Gromov-Witten generating series of elliptic curves as configuration space integrals

The generating series of Gromov-Witten invariants of elliptic curves can be expressed in terms of multi-variable elliptic functions by works of Bloch-Okounkov and Okounkov-Pandharipande. In this work we give new sum-over-partitions formulas for these generating series and show that they are configuration space integrals of cohomology classes constructed from sections of Poincare bundles. We also discuss their quasi-elliptic and quasi-modular properties.