# Mirror symmetry for Nahm branes

Emilio Franco  Emilio Franco
CMUP (Centro de Matemática da Universidade do Porto)
Universidade do Porto
Rua do Campo Alegre 1021/1055
4169-007, Porto (Portugal<EMAIL_ADDRESS>and  Marcos Jardim  Marcos
Jardim
Instituto de Matemática, Estatística e Computação Científica
Universidade Estadual de Campinas
Rua Sérgio Buarque de Holanda, 651
13083-859 Campinas/SP (Brazil<EMAIL_ADDRESS>
###### Abstract.

Using the Dirac–Higgs bundle and the morphism given by tensorization, we
consider a new class of virtual hyperholomorphic bundles over the moduli space
of $\mathrm{M}$ of degree 0 semistable Higgs bundles. This construction
generalizes the Nahm transform of a stable Higgs bundle.

In the physicist’s language, our class of hyperholomorphic vector bundles can
be seen as (virtual) space filling $\mathrm{(BBB)}$-branes on $\mathrm{M}$. We
then use the Fourier–Mukai–Nahm transform to describe the corresponding dual
branes restricted to the smooth locus of the Hitchin fibration. The dual
branes are checked to be $\mathrm{(BAA)}$-branes supported on a complex
Lagrangian multisection of the Hitchin fibration.

EF was supported by the FAPESP postdoctoral grant number 2012/16356-6, and is
currently supported by the project PTDC/MAT- GEO/2823/2014 funded by FCT
(Portugal) with national funds. MJ is partially supported by the CNPq grant
number 303332/2014-0 and the FAPESP grant number 2016/03759-6, and thanks the
University of Edinburgh for its hospitality.

###### Contents

  1. 1 Introduction
  2. 2 Geometry of the Hitchin system
    1. 2.1 Non-abelian Hodge theory
    2. 2.2 The Hitchin fibration
    3. 2.3 The universal bundle and gerbes
    4. 2.4 Fourier–Mukai and mirror symmetry for the Hitchin system
  3. 3 The Dirac–Higgs bundle and connection
  4. 4 Tensorization and spectral data
  5. 5 Nahm transform of high rank
  6. 6 The mirror brane
    1. 6.1 The Fourier–Mukai transform
    2. 6.2 A complex Lagrangian multisection
    3. 6.3 The Fourier–Mukai–Nahm transform

##  1\. Introduction

Higgs bundles were introduced by Hitchin in [H1] as solutions of the
dimensional reduction to a Riemann surface of the self-dual Yang–Mills
equations in $4$ dimensions. The moduli space $\mathrm{M}$ of $G$-Higgs
bundles has a rich geometry, in particular, can be constructed as a
hyperkähler quotient in the context of gauge theory [H1, Si0, Si1, Si2, Do, C]
inheriting a hyperkähler structure; in addition, it admits a fibration
$\mathrm{M}\to B$ over a vector space, becoming an algebraically completely
integrable system [H2] which is known as the Hitchin system. It was shown in
[HT, DG, DP] that Hitchin systems for Langlands dual groups are dual,
satisfying thereby the requirements of being _Strominger–Yau–Zaslow (SYZ)
mirror partners_ [SYZ], which allows for the identification of T-duality with
mirror symmetry between them. Since the group $G=\mathrm{GL}(n,\mathbb{C})$ is
Langlands self-dual, we obtain a self-dual Hitchin system in this case, which
is the one that we study in this paper.

Arinkin–Polishchuk [AP] gave a realization of mirror symmetry in the case of
smooth torus fibrations, by using a Fourier–Mukai–Nahm transform to obtain
holomorphic bundles from flat bundles supported on Lagrangian subvarieties of
the mirror. This was further extended by Bruzzo–Marelli–Pioli [BMP, BMP2] and
by Glazebrook, Kamber, and the second named author [GJK]. In the complex case,
the Fourier–Mukai–Nahm transform underlies a Fourier–Mukai transform, which is
the case for the locus of smooth fibres of the Hitchin system. We have not
found in the literature an extension of the Fourier–Mukai transform outside
the smooth locus of the Hitchin fibration, although Arinkin [Ar] and
Melo–Rapagnetta–Viviani [MRV1, MRV2] have extended the duality to Hitchin
fibres associated to non-reduced curves, providing the corresponding
Fourier–Mukai transform. A similar statement for the locus of non-reduced
spectral curves is still missing to our knowledge.

The rich geometry of the moduli space of Higgs bundles $\mathrm{M}$ makes it
an object of interest for theoretical physics. In [BJSV, HMS] it was shown
that the dimensional reduction of an $N=4$ Super Yang–Mills theory in $4$
dimensions gives a $2$ dimensional sigma model with hyperkähler target
$\mathrm{M}$, and, hence, S-duality in the former becomes T-duality (mirror
symmetry) in the latter. This was the starting point for the ground-breaking
article by Kapustin and Witten [KW], where they relate the Geometric Langlands
Conjecture and S-duality in the original $N=4$ super Yang–Mills theory.

Following Kapustin and Witten [KW], a $\mathrm{(BBB)}$-brane in a manifold $M$
with hyperkähler structure $(g,\Gamma^{1},\Gamma^{2},\Gamma^{3})$, is a pair
$(Z,W)$ consisting of a hyperkähler submanifold $Z\subset M$, and a
hyperholomorphic vector bundle $W\to Z$. Similarly, a $\mathrm{(BAA)}$-brane
is a triple $(\Xi,V,\nabla_{V})$ consisting of a submanifold $\Xi\subset M$
which is complex for $\Gamma^{1}$ and Lagrangian for the second and third
Kähler structures (i.e. complex Lagrangian with respect to $\Gamma^{1}$), and
a vector bundle $V\to\Xi$, holomorphic with respect to $\Gamma^{1}$, equipped
with a compatible flat connection $\nabla_{V}$. In String Theory, branes are
geometrical objects that encode the Dirichlet boundary conditions, and mirror
symmetry predicts a 1-1 correspondence between $\mathrm{(BBB)}$-branes on a
moduli space of $G$-Higgs bundles and $\mathrm{(BAA)}$-branes on a moduli
space of $G^{L}$-Higgs bundles, where $G^{L}$ denotes the Langlands dual of
$G$.

Motivated by this context, many authors have considered different ways of
constructing hyperkähler and complex Lagrangian submanifolds in moduli spaces
of Higgs bundles on Riemann surfaces, see for instance [H4, BS1, BS2, BG, HS,
BCFG, BS3] and the geometry of these objects has been intensively studied
[BGH1, BGH2, GW, Ba1, Ba2]. More generally, due to their intrinsic geometric
interest, one can also study these special submanifolds on other classes of
hyperkähler manifolds, like quiver varieties [FJMa, HSc], and moduli spaces of
stable sheaves on K3 or abelian surfaces [FJMe]. Recently, Gaiotto [Ga] and
Hitchin [H4, H5] constructed several examples of $\mathrm{(BAA)}$-branes in
the moduli space of Higgs bundles, and their dual $\mathrm{(BBB)}$-branes
arising from the Dirac–Higgs bundle. The latter, originally defined by Hitchin
in [H3], is a virtual bundle (see Section 2.3) over a moduli space of Higgs
bundles which comes equipped with a natural hyperholomorphic connection, and
therefore is an important source of examples of $\mathrm{(BBB)}$-branes.

Starting also from the Dirac–Higgs bundle, we consider in this paper a new
class of virtual hyperholomorphic bundles over the moduli space of
$\mathrm{M}$ of degree 0 semistable Higgs bundles which are given by a
generalized Nahm transform of a given stable Higgs bundle on $X$. As pointed
out before, these hyperholomorphic bundles can be seen as (virtual) space
filling $\mathrm{(BBB)}$-branes on $\mathrm{M}$. We then apply the
Fourier–Mukai–Nahm transform to the restriction of these space-filling
$\mathrm{(BBB)}$-branes to the smooth locus of the Hitchin fibration, which we
call _Nahm branes_ ; we check that the transformed object is indeed a
$\mathrm{(BAA)}$-brane supported on a complex Lagrangian multisection of the
Hitchin fibration. Since the Nahm transform of the trivial Higgs bundle
corresponds to the Dirac–Higgs bundle, we describe, as a particular case, the
behaviour under mirror symmetry of the $\mathrm{(BBB)}$-brane that constitutes
the Dirac–Higgs bundle, when restricted to the smooth locus of the Hitchin
fibration.

Beyond the generalized Nahm transform for Higgs bundles, our main contribution
is to take into account the additional structures (bundles and connections)
that constitute branes in the correspondence between $\mathrm{(BBB)}$\- and
$\mathrm{(BAA)}$-branes predicted by mirror symmetry.

This paper is organized as follows.

In Section 2 we review the properties of the Hitchin system and the theory of
Higgs bundles. In Section 4 we study the behaviour of spectral data of Higgs
bundles under tensorization, which will be crucial to understand our
generalization of the Nahm transform. Section 3 is dedicated to the study of
the Dirac–Higgs bundle and connection, the second ingredient necessary both
defining the generalized Nahm transform, described in Section 5. More
precisely, the _rank $m$ Nahm transform_ of a stable Higgs bundle
$\mathcal{E}$ is a virtual vector bundle
$\widehat{\mathbf{E}}_{m}\to\mathrm{M}_{m}$, with respect to the gerbe
$\beta_{m}$ (described in Section 2.3), on the moduli space of stable Higgs
bundles of rank $m$, equipped with a hyperholomorphic connection
$\boldsymbol{\widehat{\nabla}}_{m}^{\mathcal{E}}$. This virtual vector bundle
with the hyperholomorphic structure induced from the connection
$\boldsymbol{\widehat{\nabla}}_{m}^{\mathcal{E}}$ constitutes a virtual space-
filling $\mathrm{(BBB)}$-brane, called a _Nahm brane_.

In Section 6.1 we describe the Fourier–Mukai transform of the restriction
$\widehat{E}^{\prime}_{m}$ of $\widehat{\mathbf{E}}_{m}\to\mathrm{M}_{m}$ to
the smooth locus of the Hitchin fibration
$\mathrm{M}_{m}^{\prime}\subset\mathrm{M}_{m}$, where the gerbe $\beta_{m}$
trivializes, so that $\widehat{E}^{\prime}_{m}$ is a vector bundle in the
usual sense. We show that the Fourier–Mukai transform of
$\widehat{E}^{\prime}_{m}$ is a sheaf $\widecheck{\mathfrak{E}}^{\prime}_{m}$
supported on a multisection of the Hitchin fibration. In Section 6.2 we show
that the support of $\widecheck{\mathfrak{E}}^{\prime}_{m}$ is a complex
Lagrangian subvariety and in Section 6.3 we prove that there exists a flat
connection $\widecheck{\nabla}_{m}^{\mathcal{E}}$ on it and therefore, the
triple
$\left(\mathrm{supp}\left(\widecheck{\mathfrak{E}}^{\prime}_{m}\right),\widecheck{\mathfrak{E}}^{\prime}_{m},\widecheck{\nabla}_{m}^{\mathcal{E}}\right)$
is a $\mathrm{(BAA)}$-brane in $\mathrm{M}_{m}^{\prime}$. Furthermore, we see
in this last section that the Fourier–Mukai–Nahm transform of this triple
returns our original
$\left(\widehat{E}^{\prime}_{m},\widehat{\nabla}_{m}^{\mathcal{E}}\right)$,
restricted to the smooth locus of the Hitchin fibration.

##  2\. Geometry of the Hitchin system

###  2.1. Non-abelian Hodge theory

Given a smooth projective curve $X$ over $\mathbb{C}$ of genus $g\geq 2$, and
denoting by $\mathbb{E}_{n}$ is the (unique up to isomorphism)
$C^{\infty}$-bundle of rank $n$ over $X$, a Higgs pair [H1, Si0, Si1, Si2] of
rank $n$ on $X$ is pair $(\overline{\partial}_{E},\varphi)$ where
$\overline{\partial}_{E}$ is a Dolbeault operator on $\mathbb{E}_{n}$ fixing
an integrable complex structure on it, and $\varphi$ is an element of
$\Omega_{X}^{1,0}(\mathbb{E}_{n})$.

A Higgs bundle over $X$ is a Higgs pair $(\overline{\partial}_{E},\varphi)$
satisfying $\overline{\partial}_{E}\varphi=0$. Equivalently, a Higgs bundle is
a pair $\mathcal{E}=(E,\varphi)$, where $E$ is a holomorphic vector bundle on
$X$, and $\varphi\in H^{0}(X,\mathrm{End}(E)\otimes K_{X})$ is a holomorphic
section of the endomorphisms bundle, twisted by the canonical bundle $K_{X}$.
Recall from [H1, Si0, Si1, Ni] that $\mathcal{E}$ is said to be _(semi)stable_
if every $\varphi$-invariant sub-bundle $F\subset E$ satisfies

$\frac{\deg F}{\mathrm{rk}F}<~{}(\leq)~{}\frac{\deg E}{\mathrm{rk}E}.$

In addition, $\mathcal{E}$ is _polystable_ if it is a direct sum of semistable
bundles $\mathcal{E}_{i}=(E_{i},\varphi_{i})$, all with the same slope $\deg
E_{i}/\mathrm{rk}E_{i}$.

It is possible to construct [H1, Si0, Si1, Si2, Ni] the moduli space
$\mathrm{M}_{n}$ of rank $n$ and degree $0$ semistable Higgs bundles on $X$.
One has that

$\dim\mathrm{M}_{n}=2n^{2}(g-1)+2.$

Non-abelian Hodge theory establishes the existence of a homeomorphism [H1,
Si1, Si2, Do, C] between $\mathrm{M}_{n}$ and the moduli space of flat
connections on the trivial $C^{\infty}$-bundle of rank $n$. This is a
consequence of the construction of these moduli spaces as a hyperkähler
quotient of the space of Higgs pairs $\mathcal{A}_{n}$, which is an infinite
dimensional affine space modeled on the infinite vector space
$\Omega_{X}^{1,0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n})$, by
the gauge group $\mathcal{G}_{n}$ of complex automorphisms of
$\mathbb{E}_{n}$. Tangent to any Higgs pair
$(\overline{\partial}_{E},\varphi)$ we can consider its infinitessimal
deformations $\dot{\alpha}\in\Omega_{X}^{0,1}(\mathrm{End}(\mathbb{E}_{n}))$
and $\dot{\varphi}\in\Omega_{X}^{1,0}(\mathrm{End}(\mathbb{E}_{n}))$. The
hyperkähler structure on $\mathcal{A}_{n}$ is given by the flat metric on this
space,

$\widetilde{g}_{n}\left((\dot{\alpha}_{1},\dot{\varphi}_{1}),(\dot{\alpha}_{2},\dot{\varphi}_{2})\right)=\int_{X}\mathrm{tr}\left(\dot{\alpha}_{1}^{*}\wedge\dot{\alpha}_{2}+\dot{\alpha}_{2}^{*}\wedge\dot{\alpha}_{1}+\dot{\varphi}_{1}\wedge\dot{\varphi}_{2}^{*}+\dot{\varphi}_{2}\wedge\dot{\varphi}_{1}^{*}\right),$

and the complex structures

$\widetilde{\Gamma}^{1}_{n}(\dot{\alpha},\dot{\varphi})=\left(\,\mathrm{i}\,\dot{\alpha},\,\mathrm{i}\,\dot{\varphi}\right),$

where $\\!\,\mathrm{i}\,\\!$ denotes $\sqrt{-1}$,

$\widetilde{\Gamma}^{2}_{n}(\dot{\alpha},\dot{\varphi})=\left(-\dot{\varphi}^{*},\dot{\alpha}^{*}\right)$

and

$\widetilde{\Gamma}^{3}_{n}=\widetilde{\Gamma}^{1}_{n}\widetilde{\Gamma}^{2}_{n}.$

We denote by
$\widetilde{\omega}^{j}_{n}(\cdot,\cdot)=\widetilde{g}_{n}(\cdot,\widetilde{\Gamma}^{j}_{n}(\cdot))$
the associated Kähler forms, and by
$\widetilde{\Lambda}^{j}_{n}=\widetilde{\omega}_{n}^{j+1}+\,\mathrm{i}\,\widetilde{\omega}_{n}^{j-1}$
the corresponding holomorphic symplectic forms. In particular, one has the
following description for the first one,

(2.1)
$\widetilde{\Lambda}_{n}^{1}\left((\dot{\alpha}_{1},\dot{\varphi}_{1}),(\dot{\alpha}_{2},\dot{\varphi}_{2})\right)=\frac{1}{\pi}\int_{X}\mathrm{tr}\left(\dot{\varphi}_{1}\wedge\dot{\alpha}_{2}-\dot{\varphi}_{2}\wedge\dot{\alpha}_{1}\right).$

From each of the Kähler forms $\widetilde{\omega}^{j}_{n}$ one can construct a
moment map $\mu^{j}_{n}$ and $\mathrm{M}_{n}$ is identified with the
hyperkäler quotient

$\mathrm{M}_{n}\cong{\raisebox{1.99997pt}{\thinspace$(\mu^{1}_{n})^{-1}(0)\cap(\mu^{2}_{n})^{-1}(0)\cap(\mu^{3}_{n})^{-1}(0)$}\left/\raisebox{-1.49994pt}{
$\mathcal{G}_{n}$}\right.}.$

The complex structures $\widetilde{\Gamma}_{n}^{j}$ descend naturally to
complex structures $\Gamma_{n}^{j}$ on $\mathrm{M}_{n}$. Note that the
$2$-forms $\widetilde{\omega}^{j}_{n}$ and $\widetilde{\Lambda}^{j}_{n}$ are
gauge invariant, so they provide naturally the Kähler forms $\omega^{j}_{n}$
on $\mathrm{M}_{n}$, and the holomorphic symplectic forms $\Lambda^{j}_{n}$.

The moduli space of Higgs bundles is realized in $\mathrm{M}_{n}$ after fixing
the complex structure $\Gamma_{n}^{1}$. Given a Higgs bundle
$\mathcal{E}=(E,\varphi)$, consider the complex

$C^{\bullet}_{\mathcal{E}}:\mathrm{End}(E)\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}\mathrm{End}(E)\otimes
K_{X},$

which induces the following exact sequence

$\displaystyle
0\longrightarrow\mathbb{H}^{0}(C^{\bullet}_{\mathcal{E}})\longrightarrow
H^{0}(\mathrm{End}(E))\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}H^{0}(\mathrm{End}(E)\otimes
K_{X})\longrightarrow$
$\displaystyle\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}H^{1}(\mathrm{End}(E))\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}H^{1}(\mathrm{End}(E)\otimes
K_{X})\longrightarrow\mathbb{H}^{2}(C^{\bullet}_{\mathcal{E}})\longrightarrow
0.$

where $\mathbb{H}^{p}(C^{\bullet}_{\mathcal{E}})$ are the hypercohomology
groups for the complex $C^{\bullet}_{\mathcal{E}}$. If $\mathcal{E}$ is a
smooth point of the moduli space, the tangent space at it is given by
$T_{\mathcal{E}}\mathrm{M}_{n}=\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})$.
Thanks to Serre duality, $\varphi\in H^{0}(\mathrm{End}(E)\otimes K_{X})$ can
also be regarded as an element of the dual space $H^{1}(\mathrm{End}(E))^{*}$;
a 1-form $\theta\in T_{\mathcal{E}}^{*}\mathrm{M}_{n}^{\rm st}$ can then be
defined as the composition of $\varphi$ with the map
$\eta:\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})\to H^{1}(\mathrm{End}(E))$,
i.e. $\theta(v)=\varphi\circ\eta(v)$, for each
$v\in\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})$. It can be checked that, on
the smooth locus of $\mathrm{M}_{n}$, $d\theta$ is proportional to the the
holomorphic symplectic form $\Lambda^{1}_{n}$, induced from the hyperkähler
structure.

###  2.2. The Hitchin fibration

We recall here the spectral construction given in [H2, BNR, Si2]. Let
$(q_{1},\dots,q_{n})$ be a basis of $\mathrm{GL}(n,\mathbb{C})$-invariant
polynomials with $\deg(q_{i})=i$. The Hitchin fibration is the dominant
morphism

$\begin{array}[]{cccc}h:&\mathrm{M}_{n}&\stackrel{{\scriptstyle}}{{\longrightarrow}}&B_{n}:=\bigoplus_{i=1}^{n}H^{0}(X,K_{X}^{\otimes
i})\\\
&(E,\varphi)&\longmapsto&\left(q_{1}(\varphi),\dots,q_{n}(\varphi)\right),\end{array}$

and we refer to $B_{n}$ as the Hitchin base.

Consider the total space $\mathrm{Tot}(K_{X})$ of the canonical bundle, and
the obvious algebraic surjection $p:\mathrm{Tot}(K_{X})\to X$; let $\lambda$
be the tautological section of the pullback bundle
$p^{*}K_{X}\to\mathrm{Tot}(K_{X})$. Given an element $b=(b_{1},\dots,b_{n})\in
B_{n}$ we construct the associated spectral curve
$S_{n,b}\subset\mathrm{Tot}(K_{X})$ by considering the vanishing locus of the
section

(2.2)
$\lambda^{n}+p^{*}b_{1}\lambda^{n-1}+\dots+p^{*}b_{n-1}\lambda+p^{*}b_{n}\in
H^{0}(X,K_{X}^{\otimes n}).$

Restricting $p$ to $S_{n,b}$ yields a finite morphism $p_{n,b}:S_{n,b}\to X$
of degree $n$. Considering further the vanishing in $\mathrm{Tot}(K_{X})\times
B_{n}$ of the relative version of (2.2), we construct as well a family of
spectral curves $\mathcal{S}_{n}\subset\mathrm{Tot}(K_{X})\times B$ for which
we naturally have that
$\mathcal{S}_{n}\cap(\mathrm{Tot}(K_{X})\times\\{b\\})=S_{n,b}$. Restricting
the projection $p\times\mathbf{1}_{B}:\mathrm{Tot}(K_{X})\times B_{n}\to
X\times B$, we obtain a finite morphism of degree $n$:

$p_{n}:\mathcal{S}_{n}\to X\times B_{n}.$

For every $b\in B$, the corresponding spectral curve $S_{n,b}$ belongs to the
linear system $|nX|$, and, by Bertini’s theorem, it is generically smooth and
irreducible. Furthermore, since the canonical divisor of the symplectic
surface $\mathrm{Tot}(K_{X})$ is zero, the genus of $S_{n,b}$ is given by

(2.3) $d_{n}:=g\left(S_{n,b}\right)=1+n^{2}(g-1).$

Thanks to Riemann–Roch theorem, $p_{n,b,*}\mathcal{O}_{S_{n,b}}$ is a rank $n$
vector bundle of degree
$\deg(p_{n,b,*}\mathcal{O}_{S_{n,b}})=-(n^{2}-n)(g-1)$. This motivates the
notation

$\delta_{n}:=(n^{2}-n)(g-1).$

Following [BNR], we consider the push-forward $E_{L}:=p_{n,b,*}L$ of a torsion
free sheaf $L$ on $S_{n,b}$ of rank $1$ and degree $\delta_{n}$, which is a
vector bundle on $X$ of rank $n$ and degree
$\delta_{n}+\deg(\pi_{*}\mathcal{O}_{S_{n,b}})=0$. We consider as well the
multiplication by the restriction to $S_{n,b}$ of tautological section,
$\lambda_{b}:\mathcal{O}_{S_{n,b}}\to\mathcal{O}_{S_{n,b}}\otimes
p_{b}^{*}K_{X}$. Note that this induces the following twisted endomorphism of
$L$

$\mathrm{1}_{L}\otimes\lambda_{b}:L\to L\otimes p_{b}^{*}K_{X}.$

whose push-forward returns the Higgs field

$\varphi=p_{n,b,*}(\mathrm{1}_{L}\otimes\lambda_{b}):E_{L}\to E_{L}\otimes
K_{X},$

so that $b=h(\varphi)$.

Thanks to Simpson’s construction [Si1], one can consider the moduli space
$\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})\to
B_{n}$ of rank $1$ torsion free sheaves with relative degree $\delta_{n}$ over
the family of spectral curves $\mathcal{S}_{n}\to B_{n}$ with linearization
$p_{n}^{*}\mathcal{L}$ where $\mathcal{L}$ is the linearizarion on $X\times
B_{n}$ naturally induced from the linearization of $X$. After [Si2], there is
a canonical identification

(2.4)
$\mathrm{M}_{n}\cong\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n}),$

such that each Hitchin fibre is identified with the compactified Jacobian of
the corresponding spectral curve

$h^{-1}\left(b\right)\cong\overline{\operatorname{Jac}}^{\,\delta_{n}}(S_{n,b}).$

Fixing a point $x_{0}\in X$ in our curve, we construct a section
$\hat{\sigma}:B_{n}\to\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(S_{n})$
by considering for every $b$ the line bundle
$p_{n,b}^{*}\mathcal{O}_{X}(x_{0})^{\otimes(n-1)(g-1)}$ on $S_{n,b}$. Such
choice induces the following identification

(2.5)
$\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})\cong\overline{\operatorname{Jac}}^{\,0}_{B_{n}}(\mathcal{S}_{n}).$

Again by Simpson’s construction [Si1], one can consider
$\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})^{\vee}$
to be the moduli space of torsion free sheaves, with trivial relative degree,
over the $B_{n}$-scheme
$\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})$. We
have that
$\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})^{\vee}\to
B_{n}$ is a relative scheme over $B_{n}$ which we call the relative dual
compactified Jacobian of $\mathcal{S}_{n}$. We can even construct the product
over $B_{n}$ of
$\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})$ and
$\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})^{\vee}$,
obtaining the commutative diagram

(2.6)
$\textstyle{\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})\times_{B_{n}}\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{\pi}}$$\scriptstyle{\hat{\pi}}$$\textstyle{\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{h}}$$\textstyle{\overline{\operatorname{Jac}}^{\,\delta_{n}}_{B_{n}}(\mathcal{S}_{n})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{h}}$$\textstyle{B_{n},\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\sigma}}$$\scriptstyle{\check{\sigma}}$-----

where $\hat{\sigma}$ is the constant section considered above, and
$\check{\sigma}$ is the section given by considering the structural sheaf on
each $\operatorname{Jac}(S_{n,b})$.

###  2.3. The universal bundle and gerbes

We review in this section the universal (virtual) bundle over the moduli space
$\mathrm{M}_{n}$ of semistable Higgs bundles, as it is a fundamental piece to
describe the Dirac–Higgs (virtual) bundle.

For $n>1$, the moduli space of Higgs bundles $\mathrm{M}_{n}$ is a coarse
moduli space, not a fine one. In other words, it is not possible to construct
a universal bundle over $\mathrm{M}_{n}\times X$. The best we can do is to
construct a virtual universal bundle for a certain flat unitary gerbe.

For any algebraic variety $Y$ and any open subset $Z\subset Y$, recall that a
$\mathrm{U}(1)$-torsor on $Y$ is a sheaf of sets locally isomorphic to
$\mathrm{U}(1)$. Denote by $\mathrm{Tors}(\mathrm{U}(1),Z)$ the set of
$\mathrm{U}(1)$-torsors over $Z\subset Y$ and note that it is a group with the
tensor product. A flat unitary gerbe on $Y$ is a sheaf of categories $\beta$
over $Y$ such that $\beta|_{Z}$ is a torsor for the group
$\mathrm{Tors}(\mathrm{U}(1),Z)$. Given a covering $\\{Z_{i}\\}_{i\in I}$, a
gerbe provides a category (a groupoid indeed) for every $Z_{i}$, the natural
transformations of these categories in the intersections is realized via
tensoring by $L_{ij}\to Z_{i}\cap Z_{j}$. Therefore, a gerbe defines a set of
flat unitary line bundles over the intersections $\\{L_{ij}\to Z_{i}\cap
Z_{j}\\}_{i,j\in I}$ such that $L_{ij}\cong L_{ji}^{-1}$ and over the triple
intersections $Z_{i}\cap Z_{j}\cap Z_{k}$, one has that $L_{ij}\otimes
L_{jk}\otimes L_{ki}$ is isomorphic to the trivial bundle. When we have a fine
enough $\\{Z_{i}\\}_{i\in I}$ cover of $Y$, a virtual bundle $\mathbf{E}$
associated to the gerbe $\beta$ is a set of vector bundles $\\{E_{i}\to
Z_{i}\\}_{i\in I}$ such that on each intersection $E_{i}|_{Z_{i}\cap
Z_{j}}=L_{ij}\otimes E_{j}|_{Z_{i}\cap Z_{j}}$.

Even if one can not define a universal bundle over $\mathrm{M}_{n}$, it is
possible to construct a universal projective bundle, that we denote by
$(\mathbb{P}U_{n},\Phi_{n})\to X\times\mathrm{M}_{n}$. Define $\beta_{n}$ to
be the flat unitary gerbe of liftings of $\mathbb{P}U_{n}$ to a vector bundle.
By definition of $\beta_{n}$ one can construct a virtual vector bundle from
$\mathbb{P}U_{n}$, and therefore, one gets the virtual universal Higgs bundle
$X\times\mathrm{M}_{n}$

$\left(\mathbf{U}_{n},\boldsymbol{\Phi}_{n}\right)=\left\\{\left(U_{n,i},\Phi_{n,i}\right)\to
X\times Z_{n,i}\right\\}_{i\in I},$

where $\\{Z_{n,i}\\}_{i\in I}$ is a covering of $\mathrm{M}_{n}$. Observe that
the gerbe $\beta_{n}$ trivializes over the open subset given by the stable
locus of the moduli space, as the existence of a universal bundle is
guaranteed there.

It can be shown that the gerbe of liftings $\beta_{n}$ is the pull-back to
$X\times\mathrm{M}_{n}$ of a gerbe over $\mathrm{M}_{n}$, which we still
denote by $\beta_{n}$ by abuse of notation. Consider the obvious projections
occurring in the following commutative diagram,

(2.7)
$\textstyle{\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})\times_{B_{n}}\mathcal{S}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}_{S}}$$\scriptstyle{\widetilde{\pi}_{\operatorname{Jac}}}$$\scriptstyle{(\mathrm{1}_{\operatorname{Jac}}\times
p_{n})}$$\textstyle{\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{1}_{\operatorname{Jac}}}$$\textstyle{\mathcal{S}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n}}$$\textstyle{\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{V\times
X}}$$\scriptstyle{\pi_{\operatorname{Jac}}}$$\scriptstyle{\pi_{X}}$$\textstyle{\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})}$$\textstyle{B_{n}\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{n}}$$\textstyle{X}$

Let us denote by $\widetilde{p}_{n}:\mathcal{S}_{n}\to X$ the composition of
$p_{n}$ with the obvious projection $r_{n}$ onto the second factor.

Let us denote by $B^{\prime}_{n}\subset B_{n}$ the Zariski open subset given
by those points $b\in B_{n}$ such that $S_{n,b}$ is smooth. We denote the
restriction of $\mathcal{S}_{n}$ and $\mathrm{M}_{n}$ to $B^{\prime}_{n}$ by
$\mathcal{S}^{\prime}_{n}$ and
$\mathrm{M}^{\prime}_{n}=\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n})$.
Note that all the points of $\mathrm{M}^{\prime}_{n}$ are associated to line
bundles over smooth spectral curves, which are automatically stable.
Therefore, $\mathrm{M}^{\prime}_{n}$ is contained in the stable locus,

$\mathrm{M}^{\prime}_{n}\subset\mathrm{M}_{n}^{\mathrm{st}}.$

As a consequence, the gerbe $\beta_{n}$ trivializes over
$\mathrm{M}^{\prime}_{n}$ so the restriction of
$(\mathbf{U}_{n},\boldsymbol{\Phi}_{n})$ is a bundle in the usual sense that
we denote by

$(U^{\prime}_{n},\Phi^{\prime}_{n})\to X\times\mathrm{M}^{\prime}_{n}.$

There exists a universal line bundle associated to the family of smooth curves
$\mathcal{S}^{\prime}_{n}$ that we denote by

$P_{n}\to\mathcal{S}^{\prime}_{n}\times_{B_{n}^{\prime}}\operatorname{Jac}^{\,\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n}).$

Observe that one recovers the universal bundle by setting
$U^{\prime}_{n}=(\mathrm{1}_{\operatorname{Jac}}\times p_{n})_{*}P_{n}$. Note,
also, that $\Phi^{\prime}_{n}$ coincides with the direct image of the
multiplication by the tautological section. Observe as well that the universal
line bundle $P_{n}$ defines an embedding

$\check{\imath}_{n}:\mathcal{S}^{\prime}_{n}\hookrightarrow\operatorname{Jac}_{B^{\prime}_{n}}^{\,\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}$

given by $y\mapsto
P_{n}|_{\operatorname{Jac}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n})\times\\{y\\}}$.
Finally, since
$\operatorname{Jac}_{B^{\prime}_{n}}^{\,\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}$
is naturally isomorphic to
$\operatorname{Jac}_{B^{\prime}_{n}}^{\,\delta_{n}}(\mathcal{S}^{\prime}_{n})$,
define also

$\hat{\imath}_{n}:\mathcal{S}^{\prime}_{n}\hookrightarrow\operatorname{Jac}_{B^{\prime}_{n}}^{\,\delta_{n}}(\mathcal{S}^{\prime}_{n})$

to be the embedding defined by $\check{\imath}_{n}$ and the isomorphism
$\operatorname{Jac}_{B^{\prime}_{n}}^{\,\delta_{n}}(\mathcal{S}^{\prime}_{n})\cong\operatorname{Jac}_{B^{\prime}_{n}}^{\,\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}$.

One can consider the relative Poincaré bundle $\mathcal{P}_{n}$ over the
smooth locus of the Hitchin fibration
$\mathrm{M}^{\prime}_{n}\cong\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})$,

$\mathcal{P}_{n}\to\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\times_{B^{\prime}_{n}}\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}.$

Since $\mathcal{P}_{n}$ is a universal bundle for the topologically trivial
line bundles on
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})$
and $P_{n}$ is a family of these objects, by definition of
$\check{\imath}_{n}$ above, one has that the relation between
$\mathcal{P}_{n}$ and $P_{n}$ is

$P_{n}\cong(\check{\imath}_{n}\times
1_{\operatorname{Jac}})^{*}\mathcal{P}_{n}.$

We finish the section with a technical result that will be useful in Section
3.

######  Lemma 2.1.

In the notation of diagram (2.7), we have:

$\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}\stackrel{{\scriptstyle\lambda_{n}}}{{\longrightarrow}}\widetilde{p}_{n}^{*}K_{X})\right)\cong
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{n}^{*}K_{X}|_{\mathcal{S}^{\prime}_{n}\cap\\{\lambda_{n}=0\\}}\right)\right).$

###### Proof.

Recall that the hypercohomology groups $\mathbb{H}^{p}(C^{\bullet})$ of a
2-step complex $C^{\bullet}:C^{0}\stackrel{{\scriptstyle f}}{{\to}}C^{1}$ of
sheaves on a variety satisfies the following exact sequence

$0\to\mathbb{H}^{0}(C^{\bullet})\to H^{0}(C^{0})\stackrel{{\scriptstyle
f}}{{\to}}H^{0}(C^{1})\to\mathbb{H}^{1}(C^{\bullet})\to H^{1}(C^{0})\to\cdots$

Applying this principle to the complex of sheaves
$P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}\stackrel{{\scriptstyle\lambda_{n}}}{{\longrightarrow}}\widetilde{p}_{n}^{*}K_{X})$
on
$\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})\times_{B_{n}}\mathcal{S}_{n}^{\prime}$
as a scheme over
$\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})$, we
conclude that

$\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}\stackrel{{\scriptstyle\lambda_{n}}}{{\longrightarrow}}\widetilde{p}_{n}^{*}K_{X})\right)\cong$
$\cong\mathrm{coker}\left\\{\widetilde{\pi}_{\operatorname{Jac},*}(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}))\stackrel{{\scriptstyle\lambda_{n}}}{{\rightarrow}}\widetilde{\pi}_{\operatorname{Jac},*}(P_{n}\otimes\widetilde{\pi}_{S}^{*}\widetilde{p}_{n}^{*}K_{X})\right\\}$

since
$R^{1}\widetilde{\pi}_{\operatorname{Jac},*}(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}))=0$.

The identification on the statement of the Lemma then follows from applying
the functor $\widetilde{\pi}_{\operatorname{Jac},*}$ to the following exact
sequence of sheaves on
$\operatorname{Jac}^{\,\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n})\times_{B^{\prime}_{n}}\mathcal{S}^{\prime}_{n}$:

$0\to
P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}})\stackrel{{\scriptstyle\lambda_{n}}}{{\rightarrow}}P_{n}\otimes\widetilde{\pi}_{S}^{*}\widetilde{p}_{n}^{*}K_{X}\to
P_{n}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{n}^{*}K_{X}|_{\mathcal{S}^{\prime}_{n}\cap\\{\lambda_{n}=0\\}}\right)\to
0,$

again using that
$R^{1}\widetilde{\pi}_{\operatorname{Jac},*}(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}))=0$.
∎

###  2.4. Fourier–Mukai and mirror symmetry for the Hitchin system

Recall from the Section 2.3 that we denote by $B_{n}^{\prime}$ the locus of
the Hitchin base that parametrizes smooth spectral curves, and by
$\mathrm{M}^{\prime}_{n}=\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n})$
the smooth locus of the Hitchin fibration. Over
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\times_{B^{\prime}_{n}}\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}$
we can consider the relative Poincaré bundle $\mathcal{P}_{n}$ and the
relative Fourier–Mukai transforms

$\begin{array}[]{cccc}R\check{\mathscr{F}}_{n}:&D^{b}_{B^{\prime}_{n}}(\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n}))\cong
D^{b}_{B^{\prime}_{n}}(\mathrm{M}^{\prime}_{n})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&D^{b}_{B^{\prime}_{n}}(\operatorname{Jac}^{\delta_{n}}_{V}(\mathcal{S}^{\prime}_{n})^{\vee})\cong
D^{b}_{B^{\prime}_{n}}(\mathrm{M}^{\prime}_{n})\\\
&F^{\bullet}&\longmapsto&\check{F}^{\bullet}:=R\check{\pi}_{*}\left(\mathcal{P}_{n}\otimes\hat{\pi}^{*}F^{\bullet}\right)\end{array}$

and

$\begin{array}[]{cccc}R\hat{\mathscr{F}}_{n}:&D^{b}_{B^{\prime}_{n}}(\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n})^{\vee})\cong
D^{b}_{B^{\prime}_{n}}(\mathrm{M}^{\prime}_{n})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&D^{b}_{B^{\prime}_{n}}(\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n}))\cong
D^{b}_{B^{\prime}_{n}}(\mathrm{M}^{\prime}_{n})\\\
&G^{\bullet}&\longmapsto&\hat{G}^{\bullet}:=R\hat{\pi}_{*}\left(\mathcal{P}_{n}^{*}\otimes\check{\pi}^{*}G\right).\end{array}$

This is an equivalence of categories since

(2.8) $R\hat{\mathscr{F}}_{n}\circ
R\check{\mathscr{F}}_{n}=[d_{n}]\circ\left(\mathrm{1}_{\operatorname{Jac}}^{-1}\right)^{*},$

where $\mathrm{1}_{\operatorname{Jac}}^{-1}$ denotes the involution given by
inverting elements on each $\operatorname{Jac}^{\delta_{n}}(S_{n,b})$ under
the group structure, and $d_{n}$ is defined in (2.3).

We now recall the work of Arinkin and Polishchuk [AP] which describes mirror
symmetry for Lagrangian tori fibrations in terms of a Fourier–Mukai—Nahm
transform for bundles with connections. This approach can be applied to the
smooth locus of the Hitchin system as the fibres of
$\mathrm{M}^{\prime}_{n}\cong\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\to
B^{\prime}_{n}$ are Lagrangian with respect to $\Lambda^{1}_{n}$. Some useful
properties of the Fourier–Mukai–Nahm transform can be found in [GJK].

Corresponding to the definition of the Poincaré bundle $\mathcal{P}_{n}$,
there is a canonical connection on it, the Poincaré connection
$\nabla_{\mathcal{P}_{n}}$. Its curvature is given by

(2.9)
$\nabla_{\mathcal{P}_{n}}^{2}=2\pi\,\mathrm{i}\,\cdot\mathrm{1}_{\mathcal{P}_{n}}\otimes\rho_{n},$

where $\rho_{n}$ is fibrewise the standard holomorphic $2$-form on
$T_{L}\operatorname{Jac}(S_{n,b})\times
T_{\check{L}}\operatorname{Jac}(S_{n,b})^{\vee}$ which is canonically
isomorphic to $H^{1}(S_{n,b},\mathcal{O}_{S_{n,b}})\times
H^{0}(S_{n,b},K_{S_{n,b}})$. We observe that,
$T_{L}\operatorname{Jac}(S_{n,b})\times
T_{\check{L}}\operatorname{Jac}(S_{n,b})^{\vee}$ can be identified with the
tangent space of $\mathrm{M}^{\prime}_{n}$, and after this identification,
$\rho_{n}$ coincides with $\Lambda_{n}^{1}$, the standard holomorphic
symplectic form associated to the Dolbeault complex structure $\Gamma^{1}$.
Analogously, we can define the dual Poincaré connection
$\nabla_{\mathcal{P}_{n}^{*}}$, whose curvature is
$\nabla_{\mathcal{P}_{n}^{*}}^{2}=-2\pi\,\mathrm{i}\,\cdot\mathrm{1}_{\mathcal{P}_{n}^{*}}\otimes\rho_{n}$.

Consider a multisection $\Xi$ of the fibration
$\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}^{\prime}_{n})\to
B^{\prime}_{n}$ and the associated projections

(2.10)
$\textstyle{\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\times_{B^{\prime}_{n}}\Xi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}_{\Xi}}$$\scriptstyle{\check{\pi}_{\Xi}}$$\textstyle{\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})}$$\textstyle{\Xi}$

Given a vector bundle $V$ over $\Xi$, endowed with a connection $\nabla_{V}$,
Arinkin and Polishchuk [AP] considered its Fourier–Mukai–Nahm transform,

$\hat{\mathscr{N}}_{n}\left(\Xi,V,\nabla_{V}\right):=\left(\widehat{V},\nabla_{\widehat{V}}\right),$

where

$\widehat{V}:=\hat{\pi}_{\Xi,*}\left(\check{\pi}_{\Xi,*}\mathcal{P}^{*}_{n}\otimes\hat{\pi}_{\Xi,*}V\right)$

is the Fourier–Mukai transform, and

$\nabla_{\widehat{V}}:=\hat{\pi}_{\Xi,*}\left(\nabla_{V}\otimes\mathrm{1}_{\mathcal{P}^{*}_{n}}+\mathrm{1}_{V}\otimes\nabla_{\mathcal{P}^{*}_{n}}\right).$

Observe that, so far, we have only used the $C^{\infty}$-structure of these
Lagrangian fibrations. In fact, we could have worked with $M=T^{*}B/H$, where
$B$ is a (real) vector space and $H$ a relative lattice, with dual fibration
$M^{\vee}=TB/H^{\vee}$. Denote by $\omega$ the symplectic form defined on $M$
(coming from the canonical symplectic form in $T^{*}B$) and consider a
complexified symplectic form $\Lambda=\omega+\,\mathrm{i}\,\xi$, where $\xi$
is a covariantly constant $2$-form. Using $\omega$ and $\Lambda$ one can
define naturally a complex structure $\Gamma$ on $M$. The power of the
construction of Arinkin and Polishchuk arises in the following theorem, which
ensures, under certain circumstances, the existence of a integrable complex
structure on $\widehat{V}$ with respect to $\Gamma$.

######  Theorem 2.2 (Theorem 1.1 of [AP], see also Theorem 6.3 of [Po]).

The $(0,1)$-component of $\nabla_{\widehat{V}}$ with respect to $\Gamma$ is
flat (hence $\widehat{V}$-is holomorphic with respect to $\Gamma$) if and only
if

$\nabla_{V}^{2}:=2\pi\Lambda|_{\Xi}.$

In the Hitchin system we deal with a complexified version of the previous
situation. Observe that $\widehat{V}$ is naturally endowed with a
$\Gamma^{1}$-complex structure, and recall that
$\Lambda^{1}_{n}=\omega_{n}^{2}+\,\mathrm{i}\,\omega_{n}^{3}$. Then, a complex
Lagrangian subvariety with respect to $\Lambda_{n}^{1}$ implies that it is
Lagrangian with respect to $\omega_{n}^{2}$ and $\omega_{n}^{3}$.

######  Corollary 2.3.

The connection $\nabla_{\widehat{V}}$ defines a hyperholomorphic structure on
$\widehat{V}$ if and only if

$\nabla_{V}^{2}:=2\pi\Lambda|_{\Xi}.$

In particular when $\nabla_{V}$ is flat and $\Xi$ complex Lagrangian with
respect to $\Lambda_{n}^{1}$.

In [GJK] one can find a detailed study of the Fourier–Mukai–Nahm transform
$\hat{\mathscr{N}}_{n}$, in particular when this construction has an inverse
$\check{\mathscr{N}}_{n}$. With respect to the torus fibration
$\mathrm{M}^{\prime}_{n}\cong\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\stackrel{{\scriptstyle\hat{h}}}{{\longrightarrow}}B^{\prime}_{n}$,
one can associate an exact sequence of $1$-forms

$0\longrightarrow\hat{h}^{*}\Omega^{1}_{B^{\prime}_{n}}\longrightarrow\Omega^{1}_{\mathrm{M}^{\prime}_{n}}\longrightarrow\Omega^{1}_{\mathrm{M}^{\prime}_{n}/B^{\prime}_{n}}\longrightarrow
0,$

which splits thanks to the Gauss-Manin connection, giving

(2.11)
$\Omega_{\mathrm{M}^{\prime}_{n}}^{1}\cong\hat{h}^{*}\Omega_{B^{\prime}_{n}}^{1}\oplus\Omega_{\mathrm{M}^{\prime}_{n}/B^{\prime}_{n}}^{1}.$

Following [GJK], we introduce a bigrading on the De Rham algebra
$\Omega_{\mathrm{M}^{\prime}_{n}}^{\bullet}$,

(2.12) $\Omega_{\mathrm{M}^{\prime}_{n}}^{\langle
u,v\rangle}=\Omega_{\mathrm{M}^{\prime}_{n}}^{\langle
u,0\rangle}\otimes\Omega_{\mathrm{M}^{\prime}_{n}}^{\langle
0,v\rangle}=\hat{h}^{*}\Omega_{B^{\prime}_{n}}^{u}\otimes\Omega_{\mathrm{M}^{\prime}_{n}/B^{\prime}_{n}}^{v},$

where the index $u$ is called the basic degree and $v$ is the fibre degree.
Then, the connection $\nabla_{W}$ is adapted to the fibration if the $\langle
0,2\rangle$-component of its curvature is zero,

(2.13) $(\nabla_{W}^{2})^{\langle 0,2\rangle}=0.$

Given a Hermitian vector bundle $W$ over
$\mathrm{M}^{\prime}_{n}\cong\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})$
endowed with a connection $\nabla_{W}$, we denote

$\widetilde{\nabla}_{W}:=\nabla_{W}\otimes\mathrm{1}_{\mathcal{P}_{n}}+\mathrm{1}_{W}\otimes\nabla_{\mathcal{P}_{n}}.$

When the connection $\nabla_{W}$ is adapted, [GJK] provides the transform

$\check{\mathscr{N}}_{n}(W,\nabla_{W}):=\left(\widecheck{W},\nabla_{\widecheck{W}}\right),$

given by

(2.14)
$\widecheck{W}=R^{d_{n}}\check{\pi}_{*}\left(\ker\widetilde{\nabla}_{W}^{\langle
0,1\rangle}\right)$

and

$\nabla_{\widecheck{W}}=R^{d_{n}}\check{\pi}_{\Xi,*}\left(\widetilde{\nabla}_{W}^{\langle
1,0\rangle}|_{\ker\widetilde{\nabla}_{W}^{\langle 0,1\rangle}}\right).$

They also proved that its support, $\Xi:=\mathrm{supp}(\widecheck{W})$, is a
multisection of the dual fibration
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}$.

We can see that (2.14) coincides with the Fourier–Mukai transform.

######  Lemma 2.4.

Consider the fibration
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\to
B^{\prime}_{n}$ and its dual, and consider a Hermitian vector bundle $W$ with
an adapted connection $\nabla_{W}$. Recall that
$d_{n}=\dim(\operatorname{Jac}(S_{n,b}))$. Then, for $0\geq\ell\geq d_{n}-1$

$R^{\ell}\check{\mathscr{F}}_{n}(W)=0,$

while

$R^{d_{n}}\check{\mathscr{F}}_{n}(W)\cong\widecheck{W}$

is the sheaf constructed in (2.14).

###### Proof.

Note that $W$ equipped with an adapted connection $\nabla_{W}$ can be
understood as a family of vector bundles with flat connection over the fibres.
Then, the lemma follows easily from and in [GJK, (3.19)] and [BMP, Proposition
2.5]. ∎

Following again [GJK], an adapted connection $\nabla_{W}$ on
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})$
is Poincaré basic if the $\langle 1,1\rangle$-part of the curvature of
$\widetilde{\nabla}_{W}$ vanishes, equivalently, if

(2.15) $\left.\hat{\pi}_{\Xi}^{*}(\nabla_{W})^{\langle
1,1\rangle}\right|_{\ker\widetilde{\nabla}_{W}^{\langle
0,1\rangle}}=\left.\nabla_{\mathcal{P}_{n}}^{2}\right|_{\mathrm{M}^{\prime}_{n}\times_{B^{\prime}_{n}}\Xi},$

where we observe that $\mathrm{M}^{\prime}_{n}\times_{B^{\prime}_{n}}\Xi$ is
precisely the support of $\ker\widetilde{\nabla}_{W}^{\langle 0,1\rangle}$.

Let us denote by $\mathrm{Vect}^{\nabla}$ the category of pairs
$(W,\nabla_{W})$, where $W$ is a vector bundle on
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})$
endowed with an adapted Poncaré basic connection $\nabla_{W}$. Also, denote
$\mathrm{SpcDat}$ the category of spectral data, i.e. triples
$(\Xi,V,\nabla_{V})$ where $(\Xi,V)$ is a relative skyscraper sheaf on the
dual fibration
$\operatorname{Jac}^{\delta_{n}}_{B^{\prime}_{n}}(\mathcal{S}_{n})^{\vee}$,
and $\nabla_{V}$ is a connection on $V$.

######  Theorem 2.5 (Corollary 4.18 of [GJK]).

The Fourier–Mukai–Nahm transforms $\hat{\mathscr{N}}_{n}$ and
$\check{\mathscr{N}}_{n}$ provide an equivalence of categories between
$\mathrm{Vect}^{\nabla}$ and $\mathrm{SpcDat}$.

We refer to this functor as the Fourier–Mukai–Nahm transform. The curvature of
$\nabla_{\widecheck{W}}$ can be computed in terms of the basic component of
the starting curvature.

######  Lemma 2.6 (Lemma 4.6 of [GJK]).

The curvature of $\nabla_{\widecheck{W}}$ is

$\nabla_{\widecheck{W}}^{2}=R^{d}\check{\pi}_{\Xi,*}\left(\hat{\pi}_{\Xi}^{*}(\nabla_{W}^{2})^{\langle
2,0\rangle}|_{\ker\widetilde{\nabla}_{W}^{\langle 0,1\rangle}}\right).$

After Lemma 2.6, we say that $(W,\nabla_{W})$ is transversely flat, if

$\hat{\pi}_{\Xi}^{*}(\nabla_{W}^{2})^{\langle
2,0\rangle}|_{\ker\widetilde{\nabla}_{W}^{\langle 0,1\rangle}}=0.$

##  3\. The Dirac–Higgs bundle and connection

Let us fix a metric $\zeta$ on the rank $n$ topologically trivial
$C^{\infty}$-bundle $\mathbb{E}_{n}$. Given a Higgs vector bundle
$\mathcal{E}=(E,\varphi)$ supported on $\mathbb{E}_{n}$, we write
$\overline{\partial}_{E}$ for the associated Dolbeault operator and
$\partial_{E}$ for the $(1,0)$-part of the Chern connection constructed with
$\zeta$ and $\overline{\partial}_{E}$. Hitchin introduced in [H3] the
following _Dirac–Higgs operator_

(3.1)
$\begin{array}[]{cccc}\mathscr{D}_{\mathcal{E}}:&\Omega_{X}^{0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0}(\mathbb{E}_{n})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\Omega_{X}^{1,0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n})\\\
&(f_{1},f_{2})&\longmapsto&(-\partial_{E}f_{1}+\varphi
f_{2},-\varphi^{*}f_{1}-\overline{\partial}_{E}f_{2}),\end{array}$

with adjoint

(3.2)
$\begin{array}[]{cccc}\mathscr{D}_{\mathcal{E}}^{*}:&\Omega_{X}^{1,0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\Omega_{X}^{0}(\mathbb{E}_{n})\oplus\Omega_{X}^{1,1}(\mathbb{E}_{n})\\\
&(\psi_{1},\psi_{2})&\longmapsto&\varphi^{*}\psi_{1}+\overline{\partial}^{*}_{E}\psi_{2},-\overline{\partial}_{E}\psi_{1}-\varphi\psi_{2}).\end{array}$

Let $\mathbb{H}^{p}(\mathcal{E})$ denote the hypercohomology groups of the
complex of sheaves $E\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}E\otimes
K_{X}$. The key fact about such operators is that

$\ker\mathscr{D}_{\mathcal{E}}\simeq\mathbb{H}^{0}(\mathcal{E})\oplus\mathbb{H}^{2}(\mathcal{E})~{}~{}{\rm
and}~{}~{}\ker\mathscr{D}_{\mathcal{E}}^{*}\simeq\mathbb{H}^{1}(\mathcal{E}),$

see [H3, Section 7]. Hausel proved in [Ha, Corollary 5.1.4] that if
$\mathcal{E}$ is nontrivial stable Higgs bundle of degree 0, then
$\mathbb{H}^{0}(\mathcal{E})=\mathbb{H}^{2}(\mathcal{E})=0$, so that
$\ker\mathscr{D}_{\mathcal{E}}=0$.

If $\mathcal{E}$ be a semistable Higgs bundle of degree 0, let ${\rm
gr}(\mathcal{E})=\oplus_{j}\mathcal{E}_{j}$ be the associated graded object,
where $(\mathcal{E}_{1},\dots,\mathcal{E}_{l})$ are the stable factors of its
Jordan–Hölder filtration; we say that $\mathcal{E}$ is _without trivial
factors_ (or w.t.f., for short) if none of these factors $\mathcal{E}_{j}$ is
trivial.

######  Lemma 3.1.

Let $\mathcal{E}$ is a semistable Higgs bundle of degree 0.
$\ker\mathscr{D}_{\mathcal{E}}=0$ if and only if $\mathcal{E}$ is without
trivial factors. In addition, if $\mathcal{E}$ is without trivial factors,
then

$\ker\mathscr{D}_{\mathcal{E}}^{*}=\mathbb{H}^{1}({\rm
gr}(\mathcal{E}))\simeq\oplus_{j}\mathbb{H}^{1}(\mathcal{E}_{j}),$

where $\mathcal{E}_{j}$ are the stable factors of the Jordan–Hölder filtration
of $\mathcal{E}$.

###### Proof.

First, assume that $\mathcal{E}$ is a semistable Higgs bundle of degree 0
without trivial factors; in order to prove that
$\ker\mathscr{D}_{\mathcal{E}}=0$, it is enough to check the case $l=2$ and
use induction on the length $l$ of Jordan–Hölder filtration. So assume that
$\mathcal{E}$ fits into a short exact sequence (of Higgs bundles) of the
following form:

$0\to\mathcal{E}_{1}\to\mathcal{E}\to\mathcal{E}_{2}\to 0.$

Using the induced long exact sequence in hypercohomology, it follows that
$\mathbb{H}^{0}(\mathcal{E})=\mathbb{H}^{2}(\mathcal{E})=0$ since
$\mathbb{H}^{0}(\mathcal{E}_{j})=\mathbb{H}^{2}(\mathcal{E}_{j})=0$ for
$j=1,2$ by Hausel’s corollary. In addition, we also have that
$\mathbb{H}^{1}(\mathcal{E})\simeq\mathbb{H}^{1}(\mathcal{E}_{1})\oplus\mathbb{H}^{1}(\mathcal{E}_{2})$,
as desired.

Conversely, assume that $\mathcal{E}$ has a trivial factor in its
Jordan–Hölder filtration. It follows that there exists a monomorphism
$(\mathcal{O}_{X},0)\hookrightarrow\mathcal{E}$, implying that
$\mathbb{H}^{0}(\mathcal{E})\neq 0$, thus also
$\ker\mathscr{D}_{\mathcal{E}}\neq 0$. ∎

Since the index of $\mathscr{D}_{\mathcal{E}}$ is $-2n(g-1)$, see [Bl, Lemma
2.1.8] or [FJ, page 1226], we conclude that

(3.3) $\dim\ker\mathscr{D}_{\mathcal{E}}^{*}=2n(g-1)$

whenever $\mathcal{E}$ is a semistable Higgs bundle of degree 0 of rank $n$
without trivial factors. Furthermore, the vector space
$\ker\mathscr{D}_{\mathcal{E}}^{*}$ does not depend upon the choice of a
representative within the S-equivalence class $[\mathcal{E}]$ of
$\mathcal{E}$.

We denote by $\mathrm{M}_{n}^{\mathrm{wtf}}$ the subset of $\mathrm{M}_{n}$
consisting of the S-equivalence classes of semistable Higgs bundle without
trivial factors.

Consider now the trivial vector bundle (of infinity rank)

$\underline{\Omega}_{n}:=\left(\Omega_{X}^{1,0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n})\right)\times\mathrm{M}_{n}^{\mathrm{wtf}}.$

Since $\Omega_{X}^{1,0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n})$
has a natural metric on it, we see that
$\underline{\Omega}_{n}\to\mathrm{M}_{n}^{\mathrm{wtf}}$ comes equipped with a
metric.

One would like to define an action of $\mathscr{D}_{\mathcal{E}}^{*}$ on the
corresponding fibre of $\underline{\Omega}_{n}$ to define an index bundle. To
do so we need to provide, for each point of $\mathrm{M}_{n}^{\mathrm{wtf}}$
represented by an S-equivalence class $[\mathcal{E}]$ of Higgs bundles, a
polystable representative in this S-equivalence class. This amounts to give a
universal Higgs bundle over our moduli space and we have already seen that
such universal bundle only exists for $n=1$. For general rank, the best we can
obtain is the virtual universal bundle
$(\mathbf{U}_{n},\boldsymbol{\Phi}_{n})=\\{(U_{n,i},\Phi_{n,i})\to X\times
Z_{n,i})\\}$ as in Section 2.3, where $\\{Z_{n,i}\\}_{i\in I}$ is a covering
of $\mathrm{M}_{n}$ which is fine enough for the gerbe $\beta_{n}$. We can now
define locally in $Z_{n,i}\cap\mathrm{M}_{n}^{\mathrm{wtf}}$, the family of
Dirac-type operators $\mathscr{D}_{(U_{n,i},\Phi_{n,i})}^{*}\to Z_{n,i}$ given
by
$\left.\mathscr{D}_{(U_{n,i},\Phi_{n,i})}^{*}\right|_{v}=\mathscr{D}_{(U_{n,i},\Phi_{n,i})|_{v}}^{*}$.
Let us denote, for every $Z_{n,i}$ restricted to
$\mathrm{M}_{n}^{\mathrm{wtf}}$, the rank $2n(g-1)$ holomorphic bundle

(3.4) $D_{n,i}:=\ker\mathscr{D}_{(U_{n,i},\Phi_{n,i})}^{*}\longrightarrow
Z_{n,i}\cap\mathrm{M}_{n}^{\mathrm{wtf}},$

and consider

(3.5) $\mathbf{D}_{n}:=\left\\{D_{n,i}\to
Z_{n,i}\cap\mathrm{M}_{n}^{\mathrm{wtf}}\right\\}_{i\in I},$

which is a virtual bundle for the gerbe $\beta_{n}$ over
$\mathrm{M}^{\mathrm{wtf}}_{n}$, called the _Dirac–Higgs bundle_.

Thanks to the work of Hausel [Ha], one can describe the Dirac–Higgs bundle in
terms of the universal bundle.

######  Proposition 3.2.

Consider the obvious projection
$\pi_{\mathrm{M}}:X\times\mathrm{M}_{n}^{\mathrm{wtf}}\to\mathrm{M}_{n}^{\mathrm{wtf}}$.
The Dirac–Higgs bundle is

(3.6)
$\mathbf{D}_{n}\cong\mathbb{R}^{1}\pi_{\mathrm{M},*}\left(\mathbf{U}_{n}\stackrel{{\scriptstyle\boldsymbol{\Phi}_{n}}}{{\longrightarrow}}\mathbf{U}_{n}\otimes\pi_{X}^{*}K_{X}\right).$

###### Proof.

Considering each local universal bundle $(U_{n,i},\Phi_{n,i})$ over the open
subset $Z_{n,i}\cap\mathrm{M}_{n}^{\mathrm{wtf}}$ of $\mathrm{M}_{n}$, note
that

$\mathbb{R}^{0}\pi_{\mathrm{M},*}\left(U_{n,i}\stackrel{{\scriptstyle\Phi_{n,i}}}{{\longrightarrow}}U_{n,i}\otimes\pi_{X}^{*}K_{X}\right)=\mathbb{R}^{2}\pi_{\mathrm{M},*}\left(U_{n,i}\stackrel{{\scriptstyle\Phi_{n,i}}}{{\longrightarrow}}U_{n,i}\otimes\pi_{X}^{*}K_{X}\right)=0$

since, as observed above,
$\mathbb{H}^{0}(\mathcal{E})=\mathbb{H}^{2}(\mathcal{E})=0$ for each
$\mathcal{E}\in\mathrm{M}_{n}^{\mathrm{wtf}}$. It follows from Lemma 3.1 that

$D_{n,i}\cong\mathbb{R}^{1}\pi_{\mathrm{M},*}\left(U_{n,i}\stackrel{{\scriptstyle\Phi_{n,i}}}{{\longrightarrow}}U_{n,i}\otimes\pi_{X}^{*}K_{X}\right).$

Using our gerbe $\beta$, it follows that the virtual Dirac–Higgs bundle is
(3.6) and the proof is completed. ∎

Define

$\Xi^{0}_{n}:=\mathcal{S}_{n}\cap\left(\left(X\times\\{0\\}\right)\times
B_{n}\right).$

Note that for each $b=(b_{1},\dots,b_{n})\in B^{\prime}_{n}$ one has that
$S_{n,b}\cap X\times\\{0\\}$ is the locus where $b_{n}=0$, being $b_{n}\in
H^{0}(K_{X}^{\otimes n})$. Recall that $\deg{K_{X}^{\otimes n}}=2n(g-1)$. Then
$\Xi^{0}_{n}$ is a finite cover over $B_{n}$ of degree $2n(g-1)$,

(3.7)
$\textstyle{\Xi^{0}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2n(g-1)\,:\,1}$$\textstyle{B_{n}.}$

Recall that the gerbe $\beta_{n}$ trivializes over $\mathrm{M}^{\prime}_{n}$.
Using (3.6) we give a description of
$D^{\prime}_{n}:=\mathbf{D}_{n}|_{\mathrm{M}^{\prime}_{n}}$, which generalizes
the fibrewise description given in [H4, Section 7]. Recall the embedding
$\check{\imath}_{n}:\mathcal{S}^{\prime}_{n}\hookrightarrow\mathrm{M}^{\prime}_{n}$
defined at the end of Section 2.3.

######  Proposition 3.3.

Take the sheaf $\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}}$ on
$\mathcal{S}_{n}$ and consider its restriction to $\mathcal{S}^{\prime}_{n}$.
By abuse of notation, we denote its pushforward under $\check{\imath}_{n}$ by
$\check{\imath}_{n,*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})$,
which is a sheaf on
$\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}_{n})^{\vee}$.
Then,

$D^{\prime}_{n}\cong
R\check{\mathscr{F}}_{n}\left(\check{\imath}_{n,*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})\right).$

###### Proof.

Recall from Section 2.3 that we denote by $U^{\prime}_{n}$ the restriction of
$\mathbf{U}_{n}$ to $\mathrm{M}^{\prime}_{n}$, and by $P_{n}$ the universal
line bundle on $\mathcal{S}^{\prime}_{n}$. All the maps considered here are
the restriction to the corresponding objects given by restricting $B_{n}$ to
$B^{\prime}_{n}$.

Starting from (3.6) and the isomorphism (2.4), note that

$D^{\prime}_{n}\cong\mathbb{R}^{1}\pi_{\mathrm{M},*}\left(U^{\prime}_{n}\stackrel{{\scriptstyle\Phi_{n}}}{{\longrightarrow}}U^{\prime}_{n}\otimes\pi_{X}^{*}K_{X}\right)\cong\mathbb{R}^{1}\pi_{\operatorname{Jac},*}\left(U^{\prime}_{n}\stackrel{{\scriptstyle\Phi_{n}}}{{\longrightarrow}}U^{\prime}_{n}\otimes\pi_{X}^{*}K_{X}\right).$

Next, recalling the relation between the universal bundle and the Poincaré
bundle described in Section 2.3, and making use of the projection formula and
base change theorems for the various morphisms in the diagram (2.7), we obtain

$\displaystyle D^{\prime}_{n}$
$\displaystyle\cong\mathbb{R}^{1}\pi_{\operatorname{Jac},*}\left((\mathrm{1}_{\operatorname{Jac}}\times
p_{n})_{*}P_{n}\stackrel{{\scriptstyle\Phi_{n}}}{{\longrightarrow}}(\mathrm{1}_{\operatorname{Jac}}\times
p_{n})_{*}P_{n}\otimes\pi_{X}^{*}K_{X}\right)$
$\displaystyle\cong\mathbb{R}^{1}\pi_{\operatorname{Jac},*}\left((\mathrm{1}_{\operatorname{Jac}}\times
p_{n})_{*}\left(P_{n}\stackrel{{\scriptstyle(\mathrm{1}_{\operatorname{Jac}}\times\lambda_{n})}}{{\longrightarrow}}P_{n}\otimes(\mathrm{1}_{\operatorname{Jac}}\times
p_{n})^{*}\pi_{X}^{*}K_{X}\right)\right)$
$\displaystyle\cong\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\stackrel{{\scriptstyle(\mathrm{1}_{\operatorname{Jac}}\times\lambda_{n})}}{{\longrightarrow}}P_{n}\otimes\widetilde{\pi}_{S}^{*}\widetilde{p}_{n}^{*}K_{X}\right)$
$\displaystyle\cong\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\mathcal{O}_{\mathcal{S}^{\prime}_{n}}\stackrel{{\scriptstyle\lambda_{n}}}{{\longrightarrow}}\widetilde{p}_{n}^{*}K_{X})\right)$
$\displaystyle\cong
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{n}^{*}K_{X}|_{\mathcal{S}^{\prime}_{n}\cap\\{\lambda_{n}=0\\}}\right)\right),\textrm{by
Lemma \ref{RR1=R0}},$ $\displaystyle\cong
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})\right).$

Recall that, by the definition of $\check{\imath}_{n}$, one has that the
restriction of the Poincaré bundle
$\mathcal{P}^{*}_{n}\to\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})\times_{B^{\prime}_{n}}\operatorname{Jac}_{B^{\prime}_{n}}^{\delta_{n}}(\mathcal{S}^{\prime}_{n})^{\vee}$
to the image of the embedding
$\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{n}$, coincides with
$P_{n}$. Using this, the projection formula, and base change theorems on the
diagram (2.7), we have that

$\displaystyle D^{\prime}_{n}\cong$ $\displaystyle
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{n}\otimes\widetilde{\pi}_{S}^{*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})\right)$
$\displaystyle\cong$ $\displaystyle
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left((\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{n})^{*}\mathcal{P}^{*}_{n}\otimes\widetilde{\pi}_{S_{n}}^{*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})\right)$
$\displaystyle\cong$ $\displaystyle
R^{0}\hat{\pi}_{*}R^{0}\left(\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{n})_{*}((\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{n})^{*}\mathcal{P}^{*}_{n}\otimes\widetilde{\pi}_{S}^{*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})\right)$
$\displaystyle\cong$ $\displaystyle
R^{0}\hat{\pi}_{*}\left(\mathcal{P}^{*}_{n}\otimes
R^{0}(\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{n})_{*}\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}}\right)\right)$
$\displaystyle\cong$ $\displaystyle
R^{0}\hat{\pi}_{*}\left(\mathcal{P}^{*}_{n}\otimes\check{\pi}^{*}R^{0}\check{\imath}_{n,*}\left(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}}\right)\right)$
$\displaystyle\cong$ $\displaystyle
R^{0}\check{\mathscr{F}}_{n}\left(R^{0}\check{\imath}_{n,*}\left(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}}\right)\right).$

Since $\check{\imath}_{n}$ is an embedding, one has that
$R^{0}\check{\imath}_{n,*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})$
coincides with
$\check{\imath}_{n,*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})$.
The support of $(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})$
is $\Xi^{0}_{n}$, which is a finite $2n(g-1)$-cover of $B^{\prime}_{n}$, and
so is $\check{\imath}_{n}(\Xi^{0}_{n})$ which is the support of
$\check{\imath}_{n,*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}})$.
Therefore,
$R\check{\mathscr{F}}_{n}(\check{\imath}_{n,*}(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}}))$
is a complex supported in degree $0$. ∎

After Proposition 3.3 and equation (2.8), it is possible to study the
Fourier–Mukai transform of $D^{\prime}_{n}$.

######  Corollary 3.4.

We have that $R\hat{\mathscr{F}}_{n}\left(D^{\prime}_{n}\right)$ is a complex
entirely supported in degree $d_{n}=1+n^{2}(g-1)$, and the sheaf
$R^{d_{n}}\hat{\mathscr{F}}_{n}\left(D^{\prime}_{n}\right)$ is

$\check{\mathfrak{D}}^{\prime}_{n}:=R^{d_{n}}\hat{\mathscr{F}}_{n}\left(D^{\prime}_{n}\right)\cong\check{\imath}_{n,*}\left(\widetilde{p}_{n}^{*}K_{X}\otimes\mathcal{O}_{\Xi^{0}_{n}}\right),$

with

$\mathrm{supp}\left(\check{\mathfrak{D}}^{\prime}_{n}\right)=\check{\imath}_{n}\left(\Xi^{0}_{n}\right).$

One can also define naturally a connection on $\mathbf{D}_{n}$. Consider the
trivial connection $\underline{d}$ on the trivial bundle
$\underline{\Omega}_{n}$, and consider the embedding
$j_{n}:\mathbf{D}_{n}\hookrightarrow\underline{\Omega}_{n}$ and the projection
$\mathrm{pr}:\underline{\Omega}_{n}\to\mathbf{D}_{n}$ defined by the natural
metric on $\underline{\Omega}_{n}$. We define the Dirac–Higgs connection to be
the connection given by the composition

(3.8) $\boldsymbol{\nabla}_{n}=\mathrm{pr}_{n}\circ\underline{d}\circ j_{n}.$

The second part of the following statement can be found in Blavaand [Bl,
Theorem 2.6.3] (see also [FJ, Proposition 11]) although the description for
the curvature.

######  Proposition 3.5.

The curvature of the Dirac–Higgs connection is proportional to the gauge-
invariant $2$-form

(3.9)
$\Theta\left(\left(\dot{\alpha}_{1},\dot{\varphi}_{1}\right),\left(\dot{\alpha}_{2},\dot{\varphi}_{2}\right)\right):=\int_{X}\mathrm{tr}\left(\dot{\alpha}_{1}\wedge\dot{\alpha}_{2}^{*}-\dot{\alpha}_{2}\wedge\dot{\alpha}_{1}^{*}-\dot{\varphi}_{1}\wedge\dot{\varphi}_{2}^{*}+\dot{\varphi}_{2}\wedge\dot{\varphi}_{1}^{*}\right),$

where

$\dot{\alpha}_{i}\in\ker\left(H^{1}(\mathrm{End}(E))\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}H^{1}(\mathrm{End}(E)\otimes
K_{X})\right)$

and

$\dot{\varphi}_{i}\in\mathrm{coker}\left(H^{0}(\mathrm{End}(E))\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}H^{0}(\mathrm{End}(E)\otimes
K_{X})\right).$

In particular, it follows that the Dirac–Higgs connection is of type $(1,1)$
with respect to all complex structures on $\mathrm{M}_{n}$, hence the
Dirac–Higgs bundle $\mathbf{D}_{n}$ admits a hyperholomorphic structure, cf.
[Bl, Theorem 2.6.3] and [FJ, Proposition 11]). The new content of the previous
proposition is the explicit formula for the curvature of the Dirac–Higgs
connection, which will play an crucial role later on.

###### Proof.

Recall the definition of the Dirac–Higgs bundle given in (3.4). For a certain
point $\mathcal{E}=(E,\varphi)$ of the moduli space $\mathrm{M}_{n}$, the
projection

$\mathrm{pr}_{n}|_{\mathcal{E}}:\underline{\Omega}_{n}|_{\mathcal{E}}\twoheadrightarrow\mathbf{D}_{n}|_{\mathcal{E}}=\ker\mathscr{D}_{\mathcal{E}}^{*}$

can be expressed as

(3.10)
$\mathrm{pr}_{n}|_{\mathcal{E}}=\mathrm{1}_{\underline{\Omega}_{n}}-\mathscr{D}_{\mathcal{E}}\mathscr{G}_{\mathcal{E}}\mathscr{D}_{\mathcal{E}}^{*},$

where $\mathscr{G}_{\mathcal{E}}$ denotes the associated Green operator,
defined to be

$\mathscr{G}_{\mathcal{E}}=\left(\mathscr{D}_{\mathcal{E}}^{*}\mathscr{D}_{\mathcal{E}}\right)^{-1}.$

If
$j_{n}|_{\mathcal{E}}:\mathbf{D}_{n}|_{\mathcal{E}}=\ker\mathscr{D}_{\mathcal{E}}^{*}\hookrightarrow\underline{\Omega}_{n}|_{\mathcal{E}}$
is the natural embedding, recalling the definition of the Dirac–Higgs
connection from (3.8), we can write the curvature at $\mathcal{E}$ as

$\boldsymbol{\nabla}_{n,\mathcal{E}}^{2}=(\mathrm{pr}_{n}|_{\mathcal{E}})\,\underline{d}\,(j_{n}|_{\mathcal{E}})\,(\mathrm{pr}_{n}|_{\mathcal{E}})\,\underline{d}\,(j_{n}|_{\mathcal{E}}).$

Using (3.10), we have for every local section $\sigma$ of $\mathbf{D}_{n}$

$\displaystyle\boldsymbol{\nabla}_{n,\mathcal{E}}^{2}(\sigma)$
$\displaystyle=(\mathrm{pr}_{n}|_{\mathcal{E}})\,\underline{d}\,(j_{n}|_{\mathcal{E}})\left(\mathrm{1}_{\underline{\Omega}_{n}}-\mathscr{D}_{\mathcal{E}}\mathscr{G}_{\mathcal{E}}\mathscr{D}_{\mathcal{E}}^{*}\right)d\sigma$
$\displaystyle=-(\mathrm{pr}_{n}|_{\mathcal{E}})\,\underline{d}\,(j_{n}|_{\mathcal{E}})\left(\mathscr{D}_{\mathcal{E}}\mathscr{G}_{\mathcal{E}}\mathscr{D}_{\mathcal{E}}^{*}\right)d\sigma~{}~{},\textrm{
since }d^{2}\sigma=0;$
$\displaystyle=(\mathrm{pr}_{n}|_{\mathcal{E}})\,\underline{d}\,(j_{n}|_{\mathcal{E}})\,\mathscr{D}_{\mathcal{E}}\mathscr{G}_{\mathcal{E}}(d\mathscr{D}_{\mathcal{E}}^{*})\sigma$

because
$\mathscr{D}_{\mathcal{E}}^{*}(d\sigma)=d(\mathscr{D}_{\mathcal{E}}^{*}\sigma)-(d\mathscr{D}_{\mathcal{E}}^{*})$,
and $d(\mathscr{D}_{\mathcal{E}}^{*}\sigma)=0$ since
$\mathscr{D}_{\mathcal{E}}^{*}\sigma$ is a holomorphic section of
$\mathbf{D}_{n}$. Continuing our chain of equalities, we have

$\displaystyle\boldsymbol{\nabla}_{n,\mathcal{E}}^{2}(\sigma)$
$\displaystyle=(\mathrm{pr}_{n}|_{\mathcal{E}})\,d\left(\mathscr{D}_{\mathcal{E}}\mathscr{G}_{\mathcal{E}}(d\mathscr{D}_{\mathcal{E}}^{*})\sigma\right)$
$\displaystyle=(\mathrm{pr}_{n}|_{\mathcal{E}})\left((d\mathscr{D}_{\mathcal{E}})\mathscr{G}_{\mathcal{E}}(d\mathscr{D}_{\mathcal{E}}^{*})\sigma+\mathscr{D}_{\mathcal{E}}d\mathscr{G}_{\mathcal{E}}(d\mathscr{D}_{\mathcal{E}}^{*})\sigma\right)$
$\displaystyle=(\mathrm{pr}_{n}|_{\mathcal{E}})(d\mathscr{D}_{\mathcal{E}})\mathscr{G}_{\mathcal{E}}(d\mathscr{D}_{\mathcal{E}}^{*})\sigma$

since
$d\mathscr{D}_{\mathcal{E}}=(d\mathscr{D}_{\mathcal{E}})+\mathscr{D}_{\mathcal{E}}d$,
and finally because
$(\mathrm{pr}_{n}|_{\mathcal{E}})\mathscr{D}_{\mathcal{E}}=0$, which is a
consequence of

$(\mathrm{pr}_{n}|_{\mathcal{E}})\mathscr{D}_{\mathcal{E}}=\mathscr{D}_{\mathcal{E}}-\mathscr{D}_{\mathcal{E}}\mathscr{G}_{\mathcal{E}}\mathscr{D}_{\mathcal{E}}^{*}\mathscr{D}_{\mathcal{E}}$

and
$\mathscr{G}_{\mathcal{E}}\mathscr{D}_{\mathcal{E}}^{*}\mathscr{D}_{\mathcal{E}}=\mathrm{1}_{\underline{\Omega}_{n}}$.

Summing up, we see that the Dirac–Higgs curvature
$\boldsymbol{\nabla}_{n,\mathcal{E}}^{2}$ is the projection of
$d\mathscr{D}_{\mathcal{E}}\wedge d\mathscr{D}_{\mathcal{E}}^{*}$ in
$\left(\Omega_{X}^{1,0}(\mathbb{E}_{n})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n})\right)\times\mathcal{A}_{n}$,
where $\mathcal{A}_{n}$ is the affine space of Higgs pairs
$(\varphi,\overline{\partial}_{E})$.

Recall (3.1) and (3.2) and express $\mathscr{D}_{\mathcal{E}}$ and
$\mathscr{D}_{\mathcal{E}}^{*}$ in matricial form,

$\mathscr{D}_{\mathcal{E}}=\begin{pmatrix}-\varphi&\partial_{E}\\\
-\overline{\partial}_{E}&-\varphi^{*}\end{pmatrix}:\begin{pmatrix}\Omega_{X}^{0}(\mathbb{E}_{n})\\\
\Omega_{X}^{0}(\mathbb{E}_{n})\end{pmatrix}\longrightarrow\begin{pmatrix}\Omega_{X}^{1,0}(\mathbb{E}_{n})\\\
\Omega_{X}^{0,1}(\mathbb{E}_{n})\end{pmatrix}$

and

$\mathscr{D}_{\mathcal{E}}^{*}=\begin{pmatrix}\varphi^{*}&\overline{\partial}_{E}^{*}\\\
-\overline{\partial}_{E}&-\varphi\end{pmatrix}:\begin{pmatrix}\Omega_{X}^{1,0}(\mathbb{E}_{n})\\\
\Omega_{X}^{0,1}(\mathbb{E}_{n})\end{pmatrix}\longrightarrow\begin{pmatrix}\Omega_{X}^{0}(\mathbb{E}_{n})\\\
\Omega_{X}^{1,1}(\mathbb{E}_{n})\end{pmatrix}$

Recall that the affine space $\mathcal{A}_{n}$ is modeled on the infinite
dimensional vector space
$\Omega_{X}^{1,0}(\mathrm{End}(\mathbb{E}_{n}))\oplus\Omega_{X}^{0,1}(\mathrm{End}(\mathbb{E}_{n}))$,
and one can consider the infinitesimal deformations of $\partial_{E}$ and
$\varphi$,
$\dot{\alpha}^{*},\dot{\varphi}\in\Omega_{X}^{1,0}(\mathrm{End}(\mathbb{E}_{n}))$,
as well as
$\dot{\alpha},\dot{\varphi}^{*}\in\Omega_{X}^{0,1}(\mathrm{End}(\mathbb{E}_{n}))$
associated to $\overline{\partial}_{E}$ and $\varphi^{*}$. Then,

$d\mathscr{D}_{\mathcal{E}}=\begin{pmatrix}0&0\\\
-1&0\end{pmatrix}d\dot{\alpha}+\begin{pmatrix}0&1\\\
0&0\end{pmatrix}d\dot{\alpha}^{*}+\begin{pmatrix}-1&0\\\
0&0\end{pmatrix}d\dot{\varphi}+\begin{pmatrix}0&0\\\
0&-1\end{pmatrix}d\dot{\varphi}^{*}$

and

$d\mathscr{D}_{\mathcal{E}}^{*}=\begin{pmatrix}0&0\\\
-1&0\end{pmatrix}d\dot{\alpha}+\begin{pmatrix}0&-1\\\
0&0\end{pmatrix}d\dot{\alpha}^{*}+\begin{pmatrix}0&0\\\
0&-1\end{pmatrix}d\dot{\varphi}+\begin{pmatrix}1&0\\\
0&0\end{pmatrix}d\dot{\varphi}^{*}$

so

$\displaystyle d\mathscr{D}_{\mathcal{E}}\wedge
d\mathscr{D}_{\mathcal{E}}^{*}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}d\dot{\alpha}\wedge d\dot{\alpha}^{*}-\begin{pmatrix}1&0\\\
0&1\end{pmatrix}d\dot{\varphi}\wedge d\dot{\varphi}^{*}$ $\displaystyle\hskip
56.9055pt+\begin{pmatrix}0&0\\\ 2&0\end{pmatrix}d\dot{\varphi}^{*}\wedge
d\dot{\alpha}+\begin{pmatrix}0&2\\\ 0&0\end{pmatrix}d\dot{\varphi}\wedge
d\dot{\alpha}^{*}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\left(d\dot{\alpha}\wedge
d\dot{\alpha}^{*}-d\dot{\varphi}\wedge d\dot{\varphi}^{*}\right),$

since $d\dot{\varphi}^{*}\wedge d\dot{\alpha}$ and $d\dot{\varphi}\wedge
d\dot{\alpha}^{*}$ belong to $\Omega_{X}^{0,2}(\mathrm{End}(\mathbb{E}_{n}))$
and $\Omega_{X}^{2,0}(\mathrm{End}(\mathbb{E}_{n}))$ respectively, and these
spaces vanish over a curve.

We observe that $d\dot{\alpha}\wedge d\dot{\alpha}^{*}-d\dot{\varphi}\wedge
d\dot{\varphi}^{*}$ projects to (3.9), and this proves the first statement.

We can see that $\Theta$ is of type $(1,1)$ with respect to the complex
structure $\Gamma^{1}_{n}$. To study the type with respect to $\Gamma^{2}_{n}$
we first perform the following change of variables

$\displaystyle\dot{u}:=\frac{1}{2}\left(\dot{\alpha}+\,\mathrm{i}\,\dot{\varphi}^{*}\right),$
$\displaystyle\dot{u}^{*}:=\frac{1}{2}\left(\dot{\alpha}-\,\mathrm{i}\,\dot{\varphi}^{*}\right),$
$\displaystyle\dot{v}:=\frac{1}{2}\left(\dot{\alpha}^{*}+\,\mathrm{i}\,\dot{\varphi}\right),$
$\displaystyle\dot{v}^{*}:=\frac{1}{2}\left(\dot{\alpha}^{*}-\,\mathrm{i}\,\dot{\varphi}\right).$

One can check that $\dot{u}$ and $\dot{v}$ are holomorphic vectors with
respect to $\Gamma^{2}_{n}$, and $\dot{u}^{*}$ and $\dot{v}^{*}$
antiholomorphic. In this base, $\Theta$ can be expressed as follows

(3.11)
$\Theta\left(\left(\dot{u}_{1},\dot{v}_{1}\right),\left(\dot{u}_{2},\dot{v}_{2}\right)\right):=2\int_{X}\mathrm{tr}\left(\dot{u}_{1}^{*}\wedge\dot{v}_{2}-\dot{u}_{2}^{*}\wedge\dot{v}_{1}-\dot{v}_{1}^{*}\wedge\dot{u}_{2}+\dot{v}_{2}^{*}\wedge\dot{u}_{1}\right).$

We also see that $\Theta$ is of type $(1,1)$ with respect to $\Gamma^{2}_{n}$,
and then, it is also of type $(1,1)$ with respect to
$\Gamma^{3}_{n}=\Gamma^{1}_{n}\Gamma^{2}_{n}$. ∎

We conclude this section by showing Dirac–Higgs connection is adapted to the
Hitchin fibration, so that one can apply the Fourier–Mukai transform defined
in Section 2.4.

######  Proposition 3.6.

The Dirac–Higgs connection $\boldsymbol{\nabla}_{n}$ is adapted to the Hitchin
fibration
$\mathrm{M}_{n}\stackrel{{\scriptstyle\hat{h}}}{{\longrightarrow}}B_{n}$.

###### Proof.

Fixing a point of the smooth locus of the Hitchin base $b\in
B^{\prime}_{n}\subset B_{n}$, recall that
$\hat{h}^{-1}(b)\cong\operatorname{Jac}^{\delta_{n}}(S_{n,b})$ and denote by
$f_{b}:\operatorname{Jac}^{\delta_{n}}(S_{n,b})\hookrightarrow\mathrm{M}^{\prime}_{n}$
the corresponding embedding, which we recall that corresponds to the push-
forward under $p_{n,b}:S_{n,b}\to X$. For every $\dot{\beta}\in
H^{1}(S_{n,b},\mathcal{O}_{S_{n,b}})=H^{1}(S_{n,b},L^{*}\otimes L)$, we have
that $df_{b}\dot{\beta}=p_{n,b,*}\dot{\beta}\in H^{1}(X,p_{n,b,*}L^{*}\otimes
p_{n,b,*}L)=H^{1}(X,\mathrm{End}(E))$, which is contained indeed in
$\ker\left(H^{1}(X,\mathrm{End}(E))\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}H^{1}(X,\mathrm{End}(E)\otimes
K_{X})\right)$. Also, one has for every $\dot{\beta}^{*}\in
H^{0}(S_{n,b},K_{S_{n,b}})=H^{0}(S_{n,b},L^{*}\otimes L\otimes K_{S_{n,b}})$,
that
$df_{b}(\dot{\beta}^{*})=p_{n,b,*}(\dot{\beta}^{*})\in\mathrm{coker}\left(H^{0}(X,\mathrm{End}(E))\stackrel{{\scriptstyle[\cdot,\varphi]}}{{\longrightarrow}}H^{0}(X,\mathrm{End}(E)\otimes
K_{X})\right)$. We recall from Proposition 3.5, that the Dirac–Higgs curvature
is proportional to the holomorphic $2$-form $\Theta$, where $\Theta$ is
defined in (3.9). Taking the pull-back of $\Theta$ under $f_{b}$ we obtain,
for every two $\dot{\beta}_{1},\dot{\beta}_{2}\in
H^{1}(S_{n,b},\mathcal{O}_{S_{n,b}})$, that

$\displaystyle f_{b}^{*}\Theta\left(\dot{\beta}_{1},\dot{\beta}_{2}\right)$
$\displaystyle=\int_{X}\mathrm{tr}\left(df_{b}\dot{\beta}_{1}\wedge(df_{b}\dot{\beta}_{2})^{*}-df_{b}\dot{\beta}_{2}\wedge(df_{b}\dot{\beta}_{1})^{*}\right)$
$\displaystyle\qquad\qquad-\int_{X}\mathrm{tr}\left(df_{b}(\dot{\beta}_{1}^{*})\wedge
df_{b}(\dot{\beta}_{2}^{*})^{*}-df_{b}(\dot{\beta}_{2}^{*})\wedge
df_{b}(\dot{\beta}_{1}^{*})^{*}\right)$
$\displaystyle=\int_{X}\mathrm{tr}\left(p_{n,b,*}\dot{\beta}_{1}\wedge(p_{n,b,*}\dot{\beta}_{2})^{*}-p_{n,b,*}\dot{\beta}_{2}\wedge(p_{n,b,*}\dot{\beta}_{1})^{*}\right)$
$\displaystyle\qquad\qquad-\int_{X}\mathrm{tr}\left(p_{n,b,*}\dot{\beta}_{1}^{*}\wedge(p_{n,b,*}\dot{\beta}_{2}^{*})^{*}-p_{n,b,*}\dot{\beta}_{2}^{*}\wedge(p_{n,b,*}\dot{\beta}_{1}^{*})^{*}\right)$
$\displaystyle=\int_{S_{n,b}}\dot{\beta}_{1}\wedge\dot{\beta}_{2}^{*}-\dot{\beta}_{2}\wedge\dot{\beta}_{1}^{*}-\dot{\beta}_{1}^{*}\wedge\dot{\beta}_{2}^{**}+\dot{\beta}_{2}^{*}\wedge\dot{\beta}_{1}^{**}$
$\displaystyle=\int_{S_{n,b}}\dot{\beta}_{1}\wedge\dot{\beta}_{2}^{*}-\dot{\beta}_{2}\wedge\dot{\beta}_{1}^{*}-\dot{\beta}_{1}^{*}\wedge\dot{\beta}_{2}+\dot{\beta}_{2}^{*}\wedge\dot{\beta}_{1}$
$\displaystyle=0.$

So we observe that the Dirac–Higgs curvature vanishes when restricted to the
Hitchin fibres: $\boldsymbol{\nabla}_{n}^{2}|_{\hat{h}^{-1}(b)}=0$. In other
words, the Dirac–Higgs connection is adapted to the Hitchin fibration, cf.
equation (2.13). ∎

##  4\. Tensorization and spectral data

In this section we explore the behaviour of the spectral data under
tensorization, generalizing partial results established in [BS] for Higgs
bundles of rank $2$ and $4$.

Let us introduce in this section the tensorization of two Higgs bundles
$\mathcal{E}=(E,\varphi)$ and $\mathcal{F}=(F,\phi)$,

$\mathcal{E}\otimes\mathcal{F}:=\left(E\otimes
F,\varphi\otimes\mathrm{1}_{F}+\mathrm{1}_{E}\otimes\phi\right).$

It is well known that, if $\mathcal{E}$ and $\mathcal{F}$ are semistable, then
$\mathcal{E}\otimes\mathcal{F}$ is semistable too. Then, fixing some
$\mathcal{E}\in\mathrm{M}_{n}$, one can define a map

(4.1)
$\begin{array}[]{cccc}\tau^{\mathcal{E}}_{m}:&\mathrm{M}_{m}&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathrm{M}_{nm}\\\
&[\mathcal{F}]&\longmapsto&[\mathcal{E}\otimes\mathcal{F}].\end{array}$

######  Remark 4.1.

Note that $\tau^{\mathcal{E}}_{m}$ is hyperholomorphic, meaning that it is a
holomorphic morphism between $(\mathrm{M}_{m},\Gamma_{m}^{i})$ and
$(\mathrm{M}_{nm},\Gamma_{nm}^{i})$ for each of the $i=1$, $2$ or $3$. As it
is defined, $\tau^{\mathcal{E}}_{m}$ is clearly holomorphic for $i=1$. To see
that it is also holomorphic for $i=2$, consider the vector bundle with flat
connection $(E^{\prime},\nabla_{E^{\prime}})$ corresponding to $\mathcal{E}$
under the Hitchin–Kobayashi correspondence, and observe that
$\tau^{\mathcal{E}}_{m}$, in the complex structure $\Gamma_{2}$, sends the
vector bundle with flat connection $(F^{\prime},\nabla_{F^{\prime}})$
corresponding to $\mathcal{F}$, to $(E^{\prime}\otimes
F^{\prime},\nabla_{E^{\prime}}\otimes\mathrm{1}_{F^{\prime}}+\mathrm{1}_{E^{\prime}}\otimes\nabla_{F^{\prime}})$.
If $\tau^{\mathcal{E}}_{m}$ is holomorphic for $i=1$ and $i=2$, it is also
holomorphic for $i=3$ since this complex structure is given by the composition
of the previous two.

It will be necessary for us the understanding of the behaviour of
$\tau^{\mathcal{E}}_{m}$ under the isomorphisms
$\mathrm{M}_{n}\cong\overline{\operatorname{Jac}}_{B_{n}}^{\,\delta_{n}}(\mathcal{S}_{n})$,
which we address in the rest of this section. For the sake of simplicity, we
restrict ourselves to the case of reduced spectral curves.

Consider the map given by the sum along the fibres of the canonical line
bundle

$\sigma:K_{X}\times_{X}K_{X}\longrightarrow K_{X}.$

######  Proposition 4.2.

Denote by $S_{\mathcal{E}}$ and $S_{\mathcal{F}}$ the spectral curves
associated to the stable Higgs bundles $\mathcal{E}=(E,\varphi)$ and
$\mathcal{F}=(F,\phi)$ of trivial degree and rank $n$ and $m$ respectively.
Assume that $S_{\mathcal{E}}$ and $S_{\mathcal{F}}$ are reduced and, in the
case where $n=m$, that $S_{\mathcal{E}}\neq S_{\mathcal{F}}$. Consider the
tensor product $\mathcal{E}\otimes\mathcal{F}=(E\otimes
F,\varphi\otimes\mathrm{1}_{F}+\mathrm{1}_{E}\otimes\phi)$. Then the spectral
curve of $\mathcal{E}\otimes\mathcal{F}$ is

(4.2)
$S_{\mathcal{E}\otimes\mathcal{F}}=\sigma(S_{\mathcal{E}}\times_{X}S_{\mathcal{F}}).$

The curve $S_{\mathcal{E}\otimes\mathcal{F}}$ is reduced and singular.
Furthermore, there exists a partial desingularization
$\eta_{\mathcal{E}\otimes\mathcal{F}}:\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{E}\otimes\mathcal{F}}$ at a divisor of length $(m^{2}n^{2}+mn-
mn^{2}-nm^{2})(g-1)$, such that one has the diagrams

(4.3)
$\textstyle{\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\mathcal{E}}}$$\scriptstyle{m}$$\scriptstyle{q_{\mathcal{F}}}$$\scriptstyle{n}$$\scriptstyle{\widetilde{p}_{\mathcal{E}\otimes\mathcal{F}}}$$\scriptstyle{nm}$$\textstyle{S_{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\mathcal{E}}}$$\scriptstyle{n}$$\textstyle{S_{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\mathcal{F}}}$$\scriptstyle{m}$$\textstyle{X,}$

and

(4.4)
$\textstyle{\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{p}_{\mathcal{E}\otimes\mathcal{F}}}$$\scriptstyle{\eta_{\mathcal{E}\otimes\mathcal{F}}}$$\textstyle{S_{\mathcal{E}\otimes\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\mathcal{E}\otimes\mathcal{F}}}$$\textstyle{X,}$

commute, where $p_{\mathcal{E}\otimes\mathcal{F}}$ and
$\widetilde{p}_{\mathcal{E}\otimes\mathcal{F}}$ are $nm$-covers,
$p_{\mathcal{E}}$ and $q_{\mathcal{F}}$ $n$-covers, and $p_{\mathcal{F}}$ and
$q_{\mathcal{E}}$ are $m$-covers. Note that, by construction, all these maps
are flat morphisms except $\eta_{\mathcal{E}\otimes\mathcal{F}}$.

Finally, denote by $L_{\mathcal{E}}\to S_{\mathcal{E}}$, $L_{\mathcal{F}}\to
S_{\mathcal{F}}$ and $L_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{E}\otimes\mathcal{F}}$, the spectral data for $\mathcal{E}$,
$\mathcal{F}$ and $\mathcal{E}\otimes\mathcal{F}$ respectively. Then

$L_{\mathcal{E}\otimes\mathcal{F}}\cong\eta_{\mathcal{E}\otimes\mathcal{F},*}\left(q_{\mathcal{E}}^{*}L_{\mathcal{E}}\otimes
q_{\mathcal{F}}^{*}L_{\mathcal{F}}\right).$

###### Proof.

We start by constructing $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ and
the maps in (4.3). We have that

(4.5)
$\textstyle{\mathrm{Tot}(p_{\mathcal{E}}^{*}K_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{p}_{\mathcal{E}}}$$\scriptstyle{q_{\mathcal{E}}}$$\textstyle{\mathrm{Tot}(K_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{S_{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\mathcal{E}}}$$\textstyle{X}$

commutes. Furthermore, one has that
$q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}K_{X}\to\mathrm{Tot}(p_{\mathcal{E}}^{*}K_{X})$
has a tautological section $\gamma:\mathrm{Tot}(p_{\mathcal{E}}^{*}K_{X})\to
q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}K_{X}$. Recall that the equation
defining $S_{\mathcal{F}}$ is

(4.6)
$\lambda^{m}+p^{*}b_{1}(\phi)\lambda^{m-1}+\dots+p^{*}b_{m-1}(\phi)\lambda+p^{*}b_{m}(\phi)=0,$

where $b_{i}(\phi)\in H^{0}(X,K_{X}^{\otimes i})$. Then, one has that the
vanishing of

$\displaystyle\gamma^{m}+q_{\mathcal{E}}^{*}b_{1}(p_{\mathcal{E}}^{*}\phi)\gamma^{m-1}+\dots+q_{\mathcal{E}}^{*}b_{m-1}(p_{\mathcal{E}}^{*}\phi)\gamma+q_{\mathcal{E}}^{*}b_{m}(p_{\mathcal{E}}^{*}\phi)=,$
$\displaystyle=\gamma^{m}+q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}b_{1}(\phi)\gamma^{m-1}+\dots+q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}b_{m-1}(\phi)\gamma+q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}b_{m}(\phi)$

defines the spectral curve $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ of
the $p_{\mathcal{E}}^{*}K_{X}$-Higgs bundle $p_{\mathcal{E}}^{*}\mathcal{F}$
over $S_{\mathcal{E}}$.

Since (4.5) commutes, observe that

$q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}K_{X}\cong\widetilde{p}_{\mathcal{E}}^{*}p^{*}K_{X}.$

Also, the pull-back of the tautological section
$\lambda:\mathrm{Tot}(K_{X})\to\mathrm{Tot}(p^{*}K_{X})$ under
$\widetilde{p}_{\mathcal{E}}$ equals the tautological section
$\gamma:\mathrm{Tot}(p_{\mathcal{E}}^{*}K_{X})\to\mathrm{Tot}(q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}K_{X})$,

$\gamma=\widetilde{p}_{\mathcal{E}}^{*}\lambda.$

Then, we see that $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ coincides
with the vanishing of the pull-back under $\widetilde{p}_{\mathcal{E}}$ of the
section stated in (4.6) and whose vanishing locus defines $S_{\mathcal{F}}$,

$\displaystyle\widetilde{p}_{\mathcal{E}}^{*}\lambda^{m}+\widetilde{p}_{\mathcal{E}}^{*}p^{*}b_{1}(\phi)\widetilde{p}_{\mathcal{E}}^{*}\lambda^{m-1}+\dots+\widetilde{p}_{\mathcal{E}}^{*}p^{*}b_{m-1}(\phi)\widetilde{p}_{\mathcal{E}}^{*}\lambda+\widetilde{p}_{\mathcal{E}}^{*}p^{*}b_{m}(\phi)=,$
$\displaystyle\gamma^{m}+\widetilde{p}_{\mathcal{E}}^{*}p^{*}b_{1}(\phi)\gamma^{m-1}+\dots+\widetilde{p}_{\mathcal{E}}^{*}p^{*}b_{m-1}(\phi)\gamma+\widetilde{p}^{*}_{\mathcal{E}}p^{*}b_{m}(\phi)=,$
$\displaystyle=\gamma^{m}+q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}b_{1}(\phi)\gamma^{m-1}+\dots+q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}b_{m-1}(\phi)\gamma+q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}b_{m}(\phi).$

This proves that $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ is naturally
isomorphic to the pull-back of the finite bundle defined by
$S_{\mathcal{F}}\to X$,

$\widetilde{p}_{\mathcal{E}}^{*}(S_{\mathcal{F}}\to X),$

and therefore
$\widetilde{p}_{\mathcal{E}}(S_{p_{\mathcal{E}}^{*}\mathcal{F}})$ is equal to
$S_{\mathcal{F}}$. We denote by
$q_{\mathcal{F}}:\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{F}}$ the restriction of $\widetilde{p}_{\mathcal{E}}$, and,
abusing of notation, we denote the restriction of the other projection by
$q_{\mathcal{E}}:\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{E}}$. This proves the existence of the diagram (4.3) where we
observe that all the maps are flat by construction.

Denote by $q$ the structural morphism of $p^{*}K_{X}\to\mathrm{Tot}(K_{X})$.
Observe that $p^{*}K_{X}|_{X\times\\{0\\}}\cong K_{X}$ and consider the
associated morphism $\eta:\mathrm{Tot}(p^{*}K_{X})\to\mathrm{Tot}(K_{X})$.
Note that one has the commutative diagram

$\textstyle{\mathrm{Tot}(p^{*}K_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\eta}$$\textstyle{\mathrm{Tot}(K_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathrm{Tot}(K_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{X,}$

We denote by $S_{\mathcal{E}\otimes\mathcal{F}}$ the image of
$\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ under $\eta$, and we denote by
$\eta_{\mathcal{E}\otimes\mathcal{F}}$ the restriction. By construction, the
diagram (4.4) commutes. Since $\eta_{\mathcal{E}\otimes\mathcal{F}}$ is the
projection of a curve, it is a partial desingularization or an isomorphism.

Recalling that $S_{\mathcal{E}}\subset\mathrm{Tot}(K_{X})$ we have that
$\mathrm{Tot}(p_{\mathcal{E}}^{*}K_{X})=q^{-1}(S_{\mathcal{E}})$ embeds
naturally into $\mathrm{Tot}(p^{*}K_{X})$ in such a way that the restriction
$q$ to it equals $\widetilde{p}_{\mathcal{E}}$. Therefore, we see that
$q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}K_{X}\cong\widetilde{p}_{\mathcal{E}}^{*}p^{*}K_{X}$
can be identified with to the restriction of $q^{*}p^{*}K_{X}$ to
$q^{-1}(S_{\mathcal{E}})\subset\mathrm{Tot}(p^{*}K_{X})$. Under this
isomorphism, the restriction of the tautological section of $q^{*}p^{*}K_{X}$,
$\widetilde{\lambda}:\mathrm{Tot}(p^{*}K_{X})\to q^{*}p^{*}K_{X}$, to
$q^{-1}(S_{\mathcal{E}})$ is related to the tautological section
$\gamma:\mathrm{Tot}(p_{\mathcal{E}}^{*}K_{X})\to
q_{\mathcal{E}}^{*}p_{\mathcal{E}}^{*}K_{X}$ as follows

(4.7)
$\gamma=\widetilde{\lambda}|_{q^{-1}(S_{\mathcal{E}})}\,-\,q^{*}\lambda_{1}|_{q^{-1}(S_{\mathcal{E}})},$

where the index of $\lambda_{1}$ indicates that it is the tautological section
of $p_{1}^{*}K_{X}$. Similarly, we denote by $\lambda_{2}$ the tautological
section of $p_{2}^{*}K_{X}$. We observe that

(4.8) $\widetilde{\lambda}=\eta^{*}\lambda_{2}$

by construction of $\eta$. Then, we can rewrite the equation defining
$\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ as the system of equations in
$\mathrm{Tot}(p^{*}K_{X})$

(4.9)
$\begin{cases}0=q^{*}\lambda_{1}^{n}+q^{*}p^{*}b_{1}(\varphi)q^{*}\lambda_{1}^{n-1}+\dots+q^{*}p^{*}b_{n-1}(\varphi)q^{*}\lambda_{1}+q^{*}p^{*}b_{n}(\varphi)=0,\\\
0=(\eta^{*}\lambda_{2}-q^{*}\lambda_{1})^{m}+q^{*}p^{*}b_{1}(\phi)(\eta^{*}\lambda_{2}-q^{*}\lambda_{1})^{m-1}+\dots\\\
\qquad\dots+q^{*}p^{*}b_{m-1}(\phi)(\eta^{*}\lambda_{2}-q^{*}\lambda_{1})+q^{*}p^{*}b_{m}(\phi)=0.\end{cases}$

One naturally have that
$\mathrm{Tot}(p^{*}K_{X})\cong\mathrm{Tot}(K_{X}\times_{X}K_{X})$ and, under
this isomorphism, $q$ corresponds to the projection of the first factor while
$\eta$ corresponds to the projection of the second. Using the sum map
$\sigma:K_{X}\times_{X}K_{X}\to K_{X}$, we define

$\begin{array}[]{cccc}\widetilde{\sigma}:&\mathrm{Tot}(K_{X}\times_{X}K_{X})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathrm{Tot}(K_{X}\times_{X}K_{X})\\\
&(k_{1},k_{2})&\longmapsto&(k_{1},\sigma(k_{1},k_{2})).\end{array}$

Recall that $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ is defined by
(4.9). The first equation of the system defines $S_{\mathcal{E}}$, while the
second correspond to the points of $S_{\mathcal{F}}$ centered at
$S_{\mathcal{E}}$, so

$\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\cong\widetilde{\sigma}\left(S_{\mathcal{E}}\times_{X}S_{\mathcal{F}}\right).$

This proves the isomorphism (4.2), since taking the image under $\eta$
corresponds to projecting to the second factor.

$\eta\left(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\right)\cong\sigma\left(S_{\mathcal{E}}\times_{X}S_{\mathcal{F}}\right).$

We observe that $\sigma\left(S_{\mathcal{E}}\times_{X}S_{\mathcal{F}}\right)$
is reduced since both $S_{\mathcal{E}}$ and $S_{\mathcal{F}}$ are reduced by
hypothesis, and assumed to be $S_{\mathcal{E}}\neq S_{\mathcal{F}}$ in the
case where $n=m$. Furthermore, we observe that the tautological section on
$\sigma(S_{\mathcal{E}}\times_{X}S_{\mathcal{F}})$ is precisely $\lambda_{2}$,

(4.10)
$\lambda|_{\sigma(S_{\mathcal{E}}\times_{X}S_{\mathcal{F}})}=\lambda_{2}.$

It remains to show that $L_{\mathcal{E}\otimes\mathcal{F}}$, as described in
the hypothesis, and $S_{\mathcal{E}\otimes\mathcal{F}}$, as constructed above,
constitute indeed the spectral data for $\mathcal{E}\otimes\mathcal{F}$. The
projection formula gives

$\displaystyle\widetilde{p}_{\mathcal{E}\otimes\mathcal{F},*}(q_{\mathcal{E}}^{*}L_{\mathcal{E}}\otimes
q_{\mathcal{F}}^{*}L_{\mathcal{F}})$ $\displaystyle\cong
p_{\mathcal{E},*}q_{\mathcal{E},*}(q_{\mathcal{E}}^{*}L_{\mathcal{E}}\otimes
q_{\mathcal{F}}^{*}L_{\mathcal{F}})$ $\displaystyle\cong
p_{\mathcal{E},*}(L_{\mathcal{E}}\otimes
q_{\mathcal{E},*}q_{\mathcal{F}}^{*}L_{\mathcal{F}})$ $\displaystyle\cong
p_{\mathcal{E},*}(L_{\mathcal{E}}\otimes
p_{\mathcal{E}}^{*}p_{\mathcal{F},*}L_{\mathcal{F}})$ $\displaystyle\cong
p_{\mathcal{E},*}L_{\mathcal{E}}\otimes p_{\mathcal{F},*}L_{\mathcal{F}}$
$\displaystyle\cong E\otimes F.$

Note that in the third line we have used
$q_{\mathcal{E},*}q_{\mathcal{F}}^{*}L_{\mathcal{F}}\cong
p_{\mathcal{E}}^{*}p_{\mathcal{F},*}L_{\mathcal{F}}$, which follows from the
commutativity of

$\textstyle{\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\mathcal{F}}}$$\scriptstyle{q_{\mathcal{E}}}$$\textstyle{S_{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\mathcal{F}}}$$\textstyle{S_{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\mathcal{E}}}$$\textstyle{X,}$

and the fact that all these maps are flat. Since
$\widetilde{p}_{\mathcal{E}\otimes\mathcal{F}}=p_{\mathcal{E}\otimes\mathcal{F}}\circ\eta_{\mathcal{E}\otimes\mathcal{F}}$,

$E\otimes F\cong
p_{\mathcal{E}\otimes\mathcal{F},*}\left(\eta_{\mathcal{E}\otimes\mathcal{F},*}(q_{\mathcal{E}}^{*}L_{\mathcal{E}}\otimes
q_{\mathcal{F}}^{*}L_{\mathcal{F}})\right)\cong
p_{\mathcal{E}\otimes\mathcal{F},*}L_{\mathcal{E}\otimes\mathcal{F}}.$

Then, $L_{\mathcal{E}\otimes\mathcal{F}}\to S_{\mathcal{E}\otimes\mathcal{F}}$
is indeed the spectral data of our Higgs bundle
$\mathcal{E}\otimes\mathcal{F}$.

Since it is a spectral curve, we know that
$g(S_{\mathcal{E}\otimes\mathcal{F}})=1+n^{2}m^{2}(g-1)$. Also, thanks to
(4.3), we know that $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{F}}$ is an $n$-cover whose ramification is

$\mathrm{Ram}\left(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{F}}\right)\cong
p_{\mathcal{F}}^{*}\mathrm{Ram}\left(S_{\mathcal{E}}\to X\right).$

Since the length of
$\mathrm{Ram}\left(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{F}}\right)$ is the difference of its genus
$g(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}})$ and the genus of the
unramified $n$-cover, one has

$g(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}})-\left(n(g(S_{\mathcal{F}})-1)-1\right)=m(n^{2}-n)(g-1).$

Then, the genus of $\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}$ is

$g(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}})=1+(mn^{2}+nm^{2}-nm)(g-1).$

Then, we conclude that
$\eta_{\mathcal{E}\otimes\mathcal{F}}:\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}\to
S_{\mathcal{E}\otimes\mathcal{F}}$ is a partial desingularization at a divisor
of length

$g(S_{\mathcal{E}\otimes\mathcal{F}})-g(\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}})=(m^{2}n^{2}+mn-
mn^{2}-nm^{2})(g-1).$

∎

Consider the semistable Higgs bundle $\mathcal{E}=(E,\varphi)$ associated to
the spectral data $L_{\mathcal{E}}\to S_{\mathcal{E}}$, where
$S_{\mathcal{E}}$ is reduced. After Proposition 4.2, the morphism

$\begin{array}[]{cccc}\hat{\tau}^{\mathcal{E}}_{m}:&\operatorname{Jac}^{\,\delta_{m}}_{B^{\prime}_{m}}(\mathcal{S}^{\prime}_{m})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\overline{\operatorname{Jac}}^{\delta_{nm}}_{B_{nm}}(\mathcal{S}_{nm})\\\
&L_{\mathcal{F}}\to
S_{\mathcal{F}}&\longmapsto&\eta_{\mathcal{E}\otimes\mathcal{F},*}(q_{\mathcal{E}}^{*}L_{\mathcal{E}}\otimes
q_{\mathcal{F}}^{*}L_{\mathcal{F}})\to
S_{\mathcal{E}\otimes\mathcal{F}},\end{array}$

corresponds to $\tau^{\mathcal{E}}_{m}$.

######  Corollary 4.3.

For every semistable Higgs bundle $\mathcal{E}=(E,\varphi)$ of rank $n$, the
diagram

$\textstyle{\operatorname{Jac}_{B^{\prime}_{m}}^{\,\delta_{m}}(\mathcal{S}^{\prime}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{m,*}}$$\scriptstyle{\cong}$$\scriptstyle{\hat{\tau}^{\mathcal{E}}_{m}}$$\textstyle{\overline{\operatorname{Jac}}_{B_{nm}}^{\delta_{nm}}(\mathcal{S}_{nm})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{nm,*}}$$\scriptstyle{\cong}$$\textstyle{\mathrm{M}^{\prime}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\mathcal{E}}_{m}}$$\textstyle{\mathrm{M}_{nm},}$

commutes.

Denote by

$\mathcal{S}_{\mathcal{E}}:=S_{\mathcal{E}}\times B^{\prime}_{m}$

the constant family of curves. Define also the families of curves inside
$\mathrm{Tot}(K_{X})$ parametrized by $B^{\prime}_{m}$,

$\widetilde{\Sigma}_{m}^{\mathcal{E}}:=p_{\mathcal{E}}^{*}\left(\mathcal{S}^{\prime}_{m}\right)$

and

$\Sigma_{m}^{\mathcal{E}}:=\sigma\left(\mathcal{S}_{\mathcal{E}}\times_{X\times
B^{\prime}_{m}}\mathcal{S}^{\prime}_{m}\right),$

where $\sigma$ denotes here the fibrewise sum in $r_{m}^{*}K_{X}\to X\times
B^{\prime}_{m}$. Since we have chosen $S_{\mathcal{E}}$ smooth, and all the
curves in $\mathcal{S}^{\prime}_{m}$ are smooth by definition of
$B^{\prime}_{m}$, the curves in $\widetilde{\Sigma}_{m}^{\mathcal{E}}$ are
smooth too. From the proof of Proposition 4.2 we have the morphism

$\eta^{\mathcal{E}}_{m}:\widetilde{\Sigma}_{m}^{\mathcal{E}}\longrightarrow\Sigma_{m}^{\mathcal{E}}$

which is, fibrewise, a partial desingularization at a divisor of length
$(m^{2}n^{2}+mn-mn^{2}-nm^{2})(g-1)$. Denote by
$\delta^{\mathcal{E}}_{m}=m\delta_{n}+n\delta_{m}$. Note that we can define

$\begin{array}[]{cccc}\hat{\eta}^{\mathcal{E}}_{m}:&\operatorname{Jac}_{B^{\prime}_{m}}^{\delta_{m}^{\mathcal{E}}}(\widetilde{\Sigma}^{\mathcal{E}}_{m})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\overline{\operatorname{Jac}}_{B^{\prime}_{m}}^{\delta_{nm}}(\Sigma^{\mathcal{E}}_{m})\\\
&L\to\widetilde{S}_{\mathcal{E}\otimes\mathcal{F}}&\longmapsto&\eta^{\mathcal{E}}_{m,*}L\to
S_{\mathcal{E}\otimes\mathcal{F}}.\end{array}$

After Proposition 4.2, one has the following commutative diagram for families
given by the obvious projections

(4.11)
$\textstyle{\widetilde{\Sigma}_{m}^{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{q}_{\mathcal{E}}}$$\scriptstyle{m:1}$$\scriptstyle{q_{m}}$$\scriptstyle{n:1}$$\scriptstyle{p_{m}^{\mathcal{E}}}$$\scriptstyle{nm:1}$$\textstyle{\mathcal{S}_{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{p}_{\mathcal{E}}}$$\scriptstyle{n:1}$$\textstyle{\mathcal{S}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{m}}$$\scriptstyle{m:1}$$\textstyle{X\times
B^{\prime}_{m}}$

where $p_{m}^{\mathcal{E}}$ is a $nm$-cover,
$\widetilde{p}_{\mathcal{E}}=(p_{\mathcal{E}}\times\mathrm{1}_{B^{\prime}_{m}})$
and $q_{m}$ are $n$-covers, and $p_{m}$ and $\widetilde{q}_{\mathcal{E}}$ are
$m$-covers.

Finally, we study the relation of $\tau^{\mathcal{E}}_{m}$ and the holomorphic
$2$-forms $\Lambda^{1}_{m}$ and $\Lambda_{nm}^{1}$.

######  Lemma 4.4.

One has that

$\tau^{\mathcal{E},*}_{m}\Lambda^{1}_{nm}(\cdot,\cdot)\cong
n\Lambda^{1}_{m}(\cdot,\cdot).$

###### Proof.

Since $\tau^{\mathcal{E}}_{m}$ commutes with all the complex structures
$\Gamma^{1}$, $\Gamma^{2}$ and $\Gamma^{3}$ and one can easily check that
$\check{\tau}^{\mathcal{E},*}_{m}g_{nm}=\mathrm{rk}(\mathcal{E})g_{m}$, we
have

$\displaystyle\tau^{\mathcal{E},*}_{m}\Lambda^{1}_{nm}(\cdot,\cdot)$
$\displaystyle=\tau^{\mathcal{E},*}_{m}\omega^{2}_{nm}(\cdot,\cdot)+\,\mathrm{i}\,\tau^{\mathcal{E},*}_{m}\omega^{3}_{nm}(\cdot,\cdot)$
$\displaystyle=\tau^{\mathcal{E},*}_{m}g_{nm}(\cdot,\Gamma^{2}(\cdot))+\,\mathrm{i}\,\tau^{\mathcal{E},*}_{m}g_{nm}(\cdot,\Gamma^{3}(\cdot))$
$\displaystyle=(\tau^{\mathcal{E},*}_{m}g_{nm})(\cdot,\Gamma^{2}(\cdot))+\,\mathrm{i}\,(\tau^{\mathcal{E},*}_{m}g_{nm})(\cdot,\Gamma^{3}(\cdot))$
$\displaystyle=ng_{m}(\cdot,\Gamma^{2}(\cdot))+\,\mathrm{i}\,ng_{m}(\cdot,\Gamma^{3}(\cdot))$
$\displaystyle=n\omega^{2}_{m}(\cdot,\cdot)+\,\mathrm{i}\,n\omega^{3}_{m}(\cdot,\cdot)$
$\displaystyle=n\Lambda^{1}_{m}(\cdot,\cdot).$

∎

We conclude this section with a generalization of Lemma 2.1 to arbitrary Higgs
bundles on $X$, a technical result that will be used in Section 6.1.

######  Lemma 4.5.

Let $\mathcal{E}=(E,\varphi)$ be a Higgs bundle on $X$. In the notation of
diagram (2.7), we have:

$\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{m}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}E\stackrel{{\scriptstyle\mathrm{1}\otimes\lambda_{m}+\widetilde{p}_{m}^{*}\varphi\otimes\mathrm{1}}}{{\longrightarrow}}\widetilde{p}_{m}^{*}E\otimes\widetilde{p}_{m}^{*}K_{X}\right)\right)$
$\cong
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{m}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right).$

Note that the identification provided by Lemma 2.1 is just the one above
applied to the trivial Higgs bundle $\mathcal{E}=(\mathcal{O}_{X},0)$.

###### Proof.

The idea of the proof is the similar to the one used in Lemma 2.1. Note that
$\left(\widetilde{p}_{m}^{*}E,\widetilde{p}_{m}^{*}\varphi\right)$ is a
relative $\widetilde{p}_{m}^{*}K_{X}$-Higgs bundle over the family of curves
$\mathcal{S}_{m}\to B^{\prime}_{m}$. This point of view provides the
isomorphism

$\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{m}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}E\stackrel{{\scriptstyle\mathrm{1}\otimes\lambda_{m}+\widetilde{p}_{m}^{*}\varphi\otimes\mathrm{1}}}{{\longrightarrow}}\widetilde{p}_{m}^{*}E\otimes\widetilde{p}_{m}^{*}K_{X}\right)\right)$
$\cong\mathrm{coker}\left\\{\widetilde{p}_{m}^{*}E\stackrel{{\scriptstyle\mathrm{1}\otimes\lambda_{m}+\widetilde{p}_{m}^{*}\varphi\otimes\mathrm{1}}}{{\longrightarrow}}\widetilde{p}_{m}^{*}E\otimes\widetilde{p}_{m}^{*}K_{X}\right\\}.$

From the proof of Proposition 4.2 and recalling the commutative diagram
(4.11), we know that the (family of) spectral data is
$\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\to\Sigma^{\mathcal{E}}_{m}$.
We denote by $\gamma$ the corresponding tautological section
$\gamma:\mathrm{Tot}(\widetilde{p}_{m}^{*}K_{X})\to\mathrm{Tot}(q_{m}^{*}\widetilde{p}_{m}^{*}K_{X})$.
Recall from (4.7), (4.8) and (4.10) that

$\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}=\gamma+q_{m}^{*}\lambda_{m}$

where $\lambda_{\Sigma}$ is the tautological section of
$\Sigma_{m}^{\mathcal{E}}$. Then,

$\displaystyle\mathrm{coker}$
$\displaystyle\left\\{\widetilde{p}_{m}^{*}E\stackrel{{\scriptstyle\mathrm{1}\otimes\lambda_{m}+\widetilde{p}_{m}^{*}\varphi\otimes\mathrm{1}}}{{\longrightarrow}}\widetilde{p}_{m}^{*}E\otimes\widetilde{p}_{m}^{*}K_{X}\right\\}\cong$
$\displaystyle\cong\mathrm{coker}\left\\{q_{m,*}\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\stackrel{{\scriptstyle\mathrm{1}\otimes\lambda_{m}+q_{m,*}\gamma\otimes\mathrm{1}}}{{\longrightarrow}}q_{m,*}\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes\widetilde{p}_{m}^{*}K_{X}\right\\}$
$\displaystyle\cong
q_{m,*}\,\mathrm{coker}\left\\{\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\stackrel{{\scriptstyle\mathrm{1}\otimes
q_{m}^{*}\lambda_{m}+\mathrm{1}\otimes\gamma}}{{\longrightarrow}}\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes
q_{m}^{*}\widetilde{p}_{m}^{*}K_{X}\right\\}$ $\displaystyle\cong
q_{m,*}\,\mathrm{coker}\left\\{\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\stackrel{{\scriptstyle\mathrm{1}\otimes\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}}}{{\longrightarrow}}\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes
q_{m}^{*}\widetilde{p}_{m}^{*}K_{X}\right\\}$
$\displaystyle\cong\,q_{m,*}\left(\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes\mathrm{coker}\left\\{\mathcal{O}_{\widetilde{\Sigma}_{m}}\stackrel{{\scriptstyle\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}}}{{\longrightarrow}}q_{m}^{*}\widetilde{p}_{m}^{*}K_{X}\right\\}\right)$
$\displaystyle\cong\,q_{m,*}\left(\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes\mathcal{O}_{\\{\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}=0\\}}\otimes
q_{m}^{*}\widetilde{p}_{m}^{*}K_{X}\right)$
$\displaystyle\cong\,q_{m,*}\left(\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes\mathcal{O}_{\\{\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}=0\\}}\right)\otimes\widetilde{p}_{m}^{*}K_{X}.$

Note that $\lambda_{\Sigma}$ vanishes at

$\\{\lambda_{\Sigma}=0\\}=\Sigma^{\mathcal{E}}_{m}\cap(X\times\\{0\\})\times
B_{m}=\mathcal{S}_{m}\cap-\mathcal{S}_{\mathcal{E}}=\Xi_{m}^{\mathcal{E}}.$

It follows that

$q_{m,*}\mathcal{O}_{\\{\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}=0\\}}\cong\mathcal{O}_{\Xi_{m}^{\mathcal{E}}},$

so

$q_{m,*}\left(\widetilde{q}_{\mathcal{E}}^{*}\mathcal{L}_{\mathcal{E}}\otimes\mathcal{O}_{\\{\eta^{\mathcal{E},*}_{m}\lambda_{\Sigma}=0\\}}\right)\cong(-1)^{*}\mathcal{L}_{\mathcal{E}}\otimes\mathcal{O}_{\Xi_{m}^{\mathcal{E}}}.$

Completing the desired identification. ∎

##  5\. Nahm transform of high rank

In [FJ] the Nahm transform for Higgs bundles is considered, resulting in a
Hermitian connection over $\mathrm{M}_{1}$ of type $(1,1)$ with respect to the
complex structures $\Gamma^{1}_{1}$, $\Gamma^{2}_{1}$ and $\Gamma^{3}_{1}$.
Here, we generalize this construction to moduli spaces of arbitrary rank.

Fix a Higgs bundle $\mathcal{E}=(E,\varphi)$ of rank $n$ and degree $0$,
supported on the Hermitian $C^{\infty}$ vector bundle $\mathbb{E}_{n}$. For
every Higgs bundle $\mathcal{F}=(F,\phi)$ of rank $m$ and $\deg(F)=0$,
supported on the Hermitian $C^{\infty}$ vector bundle $\mathbb{E}_{m}$, we can
consider the Higgs bundle $\mathcal{E}\otimes\mathcal{F}=(E\otimes
F,\varphi\otimes\mathrm{1}_{F}+\mathrm{1}_{F}\otimes\phi)$ on
$\mathbb{E}_{n}\otimes\mathbb{E}_{m}\cong\mathbb{E}_{nm}$.

Associated to $\mathcal{E}=(E,\varphi)$, we can define, for each
$\mathcal{F}=(F,\phi)$ in $\mathrm{M}_{m}^{\mathrm{wtf}}$, the following
Dirac-type operators,

$\mathscr{D}^{\mathcal{E}}_{\mathcal{F}}:=\mathscr{D}_{\mathcal{E}\otimes\mathcal{F}}$

and its adjoint

$\mathscr{D}^{\mathcal{E},*}_{\mathcal{F}}:=\mathscr{D}^{*}_{\mathcal{E}\otimes\mathcal{F}}.$

Recall that if $\mathcal{E}$ and $\mathcal{F}$ are both semistable, the so is
$\mathcal{E}\otimes\mathcal{F}$; below we find conditions under which
$\mathcal{E}\otimes\mathcal{F}$ is without trivial factors.

######  Lemma 5.1.

Let $\mathcal{E}=(E,\varphi)$ and $\mathcal{F}=(F,\phi)$ be semistable Higgs
bundles of degree $0$ and rank $n$ and $m$, respectively.

  1. (1)

if $n>m$ and $\mathcal{E}$ is stable, then $\mathcal{E}\otimes\mathcal{F}$ is
without trivial factors;

  2. (2)

if $n<m$ and $\mathcal{F}$ is stable, then $\mathcal{E}\otimes\mathcal{F}$ is
without trivial factors.

In particular, if $n\neq m$ and both $\mathcal{E}$ and $\mathcal{F}$ are
stable, then $\mathcal{E}\otimes\mathcal{F}$ is without trivial factors.

###### Proof.

The (semi)stability of $\mathcal{E}$ implies the (semi)stability of
$\mathcal{E}^{*}=(E^{*},-\varphi^{t})$. If $\mathcal{E}\otimes\mathcal{F}$ has
a trivial factor, then there exists $\psi:\mathcal{O}_{X}\to E\otimes F$ such
that $(\varphi\otimes\mathrm{1}_{F}+\mathrm{1}_{E}\otimes\phi)(\psi)=0$.
Equivalently, there exists a nontrivial morphism $\psi:E^{*}\to F$ such that
$(\psi\otimes\mathrm{1}_{K_{X}})\circ(-\varphi^{t})=\phi\circ\psi$. As a
consequence, the image $\mathrm{Im}\,\psi$ is a $\phi$-invariant subbundle of
$F$ and the kernel $\ker\psi$ is a $(-\varphi^{t})$-invariant bundle of $E$.
Note that $\deg(\mathrm{Im}\,\psi)=\deg(\ker\psi)=0$. If $n\neq m$, either
$\mathrm{Im}\,\psi$ or $\ker\psi$ are proper subbundles, contradicting the
stability of $\mathcal{F}$ or $\mathcal{E}^{*}$, respectively. ∎

Denote by $\mathrm{M}_{m}^{\mathcal{E}}$ the open subset of $\mathrm{M}_{m}$
given by those Higgs bundles $\mathcal{F}$ such that
$\mathcal{E}\otimes\mathcal{F}$ is without trivial factors. Thanks Lemma 5.1
we know that

  * •

if $\mathcal{E}$ is stable and $\mathrm{rk}(\mathcal{E})>m$,

$\mathrm{M}_{m}^{\mathcal{E}}=\mathrm{M}_{m};$

  * •

if $\mathrm{rk}(\mathcal{E})<m$ or if $\mathrm{rk}(\mathcal{E})=m$ and
$\mathcal{E}$ stable,

$\mathrm{M}^{\mathrm{st}}_{m}\subset\mathrm{M}_{m}^{\mathcal{E}}.$

Let us consider the infinite rank trivial bundle on
$\mathrm{M}_{m}^{\mathcal{E}}$

$\underline{\Omega}_{m}^{\mathcal{E}}:=\left(\Omega_{X}^{1,0}(\mathbb{E}_{n}\otimes\mathbb{E}_{m})\oplus\Omega_{X}^{0,1}(\mathbb{E}_{n}\otimes\mathbb{E}_{m})\right)\times\mathrm{M}_{m}^{\mathcal{E}}.$

After Lemma 5.1 and (3.3), one has that

$\dim\ker\mathscr{D}^{\mathcal{E},*}_{\mathcal{F}}=\dim\ker\mathscr{D}_{\mathcal{E}\otimes\mathcal{F}}^{*}=2nm(g-1)$

is fixed. As we did in the definition of the Dirac–Higgs bundle, take a
covering $\\{Z_{m,i}\\}$ of $\mathrm{M}_{m}$ which is fine enough for the
gerbe $\beta_{m}$, consider the families of Dirac-type operators
$\mathscr{D}^{\mathcal{E},*}_{(U_{m,i},\Phi_{m,i})}=\mathscr{D}_{\mathcal{E}\otimes(U_{m,i},\Phi_{m,i})}$
over $Z_{m,i}\cap\mathrm{M}^{\mathcal{E}}_{m}$, and set

$\widehat{E}_{m,i}:=\ker\mathscr{D}^{\mathcal{E},*}_{(U_{m,i},\Phi_{m,i})}$

which is a holomorphic rank $2nm(g-1)$ bundle over each
$Z_{m,i}\cap\mathrm{M}^{\mathcal{E}}_{m}$. We consider, over
$\mathrm{M}_{m}^{\mathcal{E}}$, the virtual bundle for $\beta_{m}$ given by

$\widehat{\mathbf{E}}_{m}:=\left\\{\widehat{E}_{m,i}\to
Z_{m,i}\cap\mathrm{M}^{\mathcal{E}}_{m}\right\\}_{i\in I}.$

Note that there is a natural metric on $\underline{\Omega}^{n}_{m}$ and
consider the projection
$\mathrm{pr}_{m}^{\mathcal{E}}:\underline{\Omega}_{m}^{\mathcal{E}}\to\widehat{\mathbf{E}}_{m}$
defined by it. Observe that one naturally has a trivial connection
$\underline{d}$ on the trivial bundle $\underline{\Omega}_{m}^{\mathcal{E}}$,
and consider the embedding
$j^{\mathcal{E}}_{m}:\widehat{\mathbf{E}}_{m}\hookrightarrow\underline{\Omega}_{n}$.
Define also, the connection on $\widehat{\mathbf{E}}_{m}$

$\widehat{\boldsymbol{\nabla}}_{m}^{\mathcal{E}}:=\mathrm{pr}_{m}^{\mathcal{E}}\circ\underline{d}\circ
j^{\mathcal{E}}_{m}.$

We define the rank $m$ Nahm transform of the Higgs bundle $\mathcal{E}$ as the
virtual vector bundle over $\mathrm{M}_{m}^{\mathcal{E}}$ with connection

(5.1)
$\hat{\mathcal{E}}_{m}:=\left(\widehat{\mathbf{E}}_{m},\widehat{\boldsymbol{\nabla}}_{m}^{\mathcal{E}}\right).$

######  Remark 5.2.

The Nahm transform defined by Frejlich and Jardim [FJ] is precisely the case
$m=1$ of the construction above. Note that, in this case, the gerbe
$\beta_{1}$ is trivial, so $\widehat{\mathbf{E}}_{1}$ can be defined globally
as a vector bundle with connection over $\mathrm{M}_{1}$.

######  Remark 5.3.

Observe that the rank $m$ Nahm transform of the trivial Higgs bundle
$\mathcal{O}:=(\mathcal{O}_{X},0)$ coincides with the Dirac–Higgs bundle and
connection of rank $m$,

$\hat{\mathcal{O}}_{m}\cong\left(\mathbf{D}_{m},\boldsymbol{\nabla}_{m}\right).$

By definition of $\mathrm{M}^{\mathcal{E}}_{m}$, the image of the morphism
$\tau^{\mathcal{E}}_{m}$ defined (4.1) is contained in
$\mathrm{M}_{nm}^{\mathrm{wtf}}$,

$\tau^{\mathcal{E}}_{m}\,:\,\mathrm{M}_{m}^{\mathcal{E}}\longrightarrow\mathrm{M}_{nm}^{\mathrm{wtf}}.$

Such point of view allows us to give an alternative description of the rank
$m$ Nahm transform, compare with [Bl, Definition 3.0.2].

######  Proposition 5.4.

Consider the rank $m$ Nahm transform of a stable Higgs bundle $\mathcal{E}$ of
rank $n$, with $n\neq m$. One has that

(5.2) $\widehat{\mathbf{E}}_{m}\cong\tau^{\mathcal{E},*}_{m}\mathbf{D}_{nm}$

and

(5.3)
$\widehat{\boldsymbol{\nabla}}_{m}^{\mathcal{E}}\cong\tau^{\mathcal{E},*}_{m}\boldsymbol{\nabla}_{nm}.$

###### Proof.

We first observe that $\underline{\Omega}_{m}^{\mathcal{E}}$ is naturally
isomorphic to $\tau^{\mathcal{E},*}_{m}\underline{\Omega}_{nm}$ as bundles
over $\mathrm{M}_{m}^{\mathcal{E}}$. Over the open subset
$Z_{i}=Z_{m,i}\cap\mathrm{M}^{\mathcal{E}}_{m}$, we have that $\hat{E}_{m,i}$
and $\tau^{\mathcal{E},*}_{m}D_{nm,i}$ are both defined as subbundles of
$\underline{\Omega}_{m}^{\mathcal{E}}|_{Z_{i}}\cong\tau^{\mathcal{E},*}_{m}\underline{\Omega}_{nm}|_{Z_{i}}$.
Also, for every $\mathcal{F}$, one has that

(5.4)
$\left.\tau^{\mathcal{E},*}_{m}D_{nm,i}\right|_{\mathcal{F}}=\ker\mathscr{D}^{*}_{\tau_{m}^{\mathcal{E}}(\mathcal{F})}=\ker\mathscr{D}^{*}_{\mathcal{E}\otimes\mathcal{F}}=\left.\widehat{E}_{m,i}\right|_{\mathcal{F}},$

so, they coincide and (5.2) follows. Also, after (5.4), it follows that
$\tau^{\mathcal{E},*}_{m}\mathrm{pr}_{nm}=\mathrm{pr}^{\mathcal{E}}_{m}$ and
$\tau^{\mathcal{E},*}_{m}j_{nm}=j^{\mathcal{E}}_{m}$. Recalling that
$\underline{d}$ is the trivial derivation in the trivial bundles
$\underline{\Omega}_{m}^{\mathcal{E}}\cong\tau^{\mathcal{E},*}_{m}\underline{\Omega}_{nm}$,
we have

$\tau^{\mathcal{E},*}_{m}\widehat{\nabla}_{m,i}=(\tau^{\mathcal{E},*}_{m}\mathrm{pr}_{nm})\circ(\tau^{\mathcal{E},*}_{m}\underline{d})\circ(\tau^{\mathcal{E},*}_{m}\circ
j)=\mathrm{pr}^{\mathcal{E}}_{m}\circ\underline{d}\circ
j^{\mathcal{E}}_{m}=\widehat{\nabla}^{\mathcal{E}}_{nm,i},$

and (5.3) follows as well. ∎

We have seen in Remark 4.1 that $\tau^{\mathcal{E}}_{m}$ is a hyperholomorphic
morphism. Then, thanks to Proposition 3.5 and Proposition 5.4, one has that
the higher rank Nahm transform is hyperholomorphic as well.

######  Corollary 5.5.

Given a stable Higgs bundle $\mathcal{E}$, its rank $m$ Nahm transform
$\hat{\mathcal{E}}_{m}$ is a virtual hyperholomorphic vector bundle over
$\mathrm{M}_{m}^{\mathcal{E}}$ with a hyperholomorphic connection.

In other words, $\hat{\mathcal{E}}_{m}$ is a space filling, virtual
$\mathrm{(BBB)}$-brane over $\mathrm{M}_{m}^{\mathrm{st}}$, as defined in the
Introduction.

######  Remark 5.6.

One can construct other $\mathrm{(BBB)}$-branes using the objects considered
in this section. The restriction of the Dirac–Higgs bundle and connection
$(\mathbf{D}_{nm},\boldsymbol{\nabla}_{nm})$ on
$\mathrm{M}_{nm}^{\mathrm{wtf}}$ to the image of $\tau^{\mathcal{E}}_{m}$

$\left.\left(\mathbf{D}_{nm},\boldsymbol{\nabla}_{nm}\right)\right|_{\mathrm{Im}(\tau^{\mathcal{E}}_{m})},$

and the push-forward under $\tau^{\mathcal{E}}_{m}$ of the Dirac–Higgs bundle
and connection $(\mathbf{D}_{m},\boldsymbol{\nabla}_{m})$ on
$\mathrm{M}_{m}^{\mathrm{st}}$

$\tau^{\mathcal{E}}_{m,*}\left(\mathbf{D}_{m},\boldsymbol{\nabla}_{m}\right)$

are two virtual $\mathrm{(BBB)}$-branes supported on the hyperkähler
subvariety
$\mathrm{Im}(\tau^{\mathcal{E}}_{m})\subset\mathrm{M}_{nm}^{\mathrm{wtf}}$.

######  Remark 5.7.

Another class of space filling $\mathrm{(BBB)}$-branes would arise by
considering the pull-back under $\tau^{\mathcal{E}}_{m}$ either of the
hyperholomorphic line bundle constructed by Neitzke in [Ne], or of the even
exterior powers of the Dirac–Higgs bundles, which is a hyperholomorphic vector
bundle in the usual sense, see [H4].

##  6\. The mirror brane

###  6.1. The Fourier–Mukai transform

Recall that the smooth locus of the Hitchin fibration is contained in the
stable locus, $\mathrm{M}^{\prime}_{m}\subset\mathrm{M}_{m}^{\mathrm{st}}$.
Then, picking $\mathcal{E}$ stable we always have
$\mathrm{M}^{\prime}_{m}\subset\mathrm{M}^{\mathcal{E}}_{m}$. Note as well
that the gerbe $\beta_{m}$ trivializes over
$\mathrm{M}^{\prime}_{m}\subset\mathrm{M}_{m}^{\mathrm{st}}$, so the
restriction of the rank $m$ Nahm transform
$\hat{\mathcal{E}}^{\prime}_{m}=\hat{\mathcal{E}}_{m}|_{\mathrm{M}^{\prime}_{m}}$
becomes there a hyperholomorphic bundle in the usual sense,
$\hat{\mathcal{E}}^{\prime}_{m}=\left(\widehat{E}^{\prime}_{m},\widehat{\nabla}^{\mathcal{E}}_{m}\right)$.
In this section perform the Fourier–Mukai transform of
$\hat{\mathcal{E}}^{\prime}_{m}$.

Given an stable Higgs bundle $\mathcal{E}=(E,\varphi)$ with spectral data
$L_{\mathcal{E}}\to S_{\mathcal{E}}$, where $S_{\mathcal{E}}$ is a reduced
spectral curve, consider the constant family of curves
$\mathcal{S}_{\mathcal{E}}\to B^{\prime}_{m}$ and the constant family of rank
$1$ torsion free sheaves $\mathcal{L}_{\mathcal{E}}$ determined by
$L_{\mathcal{E}}\to S_{\mathcal{E}}$. Noting that
$S_{\mathcal{E}}\subset\mathrm{Tot}(K_{X})$, consider $(-1)$ to be the
(additive) inversion along the fibres of $K_{X}$ and consider the family of
curves $-\mathcal{S}_{\mathcal{E}}$ and the constant family of rank $1$
torsion free sheaves
$(-1)^{*}\mathcal{L}_{\mathcal{E}}\to-\mathcal{S}_{\mathcal{E}}$ on it. Define

$\Xi_{m}^{\mathcal{E}}:=\mathcal{S}_{m}\cap-\mathcal{S}_{\mathcal{E}}$

and consider the sheaf
$(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi^{\mathcal{E}}_{m}}$ supported on it.
Note that $\Xi_{m}^{\mathcal{E}}$ equals
$\Sigma_{m}^{\mathcal{E}}\cap(X\times\\{0\\})\times B_{m}$. Since
$\Sigma_{m}^{\mathcal{E}}$ is a family of spectral curves of the form
$S_{\mathcal{E}\otimes\mathcal{F}}$, by (3.7) we have that
$\Xi_{m}^{\mathcal{E}}$ is a $2nm(g-1)$-cover of $B_{m}$,

$\textstyle{\Xi_{m}^{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2nm(g-1)\,:\,1}$$\textstyle{B_{m}.}$

We can now provide a result analogous to Proposition 3.3 for
$\widehat{E}^{\prime}_{m}:=\widehat{\mathbf{E}}_{m}|_{\mathrm{M}^{\prime}_{m}}$.

######  Theorem 6.1.

Recall that we denoted by $\widetilde{p}_{m}:\mathcal{S}_{m}\to X$ the
composition of $p_{m}:\mathcal{S}_{m}\to B_{m}\times X$ with the obvious
projection $r_{m}:B_{m}\times X\to X$.

Let $\mathcal{E}=(E,\varphi)$ be a stable Higgs bundle with spectral data
$L_{\mathcal{E}}\to S_{\mathcal{E}}$ such that $S_{\mathcal{E}}$ is reduced.
Then,

$\widehat{E}^{\prime}_{m}\cong
R\check{\mathscr{F}}_{m}\left(\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi^{\mathcal{E}}_{m}}\right)\right).$

###### Proof.

Observe that we have the following commutative diagram

$\textstyle{X\times\mathrm{M}^{\prime}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathrm{M}_{m}}}$$\scriptstyle{\mathrm{1}_{X}\times\tau^{\mathcal{E}}_{m}}$$\textstyle{\mathrm{M}^{\prime}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\mathcal{E}}_{m}}$$\textstyle{X\times\mathrm{M}^{\mathrm{wtf}}_{nm}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\pi_{\mathrm{M}_{nm}}}$$\textstyle{\mathrm{M}^{\mathrm{wtf}}_{nm}.}$

Since $\pi_{\mathrm{M}_{nm}}$ is flat, it follows that
$\tau^{\mathcal{E},*}_{m}\circ
R^{i}\pi_{\mathrm{M}_{nm},*}=R^{i}\pi_{\mathrm{M}_{m},*}\circ(\mathrm{1}_{X}\times\tau^{\mathcal{E}}_{m})^{*}$.
Recalling (5.2), and proceeding as in the proof of Proposition 3.3, we have

$\displaystyle\widehat{E}^{\prime}_{m}$
$\displaystyle\cong\tau^{\mathcal{E},*}_{m}\,\mathbf{D}_{nm}|_{\mathrm{M}^{\prime}_{m}}$
$\displaystyle\cong\tau^{\mathcal{E},*}_{m}\left.\mathbb{R}^{1}\pi_{\mathrm{M}_{nm},*}\left(\mathbf{U}_{nm}\stackrel{{\scriptstyle\boldsymbol{\Phi}_{nm}}}{{\longrightarrow}}\mathbf{U}_{nm}\otimes\pi_{X}^{*}K_{X}\right)\right|_{\mathrm{M}^{\prime}_{m}}$
$\displaystyle\cong\left.\mathbb{R}^{1}\pi_{\mathrm{M}_{m},*}\left((\mathrm{1}_{X}\times\tau^{\mathcal{E}}_{m})^{*}(\mathbf{U}_{nm}\stackrel{{\scriptstyle\boldsymbol{\Phi}_{nm}}}{{\longrightarrow}}\mathbf{U}_{nm}\otimes\pi_{X}^{*}K_{X}\right)\right|_{\mathrm{M}^{\prime}_{m}}$
$\displaystyle\cong\left.\mathbb{R}^{1}\pi_{\operatorname{Jac},*}(\mathrm{1}_{X}\times\tau^{\mathcal{E}}_{m})^{*}\left(\mathbf{U}_{nm}\stackrel{{\scriptstyle\boldsymbol{\Phi}_{nm}}}{{\longrightarrow}}\mathbf{U}_{nm}\otimes\pi_{X}^{*}K_{X}\right)\right|_{\mathrm{M}^{\prime}_{m}}$
$\displaystyle\cong\mathbb{R}^{1}\pi_{\operatorname{Jac},*}\left(U^{\prime}_{m}\otimes\pi_{X}^{*}E\stackrel{{\scriptstyle\Phi_{m}\otimes\mathrm{1}+\mathrm{1}\otimes\pi_{X}^{*}\varphi}}{{\longrightarrow}}U^{\prime}_{m}\otimes\pi_{X}^{*}E\otimes\pi_{X}^{*}K_{X}\right)$
$\displaystyle\cong\mathbb{R}^{1}\pi_{\operatorname{Jac},*}\left((\mathrm{1}_{\operatorname{Jac}}\times
p_{m})_{*}P_{m}\otimes\pi_{X}^{*}E\stackrel{{\scriptstyle\Phi_{m}\otimes\mathrm{1}+\mathrm{1}\otimes\pi_{X}^{*}\varphi}}{{\longrightarrow}}(\mathrm{1}_{\operatorname{Jac}}\times
p_{m})_{*}P_{m}\otimes\pi_{X}^{*}(E\otimes K_{X})\right)$
$\displaystyle\cong\mathbb{R}^{1}\pi_{\operatorname{Jac},*}\left((\mathrm{1}_{\operatorname{Jac}}\times
p_{m})_{*}\left(P_{m}\otimes\widetilde{\pi}^{*}_{S}\widetilde{p}_{m}^{*}E\stackrel{{\scriptstyle(\mathrm{1}_{\operatorname{Jac}}\times\lambda_{m})\otimes\mathrm{1}+\mathrm{1}\otimes\widetilde{\pi}^{*}_{S}\widetilde{p}_{m}^{*}\varphi}}{{\longrightarrow}}P_{m}\otimes\widetilde{\pi}^{*}_{S}\widetilde{p}_{m}^{*}(E\otimes
K_{X})\right)\right)$
$\displaystyle\cong\mathbb{R}^{1}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{m}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}E\stackrel{{\scriptstyle\mathrm{1}\otimes\lambda_{m}+\widetilde{p}_{m}^{*}\varphi\otimes\mathrm{1}}}{{\longrightarrow}}\widetilde{p}_{m}^{*}E\otimes\widetilde{p}_{m}^{*}K_{X}\right)\right).$

Invoking the identification in Lemma 4.5, we then have

$\widehat{E}^{\prime}_{m}\cong
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left(P_{m}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right).$

Next, using the description of the Poincaré bundle $P_{m}$ in terms of
$\mathcal{P}_{m}$ given in Section 2.3, the projection formula and base change
theorems for the various morphism of diagram (2.7), one obtains the following
chain of identifications:

$\displaystyle\widehat{E}^{\prime}_{m}$ $\displaystyle\cong
R^{0}\widetilde{\pi}_{\operatorname{Jac},*}\left((\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{m})^{*}\mathcal{P}_{m}\otimes\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right)$
$\displaystyle\cong
R^{0}\hat{\pi}_{*}R^{0}(\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{m})_{*}\left((\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{m})^{*}\mathcal{P}_{m}\otimes\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right)$
$\displaystyle\cong R^{0}\hat{\pi}_{*}\left(\mathcal{P}_{m}\otimes
R^{0}(\mathrm{1}_{\operatorname{Jac}}\times\check{\imath}_{m})_{*}\widetilde{\pi}_{S}^{*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right)$
$\displaystyle\cong
R^{0}\hat{\pi}_{*}\left(\mathcal{P}_{m}\otimes\check{\pi}^{*}R^{0}\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right)$
$\displaystyle\cong
R^{0}\check{\mathscr{F}}_{m}\left(R^{0}\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right).$

Finally, as in the last part of the proof of Proposition 3.3, due to the fact
that $\Xi_{m}^{\mathcal{E}}$ is a finite $2nm(2g-2)$-cover of
$B^{\prime}_{n}$,
$R\check{\mathscr{F}}_{m}\left(\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi_{m}^{\mathcal{E}}}\right)\right)$
is a complex supported in degree $0$. ∎

Thanks to Theorem 6.1 and (2.8) we can describe the Fourier–Mukai transform

$\widecheck{\mathfrak{E}}^{\prime}_{m}:=R\hat{\mathscr{F}}_{m}\left(\widehat{E}^{\prime}_{m}\right),$

restricted to the smooth locus of the Hitchin fibration
$\mathrm{M}^{\prime}_{m}\subset\mathrm{M}_{m}$, given by the subset
$B^{\prime}_{m}\subset B_{m}$ of smooth spectral curves.

######  Corollary 6.2.

We have that $\widecheck{\mathfrak{E}}^{\prime}_{m}$ is a complex entirely
supported in degree $d_{m}=1+m^{2}(g-1)$, and as a sheaf over
$\operatorname{Jac}_{B^{\prime}_{m}}^{\delta_{m}}(\mathcal{S}^{\prime}_{m})^{\vee}$
is

$\widecheck{\mathfrak{E}}^{\prime}_{m}\cong\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes\mathcal{L}_{\mathcal{E}}|_{-\Xi^{\mathcal{E}}_{m}}\right),$

with

$\mathrm{supp}\left(\widecheck{\mathfrak{E}}^{\prime}_{m}\right)=\check{\imath}_{m}\left(-\Xi^{\mathcal{E}}_{m}\right).$

###### Proof.

Observe that

$\displaystyle\widecheck{\mathfrak{E}}^{\prime}_{m}\cong$ $\displaystyle
R^{d}\hat{\mathscr{F}}_{m}R^{0}\check{\mathscr{F}}_{m}\left(\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi^{\mathcal{E}}_{m}}\right)\right)$
$\displaystyle\cong$
$\displaystyle\left(\mathrm{1}_{\operatorname{Jac}_{V}(S)}^{-1}\right)^{*}\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes(-1)^{*}\mathcal{L}_{\mathcal{E}}|_{\Xi^{\mathcal{E}}_{m}}\right)$
$\displaystyle\cong$
$\displaystyle\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes\mathcal{L}_{\mathcal{E}}|_{-\Xi^{\mathcal{E}}_{m}}\right).$

∎

######  Remark 6.3.

Observe that Proposition 4.2 implies that
$\mathrm{Im}(\tau^{\mathcal{E}}_{m})$ lies outside the smooth locus of the
Hitchin fibration. Therefore, the method used in Section 6.1 is not suited to
study the behaviour of the $\mathrm{(BBB)}$-branes defined in Remark 5.6 under
mirror symmetry. For that we would need to consider Fourier–Mukai transforms
over compactified jacobians of singular curves as in [Ar, MRV1, MRV2].

###  6.2. A complex Lagrangian multisection

We study in this section the support of
$\widecheck{\mathfrak{E}}^{\prime}_{m}$, the Fourier–Mukai transform of the
higher rank Nahm transform, restricted to the smooth locus of the Hitchin
fibration $\mathrm{M}_{m}^{\prime}$.

######  Theorem 6.4.

The subvariety $\hat{\imath}_{m}\left(-\Xi^{\mathcal{E}}_{m}\right)$ is a
$2nm(g-1)$-section for the Hitchin fibration and it is Lagrangian with respect
to $\Lambda_{m}^{1}$.

###### Proof.

Since $\check{\imath}_{m}$ is an embedding and $\Xi^{\mathcal{E}}_{m}$ is a
$2nm(g-1)$-cover of $B^{\prime}_{m}$, we have that
$\check{\imath}_{m,i}(\Xi^{\mathcal{E}}_{m})$ is a $2nm(g-1)$-section of
$\mathrm{M}^{\prime}_{m}\to B^{\prime}_{m}$.

Now, we have to prove that $\hat{\imath}_{m}(\Xi^{\mathcal{E}}_{m})$ is
Lagrangian with respect to $\Lambda_{m}^{1}$. Recall from Section 2.1, that
$\Lambda_{m}^{1}$ is defined as the exterior derivative $d\theta$ of a certain
$1$-form $\theta$. Then, we see that $\hat{\imath}_{m}(\Xi^{\mathcal{E}}_{m})$
is Lagrangian if and only if $\theta$ is a constant 1-form along
$\hat{\imath}_{m}(\Xi^{\mathcal{E}}_{m})$. Since $\hat{\imath}_{m}$ is an
embedding, it suffices to prove that
$\hat{\imath}_{m}^{*}\theta|_{\Xi^{\mathcal{E}}_{m,i}}$ is constant.

We recall the definition of $\theta$. We recall that
$\mathrm{M}^{\prime}_{m}\subset\mathrm{M}^{\mathrm{st}}_{m}$ so all the points
are smooth and represented by the stable Higgs bundle
$\mathcal{E}=(E,\varphi)$, the tangent space is
$T_{\mathcal{E}}\mathrm{M}_{m}=\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})$,
which comes naturally equipped with the map
$t:\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})\to H^{1}(X,\mathrm{End}(E))$. By
Serre duality, the Higgs field $\varphi\in H^{0}(\mathrm{End}(E)\otimes
K_{X})$ is an element of the dual space of $H^{1}(\mathrm{End}(E))$ and recall
that we defined $\theta(v)=\langle\varphi,t(v)\rangle$, for each
$v\in\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})$.

We now study the description of $\theta$ over
$\operatorname{Jac}_{B^{\prime}_{m}}^{\,\delta_{m}}(\mathcal{S}^{\prime}_{m})$.
By the spectral correspondence, given the spectral data $L\to S_{m,b}$, one
has that $E=(p_{b})_{*}L$ and $\varphi=(p_{b})_{*}\lambda_{b}$ where
$\lambda_{b}:L\to L\otimes p_{b}^{*}K_{X}$ is given by tensorizing by the
restriction to $S_{m,b}$ of the tautological section
$\lambda:\mathrm{Tot}(K_{X})\to p^{*}K_{X}$. Note that $p_{b}^{*}K_{X}$ is a
subsheaf of the canonical bundle $K_{S_{m,b}}$, then $\lambda_{b}$ gives
naturally an element of $H^{0}(S_{m,b},K_{S_{m,b}})$. The isomorphism
$\mathrm{M}^{\prime}_{m}\cong\operatorname{Jac}_{B^{\prime}_{m}}^{\,\delta_{m}}(\mathcal{S}^{\prime}_{m})^{\vee}$,
given by the push-forward under $p_{b}$, provides as well the isomorphism
between $\mathbb{H}^{1}(C^{\bullet}_{\mathcal{E}})$ and
$\mathrm{Ext}^{1}_{\mathrm{Tot}(K_{X})}(L,L)$ and between
$H^{1}(X,\mathrm{End}(E))$ and $\mathrm{Ext}^{1}_{S_{m,b}}(L,L)\cong
H^{1}(S_{m,b},\mathcal{O}_{S_{m,b}})$. Then, we can express $\theta(v)$ to be
$\langle\lambda_{b},t^{\prime}(v)\rangle$ given by Serre duality, where now
$v\in
T_{L}\operatorname{Jac}_{B^{\prime}_{m}}^{\,\delta_{m}}(\mathcal{S}^{\prime}_{m})\cong\mathrm{Ext}^{1}_{\mathrm{Tot}(K_{X})}(L,L)$,
the section $\lambda_{b}\in H^{0}(S_{m,b},K_{S_{m,b}})$ is defined by the
restriction of the tautological section $\lambda:\mathrm{Tot}(K_{X})\to
p^{*}K_{X}$ to $S_{m,b}\subset\mathrm{Tot}(K_{X})$, and
$t^{\prime}:\mathrm{Ext}^{1}_{\mathrm{Tot}(K_{X})}(L,L)\to\mathrm{Ext}^{1}_{S_{m,b}}(L,L)\cong
H^{1}(S_{m,b},\mathcal{O}_{S_{m,b}})$ is the projection to those deformations
that preserve the support.

Note that, for every $L_{1},L_{2}\in\operatorname{Jac}(S_{m,b})$, one has
naturally that $\mathrm{Ext}^{1}_{S_{m,b}}(L_{1},L_{1})\cong
H^{1}(S_{m,b},\mathcal{O}_{S_{m,b}})\cong\mathrm{Ext}^{1}_{S_{m,b}}(L_{2},L_{2})$.
We observe that $\theta$ is a 1-form which is constant along the fibres
$\overline{\operatorname{Jac}}^{\delta_{m}}(S_{m,b})$. On the other hand, the
1-form $\hat{\imath}_{m}^{*}\theta$ in $\mathcal{S}^{\prime}_{m}$ depends on
the embedding $d\hat{\imath}_{m}:T_{s}S_{m,b}\hookrightarrow
T_{\mathcal{O}(s-s_{i})}\operatorname{Jac}^{\delta_{m}}(S_{m,b})\cong
H^{1}(S_{m,b},\mathcal{O}_{S_{m,b}})$. Recall that $\hat{\imath}_{m}$ sends
the point $s\in S_{m,b}$ to the the line bundle associated whose meromorphic
sections have pole at $s\in S_{m,b}$ and a zero at $s_{i}$. Since Serre
duality $\langle\cdot,\cdot\rangle:H^{0}(K_{S_{m,b}})\times
H^{1}(\mathcal{O}_{S_{m,b}})\to k$ sends $\langle\lambda,\xi\rangle$ to the
sum of residues of the meromorphic differential $\lambda\xi$, one has that

$\hat{\imath}_{m}^{*}\theta|_{s}\cong\hat{\imath}_{m}^{*}\langle\lambda_{b},\cdot\rangle|_{s}\cong\lambda_{b}(s).$

So, $\hat{\imath}_{m}^{*}\theta$ is the one form defined by the tautological
section $\lambda:\mathrm{Tot}(K_{X})\to p^{*}K_{X}$.

Obviously, the tautological section $\lambda$ restricted to
$X\times\\{0\\}\subset\mathrm{Tot}(K_{X})$, is the zero section. Recall that
we have defined $\Xi^{\mathcal{E}}_{m}$ as the intersection of
$\mathcal{S}_{\mathcal{E}}$ and $-\mathcal{S}_{m}$ inside
$\mathrm{Tot}(K_{X})$. But this is equivalent to the intersection of
$\Sigma_{m}^{\mathcal{E}}$ with $X\times\\{0\\}$. Therefore, $\lambda$ is
constantly $0$ along $\Xi^{\mathcal{E}}_{m}$, that is
$\hat{\imath}_{m}^{*}\theta|_{\Xi^{\mathcal{E}}_{m}}=0$, and this concludes
the proof. ∎

Since
$\operatorname{Jac}_{B^{\prime}_{m}}^{\,\delta_{m}}(\mathcal{S}^{\prime}_{m})$
and
$\operatorname{Jac}_{B^{\prime}_{m}}^{\,\delta_{m}}(\mathcal{S}^{\prime}_{m})^{\vee}$
are naturally isomorphic, one has a similar statement on the dual fibration,
where $\check{\imath}_{m}(\Xi^{\mathcal{E}}_{m})$ coincides with the support
of $\widecheck{\mathfrak{E}}^{\prime}_{m}$, the Fourier–Mukai transform of the
vector bundle underlying the rank $m$ Nahm transform
$\hat{\mathcal{E}}^{\prime}_{m}$.

######  Corollary 6.5.

The support of the Fourier–Mukai transform
$\widecheck{\mathfrak{E}}^{\prime}_{m}$ of the rank $m$ Nahm transform,
$\check{\imath}_{m}\left(\Xi^{\mathcal{E}}_{m}\right)$, is a Lagrangian
$2nm(g-1)$-section of the Hitchin fibration.

###  6.3. The Fourier–Mukai–Nahm transform

In this section we will apply the Fourier–Mukai–Nahm transform
$\check{\mathscr{N}}_{n}$ to $\hat{\mathcal{E}}^{\prime}_{m}$, the restriction
of the smooth locus of the Hitchin fibration of the $\mathrm{(BBB)}$-brane
which constitutes the higher rank Nahm transform of a stable Higgs bundle
$\mathcal{E}$. By doing so, we will obtain a mirror $\mathrm{(BAA)}$-brane in
$\mathrm{M}^{\prime}_{m}$.

######  Theorem 6.6.

The connection $\widehat{\boldsymbol{\nabla}}^{\mathcal{E}}_{m}$ is adapted,
Poincaré basic and transversally flat. Hence, one can consider the
Fourier–Mukai–Nahm transform of $\hat{\mathcal{E}}_{m}$, giving

$\hat{\mathscr{N}}_{n}\left(\widehat{E}^{\prime}_{m},\widehat{\nabla}^{\mathcal{E}}_{m}\right)=\left(\check{\imath}_{m}\left(-\Xi^{\mathcal{E}}_{m}\right),\widecheck{\mathfrak{E}}^{\prime}_{m},\widecheck{\nabla}_{m}^{\mathcal{E}}\right)$

where

$\widecheck{\mathfrak{E}}^{\prime}_{m}\cong\check{\imath}_{m,*}\left(\widetilde{p}_{m}^{*}K_{X}\otimes\mathcal{L}_{\mathcal{E}}|_{-\Xi^{\mathcal{E}}_{m}}\right),$

its support,

$\check{\imath}_{m}\left(-\Xi^{\mathcal{E}}_{m}\right)=\check{\imath}_{m}\left(\mathcal{S}_{\mathcal{E}}\cap-\mathcal{S}_{m}\right)$

is a Lagrangian $2nm(g-1)$-section of the Hitchin fibration, and the
connection on $\widecheck{\mathfrak{E}}^{\prime}_{m}$,