Discreteness of silting objects and t-structures in triangulated categories

TL;DR

This paper explores the relationship between silting objects and t-structures in triangulated categories, establishing conditions for bijections and contractibility of stability spaces, with applications to Calabi-Yau categories.

Contribution

It introduces ST-pairs and demonstrates a bijective correspondence between silting objects and t-structures, linking silting-discreteness to the contractibility of stability spaces.

Findings

Bijective map between silting objects and t-structures in ST-pairs.

Silting-discrete categories have contractible stability spaces.

Applied results to Calabi-Yau categories associated with Dynkin quivers.

Abstract

We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we construct an order-preserving map from silting objects in $\C$ to bounded $t$-structures on $\D$ and show that the map is bijective if and only if $\C$ is silting-discrete if and only if $\D$ is $t$-discrete. Based on a work of Qiu and Woolf, the above result is applied to show that if $\C$ is silting-discrete then the stability space of $\D$ is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi--Yau triangulated categories associated to Dynkin quivers.