A-infinity algebras associated with elliptic curves and Eisenstein-Kronecker series

TL;DR

This paper computes the A-infinity algebra structure on certain vector bundles over elliptic curves, expressing the results via Eisenstein-Kronecker numbers and deriving new identities and series representations.

Contribution

It provides the first explicit A-infinity structure for these bundles and introduces new identities and series for Eisenstein-Kronecker numbers.

Findings

Derived quadratic identities between Eisenstein-Kronecker numbers.

Expressed Eisenstein-Kronecker numbers using rapidly converging series.

Computed the A-infinity structure explicitly for specific vector bundles.

Abstract

We compute the A-infinity structure on the self-Ext algebra of the vector bundle $G$ over an elliptic curve of the form $G=\bigoplus_{i=1}^r P_i\oplus \bigoplus_{j=1}^s L_j$, where $(P_i)$ and $(L_j)$ are line bundles of degrees 0 and 1, respectively. The answer is given in terms of Eisenstein-Kronecker numbers $(e^*_{a,b}(z,w))$. The A-infinity constraints lead to quadratic polynomial identities between these numbers, allowing to express them in terms of few ones. Another byproduct of the calculation is the new representation for $e^*_{a,b}(z,w)$ by rapidly converging series.