Mathematics Area – PhD course in

Geometry and Mathematical Physics

The extended tropical vertex group

Candidate:

Veronica Fantini

Advisor:

Prof. Jacopo Stoppa

Academic Year 2019-20

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ABSTRACT. In this thesis we study the relation between scattering diagrams and defor-

mations of holomorphic pairs, building on a recent work of Chan–Conan Leung–Ma

[CCLM17a]. The new feature is the extended tropical vertex group where the scattering

diagrams are defined. In addition, the extended tropical vertex provides interesting

applications: on one hand we get a geometric interpretation of the wall-crossing for-

mulas for coupled 2d -4d systems, previously introduced by Gaiotto–Moore–Neitzke

[GMN12]. On the other hand, Gromov–Witten invariants of toric surfaces relative to

their boundary divisor appear in the commutator formulas, along with certain absolute

invariants due to Gross–Pandharipande–Siebert [GPS10], which suggests a possible

connection to open/closed theories in geometry and mathematical physics.

Ai nonni

Sesta e Olindo,

Aldo e Rina

Acknowledgements

I am grateful to my advisor Jacopo Stoppa, for his constant support, fruitful discus-

sions, suggestions and corrections.

I wish to thank Kwokwai Chan, Mark Gross and Andrew Neitzke for suggesting

corrections and further directions and applications of the results of the thesis.

I thank SISSA for providing a welcoming and professional enviroment for PhD studies

and research. I thank the PhD students for innumerous discussions, and among them

Guilherme Almeida, Nadir Fasola, Xiao Han, Vitantonio Peragine, Carlo Scarpa, Michele

Stecconi and Boris Stupovski, with whom I started the PhD at SISSA.

Last but not least, I would like to thank my family and Angelo, my flatmates Monica

Nonino e Federico Pichi, and my friends Matteo Wauters, Andrea Ricolfi, Luca Franzoi,

Raffaele Scandone, Alessio Lerose, Matteo Zancanaro, Maria Strazzullo, Alessandro

Nobile, Saddam Hijazi and Daniele Agostinelli. They have always encouraged me in my

studies and I have spent great time with them in Trieste.

v

Contents

Chapter 1. Introduction 1

1.1 Wall Crossing Formulas 2

1.2 Relative Gromov–Witten Invariants 3

1.2.1 Relation to open invariants 4

1.3 Main results 5

1.4 Plan of the thesis 8

Chapter 2. Preliminaries 9

2.1 Deformations of complex manifolds and holomorphic pairs 9

2.1.1 Infinitesimal deformations of holomorphic pairs 15

2.2 Scattering diagrams 21

2.2.1 Extension of the tropical vertex group 24

Chapter 3. Holomorphic pairs and scattering diagrams 27

3.1 Symplectic DGLA 28

3.1.1 Relation with the Lie algebra h 31

3.2 Deformations associated to a single wall diagram 33

3.2.1 Ansatz for a wall 33

3.2.2 Gauge fixing condition and homotopy operator 38

3.2.3 Asymptotic behaviour of the gauge ϕ 41

3.3 Scattering diagrams from solutions of Maurer-Cartan 45

3.3.1 From scattering diagram to solution of Maurer-Cartan 45

3.3.2 From solutions of Maurer-Cartan to the consistent diagram D∞ 53

Chapter 4. Relation with the wall-crossing formulas in coupled 2d -4d systems 61

4.0.1 Example 1 68

4.0.2 Example 2 70

Chapter 5. Gromov–Witten invariants in the extended tropical vertex 72

5.1 Tropical curve count 79

vi

5.2 Gromov–Witten invariants 88

5.2.1 Gromov–Witten invariants for (Ym,∂ Ym) 88

5.2.2 Gromov–Witten invariants for (Y m,∂ Y m) 89

5.3 Gromov–Witten invariants from commutators in V 91

5.3.1 The generating function of N0,w(Y m) 93

Bibliography 101

vii

1Introduction

In this thesis we are going to apply many techniques and ideas which have been devel-

oped by studying mirror symmetry with different approaches. This introduction aims

to present this circle of ideas, paying special attention to wall crossing formulas and

Gromov–Witten invariants.

Mirror symmetry predicts that Calabi–Yaus come in pairs, i.e. that type IIB string theory

compactified on a Calabi–Yau X gives the same physical theory of type IIA string theory

compactified on the mirror Calabi–Yau X . Calabi–Yau manifolds are complex Kähler

manifolds with trivial canonical bundle, and they admit a Ricci flat Kähler metric. In

particular, type A string refers to X as a symplectic manifold, while type B refers to X as a

complex manifold. Hence mirror symmetry can be considered as map between X and X

which exchanges the two structures. The first intrinsic formulation of mirror symmetry

was the Homological Mirror Symmetry proposed by Kontsevich [Kon95], who conjec-

tures existence of an equivalence between the derived category of coherent sheaves on

X and the Fukaya category of Lagrangian submanifolds on X . The Strominger–Yau–

Zaslow Mirror Symmetry [SYZ96] asserts that mirror symmetry is T-duality, i.e. that if

X has a mirror and it has a special Lagrangian torus fibrations, then the moduli space

of Lagrangian torus fibration with flat U (1) connections is the mirror X . Actually this

statement holds in the large radius limit: indeed the metric on the moduli space has

to be modified by adding quantum corrections (coming from open Gromov–Witten

invariants). In particular, these corrections exponentially decay in the large radius limit

at generic points, but near the singular points the corrections can not be neglected. This

limit version of mirror symmetry was then studied by Fukaya [Fuk05]. In particular,

assuming there are mirror pairs of dual torus fibrations X →M and X →M he proposed

that in the large volume limit (i.e. rescaling the symplectic structure on X by ħh−1 and

taking the limit as ħh→ 0), quantum corrections arise from studying the semi-classical

1

limit of the Fourier expansion along the torus fibers of deformations of the complex

structure Jħh , of X . In order to study deformations of the complex structure of X he

adopted the analytic approach of studying solutions of the Maurer–Cartan equation in

Kodaira–Spencer theory.

To overcome convergence issues, Kontsevich–Soibelman [KS01] and Gross–Siebert

[GS06], [GS10] come up with new approaches. On one side Kontsevich–Soibelman study

the Gromov–Hausdorff collapse of degenerating families of Calabi–Yaus obtaining a limit

structure which is an integral affine structure either coming from an analytic manifold

overC((t )) (B model) or from the base of a fibration of a Calabi–Yau by Lagrangian tori

(A model). In addition, in [KS06] the authors study the problem of reconstructing the

family from the integral affine structure of the base and they solve it for analytic K3

surfaces. In particular they define combinatorial objects, called scattering diagrams,

which prescribe how to deform the sheaf of functions on the smooth locus and how to

perform the gluing near the singularities.

On the other side Gross–Siebert apply techniques of logarithmic geometry and from

an integral affine manifold with singularities B together with a polyhedral decomposition

P , they recover a degeneration of toric Calabi–Yau varieties (i.e. with singular fibers

which are union of toric varietis). In this contest they define mirror pair of polarized

log Calabi–Yau (X ,L) and (X , L) such that their degeneration data (B ,P) and (B , P) are

mirror under a discrete Legendre transform. They also verify the expected computation

of Hodge numbers for K3 surfaces.

A common feature of Kontsevich–Soibelman and Gross–Siebert approach is the

combinatorial structure which governs the construction of the mirror manifold. Apart

from the reconstruction problem, scattering diagrams have been studied in relation with

some enumerative problems, such as wall crossing formulas for Donaldson–Thomas

invariants [KS10] and Gromov–Witten invariants for toric surfaces [GPS10], [Bou18].

We are going to explain these relations in the following sections.

1.1 Wall Crossing Formulas

The first appearance of wall-crossing-formulas (WCFs) has been in the context of

studying certain classes of two dimensional N = 2 supersymmetric fields theories, by

Cecotti and Vafa in [CV93]. In particular, their WCFs compute how the number of Bogo-

molony solitons jumps when the central charge crosses a wall of marginal stability: let

i , j , k ... be the vacua, if deforming the central charge the j-th critical value crosses the

lines which connect the i-th and the k-th vacua, then the number of solitonsµi k between

i and k becomes µi k ±µi jµ j k . From a mathematical viewpoint, these WCFs look like

2

braiding identities for the matrices representing monodromy data (see e.g. [Dub93]).

Indeed deformations of massive N = 2 SCFTs can be studied via isomonodromic defor-

mations of a linear operator with rational coefficients, and the BPS spectrum is encoded

in the Stokes matrices of the ODE associated to the linear operator.

Later on Kontsevich–Soibelman, studying numerical Donaldson–Thomas invariants

on 3 dimensional Calabi–Yau categories, come up with new WCFs ([KS08]). In particular,

their WCFs encode the jumping behaviour of some semistable objects when stability

conditions cross a codimension one subvariety (wall). It has been remarkably studied

in a series of papers by Gaiotto–Moore–Neitzke [GMN10], [GMN13b], [GMN13a] that

Kontsevich–Soibelman WCFs have the same algebraic structure as WCFs for BPS states

in four dimensional N = 2 SCFTs. These 4d WCFs can be defined starting from the

datum of the (“charge") lattice Γ endowed with an antisymmetric (“Dirac") pairing

⟨·, ·⟩D : Γ × Γ → Z, and a graded Lie algebra closely related to the Poisson algebra of

functions on the algebraic torus (C∗)rkΓ . Then WCFs are expressed in terms of formal

Poisson automorphisms of this algebraic torus.

For the purpose of this work, we are interested in WCFs of so called coupled 2d -4d

systems, namely of N = 2 super symmetric field theories in four dimension coupled with

a surface defect, introduced by Gaiotto–Moore–Neitzke in [GMN12]. This generalizes

both the formulas of Cecotti–Vafa in the pure 2d case and those of Kontsevich–Soibelman

in the pure 4d case. In the coupled 2d -4d the setting becomes rather more complicated:

the lattice Γ is upgraded to a pointed groupoidG, whose objects are indices i , j , k · · · and whose morphisms include the charge lattice Γ as well as arrows parameterized by

qi , j Γi j , where Γi j is a Γ -torsor. Then the relevant wall-crossing formulas involve two

types of formal automorphisms of the groupoid algebra C[G]: type S , corresponding

to Cecotti–Vafa monodromy matrices, and type K , which generalize the formal torus

automorphisms of Kontsevich–Soibelman. The main new feature is the non trivial

interaction of automorphisms of type S and K .

It is worth mentioning that, despite the lack of a categorical description for pure 2d

WCFs, the 2d -4d formulas have been recently studied within a categorical framework

by Kerr and Soibelman in [KS17].

1.2 Relative Gromov–Witten Invariants

From an algebraic geometric view point, Gromov–Witten theory usually requires the

definition of a compact moduli space (or a proper, separated Deligne–Mumford stack

with a virtual fundamental class) parameterizing smooth curves of genus g and class

β ∈H2(X ,Z): Kontsevich introduced the notion of stable maps to compactify the moduli

3

space of genus g , degree d curves inP2. Here we are interested in target manifolds which

are either complete toric surfaces X toric or log Calabi–Yau pairs (X , D ).1 In addition we

are going to count rational curves with tangency conditions along the boundary divisor

(namely, either of the union of all toric boundary divisors ∂ X toric or of the divisor D ).

A first approach to properly define Gromov–Witten invariants for X toric and (X , D )

comes with the notion of relative stable morphisms2 introduced by Li [Li01][Li04]. How-

ever Li’s theory requires D to be smooth, hence in the toric case it can be applied on

the open locus (X toric,o ,∂ X toric,o ) where the zero torus orbits have been removed. In

addition the moduli space of relative stable maps is not proper in general, and Li intro-

duces the notion of expanded degenerations, which require changing the target variety

blowing it up. In [GPS10], relative stable maps have been used to define genus zero

Gromov–Witten invariants for open toric surface X toric,o with tangency conditions rela-

tive to the toric boundary divisors ∂ X toric,o . In addition by Li’s degeneration formula

[Li02] the authors get an expression to compute genus zero Gromov–Witten for the

projective surface (X , D ) in terms of the invariants of (X toric,o ,∂ X toric,o ), where (X , D )

is the blowup of the toric surface along a fixed number of generic points on the toric

divisors ∂ X toric,o and D is the strict transform of ∂ X toric,o .

An alternative approach introduced by Gross–Siebert [GS13] relies on logarithmic

geometry. Compared to Li’s theory, log theory allows to consider (X , D )with D a reduced

normal crossing divisor. In addition every complete toric variety has a log structure over

the full toric boundary divisor (without removing the zero torus orbits).

1.2.1 Relation to open invariants

Parallel to the existence of open and closed string theories, on one hand open

Gromov–Witten theory concerns “counting” of holomorphic maps from a genus g curve

with boundary components to a target manifold X which admits a Lagrangian submani-

fold L , such that the image of boundary of the source lies on a Lagrangian fiber of the

target. On the other hand closed Gromov–Witten invariants aims to “count” holomorphic

maps from a genus g projective curve to a target X . In particular, open Gromov–Witten

are rare in algebraic geometry, because it is not clear in general how to construct moduli

space of maps between manifold with boundaries. However in [LS06] by using relative

1A log Calabi Yau pair (X , D ) is by definition (Definition 2.1 [HK18]) a pair of a smooth projective

variety X and a reduced normal crossing divisor D ⊂ X such that D +KX = 0.2A relative map f : (Cg , p1, ..., pn , q1, ..., qs ) → (X , D ) is such that Cg is smooth genus g curve with

p1, ..., pn , q1, ..., qs marked points and f (qi ) ∈ D . A relative map ( f , Cg , p1, ..., pn , q1, ..., qs ) is stable if it is

stable map and f −1(D ) =∑s

j=1 w j q j for some weight w j .

4

stable maps the authors obtain the analogue results of the open string amplitudes com-

puted by Ooguri–Vafa [OV00]. Furthermore in [Bou18], [Bou20] Bousseau explains how

higher genus Gromov–Witten for log Calabi–Yau surfaces (which are higher genus gener-

alizations of invariants for the blow-up surface of [GPS10]) offer a rigorous mathematical

interpretation of the open topological string amplitudes computed in [CV09].

From a different perspective, there have been many successfully results on comput-

ing open Gromov–Witten invariants in terms of holomorphic discs with boundary on

a Lagrangian fiber (see e.g. [CLLT17] and the reference therein). The main advantage

is the existence of a well defined notion of moduli space of stable discs, introduced by

[FOOO10].

1.3 Main results

Let M be an affine tropical two dimensional manifold. Let Λ be a lattice subbundle

of T M locally generated by ∂∂ x1

, ∂∂ x2

, for a choice of affine coordinates x = (x1, x2) on a

contractible open subset U ⊂M . We denote by Λ∗ =HomZ(Λ,Z) the dual lattice and by

⟨·, ·⟩: Λ∗×Λ→C

the natural pairing.

Define X ..= T M /Λ to be the total space of the torus fibration p : X →M and similarly

define X ..= T ∗M /Λ∗ as the total space of the dual torus fibration p : X →M . Then, let

y1, y2 be the coordinates on the fibres of X (U )with respect to the basis ∂∂ x1

, ∂∂ x2

, and

define a one-parameter family of complex structures on X :

Jħh =

0 ħh I

−ħh−1I 0

!

with respect to the basis¦

∂∂ x1

, ∂∂ x2

, ∂∂ y1

, ∂∂ y2

©

, parameterized by ħh ∈R>0. Notice that a set

of holomorphic coordinates with respect to Jħh is defined by

z j..= yj + iħh x j

j = 1,2; in particular we will denote by w j..= e 2πi z j . On the other hand X is endowed

with a natural symplectic structure

ωħh..= ħh−1d yj ∧d x j

where yj are coordinates on the fibres of X (U ).

Motivated by Fukaya approach to mirror symmetry in [CCLM17a] the authors show

how consistent scattering diagrams, in the sense of Kontsevich–Soibelman and Gross–

Siebert can be constructed via the asymptotic analysis of deformations of the complex

5

manifold X . Since the complex structure depends on a parameter ħh , the asymptotic

analysis is performed in the semiclassical limit ħh → 0. The link between scattering

diagrams and solutions of Maurer–Cartan equations comes from the fact that the gauge

group acting on the set of solutions of the Maurer–Cartan equation (which governs

deformations of X ) contains the tropical vertex groupVof Gross–Pandharipande–Siebert

[GPS10].

In [Fan19] and in the thesis (Section 2.2.1), we introduce the extended tropical vertex

group Vwhich is an extension of the tropical vertex groupV via the general linear group

G L (r,C). Hence the elements of V are pairs with a matrix component and a derivation

component. Moreover, V is generated by a Lie algebra h, with a twisted Lie bracket and

its definition is modelled on the deformation theory of holomorphic pairs (X , E ). In our

applications, X is defined as above and E is a holomorphically trivial vector bundle on

X . We always assume X has complex dimension 2, but we believe that this restriction

can be removed along the lines of [CCLM17a]. Our first main result gives the required

generalization of the construction of Chan, Conan Leung and Ma.

Theorem 1.3.1 (Theorem 3.3.16, Theorem 3.3.21). Let D be an initial scattering diagram,

with values in the extended tropical vertex group V, consisting of two non-parallel walls.

Then there exists an associated solutionΦ of the Maurer-Cartan equation, which governs

deformations of the holomorphic pair (X , E ), such that the asymptotic behaviour of Φ

as ħh → 0 defines uniquely a scattering diagram D∞ with values in V. The scattering

diagram D∞ is consistent.

Elements of the tropical vertex group are formal automorphisms of an algebraic torus

and are analogous to the type K automorphisms, and consistent scattering diagrams with

values the tropical vertex group reproduce wall-crossing formulas in the pure 4d case.

This is a motivation for our second main result, namely the application to 2d -4d wall-

crossing. As we mentioned WCFs for coupled 2d -4d systems involve automorphisms

of type S and K . By considering their infinitesimal generators (i.e. elements of the Lie

algebra of derivations of Aut(C[G][[t ]])), we introduce the Lie ring LΓ which they generate

as aC[Γ ]-module. On the other hand we construct a Lie ring L generated asC[Γ ]-module

by certain special elements of the extended tropical vertex Lie algebra for holomorphic

pairs, h. Then we compare these two Lie rings:

Theorem 1.3.2 (Theorem 4.0.6). Let

LΓ , [−,−]Der(C[G])

and

L, [·, ·]h

be theC[Γ ]-modules

discussed above (see Section 4). Under an assumption on the BPS spectrum, there exists

a homomorphism of C[Γ ]-modules and of Lie rings Υ : LΓ → L.

6

This result shows that a consistent scattering diagram with values in (the formal

group of) L is the same as a 2d -4d wall-crossing formula. Thus, applying our main

construction with suitable input data, we can recover a large class of WCFs for coupled

2d -4d systems from the deformation theory of holomorphic pairs (X , E ).

In [GPS10] the authors show that computing commutators in the tropical vertex

group allows one to compute genus zero Gromov–Witten invariants for weighted pro-

jective surfaces. Recently, Bousseau defines the quantum tropical vertex [Bou18] and he

shows that higher genus invariants (with insertion of Lambda classes) can be computed

from commutators in the quantum tropical vertex. In the spirit of the previous works, we

show that genus zero, relative Gromov–Witten invariants for some toric surfaces appear

in the matrix component of the automorphisms of consistent scattering diagrams in

V. Let m= ((−1, 0), (0,−1), (a , b )) ∈Λ3 and let Y m be the toric surface associated to the

complete fan generated by (−1,0), (0,−1), (a , b ), where (a , b ) is a primitive vector. In

section 5.2.2 we define relative Gromov–Witten invariants N0,w(Y m) counting curves

of class βw with tangency conditions at the boundary divisors, specified by the vector

w ∈ Λs for some positive integer s . Then we consider the blow-up surface Ym along a

finite number of points on the toric boundary divisors associated to (−1, 0) and (0,−1).

Let N0,P(Ym) be the Gromov–Witten invariants with full tangency at a point on the strict

transform of the boundary divisor. Then our main result is the following:

Theorem 1.3.3 ( Theorem 5.3.1). Let D be a standard scattering diagram in V which

consists of n initial walls. Assuming the matrix contributions of the initial scattering

diagram are all commuting, then the automorphism associated to a ray (a , b ) in the

consistent scattering diagram D∞ is explicitly determined by N0,P(Ym) and N0,w(Y m).

Furthermore we conjecture that the automorphism associated to a ray (a , b ) in

the consistent scattering diagram D∞ is a generating function of N0,w(Y m) (Conjec-

ture 5.3.4), i.e. it determines them completely. We show this in some special cases

(see Theorem 5.3.10). Thus scattering diagrams in the extended tropical vertex group

are closely related to both relative and absolute invariants. It seems that this may be

compared with a much more general expectation in the physical literature that “holo-

morphic Chern–Simons/BCOV” theories, coupling deformations of a complex structure

to an auxiliary bundle, contain the information of certain open/closed Gromov–Witten

invariants [CL15], [CL12].

7

1.4 Plan of the thesis

The first chapter is organized as follows: in Section 2.1 we provide some background

on deformations of complex manifolds and of holomorphic pairs in terms of differential

graded Lie algebras. Then in Section 2.2 we recall definitions and properties of scattering

diagrams. In Chapter 3 we introduce the tools which finally lead to the proof of Theorem

1.3.1 in Section 3.3.

Then in Chapter 4 we recall the setting of wall-crossing formulas in coupled 2d -4d

systems and prove Theorem 1.3.2. We also include two examples to show how the

correspondence works explicitly.

Finally in Chapter 5 we exploit the relation between scattering diagrams in the extended

tropical vertex group and relative Gromov–Witten invariants. In particular, in Section

5.1 we give a first interpretation of commutators in the extended tropical vertex in terms

of tropical curves counting. Then in Section 5.2 we review definitions and properties

of relative Gromov–Witten invariants for toric surfaces and for log Calabi–Yau surfaces.

Finally the proof of the Theorem 1.3.3 is given in Section 5.3.

8

2Preliminaries

2.1 Deformations of complex manifolds and holomorphic pairs

In this section we review some background materials about infinitesimal defor-

mations of complex manifolds and of holomorphic pairs with a differential geometric

approach. We try to keep the material self-consistent and we refer the reader to Chapter

6 of Huybrechts’s book [Huy05], Manetti’s lectures note [Man04] and Chan–Suen’s paper

[CS16] for more detailed and complete discussions.

Classically infinitesimal deformations of compact complex manifolds were studied

as small parametric variations of their complex structure. Let B ⊂Cm be an open subset

which contains the origin and let X be a compact complex manifold of dimension n .

DEFINITION 2.1.1. A deformation of X is a proper holomorphic submersion π: X →B such that:

• X is a complex manifold;

• π−1(0) = X ;

• π−1(t ) =: X t is a compact complex manifold.

Two deformations π: X → B and π′ : X ′→ B over the same base B are isomorphic

if and only if there exists a holomorphic morphism f : X → X ′ which commutes with

π,π′. A deformation X → B is said trivial if it is isomorphic to the product X ×B → B .

If X → B is a deformation of X , then the fibers X t are diffeomorphic to X (in general

they are not biholomorphic, see Ehresmann’s theorem, Proposition 6.2.2 [Huy05]).

A complex structure on X is an integrable almost complex structure J ∈ End(T X )

such that J 2 = id and the holomorphic tangent bundle T 1,0X ⊂ TCX ..= T X ⊗RC is an

integrable distribution [T 1,0X , T 1,0X ]⊂ T 1,0X , where [−,−] is the Lie bracket on vector

fields in TCX . Analogously a complex structure on X t is an integrable almost complex

9

structure Jt , and we denote by T 1,0t X (T 0,1

t X ) the holomorphic (antiholomorphic) tan-

gent bundle with respect to the splitting induced by Jt . If t is small enough, then the

datum of Jt is equivalent to the datum ofφ(t ): T 0,1X → T 1,0X with v +φ(t )v ∈ T 0,1t X

such that φ(0) = 01. In addition, the integrability of Jt is equivalent to the so called

Maurer–Cartan equation, namely

(2.1) ∂ φ(t ) +1

2[φ(t ),φ(t )] = 0

where ∂ is the Dolbeaut differential with respect to the complex structure J , and [−,−]is the standard Lie bracket on TCX (see Theorem 1.1 Chapter 4 [MK06]).

Let us assumeφ(t ) has a formal power expansion in t ∈ Bε ⊂Cm ,φ(t ) =∑

k≥1φk t k ,

where t k is a homogeneous polynomial of degree k in t1, ..., tm : the Maurer–Cartan

equation can be written order by order in t as follows:

∂ φ1 = 0

∂ φ2+1

2[φ1,φ1] = 0

...

∂ φ j +1

2

j−1∑

i=1

[φi ,φ j−i ] = 0

(2.2)

In particular, the first equation says that φ1 is a ∂ -closed 1-form, hence it defines a

cohomology class [φ1] ∈H 1(X , T 1,0X ).

Let X → Bε and X ′→ Bε be two isomorphic deformations of X and let f : X →X ′

be a holomorphic morphism such that f |X = idX . In particular, for t small enough, we

denote by ft the one parameter family of diffeomorphisms of X such that ft : X t → X ′twith f0 = id. If the complex structures of X t , X ′t are Jt , J ′t respectively, then J ′t ft = ft Jt .

Letφ(t ),φ(t )′ ∈Ω0,1(X , T 1,0X ) be such that Jt = ∂ +φ(t )ù∂ and J ′t = ∂ +φ(t )′ù∂ . Then

the difference between two isomorphic first order deformations is

d

d t

J ′t − Jt

t=0=

d

d t

φ(t )′−φ(t )

t=0ù∂ =

φ′1−φ1

ù∂ .

In addition,

d

d t

J ′t ft

t=0=

d

d t

ft Jt

t=0

1Indeed if Jt is know, thenφ(t ) =−p rT 1,0t X j where p rT 1,0 : T 0,1→ TCX is the projection and j : T 0,1 ,→

TCX is the inclusion. Conversely ifφ(t ) is given, then T 1,0t

..=

id+φ(t )

T 0,1X .

10

which is equivalent to

d

d t

J ′t

t=0 f0+ J ′0

d

d t

ft

t=0=

d

d t

ft

t=0 J0+ f0

d

d t(Jt )

t=0

d

d t

J ′t

t=0+ ∂

d

d t

ft

t=0=

d

d t

ft

t=0 ∂ +

d

d t(Jt )

t=0.

Since dd t

ft

t=0∈ Ω0(X , T X ), we define h ..=

d ftd t

t=0

1,0∈ Ω0(X , T 1,0X ) and from the

previous computations we get

φ′1−φ1

T 1,0X =

d

d t

J ′t − Jt

t=0

T 1,0X =

d

d t

ft

t=0 ∂ − ∂

d

d t

ft

t=0

T 1,0X = ∂ hT 1,0X .

We have proved the following proposition:

Proposition 2.1.2. LetX → Bε andX ′→ Bε be two isomorphic deformations of X . Then

their first order deformations differ from a ∂ -exact form.

Hence first order deformations are characterized as follows:

Proposition 2.1.3. There exists a natural bijection between first order deformations of a

compact complex manifold X (up to isomorphism) and H 1(X , T 1,0X ).

In particular, if H 1(X , T 1,0X ) = 0 then every deformation is trivial.

Let us now study the existence of solution of the Maurer–Cartan equation. Let g

be a hermitian metric on X , define the formal adjoint ∂ ∗ of the Dolbeaut operator ∂

with respect to the metric g and the Laplace operator∆∂..= ∂ ∂ ∗+ ∂ ∗∂ . Then we denote

the set of harmonic form as Hp (X ) ..= α ∈ Ωp (X )|∆∂ α = 0 and let us choose a basis

η1, ...,ηm ofH1(X ), m = dimCH1(X ). Let L p be the completion of Ωp (X , T 1,0X ) with

respect to the metric g and recall that the Green operator G : L q (X )→ L q (X ) is a linear

operator such that

id=H+∆∂G

where H: L q (X )→Hq (X ) is the harmonic projector.

Lemma 2.1.4 (Kuranishi’s method). Let η =∑m

j=1η j t j be a harmonic form η ∈H1(X ).

Then there exists a uniqueφ(t ) ∈Ω0,1(X , T 1,0X ) such that

φ(t ) =η−1

2∂ ∗G ([φ(t ),φ(t )])(2.3)

and for |t | small enough it is holomorphic in t .

In addition, suchφ(t ) is a solution of the Maurer–Cartan equation (2.1) if and only

if H([φ(t ),φ(t )]) = 0.

11

PROOF. Let φ(t ) be a solution of (2.3) and assume it is a solution of the Maurer–

Cartan equation. Then by property of Green’s operator G :

[φ(t ),φ(t )] =H([φ(t ),φ(t )])+∆∂G ([φ(t ),φ(t )])

which is equivalent to

−2∂ φ(t ) =H([φ(t ),φ(t )] + ∂ ∗∂G ([φ(t ),φ(t )])∂ ∂ ∗G ([φ(t ),φ(t )])

=H([φ(t ),φ(t )]+ ∂ ∗G ∂ ([φ(t ),φ(t )])∂ (η−φ(t ))

=H([φ(t ),φ(t )]−2∂ (φ(t ))

hence H([φ(t ),φ(t )]) = 0.

Conversely, assumeφ(t ) is a solution of (2.3) and H([φ(t ),φ(t )]) = 0. Then

∂ φ(t ) = ∂ (η)−1

2∂ ∂ ∗G ([φ(t ),φ(t )])

=−1

2∆∂G ([φ(t ),φ(t )])+

1

2∂ ∗∂G ([φ(t ),φ(t )])

=1

2H([φ(t ),φ(t )])−

1

2[φ(t ),φ(t )] +

1

2∂ ∗G ∂ ([φ(t ),φ(t )])

=−1

2[φ(t ),φ(t )]+ ∂ ∗G ([∂ φ(t ),φ(t )])

and by Jacobi identity

∂ φ(t ) +1

2[φ(t ),φ(t )] = ∂ ∗G ([∂ φ(t ) +

1

2[φ(t ),φ(t )],φ(t )]).

Letψ(t ) = ∂ φ(t ) 12 [φ(t ),φ(t )] and let us introduce the Holder norm || • ||k ,α with respect

to the metric g . From analytic estimates in the Holder norm (see Chapter 4, Proposition

2.2, Proposition 2.3, Proposition 2.4 [MK06]) it follows that

||ψ(t )||k ,α = ||∂ ∗G ([ψ(t ),φ(t )])||k ,α ≤C1||G ([ψ(t ),φ(t )])||k+1,α ≤C2||[ψ(t ),φ(t )]||k−1,α

≤C3||ψ(t )||k ,α||φ(t )||k ,α

and by choosing t small enough such that C3||φ(t )||k ,α < 1 (Proposition 2.4 [MK06]), we

get a contradiction unlessψ(t ) = 0.

Existence and uniqueness ofφ(t ) solution of (2.3) relies on implicit function theorem

for Banach spaces and some analytic estimates in the Holder norm (we refer to [Kur65]).

DEFINITION 2.1.5. A deformationπ: X → B of a compact complex manifold X is said

complete if any other deformation π′ : X ′→ B ′ is the pull-back under some f : B ′→ B ,

namely X ′ =X ×B B ′. Moreover if d f0 : T0′B′→ T0B is always unique, the deformation

X → B is called versal.

12

Deformations of X are called unobstructed if X admits a versal deformation X → B

and B is smooth.

Theorem 2.1.6 ([Kur65]). Any compact complex manifold admits a versal deformation.

DEFINITION 2.1.7. Let S⊂Cm be the set S ..= t ∈Cm ||t |< ε, H([φ(t ),φ(t )]) = 0. A

family X → S parametrizing deformations of the manifold X is the so called Kuranishi

family.

In particular if H 2(X , T 1,0X ) = 0 then the solutionφ(t )defined in (2.3) solves Maurer–

Cartan euqation (2.1) and deformations are unobstructed (this result was originally

proved by Kodaira, Nirenberg and Spencer, see Chapter 4, Theorem 2.1 [MK06]). How-

ever, even if H 2(X , T 1,0X ) 6= 0, deformations may be unobstructed. Indeed let X be

Calabi–Yau manifolds, namely compact Kähler manifolds with trivial canonical bundle

and let ΩX be the holomorphic volume form of X , then deformations are unobstructed:

Theorem 2.1.8 (Bogomolov–Tian–Todorov2). Let X be a Calabi–Yau and let g be the

Kähler–Einstein metric. Let η ∈H1(X ) be a harmonic form, then there exists a unique

convergent power seriesφ(t ) =∑

j≥1φ j t j ∈Ω0,1(X , T 1,0X ) such that for |t |< ε

(a ) [φ1] =η;

(b ) ∂ ∗φ(t ) = 0;

(c ) φ j ùΩX is ∂ -exact for every j > 1;

(d ) ∂ φ(t ) + 12 [φ(t ),φ(t )] = 0.

The proof is by induction on the order of the power series, the convergence follows

by analytic estimates and it realyies on the Kuranishi method (see [Tod89]). If we relax

the assumption on the metric and we let g be a generic Kähler metric, then φ(t ) will

be a formal power series satisfying (a ), (c ) and (d ) (see Huybrechts Proposition 6.1.11

[Huy05]). Indeed first order deformation φ1 ∈Ω1,0(X , T 1,0X )must be ∂ -closed hence

by Hodge theorem

[φ1] ∈H 0,1(X , T 1,0X )'H 1∂(X )'H1(X )

and we can chooseφ1 =η. Then at order two in the formal parameter t , we need to find

a solutionφ2 ∈Ω0,1(X , T 1,0X ) such that

(d ) ∂ (φ2) =− 12 [φ1,φ1] and

(c ) φ2ùΩX is ∂ -exact.

2This result was first annaunced by Bogomolov [Bog78] and then it has been proved independently by

Tian [Tia87] and Todorov [Tod89].

13

These are consequences of the Tian–Todorov lemma3: indeed it follows that ∂ ([φ1,φ1]) =

0 and [φ1,φ1]ùΩX is ∂ -exact. Hence, by Hodge decomposition, [φ1,φ1]has no non-trivial

harmonic part and it is ∂ -exact. Moreoverφ2 can be chosen such thatφ2ùΩX is ∂ -exact;

indeed by ∂ ∂ -lemma there exists γ ∈Ωn−2,0(X , T 1,0X ) such that ∂ ∂ γ=φ2ùΩX hence we

chooseφ2ùΩX = ∂ γ.

A modern approach to study deformations is via differential graded Lie algebras

(DGLA) and we are going to define the Kodaira–Spencer DGLA which govern deforma-

tions of a complex manifold X .

DEFINITION 2.1.9. A differential graded Lie algebra is the datum of a differential

graded vector space (L , d) together a with bilinear map [−,−]: L × L→ L (called bracket)

of degree 0 such that the following properties are satisfied:

− (graded skewsymmetric) [a , b ] =−(−1)deg(a )deg(b )[b , a ]

− (graded Jacobi identity) [a , [b , c ]] = [[a , b ], c ] + (−1)deg(a )deg(b )[b , [a , c ]]

− (graded Leibniz rule) d[a , b ] = [d a , b ] + (−1)deg(a )[a , d b ].

Let Art be the category of Artinian rings and for every A ∈Art let mA be the maximal

ideal of A. Then we define the functor of deformations of a DGLA:

DEFINITION 2.1.10. Let (L , d, [−,−]) be a DGLA, deformations of (L , d, [−,−]) are

defined to be a functor

DefL : Art→ Sets

from the category of Aritinian rings to the category of sets, such that

(2.4) DefL (A) ..=

φ ∈ L 1⊗mA |dφ+ [φ,φ] = 0

/gauge

where the gauge action is defined by h ∈ L 0⊗mA such that

(2.5) e h ∗φ ..=φ−∑

k≥0

1

(k +1)!adk

h (d h − [h ,φ]),

and adh (−) = [h ,−].

3The Tian–Todorov lemma says the following: Let α ∈Ω0,p (X , T 1,0X ) and β ∈Ω0,q (X , T 1,0X ), then

(−1)p [α,β ] =∆(α∧β )−∆(α)∧β − (−1)p+1α∧∆(β )

where∆: Ω0,q∧p T 1,0 x

→Ω0,q∧p−1 T 1,0X

is defined as

∆: Ω0,q

p∧

T 1,0X

ùΩX−→Ωn−p ,q

X ∂−→Ωn−p+1,q

X ùΩX−→Ω0,q

p−1∧

T 1,0X

.

In addition the operator ∆ anti-commutes with the differential ∂ , i.e. ∂ ∆ = −∆ ∂ (see Lemma 6.1.8

[Huy05]).

14

We usually restrict to A = C[[t ]] so that φ ∈ L 1 ⊗mt can be expanded as a formal

power series in the formal parameter t . Then the Kodaira–Spencer DGLA KS(X )which

governs deformations of the complex manifold X is defined as follows: let ∂X be the

Dolbeaut operator of the complex manifold X and let [−,−] be the Lie bracket such that

[αJ d z J ,βK d zK ] = d z J ∧d zK [αJ ,βK ] for every αJ d z J ∈ Ω0,p (X , T 1,0X ) and βK d zK ∈Ω0,q (X , T 1,0X )with |J |= p and |K |= q .

DEFINITION 2.1.11. The Kodaira–Spencer DGLA is

(2.6) KS(X ) ..=

Ω0,•(X , T 1,0X ), ∂X , [−,−]

.

The gauge group action (2.5) corresponds to the infinitesimal action of the diffeo-

morphisms group of X : indeed let h =

d ftd t |t=0

1,0∈ Ω0(X , T 1,0X ) be a one parameter

family of diffeomorphism ft , then

e h (∂X +φ(t )) e −h =∑

j≥0

1

j !h k

∂X +φ(t )

∑

l≥0

1

l !(−h )l

=φ(t ) +∑

n≥1

1

n !

n∑

j=0

n

j

h n− j

∂X +φ(t )

(−h ) j

=φ(t ) +∑

n≥1

1

n !adn

h

∂X +φ(t )

=φ(t ) +∑

n≥0

1

(n +1)!adn

h

h , ∂X +φ(t )

=φ(t ) +∑

n≥0

1

(n +1)!adn

h

−∂X h +

h ,φ(t )

= e h ∗φ(t ).

Hence

DefKS(X )(C[[t ]]) =

(

φ(t ) =∑

j≥1

φ j t j ∈Ω0,1(X , T 1,0X )[[t ]]

∂Xφ(t ) +

φ(t ),φ(t )

= 0

)

/gauge.

2.1.1 Infinitesimal deformations of holomorphic pairs

Let E be a rank r holomorphic vector bundle on a compact complex manifold X

with fixed hermitian metric hE . Then let ∂E be the complex structure on E .

DEFINITION 2.1.12. A holomorphic pair (X , E ) is the datum of a complex manifold

(X , ∂X ) and of a holomorphic vector bundle (E , ∂E ) on X .

Then let ∇E be the Chern connection on E with respect to (hE , ∂E ) and let FE be

the Chern curvature. The class [FE ] ∈H 1,1(X , End E ) is called the Atiyah class of E and

it does not depend on the metric hE . Moreover it allows to define an extension A(E )

15

of End E by T 1,0X : indeed A(E ) ..= End E ⊕ T 1,0X as a complex vector bundle on X

and it has an induced a holomorphic structure defined by ∂A(E ) =

∂E B

0 ∂X

!

, where

B : Ω0,q (X , T 1,0X )→Ω0,q (X , End E ) acts on ϕ ∈Ω0,q (X , T 1,0X ) as Bϕ ..=ϕùFE .

DEFINITION 2.1.13 (Definition 3.4 [CS16]). Let (X , E ) be a holomorphic pair. A

deformation of (X , E ) consists of a holomorphic proper submersion π: X → Bε such

that

- π−1(0) = X

- π−1(t ) = X t is a compact complex manifold

and of a holomorphic vector bundle E→X such that

- E |π−1(0) = E

- for every t ∈ Bε the pair (X t , Et ) is the holomorphic pair parametrized by t .

In particular, deformations of a holomorphic pair (X , E ) are deformations of both

the complex structure on X and on E .

DEFINITION 2.1.14. Two deformations E→X and E ′→X ′ of (X , E ) on the same base

B are isomorphic if there exists a diffeomorphism f : X →X ′ and a bundle isomorphism

Φ: E→ E ′ such that f |X = id, Φ|E = id and the following diagram

E E ′

X X ′

←→Φ

←→ ←→

←→f

commutes.

We have already discussed how to characterize deformations of the complex struc-

ture of X . Indeed for t small enough we can assume the X t = t × X and the complex

structure on X t is parametrized byφ(t ) ∈Ω0,1(X , T 1,0X ). In addition we can trivialize Eas Bε×E so that the holomorphic structure on Et = t ×E is induced from E |X t

. Hence,

following [CS16]we define a differential operator on End E which defines a deformation

of ∂E in terms of deformations of ∂X .

DEFINITION 2.1.15. Let (X , E )be a holomorphic pair and let (A(t ),φ(t )) ∈Ω0,1(X , A(E )),

then define

D t..= ∂E +At +φt ù∇E

Lemma 2.1.16. D t : Ω0,q (X , E )→Ω0,q+1(X , E ) is a well-defined operator and it satisfies

the Leibniz rule.

16

PROOF. Let s be a section of E and let f be a smooth function on X , then

D t ( f s ) = ∂E ( f s ) +At f s +φ(t )ù∇E ( f s )

= (∂X f )s + f ∂E s + f At s + (φ(t )ùd f )s + f φ(t )∇E s

= (∂X f +φ(t )ù∂ f )s + f D t s

= (∂t f )s + f D t s .

Theorem 2.1.17 (Theorem 3.12 [CS16]). Let (A(t ),φ(t )) ∈Ω0,1(X , A(E )) and assumeφ(t )

is a deformation of X . If D2t = 0 then it induces a holomorphic structure on E over X t

which is denoted by Et → X t .

DEFINITION 2.1.18. Deformations of a holomorphic pair (X , E ) are defined by pairs

(A(t ),φ(t )) ∈Ω0,1(X , A(E )) such thatφ(t ) is a deformation of X and D2t = 0.

Lemma 2.1.19. Let (A(t ),φ(t )) ∈Ω0,1(X , A(E )) and assume

∂Xφ(t ) +1

2[φ(t ),φ(t )] = 0

then integrability of D t is equivalent to the following system of equations:

(2.7)

∂E A(t ) +φ(t )ùFE +12 [A(t ), A(t )]+φ(t )ù∇E A(t ) = 0

∂Xφ(t ) +12 [φ(t ),φ(t )] = 0

PROOF. Let ek be a holomrphic frame for (E , ∂E ) and let s = s k ek be a smooth

section of E , then

D2t s = (∂E +A(t ) +φ(t )ù∇E )(∂E s +A(t )s +φ(t )ù∇E s )

= ∂ 2E s + ∂E (A(t )s ) + ∂E (φ(t )ù∇E s ) +A(t )∧ ∂E (s ) +A(t )∧A(t )s +A(t )∧φ(t )ù∇E s+

+φ(t )ù∇E (∂E s ) +φ(t )ù∇E (A(t )s ) +φ(t )ù∇E (φ(t )ù∇E s )

= ∂E (A(t ))s −A(t )∧ ∂E s + (∂ φ(t ))ù∇E s +φ(t )ù∂E∇E s +A(t )∧ ∂E (s ) +A(t )∧A(t )s+

+A(t )∧φ(t )ù∇E s +φ(t )ù∇E (∂E s ) +φ(t )ù∇E (A(t )s ) +φ(t )ù∇E (φ(t )ù∇E s )

= ∂E (A(t ))s + (∂ φ(t ))ù∇E s +φ(t )ù(∇E ∂E s + ∂E∇E s ) +A(t )∧A(t )s +A(t )∧φ(t )ù∇E s+

+φ(t )ù∇E (A(t ))s −φ(t )ùA(t )∧∇E (s ) +φ(t )ù∇E (φ(t )ù∇E s )

= ∂E (A(t ))s + (∂ φ(t ))ù∇E s +φ(t )ùF E s +A(t )∧A(t )s +φ(t )ù∇E (A(t ))s +φ(t )ù∇E (φ(t )ù∇E s )

=

∂E (A(t ))+φ(t )ùF E +A(t )∧A(t ) +φ(t )ù∇E (A(t ))

s +φ(t )ù∇E (φ(t )ù∇E s ) + (∂ φ(t ))ù∇E s .

Now we claimφ(t )ù∇E (φ(t )ù∇E s ) = [φ(t ),φ(t )]ù∇E s and sinceφ(t ) is a solution of the

Maurer–Cartan equation, we get the expected equivalence. Let us prove the claim in

17

local coordinates: letφ(t ) =φi j d zi ∂ j and let∇E = d +Θl d zl (where we are summing

over repeated indexes), then

φ(t )ù∇E (φ(t )ù∇E s ) =φ(t )ù∇E

φi j d zi

∂ s a

∂ z j+Θ j

a b s b

ea

=φml

∂ φi j

∂ zl

∂ s a

∂ z j+φi j

∂ 2s a

∂ z j ∂ zl+∂ φi j

∂ zlΘ

ja b s b

d zm ∧d zi ea+

+φml

φi j∂ Θ

ja b

∂ zls b +φi jΘ

ja b

∂ s b

∂ zl

d zm ∧d zi ea

+φmlΘlhk

φi j∂ s k

∂ z j+φi jΘ

jk b s b

d zm ∧d zi eh

=φml

∂ φi j

∂ zl

∂ s a

∂ z j+Θ j

a b s b

d zm ∧d zi ea+

+φmlφi j

∂ Θja b

∂ zl+Θl

a kΘjk b

s b d zm ∧d zi ea

where in the last step we notice thatφmlφi j∂ 2s a

∂ z j ∂ zld zm ∧d zi ea = 0. Finally we have

φml

∂ φi j

∂ zl

∂ s a

∂ z j+Θ j

a b s b

d zm ∧d zi ea = 2[φml d z m∂l ,φi j d z i ∂ j ]ù

∂ s a

∂ zpea +Θ

pa b s b ea d zp

= 2

φ(t ),φ(t )

ù∇E s

while, since FE is of type (1, 1),∂ Θ

ja b

∂ zl+Θl

a kΘjk b = 0 and the claim is proved.

Recall the definition of ∂A(E ), then we define the following operator:

[−,−]: Ω0,q (X , A(E ))×Ω0,r (X , A(E ))→Ω0,q+r (X , A(E ))

(A(t ),φ(t )), (A′(t ),φ′(t ))

..=

[A(t ), A′(t )]+φ(t )ù∇E A′(t ) +φ′(t )ù∇E A(t ), [φ(t ),φ′(t )]

.

It is possible to prove that [−,−] is indeed a Lie bracket. Therefore deformations of a

holomorphic pair (X , E ) are governed by (A(t ),φ(t )) ∈Ω0,1(X , A(E ))which is a solution

of the Maurer–Cartan equation

(2.8) ∂A(E )(A(t ),φ(t ))+1

2

(A(t ),φ(t )), (A(t ),φ(t ))

= 0.

These lead to the following definition:

DEFINITION 2.1.20. The Kodaira-Spencer DGLA which governs deformations of a

holomorphic pair (X , E ) is defined as follows

(2.9) KS(X , E ) ..= (Ω0,•(X , A(E )), ∂A(E ), [−,−]).

18

The proof that KS(X , E ) is indeed a DGLA follows from the definitions of ∂A(E ) and

of the Lie bracket [−,−]. Moreover, it can be proved that KS(X , E ) does not depend on

the choice of the hermitian metric hE (see Appendix A [CS16]). Thus deformations of

KS(X , E ) are defined by

(2.10) DefKS(X ,E )(C[[t ]]) =¦

(A(t ),φ(t )) ∈Ω0,1(X , A(E ))[[t ]]

∂A(E )(A(t ),φ(t ))+

A(t ),φ(t )

,

A(t ),φ(t )

= 0©

/gauge

where the gauge group acting on the set of solutions of (2.8) is defined by (Θ, h ) ∈Ω0(X , End E ⊕T 1,0X )[[t ]] such that it acts on (A,φ) ∈Ω0,1(X , End E ⊕T 1,0X )[[t ]] as follows

(2.11) e (Θ,h ) ∗ (A,φ) ..=

A−∞∑

k=0

adkΘ

(k +1)!(∂ Θ− [Θ, A]),φ−

∞∑

k=0

adkh

(k +1)!(∂ h − [h ,φ])

.

Let us now assume (A(t ),φ(t )) =∑

j≥1(A j ,φ j )t j , then from the Maurer–Cartan

equation (2.8) it is immediate that first order deformations (A1,φ1) ∈ Ω0,1(X , T 1,0X )

are ∂A(E )-closed 1-forms hence they define a cohomology class [(A1,φ1)] ∈H 1(X , A(E )).

Furthermore we claim that two isomorphic deformations differ by a ∂A(E )-exact form.

Indeed let E → X and E ′ → X ′ be two isomorphic families of deformations of (X , E )

which are represented by (A(t ),φ(t )), (A(t )′,φ(t )′) ∈Ω0,1(X , A(E )) respectively. Then, for

t small enough, we denote by ft : X t → X ′t a one parameter family of diffeomorphisms of

X such that f0 = idX and byΦt : Et → E ′t a one parameter family of bundle isomorphisms

such that Φ0 = idE and they cover ft , i.e. Φt : Ex → E ft (x ) for every x ∈ X where Ex is the

fiber over x . In addition if D′t and D t denote the deformed complex structure of Et and

E ′t respectively, which are defined in terms of (A(t ),φ(t )) and (A′(t ),φ′(t )), then

(2.12) D′tΦt =Φt D t .

Let x ∈ X and denote by Pγx (t ) : Ex → E ft (x ) the parallel transport along γx (t ) = ft (x )

associated with the connection∇E . Then we define Θt..=P−1

γx (t )Φt : Ex → Ex . Since Φt is

a holomorphic bundle isomorphism, it follows that Θt is a well defined holomorphic

bundle isomorphism too and Θ0 = idE . We claim that

(2.13) (A′1,φ′1)− (A1,φ1) =

∂EΘ1+hùFE , ∂X h

= ∂A(E )(Θ1, h ).

Recall that two first order deformations of X differ by a ∂ -exact form, i.e. there is

h ∈Ω0(X , T 1,0X ) such that ∂X h =φ′1−φ1. Let us now compute A′1−A1: from definition

of D′t and D t we have

A′1−A1 =d

d t

D′t −D t

t=0− ∂X hù∇E .

19

In addition, taking derivatives from (2.12) we get

d

d t

D′t

t=0Φ0+ ∂E

d

d t(Φt )

t=0=

d

d t(Φt )

t=0∂E +Φ0

d

d t

D t

t=0

hence dd t

D′t −D t

t=0= d

d t (Φt )

t=0∂E − ∂E

dd t (Φt )

t=0. Furthermore, if st is a holomor-

phic section of Et , then Φt s ′t =Pγ(t )Θt st and

d

d t(Φt st )

t=0=

d

d t

Pγ(t )Θt st

t=0

d

d t(Φt )

t=0s0+

d

d t(st )

t=0=−hù∇E s0+Θ1s0+

d

d t(st )

t=0

thus dd t (Φt )

t=0s0 =−hù∇E s0+Θ1s0. Collecting these results together we end up with

(A′1−A1)s0 =−∂Ed

d t(Φt )

t=0s0− ∂X hù∇E s0

=−∂E

−hù∇E s0+Θ1s0

− ∂X hù∇E s0

= ∂X hù∇E s0+hù∂E∇E s0+ ∂EΘ1s0− ∂X hù∇E s0

= hùFE s0+ ∂EΘ1s0

and we have proved the claim (2.13).

Lemma 2.1.21. There is a bijective correspondence between the first order deformations

up to gauge equivalence and the first cohomology group H 1(X , A(E )).

The obstruction to extend the solution of Maurer–Cartan to higher order in the

formal parameter t is encoded in the second cohomology H 2(X , A(E )). In particular, let

us fix an hermintian metric gA(E ) on A(E ), then the Kuranishi method applies:

Proposition 2.1.22. Let η=∑m

j=1η j t j ∈H1(A(E )) be a harmonic form, where η1, ...ηmis a basis forH1(X ). There exists a unique solution (A(t ),φ(t )) ∈Ω0,1(X , A(E )) of

(A(t ),φ(t )) =η+ ∂ ∗A(E )G ([(A(t ),φ(t )), (A(t ),φ(t )))]

In addition it solves the Maurer–Cartan equation (2.8) if and only if

H([(A(t ),φ(t )), (A(t ),φ(t ))]) = 0

where H is the harmonic projection on H(A(E )) and G is the Green operator of the

Laplacian∆∂A(E )= ∂A(E )∂

∗gA(E )

A(E ) + ∂∗gA(E )

A(E ) ∂A(E ).

As a consequence of Bogomolov–Tian–Todorov theorem the following result has

been proved in [CS16]:

20

Theorem 2.1.23 (Proposition 7.7 [CS16]). Let X be a compact Calabi–Yau surface and

let E be a holomorphic vector bundle on X such that c1(E ) 6= 0 and the trace free second

cohomology vanishes H 2(X , End0 E ) = 0. Then infinitesimal deformations of the pair

(X , E ) are unobstructed.

2.2 Scattering diagrams

Scattering diagrams have been introduced by Kontsevich–Soibelman in [KS06] and

they usually encode a combinatorial structure. Naively we may define scattering dia-

grams as a bunch of co-dimension one subspace inRn decorated with some automor-

phisms. In our application we will restrict to R2 and the automorphisms group will be a

generalization of the group of formal automorphisms of an algebraic torus C∗×C∗. Let

us first introduce some notation:

NOTATION 2.2.1. Let Λ'Z2 be a rank two lattice and choose e1 and e2 being a basis for Λ.

Then the group ring C[Λ] is the ring of Laurent polynomial in the variable z m , with the

convention that z e1 =: x and z e2 =: y .

We define the Lie algebra g as follows:

(2.14) g ..=mt

C[Λ]⊗CC[[t ]]

⊗ZΛ∗

where every n ∈Λ∗ is associated to a derivation ∂n such that ∂n (z m ) = ⟨m , n⟩z m and the

natural Lie bracket on g is

(2.15) [z m∂n , z m ′∂n ′ ] ..= z m+m ′

∂⟨m ′,n⟩n ′−⟨m ,n ′⟩n .

In particular g has a Lie sub-algebra h⊂ g defined by:

(2.16) h ..=⊕

m∈Λr0z m ·

mt ⊗m⊥,

where m⊥ ∈ Λ∗ is identified with the derivation ∂n and n the unique primitive vector

such that ⟨m , n⟩= 0 and it is positive oriented according with the orientation induced

by ΛR ..=Λ⊗ZR.

DEFINITION 2.2.2. The tropical vertex groupV is the sub-group of AutC[[t ]]

C[Λ]⊗CC[[t ]]

,

such that V ..= exp(h). The product on V is defined by the Baker-Campbell-Hausdorff

(BCH) formula, namely

(2.17) g g ′ = exp(h ) exp(h ′) ..= exp(h •h ′) = exp(h +h ′+1

2[h , h ′] + · · · )

where g = exp(h ), g ′ = exp(h ′) ∈V.

21

The tropical vertex group has been introduced by Kontsevich–Soibelman in [KS06]

and in the simplest case its elements are formal one parameter families of symplecto-

morphisms of the algebraic torus C∗ ×C∗ = Spec C[x , x−1, y , y −1] with respect to the

holomorphic symplectic formω= d xx ∧

d yy . Indeed let f(a ,b ) = 1+ t x a y b · g (x a y b , t ) ∈

C[Λ]⊗C C[[t ]] for some (a , b ) ∈ Z2 and for a polynomial g (x a y b , t ), then θ(a ,b ), f(a ,b )is

defined as follows:

(2.18) θ(a ,b ), f(a ,b )(x ) = f −b x θ(a ,b ), f(a ,b )

(y ) = f a y .

In particular θ ∗(a ,b ), f(a ,b )ω=ω.

We can now state the definition of scattering diagrams according to [GPS10]:

DEFINITION 2.2.3 (Scattering diagram). A scattering diagram D is a collection of

walls wi = (mi ,di ,θi ), where

• mi ∈Λ,

• di can be either a line through ξ0, i.e. di = ξ0 −miR or a ray (half line) di =

ξ0−miR≥0,

• θi ∈V is such that log(θi ) =∑

j ,k a j k t j z k mi ∂ni.

Moreover for any k > 0 there are finitely many θi such that θi 6≡ 1 mod t k .

As an example, the scattering diagram

D= w1 =

m1 = (1, 0),d1 =m1R,θ1

,w2 =

m2 = (0, 1),d2 =m2R,θ2

can be represented as if figure 2.1.

00

θ1

θ2

FIGURE 2.1. A scattering diagram with only two walls D= w1,w2

Denote by Sing(D) the singular set of D:

Sing(D) ..=⋃

w∈D∂ dw ∪

⋃

w1,w2

dw1∩dw2

22

where ∂ dw = ξ0 if dw is a ray and zero otherwise. There is a notion of ordered product

for the automorphisms associated to each lines of a given scattering diagram, and it is

defined as follows:

DEFINITION 2.2.4 (Path ordered product). Let γ : [0, 1]→Λ⊗ZR\Sing(D) be a smooth

immersion with starting point that does not lie on a ray of the scattering diagram D and

such that it intersects transversally the rays of D (as in figure 2.2). For each power k > 0,

there are times 0<τ1 ≤ · · · ≤τs < 1 and rays di ∈D such that γ(τ j )∩d j 6= 0. Then, define

Θkγ,D

..=∏s

j=1θ j . The path ordered product is given by:

(2.19) Θγ,D..= lim

k→∞Θkγ,D

0θ1

θ2

θ−11

θ−12

θm

γ

FIGURE 2.2. Θγ,D∞ = θ1 θm θ2 θ−11 θ

−12

DEFINITION 2.2.5 (Consistent scattering diagram). A scattering diagram D is consis-

tent if for any closed path γ intersecting D generically, Θγ,D = idV.

The following theorem by Kontsevich and Soibelman is an existence (and unique-

ness) result of consistent scattering diagrams:

Theorem 2.2.6 ([KS06]). LetDbe a scattering diagram with two non parallel walls. There

exists a unique minimal scattering diagram4 D∞ ⊇D such that D∞ \D consists only of

rays, and it is consistent.

PROOF. Let D0 =D and let Dk be the order k scattering diagram obtained from Dk−1

adding only rays emanating from p ∈ Sing(Dk−1) such that for any generic loop around

p , Θγ,Dk= id mod t k+1. The proof goes by induction on k , and we need to prove that

there exists a scattering diagram Dk . Assume Dk−1 satisfies the inductive assumption,

then Dk is constructed as follows: let p ∈ Sing(Dk−1) and compute Θγp ,Dk−1for a generic

4The diagram is minimal meaning that we do not consider rays with trivial automorphisms and no

rays with the same support.

23

loop γp around p . By inductive assumption Θγp ,Dk−1= exp(

∑si=1 ai (p )z mi ∂ni

)mod t k+1,

for some ai ∈C[[t ]]/(t )k , mi ∈Λ and ⟨mi , ni ⟩= 0. Hence

Dk =Dk−1 ∪⋃

p∈Sing(Dk−1)

di = p +miR≥0,θi = exp(−ai (p )zmi ∂ni

)

|i = 1, ..., s

.

Notice that we have to chose the opposite sign for the automorphisms of Dk in order to

cancel the contribution from Θγp ,Dk−1.

There are examples in which the final configuration of rays (for which the diagram is

consistent) it is known explicitly. For instance let D as in Figure 2.1 with automorphisms

θ1,θ2 ∈V

θ1 : x → x θ2 : x → x/(1+ t y )

y → y (1+ t x ) y → y

Then the consistent scattering diagram D∞ consists of one more ray d = (1,1) with

automorphism θ such that

θm : x → x/(1+ t 2 x y )

y → y (1+ t 2 x y )

as it is represented in Figure 2.2.

2.2.1 Extension of the tropical vertex group

In [CCLM17a] the authors prove that some elements of the gauge group acting on

Ω0,1(X , T 1,0X ) can be represented as elements of the tropical vertex groupV. Here we

are going to define an extension of the Lie algebra h, which will be related with the

infinitesimal generators of the gauge group acting on Ω0,1(X , End E ⊕T 1,0X ). Let gl(r,C)be the Lie algebra of the Lie group GL(r,C), then we define

(2.20) h ..=⊕

m∈Λr0Cz m ·

mt gl(r,C)⊕

mt ⊗m⊥ .

Lemma 2.2.7.

h, [·, ·]∼

is a Lie algebra, where the bracket [·, ·]∼ is defined by:

(2.21)

[(A,∂n )zm , (A′,∂n ′ )z

m ′]∼ ..= ([A, A′]glz

m+m ′+A′⟨m ′, n⟩z m+m ′

−A⟨m , n ′⟩z m+m ′, [z m∂n , z m ′

∂n ′ ]h).

The definition of the Lie bracket [·, ·]∼ is closely related with the Lie bracket of

KS(X , E ) and we will explain it below, in (3.1.1).

DEFINITION 2.2.8. The extended tropical vertex group V is the sub-group of GL(r,C)×AutC[[t ]]

C[Λ]⊗CC[[t ]]

, such that V ..= exp(h). The product on V is defined by the BCH

formula.

24

PROOF. First of all the the bracket is antisymmetric:

[(A,∂n )zm , (A′,∂n ′ )z

m ′]∼ = ([A, A′]glz

m+m ′+A′⟨m ′, n⟩z m+m ′

−A⟨m , n ′⟩z m+m ′, [z m∂n , z m ′

∂n ′ ]h)

= (−[A′, A]glzm+m ′

+A′⟨m ′, n⟩z m+m ′−A⟨m , n ′⟩z m+m ′

,−[z m ′∂n ′ , z m∂n ]h)

=−([A′, A]glzm+m ′

+A⟨m , n ′⟩z m+m ′−A′⟨m ′, n⟩z m+m ′

, [z m ′∂n ′ , z m∂n ]h).

Moreover the Jacobi identity is satisfied:

[(A1,∂n1)z m1 , (A2,∂n2

)z m2 ]∼, (A3,∂n3)z m3

∼ =

[A1, A2]gl+A2⟨m2, n1⟩−A1⟨m1, n2⟩,

∂⟨m2,n1⟩n2−⟨m1,n2⟩n1

z m1+m2 , (A3,∂n3)z m3

∼

=

([A1, A2]gl+A2⟨m2, n1⟩−A1⟨m1, n2⟩), A3

gl+A3⟨m2, n1⟩⟨m3, n2⟩−A3⟨m1, n2⟩⟨m3, n1⟩+

−

[A1, A2] +A2⟨m2, n1⟩−A1⟨m1, n2⟩

⟨m1+m2, n3⟩, (m1+m2+m3)⊥)

z m1+m2+m3

=

[[A1, A2]gl, A3]gl+ [A2, A3]gl⟨m2, n1⟩− ⟨m1, n2⟩[A1, A3]gl− [A1, A2]gl⟨m1+m2, n3⟩+

+A3⟨m2, n1⟩⟨m3, n2⟩−A3⟨m1, n2⟩⟨m3, n1⟩−A2⟨m2, n1⟩⟨m1+m2, n3⟩+A1⟨m1, n2⟩⟨m1+m2, n3⟩,

(m1+m2+m3)⊥

z m1+m2+m3

Then by cyclic permutation we compute also the other terms:

(A2,∂n2)z m2 , (A3,∂n3

)z m3

∼ , (A1,∂n1)z m1

∼ =

[A2, A3]gl, A1]

+ [A3, A1]gl⟨m3, n2⟩+

−⟨m2, n3⟩[A2, A1]gl− [A2, A3]gl⟨m3+m2, n1⟩+A1⟨m3, n2⟩⟨m1, n3⟩+

−A1⟨m2, n3⟩⟨m1, n2⟩−A3⟨m3, n2⟩⟨m2+m3, n1⟩+A2⟨m2, n3⟩⟨m2+m3, n1⟩,

(m1+m2+m3)⊥

z m1+m2+m3

(A1,∂n3)z m3 , (A1,∂n1

)z m1

∼ , (A2,∂n2)z m2

∼ =

[A3, A1]gl, A2]

+ [A1, A2]⟨m1, n3⟩

− ⟨m3, n1⟩[A3, A2]gl− [A3, A1]gl⟨m1+m3, n2⟩+A2⟨m1, n3⟩⟨m2, n1⟩+

−A2⟨m3, n1⟩⟨m2, n3⟩−A1⟨m1, n3⟩⟨m3+m1, n2⟩+A3⟨m3, n1⟩⟨m3+m1, n2⟩,

(m1+m2+m3)⊥

z m1+m2+m3

Since Jacobi identity holds for [·, ·]gl and [·, ·]h, we are left to check that the remaining terms

sum to zero. Indeed the coefficient of [A2, A3]gl is ⟨n1, m2⟩−⟨n1, m2+m3⟩−⟨n1, m3⟩, and

it is zero. By permuting the indexes, the same hold true for the coefficients in front of the

other bracket [A1, A3]gl and [A2, A1]gl. In addition the coefficient of A3 is ⟨m2, n1⟩⟨m3, n2⟩−⟨m1, n2⟩⟨m3, n1⟩−⟨m3, n2⟩⟨m2, n1⟩−⟨m3, n2⟩⟨m3, n1⟩+⟨m3, n1⟩⟨m1, n2⟩+⟨m3, n1⟩⟨m3, n2⟩

25

and it is zero. By permuting the indexes the same holds true for the coefficient in front

of A1 and A2.

NOTATION 2.2.9. Let di = ξi +miR≥0 and let−→f i be the function

−→f i

..=

1+Ai ti z mi , fi

fi = 1+ ci ti z mi

such that log−→f i

..=

log(1+Ai ti z mi ), log fi ∂ni

∈ h, where ni is the unique primitive

vector in Λ∗ orthogonal to mi and positively oriented.

We define scattering diagrams in the extended tropical vertex group V by replacing

the last assumption of Definition 2.2.3 with−→f i such that

log−→f i =

log(1+Ai ti z mi ), log fi ∂ni

∈ h.

In Chapter 5 we will introduce other definitions about scattering diagrams which gen-

eralizes that of [GPS10] to scattering diagrams defined in the extended tropical vertex

group.

26

3Holomorphic pairs and scattering diagrams

This section is devoted to study the relation between scattering diagrams in the extended

tropical vertex group and the asymptotic behaviour of the solutions of the Maurer–Cartan

equation which governs deformations of holomorphic pairs.

We briefly highlight the main steps of the construction, which follows closely that of

[CCLM17a], adapting it to pairs (X , E ).

Step 1 We first introduce a symplectic DGLA as the Fourier-type transform of the

Kodaira-Spencer DGLAKS(X , E )which governs deformation of the pair (X , E ). Although

the two DGLAs are isomorphic, we find that working on the symplectic side makes the re-

sults more transparent. In particular we define the Lie algebra h as a subalgebra, modulo

terms which vanish as ħh→ 0, of the Lie algebra of infinitesimal gauge transformations

on the symplectic side.

Step 2.a Starting from the data of a wall in a scattering diagram, namely from the

automorphism θ attached to a line d, we construct a solutionΠ supported along the wall,

i.e. such that there exists a unique normalised infinitesimal gauge transformation ϕ

which takes the trivial solution to Π and has asymptotic behaviour with leading ordered

term given by log(θ ) (see Proposition 3.2.17). The gauge-fixing condition ϕ is given by

choosing a suitable homotopy operator H .

Step 2.b Let D= w1,w2 be an initial scattering diagram with two non-parallel walls.

By Step 2.a., there are Maurer-Cartan solutionsΠ1,Π2, which are respectively supported

along the walls w1, w2. Using Kuranishi’s method we construct a solution Φ taking as

input Π1+Π2, of the form Φ=Π1+Π2+Ξ, where Ξ is a correction term. In particular Ξ

is computed using a different homotopy operator H.

Step 2.c By using labeled ribbon trees we write Φ as a sum of contributions Φa over

a ∈

Z2≥0

prim, each of which turns out to be independently a Maurer-Cartan equation

(Lemma 3.3.15). Moreover we show that each Φa is supported on a ray of rational slope,

27

meaning that for every a , there is a unique normalised infinitesimal gauge transforma-

tionϕa whose asymptotic behaviour is an element of our Lie algebra h (Theorem 3.3.16).

The transformationsϕa allow us to define the scattering diagram D∞ (Definition 3.3.20)

from the solution Φ.

Step 3 The consistency of the scattering diagram D∞ is proved by a monodromy

argument.

Note that in fact the results of [CCLM17a] have already been extended to a large

class of DGLAs (see [CLMY19]). For our purposes however we need a more ad hoc study

of a specific differential-geometric realization of KS(X , E ): for example, there is a back-

ground Hermitian metric on E which needs to be chosen carefully.

3.1 Symplectic DGLA

In order to construct scattering diagrams from deformations of holomorphic pairs,

it is more convenient to work with a suitable Fourier transformF of the DGLA KS(X , E ).

Following [CCLM17b]we start with the definition ofF (KS(X , E )) (see also Section 3.2.1

of [Ma19]). Let L be the space of fibre-wise homotopy classes of loops with respect to

the fibration p : X →M and the zero section s : M → X

L=⊔

x∈M

π1(p−1(x ), s (x )).

Define a map ev: L→ X , which maps a homotopy class [γ] ∈L to γ(0) ∈ X and define

pr: L→M the projection, such that the following diagram commutes:

L X

M

← →ev

←

→pr←→

p

In particular pr is a local diffeomorphism and on a contractible open subset U ⊂M

it induces an isomorphism Ω•(U , T M ) ∼= Ω•(Um, T L), where Um..= m ×U ∈ pr−1(U ),

m ∈ Λ. In addition, there is a one-to-one correspondence between Ω0(U , T 1,0X ) and

Ω0(U , T M ),∂

∂ z j←→ħh

4π

∂

∂ x j

which leads us to the following definition:

DEFINITION 3.1.1. The Fourier transform is a mapF : Ω0,k (X , T 1,0X )→Ωk (L, T L),such that

(3.1)

F (ϕ)

m(x )..=

4π

ħh

|I |−1∫

p−1(x )ϕI

j (x , y )e −2πi (m, y )d y d xI ⊗∂

∂ x j,

28

where m ∈Λ represents an affine loop in the fibre p−1(x )with tangent vector∑2

j=1 m j∂∂ yj

and ϕ is locally given by ϕ =ϕIj (x , y )d zI ⊗ ∂

∂ z j, |I |= k .

The inverse Fourier transform is then defined by the following formula, providing

the coefficients have enough regularity:

(3.2) F−1

α

(x , y ) =

4π

ħh

−|I |+1 ∑

m∈ΛαI

j ,me 2πi (m, y )d zI ⊗∂

∂ z j

where αIj ,m(x )d xI ⊗ ∂

∂ x j ∈ Ωk (Um, T L) is the m-th Fourier coefficient of α ∈ Ωk (L, T L)and |I |= k .

The Fourier transform can be extended to KS(X , E ) as a map

F : Ω0,k (X , End E ⊕T 1,0X )→Ωk (L, End E ⊕T L)

F

AI d zI ,ϕIj d zI ⊗

∂

∂ z j

m

..=

4π

ħh

|I |

∫

p−1(x )AI (x , y )e −2πi (m, y )d y d xI ,

4π

ħh

−1∫

p−1(x )ϕI

j (x , y )e −2πi (m, y )d y d xI ⊗∂

∂ x j

(3.3)

where the first integral is meant on each matrix element of AI .

In order to define a DGLA isomorphic to KS(X , E ), we introduce the so called Witten

differential dW and the Lie bracket ·, ·∼, acting onΩ•(L, End E ⊕T L). It is enough for us

to consider the case in which E is holomorphically trivial E =OX ⊕OX ⊕· · ·⊕OX and the

hermitian metric hE is diagonal, hE = diag(e −φ1 , · · · , e −φr ) andφ j ∈Ω0(X ) for j = 1, ..., r .

The differential dW is defined as follows:

dW : Ωk (L, End E ⊕T L)→Ωk+1(L, End E ⊕T L)

dW..=

dW ,E B

0 dW ,L

!

.

29

In particular, dW ,E is defined as:

dW ,E

A J d x J

n..=F (∂E (F−1(A J d x J ))n

=F

∂E

4π

ħh

−|J |∑

m∈Λe 2πi (m, y )A J

md z J

n

=

4π

ħh

−|J |F

∑

m

e 2πi (m, y )

2πi mk A Jm+ iħh

∂ A Jm

∂ xk

d zk ∧d z J

n

=4π

ħh

∫

p−1(x )

∑

m

e 2πi (m, y )

2πi mk A Jm+ iħh

∂ A Jm

∂ xk

e −2πi (n, y )

d y d xk ∧d x J

=4π

ħh

2πi nk A Jn+ iħh

∂ A Jn

∂ xk

d xk ∧d x J .

(3.4)

The operator B is then defined by

B (ψIj d xI ⊗

∂

∂ x j)

n

..=F (F−1(ψIj d xI ⊗

∂

∂ x j)ùFE )

=

4π

ħh

1−|I |F∑

m

ψIm, j e 2πi (m, y )d zI ⊗

∂

∂ z jùFp q (φ)d zp ∧d zq )

=

4π

ħh

1−|I |4π

ħh

1+|I |∫

p−1(x )

∑

m

ψIm, j e 2πi (m−n, y )Fj q (φ)d y d xI ∧d xq

(3.5)

where Fp q (φ) is the curvature matrix. Then the dW ,L is defined by:

(3.6)

dW ,L(ψIj d xI ⊗

∂

∂ x j)

n

..= e −2πħh−1(n,x )d

ψIj d xI ⊗

∂

∂ x je 2πħh−1(n,x )

,

where d is the de Rham differential on the base M . Notice that by definition dW =

F (∂ )F−1.

Analogously we define the Lie bracket ·, ·∼ ..=F ([·, ·]∼)F−1. If we compute it explic-

itly in local coordinates we find:

·, ·∼ : Ωp (L, End E ⊕T L)×Ωq (L, End E ⊕T L)→Ωp+q (L, End E ⊕T L)

(A,ϕ), (N ,ψ)∼ =

A, N +ad(ϕ, N )− (−1)p q ad(ψ, A),ϕ,ψ

.

In particular locally on Um we consider

(A,ϕ) =

AImd xI ,ϕI

j ,md xI ⊗∂

∂ x j

∈Ωp (Um, End E ⊕T L)

and on Um′ we consider

(N ,ψ) = (N Jm′d x J ,ψJ

J ,m′d x J ⊗∂

∂ x _m) ∈Ωq (Um′ , End E ⊕T L)

30

then

(3.7) A, N n ..=∑

m+m’=n

[AIm, N J

m’]d xI ∧d x J

where the sum over m+m’=n makes sense under the assumption of enough regular-

ity of the coefficients. The operator ad: Ωp (L, T L)×Ωq (L, End E )→ Ωp+q (L, End E ) is

explicitly

ad

ϕ, N

n..=

4π

ħh

∫

p−1(x )

∑

m+m′=n

ϕjI ,m

∂ N Jm′

∂ z j+2πi m′ j +A j (φ)N

Jm′

·

· e 2πi (m+m′−n, y )

d y

d xI ∧d x J

where A j (φ)d z j is the connection∇E one-form matrix. Finally the Lie bracket ψ,φ∼is

ϕ,ψn ..=

∑

m′+m=n

e −2πħh−1(n,x )

ϕIj ,me 2π(m,x )∇ ∂

∂ x j

e 2π(m′,x )ψJk ,m′

∂

∂ xk

− (−1)p qψJk ,m′e

2πħh−1(m′,x )∇ ∂∂ xk

e 2πħh−1(m,x )ϕIj ,m

∂

∂ x j

d xI ∧d x J

(3.8)

where∇ is the flat connection on M .

DEFINITION 3.1.2. The symplectic DGLA is defined as follows:

G ..= (Ω•(L, End E ⊕T L), dW ,·, ·∼)

and it is isomorphic to KS(X , E ) viaF .

As we mention above, the gauge group on the symplectic side Ω0(L, End E ⊕T L) is

related with the extended Lie algebra h. However to figure it out, some more work has to

be done, as we prove in the following subsection.

3.1.1 Relation with the Lie algebra h

Let A f f ZM be the sheaf of affine linear transformations over M defined for any open

affine subset U ⊂M by fm (x ) = (m , x ) + b ∈ A f f ZM (U ) where x ∈U , m ∈ Λ and b ∈ R.

Since there is an embedding of A f f ZM (U ) into OX (p−1(U ))which maps fm (x ) = (m , x )+

b ∈ A f f ZM (U ) to e 2πi (m ,z )+2πi b ∈ OX (p−1(U )), we define Oa f f the image sub-sheaf of

A f f ZM in OX . Then consider the embedding of the dual lattice Λ∗ ,→ T 1,0X which maps

n→ n j ∂

∂ z j=: ∂n .

It follows that the Fourier transformF maps

N e 2πi ((m ,z )+b ), e 2πi ((m ,z )+b )∂n

∈Oa f f

p−1(U ),gl(r,C)⊕T 1,0X

31

to

N e 2πi bwm ,ħh

4πe 2πi bwm n j ∂

∂ x j

∈wm ·C (Um ,gl(r,C)⊕T L)

where wm ..=F (e 2πi (m ,z )), i.e. on Uk

wm =

e 2πħh−1(m ,x ) if k =m

0 if k 6=m

and we define

∂n..=ħh

4πn j ∂

∂ x j.

Let G be the sheaf over M defined as follows: for any open subset U ⊂M

G(U ) ..=⊕

m∈Λ\0wm ·C(U ,gl(r,C)⊕T M ).

In particular h is a subspace of G(U ) once we identify z m with wm and m⊥ with ∂m⊥ . In

order to show how the Lie bracket on h is defined, we need to make another assumption

on the metric: assume that the metric hE is constant along the fibres of X , i.e. in an

open subset U ⊂M φ j =φ j (x1, x2), j = 1, · · · , r . Hence, the Chern connection becomes

∇E = d+ħh A j (φ)d z j while the curvature becomes FE = ħh 2Fj k (φ)d z j∧d zk . We now show

G(U ) is a Lie sub-algebra of

Ω0(pr−1(U )), End E ⊕T L),·, ·∼

⊂G (U ) and we compute

the Lie bracket ·, ·∼ explicitly on functions of G(U ).

Awm ,wm∂n

,

N wm ′,wm ′

∂n ′

∼ =

[A, N ]wm+m ′+ad(wm∂n , N wm ′

)−ad(wm ′∂n ′ , Awm ),

wm∂n ,wm ′∂n ′

ad(wm∂n , N wm ′)s =

4π

ħh

∑

k+k ′=s

wm ħh4π

n j

∂ N wm ′

∂ x j+2πi m ′

j +ħh A j (φ)N

=wm+m ′n j

2πi m ′j + iħh

∂ φ

∂ x jN

=wm+m ′ 2πi ⟨m ′, n⟩+ iħhn j A j (φ)N

(3.9)

where in the second step we use the fact that wm is not zero only on Um , and in the last

step we use the pairing of Λ and Λ∗ given by ⟨m , n ′⟩=∑

j m j n ′ j . Thus

Awm ,wm∂n

,

N wm ′,wm ′

∂n ′

∼ = ([A, N ]wm+m ′+N (2πi m ′

j )njwm+m ′

+

+ħhN A j (φ)njwm+m ′

−A(2πi m j )n′ jwm+m ′

+

−ħh AA j (φ)n′ jwm+m ′

,wm∂n ,wm ′∂n ′)

32

Taking the limit as ħh→ 0, the Lie bracket (Awm ,wm∂n ) ,

N wm ′,wm ′

∂n ′

∼ converges to

[A, N ]wm+m ′+2πi ⟨m ′, n⟩N wm+m ′

−2πi ⟨m , n ′⟩Awm+m ′,wm∂n ,wm ′

∂n ′

and we finally recover the definition of the Lie bracket of [·, ·]h (2.21), up to a factor of

2πi . Hence (h, [·, ·]∼) is the asymptotic subalgebra of

Ω0(pr−1(U )), End E ⊕T L),·, ·∼

.

3.2 Deformations associated to a single wall diagram

In this section we are going to construct a solution of the Maurer-Cartan equation

from the data of a single wall. We work locally on a contractible, open affine subset

U ⊂M .

Let (m ,dm ,θm )be a wall and assume log(θm ) =∑

j ,k

A j k t jwk m , a j k t jwk m∂n

, where

A j k ∈ gl(r,C) and a j k ∈C, for every j , k .

NOTATION 3.2.1. We need to introduce a suitable set of local coordinates on U , namely

(um , um ,⊥), where um is the coordinate in the direction of dm , while um⊥ is normal to

dm , according with the orientation of U . We further define Hm ,+ and Hm ,− to be the half

planes in which dm divides U , according with the orientation.

NOTATION 3.2.2. We will denote by the superscript CLM the elements already introduced

in [CCLM17a].

3.2.1 Ansatz for a wall

Let δm..= e −

u2m⊥ħhpπħh d um⊥ be a normalized Gaussian one-form, which is supported on

dm .Then, let us define

Π ..= (ΠE ,ΠC LM )

where ΠE =−∑

j ,k A j k t jδmwk m and ΠC LM =−∑

j ,k≥1 ak jδm t jwk m∂n .

From section 4 of [CCLM17a] we are going to recall the definition of generalized

Sobolev space suitably defined to compute the asymptotic behaviour of Gaussian k-

forms like δm which depend on ħh . Let Ωkħh (U ) denote the set of k -forms on U whose

coefficients depend on the real positive parameter ħh .

DEFINITION 3.2.3 (Definition 4.15 [CCLM17a]).

W−∞k (U ) ..=

α ∈Ωkħh (U )|∀q ∈U ∃V ⊂U , q ∈V s.t. sup

x∈V

∇ jα(x )

≤C ( j , V )e −cVħh , C ( j , V ), cV > 0

is the set of exponential k-forms.

33

DEFINITION 3.2.4 (Definition 4.16 [CCLM17a]).

W∞k (U )

..=

α ∈Ωkħh (U )|∀q ∈U ∃V ⊂U , q ∈V s.t. sup

x∈V

∇ jα(x )

≤C ( j , V )ħh−Nj ,V , C ( j , V ), Nj ,V ∈Z>0

is the set of polynomially growing k-forms.

DEFINITION 3.2.5 (Definition 4.19 [CCLM17a]). Let dm be a ray in U . The setW sdm(U )

of 1-forms α which have asymptotic support of order s ∈ Z on dm is defined by the

following conditions:

(1) for every q∗ ∈U \ dm , there is a neighbourhood V ⊂U \ dm such that α|V ∈W−∞

1 (V );

(2) for every q∗ ∈ dm there exists a neighbourhood q∗ ∈ W ⊂ U where in local

coordinates uq = (uq ,m , uq ,m⊥ ) centred at q∗, α decomposes as

α= f (uq ,ħh )d uq ,m⊥ +η

η ∈W−∞1 (W ) and for all j ≥ 0 and for all β ∈Z≥0

(3.10)

∫

(0,uq ,m⊥ )∈W

(um⊥ )β

sup(uq ,m ,um⊥ )∈W

∇ j ( f (uq ,ħh ))

d um⊥ ≤C ( j , W ,β )ħh−j+s−β−1

2

for some positive constant C (β , W , j ).

REMARK 3.2.6. A simpler way to figure out what is the spaceW sdm(U ), is to understand

first the case of a 1-form α ∈Ω1ħh (U )which depends only on the coordinate um⊥ . Indeed

α = α(um⊥ ,ħh )d um⊥ has asymptotic support of order s on a ray dm if for every q ∈ dm ,

there exists a neighbourhood q ∈W ⊂U such that∫

(0,uq ,m⊥ )∈W

uβq ,m⊥

∇ jα(uq ,m⊥ ,ħh )

d uq ,m⊥ ≤C (W ,β , j )ħh−β+s−1− j

2

for every β ∈Z≥0 and j ≥ 0.

In particular for β = 0 the estimate above reminds to the definition of the usual

Sobolev spaces Lj1 (U ).

Lemma 3.2.7. The one-form δm defined above, has asymptotic support of order 1 along

dm , i.e. δm ∈W1dm(U ).

PROOF. We claim that

(3.11)

∫ b

−a

(um⊥ )β∇ j

e −u2

m⊥ħh

pħhπ

d um⊥ ≤C (β , W , j )ħh−j−β

2

for every j ≥ 0, β ∈ Z≥0, for some a , b > 0. This claim holds for β = 0 = j , indeed

∫ b

−ae −

u2m⊥ħhpħhπ d um⊥ is bounded by a constant C =C (a , b )> 0.

34

Then we prove the claim by induction on β , at β = 0 it holds true by the previous

computation. Assume that

(3.12)

∫ b

−a

(um⊥)β e −

u2m⊥ħh

pħhπ

d um⊥ ≤C (β , a , b )ħhβ/2

holds for β , then

∫ b

−a

(um⊥ )β+1 e −u2

m⊥ħh

pħhπ

d um⊥ =−ħh2

∫ b

−a

(um⊥ )β

−2um⊥

ħhe −

u2m⊥ħh

pħhπ

d um⊥

=−ħh2

(um⊥ )βe −

u2m⊥ħh

pħhπ

b

−a

+βħh2

∫ b

−a

(um⊥ )β−1 e −u2

m⊥ħh

pħhπ

d um⊥

≤C (β , a , b )ħh12 + C (β , a , b )ħh 1+ β−1

2

≤C (a , b ,β )ħhβ+1

2 .

(3.13)

Analogously let us prove the estimate by induction on j . At j = 0 it holds true, and

assume that

(3.14)

∫ b

−a

(um⊥ )β∇ j

e −u2

m⊥ħh

pħhπ

d um⊥ ≤C (a , b ,β , j )ħh−j−β

2

holds for j . Then at j +1 we have the following

∫ b

−a

(um⊥ )β∇ j+1

e −u2

m⊥ħh

pħhπ

d um⊥ =

uβm⊥∇ j

e −u2

m⊥ħh

pħhπ

b

−a

−β∫

NV

(um⊥ )β−1∇ j

e −u2

m⊥ħh

pħhπ

d um⊥

≤ C (β , a , b , j )ħh− j− 12 +C (a , b ,β , j )ħh−

j−β+12

≤C (a , b ,β , j )ħh−j+1−β

2

This ends the proof.

NOTATION 3.2.8. We say that a function f (x ,ħh ) on an open subset U ×R≥0 ⊂M ×R≥0

belongs to Ol o c (ħh l ) if it is bounded by CK ħh l on every compact subset K ⊂U , for some

constant CK (independent on ħh), l ∈R.

In order to deal with 0-forms “asymptotically supported on U ”, we define the follow-

ing space W s0 :

35

DEFINITION 3.2.9. A function f (uq ,ħh ) ∈Ω0ħh (U ) belongs to W s

0 (U ) if and only if for

every q∗ ∈U there is a neighbourhood q∗ ∈W ⊂U such that

supq∈W

∇ j f (uq ,ħh )

≤C (W , j )ħh−s+ j

2

for every j ≥ 0.

NOTATION 3.2.10. Let us denote by Ωkħh (U , T M ) the set of k -forms valued in T M , which

depends on the real parameter ħh and analogously we denote by Ωkħh (U , End E ) the set

of k -forms valued in End E which also depend on ħh . We say that α = αK (x ,ħh )d x K ⊗∂n ∈Ωk

ħh (U , T M ) belongs to W sP (U , T M )/W∞

k (U , T M )/W−∞k (U , T M ) if αK (x ,ħh )d x K ∈

W sP (U )/W∞

k (U )/W−∞k (U ). Analogously we say that A = AK (x ,ħh )d x K ∈ Ωk

ħh (U , End E )

belongs to W sP (U , End E )/W∞

k (U , End E )/W−∞k (U , End E ) if for every p , q = 1, · · · , r

then (AK )i j (x ,ħh )d x K ∈W sP (U )/W∞

k (U )/W−∞k (U ).

Proposition 3.2.11. Π is a solution of the Maurer-Cartan equation dWΠ+12Π,Π∼ = 0,

up to higher order term in ħh , i.e. there exists ΠE ,R ∈Ω1(U , End E ⊕T L) such that Π ..=

(ΠE +ΠE ,R ,ΠC LM ) is a solution of Maurer-Cartan and ΠE ,R ∈W−1dm(U ).

PROOF. First of all let us compute dWΠ:

dWΠ=

dW ,EΠE + BΠC LM , dW ,LΠC LM

=

−A j k t jwk m dδm + BΠC LM ,−a j k t jwk m d (δm )⊗ ∂n

and notice that d (δm ) = 0. Then, let us compute BΠC LM :

BΠC LM =F (F−1(ΠC LM )ùFE )

=−F

4π

ħh

−1

a j k t j wk m δm ⊗ ∂nù

ħh 2Fq

j (φ)d z j ∧d zq

=−

4π

ħh

−1

F

a j k t j wk m n l ħh 2Fj q (φ)δm ∧d z q

=−ħh 24π

ħh

−14π

ħh

2a j k t jwk m n l Fl q (φ)δm ∧d x q

=−4πħh (a j k t jwk m n l Fl q (φ)δm ∧d x q )

where we denote by δm the Fourier transform of δm . Notice that BΠC LM is an exact

two form, thus since Fl q (φ)d x q = d Al (φ) (recall that the hermitian metric on E is

diagonal)we define

ΠE ,R..= 4πħh (a j k t jwk m n l Al (φ)δm )

i.e. as a solution of dWΠE ,R =−BΠC LM .

36

In particular, since δm ∈ W1dm(U ) then ħhδm ∈ W−1

dm(U ). Therefore ΠE ,R has the

expected asymptotic behaviour and dW Π= 0. Let us now compute the commutator:

Π, Π∼ =

2F

F−1ΠC LM ù∇EF−1(ΠE +ΠE ,R )

+ ΠE +ΠE ,R ,ΠE +ΠE ,R ,ΠC LM ,ΠC LM

=

2F

F−1ΠC LM ù∇EF−1(ΠE +ΠE ,R )

+2(ΠE +ΠE ,R )∧ (ΠE +ΠE ,R ), 0

Notice that, since both ΠE and ΠE ,R are matrix valued one forms where the form part

is given by δm , the wedge product (ΠE +ΠE ,R )∧ (ΠE +ΠE ,R ) vanishes as we explicitly

compute below

(ΠE +ΠE ,R )∧ (ΠE +ΠE ,R ) = A j k Ar s t j+r wk m+s mδm ∧δm+

+8πħha j k t j+r wk m+s m n l Al (φ)Ar sδm ∧δm +4πħh (a j k t jwk m n l Al (φ))2δm ∧δm = 0.

Hence we are left to computeF

F−1ΠC LM ù∇EF−1

(ΠE +ΠE ,R )

:

F

F−1ΠC LM ù∇EF−1(ΠE +ΠE ,R )

=

=F

4π

ħh

−1a j k t j wk m δm ∂nùd

4π

ħh

−1At wm δm +4πħhar s t r ws m n l Al (φ)δm

+

+4π

ħh

−1a j k t j wk m δm ∂nù

iħh Aq (φ)d z q ∧4π

ħh

−1At wm δm +4πħhar s t r ws m n l Al (φ)δm

=4π

ħh

−1F

a j k t j wk m δm ∂nù

At ∂l (wm )d z l ∧ δm +At wm d (δm )+

+4πħhar s t r ∂l (nq Aq (φ)w

s m )δm +4πħhar s t r n q Aq (φ)ws m d (δm )

=4π

ħh

−1F

a j k At j+1wk m δm n l ∂l (wm )∧ δm +a j k At j+1wk m+m n l γl (iħh−1γp z p )δm ∧ δm+

+4πħha j k ar s t j+r wk m δm n l ∂l (nq Aq (φ)w

s m )δm+

+4πħha j k ar s t j+r wk m+s m n q Aq (φ)nl γl (iħh−1γp z p )δm ∧ δm

= 0

where δm =e −

u2m⊥ħhpπħh γp d z p for some constant γp such that um⊥ = γ1 x 1 + γ2 x 2, and ∂l

is the partial derivative with respect to the coordinate z l . In the last step we use that

δm ∧ δm = 0.

REMARK 3.2.12. In the following it will be useful to consider Π in order to compute

the solution of Maurer-Cartan from the data of two non-parallel walls (see section (3.3)).

However, in order to compute the asymptotic behaviour of the gauge it is enough to

consider Π.

Since X (U ) ∼=U ×C/Λ has no non trivial deformations and E is holomorphically

trivial, then also the pair (X (U ), E ) has no non trivial deformations. Therefore there is a

37

gauge ϕ ∈Ω0(U , End E ⊕T M )[[t ]] such that

(3.15) e ϕ ∗0= Π

namely ϕ is a solution of the following equation

(3.16) dWϕ =−Π−∑

k≥0

1

(k +1)!adkϕdWϕ.

In particular the gauge ϕ is not unique, unless we choose a gauge fixing condition (see

Lemma 3.2.14). In order to define the gauge fixing condition we introduce the so called

homotopy operator.

3.2.2 Gauge fixing condition and homotopy operator

Since L(U ) =⊔

m∈ΛUm , it is enough to define the homotopy operator Hm for every

frequency m . Let us first define morphisms p ..=⊕

m∈Λ\0pm and ι ..=⊕

m∈Λ\0 ιm . We

define pm : wm ·Ω•(U )→wm ·H •(U )which acts as pm (αwm ) =α(q0)wm if α ∈Ω0(U ) and

it is zero otherwise.

Then ιm : wm ·H •(U )→wmΩ•(U ) is the embedding of constant functions on Ω•(U )

at degree zero, and it is zero otherwise. Then let q0 ∈H− be a fixed base point, then since

U is contractible, there is a homotopy % : [0,1]×U →U which maps (τ, um , um ,⊥) to

(%1(τ, um , um ,⊥),%2(τ, um , um ,⊥)) and such that %(0, ·) = q0 = (u 10 , u 2

0 ) and %(1, ·) = Id. We

define Hm as follows:

Hm : wm ·Ω•(U )→wm ·Ω•(U )[−1]

Hm (wmα) ..=wm

∫ 1

0

dτ∧∂

∂ τù%∗(α)

(3.17)

Lemma 3.2.13. The morphism H is a homotopy equivalence of idΩ• and ι p , i.e. the

identity

(3.18) id− ι p = dW H +H dW

holds true.

PROOF. At degree zero, let f ∈ Ω0(U ): then ιm pm ( f wm ) = f (q0)wm . By degree

reason Hm ( f wm ) = 0 and

Hm dW (wm f ) =wm

∫ 1

0

dτ∧∂

∂ τù(dM ( f (%))+dτ

∂ f (%)∂ τ

) =wm

∫ 1

0

dτ∂ f (%)∂ τ

=wm ( f (q )− f (q0)).

38

At degree k = 1, let α= fi d x i ∈Ω1(U ) then: ιm pm (αwm ) = 0,

Hm dW (αwm ) =wm

∫ 1

0

dτ∧∂

∂ τù(d (%∗(α))

=wm

∫ 1

0

dτ∧∂

∂ τù(dM (%

∗(α))+dτ∧∂

∂ τ( fi (%)

∂ %i

∂ x i)d x i )

=−wm dM

∫ 1

0

dτ∧∂

∂ τù(%∗(α))

+wm

∫ 1

0

dτ∂

∂ τ( fi (%)

∂ %i

∂ x i)d x i

=−wm dM

∫ 1

0

dτ∧∂

∂ τù(%∗(α))

+wmα

and

dW Hm (wmα) =wm dM

∫ 1

0

dτ∧∂

∂ τù%∗(α)

.

Finally let α ∈Ω2(U ), then pm (αwm ) = 0 and dW (αwm ) = 0. Then it is easy to check that

dW Hm (wmα) =α.

Lemma 3.2.14 (Lemma 4.7 in [CCLM17a]). Among all solution of e ϕ ∗0= Π, there exists

a unique one such that p (ϕ) = 0.

PROOF. First of all, let σ ∈ Ω0(U ) such that dσ = 0. Then e ϕ•σ ∗ 0 = Π, indeed

eσ ∗ 0 = 0−∑

k[σ,·]k

k ! (dσ) = 0. Thus e ϕ•σ ∗ 0 = e ϕ ∗ (eσ ∗ 0) = e ϕ ∗ 0 = Π. Thanks to the

BCH formula

ϕ •σ=ϕ+σ+1

2ϕ,σ∼+ · · ·

we can uniquely determineσ such that p (ϕ •σ) = 0. Indeed working order by order in

the formal parameter t , we get:

(1) p (σ1+ϕ1) = 0, hence by definition of p ,σ1(q0) =−ϕ1(q0);

(2) p (σ2+ϕ2+12ϕ1,σ1∼) = 0, henceσ2(q0) =−

ϕ2(q0) +12ϕ1,σ1∼(q0)

;

and any further order is determined by the previous one.

Now that we have defined the homotopy operator and the gauge fixing condition (as

in Lemma 3.2.14), we are going to study the asymptotic behaviour of the gauge ϕ such

that it is a solution of (3.16) and p (ϕ) = 0. Equations (3.16), (3.18) and p (ϕ) = 0 together

say that the unique gauge ϕ is indeed a solution of the following equation:

(3.19) ϕ =−H dW (ϕ) =−H

Π+∑

k

adkϕ

(k +1)!dWϕ

.

Up to now we have used a generic homotopy %, but from now on we are going to

choose it in order to get the expected asymptotic behaviour of the gaugeϕ. In particular

39

we choose the homotopy % as follows: for every q = (uq ,m , uq ,m⊥ ) ∈U

(3.20) %(τ, uq ) =

(1−2τ)u 01 +2τuq ,m , u 0

2

ifτ ∈ [0, 12 ]

uq ,m , (2τ−1)uq ,m⊥ + (2−2τ)u 02

ifτ ∈ [ 12 , 1]

where (u 10 , u 2

0 ) are the coordinates for the fixed point q0 on U . Then we have the

following result:

Lemma 3.2.15. Let dm be a ray in U and let α ∈ W sdm(U ). Then H (αwm ) belongs to

W s−10 (U ).

PROOF. Let us first consider q∗ ∈U \dm . By assumption there is a neighbourhood of

q∗, V ⊂U such that α ∈W−∞1 (V ). Then by definition

H (αwm ) =wm

∫ 1

0

dτ∧∂

∂ τù%∗(α) =

∫ 1

0

dτα(%)

∂ %1

∂ τ+∂ %2

∂ τ

hence, since % does not depend on ħh

supq∈V∇ j

∫ 1

0

dτα(%)

∂ %1

∂ τ+∂ %2

∂ τ

≤∫ 1

0

dτsupq∈V

∇ j (α(%)

∂ %1

∂ τ+∂ %2

∂ τ

)

≤C (V , j )e −cvħh .

Let us now consider q∗ ∈ dm . By assumption there is a neighbourhood of q , W ⊂U

such that for all q = (uq ,m , uq ,m⊥ ) ∈ W α = h (uq ,ħh )d um⊥q+η and η ∈W−∞

1 (W ). By

definition

H (αwm ) =wm

∫ 1

0

dτ∧∂

∂ τù%∗(α)

= 2

∫ 1

12

dτh (uq ,m , (2τ−1)uq ,m⊥ + (2−2τ)u 20 )(uq ,m⊥ −u 2

0 ) +

∫ 1

0

dτη(%)∂ %1

∂ τ

=

∫ uq ,m⊥

u 20

d u⊥m h (uq ,m , um⊥ ) +

∫ 1

0

dτη(%)∂ %1

∂ τ

and since η ∈W−∞1 (W ) the second term

∫ 1

0dτη(%) ∂ %1

∂ τ belongs to W∞0 . The first

term is computed below:

40

supq∈W

∇ j

∫ uq ,m⊥

u 20

d u⊥m h (uq ,m , um⊥ )

= supq∈W

∫ uq ,m⊥

u 20

d u⊥m∇j (h (uq ,m , um⊥ ))+

+

∂ j−1

∂ uj−1m⊥

(h (uq ,m , um⊥))

um⊥=uq ,m⊥

≤ supuq ,m⊥

∫ uq ,m⊥

u 20

d u⊥m supuq ,m

∇ j (h (uq ,m , um⊥ ))

+

+

supq∈W

∂ j−1

∂ uj−1m⊥

(h (uq ,m , um⊥))

um⊥=u 20

≤C ( j , W )ħh−s+ j−1

2

(3.21)

where in the last step we use that

∂ j−1

∂ uj−1

m⊥(h (uq ,m , um⊥))

um⊥=u 20

is outside the support of

dm .

Corollary 3.2.16. Let dm be a ray in U , then H (δmwm ) ∈W00 (U )w

m .

3.2.3 Asymptotic behaviour of the gaugeϕ

We are going to compute the asymptotic behaviour ofϕ =∑

j ϕ( j )t j ∈Ω0(U , End E ⊕

T M )[[t ]] order by order in the formal parameter t . In addition since ΠE ,R gives a higher

ħh-order contribution in the definition of Π we get rid of it by replacing Π with Π in

equation (3.19).

Proposition 3.2.17. Let (m ,dm ,θm )be a wall with logθm =∑

j ,k≥1

A j k t jwk m , a j kwk m t j ∂n

.

Then, the unique gauge ϕ = (ϕE ,ϕC LM ) such that e ϕ ∗0=Π and P (ϕ) = 0, has the fol-

lowing asymptotic jumping behaviour along the wall, namely

(3.22)

ϕ(s+1) ∈

∑

k≥1

As+1,k t s+1wk m , as+1,kwk m t s+1∂n

+⊕

k≥1W−10 (U , End E ⊕T M )wk m t s+1 Hm ,+

⊕

k≥1W−∞0 (U , End E ⊕T M )wk m t s+1 Hm ,−.

Before giving the proof of Proposition 3.2.17, let us introduce the following Lemma

which are useful to compute the asymptotic behaviour of one-forms asymptotically

supported on a ray dm .

Lemma 3.2.18. Let dm be a ray in U . Then W sdm(U )∧W r

0 (U )⊂W r+sdm(U ).

41

PROOF. Let α ∈W sdm(U ) and let f ∈W r

0 (U ). Pick a point q∗ ∈ dm and let W ⊂U be a

neighbourhood of q∗ where α= h (uq ,ħh )d um⊥ +η, we claim

(3.23)

∫ b

−a

uβm⊥ sup

um

∇ j (h (uq ,ħh ) f (uqħh ))

d um⊥ ≤C (a , b , j ,β )ħh−r+s+ j−β−1

2

for every β ∈Z≥0 and for every j ≥ 0.

∫ b

−a

uβm⊥ sup

um

∇ j

h (uq ,ħh ) f (uqħh )

d um⊥ =

=∑

j1+ j2= j

∫ b

−a

uβm⊥ sup

um

∇ j1 (h (uq ,ħh ))∇ j2 ( f (uqħh ))

d um⊥

≤∑

j1+ j2= j

C (a , b , j2)ħh−r+ j2

2

∫ b

−a

uβm⊥ sup

um

∇ j1 (h (uq ,ħh ))

d um⊥

≤∑

j1+ j2= j

C (a , b , j2, j1)ħh−r+ j2

2 ħh−s+ j1−β−1

2

≤C (a , b , j )ħh−r+s+ j−β−1

2

Finally, since η ∈W−∞1 (W ) also f (x ,ħh )η belongs to W−∞

1 (W ).

Lemma 3.2.19. Let dm be a ray in U . If (Awm ,ϕwm∂n ) ∈W rdm(U , End E ⊕T M )wm for

some r ≥ 0 and (T wm ,ψwm∂n ) ∈W s0 (U , End E ⊕T M )wm for some s ≥, then

(3.24) (Awm ,αwm∂n ), (T wm , f wm∂n )∼ ∈W r+sdm(U , End E ⊕T M )w2m .

PROOF. We are going to prove the following:

(1)Awm , T wmEnd E ⊂W r+sdm(U , End E )w2m

(2)ad

ϕwm∂n , T wm

⊂W r+s−1dm

(U , End E )w2m

(3)ad

ψwm∂n , Awm

⊂W r+s−1dm

(U , End E )w2m

(4)ϕwm∂n ,ψwm∂n ⊂W r+sdm(U , T M )wm .

The first one is a consequence of Lemma 3.2.18, indeed by definition

Ak (x )d x k , T (x )End E = [Ak (x ), T (x )]d x k

which is an element in End E with coefficients in W r+sdm(U ).

42

The second one is less straightforward and need some explicit computations to be

done.

ad(ϕk (x )wm d x k∂n , T (x )wm ) =F

F−1(ϕk (x )wm d x k ⊗ ∂n )ù∇EF−1(T (x )wm )

=F

4π

ħh

−1ϕk (x )w

m d z k ⊗ ∂nù∇E (T (x )wm )

=4π

ħh

−1F

ϕk (x )wm d z k ∂nù

∂ j (T (x )wm )d z j

+ iħh4π

ħh

−1F

ϕk (x )wm d z k ∂nù

A j (φ)T (x )wm d z j

=4π

ħh

−1F

ϕk (x )wm d z k n l ∂

∂ zlù

∂ j (T (x ))wm d z j +m j A(x )wm d z j

+ iħh4π

ħh

−1F

ϕk (x )wm d z k n l ∂

∂ zlù

A j (φ)T (x )wm d z j

=4π

ħh

−1F

ϕk (x )d z k n l ml T (x )w2m

+

+ iħh4π

ħh

−1F

ϕk (x )d z k

n l T (x )Al (φ) +n l ∂ T (x )∂ x l

w2m

= iħhϕk (x )

n l T (x )Al (φ) +n l ∂ T (x )∂ x l

w2m d x k

Notice that as a consequence of Lemma 3.2.18, ħhϕk (x )d x k Al (φ)T (x ) ∈ W s+r−2dm

(U )

while ħhϕk (x )∂ T (x )∂ x l d x k ∈W r+s−1

dm.

The third one is

ad(ψ(x )wm∂n , Ak (x )wm d x k ) =

=F

F−1(ψ(x )∂n )ù∇EF−1(Ak (x )wm d x k )

=F

ψ(x )wm ∂nù∇E (4π

ħh

−1Ak (x )w

m d z k )

=4π

ħh

−1F

ψ(x )wm ∂nù

∂ j (Ak (x )wm )

d z j ∧d z k

+ iħh4π

ħh

−1F

ψ(x )wm ∂nù

A j (φ)Ak (x )wm

∧d z k

=4π

ħh

−1F

ψ(x )wm n l ∂

∂ zlù

∂ j (Ak (x ))wm d z j +m j Ak (x )w

m d z j

∧d z k

+ iħh4π

ħh

−1F

ψ(x )wm ∂nù

A j (φ)Ak (x )wm d z j

∧d z k

=4π

ħh

−1F

ψ(x )w2m n l ml A(x )d z k

+

+ iħh4π

ħh

−1F

ψ(x )w2m n l Ak (x )Al (φ) +ψ(x )nl ∂ Ak (x )∂ x l

w2m d z k

= iħhψ(x )

n l Ak (x )Al (φ) +n l ∂ Ak (x )∂ x l

w2m d x k

43

Notice that ħhψ(x )Ak (x )A(φ)d x k ∈W r+s−2dm

(W ) and ħhψ(x ) ∂ Ak (x )∂ x l d x k ∈W r+s−1

dm(W ).

In the end ϕwm∂n ,ψwm∂n is equal to zero, indeed by definition

ϕk (x )d x k∂n ,ψ(x )∂n=

ϕk d x k ∧ψ

[wm∂n ,wm∂n ]

and [wm∂n ,wm∂n ] = 0.

PROOF. (Proposition 3.2.17)

It is enough to show that for every s ≥ 0

(3.25)∑

k≥1

adkϕs

(k +1)!dWϕ

s(s+1)

∈W0dm(U , End E ⊕T M ),

where ϕs =∑s

j=1ϕ( j )t j . Indeed from equation (3.19), at the order s + 1 in the formal

parameter t , the solution ϕ(s+1) is:

(3.26) ϕ(s+1) =−H (Π(s+1))−H

∑

k≥1

adkϕs

(k +1)!dWϕ

s

!(s+1)

.

In particular, if we assume equation (3.25) then by Lemma 3.2.15

H

∑

k≥1

adkϕs

(k +1)!dWϕ

s

!(s+1)

∈W−10 (U , End E ⊕T M ).

By definition of H ,

H (Π(s+1)) =∑

k

(As+1,k t s+1H (wk mδm ), as+1,k t s+1H (wk mδm )∂n )

and by Corollary 3.2.16 H (δmwk m ) ∈W00 (U , End E ⊕T M )wk m for every k ≥ 1. Hence

H (Π(s+1)) is the leading order term and ϕ(s+1) has the expected asymptotic behaviour.

Let us now prove the claim (3.25) by induction on s . At s = 0,

(3.27) ϕ(1) =−H (Π(1))

and there is nothing to prove. Assume that (3.25) holds true for s , then at order s +1

we get contributions for every k = 1, · · · , s . Thus let start at k = 1 with adϕs dWϕs :

adϕs dWϕs = ϕs , dWϕ

s ∼

∈ H (Πs ) +W−10 (U ),Π

s +W0dm(U )∼

= H (Πs ),Πs ∼+ H (Πs ),W0dm(U )∼+ W−1

0 (U ),Πs ∼+ W−1

0 (U ),W0dm(U )∼

∈ H (Πs ),Πs ∼+W0dm(U )

(3.28)

where in the first step we use the inductive assumption onϕs and dWϕs and the identity

(3.18). In the last step since H (Πs ) ∈W00 (U ) then by Lemma 3.2.19 H (Πs ),W0

dm(U )∼ ∈

44

W0dm(U ). Then, since Πs ∈W1

dm(U ), still by Lemma 3.2.19 W−1

0 (U ),Πs ∼ ∈W0

dm(U ) and

W−10 (U ),W0

dm(U )∼ ∈W−1

dm(U )⊂W0

dm(U ). In addition H (Πs ),Πs ∼ ∈W0

dm(U ), indeed

H (Πs ),Πs ∼ =

H (ΠsE ),Π

sE End E +ad(H (ΠC LM ,s ),Πs

E )−ad(ΠC LM ,s , H (ΠsE )),

H (ΠC LM ,s ),ΠC LM ,s

Notice that since [A, A] = 0 then H (ΠsE ),Π

sE End E = 0 and because of the grading

H (ΠC LM ,s ),ΠC LM ,s = 0.

Then by the proof of Lemma 3.2.19 identities (2) and (3)we get

ad(H (ΠC LM ,s ),ΠsE ),ad(Π

C LM ,s , H (ΠsE )) ∈W

0dm(U )

therefore

(3.29) H (Πs ),Πs ∼ ∈W0dm(U ).

Now at k > 1 we have to prove that:

(3.30) adϕs · · ·adϕs dWϕs ∈W0

dm(U )

By the fact that H (Πs ) ∈W00 (U ), applying Lemma 3.2.19 k times we finally get:

(3.31)

adϕs · · ·adϕs dWϕs ∈ H (Πs ), · · · ,H (Πs ),H (Πs ),Πs ∼∼ · · · ∼+W0

dm(U ) ∈W0

dm(U ).

3.3 Scattering diagrams from solutions of Maurer-Cartan

In this section we are going to construct consistent scattering diagrams from solu-

tions of the Maurer-Cartan equation. In particular we will first show how to construct a

solution Φ of the Maurer-Cartan equation from the data of an initial scattering diagram

D with two non parallel walls. Then we will define its completion D∞ by the solution Φ

and we will prove it is consistent.

3.3.1 From scattering diagram to solution of Maurer-Cartan

Let the initial scattering diagram D = w1,w2 be such that w1 = (m1,d1,θ1) and

w2 = (m2,d2,θ2) are two non-parallel walls and

log(θi ) =∑

ji ,ki

A ji ,kiwki mi t ji , a ji ,ki

wki mi t ji ∂ni

for i = 1,2. As we have already done in Section 3.2, we can define Π1 and Π2 to be

solutions of Maurer-Cartan equation, respectively supported on w1 and w2.

45

Although Π ..= Π1 + Π2 is not a solution of Maurer-Cartan, by Kuranishi’s method

we can construct Ξ=∑

j≥2Ξ( j )t j such that the one form Φ ∈Ω1(U , End E ⊕T M )[[t ]] is

Φ= Π+Ξ and it is a solution of Maurer-Cartan up to higher order in ħh . Indeed let us we

write Φ as a formal power series in the parameter t , Φ=∑

j≥1Φ( j )t j , then it is a solution

of Maurer-Cartan if and only if:

dW Φ(1) = 0

dW Φ(2)+

1

2Φ(1),Φ(1)∼ = 0

...

dW Φ(k )+

1

2

k−1∑

s=1

Φ(s ),Φ(k−s )∼

= 0

Moreover, recall from (3.2.11) that Πi..=

ΠE ,i +ΠE ,R ,i ,ΠC LMi

, i = 1, 2 are solutions of the

Maurer-Cartan equation and they are dW -closed. Therefore at any order in the formal

parameter t , the solution Φ= Π+Ξ is computed as follows:

Φ(1) = Π(1)

Φ(2) = Π(2)+Ξ(2), where dW Ξ(2) =−

1

2(Φ(1),Φ(1)∼)

Φ(3) = Π(3)+Ξ(3), where dW Ξ(3) =−

1

2

Φ(1), Π(2)+Ξ(2)∼+ Π(2)+Ξ(2),Φ(1)∼

...

Φ(k ) = Π(k )+Ξ(k ), where dW Ξ(k ) =−

1

2(Φ,Φ∼)(k ) .

(3.32)

In order to explicitly compute Ξ we want to “invert” the differential dW and this can

be done by choosing a homotopy operator. Let us recall that a homotopy operator is a

homotopy H of morphisms p and ι, namely H : Ω•(U )→Ω•[−1](U ), p : Ω•(U )→H •(U )

and ι : H •(U )→Ω•(U ) such that idΩ• − ι p = dW H +H dW . Let us now explicitly define

the homotopy operator H. Let U be an open affine neighbourhood of ξ0 = d1∩d2, and fix

q0 ∈

H−,m1∩H−,m2

∩U . Then choose a set of coordinates centred in q0 and denote by

(um , um⊥ ) a choice of such coordinates such that with respect to a ray dm = ξ0+R≥0m ,

um⊥ is the coordinate orthogonal to dm and um is tangential to dm . Moreover recall

the definition of morphisms p and ι, namely p ..=⊕

m pm and pm maps functions

αwm ∈ Ω0(U )wm to α(q0)wm , while ι ..=⊕

m ιm and ιm is the embedding of constant

function at degree zero, and it is zero otherwise.

DEFINITION 3.3.1. The homotopy operator H=⊕

m Hm :⊕

m Ω•(U )wm →

⊕

m Ω•(U )[−1]wm

is defined as follows. For any 0-form α ∈Ω0(U ), H(αwm ) = 0, since there are no degree

46

−1-forms. For any 1-form α ∈Ω1(U ), in local coordinates we have α= f0(um , um⊥ )d um +

f1(um , um⊥ )d um⊥ and

H(αwm ) ..=wm

∫ um

0

f0(s , um⊥ )d s +

∫ um⊥

0

f1(0, r )d r

Finally since any 2-formsα ∈Ω2(U ) in local coordinates can be writtenα= f (um , um⊥ )d um∧d um⊥ , then

H(αwm ) ..=wm

∫ um

0

f (s , um⊥ )d s

d um⊥ .

The homotopy H seems defined ad hoc for each degree of forms, however it can be

written in an intrinsic way for every degree, as in Definition 5.12 [CCLM17a]. We have

defined H in this way because it is clearer how to compute it in practice.

Lemma 3.3.2. The following identity

(3.33) idΩ• − ιm pm =Hm dW +dW Hm

holds true for all m ∈Λ.

PROOF. We are going to prove the identity separately for 0, 1 and 2 forms.

Let α= α0 be of degree zero, then by definition Hm (αwm ) = 0 and ιm pm (αwm ) =

α0(q0). Then dW (αwm ) =wm ∂ α0∂ um

d um +∂ α0∂ um⊥

d um⊥

. Hence

Hm dW (α0wm ) =wm

∫ um

0

∂ α0(s , um⊥ )∂ s

d s +wm

∫ um⊥

0

∂ α0

∂ um⊥(0, r )d r

=wm [α0(um , um⊥ )−α0(0, um⊥ ) +α0(0, um⊥ )−α0(0, 0)].

Then consider α ∈Ω1(U )wm . By definition ιm pm (αwm ) = 0 and

Hm dW (αwm ) =HdW

wm ( f0d um + f1d um⊥ )

=Hm

wm

∂ f0

∂ um⊥d um⊥ ∧d um +

∂ f1

∂ umd um ∧d um⊥

=wm

∫ um

0

−∂ f0

∂ um⊥+∂ f1

∂ s

d s

d um⊥

=

−∫ um

0

∂ f0

∂ um⊥(s , um⊥ )d s + f1(um , um⊥ )− f1(0, um⊥ )

wm d um⊥

47

dW Hm (αwm ) = dW H(wm

f0d um + f1d um⊥

)

= dW

wm

∫ um

0

f0(s , um⊥ )d s +

∫ um⊥

0

f1(0, r )d r

=wm d

∫ um

0

f0(s , um⊥ )d s +

∫ um⊥

0

f1(0, r )d r

=wm

f0(um , um⊥ )d um +∂

∂ um⊥

∫ um

0

f0(s , um⊥ )d s

d um⊥ + f1(0, um⊥ )d um⊥

.

We are left to prove the identity whenα is of degree two: by degree reasons dW (αwm ) =

0 and ιm pm (αwm ) = 0. Then

dW Hm (αwm ) =wm d

∫ um

0

f (s , um⊥ )d s

d um⊥

=wm f (um , um⊥ )d um ∧d um⊥ .

Proposition 3.3.3 (see prop 5.1 in [CCLM17a]). Assume that Φ is a solution of

(3.34) Φ= Π−1

2H (Φ,Φ∼)

Then Φ is a solution of the Maurer-Cartan equation.

PROOF. First notice that by definition p (Φ,Φ∼) = 0 and by degree reasons dW (Φ,Φ∼) =0 too. Hence by identity (3.33) we get that Φ,Φ,∼ = dW H(Φ,Φ∼), and if Φ is a solution

of equation (3.34) then dW Φ= dW Π− 12 dW H(Φ,Φ∼) =− 1

2 dW H(Φ,Φ∼).

From now on we will look for solutionsΦ of equation (3.34) rather than to the Maurer-

Cartan equation. The advantage is that we have an integral equation instead of a dif-

ferential equation, and Φ can be computed by its expansion in the formal parameter t ,

namely Φ=∑

j≥1Φ( j )t j .

NOTATION 3.3.4. Let dm1= ξ0−m1R and dm2

= ξ0−m2R and let (um1, um⊥

1) and (um2

, um⊥2)

be respectively two basis of coordinates in U , centred in q0 as above. Let ma..= a1m1+

a2m2, consider the raydma..= ξ0−maR≥0 and choose coordinates uma

..=

−a2um⊥1+a1um⊥

2

and um⊥a

..=

a1um⊥1+a2um⊥

2

.

REMARK 3.3.5. If α=δm1∧δm2

, then by the previous choice of coordinates

δm1∧δm2

=e −

u2m⊥1+u2

m⊥2ħh

ħhπd um⊥

1∧d um⊥

2=

e−

u2ma +u2

m⊥a(a 2

1+a 22 )ħh

ħhπd uu⊥ma

∧d uma.

48

In particular we explicitly compute H(δm1∧δm2

wl ma ):

H(αwl ma ) =wl ma

∫ uma

0

e−

s 2+u2m⊥a

(a 21+a 2

2 )ħh

ħhπd s

d uu⊥a=wl ma

∫ uma

0

e− s 2

(a 21+a 2

2 )ħh

pħhπ

d s

e−

u2m⊥a

(a 21+a 2

2 )ħh

pħhπ

d um⊥a

(3.35)

Hence H

δm1∧δm2

wl ma

= f (ħh , uma)δma

where f (ħh , uma) =

∫ uma

0e− s 2

(a 21+a 2

2 )ħhpħhπ d s ∈Ol o c (1).

In order to construct a consistent scattering diagram from the solution Φwe intro-

duce labeled ribbon trees. Indeed via the combinatorial of such trees we can rewrite Φ

as a sum over primitive Fourier mode, coming from the contribution of the out-going

edge of the trees.

Labeled ribbon trees

Let us briefly recall the definition of labeled ribbon trees, which was introduced in

[CCLM17a].

DEFINITION 3.3.6 (Definition 5.2 in [CCLM17a]). A k-tree T is the datum of a finite set

of vertices V , together with a decomposition V =Vi n tV0tvT , and a finite set of edges

E , such that, given the two boundary maps ∂i n ,∂o u t : E →V (which respectively assign

to each edge its incoming and outgoing vertices), satisfies the following assumption:

(1) #Vi n = k and for any vertex v ∈Vi n , #∂ −1i n (v ) = 0 and #∂ −1

o u t (v ) = 1;

(2) for any vertex v ∈V0, #∂ −1i n (v ) = 1 and #∂ −1

o u t (v ) = 2;

(3) vT is such that #∂ −1i n (vT ) = 0 and #∂ −1

o u t (v ) = 1.

We also define eT = ∂ −1i n (vT ).

vT

v1 v2

v3

v4

v5

•

• •

• •

•

eT

FIGURE 3.1. This is an example of a 3-tree, where the set of vertices is

decomposed by Vi n = v1, v2, v4, V0 = v3, v5.

49

Two k-trees T and T ′ are isomorphic if there are bijections V ∼= V ′ and E ∼= E ′

preserving the decomposition V0∼= V ′0 , Vi n

∼= V ′i n and vT ∼= vT ′ and the boundary

maps ∂i n ,∂o u t .

It will be useful in the following to introduce the definition of topological realization

T (T ) of a k-tree T , namely T (T ) ..=∐

e∈E [0, 1]

/s, where s is the equivalence relation

that identifies boundary points of edges with the same image in V \ vT .Since we need to keep track of all the possible combinations while we compute com-

mutators (for instance for Φ(3) there is the contribution of Φ(1),Φ(2)∼ and Φ(2),Φ(1)∼),

we introduce the notion of ribbon trees:

DEFINITION 3.3.7 (Definition 5.3 in [CCLM17a]). A ribbon structure on a k-tree is a

cyclic ordering of the vertices. It can be viewed as an embedding T (T ) ,→D, whereD is

the disk inR2, and the cyclic ordering is given according to the anticlockwise orientation

of D.

Two ribbon k-trees T and T ′ are isomorphic if they are isomorphic as k-trees and

the isomorphism preserves the cyclic order. The set of all isomorphism classes of rib-

bon k-trees will be denoted by RTk . As an example, the following two 2-trees are not

isomorphic:

vT

v1 v2

v3

•

• •

•

eT

v ′T

v2 v1

v4

•

• •

•

e ′T

In order to keep track of the ħh behaviour while we compute the contribution from

the commutators, let us decompose the bracket on the DGLA as follows:

DEFINITION 3.3.8. Let (A,α) = (A J d x J wm1 ,αJ d x J wm1∂n1) ∈Ωp (U , End E ⊕T M )wm1

and (B ,β ) = (BK d x K wm2 ,βK d x K wm2∂n2) ∈Ωq (U , End E ⊕T M )wm2 . Then we decom-

pose ·, ·∼ as the sum of:

\ (A,α), (B ,β )\ ..=

α∧B ⟨n1, m2⟩−β ∧A⟨n2, m1⟩+ A, B End E ,α,β

[ (A,α), (B ,β )[ ..=

iħhβK nq2∂ A J∂ xq

d x J ∧d x K wm1+m2 ,β (∇∂n2α)wm1+m2∂n1

] (A,α), (B ,β )] ..=

iħhαJ nq1∂ BK∂ xq

d x K ∧d x J wm1+m2 ,α(∇∂n1β )wm1+m2∂n2

? (A,α), (N ,β )? ..= iħh

αJ nq1 BK Aq (φ)d x J ∧d x K −n

q2 βK Aq (φ)A J d x K ∧d x J , 0

.

The previous definition is motivated by the following observation: the label \ con-

tains terms of the Lie bracket ·, ·∼ which leave unchanged the behaviour in ħh . Then

50

both the labels [ and ] contain terms which contribute with an extra ħh factor and at the

same time contain derivatives. The last label ? contains terms which contribute with an

extra ħh but do not contain derivatives.

DEFINITION 3.3.9. A labeled ribbon k-tree is a ribbon k-tree T together with:

(i) a label \, ], [, ? -as defined in Definition 3.3.8- for each vertex in V0;

(ii) a label (me , je ) for each incoming edge e , where me is the Fourier mode of the

incoming vertex and je ∈Z>0 gives the order in the formal parameter t .

There is an induced labeling of all the edges of the trees defined as follows: at any

trivalent vertex with incoming edges e1, e2 and outgoing edge e3 we define (me3, je3) =

(me1+me2

, je1+ je2

). We will denote by (mT , jT ) the label corresponding to the unique

incoming edge of νT . Two labeled ribbon k-trees T and T ′ are isomorphic if they are

isomorphic as ribbon k-trees and the isomorphism preserves the labeling. The set of

equivalence classes of labeled ribbon k-trees will be denoted byLRTk . We also introduce

the following notation for equivalence classes of labeled ribbon trees:

NOTATION 3.3.10. We denote byLRTk ,0 the set of equivalence classes of k labeled ribbon

trees such that they have only the label \. We denote by LRTk ,1 the complement set,

namely LRTk ,1 =LRTk −LRTk ,0.

Let us now define the operator tk ,T which allows to write the solution Φ in terms of

labeled ribbon trees.

DEFINITION 3.3.11. Let T be a labeled ribbon k-tree, then the operator

(3.36) tk ,T :Ω1(U , End E ⊕T M )⊗k →Ω1(U , End E ⊕T M )

is defined as follows: it aligns the input with the incoming vertices according with the

cyclic ordering and it labels the incoming edges (as in part (ii) of Definition 3.3.9). Then

it assigns at each vertex in V0 the commutator according with the part (i) of Definition

3.3.8. Finally it assigns the homotopy operator −H to each outgoing edge.

In particular the solution Φ of equation (3.34) can be written as a sum on labeled

ribbon k-trees as follows:

(3.37) Φ=∑

k≥1

∑

T ∈LRTk

1

2k−1tk ,T (Π, · · · , Π).

Recall that by definition

A,α∂n1

wk1m1 ,

B ,β∂n2

wk2m2∼ =

C ,γ∂⟨k2m2,n1⟩n2−⟨k1m1,n2⟩n1

wk1m1+k2m2

51

for some (A,α) ∈Ωs (U , End E⊕T M ), (B ,β )Ωr (U , End E⊕T M )and (C ,γ) ∈Ωr+s (U , End E⊕T M ), hence the Fourier mode of any labeled brackets has the same frequency me =

k1m1+k2m2 independently of the label \, [, ],?. In particular each me can be written as

me = l (a1m1+a2m2) for some primitive elements (a1, a2) ∈

Z2≥0

prim. Let us introduce

the following notation:

NOTATION 3.3.12. Let a = (a1, a2) ∈

Z2≥0

primand define ma

..= a1m1+a2m2. Then we

define Φa to be the sum over all trees of the contribution to tk ,T (Π, · · · , Π)with Fourier

mode wl ma for every l ≥ 1. In particular we define Φ(1,0)..= Π1 and Φ(0,1)

..= Π2.

It follows that the solution Φ can be written as a sum on primitive Fourier mode as

follows:

(3.38) Φ=∑

a∈

Z2≥0

prim

Φa .

As an example, let us consider Φ(2). From equation (3.34) we get

Φ(2) = Π(2)−1

2H

Π(1), Π(1)∼

and the possible 2-trees T ∈LRT2, up to choice of the initial Fourier modes, are repre-

sented in figure 3.2. Hence

vT\

Π(1) Π(1)

Π(1), Π(1)\

•

• •

•

−H

vT]

Π(1) Π(1)

Π(1), Π(1)]

•

• •

•

−H

vT[

Π(1) Π(1)

Π(1), Π(1)[

•

• •

•

−H

vT?

Π(1) Π(1)

Π(1), Π(1)?

•

• •

•

−H

FIGURE 3.2. 2-trees labeled ribbon trees, which contribute to the solu-

tion Φ.

52

Φ(2) = Π(2)1 + Π(2)2 −

1

2H

Π(1), Π(1)\+ Π(1), Π(1)]+ Π(1), Π(1)[+ Π(1), Π(1)?

=Φ(2)(1,0)+Φ(2)(0,1)+

−

a1,k1A1,k2

⟨n1, k2m2⟩−a1,k2A1,k1

⟨n2, k1m1⟩+ [A1,k1, A1,k2

]

, a1,k1a1,k2

∂⟨k2m2,n1⟩n2−⟨k1m1,n2⟩n1

·

·H(δm1∧δm2

wk1m1+k2m2 )

+

a1,k2A1,k1

,−a1,k2a1,k1

∂n1

H

iħhnq2

∂ δm1

∂ xq∧δm2

wk1m1+k2m2

+

a1,k1A1,k2

, a1,k1a1,k2

∂n2

H

iħhnq1

∂ δm2

∂ xq∧δm1

wk1m1+k2m2

+ iħh

a1,k1A1,k2

nq1 H

Aq (φ)wk1m1+m2δm1

∧δm2

−a1,k2A1n

q2 H

Aq (φ)wk1m1+k2m2δm1

∧δm2

, 0

By Remark 3.3.5

H(δm1∧δm2

wk1m1+k2m2 ) = f (ħh , uma)wl maδma

and f (ħh , uma) ∈Ol o c (1), for k1m1+k2m2 = l ma . Analogously

H

Aq (φ)wk1m1+k2m2δm1

∧δm2

= f (ħh , A(φ), uma)wl maδma

and f (ħh , A(φ), uma) ∈Ol o c (1). Then

H

ħh∂ δm1

∂ xq∧δm2

wk1m1+k2m2

=wl ma f (ħh , uma)δma

and f (ħh , uma) ∈Ol o c (ħh 1/2). This shows that every term in the sum above, is a function

of some order in ħh times a delta supported along a ray of slope m(a1,a2) = a1m1+a2m2.

For any given a ∈

Z2≥0

prim, these contributions are by definition Φ(2)(a1,a2)

, hence

Φ(2) =Φ(2)(1,0)+∑

a∈(Z2≥0)prim

Φ(2)(a1,a2)+Φ(2)(0,1).

In general, the expression of Φa will be much more complicated, but as a consequence

of the definition of H it always contains a delta supported on a ray of slope ma .

3.3.2 From solutions of Maurer-Cartan to the consistent diagram D∞

Let us first introduce the following notation:

NOTATION 3.3.13. Let A ..=U \ ξ0 be an annulus and let A be the universal cover of A

with projection$: A→ A. Then let us denote by Φ the pullback of Φ by$, in particular

by the decomposition in its primitive Fourier mode Φ=∑

a∈(Z2≥0)$

∗(Φa ) ..=∑

a∈(Z2≥0) Φa .

NOTATION 3.3.14. We introduce polar coordinates in ξ0, centred in ξ0 = dm1∪dm2

, de-

noted by (r,ϑ) and we fix a reference angle ϑ0 such that the ray with slope ϑ0 trough ξ0

contains the base point q0 (see Figure 3.3).

53

0

q0•

ϑ0

FIGURE 3.3. The reference angle ϑ0.

Then for every a ∈

Z2≥0

primwe associate to the ray dma

..= ξ0 +R≥0ma an angle

ϑa ∈ (ϑ0,ϑ0+2π). We identify dma∩A with its lifting in A and by abuse of notation we

will denote it by dma. We finally define A0

..= (r,ϑ)|ϑ0−ε0 <ϑ <ϑ0+2π, for some small

positive ε0.

Lemma 3.3.15 (see Lemma 5.40 [CCLM17a]). Let Φ be a solution of equation (3.34)

which has been decomposed as a sum over primitive Fourier mode, as in (3.38). Then

for any a ∈

Z2≥0

prim, Φa is a solution of the Maurer-Cartan equation in A0, up to

higher order in ħh , namely Φa , Φa ′∼ ∈W−∞2 (A0, End E ⊕T M )wk ma+k ′ma ′ and dW Φa ∈

W−∞2 (A0, End E ⊕T M )wk ma for some k , k ′ ≥ 1.

PROOF. Recall that Φ is a solution of Maurer-Cartan dW Φ+ Φ,Φ∼ = 0, hence its

pullback$∗(Φ) is such that∑

a∈

Z2≥0

prim

dW Φa +∑

a ,a ′∈

Z2≥0

prim

Φa , Φa ′∼ = 0. Looking at

the bracket there are two possibilities: first of all, if a 6= a ′ then Φa , Φa ′∼ is proportional

to δma∧δma ′w

k m+k ′m ′. Since dma

∩ dm ′a∩ A = ∅ then δma

∧δma ′ ∈W−∞2 (A0), indeed

writingδma∧δma ′ in polar coordinates it is a 2-form with coefficient a Gaussian function

in two variables centred in ξ0 6∈ A0. Hence it is bounded by e −cVħh in the open subset

V ⊂ A0. Secondly, if a = a ′ then by definition Φa , Φa ∼ = 0. Finally, by the fact that

dW Φa =−∑

a ′,a ′′Φa ′ , Φa ′′∼ it follows that dW Φa ∈W−∞2 (A0)wk ma for some k ≥ 1.

Now recall that the homotopy operator we have defined in Section 3.2 gives a gauge

fixing condition, hence for every a ∈

Z2≥0

primthere exists a unique gauge ϕa such

that e ϕa ∗0= Φa and p (ϕa ) = 0. To be more precise we should consider p ..=$∗(p ) as

gauge fixing condition and similarly ι ..=$∗(ι) and H ..=$∗(H ) as homotopy operator,

however if we consider affine coordinates on A, these operators are equal to p , ι and H

respectively. In addition in affine coordinates on A the solution Φa is also equal to Φa .

Therefore in the following computations we will always use the original operators and

the affine coordinates on A. We compute the asymptotic behaviour of the gauge ϕa in

the following theorem:

54

Theorem 3.3.16. Let ϕa ∈Ω0(A0, End E ⊕T M ) be the unique gauge such that p (ϕa ) = 0

and e ϕa ∗0= Φa . Then the asymptotic behaviour of ϕa is

ϕ(s )a ∈

∑

l

Bl , bl ∂na

t swl ma +⊕

l≥1W00 (A0, End E ⊕T M )wl ma on Hma ,+

⊕

l≥1W−∞0 (A0, End E ⊕T M )wl ma on Hma ,−

for every s ≥ 0, where

Bl , bl ∂na

wl ma ∈ h.

REMARK 3.3.17. Notice that, from Theorem 3.3.16 the gauge ϕa is asymptotically an

element of the DGLA h. Hence the saturated scattering diagram (see Definition 3.3.20)

is strictly contained in the mirror DGLA G (see Definition ??).

We first need the following lemma (for a proof see Lemma 5.27 [CCLM17a]) which

gives the explicit asymptotic behaviour of each component of the Lie bracket ·, ·∼:

Lemma 3.3.18. Let dm and dm ′ be two rays on U such that dma∩dma ′ = ξ0.

If (Awm ,αwm∂n ) ∈W s0 (U , End E ⊕T M )wm and (Bwm ′

,βwm ′∂n ′ ) ∈W r

0 (U , End E ⊕T M )wm ′

, then

H

(Aδmwm ,αδmwm∂n ), (Bδm ′wm ′,βδm ′wm ′

∂n ′ )\

∈W s+r+1dm+m ′

(U , End E ⊕T M )wm+m ′

H

(δmwm ,αδmwm∂n ), (Bδm ′wm ′,βδm ′wm ′

∂n ′ )[

∈W s+rdm+m ′

(U , End E ⊕T M )wm+m ′

H

(Aδmwm ,αδmwm∂n ), (Bδm ′wm ′,βδm ′wm ′

∂n ′ )]

∈W s+rdm+m ′

(U , End E ⊕T M )wm+m ′

H

(Aδmwm ,αδmwm∂n ), (Bδm ′wm ′,βδm ′wm ′

∂n ′ )?

∈W s+r−1dm+m ′

(U , End E ⊕T M )wm+m ′.

REMARK 3.3.19. The homotopy operators H and H are different. However it is not

a problem because the operator H produce a solution of Maurer-Cartan and not of

equation (3.34).1

PROOF. [Theorem 3.3.16] First of all recall that for every s ≥ 0

(3.39) ϕ(s+1)a =−H

Φa +∑

k≥1

adkϕs

a

k !dWϕ

sa

!(s+1)

where H is the homotopy operator defined in (3.17) with the same choice of the path %

as in (3.20). In addition as in the proof of Proposition 3.2.17,

(3.40) −H

∑

k≥1

adkϕs

a

k !dWϕ

sa

!(s+1)

∈⊕

l≥1

W00 (A0, End E ⊕T M )wl ma

1Recall that we were looking for a solutionΦof Maurer-Cartan of the formΦ= Π+Ξ and since dW (Π) = 0,

the correction term Ξ is a solution of dW Ξ = − 12 Φ,Φ∼. At this point we have introduced the homotopy

operator H in order to compute Ξ and we got Ξ=− 12 H(Φ,Φ∼).

55

hence we are left to study the asymptotic of H (Φ(s+1)a ). By definition Φ(s+1)

a is the sum

over all k ≤ s -trees such that they have outgoing vertex with label mT = l ma for some

l ≥ 1.

We claim the following:

(3.41) H (Φ(s+1)a ) ∈H (νT\ ) +

⊕

l≥1

W r0 (A0, End E ⊕T M )wl ma

for every T ∈LRTk ,0 such that tk ,T (Π, · · · , Π) has Fourier mode mT = l ma , for every k ≤ s

and for some r ≤ 0. Indeed if k = 1 the tree has only one root and there is nothing to

prove because there is no label. In particular

H (νT ) =H (Π(s+1)) =H (Φ(s+1)(1,0) + Φ

(s+1)(0,1) )

and we will explicitly compute it below. Then at k ≥ 2, every tree can be considered as a

2-tree where the incoming edges are the roots of two sub-trees T1 and T2, not necessary

in LRT0, such that their outgoing vertices look like

νT1= (Akδma ′w

k ma ′ ,αkδma ′wk ma ′∂na ′ ) ∈

⊕

k≥1

W r ′

dma ′(A0, End E ⊕T M )wk ma ′

νT2= (Bk ′′δma ′′w

k ′′ma ′′ ,βk ′′δma ′′wk ′′ma ′′∂na ′′ ) ∈

⊕

k ′′≥1

W r ′′

dma ′′(A0, End E ⊕T M )wk ′′ma ′′

where k ′ma ′ + k ′′ma ′′ = l ma . Thus it is enough to prove the claim for a 2-tree with

ingoing vertex νT1and νT2

as above. If T ∈LRT2, then νT = νT\ +νT[ +νT] +νT? and we

explicitly compute H (νT[ ), H (νT] ) and H (νT? ).

H (νT[ ) =−1

2H

H

νT1,νT2[

=−1

2H

H

(Akδma ′wk ma ′ ,αkδma ′w

k ma ′δna ′ ), (Bk ′′δma ′′wk ′′ma ′′ ,βk ′′δma ′′w

k ′′ma ′′∂na ′′ )[

=−1

2iħhH (H

βk ′′nq2

∂

∂ xq(Akδma ′′ )∧δma ′w

k ma ′+k ′′ma ′′ ,

βk ′′nq2 δma ′′ ∧

∂

∂ xq

αkδma ′

wk ma ′+k ′′ma ′′∂n1

)

=1

2iħhH (

∫ uma

0

βk ′′nq2

∂ Ak

∂ xq

e −s 2

ħhpπħh

d s

!

δmawl ma ,

∫ uma

0

βk ′′∂ αk

∂ xqn

q2

e −s 2

ħhpπħh

d s

!

δmawk ma ′+k ′′ma ′′∂na ′

)+

+1

2iħhH (

∫ uma

0

e −s 2

ħhpπħh

βk ′′nq2 Ak ,βk ′′n

q2 ∂na ′

2γq (s )

ħhd s

!

δmawl ma )

∈⊕

l≥1

W r ′+r ′′−20 (A0)w

l ma +W r ′+r ′′−10 (A0)w

l ma

56

where we assume k ′ma ′+k ′′ma ′′ = l ma and in the last step we use Lemma 3.2.15 to com-

pute the asymptotic behaviour of H (δma).We denote by γq (s ) the coordinates um⊥

awrit-

ten as functions of x q (s ). In particular, since Φ(1)(1,0) ∈⊕

k1≥1W1dm1(A0, End E ⊕T M )wk1m1

and Φ(1)(0,1) ∈⊕

k2≥1W1dm2(A0, End E ⊕T M )wk2m2 , we have H (Φ(2)(a1,a2)

) ∈⊕

l≥1W00w

l ma . The

same holds true for H (νT] ) by permuting A,α and B ,β .

Then we compute the behaviour of H (νT? ):

H (νT? ) =−1

2H

H

νT1,νT2?

=H

H

(Akδma ′wk ma ′ ,αkδma ′w

k ma ′δna ′ ), (Bk ′′δma ′′wk ′′ma ′′ ,βk ′′δma ′′w

k ′′ma ′′∂na ′′ )?

= iħhH (H

(αk nq1 Bk ′′Aq (φ)δma ′′ ∧δma ′ −n

q2 βk ′′Aq (φ)Akδma ′ ∧δma ′′ , 0)

)

= iħhH

∫ uma

0

αk nq1 Bk ′′Aq (φ)

e −s 2

ħhpπħh

d s −∫ uma

0

nq2 βk ′′Aq (φ)Ak

e −s 2

ħhpπħh

d s

δmawl ma , 0

∈⊕

l≥1

W r ′+r ′′−2dma

(A0)wl ma

where we denote by ma the primitive vector such that for some l ≥ 1 l ma = k ′ma ′ +

k ′′ma ′′ . Finally let us compute H (νT\ ): at k = 1 there is only a 1-tree, hence H (νT\ ) =

H (Φ(s+1)(1,0) + Φ

(s+1)(0,1) ) and for all k1, k2 ≥ 1

H (Φ(s+1)(1,0) ) ∈ (As+1,k1

t s+1, as+1,k1t s+1∂n1

)H (δm1wk1m1 ) +W0

dm1(A0, End E ⊕T M )wk1m1

∈ (As+1,k1t s+1, as+1,k1

t s+1∂n1)wk1m1 +W−1

dm1(A0 End E ⊕T M )wk1m1

H (Φ(s+1)(0,1) ) ∈ (As+1,k2

t , as+1,k2t ∂n2)H (δm2

wk2m2 ) +W−1dm2(A0, End E ⊕T M )wk2m2

∈ (As+1,k2t s+1, as+1,k2

t s+1∂n2)wk2m2 +W−1

dm2(A0, End E ⊕T M )wk2m2

Then every other k -tree (k ≤ s ) can be decomposed in two sub-trees T1 and T2 as above,

and we can further assume T1, T2 ∈LRT0, because if either T1 or T2 contains at least a

label different from \ then by Lemma 3.3.18 their asymptotic behaviour is of higher order

in ħh . We explicitly compute H (νT\ ) at s = 1:

H (νT\ ) =H (−1

2H

Π(1), Π(1)\

)

=−1

2H

H

Π(1)1 , Π(1)2 \+ Π(1)2 , Π(1)1 \

=−H

H

a1,k1A1,k2

⟨n1, k2m2⟩−a1,k2A1,k1

⟨n2, k1m1⟩+ [A1,k1, A1,k2

],

a1,k1a1,k2

∂⟨n1,k2m2⟩n2+⟨n2,k1m1⟩n1

t 2δm1∧δm2

wk1m1+k2m2 )

57

=−H

a1,k1A1,k2

⟨n1, k2m2⟩−a1,k2A1,k1

⟨n2, k1m1⟩+ [A1,k1, A1,k2

],

a1,k1a1,k2

∂⟨n1,k2m2⟩n2+⟨n2,k1m1⟩n1

t 2

∫ uma

0

e −s 2

ħhpħhπ

d s

!

δmawl ma

=−

a1,k1A1,k2

⟨n1, k2m2⟩−a1,k2A1,k1

⟨n2, k1m1⟩+ [A1,k1, A1,k2

],

a1,k1a1,k2

∂⟨n1,k2m2⟩n2+⟨n2,k1m1⟩n1

t 2H

∫ uma

0

e −s 2

ħhpħhπ

d s

!

δmawl ma

!

=−

a1,k1A1,k2

⟨n1, k2m2⟩−a1,k2A1,k1

⟨n2, k1m1⟩+ [A1,k1, A1,k2

],

a1,k1a1,k2

∂⟨n1,k2m2⟩n2+⟨n2,k1m1⟩n1

t 2wl ma Hl ma

∫ uma

0

e −s 2

ħhpħhπ

d s

!

δma

!

Now

wl ma Hl ma

∫ uma

0

e −s 2

ħhpħhπ

d s

!

δma

!

∈wl ma +W00 (A0)w

l ma

Therefore the leading order term of H (νT\ )with labels mT = l ma = k1m1+k2m2 at s = 1

is

a1,k1A1,k2

⟨n1, k2m2⟩−a1,k1A1,k2

⟨n2, k1m1⟩+[A1,k1, A1,k2

], a1,k1a1,k2

∂⟨n1,k2m2⟩n2+⟨n2,k1m1⟩n1

t 2wl ma

At s ≥ 2, every other k -tree T (k ≤ s ) can be decomposed in two sub-trees, say T1 and T2

such that νT\ =−H(νT1,\,νT2,\

\) +⊕

l≥1W1dma(A0)wl ma .

Notice that the leading order term of H

H(Π(1)1 , Π(1)2 \)

is the Lie bracket of

(A1,k1wk1m1 , a1,k1

wk1m1∂n1), (A1,k2

wk2m2 , a1,k2wk2m2∂n2

)h

hence the leading order term of H (νT\ ) belongs to h.

Notice that at any order in the formal parameter t , there are only a finite number

of terms which contribute to the solution Φ in the sum (3.37), hence we define the set

W(N ) as

(3.42)

W(N ) ..= a ∈

Z2≥0

prim|l ma =mT for some l ≥ 0and T ∈LRTk with 1≤ jT ≤N .

DEFINITION 3.3.20 (Scattering diagram D∞). The order N scattering diagram DN

associated to the solution Φ is

DN..=

w1,w2

∪

wa =

ma ,dma,θa

a∈W(N )

where

58

• ma = a1m1+a2m2;

• dma= ξ0+maR≥0

• log(θa ) is the leading order term of the unique gauge ϕa , as computed in Theo-

rem (3.3.16).

The scattering diagram D∞ ..= lim−→NDN .

Consistency of D∞

We are left to prove consistency of the scattering diagram D∞ associated to the

solution Φ. In order to do that we are going to use a monodromy argument (the same

approach was used in [CCLM17a]).

Let us define the following regions

A ..= (r,ϑ)|ϑ0−ε0+2π<ϑ <ϑ0+2π(3.43)

A−2π ..= (r,ϑ)|ϑ0−ε0 <ϑ <ϑ0.(3.44)

for small enough ε0 > 0, such that A−2π is away from all possible walls in D∞.

Theorem 3.3.21. Let D∞ be the scattering diagram defined in (3.3.20). Then it is con-

sistent, i.e. ΘD∞,γ = Id for any closed path γ embedded in U \ξ0, which intersects D∞

generically.

PROOF. It is enough to prove that DN is consistent for any fixed N > 0. First of all

recall that Θγ,DN=∏γ

a∈W(N )θa . Then let us prove that the following identity

(3.45)γ∏

a∈W(N )e ϕa ∗0=

∑

a∈W(N )Φa

holds true. Indeed

e ϕa ∗ e ϕa ′

∗0= e ϕa ∗

Φa ′

= Φa ′ −∑

k

adkϕa

k !

dWϕa + ϕa , Φa ′∼

.

For degree reason ϕa , Φa ′∼ = 0 and by definition

−∑

k

adkϕa

k !

dWϕa

= e ϕa ∗0= Φa .

Iterating the same procedure for more than two rays, we get the result.

Recall that ifϕ is the unique gauge such that p (ϕ) = 0 and e ϕ ∗0=Φ, then e$∗(ϕ)∗0=

$∗(Φ) on A. Hence e$∗(ϕ) ∗0=$∗(Φ) =

∑

a∈W(N )$∗(Φa ) =

∑

a∈W(N ) Φa and by equation

(3.45)

e$∗(ϕ) ∗0=

γ∏

a∈W(N )e ϕa ∗0.

59

In particular, by uniqueness of the gauge, e$∗(ϕ) =

∏γa∈W(N ) e

ϕa . Since$∗(ϕ) is defined

on all U , e$∗(ϕ) is monodromy free, i.e.

γ∏

a∈W(N )e ϕa |A =

γ∏

a∈W(N )e ϕa |A−2π.

Notice that A−2π does not contain the support of ϕa ∀a ∈

Z2≥0

prim, therefore

γ∏

a∈W(N )e ϕa |A−2π =

γ∏

a∈W(N )e 0 = Id.

60

4Relation with the wall-crossing formulas in cou-

pled 2d -4d systems

We are going to show how wall-crossing formulas in coupled 2d -4d systems, introduced

by Gaiotto, Moore and Nietzke in [GMN12], can be interpreted in the framework we were

discussing before. Let us first recall the setting for the 2d -4d WCFs:

• let Γ be a lattice, whose elements are denoted by γ;

• define an antisymmetric bilinear form ⟨·, ·⟩D : Γ ×Γ →Z, called the Dirac pairing;

• let Ω: Γ →Z be a homomorphism;

• denote by V a finite set of indices, V = i , j , k , · · · ;• define a Γ -torsor Γi , for every i ∈ V . Elements of Γi are denoted by γi and the

action of Γ on Γi is γ+γi = γi +γ;

• define another Γ -torsor Γi j..= Γi − Γ j whose elements are formal differences

γi j..= γi −γ j up to equivalence, i.e. γi j = (γi +γ)−(γ j +γ) for every γ ∈ Γ . If i = j ,

then Γi i is identified with Γ . The action of Γ on Γi j is γi j +γ= γ+γi j . Usually

it is not possible to sum elements of Γi j and Γk l , for instance γi j +γk l is well

defined only if j = k and in this case it is an element of Γi l ;

• let Z : Γ →C be a homomorphism and define its extension as an additive map

Z : qi∈V Γi →C, such that Z (γ+γi ) = Z (γ) +Z (γi ). In particular, by additivity

Z is a map from qi , j∈VΓi j to C, namely Z (γi j ) = Z (γi )−Z (γ j ). The map Z is

usually called the central charge;

• letσ(a , b ) ∈ ±1 be a twisting function defined whenever a + b is defined for

a , b ∈ Γ tqi Γi tqi 6= j Γi j and valued in ±1. Moreover it satisfies the following

61

conditions:

(i) σ(a , b + c )σ(b , c ) =σ(a , b )σ(a + b , c )

(ii) σ(a , b ) =σ(b , a ) if both a + b and b +a are defined

(iii) σ(γ,γ′) = (−1)⟨γ,γ′⟩D ∀γ,γ′ ∈ Γ ;

(4.1)

• let Xa denote formal variables, for every a ∈ Γ tqi Γi tqi 6= j Γi j . There is a notion

of associative product:

Xa Xb..=

σ(a , b )Xa+b if the sum a + b is defined

0 otherwise

The previous data fit well in the definition of a pointed groupoidG, as it is defined

in [GMN12]. In particular Ob(G) =V to and Mor(G) =qi , j∈Ob(G)Γi j , where the torsor

Γi is identified with Γi o and elements of Γ are identified with qi Γi i . The composition of

morphism is written as a sum, and the formal variables Xa are elements of the groupoid

ring C[G]. In this setting, BPS rays are defined as

li j..= Z (γi j )R>0

l ..= Z (γ)R>0

and they are decorated with automorphisms of C[G][[t ]] respectively of type S and of

type K , defined as follows: let Xa ∈C[G], then

(4.2) Sµγi j(Xa ) ..=

1−µ(γi j )t Xγi j

Xa

1+µ(γi j )t Xγi j

where µ: qi , j∈V Γi j →Z is a homomorphism;

(4.3) K ωγ (Xa ) ..=

1−Xγt−ω(γ,a )

Xa

whereω: Γ×qi∈VΓi →Z is a homomorphism such thatω(γ,γ′) =Ω(γ)⟨γ,γ′⟩D andω(γ, a+

b ) =ω(γ, a ) +ω(γ, b ) for a , b ∈G.

In particular under the previous assumption, the action of Sµγi j

and K ωγ can be

explicitly computed on variables Xγ and Xγkas follows:

Sµγi j: Xγ′→ Xγ′

Sµγi j: Xγk

→

Xγkif k 6= j

Xγ j−µ(γi j )t Xγi j

Xγ jif k = j

K ωγ : Xγ′→ (1− t Xγ)

−ω(γ,γ′)Xγ′

K ωγ : Xγk

→ (1− t Xγ)−ω(γ,γk )Xγk

(4.4)

62

In order to interpret the automorphisms S and K as elements of exp(h)we are going

to introduce their infinitesimal generators. Let Der (C[G]) be the Lie algebra of the

derivations of C[G] and define:

(4.5) dγi j..= adXγi j

where dγi j(Xa ) ..=

Xγi jXa −Xa Xγi j

, for every Xa ∈C[G];

(4.6) dγ..=ω(γ, ·)Xγ

where dγ(Xa ) ..=

ω(γ, a )XγXa

, for every Xa ∈C[G].

DEFINITION 4.0.1. Let LΓ be the C[Γ ]- module generated by dγi jand dγ, for every

i 6= j ∈V , γ ∈ Γ .

For instance a generic element of LΓ is given by

∑

i , j∈V

∑

l≥1

c(γi j )l Xal

dγi j+∑

γ∈Γ

∑

l≥1

c(γ)l Xal

dγ

where c (•)l Xal∈C[Γ ].

Lemma 4.0.2. Let LΓ be the C[Γ ]-module defined above. Then, it is a Lie ring1 with the

the Lie bracket [·, ·]Der(C[G]) induced by Der(C[G])2.

PROOF. It is enough to prove that LΓ is a Lie sub-algebra of Der(C[G]), i.e. it is closed

under [·, ·]Der(C[G]). By C-linearity it is enough to prove the following claims:

(1) [Xγdγi j, Xγ′dγk l

] ∈ LΓ : indeed

[Xγdγi j, Xγ′dγk l

] = Xγadγi j(X ′γadγk l

)−Xγ′adγk l(Xγadγi j

)

= XγXγ′adγi j(adγk l

)−Xγ′Xγadγk l(adγi j

)

=σ(γi j ,γk l )XγXγ′adγi j+γk l−σ(γk l ,γi j )XγXγ′adγk l+γi j

;

1A Lie ring LΓ is an abelian group (L ,+)with a bilinear form [, ]: : L × L→ L such that

(1) [a , b + c ] = [a , b ] + [a , c ] and [a + b , c ] = [a , c ] + [b , c ];

(2) [·, ·] is antisymmetric, i.e. [a , b ] =−[b , a ];

(3) [·, ·] satisfy the Jacobi identity.

2LΓ is not a Lie algebra over C[Γ ] because the bracket induced from Der(C[G]) is not C[Γ ]−linear.

63

(2) [Xγdγi j, Xγ′dγ′′ ] ∈ LΓ :indeed for every Xa ∈C[G]

[Xγdγi j, Xγ′dγ′′ ]Xa = Xγdγi j

Xγ′dγ′′Xa

−Xγ′dγ′′

Xγdγi jXa

= Xγdγi j

Xγ′ω(γ′′, a )Xγ′′Xa

−Xγ′dγ′′

XγXγi jXa −XγXa Xγi j

=ω(γ′′, a )Xγ

Xγi jXγ′Xγ′′Xa −Xγ′Xγ′′Xa Xγi j

−Xγ′ω(γ′′,γ+γi j +a )Xγ′′XγXγi j

Xa +Xγ′ω(γ′′,γ+a +γi j )Xγ′′XγXa Xγi j

=

ω(γ′′, a )−ω(γ′′,γ+γi j )−ω(γ′′, a )

XγXγ′Xγ′′Xγi jXa

−

ω(γ′′, a )−ω(γ′′, a )−ω(γ′′,γ+γi j )

XγXγ′Xγ′′Xa Xγi j

=−ω(γ′′,γ+γi j )XγXγ′Xγ′′dγi j(Xa );

(3) [Xγdγ′ , Xγ′′dγ′′′ ] ∈ L ;indeed for every Xa ∈C[G]

[Xγdγ′ , Xγ′′dγ′′′ ]Xa = Xγdγ′

Xγ′′ω(γ′′′, a )Xγ′′′Xa

−Xγ′′dγ′′′

ω(γ′, a )Xγ′Xa

=ω(γ′′′, a )ω(γ′,γ′′′+γ′′+a )XγXγ′Xγ′′Xγ′′′Xa+

−ω(γ′, a )ω(γ′′′,γ+γ′+a )Xγ′′Xγ′′′XγXγ′Xa

=

ω(γ′′′, a )

ω(γ′, a ) +ω(γ′,γ′′′+γ′′)

XγXγ′Xγ′′Xγ′′′Xa+

−

ω(γ′, a )

ω(γ′′′, a ) +ω(γ′′′,γ+γ′)

XγXγ′Xγ′′Xγ′′′Xa

=ω(γ′,γ′′′+γ′′)Xγ′Xγ′′Xγdγ′′′ (Xa )−ω(γ′′′,γ+γ′)Xγ′′′Xγ′′Xγdγ′ (Xa ).

We can now define the infinitesimal generators of Sµγi j

and K ωγ as elements of LΓ : we

first define

(4.7) sγi j..=−µ(γi j )t dγi j

then exp(sγi j) = S

µγi j

, indeed

exp(sγi j)(Xa ) =

∑

k≥0

1

k !skγi j(Xa )

=∑

k≥0

(−1)k

k !t kµ(γi j )

kadkXγi j(Xa )

= Xa −µ(γi j )t adγi j(Xa ) +

1

2t 2µ(γi j )

2ad2γi j(Xa ),

where adγi j(Xa ) = Xγi j

Xa −Xa Xγi j. Hence

ad2γi j(Xa ) =−2Xγi j

Xa Xγi j− t 2Xγi j

Xa Xγi j

64

and since γi j can not be composed with γi j +a +γi j , then ad3γi j(Xa ) = 0. Moreover

if a ∈ Γ then adγi jXa = 0, while if a = γo k then ad2Xa = 0 and we recover the formulas

(4.4).

Then we define

(4.8) kγ..=∑

l≥1

1

lt l X (l−1)

γ dγ

and we claim exp(kγ) = K ωγ , indeed

exp(kγ)(Xa ) =∑

k≥0

1

k !kkγ (Xa )

=∑

k≥0

1

k !

∑

lk≥1

1

lkt lk X lk

γ ω(γ, ·)

· · ·

∑

l2≥1

t l21

l2X l2γ ω(γ, ·)

∑

l1≥1

1

l1t l1ω(γ, a )X l1γXa

!!

· · ·

!!

=∑

k≥0

1

k !

∑

lk≥1

1

lkt lk X lk

γ ω(γ, ·)

· · ·

∑

l2≥1

∑

l1≥1

1

l1l2t l1+l2ω(γ, l1γ+a )ω(γ, a )X l2

γ X l1γ Xa

!

· · ·

!!

=∑

k≥0

1

k !

∑

lk≥1

1

lkt lk X lk

γ ω(γ, ·)

· · ·

∑

l2≥1

∑

l1≥1

1

l1l2t l1+l2ω(γ, a )ω(γ, a )X l2+l1

γ Xa

!

· · ·

!!

=∑

k≥0

1

k !ω(γ+a )k

∑

l≥1

1

lt l X l

γ

k

Xa

= exp

−ω(γ, a ) log(1− t Xγ)

Xa .

From now on we are going to assume that Γ ∼= Z2 ∼= Λ. We distinguish between

polynomial inC[Γ ] andC[Λ] by writing Xγ for a variable inC[Γ ] and wγ as a variable in

C[Λ].

REMARK 4.0.3. The group ring C[Γ ] is isomorphic to C[Λ] even if there two different

products: onC[Γ ] the product is XγXγ′ ..=σ(γ,γ′)Xγ+γ′ = (−1)⟨γ,γ′⟩D Xγ+γ′ while the prod-

uct inC[Λ] is defined by wγwγ′=wγ+γ

′. In particular the isomorphism depends on the

choice ofσ.

Let us choose an element ei j ∈ Γi j for every i 6= j ∈ V and set ei i..= 0 ∈ Γ for every

i ∈ V . Under this assumption LΓ turns out to be generated by dei jfor all i 6= j ∈ V and

by dγ for every γ ∈ Γ . Indeed every γi j ∈ Γi j can be written as ei j +γ for some γ ∈ Γ and

dγi j= dei j+γ = Xγdei j

. Then, we define an additive map

m : qi , j∈V Γi j → Γ

m (γi j ) ..= γi j − ei j

In particular, notice that m (γi i ) = γi i − ei i = γi i , hence, since Γ =qi Γi i , m (Γ ) = Γ .

65

We now define a C[Γ ]-module in the Lie algebra h:

DEFINITION 4.0.4. Define L as theC[Λ]-module generated by lγi j..=

Ei jwm (γi j ), 0

for

every i 6= j ∈V and lγ..=

0,Ω(γ)wγ∂nγ

for every γ ∈ Γ , where Ei j ∈ gl(r ) is an elementary

matrix with all zeros and a 1 in position i j .

Lemma 4.0.5. The C[Λ]-module L is a Lie ring with respect to the Lie bracket induced

by h (see Definition (2.21)).3

PROOF. As we have already comment in the proof of Lemma 4.0.2, since the bracket

is induced by the Lie bracket [·, ·]h, we are left to prove that L is closed under [·, ·]h. In

particular by C-linearity it is enough to show the following:

(1) [wγ

Ei jwm (γi j ), 0

,wγ′

Ek l wm (γk l ), 0

] ∈ L

(2) [wγ

Ei jwm (γi j ), 0

,wγ′

0,Ω(γ)wγ′∂nγ′

] ∈ L

(3) [wγ

0,Ω(γ)wγ′t ∂nγ′

,wγ′′

0,Ω(γ′′′)wγ′′′∂nγ′′′

] ∈ L

and they are explicitly computed below:

(1) [wγ

Ei j t wm (γi j ), 0

,wγ′

Ek l t wm (γk l ), 0

] =

wγwγ′[Ei j , Ek l ]gl(n )w

m (γi j )wm (γk l ), 0

(2) [wγ

Ei jwm (γi j ), 0

,wγ′′

0,Ω(γ)wγ′∂nγ′

] =

−Ei jΩ(γ′)⟨γ+m (γi j ), nγ′⟩wm (γi j )wγwγ

′′wγ

′, 0

(3) [wγ

0,Ω(γ)wγ′∂nγ′

,wγ′′

0,Ω(γ′′′)wγ′′′∂nγ′′′

] =

0,Ω(γ)Ω(γ′′′)wγwγ′wγ

′′wγ

′′′·

·

⟨γ′′+γ′′′, nγ′⟩∂nγ′′′ −⟨γ+γ′, nγ′′′⟩∂nγ′

.

Theorem 4.0.6. Let

LΓ , [·, ·]Der(C[G])

and

L, [·, ·]h

be the C[Γ ]-modules defined before.

Assumeω(γ, a ) =Ω(γ)⟨a , nγ⟩, then there exists a homomorphism of C[Γ ]-modules and

of Lie rings Υ : LΓ → L, which is defined as follows:

Υ (Xγdγi j) ..=wγ

−Ei jwm (γi j ), 0

,∀i 6= j ∈V ,∀γ ∈ Γ ;

Υ (Xγ′dγ) ..=wγ′

0,Ω(γ)wγ∂nγ

,∀γ′,γ ∈ Γ(4.9)

REMARK 4.0.7. The assumption onω is compatible with its Definition (4.3), indeed

by linearity of the pairing ⟨·, ·⟩,

ω(γ, a + b ) =Ω(γ)⟨a + b , nγ⟩=Ω(γ)⟨a , nγ⟩+Ω(γ)⟨b , nγ⟩=ω(γ, a ) +ω(γ, b ).

Moreover notice that by the assumption onω, LΓ turns out to be the C[Γ ]-module gen-

erated by dei jfor every i 6= j ∈ V and by dγ for every primitive γ ∈ Γ . Indeed if γ′ is not

primitive, then there exists a γ ∈ Γprim such that γ′ = kγ. Hence dkγ =ω(kγ, ·)X (k−1)γ Xγ =

3L is not a Lie sub-algebra of h because the Lie bracket is not C[Λ]−linear.

66

C X (k−1)γdγ, where C = kΩ(kγ)Ω(γ) . In particular, if γ,γ′ are primitive vectors in Γ , then

ω(γ,γ′) =Ω(γ)⟨γ′, nγ⟩=Ω(γ)⟨γ,γ′⟩D .

PROOF. We have to prove that Υ preserves the Lie-bracket, i.e. that for every l1, l2 ∈ L ,

then Υ

[l1, l2]LΓ

= [Υ (l1),Υ (l2)]L. In particular, by C-linearity it is enough to prove the

following identities:

(1)Υ

Xγdγi j, Xγ′dγk l

LΓ

=

Υ (Xγdγi j),Υ (Xγ′dγk l

)

L

(2)Υ

Xγdγi j, Xγ′dγ′′

LΓ

=

Υ (Xγdγ′ ),Υ (Xγ′′dγk l)

L

(3)Υ

Xγdγ′ , Xγ′′dγ′′′

LΓ

=

Υ (Xγdγ′ ),Υ (Xγ′′dγ′′′ )

L .

The identity (1) is proved below:

LHS= Υ

XγXγ′σ(γi j ,γk l )adγi j+γk l−XγXγ′σ(γk l ,γi j )adγk l+γi j

=wγwγ′

Ei j Ek l wm (γi j )wm (γk l )−Ek l Ei jw

m (γk l )wm (γi j ), 0

=

wγwγ′[Ei jw

m (γi j ), Ek l wm (γk l )], 0

RHS=

wγEi jwm (γi j ), 0

,

wγ′Ek l w

m (γk l ), 0

h

=

wγwγ′[Ei jw

m (γi j ), Ek l wm (γk l )]gl(n ), 0

=

wγwγ′[Ei jw

m (γi j ), Ek l wm (γk l )], 0

.

Then the second identity can be proved as follows:

LHS= Υ

ω(γ′′,γ+γi j )Xγ+γ′+γ′′dγi j

=−ω(γ′′,γ+γi j )wγwγ

′wγ

′′ Ei jw

m (γi j ), 0

RHS=

wγEi jwm (γi j ), 0

,

0,wγ′Ω(γ′′)wγ

′′∂nγ′′

h

=−

wγwγ′Ei jΩ(γ

′′)⟨m (γi j ) +γ, nγ′′⟩wm (γi j )wγ′′, 0

.

Finally the third identity is proved below:

LHS= Υ

ω(γ′,γ′′+γ′′′)XγXγ′′Xγ′′′dγ′′′ −ω(γ′′′,γ+γ′)Xγ′′′Xγ′′Xγdγ′

=

0,ω(γ′,γ′′+γ′′′)wγwγ′wγ

′′Ω(γ′′′)wγ

′′′∂nγ′′′ −ω(γ

′′′,γ+γ′)wγ′′′wγ

′′wγΩ(γ′)wγ

′∂nγ′

RHS=

0,wγΩ(γ′)wγ′∂nγ′

,

0,wγ′′Ω(γ′′′)wγ

′′′∂nγ′′′

h

=

0,Ω(γ′)Ω(γ′′′)

wγwγ′∂nγ′ ,w

γ′′wγ′′′∂nγ′′′

h

=

0,Ω(γ′)Ω(γ′′′)wγwγ′wγ

′′wγ

′′′

⟨γ′′+γ′′′, nγ′⟩∂nγ′′′ −⟨γ+γ′, nγ′′′⟩∂nγ′

.

67

Let us now show which is the correspondence between WCFs in coupled 2d -4d

systems and scattering diagrams which come from solutions of the Maurer-Cartan

equation for deformations of holomorphic pairs:

(1) to every BPS ray la = Z (a )R>0 we associate a ray da =m (a )R>0 if either µ(a ) 6=µ(a )′ orω(a , ·) 6=ω(a , ·)′. Conversely we associate a line da =m (a )R;

(2) to the automorphism Sµγi j

we associate an automorphism θS ∈ exp(h) such that

log(θS ) = Υ (sγi j) =

−µ(γi j )t Ei jwm (γi j ), 0

;

(3) to the automorphism K ωγ we associate an automorphism θK ∈ exp(h) such that

log(θK ) = Υ (kγ) =

0,Ω(γ)∑

l1l t l wl γ∂nγ

.

REMARK 4.0.8. If m (γi j ) =m (γi l +γl j ) then log(θS ) = Υ (sγi j) = Υ (sγi l

)Υ (sγl j). Analo-

gously if m (γ′i j ) =m (γi j ) +kγ then log(θ ′S ) = Υ (sγ′i j) = t kΥ (sγi j

).

In the following examples we will show this correspondence in practice: we consider

two examples of WCFs computed in [GMN12] and we construct the corresponding

consistent scattering diagram.

4.0.1 Example 1

Let V = i , j , k = l and set γk k = γ ∈ Γ . Assumeω(γ,γi j ) =−1 and µ(γi j ) = 1, then

the wall-crossing formula (equation 2.39 in [GMN12]) is

(4.10) K ωγ Sµγi j

= Sµ′

γi jSµ′

γi j+γK ω′

γ

with µ′(γi j ) = 1, µ′(γ+γi j ) =−1 andω′ =ω.

Since µ′(γi j ) = µ(γi j ) and ω′ = ω the initial scattering diagram has two lines. In

addition, since −1=ω(γ,γi j ) =Ω(γ)⟨m (γi j ), nγ⟩, we can assume Ω(γ) = 1, m (γi j ) = (1, 0)

and γ= (0, 1). Therefore the initial scattering diagram is

D=

wS =

mS =m (γi j ),dS ,θS

,wK =

mK = γ,dK ,θK

where logθS =

−t Ei jwm (γi j ), 0

and logθK =

0,∑

l≥11l t l wl γ∂nγ

. Then the wall crossing

formula says that the complete scattering diagram D∞ has one more S-ray, dS+K =

(γ+m (γi j ))R≥0 and wall-crossing factor logθS+K =

t 2Ei jwγ+m (γi j ), 0

.

We can check that D∞ is consistent (see Definition 2.2.4). In particular we need to

prove the following identity:

(4.11) θK θS θK −1 = θS θS+K

68

0 θS

θK

θ−1K

θ−1S

θS+K

FIGURE 4.1. The complete scattering diagram with K and S rays.

RHS= θS θS+K

= exp(logθS ) exp(logθS+K )

= exp(logθS•BCH logθS+K )

= exp

logθS + logθS+K

LHS= θK θS θK −1

= exp(logθK •BCH logθS •BCH logθK −1 )

= exp

logθS +∑

l≥1

1

l !adl

logθKlogθS

= exp

logθS + [logθK , logθs ] +∑

l≥2

1

l !adl

logθKlogθS

= exp

logθS −

Ei j

∑

k≥1

1

kwm (γi j )+kγ⟨m (γi j ), nγ⟩, 0

+∑

l≥2

1

l !adl

logθKlogθS

= exp

logθS +

t 2Ei jwm (γi j )+γ, 0

+

Ei j

∑

k≥2

1

kt k+1wm (γi j )+kγ, 0

+∑

l≥2

1

l !adl

logθKlogθS

.

We claim that

(4.12) −

Ei j

∑

k≥2

1

kt k+1wm (γi j )+kγ, 0

=∑

l≥2

1

l !adl

logθKlogθS

69

and we compute it explicitly:

∑

l≥2

1

l !adl

logθKlogθS =

∑

l≥2

1

l !

−t Ei j (−1)l∑

k1,··· ,kl≥1

1

k1 · · ·klt k1+···+kl w(k1+···+kl )γ+m (γi j ), 0

!

=−∑

l≥2

(−1)l

l !

t Ei jwm (γi j )

∑

k1,··· ,kl≥1

1

k1 · · ·klt k1+···+kl w(k1+···+kl )γ

!

, 0

!

=−∑

l≥2

(−1)l

l !

t Ei jwm (γi j )

∑

k≥1

1

kt kwkγ

l

, 0

!

=−

t Ei jwm (γi j )

∑

l≥2

(−1)l

l !

∑

k≥1

1

kt kwkγ

l

, 0

!

=−

t Ei jwm (γi j )

exp

−∑

k≥1

1

kt kwkγ

+∑

k≥1

1

kt kwkγ−1

, 0

=−

t Ei jwm (γi j )

1− t wγ

+∑

k≥1

1

kt kwkγ−1

, 0

=−

t Ei jwm (γi j )

∑

k≥2

1

kt kwkγ, 0

.

(4.13)

4.0.2 Example 2

Finally let us give a example with only S-rays: assume V = i = l , j = k , then the

wall-crossing formula (equation (2.35) in [GMN12]) is

(4.14) Sµγi jSµγi l

Sµγ j l= Sµγ j l

Sµ′

γi lSµγi j

with γi l..= γi j+γ j l andµ′(γi l ) =µ(γi l )−µ(γi j )µ(γ j l ). Let us further assume thatµ(i l ) = 0,

then the associated initial scattering has two lines:

D=

w1 =

m1 =m (γi j ),d1 =m1+R,θS1

,w2 =

m2 =m (γ j k ),d2 =m2R,θS2

with

logθS1=−

µ(γi j )t Ei jwm (γi j ), 0

logθS2=−

µ(γ j l )t E j l wm (γ j l ), 0

.

Its completion has one more rayd′3 = (m1+m2)R≥0 decorated with the automorphism

θ ′3 such that

logθ ′3 =

µ(γi j )µ(γ j l )t2Ei l w

m (γi l ), 0

.

Since the path order product involves matrix commutators, the consistency of D∞

can be easily verified.

70

0

θS1

θS2

θ−1S2

θ−1S1

θ ′S3

FIGURE 4.2. The complete scattering diagram with only S rays.

REMARK 4.0.9. In the latter example we assume µ(γi l ) = 0 in order to have an initial

scattering diagram with only two rays, as in our construction of solution of Maurer–

Cartan equation in Section 3.3. However the general formula can be computed with the

same rules, by adding the wall

w=

−(m1+m2), logθ3 =

−µ(γi l )t2Ei l w(γi l )

, 0

to the initial scattering diagram.

71

5Gromov–Witten invariants in the extended trop-

ical vertex

In this chapter all scattering diagrams are defined in the extended tropical vertex group

V. We start with some preliminaries, following the approach of [GPS10].

NOTATION 5.0.1. We denote gl(r,C)e ⊂ gl(r,C) the subset of elementary matrices in

gl(r,C), namely the matrices with only a non zero entry in position i j for i < j .

Deformation technique: let D =¦

(di =miR,−→f i )|i = 1, ..., n

©

be a scattering dia-

gram which consists of walls through the origin, as in figure 2.2. Assume each−→f i can be

factored as−→f i =

∏

j

−→f i j =

∏

j

(1+Ai j z m ′i ), fi j

. Then replace each line (di ,−→f i )with

the collection of lines (ξi j +di ,−→f i j ), where ξi j ∈ΛR is chosen generically. This new

scattering diagram is denoted by D.

As an example let us consider the scattering diagram:

D=

(1, 0)R, (1, (1+u11 x )) (1, (1+u12 x ))

1, (1−u11u12 x 2)

,

(0, 1)R,

(1+Au21 y ), (1+u21 y )

(1+Au22 y ), (1+u22 y )

1, (1−u21u22 y 2)

where A ∈ gle (r,C). Then if we apply the deformation techniques we get the following

diagram of Figure 5.1.

DEFINITION 5.0.2. Let D be a scattering diagram, then the asymptotic scattering dia-

gram Da s is obtained by replacing each ray (ξi +miR≥0,−→f i )with (R≥0mi ,

−→f i ) and every

line (ξi +miR,−→f i )with a parallel line through the origin (miR,

−→f i ). If two lines/rays in

D have the same slope given by the vector mi and functions−→f i ,−→f i ′ , then the function

−→f attached to the line/ray through the origin is the product of the

−→f i and

−→f i ′ .

NOTATION 5.0.3. Let m1, m2 be two primitive vectors inΛwith coordinates m1 = (m11, m12)

and m2 = (m21, m22). Let also choose the anticlockwise orientation on ΛR and assume

72

(1+Au21 y ), (1+u21 y )

(1+Au22 y ), (1+u22 y )

1, (1−u21 u22 y 2)

(1, (1+u11 x ))

(1, (1+u12 x ))

1, (1−u11 u12 x 2)

FIGURE 5.1. Deformed scattering diagram

m1 and m2 are positive oriented. We denote by

|m1 ∧m2|= |m11m22−m12m21|(5.1)

Moreover, let ni = (−mi 2, mi 1) be the primitive vector orthogonal to mi and positive

oriented, then

⟨m2, n1⟩=−⟨m1, n2⟩= |m1 ∧m2|

where ⟨−,−⟩ is the pairing between Λ and Λ∗. Finally if m ′ =w m for some primitive m

and some integer w , then w is called index of m ′.

NOTATION 5.0.4. Let d1,d2 be two rays (or lines) in Dk such that d1 ∩d2 6= ; and let dout a

third ray emanating from d1 and d2. Then we define

(5.2) Parents(dout) =

d1,d2 if d1,d2 are rays

; otherwise

in which case dout ∈Child(d1), Child(d2);

(5.3) Ancestors(dout) = dout∪⋃

d′∈Parents (dout)

Ancestors(d′)

and

(5.4) Leaves(dout) = d′ ∈Ancestor (dout)|d′ is a line.

DEFINITION 5.0.5. A standard scattering diagram D= wi = (di ,θi ) , 1≤ i ≤ n over

R = C[[t1, ..., tn ]], consists of finite collection of lines di =miR through the origin and

73

automorphisms θi such that logθi ∈ C[z mi ][[ti ]] ·

gl(r,C),∂ni

. We allow m j =m j ′ for

j 6= j ′, but we keep t j 6= t j ′ .

Given a standard scattering diagram, there is a method to compute its completion

iteratively over the ring RN = C[t1, ..., tn ]/(t N+11 , ..., t N+1

n ). We start with a preliminary

lemma

Lemma 5.0.6. Let

D=¦

m1R,−→f 1 =

1+A1t1z l1m1 , 1+ c1t1z l1m1

,−→f 2 =

m2R,

1+A2t2z l2m2 , 1+ c2t2z l2m2

©

with A1, A2 ∈ gl(r,C), m1, m2 primitive vector in Λ, l1, l2 ∈ Z>0 and c1, c2 ∈C. Then the

consistent scattering diagram D∞ on R2 is obtained by adding a single wall

wout =

dout =R≥0(l1m1+ l2m2),−→f out

with function

−→f out =

1+ ([A1, A2] +A2c1l2|m1 ∧m2|+A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

1+ c1c2t1t2lout|m1 ∧m2|z l1m1+l2m2

(5.5)

where lout is the index of l1m1+l2m2. If mout = 0, then no wall is added since |m1∧m2|= 0

and Ai commutes with itself.

PROOF. Let γ be a generic loop around the origin, such that the path order product

Θγ,D∞ = θ−12 θ

−11 θ2θ1, where

logθ1 = (

A1t1z l1m1 , c1t1z l1m1∂n1

logθ2 =

A2t2z l2m2 , c2t2z l2m2∂n2

.

We now compute θ−12 θ

−11 θ2θ1: we begin with θ−1

1 θ2θ1

θ−11 θ2θ1 = exp

− logθ1 • logθ2 • logθ1

= exp(

A2t2z l2m2 , c2t2z l2m2∂n2

+∑

l≥1

1

l !adl− logθ1

A2t2z l2m2 , c2t2z l2m2∂n2

)

74

= exp(

A2t2z l2m2 , c2t2z l2m2∂n2

+

+ ((−[A1, A2]−A2c1l2|m1 ∧m2| −A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

− c1c2t1t2wout|m1 ∧m2|z l1m1+l2m2 )+

+∑

l≥2

1

l !adl(− logθ1)

A2t2z l2m2 , c2t2z l2m2∂n2

))

exp

A2t2z l2m2 , c2t2z l2m2∂n2

+

+

(−[A1, A2]−A2c1l2|m1 ∧m2| −A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

− c1c2t1t2lout|m1 ∧m2|z l1m1+l2m2∂nout

where in the last step∑

l≥21l !ad

l(− logθ1)

A2t2z l2m2 , c2t2z l2m2∂n2

vanish on R2. Then

θ−12 θ

−11 θ2θ1 = exp

− logθ2 •

A2t2z l2m2 , c2t2z l2m2∂n2

+

+

(−[A1, A2]−A2c1l2|m1 ∧m2| −A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

− c1c2t1t2lout|m1 ∧m2|z l1m1+l2m2∂nout

exp

−

A2t2z l2m2 , c2t2z l w2m2∂n2

+

A2t2z l2m2 , c2t2z l2m2∂n2

+

−

([A1, A2] +A2c1l2|m1 ∧m2|+A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

c1c2t1t2lout|m1 ∧m2|z l1m1+l2m2∂nout

= exp

−

([A1, A2] +A2c1l2|m1 ∧m2|+A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

c1c2t1t2lout|m1 ∧m2|z l1m1+l2m2∂nout

where we omit the other terms in the BCH product because they vanish on R2. Hence

the automorphism of the new wall is

logθout =

([A1, A2] +A2c1l2|m1 ∧m2|+A1c2l1|m1 ∧m2|)t1t2z l1m1+l2m2 ,

c1c2t1t2lout|m1 ∧m2|z l1m1+l2m2∂nout

(5.6)

and the corresponding function−→f out is as (5.5).

Let us introduce the ring RN

(5.7) RN =C[ui j |1≤ i ≤ n , 1≤ j ≤N ]⟨u 2

i j |1≤ i ≤ n , 1≤ j ≤N ⟩

75

while−→f i may not factor on RN , it does on RN by replacing ti with ti =

∑Nj=1 ui j . For

instance, let

(5.8) log−→f i =

N∑

j=1

∑

l≥1

Ai j l tj

i z l mi , ai j l tj

i z l mi

on RN

then, since

tj

i =∑

J⊂1,...N |J |= j

j !∏

j ′∈J

ui j ′

we get:

(5.9) log−→f i =

N∑

j=1

∑

l≥1

∑

J⊂1,...,N |J |= j

j !

Ai j l

∏

j ′∈J

ui j ′

!

z l mi , ai j l

∏

j ′∈J

ui j ′

!

z l mi

!

on RN

Hence we can define a new scattering diagram

DN =

(di J l ,θi J l )|1≤ i ≤ n , l ≥ 1, J ⊂ 1, ..., N , #J ≥ 1

such that for a generic choice of ξi l ∈R2, di J l = ξi l +miR is a line parallel to di and the

associated function−→f i J l is a single term in Equation (5.9), i.e.

−→f i j l =

1+ (#J )!Ai (#J )l

∏

j ′∈J

ui j ′

!

z l mi , 1+ (#J )!ai (#J )l

∏

j ′∈J

ui j ′

!

z l mi

!

.

Then we produce a sequence of scattering diagrams DN = D0N ,D1

N ,D2N , ... which even-

tually stabilizes, and D∞N = DiN for i large enough. Inductively we assume the following:

(1) each wall in DiN is of the form w=

d,−→f d

with

log−→f d =

AduI (d)zmd , aduI (d)z

md∂nd

with uI (d) =∏

(i , j ′)∈I (d) ui j ′ for some index set I (d)⊂ 1, ..., n× 1, ..., N ;(2) for each ξ ∈ Sing(Di

k ), there are no more than two rays emanating from ξ and

if d1,d2 are these rays, then I (d1)∩ I (d2) = ;.

These assumptions hold for D0N . Then to construct Di+1

N from DiN we look at every pair

of lines or rays d1,d2 ∈ DiN such that

(i) d1,d2 /∈ Di−1N ;

(ii) d1 ∩d2 = ξ and ξ 6= ∂ d1,∂ d2;(iii) I (d1)∩ I (d2) = ;.

Notice that by Lemma 5.0.6, if d1 ∩d2 = ξ and I (d1)∩ I (d2) = ;, then there is only one

ray emanating from ξ: dout = d(d1,d2) = ξ+

ld1md1+ ld2

md2

R≥0 = ξ+ ldoutmdoutR≥0 with

76

function

−→f out =

1+ ([Ad1, Ad2] +Ad2

ad1ld2|md1

∧md2|+Ad1

ad2ld1|md1

∧md2|)uI (d1)∪I (d2)z

ld1 md1+ld2 md2 ,

1+ad1ad2

uI (d1)∪I (d2)ldout|md1

∧md2|z ld1 md1+ld2 md2 )

.

(5.10)

Hence we take

(5.11) Di+1N = Di

N ∪¦

w=

d(d1,d2),−→f d

|d1,d2satsify (i)-(iii)©

.

Of course Di+1N satisfies assumption (1) with I (d) =

⋃

d′∈Parents(d) I (d′) for any d ∈

Di+1N \ Di

N . Then let ξ ∈ Sing(Di+1N ) and assume that (2) does not hold, i.e. there are at

least 3 rays from ξ. Since I (d1)∩ I (d2) = ; then Leaves(d1)∩Leaves(d1) = ; hence their

lines can be slightly moved independently. In particular by the assumption DN is generic

we can move the lines from the point ξ ∈ d1∩d2 violating assumption (2). This procedure

stops since at any step I (d) is the union of the I (di )with di ∈ Leaves(d) and the maximal

cardinality of I (d) is nN .

Let us explain how the previous construction works in practice in the following

example.

EXAMPLE 5.0.7. Let us consider the initial scattering diagram

D=

d1 = (0, 1)R,

1+At1 y , 1+ t1 y

, (d2 = (1, 0)R, (1, 1+ t2 x ))

with A ∈ gle (r,C). By the deformation techniques we get D2 = D02 as in Figure 5.1.

In order to construct D12 we consider the marked points in Figure 5.2, that are inter-

section of lines which satisfy conditions (i)-(iii). For each such point we draw a new ray

as prescribed by Lemma 5.0.6. We collect the functions−→f i associated to the new rays

below:

Slope 1 (1+Au21u11 x y , 1+u21u11 x y ), (1+Au21u12 x y , 1+u21u12 x y ), (1+Au22u11 x y , 1+

u22u11 x y ), (1+Au22u12 x y , 1+u22u12 x y ) and (1, 1+2u21u11u22u12 x 2 y 2).

Slope 1/2 (1−Au21u22u11 x 2 y , 1−u21u22u11 x 2 y )and (1−Au22u21u12 x 2 y , 1−u21u22u12 x 2 y ).

Slope 2 (1, 1−u21u11u12 x y 2) and (1, 1−u22u12u11 x y 2).

There is one step we can go further, since marked points in Figure 5.3 are the only

intersection points of either lines or rays for which conditions (i)-(iii) hold true. The

functions−→f i associate to the new rays are:

Slope 1 (1, 1−4u21u22u11u12 x 2 y 2)and (1−4Au21u22u11u12 x 2 y 2, 1−4u21u22u11u12 x 2 y 2)

Slope 1/2 (1+Au21u22u11 x 2 y , 1+u21u11u22 x 2 y )and (1+Au21u12u22 x 2 y , 1+u21u12u22 x 2 y )

Slope 2 (1+2Au21u11u12 x y 2, 1+u21u11u12 x y 2)and (1+2Au12u11u22 x y 2, 1+u12u11u22 x y 2).

77

(1+Au11 y ), (1+u11 y )

(1+Au12 y ), (1+u12 y )

1, (1−u11 u12 y 2)

(1, (1+u21 x ))

(1, (1+u22 x ))

1, (1−u21 u22 x 2)

•

•

•

•

•

•

•

•

•

FIGURE 5.2. Construction of D12. We color distinctly rays with different

slope: slope 1 are red, slope 2 are cyan and slope 1/2 are blue.

(1+Au11 y ), (1+u11 y )

(1+Au12 y ), (1+u12 y )

1, (1−u11 u12 y 2)

(1, (1+u21 x ))

(1, (1+u22 x ))

1, (1−u21 u22 x 2)

• • •

•

•

•

FIGURE 5.3. Construction of D22. We color distinctly rays with different

slope: slope 1 are red, slope 2 are cyan and slope 1/2 are blue. Last rays

added are thicker.

The last step is represented as in Figure 5.4 and it contains one more ray of slope

one and with function

(5.12)

1+4Au21u22u11u12 x 2 y 2, 1+4u11u12u21u22 x 2 y 2

.

78

(1+Au11 y ), (1+u11 y )

(1+Au12 y ), (1+u12 y )

1, (1−u11 u12 y 2)

(1, (1+u21 x ))

(1, (1+u22 x ))

1, (1−u21 u22 x 2)

•

FIGURE 5.4. Construction of D32. We color distinctly rays with different

slope: slope 1 are red, slope 2 are cyan and slope 1/2 are blue. Last ray

added is thicker.

Now if we take the product of all functions many factors cancel and we finally get

the scattering diagram D32 with two more rays:

(5.13) D32 \D=

(1, 1)R≥0, (1+At1t2 x y , 1+ t1t2 x y )

,

(1, 2)R≥0, (1+Ax y 2t2t 21 , 1)

Notice that the ray of slope 2 has only the matrix contribution, which was not there in

the tropical vertex group V (see the analogous Example 1.11 of [GPS10]).

0

FIGURE 5.5. The asymptotic scattering diagram D32.

5.1 Tropical curve count

NOTATION 5.1.1. Let Γ be a weighted, connected, finite graph without divalent vertices

and denote the set of vertices by Γ[0]

and the set of edges by Γ[1]

. The weight function

79

wΓ : Γ[1]→Z>0 assigns a weight to each edge. We denote the set of univalent vertices by

Γ[0]∞ and we define Γ = Γ \Γ [0]∞. The set of edges and vertices of Γ is denoted by Γ [0] and Γ [1]

respectively. The edges of Γ which are not compact are called unbounded and denoted

by Γ [1]∞ ⊂ Γ [1].

DEFINITION 5.1.2. A proper map h : Γ →ΛR is called a parametrized tropical curve if

it satisfies the following conditions:

(1) for every edge E ∈ Γ [1], h |E in an embedding with image h (E ) contained in an

affine line with rational slope;

(2) for every vertex V ∈ Γ [0], the following balancing condition holds true: let

E1, ..., Es ∈ Γ [1] be the adjacent edges of V , let w1, ...ws be the weights of E1, ..., Es

and let m1, ..., ms be the primitive integer vectors at the point h (V ) in the direc-

tion of h (E1), ..., h (Es ), then

(5.14)n∑

j=1

wΓ (E j )m j = 0

is the balancing condition.

Two parametrized tropical curves h : Γ →ΛR and h ′ : Γ ′→ΛR are isomorphic if there

exists a homeomorphism Φ: Γ → Γ ′ respecting the weights such that h ′ =Φ h .

DEFINITION 5.1.3. A tropical curve is an isomorphism class of parametrized tropical

curves.

DEFINITION 5.1.4. The genus of a tropical curve h : Γ →ΛR is the first Betti number

of the underling graph Γ . Genus zero tropical curves are called rational.

DEFINITION 5.1.5. Let h : Γ →ΛR be a tropical curve such that Γ has only vertices of

valency one and three. The multiplicity of a vertex V ∈ Γ [0] in h is

(5.15) MultV (h ) =w1w2|m1 ∧m2|=w1w3|m1 ∧m3|=w2w3|m2 ∧m3|

where E1, E2, E3 ∈ Γ [1] are the edges containing V with wi = wΓ (Ei ) and mi ∈ Λ is a

primitive vector in the direction h (Ei ) emanating from h (V ). The equality of three

expressions follows from the balancing condition.

DEFINITION 5.1.6. The multiplicity of a tropical curve h is

Mult(h ) =∏

V ∈Γ [0]MultV (h )

80

Theorem 5.1.7. Let DN =¦

(di J l ,−→f i J l )|1≤ i ≤ n , l ≥ 1, J ∈ 1, ..., N #J ≥ 1

©

be a scatter-

ing diagram, such that

(5.16)−→f i J l =

1+ (#J )!Ai (#J )l z l mi

∏

j∈J

ui j , 1+ (#J )!ai (#J )l l∏

j∈J

ui j z l mi

!

and assume [Ai (#J )l , Ai ′(#J ′)l ′ ] = 0 for all i , i ′ ∈ 1, ..., n.Then there is a bijective correspondence between elements in the complete scatter-

ing diagram D∞N and rational tropical curve h : Γ →ΛR such that:

(1) there is an edge Eout ∈ Γ[1]∞ with h (Eout) = d;

(2) if E ∈ Γ [1]∞\Eoutor if Eout is the only edge of Γ (in which case E = Eout), then h (E )

is contained in some di J l , where 1 ≤ i ≤ n , J ⊂ 1, ..., N and l ≥ 1. Moreover

if E 6= Eout, the unbounded direction of h (E ) is given by −mi and its weight is

wΓ (E ) = l ;

(3) if E , E ′ ∈ Γ [1]∞ \ Eout and h (E )⊂ di J l and h (E ′)⊂ di J ′l ′ , then J ∩ J ′ = ;.

If d is a ray, the corresponding curve h is trivalent and

(5.17)

−→f d =

1+Mult(h )

∑

ip Jp lp∈Leaves(d)

(#Jp )!Ai #Jp lp

∏

iq Jq lq∈Leaves(d)q 6=p

(#Jq )!aiq (#Jq )lq

∏

j∈Jq

uiq j

∏

j∈Jp

uip j

z ldmd , fd

where fd is given by

(5.18) fd = 1+ ldMult(h )

∏

di J l ∈Leaves(d)

(#J )!ai J l

∏

j ′∈J

ui j ′

!

z ldmd

and ldmd =∑n

i=1

∑

l≥1 l mi , with md primitive vector in Λ and ld =wΓ (Eout).

This theorem is a generalization of Theorem 2.4 in [GPS10] to scattering diagram

in the extended tropical vertex group V, thus in the proof we follow the same path of

[GPS10].

PROOF. Let d= di J l be a line in DN , then Γ [1]∞ is defined by a single edge Ed such that

h (Ed) has slope −mi and weight wΓ (Ed) = l . Conversely, if h : Γ →ΛR is a tropical curve

satisfying (1)− (3), which consists of a single edge Eout = Γ[1]∞, then from (2) h (Eout ⊂ di J l ,

and by assumption (di J l ,−→f i J l ) ∈ D∞N . Now let d ∈ D∞N \ DN be a ray, then we define the

graph Γ by

Γ [0] =

Vd′ |d′ ∈Ancestors(d) and d′ ray

Γ [1] =

Ed′ |d′ ∈Ancestors(d)

.

If d′ ∈Ancestors(d) then one of the following options can occur:

81

(a) d′ 6= d and d′ is a ray, then Ed′ have vertices Vd′ and VChild(d′);

(b) d= d′, then Ed′ is an unbounded edge with vertex Vd;

(c) d′ is a line, then Ed′ is an unbounded edge with vertex VChild(d′).

Thus, for any d′ ∈Ancestors(d), fd′ = 1+ cd′zld′md′ with md′ ∈Λ primitive, and we define

the weight wΓd (Ed′ ) = ld′ . The tropical curve h is defined by mapping Ed′ in

- a line of slope md′ joining Init(d′) and Init(Child(d)), if d′ in case (a);

- the line d, if d′ in case (b);

- the ray Init(Child)(d′) +R≥0md′ , if d′ in case (c).

Since Γd is trivalent, the genus zero condition is satisfied. The balancing condition

follows from equation (5.10): indeed let Parents(d) = d1,d2, then ldmd = ld1md1+ ld2

md2

and since h (Ed1), h (Ed2

) are incoming edges while h (Ed) is outgoing, we get

(5.19) wΓd (Ed)md =−wΓd (Ed1)md1

−wΓd (Ed2)md2

.

We can prove the expression (5.17) by induction: indeed if d= di J l is a line in DN ,

then Γd has only an edge Eout = Γ[1]∞, hence Mult(h ) = 1 because no trivalent vertexes

occur. The inductive step is the following: let d be a ray and assume (5.17) holds for

d1,d2 ∈ Parents(d), and let h1, h2 be the tropical curves associated respectively to d1,d2.

Then, by equation (5.10)

−→f d =

1+Mult(h1)Mult(h2)∑

dir Jr lr ∈Leaves(d1)

∑

di ′r J ′r l ′r∈

Leaves(d2)

(#Jr )!(#J ′r )!

Air (#Jr )lr, Ai ′r (#J ′r )l ′r

∏

iq Jq lq

q 6=r

(#Jq )!aiq (#Jq )lq·

·∏

i ′q (#J ′q )l′q

q ′ 6=r ′

(#J ′q )!ai ′q (#J ′q )l ′q

∏

j∈Jq

uiq j

∏

j∈Jr

uir j

∏

j ′∈J ′q

ui ′q j ′

∏

j ′∈J ′r

ui ′r j ′+

+Mult(h1)Mult(h2)

∑

dir Jr lr ∈Leaves(d2)

Air (#Jr )lr(#Jr )!

∏

iq Jq lq

q 6=r

(#Jq )!aiq (#Jq )lq

∏

j∈Jq

uiq j

∏

j∈Jr

uir j

·

·

ld1

∏

di J l ∈Leaves(d1)

(#J )!ai J l

∏

j∈J

ui j

!

ld2|md1

∧md2|z ld1 md1+ld2 md2+

+Mult(h1)Mult(h2)

∑

dir Jr lr ∈Leaves(d1)

Air (#Jr )lr(#Jr )!

∏

iq Jq lq

q 6=r

(#Jq )!aiq (#Jq )lq

∏

j∈Jq

uiq j

∏

j∈Jr

uir j

·

·

ld2

∏

di J l ∈Leaves(d2)

(#J )!ai (#J )l

∏

j∈J

ui j

!

ld1|md1

∧md2|z ld1 md1+ld2 md2 ,

82

1+ ld1Mult(h1)

∏

di J l ∈Leaves(d1)

(#J )!ai (#J )l

∏

j∈J

ui j

!

ld2Mult(h2)

∏

di J l ∈Leaves(d2)

(#J )!ai (#J )l

∏

j∈J

ui j

!

·

· ld|md1∧md2

|z ld1 md1+ld2 md2

=

1+Mult(h1)Mult(h2)MultVd(h )

∑

dir Jr lr ∈Leaves(d2)

(#Jr )!Air (#Jr )lr

∏

iq Jq lq

∈Leaves(d)q 6=r

(#Jq )!aiq (#Jq )lq

∏

j∈Jq

uiq j

∏

j∈Jr

uir j

+

+

∑

dir Jr lr ∈Leaves(d1)

(#Jr )!Air (#Jr )lr

∏

diq Jq lq ∈Leaves(d)q 6=r

(#Jq )!aiq (#Jq )lq

∏

j∈Jq

uiq j

∏

j∈Jr

uir j

z ld1 md1+ld2 md2 ,

1+Mult(h1)Mult(h2)MultVd(h )ld

∏

di J l ∈Leaves(d)

(#J )!ai (#J )l

∏

j∈J

ui j

!

z ld1 md1+ld2 md2

=

1+Mult(h )

∑

dir Jr lr ∈Leaves(d)

(#Jr )!Air (#Jr )lr

∏

diq Jq lq

q 6=r

(#Jq )!aiq (#Jq )lq

∏

j∈Jq

uiq j

∏

j∈Jr

uir j

z ld1 md1+ld2 md2 ,

1+Mult(h )lout(d)

∏

di J l ∈Leaves(d)

(#J )!ai (#J )l

∏

j∈J

ui j

!

z ld1 md1+ld2 md2

where in the second step we use that [Air (#Jr )lr, Ai ′r (#J ′r )l ′r ] vanishes by assumption.

Conversely, if h : Γ →ΛR is a tropical curve satisfying (1)− (3) and d is a ray, then by

the previous computations we have (d,−→f d) ∈ D∞N .

We are now going to introduce invariants to count tropical curves: let w= (w1, ..., ws )

be a s-tuple of non-zero vectors wi ∈ Λ and fix a set of points ξ = (ξ1, ...,ξs ). Then a

parametrized tropical curve h : Γ →ΛR of type (w,ξ) is the datum of

• Γ [1]∞ = Er |1≤ r ≤ s ∪ Eout;• h (Er ) asymptotically coincide with the ray dr = ξr −R≥0wr and wΓ (Er ) = |wr |;• h (Eout) pointing in the direction of wout

..=∑s

r=1 wr and wΓ (Eout) = |wout|.

We denote by Tw,ξ the set of tropical curves h : Γ →ΛR as above. Then we define

(5.20) N tropw

..=∑

h∈Tw,ξ

Mult(h )

as the number of tropical curve in Tw,ξ counted with multiplicity.

83

The definition is well-posed since according to to Proposition 4.13 of [Mik05] the

set Tw,ξ is finite. In addition the number N tropw does not depend on the generic choice of

the vectors ξ j .

NOTATION 5.1.8. Let P = (P1, ..., Pn ) be a n-tuple of integer, and let k = (k1, ..., kn ) be a

n-tuple of vectors such that k j = (k j l )l≥1 is a partition of Pj with∑

l≥1 l k j l = Pj . We

denote the partition k by k `P.

Given a partition k `P, we define

(5.21) s (k) ..=n∑

j=1

∑

l≥1

k j l

and the s (k)-tuple w(k) =

w1(k), ..., ws (k)(k)

of non zero vectors in Λ, such that

wr (k) ..= l m j ,

for every 1+∑ j

j ′=1

∑l−1l ′=1 kl ′ j ′ ≤ r ≤ k j l +

∑ jj ′=1

∑l−1l ′=1 kl ′ j ′ . Notice that

∑s (k)r=1 wr (k) =

ldmd.

We can now state the first result which provide a link between consistent scattering

diagrams in the extended tropical vertex group V and tropical curves count.

Theorem 5.1.9. Let D=¦

wi = (di =miR,−→f i )|1≤ i ≤ n

©

such that for every i = 1, ..., n

−→f i =

1+Ai ti z mi , 1+ ti z mi

on C[[t1, ..., tn ]] and assume [Ai , Ai ′ ] = 0 for all index i , i ′ ∈ 1, ..., n.Then for every wall

d=mdR≥0,−→f d

∈D∞ \D where md ∈Λ is a primitive non-zero

vector, the function−→f d is explicitly given by the following expression:

(5.22)

log−→f d =

∑

ld≥1

∑

P

∑

k`P

N tropw(k)

n∑

i=1

∑

l≥1

l Ali ki l

∏

1≤i≤n

∏

l ′≥1

(−1)l′−1

(l ′)2

ki j ′ 1

ki l ′ !t Pi

i

z∑

i Pi mi ,

ld

∏

1≤i≤n

∏

l≥1

(−1)l−1

l 2

ki l 1

ki l !t Pi

i

z∑

i Pi mi

where the second sum is over all n-tuple P= (P1, ..., Pn ) ∈Nn such that∑n

i=1 Pi mi =

ldmd.

PROOF. Recall that since D is a standard scattering diagram we can consider the

associate deformed diagram D, and on RN the complete scattering diagrams D∞N and

84

(D∞N )a s are equivalent. In particular DN is defined by taking the logarithmic expansion

of−→f i ∈DN and substituting t l

i =∑

Ji l ∈1,...,N #Ji l=l

l !∏

j∈Ji lui j :

−→f i Ji l l =

1+(−1)l−1

lAl

i (l )!∏

j ′∈Ji l

ui j ′zl mi , 1+ l

(−1)l−1

l 2(l )!

∏

j ′∈Ji l

ui j ′zl mi

!

and the diagram DN is

DN =¦

(di Ji l l ,−→f i Ji l l )|i = 1, ..., n , 1≤ l ≤N , Ji l ⊂ 1, ..., N , #Ji l = l

©

where di Ji l l..= ξi Ji l l −R≥mi for a generic choice ξi Ji l l . Now, for every d′ ∈ D∞N \ DN

there is a unique tropical curve h : Γ → ΛR defined as in Theorem 5.1.7. Assuming

d′ = ξ′ +md′R≥0 for some generic ξ′ ∈ ΛR and some primitive vector md′ ∈ Λ, and

wΓ (Eout) = ld′ ≥ 1, let us define P = (P1, ..., Pn ) ∈ Nn such that∑n

j=1 Pj m j = ld′md′ and

k `P, as in Notation 5.1.8. Then the function−→f d′ is

(5.23) log−→f d′ =

Mult(h )n∑

i=1

∑

l≥1

∑

Ji l

(l )!Ali

(−1)l−1

l

∏

1≤i ′≤n

∏

l ′≥1

∏

Ji ′l ′Ji ′l ′∩Ji l=;

(l ′)!(−1)l

′−1

(l ′)2

·

·

∏

j ′∈Ji ′l ′

ui ′ j ′

∏

j∈J1l

ui j

!

z∑

i Pi mi , ldMult(h )

∏

1≤i≤n

∏

l≥1

∏

Ji l

(l )!(−1)l−1

l 2

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

where both the products and the sums over Ji l are on the subset Ji l ⊂ 1, ..., N of size l

such that di Ji l l ∈Leaves(d′).

For every 1≤ i ≤ n and l ≥ 1, ki l counts how many subset Ji l ⊂ 1, ..., N of size l are

in Leaves(d′), hence we can write log−→f d′

log−→f d′ =

Mult(h )n∑

i=1

∑

l≥1

l Ali

(l )!(−1)l−1

l 2

∑

Ji l

∏

l ′≥1

∏

Ji l ′Ji l ′∩Ji l=;

(l ′)!(−1)l

′−1

(l ′)2

∏

j ′∈Ji l ′

ui j ′

∏

j∈Ji l

ui j ·

·

∏

i ′ 6=i

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′∏

Ji ′ j ′

∏

j ′∈Ji ′l ′

ui ′ j ′

!

z∑

i Pi mi ,

ldMult(h )

∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l∏

Ji l

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

85

=

Mult(h )n∑

i=1

∑

l≥1

l Ali

(l )!(−1)l−1

l 2

(l )!(−1)l−1

(l )2

ki l−1∑

Ji l

∏

J ′i lJ ′i l ∩Ji l=;

∏

j ′∈J ′i l

ui j ′

∏

j∈Ji l

ui j ·

·

∏

l ′ 6=l

(l ′)!(−1)l

′−1

(l ′)2

ki l ′∏

j ′∈Ji l ′

ui j ′

!

∏

i ′ 6=i

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′∏

Ji ′ j ′

∏

j ′∈Ji ′l ′

ui ′ j ′

!

z∑

i Pi mi ,

ldMult(h )

∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l∏

Ji l

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

=

Mult(h )n∑

i=1

∑

l≥1

l Ali

(l )!(−1)l−1

l 2

ki l

ki l

∏

Ji l

∏

j ′∈Ji l

ui j ′

!

∏

l ′ 6=l

(l ′)!(−1)l

′−1

(l ′)2

ki l ′∏

j ′∈Ji l ′

ui j ′

!

·

·

∏

i ′ 6=i

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′∏

Ji ′ j ′

∏

j ′∈Ji ′l ′

ui ′ j ′

!

z∑

i Pi mi ,

ldMult(h )

∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l∏

Ji l

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

=

Mult(h )n∑

i=1

∑

l≥1

l Ali ki l

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki l ′∏

Ji l ′

∏

j ′∈Ji l ′

ui j ′

!

·

·

∏

i ′ 6=i

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′∏

Ji ′ j ′

∏

j ′∈Ji ′l ′

ui ′ j ′

!

z∑

i Pi mi ,

ldMult(h )

∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l∏

Ji l

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

=

Mult(h )n∑

i=1

∑

l≥1

l Ali ki l

∏

1≤i ′≤n

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′∏

Ji ′ j ′

∏

j ′∈Ji ′l ′

ui ′ j ′

!

z∑

i Pi mi ,

ldMult(h )

∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l∏

Ji l

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

In addition, defining the s (k)-tuple of vectors w(k) as in Notation 5.1.8, then

wΓ (Eout) =s (k)∑

r=1

wr (k)

and there are ki l unbounded edges Ei Ji l l such that h (Ei Ji l l )⊂ ξi Ji l l + l miR≥0; thus Γ is

of type (w(k),ξ).

Recall, by construction of D∞N that for every d ∈D∞N \DN such that d=md′R≥0

86

log−→f d =

∑

ld≥1

∑

d′∈D∞Nd′=ξ′+ldmdR≥0

log−→f d′ .

At fixed P and k ` P: for every 1 ≤ i ≤ n , l ≥ 1 let Ai l be the set of ki l disjoint subsets

Ji l ⊂ 1, ..., N of size l . Then, by plugging-in the contribution from each d′ we have

log−→f d =

∑

ld≥1

∑

P

∑

k`P

∑

Ai l

∑

h∈Tw(k),ξ′

Mult(h )n∑

i=1

∑

l≥1

l Ali ki l

∏

1≤i ′≤n

∏

l ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′

·

·∏

Ji ′ j ′∈Ai ′l ′

∏

j ′∈Ji ′l ′

ui ′ j ′

z∑

i Pi mi ,

ldMult(h )

∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l∏

Ji l ∈Ai l

∏

j ′∈Ji j

ui j ′

!

z∑

i Pi mi

where the second sum is over all n-tuple P= (P1, ..., Pn ) ∈Nn such that∑

i Pi mi = ldmd.

Given Ai l , we can rearrange the sum in the following way: let Bi..=⋃

Ji l ∈Ai lJi l , then Bi

is of size∑

l≥1 l ki l = Pi and there are

Pi !∏

l≥1 ki l !(l !)ki l

different ways of writing Bi as disjoint union of ki l subsets of size l . Therefore, log−→f d

can be rewritten as

log−→f d =

∑

ld≥1

∑

P

∑

k`P

N tropw(k)

n∑

i=1

∑

l≥1

l Ali ki l

∏

1≤i ′≤nl ′≥1

(l ′)!(−1)l

′−1

(l ′)2

ki ′ j ′ Pi ′ !∏

l≥1 ki ′l !(l !)ki ′l·

·

∑

Bi ′⊂1,...,N |Bi ′ |=Pi ′

∏

j ′∈Bi ′

ui ′ j ′

z∑

i Pi mi ,

ld∏

1≤i≤n

∏

l≥1

(l )!(−1)l−1

l 2

ki l Pi !∏

l≥1 ki l !(l !)ki l

∑

Bi⊂1,...,N |Bi |=Pi

∏

j ′∈Bi

ui j ′

z∑

i Pi mi

and by recalling the combinatorics of t Pii as sum of ui j we finally get the expected result:

87

(5.24)

log−→f d =

∑

ld≥1

∑

P

∑

k`P

N tropw(k)

n∑

i=1

∑

l≥1

l Ali ki l

∏

1≤i ′≤n

∏

l ′≥1

(−1)l′−1

(l ′)2

ki ′l ′ 1

ki ′l ′ !t Pi ′

i ′

z∑

i Pi mi ,

ld

∏

1≤i≤n

∏

l≥1

(−1)l−1

l 2

ki l 1

ki l !t Pi

i

z∑

i Pi mi

.

5.2 Gromov–Witten invariants

Let m= (m1, ..., mn ) be an n-tuple of primitive non-zero vectors m j ∈Λ and consider

the toric surface Y m whose fan in ΛR has rays −R≥0m1, ...,−R≥0mn . If the fan is not

complete (i.e. its rays do not span ΛR) we can add some more rays and we still denote

the compact toric surface by Y m1. Let Dm1

, ..., Dmnbe the toric divisors corresponding to

the rays −R≥0m1, ...,−R≥0mn : we blow-up a point ξ j in general position on the divisor

Dm j, for j = 1, ..., n . Since we allow m j =mk for k 6= j , we may blow-up more than one

distinct points on the same toric divisor Dm j. We denote by E j the exceptional divisor

of the blow-up of the point ξ j and by Ym the resulting projective surface. The strict

transform of the toric boundary divisor ∂ Ym is an anti-canonical cycle of rational curves

and the pair (Ym,∂ Ym) is a log Calabi–Yau pair.

Following [Bou18]we introduce genus 0 Gromov–Witten invariants both for a pro-

jective surface Ym relative to the divisor ∂ Ym and for the toric surface Y m relative to the

full toric boundary divisor ∂ Ym =D1 ∪ ...∪Dn . In the following sections, we review the

definitions and we recall the relation with tropical curve count (Proposition 5.2.1) and

the degeneration formula (Proposition 5.2.2).

5.2.1 Gromov–Witten invariants for (Ym,∂ Ym)

Let P= (P1, ..., Pn ) be a n-tuple P ∈Nn . We assume

lPmP..=

n∑

j=1

Pj m j 6= 0

for some primitive mP ∈Λ. The vector mP can be written as a combination of vectors

mi which generates the fan of Y m:

mP = aL ,PmL +aR ,PmR

1Adding extra rays is irrelevant, because Gromov–Witten invariants are equivalent under birational

transformations.

88

aL ,P, aR ,P ∈C. Then, denote by DmLand DmR

the toric divisors corresponding to mL and

mR . We are going to define a class βP ∈H2(Y m,Z)which represents curves in Ym with

tangency conditions prescribed by P: let β be the class whose intersection numbers are:

• for every divisor Dm j, j = 1, ..., n , distinct from DmR

and DmL

β ·Dm j=

∑

j ′:Dm j ′=Dm j

Pj ′

• for DmL

β ·DmL= lPaL ,P+

∑

j ′:Dm j ′=DmL

Pj ′

• for DmR

β ·DmR= lPaR ,P+

∑

j ′:Dm j ′=DmR

Pj ′

• for every divisor D different from all DmL, DmR

and all Dmi, then β ·D = 0.

Then we define βP ∈H2(Ym,Z) as

βP..= ν∗β −

n∑

j=1

Pj [E j ]

where ν: Y m→ Ym is the blow-up morphism. Let Mg ,P(Ym/∂ Ym) be the moduli space of

stable log maps of genus g and class βP to a target log space Ym that is log smooth over

∂ Ym. In [GS13], the authors prove that Mg ,P(Ym/∂ Ym) is a proper Deligne–Mumford

stack and it admits a virtual fundamental class

[Mg ,P(Ym/∂ Ym)]vir ∈ Ag (Mg ,P(Ym,∂ Ym),Q).

In particular for genus 0, M0,P(Ym) has virtual dimension zero, hence the log Gromov–

Witten invariants of Ym are defined as

(5.25) N0,P(Ym) ..=

∫

[M0,P(Ym/∂ Ym)]vir

1

where 1 ∈ A0(M 0,P(Ym/∂ Ym),Q) is the dual class of a point.

5.2.2 Gromov–Witten invariants for (Y m,∂ Y m)

We now introduce Gromov–Witten invariants for the toric surface Y m. Let s be

an integer number and let w= (w1, ..., ws ) be a s -tuple of weight vectors such that for

every r = 1, ..., s there is an index i ∈ 1, ..., n such that −miR≥0 =−wrR≥0. In particular

−wrR≥0 is contained in the fan of Y m and we denote by Dwrthe corresponding divisor

in ∂ Y m. We also assume∑s

r=1 wi 6= 0. In order to “count” curves in Y m meeting ∂ Y m

in s prescribed points with multiplicity |wr | and in a single unprescribed point with

89

multiplicity |∑s

r=1 wr |, we are going to define a suitable curve class βw ∈H2(Y m,Z). Let

mw ∈Λ be a primitive vector in Λ, such that

s∑

r=1

wr = lwmw

where lw =

∑sr=1 wr

. In particular mw belongs to a cone of the fan of Y m and it can be

written uniquely as

mw = aL mL +aR mR

where mL , mR are primitive generator of the rays of the fan and aL , aR ∈N.

Then βw is determined by the following intersection numbers:

• for every divisor Dwr, r = 1, ..., s , distinct from DmR

and DmL

βw ·Dwr=

∑

r ′:Dwr ′=Dwr

|w ′r |

• for both the divisors DmRand DmL

βw ·DmL= lwaL +

∑

r ′:Dwr ′=DmL

|w ′r |

βw ·DmR= lwaR +

∑

r ′:Dwr ′=DmR

|w ′r |

• for every divisor D different from all Dwrand all Dmi

, then βw ·D = 0.

The existence of this class follows from toric geometry: since Y m is complete, A1(Y m) is

generated by the class of the divisors associated to the rays of the fan. Let M0,w(Y m,∂ Y m)

be the moduli space of stable log maps of genus 0 and class βw; it is a proper Deligne–

Mumford stack of virtual dimension s . In addition, there are s evaluation maps ev1, ..., evs

such that

evr : M0,w(Y m,∂ Y m)→Dwr

and the Gromov–Witten invariants are defined as follows

(5.26) N0,w(Y m) ..=

∫

[M0,w(Y m,∂ Y m)]vir

s∏

r=1

ev∗r (ptr )

where ptr ∈ A1(Dwr) is the dual class of a point.

Proposition 5.2.1. Let m= (m1..., mn ) be a n-tuple of non-zero primitive vectors in Λ.

Then for every n-tuple P= (P1, ..., Pn ) ∈Nn and every partition k `P

(5.27) N trop0,w(k) =N0,w(k)(Y m)

n∏

r=1

∏

l≥1

l kr l

90

Proposition 5.2.2. Let m= (m1, ...mn ) be a n-tuple of primitive, non zero vectors in Λ,

and let P= (p1, ..., pn ) ∈Nn be a n-tuple of positive integers, then

(5.28) N0,P(Ym) =∑

k`P

N0,w(k)(Y m)n∏

j=1

l j∏

l=1

l k j l

k j l !(Rl )

k j l

where the sum is over all partition k of P and Rl =(−1)l−1

l 2 .

A first proof of this result is Proposition 5.3 [GPS10], and it has been computed by

applying Li’s degeneration formula. In [Bou18], the author proves a general version of

(5.28) (see Proposition 11 of [Bou18]) using a more sophisticated approach. However in

genus 0 the two approaches give the same formula, as Gromov–Witten invariants are

the same in Li’s theory and in the logarithmic theory.

5.3 Gromov–Witten invariants from commutators in V

In this section we finally collect together the previous results and we get a generating

function for genus 0 relative Gromov–Witten invariants in terms of consistent scattering

diagrams in the extended tropical vertex group.

Theorem 5.3.1. Let m= (m1, ..., mn ) be n primitive non zero vectors in Λ and let D be a

standard scattering diagram over C[[t1, ..., tn ]]with

D=¦

di =miR,−→f di=

1+Ai ti z mi , 1+ ti z mi

|1≤ i ≤ n©

where Ai ∈ gl(r,C) and assume [Ai , Ai ′ ] = 0 for all i , i ′ ∈ 1, ..., n. Then for every wall

(d=mdR≥0,−→f d) ∈D∞ \D:

(5.29)

log−→f d =

∑

ld≥1

∑

P

∑

k`P

N0,w(k)(Y m)n∑

i=1

∑

l≥1

l Ali ki l

n∏

i ′=1

∏

l ′≥1

(−1)l′−1

l ′

ki ′l ′ 1

ki ′l ′ !

n∏

i=1

t Pii z

∑

i Pi mi ,

∑

ld≥1

∑

P

ldN0,P(Ym)

n∏

i=1

t Pii

z∑

i Pi mi

where the second is over all n-tuples P= (P1, ..., Pn ) satisfying∑n

i=1 Pi mi = ldmd.

PROOF. The proof is a consequence of Theorem 5.1.9, Theorem 5.2.2 and Theorem

5.2.1. Indeed from equation (5.28) and Proposition 5.2.1

91

N0,P(Ym) =∑

k`P

N0,w(k)(Y m)n∏

i=1

∏

l≥1

l ki l

ki l !R ki l

li(5.30)

=∑

k`P

N tropw(k)

n∏

i=1

∏

l≥1

1

ki l !R ki l

l(5.31)

Then, for every every wall (d=mdR≥0,−→f d) ∈D∞ \D is explicitly given by (5.22):

(5.32)

log−→f d =

∑

ld≥1

∑

P

∑

k`P

N tropw(k)

n∑

i=1

∑

l≥1

l Ali ki l

n∏

i ′=1

∏

l ′≥1

(−1)l′−1

(l ′)2

ki ′l ′ 1

ki ′l ′ !t Pi ′

i ′

z∑

i Pi mi ,

ld

n∏

i=1

∏

l≥1

(−1)l−1

l 2

ki l 1

ki l !t Pi

i

z∑

i Pi mi

and plugging-in equation 5.27, we get

log−→f d =

∑

ld≥1

∑

P

∑

k`P

N0,w(k)(Y m)n∑

i=1

∑

l≥1

l Ali ki l

n∏

i ′=1

∏

l ′≥1

(−1)l′−1

l ′

ki ′l ′ 1

ki ′l ′ !t Pi ′

i ′

z∑

i Pi mi ,

∑

ld≥1

∑

P

ldN0,P(Ym)

n∏

i=1

t Pii

z∑

i Pi mi

where we use the relation between tropical curve counting and open Gromov–Witten of

Theorem 5.2.1, namely N trop0,w(k)

∏nr=1

∏

l≥11

l kr l=N0,w(k)(Y m).

EXAMPLE 5.3.2. Let m1 = (0, 1), m2 = (1, 0) and consider the scattering diagram

D=¦

d1 =m1R,−→f d1=

1+A1t1z m1 , 1+ t1z m1

,

d2 =m2R,−→f d2=

1, 1+ t2z m2

©

with A1 ∈ gl(r,C)e , which corresponds to the initial standard scattering diagram of

Example 5.0.7. Let us consider the ray d= (1,2)R≥0 ∈D∞ \D. From Theorem 5.3.1 we

have

(5.33)

log−→f (1,2) =

∑

ld≥1

∑

P

∑

k`P

A1k11N0,w(k)(Y m)2∏

i=1

∏

l ′≥1

(−1)l′−1

l ′

ki l ′ 1

ki l ′ !

t Pii z

∑

i Pi mi ,

ldN0,P(Ym)

2∏

i=1

t Pii

z∑

i Pi mi

where the second sum is over all n-tuples P = (2ld, ld). Now as a consequence of the

pentagon identity for the di–logarithm we know that N0,P(Ym) = 0 unless P=α(1,1) for

some α≥ 1, hence we have

92

log−→f (1,2) =

∑

ld≥1

∑

P=(2ld,ld)

∑

k`P

A1k11N0,w(k)(Y m)2∏

i=1

∏

l ′≥1

(−1)l′−1

l ′

ki l ′ 1

ki l ′ !t 2ld

1 t ld2 x ld y 2ld , 0

!

.

For ld = 1 we can explicitly compute the partitions k ` (2, 1) and the weights w(k):

(a) k1 = (k1l )l≥1 = (2) and k2 = (k2l )l≥1 = (1)with weight w(k) = (m1, m1, m2);

(b) k1 = (k1l )l≥1 = (0, 1) and k2 = (k2l )l≥1 = (1)with weight w(k) = (2m1, m2).

Therefore the non trivial contributions to log−→f (1,2) are

A1

·2 ·N0,w(k)(a ) (Y m)1

2!·1+0 ·N0,w(k)(b ) (Y m)

(−1)2·1

t 21 t2 x y 2

= A1N0,w(k)(a ) (Y m)t12t2 x y 2

and comparing it with the function−→f (1,2) of Example 5.0.7 we get N0,w(k)(a ) (Y m) = 1. In

this simple case N0,w(k)(a ) (Y m) can be also explicitly computed by applying formula (5.30):

indeed

N0,w(k)(a ) (Y m) =N trop

w(k)(a )∏

1≤i≤2

∏

l≥1 l ki l=

N tropw(k)(a )

1 ·1=N trop

w(k)(a )

and the there is a unique tropical curve, counted with multiplicity with weight w(k)(a ) =

(m1, m1, m2), drawn in Figure 5.6.

w1 = 1

w3 = 1

w2 = 1

wout = 1

FIGURE 5.6. Case (a )

In conclusion, the non trivial contribution in the matrix component of−→f (1,2) allow

to compute a Gromov–Witten invariant for the toric surface associated to the fan with

rays (−1, 0), (0,−1), (1, 2).

5.3.1 The generating function of N0,w(Y m)

In this section we introduce a new setting in order to study when the automorphism

associated to a ray of the consistent scattering diagram is a generating function for the

93

invariants N0,w(Y m) (as in Theorem 5.3.1). In particular, let `1,`2 ≥ 1 and let us now

consider the initial scattering diagram

D=¦

di =m1R,−→f di=

1+Ai ti z m1 , 1+ ti z m1

,

di =m2R,−→f d j=

1+Q j s j z m2 , 1+ s j z m2

©

1≤i≤`11≤ j≤`2

with [Ai ,Q j ] = [Ai , Ai ′ ] = [Q j ,Q j ′ ] = 0 for all i , i ′ = 1, ...,`1 and j , j ′ = 1, ...,`2. We will not

put any restrictions on the rank of the matrices Ai ,Q j . As a consequence of Theorem

5.3.1, for every ray (d=mdR,−→f d) ∈D∞ \D, the function

−→f d can be written as follows:

(5.34) log−→f d =

∑

ld≥1

∑

P

∑

k`P

N0,w(k)(Y m)

`1∑

i=1

∑

l≥1

l Ali ki l +

`2∑

i=1

∑

l≥1

l Q li k(i+`1)l

·

`1+`2∏

i ′=1

∏

l ′≥1

(−1)l′−1

l ′

ki ′l ′ 1

ki ′l ′ !

`1∏

i=1

t Pii

`2∏

i=1

sPi+`1i z

∑

i Pi mi ,

∑

ld≥1

∑

P

ldN0,P(Ym)`1∏

i=1

t Pii

`2∏

i=1

sPi+`1i z

∑

i Pi mi

where for every ld,∑`1

i=1 Pi m1+∑`2

i=1 P`1+i m2 = ldmd. In particular, log−→f d is a generating

function for the invariants N0,P(Ym). We also expect that log−→f d is a generating function

for N0,w(Y m).

Since the tangency conditions for N0,w(Y m) are specified by weight vectors w, it

is natural to rewrite (5.34) summing over all w rather than P. We do it below: let w =

(w1, ..., ws ) such that∑s

i=1 wi = ldmd and wi = |wi |m1 for i = 1, ..., s1 and wi = |wi |m2 for

i = s1+1, ..., s2+ s1. Then we can rewrite (5.34) as follows:

(5.35) log−→f d =

∑

ld≥1

∑

w

N0,w(Y m)∑

k:w(k)=w

`1∑

i=1

∑

l≥1

l Ali ki l +

`2∑

i=1

∑

l≥1

l Q li k(i+`1)l

·

·

`1+`2∏

i ′=1

∏

l ′≥1

(−1)l′−1

l ′

ki ′l ′ 1

ki ′l ′ !

`1∏

i=1

t∑

l≥1 l ki l

i

`2∏

i=1

s∑

l≥1 l ki+s1l

i z∑

i wi ,

∑

ld≥1

∑

P

ldN0,P(Ym)`1∏

i=1

t Pii

`2∏

i=1

sPi+`1i z

∑

i Pi mi

where for every ld,∑`1

i=1 Pi m1+∑`2

i=1 P`1+i m2 = ldmd.

Let us define vectors ek(w) ∈ gl(r,C)⊗CC[t1, ..., t`1, s1, ..., s`2

]

(5.36) ek(w)..=

`1∑

i=1

∑

l≥1

l Ali ki l +

`2∑

i=1

∑

l≥1

l Q li k(i+`1)l

`1∏

i=1

t∑

l≥1 l ki l

i

`2∏

i=1

s∑

l≥1 l ki+s1l

i

94

and set

Vw..=

∑

k:w(k)=w

`1+`2∏

i ′=1

∏

l ′≥1

(−1)l′−1

l ′

ki ′l ′ 1

ki ′l ′ !

ek(w).

To simplify the notation we rewrite the vectors Vw, ek(w) as follows:

NOTATION 5.3.3. Assume ldmd is fixed, then there are wi i=1,...,s vectors such that∑s

j=1 w j =

ldmd. Moreover for every wi there are k j = k j (wi ) such that wi (k j ) =wi , for j = 1, ..., si .

In particular, for every i = 1, ..., s there are si vectors ek j (wi ) which we denote by e (i )j . The

vectors Vwiare rewritten as follows:

Vwi=

si∑

j=1

λ(i )j e (i )j

where λ(i )j =λk j (wi ) =

∏`1+`2i ′=1

∏

l ′≥1

(−1)l′−1

l ′

ki ′l ′ 1ki ′l ′ !

.

CONJECTURE 5.3.4. If `1,`2,Ai 1≤i≤`1,Q j 1≤ j≤`2

are regarded as arbitrary param-

eters, and ldmd is chosen such that d = ldmdR≥0 is a ray in the consistent scattering

diagram D∞. Then−→f d is a generating function for N0,w(Y m).

REMARK 5.3.5. If the collection of e (i )j i=1,...,sj=1,...,si

is a basis, then Conjecture 5.3.4 will

follow immediately; indeed for every ci i=1,...,s ∈Q

0=s∑

i=1

ci Vwi=

s∑

i=1

ci

si∑

j=1

λ(i )j e (i )j =s∑

i=1

si∑

j=1

ciλ(i )j e (i )j ⇔ ciλ

(i )j = 0 ∀i = 1, ..., s , ∀ j = 1, ..., si .

In particular since λ(i )j 6= 0 for all j = 1, ..., si , then ci = 0 for all wi , i = 1, ..., s .

We do not have a proof of the conjecture yet, but in the following examples we show

that even if for `1 = `2 = 1 the vectors ek(w) are linearly dependent, then for `1 = `2 = 2

they are indeed a basis. Then we conclude the section with a partial result, which states

how to compute some invariants from log−→f d (see Theorem 5.3.10).

We are going to consider the initial scattering diagram

D=¦

di =m1R,−→f di=

1+Ai ti z m1 , 1+ ti z m1

,

di =m2R,−→f d j=

1+Q j s j z m2 , 1+ s j z m2

©

1≤i≤`11≤ j≤`2

with m1 = (1,0) and m2 = (0,1) and such that [Ai , Ai ′ ] = [Q j ,Q j ′ ] = [Ai ,Q j ] = 0 for every

i , i ′ = 1, ...,`1, j , j ′ = 1, ...,`2. We remark that Ai 1≤i≤`1,Q j 1≤ j≤`2

are regarded as formal

parameters.

EXAMPLE 5.3.6. Let `1 = `2 = 1, then the consistent scattering diagram D∞ has the

ray d= (1, 1)R≥0. Let ld = 2: the possible vector w such that∑

j w j = (2, 2) are

(a ) w= (m1, m1, m2, m2);

(b ) w= (2m1, m2, m2);

95

(c ) w= (m1, m1, 2m2);

(d ) w= (2m1, 2m2)

and the partition k= (k1, k2) such that w(k) =w are respectively

(a ) k1 = (2) and k2 = (2);

(b ) k1 = (0, 1) and k2 = (2);

(c ) k1 = (2) and k2 = (0, 1);

(d ) k1 = (0, 1) and k2 = (0, 1).

Hence from equation (5.35) we get

log−→f (2,2) =Nw(a )(2A1+2Q1)

1

4

t 21 s 2

1 +Nw(b )(2A21+2Q1)

−1

4

t 21 s 2

1+

+Nw(c )(2A1+2Q 21 )−1

4

t 21 s 2

1 +Nw(d )(2A21+2Q 2

1 )

1

4

t 21 s 2

1 .

According to Notation 5.3.3 the vectors ek(w) are respectively:

e(a ) = (2A1+2Q1)t21 s 2

1 ; e(b ) = (2A21+2Q1)t

21 s 2

1 ;

e(c ) = (2A1+2Q 21 )t

21 s 2

1 ; e(d ) = (2A21+2Q 2

1 )t21 s 2

1 ,

but they are not linearly independent vectors (e(a ) =−e(d )− e(c )− e(d )) hence log−→f (2,2) is

not a generating function for Nw.

EXAMPLE 5.3.7. Let `1 = `2 = 2, the consistent scattering diagram D has many more

rays, and we consider d= (1, 1)R≥0 with ld = 2 as in Example 5.3.6. The vectors w are the

same as before but there are more combinations for the vectors k= (k (1)1 , k (2)1 , k (1)2 , k (2)2 ),

where we denote by k (i )j the partition which corresponds to Ai if j = 1, Qi if j = 2:

(a ) k (i )1 = (2) and k (i′)

2 = (2) for i , i ′ = 1,2, k (1)1 = (1), k (2)1 = (1) and k (i )2 = (2) for

i = 1,2, k (i )1 = (2) and k (1)2 = (1), k (2)2 = (1) for i = 1,2, k (1)1 = (1), k (2)1 = (1) and

k (1)2 = (1), k (2)2 = (1);

(b ) k (i )1 = (0,1) and k (i′)

2 = (2) for i , i ′ = 1,2, k (i )1 = (0,1) and k (1)2 = (1), k (2)2 = (1) for

i = 1, 2 ;

(c ) k (i )1 = (2) and k (i′)

2 = (0,1) for i , i ′ = 1,2, k (1)1 = (1), k (2)1 = (1) and k (i′)

2 = (0,1) for

i ′ = 1, 2 ;

(d ) k (i )1 = (0, 1) and k (i′)

2 = (0, 1) for i , i ′ = 1, 2.

96

Hence the vector ek(w) are the following:

e(a ) =∑

i ,i ′=1,2

(2Ai +2Qi ′ )t2i s 2

i ′ +∑

i=1,2

(A1+A2+2Qi )t1t2s 2i +

∑

i=1,2

(2Ai +Q1+Q2)t2i s1s2+

+ (A1+A2+Q1+Q2)t1t2s1s2,

e(b ) =∑

i ,i ′=1,2

(2A2i +2Qi ′ )t

2i s 2

i ′ +∑

i=1,2

(2A2i +Q1+Q2)t

2i s1s2,

e(c ) =∑

i ,i ′=1,2

(2Ai +2Q 2i ′ )t

2i s 2

i ′ +∑

i=1,2

(A1+A2+2Q 2i )t1t2s 2

i ,

e(d ) =∑

i ,i ′=1,2

(2A2i +2Q 2

j )t2i s 2

i ′ .

These vectors are linearly independent, indeed the monomial A1t1t2s1s2 is only in e(a ),

the monomial Q1t 21 s1s2 is only in e(b ) and the monomial A1t1t2s 2

1 is only in e(c ).

EXAMPLE 5.3.8. Let `1 = `2 = 1 and consider the ray d = (1,2)R≥0 with ld = 2. The

vectors w and the possible vector k= (k1, k2) are written in the following tables:

2m1/4m2 (m2, m2, m2, m2) (m2, 3m2) (m2, m2, 2m2) (2m2, 2m2) (4m2)

(m1, m1) w(a ) w(b ) w(c ) w(d ) w(e )

(2m1) w( f ) w(g ) w(h ) w(i ) w(l )

k1/k2 (4) (1, 0, 1) (2, 1) (0, 2) (0, 0, 0, 1)

(2) k(w(a )) k(w(b )) k(w(c )) k(w(d )) k(w(e ))

(0, 1) k(w( f )) k(w(g )) k(w(h )) k(w(i )) k(w(l ))

In this example we have 10 vectors e•

e(a ) = (2A1+4Q1)t21 s 4

1 ; e(b ) = (2A1+Q1+3Q 31 )t

21 s 4

1 ; e(c ) = (2A1+2Q1+2Q 21 )t

21 s 4

1 ;

e(d ) = (2A1+4Q 21 )t

21 s 4

1 ; e(e ) = (2A1+4Q 41 )t

21 s 4

1 ; e( f ) = (2A21+4Q1)t

21 s 4

1 ;

e(g ) = (2A21+Q1+3Q 3

1 )t21 s 4

1 ; e(h ) = (2A21+2Q1+2Q 2

1 )t21 s 4

1 ; e(i ) = (2A21+4Q 2

1 )t21 s 4

1 ;

e(l ) = (2A21+4Q 4

1 )t21 s 4

1

but they are linear combination of 6 monomials (namely A1t 21 s 4

1 , Q1t 21 s 4

1 , A21s 4

1 t 21 , Q 2

1 t 21 s 4

1 ,

Q 31 t 2

1 s 41 , Q 4

1 t 21 s 4

1 ), hence they are linearly dependent (e.g. e(l ) = e(e )− e(i )− e(d )).

EXAMPLE 5.3.9. Let `1 = `2 = 2 and consider the same vector ldmd = (2,4) as in

Example 5.3.8. The vectors w• are as before, while there are more possibilities for the

97

vectors k= (k (1)1 , k (2)1 , k (1)2 , k (2)2 ) hence e• are listed below:

e(a ) =∑

i ,i ′=1,2

(2Ai +4Qi ′ )t2i s 4

i ′ +∑

i=1,2

(2Ai +Q1+3Q2)t2i s1s 3

2 +∑

i=1,2

(2Ai +Q2+3Q1)t2i s 3

1 s2+

+∑

i=1,2

(2Ai +2Q1+2Q2)t2i s 2

1 s 22 +

∑

i ′=1,2

(A1+A2+4Qi ′ )t1t2s 4i ′ + (A1+A2+Q1+3Q2)t1t2s1s 3

2+

+ (A1+A2+Q2+3Q1)t1t2s 31 s2+ (A1+A2+2Q1+2Q2)t1t2s 2

1 s 22 ;

e(b ) =∑

i ,i ′=1,2

(2Ai +Qi ′ +3Q 3i ′ )t

2i s 4

i ′ +∑

i=1,2

(2Ai +Q1+3Q 32 )t

2i s1s 3

2 +∑

i=1,2

(2Ai +Q2+3Q 31 )t

2i s 3

1 s2

+∑

i ′=1,2

(A1+A2+Qi ′ +3Q 3i ′ )t1t2s 4

i ′ + (A1+A2+Q1+3Q 32 )t1t2s1s 3

2 + (A1+A2+Q2+3Q 31 )t1t2s 3

1 s2;

e(c ) =∑

i ,i ′=1,2

(2Ai +2Qi ′ +2Q 2i ′ )t

2i s 4

i ′ +∑

i=1,2

(2Ai +2Q1+2Q 22 )t

2i s 2

1 s 22 +

∑

i=1,2

(2Ai +2Q2+2Q 21 )t

2i s 2

1 s 22+

+∑

i ′=1,2

(A1+A2+2Qi ′ +2Q 2i ′ )t1t2s 4

i ′ + (A1+A2+2Q1+2Q 22 )t1t2s 2

1 s 22+

+ (A1+A2+2Q2+2Q 21 )t1t2s 2

1 s 22 + (A1+A2+Q1+Q2+2Q 2

2 )t1t2s1s 32+

+ (A1+A2+Q2+Q1+2Q 21 )t1t2s 3

1 s2+∑

i=1,2

(2Ai +Q1+Q2+2Q 22 )t

2i s1s 3

2+

+∑

i=1,2

(2Ai +Q2+Q1+2Q 21 )t

2i s 3

1 s2;

e(d ) =∑

i ,i ′=1,2

(2Ai +4Q 2i ′ )t

2i s 4

i ′ +∑

i=1,2

(2Ai +2Q 21 +2Q 2

2 )t2i s 2

1 s 22 +

∑

i ′=1,2

(A1+A2+4Q 2i ′ )t1t2s 4

i ′+

+ (A1+A2+2Q 21 +2Q 2

2 )t1t2s 21 s 2

2 ;

e(e ) =∑

i ,i ′=1,2

(2Ai +4Q 4i ′ )t

2i s 4

i ′ +∑

i ′=1,2

(A1+A2+4Q 4i ′ )t1t2s 4

i ′ ;

e( f ) =∑

i ,i ′=1,2

(2A2i +4Qi ′ )t

2i s 4

i ′ +∑

i=1,2

(2A2i +Q1+3Q2)t

2i s1s 3

2 +∑

i=1,2

(2A2i +3Q1+Q2)t

2i s 3

1 s2+

+∑

i=1,2

(2A2i +2Q1+2Q2)t

2i s 2

1 s 22 ;

e(g ) =∑

i ,i ′=1,2

(2A2i +Qi ′ +3Q 3

i ′ )t2i s 4

i ′ +∑

i=1,2

(2A2i +Q1+3Q 3

2 )t2i s1s 3

2 +∑

i=1,2

(2A2i +Q2+3Q 3

1 )t2i s 3

1 s2;

e(h ) =∑

i ,i ′=1,2

(2A2i +2Qi ′ +2Q 2

i ′ )t2i s 4

i ′ +∑

i=1,2

(2A2i +2Q1+2Q 2

2 )t2i s 2

1 s 22 +

∑

i=1,2

(2A2i +2Q2+2Q 2

1 )t2i s 2

1 s 22+

+∑

i=1,2

(2A2i +Q1+Q2+2Q 2

2 )t2i s1s 3

2 +∑

i=1,2

(2A2i +Q2+Q1+2Q 2

1 )t2i s 3

1 s2;

e(i ) =∑

i ,i ′=1,2

(2A2i +4Q 2

i ′ )t2i s 4

i ′ +∑

i=1,2

(2A2i +2Q 2

1 +2Q 22 )t

2i s 2

1 s 22 ;

e(l ) =∑

i ,i ′=1,2

(2A2i +4Q 4

i ′ )t2i s 4

i ′ .

It is possible to check they are all linearly independent vectors.

98

Theorem 5.3.10. Let `1,`2 be positive integer greater or equal than 1 and let

D=¦

di =m1R,−→f di=

1+Ai ti z m1 , 1+ ti z m1

,

di =m2R,−→f d j=

1+Q j s j z m2 , 1+ s j z m2

©

1≤i≤`11≤ j≤`2

be the initial scattering diagram with [Ai ,Q j ] = 0 for every i = 1, ...,`1, j = 1, ...,`2. If

d =mdR≥0 is a ray in the consistent scattering diagram D∞ and ldmd = `1m1 + `2m2,

then−→f d allows to compute the relative Gromov–Witten invariants N0,w(Y m)with the

following tangency conditions:

(a ) w(a ) = (m1, ..., m1︸ ︷︷ ︸

`1−times

, m2, ..., m2︸ ︷︷ ︸

`2−times

), namely rational curve through `1 distinct points

on the divisor Dm1, `2 distinct points on the divisor Dm2

and tangent of order `1

to an unspecified point on Dm1and of order `2 to an unspecified point on Dm2

;

(b j ) w(b j ) = ( j m1, m1, ..., m1︸ ︷︷ ︸

(`1− j )−times

, m2, ..., m2︸ ︷︷ ︸

`2−times

) for j = 2, ...,`1. These are rational curves

through `1− j distinct points on the divisor Dm1, `2 distinct points on the divisor

Dm2, tangent of order j to a given point on Dm1

(distinct from the previous `1− j

points), tangent of order `1 to an unspecified point on Dm1and of order `2 to

an unspecified point on Dm2;

(c j ) w(c j ) = (m1, ..., m1︸ ︷︷ ︸

`1−times

, j m2, m2, ..., m2︸ ︷︷ ︸

(`2− j )−times

) for j = 2, ...,`2. These are rational curves

through `1 distinct points on the divisor Dm1, `2− j distinct points on the divisor

Dm2, tangent of order j to a given point on Dm2

(distinct from the previous `2− j

points), tangent of order `1 to an unspecified point on Dm1and of order `2 to

an unspecified point on Dm2.

PROOF. Being an element of h, log−→f d is a polynomial in the variables A1, ..., A`1

,

Q1, ...,Q`2, t1, ..., t`1

, s1, ..., s`2and, according to formula (5.35), the invariants N0,w(Y m)

are the coefficients of the polynomial Vw. In particular there are some monomials which

only appear in Vw for a given w, hence from the expression of log−→f d we can compute the

invariants N0,w(Y m) looking at the coefficient of these monomials. From the definition

of Vw, the vectors k (such that k(w) =w) govern the possible polynomials which appears

in Vw. We collect the possible partitions of `1 and `2 in the following table (see Figure 5.7).

Then t1 · ... · t`1can only appear when w= (m1, ..., m1

︸ ︷︷ ︸

`1−times

,∗) because they are all distinct, and

for the same reason s1 · ... · s`2can only appear when w= (∗, m2, ..., m2

︸ ︷︷ ︸

`1−times

). In addition this

forces A1, ..., A`1and Q1, ...,Q`2

not being of higher powers. Hence A1t1 · ... · t`1s1 · ... · s`2

only contributes to N0,w(a ).

99

`1m1/`2m2 (m2, ..., m2︸ ︷︷ ︸

`2−times

) (2m2, m2, ..., m2︸ ︷︷ ︸

(`2−2)−times

) · · · (`2m2) (2m2, 2m2, m2, ..., m2︸ ︷︷ ︸

(`2−4)−times

) · · ·

(m1, ..., m1︸ ︷︷ ︸

`1−times

) w(a ) w(c2) · · · w(c`2) w(♠)

(2m1, m1, ..., m1︸ ︷︷ ︸

(`1−2)−times

) w(b2) w(∗) · · · w(∗)

......

......

(`1m1) w(b`1) w(∗) · · · w(∗)

(2m1, 2m1, m1, ..., m1︸ ︷︷ ︸

(`1−4)−times

) w(♣)

...

FIGURE 5.7. Table of the possible invariants appearing in the commuta-

tor formula.

As soon as we consider tj

1 t2 · ... · t`1− j for j ∈ 2, ...,`1 it is not possible to uniquely de-

termine the corresponding partition of `1: these monomials appear in (r m1, m1, ..., m1︸ ︷︷ ︸

`1−r

)

for r ∈ 1, ..., j −1 by choosing non distinct values (e.g. if r = 1 and k (1)1 = ( j ), k (i )1 = (1) for

i = 2, ...,`1− j ). However Aj1 t

j1 t2 · ... · t`1− j can only occur when w= ( j m1, m1, ..., m1

︸ ︷︷ ︸

(`1− j )−times

,∗)

because j m1 forces k1 = (`1− j , ..., 1)with 1 in the j -th position and consequently Aj1 t

j1

comes from k (1)1 = (0, ...,1) with 1 in the j -th position. If w = (r m1, m1, ..., m1︸ ︷︷ ︸

`1−r

,∗) for

r ∈ 1, ..., j − 1 then the higher power of A1 would have been Ar1 . Therefore A

j1 t

j1 t2 ·

... · t`1− j s1 · ... · s`2only contributes to N0,w(b j ). Reversing the role of k1 and k2 the same

arguments apply to N0,w(c j ) which is the coefficient of Qj

1 t1 · ... · t`1s

j1 s2 · ... · s`2− j .

What goes wrong with the other N0,w is that we are not able to isolate a unique

monomial which corresponds to w: for the w(∗) in the table, we would need both Aj1 t

j1 t2 ·

...·t`1− j and Q r1 s r

1 s2 ·...·s`2−r but

Aj1 t

j1 t2 · ... · t`1− j ·Q r

1 s r1 s2 · ... · s`2−r ,•

is not a monomial

in log−→f d ⊂ h. For the w(♣) we may choose t 2

1 t 22 t3, ..., t`1−4 but as for w(b2) we need to

consider also A21 and A2

2; however

A21A2

2t 21 t 2

2 t3 · ... · t`1−4 · s1 · ... · s`2,•

is not a monomial

in log−→f d ⊂ h. The same argument applies to Nw(♠).

100

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