Mirror Symmetry and Landau Ginzburg Calabi-Yau Superpotentials in F-theory Compactifications

TL;DR

This paper explores mirror symmetry in Landau Ginzburg models related to toric Calabi-Yau manifolds within F-theory, deriving new superpotential constraints and examining specific examples including elliptic K3 and ADE hypersurfaces.

Contribution

It introduces new constraint equations for LG superpotentials and provides explicit mirror symmetry examples for Calabi-Yau threefolds and fourfolds in F-theory.

Findings

Derived new superpotential constraint equations.

Established mirror symmetry for the canonical bundle over Hirzebruch surfaces.

Presented a geometric realization for ADE hypersurfaces.

Abstract

We study Landau Ginzburg (LG) theories mirror to 2D N=2 gauged linear sigma models on toric Calabi-Yau manifolds. We derive and solve new constraint equations for Landau Ginzburg elliptic Calabi-Yau superpotentials, depending on the physical data of dual linear sigma models. In Calabi-Yau threefolds case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces $\bf F_n$. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibered K3. This geometry leads to a pure N=1 six dimensional SO(8) gauge model from the F-theory compactification. For Calabi-Yau fourfolds, we give a new algebraic realization for ADE hypersurfaces.