Skip to main content

Seiberg—Witten on four-manifolds

  • Chapter
Seiberg-Witten Gauge Theory

Part of the book series: Texts and Readings in Mathematics ((TRM,volume 17))

  • 149 Accesses

Abstract

We introduce here the basic concepts that are needed in order to define the Seiberg-Witten equations and invariants. This introduction will be rather sketchy: occasionally we will refer to more detailed references listed in the bibliography.

You have the glow of a man who knows brahman! Tell me -who taught you? ‘Other than human beings’ he acknowledged. ‘But if it pleases you, sir, you should teach it to me yourself for I have heard from people of your eminence that knowledge leads one most securely to the goal only when it is learnt from a teacher’.

Chāndogya Upaniṣad, 4.9.2–3

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 52.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. T. Aubin, Non-linear analysis on manifolds: Monge-Ampere equations, Springer 1982.

    Book  MATH  Google Scholar 

  2. B. Boos, D.D. Bleecker, Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic Physics, Springer 1985.

    Book  Google Scholar 

  3. K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser 1993.

    Book  MATH  Google Scholar 

  4. S. Cordes, G. Moore, S. Ramgoolam, Lectures on 2D Yang-Mills theory, equivariant cohomology, and topological field theories, Nuclear Phys. B Proc. Suppl. 41 (1995), 184–244.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Cordes, G. Moore, S. Ramgoolam, Large N 2D Yang-Mills theory and topological string theory, Comm. Math. Phys. 185 (1997), no. 3, 543–619.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Donaldson, P. Kronheimer, The geometry of four-manifolds, Oxford 1990.

    MATH  Google Scholar 

  7. R. Friedman, J.W. Morgan, Algebraic Surfaces and Seiberg-Witten Invariants, J. Alg. Geom. 6 (1997) N.3 445–479.

    MathSciNet  MATH  Google Scholar 

  8. D. Freed, K.K. Uhlenbeck, Instantons and four-manifolds, Springer 1984. Second edition. MSRI Publications, 1. Springer, 1991.

    Book  MATH  Google Scholar 

  9. R. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), N.3, 527–595.

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Hirzebruch, H. Hopf, Felder von Flachenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Annalen 136 (1958) 156–172.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Kesavan, Topics in functional analysis and applications, Wiley 1989.

    MATH  Google Scholar 

  12. J. Jost, X. Peng, G. Wang, Variational aspects of the Seiberg-Witten functional, Calc. Var. Partial Differential Equations 4, N.3 (1996) 205–218.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Kronheimer, T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Research Letters 1 (1994) 797–808.

    Article  MathSciNet  MATH  Google Scholar 

  14. H.B. Lawson, M.L. Michelsohn, Spin geometry, Princeton University Press, 1989.

    MATH  Google Scholar 

  15. T.J. Li, A. Liu, General wall crossing formula, Math. Res. Lett. (1995) N.2, 797–810.

    Google Scholar 

  16. R.B. Lockhart, R.C. McOwen, Elliptic operators on non-compact manifolds, Ann. Sci. Norm. Sup. Pisa, IV-12 (1985), 409–446.

    MATH  Google Scholar 

  17. D. McDuff, Examples of simply-connected symplectic manifolds, J. Diff. Geom. 20 (1984), 267–278.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. McDuff, D. Salamon, Introduction to symplectic topology, Oxford 1995.

    MATH  Google Scholar 

  19. J.W. Morgan, T.S. Mrowka, D. Ruberman, The L 2-moduli space and a vanishing theorem for Donaldson polynomial invariants, International Press 1994.

    MATH  Google Scholar 

  20. M.S. Narasimhan, T.R. Ramadas, Geometry of SU(2) gauge fields, Comm. Math. Phys. 67 (1979), no. 2, 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Roe, Elliptic operators, topology, and asymptotic methods, Longman Sc. 1988.

    MATH  Google Scholar 

  22. C.H. Taubes, On the equivalence of first and second order equations for gauge theories, Commun. Math. Phys. 75 (1980), 207–227.

    Article  MathSciNet  MATH  Google Scholar 

  23. K.K. Uhlenbeck, Connections with L p bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Witten, Monopoles and Four-Manifolds, Math. Research Letters 1 (1994) 769–796.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Hindustan Book Agency

About this chapter

Cite this chapter

Marcolli, M. (1999). Seiberg—Witten on four-manifolds. In: Seiberg-Witten Gauge Theory. Texts and Readings in Mathematics, vol 17. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-00-2_2

Download citation

Publish with us

Policies and ethics