Spectral Networks and Fenchel–Nielsen Coordinates

Lotte Hollands, Andrew Neitzke

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)
65 Downloads (Pure)


It is known that spectral networks naturally induce certain coordinate systems on moduli spaces of flat SL(K)-connections on surfaces, previously studied by Fock and Goncharov. We give a self-contained account of this story in the case K = 2 and explain how it can be extended to incorporate the complexified Fenchel–Nielsen coordinates. As we review, the key ingredient in the story is a procedure for passing between moduli of flat SL(2)-connections on C (equipped with a little extra structure) and moduli of equivariant GL(1)-connections over a covering Σ→CΣ→C; taking holonomies of the equivariant GL(1)-connections then gives the desired coordinate systems on moduli of SL(2)-connections. There are two special types of spectral network, related to ideal triangulations and pants decompositions of C; these two types of network lead to Fock–Goncharov and complexified Fenchel–Nielsen coordinate systems, respectively.
Original languageEnglish
Pages (from-to)811-877
Number of pages67
JournalLetters in Mathematical Physics
Issue number6
Early online date2 May 2016
Publication statusPublished - Jun 2016


  • spectral networks
  • flat connections
  • Darboux coordinates

ASJC Scopus subject areas

  • Mathematics(all)


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