M. P. Appell's function and vector bundles of rank 2 on elliptic curve

Abstract

This paper is devoted to the function introduced by M. P. Appell in connection with decomposition of elliptic functions of the third kind into simple elements. We show that this function (which appeared as a Fukaya triple product in math.AG/9803017) is related to global sections of rank-2 vector bundles on elliptic curves. We derive analogues of theta-identities for this function and prove the divisibility property for the action of modular group which should be considered as a replacement for functional equation.