Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family

Abstract

We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces OP2=E6sc/P6 and E7sc/P7, known respectively as the Cayley plane and Freudenthal variety. These models are defined on the complement Xcan∨ of an anti-canonical divisor of the “Langlands dual homogeneous spaces” X∨=P∨﹨G∨ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in [20,25,23]. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in [26] using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in [28]. We also give a cluster structure on C[X∨], prove that the Plücker coordinates form a Khovankii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families.