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Asymptotics of Higgs bundles in the Hitchin component

Collier, Brian and Li, Qiongling (2017) Asymptotics of Higgs bundles in the Hitchin component. Advances in Mathematics, 307 . pp. 488-558. ISSN 0001-8708. https://resolver.caltech.edu/CaltechAUTHORS:20170707-095300017

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Abstract

In this paper we pursue a more geometric approach to compactification of the Hitchin component. Our main motivation is Wolf's harmonic map interpretation of Thurston's compactification of Teichmüller space with measured foliations. Using Hitchin's parameterization of the Hitchin component by holomorphic differentials, we study asymptotics of certain rays of representations. More precisely, along these rays we solve the Hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection. The asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space SL(n,R)/SO(n,R) proves a conjecture of Katzarkov, Noll, Pandit and Simpson [17] on the Hitchin WKB problem in our setting.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.aim.2016.11.031DOIArticle
http://www.sciencedirect.com/science/article/pii/S0001870816315997PublisherArticle
https://arxiv.org/abs/1405.1106arXivDiscussion Paper
Additional Information:© 2016 Elsevier Inc. Received 13 July 2015, Revised 10 November 2016, Accepted 21 November 2016, Available online 28 November 2016. The authors would like to thank their advisors Steve Bradlow and Mike Wolf for encouraging this collaboration and for their instructive comments. We also thank John Loftin and Jakob Blaavand for their many helpful comments, and Andy Sanders and Daniele Alessandrini for numerous enlightening conversations. We also graciously thank the referee for many useful comments and corrections. Both authors acknowledge the support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). Our collaboration would not have been possible without this support. The second author is supported by the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95).
Funders:
Funding AgencyGrant Number
NSFDMS-1107452
NSFDMS-1107263
NSFDMS-1107367
Danish National Research FoundationDNRF95
Subject Keywords:Higgs bundles; Hitchin representations; Asymptotics; Harmonic maps; Toda lattice
Record Number:CaltechAUTHORS:20170707-095300017
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170707-095300017
Official Citation:Brian Collier, Qiongling Li, Asymptotics of Higgs bundles in the Hitchin component, Advances in Mathematics, Volume 307, 5 February 2017, Pages 488-558, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2016.11.031. (http://www.sciencedirect.com/science/article/pii/S0001870816315997)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78840
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:07 Jul 2017 17:06
Last Modified:03 Oct 2019 18:12

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