Gentle algebras arising from surfaces with orbifold points, Part II: Locally free Caldero-Chapoton functions

Abstract

We prove that in the skew-symmetrizable cluster algebras associated by Felikson-Shapiro-Tumarkin to unpunctured surfaces with orbifold points of order $2$ and a specific choice of weights, the Laurent expansion of any cluster variable with respect to any cluster coincides with the locally free Caldero-Chapoton function of a $τ$-rigid representation of a gentle algebra. These cluster algebras are typically non-acyclic and of infinite type, whereas for polygons with one orbifold point one recovers cluster algebras of finite type $C$; so, our result is an ample extension of a seminal result established by Geiss-Leclerc-Schröer for skew-symmetrizable cluster algebras of finite type and acyclic initial seeds. As the main means to achieve the result, we provide a generalization of Derksen-Weyman-Zelevinsky's mutation theory of loop-free quivers with potential to the quivers-with-loops with potential we associate to the triangulations of unpunctured surfaces with orbifold points, and study the relation with $τ$-tilting theory. As a result of independent interest, we compute the aforementioned $τ$-rigid representations explicitly. To this end, we show that the indecomposable $τ$-rigid string modules arising from arcs on the surface, and the quasi-simple band modules arising from simple closed curves, are well-behaved under the mutations of representations we define in the paper, thus extending results of the first author's Ph.D. thesis.