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 $\mathrm{ Lie}(\mathsf{ G})$. We prove that the extended Lie algebra generated by $\mathrm{ Lie}(\mathsf{ G})$ together with modular vector fields on $\mathsf{ T}$ is isomorphic to $\mathrm{ sl}_2(\mathbb{ C})\oplus\mathrm{ sl}_2(\mathbb{ C})$.
Categories: math.AG hep-th
