Alexander Thomas

HIGHER COMPLEX STRUCTURES AND FLAT CONNECTIONS

, Vladimir Fock and the author introduced a new geometric structure on surfaces, called higher complex structure, whose moduli space is conjecturally diffeomorphic to Hitchin's component. This would give a new geometric approach to higher Teichmüller theory. In this paper, we prove several steps towards this conjecture and give a precise picture what has to be done.

We show that higher complex structures can be deformed to flat connections. More precisely we show that the cotangent bundle of the moduli space of higher complex structures can be included into a 1-parameter family of spaces of flat connections.

Introduction

In [START_REF] Hitchin | Lie Groups and Teichmüller Space[END_REF], Nigel Hitchin describes, for a Riemann surface S, a connected component of the character variety Rep(π 1 (S), PSL n (R)) = Hom(π 1 (S), PSL n (R)) PSL n (R) which he parametrizes by holomorphic differentials. These components are called Hitchin components and their study higher Teichmüller theory. His approach uses Higgs bundle theory, more precisely the hyperkähler structure of the moduli space of polystable Higgs bundles. These components can also be described by representation-theoretic methods.

For PSL 2 (R), Hitchin's component is Teichmüller space, which is the moduli space of various geometric structures on the underlying smooth surface Σ, for example complex structures or hyperbolic structures. Thus, the question naturally arises whether there is a geometric structure on Σ whose moduli space gives Hitchin's component for higher rank.

In [START_REF] Fock | Higher complex structures[END_REF] a candidate for such a geometric structure is constructed, called the higher complex structure or n-complex structure, since it generalizes the complex structure. The higher complex structure can be seen as a special sl n -valued 1-form. In local coordinates it is given by Φ = Φ 1 dz + Φ 2 dz where (Φ 1 , Φ 2 ) is a pair of commuting nilpotent matrices. The construction of the n-complex structure uses the punctual Hilbert scheme of the plane and is reviewed in section 2.

The group of symplectomorphisms of T * Σ acts on 1-forms, so on the higher complex structure. We denote by T n the moduli space of higher complex structures. A prominent role is played by the cotangent bundle T * T n . Its elements consist of a higher complex structure and a cotangent vector, described by a set of holomorphic differentials.

In this paper we prove several steps towards a canonical diffeomorphism between the moduli space of higher complex structures T n and Hitchin's component. Before giving the structure of the paper, we give a comparison to Hitchin's approach, which motivates and clarifies our ideas.

1.1. Comparison to Hitchin's approach. Hitchin's approach to construct components in the character variety is to use the hyperkähler structure of the moduli space of Higgs bundles

1 M H 0 M H ∞ ↓ ⊕ n i=2 H 0 (K i ) Hitchin section Hitchin fibration Rep(π 1 Σ, G C ) ≅ A G T * T n 0 T * T n ∞ ↓ T n zero- section canonical projection Rep(π 1 Σ, G C ) ≅ (A P) Symp 0 Figure 1.1.
Twistor space for Higgs bundles and T * T n M H . One starts from a Riemann surface S, i.e. a smooth surface Σ equipped with a fixed complex structure. Then one considers Higgs bundles on S, i.e. pairs of a holomorphic bundle V and a holomorphic End(V )-valued 1-form Φ, the Higgs field. We summarize this approach in one picture: the twistor space of M H , which encodes all Kähler structures at once.

To a hyperkähler manifold M one associates the twistor space X M = CP 1 × M endowed with the complex structure at the point (λ, m) given by I λ,m = (I 0 , I λ ) where I 0 is the standard structure of CP 1 and I λ is the complex structure of M associated to λ ∈ CP 1 . The projection X M → CP 1 is holomorphic and a holomorphic section is called a twistor line. With some extra data, it is possible to reconstruct the hyperkähler manifold M as the space of all real twistor lines. This is a result of [HKLR] (theorem 3.3).

On the left hand side of figure 1.1 we draw the twistor space of the moduli space of Higgs bundles M H . In one complex structure, say at λ = ∞, we have the moduli space of Higgs bundles M H (with its complex structure coming from the one of S). For λ = 0, we see the conjugated complex structure. In all other λ, we see the complex structure of the character variety Rep(π 1 (Σ), G C ), which can be seen as hamiltonian reduction of the space of all connections A by all the gauge transformations G (Atiyah-Bott reduction for unitary gauge). Going from λ = 0 to λ = 1 is the non-abelian Hodge correspondence. Finally, there is the Hitchin fibration going from M H to a space of holomorphic differentials. This fibration admits a section whose monodromy, via the non-abelian Hodge correspondence, is in the split real form. For G = SL n (C), we get flat PSL n (R)-connections.

In our approach, we start from a smooth surface Σ which we equip with a higher complex structure (which can vary). This structure is locally given by a 1-form Φ = Φ 1 dz + Φ 2 dz where (Φ 1 , Φ 2 ) is a pair of commuting nilpotent matrices. The role of M H is played by the cotangent bundle T * T n to the moduli space of higher complex structures, which is conjecturally hyperkähler near the zero-section (see discussion around conjecture 2.6).

On the right-hand side of figure 1.1, we draw the conjectural twistor space of T * T n . In complex structure at λ = ∞, we see the cotangent bundle T * T n . At the opposite point λ = 0 we see some conjugated structure T * T n . In all other complex structures, we see the character variety Rep(π 1 (Σ), G C ), this time obtained as a double reduction of the space of connections A, by some parabolic subgroup P of all gauges, and then by higher diffeomorphisms Symp 0 (details in section 3). The analog of the Hitchin fibration is simply the projection map T * T n → T n and the analog of the Hitchin section is the zero-section T n ⊂ T * T n . A flat connection associated to a point of the zero-section T n ⊂ T * T n should have real monodromy.

We stress again that most of the right-hand side is conjectural. In this paper we • describe the double reduction space (A P) Symp 0 and show that it is a space of flat connections, • include this space into a family of flat h-connections (h = λ -1 ) • show that at the limit λ → ∞ we get T * T n , • give partial results for the existence of twistor lines, i.e. a canonical deformation of T * T n to flat connections, • prove the diffeomorphism between T n and Hitchin's component assuming the existence of twistor lines.

1.2. Summary and structure. In section 2, we review the construction of the higher complex structure. In particular, we describe the cotangent bundle T * T n in subsection 2.2. Then we give some new aspects: we describe a bundle induced by the n-complex structure in 2.3 and the conjugated structure in 2.4. The space of flat connections is constructed in section 3 by a double hamiltonian reduction: starting from the space of all connections A, we reduce with respect to a parabolic subgroup P of the gauge group, those which fix a given direction. We show that symplectomorphisms of T * Σ act by gauge on A P and that the double reduction is a space of flat connections (see corollary 3.4). It can also be described as a space of pairs of commuting differential operators. We perform this double reduction for h-connections in section 4, such that in the limit when h goes to zero we get T * T n (see theorem 4.6).

We then investigate how to get a flat connection from a higher complex structure. We first put the connections we look at in some standard form in 5.1. We then give partial results and ideas of the existence of a canonical deformation of T * T n to flat connections in 5.4. Finally under the assumption that this canonical deformation exists, we prove that our moduli space T n is diffeomorphic to Hitchin's component in theorem 5.10.

We include three appendices: in the first appendix A we give some facts about the punctual Hilbert scheme of the plane. In appendix B and C we prove two technical points.

Notations. Throughout the paper, Σ denotes a smooth closed surface of genus g ≥ 2. A complex local coordinate system on Σ is denoted by (z, z) and its conjugate coordinates on T * C Σ by p and p. The canonical bundle is K = T * (1,0) Σ. The space of sections of a bundle B is denoted by Γ(B). The hamiltonian reduction (or symplectic reduction, or Marsden-Weinstein quotient) of a symplectic manifold X by a group G is denoted by X G where the reduction is over the zero-coadjoint orbit. The equivalence class of some element a is denoted by [a].

Acknowledgments. I warmly thank Vladimir Fock for all the ideas and discussions he shared with me.

Higher complex structures

We recall here the construction of higher complex structures, their moduli space and the cotangent bundle to its moduli space. All details can be found in [START_REF] Fock | Higher complex structures[END_REF]. The main ingredient for the higher complex structure is the punctual Hilbert scheme of the plane (see also appendix A). We also briefly discuss a bundle induced by the higher complex structure and a conjugated structure.

2.1. Higher complex structures. A complex structure on a surface is characterized by the Beltrami differential µ ∈ Γ(K -1 ⊗ K) where K is the canonical bundle. It determines the notion of a local holomorphic function f by the condition ( ∂ -µ∂)f = 0. The Beltrami differential determines a linear direction in T C Σ, the direction generated by the vector ∂ -µ∂. Since ∂-µ∂ and ∂-μ ∂ have to be linearly independent, we get the condition µμ ≠ 1. Replacing the tangent bundle T Σ by the cotangent bundle T * Σ, we can say that the complex structure is entirely encoded in a section of P(T * C Σ).

The idea of higher complex structures is to replace the linear direction by a polynomial direction, or more precisely a n-jet of a curve inside T * C Σ. To get a precise definition, we use the punctual Hilbert scheme of the plane, denoted by Hilb n (C 2 ) which is defined by

Hilb n (C 2 ) = {I ideal of C[x, y] dim C[x, y] I = n}.
A generic point in Hilb n (C 2 ) is an ideal whose algebraic variety is a collection of n distinct points in C 2 . A generic ideal can be written as

⟨-x n + t 1 x n-1 + ... + t n , -y + µ 1 + µ 2 x + ... + µ n x n-1 ⟩.
Moving around in Hilb n (C 2 ) corresponds to a movement of n particles in C 2 . But whenever k particles collide the Hilbert scheme retains an extra information: the (k -1)-jet of the curve along which the points entered into collision. The zero-fiber, denoted by Hilb n 0 (C 2 ), consists of those ideals whose support is the origin. A generic point in Hilb n 0 (C 2 ) is of the form ⟨x n , -y + µ 2 x + µ 3 x 2 + ... + µ n x n-1 ⟩ which can be interpreted as a (n -1)-jet of a curve at the origin (see appendix A for details).

We can now give the definition of the higher complex structure:

Definition 2.1 (Def.2 in [START_REF] Fock | Higher complex structures[END_REF]). A higher complex structure of order n on a surface Σ, in short n-complex structure, is a section I of Hilb n 0 (T * C Σ) such that at each point z ∈ Σ we have I(z) + Ī(z) = ⟨p, p⟩, the maximal ideal supported at the origin of T * C z Σ. Notice that we apply the punctual Hilbert scheme pointwise, giving a Hilbert scheme bundle over Σ. The condition on I + Ī ensures that I is a generic ideal, so locally it can be written as

I(z, z) = ⟨p n , -p + µ 2 (z, z)p + µ 3 (z, z)p 2 ... + µ n (z, z)p n-1 ⟩.
The coefficients µ k are called higher Beltrami differentials. A direct computation gives

µ k ∈ Γ(K 1-k ⊗ K).
The coefficient µ 2 is the usual Beltrami differential. In particular for n = 2 we get the usual complex structure.

The punctual Hilbert scheme admits an equivalent description as a space of pairs of commuting operators. To an ideal I of C[x, y] of codimension n, one can associate the multiplication operators by x and by y in the quotient C[x, y] I, denoted by M x and M y . This gives a pair of commuting operators. Conversely, to two commuting operators (A, B) we can associate the ideal I(A, B) = {P ∈ C[x, y] P (A, B) = 0}. For details see A.2 in the appendix. The zero-fiber Hilb n 0 (C 2 ) corresponds to nilpotent commuting operators. From this point of view, a higher complex structure is a gauge class of special matrix-valued 1-forms locally of the form Φ 1 dz + Φ 2 dz where (Φ 1 , Φ 2 ) is a pair of commuting nilpotent matrices with Φ 1 principal nilpotent (which means of maximal rank n -1).

To define a finite-dimensional moduli space of higher complex structures, we have to define some equivalence relation. It turns out that the good notion is the following: Definition 2.2 (Def.3 in [START_REF] Fock | Higher complex structures[END_REF]). A higher diffeomorphism of a surface Σ is a hamiltonian diffeomorphism of T * Σ preserving the zero-section Σ ⊂ T * Σ setwise. The group of higher diffeomorphisms is denoted by Symp 0 (T * Σ).

Symplectomorphisms act on T * C Σ, so also on 1-forms. This is roughly how higher diffeomorphisms act on the n-complex structure, considered as the limit of an n-tuple of 1-forms.

We then consider higher complex structures modulo higher diffeomorphisms, i.e. two structures are equivalent if one can be obtained by the other by applying a higher diffeomorphism. Locally, all n-complex structures are equivalent: Theorem 2.3 (Theorem 1 in [START_REF] Fock | Higher complex structures[END_REF]). The n-complex structure can be locally trivialized, i.e. there is a higher diffeomorphism which sends the structure to (µ 2 (z, z), ..., µ n (z, z)) = (0, ..., 0) for all small z ∈ C.

We define the moduli space of higher complex structures, denoted by T n , as the space of n-complex structures modulo higher diffeomorphisms. The main properties are given in the following theorem: Theorem 2.4 (Theorem 2 in [START_REF] Fock | Higher complex structures[END_REF]). For a surface Σ of genus g ≥ 2 the moduli space T n is a contractible manifold of complex dimension (n 2 -1)(g -1). Its cotangent space at any point µ = (µ 2 , ..., µ n ) is given by

T * µ T n = n ⊕ m=2 H 0 (Σ, K m ).
In addition, there is a forgetful map T n → T n-1 and a copy of Teichmüller space T 2 → T n .

The forgetful map in coordinates is just given by forgetting the last Beltrami differential µ n . The copy of Teichmüller space is given by µ 3 = ... = µ n = 0 (this relation is unchanged under higher diffeomorphisms).

We notice the similarity to Hitchin's component, especially the contractibility, the dimension and the copy of Teichmüller space inside. At the end of the paper in section 5 we indicate how to link T n to Hitchin's component. Assuming a strong conjecture (an analog of the nonabelian Hodge correspondence in our setting), we prove that T n is canonically diffeomorphic to Hitchin's component in theorem 5.10. 2.2. Cotangent bundle of higher complex structures. The main object to link higher complex structures to character varieties is the total cotangent bundle T * T n which we describe here in detail.

The punctual Hilbert scheme inherits a complex symplectic structure from C 2 . It can be described as follows: to an ideal I ∈ Hilb n (C 2 ), associate the two multiplication operators M x and M y . The symplectic structure ω is given by ω = tr dM x ∧ dM y .

The zero-fiber is an isotropic subspace of dimension n -1. The dimension of Hilb n (C 2 ) being 2n, the zero-fiber cannot be Lagrangian. The subspace Hilb n red (C 2 ), called reduced Hilbert scheme, consisting of those ideals I whose support has barycenter the origin (generically n points with barycenter equal to the origin), is a symplectic submanifold of Hilb n (C 2 ) and the zero-fiber is Lagrangian inside the reduced Hilbert scheme. Hence its cotangent bundle is isomorphic to its normal bundle (using the symplectic form):

T * Hilb n 0 (C 2 ) ≅ T normal Hilb n 0 (C 2 ) ≈ Hilb n red (C 2
). Near the zero-section, the normal bundle is isomorphic to the whole space, here the reduced Hilbert scheme.

There is a general fact stating that the cotangent bundle to a quotient space X G (where X is a manifold and G a Lie group) is a hamiltonian reduction: T * (X G) ≅ T * X G. Using this we can compute

T * T n = T * Γ(Hilb n 0 (T * C Σ)) Symp 0 (T * Σ) = Γ(T * Hilb n 0 (T * C Σ)) Symp 0 (T * Σ) = Γ(T normal Hilb n 0 (T * C Σ)) Symp 0 (T * Σ) = Γ(Hilb n red (T * C Σ)) Symp 0 (T * Σ) mod t 2 . (2.1)
We see that T * T n is obtained by a hamiltonian reduction of Hilb n red (T * C Σ). An element of the latter Hilbert scheme bundle is an ideal

I = ⟨p n -t 2 p n-2 -... -t n , -p + µ 1 + µ 2 p + ... + µ n p n-1 ⟩.
The coefficient µ 1 is an explicit function of the other variables. So the 2n -2 variables (t k , µ k ) 2≤k≤n form a coordinate system. The fact that the normal bundle is only the total space near the zero-section is expressed by "modulo t 2 ", meaning that all quadratic or higher terms in the t k have to be dropped.

To compute the moment map, we have to understand with more detail the action of higher diffeomorphisms on the Hilbert scheme bundle. The ideal I has two generators which we put into the form p n -P (p) and -p+Q(p) where P (p) = t 2 p n-2 +...+t n and Q(p) = µ 2 p+...+µ n p n-1 . A higher diffeomorphism generated by some Hamiltonian H acts on I by changing the two polynomials. Their infinitesimal variations δP and δQ are given by δP = {H, p n -P (p)} mod I δQ = {H, -p + Q(p)} mod I.

(2.2) Remark. One can easily show that only the class H mod I acts, i.e. H 1 and H 2 with H 1 = H 2 mod I have the same action. △ Using these variation formulas, one can compute the moment map which gives:

Theorem 2.5 (Theorem 3 in [START_REF] Fock | Higher complex structures[END_REF]). The cotangent bundle to the moduli space of n-complex structures is given by

T * T n = (µ 2 , ..., µ n , t 2 , ..., t n ) µ k ∈ Γ(K 1-k ⊗ K), t k ∈ Γ(K k ) and ∀k (-∂ +µ 2 ∂ +k∂µ 2 )t k + n-k l=1 ((l+k)∂µ l+2 + (l+1)µ l+2 ∂)t k+l = 0 Symp 0 (T * Σ)
We call the condition coming from the moment map condition (C). It is a generalized holomorphicity condition: for µ k = 0 for all k, we simply get ∂t k = 0.

The punctual Hilbert scheme Hilb n (C 2 ) inherits a hyperkähler structure from C 2 (see [START_REF] Nakajima | Lectures on Hilbert Schemes of Points on Surfaces[END_REF]). This should induce a hyperkähler structure on T * T n :

Conjecture 2.6. The cotangent bundle T * T n admits a hyperkähler structure near the zerosection.

There are three good reasons to believe in the conjecture:

• Construction by hyperkähler quotient: Equation 2.1 points towards a possible hyperkähler reduction. Indeed under some mild conditions a complex symplectic reduction X G C is isomorphic to a hyperkähler quotient X G R . In our case X = Γ(Hilb n red (T * C Σ)) is hyperkähler, since Hilb n red (C 2 ) is. So it is plausible that T * T n can be obtained as HK quotient of Γ(Hilb n red (T * C Σ)) by the real group Symp 0 (T * Σ), so it gets a hyperkähler structure itself. Notice that the complexified Lie algebra of Symp 0 (T * Σ), i.e. the space of smooth complex-valued functions on T * Σ, has the same action on X as the real Lie algebra since one can prove that a Hamiltonian H acts the same as H mod I.

• Feix-Kaledin structure: If Hitchin's component and our moduli space T n are diffeomorphic, Hitchin's component gets a complex structure. With its Goldman symplectic structure, there is good hope to get a Kähler structure. A general result of Feix and Kaledin (see [START_REF] Feix | Hyperkähler metrics on cotangent bundles[END_REF] and [START_REF] Kaledin | Hyperkähler structures on total spaces of holomorphic cotangent bundles[END_REF]) asserts that for a Kähler manifold X, there is a neighborhood of the zero-section in T * X which admits a hyperkähler structure. • Construction by twistor approach: The 1-parameter deformation of T * T n described in this paper is a good candidate to be the twistor space of T * T n (see figure 1.1).

Induced bundle.

To any point in the Hilbert scheme bundle Hilb n (T * C Σ), we can canonically associate a vector bundle V of rank n whose fiber over a point z is C[p, p] I(z).

We can glue these fibers together which gives a n-dimensional vector bundle over Σ. Locally, there is basis of the form (s, ps, ..., p n-1 s) for some generic section s. Under a coordinate change z ↦ w(z), this basis transforms in a diagonal way since p k ↦ ( dw dz ) k p k , so we get a bundle which is a direct sum of line bundles.

The matrix viewpoint of the punctual Hilbert scheme gives a sl n -valued 1-form M p dz+M pdz which acts on this bundle (locally M p by multiplication by p, M p by multiplication by p). If I is a higher complex structure, then we get the 1-form Φ 1 dz + Φ 2 dz where (Φ 1 , Φ 2 ) is a pair of commuting nilpotent matrices. In that case there is a preferred direction in each fiber : the common kernel of both Φ 1 and Φ 2 . In our local basis it is generated by p n-1 s.

The idea of this paper is to deform the 1-form Φ 1 dz +Φ 2 dz to a flat connection. The bundle V itself gets deformed under this procedure. This is how we will associate a flat connection to a higher complex structure.

Conjugated higher complex structures.

There is a natural notion of conjugated space to T * T n using the natural complex conjugation on the complexified cotangent bundle T * C Σ. We associate to an ideal I ∈ Hilb n red (T * C Σ) the ideal Ī and then take Symp 0 -equivalence classes.

In coordinates, we start from

I = ⟨p n -t 2 p n-2 -... -t n , -p + µ 1 + µ 2 p + ... + µ n p n-1 ⟩.
To get the conjugated structure, we have to express Ī in the same form as I, i.e. as

Ī = ⟨p n -t2 pn-2 -... -tn , -p + μ1 + μ2 p + ... + μn pn-1 ⟩ = ⟨p n -2 tp n-2 -... -n t, -p + 1 µ + 2 µp + ... + n µp n-1 ⟩.
where ( k t, k µ) are the parameters of the conjugate to T * T n . It is possible to explicitly express the conjugated coordinates ( k t, k µ) k in terms of (t k , µ k ) k . For example one gets 2 µ = 1 μ2 and n t = μn 2 tn .

Parabolic connections and reduction

In this section, we describe the generic fiber of the twistor space of T * T n from figure 1.1 which is a space of flat connections.

The idea about the deformation of T * T n is to replace the polynomial functions on T * Σ by differential operators. The higher complex structure is given by two polynomials (the generators of I), so in the deformation one gets a pair of differential operators.

The space of pairs of differential operators can be obtained by a reduction of all connections by some specific parabolic gauge. This procedure was first introduced by Bilal, Fock and Kogan in [START_REF] Bilal | On the origins of W-algebras[END_REF]. In that paper, the authors also describe some ideas for generalized complex and projective structures. Our higher complex structures are the mathematically rigorous version of their ideas. Our treatment of the parabolic reduction is independent of their paper and follows some other notation. The question about how to impose the commutativity condition on the differential operators remained open in their paper. We show that the answer is given by a second reduction with respect to the group of higher diffeomorphisms.

3.1. Atiyah-Bott reduction. Before going to the parabolic reduction, we recall the classical reduction of connections by gauge transforms, developed by Atiyah and Bott in their famous paper [START_REF] Michael | The Yang-Mills equations over Riemann surfaces[END_REF].

Let Σ be a surface and G be a semisimple Lie group with Lie algebra g. Let E be a trivial G-bundle over Σ. Denote by A the space of all g-connections on E. It is an affine space modeled over the vector space of g-valued 1-forms Ω 1 (Σ, g). Further, denote by G the space of all gauge transforms, i.e. bundle automorphisms. We can identify the gauge group with G-valued functions: G = Ω 0 (Σ, G).

On the space of all connections A, there is a natural symplectic structure given by

ω = Σ tr δA ∧ δA
where tr denotes the Killing form on g (the trace for matrix Lie algebras). Since A is an affine space, its tangent space at every point is canonically isomorphic to Ω 1 (Σ, g). So given

A ∈ A and A 1 , A 2 ∈ T A A ≅ Ω 1 (Σ, g), we have ωA (A 1 , A 2 ) = ∫ Σ tr A 1 ∧ A 2 . Note that ω is constant (independent of A) so dω = 0.
Further, the 2-form ω is clearly antisymmetric and non-degenerate (since the Killing form is). Remark finally that this construction only works on a surface. We can now state the famous theorem of Atiyah-Bott (see end of chapter 9 in [START_REF] Michael | The Yang-Mills equations over Riemann surfaces[END_REF] for unitary case, see section 1.8 in Goldman's paper [START_REF] William | The symplectic nature of fundamental groups of surfaces[END_REF] for the general case): the action of gauge transforms on the space of connections is hamiltonian and the moment map is the curvature. Thus, the hamiltonian reduction A G is the moduli space of flat connections.

Let us explain the moment map with more detail: the moment map m is a map from A to Lie(G) * . The Lie algebra Lie(G) is equal to Ω 0 (Σ, g), so its dual is isomorphic to Ω 2 (Σ, g) via the pairing ∫ Σ tr. On the other hand, given a connection A, its curvature F (A) is a g-valued 2-form, i.e. an element of Ω 2 (Σ, g). Hence, the map m is well-defined.

Parabolic reduction.

3.2.1. Setting and coordinates. In subsection 2.3 we have seen how to associate a rank nbundle V over Σ to a higher complex structure. Moreover we have seen that there is a privileged direction in each fiber, the common kernel of Φ 1 and Φ 2 . This gives a line-subbundle L in V . We want to mimic the Atiyah-Bott reduction with the extra constraint of fixing L. That is why we consider the subspace of gauge transformations fixing the subbundle L (more precisely its dual).

Let us take the same setting as for the Atiyah-Bott reduction with G = SL n (C). But instead of all gauge transforms G, we consider the subgroup P ⊂ G consisting of matrices of the form

⎛ ⎜ ⎜ ⎜ ⎝ * ⋯ * * ⋮ ⋮ ⋮ * ⋯ * * 0 ⋯ 0 * ⎞ ⎟ ⎟ ⎟ ⎠
i.e. preserving the last direction in the dual space. We want to compute and analyze the hamiltonian reduction A P, which we call space of parabolic connections.

Remark. The reason to consider a fixed direction in the dual bundle and not in the bundle itself is purely of technical advantage. △

Since P ⊂ G, we know by the Atiyah-Bott theorem that the action of P on the space of connections A is hamiltonian with moment map m ∶ A ↦ i * F (A) where i ∶ P ↪ G is the inclusion and i * ∶ Lie(G) * ↠ Lie(P) * the induced surjection on the dual Lie algebras. Since G = SL n (C), the map i * is explicitly given by forgetting the first n -1 entries in the last column. This means that m -1 ({0}) is the space of all A ∈ A such that the curvature

F (A) is of the form ⎛ ⎜ ⎜ ⎜ ⎝ 0 ⋯ 0 ξ n ⋮ ⋮ ⋮ ⋮ ⋮ ξ 2 0 ⋯ 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ .
In order to give a description in coordinates of the hamiltonian reduction A P, we fix a reference complex structure on the surface Σ. We take a connection A ∈ A and decompose it into its holomorphic and anti-holomorphic parts: A = A 1 + A 2 . As a covariant derivative, we set ∇ = ∂ + A 1 and ∇ = ∂ + A 2 . Using the parabolic gauge, it is possible to reduce A 1 locally to a companion matrix:

(3.1) A 1 ∼ ⎛ ⎜ ⎜ ⎜ ⎝ tn 1 ⋮ ⋱ t2 1 0 ⎞ ⎟ ⎟ ⎟ ⎠ dz.
The existence of such a gauge is proven in the appendix B. Reducing A 1 to the above form means that we choose a basis of the form B = (s, ∇s, ∇ 2 s, ..., ∇ n-1 s). This takes all the gauge freedom. A connection in

A P verifies [∇, ∇]∇ i s = 0 for i = 0, 1, ..., n -2 since [∇, ∇] = F (A)
is the curvature which is concentrated on the last column. It follows that ∇∇ i s = ∇ i ∇s for all i = 1, ..., n -1. Thus, the connection is fully described by ∇ n s and ∇s. We can write these expressions in the basis B:

(3.2) ∇ n s = tn s + tn-1 ∇s + ... + t2 ∇ n-2 s = P (∇)s (3.3) ∇s = μ1 s + μ2 ∇s + ... + μn ∇ n-1 s = Q(∇)s.
Notice that t1 = 0 since tr A 1 = 0. The second part A 2 is uniquely determined by its first column given by equation (3.3). Since ∇∇ i s = ∇ i ∇s for i = 1, ..., n -1, the i-th column of A 2 is given by applying (i -1) times ∇ to the first column. We get a 1-form of the following type:

(3.4) A 2 ∼ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ μ1 ∂μ 1 + μn tn ⋯ μ2 μ1 + ∂μ 2 + μn tn-1 ⋯ ⋮ ⋮ ⋮ μn-1 μn-2 + ∂μ n-1 + μn t2 ⋯ μn μn-1 + ∂μ n ⋯ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ dz.
Remark. Notice that modulo ∂ (meaning that you drop all terms with a partial derivative), equations (3.2) and (3.3) become the relations of p n and p in a generic ideal of Hilb n red (C 2 ). So A 1 and A 2 become the multiplication operators by p and p respectively. △

The functions (μ 2 , ..., μn , t2 , ..., tn ) completely parameterize A P since it is possible to express μ1 in terms of these using that the second matrix is traceless. We call an element of A P a parabolic connection. We consider A P as a subspace of A by using the representative A 1 +A 2 with A 1 of the local form 3.1 and A 2 like in 3.4. Its parabolic curvature is concentrated on the last column:

[∇, ∇]∇ n-1 s = ξ n s + ξ n-1 ∇s + ... + ξ 2 ∇ n-2 s.
The following proposition allows to compute the parabolic curvature easily.

Proposition 3.1 (Parabolic curvature). [∇ n , ∇]s = ∑ n k=2 ξ k ∇ n-k s.
Proof. Since the first n -1 columns of the curvature F (A) are 0, we have [∇, ∇]∇ i s = 0 for i = 0, 1, ..., n -2. Using Leibniz's rule and induction on k, we can prove that [∇ k , ∇]s = 0 for k = 1, ..., n -1. Indeed, it is true for k = 1 and we have

[∇ k+1 , ∇]s = ∇[∇ k , ∇]s + [∇, ∇]∇ k s = 0 whenever k ≤ n -2.
Therefore, we get

[∇ n , ∇]s = ∇[∇ n-1 , ∇]s + [∇, ∇]∇ n-1 s = [∇, ∇]∇ n-1 s = n k=2 ξ k ∇ n-k s
by the last column of the curvature.

Inside the non-commutative ring of differential operators, we define the left-ideal Î = ⟨∇ n -P , -∇ + Q⟩ where P and Q are defined in equations (3.2) and (3.3) respectively. We can express the previous proposition as

[∇ n , ∇] = n k=2 ξ k ∇ n-k mod Î.
Notice finally that our coordinates (μ 2 , ..., μn , t2 , ..., tn ) do not behave like tensors under coordinate change z ↦ w(z). We will see in the following section 4 that if we introduce a parameter λ we get at the semiclassical limit tensors out of our coordinates.

3.2.2.

Example n = 2. Consider a parabolic SL(2, C)-connection locally written as A = A 1 dz + A 2 dz. The first matrix A 1 is a companion matrix of the form 0 t2 1 0 . Let us compute the transformed matrix A 2 . It is the image of the operator ∇ in a basis (s, ∇s). Put ∇s = μ1 s + μ2 ∇s. The second column can be computed using ∇∇s = ∇ ∇s -[∇, ∇]s = ∇ ∇s and ∇ 2 s = t2 s. Since the trace of the matrix is zero, we get μ1 = -1 2 ∂μ 2 . Hence

A 2 = -1 2 ∂μ 2 -1 2 ∂ 2 μ2 + t2 μ2 μ2 1 2 ∂μ 2 .
The curvature is of the form 0 ξ 2 0 0 where

ξ 2 = ( ∂ -μ2 ∂ -2∂μ 2 ) t2 + 1 2 ∂ 3 μ2 .
Suppose that the curvature ξ 2 is 0. We can then look for flat sections Ψ = (ψ 1 , ψ 2 ). The first condition (∂ + A 1 )Ψ = 0 gives ψ 1 = -∂ψ 2 and

(∂ 2 -t2 )ψ 2 = 0.
The second condition ( ∂ + A 2 )Ψ = 0 only gives one extra condition:

( ∂ -μ2 ∂ + 1 2 ∂μ 2 )ψ 2 = 0.
For μ2 = 0 this just means that ψ 2 is holomorphic and we get an ordinary differential equation (∂ 2 -t2 )ψ 2 = 0. For μ2 ≠ 0, the second condition is still a holomorphicity condition, but with respect to another complex structure. For general n, a flat section Ψ = (ψ k ) 1≤k≤n is of the form ψ n-k = ∂ k ψ n and there are two equations on ψ n . The first equation comes from the last column in A 1 , so directly generalizes to (∂ n -t1 ∂ n-1 -... -tn )ψ n = 0. The generalized holomorphicity condition comes from the last row in A 2 :

(3.5) (-∂ + αnn + αn,n-1 ∂ + ... + αn,1 ∂ n-1 )ψ n = 0
where αij denote the entries of A 2 which have an explicit but complicated expression in terms of the μk and tk .

3.3. Higher diffeomorphisms and flat connections. To get from parabolic connections to flat connections, we define an action of higher diffeomorphisms on the space of parabolic connections A P. We prove that this action is hamiltonian and show that the double reduction A P Symp 0 (T * Σ) is a space of flat connections.

3.3.1. Action of higher diffeomorphisms. Recall that the description in coordinates of the space of parabolic connections relies on a basis B of the form (s, ∇s, ..., ∇ n-1 s). A variation δs of the section s can be expressed in this basis:

δs = v 1 s + v 2 ∇s + ... + v n ∇ n-1 s = Ĥs where Ĥ = v 1 + v 2 ∇ + ... + v n ∇ n-1 is a differential operator of degree n -1.
The Lie algebra of higher diffeomorphisms Lie(Symp 0 (T * Σ)) is the space of functions on T * Σ which can be deformed to differential operators on Σ. The infinitesimal action of higher diffeomorphisms on parabolic connections is given by a base change induced by s ↦ s+εδs such that the basis B preserves its form. More specifically, to a higher diffeomorphism generated by

H = v 2 p + v 3 p 2 + ... + v n p n-1 we associate the variation Ĥ = v 1 + v 2 ∇ + v 3 ∇ 2 + ... + v n ∇ n-1
where v 1 is uniquely determined by the other v i by the condition that the infinitesimal gauge transform is of trace zero.

Remark. This only defines the infinitesimal action. The question about how to integrate the action to the whole group Symp 0 , or maybe to a deformation of it, has to be worked out. △ Let us describe how to compute the matrix X describing the infinitesimal base change induced by a higher diffeomorphism. Write the base change as (s, ∇s, ..., ∇ n-1 s) ↦ (s, ∇s, ..., ∇ n-1 s) + ε(δs, ∇δs, ..., ∇ n-1 δs).

So the first column of X is just given by Xs = δs = v 1 s + v 2 ∇s + ... + v n ∇ n-1 s. The second is given by X∇s = ∇δs = ∇(v 1 s + v 2 ∇s + ... + v n ∇ n-1 s). We notice that the construction of this matrix X is exactly the same as for the matrix A 2 (see equation (3.4)) with the only difference that the variables in A 2 are called μk instead of v k . Since both matrices are traceless, even the terms v 1 and μ1 coincide. Notice that if X is a parabolic gauge, i.e. the (n -1) first entries of the last column are zero, then v k = 0 ∀k, so X = 0.

Proposition 3.2. The matrix X of the gauge coming from a higher diffeomorphism is given by X = A 2 μk ↦v k .

Let us indicate how to compute the action of a higher diffeomorphism on our coordinates ( tk , μk ). The coordinates tk are given by the relation ∇ n s = P s where P = t2 ∇ n-2 + ... + tn . The variation δ P satisfies

∇ n (s + ε Ĥs) = ( P + εδ P )(s + ε Ĥs) which gives (3.6) δ P = [ Ĥ, -∇ n + P ] mod Î where Î = ⟨∇ n -P , -∇ + Q⟩ is a left ideal of differential operators.
Similarly, the coordinates μk are given by ∇s = Qs where Q = μ1 + μ2 ∇... + μn ∇ n-1 . We can easily compute the variation of Q to be

(3.7) δ Q = [ Ĥ, -∇ + Q] mod Î.
Remark. In the case of the Hilbert scheme, we used the variation formula from symplectic geometry δf = df dt = {H, f } (see equation (2.2)). Here we find the deformed version of this: δ P = [ Ĥ, P ] where Ĥ is the quantum Hamiltonian and P some operator. In the next section, we introduce a deformation parameter h and we get δ P (h) = 1 h [ Ĥ(h), P (h)]. △ 3.3.2. Double reduction to flat connections. We have just seen that higher diffeomorphisms act on the space of parabolic connections by gauge transforms. Since we see A P as a subset of A and since the gauge action on A is hamiltonian, we see that the action of higher diffeomorphisms on A P is also hamiltonian. It is not surprising that the moment map is nothing else than the parabolic curvature:

Theorem 3.3. The infinitesimal action of higher diffeomorphisms Symp 0 (T * Σ) on the space of parabolic connections A P is hamiltonian with moment map

m( ti , μj ).(v 2 , ..., v n ) = Σ n i=1
x n,n+1-i ξ i where x i,j are the matrix elements of the gauge X and ξ i is the parabolic curvature of the parabolic connection described by ( ti , μi ) 2≤i≤n .

Some explanation for the moment map is necessary: m goes from the space A P, which is described by coordinates ( ti , μj ), into Lie(Symp 0 ) * , the dual to the Lie algebra of higher diffeomorphisms. The Lie algebra of higher diffeomorphisms is described by Hamiltonians of the form v 2 p + ... + v n p n-1 . To such a function, we compute the associated matrix X (see proposition 3.2) from which we take the last row for computing m. All elements of X are functions depending on the v k and the tk . The parabolic curvature described by the ξ i is a function of ( ti , μj ).

Proof. Our computation is analogous to the Atiyah-Bott reduction. An infinitesimal gauge transform given by X affects A 1 and A 2 by

χ(A 1 ) = [X, A 1 ] -∂X χ(A 2 ) = [X, A 2 ] -∂X
The symplectic form on A P is the restriction of the one on A, so we can compute

ι χ ω A P = tr (χ(A 1 )δA 2 -χ(A 2 )δA 1 ) = tr([X, A 1 ] -∂X)δA 2 -([X, A 2 ] -∂X)δA 1 = tr([A 1 , δA 2 ] + δ∂A 2 -[A 2 , δA 1 ] -δ ∂A 1 )X = tr δ(∂A 2 -∂A 1 + [A 1 , A 2 ])X = δ tr F (A)X = δ n i=1 x n,n+1-i ξ i . Therefore m = Σ n i=1
x n,n+1-i ξ i .

Corollary 3.4. The double reduction A P Symp 0 gives a space of flat connections:

A P Symp 0 ≅ {A 1 + A 2 ∈ A P ξ i = 0 ∀i} Symp 0
with A 1 locally of the form 3.1 and A 2 like in 3.4.

The corollary directly follows from the previous theorem since m( ti , μj )(v 2 , ..., v n ) = 0 for all v 2 , ..., v n implies ξ i = 0 for all i.

We call the double reduction space A P Symp 0 the space of flat parabolic connections. For n = 2 we get those flat SL 2 (C)-connections whose monodromy is the developing map of a complex projective structure on Σ. For general n, we probably get a complicated subset of the space of all flat connections.

Parabolic reduction of h-connections

In this section, we study the parabolic reduction on h-connections to get the twistor space description from figure 1.1. The main idea is the following: a point in T * T n is a Symp 0equivalence class of ideals of the form

I = ⟨-p n + t 2 p n-2 + ... + t n , -p + µ 1 + µ 2 p + ... + µ n p n-1 ⟩.
Replace the polynomials by h-connections using the rule p ↦ ∇ = h∂ + A 1 (h) and p ↦ ∇ = h ∂ + A 2 (h) where h is a formal parameter. This corresponds to the deformation of a higher complex structure Φ to Φ + hd

+ hA + h 2 Φ * = h(d + λΦ + A + λ -1 Φ * ) where λ = h -1 .
For h ≠ 0 we divide the connection by h to get a usual connection with parameter λ. For all λ ∈ C * fixed, we get the same space as described in the previous section 3, i.e. the space of flat parabolic connections. For λ → ∞ we get the cotangent bundle T * T n . For λ → 0 we get the space of conjugated structures [( k t, k µ)] (see subsection 2.4). 4.1. Parametrization. Take A(λ) = λΦ+A+λ -1 Φ * where Φ is in the Hilbert scheme bundle Hilb n 0 (T * C Σ) and Φ * in the conjugated bundle. Recall that this means that locally Φ(z, z

) = Φ 1 (z, z)dz+Φ 2 (z, z)dz with Φ 1 ∈ sl n is a principal nilpotent element and Φ 2 is in the centralizer of Φ 1 , i.e. [Φ 1 , Φ 2 ] = 0. We define A 1 (λ) = λΦ 1 + A 1 + λ -1 Φ * 2 and A 2 (λ) = λΦ 2 + A 2 + λ -1 Φ * 1 , i.e.
the (1, 0)-part and (0, 1)-part of A(λ). We also define

∇ = ∂ + A 1 (λ) and ∇ = ∂ + A 2 (λ).
As for the case without parameter, there is a parabolic gauge which transforms A(λ) locally to (4.1)

⎛ ⎜ ⎜ ⎜ ⎝ tn (λ) 1 ⋮ ⋱ t2 (λ) 1 0 ⎞ ⎟ ⎟ ⎟ ⎠ dz + ⎛ ⎜ ⎜ ⎜ ⎝ μ1 (λ) μ2 (λ) αij (λ) ⋮ μn (λ) ⎞ ⎟ ⎟ ⎟ ⎠ dz
where αij (λ) and μ1 (λ) are explicit functions of the other variables. Thus, the space is parametrized by ( ti (λ), μi (λ)) i=2,...,n .

This local representative comes from a basis of the form (s, ∇s, ..., ∇ n-1 s) for some section s. We then get our coordinates by (4.2)

∇ n s = tn (λ)s + tn-1 (λ)∇s + ... + t2 (λ)∇ n-2 s (4.3) ∇s = μ1 (λ)s + μ2 (λ)∇s + ... + μn (λ)∇ n-1 s.

You can compute the αij (λ) using ∇∇ k s = ∇ k ∇s for k ≤ n -1 which holds since the curvature [∇, ∇] is concentrated in the last column.

Example 4.1. Take n = 2 and consider

Φ 1 = 0 0 b 1 0 , A 1 = ( a 0 a 1 a 2 -a 0 ), Φ 2 = µ 2 Φ 1 and A 2 = -A 1 . So we have A 1 (λ) = a 0 a 1 + λ -1 μ2 b1 a 2 + λb 1 -a 0 and A 2 (λ) = -ā 0 -ā 2 + λ -1 b1 -ā 1 + λµ 2 b 1 ā0 .
We look for P =

p 1 p 2 0 1 p 1 such that P A 1 (λ)P -1 + P ∂P -1 = 0 t2 (λ) 1 0 .
Multiplying by P from the right, one can solve the system. One finds

p 1 = (λb 1 + a 2 ) 1 2 and p 2 = -a 0 p 1 + ∂p 1 p 2 1 . Hence t2 (λ) = λa 1 b 1 + constant term + λ -1 μ2 a 2 b1 .
Transforming A 2 (λ) with P we get

μ2 (λ) = -ā 1 + λµ 2 b 1 λb 1 + a 2 = -ā 1 + λµ 2 b 1 λb 1 -μ2 ā1
where we used a 2 = -μ 2 ā1 coming from the flatness of A(λ).

For λ → ∞, we can develop the rational expression of μ2 (λ) to get

μ2 (λ) = µ 2 + (µ 2 μ2 -1) ∞ k=1 μk-1 2 āk 1 b k 1 λ -k .
For λ → 0, we get

μ2 (λ) = 1 μ2 + (1 -µ 2 μ2 ) ∞ k=1 b k 1 μk+1 2 āk 1 λ k .
Notice that we get 2 µ = 1 μ2 as leading term (see section 2.4). △

The example shows several phenomena which are true in general:

Proposition 4.2. The μk (λ) are rational functions in λ. The highest term in λ when λ → ∞ is λ 2-k µ k where µ k is the higher Beltrami differential from the n-complex structure. For λ → 0 we get as lowest term λ k-2 k µ where k µ is the conjugated n-complex structure. The tk (λ) are also rational functions in λ. For λ → ∞, the highest term is given by λ k-1 t k , and the lowest term for λ → 0 is given by λ 1-k k t where

t k = tr A 1 Φ k-1 1 and k t = tr A 1 (Φ * 2 ) k-1 .
We will see later that (µ k , t k ) is a point of the cotangent bundle T * T n , and that ( k µ, k t) is the conjugated structure, which justifies the notation.

Proof. The whole point is to analyze equations (4.2) and (4.3) in detail. Let us start with ∇s = μ1 (λ)s + μ2 (λ)∇s + ... + μn (λ)∇ n-1 s.

Since ∇s = ( ∂ + λΦ

2 + A 2 + λ -1 Φ *
1 )s the highest λ-term is λΦ 2 s = λµ 2 Φ 1 s + ... + λµ n Φ n-1 1 s. On the other side, the highest term of ∇ k s is λ k Φ k 1 s for 0 ≤ k ≤ n -1. For generic s the set (s, Φ 1 s, ..., Φ n-1 1 s) is a basis. Hence, we can compare the highest terms and deduce that for λ → ∞:

μk (λ) = λ 2-k µ k + lower terms.
Similarly, the set (s, Φ * 2 s, ..., Φ * (n-1) 2 s) is generically a basis. Comparing highest terms and using Φ * 1 = 2 µΦ * 2 + ... + n µΦ * (n-1) 2

, we get for λ → 0:

μk (λ) = λ k-2 k µ + higher terms.
In any case, we can decompose ∇ k s and ∇s in the basis (s, Φ 1 s, ..., Φ n-1 1 s) and notice that the defining equations for μk is a quotient of two polynomials in λ, i.e. μk is a rational function in λ. The same decomposition gives that tk is a rational function in λ.

The last thing is to study the asymptotic behavior of tk . For that, we have to study

∇ n s = tn (λ)s + tn-1 (λ)∇s + ... + t2 (λ)∇ n-2 s.
The highest term of ∇ n s is not λ n Φ n 1 since Φ n 1 = 0. The next term is given by

λ n-1 n-1 l=0 Φ l 1 ○ (∂ + A 1 ) ○ Φ n-1-l 1 s
where ○ denotes the composition of differential operators. On the other side, the highest terms are given by tk λ n-k Φ n-k 1 s. When λ goes to infinity, we compare coefficients in the basis (s, Φ 1 s, ..., Φ n-1 1 s) as before. Using Dirac's "bra-ket" notation, we get

λ n-k tk = λ n-1 ⟨Φ n-k 1 s n-1 l=0 Φ l 1 ○ (∂ + A 1 ) ○ Φ n-1-l 1 s⟩ = λ n-1 n-k l=0 ⟨Φ n-k-l 1 s (∂ + A 1 ) ○ Φ n-1-l 1 s⟩ = λ n-1 n-k l=0 ⟨Φ n-k-l 1 s (∂ + A 1 ) ○ Φ k-1 1 Φ n-k-l 1 s⟩ = λ n-1 tr(∂ + A 1 ) ○ Φ k-1 1 = λ n-1 tr A 1 Φ k-1
1 . In the last line, we used that tr ∂ ○ Φ k-1 1 = 0 since Φ 1 is strictly lower triangular which is preserved under derivation. This precisely gives the expression for t k as stated in the proposition. The same analysis goes through for λ → 0.

At the end of subsection 3.2.1 we have noticed that tk and μk do not transform as tensors. We now show that the highest terms, t k and µ k , are tensors. Recall that K = T * (1,0) Σ is the canonical bundle and that Γ(.) denotes the space of sections. Proposition 4.3. We have

t i ∈ Γ(K i ) and µ i ∈ Γ(K 1-i ⊗ K).
Proof. Consider a holomorphic coordinate change z ↦ w(z). We compute how µ i (z) and t i (z) change.

For µ i , notice that Φ 1 dz ↦ Φ 1 dz dw dw, so using Φ 2 dz = µ 2 (z)Φ 1 dz + ... + µ n Φ n-1 1 dz n-1 we easily get µ i (z) = dz d w (dz dw) i-1 µ i (w). For t i , we use t i = tr(Φ i-1 1 A 1 ) where Φ 1 and A 1 are both (1, 0)-forms, thus t i is a (i, 0)-form, i.e. a section of K i . 4.2. Action of higher diffeomorphisms. In 3.3 we have described an infinitesimal action of Symp 0 (T * Σ) on the space of parabolic connections A P. Recall that to write a representative of an element of A P, we use a basis of the form (s, ∇s, ..., ∇ n-1 s). A higher diffeomorphism changes the section s and thus the whole basis. The same action holds for the parabolic h-connections. In particular, corollary 3.4 about flat parabolic connections stays true.

Here we analyze the infinitesimal action of Symp 0 on A(h) P, in particular what it does on the highest terms µ k and t k . There are two steps: a local analysis and a global analysis. 4.2.1. Local analysis. We prove that the action of higher diffeomorphisms on the highest terms µ k of the parabolic reduction is precisely the action on the n-complex structure. So we can trivialize it locally.

Take a change of section δs = v1 s + v2 ∇s + ... + vn ∇ n-1 s = Ĥs. We have previously seen in equation (3.7) that the change of coordinates δμ k can be computed by

δ Q = [ Ĥ, Q] mod Î
where Î = ⟨-∇ n + t2 ∇ n-2 + ... + tn , -∇ + μ1 + μ2 ∇ + ... + μn ∇ n-1 ⟩ is a left-ideal in the space of differential operators.

Since we have a parameter λ in our setting, the variations vk also depend on λ. More precisely, for k ≥ 2 we have that vk (λ) is a rational function in λ with highest term λ 2-k v k when λ → ∞. Notice that v1 is not a free parameter, but depends on the others. It assures that the trace of the gauge transform is zero. One can compute that v1 has highest term of degree 0.

Remark. It is not clear for the moment how to determine the precise expression for vk (λ) from a higher diffeomorphism generated by some Hamiltonian H. The highest λ-terms in vk (λ) are given by the coefficients of H = v 2 p + ... + v n p n-1 . △

We can now state:

Theorem 4.4. The infinitesimal action of Symp 0 (T * Σ) on the highest terms µ k of the coordinates μk (λ) of the space of parabolic connections with parameter is the same as the infinitesimal action of higher diffeomorphisms on the n-complex structure.

The reason for the theorem to be true is roughly speaking that the Poisson bracket is the semi-classical limit of commutators of differential operators. The strategy of the proof is the following: we prove the theorem first for µ 2 , and then for µ k (k > 2) supposing µ 2 = ... = µ k-1 = 0 which simplifies the computations. From [START_REF] Fock | Higher complex structures[END_REF] proposition 3, we know that the infinitesimal action of a Hamiltonian H = v 2 p + ... + v n p n-1 on the higher Beltrami differentials is given by

δµ 2 = ( ∂ -µ 2 ∂ + ∂µ 2 )v 2
for µ 2 and for µ k , supposing µ 2 = ... = µ k-1 = 0, we simply have

δµ k = ∂v k .
Proof. First, we compute the variation of µ 2 using equation (3.7):

δμ 1 + δμ 2 ∇ + ... + δμ n ∇ n-1 = [v 1 + v2 ∇ + ... + vn ∇ n-1 , -∇+μ 1 +μ 2 ∇ + ... + μn ∇ n-1 ] mod Î.
Since the highest λ-term of μ2 is of degree 0, we are interested in the part of degree 0 of the coefficient of

∇ in [v 1 + v2 ∇ + ... + vn ∇ n-1 , -∇ + μ1 + μ2 ∇ + ... + μn ∇ n-1 ] mod Î.
We first look on contributions coming from [v k ∇ k-1 , μl ∇ l-1 ] for k, l ≥ 2: If k + l -3 < n then we do not reduce modulo Î, so the highest term in λ is of degree 4 -(k + l). Since we have k, l ≥ 2, the highest term comes from k = l = 2, which gives v 2 ∂µ 2µ 2 ∂v 2 . If k + l -3 ≥ n, we can have terms with ∇ m with n ≤ m ≤ k+l-3. So we have to use Î to reduce it. This reduction gives ∇ m = c(λ)∇ + other terms, and the highest term of c(λ) is of degree m -2 ≤ k + l -5. Hence, the highest term for

[v k ∇ k-1 , μl ∇ l-1 ] is 4 -(k + l) + k + l -5 = -1.
The contributions from μ1 and v1 also have degree at most -1. There is one more contribution in degree 0 coming from [v 2 ∇, -∇], which gives ∂v 2 . Therefore, we have

δµ 2 = ( ∂ -µ 2 ∂ + ∂µ 2 )v 2 .
Now, suppose µ 2 = ... = µ k-1 = 0 and compute the variation δµ k under an action generated by vk ∇ k-1 + ... + vn ∇ n-1 . From

δμ k ∇ k-1 + ... + δμ n ∇ n-1 = [v k ∇ k-1 + ... + vn ∇ n-1 , -∇ + μ1 + μ2 ∇ + ... + μn ∇ n-1 ] mod Î
we can analyze as above the contribution to the term of degree 2k of the coefficient of ∇ k-1 . Since vl is of degree at most 2l and μl of degree at most 1l for l < k (since we suppose that µ l = 0), we can see that [v l ∇ l-1 , μm ∇ m-1 ] cannot contribute to the highest degree. The only contribution comes from the term with -∇. Thus,

δµ k = ∂v k .
This concludes the proof since the action of higher diffeomorphisms on the n-complex structure has the same expression.

Corollary 4.5. Under the action of higher diffeomorphisms, we can locally render Φ 2 = 0.

The corollary directly follows from the previous theorem and the fact that the higher complex structure can be locally trivialized (theorem 2.3), i.e. we can render µ 2 = ... = µ n = 0 locally and since

Φ 2 = µ 2 Φ 1 + ... + µ n Φ n-1 1 this implies Φ 2 = 0.
Remark. We see that a term vk ∇ k-1 can influence μi with i < k (unlike the case higher complex structures where H acts like H mod I), but it does not influence the highest term µ i . In the same vein, a term vk ∇ k-1 with k > n acts on parabolic connections, but not on the highest terms. △ 4.2.2. Global analysis. We show that the highest term in λ in the zero-curvature condition relates (µ k , t k ) to the cotangent bundle T * T n . We know that the moment map of the hamiltonian action of Symp 0 (T * Σ) on A P is given by ξ k = 0, i.e. the remaining curvature of a parabolic connection has to vanish. For connections with parameter λ, this gives ξ k (λ) = 0.

Theorem 4.6. The highest term in λ of ξ k (λ) = 0 gives the condition (C) of the cotangent bundle T * T n (see theorem 2.5).

The proof strategy is to reduce the analysis of the highest term in the parabolic curvature to the expression ξ k mod t2 mod ∂ 2 . The following lemma shows that we then get condition (C).

Lemma 4.7. The parabolic curvature modulo t2 and ∂ 2 gives condition (C) on T * T n :

ξ k = ( ∂ -μ 2 ∂ -k∂μ k ) tk - n-k l=1 ((l+k)∂μ l+2 + (l+1)μ l+2 ∂) tk+l mod t2 mod ∂ 2 .
You find the proof of this technical lemma in appendix C. Using the lemma, we can prove theorem 4.6:

Proof. From the explicit expression of ξ k (λ), we know that only derivatives, tk 's and μk 's appear. Since we are only interested in the highest term, we can replace tk by λ k-1 t k and μk by λ 2-k µ k . Hence, we get an expression which is a tensor, since both t k and µ k are tensors (by proposition 4.3). Since one term is ∂t k , we know that the highest term of ξ k (λ) is a section of K k ⊗ K and is of degree k -1 in λ.

In addition, we know that every term in ξ k , apart from ∂t k , has at least one partial derivative ∂, which adds a K-factor to the tensor. The rest is thus at most of type K k-1 ⊗ K. The K-factor comes from a unique µ m in each term. Once this µ m fixed, only partial derivatives ∂ and t k 's contribute to the K-factor.

Since t k comes with a factor λ k-1 , we see that whenever there is a term with a factor t i t j , the contribution in λ is λ i+k-2 which is not optimal, since t i+j would contribute with λ i+j-1 . In the same vein, whenever there is a term with at least two ∂, so that the rest is a tensor of type at most K k-2 ⊗ K, this term does not have an optimal contribution in λ.

Therefore, the highest term in ξ k (λ) is the same as in ξ k (λ) mod t2 mod ∂ 2 . Finally, the statement of the previous lemma 4.7 concludes the proof of theorem 4.6.

With the previous theorem, we now understand the global meaning of the highest terms (µ k , t k ): the µ k are the higher Beltrami differentials coming from the higher complex structure, whereas the t k are a cotangent vector to that higher complex structure. We can say that the semi-classical limit of A P Symp 0 is T * T n , which confirms the twistor space picture 1.1.

Remark. We have seen in proposition 4.2 that t k = tr Φ k-1 1 A 1 . The previous theorem applied for trivial n-complex structure µ k = 0 ∀k gives ∂t k = 0. It can be checked directly that ∂ tr Φ k-1

1 A 1 = 0 using the flatness of A(λ). △

The question remains how to determine the coefficients of lower degree in μk and tk . This will be discussed in the next section.

Conjectural geometric approach to Hitchin components

In this section, we try to construct an analog to the non-abelian Hodge correspondence in our setting: the existence and uniqueness of real twistor lines. We give partial results and conjectures. Assuming the existence of real twistor lines, we prove a canonical diffeomorphism between higher complex structures and Hitchin components.

Consider A(λ) = λΦ + A + λ -1 Φ * where Φ = Φ 1 + Φ 2 is given by an n-complex structure. Now, we look at A(λ) as a twistor line, i.e. a section of the twistor space. We impose the reality condition -A(-1 λ) * = A(λ).

Notice that -1 λ is the diametrically opposed point of λ in CP 1 . For trivial n-complex structure the * -operator is the hermitian conjugate M * = M = M ⊺ . Intrinsically, the operation A ↦ -A * is an antiholomorphic involution which corresponds to the compact real form of sl n .

Remark. For general higher complex structure, the real structure * has to be defined in such a way that Symp 0 preserves it. Notice also that we need a hermitian structure on the bundle. △ 5.1. Standard form. We start with A(λ) = λΦ + A + λ -1 Φ * as above and reduce it to a standard form.

Lemma 5.1. There is a unitary gauge such that Φ 1 becomes lower triangular with entries of coordinates (i + 1, i) given by positive real numbers of the form e ϕ i for all i = 1, ..., n -1.

Proof. The gauge acts by conjugation on Φ 1 (z). Since Φ 1 (z) is nilpotent, for every z ∈ Σ, there is an invertible matrix G(z) ∈ GL n (C) such that GΦ 1 G -1 is strictly lower triangular. Since Φ(z) varies smoothly with z, so does G(z). We omit the dependence in z in the sequel of the proof. We decompose G as G = T U where T is lower triangular (not strict) and U is unitary (Gram-Schmidt). Then the matrix U Φ 1 U -1 = T -1 (GΦ 1 G -1 )T is already lower triangular. So we have conjugated Φ 1 to a lower triangular matrix via a unitary gauge.

Finally, we use a diagonal unitary gauge to change the arguments of the matrix elements with coordinates (i + 1, i) to zero. Since Φ 1 is principal nilpotent, all these elements are non-zero, so strictly positive real numbers which can be written as e ϕ i with ϕ i ∈ R.

Notice that the unitary gauge preserves the operation * , so the form λΦ + A + λ -1 Φ * is preserved. Now, we show that for µ = 0, the matrix A 1 is upper triangular. Notice the importance of Φ 1 being principal nilpotent.

Lemma 5.2. For Φ 2 = 0 (trivial higher complex structure) and Φ 1 lower triangular, the flatness of A(λ) implies that A 1 is upper triangular.

Proof. We write A 1 = A l + A u where A l and A u are respectively the strictly lower and the (not strictly) upper part of A 1 . Thus we have A 2 = -A * l -A * u . The flatness condition at the term λ gives

0 = ∂Φ 1 + [Φ 1 , A * u ] + [Φ 1 , A * l ].
Since the first two terms are lower triangular (the operation * exchanges upper and lower triangular matrices), so is the third term [Φ 1 , A * l ]. A simple computation shows that a commutator between a principal nilpotent lower triangular matrix and a non-zero strictly upper triangular matrix can never be strictly lower triangular. Thus, A l = 0. 5.2. Case n = 2 and n = 3. Let us study the examples of smallest rank, those with n = 2 and n = 3. We work locally, so we can suppose that the n-complex structure is trivial, i.e. µ k = 0 for k = 2, 3. We use the standard form from subsection 5.1.

For n = 2, write Φ 1 = 0 0 e ϕ 0 , A 1 = ( a 0 a 1 a 2 -a 0 ) and A 2 = -A 1 . So we have

A(λ) = a 0 a 1 a 2 + λe ϕ -a 0 dz + -ā 0 -ā 2 + λ -1 e ϕ -ā 1 ā0 dz.
Notice that this is example 4.1 with µ 2 = 0 and b 1 = e ϕ . The flatness equation gives

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a 2 e ϕ = 0 ∂ϕ = -2ā 0 ∂a 1 = 2ā 0 a 1 ∂ā 0 + ∂a 0 = -a 1 ā1 -e 2ϕ .
The first equation gives a 2 = 0, the second a 0 = -∂ϕ 2 , the third is automatic once we write a 1 = t 2 e -ϕ , where t 2 = tr Φ 1 A 1 is the holomorphic quadratic differential. Finally, the last equation gives

∂ ∂ϕ = e 2ϕ + t 2 t2 e -2ϕ
which is the so-called cosh-Gordon equation, which is elliptic for small t 2 . So we see that the flat connection is uniquely determined by µ 2 = 0, t 2 and a solution to the cosh-Gordon equation. More details for this case can be found in [START_REF] Fock | Cosh-Gordon equation and quasi-Fuchsian groups[END_REF], in particular a link to minimal surface sections in Σ × R.

For n = 3, take Φ 1 = c 1 b 2 c 2 . As for n = 2 the matrix A 1 is upper triangular. Thus, we get

A(λ) = ⎛ ⎜ ⎝ a 0 b 0 c 0 λc 1 a 1 b 1 λb 2 λc 2 a 2 ⎞ ⎟ ⎠ dz + ⎛ ⎜ ⎝ -ā 0 λ -1 c1 λ -1 b2 -b0 -ā 1 λ -1 c2 -c 0 -b1 -ā 2 ⎞ ⎟ ⎠ dz.
With a diagonal gauge, we can suppose c 1 = e ϕ 1 , c 2 = e ϕ 2 ∈ R + . Further, the expressions for the holomorphic differentials are

t 3 = tr Φ 2 1 A 1 = c 0 c 1 c 2 and t 2 = tr Φ 1 A 1 = b 0 c 1 + b 1 c 2 + b 2 c 0 , hence c 0 = t 3 e -ϕ 1 -ϕ 2 and b 1 = -e ϕ 1 -ϕ 2 b 0 -b 2 t 3 e -2ϕ 2 -ϕ 1 .
The flatness condition and the zero trace condition then give a 0 = -2 3 ∂ϕ 1 -1 3 ∂ϕ 2 , a 1 = 1 3 ∂ϕ 1 -1 3 ∂ϕ 2 and a 2 = -a 0a 1 . Let us consider the case where t 2 = t 3 = 0. Then c 0 = 0 and b 1 = -e ϕ 1 -ϕ 2 b 0 . The remaining equations of the flatness are

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂b 2 = b 2 ( ∂ϕ 1 + ∂ϕ 2 ) -b0 (e ϕ 2 + e 2ϕ 1 -ϕ 2 ) -∂b 0 = b 0 ∂ϕ 1 + b2 e ϕ 2 2∂ ∂ϕ 1 = 2e 2ϕ 1 -e 2ϕ 2 + b 2 b2 + b 0 b0 (2 -e 2ϕ 1 -2ϕ 2 ) 2∂ ∂ϕ 2 = 2e 2ϕ 2 -e 2ϕ 1 + b 2 b2 + b 0 b0 (-1 + 2e 2ϕ 1 -2ϕ 2 ).
For b 0 = b 2 = 0 we get the Toda integrable system for sl 3 . This is the same solution as the one obtained from the non-abelian Hodge correspondence applied to the principal nilpotent Higgs field. We see that we need some extra data in order to impose b 0 = b 2 = 0. The two variables b 0 and b 2 are solutions to a system of differential equations. Thus, we only need some initial conditions.

For t 2 = 0 and t 3 ≠ 0, if we impose b 0 = b 1 = b 2 = 0 and ϕ 1 = ϕ 2 = ϕ, the flatness becomes T ¸it ¸eica's equation

(5.1) 2∂ ∂ϕ = e 2ϕ + t 3 t3 e -4ϕ .
From [START_REF] Loftin | Cubic Differentials in the Differential Geometry of Surfaces in Handbook of Teichmüller Theory[END_REF], we know that T ¸it ¸eica's equation is linked to affine spheres, minimal embeddings and Hitchin representations. Before going to the general case, we push the similarity to Higgs bundles further by choosing a special gauge. 5.3. Higgs gauge. Up to now, we have seen the flat connection A(λ) in two gauges. The first, which we call symmetric gauge, is the form A(λ) = λΦ + A + λ -1 Φ * where A 2 = -A * 1 and the * -operator is the hermitian conjugate. The second, which we call parabolic gauge and which in the literature is sometimes called W -gauge or Drinfeld-Sokolov gauge, is the form described in equation (4.1) where our parameters tk (λ) and μk (λ) appear. The existence of parabolic gauge (see subsection B.1) assures that one can go from the symmetric to the parabolic gauge. In Higgs theory, there is a third gauge used, which we call Higgs gauge, characterized by A 2 = 0 and by the fact that Φ 1 is a companion matrix. Here we show that for trivial higher complex structure, there exists the Higgs gauge in our setting.

We start with the existence of the Higgs gauge for trivial higher complex structure. We denote by E -the sum of the negative simple roots, i.e.

E -= 0 1 0 ⋱ ⋱ 1 0 .
Proposition 5.3. For µ = 0 and a flat connection λΦ + A + λ -1 Φ * in symmetric gauge, there is a gauge P which is lower triangular transforming Φ 1 to E -and A 2 to 0.

Proof. The statement is equivalent to the following two equations:

P Φ 1 = E -P and P A 2 -∂P = 0.
The first matrix equation allows to express all entries p i,j of P in terms of the last row (p n,k ) 1≤k≤n .

We then put Φ 1 = P -1 E -P into the flatness equation 0

= ∂Φ 1 + [A 2 , Φ 1 ].
After some manipulation, we get 0 = [E -, ( ∂P )P -1 -P A 2 P -1 ].

We know that the centralizer of E -are polynomials in E -. Hence we get

∂P -P A 2 = ⎛ ⎜ ⎜ ⎜ ⎝ 0 w 2 0 ⋮ ⋱ ⋱ w n ⋯ w 2 0 ⎞ ⎟ ⎟ ⎟ ⎠ P.
Looking at the n equations given by the last row, we can choose (p n,k ) 1≤k≤n such that w 2 = ... = w n = 0. Therefore ∂P = P A 2 , i.e. A 2 is transformed to 0.

In the Higgs gauge, our flat connection takes the following form:

Proposition 5.4. We suppose µ = 0. The flat connection A(λ) in Higgs gauge is locally given by (λE -+ A)dz + λ -1 E * -dz where the * -operation is given by M * = HM H -1 for some hermitian matrix H. Further, we have tr E k -A = t k+1 and A = -(∂H)H -1 . Proof. From the existence of Higgs gauge, we know that Φ 1 = E -and A 2 = 0. Since µ = 0, we also have Φ 2 = 0. A direct computation shows that if P denotes the matrix from the Higgs gauge, the matrix Φ * 1 transforms to P P E -(P P ) -1 . So H = P P which is indeed a hermitian matrix. Since P is lower triangular, t k+1 = tr Φ k 1 A 1 transforms to t k+1 = tr E k -A. Finally, since A 2 = P -1 ∂P and A 2 = -A 1 , we get A 1 = -(∂P )P -1 which transforms under P to A = -∂(P P )(P P ) -1 = -(∂H)H -1 . We see that A(λ) in the Higgs gauge becomes close to a Higgs bundle. But in our setting the holomorphic differentials are in A, and not in the Higgs field. We illustrate the similarity for n = 2.

Example 5.5. For n = 2 and µ = 0, we have seen in the previous subsection 5.2 that in symmetric gauge, our connection reads

A(λ) = -∂ϕ 2 t 2 e -ϕ λe ϕ ∂ϕ 2 dz + ⎛ ⎝ ∂ϕ 2 λ -1 e ϕ -t2 e -ϕ -∂ϕ 2 ⎞ ⎠ dz.
The flatness condition is equivalent to the cosh-Gordon equation ∂ ∂ϕ = e 2ϕ + t 2 t2 e -2ϕ . A direct computation gives the form in parabolic gauge:

A(λ) = 0 t2 (λ) 1 0 dz + -1 2 ∂μ 2 -1 2 ∂ 2 μ2 + t2 μ2 μ2 (λ) 1 2 ∂μ 2 dz where t2 (λ) = λt 2 + (∂ϕ) 2 -∂ 2 ϕ and μ2 (λ) = -λ -1
t2 e -2ϕ . In Higgs gauge, we get

A(λ) = -∂ϕ -t 2 p 2 e -ϕ 2 t 2 λ -a 1 ∂ϕ + t 2 p 2 e -ϕ 2 dz + -λ -1 p 2 e 3ϕ 2 λ -1 e 2ϕ -λ -1 p 2 2 e ϕ λ -1 p 2 e 3ϕ 2 dz
where a 1 = (∂p 2 + 3 2 p 2 ∂ϕ + t 2 p 2 2 e -ϕ 2 )e -ϕ 2 and p 2 comes from the matrix of the Higgs gauge and satisfies ∂p 2 = -t2 e -3ϕ 2 + p 2 ∂ϕ 2 .

Finally, we can compare to the non-abelian Hodge correspondence which gives

A(λ) = -∂ϕ 0 λ ∂ϕ dz + 0 λ -1 e 2ϕ 0 0 dz.
The flatness condition is equivalent to Liouville's equation ∂ ∂ϕ = e 2ϕ . Notice that we get this connection in our setting in the Higgs gauge for t 2 = 0 (then p 2 = 0). △

For non-trivial n-complex structure µ ≠ 0, there is no Higgs gauge. Even for n = 2, one can check that there is no P satisfying P Φ 1 = E -P and P A 2 -∂P = 0. 5.4. General case. Set t = (t 2 , ..., t n ) and µ = (µ 2 , ..., µ n ). To examine the existence of an analog to the non-abelian Hodge correspondence, we discuss the cases when t = 0 or µ = 0.

Case t = 0 and µ = 0. For the trivial structure we find the following result, generalizing the observations for n = 2 and n = 3 from the previous subsection 5.2. Proposition 5.6. For Φ 2 = 0 and t = 0, the flat connection A(λ) is uniquely determined up to some finite initial data. There is a choice of initial data such that the flatness equations are equivalent to the Toda integrable system. In particular A(λ) is the same as the connection given by the non-abelian Hodge correspondence applied to a principal nilpotent Higgs field.

Proof. Using lemmas 5.1 and 5.2, we can write A 1 (λ) in the following form:

A 1 (λ) = a 0 + a 1 T + ... + a n T n
where a i are diagonal matrices and T is given by (5.2)

T = ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⋱ 1 λ ⎞ ⎟ ⎟ ⎟ ⎠ .
We denote by a i,j the j-th entry of the diagonal matrix a i and a ′ i the shifted matrix with a ′ i,j = a i,j+1 . We write a (k) for the shift applied k times. Notice that aT = T a (n-1) . We can then write

A 2 (λ) = a * 0 + T -1 a * 1 + ... + T -n a * n
where a * i,j = ±ā i,j , the sign depends on whether the coefficient comes with a λ or not in A 2 (λ). By the standard form (lemma 5.1) we can further impose a n,i = e ϕ i for i = 1, ..., n -1 and a n,0 = 0 since 0 = t n = ∏ i a n,i . One of the flatness equations gives ∂a n = a n (a (n-1) 0 a 0 ). Together with the condition that the trace is 0, we can compute a 0 . We get

(5.3) a 0,i = i-1 k=1 k n ∂ϕ k - n-1 k=i n -k k ∂ϕ k .
The other equations give a system of differential equations in a 1 , ..., a n-1 which is quadratic. It allows the solution a i = 0 for all i = 1, ..., n -1. In that case, using a diagonal gauge diag(1, λ, ..., λ n-1 ) the connection A(λ) becomes

(5.4)

A(λ) = ⎛ ⎜ ⎜ ⎜ ⎝ * e ϕ 1 * ⋱ * e ϕ n-1 * ⎞ ⎟ ⎟ ⎟ ⎠ dz + ⎛ ⎜ ⎜ ⎜ ⎝ * e ϕ 1 * ⋱ * e ϕ n-1 * ⎞ ⎟ ⎟ ⎟ ⎠ dz
where on the diagonals are the a 0,i and -ā 0,i given by equation (5.3). This is precisely the form of the Toda system. It is known that the Hitchin equations for a principal nilpotent Higgs field are the Toda equations for sl n (see [START_REF] Aldrovandi | Geometry of Higgs and Toda Fields on Riemann Surfaces[END_REF], proposition 3.1).

Notice that in particular the gauge class of the connection A(λ) is independent of λ ∈ C * (i.e. we have a variation of Hodge structure). This is an intrinsic property which might be used to fix the initial data.

Putting (5.4) in parabolic gauge, we get the following explicit formula for our coordinates t(λ) and μ(λ) (see also proposition 3.1 and 4.4 in [START_REF] Aldrovandi | Geometry of Higgs and Toda Fields on Riemann Surfaces[END_REF]):

Proposition 5.7. For µ = 0 and t = 0, one can choose initial conditions such that μk (λ) = 0 and tk (λ) = w k for all k, where the w k are given by det(∂ -A 1 ) = ∏ i (∂a 0,i ) = ∂ n + w 2 ∂ n-2 + ... + w n (a "Miura transform"). Furthermore, A 1 is diagonal given by equation (5.3) and the parabolic gauge is upper triangular.

Case t = 0. We get the following result: Proposition 5.8. For t = 0, the connection A(λ) is determined by the flatness condition and by some initial conditions. Its monodromy is in PSL n (R).

The idea of the proof is the following: locally, one can trivialize the higher complex structure, so we are led to µ = 0 and t = 0. Thus A(λ) is given by the non-abelian Hodge correspondence and we can apply Hitchin's strategy to prove real monodromy, which is a local argument.

Proof. By theorem 4.4, we know that we can locally render Φ 2 = 0 by trivializing the ncomplex structure. Thus we can choose the initial conditions such that A(λ) is given by the non-abelian Hodge correspondence applied to the nilpotent Higgs field Φ 1 (see proposition 5.6).

In [START_REF] Hitchin | Lie Groups and Teichmüller Space[END_REF], Hitchin constructs a real form τ , associated with the split real form, which for sl n is given by a rotation of the matrix by 180 degrees composed with complex conjugation. He shows that τ * Φ 1 = Φ * 1 and that τ * A(λ) = A(λ). This is a local statement, therefore the monodromy of A(λ) has to be in the fixed point set of τ , so in PSL n (R).

Case µ = 0. For trivial n-complex structure, the standard form from lemmas 5.1 and 5.2 allow to consider A(λ) as an affine connection with special properties. We denote by L(sl n ) the loop algebra of sl n . It is defined by L(sl n ) = sl n ⊗ C[λ, λ -1 ], the space of Laurent polynomials with matrix coefficients. There is another way to think of elements of L(sl n ): as an infinite periodic matrix (M i,j ) i,j∈Z with M i,j = M i+n,j+n and finite width (i.e. M i,j = 0 for all i + j big enough). The isomorphism is given as follows: to ∑ N i=-N N i λ i we associate M i,j = (N k j -k i ) r i ,r j where i = k i n + r i and j = k j n + r j are the Euclidean divisions of i and j by n (so 0 ≤ r i , r j < n), see also figure 5.1.

In the second viewpoint, a connection λΦ + A + λ -1 Φ * with Φ 1 lower triangular, Φ 2 = 0 and thus A 1 upper triangular, is precisely an infinite matrix with period n and width n (shown in figure 5.1 by dashed lines). The (1, 0)-part A 1 (λ) is upper triangular (Φ 1 is lower triangular but λΦ 1 is upper triangular in the infinite matrix) and the (0, 1)-part A 2 (λ) is lower triangular.

⋯ ⋯ ⋮ ⋮ ⋱ ⋱ N 0 N 0 N 0 N 1 N 1 N 2 N -1 N -1 N -2
Figure 5.1. Affine matrix as infinite periodic matrix Thus, the flatness of A(λ) is a generalized Toda system, replacing the tridiagonal property by "width equal to periodicity". For t i = 0 for i = 2, ..., n -1 but t n ≠ 0, we should get the usual affine Toda system for L(sl n ).

Remark. In order to describe h-connections, we can include parameters into L(sl n ) by considering its central extension ŝl n or central coextension.

△

Since for t = 0 we get an elliptic system, the system stays elliptic for at least small t ≠ 0, since ellipticity is an open condition (Cauchy-Kowalewskaya theorem). So the generalized Toda system can be solved for small t.

The study of this generalized Toda system is subject of future research.

General case. For µ ≠ 0 and t ≠ 0, the system is still elliptic at least for small t, since it is for t = 0. We should get a generalized Toda system with differentials t k satisfying the higher holomorphicity condition (C).

We conjecture that the connection A(λ) = λΦ + A + λ -1 Φ * is uniquely determined by µ and t. To be more precise: Conjecture 5.9. Given an element [(µ k , t k )] ∈ T * T n and some finite extra data (initial conditions to differential equations), there is a unique (up to unitary gauge) flat connection

A(λ) = λΦ + A + λ -1 Φ * satisfying (1) Locally, Φ = Φ 1 dz + Φ 2 dz with Φ 1 principal nilpotent and Φ 2 = µ 2 Φ 1 + ... + µ n Φ n-1 1 (2) Reality condition: -A(-1 λ) * = A(λ) (3) t k = tr Φ k-1 1 A 1 . In addition, if t k = 0 for all k, then the monodromy of A(λ) is in PSL n (R).
Assuming this conjecture, we get the desired link to Hitchin's component: Theorem 5.10. If conjecture 5.9 holds true, there is a canonical diffeomorphism between our moduli space T n and Hitchin's component T n .

Proof. With conjecture 5.9 we get a canonical way to associate a flat connection A(λ = 1) to a point in T * T n . By proposition 5.8 the monodromy of A(λ) for t = 0 is in PSL n (R). Following Hitchin's argument from theorem 7.5 in [START_REF] Hitchin | Lie Groups and Teichmüller Space[END_REF], we prove that the zero-section in T * T n where t = 0 describes a connected component of Rep(π 1 (Σ), PSL n (R)).

Since T n is closed in T * T n , the image of the map s ∶ T n → Rep(π 1 (Σ), PSL n (R)) is a closed submanifold. Furthermore both spaces have the same dimension by theorem 2.4. Therefore the image of s is an open and closed submanifold, i.e. a connected component.

Finally, for µ = 0 we get the same connection A(λ) as by the non-abelian Hodge correspondence of the principal nilpotent Higgs field. So the component described by T n and Hitchin's component T n coincide.

Notice that the map between T n and T n is something like an exponential map. For n = 2 Hitchin's description of Teichmüller space is exactly via the exponential map identifying a fiber of the cotangent bundle T * µ T 2 to T 2 . Corollary 5.11. Hitchin's component has a natural complex structure. Further, there is a natural action of the mapping class group on it, preserving the complex structure.

The first statement follows from theorem 2.4 since we explicitly know the cotangent space at a point. The second simply follows by the description of Hitchin's component as moduli space of some geometric structure on the surface. Labourie describes this action in [START_REF] Labourie | Cross Ratios, Anosov Representations and the Energy Functional on Teichmüller Space[END_REF] and shows that it is properly discontinuous using cross ratios.

Appendix A. Punctual Hilbert schemes revisited

In this appendix, we review some aspects of the punctual Hilbert scheme of the plane. Main references are Nakajima's book [Na99] and Haiman's paper [START_REF] Haiman | t, q-Catalan numbers and the Hilbert scheme[END_REF].

A.1. Definition. Consider n points in the plane C 2 as an algebraic variety, i.e. defined by some ideal I in C[x, y]. Its function space C[x, y] I is of dimension n, since a function on n points is defined by its n values. So the ideal I is of codimension n. The space of all such ideals, or in more algebraic language, the space of all zero-subschemes of the plane of given length, is the punctual Hilbert scheme: Definition A.1. The punctual Hilbert scheme Hilb n (C 2 ) of length n of the plane is the set of ideals of C [x, y] of codimension n:

Hilb n (C 2 ) = {I ideal of C [x, y] dim(C [x, y] I) = n}.
The subspace of Hilb n (C 2 ) consisting of all ideals supported at the origin, i.e. whose associated algebraic variety is (0, 0), is called the zero-fiber of the punctual Hilbert scheme and is denoted by Hilb n 0 (C 2 ). A theorem of Grothendieck and Fogarty asserts that Hilb n (C 2 ) is a smooth and irreducible variety of dimension 2n (see [START_REF] Fogarty | Algebraic families on an algebraic surface[END_REF]). The zero-fiber Hilb n 0 (C 2 ) is an irreducible variety of dimension n -1, but it is in general not smooth.

A generic element of Hilb n (C 2 ), geometrically given by n distinct points, is given by

I = ⟨-x n + t 1 x n-1 + ⋯ + t n , -y + µ 1 + µ 2 x + ... + µ n x n-1 ⟩ .
The second term can be seen as the Lagrange interpolation polynomial of the n points.

A generic element of the zero-fiber is given by I = ⟨x n , -y + µ 2 x + ... + µ n x n-1 ⟩ .

In particular, we see that for n = 2, we get projective space:

Hilb 2 0 (C 2 ) ≅ P(C 2 ) = CP 1 . Given an ideal I of codimension n, we can associate its support, the algebraic variety defined by I, which is a collection of n points (counted with multiplicity). The order of the points does not matter, so there is a map, called the Chow map, from Hilb n (C 2 ) to Sym n (C 2 ) ∶= (C 2 ) n S n , the configuration space of n points (S n denotes the symmetric group). A theorem of Fogarty asserts that the punctual Hilbert scheme is a minimal resolution of the configuration space.

A.2. Matrix viewpoint. To an ideal I of codimension n, we can associate two matrices: the multiplication operators M x and M y , acting on the quotient C[x, y] I by multiplication by x and y respectively. To be more precise, we can associate a conjugacy class of the pair:

[(M x , M y )].
The two matrices M x and M y commute and they admit a cyclic vector, the image of 1 ∈ C[x, y] in the quotient (i.e. 1 under the action of both M x and M y generate the whole quotient).

Proposition A.2. There is a bijection between the Hilbert scheme and conjugacy classes of certain commuting matrices:

Hilb n (C 2 ) ≅ {(A, B) ∈ gl 2 n [A, B] = 0, (A, B) admits a cyclic vector} GL n The inverse construction goes as follows: to a conjugacy class [(A, B)], associate the ideal I = {P ∈ C[x, y] P (A, B) = 0}, which is well-defined and of codimension n (using the fact that (A, B) admits a cyclic vector). For more details see [START_REF] Nakajima | Lectures on Hilbert Schemes of Points on Surfaces[END_REF].

The zero-fiber of the Hilbert scheme corresponds to nilpotent commuting matrices.

A.3. Reduced Hilbert scheme. We wish to define a subspace of Hilb n (C 2 ) corresponding to matrices in sl n in the matrix viewpoint. A generic point should be a pair of points in the Cartan subalgebra h of sl n modulo order. This corresponds to n points in the plane with barycenter 0.

Definition A.3. The reduced Hilbert scheme Hilb n red (C 2 ) is the space of all elements of Hilb n (C 2 ) whose image under the Chow map (n points with multiplicity modulo order) has barycenter 0.

With this definition, we get

Proposition A.4.

Hilb n red (C 2 ) ≅ {(A, B) ∈ sl 2 n [A, B] = 0, (A, B) admits a cyclic vector} SL n . Finally, it can be proven that the reduced Hilbert scheme is symplectic and that the zerofiber Hilb n 0 (C 2 ) is a Lagrangian subspace of Hilb n red (C 2 ).

Appendix B. Existence of parabolic gauge

In this rather technical appendix, we prove the existence of a parabolic gauge (see subsection 3.2.1).

Proposition B.1. For a generic connection A = A 1 +A 2 , there is a gauge P ∈ C ∞ (Σ, SL(n, C)) with last row zero except for the last entry (parabolic gauge) such that A 1 is locally transformed into a companion matrix.

Proof. We begin by setting up notations. The matrix P we look for, is of the following form: where L k denotes the k th line in the matrix without the last entry (i.e. a row vector of length n -1). Denote the entries of A 1 by a ij . We adopt Einstein's summation convention in this section (automatic summation over repeated indices). We want that P transforms A 1 into a companion matrix under gauge transform P A 1 P -1 + P ∂(P -1 ). Since P is of determinant 1, A 1 stays traceless. Using P ∂P -1 = -∂P P -1 , we get (B.1)

P = ⎛ ⎜ ⎜ ⎜ ⎝
P A 1 -∂P = ⎛ ⎜ ⎜ ⎜ ⎝ * 1 ⋮ ⋱ * 1 0 ⎞ ⎟ ⎟ ⎟ ⎠ P = ⎛ ⎜ ⎜ ⎜ ⎝ 0 * L 1 * ⋮ ⋮ L n-1 * ⎞ ⎟ ⎟ ⎟ ⎠ .
This gives n 2n equations by the first n -1 columns.

Our strategy is the following: we express p ij for 1 ≤ i ≤ n -1 and 1 ≤ j ≤ n -1 in terms of a ij (the "constants") and the (p kn ) k=1,...,n (and their derivatives). Then we get an expression of p nn in terms of a ij . Finally we compute p kn for k = 1, ..., n -1.

The matrix equation (B.1) above gives (B.2)

n k=1
p ik a kj -∂p ij = p i-1,j ∀1 ≤ i ≤ n and ∀1 ≤ j ≤ n -1 where we have put p 0j = 0. Setting i = n (and j < n), we get p n-1,j = a nj p nn , i.e. L n-1 is p nn times the last row of A 1 . Setting i = n -1 (and j < n), we get p n-2,j = p n-1,k a kj -∂p n-1,j = p nn (a nk a kj -∂a nj )a nj ∂p nn + p n-1,n a nj .

By continuing, we see that for 2 ≤ i ≤ n, we get our first goal: the equations (B.2) express the p ij for 1 ≤ i ≤ n -1 and 1 ≤ j ≤ n -1 in terms of a ij and the (p kn ) k=1,...,n .

To achieve our second goal, we prove the following:

Lemma B.2. Denote by P 0 the square-submatrix of the p ij for 1 ≤ i ≤ n -1 and 1 ≤ j ≤ n -1.

We then have det(P 0 ) = Ap n-1 nn where A is some constant only depending on the a ij .

Proof. We interpret the equation for P A 1 -∂P as a condition on the covariant derivative ∇ = (-∂ + A 1 ) on P , acting from the right. The factor p nn is interpreted as a scalar denoted by f . Put a = (a n1 , a n2 , ..., a n,n-1 ) the last row of A 1 which we consider as a row vector. We already noticed that L n-1 = f a. In what follows we write ∇a for -∂a + aA 1 . The other equations of (B.2) now successively give With this expression for L n-k , we see that P 0 in the basis (∇ n-2 a, ..., ∇a, a) is uppertriangular with f on the diagonal. Hence, its determinant is det(P 0 ) = f n-1 det(∇ n-2 a, ..., ∇a, a) = p n-1 nn A.

Now, we are ready to conclude. By a direct computation, we can see that modulo t2 , ∂ 2 we can replace αn,n+1-l by μl in point 2. Define D 1 (h) = h n ∂ n -t2 h n-2 ∂ n-2 -... -tn and D 2 (h) = -h ∂ + μ1 + μ2 h∂ + ... + μn h n-1 ∂ n-1 . Then using 1. to 4. and computing modulo t2 and ∂ 2 , we get: Comparing coefficients, we get the lemma.

[Tr19] Samuel Trautwein: The hyperkähler metric on the almost-Fuchsian moduli space, 2019, arxiv:1809.00869

Université de Strasbourg, IRMA UMR 7501, 67084 Strasbourg, France Email address: thomas@math.unistra.fr

  n-1,1 ... p n-1,n-1 p n-1,n

L

  n-2 = ∇(f a) + p n-1,n a L n-3 = ∇L n-2 + p n-2,n a = ∇(∇(f a) + p n-1,n a) + p n-2,n a L n-k = ∇L n-k+1 + p n-k+1,n a (B.3) Thus, we can write L n-k = ∑ k-1 l=0 α l,k ∇ l a with α k-1,k = f. The other α l,k are functions of a ij and the (p kn ) k=1,...,n-1 .

  n -t2 p n-2 -... -tn , -p + μ1 + μ2 p + ... + μn p n-1 } mod I = n k=2 ( ∂ -μ 2 ∂ -k∂μ k ) tkn-k l=1 ((l+k)∂μ l+2 + (l+1)μ l+2 ∂) tk+l p n-k

Corollary B.3. Since 1 = det P = p nn det P 0 , we get

We see in particular the condition under which the parabolic gauge exists: we need that A = det(∇ n-2 a, ..., ∇a, a) ≠ 0.

To finish, take equations (B.3) for i = 1 which give

with α l,n = p l+1,n + terms with p k,n with k > l + 1. Then, we express ∇ n-1 a in the basis (a, ∇a, ..., ∇ n-2 a): ∇ n-1 a = β 0 a + β 1 ∇a + ... + β n-2 ∇ n-2 a (thus the β k depend only on the a ij ). By the freedom of (a, ∇a, ..., ∇ n-2 a), we get out of (B.4)

Therefore, we can successively express p 1n , p 2n , ... up to p n-1,n in terms of a ij and p nn (which we already expressed in terms of the a ij ). This proves the existence of the parabolic gauge.

Appendix C. Proof of lemma 4.7

Lemma. The parabolic curvature modulo t2 and ∂ 2 gives condition (C) on T * T n :

Proof. The proof is a combination of several formulas:

(1) Proposition 3.1 together with the expression of the differential operators (see (3.5)) give

where αij are the entries of the matrix A 2 . (2) Link between Poisson bracket and commutator:

(3) Link between mod ∂ 2 and brackets: for h-connections D 1 (h) and D 2 (h), we have