Some Landau--Ginzburg models viewed as rational maps

TL;DR

This paper explores the extension of superpotentials in Landau-Ginzburg models associated with adjoint orbits, connecting algebraic geometry and Lie theory to enhance understanding of their symplectic and geometric structures.

Contribution

It introduces methods to extend superpotentials to compactifications, bridging algebraic geometry and Lie theory in the study of Landau-Ginzburg models.

Findings

Extended superpotentials to compactified adjoint orbits.

Connected symplectic Lefschetz fibrations with Lie algebra structures.

Provided new geometric insights into Landau-Ginzburg models.

Abstract

[GGSM2] showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.