Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf E$-clusters

TL;DR

This paper extends geometric models and mutation concepts to continuous type A cluster structures, introduces isomorphisms of cluster theories, and explores classification and mutation spaces in a continuous setting.

Contribution

It generalizes geometric models and mutation to continuous type A cluster structures, introducing isomorphisms and the mutation space for the first time.

Findings

Established geometric models for continuous type A clusters.

Defined a continuous mutation process encompassing infinite sequences.

Introduced the space of mutations generalizing the exchange graph.

Abstract

This if the final paper in the seriesContinuous Quivers of Type $A$. In this part, we generalize existing geometric models of type $A$ cluster structures to the new $\mathbf E$-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use thes geometric models and isomorphisms of cluster theories to begin classifying continuous type $A$ cluster structures. We also introduce a continuous generalization of mutation. This encompasses mutation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure.