N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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34 Chapter 2. The Vector-Tensor Multiplet Since v does not depend on ¯ Z and u is real, we conclude that there is a function w(Z) such that u(Z, ¯ Z) = w(Z) + ¯w( ¯ Z) , v(Z) = ∂w(Z) . (2.76) Now let us perform a field redefinition (2.72) in eq. 25). With the transformation of C as in eq. (2.54), we calculate ˆL + ZĈ + ¯ Z ˆ¯ 1 C = L ′ L − f − f¯ Z + L ′ Z¯ C − ∂f + L ′ = 1 L ′ w + ¯w − f − f¯ − Z∂f − Z¯ ∂¯ f¯ . The same redefinition applied to eq. 26) gives ¯C − ¯ ∂ ¯ f Z ˆ D + ¯ Z ˆ Ē = 1 L ′ ZD + ZĒ ¯ + ∂f (ZA + ¯ Z ¯ B) + ∂f ∂f (ZF + ¯ Z ¯ G) − Z∂ 2 f = 1 L ′ ∂w − f − Z∂f , where eqs. 23) and 24) have been used. So provided there is a function g(Z) such that ∂g = w, a field redefinition <strong>with</strong> f = g/Z yields (omitting the hats) 25) 0 = L + ZC + ¯ Z ¯ C 26) 0 = ZD + ¯ ZĒ . (2.77) Note that there remains a residual gauge freedom L = κ ˆ L + ϱ Z + ¯ϱ ¯Z , κ ∈ R∗ , ϱ ∈ C , (2.78) for g may be shifted by a complex constant ϱ. Working in a gauge where u = v = 0, the 26 consistency conditions can be reduced to a set of 11 independent equations, 1) 0 = 2∂F − ∂LA 2) 0 = ∂LC − CF − 1 2 A 3) 0 = ∂C − 1 2 AC + ¯ Z Z Ē 4) 0 = ∂L ¯ F − F ¯ F 5) 0 = 2∂ ¯ F − A + 2 ¯F Z 6) 0 = ∂LE − EF (2.79) 7) 0 = ∂E − 1 2AE 1 9) 0 = ∂Ā − A + 2 2 2 Ā + Z ¯Z 10) 0 = ∂A − 1 2A2 − 2ĒF − ¯ F ¯ Z Z 19) 0 = ∂LA − A + 2 ¯F Z
2.3. The Ansatz 35 25) 0 = L + ZC + ¯ Z ¯ C . The trivial solution M ij = 0 = N ij α ˙α is now excluded as some of the equations are inhomogeneous. Eq. 4) in (2.79) we have already solved in the previous section, only now the integration constants may depend on Z and ¯ Z. The solution that corresponds to F3 we discard again since, as mentioned, it leads to inconsistent constraints, and the situation certainly does not improve when gauging the central charge. This leaves the two possibilities 1 F1 = 0 , F2 = − L + h(Z, ¯ Z) , h real . (2.80) In the following two chapters we shall explore the consequences of each in detail.
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 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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