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We compute the Donaldson-Thomas invariants of a local elliptic surface withsection. We introduce a new computational technique which is a mixture ofmotivic and toric methods. This allows us to write the partition function forthe invariants in terms of the topological vertex. Utilizing identities for thetopological vertex proved in arXiv:1603.05271, we derive product formulas forthe partition functions. The connected version of the partition function iswritten in terms of Jacobi forms. In the special case where the ellipticsurface is a K3 surface, we get a derivation of the Katz-Klemm-Vafa formula forprimitive curve classes which is independent of the computation ofKawai-Yoshioka.

Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex