Abstract: An Enriques surface is the quotient of a K3 surface by a fixed-point free involution. Klemm and Marino conjectured that the Gromov-Witten invariants (which are virtual counts of curves) of the local Enriques surface are certain orthogonal modular forms. In particular the genus 1 series recovers Borcherds famous automorphic form on the moduli space of Enriques surface. However, not much is known about the more general descendent Gromov-Witten theory of the Enriques surface. In this talk I will first discuss and sketch the proof of the Klemm-Marino formula and then explain a conjecture which links the descendent Gromov-Witten invariants of the Enriques to orthogonal modular forms in general. If time permits I will give some explicit formulas for point constraints and their descendents.

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