Given a generic stable strongly parabolic SL(2, ℂ)-Higgs bundle (E,φ) we describe the family of harmonic metrics h_t for the ray of Higgs bundles (E,tφ) for t≫0 by perturbing from an explicitly constructed family of approximate solutions h_t^app. We then describe the natural hyperkähler metric on ℳ by comparing it to a simpler ``semiflat'' hyperkäahler metric. We prove that g_L²-g_sf=O(e^-γt) along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to the weakly parabolic SL(2, ℂ)-Higgs bundles as well. In the case of the four-punctured sphere, we describe the moduli space and metric more explicitly. In this case, we prove that the hyperkähler metric is ALG and show that the rate of of exponential decay is the conjectured optimal one, γ=4L, where L is the length of the shortest geodesic on the base curve measured in the singular flat metric |det φ|.

We establish a correspondence between the tt^∗ flat connections describing the variation of Hodge structure of the middle dimensional cohomology of families of Calabi-Yau manifolds with one-dimensional moduli spaces and Hitchin's equations associated to parabolic Higgs bundle associated to the compactified moduli space of the family. We determine the parabolic degrees of the underlying parabolic bundles in terms of the exponents of the Picard-Fuchs equations obtained from the Variation of Hodge structure. We prove in these setups that the flat non-abelian Hodge, or tt^∗ connection is gauge equivalent to an oper which is determined by the corresponding Picard-Fuchs equations. This gauge equivalence puts forward a new derivation of non-linear differential relations between special functions on the moduli space which generalize Ramanujan's relations for the differential ring of quasi-modular forms.

We consider Hitchin's hyperkähler metric g_L2 on the SU(2)-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric g_L2 and a simpler ``semiflat'' hyperkäahler metric g_sf is exponentially-decaying along generic rays in the Hitchin moduli space, as conjectured by Gaiotto-Moore-Neitzke.

Given a generic ray of Higgs bundles (∂_E, tφ), we describe the corresponding family of hermitian metrics h_t solving Hitchin's equations via gluing methods. In the process, we construct a family of approximate solutions h_t^app which differ from the actual harmonic metrics h_t by error terms of size e^-δt. Such families of explicit approximate solutions have already proved useful for answering finer questions about the asymptotic geometry of the Hitchin moduli space.

We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.

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This brief survey aims to set the stage and summarize some of the ideas under discussion at the Workshop on Singular Geometry and Higgs Bundles in String Theory, to be held at the American Institute of Mathematics from October 30th to November 3rd, 2017. One of the most interesting aspects of the duality revolution in string theory is the understanding that gauge fields and matter representations can be described by intersection of branes. Since gauge theory is at the heart of our description of physical interactions, it has opened the door to the geometric engineering of many physical systems, and in particular those involving Higgs bundles. This note presents a curated overview of some current advances and open problems in the area, with no intention of being a complete review of the whole subject.

We study a set M_K,N parameterizing filtered SL(K)-Higgs bundles over ℂℙ¹ with an irregular singularity at z = ∞, such that the eigenvalues of the Higgs field grow like |λ|~ |z|^N/Kdz, where K and N are coprime. M_K,N carries a ℂ^×-action analogous to the famous ℂ^×-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this ℂ^×-action on M_K,N involves the rotation automorphism of the base ℂℙ¹. We classify the fixed points of this ℂ^×-action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra W_K; in particular we have the relation μ = 1/12(K - 1 - c_eff), where μ is a regulated version of the L² norm of the Higgs field, and c_eff is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Bialynicki-Birula-type stratification of M_K,N, where the strata are labeled by isomorphism classes of the underlying filtered vector bundles..

Here is the version of the paper.

We use Coulomb branch indices of Argyres-Douglas theories on S¹ × L(k,1) to quantize moduli spaces M_H of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of M_H, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in M_H under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the modular transform ST^kS in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising. .

Here is the version of the paper.

For a complex simple simply-connected Lie group G, and a compact Riemann surface C, we consider two sorts of families of flat G-connections over C. Each family is determined by a point u of the base of Hitchin's integrable system for (G,C). One family ∇(ℏ) consists of G-opers, and depends on a non-zero complex parameter, ℏ. The other family ∇(R,ζ )is built from solutions of Hitchin's equations, and depends on non-zero complex parameter, ζ, and positive parameter, R. We show that in the scaling limit R→0, ζ=ℏ R, we have ∇(R,ζ ) → ∇(ℏ). This establishes and generalizes a conjecture formulated by Gaiotto in 1403.6137.

Here is the version of the paper.