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Torsion functions on moduli spaces in view of the cluster algebra

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Abstract

We introduce non-acyclic \(PGL_n({\mathbb {C}})\)-torsion of a \(3\)-manifold with toroidal boundary as an extension of J. Porti’s \(PGL_2({\mathbb {C}})\)-torsion, and present an explicit formula of the \(PGL_n({\mathbb {C}})\)-torsion of a mapping torus for a surface with punctures, by using the higher Teichmüler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.

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Acknowledgments

The authors would like to thank H. Fuji, K. Nagao, Y. Yamaguchi and M. Yamazaki for valuable conversations. The authors also wishes to express their thanks to the anonymous referee for several useful comments in revising the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

    Takahiro Kitayama

  2. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8550, Japan

    Yuji Terashima

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  1. Takahiro Kitayama
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  2. Yuji Terashima
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Correspondence to Yuji Terashima.

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Kitayama, T., Terashima, Y. Torsion functions on moduli spaces in view of the cluster algebra. Geom Dedicata 175, 125–143 (2015). https://doi.org/10.1007/s10711-014-0032-x

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  • DOI: https://doi.org/10.1007/s10711-014-0032-x

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