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

TL;DR

This paper proves the non-degeneracy of quivers with potentials from triangulated surfaces with marked points, including those with boundary, by establishing compatibility with flips and introducing the Popping Theorem.

Contribution

It extends non-degeneracy results to surfaces with boundary and arbitrary punctures, and introduces the Popping Theorem to handle self-folded triangles.

Findings

Non-degeneracy of QPs for surfaces with boundary.

Compatibility of QP-mutations with flips in tagged triangulations.

Introduction of the Popping Theorem to fix potential asymmetries.

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.