Canonical Landau-Ginzburg models for cominuscule homogeneous spaces

Abstract

We present a type-independent Landau-Ginzburg (LG) model $(X_{\mathrm{can}},{\mathcal{W}}_{\mathrm{can}})$ for any cominuscule homogeneous space $X=G/P$. We give a fully combinatorial construction for our superpotential ${\mathcal{W}}_{\mathrm{can}}$ as a sum of $n+1$ rational functions in the (generalized) Plücker coordinates on the "Langlands dual" minuscule homogeneous space $\mathbb{X}=P^{\vee }\backslash G^{\vee }$. Explicitly, we define the denominators ${\mathcal{D}}_{i_{\ast }}$ of these rational functions using the combinatorics of order ideals of the corresponding minuscule poset, which can be interpreted as (generalized) Young diagrams, by a process that can be described by "moving boxes" and hence is easily implemented. To construct the corresponding numerators, we define derivations ${\delta }_{i_1}$ on $\mathbb{C}[\mathbb{X}]$ that act by "adding an appropriate box if possible" and then we apply each ${\delta }_{i_1}$ to the corresponding ${\mathcal{D}}_{i_{\ast }}$. By studying certain Weyl orbits in the fundamental representations of ${\tilde{G}}^{\vee }$ and exploiting the existence of a certain dense algebraic torus in $\mathbb{X}$, we show that the polynomials ${\mathcal{D}}_{i_{\ast }}$ coincide with the generalized minors ${\varphi }_{i_{\ast }}$ appearing in the cluster structures for homogeneous spaces studied by Geiß-Leclerc-Schröer in arXiv:math/0609138. We then define the mirror variety $X_{\mathrm{can}}=\mathbb{X}\setminus D_{\mathrm{ac}}$ to be the complement of the anticanonical divisor $D_{\mathrm{ac}}={\sum }_{i_{\ast }}\{{\mathcal{D}}_{i_{\ast }}=0\}$ formed by the ${\mathcal{D}}_{i_{\ast }}$. Moreover, we show that the LG models $(X_{\mathrm{can}},{\mathcal{W}}_{\mathrm{can}})$ are isomorphic to the Lie-theoretic LG-models $(X_{\mathrm{Lie}},{\mathcal{W}}_{\mathrm{Lie}})$ constructed by Rietsch in arXiv:math/0511124 and our models naturally generalize the type-dependent Plücker coordinate LG-models previously studied by various authors.

72 pages