Twisted spectral correspondence and torus knots

TL;DR

This paper develops a spectral correspondence framework for twisted wild character varieties linked to torus knots, extending previous results and connecting cohomological invariants with enumerative geometry and refined Chern-Simons theory.

Contribution

It introduces a new spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes, generalizing untwisted cases to twisted wild character varieties associated with (l, kl-1) torus knots.

Findings

Derived cohomological invariants from enumerative Calabi-Yau geometry.

Established a spectral correspondence for twisted wild character varieties.

Provided a colored generalization of existing knot and character variety results.

Abstract

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi-Yau geometry and refined Chern-Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to (l, kl-1) torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.