Mathematics – Algebraic Geometry
Scientific paper
2011-11-03
Mathematics
Algebraic Geometry
Adjust the statement of simply connectedness in Calabi-Yau case (Theorem 4.5 and Corollary 4.6)
Scientific paper
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 invariants 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; moreover we provide a topological realization of almost completed cluster tilting objects.
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