Skip to main content

Instantons and Special Geometry

  • Chapter
  • First Online:
Special Metrics and Group Actions in Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 23))

Abstract

We survey and discuss constructions of instantons on non-compact complete manifolds of special holonomy from the viewpoint of evolution equations and give several explicit examples.

To Simon Salamon on the occasion of his 60th birthday

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Atiyah, N. Hitchin, The Geometry and Dynamics of Magnetic Monopoles (M. B. Porter Lectures) (Princeton University Press, Princeton, NJ, 1988)

    Google Scholar 

  2. A. Brandhuber, J. Gomis, S.S. Gubser, S. Gukov, Gauge theory at large \(N\) and new \(\mathop{\mathrm{\mathrm{G}}}\nolimits _{2}\) holonomy metrics. Nucl. Phys. B 611(1–3), 179–204 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. R.L. Bryant, Non-embedding and non-extension results in special holonomy, in The Many Facets of Geometry (Oxford University Press, Oxford, 2010), pp. 346–367

    MATH  Google Scholar 

  4. R.L. Bryant, S.M. Salamon, On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58(3), 829–850 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Clarke, Instantons on the exceptional holonomy manifolds of Bryant and Salamon. J. Geom. Phys. 82, 84–97 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Conti, Special holonomy and hypersurfaces, PhD thesis, Scuola Normale Superiore, Pisa, 2005

    Google Scholar 

  7. D. Conti, Invariant forms, associated bundles and Calabi-Yau metrics. J. Geom. Phys. 57(12), 2483–2508 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Conti, T.B. Madsen, Harmonic structures and intrinsic torsion. Transform. Groups 20(3), 699–723 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Cvetič, G.W. Gibbons, H. Lü, C.N. Pope, New complete noncompact Spin(7) manifolds. Nucl. Phys. B 620(1–2), 29–54 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. J.-H. Eschenburg, M.Y. Wang, The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10(1), 109–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. D.B. Fairlie, J. Nuyts, Spherically symmetric solutions of gauge theories in eight dimensions. J. Phys. A 17(14), 2867–2872 (1984)

    Article  MathSciNet  Google Scholar 

  12. M. Fernández, A classification of Riemannian manifolds with structure group Spin(7). Ann. Mat. Pura Appl. (4) 143, 101–122 (1986)

    Google Scholar 

  13. M. Fernandez, A. Gray, Riemannian manifolds with structure group \(\mathop{\mathrm{\mathrm{G}}}\nolimits _{2}\). Ann. Mat. Pura Appl. (4) 132, 19–45 (1982)

    Google Scholar 

  14. S. Fubini, H. Nicolai, The octonionic instanton. Phys. Lett. B 155(5–6), 369–372 (1985)

    Article  MathSciNet  Google Scholar 

  15. A. Gambioli, SU(3)-manifolds of cohomogeneity one. Ann. Glob. Anal. Geom. 34(1), 77–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. G.W. Gibbons, P.J. Ruback, The hidden symmetries of multicentre metrics. Commun. Math. Phys. 115(2), 267–300 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Gukov, J. Sparks, M-theory on \(\mathop{\mathrm{\mathrm{Spin}}}\nolimits (7)\) manifolds. Nucl. Phys. B 625(1–2), 3–69 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Günaydin, H. Nicolai, Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton. Phys. Lett. B 351(1–3), 169–172 (1995)

    Article  MathSciNet  Google Scholar 

  19. N.J. Hitchin, The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55, 59–126 (1987)

    Google Scholar 

  20. N.J. Hitchin, Stable forms and special metrics, in Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Bilbao, 2000). Contemporary Mathematics, vol. 288 (American Mathematical Society, Providence, RI, 2001), pp. 70–89

    Google Scholar 

  21. T.A. Ivanova, A.D. Popov, (Anti)self-dual gauge fields in dimension \(d\geqslant 4\). Teor. Mat. Fiz. 94(2), 316–342 (1993)

    Google Scholar 

  22. H. Kanno, Y. Yasui, Octonionic Yang-Mills instanton on quaternionic line bundle of \(\mathop{\mathrm{\mathrm{Spin}}}\nolimits (7)\) holonomy. J. Geom. Phys. 34(3–4), 302–320 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Mamone Capria, S.M. Salamon, Yang-Mills fields on quaternionic spaces. Nonlinearity 1(4), 517–530 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Oliveira, Monopoles on the Bryant-Salamon \(\mathop{\mathrm{\mathrm{G}}}\nolimits _{2}\)-manifolds. J. Geom. Phys. 86, 599–632 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Reidegeld, Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces. Proc. Lond. Math. Soc. (3) 102(6), 1127–1160 (2011)

    Google Scholar 

  26. R. Reyes Carrión, A generalization of the notion of instanton. Differ. Geom. Appl. 8(1), 1–20 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

JDL was partially supported by EPSRC grant EP/K010980/1. TBM gratefully acknowledges financial support from Villum Fonden.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Thomas Bruun Madsen .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Lotay, J.D., Madsen, T.B. (2017). Instantons and Special Geometry. In: Chiossi, S., Fino, A., Musso, E., Podestà, F., Vezzoni, L. (eds) Special Metrics and Group Actions in Geometry. Springer INdAM Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-67519-0_10

Download citation

Publish with us

Policies and ethics