The algebra of derivations of quasi-modular forms from mirror symmetry

We study moduli spaces of mirror non-compact Calabi–Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed Hodge structures which is equipped with a flat Gauss–Manin connection. We construct graded differential rings of special functions on these moduli spaces and show that they contain rings of quasi-modular forms. We show that the algebra of derivations of quasi-modular forms can be obtained from the Gauss–Manin connection contracted with vector fields on the enhanced moduli spaces. We provide examples for this construction given by the mirrors of the canonical bundles of $\mathbb{P}^2$ and $\mathbb{F}_2$.

Keywords

Calabi–Yau, modular forms, mixed Hodge structure, Gauss–Manin connection, mirror symmetry

2010 Mathematics Subject Classification

Primary 14D07, 14J15, 14J32. Secondary 14J33.

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Research supported by DFG Emmy-Noether grant on “Building blocks of physical theories from the geometry of quantization and BPS states”, number AL 1407/2-1.

Received 29 December 2021

Received revised 18 February 2022

Accepted 23 February 2022

Published 24 July 2022