Abstract: Last time we introduced nested Hilbert schemes and how one can express vertical Vafa-Witten as intersection numbers on nested Hilbert schemes.
After reviewing this we will the sketch of proof of Laarakker's structure theorem for the vertical Vafa-Witten invariants, expressing them in terms of universal generating functions and Seiberg-Witten invariants. The proof uses the cobordism invariants of intersection numbers on Hilbert schemes of points.
Then we will show how one can use this to explicitely determine the generating function for the Vertical-Vafa-Witten invariants, by first reducing to the case of toric surfaces and then localizing on Hilbert schemes of points on toric surfaces.
In the remaining two (2) lectures we will
(1) finish explaining the ingrediends of this computation, and present the formulas for the vertical-Vafa-Witten invariants, in terms of modular forms
(2) state Mochizuki's formula for computing the horizontal Vafa-Witten invariants and use it to compute horizontal Vafa-Witten invariants.
Previous Lecture held on 6th of April:
Title: Vertical Vafa-Witten invariants and nested Hilbert schemes
Abstract: We state the structure theorem of Laarakker for the vertical Vafa-Witten invariants of a projective surface S. We introduce nested Hilbert schemes (an incidence variety in products of Hilbert schemes of points and curves on the surface S), and their relation to vertical components of Vafa-Witten moduli spaces.
We describe how the vertical Vafa-Witten invariants can be computed in terms of nested Hilbert schemes. We sketch the proof of Laarakker's structure theorem.
This will be a hybrid seminar. All are very welcome to join either online or in person (if provided with a green pass). Venue: Euler Lecture Room (ICTP Leonardo Da Vinci Building), for those wishing to attend in person.