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Higgs Bundles – Recent Applications

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Higgs Bundles – Recent Applications Higgs bundles { Recent applications Laura P. Schaposnik ∗ September 25, 2019 Introduction Some prominent examples where these moduli spaces appear in mathematics and physics are: This note is dedicated to introducing Higgs bundles • Through the Hitchin fibration, MG give ex- and the Hitchin fibration, with a view towards their C appearance within different branches of mathematics amples of hyperk¨ahlermanifolds which are in- and physics, focusing in particular on the role played tegrable systems, leading to remarkable applica- by the integrable system structure carried by their tions in physics which we shall discuss later on. moduli spaces. On a compact Riemann surface Σ of • Building on the work of Hausel and Thaddeus re- genus g ≥ 2, Higgs bundles are pairs (E; Φ) where lating Higgs bundles to Langlands duality, Don- • E is a holomorphic vector bundle on Σ; agi and Pantev presented MG as a fundamental C example of mirror symmetry for CY manifolds. • the Higgs field Φ : E ! E ⊗ K is a holomorphic section, for K := T ∗Σ. • Within the work of Kapustin and Witten, Higgs bundles were used to obtain a physical deriva- Since their origin in the late 80's in work of Hitchin tion of the geometric Langlands correspondence and Simpson, Higgs bundles manifest as fundamental through mirror symmetry and soon after, Ng^o objects which are ubiquitous in contemporary math- found Higgs bundles to be key ingredients when ematics, and closely related to theoretical physics. proving the Fundamental Lemma, which led him For G a complex semisimple Lie group, the Dol- C to the Fields Medal a decade ago. beault moduli space of G -Higgs bundles MG has C C a hyperk¨ahlerstructure, and via different complex Higgs bundles and Hitchin systems have been an structures it can be seen as different moduli spaces: increasingly vibrant area, and thus there are several • Via the non-abelian Hodge correspondence de- expository articles some of which we shall refer to: veloped by Corlette, Donaldson, Simpson and from the Notices' article \What is a Higgs bundle?" Hitchin, and in the spirit of Uhlenbeck-Yau's [BGPG07], to several graduate notes on Higgs bun- work for compact groups, the moduli space dles (e.g., the author's recent [Sch19]), to more ad- is diffeomorphic as a real manifold to the De vance reviews such as Ng^o's2010 ICM Proceedings arXiv:1909.10543v1 [math.DG] 23 Sep 2019 article [Ch^a10]. Hoping to avoid repeating material Rahm moduli space MdR of flat connections on a smooth complex bundle. nicely covered in other reviews, whilst still attempt- ing to engage the reader into learning more about • Via the Riemann-Hilbert correspondence there the subject, we shall take this opportunity to focus is an analytic correspondence between the de on some of the recent work done by leading young 1 Rham space and the Betti moduli space MB of members of the community . surface group representations. 1As in other similar reviews, the number of references is ∗The author is a professor of mathematics at the University limited to twenty, and thus we shall refer the reader mostly to of Illinois at Chicago. Her email address is [email protected]. survey articles where precise references can be found. 1 Higgs bundles Example 2. SL(2; R)-Higgs bundles are pairs Higgs bundles arise as solutions to self-dual Yang- 0 β E = L ⊕ L∗; Φ = ; Mills equations, a non-abelian generalization of γ 0 Maxwell's equations which recurs through much of modern physics. Recall that instantons are solutions for L a line bundle on Σ. Hence, in Example 1. one to Yang-Mills self-duality equations in Euclidean 4d has a family (E; Φa) of SL(2; R)-Higgs bundles. space, and when these equations are reduced to Eu- In order to define the moduli space of Higgs bun- clidean 3d space by imposing translational invariance dles, one needs to incorporate the notion of stability. in one dimension, one obtains monopoles as solutions. For this, recall that holomorphic vector bundles E on Higgs bundles were introduced by Hitchin in [Hit87] Σ are topologically classified by their rank rk(E) and as solutions of the so-called Hitchin equations, the 2- their degree deg(E), though which one may define dimensional reduction of the Yang-Mills self-duality their slope as µ(E) := deg(E)=rk(E): Then, a vector equations, given by bundle E is stable (or semi-stable) if for any proper, non-zero sub-bundle F ⊂ E one has µ(F ) < µ(E) (or F + [Φ; Φ∗] = 0; (1) A µ(F ) ≤ µ(E)). It is polystable if it is a direct sum of @AΦ = 0; (2) stable bundles whose slope is µ(E). One can generalize the stability condition to Higgs where FA is the curvature of a unitary connection bundles (E; Φ) by considering Φ-invariant subbun- rA = @A + @A associated to the @-connection @A dle F of E, a vector subbundle F of E for which on a principal GC bundle P . Concretely, principal G -Higgs bundles are pairs (P; Φ), where Φ(F ) ⊂ F ⊗K. Then, a Higgs bundle (E; Φ) is said to C be stable (semi-stable) if for each proper Φ-invariant • P is a principal GC-bundle, and F ⊂ E one has µ(F ) < µ(E)(equiv: ≤). Then, the moduli space MG of stable G -Higgs bundles • Φ a holomorphic section of ad(P ) ⊗ K. C C up to holomorphic automorphisms of the pairs can Throughout these notes, we shall refer to classical be constructed (also denoted MDol). Going back Higgs bundles as the Higgs bundles described in the to Hitchin's equations, one of the most important Introduction, and consider GC-Higgs bundles in their characterisations of stable Higgs bundles is given in vector bundle representation, through which they can the work of Hitchin and Simpson, and which carries be seen as classical Higgs bundles satisfying some through to more general settings: If a Higgs bun- extra conditions reflecting the nature of the group dle (E; Φ) is stable and deg E = 0, then there is a GC. For instance when considering GC = SL(n; C), unique unitary connection A on E, compatible with a GC-Higgs bundle (E; Φ) is composed of a holomor- the holomorphic structure, satisfying (1)-(2). phic rank n vector bundle E with trivial determinant Finally, Hitchin showed that the moduli space of ΛnE =∼ O, and a Higgs field satisfies Tr(Φ) = 0, for Higgs bundles is a hyperk¨ahlermanifold with nat- 0 which we shall write Φ 2 H (Σ; End0(E) ⊗ K). ural symplectic form ! defined on the infinitesimal Example 1. Choosing a square root of K, consider deformations (A;_ Φ)_ of a Higgs bundle (E; Φ) by the vector bundle E = K1=2 ⊕K−1=2. Then, a family Z of SL(2; )-Higgs bundles (E; Φ ) parametrized by C a !((A_ 1; Φ_ 1); (A_ 2; Φ_ 2)) = tr(A_ 1Φ_ 2 − A_ 2Φ_ 1); (4) 0 2 quadratic differentials a 2 H (Σ;K ) is given by Σ 0 a _ 0;1 _ 1;0 E = K1=2 ⊕ K−1=2; Φ = : (3) where A 2 Ω (End0E) and Φ 2 Ω (End0E). a 1 0 Moreover, he presented a natural way of studying the moduli spaces MG of G -Higgs bundles through One may also consider G-Higgs bundles, for G a C C what is now called the Hitchin fibration, which we real form of G , which in turn correspond to the Betti C shall consider next. moduli space of representations into G. 2 Integrable systems • the spectral curve π : S ! Σ, defining a point in the Hitchin base, since the coefficients of Given fp1; : : : ; pkg a homogeneous basis for the alge- fdet(Φ − η) = 0g give a basis of invariant poly- bra of invariant polynomials on the Lie algebra gc of nomials for the Lie algebra, and GC, we denote by di the degree of pi. The Hitchin fibration, introduced in [Hit87b], is then given by • a line bundle on S, defining a point in the Hitchin fibre and obtained as the eigenspace of Φ. k M 0 di h : MG −! AG := H (Σ;K ); For classical Higgs bundles, the smooth fibres are C C ∗ i=1 Jac(S), and for η the tautological section of π K, (E; Φ) 7! (p1(Φ); : : : ; pk(Φ)): one recovers (E; Φ) up to isomorphism from the curve (S; L 2 Jac(S)) by taking E = π∗L and Φ = π∗η. where the map h is referred to as the Hitchin map: it When considering GC-Higgs bundles, one has to is a proper map for any choice of basis and makes the require appropriate conditions on the spectral curve moduli space into an integrable system. In what fol- and the line bundle reflecting the nature of GC. This lows we shall restrict our attention to GL(n; C)-Higgs approach originates in the work of Hitchin and of bundles, which are those Higgs bundles introduced in Beauville, Narasimhan and Ramanan (see [Sch19] for the first paragraph of these notes, and whose Hitchin references), and we shall describe here an example to fibration is depicted in Figure 1. illustrate the setting. Consider SL(n; C)-Higgs bun- dles, for which the coefficient in (5) corresponding to Tr(Φ) = 0, and the generic fibres of the Hitchin fibra- tion are isomorphic to Prym varieties Prym(S; Σ). Example 3. For rank two Higgs bundles, we return to the example in the previous page in which the Hitchin fibration is over H0(Σ;K2), and the Hitchin map is h :(E; Φ) 7! det(Φ): Then, the family (E; Φa) gives a section of the Hitchin fibration: a smooth map from the Hitchin base to the fibres, known as the Hitchin section.
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