Duke Mathematical Journal

Ends of the moduli space of Higgs bundles

Rafe Mazzeo, Jan Swoboda, Hartmut Weiss, and Frederik Witt

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We associate to each stable Higgs pair (A0,Φ0) on a compact Riemann surface X a singular limiting configuration (A,Φ), assuming that detΦ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions (At,tΦt) to Hitchin’s equations, which converge to this limiting configuration as t. This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

Article information

Source
Duke Math. J., Volume 165, Number 12 (2016), 2227-2271.

Dates
Received: 8 July 2014
Revised: 9 August 2015
First available in Project Euclid: 6 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1473186400

Digital Object Identifier
doi:10.1215/00127094-3476914

Mathematical Reviews number (MathSciNet)
MR3544281

Zentralblatt MATH identifier
1352.53018

Subjects
Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]
Secondary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]

Keywords
Hitchin’s equations Higgs bundles limiting configurations

Citation

Mazzeo, Rafe; Swoboda, Jan; Weiss, Hartmut; Witt, Frederik. Ends of the moduli space of Higgs bundles. Duke Math. J. 165 (2016), no. 12, 2227--2271. doi:10.1215/00127094-3476914. https://projecteuclid.org/euclid.dmj/1473186400


Export citation

References

  • [1] C. Bär, The Dirac operator on hyperbolic manifolds of finite volume, J. Differential Geom. 54 (2000), 439–488.
  • [2] O. Biquard and P. Boalch, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179–204.
  • [3] S. Cecotti and C. Vafa, On classification of $N=2$ supersymmetric theories, Comm. Math. Phys. 158 (1993), 569–644.
  • [4] L. Fredrickson, Asymptotic limits in the Hitchin moduli space, Ph.D. dissertation, University of Texas at Austin, Austin, Tex., 2016.
  • [5] D. S. Freed, Special Kähler manifolds, Comm. Math. Phys. 203 (1999), 31–52.
  • [6] D. Gaiotto, G. W. Moore, and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), 163–224.
  • [7] D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239–403.
  • [8] W. M. Goldman and E. Z. Xia, Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces, Mem. Amer. Math. Soc. 193 (2008), no. 904.
  • [9] P. B. Gothen, “Representations of surface groups and Higgs bundles” in Moduli Spaces, London Math. Soc. Lecture Note Ser. 411, Cambridge Univ. Press, Cambridge, 2014, 151–178.
  • [10] T. Hausel, Vanishing of intersection numbers on the moduli space of Higgs bundles, Adv. Theor. Math. Phys. 2 (1998), 1011–1040.
  • [11] T. Hausel, E. Hunsicker, and R. Mazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), 485–548.
  • [12] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), 59–126.
  • [13] N. J. Hitchin, “Riemann surfaces and integrable systems” in Integrable Systems (Oxford, 1997), Oxf. Grad. Texts Math. 4, Oxford Univ. Press, New York, 1999, 11–52.
  • [14] N. J. Hitchin, $L^{2}$-cohomology of hyperkähler quotients, Comm. Math. Phys. 211 (2000), 153–165.
  • [15] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publ. Math. Soc. Japan 15, Princeton Univ. Press, Princeton, 1987.
  • [16] J. Le Potier, Fibrés de Higgs et systèmes locaux, Astérisque 201–203 (1991), 221–268, Séminaire Bourbaki 1990/1991, no. 737.
  • [17] L. J. Mason and N. M. J. Woodhouse, Self-duality and the Painlevé transcendents, Nonlinearity 6 (1993), 569–581.
  • [18] R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations 16 (1991), 1615–1664.
  • [19] R. Mazzeo and G. Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra, J. Differential Geom. 87 (2011), 525–576.
  • [20] R. Mazzeo, J. Swoboda, H. Weiß, and F. Witt, Limiting configurations for solutions of Hitchin’s equation, Sémin. Théor. Spectr. Géom. 31 (2012–2014), 91–116.
  • [21] B. M. McCoy, C. A. Tracy, and T. T. Wu, Painlevé functions of the third kind, J. Math. Phys. 18 (1977), 1058–1092.
  • [22] J. Milnor, “Remarks on infinite-dimensional Lie groups” in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, 1007–1057.
  • [23] A. Neitzke, personal communication, August 2013.
  • [24] L. I. Nicolaescu, Notes on Seiberg-Witten Theory, Grad. Stud. Math. 28, Amer. Math. Soc., Providence, 2000.
  • [25] A. Sen, Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and $\mathrm{SL}(2,\mathbb{Z})$-invariance in string theory, Phys. Lett. B 329 (1994), 217–221.
  • [26] C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867–918.
  • [27] C. T. Simpson, Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci. 75 (1992), 5–95.
  • [28] C. H. Taubes, Compactness theorems for $\mathrm{SL}(2;\mathbb{C})$ generalizations of the $4$-dimensional anti-self dual equations, I, preprint, arXiv:1307.6447v4 [math.DG].
  • [29] C. H. Taubes, The zero loci of $\mathbb{Z}/2$ harmonic spinors in dimension 2, 3 and 4, preprint, arXiv:1407.6206 [math.DG].
  • [30] R. Wells, Differential Analysis on Complex Manifolds, with a new appendix by O. Garcia-Prada, Grad. Texts in Math. 65, Springer, New York, 2008.
  • [31] H. Widom, On the solution of a Painlevé III equation, Math. Phys. Anal. Geom. 3 (2000), 375–384.