Moduli of Higgs bundles over the five punctured sphere

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, \phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $\phi$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, \phi)$ together with a section $\theta\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $\theta(x)$ is compatible with the framing $\phi$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}_{FH}(G)$ to the moduli space $\mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, \theta)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, \{\text{I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}_{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.

We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

We define a functional [Formula: see text] for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that [Formula: see text] is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite–Yang–Mills metrics. We derive a formula relating [Formula: see text] and another functional [Formula: see text], closely related to the Yang–Mills–Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of [Formula: see text], which is expressed as a certain [Formula: see text]-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of [Formula: see text] if and only if the corresponding Hitchin–Simpson mean curvature is parallel with respect to the Hitchin–Simpson connection.

Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard--Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov--Witten invariants

Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 . The geometry of the moduli space ℳ ( Spin ( 8 , C ) ) {\mathcal{ {\mathcal M} }}\left({\rm{Spin}}\left(8,{\mathbb{C}})) of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) -Higgs bundles over X X is of great interest both in algebraic geometry and mathematical physics. Consequently, several works have studied subvarieties of fixed points of this moduli space, especially those arising from the automorphism induced by the action of triality. In this work, fixed points of automorphisms of ℳ ( Spin ( 8 , C ) ) {\mathcal{ {\mathcal M} }}\left({\rm{Spin}}\left(8,{\mathbb{C}})) induced by outer automorphisms of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) are studied. This is done by giving explicit descriptions of the spectral data of these fixed points induced by the Hitchin fibration, under certain technical conditions that must be required. Specifically, it is proved that stable Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) -Higgs bundles that admit nontrivial automorphisms reduce their structure group to a subgroup isomorphic to SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) {\rm{SL}}{\left(2,{\mathbb{C}})}^{4}={\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}}) . Subsequently, the spectral data of these reductions are described and the manner in which the outer automorphisms of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) act on them is analyzed. The final application of these results allows a description of the mentioned fixed points through their spectral data.

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus$g$defined over a finite field, when the twisting line bundle degree is at least$2g-2$(this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$(finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable$U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.

On vanishing theorems for Higgs bundles

In this paper, we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study the generic and non-generic fibres.

Given a generic stable strongly parabolic SL ( 2 , C ) $\operatorname{SL}(2,\mathbb {C})$ -Higgs bundle ( E , φ ) $({\mathcal {E}}, \varphi )$ , we describe the family of harmonic metrics h t $h_t$ for the ray of Higgs bundles ( E , t φ ) $({\mathcal {E}}, t \varphi )$ for t ≫ 0 $t\gg 0$ by perturbing from an explicitly constructed family of approximate solutions h t app $h_t^{\mathrm{app}}$ . We then describe the natural hyperkähler metric on M $\mathcal {M}$ by comparing it to a simpler ‘semiflat’ hyperkähler metric. We prove that g L 2 − g sf = O ( e − γ t ) $g_{L^2} \,{-}\, g_{\mathrm{sf}} \,{=}\, O({\mathrm{e}}^{-\gamma t})$ along a generic ray, proving a version of Gaiotto–Moore–Neitzke's conjecture. Our results extend to weakly parabolic SL ( 2 , C ) $\operatorname{SL}(2,\mathbb {C})$ -Higgs bundles as well. A centerpiece of this paper is our explicit description of the moduli space and its L 2 $L^2$ metric for the case of the four-punctured sphere. We prove that the hyperkähler manifold in this case is a gravitational instanton of type ALG and that its rate of exponential decay to the semiflat metric is the conjectured optimal one, γ = 4 L $\gamma =4L$ , where L $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric | det φ | $|\det \varphi |$ .

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.Comment: 29 pages. Final version

In this article, we study the Hitchin morphism over a smooth projective variety. The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which in general not surjective when the dimension of the variety is greater than one. Chen–Ngô introduced the spectral base, which is a closed subvariety of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen–Macaulayfications of the spectral varieties. For rank two Higgs bundles over a projective manifold, we explicitly construct a finite Cohen–Macaulayfication of the spectral variety as a double branched covering, thereby confirming Chen–Ngô's conjecture. Moreover, using this Cohen–Macaulayfication, we can construct the Hitchin section for rank two Higgs bundles, which allows us to study the rigidity problem of the character variety and also to explore a generalization of the Milnor–Wood‐type inequality.

Higgs bundles and holomorphic forms

Geometric structures on manifolds became popular when Thurston used them in\nhis work on the geometrization conjecture. They were studied by many people and\nthey play an important role in higher Teichm\\"uller theory. Geometric\nstructures on a manifold are closely related with representations of the\nfundamental group and with flat bundles. Higgs bundles can be very useful in\ndescribing flat bundles explicitly, via solutions of Hitchin's equations.\nBaraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to\nconstruct geometric structures in some interesting cases. In this paper, we\nwill explain the main ideas behind this theory and we will survey some recent\nresults in this direction, which are joint work with Qiongling Li.\n

In this research, we study the geometry of the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface through the analysis of singular spectral curves and the triality automorphism of Spin(8,C). We establish a characterization of triality invariance, proving that a Spin(8,C)-Higgs bundle admits a reduction to the exceptional group G2 if and only if its spectral curve is invariant under the induced triality action. This transforms the problem of detecting G2-structures into a question about spectral data. We decompose the discriminant locus of the Hitchin fibration into two disjoint strata: a fixed stratum arising from G2-Higgs bundles with singular spectral curves and a free stratum consisting of orbits of size three under triality. We prove the existence of non-abelian spectral data compatible with triality symmetry, showing that non-abelian phenomena persist in free triality orbits. To quantify symmetry breaking, we introduce a triality defect invariant, which measures the dimension of the quotient of the Prym variety by its triality-invariant sublocus, and we prove that Higgs bundles with positive defects form a Zariski open dense subset.