Exact formulae and Tur\'{a}n inequalities for Vafa-Witten invariants of $K3$ surfaces

TL;DR

This paper derives exact formulas and proves Turán inequalities for Vafa-Witten invariants of K3 surfaces, revealing deep connections between these invariants and modular forms.

Contribution

It provides explicit formulas for Vafa-Witten invariants of K3 surfaces and establishes their asymptotic Turán inequalities using properties of Dedekind eta-functions.

Findings

Exact formulas for Vafa-Witten invariants

Asymptotic satisfaction of Turán inequalities

Connection to modular forms and eta-functions

Abstract

We consider three different families of Vafa-Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa-Witten invariants is given by combinations of twisted Dedekind $\eta$-functions. By utilising known properties of these $\eta$-functions, we obtain exact formulae for each of the invariants and prove that they asymptotically satisfy all higher-order Tur\'{a}n inequalities.