N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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52 Chapter 3. The Linear Case contains the nonabelian Chern-Simons form that lends its name to this section. It originates from the term F I µν ˜ F µνI , which can be written as a total derivative using the Jacobi identity and the antisymmetry of the structure constants fIJ K δKL, 1 2 εµνρσF I µνF I ρσ = 2ε µνρσ ∂ µA I ν ∂ρA I σ − fJK I A J µA K ν ∂ρA I σ + 1 4AJ µA K ν A L ρ A M σ fJK I fLM I I Aν∂ρA I σ − 1 = 2ε µνρσ ∂µ 3 AI νA J ρ A K σ fJK I . Again, one can replace Λµ with ˆ Λµ in eq. (3.29), and we conclude that for the function (3.78) the constraints (3.67) are consistent, since the Bianchi identities can be solved exactly as in section 3.2. Note however that, although ˆ Λ µ is δI-invariant, a full gauge transformation ∆ g yields ∆ g (C) ˆ Λ µ = 2e ε µνρσ ∂ν(C I ∂ ρA I σ) . (3.81) Hence, in order to render the solution (3.30) to the first Bianchi identity ∆ g -invariant, we have to cancel the contribution from ˆ Λ µ by assigning to Bµν the nontrivial transformation law ∆ g (C) Bµν = −4e C I ∂ [µA I ν] . (3.82) Vµ on the other hand remains gauge invariant. What happens when we choose eφ = Z, i.e. F = Z 2 /2e? This corresponds to the function h = −2iZ/e in the coefficients (3.66), and by a field redefinition L = ˆ L + i e (Z − ¯ Z) (3.83) we can achieve ˆ h = 0. Thus the Bianchi identities have singled out a function F(Z) that may be gauged away, which shows that, modulo field redefintions, the constraints (3.19) uniquely describe the linear vector-tensor multiplet with gauged central charge. An invariant action can now easily be written down, as in the previous section we have used only the two properties N ij α ˙α = 0 and ¯ M ij = −M ij in the derivation of the Lagrangian (3.58), and these are also valid in the case at hand. Therefore, we just need to determine LM according to eq. (3.59). As all the ingredients have been given above, this is merely a matter of inserting and combining terms, and we proceed immediately to the final Lagrangian, which reads (modulo a total derivative) µ L = LlinVT Λ → Λ ˆ µ , Σµν → ˆ Σµν + Lcc − 2e iF I µνψiσ µνχ iI − ψ i σ µ ¯χ I i Dµ ¯ φ I + (φ − ¯ φ) I ψ i σ µ Dµ ¯χI i + iD ijI ψi χI j + i 2 ψiχ J i ¯ φ K fJK I φ I + c.c. − e ∂µL (φ + ¯ φ) I ∂ µ (φ + ¯ φ) I + (eL + 1/4g 2 ) 2D µ φ I Dµ ¯ φ I − F µνI F I µν − iχiI ↔ µ σ Dµ ¯χI + χ iIχ J i ¯ φ K − ¯χ I i ¯χiJ φ K fJK I + 1 2 i + D I ijD ijI φ J ¯ φ K fJK I 2 . (3.84) LlinVT has been given in eq. (3.61) and depends on Λ µ and Σµν through the composite fields W µ and Gµν. Upon replacing Λ µ with ˆ Λ µ in the generalized field strength H µ ,
3.5. Henneaux-Knaepen Models 53 eq. (3.32), a coupling of the Chern-Simons form to the tensor gauge field Bµν emerges to zeroth order in gz, cf. (3.63). For the pure gauge field part we find where L = − 1 4 V µν Vµν − 1 2E ( ˆ H µ + V µν Aν) 2 + 1 2E (Aµ ˆ H µ ) 2 − 1 4g2 F z µν Fµν − 1 4g2 F µνI F I µν , (3.85) ˆH µ = 1 2 εµνρσ ∂νBρσ − 4e A I ν∂ ρA I σ − 1 3 AI νA J ρ A K σ fJK I . (3.86) 3.5 Henneaux-Knaepen Models We conclude the chapter by showing how the special gauge couplings we have found as a result of supersymmetry fit into a more general scheme devised by Henneaux and Knaepen in [25]. When formulated in D spacetime dimensions, these models involve interactions of (D−2)-form gauge fields with gauge potentials of lesser form degree and include as a subset the so-called Freedman-Townsend models [26, 27], which describe nonpolynomial self-couplings of (D − 2)-forms. In four dimensions the field content consists of 2-form and ordinary 1-form gauge potentials. While it has been shown by Brandt and the author in [28] that every four-dimensional Henneaux-Knaepen model admits an N = 1 supersymmetric generalization, the only known example of such a model possessing two supersymmetries is the (linear) vector-tensor multiplet with gauged central charge. Let us now collectively denote the antisymmetric tensors as BµνA and the vector fields as A a µ, with field strengths H µ 1 A = 2εµνρσ∂νBρσA , F a µν = ∂µA a ν − ∂νA a µ . (3.87) In the case of the vector-tensor multiplet with gauged central charge we would have two 1-forms A 1 µ, A 2 µ, one of which being identical to what we used to call Vµ, and just one 2-form Bµν. A Lagrangian that is invariant under abelian gauge transformations ∆ z (C) A a µ = −∂µC a , ∆ B (Ω) BµνA = −2 ∂[µΩν]A (3.88) is given simply in terms of the field strengths, L = − 1 2 Hµ AHA µ − 1 µν F a F 4 a µν . (3.89) The key observation is that the action has in addition global symmetries generated by where the T a A b currents read δaA b µ = −H A µ T b A a , δaBµνA = − 1 ρσ εµνρσF 2 b T b A a , (3.90) are, at this stage, arbitrary real constants. The corresponding Noether J µ a = T b A aF µν b HA ν . (3.91)
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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
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 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: 3.4. Chern-Simons Couplings 51 Here
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.
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N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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