# Polygonal Quivers

## TL;DR

This paper explores how Fano lattice polygons generate balanced quivers with special properties, linking their combinatorics to surface singularities and algebraic hypersurfaces, and introduces generalized mutations preserving these structures.

## Contribution

It introduces a new class of balanced quivers derived from Fano polygons, connecting combinatorics, algebraic geometry, and mutations, with extensions to higher dimensions.

## Key findings

- Fano polygons define balanced quivers with specific properties.
- These quivers relate to singularities of toric Fano surfaces.
- A family of algebraic hypersurfaces is associated with each Fano polygon.

## Abstract

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano polygon defines a point on a certain family of algebraic hypersurfaces. Our quivers admit a generalized mutation which preserves balancing and coincides with combinatorial mutation of Fano polygons whenever both operations are defined. We characterize balanced quivers arising from Fano polygons and discuss generalizations to higher dimensions.