A geometric realization of silting theory for gentle algebras

TL;DR

This paper provides a geometric interpretation of silting theory for gentle algebras by relating silting mutations and reductions to surface dissections and flips, extending surface-based models of derived categories.

Contribution

It introduces a geometric framework for silting theory in gentle algebras, linking mutations and reductions to surface manipulations like arc flips and surface cutting.

Findings

Silting mutation corresponds to flipping graded arcs on the surface.

Silting reduction is modeled by cutting the underlying surface.

The geometric interpretation extends to octahedral axioms and surface dissections.

Abstract

A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyckerhoff-Kapranov in the context of triangulated surfaces. We also show that silting reduction corresponds to the cutting of the underlying surface as is the case for Calabi-Yau reduction of surface cluster categories as shown by Marsh-Palu and Qiu-Zhou.