Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 39–53 | Cite as

Decorated marked surfaces (part B): topological realizations

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Abstract

We study categories associated to a decorated marked surface \({\mathbf {S}}_\bigtriangleup \), which is obtained from an unpunctured marked surface \(\mathbf {S}\) by adding a set of decorating points. For any triangulation \(\mathbf {T}\) of \({\mathbf {S}}_\bigtriangleup \), let \(\Gamma _\mathbf {T}\) be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in \({\mathbf {S}}_\bigtriangleup \) and the reachable rigid indecomposables in the perfect derived category \({\text {per}}\,\Gamma _\mathbf {T}\). This is the dual of the bijection, between simple closed arcs in \({\mathbf {S}}_\bigtriangleup \) and reachable spherical objects in the 3-Calabi-Yau category \({\mathcal {D}}_{fd}(\Gamma _\mathbf {T})\), constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient \({\text {per}}\,\Gamma _\mathbf {T}/{\mathcal {D}}_{fd}(\Gamma _\mathbf {T})\) that defines the generalized cluster categories corresponds to the forgetful map \({\mathbf {S}}_\bigtriangleup \rightarrow \mathbf {S}\) (forgetting the decorating points) in a suitable sense.

Keywords

Calabi-Yau categories Spherical twists Quivers with potential Silting objects Cluster categories 

Notes

Acknowledgements

I would like to thank Alastair King, Tom Bridgeland, Ivan Smith and Yu Zhou for the help on the Decorated Marked Surfaces project.

References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong

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