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Electromagnetic Duality

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Dualities, Helicity Amplitudes, and Little Conformal Symmetry

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Abstract

Electromagnetism developed with two simultaneous gauge potentials naturally allows for a quantum field theory that includes magnetic monopoles and dyons. Exploiting this symmetry and SL(2,Z) duality, we can examine the spectrum of dyons (including the Witten effect), charge fractionalization, and the impact on electric dipole moments.

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Notes

  1. 1.

    For a review of electromagnetic field tensors, see Chaps. 11.9 and 12 of [1]. For a review of the quantum theory, see Chap. 8 of [2], Chap. 5 of [3], and Sects. 54–68 of [4].

  2. 2.

    We define the dual field strength \(\widetilde{F}^{\mu \nu } \equiv \frac{1} {2}\varepsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }\), also occasionally referred to as F d.

  3. 3.

    The minimal charge g D = 2πℏ∕e is called the Dirac magnetic charge.

  4. 4.

    Defining (a ∧ b) μν = a μ b ν − a ν b μ , (a ⋅ G)ν = a μ G μν = −G νμ a μ = −(G ⋅ a)ν, and \(\left [a \cdot (b \wedge c)^{d}\right ]^{\nu } = a_{\mu }\varepsilon ^{\mu \nu \rho \sigma }b_{\rho }c_{\sigma }\) for four-vectors a, b, c and antisymmetric tensor G. Note (n ⋅ ∂)(n ⋅ ∂)−1(x) = δ (4)(x).

  5. 5.

    Since \(\widetilde{F} = {\ast}F\) is the Hodge dual, we must remember that ∗2 = (−1)k(n−k)+q for a k-form in n-dimensional space with signature (p, q). 2-forms in 4D Minkowski space have ∗2 = −1.

  6. 6.

    Alternatively, the algebra may be generated by K, G, and ɛ, with the relations

    $$\displaystyle\begin{array}{rcl} K^{2} = K,\quad \varepsilon ^{2} = -G,\quad \varepsilon K\varepsilon = E,\quad K -\varepsilon K\varepsilon = G.& & {}\end{array}$$
    (1.3.33)
  7. 7.

    For \(\mathcal{N} = 4\) supersymmetric Yang-Mills \(SL(2, \mathbb{Z})\) is not just a duality, but an actual invariance of the spectrum.

  8. 8.

    Technically only the projective group \(PSL(2, \mathbb{Z}) = SL(2, \mathbb{Z})/\{ \pm 1\}\) acts on τ, but we (and much of the literature) will ignore this distinction.

  9. 9.

    We will use the shorthand F 2 = F μν F μν frequently. There are many relations between the helicity field strengths (1.5.9) and other combinations of field strengths catalogued in Appendix A.4.

  10. 10.

    A half-plane since the gauge coupling is always real and positive.

  11. 11.

    We may also be concerned about global U(1) X or additional SU(N) gauge anomalies for more complicated theories, which places additional restrictions on the allowed charges.

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  1. Physics, Diablo Valley College, Pleasant Hill, CA, USA

    Kitran Macey M. Colwell

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Colwell, K.M.M. (2017). Electromagnetic Duality. In: Dualities, Helicity Amplitudes, and Little Conformal Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-67392-9_1

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