Stability conditions and quantum dilogarithm identities for Dynkin quivers

TL;DR

This paper investigates the topology of stability condition spaces for derived categories of Dynkin quivers and associated Ginzburg algebras, linking quantum dilogarithm identities to Donaldson-Thomas invariants.

Contribution

It proves the simply connectedness of stability condition spaces for these categories and relates Donaldson-Thomas invariants to quantum dilogarithm functions.

Findings

Space of stability conditions for D(Q) is simply connected

Donaldson-Thomas invariants computed via quantum dilogarithm functions

Faithfulness of braid group action implies stability space properties

Abstract

We study fundamental group of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(\Gamma_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected; as applications, we show that its Donanldson-Thomas invariant can be calculated via a quantum dilogarithm function on exchange graphs. In the case of D(\Gamma_N Q), we show that faithfulness of the Seidel-Thomas braid group action (which is known for Q of type A or N = 2) implies the simply connectedness of its space of stability conditions.