Searcharxiv⌕ Search

arXiv subjects

Laura Fredrickson

Publications and source records attributed to Laura Fredrickson.

14 recordsLinked to original sources
arXiv2026

ALG gravitational instantons and Hitchin moduli spaces, I: Torelli parameters

This is the first of two papers which together prove that the $12$-parameter family of parabolic $SU(2)$-Hitchin moduli spaces on the four-punctured sphere are all ALG gravitational instantons of type D4, and hence are asymptotic to $(\mathbb{C} \times T^2_\tau)/\mathbb{Z}_2$ at infinity. The elliptic modulus $\tau$ is determined by the cross-ratio of the four points. In this first paper, we consider each Hitchin moduli space corresponding to an allowable set of parabolic data and compute its Torelli parameters. There is a $12$-parameter family of Hitchin moduli spaces corresponding to different parabolic data, and we show that these realize all possible allowable Torelli parameters. In the companion paper, we we will show there that all of the Hitchin moduli spaces studied here are indeed ALG of type $D_4$, and consequently that every ALG-$D_4$ gravitational instanton can be realized as a Hitchin moduli space. Altogether, this will give the first verification of any case of the Modularity Conjecture: that all ALG gravitational instantons with tangent cone $\mathbb{C}/\mathbb{Z}_2$ can be realized as Hitchin moduli spaces with their natural associated $L^2$ metrics.

math.DG↗
arXiv2026

Hyperk\"ahler Degenerations from Parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs Bundles Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces

Complete hyperk\"ahler 4-manifolds of finite energy are grouped into ALE, ALF, ALG$^{(*)}$, ALH$^{(*)}$, each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG-$D_4$ spaces are modeled by certain moduli spaces of strongly parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles on the Riemann sphere with $n=4$ punctures. Meanwhile, a family of ALE-$D_4$ spaces are modeled by certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces. There is a map from hyperpolygon space to the moduli space of strong parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles that is a diffeomorphism onto its open and dense image. We show that under a fine-tuned degenerate limit, the pullback of a family of ALG-$D_4$ metrics parameterized by $R$ converges pointwise to the ALE-$D_4$ metric as $R \to 0$. While the connection to gravitational instantons occurs in the $n=4$ case, we prove our result for any finite $n$.

math.DG↗
arXiv2025

Hyper-K\"ahler manifolds from Riemann-Hilbert problems I: Ooguri-Vafa-like model geometries

We construct model hyper-K\"ahler geometries that include and generalize the multi-Ooguri-Vafa model using the formalism of Gaitto, Moore, and Neitzke. This is the first paper in a series of papers making rigorous Gaiotto--Moore--Neitzke's formalism for constructing hyper-K\"ahler metrics near semi-flat limits. In that context, this paper describes the assumptions we will make on a sequence of lattices $0 \to \Gamma_{f} \to \widehat{\Gamma} \to \Gamma \to 0$ over a complex manifold $\mathcal{B}'=\mathcal{B} - \mathcal{B}''$ near the singular locus, $\mathcal{B}''$, in order to define a smooth manifold $\mathcal{M} \to \mathcal{B}$ and hyper-K\"ahler model geometries on neighborhoods of points of the singular locus. In follow-up papers, we will use a modified version of Gaiotto-Moore-Neitzke's iteration scheme starting at these model geometries to produce true global hyper-K\"ahler metrics on $\mathcal{M}$.

math.DG↗
arXiv2024

A Concrete Variant of the Twistor Theorem

In this note, we prove a concrete variant of the twistor theorem of Hitchin--Karlhede--Lindstr\"om--Ro\v{c}ek which applies when one already has the real manifold on which one wishes to construct a hyper-K\"ahler structure, and so one does not need to construct it as a parameter space of twistor lines.

math.DG↗
arXiv2024

Conformal limits for parabolic SL(n,C)-Higgs bundles

In this paper we generalize the conformal limit correspondence between Higgs bundles and holomorphic connections to the parabolic setting. Under mild genericity assumptions on the parabolic weights, we prove that the conformal limit always exists and that it defines holomorphic sections of the space of parabolic lambda-connections which preserve a natural stratification and foliate the moduli space. Along the way, we give a careful gauge theoretic construction of the moduli space of parabolic Higgs bundles with full flags which allows the eigenvalues of the residues of the Higgs field to vary. A number of new phenomena arise in the parabolic setting. In particular, in the generality we consider, unlike the nonparabolic case, the nonabelian Hodge correspondence does not define sections of the space of logarithmic lambda-connections, and the conformal limit does not define a one-parameter family in any given moduli space.

math.DG↗
arXiv2020

Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle $(\mathcal{E}, \varphi)$, we describe the family of harmonic metrics $h_t$ for the ray of Higgs bundles $(\mathcal{E}, t \varphi)$ for $t\gg0$ by perturbing from an explicitly constructed family of approximate solutions $h_t^{\mathrm{app}}$. We then describe the natural hyperK\"ahler metric on $\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperK\"ahler metric. We prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-\gamma t})$ along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as well. In the case of the four-puncture sphere, we describe the moduli space and metric more explicitly. In this case, we prove that the hyperk\"ahler metric is ALG and show that the rate of exponential decay is the conjectured optimal one, $\gamma=4L$, where $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric $|\mathrm{det}\, \varphi|$.

math.DG↗
arXiv2019

Parabolic Higgs bundles, $tt^*$ connections and opers

The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This setup enables us to apply techniques from mirror symmetry. For example, the corresponding Higgs bundles extend to parabolic Higgs bundles to the compactification of the base of the families. We determine the parabolic degrees of the underlying parabolic bundles in terms of the exponents of the Picard-Fuchs equations obtained from the variations of Hodge structure. Moreover, we prove in this setup 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.

math.AG↗
arXiv2018

Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$-Higgs Bundles

Given a generic ray of Higgs bundles $(\overline{\partial}_E, t\varphi)$, 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 $h_t^{\mathrm{app}}$ which differ from the actual harmonic metrics $h_t$ by error terms of size $\mathrm{e}^{-\delta t}$. Such families of explicit approximate solutions have already proved useful for answering finer questions about the asymptotic geometry of the Hitchin moduli space.

math.DG↗
arXiv2018

Exponential Decay for the Asymptotic Geometry of the Hitchin Metric

We consider Hitchin's hyperk\"ahler metric $g_{L^2}$ on the $SU(n)$-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric $g_{L^2}$ and a simpler "semiflat" hyperk\"ahler metric $g_{\mathrm{sf}}$ is exponentially-decaying along generic rays in the Hitchin moduli space, as conjectured by Gaiotto-Moore-Neitzke.

math.DG↗
arXiv2017

Singular Geometry and Higgs Bundles in String Theory

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.

math.DG↗
arXiv2017

From $S^1$-fixed points to $\mathcal{W}$-algebra representations

We study a set $\mathcal{M}_{K,N}$ parameterizing filtered $SL(K)$-Higgs bundles over $\mathbb{CP}^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\lvert \lambda \rvert \sim \lvert z ^{N/K} \mathrm{d} z \rvert$, where $K$ and $N$ are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{CP}^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious $1$-$1$ correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$; in particular we have the relation $\mu = \frac{1}{12} \left(K - 1 - c_{\mathrm{eff}} \right)$, where $\mu$ is a regulated version of the $L^2$ norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding $W$-algebra representation. We also discuss a Bialynicki-Birula-type stratification of $\mathcal{M}_{K,N}$, where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

math.DG↗
arXiv2017

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters

We use Coulomb branch indices of Argyres-Douglas theories on $S^1 \times L(k,1)$ to quantize moduli spaces ${\cal M}_H$ 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 ${\cal M}_H$, 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 ${\cal M}_H$ 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.

hep-th↗
arXiv2016

Opers versus nonabelian Hodge

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 ${\mathbf u}$ of the base of Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,\zeta,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $\zeta \in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $\zeta = \hbar R$, we have $\nabla_{R,\zeta,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

math.DG↗