The Donaldson-Thomas theory of $K3\times E$ via the topological vertex

Abstract

Oberdieck and Pandharipande conjectured that the curve counting invariants of $S\times E$, the product of a $K3$ surface and an elliptic curve, is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive curve class in $S$ of square $2h-2$, their conjecture predicts that the corresponding partition functions are given by meromorphic Jacobi forms of weight $-10$ and index $h-1$. We calculate the partition functions for primitive classes of square -2 and of square 0. Our computation uses reduced Donaldson-Thomas invariants which are defined as the Behrend function weighted Euler characteristics of the quotient of the Hilbert scheme of curves in $S\times E$ by the action of $E$. Our technique is a mixture of motivic and toric methods (developed with Martijn Kool) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute the partition functions for both Behrend function weighted Euler characteristics and for unweighted Euler characteristics. The results for the Behrend function weighted case depends on Conjecture 18 from https://arxiv.org/abs/1608.07369