Harmonic bundles and Toda lattices

with opposite sign

Takuro Mochizuki

RIMS, Kyoto University

2013 February

Outline

1. Introduction

1.1 Semi-infinite variation of Hodge structure

1.2 Toda lattice with opposite sign

2. Classification of R-valued solutions

3. Integral structure

Semi-infinite variation of Hodge structure

What is semi-infinite variation of Hodge structure?

Construction of semi-infinite variation of Hodge structure by

the method of the Kobayashi-Hitchin correspondence

Variation of Hodge structure (VHS)

Flat bundle (V,∇) on a complex manifold X

Decreasing filtration (F i | i ∈ Z) of V by holomorphic subbundles

satisfying the Griffiths transversality ∇(F i)⊂ F i−1⊗Ω1X

Real structure, pure, mixed, polarization, etc.

Semi-infinite variation of Hodge structure (∞2 -VHS)

Holomorphic vector field v on X

Holomorphic vector bundle V on X := Cλ ×X with a

meromorphic flat connection of V such that

- ∇(V )⊂ V ⊗Ω1X

(logX 0)⊗OX (X 0), where X 0 := 0×X- ∇v+λ ∂λ

(V )⊂ V (stronger than ∇v+λ∂λ(V )⊂ V ⊗O(X 0))

Real structure, pure, mixed, polarization, etc.

VHS on X induces ∞2 -VHS on X with v = 0 by the Rees construction.

Construction of VHS

There are two methods to construct VHS.

Motivic construction (geometric)

Let f : X −→ Y be a smooth projective morphism of complex manifolds.

Then, Ri f+(OX ) is naturally a variation of Hodge structure.

Kobayashi-Hitchin correspondence (solve non-linear PDE, Simpson, M)

Let X be a smooth projective variety with simply normal crossing hy-

persurface D. Let (P∗E,θ ) be a regular filtered Higgs bundle on (X ,D)

such that

P∗E is equipped with a C∗-action ρ.

θ is homogeneous in the sense ρ−1t θ ρt = t ·θ .

(P∗E,θ ) is stable, and par-c1(P∗E) = par-ch2(P∗E) = 0.

Then, we obtain a polarized VHS on X \D.

Harmonic bundle

(E,∂ E ,θ ) a Higgs bundle (θ ∈ End(E)⊗Ω1,θ 2 = 0)

h a hermitian metric of E

=⇒ Chern connection ∇h = ∂ E +∂E,h and the adjoint θ †h .

Definition (E,∂ E ,θ ,h) is called harmonic bundle, if the connection

∇1 := ∇h +θ +θ †h

is flat. In that case, h is called pluri-harmonic metric.

Kobayashi-Hitchin correspondence for regular filtered Higgs bundles

Let X be a smooth projective variety with a simply normal crossing hypersurface

D =⋃

i∈Λ Di. Let (P∗E,θ ) be a regular filtered Higgs bundle on (X ,D), i.e.,

P∗E be an increasing sequence of OX -locally free modulesPaE

∣∣a ∈RΛ

satisfying Pa(E)|X\D = Pb(E)|X\D =: E and some conditions.

θ (PaE)⊂Pa(E)⊗Ω1X (logD) for ∀a ∈ RΛ.

If (P∗E,θ ) is stable with par-c1(P∗E) = par-ch2(P∗) = 0, then (E,θ ) has an

essentially unique pluri-harmonic metric h adapted to the parabolic structure.

Construction of VHS by KH-correspondence

Kobayashi-Hitchin correspondence for VHS

Let X be a smooth projective variety with simply normal crossing hypersurface

D. Let (P∗E,θ ) be a regular filtered Higgs bundle such that

P∗E is equipped with a C∗-action ρ.

ρ−1t θ ρt = t ·θ .

(P∗E,θ ) is stable, and par-c1(P∗E) = par-ch2(P∗E) = 0.

Then, we obtain a polarized VHS on X \D.

By KH-correspondence for regular filtered Higgs bundles, we have a pluri-harmonic

metric h of (E,∂ E ,θ ) adapted to P∗E.

We have ρ∗t h = h for t ∈ S1 =

t ∈C∣∣ |t|= 1

.

ρ∗t h is a pluri-harmonic metric for (P∗E,t θ )≃ ρ∗t (P∗E,θ ). Hence, ρ∗t h is a

pluri-harmonic metric of (E,∂ E ,θ ) if |t|= 1. The uniqueness of the adapted

pluri-harmonic metric implies ρ∗t h = h.

Polarized VHS ⇐⇒ (E,∂ E ,θ ,h) with an S1-action ρ on (E,∂ E ,h) s.t. ρ∗t θ = tθ .

The weight decomposition E =⊕

Ei of the C∗-action is orthogonal with

respect to h. (t •v = t iv on E i.)

∇h preserves the decomposition E =⊕

Ei.

We have θ (E i)⊂ E i−1⊗Ω1,0X . Hence, we obtain θ †(E i)⊂ E i+1⊗Ω0,1.

We set F i :=⊕

j≥i E j. We have

(∂ E +θ †)F i ⊂ F i⊗Ω0,1 (∂E +θ )F i ⊂ F i−1⊗Ω1,0

(Note ∇1 = (∂ E +θ †)+(∂E +θ ).)

It means the filtration F i | i ∈ Z| gives a Hodge structure of (E,∇1).

Moreover, h gives a polarization.

(Polarized

VHS

)formal←→

(Homogeneous

harmonic bundle

)KH←→

(Homogeneous filtered

Higgs bundle (+α)

)

Construction of ∞2 -VHS

There are two methods to construct ∞2 -VHS.

“Motivic” construction

Quantum cohomology, Landau-Ginzburg model,...

Kobayashi-Hitchin correspondence (analytic, solve non-linear PDF)

Suppose that (X ,D) is equipped with a C∗-action ρ. Let (P∗E,θ ) be a

good filtered Higgs bundle on (X ,D) such that

P∗E is C∗-equivariant.

ρ∗t θ = tmθ (m 6= 0).

(P∗E,θ ) is stable, and par-c1(P∗E) = par-ch2(P∗) = 0.

Then, we obtain a ∞2 -VHS on X \D.

Kobayashi-Hitchin correspondence for good filtered Higgs bundles

Let X be a smooth projective variety with a simply normal crossing hypersurface

D =⋃

i∈Λ Di. Let (P∗E,θ ) be a good filtered Higgs bundle on (X ,D). If (P∗E,θ )

is stable with par-c1(P∗E) = par-ch2(P∗) = 0, then (E,θ ) has an essentially unique

pluri-harmonic metric h adapted to the parabolic structure.

Although the proof has not yet written down in the case that (P∗E,θ ) is not

tame, it can be given by the arguments for the cases of regular filtered Higgs

bundles and good filtered flat bundles.

Suppose P∗E is C∗-equivariant, and ρ∗t θ = tmθ . Then we have ρ∗t h = h for t ∈ S1.

Let v′ be the fundamental vector field of the S1-action. We have a natural

v′-action on E. We have

v′(h) = 0, v

′(θ ) =√−1mθ , [v′,∇h] = 0, v

′(θ †) =−√−1mθ †.

Such (E,∂ E ,θ ,h) is called homogeneous harmonic bundle.

Polarized ∞2 -VHS ⇐⇒ homogeneous harmonic bundle

Associated to the harmonic bundle (E,∂ E ,θ ,h), we obtain a holomorphic

vector bundle E =(

p−1E, p∗∂ E +λθ † +∂ λ)

with a meromorphic flat relative

connection ∇λ := ∇h +λθ † +λ−1θ on X := C×X (recall X 0 := 0×X):

∇λ : E −→ E ⊗Ω1X /C⊗O(X 0).

We set u :=√−1λ∂λ . We have a natural action of v′+mu on E for which[

v′+mu,∇λ ]= 0:

[v′,∇h] = [u,∇h] = 0, [v′,λ−1θ ] =√−1mθ =−[mu,λ−1θ ], [v′,λθ †] =−[mu,λθ †]

Then, ∇λ is extended to a meromorphic flat connection

∇ : E −→ E ⊗Ω1X

(logX0)⊗O(X 0)

by setting ∇λ∂λ(s) = (

√−1m)−1

((v′+mu)s−∇λ

v′ (s)). By construction, we have

∇v′+muE ⊂ E .

For an appropriate normalization of v′, we obtain ∞2 -VHS.

The hermitian metric h induces a polarization.

(Polarized

∞2 -VHS

)formal←→

(Homogeneous

harmonic bundle

)KH←→

(Homogeneous filtered

Higgs bundle (+α)

)

We can use general theory of harmonic bundle (such as limit mixed twistor

structure) to investigate ∞2 -VHS.

We shall apply it to study Toda equation.

Toda lattice with opposite sign

The original Toda equation is a non-linear ordinary differential

equation, which was introduced as a model of a mass-spring system:

d2

dt2 α j =−eα j−α j+1 + eα j−1−α j ( j ∈ Z)

αj−2•

αj−1•

αj•

αj+1•

αj+2•

• • • • •

eαj−αj+1

eαj−1−αj

Various versions of Toda equations have been studied from many

viewpoints.

Toda lattice with opposite sign

We would like to consider the following Toda equation:

∂z∂zwi− ewi−wi−1 + ewi+1−wi = 0 (i = 1, . . . ,n)

Here, wi : C∗ −→R, and wi = wi+n. (C∗ = C\ 0.)

It is natural to require some additional conditions:

(i) wi depend only on |z|.

(ii) wi + wn−i+1 = 0.

We call the equation “Toda lattice with opposite sign” by following M. Guest

and C.-S. Lin, although it might be the standard 2-dimensional Toda equation

for some people. It seems that the equation ∂z∂zwi + ewi−wi−1− ewi+1−wi = 0 has

been studied more intensively in differential geometry.

This equation appeared in the study of the physicist S. Cecotti and C.

Vafa in their study of the tt∗-equation for Frobenius manifolds.

Frobenius manifold

A complex manifold X whose tangent sheaf ΘX is equipped with

- Multiplication ΘX ×ΘX −→ΘX , (u,v) 7−→ u · v which is

commutative and associative.

- Non-degenerate inner product ΘX ×ΘX −→OX , (u,v) 7−→ 〈u,v〉such that 〈u · v,w〉= 〈u,v ·w〉.

- The multiplication is flat with respect to the Levi-Civita

connection of 〈·, ·〉.

∃ unity vector field, ∃ Euler field, satisfying some conditions.

If a Frobenius manifold comes from a significant model in physics, it

should be equipped with a natural real structure (a hermitian metric),

satisfying some non-linear differential equation (tt∗-equation).

Frobenius manifold is equipped with a flow given by the Euler field.

Cecotti and Vafa considered the restriction of the tt∗-equation to a

significant special orbit. They observed, the Toda equations appear from

several significant models.

(Example) From the An-minimal model, they obtained the following equation:

∂t∂tϕ1 +eϕ2−ϕ1−|t|2eϕ1−ϕn = 0

∂t∂tϕi +eϕi+1−ϕi −eϕi−ϕi−1 = 0 (i = 2, . . . ,n−1)

∂t∂tϕn + |t|2eϕ1−ϕn −eϕn−ϕn−1 = 0

It is transformed to the Toda lattice with opposite sign by the following change:

ϕi = wi +2i−n−1

2nlog|t|2, z =

nn+1

t(n+1)/n

Suppose n = 2, ϕ1 +ϕ2 = 0, and that ϕ2 depends only on x = |t|2. Let Y be

determined by√

xY((4/3)x3/4

)2= e2ϕ2(x), then the equation is transformed to

Y ′′ =(Y ′)2

Y− Y ′

z+Y 3− 1

Y

This is obtained from the following by setting α = β = 0, γ =−δ = 1:

Y ′′ =(Y ′)2

Y− Y ′

z+

1z(αY 2 +β )+ γY 3 +

δY

Based on the deep analysis on Painleve III, Cecotti-Vafa mathematically verified

the existence and uniqueness of the desired solutions in some cases.

Cecotti and Vafa asked when the associated ∞2 -VHS has a Z-structure.

A solution of the Toda equation induces a ∞2 -VHS, and it makes sense to ask

whether the underlying meromorphic flat bundle has a Z-structure.

If a solution comes from a physics, then the induced ∞2 -VHS should have a

Z-structure, as the Gauss-Manin connection has a Z-structure. So, the

classification might be useful for physics.

M. Guest and C.-S. Lin systematically studied the classification of the

real valued solutions of the Toda equation, and the existence of Z-

structure. They classified the solutions in the case that the equation

can be reduced to some equation with two unknown functions. And, col-

laborating with A. Its, they studied when the solution has a Z-structure.

Generalization (application of the theory of harmonic bundles, M)

We give a classification of R-valued solutions of the Toda lattice

with opposite sign by the parabolic weights.

We use the Kobayashi-Hitchin correspondence.

We give an algebraic criterion in terms of the parabolic weights,

for the existence of Z-structure.

We use the limit mixed twistor structure for harmonic bundle.

Remark

There are a huge amount of the studies on the Toda equation.

Although there is a systematic way to construct C-valued solutions of the

Toda equation, at this moment, it is not clear to me whether we can use

the method to classify the R-valued solutions.

According to Guest-Lin, there is a systematic way to construct R-valued

solutions for the equation ∂z∂zwi +ewi−wi−1−ewi+1−wi = 0. But, for the Toda

lattice with opposite sign, the method does not give global solutions.

Classification of the real valued solutions

Theorem Let a ∈R. We have a bijective correspondence between

solutions w = (wi) of the Toda lattice with opposite sign satisfying

∑ni=1 wi =

(−n(n−1)/2−a

)log|z|2

(ai | i = 1, . . . ,n) ∈ Rn such that

(i) a1≥ a2≥ ·· · ≥ an ≥ a1−n

(ii) ∑ni=1ai = a

Let w = (wi) be a solution of the Toda lattice with opposite sign as above. We

have the following estimate for some ai ∈ R and ki ∈ Z:

wi +(ai + i−1) log |z|2− ki

2log(− log|z|2

)= O(1) (|z| → 0)

The correspondence is given by (wi) 7−→ (ai).

We set E0 :=⊕n

i=1OX vi. Let θ0 be the Higgs field given by

θ0(vi) = vi+1dz (i = 1, . . . ,n−1), θ0(vn) = v1dz

Lemma We have a natural correspondence between

harmonic metrics of h0 of (E0,θ0) such that h0(vi,v j) = 0 (i 6= j)

solutions of the Toda equation (wi)

The correspondence is given by wi = logh(vi,vi).

Actually, it is easy to check that the Hitchin equation is exactly the Lax

form of the Toda equation.

(n = 3) We set wi := logh(vi,vi).

θ0(v1,v2,v3) = (v1,v2,v3)

0 0 1

1 0 0

0 1 0

dz

θ †0(v1,v2,v3) = (v1,v2,v3)

0 0 e−w3+w1

e−w1+w2 0 0

0 e−w2+w3 0

dz

∂h(v1,w2,w3) = (v1,v2,v3)

∂w1 0 0

0 ∂w2 0

0 0 ∂w3

The Hitchin equation[∂h +θ ,∂ +θ †

]= 0 ⇐⇒

∂z +

0 0 e−w3+w1

e−w1+w2 0 0

0 e−w2+w3 0

, ∂z +

∂zw1 0 1

1 ∂zw2 0

0 1 ∂zw3

= 0

⇐⇒ the Toda equation ∂z∂zwi−ewi−wi−1 +ewi+1−wi = 0 (i = 1, . . . ,n).

((wi) =⇒ (ai)) We set ei := zi−1vi. We have

θ0(ei) = ei+1dzz

, (i = 1, . . . ,n−1) θ0(en) = e1zn dzz

By a general theory of the asymptotic behaviour of tame harmonic bundles,

there exist ai ∈R and ki ∈ Z (i = 1, . . . ,n) such that

logh(ei,ei)+ai log|z|2−ki log(− log|z|2

)= O(1) (|z| → 0)

⇐⇒ wi +(ai + i−1) log |z|2− ki

2log(− log|z|2

)= O(1) (|z| → 0)

Because θ is bounded with respect to h and the Poincare metric around 0, we

have

a1 ≥ ·· · ≥ ai ≥ ai+1 ≥ ·· · ≥ an ≥ a1−n

We also have the following estimate around ∞, which follows from the vanishing of

the parabolic degree:

logh(ei,ei)+(−a/n− (n−1)/2− (i−1)

)log|z|2 = O(1) (|z| → ∞)

wi +(a/n+(n−1)/2

)log|z|2 = O(1) (|z| → ∞)

(ai) =⇒ (wi) We put a∞i :=−a/n− (n−1)/2+(i−1). For (c0,c∞) ∈ R2, we set

P(ai)c0,c∞ E0 :=

r⊕

i=1

OP1

([c0−ai] · 0+[c∞−a∞

i ] · ∞)

ei

The filtered Higgs bundle (P(ai)∗ E0,θ0) over (P1,0,∞) satisfies

(i) its restriction to C∗ is (E0,θ0),

(ii) it is regular at 0 and good at ∞,

(iii) a j = min

c∣∣e j ∈P

(ai)c E0 around 0

,

(iv) P(ai)∗ E0 is σ-equivariant, where σ is the automorphism of E0 given by

σ(e j) = τ je j for a primitive n-th root τ,

(v) deg(P(ai)∗ E0) = 0.

The conditions (i)–(v) characterize (P(ai)∗ E0,θ0) uniquely.

We consider an action of µn =

κ∣∣κn = 1

on P1 by (κ,z) 7−→ κz.

The filtered Higgs bundle (P(ai)∗ E0,θ0) is µn-equivariant by κ∗ei = ei. We obtain

the descent (P∗E ′,θ ′) on the quotient space P1/µn ≃ P1. It is stable, because

it is irreducible as a Higgs bundle.

According to the Kobayashi-Hitchin correspondence for wild harmonic

bundles on curves (Biquard-Boalch, Simpson), there exists a unique

pluri-harmonic metric h′ of (E ′,θ ′) adapted to P∗E ′.

It induces a pluri-harmonic metric h0 of (E0,θ0), adapted to the filtered

bundle P(ai)∗ E0.

Because σ−1 θ0 σ = τ θ0 with |τ|= 1, σ∗h is also a pluri-harmonic metric of

(E0,θ0) adapted to P∗E0. By the uniqueness, we obtain h = σ∗h.

It implies the orthogonality h(vi,v j) = 0 (i 6= j).

We have the C∗-action on P1, given by (t,z) 7−→ tz.

P∗E0 is naturally C∗-equivariant by t∗ei = t i−1ei. We have t∗θ0 = t θ0.

We have the associated ∞2 -VHS. (More precisely, we need to deal with

the poly-stable case.)

Integral structure

Let ~a = (ai) ∈Qn satisfying a1≥ a2≥ ·· · ≥ an ≥ a1−n. We have the

corresponding harmonic bundle (E0,∂ E0,θ0,h~a). We have the associated

meromorphic flat bundle (E~a, ∇~a).

Theorem (E~a, ∇~a) has a Z-structure, if and only if there exists γ ∈ C∗

such that

γn ∈Q

∏ni=1(T − γe2π

√−1ai/n) ∈ Z[T ].

If n is an odd prime, the condition is equivalent to ∏ni=1(T −e

√−1ai/n) ∈ Z[T ].

The proof consists of two parts.

Step 1 Explicit description of (E~a,∇~a).

Step 2 Explicit computation of the Stokes structure.

Step 1. Explicit description of (E~a,∇~a)

Let p : Cλ ×C∗z −→ C∗z be the projection. We have the following meromorphic flat

connection ∇ on p∗E0:

z∇z(e1, . . . ,en) = (e1, . . . ,en)

(−diag[a1, . . . ,an]+

1λ

K(n)(z)

)

λ ∇λ (e1, . . . ,en) = (e1, . . . ,en)(−diag[1,2, . . . ,n]+diag[a1, . . . ,an]−

1λ

K(n)(z)

)

Here, diag[a1, . . . ,an] denotes the diagonal matrix whose (i, i)-entries are ai, and

K (n)(z) is a matrix K(n)

i j = 1 (i = j +1), K(n)

1n = zn, and K(n)

i j = 0 otherwise.

K(3)(z) =

0 0 z3

1 0 0

0 1 0

Theorem We have an isomorphism (E~a, ∇~a)≃ (p∗E0, ∇).

We set X := C×P1, Da := C×a, and X 0 := 0×P1.

According to the general theory of the asymptotic behaviour of wild harmonic

bundles, we obtain a locally free OX (∗D∞)-module Q0E~a.

It is equipped with a meromorphic flat connection ∇~a. It is logarithmic along

D0, and it has Poincare rank 2 along X 0.

We set X := P1×P1, D∞ := P1×∞ and Xλ := λ×P1. We have X = X ∪X∞.

Proposition Q0E~a is extended to a locally free OX(∗D∞)-module Q0E~a such that

(tr-TLE-structure)

The restrictions to P1×z (z ∈C) are isomorphic to OrankP1 .

∇~a is logarithmic along X∞ with respect to Q0E~a.

We obtain the limit mixed twistor structure from the data of Q0E~a at 0.

In this case, the mixed twistor structure is naturally equipped with a

C∗-action. Hence, it comes from a mixed Hodge structure.

Once we obtain the mixed Hodge structure, by using Deligne’s canonical

decomposition of the mixed Hodge structure, we obtain the opposite

filtration, and we can construct the tr-TLE structure.

Proposition Q0E~a is extended to a locally free OX(∗D∞)-module Q0E~a such that

(tr-TLE-structure)

The restrictions to P1×z (z ∈C) are isomorphic to O rankP1 .

∇~a is logarithmic along X∞ with respect to Q0E~a.

Let p : P1×C∗ −→ C∗ denote the projection.

The proposition implies Q0E~a|P1×C∗∃ ≃ p∗E0, under which

∇λ~a = d +C +λ−1K (n)(z).

We obtain that C is diagonal by using the symmetry of ∇λ~a with respect to σ .

By using the relation of the monodromy and the parabolic weight, we can

determine C.

Because the derivative in the λ -direction is determined by the C∗-action and

∇λ , we can conclude that the meromorphic flat connection has the desired

form.

Step 2. Explicit computation of the Stokes structure

(Q0E~a, ∇~a)|P1×1 and (Q0E~a, ∇~a)|1×P1 are isomorphic.

(Q0E~a, ∇~a)|1×P1 is isomorphic to the meromorphic flat bundle (V ,∇)

on P1 given by

∇(e1, . . . ,en) = (e1, . . . ,en)(−diag[a1, . . . ,an]+K

(n)(z))

We consider when it has a Z-structure, i.e., the underlying local system

has a Z-structure compatible with the Stokes structure.

We have an action of µn = κ ∈ C |κn = 1 on P1. The meromorphic flat

bundle (V ,∇) is µn-equivariant by κ∗ei = ei.

Proposition (V ,∇) has a µn-equivariant Z-structure, if and only if

∏ni=1

(T − e2π

√−1ai/n

)∈ Z[T ].

The “only if” part is clear. If (V ,∇) has a µn-equivariant Z-structure, then

the descent on P1/µn also has a Z-structure. Because ∏ni=1

(T − e2π

√−1ai/n

)

is the characteristic polynomial of the monodromy, it is contained in Z[T ].

To prove the “if” part, we shall compute the Stokes factor.

Formal structure

We put τ := e2π√−1/n. We set b =−∑a j− (r−1)/2, ω1 := e2π

√−1b/n and

ω := e2π√−1b. Let C1 be the matrix such that

(C1)i j =

1 (i− j = 1)

ω (i, j = (n,1))

0 (otherwise)

C1 =

0 0 ω1

1 0 0

0 1 0

(n = 3)

There exists a formal frame ( p1, . . . , pn) of (V ,∇)|∞ such that

∇( p1, . . . , pn) = ( p1, . . . , pn)(

diag[τ−1, . . . ,τ−r ]+diag[b, . . . ,b]z−1)

dz

(τ−1)∗( p1, . . . , pn) = ( p1, . . . , pn)C1 ω−11

In the following, we assume b = 0 for simplicity.

C =

0 0 1

1 0 0

0 1 0

Stokes factor We consider the following sectors

S1 :=

z∣∣ |arg(z)− (π/2− ε)| < π/2

S2 :=

z∣∣ |arg(z)− (π/2+2π/n− ε)| < π/2

S1 S2

Each (V ,∇)|Sihas a flat frame ( pSi,1, . . . , pSi,n) such that

they are compatible with the Stokes filtrations,

(τ−1)∗( pS1,1, . . . , pS1,n) = ( pS2,1, . . . , pS2,n)C.

We have the following equalities for some α j,β j ∈C:

p j,S1 =

p j,S2 (0≤ j ≤ [(n−1)/2])

p j,S2 +α− j p− j,S2 +β− j p− j−1,S2 (−[n/2]≤ j < 0)

(We set α[n/2] = 0, if n is even.)

Re(τ iz) = Re(τ−iz)

Re(τ i−1z) = Re(τ−iz)

Lemma

n

∏i=1

(T −e2π√−1ai/n) = T n−

[(n−1)/2]

∑j=1

α jTn−2 j−

[n/2]

∑j=1

β jTn−2 j+1−1

We have the relations:

(τ−1)∗( pS1,1, . . . , pS1,n) = ( pS2,1, . . . , pS2,n)C

( pS1,1, . . . , pS1,n) = (τ−1)∗( pS1,1, . . . , pS1,n)M

( pS1,1, . . . , pS1,n) = ( pS2,1, . . . , pS2,n)A

C is a cyclic matrix, M represents the monodromy of the descent, A is the Stokes

factor. From M = C−1A, we obtain

L.H.S= det(T id−M

)= det

(T id−C−1A

)= R.H.S

If ∏ni=1(T−e2π

√−1ai/n)∈Z[T ], by using the frames (τ− j)∗( pS1,1, . . . , pS1,n) on τ j(S1),

we can construct a µn-equivariant Z-structure of (V ,∇) which is compatible with

the Stokes structure.