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 , which we discuss in detail here.
@article{TSG_2012-2014__31__91_0, author = {Mazzeo, Rafe and Swoboda, Jan and Wei{\ss}, Hartmut and Witt, Frederik}, 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 = {http://www.numdam.org/articles/10.5802/tsg.296/} }
TY - JOUR AU - Mazzeo, Rafe AU - Swoboda, Jan AU - Weiß, Hartmut AU - Witt, Frederik 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 - http://www.numdam.org/articles/10.5802/tsg.296/ DO - 10.5802/tsg.296 LA - en ID - TSG_2012-2014__31__91_0 ER -
%0 Journal Article %A Mazzeo, Rafe %A Swoboda, Jan %A Weiß, Hartmut %A Witt, Frederik %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 http://www.numdam.org/articles/10.5802/tsg.296/ %R 10.5802/tsg.296 %G en %F TSG_2012-2014__31__91_0
Mazzeo, Rafe; Swoboda, Jan; Weiß, Hartmut; Witt, Frederik. Limiting configurations for solutions of Hitchin’s equation. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 91-116. doi : 10.5802/tsg.296. http://www.numdam.org/articles/10.5802/tsg.296/
[1] 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] Spectral curves and the generalised theta divisor, J. Reine Angew. Math., Volume 398 (1989), pp. 169-179 | DOI | MR | Zbl
[3] Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992, pp. xiv+330 | DOI | MR | Zbl
[4] Asymptotic behaviour of monopole metrics, J. Reine Angew. Math., Volume 468 (1995), pp. 139-165 | DOI | MR | Zbl
[5] Monopoles and the Gibbons-Manton metric, Comm. Math. Phys., Volume 194 (1998) no. 2, pp. 297-321 | DOI | MR | Zbl
[6] Monopoles and clusters, Comm. Math. Phys., Volume 284 (2008) no. 3, pp. 675-712 | DOI | MR | Zbl
[7] Representations of orbifold groups and parabolic bundles, Comment. Math. Helv., Volume 66 (1991) no. 3, pp. 389-447 | DOI | MR | Zbl
[8] Moduli spaces of parabolic Higgs bundles and parabolic pairs over smooth curves. I, Internat. J. Math., Volume 7 (1996) no. 5, pp. 573-598 | DOI | MR | Zbl
[9] Flat -bundles with canonical metrics, J. Differential Geom., Volume 28 (1988) no. 3, pp. 361-382 http://projecteuclid.org/euclid.jdg/1214442469 | MR | Zbl
[10] 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] 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] Hyperkähler metrics on cotangent bundles, J. Reine Angew. Math., Volume 532 (2001), pp. 33-46 | DOI | MR | Zbl
[13]
, University of Texas at Austin (in preparation) (Ph. D. Thesis)[14] Special Kähler manifolds, Comm. Math. Phys., Volume 203 (1999) no. 1, pp. 31-52 | DOI | MR | Zbl
[15] Seifert fibred homology -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] Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys., Volume 299 (2010) no. 1, pp. 163-224 | DOI | MR | Zbl
[17] Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., Volume 234 (2013), pp. 239-403 | DOI | MR
[18] 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] Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496 (Graduate Texts in Mathematics, No. 52) | MR | Zbl
[20] 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] Hodge cohomology of gravitational instantons, Duke Math. J., Volume 122 (2004) no. 3, pp. 485-548 | DOI | MR | Zbl
[22] 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] 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] 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] Stable bundles and integrable systems, Duke Math. J., Volume 54 (1987) no. 1, pp. 91-114 | DOI | MR | Zbl
[26] -cohomology of hyperkähler quotients, Comm. Math. Phys., Volume 211 (2000) no. 1, pp. 153-165 | DOI | MR | Zbl
[27] Limiting configurations, private communication, 2014 | MR
[28] Vortices and monopoles, Progress in Physics, 2, Birkhäuser, Boston, Mass., 1980, pp. v+287 (Structure of static gauge theories) | MR | Zbl
[29] 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] 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] Ends of the moduli space of Higgs bundles (2014) (http://arxiv.org/abs/1405.5765)
[32] Moduli of vector bundles on curves with parabolic structures, Math. Ann., Volume 248 (1980) no. 3, pp. 205-239 | DOI | MR | Zbl
[33] 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] 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] Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), Volume 82 (1965), pp. 540-567 | MR | Zbl
[36] 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] 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] 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] Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and invariance in string theory, Phys. Lett. B, Volume 329 (1994) no. 2-3, pp. 217-221 | DOI | MR | Zbl
[40] Moduli of vector bundles on curves with parabolic structures, Bull. Amer. Math. Soc., Volume 83 (1977) no. 1, pp. 124-126 | MR | Zbl
[41] 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] Harmonic bundles on noncompact curves, J. Amer. Math. Soc., Volume 3 (1990) no. 3, pp. 713-770 | DOI | MR | Zbl
[43] Compactness theorems for generalizations of the -dimensional anti-self dual equations, Part I (2013) (http://arxiv.org/abs/1307.6447)
[44] Compactness theorems for generalizations of the -dimensional anti-self dual equations, Part II (2013) (http://arxiv.org/abs/1307.6451)
[45] Hyperkahler manifolds, Mathematical Physics (Somerville), 12, International Press, Somerville, MA, 1999, pp. iv+257 | MR | Zbl
[46] 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
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