$\mathrm{X}$-evolution flows/foliations for cluster algebras: partial Bongartz (co-)completions and acyclicity

Abstract

We first generalize the notion of $\mathrm{X}$-foliation on cluster complexes to the context of cluster algebras. This is a continuous generalization of Keller's green mutation, which was previously introduced in the prequel within a categorical setting. We show that the wall-crossing structure (of possibly higher co-dimension) is encoded by $\mathrm{X}$-supports, which form a partial version of Cao-Gyoda-Yurikusa's Bongartz (co-)completions for cluster algebras. Moreover, we prove that for any cell transversed nontrivially by a fixed $\mathrm{X}$-leaf, its rank is an invariant bounded below by the rank of $\mathrm{X}$. Finally, we prove the acyclicity conjecture for $\mathrm{X}$-foliation, which also implies the compactness conjecture for any $\mathrm{X}$-foliation in finite type cases.