Communications in Mathematical Physics

, Volume 341, Issue 2, pp 699–728 | Cite as

Calabi–Yau Monopoles for the Stenzel Metric

  • Goncalo Oliveira


We construct the first nontrivial examples of Calabi–Yau monopoles. Our main interest in these comes from Donaldson and Segal’s suggestion (Geometry of Special Holonomy and Related Topics. Surveys in Differential Geometry, vol 16, pp 1–41, 2011) that it may be possible to define an invariant of certain noncompact Calabi–Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the 3-sphere \({T^* \mathbb{S}^3}\) and study monopoles under a symmetry assumption. Our main result constructs the moduli of these symmetric monopoles and shows that these are parametrized by a positive real number known as the mass of the monopole. In other words, for each fixed mass we show that there is a unique monopole that is invariant in a precise sense. Moreover, we also study the large mass limit under which we give precise results on the bubbling behavior of our monopoles. Towards the end, an irreducible SU(2) Hermitian–Yang–Mills connection on the Stenzel metric is constructed explicitly.


Zero Section Asymptotically Conical Complex Line Bundle Ordinary Double Point Invariant Connection 
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  1. 1.
    Candelas P., de la Ossa X.C.: Comments on conifolds. Nucl. Phys. B 342, 246–268 (1990)CrossRefADSGoogle Scholar
  2. 2.
    Conlon R.J., Hein H.J.: Asymptotically conical Calabi–Yau manifolds, I. Duke Math. J. 162, 2855–2902 (2013)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Donaldson, S.K., Segal, E.P.: Gauge theory in higher dimensions, II. In: Geometry of Special Holonomy and Related Topics. Surveys in Differential Geometry, vol. 16, pp. 1–41 (2011)Google Scholar
  4. 4.
    Joyce, D.: On counting special Lagrangian homology 3-spheres. In: Topology and Geometry: Commemorating SISTAG. Contemporary Mathematics, vol. 314, pp. 125–151 (2002)Google Scholar
  5. 5.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry, vol. 1. Interscience, New-York (1963)Google Scholar
  6. 6.
    McLean R.C.: Deformations of calibrated submanifolds. Commun. Anal. Geom. 6, 705–747 (1998)MATHMathSciNetGoogle Scholar
  7. 7.
    Oliveira, G.: Monopoles in Higher Dimensions, Ph.D. thesis, Imperial College London (2014). Available online at
  8. 8.
    Oliveira G.: Monopoles on the Bryant–Salamon G 2-manifolds. J. Geom. Phys. 86, 599–632 (2014)MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Prasad M.K., Sommerfield C.M.: Exact classical solution for the’t Hooft monopole and the Julia-Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975)CrossRefADSGoogle Scholar
  10. 10.
    Sibner L.M., Sibner R.J.: Removable singularities of coupled Yang–Mills fields in \({\mathbb{R}^3}\). Commun. Math. Phys. 93, 1–17 (1984)MATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Stenzel M.B.: Ricci-flat metrics on the complexification of a compact rank one symmetric space. Manuscripta mathematica 80, 151–163 (1993)MATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK
  2. 2.Duke UniversityDurhamUSA

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