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Stability conditions and quantum dilogarithm identities for Dynkin quivers

Yu Qiu Tsinghua University

Representation Theory mathscidoc:1906.01001

Advances in Mathematics, 269, 220–264, 2015
We study the fundamental groups 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 an application, we show that the Donaldson–Thomas type invariant associated to Q can be calculated as a quantum dilogarithm function on its exchange graph. In the case of D(Gamma_N Q), we show that the faithfulness of the Seidel–Thomas braid group action (which is known for Q of type A or N=2) implies the simply connectedness of the space of stability conditions.
Space of stability conditions Calabi–Yau categories Higher cluster categories Donaldson–Thomas invariants Quantum dilogarithm identities
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@inproceedings{yu2015stability,
  title={Stability conditions and quantum dilogarithm identities for Dynkin quivers},
  author={Yu Qiu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190625201717997928357},
  booktitle={Advances in Mathematics},
  volume={269},
  pages={220–264},
  year={2015},
}
Yu Qiu. Stability conditions and quantum dilogarithm identities for Dynkin quivers. 2015. Vol. 269. In Advances in Mathematics. pp.220–264. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190625201717997928357.
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