Gromov-Witten invariants of elliptic curves revisited

Elliptic curves play an important role in understanding the modern aspects of enumerative geometry. In this talk I will start by reviewing some results of Okounkov-Pandharipande and Bloch-Okounkov on the generating series of Gromov-Witten invariants of elliptic curves. I will then report my recent results on this topic from the perspective of the mirror elliptic curve. These include:
1. a formulation of the (all-genus) generating series as period integrals on the configuration space of the mirror curve,
2. new combinatorial formulas for the generating series,
3. a connection between a slightly modified but equivalent version of the generating series and mixed Hodge structures on the cohomology of configuration spaces of the mirror curve.
The talk is based on my recent works arXiv:2304.03912, arXiv:2310.08018, arXiv:2305.12362.