Cluster deep loci and mirror symmetry

Abstract

Affine cluster varieties are covered up to codimension 2 by open algebraic tori. We put forth a general conjecture (based on earlier conversation between Vivek Shende and the last author) characterizing their deep locus, i.e. the complement of all cluster charts, as the locus of points with non-trivial stabilizer under the action of cluster automorphisms. We use the diagrammatics of Demazure weaves to verify the conjecture for skew-symmetric cluster varieties of finite cluster type with arbitrary choice of frozens and for the top open positroid strata of Grassmannians $\mathrm{Gr}(2,n)$ and $\mathrm{Gr}(3,n)$. We illustrate how this already has applications in symplectic topology and mirror symmetry, by proving that the Fukaya category of Grassmannians $\mathrm{Gr}(2,2n+1)$ is split-generated by finitely many Lagrangian tori, and homological mirror symmetry holds with a Landau--Ginzburg model proposed by Rietsch. Finally, we study the geometry of the deep locus, and find that it can be singular and have several irreducible components of different dimensions, but they all are again cluster varieties in our examples in really full rank cases.

Figures (11)

• Figure 1: A complete weave $\mathfrak{w}: \beta \to \delta(\beta)$. This weave depicts the path $\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1\sigma_2\sigma_1\sigma_2 \to \sigma_1\sigma_1\sigma_2\sigma_1\sigma_2\sigma_1\sigma_2 \to \sigma_1\sigma_1\sigma_2\sigma_2\sigma_1\sigma_2\sigma_2 \to \sigma_1\sigma_1\sigma_2\sigma_1\sigma_2\sigma_2 \to \sigma_1\sigma_1\sigma_2\sigma_1\sigma_2 \to \sigma_1\sigma_1\sigma_1\sigma_2\sigma_1 \to \sigma_1\sigma_1\sigma_2\sigma_1 \to \sigma_1\sigma_2\sigma_1$.
• Figure 2: Pictorial interpretation of Lemma \ref{['lem:easy3valent']}. Note that a variable to the right of $z_1$, if any, will simply be multiplied by a power of $w$.
• Figure 3: Pictorial interpretation of Lemma \ref{['lem:hard3valent']}. Note that a variable to the right of $z_3$, if any, will simply be multiplied by a power of $w$. Also note that we have a similar diagram with the colors blue and green interchanged.
• Figure 4: We can form this weave if $z_{a+3} \neq 0$. Note also that we can form a similar weave when $z_{a+3+3k} \neq 0$ for $k \geq 0$.
• Figure 5: We can form this weave if $z_{a+4} \neq 0$. We can form a similar weave if $z_{a+4+3k} \neq 0$ for $k \geq 0$.

Theorems & Definitions (189)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5