Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

by William Mark Goldman, Eugene Zhu Xia
     
 

ISBN-10: 082184136X

ISBN-13: 9780821841365

Pub. Date: 04/27/2008

Publisher: American Mathematical Society

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The

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Overview

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkahler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$. This is the simplest case of the theory developed by Hitchin, Simpson and others. The authors emphasize its formal aspects that generalize to higher rank Higgs bundles over higher dimensional Kahler manifolds.

Product Details

ISBN-13:
9780821841365
Publisher:
American Mathematical Society
Publication date:
04/27/2008
Series:
Memoirs of the American Mathematical Society Series, #193
Pages:
69
Product dimensions:
6.90(w) x 9.90(h) x 0.20(d)

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