The Donaldson-Thomas Theory of K3 × E via the Topological Vertex

Abstract

We give a general overview of the Donaldson-Thomas invariants of elliptic fibrations and their relation to Jacobi forms. We then focus on the specific case of where the fibration is S × E, the product of a K3 surface and an elliptic curve. Oberdieck and Pandharipande conjectured (Oberdieck and Pandharipande, K3 Surfaces and Their Moduli, Progress in Mathematics, vol. 315 (Birkhauser/Springer, Cham, 2016), pp. 245–278, arXiv:math/1411.1514) that the partition function of the Gromov-Witten/Donaldson-Thomas invariants of S × E 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 Donaldson-Thomas partition function for primitive classes of square − 2 and of square 0, proving strong evidence for their conjecture. 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 × E by the action of E. Our technique is a mixture of motivic and toric methods (developed with Kool in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369)) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute both versions of the invariants: unweighted and Behrend function weighted Euler characteristics. Our Behrend function weighted computation requires us to assume Conjecture 18 in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369).