topological computation of stokes matrices of some …...anna-laura sattelberger topological...

manuscripta math. 164, 327–347 (2021) © The Author(s) 2020

Anna-Laura Sattelberger

Topological computation of Stokes matrices of someweighted projective lines

Received: 6 November 2019 / Accepted: 13 March 2020 / Published online: 5 April 2020

Abstract. By mirror symmetry, the quantum connection of a weighted projective line isclosely 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 Stokesmatrices for the latter at ∞ for the cases P(1, 3) and P(2, 2) by purely topological methods.We compare them to the Gram matrix of the Euler–Poincaré pairing on Db(Coh(P(1, 3)))and Db(Coh(P(2, 2))), respectively. This article is based on the doctoral thesis of the author.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3271. Gauß–Manin system and its Fourier–Laplace transform . . . . . . . . . . . . . 3292. Topological computation of the Stokes matrices . . . . . . . . . . . . . . . . . 3303. Quantum connection and Dubrovin’s conjecture . . . . . . . . . . . . . . . . . 335

3.1. Quantum connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3353.2. Dubrovin’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3393.3. Comparison of the Gram and Stokes matrix . . . . . . . . . . . . . . . . . 340

4. Non-coprime parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Introduction

In [6], D’Agnolo et al. describe how to compute the Stokesmatrices of the enhancedFourier–Sato transform of a perverse sheaf on the affine line by purely topolog-ical methods. To a regular singular holonomic D-module M ∈ Modrh

(DA1)on

the affine line, one associates a perverse sheaf via the regular Riemann–Hilbertcorrespondence

RHomDanA1

((•)an,Oan

A1

) [1] : Modrh(DA1

) �−→ Perv(CA1

).

A.-L. Sattelberger (B): Max Planck Institute for Mathematics in the Sciences, Leipzig,Germany. e-mail: [email protected]

Mathematics Subject Classification: 32C38 · 34M40 · 53D37

https://doi.org/10.1007/s00229-020-01193-3

http://crossmark.crossref.org/dialog/?doi=10.1007/s00229-020-01193-3&domain=pdf

http://orcid.org/0000-0002-2308-4070

328 A.-L. Sattelberger

Let � ⊂ A1 denote the set of singularities of M. Following [6, Sect. 4.2], after

suitably choosing a total order on�, the resulting perverse sheaf F ∈ Perv�

(CA1

)

can be described by linear algebra data, namely its quiver

(�(F),�σ (F), uσ , vσ )σ∈� ,

where �(F) and �σ (F) are finite dimensional C-vector spaces and uσ : �(F) →�σ (F) and vσ : �σ (F) → �(F) are linear maps such that 1 − uσ vσ is invert-ible for any σ . The main result in [6] is a determination of the Stokes matrices ofthe enhanced Fourier–Sato transform of F and therefore of the Fourier–Laplacetransform of M in terms of the quiver of F . This result builds on the irregu-lar Riemann–Hilbert correspondence of D’Agnolo and Kashiwara [7], which pro-vides a topological description of holonomicD-modules. As proven by Kashiwaraand Schapira [14], this correspondence intertwines the Fourier–Laplace with the(enhanced) Fourier–Sato transform.

Mirror symmetry connects the weighted projective line P(1, 3) with theLandau–Ginzburg model

(Gm, f = x + x−3

).

The quantum connection of P(1, 3) is closely related to the Fourier–Laplace trans-form of the Gauß–Manin system H0(

∫f O) of f . We compute that

F := R f∗C[1] ∈ Perv�

(CA1

),

where � denotes the set of singular values of f , is the perverse sheaf associatedto H0(

∫f O) by the Riemann–Hilbert correspondence. In Sect. 1, we compute the

localized Fourier–Laplace transform of H0(∫f O). In Sect. 2, analogous to the

examples in [6, Sect. 7], we carry out the topological computation of the Stokesmatrices of the Fourier–Laplace transform of H0(

∫f O). In Sect. 3, we compare

the Stokes matrix Sβ , that we obtained from our topological computations, to theGram matrix of the Euler–Poincaré pairing on Db(Coh(P(1, 3))) with respect toa suitable full exceptional collection. Following Dubrovin’s conjecture about theStokes matrix of the quantum connection, proven for the weighted projective spaceP (ω0, . . . , ωn) by Tanabé and Ueda in [19] and by Cruz Morales and van derPut in [5], they are known to be equivalent after appropriate modifications. Wegive the explicit braid of the braid group B4 that deforms the Gram matrix intothe Stokes matrix Sβ . Section 4 tackles the computations for the case of non-coprimeparameters. In comparison to the case of coprimeparameters, this requires aslightlymodified approach.We compute the Stokesmatrices of the Fourier–Laplacetransform of the Gauß–Manin system of the Landau–Ginzburg model of P(2, 2)and set it into relation with the Gram matrix of the Euler–Poincaré pairing onDb(Coh(P(2, 2))).

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

Topological computation of Stokes matrices of some P(a, b) 329

1. Gauß–Manin system and its Fourier–Laplace transform

Let X be affine and f a regular function f : X → A1 on X . Denote by

∫f (•)

the direct image in the category of D-modules and by M := H0(∫f OX ) ∈

Modrh(DA1

)the zeroth cohomology of the Gauß–Manin system of f . Follow-

ing [9, Sect. 2.c], it is given by

M = �n(X) [∂t ] /(d − ∂t d f ∧)�n−1(X) [∂t ] .

Denote by G := M̂[τ−1] the Fourier–Laplace transform of M , localized at τ = 0.It is given by

G = �n(X)[τ, τ−1

]/ (d − τd f ∧)�n−1(X)

[τ, τ−1

].

G is a free C[τ, τ−1]-module of finite rank. Rewriting in the variable θ = τ−1

gives the C[θ, θ−1]-module

G = �n(X)[θ, θ−1

]/ (θd − d f ∧)�n−1(X)

[θ, θ−1

].

G is endowed with a flat connection given as follows. For γ = [∑k∈Z ωkθ

k] ∈ G,

where �n(X) ωk = 0 for almost all k, the connection is given by (cf. [12,Definition 2.3.1]):

θ2∇ ∂∂θ

(γ ) =[∑

k

f ωkθk +

∑

k

kωkθk+1

]

.

It is known that (G,∇) has a regular singularity at θ = ∞ and possibly an irregularone at θ = 0.

We now consider the Laurent polynomial f = x + x−3 ∈ C[x, x−1

], being

a regular function on the multiplicative group Gm . For our computations we passto the variable θ = τ−1. We compute that for the given f , G is given by the freeC[θ, θ−1

]-module

G = C

[x, x−1

]dx[θ, θ−1

]/(θd −

(dx − 3x−4dx

)∧)C

[x, x−1

] [θ, θ−1

]

with basis over C[θ, θ−1

]given by

[ dxx

],[dxx2

],[dxx3

],[dxx4

]. In this basis, the

connection is given by

θ∇ ∂∂θ

= θ∂θ +

⎛

⎜⎜⎝

0 43θ 0 0

0 13

43θ 0

0 0 23

43θ

4θ

0 0 1

⎞

⎟⎟⎠ . (1)

Via the cyclic vector m = (1, 0, 0, 0)t , we compute the relation

∇4θ∂θ

m + 4∇3θ∂θ

m + 32

9∇2

θ∂θm − 256

27θ4m = 0


NP (P )

0

1

1

Fig. 1. Newton polygon of P

and therefore associate the differential operator

P = (θ∂θ )4 + 4 (θ∂θ )

3 + 32

9(θ∂θ )

2 − 256

27θ4∈ C

[θ, θ−1

]〈∂θ 〉 = DGm .

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

2. Topological computation of the Stokes matrices

We consider the Laurent polynomial f = x + x−3 : Gm → A1. Its critical points

are given by{± 4

√3,± 4

√3i}. The critical values of f are given by

� ={± 4

4√27

,± 4i4√27

}⊂ A

1.

The preimages of

• 44√27

are 4√3 (double), −1−√

2i4√27

and −1+√2i

4√27,

• − 44√27

are − 4√3 (double), 1−√

2i4√27

and 1+√2i

4√27,

• 4i4√27

are 4√3i (double), −√

2−i4√27

and√2−i4√27

,

• − 4i4√27

are − 4√3i (double),

√2+i4√27

and −√2+i

4√27.

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

RHomDan

((∫

fO)an

,Oan)

� R f an∗ RHomDan

(Oan, f †Oan

)� R f an∗ C.


Fig. 2. LHS: {x | �( f (x)) ≥ 0}, RHS: {x | �( f (x)) ≥ 0}

Since f is semismall, R f∗C[1] ∈ Perv(CA1) is a perverse sheaf (cf. [8]). Outsideof�, f is a covering of degree 4, therefore R f∗C[1] ∈ Perv�(CA1). By the regularRiemann–Hilbert correspondence

Sol(•)[dim X ]:=RHomDanX

((•)an ,Oan

X

)[dim X ] : Modrh (DX )

�−→Perv(CXan),

we associate to H0(∫f O) the perverse sheaf F := R f∗C[1].

Wefixα = eπ i8 ∈ A

1, β = e3π i8 ∈ (A1

)∨, such that�(〈α, β〉) = 0, �(〈α, β〉) = 1.

This induces the following order on � (cf. [6, Sect. 4]):

σ1 := 4i4√27

<β σ2 := − 44√27

<β σ3 := 44√27

<β σ4 := − 4i4√27

.

In Fig. 4, the σi are depicted in the following colors:

• σ1: green, • σ2: red, • σ3: purple, • σ4: orange.

The blue area in Fig. 2 shows where f has real (resp. imaginary) part greater thanor equal to 0. In Fig. 3, the preimage of the imaginary (resp. real) axis under fis plotted in blue (resp. red) color. We consider lines passing through the singularvalues with phase π

8 , as depicted in Fig. 4. The preimages of these lines are plottedin Fig. 5.We fix a base point ewith�(e) > � (σi ) for all i and denote its preimagesby e1, e2, e3, e4, 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

�σi (F) := R�c

(A1;C�×

σi⊗ F

)� H0R�c

(�×σi

; F) ∼=⊕

e j∈ f −1(e)

Ce j∼= C

4,

�(F) := R�c

(A1;CA1\��

⊗ F)

[1] � �σi (F) ∼= C4.

Furthermore, we fix isomorphisms i−1σi

F[−1] ∼= ⊕σ

ji ∈ f −1(σi )

Cσ

ji

∼= C3.


Fig. 3. Preimage of the imaginary (resp. real) axis in blue (resp. red) color

Fig. 4. Lines passing through σi with phaseπ8

The exponential components at ∞ of the Fourier–Laplace transform ofH0(

∫f O) are known to be of linear type with coefficients given by the σi ∈ �.

The Stokes rays are therefore given by

{0,±π

4,±π

2,±3π

4, π

}.

We consider loops γσi , starting at e and running around the singular value σi in

counterclockwise orientation,1 as depicted in Fig. 6.We denote by γj

σi the preimageof γσi starting at e j , j = 1, 2, 3, 4. The figure constitutes a rough drawing of thepreimages of γσi . By taking into account the preimages of the different segments of

the axes and the intersections of γσi with them, one recovers the γj

σi as depicted in

1 Counterclockwise orientation since the imaginary part of 〈α, β〉 is positive.


Fig. 5. Preimages under f

the figure. From Fig. 6 we read, in the ordered basis e1, e2, e3, e4, the monodromies

Tσ1 =

⎛

⎜⎜⎝

0 1 0 01 0 0 00 0 1 00 0 0 1

⎞

⎟⎟⎠ , Tσ2 =

⎛

⎜⎜⎝

1 0 0 00 0 1 00 1 0 00 0 0 1

⎞

⎟⎟⎠ ,

Tσ3 =

⎛

⎜⎜⎝

0 0 0 10 1 0 00 0 1 01 0 0 0

⎞

⎟⎟⎠ , Tσ4 =

⎛

⎜⎜⎝

0 0 1 00 1 0 01 0 0 00 0 0 1

⎞

⎟⎟⎠ .

In order to obtain the maps bσi , we consider the half-lines �σi := σi + αR≥0. We

denote their preimages under f by {� jσi } j=1,2,3,4, depending on which γ

jσi they

intersect. We label the preimages of σi by σ 1i , σ 2

i , σ 3i , as depicted in Fig. 7. By

the derivation of the short exact sequence of quivers [6, (7.1.3)] and passing toBorel–Moore homology as described in [6, Lemma 5.3.1.(i)], bσi is induced fromthe corresponding boundary value map from �σi to its origin σi . Therefore, bσi

encodes which lift of �σi starts at which preimage of σi . Namely, from Fig. 7 weread the following:

σ1: �1σ1 �→ σ 11 , �2σ1 �→ σ 1

1 , �3σ1 �→ σ 21 , �4σ1 �→ σ 3

1 .

Therefore, bσ1 is the transpose of

⎛

⎝1 1 0 00 0 1 00 0 0 1

⎞

⎠.

σ2: �1σ2 �→ σ 32 , �2σ2 �→ σ 1

2 , �3σ2 �→ σ 12 , �4σ2 �→ σ 2

2 .


⎛

⎝0 1 1 00 0 0 11 0 0 0

⎞

⎠.


σ3: �1σ3 �→ σ 13 , �2σ3 �→ σ 2

3 , �3σ3 �→ σ 33 , �4σ3 �→ σ 1

3 .


⎛

⎝1 0 0 10 1 0 00 0 1 0

⎞

⎠.

σ4: �1σ4 �→ σ 14 , �2σ4 �→ σ 3

4 , �3σ4 �→ σ 14 , �4σ4 �→ σ 2

4 .


⎛

⎝1 0 1 00 0 0 10 1 0 0

⎞

⎠.

We obtain, in the ordered bases σ 1i , σ 2

i , σ 3i and �1σi , �

2σi

, �3σi , �4σieach:

bσ1 =

⎛

⎜⎜⎝

1 0 01 0 00 1 00 0 1

⎞

⎟⎟⎠ , bσ2 =

⎛

⎜⎜⎝

0 0 11 0 01 0 00 1 0

⎞

⎟⎟⎠ ,

bσ3 =

⎛

⎜⎜⎝

1 0 00 1 00 0 11 0 0

⎞

⎟⎟⎠ , bσ4 =

⎛

⎜⎜⎝

1 0 00 0 11 0 00 1 0

⎞

⎟⎟⎠ .

Denote by ui := uσi , vi := vσi , Ti := Tσi and �i := �σi . As described in [6,

Sect. 7], we obtain �i (F) �(F)vi

uias the cokernels of the following dia-

grams:

i−1σi

F[−1] �(F)

0 �(F)

bσi

1−Ti1

We identify the cokernels of bσi in the following way:

• coker(bσ1) � C via

⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1a2a3a4

⎞

⎟⎟⎠

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1 − a2000

⎞

⎟⎟⎠

⎤

⎥⎥⎦ ,


⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1a2a3a4

⎞

⎟⎟⎠

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

⎛

⎜⎜⎝

0a2 − a3

00

⎞

⎟⎟⎠

⎤

⎥⎥⎦ ,


⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1a2a3a4

⎞

⎟⎟⎠

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1 − a4000

⎞

⎟⎟⎠

⎤

⎥⎥⎦ ,



⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1a2a3a4

⎞

⎟⎟⎠

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

⎛

⎜⎜⎝

a1 − a3000

⎞

⎟⎟⎠

⎤

⎥⎥⎦ .

We obtain that (�i (F) �(F))vi

ui� (C C

4),vi

uiwhere

u1 = (1 −1 0 0

), u2 = (

0 1 −1 0),

u3 = (1 0 0 −1

), u4 = (

1 0 −1 0),

and vi = uti . By [6, Theorem 5.2.2], we obtain the following

Theorem. Under the choices made, the Stokes matrices of the Fourier–Laplacetransform of H0(

∫f O) at ∞ are given as

Sβ =

⎛

⎜⎜⎝

1 u1v2 u1v3 u1v40 1 u2v3 u2v40 0 1 u3v40 0 0 1

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

1 −1 1 10 1 0 10 0 1 10 0 0 1

⎞

⎟⎟⎠ ,

S−β =

⎛

⎜⎜⎝

T1 0 0 0−u2v1 T2 0 0−u3v1 −u3v2 T3 0−u4v1 −u4v2 −u4v3 T4

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

−1 0 0 01 −1 0 0

−1 0 −1 0−1 −1 −1 −1

⎞

⎟⎟⎠ = −Stβ,

where Ti := 1 − uivi . S±β describes crossing h±β from Hα to H−α , where

Hα ={w | arg(w) ∈

[−5π

8,3π

8

]},

H−α ={w ∈| arg(w) ∈

[3π

8,11π

8

]}⊂(A1)∨

denote the closed sectors at∞ and h±β = ±R>0β ⊂ (A1)∨, such that Hα∩H−α =

hβ ∪ h−β . ��

3. Quantum connection and Dubrovin’s conjecture

3.1. Quantum connection

The quantum connection of a Fano variety (resp. an orbifold) X is a connection onthe trivial vector bundle over P1 with fiber H∗(X,C) (resp. H∗

orb(X,C)), where zdenotes the standard inhomogeneous coordinate at∞. By [11, (2.2.1)], the quantumconnection is the connection given by

∇z∂z = z∂

∂z− 1

z(−KX◦) + μ,


Fig. 6. �σi and their preimages under f


Fig. 7. �σi and their preimages under f


where the first term on the right hand side is ordinary differentiation, the secondone is pointwise quantummultiplication by (−KX ), and the third one is the gradingoperator

μ(a) :=(i

2− dim X

2

)a for a ∈ Hi (X,C).

The quantum connection is regular singular at z = ∞ and irregular singular atz = 0.

For the weighted projective line P(a, b), the orbifold cohomology ring is givenby (cf. [16, Example 3.20])

H∗orb(P(a, b),C) = C [x, y, ξ ] /〈xy, ax a

d − bybd ξn−m, ξd − 1〉,

where d = gcd(a, b) andm, n ∈ Z such that am+bn = d. The grading is given asfollows (cf. [1, Sect. 9]): deg x = 1

A , deg y = 1B , deg ξ = 0, where A = a

d , B = bd .

Quantum multiplication2 is computed in

QH∗orb = C [x, y, ξ ] /〈xy − 1, ax

ad − by

bd ξn−m, ξd − 1〉.

For gcd(a, b) = 1,−KP(a,b) is given by the element [xa + yb] ∈ H1orb(P(a, b),C).

Taking into account that the grading is scaled by 2, the grading operator is definedby μ(a) = (

i − dim X2

)a for a ∈ Hi

orb(X,C).We obtain the quantum connection of P(1, 3) as follows.

H∗orb(P(1, 3),C) = C [x, y] /〈xy, x − 3y3〉

with grading given by deg x = 1, deg y = 13 .A basis overC is given by 1, y, y2, y3.

Quantum multiplication by −KP(1,3) = [x + y3

] = [4y3

]in this basis is given by

the matrix⎛

⎜⎜⎝

0 43 0 0

0 0 43 0

0 0 0 43

4 0 0 0

⎞

⎟⎟⎠ .

The grading μ is given by the matrix⎛

⎜⎜⎝

− 12 0 0 00 − 1

6 0 00 0 1

6 00 0 0 1

2

⎞

⎟⎟⎠ .

Therefore, the quantum connection of P(1, 3) is given by

∇z∂z = z∂z − 1

z

⎛

⎜⎜⎝

0 43 0 0

0 0 43 0

0 0 0 43

4 0 0 0

⎞

⎟⎟⎠+

⎛

⎜⎜⎝

− 12 0 0 00 − 1

6 0 00 0 1

6 00 0 0 1

2

⎞

⎟⎟⎠ . (2)

2 We always consider the case q = 1.


Observation. By the gauge transformation h = diag(θ− 12 , θ− 1

2 , θ− 12 , θ− 1

2 ),which subtracts 1

2 on the diagonal entries, and passing to −θ , connection (1)arising from the Landau–Ginzburg model is exactly the quantum connection (2) ofP(1, 3), as predicted by mirror symmetry.

3.2. Dubrovin’s conjecture

Let X be a Fano variety (or an orbifold), such that the bounded derived cate-gory Db(Coh(X)) of coherent sheaves on X admits a full exceptional collection〈E1, . . . , En〉, where the collection 〈E1, . . . , En〉 is called• exceptional if RHom(Ei , Ei ) = C for all i and RHom(Ei , E j ) = 0 for i > j ,• full if Db(Coh(X)) is the smallest full triangulated subcategory of Db(Coh(X))

containing E1, . . . , En .

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

χ(E, F) :=∑

k

(−1)k dimC Extk(E, F), E, F ∈ Db(Coh(X)).

TheGrammatrix ofχ with respect to a full exceptional collection is upper triangularwith ones on the diagonal.

For P(a, b), 〈O,O(1), . . . ,O(a + b − 1)〉 is a full exceptional collection ofDb(Coh(P(a, b))) (cf. [2, Theorem 2.12]). Following [3, Theorem 4.1], the coho-mology of the twisting sheaves for k ∈ Z is given by

• H0 (P(a, b),O(k)) = ⊕(m,n)∈I0 Cx

m yn, where

I0 = {(m, n) ∈ Z≥0 × Z≥0 | am + bn = k

},

• H1 (P(a, b),O(k)) = ⊕(m,n)∈I1 Cx

m yn, where

I1 = {(m, n) ∈ Z<0 × Z<0 | am + bn = k} ,

• Hi (P(a, b),O(k)) = 0 for all i ≥ 2.

We only need to compute Extk(O(i),O( j)) for i < j , which is given byHk (O ( j − i)) (cf. [17, Lemma 4.5]). Therefore, the zeroth cohomologies of thetwisting sheaves O ( j − i) are the only ones that contribute to the Gram matrixof χ . For P(1, 3) we obtain the cohomology groups

H0(O(1)) ∼= C, H0(O(2)) ∼= C, H0(O(3)) ∼= C2


and therefore the Gram matrix of the Euler–Poincaré pairing on Db(Coh(P(1, 3)))with respect to the full exceptional collection E := 〈O,O(1),O(2),O(3)〉 is givenby

SGram =

⎛

⎜⎜⎝

1 1 1 20 1 1 10 0 1 10 0 0 1

⎞

⎟⎟⎠ . (3)

3.3. Comparison of the Gram and Stokes matrix

Mirror symmetry relates the Laurent polynomial f = x + x−3 to the weightedprojective line P(1, 3). The pair (Gm, f = x + x−3) is a Landau–Ginzburg modelof the weighted projective line P(1, 3). According to Dubrovin’s conjecture, theStokes matrix of the quantum connection of P(1, 3) equals the Gram matrix ofthe Euler–Poincaré pairing with respect to some full exceptional collection ofDb(Coh(P(1, 3))). Note that there is a natural action of the braid group on theStokes matrix reflecting variations in the choices involved to determine the Stokesmatrix (cf. [13]). In our case, we have to consider the braid group on four strands,namely

B4 = 〈β1, β2, β3 | β1β3β1 = β3β1β3, β1β2β1 = β2β1β2, β2β3β2 = β3β2β3〉.Proposition. SGram and Sβ correspond to each other under the action of the ele-mentary braid β1 ∈ B4.

Proof. We computed that the Gram matrix of χ with respect to the full exceptionalcollection E is given by (3). Following [13, Sect. 6], the braid β1 acts on the Grammatrix as

SGram �→ Sβ1Gram := Aβ1 (SGram) · SGram · (Aβ1 (SGram)

)t,

where Aβ1 (SGram) is given by

Aβ1(SGram) =

⎛

⎜⎜⎝

0 1 0 01 −1 0 00 0 1 00 0 0 1

⎞

⎟⎟⎠ .

We obtain that

Sβ1Gram =

⎛

⎜⎜⎝

1 −1 1 10 1 0 10 0 1 10 0 0 1

⎞

⎟⎟⎠ = Sβ.

��Remark. Sβ1

Gram = Sβ is theGrammatrix of the Euler–Poincaré pairingwith respectto the right mutation R1E of the full exceptional collection E (cf. [4, Proposi-tion 13.1]). In our topological computations, the action of the braid β1 ∈ B4 shouldcorrespond to a counterclockwise rotation of β.


Fig. 8. LHS: {x | � ( f1(x)) ≥ 0}, RHS: {x | � ( f1(x)) ≥ 0}

4. Non-coprime parameters

In this section, we consider the weighted projective line P(2, 2) as an example forthe case of non-coprime parameters. The topological computation of the Stokesmatrices of the quantum connection at ∞ requires some adaptions.

A Landau–Ginzburg model of P(2, 2) is given by the curve{x2y2 = 1

} ⊂ G2m

together with the potential f = x + y. This splits into two disjoint componentsU1 := {xy+1 = 0} andU2 := {xy−1 = 0}. f restricts to f1 = x−x−1 onU1 andto f2 = x + x−1 onU2, where we identified y = −x−1 and y = x−1, respectively.The blue area in Fig. 8 shows where f1 has real (resp. imaginary) part greater thanor equal to 0. The blue area in Fig. 9 shows where f2 has real (resp. imaginary) partgreater than or equal to 0. In Fig. 10, the preimages of the real (resp. imaginary)axis under f1 and f2 are plotted.f has singular fibers at� := {±2i,±2}. For our topological computations, we con-sider the perverse sheaf F = R f∗C[1] ∈ Perv�

(A1). The exponential components

at ∞ of the Fourier–Sato transform of F are of linear type, with coefficients givenby the σi ∈ �. The Stokes rays are therefore given by

{0,±π

4 ,±π2 ,± 3π

4 , π}.

• f −1(2) = {(1, 1) ∈ U2, (1 − √2, 1 + √

2) ∈ U1, (1 + √2, 1 − √

2) ∈ U1},(1, 1) being the double inverse image,

• f −1(−2) = {(−1,−1) ∈ U2, (−1 − √2,−1 + √

2) ∈ U1, (−1 + √2,−1 −√

2) ∈ U1}, (−1,−1) being the double inverse image,• f −1(2i) = {(i, i) ∈ U1, (i + √

2i, i − √2i) ∈ U2, (i − √

2i, i + √2i) ∈ U2},

(i, i) being the double inverse image,• f −1(−2) = {(−i,−i) ∈ U1, (−i +√

2i,−i −√2i), (−i −√

2i,−i +√2i) ∈

U2}, (−i,−i) being the double inverse image.

We choose α = e3π i/8, β = e9π i/8. This induces the following order on �:

σ1 := 2 <β σ2 := −2i <β σ3 := 2i <β σ4 := −2.

Denote by �σi = σi + R≥0α. Their preimages are depicted in Figs. 11 and 12.


Fig. 9. LHS: {x | �( f2(x)) ≥ 0}, RHS: {x | �( f2(x)) ≥ 0}

Fig. 10. Preimage of the real (resp. imaginary) axis in blue (resp. red) color under f1 (LHS)and f2 (RHS)

Fig. 11. Preimages under f1 of lines passing through σ2 and σ3 with phase 3π/8


Fig. 12. Preimages under f2 of lines passing through σ1 and σ4 with phase 3π/8

Fig. 13. Preimages of γσi under f1 (LHS) and f2 (RHS)


Fig. 14. Preimages of �σi under f1 (LHS) and f2 (RHS)

As in the previous example, only the lifts of γσi and �σi around the doublepreimages of σi , which we denote by σ 1

i , contribute to the monodromy and thecokernel of bσi . Therefore, in our figures, we restricted to this information.From Fig. 13 we read the monodromies in the ordered basis e1, e2, e3, e4 to be

Tσ1 =

⎛

⎜⎜⎝

0 0 1 00 1 0 01 0 0 00 0 0 1

⎞

⎟⎟⎠ , Tσ2 =

⎛

⎜⎜⎝

1 0 0 00 0 0 10 0 1 00 1 0 0

⎞

⎟⎟⎠ ,

Tσ3 =

⎛

⎜⎜⎝

1 0 0 00 0 0 10 0 1 00 1 0 0

⎞

⎟⎟⎠ , Tσ4 =

⎛

⎜⎜⎝

0 0 1 00 1 0 01 0 0 00 0 0 1

⎞

⎟⎟⎠ .


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

• bσ1 with C via[(a1, a2, a3, a4

)]t = [(a1 − a3, 0, 0, 0

)]t ,• bσ2 with C via

[(a1, a2, a3, a4

)]t = [(0, a2 − a4, 0, 0


[(a1, a2, a3, a4

)]t = [(0, a2 − a4, 0, 0


[(a1, a2, a3, a4

)]t = [(a1 − a3, 0, 0, 0

)]t .

We therefore obtain

uσ1 = (1 0 −1 0) = uσ4 ,

uσ2 = (0 1 0 −1) = uσ3 ,

and vσi = utσi . In summary, we obtain the following

Theorem. The Stokes matrices of the Fourier–Laplace transform of H0(∫f O) in

the chosen bases are given by

Sβ =

⎛

⎜⎜⎝

1 0 0 20 1 2 00 0 1 00 0 0 1

⎞

⎟⎟⎠ , S−β = −Stβ. (4)

S±β describes passing ±βR>0 ⊂ (A1)∨ \ {0} from Hα to H−α , where

Hα ={w | arg(w) ∈

[−7π

8,π

8

]},

H−α ={w | arg(w) ∈

[π

8,9π

8

]}⊂(A1)∨ \ {0}.

��In the non-coprime case gcd(a, b) �= 1, the computation of the orbifold coho-

mology of P(a, b) is more subtle. We refer to [16] for precise formulae and thecorrespondence of the quantum connection and the Fourier–Laplace transform ofthe Gauß–Manin connection of the Landau–Ginzburg model.

For P(2, 2) we get the cohomology groups

H0(O(1)) ∼= H0(O(3)) ∼= 0, H0(O(2)) ∼= C2

and therefore the Gram matrix of the Euler–Poincaré pairing on Db(Coh(P(2, 2)))with respect to E = 〈O,O(1),O(2),O(3)〉 is given by

SGram =

⎛

⎜⎜⎝

1 0 2 00 1 0 20 0 1 00 0 0 1

⎞

⎟⎟⎠ . (5)

Proposition. SGram and Sβ correspond to each other under the action of S4.


Proof. By the permutation

P =

⎛

⎜⎜⎝

1 0 0 00 1 0 00 0 0 10 0 1 0

⎞

⎟⎟⎠ ∈ S4,

acting on the Grammatrix SGram as P · SGram · P−1 (cf. [13, Sect. 6.c]), we find thatthe Gram matrix SGram (5) is transformed into the topologically computed Stokesmatrix Sβ (4). ��

Acknowledgements Open access funding provided by Projekt DEAL. I am grateful toMarcoHien andMaximSmirnov for supportingme throughout thework on this article. I am thankfulto Étienne Mann and Thomas Reichelt for useful discussions and hints.

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