I will describe an approach to studying meromorphic connections on vector bundles called abelianisation. This technique has its origins in the works of Fock-Goncharov (2006) and Gaiotto-Moore-Neitzke (2013)\, as well as th e WKB analysis. Its essence is to put rank-n connections on a complex curv e X in correspondence with much simpler objects: connections on line bundl es over an n-fold cover \;&Sigma\; ->\; X. The point of view is similar in spirit to abelianisation of Higgs bundles\, aka the spectral correspondence: Higgs bundles on X are pu t in correspondence with rank-one Higgs line bundles on a spectral cover < span style="font-size:12.0pt\;font-family:"\;Times New Roman"\;\,s erif\;mso-fareast-font-family:"\;Times New Roman"\;\;mso-ansi-lang uage:EN-US\;mso-fareast-language:EN-US\;mso-bidi-language:AR-SA"> \;&Sigma\; ->\; X. However\, u nlike Higgs bundles\, abelianisation of connections requires the introduct ion of a new object\, which we call the Voros cocycle. The Voros cocycle i s a cohomological way to encode objects such as ideal triangulations that appeared in Fock-Goncharov\, spectral networks that appeared in Gaiotto-Mo ore-Neitzke\, as well as the connection matrices appearing in the WKB anal ysis. By focusing our attention on the simplest case of logarithmic singul arities with generic residues\, I will describe an equivalence of categori es\, which I call the abelianisation functor\, between sl(2)-connections o n X satisfying a certain transversality condition and rank-one connections on an appropriate 2-fold spectral cover \; &nb sp\;&Sigma\; ->\; X. This presentation is based on the wo rk completed in my thesis (2018) and recent extensions thereof.

LOCATION:Big Seminar room Ground floor / Office Bldg West (I21.EG.101)\, IS T Austria ORGANIZER:jdeanton@ist.ac.at SUMMARY:Abelianisation of Logarithmic sl(2)-Connections URL:https://talks-calendar.app.ist.ac.at/events/1378 END:VEVENT END:VCALENDAR