N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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66 Chapter 4. The Nonlinear Case where for consistency the fermions contained in the composite fields W µ and Gµν have to be set to zero. The latter depends on the former according to eq. (4.21), so we first replace Gµν, − 1 4 GµνGµν + 4AµWν + 2εµνρσ ∂ ρ (LA σ ) = = − 1 4 V µν Vµν − ˜ V µν ∂µ(LAν) − LWµ ˜ F µν Aν + A µ W ν A[µWν] . Then the terms linear in W µ cancel, while the bilinear ones combine into W µ KµνW ν just as for the linear vector-tensor multiplet, 1 LnlinVT ϱL 2 |Z|2∂ µ L ∂µL + EU 2 − 1 2 W µ KµνW ν − 1 4 V µν Vµν − 1 4 L ˜ F µν Vµν + 1 12 L2F µν Fµν − 1 6 L2 (Z ¯ Z + ¯ ZZ) − 1 12 L2 Y ij Yij + ϱ 6 ∂µ 3 µν 2L F Aν − 3L 2 V ˜ µν Aν . (4.47) Again the nonpolynomial interactions arise from inverting Kµν. Using eqs. (3.38) and (4.29), the substitution of W µ gives − ϱ 2 L W µ KµνW ν = − ϱ 2L (LKµρWρ) (K −1 )µν (LK νσ Wσ) = − ϱ µ 1 H − 2 2LE ˜ V µν Vν + 1 2L2F ˜µν Aν + (LV µν + Π µν )Aν ϱ + 2LE|Z| 2 AµH µ − 1 2Aµ ˜ V µν 2 Vν . 2 (4.48) At last, let us neglect also fluctuations of the scalars around their background values 〈Z〉 = i and 〈L〉 = 1/ϱ. Then only the gauge potentials Vµ, Bµν and Aµ remain, and after rescaling Aµ → gzAµ , Bµν → Bµν/ϱ , such that both fields have canonical dimension 1, we find 2 (dropping a total derivative) L = − 1 4 V µν Vµν − 1 2 1 − g 4 z/3ϱ 2 F µν Fµν − 1 µ 1 H − 2 2E ϱ ˜ V µν Vν + gzV µν Aν + 1 2ϱ−1g 2 z ˜ F µν Aν AµH µ − 1 2ϱAµ ˜ V µν 2 Vν . + g2 z 2E 2 (4.49) Apart from the normalisation of Aµ, this Lagrangian follows from the one in eq. (3.64), when in the latter we make the substitution H µ → H µ − 1 2 ϱ ˜ V µν Vν + 1 2 ϱ−1 g 2 z ˜ F µν Aν , (4.50) 2 Evidently, positivity of the kinetic energies requires 3ϱ 2 > g 2 z.
4.3. The Lagrangian 67 which introduces couplings of H a to Chern-Simons terms of both Vµ and Aµ. At first glance, these seem to fit into the structure of the Henneaux-Knaepen models, eq. (3.103). However, it is easily verified that no choice of the parameters cab, which govern the Chern-Simons couplings, can result in the specific combination H µ = H µ − 1 2 ϱ ˜ V µν Vν + 1 2 ϱ−1 g 2 z ˜ F µν Aν + gzV µν Aν (4.51) as displayed above: A comparison <strong>with</strong> eq. (3.103) shows that again the second column of the matrix T a 1 b vanishes (otherwise at least one of the two terms V µν Vν, F µν Vν would be present), hence the coefficients cabT a 1 c can at most yield a Chern-Simons term for A 1 µ = Aµ. We conclude that the Henneaux-Knaepen models, in their current formulation, cannot account for the type of gauge field interactions described by the nonlinear vectortensor multiplet <strong>with</strong> local central charge. Most likely, however, the former admit a generalization which then includes also the case presented here. Work in this direction is in progress, but as yet we cannot report on results.
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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
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1.2. The Linear Multiplet 11 It wil
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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 73: 4.3. The Lagrangian 65 This of cour
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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