sector, the dominance relation $|e^{zu_{4}}|<|e^{zu_{5}}|$ holds. Thus,

$\xi_{(4),1}^{R}=-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\frac{\pi}{2}})+v\xi_{(5),1}^{R},$
(6.28)

for some complex number $v\in\mathbb{C}$, to be determined. Analogously, we
observe that

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

The sub-sector $-\pi/4<\arg z<\pi/4$ of $\mathcal{S}_{\rm right}$ is not
covered by the sector where the asymptotic behaviour holds. Now, the following
dominance relations hold: $|e^{zu_{4}}|<|e^{zu_{i}}|$, for $i=1,2,3,6$, in
$0<\arg z<\pi/4$; for $i=6$ in $-\pi/4<\arg z<0$. Thus

$\xi_{(4),1}^{R}=-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{i\frac{3\pi}{2}})+\gamma_{1}\xi_{(1),1}^{R}+\gamma_{3}\xi_{(3),1}^{R}+\gamma_{6}\xi_{(6),1}^{R}$
(6.29)

for some complex number $\gamma_{1},\gamma_{3},\gamma_{6}\in\mathbb{C}$, to be
determined212121There is no need to include a term
$+\gamma_{2}\xi_{(2),1}^{R}$ in the linear combination, since
$\xi_{(1),1}^{R}=\xi_{(2),1}^{R}$.. The above (6.28) and (6.29) become a
6-terms linear relation between functions $\Phi_{2}(ze^{i\frac{k\pi}{2}})$, as
follows

$-\Phi_{1}(ze^{-i\frac{\pi}{2}})+v\Phi_{1}(z)=-\Phi_{1}(ze^{i\frac{3\pi}{2}})+\frac{\gamma_{1}}{\pi}\Phi_{2}(ze^{i\frac{\pi}{2}})-\gamma_{3}\Phi_{1}(ze^{i\frac{\pi}{2}})+\gamma_{6}\Phi_{1}(ze^{i\pi}),$
$\Phi_{1}(z)=\frac{1}{2\pi}\left[\Phi_{2}(z)+\Phi_{2}(ze^{i\frac{\pi}{2}})\right].$

At this step, some further information is needed. The equation
$\Theta^{5}\Phi-1024z^{4}\Theta\Phi-2048z^{4}\Phi=0$ admits the symmetry
$z\mapsto ze^{i\frac{\pi}{2}}$. This means that if $\Phi$ is a solution of the
equation then also $\Phi(ze^{i\frac{\pi}{2}})$ is. Such a symmetry defines a
linear map on the vector space of solutions of the equation defined in a
neighborhood of $z=0$. Because of this symmetry, the form (6.22) can be
refined as

$\Phi(z)=\sum_{n\geq 0}z^{4n}\left(a_{n}+b_{n}\log
z+c_{n}\log^{2}z+d_{n}\log^{3}z+e_{n}\log^{4}z\right),$ (6.30)

where $a_{0},b_{0},c_{0},d_{0},e_{0}$ are arbitrary constants, and successive
coefficients can be obtained recursively. In the basis of solutions of the
form (6.30) with $(a_{0},b_{0},c_{0},d_{0},e_{0})=(1,0,...,0)$,
$(0,1,0,...,0)$ and so on, the matrix of the operator

$\left(A\Phi\right)(z):=\Phi(ze^{i\frac{\pi}{2}})$

is of triangular form with 1’s on the diagonal. Hence, by Cayley-Hamilton
Theorem we deduce that

$(A-\mathbbm{1})^{5}=0,$

i.e.

$A^{5}-5A^{4}+10A^{3}-10A^{2}+5A-\mathbbm{1}=0.$

This proves the following

######  Lemma 6.5.

The solutions of the equation
$\Theta^{5}\Phi-1024z^{4}\Theta\Phi-2048z^{4}\Phi=0$ satisfy the identity

$\Phi(ze^{i\frac{5\pi}{2}})-5\Phi(ze^{2\pi
i})+10\Phi(ze^{i\frac{3\pi}{2}})-10\Phi(ze^{i\pi})+5\Phi(ze^{i\frac{\pi}{2}})-\Phi(z)=0.$
(6.31)

The relation (6.31) applied to $\Phi_{2}$ determines $v$, $\gamma_{1}$,
$\gamma_{3}$, $\gamma_{6}$. For example, $v=6$. This determines
$\xi_{(4),1}^{R}$ through formula (6.28). The fourth column of $\Xi_{\rm
right}$ is then constructed with formula (6.2.3) applied to $\xi_{(4),1}^{R}$
(with $h=0$). The value $v=6$ will be determined again in Section 6.4.3 making
use of the constraint (2) of Theorem 2.11.

Proceeding in the same way, we also determine $\xi_{(3),1}^{L}$. One observes
that

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

The first asymptotic relation does not hold in the sub-sector $-\pi/4<\arg
z<\pi/4$ of $\mathcal{S}_{\rm left}$. The second one does not hold in
$3\pi/4<\arg z<5\pi/4$. Then, the dominance relations in these sub-sectors
generate a 6-terms linear relation with unknown coefficients. The coefficients
are determined by (6.31).

Once $\Xi_{\rm left/right}$ has been determined, $S$ can be computed by direct
comparison of the two fundamental matrices (formula (6.31) need to be used at
some point of the comparison). The final result is the Stokes matrix $S$ of
formula (6.33) below with $v=6$.

$\Lambda_{1/2}$deformed path Figure 15. Deformation of the path
$\Lambda_{1/2}$, in order to apply residue theorem. Poles are represented.

####  6.4.3. Computation of Stokes and Central Connection Matrices, using
constraint (2) of Theorem 2.11

We start from formula (6.28):

$\xi_{(4),1}^{R}=-\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\Phi_{1}(ze^{-i\frac{\pi}{2}})+v\xi_{(5),1}^{R}\equiv\frac{c^{\frac{1}{2}}}{2\pi^{\frac{3}{2}}}z^{2}\left(-\Phi_{1}(ze^{-i\frac{\pi}{2}})+v\Phi_{1}(z)\right).$

We show that the constraint (2) of Theorem 2.11 suffices to determine the
value of $v$ and reconstruct both the Stokes and the central connection
matrices, as follows.

The definition of the central connection matrix $C$ and the transformation
(6.10) imply that

$\Xi_{\rm right}=\Xi_{0}C.$

The matrix $C$ can be obtained by comparing the leading behaviours of
$\Xi_{\rm right}$ and $\Xi_{0}$ near $z=0$. The leading behaviour of $\Xi_{0}$
in (6.21) is $\eta z^{\mu}z^{R}$. In order to find the behaviour of $\Xi_{\rm
right}$, we need to compute the behaviour of $\Phi_{1}$ and $\Phi_{2}$ near
$z=0$. To this end, we consider the integral representations in Lemma 6.1, and
deform both paths $\Lambda_{1}$ and $\Lambda_{2}$ to the left, as shown in
Figure 15. By residue theorem, we obtain a representations of $\Phi_{1}$ and
$\Phi_{2}$ as a series of residues at the poles $s=0,-1,-2\dots$. Then, by the
reconstruction dictated by equations (6.2.3),(6.12),(6.13),(6.14), for each
entry of the matrix $\Xi_{\rm right}$ we obtain an expansion in $z$ and $\log
z$, converging for small $|z|$.

For example, let us compute the first and second columns of the matrix $C$: by
deformation of the path $\Lambda_{2}$ we obtain that for small $z$ the
following series expansions hold:

$\displaystyle\xi_{(1),1}^{R}=\xi_{(2),1}^{R}$
$\displaystyle=\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}z^{2}\Phi_{2}(ze^{i\frac{\pi}{2}})$
$\displaystyle=\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}z^{2}\sum_{n=0}^{\infty}\underset{s=-n}{\text{res}}\left(\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{-i\pi
s}4^{-s}z^{-4s}\right)$
$\displaystyle=\alpha_{1}z^{2}\log^{4}z+\alpha_{2}z^{2}\log^{3}z+\alpha_{3}z^{2}\log^{2}z+\alpha_{4}z^{2}\log
z+\alpha_{5}z^{2}+O(z^{4}),$

where $\alpha_{i}$ can be explicitly computed. By comparison with the first
row of $\eta z^{\mu}z^{R}$ we determine the entries

$\displaystyle C_{11}=C_{12}$
$\displaystyle=\frac{3}{64c}\alpha_{1},\quad\quad
C_{21}=C_{22}=\frac{3}{64c}\alpha_{2},$ $\displaystyle C_{51}=C_{52}$
$\displaystyle=\frac{1}{4c}\alpha_{4},\quad\quad
C_{61}=C_{62}=\frac{1}{c}\alpha_{5}.$

For the other entries we have to consider expansions of
$\xi^{R}_{(1),3},\xi^{R}_{(2),3},\xi^{R}_{(1),4},\xi^{R}_{(2),4}$. For
example,

$\displaystyle\xi^{R}_{(1),3}=\xi^{R}_{(2),4}$
$\displaystyle=\frac{c^{\frac{1}{2}}}{\pi^{\frac{5}{2}}}\cdot\frac{1}{32}(z\Phi^{\prime}_{2}(ze^{i\frac{\pi}{2}})+z^{2}\Phi^{\prime\prime}_{2}(ze^{i\frac{\pi}{2}}))-\frac{c^{\frac{1}{2}}}{2}$
$\displaystyle=-\frac{c^{\frac{1}{2}}}{2}+\frac{c^{\frac{1}{2}}}{2\pi^{\frac{5}{2}}}\sum_{n=0}^{\infty}\underset{s=-n}{\text{res}}\left(\Gamma(s)^{5}\Gamma\left(\frac{1}{2}-s\right)e^{-i\pi
s}4^{-s}s^{2}z^{-4s}\right)$ $\displaystyle=\beta_{1}\log^{2}z+\beta_{2}\log
z+\beta_{3}+O(z^{4}),$

where $\beta_{i}$ can be explicitly computed. So, by comparison of the third
row of $gz^{\mu}z^{R}$ we obtain

$C_{31}=C_{42}=\frac{\beta_{3}}{c}.$

Analogously one obtains $C_{32}=C_{41}$. Note that the other entries $C_{ij}$,
with $j=3,4,5,6$, are uniquely determined only by the expansion of
$\xi^{R}_{(j),i}$ because of (6.14). The compuation for all the other entries
of $C$ can be done in the same way, so it will not be repeated here. Due to
the length of the result, we write the whole $C$ in Appendix A. As it can be
seen in Appendix A, only the fifth column of $C$ is expressed in terms of the
constant $v$. This $v$ will now be determined.

Since $S$ and $C$ are associated with a Frobenius manifold, the constraint (2)
of Theorem 2.11 holds:

$S=C^{-1}e^{-\pi iR}e^{-\pi i\mu}\eta^{-1}(C^{T})^{-1}.$ (6.32)

Substituting $C$ of Appendix A with an indeterminate $v$ in the above
constraint, we obtain the Stokes matrix

$S=\left(\begin{array}[]{cccccc}1&0&4&0&0&4\\\ 0&1&4&0&0&4\\\ 0&0&1&0&0&6\\\
-4&-4&-16&1&6-v&-6\\\ 4(v-1)&4(v-1)&16v-26&-v&(v-6)v+1&6v-16\\\ 0&0&0&0&0&1\\\
\end{array}\right).$ (6.33)

By a direct comparison with the expected matrix form (6.25), which dictates
that $S_{45}=0$ and $S_{55}=1$, we conclude that necessarily

$v=6.$

In this way we have completely determined both the Stokes and central
connection matrices as well as the fundamental matrix $\Xi_{\rm right}$. See
also (6.39) below.

###  6.5. Monodromy data and Exceptional collections in
$\mathcal{D}^{b}(\mathbb{G})$

The monodromy data $R$ and $C$ computed above can be read as characteristic
classes of objects of an exceptional collection in
$\mathcal{D}^{b}(\mathbb{G})$, as it has been conjectured by one of the
authors ([Dub98]), though the formulation for the central connection matrix
was not well understood then. Following [KKP08] where the role of the
$\widehat{\Gamma}^{\pm}$-classes (characteristic classes obtained by the
Hirzebruch’s procedure starting from the series expansion of the functions
$\Gamma(1\pm t)$ near $t=0$) was pointed out, we claim that the central
connection matrix (for canonical coordinates in triangular/lexicographical
order) can be identified with the matrix of the $\mathbb{C}$-linear morphism

$\displaystyle\mathfrak{X}_{\mathbb{G}}^{\pm}\colon K_{0}(\mathbb{G})$
$\displaystyle\otimes_{\mathbb{Z}}\mathbb{C}\to
H^{\bullet}(\mathbb{G};\mathbb{C})$ $\displaystyle E$
$\displaystyle\mapsto\frac{1}{(2\pi)^{2}c^{\frac{1}{2}}}\widehat{\Gamma}^{\pm}(\mathbb{G})\cup\operatorname{Ch}(E)$
$\widehat{\Gamma}^{\pm}(\mathbb{G}):=\prod_{j}\Gamma(1\pm\delta_{j})\quad\text{where
}\delta_{j}\text{'s are the Chern roots of }T\mathbb{G},$
$\operatorname{Ch}(V):=\sum_{k}e^{2\pi ix_{k}}\quad\text{ $x_{k}$'s are the
Chern roots of a vector bundle $V$},$

expressed w.r.t.

  * •

an exceptional basis $(\varepsilon_{i})_{i}$ of
$K_{0}(\mathbb{G})\otimes_{\mathbb{Z}}\mathbb{C}$, i.e. satisfying
$\chi(\varepsilon_{i},\varepsilon_{i})=1$, and the Grothendieck–Euler–Poincaré
orthogonality conditions $\chi(\varepsilon_{i},\varepsilon_{j})=0$ for $i>j$,
obtained by projection of a full exceptional collection $(E_{i})_{i}$ in
$\mathcal{D}^{b}(\mathbb{G})$;

  * •

a basis in $H^{\bullet}(\mathbb{G};\mathbb{C})$ related to
$(\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{1,1},\sigma_{2,1},\sigma_{2,2})$
(the Schubert basis we have fixed) by a $(\eta,\mu)$-orthogonal-parabolic $G$
endomorphism (as described in Section 2.1) which commutes with the operator of
classical $\cup$-multiplication $c_{1}(\mathbb{G})\cup-\colon
H^{\bullet}(\mathbb{G};\mathbb{C})\to H^{\bullet}(\mathbb{G};\mathbb{C})$.

By application of the constraint (6.32) and the
Grothendieck–Hirzebruch–Riemann–Roch Theorem, one can prove that the Stokes
matrix (in triangular/lexicographical order) is equal to the inverse of the
Gram matrix:

$(S^{-1})_{ij}=\chi(\varepsilon_{i},\varepsilon_{j}).$

See [CDG18] for a proof.

######  Remark 6.2.

As it was formulated in Theorem 1.2 in the Introduction and in Section 3, some
natural transformations are allowed, such as

  * •

the left action of the group $\mathcal{C}_{0}(\eta,\mu,R)$:

$\text{no action on }S,\quad\quad C\longmapsto G~{}C,$ (6.34)

where $G\in\mathcal{C}_{0}(\eta,\mu,R)$ and has the form prescribed by
Proposition 6.1;

  * •

the right action of the group $(\mathbb{Z}/2\mathbb{Z})^{\times 6}$:

$S\longmapsto\mathcal{I}~{}S~{}\mathcal{I},\quad\quad C\longmapsto
C~{}\mathcal{I},$ (6.35)

where $\mathcal{I}$ is a diagonal matrix of $1$’s and $-1$’s;

  * •

the right action of the braid group $\mathcal{B}_{6}$:

$S\longmapsto A^{\beta}~{}S~{}(A^{\beta})^{T},\quad\quad C\longmapsto
C~{}(A^{\beta})^{-1},$ (6.36)

as in formulae (3.4) and (3.5).

The above actions naturally manifest respectively on the space
$H^{\bullet}(\mathbb{G};\mathbb{C})$, on the set of full exceptional
collections in the category $\mathcal{D}^{b}(\mathbb{G})$, and/or on the set
of exceptional bases of the complexified Grothendieck group
$K_{0}(\mathbb{G})\otimes_{\mathbb{Z}}\mathbb{C}$. More precisely,

  * •

$\mathcal{C}_{0}(\eta,\mu,R)$ acts on $H^{\bullet}(\mathbb{G};\mathbb{C})$ as
$(\eta,\mu)$-orthogonal-parabolic endomorphisms commuting with the classical
$\cup$-product by the first Chern class $c_{1}(\mathbb{G})$;

  * •

the action of the shift functor
$[1]\colon\mathcal{D}^{b}(\mathbb{G})\to\mathcal{D}^{b}(\mathbb{G})$ on the
objects of a full exceptional collection projects as an action of
$(\mathbb{Z}/2\mathbb{Z})^{\times 6}$ on
$K_{0}(\mathbb{G})\otimes_{\mathbb{Z}}\mathbb{C}$ by changing of signs of the
elements of the corresponding exceptional basis;

  * •

the braid group $\mathcal{B}_{6}$ acts on the set of exceptional collections
(and the corresponding exceptional bases) as follows: the generator
$\beta_{i,i+1}$ ($1\leq i\leq 5$) transforms the collection
$(E_{1},\dots,E_{i-1},E_{i},E_{i+1},E_{i+2}\dots,E_{6})$ into
$(E_{1},\dots,E_{i-1},L_{E_{i}}E_{i+1},E_{i},E_{i+2},\dots,E_{6})$, where the
object $L_{E_{i}}E_{i+1}$ is defined, up to unique isomorphism, by the
distinguished triangle

$L_{E_{i}}E_{i+1}[-1]\to\operatorname{Hom}^{\bullet}(E_{i},E_{i+1})\otimes
E_{i}\to E_{i+1}\to L_{E_{i}}E_{i+1}.$

Notice that our definition of braid mutations of exceptional objects differs
from the one given, for example, in [GK04] by a shift: this difference is
important in order to obtain the coincidence of the braid group action on the
matrix representing the morphism $\mathfrak{X}_{\mathbb{G}}^{\pm}$ with the
action on the central connection matrix.

######  Remark 6.3.

The conjecture we are discussing was also formulated in [GGI16] in the same
time as [Dub13] for any Fano manifold $X$. In [GGI16] the authors stress the
relevance of the class $\widehat{\Gamma}^{+}(X)$, while in [Dub13] it was
conjectured that $\widehat{\Gamma}^{-}(X)$ is the relevant characteristic
class. As we will show below, $\widehat{\Gamma}^{+}(X)$ and
$\widehat{\Gamma}^{-}(X)$ can be interchanged by the action (6.34) of the
group $\mathcal{C}_{0}(\eta,\mu,R)$.

We now show that the monodromy data computed in the previous Section are of
the above form for an exceptional collection in the same orbit of the Kapranov
collection, under the action of the braid group. The Kapranov exceptional
collection for $\mathbb{G}$ is formed by vector bundles
$\mathbb{S}^{\lambda}(\mathcal{S}^{*})$ ($\mathcal{S}$ is the tautological
bundle), where $\mathbb{S}^{\lambda}$ denotes the Schur functor corresponding
to the Young diagram $\lambda$ 222222The reader can find the definition of
Schur functors as endo-functors of the category of vector spaces in [FH91].
The definition easily extends to the category of vector bundles.. In the
general case of $G_{\mathbb{C}}(k,n)$, the graded Chern character of these
bundles is given by

$\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=s_{\lambda}(e^{2\pi\sqrt{-1}x_{1}},\dots,e^{2\pi\sqrt{-1}x_{k}}):=\frac{\det(e^{2\pi\sqrt{-1}x_{i}(\lambda_{j}+r-j)})_{1\leq
i,j\leq k}}{\prod_{i<j}(e^{2\pi\sqrt{-1}x_{i}}-e^{2\pi\sqrt{-1}x_{j}})}$

i.e. the Schur polynomial calculated at the Chern roots $x_{1},\dots,x_{k}$ of
$\mathcal{S}^{*}$. In our case we obtain the following classes: posing
$a:=e^{2\pi ix_{1}}$ and $b:=e^{2\pi ix_{2}}$ with $x_{1}+x_{2}=\sigma_{1}$
and $x_{1}x_{2}=\sigma_{1,1}$ we have that

$\text{for
}\lambda=0\quad\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=1,$
$\text{for
}\lambda=\tiny{\yng(1)},\normalsize\quad\quad\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=a+b,$
$\text{for
}\lambda=\tiny\yng(2),\normalsize\quad\quad\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=(a+b)^{2}-ab,$
$\text{for
}\lambda=\tiny\yng(1,1),\normalsize\quad\quad\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=ab,$
$\text{for
}\lambda=\tiny\yng(2,1),\normalsize\quad\quad\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=(a+b)ab,$
$\text{for
}\lambda=\tiny\yng(2,2),\normalsize\quad\quad\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)=a^{2}b^{2}.$

Observing that

$ab=1+2\pi
i\sigma_{1}-2\pi^{2}(\sigma_{2}+\sigma_{1,1})-\frac{8}{3}i\pi^{3}\sigma_{2,1}+\frac{4}{3}\pi^{4}\sigma_{2,2},$
$a+b=2+2\pi
i\sigma_{1}-2\pi^{2}\sigma_{2}+2\pi^{2}\sigma_{1,1}+\frac{4}{3}i\pi^{3}\sigma_{2,1},$

after some computations one obtains all graded Chern characters. Recalling the
value of the $\widehat{\Gamma}^{\mp}$-class

$\displaystyle\widehat{\Gamma}^{\mp}(\mathbb{G})=1$ $\displaystyle\pm
4\gamma\sigma_{1}+\frac{1}{6}\left(48\gamma^{2}+\pi^{2}\right)(\sigma_{1,1}+\sigma_{2})\pm\frac{4}{3}(16\gamma^{3}+\gamma\pi^{2}-\zeta(3))\sigma_{2,1}$
$\displaystyle+\frac{1}{36}\left(768\gamma^{4}+96\gamma^{2}\pi^{2}-\pi^{4}-192\gamma\zeta(3)\right)\sigma_{2,2}$

we can explicitly compute all the classes

$\frac{1}{4\pi^{2}c^{\frac{1}{2}}}\left(\widehat{\Gamma}^{\mp}(\mathbb{G})\wedge\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right)\right).$

We denote by $C^{\mp}_{\text{Kap}}$ the matrix obtained in this way: in
appendix A the reader can find the entries of the matrix $C^{-}_{\text{Kap}}$.

The Stokes matrix can be put in triangular form by a suitable permutation of
$(u_{1},...,u_{6})$, to which a permutation matrix $P$ is associated,
according to the transformations (3.1). There are two permutations which yield
$PSP^{-1}$ in triangular form, namely

$\tau_{1}\colon(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\mapsto(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime},u_{4}^{\prime},u_{5}^{\prime},u_{6}^{\prime}):=(u_{5},u_{4},u_{2},u_{1},u_{3},u_{6}),$
(6.37)
$\tau_{2}\colon(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\mapsto(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime},u_{4}^{\prime},u_{5}^{\prime},u_{6}^{\prime}):=(u_{5},u_{4},u_{1},u_{2},u_{3},u_{6}).$
(6.38)

In both cases, the Stokes matrix $S$ in (6.33), with $v=6$, becomes

$S\longmapsto PSP^{-1}=\left(\begin{array}[]{cccccc}1&-6&20&20&70&20\\\
0&1&-4&-4&-16&-6\\\ 0&0&1&0&4&4\\\ 0&0&0&1&4&4\\\ 0&0&0&0&1&6\\\
0&0&0&0&0&1\\\ \end{array}\right).$ (6.39)

The matrix $C$ in Appendix A, with $v=6$, becomes

$C\mapsto CP^{-1}$ (6.40)

A direct computation proves the following:

######  Theorem 6.2.

Consider the monodromy data of the quantum cohomology of the Grassmannian
$\mathbb{G}$ at $0\in QH^{\bullet}(\mathbb{G})$, as computed in Section 6.4.3
with respect to an admissible line232323The computations have been done for
$\phi=\pi/6$, but nothing changes if $0<\phi<\frac{\pi}{4}$, since the sectors
where the asymptotic behaviours are studied always are the same
$\mathcal{S}_{\rm left/right}$. $\ell=\ell(\phi)$ of slope
$0<\phi<\frac{\pi}{4}$ and w.r.t. the basis of solutions (6.21). These are the
matrix $S$ in formula (6.33) and the matrix $C$ in Appendix A, with $v=6$.
Arrange $S$ in triangular form as in (6.39), with $P$ associated with one of
the above permutations $\tau_{1}$ or $\tau_{2}$ above, and transform $C$ as in
(6.40). The data so obtained are related to the Kapranov exceptional
collection by a finite sequence of natural transformations (6.34), (6.35),
(6.36). More precisely, the following sequence transforms $CP^{-1}$ into
$C^{-}_{\text{Kap}}$:

  * •

(1) the change of signs in the normalised idempotents vector fields,
determined by the action (6.35) of the diagonal matrix
$\mathcal{I}:=\operatorname{diag}(1,-1,-1,1,-1,1)$ (if we start from the cell
where $\tau_{1}$ is lexicographical), or
$\mathcal{I}:=\operatorname{diag}(1,-1,1,-1,-1,1)$ (if we start from the cell
where $\tau_{2}$ is lexicographical),

  * •

(2) change of solution at the origin through the action (6.34), with $G$ equal
to

$A=\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ 2i\pi&1&0&0&0&0\\\
-2\pi^{2}&2i\pi&1&0&0&0\\\ -2\pi^{2}&2i\pi&0&1&0&0\\\
-\frac{1}{3}\left(8i\pi^{3}\right)&-4\pi^{2}&2i\pi&2i\pi&1&0\\\
\frac{4\pi^{4}}{3}&-\frac{1}{3}\left(8i\pi^{3}\right)&-2\pi^{2}&-2\pi^{2}&2i\pi&1\\\
\end{array}\right)\in\mathcal{C}_{0}(\eta,\mu,R),$

  * •

(3) the action (6.36) with either the braid
$\beta_{12}\beta_{56}\beta_{45}\beta_{23}\beta_{34}$ (if we start from the
cell where $\tau_{1}$ is lexicographical), or the braid
$\beta_{34}\beta_{12}\beta_{56}\beta_{45}\beta_{23}\beta_{34}$ (if we start
from the cell where $\tau_{2}$ is lexicographical).

Moreover, $CP^{-1}$ in (6.40) is transformed into $C^{+}_{\text{Kap}}$ if,
after the sequence of the above transformations (1),(2),(3) above, the
following transformation is further applied:

  * •

(4) the action (6.34), with matrix $G$ equal to

$B=\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ -8\gamma&1&0&0&0&0\\\
32\gamma^{2}&-8\gamma&1&0&0&0\\\ 32\gamma^{2}&-8\gamma&0&1&0&0\\\
\frac{8}{3}\left(\zeta(3)-64\gamma^{3}\right)&64\gamma^{2}&-8\gamma&-8\gamma&1&0\\\
\frac{64}{3}\left(16\gamma^{4}-\gamma\zeta(3)\right)&\frac{8}{3}\left(\zeta(3)-64\gamma^{3}\right)&32\gamma^{2}&32\gamma^{2}&-8\gamma&1\\\
\end{array}\right)\in\mathcal{C}_{0}(\eta,\mu,R).$

The inverse of the Stokes matrix obtained from $PSP^{-1}$ in (6.39) by either
the sequence (1),(2),(3) or (1)(2)(3)(4) (recall that steps (2) and (4) act
trivially on $S$) coincides with the following Gram matrix of the Kapranov
exceptional collection

$G_{\text{\rm Kap}}=\left(\begin{array}[]{cccccc}1&4&10&6&20&20\\\
0&1&4&4&16&20\\\ 0&0&1&0&4&10\\\ 0&0&0&1&4&6\\\ 0&0&0&0&1&4\\\ 0&0&0&0&0&1\\\
\end{array}\right).$ (6.41)

Equivalently, the Stokes and connection matrices at $0\in
QH^{\bullet}(\mathbb{G})$ correspond to the exceptional block collections
obtained from the Kapranov block collection by mutations under the inverse of
the braid $\beta_{12}\beta_{56}\beta_{45}\beta_{23}\beta_{34}$ or the braid
$\beta_{34}\beta_{12}\beta_{56}\beta_{45}\beta_{23}\beta_{34}$ (the action of
$\beta_{34}$ acting just as a permutation of the third and fourth elements of
the block).

######  Remark 6.4.

In both cases $C^{+}_{\text{Kap}}$ and $C^{-}_{\text{Kap}}$, the relation
(6.32) holds between $C^{\pm}_{\text{Kap}}$ and $G_{\text{\rm Kap}}^{-1}$.

The algebro-geometric meaning of the matrices $A$ and $B$ of Theorem 6.2 will
be thoroughly explained in our [CDG18].

###  6.6. Reconstruction of Monodromy Data along the Small Quantum locus

In this section we reconstruct the monodromy data at all other points of the
small quantum cohomology of $\mathbb{G}$, by applying the procedure described
in Section 4.1, and already illustrated in Section 5.

We identify the small quantum cohomology with the set of points
$t=(0,t^{2},0,...,0)$. These points can be represented on the real plane
$(\operatorname{Re}t^{2},\operatorname{Im}t^{2})$. At a point
$(0,t^{2},0,...,0)$, the canonical coordinates are (6.3), so that the Stokes
rays are

$R_{ij}(t^{2})=e^{\overline{t^{2}}/4}R_{ij}(0)\equiv
e^{-i\operatorname{Im}{t^{2}}/4}R_{ij}(0),$

where $R_{ij}(0)$ are the rays $R_{ij}$ of Section 6.4.2. Let $\ell$ be a line
of slope $\varphi\in]0,\pi/4[$, admissible for $t^{2}=0$, i.e. for the Stokes
rays $R_{ij}(0)$. Then, whenever
$\operatorname{Im}t^{2}\in\pi\cdot\mathbb{Z}-4\phi$, at least a pair of rays
$R_{ij}(t^{2})$ and $R_{ji}(t^{2})$ lie along the line $\ell$, for some
$(i,j)$. This means that the small quantum cohomology of $\mathbb{G}$ is split
into the following horizontal bands of the
$(\operatorname{Re}t^{2},\operatorname{Im}t^{2})$-plane:

$\mathcal{H}_{k}:=\\{t^{2}:~{}k\pi-4\phi<\operatorname{Im}t^{2}<(k+1)\pi-4\phi\\},\quad
k\in\mathbb{Z}.$

If $t^{2}$ varies along a curve connecting two neighbouring bands, at least a
pair of opposite rays $R_{ij}(t^{2})$ and $R_{ji}(t^{2})$ cross $\ell$ in
correspondence with $t^{2}$ crossing the border between the bands.

A point $(0,t^{2},0,...,0)$, such that $t^{2}$ is interior to a band, is a
semisimple coalescence point, where Theorem 4.1 applies. The polydisc
$\mathcal{U}_{\epsilon_{1}}(u(0,t^{2},...,0))$ is split into two $\ell$-cells.
Each cell corresponds, through the coordinate map $p\mapsto u(p)$, to the
closure of an open connected subset of an $\ell$-chamber of
$QH^{\bullet}(\mathbb{G})$, as explained in Section 4.1. Therefore, each band
$\mathcal{H}_{k}$ precisely belongs to the boundary of two $\ell$-chambers
corresponding to the two cells, while each line
$\operatorname{Im}t^{2}=k\pi-4\phi$ between two bands $\mathcal{H}_{k-1}$ and
$\mathcal{H}_{k}$ belongs to the intersection of the boundaries of four
neighbouring chambers of $QH^{\bullet}(\mathbb{G})$. As explained in Section
4.1, the monodromy data computed via Theorem 4.1 in
$\mathcal{U}_{\epsilon_{1}}(u(0,t^{2},...,0))$ are the data of the two
chambers sharing the boundary $\mathcal{H}_{k}$. In particular, as a necessary
consequence of Theorem 4.1, these data are the data at each point of
$\mathcal{H}_{k}$. This means that the monodromy data are constant in each
band $\mathcal{H}_{k}$.

In order to compute the monodromy data in every chamber of
$QH^{\bullet}(\mathbb{G})$ is sufficies to apply the procedure of Section 4.1
starting from the data $C$, $S$ computed at $t=0$ in Section 6.4.3.
Preliminarily, by a permutation $P$, we have obtained upper triangular
$PSP^{-1}$ and the corresponding $CP^{-1}$ in (6.39) and (6.40), which are the
monodromy data in the cell of
$\mathcal{U}_{\epsilon_{1}}(u^{\prime}(0,0,...,0))$ where
$u_{1}^{\prime}(0,0,...,0),...,u_{6}^{\prime}(0,0,...,0)$ are in
lexicographical order as in (6.37) or (6.38). Thus, they are the data of the
band $\mathcal{H}_{0}$. Then, the braid group actions (3.4) and (3.5) can be
applied. In particular, we have computed the action of those braids which
allow to pass from the chamber (with lexicographical order) whose boundary
contains $\mathcal{H}_{0}$, to the chambers whose boundary contains
$\mathcal{H}_{k}$, for $k=1,2,...,8$. The values of $S$ and $C$ so obtained
are, as explained above, the constant monodromy data for $\mathcal{H}_{0}$,
$\mathcal{H}_{1}$, …., $\mathcal{H}_{8}$. They are reported in Table
LABEL:30luglio2016-6. From the table, we can read the monodromy data for the
whole small quantum cohomology, since for any $k\in\mathbb{Z}$, the data for
$\mathcal{H}_{k+8}$ are the same as for $\mathcal{H}_{k}$, as will be clear
from the explanation below.

We need to determine the braid connecting neighbouring bands
$\mathcal{H}_{k}$’s, from $\mathcal{H}_{0}$ to $\mathcal{H}_{1}$, from
$\mathcal{H}_{1}$ to $\mathcal{H}_{2}$, and so on. The passage from
$\mathcal{H}_{k}$ to $\mathcal{H}_{k+1}$ is achieved by increasing
$\operatorname{Im}(t^{2})$, to which a clockwise rotation of the Stokes rays
corresponds. In order to identify the corresponding braid, we have to keep
track of the rays which cross a fixed $\ell$ with slope $\phi\in]0,\pi/4[$.
Equivalently, we can consider a fixed configuration
$u_{1}(0,t^{2},...,0),...,u_{6}(0,t^{2},...,0)$ in lexicographical order,
corresponding to a fixed $t=(0,t^{2},0,..,0)$ in $\mathcal{H}_{0}$, so that
the corresponding rays $R_{ij}(t^{2})$ are fixed. Then, we let $\ell$ rotate
counter-clockwise increasing $\phi$, with the consequent gliding of the
$\ell$-horizontal bands towards $\operatorname{Im}(t^{2})\to-\infty$, and we
keep track of the rays which are crossed by $\ell$.

In order to apply the procedure explained in Section 4.1, we actually need to
start with a fixed configuration $u_{1}(t),...,u_{6}(t)$ of distinct canonical
coordinates. This is achieved by taking $t$ slightly away from
$\mathcal{H}_{0}$, in the interior of one of the two chambers whose boundaries
contain $\mathcal{H}_{0}$, so that the two coalescing canonical coordinates
(which equal $0$ in any $\mathcal{H}_{0}$) slightly split. It is after this
splitting that we let $\ell$ rotate and keep track of the rays which are
crossed by $\ell$.

This process is shown in Figure 16. The two canonical coordinates close to $0$
(the centres of the circles) come from the splitting of the two coalescing
eigenvalues $0$. The rays $L_{j}(\phi)$ defined in (3.2) are represented.
Their clockwise rotation corresponds to the counter-clockwise rotation of
$\ell$. A Stokes ray $R_{j,j+1}(t)$ is crossed by $\ell$ every time a ray
$L_{j}(\phi)$ aligns with $L_{j+1}(\phi)$. This yields the braid
$\beta_{j,j+1}$. The order $u_{j},u_{j+1}$ is $\ell$-lexicographical with
reference to $\ell$ just before the crossing. Each time $\ell$ has just
crossed a ray $R_{j,j+1}(t)$ (i.e. $L_{j}(\phi)$ has just crossed
$L_{j+1}(\phi)$), the coordinates $u_{j}$’s must be relabelled in the
$\ell$-lexicographical order corresponding to $\ell$ just after the crossing.

As it can be read in Figure 16 and Table LABEL:30luglio2016-6, the passage
from $\mathcal{H}_{0}$ to $\mathcal{H}_{k}$, for $k=1,2,...,8$, is obtained by
composition of the braids

$\omega_{1}:=\beta_{12}\beta_{56},\quad\quad\omega_{2}:=\beta_{23}\beta_{45}\beta_{34}\beta_{23}\beta_{45},\quad\quad\widehat{\omega}_{1}:=\beta_{12}\beta_{34}\beta_{56},$

in the form of products of increasing length

$\omega_{1},\quad\omega_{1}\omega_{2},\quad\omega_{1}\omega_{2}\widehat{\omega}_{1},\quad\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2},\quad\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1},$
$\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2},\quad\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}\widehat{\omega}_{1},\quad\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}.$

The result of Table LABEL:30luglio2016-6 is obtained applying formula (3.4).

The last braid
$\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}$
corresponds to a complete $2\pi$-rotation of the admissible line $\ell$. It is
not difficult to show that

$\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}=(\beta_{12}\beta_{23}\beta_{34}\beta_{45}\beta_{56})^{6},$

the right hand-side being the generator of the center of the braid group
$\mathcal{B}_{6}$ in Lemma 3.1 (the proof can be done by graphically
representing the two braids in the l.h.s. and r.h.s. respectively, and
noticing that they can be deformed the one into the other). This corresponds
to the cyclical repetition of the same Stokes matrix in $\mathcal{H}_{k}$ and
$\mathcal{H}_{k+8}$ (while the central connection matrix $C$ is transformed to
$M_{0}^{-1}C$).

Notice that the action (3.3) of $\beta_{34}$ in $\widehat{\omega}_{1}$ is a
permutation of the third and fourth rows and columns of the Stokes matrix,
because the entry (3,4) is zero (see (4.7)).

######  Remark 6.5.

There is a remarkable similarity between the above cyclical repetition and the
fact that exceptional collections are organised in algebraic structures called
_helices_ , introduced in [Gor89] [GR87], and extensively developed in [Gor90]
[Gor94] [GK04]. This will be thoroughly explained in our paper [CDG18].

$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$2$$3$$4$$5$$6$$\beta_{12}\beta_{56}$$\beta_{12}\beta_{34}\beta_{56}$$\beta_{12}\beta_{56}$$\beta_{12}\beta_{34}\beta_{56}$$\beta_{23}\beta_{45}$$\beta_{23}\beta_{45}$$\beta_{23}\beta_{45}$$\beta_{23}\beta_{45}$$\beta_{23}\beta_{45}$$\beta_{23}\beta_{45}$$\beta_{23}\beta_{45}$

$\beta_{23}\beta_{45}$

$\beta_{34}$$\beta_{34}$$\beta_{34}$$\beta_{34}$ Figure 16. The picture, to be read in boustrophedon order, shows the braids corresponding to the passage from one band $\mathcal{H}_{k}$ to $\mathcal{H}_{k+1}$ . Starting from the configuration of the canonical coordinates at $0\in QH^{\bullet}(\mathbb{G})$, we slightly split the coalescence as described in the first red picture in the first line. The numbers represent the lexicographical order of the canonical coordinates w.r.t. the admissible line. Letting the admissible line $\ell$ continuously rotate by increasing its slope, we determine all elementary braids acting in the mutation up to the next red configuration (the counter-clockwise rotation of $\ell$ is visualised by the clockwise rotation of the rays $L_{j}(\phi)$ defined in (3.2)). By coalescence of the points $u_{3},u_{4}$ in a red picture we obtain a configuration of canonical coordinates in the locus of small quantum cohomology (i.e. in a strip $\mathcal{H}_{k}$). Thus we deduce that successive bands of the small quantum cohomology are related by alternate compositions of the braids $\omega_{1}:=\beta_{12}\beta_{56}$, $\omega_{2}:=\beta_{23}\beta_{45}\beta_{34}\beta_{23}\beta_{45}$ and $\hat{\omega}_{1}:=\beta_{12}\beta_{34}\beta_{56}$. Table 3. List of Stokes matrices for all bands decomposing the small quantum cohomology of $\mathbb{G}$: the computation is done at a point $(0,t^{2},0,\dots,0)$ w.r.t. a line $\ell$ of slope $\phi\in]0,\pi/4[$, admissible for $t=0$. The starting matrix $S_{\text{lex}}$ in $\mathcal{H}_{0}$ is $PSP^{-1}$ of formula (6.39), with signs changed by (6.35) with $\mathcal{I}=\operatorname{diag}(-1,1,1,-1,1,-1)$. The braids acting on the monodromy data are $\omega_{1}:=\beta_{12}\beta_{56}$, $\omega_{2}:=\beta_{23}\beta_{45}\beta_{34}\beta_{23}\beta_{45}$ and $\widehat{\omega}_{1}:=\beta_{12}\beta_{34}\beta_{56}$. The table is computed using (3.4), namely by successively applying (3.3) for the elementary braids. Band $\mathcal{H}_{k}$, $0\leq k\leq 8$ | $S_{\text{lex}}$ | Braid
---|---|---
|  |
$0<\operatorname{Im}(t^{2})+4\phi<\pi$ | $\left(\begin{array}[]{cccccc}1&6&-20&20&-70&20\\\ 0&1&-4&4&-16&6\\\ 0&0&1&0&4&-4\\\ 0&0&0&1&-4&4\\\ 0&0&0&0&1&-6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $id$
$\pi<\operatorname{Im}(t^{2})+4\phi<2\pi$ | $\left(\begin{array}[]{cccccc}1&-6&-4&4&6&20\\\ 0&1&4&-4&-16&-70\\\ 0&0&1&0&-4&-20\\\ 0&0&0&1&4&20\\\ 0&0&0&0&1&6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}$
$2\pi<\operatorname{Im}(t^{2})+4\phi<3\pi$ | $\left(\begin{array}[]{cccccc}1&6&20&-20&-70&20\\\ 0&1&4&-4&-16&6\\\ 0&0&1&0&-4&4\\\ 0&0&0&1&4&-4\\\ 0&0&0&0&1&-6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}$
$3\pi<\operatorname{Im}(t^{2})+4\phi<4\pi$ | $\left(\begin{array}[]{cccccc}1&-6&-4&4&6&20\\\ 0&1&4&-4&-16&-70\\\ 0&0&1&0&-4&-20\\\ 0&0&0&1&4&20\\\ 0&0&0&0&1&6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}\widehat{\omega}_{1}$
$4\pi<\operatorname{Im}(t^{2})+4\phi<5\pi$ | $\left(\begin{array}[]{cccccc}1&6&20&-20&-70&20\\\ 0&1&4&-4&-16&6\\\ 0&0&1&0&-4&4\\\ 0&0&0&1&4&-4\\\ 0&0&0&0&1&-6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}$
$5\pi<\operatorname{Im}(t^{2})+4\phi<6\pi$ | $\left(\begin{array}[]{cccccc}1&-6&4&-4&6&20\\\ 0&1&-4&4&-16&-70\\\ 0&0&1&0&4&20\\\ 0&0&0&1&-4&-20\\\ 0&0&0&0&1&6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}$
$6\pi<\operatorname{Im}(t^{2})+4\phi<7\pi$ | $\left(\begin{array}[]{cccccc}1&6&-20&20&-70&20\\\ 0&1&-4&4&-16&6\\\ 0&0&1&0&4&-4\\\ 0&0&0&1&-4&4\\\ 0&0&0&0&1&-6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}$
$7\pi<\operatorname{Im}(t^{2})+4\phi<8\pi$ | $\left(\begin{array}[]{cccccc}1&-6&4&-4&6&20\\\ 0&1&-4&4&-16&-70\\\ 0&0&1&0&4&20\\\ 0&0&0&1&-4&-20\\\ 0&0&0&0&1&6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}\widehat{\omega}_{1}$
$8\pi<\operatorname{Im}(t^{2})+4\phi<9\pi$ | $\left(\begin{array}[]{cccccc}1&6&-20&20&-70&20\\\ 0&1&-4&4&-16&6\\\ 0&0&1&0&4&-4\\\ 0&0&0&1&-4&4\\\ 0&0&0&0&1&-6\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ | $\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}\omega_{1}\omega_{2}\widehat{\omega}_{1}\omega_{2}$

##  7\. A note on the topological solution for Fano manifolds

For quantum cohomologies of smooth projective varieties, a fundamental system
of solutions of the equation for gradients of deformed flat coordinates

$\left\\{\begin{array}[]{@{}l@{}}\partial_{\alpha}\zeta=z\mathcal{C}_{\alpha}\zeta,\\\
\\\
\partial_{z}\zeta=\left(\mathcal{U}+\frac{1}{z}\mu\right)\zeta,\end{array}\right.$
(7.1)

can be expressed in _enumerative-topological_ terms, namely the genus 0
correlations functions.

######  Proposition 7.1.

For a sufficiently small $R>0$, it is defined an analytic function

$\Theta\colon
B_{\mathbb{C}}(0;R)\times\Omega\to\operatorname{End}(H^{\bullet}(X;\mathbb{C}))$

with series expansion

$\displaystyle\Theta(z,t):$
$\displaystyle=\operatorname{Id}+\sum_{\alpha=0}^{N}\mathopen{\hbox{${\bigg{\langle}}$}\kern-1.94444pt\leavevmode\hbox{${\bigg{\langle}}$}}\frac{z\cdot(-)}{1-z\psi},T_{\alpha}\mathclose{\hbox{${\bigg{\rangle}}$}\kern-1.94444pt\leavevmode\hbox{${\bigg{\rangle}}$}}_{0}(t)T^{\alpha}$
$\displaystyle=\operatorname{Id}+\sum_{n=0}^{\infty}z^{n+1}\sum_{\alpha=0}^{N}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{n}(-),T_{\alpha}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}(t)T^{\alpha}.$

This function $\Theta$ satisfies the following properties:

  1. (1)

for any $\phi\in H^{\bullet}(X;\mathbb{C})$, the vector field

$\displaystyle\Theta_{\phi}:=\Theta(z,t)\phi$
$\displaystyle=\phi+\sum_{\alpha=0}^{N}\mathopen{\hbox{${\bigg{\langle}}$}\kern-1.94444pt\leavevmode\hbox{${\bigg{\langle}}$}}\frac{z\phi}{1-z\psi},T_{\alpha}\mathclose{\hbox{${\bigg{\rangle}}$}\kern-1.94444pt\leavevmode\hbox{${\bigg{\rangle}}$}}_{0}(t)T^{\alpha}$
$\displaystyle=\phi+\sum_{n=0}^{\infty}z^{n+1}\sum_{\alpha=0}^{N}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{n}\phi,T_{\alpha}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}(t)T^{\alpha}$

satisfies the equations

$\partial_{\alpha}\Theta_{\phi}=z\partial_{\alpha}*\Theta_{\phi};$

  2. (2)

when restricted to the small quantum locus $\Omega\cap H^{2}(X;\mathbb{C})$,
i.e. $t^{i}=0$ for $i=0,r+1,\dots,N$, then

$\Theta_{\phi}=e^{z\delta}\cup\phi+\sum_{\beta\neq
0}\sum_{\alpha=0}^{N}e^{\int_{\beta}\delta}\bigg{\langle}\frac{ze^{z\delta}\cup\phi}{1-z\psi},T_{\alpha}\bigg{\rangle}^{X}_{0,2,\beta}T^{\alpha},\quad\delta:=\sum_{i=1}^{r}t^{i}T_{i}\in
H^{2}(X;\mathbb{C});$

  3. (3)

for any $\phi_{1},\phi_{2}\in H^{\bullet}(X;\mathbb{C})$ we have

$\eta\left(\Theta(-z,t)\phi_{1},\Theta(z,t)\phi_{2}\right)=\eta\left(\phi_{1},\phi_{2}\right);$

  4. (4)

for any $\phi\in H^{\bullet}(X;\mathbb{C})$, the vector field

$\widetilde{\Theta}_{\phi}:=\left(\Theta(z,t)\circ
z^{\mu}z^{c_{1}(X)\cup(-)}\right)\phi$

is a solution of the system (7.1), i.e.

$\partial_{\alpha}\widetilde{\Theta}_{\phi}=z\partial_{\alpha}*\widetilde{\Theta}_{\phi},\quad\quad\partial_{z}\widetilde{\Theta}_{\phi}=\left(\mathcal{U}+\frac{1}{z}\mu\right)\widetilde{\Theta}_{\phi}.$

Thus, the vector fields $\widetilde{\Theta}_{T_{\alpha}}$’s are gradients of
deformed flat coordinates: if
$(\Theta^{\alpha}_{\beta})_{\alpha,\beta},(\widetilde{\Theta}^{\alpha}_{\beta})_{\alpha,\beta}$
are the matrices representing the two
$\operatorname{End}(H^{\bullet}(X;\mathbb{C}))$-valued functions $\Theta$ and
$\widetilde{\Theta}$ w.r.t. the basis $(T_{\alpha})_{\alpha}$, i.e.

$\Theta(z,t)T_{\beta}=\sum_{\alpha=0}^{N}\Theta^{\alpha}_{\beta}(z,t)T_{\alpha},\quad\widetilde{\Theta}(z,t)T_{\beta}=\sum_{\alpha=0}^{N}\widetilde{\Theta}^{\alpha}_{\beta}(z,t)T_{\alpha},$

then there exist analytic functions
$(\widetilde{t}_{\alpha}(z,t))_{\alpha},(h_{\alpha}(z,t))_{\alpha}$ on
$B_{\mathbb{C}}(0;R)\times\Omega$ such that

$\widetilde{\Theta}^{\alpha}_{\beta}(z,t)=(\operatorname{grad}\tilde{t}_{\beta}(z,t))^{\alpha},\quad(\tilde{t}_{0},\tilde{t}_{1},\dots,\tilde{t}_{N})=(h_{0},h_{1},\dots,h_{N})\cdot
z^{\mu}z^{R},$
$\Theta^{\alpha}_{\beta}(z,t)=(\operatorname{grad}h_{\beta}(z,t))^{\alpha},\quad\Theta^{T}(-z,t)\eta\Theta(z,t)=\eta,$
$h_{\alpha}(z,t):=\sum_{p=0}^{\infty}h_{\alpha,p}(t)z^{p},\quad
h_{\alpha,0}(t)=t_{\alpha}\equiv t^{\lambda}\eta_{\lambda\alpha}.$

###### Proof.

Notice that

$Y(z,t):=H(z,t)z^{\mu}z^{R},\quad
H(z,t)=\sum_{p=0}^{\infty}H_{p}(t)z^{p},\quad H_{0}(t)\equiv\mathbbm{1}$

is a fundamental solution of (7.1) if and only if $H(z,t)$ satisfies the
system

$\left\\{\begin{array}[]{@{}l@{}}\partial_{\alpha}H=z\mathcal{C}_{\alpha}H,\\\
\\\ \partial_{z}H=\mathcal{U}H+\frac{1}{z}[\mu,H]-HR.\end{array}\right.$

Because of the symmetry of $c_{\alpha\beta\gamma}$, the columns of $H$ are the
components w.r.t. $(\partial_{\alpha})_{\alpha}$ of the gradients of some
functions:

$h_{\alpha}(z,t):=\sum_{p=0}^{\infty}h_{\alpha,p}(t)z^{p},\quad
h_{\alpha,0}(t)=t_{\alpha},$
$H_{\beta}^{\alpha}(z,t)=(\operatorname{grad}h_{\beta})^{\alpha},\quad
H_{\beta,p}^{\alpha}(z,t)=(\operatorname{grad}h_{\beta,p})^{\alpha}.$

The above system for $H$ is equivalent to the following recursion relations on
$h_{\alpha,p}$’s functions:

$\partial_{\alpha}\partial_{\beta}h_{\gamma,p}(t)=c_{\alpha\beta}^{\nu}\partial_{\nu}h_{\gamma,p-1}(t),\quad
p\geq 1,$ (7.2)
$\mathfrak{L}_{E}(\operatorname{grad}h_{\alpha,p})=\left(p+\frac{\dim_{\mathbb{C}}X-2}{2}+\mu_{\alpha}\right)\operatorname{grad}h_{\alpha,p}+\sum_{\beta=0}^{N}(\operatorname{grad}h_{\beta,p-1})R^{\beta}_{\alpha},\quad
p\geq 1.$

The last equation is equivalent to the recursion relations on the
differentials

$\mathfrak{L}_{E}(dh_{\alpha,p})=\left(p-\frac{\dim_{\mathbb{C}}X-2}{2}+\mu_{\alpha}\right)dh_{\alpha,p}+\sum_{\beta=0}^{N}dh_{\beta,p-1}R^{\beta}_{\alpha},\quad
p\geq 1.$ (7.3)

In our case we have

$H(z,t)=(\Theta^{\alpha}_{\beta}(z,t))_{\alpha,\beta},\quad\partial_{\alpha}h_{\beta,p}(t)=\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{p-1}T_{\beta},T_{\alpha}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}(t).$

The recursion relations (7.2) then reads

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}T_{\alpha},T_{\beta},T_{\gamma}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}=\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}T_{\alpha},T_{\beta},T^{\nu}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}\eta_{\nu\gamma}\quad\text{for
}p=1,$
$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}T_{\alpha},\tau_{p-1}T_{\gamma},T_{\beta}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}=\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}T_{\alpha},T_{\beta},T^{\nu}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{p-2}T_{\gamma},T_{\nu}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}\quad\text{for
}p\geq 2.$

These are exactly the _topological recursion relations in genus 0_.

Let us now prove that also the recursion relations (7.3) hold. K. Hori
([Hor95], see also [EHX97]) proved that, for any $\omega\in
H^{2}(X;\mathbb{C})$, we have the following constraint on the genus $g$ free
energy

$\omega^{\alpha}\frac{\partial\mathcal{F}^{X}_{g}}{\partial
t^{\alpha,0}}=\sum_{\beta\in\operatorname{Eff}(X)}\left(\int_{\beta}\omega\right)\mathcal{F}^{X}_{g,\beta}+\sum_{n,\alpha,\atop\sigma,\nu}\omega^{\sigma}c_{\sigma\alpha}^{\nu}t^{\alpha,n}\frac{\partial\mathcal{F}^{X}_{g}}{\partial
t^{\nu,n-1}}+\frac{\delta^{0}_{g}}{2}\sum_{\alpha,\nu,\sigma}\omega^{\sigma}c_{\sigma\alpha\nu}t^{\alpha,0}t^{\nu,0}-\frac{\delta^{1}_{g}}{24}\int_{X}\omega\cup
c_{\dim X-1}(X),$ (7.4)

where $\mathcal{F}^{X}_{g,\beta}$ is the $(g,\beta)$-free energy

$\mathcal{F}^{X}_{g,\beta}:=\sum_{n=0}^{\infty}\frac{1}{n!}\langle\underbrace{\gamma.\dots,\gamma}_{n\text{
times}}\rangle_{g,n,\beta}^{X}.$

By dimensional consideration, one obtains also the selection rule

$\sum_{n,\alpha}(n+q_{\alpha}-1)t^{\alpha,n}\frac{\partial\mathcal{F}^{X}_{g}}{\partial
t^{\alpha,n}}=\sum_{\beta\in\operatorname{Eff}(X)}\left(\int_{\beta}\omega\right)\mathcal{F}^{X}_{g,\beta}+(3-\dim
X)(g-1)\mathcal{F}^{X}_{g}.$ (7.5)

If we introduce the perturbed first Chern class

$\mathcal{E}(\mathbb{t}):=c_{1}(X)+\sum_{m,\sigma}(1-q_{\sigma}-m)t^{\sigma,m}\tau_{m}(T_{\sigma})-\sum_{m,\sigma}t^{\sigma,m}\tau_{m-1}(c_{1}(X)\cup
T_{\sigma}),$

and using the selection rule (7.5), the Hori’s constraint (7.4) (specialized
to $g=0$ and $\omega=c_{1}(X)$) can be reformulated as

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{E}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}=(3-\dim
X)\mathcal{F}^{X}_{0}+\frac{1}{2}t^{\sigma,0}t^{\rho,0}\int_{X}c_{1}(X)\cup
T_{\sigma}\cup T_{\rho}.$

Taking the derivative w.r.t. $t^{\alpha,n},t^{\beta,0}$ we obtain

$\displaystyle\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{E},\tau_{n}T_{\alpha},T_{\beta}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}$
$\displaystyle-(n+q_{\alpha}+q_{\beta}-2)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{n}T_{\alpha},T_{\beta}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}-\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\tau_{n-1}(c_{1}(X)\cup
T_{\alpha}),T_{\beta}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}$
$\displaystyle=(3-\dim
X)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{\tau}_{n}T_{\alpha},T_{\beta}\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}_{0}+\delta_{n,0}\int_{X}c_{1}(X)\cup
T_{\alpha}\cup T_{\beta}.$

These recursion relations, restricted to the small phase space, are easily
seen to be equivalent to (7.3). This proves (1), (4) and the convergence of
$\Theta(z,t)$ for $|z|$ small enough, because of the regular feature of the
singularity $z=0$. The proof of (2) can be found in [CK99]. Condition (3)
follows from WDVV and string equation, as shown in [Giv98]. ∎

In the case of Fano manifolds, we have the following analytic characterization
of the fundamental solution
$\left(\widetilde{\Theta}^{\alpha}_{\beta}\right)_{\alpha,\beta}$.
Furthermore, because of Proposition 2.1, we obtain another proof of (3) in the
previous Proposition.

######  Proposition 7.2.

If $X$ is a Fano manifold, among all fundamental matrix solutions of the
system (7.1) for deformed flat coordinates,242424Throughout the paper,
$Y(z,t)=H(z,t)z^{\mu}z^{R}$ has been denoted $Y(z,t)=\Phi(z,t)z^{\mu}z^{R}$.
there exists a unique solution such that, on the small quantum locus (i.e.
$t^{i}=0$ for $i=0,r+1,\dots,N$) the function $z^{-\mu}H(z,t)z^{\mu}$ is
holomorphic at $z=0$, with series expansion

$z^{-\mu}H(z,t)z^{\mu}=e^{t\cup}+zK_{1}(t)+z^{2}K_{2}(t)+\dots,\quad
t^{i}=0\text{ for }i=0,r+1,\dots,N,$

This solution coincides with the solution
$\left(\widetilde{\Theta}^{\alpha}_{\beta}(z,t)\right)_{\alpha,\beta}$.

###### Proof.

We already know from Proposition 2.1 that such a solution is unique. Let us
now prove the main statement. In what follows, we will denote the degree $\deg
T_{\alpha}$ just by $|\alpha|$ for brevity. By point $(2)$ of Proposition 7.1,
we have that

$z^{-\mu}\left(\Theta_{(z^{\mu}\phi)}\right)=z^{-\mu}\left(e^{z\delta}\cup
z^{\mu}\phi+\sum_{\beta\neq
0}\sum_{\alpha=0}^{N}e^{\int_{\beta}\delta}\bigg{\langle}\frac{ze^{z\delta}\cup
z^{\mu}\phi}{1-z\psi},T_{\alpha}\bigg{\rangle}^{X}_{0,2,\beta}T^{\alpha}\right),$

with $\delta:=\sum_{i=1}^{r}t^{i}T_{i}\in H^{2}(X;\mathbb{C})$. Specialising
to $\phi=T_{\sigma}$, we have

$z^{-\mu}\left(\Theta_{(z^{\mu}T_{\sigma})}\right)=e^{\delta}\cup
T_{\sigma}+\sum_{\beta\neq
0}\sum_{\alpha,\lambda=0}^{N}\sum_{n,k=0}^{\infty}\frac{e^{\int_{\beta}\delta}}{k!}z^{n+1+k+\mu_{\sigma}-\mu_{\lambda}}\langle\tau_{n}(\delta^{\cup
k}\cup
T_{\sigma}),T_{\alpha}\rangle^{X}_{0,2,\beta}\eta^{\alpha\lambda}T_{\lambda}.$

In the second addend, we have non-zero terms only if

  * •

$|\alpha|+|\lambda|=2\dim_{\mathbb{C}}X$,

  * •

$2n+2k+|\sigma|+|\alpha|=\operatorname{vir\ dim}_{\mathbb{R}}X_{0,2,\beta}$.

By putting together these conditions, we obtain

$n+1+k+\frac{1}{2}(|\sigma|-|\lambda|)=-\int_{\beta}\omega_{X}.$

The assumption of being Fano is equivalent to the requirement that the
functional $\beta\mapsto-\int_{\beta}\omega_{X}$ is positive on the closure of
the effective cone. This proves the Proposition, the l.h.s. being exactly the
exponents of $z$ which appear in the above series expansion. ∎

######  Example 7.1.

Notice that the solution (6.21) that we considered in the previous section for
the computation of the monodromy data fo $QH^{\bullet}(\mathbb{G})$ satisfies
the condition

$z^{-\mu}\left(\eta^{-1}S(0,z)\eta\right)z^{\mu}\text{ is holomorphic near
}z=0,$
$z^{-\mu}\left(\eta^{-1}S(0,z)\eta\right)z^{\mu}=\left(\begin{array}[]{cccccc}1-2z^{4}&2z^{4}&-z^{4}&-z^{4}&z^{4}&z^{8}\\\
0&4z^{4}+1&-z^{4}&-z^{4}&0&z^{4}\\\ 0&0&1&0&-z^{4}&z^{4}\\\
0&0&0&1&-z^{4}&z^{4}\\\ 0&0&0&0&1-4z^{4}&2z^{4}\\\ 0&0&0&0&0&2z^{4}+1\\\
\end{array}\right)+O(z^{9}).$

This means that $\left(\eta^{-1}S(0,z)\eta\right)z^{\mu}z^{R}$ coincides with
the topological solution $\widetilde{\Theta}(0,z)$.

## Appendix A

Here we summarize the explicit values for the columns of the central
connection matrix $C=(C_{ij})$, computed in Section 6.4.3, where $v$ is
indicated. The correct value is $v=6$ ($v$ was first introduced in (6.28)).

$C_{i1}=\left(\begin{array}[]{c}\frac{1}{2\sqrt{c}\pi^{2}}\\\
\frac{4\gamma+i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+24i\gamma\pi-5\pi^{2}}{12\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+24i\gamma\pi+7\pi^{2}}{12\sqrt{c}\pi^{2}}\\\
\frac{64\gamma^{3}+48i\gamma^{2}\pi+4\gamma\pi^{2}+3i\pi^{3}-4\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+768i\gamma^{3}\pi+96\gamma^{2}\pi^{2}+144i\gamma\pi^{3}-\pi^{4}-48(4\gamma+i\pi)\zeta(3)}{72\sqrt{c}\pi^{2}}\\\
\end{array}\right),$
$C_{i2}=\left(\begin{array}[]{c}\frac{1}{2\sqrt{c}\pi^{2}}\\\
\frac{4\gamma+i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+24i\gamma\pi+7\pi^{2}}{12\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+24i\gamma\pi-5\pi^{2}}{12\sqrt{c}\pi^{2}}\\\
\frac{64\gamma^{3}+48i\gamma^{2}\pi+4\gamma\pi^{2}+3i\pi^{3}-4\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+768i\gamma^{3}\pi+96\gamma^{2}\pi^{2}+144i\gamma\pi^{3}-\pi^{4}-48(4\gamma+i\pi)\zeta(3)}{72\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$C_{i3}=\left(\begin{array}[]{c}-\frac{1}{4\sqrt{c}\pi^{2}}\\\
\frac{-2\gamma-i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{-48\gamma^{2}-48i\gamma\pi+11\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{-48\gamma^{2}-48i\gamma\pi+11\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{2\zeta(3)-(2\gamma+i\pi)(4\gamma+i\pi)(4\gamma+3i\pi)}{6\sqrt{c}\pi^{2}}\\\
\frac{-768\gamma^{4}-1536i\gamma^{3}\pi+1056\gamma^{2}\pi^{2}-23\pi^{4}+96i\pi\zeta(3)+96\gamma\left(3i\pi^{3}+2\zeta(3)\right)}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right),$
$C_{i4}(v)=\left(\begin{array}[]{c}\frac{v-1}{4\sqrt{c}\pi^{2}}\\\
\frac{2\gamma(v-1)+i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}(v-1)+48i\gamma\pi+(v+11)\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}(v-1)+48i\gamma\pi+(v+11)\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{32\gamma^{3}(v-1)+48i\gamma^{2}\pi+2\gamma(v+11)\pi^{2}-3i\pi^{3}-2(v-1)\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}(v-1)+1536i\gamma^{3}\pi+96\gamma^{2}(v+11)\pi^{2}-(v+23)\pi^{4}-96i\pi\zeta(3)+96\gamma\left(-3i\pi^{3}-2(v-1)\zeta(3)\right)}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$C_{i4}(6)=\left(\begin{array}[]{c}\frac{5}{4\sqrt{c}\pi^{2}}\\\
\frac{10\gamma+i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{240\gamma^{2}+48i\gamma\pi+17\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{240\gamma^{2}+48i\gamma\pi+17\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{160\gamma^{3}+48i\gamma^{2}\pi+34\gamma\pi^{2}-3i\pi^{3}-10\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{3840\gamma^{4}+1536i\gamma^{3}\pi+1632\gamma^{2}\pi^{2}-288i\gamma\pi^{3}-29\pi^{4}-960\gamma\zeta(3)-96i\pi\zeta(3)}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$C_{i5}=\left(\begin{array}[]{c}\frac{1}{4\sqrt{c}\pi^{2}}\\\
\frac{\gamma}{\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{-\zeta(3)+16\gamma^{3}+\gamma\pi^{2}}{3\sqrt{c}\pi^{2}}\\\
-\frac{192\gamma\zeta(3)-768\gamma^{4}+\pi^{4}-96\gamma^{2}\pi^{2}}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$C_{i6}=\left(\begin{array}[]{c}\frac{1}{4\sqrt{c}\pi^{2}}\\\
\frac{\gamma+i\pi}{\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+96i\gamma\pi-47\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{48\gamma^{2}+96i\gamma\pi-47\pi^{2}}{24\sqrt{c}\pi^{2}}\\\
\frac{(\gamma+i\pi)(4\gamma+3i\pi)(4\gamma+5i\pi)-\zeta(3)}{3\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+3072i\gamma^{3}\pi-4512\gamma^{2}\pi^{2}-2880i\gamma\pi^{3}+671\pi^{4}-192(\gamma+i\pi)\zeta(3)}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right).$

We write now entries of the matrix $C^{-}_{\text{Kap}}$ whose columns are
given by the components of the characteristic classes

$\frac{1}{4\pi
c^{\frac{1}{2}}}\widehat{\Gamma}^{-}(\mathbb{G})\cup\operatorname{Ch}\left(\mathbb{S}^{\lambda}(\mathcal{S}^{*})\right);$

the order of the column is given by $\lambda=0$, $\lambda=1$, $\lambda=2$,
$\lambda=(1,1)$, $\lambda=(2,1)$ and $\lambda=(2,2)$.

$\left(C^{-}_{\text{Kap}}\right)_{0}=\left(\begin{array}[]{c}\frac{1}{4\sqrt{c}\pi^{2}}\\\
\frac{\gamma}{\sqrt{c}\pi^{2}}\\\
\frac{\frac{1}{24}+\frac{2\gamma^{2}}{\pi^{2}}}{\sqrt{c}}\\\
\frac{\frac{1}{24}+\frac{2\gamma^{2}}{\pi^{2}}}{\sqrt{c}}\\\
\frac{-\zeta(3)+16\gamma^{3}+\gamma\pi^{2}}{3\sqrt{c}\pi^{2}}\\\
-\frac{192\gamma\zeta(3)-768\gamma^{4}+\pi^{4}-96\gamma^{2}\pi^{2}}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$\left(C^{-}_{\text{Kap}}\right)_{\tiny\yng(1)}=\left(\begin{array}[]{c}\frac{1}{2\sqrt{c}\pi^{2}}\\\
\frac{4\gamma+i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{\frac{2\gamma(2\gamma+i\pi)}{\pi^{2}}-\frac{5}{12}}{\sqrt{c}}\\\
\frac{\frac{2\gamma(2\gamma+i\pi)}{\pi^{2}}+\frac{7}{12}}{\sqrt{c}}\\\
\frac{64\gamma^{3}+48i\gamma^{2}\pi+4\gamma\pi^{2}+3i\pi^{3}-4\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+768i\gamma^{3}\pi+96\gamma^{2}\pi^{2}+144i\gamma\pi^{3}-\pi^{4}-48(4\gamma+i\pi)\zeta(3)}{72\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$\left(C^{-}_{\text{Kap}}\right)_{\tiny\yng(2)}=\left(\begin{array}[]{c}\frac{3}{4\sqrt{c}\pi^{2}}\\\
\frac{3(2\gamma+i\pi)}{2\sqrt{c}\pi^{2}}\\\
\frac{\frac{6\gamma(\gamma+i\pi)}{\pi^{2}}-\frac{19}{8}}{\sqrt{c}}\\\
\frac{\frac{6\gamma(\gamma+i\pi)}{\pi^{2}}+\frac{13}{8}}{\sqrt{c}}\\\
\frac{32\gamma^{3}+48i\gamma^{2}\pi-6\gamma\pi^{2}+5i\pi^{3}-2\zeta(3)}{2\sqrt{c}\pi^{2}}\\\
\frac{-6\gamma^{2}+\frac{16\gamma^{4}}{\pi^{2}}+\frac{32i\gamma^{3}}{\pi}+10i\gamma\pi+\frac{7\pi^{2}}{48}-\frac{2(2\gamma+i\pi)\zeta(3)}{\pi^{2}}}{\sqrt{c}}\\\
\end{array}\right),$

$\left(C^{-}_{\text{Kap}}\right)_{\tiny\yng(1,1)}=\left(\begin{array}[]{c}\frac{1}{4\sqrt{c}\pi^{2}}\\\
\frac{2\gamma+i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{\frac{2\gamma(\gamma+i\pi)}{\pi^{2}}-\frac{11}{24}}{\sqrt{c}}\\\
\frac{\frac{2\gamma(\gamma+i\pi)}{\pi^{2}}-\frac{11}{24}}{\sqrt{c}}\\\
\frac{(2\gamma+i\pi)(4\gamma+i\pi)(4\gamma+3i\pi)-2\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+1536i\gamma^{3}\pi-1056\gamma^{2}\pi^{2}-288i\gamma\pi^{3}+23\pi^{4}-96(2\gamma+i\pi)\zeta(3)}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$\left(C^{-}_{\text{Kap}}\right)_{\tiny\yng(2,1)}=\left(\begin{array}[]{c}\frac{1}{2\sqrt{c}\pi^{2}}\\\
\frac{4\gamma+3i\pi}{2\sqrt{c}\pi^{2}}\\\
\frac{\frac{2\gamma(2\gamma+3i\pi)}{\pi^{2}}-\frac{29}{12}}{\sqrt{c}}\\\
\frac{\frac{2\gamma(2\gamma+3i\pi)}{\pi^{2}}-\frac{17}{12}}{\sqrt{c}}\\\
\frac{(4\gamma+i\pi)(4\gamma+3i\pi)(4\gamma+5i\pi)-4\zeta(3)}{6\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+2304i\gamma^{3}\pi-2208\gamma^{2}\pi^{2}-720i\gamma\pi^{3}+47\pi^{4}-48(4\gamma+3i\pi)\zeta(3)}{72\sqrt{c}\pi^{2}}\\\
\end{array}\right),$

$\left(C^{-}_{\text{Kap}}\right)_{\tiny\yng(2,2)}=\left(\begin{array}[]{c}\frac{1}{4\sqrt{c}\pi^{2}}\\\
\frac{\gamma+i\pi}{\sqrt{c}\pi^{2}}\\\
\frac{\frac{2\gamma(\gamma+2i\pi)}{\pi^{2}}-\frac{47}{24}}{\sqrt{c}}\\\
\frac{\frac{2\gamma(\gamma+2i\pi)}{\pi^{2}}-\frac{47}{24}}{\sqrt{c}}\\\
\frac{(\gamma+i\pi)(4\gamma+3i\pi)(4\gamma+5i\pi)-\zeta(3)}{3\sqrt{c}\pi^{2}}\\\
\frac{768\gamma^{4}+3072i\gamma^{3}\pi-4512\gamma^{2}\pi^{2}-2880i\gamma\pi^{3}+671\pi^{4}-192(\gamma+i\pi)\zeta(3)}{144\sqrt{c}\pi^{2}}\\\
\end{array}\right).$

By application of the constraint

$S=\left(C^{-}_{\text{Kap}}\right)^{-1}e^{-\pi iR}e^{-\pi
i\mu}\eta^{-1}\left((C^{-}_{\text{Kap}})^{T}\right)^{-1},$

we find

$S_{\text{Kap}}=\left(\begin{array}[]{cccccc}1&-4&6&10&-20&20\\\
0&1&-4&-4&16&-20\\\ 0&0&1&0&-4&6\\\ 0&0&0&1&-4&10\\\ 0&0&0&0&1&-4\\\
0&0&0&0&0&1\\\ \end{array}\right),\quad\quad\quad
S_{\text{Kap}}^{-1}=\left(\begin{array}[]{cccccc}1&4&10&6&20&20\\\
0&1&4&4&16&20\\\ 0&0&1&0&4&10\\\ 0&0&0&1&4&6\\\ 0&0&0&0&1&4\\\ 0&0&0&0&0&1\\\
\end{array}\right).$

Now, $S_{\text{Kap}}^{-1}$ coincides with the Gram matrix $G_{\rm
Kap}=\left(\chi(\mathbb{S}^{\lambda}\mathcal{S}^{*},\mathbb{S}^{\mu}\mathcal{S}^{*})\right)_{\lambda,\mu}$
of the Kapranov exceptional collection.

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