Higgs Bundles— Recent Applications

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.

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