An Expansion Formula for Decorated Super-Teichmüller Spaces
Motivated by the definition of super-Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichmüller spaces, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super λ-lengths associated to arcs in a borde...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211343 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | An Expansion Formula for Decorated Super-Teichmüller Spaces. Gregg Musiker, Nicholas Ovenhouse and Sylvester W. Zhang. SIGMA 17 (2021), 080, 34 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862545916469182464 |
|---|---|
| author | Musiker, Gregg Ovenhouse, Nicholas Zhang, Sylvester W. |
| author_facet | Musiker, Gregg Ovenhouse, Nicholas Zhang, Sylvester W. |
| citation_txt | An Expansion Formula for Decorated Super-Teichmüller Spaces. Gregg Musiker, Nicholas Ovenhouse and Sylvester W. Zhang. SIGMA 17 (2021), 080, 34 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Motivated by the definition of super-Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichmüller spaces, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super λ-lengths associated to arcs in a bordered surface. In the special case of a disk, we can give combinatorial expansion formulas for the super λ-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's -path formulas for type cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type ₙ. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the μ-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.
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| first_indexed | 2026-03-13T00:19:07Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211343 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T00:19:07Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Musiker, Gregg Ovenhouse, Nicholas Zhang, Sylvester W. 2025-12-30T15:51:03Z 2021 An Expansion Formula for Decorated Super-Teichmüller Spaces. Gregg Musiker, Nicholas Ovenhouse and Sylvester W. Zhang. SIGMA 17 (2021), 080, 34 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 17A70; 30F60 arXiv:2102.09143 https://nasplib.isofts.kiev.ua/handle/123456789/211343 https://doi.org/10.3842/SIGMA.2021.080 Motivated by the definition of super-Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichmüller spaces, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super λ-lengths associated to arcs in a bordered surface. In the special case of a disk, we can give combinatorial expansion formulas for the super λ-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's -path formulas for type cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type ₙ. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the μ-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra. The authors would like to thank the support of the NSF grant DMS-1745638 and the University of Minnesota UROP program. We would also like to thank Misha Shapiro and Leonid Chekhov for inspiring conversations, as well as the anonymous referees for their helpful feedback. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications An Expansion Formula for Decorated Super-Teichmüller Spaces Article published earlier |
| spellingShingle | An Expansion Formula for Decorated Super-Teichmüller Spaces Musiker, Gregg Ovenhouse, Nicholas Zhang, Sylvester W. |
| title | An Expansion Formula for Decorated Super-Teichmüller Spaces |
| title_full | An Expansion Formula for Decorated Super-Teichmüller Spaces |
| title_fullStr | An Expansion Formula for Decorated Super-Teichmüller Spaces |
| title_full_unstemmed | An Expansion Formula for Decorated Super-Teichmüller Spaces |
| title_short | An Expansion Formula for Decorated Super-Teichmüller Spaces |
| title_sort | expansion formula for decorated super-teichmüller spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211343 |
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