Topological computation of Stokes matrices of some weighted projective lines

Abstract

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at \(\infty \) for the cases \({\mathbb {P}}(1,3)\) and \({\mathbb {P}}(2,2)\) by purely topological methods. We compare them to the Gram matrix of the Euler–Poincaré pairing on \(D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))\) and \(D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))\), respectively. This article is based on the doctoral thesis of the author.

Introduction

In [6], D’Agnolo et al. describe how to compute the Stokes matrices of the enhanced Fourier–Sato transform of a perverse sheaf on the affine line by purely topological methods. To a regular singular holonomic \({\mathcal {D}}\)-module \({\mathcal {M}} \in \mathrm{Mod}_{\mathrm {rh}} \left( {\mathcal {D}}_{{\mathbb {A}}^1}\right) \) on the affine line, one associates a perverse sheaf via the regular Riemann–Hilbert correspondence

$$\begin{aligned} R{\mathcal {H}}om_{{\mathcal {D}}_{{\mathbb {A}}^1}^{\mathrm {an}}} \left( (\bullet )^{\mathrm {an}}, {\mathcal {O}}_{{\mathbb {A}}^1}^{\mathrm {an}}\right) [1] :\mathrm{Mod}_{\mathrm {rh}} \left( {\mathcal {D}}_{{\mathbb {A}}^1}\right) {\mathop {\longrightarrow }\limits ^{\simeq }} \mathrm{Perv}\left( {\mathbb {C}}_{{\mathbb {A}}^1}\right) . \end{aligned}$$

Let \(\Sigma \subset {\mathbb {A}}^1\) denote the set of singularities of \({\mathcal {M}}\). Following [6, Sect. 4.2], after suitably choosing a total order on \(\Sigma \), the resulting perverse sheaf \(F\in \mathrm{Perv}_{\Sigma }\left( {\mathbb {C}}_{{\mathbb {A}}^1}\right) \) can be described by linear algebra data, namely its quiver

$$\begin{aligned} \left( \Psi (F),\Phi _{\sigma }(F), u_{\sigma },v_{\sigma } \right) _{\sigma \in \Sigma }, \end{aligned}$$

where \(\Psi (F)\) and \(\Phi _{\sigma }(F)\) are finite dimensional \({\mathbb {C}}\)-vector spaces and \(u_{\sigma } :\Psi (F) \rightarrow \Phi _{\sigma }(F)\) and \(v_{\sigma } :\Phi _{\sigma }(F) \rightarrow \Psi (F)\) are linear maps such that \(1-u_{\sigma }v_{\sigma }\) is invertible for any \(\sigma \). The main result in [6] is a determination of the Stokes matrices of the enhanced Fourier–Sato transform of F and therefore of the Fourier–Laplace transform of \({\mathcal {M}}\) in terms of the quiver of F. This result builds on the irregular Riemann–Hilbert correspondence of D’Agnolo and Kashiwara [7], which provides a topological description of holonomic \({\mathcal {D}}\)-modules. As proven by Kashiwara and Schapira [14], this correspondence intertwines the Fourier–Laplace with the (enhanced) Fourier–Sato transform.

Mirror symmetry connects the weighted projective line \({\mathbb {P}}(1,3)\) with the Landau–Ginzburg model

$$\begin{aligned} \left( {\mathbb {G}}_m, f=x+x^{-3} \right) . \end{aligned}$$

The quantum connection of \({\mathbb {P}}(1,3)\) is closely related to the Fourier–Laplace transform of the Gauß–Manin system \(H^0(\int _f {\mathcal {O}})\) of f. We compute that

$$\begin{aligned} F:=\mathrm{R}f_{*}{\mathbb {C}}[1]\in \mathrm{Perv}_{\Sigma }\left( {\mathbb {C}}_{{\mathbb {A}}^1}\right) , \end{aligned}$$

where \(\Sigma \) denotes the set of singular values of f, is the perverse sheaf associated to \(H^0(\int _f {\mathcal {O}})\) by the Riemann–Hilbert correspondence. In Sect. 1, we compute the localized Fourier–Laplace transform of \(H^0(\int _f {\mathcal {O}})\). In Sect. 2, analogous to the examples in [6, Sect. 7], we carry out the topological computation of the Stokes matrices of the Fourier–Laplace transform of \(H^0(\int _f {\mathcal {O}})\). In Sect. 3, we compare the Stokes matrix \(S_{\beta }\), that we obtained from our topological computations, to the Gram matrix of the Euler–Poincaré pairing on \(D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))\) with respect to a suitable full exceptional collection. Following Dubrovin’s conjecture about the Stokes matrix of the quantum connection, proven for the weighted projective space \({\mathbb {P}}\left( \omega _0,\ldots ,\omega _n\right) \) by Tanabé and Ueda in [19] and by Cruz Morales and van der Put in [5], they are known to be equivalent after appropriate modifications. We give the explicit braid of the braid group \(B_4\) that deforms the Gram matrix into the Stokes matrix \(S_{\beta }\). Section 4 tackles the computations for the case of non-coprime parameters. In comparison to the case of coprime parameters, this requires a slightly modified approach. We compute the Stokes matrices of the Fourier–Laplace transform of the Gauß–Manin system of the Landau–Ginzburg model of \({\mathbb {P}}(2,2)\) and set it into relation with the Gram matrix of the Euler–Poincaré pairing on \(D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))\).

This article is based on the doctoral thesis [18] of the author. The figures in Sects. 2 and 4 were mainly produced in SAGE. In the online version of this article, the figures are provided in color.

Gauß–Manin system and its Fourier–Laplace transform

Let X be affine and f a regular function \(f :X \rightarrow {\mathbb {A}}^1\) on X. Denote by \(\int _f (\bullet )\) the direct image in the category of \({\mathcal {D}}\)-modules and by \(M:=H^0 (\int _f {\mathcal {O}}_X )\in \mathrm{Mod}_{\mathrm {rh}}\left( {\mathcal {D}}_{{\mathbb {A}}^1}\right) \) the zeroth cohomology of the Gauß–Manin system of f. Following [9, Sect. 2.c], it is given by

$$\begin{aligned} M= \Omega ^n (X)\left[ \partial _t \right] / (d-\partial _t df \wedge )\Omega ^{n-1} (X)\left[ \partial _t \right] . \end{aligned}$$

Denote by \(G:={\widehat{M}}[\tau ^{-1}]\) the Fourier–Laplace transform of M, localized at \(\tau = 0\). It is given by

$$\begin{aligned} G = \Omega ^n(X)\left[ \tau ,\tau ^{-1}\right] / \left( d - \tau df\wedge \right) \Omega ^{n-1}(X)\left[ \tau ,\tau ^{-1}\right] . \end{aligned}$$

G is a free \({\mathbb {C}} [\tau , \tau ^{-1}]\)-module of finite rank. Rewriting in the variable \(\theta =\tau ^{-1}\) gives the \({\mathbb {C}} [\theta ,\theta ^{-1}]\)-module

$$\begin{aligned} G = \Omega ^n(X)\left[ \theta ,\theta ^{-1}\right] / \left( \theta d - df\wedge \right) \Omega ^{n-1}(X)\left[ \theta ,\theta ^{-1}\right] . \end{aligned}$$

G is endowed with a flat connection given as follows. For \(\gamma = \left[ \sum _{k \in {\mathbb {Z}}} \omega _k \theta ^k \right] \in G\), where \(\Omega ^n(X)\ni \omega _k =0\) for almost all k, the connection is given by (cf. [12, Definition 2.3.1]):

$$\begin{aligned} \theta ^2 \nabla _{ \frac{\partial }{\partial \theta }} \left( \gamma \right) = \left[ \sum _k f \omega _k \theta ^k + \sum _k k \omega _k \theta ^{k+1} \right] . \end{aligned}$$

It is known that \((G,\nabla )\) has a regular singularity at \(\theta = \infty \) and possibly an irregular one at \(\theta = 0\).

We now consider the Laurent polynomial \(f=x+x^{-3}\in {\mathbb {C}}\left[ x,x^{-1}\right] \), being a regular function on the multiplicative group \({\mathbb {G}}_m\). For our computations we pass to the variable \(\theta =\tau ^{-1}\). We compute that for the given f, G is given by the free \({\mathbb {C}} \left[ \theta ,\theta ^{-1}\right] \)-module

$$\begin{aligned} G={\mathbb {C}} \left[ x,x^{-1}\right] dx\left[ \theta ,\theta ^{-1}\right] /\left( \theta d-\left( dx-3x^{-4}dx\right) \wedge \right) {\mathbb {C}} \left[ x,x^{-1}\right] \left[ \theta ,\theta ^{-1}\right] \end{aligned}$$

with basis over \({\mathbb {C}} \left[ \theta ,\theta ^{-1} \right] \) given by \(\left[ \frac{dx}{x}\right] , \left[ \frac{dx}{x^2}\right] ,\left[ \frac{dx}{x^3}\right] , \left[ \frac{dx}{x^4}\right] \). In this basis, the connection is given by

$$\begin{aligned} \theta \nabla _{ \frac{\partial }{\partial \theta } }= \theta \partial _{\theta } + \begin{pmatrix} 0 &{}\quad \frac{4}{3\theta } &{}\quad 0 &{}\quad 0\\ 0 &{}\quad \frac{1}{3} &{}\quad \frac{4}{3\theta } &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \frac{2}{3} &{}\quad \frac{4}{3\theta }\\ \frac{4}{\theta } &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$
(1)

Via the cyclic vector \(m=(1,0,0,0)^{\mathrm {t}}\), we compute the relation

$$\begin{aligned} \nabla _{\theta \partial _{\theta }}^4m + 4 \nabla _{\theta \partial _{\theta }}^3m + \frac{32}{9}\nabla _{\theta \partial _{\theta }}^2m-\frac{256}{27\theta ^4}m =0 \end{aligned}$$

and therefore associate the differential operator

$$\begin{aligned} P = \left( \theta \partial _{\theta }\right) ^4 + 4 \left( \theta \partial _{\theta }\right) ^3 + \frac{32}{9}\left( \theta \partial _{\theta }\right) ^2 - \frac{256}{27\theta ^4}\in {\mathbb {C}}\left[ \theta , \theta ^{-1} \right] \langle \partial _{\theta } \rangle = {\mathcal {D}}_{{\mathbb {G}}_m}. \end{aligned}$$
Fig. 1
figure1

Newton polygon of P

As it is well known, one can read the type of the singularities at 0 and \(\infty \) from the Newton polygon in the sense of Ramis (cf. [15, Chapter V]). The Newton polygon in Fig. 1 confirms that P—and therefore system (1)—has the nonzero slope 1 and therefore is irregular singular at \(\theta = 0\) and regular singular at \(\theta = \infty \).

Topological computation of the Stokes matrices

We consider the Laurent polynomial \(f=x+x^{-3}:{\mathbb {G}}_m \rightarrow {\mathbb {A}}^1\). Its critical points are given by \(\left\{ \pm \root 4 \of {3}, \pm \root 4 \of {3} i \right\} \). The critical values of f are given by

$$\begin{aligned} \Sigma = \left\{ \pm \frac{4}{\root 4 \of {27}}, \pm \frac{4i}{\root 4 \of {27}} \right\} \subset {\mathbb {A}}^1. \end{aligned}$$

The preimages of 

  • \(\frac{4}{\root 4 \of {27}}\) are \( \root 4 \of {3}\) (double), \(\frac{-1-\sqrt{2}i}{\root 4 \of {27}}\) and \(\frac{-1+\sqrt{2}i}{\root 4 \of {27}}\),

  • \(-\frac{4}{\root 4 \of {27}}\) are \( - \root 4 \of {3}\) (double), \(\frac{1-\sqrt{2}i}{\root 4 \of {27}}\) and \(\frac{1+\sqrt{2}i}{\root 4 \of {27}}\),

  • \(\frac{4i}{\root 4 \of {27}}\) are \( \root 4 \of {3}i\) (double), \(\frac{ -\sqrt{2}-i}{\root 4 \of {27}}\) and \(\frac{\sqrt{2}-i}{\root 4 \of {27}}\),

  • \(-\frac{4i}{\root 4 \of {27}}\) are \( -\root 4 \of {3}i\) (double), \(\frac{\sqrt{2}+i}{\root 4 \of {27}}\) and \(\frac{-\sqrt{2}+i}{\root 4 \of {27}}\).

Since f is proper, we compute by the adjunction formula that

$$\begin{aligned} R{\mathcal {H}}om_{{\mathcal {D}}^{\mathrm {an}} }\left( \left( \int _f {\mathcal {O}}\right) ^{\mathrm {an}} ,{\mathcal {O}}^{\mathrm {an}}\right) \simeq \mathrm{R}f^{\mathrm {an}}_{*}R{\mathcal {H}}om_{{\mathcal {D}}^{\mathrm {an}} } \left( {\mathcal {O}}^{\mathrm {an}} ,f^{\dagger }{\mathcal {O}}^{\mathrm {an}}\right) \simeq \mathrm{R}f^{\mathrm {an}}_{*}{\mathbb {C}}. \end{aligned}$$

Since f is semismall, \(\mathrm{R}f_{*} {\mathbb {C}}[1]\in \mathrm{Perv}({\mathbb {C}}_{{\mathbb {A}}^1})\) is a perverse sheaf (cf. [8]). Outside of \(\Sigma \), f is a covering of degree 4, therefore \(\mathrm{R}f_{*} {\mathbb {C}}[1]\in \mathrm{Perv}_{\Sigma }({\mathbb {C}}_{{\mathbb {A}}^1})\). By the regular Riemann–Hilbert correspondence

$$\begin{aligned} {\mathcal {S}}ol(\bullet )[\dim X] {:=} R{\mathcal {H}}om_{{\mathcal {D}}_X^{\mathrm {an}}}\left( \left( \bullet \right) ^{\mathrm {an}}, {\mathcal {O}}^{\mathrm {an}}_X\right) \left[ \dim X\right] {:} \mathrm{Mod}_{\mathrm {rh}}\left( {\mathcal {D}}_X\right) {{\mathop {\longrightarrow }\limits ^{\simeq }}} \mathrm{Perv}({\mathbb {C}}_{X^{\mathrm {an}}}), \end{aligned}$$

we associate to \(H^0 (\int _f {\mathcal {O}})\) the perverse sheaf \(F :=\mathrm{R}f_{*}{\mathbb {C}}[1]\).

We fix \(\alpha = e^{\frac{\pi i}{8}}\in {\mathbb {A}}^1,\ \beta = e^{\frac{3 \pi i}{8}}\in \left( {\mathbb {A}}^1\right) ^{\vee }\), such that \(\mathfrak {R}(\langle \alpha , \beta \rangle )=0, \ \mathfrak {I}(\langle \alpha , \beta \rangle )=1\). This induces the following order on \(\Sigma \) (cf. [6, Sect. 4]):

$$\begin{aligned} \sigma _1:=\frac{4i}{\root 4 \of {27}}<_{\beta } \ \sigma _2:=-\frac{4}{\root 4 \of {27}}<_{\beta } \ \sigma _3:=\frac{4}{\root 4 \of {27}} <_{\beta } \ \sigma _4:=-\frac{4i}{\root 4 \of {27}}. \end{aligned}$$

In Fig. 4, the \(\sigma _i\) are depicted in the following colors:

\(\bullet \) : green,   \(\bullet \) : red,   \(\bullet \) : purple,    \(\bullet \) : orange.

Fig. 2
figure2

LHS: \(\{ x \mid \mathfrak {R}(f(x))\ge 0 \}\), RHS: \(\{ x \mid \mathfrak {I}(f(x))\ge 0 \}\)

Fig. 3
figure3

Preimage of the imaginary (resp. real) axis in blue (resp. red) color

Fig. 4
figure4

Lines passing through \(\sigma _i\) with phase \(\frac{\pi }{8}\)

The blue area in Fig. 2 shows where f has real (resp. imaginary) part greater than or equal to 0. In Fig. 3, the preimage of the imaginary (resp. real) axis under f is plotted in blue (resp. red) color. We consider lines passing through the singular values with phase \(\frac{\pi }{8}\), as depicted in Fig. 4. The preimages of these lines are plotted in Fig. 5. We fix a base point e with \(\mathfrak {R}(e) > \mathfrak {R}\left( \sigma _i\right) \) for all i and denote its preimages by \(e_1,e_2,e_3,e_4,\) as depicted in Fig. 6. In the following, we adopt the notation of [6, Sect. 4]. The nearby and global nearby cycles of F are given by 

$$\begin{aligned} \Psi _{\sigma _i}(F)&:=\mathrm{R}\Gamma _c\left( {\mathbb {A}}^1;{\mathbb {C}}_{\ell _{\sigma _i}^{\times }}\otimes F\right) \simeq H^0\mathrm{R}\Gamma _c \left( \ell _{\sigma _i}^{\times };F\right) \cong \bigoplus _{ e_j \in f^{-1}(e) } {\mathbb {C}}_{e_j} \cong {\mathbb {C}}^4,\\ \Psi (F)&:=\mathrm{R}\Gamma _c\left( {\mathbb {A}}^1;{\mathbb {C}}_{{{\mathbb {A}}^1\setminus \ell _{\Sigma }}}\otimes F\right) [1] \simeq \Psi _{\sigma _i}(F) \cong {\mathbb {C}}^4. \end{aligned}$$

Furthermore, we fix isomorphisms \(i_{\sigma _i}^{-1}F[-1] \cong \bigoplus _{ \sigma _i^j \in f^{-1}\left( \sigma _i\right) } {\mathbb {C}}_{\sigma _i^j} \cong {\mathbb {C}}^3.\)

The exponential components at \(\infty \) of the Fourier–Laplace transform of \(H^0(\int _f {\mathcal {O}})\) are known to be of linear type with coefficients given by the \(\sigma _i\in \Sigma \). The Stokes rays are therefore given by 

$$\begin{aligned} \left\{ 0,\pm \frac{\pi }{4},\pm \frac{\pi }{2},\pm \frac{3\pi }{4},\pi \right\} . \end{aligned}$$

We consider loops \(\gamma _{\sigma _i}\), starting at e and running around the singular value \(\sigma _i\) in counterclockwise orientation,Footnote 1 as depicted in Fig. 6. We denote by \(\gamma _{\sigma _i}^j\) the preimage of \(\gamma _{\sigma _i}\) starting at \(e_j\), \(j=1,2,3,4\). The figure constitutes a rough drawing of the preimages of \(\gamma _{\sigma _i}\). By taking into account the preimages of the different segments of the axes and the intersections of \(\gamma _{\sigma _i}\) with them, one recovers the \(\gamma _{\sigma _i}^j\) as depicted in the figure. From Fig. 6 we read, in the ordered basis \(e_1,e_2,e_3,e_4\), the monodromies

$$\begin{aligned} T_{\sigma _1} = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix}, \quad T_{\sigma _2} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix},\\ T_{\sigma _3} = \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{pmatrix},\quad T_{\sigma _4} = \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1&{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix}. \end{aligned}$$
Fig. 5
figure5

Preimages under f

In order to obtain the maps \(b_{\sigma _i}\), we consider the half-lines . We denote their preimages under f by \(\{ \ell _{\sigma _i}^j \}_{j=1,2,3,4}\), depending on which \(\gamma _{\sigma _i}^j\) they intersect. We label the preimages of \(\sigma _i\) by \(\sigma _i^1,\sigma _i^2,\sigma _i^3\), as depicted in Fig. 7. By the derivation of the short exact sequence of quivers [6, (7.1.3)] and passing to Borel–Moore homology as described in [6, Lemma 5.3.1.(i)], \(b_{\sigma _i}\) is induced from the corresponding boundary value map from \(\ell _{\sigma _i}\) to its origin \(\sigma _i\). Therefore, \(b_{\sigma _i}\) encodes which lift of \(\ell _{\sigma _i}\) starts at which preimage of \(\sigma _i\). Namely, from Fig. 7 we read the following:

::

\(\ell _{\sigma _1}^1 \mapsto \sigma _1^1, \ \ell _{\sigma _1}^2 \mapsto \sigma _1^1, \ \ell _{\sigma _1}^3 \mapsto \sigma _1^2, \ \ell _{\sigma _1}^4 \mapsto \sigma _1^3\).

Therefore, \(b_{\sigma _1}\) is the transpose of \(\begin{pmatrix} 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix}\).

::

\(\ell _{\sigma _2}^1 \mapsto \sigma _2^3, \ \ell _{\sigma _2}^2 \mapsto \sigma _2^1, \ \ell _{\sigma _2}^3 \mapsto \sigma _2^1, \ \ell _{\sigma _2}^4 \mapsto \sigma _2^2\).

Therefore, \(b_{\sigma _2}\) is the transpose of \(\begin{pmatrix} 0 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{pmatrix}\).

::

\(\ell _{\sigma _3}^1 \mapsto \sigma _3^1, \ \ell _{\sigma _3}^2 \mapsto \sigma _3^2, \ \ell _{\sigma _3}^3 \mapsto \sigma _3^3, \ \ell _{\sigma _3}^4 \mapsto \sigma _3^1\).

Therefore, \(b_{\sigma _3}\) is the transpose of \(\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1&{}\quad 0\\ \end{pmatrix}\).

::

\(\ell _{\sigma _4}^1 \mapsto \sigma _4^1, \ \ell _{\sigma _4}^2 \mapsto \sigma _4^3, \ \ell _{\sigma _4}^3 \mapsto \sigma _4^1, \ \ell _{\sigma _4}^4 \mapsto \sigma _4^2\).

Therefore, \(b_{\sigma _4}\) is the transpose of \(\begin{pmatrix} 1 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ \end{pmatrix}\).

We obtain, in the ordered bases \(\sigma _i^1,\sigma _i^2,\sigma _i^3\) and \(\ell _{\sigma _i}^1,\ell _{\sigma _i}^2,\ell _{\sigma _i}^3,\ell _{\sigma _i}^4\) each:

$$\begin{aligned} b_{\sigma _1} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix}, \quad b_{\sigma _2} = \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0\\ \end{pmatrix},\\ b_{\sigma _3} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 0&{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0&{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0\\ \end{pmatrix},\quad b_{\sigma _4} = \begin{pmatrix} 1&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0\\ 0&{}\quad 1 &{}\quad 0\\ \end{pmatrix}. \end{aligned}$$

Denote by \(u_i:=u_{\sigma _i}\), \(v_i:=v_{\sigma _i}\), \(T_i:=T_{\sigma _i}\) and \(\Phi _i :=\Phi _{\sigma _i}\). As described in [6, Sect. 7], we obtain

figurei

as the cokernels of the following diagrams:

figurej

We identify the cokernels of \(b_{\sigma _i}\) in the following way:

  • coker\((b_{\sigma _1}) \simeq {\mathbb {C}} \) via \(\left[ \begin{pmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{pmatrix} \right] = \left[ \begin{pmatrix} a_1-a_2\\ 0\\ 0\\ 0 \end{pmatrix} \right] ,\)

  • coker\((b_{\sigma _2}) \simeq {\mathbb {C}} \) via \(\left[ \begin{pmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{pmatrix} \right] = \left[ \begin{pmatrix} 0\\ a_2-a_3\\ 0\\ 0 \end{pmatrix} \right] ,\)

  • coker\((b_{\sigma _3}) \simeq {\mathbb {C}} \) via \(\left[ \begin{pmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{pmatrix} \right] = \left[ \begin{pmatrix} a_1-a_4\\ 0\\ 0\\ 0 \end{pmatrix} \right] ,\)

  • coker\((b_{\sigma _4}) \simeq {\mathbb {C}} \) via \(\left[ \begin{pmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{pmatrix} \right] = \left[ \begin{pmatrix} a_1-a_3\\ 0\\ 0\\ 0 \end{pmatrix} \right] .\)

We obtain that

figurek

\(\simeq \)

figurel

where

$$\begin{aligned} u_1= \begin{pmatrix} 1 &{}\quad -1 &{}\quad 0 &{}\quad 0\\ \end{pmatrix}, \quad u_2=\begin{pmatrix} 0 &{}\quad 1 &{}\quad -1 &{}\quad 0\\ \end{pmatrix}, \\ u_3=\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad -1\\ \end{pmatrix}, \quad u_4=\begin{pmatrix} 1 &{}\quad 0 &{}\quad -1 &{}\quad 0\\ \end{pmatrix}, \end{aligned}$$

and \(v_i=u_i^{\mathrm {t}}\). By [6, Theorem 5.2.2], we obtain the following

Fig. 6
figure6

\(\ell _{\sigma _i}\) and their preimages under f

Fig. 7
figure7

\(\ell _{\sigma _i}\) and their preimages under f

Theorem

Under the choices made, the Stokes matrices of the Fourier–Laplace transform of \(H^0(\int _f {\mathcal {O}})\) at \(\infty \) are given as

$$\begin{aligned} S_{\beta }&= \begin{pmatrix} 1 &{}\quad u_1v_2&{}\quad u_1v_3&{}\quad u_1v_4\\ 0 &{}\quad 1&{}\quad u_2v_3&{}\quad u_2v_4\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad u_3v_4\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix} = \begin{pmatrix} 1 &{}\quad -1&{}\quad 1 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix},\\ S_{-\beta }&= \begin{pmatrix} {\mathbb {T}}_{1} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -u_2v_1 &{}\quad {\mathbb {T}}_2 &{}\quad 0 &{}\quad 0\\ -u_3v_1 &{}\quad -u_3v_2 &{}\quad {\mathbb {T}}_3 &{}\quad 0\\ -u_4v_1 &{}\quad -u_4v_2 &{}\quad -u_4v_3 &{}\quad {\mathbb {T}}_4\\ \end{pmatrix} = \begin{pmatrix} -1 &{}\quad 0 &{}\quad 0&{}\quad 0\\ 1 &{}\quad -1 &{}\quad 0 &{}\quad 0\\ -1 &{}\quad 0 &{}\quad -1 &{}\quad 0\\ -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 \end{pmatrix} = -S_{\beta }^{\mathrm {t}}, \end{aligned}$$

where \({\mathbb {T}}_i :=1-u_iv_i\). \(S_{\pm \beta }\) describes crossing \(h_{\pm \beta }\) from \(H_{\alpha }\) to \(H_{-\alpha }\), where

$$\begin{aligned} H_{\alpha }=&\left\{ w \mid \arg (w) \in \left[ - \frac{5 \pi }{8}, \frac{ 3\pi }{8} \right] \right\} ,\ \\ H_{-\alpha }=&\left\{ w \in \mid \arg (w) \in \left[ \frac{ 3 \pi }{8}, \frac{11 \pi }{8} \right] \right\} \subset \left( {\mathbb {A}}^1\right) ^{\vee } \end{aligned}$$

denote the closed sectors at \(\infty \) and \(h_{\pm \beta }= \pm {\mathbb {R}}_{>0} \beta \subset \left( {\mathbb {A}}^1\right) ^{\vee }\), such that \( H_{\alpha }\cap H_{-\alpha } = h_{\beta } \cup h_{-\beta }\). \(\square \)

Quantum connection and Dubrovin’s conjecture

Quantum connection

The quantum connection of a Fano variety (resp. an orbifold) X is a connection on the trivial vector bundle over \({\mathbb {P}}^1\) with fiber \(H^*(X, {\mathbb {C}})\) (resp. \(H_{\mathrm {orb}}^*(X, {\mathbb {C}})\)), where z denotes the standard inhomogeneous coordinate at \(\infty \). By [11, (2.2.1)], the quantum connection is the connection given by

$$\begin{aligned} \nabla _{z \partial _z} = z \frac{\partial }{\partial z} - \frac{1}{z} \left( -K_X \circ \right) + \mu , \end{aligned}$$

where the first term on the right hand side is ordinary differentiation, the second one is pointwise quantum multiplication by \((-K_X)\), and the third one is the grading operator

$$\begin{aligned} \mu (a) :=\left( \frac{i}{2} - \frac{\dim X}{2} \right) a \quad \mathrm {for}\ a \in H^i(X, {\mathbb {C}}). \end{aligned}$$

The quantum connection is regular singular at \(z=\infty \) and irregular singular at \(\text{ z }=0\).

For the weighted projective line \({\mathbb {P}}(a,b)\), the orbifold cohomology ring is given by (cf. [16, Example 3.20])

$$\begin{aligned} H^{*}_{\mathrm {orb}}({\mathbb {P}}(a,b),{\mathbb {C}} ) = {\mathbb {C}} \left[ x,y,\xi \right] / \langle xy, ax^{\frac{a}{d}}-by^{\frac{b}{d}}\xi ^{n-m},\xi ^d-1 \rangle , \end{aligned}$$

where \(d=\gcd (a,b)\) and \(m,n\in {\mathbb {Z}}\) such that \(am+bn=d\). The grading is given as follows (cf. [1, Sect. 9]): \(\deg x = \frac{1}{A}\), \(\deg y = \frac{1}{B}\), \(\deg \xi = 0\), where \(A=\frac{a}{d} \), \(B=\frac{b}{d} \). Quantum multiplicationFootnote 2 is computed in

$$\begin{aligned} QH_{\mathrm {orb}}^{*} = {\mathbb {C}} \left[ x,y,\xi \right] / \langle xy-1, ax^{\frac{a}{d}}-by^{\frac{b}{d}}\xi ^{n-m},\xi ^d-1 \rangle . \end{aligned}$$

For \(\gcd (a,b)=1\), \(-K_{{\mathbb {P}}(a,b)}\) is given by the element \([x^a+y^b]\in H^1_{\mathrm {orb}}({\mathbb {P}}(a,b),{\mathbb {C}})\). Taking into account that the grading is scaled by 2, the grading operator is defined by \(\mu (a)= \left( i - \frac{\dim X}{2}\right) a\) for \(a\in H^i_{\mathrm {orb}}(X,{\mathbb {C}})\).

We obtain the quantum connection of \({\mathbb {P}}(1,3)\) as follows.

$$\begin{aligned} H^{*}_{\mathrm {orb}}({\mathbb {P}}(1,3),{\mathbb {C}} ) = {\mathbb {C}} \left[ x,y\right] / \langle xy,x-3y^3 \rangle \end{aligned}$$

with grading given by \(\deg x=1,\ \deg y=\frac{1}{3}\). A basis over \({\mathbb {C}} \) is given by \(1,y,y^2,y^3\). Quantum multiplication by \(-K_{{\mathbb {P}}(1,3)}=\left[ x+y^3\right] =\left[ 4y^3\right] \) in this basis is given by the matrix

$$\begin{aligned} \begin{pmatrix} 0 &{}\quad \frac{4}{3} &{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad \frac{4}{3} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{4}{3}\\ 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} . \end{aligned}$$

The grading \(\mu \) is given by the matrix

$$\begin{aligned} \begin{pmatrix} -\frac{1}{2} &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0&{}\quad -\frac{1}{6} &{}\quad 0 &{}\quad 0\\ 0&{}\quad 0 &{}\quad \frac{1}{6} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1}{2} \end{pmatrix}. \end{aligned}$$

Therefore, the quantum connection of \({\mathbb {P}}(1,3)\) is given by

$$\begin{aligned} \nabla _{z \partial _z} = z \partial _z - \frac{1}{z} \begin{pmatrix} 0 &{}\quad \frac{4}{3} &{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad \frac{4}{3} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{4}{3}\\ 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} + \begin{pmatrix} -\frac{1}{2} &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0&{}\quad -\frac{1}{6} &{}\quad 0 &{}\quad 0\\ 0&{}\quad 0 &{}\quad \frac{1}{6} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1}{2} \end{pmatrix}. \end{aligned}$$
(2)

Observation

By the gauge transformation \(h=\text {diag}(\theta ^{-\frac{1}{2}},\theta ^{-\frac{1}{2}},\theta ^{-\frac{1}{2}},\theta ^{-\frac{1}{2}})\), which subtracts \(\frac{1}{2}\) on the diagonal entries, and passing to \(-\theta \), connection (1) arising from the Landau–Ginzburg model is exactly the quantum connection (2) of \({\mathbb {P}}(1,3)\), as predicted by mirror symmetry.

Dubrovin’s conjecture

Let X be a Fano variety (or an orbifold), such that the bounded derived category \(D^b(\mathrm{Coh}(X))\) of coherent sheaves on X admits a full exceptional collection \(\langle E_1,\ldots , E_n \rangle \), where the collection \(\langle E_1,\ldots , E_n\rangle \) is called

  • exceptional if \(R{\mathcal {H}}om(E_i,E_i)={\mathbb {C}} \) for all i and \(R{\mathcal {H}}om(E_i,E_j)=0\) for \(i>j\),

  • full if \(D^b(\mathrm{Coh}(X))\) is the smallest full triangulated subcategory of \(D^b(\mathrm{Coh}(X))\) containing \(E_1,\ldots ,E_n\).

In [10], Dubrovin conjectured that, under appropriate choices, the Stokes matrix of the quantum connection of X equals the Gram matrix of the Euler–Poincaré pairing with respect to some full exceptional collection—modulo some action of the braid group, sign changes and permutations (cf. [4, Sect. 2.3]). Then the second Stokes matrix is the transpose of the first one. The Euler–Poincaré pairing is given by the bilinear form

$$\begin{aligned} \chi (E,F) :=\sum _k (-1)^k \dim _{{\mathbb {C}}} \mathrm {Ext}^k(E,F), \quad E,F \in D^b(\mathrm{Coh}(X)). \end{aligned}$$

The Gram matrix of \(\chi \) with respect to a full exceptional collection is upper triangular with ones on the diagonal.

For \({\mathbb {P}}(a,b)\), \(\langle {\mathcal {O}},{\mathcal {O}}(1), \ldots , {\mathcal {O}}(a+b-1)\rangle \) is a full exceptional collection of \(D^b(\mathrm{Coh}({\mathbb {P}}(a,b)))\) (cf. [2, Theorem 2.12]). Following [3, Theorem 4.1], the cohomology of the twisting sheaves for \(k\in {\mathbb {Z}}\) is given by

  • \(H^0\left( {\mathbb {P}}(a,b),{\mathcal {O}}(k)\right) = \bigoplus _{(m,n)\in I_0} {\mathbb {C}} x^my^n, \) where

    $$\begin{aligned} I_0 = \left\{ (m,n) \in {\mathbb {Z}}_{\ge 0} \times {\mathbb {Z}}_{\ge 0} \mid am+bn=k \right\} , \end{aligned}$$
  • \(H^1 \left( {\mathbb {P}}(a,b),{\mathcal {O}}(k) \right) = \bigoplus _{(m,n) \in I_1} {\mathbb {C}} x^my^n, \) where

    $$\begin{aligned} I_1 = \left\{ (m,n)\in {\mathbb {Z}}_{<0} \times {\mathbb {Z}}_{<0} \mid am+bn=k \right\} , \end{aligned}$$
  • \(H^i \left( {\mathbb {P}}(a,b),{\mathcal {O}}(k) \right) = 0 \) for all \(i\ge 2\).

We only need to compute \(\mathrm {Ext}^k({\mathcal {O}}(i),{\mathcal {O}}(j))\) for \(i<j\), which is given by \(H^k\left( {\mathcal {O}}\left( j-i\right) \right) \) (cf. [17, Lemma 4.5]). Therefore, the zeroth cohomologies of the twisting sheaves \({\mathcal {O}}\left( j-i\right) \) are the only ones that contribute to the Gram matrix of \(\chi \). For \({\mathbb {P}}(1,3)\) we obtain the cohomology groups

$$\begin{aligned} H^0({\mathcal {O}}(1))\cong {\mathbb {C}},\ H^0({\mathcal {O}}(2))\cong {\mathbb {C}},\ \ H^0({\mathcal {O}}(3))\cong {\mathbb {C}}^2 \end{aligned}$$

and therefore the Gram matrix of the Euler–Poincaré pairing on \(D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))\) with respect to the full exceptional collection is given by

$$\begin{aligned} S_{\text {Gram}}= \begin{pmatrix} 1 &{}\quad 1 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$
(3)

Comparison of the Gram and Stokes matrix

Mirror symmetry relates the Laurent polynomial \(f=x + x^{-3}\) to the weighted projective line \({\mathbb {P}}(1,3)\). The pair \(({\mathbb {G}}_m, f=x + x^{-3})\) is a Landau–Ginzburg model of the weighted projective line \({\mathbb {P}}(1,3)\). According to Dubrovin’s conjecture, the Stokes matrix of the quantum connection of \({\mathbb {P}}(1,3)\) equals the Gram matrix of the Euler–Poincaré pairing with respect to some full exceptional collection of \(D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))\). Note that there is a natural action of the braid group on the Stokes matrix reflecting variations in the choices involved to determine the Stokes matrix (cf. [13]). In our case, we have to consider the braid group on four strands, namely

$$\begin{aligned} B_4 = \langle \beta _1, \beta _2, \beta _3 \mid \beta _1\beta _3\beta _1=\beta _3\beta _1\beta _3,\ \beta _1\beta _2\beta _1= \beta _2\beta _1\beta _2,\ \beta _2\beta _3\beta _2=\beta _3\beta _2\beta _3 \rangle . \end{aligned}$$

Proposition

\(S_{\text {Gram}}\) and \(S_{\beta }\) correspond to each other under the action of the elementary braid \(\beta _1\in B_4\).

Proof

We computed that the Gram matrix of \(\chi \) with respect to the full exceptional collection \({\mathcal {E}} \) is given by (3). Following [13, Sect. 6], the braid \(\beta _1\) acts on the Gram matrix as

where \(A^{\beta _1}\left( S_{\text {Gram}}\right) \) is given by

$$\begin{aligned} A^{\beta _1}(S_{\text {Gram}}) = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad -1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix} . \end{aligned}$$

We obtain that

$$\begin{aligned} S_{\text {Gram}}^{\beta _1} = \begin{pmatrix} 1 &{}\quad -1 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix} =S_{\beta }. \end{aligned}$$

\(\square \)

Remark

\(S_{\text {Gram}}^{\beta _1}=S_{\beta }\) is the Gram matrix of the Euler–Poincaré pairing with respect to the right mutation \({\mathbb {R}}_1{\mathcal {E}}\) of the full exceptional collection \({\mathcal {E}}\) (cf. [4, Proposition 13.1]). In our topological computations, the action of the braid \(\beta _1\in B_4\) should correspond to a counterclockwise rotation of \(\beta \).

Non-coprime parameters

In this section, we consider the weighted projective line \({\mathbb {P}}(2,2)\) as an example for the case of non-coprime parameters. The topological computation of the Stokes matrices of the quantum connection at \(\infty \) requires some adaptions.

A Landau–Ginzburg model of \({\mathbb {P}} (2,2)\) is given by the curve \(\left\{ x^2y^2=1 \right\} \subset {\mathbb {G}}_m^{2}\) together with the potential \(f=x+y\). This splits into two disjoint components \(U_1 :=\{xy+1=0 \} \) and \(U_2 :=\{ xy -1 =0 \}\). f restricts to \(f_1=x-x^{-1}\) on \(U_1\) and to \(f_2=x+x^{-1}\) on \(U_2\), where we identified \(y=-x^{-1}\) and \(y=x^{-1}\), respectively. The blue area in Fig. 8 shows where \(f_1\) has real (resp. imaginary) part greater than or equal to 0. The blue area in Fig. 9 shows where \(f_2\) has real (resp. imaginary) part greater than or equal to 0. In Fig. 10, the preimages of the real (resp. imaginary) axis under \(f_1\) and \(f_2\) are plotted.

f has singular fibers at \(\Sigma :=\{ \pm 2 i,\pm 2\}\). For our topological computations, we consider the perverse sheaf \(F=\mathrm{R}f_{*}{\mathbb {C}}[1] \in \mathrm{Perv}_{\Sigma }\left( {\mathbb {A}}^1\right) \). The exponential components at \(\infty \) of the Fourier–Sato transform of F are of linear type, with coefficients given by the \(\sigma _i\in \Sigma \). The Stokes rays are therefore given by \(\left\{ 0,\pm \frac{\pi }{4},\pm \frac{\pi }{2},\pm \frac{3\pi }{4},\pi \right\} \).

  • \(f^{-1}(2)=\{ (1,1)\in U_2, (1-\sqrt{2},1+\sqrt{2}) \in U_1, (1+\sqrt{2},1-\sqrt{2})\in U_1 \}\), (1, 1) being the double inverse image,

  • \(f^{-1}(-2)= \{ (-1,-1)\in U_2, (-1-\sqrt{2},-1+\sqrt{2})\in U_1, (-1+\sqrt{2},-1-\sqrt{2})\in U_1 \}\), \((-1,-1)\) being the double inverse image,

  • \(f^{-1}(2i)=\{ (i,i)\in U_1, (i+\sqrt{2}i,i-\sqrt{2}i)\in U_2, (i-\sqrt{2}i,i+\sqrt{2}i)\in U_2 \}\), (ii) being the double inverse image,

  • \(f^{-1}(-2)= \{ (-i,-i)\in U_1, (-i+\sqrt{2}i,-i-\sqrt{2}i), (-i-\sqrt{2}i,-i+\sqrt{2}i)\in U_2 \}\), \((-i,-i)\) being the double inverse image.

Fig. 8
figure8

LHS: \(\left\{ x \mid \mathfrak {R}\left( f_1(x)\right) \ge 0 \right\} \), RHS: \(\left\{ x \mid \mathfrak {I}\left( f_1(x)\right) \ge 0 \right\} \)

Fig. 9
figure9

LHS: \(\{ x \mid \mathfrak {R}(f_2(x))\ge 0 \}\), RHS: \(\{ x \mid \mathfrak {I}(f_2(x))\ge 0 \}\)

Fig. 10
figure10

Preimage of the real (resp. imaginary) axis in blue (resp. red) color under \(f_1\) (LHS) and \(f_2\) (RHS)

We choose \(\alpha = e^{3\pi i/8},\ \beta = e^{9 \pi i /8}\). This induces the following order on \(\Sigma \):

$$\begin{aligned} {\sigma _1} :=2<_{\beta } {\sigma _2} :=-2i<_{\beta } {\sigma _3} :=2i <_{\beta } {\sigma _4} :=-2. \end{aligned}$$

Denote by \(\ell _{\sigma _i} = \sigma _i + {\mathbb {R}}_{\ge 0} \alpha \). Their preimages are depicted in Figs. 11 and 12.

Fig. 11
figure11

Preimages under \(f_1\) of lines passing through \(\sigma _2\) and \(\sigma _3\) with phase \(3\pi /8\)

Fig. 12
figure12

Preimages under \(f_2\) of lines passing through \(\sigma _1\) and \(\sigma _4\) with phase \(3\pi /8\)

Fig. 13
figure13

Preimages of \(\gamma _{\sigma _i}\) under \(f_1\) (LHS) and \(f_2\) (RHS)

Fig. 14
figure14

Preimages of \(\ell _{\sigma _i}\) under \(f_1\) (LHS) and \(f_2\) (RHS)

As in the previous example, only the lifts of \(\gamma _{\sigma _i}\) and \(\ell _{\sigma _i}\) around the double preimages of \(\sigma _i\), which we denote by \(\sigma _i^1\), contribute to the monodromy and the cokernel of \(b_{\sigma _i}\). Therefore, in our figures, we restricted to this information.

From Fig. 13 we read the monodromies in the ordered basis \(e_1,e_2,e_3,e_4\) to be

$$\begin{aligned} T_{\sigma _1} = \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix},\quad T_{\sigma _2} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \end{pmatrix},\\ T_{\sigma _3} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \quad T_{\sigma _4} = \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$

Taking into account Fig. 14, we identify the cokernel of 

  • \(b_{\sigma _1}\) with \({\mathbb {C}}\) via \( \left[ \begin{pmatrix} a_1 ,a_2,a_3, a_4 \end{pmatrix}\right] ^{\mathrm {t}}= \left[ \begin{pmatrix} a_1-a_3, 0, 0, 0 \end{pmatrix}\right] ^{\mathrm {t}}\),

  • \(b_{\sigma _2}\) with \({\mathbb {C}}\) via \(\left[ \begin{pmatrix} a_1 ,a_2,a_3,a_4 \end{pmatrix} \right] ^{\mathrm {t}}= \left[ \begin{pmatrix} 0,a_2-a_4,0,0 \end{pmatrix} \right] ^{\mathrm {t}}\),

  • \(b_{\sigma _3}\) with \({\mathbb {C}}\) via \(\left[ \begin{pmatrix} a_1 ,a_2,a_3,a_4 \end{pmatrix} \right] ^{\mathrm {t}}= \left[ \begin{pmatrix} 0,a_2-a_4,0,0 \end{pmatrix} \right] ^{\mathrm {t}}\),

  • \(b_{\sigma _4}\) with \({\mathbb {C}}\) via \(\left[ \begin{pmatrix} a_1 ,a_2,a_3,a_4 \end{pmatrix} \right] ^{\mathrm {t}}= \left[ \begin{pmatrix} a_1-a_3,0,0,0 \end{pmatrix} \right] ^{\mathrm {t}}\).

We therefore obtain

$$\begin{aligned} u_{\sigma _1}=&\begin{pmatrix} 1&\quad 0&\quad -1&\quad 0 \end{pmatrix}=u_{\sigma _4}, \\ u_{\sigma _2} =&\begin{pmatrix} 0&\quad 1&\quad 0&\quad -1 \end{pmatrix} = u_{\sigma _3}, \end{aligned}$$

and \(v_{\sigma _i}=u_{\sigma _i}^{\mathrm {t}}\). In summary, we obtain the following

Theorem

The Stokes matrices of the Fourier–Laplace transform of \(H^0(\int _f {\mathcal {O}})\) in the chosen bases are given by

$$\begin{aligned} S_{\beta } = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 2\\ 0 &{}\quad 1 &{}\quad 2 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}, \quad S_{-\beta } = -S_{\beta }^{\mathrm {t}}. \end{aligned}$$
(4)

\(S_{\pm \beta }\) describes passing \(\pm \beta {\mathbb {R}}_{>0}\subset \left( {\mathbb {A}}^1\right) ^{\vee }\setminus \{ 0 \}\) from \(H_{\alpha }\) to \(H_{-\alpha }\), where

$$\begin{aligned} H_{\alpha }= & {} \left\{ w\mid \arg (w) \in \left[ -\frac{7\pi }{8},\frac{\pi }{8}\right] \right\} ,\\ \ H_{-\alpha }= & {} \left\{ w \mid \arg (w) \in \left[ \frac{\pi }{8},\frac{9\pi }{8}\right] \right\} \subset \left( {\mathbb {A}}^1\right) ^{\vee }\setminus \{ 0 \} . \end{aligned}$$

\(\square \)

In the non-coprime case \(\gcd (a,b)\ne 1\), the computation of the orbifold cohomology of \({\mathbb {P}}(a,b)\) is more subtle. We refer to [16] for precise formulae and the correspondence of the quantum connection and the Fourier–Laplace transform of the Gauß–Manin connection of the Landau–Ginzburg model.

For \({\mathbb {P}}(2,2)\) we get the cohomology groups

$$\begin{aligned} H^0({\mathcal {O}}(1))\cong H^0({\mathcal {O}}(3))\cong 0,\ H^0({\mathcal {O}}(2))\cong {\mathbb {C}}^2 \end{aligned}$$

and therefore the Gram matrix of the Euler–Poincaré pairing on \(D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))\) with respect to \({\mathcal {E}} =\langle {\mathcal {O}},{\mathcal {O}}(1),{\mathcal {O}}(2),{\mathcal {O}}(3)\rangle \) is given by

$$\begin{aligned} S_{\text {Gram}}= \begin{pmatrix} 1 &{}\quad 0 &{}\quad 2 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 2\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$
(5)

Proposition

\(S_{\text {Gram}}\) and \(S_{\beta }\) correspond to each other under the action of \(S_4\).

Proof

By the permutation 

$$\begin{aligned} P=\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0&{}\quad 0 &{}\quad 1 &{}\quad 0 \end{pmatrix}\in S_4, \end{aligned}$$

acting on the Gram matrix \(S_{\text {Gram}}\) as \(P\cdot S_{\text {Gram}} \cdot P^{-1}\) (cf. [13, Sect. 6.c]), we find that the Gram matrix \(S_{\text {Gram}}\) (5) is transformed into the topologically computed Stokes matrix \(S_{\beta }\) (4). \(\square \)

Notes

  1. 1.

    Counterclockwise orientation since the imaginary part of \(\langle \alpha , \beta \rangle \) is positive.

  2. 2.

    We always consider the case \(q=1\).

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Acknowledgements

Open access funding provided by Projekt DEAL. I am grateful to Marco Hien and Maxim Smirnov for supporting me throughout the work on this article. I am thankful to Étienne Mann and Thomas Reichelt for useful discussions and hints.

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Sattelberger, AL. Topological computation of Stokes matrices of some weighted projective lines. manuscripta math. 164, 327–347 (2021). https://doi.org/10.1007/s00229-020-01193-3

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Mathematics Subject Classification

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