Starts 22 Apr 2022 14:00

Ends 22 Apr 2022 15:30

Central European Time

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ICTP/IGAP Algebraic Geometry Seminar "Computation of vertical Vafa-Witten invariants"

Starts 22 Apr 2022 14:00

Ends 22 Apr 2022 15:30

Central European Time

Online

Leonardo Building - Euler Lecture Hall

Register in advance for this meeting:

https://zoom.us/meeting/register/tJEpfuCvpjguHdKtzcES0xzHegVySNQN0hC3** **

After registering, you will receive a confirmation email containing information about joining the meeting.

**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.

https://zoom.us/meeting/register/tJEpfuCvpjguHdKtzcES0xzHegVySNQN0hC3

After registering, you will receive a confirmation email containing information about joining the meeting.

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.

Title:

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.