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We develop the quantum Kodaira-Spencer theory on the elliptic curve andestablish the corresponding higher genus B-model. We show that the partitionfunctions of the higher genus B-model on the elliptic curve are almostholomorphic modular forms, which can be identified with partition functions ofdescendant Gromov-Witten invariants on the mirror elliptic curve. This givesthe first compact Calabi-Yau example where mirror symmetry is established atall genera.

BCOV theory on the elliptic curve and higher genus mirror symmetry