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

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.