Instanton geometry and quantum A∞ structure on the elliptic curve

We first determine and then study the complete set of non-vanishing A-model correlation functions associated with the 'long-diagonal branes' on the elliptic curve. We verify that they satisfy the relevant A∞ consistency relations at both classical and quantum levels. In particular we find that the A∞ relation for the annulus provides a reconstruction of annulus instantons out of disk instantons. We note in passing that the naive application of the Cardy-constraint does not hold for our correlators, confirming expectations. Moreover, we analyze various analytical properties of the correlators, including instanton flops and the mixing of correlators with different numbers of legs under monodromy. The classical and quantum A∞ relations turn out to be compatible with such homotopy transformations. They lead to a non-invariance of the effective action under modular transformations, unless compensated by suitable contact terms which amount to redefinitions of the tachyon fields. (orig.)