Stability conditions and quantum dilogarithm identities for Dynkin quivers

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.Comment: Journal (almost) equivalent versio

Crossed simplicial groups and structured surfaces

We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.Comment: 86 pages, v2: revised versio