Gauss-Manin connection in disguise

In this talk I will first remind how to derive the Ramanujan relations between Eisenstein series and the
Darboux-Halphen differential equation from the Gauss-Manin connection of families of elliptic curves.
Then I will explain a generalization of this fact in  the case of Calabi-Yau threefolds. In this way one gets an algebra
which generalizes the algebra of quasi-modular forms. Genus g topological string partition functions turn
out to be elements of this new algebra and the corresponding Bershadsky-Cecotti-Ooguri-Vafa anomaly equation
can be written in terms of certain vector fields derived from the Gauss-Manin connection. The talk is based on
the papers  arXiv:1111.0357 [4] and  arXiv:1110.3664 [5] and a joint work under preparation with
M. Alim, E. Scheidegger, S.T. Yau.