Matrix formulae for decorated super Teichmüller spaces

Gregg Musiker, Nick Ovenhouse, Sylvester W Zhang

Research output: Contribution to journalArticlepeer-review


For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup OSp(1|2), which defines a flat OSp(1|2)-connection on the surface. We show that our matrix formula of an arc yields its super λ-length in Penner-Zeitlin's decorated super Teichmüller space. This generalizes the matrix formula of Fock-Goncharov and Musiker-Williams. We also prove that our matrix formulas agree with the combinatorial formulas given in the authors' previous works. As an application, we use our matrix formula in the case of an annulus to obtain new results on super Fibonacci numbers.

Original languageEnglish (US)
Article number104828
JournalJournal of Geometry and Physics
StatePublished - Jul 2023

Bibliographical note

Funding Information:
The authors would like to acknowledge the support of the NSF grants DMS-1745638 and DMS-1854162 .

Publisher Copyright:
© 2023 Elsevier B.V.


  • Cluster algebras
  • Superalgebras
  • Supergeometry
  • Teichmuller theory


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