Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces

TL;DR

This paper constructs explicit Laurent polynomial Landau-Ginzburg models for various cominuscule homogeneous spaces, generalizing previous models and providing a systematic enumeration method based on quivers.

Contribution

It introduces a new, type-independent enumeration method for quantum summands in Laurent polynomial potentials for cominuscule spaces.

Findings

Constructed Laurent polynomial models for orthogonal Grassmannians, Cayley plane, and Freudenthal variety.

Generalized the enumeration of quantum terms using quivers, extending Young diagram techniques.

Models match known results for quadrics and Lagrangian Grassmannians.

Abstract

In this article we construct Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in arXiv:math/0511124. The Laurent polynomial takes a similar shape to the one given in arXiv:alg-geom/9603021 for projective complete intersections, i.e. it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in arXiv:math/0607492, associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in arXiv:1404.4844 and arXiv:1304.4958 for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau-Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.