##### Department of Mathematics,

University of California San Diego

****************************

### Algebraic Geometry Seminar

## Martijn Kool

#### Utrecht University

## New directions in Vafa-Witten theory

##### Abstract:

In the 1990's, Vafa-Witten tested S-duality of N=4 SUSY Yang-Mills theory on a complex algebraic surface X by studying modularity of a certain partition function. In 2017, Tanaka-Thomas defined Vafa-Witten invariants by constructing a symmetric perfect obstruction theory on the moduli space of Higgs pairs (E,$\phi$) on X. The instanton contribution ($\phi=0$) to these invariants is the virtual Euler number of moduli space of sheaves. I outline a method to calculate this contribution, when X is of general type, by reducing to descendent Donaldson invariants. For rank 2, this leads to verifications of a formula from Vafa-Witten. The method can be ``refined'' to virtual $\chi_y$ genus, elliptic genus, and cobordism class, which involves weak Jacobi forms and Borcherds lifts thereof. I also give a new formula for rank 3 VW invariants on general type surfaces, correcting an error in the physics literature. Joint with G$\ddot{\text{o}}$ ttsche.

Host: Dragos Oprea

### May 18, 2018

### 2:30 PM

### AP&M 5829

****************************