Limiting configurations for solutions of Hitchin’s equation

Rafe Mazzeo; Jan Swoboda; Hartmut Weiß; Frederik Witt

doi:10.5802/tsg.296

Rafe Mazzeo ¹; Jan Swoboda ²; Hartmut Weiß ³; Frederik Witt ⁴

¹ Department of Mathematics Stanford University Stanford, CA 94305 (USA)
² Mathematisches Institut der LMU München Theresienstraße 39 D–80333 München (Germany)
³ Mathematisches Seminar der Universität Kiel Ludewig-Meyn Straße 4 D–24098 Kiel (Germany)
⁴ Mathematisches Institut der Universität Münster Einsteinstraße 62 D–48149 Münster (Germany)

Séminaire de théorie spectrale et géométrie, Volume 31 (2012-2014), pp. 91-116.

Abstract

We review recent work on the compactification of the moduli space of Hitchin’s self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top boundary stratum of this space of limiting configurations to a Prym variety. A key role is played by the family of rotationally symmetric solutions to the self-duality equation on $ℂ$ , which we discuss in detail here.

DOI: 10.5802/tsg.296

Author's affiliations:

Rafe Mazzeo ¹; Jan Swoboda ²; Hartmut Weiß ³; Frederik Witt ⁴

@article{TSG_2012-2014__31__91_0,
     author = {Rafe Mazzeo and Jan Swoboda and Hartmut Wei{\ss} and Frederik Witt},
     title = {Limiting configurations for solutions of {Hitchin{\textquoteright}s} equation},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {91--116},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     year = {2012-2014},
     doi = {10.5802/tsg.296},
     language = {en},
     url = {https://tsg.centre-mersenne.org/articles/10.5802/tsg.296/}
}

TY  - JOUR
AU  - Rafe Mazzeo
AU  - Jan Swoboda
AU  - Hartmut Weiß
AU  - Frederik Witt
TI  - Limiting configurations for solutions of Hitchin’s equation
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2012-2014
SP  - 91
EP  - 116
VL  - 31
PB  - Institut Fourier
PP  - Grenoble
UR  - https://tsg.centre-mersenne.org/articles/10.5802/tsg.296/
UR  - https://doi.org/10.5802/tsg.296
DO  - 10.5802/tsg.296
LA  - en
ID  - TSG_2012-2014__31__91_0
ER  -

%0 Journal Article
%A Rafe Mazzeo
%A Jan Swoboda
%A Hartmut Weiß
%A Frederik Witt
%T Limiting configurations for solutions of Hitchin’s equation
%J Séminaire de théorie spectrale et géométrie
%D 2012-2014
%P 91-116
%V 31
%I Institut Fourier
%C Grenoble
%U https://doi.org/10.5802/tsg.296
%R 10.5802/tsg.296
%G en
%F TSG_2012-2014__31__91_0

Rafe Mazzeo; Jan Swoboda; Hartmut Weiß; Frederik Witt. Limiting configurations for solutions of Hitchin’s equation. Séminaire de théorie spectrale et géométrie, Volume 31 (2012-2014), pp. 91-116. doi : 10.5802/tsg.296. https://tsg.centre-mersenne.org/articles/10.5802/tsg.296/

References
Cited by

[1] M. F. Atiyah; R. Bott The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A, Volume 308 (1983) no. 1505, pp. 523-615 | DOI | MR | Zbl

[2] Arnaud Beauville; M. S. Narasimhan; S. Ramanan Spectral curves and the generalised theta divisor, J. Reine Angew. Math., Volume 398 (1989), pp. 169-179 | DOI | MR | Zbl

[3] Riccardo Benedetti; Carlo Petronio Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992, pp. xiv+330 | DOI | MR | Zbl

[4] Roger Bielawski Asymptotic behaviour of $S U (2)$ monopole metrics, J. Reine Angew. Math., Volume 468 (1995), pp. 139-165 | DOI | MR | Zbl

[5] Roger Bielawski Monopoles and the Gibbons-Manton metric, Comm. Math. Phys., Volume 194 (1998) no. 2, pp. 297-321 | DOI | MR | Zbl

[6] Roger Bielawski Monopoles and clusters, Comm. Math. Phys., Volume 284 (2008) no. 3, pp. 675-712 | DOI | MR | Zbl

[7] Hans U. Boden Representations of orbifold groups and parabolic bundles, Comment. Math. Helv., Volume 66 (1991) no. 3, pp. 389-447 | DOI | MR | Zbl

[8] Hans U. Boden; Kôji Yokogawa Moduli spaces of parabolic Higgs bundles and parabolic $K (D)$ pairs over smooth curves. I, Internat. J. Math., Volume 7 (1996) no. 5, pp. 573-598 | DOI | MR | Zbl

[9] Kevin Corlette Flat $G$ -bundles with canonical metrics, J. Differential Geom., Volume 28 (1988) no. 3, pp. 361-382 http://projecteuclid.org/euclid.jdg/1214442469 | MR | Zbl

[10] S. K. Donaldson A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom., Volume 18 (1983) no. 2, pp. 269-277 http://projecteuclid.org/euclid.jdg/1214437664 | MR | Zbl

[11] S. K. Donaldson Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. (3), Volume 55 (1987) no. 1, pp. 127-131 | DOI | MR | Zbl

[12] Birte Feix Hyperkähler metrics on cotangent bundles, J. Reine Angew. Math., Volume 532 (2001), pp. 33-46 | DOI | MR | Zbl

[13] Laura Fredrickson, University of Texas at Austin (in preparation) (Ph. D. Thesis)

[14] Daniel S. Freed Special Kähler manifolds, Comm. Math. Phys., Volume 203 (1999) no. 1, pp. 31-52 | DOI | MR | Zbl

[15] Mikio Furuta; Brian Steer Seifert fibred homology $3$ -spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math., Volume 96 (1992) no. 1, pp. 38-102 | DOI | MR | Zbl

[16] Davide Gaiotto; Gregory W. Moore; Andrew Neitzke Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys., Volume 299 (2010) no. 1, pp. 163-224 | DOI | MR | Zbl

[17] Davide Gaiotto; Gregory W. Moore; Andrew Neitzke Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., Volume 234 (2013), pp. 239-403 | DOI | MR

[18] R. C. Gunning Lectures on vector bundles over Riemann surfaces, University of Tokyo Press, Tokyo; Princeton University Press, Princeton, N.J., 1967, pp. v+243 | MR | Zbl

[19] Robin Hartshorne Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[20] Tamás Hausel Vanishing of intersection numbers on the moduli space of Higgs bundles, Adv. Theor. Math. Phys., Volume 2 (1998) no. 5, pp. 1011-1040 | MR | Zbl

[21] Tamás Hausel; Eugenie Hunsicker; Rafe Mazzeo Hodge cohomology of gravitational instantons, Duke Math. J., Volume 122 (2004) no. 3, pp. 485-548 | DOI | MR | Zbl

[22] N. J. Hitchin The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Volume 55 (1987) no. 1, pp. 59-126 | DOI | MR | Zbl

[23] N. J. Hitchin; A. Karlhede; U. Lindström; M. Roček Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys., Volume 108 (1987) no. 4, pp. 535-589 http://projecteuclid.org/euclid.cmp/1104116624 | MR | Zbl

[24] N. J. Hitchin; G. B. Segal; R. S. Ward Integrable systems, Oxford Graduate Texts in Mathematics, 4, The Clarendon Press, Oxford University Press, New York, 1999, pp. x+136 (Twistors, loop groups, and Riemann surfaces, Lectures from the Instructional Conference held at the University of Oxford, Oxford, September 1997) | MR | Zbl

[25] Nigel Hitchin Stable bundles and integrable systems, Duke Math. J., Volume 54 (1987) no. 1, pp. 91-114 | DOI | MR | Zbl

[26] Nigel Hitchin $L^{2}$ -cohomology of hyperkähler quotients, Comm. Math. Phys., Volume 211 (2000) no. 1, pp. 153-165 | DOI | MR | Zbl

[27] Nigel Hitchin Limiting configurations, private communication, 2014 | MR

[28] Arthur Jaffe; Clifford Taubes Vortices and monopoles, Progress in Physics, 2, Birkhäuser, Boston, Mass., 1980, pp. v+287 (Structure of static gauge theories) | MR | Zbl

[29] Shoshichi Kobayashi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987, pp. xii+305 (Kanô Memorial Lectures, 5) | DOI | MR | Zbl

[30] L. J. Mason; N. M. J. Woodhouse Self-duality and the Painlevé transcendents, Nonlinearity, Volume 6 (1993) no. 4, pp. 569-581 http://stacks.iop.org/0951-7715/6/569 | MR | Zbl

[31] R Mazzeo; J Swoboda; H Weiß; F Witt Ends of the moduli space of Higgs bundles (2014) (http://arxiv.org/abs/1405.5765)

[32] V. B. Mehta; C. S. Seshadri Moduli of vector bundles on curves with parabolic structures, Math. Ann., Volume 248 (1980) no. 3, pp. 205-239 | DOI | MR | Zbl

[33] David Mumford Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965, pp. vi+145 | MR | Zbl

[34] David Mumford Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325-350 | MR | Zbl

[35] M. S. Narasimhan; C. S. Seshadri Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), Volume 82 (1965), pp. 540-567 | MR | Zbl

[36] Ben Nasatyr; Brian Steer Orbifold Riemann surfaces and the Yang-Mills-Higgs equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 22 (1995) no. 4, pp. 595-643 | Numdam | MR | Zbl

[37] P. E. Newstead Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 51, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978, pp. vi+183 | MR | Zbl

[38] Nitin Nitsure Moduli space of semistable pairs on a curve, Proc. London Math. Soc. (3), Volume 62 (1991) no. 2, pp. 275-300 | DOI | MR | Zbl

[39] Ashoke Sen Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and $S L (2, Z)$ invariance in string theory, Phys. Lett. B, Volume 329 (1994) no. 2-3, pp. 217-221 | DOI | MR | Zbl

[40] C. S. Seshadri Moduli of vector bundles on curves with parabolic structures, Bull. Amer. Math. Soc., Volume 83 (1977) no. 1, pp. 124-126 | MR | Zbl

[41] Carlos T. Simpson Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., Volume 1 (1988) no. 4, pp. 867-918 | DOI | MR | Zbl

[42] Carlos T. Simpson Harmonic bundles on noncompact curves, J. Amer. Math. Soc., Volume 3 (1990) no. 3, pp. 713-770 | DOI | MR | Zbl

[43] C. H. Taubes Compactness theorems for $SL (2; ℂ)$ generalizations of the $4$ -dimensional anti-self dual equations, Part I (2013) (http://arxiv.org/abs/1307.6447)

[44] C. H. Taubes Compactness theorems for $SL (2; ℂ)$ generalizations of the $4$ -dimensional anti-self dual equations, Part II (2013) (http://arxiv.org/abs/1307.6451)

[45] Misha Verbitsky; Dmitri Kaledin Hyperkahler manifolds, Mathematical Physics (Somerville), 12, International Press, Somerville, MA, 1999, pp. iv+257 | MR | Zbl

[46] Raymond O. Wells Differential analysis on complex manifolds, Graduate Texts in Mathematics, 65, Springer, New York, 2008, pp. xiv+299 (With a new appendix by Oscar Garcia-Prada) | DOI | MR | Zbl

Cited by Sources: