Species with potential arising from surfaces with orbifold points of order 2, Part II: arbitrary weights

TL;DR

This paper constructs species and potentials from colored triangulations of surfaces with orbifold points of order 2, establishing mutation relations and analyzing the structure of the flip graph, with implications for cluster algebras.

Contribution

It introduces a new framework for associating species and potentials to colored triangulations with orbifold points, extending previous work to arbitrary weights and analyzing mutation and cohomology.

Findings

The flip graph is disconnected for non-contractible surfaces.

Isomorphism of Jacobian algebras corresponds to cohomology classes of cocycles.

Every SP-realization over certain fields is right-equivalent to a constructed SP.

Abstract

Let $\mathbf{\Sigma}=(\Sigma,M,O)$ be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and $\omega:O\rightarrow\{1,4\}$ a function. For each triangulation $\tau$ of $\mathbf{\Sigma}$ we construct a cochain complex $C^\bullet(\tau,\omega)$. A colored triangulation is defined to be a pair consisting of a triangulation $\tau$ and a 1-cocycle of $C^\bullet(\tau,\omega)$; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a colored flip have SPs related by the corresponding SP-mutation. We define the flip graph of $(\Sigma,M,O,\omega)$, whose vertices are the pairs $(\tau,x)$ with $\tau$ a triangulation and $x$ a cohomology class in $H^1(C^\bullet(\tau,\omega))$, with an edge between $(\tau,x)$ and $(\sigma,z)$ iff $(\tau,\xi)$ and $(\sigma,\zeta)$ are related by a colored flip for some cocycles $\xi$ and $\zeta$ respectively representing $x$ and $z$. We prove that this graph is disconnected if $\Sigma$ is not contractible. For unpunctured surfaces we show that $(\tau,\xi)$ and $(\tau,\xi')$ yield isomorphic Jacobian algebras if and only if $[\xi]=[\xi']$ in cohomology. We prove that every SP-realization of any $(\tau,\omega)$ via a non-degenerate SP over a cyclic Galois extension with certain roots of unity is right-equivalent to one of the SPs we construct here. The species constructed here are species realizations of the $2^{|O|}$ skew-symmetrizable matrices assigned by Felikson-Shapiro-Tumarkin to any given $\tau$. In the prequel to this paper we realized only one of these matrices via species, but therein we allowed the presence of arbitrarily many punctures.