Virtual refinements of the Vafa-Witten formula

開催日時
2017/10/20 金 10:30 - 12:00
場所
3号館152号室
講演者
Lothar Göttsche
講演者所属
International Centre for Theoretical Physics, Trieste
概要

(joint work with Martijn Kool) Vafa and Witten made predictions about the Euler numbers of moduli spaces of sheaves on surfaces. They give explicit generating functions in terms of modular forms. These moduli spaces are in general very singular, but they have a perfect obstruction theory, and thus a virtual fundamental class and a virtual Tangent bundle, and thus virtual Chern numbers and in particular a virtual Euler number. We interpret the prediction as being about the virtual Euler numbers. Then a formula of Mochizuki allows to compute the virtual Euler numbers in terms of integrals on Hilbert schemes of points, which we do via reduction to toric surfaces and virtual localization. This allows to check the conjecture in a wide variety of cases up to high expected dimensions of the moduli spaces. We then extend the conjecture first to the $\chi_y$-genus and then to the elliptic genus, where we obtain generating functions similar to that of Dijkgraaf-Moore-Verlinde-Verlinde for Hilbert schemes of points. Finally we extend the conjectures to the virtual cobordism class of the moduli spaces.