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OnS-duality in Abelian gauge theory

Abstract

U(1) gauge theory onR 4 is known to possess an electric-magnetic duality symmetry that inverts the coupling constant and extends to an action ofSL(2,Z). In this paper, the duality is studied on a general four-manifold and it is shown that the partition function is not a modular-invariant function but transforms as a modular form. This result plays an essential role in determining a new low-energy interaction that arises whenN=2 supersymmetric Yang-Mills theory is formulated on a four-manifold; the determination of this interaction gives a new test of the solution of the model and would enter in computations of the Donaldson invariants of four-manifolds with b 2 + ≤1. Certain other aspects of abelian duality, relevant to matters such as the dependence of Donaldson invariants on the second Stieffel-Whitney class, are also analyzed.

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References

  1. [1]

    C. Vafa and E. Witten.A Strong Coupling Test Of S-Duality. Nucl. Phys.,B431 (1994) 3.

  2. [2]

    C. Montonen and D. Olive.Magnetic Monopoles As Gauge Particles? Phys. Lett.,72B (1977) 117.

  3. [3]

    J. Cardy and E. Rabinovici.Phase Structure Of Z p Models In The Presence Of A Theta Parameter. Nucl. Phys.,B205 (1982) 1.

  4. [4]

    J. Cardy.Duality And The Theta Parameter In Abelian Lattice Models. Nucl. Phys.,B205 (1982) 17.

  5. [5]

    A. Shapere and F. Wilczek.Self-Dual Models With Theta Terms. Nucl. Phys.,B320 (1989) 669.

  6. [6]

    T.H. Buscher.Path Integral Derivation Of Quantum Duality In Nonlinear Sigma Models. Phys. Lett.,201B (1988) 466.

  7. [7]

    A. Giveon, M. Porrati, and E. Rabinovici.Target-Space Duality In String Theory. Phys. Rept.,244 (1994) 77.

  8. [8]

    N. Seiberg and E. Witten.Monopoles, Duality, And Chiral Symmetry Breaking In N=2 Supersymmetric QCD. Nucl. Phys.,B431 (1994) 484.

  9. [9]

    N. Seiberg and E. Witten.Electric-Magnetic Duality, Monopole Condensation, and Confinement In N=2 Supersymmetric Yang-Mills Theory. Nucl. Phys.,B426 (1994) 19.

  10. [10]

    E. Witten.Monopoles And Four-Manifolds. Math. Res. Lett.,1 (1994) 769.

  11. [11]

    K.S. Narain.New Heterotic String Theories In Uncompactified Dimension<10. Phys. Lett.,169B (1986) 41; K.S. Narain, M.H. Sarmadi, and E. Witten.A Note On Toroidal Compactification Of Heterotic String Theory. Nucl. Phys.,B279 (1987) 369.

  12. [12]

    M. Rocek and E. Verlinde.Duality, Quotients, And Currents. Nucl. Phys.,B373 (1992) 630.

  13. [13]

    S. Donaldson and P. Kronheimer.The Geometry Of Four-Manifolds. Oxford University Press, 1990.

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Witten, E. OnS-duality in Abelian gauge theory. Selecta Mathematica, New Series 1, 383–410 (1995). https://doi.org/10.1007/BF01671570

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Keywords

  • Gauge Theory
  • Partition Function
  • Essential Role
  • Modular Form
  • Abelian Gauge