Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces

Authors:
Indranil Biswas, Pablo Arés-Gastesi and Suresh Govindarajan

Journal:
Trans. Amer. Math. Soc. **349** (1997), 1551-1560

MSC (1991):
Primary 58E15, 58E20, 32G15; Secondary 14E99

DOI:
https://doi.org/10.1090/S0002-9947-97-01870-9

MathSciNet review:
1407481

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Abstract: In this paper we study the relation between parabolic Higgs vector bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL(). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hitchin, and which in the case of is the usual Teichmüller space of the punctured surface.

**1.**I. Biswas,*Parabolic ample bundles*, Math. Annalen**307**(1997).**2.**K. Corlette,*Flat G-bundles with canonical metrics*, J. Differential Geom.**28**(1988), 361-382. MR**89k:58066****3.**N. Hitchin,*The self-duality equation on a Riemann surface*, Proc. London Math. Soc.**55**(1987), 59-126. MR**89a:32021****4.**-,*Lie groups and Teichmüller spaces*, Topology**31**(1992), 449-473. MR**93e:32023****5.**A. Jaffe and C. Taubes,*Vortices and Monopoles*, Progress in Physics, vol. 2, Birkhäuser, Boston, Basel and Stuttgart, 1980. MR**82m:81051****6.**V. Mehta and C. Seshadri,*Moduli of vector bundles on curves with parabolic structures*, Math. Ann.**248**(1980), 205-239. MR**81i:14010****7.**C. Simpson,*Constructing variations of Hodge structure using Yang-Mills theory and application to uniformization*, J. Amer. Math. Soc.**1**(1988), 867-918. MR**90e:58026****8.**-,*Harmonic bundles on noncompact curves*, J. Amer. Math. Soc.**3**(1990), 713-770. MR**91h:58029****9.**Balaji Srinivasan, Ph.D. thesis, University of Chicago, 1994.

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Additional Information

**Indranil Biswas**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Bombay, India

Email:
indranil@math.tifr.res.in

**Pablo Arés-Gastesi**

Email:
pablo@math.tifr.res.in

**Suresh Govindarajan**

Affiliation:
Department of Physics, Indian Institute of Technology, Madras, India

Email:
suresh@imsc.ernet.in

DOI:
https://doi.org/10.1090/S0002-9947-97-01870-9

Keywords:
Higgs bundles,
parabolic bundles,
hermitian metric

Received by editor(s):
October 24, 1995

Article copyright:
© Copyright 1997
American Mathematical Society