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

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 aK3 surface and an elliptic curve. Oberdieck and Pandharipande conjectured [11] that the partition function of the Gromov-Witten/DonaldsonThomas 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 [4]) 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 [4]. 1. AN OVERVIEW OF DONALDSON-THOMAS THEORY FOR ELLIPTIC FIBRATIONS. Let X be a quasi-projective non-singular Calabi-Yau threefold over C. The DonaldsonThomas invariants are a virtual count of sheaves on X . They are invariant under deformations and they are the mathematical counterpart of counting BPS states in type IIB topological string theory compactified on X . Of particular importance is the case where the sheaves are ideal sheaves of curves. Then the Donaldson-Thomas invariants “count curves”, encoding subtle information about the enumerative geometry of X . We can identify the moduli space of ideal sheaves of curves with Hilb(X) = {Z ⊂ X : [Z] = β ∈ H2(X), n = χ(OZ)} the Hilbert scheme of proper subschemes ofX with fixed homology class and holomorphic Euler characteristic. To obtain a curve count, we would like to determine the “number of points” in Hilb(X). One way to assign an integer to a C-scheme, which generalizes the number of points in a zero dimensional scheme, is the topological Euler characteristic of the scheme with its complex analytic topology. This gives rise to the “naı̈ve DonaldsonThomas invariant” which we denote by D̂T β,n(X) = e(Hilb (X)). Date: January 30, 2018. 1