Gauss-Manin Lie algebra of mirror elliptic K3 surfaces

We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type $E_6$, $E_7$ and $E_8$. We consider a moduli space $\mathsf{T}$ of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on $\mathsf{T}$ are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce an algebraic group $\mathsf{G}$ which acts on $\mathsf{T}$ from the right and construct its Lie algebra $\operatorname{Lie}(\mathsf{G})$.We prove that the extended Lie algebra generated by $\operatorname{Lie}(\mathsf{G})$ together with modular vector fields on $\mathsf{T}$ is isomorphic to $\operatorname{sl}_2( C) \oplus \operatorname{sl}_2 (C)$.

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This research is 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 6 April 2019

Accepted 12 July 2020

Published 2 June 2021