# Gauss-Manin Lie algebra of mirror elliptic K3 surfaces

## TL;DR

This paper explores the mirror symmetry of elliptic K3 surfaces, revealing that the associated moduli space's structure is governed by a Lie algebra isomorphic to a direct sum of two sl_2(C) algebras, linked to quasi modular forms.

## Contribution

It introduces a new algebraic group action on the moduli space of mirror elliptic K3 surfaces and characterizes its Lie algebra in terms of modular vector fields and quasi modular forms.

## Key findings

- Coordinates on the moduli space are given by quasi modular forms in two variables.
- The algebraic group acts on the moduli space, with its Lie algebra explicitly constructed.
- The extended Lie algebra is isomorphic to sl_2(C) ⊕ sl_2(C).

## Abstract

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 $\mathrm{sl}_2(\mathbb{C})\oplus\mathrm{sl}_2(\mathbb{C})$.