Singular Geometry and Higgs Bundles in String Theory

Singular Geometry and Higgs Bundles
in String Theory

Lara B. ANDERSON ${}^{{\dagger}^{1}}$, Mboyo ESOLE ${}^{{\dagger}^{2}}$, Laura
FREDRICKSON ${}^{{\dagger}^{3}}$
and Laura P. SCHAPOSNIK ${}^{{\dagger}^{4}{\dagger}^{5}}$

L.B. Anderson, M. Esole, L. Fredrickson and L.P. Schaposnik

${}^{{\dagger}^{1}}$ Department of Physics and Department of Mathematics,
Virginia Tech,
${}^{{\dagger}^{1}}$ Blacksburg, VA 24061, USA<EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Department of Mathematics, Northeastern University,
Boston, MA 02115, USA<EMAIL_ADDRESS>
${}^{{\dagger}^{3}}$ Department of Mathematics, Stanford University, Stanford,
CA 94305, USA<EMAIL_ADDRESS>
${}^{{\dagger}^{4}}$ Department of Mathematics, University of Illinois at
Chicago, 60607 Chicago, USA<EMAIL_ADDRESS>${}^{{\dagger}^{5}}$ Department of
Mathematics, FU Berlin, 14195 Berlin, Germany

Received November 22, 2017, in final form April 13, 2018; Published online
April 18, 2018

This brief survey aims to set the stage and summarize some of the ideas under
discussion at the Workshop on Singular Geometry and Higgs Bundles in String
Theory, to be held at the American Institute of Mathematics from October 30th
to November 3rd, 2017. One of the most interesting aspects of the duality
revolution in string theory is the understanding that gauge fields and matter
representations can be described by intersection of branes. Since gauge theory
is at the heart of our description of physical interactions, it has opened the
door to the geometric engineering of many physical systems, and in particular
those involving Higgs bundles. This note presents a curated overview of some
current advances and open problems in the area, with no intention of being a
complete review of the whole subject.

Higgs bundles; Hitchin fibration; mirror symmetry; F-theory; Calabi–Yau;
singular curves; singularities

14D20; 14D21; 53C07; 14H70; 14P25

##  1 Introduction

One of the most interesting aspects of the duality revolution in string theory
is the understanding that gauge fields and matter representations can be
described by the intersection of branes. Since gauge theory is at the heart of
our description of physical interactions, it has opened the door to the
geometric engineering of many physical systems, and in particular those
arising from Higgs bundles, whose moduli spaces have become a source of many
interesting branes.

In an effort to consolidate and disseminate the variety of different
techniques, heuristics, and approaches that have been applied to the study of
Higgs bundles and spectral data in recent years by the mathematics and physics
communities, we present here a short survey on these subjects, as well as a
collection of open problems and ideas revolving around them. This note focuses
on two interrelated themes concerning Higgs bundles and the Hitchin fibration,
and their interactions with mathematical physics:

  * (A)

Higgs bundles and algebraic geometry: invariants of singular spaces, and in
particular singular fibers of the Hitchin fibration (Section 2); the effect of
these fibers on the geometry of the moduli spaces of Higgs bundles, including
limits within the Hitchin fibration (Section 3), and the appearance of Higgs
bundles on singular curves (Section 4).

  * (B)

Hitchin systems and T-branes: the study of the moduli space of Higgs bundles
and its branes through the Hitchin fibration (Section 5), and their appearance
within the broader setting of string/F-theory (Section 6), and Calabi–Yau
elliptic fibrations (Section 7).

Although these two themes are closely related, correspondences between them
are just in their infancy. In particular, obtaining a global understanding of
Higgs bundles over singular curves, and of Higgs bundles which have singular
spectral data, would be most beneficial from the perspective of F-theory and
superconformal theories in diverse dimensions. We hope these notes will help
to further clarify the role that spectral curves and spectral data play in
string theory, both for those studying Higgs bundles on Riemann surfaces and
for those studying Higgs bundles on higher dimensional spaces.

This short survey is not intended to be a complete overview of the research
done in the area, but rather a concise description of certain particular paths
of research that are currently receiving much attention, and that present open
problems that could be tackled by researchers in different areas of
mathematics and physics.

##  2 Higgs bundles and the Hitchin fibration

Throughout the paper we shall consider a compact Riemann surface $\Sigma$ of
genus $g\geq 2$ with canonical bundle $K=T^{*}\Sigma$. In what follows, we
recall some of the main properties of complex and real Higgs bundles, as well
as the associated Hitchin fibration whose structure groups are real or complex
subgroups of ${\rm GL}(n,\mathbb{C})$.

###  2.1 Higgs bundles

We begin by briefly reviewing the notions of Higgs bundles for real and
complex groups which are relevant to this paper. Further details can be found
in standard references such as Hitchin [112, 113] and Simpson [169, 170, 171].
Recall that ${\rm GL}(n,\mathbb{C})$-Higgs bundles of degree 0 on $\Sigma$ are
pairs $(E,\Phi)$ where

  * •

$E\rightarrow\Sigma$ is a holomorphic vector bundle of rank $n$ and degree
$0$,

  * •

the Higgs field $\Phi\colon E\rightarrow E\otimes K$, is a holomorphic
$K$-valued endomorphism.

By the work of Hitchin and Simpson, given a polystable Higgs bundle, there is
a unique hermitian metric $h$ on $E$, known as the harmonic metric, solving
the so-called Hitchin equations:

$\displaystyle
F_{D(\overline{\partial}_{E},h)}+[\Phi,\Phi^{*_{h}}]=0,\qquad\overline{\partial}_{E}\Phi=0,$

where $D(\overline{\partial}_{E}{,}h)$ is the Chern connection, i.e., the
unique $h$-unitary connection such that $D^{0,1}{=}\overline{\partial}_{E}$,
the curvature of the Chern connection is denoted by
$F_{D(\overline{\partial}_{E},h)}$, and $\Phi^{*_{h}}$ represents the
hermitian adjoint of $\Phi$ with respect to the hermitian metric $h$. The
correspondence between pairs $(E,\Phi)$ and triples $(E,\Phi,h)$ is known as
the nonabelian Hodge correspondence. More generally, for a complex reductive
Lie group $G_{\mathbb{C}}$, one has the following [113]:

######  Definition 2.1.

A $G_{\mathbb{C}}$-Higgs bundle is a pair $(P,\Phi)$, where $P$ is a
holomorphic principal $G_{\mathbb{C}}$-bundle, and $\Phi$ is a holomorphic
section of ${\rm ad}(P)\otimes K$, where ${\rm ad}(P)$ is the adjoint bundle
of $P$.

In this setting, there is a similar nonabelian Hodge correspondence, where the
notion of a hermitian metric is replaced by a reduction of structure of $P$ to
the maximal compact subgroup of $G_{\mathbb{C}}$. By considering appropriate
stability conditions, one may define the Hitchin moduli space
$\mathcal{M}_{G_{\mathbb{C}}}$ of isomorphism classes of polystable
$G_{\mathbb{C}}$-Higgs bundles, which was introduced by Hitchin in [112]. Up
to gauge equivalence, the points of the moduli space
$\mathcal{M}_{G_{\mathbb{C}}}$ represent polystable $G_{\mathbb{C}}$-Higgs
bundles on $\Sigma$. Moreover, the through nonabelian Hodge correspondence,
points of the moduli space represent solutions of the
$G_{\mathbb{C}}$-Hitchin’s equations.

Given a real form $G$ of the complex reductive lie group $G_{\mathbb{C}}$, we
may define $G$-Higgs bundles as follows. Let $H$ be the maximal compact
subgroup of $G$ and consider the Cartan decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ of $\mathfrak{g}$, where
$\mathfrak{h}$ is the Lie algebra of $H$, and $\mathfrak{m}$ its orthogonal
complement. This induces a decomposition of the Lie algebra
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{h}^{\mathbb{C}}\oplus\mathfrak{m}^{\mathbb{C}}$
of $G_{\mathbb{C}}$. Note that the Lie algebras satisfy
$[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h}$,
$[\mathfrak{h,\mathfrak{m}}]\subset\mathfrak{m}$,
$[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h},$ and there is an induced
isotropy representation ${\rm Ad}|_{H^{\mathbb{C}}}\colon
H^{\mathbb{C}}\rightarrow{\rm GL}(\mathfrak{m}^{\mathbb{C}})$.

######  Definition 2.2.

A principal $G$-Higgs bundle is a pair $(P,\Phi)$ where

  * •

$P$ is a holomorphic principal $H^{\mathbb{C}}$-bundle on $\Sigma$,

  * •

$\Phi$ is a holomorphic section of
$P\times_{\mathrm{Ad}}\mathfrak{m}^{\mathbb{C}}\otimes K$.

Similarly to the case of Higgs bundles for complex groups, there are notions
of stability, semistability and polystability for $G$-Higgs bundles. One can
see that the polystability of a $G$-Higgs bundle for a group $G\subset{\rm
GL}(n,\mathbb{C})$ is equivalent to the polystability of the corresponding
${\rm GL}(n,\mathbb{C})$-Higgs bundle. However, it should be noted that a
$G$-Higgs bundle can be stable as a $G$-Higgs bundle but not as a ${\rm
GL}(n,\mathbb{C})$-Higgs bundle. The moduli space of polystable $G$-Higgs
bundles on the compact Riemann surface $\Sigma$ shall be denoted by
$\mathcal{M}_{G}$.

###  2.2 The Hitchin fibration

The moduli space $\mathcal{M}_{G_{\mathbb{C}}}$ of $G_{\mathbb{C}}$-Higgs
bundles admits a natural complete hyperkähler metric over its smooth points,
and a way of studying it is through the Hitchin fibration [113]. This
fibration maps $(E,\Phi)$ to the eigenvalues of $\Phi$ encoded in the
characteristic polynomial $\det(\Phi-\eta\operatorname{Id})$ of $\Phi$, and is
obtained as follows. Let $\\{p_{1},\ldots,p_{k}\\}$ be a homogeneous basis for
the algebra of invariant polynomials on the Lie algebra $\mathfrak{g}_{c}$ of
$G_{\mathbb{C}}$, and let $d_{i}$ denote the degree of $p_{i}$. The Hitchin
fibration is then given by

$\displaystyle\mathrm{Hit}\colon\ \mathcal{M}_{G_{\mathbb{C}}}$
$\displaystyle\longrightarrow\mathcal{A}_{G_{\mathbb{C}}}:=\bigoplus_{i=1}^{k}H^{0}\big{(}\Sigma,K^{d_{i}}\big{)},$
(2.1) $\displaystyle(E,\Phi)$
$\displaystyle\mapsto(p_{1}(\Phi),\ldots,p_{k}(\Phi)),$

where $\mathrm{Hit}$ is referred to as the Hitchin map. It is a proper map for
any choice of basis111In particular, it can be expressed in terms of the
coefficients of $\det(\Phi-\eta\operatorname{Id})$., its generic fibers are
abelian varieties, and makes the moduli space into a complex integrable system
[113].

Each connected component of a generic fiber of the Hitchin map is an abelian
variety. In the case of $G_{\mathbb{C}}$-Higgs bundles this can be seen using
spectral data [26, 113]. Through the characteristic polynomial of the Higgs
field of a $G_{\mathbb{C}}$-Higgs bundle $(E,\Phi)$, one may define an
algebraic curve, called the spectral curve of $(E,\Phi)$, which is generically
smooth222When considering classical groups $G_{\mathbb{C}}$, only for ${\rm
SO}(2p,\mathbb{C})$ one needs to consider a normalization of the curve.:

$\displaystyle S=\\{{\rm det}(\Phi-\eta\operatorname{Id})=0\\}\subset{\rm
Tot}(K),$ (2.2)

where ${\rm Tot}(K)$ is the total space of $K$, the map $\eta$ is the
tautological section of $K$ on ${\rm Tot}(K)$, and by abuse of notation, we
consider $\Phi$ as its pull-back to ${\rm Tot}(K)$ (the reader should refer to
[26] for thorough details on the construction). We say that $(E,\Phi)$ lies in
the regular locus of $\mathcal{M}_{G_{\mathbb{C}}}$ if the curve $S$ is non-
singular, and denote the regular locus of the moduli space by
$\mathcal{M}_{G_{\mathbb{C}}}^{\prime}$. Let $\pi\colon S\to\Sigma$ be the
natural projection to $\Sigma$, and let $\eta\in H^{0}(S,\pi^{*}(K))$ denote
the restriction of the tautological section of $K$ to $S$. If $(E,\Phi)$ is in
the regular locus, then there exists a line bundle $L\to S$ for which
$E=\pi_{*}L$, and $\Phi$ is obtained by pushing down the map $\eta\colon L\to
L\otimes\pi^{*}(K)$. In this way, one recovers the pair $(E,\Phi)$ from the
pair $(S,L)$, which is referred to as the spectral data associated to the pair
$(E,\Phi)$.

Note that the spectral curve $S$ of the pair $(E,\Phi)$ depends only on the
characteristic polynomial of $\Phi$ and hence it only depends on the image of
$(E,\Phi)$ under the Hitchin map. Therefore any point
$a\in\mathcal{A}_{G_{\mathbb{C}}}$ has an associated spectral curve $S_{a}$.
If $a$ is in the regular locus of $\mathcal{A}_{G_{\mathbb{C}}}$, in other
words, if the associated spectral curve $S_{a}$ is smooth, then the spectral
data construction identifies the fiber $\mathrm{Hit}^{-1}(a)$ of the Hitchin
system with some subspace of $\mathrm{Pic}(S_{a})$, the Picard variety of the
spectral curve $S$. The connected components of $\mathrm{Pic}(S_{a})$ are
isomorphic to copies of $\mathrm{Jac}(S_{a})$, the Jacobian of $S_{a}$, and
are labeled by the degree of the vector bundle $E$ of a Higgs pair $(E,\Phi)$.
In particular, for $G_{\mathbb{C}}={\rm GL}(n,\mathbb{C})$, the generic fibers
are isomorphic to $\mathrm{Jac}(S_{a})$, and one can see from here that the
components of the regular fibers are abelian varieties. While much is known
about the generic fibers of the $G_{\mathbb{C}}$-Hitchin fibration, there are
still several interesting open questions. In particular, it would be
interesting to understand the geometry of the generic fibers of the Hitchin
fibration stated in the open problems of [160]. For instance, it is
interesting to consider the following:

######  Open Question 2.3.

Considering the notion of “strong real form”333The notion of strong real form
is a refinement of the notion of real form. For example for ${\rm
SL}(2,\mathbb{C})$, there are three equivalence classes of strong real forms
corresponding to ${\rm SU}(2,0),~{}{\rm SU}(0,2)$ and ${\rm SU}(1,1)$. from
[1], describe the corresponding Higgs bundles and determine which ones define
singular spectral curves.

When considering arbitrary groups $\mathcal{G}$, the algebraic curve defined
by the characteristic polynomial of $\Phi$ is not always generically smooth
(for real groups, see for instance the case of $\mathcal{G}={\rm SU}(p,q)$ and
the spectral data described in [157]). In this case, one may consider cameral
covers [60]: these are $K$ valued covers of $\Sigma$ with an action of
$\mathcal{W}$, the Weyl group of $\mathcal{G}$. The fiber of the associated
fibration can be described in terms of these covers, and over a generic point
of the base the cover is a $\mathcal{W}$-Galois cover (e.g., see [59, 60, 61,
80, 164], and [65] for further references). In this set up, there is a natural
discriminant locus in the Hitchin base, away from which the connected
component of the fiber is isomorphic to a certain abelian variety which can be
described as a generalized Prym variety of the cameral cover. However, the
study of the singular fibers, even from the perspective of cameral covers, is
not fully understood.

######  Open Question 2.4.

Give a comparison of what is known for singular fibers of the Hitchin
fibration from the perspective of cameral covers and of spectral data.

Cameral covers have shown to be very useful tools to understand the moduli
spaces of principal Higgs bundles and their relation to many other fields.
However, the abstraction of the method and the constructions of the covers can
sometimes make certain properties of the moduli spaces very difficult to
discern. Although most objects are defined in the above papers in a general
way, their description and study for particular groups is still being done by
many researchers (e.g., see recent developments for real Higgs bundles in
[153, 154], where interesting comparisons with classical spectral data are
carefully explained).

######  Open Question 2.5.

Extend the cameral cover methods of [153] for ${\rm SU}(p,p+1)$-Higgs bundles
to all other real Higgs bundles which lie completely over the singular locus
of the Hitchin fibration, i.e., to those Higgs bundles whose characteristic
polynomial defines singular curves through (2.2).

###  2.3 The singular locus of the Hitchin fibration

As mentioned before, the fiber $\mathrm{Hit}^{-1}(a)$ over
$a\in\mathcal{A}_{G_{\mathbb{C}}}$ is said to be singular when the
corresponding spectral curve $S_{a}$ defined as in (2.2) is singular. The most
singular fiber is the nilpotent cone444The name was given by Laumon [130], to
emphasize the analogy with the nilpotent cone in Lie algebra., which sits over
$\mathbf{0}\in\mathcal{A}_{G_{\mathbb{C}}}$. One of the tools to study the
nilpotent cone is the moment map $\mu$ of the $S^{1}$ action
$(E,\Phi,h)\to(E,\mathrm{e}^{i\theta}\Phi,h)$ [105, 112]. Moreover, the
nilpotent cone is preserved by the flow by $\mu$ [104, Theorem 5.2], and since
points of $\mathcal{M}_{G_{\mathbb{C}}}$ flow towards the nilpotent cone, it
encodes the topology of the moduli space.

The nilpotent cone has primarily been studied for ${\rm SL}(n,\mathbb{C})$ and
${\rm GL}(n,\mathbb{C})$, and much of its geometry remains unknown for the
moduli spaces of $G_{\mathbb{C}}$-Higgs bundles. For ${\rm SL}(n,\mathbb{C})$
and ${\rm GL}(n,\mathbb{C})$-Higgs bundles, the irreducible components of the
nilpotent cone are labeled by connected components of the fixed point set of
the $S^{1}$ action. Among these components is the moduli space $\mathcal{N}$
of semistable bundles555Given a stable bundle $E$, take $\Phi=0$, then the
Higgs bundle $(E,0)$ is stable and trivially fixed by the $S^{1}$-action..

Other singular fibers have been the subject of more recent research (e.g., see
[25, 96, 116]). In particular, in the case of ${\rm GL}(n,\mathbb{C})$-Higgs
bundles while when the spectral curve $S$ is smooth, the corresponding fiber
$\mathrm{Hit}^{-1}(S)$ can be identified with the Jacobian $\mathrm{Jac}(S)$
of all line bundles $L\rightarrow S$ of degree $0$, when the spectral curve
$S$ is not smooth, the corresponding fiber is seen to be the _compactified
Jacobian_ [26, 162] (see also [139, Fact 10.3] for a clear explanation). The
compactified Jacobian $\overline{\mathrm{Jac}}(S)$ is the moduli space of all
torsion-free rank-1 sheaves on $S$, where the usual “locally-free” condition
is missing. Moreover, when $S$ is not integral, the fine moduli space needs to
be considered. A more intuitive definition of the compactified Jacobian is the
following: consider a path of smooth curves $S_{t}$ approaching a singular
curve $S_{0}$; since the limit of $\mathrm{Jac}(S_{t})$ does not depend on the
choice of smooth family [119], this limit is $\overline{\mathrm{Jac}}(S)$.

In the case of ${\rm SL}(2,\mathbb{C})$-Higgs bundles, much work has been done
on the singular fibers of the corresponding Hitchin fibration. For example,
see [89, Section 5.2.2] for connectedness of the fiber of $\mathcal{M}_{{\rm
SL}(2,\mathbb{C})}$ when $S$ is irreducible and has only simple nodes; see
[96] for a fuller description of the singular fibers666The authors of [96]
actually study the slightly more general situation of “twisted Higgs bundles”
where the canonical bundle $K$ is replaced by a line bundle $L$ with
$\deg(L)>0$. For this setting of $L$-twisted ${\rm SL}(2,\mathbb{C})$-Higgs
bundles, the monodromy action was considered in [24]. of $\mathcal{M}_{{\rm
SL}(2,\mathbb{C})}$, and see [156] for the monodromy action around singular
fibers of the Hitchin fibration.

######  Open Question 2.6.

Building on the results for ${\rm SL}(2,\mathbb{C})$-Higgs bundles, describe
the singular fibers of the Hitchin fibration for arbitrary $G_{\mathbb{C}}$.

When the singular fiber lies above some particular types of spectral curves,
one may describe the fibers by considering some modified version of spectral
data, leading to the following natural question:

######  Open Question 2.7.

Extending on [116], obtain a geometric description of the fibers of the
Hitchin fibration of $G_{\mathbb{C}}$-Higgs bundles which lie over points of
the Hitchin base defining curves through (2.2) with equation
$\det(\Phi-\eta\operatorname{Id})=P^{k}(\eta)$ for $k\geq 2$, and for which
$\\{P(\eta)=0\\}$ is generically smooth $($and thus defines itself a smooth
spectral curve$)$.

When the spectral curve has defining equation
$\det(\Phi-\eta\operatorname{Id})=P^{2}(\eta)$, components of the fiber were
studied in [116], and the full fibers of the Hitchin fibration with that base
point are described in [43]. From a different perspective, in terms of fiber
products of spectral curves, certain singular spectral curves were considered
in [42]. While not much is known about the singular fibers of the Hitchin
fibration for $G_{\mathbb{C}}$-Higgs bundles, one may deduce properties of the
whole moduli space by considering the monodromy action of the natural
Gauss–Manin connection of the fibration. In the case of ${\rm
SL}(2,\mathbb{C})$-Higgs bundles, the study of the monodromy was done in
[156], where an explicit formula was used to understand connectivity of the
moduli space. The work was later extended to twisted rank 2 Higgs bundles in
[24], and to all ${\rm SL}(n,\mathbb{C})$-Higgs bundles in [18]. However, the
general understanding of the monodromy action for other groups remains open:

######  Open Question 2.8.

Give a geometric description of the monodromy action for the Hitchin fibration
of $G_{\mathbb{C}}$-Higgs bundles.

Finally, since the moduli spaces $\mathcal{M}_{G_{\mathbb{C}}}$ are often not
smooth, it is important to understand the singularities of
$\mathcal{M}_{G_{\mathbb{C}}}$. For a beautiful survey on recent developments
in the theory of moduli spaces of sheaves on projective varieties, and
implications for Higgs bundles, the reader may refer to [140]. In the case of
parabolic Higgs bundles, a description of the Hitchin fibration was given
recently in [21], and it would be very interesting to understand the above
considerations and open questions in this other setting.

##  3 Higgs bundles and limiting structures

Many conjectures from mathematics and physics about
$\mathcal{M}_{G_{\mathbb{C}}}$ remain open because they require a finer
knowledge of the ends of the moduli space than what is provided by traditional
algebro-geometric techniques. In this section we shall restrict our attention
to ${\rm SL}(2,\mathbb{C})$-Higgs bundles, and note that for other groups most
of the questions mentioned here remain open. In this setting, one has the
following conjecture of Hausel:

######  Conjecture 3.1 ([105, Conjecture 1]).

There are no non-trivial $L^{2}$ harmonic forms on the Hitchin moduli space.

There is a similar conjecture for the moduli space of monopoles which is
called the Sen Conjecture. By analogy, the conjecture for the Hitchin moduli
space is sometimes called the Sen Conjecture as well. In order to obtain a
finer knowledge of the ends of $\mathcal{M}_{{\rm SL}(2,\mathbb{C})}$, finer
descriptions of solutions of Hitchin’s equations near the ends are needed. A
number of recent results [136, 137, 138, 142] demonstrate the power of
constructive analytic techniques for describing the ends of the Hitchin moduli
space.

Fixing a stable Higgs bundle $(E,\Phi)$ in $\mathcal{M}_{{\rm
SL}(2,\mathbb{C})}$, the ray of Higgs bundles with harmonic metric
$(E,t\Phi,h_{t})$ approaches the ends of the moduli space as $t\to\infty$. In
order to understand what the behavior of the harmonic metrics $h_{t}$ as
$t\to\infty$ is, note that in the limit the curvature
$F_{D(\overline{\partial}_{E},h_{t})}$ concentrates at the ramification points
$Z\subset\Sigma$ of $\pi\colon S\rightarrow\Sigma$ and vanishes everywhere
else. The decay is exponential in $t$, leading to the following result:

######  Theorem 3.2 ([142, Theorem 2.7]).

On a compact subset $\overline{U}$ of $\Sigma-Z$, there exist positive
constants $c_{0}$ and $\epsilon_{0}$ such at any point in $\overline{U}$

$\displaystyle\big{|}[\varphi,\varphi^{\dagger_{h_{t}}}]\big{|}_{h_{t},g_{\Sigma}}\leq
c_{0}\exp(-\epsilon_{0}t).$

Consequently, the limiting hermitian metric is singular at the ramification
points $Z\subset\Sigma$ and

$\displaystyle
F_{D(\overline{\partial}_{E},h_{\infty})}=0,\qquad[\Phi,\Phi^{*h_{\infty}}]=0,\qquad\overline{\partial}_{E}\Phi=0.$
(3.1)

It is often said that Hitchin’s equations “abelianize” asymptotically. The
vanishing of the Lie bracket in (3.1) reflects the deeper expectation that the
metric $h_{\infty}$ is the pushforward of a singular harmonic metric $h_{L}$
on the spectral data $L\rightarrow S$. This has been proved when $S$ is
smooth, i.e., $(E,\Phi)\in\mathcal{M}^{\prime}_{{\rm SL}(n,C)}$, by
Mazzeo–Swoboda–Weiss–Witt when $n=2$ [137] and generalized to any rank by
Fredrickson [85]. In [142, Theorem 5.1] Mochizuki proves this for all of
$\mathcal{M}_{{\rm SL}(2,\mathbb{C})}$, making no assumptions about the
smoothness of $S$.

###  3.1 The ends of the regular locus

More is known about the ends of the regular locus $\mathcal{M}^{\prime}_{{\rm
SL}(2,\mathbb{C})}$. For $t$ large but finite, the harmonic metric $h_{t}$ is
close to an approximate harmonic metric $h_{t}^{\mathrm{approx}}$, constructed
by desingularizing $h_{\infty}$ [137]. On small disks around points in
$Z\subset\Sigma$, the approximate metric is equal to a smooth local model
solution

$\displaystyle\overline{\partial}_{E}=\overline{\partial},\qquad
t\Phi=t\begin{pmatrix}0&1\\\ z&0\end{pmatrix}\mathrm{d}z,\qquad
h_{t}^{\mathrm{model}}=\begin{pmatrix}|z|^{1/2}{\mathrm{e}}^{u_{t}(|z|)}&\\\
&|z|^{-1/2}{\mathrm{e}}^{-u_{t}(|z|)}\end{pmatrix},$

where $u_{t}(|z|)$ comes as a solution of a $t$-rescaled Painlevé III ODE with
boundary conditions given by $u_{t}(|z|)\sim-\frac{1}{2}\log(|z|)$ near
$|z|=0$ (so that $h_{t}$ is smooth), and
$\lim\limits_{|z|\to\infty}u_{t}(|z|)=0$. Note that in this same local gauge,
the singular limiting metric $h_{\infty}$ would be equal to

$\displaystyle h_{\infty}=\begin{pmatrix}|z|^{1/2}&\\\
&|z|^{-1/2}\end{pmatrix}.$

Outside of small disks around $Z\subset\Sigma$, the approximate harmonic
metric $h_{t}^{\mathrm{approx}}$ is equal to $h_{\infty}$.

The approximate description of $h_{t}$ by $h_{t}^{\mathrm{approx}}$ has
already been useful in [138] for describing the hyperkähler metric on
$\mathcal{M}^{\prime}_{{\rm SL}(2,\mathbb{C})}$ near the ends. There are two
natural hyperkähler metrics on $\mathcal{M}^{\prime}_{{\rm
SL}(2,\mathbb{C})}$: first, the hyperkähler metric $g_{L^{2}}$ on
$\mathcal{M}_{{\rm SL}(2,\mathbb{C})}$ restricts to
$\mathcal{M}^{\prime}_{{\rm SL}(2,\mathbb{C})}$; second, there is a metric
$g_{\mathrm{sf}}$ on $\mathcal{M}^{\prime}_{{\rm SL}(2,\mathbb{C})}$, known as
the semiflat metric because $g_{\mathrm{sf}}$ is flat on the half-dimensional
torus fibers [87]. The metric $g_{L^{2}}$ comes from taking the $L^{2}$ metric
on triples $(E,t\Phi,h_{t})$, while the semiflat metric $g_{\mathrm{sf}}$
comes from taking the $L^{2}$ metric on the moduli space of triples
$(E,t\Phi,h_{\infty})$ [138]. Consequently, Mazzeo–Swoboda–Weiss–Witt are able
to describe the difference between $g_{L^{2}}$ and $g_{\mathrm{sf}}$ using
their careful description of $h_{t}$ and $h_{\infty}$. They prove

######  Theorem 3.3 ([138, Theorem 1.2]).

The metric $g_{L^{2}}$ admits an asymptotic expansion

$\displaystyle
g_{L^{2}}=g_{\mathrm{sf}}+\sum_{j=0}^{\infty}t^{\frac{4-j}{3}}G_{j}+\mathcal{O}\big{(}e^{-\beta
t}\big{)}$ (3.2)

as $t\to\infty$. Here each $G_{j}$ is a symmetric two-tensor.

######  Open Question 3.4.

Are the polynomial correction terms $G_{j}$ non-zero?

Separate from this description of the ends with PDE techniques, a remarkable
conjectural picture of the asymptotic geometry of $\mathcal{M}_{{\rm
SL}(n,\mathbb{C})}$ has emerged from physics in the work of Gaiotto, Moore,
and Neitzke [91, 150]. Their starting point is the semiflat metric
$g_{\mathrm{sf}}$ on $\mathcal{M}^{\prime}_{{\rm SL}(n,\mathbb{C})}$ which is
too homogeneous to extend to all of $\mathcal{M}_{{\rm SL}(n,\mathbb{C})}$.
They give a recipe for constructing a complete hyperkähler metric
$g_{\mathrm{GMN}}$ on $\mathcal{M}_{{\rm SL}(n,\mathbb{C})}$ differing from
$g_{\mathrm{sf}}$ by “quantum corrections” which are computed by counting
certain BPS states in supersymmetric field theory. In particular, the quantum
corrections have the following size

$\displaystyle g_{\mathrm{GMN}}=g_{\mathrm{sf}}+O\left(\sum_{\gamma\in
H_{1}(S_{a},\mathbb{Z})}\Omega(\gamma,a)\mathrm{e}^{-t\big{|}\int_{\gamma}\eta\big{|}}\right)$
(3.3)

as $t\to\infty$. In this formula, as in the previous sections, $a$ is a point
in $\mathcal{A}_{{\rm SL}(n,\mathbb{C})}$, the corresponding spectral curve is
$S_{a}$ with tautological one-form $\eta$; and the sum is over all loops
$\gamma$ in $S_{a}$. The $\Omega(\gamma,a)$ are BPS counts in supersymmetric
field theory. These are $\mathbb{Z}$-valued and piecewise-constant, jumping
across certain walls in the parameter space. The jumps are constrained to
satisfy the Kontesevich–Soibelman wall-crossing formula [128], and thus
$g_{\mathrm{GMN}}$ is smooth. Moreover, Gaiotto–Moore–Neitzke conjecture

######  Conjecture 3.5 ([91]).

The hyperkähler metric $g_{\mathrm{GMN}}$ on the moduli space is the natural
hyperkähler metric $g_{L^{2}}$ on $\mathcal{M}_{{\rm SL}(n,\mathbb{C})}$.

If the conjecture of Gaiotto–Moore–Neitzke is correct, then all of the
symmetric two-tensors $G_{j}$ appearing in (3.2) vanish, answering Open
Question 3.4. Note that there is already evidence that this happens on the
Hitchin section [68]. As $a\in\mathcal{A}_{{\rm SL}(n,\mathbb{C})}$ approaches
the singular locus $\mathcal{A}^{\mathrm{sing}}_{{\rm SL}(n,\mathbb{C})}$, the
spectral curves $S_{a}$ become singular. In particular, there is at least one
loop $\gamma_{0}$ on $S_{a}$ which pinches; hence the “quantum correction” in
(3.3) corresponding to $\gamma_{0}$ is not exponentially suppressed as $t$
approaches $\infty$. While we have focused here on ${\rm
SL}(2,\mathbb{C})$-Higgs bundles, equivalent questions (and conjectures) may
be asked for more general $G_{\mathbb{C}}$-Higgs bundles, providing several
new lines of research:

######  Open Question 3.6.

Generalize the above results to the case of $G_{\mathbb{C}}$-Higgs bundles for
arbitrary $G_{\mathbb{C}}$.

##  4 Higgs bundles on singular curves

While we have considered before $G_{\mathbb{C}}$-Higgs bundles on a compact
Riemann surface $\Sigma$, principal Higgs bundles can also be defined over
singular spaces $X$, and in particular, over singular curves. For simplicity,
we shall begin by considering nodal curves $X$, i.e., irreducible projective
curves $X$ whose singularities are nodes. In order to generalize the notion of
vector bundles on smooth projective curves, one may consider the torsion-free
sheaves on the nodal curve $X$. Through the work of Bhosle in [31], the
category of torsion-free sheaves on a nodal curve $X$ and category of
generalized parabolic bundles over its normalization $\widetilde{X}$ are
equivalent. Moreover, a first general construction of compactified moduli
spaces for semistable $G_{\mathbb{C}}$-bundles on an irreducible complex
projective curve $X$ with exactly one node was given in [163].

###  4.1 Singular principal $\boldsymbol{G_{\mathbb{C}}}$-Higgs bundles

Through the work of [163], it was shown in [95] that one can treat a principal
$G_{\mathbb{C}}$-Higgs bundle over a nodal curve $X$ as a particular type of
vector bundle on the normalization of the curve called a descending bundle,
objects which are in one-to-one correspondence with the following singular
principal $G_{\mathbb{C}}$-Higgs bundles.

######  Definition 4.1.

A singular principal $G_{\mathbb{C}}$-Higgs bundle is a triple
$(\mathcal{E},\tau,\Phi)$ where

  * •

$\mathcal{E}$ is a locally free sheaf;

  * •

$\tau\colon{\rm Sym}^{*}(\mathcal{E}\otimes
V)^{G_{\mathbb{C}}}\rightarrow\mathcal{O}_{X}$, for a fixed faithful
representation $G_{\mathbb{C}}\rightarrow{\rm GL}(V)$ of $G_{\mathbb{C}}$;

  * •

$\Phi\colon X\rightarrow{\rm End}(\mathcal{E})\otimes\Omega_{X}^{1}$ is a
section;

######  Open Question 4.2.

Give, if possible, a notion of the Hitchin fibration for these singular
principal $G_{\mathbb{C}}$-Higgs bundles, and describe the geometry of the
smooth and singular fibers.

While Schmitt [163] and Bhosle [31] proved that there is a moduli space of
singular $G_{\mathbb{C}}$-bundles on $X$ with good specialization properties,
Seshadri gave a further study of the spaces of torsion-free sheaves on nodal
curves and generalizations to, among others, ramified $G_{\mathbb{C}}$-bundles
[167]. When considering degenerations of moduli spaces of vector bundles on
curves, which are closely related to the singular $G_{\mathbb{C}}$-bundles
mentioned above, the reader may want to consider the conjectures presented in
[166].

Just as one may define parabolic Higgs bundles on Riemann surfaces to consider
Higgs bundles on marked curves, one may extend these objects to singular
curves. Extending the notion of a parabolic vector bundle on a smooth curve,
Bhosle defined generalized parabolic sheaves (GPS) on any integral projective
curve $X$ [32] and generalized parabolic bundles (GPB) [33]. She constructed
the moduli spaces of GPS and GPB, and studied the correspondences appearing
when curves $X$ are obtained from blowing up finitely many nodes in a space
$Y$. Moreover, Bhosle also extended the notion of parabolic Higgs bundles to
that of generalized parabolic Higgs bundles (GPH) on the normalization $X$ of
an integral projective curve $Y$ [32, 33, 34]. In particular, she constructed
a birational morphism from the moduli space of good GPH on $X$ to the moduli
space of Higgs bundles on $Y$, and defined a proper Hitchin map on the space
of GPH. In this context, the following question is natural.

######  Open Question 4.3.

Generalize [34] to define generalized parabolic $G_{\mathbb{C}}$-Higgs bundles
on $X$, as well as a Hitchin fibration.

Moreover, the open questions mentioned for the moduli spaces of classical
$G_{\mathbb{C}}$-Higgs bundles may also be considered both for parabolic Higgs
bundles on Riemann surfaces $\Sigma$, and for parabolic Higgs bundles on
integral projective curves $X$. In particular, Bhosle studied recently the
relationship between Higgs bundles and the compactified Jacobian of a spectral
curve [34], considering Higgs bundles on the normalization $X$ of integral
projective curves $Y$, leading to an analogous question to that stated for
classical Higgs bundles:

######  Open Question 4.4.

Obtain a geometric description of the singular fibers of the Hitchin fibration
for generalized parabolic $G_{\mathbb{C}}$-Higgs bundles on $X$.

The study of Higgs bundles on singular curves may also be considered in a
limiting setting, where one begins with Higgs bundles on a smooth curve and
parametrically tunes the curve to degenerate to a singular curve. The
particular case of vector bundles on smooth curves degenerating to an
irreducible curve with one double point was considered in [122]. The case of
the degeneration of the moduli space of Higgs bundles on smooth projective
curves when the curve degenerates to an irreducible curve with a single node
was studied in [17].

######  Open Question 4.5.

Obtain equivalent degenerations to those in [17, 122] for the moduli spaces of
$G_{\mathbb{C}}$-Higgs bundles.

In particular, as explained in [17], their degeneration is analogous to the
models constructed by Gieseker and Nagaraj–Seshadri for the case of the moduli
spaces for which the Higgs structure is trivial. It should be noted that in
[17] the authors also construct a corresponding canonical relative proper
Hitchin map, whose fiber provides a new compactification of the Picard variety
of smooth curves with normal crossing singularities. In their setting, the
single node on the base curve leads to an irreducible vine curve with
$n$-nodes appearing as the spectral curve. It would then seem natural that the
quasi-abelianization of [17] (resembling Hitchin’s classical abelianization)
could be generalized.

######  Open Question 4.6.

Describe the quasi-abelianization of the moduli space of
$G_{\mathbb{C}}$-Higgs bundles for different degenerations, following the
techniques of [17].

Finally, since it is important to find natural compactifications of open
moduli, and torsion-free sheaves on nodal curves play an important role in
[152] within the study of the compactification of the universal moduli space
of slope-semistable vector bundles over the compactification
$\overline{M}_{g}$ of the moduli space of genus $g$ curves, it is natural to
ask the following:

######  Open Question 4.7.

Understand the relation between the degenerations of moduli spaces of Higgs
bundles above, and the known compactifications of
$\mathcal{M}_{G_{\mathbb{C}}}$.

Since one has the correspondence between Langlands dual groups
$G_{\mathbb{C}}$ and ${}^{L}G_{\mathbb{C}}$, once the corresponding moduli
spaces are understood, and the Hitchin fibrations are shown to exist, one may
also want to consider, if possible, the duality between the fibrations. In
particular, the work of Arinkin [11] for rank 2 Higgs bundles on the auto-
duality of compactified Jacobians for curves with plane singularities would
allow one to understand both this setting, as well as the one of singular
fibers of the classical Hitchin fibration, leading to an intermediate
question:

######  Open Question 4.8.

Extend the constructions of [11] to the setting of generalized parabolic
$G_{\mathbb{C}}$-Higgs bundles à la Bhosle [34].

##  5 Higgs bundles and branes within singular fibers

The appearance of Higgs bundles (and flat connections) within string theory
and the geometric Langlands program has led researchers to study the derived
category of coherent sheaves and the Fukaya category of these moduli spaces.
Therefore, it has become fundamental to understand Lagrangian submanifolds of
the moduli space of Higgs bundles supporting holomorphic sheaves ($A$-branes),
and their dual objects ($B$-branes). For ${}^{L}G_{\mathbb{C}}$ the Langlands
dual group of $G_{\mathbb{C}}$, there is a correspondence between invariant
polynomials for $G_{\mathbb{C}}$ and ${}^{L}G_{\mathbb{C}}$ giving an
identification
$\mathcal{A}_{G_{\mathbb{C}}}\simeq\mathcal{A}_{{}^{L}G_{\mathbb{C}}}$ of the
Hitchin bases.

###  5.1 Construction of branes

Through the Hitchin fibrations, the two moduli spaces
$\mathcal{M}_{G_{\mathbb{C}}}$ and $\mathcal{M}_{{}^{L}G_{\mathbb{C}}}$ are
then torus fibrations over a common base and their non-singular fibers are
dual abelian varieties [65, 106], answering some of the conjectures presented
in [172]. Kapustin and Witten give a physical interpretation of this in terms
of S-duality, using it as the basis for their approach to the geometric
Langlands program [121]. In this approach a crucial role is played by the
various types of branes and their transformation under mirror symmetry.
Adopting the language of physicists, a Lagrangian submanifold of a symplectic
manifold supporting a hyperholomorphic sheaf is called (the base of) an
A-brane, and a complex submanifold supporting a hyperholomorphic sheaf is (the
base of) a B-brane. A submanifold of a hyperkähler manifold may be of type $A$
or $B$ with respect to each of the complex or symplectic structures, and thus
choosing a triple of structures one may speak of branes of type $(B,B,B)$,
$(B,A,A)$, $(A,B,A)$ and $(A,A,B)$777One should note that since the complex
structures satisfy the quaternionic equations, and the symplectic forms are
obtained through them, branes of types $(A,A,A)$, $(A,B,B)$, $(B,A,B)$,
$(B,B,A)$ do not exist.. Throughout these notes we shall follow the convention
in [121] and fix the three complex structures $I$, $J$ and $K$, such that $I$
is induced from the Riemann surface $\Sigma$, and $J$ from the complex group
$G_{\mathbb{C}}$.

It is hence natural to seek constructions of different families of branes
inside the moduli space $\mathcal{M}_{G_{c}}$, understand their appearance
within the Hitchin fibration, and describe their mirror families of branes. In
the context of Higgs bundles, branes were first considered by Kapustin and
Witten in 2006 in [121], where much attention was given to the $(B,A,A)$-brane
of $G$-Higgs bundles inside $\mathcal{M}_{G_{\mathbb{C}}}$, where $G$ is a
real form of the complex Lie group $G_{\mathbb{C}}$. Soon after, examples of
brane dualities were considered in [100]; in particular the case of $G$-Higgs
bundles for compact real forms $G$ of low rank was considered, in which case
the $(B,A,A)$-brane lies completely inside the nilpotent cone. While partial
results exist for these branes over singular fibers, the more global picture
remains unknown.

######  Open Question 5.1.

Give a geometric description of all $(B,A,A)$-branes of $G$-Higgs bundles
which live completely inside the most singular fiber of the Hitchin fibration,
the nilpotent cone of $\mathcal{M}_{G_{\mathbb{C}}}$.

The study of branes within moduli spaces of Higgs bundles continued evolving
slowly, until natural generic methods to construct families of all types of
branes in $\mathcal{M}_{G_{\mathbb{C}}}$ were introduced in [23]. These branes
were constructed as fixed point sets of certain families of involutions on the
moduli spaces of complex Higgs bundles. Consider $\sigma$ an anti-holomorphic
involution fixing a real form $G$ of $G_{\mathbb{C}}$, and $\rho$ the anti-
holomorphic involution fixing the compact real form of $G_{\mathbb{C}}$. Then,
through the Cartan involution $\theta=\sigma\circ\rho$ of a real form $G$ of
$G_{\mathbb{C}}$, one may define

$\displaystyle
i_{1}\big{(}\bar{\partial}_{E},\Phi\big{)}:=\big{(}\theta(\bar{\partial}_{E}\big{)},-\theta(\Phi)).$

Moreover, a real structure $f\colon\Sigma\to\Sigma$ on $\Sigma$ induces an
involution

$\displaystyle
i_{2}(\bar{\partial}_{E},\Phi):=\big{(}f^{*}(\partial_{E}),f^{*}(\Phi^{*})\big{)}=\big{(}f^{*}\big{(}\rho\big{(}\bar{\partial}_{E}\big{)}\big{)},-f^{*}(\rho(\Phi))\big{)}.$

Lastly, by setting $i_{3}:=i_{1}\circ i_{2}$, one may define a third
involution:

$\displaystyle
i_{3}(\bar{\partial}_{E},\Phi)=\big{(}f^{*}\sigma\big{(}\bar{\partial}_{E}\big{)},f^{*}\sigma(\Phi)\big{)}.$

The fixed point sets of the induced involutions $i_{1}$, $i_{2}$, $i_{3}$
introduced in [23] are branes of type $(B,A,A),$ $(A,B,A)$ and $(A,A,B)$
respectively, and through the associated spectral data their topological
invariants can be described using $KO$, $KR$ and equivariant $K$-theory. In
particular, it was shown that among the fixed points of $i_{1}$ are solutions
to the Hitchin equations with holonomy in $G$. Moreover, those fixed by
$i_{2}$ were shown to give real integrable systems, fibered as a Lagrangian
fibration over a real slice of the Hitchin base [22]. In order to construct
the fourth type of branes, $(B,B,B)$-branes, one may consider Higgs bundles
for a complex subgroup of $G_{\mathbb{C}}$, but these branes would not appear
through a symmetry in the spirit of the above constructions888An instance of
this setting was recently explored in [84], where Langlands duality was
studied for branes appearing though Borel subgroups.. On the other hand, it is
shown in [111] that one may construct $(B,B,B)$-branes by considering the
subspaces of $\Gamma$-equivariant Higgs bundles for $\Gamma$ a finite group
acting on the Riemann surface $\Sigma$. In particular, it was shown in [111]
that for $G_{\mathbb{C}}={\rm SL}(2,\mathbb{C})$, these branes would be mid-
dimensional only under very restrictive conditions, and no equivalent result
has been shown for higher rank groups.

######  Open Question 5.2.

Describe the mid-dimensional $(B,B,B)$-branes appearing through
$\Gamma$-equivariant Higgs bundles when $\Gamma$ is a group of any rank, and
classify those components completely contained in the singular locus of the
Hitchin fibration.

The construction of branes following the procedures of [22, 23] have recently
been generalized to the space of framed instantons [82], Higgs bundles over K3
surfaces [83], Higgs bundles over elliptic curves [37], quiver varieties [117,
118], more general hyperkähler spaces [38], and principal Schottky bundles
[45]. Moreover, many of the geometric properties of the branes in [22, 23] are
yet unknown, and researchers continue to study them (e.g., see [16, 38]). In
the case of finite group actions, the branes introduced in [111] were later
studied in [94] from the perspective of character varieties, and many of their
properties remain unknown.

######  Open Question 5.3.

In the spirit of [19] and [20], describe the Brauer groups and automorphism
groups of the branes mentioned above.

Finally, it should be mentioned that in the last couple of years researchers
have found other novel ways in which branes can be constructed within the
moduli space of Higgs bundles, and which are yet to be generalized to other
settings. Examples of these are Nahm branes [81], branes appearing through
spinors [115], through moment maps [90], and through Borel subgroups [84].
However, since Lagrangian branes can appear in any of three types, it is of
interest to understand families of Lagrangian branes of each types which are
related in some geometric fashion, but it is not yet known of other triples of
families of branes appearing within hyperkähler spaces other than the ones
obtained thought the methods of [23].

######  Open Question 5.4.

Construct natural triples of families of branes in
$\mathcal{M}_{G_{\mathbb{C}}}$, and more generally, within hyperkähler spaces.

Considering the appearance of branes through real structures on Riemann
surfaces, one should also be able to impose other structures on the surfaces
to construct novel branes. A canonical example of such structure would be that
of a log-symplectic structure on $\Sigma$, also called a b-Poisson structure.
These structures are given by Poisson structures
$\pi\in\mathfrak{X}^{2}(\Sigma)$ for which $\pi$ has only non-degenerate
zeros. In particular, $\pi$ is generically symplectic. These structures were
completely classified by O. Radko [155], where she noted that every surface
(orientable or not) has a log-symplectic structure. The sets of invariants of
log-symplectic structures are:

  * •

The zero curves $\gamma_{1},\ldots,\gamma_{n}$, taken with orientation defined
by $\pi$;

  * •

The periods associated to each $\gamma_{i}$;

  * •

The volume invariant of $\pi$.

Since there is a natural relation between the data defining log-symplectic
structures $\pi$ on $\Sigma$ as in [155], and the real structures
$f\colon\Sigma\rightarrow\Sigma$ considered in [22] to define
$(A,B,A)$-branes, a natural question is the following.

######  Open Question 5.5.

Which branes of Higgs bundles are characterized by log-symplectic structures,
and how do these relate to the $(A,B,A)$-branes introduced in [22, 23]?

###  5.2 Langlands duality

While it is understood that Langlands duality exchanges brane types, the exact
correspondence is not yet known999Strictly speaking, Langlands is a
correspondence between local systems on $\Sigma$ (or more precisely, coherent
sheaves on the moduli space), and $D$-modules over the moduli stack Bun of
bundles on $\Sigma$.. As mentioned before, the first instances of the
correspondence being studied for low rank Higgs bundles appeared in [121] and
[100], but no proof has yet been given of a pair of branes of Higgs bundles
being dual. In the case of $(B,A,A)$-branes of $G$-Higgs bundles, it was
conjectured in [23] how the duality should appear:

######  Conjecture 5.6 ([23, Section 7]).

The support of the dual $(B,B,B)$-brane in
$\mathcal{M}_{{}^{L}G_{\mathbb{C}}}$ to the $(B,A,A)$-brane
$\mathcal{M}_{G}\subset\mathcal{M}_{G_{\mathbb{C}}}$ is the moduli space
$\mathcal{M}_{\check{H}}\subset\mathcal{M}_{{}^{L}G_{\mathbb{C}}}$ of
$\check{H}$-Higgs bundles for $\check{H}$ the group associated to the Lie
algebra $\check{\mathfrak{h}}$ in [148, Table 1].

Support for this conjecture is given in [114] for the group $G=U(m,m)$ by
considering the spectral data description of the brane in [158], and in [25,
161] for the groups $G={\rm SO}(p+q,p)$ and ${\rm Sp}(2p+2q,2p)$. One should
note that, in contrast with the $(A,B,A)$ and $(A,A,B)$ branes considered in
[23], for any $q>1$ the $(B,A,A)$-branes studied in [25] lie completely over
the singular locus of the Hitchin fibrations. For these branes of orthogonal
Higgs bundles, support for Conjecture 5.6 is obtained from the description of
how the brane intersects the most generic fibers of the Hitchin fibration:
indeed the rank of the hyperholomorphic sheaf depends on the number of
components in this intersection, which remains constant for different $q$,
leading to the following conjecture.

######  Conjecture 5.7 ([25, Section 8]).

For all $q$ even $($and for all $q$ odd$)$, the $(B,A,A)$-brane of ${\rm
SO}(p+q,p)$-Higgs bundles has dual $(B,B,B)$-brane obtained by considering the
same base of Conjecture 5.6 and the same hyperholomorphic bundle supported on
it101010This hyperholomorphic bundle being, for example, the one introduced by
Hitchin in [114]., and it is only the way in which these spaces are embedded
into the different Langlands dual moduli spaces which depends on $q$.

In particular, the support of branes for $q$ odd and $q$ even are dual to each
other as hyperkähler moduli spaces of complex Higgs bundles. From the
description of the invariant polynomials appearing for $(B,A,A)$-branes of
$G$-Higgs bundles in [157], one can see that the majority of these branes lie
over the singular locus of the Hitchin fibration. However, it is still
possible to describe the intersection of these branes with the most regular of
singular fibers. For example, it was shown in [116] that the generic
intersections of the $(B,A,A)$-branes of ${\rm SL}(m,\mathbb{H})$, ${\rm
SO}(2n,\mathbb{H})$ and ${\rm Sp}(2m,2m)$-Higgs bundles with the fibers of the
Hitchin fibrations are not abelian varieties, but are instead moduli spaces of
rank 2 bundles on a spectral curve, satisfying certain natural stability
conditions. In order to fully understand Langlands duality for branes one
would need to understand how different branes in a moduli space intersect, and
thus the particular case of branes within the nilpotent cone is of much
importance.

######  Open Question 5.8.

Describe the intersections and relations between all $(B,A,A)$-branes in the
nilpotent cone of $G_{\mathbb{C}}$-Higgs bundles, for $G_{\mathbb{C}}$ an
arbitrary group.

While a first step towards an answer would be to consider $(B,A,A)$-branes of
$G$-Higgs bundles, or those constructed in the papers mentioned above, a more
general perspective considering generators of the corresponding Fukaya
category would be ideal. For a short review of the open problems and
literature of this section, the reader may refer to [159] and references
therein. When studying the nilpotent cone, a few questions arise from the work
of Gukov and his colleagues in relation to quantization:

######  Open Question 5.9.

What is the brane quantization à la [100] of the branes in the nilpotent cone
mentioned in this section, and how does this relate to the curve quantization
à la [69] of the spectral curves defined by Higgs bundles in those branes?

######  Open Question 5.10.

Use the above methods to construct branes for wild Hitchin systems, and
approach Langlands duality as appearing in [101].

###  5.3 Surface group representations and $\boldsymbol{GW}$-components

Finally, it should be noted that branes which lie completely over the singular
locus of the Hitchin fibration also play an important role in representation
theory. In particular, the following has been predicted by Guichard and
Wienhard:

######  Conjecture 5.11 ([99, Conjecture 5.6]).

Additional connected components coming from positive representations
$($through the notion of $\Theta$-positivity$)$, giving further families of
higher Teichmüller spaces, appear in the moduli space of surface group
representations into ${\rm SO}(p+q,p)$ for $q\geq 1$.

In the case of the moduli space $\mathcal{M}_{{\rm SO}(p+1,p)}$, the existence
of the extra Guichard–Wienhard components (or simply GW-components) as
predicted in [99, Conjecture 5.6] is known to be true [13, 48, 49], and
moreover it was shown in [49] that those components indeed contain
$\Theta$-positive representations. From the perspective of the spectral data
description of the $(B,A,A)$-branes of ${\rm SO}(p+q,p)$-Higgs bundles of [25,
161], natural candidates for $GW$-components for arbitrary $q\in\mathbb{N}$
are the following:

######  Conjecture 5.12 ([25, Section 7]).

The natural candidates for the $GW$-components of Conjecture 5.11 conjectured
to exist by Guichard and Wienhard [99, Conjecture 5.6] are those containing
Higgs bundles whose spectral data111111The spectral data is a triple
$(L,M,\tau)$ consisting of a line bundle $L$, an orthogonal bundle $M$ (on an
auxiliary curve) and an extension class $\tau$. $(L,M,\tau)$ in [25] has the
form $(\mathcal{O},\mathcal{O}^{q},\tau)$. Alternatively, this is equivalent
to taking ${\rm SO}(p+q,p)$-Higgs bundles whose vector bundle is of form
$(W,V\oplus\mathcal{O}^{q-1})$, where the pair $(W,V)$ gives the vector
bundles of one of the ${\rm SO}(p+1,p)$-Higgs bundles in the $GW$-components
known to exist.

To prove that this actually gives the $GW$-components, the monodromy action à
la [24, 156] should be taken into consideration as well as the behavior over
singular fibers. On the symplectic side, from the study of spectral data in
[25], one can see the geometric reason for the absence of any extra
$GW$-components in the $(B,A,A)$-brane of ${\rm Sp}(2p+2q,2p)$-Higgs bundles.

##  6 Higgs bundles and Calabi–Yau geometry

Higgs bundles have played an important role in string theory in a wide range
of contexts. But one recent application has provided some of the perhaps most
surprising connections between previously unrelated aspects of geometry – that
is, the links between Hitchin systems and the geometry of singular $3$\- and
$4$-(complex)dimensional Calabi–Yau (CY) varieties.

###  6.1 Calabi–Yau integrable systems and Hitchin Systems

The first hint of such a connection appeared in [56] in which links were
developed between Calabi–Yau integrable systems and Hitchin Systems. Briefly,
as described in Section 2, the Hitchin system forms an integrable system
through the definition of the Hitchin fibration (2.1), whose generic fibers
are even abelian varieties obtained through branched coverings of an
underlying Riemann surface $\Sigma$ with genus $g\geq 2$. On the other hand,
_Calabi–Yau_ integrable systems were first explored for families of Calabi–Yau
$3$-folds in [63, 64], where the base of the system was formed by the moduli
space of Calabi–Yau varieties in the family, and the fibers were formed by the
Deligne cohomology groups of the intermediate Jacobians:

$\displaystyle
J^{2}(X)=H^{3}(X,\mathbb{C})/\big{(}F^{2}H^{3}(X,\mathbb{C})+H^{3}(X,\mathbb{Z})\big{)}$

of the Calabi–Yau 3-folds $X$. Fiber and base fit together into a total space
carrying a holomorphic symplectic form and the fibers are Lagrangian [63].
Furthermore, in remarkable work [56] Diaconescu, Donagi, and Pantev developed
an isomorphism between Calabi–Yau integrable systems and those of Hitchin, the
DDP correspondence. More precisely, by considering a smooth projective complex
curve $\Sigma$ and an ADE group $G$, for a fixed pair $(\Sigma,G)$ they
constructed a family of quasi-projective (i.e., _non-compact_) CY $3$-folds
(defined as $\operatorname{Tot}(V)$ for a rank 2 vector bundle, $V$,
satisfying $\det(V)=K_{\Sigma}$).

Treating the moduli space of the non-compact CY manifold as the base of a
Hitchin integrable system for the group $G$, a correspondence between the CY
integrable system (whose fibers are the intermediate Jacobians of a family of
non-compact CY 3-folds) and that of the Hitchin system (whose fibers are Prym
varieties of the corresponding spectral covers) was explicitly laid out for
the Lie groups $A_{k}$, but the description is only valid away from the
discriminant. This mapping between Hitchin and CY integrable systems was
nicely generalized via a sheaf-theoretic approach to the remaining simple Lie
groups $B_{k}$, $C_{k}$, $F_{4}$ and $G_{2}$ in [27].

This important correspondence found a ready audience within string theory in
the context of F-theory [177] – a geometric approach to compactifications of
the type IIB string with non-trivial axio-dilaton backgrounds – in which the
effective physics of the type IIB compactification to $(12-2n)$ spacetime
dimensions is encoded in the geometry of an elliptically fibered (or more
generally genus one fibered), complex Calabi–Yau $n$-fold, $\pi\colon X_{n}\to
B_{n-1}$. The degeneration of the elliptic fibers encode information about a
Lie group, $G$, corresponding to D7-branes wrapping the discriminant locus of
the elliptic fibration (see Section 7 for further details). From a physical
perspective, the Calabi–Yau geometry is a tool to investigate intersecting
brane theories, which are innately linked to Higgs bundles. Within a “local”
description of F-theory, intersecting branes (wrapping a sub-variety
$\Sigma\subset X_{n}$) come equipped with an adjoint field $\Phi$ (the Higgs
field) which parametrizes normal motions of a stack of branes. Matter fields
are fluctuations around the background $\langle\Phi\rangle$ and Yukawa
couplings measure obstructions to extend these solutions beyond the linear
order. Usually the Higgs field $\Phi$ is taken to live in a Cartan subspace of
the Lie algebra so that only the eigenvalues of $\Phi$ are relevant. But this
seems to be an incomplete description in many situations relevant to
interesting physical models.

###  6.2 T-branes and Hitchin systems

Usually the connection with F-theory is made using the spectral cover, defined
through $\\{\det(\Phi-\eta\operatorname{Id})=0\\}$, and which in the case that
$\Sigma\subset X$ (with $X$ a CY $n$-fold) locally defines the CY as the
normal cone of $\Sigma$. However, the spectral cover will not accurately
parameterize the local geometry of $X$ when the Higgs field is non-
diagonalizable (for example when $\Phi$ is upper triangular) [62]. So-called
_T-branes_ are non-Abelian bound states that generalize intersecting branes
and admit a matrix of normal deformations (or Higgs field) that is nilpotent
over some loci [47, 66, 67]. Mathematically, T-branes correspond to singular
fibers in the Hitchin fibration.

As they first originated in the physics literature, the Hitchin systems
corresponding to T-branes were not explicitly linked geometrically to the
background Calabi–Yau elliptic fibrations of F-theory. A first step in this
direction was taken in [8] which attempted to extend some of the links
developed in [56] to _compact Calabi–Yau varieties_. A limiting mixed Hodge
structure analysis was employed to study the form of the intermediate Jacobian
of CY 3-folds in the limit that the geometry became singular. In certain
singular limits of the elliptic fibration, the degeneration of the
intermediate Jacobian, $J^{2}(X)$, leads to _an emergent Hitchin system_ –
i.e., generates the moduli space of Higgs bundles defined over the
discriminant locus of the fibration [8, 64]. In [8] T-branes were explored in
the context of six-dimensional F-theory vacua, that is using compactifications
of F-theory on _singular_ elliptically fibered Calabi–Yau $3$-folds,
$\pi\colon X_{3}\to B_{2}$.

The intrinsic intersecting brane Hitchin system was defined over a curve
$\Sigma\subset B$ in the base of the elliptic fibration, obtained through a
component of the discriminant locus ($\Delta=0$) describing degenerating
fibers. Upon a crepant resolution of the singular variety, it was argued that
the geometric remnants of T-branes correspond to periods of the three-form
potential of F-theory valued in the intermediate Jacobian of a now smooth
Calabi–Yau 3-fold. Moreover, in [8] a partial compactification of the DDP
correspondence was established and it was demonstrated that the Hitchin system
defined on the discriminant locus is contained in the local part of the
(compact) Calabi–Yau integrable system:

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.39583pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 23.39583pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi^{*}\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.58478pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
8.39583pt\raise-22.0882pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
35.22931pt\raise-18.77776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.76418pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.76418pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
75.34752pt\raise-14.49998pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\rm Hit}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
75.34752pt\raise-19.16667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
103.63225pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-8.39583pt\raise-28.99998pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
26.7293pt\raise-28.99998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
26.7293pt\raise-28.99998pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\widetilde{M}_{\rm
cx}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.29332pt\raise-24.49304pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
62.06279pt\raise-28.99998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
62.06279pt\raise-28.99998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{M_{\rm
loc},}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$ (6.1)

where $\mathcal{M}$ and $M$ are the full Hitchin and Calabi–Yau moduli spaces,
respectively, and $\widetilde{M}_{\rm cx}$ and $M_{\rm loc}$ the complex
structure moduli spaces of the resolved Calabi–Yau geometry and local
deformations of the singular Calabi–Yau variety (preserving the form of the
singular elliptic fibers). The maps are defined such that the right map is the
Hitchin fibration and the upper left (diagonal) map is an inclusion. This
correspondence was established for singular CY 3-folds with $A_{n}$-type
singularities. See [8], Section A.4 for details.

######  Conjecture 6.1 ([8]).

The correspondence described in (6.1) can be extended to any compact,
singular, elliptically fibered CY $3$-fold with singular fibers associated to
${\mathcal{G}}$-symmetry $($co-dimension $1$ over the base and admitting a
crepant resolution, see Section 7 for details$)$, and ${\mathcal{H}}$-type
Hitchin system defined over the discriminant locus of the elliptic fibration
such that ${\mathcal{H}}\subset{\mathcal{G}}$.

Although we will not explore it in detail here, it is also expected that
correspondences between Hitchin and CY moduli spaces (in either the compact or
non-compact CY setting) should extend to the Deligne cohomology of Calabi–Yau
$4$-fold geometries [35] (see also [28, 50, 51, 52, 132] for recent progress
on T-branes) and Higgs bundles defined over complex surfaces [168].

###  6.3 Wild Hitchin systems and F-theory

An intrinsic difficulty with the Hitchin systems arising within F-theory comes
from the fact that the Higgs bundles are defined on the discriminant locus of
the elliptically fibered CY manifold – and hence not on smooth Riemann
surfaces, but rather on complex curves that are in general singular (including
sometimes non-reduced and reducible). At the singular/intersection points of
such a curve, the physical theory suggests that the associated Higgs bundles
should also exhibit singularities. That is, in this context, it is natural to
also consider stable pairs with _singular connections_ (see also the
discussion in Section 4).

Thus far, work has focused primarily on so-called parabolic Hitchin systems
[127, 168, 169] which accommodate the possibility of simple poles in the gauge
and Higgs fields at marked points on a Riemann surface. However, many
questions – of both mathematical as well as physical interest – require the
consideration of higher order singularities in the gauge fields.
Wild/irregular Higgs bundles [6, 39, 41, 86] extend this formalism to include
stable, integrable connections with irregular singularities of the form

$\displaystyle d+A_{n}\frac{dz}{z^{n}}+\cdots+A_{1}\frac{dz}{z},$

with $n>1$ and stable parabolic Higgs pairs $(E,\Phi)$ where the Higgs field
has polar parts, e.g.,

$\displaystyle T_{n}\frac{dz}{z^{n}}+\cdots+T_{1}\frac{dz}{z}.$

As in Simpson’s construction [169] for parabolic Higgs bundles, a natural
assumption for this study is that the connections and Higgs fields are
holomorphically gauge equivalent to ones with diagonal polar parts (this is
weakened slightly in [86]). This leads to a correspondence between
singularities (after diagonalizing) of the form $T_{i}=\frac{1}{2}A_{i}$ for
$i\geq 2$ [36]. The moduli space of such wild Higgs bundles was described in
[39, 40] as a hyperkähler quotient. Already these irregular Hitchin systems
have played a significant role in the geometric Langlands program [88, 101],
and string applications including topologically twisted $\mathcal{N}=4$ super
Yang–Mills theories [101, 121], particularly in so-called ’Stokes phenomena’
[178, 180] (which describe how the asymptotical behavior of the solutions
changes in different angular regions around the singularity. Stokes matrices
link the solutions in different regions and define a generalized monodromy
which plays a central role in describing a wild Hitchin moduli space).

Recent progress [9] has demonstrated that the study of ordinary smooth Hitchin
systems is insufficient in the context of $6$-dimensional F-theory
compactifications. Not only should generic CY $3$-folds have a correspondence
to parabolic or wild Hitchin systems, but in general deformations of the
singular variety ($M_{\rm loc}$ in (6.1) above) can dynamically change the
pole order of the relevant singular complexified connections appearing in the
Higgs bundles (see [7] and [9, Section 5.1] for an example). That is, by
varying the complex structure moduli of a singular CY variety, the location of
singularities in the discriminant locus $\Delta$ can be tuned to coincide.
This tuning, when viewed from the intersecting brane models, should correspond
to a parametric deformation of a Hitchin System in which the location of
simple poles are tuned and forced to coincide into higher order poles. In many
instances it seems this tuning can be done without changing the dimension of
the underlying CY/Hitchin moduli space.

######  Conjecture 6.2 ([9]).

There exists a flat morphism between the moduli spaces ${\mathcal{M}}_{\rm
par}$ and ${\mathcal{M}}_{\rm wild}$ in the case of a singular parabolic Higgs
bundle with $n$-simple poles in its connection and that of a wild Higgs bundle
with a single, higher order pole of order $n$.

This limiting process in the context of CY varieties also leads to the open
question:

######  Open Question 6.3 ([9]).

Does the Stokes phenomenon exhibited by wild Higgs bundles have an analog in
the moduli space of singular, elliptically fibered CY geometries or CY
integrable systems?

###  6.4 Singular CY varieties

Finally, it should be noted that there are likely many unexplored links
between classification problems in parabolic/wild Hitchin Systems and singular
CY varieties. In general, the criteria for CY 3-folds to exhibit a generic
singularity everywhere in its complex structure moduli space has attracted
interest from the physics community in the context of “non-Higgsable clusters”
[10, 103, 143, 146]. The maps in (6.1) embedding the Hitchin moduli space into
that of the singular CY variety indicate that in such cases the highly
constrained form of the singular CY geometry must correspond to an equally
constrained Hitchin system. From the underlying effective physics of F-theory,
this correspondence is linked to an ${\rm SU}(2)$ R-symmetry which can rotate
components of hypermultiplets of the $6$-dimensional effective theory [14].
Here these halves of hypermultiplets correspond to complex structure moduli
and degrees of freedom in the intermediate Jacobian of the CY variety (the so-
called “RR-moduli”), respectively. Thus, any T-brane solution (or more
generally Higgs bundle on the brane with associated spectral cover) must
correspond under hypermultiplet rotation to a deformation of complex structure
of the singular 3-fold [8]. In particular, in the case of non-Higgsable CY
$3$-fold geometries an open question is to understand the following:

######  Open Question 6.4.

In the case that a non-Higgsable CY manifold exhibits ${\mathcal{G}}$ singular
elliptic fibers over a $($possibly singular$)$ curve $\Sigma$, does this
correspond to a trivial/empty Hitchin moduli space $($including parabolic or
wild Hitchin moduli space depending on the singularities of $\Sigma)$ of
${\mathcal{H}}$-Higgs bundles over $\Sigma$ where
${\mathcal{H}}\subset{\mathcal{G}}$?

One example where this open question can be confirmed in the affirmative is in
the case of CY $3$-folds $\pi\colon X\to\mathbb{F}_{n}$, defined as a
Weierstrass model (see Section 7) over a Hirzebruch surface, $\mathbb{F}_{n}$.
For each $n>2$, such elliptically fibered geometries are generically singular
[29, 147]. For $n=3$ for example there is a generic ${\rm SU}(3)$ singularity
(more specifically Kodaira type IV fibers) over a discriminant locus
$\Delta\subset\mathbb{F}_{3}$ which takes the form of a smooth curve of genus
zero (one of the sections of the rational fibration of $\mathbb{F}_{3}$). The
fact that this symmetry is “un-Higgsable” in the physical theory corresponds
to the triviality of ${\rm SL}(N,\mathbb{C})$-Higgs bundles over
$\mathbb{P}^{1}$ with $N=2,3$. See [8] for further examples involving
parabolic Hitchin systems over $\mathbb{P}^{1}$ with marked points.

In general, there are a large number of possible connections between Higgs
bundles and the effective theories and geometry arising within
string/M-/F-theory. One new correspondence has recently arisen within the
context of $4$-dimensional compactifications of F-theory in ${\mathcal{N}}=4$
supersymmetric Yang–Mills theories (with unity gauge groups) which are
quotiented by particular combinations of $R$-symmetry and ${\rm
SL}(2,\mathbb{Z})$ automorphisms (such theories can arise as D3-branes probing
terminal singularities in F-theory) [4, 5, 12, 92, 93].

######  Open Question 6.5.

What generalizations of Higgs bundles correspond to the new ${\mathcal{N}}=3$
supersymmetric theories recently discovered in [92]?

The self duality equations of ${\mathcal{N}}=4$ supersymmetric Yang–Mills
theories have led to a rich interplay between theories of branes arising in
string theory and Higgs bundles. It would be intriguing to understand whether
such links could arise between ${\mathcal{N}}=3$ theories and “cousins” of the
Hitchin system over Riemann surfaces. Finally, it should be noted that within
F-theory and the subject of T-branes there remain many open questions linking
so-called ”matrix factorization” techniques, K-theory, Hitchin systems and
Calabi–Yau geometry (see, e.g., [28]).

##  7 Elliptic fibrations, Weierstrass models,
and Calabi–Yau resolutions

Since elliptic fibrations play an important role when studying the relations
between Higgs bundles and F-theory, we shall conclude these notes with a
review of some of the basic ideas and recent advances. The reader should not
take this a thorough review, but rather a brief, curated overview of some
essential aspects of the underlying geometry.

######  Definition 7.1.

A surjective proper morphism $\varphi\colon Y\to B$ between two algebraic
varieties $Y$ and $B$ is called an elliptic fibration if the generic fiber of
$\varphi$ is a smooth projective curve of genus one and $\varphi$ has a
rational section. When $B$ is a curve, $Y$ is called an elliptic surface.
Moreover, when $B$ is a surface, $Y$ is said to be an elliptic $3$-fold. In
general, if $B$ has dimension $n-1$, $Y$ is called an elliptic $n$-fold.

###  7.1 Classification of singular fibers

The locus of singular fibers of an elliptic fibration, $\varphi\colon Y\to B$,
is called the discriminant locus, and is denoted by $\Delta(\varphi)$, or
simply $\Delta$ when the context is clear. If the base $B$ is smooth, the
discriminant locus is a divisor [58]. In the early 1960s, Kodaira classified
singular fibers of minimal elliptic surfaces in terms of numerical invariants
showing that there are 8 possibilities including two infinite series and 6
exceptional cases [125, 126]. Soon after, Néron obtained an equivalent
classification in an arithmetic setting using explicit regularizations of
singular Weierstrass models [151]. Based on Néron’s analysis, Tate proposed an
algorithm that allows (among other things) the determination of the type of
singular fibers of a Weierstrass model by analyzing the valuation of its
coefficients [176]. Under appropriate conditions, Kodaira’s classification of
singular fibers of an elliptic surface and Tate’s algorithm can be used to
describe the possible singular fibers and monodromies of an elliptically
fibered $n$-fold over points in codimension-1 in the base.

However, over points in codimension-2, new fibers not in Kodaira’s list are
known to occur [141, 173]. These are frequently referred to as collisions of
singular fibers as they usually appear at the intersections of two divisors of
the discriminant locus of the elliptic fibration. In general however, no
classification exists for the singular fibers of elliptic $3$-folds and
$4$-folds, leading to the broad question:

######  Open Question 7.2.

How can one geometrically classify non-Kodaira fibers?

Under some assumptions, the answer is known for the elliptic $n$-folds called
Miranda models [141, 173]. More generally, in the case of flat elliptic
fibrations obtained by crepant resolutions of Weierstrass models, non-Kodaira
fibers are expected to be contractions of usual Kodaira fibers. This is proven
for elliptic 3-folds [46] and confirmed in all known examples of non-Kodaira
fibers appearing in the F-theory literature [70, 76, 77, 78, 79, 131, 144,
175].

In view of the links described in Section 6 between elliptic CY geometry and
Hitchin systems, this leads naturally to the conjecture that two
classification problems might be linked:

######  Open Question 7.3.

How is the classification of non-Kodaira fibers of an elliptically fibered
Calabi–Yau $3$-fold related to the classification of parabolic or wild Hitchin
systems defined over the discriminant locus?

###  7.2 Weierstrass models

Since an elliptic fibration over a smooth base is birational to a (possibly
singular) Weierstrass model [55], the starting point of such an analysis will
usually be a Weierstrass model. We shall review here the main features of
these models, following the notation of Deligne [55]. Let $\mathcal{L}$ be a
line bundle over a quasi-projective variety $B$. We define the following
projective bundle (of lines):

$\displaystyle\pi\colon\
X_{0}=\mathbb{P}_{B}\big{[}\mathcal{O}_{B}\oplus\mathcal{L}^{\otimes
2}\oplus\mathcal{L}^{\otimes 3}\big{]}\longrightarrow B.$

We denote by $\mathcal{O}_{X_{0}}(1)$ the dual of the tautological line bundle
of the projective bundle $X_{0}$. The relative projective coordinates of
$X_{0}$ over $B$ are denoted $[z:x:y]$, where $z$, $x$, and $y$ are defined
respectively by the natural injection of $\mathcal{O}_{B}$,
$\mathcal{L}^{\otimes 2}$, and $\mathcal{L}^{\otimes 3}$ into
$\mathcal{O}_{B}\oplus\mathcal{L}^{\otimes 2}\oplus\mathcal{L}^{\otimes 3}$.
Hence, $z$ is a section of $\mathcal{O}_{X_{0}}(1)$, $x$ is a section of
$\mathcal{O}_{X_{0}}(1)\otimes\pi^{\ast}\mathcal{L}^{\otimes 2}$, and $y$ is a
section of $\mathcal{O}_{X_{0}}(1)\otimes\pi^{\ast}\mathcal{L}^{\otimes 3}$.
The most general Weierstrass equation is then the zero locus of the following
section of $\mathcal{O}(3)\otimes\pi^{\ast}\mathcal{L}^{\otimes 6}$ in $X_{0}$

$\displaystyle
F=y^{2}z+a_{1}xyz+a_{3}yz^{2}-\big{(}x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}\big{)},$

where $a_{i}$ is a section of $\pi^{\ast}\mathcal{L}^{\otimes i}$. The line
bundle $\mathcal{L}$ is called the fundamental line bundle of the Weierstrass
model $\varphi\colon Y\to B$ and can be defined directly from the elliptic
fibration $Y$ as $\mathcal{L}=R^{1}\varphi_{\ast}\mathcal{O}_{Y}$. The
Weierstrass model has a trivial canonical class when the fundamental line
bundle $\mathcal{L}$ is the anti-canonical line bundle of $B$.

Each crepant resolution of a singular Weierstrass model is a relative minimal
model (in the sense of the Minimal Model Program) over the Weierstrass model
[134]. When the base of the fibration is a curve, the Weierstrass model has a
unique crepant resolution. On the other hand, when the base is of dimension
two or higher, a crepant resolution does not always exist; furthermore, when
it does, it is not necessarily unique. Different crepant resolutions of the
same Weierstrass model are connected by a finite sequence of flops (see for
example [73, 75, 77, 78, 79, 107, 129, 134]). Crepant resolutions of
Weierstrass models have the same Euler characteristic, and these have recently
been computed in [74].

Following F-theory, we can attach to a given elliptic fibration a Lie algebra
$\mathfrak{g}$, a representation $\mathbf{R}$ of $\mathfrak{g}$, and a
hyperplane arrangement ${\rm I}(\mathfrak{g},\mathbf{R})$. The Lie algebra
$\mathfrak{g}$ and the representation $\mathbf{R}$ are determined by the
fibers over codimension-1 and codimension-2 points, respectively, of the base
in the discriminant locus. The hyperplane arrangement ${\rm
I}(\mathfrak{g},\mathbf{R})$ is defined inside the dual fundamental Weyl
chamber of $\mathfrak{g}$ (i.e., the dual cone of the fundamental Weyl chamber
of $\mathfrak{g}$), and its hyperplanes are the set of kernels of the weights
of $\mathbf{R}$. Moreover, one may study the network of flops using the
hyperplane arrangement ${\rm I}(\mathfrak{g},\mathbf{R})$ inspired from the
theory of Coulomb branches of five-dimensional supersymmetric gauge theories
with eight supercharges [120].

The network of crepant resolutions is isomorphic to the network of chambers of
the hyperplane arrangement I$(\mathfrak{g},\mathbf{R})$ defined by splitting
the dual fundamental Weyl chamber of the Lie algebra $\mathfrak{g}$ by the
hyperplanes dual to the weights of $\mathbf{R}$. The hyperplane arrangement
I$(\mathfrak{g},\mathbf{R})$, its relation to the Coulomb branches of
supersymmetric gauge theories and the network of crepant resolutions are
studied, among others, in [44, 57, 71, 72, 73, 75, 77, 78, 97, 107, 108, 120].

The representation $\mathbf{R}$ attached to an elliptic fibration can be
derived systematically using intersection theory [15]. Indeed, let $C$ be a
vertical curve, i.e., a curve contained in a fiber of the elliptic fibration.
Let $S$ be an irreducible component of the reduced discriminant of the
elliptic fibration $\varphi\colon Y\to B$. The pullback of $\varphi^{*}S$ has
irreducible components $D_{0},D_{1},\ldots,D_{n}$, where $D_{0}$ is the
component touching the section of the elliptic fibration. The weight vector of
$C$ over $S$ is by definition the vector ${\varpi}_{S}(C)=(-D_{1}\cdot
C,\ldots,-D_{n}\cdot C)$ of intersection numbers $D_{i}\cdot C$ for
$i=1,\ldots,n$. To an elliptic fibration, we associate a representation
$\mathbf{R}$ of the Lie algebra $\mathfrak{g}$ as follows. The weight vectors
of the irreducible vertical rational curves of the fibers over codimension-2
points form a set $\Pi$ derived by intersection theory. The saturation of
$\Pi$ (by adding and subtracting roots) defines uniquely a representation
$\mathbf{R}$. This method due to Aspinwall and Gross [15, Section 4] explains
how the representation $\mathbf{R}$ can be deduced even in presence of non-
Kodaira fibers [133]. The method can be formalized using the notion of
saturation set of weights borrowed from Bourbaki [73, 75].

One interesting property of the derivation of the representation $\mathbf{R}$
from intersection theory is that it does not assume the Calabi–Yau condition
nor relies on anomaly cancellations. Hence, from that point of view, the
representation attached to an elliptic fibration is purely a geometric data of
the elliptic fibration that also controls aspects of its birational geometry
via the hyperplane arrangement ${\rm I}(\mathfrak{g},\mathbf{R})$. There are
subtleties in presence of exotic matter [7], when the component of the
discriminant supporting the gauge group is singular [124], in presence of a
non-trivial Mordell–Weil group [135], when the codimension-two fibers are non-
split [73, 75], or when the fibration is non-flat [131]. Although we
understand the structure of the hyperplane arrangement ${\rm
I}(\mathfrak{g},\mathbf{R})$ for most of the F-theory models with simple
groups (see for example [57, 71, 72, 73, 75, 107, 120]), the structure in
presence of semi-simple groups is still not well explored. This lead to the
following question.

######  Open Question 7.4.

What are the intersection properties of $($exotic$)$ representations appearing
in F-theory and the structure of their associated hyperplane arrangements?

###  7.3 Superconformal field theories in the context of F-theory

Finally, it should be noted that there is a rich array of open questions that
have arisen form the recent investigations into superconformal field theories
in the context of F-theory. The superconformal algebra is a graded Lie algebra
that combines the conformal Poincaré algebra and supersymmetry. Some of the
most basic data characterizing a superconformal field theory (SCFT) is the
number of spacetime dimensions in which it is defined, and the amount of
supersymmetry generators and their chirality.

Recently, substantial interest has centered on six-dimensional SCFTs with
$(2,0)$ and $(1,0)$ supersymmetry. According to the seminal work of Werner
Nahm, SCFTs are only possible for spacetime dimensions 2, 3, 4, 5 and 6 [149].
In particular, the (2,0) theories in $d=6$ are the SCFTs with the maximal
amount of supersymmetry in the highest dimension [174]. The six-dimensional
superconformal field theories with (1,0) supersymmetry are among the least
understood quantum field theories, for example, they do not always have a
Lagrangian formulation [165, 179]. They are connected to questions in broad
areas such as Donaldson-Thomas theory of Calabi–Yau manifolds, modular and
automorphic forms [98, 102], singularities [54, 109, 110], quivers [3, 30,
123], and representation theory [2]. As they arise in F-theory and CY elliptic
fibrations, it is then natural to ask:

######  Open Question 7.5.

What is the geometry of elliptic fibrations used to model $(1,0)$ theories?
How are the conformal matter connected to the structure of Higgs bundles
appearing in F-theory?

The crepant resolutions of the singularities of CY elliptic fibrations
exhibiting SCFT loci provide a beautiful connection between the mechanism of
anomaly cancellation as seen in physics and topological quantities that have
been recently discovered. An understanding of the SCFT geometry must be linked
to the simplest building blocks of (1,0) theories, the so called non-Higgsable
clusters [109, 110, 143, 145]. First steps towards the analysis of the crepant
resolutions of such SCFT loci (including so-called ”matter transitions” in
F-theory [7]) is already underway [53, 73, 75, 102, 131].

### Acknowledgements

The authors would like to thank the American Institute of Mathematics for the
support and hospitality which made the 2017 workshop on Singular Geometry and
Higgs Bundles in String Theory possible (and on which this survey of
ideas/open questions is based). In addition we would like to thank Steven
Rayan for helpful comments on the manuscript. The work of L.B. Anderson is
supported in part by NSF grant PHY-1720321 and is part of the working group
activities of the 4-VA initiative “A Synthesis of Two Approaches to String
Phenomenology”. M. Esole is supported in part by the National Science
Foundation (NSF) grant DMS-1701635 “Elliptic Fibrations and String Theory”.
The work of L.P. Schaposnik is partially supported by the NSF grant
DMS-1509693, and by the Alexander von Humboldt Foundation.

## References

  * [1] Adams J., Strong real forms and the Kac classification, Atlas of Lie Groups and Representations, 2005, available at http://www.liegroups.org/papers/realforms.pdf.
  * [2] Aganagic M., Frenkel E., Okounkov A., Quantum $q$-Langlands correspondence, arXiv:1701.03146.
  * [3] Aganagic M., Okounkov A., Elliptic stable envelope, arXiv:1604.00423.
  * [4] Aharony O., Evtikhiev M., On four dimensional $N=3$ superconformal theories, J. High Energy Phys. 2016 (2016), no. 4, 040, 13 pages, arXiv:1512.03524.
  * [5] Aharony O., Tachikawa Y., S-folds and 4d ${\mathcal{N}}=3$ superconformal field theories, J. High Energy Phys. 2016 (2016), no. 6, 044, 26 pages, arXiv:1602.08638.
  * [6] Aker K., Szabó S., Algebraic Nahm transform for parabolic Higgs bundles on $\mathbb{P}^{1}$, Geom. Topol. 18 (2014), 2487–2545, math.AG/0610301.
  * [7] Anderson L.B., Gray J., Raghuram N., Taylor W., Matter in transition, J. High Energy Phys. 2016 (2016), no. 4, 080, 103 pages, arXiv:1512.05791.
  * [8] Anderson L.B., Heckman J.J., Katz S., T-branes and geometry, J. High Energy Phys. 2014 (2014), no. 5, 080, 68 pages, arXiv:1310.1931.
  * [9] Anderson L.B., Heckman J.J., Katz S., Schaposnik L.P., T-branes at the limits of geometry, J. High Energy Phys. 2017 (2017), no. 10, 058, 56 pages, arXiv:1702.06137.
  * [10] Anderson L.B., Taylor W., Geometric constraints in dual F-theory and heterotic string compactifications, J. High Energy Phys. 2014 (2014), no. 8, 025, 81 pages, arXiv:1405.2074.
  * [11] Arinkin D., Autoduality of compactified Jacobians for curves with plane singularities, J. Algebraic Geom. 22 (2013), 363–388, arXiv:1001.3868.
  * [12] Arras P., Grassi A., Weigand T., Terminal singularities, Milnor numbers, and matter in F-theory, J. Geom. Phys. 123 (2018), 71–97, arXiv:1612.05646.
  * [13] Arroyo M.A., The geometry of ${\rm SO}(p,q)$-Higgs bundles, Ph.D. Thesis, Universidad de Salamanca, Spain, 2009.
  * [14] Aspinwall P.S., Donagi R.Y., The heterotic string, the tangent bundle and derived categories, Adv. Theor. Math. Phys. 2 (1998), 1041–1074, hep-th/9806094.
  * [15] Aspinwall P.S., Gross M., The ${\rm SO}(32)$ heterotic string on a $K3$ surface, Phys. Lett. B 387 (1996), 735–742, hep-th/9605131.
  * [16] Baird T.J., Symmetric products of a real curve and the moduli space of Higgs bundles, arXiv:1611.09636.
  * [17] Balaji V., Barik P., Nagaraj D.S., A degeneration of moduli of Hitchin pairs, Int. Math. Res. Not. 2016 (2016), 6581–6625, arXiv:1308.4490.
  * [18] Baraglia D., Monodromy of the ${\rm SL}(n)$ and ${\rm GL}(n)$ Hitchin fibrations, arXiv:1612.01583.
  * [19] Baraglia D., Biswas I., Schaposnik L.P., Automorphisms of $\mathbb{C}^{*}$ moduli spaces associated to a Riemann surface, SIGMA 12 (2016), 007, 14 pages, arXiv:1508.06587.
  * [20] Baraglia D., Biswas I., Schaposnik L.P., On the Brauer group of Higgs bundles and connections, in Proceedings of Hitchin 70, Oxford University Press, to appear, arXiv:1609.00454.
  * [21] Baraglia D., Kamgarpour M., On the image of the parabolic Hitchin map, arXiv:1703.09886.
  * [22] Baraglia D., Schaposnik L.P., Higgs bundles and $(A,B,A)$-branes, Comm. Math. Phys. 331 (2014), 1271–1300, arXiv:1305.4638.
  * [23] Baraglia D., Schaposnik L.P., Real structures on moduli spaces of Higgs bundles, Adv. Theor. Math. Phys. 20 (2016), 525–551, arXiv:1309.1195.
  * [24] Baraglia D., Schaposnik L.P., Monodromy of rank $2$ twisted Hitchin systems and real character varieties, Trans. Amer. Math. Soc., to appear, arXiv:1506.00372.
  * [25] Baraglia D., Schaposnik L.P., Cayley and Langlands type correspondences for orthogonal Higgs bundles, Trans. Amer. Math. Soc., to appear, arXiv:1708.08828.
  * [26] Beauville A., Narasimhan M.S., Ramanan S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169–179.
  * [27] Beck F., Hitchin and Calabi–Yau integrable systems via variations of Hodge structures, arXiv:1707.05973.
  * [28] Bena I., Blåbäck J., Savelli R., T-branes and matrix models, J. High Energy Phys. 2017 (2017), no. 6, 009, 15 pages, arXiv:1703.06106.
  * [29] Bershadsky M., Intriligator K., Kachru S., Morrison D.R., Sadov V., Vafa C., Geometric singularities and enhanced gauge symmetries, Nuclear Phys. B 481 (1996), 215–252, hep-th/9605200.
  * [30] Bhardwaj L., Classification of 6d ${\mathcal{N}}=(1,0)$ gauge theories, J. High Energy Phys. 2015 (2015), no. 11, 002, 26 pages, arXiv:1502.06594.
  * [31] Bhosle U.N., Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves, Ark. Mat. 30 (1992), 187–215.
  * [32] Bhosle U.N., Generalized parabolic sheaves on an integral projective curve, Proc. Indian Acad. Sci. Math. Sci. 102 (1992), 13–22.
  * [33] Bhosle U.N., Generalized parabolic bundles and applications. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 403–420.
  * [34] Bhosle U.N., Generalized parabolic Hitchin pairs, J. Lond. Math. Soc. 89 (2014), 1–23.
  * [35] Bies M., Mayrhofer C., Pehle C., Weigand T., Chow groups, Deligne cohomology and massless matter in F-theory, arXiv:1402.5144.
  * [36] Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179–204, math.DG/0111098.
  * [37] Biswas I., Calvo L.A., Franco E., García-Prada O., Involutions of the moduli spaces of $G$-Higgs bundles over elliptic curves, arXiv:1612.08364.
  * [38] Biswas I., Wilkin G., Anti-holomorphic involutive isometry of hyper-Kähler manifolds and branes, J. Geom. Phys. 88 (2015), 52–55, arXiv:1410.6616.
  * [39] Boalch P., Hyperkähler manifolds and nonabelian Hodge theory of (irregular) curves, arXiv:1203.6607.
  * [40] Boalch P., Poisson varieties from Riemann surfaces, Indag. Math. (N.S.) 25 (2014), 872–900, arXiv:1309.7202.
  * [41] Boden H.U., Yokogawa K., Moduli spaces of parabolic Higgs bundles and parabolic $K(D)$ pairs over smooth curves. I, Internat. J. Math. 7 (1996), 573–598, alg-geom/9610014.
  * [42] Bradlow S.B., Schaposnik L.P., Higgs bundles and exceptional isogenies, Res. Math. Sci. 3 (2016), 14, 28 pages, arXiv:1508.02650.
  * [43] Branco L.C., Higgs bundles, Lagrangians and mirror symmetry, Ph.D. Thesis, University of Oxford, 2017, arXiv:1803.01611.
  * [44] Braun A.P., Schäfer-Nameki S., Box graphs and resolutions I, Nuclear Phys. B 905 (2016), 447–479, arXiv:1407.3520.
  * [45] Casimiro A.C., Ferreira S., Florentino C., Principal Schottky bundles over Riemann surfaces, arXiv:1612.08662.
  * [46] Cattaneo A., Crepant resolutions of Weierstrass threefolds and non-Kodaira fibres, arXiv:1307.7997.
  * [47] Cecotti S., Córdova C., Heckman J.J., Vafa C., T-branes and monodromy, J. High Energy Phys. 2011 (2011), no. 7, 030, 93 pages, arXiv:1010.5780.
  * [48] Collier B., Finite order automorphisms of Higgs bundles: theory and application, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2016.
  * [49] Collier B., ${\rm SO}(n,n+1)$-surface group representations and their Higgs bundles, arXiv:1710.01287.
  * [50] Collinucci A., Giacomelli S., Savelli R., Valandro R., T-branes through 3d mirror symmetry, J. High Energy Phys. 2016 (2016), no. 7, 093, 41 pages, arXiv:1603.00062.
  * [51] Collinucci A., Giacomelli S., Valandro R., T-branes, monopoles and S-duality, J. High Energy Phys. 2017 (2017), no. 10, 113, 62 pages, arXiv:1703.09238.
  * [52] Collinucci A., Savelli R., F-theory on singular spaces, J. High Energy Phys. 2015 (2015), no. 9, 100, 39 pages, arXiv:1410.4867.
  * [53] Del Zotto M., Heckman J.J., Morrison D.R., 6D SCFTs and phases of 5D theories, J. High Energy Phys. 2017 (2017), no. 9, 147, 38 pages, arXiv:1703.02981.
  * [54] Del Zotto M., Heckman J.J., Tomasiello A., Vafa C., 6d conformal matter, J. High Energy Phys. 2015 (2015), no. 2, 054, 56 pages, arXiv:1407.6359.
  * [55] Deligne P., Courbes elliptiques: formulaire d’après J. Tate, in Modular Functions of One Variable, IV (Proc. Internat. Summer School, University Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, 53–73.
  * [56] Diaconescu D.-E., Donagi R., Pantev T., Intermediate Jacobians and ADE Hitchin systems, hep-th/0607159.
  * [57] Diaconescu D.-E., Entin R., Calabi–Yau spaces and five-dimensional field theories and exceptional gauge symmetry, Nuclear Phys. B 538 (1999), 451–484, hep-th/9807170.
  * [58] Dolgačev I.V., On the purity of the degeneration loci of families of curves, Invent. Math. 8 (1969), 34–54.
  * [59] Donagi R., Decomposition of spectral covers, Astérisque 218 (1993), 145–175.
  * [60] Donagi R., Spectral covers, in Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge University Press, Cambridge, 1995, 65–86, alg-geom/9505009.
  * [61] Donagi R., Gaitsgory D., The gerbe of Higgs bundles, Transform. Groups 7 (2002), 109–153, math.AG/0005132.
  * [62] Donagi R., Katz S., Sharpe E., Spectra of D-branes with Higgs vevs, Adv. Theor. Math. Phys. 8 (2004), 815–860, hep-th/0309270.
  * [63] Donagi R., Markman E., Cubics, integrable systems, and Calabi–Yau threefolds, in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan University, Ramat Gan, 1996, 199–221, alg-geom/9408004.
  * [64] Donagi R., Markman E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 1–119, alg-geom/9507017.
  * [65] Donagi R., Pantev T., Langlands duality for Hitchin systems, Invent. Math. 189 (2012), 653–735, arXiv:1311.3624.
  * [66] Donagi R., Wijnholt M., Gluing branes – I, J. High Energy Phys. 2013 (2013), no. 5, 068, 46 pages, arXiv:1104.2610.
  * [67] Donagi R., Wijnholt M., Gluing branes – II: flavour physics and string duality, J. High Energy Phys. 2013 (2013), no. 5, 092, 50 pages, arXiv:1112.4854.
  * [68] Dumas D., Neitzke A., Asymptotics of Hitchin’s metric on the Hitchin section, arXiv:1802.07200.
  * [69] Dumitrescu O., Mulase M., Quantum curves for Hitchin fibrations and the Eynard–Orantin theory, Lett. Math. Phys. 104 (2014), 635–671, arXiv:1310.6022.
  * [70] Esole M., Fullwood J., Yau S.-T., $D_{5}$ elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory, Commun. Number Theory Phys. 9 (2015), 583–642, arXiv:1110.6177.
  * [71] Esole M., Jackson S., Jagadeesan R., Noël A.G., Incidence geometry in a Weyl chamber I: ${\rm GL}_{n}$, arXiv:1508.03038.
  * [72] Esole M., Jackson S., Jagadeesan R., Noël A.G., Incidence geometry in a Weyl chamber II: ${\rm SL}_{n}$, arXiv:1601.05070.
  * [73] Esole M., Jagadeesan R., Kang M.J., The geometry of $G_{2}$, ${\rm Spin}(7)$, and ${\rm Spin}(8)$-models, arXiv:1709.04913.
  * [74] Esole M., Jefferson P., Kang M.J., Euler characteristics of crepant resolutions of Weierstrass models, arXiv:1703.00905.
  * [75] Esole M., Jefferson P., Kang M.J., The geometry of F4-models, arXiv:1704.08251.
  * [76] Esole M., Kang M.J., Yau S.-T., A new model for elliptic fibrations with a rank one Mordell–Weil group I. Singular fibers and semi-stable degenerations, arXiv:1410.0003.
  * [77] Esole M., Shao S.-H., Yau S.-T., Singularities and gauge theory phases, Adv. Theor. Math. Phys. 19 (2015), 1183–1247, arXiv:1402.6331.
  * [78] Esole M., Shao S.-H., Yau S.-T., Singularities and gauge theory phases II, Adv. Theor. Math. Phys. 20 (2016), 683–749, arXiv:1407.1867.
  * [79] Esole M., Yau S.-T., Small resolutions of $\rm SU(5)$-models in F-theory, Adv. Theor. Math. Phys. 17 (2013), 1195–1253, arXiv:1107.0733.
  * [80] Faltings G., Stable $G$-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507–568.
  * [81] Franco E., Jardim M., Mirror symmetry for Nahm branes, arXiv:1709.01314.
  * [82] Franco E., Jardim M., Marchesi S., Branes in the moduli space of framed instantons, arXiv:1504.05883.
  * [83] Franco E., Jardim M., Menet G., Brane involutions on irreducible holomorphic symplectic manifolds, arXiv:1606.09040.
  * [84] Franco E., Peón-Nieto A., The Borel subgroup and branes on the Higgs moduli space, arXiv:1709.03549.
  * [85] Fredrickson L., Generic ends of the moduli space of ${\rm SL}(n,\mathbb{C})$-Higgs bundles, in preparation.
  * [86] Fredrickson L., Neitzke A., From $S^{1}$-fixed points to ${\mathcal{W}}$-algebra representations, arXiv:1709.06142.
  * [87] Freed D.S., Special Kähler manifolds, Comm. Math. Phys. 203 (1999), 31–52, hep-th/9712042.
  * [88] Frenkel E., Ramifications of the geometric Langlands program, in Representation Theory and Complex Analysis, Lecture Notes in Math., Vol. 1931, Springer, Berlin, 2008, 51–135, math.QA/0611294.
  * [89] Frenkel E., Witten E., Geometric endoscopy and mirror symmetry, Commun. Number Theory Phys. 2 (2008), 113–283, arXiv:0710.5939.
  * [90] Gaiotto D., S-duality of boundary conditions and the geometric Langlands, arXiv:1609.09030.
  * [91] Gaiotto D., Moore G.W., Neitzke A., Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), 163–224, arXiv:0807.4723.
  * [92] García-Etxebarria I., Regalado D., $\mathcal{N}=3$ four dimensional field theories, J. High Energy Phys. 2016 (2016), no. 3, 083, 21 pages, arXiv:1512.06434.
  * [93] García-Etxebarria I., Regalado D., Exceptional ${\mathcal{N}}=3$ theories, J. High Energy Phys. 2017 (2017), no. 12, 042, 21 pages, arXiv:1611.05769.
  * [94] García-Prada O., Wilkin G., Action of the mapping class group on character varieties and Higgs bundles, arXiv:1612.02508.
  * [95] Giudice A.L., Pustetto A., A compactification of the moduli space of principal Higgs bundles over singular curves, arXiv:1110.0632.
  * [96] Gothen P.B., Oliveira A.G., The singular fiber of the Hitchin map, Int. Math. Res. Not. 2013 (2013), 1079–1121, arXiv:1012.5541.
  * [97] Grimm T.W., Hayashi H., F-theory fluxes, chirality and Chern-Simons theories, J. High Energy Phys. 2012 (2012), no. 3, 027, 54 pages, arXiv:1111.1232.
  * [98] Gu J., Huang M.-X., Kashani-Poor A.-K., Klemm A., Refined BPS invariants of 6d SCFTs from anomalies and modularity, J. High Energy Phys. 2017 (2017), no. 5, 130, 62 pages, arXiv:1701.00764.
  * [99] Guichard O., Wienhard A., Positivity and higher Teichmüller theory, arXiv:1802.02833.
  * [100] Gukov S., Quantization via mirror symmetry, Jpn. J. Math. 6 (2011), 65–119, arXiv:1011.2218.
  * [101] Gukov S., Witten E., Gauge theory, ramification, and the geometric Langlands program, in Current Developments in Mathematics, 2006, Int. Press, Somerville, MA, 2008, 35–180, hep-th/0612073.
  * [102] Haghighat B., Klemm A., Lockhart G., Vafa C., Strings of minimal 6d SCFTs, Fortschr. Phys. 63 (2015), 294–322, arXiv:1412.3152.
  * [103] Halverson J., Taylor W., ${\mathbb{P}}^{1}$-bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models, J. High Energy Phys. 2015 (2015), no. 9, 086, 59 pages, arXiv:1506.03204.
  * [104] Hausel T., Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998), 169–192, math.AG/9804083.
  * [105] Hausel T., Geometry of the moduli space of Higgs bundles, math.AG/0107040.
  * [106] Hausel T., Thaddeus M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197–229, math.AG/0205236.
  * [107] Hayashi H., Lawrie C., Morrison D.R., Schäfer-Nameki S., Box graphs and singular fibers, J. High Energy Phys. 2014 (2014), no. 5, 048, 92 pages, arXiv:1402.2653.
  * [108] Hayashi H., Lawrie C., Schäfer-Nameki S., Phases, flops and F-theory: ${\rm SU}(5)$ gauge theories, J. High Energy Phys. 2013 (2013), no. 10, 046, 43 pages, arXiv:1304.1678.
  * [109] Heckman J.J., Morrison D.R., Rudelius T., Vafa C., Atomic classification of 6D SCFTs, Fortschr. Phys. 63 (2015), 468–530, arXiv:1502.05405.
  * [110] Heckman J.J., Morrison D.R., Vafa C., On the classification of 6D SCFTs and generalized ADE orbifolds, J. High Energy Phys. 2014 (2014), no. 5, 028, 49 pages, arXiv:1312.5746.
  * [111] Heller S., Schaposnik L.P., Branes through finite group actions, J. Geom. Phys. 129 (2018), 279–293, arXiv:1611.00391.
  * [112] Hitchin N., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59–126.
  * [113] Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.
  * [114] Hitchin N., Higgs bundles and characteristic classes, in Arbeitstagung Bonn 2013, Progr. Math., Vol. 319, Birkhäuser/Springer, Cham, 2016, 247–264, arXiv:1308.4603.
  * [115] Hitchin N., Spinors, Lagrangians and rank 2 Higgs bundles, Proc. London Math. Soc. 115 (2017), 33–54, arXiv:1605.06385.
  * [116] Hitchin N., Schaposnik L.P., Nonabelianization of Higgs bundles, J. Differential Geom. 97 (2014), 79–89, arXiv:1307.0960.
  * [117] Hoskins V., Schaffhauser F., Group actions on quiver varieties and applications, arXiv:1612.06593.
  * [118] Hoskins V., Schaffhauser F., Rational points of quiver moduli spaces, arXiv:1704.08624.
  * [119] Igusa J.-I., Fibre systems of Jacobian varieties, Amer. J. Math. 78 (1956), 171–199.
  * [120] Intriligator K., Morrison D.R., Seiberg N., Five-dimensional supersymmetric gauge theories and degenerations of Calabi–Yau spaces, Nuclear Phys. B 497 (1997), 56–100, hep-th/9702198.
  * [121] Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1–236, hep-th/0604151.
  * [122] Kausz I., A Gieseker type degeneration of moduli stacks of vector bundles on curves, Trans. Amer. Math. Soc. 357 (2005), 4897–4955, math.AG/0201197.
  * [123] Kimura T., Pestun V., Quiver elliptic W-algebras, arXiv:1608.04651.
  * [124] Klevers D., Taylor W., Three-index symmetric matter representations of ${\rm SU}(2)$ in F-theory from non-Tate form Weierstrass models, J. High Energy Phys. 2016 (2016), no. 6, 171, 32 pages, arXiv:1604.01030.
  * [125] Kodaira K., On compact analytic surfaces. II, Ann. of Math. 77 (1963), 563–626.
  * [126] Kodaira K., On compact analytic surfaces. III, Ann. of Math. 78 (1963), 1–40.
  * [127] Konno H., Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface, J. Math. Soc. Japan 45 (1993), 253–276.
  * [128] Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv:0811.2435.
  * [129] Krause S., Mayrhofer C., Weigand T., $G_{4}$-flux, chiral matter and singularity resolution in F-theory compactifications, Nuclear Phys. B 858 (2012), 1–47, arXiv:1109.3454.
  * [130] Laumon G., Un analogue global du cône nilpotent, Duke Math. J. 57 (1988), 647–671.
  * [131] Lawrie C., Schäfer-Nameki S., The Tate form on steroids: resolution and higher codimension fibers, J. High Energy Phys. 2013 (2013), no. 4, 061, 66 pages, arXiv:1212.2949.
  * [132] Marchesano F., Savelli R., Schwieger S., Compact T-branes, J. High Energy Phys. 2017 (2017), no. 9, 132, 28 pages, arXiv:1707.03797.
  * [133] Marsano J., Schäfer-Nameki S., Yukawas, $G$-flux, and spectral covers from resolved Calabi–Yau’s, J. High Energy Phys. 2011 (2011), no. 11, 098, 59 pages, arXiv:1108.1794.
  * [134] Matsuki K., Weyl groups and birational transformations among minimal models, Mem. Amer. Math. Soc. 116 (1995), vi+133 pages.
  * [135] Mayrhofer C., Morrison D.R., Till O., Weigand T., Mordell–Weil torsion and the global structure of gauge groups in F-theory, J. High Energy Phys. 2014 (2014), no. 10, 016, 47 pages, arXiv:1405.3656.
  * [136] Mazzeo R., Swoboda J., Weiss H., Witt F., Limiting configurations for solutions of Hitchin’s equation, Semin. Theor. Spectr. Geom. 31 (2012–2014), 91–116, arXiv:1502.01692.
  * [137] Mazzeo R., Swoboda J., Weiss H., Witt F., Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016), 2227–2271, arXiv:1405.5765.
  * [138] Mazzeo R., Swoboda J., Weiss H., Witt F., Asymptotic geometry of the Hitchin metric, arXiv:1709.03433.
  * [139] Melo M., Rapagnetta A., Vivani F., Fourier–Mukai and autoduality for compactified Jacobians, I, arXiv:1207.7233.
  * [140] Mestrano N., Simpson C., Moduli of sheaves, in Development of Moduli Theory – Kyoto 2013, Adv. Stud. Pure Math., Vol. 69, Math. Soc. Japan, Tokyo, 2016, 77–172.
  * [141] Miranda R., Smooth models for elliptic threefolds, in The Birational Geometry of Degenerations (Cambridge, Mass., 1981), Progr. Math., Vol. 29, Birkhäuser, Boston, Mass., 1983, 85–133.
  * [142] Mochizuki T., Asymptotic behavior of certain families of harmonic bundles on Riemann surfaces, J. Topology 9 (2016), 1021–1073, arXiv:1508.05997.
  * [143] Morrison D.R., Taylor W., Classifying bases for 6D F-theory models, Cent. Eur. J. Phys. 10 (2012), 1072–1088, arXiv:1201.1943.
  * [144] Morrison D.R., Taylor W., Matter and singularities, J. High Energy Phys. 2012 (2012), no. 1, 022, 54 pages, arXiv:1106.3563.
  * [145] Morrison D.R., Taylor W., Toric bases for 6D F-theory models, Fortschr. Phys. 60 (2012), 1187–1216, arXiv:1204.0283.
  * [146] Morrison D.R., Taylor W., Non-Higgsable clusters for 4D F-theory models, J. High Energy Phys. 2015 (2015), no. 5, 080, 37 pages, arXiv:1412.6112.
  * [147] Morrison D.R., Vafa C., Compactifications of $F$-theory on Calabi–Yau threefolds. II, Nuclear Phys. B 476 (1996), 437–469, hep-th/9603161.
  * [148] Nadler D., Perverse sheaves on real loop Grassmannians, Invent. Math. 159 (2005), 1–73, math.AG/0202150.
  * [149] Nahm W., Supersymmetries and their representations, Nuclear Phys. B 135 (1978), 149–166.
  * [150] Neitzke A., Notes on a new construction of hyperkähler metrics, arXiv:1308.2198.
  * [151] Néron A., Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 5–125.
  * [152] Pandharipande R., A compactification over $\overline{M}_{g}$ of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), 425–471, alg-geom/9502020.
  * [153] Peón-Nieto A., Higgs bundles, real forms and the Hitchin fibration, Ph.D. Thesis, Universidad Autónoma de Madrid, 2013.
  * [154] Peón-Nieto A., Cameral data for ${\rm SU}(p,p+1)$-Higgs bundles, arXiv:1506.01318.
  * [155] Radko O., A classification of topologically stable Poisson structures on a compact oriented surface, J. Symplectic Geom. 1 (2002), 523–542, math.SG/0110304.
  * [156] Schaposnik L.P., Monodromy of the ${\rm SL}_{2}$ Hitchin fibration, Internat. J. Math. 24 (2013), 1350013, 21 pages, arXiv:1111.2550.
  * [157] Schaposnik L.P., Spectral data for G-Higgs bundles, Ph.D. Thesis, University of Oxford, 2013, arXiv:1301.1981.
  * [158] Schaposnik L.P., Spectral data for $U(m,m)$-Higgs bundles, Int. Math. Res. Not. 2015 (2015), 3486–3498, arXiv:1307.4419.
  * [159] Schaposnik L.P., Higgs bundles and applications, Oberwolfach Reports, 2015, 31–33, arXiv:1603.06691.
  * [160] Schaposnik L.P., An introduction to spectral data for Higgs bundles, Lecture Note Series, Vol. 65, Institute for Mathematical Sciences, National University of Singapore, 2017, 38 pages, arXiv:1408.0333.
  * [161] Schaposnik L.P., A geometric approach to orthogonal Higgs bundles, Eur. J. Math., to appear, arXiv:1608.00300.
  * [162] Schaub D., Courbes spectrales et compactifications de jacobiennes, Math. Z. 227 (1998), 295–312.
  * [163] Schmitt A.H.W., Singular principal $G$-bundles on nodal curves, J. Eur. Math. Soc. 7 (2005), 215–251.
  * [164] Scognamillo R., An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups, Compositio Math. 110 (1998), 17–37, alg-geom/9412020.
  * [165] Seiberg N., Witten E., Comments on string dynamics in six dimensions, Nuclear Phys. B 471 (1996), 121–134, hep-th/9603003.
  * [166] Seshadri C., Degenerations of the moduli spaces of vector bundles on curves, Lecture given at the School on Algebraic Geometry, 1999.
  * [167] Seshadri C.S., Moduli spaces of torsion free sheaves on nodal curves and generalisations. I, in Moduli Spaces and Vector Bundles, London Math. Soc. Lecture Note Ser., Vol. 359, Cambridge University Press, Cambridge, 2009, 484–505.
  * [168] Simpson C.T., Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867–918.
  * [169] Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992), 5–95.
  * [170] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47–129.
  * [171] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5–79.
  * [172] Strominger A., Yau S.-T., Zaslow E., Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), 243–259, hep-th/9606040.
  * [173] Szydlo M.G., Flat regular models of elliptic schemes, Ph.D. Thesis, Harvard University, 1999.
  * [174] Tachikawa Y., ${\mathcal{N}}=2$ supersymmetric dynamics for pedestrians, Lecture Notes in Physics, Vol. 890, Springer, Cham, Hindustan Book Agency, New Delhi, 2015, arXiv:1312.2684.
  * [175] Tatar R., Walters W., GUT theories from Calabi-Yau 4-folds with ${\rm SO}(10)$ singularities, J. High Energy Phys. 2012 (2012), no. 12, 092, 24 pages, arXiv:1206.5090.
  * [176] Tate J., Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable, IV (Proc. Internat. Summer School, University Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, 33–52.
  * [177] Vafa C., Evidence for $F$-theory, Nuclear Phys. B 469 (1996), 403–415, hep-th/9602022.
  * [178] Witten E., Gauge theory and wild ramification, arXiv:0710.0631.
  * [179] Witten E., Conformal field theory in four and six dimensions, in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., Vol. 308, Cambridge University Press, Cambridge, 2004, 405–419, arXiv:0712.0157.
  * [180] Xie D., General Argyres-Douglas theory, J. High Energy Phys. 2013 (2013), no. 1, 100, 52 pages, arXiv:1204.2270.