Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions

Abstract

We prove that the quivers with potentials associated to triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly 5 punctures. This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related by a flip, their associated QPs are related by the corresponding QP-mutation. As a byproduct, for (arbitrarily punctured) surfaces with non-empty boundary we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series. The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called \emph{Popping Theorem}, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence.

v5: Final version. Published by Selecta Mathematica (New Series). Published version contains a few minor redaction errors. E.g., "related by a flip" and "related by a QP-mutation" were incorrectly replaced by "related to a flip" and "related to a QP-mutation" therein. Changes following suggestions by the referee. 36 pages, 21 figures. Dedicated to the memory of Professor Andrei Zelevinsky

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Cited by in corpus (23)

• The representation type of Jacobian algebras
• Cluster algebras and continued fractions
• On Jacobian algebras from closed surfaces
• Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
• Braid groups and quiver mutation
• On cluster algebras from once punctured closed surfaces
• Density of $g$-vector cones from triangulated surfaces
• Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules
• Exact WKB analysis and cluster algebras
• Derived invariants for surface cut algebras II: the punctured case
• From groups to clusters
• Hochschild cohomology of Jacobian algebras from unpunctured surfaces: A geometric computation
• On finite dimensional Jacobian Algebras
• Tame algebras have dense $\mathbf{g}$-vector fans
• Acyclic cluster algebras with dense $g$-vector fans
• Strongly primitive species with potentials I: Mutations
• Stability conditions and Teichmüller space
• Laminations of punctured surfaces as $\tau$-regular irreducible components
• Quivers with potentials associated to triangulations of closed surfaces with at most two punctures
• Non-degenerate potentials on the quiver $X_7$
• Denominator vectors and dimension vectors from triangulated surfaces
• Compatibility degree of cluster complexes
• Stability scattering diagrams and quiver coverings