The Katz–Klemm–Vafa product formula for BPS counts of K3 surfaces

Let $r_{g,h}$ denote the BPS count associated to a class $\beta$ on a K3 surface whose norm square satisfies

$$\langle\beta,\beta\rangle=2h-2.$$

Let $q$ and $y$ be formal variables. Katz–Klemm–Vafa product formula. The BPS counts $r_{g,h}$ are uniquely determined by

$$\sum_{g\geq 0} \sum_{h\geq 0} (-1)^g r_{g,h}(y^{\frac{1}{2}} - y^{-\frac{1}{2}})^{2g}q^h = \prod_{n\geq 1} \frac{1}{(1-q^n)^{20} (1-yq^n)^2 (1-y^{-1}q^n)^2}.$$

This formula is the generating-series form of the Katz–Klemm–Vafa prediction and packages the BPS counts for all genera and arithmetic genera. It follows from the preceding conjectural independence from divisibility and is the central product formula for K3 BPS invariants.