Wallcrossing and cohomology of the moduli space of Hitchin pairs

Abstract

A conjectural recursive relation for the Poincaré polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a a curve. A doubly refined generalization of this theory is also conjectured and shown to similarly determine the Hodge polynomial of the same moduli space.

On some computations of higher rank refined Donaldson-Thomas invariants

“…We also define the rational refined Donaldson-Thomas invariants by the refined multicover formula [7] DT(γ; y) =…”

“…Comparing the coefficients ofĝ α and using the rank one refined wall-crossing formula introduced in [7], we obtain the rank two refined wall-crossing formula:…”

Local curves, wild character varieties, and degenerations

“…At the same the topology of wild character varieties has been related to ploynomial invariants of legendrian knots in [57,56]. Finally, a string theoretic framework for this problem has been developed in [11,12,10] based on an identification of perverse Betti numbers of Higgs moduli spaces with degeneracies of spinning BPS states in M-theory. In particular the conjectural formulas derived by Hausel and Rodriquez-Villegas [26], Hausel, Letellier and Rogriguez-Villegas [24] are identified with Gopakumar-Vafa expansions in the refined stable pair theory of certain Calabi-Yau threefolds.…”

Quiver bundles and wall crossing for chains

“…Recent important examples of the study of wall crossing of chains and applications to moduli of Higgs bundles include the work of García-Prada-Heinloth-Schmitt [9], García-Prada-Heinloth [8], and Heinloth (see also Bradlow-García-Prada-Gothen-Heinloth [7] for an application to U(p, q)-Higgs bundles). We should mention here that recently alternative approaches to the study of the cohomology of Higgs bundle moduli have been highly succesful: see Schiffman [17], Mozgovoy-Schiffman [15] and Mellit [14]; also, Maulik-Pixton have announced a proof of a conjecture of Chuang-Diaconescu-Pan [4] which leads to a calculation of the motivic class of the moduli space of twisted Higgs bundles.…”