Higgs Bundles— Recent Applications

Laura P. Schaposnik

Introduction • Via the nonabelian Hodge correspondence de- This note is dedicated to introducing Higgs bundles and veloped by Corlette, Donaldson, Simpson, and the Hitchin fibration, with a view towards their appearance Hitchin and in the spirit of Uhlenbeck–Yau’s work within different branches of mathematics and physics, fo- for compact groups, the moduli space is analyti- cusing in particular on the role played by the integrable cally isomorphic as a real manifold to the de Rahm system structure carried by their moduli spaces. On a com- moduli space ℳ푑푅 of flat connections on a smooth pact Riemann surface Σ of genus 푔 ≥ 2, Higgs bundles are complex bundle. pairs (퐸, Φ) where • Via the Riemann–Hilbert correspondence there is a complex analytic isomorphism between the de • 퐸 is a holomorphic vector bundle on Σ, and Rham space and the Betti moduli space ℳ of sur- • the Higgs field Φ ∶ 퐸 → 퐸 ⊗ 퐾 is a holomorphic 퐵 face group representations 휋 (Σ) → 퐺 . map for 퐾 ≔ 푇∗Σ. 1 ℂ Since their origin in the late 1980s in work of Hitchin Some prominent examples where these moduli spaces ap- and Simpson, Higgs bundles manifest as fundamental ob- pear in mathematics and physics are: jects that are ubiquitous in contemporary mathematics • Through the Hitchin fibration, ℳ퐺ℂ gives exam- and closely related to theoretical physics. For 퐺ℂ a com- ples of hyperkähler manifolds that are integrable plex semisimple Lie group, the Dolbeault moduli space of systems, leading to remarkable applications in

퐺ℂ-Higgs bundles ℳ퐺ℂ has a hyperkähler structure, and physics, which we shall discuss later on. via different complex structures it can be seen as different • Building on the work of Hausel and Thaddeus re- moduli spaces: lating Higgs bundles to Langlands duality, Donagi

and Pantev presented ℳ퐺ℂ as a fundamental ex- Laura P. Schaposnik is a professor of mathematics at the University of Illinois ample of mirror symmetry. [email protected] at Chicago. Her email address is . • Within the work of Kapustin and Witten, Higgs The author was supported in part by the NSF grant DMS-1509693 and the NSF bundles were used to obtain a physical derivation CAREER Award DMS-1749013. This material is based upon work supported by the NSF under Grant No. DMS1440140 while the author was in residence of the geometric Langlands correspondence through at the Mathematical Sciences Research Institute in Berkeley, California, during mirror symmetry. Soon after, Ngô found Higgs the Fall 2019 semester. bundles to be key ingredients when proving the Communicated by Notices Associate Editor Chikako Mese. fundamental lemma of the Langlands program, For permission to reprint this article, please contact: which led him to the Fields Medal a decade ago. [email protected]. Higgs bundles and the corresponding Hitchin inte- DOI: https://doi.org/10.1090/noti2074 grable systems have been an increasingly vibrant area, and

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 625 thus there are several expository articles, some of which we SL(2, ℂ)-Higgs bundles (퐸, Φ푎) parametrized by quadratic shall refer to: from the Notices article “What is... a Higgs differentials 푎 ∈ 퐻0(Σ, 퐾2) is given by bundle?” [3], to several graduate notes on Higgs bundles (e.g., the author’s recent [18]), to more advanced reviews 0 푎 (퐸 = 퐾1/2 ⊕ 퐾−1/2,Φ = ( )) . (3) such as Ngô’s 2010 ICM Proceedings article [4]. Hoping 푎 1 0 to avoid repeating material nicely covered in other reviews whilst still attempting to inspire the reader to learn more One may also consider 퐺-Higgs bundles for 퐺 a real about the subject, we shall take this opportunity to focus form of 퐺ℂ, which in turn correspond to the Betti moduli on some of the recent work done by leading young mem- space of representations 휋1(Σ) → 퐺. For example, SL(2, ℝ)- 1 bers of the community. Higgs bundles are pairs (퐸 = 퐿 ⊕ 퐿∗, Φ) for 퐿 a line bundle and Φ off diagonal, a family of which is described in Ex- Higgs Bundles ample 1. Higgs bundles arise as solutions to self-dual Yang–Mills In order to define a Hausdorff moduli space of Higgs equations, a nonabelian generalization of Maxwell’s equa- bundles, one needs to incorporate the notion of stability. tions that recurs through much of modern physics. Solu- For this, recall that holomorphic vector bundles 퐸 on Σ tions to Yang–Mills self-duality equations in Euclidean 4D are topologically classified by their rank 푟푘(퐸) and their space are called instantons, and when these equations are degree deg(퐸), through which one may define their slope reduced to Euclidean 3D space by imposing translational as 휇(퐸) ≔ deg(퐸)/푟푘(퐸). Then, a vector bundle 퐸 is stable invariance in one dimension, one obtains monopoles as (or semistable) if for any proper subbundle 퐹 ⊂ 퐸 one has solutions. Higgs bundles were introduced by Hitchin in that 휇(퐹) < 휇(퐸) (or 휇(퐹) ≤ 휇(퐸)). It is polystable if it is a [10] as solutions of the so-called Hitchin equations, the 2- direct sum of stable bundles whose slope is 휇(퐸). dimensional reduction of the Yang–Mills self-duality equa- One can generalize the stability condition to Higgs bun- tions given by dles (퐸, Φ) by considering Φ-invariant subbundles 퐹 of 퐸, vector subbundles 퐹 ⊂ 퐸 for which Φ(퐹) ⊂ 퐹 ⊗ 퐾. A Higgs 퐹 + [Φ, Φ∗] = 0, 휕 Φ = 0, (1) 퐴 퐴 bundle (퐸, Φ) is said to be stable (semistable) if for each where 퐹퐴 is the curvature of a unitary connection ∇퐴 = proper Φ-invariant 퐹 ⊂ 퐸 one has 휇(퐹) < 휇(퐸) (equiv. ≤). 휕퐴 + 휕퐴 associated to a Dolbeault operator 휕퐴 on a holo- Then, by imposing stability conditions, one can construct ℳ 퐺 morphic principal 퐺ℂ bundle 푃. The equations give a flat the moduli space 퐺ℂ of stable ℂ-Higgs bundles up to connection holomorphic automorphisms of the pairs (also denoted ℳ ). Going back to Hitchin’s equations, one of the ∗ 퐷표푙 ∇퐴 + Φ + Φ (2) most important characterizations of stable Higgs bundles and express the harmonicity condition for a metric in the is given in the work of Hitchin and Simpson and which carries through to more general settings: If a Higgs bun- resulting flat bundle. Concretely, principal 퐺ℂ-Higgs bundles are pairs (푃, Φ) where dle (퐸, Φ) is stable and deg 퐸 = 0, then there is a unique unitary connection ∇ on 퐸, compatible with the holo- • 푃 is a principal 퐺 -bundle, and 퐴 ℂ morphic structure, satisfying (1). • Φ is a holomorphic section of ad(푃) ⊗ 퐾. Finally, Hitchin showed that the underlying smooth We shall refer to classical Higgs bundles as those described manifold of solutions to (1) is a hyperkähler manifold, in the introduction and consider 퐺ℂ-Higgs bundles in their with a natural symplectic form 휔 defined on the infinitesi- vector bundle representation: seen as classical Higgs bun- mal deformations (퐴,̇ Φ)̇ of a Higgs bundle (퐸, Φ) by dles satisfying some extra conditions reflecting the nature of 퐺ℂ, dictated by the need for the (projectively) flat con- 휔((퐴̇ , Φ̇ ), (퐴̇ , Φ̇ )) = ∫ (퐴̇ Φ̇ − 퐴̇ Φ̇ ), nection to have holonomy in 퐺ℂ. For instance, when 1 1 2 2 tr 1 2 2 1 (4) Σ 퐺ℂ = SL(푛, ℂ), a 퐺ℂ-Higgs bundle (퐸, Φ) is composed of a holomorphic rank 푛 vector bundle 퐸 with trivial deter- 퐴̇ ∈ Ω0,1( 퐸) Φ̇ ∈ Ω1,0( 퐸) minant Λ푛퐸 ≅ 풪 and a Higgs field satisfying Tr(Φ) = 0, where End0 and End0 . Moreover, 0 he presented a natural way of studying the moduli spaces for which we shall write Φ ∈ 퐻 (Σ, End0(퐸) ⊗ 퐾). ℳ퐺ℂ of 퐺ℂ-Higgs bundles through what is now called the Example 1. Choosing a square root of 퐾, consider the Hitchin fibration, which we shall consider next. vector bundle 퐸 = 퐾1/2 ⊕ 퐾−1/2. Then, a family of Integrable Systems {푝 , … , 푝 } 1As in other similar reviews, the number of references is limited to twenty, and Given a homogeneous basis 1 푘 for the ring of in- thus we shall refer the reader mostly to survey articles where precise references variant polynomials on the Lie algebra 픤ℂ of 퐺ℂ, we denote can be found. by 푑푖 the degree of 푝푖. The Hitchin fibration, introduced in

626 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 [11], is then given by the line bundle reflecting the nature of 퐺ℂ. This ap- 푘 proach originates in the work of Hitchin and of Beauville, ℎ ∶ ℳ ⟶ 풜 ≔ 퐻0(Σ, 퐾푑푖 ), Narasimhan, and Ramanan (see [18] for references), and 퐺ℂ 퐺ℂ ⨁ 푖=1 we shall describe here an example to illustrate the setting. For SL(푛, ℂ)-Higgs bundles, the linear term in (5) vanishes (퐸, Φ) ↦ (푝1(Φ), … , 푝푘(Φ)). since Tr(Φ) = 0, and the generic fibres are isomorphic to ℎ The map is referred to as the Hitchin map: it is a proper Prym varieties Prym(푆, Σ) since Λ푛퐸 ≅ 풪. map for any choice of basis and makes the moduli space into an integrable system whose base and fibres have di- Example 2. For rank two Higgs bundles, we return to Ex- ample 1 in which the Hitchin fibration is over 퐻0(Σ, 퐾2) mension dim(ℳ퐺ℂ )/2. In what follows we shall restrict our attention to GL(푛, ℂ)-Higgs bundles, which are those and the Hitchin map is ℎ ∶ (퐸, Φ) ↦ −det(Φ). The family Higgs bundles introduced in the first paragraph of these (퐸, Φ푎) gives a section of the Hitchin fibration: a smooth notes and whose Hitchin fibration in low dimension is de- map from the Hitchin base to the fibres, known as the picted in Figure 1. Hitchin section. Moreover, this comprises a whole component of the moduli space of real SL(2, ℝ)-Higgs bundles in-

side ℳ퐺ℂ , which Hitchin identified with Teichmüller space and which is now referred to as a Hitchin component or Te- ichmüller component. Recall that the Teichmüller space 풯(푆) of the underlying surface 푆 of Σ is the space of marked conformal classes of Riemannian metrics on 푆. In the early 1990s Hitchin showed that for any split group 퐺, e.g., for the split form SL(푛, ℝ) of SL(푛, ℂ), the above components are homeomorphic to a vector space of dimension dim(퐺)(2푔 − 2) and conjectured that they should parametrize geometric structures. These spaces pre- Figure 1. An example of a Hitchin ﬁbration. sented the first family of higher Teichmüller spaces within the Betti moduli space of reductive surface group representa- ℳ (퐺) The generic or regular fibre of the Hitchin fibration— tions 퐵 , which leads us to applications of Higgs bun- 퐺 퐺 appearing in violet in Figure 1—is an abelian variety, dles within higher Teichmüller theory for real forms of ℂ. leading to what is referred to as the abelianization of the Higher Teichmüller Theory moduli space of Higgs bundles and which can be seen ge- The moduli spaces of 퐺-Higgs bundles have several con- ometrically by considering eigenvalues and eigenspaces of nected components. For a split real form 퐺 of 퐺 , the the Higgs field. Indeed, a Higgs bundle (퐸, Φ) defines a ℂ Hitchin component of 퐺-Higgs bundles, or equivalently ramified cover 휋 ∶ 푆 → Σ of the Riemann surface given of surface group representations, can be defined as the by its eigenvalues and obtained through its characteristic connected component of the Betti moduli space ℳ (퐺) equation, 퐵 containing Fuchsian representations in 퐺, which are rep- 푆 = {det(Φ − 휂) = 0} ⊂ Tot퐾, (5) resentations obtained by composing a discrete and faith- for 휂 the tautological section of 휋∗퐾. This cover allows one ful representation 휌 ∶ 휋1(Σ) → SL(2, ℝ) (classically called to construct the spectral data associated to generic (퐸, Φ) Fuchsian) with the unique (up to conjugation) irreducible given by: representation SL(2, ℝ) → 퐺. Moreover, as mentioned be- • the spectral curve 푆 from (5), generically smooth, fore, these representations, called Hitchin representations, defining a generic point in the Hitchin base, since are considered the first example of higher Teichmüller the coefficients of {det(Φ − 휂) = 0} give a basis of space for surfaces: a component of the set of representa- invariant polynomials, and tions of discrete groups into Lie groups of higher ranks • a line bundle on 푆, defining a point in the Hitchin consisting entirely of discrete and faithful elements. In fibre and obtained as the eigenspace of Φ. order to give a geometric description of Hitchin representations and motivated by dynamical properties, Labourie For classical Higgs bundles, the smooth fibres are Jacobian introduced the notion of Anosov representations, which can varieties Jac(푆), and one recovers (퐸, Φ) up to isomorphism be thought of as a generalization of convex-cocompact rep- from the data (푆, 퐿 ∈ Jac(푆)) by taking the direct images resentations to Lie groups 퐺 of higher real rank.2 퐸 = 휋∗퐿 and Φ = 휋∗휂. When considering 퐺ℂ-Higgs bundles, one has to re- 2For example, for representations in SL(2, ℂ), these are quasi-Fuchsian quire appropriate conditions on the spectral curve and representations.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 627 As beautifully described in Wienhard’s ICM Proceed- coordinate charts on 푀 with values in 푋 such that the tran- ings article [20], building on Labourie’s work, higher sition maps are given by elements of 퐺. Higgs bundles Teichmüller theory recently emerged as a new field in have played a key role in describing the closed manifold mathematics, closely related to Higgs bundles (see also on which (퐺, 푋)-structures live. For example, Higgs bun- [5, 13]). There are two known families of higher Teich- dles were used to show that maximal representations to müller spaces, giving the only known examples of compo- 푃푂(2, 3) give rise to (퐺, 푋)-manifolds when 푋 is the space nents that consist entirely of Anosov representations for of null geodesics (photons) in particular Einstein mani- surfaces: folds and when 푋 = ℙ(ℝ5) (e.g., see [5]). For an excellent (I) the space of Hitchin representations into a real review of geometric structures, see Kassel’s ICM Proceed- split simple Lie group 퐺 and ings [13]. (II) the space of maximal representations into a Her- Harmonic Metrics mitian Lie group 퐺. Equivariant harmonic maps play an important role in A representation 휌 ∶ 휋 (Σ) → 퐺 is maximal if it maximizes 1 the nonabelian Hodge correspondence mentioned before the Toledo invariant 푇(휌), a topological invariant defined (and beautifully reviewed in [3]), and thus we shall devote for any simple Lie group 퐺 of Hermitian type as this section to some of the advances made in this direc- 1 ∫ 푓∗휔 (6) tion. In our setting, from the work of Corlette and Don- 2휋 Σ aldson, any reductive representation 휌 ∶ 휋1(Σ) → 퐺ℂ has for 휔 the invariant Kähler form on the Riemannian sym- associated a 휌-equivariant harmonic map 푓 from the uni- metric space, and 푓 ∶ Σ̃ → 푋 any 휌-equivariant smooth versal cover Σ̃ of Σ to the corresponding symmetric space map. of 퐺ℂ, which in turn defines a Higgs bundle (퐸, Φ). Re- call that a map 푓 ∶ Σ̃ → 푀 is called 휌-equivariant if Example 3. The Toledo invariant can be expressed in 푓(훾 ⋅ 푥) = 휌(훾) ⋅ 푓(푥) for all 푥 ∈ Σ̃ and 훾 ∈ 휋1(Σ). Moreover, terms of Higgs bundles. For example, for SL(2, ℝ)-Higgs Σ ∗ through a choice of metric on , one may define the energy bundles (퐿 ⊕ 퐿 , Φ), the Toledo invariant is 2 deg(퐿) and density satisfies 0 ≤ |2 deg(퐿)| ≤ 2푔 − 2. Hence, the family (퐸, Φ푎) from Example 1 is maximal. 1 푒(푓) = ⟨푑푓, 푑푓⟩ ∶ Σ̃ → ℝ, (7) 2 The existence of spaces other than those in (I) and (II) with similar properties to Teichmüller space is a topic of which is 휌-invariant and descends to Σ. Then, the energy of significant investigation. Expected candidates are spaces 푓 is defined as of 휃-positive representations conjectured by Guichard– Wienhard, some of which were shown to exist via Higgs 퐸(푓) = ∫ 푒(푓)푑Vol. (8) bundles [1]. Σ Whilst Anosov representations give a clear link be- It depends only on the conformal class and is finite since tween discrete and faithful representations and geometric Σ is compact. The map 푓 is harmonic if it is a critical point structures, there is no known Higgs bundle characteriza- of the energy functional 퐸(푓) in (8). tion of Anosov representations, and very little is known Conversely, through the work of Hitchin and Simpson, about which explicit geometric structures correspond to a polystable Higgs bundle admits a Hermitian metric ℎ these spaces. For instance, work of Choi and Goldman on the bundle such that the associated Chern connection shows that the holonomy representations of convex pro- 퐴 solves the Hitchin equations (1), and such a metric is jective structures are the Hitchin representations when 퐺 = called harmonic. Moreover, the harmonic metric induces PSL(3, ℝ). a completely reducible representation 휌 ∶ 휋 (Σ) → 퐺 Whilst there is no Higgs bundle characterization of 1 ℂ and a 휌-equivariant harmonic map into the correspond- Anosov representations,3 Higgs bundles have been an ef- ing symmetric space. These two directions together give fective tool for describing these structures. This brings the celebrated nonabelian Hodge correspondence. us to one of the fundamental problems in modern ge- Understanding the geometric and analytic properties of ometry: the classification of geometric structures admit- the harmonic maps arising from Hitchin’s equations (1) ted by a manifold 푀. Recall that a model geometry is a is of significant importance. For instance, one may ask pair (퐺, 푋) where 푋 is a manifold (model space) and 퐺 is how these metrics behave at the boundaries of the mod- a Lie group acting transitively on 푋 (group of symmetries). uli space or how the energy densities of the corresponding A (퐺, 푋)-structure on a manifold 푀 is a maximal atlas of harmonic maps at different points of the Hitchin fibration 3Anosov representations are holonomy representations of geometric structures relate to each other (the reader may be interested in the on certain closed manifolds. reviews [15] and [7] and references therein).

628 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 From Hitchin’s work, the moduli space of Higgs bun- underlying surface 푆 in which the 휌-equivariant harmonic dles has a natural ℂ∗-action 휆 ⋅ (퐸, Φ) = (퐸, 휆Φ), whose metric is a minimal immersion. In particular, Labourie fixed point sets allow one to study different aspects ofthe showed that for all Anosov representations such a con- topology and the geometry of the space, as done in [11] formal structure exists, but the difficulty lies in proving (see also [5, 16]). Moreover, as shown by Simpson, the uniqueness; the conjecture has been established only for fixed points of this action are complex variations ofHodge Lie groups of rank two ([5, 14]). To understand this prob- structure (VHS). Recall that a VHS is a ℂ∞ vector bundle lem, one may consider the study of deformations of con- 푝,푞 푉 with a Hodge decomposition 푉 = ⨁푝+푞 푉 , a ratio- formal structures on surfaces and the corresponding har- nal structure, and a flat connection satisfying the axioms monic metric. of Griffiths transversality and existence of a polarization. Some of these deformations can be seen through the hy- From the above, one may ask how the energy density of perkähler structure of the moduli space, by virtue of which harmonic maps changes along the ℂ∗-flow on the moduli it has a ℂℙ1-worth of complex structures labelled by a pa- space of Higgs bundles. Whilst this remains a challenging rameter 휉. Indeed, we can think of a hyperkähler mani- open question in the area, a better understanding might fold as a manifold whose tangent space admits an action come from the following conjectural picture of Dai–Li de- of three complex structures 퐼, 퐽, and 퐾 satisfying the quater- scribed in Figure 2 and through which the harmonic map nionic equations and compatible with a single metric. In ∗ our case, 퐼 arises from the complex structure on the Rie- of a fixed point set of the ℂ action on ℳ퐺ℂ gives rise to two other related harmonic maps. mann surface Σ, while 퐽 is from the complex structure on the group 퐺ℂ. In this setting, one has the following moduli spaces: • for 휉 = 0 the space of Higgs bundles, • for 휉 ∈ ℂ× the space of flat connections4 −1 ̄ ∗ ∇휉 = 휉 Φ + 휕퐴 + 휕퐴 + 휉Φ , (10) • for 휉 = ∞ the space of “anti-Higgs bundles.” The hyperkähler metric on Hitchin moduli space is expected to be of type “quasi-ALG,” which is some expected generalization of ALG. A far-reaching open question is the understanding of the behavior of the metrics at the bound- Figure 2. The nilpotent cone in red over the 0 and the points 퐴, 퐵, and 퐶 lying over the ℂ∗-ﬂow and over the Hitchin section, aries of the space, for instance along a path in the Hitchin respectively. base via the limit ̄ lim(휕퐴, 푡Φ). 푡→∞ A point 퐴 within the Hitchin fibration naturally deter- Almost a decade ago Gaiotto–Moore–Neitzke gave a mines two other points: the point 퐵, which is the limit of conjectural description of the hyperkähler metric on ℳ퐺ℂ the ℂ∗-flow 휆 ⋅ 퐴 as 휆 → 0 in the nilpotent cone, and the near infinity, which surprisingly suggests that much of the point 퐶, which is the intersection point of the Hitchin fi- asymptotic geometry of the moduli space can be derived bre containing 퐴 and the Hitchin section. Then Dai–Li’s from the abelian spectral data. Recent progress has been conjecture states that the energy densities defined as in (7) made by Mazzeo–Swoboda–Weiss–Witt, Dumas–Neitzke, of the corresponding harmonic maps 푓퐴, 푓퐵, 푓퐶 satisfy and Fredrickson, but the global picture remains open (see [7]). 푒(푓 ) < 푒(푓 ) < 푒(푓 ). (9) 퐵 퐴 퐶 Finally, one further type of limiting structure we would As evidence for the above conjecture, one can consider the like to mention is that of opers, appearing as certain limits integral version (through (8)) for which Hitchin showed of Higgs bundles in the Hitchin components. To see this, that 퐸(푓퐵) < 퐸(푓퐴), but where the other corresponding note that for a solution of (1) in the SL(푛, ℂ)-Hitchin sec- inequality in (9) remains open. tion, one can add a real parameter 푅 > 0 to (10) to obtain a natural family of connections5 Limiting Structures ∗ ∇(휉, 푅) ≔ 휉−1푅Φ + 휕̄ + 휕 + 휉푅Φ . (11) The study of 휌-equivariant harmonic metrics and higher 퐴 퐴 Teichmüller theory through Higgs bundles has received Gaiotto conjectured that the space of opers (a gener- much attention in recent years and brings us to one of alization of projective structures that, like the Hitchin the most important conjectures in the area. This conjec- 4In particular, for 휉 = 1 we recover (2). ture, due to Labourie, states that for each Hitchin repre- 5 ∗ Note that 휕퐴 and Φ depend on the scaling parameter 푅. Indeed, one needs to sentation 휌 there is a unique conformal structure 푋휌 on the use the metric solving the Hitchin equation for 푅Φ to get a flat connection.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 629 section, is parametrized by the Hitchin base) should be obtained as the ℏ-conformal limit of the Hitchin section: taking 푅 → 0 and 휉 → 0 simultaneously while holding the ratio ℏ = 휉/푅 fixed. The conjecture was recently established for general simple Lie groups by Dumistrescu–Fredrickson–Kydonakis–Mazzeo– Mulase–Neitzke, who also conjectured that this oper is the quantum curve in the sense of Dumitrescu–Mulase, a quantization of the spectral curve 푆 of the corresponding Higgs bundle by topological recursion (see [6]). More re- Figure 3. A star-shaped quiver. cently, Collier–Wentworth showed that the above conformal limit exists in much more generality and gives a cor- hyperpolygons as in Figure 3 may be identified with a mod- respondence between (Lagrangian) strata for every stable uli space of certain rank 푟 parabolic Higgs bundles on ℙ1. VHS—and not only the Hitchin components. Specifically, In this setting, one has to puncture ℙ1 along a positive they constructed a generalization of the Hitchin section divisor 퐷 and then regard the Higgs field as being valued in 0 by considering stable manifolds 풲 (퐸0,Φ0) arising from 풪(푞) = 퐾⊗풪(퐷), with poles along 퐷 and satisfying certain each VHS (퐸0,Φ0) given by conditions on its residues at the poles. This takes us to a generalization of Higgs bundles on higher genus surfaces {(퐸, 휙) ∈ ℳ퐺ℂ | lim 푡 ⋅ (퐸, Φ) = (퐸0,Φ0)}. (12) 푡→0 obtained by allowing the Higgs field to have poles, leading The analog of the Hitchin section is then obtained by pa- to the moduli spaces of tame or parabolic Higgs bundles 0 rameterizing 풲 (퐸0,Φ0) with a slice in the space of Higgs (for logarithmic singularities) initiated by Simpson [19] or bundles through a global slice theorem, analogous to the of wild Higgs bundles (for higher order poles) initiated by definition of the Hitchin section. Boalch and Biquard; see references in [2] to learn more about these other settings. Understanding the more gen- Correspondences eral appearance of parabolic (and wild) Higgs bundles on The appearance of Higgs bundles as parameter spaces for higher genus Riemann surfaces in terms of hyperpolygons geometric structures is an example of the study of corre- remains an open question. spondences between solutions to Hitchin’s equations (1) In a different direction, given a fixed Riemann sur- and different mathematical objects. In what follows we face Σ and a homomorphism between two Lie groups ′ shall restrict our attention to a few correspondences be- Ψ ∶ 퐺ℂ → 퐺ℂ, there is a naturally induced map between tween Higgs bundles and two classes of mathematical ob- the Betti moduli spaces of representations jects: quiver varieties and hyperpolygons (e.g., see refer- Ψ ∶ ℳ (퐺 ) → ℳ (퐺′ ). ences in [12, 16]). 퐵 ℂ 퐵 ℂ Recall that a quiver 푄 = (푉, 퐴, ℎ, 푡) is an oriented graph It follows that there must be a corresponding induced map consisting of a finite vertex set 푉, a finite arrow set 퐴, and between the Higgs bundle moduli spaces, but this does head and tail maps ℎ, 푡 ∶ 퐴 → 푉. A Nakajima repre- not transfer readily to the induced map on the Hitchin fi- sentation of a quiver 푄 can be written as families 푊 ≔ brations, in particular since the image might be over the ((푊푣), 휙푎, 휓푎) for 푎 ∈ 퐴 and 푣 ∈ 푉, where 푊푣 is a finite- singular locus of the base. When the maps arise through dimensional vector space, the map 휙푎 ∶ 푊푡(푎) → 푊ℎ(푎) is a isogenies, together with Bradlow and Branco, the author linear map for all 푎 ∈ 퐴, and 휓푎 is in the cotangent space obtained a description of the map for spectral data in terms to Hom(푊푡(푎), 푊ℎ(푎)) at 휙푎. In particular, a hyperpolygon is of fibre products of spectral curves [18], but of much in- a representation of the star-shaped quiver, an example of terest is the understanding of other maps arising in this which appears in Figure 3. manner. For the star-shaped quiver in Figure 3, for which the di- mensions of 푊푣 are indicated in each vertex, the cotangent space 푇∗Rep(푄) of representations of 푄 is 푛 푇∗ ( Hom(ℂ, ℂ푟)) = 푇∗ (Hom(ℂ푛,ℂ푟)) . Figure 4. A degeneration of the Riemann surface. ⨁ 푖=1 Konno showed that hyperpolygon spaces are hyper- Finally, when considering compactifications of the kähler analogs of polygon spaces, which are representa- moduli space, one may ask how the moduli spaces trans- tion spaces of the star-shaped quivers with simple arrows. form when the base Riemann surface Σ changes (for in- Moreover, through the work of Fisher–Rayan, the space of stance, when degenerating the surface Σ as in Figure 4), a

630 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 question closely related to the relation between Higgs bun- • Abelianization—zero-dimensional intersection. dles and singular geometry, which we shan’t touch upon When 퐺 is a split real form, the author showed here; see [2] for a survey and open problems in this direc- that the (퐵, 퐴, 퐴)-brane intersects the Hitchin fibration. tion in torsion two points. • Abelianization—positive dimensional intersection. Mirror Symmetry and Branes Moreover, we can also show that for other real One of the most interesting correspondences of Hitchin groups such as SU(푛, 푛), the intersection has pos- 퐿 systems arises through mirror symmetry. For 퐺ℂ the Lang- itive dimension but may still be described solely lands dual group of 퐺ℂ, there is an identification of the in terms of abelian data. 퐿 Hitchin bases 풜퐺ℂ ≃ 풜 퐺ℂ . The two moduli spaces ℳ퐺ℂ • Cayley/Langlands type correspondences. 퐿 and ℳ 퐺ℂ are then torus fibrations over a common base, Surprisingly, many spaces of Higgs bundles cor- and through the famous SYZ conjecture, mirror symmetry responding to nonabelian real gauge theories should manifest as a duality between the spaces of Higgs do admit abelian parametrizations via auxiliary bundles for Langlands dual groups fibred over the same spectral curves, as shown with Baraglia through base via the Hitchin fibration. As first observed by Hausel– Cayley/Langlands type correspondences for the Thaddeus for SL(푛, ℂ) and PGL(푛, ℂ)-Higgs bundles and groups 퐺 = SO(푝 + 푞, 푝) and 퐺 = Sp(2푝 + 2푞, 2푝). shown by Donagi and Pantev for general pairs of Lang- • Nonabelianization. lands dual reductive groups, the nonsingular fibres are in- Together with Hitchin we initiated the study deed dual abelian varieties. Kapustin and Witten gave a of branes that don’t intersect the regular lo- physical interpretation of this in terms of S-duality, using it cus through the nonabelianization of Higgs bun- as the basis for their approach to the geometric Langlands dles, which characterized the branes for 퐺 = program. SL(푛, ℍ), SO(푛, ℍ), and Sp(푛, 푛) in terms of non- The appearance of Higgs bundles (and flat connections) abelian data given by spaces of rank two vector within string theory and the geometric Langlands program bundles on the spectral curves. has led researchers to study the derived category of coherent Moreover, it has been conjectured (Baraglia–Schapos- sheaves (퐵-model) and the Fukaya category (퐴-model) of nik) that the Langlands dual in (13) to the above (퐵, 퐴, 퐴)- these moduli spaces. It then becomes fundamental to un- branes should correspond to the (퐵, 퐵, 퐵)-branes of Higgs derstand Lagrangian submanifolds of ℳ (the support of 퐺ℂ bundles with structure group the Nadler group [17]. More 퐴-branes), and their dual objects (the support of 퐵-branes). generally, branes of Higgs bundles have proven notori- By considering the support of branes, we shall refer to a ously difficult to compute in practice, and very few broad submanifold of a hyperkähler manifold as being of type 퐴 classes of examples are known; e.g., see [18] for a partial or 퐵 with respect to each of the complex structures (퐼, 퐽, 퐾). list of examples. We shall next describe a family of branes Hence one may study branes of the four possible types: defined by the author and Baraglia, obtained by imposing (퐵, 퐵, 퐵), (퐵, 퐴, 퐴), (퐴, 퐵, 퐴), and (퐴, 퐴, 퐵), whose dual part- symmetries to the solutions of (1); see [17] and references ner is predicted by Kontsevich’s homological mirror sym- therein. metry to be Higgs Bundles and 3-Manifolds (퐵, 퐴, 퐴) ⟷ (퐵, 퐵, 퐵), (13) By considering actions on the Riemann surface Σ and on (퐴, 퐴, 퐵) ⟷ (퐴, 퐴, 퐵), (14) the Lie group 퐺ℂ, one can induce actions on the moduli (퐴, 퐵, 퐴) ⟷ (퐴, 퐵, 퐴). (15) space of Higgs bundles and on the Hitchin fibration and study their fixed point sets. Indeed, for 휌 the compact struc- In view of the SYZ conjecture, it is crucial to obtain ture of 퐺 and 휎 a real form of 퐺 , together with Baraglia the duality between branes within the Hitchin fibration. ℂ ℂ we defined the following: In particular, understanding the correspondence between branes contained completely within the irregular fibres has (i) Through the Cartan involution 휃 of a real form 퐺 ̄ ̄ remained a very fruitful direction of research for decades. of 퐺ℂ one obtains 푖1(휕퐴, Φ) = (휃(휕퐴), −휃(Φ)). In 2006 Gukov, Kapustin, and Witten introduced the first (ii) A real structure 푓 ∶ Σ → Σ on Σ induces ̄ ∗ ̄ ∗ studies of branes of Higgs bundles in relation to the geo- 푖2(휕퐴, Φ) = (푓 (휌(휕퐴)), −푓 (휌(Φ))). metric Langlands program and electromagnetic duality (iii) Lastly, by looking at 푖3 = 푖1 ∘ 푖2, one obtains ̄ ∗ ̄ ∗ where the (퐵, 퐴, 퐴)-branes of real 퐺-Higgs bundles were 푖3(휕퐴, Φ) = (푓 휎(휕퐴), 푓 휎(Φ)). considered. These branes, which correspond to surface The fixed point sets of 푖1, 푖2, 푖3 are branes of group representations into the real Lie group 퐺, may in- type (퐵, 퐴, 퐴), (퐴, 퐵, 퐴), and (퐴, 퐴, 퐵), respectively. The tersect the regular fibres of the Hitchin fibration invery topological invariants can be described using 퐾푂, 퐾푅, and different ways (see [17, 18] for references): equivariant 퐾-theory [17]. In particular, the fixed points of

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 631 푖1 give the (퐵, 퐴, 퐴)-brane of 퐺-Higgs bundles mentioned the 3-manifolds in the previous section, an example of which appears in Σ × [−1, 1] 푀 = , (16) Figure 5 and which one can study via the monodromy ac- (푥, 푡) ↦ (푓(푥), −푡) tion on the Hitchin fibration (e.g., see [18]). for which 휕푀 = Σ (e.g., a handle body). In this setting, together with Baraglia, we were able to show that a connection solving the Hitchin equations (1) on Σ extends over 푀 given in (16) as a flat connection if and only if the Higgs ̃0 bundle (퐸, Φ) is fixed by 푖2 and the class [퐸] ∈ 퐾ℤ2 (Σ) in reduced equivariant 퐾-theory is trivial. That is, the Higgs bundles that will extend are only those whose vector bundle is preserved by the lift of the involution 푖2 and for Figure 5. A real slice fixed by 푖1 of the moduli space of SL(2, ℂ)-Higgs bundles, from two different angles, obtained which the action of 푖2 over the fibres of 퐸 is trivial when through Hausel’s 3D prints of slices of ℳ퐺ℂ . restricted to each fixed circle. Global Topology The fixed points of 푖 are pseudo-real Higgs bundles. To 3 The computation of topological invariants of Higgs bundle describe the fixed points of the involution 푖 , note that a 2 moduli spaces has received vast attention from researchers real structure (or anticonformal map) on a compact con- who have tackled this problem with a diverse set of mathe- nected Riemann surface Σ is an antiholomorphic involu- matical tools (see references in [9,16]). One of the central tion 푓 ∶ Σ → Σ. The classification of real structures on questions considered for Higgs bundles and their general- compact Riemann surfaces is a classical result of Klein, izations is what the Poincarépolynomial of the space is. A who showed that all such involutions on Σ may be char- useful fact is that the total space of the Hitchin fibration de- acterized by two integer invariants (푛, 푎): the number 푛 of formation retracts onto the nilpotent cone ℎ−1(0) via the disjoint union of copies of the circle embedded in Σ fixed gradient flow of the moment map ofthe ℂ∗-action intro- by 푓 and 푎 ∈ ℤ determining whether the complement of 2 duced in the harmonic metrics section. The cohomology the fixed point set has one (푎 = 1) or two (푎 = 0) compo- ring localizes to the fixed-point locus inside ℎ−1(0): as first nents; e.g., see Figure 6. seen by Morse-theoretic methods in the work of Hitchin, the Poincaréseries that generates the Betti numbers of the • rational cohomology 퐻 (ℳ퐺ℂ , ℚ) is a weighted sum of the Poincaréseries of the connected components of the fixed- point locus.

Figure 6. A genus 2 Riemann surface and its fixed point sets Example 4. As shown by Hitchin, for the family of under an antiholomorphic involution with invariants SL(2, ℂ)-Higgs bundles in Example 1, when the genus of (푛, 푎) = (3, 0). Σ is 푔 = 2, the Poincaréseries is 1 + 푡2 + 4푡3 + 2푡4 + 34푡5 + 2푡6. (17) A real structure 푓 on the Riemann surface Σ induces involutions on the moduli space of representations 휋 (Σ) → Using Morse theory, it has been possible to compute 1 Poincarépolynomials only for low rank groups, and ex- 퐺ℂ of flat connections and of 퐺ℂ-Higgs bundles on Σ, and the fixed point sets define the (퐴, 퐵, 퐴)-branes of Higgs tending this to higher rank has been a challenging open bundles in (ii). These branes can be shown to be real in- problem for some time. More recently, interesting alterna- tegrable systems, given by (possibly singular) Lagrangian tive techniques have been used to access the higher-rank fibrations. Poincarépolynomials by Mozgovoy, Schiffmann, Mellit, From a representation theoretic point of view, one may and others. One may further ask about the structure of the ring 퐻•(ℳ , ℚ) itself: for instance, Heinloth recently ask which interesting representations these branes corre- 퐺ℂ spond to, a question closely related to the understanding proved that the intersection pairing in the middle dimen- of which representations of 휋 (Σ) extend to 휋 (푀) for 푀 a 3- sion for the smooth moduli space vanishes in all dimen- 1 1 퐺 = (푛, ℂ) manifold whose boundary is Σ. Whilst this question in its sions for ℂ PGL , and Cliff–Nevins–Shen proved full generality remains an important open problem, we that the Kirwan map from the cohomology of the mod- 퐺 can consider some particular cases in which the answer be- uli stack of -bundles to the moduli stack of semistable 퐺 comes clear from the perspective of Higgs bundles. For -Higgs bundles fails to be surjective. this, as seen in [17] and references therein, we considered One of the most important cohomological conjectures in the area is de Cataldo–Hausel–Migliorini’s P=W conjecture, which gives a correspondence between the

632 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 weight filtration and the perverse filtration on the coho- References mology of ℳ퐵 and ℳ퐷표푙, respectively, obtained via non- [1] Marta Aparicio-Arroyo, Steven Bradlow, Brian Col- abelian Hodge theory. Only certain special cases are lier, Oscar Garc´ıa-Prada, Peter B. Gothen, and Andr´e known, e.g., for rank two Higgs bundles, shown by de Oliveira, SO(푝, 푞)-Higgs bundles and higher Teichmüller com- Cataldo–Hausel–Migliorini (see [9]), and for certain mod- ponents, Invent. Math. 218 (2019), no. 1, 197–299, DOI uli spaces of wild Higgs bundles, proven recently by Shen– 10.1007/s00222-019-00885-2. MR3994589 [2] Lara B. Anderson, Mboyo Esole, Laura Fredrickson, and Zhang and Szabo. Laura P. Schaposnik, Singular geometry and Higgs bun- Inspired by the SYZ conjecture mentioned before, dles in string theory, SIGMA Symmetry Integrability Geom. Hausel–Thaddeus conjectured that mirror moduli spaces Methods Appl. 14 (2018), Paper No. 037, 27 pp., DOI of Higgs bundles present an agreement of appropriately 10.3842/SIGMA.2018.037. MR3787793 defined Hodge numbers: [3] Steven B. Bradlow, Oscar Garc´ıa-Prada, and Peter B. Gothen, What is… a Higgs bundle?, Notices Amer. Math. 푝,푞 푝,푞 퐿 ℎ (ℳ퐺ℂ ) = ℎ (ℳ 퐺ℂ ). (18) Soc. 54 (2007), no. 8, 980–981. MR2343296 [4] Ngô Bao Châu, Endoscopy theory of automorphic forms, Very recently, the first proof of this conjecture was estab- Proceedings of the International Congress of Mathemati- lished for the moduli spaces of SL(푛, ℂ) and PGL(푛, ℂ)- cians. Volume I, Hindustan Book Agency, New Delhi, 2010, Higgs bundles by Groechenig–Wyss–Ziegler in [8], where pp. 210–237. MR2827891 they established the equality of stringy Hodge numbers us- [5] Brian Collier, Studying deformations of Fuchsian representations with Higgs bundles, SIGMA Symmetry Integrability ing 푝-adic integration relative to the fibres of the Hitchin Geom. Methods Appl. 15 (2019), Paper No. 010, 32 pp., fibration and interpreted canonical gerbes present on these DOI 10.3842/SIGMA.2019.010. MR3910865 moduli spaces as characters on the Hitchin fibres. [6] Olivia Dumitrescu, A journey from the Hitchin section to the

Further combinatorial properties of ℳ퐺ℂ can be oper moduli, String-Math 2016, Proc. Sympos. Pure Math., glimpsed through their twisted version, consisting of vol. 98, Amer. Math. Soc., Providence, RI, 2018, pp. 107– Higgs bundles (퐸, Φ) on Σ with Φ ∶ 퐸 → 퐸 ⊗ ℒ, where 138. MR3821751 Σ now has any genus, 퐿 is a line bundle with deg(퐿) > [7] Laura Fredrickson, Perspectives on the asymptotic geometry deg(퐾), but without any punctures or residues being fixed. of the Hitchin moduli space, SIGMA Symmetry Integrability The corresponding moduli spaces carry a natural ℂ∗-action Geom. Methods Appl. 15 (2019), Paper No. 018, 20 pp., DOI 10.3842/SIGMA.2019.018. MR3921918 but are not hyperkähler, and there is no immediate rela- [8] Michael Groechenig, Dimitri Wyss, and Paul Ziegler, Mir- tionship to a character variety. Hence, there is no obvi- ror symmetry for moduli spaces of Higgs bundles via p-adic in- ous reason for the Betti numbers to be invariant with re- tegration, arXiv:1707.06417, 2017. gard to the choice of deg(퐸), which in the classical setting [9] Tam´asHausel, Global topology of the Hitchin system, Hand- would follow from nonabelian Hodge theory. However, book of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, the independence holds in direct calculations of the Betti Int. Press, Somerville, MA, 2013, pp. 29–69. MR3184173 numbers in low rank and was recently shown for GL(푛, ℂ) [10] N. J. Hitchin, The self-duality equations on a Riemann sur- and SL(푛, ℂ)-Higgs bundles by Groechenig–Wyss–Ziegler face, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59– in [8]. This suggests that some topological properties of 126, DOI 10.1112/plms/s3-55.1.59. MR887284 [11] Nigel Hitchin, Stable bundles and integrable systems, Duke Hitchin systems are independent of the hyperkähler geom- Math. J. 54 (1987), no. 1, 91–114, DOI 10.1215/S0012- etry (see references in [9, 16] for more details). 7094-87-05408-1. MR885778 Finally, it should be mentioned that an alternative de- [12] Victoria Hoskins, Parallels between moduli of quiver repre- scription of the Hitchin fibration can be given through sentations and vector bundles over curves, SIGMA Symmetry cameral data, as introduced by Donagi and Gaitsgory, and Integrability Geom. Methods Appl. 14 (2018), Paper No. this perspective presents many advantages, in particular 127, 46 pp., DOI 10.3842/SIGMA.2018.127. MR3882963 when considering correspondences arising from mirror [13] Fanny Kassel, Geometric structures and representations of symmetry and Langlands duality, as those mentioned in discrete groups, Proceedings of the International Congress previous sections studied by Donagi–Pantev. of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1115– 1151. MR3966802 ACKNOWLEDGMENTS. The author would like to [14] François Labourie, Cyclic surfaces and Hitchin components thank Steve Bradlow, Laura Fredrickson, Tamas Hausel, in rank 2, Ann. of Math. (2) 185 (2017), no. 1, 1–58, DOI Nigel Hitchin, Qiongling Li, Sara Maloni, Du Pei, James 10.4007/annals.2017.185.1.1. MR3583351 Unwin, Richard Wentworth, Anna Wienhard, the re- [15] Qiongling Li, An introduction to Higgs bundles via har- viewers of the paper, and very especially Steve Rayan monic maps, SIGMA Symmetry Integrability Geom. Meth- for useful comments on a draft. ods Appl. 15 (2019), Paper No. 035, 30 pp., DOI 10.3842/SIGMA.2019.035. MR3947036

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 633 FEATURED TITLES FROM THE [16] Steven Rayan, Aspects of the topology and combinatorics of EUROPEAN Higgs bundle moduli spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 129, 18 pp., MATHEMATICAL DOI 10.3842/SIGMA.2018.129. MR3884746 [17] Laura P. Schaposnik, Higgs bundles and applications, Ober- SOCIETY wolfach Reports 31 (2015), 31–33, available at arXiv: 1603.06691. [18] Laura P. Schaposnik, Advanced topics in gauge theory: Ge- Gösta Mittag-Leffler ometry and physics of Higgs bundles (2019), available at and Vito Volterra arXiv:1907.09800. 40 Years of [19] Carlos T. Simpson, Higgs bundles and local systems, Correspondence Inst. Hautes Etudes´ Sci. Publ. Math. 75 (1992), 5–95. Frédéric Jaëck, Ecole Normale MR1179076 Supérieure, Paris, France, Laurent [20] Anna Wienhard, An invitation to higher Teichmüller Mazliak, Université Pierre et theory, Proceedings of the International Congress of Marie Curie, Paris, France, Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lec- Emma Sallent Del Colombo, tures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1013– Universitat de Barcelona, Spain, 1039. MR3966798 and Rossana Tazzioli, Université Lille 1, Villeneuve-d’Ascq, France, Editors Volterra and Mittag-Leffler’s exchanges illustrate how general analysis, especially functional analysis, gained a dramatic momentum during those years, and how Volterra became one of the major leaders of the field, opening the path for several fundamental developments over the following decades. Heritage of European Mathematics, Volume 12; 2019; 438 pages; Hardcover; ISBN: 978-3-03719-199-6; List US$98; AMS members US$78.40; Order code EMSHEM/12 Laura P. Schaposnik Microlocal Analysis of Quantum Credits Fields on Curved All figures are by the author. Spacetimes Photo of the author is by James Unwin. Christian Gérard, Université de Paris 11, Orsay, France This monograph focuses on free fields and the corresponding quasi-free states, and, more precisely, on Klein Gordon fields and Dirac fields. It will be useful to both mathematicians and mathematical physicists. ESI Lectures in Mathematics and Physics, Volume 10; 2019; 228 pages; Softcover; ISBN: 978-3-03719-094-4; List US$58; Individual member US$46.40; Order code EMSESILEC/10

Explore more titles at bookstore.ams.org. Publications of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

634 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5