Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces
This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinat...
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| citation_txt | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces / S. Rayan // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 67 назв. — англ. |
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| description | This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 129, 18 pages
Aspects of the Topology and Combinatorics
of Higgs Bundle Moduli Spaces
Steven RAYAN
Department of Mathematics & Statistics, McLean Hall, University of Saskatchewan,
Saskatoon, SK, Canada S7N 5E6
E-mail: rayan@math.usask.ca
URL: https://www.math.usask.ca/~rayan/
Received September 23, 2018, in final form December 04, 2018; Published online December 07, 2018
https://doi.org/10.3842/SIGMA.2018.129
Abstract. This survey provides an introduction to basic questions and techniques sur-
rounding the topology of the moduli space of stable Higgs bundles on a Riemann surface.
Through examples, we demonstrate how the structure of the cohomology ring of the moduli
space leads to interesting questions of a combinatorial nature.
Key words: Higgs bundle; Morse–Bott theory; localization; Betti number; moduli space;
stability; quiver; partition problem
2010 Mathematics Subject Classification: 14D20; 46M20; 57N65; 05A19
1 Introduction
Nonabelian Hodge theory realizes an equivalence between three types of objects in geometry
and topology: representations of the fundamental group of a complex projective manifold, flat
connections on that manifold, and Higgs bundles on that same manifold. The first type of
object is topological, the second records the smooth geometry of the manifold, and the third is
holomorphic. The nonabelian Hodge correspondence can be formulated into a diffeomorphism
of appropriately-defined moduli spaces of these objects. One of the nice features of working
on the “Higgs” side is the existence of a Hamiltonian U(1)-action – equivalently, an algebraic
C?-action, depending on how exactly one constructs the moduli space. By localization, one can
at least in principle compute numerical topological invariants of the Higgs bundle moduli space
using this action and then possess, by virtue of nonabelian Hodge theory, these invariants for
all three moduli spaces.
While the U(1)-action provides a place to get started, the localization calculation does not
scale easily, with explicit results revealing themselves readily only in low rank, even when we
restrict to Riemann surfaces. That being said, the structure of the fixed-point locus hints at
interesting combinatorics lurking in the cohomology ring of the moduli space, some of which
we see below. The fact that the cohomology ring lies at the centre of a number of conjectures
in mirror symmetry [35] (some of which have been recently addressed [27, 28]) makes these
combinatorial questions even more intriguing.
In this article, we present some basic concepts and examples surrounding the problem of
computing topological invariants of Higgs bundle moduli spaces. For simplicity, we restrict
to the Betti numbers of the rational cohomology ring. The article is based more or less on
a mini-course given by the author at the first “Workshop on the Geometry and Physics of
Higgs Bundles”, held in October 2016 at the University of Illinois at Chicago. The mini-course
This paper is a contribution to the Special Issue on Geometry and Physics of Hitchin Systems. The full
collection is available at https://www.emis.de/journals/SIGMA/hitchin-systems.html
mailto:rayan@math.usask.ca
https://www.math.usask.ca/~rayan/
https://doi.org/10.3842/SIGMA.2018.129
https://www.emis.de/journals/SIGMA/hitchin-systems.html
2 S. Rayan
consisted of three lectures and three problem sessions. The presentation in this article, much
as in the mini-course, is somewhat bare bones and involves only traditional Morse–Bott theory.
For Higgs bundles this is by now “old hat”, having been supplanted by a number of refinements
or wholly different techniques, including arithmetic harmonic analysis; wall-crossing techniques;
and motivic and p-adic integration. These techniques have led to explicit results about the
cohomology that once seemed quite far away. It is difficult to provide a complete list of references
on these developments, although here are some that reflect the evolution of these developments:
[22, 28, 31, 32, 33, 47, 48, 49, 50, 58].
The mini-course had been delivered for an audience of mostly beginning graduate students.
This survey has been written with similar considerations in mind. We imagine that the reader
possessing some basic Riemann surface theory – including Jacobians, Čech cohomology, Serre
duality, and the Riemann–Roch theorem for holomorphic vector bundles – will get the most
from these notes. We have included a few basic exercises to capture some of the spirit of the
problem sessions.
2 Background on Higgs bundles
Higgs bundles originated within mathematical inquiries into gauge theories in the 1970s and
1980s but can also be understood in a mostly algebraic way. We briefly examine both points of
view here, with an aim to understanding roughly the geometric features of the moduli space of
Higgs bundles.
2.1 Gauge theory
From this point forward, X is a smooth compact Riemann surface. For now, the genus g of X is
at least 2. We use the symbols OX and ωX for the trivial line bundle and cotangent bundle of X,
respectively. Higgs bundles originally arose as solutions of the Hitchin equations or “self-duality
equations” on X [38]. These are self-dual, dimensionally-reduced Yang–Mills equations written
on a smooth Hermitian bundle of rank r ≥ 1 and degree 0 on X. We will use E for this bundle
and h for the metric. The equations take the form
F (A) + φ ∧ φ∗ = 0, (2.1)
∂Aφ = 0. (2.2)
In the equations, A is a connection on the bundle (unitary with regards to h), F is its curvature,
and φ is a smooth bundle map from E to E ⊗ ωX , called a Higgs field. The equations are
trivially satisfied by a flat connection A with φ = 0. Equation (2.1) says that, whenever A is
not flat, its curvature (1, 1)-form should be expressible in terms of φ and its Hermitian adjoint.
Equation (2.2) says that φ should be holomorphic with respect to the holomorphic structure
on E induced by A. The equations can be altered appropriately, involving a constant central
curvature term on the right side of (2.1), in order to accommodate an arbitrary degree d ∈ Z.
Throughout, we will assume that r and d are coprime.
Now, assume that E is a holomorphic bundle on X together with a holomorphic section
φ ∈ H0(X,End(E) ⊗ ωX). We refer to such a pair as a Higgs bundle. One can ask: when does
the data (E , φ) arise from a solution to the Hitchin equations? In other words, when does there
exist a Hermitian metric h on the underlying smooth bundle and a unitary connection A such
that the holomorphic structure on E is induced by (h,A) and (A, φ) is a solution of the Hitchin
equations for h? The answer is a numerical condition on the pair (E , φ), asking that the following
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 3
inequality holds: for each subbundle 0 ( U ( E for which φ(U) ⊆ U ⊗ ωX , we must have
deg(U)
rank(U)
<
deg(E)
rank(E)
.
Such U are said to be φ-invariant and the ratio in question is referred to as the slope of U . If
the inequality is satisfied for all such U , we say that the Higgs bundle (E , φ) is stable. (The edge
case where equality is permitted, known as semistability, is eliminated by the earlier coprime
assumption.)
This correspondence is an example of what are now generally referred to as Kobayashi–
Hitchin correspondences, relating bundles with special metrics to ones with algebro-geometric
restrictions. As an equivalence of moduli spaces, on one side we have the space of solutions (A, φ)
of (2.1) and (2.2) for (E, h) taken up to gauge equivalence, which are orbits of the conjugation
action of the group of smooth unitary diffeomorphisms of E. This quotient has the structure
of a smooth, non-compact manifold. On the Higgs bundle side, we have the space of all stable
pairs (E , φ) with underlying smooth bundle E taken up to isomorphism, which is given by the
conjugation action of the group of holomorphic automorphisms of E . This quotient has the
structure of a non-singular, quasiprojective variety.
The gauge-theoretic side can be interpreted as an infinite-dimensional hyperkähler quotient,
in the sense of [40]. Here, the hyperkähler moment maps are the left side of (2.1) and the real and
imaginary parts of the left side of (2.2). The quotient inherits a hyperkähler metric, compatible
with three quaterionically-commuting complex structures. It is an immediate consequence that
the moduli space is Calabi–Yau, although it is not compact. The moduli variety on the other
side of the correspondence, which we denote by MX(r, d), can be interpreted as a geometric-
invariant theory quotient, with its stability condition given by our notion of “stable” above.
Indeed, this is exactly the condition required to form a Hausdorff moduli space here.
This correspondence generalizes the earlier one of Narasimhan–Seshadri [51], which relates
flat bundles to stable holomorphic bundles. At the same time, the Kobayashi–Hitchin corre-
spondence can be viewed as a “fourth corner” in nonabelian Hodge theory, extending the equiv-
alence to one between flat connections, representations of π1(X), Higgs bundles, and solutions
of Hitchin’s equations.
For our purposes (and until we introduce some tools from differential topology in Section 3),
we will lean in an algebro-goemetric direction and concentrate on Higgs bundles and MX(r, d).
For a deeper discussion of the gauge theory, including an exploration of recent results concerning
the global properties of the hyperkähler metric, we refer the reader to [21] in the same collection
of mini-course articles – as well as of course Hitchin’s original article [38]. Regarding nonabelian
Hodge theory in particular, we refer the reader to works of Simpson [61, 62] and to recent surveys
such as [23, 65].
One common preference, which is useful for instance when going from Higgs bundles to
representations of surface groups, is to fix the determinant of the Higgs bundle, which means
taking ∧rE to be some fixed degree-d line bundle. This takes us from the vector bundle (i.e.,
GL(r,C)) situation to principal SL(r,C)-Higgs bundles. Accordingly, the Higgs field is taken to
be trace-free, which we denote by φ ∈ H0(X,End0(E) ⊗ ωX). We will use M0
X(r, d) to denote
this moduli space, i.e., that of stable SL(r,C)-Higgs bundles with fixed determinant of degree d.
2.2 Deformation theory
The first piece of topological information to compute about MX(r, d) is its dimension. For
this, we can use deformation theory. Let us assume, to begin with, that we are working with
SL(r,C)-Higgs bundles. To such a Higgs bundle (E , φ), we can associate a deformation complex
determined by the Čech co-differential δ on E and the Higgs field itself. We can view the Higgs
4 S. Rayan
field as a map that acts on Lie-algebra-valued forms by the Lie bracket on the Lie algebra part
and by the wedge product on the form part. In our situation, where the Higgs field is a section
of ad(E) ⊗ ωX ∼= End0(E) ⊗ ωX , the fact that ωX ∧ ωX = 0 on a curve means that the map
(∧φ)2 is always zero and hence is a co-differential for our purposes. (For X of higher dimension,
this is one motivation for including an extra condition on Higgs bundles, namely that φ satisfies
φ ∧ φ = 0.)
By analogy with the fact that the tangent space to the moduli space of stable bundles at
a point E is the cohomology H1(X,End0(E)) of the complex associated to δ, the tangent space
to the moduli space at a stable pair (E , φ) is the hypercohomology H1 of the double complex
associated to the two co-differentials, δ and ∧φ [9]. By working with the double complex as
in [9], we find that dimCH1 is a sum of two numbers. The first is the dimension of
kerH1(X,End0(E))
∧φ−→ H1(X,End0(E)⊗ ωX),
which is a subspace of the usual tangent space to the moduli space of stable bundles. Here,
we only want deformations of the holomorphic structure on the bundle for which φ is still
holomorphic itself. The second number is the dimension of
H0(X,End0(E)⊗ ωX)
imH0(X,End0(E))
∧φ−→ H0(X,End0(E)⊗ ωX)
,
which captures deformations of the Higgs field.
It is a consequence of stability that the map
∧φ : H0(X,End0(E))−→H0(X,End0(E)⊗ ωX)
is injective. (See, for instance, [65, Remark 2.8].) It then follows by duality that the map
∧φ : H1(X,End0(E))−→H1(X,End0(E)⊗ ωX)
is surjective.
Exercise 2.1. Show that dimCM0
X(r, d) = 2(r2 − 1)(g − 1).1
With this in place, it is easy to reason in a number of ways that dimCMX(r, d) = 2r2(g−1)+2.
The difference between the two dimensions is 2g, which is the sum of the dimension of the
Jacobian of X and number of linearly independent 1-forms on X – the latter accounts for
removing the trace from φ.
2.3 Examples
The Kobayashi–Hitchin correspondence allows us to construct examples of solutions to Hitchin’s
equations as Higgs bundles, simply by combining a holomorphic bundle with a Higgs field φ
that fails to preserve “bad” subbundles with excess slope. One can achieve this by constructing
a Higgs field that leaves no proper subbundle invariant whatsoever. In fact, if E = L is a holo-
morphic line bundle on X, then any φ has this property, and so a line bundle with a section
φ ∈ H0(X,L⊗L∗ ⊗ ωX) = H0(X,ωX), which is nothing more than a holomorphic one-form, is
an example of a Higgs bundle.
Exercise 2.2. Show thatMX(1, d) is homeomorphic to R2g×
(
S1
)2g
and thatM0
X(1, d) is just
a point.
1Hint : Each of the two numbers that must be summed to give dimC H1 can be expressed as a difference,
owing to the injectivity and surjectivity properties. These differences can be rearranged in such a way that
Riemann–Roch can be applied.
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 5
A more interesting example comes from considering the rank-2, degree-0 split bundle E ∼=
ω
1/2
X ⊕ ω−1/2X , where ω
1/2
X is a choice of holomorphic square root of ωX . (There are 22g such line
bundles on X.) The anti-diagonal Higgs field
φ =
(
0 α
1 0
)
preserves neither summand of E , and so is stable. Here, 1 is interpreted as the identity endomor-
phism for ω
1/2
X . The section α is a quadratic differential on X. Hence, we have injective maps
from H0
(
X,ω⊗2X
)
into M0
X(2, 0) and MX(2, 0). Through the Hitchin equations, the existence
of this particular family of Higgs bundles induces a uniformizing metric on X, as in Hitchin’s
paper [38].
2.4 Hitchin fibration
The principal tool for understanding the structure of MX(r, d) is the Hitchin map, which is
nothing more than the map that assigns to each Higgs bundle the characteristic polynomial
(interpreted correctly) of its Higgs field. We write
Θ: MX(r, d) −→ Ar :=
r⊕
i=1
H0
(
X,ω⊗iX
)
defined by sending the isomorphism class of (E , φ) to the r-tuple of coefficients of the character-
istic polynomial, each of which is a section of a respective tensor power of ωX . The codomain Ar
is an affine space called the Hitchin base. The map Θ is proper and thus fibres MX(r, d) by
compact subvarieties, the Hitchin fibres. This properness result was established for the space
MX(2, d) by Hitchin [38]. In general, see [52].
This gives us a very coarse idea of how the moduli space “looks”: it is an affine space
populated by compact fibres, the generic ones certainly being smooth. Can we sharpen this?
To do so, we take a closer look at the characteristic polynomial of a given φ – namely, we want
to understand the geometry of its roots. Denote by |ωX | the total space of ωX ; by (x, y(x)),
a local coordinate on |ωX | (x is “horizontal” and y is “vertical”); and by p, the bundle projection
ωX → X. The bundle ρ∗ωX on |ωX | has a natural section w given by w(x, y(x)) = y(x), where
the output value is seen as living in the copy of the fibre (ωX)x attached to itself at y(x) in
the pullback bundle. This is the so-called Seiberg–Witten differential. These objects allow us to
define:
Definition 2.3. The spectral curve determined by a = (a1, . . . , ar) ∈ Ar is the 1-dimensional
subvariety Xa ⊂ |ωX | given by the zero locus of the polynomial
wr(y) + a1(p(y))wr−1(y) + · · ·+ ar(p(y)).
For a sufficiently general choice of a, Xa is a non-singular curve ramified over X with order r.
In other words, it is an r : 1 branched cover and so we have fashioned a new Riemann surface,
related to X, from data in the Hitchin base Ar. Now, consider any line bundle L on Xa. The
direct image p∗L is a locally-free sheaf of rank r and hence can be identified as the sheaf of
sections of a holomorphic bundle E → X. The Seiberg–Witten differential, thought of as acting
by
w|Xa : L −→ L⊗ p∗ωX ,
s 7−→ s · y
6 S. Rayan
on the line bundle, pushes forward to a linear map between the sheaves E and E ⊗ωX . In other
words, we have constructed a Higgs field φ for the bundle E , and so the data of a line bundle onXa
leads to a Higgs bundle on X. In the opposite direction, a Higgs bundle (E , φ) on X determines
a tuple a ∈ Ar through the Hitchin map. This tuple generates a spectral curve Xa, which is
exactly the spectrum of φ, producing distinct eigenvalues at most points x ∈ X (corresponding
to the r sheets of Xa, branching wherever there are repeated eigenvalues). The eigenspaces of φ,
which are generically 1-dimensional, form a sheaf L on Xa, which can be shown to be a line
bundle. (See Proposition 4.2(2) in Chapter 2 of [41].)
Essentially, we have that an isomorphism class of holomorphic line bundles [L] on Xa is
equivalent to the data of an isomorphism class of Higgs bundles [(E , φ)] on X. This is the spectral
correspondence as developed in [6, 16, 17, 39]. It follows from it that the generic fibre Θ−1(a)
is isomorphic to the Jacobian variety of Xa. This Jacobian, however, is not typically the space
of degree 0 line bundles on Xa. Rather, their degree is shifted by the ramification. The actual
degree e is given by
e = d− (1− g′) + r(1− g),
where g′ is the genus of Xa. We denote this Jacobian by Jace(Xa) – it has the same dimension
regardless of the value of e.
Exercise 2.4. Derive the above formula for e.2
Since the genus g′ of Xa is equal to the complex dimension of its Jacobian and since Θ−1(a) ∼=
Jace(Xa) for generic a ∈ Ar, we can obtain the genus of the generic spectral curve by subtracting
the dimension of Ar from the dimension of the moduli space. For each power of ωX , Riemann–
Roch reads as
h0
(
X,ω⊗iX
)
− h0
(
X,ω⊗1−iX
)
= (2i− 1)(g − 1).
For each i > 1, ω⊗1−iX has degree (1 − i)(2g − 2) > 0 and so h0
(
X,ω⊗1−iX
)
vanishes, leaving us
with
h0
(
X,ω⊗iX
)
=
{
g if i = 1,
(2i− 1)(g − 1) if i > 1.
It follows that
dimCAr = r2(g − 1) + 1.
We observe that this is exactly half the dimension ofMX(r, d), and so g′ is also r2(g−1)+1. In
the SL(r,C) case, we subtract h0(X,ωX) = g from the dimension of Ar (to remove the trace).
We denote this reduced based by A0
r . At the same time, we recall that we subtract 2g from
the dimension of MX(r, d) to get that of M0
X(r, d), and so the half-dimensionality of the base
persists here. (The spectral curve has the same genus as in the GL(r,C) case, but the Jacobian
is replaced with a smaller-dimensional Prym variety.)
For an example, let us examine the moduli spaceM0
X(2, 0). According to the formulas derived
above, it has dimension 6g − 6; the generic spectral curve has genus g′ = 4g − 3, which is also
the dimension of the base A0
2; and the degree of the relevant line bundles on the spectral curve
is e = 3g − 6. The Hitchin base is just H0
(
X,ω2
X
)
, the space of quadratic differentials, which
are the possible determinants of φ. If we take X of genus g = 2 specifically, then the moduli
2Hint : Use the Riemann–Roch theorem in combination with properties of the pushforward operation between
two smooth curves, one a branched cover of the other.
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 7
space is 6-dimensional, fibering over a 3-dimensional base, with X covered 2 : 1 by a smooth
genus g′ = 5 curve Xa for each generic a ∈ H0
(
X,ω2
X
)
. By the spectral correspondence, line
bundles of degree e = 0 push forward from Xa to produce stable Higgs bundles on X. Recall
now the family of Higgs bundles E ∼= ω
1/2
X ⊕ ω−1/2X with
φ =
(
0 α
1 0
)
that live in this moduli space. The map Θ sends φ = ( 0 α
1 0 ) to −α ∈ H0
(
X,ω2
X
)
. These Higgs
fields form the Hitchin section, intersecting each Hitchin fibre in exactly one point. From the
spectral point of view, there is a special line bundle on each Xa that pushes forward to produce
an element of this family.
2.5 Integrable system
The moduli space is a fibration in a different way. If NX(r, d) is the moduli space of stable
bundles of rank r and degree d (stable here means that all proper subbundles must satisfy the
slope condition), then the tangent space TE(NX(r, d)) at some bundle E is
H1(X,End(E))
Serre∼= H0(X,End(E)⊗ ωX)∗
and so the cotangent bundle to NX(r, d) is contained inside the moduli space of Higgs bundles.
It is important to note there are stable Higgs bundles (E , φ) for which the vector bundle E
alone is unstable and so the projectionMX(r, d) −→ NX(r, d) is only defined above those Higgs
bundles with stable underlying bundle. The symplectic form on T ∗NX(r, d) can, however, be
canonically extended to one on MX(r, d). (The complex structure on T ∗NX(r, d) also extends
to MX(r, d) in a compatible way, producing one of the complex structures making up the
hyperkähler structure on the moduli space.)
Hitchin proved in [39] that this symplectic structure on MX(r, d) is an algebraically com-
pletely integrable Hamiltonian system. In particular, the real and imaginary parts of the com-
ponents of the Hitchin map Θ are functionally-independent, Poisson-commuting functions, of
which there are sufficiently-many due to the half-dimensionality of Ar, thereby providing a com-
plete set of Hamiltonians. The Hitchin fibres are the Liouville tori of the dynamical system.
Many known integrable systems can be realized as Hitchin systems, with flows linearizing on
the Hitchin fibres. (It is often necessary to allow the genus to be 0 or 1 and to puncture X
so that φ develops poles at the punctures. This leads naturally to the parabolic Higgs bundle
story, cf. [1, 10]. See also for [45] for Hitchin-type integrable systems in which ωX is replaced
with other line bundles.)
3 U(1)-action
The coarse description above is not enough to tell us the global topology of the Hitchin fibration.
The fibration is nontrivial, due to the presence of special degenerate fibres, and so the global
topology is not simply that of a generic torus fibre (unless r = 1 – see Exercise 2.2). It turns
out that only one special fibre really matters: this is the one that we call the “nilpotent cone”,
as we will see below.
To study the topology, we could regard the moduli space as the gauge-theoretic moduli
space of solutions to Hitchin’s equations, in which case we would employ Morse theory for
a suitable height function. For us, this would be the L2-norm on MX(r, d), which is a multiple
of f(E , φ) = ‖φ‖2 coming from the Kähler metric associated to the complex structure extended
from T ∗NX(r, d) (cf. [13, 24, 38, 66, 67]). Here, we are concerned with critical points of f . If
8 S. Rayan
we regard the moduli space as the quasiprojective variety MX(r, d), as we have been doing up
until now, then we can employ Bia lynicki-Birula theory [7] for an algebraic group action. For
us, this is the action
λ · (E , φ) = (E , λ · φ)
of C?. Here, we are concerned with fixed points of the action. The two approaches are connected
by the following fact: all of the fixed points of the action are fixed points of the compact group
U(1) ⊂ C?. Moreover, the height function is a moment map for the U(1)-action and the fixed
points of the U(1)-action are critical points of f [38].
We denote byMX(r, d)U(1) the fixed points of the U(1)-action. A stable Higgs bundle (E , φ)
belongs toMX(r, d)U(1) if and only if there exists a automorphismAλ of E so thatAλφA
−1
λ = eiθφ
for each λ ∈ [0, 2π). In other words, a Higgs bundle is fixed if and only there is a change of basis
that undoes the action of U(1). We would like to have a useful description of these fixed points.
3.1 Holomorphic chains
Now, suppose that (E , φ) ∈MX(r, d)U(1). If Aλ is the one-parameter family of transformations
that corrects for the action, then there is a limiting endomorphism Λ that generates this family
infinitesimally, i.e.,
Λ := Dλ(Aλ)|λ=0,
where Dλ is a suitably-defined derivative.
Exercise 3.1. Show that [Λ, φ] = iφ.3
It is also possible to argue that, if ∂A is a C-linear operator that determines the holomorphic
structure on E , e.g., an operator induced by the unitary connection A satisfying Hitchin’s equa-
tions, then ∂A and Λ must be simultaneously diagonalizable. (This comes from the fact that
automorphisms Aλ act trivially by conjugation on the holomorphic structure, by definition of
the U(1)-action.) It follows that E decomposes into eigenspaces of Λ.
We will call these eigenspaces B1, . . . ,Bn. Geometrically speaking, these are holomorphic sub-
bundles of E . Likewise, the eigenvalues of Λ are global holomorphic functions on X: s1, . . . , sn,
respectively. Now, we take some Bk and apply both sides of the identity from Exercise 3.1 to it.
We find
Λ(φBk) = (sk + i)(φBk),
where i =
√
−1. This indicates that the image of Bk under the Higgs field is a subbundle of
the eigen-bundle for eigenvalue sk + i. In turn, this implies that the eigenspaces are grouped
into sequences, with their eigenvalues ordered as sk, sk + i, sk + 2i, and so on. These sequences
terminate when the image of an eigen-bundle under φ is zero (or when we reach the last eigen-
bundle). It can be shown that the existence of multiple, disconnected sequences for a fixed point
would violate stability, as stable Higgs bundles are irreducible in the sense that they cannot
decompose into proper, nonzero Higgs subbundles. Hence, it follows that for a rank-r Higgs
bundle (E , φ) ∈MX(r, d)U(1), there exists a number n such that E =
⊕n
k=1 Bk and
B1
φ1−→ B2 ⊗ ωX
φ2−→ · · · φn−1−→ Bn ⊗ (ωX)⊗(n−1)
φn−→ 0,
where φk = φ|Bk and φk is not identically zero for k < n.
3Hint : Differentiate the fixed-point equation AλφA
−1
λ = eiθφ using the same derivative.
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 9
A Higgs bundle admitting a description such as above is referred to as a holomorphic chain,
cf. [2, 3, 12, 22]. Equivalently, such Higgs bundles can be regarded as complex variations of
Hodge structure – see [61].
This description says that we can write a fixed point in a basis of sections where φ has the
blocks φi arranged sub-diagonally:
φ =
0 0 · · · 0 0
φ1 0 · · · 0 0
0 φ2 · · · 0 0
. . .
0 0 · · · φn−1 0
.
Such a matrix is nilpotent and so every fixed point belongs to the Hitchin fibre Θ−1(0), which is
what we refer to as the nilpotent cone. In general, not every point in the nilpotent cone is fixed:
only those admitting a strict block sub-diagonal (or super-diagonal) description are fixed.
Exercise 3.2. Show that a Higgs bundle (E , φ) with strict block sub-diagonal Higgs field is
necessarily fixed under the U(1)-action.
If (E , φ) ∈ MX(r, d)U(1), then there is a well-defined n-tuple (r1, . . . , rn) that encodes the
ranks of the Bk subbundles – this is the rank vector of the fixed point.
3.2 Localization
The key result for us is that the total space of the Hitchin fibration MX(r, d) deformation
retracts, via the gradient flow of the moment map of the U(1)-action, onto Θ−1(0) [30]. In terms
of invariants, the cohomology ring localizes to the fixed-point locus inside Θ−1(0). The Poincaré
series P[MX(r, d)] that generates the Betti numbers of the rational cohomologyH•(MX(r, d),Q)
will be a weighted sum of the Poincaré series P[Ci] of the connected components Ci, i ∈ I, of the
fixed-point locus. Also, let
ι : MX(r, d)U(1) → N
be the function that assigns to each fixed point the number of negative eigenvalues of the Hessian
of f at that point, where f is again the moment map. This function ι is constant on each Ci as
per Lemma 9.2 in [35] and so the natural number ι(Ci) is well-defined. It is also worth noting
that the rank vector (r1, . . . , rn) is constant on connected components of the fixed-point locus,
as are the degrees of the Bk’s.
Computing ι will be an important ingredient in the weighted sum that yields P[MX(r, d)].
Thinking of ι as the dimension of the “downward” subbundle of the normal bundle toMX(r,d)U(1)
at a fixed point, we can obtain the value of ι by taking a deeper look at the deformation theory
from Section 2.2 in the case of a fixed point (cf. Section 2.1 of [55]). When (E , φ) is fixed, so
that a decomposition into an ordered sequence of subbundles Bk exists, the action of φ is with
weight 1 with respect to this sequence, i.e.,
φk : Bk −→ Bk+1 ⊗ ωX .
In other words, elements
θ ∈ H0(X,End0(E)⊗ ωX)
imH0(X,End0(E))
∧φ−→ H0(X,End0(E)⊗ ωX)
10 S. Rayan
that act with weight ` = 1 with respect to the sequence form part of the tangent space at (E , φ)
to MX(r, d)U(1). The other part comes from the elements
β ∈ kerH1(X,End0(E))
∧φ−→ H1(X,End0(E)⊗ ωX)
that act with weight m = 0 on the sequence, preserving the holomorphic structure of each Bk.
(Since the Higgs field is nilpotent, we can use End0 here regardless of whether the group is
GL(r,C) or SL(r,C).) The downward flow comes from weights (`,m) with ` ≥ 2 and m ≥ 1.
These weights shorten the holomorphic chain until its length is n = 1 and the Higgs field is zero,
taking us to the “bottom” of the nilpotent cone. Out of this comes something computational:
ι(Ci) is the sum of the (real) dimensions of the respective ` ≥ 2 and m ≥ 1 subspaces of the
tangent space.
With all of this in place, the localization identity takes the precise form:
Theorem 3.3 (Hitchin [38]). P[MX(r, d)](t) =
∑
i∈I
tι(Ci)P[Ci](t).
Were the moduli space compact, we would have P[Ci](t) = 1 for each i ∈ I, as in standard
Morse theory, and so the Poincaré series would reduce to
∑
i∈I
tι(Ci). However, in our case the Ci
are generally positive-dimensional with nontrivial contributions to the cohomology ring. For
example, the downward flow of f terminates at the points with ι = 0, which is also where
‖φ‖2 = 0. These global minimizers are precisely the stable Higgs bundles of the form (E , 0),
which is the set of fixed points with rank vector (r). This component is in fact the moduli space
of stable bundles, NX(r, d), which is positive-dimensional for g ≥ 1. For example, if we consider
the SL(2,C) case with fixed determinant of odd degree d, then the Poincaré polynomial of this
component is known by [4, 29] to be
P
[
N 0
X(2, d)
]
(t) =
(
1 + t3
)2g − t2g(1 + t)2g(
1− t2
)(
1− t4
) .
Like the presentation here, [4] also takes a Morse-theoretic approach. The Poincaré series of
NX(r, d) factors as the product of P
[
N 0
X(2, d)
]
(t) and that of the Jacobian of X (cf. [4]), and
so we have
P[NX(2, d)](t) = (1 + t)2g
(
1 + t3
)2g − t2g(1 + t)2g(
1− t2
)(
1− t4
) .
The connected components with higher values of ι, for which less is immediately known, are
an obstruction to determining P[MX(r, d)] in high rank, although much recent progress has been
achieved via other means as highlighted in the introduction. To shed some light on the difficulty,
we recognize that the fixed points can be thought of as representations of A-type quivers, with
lengths and labels determined by partitions of r and d:
•r1,d1 −→ •r2,d2 −→ · · · −→ •rn,dn .
However, we are not looking at representations in the usual category of vector spaces; rather,
we are in the category of bundles on a fixed curve X with ωX -twisted morphisms. These
representations are also known as quiver bundles, cf. [25, 26, 55, 56, 59]. The moduli space
of stable bundles is the solution to the simplest version of this problem, where the quiver has
a single node:
•r,d.
Nevertheless, we wish to exhibit a couple of sample calculations in low rank where we can
determine this polynomial completely.
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 11
4 Calculations
4.1 Rank r = 1
We start off with the simplest possible example, just to have an instance where the answer is
readily seen to be correct. The only partition of r = 1 is the rank vector (1). The entire fibre
Θ−1(0) of MX(1, d), which is the submanifold {(L, 0) : L ∈ Jacd(X)}, is fixed by the U(1)-
action. Hence, there is a single connected component of the fixed-point locus and the number ι
is 0 – there are no further components to which to flow down. It follows that
P[MX(1, d)](t) = P
[
Jacd(X)
]
(t) = (1 + t)2g,
agreeing exactly with Exercise 2.2. (Of course, for M0
X(1, d) the moduli space is just a point
and the result is even more trivial.)
4.2 Rank r = 2
Now, we look at MX(2, d) for some odd d. For convenience, we take d = 1. Here, we mostly
follow Hitchin in [38], although there are a few notable differences: we do the GL(2,C) case rather
than SL(2,C) and our calculation of ι will use the approach outlined in the preceding section.
The elements of the fixed point set are of two types, (2) and (1, 1). Those with rank vector (2)
correspond to the moduli space of stable bundles on X, as mentioned earlier. These are the
fixed points with ι = 0, as per the previous section. Therefore, the contribution to the Poincaré
series is
t0(1 + t)2g
(
1 + t3
)2g − t2g(1 + t)2g(
1− t2
)(
1− t4
) .
Now, each holomorphic chain of type (1, 1) consists of two line bundles B1 and B2 together
with a map φ1 : B1 → B2 ⊗ ωX . Let b = degB1, in which case degB2 = 1 − b. Note that B2 is
annihilated by the overall Higgs field, and so we must have 1− b strictly less than the slope of
E = B1 ⊕ B2. Hence, b ≥ 1. On the other hand, if φ1 = 0, then B1 would be invariant, which
violates stability as b would exceed the slope of E . Having φ 6= 0 requires that
deg(B∗1 ⊗ B2 ⊗ ωX) = 2g − 2b− 1
is nonnegative. Taking these together, we have 1 ≤ b ≤ g − 1.
Certainly, two choices of B1 with different degrees cannot lie in the same connected component
of MX(2, 1)U(1). Therefore, let us fix a value of b in the range above. The data is thus a triple
of a line bundle in Jacb(X), another in Jac1−b(X), and a map in H0(X,B∗1 ⊗ B2 ⊗ ωX). The
dimension of the third space depends on B1 and B2. To clarify this, suppose B2 is fixed. Instead
of keeping track of B1, we can instead deal with D = B∗1⊗B2⊗ωX . The choice of B1 determines D
and vice-versa. The relevant data is now the pair (D, φ1) in which D is a line bundle of degree
−2b+ 2g−1 and φ1 is a holomorphic section of this line bundle. Since φ1 is not identically zero,
this data determines an effective divisor of degree −2b+ 2g−1 on X, which is an element of the
(−2b+2g−1)-fold symmetric product of X with itself: S−2b+2g−1(X). Notice that for g ≥ 2 and
1 ≤ b ≤ g − 1, the order of this product is always positive – in other words, we are considering
divisors of at least 1 point. An element of this symmetric product determines a line bundle D
together with a nonzero section φ1 vanishing on the divisor. This section is determined only up
to scale, i.e., φ1 ∈ PH0(X,D). However, since we are working inside the moduli spaceMX(2, 1),
we are only considering holomorphic chains up to equivalence by automorphisms of E = B1⊕B2
that preserve the structure of a (1, 1) chain. In other words, we are free to use the action of
C∗ × C∗ ⊂ Aut(E) to put a given chain into a representative form. We can use either C∗ to
12 S. Rayan
identify any two φ1’s that differ only by scale, and so the projective representatives given by the
divisor coincide exactly with the equivalence classes of pairs (D, φ1) in the moduli space.
Hence, MX(r, d)U(1) has g connected components: the moduli space of stable bundles to-
gether with g−1 components coming from fixed points with rank vector (1, 1). By the argument
above, components of the latter type are indexed by b in 1 ≤ b ≤ g − 1 and each component is
a bundle over S−2b+2g−1(X) with fibre Jac1−b(X), where the Jacobian accounts for the choice
of B2. For each b we need the Poincaré series of the respective (−2b + 2g − 1)-fold symmetric
product of X. These generating functions are due to Macdonald [44]. Specifically, the Poincaré
polynomial, in t, of SnX is the coefficient of sn in the Taylor–Maclaurin series expansion of
(1 + st)2g
(1− s)
(
1− st2
) .
Now, regarding the indices ι for the type (1, 1) components, we note that the only element θ in
H0(X,End0(E)⊗ ωX)
imH0(X,End0(E))
∧φ−→ H0(X,End0(E)⊗ ωX)
acting with weight 2 or higher on the sequence (B1,B2) is θ = 0, as there are only two bundles
in the sequence. Hence, we need only account for elements β of weight at least 1 in
kerH1(X,End0(E))
∧φ−→ H1(X,End0(E)⊗ ωX).
For the same reasons, there are no elements of weight 2 or higher, and so we seek the elements
of weight exactly 1. Before the action of ∧φ, the weight 1 elements form H1(X,B∗1 ⊗ B2). The
map ∧φ sends these to weight 2 elements in H1(X,B∗1⊗B2⊗ωX). Since the only weight 2 element
is the zero element, we have that all weight 1 elements are in the kernel of ∧φ. Our calculation
of ι thereby reduces to the real dimension of H1(X,B∗1 ⊗B2). Since deg(B∗1 ⊗B2) = 1− 2b < 0,
we have that H0(X,B∗1 ⊗ B2) vanishes. Then, by Riemann–Roch we have
ι(E , φ) = 4b− 4 + 2g.
Taking all of this together, we get that the Poincaré series of MX(2, 1) is
P[MX(2, 1)][t]
= (1 + t)2g
((
1 + t3
)2g − t2g(1 + t)2g(
1− t2
)(
1− t4
) +
g−1∑
b=1
t4b−4+2gP
[
S−2b+2g−1(X)
]
(t)
)
,
where the Poincaré polnyomials for the symmetric products come from Macdonald’s function.
Exercise 4.1. Using the results above, check that when g = 2, we have that
P[MX(2, 1)][t] = (1 + t)4
(
1 + t2 + 4t3 + 2t4 + 4t5 + 2t6
)
.
Exercise 4.2. Using the results above, check that when g = 3, we have that
P[MX(2, 1)][t] = (1 + t)6
(
1 + t2 + 6t3 + 2t4 + 6t5 + 17t6 + 12t7 + 18t8 + 32t9
+ 18t10 + 12t11 + 3t12
)
.
Notice that the Poincaré polynomials above are not palindromes, even though the moduli
spaces are smooth. This is of no concern, given that the moduli spaces are non-compact. For
example, in g = 3 the unequal Betti numbers in degrees 0 and 18 tell us that, whileMX(2, 1) is
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 13
topologically connected (b0 = 1), the space has a number of irreducible or “algebraic” compo-
nents (b18 = 3 of them). It is also worth noting that the highest power of t in each case is equal
to 2r2(g− 1) + 2, which is the real dimension of the fibre of the Hitchin map. This is consistent
with the fact that the Hitchin base is contractible and the nontrivial topology lies in Θ−1(0).
A reasonable question is whether P[MX(2, 1)](t)/(1+ t)2g is the Poincaré series ofM0
X(2, 1),
the SL(2,C) moduli space. In general, this is not the case. Rather, the quotient is the generating
function for the Betti numbers of the Langlands dual moduli space; that is, the PGL(2,C) moduli
space. The issue is that there is a nontrivial action of the finite group Γ of 2-torsion line bundles –
the line bundles P with P⊗2 = OX – on M0
X(2, 1). As a result, there is a variant cohomology
and an invariant cohomology with regards to this action. The quotient ofM0
X(2, 1) by Γ, which
has order 22g, is the PGL(2,C) moduli space. It possesses only the invariant cohomology, whose
ranks are given by the coefficients of P[MX(2, 1)](t)/(1 + t)2g. For genus g = 2, this invariant
part is
1 + t2 + 4t3 + 2t4 + 4t5 + 2t6,
as in the exercise above. In contrast, the Poincaré series of M0
X(2, 1) for g = 2 is
1 + t2 + 4t3 + 2t4 + 34t5 + 2t6
as computed by Hitchin in [38]. Here, we can see the Γ-variant cohomology concentrating in the
degree 5 part of the cohomology ring. In terms of the calculations, the main difference relative
to above is that we are fixing the determinant of E to be some fixed line bundle V, from which B1
and B2 are related by B2 = B∗1 ⊗ V. Then, to bring in divisors, we need to define a line bundle
D = (B∗1)2⊗V⊗ωX . It follows that instead of symmetric products of X, we get 22g-fold covers of
symmetric products, with fibres consisting of the line bundles B1 whose squares are isomorphic
to one another. Here, we see the action of Γ working itself into the cohomology.
For further information on the variant versus invariant cohomology, we refer the reader
to [33, 35]. It is also perhaps crucial to point out that the appearance of Langlands duality
here is neither superficial nor a red herring. For how Langlands duality manifests in Higgs bun-
dle moduli spaces – and how it relates to mirror symmetry – we refer the reader to the same
reference in addition to [18, 19, 43].
The next logical step would be to try our hand at rank 3. The calculation using Morse
theory is noticeably more difficult, because of fixed points with rank vectors (1, 2) and (2, 1).
The type (3) case remains the moduli space of bundles, whose topological contribution we
already know as per above, while the type (1, 1, 1) fixed points involve symmetric products of X
in an analogous way to the preceding calculations. For (1, 2) and (2, 1), the data of the fixed
point can be converted into a pair (D, θ) in which D is a rank 2 bundle related to the bundles
in the chain and θ is a section of D. The issue now is to understand the moduli space of such
pairs on X. Gothen’s approach [24] uses Thaddeus’ strategy of varying a stability parameter and
then constructing the moduli space in steps by keeping track of birational transformations as the
parameter is deformed [64]. This stability parameter, which is natural in quiver bundle moduli
problems, originates in [11]. The rank 4 Poincaré series was computed in [22] using a method that
is formally similar to the Morse localization above, but which is rooted in motivic considerations.
Notably, the (2, 2) case had not submitted readily to the variation-of-stability approach, but was
resolved via the motivic approach.
We can also ask about the exact structure of the ring H•(MX(r, d),Q) itself. For r = 2, the
generators and relations are worked out in [36, 37, 46]. For the status of this in higher rank, we
refer the reader to [15, 14]. For examples of Betti numbers over other fields, we refer the reader
to [5] where the Z2 Betti numbers are calculated for rank 2 Higgs bundles
14 S. Rayan
5 Combinatorial questions
In the Morse-theoretic calculations of the preceding section, the degree d of the Higgs bundles
enters the calculations explicitly when we work with stable holomorphic chains. However, non-
abelian Hodge theory forces the Betti numbers ofMX(r, d) to be independent of d ∈ Z, at least
when d is coprime to r as we have been assuming all along. This is due to the fact that the
Poincaré series of the GL(r,C) character variety of X is insensitive to d, where d is used to
define twisted representations of π1(X) [33]. This is combinatorially interesting because there
is nothing at first glance to say that corresponding connected components ofMX(r, d)U(1) have
identical Poincaré polynomials – or even that there are the same number of components.
The d-independence of Betti numbers leads to a number of combinatorial observations. We
offer a small sample. For our purposes, these are easier to see if we permit X to have genus
g = 0 and if permit Higgs fields twisted by a line bundle other than ωX . Namely, we wish to
consider “twisted” Higgs bundles of the form (E , φ) with E a vector bundle on the projective
line P1 and
φ : E −→ E ⊗O(q),
where O(q) is the unique (up to isomorphism) line bundle on P1 of degree q > 0. (The cotangent
bundle ωP1 is unsuitable here, as we will then have q = −2 and all Higgs bundles of rank r > 1
and coprime degree d will be unstable.) These Higgs bundles do not rise in the same natural
way in gauge theory, but they are nonetheless useful as a test case here. In particular, these
moduli spaces, which are constructed using slope stability in exactly the same way asMX(r, d),
have the same natural U(1)-action [55, 56].
Interestingly, this moduli space does not fit in a natural way into nonabelian Hodge theory –
one would have to puncture P1 along a divisor D and then regard φ as being valued in O(q) =
ωX ⊗O(D) with poles along D, with certain conditions on the residues of φ at the poles [8, 60].
However, this changes the topology of the moduli space in a significant way and reintroduces
the bundle moduli (as we are now keeping track of data in the fibres of E at the poles). Keeping
our definition the way it is, i.e., holomorphic bundles with holomorphic O(q)-valued Higgs fields,
there is no immediate relationship to a character variety and, as such, no obvious reason for
degree independence of the Betti numbers. Yet, it seems to hold in direct calculations of the
Betti numbers in low rank, as in [8, 48, 54, 60].
In this setting, because of the relative lack of vector bundle moduli, we attain fairly clear
combinatorial descriptions for certain Betti numbers. It is possible for this moduli space to
establish via Morse theory that the top Betti number – that is, the coefficient of the highest
power of t appearing in the Poincaré series – is precisely the number of connected components
of the fixed-point locus coming from fixed points of type (1, . . . , 1). This can be shown in turn
to be the number of solutions (d1, . . . , dr) ∈ Zr to the equation
d1 + · · ·+ dr = d
subject to di−di−1 ≤ q and, if r > 1, (dj + · · ·+dr)/(r− j+ 1) < d/r for all 2 ≤ j ≤ r. Because
the dj ’s are degrees of line bundles, they are permitted to be negative, and so the equation
d1 + · · · + dr = d alone is an unbounded integer partition problem. The problem becomes
well-posed precisely because of stability.
The degree independence of the Betti numbers would, as a corollary, make the solution of
this partition problem independent of d, again assuming coprimality with regards to r. If we
fix, say, q = 1 and then compute the solutions of the above partition problem for increasing r,
we find the following sequence regardless of which (coprime) d we choose:
1, 1, 1, 2, 5, 13, 35, 100, 300, 925, 2915, 9386, 30771, 102347, 344705, . . . .
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces 15
Interestingly, this sequence appears in the OEIS database as A131868 [42]. The entry gives the
following function that yields these numbers for each r:
Ω(r) =
1
2r2
∑
e|r
µ(r/e)
(
2e
e
)
(−1)e+1,
where µ is the Möbius function. By examining type (1, . . . , 1) fixed points for other values of q
and experimenting with the function Ω, it is not hard to make an educated guess as to a more
general version of this function for any q:
Ω(r, q) =
1
(q + 1)r2
∑
e|r
µ(r/e)
(
(q + 1)e
e
)
(−1)qe+1.
That this is the correct function for all r > 0, q > 0 for our counting problem is actually
established by Reineke in [57]. This also establishes the d independence.
The OEIS entry provides a combinatorial interpretation for the top Betti numbers of the
q = 1 moduli spaces that, while similar in spirit, is not exactly the same as the ours: r ·Ω(r, 1) is
the number of size r subsets of {1, . . . , 2r − 1} that sum to 1 modulo r. Right away, the degree
independence means that we can replace 1 mod r in this problem with dmod r without changing
the solutions. This problem falls into a set of related combinatorial problems studied by Erdös–
Ginzburg–Ziv [20]; in some of these, it is known that one can shift the interval {1, . . . , 2r − 1}
freely to any consecutive 2r− 1 numbers (cf. the related entry, A145855 [53]). That being said,
the partition problem of type (1, . . . , 1) fixed points is one in which the differences between
consecutive parts of the partition are bounded, rather than overall interval in which the parts
are allowed to lie.
We can also examine the Poincaré series itself as r and q grow. With r fixed and q allowed
to grow indefinitely, the Poincaré series can be seen to tend to that of the classifying space of
the gauge group of the underlying smooth bundle. If we fix q and drive r to larger values – or
drive both to infinity – the series tends to
1 + t2 + 3t4 + 5t6 + 10t8 + 16t10 + 29t12 + 45t14 + 75t16 + 115t18 + · · · ,
whose coefficients are captured in A000990 [63]. If the equivalence of counting problems is
correct, this would say that the coefficient of t2n is the number of plane partitions of n with at
most 2 rows. This is especially interesting because it provides a combinatorial interpretation
for each Betti number individually, while Morse theory builds each coefficient from potentially
many separate combinatorial problems as data from different components of the fixed-point
locus contribute to the same coefficient.
Finally, it is worth commenting that in all of these cases – the ordinary Higgs bundles of
the preceding sections and the twisted ones on P1 here – that the lack of palindromy in the
Poincaré series is skewed in such a way that the largest Betti number lies to the “right” of
the middle coefficients, i.e., between the middle and the top Betti number. This phenomenon
is studied in [34] in the context of non-compact, hyperkähler semiprojective moduli spaces X .
Here, “semiprojective” refers to the property of the having an algebraic C?-action with projective
fixed-point set with the limit lim
λ→0
λx existing for all x ∈ X . The fact that this persists for the
twisted Higgs bundle moduli spaces on P1, which are semiprojective but have no hyperkähler
structure, suggests there could be a combinatorial explanation for the phenomenon, independent
of the geometry.
In general, we see that for Higgs-bundle-type moduli spaces there is a complicated dance
between geometry and combinatorics playing out within the cohomology ring, with geometric
phenomena forcing combinatorial identities to emerge and with combinatorial identities express-
ing themselves geometrically in surprising ways. Throughout, topology is the conduit.
https://oeis.org/A131868
https://oeis.org/A145855
https://oeis.org/A000990
16 S. Rayan
Acknowledgements
I thank Laura Schaposnik for organizing the series of workshops in which the mini-course took
place, and both her and Lara Anderson for encouraging the preparation of this survey. With
regards to the workshops, I acknowledge support from UIC NSF RTG Grant DMS-1246844,
the UIC Start-Up Fund of L. Schaposnik, and the grants NSF DMS 1107452, 1107263, 1107367
RNMS: GEometric structures And Representation varieties (the GEAR Network). I am grateful
to Marina Logares, who gave a mini-course in parallel to mine, for insightful discussions as well
as to Laura Fredrickson for useful comments on the manuscript during its preparation. I thank
the referees for helpful remarks and corrections that led to the final version of this article.
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https://arxiv.org/abs/math.DG/0611113
1 Introduction
2 Background on Higgs bundles
2.1 Gauge theory
2.2 Deformation theory
2.3 Examples
2.4 Hitchin fibration
2.5 Integrable system
3 U(1)-action
3.1 Holomorphic chains
3.2 Localization
4 Calculations
4.1 Rank r=1
4.2 Rank r=2
5 Combinatorial questions
References
|
| id | nasplib_isofts_kiev_ua-123456789-209875 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:19:37Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rayan, S. 2025-11-28T09:38:21Z 2018 Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces / S. Rayan // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 67 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14D20; 46M20; 57N65; 05A19 arXiv: 1809.05732 https://nasplib.isofts.kiev.ua/handle/123456789/209875 https://doi.org/10.3842/SIGMA.2018.129 This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature. I thank Laura Schaposnik for organizing the series of workshops in which the mini-course took place, and both her and Lara Anderson for encouraging the preparation of this survey. With regards to the workshops, I acknowledge support from UIC NSF RTG Grant DMS-1246844, the UIC Start-Up Fund of L. Schaposnik, and the grants NSF DMS 1107452, 1107263, 1107367 RNMS: GEometric structures And Representation varieties (the GEAR Network). I am grateful to Marina Logares, who gave a mini-course in parallel to mine, for insightful discussions, as well as to Laura Fredrickson for useful comments on the manuscript during its preparation. I thank the referees for their helpful remarks and corrections that led to the final version of this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces Article published earlier |
| spellingShingle | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces Rayan, S. |
| title | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces |
| title_full | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces |
| title_fullStr | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces |
| title_full_unstemmed | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces |
| title_short | Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces |
| title_sort | aspects of the topology and combinatorics of higgs bundle moduli spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209875 |
| work_keys_str_mv | AT rayans aspectsofthetopologyandcombinatoricsofhiggsbundlemodulispaces |