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

Abstract

We compute the A-infinity structure on the self-Ext algebra of the vector bundle $G$ over an elliptic curve of the form $G={⨁}_{i=1}^r{P}_i\oplus {⨁}_{j=1}^s{L}_j$, 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 $(e_{a,b}^{\ast }(z,w))$. 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 $e_{a,b}^{\ast }(z,w)$ by rapidly converging series.

v1:19 pages; v2: 20 pages, references added

References in corpus (2)

• Moduli of curves as moduli of A-infinity structures
• A modular compactification of ${\mathcal{M}}_{1,n}$ from $A_{\infty }$-structures

Cited by in corpus (1)

• Fukaya categories of two-tori revisited