Species with potential arising from surfaces with orbifold points of order 2, Part I: one choice of weights

TL;DR

This paper defines mutations for species with potential over cyclic Galois extensions, constructs species with potential from surface triangulations with orbifold points, and proves their invariance under flips, extending previous work.

Contribution

It introduces a mutation framework for species with potential applicable to skew-symmetrizable matrices with orbifold points, and establishes invariance under surface triangulation flips.

Findings

Species with potential are related by mutation under flips of triangulations.

Construction applies to surfaces with orbifold points of order 2.

Results generalize previous work without orbifold points.

Abstract

We present a definition of mutations of species with potential that can be applied to the species realizations of any skew-symmetrizable matrix B over cyclic Galois extensions E/F whose base field F has a primitive [E:F]-th root of unity. After providing an example of a globally unfoldable skew-symmetrizable matrix whose species realizations do not admit non-degenerate potentials, we present a construction that associates a species with potential to each tagged triangulation of a surface with marked points and orbifold points of order 2. Then we prove that for any two tagged triangulations related by a flip, the associated species with potential are related by the corresponding mutation (up to a possible change of sign at a cycle), thus showing that these species with potential are non-degenerate. In the absence of orbifold points, the constructions and results specialize to previous work by the second author. The species constructed here for each triangulation $\tau$ is a species realization of one of the several matrices that Felikson-Shapiro-Tumarkin have associated to $\tau$, namely, the one that in their setting arises from choosing the number 1/2 for every orbifold point.