Abstract
Let ${\boldsymbol{\Sigma }}=(\Sigma ,\mathbb{M},\mathbb{O})$ be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair $(\tau ,\omega )$ consisting of a triangulation $\tau $ of ${\boldsymbol{\Sigma }}$ and a function $\omega :\mathbb{O}\rightarrow \{1,4\}$, we define a chain complex $C_\bullet (\tau , \omega )$ with coefficients in $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$. Given ${\boldsymbol{\Sigma }}$ and $\omega $, we define a colored triangulation of ${\boldsymbol{\Sigma }_\omega }=(\Sigma ,\mathbb{M},\mathbb{O},\omega )$ to be a pair $(\tau ,\xi )$ consisting of a triangulation of ${\boldsymbol{\Sigma }}$ and a 1-cocycle in the cochain complex that is dual to $C_\bullet (\tau , \omega )$; 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 flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [25]. We define the flip graph of ${\boldsymbol{\Sigma }_\omega }$ as the graph whose vertices are the pairs $(\tau ,x)$ consisting of a triangulation $\tau $ and a cohomology class $x\in H^1(C^\bullet (\tau , \omega ))$, with an edge connecting two such pairs, $(\tau ,x)$ and $(\sigma ,z),$ if and only if there exist 1-cocycles $\xi \in x$ and $\zeta \in z$ such that $(\tau ,\xi )$ and $(\sigma ,\zeta )$ are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface $\Sigma $ is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair $(\tau ,\omega )$ over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the $2^{|\mathbb{O}|}$ skew-symmetrizable matrices that Felikson–Shapiro–Tumarkin associated in [17] to any given triangulation of ${\boldsymbol{\Sigma }}$. In the prequel [25] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.
Cluster algebras and symmetrizable matrices
