N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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62 Chapter 4. The Nonlinear Case results in DµW µ = − 1 4 Gµν ˜ G µν + 1 2 Fµν(LG µν + 1 2 L2 ˜ F µν + Π µν ) . (4.27) Let us now split the covariant derivative and express δzW µ by means of eq. (4.23), ∂µW µ = DµW µ − AµδzW µ = ( 1 2 Fµν − Aµ∂ν) (LG µν + 1 2 L2 ˜ F µν + Π µν ) − 1 4 (Gµν + 4AµWν) ˜ G µν = ∂µ(LG µν Aν + 1 2 L2 ˜ F µν Aν + Π µν Aν) − 1 2 εµνρσ (∂µVν + AµWν) (∂ρVσ − AρWσ) = ∂µ(LG µν Aν + 1 2 L2 ˜ F µν Aν + Π µν Aν − 1 2 εµνρσ Vν∂ρVσ) . Here we have inserted the solution for Gµν in the third step. The first Bianchi identity may thus be solved in terms of an antisymmetric tensor gauge field Bµν, W µ = 1 2 εµνρσ (∂νBρσ − Vν∂ρVσ) + (LG µν + 1 2 L2 ˜ F µν + Π µν )Aν , (4.28) which proves that the constraints (4.13), and their flat limit (2.68) in particular, are indeed consistent. We observe that they give rise to abelian Chern-Simons terms both for the gauge potential Vµ of the vector-tensor multiplet and for the vector field Aµ associated with the central charge. We are interested in W µ rather than in W µ , of course, as the former determines the central charge transformation of Vµ (and also the one of Bµν, see below), while the latter had been introduced merely as an auxiliary means to simplify our calculations. When Gµν is replaced in eq. (4.28), we find that the prefactors of W µ combine into the matrix K µν given in eq. (3.37) just as for the linear vector-tensor multiplet, LK µν Wν = H µ − 1 2 ˜ V µν Vν + 1 2 L2 ˜ F µν Aν + (LV µν + Π µν )Aν , (4.29) where H µ is now defined by Inverting K µν then yields H µ ≡ 1 2 εµνρσ ∂νBρσ + |Z| 2 ψ i σ µ ¯ ψi − iL (Zψ i σ µ¯ λi − ¯ Zλ i σ µ ¯ ψi) − i 2 L2 (Z∂ µ ¯ Z − ¯ Z∂ µ Z + iλ i σ µ¯ λi) . W µ = 1 µ 1 H − 2 LE ˜ V µν Vν + 1 − |Z| −2 A µ AνH ν + 1 2 |Z|−2A µ V˜ νρ AνVρ , 2 L2 ˜ F µν Aν + (LV µν + Π µν )Aν (4.30) (4.31) with E as in eq. (1.55). It remains to determine the transformations of the gauge field Bµν. The central charge transformation we obtain most easily by applying ∆z (C) to eq. (4.28) and comparing the result with CδzW µ as it follows from eq. (4.23). This gives 0 = ε µνρσ z ∂ν ∆ (C)Bρσ + C(L ˜ Gρσ − 1 2L2Fρσ + ˜ Πρσ − VρWσ) − ε µνρσ C Wν(Gρσ − Vρσ) ,
4.2. Transformations and Bianchi Identities 63 and as the terms proportional to C vanish by virtue of the antisymmetry of the ε-tensor, we conclude that δzBµν = V[µWν] − L εµνρσ(∂ ρ V σ − A ρ W σ ) + 1 2 L2 Fµν − ˜ Πµν . (4.32) As we had seen in section 3.4, the occurence of a Chern-Simons term in the generalized field strength of Bµν requires the gauge transformation associated with the corresponding vector field to act nontrivially on Bµν in order to render the field strength gauge invariant. According to eq. (4.28), the change of Vµ by the gradient of some scalar field Θ is to be accompanied by the transformation ∆ V (Θ) Bµν = − 1 2 ΘVµν . (4.33) At last, the supersymmetry transformation of Bµν follows from the one of δzBµν, δzD i αBµν = D i α V[µWν] − L εµνρσD ρ V σ + 1 2L2Fµν − ˜ Πµν + [ δz , D i α ] Bµν = δz V[µσν] (i ¯ Z ¯ ψ i + 1 2L¯ λ i ) − V[µAν] ψ i − ¯ ZLσµν (2i Zψ i − Lλ i ) − iLA[µσν] (2i ¯ Z ¯ ψ i + 1 2L¯ λ i ) + ∂[µ Vν]ψ α i + 2 ¯ ZLσν] ¯ ψ i − i 2L2σν] ¯ λ i α + 1 2ψi αVµν + [ δz , D i α ] Bµν . From this we infer that (modulo δz-invariant terms, which can be neglected however) D i αBµν = V[µσν] (i ¯ Z ¯ ψ i + 1 2L¯ λ i ) − V[µAν] ψ i − ¯ ZLσµν (2i Zψ i − Lλ i ) − iLA[µσν] (2i ¯ Z ¯ ψ i + 1 2L¯ λ i ) α , (4.34) and it is easily verified that on all the component fields supersymmetry and central charge transformations commute modulo gauge transformations, [ ∆ z (C) , ∆(ξ) ] = ∆ V (Θ) + ∆ B (Ω) , where ∆ V now acts on both Vµ and Bµν. Here the parameters read explicitly Θ = C(ξiψ i + ¯ ξ i ¯ ψi) , Ωµ = C Re(Vµξiψ i + 2 ¯ ZL ξiσµ ¯ ψ i − i 2 L2 ξiσµ ¯ λ i ) . (4.35) Finally, a straightforward though tedious computation of the supersymmetry commutation relations on Bµν results in {D i α , D j β } Bµν = εαβ ε ij ZδzBµν ¯ − ∂[µ(iVν] ¯ ZL + Aν] ¯ ZL 2 ) − i 2 ¯ ZLVµν {D i α , ¯ D ˙αj} Bµν = −iδ i j Dα ˙αBµν + 2 ∂[µ(Bν]ρ − 1 2ην]ρ|Z| 2 L 2 )σ ρ 1 α ˙α − 2Vα (4.36) ˙αVµν , which implies that the parameters ɛ µ , C and Θ on the right-hand side of the equation [ ∆(ξ) , ∆(ζ) ] = ɛ µ ∂µ + ∆ z (C) + ∆ V (Θ) + ∆ B (Ω) coincide with those for the linear vector-tensor multiplet, eqs. (3.47), while Ωµ reads Ωµ = 1 2 ɛµ|Z| 2 L 2 − Bµνɛ ν − i 2 VµL(ξiζ i ¯ Z + ¯ ξ i ¯ ζiZ) − 1 2 AµL 2 (ξiζ i ¯ Z − ¯ ξ i ¯ ζiZ) . (4.37)
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N=2 Supersymmetric Gauge Theories w
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Abstract In this thesis the central
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Acknowledgments I would like to exp
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Introduction Despite the lack of ex
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Chapter 1 Gauging the Central Charg
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1.1. The N=2 Supersymmetry Algebra
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1.1. The N=2 Supersymmetry Algebra
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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 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 69: 4.2. Transformations and Bianchi Id
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.
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N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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