Solutions of the Motivic ADHM Recursion Formula

Abstract

We give an explicit solution of the ADHM recursion formula conjectured by Chuang, Diaconescu, and Pan. This solution is closely related to the formula for the Hodge polynomials of Higgs moduli spaces conjectured by Hausel and Rodriguez-Villegas. We solve also the twisted motivic ADHM recursion formula. As a byproduct we obtain a conjectural formula for the motives of twisted Higgs moduli spaces, which generalizes the conjecture of Hausel and Rodriguez-Villegas.

Parabolic Refined Invariants and Macdonald Polynomials

“…Assuming that the invariants Ω(m, e; y) are independent of the degree e ∈ Z for any m, it will be shown below that Ω(m, e; y) = P m (1, y) (7.6) for all (m, e). The proof is entirely analogous to the proof of [49,Thm. 4.6], some details being presented below for completeness.…”

“…For simplicity, consider local curves on type (0, 2g − 2) in the following. Using the same notation as [49], the refined partition function (4.7) will be denoted by A ∞ (q, y, x). Hence As shown in Section 6, geometric engineering predicts that A ∞ (q, y, x) is determined by equation (6.1)…”

Sur une variante des troncatures d’Arthur

“…Cette dernière est elle-même un raffinement de conjectures de Hausel-Rodriguez-Villegas et Mozgovoy (cf. [14] et [19]). Elle a une certaine ressemblance formelle avec le théorème 10.1.1.…”

The quiver at the bottom of the twisted nilpotent cone on $$\mathbb P^1$$ P 1

“…For a large range of r and t values that have been inspected by computer, Theorem 6.1 verifies the conjectural lowest Betti numbers for M t (r, −d) coming from Mozgovoy's twisted version of the ADHM recursion formula [23]. These conjectures can presumably be checked using alternative recent results, namely by extracting the Betti numbers from the Donaldson-Thomas invariants for twisted Higgs bundle moduli spaces computed in [24] or by making appropriate modifications to the arguments for ordinary Higgs bundles over finite fields in [28], so that the closedform Poincaré series obtained in that paper for ordinary Higgs bundles on Riemann surfaces generalizes to g = 0 and twisted Higgs bundles.…”

“…Finally, we remark that our calculation of the lowest nonzero Betti number of M t (r, −d), which is always 1 whenever gcd(r, d) = 1, verifies the conjectural lowest Betti number coming from the twisted ADHM recursion formula for a large range of ranks and twists that can be checked by computer. The twisted ADHM recursion formula was posed and studied by Mozgovoy [23] as a generalization of the Chuang-Diaconescu-Pan ADHM recursion formula coming from physics [6] (see also [5]). Regarding its solutions, we should add that the ADHM Betti numbers depend on two parameters which can be identified with r and t. There is no dependence on d, which is consistent with the fact that the Betti numbers of ordinary Higgs bundle moduli spaces are independent of the degree, as proved in [16], at least when gcd(r, d) = 1.…”