $t=s+1$, we can rewrite

$\Theta^{5}\Phi_{2}=\frac{4^{5}}{2\pi
i}\int_{\Lambda_{2}}-\left(t-\frac{1}{2}\right)\Gamma(t)^{5}\Gamma\left(\frac{1}{2}-t\right)e^{i\pi
t}4^{-(t-1)}w^{-4(t-1)}dt=4^{5}w^{4}\Theta\Phi_{2}+2\cdot 4^{5}w^{4}\Phi_{2}.$

This shows that effectively $\Phi_{1}$ and $\Phi_{2}$ are solutions.∎

Note that solutions $\Phi_{1}$ and $\Phi_{2}$ are $\mathbb{C}$-linearly
independent, since their Mellin transforms are. However we have the following
identities

######  Lemma 6.2.

By analytic continuation of the functions $\Phi_{1}$ and $\Phi_{2}$, we have

$\displaystyle\Phi_{2}(we^{i\frac{\pi}{2}})$
$\displaystyle=2\pi\Phi_{1}(w)-\Phi_{2}(w)$ (6.16)
$\displaystyle\Phi_{2}(we^{-i\frac{\pi}{2}})$
$\displaystyle=2\pi\Phi_{1}(we^{-i\frac{\pi}{2}})-\Phi_{2}(w)$ (6.17)
$\displaystyle\Phi_{2}(we^{i\frac{\pi}{2}})$
$\displaystyle=2\pi\Phi_{1}(w)+\Phi_{2}(we^{-i\frac{\pi}{2}})-2\pi\Phi_{1}(we^{-i\frac{\pi}{2}})$
(6.18)

###### Proof.

We have that

$\Gamma\left(\frac{1}{2}+s\right)\Gamma\left(\frac{1}{2}-s\right)=\frac{\pi}{\sin\left(\pi\left(\frac{1}{2}+s\right)\right)}=\frac{2\pi
e^{\pm i\pi s}}{e^{\pm 2i\pi s}+1}$

for a coherent choice of the sign. So

$e^{\pm 2i\pi s}=\frac{2\pi e^{\pm i\pi
s}}{\Gamma\left(\frac{1}{2}+s\right)\Gamma\left(\frac{1}{2}-s\right)}-1.$

First let us choose the identity with $(-)$: we find that

$\displaystyle\Phi_{2}(we^{i\frac{\pi}{2}})$ $\displaystyle=\frac{1}{2\pi
i}\int_{\Lambda_{2}}\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{i\pi
s}\left(\frac{2\pi e^{-i\pi
s}}{\Gamma\left(\frac{1}{2}+s\right)\Gamma\left(\frac{1}{2}-s\right)}-1\right)4^{-s}w^{-4s}ds$
$\displaystyle=2\pi\Phi_{1}(w)-\Phi_{2}(w),$

which is the first identity. The second one can be deduce analogously using
the formula with (+) sign. Finally the third identity is the difference of
(6.16) and (6.17). ∎

The following lemma gives the asymptotic expansions of $\Phi_{1}(w)$ and
$\Phi_{1}(w)$ when $w\to\infty$ in certain sectors. Later, in Lemma 6.4 we
will be able to prove the same asymptotics in larger sectors.

######  Lemma 6.3.

The following asymptotic expansions hold for $w\to\infty$ in the sectors
specified below:

$\displaystyle\Phi_{1}(w)=(2\pi)^{\frac{3}{2}}\frac{e^{-4\sqrt{2}w}}{4w^{2}}\left(1+O\left(\frac{1}{w}\right)\right),\quad\text{
for }-\frac{\pi}{2}<\arg w<\frac{\pi}{2},$
$\displaystyle\Phi_{2}(w)=\frac{i\pi^{\frac{5}{2}}}{2w^{2}}\left(1+O\left(\frac{1}{w}\right)\right),\quad\text{
for }-\frac{\pi}{2}<\arg w<\pi.$

The sectors where the asymptotics is valid will be enlarged in Lemma 6.4.

###### Proof.

By Stirling’s formula we have that

$\Phi_{1}(w)=\frac{(2\pi)^{2}}{2\pi i}\int_{\Lambda_{1}}e^{\phi(s)}ds,$

where

$\phi(s)=-5s+5\left(s-\frac{1}{2}\right)\log
s+s+\frac{1}{2}-s\log\left(s+\frac{1}{2}\right)-s\log 4-4s\log
w+O\left(\frac{1}{|s|}\right)$

for $s\to\infty$ and where $\log$ stands for the principal branch of
logarithm. Let us find stationary points of $\phi(s)$ for large values of
$|s|,|w|$. The derivative $\phi^{\prime}$ is

$\phi^{\prime}(s)=-4+5\log
s+\frac{10s-5}{2s}-\log\left(s+\frac{1}{2}\right)-\frac{s}{s+\frac{1}{2}}-\log
4-4\log w+O\left(\frac{1}{|s|}\right).$

For $|s|$ large enough, we have

$\frac{10s-5}{2s}\sim 5-\frac{5}{2s},\quad\quad\frac{s}{s+\frac{1}{2}}\sim
1-\frac{1}{2s},\quad\quad\log\left(s+\frac{1}{2}\right)=\log
s+\log\left(1+\frac{1}{2s}\right)\sim\log s+\frac{1}{2s}.$

Substituting these identities in $\phi^{\prime}$, we find that the critical
point $\bar{s}(w)$ in functions of $w$ (for $|w|$ large)

$\bar{s}(w)=\sqrt{2}w+\frac{5}{8}+O\left(\frac{1}{|w|}\right).$ (6.19)

Note that for $-\frac{\pi}{2}<\arg w<\frac{\pi}{2}$, the point $\bar{s}(w)$ is
in the half-plane $\text{Re }s>0$, region in which there are no poles of the
integrand functions in $\Phi_{1}$. So we can shift the line $\Lambda_{1}$ in
order that it passes through $\bar{s}$. In this way we obtain

$\Phi_{1}(w)=\frac{(2\pi)^{2}}{2\pi
i}e^{\phi(\bar{s})}\int_{\Lambda_{1}}e^{\phi(s)-\phi(\bar{s})}ds~{}\sim\frac{(2\pi)^{2}}{2\pi
i}e^{\phi(\bar{s})}\int_{\Lambda_{1}}e^{\frac{\phi^{\prime\prime}(\bar{s})}{2}(s-\bar{s})^{2}}ds.$

The computation of this Gaussian integral shows that

$\Phi_{1}(w)\sim\frac{(2\pi)^{2}}{2\pi}e^{\phi(\bar{s})}\frac{\sqrt{2\pi}}{\sqrt{\phi^{\prime\prime}(\bar{s})}}=(2\pi)^{\frac{3}{2}}\frac{e^{\phi(\bar{s})}}{\sqrt{\phi^{\prime\prime}(\bar{s})}},$

where $\text{Re }\sqrt{\phi^{\prime\prime}(\bar{s})}>0$. An explicit series
expansion shows that

$\phi(\bar{s}(w))\sim-4\sqrt{2}w-\frac{5}{2}\log w-\frac{5}{8}\log
4+O\left(\frac{1}{|w|}\right),$

whereas

$\phi^{\prime\prime}(\bar{s}(w))\sim\frac{2\sqrt{2}}{w}+O\left(\frac{1}{|w|^{3}}\right)$

and from this we deduce that

$\Phi_{1}(w)\sim(2\pi)^{\frac{3}{2}}\frac{e^{-4\sqrt{2}w}}{4w^{2}}\left(1+O\left(\frac{1}{w}\right)\right).$

Let us now focus on $\Phi_{2}(w)$. From Theorem 6.1 we deduce that

$\Gamma(-s)=e^{-(s+\frac{1}{2})\log s}e^{-i\pi
s}e^{s}(-i\sqrt{2\pi})\left(1+O\left(\frac{1}{|s|}\right)\right)$

for $s\to\infty$ and $s\notin\mathbb{R}_{+}$. So,

$\Phi_{2}(w)=\frac{(2\pi)^{3}}{2\pi i}\int_{\Lambda_{2}}e^{\phi(s)}ds,$

where

$\phi(s)=5\left(s-\frac{1}{2}\right)\log
s-5s-s\log\left(s-\frac{1}{2}\right)+s-\frac{1}{2}-s\log 4-4s\log
w+O\left(\frac{1}{w}\right),$

for $w\to\infty$. By computations analogous to those of the previous case, we
find that $\phi$ has a critical point at

$\bar{s}(w)=\sqrt{2}w+\frac{5}{4\sqrt{2}}+O\left(\frac{1}{w}\right)$

for large values of $|w|$. Note explicitly that for $-\frac{\pi}{2}<\arg
w<\frac{\pi}{2}$ this critical point is in the half-plane $\text{Re }s>0$.

$\Lambda_{2}$$\Lambda_{2}^{\prime}$poles of $\Phi_{2}$$\bar{s}(w)$ Figure 14.
Deformation of the path $\Lambda_{2}$

By modifying the path of integration as in Figure 14, in order that it passes
through the critical point, by Cauchy Theorem we have

$\Phi_{2}(w)=\frac{(2\pi)^{3}}{2\pi
i}\int_{\Lambda_{2}^{\prime}}e^{\phi(s)}ds-\sum_{p\in
P}\underset{s=p}{\text{res}}\ \mathcal{I}_{2}(s),$

where $P$ stands for the set of poles in the region between $\Lambda_{2}$ and
$\Lambda_{2}^{\prime}$ and

$\mathcal{I}_{2}(s)=\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{i\pi
s}4^{-s}w^{-4s}$

is the integrand in $\Phi_{2}(w)$. For the first summand we have an asymptotic
behavior as before

$\frac{(2\pi)^{3}}{2\pi
i}\int_{\Lambda_{2}^{\prime}}e^{\phi(s)}ds\sim\alpha\frac{e^{-4\sqrt{2}w}}{w^{2}}$

with $\alpha$ a certain constant we will not need to specify. For the second
summand, keeping into account that $\underset{z=n}{\text{res}}\
\Gamma(-z)=(-1)^{n+1}/n!$ for $n\in\mathbb{N}$, and that
$\Gamma(z+\frac{1}{2})=\pi^{1/2}2^{-2z+1}\Gamma(2z)\Gamma(z)^{-1}$, we receive

$\displaystyle\underset{s=n+\frac{1}{2}}{\text{res}}\ \mathcal{I}_{2}(s)$
$\displaystyle=\frac{(-1)^{n+1}}{n!}\Gamma\left(n+\frac{1}{2}\right)^{5}e^{i\pi\left(n+\frac{1}{2}\right)}4^{-n-\frac{1}{2}}w^{-4n-2}$
$\displaystyle=-\frac{i}{n!}\left(\frac{(2n-1)!!}{2^{n}}\pi^{\frac{1}{2}}\right)^{5}4^{-n-\frac{1}{2}}w^{-4n-2}.$

Therefore,

$\sum_{p\in P}\underset{s=p}{\text{res}}\
\mathcal{I}_{2}(s)=-\frac{i\pi^{\frac{5}{2}}}{2w^{2}}-\frac{i\pi^{\frac{5}{2}}}{256w^{6}}+O\left(\frac{1}{w^{10}}\right).$

In conclusion,

$\Phi_{2}(w)\sim\frac{i\pi^{\frac{5}{2}}}{2w^{2}}\left(1+O\left(\frac{1}{w}\right)\right),\quad\text{for
$-\frac{\pi}{2}<\arg w<\frac{\pi}{2}$}.$

Let us now use the identity (6.16) in the following form:

$\Phi_{2}(w)=2\pi\Phi_{1}(we^{-i\frac{\pi}{2}})-\Phi_{2}(we^{-i\frac{\pi}{2}}),\quad\quad-\frac{\pi}{2}<\arg(we^{-i\frac{\pi}{2}})<\frac{\pi}{2}.$

It implies that

$\Phi_{2}(w)\sim\frac{i\pi^{\frac{5}{2}}}{2w^{2}}\left(1+O\left(\frac{1}{w}\right)\right)\quad\text{
on the whole sector }-\frac{\pi}{2}<\arg w<\pi.$

∎

###  6.4. Computation of Monodromy Data

####  6.4.1. Solution at the origin and computation of
$\mathcal{C}_{0}(\eta,\mu,R)$

Monodromy data at the origin $z=0$ are determined by the action of the first
Chern class $c_{1}(\mathbb{G})=4\sigma_{1}$ on the classical cohomology ring.
So,

$R=\begin{pmatrix}0&0&0&0&0&0\\\ 4&0&0&0&0&0\\\ 0&4&0&0&0&0\\\ 0&4&0&0&0&0\\\
0&0&4&4&0&0\\\ 0&0&0&0&4&0\end{pmatrix}.$ (6.20)

By Theorem 2.2 and Theorem 2.4, there exists a fundamental matrix solution
(2.9)

$Y(z)=\Phi(t^{2},z)z^{\mu}z^{R},$

for some appropriate converging power series $\Phi(t^{2},z)=\mathbbm{1}+O(z)$
such that

$\Phi^{T}(t^{2},-z)~{}\eta~{}\Phi(t^{2},z)=\eta.$

Thus, a fundamental matrix for our problem is given by

$\Xi_{0}(z)=\eta~{}\Phi(t^{2},z)~{}z^{\mu}z^{R}~{}=\Phi^{T}(t^{2},-z)^{-1}\eta~{}z^{\mu}z^{R}.$

By applying the iterative procedure in [Dub99b] for the proof of Theorem 2.2,
at $t^{2}=0$ one finds the following fundamental solution

$\Xi_{0}(0,z)=S(0,z)\eta z^{\mu}z^{R},$ (6.21)

$S(0,z)=\left(\begin{array}[]{cccccc}2z^{4}+1&0&0&0&0&0\\\
2z^{3}&1-4z^{4}&0&0&0&0\\\ z^{2}&-z^{3}&1&0&0&0\\\ z^{2}&-z^{3}&0&1&0&0\\\
z&0&-z^{3}&-z^{3}&4z^{4}+1&0\\\ z^{4}&z&-z^{2}&-z^{2}&2z^{3}&1-2z^{4}\\\
\end{array}\right)+O(z^{5}).$

Notice that the leading term of the solution $\Xi_{0}$ in (6.21) is exactly

$\eta~{}z^{\mu}z^{R}=c\left(\begin{array}[]{cccccc}\frac{64}{3}z^{2}\log^{4}(z)&\frac{64}{3}z^{2}\log^{3}(z)&8z^{2}\log^{2}(z)&8z^{2}\log^{2}(z)&4z^{2}\log(z)&z^{2}\\\
\frac{64}{3}z\log^{3}(z)&16z\log^{2}(z)&4z\log(z)&4z\log(z)&z&0\\\
8\log^{2}(z)&4\log(z)&1&0&0&0\\\ 8\log^{2}(z)&4\log(z)&0&1&0&0\\\
\frac{4\log(z)}{z}&\frac{1}{z}&0&0&0&0\\\ \frac{1}{z^{2}}&0&0&0&0&0\\\
\end{array}\right).$

From the first row, we deduce that near $z=0$ any solution of the equation
(6.9), i.e.

$\Theta^{5}\Phi-1024z^{4}\Theta\Phi-2048z^{4}\Phi=0$

is of the form

$\Phi(z)=\sum_{n\geq 0}z^{n}\left(a_{n}+b_{n}\log
z+c_{n}\log^{2}z+d_{n}\log^{3}z+e_{n}\log^{4}z\right),$ (6.22)

where $a_{0},b_{0},c_{0},d_{0},e_{0}$ are arbitrary constants, and successive
coefficients can be obtained recursively.

######  Proposition 6.1.

Let $R$ be as in (6.20). Then, $\mathcal{C}_{0}(\eta,\mu,R)$ is the algebraic
abelian group of complex dimension 3 given by

$\mathcal{C}_{0}(\eta,\mu,R)=\left\\{\begin{pmatrix}1&0&0&0&0&0\\\
\alpha_{1}&1&0&0&0&0\\\ \alpha_{2}&\alpha_{1}&1&0&0&0\\\
\alpha_{3}&\alpha_{1}&0&1&0&0\\\
\alpha_{4}&\alpha_{2}+\alpha_{3}&\alpha_{1}&\alpha_{1}&1&0\\\
\alpha_{5}&\alpha_{4}&\alpha_{3}&\alpha_{2}&\alpha_{1}&1\end{pmatrix}\colon\alpha_{i}\in\mathbb{C}\text{
s.t.
}\left\\{\begin{array}[]{@{}l@{}}\alpha_{1}^{2}-\alpha_{2}-\alpha_{3}=0\\\
\alpha_{2}^{2}+\alpha_{3}^{2}-2\alpha_{1}\alpha_{4}+2\alpha_{5}=0\end{array}\right.\right\\}.$

In particular, if $F(t)\in\mathbb{C}[\\![t]\\!]$ is a formal power series of
the form $F(t)=1+F_{1}t+F_{2}t^{2}+\dots$, then the matrix (computed w.r.t.
the chosen Schubert basis
$\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{1,1},\sigma_{2,1},\sigma_{2,2}$)
representing the endomorphism

$\lambda_{F}\cup(-)\colon H^{\bullet}(\mathbb{G};\mathbb{C})\to
H^{\bullet}(\mathbb{G};\mathbb{C}),$

where $\lambda_{F}\in H^{\bullet}(\mathbb{G};\mathbb{C})$ is such that

$\widehat{F}(T\mathbb{G})\cup\lambda_{F}=\widehat{F}(T^{*}\mathbb{G}),$

is an element of $\mathcal{C}_{0}(\eta,\mu,R)$. Here $\widehat{F}(V)$ denotes
the Hirzebruch multiplicative characteristic class of the vector bundle
$V\to\mathbb{G}$ associated with the formal power series $F(t)$ (see [Hir78]).

###### Proof.

The equations defining the group $\mathcal{C}_{0}(\eta,\mu,R)$ are obtained by
direct computation from the requirement that $P(z):=z^{\mu}z^{R}\cdot C\cdot
z^{-R}z^{-\mu}$ is a polynomial of the form
$P(z)=\mathbbm{1}+A_{1}z+A_{2}z^{2}+\dots$, together with the orthogonality
condition $P(-z)^{T}\eta P(z)=\eta$. Notice that the polynomial for the
generic matrix of the above form is equal to

$P(z)=\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ z\alpha_{1}&1&0&0&0&0\\\
z^{2}\alpha_{2}&z\alpha_{1}&1&0&0&0\\\ z^{2}\alpha_{3}&z\alpha_{1}&0&1&0&0\\\
z^{3}\alpha_{4}&z^{2}\left(\alpha_{2}+\alpha_{3}\right)&z\alpha_{1}&z\alpha_{1}&1&0\\\
z^{4}\alpha_{5}&z^{3}\alpha_{4}&z^{2}\alpha_{3}&z^{2}\alpha_{2}&z\alpha_{1}&1\\\
\end{array}\right).$

We leave as an exercise to show that such a matrix group is abelian. Let
$\delta_{1},\dots,\delta_{6}$ be the Chern roots of $T\mathbb{G}$. Then, for
some complex constants $a_{i,j}\in\mathbb{C}$, we have

$\widehat{F}(T\mathbb{G}):=\prod_{j=1}^{6}F(\delta_{j})=1+a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{1,1}\sigma_{1,1}+a_{2,1}\sigma_{2,1}+a_{2,2}\sigma_{2,2},$
(6.23)
$\widehat{F}(T^{*}\mathbb{G}):=\prod_{j=1}^{6}F(-\delta_{j})=1-a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{1,1}\sigma_{1,1}-a_{2,1}\sigma_{2,1}+a_{2,2}\sigma_{2,2}.$
(6.24)

Thus, if

$\lambda_{F}=1+x_{1}\sigma_{1}+x_{2}\sigma_{2}+x_{3}\sigma_{1,1}+x_{4}\sigma_{2,1}+x_{5}\sigma_{2,2},$

from the condition
$\widehat{F}(T\mathbb{G})\cup\lambda_{F}=\widehat{F}(T^{*}\mathbb{G})$ we
obtain the constraints

$\left\\{\begin{array}[]{@{}l@{}}x_{1}=-2a_{1},\\\ x_{2}=2a_{1}^{2},\\\
x_{3}=2a_{1}^{2},\\\ x_{4}=2a_{1}(a_{2}+a_{1,1})-4a_{1}^{3}-2a_{2,1},\\\
x_{5}=4a_{1}a_{2,1}-4a_{1}^{2}(a_{2}+a_{1,1})+4a_{1}^{4}.\end{array}\right.$

From this it is immediately seen that $x_{1}^{2}-x_{2}-x_{3}=0$ and
$x_{2}^{2}+x_{3}^{2}-2x_{1}x_{4}+2x_{5}=0$. ∎

####  6.4.2. Stokes rays and computation of $\Xi_{\rm left}$, $\Xi_{\rm
right}$

According to Theorem 4.1, monodromy data of $QH^{\bullet}(\mathbb{G})$ can be
computed starting from a point $(0,t^{2},0,...,0)$ of the small quantum
cohomology. Moreover, thanks to the Isomonodromy Theorems, it suffices to do
the computation at $t^{2}=0$, i.e. $q=1$, where the canonical coordinates
(6.3) are

$u_{1}=u_{2}=0,\quad u_{3}=-4i\sqrt{2},\quad u_{4}=4i\sqrt{2},\quad
u_{5}=-4\sqrt{2},\quad u_{6}=4\sqrt{2}.$

The Stokes rays (2.20) are easily seen to be

$\displaystyle R_{13}=R_{23}$ $\displaystyle=\left\\{-\rho\colon\rho\geq
0\right\\},$ $\displaystyle R_{14}=R_{24}=R_{34}$
$\displaystyle=\left\\{\rho\colon\rho\geq 0\right\\},$ $\displaystyle
R_{15}=R_{25}$ $\displaystyle=\left\\{-i\rho\colon\rho\geq 0\right\\},$
$\displaystyle R_{16}=R_{26}=R_{56}$
$\displaystyle=\left\\{i\rho\colon\rho\geq 0\right\\},$ $\displaystyle
R_{35}=\left\\{\rho e^{-i\frac{\pi}{4}}\colon\rho\geq 0\right\\},\quad R_{36}$
$\displaystyle=\left\\{\rho e^{i\frac{\pi}{4}}\colon\rho\geq 0\right\\},$
$\displaystyle R_{45}=\left\\{-\rho e^{i\frac{\pi}{4}}\colon\rho\geq
0\right\\},\quad R_{46}$ $\displaystyle=\left\\{-\rho
e^{-i\frac{\pi}{4}}\colon\rho\geq 0\right\\},\quad R_{ji}=-R_{ij}.$

We fix the admissible line $\ell$

$\ell:=\left\\{\rho e^{i\frac{\pi}{6}}\colon\rho\in\mathbb{R}\right\\},$

so that the sectors for the asymptotic expansion, containing $\Pi_{\rm
left/right}$ and extending up to the nearest Stokes rays are

$\mathcal{S}_{\rm right}=\\{z:~{}-\pi<\arg
z<\pi/4\\}\quad\quad\mathcal{S}_{\rm left}=\\{z:~{}-0<\arg z<\pi+\pi/4\\}.$

For such a choice of the line, according to Theorem 2.10, the structure of the
Stokes matrix is

$S=\begin{pmatrix}1&0&*&0&0&*\\\ 0&1&*&0&0&*\\\ 0&0&1&0&0&*\\\ *&*&*&1&0&*\\\
*&*&*&*&1&*\\\ 0&0&0&0&0&1\end{pmatrix}.$ (6.25)

We use the following notation for fundamental matrices

$\Xi_{\rm
right}=\begin{pmatrix}\xi_{(1),1}^{R}&\xi_{(2),1}^{R}&\xi_{(3),1}^{R}&\xi_{(4),1}^{R}&\xi_{(5),1}^{R}&\xi_{(6),1}^{R}\\\
\\\
\xi_{(1),2}^{R}&\xi_{(2),2}^{R}&\xi_{(3),2}^{R}&\xi_{(4),2}^{R}&\xi_{(5),1}^{R}&\xi_{(6),2}^{R}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\xi_{(1),6}^{R}&\xi_{(2),6}^{R}&\xi_{(3),6}^{R}&\xi_{(4),6}^{R}&\xi_{(5),6}^{R}&\xi_{(6),6}^{R}\\\
\end{pmatrix},\quad\Xi_{\rm
left}=\begin{pmatrix}\xi_{(1),1}^{L}&\xi_{(2),1}^{L}&\xi_{(3),1}^{L}&\xi_{(4),1}^{L}&\xi_{(5),1}^{L}&\xi_{(6),1}^{L}\\\
\\\
\xi_{(1),2}^{L}&\xi_{(2),2}^{L}&\xi_{(3),2}^{L}&\xi_{(4),2}^{L}&\xi_{(5),1}^{L}&\xi_{(6),2}^{L}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\xi_{(1),6}^{L}&\xi_{(2),6}^{L}&\xi_{(3),6}^{L}&\xi_{(4),6}^{L}&\xi_{(5),6}^{L}&\xi_{(6),6}^{L}\\\
\end{pmatrix}$

Note, in particular, that (6.25) implies that the fifth columns of $\Xi_{\rm
right}$ and $\Xi_{\rm left}$ coincide. Then $\xi_{(5),1}^{L}$ is the
analytical continuation of $\xi_{(5),1}^{R}$ on $\mathcal{S}_{\rm left}$.
Moreover, the exponential $e^{zu_{5}}$ dominates all others $e^{zu_{j}}$’s in
the sector between the rays $R_{45}$ and $R_{46}$, i.e. for $-\pi-\pi/4<\arg
z<-\pi+\pi/4$. This implies that the asymptotics

$\xi_{(5),1}^{L}=\xi_{(5),1}^{R}=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{zu_{5}}\left(1+O\left(\frac{1}{z}\right)\right),$

is valid in the whole sector $-\pi-\pi/4<\arg z<\pi+\pi/4$. By lemma 6.3,

$\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(z)=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{zu_{5}}\left(1+O\left(\frac{1}{z}\right)\right),\quad\text{
for }-\frac{\pi}{2}<\arg z<\frac{\pi}{2}.$

Since the exponential $e^{zu_{5}}$ is dominated by all others exponentials
$e^{zu_{j}}$ in the region between $R_{35}$ and $R_{36}$, namely for
$-\pi/4<\arg z<\pi/4$, we conclude necessarily that

$\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(z)=\xi_{(5),1}^{L/R}(z).$

This determines the 5-th column of $\Xi_{\rm right}$ and $\Xi_{\rm left}$ in
terms of $\Phi_{1}$, using equations (6.2.3),(6.14). We also obtain an
improvement of Lemma 6.3:

######  Lemma 6.4.

$\Phi_{1}$ and $\Phi_{2}$ have the following asymptotic behaviour for
$w\to\infty$ in the sectors below

$\displaystyle\Phi_{1}(w)=(2\pi)^{\frac{3}{2}}\frac{e^{-4\sqrt{2}w}}{4w^{2}}\left(1+O\left(\frac{1}{w}\right)\right),$
$\displaystyle\quad\text{ for }-\pi-\frac{\pi}{4}<\arg w<\pi+\frac{\pi}{4}$
$\displaystyle\Phi_{2}(w)=\frac{i\pi^{\frac{5}{2}}}{2w^{2}}\left(1+O\left(\frac{1}{w}\right)\right),$
$\displaystyle\quad\text{ for }-\frac{\pi}{2}<\arg w<\pi.$

We are ready to determine the other columns of $\Xi_{\rm left/right}$. By
Lemma 6.4,

$-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{i\frac{\pi}{2}})=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{zu_{3}}\left(1+O\left(\frac{1}{z}\right)\right),\quad\text{for
}-2\pi+\frac{\pi}{4}<\arg z<\frac{3\pi}{4},$ (6.26)
$\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{i\pi})=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{zu_{6}}\left(1+O\left(\frac{1}{z}\right)\right),\quad\text{for
}-2\pi-\frac{\pi}{4}<\arg z<\frac{\pi}{4}.$ (6.27)

We consider first (6.26). Being solutions of a differential equation, the
following holds:

$-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{i\frac{\pi}{2}})=\text{
linear combination of the }\xi_{(1),i}^{R},\quad 1\leq i\leq 6.$

On the other hand, $e^{zu_{3}}$ is dominated by all other $e^{zu_{i}}$’s in
the sector $-\pi+\pi/4<\arg z<-\pi/2$ between $R_{45}$ and $R_{35}$. This
requires that the linear combination necessarily reduces to

$-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{i\frac{\pi}{2}})=\xi_{(3),1}^{R}$

Now we consider (6.27). As above, since $e^{zu_{6}}$ is dominated by all the
other $e^{zu_{i}}$’s in the sector $-5\pi/4<\arg z<-3\pi/4$ between $R_{46}$
and $R_{45}$, we conclude that

$\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{i\pi})=\xi_{(6),1}^{R}.$

Analogously we find that

$-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\frac{\pi}{2}})=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{zu_{4}}\left(1+O\left(\frac{1}{z}\right)\right)\quad\text{for
}-\frac{3\pi}{4}<\arg z<\pi+\frac{3\pi}{4},$
$\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\pi})=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{zu_{6}}\left(1+O\left(\frac{1}{z}\right)\right)\quad\text{on
}-\frac{\pi}{4}<\arg z<2\pi+\frac{\pi}{4}.$

By dominance considerations as above, we conclude that

$\xi_{(4),1}^{L}=-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\frac{\pi}{2}}),\quad\xi_{(6),1}^{L}=\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\pi}).$

The above results reconstruct (using identities (6.2.3),(6.14)) three columns
of matrices $\Xi_{\rm right}$ and $\Xi_{\rm left}$ respectively. As far as the
first two columns are concerned, we invoke again Lemma 6.4 for $\Phi_{2}$,
which yields

$\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}z^{2}\Phi_{2}(ze^{i\frac{\pi}{2}})=-\frac{ic^{\frac{1}{2}}}{2}\left(1+O\left(\frac{1}{z}\right)\right)\quad\text{on
}-\pi<\arg z<\frac{\pi}{2},$
$\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}z^{2}\Phi_{2}(ze^{-i\frac{\pi}{2}})=-\frac{ic^{\frac{1}{2}}}{2}\left(1+O\left(\frac{1}{z}\right)\right)\quad\text{on
}0<\arg z<\frac{3\pi}{2}.$

Exactly as before, dominance relations of the exponentials $e^{zu_{i}}$ yield

$\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}z^{2}\Phi_{2}(ze^{i\frac{\pi}{2}})=\xi_{(1),1}^{R}=\xi_{(2),1}^{R},\quad\quad\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}z^{2}\Phi_{2}(ze^{-i\frac{\pi}{2}})=\xi_{(1),1}^{L}=\xi_{(2),1}^{L},$

Using (6.2.3),(6.12),(6.13), the first two columns are contructed.
Summarizing, we have determined the following columns in terms of $\Phi_{1}$
and $\Phi_{2}$.

$\Xi_{\rm
right}=\begin{pmatrix}\xi_{(1),1}^{R}&\xi_{(2),1}^{R}&\xi_{(3),1}^{R}&\text{\rm
unknown}&\xi_{(5)}^{R}&\xi_{(6),1}^{R}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\end{pmatrix},$ $\Xi_{\rm
left}=\begin{pmatrix}\xi_{(1),1}^{L}&\xi_{(2),1}^{L}&\text{\rm
unknown}&\xi_{(4),1}^{L}&\xi_{(5),1}^{L}&\xi_{(6),1}^{L}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\end{pmatrix}.$

In Section 6.4.3 we show that the above partial information and the constraint
(2) in Theorem 2.11 are sufficient to determine the Stokes and central
connection matrices simultaneously. Since constraint (2) holds only in case
$S$ and $C$ are related to Frobenius manifolds, we sketch below – for the sake
of completeness – the general method to obtain the missing columns of
$\Xi_{\rm left/right}$ and $S$, in a pure context of asymptotic analysis of
differential equations.

We observe that

$-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\frac{\pi}{2}})=\frac{c^{\frac{1}{2}}}{2\sqrt{2}}e^{u_{4}z}(1+O(1/z)),\quad\text{
for }-\pi+\frac{\pi}{4}<\arg z<\pi+\frac{3\pi}{4}.$

The sub-sector $-\pi<\arg z<-3\pi/4$ of $\mathcal{S}_{\rm right}$ is not
covered by the sector where the above asymptotic behaviour holds. On the sub-