N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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42 Chapter 3. The Linear Case Again the terms in parentheses will turn out to be the covariant derivative of the potential Vµ. For Z = i the equation reduces to Gµν = Vµν. We are not done yet, as the equations for W µ and Gµν are still coupled. To simplify the following calculations let us introduce the abbreviations H µ ≡ 1 Vµν ≡ I |Z| 2 2εµνρσ∂νBρσ + Λ µ Vµν + ˜ Σµν Then we first solve eq. (3.31) for Gµν, and insert this into eq. (3.30), (3.32) R + |Z| 2 ˜Vµν − Σµν . (3.33) Gµν = Vµν − 2I |Z| 2 A[µWν] − R |Z| 2 εµνρσA ρ W σ , (3.34) IW µ = H µ + V µν Aν − 2I |Z| 2 A[µ W ν] Aν . (3.35) Collecting the terms <strong>with</strong> W µ , this can be written as where the field dependent matrix K µν is given by IK µν Wν = |Z| 2 (H µ + V µν Aν) , (3.36) K µν = η µν E + A µ A ν , (3.37) E being the expression (1.55) that we have already encountered in the central charge transformation of the spinors ψ i . To solve for W µ , we need to invert K µν . It can be easily checked that (K −1 )µν = 1 E ηµν − |Z| −2 AµAν . (3.38) Due to the appearance of E in the denominator, this expression is nonpolynomial in the gauge field Aµ. Having determined W µ , we obtain Gµν from eq. (3.34). The final solution to the Bianchi identities then reads W µ = |Z|2 µ µν H + V Aν − |Z| IE −2 A µ AνH ν (3.39) Gµν = Vµν − 2 E A[µ Hν] + Vν]ρA ρ − R IE εµνρσA ρ H σ + V σλ Aλ . (3.40) The fundamental fields of course are the gauge potentials Vµ and Bµν rather than W µ and Gµν, so we now have to determine their supersymmetry and central charge transformations as well. These can be obtained most easily from eqs. (3.30) and (3.31). Applying ∆ z to the former we have ∆ z (C) (IW µ − Λ µ ) = 1 2εµνρσ z µν ∂ν ∆ (C) Bρσ − G ∂νC + CAν δzG µν .
3.2. Transformations and Bianchi Identities 43 On the other hand it follows from eq. (3.29) that ∆ z (C) (IW µ − Λ µ ) = CDνG µν = C∂νG µν + CAν δzG µν , and comparing the two expressions we find ε µνρσ z ∂ν ∆ (C) Bρσ + C ˜ Gρσ = 0 . It comes as no surprise that the action of ∆z on Bµν is determined only modulo a gauge transformation ∆B . We are free to choose a homogeneous transformation law, however, which is then generated by δzBµν = − 1 2εµνρσ G ρσ , (3.41) and the terms in parantheses in eq. (3.30) constitute a proper covariant derivative of Bµν. The central charge transformation of Vµ is derived in like manner. From the eqs. (3.31) and (3.28) we obtain ε µνρσ z ∂ρ ∆ (C) Vσ + CWσ = 0 , so that we set δzVµ = −Wµ . (3.42) Thus, formally the action of the central charge generator δz on Vµ and Bµν has not changed upon gauging the symmetry, cf. equation (2.9). The difference is that now W µ and Gµν are not merely the field strengths but composite expressions that are moreover nonpolynomial in the gauge field Aµ. Therefore we expect that also the action will be nonpolynomial. Since E contains no derivatives this should not spoil locality. The supersymmetry transformations of the gauge potentials can be determined in the same way as demonstrated for the central charge transformations. As this is a lengthy calculation we just give the result, again choosing the simplest form possible by neglecting any contribution that is a gauge transformation, D i αVµ = − i ¯ Zσµ ¯ ψ i + 1 2Lσµ ¯ λ i − Aµψ i α D i αBµν = −2i Iσµνψ i + 1 2Lσµνλ i + A[µσν] ¯ ψ i α (3.43) . (3.44) There is a short cut, however, that immediately yields the supersymmetry transformations modulo possible δz-invariant terms: Using eqs. (3.22), we calculate δz D i αVµ = −D i αWµ + [ δz , D i α ] Vµ = −δz iZσµ ¯ ¯ ψ i + 1 2Lσµ ¯ λ i − Aµψ i α + ∂µψ i α + [ δz , D i α ] Vµ , and similarly for Bµν, δz D i αBµν = −D i α ˜ Gµν + [ δz , D i α ] Bµν = −2i δz Iσµνψ i + 1 2Lσµνλ i + A[µσν] ¯ ψ i α − 2i ∂[µ(σν] ¯ ψ i )α + [ δz , D i α ] Bµν . Comparing the δz-exact terms on the left and on the right, one obtains the previously found relations. In addition, the equations show that central charge transformations
Page 1 and 2: N=2 Supersymmetric Gauge Theories w
Page 3: Abstract In this thesis the central
Page 6 and 7: Acknowledgments I would like to exp
Page 9 and 10: Introduction Despite the lack of ex
Page 11 and 12: Chapter 1 Gauging the Central Charg
Page 13 and 14: 1.1. The N=2 Supersymmetry Algebra
Page 15 and 16: 1.1. The N=2 Supersymmetry Algebra
Page 17 and 18: 1.2. The Linear Multiplet 9 1.2 The
Page 19 and 20: 1.2. The Linear Multiplet 11 It wil
Page 21 and 22: 1.3. The Hypermultiplet 13 second e
Page 23: 1.3. The Hypermultiplet 15 so in th
Page 26 and 27: 18 Chapter 2. The Vector-Tensor Mul
Page 45 and 46: Chapter 3 The Linear Case In this c
Page 47 and 48: 3.2. Transformations and Bianchi Id
Page 49: 3.2. Transformations and Bianchi Id
Page 53 and 54: 3.3. The Lagrangian 45 7) 0 = ∂
Page 55 and 56: 3.3. The Lagrangian 47 compute the
Page 57 and 58: 3.4. Chern-Simons Couplings 49 To c
Page 59 and 60: 3.4. Chern-Simons Couplings 51 Here
Page 61 and 62: 3.5. Henneaux-Knaepen Models 53 eq.
Page 63 and 64: 3.5. Henneaux-Knaepen Models 55 whe
Page 65 and 66: Chapter 4 The Nonlinear Case In sec
Page 73 and 74: 4.3. The Lagrangian 65 This of cour
Page 75: 4.3. The Lagrangian 67 which introd
Page 78 and 79: 70 Conclusions and Outlook likely t
Page 80 and 81: 72 Appendix A. Conventions and the
Page 82 and 83: 74 Appendix A. Conventions σ µν
Page 84 and 85: 76 References [20] N. Dragon and S.

N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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