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FIELD
Mathematics
DATE
Dec 06 (Tue), 2022
TIME
14:00 ~ 15:00
PLACE
1423
SPEAKER
Arbesfeld, Noah
HOST
Campo, Livia
INSTITUTE
Kavli IPMU
TITLE
Nested Hilbert schemes and instanton moduli spaces
ABSTRACT
Moduli of Higgs pairs on algebraic curves are of central importance in mirror symmetry and mathematical physics. Recently, moduli of Higgs pairs on algebraic surfaces have also emerged as a subject of active research. Tanaka-Thomas use moduli spaces of stable Higgs pairs on surfaces to formulate an algebro-geometric definition of Vafa-Witten invariants. The invariants are defined by integration over a virtual fundamental class and are formed from contributions of components; the physical notion of S-duality translates to conjectural relationships between these contributions. One component, the so-called ``vertical component,’’ can be realized as a nested Hilbert scheme of curves and points on a surface. I will explain work in progress with M. Kool and T. Laarakker, in which we express refined invariants of the vertical component in terms of invariants of a well-studied quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{CP}^2$. We then use a recent blow-up formula of Kuhn-Leigh-Tanaka to obtain formulas conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for vertical refined Vafa-Witten invariants in rank 2, verifying a conjecture of Göttsche-Kool.
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