# Fock bundles and Hitchin components

Georgios Kydonakis, Charlie Reid, Alexander Thomas

###### Abstract

We introduce the concept of a Fock bundle, a smooth principal bundle over a
surface equipped with a special kind of adjoint-valued 1-form, as a new tool
for studying character varieties of surface groups. Although similar to Higgs
bundles, the crucial difference is that no complex structure is fixed on the
underlying surface. Fock bundles are the gauge-theoretic realization of higher
complex structures. We construct a canonical connection to a Fock bundle
equipped with compatible symmetric pairing and hermitian structure. The space
of flat Fock bundles maps to the character variety of the split real form.
Determining the hermitian structure such that this connection is flat gives a
non-linear PDE similar to Hitchin’s equation. We explicitly construct
solutions for Fock bundles in the Fuchsian locus. Ellipticity of the relevant
linear operator provides a map from a neighborhood of the Fuchsian locus in
the space of higher complex structures modulo higher diffeomorphisms to a
neighborhood of the Fuchsian locus in the Hitchin component.

###### Contents

  1. 1 Introduction and results
    1. 1.1 Context
    2. 1.2 Results and structure
  2. 2 Preliminaries
    1. 2.1 Higher complex structures
    2. 2.2 Nonabelian Hodge theory and twistor approach
    3. 2.3 Principal nilpotent elements
    4. 2.4 Involutions and reductions of structure group
  3. 3 Fock bundles
    1. 3.1 Fock bundles and higher complex structures
    2. 3.2 Fuchsian Locus
    3. 3.3 Variations of Fock bundles
  4. 4 Connections associated to Fock bundles
    1. 4.1 Three-term connections
    2. 4.2 Transversality
    3. 4.3 Canonical connection from transverse Fock fields
    4. 4.4 Unitary case
    5. 4.5 Main Conjecture
    6. 4.6 Flat Fock bundles and the Hitchin component
  5. 5 Neighborhood of the Fuchsian locus
    1. 5.1 Linearized Equation
    2. 5.2 Application of the implicit function theorem
    3. 5.3 Solution in a neighborhood
  6. 6 Higher diffeomorphisms
    1. 6.1 Special $\lambda$-dependent gauge transformations
    2. 6.2 First variation formula
    3. 6.3 Constant monodromy along paths of higher diffeomorphisms
  7. 7 Covectors
    1. 7.1 Definition of covectors
    2. 7.2 Canonical connection with covectors
    3. 7.3 $\mu$-holomorphicity
    4. 7.4 Higher diffeomorphism action on covectors

##  1 Introduction and results

###  1.1 Context

Let $S$ be a smooth closed orientable surface of genus at least 2. The moduli
space of complex structures on $S$ modulo diffeomorphisms isotopic to the
identity is famously known as the _Teichmüller space_. The Poincaré–Koebe
uniformization theorem implies that the Teichmüller space also describes all
marked hyperbolic structures on $S$. The holonomy representation of a
hyperbolic structure leads to a discrete and faithful representation
$\pi_{1}S\to\mathrm{PSL}_{2}(\mathbb{R})$, thus allowing for an algebraic
realization of the Teichmüller space of $S$ as a connected component of the
character variety $\mathcal{X}(\pi_{1}S,\mathrm{PSL}_{2}(\mathbb{R}))$.
Generalizing the uniformization theorem to higher rank Lie groups is the main
motivation of this work.

Representations of fundamental groups and their links to geometric structures
are captured by the character variety. For a Lie group $G$, the $G$-_character
variety_

$\mathcal{X}(\pi_{1}S,G)=\mathrm{Hom}(\pi_{1}S,G)/G$

is defined as the space of isomorphism classes of completely reducible
representations, where $G$ acts by conjugation.

Higher Teichmüller theory concerns the study of connected components of
character varieties for more general Lie groups than
$\mathrm{PSL}_{2}(\mathbb{R})$ which share analogous properties to the
classical Teichmüller space. The first step in this direction was taken by
Hitchin in [14], where he found a connected component of
$\chi(\pi_{1}S,\mathrm{PSL}_{n}(\mathbb{R}))$ (and more generally for any
adjoint group of the split real form of a complex simple Lie group) which is
contractible, and naturally contains a copy of Teichmüller space.
Representations parametrized by these components, now most often called
_Hitchin components_ , were later shown to be discrete and faithful [17, 6].

The _mapping class group_ $\mathrm{Mod}(S)$ of the surface $S$, that is, the
group of all orientation-preserving diffeomorphisms of $S$ modulo the ones
which are isotopic to the identity, acts naturally on the Teichmüller space by
changing the marking. This action is properly discontinuous and the resulting
quotient is the moduli space of Riemann surfaces with the same topological
type as $S$. Similarly, $\mathrm{Mod}(S)$ acts on any character variety by
precomposition.

Fixing a complex structure on $S$, the nonabelian Hodge correspondence
identifies $\chi(\pi_{1}S,G)$, for any reductive Lie group $G$, with the
moduli space of polystable $G$-Higgs bundles. The nonabelian Hodge
correspondence is very effective in providing complex analytic descriptions of
character varieties $\mathcal{X}(\pi_{1}S,G)$, their Hitchin components, and
other Teichmüller space-like components, now called _positive_ components.
However, a major drawback in this correspondence is that one has to fix a
complex structure on $S$, and so $\mathrm{Mod}(S)$ does not act on the moduli
of Higgs bundles.

In [19], Labourie proposed a remedy which, if generally applicable, could
provide a canonical way to associate a complex structure on $S$ to a given
point in the Hitchin component. This analytic method involving harmonic maps
and minimal surfaces has received considerable attention and has proven to be
effective in several cases of higher Teichmüller spaces but only for groups of
rank 2; we refer to [20] and [2] for general accounts on split real simple Lie
groups of rank 2 and Hermitian Lie groups of rank 2 respectively. Yet, it
recently became clear that this approach via minimal surfaces towards
establishing a canonical choice of complex structure on $S$ and thus obtaining
a mapping class group equivariant parametrization of the associated higher
Teichmüller spaces in terms of holomorphic data, fails for groups of rank
greater than 2. In [24], Sagman and Smillie demonstrated the existence of
numerous equivariant minimal surfaces for certain Hitchin representations
$\rho:\pi_{1}(S)\to G$, when $G$ is a split real semisimple Lie group of rank
at least 3, building on the breakthrough accomplished by Marković [22], who
applied his New Main Inequality to get an analogous statement in the case of
$\mathrm{PSL}_{2}(\mathbb{R})^{3}$.

A new approach towards realizing an equivariant parametrization of the Hitchin
components was proposed by Vladimir Fock and the third author [8] in terms of
newly introduced geometric structures called _higher complex structures_.
These were originally defined using the punctual Hilbert scheme of the plane
and a moduli space of higher complex structures was defined in [8] as a
quotient by Hamiltonian diffeomorphisms of the cotangent bundle of the surface
preserving the zero section $S\subset T^{*}S$ setwise. Such diffeomorphisms
were called _higher diffeomorphisms_ and the resulting quotient space, called
the _geometric Hitchin space_ and denoted by $\hat{\mathcal{T}}^{n}(S)$,
recovers the classical Teichmüller space when $n=2$. Moreover, this space is a
manifold of complex dimension equal to that of the Hitchin component for
$G=\mathrm{PSL}_{2}(\mathbb{R})$ [8, Theorem 2]. Its cotangent bundle can be
parametrized by a set of tensors on the underlying surface, satisfying a
certain compatibility condition called _$\mu$ -holomorphicity_ [8, Theorem 3].
The main conjecture suggested in [8] was that the geometric Hitchin space is
canonically diffeomorphic to the Hitchin component.

In [23], Nolte introduced the notion of _harmonic_ representatives for higher
complex structures and studied in detail the group of higher diffeomorphisms.
There, the geometric Hitchin space is denoted by $\mathcal{T}^{n}(S)$ and is
called the degree-n Fock–Thomas space. Moreover, a canonical diffeomorphism
from $\mathcal{T}^{3}(S)$ to the $\mathrm{PSL}_{3}(\mathbb{R})$-Hitchin
component was constructed. This construction is, in fact,
$\mathrm{Mod}(S)$-equivariant. More generally, the space $\mathcal{T}^{n}(S)$
is diffeomorphic to a ball of complex dimension $(g-1)(n^{2}-1)$, meaning that
it is abstractly diffeomorphic to the Hitchin component; see [8, Theorem 2]
and [23, Corollary 1.7]. Nolte’s method, however, for getting this canonical
$\mathrm{Mod}(S)$-equivariant diffeomorphism uses the affine spheres
perspective on the $\mathrm{PSL}_{3}(\mathbb{R})$-Hitchin component by
Labourie [18] and Loftin [21], and so the technique does not directly apply
for higher rank.

In this article, we introduce a gauge-theoretic realization of higher complex
structures which allows us to make progress on their link to character
varieties, in particular to Hitchin components. In addition, central
associated notions such as higher diffeomorphisms and the $\mu$-holomorphicity
condition become more transparent.

We introduce a new type of object $(P,\Phi,\sigma)$, that we call _$G$ -Fock
bundle_, consisting of a smooth principal $G$-bundle $P$ together with an
adjoint-valued 1-form $\Phi$, called a _Fock field_ , that satisfies certain
conditions. In this triple, $\sigma$ is an involution on $P$ generalizing the
notion of a symmetric pairing on a vector bundle. This notion is similar to a
$G$-Higgs bundle, with the crucial difference that it does not involve fixing
a priori any complex structure on the underlying surface $S$. Isomorphism
classes of $\operatorname{PSL}_{n}(\mathbb{C})$-Fock bundles are equivalent to
higher complex structures of order $n$. A $G$-Fock field $\Phi$ as above
naturally induces a complex structure on $S$. Varying the Fock field also
varies this complex structure.

We then set to establish a passage from $G$-Fock bundles to connections
analogously to the nonabelian Hodge correspondence. Our theory associates to a
Fock bundle $(P,\Phi,\sigma)$ equipped with a compatible hermitian structure
$\rho$, a connection $\nabla=\Phi+d_{A}+\Phi^{*}$, where
$\Phi^{*}=-\rho(\Phi)$ is the hermitian conjugate of $\Phi$ and $d_{A}$ is the
unique unitary $\sigma$-invariant connection satisfying $d_{A}\Phi=0$. We
conjecture that there exists a hermitian structure such that $\nabla$ is flat.

The flatness equation for $\nabla$ is similar to Hitchin’s self-duality
equations over a Riemann surface [12] and, in fact, coincides with Hitchin’s
equation in the most trivial examples, the so-called Fuchsian locus (Fock
bundles coming from uniformizing Higgs bundles). Moreover we prove the
conjecture in a neighborhood of the Fuchsian locus. For the family of flat
$G$-Fock bundles, we show that the monodromy of the connection is always in
the split real form of $G$.

Rudiments of the approach towards establishing a passage to a canonical family
of flat connections have appeared in the previous articles [29, 30] and [31]
by the third author. In particular, it was shown in [29] that the cotangent
bundle of the moduli space of higher complex structures can be included into a
1-parameter family of spaces whose sections are flat formal connections. The
theory of $\mathfrak{g}$-complex structures was introduced in [31], extending
the case of $\mathfrak{sl}_{n}(\mathbb{C})$ to a general complex simple Lie
algebra $\mathfrak{g}$, and constitutes the starting point for our definition
of $G$-Fock bundles.

The introduction of Fock bundles and their relationship to families of
connections points towards a broader framework to study character varieties,
which includes both the theory of Higgs and Fock bundles. Fixing a complex
structure on $S$, one may consider a $G$-principal bundle $P$ on $S$, together
with a connection $d_{A}$ and a field $\Phi\in\Omega^{1}(S,\mathfrak{g}_{P})$
satisfying $[\Phi\wedge\Phi]=0$, $d_{A}(\Phi)=0$ and with fixed conjugacy
class of $\Phi^{0,1}$. Here we used the complex structure to decompose $\Phi$
into Hodge types. Under this setup, then whenever $\Phi^{0,1}=0$, the field
$\Phi$ is holomorphic with respect to the holomorphic structure on $P$ given
by $d_{A}^{0,1}$, thus leading to the theory of Higgs bundles. On the other
hand, whenever $\Phi^{0,1}$ is principal nilpotent, this leads to our theory
of Fock bundles. We hope that to such data, one can always associate a flat
connection, in analogy to the nonabelian Hodge correspondence. Alternative
conditions for the conjugacy class of $\Phi^{0,1}$ can possibly lead to
alternative approaches to character varieties, yet the case when $\Phi^{0,1}$
is principal nilpotent describes the only possible conjugacy class which stays
invariant under coordinate change on $S$. In other words, _the approach via
Fock bundles within this broader framework is the only one which is
independent of the complex structure on $S$_.

###  1.2 Results and structure

We now make the statements appearing in the article more precise. Throughout
the whole paper we denote by $S$ a smooth closed connected orientable surface
of genus at least 2 and by $G$ a complex simple Lie group with Lie algebra
$\mathfrak{g}$. The main new object we introduce is the notion of a $G$-Fock
bundle. We restrict to the case $G=\mathrm{SL}_{n}(\mathbb{C})$ here for
simplicity and give the full definition in Section 3.

######  Definition (Definition 3.1).

An _$\mathrm{SL}_{n}(\mathbb{C})$ -Fock bundle_ over $S$ is a triple
$(E,\Phi,g)$ where $E$ is a complex vector bundle over $S$ with fixed volume
form, non-degenerate symmetric pairing $g:E\times E\to\underline{\mathbb{C}}$,
where $\underline{\mathbb{C}}$ denotes the trivial line bundle, and
$\Phi\in\Omega^{1}(S,\mathfrak{sl}(E))$ satisfying

  1. 1.

$\Phi\wedge\Phi=0$,

  2. 2.

$\Phi(v)(z)$ is principal nilpotent for all $z\in S$ and all vectors $v\in
T_{z}S$.

  3. 3.

$\Phi$ is $g$-self-adjoint.

We refer to $\Phi$ as the _Fock field_.

This generalizes easily to general $G$: a $G$-Fock bundle is a principal
$G$-bundle $P$ with a certain involution $\sigma$ playing the role of the
symmetric pairing, and an adjoint-valued 1-form $\Phi$ satisfying the
analogous three properties above. An important example of $G$-Fock bundles,
the so-called _Fuchsian locus_ , arises from the underlying smooth principal
bundle to the uniformizing $G$-Higgs bundle when we equip $S$ with a complex
structure (see Section 3.2). The link to higher complex structures is given by
the following:

######  Proposition A (Proposition 3.5).

An $\mathrm{SL}_{n}(\mathbb{C})$-Fock bundle induces a higher complex
structure of order $n$ on $S$. Isomorphism classes of
$\mathrm{PSL}_{n}(\mathbb{C})$-Fock bundles are equivalent to higher complex
structures.

In Subsection 3.3 we describe variations of Fock bundles via the cohomology of
a certain chain complex. In particular, we prove that a Fock bundle has no
infinitesimal automorphisms, hence is stable in that sense.

The main result of Section 4 is the construction of a canonical connection
associated to a Fock bundle equipped with a compatible positive hermitian
structure. A hermitian structure $\rho$ is an involution on $P$ associated to
the compact real form of $G$. It is said to be _compatible_ with a Fock bundle
if $\rho$ and $\sigma$ commute. The property of being _positive_ is a certain
open condition, see Definition 4.10 for the precise statement.

######  Theorem B (Theorem 4.6).

For a $G$-Fock bundle $(P,\Phi,\sigma)$ equipped with a compatible, positive
hermitian structure $\rho$, there is a unique unitary, $\sigma$-invariant
connection $d_{A}$ satisfying $d_{A}\Phi=0$.

This result is analogous to the existence of a Chern connection on a
holomorphic bundle induced by a hermitian structure. Thus, to the data
$(P,\Phi,\sigma,\rho)$ as above we can associate a connection
$\Phi+d_{A}+\Phi^{*}$, where $\Phi^{*}=-\rho(\Phi)$, which preserves a split-
real structure, and has curvature

$F(A)+[\Phi\wedge\Phi^{*}].$ (1.1)

Our main conjecture states:

######  Conjecture C (Conjecture 4.14).

For a $G$-Fock bundle $(P,\Phi,\sigma)$, there exists a unique compatible
positive hermitian structure $\rho$ such that the associated connection
$\Phi+d_{A}+\Phi^{*}$ is flat, in other words, such that
$F(A)+[\Phi\wedge\Phi^{*}]=0$.

This would give a map from the space of Fock bundles to the space of Hitchin
representations. The conjecture is true for examples of Fock bundles from the
Fuchsian locus obtained by the nonabelian Hodge correspondence (the
uniformizing Higgs bundles).

Section 5 analyzes the conjecture near the Fuchsian locus. The following
result provides strong evidence towards the validity of the conjecture in
general:

######  Theorem D (Theorem 5.2).

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a compatible positive
hermitian structure $\rho$. The derivative of the map from the
$\operatorname{Aut}(P,\sigma)$-conjugacy class of $\Phi$ to the curvature of
the resulting connection $F(A)+[\Phi\wedge\Phi^{*}]$ is an elliptic
isomorphism.

We then use the implicit function theorem in Section 5.2 to conclude that the
space of solutions to Equation (1.1) is a Banach manifold which maps locally
diffeomorphically to the space of Fock bundles. In particular, we get a map
from a neighborhood of the Fuchsian locus in the space of higher complex
structures to the Hitchin component. In [23] it is shown that on each higher
diffeomorphism orbit there is a harmonic representative unique up to isotopy.
By restricting our map to this slice, we get the following:

######  Theorem E (Theorem 5.6).

There is an open neighborhood of the Fuchsian locus in the moduli space
$\mathcal{T}^{n}(S)$ of higher complex structures of order $n$, which has a
canonical map to the $\operatorname{PSL}_{n}(\mathbb{R})$-Hitchin component,
via solution of Equation (1.1).

It is expected that the restriction to harmonic higher complex structures is
not actually necessary because our map to the character variety should be
constant along higher diffeomorphism orbits. In Section 6, we analyse the
action of special $\lambda$-dependent gauge transformations on a family of
flat connections of the form $\lambda^{-1}\Phi+d_{A}+\lambda\Phi^{*}$ where
$\lambda\in\mathbb{C}^{*}$ is a parameter. In the case of
$\mathrm{SL}_{n}(\mathbb{C})$, these special gauge transformations relate to
higher diffeomorphisms:

######  Theorem F (Theorem 6.4).

The variation on an $\mathrm{SL}_{n}(\mathbb{C})$-Fock field $\Phi$ induced by
an infinitesimal gauge transformation $\lambda^{-1}\eta$ with
$\eta=\Phi(v_{1})\cdots\Phi(v_{k})$ is equivalent to the infinitesimal action
of the Hamiltonian $H=v_{1}\cdots v_{k}$ on the higher complex structure
induced by $\Phi$.

This theorem gives a clear gauge-theoretic meaning to higher diffeomorphisms,
which in the theory of higher complex structures are Hamiltonian
diffeomorphisms of $T^{*}S$ preserving the zero-section $S\subset T^{*}S$
setwise. This realization of higher diffeomorphisms as gauge transformations,
together with our ellipticity result gives the following:

######  Proposition G (Proposition 6.5).

A differentiable family $\Phi_{t}$ of solutions to Equation (1.1) for
$G=\operatorname{SL}_{n}(\mathbb{C})$ which induces a family of higher
diffeomorphic higher complex structures maps to a constant path of
representations.

Together with the main Conjecture C, this would give the desired map from
$\mathcal{T}^{n}(S)$ to the $\operatorname{PSL}_{n}(\mathbb{R})$-Hitchin
component, without the need for harmonic representatives.

In the final Section 7 we consider $G$-Fock bundles $(P,\Phi,\sigma,\rho)$
equipped with a hermitian structure $\rho$ not necessarily commuting with
$\sigma$. We parametrize the space of unitary connections $d_{A}$ satisfying
$d_{A}\Phi=0$ by so-called _covectors_. In the case of
$\mathrm{SL}_{n}(\mathbb{C})$, covectors can be identified with cotangent
vectors to the space of higher complex structures. We conjecture that we can
still solve the equation $F(A)+[\Phi\wedge\Phi^{*}]=0$ for small and
$\mu$-holomorphic covectors and that the monodromies of the flat connections
$\Phi+d_{A}+\Phi^{*}$ describe a tubular neighborhood of the $G$-Hitchin
component inside the complex character variety $\chi(\pi_{1}S,G)$. This
picture generalizes the work of Donaldson [5] and Trautwein [32] on the space
of almost-Fuchsian representations in the $\mathrm{SL}_{2}(\mathbb{C})$-case.
The $\mu$-holomorphicity condition from the theory of higher complex
structures gets a precise gauge-theoretical interpretation:

######  Theorem H (Theorem 7.13).

For an $\mathrm{SL}_{n}(\mathbb{C})$-Fock bundle, the condition
$F(A)\in\mathrm{Im}(\operatorname{ad}_{\Phi})\subset\Omega^{2}(S,\mathfrak{g}_{P})$
is equivalent to the $\mu$-holomorphicity condition (2.2).

Acknowledgements. We warmly thank Abdelmalek Abdesselam, Andrea Bianchi, Jeff
Danciger, Vladimir Fock, Dan Freed, Andy Neitzke and Alex Nolte for many
fruitful discussions and insights. We are also grateful to the Alexander von
Humboldt Foundation for support towards completing this project, to the
National Science Foundation (grants DMS-1937215, and DMS-1945493), to the SPP
2026 Geometry at Infinity for a travel grant and to the University of
Heidelberg and the University of Texas at Austin where most of the work has
been carried out.

##  2 Preliminaries

We gather necessary material for the main part of the text. We review higher
complex structures in Subsection 2.1, the nonabelian Hodge correspondence in
2.2, the Lie theoretic background in 2.3 (especially principal nilpotent
elements) and involutions on principal $G$-bundles in 2.4. Part of the
material in the Lie-theoretic section seems new, the rest is well-known.

###  2.1 Higher complex structures

Motivated by the aspiration to describe components of real character varieties
as moduli spaces of geometric structures, Vladimir V. Fock and the third
author introduced higher complex structures [8] and conjectured that they
parametrize $\mathrm{PSL}_{n}(\mathbb{R})$-Hitchin components. These were, in
turn, generalized to $\mathfrak{g}$-complex structures where $\mathfrak{g}$ is
a complex simple Lie algebra in [31]. We give here a brief account,
concentrating on the properties we need in the subsequent sections.

####  2.1.1 Definitions

Let $S$ be a closed orientable surface of genus at least 2. A complex
structure on $S$ is equivalent to an _almost complex structure_ , that is, an
automorphism $J$ of $TS$ satisfying $J^{2}=-\mathrm{id}$. The complexified
tangent bundle then decomposes as

$T^{\mathbb{C}}S=T^{1,0}S\oplus T^{0,1}S,$

where $T^{1,0}S$ and $T^{0,1}S$ are pointwise the eigendirections of $J$.
Since $T^{1,0}S$ is the complex conjugate of $T^{0,1}S$, the complex structure
is uniquely encoded by $T^{1,0}S$ which is a non-real direction of
$T^{\mathbb{C}}S$. This in turn is uniquely encoded by a certain ideal $I$ in
$\mathrm{Sym}(T^{\mathbb{C}}S)$ generated by $T^{0,1}S$ and $(T^{1,0}S)^{2}$.
Geometrically, $\mathrm{Sym}(T^{\mathbb{C}}S)$ is the ring of functions on
$T^{*\mathbb{C}}S$ which are polynomial on fibers, and $I$ cuts out an
infinitesimal thickening of the zero section of $T^{\mathbb{C}}S$ in the
direction of $T^{1,0}S$.

To describe this equivalence more explicitly, fix a reference complex
structure on $S$ and local coordinates $(z,\bar{z})$. Denote by $(p,\bar{p})$
the linear coordinates on $T^{*\mathbb{C}}S$ corresponding to vector fields
$\partial_{z}$ and $\partial_{\bar{z}}$. Any other complex structure can be
then described by an ideal $I$ locally of the form

$I=\langle p^{2},\bar{p}-\mu(z,\bar{z})p\rangle,$

where $\mu$ is known as the _Beltrami differential_ which satisfies
$\lvert\mu\rvert\neq 1$. The holomorphic cotangent bundle of the new complex
structure is defined by the equation $\bar{p}=\mu(z,\bar{z})p$.

######  Definition 2.1.

A _higher complex structure of order $n$_ (or of rank $n-1$) on $S$ is a
special ideal $I$ in $\mathrm{Sym}(T^{\mathbb{C}}S)$ locally of the form

$I=\langle
p^{n},-\bar{p}+\mu_{2}(z,\bar{z})p+\mu_{3}(z,\bar{z})p^{2}+...+\mu_{n}(z,\bar{z})p^{n-1}\rangle,$
(2.1)

where $\mu_{2}=\mu$ is the usual Beltrami differential and $\mu_{\ell}$ for
$\ell=3,...,n$ are called _higher Beltrami differentials_ (see [8, Proposition
1]).

Globally, $\mu_{\ell}$ is a smooth section of $K^{1-\ell}\otimes\bar{K}$ where
$K$ denotes the canonical bundle. A usual complex structure is a higher
complex structure of order 2, i.e. of rank 1.

An important feature of these structures is the forgetful map, which
associates to a higher complex structure of order $n$ a structure of order
$n-1$ by forgetting the last Beltrami differential $\mu_{n}$. In particular,
_any higher complex structure induces a complex structure on $S$_.

######  Remark.

The space of higher complex structures as defined above has two connected
components. A higher complex structure induces a complex structure which in
turn induces an orientation on $S$. This orientation coincides with the one
induced from the reference complex structure iff $\lvert\mu_{2}\rvert<1$.
Changing the reference complex structure to the complex conjugate one changes
$\mu_{2}$ to $1/\bar{\mu}_{2}$ which is of norm strictly bigger than 1.

More generally, let $\mathfrak{g}$ be a complex simple Lie algebra. An element
$x\in\mathfrak{g}$ is called _principal nilpotent_ if $\operatorname{ad}_{x}$
is nilpotent and $\dim Z(x)=\mathrm{rk}\,\mathfrak{g}$, where
$Z(x)=\\{y\in\mathfrak{g}\mid[x,y]=0\\}$ denotes the centralizer (see also
Section 2.3).

######  Definition 2.2 (Definition 4.1 in [31]).

A _$\mathfrak{g}$ -complex structure_ on $S$ is a $G$-conjugacy class of
fields $\Phi\in\Omega^{1}(S,\mathfrak{g})$ locally of the form
$\Phi=\Phi_{1}(z,\bar{z})dz+\Phi_{2}(z,\bar{z})d\bar{z}$ such that $\Phi_{1}$
is principal nilpotent and $\Phi_{2}\in Z(\Phi_{1})$ commutes with $\Phi_{1}$
satisfying a certain inequality explained below.

There is a unique linear combination of the form $\mu_{2}\Phi_{1}-\Phi_{2}$
which is not principal nilpotent [31, Proposition 2.21]. The inequality we
impose is $\lvert\mu_{2}\rvert<1$. This allows to identify $\mu_{2}$ with a
Beltrami differential of a complex structure on $S$. Therefore a
$\mathfrak{g}$-complex structure induces a complex structure on $S$.

For $\mathfrak{g}=\mathfrak{sl}_{n}(\mathbb{C})$ we get the notion of higher
complex structure as follows: principal nilpotent elements in $\mathfrak{g}$
form a single conjugacy orbit. So we can fix a principal nilpotent element
$F\in\mathfrak{sl}_{n}(\mathbb{C})$ and use the gauge freedom to fix
$\Phi_{1}=F$. Using the standard representation of
$\mathfrak{sl}_{n}(\mathbb{C})$ on $\mathbb{C}^{n}$, the centralizer of $F$ in
$\mathfrak{g}$ is generated by all polynomials in $F$ without constant term.
Hence the ideal of polynomials $P\in\mathbb{C}[p,\bar{p}]$ such that
$P(\Phi_{1},\Phi_{2})=0$ (which makes sense since $\Phi_{1}$ and $\Phi_{2}$
commute) is of the form (2.1).

####  2.1.2 Moduli space

For complex structures, the associated moduli space is the _Teichmüller space_
, where complex structures are considered modulo diffeomorphisms of $S$
isotopic to the identity. For higher complex structures, one needs to mod-out
by a larger group in order to get a finite-dimensional moduli space. We
restrict exposition to the case $\mathfrak{g}=\mathfrak{sl}_{n}(\mathbb{C})$
here. For $\mathfrak{g}$ of classical type, see [31, Section 4]. For general
complex simple $\mathfrak{g}$, the moduli space of $\mathfrak{g}$-complex
structures is not constructed yet.

######  Definition 2.3 (Definition 3 in [8]).

A _higher diffeomorphism_ of a surface $S$ is a hamiltonian diffeomorphism of
$T^{*}S$ preserving the zero-section $S\subset T^{*}S$ setwise. The group of
higher diffeomorphisms is denoted by $\mathrm{Ham}_{0}(T^{*}S)$.

Diffeomorphisms of $T^{*}S$ fixing the zero-section act on the completed
symmetric algebra $\widehat{\mathrm{Sym}}(T^{\mathbb{C}}S)$ of power series.
The map from ideals in $\mathrm{Sym}(T^{\mathbb{C}}S)$ to ideals in
$\widehat{\mathrm{Sym}}(T^{\mathbb{C}}S)$ is injective on those which contain
some power of the ideal
$(T^{\mathbb{C}}S)\subset\mathrm{Sym}(T^{\mathbb{C}}S)$. This includes higher
complex structures, so we can just as well view higher complex structures as
ideals in $\widehat{\mathrm{Sym}}(T^{\mathbb{C}}S)$ where diffeomorphisms
naturally act. A precise study of the space $\mathrm{Ham}_{0}(T^{*}S)$ and the
action by diffeomorphisms appears in [23, Section 7].

######  Definition 2.4.

The _moduli space of higher complex structures of order $n$_, denoted by
$\mathcal{T}^{n}(S)$, is the space of higher complex structures of order $n$,
denoted by $\mathbb{M}^{n}(S)$, modulo higher diffeomorphisms.

The _Fuchsian locus_ in $\mathbb{M}^{n}(S)$ consists in those higher complex
structures with trivial higher Beltrami differentials. The Fuchsian locus in
$\mathcal{T}^{n}(S)$ is the image under the projection
$\mathbb{M}^{n}(S)\to\mathcal{T}^{n}(S)$.

This moduli space is finite-dimensional of dimension $(n^{2}-1)(2g-2)$,
contractible and allows a forgetful map
$\mathcal{T}^{n}(S)\to\mathcal{T}^{n-1}(S)$, see [8, Theorem 2] and [23,
Theorem 1.1]. The Fuchsian locus inside $\mathcal{T}^{n}(S)$ is a copy of
Teichmüller space, which is $\mathcal{T}^{2}(S)$. The main conjecture within
this theory concerns the existence of a canonical diffeomorphism between
$\mathcal{T}^{n}(S)$ and the $\mathrm{PSL}_{n}(\mathbb{R})$-Hitchin component
which is equivariant with respect to the natural mapping class group action.
This conjecture has been proven for $n=3$ by Nolte in [23] using techniques
which are special to $n=3$ and are analogous to a positive resolution to the
Labourie Conjecture for $n=3$ [19], but are known to fail for higher $n$ (see
[24]).

Finally, we present the description of the total cotangent bundle
$T^{*}\mathcal{T}^{n}(S)$.

######  Theorem 2.5 (Theorem 3 in [8]).

The cotangent bundle $T^{*}\mathcal{T}^{n}(S)$ is an
$\mathrm{Ham}_{0}(T^{*}S)$-equivalence class of tensors
$\mu_{\ell}\in\Gamma(K^{1-\ell}\otimes\bar{K})$ and
$t_{\ell}\in\Gamma(K^{\ell})$ for $\ell=2,...,n$ satisfying

$-\bar{\partial}t_{k}\\!+\\!\mu_{2}\partial
t_{k}\\!+\\!kt_{k}\partial\mu_{2}+\sum_{l=1}^{n-k}((l\\!+\\!k)t_{k+l}\partial\mu_{l+2}+(l\\!+\\!1)\mu_{l+2}\partial
t_{k+l})=0.$ (2.2)

We refer to this condition as the _$\mu$ -holomorphicity condition_.

The tensors $\mu_{\ell}$ are the higher Beltrami differentials from (2.1). The
tensors $t_{\ell}$ describe the covector. For a trivial higher complex
structure, that is, when $\mu_{k}=0$ for all $k\in\\{2,...,n\\}$, the
condition simply reduces to $\bar{\partial}t_{k}=0$. Thus,
$\mu$-holomorphicity can be seen as _a generalization of the usual
holomorphicity condition_.

###  2.2 Nonabelian Hodge theory and twistor approach

Building on the fundamental theorems by Narasimhan–Seshadri and Eells–Sampson,
nonabelian Hodge theory provides an abundance of methods that can be used for
the study of character varieties via holomorphic techniques and Higgs bundles,
as well as analytic techniques from the theory of harmonic maps. We provide a
brief overview of this holomorphic viewpoint and the twistor space framework
to character varieties.

####  2.2.1 The nonabelian Hodge correspondence

Let $X$ be a compact Kähler manifold and let $G$ be a connected complex
reductive Lie group. For $H\subseteq G$ a maximal compact subgroup of $G$, its
complexification $H^{\mathbb{C}}$ is isomorphic to $G$, and the Cartan
decomposition for the corresponding Lie algebras
$\mathfrak{g}:=\mathrm{Lie}(G)$ and $\mathfrak{h}:=\mathrm{Lie}(H)$ reads in
this case as

$\mathfrak{g}=\mathfrak{h}\oplus i\mathfrak{h}.$

A flat principal $G$-bundle $(P,\Theta)$ over $X$ is equivalent to a reductive
representation $\rho:\pi_{1}(X)\to G$ via the _Riemann–Hilbert correspondence_
, for $\Theta$ a 1-form on $P$ with values in $\mathfrak{g}$ (a principal
connection on $P$). The flatness condition means that $\Theta$ satisfies the
equation

$d\Theta+\tfrac{1}{2}[\Theta\wedge\Theta]=0.$

Since $\rho$ is assumed to be reductive, Corlette’s Theorem [4] provides the
existence of a $\rho$-equivariant harmonic map from the universal cover
$\widetilde{X}$ of $X$ to the associated symmetric space,

$f:\widetilde{X}\to G/H.$

An equivariant map from $\widetilde{X}$ to $G/H$ is the same as a reduction of
structure group of the principal $G$-bundle $P$ to the maximal compact $H$.
This means there is an $H$-bundle $P_{H}$, and an $H$-equivariant map
$\iota:P_{H}\to P$, and the flat connection ${{\iota}^{*}}\Theta$ on $P_{H}$
now splits as

${{\iota}^{*}}\Theta=A+\psi,$

where $A$ is a connection on $P_{H}$, and $\psi$ descends to a 1-form on $X$
with values in the bundle associated to $P_{H}$ via the isotropy
representation $\mathrm{Ad}:H\to\mathrm{GL}(i\mathfrak{h})$. The flatness of
the connection ${{\iota}^{*}}\Theta$ implies the two equations

$\displaystyle{{F}_{A}}+\tfrac{1}{2}\left[\psi\wedge\psi\right]$
$\displaystyle=0$ $\displaystyle{{d}_{A}}\psi$ $\displaystyle=0,$

and harmonicity of the map $f$ gives the additional equation

$d_{A}^{*}\psi=0.$

In the case when $G=\mathrm{SL}_{n}(\mathbb{C})$, a reduction to a maximal
compact is the same as a unit volume hermitian metric on the associated vector
bundle. In the light of this equivalence, we call the special metric provided
by Corlette’s Theorem, a _harmonic hermitian metric_.
An application of the Siu–Sampson Theorem [25, Theorem 1] now gives that for a
reduction of structure group as above, then the $(0,1)$-part
${{\bar{\partial}}_{{{P}_{H}}}}$ of the connection $A$ and the $(1,0)$-part
$\varphi$ of the 1-form $\psi$ satisfy the equations

$\bar{\partial}^{2}_{P_{H}}\varphi=0\;,\;\;\bar{\partial}_{P_{H}}\varphi=0\;,\;\;[\varphi\wedge\varphi]=0.$

The $(0,1)$-form ${{\bar{\partial}}_{{{P}_{H}}}}$ defines a holomorphic
structure on the ${{C}^{\infty}}$-principal bundle $P_{H}$; extending the
structure group from $H$ to ${{H}^{\mathbb{C}}}$ we finally have:

######  Definition 2.6.

For a compact Kähler manifold $X$ and a connected complex reductive Lie group
$G$, a _$G$ -Higgs bundle_ over $X$ is a pair $(P,\varphi)$, where

  * •

$P$ is a holomorphic principal $G$-bundle over $X$, and

  * •

$\varphi\in{{\Omega}^{1,0}}\left(X,\mathfrak{g}_{P}\right)$ satisfying
${{\bar{\partial}}_{{{P}_{H}}}}\varphi=0$ and $[\varphi\wedge\varphi]=0$.

Equivalently, we can be thinking of the Higgs field $\varphi$ as a holomorphic
section $\varphi\in\mathrm{H}^{0}\left(X,\mathfrak{g}_{P}\otimes K\right)$,
where $\mathfrak{g}_{P}$ denotes the adjoint bundle of $P$.

######  Remark.

In the case when $X$ is a compact Riemann surface, the two conditions
$\bar{\partial}^{2}_{P_{H}}\varphi=0$ and $[\varphi\wedge\varphi]=0$ are
automatically satisfied.

######  Remark.

When $G\subset\mathrm{GL}_{n}(\mathbb{C})$, a $G$-Higgs bundle can be
naturally interpreted as a Higgs bundle $(E,\Phi)$ in the original sense of
Hitchin [12], [26] together with some additional structure reflecting the
structure of the group $G$. In particular, when
$G=\mathrm{SL}_{n}(\mathbb{C})$, an $\mathrm{SL}_{n}(\mathbb{C})$-Higgs bundle
is described by a pair $(E,\Phi)$, where $E$ is a holomorphic rank $n$ vector
bundle with trivial determinant and the Higgs field $\Phi$ is a holomorphic
section $\Phi\in\mathrm{H}^{0}\left(X,\mathrm{End}(E)\otimes K\right)$ with
$\mathrm{tr}(\Phi)=0$.

The flatness of the connection $\Theta$ finally gives the so-called _Hitchin
equation_ for the $G$-Higgs bundle $\left(P,\varphi\right)$ constructed above,

$F(A)+[\varphi\wedge\varphi^{*}]=0,$ (2.3)

where $\varphi^{*}$ denotes the hermitian adjoint coming from the Cartan
involution of $\mathfrak{g}$ extended to
$\mathfrak{g}_{P}\otimes\Omega^{1}(S)$.

So far we have described the passage from a reductive representation to a
Higgs bundle; the opposite direction is provided by Simpson’s Theorem [26] and
its generalizations [9], which says that for a $G$-Higgs bundle
$\left(P,\varphi\right)$ with vanishing Chern classes, there exists a
reduction of structure group of $P$ such that Equation (2.3) holds if and only
if $\left(P,\varphi\right)$ is polystable. The polystability condition is a
condition appropriately extending Mumford’s stability condition for vector
bundles, which is implemented in order to construct the _moduli space of
$G$-Higgs bundles_, or Dolbeault moduli space, $\mathcal{M}_{Dol}(X,G)$ as a
GIT quotient. The nonabelian Hodge correspondence for a Kähler manifold $X$
with underlying topological manifold $M$, therefore, is describing a bijection
between this Dolbeault moduli space and the character variety, or Betti moduli
space, $\mathcal{M}_{B}(M,G)$ of reductive fundamental group representations:

######  Theorem 2.7 (Nonabelian Hodge correspondence).

There is a real-analytic isomorphism
$\mathcal{M}_{Dol}(X,G)\cong\mathcal{M}_{B}(M,G)$.

####  2.2.2 The twistor approach

We now restrict attention to a compact Riemann surface $X$ with underlying
smooth surface $S$. For a polystable Higgs bundle
$\left({{{\bar{\partial}}}_{{{E}_{H}}}},\varphi\right)$ over $X$, Hitchin [12]
interpreted Equation (2.3) together with the condition of holomorphicity for
the Higgs field $\varphi$, ${{{\bar{\partial}}}_{{{E}_{H}}}}(\varphi)=0$, in
terms of a set of three moment maps for the action of the unitary gauge group.
Following symplectic reduction techniques, the moduli space
$\mathcal{M}_{Hit}$ of solutions to this set of equations was constructed as a
_hyperkähler quotient_. This means that $\mathcal{M}_{Hit}$ is a
$4n$-dimensional Riemannian manifold equipped with three covariant constant
(with respect to the Levi-Civita connection) orthogonal automorphisms $I,J$
and $K$ of the tangent bundle $T\mathcal{M}_{Hit}$ which satisfy the
quaternionic identities

${{I}^{2}}={{J}^{2}}={{K}^{2}}=IJK=-1.$

In fact, any linear combination of the form

$i(a,b,c)=aI+bJ+cK$ (2.4)

satisfies $i^{2}=-\mathrm{id}$ if and only if $a^{2}+b^{2}+c^{2}=1$. Taking
$\lambda=(a,b,c)\in\mathbb{CP}^{1}$, we have a 1-parameter family of complex
structures on $\mathcal{M}_{Hit}$. Then, using the Riemannian metric on
$\mathcal{M}_{Hit}$, one gets a family of symplectic structures which,
combined with the complex structures above, give a 1-parameter family of
Kähler structures.

In the light of [13], the _twistor approach_ allows one to incorporate all
complex structures in the 1-parameter family described above into a single
complex structure on a larger manifold, the _twistor space of_
$\mathcal{M}_{Hit}$. As a smooth manifold, this is defined as the product
manifold

$Z=\mathcal{M}_{Hit}\times\mathbb{CP}^{1}.$

One can equip $Z$ with a complex structure as follows: at a point
$(m,\lambda)$ of $Z$ it is given by the pair $(i_{\lambda},i_{0})$, where
$i_{0}$ denotes the standard complex structure on $\mathbb{CP}^{1}$ and
$i_{\lambda}$ is as in Equation (2.4), for
$\lambda=(a,b,c)\in\mathbb{CP}^{1}$. This defines an almost-complex structure
and it is not hard to prove the integrability. The projection
$p:Z\to\mathbb{CP}^{1}$ is holomorphic and each copy $(m,\mathbb{CP}^{1})$ of
the projective line is a holomorphic section of this projection, called a
_twistor line_.

Using ideas of Deligne, Simpson [27] constructed the twistor space for
$\mathcal{M}_{Hit}$ as the moduli space of $\lambda$-connections which he
called _Hodge moduli space_ $\mathcal{M}_{Hod}$. In this broader picture, the
Betti moduli space $\mathcal{M}_{B}(S,G)$ and the Dolbeault moduli space
$\mathcal{M}_{Dol}(X,G)$ are two special fibers of the holomorphic fiber
bundle $\mathcal{M}_{Hod}\to\mathbb{CP}^{1}$. One possible way to describe
Deligne’s $\lambda$-connections is via what we call three-term connections,
which is discussed next.

####  2.2.3 Three-term connections

For a complex reductive Lie group $G$, the work of Hitchin [12] and its
extensions provide that the cotangent bundle of the space of holomorphic
$G$-bundles on a given Riemann surface $X$ can be mapped isomorphically to the
space of families of connections of the form

$\mathcal{A}(\lambda)=\lambda^{-1}\Phi+d_{A}+\lambda\Phi^{*},$

where $\lambda\in\mathbb{C}^{*}$ is a parameter, $\Phi$ a holomorphic Higgs
field, $\Phi^{*}$ the hermitian conjugate with respect to a harmonic metric
$h$, and $d_{A}$ the associated Chern connection. In the Higgs bundle setting,
this is a family of _flat connections_ , the $(0,1)$-part of the background
connection $d_{A}$ defines a holomorphic structure on the bundle and the Higgs
field $\Phi$ defines a cotangent vector to the space of holomorphic bundles.
The family $\mathcal{A}(\lambda)$ gives a family of maps from the moduli space
of flat $G$-connections to itself depending on the parameter
$\lambda\in\mathbb{C}^{*}$. In the limit $\lambda\to 0$ (resp.
$\lambda\to\infty$) these structures tend to the moduli space of $G$-Higgs
bundles on $X$ (resp. the complex conjugate $\bar{X}$) with its Kähler
structure.

Another important example of a 1-parameter family of flat connections
depending on a parameter $\lambda\in\mathbb{C}^{*}$ analogous to the Hitchin
family of flat connections described above was given by Fock in [7]. These
flat connections are determined by solutions of the cosh-Gordon equation and
describe a candidate for the twistor space of almost-Fuchsian representations.
In the approach of [7], the complex structure on the surface is a function of
a background connection determined as above, and is not fixed once for all
connections in the family as in Hitchin’s case.

###  2.3 Principal nilpotent elements

Let $\mathfrak{g}$ be a complex simple Lie algebra and $G$ be a Lie group with
Lie algebra $\mathfrak{g}$. For $x\in\mathfrak{g}$ we denote by
$Z(x)=\\{y\in\mathfrak{g}\mid[x,y]=0\\}$ its centralizer in $\mathfrak{g}$. We
also denote by $Z_{G}(x)$ the centralizer of $x$ in $G$ and by $Z(G)$ the
center of $G$.

######  Definition 2.8.

An element of $\mathfrak{g}$ is called _regular_ if the dimension of its
centralizer equals the rank of the Lie algebra. A regular nilpotent element is
called _principal nilpotent_.

A more general statement holds about the dimension of the centralizer: for any
$x\in\mathfrak{g}$, we have $\dim Z(x)\geq\operatorname{rk}(\mathfrak{g})$
(see for example Lemma 2.1.15. in [3]). So the regular elements minimize this
dimension. For $\mathfrak{g}=\mathfrak{sl}_{n}(\mathbb{C})$, a nilpotent
element is principal nilpotent iff it has maximal rank, i.e. it is of rank
$n-1$.

######  Theorem 2.9.

[16, Corollary 5.5.] All principal nilpotent elements are conjugate under the
adjoint action of the Lie group $G$.

For a principal nilpotent element $F$, its centralizer has properties quite
analogous to a Cartan subalgebra (the centralizer of a regular semisimple
element):

######  Theorem 2.10.

For $F$ a principal nilpotent element, its centralizer $Z(F)$ is abelian and
nilpotent.

Using a limit argument one can show even more: for any element
$x\in\mathfrak{g}$, there is an abelian subalgebra of $Z(x)$ of dimension
$\operatorname{rk}(\mathfrak{g})$; see [16, Theorem 5.7]. The nilpotency of
$Z(f)$ can be found in [28, Corollary in Section 3.7].

The _Jacobson–Morozov lemma_ states that any non-zero nilpotent element
$F\in\mathfrak{g}$ can be included into an $\mathfrak{sl}_{2}$-subalgebra, the
image of an injective homomorphism from $\mathfrak{sl}_{2}(\mathbb{C})$ into
$\mathfrak{g}$. An $\mathfrak{sl}_{2}$-subalgebra is called _principal_ if it
contains a principal nilpotent element. It follows from Theorem 2.9 that all
principal $\mathfrak{sl}_{2}$-subalgebras are conjugate. Given any
$\mathfrak{sl}_{2}$-subalgebra, the Lie algebra $\mathfrak{g}$ splits into
irreducible $\mathfrak{sl}_{2}$-modules. For a principal one, none of these
modules is trivial and exactly one module is of dimension 3 (the principle
$\mathfrak{sl}_{2}$-subalgebra itself).

######  Proposition 2.11.

For a principal nilpotent element $F\in\mathfrak{g}$, we have
$Z(F)\subset\mathrm{Im}(\operatorname{ad}_{F})$.

To see this, include $F$ into a principal $\mathfrak{sl}_{2}$-subalgebra,
decompose $\mathfrak{g}$ into irreducible $\mathfrak{sl}_{2}$-modules and use
$\mathfrak{sl}_{2}$-representation theory. The important point is that there
is no trivial $\mathfrak{sl}_{2}$-module.

######  Proposition 2.12.

The $\mathfrak{sl}_{2}$-subalgebras containing a given principal nilpotent $F$
are acted on transitively by $Z_{G}(F)\cong\exp(Z(F))\times Z(G)$ with
stabilizer $\exp(F)\times Z(G)$. In particular, the set of
$\mathfrak{sl}_{2}$-subalgebras containing $F$ is a torsor for the quotient
group which is isomorphic to $\mathbb{C}^{\mathrm{rk}(\mathfrak{g})-1}$. In
particular, this space is contractible.

The structure of the centralizer $Z_{G}(F)$ is described in Lemma 3.7.3 in
[3]. It states that for a non-zero nilpotent element $x$ the centralizer
$Z_{G}(x)$ is a semidirect product between
$\exp(Z(x)\cap\mathrm{Im}(\operatorname{ad}_{x}))$, which is $\exp(Z(F))$ for
$x=F$ by Proposition 2.11, and the centralizer of an
$\mathfrak{sl}_{2}$-subalgebra containing $x$, which in the principal case is
$Z(G)$.

A basis $(F,H,E)$ of an $\mathfrak{sl}_{2}$-subalgebra, satisfying the
standard relations $[H,E]=2E$, $[H,F]=-2F$ and $[E,F]=H$, is called an
_$\mathfrak{sl}_{2}$ -triple_. Fix a principal $\mathfrak{sl}_{2}$-triple
$(F,H,E)$. To this triple, Hitchin [14] associates a Lie algebra involution
$\sigma_{0}:\mathfrak{g}\to\mathfrak{g}$ as follows. Using the decomposition
of $\mathfrak{g}$ into irreducible $\mathfrak{sl}_{2}$-modules, the involution
$\sigma_{0}$ is uniquely determined by negating all highest and lowest weight
vectors [14, Proposition 6.1]. He also defines a compact real form
$\rho_{0}:\mathfrak{g}\to\mathfrak{g}$ which extends

$E\mapsto-F,\;\;\;\;H\mapsto-H,\;\;\;\;F,\mapsto-E$

and commutes with $\sigma_{0}$. Finally, he shows that
$\tau_{0}:=\sigma_{0}\rho_{0}$ is a split real structure. In fact,
$\sigma_{0}$ can be defined up to conjugation by inner automorphisms as the
product of commuting split and compact real forms.

It will be useful to understand the set of involutions conjugate to
$\sigma_{0}$ which negate a given principal nilpotent element.

######  Lemma 2.13.

The collection of involutions $\sigma$ conjugate to $\sigma_{0}$ which negate
a given principal nilpotent $F$ is acted on simply transitively by
$\exp(Z(F))$. In particular, this space is contractible.

###### Proof.

Let $\sigma,\sigma^{\prime}$ be two such involutions which negate $F$. They
differ by an inner automorphism:
$\sigma^{\prime}=\sigma\circ\operatorname{Ad}_{\gamma}$ where $\gamma\in G$.
We see that $\operatorname{Ad}_{\gamma}F=\sigma\sigma^{\prime}\cdot F=F$, so
$\gamma\in Z_{G}(F)$. Since $Z_{G}(F)\cong\exp(Z(F))\times Z(G)$, we see that
$\exp(Z(F))$ must act transitively. From this fact, we deduce Corollary 2.14
below, stating that $\sigma$ negates the whole centralizer $Z(F)$. Hence it
inverts $\exp(Z(F))$. We have to check that if $z\in\exp(Z(F))$ is different
from the identity, then $\sigma\circ\operatorname{Ad}_{z}\neq\sigma$. For
$X\in\mathfrak{g}$ we have:

$\sigma(\operatorname{Ad}_{z}X)=\operatorname{Ad}_{\sigma(z)}\sigma(X)=\operatorname{Ad}_{z^{-1}}\sigma(X).$

If this was always equal to $\sigma(X)$, then $z$ would be central, which
contradicts the fact that it is the exponential of a nilpotent element. ∎

######  Corollary 2.14.

Any $\sigma$ conjugate to $\sigma_{0}$ which negates $F$, actually negates all
of $Z(F)$.

###### Proof.

This is true for the involution $\sigma$ of Hitchin’s construction, and this
property is unchanged under inner automorphisms by elements in $\exp(Z(F))$. ∎

######  Lemma 2.15.

Let $\sigma$ be an involution conjugate to $\sigma_{0}$ and let
$F\in\mathfrak{g}$ be principal nilpotent with $\sigma(F)=-F$. Then there is a
unique $\sigma$-invariant $\mathfrak{sl}_{2}$-subalgebra containing $F$.

###### Proof.

Proposition 2.12 shows that $\exp(Z(F))$ acts transitively on the space of
$\mathfrak{sl}_{2}$-subalgebras containing $F$. By Corollary 2.14, we know
that $\sigma$ inverts $\exp(Z(F))$. The involution $\sigma$ acts on the set of
principal $\mathfrak{sl}_{2}$-subalgebras containing $F$, intertwining the
action of $\exp(Z(F))$ with its negative. More explicitly, let
$\mathfrak{p}\subset\mathfrak{g}$ be a principal
$\mathfrak{sl}_{2}$-subalgebra containing $F$, and let $z\in\exp(Z(F))$. Then:

$\sigma(\operatorname{Ad}_{z}\mathfrak{p})=\operatorname{Ad}_{z^{-1}}\sigma(\mathfrak{p}).$

It will follow from this property that there is a unique fixed point of the
action by $\sigma$, which we can construct as a kind of midpoint. Let $z$ be
the unique element of $\exp(Z(F))$ such that
$\operatorname{Ad}_{z^{2}}(\mathfrak{p})=\sigma(\mathfrak{p})$. Moreover,

$\sigma(\operatorname{Ad}_{z}(\mathfrak{p}))=\operatorname{Ad}_{z^{-1}}(\sigma(\mathfrak{p}))=\operatorname{Ad}_{z^{-1}}(\operatorname{Ad}_{z^{2}}(\mathfrak{p}))=\operatorname{Ad}_{z}(\mathfrak{p}).$

Therefore, $\operatorname{Ad}_{z}(\mathfrak{p})$ is an
$\mathfrak{sl}_{2}$-subalgebra fixed by $\sigma$. If there were two fixed
subalgebras $\mathfrak{p},\mathfrak{p}^{\prime}$, then
$\mathfrak{p}^{\prime}=\operatorname{Ad}_{z}\mathfrak{p}$ for some $z$. Such
$z$ satisfies $\sigma(z)=z$, thus is the identity. ∎

Finally, we will need the following lemma:

######  Lemma 2.16.

For a principal nilpotent element $F\in\mathfrak{g}$ and an element
$F^{\prime}\in Z(F)$, we have
$\mathrm{Im}(\operatorname{ad}_{F^{\prime}})\subset\mathrm{Im}(\operatorname{ad}_{F})$.

###### Proof.

By the Jacobson–Morozov lemma we can complete $F$ into a principal
$\mathfrak{sl}_{2}$-triple $(F,H,E)$. Choose a compact real form $\rho$ such
that $E=-\rho(F)$. Let $E^{\prime}=-\rho(F^{\prime})$. Since $F^{\prime}\in
Z(F)$ we get $E^{\prime}\in Z(E)$. The commutator $Z(E)$ is abelian, hence we
have $Z(E)\subset Z(E^{\prime})$. With respect to the hermitian inner product
$\operatorname{tr}(\rho(.).)$ on $\mathfrak{g}$, $E$ and $F$ are adjoint, and
so are $E^{\prime}$ and $F^{\prime}$. This implies that
$\mathrm{Im}(\operatorname{ad}_{F})$ is the perp of
$\ker(\operatorname{ad}_{E})=Z(E)$, and
$\mathrm{Im}(\operatorname{ad}_{F}^{\prime})$ is the perp of
$\ker(\operatorname{ad}_{E^{\prime}})=Z(E^{\prime})$. This concludes the proof
since $Z(E)\subset Z(E^{\prime})$. ∎

###  2.4 Involutions and reductions of structure group

There are three basic structures often put on a complex vector bundle: a
hermitian structure, a symmetric pairing, and a real structure. In this
section we explain precisely how to generalize these notions to principal
$G$-bundles for connected complex simple $G$.

Fix an antiholomorphic involution $\rho_{0}:G\to G$ whose fixed point locus is
a maximal compact subgroup, and another commuting antiholomorphic involution
$\tau_{0}$ whose fixed point locus is a split real subgroup. Call their
composition $\sigma_{0}:=\rho_{0}\tau_{0}$. These involutions induce, (and are
determined by) anti-linear involutions of the lie algebra $\mathfrak{g}$ which
we will refer to by the same symbols.

######  Remark.

The involution $\sigma_{0}$ can equivalently be obtained by Hitchin’s
construction using a principal $\mathfrak{sl}_{2}$-triple [14, Proposition
6.1].

######  Definition 2.17.

Let $G$ be a group, let $\epsilon_{0}:G\to G$ be an involution, and let $P$ be
a principal $G$-bundle on a manifold $M$. An _$\epsilon_{0}$ -structure_ on
$P$ is an involution $\epsilon:P\to P$ such that

$\epsilon(p.g)=p.\epsilon_{0}(g)\;\;\;\forall\,p\in P,g\in G.$

An $\epsilon_{0}$-structure on a $G$-bundle $P\to M$ is equivalent to a
reduction of structure group of $P$ from $G$ to the group of fixed points
$G^{\epsilon_{0}}$. Indeed, the fixed point locus of $\epsilon$ is naturally a
$G^{\epsilon_{0}}$-bundle.

The special case of $\rho_{0}$-structures, which we will refer to as a
_hermitian structure_ , can be understood more concretely. A hermitian
structure $\rho$ on a $G$-bundle $P$ induces an involution (which we also call
$\rho$) on the adjoint bundle $\mathfrak{g}_{P}$. The eigenspaces of $\rho$
give a fiberwise Cartan decomposition of $\mathfrak{g}_{P}$. Since
$G^{\rho_{0}}$ is precisely the subgroup of $G$ which preserves a Cartan
decomposition, the reduction to $G^{\rho_{0}}$ is fully specified by this
involution of $\mathfrak{g}_{P}$.

In the case of $\sigma_{0}$\- and $\tau_{0}$-structures, we also get
involutions $\sigma$ and $\tau$ of $\mathfrak{g}_{P}$, but there is slightly
more data in the structure. This is because the subgroup of $G$ commuting with
$\tau_{0}$ is not just $G^{\tau_{0}}$, but also the center of $G$. So just
specifying an involution $\tau$ of $\mathfrak{g}_{P}$, which is conjugate to
$\tau_{0}$ in each fiber, only gives a reduction to a slightly larger group.
The same is true for $\sigma$. In the case $G=G_{ad}$ where the center is
trivial, this is not an issue, and we may think of $\tau_{0}$\- and
$\sigma_{0}$-structures purely as involutions of the adjoint bundle.

##  3 Fock bundles

In this section, we introduce Fock bundles, analyze the so-called Fuchsian
locus and describe the variation in the Fock field.

###  3.1 Fock bundles and higher complex structures

Fix a smooth closed orientable surface $S$ with genus at least 2. Consider a
complex simple Lie group $G$ with associated Lie algebra $\mathfrak{g}$.
Throughout, we will fix commuting involutions $\rho_{0},\tau_{0}$ and
$\sigma_{0}$ of $G$ such that $\rho_{0}$ is a compact real form, $\tau_{0}$ is
a split real form, and $\sigma_{0}=\rho_{0}\tau_{0}$. For a principal
$G$-bundle $P$, denote by $\mathfrak{g}_{P}$ the associated
$\mathfrak{g}$-bundle using the adjoint action of $G$ on $\mathfrak{g}$.

Recall the notions of a principal nilpotent element and a
$\sigma_{0}$-structure from the previous Subsections 2.3 and 2.4. The main
notion we introduce in this article is the following:

######  Definition 3.1.

A _$G$ -Fock bundle_ over $S$ is a triple $(P,\Phi,\sigma)$, where $P$ is a
principal $G$-bundle over $S$, $\sigma$ is a $\sigma_{0}$-structure, and
$\Phi\in\Omega^{1}(S,\mathfrak{g}_{P})$ is a $\mathfrak{g}_{P}$-valued 1-form
satisfying

  1. 1.

$[\Phi\wedge\Phi]=0$,

  2. 2.

$\Phi(v)(z)$ is principal nilpotent for all $z\in S$ and all non-zero vectors
$v\in T_{z}S$.

  3. 3.

$\sigma(\Phi)=-\Phi$.

We shall call a field $\Phi$ defined as above a _Fock field_ and will often
refer to the second condition above as the _nilpotency condition_. An
isomorphism of Fock bundles
$(P,\Phi,\sigma)\to(P^{\prime},\Phi^{\prime},\sigma^{\prime})$ is a $G$-bundle
isomorphism $P\to P^{\prime}$ taking $\Phi$ to $\Phi^{\prime}$ and $\sigma$ to
$\sigma^{\prime}$.

Note that all conditions in Definition 3.1 are given pointwise. We stress that
no term in the definition of a Fock bundle is considered to be holomorphic.

There are two main cases one should highlight:

  * •

An $\operatorname{SL}_{n}(\mathbb{C})$-Fock bundle is a vector bundle $E$ of
rank $n$ with fixed volume form, equipped with a symmetric pairing $g$ (a
complex bilinear non-degenerate symmetric form) and a Fock field $\Phi$
satisfying the three conditions above.

  * •

For $G=G_{ad}$ the adjoint group, a $G$-Fock bundle is specified up to
isomorphism by the pair $(P,\Phi)$.

The data contained in $\sigma$ is actually very little: We will see later in
Proposition 3.22 that a $\sigma_{0}$-structure $\sigma$ which negates $\Phi$
always exists locally, and always exists globally if $G=G_{ad}$ is the adjoint
group. For any $G$, any two choices of $\sigma$ are conjugate by an
automorphism fixing $\Phi$, thus give isomorphic Fock bundles. That is why we
will sometimes write $(P,\Phi)$ for a $G$-Fock bundle.

By [31, Proposition 2.21], we know that there is a unique complex line
$L\subset T^{\mathbb{C}}_{z}S$ such that for all $v\in L$ the matrix
$\Phi(v)(z)$ is not principal nilpotent. Note that the uniqueness needs the
Lie algebra $\mathfrak{g}$ to be simple. The nilpotency condition in
Definition 3.1 then implies that this direction $L$ is avoiding the real locus
and hence encodes a complex structure on $S$ (see Section 2.1).

######  Proposition 3.2.

A $G$-Fock bundle induces a complex structure on $S$.

In the sequel, unless stated explicitly otherwise, whenever we work with
complex local coordinates on $S$, we use the complex structure induced by the
Fock bundle. For such a complex coordinate $z$ on $S$, we can locally write
$\Phi=\Phi_{1}dz+\Phi_{2}d\bar{z}$. The condition $[\Phi\wedge\Phi]=0$ for a
Fock field can be then written as $[\Phi_{1},\Phi_{2}]=0$. By construction of
the complex structure on $S$, we know that $\Phi_{2}$ is not principal
nilpotent. Hence $\Phi_{1}$ has to be principal nilpotent.

We now describe more explicitly what $\mathrm{SL}_{n}(\mathbb{C})$-Fock
bundles look like locally. We start with $n=2$. Let $E$ be a complex vector
bundle of degree zero and rank $2$ over $S$ with a fixed volume form $\nu$.

######  Lemma 3.3.

Fix an arbitrary complex coordinate on $S$. An
$\mathrm{SL}_{2}(\mathbb{C})$-Fock field on $E$ locally is of the form

$\Phi=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}dz+\begin{pmatrix}0&0\\\
\mu_{2}&0\end{pmatrix}d\bar{z},$

where $\mu_{2}$ is a local complex function on $S$ with
$\mu_{2}\bar{\mu}_{2}\neq 1$.

Note that globally, $\mu_{2}$ is the Beltrami differential of the complex
structure on $S$ induced by the Fock bundle.

###### Proof.

There is a local trivialization of $E$ in which $\Phi_{1}$ is given as in the
statement of the lemma since all principal nilpotent elements are conjugate.
The condition $[\Phi_{1},\Phi_{2}]=0$ then implies the form of $\Phi_{2}$.
Finally, the nilpotency condition implies that for any non-zero real tangent
vector $v\partial+\bar{v}\bar{\partial}$ (where $v$ is a complex function) the
combination $v\Phi_{1}+\bar{v}\Phi_{2}$ is principal nilpotent. This means
that $v+\mu_{2}\bar{v}\neq 0$ for all $v\neq 0$. This is equivalent to the
condition $\lvert\mu_{2}\rvert\neq 1$. ∎

We can generalize this local description to higher rank. Let $E$ be a complex
vector bundle of degree zero and rank $n$ over $S$ with a fixed volume form
$\nu$.

######  Lemma 3.4.

Fix an arbitrary complex coordinate on $S$. An
$\mathrm{SL}_{n}(\mathbb{C})$-Fock field on $E$ locally is of the form
$\Phi=\Phi_{1}dz+\Phi_{2}d\bar{z}$ with

$\Phi_{1}=\sum_{i=1}^{n-1}E_{i+1,i}\;\text{ and
}\;\Phi_{2}=\mu_{2}\Phi_{1}+\mu_{3}\Phi_{1}^{2}+...+\mu_{n}\Phi_{1}^{n-1},$

where the $\mu_{k}$ are local complex functions on $S$ with
$\mu_{2}\bar{\mu}_{2}\neq 1$.

The proof is similar to the one given above; there is only one conjugacy class
of principal nilpotent elements which explains the form of $\Phi_{1}$. The
centralizer of $\Phi_{1}$ is the set of polynomials in $\Phi_{1}$ which
provides the form of $\Phi_{2}$. In the complex structure induced by the Fock
field, we always have $\mu_{2}=0$. The case when all higher Beltrami
differentials vanish is the so-called _Fuchsian locus_ which will be analysed
in Section 3.2 below.

Comparing Definition 3.1 of a $G$-Fock bundle to Definition 2.2 of a
$\mathfrak{g}$-complex structure, we immediately see the following:

######  Proposition 3.5.

Any $G$-Fock bundle induces a $\mathfrak{g}$-complex structure on $S$. For the
adjoint group $G_{ad}$, the isomorphism class of a $G_{ad}$-Fock bundle is
equivalent to a $\mathfrak{g}$-complex structure.

The second assertion follows directly from Corollary 3.15 below. For
$G=\mathrm{SL}_{n}(\mathbb{C})$, we get a direct link between Fock bundles and
the ideals describing higher complex structures of order $n$:

######  Proposition 3.6.

Let $(E,\Phi)$ be an $\mathrm{SL}_{n}(\mathbb{C})$-Fock bundle over a surface
$S$. The map

$p:\left\\{\begin{array}[]{ccc}\mathrm{Sym}(T^{\mathbb{C}}S)&\longrightarrow&\mathrm{End}(E)\\\
v_{1}\cdots v_{k}&\mapsto&\Phi(v_{1})\cdots\Phi(v_{k})\end{array}\right.$

is well-defined and the kernel of $p$ defines a higher complex structure.

###### Proof.

To prove that $p$ is well-defined, one has to show that the expression
$\Phi(v_{1})\cdots\Phi(v_{k})$ remains unchanged under permutation of
$(v_{1},...,v_{k})$. This follows from $\Phi\wedge\Phi=0$. The matrix
viewpoint of higher complex structures analyzed in [31, Section 4.2] implies
that the kernel of $p$ is a higher complex structure. ∎

######  Proposition 3.7.

A $G$-Fock bundle $(P,\Phi,\sigma)$ has no infinitesimal automorphisms. Thus,
all Fock bundles are stable in this sense.

###### Proof.

Consider an infinitesimal gauge transformation
$\eta\in\Omega^{0}(S,\mathfrak{g}_{P})$. In order to preserve $\Phi$, we need
$\eta\in Z(\Phi)$. Since $Z(\Phi)$ is negated by $\sigma$ (see Corollary
2.14), the only way for $\eta$ to preserve $\sigma$ is to be zero. ∎

For $\operatorname{SL}_{n}(\mathbb{C})$, the Fock field $\Phi$ induces a
natural filtration $\mathcal{F}$ on the bundle $E$:

######  Proposition 3.8.

An $\operatorname{SL}_{n}(\mathbb{C})$-Fock bundle has a natural increasing
filtration given by $\mathcal{F}_{k}=\ker\Phi(v)^{k}$, where $v$ is a local
non-vanishing vector field. The filtration is independent of the choice of
$v$.

The proposition follows directly from the fact that $\Phi(v)$ is a principal
nilpotent element. Independence follows from $\Phi\wedge\Phi=0$ and the fact
that $Z(\Phi)$ is abelian.

###  3.2 Fuchsian Locus

The Fuchsian locus in a Hitchin component, for an adjoint group, is the set of
representations which factor through $\operatorname{PSL}_{2}(\mathbb{R})$.
Similarly, we will introduce the Fuchsian locus in the space of Fock bundles
for an adjoint group as the subset induced from
$\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundles. When $G$ has center, we
call a Fock bundle Fuchsian if the associated Fock bundle for $G_{ad}$ is
Fuchsian. It will turn out that every Fock bundle with $\Phi_{2}=0$ is
Fuchsian, so all Fock bundles are deformable to the Fuchsian locus. Every Fock
bundle in the Fuchsian locus can be equipped with a holomorphic structure,
making it into a Higgs bundle. This is the uniformizing Higgs bundle.

######  Proposition 3.9.

Any $\operatorname{SL}_{2}(\mathbb{C})$-Fock bundle is of the form
$(E,\Phi,g)$ where $E=K^{1/2}\oplus K^{-1/2}$, where we use the complex
structure on $S$ induced from the Fock bundle, $g$ is the apparent symmetric
pairing in which these line subbundles are isotropic, and

$\Phi=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$

where $1$ denotes the canonical 1-form valued in
$\operatorname{Hom}(K^{1/2},K^{-1/2})\cong K^{-1}$.

###### Proof.

Let $(E,\Phi,g)$ be an $\operatorname{SL}_{2}(\mathbb{C})$-Fock bundle. The
symmetric pairing $g$, gives an isotropic decomposition into dual line bundles
$E=L\oplus L^{-1}$. We get a decomposition

$\mathfrak{sl}(E)=L^{-2}\oplus\underline{\mathbb{C}}\oplus L^{2}$

in which $\sigma:X\mapsto-X^{*_{g}}$ acts by $(-1,1,-1)$. The Fock field
$\Phi$ must be valued in the $-1$ eigenspace $L^{2}\oplus L^{-2}$. Since
$\Phi$ is nilpotent, it is valued in only one of these line bundles at any
given point, and since it is nowhere vanishing, it is valued in only one of
the line bundles globally. Without loss of generality, suppose it is valued in
$L^{-2}$. Since it is nowhere vanishing, $\Phi$ is an isomorphism from the
holomorphic tangent bundle of $S$ to $L^{-2}$, so $L$ must be a square root of
the canonical bundle. ∎

As a corollary, any $\operatorname{SL}_{2}(\mathbb{C})$-Fock bundle has a
natural upgrade to a Higgs bundle for the induced complex structure, because
$K^{1/2}\oplus K^{-1/2}$ is naturally a holomorphic vector bundle. We see that
an $\operatorname{SL}_{2}(\mathbb{C})$-Fock bundle is equivalent to a complex
structure together with a spin structure. Similarly, a
$\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundle is equivalent to a complex
structure.

######  Proposition 3.10.

Any $\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundle is of the form
$(P,\Phi,\sigma)$ where
$P=K_{*}\times_{\mathbb{C}^{*}}\operatorname{PSL}_{2}(\mathbb{C})$ with
adjoint bundle
$\mathfrak{sl}_{2}(\mathbb{C})_{P}=K^{-1}\oplus\underline{\mathbb{C}}\oplus
K$, $\Phi$ is the canonical $K^{-1}$-valued $1$-form, and $\sigma$ acts by
$(-1,1,-1)$ on the adjoint bundle. Here, $K_{*}$ denotes the
$\mathbb{C}^{*}$-bundle of nonzero covectors.

In particular, $P$ is topologically trivial because $K$ has even degree. Any
$\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundle can be upgraded to an
$\operatorname{SL}_{2}(\mathbb{C})$-Fock bundle by choosing a square root of
$K$.

Now restrict attention to an adjoint group $G_{ad}$. Then, the principal
3-dimensional subgroup is always $\operatorname{PSL}_{2}(\mathbb{C})$. This is
because the $\mathfrak{sl}_{2}$-representations appearing in $\mathfrak{g}$
are always odd dimensional. Fix a principal embedding
$\operatorname{PSL}_{2}(\mathbb{C})\to G_{ad}$ such that the diagram

${\operatorname{PSL}_{2}(\mathbb{C})}$${G_{ad}}$${\operatorname{PSL}_{2}(\mathbb{C})}$${G_{ad}}$$\scriptstyle{\sigma_{0}}$$\scriptstyle{\sigma_{0}}$

commutes.

######  Definition 3.11.

If $(P,\Phi,\sigma)$ is a $\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundle,
then the _induced $G_{ad}$-Fock bundle_ is the triple
$(P^{\prime},\Phi^{\prime},\sigma^{\prime})$ where

  * •

$P^{\prime}$ is the induced bundle
$P\times_{\operatorname{PSL}_{2}(\mathbb{C})}G_{ad}$, or alternatively
$P^{\sigma}\times_{\operatorname{PSL}_{2}(\mathbb{C})^{\sigma_{0}}}G_{ad}$.

  * •

$\Phi^{\prime}$ is the composition of $\Phi$ with the inclusion
$\mathfrak{sl}_{2}(\mathbb{C})_{P}\to\mathfrak{g}_{P^{\prime}}$, and

  * •

$\sigma^{\prime}$ is $\sigma_{0}$ acting on the right factor $G_{ad}$ in the
second description of $P^{\prime}$.

The same definition will work for general $G$ with
$\operatorname{PSL}_{2}(\mathbb{C})$ possibly replaced by
$\operatorname{SL}_{2}(\mathbb{C})$.

######  Definition 3.12.

A $G_{ad}$-Fock bundle is in the _Fuchsian locus_ if it is induced from a
$\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundle. For general $G$, a $G$-Fock
bundle is in the Fuchsian locus if the associated $G_{ad}$-Fock bundle is in
the Fuchsian locus.

######  Example 3.13.

The $\operatorname{SL}_{n}(\mathbb{C})$-Fock bundle induced from an
$\operatorname{SL}_{2}(\mathbb{C})$-Fock bundle is the pair $(E,\Phi)$ given
by

$E=K^{(n-1)/2}\oplus K^{(n-3)/2}\oplus...\oplus K^{(1-n)/2}\;\;\text{ and
}\;\;\Phi=\sum_{i=1}^{n-1}E_{i+1,i},$

where $K$ denotes the canonical bundle on $S$, $K^{1/2}$ is a choice of a
square-root and $E_{i+1,i}$ is the canonical identity 1-form valued in
$\mathrm{Hom}(K^{(n-k)/2},K^{(n-k-2)/2})=K^{-1}$. Again, if we view $E$ as a
smooth vector bundle, this is a Fock bundle, but if we view it as a
holomorphic bundle, it becomes the uniformizing
$\mathrm{SL}_{n}(\mathbb{C})$-Higgs bundle.

######  Proposition 3.14.

A Fock bundle $(P,\Phi,\sigma)$ is in the Fuchsian locus if and only if
$\Phi_{2}=0$.

###### Proof.

A Fock field in the Fuchsian locus clearly has no $(0,1)$-part. For the
converse, by Lemma 2.15, we know that there is a unique $\sigma$-invariant
$\mathfrak{sl}_{2}$-subbundle $\mathfrak{p}$ of $\mathfrak{g}_{P}$ containing
the image of $\Phi$, because the image of $\Phi$ is a line bundle of principal
nilpotents negated by $\sigma$.

The normalizer in $G_{ad}$ of a principal $\mathfrak{sl}_{2}$-subalgebra is
just the principal $\operatorname{PSL}_{2}(\mathbb{C})$ it generates. To see
this, first recall that the centralizer of a principal $\mathfrak{sl}_{2}$ is
trivial, then note that anything in the normalizer can be multiplied by an
element of $\operatorname{PSL}_{2}(\mathbb{C})$ to be in the centralizer. The
subbundle $\mathfrak{p}$ thus gives a reduction of structure group of $G_{ad}$
to $\operatorname{PSL}_{2}(\mathbb{C})$. Call this reduction of structure
group $Q\subset P$. The induced map of adjoint bundles is simply the inclusion
$\mathfrak{p}\to\mathfrak{g}_{P}$. The involution $\sigma$ on
$\mathfrak{g}_{P}$ restricts to an involution on $\mathfrak{p}$, and in fact
is the unique extension of this involution conjugate to $\sigma_{0}$. We see
that $(P,\Phi,\sigma)$ is the induction from
$\operatorname{PSL}_{2}(\mathbb{C})$ to $G$ of the Fock bundle
$(Q,\Phi,\sigma|_{Q})$. ∎

######  Corollary 3.15.

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle. Then $P$ is topologically trivial.

###### Proof.

We can deform $\Phi$ continuously to get to the Fuchsian locus where
$\Phi_{2}=0$. This does not alter the topology of $P$. We then consider the
induced $G_{ad}$-bundle $P_{ad}$. If $P_{ad}$ is trivial, it follows the same
for $P$. We know that $P_{ad}$ is induced from a
$\operatorname{PSL}_{2}(\mathbb{C})$-Fock bundle. We have seen in Proposition
3.10 that those are topologically trivial. ∎

As with the $\operatorname{SL}_{n}(\mathbb{C})$-case, $G$-Fock bundles in the
Fuchsian locus are exactly those obtained by uniformizing $G$-Higgs bundles by
forgetting the holomorphic structure. For any $G$, there is a finite
collection of $G$-Fock bundles which induce a given $G_{ad}$-Fock bundle.

######  Proposition 3.16.

Let $A$ be the subgroup of the center of $G$ which is fixed by $\sigma$. There
is a simply transitive action of $\mathrm{H}^{1}(S,A)$ on the set of $G$-Fock
bundles which induce a given $G_{ad}$-Fock bundle.

The proof is once more abstract bundle theory, which we leave to the reader.
Instead, we describe the case of $\operatorname{SL}_{n}(\mathbb{C})$. The
center of $\operatorname{SL}_{n}(\mathbb{C})$ is the $n$-th roots of unity and
$\sigma$ acts by inversion, so for $n$ even $A$ is $\\{1,-1\\}$ whereas for
$n$ odd $A$ is trivial. This means that for $n$ even there are $2^{2g}$
$\operatorname{SL}_{n}(\mathbb{C})$-Fock bundles for every
$\mathrm{PSL}_{n}(\mathbb{C})$-Fock bundle, whereas for $n$ odd
$\mathrm{SL}_{n}(\mathbb{C})$\- and $\mathrm{PSL}_{n}(\mathbb{C})$-Fock
bundles are the same.

###  3.3 Variations of Fock bundles

Simpson [26] studied the infinitesimal deformation space of stable Higgs
bundles via a hypercohomology group for suitable chain complexes. In a similar
way, we study the variations of $G$-Fock bundles $(P,\Phi,\sigma)$. We will
first forget about $\sigma$ and study the variations of Fock fields, then see
what happens when we introduce $\sigma$.

####  3.3.1 Variation complex without $\sigma_{0}$-structure

Consider the complex $\Omega^{\bullet}(S,\mathfrak{g}_{P})$ of
$\mathfrak{g}$-valued differential forms, with differential given by the
adjoint action of the Fock field $\Phi$:

$\Omega^{0}(S,\mathfrak{g}_{P})\xrightarrow{\operatorname{ad}_{\Phi}\;}\Omega^{1}(S,\mathfrak{g}_{P})\xrightarrow{\operatorname{ad}_{\Phi}\;}\Omega^{2}(S,\mathfrak{g}_{P}).$
(3.1)

######  Proposition 3.17.

The complex (3.1) is a chain complex.

###### Proof.

Indeed, using the Jacobi identity and $[\Phi\wedge\Phi]=0$, we get for all
$A\in\Omega^{0}(S,\mathfrak{g}_{P})$:

$[\Phi\wedge[\Phi,A]]=-[\Phi\wedge[\Phi,A]]+[A,[\Phi\wedge\Phi]],$

which implies $\operatorname{ad}_{\Phi}^{2}(A)=0$. ∎

We call the cohomology groups defined for this chain complex, _$\Phi$
-cohomology groups_, and denote them by $\mathrm{H}^{\bullet}(\Phi)$. The
zero-th cohomology group $\mathrm{H}^{0}(\Phi)$ describes the centralizer of
$\Phi$. To be more precise, the centralizers of all $\Phi(v)(z)$ for a non-
zero real vector $v\in T_{z}S$ are all equal. This follows from
$[\Phi\wedge\Phi]=0$, the nilpotency condition and the general fact that the
centralizer of a principal nilpotent element is abelian. This is why we often
write $Z(\Phi)\subset\Omega^{0}(S,\mathfrak{g}_{P})$ for any of these
centralizers. We have the following:

######  Proposition 3.18.

The dimensions of $\mathrm{H}^{0}(\Phi),\mathrm{H}^{1}(\Phi)$ and
$\mathrm{H}^{2}(\Phi)$ are respectively $\mathrm{rk}(\mathfrak{g})$,
$2\mathrm{rk}(\mathfrak{g})$ and $\mathrm{rk}(\mathfrak{g})$.

###### Proof.

We have $\mathrm{H}^{0}(\Phi)=\ker\,\operatorname{ad}_{\Phi}=Z(\Phi)$, which
by the nilpotency condition is of dimension $\mathrm{rk}(\mathfrak{g})$. The
natural pairing between
$a\in\ker\,\operatorname{ad}_{\Phi}\subset\Omega^{0}(S,\mathfrak{g}_{P})$ and
$b\in\Omega^{2}(S,\mathfrak{g}_{P})$ given by $\int_{S}\operatorname{tr}(ab)$,
where $\operatorname{tr}$ denotes the Killing form on $\mathfrak{g}$, descends
to cohomology. This follows from the cyclicity property
$\operatorname{tr}([a,b]c)=\operatorname{tr}([b,c]a)$ of the Killing form.
Hence $\mathrm{H}^{2}(\Phi)\cong\mathrm{H}^{0}(\Phi)^{*}$ which gives
$\dim\mathrm{H}^{2}(\Phi)=\dim\mathrm{H}^{0}(\Phi)=\mathrm{rk}(\mathfrak{g})$.
Finally, the dimension of $\mathrm{H}^{1}(\Phi)$ can be computed via the Euler
characteristic of the complex which is zero. ∎

The first $\Phi$-cohomology group describes variations of $G$-Fock fields
which do not necessarily preserve the nilpotency condition:

######  Proposition 3.19.

The first $\Phi$-cohomology group $\mathrm{H}^{1}(\Phi)$ describes variations
of a $G$-Fock field $\Phi$ leaving the condition $[\Phi\wedge\Phi]=0$
invariant, modulo gauge transformations.

###### Proof.

A variation $\delta\Phi$ preserves the condition $[\Phi\wedge\Phi]=0$ if and
only if $[\Phi\wedge\delta\Phi]=0$, that is, if and only if $\delta\Phi$ is a
cocycle. An infinitesimal gauge transformation
$\eta\in\Omega^{0}(S,\mathfrak{g}_{P})$ induces $\delta\Phi=[\Phi,\eta]$ which
is a coboundary. ∎

We next analyze variations of Fock bundles including the nilpotency condition.
The space $\Omega^{1}(S,\mathfrak{g}_{P})$ has a natural symplectic structure
$\omega$ defined by

$\omega(\alpha,\beta)=\int_{S}\operatorname{tr}\,\alpha\wedge\beta,$ (3.2)

where $\operatorname{tr}$ denotes the Killing form of $\mathfrak{g}$. The
symplectic structure $\omega$ descends to a symplectic structure on
$\mathrm{H}^{1}(\Phi)$. Indeed, for
$\alpha,\beta\in\Omega^{1}(S,\mathfrak{g}_{P})$ representing classes in
$\mathrm{H}^{1}(\Phi)$, adding a coboundary $[\gamma,\Phi]$ to $\alpha$ adds

$\mathrm{tr}([\gamma,\Phi]\wedge\beta)=\mathrm{tr}(\gamma[\Phi\wedge\beta])=0.$

The same holds when adding a coboundary to $\beta$. Hence the symplectic form
only depends on the cohomology classes. Denote by $\mathrm{Var}(\Phi)$ the
variations of $\Phi$ which both preserve $[\Phi\wedge\Phi]=0$ and the
nilpotency condition.

######  Proposition 3.20.

The variations of Fock fields $\mathrm{Var}(\Phi)$ are described by the direct
sum

$\mathrm{Var}(\Phi)\cong\mathrm{Im}(\operatorname{ad}_{\Phi})\oplus
Z(\Phi)\bar{K},$

where $Z(\Phi)\subset\Omega^{0}(S,\mathfrak{g}_{P})$ denotes the kernel of
$\operatorname{ad}_{\Phi}$ and $\bar{K}$ the conjugated canonical bundle.

###### Proof.

Using the complex structure on $S$ induced from the Fock bundle, we can
locally write $\Phi=\Phi_{1}dz+\Phi_{2}d\bar{z}$, where $\Phi_{1}$ is
principal nilpotent. Since all principal nilpotent matrices are conjugate, any
variation preserving the nilpotency condition is equivalent modulo
$\mathrm{Im}\operatorname{ad}_{\Phi}$ to a a variation which changes
$\Phi_{2}$ without changing $\Phi_{1}$. Such a variation is of the form
$Z(\Phi_{1})d\bar{z}$ since $[\Phi\wedge\delta\Phi]=0$. ∎

######  Corollary 3.21.

For a $G$-Fock bundle $(P,\Phi)$ over $S$, $\mathrm{Var}(\Phi)$ forms an
isotropic subspace in $(\Omega^{1}(S,\mathfrak{g}_{P}),\omega)$. Modulo
$\mathrm{Im}(\operatorname{ad}_{\Phi})$, it descends to an isotropic subspace
of $\mathrm{H}^{1}(\Phi)$.

###### Proof.

Take the variations of two Fock fields
$\delta\Phi=\operatorname{ad}_{\Phi}(\eta)+Cd\bar{z}$ and
$\delta\Phi^{\prime}=\operatorname{ad}_{\Phi^{\prime}}(\eta^{\prime})+C^{\prime}d\bar{z}$,
with $C\in Z(\Phi_{1})$, $C^{\prime}\in Z(\Phi^{\prime}_{1})$ and
$\eta,\eta^{\prime}\in\Omega^{0}(S,\mathfrak{g}_{P})$. One now checks that the
symplectic form $\omega(\delta\Phi,\delta\Phi^{\prime})$ defined by (3.2)
vanishes. Indeed, since $\operatorname{ad}_{\Phi}$ squares to zero, we get
$\mathrm{tr}\operatorname{ad}_{\Phi}(\eta)\wedge\operatorname{ad}_{\Phi^{\prime}}(\eta^{\prime})=0$.
Moreover, $Cd\bar{z}\wedge C^{\prime}d\bar{z}=0$ and the wedge product of the
crossed terms also vanishes since $C\in Z(\Phi_{1})$ and $C^{\prime}\in
Z(\Phi^{\prime}_{1})$. ∎

Note that pointwise, $\mathrm{Var}(\Phi)$ is a Lagrangian in
$T^{*}S\otimes\mathfrak{g}$ (where we choose a volume form at a point to
identify $\Lambda^{2}(TS)$ with $\mathbb{C}$).

####  3.3.2 Introduction of $\sigma_{0}$-structures

We first analyse the possible $\sigma_{0}$-structures $\sigma$ one can put on
a pair $(P,\Phi)$ where $\Phi$ satisfies the first two conditions of
Definition 3.1.

######  Proposition 3.22.

For any such pair $(P,\Phi)$, there locally exist $\sigma_{0}$-structures
making it into a Fock bundle. If $G=G_{ad}$, they exist globally.

###### Proof.

Recall that $\Phi_{1}$ is valued in principal nilpotent elements. By Lemma
2.13 there is a contractible space of choices of involution of each fiber of
the adjoint bundle which negate the image of $\Phi_{1}$. We can thus choose
$\bar{\sigma}:{\mathfrak{g}_{P}}\to\mathfrak{g}$ which negates $\Phi_{1}$
globally. By Lemma 2.14, $\bar{\sigma}$ will also negate the centralizer of
$\Phi_{1}$, so must negate $\Phi_{2}$ as well. When $G$ is adjoint,
$\bar{\sigma}$ uniquely determines a $\sigma_{0}$ structure $\sigma:P\to P$.

When $G$ is not adjoint, there is slightly more data to a
$\sigma_{0}$-structure. The involution $\bar{\sigma}$ gives a reduction to
$\pi^{-1}(G_{ad}^{\sigma})$ where $\pi:G\to G_{ad}$ is the projection. A
$\sigma_{0}$-structure on the other hand is a reduction to $G^{\sigma}$ which
is the identity component of this group. Reductions of a principal bundle to
its identity component are sections of a bundle with discrete fibers, so they
always exist locally. ∎

######  Proposition 3.23.

The infinitesimal variations of $\sigma_{0}$-structures negating $\Phi$ are
described by $\mathrm{H}^{0}(\Phi)$.

###### Proof.

Consider a $\sigma_{0}$-structure $\sigma$ satisfying $\sigma(\Phi)=-\Phi$.
Since all $\sigma_{0}$-structures are conjugate, any infinitesimal variation
$\delta\sigma$ is described by $\xi\in\mathfrak{g}^{-\sigma}$ and given by
$\delta\sigma(x)=[\xi,x]$ for $x\in\mathfrak{g}$. The variations negating
$\Phi$ have to satisfy $\delta\sigma(\Phi)=0$, hence
$\xi\in\ker\operatorname{ad}_{\Phi}$, i.e. $\xi\in\mathrm{H}^{0}(\Phi)$. ∎

Actually a stronger statement is true: by Lemma 2.13 the exponential of
$\mathrm{H}^{0}(\Phi)=Z(\Phi)$ acts simply transitively on the space of
choices of $\sigma_{0}$-structure. In particular, all $\sigma_{0}$ structures
are conjugate.

We can now use the $\sigma_{0}$-structure $\sigma$ in the $\Phi$-cohomology.
Define $\Omega^{k,\sigma}(S,\mathfrak{g}_{P})$ to be the space of
$\sigma$-invariant $\mathfrak{g}_{P}$-valued $k$-forms, and similarly
$\Omega^{k,-\sigma}(S,\mathfrak{g}_{P})$ to be the space of $\sigma$-anti-
invariant forms. Since $\sigma(\Phi)=-\Phi$, the $\Phi$-cohomology complex
splits as a direct sum of two complexes

$\Omega^{0,\pm\sigma}(S,\mathfrak{g}_{P})\xrightarrow{\operatorname{ad}_{\Phi}\;}\Omega^{1,\mp\sigma}(S,\mathfrak{g}_{P})\xrightarrow{\operatorname{ad}_{\Phi}\;}\Omega^{2,\pm\sigma}(S,\mathfrak{g}_{P}),$

and we may define $\mathrm{H}^{k,\sigma}(\Phi)$ and
$\mathrm{H}^{k,-\sigma}(\Phi)$ as the subspaces represented by
$\sigma$-invariant and respectively $\sigma$-anti-invariant elements of these
complexes. Equivalently, we can define them as the $\sigma$-invariant and
$\sigma$-anti-invariant parts of $\mathrm{H}^{k}(\Phi)$:

$\mathrm{H}^{k}(\Phi)\cong\mathrm{H}^{k,\sigma}(\Phi)\oplus\mathrm{H}^{k,-\sigma}(\Phi).$

Moreover, one sees that the Lie bracket respects the $\sigma$-grading, because
$\sigma$ commutes with the Lie bracket.

######  Proposition 3.24.

We have $\mathrm{H}^{k,\sigma}(\Phi)=0$, for $k=0,1,2$.

###### Proof.

One needs to show that $\sigma$ acts by $-1$ on $\Phi$-cohomology. This is a
pointwise statement. For $k=0$, the 0-th cohomology is simply the center
$Z(\Phi)$, on which $\sigma$ acts by $-1$. The natural pairing between 0-forms
and 2-forms is $\sigma$-invariant. Hence $\sigma$ also acts by $-1$ on
$\mathrm{H}^{2}(\Phi)$.

For $k=1$, the centralizer $Z(\Phi)\bar{K}$ descends to a Lagrangian subspace
in $\mathrm{H}^{1}(\Phi)$. This subspace is negated by $\sigma$, so is its
complement under the symplectic pairing between 1-forms. This describes the
entire space $\mathrm{H}^{1}(\Phi)$. ∎

In view of Proposition 3.24, we have
$\mathrm{H}^{k}(\Phi)=\mathrm{H}^{k,-\sigma}(\Phi)$, hence
$[\mathrm{H}^{k}(\Phi),\mathrm{H}^{\ell}(\Phi)]\subset\mathrm{H}^{k+\ell,\sigma}(\Phi)=0$.
We thus have the following:

######  Corollary 3.25.

For representatives $\alpha,\beta$ of cohomology classes
$[\alpha]\in\mathrm{H}^{k}(\Phi)$ and $[\beta]\in\mathrm{H}^{\ell}(\Phi)$, the
bracket $[\alpha,\beta]$ is a coboundary.

##  4 Connections associated to Fock bundles

In nonabelian Hodge theory, one associates to polystable Higgs bundles, gauge
equivalence classes of Hermitian-Yang-Mills metrics and subsequently flat
bundles over a Riemann surface. In this section, we will associate to Fock
bundles a certain family of connections. Our main conjecture is that we can
moreover choose these connections to be flat. To begin with, we define a
compatible connection on a Fock bundle:

######  Definition 4.1.

On a $G$-Fock bundle $(P,\Phi,\sigma)$, a connection $d_{A}$ is called _$\Phi$
-compatible_ if $d_{A}\Phi=0$.

Note that $d_{A}\Phi=0$ is an affine equation in the connection matrix $A$,
thus easily provides the existence of $\Phi$-compatible connections. Indeed,
local solutions can be found by choosing a local trivialization of $P$ in
which the centralizer of $\Phi$ is constant. In this trivialization, $d\Phi$
will be valued in the centralizer of $\Phi$, thus it will be in the image of
$\operatorname{ad}_{\Phi}$ (see Proposition 2.11), allowing us to find a
matrix valued $1$-form $A_{0}$ with $d\Phi+[A_{0}\wedge\Phi]=0$. These local
solutions can be now patched together using a partition of unity. We thus have
the following:

######  Proposition 4.2.

There exist $\Phi$-compatible connections on any given $G$-Fock bundle.

###  4.1 Three-term connections

In Section 2.2.3 we reviewed two sorts of families of 3-term connections from
the works of Hitchin [12] and Fock [7] subject to a parameter
$\lambda\in\mathbb{C}^{*}$. The form of these families comes from the general
theory of hyperkähler manifolds and their twistor space [13]. Within this
framework, we consider a family of connections on a principal $G$-bundle $P$
over a surface $S$ of the form

$\mathcal{A}(\lambda)=\lambda^{-1}\Phi+d_{A}+\lambda\Psi,$ (4.1)

where $\lambda\in\mathbb{C}^{*}$ is a parameter, $d_{A}$ is a fixed background
connection on $P$ and $\Phi,\Psi\in\Omega^{1}(S,\mathfrak{g}_{P})$. The
curvature of a connection in this family is a Laurent polynomial in $\lambda$
given by:

$\displaystyle F(\mathcal{A}(\lambda))$
$\displaystyle=[\mathcal{A}(\lambda)\wedge\mathcal{A}(\lambda)]$
$\displaystyle=\lambda^{-2}[\Phi\wedge\Phi]+\lambda^{-1}d_{A}(\Phi)+F(A)+[\Phi\wedge\Psi]+\lambda
d_{A}(\Psi)+\lambda^{2}[\Psi\wedge\Psi],$ (4.2)

where $F(A)$ denotes the curvature of the fixed background connection $d_{A}$.

In order to have a family $\mathcal{A}(\lambda)$ of flat connections, i.e.
flat for all values of $\lambda$, all five coefficients of the Laurent
polynomial in (4.1) have to vanish. Note that if $\Phi$ and $\Psi$ are Fock
fields, then the terms $\lambda^{-2}[\Phi\wedge\Phi]$ and
$\lambda^{2}[\Psi\wedge\Psi]$ in (4.1) do vanish. If, in addition, the
background connection $d_{A}$ is a connection compatible with both $\Phi$ and
$\Psi$, then the curvature $F(\mathcal{A}(\lambda))$ becomes independent of
the parameter $\lambda$. In this situation, the flatness of
$\mathcal{A}(\lambda)$ is equivalent to the equation

$F(A)+[\Phi\wedge\Psi]=0.$ (4.3)

Note the similarity of this equation to the form of the well-known Hitchin
equation from [12].

In Section 4.3 below, we will show the existence of a connection $d_{A}$
compatible with both Fock fields $\Phi$ and $\Psi$. For this to be possible,
the two fields have to satisfy a certain transversality condition.

###  4.2 Transversality

We next define the notion of transverse Fock fields.

######  Definition 4.3.

For a pair of Fock fields $\Phi,\Psi\in\Omega^{1}(S,\mathfrak{g}_{P})$, we
call $\Phi$ and $\Psi$ _transverse_ if
$\Omega^{1}(S,\mathfrak{g}_{P})=\ker(\operatorname{ad}_{\Phi})\oplus\mathrm{Im}(\operatorname{ad}_{\Psi})$
and
$\Omega^{1}(S,\mathfrak{g}_{P})=\ker(\operatorname{ad}_{\Psi})\oplus\mathrm{Im}(\operatorname{ad}_{\Phi})$.

Figure 4.1 illustrates the transversality condition; note that the figure
shows the decomposition of $\Omega^{1}(S,\mathfrak{g}_{P})$ as a direct sum,
not as a union.

Figure 4.1: Decomposition of $\Omega^{1}(S,\mathfrak{g}_{P})$ for transverse
$\Phi$ and $\Psi$.

######  Proposition 4.4.

For transverse $\Phi$ and $\Psi$, the cohomology $\mathrm{H}^{1}(\Phi)$ can be
identified with
$\ker(\operatorname{ad}_{\Phi})\cap\ker(\operatorname{ad}_{\Psi})$.

###### Proof.

Let $[\alpha]\in\mathrm{H}^{1}(\Phi)$ be represented by
$\alpha\in\ker(\operatorname{ad}_{\Phi})$. Using the transversality condition
$\Omega^{1}(S,\mathfrak{g}_{P})=\ker(\operatorname{ad}_{\Psi})\oplus\mathrm{Im}(\operatorname{ad}_{\Phi})$,
we see that the projection onto $\ker(\operatorname{ad}_{\Psi})$ changes
$\alpha$ only by a coboundary. Hence we can use this projection as a preferred
representative for the class $[\alpha]$. ∎

Moreover, in the setting of $\sigma$-splitted $\Phi$-cohomology from Section
3.3.2, we get the following corollary:

######  Corollary 4.5.

For a pair of transverse Fock fields
$\Phi,\Psi\in\Omega^{1}(S,\mathfrak{g}_{P})$, one has:

$\Omega^{1}(S,\mathfrak{g}_{P})^{\sigma}=\mathrm{Im}(\operatorname{ad}_{\Phi})^{\sigma}\oplus\mathrm{Im}(\operatorname{ad}_{\Psi})^{\sigma}.$

This follows from the fact that $\mathrm{H}^{1,\sigma}(\Phi)=0$ (see
Proposition 3.24) and the identification of
$\ker(\operatorname{ad}_{\Phi})\cap\ker(\operatorname{ad}_{\Psi})$ with
$\mathrm{H}^{1}(\Phi)$.

The transversality condition is an open condition. In the unitary setting
(where the bundle $P$ is equipped with a hermitian structure $\rho$), we will
see that the Fock field $\Psi=\Phi^{*}$ can be made transverse to $\Phi$, by a
suitable diagonal gauge transformation.

###  4.3 Canonical connection from transverse Fock fields

We now associate a canonical connection $d_{A}$ to a given pair of transverse
Fock fields $\Phi$ and $\Psi$. This allows to get a family of 3-term
connections $\lambda^{-1}\Phi+d_{A}+\lambda\Psi$. The existence and uniqueness
of $d_{A}$ brings to mind the Chern connection on a holomorphic bundle with
hermitian structure.

######  Theorem 4.6.

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a second Fock field
$\Psi$, transverse to $\Phi$. Then there is a unique $\sigma$-invariant
connection $d_{A}$ on $P$ which is compatible with both $\Phi$ and $\Psi$,
i.e. solving the equations

$d_{A}\Phi=d_{A}\Psi=0.$

In the sequel, we call $d_{A}$ the _canonical connection_ and refer to the
construction of $d_{A}$ as the _filling-in procedure_ (since it gives the
middle term in the 3-term connection). The associated 3-term connection
$\Phi+d_{A}+\Psi$ will be called the _canonical 3-term connection_.

###### Proof.

_Existence_ : By Proposition 4.2, we first find a compatible connection
$d_{A_{0}}$ on $P$, i.e. such that $d_{A_{0}}\Phi=0$. The existence of
$d_{A_{0}}$ relies on the fact that for appropriate local trivializations,
$d\Phi$ lies in the image of $\operatorname{ad}_{\Phi}$. Note that we can
choose $d_{A_{0}}$ to be $\sigma$-invariant since $\sigma(\Phi)=-\Phi$.

The same argument as for $\Phi$ shows that
$d_{A_{0}}\Psi=d\Psi+[A_{0}\wedge\Psi]$ lies in the image of
$\operatorname{ad}_{\Psi}$. Hence there is a section
$R\in\Omega^{1}(S,\mathfrak{g}_{P})$ such that $d_{A_{0}}\Psi=[R\wedge\Psi]$.
Since both $\Psi$ and $d_{A_{0}}\Psi$ are $\sigma$-anti-invariant, we can
choose $R$ to be $\sigma$-invariant. By Corollary 4.5 (since $\Phi$ and $\Psi$
are transverse) we can decompose $R=R_{1}+R_{2}$ where
$R_{1}\in\mathrm{Im}(\operatorname{ad}_{\Phi})^{\sigma}$ and
$R_{2}\in\mathrm{Im}(\operatorname{ad}_{\Psi})^{\sigma}$. Since
$\operatorname{ad}_{\Psi}^{2}=0$, we have $[R\wedge\Psi]=[R_{1}\wedge\Psi]$.

We claim that $d_{A}:=d_{A_{0}}-R_{1}$ is a solution. Indeed, $d_{A}\Psi=0$ by
definition of $R_{1}$ and $d_{A}\Phi=-[R_{1}\wedge\Phi]=0$ since
$R_{1}\in\mathrm{Im}(\operatorname{ad}_{\Phi})$ and
$\operatorname{ad}_{\Phi}^{2}=0$. Finally, both $d_{A_{0}}$ and $R_{1}$ are
$\sigma$-invariant. Therefore $d_{A}$ is indeed a solution.

_Uniqueness_ : The space of solutions to the equations $d_{A}\Phi=d_{A}\Psi=0$
is an affine subspace of the space of connections. If $d_{A}$ is one such
solution, then all solutions are of the form $d_{A}+C$, where
$C\in\ker(\operatorname{ad}_{\Phi})\cap\ker(\operatorname{ad}_{\Psi})$. Lemma
3.24 shows that the involution $\sigma$ acts by $-1$ on the $\Phi$-cohomology,
so acts by $-1$ on
$\ker(\operatorname{ad}_{\Phi})\cap\ker(\operatorname{ad}_{\Psi})$. Since
$d_{A}+C$ and $d_{A}$ are $\sigma$-invariant, $C$ has to be
$\sigma$-invariant. Therefore $C=0$. ∎

###  4.4 Unitary case

We equip a $G$-Fock bundle $(P,\Phi,\sigma)$ with a compatible hermitian
structure $\rho$. Ideally we would like to apply the construction of a
canonical connection from Theorem 4.6 for the pair of Fock fields $\Phi$ and
$\Psi=\Phi^{*}=-\rho(\Phi)$. However, the construction can not be used
directly since the Fock fields $\Phi$ and $\Phi^{*}$ need not be necessarily
transversal. We will see that we can always conjugate $\Phi$ so that $\Phi$
and $\Phi^{*}$ become transversal.

######  Definition 4.7.

A hermitian structure $\rho$ on a $G$-Fock bundle $(P,\Phi,\sigma)$ is called
_compatible_ if the involutions $\rho$ and $\sigma$ of $\mathfrak{g}_{P}$
commute.

In the case of an $\mathrm{SL}_{n}(\mathbb{C})$-Fock bundle $(E,\Phi,g)$, this
is equivalent to the existence of a real vector bundle $E^{\mathbb{R}}$
equipped with a real metric $g^{\mathbb{R}}$ such that the complexification of
$E^{\mathbb{R}}$ gives the bundle $E$, the complex linear extension of the
real metric $g^{\mathbb{R}}$ gives $g$ and the hermitian extension gives the
hermitian form $h$.

######  Proposition 4.8.

For any $G$-Fock bundle $(P,\Phi,\sigma)$, there is a compatible hermitian
structure.

###### Proof.

The statement is pointwise, so we can reduce to a problem of Lie algebra
involutions on $\mathfrak{g}$. The proposition directly follows from the fact
that for any Lie algebra involution (here $\sigma$), there is a compact real
form which commutes with it [15, Theorem 6.16]. ∎

######  Proposition 4.9.

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a compatible
hermitian structure $\rho$. Then $(P,\Phi^{*},\sigma)$ is again a $G$-Fock
bundle.

###### Proof.

Applying the hermitian conjugation to $[\Phi\wedge\Phi]=0$ we get the same
condition for $\Phi^{*}$. Since the set of principal nilpotent matrices is
invariant under hermitian conjugation, $\Phi^{*}$ satisfies the nilpotency
condition. Finally, $\Phi^{*}$ is negated by $\sigma$:

$\sigma(\Phi^{*})=-\sigma\circ\rho(\Phi)=-\rho\circ\sigma(\Phi)=\rho(\Phi)=-\Phi^{*}.$

∎

In order to use the filling-in procedure of Theorem 4.6, we need $\Phi$ and
$\Phi^{*}$ to be transverse. For that, note that the spaces
$\mathrm{Im}(\operatorname{ad}_{\Phi})$ and
$\ker(\operatorname{ad}_{\Phi^{*}})$ are orthogonal complements with respect
to the (pseudo-)hermitian form on $T^{*}S\otimes\mathfrak{g}_{P}$ given by

$\langle\alpha,\beta\rangle d\bar{z}\wedge
dz=\mathrm{tr}(\alpha^{*}\wedge\beta),$

where $\operatorname{tr}$ denotes the Killing form on $\mathfrak{g}$. These
are indeed orthogonal complements because $\operatorname{ad}_{\Phi}$ and
$\operatorname{ad}_{\Phi^{*}}$ are adjoint with respect to the inner product
on the entire space $\Lambda^{\bullet}T^{*}S\otimes\mathfrak{g}_{P}$ given by
this same formula. Note that we chose a volume form to define this, but
different choices of volume form will lead to metrics which differ by a
scalar, so the notions of adjoint, and orthogonal complement do not depend on
this choice. This hermitian form is not positive definite though, in fact, the
norm of $a\,dz+b\,d\bar{z}$ is given by
$\mathrm{tr}(a^{*}a)-\mathrm{tr}(b^{*}b)$.

On $\Omega^{1}(S,\mathfrak{g}_{P})$, the pseudo-hermitian form is given by the
formula

$\langle\alpha,\beta\rangle=-\tfrac{i}{2}\int_{S}\operatorname{tr}\alpha^{*}\wedge\beta.$
(4.4)

Recall that any subspace $W$ of a vector space equipped with an indefinite
form is complementary to a subspace $W^{\perp}$, precisely whenever the form
is non-degenerate when restricted to $W$. This motivates the following:

######  Definition 4.10.

For a $G$-Fock bundle $(P,\Phi)$ with hermitian form $\rho$, the Fock field
$\Phi$ is called _positive_ if
$\mathrm{Im}(\operatorname{ad}_{\Phi})\subset\Omega^{1}(S,\mathfrak{g}_{P})$
is positive-definite with respect to the pseudo-hermitian form in (4.4). The
space of positive Fock fields is denoted by $\mathcal{P}$. We also refer to
$\rho$ being a positive hermitian structure.

Note that for a positive Fock field $\Phi\in\mathcal{P}$, the fields $\Phi$
and $\Phi^{*}$ are transverse. Note further that the Fuchsian locus, as
described in Section 3.2, is included in $\mathcal{P}$ since
$\mathrm{Im}(\operatorname{ad}_{\Phi})$ consists only of $(1,0)$-forms.

######  Lemma 4.11.

For every Fock field $\Phi\in\Omega^{1}(S,\mathfrak{g}_{P})$ there is a
diagonal gauge transformation $a\in\mathrm{Aut}(P,\sigma)$ such that
$a^{-1}\Phi a$ is positive.

###### Proof.

Consider a $G$-Fock bundle $(P,\Phi,\sigma)$. Decompose
$\Phi=\Phi_{1}+\Phi_{2}$. There is an $\mathfrak{sl}_{2}$-subbundle
$\mathfrak{p}\subset\mathfrak{g}_{P}$ obtained by completing $\Phi_{1}$ into a
principal $\mathfrak{sl}_{2}$-triple. Let $H$ be a non-vanishing section of
the line subbundle $\mathfrak{l}\subset\mathfrak{p}$ consisting of the
semisimple elements of the principal $\mathfrak{sl}_{2}$-triples (acting on
the nilpotent part by scaling). Note that $\mathfrak{l}$ is a degree zero line
bundle, so this section indeed exists.

Consider the gauge transformation $a_{t}=\exp(tH)$. Since $H$ is
$\sigma$-invariant, we have $a_{t}\in\mathrm{Aut}(P,\sigma)$. The element $H$
allows to decompose the bundle
$\mathfrak{g}_{P}\cong\oplus_{k\in\mathbb{Z}}\,\mathfrak{g}_{P,k}$ where
elements in $\mathfrak{g}_{P,k}$ are eigenvectors of $\operatorname{ad}_{H}$
with eigenvalue $k$. We know that $\Phi_{1}\in\mathfrak{g}_{P,-2}$ since
$\Phi_{1}$ and $H$ are part of the same $\mathfrak{sl}_{2}$-triple. We have
$\Phi_{2}\in Z(\Phi_{1})$, i.e. $\Phi_{2}$ is a linear combination of lowest
weight vectors. In addition, $\Phi_{2}$ is not regular since we work in the
complex structure induced from $(P,\Phi)$. This implies that
$\Phi_{2}\in\oplus_{k\leq-4}\mathfrak{g}_{P,k}$ since in the decomposition of
$\mathfrak{g}_{P}$ into irreducible $\mathfrak{sl}_{2}$-modules (via adjoint
action by $\mathfrak{p}$), only one module is of dimension 3.

Therefore the quantity $e^{2t}a_{t}^{-1}\Phi a_{t}$ converges to $\Phi_{1}$
for $t\to\infty$. Since $\Phi_{1}$ is clearly positive and positive Fock
fields form an open set, we conclude. ∎

For transverse $\Phi$ and $\Phi^{*}$ we can use the filling-in procedure from
Theorem 4.6 to get a canonical connection $d_{A}$.

######  Proposition 4.12.

For a Fock bundle $(P,\Phi,\sigma)$ equipped with a compatible positive
hermitian structure $\rho$, then the canonical connection $d_{A}$ is unitary.

###### Proof.

One can first check that both $d_{A}$ and $\rho(d_{A})$ are $\sigma$-invariant
solutions to the equations $d_{A}(\Phi)=0$ and $d_{A}(\Phi^{*})=0$. Indeed,
since $\sigma$ and $\rho$ commute, $\rho(d_{A})$ is $\sigma$-invariant.
Applying $\rho$ to the equation $d_{A}\Phi=0$ gives $\rho(d_{A})(\Phi^{*})=0$.
Similarly, applying $\rho$ to $d_{A}(\Phi^{*})=0$ gives $\rho(d_{A})(\Phi)=0$.
Hence $\rho(d_{A})$ is also a solution. By uniqueness from Theorem 4.6, we get
$\rho(d_{A})=d_{A}$, thus the connection $d_{A}$ is unitary. ∎

An important example of positive Fock fields is the Fuchsian locus:

######  Proposition 4.13.

If $(P,\Phi,\sigma)$ lies in the Fuchsian locus, then there is a compatible
positive hermitian structure $\rho$. In addition, the canonical connection
$d_{A}$ is the Chern connection.

###### Proof.

From Proposition 3.14, we know that a Fock bundle in the Fuchsian locus comes
from some uniformizing $G$-Higgs bundle $(V,\Phi)$. From [14] we know that $V$
can be equipped with a compatible hermitian structure $\rho$ and
$\sigma_{0}$-structure $\sigma$. Since
$\Phi\in\Omega^{1,0}(S,\mathfrak{g}_{V})$, the image of
$\operatorname{ad}_{\Phi}$ is positive definite for the form defined in (4.4),
i.e. $\Phi$ is positive.

From nonabelian Hodge theory, we know that the Chern connection $d_{A}$
satisfies $d_{A}\Phi=0$ and $\sigma(d_{A})=d_{A}$ [14, Equation (7.2)]. Hence
the Chern connection is the result of the filling-in procedure by the
uniqueness part of Theorem 4.6. ∎

###  4.5 Main Conjecture

We have seen that on a $G$-Fock bundle $(P,\Phi,\sigma)$ with compatible
positive hermitian structure $\rho$, there is a canonical 3-term connection
$\Phi+d_{A}+\Phi^{*}$. Our main conjecture is that we can choose $\rho$ to get
a _flat connection_. We give evidence in favor of the conjecture.

Instead of varying $\rho$, we can equivalently fix the hermitian structure
$\rho$ and vary the gauge class of the Fock field.

######  Conjecture 4.14.

Let $(P,\Phi_{0},\sigma)$ be a $G$-Fock bundle over $S$ equipped with a
compatible hermitian structure $\rho$. Then there exists a unique hermitian
endomorphism-valued function

$\eta\in\Omega^{0}(S,\mathfrak{g}_{P}^{\sigma,-\rho})$

such that $\Phi=e^{-\eta}\Phi_{0}e^{\eta}$ is positive, and the corresponding
3-term connection $\Phi+d_{A}+\Phi^{*}$ is flat, that is, it satisfies the
equation

$F(A)+[\Phi\wedge\Phi^{*}]=0.$ (4.5)

The conjecture is appealingly similar to nonabelian Hodge theory associating a
flat connection to the gauge equivalent class of any polystable Higgs bundle.
In our setting, Fock bundles are automatically stable, see Proposition 3.7.
Apart from the similarity with nonabelian Hodge theory, there is strong
evidence for the conjecture which is discussed below:

  1. 1.

There is a class of Fock bundles for which Conjecture 4.14 is indeed true,
namely, for Fock bundles in the Fuchsian locus from Section 3.2. Fix a complex
structure on $S$ and consider the uniformizing $G$-Higgs bundle $(V,\Phi)$,
i.e. the Higgs bundle in the Hitchin section with principal nilpotent $\Phi$.
Forgetting about the holomorphic structure, we get a $G$-Fock bundle
$(P,\Phi)$. Proposition 4.13 tells us that $\Phi$ and $\Phi^{*}$ are
transverse and that the Chern connection is the result from our construction
of 3-term connections. Therefore, Conjecture 4.14 follows from the nonabelian
Hodge correspondence in this case.

  2. 2.

The strongest argument in favor of the conjecture is that the linearization is
an elliptic isomorphism. This is developed below in Section 5.1. We can then
conclude that the subset of Fock fields for which there exists a solution is
open.

  3. 3.

Another interesting class of examples pertains to _harmonic higher complex
structures_ as introduced in [23]. Fix a complex structure on $S$ giving a
hyperbolic metric $g_{S}$ (via uniformization). A higher complex structure is
called _harmonic_ if its induced complex structure on $S$ coincides with the
fixed one and if

$\bar{\partial}(\bar{\mu}_{k}g_{S}^{k-1})=0,$ (4.6)

for all $k\in\\{3,...,n\\}$. The gauge-theoretic meaning of condition (4.6)
can be given as follows: denote by $h_{S}$ the hermitian structure on
$V=K^{(n-1)/2}\oplus...\oplus K^{(1-n)/2}$ induced from the hyperbolic metric
on $S$. Denote again by $E$ the underlying smooth complex vector bundle of
$V$. Then a Fock field $\Phi$, with $(1,0)$-part fixed to be the Fuchsian Fock
field in Example 3.13, corresponds to a harmonic higher complex structure (via
Proposition 3.6) if and only if $\bar{\partial}(\Phi^{*_{h_{S}}})=0$. Indeed,
the hermitian metric $h_{S}$ is diagonal and is given by
$h_{S}=\mathrm{diag}(g_{S}^{(1-n)/2},g_{S}^{(3-n)/2},...,g_{S}^{(n-1)/2})$.
The non-zero entries of $(\Phi^{0,1})^{*_{h_{S}}}$ are given by
$\bar{\mu}_{k}g_{S}^{k-1}$.

One of the main results of [23] is that every equivalence class of higher
complex structures (modulo higher diffeomorphisms) has a harmonic
representative (unique up to the action of usual diffeomorphisms isotopic to
the identity). In Section 6 below, we give the gauge-theoretic meaning of
higher diffeomorphisms as special gauge transformations. Hence it seems that
we can reduce Conjecture 4.14 to harmonic higher complex structures, in which
the elliptic condition $\bar{\partial}(\Phi^{*_{h_{S}}})=0$ holds. We are very
grateful to Alexander Nolte for his insight in this description.

  4. 4.

In the case of an $\mathrm{SL}_{n}(\mathbb{C})$-Fock bundle $(E,\Phi,g)$
equipped with a hermitian structure $h$, we have two filtrations $\mathcal{F}$
and $\mathcal{F}^{*}$ associated to $(E,\Phi)$ and $(E,\Phi^{*})$ respectively
by Proposition 3.8. They are transverse since $\mathcal{F}_{k}$ is
$h$-orthogonal to $\mathcal{F}_{n-k}^{*}$. Therefore by considering
$L_{k}=\mathcal{F}_{k}\cap\mathcal{F}_{n-k}^{*}$, we get a line decomposition
of $E$ for which $\Phi$ is lower triangular, $\Phi^{*}$ is upper triangular
and $h$ is diagonal. This line decomposition might help to find estimates.

  5. 5.

Further evidence for the validity of the conjecture comes from a symplectic
viewpoint. The Atiyah–Bott symplectic form on the space of all connections
gives a presymplectic form on the space of appropriate Fock bundles. The gauge
group action gives a moment map which is precisely given by
$F(A)+[\Phi\wedge\Phi^{*}]$ in this case. Then Conjecture 4.14 becomes
equivalent to an infinite-dimensional symplectic reduction. The difficulty
here comes from the fact that the presymplectic form has degenerate
directions, in particular, the orbits of the higher diffeomorphism action.
Along the complex gauge orbits, the form is non-degenerate though. This is
work in progress.

###  4.6 Flat Fock bundles and the Hitchin component

In this section, Fock bundles with a flat canonical 3-term connection are
linked to the Hitchin component. We show that the monodromy is always in the
split real form, and assuming Conjecture 4.14, one then gets a map from Fock
bundles to the Hitchin component.

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a compatible positive
hermitian structure $\rho$.

######  Proposition 4.15.

Suppose the canonical 3-term connection $\nabla=\Phi+d_{A}+\Phi^{*}$
associated to the tuple $(P,\Phi,\sigma,\rho)$ as above is flat. Then the
monodromy of $\nabla$ lies in the split real form of the Lie algebra
$\mathfrak{g}$.

###### Proof.

We simply show that $\Phi+d_{A}+\Phi^{*}$ is invariant under the involution
$\tau=\sigma\rho=\rho\sigma$. Recall that $\sigma(\Phi)=-\Phi$. Hence

$\tau(\Phi)=\rho\circ\sigma(\Phi)=\rho(-\Phi)=\Phi^{*}.$

Since the canonical connection $d_{A}$ is unitary, we have
$\rho(d_{A})=d_{A}$. Since $d_{A}$ is also $\sigma$-invariant, we get
$\tau(d_{A})=\rho(\sigma(d_{A}))=d_{A}$. Therefore, $\Phi+d_{A}+\Phi^{*}$ is
$\tau$-invariant. By [14, Proposition 6.1], the monodromy of the connection
$\nabla$ lies in the split real form of $\mathfrak{g}$. ∎

######  Proposition 4.16.

Assume the main conjecture 4.14 is true. Then we get a map from the space of
$G$-Fock bundles equipped with compatible hermitian structure to the Hitchin
component.

###### Proof.

Consider a $G$-Fock bundle $(P,\Phi,\sigma)$ together with a compatible
hermitian structure $\rho$. The main conjecture associates a flat connection
$\Phi+d_{A}+\Phi^{*}$ which by the previous proposition has monodromy in the
split real form.

We can continuously deform any Fock field $\Phi$ to a Fock field with
vanishing $(0,1)$-part, which is a point in the Fuchsian locus by Proposition
3.14. Using the main conjecture, this gives a continuous path in the character
variety for the split real form. Since the monodromy of any point in the
Fuchsian locus is in the Hitchin component, the same has to be true for
$(P,\Phi,\sigma)$. ∎

##  5 Neighborhood of the Fuchsian locus

We prove the Main Conjecture 4.14 in a neighborhood of the Fuchsian locus. In
the whole section, let $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a
compatible positive hermitian structure $\rho$.

###  5.1 Linearized Equation

We show that the differential of the map from positive Fock fields to
curvature $F(A)+[\Phi\wedge\Phi^{*}]$, is an elliptic operator of order two
and, in fact, is an isomorphism of appropriate Sobolev spaces. We compute the
change in curvature as we infinitesimally conjugate $\Phi$ by a hermitian
endomorphism.

Let $(P,\sigma,\rho)$ be a principal $G$-bundle equipped with
$\sigma_{0}$-structure $\sigma$ and compatible hermitian structure $\rho$.
Define $F$ to be the map $\mathcal{P}\to\Omega^{2}(S,\mathfrak{g}_{P})$ given
by the curvature of the canonical 3-term connection:

$F(\Phi):=F(A)+[\Phi\wedge\Phi^{*}].$ (5.1)

######  Lemma 5.1.

Let $\Phi_{t}$ be a smooth path of positive Fock fields on a
$G^{\sigma,\rho}$-bundle with $\Phi_{0}=\Phi$, and whose derivative at $t=0$
is

$\dot{\Phi}:=[\Phi,\eta]$

for some $\eta\in\Omega^{0}(S,\mathfrak{g}_{P}^{\sigma,-\rho})$. Let $A_{t}$
be the path of connections obtained by the filling-in procedure of Theorem
4.6. We have

$\dot{A}=Qd_{A}\eta,$

where $Q$ is the involution on
$\Omega^{1}(S,\mathfrak{g}_{P})^{\sigma}\cong\mathrm{Im}(\operatorname{ad}_{\Phi})^{\sigma}\oplus\mathrm{Im}(\operatorname{ad}_{\Phi^{*}})^{\sigma}$
defined by $-1$ on $\mathrm{Im}(\operatorname{ad}_{\Phi})^{\sigma}$ and by $1$
on $\mathrm{Im}(\operatorname{ad}_{\Phi^{*}})^{\sigma}$. Note that we used
Corollary 4.5 here.

###### Proof.

It suffices to check that the derivatives of $d_{A}\Phi$ and $d_{A}\Phi^{*}$
vanish at $t=0$. One sees

$\frac{d}{dt}d_{A}\Phi=d_{A}\dot{\Phi}+[\dot{A}\wedge\Phi]=-[\Phi\wedge
d_{A}\eta]+[\dot{A}\wedge\Phi]=[\Phi\wedge(\dot{A}-d_{A}\eta)]$

and similarly

$\frac{d}{dt}d_{A}\Phi^{*}=[\Phi^{*}\wedge(\dot{A}+d_{A}\eta)].$

The fact that $\operatorname{ad}_{\Phi}^{2}=0$ and
$\operatorname{ad}_{\Phi^{*}}^{2}=0$, implies that $\dot{A}=Qd_{A}\eta$ indeed
solves both of these equations. ∎

We can now easily calculate the variation of curvature as we move the Fock
field in the direction of $\eta$.

######  Theorem 5.2.

The derivative of the map $F$ from (5.1) is the differential operator
$L:\Omega^{0}(S,\mathfrak{g}_{P}^{\sigma,-\rho})\to\Omega^{2}(S,\mathfrak{g}_{P}^{\sigma,\rho})$
given by

$L\eta=d_{A}Qd_{A}\eta+[[\Phi,\eta]\wedge\Phi^{*}]-[\Phi\wedge[\Phi^{*},\eta]].$

The operator $L$ is elliptic, and extends to a continuous linear isomorphism
of Sobolev spaces

$H_{l+2}(S,\mathfrak{g}_{P}^{\sigma,-\rho})\to
H_{l}(S,\Lambda^{2}T^{*}S\otimes\mathfrak{g}_{P}^{\sigma,\rho}).$

for all $l\in\mathbb{N}$, where $H_{l}$ denotes the Sobolev space with $l$
square integrable derivatives.

###### Proof.

To compute the symbol of $L$, we only need to look at the highest order term
$d_{A}Qd_{A}$. This is a composition of three operators, so its symbol is the
composition of three symbols. The symbol $\sigma_{d_{A}}$ of $d_{A}$ is given
by the formula

$\sigma_{d_{A}}(\alpha)\eta=\alpha\wedge\eta.$

This is a map
$T^{*}S\times\Lambda^{\bullet}T^{*}S\otimes\mathfrak{g}_{P}\to\Lambda^{\bullet}T^{*}S\otimes\mathfrak{g}_{P}$.
Note that this does not actually depend on the connection $d_{A}$. Since $Q$
is an operator of order zero, the symbol of $Q$ is just itself. Putting these
together, we get the symbol of $L$

$\sigma_{L}(\alpha)\eta=\alpha\wedge Q(\alpha\eta).$

The operator $L$ is elliptic if this is an isomorphism from
$\mathfrak{g}_{P}^{\sigma,-\rho}$ to
$\Lambda^{2}T^{*}S\otimes\mathfrak{g}_{P}^{\sigma,\rho}$ for all non-zero
$\alpha$. This is implied if $-\frac{i}{2}\operatorname{tr}(\eta\alpha\wedge
Q(\alpha\eta))$ is a definite form in $\eta$ for all non-zero $\alpha$. This
is indeed true, because if we write $\alpha\eta=\xi+\xi^{*}$ where $\xi$ is in
$\mathrm{Im}(\operatorname{ad}_{\Phi})$, we get

$-\tfrac{i}{2}\operatorname{tr}(\eta\alpha\wedge
Q(\alpha\eta))=-\tfrac{i}{2}\operatorname{tr}((\xi+\xi^{*})\wedge(-\xi+\xi^{*}))=i\operatorname{tr}(\xi^{*}\wedge\xi)=-2\lVert\xi\rVert^{2},$

which is negative-definite on $\mathcal{P}$ by definition 4.10.

Now we prove that $L$ induces an isomorphism $H_{1}\to H_{-1}$. The essential
point is that the bilinear form $\int_{S}\operatorname{tr}(\eta L\eta)$ is
equivalent to the standard $H_{1}$-norm (defined using any choice of metrics).
Recall the pseudo-hermitian product given by Equation (4.4). We use
integration by parts, and conjugation invariance of trace to write

$\displaystyle-\tfrac{i}{2}\int_{S}\operatorname{tr}(\eta L\eta)$
$\displaystyle=-\tfrac{i}{2}\int_{S}\operatorname{tr}(\eta
d_{A}Qd_{A}\eta)+\operatorname{tr}(\eta[[\Phi,\eta]\wedge\Phi^{*}])-\operatorname{tr}(\eta[\Phi\wedge[\Phi^{*},\eta]])$
$\displaystyle=-\tfrac{i}{2}\int_{S}-\operatorname{tr}(d_{A}\eta\wedge
Qd_{A}\eta)+\operatorname{tr}([\eta,\Phi^{*}]\wedge[\Phi,\eta])-\operatorname{tr}([\eta,\Phi]\wedge[\Phi^{*},\eta])$
$\displaystyle=-\tfrac{i}{2}\int_{S}2\operatorname{tr}(\pi_{\mathrm{Im}\,\operatorname{ad}_{\Phi}}(d_{A}\eta)^{*}\wedge\pi_{\mathrm{Im}\,\operatorname{ad}_{\Phi}}(d_{A}\eta)+2\operatorname{tr}([\eta,\Phi^{*}]\wedge[\Phi,\eta])$
$\displaystyle=2\lVert\pi_{\mathrm{Im}\,\operatorname{ad}_{\Phi}}(d_{A}\eta)\rVert^{2}+2\lVert[\Phi,\eta]\rVert^{2}.$

This is a norm on $\Omega^{1}(S,\mathfrak{g}^{\sigma,-\rho})$ which is the
integral of the sum of a positive definite norm of the first derivatives of
$\eta$, and a positive definite norm of $\eta$ pointwise. This means it is
equivalent to the standard $H_{1}$-norm. Recall that
$H_{-1}(S,E\otimes\Lambda^{2}T^{*}S)$ can be defined as the topological dual
to $H_{1}(E)$ for any vector bundle $E$. The norms being equivalent means that
$L$ induces an isomorphism $H_{1}\to H_{-1}$. This implies that the induced
maps $H_{l}\to H_{l-2}$ are all isomorphisms for $l\in\mathbb{N},l\geq 1$. ∎

###  5.2 Application of the implicit function theorem

In this section we use standard techniques for nonlinear elliptic PDE to show
that for Fock fields with small enough $(0,1)$-part, i.e. near the Fuchsian
locus, there is a hermitian structure such that the canonical connection is
flat. We start by recalling the implicit function theorem:

######  Theorem 5.3 (Implicit function theorem).

Let $W\subset X\times Y$ be an open subset of a product of differentiable
Banach manifolds. Let $f:W\to Z$ be a continuously differentiable function to
a third differentiable Banach manifold. Suppose that $f(x,y)=z$, and that
$df_{(x,y)}$ restricts to an isomorphism from $T_{y}Y$ to $T_{z}Z$. Then there
is a function $g$ from a neighborhood $U$ of $x$ to $Y$ satisfying

$f(x,g(x))=0$

for all $x\in U$. Furthermore, $g$ is continuously differentiable.

We will apply this where $X$ is the space of Fock fields for a principle
bundle $P$ equipped with $\sigma$ and $\rho$, $Y$ is the space of sections of
$\mathfrak{g}_{P}^{\sigma,-\rho}$, $W\subset X\times Y$ is the subset of
$(\phi,\eta)$ where $e^{-\eta}\Phi e^{\eta}\in\mathcal{P}$, $Z$ is the space
of two-forms valued in $\mathfrak{g}^{\sigma,\rho}$, and $f$ is the map taking
$(\phi,\eta)$ to the curvature of the canonical connection $d_{A}$ associated
to the conjugated Fock field:

$f:\left\\{\begin{array}[]{ccc}W&\longrightarrow&\Omega^{2}(S,\mathfrak{g}_{P}^{\sigma,\rho})\\\
(\phi,\eta)&\mapsto&F(e^{-\eta}\Phi e^{\eta})\end{array}\right.$

where $F$ is the map from Equation (5.1). We need to equip all these spaces
with differentiable Banach manifold structures such that the criteria of the
implicit function theorem are satisfied. To do this we should understand the
derivative of the map from Fock fields to curvature.

Let $\dot{\Phi}\in\mathrm{Var}(\Phi)^{-\sigma}$ be a tangent vector at $\Phi$
to the space of Fock fields. Let $\dot{A}$ be the corresponding variation in
canonical connection. The derivative of $d_{A}\Phi$ and $d_{A}\Phi^{*}$ must
be zero, giving us two equations on $\dot{A}$.

$d_{A}\dot{\Phi}+[\dot{A}\wedge\Phi]=0$
$d_{A}\dot{\Phi}^{*}+[\dot{A}\wedge\Phi^{*}]=0$

By transversality, we know that $ad_{\Phi}$ restricts to an isomorphism from
$\mathrm{Im}(\operatorname{ad}_{\Phi^{*}})^{\sigma}\subset\Omega^{1}(S,\mathfrak{g}_{P})$
to
$\mathrm{Im}(\operatorname{ad}_{\Phi})^{-\sigma}\subset\Omega^{2}(S,\mathfrak{g}_{P})$.
Let
$Y:\mathrm{Im}(\operatorname{ad}_{\Phi})^{-\sigma}\to\mathrm{Im}(\operatorname{ad}_{\Phi^{*}})^{\sigma}$
denote the inverse of this map. We can express
$\dot{A}\in\Omega^{1}(S,\mathfrak{g}_{P})^{\sigma,\rho}$ in terms of $Y$,
using Corollary 4.5.

######  Lemma 5.4.

The variation of the connection $d_{A}$ is given by

$\dot{A}=-Y(d_{A}\dot{\Phi})+Y(d_{A}\dot{\Phi})^{*}.$

It follows that the total change in the curvature $F(\Phi)$ is

$dF_{\Phi}(\dot{\Phi})=-d_{A}Y(d_{A}\dot{\Phi})+d_{A}Y(d_{A}\dot{\Phi})^{*}+[\dot{\Phi}\wedge\Phi^{*}]+[\Phi\wedge\dot{\Phi}^{*}].$

All we will use here is that this is a second order operator which varies
continuously (in the operator norm topology) with respect to variations of
$\Phi$ in the $C^{0}$-topology. This is because the linear maps $Y$ depend
continuously on $\Phi$.

Since $dF_{\Phi}$ is a second order operator, it extends to a continuous map
$H_{k}(\mathrm{Var}(\Phi))\to
H_{k-2}(\Lambda^{2}T^{*}S\otimes\mathfrak{g}^{\rho,\sigma})$ where $H_{k}$
denotes the Sobolev Hilbert space of sections with $k$ square-integrable
derivatives, where $k\in\mathbb{N}$ with $k\geq 2$. The Sobolev embedding
theorem tells us that in two dimensions, $H_{k}$ is in $C^{0}$. If we equip
$\mathcal{P}$ with the $H_{k}$-topology, and
$\Omega^{2}(S\mathfrak{g}^{\sigma,\rho})$ with the $H_{k-2}$-topology, then
$F$ is continuously differentiable. A Sobolev inequality tells us that
multiplication $H_{k}\times H_{k}\to H_{k}$ is continuous in dimension 2, so
the map $(\Phi,\eta)\mapsto e^{-\eta}\Phi e^{\eta}$ is continuous (and
continuously differentiable). Finally, by Theorem 5.2 we know that the partial
derivative of $f$ with respect to $\eta$ is an isomorphism. All the axioms of
the implicit function theorem are satisfied, so we get the following:

######  Proposition 5.5.

Let $k\geq 2$. The set of Fock fields $\Phi$ of class $H_{k}$, such that there
exists $\eta\in H_{k}(S,\mathfrak{g}_{P}^{\sigma,-\rho})$ with flat canonical
connection associated to $e^{-\eta}\Phi e^{\eta}$, is open. Furthermore, the
map from $\Phi$ to this solution $\eta$ is differentiable.

Note that solutions $\eta$ for a given $\Phi$ are isolated, but at this point
we have not proven them to be unique.

Since the operator $L$ is elliptic in the variable $\eta$, and continuously
differentiable with respect to the $C^{0}$-topology (thus also the
$H_{k}$-topology,) it can be approximated by its linearization for small
continuous fluctuations of $\eta$. Standard elliptic regularity arguments then
imply that if $\Phi$ is smooth, then $\eta$ must be as well.

###  5.3 Solution in a neighborhood

Now we restrict to $\mathrm{SL}_{n}(\mathbb{C})$ to get a link from a
neighborhood of the Fuchsian locus in the moduli space of higher complex
structures $\mathcal{T}^{n}(S)$ to the Hitchin component.

Proposition 5.5 gives us a map from a neighborhood $U\subset\mathbb{M}^{n}(S)$
of the Fuchsian locus in the space of higher complex structures to the Hitchin
component. In Section 6 we will see that this map is locally constant along
higher diffeomorphism orbits. Unfortunately, this is not quite enough to
conclude that we have a canonical map from a neighborhood of the Fuchsian
locus in $\mathcal{T}^{n}(S)$ to the Hitchin component, because we can not
guarantee that the intersection of $U$ with higher diffeomorphism orbits is
connected.

Luckily there is a convenient slice of the higher diffeomorphism action,
namely the harmonic higher complex structures defined in [23]. Let
$\mathcal{H}\mathbb{M}^{n}(S)$ denote the space of harmonic higher complex
structures of order $n$ on $S$. A harmonic higher complex structure amounts to
a complex structure, and list of tensors $(\mu_{3},...,\mu_{n})$ with
$\mu_{k}$ a section of $K^{1-n}\otimes\bar{K}$ satisfying
$\bar{\partial}(\bar{\mu}_{k}g_{S}^{k-1})=0$, where $g_{S}$ denotes the
hyperbolic metric on $S$ associated to the complex structure. Theorem 8.2 in
[23] states that any higher complex structure modulo higher diffeomorphism
allows a harmonic representative, unique up to isotopy:

$\mathbb{M}^{n}(S)/\mathrm{Ham}_{0}(T^{*}S)\cong\mathcal{H}\mathbb{M}^{n}(S)/\mathrm{Diff}_{0}(S).$

There is a natural action of $\mathbb{R}^{*}$ on higher complex structures by

$t\cdot(\mu_{3},...,\mu_{n})=(t\mu_{3},...,t^{n-2}\mu_{n})$

coming from scalar multiplication by $t$ in $T^{*}S$. This action clearly
preserves $\mathcal{H}\mathbb{M}^{n}(S)$. Let
$\mathcal{H}\mathbb{M}^{n}_{*}(S)\subset\mathcal{H}\mathbb{M}^{n}(S)$ denote
the subset for which there is a path of solutions to (4.5), for
$\\{(t\mu_{3},...,t^{n-2}\mu_{n})|\,\,t\in[0,1]\\}$ starting at the Fuchsian
solution. Proposition 5.5 gives that this path of solutions is unique if it
exists, so we have a natural map from $\mathcal{H}\mathbb{M}^{n}_{*}(S)$ to
the Hitchin component via evaluating at the end of the path.

Proposition 5.5 also implies that $\mathcal{H}\mathbb{M}^{n}_{*}(S)$ is open.
Also, note that $\mathcal{H}\mathbb{M}^{n}_{*}(S)$ is necessarily
$\mathrm{Diff}_{0}(S)$-invariant because we can pull-back solutions by
diffeomorphisms. We can define
$\mathcal{T}^{n}_{*}(S)\subset\mathcal{T}^{n}(S)$ to be the quotient of
$\mathcal{H}\mathbb{M}^{n}_{*}(S)$ by $\mathrm{Diff}_{0}(S)$. The map from
$\mathcal{H}\mathbb{M}^{n}_{*}(S)$ to the Hitchin component factors through
$\mathcal{T}^{n}_{*}(S)$ because diffeomorphisms isotopic to the identity
cannot change monodromy. Therefore, we have the following result:

######  Theorem 5.6.

There is an open neighborhood of the Fuchsian locus, namely
$\mathcal{T}^{n}_{*}(S)$, which has a canonical map to the Hitchin component.

Of course it is the nature of the map that is interesting, not its mere
existence.

##  6 Higher diffeomorphisms

In this section, we exhibit a gauge-theoretic interpretation for higher
diffeomorphisms. These can be described by special $\lambda$-dependent gauge
transformations acting on a family of flat 3-term connections
$\lambda^{-1}\Phi+d_{A}+\lambda\Psi$. For $\Psi=\Phi^{*}$, this action does
not change the associated point in the Hitchin component since a gauge
transformation does not change the monodromy.

###  6.1 Special $\lambda$-dependent gauge transformations

Consider a $G$-Fock bundle $(P,\Phi,\sigma)$ equipped with a second Fock field
$\Psi$. Assume the existence of a connection $d_{A}$ on $P$ such that

$\mathcal{A}(\lambda)=\lambda^{-1}\Phi+d_{A}+\lambda\Psi$

is flat for all $\lambda\in\mathbb{C}^{*}$. The space of all compatible
triples $(\Phi,\Psi,d_{A})$ on $(P,\sigma)$ such that the associated
connection is flat is denoted by $\mathcal{C}^{fl}(P,\sigma)$.

Consider a 3-term infinitesimal gauge transformation

$\eta(\lambda)=\lambda^{-1}\eta_{-1}+\eta_{0}+\lambda\eta_{1}\;\text{ where
}\;\eta_{-1},\eta_{0},\eta_{1}\in\Omega^{0}(S,\mathfrak{g}_{P}).$

The action on $\mathcal{A}(\lambda)$ is given by

$\displaystyle\eta(\lambda).\mathcal{A}(\lambda)$
$\displaystyle=d\eta(\lambda)+[\mathcal{A}(\lambda),\eta(\lambda)]$
$\displaystyle=\lambda^{-2}[\Phi,\eta_{-1}]+\lambda^{-1}(d_{A}(\eta_{-1})+[\Phi,\eta_{0}])+d_{A}(\eta_{0})+[\Phi,\eta_{1}]+[\Psi,\eta_{-1}]$
$\displaystyle\;\;\;+\lambda(d_{A}(\eta_{1})+[\Psi,\eta_{0}])+\lambda^{2}[\Psi,\eta_{1}].$
(6.1)

The action of $\eta_{0}$ being the usual complex gauge action, we consider
$\eta_{0}=0$ here to concentrate on the new $\lambda$-dependent gauge action.
In order to preserve the space of 3-term connections, the lowest and highest
term in $\lambda$ have to vanish. This gives that $[\Phi,\eta_{-1}]=0$ and
$[\Psi,\eta_{1}]=0$. Hence $\eta_{-1}\in Z(\Phi)$ and $\eta_{1}\in Z(\Psi)$.
The variation of $\Phi$ is then given by

$\delta\Phi=d_{A}(\eta_{-1}).$ (6.2)

Similarly we get $\delta\Psi=d_{A}(\eta_{1})$ and $\delta
A=[\eta_{-1},\Psi]+[\eta_{1},\Phi]$. Note that the variation $\delta A$,
coming from the gauge transformation, is the same as the one induced from the
filling-in procedure 4.6, since gauge transformations preserve curvature.

######  Definition 6.1.

A _special $\lambda$-dependent gauge_ of type
$(P,\Phi,\Psi)\in\mathcal{C}^{fl}(P,\sigma)$ is an element of
$\Omega^{0}(S,\mathfrak{g}_{P}\otimes\mathbb{C}[\lambda^{\pm 1}])$ of the form
$\lambda^{-1}\eta_{-1}+\lambda\eta_{1}$ such that $\eta_{-1}\in Z(\Phi)$ and
$\eta_{1}\in Z(\Psi)$.

Note that the notion of special $\lambda$-dependent gauge depends explicitly
on $\Phi$ and $\Psi$. Considering Fock fields up to these transformations
gives equivalence classes, but this equivalence relation does not come from a
group action.

In the unitary setting, where $(P,\Phi)$ is equipped with a compatible
hermitian structure $\rho$ and $\Psi=\Phi^{*}=-\rho(\Phi)$, we ask for
$\eta_{1}=\eta_{-1}^{*}$ in order to preserve $\rho$.

######  Proposition 6.2.

Variations coming from special $\lambda$-dependant gauge transformations are
tangent to $\mathcal{C}^{fl}(P,\sigma)$.

###### Proof.

Let $\mathcal{A}(\lambda)\in\mathcal{C}^{fl}(P,\sigma)$. Since gauge
transformations preserve curvature, the flatness of $\mathcal{A}(\lambda)$
remains. We have also already seen that the space of 3-term connections is
preserved. It remains to show that $\Phi$ and $\Psi$ stay Fock fields. It is
enough to prove it for $\Phi$. Let us show that $\sigma$ negates the variation
$\delta\Phi=d_{A}(\eta_{-1})$. Since $\eta_{-1}\in Z(\Phi)$, it is negated by
$\sigma$. The canonical connection $d_{A}$ being $\sigma$-invariant, we get
that $\delta\Phi$ is negated by $\sigma$.

The only remaining point is that the nilpotency condition is preserved. This
is a bit more delicate and needs Lemma 2.16. Consider a local chart in which
$\Phi=Fdz+\Phi_{2}d\bar{z}$, where $F$ is a fixed principal nilpotent element.
We have to show that $\delta\Phi=d_{A}(\eta_{-1})\in\mathrm{Var}(\Phi)$. By
Proposition (3.20) we know that
$\mathrm{Var}(\Phi)\cong\mathrm{Im}(\operatorname{ad}_{\Phi})\oplus
Z(\Phi)\bar{K}$. The (1,0)-part of $d_{A}(\eta_{-1})$ is
$\partial\eta_{-1}+[A_{1},\eta_{-1}]$. We have $\partial\eta_{-1}\in Z(F)$ and
$Z(F)\subset\mathrm{Im}(\operatorname{ad}_{F})$ since $Z(F)$ describes the
space generated by lowest weight vectors. From $[\eta_{-1},\Phi]=0$, we get
$\eta_{-1}\in Z(F)$, hence
$[A_{1},\eta_{-1}]\in\mathrm{Im}(\operatorname{ad}_{\eta_{-1}})\subset\mathrm{Im}(\operatorname{ad}_{F})$
by Lemma 2.16. Therefore, locally there is
$R\in\Omega^{0}(S,\mathfrak{g}_{P})$ such that
$[R,F]=\partial\eta_{-1}+[A_{1},\eta_{-1}]$.

We show that $d_{A}(\eta_{-1})-[R,\Phi]\in Z(\Phi)\bar{K}$. By definition of
$R$ this difference has no (1,0)-part. It is sufficient to prove
$\bar{\partial}\eta_{-1}+[A_{2},\eta_{-1}]-[R,\Phi_{2}]\in Z(F)$. We compute

$[F,\bar{\partial}\eta_{-1}+[A_{2},\eta_{-1}]-[R,\Phi_{2}]]=[\eta_{-1},[A_{2},F]]-[\Phi_{2},[R,F]],$
(6.3)

where we used $[F,\eta_{-1}]=0$ (hence also $[F,\bar{\partial}\eta_{-1}]=0$),
the Jacobi identity and $[F,\Phi_{2}]=0$. Now the identity $d_{A}\Phi=0$ gives

$[A_{2},F]=\bar{\partial}\Phi_{2}+[A_{1},\Phi_{2}].$

Together with $[R,F]=\partial\eta_{-1}+[A_{1},\eta_{-1}]$ (from definition of
$R$), we continue Equation (6.3):

$[\eta_{-1},[A_{2},F]]-[\Phi_{2},[R,F]]=[\eta_{-1},\bar{\partial}\Phi_{2}]-[\Phi_{2},\partial\eta_{-1}]=0,$

where some cancellation happened and we use that $[\eta_{-1},\Phi_{2}]=0$. ∎

######  Remark.

At first glance it might be surprising that the action is only defined on the
space of _flat_ 3-term connections. In fact, the condition $d_{A}\Phi=0$ is
needed in order to get a variation $\delta\Phi$ which preserves
$[\Phi\wedge\Phi]=0$, i.e. which satisfies $[\Phi\wedge\delta\Phi]=0$. Indeed,
since $\delta\Phi=d_{A}(\eta_{-1})$ and $[\Phi,\eta_{-1}]=0$, we get

$[\Phi\wedge\delta\Phi]=[\Phi\wedge
d_{A}(\eta_{-1})]=[d_{A}(\Phi),\eta_{-1}]=0.$

The condition $F(A)+[\Phi\wedge\Psi]=0$ assures the preservation of
$d_{A}(\Phi)=0$:

$\delta(d_{A}(\Phi))=d_{A}(d_{A}(\eta_{-1}))+[([\eta_{-1},\Psi]+[\eta_{1},\Phi])\wedge\Phi]=[F(A)+[\Phi\wedge\Psi],\eta_{-1}]=0.$

The variation $\delta\Phi=d_{A}(\eta_{-1})$ seems to depend on the connection
$d_{A}$. We show that this is not really the case:

######  Proposition 6.3.

Up to usual gauge transformations, the variation $\delta\Phi=d_{A}(\eta_{-1})$
does not depend on the $\Phi$-compatible connection $d_{A}$.

The main ingredient of the proof is Corollary 3.25 about the vanishing of
brackets in $\Phi$-cohomology.

###### Proof.

If $\lambda^{-1}\Phi+d_{B}+\lambda\Psi$ is another flat 3-term connection,
then we prove that there exists a usual gauge transformation $R$ such that
$d_{A}(\eta_{-1})-d_{B}(\eta_{-1})=[R,\Phi].$ This means that the variation of
$\Phi$ in (6.2) does not depend on $d_{A}$ modulo gauge transformations.

Note that $d_{A}\Phi=0$ and $d_{B}\Phi=0$ implies $[A-B,\Phi]=0$, and also
$d_{A}(\eta_{-1})-d_{B}(\eta_{-1})=[A-B,\eta_{-1}]$. Hence in the language of
$\Phi$-cohomology, $A-B$ is a 1-cocycle and $\eta_{-1}$ is a 0-cocycle. We
have seen in Corollary 3.25 that the bracket of cohomology classes of Fock
fields is always a coboundary. This gives the existence of $R$. ∎

###  6.2 First variation formula

Now we restrict to the case of $\mathrm{SL}_{n}(\mathbb{C})$-Fock bundles
$(E,\Phi,g)$. We prove that the induced infinitesimal action of special
$\lambda$-gauge transformations on the higher complex structure is given by
the infinitesimal action by higher diffeomorphisms.

Here we concentrate on the action on $\Phi$ and consider a $\lambda$-dependent
gauge transformation $\lambda^{-1}\xi$ (we slightly change notation to the
previous subsection where $\xi=\eta_{-1}$), where
$\xi\in\Omega^{0}(S,\mathfrak{sl}(E))$ with $[\Phi,\xi]=0$. This implies that
$\xi$ is a polynomial in $\Phi$, evaluated in various vector fields. Recall
the variation of $\Phi$ from Equation (6.2):

$\delta\Phi=d_{A}\xi.$

######  Theorem 6.4.

The variation on a Fock field $\Phi$ induced by a gauge transformation
$\lambda^{-1}\xi$ with $\xi=\Phi(v_{1})\cdots\Phi(v_{k})$ is equivalent to the
action of the Hamiltonian $H=v_{1}\cdots v_{k}$ on the higher complex
structure induced by $\Phi$.

The proof is a direct computation using local coordinates and the condition
$d_{A}\Phi=0$.

###### Proof.

The statement is local. Hence consider a coordinate system $(z,\bar{z})$ on
$S$ and a gauge fixing in which the Fock field $\Phi$ is
$\Phi=F\,dz+Q(F)d\bar{z}$, where $F$ is a fixed principal nilpotent element
and $Q$ is a polynomial without constant term. We can decompose
$\xi=\Phi(v_{1})\cdots\Phi(v_{k})$ into monomials in $F$. So let us suppose
that $\xi=w_{k}(z,\bar{z})F^{k}$. We then get

$\delta\Phi=d_{A}\xi=d\xi+[A,\xi]=\left(\partial
w_{k}F^{k}+w_{k}[A_{1},F^{k}]\right)dz+\left(\bar{\partial}w_{k}F^{k}+w_{k}[A_{2},F^{k}]\right)d\bar{z},$

where we used $A=A_{1}dz+A_{2}d\bar{z}$. With
$\Phi=\Phi_{1}dz+\Phi_{2}d\bar{z}$, we get

$\delta\Phi_{1}=\partial w_{k}F^{k}+w_{k}[A_{1},F^{k}]\;\;\text{ and