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We show that the elliptic genus of the higher rank E-strings can be computedbased solely on the genus 0 Gromov-Witten invariants of the correspondingelliptic geometry. To set up our computation, we study the structure of thetopological string free energy on elliptically fibered Calabi-Yau manifoldsboth in the unrefined and the refined case, determining the maximal amount ofthe modular structure of the partition function that can be salvaged. In thecase of fibrations exhibiting only isolated fibral curves, we show that theprincipal parts of the topological string partition function at givenbase-wrapping can be computed from the knowledge of the genus 0 Gromov-Witteninvariants at this base-wrapping, and the partition function at lowerbase-wrappings. For the class of geometries leading to the higher rankE-strings, this leads to the result stated in the opening sentence.

Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants