Local curves, wild character varieties, and degenerations

Abstract

Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate Calabi-Yau threefolds. A conjectural formula for E-polynomials is derived from the Gromov-Witten theory of local Calabi-Yau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincaré polynomials of wild character varieties.

Motivic classes of moduli of Higgs bundles and moduli of bundles with connections

“…Let Coh (0,d),(r,e) be the open substack of Coh (0,d),(r,e) classifying pairs of sheaves (F 1 ⊂ F ) such that F 1 is torsion free. Similarly, let Coh (r,e),(0,d) be the open substack of Coh (r,e),(0,d) classifying pairs of sheaves (F 1 ⊂ F ) such that F/F 1 is torsion free.…”

Intersection cohomology of character varieties for punctured Riemann surfaces

“…Chuang-Diaconescu-Pan [CDP] and Chuang-Diaconescu-Donagi-Pantev [CDDP] proposed a string theoretic interpretation of the conjecture. This string theoretic approach was also applied to wild character varieties by Diaconescu [Dia17] and Diaconescu-Donagi-Pantev [DDP18]. Another approach uses recursive relations for various genus.…”

Twisted spectral correspondence and torus knots

“…For unrefined invariants, a higher genus generalization of the formula (1.5) can be derived using degenerate Gromov-Witten theory by analogy with [Dia17]. In this approach the number m of marked points can be arbitrary although the triples (n, ℓ, r) are the same for all marked points.…”

“…This is the latest installment in a series of papers [CDP11, CDP14, CDDP15, DDP18,Dia17] developing a string theoretic approach to the cohomology of character varieties that is an important problem in geometry, topology and number theory, as shown by the fairly extensive mathematical literature. There have been several approaches from different viewpoints to this question, by using arithmetic methods for character varieties [HRV08, HLRV11, HLRV13, HMW16, Let15, Haua, Haub], and by using arithmetic and/or motivic methods for Higgs bundles [Sch16, MS14, DGT16, Let16, FSS17, GPHS14, GPH13].…”