Stability conditions and quantum dilogarithm identities for Dynkin quivers

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(Γ_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(Γ_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.

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