Constructs hyper-Kähler models using Riemann-Hilbert problems.
problem Creating smooth hyper-Kähler metrics near singularities.
method Using Gaitto, Moore, and Neitzke's formalism and assumptions on lattices.
result Defines hyper-Kähler model geometries including the Ooguri-Vafa model.
Certain six-dimensional (1,0) supersymmetric little string theories, when compactified on T3, have moduli spaces of vacua given by smooth K3 surfaces. Using ideas of Gaiotto-Moore-Neitzke, we show that this provides a systematic procedure for determining the Ricci-flat metric on a smooth K3 surface in terms of BPS d…
We consider Hitchin's hyperkähler metric gL2 on the SU(n)-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric gL2 and a simpler "semiflat" hyperkähler metric gsf is exponentially-decaying along generic rays in the Hitchin moduli s…
Study describes metrics on moduli space of Higgs bundles, proving exponential decay rate.
problem Analyzing the geometry of moduli space of Higgs bundles.
method Perturbing from approximate solutions to describe metrics, comparing to semi-flat metrics.
result Proves exponential decay rate of gL2−gsf=O(e−γt). Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an A-type X-cluster structure. We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topologic…
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
Study polynomial cubic differentials on Riemann surfaces using spectral networks.
problem Characterize polynomial cubic differentials with saddle connections or critical tripods.
method Introduced spectral core, refined classical core concept, and applied Gaiotto-Moore-Neitzke's algorithm.
result Completely characterized polynomial cubic differentials up to degree 3, including wall-and-chamber structure.