$\big{(}\mathtt{i}^{-1}\mathtt{j}_{*}\operatorname{\mathtt{s}}_{>0}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q_{0}}$
given by
$\\{\Gamma_{\mathbb{R}}(\operatorname{\mathtt{s}}_{t}(q))\\}_{t>0,q}$. ∎

###  7.5 Functoriality

Let $\mathcal{V}_{0}$ be an open neighbourhood of
$\mathcal{C}\setminus\\{0_{+},0_{-}\\}$ as in Section 7.4.1. We consider the
open dense subset $\mathcal{V}^{\rm ss}_{0}$ of $\mathcal{V}_{0}$:

$\mathcal{V}^{\rm
ss}_{0}:=\\{q\in\mathcal{V}_{0}\cap\mathcal{M}^{\times}\colon\text{$F_{q}=F|_{\operatorname{pr}^{-1}(q)}$
has only non-degenerate critical points}\\}.$

Take $q_{0}\in\mathcal{V}^{\rm ss}_{0}$ and choose an admissible phase $\phi$
for the critical values of $F_{q_{0}}$. By composing the formal decomposition
of the analytified Brieskorn module in Proposition 6.10 and (the non-
equivariant version of) the mirror isomorphism in Theorem 4.34, we get formal
decompositions of the pulled-back quantum D-modules over an analytic open
neighbourhood $B$ of $q_{0}$:

$\operatorname{mir}_{\pm}^{*}\overline{\operatorname{QDM}}^{\rm
an}(\mathfrak{X}_{\pm})\big{|}_{B}\cong\bigoplus_{i=1}^{N_{\pm}}(\mathcal{O}_{B}[\\![z]\\!],d+d(\mathbf{u}_{i}/z),P_{\rm
std}),$

where $N_{\pm}=\dim H^{*}_{\rm CR}(\mathfrak{X}_{\pm})$ and $\mathbf{u}_{i}$
are relative critical values of $F$. By Proposition 6.5, by shrinking $B$ if
necessary, we have analytic lifts of these formal decompositions over $B\times
I_{\phi}$, where
$I_{\phi}=\big{\\{}\big{(}r,e^{\boldsymbol{\mathrm{i}}\theta}\big{)}\in{\widetilde{\mathbb{C}}}\colon|\theta-\phi|<\frac{\pi}{2}+\epsilon\big{\\}}$
(for some small $\epsilon>0$):

$\pi^{*}\operatorname{mir}_{\pm}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{\pm})\big{|}_{B\times
I_{\phi}}\cong\bigoplus_{i=1}^{N_{\pm}}(\mathcal{A}_{B\times
I_{\phi}},d+d(\mathbf{u}_{i}/z)),$ (7.14)

where
$\pi\colon\mathcal{M}^{\times}\times{\widetilde{\mathbb{C}}}\to\mathcal{M}^{\times}\times\mathbb{C}$
denotes the oriented real blowup along $\mathcal{M}\times\\{0\\}$. Summands of
this sectorial decomposition are indexed by relative critical points of $F$;
those for $\mathfrak{X}_{+}$ are indexed by all critical points and those for
$\mathfrak{X}_{-}$ are indexed by the subset of convergent critical points.
Combining these two decompositions, we get a sectorial decomposition:

$\pi^{*}\operatorname{mir}_{+}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})\big{|}_{B\times
I_{\phi}}\cong\pi^{*}\operatorname{mir}_{-}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})\big{|}_{B\times I_{\phi}}\oplus\mathcal{A}_{B\times
I_{\phi}}^{\oplus(N_{+}-N_{-})}.$ (7.15)

This gives an analytic lift of the formal decomposition in Theorem 5.16. We
consider the inclusion

$\pi^{*}\operatorname{mir}_{-}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})\big{|}_{B\times
I_{\phi}}\hookrightarrow\pi^{*}\operatorname{mir}_{+}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})\big{|}_{B\times I_{\phi}}$ (7.16)

induced by (7.15). Note that the maps (7.15), (7.16) are associated with a
semisimple point $q_{0}$ and an admissible direction
$e^{\boldsymbol{\mathrm{i}}\phi}$.

######  Theorem 7.25.

Let $\varphi\colon\mathfrak{X}_{+}\to\mathfrak{X}_{-}$ be a toric birational
morphism as in Section 7.1. For each point $q_{*}\in\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm
sm}^{\times\circ}\subset\mathcal{M}^{\rm sm}_{\mathbb{R}}$, there exists a
contractible open neighbourhood $W_{*}$ of $q_{*}$ in $\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times\circ}$ and positive
numbers $t_{*}\in(0,1)$, $\alpha_{0}\in\big{(}0,\frac{\pi}{2J}\big{)}$ such
that the following holds. For each
$q_{0}\in\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\cap\mathcal{V}^{\rm
ss}_{0}$ and each $\phi\in(-\alpha_{0},\alpha_{0})$ that is admissible for the
critical values of $F_{q_{0}}$, the inclusion (7.16) associated with $q_{0}$
and $e^{\boldsymbol{\mathrm{i}}\phi}$ is induced by the pull-back in
$K$-theory via the $\widehat{\Gamma}$-integral structure, i.e., it sends
$\mathfrak{s}_{V}$ to $\mathfrak{s}_{\varphi^{*}V}$ for $V\in
K(\mathfrak{X}_{-})$.

Recall from Section 7.3.2 that $\mathcal{V}^{\rm sm,ss}_{-,\mathbb{R}}$ is the
intersection of the semisimple locus $\mathcal{V}^{\rm ss}_{-}$ of
$\mathcal{V}_{-}$ and the real positive locus $\mathcal{M}^{\rm
sm}_{\mathbb{R}}$ of $\mathcal{M}^{\rm sm}$. Also, $\mathbb{D}_{\rm
sm}^{\times\circ}$ denotes the interior of $\mathbb{D}_{\rm
sm}^{\times}=\mathbb{D}_{\rm sm}\cap\mathcal{M}^{\rm sm,\times}$ as a subset
of $\mathcal{M}^{\rm sm}$. Recall that $\operatorname{\mathtt{s}}_{t}$ is the
sliding map from Definition 7.14 and $J\geq 1$ is the natural number in
Proposition 7.2. Note that
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\cap\mathcal{V}^{\rm
ss}_{0}$ is non-empty because $\mathcal{V}^{\rm
ss}_{0}\cap\mathcal{M}_{\mathbb{R}}$ is open dense in
$\mathcal{V}_{0}\cap\mathcal{M}_{\mathbb{R}}$ and
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$ is an open
subset of $\mathcal{V}_{0}\cap\mathcal{M}_{\mathbb{R}}$ by Lemma 7.15(2).

######  Remark 7.26.

As we explained in Remark 7.7, we have a standard identification between
$K$-classes of $\mathfrak{X}_{+}$ (resp. $\mathfrak{X}_{-}$) and flat sections
over $(\mathcal{M}_{\mathbb{R}}\cap\mathcal{V}_{+})\times\mathbb{R}_{>0}$
(resp. $(\mathcal{M}^{\rm
sm}_{\mathbb{R}}\cap\mathcal{V}_{-})\times\mathbb{R}_{>0}$). We use these
identifications in the above theorem; we also use the analytic continuation
along the sliding homotopy $\operatorname{\mathtt{s}}_{t}$ for
$\mathfrak{X}_{-}$.

######  Remark 7.27.

By the duality of the analytic lift discussed in Remark 6.7, if the inclusion
(7.16) corresponds to the pull-back $\varphi^{*}\colon K(\mathfrak{X}_{-})\to
K(\mathfrak{X}_{+})$ in $K$-theory, the projection associated with the
opposite direction
$-e^{\boldsymbol{\mathrm{i}}\phi}=e^{\boldsymbol{\mathrm{i}}(\phi+\pi)}$

$\pi^{*}\operatorname{mir}_{+}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})\big{|}_{B\times
I_{\phi+\pi}}\twoheadrightarrow\pi^{*}\operatorname{mir}_{-}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})\big{|}_{B\times I_{\phi+\pi}}$

corresponds to the push-forward $\varphi_{*}\colon K(\mathfrak{X}_{+})\to
K(\mathfrak{X}_{-})$ in $K$-theory.

######  Lemma 7.28.

Let $q_{*}$ be a point in $\mathcal{M}^{\rm
sm}_{\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times}$. We have a commutative
diagram

$\begin{aligned} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.1764pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{Lef}}^{\rm
sm}_{q_{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.13472pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\iota}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.1764pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-20.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.1764pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\big{(}\mathtt{i}^{-1}\mathtt{j}_{*}\operatorname{\mathtt{s}}_{>0}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q_{*}}\cong\big{(}\operatorname{\mathtt{s}}_{t}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q_{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
89.49396pt\raise-20.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 89.49396pt\raise-30.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K(\mathfrak{X}_{-})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.4574pt\raise-35.34723pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.29167pt\hbox{$\scriptstyle{\varphi^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 72.0731pt\raise-41.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.0731pt\raise-41.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K(\mathfrak{X}_{+})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{aligned}\qquad\text{with
$t\in(0,1]$},$ (7.17)

where the vertical arrows are induced by the isomorphisms in Theorems 7.5 and
7.12, the top horizontal arrow is the inclusion in Theorem 7.22.

###### Proof.

Note that the left vertical arrow is induced, via the
$\widehat{\Gamma}$-integral structure, by the isomorphism of local systems
over $\mathbb{D}_{\rm sm}^{\times}$:

$\operatorname{\mathsf{Lef}}^{\rm sm}\otimes\mathbb{C}\big{|}_{\mathbb{D}_{\rm
sm}^{\times}}\cong\big{(}\operatorname{mir}_{-}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})|_{\mathbb{D}_{\rm
sm}^{\times}\times\\{1\\}}\big{)}^{\nabla}$ (7.18)

sending a class $[\Gamma]\in H_{n}(\mathcal{Y}_{q},\\{\Re(F_{q})\gg 0\\})$ to
a flat section $s_{\Gamma}$ of
$\operatorname{mir}_{-}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})|_{\mathbb{D}_{\rm sm}^{\times}\times\\{1\\}}$ satisfying

$\frac{1}{(2\pi)^{n/2}}\int_{\Gamma}e^{-F_{q}}\Omega_{q,-1}=P(\operatorname{Mir}(\Omega),s_{\Gamma})$

for every local section $\Omega_{q,z}$ of $\operatorname{Bri}^{\rm sm}(F)$.
Here $(\cdots)^{\nabla}$ denotes the local system defined by the kernel of
$\nabla$. Also, the right vertical arrow in (7.17) is induced by the
isomorphism of local systems over
$\operatorname{Im}(\operatorname{\mathtt{s}}_{>0})$:

$\operatorname{\mathsf{Lef}}\otimes\mathbb{C}\big{|}_{\operatorname{Im}(\operatorname{\mathtt{s}}_{>0})}\cong\big{(}\operatorname{mir}_{+}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})|_{\operatorname{Im}(\operatorname{\mathtt{s}}_{>0})\times\\{1\\}}\big{)}^{\nabla},$
(7.19)

which is defined similarly. We conclude the lemma by studying the monodromy of
these local systems. Let
$\mathbb{L}^{\prime}=\operatorname{Ker}\big{(}\mathbb{Z}^{S_{-}}\to\mathbf{N}\big{)}$
be as in Section 6.3.3. The inclusion $\mathbb{D}_{\rm
sm}^{\times}\subset\mathcal{M}^{\rm
sm,\times}=(\mathbb{L}^{\prime})^{\star}\otimes\mathbb{C}^{\times}$ induces an
isomorphism $\pi_{1}(\mathbb{D}_{\rm
sm}^{\times})\cong(\mathbb{L}^{\prime})^{\star}$; an element
$\xi\in(\mathbb{L}^{\prime})^{\star}$ corresponds to the loop
$[0,2\pi]\ni\theta\mapsto e^{\boldsymbol{\mathrm{i}}\theta\xi}\cdot q_{*}$
based at $q_{*}\in\mathbb{D}_{\rm sm}^{\times}$. We claim that the monodromy
of $\operatorname{\mathsf{Lef}}^{\rm sm}$ around the loop
$e^{\boldsymbol{\mathrm{i}}\theta\xi}\cdot q_{*}$ corresponds to tensoring by
$L_{\xi}^{-1}$ in $K(\mathfrak{X}_{-})$ on the $\widehat{\Gamma}$-integral
structure under the isomorphism (7.18), where $L_{\xi}$ denotes the orbi-line
bundle as in Section 3.1.2. We also claim that the monodromy of
$\operatorname{\mathsf{Lef}}$ around
$\operatorname{\mathtt{s}}_{t}(e^{\boldsymbol{\mathrm{i}}\theta\xi}\cdot
q_{*})=(e^{\boldsymbol{\mathrm{i}}\theta\xi}\cdot q_{*},t)$ corresponds to
tensoring by $\varphi^{*}L_{\xi}^{-1}$ in $K(\mathfrak{X}_{+})$ under the
isomorphism (7.19). These two claims prove the lemma. In fact, since the map
$\iota$ is monodromy-equivariant, the composition

$K(\mathfrak{X}_{-})\cong\operatorname{\mathsf{Lef}}^{\rm
sm}_{q_{*}}\overset{\iota}{\longrightarrow}\big{(}\operatorname{\mathtt{s}}_{t}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q_{*}}\cong
K(\mathfrak{X}_{+})$ (7.20)

identifies tensoring by $L_{\xi}$ with tensoring by $\varphi^{*}L_{\xi}$; also
this map (7.20) sends the structure sheaf to the structure sheaf since $\iota$
sends the positive real Lefschetz thimble $\Gamma_{\mathbb{R}}(q_{*})$ to the
positive-real one $\Gamma_{\mathbb{R}}(\operatorname{\mathtt{s}}_{t}(q_{*}))$
(see Theorem 7.22); since $K(\mathfrak{X}_{-})$ is generated by line bundles
[16], we conclude that the map (7.20) equals $\varphi^{*}$.

It remains to prove the above two claims. Let $[\xi]$ denote the class in
$\operatorname{Pic}(\mathfrak{X}_{-})$ of $L_{\xi}$. By [62, (61)], the mirror
map satisfies

$\operatorname{mir}_{-}\big{(}e^{2\pi\boldsymbol{\mathrm{i}}\xi}\cdot
q_{*}\big{)}=g(-[\xi])\operatorname{mir}_{-}(q_{*})$

(see Section 2.2 for $g(-[\xi])$) and hence the flat section
$\mathfrak{s}_{V}(\operatorname{mir}_{-}(q_{*}),z=1)$ is analytically
continued, along the path $\theta\mapsto
e^{\boldsymbol{\mathrm{i}}\theta\xi}\cdot q_{*}$, to
$\mathfrak{s}_{V}(g(-[\xi])\operatorname{mir}_{-}(q_{*}),1)$, which is
identified with
$dg(-[\xi])^{-1}\mathfrak{s}_{V}(g(-[\xi])\operatorname{mir}_{-}(q_{*}),1)=\mathfrak{s}_{V\otimes
L_{\xi}^{-1}}(\operatorname{mir}_{-}(q_{*}),1)$ by the Galois action in
Section 2.3; here we used the formula (2.12). This proves the first claim. To
prove the second claim, it suffices to show that the homotopy class
${\hat{\xi}}\in\mathbb{L}^{\star}\cong\pi_{1}(\mathcal{M}^{\times})$ of the
loop $\operatorname{\mathtt{s}}_{t}(e^{\boldsymbol{\mathrm{i}}\theta\xi}\cdot
q_{*})$ defines a line bundle $L_{\hat{\xi}}$ isomorphic to
$\varphi^{*}L_{\xi}$. Observe that we have a direct sum decomposition

$\mathbb{L}=\mathbb{L}^{\prime}\oplus\mathbb{Z}\delta_{\hat{b}}^{\mathbf{\Sigma}_{-}},$
(7.21)

where
$\delta_{\hat{b}}^{\mathbf{\Sigma}_{-}}=e_{\hat{b}}-\Psi_{-}({\hat{b}})=-\mathbf{w}$
is an element introduced in Section 3.3 (see also Section 7.1) which
corresponds to the co-ordinate $t=t_{\hat{b}}$. The class
${\hat{\xi}}\in\mathbb{L}^{\star}$ is a unique lift of
$\xi\in(\mathbb{L}^{\prime})^{\star}$ such that
${\hat{\xi}}\cdot\delta_{\hat{b}}^{\mathbf{\Sigma}_{-}}=0$. The map (7.1)
inducing the birational morphism
$\varphi\colon\mathfrak{X}_{+}\to\mathfrak{X}_{-}$ is equivariant with respect
to the homomorphism
$\mathbb{L}\otimes\mathbb{C}^{\times}\to\mathbb{L}^{\prime}\otimes\mathbb{C}^{\times}$,
$\exp(\lambda)\mapsto\exp(\lambda+(D_{\hat{b}}\cdot\lambda)\mathbf{w})$ (where
$\lambda\in\mathbb{L}_{\mathbb{C}}$), which is dual to the above lift
$(\mathbb{L}^{\prime})^{\star}\to\mathbb{L}^{\star}$, $\xi\mapsto{\hat{\xi}}$.
In view of the definition of $L_{\hat{\xi}}$ in Section 3.1.2, we see that
$L_{\hat{\xi}}\cong\varphi^{*}L_{\xi}$. The lemma is proved. ∎

We show that $\alpha_{0}$ appearing in Proposition 7.8 can be chosen
independently of $q_{0}\in\mathcal{V}^{\rm ss}_{+,\mathbb{R}}$ if $q_{0}$ is
close to a given point $q_{*}$ in $\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times\circ}$.

######  Lemma 7.29.

Let $q_{*}$ be a point in $\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times\circ}$. There exist a
contractible open neighbourhood $W_{*}$ of $q_{*}$ in $\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times\circ}$ and positive
numbers $t_{*}\in(0,1)$, $\alpha_{0}\in\big{(}0,\frac{\pi}{2J}\big{)}$ such
that the conclusion of Proposition 7.8 holds for all
$q_{0}\in\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\cap\mathcal{V}^{\rm
ss}_{0}\subset\mathcal{V}^{\rm ss}_{+,\mathbb{R}}$ with this same
$\alpha_{0}$.

###### Proof.

Let $\operatorname{cr}_{1}(q),\dots,\operatorname{cr}_{N_{-}}(q)$ be branches
of “convergent” critical points of $F_{q}$ contained in $\mathcal{B}_{-}$ and
defined in a neighbourhood of $q=q_{*}$ in $\mathcal{V}^{\rm ss}_{-}$. We
assume that
$\operatorname{cr}_{1}(q_{*})=\operatorname{cr}_{\mathbb{R}}(q_{*})$ and write
$\mathbf{u}_{i}(q)=F_{q}(\operatorname{cr}_{i}(q))$ as before. We may also
assume that $\mathbf{u}_{i}(q_{*})\in\mathbf{u}_{1}(q_{*})+\mathbb{R}_{\geq
0}$ if and only if $1\leq i\leq k$. Choose
$\alpha_{0}\in\big{(}0,\frac{\pi}{2J}\big{)}$ so that the closed sector
$I_{*}:=\mathbf{u}_{1}(q_{*})+\big{\\{}re^{\boldsymbol{\mathrm{i}}\theta}\colon
r\geq 0,|\theta|\leq\alpha_{0}\big{\\}}$ with vertex at
$\mathbf{u}_{1}(q_{*})$ does not contain critical values other than
$\mathbf{u}_{1}(q_{*}),\dots,\mathbf{u}_{k}(q_{*})$, see Fig. 12. We can find
a contractible open neighbourhood $W_{*}$ of $q_{*}$ in $\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times\circ}$, $0<t_{*}<1$, and
$\epsilon>0$ such that for all $q\in\bigcup\limits_{0\leq
t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$,
$\mathbf{u}_{i}(q)\in-\epsilon+I_{*}$ if and only if $1\leq i\leq k$. Note
that $W_{*}\subset\mathcal{V}^{\rm sm,ss}_{-}\subset\mathcal{M}^{\rm sm,nd}$
and
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\subset\mathcal{V}_{0}\subset\mathcal{M}^{\rm
nd}$ by Lemma 7.15 and Remark 7.18.

Figure 12: Critical values at $q_{*}$: only
$\mathbf{u}_{1},\dots,\mathbf{u}_{k}$ (critical values on the half-line
$\mathbf{u}_{1}+\mathbb{R}_{\geq 0}$) lie in the sector
$-\alpha_{0}\leq\arg(\mathbf{u}-\mathbf{u}_{1})\leq\alpha_{0}$ ($k=5$).

We also have divergent critical points over
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$; let
$\operatorname{cr}_{j}(q)$, $j=N_{-}+1,\dots,N_{+}$ denote divergent critical
branches over $\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$
(they can be ramified and may not be single-valued). By taking smaller
$t_{*}>0$ if necessary, by Lemma 7.15(3), we may assume that the divergent
critical values $\mathbf{u}_{j}(q)=F_{q}(\operatorname{cr}_{j}(q))$,
$N_{-}+1\leq j\leq N_{+}$ do not lie in the sector $-\epsilon+I_{*}$ as long
as $q\in\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$.

Let $\Gamma^{\phi}_{i}(q)$ denote the Lefschetz thimble (6.12) of $F_{q}$
associated with the critical point $\operatorname{cr}_{i}(q)$ and the phase
$\phi$. Here $q$ can be either in $W_{*}$ or in
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$ – we have
$1\leq i\leq N_{-}$ in the former case and $1\leq i\leq N_{+}$ in the latter
case. In view of the proof of Proposition 7.8, it suffices to show that the
relative homology class

$\big{[}\Gamma_{1}^{\pi+\phi}(q)\big{]}\in
H_{n}\big{(}\mathcal{Y}_{q},\\{\Re(F_{q})\gg 0\\};\mathbb{Z}\big{)}$

represented by the thimble $\Gamma_{1}^{\pi+\phi}(q)$ is constant as
$(q,\phi)$ varies in
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\times(-\alpha_{0},\alpha_{0})$.
The homology class $\big{[}\Gamma_{1}^{\pi+\phi}(q)\big{]}$ can jump by the
Picard–Lefschetz transformation as $(q,\phi)$ varies (see, e.g., [7, Chapter
I]). By our choice of $W_{*}$, $t_{*}$ and $\alpha_{0}$, it suffices to check
that the intersection numbers of vanishing cycles at
$\operatorname{cr}_{1}(q)$ and $\operatorname{cr}_{i}(q)$ (with $2\leq i\leq
k$) are zero at some
$q\in\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$; here we
consider vanishing cycles associated with paths from $\mathbf{u}_{i}(q)$ to a
base point $\mathbf{u}_{0}\gg 0$ _inside_ the sector $-\epsilon+I_{*}$. At
$(q,\phi)=(q_{*},0)$, the Lefschetz thimble
$\Gamma^{\pi}_{1}(q_{*})=\Gamma_{\mathbb{R}}(q_{*})$ lying over
$\mathbf{u}_{1}(q_{*})+\mathbb{R}_{\geq 0}$ does not contain the critical
points $\operatorname{cr}_{i}(q_{*})$, $2\leq i\leq k$, and therefore the
vanishing cycles at $\operatorname{cr}_{1}(q_{*})$ and
$\operatorname{cr}_{i}(q_{*})$ (with $2\leq i\leq k$) do not intersect. For
$1\leq i\leq k$, choose a path $\gamma_{i}$ starting from
$\mathbf{u}_{i}(q_{*})$ and ending at $\mathbf{u}_{0}$ which avoids other
critical points, and let $\Gamma_{i}$ denote the finite Lefschetz thimble
(with boundary in $F_{q_{*}}^{-1}(\mathbf{u}_{0})$) associated with
$\operatorname{cr}_{i}(q_{*})$ and the path $\gamma_{i}$. We choose a
sufficiently big $\eta_{0}>0$ such that $\Gamma_{i}$’s are contained in
$A_{q_{*}}(\eta_{0})$ and that $\partial A_{q_{*}}(\eta_{0})\pitchfork
F_{q_{*}}^{-1}(\mathbf{u}_{0})$. Arguing as in Parts (1), (4) in the proof of
Theorem 7.22, we see that $\Gamma_{i}$ can be continuously deformed to a
relative cycle $\Gamma_{i}(q)$ of $\big{(}A_{q}(\eta_{0}),A_{q}(\eta_{0})\cap
F_{q}^{-1}(\mathbf{u}_{0})\big{)}$ as $q$ varies in a small neighbourhood of
$q_{*}$ in $\mathcal{V}^{\rm ss}_{-}$. This shows that the vanishing cycles
$\partial\Gamma_{1}(q)$ and $\partial\Gamma_{i}(q)$ (with $2\leq i\leq k$)
have zero intersection number. The lemma is proved. ∎

###### Proof of Theorem 7.25.

Choose $q_{*}\in\mathcal{V}^{\rm sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm
sm}^{\times\circ}$. Let $W_{*}\subset\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm sm}^{\times\circ}$, $t_{*}\in(0,1)$,
$\alpha_{0}\in\big{(}0,\frac{\pi}{2J}\big{)}$ as in Lemma 7.29. By taking
smaller $\alpha_{0}$ if necessary, we may assume that the conclusion of
Proposition 7.13 holds for the same $\alpha_{0}$ at $q_{0}=q_{*}$. As in the
proof of Lemma 7.29, let
$\operatorname{cr}_{1}(q),\dots,\operatorname{cr}_{N_{-}}(q)$ denote the
convergent critical branches over a neighbourhood of $q_{*}$ in
$\mathcal{V}^{\rm ss}_{-}$ such that $\operatorname{cr}_{1}(q)$ is the
conifold point when $q\in\mathcal{M}_{\mathbb{R}}\cup\mathcal{M}^{\rm
sm}_{\mathbb{R}}$. Also, let
$\operatorname{cr}_{N_{-}+1}(q),\dots,\operatorname{cr}_{N_{+}}(q)$ denote
divergent critical branches over a neighbourhood of $q_{*}$, defined away from
$\mathcal{M}^{\rm sm}$ (they can be ramified and may not be single-valued). We
write $\mathbf{u}_{i}(q)=F_{q}(\operatorname{cr}_{i}(q))$ for the
corresponding critical value. Let $\phi_{*}\in(-\alpha_{0},\alpha_{0})$ be an
admissible phase for
$\\{\mathbf{u}_{1}(q_{*}),\dots,\mathbf{u}_{N_{-}}(q_{*})\\}$. Proposition
7.13 gives an analytic decomposition

$\widetilde{\Phi}_{\phi_{*}}^{-}\colon\
\pi^{*}\operatorname{mir}_{-}^{*}\operatorname{QDM}(\mathfrak{X}_{-})\big{|}_{B^{\prime}\times
I_{\phi_{*}}}\cong\bigoplus_{i=1}^{N_{-}}(\mathcal{A}_{B^{\prime}\times
I_{\phi_{*}}},d+d(\mathbf{u}_{i}/z)),$

where $B^{\prime}$ is an open neighbourhood of $q_{*}$ in $\mathcal{V}^{\rm
ss}_{-}$,
$I_{\phi_{*}}=\big{\\{}\big{(}r,e^{\boldsymbol{\mathrm{i}}\theta}\big{)}\colon|\theta-\phi_{*}|<\frac{\pi}{2}+\epsilon\big{\\}}$
(for some $\epsilon>0$) and $\pi\colon\mathcal{V}^{\rm
ss}_{-}\times{\widetilde{\mathbb{C}}}\to\mathcal{V}^{\rm
ss}_{-}\times\mathbb{C}$ is the oriented real blowup, with the following
property: there exists a basis $V_{1}^{-},\dots,V_{N_{-}}^{-}$ of
$K(\mathfrak{X}_{-})$ such that $V_{1}^{-}$ is the structure sheaf of
$\mathfrak{X}_{-}$ and that
$\widetilde{\Phi}^{-}_{\phi}\big{(}e^{-\mathbf{u}_{j}(q)/z}s^{-}_{j}\big{)}=e_{j}$,
where $s^{-}_{j}$ is the flat section
$\mathfrak{s}_{V_{j}^{-}}(\operatorname{mir}_{-}(q),z)$ associated with
$V_{j}^{-}$. Choose
$\phi_{1},\phi_{2}\in\big{(}\phi_{*}-\frac{\epsilon}{2},\phi_{*}+\frac{\epsilon}{2}\big{)}$
(where $\epsilon$ is the one appearing in $I_{\phi_{*}}$) such that
$\phi_{1}<\phi_{2}$ and that all phases in $[\phi_{1},\phi_{2}]$ are
admissible for $\\{\mathbf{u}_{1}(q_{*}),\dots,\mathbf{u}_{N_{-}}(q_{*})\\}$.
By shrinking $W_{*}$ and taking smaller $t_{*}>0$ if necessary, we may assume
that

  * (a)

$\bigcup\limits_{0\leq t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\subset
B^{\prime}$;

  * (b)

$\mathbf{u}_{i}(q)-\mathbf{u}_{j}(q)\notin
e^{\boldsymbol{\mathrm{i}}\theta}\mathbb{R}$ for all
$q\in\bigcup\limits_{0\leq t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$,
$\theta\in[\phi_{1},\phi_{2}]$, $i,j\in\\{1,2,\dots,N_{-}\\}$ provided that
$\mathbf{u}_{i}(q_{*})\neq\mathbf{u}_{j}(q_{*})$;

  * (c)

$\bigcup\limits_{1\leq i\leq
N_{-}}\mathbf{u}_{i}(q)+\big{\\{}re^{\boldsymbol{\mathrm{i}}\theta}\colon
r\geq 0,|\theta|\leq\alpha_{0}\big{\\}}$ does not contain divergent critical
values $\mathbf{u}_{j}(q)$ with $N_{-}+1\leq j\leq N_{+}$ for
$q\in\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$.

The third point (c) is possible by Lemma 7.15(3) and the fact that
$\alpha_{0}<\frac{\pi}{2J}$.

Take
$q_{0}\in\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})\cap\mathcal{V}^{\rm
ss}_{0}$ and a phase $\phi\in(-\alpha_{0},\alpha_{0})$ which is admissible for
$\\{\mathbf{u}_{1}(q_{0}),\dots,\allowbreak\mathbf{u}_{N_{+}}(q_{0})\\}$.
First we prove the conclusion of the theorem _assuming that $\phi$ lies in
$(\phi_{1},\phi_{2})\subset(-\alpha_{0},\alpha_{0})$_. The discussion for any
admissible $\phi\in(-\alpha_{0},\alpha_{0})$ will be postponed until the last
paragraph of the proof. Since $\phi\in(-\alpha_{0},\alpha_{0})$, Lemma 7.29
(Proposition 7.8) gives an analytic decomposition
$\widetilde{\Phi}^{+}_{\phi}$

$\displaystyle\widetilde{\Phi}^{+}_{\phi}\colon\
\pi^{*}\operatorname{mir}_{+}^{*}\operatorname{QDM}(\mathfrak{X}_{+})\big{|}_{B\times
I_{\phi}}\cong\bigoplus_{i=1}^{N_{+}}(\mathcal{A}_{B\times
I_{\phi}},d+d(\mathbf{u}_{i}/z)),$

where $B$ is a neighbourhood of $q_{0}$ in $\mathcal{V}^{\rm ss}_{-}$ and
$I_{\phi}$ is a sector of the form
$\big{\\{}\big{(}r,e^{\boldsymbol{\mathrm{i}}\theta}\big{)}\in{\widetilde{\mathbb{C}}}\colon|\theta-\phi|<\frac{\pi}{2}+\delta\big{\\}}$
(for some $0<\delta\leq\frac{\epsilon}{2}$) with the following property: there
exists a basis $V_{1}^{+},\dots,V_{N_{+}}^{+}$ of $K(\mathfrak{X}_{+})$ such
that $V_{1}^{+}$ is the structure sheaf of $\mathfrak{X}_{+}$ and that
$\widetilde{\Phi}^{+}_{\phi}\big{(}e^{-\mathbf{u}_{j}(q)/z}s_{j}^{+}\big{)}=e_{j}$,
where $s_{j}^{+}$ is the flat section
$\mathfrak{s}_{V_{j}^{+}}(\operatorname{mir}_{+}(q),z)$ associated with
$V_{j}^{+}$. We may assume that $B\subset B^{\prime}$ by the condition (a)
above. We have $I_{\phi}\subset I_{\phi_{*}}$ because
$\phi\in(\phi_{1},\phi_{2})\subset\big{(}\phi_{*}-\frac{\epsilon}{2},\phi_{*}+\frac{\epsilon}{2}\big{)}$.
Therefore $\widetilde{\Phi}^{-}_{\phi_{*}}$ gives the analytic lift associated
with the point $q_{0}$ and the phase $\phi$. In particular, the decomposition
(7.15) is induced by $\widetilde{\Phi}^{+}_{\phi}$ and
$\widetilde{\Phi}^{-}_{\phi_{*}}$, and the inclusion (7.16) is induced by the
map $\kappa\colon K(\mathfrak{X}_{-})\to K(\mathfrak{X}_{+})$ sending
$V_{i}^{-}$ to $V_{i}^{+}$ for $1\leq i\leq N_{-}$. It now suffices to show
that $V_{i}^{+}=\varphi^{*}V_{i}^{-}$. We already know that this is true for
$i=1$.

Recall from the proof of Propositions 7.8, 7.13 that $V_{i}^{+}$ (resp.
$V_{i}^{-}$) corresponds to the Lefschetz thimble
$\Gamma_{i}^{\pi+\phi}(q_{0})$ of $F_{q_{0}}$ (resp. the Lefschetz thimble
$\Gamma_{i}^{\pi+\phi}(q_{*})$ of $F_{q_{*}}$) under the isomorphism in
Theorem 7.5 (resp. Theorem 7.12). We claim that the map sending
$\Gamma_{i}^{\pi+\phi}(q_{*})\in\operatorname{\mathsf{Lef}}^{\rm sm}_{q_{*}}$
to $\Gamma_{i}^{\pi+\phi}(q_{0})\in\operatorname{\mathsf{Lef}}_{q_{0}}$
coincides with the map $\iota$ from Theorem 7.22. Here we regard
$\Gamma_{i}^{\pi+\phi}(q_{0})$ as a germ of
$\mathtt{i}^{-1}\mathtt{j}_{*}\operatorname{\mathtt{s}}_{>0}^{-1}\operatorname{\mathsf{Lef}}$
at $q_{*}$ by extending it to a Gauss–Manin flat section over
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$.

Fix a sufficiently large $\mathbf{u}_{0}\gg 0$. Under the isomorphism

$H_{n}\big{(}\mathcal{Y}_{q_{*}},\\{\Re(F_{q_{*}})\gg 0\\}\big{)}\cong
H_{n}\big{(}\mathcal{Y}_{q_{*}},F_{q_{*}}^{-1}(\mathbf{u}_{0})\big{)},$

the class of $\Gamma_{i}^{\pi+\phi}(q_{*})$ corresponds to the class of a
finite Lefschetz thimble fibred over a bent ray as shown in Fig. 13(ii). We
move $q_{*}$ along the sliding map. For sufficiently small $t>0$, these finite
Lefschetz thimbles can be continuously deformed to finite thimbles for
$F_{\operatorname{\mathtt{s}}_{t}(q_{*})}$ with boundary in
$F_{\operatorname{\mathtt{s}}_{t}(q_{*})}^{-1}(\mathbf{u}_{0})$, as discussed
in Parts (1) and (4) in the proof of Theorem 7.22. By straightening these
paths in the direction $e^{\boldsymbol{\mathrm{i}}\phi}$ again, we see that
these relative cycles in
$\big{(}\mathcal{Y}_{\operatorname{\mathtt{s}}_{t}(q_{*})},F_{\operatorname{\mathtt{s}}_{t}(q_{*})}^{-1}(\mathbf{u}_{0})\big{)}$
correspond to the Lefschetz thimbles
$\Gamma_{i}^{\pi+\phi}(\operatorname{\mathtt{s}}_{t}(q_{*}))$ over the
straight ray
$\mathbf{u}_{i}(\operatorname{\mathtt{s}}_{t}(q_{*}))+e^{\boldsymbol{\mathrm{i}}\phi}\mathbb{R}_{\geq
0}$. Therefore, the map $\iota$ sends $\Gamma_{i}^{\pi+\phi}(q_{*})$ to
$\Gamma_{i}^{\pi+\phi}(\operatorname{\mathtt{s}}_{t}(q_{*}))$ for sufficiently
small $t>0$. The assumptions (b), (c) above ensure that the homology class of
$\Gamma_{i}^{\pi+\phi}(q)$ stays constant as $q$ varies inside
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$; the
Picard–Lefschetz transformation can only arise from the intersection of
vanishing cycles at $\operatorname{cr}_{i}(q)$ and $\operatorname{cr}_{j}(q)$
with $i\neq j$, $\mathbf{u}_{i}(q_{*})=\mathbf{u}_{j}(q_{*})$,
$i,j\in\\{1,\dots,N_{-}\\}$, but these intersection numbers vanish since the
vanishing cycles do not intersect at $q_{*}$. This proves the claim.

Figure 13: Deforming Lefschetz thimbles: (i) semi-infinite Lefschetz thimbles
fibred over the ray $\mathbf{u}_{j}(q_{*})+\mathbb{R}_{\geq
0}e^{\boldsymbol{\mathrm{i}}\phi}$; (ii) bent rays passing through
$\mathbf{u}_{0}$; (iii) moving from $q_{*}$ to a nearby point in
$\bigcup\limits_{0<t<t_{*}}\operatorname{\mathtt{s}}_{t}(W_{*})$; (iv)
straightening the paths again. In (iii) and (iv), the paths from
$\mathbf{u}_{4}$ and $\mathbf{u}_{5}$ overlaps, but it does not matter since
the relevant vanishing cycles do not intersect.

The above discussion gives a commutative diagram

$\textstyle{\operatorname{\mathsf{Lef}}^{\rm
sm}_{q_{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\cong}$$\textstyle{\big{(}\mathtt{i}^{-1}\mathtt{j}_{*}\operatorname{\mathtt{s}}_{>0}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q_{*}}\cong\big{(}\operatorname{\mathtt{s}}_{t}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q_{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K(\mathfrak{X}_{-})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{K(\mathfrak{X}_{+}),}$

where the vertical arrows are induced by the isomorphisms in Theorems 7.5 and
7.12, the top horizontal arrow is the inclusion in Theorem 7.22 and the bottom
arrow $\kappa$ sends $V_{i}^{-}$ to $V_{i}^{+}$ for $1\leq i\leq N_{-}$.
Comparing this with the commutative diagram in Lemma 7.28, we conclude that
$\kappa=\varphi^{*}$ as required.

Finally, we explain that the conclusion of the theorem holds for every
admissible phase $\phi\in(-\alpha_{0},\alpha_{0})$ for
$\\{\mathbf{u}_{1}(q_{0}),\dots,\mathbf{u}_{N_{+}}(q_{0})\\}$. As we reviewed
in Section 6.2, the analytic decompositions (7.14) for $\mathfrak{X}_{\pm}$
change by mutation as the phase $\phi$ varies; see [44, Sections 2.6, 4.2 and
4.3]. In the case at hand, the analytic decompositions are given by the basis
$\\{s_{j}^{\pm}\\}$ of flat sections associated with the $K$-classes
$\\{V_{j}^{\pm}\\}$; they give rise to an asymptotic basis in the sense of
Section 6.2. Their all possible mutations are completely determined by the
configuration $\\{\mathbf{u}_{1}(q_{0}),\dots,\mathbf{u}_{N_{\pm}}(q_{0})\\}$
of the critical values and the Euler pairings $\chi(V_{i}^{\pm},V_{j}^{\pm})$.
We have $\chi(V_{i}^{+},V_{j}^{+})=\chi(V_{i}^{-},V_{j}^{-})$ for $1\leq
i,j\leq N_{-}$ since $V_{i}^{+}=\varphi^{*}V_{i}^{-}$ and $\varphi$ is
birational. The condition (c) above ensures that the basis
$\\{s_{1}^{-},\dots,s_{N_{-}}^{-}\\}$ of flat sections for $\mathfrak{X}_{-}$
and the part $\\{s_{1}^{+},\dots,s_{N_{-}}^{+}\\}$ of the basis of flat
sections for $\mathfrak{X}_{+}$ undergo the same mutation when $\phi$ varies
in $(-\alpha_{0},\alpha_{0})$. Hence the relation
$V_{i}^{+}=\varphi^{*}V_{i}^{-}$ is preserved under mutation. The theorem is
proved. ∎

######  Remark 7.30.

We hope that we can analyze general discrepant transformations
$\mathfrak{X}_{+}\leftarrow\widehat{\mathfrak{X}}\rightarrow\mathfrak{X}_{-}$
by using the functoriality under blowups (Theorem 7.25) twice. In the case of
crepant toric wall crossings, we can see how a Fourier–Mukai transformation
(as discussed in [15, 31]) arises from this result, see the slides from [63].

###  7.6 Orlov’s decomposition and analytic lift

In the previous section, we have observed that the Lefschetz thimbles
associated with ‘convergent’ critical points (at some point
$q_{0}\in\mathcal{V}^{\rm ss}_{0}\cap\mathcal{M}_{\mathbb{R}}$ and for some
phase $\phi$) correspond to $K$-classes from
$\varphi^{*}(K(\mathfrak{X}_{-}))\subset K(\mathfrak{X}_{+})$. In this
section, we see that the remaining ‘divergent’ critical points correspond to
$K$-classes supported on the exceptional divisor of
$\varphi\colon\mathfrak{X}_{+}\to\mathfrak{X}_{-}$. We will see the
correspondence between Orlov’s semiorthogonal decomposition of the derived
category $D^{b}(\mathfrak{X}_{+})$ and the analytic decomposition (7.15) of
the quantum D-module of $\mathfrak{X}_{+}$.

####  7.6.1 Decomposition of the relative homology mirror to Orlov’s
decomposition

Recall from Lemma 7.15(3) that convergent critical values over
$\operatorname{\mathtt{s}}_{t}(\mathbb{D}_{\rm sm})$ are contained in
$B_{\rho_{2}}(0)$ and divergent critical values over
$\operatorname{\mathtt{s}}_{t}(\mathbb{D}_{\rm sm})$ are contained in the $J$
‘satellite’ discs $\mathfrak{B}_{k}(t):=t^{-1/J}\cdot
B_{\rho_{1}}\big{(}3\rho_{0}e^{(-2k-1)\pi\boldsymbol{\mathrm{i}}/J}\big{)}$,
$k\in\mathbb{Z}/J\mathbb{Z}$ (see Fig. 10). We choose $t_{1}>0$ small enough
so that $\overline{B_{\rho_{2}}(0)}+\mathbb{R}_{\geq 0}$ does not intersect
any other satellite discs $\overline{\mathfrak{B}_{k}(t)}$ when $t$ is real
and $0<t\leq t_{1}$. Then, if $|\phi|$ is sufficiently small, we have that

$\mathcal{S}_{\rm conv}:=\overline{B_{\rho_{2}}(0)}+\mathbb{R}_{\geq
0}e^{\boldsymbol{\mathrm{i}}\phi}$

does not intersect with any $\overline{\mathfrak{B}_{k}(t)}$ when $0<t\leq
t_{1}$. We choose a number $h\in\\{0,1,2,\dots,J\\}$ and let
$\mathcal{S}_{k}\subset\mathbb{C}$, $k=-h,-h+1,\dots,J-h-1$ be mutually
disjoint strip regions as in Fig. 14; each $\mathcal{S}_{k}$ is a closed
region disjoint from $\mathcal{S}_{\rm conv}$, emanating from
$\overline{\mathfrak{B}_{k}(t)}$, going around the origin by the angle
$(2k-1)\pi/J+\phi$ anticlockwise, and extending straight toward the direction
$e^{\boldsymbol{\mathrm{i}}\phi}$ near the end. These regions
$\mathcal{S}_{\rm conv}$, $\mathcal{S}_{k}$ depend on $t$, $\phi$ and are
defined when $0<t\leq t_{1}$ and $|\phi|$ is sufficiently small.

Figure 14: Strip regions $\mathcal{S}_{\rm conv}$ and $\mathcal{S}_{k}$. In
this picture, $J=5$, $h=3$ and $\phi=\frac{\pi}{30}$.

For sufficiently large $M\gg 0$,
$\bigcup\limits_{k=-h}^{J-h-1}\mathcal{S}_{k}\cup\mathcal{S}_{\rm
conv}\cup\\{\Re(\mathbf{u})\geq M\\}$ is a strong deformation retract of
$\mathbb{C}$. The fibration given by $F_{q,t}$ is smoothly trivial outside
this region by Corollary 7.21, and hence we get the decomposition of the
relative homology $\operatorname{\mathsf{Lef}}_{q,t}$ (see (7.10)):

$\operatorname{\mathsf{Lef}}_{q,t}=\operatorname{\mathsf{Lef}}^{(-h)}_{q,t}\oplus\cdots\oplus\operatorname{\mathsf{Lef}}^{(-1)}_{q,t}\oplus\operatorname{\mathsf{Lef}}_{q,t}^{\rm
conv}\oplus\operatorname{\mathsf{Lef}}_{q,t}^{(0)}\oplus\cdots\oplus\operatorname{\mathsf{Lef}}_{q,t}^{(J-h-1)}$
(7.22)

for $q\in\mathbb{D}_{\rm sm}^{\times}$ and $0<t\leq t_{1}$, where we use the
notation as in Sections 7.4.2–7.4.4 and

$\displaystyle\operatorname{\mathsf{Lef}}_{q,t}^{\rm
conv}=H_{n}\big{(}F_{q,t}^{-1}(\mathcal{S}_{\rm
conv}),F_{q,t}^{-1}(\mathcal{S}_{\rm conv})\cap\\{x\colon\Re(F_{q,t}(x))\geq
M\\};\mathbb{Z}\big{)},$
$\displaystyle\operatorname{\mathsf{Lef}}_{q,t}^{(k)}=H_{n}\big{(}F_{q,t}^{-1}(\mathcal{S}_{k}),F_{q,t}^{-1}(\mathcal{S}_{k})\cap\\{x\colon\Re(F_{q,t}(x))\geq
M\\};\mathbb{Z}\big{)}.$

This decomposition is independent of $\phi$ (with $|\phi|$ sufficiently small)
and is preserved by the Gauss–Manin connection. We say that a direct sum
decomposition $A=A_{1}\oplus\cdots\oplus A_{m}$ is _semiorthogonal_ with
respect to a (not necessarily symmetric) pairing $[\cdot,\cdot)$ on $A$ if
$[a_{i},a_{j})=0$ for $a_{i}\in A_{i}$, $a_{j}\in A_{j}$ whenever $i>j$. Note
that the decomposition (7.22) is semiorthogonal with respect to the pairing
$\\#\big{(}e^{-\pi\boldsymbol{\mathrm{i}}}\Gamma_{1}\cdot\Gamma_{2}\big{)}$
discussed in Remark 7.6.

Let $E\subset\mathfrak{X}_{+}$ be the exceptional divisor of the morphism
$\varphi\colon\mathfrak{X}_{+}\to\mathfrak{X}_{-}$; this is the toric divisor
corresponding to ${\hat{b}}\in R_{+}$. Let $Z=\varphi(E)$ denote the image of
$E$; this is a toric substack corresponding to the cone $\sigma_{M_{+}}$ of
$\mathbf{\Sigma}_{-}$ (see (7.1)). We write $\varphi_{E}\colon E\to Z$ for the
restriction of $\varphi$ to $E$ and $i_{E}\colon E\to\mathfrak{X}_{+}$ for the
inclusion.

######  Theorem 7.31.

For $q\in\mathbb{D}_{\rm sm}^{\times}\cap\mathcal{M}^{\rm sm}_{\mathbb{R}}$
and $0<t\leq t_{1}$, the decomposition (7.22) of the relative homology
corresponds to the decomposition of the $K$-group

$K(\mathfrak{X}_{+})=K(Z)_{-h}\oplus\cdots\oplus
K(Z)_{-1}\oplus\varphi^{*}K(\mathfrak{X}_{-})\oplus K(Z)_{0}\oplus\cdots\oplus
K(Z)_{J-h-1}$ (7.23)

under the isomorphism $\operatorname{\mathsf{Lef}}_{q,t}\cong
K(\mathfrak{X}_{+})$ in Theorem $\ref{thm:main_intpaper}(1)$, where $K(Z)_{k}$
denotes the subgroup $\mathcal{O}(-kE)\otimes i_{E*}\varphi_{E}^{*}K(Z)\subset
K(\mathfrak{X}_{+})$.

######  Remark 7.32.

The decomposition (7.23) is semiorthogonal with respect to the Euler pairing
by Remark 7.6 and the above theorem. In view of this, we expect that
$D^{b}(\mathfrak{X}_{+})$ admits a semiorthogonal decomposition:

$D^{b}(\mathfrak{X}_{+})=\big{\langle}D^{b}(Z)_{-h},\cdots,D^{b}(Z)_{-1},\varphi^{*}D^{b}(\mathfrak{X}_{-}),D^{b}(Z)_{0},\dots,D^{b}(Z)_{J-h-1}\big{\rangle},$

where $D^{b}(Z)_{j}$ denotes the full subcategory of $D^{b}(\mathfrak{X}_{+})$
which is the image of the functor $\mathcal{O}(-jE)\otimes
i_{E*}\varphi_{E}^{*}\colon D^{b}(Z)\to D^{b}(\mathfrak{X}_{+})$. When
$\mathfrak{X}_{+}$ is the blowup of a smooth _variety_ $\mathfrak{X}_{-}$
along a smooth centre $Z$, this is the semiorthogonal decomposition proved by
Orlov [87, Theorem 4.3]; Orlov stated the result for $h=J$, but the other
cases $0\leq h\leq J-1$ follow from this case by using the fact that
$D^{b}(Z)_{-h}=\mathbb{S}\big{(}D^{b}(Z)_{J-h}\big{)}$, where $\mathbb{S}$ is
the Serre functor for $D^{b}(\mathfrak{X}_{+})$.

###### Proof of Theorem 7.31.

Observe that the subgroup $\operatorname{\mathsf{Lef}}^{\rm
conv}_{q,t}\subset\operatorname{\mathsf{Lef}}_{q,t}$ is the image of the map
$\iota\colon\operatorname{\mathsf{Lef}}^{\rm
sm}_{q}\to\big{(}\operatorname{\mathtt{s}}_{t}^{-1}\operatorname{\mathsf{Lef}}\big{)}_{q}$
in Theorem 7.22 (e.g., by considering the case $\phi=0$). Hence it corresponds
to $\varphi^{*}(K(\mathfrak{X}_{-}))\subset K(\mathfrak{X}_{+})$ by Lemma
7.28. It suffices to show that $\operatorname{\mathsf{Lef}}_{q,t}^{(k)}$
corresponds to $K(Z)_{k}$ for $-h\leq k\leq J-h-1$. Let $K^{(k)}\subset
K(\mathfrak{X}_{+})$ denote the subgroup corresponding to
$\operatorname{\mathsf{Lef}}_{q,t}^{(k)}\subset\operatorname{\mathsf{Lef}}_{q,t}$.
It suffices to show that

  * (a)

$K(Z)_{k}\subset K^{(k)}$;

  * (b)

the decomposition (7.23) of the $K$-group holds true.

As discussed, over $\operatorname{\mathtt{s}}_{t}(\mathbb{D}_{\rm sm})$,
convergent critical values are contained in $B_{\rho_{2}}(0)$ and divergent
critical values are contained in the $J$ satellite discs
$\mathfrak{B}_{k}(t)$, $k\in\mathbb{Z}/J\mathbb{Z}$; these statement still
hold for a non-zero complex number $t$ with $|t|\leq 1$ by naturally extending
the definition of $\operatorname{\mathtt{s}}_{t}$ and $\mathfrak{B}_{k}(t)$ to
$t\in\mathbb{C}$ (see the proof of Lemma 7.15). Consider the class
$[\Gamma_{\mathbb{R}}]\in\operatorname{\mathsf{Lef}}_{q,t}$ of the positive
real Lefschetz thimble. It corresponds to the class $[\mathcal{O}]\in
K(\mathfrak{X}_{+})$ of the structure sheaf. We study the monodromy action on
$\operatorname{\mathsf{Lef}}_{q,t}$ along the loop
$[0,2\pi]\ni\theta\mapsto\big{(}q,e^{\boldsymbol{\mathrm{i}}\theta}t\big{)}$.
The monodromy corresponds to the tensor product by $\mathcal{O}(-E)$ on
$K(\mathfrak{X}_{+})$; this follows from the same argument as in the proof of
Lemma 7.28 and the fact that this loop corresponds to the element of
$\mathbb{L}^{\star}$ given by the second projection in the decomposition
(7.21) (which equals $D_{\hat{b}}\in\mathbb{L}^{\star}$). Under this
monodromy, $[\Gamma_{\mathbb{R}}]$ undergoes the Picard–Lefschetz
transformation. Among the satellite discs, only $\mathfrak{B}_{-1}(t)$ passes
through the strip region $\mathcal{S}_{\rm conv}$ as $t$ rotates; here
$\mathfrak{B}_{-1}(t)$ becomes
$\mathfrak{B}_{-1}\big{(}e^{2\pi\boldsymbol{\mathrm{i}}}t\big{)}=\mathfrak{B}_{0}(t)$
after rotation. Thus by the Picard–Lefschetz formula, the monodromy transform
of $[\Gamma_{\mathbb{R}}]$ equals $[\Gamma_{\mathbb{R}}]-\alpha_{0}$ for some
$\alpha_{0}\in\operatorname{\mathsf{Lef}}_{q,t}^{(0)}$. Then we find that
$\alpha_{0}$ corresponds to
$[\mathcal{O}]-[\mathcal{O}(-E)]=[\mathcal{O}_{E}]\in K(\mathfrak{X}_{+})$ and
in particular that it is non-zero. Similarly, by considering the monodromy
around the inverse loop
$\theta\mapsto(q,e^{-\boldsymbol{\mathrm{i}}\theta}t)$, we find an element
$\beta_{1}\in\operatorname{\mathsf{Lef}}_{q,t}^{(-1)}$ that corresponds to
$[\mathcal{O}(E)]-[\mathcal{O}]=[\mathcal{O}_{E}(E)]\in K(\mathfrak{X}_{+})$.
By considering further monodromy actions on $\alpha_{0},\beta_{1}$, we find
elements $\alpha_{i}\in\operatorname{\mathsf{Lef}}_{q,t}^{(i)}$ with $0\leq
i\leq J-h-1$ and $\beta_{j}\in\operatorname{\mathsf{Lef}}_{q,t}^{(-j)}$ with
$1\leq j\leq h$ such that $\alpha_{i}$ corresponds to $[\mathcal{O}_{E}(-iE)]$
and that $\beta_{j}$ corresponds to $[\mathcal{O}_{E}(jE)]$. Hence we have

$[\mathcal{O}_{E}(-kE)]=\mathcal{O}(-kE)\otimes[\mathcal{O}_{E}]\in K^{(k)}$
(7.24)

for $-h\leq k\leq J-h-1$. Note that the decomposition (7.22) is invariant
under monodromy in $q\in\mathbb{D}_{\rm sm}^{\times}$. As discussed in the
proof of Lemma 7.28, the monodromy around loops in $\mathbb{D}_{\rm
sm}^{\times}$ corresponds to the tensoring $\varphi^{*}L$ with
$L\in\operatorname{Pic}(\mathfrak{X}_{-})$. Since $K(\mathfrak{X}_{-})$ is
generated by line bundles, we conclude from (7.24) that
$\mathcal{O}(-kE)\otimes\mathcal{O}_{E}\otimes\varphi^{*}V\in K^{(k)}$ for all
$V\in K(\mathfrak{X}_{-})$. Part (a) follows from the fact that $K(Z)$ is also
generated by line bundles and that the natural map
$\operatorname{Pic}(\mathfrak{X}_{-})\to\operatorname{Pic}(Z)$ is surjective.

It remains to prove Part (b). In view of part (a) and the decomposition
(7.22), it suffices to show that $K(\mathfrak{X}_{+})$ is generated by the
classes of $\mathcal{O}$ and $\mathcal{O}_{E}(-kE)$, $-h\leq k\leq J-h-1$ as a
$K(\mathfrak{X}_{-})$-module, where the module structure is given by
$\varphi^{*}\colon K(\mathfrak{X}_{-})\to K(\mathfrak{X}_{+})$. Let
$L_{b}=L_{-D_{b}}\in K(\mathfrak{X}_{+})$ denote the class of the line bundle
associated with $-D_{b}\in\mathbb{L}^{\star}$ for $b\in S$ (see Section 3.1.2
for $L_{-D_{b}}$); $1-L_{b}$ is the class of the structure sheaf of the toric
divisor associated with $b$ when $b\in R_{+}$. Similarly, let $L_{b}^{-}\in
K(\mathfrak{X}_{-})$ denote the class of the line bundle associated with
$-D_{b}\in(\mathbb{L}^{\prime})^{\star}$ for $b\in S_{-}$. We also set
$L:=L_{{\hat{b}}}=[\mathcal{O}(-E)]\in K(\mathfrak{X}_{+})$. Recall from the
proof of Lemma 7.28 that the splitting
$(\mathbb{L}^{\prime})^{\star}\to\mathbb{L}^{\star}$, $\xi\mapsto{\hat{\xi}}$
induced by the decomposition (7.21) corresponds to the pull-back of line
bundles, i.e., $\varphi^{*}L_{\xi}\cong L_{{\hat{\xi}}}$. Since the splitting
$(\mathbb{L}^{\prime})^{\star}\to\mathbb{L}^{\star}$ sends $D_{b}$ to
$D_{b}+k_{b}D_{\hat{b}}$ (with
$k_{b}=-D_{b}\cdot\delta_{\hat{b}}^{\mathbf{\Sigma}_{-}}$ as before), we have

$\varphi^{*}L_{b}^{-}=L_{b}\cdot L^{k_{b}}.$ (7.25)

On the other hand, since the intersection of the toric divisors corresponding
to rays $b\in M_{+}$ is empty in $\mathfrak{X}_{+}$, we have the following
relation in $K(\mathfrak{X}_{+})$ (see [16]):

$\prod_{b\in M_{+}}(1-L_{b})=0.$ (7.26)

Combining the two relations (7.25), (7.26), we obtain

$\prod_{b\in M_{+}}\big{(}L^{k_{b}}-\varphi^{*}L_{b}^{-}\big{)}=0.$ (7.27)

The left-hand side of the relation is a monic polynomial in $L$ of degree
$J+1=\sum\limits_{b\in M_{+}}k_{b}$ with coefficients in $K(\mathfrak{X}_{-})$
whose constant term is an invertible element in $K(\mathfrak{X}_{-})$. Note
that $\operatorname{Pic}(\mathfrak{X}_{+})$ is generated by
$\varphi^{*}\operatorname{Pic}(\mathfrak{X}_{-})$ and $L^{\pm}$; therefore any
element in $K(\mathfrak{X}_{+})$ can be written as a Laurent polynomial of $L$
with coefficients in $K(\mathfrak{X}_{-})$. Using the above relation (7.27),
we find that every element of $K(\mathfrak{X}_{+})$ can be written in the form

$\sum_{k=-h}^{J-h}a_{k}\cdot L^{k}\qquad\text{with $a_{k}\in
K(\mathfrak{X}_{-})$},$

which can also be rewritten as a $K(\mathfrak{X}_{-})$-linear combination of
$1$ and $(1-L)L^{k}=[\mathcal{O}_{E}(-kE)]$ with $-h\leq k\leq J-h-1$. Part
(b) follows. The theorem is proved. ∎

####  7.6.2 Orlov’s decomposition as a sectorial decomposition of the quantum
D-module

Next we see that this Orlov-type decomposition actually arises from a
sectorial decomposition of the quantum D-module of $\mathfrak{X}_{+}$ – as
appears in Proposition 6.5 – associated with _some_ $\tau_{+}\in H^{*}_{\rm
CR}(\mathfrak{X}_{+})$; the point $\tau_{+}$ can be very far from the large
radius limit point and is not explicit in general (see however Example 7.34).

The analytic quantum D-module (6.1) is originally defined in a neighbourhood
of the large radius limit point. In the following theorem, we consider
analytic continuation of the quantum D-module along certain paths in
$H^{*}_{\rm CR}(\mathfrak{X}_{\pm})$; we choose a real class
$\tau_{\star,\pm}\in H^{*}_{\rm CR}(\mathfrak{X}_{\pm};\mathbb{R})$ which is
sufficiently close to the large radius limit point as a base point (see Remark
2.8).

######  Theorem 7.33.

There exist paths from $\tau_{\star,\pm}$ to $\tau_{\pm}\in H^{*}_{\rm
CR}(\mathfrak{X}_{\pm})$ and analytic continuation of the quantum D-modules
$\operatorname{QDM}^{\rm an}(\mathfrak{X}_{\pm})$ along these paths with the
following properties.

  * (1)

The eigenvalues of the Euler multiplication $E\star_{\tau_{+}}$ at $\tau_{+}$
have mutually distinct imaginary parts, and we have the following sectorial
decomposition as in Proposition 6.5

$\Phi^{+}\colon\ \pi^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})\big{|}_{W\times
I}\cong\bigoplus_{i=1}^{N_{+}}(\mathcal{A}_{W\times I},d+d(\mathbf{u}_{i}/z))$

over an open neighbourhood $W$ of $\tau_{+}$ in $H_{\rm
CR}^{*}(\mathfrak{X}_{+})$, where $\pi\colon H^{*}_{\rm
CR}(\mathfrak{X}_{+})\times{\widetilde{\mathbb{C}}}\to H^{*}_{\rm
CR}(\mathfrak{X}_{+})\times\mathbb{C}$ is the oriented real blow-up,
$I=\big{\\{}\big{(}z,e^{\boldsymbol{\mathrm{i}}\theta}\big{)}\in{\widetilde{\mathbb{C}}}\colon|\theta|<\frac{\pi}{2}+\epsilon\big{\\}}$
$($for some $\epsilon>0)$ and $\mathbf{u}_{1},\dots,\mathbf{u}_{N_{+}}$ are
the eigenvalues of the Euler multiplication.

  * (2)

Moreover, there exists a holomorphic submersion $f\colon W\to H^{*}_{\rm
CR}(\mathfrak{X}_{-})$ with $f(\tau_{+})=\tau_{-}$ such that we have the
similar sectorial decomposition for $\mathfrak{X}_{-}$

$\Phi^{-}\colon\ \pi^{*}f^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})\big{|}_{W\times
I}\cong\bigoplus_{i=1}^{N_{-}}(\mathcal{A}_{W\times
I},d+d(\mathbf{u}_{i}/z)),$

where $W$ and $I$ are the same as Part $(1)$ and
$\mathbf{u}_{1},\dots,\mathbf{u}_{N_{-}}$ are the pull-backs along $f$ of the
eigenvalues of the Euler multiplication in the quantum cohomology of
$\mathfrak{X}_{-}$ which form a subset of
$\\{\mathbf{u}_{1},\dots,\mathbf{u}_{N_{+}}\\}$ in Part $(1)$.

  * (3)

The eigenvalues $\\{\mathbf{u}_{1},\dots,\mathbf{u}_{N_{+}}\\}$ in Part $(1)$
are divided into $J+1$ groups
$\bigsqcup\limits_{k=-h}^{J-h-1}\big{\\{}\mathbf{u}^{(k)}_{i}\colon 1\leq
i\leq r\big{\\}}\sqcup\\{\mathbf{u}_{1},\dots,\mathbf{u}_{N_{-}}\\}$
satisfying

$\Im\big{(}\mathbf{u}^{(-h)}_{i_{1}}\big{)}>\cdots>\Im\big{(}\mathbf{u}^{(-1)}_{i_{h}}\big{)}>\Im(\mathbf{u}_{j})>\Im\big{(}\mathbf{u}^{(0)}_{i_{h+1}}\big{)}>\cdots>\Im\big{(}\mathbf{u}^{(J-h-1)}_{i_{J}}\big{)}$

for all $i_{k}\in\\{1,\dots,r\\}$ $($with $1\leq k\leq J)$ and
$j\in\\{1,\dots,N_{-}\\}$.

  * (4)

There exist $K$-classes $V_{i}^{\pm}\in K(\mathfrak{X}_{\pm})$, $1\leq i\leq
N_{\pm}$ such that
$\Phi^{\pm}\big{(}e^{\mathbf{u}_{i}/z}s_{i}^{\pm}\big{)}=e_{i}$, where
$s_{i}^{\pm}$ is the flat section associated with $V_{i}^{\pm}$ via the
$\widehat{\Gamma}$-integral structure $($Definition $\ref{def:s})$ $($which is
analytically continued from $\tau_{\star,\pm}$ through the specified paths$)$
and $e_{i}$ denotes the $i$th standard basis. Moreover, we have

    * $({\rm a})$

$V_{i}^{-}$, $1\leq i\leq N_{-}$ form a basis of $K(\mathfrak{X}_{-})$;

    * $({\rm b})$

$\big{\\{}V_{i}^{+}\colon\text{$\mathbf{u}_{i}=\mathbf{u}^{(k)}_{j}$ for some
$j$}\big{\\}}$ gives a basis of $K(Z)_{k}\subset K(\mathfrak{X}_{+})$;

    * $({\rm c})$

$V_{i}^{+}=\varphi^{*}V_{i}^{-}$ for $1\leq i\leq N_{-}$.

In particular, we have a sectorial decomposition

$\displaystyle\pi^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})\big{|}_{W\times I}\\!\cong\mathscr{R}^{\rm
an}_{-h}\oplus\cdots\oplus\mathscr{R}^{\rm
an}_{-1}\oplus\pi^{*}f^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})\big{|}_{W\times I}\oplus\mathscr{R}^{\rm
an}_{0}\oplus\cdots\oplus\mathscr{R}^{\rm an}_{J-h-1},$

which corresponds to the decomposition (7.23) of the $K$-group via the
$\widehat{\Gamma}$-integral structure, where $\mathscr{R}^{\rm
an}_{k}=\bigoplus\limits_{i=1}^{r_{k}}\big{(}\mathcal{A}_{W\times
I},d+d\big{(}\mathbf{u}_{i}^{(k)}/z\big{)}\big{)}$.

###### Proof.

Throughout Section 7, we have assumed that $S_{-}=S\cap\Delta_{-}$ and
$S=S_{-}\cup\\{{\hat{b}}\\}$ so that $\mathcal{M}$ is the small quantum
cohomology locus of $\mathfrak{X}_{+}$. The construction of the sectorial
decompositions (7.14), however, does not require this assumption because $S$
can be arbitrarily large in the discussion of Sections 5–6. We choose a large
enough finite set ${\widehat{S}}\subset\mathbf{N}\cap\Pi$ containing $S$ such
that the corresponding mirror map $\operatorname{mir}_{+}$ is generically
submersive (this is possible by the argument in [27, Section 7.4]). Let
$\widehat{\mathcal{M}}\supset\mathcal{M}$ denote the base of the LG model
corresponding to ${\widehat{S}}$. The discussion at the beginning of Section
7.5 yields, for any $q_{0}\in\mathcal{V}^{\rm ss}_{0}$ and an admissible phase
$\phi$ for the critical values of $F_{q_{0}}$, the following decompositions
(similarly to (7.14))

$\displaystyle\pi^{*}\operatorname{mir}_{\pm}^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{\pm})\big{|}_{\widehat{B}\times I_{\phi}}$
$\displaystyle\cong\bigoplus_{i=1}^{N_{\pm}}(\mathcal{A}_{\widehat{B}\times
I_{\phi}},d+d(\mathbf{u}_{i}/z))$ (7.28)

over a neighbourhood $\widehat{B}$ of $q_{0}$ in $\widehat{\mathcal{M}}$. Here
$(\mathbf{u}_{1},\dots,\mathbf{u}_{N_{+}})$ are the eigenvalues of
$E\star_{\tau}$ in the quantum cohomology of $\mathfrak{X}_{+}$ at
$\tau=\operatorname{mir}_{+}(q)$, and first $N_{-}$
$(\mathbf{u}_{1},\dots,\mathbf{u}_{N_{-}})$ of them are the eigenvalues of
$E\star_{\tau}$ in the quantum cohomology of $\mathfrak{X}_{-}$ at
$\tau=\operatorname{mir}_{-}(q)$, where $q$ varies in $\widehat{B}$. By the
choice of ${\widehat{S}}$, the eigenvalues
$\mathbf{u}_{1},\dots,\mathbf{u}_{N_{+}}$ are pairwise distinct at generic
points in $\widehat{B}$ (see Remark 6.2).

Suppose now that $q_{0}$ is a point from
$\bigcup\limits_{0<t<t_{1}}\operatorname{\mathtt{s}}_{t}\big{(}\mathcal{V}^{\rm
sm,ss}_{-,\mathbb{R}}\cap\mathbb{D}_{\rm
sm}^{\times\circ}\big{)}\cap\mathcal{V}^{\rm ss}_{0}$ and $|\phi|$ is
sufficiently small so that the conclusion of Theorem 7.25 holds; then there
exist $K$-classes $V_{i}^{\circ\pm}\in K(\mathfrak{X}_{\pm})$ such that the
associated flat sections $s_{i}^{\pm}$ (via the $\widehat{\Gamma}$-integral
structure) correspond to $e^{-\mathbf{u}_{i}/z}e_{i}$ under the decomposition
(7.28) and that $V_{i}^{\circ+}=\varphi^{*}V_{i}^{\circ-}$ for $1\leq i\leq
N_{-}$. Choose a point $q^{\circ}\in\widehat{B}$ such that the corresponding
eigenvalues $\mathbf{u}_{1}^{\circ},\dots,\mathbf{u}_{N_{+}}^{\circ}$ are
pairwise distinct, and set
$\tau^{\circ}_{\pm}=\operatorname{mir}_{\pm}(q^{\circ})$; we regard
$\tau^{\circ}_{\pm}$ as elements of $H_{\rm CR}^{*}(\mathfrak{X}_{\pm})$
(rather than their images in $[\mathcal{M}_{\rm
A}(\mathfrak{X}_{\pm})/\operatorname{Pic}^{\rm st}(\mathfrak{X}_{\pm})]$). Let
$C_{N}$ denote the configuration space of distinct $N$ points in $\mathbb{C}$:

$C_{N}:=\big{\\{}(\mathbf{u}_{1},\dots,\mathbf{u}_{N})\in\mathbb{C}^{N}\colon\text{$\mathbf{u}_{i}\neq\mathbf{u}_{j}$
if $i\neq j$}\big{\\}}/\mathfrak{S}_{N}.$

Since the eigenvalues of $E\star_{\tau}$ form a local co-ordinate system on
$H^{*}(\mathfrak{X}_{\pm})$ near a semisimple point, we can identify a
neighbourhood of
$\big{(}\mathbf{u}_{1}^{\circ},\dots,\mathbf{u}_{N_{\pm}}^{\circ}\big{)}$ in
$C_{N_{\pm}}$ with a neighbourhood of $\tau_{\pm}^{\circ}$ in $H^{*}_{{\rm
CR}}(\mathfrak{X}_{\pm})$. Let $\widetilde{C}_{N}$ denote the universal cover
of $C_{N}$. By isomonodromic deformation [40, Theorem 4.7], there exist
analytic hypersurfaces232323In the original version of the present paper, we
did not delete the hypersurfaces $H_{\pm}$ from $\widetilde{C}_{N_{\pm}}$. The
need for the deletion was pointed out by one of the referees.
$H_{\pm}\subset\widetilde{C}_{N_{\pm}}$ such that the quantum connection in a
neighbourhood of $\tau_{\pm}^{\circ}\in H^{*}_{\rm CR}(\mathfrak{X}_{\pm})$
can be extended to a meromorphic flat connection $\nabla$ on the trivial
bundle $H^{*}_{\rm
CR}(\mathfrak{X}_{\pm})\times\big{(}\widetilde{C}_{N_{\pm}}^{\circ}\times\mathbb{C}\big{)}\to\widetilde{C}_{N_{\pm}}^{\circ}\times\mathbb{C}$
of the form

$\nabla=d+\frac{1}{z}A+\left(-\frac{1}{z^{2}}U+\frac{1}{z}V\right)dz,$

where $\widetilde{C}_{N_{\pm}}^{\circ}:=\widetilde{C}_{N_{\pm}}\setminus
H_{\pm}$, $A$ is an $\operatorname{End}(H^{*}_{\rm
CR}(\mathfrak{X}_{\pm}))$-valued 1-form on $\widetilde{C}_{N_{\pm}}^{\circ}$,
$U$ and $V$ are $\operatorname{End}(H^{*}_{\rm
CR}(\mathfrak{X}_{\pm}))$-valued functions on
$\widetilde{C}_{N_{\pm}}^{\circ}$ and $A$, $U$, $V$ are independent of $z$.
Here the eigenvalues of $U$ give the co-ordinates
$(\mathbf{u}_{1},\dots,\mathbf{u}_{N_{\pm}})$ on $\widetilde{C}_{N_{\pm}}$.
Moreover, by taking bigger hypersurfaces $H_{\pm}$ if necessary, we may assume
that this defines a Frobenius manifold structure on
$\widetilde{C}^{\circ}_{N_{\pm}}$. By choosing a basis $\\{\phi_{i}\\}$ of
$H^{*}_{\rm CR}(\mathfrak{X}_{\pm})$, we can determine the flat vector fields
$\frac{\partial}{\partial\tau^{i}}$ by
$A_{\partial/\partial\tau^{i}}1=\phi_{i}$. In particular, we have a flat co-
ordinate system $\big{(}\widetilde{C}^{\circ}_{N_{\pm}}\big{)}^{~}\to
H^{*}_{\rm CR}(\mathfrak{X}_{\pm})$ on the universal cover
$\big{(}\widetilde{C}^{\circ}_{N_{\pm}}\big{)}^{~}$ which equals
$\tau^{\circ}_{\pm}$ at
$\big{(}\mathbf{u}^{\circ}_{1},\dots,\mathbf{u}^{\circ}_{N_{\pm}}\big{)}$. We
also have a submersion $g\colon\widetilde{C}_{N_{+}}\to\widetilde{C}_{N_{-}}$
sending $(\mathbf{u}_{1},\dots,\mathbf{u}_{N_{+}})$ to
$(\mathbf{u}_{1},\dots,\mathbf{u}_{N_{-}})$; this induces a submersion
$g\colon\widetilde{C}_{N_{+}}^{\circ\circ}\to\widetilde{C}_{N_{-}}^{\circ}$,
where
$\widetilde{C}_{N_{+}}^{\circ\circ}:=\widetilde{C}_{N_{+}}\setminus\big{(}H_{+}\cup
g^{-1}(H_{-})\big{)}\subset\widetilde{C}_{N_{+}}^{\circ}$.

We may assume that $q^{\circ}$ is sufficiently close to $q_{0}$ so that
$\big{\\{}\mathbf{u}_{1}^{\circ},\dots,\mathbf{u}_{N_{+}}^{\circ}\big{\\}}$
are contained in
$B_{\rho_{2}}(0)\sqcup\bigsqcup\limits_{k\in\mathbb{Z}/J\mathbb{Z}}\mathfrak{B}_{k}(t)$,
where $0<t<t_{1}$ is the number such that
$q_{0}\in\operatorname{Im}(\operatorname{\mathtt{s}}_{t})$. Moving
$\mathbf{u}_{i}^{\circ}$ along the strip regions $\mathcal{S}_{\rm
conv}\cup\mathcal{S}_{-h}\cup\cdots\cup\mathcal{S}_{J-h-1}$, we can connect
$\big{(}\mathbf{u}_{1}^{\circ},\dots,\mathbf{u}_{N_{+}}^{\circ}\big{)}$ by a
continuous path $\gamma$ inside $\widetilde{C}_{N_{+}}^{\circ\circ}$ with a
point
$\big{(}\mathbf{u}^{\Diamond}_{1},\dots,\mathbf{u}^{\Diamond}_{N_{+}}\big{)}\in\widetilde{C}_{N_{+}}^{\circ\circ}$
having mutually distinct imaginary parts: see Fig. 15. The numbers
$\mathbf{u}^{\Diamond}_{1},\dots,\mathbf{u}^{\Diamond}_{N_{+}}$ are divided
into $J+1$ groups as in Part (3) of the statement, depending on which strip
regions they belong to. Let $\tau_{+}\in H^{*}_{\rm CR}(\mathfrak{X}_{+})$ be
the analytically continued (along the path $\gamma$) flat co-ordinate at the
point
$\big{(}\mathbf{u}^{\Diamond}_{1},\dots,\mathbf{u}^{\Diamond}_{N_{+}}\big{)}\in\widetilde{C}_{N_{+}}^{\circ\circ}$
and $\tau_{-}\in H^{*}_{\rm CR}(\mathfrak{X}_{-})$ be the analytically
continued (along the path $g(\gamma)$) flat co-ordinate at the point
$g\big{(}\mathbf{u}^{\Diamond}_{1},\dots,\mathbf{u}^{\Diamond}_{N_{+}}\big{)}=\big{(}\mathbf{u}^{\Diamond}_{1},\dots,\mathbf{u}^{\Diamond}_{N_{-}}\big{)}\in\widetilde{C}_{N_{-}}^{\circ}$.
By the above construction, the quantum D-modules of $\mathfrak{X}_{\pm}$ are
analytically continued to $\tau_{\pm}$ along certain paths, and the submersion
$g$ induces, when written in flat co-ordinates, a submersion $f$ from a
neighbourhood of $\tau_{+}$ in $H^{*}_{\rm CR}(\mathfrak{X}_{+})$ to a
neighbourhood of $\tau_{-}$ in $H^{*}_{\rm CR}(\mathfrak{X}_{-})$ with
$f(\tau_{+})=\tau_{-}$.

Figure 15: A path from
$\big{(}\mathbf{u}_{1}^{\circ},\dots,\mathbf{u}_{N_{+}}^{\circ}\big{)}$ to
$\big{(}\mathbf{u}_{1}^{\Diamond},\dots,\mathbf{u}_{N_{+}}^{\Diamond}\big{)}$:
we move the left configuration $\big{(}\mathbf{u}_{i}^{\circ}\big{)}$ to the
right one $\big{(}\mathbf{u}_{i}^{\Diamond}\big{)}$ along the strip regions.

We claim that the statement of the theorem holds for $\tau_{\pm}$, $f$ and a
sufficiently small neighbourhood $W$ of $\tau_{+}$. Parts (1), (2) follow from
the construction and the Hukuhara–Turrittin decomposition (Proposition 6.5);
note that $0$ is an admissible phase for
$\big{(}\mathbf{u}_{1}^{\Diamond},\dots,\mathbf{u}_{N_{\pm}}^{\Diamond}\big{)}$.
Part (3) has been already achieved. We show Part (4). Recall that the
decomposition (7.28) corresponds to the basis
$\big{\\{}V_{i}^{\circ\pm}\big{\\}}$ of $K(\mathfrak{X}_{\pm})$ under the
$\widehat{\Gamma}$-integral structure and it gives rise to the asymptotic
basis (see Section 6.2) associated with $\tau_{\pm}^{\circ}$ and $\phi$. As we
move $(\mathbf{u}_{1},\dots,\mathbf{u}_{N_{\pm}})$ along the path $\gamma$ (or
$g(\gamma)$) and change the phase from $\phi$ to zero, this asymptotic basis
undergoes mutation as explained in Section 6.2. The basis
$\big{\\{}V_{i}^{\pm}\big{\\}}\subset K(\mathfrak{X}_{\pm})$ in Part (4)
arises from $\big{\\{}V_{i}^{\circ\pm}\big{\\}}$ by this mutation. Part (4-a)
is obvious from this construction. By the same argument as in the last
paragraph of the proof of Theorem 7.25, we see that the relation
$V_{i}^{\circ+}=\varphi^{*}V_{i}^{\circ-}$ is preserved by mutation, and we
get $V_{i}^{+}=\varphi^{*}V_{i}^{-}$ for $1\leq i\leq N_{-}$ as required in
Part (4-c). Let $\Gamma_{i}^{\pi+\phi}$ denote the Lefschetz thimbles (6.12)
of $F_{q_{0}}$ corresponding to the $i$th component of the decomposition
(7.28) (recall that components in (7.28) are naturally indexed by critical
points of $F_{q_{0}}$; see Proposition 6.18). As discussed in the proof of
Proposition 7.8, $V_{i}^{\circ+}$ equals the $K$-class
$V\big{(}\Gamma_{i}^{\pi+\phi}\big{)}$ associated with $\Gamma_{i}^{\pi+\phi}$
under the correspondence in Theorem 7.5. Note by Remark 7.6 that

$\chi\big{(}V_{i}^{\circ+},V_{j}^{\circ+}\big{)}=(-1)^{n(n-1)/2}\\#\big{(}e^{-\pi\boldsymbol{\mathrm{i}}}\Gamma_{i}^{\pi+\phi}\cdot\Gamma_{j}^{\pi+\phi}\big{)}.$

Therefore, mutation of $\big{\\{}V_{i}^{\circ+}\big{\\}}$ corresponds to
Picard–Lefschetz transformation (see [7, Chapter I]) of thimbles
$\big{\\{}\Gamma_{i}^{\pi+\phi}\big{\\}}$; here we consider Picard–Lefschetz
transformation with $q_{0}\in\mathcal{M}$ fixed. We transform
$\big{\\{}\Gamma_{i}^{\pi+\phi}\big{\\}}$ by the sequence of Picard–Lefschetz
transformations corresponding to the sequence of mutations that
$\big{\\{}V_{i}^{\circ+}\big{\\}}$ undergoes and obtain a new basis
$\\{\Gamma_{i}\\}$ of $\operatorname{\mathsf{Lef}}_{q_{0}}$. The basis
$\\{\Gamma_{i}\\}$ is the union of bases of
$\operatorname{\mathsf{Lef}}^{(k)}_{q_{0}}$ and
$\operatorname{\mathsf{Lef}}^{\rm conv}_{q_{0}}$;
$\big{\\{}\Gamma_{i}\colon\text{$\mathbf{u}_{i}=\mathbf{u}_{j}^{(k)}$ for some
$j$}\big{\\}}$ gives a basis of $\operatorname{\mathsf{Lef}}_{q_{0}}^{(k)}$.
Since we have $V_{i}^{+}=V(\Gamma_{i})$, Part (4-b) follows from Theorem 7.31.
The theorem is proved. ∎

######  Example 7.34.

We give an example where the Orlov-type decomposition occurs at an explicit
$\tau_{+}$. Set $X_{-}:=\mathbb{P}^{4}$ and let $X_{+}$ be the blowup of
$X_{-}$ along a line $Z=\mathbb{P}^{1}\subset X_{-}$. Both $X_{+}$ and $X_{-}$
are Fano, and their small quantum cohomologies are defined over polynomial
rings. Planes in $X_{-}=\mathbb{P}^{4}$ containing the line $Z$ are
parametrized by $\mathbb{P}^{2}$, and hence we have a natural projection
$X_{+}\to\mathbb{P}^{2}$. Thus $X_{+}$ is a $\mathbb{P}^{2}$-bundle over
$\mathbb{P}^{2}$:
$X_{+}\cong\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-1))$.
Let $p_{1}$ be the pull-back of the ample class in
$H^{2}\big{(}\mathbb{P}^{2};\mathbb{Z}\big{)}$ and let $p_{2}=\varphi^{*}(p)$
be the pull-back of the ample class $p$ in $H^{2}(X_{-};\mathbb{Z})$. They
form a nef basis of $H^{2}(X_{+};\mathbb{Z})$. The class of the exceptional
divisor is given by $[E]=p_{1}-p_{2}$. The uncompactified LG mirror of $X_{+}$
is given by the family of Laurent polynomials

$F_{q_{1},q_{2}}=x_{1}+x_{2}+x_{3}+x_{4}+\frac{q_{1}q_{2}}{x_{1}x_{2}x_{3}x_{4}}+\frac{1}{q_{1}}x_{1}x_{2}x_{3}$

parametrized by $(q_{1},q_{2})\in(\mathbb{C}^{\times})^{2}$. The large radius
limit (LRL) point for $X_{+}$ corresponds to $q_{1}=q_{2}=0$. The mirror map
for $X_{+}$ is trivial: $\operatorname{mir}_{+}(q_{1},q_{2})=p_{1}\log
q_{1}+p_{2}\log q_{2}$. On the other hand, the uncompactified LG mirror of
$X_{-}$ is given by the family

$F_{q,t}=x_{1}+x_{2}+x_{3}+x_{4}+\frac{q}{x_{1}x_{2}x_{3}x_{4}}+tx_{1}x_{2}x_{3},$

where the LRL point for $X_{-}$ is $q=t=0$. The two families are related by
the change of co-ordinates $q=q_{1}q_{2}$, $t=q_{1}^{-1}$. The small quantum
cohomology locus for $X_{-}$ is $t=0$. It is not easy to find a closed formula
for the mirror map of $X_{-}$, but we know that it has the asymptotic form
$\operatorname{mir}_{-}(q,t)\sim p\log q+t\cdot p^{3}$ as $(q,t)\to(0,0)$ (see
Remark 4.10) and that $\operatorname{mir}_{-}(q,0)=p\log q$. The Euler vector
field gives a grading $\deg q_{1}=2$, $\deg q_{2}=3$ and $\deg t=-2$, $\deg
q=5$ (in complex unit). Define a dimensionless parameter
$\lambda:=q_{1}^{-\frac{3}{2}}q_{2}=t^{\frac{5}{2}}q$. Critical points/values
of $F_{q_{1},q_{2}}=F_{q,t}$ are given by

$x_{1}=x_{2}=x_{3}=t^{-\frac{1}{2}}x,\qquad
x_{4}=t^{-\frac{1}{2}}\big{(}x+x^{3}\big{)},\qquad
F_{q,t}(\operatorname{cr})=t^{-\frac{1}{2}}\big{(}5x+3x^{3}\big{)},$

where $x$ is a root of $x^{5}\big{(}x^{2}+1\big{)}^{2}=\lambda$. The
discriminant locus (where $x$ has multiple roots) is $\lambda=0$ or $\pm
400\sqrt{5}\boldsymbol{\mathrm{i}}/3^{9}$. Therefore the quantum cohomology is
semisimple over the positive real locus $q_{1}>0$, $q_{2}>0$. The critical
values have the asymptotics

$F_{q,t}(\operatorname{cr})=\begin{cases}t^{-\frac{1}{2}}\big{(}5\lambda^{\frac{1}{5}}+\lambda^{\frac{3}{5}}+O(\lambda)\big{)},\\\
t^{-\frac{1}{2}}\big{(}2\boldsymbol{\mathrm{i}}\pm
2\sqrt{\boldsymbol{\mathrm{i}}}\lambda^{\frac{1}{2}}+O(\lambda)\big{)},\\\
t^{-\frac{1}{2}}\big{(}{-}2\boldsymbol{\mathrm{i}}\pm
2\sqrt{-\boldsymbol{\mathrm{i}}}\lambda^{\frac{1}{2}}+O(\lambda)\big{)}\end{cases}=\begin{cases}5q^{\frac{1}{5}}+tq^{\frac{3}{5}}+O\big{(}t^{2}\big{)},\\\
2\boldsymbol{\mathrm{i}}t^{-\frac{1}{2}}\pm
2\sqrt{\boldsymbol{\mathrm{i}}}q^{\frac{1}{2}}t^{\frac{3}{4}}+O\big{(}t^{2}\big{)},\\\
-2\boldsymbol{\mathrm{i}}t^{-\frac{1}{2}}\pm
2\sqrt{-\boldsymbol{\mathrm{i}}}q^{\frac{1}{2}}t^{\frac{3}{4}}+O\big{(}t^{2}\big{)}.\end{cases}$
(7.29)

as $\lambda\to 0$ (or $t\to 0$ with $q$ fixed) and

$\displaystyle
F_{q,t}(\operatorname{cr})=t^{-\frac{1}{2}}\big{(}3\lambda^{\frac{1}{3}}+3\lambda^{\frac{1}{9}}+O(\lambda^{-\frac{1}{9}})\big{)}=3q_{2}^{\frac{1}{3}}+3q_{1}^{\frac{1}{3}}q_{2}^{\frac{1}{9}}+O\big{(}q_{1}^{\frac{2}{3}}\big{)}$

as $\lambda\to\infty$ (or $q_{1}\to 0$ with $q_{2}$ fixed). The first line of
(7.29) gives 5 ‘convergent’ critical values (depending on the choice of
$q^{\frac{1}{5}}$) and the next two lines give 4 ‘divergent’ critical values
around $t=0$. They have mutually distinct imaginary parts when $t>0$ is
sufficiently close to zero and $q>0$. We will identify the corresponding
elements in the $K$-group for the phase $\phi=0$.

Figure 16: Move of critical values: the trajectories of the critical values
are shown in the middle picture; $\mathcal{E}$ is the left mutation of
$\mathcal{O}(-H_{1})$ with respect to $\mathcal{O}(-H_{2})$; similarly
$\mathcal{F}$ is the right mutation of $\mathcal{O}(H_{1})$ with respect to
$\mathcal{O}(H_{2})$; $\mathcal{G}$ is the right mutation of
$\mathcal{O}(-2H_{2})$ with respect to $\mathcal{O}(-H_{2})$.

We consider a path where the parameter $\lambda\in\mathbb{R}$ increases from
$0$ to $\infty$. It is convenient to take
$t=t(\lambda)=\lambda^{\frac{2}{3}}+\lambda^{\frac{2}{5}}$ so that
$(q_{1},q_{2})\sim\big{(}\lambda^{-\frac{3}{2}},1\big{)}$ as
$\lambda\to\infty$ and $(q,t)\sim\big{(}1,\lambda^{\frac{2}{5}}\big{)}$ as
$\lambda\to 0$. The move of the critical values is shown in Fig. 16. When
$\lambda$ is large, say $\lambda=12.5$, we can determine the mirror partners
(in the $K$-group) of some Lefschetz thimbles by using the monodromy action on
the $(q_{1},q_{2})$-space: see the rightmost picture of Fig. 16. Here $H_{1}$,
$H_{2}$ denote the pull-backs of the hyperplanes on $\mathbb{P}^{2}$ and
$\mathbb{P}^{4}$ respectively (so that $p_{i}$ is the Poincaré dual of
$H_{i}$). A numerical calculation on computer shows that, as $\lambda$
decreases from $\lambda=12.5$ to $\lambda=0.0009$, the thimble corresponding
to $\mathcal{O}(-H_{1})$ undergoes a Picard–Lefschetz transformation with
respect to that corresponding to $\mathcal{O}(-H_{2})$. As we saw in the proof
of Theorem 7.33, this corresponds to mutation in $K$-group. After (left)
mutation, $\mathcal{O}(-H_{1})$ becomes

$\mathcal{E}=\operatorname{Cone}\big{(}\operatorname{Hom}^{\bullet}(\mathcal{O}(-H_{2}),\mathcal{O}(-H_{1}))\otimes\mathcal{O}(-H_{2})\to\mathcal{O}(-H_{1})\big{)}=\mathcal{O}_{E}(E-H_{2}).$

Considering the monodromy action near $t=0$ and performing further mutation,
we find that the exceptional collection (adapted to the decomposition (7.23)
with $J=2$, $h=1$)

$\displaystyle\mathcal{E}=\mathcal{O}_{E}(E-H_{2}),\quad\mathcal{O}_{E}(E),\quad\mathcal{O}(-H_{2}),\quad\mathcal{G}=\mathcal{O}(-2H_{2})\otimes\varphi^{*}T_{\mathbb{P}^{4}}[-1],\quad\mathcal{O},$
$\displaystyle\mathcal{H}=\mathcal{O}(2H_{2})\otimes\varphi^{*}\Omega^{1}_{\mathbb{P}^{4}},\quad\mathcal{O}(H_{2}),\quad\mathcal{O}_{E}[-1],\quad\mathcal{F}=\mathcal{O}_{E}(H_{2})[-1]$

corresponds to the Lefschetz thimbles
$\Gamma_{1}^{\pi},\dots,\Gamma_{9}^{\pi}$ (6.12) ordered in such a way that
the imaginary parts of the corresponding critical values decrease
$\Im(F(\operatorname{cr}_{1}))>\Im(F(\operatorname{cr}_{2}))>\cdots>\Im(F(\operatorname{cr}_{9}))$.
These are the classes $V_{i}^{+}$ in Theorem 7.33(4) in this case. Note that
the corresponding sectorial decomposition occurs at a point $\tau_{+}=[E]\log
t+p_{2}\log q\in H^{2}(X_{+})$, $q,t>0$ such that $t$ is sufficiently small
(in the leftmost picture of Fig. 16, $t\approx 0.0698$, $q\approx 0.698$).

######  Remark 7.35.

We can use Theorem 7.33 to prove Gamma conjecture II [44, Section 4.6] in some
cases. By applying Theorem 7.33 to the case where $Z$ is a (non-stacky) point,
we know that if the Gamma conjecture II holds for a weak-Fano compact toric
stack $\mathfrak{X}_{-}$, then it also holds for a weighted blowup
$\mathfrak{X}_{+}$ of $\mathfrak{X}_{-}$ at a non-stacky torus-fixed point (as
long as $\mathfrak{X}_{+}$ is weak-Fano).

##  8 Conjecture and discussion

###  8.1 General conjecture

In view of our main results (Theorems 5.16, 5.19, 7.25, 7.31, and 7.33), we
conjecture the following phenomena for more general discrepant birational
transformations. Suppose that we have a birational transformation
$\varphi\colon\mathfrak{X}_{+}\dasharrow\mathfrak{X}_{-}$ between smooth (not
necessarily toric) DM stacks which fits into the diagram

$\textstyle{\widehat{\mathfrak{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{+}}$$\scriptstyle{f_{-}}$$\textstyle{\mathfrak{X}_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathfrak{X}_{-}}$

with $\widehat{\mathfrak{X}}$ smooth and $f_{\pm}$ projective birational
morphisms, such that
$f_{+}^{*}K_{\mathfrak{X}_{+}}-f_{-}^{*}K_{\mathfrak{X}_{-}}$ is an effective
divisor. We assume that the coarse moduli spaces of $\mathfrak{X}_{\pm}$ are
projective, but some of the discussions below can be also adapted to non-
compact cases.

We choose a base point $\tau_{\star,\pm}\in H^{*}_{\rm
CR}(\mathfrak{X}_{\pm};\mathbb{R})$ which is real and sufficiently close to
the large radius limit point.

######  Conjecture 8.1 (formal decomposition).

There exist paths from $\tau_{\star,\pm}$ to $\tau_{\pm}\in H^{*}_{\rm
CR}(\mathfrak{X}_{\pm})$ and a holomorphic map $f$ from a neighbourhood $W$ of
$\tau_{+}$ in $H^{*}_{\rm CR}(\mathfrak{X}_{+})$ to $H^{*}_{\rm
CR}(\mathfrak{X}_{-})$ with $f(\tau_{+})=\tau_{-}$ such that the quantum
D-modules $\operatorname{QDM}^{\rm an}(\mathfrak{X}_{\pm})$ are analytically
continued along these paths, and that we have the decomposition of the quantum
D-modules completed in $z$ $($see (6.2)$)$:

$\overline{\operatorname{QDM}^{\rm
an}}(\mathfrak{X}_{+})|_{W}\xrightarrow{\cong}f^{*}\overline{\operatorname{QDM}^{\rm
an}}(\mathfrak{X}_{-})|_{W}\oplus\mathscr{R}.$ (8.1)

Here $\mathscr{R}$ is a locally free $\mathcal{O}_{W}[\\![z]\\!]$-module
equipped with a meromorphic flat connection $\nabla^{\mathscr{R}}$ and a
$z$-sesquilinear pairing $P^{\mathscr{R}}$, and the formal decomposition
respects both the connection and the pairing $($in particular it is orthogonal
with respect to the pairing (2.9)$)$.

Building on the above conjecture, we can also state a conjecture comparing
higher genus Gromov–Witten theories. This involves quantization of the formal
decomposition (8.1). See Section 5.5 for the notation.

######  Conjecture 8.2 (comparison in all genera).

The ancestor potentials $\mathscr{A}_{\pm,\tau}$ of $\mathfrak{X}_{\pm}$ can
be analytically continued along the paths from $\tau_{\star,\pm}$ to
$\tau_{\pm}$ in Conjecture 8.1. There exist a family
$\mathscr{A}^{\prime}_{\tau}\in\operatorname{\mathfrak{AFock}}(V_{\tau},\mathbf{D}_{\tau})$
of tame functions such that

$T_{\mathbf{s}}{\widehat{U}}_{\tau}\mathscr{A}_{+,\tau}=\mathscr{A}_{-,f(\tau)}\otimes\mathscr{A}^{\prime}_{\tau}$

for $\tau\in W$, where
$\mathbf{s}=(-z\phi_{0},\mathbf{D}_{\tau})+U_{\tau}(z\phi_{0})$,
$V_{\tau}\subset\mathscr{R}_{\tau}$ is a $\mathbb{C}$-vector subspace such
that $\mathscr{R}_{\tau}=V_{\tau}[\\![z]\\!]$ and that $P^{\mathscr{R}}$
restricts to the $\mathbb{C}$-valued pairing $V_{\tau}\times
V_{\tau}\to\mathbb{C}$, $\mathbf{D}_{\tau}\in z\mathscr{R}_{\tau}$ and
$U_{\tau}\colon H^{*}_{\rm CR}(\mathfrak{X}_{+})[\\![z]\\!]\cong H^{*}_{\rm
CR}(\mathfrak{X}_{-})[\\![z]\\!]\oplus V_{\tau}[\\![z]\\!]$ is the unitary
isomorphism obtained from (8.1) by restricting to $\tau$ and flipping the sign
of $z$.

In this conjecture we implicitly assume that the action of the operator
$T_{\mathbf{s}}$ on ${\widehat{U}}_{\tau}\mathscr{A}_{+,\tau}$ is well-
defined; this holds if $\mathscr{A}_{+,\tau}$ is rational (see Section 5.5.2).
Note also that the space
$\operatorname{\mathfrak{AFock}}(V_{\tau},\mathbf{D}_{\tau})$ itself depends
only on $\mathscr{R}_{\tau}$ and $\mathbf{D}_{\tau}$ and does not depend on
the choice of $V_{\tau}$, but that $\mathscr{A}^{\prime}_{\tau}$ depends on
the choice of $V_{\tau}$.

Next we state a conjecture relating the analytic lift of the formal
decomposition (8.1) with a semiorthogonal decomposition of the $K$-group.
Recall from (2.11) that the action of $-\nabla_{z^{2}\partial_{z}}$ on
$\operatorname{QDM}^{\rm an}(\mathfrak{X}_{\pm})|_{z=0}$ equals the Euler
multiplication $E\star_{\tau}$. In particular, the formal decomposition (8.1)
implies that $E\star_{\tau}$ on $H_{\rm CR}^{*}(\mathfrak{X}_{+})$ is
conjugate to
$E\star_{f(\tau)}\oplus\big{(}{-}\nabla^{\mathscr{R}}_{z^{2}\partial_{z}}\big{)}$
on $H^{*}_{\rm CR}(\mathfrak{X}_{-})\oplus\mathscr{R}|_{z=0}$.

######  Conjecture 8.3 (analytic lift).

We can arrange $\tau_{\pm}$ and the paths from $\tau_{\star,\pm}$ to
$\tau_{\pm}$ in Conjecture 8.1 so that $($in addition to Conjecture
$\ref{conj:formal})$ the following holds.

  * (1)

We have a decomposition
$\big{(}\mathscr{R},\nabla^{\mathscr{R}},P^{\mathscr{R}}\big{)}=\big{(}\mathscr{R}_{1},\nabla^{\mathscr{R}_{1}},P^{\mathscr{R}_{1}}\big{)}\oplus\big{(}\mathscr{R}_{2},\nabla^{\mathscr{R}_{2}},P^{\mathscr{R}_{2}}\big{)}$,
where $\mathscr{R}_{i}$ is a locally free $\mathcal{O}_{W}[\\![z]\\!]$-module
equipped with a flat connection $\nabla^{\mathscr{R}_{i}}$ and a
$z$-sesquilinear pairing $P^{\mathscr{R}_{i}}$. There exist a phase
$\phi\in\mathbb{R}$ and real numbers $l_{2}<l_{1}$ such that

    * (a)

all eigenvalues $\mathbf{u}$ of
$\big{(}{-}\nabla^{\mathscr{R}_{1}}_{z^{2}\partial_{z}}\big{)}$ on
$\mathscr{R}_{1}|_{z=0}$ satisfy
$\Im\big{(}e^{-\boldsymbol{\mathrm{i}}\phi}\mathbf{u}\big{)}>l_{1}$,

    * (b)

all eigenvalues $\mathbf{u}$ of
$\big{(}{-}\nabla^{\mathscr{R}_{2}}_{z^{2}\partial_{z}}\big{)}$ on
$\mathscr{R}_{2}|_{z=0}$ satisfy
$\Im\big{(}e^{-\boldsymbol{\mathrm{i}}\phi}\mathbf{u}\big{)}<l_{2}$ and

    * (c)

all eigenvalues $\mathbf{u}$ of $E\star_{f(\tau)}$ on $H^{*}_{\rm
CR}(\mathfrak{X}_{-})$ satisfy
$l_{2}<\Im\big{(}e^{-\boldsymbol{\mathrm{i}}\phi}\mathbf{u}\big{)}<l_{1}$.

  * (2)

Moreover, we have a sector
$I_{\phi}=\big{\\{}\big{(}r,e^{\boldsymbol{\mathrm{i}}\theta}\big{)}\in{\widetilde{\mathbb{C}}}\colon|\theta-\phi|<\frac{\pi}{2}+\epsilon\big{\\}}$
with some $\epsilon>0$ and an analytic decomposition

$\pi^{*}\operatorname{QDM}^{\rm an}(\mathfrak{X}_{+})\big{|}_{W\times
I_{\phi}}\cong\mathscr{R}^{\rm
an}_{1}\oplus\pi^{*}f^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})\big{|}_{W\times I_{\phi}}\oplus\mathscr{R}^{\rm
an}_{2},$ (8.2)

where $\pi\colon W\times{\widetilde{\mathbb{C}}}\to W\times\mathbb{C}$ is the
oriented real blow-up and $\mathscr{R}^{\rm an}_{i}$ is a locally free
$\mathcal{A}_{W\times I_{\phi}}$-module equipped with a flat connection, such
that $\mathscr{R}_{i}\cong\mathscr{R}^{\rm
an}_{i}\otimes_{\mathcal{A}_{W\times I_{\phi}}}\mathcal{O}_{W}[\\![z]\\!]$,
$i=1,2$ and that the formal decomposition (8.1) is induced by (8.2).

  * (3)

Via the $\widehat{\Gamma}$-integral structure for $\mathfrak{X}_{+}$ and
$\mathfrak{X}_{-}$, the decomposition (8.2) is induced by a semiorthogonal
decomposition of the topological $K$-groups:

$K(\mathfrak{X}_{+})\cong K_{1}\oplus K(\mathfrak{X}_{-})\oplus K_{2}$ (8.3)

such that the associated inclusion $K(\mathfrak{X}_{-})\hookrightarrow
K(\mathfrak{X}_{+})$ respects the Euler pairing.

######  Remark 8.4.

We expect that the semiorthogonal decomposition (8.3) arises naturally from
geometry. In our setting, for example, we could hope that there is a
semiorthogonal decomposition of the bounded derived category of
$\mathfrak{X}_{+}$ of the form (see [9, Conjecture 4.3.7]):

$D^{b}(\mathfrak{X}_{+})=\big{\langle}\mathscr{C}_{1},D^{b}(\mathfrak{X}_{-}),\mathscr{C}_{2}\big{\rangle}.$
(8.4)

When $\mathfrak{X}_{\pm}$ are toric stacks arising from the variation of GIT
quotients as in Section 5, Ballard–Favero–Katzarkov [9, Theorem 5.2.1] showed
that such a semiorthogonal decomposition exists (see also [55, 71]). A
semiorthogonal summand of $D^{b}(\mathfrak{X}_{+})$ gives a $K$-motive (see
[52, Section 4]), which in turn defines a direct summand of the topological
$K$-group of $\mathfrak{X}_{+}$242424I thank Sergey Galkin for this remark and
pointing to the reference [52].. We expect that (8.3) arises from (8.4) in
this way (cf. [76] for the discussion on Hochschild homology). On the other
hand, in view of the deformation invariance of Gromov–Witten invariants, it is
more natural to state conjectures in terms of topological $K$-groups instead
of derived categories.

###  8.2 Functoriality and Riemann–Hilbert problem

We discuss how to recover the quantum cohomology of $\mathfrak{X}_{-}$ from
the quantum cohomology of $\mathfrak{X}_{+}$, assuming Conjecture 8.3. This
involves solving a Riemann–Hilbert boundary value problem.

######  Proposition 8.5.

Assume that the quantum D-modules $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{\pm})$ are of exponential type. Suppose that we are given
the analytic continuation of the quantum D-module $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})$ to a neighbourhood $W$ of $\tau_{+}\in H^{*}_{\rm
CR}(\mathfrak{X}_{+})$ and its formal decomposition

$\overline{\operatorname{QDM}^{\rm
an}}(\mathfrak{X}_{+})|_{W}=\mathscr{Q}\oplus\mathscr{R}$

corresponding to the decomposition (8.1) for which Conjecture 8.3 holds. Then
we can recover the map $f$ in Conjecture 8.3 and the quantum D-module
$f^{*}\operatorname{QDM}^{\rm an}(\mathfrak{X}_{-})$ of $\mathfrak{X}_{-}$
$($trivialized as a vector bundle over $W\times\mathbb{C})$ together with an
isomorphism $f^{*}\overline{\operatorname{QDM}^{\rm
an}}(\mathfrak{X}_{-})|_{W}\cong\mathscr{Q}$.

Before giving a proof of this proposition, we review the exponential type
assumption (see [70, Definition 2.12]). This was originally introduced by
Hertling–Sevenheck [57, Definition 8.1] under the name “require no
ramification”. We also review a mutation system of Sanda–Shamoto [101,
Definition 2.30].

The quantum D-module of $\mathfrak{X}=\mathfrak{X}_{\pm}$ is of exponential
type if for each $\tau\in H^{*}_{\rm CR}(\mathfrak{X})$, we have a formal
decomposition of the quantum D-module $\operatorname{QDM}^{\rm
an}(\mathfrak{X})_{\tau}:=\operatorname{QDM}^{\rm
an}(\mathfrak{X})|_{\\{\tau\\}\times\mathbb{C}}$ (see [57, Lemma 8.2]):

$\overline{\operatorname{QDM}^{\rm
an}}(\mathfrak{X})_{\tau}\cong\bigoplus_{\mathbf{u}\in\operatorname{Spec}(E\star_{\tau})}\big{(}e^{\mathbf{u}/z}\otimes\mathcal{F}_{\mathbf{u}}\big{)}\otimes_{\mathbb{C}\\{z\\}}\mathbb{C}[\\![z]\\!],$
(8.5)

where $\operatorname{Spec}(E\star_{\tau})$ denotes the set of (mutually
distinct) eigenvalues of $E\star_{\tau}$, $e^{\mathbf{u}/z}$ denotes the rank
one connection $(\mathbb{C}\\{z\\},d+d(\mathbf{u}/z))$ and
$\mathcal{F}_{\mathbf{u}}$ is a free $\mathbb{C}\\{z\\}$-module equipped with
a regular singular meromorphic connection. The decomposition is
(automatically) orthogonal with respect to the pairing $P$, and induces a
$z$-sesquilinear pairing $P_{\mathbf{u}}$ on each $\mathcal{F}_{\mathbf{u}}$;
$\mathcal{F}_{\mathbf{u}}$ is called the _regular singular piece_ in [57,
Lemma 8.2]. Moreover, the Hukuhara–Turrittin theorem (see [57, Lemma 8.3] in
this context) implies that, for each phase $\phi$ admissible for
$\operatorname{Spec}(E\star_{\tau})$, the formal decomposition admits a unique
analytic lift over the sector
$I_{\phi}=\big{\\{}\big{(}r,e^{\boldsymbol{\mathrm{i}}\theta}\big{)}\in{\widetilde{\mathbb{C}}}\colon|\theta-\phi|<\frac{\pi}{2}+\epsilon\big{\\}}$
(for some $\epsilon>0$)

$\pi^{*}\left(\operatorname{QDM}^{\rm
an}(\mathfrak{X})_{\tau}\right)\big{|}_{I_{\phi}}\cong\bigoplus_{\mathbf{u}\in\operatorname{Spec}(E\star_{\tau})}\pi^{*}(e^{\mathbf{u}/z}\otimes\mathcal{F}_{\mathbf{u}})\big{|}_{I_{\phi}},$
(8.6)

where $\pi\colon{\widetilde{\mathbb{C}}}\to\mathbb{C}$ is the oriented real
blowup. In Section 6.1, we discussed in details the special case of these
decompositions where the quantum cohomology is semisimple.

The analytic germ at $z=0$ of the quantum D-module $\operatorname{QDM}^{\rm
an}(\mathfrak{X})_{\tau}$ can be determined by the formal decomposition (8.5)
and the Stokes data. Sanda–Shamoto [101, Definition 2.30] encoded the Stokes
data in linear-algebraic data which they called a _mutation system_. This can
be viewed as a generalization of a marked reflection system in Section 6.2.
Let $V$ denote the space of flat sections of $\operatorname{QDM}^{\rm
an}(\mathfrak{X})_{\tau}$ over the sector
$I_{\phi}^{\times}=\big{\\{}z\in\mathbb{C}^{\times}\colon|\arg
z-\phi|<\frac{\pi}{2}+\epsilon\big{\\}}$. We define a pairing $[\cdot,\cdot)$
on $V$ by (cf. Section 6.2)

$[s_{1},s_{2})=P\big{(}s_{1}\big{(}e^{-\pi\boldsymbol{\mathrm{i}}}z\big{)},s_{2}(z)\big{)},$

where $s_{1}\big{(}e^{-\pi\boldsymbol{\mathrm{i}}}z\big{)}$ denotes the
analytic continuation of $s_{1}(z)$ along the path $[0,\pi]\ni\theta\mapsto
e^{-\boldsymbol{\mathrm{i}}\theta}z$. The mutation system (see [101,
Proposition 2.5, Section 2.7]) for $\operatorname{QDM}^{\rm
an}(\mathfrak{X})_{\tau}$ associated with the admissible phase $\phi$ is given
by the data252525We omitted the data of a labelling $\tau$ in [101, Definition
2.30] since it can be recovered from the phase $\phi$.

  * •

the tuple $(V,[\cdot,\cdot))$ of a vector space and a pairing;

  * •

the decomposition
$V\cong\bigoplus\limits_{\mathbf{u}\in\operatorname{Spec}(E\star_{\tau})}V_{\mathbf{u}}$
induced by the analytic lift (8.6), where $V_{\mathbf{u}}$ is the space of
flat sections of $e^{\mathbf{u}/z}\otimes\mathcal{F}_{\mathbf{u}}$ over the
sector $I_{\phi}^{\times}$

satisfying the semiorthogonality:

$[v_{1},v_{2})=0\qquad\text{if $v_{1}\in V_{\mathbf{u}_{1}}$, $v_{2}\in
V_{\mathbf{u}_{2}}$ and
$\Im\big{(}e^{-\boldsymbol{\mathrm{i}}\phi}\mathbf{u}_{1}\big{)}<\Im\big{(}e^{-\boldsymbol{\mathrm{i}}\phi}\mathbf{u}_{2}\big{)}$}.$

The pairing $[\cdot,\cdot)$ restricted to $V_{\mathbf{u}}$ is induced by the
pairing $P_{\mathbf{u}}$ on $\mathcal{F}_{\mathbf{u}}$ [57, Lemma 8.4]. It was
shown [101, Proposition 2.5, Section 2.5] that a mutation system is equivalent
to Stokes data (or a Stokes filtered local system) equipped with a pairing. By
the Riemann–Hilbert correspondence (see [101, Section 2.7] in this context),
the formal structure (8.5) and the mutation system together recover the
analytic germ at $z=0$ of $\operatorname{QDM}(\mathfrak{X})_{\tau}$.

###### Proof of Proposition 8.5.

Fix $\tau\in W$. We may assume that $\phi$ in Conjecture 8.3 is admissible for
the set $\operatorname{Spec}(E\star_{\tau})$ of eigenvalues of $E\star_{\tau}$
on $H^{*}_{\rm CR}(\mathfrak{X}_{+})$, by perturbing $\phi$ if necessary.
Under the assumption that $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})_{\tau}$ is of exponential type, the summands
$\mathscr{Q},\mathscr{R}_{1},\mathscr{R}_{2}$ of
$\overline{\operatorname{QDM}^{\rm an}}(\mathfrak{X}_{+})_{\tau}$ admit
decompositions similar to (8.5). We write $\operatorname{Spec}(\mathscr{Q})$
(resp. $\operatorname{Spec}(\mathscr{R}_{i})$) for the set of the eigenvalues
of the operators $-\nabla_{z^{2}\partial_{z}}$ on $\mathscr{Q}|_{z=0}$ (resp.
on $\mathscr{R}_{i}|_{z=0}$). By Conjecture 8.3(1), the sets
$\operatorname{Spec}(\mathscr{Q})$, $\operatorname{Spec}(\mathscr{R}_{1})$,
$\operatorname{Spec}(\mathscr{R}_{2})$ are mutually distinct; therefore
$\mathscr{Q}$, $\mathscr{R}_{1}$, $\mathscr{R}_{2}$ are partial sums of the
right-hand side of the formal decomposition (8.5) for
$\mathfrak{X}=\mathfrak{X}_{+}$. Writing
$V\cong\bigoplus\limits_{\mathbf{u}\in\operatorname{Spec}(E\star_{\tau})}V_{\mathbf{u}}$
for the mutation system of $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{+})_{\tau}$, we can decompose $V$ as

$V\cong V_{\mathscr{R}_{1}}\oplus V_{\mathscr{Q}}\oplus V_{\mathscr{R}_{2}}$
(8.7)

with
$V_{\mathscr{Q}}=\bigoplus\limits_{\mathbf{u}\in\operatorname{Spec}(\mathscr{Q})}V_{\mathbf{u}}$,
$V_{\mathscr{R}_{i}}=\bigoplus\limits_{\mathbf{u}\in\operatorname{Spec}(\mathscr{R}_{i})}V_{\mathbf{u}}$.
By Conjecture 8.3(2), the decomposition

$V_{\mathscr{Q}}=\bigoplus_{\mathbf{u}\in\operatorname{Spec}(\mathscr{Q})}V_{\mathbf{u}}$

gives the mutation system for $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})_{f(\tau)}$. The $\widehat{\Gamma}$-integral structure
identifies $(V,[\cdot,\cdot))$ with
$(K(\mathfrak{X}_{+})\otimes\mathbb{C},\chi)$ and the decomposition (8.7) is
identified with that (8.3) of $K(\mathfrak{X}_{+})$ by Conjecture 8.3(3). Then
we obtain an isomorphism

$\Phi\colon V_{\mathscr{Q}}\cong K(\mathfrak{X}_{-})\otimes\mathbb{C}.$ (8.8)

Since the inclusion $K(\mathfrak{X}_{-})\to K(\mathfrak{X}_{+})$ respects the
Euler pairing, we see that the restriction of the pairing $[\cdot,\cdot)$ on
$V$ to $V_{\mathscr{Q}}$ coincides with the pairing on $V_{\mathscr{Q}}$ as a
mutation system for $\operatorname{QDM}^{\rm an}(\mathfrak{X}_{-})_{f(\tau)}$.
Therefore the mutation system for $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})_{f(\tau)}$ can be recovered as a summand of the mutation
system for $\operatorname{QDM}^{\rm an}(\mathfrak{X}_{+})_{\tau}$. Thus we
recover the analytic germ at $z=0$ of $\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})_{f(\tau)}$ by the Riemann–Hilbert correspondence. We
write $\mathscr{Q}^{\rm an}$ for the germ so reconstructed.

We extend $\mathscr{Q}^{\rm an}$ to the trivial vector bundle
$\operatorname{QDM}^{\rm an}(\mathfrak{X}_{-})_{f(\tau)}$ over
$\mathbb{C}_{z}$ and construct the fundamental solution $L(f(\tau),z)$ for
$\mathfrak{X}_{-}$ (see Section 2.4) and the map $f$. Consider the trivial
bundle
$Q^{(\infty)}:=H^{*}(\mathfrak{X}_{-})\times\big{(}\mathbb{P}^{1}\setminus\\{0\\}\big{)}\to\big{(}\mathbb{P}^{1}\setminus\\{0\\}\big{)}$
equipped with the meromorphic connection

$\nabla^{(\infty)}_{z\partial_{z}}=z\frac{\partial}{\partial
z}-\frac{c_{1}(\mathfrak{X}_{-})\cup}{z}+\mu.$

This has $z^{-\mu}z^{c_{1}(\mathfrak{X})}$ as a fundamental solution and the
facts recalled in Section 2.4 imply that the quantum connection on
$\\{f(\tau)\\}\times\mathbb{C}_{z}$ is gauge equivalent to $\nabla^{(\infty)}$
via the gauge transformation by $L(f(\tau),z)$ (which is regular and the
identity at $z=\infty$). We glue this trivial bundle $Q^{(\infty)}$ with the
germ $\mathscr{Q}^{\rm an}$ of vector bundle at $z=0$ to get a vector bundle
$\widehat{Q}$ over $\mathbb{P}^{1}$. The gluing is given by sending a flat
section $s\in V_{\mathscr{Q}}$ over the sector $I_{\phi}^{\times}$ to the flat
section for $\nabla^{(\infty)}$:

$(2\pi)^{-n/2}z^{-\mu}z^{c_{1}(\mathfrak{X}_{-})}\big{(}\widehat{\Gamma}_{\mathfrak{X}_{-}}\cdot(2\pi\boldsymbol{\mathrm{i}})^{\deg_{0}/2}\operatorname{inv}^{*}\widetilde{\operatorname{ch}}(\Phi(s))\big{)}$

with $n=\dim\mathfrak{X}_{-}$, where $\Phi$ is the isomorphism in (8.8). In
view of the definition of the $\widehat{\Gamma}$-integral structure, this
glued bundle $\widehat{Q}$ must be isomorphic to the trivial extension of
$\operatorname{QDM}^{\rm an}(\mathfrak{X}_{-})_{f(\tau)}$ to $\mathbb{P}^{1}$
(with respect to the given trivialization). In particular, $\widehat{Q}$ is
trivial, and the identification $Q^{(\infty)}|_{\infty}\cong H^{*}_{\rm
CR}(\mathfrak{X}_{-})$ at infinity induces a global trivialization
$\widehat{Q}\cong H^{*}_{\rm CR}(\mathfrak{X}_{-})\times\mathbb{P}^{1}$. The
trivial bundle $\widehat{Q}$ equipped with a meromorphic connection gives the
quantum D-module $\operatorname{QDM}^{\rm an}(\mathfrak{X}_{-})_{f(\tau)}$.
Moreover, the isomorphism $\widehat{Q}|_{\mathbb{P}^{1}\setminus\\{0\\}}\cong
Q^{(\infty)}$ written in the respective trivializations gives the fundamental
solution $L(f(\tau),z)$ as an $\operatorname{End}(H^{*}_{\rm
CR}(\mathfrak{X}_{-}))$-valued function. Varying $\tau$ in $W$, we recover the
full quantum connection for $f^{*}\operatorname{QDM}^{\rm
an}(\mathfrak{X}_{-})$ from $L(f(\tau),z)$. We also recover $f(\tau)$ from the
expansion

$L(f(\tau),z)^{-1}1=1+\frac{f(\tau)}{z}+O\big{(}z^{-2}\big{)}.$

The proposition is proved. ∎

######  Remark 8.6.

Choosing fundamental solutions for regular singular pieces, we can formulate
the above reconstruction in terms of a Riemann–Hilbert boundary value problem
for the triple $(Y,Y^{-},L)$ of fundamental solutions, where $Y$ is a
fundamental solution on the sector $I_{\phi}^{\times}$ with prescribed
asymptotics, $Y^{-}$ is a fundamental solution on the opposite sector
$I_{\phi+\pi}^{\times}$ with prescribed asymptotics and $L$ is the fundamental
solution around $z=\infty$. In the semisimple case, this was explained in
details by Dubrovin [40, Lecture 4] and the method there applies to the
situation of our main Theorem 7.33. In the semisimple case, the formal
structure (8.5) is determined only by the eigenvalues of $E\star_{\tau}$, and
therefore the asymptotic basis (or, if any, the corresponding exceptional
collection) reconstructs the quantum D-module.

######  Remark 8.7.

When a candidate formal decomposition $\overline{\operatorname{QDM}^{\rm
an}}(\mathfrak{X}_{+})_{\tau}\cong\mathscr{R}_{1}\oplus\mathscr{Q}\oplus\mathscr{R}_{2}$
and a decomposition (8.3) of the $K$-group (corresponding to the analytic lift
via the $\widehat{\Gamma}$-integral structure) are given, what is non-trivial
in the above reconstruction is the triviality of the glued bundle
$\widehat{Q}$.

######  Remark 8.8.

The fundamental solution $L(\tau,z)$ in Section 2.4 is called a _calibration_
in the theory of Frobenius manifolds. A calibration of a Frobenius manifold is
not unique in general and its ambiguity was discussed in [40, Lemma 4.1]. The
above procedure recovers, not only (the germ of) the quantum cohomology
Frobenius manifold of $\mathfrak{X}_{-}$ (if $f$ is submersive), but also its
calibration.262626I thank Vasily Golyshev for asking me a question which led
me to this observation. Since the $\widehat{\Gamma}$-integral structure was
normalized at the large radius limit point, it somehow ‘remembers’ the limit
point.

######  Remark 8.9.

Conjecture 8.3 is closely related to Dubrovin’s conjecture [39, Conjecture
4.2.2] (or Gamma conjecture [44, Section 4.6] or Dubrovin-type conjecture
[101, Definition 5.2]). It can be viewed as a relative version of these
conjectures.

##  Appendix A The Brieskorn module in the weak Fano case

We discuss the coherence and the locally-freeness of the equivariant Brieskorn
module $\operatorname{Bri_{\mathbb{T}}}(F)$ mirror to the small quantum
cohomology of a weak-Fano smooth toric DM stack $\mathfrak{X}$, near the large
radius limit point $0_{\mathbf{\Sigma}}$. We stated this in Proposition 6.15.
We can describe the Brieskorn module as a certain variant of the GKZ system
(see [27, Section 5.2]). The coherence of the relevant GKZ system near the
large radius limit point has been discussed by the author [62, Proposition
4.4] (on a neighbourhood of $0_{\mathbf{\Sigma}}$ with the logarithmic locus
deleted), Reichelt–Sevenheck [91, Theorem 3.7] (on a neighbourhood of
$0_{\mathbf{\Sigma}}$ for toric manifolds) and Mann–Reichelt [81, Theorem
4.10] (similarly for toric orbifolds). We adapt the argument of [81] to our
Brieskorn module with minor modifications. Note that we impose only minimal
assumptions on $\mathfrak{X}$ and $S$; we do not assume compactness of
$\mathfrak{X}$ nor a generation condition for $S$. We also deal with the
equivariant case. In this sense our result is slightly more general than [81].

Let $\mathfrak{X}=\mathfrak{X}_{\mathbf{\Sigma}}$ be a semiprojective smooth
toric DM stack from Section 3.1.1 which is weak Fano, i.e.,
$-K_{\mathfrak{X}}$ is nef. Let $\Delta\subset\mathbf{N}_{\mathbb{R}}$ denote
the fan polytope, that is, the union of simplices spanned by
$\\{0\\}\cup\\{b\in R(\mathbf{\Sigma})\colon\overline{b}\in\sigma\\}$ over all
maximal cones $\sigma$ of $\mathbf{\Sigma}$. The weak Fano assumption implies
that $\Delta$ is a convex polytope. We assume that the set $S$ (which is used
to construct the LG mirror) is contained in the fan polytope:

$S\subset\mathbf{N}\cap\Delta.$ (A.1)

This means that the base space $\mathcal{M}$ of the LG model corresponds to
the _small quantum cohomology locus_ of $\mathfrak{X}$ in Section 6.3. The
definition of the Brieskorn module on the affine chart
$\operatorname{Spec}\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$
does not rely on Assumption 3.1 and we do not need this assumption in the
following discussion.

The equivariant Brieskorn module $\operatorname{Bri_{\mathbb{T}}}(F)$ on the
affine chart
$\operatorname{Spec}\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$ is
given by the module

$\mathsf{G}:=\mathbb{C}[\mathbb{O}(\mathbf{\Sigma})_{+}][z].$

Recall from Section 4.1 that, by choosing a splitting
$\mathbb{L}_{\mathbb{C}}^{\star}\to\big{(}\mathbb{C}^{S}\big{)}^{\star}$,
$\xi\mapsto{\hat{\xi}}$ of (3.4), $\mathsf{G}$ has the structure of a module
over the ring of differential operators:

$\mathsf{D}:=R_{\mathbb{T}}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}][z]\left\langle
z\xi q\frac{\partial}{\partial
q}\colon\xi\in\mathbb{L}^{\star}_{\mathbb{C}}\right\rangle,$

where $\chi_{i}\in R_{\mathbb{T}}$ acts by $zx_{i}\frac{\partial}{\partial
x_{i}}+x_{i}\frac{\partial F}{\partial x_{i}}$ and $z\xi
q\frac{\partial}{\partial q}$ acts by $z{\hat{\xi}}u\frac{\partial}{\partial
u}+{\hat{\xi}}u\frac{\partial F}{\partial u}$. For convenience, we choose co-
ordinates
$q_{1},\dots,q_{m}\in\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})]$
corresponding to a $\mathbb{Z}$-basis of
$\boldsymbol{\Lambda}(\mathbf{\Sigma})$ and write
$\theta_{i}=q_{i}\frac{\partial}{\partial q_{i}}$; then we have

$\mathsf{D}=R_{\mathbb{T}}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}][z]\langle
z\theta_{1},\dots,z\theta_{m}\rangle.$

Let $\mathscr{G}$ and $\mathscr{D}$ denote the sheaves on

$B:=\operatorname{Spec}(R_{\mathbb{T}}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}][z])=\operatorname{Spec}(\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}])\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$

corresponding to $\mathsf{G}$ and $\mathsf{D}$ respectively. The main result
in the appendix is the following.

######  Proposition A.1.

Suppose, as above, that $\mathfrak{X}$ is a weak Fano semiprojective toric
stack and that $S$ satisfies (A.1). There exists a Zariski open subset $U$ of
$\operatorname{Spec}(\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}])$
containing the large radius limit point $0_{\mathbf{\Sigma}}$ such that
$\mathscr{G}|_{U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}}$ is a
locally free coherent
$\mathcal{O}_{U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}}$-module
of rank $\dim H^{*}_{\rm CR}(\mathfrak{X})$.

###  A.1 Generators and relations

By Remark 3.17, the equivariant Brieskorn module $\mathsf{G}$ is generated by
$w_{v}:=u^{(\Psi^{\mathbf{\Sigma}}(\overline{v}),v)}$ with
$v\in\mathbf{N}\cap\Pi$ as a
$\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}][z]$-module. Relations
among these generators as a $\mathsf{D}$-module are given as follows.

######  Lemma A.2.

For $b\in S$, we write $e_{b}^{\star}=\widehat{D}_{b}+\chi(b)$ with
$\chi(b)\in\mathbf{M}$ $($recall that
$D_{b}=D(e_{b}^{\star})\in\mathbb{L}^{\star})$. We have the following relation
in $\mathsf{G}$:

$\displaystyle\left(\prod_{b\in
S}\prod_{c=0}^{\nu_{b}-1}\left(zD_{b}q\frac{\partial}{\partial
q}+\chi(b)-\big{(}\Psi^{\mathbf{\Sigma}}_{b}(v)+c\big{)}z\right)\right)w_{v}$
$\displaystyle\qquad{}=\left(\prod_{b\in
S}u_{b}^{\nu_{b}}\right)w_{v}=q^{\Psi(v)+\sum_{b\in
S}\nu_{b}e_{b}-\Psi(v+\sum_{b\in S}\nu_{b}b)}w_{v+\sum_{b\in S}\nu_{b}b}$

for every $(\nu_{b})_{b\in S}\in(\mathbb{Z}_{\geq 0})^{S}$ and
$v\in\mathbf{N}\cap\Pi$.

###### Proof.

By definition, $zD_{b}q\frac{\partial}{\partial q}+\chi(b)$ acts on
$\mathsf{G}$ by $zu_{b}\frac{\partial}{\partial u_{b}}+u_{b}$. The first
equality follows from this and $u_{b}\frac{\partial}{\partial
u_{b}}w_{v}=\Psi_{b}(v)w_{v}$. The second equality is just by definition, see
Section 3.5. ∎

When $b\in R(\mathbf{\Sigma})$ and $v\in\mathbf{N}\cap\Pi$ lie in the same
cone of $\Sigma$, we have a relation

$w_{v+b}=\left(zD_{b}q\frac{\partial}{\partial
q}+\chi(b)-z\Psi_{b}^{\mathbf{\Sigma}}(v)\right)w_{v}$

in $\mathsf{G}$ by Lemma A.2. From this we can see that $\mathsf{G}$ is
generated by finitely many $w_{v}$ as a $\mathsf{D}$-module. For example, the
set $\\{w_{v}\colon v\in\operatorname{Box}(\mathbf{\Sigma})\\}$ generates
$\mathsf{G}$. By Lemma A.2, $w_{v}$ is annihilated by
$P_{v,\lambda}\in\mathsf{D}$

$\displaystyle P_{v,\lambda}:=\prod_{b\in
S\colon\lambda_{b}>0}\prod_{c=0}^{\lambda_{b}-1}\left(zD_{b}q\frac{\partial}{\partial
q}+\chi(b)-\big{(}\Psi^{\mathbf{\Sigma}}_{b}(v)+c\big{)}z\right)$
$\displaystyle\hphantom{P_{v,\lambda}:=}{}-q^{\lambda}\prod_{b\in
S\colon\lambda_{b}<0}\prod_{c=0}^{-\lambda_{b}-1}\left(zD_{b}q\frac{\partial}{\partial
q}+\chi(b)-\big{(}\Psi^{\mathbf{\Sigma}}_{b}(v)+c\big{)}z\right)$ (A.2)

for any
$\lambda\in\mathbb{L}\cap\operatorname{\widehat{NE}}(\mathfrak{X}_{\mathbf{\Sigma}})\subset\mathbb{Z}^{S}$,
cf. [27, Section 5.1].

###  A.2 Characteristic variety and coherence

Define an increasing filtration $\mathcal{F}_{l}(\mathscr{D})$ of
$\mathscr{D}$ by the rank of differential operators, i.e.,
$\mathcal{F}_{l}(\mathscr{D})$ consists of differential operators of the form

$\sum_{k_{1}+\cdots+k_{m}\leq
l}a_{k}(q,\chi,z)(z\theta_{1})^{k_{1}}\cdots(z\theta_{m})^{k_{m}}.$

Choose generators $w_{v_{1}},\dots,w_{v_{k}}$ of $\mathsf{G}$ as a
$\mathsf{D}$-module and introduce a filtration on $\mathscr{G}$ by

$\mathcal{F}_{l}(\mathscr{G}):=\sum_{i=1}^{k}\mathcal{F}_{l}(\mathscr{D})w_{v_{i}}.$

An easy argument shows that if
$\operatorname{gr}_{\mathcal{F}}(\mathscr{G})|_{V}$ is finitely generated as
an $\mathcal{O}_{V}$-module for an open set $V\subset B$, then
$\mathscr{G}|_{V}$ is also finitely generated as an $\mathcal{O}_{V}$-module.
We shall show that $\operatorname{gr}_{\mathcal{F}}(\mathscr{G})$ is finitely
generated on a neighbourhood of
$\\{0_{\mathbf{\Sigma}}\\}\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$.

The associated graded module $\operatorname{gr}_{\mathcal{F}}(\mathscr{G})$ is
a $\operatorname{gr}_{\mathcal{F}}(\mathscr{D})$-module generated by
$w_{v_{1}},\dots,w_{v_{k}}$. We have
$\operatorname{gr}_{\mathcal{F}}(\mathscr{D})=\mathcal{O}_{B}[\xi_{1},\dots,\xi_{m}]$,
where $\xi_{i}$ denotes the image of
$z\theta_{i}\in\mathcal{F}_{1}(\mathscr{D})$. The _characteristic variety_
$\operatorname{Ch}(\mathscr{G})$ of $\mathscr{G}$ is defined to be the support
of $\operatorname{gr}_{\mathcal{F}}(\mathscr{G})$ as an
$\mathcal{O}_{B}[\xi_{1},\dots,\xi_{m}]$-module. It is a closed subset of
$B\times\mathbb{C}^{m}$ invariant under the dilation
$(\xi_{1},\dots,\xi_{m})\mapsto(\lambda\xi_{1},\dots,\lambda\xi_{m})$ with
$\lambda\in\mathbb{C}^{\times}$. Then
$\operatorname{gr}_{\mathcal{F}}(\mathscr{G})|_{V}$ is a finitely generated
$\mathcal{O}_{V}$-module over the following set $V$:

$V=\\{x\in
B\colon(x,\xi)\in\operatorname{Ch}(\mathscr{G})\Longrightarrow\xi=0\\}.$ (A.3)

Note that $\operatorname{Ch}(\mathscr{G})$ induces a closed subset $C\subset
B\times\mathbb{P}^{m-1}$ and $V$ is the complement of $\pi(C)$, where
$\pi\colon B\times\mathbb{P}^{m-1}\to B$ is the projection; thus $V$ is
Zariski-open.

For a differential operator
$a\in\mathcal{F}_{l}(\mathscr{D})\setminus\mathcal{F}_{l-1}(\mathscr{D})$, its
_principal symbol_ $\sigma(a)$ is the image of $a$ in
$\operatorname{gr}_{\mathcal{F}}(\mathscr{D})$; explicitly

$\displaystyle\sigma(a):=\sum_{k_{1}+\cdots+k_{m}=l}a_{k}(q,\chi,z)\xi_{1}^{k_{1}}\cdots\xi_{m}^{k_{m}}\qquad\text{if
$a=\sum\limits_{k_{1}+\dots+k_{m}\leq
l}a_{k}(q,\chi,z)(z\theta_{1})^{k_{1}}\cdots(z\theta_{m})^{k_{m}}$.}$

The principal symbol of the relation $P_{v,\lambda}$ in (A.2) is given by

$\sigma(P_{v,\lambda})=\begin{cases}\displaystyle\prod_{b\in
S\colon\lambda_{b}>0}D_{b}(\xi)^{\lambda_{b}}-q^{\lambda}\prod_{b\in
S\colon\lambda_{b}<0}D_{b}(\xi)^{-\lambda_{b}}&\displaystyle\text{if}\
\sum\limits_{b\in S}\lambda_{b}=0,\\\ \displaystyle\prod_{b\in
S\colon\lambda_{b}>0}D_{b}(\xi)^{\lambda_{b}}&\displaystyle\text{if}\
\sum\limits_{b\in S}\lambda_{b}>0,\\\ \displaystyle-q^{\lambda}\prod_{b\in
S\colon\lambda_{b}<0}D_{b}(\xi)^{-\lambda_{b}}&\displaystyle\text{if}\
\sum\limits_{b\in S}\lambda_{b}<0,\end{cases}$

where $D_{b}(\xi):=\sigma\big{(}zD_{b}q\frac{\partial}{\partial q}\big{)}$ is
a linear form in $\xi_{1},\dots,\xi_{m}$. Because $\sigma(P_{v,\lambda})$ is
independent of $v$, it is an annihilator of
$\operatorname{gr}_{\mathcal{F}}(\mathscr{G})$. Therefore
$\operatorname{Ch}(\mathscr{G})$ is contained in the closed subset of
$B\times\mathbb{C}^{m}$ defined by $\sigma(P_{v,\lambda})=0$ for all
$\lambda\in\mathbb{L}\cap\operatorname{\widehat{NE}}(\mathfrak{X})$.

######  Lemma A.3.

There exists a Zariski-open subset $U$ of
$\operatorname{Spec}\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$
containing $0_{\mathbf{\Sigma}}$ such that
$U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$ is contained in the
locus $V$ in (A.3).

###### Proof.

Note that $\sigma(P_{v,\lambda})$ does not depend on
$(\chi,z)\in\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$. Therefore it
suffices to show that $\xi=0$ if $\xi\in\mathbb{C}^{m}$ satisfies

$\sigma(P_{v,\lambda})\big{|}_{q=0_{\mathbf{\Sigma}}}=0\qquad\text{for all
$\lambda\in\mathbb{L}\cap\operatorname{\widehat{NE}}(\mathfrak{X})$.}$ (A.4)

Suppose that $\xi\in\mathbb{C}^{m}$ satisfies (A.4). We first show that there
exists a cone $\sigma\in\Sigma$ such that $\\{b\in S\colon D_{b}(\xi)\neq
0\\}=R(\mathbf{\Sigma})\cap\sigma$. Let $\\{b_{1},\dots,b_{s}\\}$ be the set
of $b\in S$ such that $D_{b}(\xi)\neq 0$. The relative interior of the convex
hull of $\\{b_{1},\dots,b_{s}\\}$ intersects with the relative interior of
some cone $\sigma$ of $\Sigma$. Hence we get a relation of the form

$\sum_{i=1}^{s}c_{i}\overline{b}_{i}-\sum_{b\in
R(\mathbf{\Sigma})\cap\sigma}f_{b}\overline{b}=0$

for some $c_{i}>0$ satisfying $\sum\limits_{i=1}^{s}c_{i}=1$ and some
$f_{b}>0$. The convexity of $\Delta$ together with (A.1) implies that
$\sum\limits_{b\in R(\mathbf{\Sigma})\cap\sigma}f_{b}\leq 1$. We may further
assume that $c_{i}$ and $f_{b}$ are rational numbers. Then for some positive
integer $l$,

$\lambda:=\sum_{i=1}^{s}lc_{i}e_{b_{i}}-\sum_{b\in
R(\mathbf{\Sigma})\cap\sigma}lf_{b}e_{b}\in\mathbb{Z}^{S}$

belong to $\mathbb{L}\cap\operatorname{\widehat{NE}}(\mathfrak{X})$ (recall
the definition of $\operatorname{\widehat{NE}}(\mathfrak{X})$ around (3.14)).
Note that

$\sum\limits_{b\in
S}\lambda_{b}=l\left(\sum\limits_{i=1}^{s}c_{i}-\sum\limits_{b\in
R(\mathbf{\Sigma})\cap\sigma}f_{b}\right)\geq 0.$

Therefore, if $\lambda\neq 0$, we have the relation

$0=\sigma(P_{v,\lambda})\big{|}_{q=0_{\mathbf{\Sigma}}}=\prod_{i=1}^{s}D_{b_{i}}(\xi)^{lc_{i}}.$

This contradicts the fact that $D_{b_{1}}(\xi)\neq 0,\dots,D_{b_{s}}(\xi)\neq
0$. Hence $\lambda=0$ and we conclude that
$\\{b_{1},\dots,b_{s}\\}=R(\mathbf{\Sigma})\cap\sigma$.

Now we have $D_{b}(\xi)=0$ for all $b\notin R(\mathbf{\Sigma})\cap\sigma$.
Since $\\{D_{b}\in\mathbb{L}_{\mathbb{C}}^{\star}\colon b\notin
R(\Sigma)\cap\sigma\\}$ spans $\mathbb{L}^{\star}_{\mathbb{C}}$, we have
$\xi=0$. The lemma is proved. ∎

######  Corollary A.4.

There exists a Zariski-open neighbourhood $U$ of $0_{\mathbf{\Sigma}}$ in
$\operatorname{Spec}\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$
such that
$\mathscr{G}|_{U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}}$ is a
coherent
$\mathcal{O}_{U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}}$-module.

###  A.3 Locally freeness and rank

We complete the proof of Proposition A.1. Recall from Theorem 4.8 that the
completed equivariant Brieskorn module is isomorphic to the quantum D-module
of $\mathfrak{X}$. Thus we have

$\mathsf{G}/\mathfrak{m}_{\mathbf{\Sigma}}\mathsf{G}\cong\widehat{\mathsf{G}}/\mathfrak{m}_{\mathbf{\Sigma}}\widehat{\mathsf{G}}\cong
H^{*}_{{\rm CR},\mathbb{T}}(\mathfrak{X})[z],$

where
$\mathfrak{m}_{\mathbf{\Sigma}}\subset\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$
denotes the maximal ideal corresponding to $0_{\mathbf{\Sigma}}$ and
$\widehat{\mathsf{G}}$ denotes the $\mathfrak{m}_{\mathbf{\Sigma}}$-adic
completion as discussed in Section 4.2. This implies that the restriction
$\mathscr{G}|_{\\{0_{\mathbf{\Sigma}}\\}\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}}$
is free of rank $\dim H^{*}_{\rm CR}(\mathfrak{X})$. Hence by coherence,
$\mathscr{G}$ is generated by $\dim H^{*}_{\rm CR}(\mathfrak{X})$ many
sections in a neighbourhood of
$\\{0_{\mathbf{\Sigma}}\\}\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$.
On the other hand, the localization map “$\operatorname{Loc}$” appearing in
[27, Definition 4.17] gives $\dim H^{*}_{\rm CR}(\mathfrak{X})$ many linearly
independent solutions of $\mathscr{G}$ for a generic
$(\chi,z)\in\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$; the weak Fano
condition ensures the convergence of the power series solution
$\operatorname{Loc}$. This implies that $\mathscr{G}$ is locally free of rank
$\dim H_{\rm CR}^{*}(\mathfrak{X})$ in a neighbourhood of
$\\{0_{\mathbf{\Sigma}}\\}\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$.

To see that $\mathscr{G}$ is locally free on an open set of the form
$U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$ (for some open set
$U\subset\operatorname{Spec}\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$
containing $0_{\mathbf{\Sigma}}$), we use the grading operator (4.1). The
grading operator makes $\mathscr{G}$ a $\mathbb{C}^{\times}$-equivariant
sheaf; the induced $\mathbb{C}^{\times}$-action on the base is given by the
same grading operator on
$\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}][z]$ and the weight one
$\mathbb{C}^{\times}$-action on $\operatorname{Lie}\mathbb{T}$.

######  Lemma A.5 (cf. [62, Section 3.1.4]).

When $\mathfrak{X}$ is weak Fano,
$\mathbb{C}[\mathbb{O}(\mathbf{\Sigma})_{+}]$ is non-negatively graded with
respect to the grading operator (4.1).

###### Proof.

It suffices to show that $\sum\limits_{b\in S}\lambda_{b}\geq 0$ for any
$(\lambda,v)\in\mathbb{O}(\mathbf{\Sigma})_{+}$. By the definition (3.15) of
$\mathbb{O}(\mathbf{\Sigma})_{+}$, it suffices to show that $\sum\limits_{b\in
S}\lambda_{b}\geq 0$ for all maximal cones $\sigma\in\Sigma$ and
$\lambda\in\widetilde{C}_{\mathbf{\Sigma},\sigma}$ (see (3.14)). Define a
linear function $h\colon\mathbf{N}_{\mathbb{R}}\to\mathbb{R}$ by
$h(\overline{b})=1$ for all $b\in R(\mathbf{\Sigma})\cap\sigma$. Then the weak
Fano condition (i.e., the convexity of $\Delta$) together with (A.1) implies
that $h(\overline{b})\leq 1$ for all $b\in S$. Hence, for
$\lambda\in\widetilde{C}_{\mathbf{\Sigma},\sigma}$, we have

$0\leq h(\beta(\lambda))=\sum_{b\in
R(\mathbf{\Sigma})\cap\sigma}\lambda_{b}+\sum_{b\notin
R(\mathbf{\Sigma})\cap\sigma}\lambda_{b}h(\overline{b})\leq\sum_{b\in
S}\lambda_{b}.$

This proves the lemma. ∎

In particular, $\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$ is non-
negatively graded. Because the locus where $\mathscr{G}$ is locally free is
preserved by the $\mathbb{C}^{\times}$-action and contains a neighbourhood of
$(0_{\mathbf{\Sigma}},0,0)\in B$, it follows that $\mathscr{G}$ is locally
free on an open set of the form
$U\times\operatorname{Lie}\mathbb{T}\times\mathbb{C}_{z}$. The proof of
Proposition A.1 is now complete.

######  Remark A.6.

The generic rank of the GKZ system has been studied by many people, notably by
Gelfand–Kapranov–Zelevinsky [46], Adolphson [6], Matusevich–Miller–Walther
[83] and has been identified with the volume272727The volume is normalized so
that the standard simplex has volume 1. When we allow $\mathbf{N}$ to have
torsions, the generic rank is $|\mathbf{N}_{\rm
tor}|\times\operatorname{vol}(\Delta)$. of $\Delta$ (when $\chi$ is not
special). Over the open torus $(\mathbb{C}^{\times})^{m}$ contained in
$\operatorname{Spec}\mathbb{C}[\boldsymbol{\Lambda}(\mathbf{\Sigma})_{+}]$,
the Brieskorn module in this paper corresponds to the better-behaved GKZ
system of Borisov–Horja [17]; they showed that the generic rank of the better-
behaved GKZ system equals $\operatorname{vol}(\Delta)$ (independently of
$\chi$).

##  Appendix B Proof of Lemma 7.24

We only prove Part (1) of the lemma; the argument for Part (2) is the same. It
is easy to see that $F_{q,0}^{-1}(\mathbf{u})$ and $\partial A_{q,0}(\eta)$
intersect transversally at $x\in F_{q,0}^{-1}(\mathbf{u})\cap\partial
A_{q,0}(\eta)$ if and only if $\operatorname{grad}F_{q,0}(x)$ and
$\operatorname{grad}H(x)$ are linearly independent over $\mathbb{C}$, where we
set

$\operatorname{grad}f(x)=\left(\overline{x_{i}\frac{\partial f}{\partial
x_{i}}}\right)_{i=1}^{n}.$

Suppose that the lemma is not true. Then we can find sequences
$(q(k),\mathbf{u}(k))\in K$ and $x(k)\in\mathcal{Y}^{\rm sm}_{q(k)}$ such that
the following holds:

  * •

$\eta_{k}=H(x(k))\to\infty$ as $k\to\infty$;

  * •

$F_{q(k),0}(x(k))=\mathbf{u}(k)$;

  * •

$\operatorname{grad}F_{q(k),0}(x(k))$ and $\operatorname{grad}H(x(k))$ are
linearly dependent over $\mathbb{C}$, i.e., there exists
$\alpha_{k}\in\mathbb{C}$ such that
$\operatorname{grad}F_{q(k),0}(x(k))=\alpha_{k}\operatorname{grad}H(x(k))$.

Define an $\mathbb{R}^{n}$-valued function $\mathbf{v}(x)$ by

$\mathbf{v}(x)=\sum_{b\in S_{-}}|x^{b}|^{2}\overline{b}.$

Then we have

$\operatorname{grad}H(x)=\frac{\mathbf{v}(x)}{2H(x)}.$

Writing
$\mathbf{v}\cdot\mathbf{w}=\sum\limits_{i=1}^{n}\mathbf{v}_{i}\mathbf{w}_{i}$
for the $\mathbb{C}$-bilinear scalar product, the third condition above can be
written as

$\operatorname{grad}F_{q(k),0}(x(k))=\frac{\operatorname{grad}F_{q(k),0}(x(k))\cdot\mathbf{v}(x(k))}{\|\mathbf{v}(x(k))\|^{2}}\mathbf{v}(x(k)).$

By the curve selection lemma in [85, Lemma 2], we can find a real analytic
curve $(0,\epsilon)\ni s\mapsto(q(s),\mathbf{u}(s),x(s))$ admitting a Laurent
expansion at $s=0$ such that for $0<s<\epsilon$, $(q(s),\mathbf{u}(s))\in K$,
$F_{q(s),0}(x(s))=\mathbf{u}(s)$,

$\operatorname{grad}F_{q(s),0}(x(s))=\frac{\operatorname{grad}F_{q(s),0}(x(s))\cdot\mathbf{v}(x(s))}{\|\mathbf{v}(x(s)\|^{2}}\mathbf{v}(x(s)),$
(B.1)

and $\lim\limits_{s\to+0}H(x(s))=\infty$. Since $K$ is compact,
$(q(0),\mathbf{u}(0)):=\lim\limits_{s\to+0}(q(s),\mathbf{u}(s))$ exists in
$K$. We write $x_{i}(s)=s^{\xi_{i}}a_{i}(1+O(s))$ with $a_{i}\neq 0$ and
$\xi_{i}\in\mathbb{Z}$. Then the leading term for $H(x(s))^{2}$ is

$H(x(s))^{2}=\left(\sum_{b\in\sigma}|a|^{2b}\right)s^{-2m}+\text{higher order
terms},$ (B.2)

where we set $\sigma:=\\{b\in S_{-}\colon\xi\cdot\overline{b}=-m\\}$ with
$\xi=(\xi_{1},\dots,\xi_{n})$ and $-m:=\min\\{\xi\cdot\overline{b}\colon b\in
S_{-}\\}$, and $|a|^{2b}=\prod\limits_{i=1}^{n}|a_{i}|^{2b_{i}}$. Note that
elements of $\sigma$ spans a face of the polytope $\Delta_{-}$. Since
$\lim\limits_{s\to+0}H(x(s))=\infty$, we have $m>0$. We calculate:

$\displaystyle\mathbf{v}(s)=\left(\sum_{b\in\sigma}|a|^{2b}\overline{b}\right)s^{-2m}+\text{higher
order terms},$ $\displaystyle\mathbf{v}(s)\cdot\frac{d\log
x(s)}{ds}=-m\left(\sum_{b\in\sigma}|a|^{2b}\right)s^{-2m-1}+\text{higher order
terms},$
$\displaystyle\overline{\operatorname{grad}F_{q(s),0}(x(s))}=\sum_{b\in
S_{-}}q(s)^{\ell_{b}}x(s)^{b}\overline{b}=\left(\sum_{b\in\sigma}a^{b}q(0)^{\ell_{b}}\overline{b}\right)s^{-m}+\text{higher
order terms}.$

By differentiating the equality $\mathbf{u}(s)=F_{q(s),0}(x(s))$ in $s$, we
get

$\displaystyle\frac{du(s)}{ds}-\sum_{i=1}^{n}\frac{\partial
F_{q(s),0}(x(s))}{\partial
q^{i}}\frac{dq^{i}(s)}{ds}=\overline{\operatorname{grad}F_{q(s),0}(x(s))}\cdot\frac{d\log
x(s)}{ds}$
$\displaystyle\qquad{}=\left(\overline{\operatorname{grad}F_{q(s),0}(x(s))}\cdot\frac{\mathbf{v}(x(s))}{\|\mathbf{v}(x(s)\|^{2}}\right)\cdot\left(\mathbf{v}(x(s))\cdot\frac{d\log
x(s)}{ds}\right),$ (B.3)

where $\\{q^{i}\\}$ denotes a local co-ordinate system on $\mathcal{M}^{\rm
sm,\times}$ and we used (B.1) in the second line. We compare the leading order
terms as $s\to+0$. By the above calculation, the right-hand side of (B.3) has
the leading term

$c\left(\sum_{b\in\sigma}a^{b}q(0)^{\ell_{b}}\overline{b}\right)\cdot\left(\sum_{b\in\sigma}|a|^{2b}\overline{b}\right)s^{-m-1}$
(B.4)

with $c=-m\sum\limits_{b\in\sigma}|a|^{2b}\neq 0$. On the other hand, because
$q(s)$, $\mathbf{u}(s)$ are regular at $s=0$ and

$\left|\frac{\partial F_{q,0}(x(s))}{\partial q^{i}}\right|=\left|\sum_{b\in
S}\frac{\partial q(s)^{\ell_{b}}}{\partial q^{i}}x^{b}\right|\leq C\cdot
H(x(s))\qquad\text{for some $C>0$},$

the left-hand side of (B.3) has poles of order at most $m$ by (B.2). Hence the
quantity (B.4) must vanish. This together with (B.1) implies:

$\displaystyle\overline{\operatorname{grad}F_{q(s),0}(x(s))}=\frac{\overline{\operatorname{grad}F_{q(s),0}(x(s))}\cdot\mathbf{v}(x(s))}{\|\mathbf{v}(x(s)\|^{2}}\mathbf{v}(x(s))=O\big{(}s^{-m+1}\big{)}.$

On the other hand, Proposition 7.19 and (B.2) give an estimate of the form

$\|\operatorname{grad}F_{q(s),0}(x(s))\|\geq\epsilon_{1}H(x(s))\geq\epsilon_{2}s^{-m}$

for sufficiently small $s>0$ (for some $\epsilon_{1},\epsilon_{2}>0$). These
two estimates contradict each other. Lemma 7.24 is proved.

### Acknowledgements

I would like to thank Pedro Acosta, Arend Bayer, Andrea Brini, Tom Coates,
Alessio Corti, Sergey Galkin, Vasily Golyshev, Eduardo González, Claus
Hertling, Yuki Hirano, Paul Horja, Yunfeng Jiang, Yuan-Pin Lee, Chiu-Chu
Melissa Liu, Wanmin Liu, Thomas Reichelt, Yongbin Ruan, Kyoji Saito, Fumihiko
Sanda, Christian Sevenheck, Yota Shamoto, Mark Shoemaker, Takuro Mochizuki,
Mauricio Romo, Hsian-Hua Tseng, Chris Woodward for many insightful discussions
and explanations. I also thank the anonymous referees for their careful
reading and helpful comments. This work is supported by EPSRC grant
EP/E022162/1, and JSPS Kakenhi Grants Number 22740042, 23224002, 24224001,
25400069, 26610008, 16K05127, 16H06335, 16H06337 and 17H06127. Part of this
work was done while I was in residence at the Mathematical Sciences Research
Institute in Berkeley, California, during the Spring semester of 2018 and the
stay was supported by the National Science Foundation under Grant No.
DMS-1440140. I thank the organizers and participants of the programme for many
stimulating discussions.

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