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Especies con potencial y superficies con puntos orbifold

Ponente: Daniel Labardini
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo 13/03/2017
de 16:00 a 17:45
Dónde Salón 1 de seminarios
Agregar evento al calendario vCal

Let \( (\Sigma,M,O) \) be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and let \(\omega\) be a function from O to {1,4}. For each triangulation \(\tau\) of \((\Sigma,M,O)\) we construct a cochain complex \(C^\bullet(\tau,\omega)\) with coefficients in the field with two elements. A colored triangulation of \((\Sigma,M,O,\omega) \) is defined to be a pair \((\tau,\xi)\) consisting of a triangulation of \((\Sigma,M,O)\) 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 as defined in the prequel to this paper. We define the flip graph of colored triangulations of \((\Sigma,M,O,\omega)\) as the graph whose vertices are the pairs \((\tau[\xi])\) consisting of a triangulation \tau and a cohomology class in H^1(C^\bullet(\tau,\omega)), with an edge between \((\tau,[\xi])\) and \((\sigma,[\zeta])\) whenever \((\tau,\xi')\) and \((\sigma,\zeta')\) are related by a colored flip for some cocycles \(\xi'\) and \(\zeta'\) respectively homologous to \(\xi\) and \(\zeta\); we prove that this graph is disconnected if \(\Sigma\) is not contractible. Furthermore, we show that if \((\tau,\xi)\) and \((\tau,\xi')\) satisfy \([\xi]=[\xi']\), then the arising Jacobian algebras are isomorphic. We also classify the non-degenerate potentials on the species associated to a given \((\tau,\xi)\).
The species constructed here are species realizations of the \(2^{|O|}\) skew-symmetrizable matrices assigned by Felikson-Shapiro-Tumarkin to any given \(\tau\).