N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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58 Chapter 4. The Nonlinear Case We observe that the combination L + h is, modulo the constant µ, precisely of the form (2.78), thus we can achieve ϱ = 0 by a field redefinition, thereby fixing the gauge modulo rescalings of L by a constant parameter. Next we insert A and F into eq. 2), the general solution to which is given by Eq. 25) then implies (L + µ)∂LC + C = − 1 L + µ , (4.5) Z C = v(Z, ¯ Z) L + µ − L + µ 2Z . (4.6) µ(L + µ) = Zv + ¯ Z¯v ⇒ µ = 0 . (4.7) From eqs. 6) and 7) we obtain the dependence of E on L and Z, respectively, L∂LE + E = 0 = Z∂E + E ⇒ E = ¯ ∂¯g ZL where ¯g( ¯ Z) is independent of Z. At last, we consider eq. 3), which requires By differentiating eq. (4.7) <strong>with</strong> respect to Z, we find , (4.8) Z∂v + v = −∂g . (4.9) 0 = Z∂v + v + ¯ Z∂¯v = ∂( ¯ Z¯v − g) ⇒ ¯v = g ¯Z + ū( ¯ Z) , (4.10) and finally from eq. (4.7) the relation u = −g/Z. The coefficient functions thus read A = − 2 Z , C = − L 2Z F = − 1 L , G = ¯ Z ZL 1 ∂¯g ¯ − g − ¯g , E = ZL ZL , a = b = c = B = 0 , , D = − ∂g ZL (4.11) where g(Z) is some arbitrary holomorphic function. Similar to the case of the linear vector-tensor multiplet, the g-dependent terms combine to − 1 i j D (g D Z) − D¯ i (¯g D¯ j Z) ¯ 1 i j = − D D f(Z) − ¯ i D ¯ j D f( ¯ Z) ¯ , (4.12) ZL ZL provided g can be integrated, ∂f = g. In the following we consider the case g = 0 only, for it can be shown [2] that the Bianchi identities again single out a function g(Z) which may be removed by a superfield redefinition. We shall not generalize the model to include Chern-Simons couplings to nonabelian vector multiplets (this can be found in the reference just mentioned), but Chern-Simons-like terms for Vµ and Aµ arise automatically, as we will see. The constraints we are now going to investigate read D (i α ¯ D j) ˙α L = 0 , D (i D j) L = − 1 (i j) 1 2L D Z D L + 2 ZL L2 D i D j Z + ZD i L D j L − ¯ Z ¯ D i L ¯ D j L . (4.13)
4.2. Transformations and Bianchi Identities 59 4.2 Transformations and Bianchi Identities To determine the Bianchi identities, we need to calculate the action of the supersymmetry generators on the deformation M ij = 1 (i j) 2 ij i j 2i Lλ ψ − L Y + Zψ ψ − Z¯ ¯i ψ ¯j ψ ZL . (4.14) Note that contrary to the case of the linear vector-tensor multiplet, M ij is neither real nor imaginary, which makes things a little more complicated. Applying Dαj, we obtain DαjM ij = 3 ij 2iL Y ψj − iL 2ZL 2 σ µ ∂µ ¯ λ i + ¯ ZLUλ i + Gµνσ µν (iZψ i − Lλ i ) + 2iL Fµνσ µν ψ i − ¯ Z(DµL − iWµ) σ µ ψ¯ i ij + M (iZψj − Lλj) + (ψ i ψ j )λj + (λ i ψ j )ψj α , (4.15) while the action of ¯ D ˙αj now cannot be derived by complex conjugation of the above expression but needs to be computed separately. One finds ¯D ˙αjM ij = − 3 i µ 2Lψ σ ∂µZ + iL Dµ(Lλ 2ZL i σ µ ) − LWµλ i σ µ + i|Z| 2 U ¯ ψ i + Z(DµL + iWµ) ψ i σ µ + i ¯ Z Gµν ¯ ψ i ¯σ µν − i ¯ Z ¯ M ij ψj ¯ + ( ¯ ψ i ψ¯ j ) λj ¯ + ( ¯ λ i ψ¯ j ) ψj ¯ ˙α . (4.16) The central charge transformation of W µ and the Bianchi identity for Gµν involve the real and imaginary part of ZDiσ µ ¯ DjM ij , respectively. After a lengthy calculation, we arrive at 1 6 Z Diσ µ DjM ¯ ij = U(i ¯ Z∂ µ Z + λ i σ µ¯ λi) − ( ˜ G µν + iG µν )∂νZ − (i ˜ F µν − F µν )Wν − Dν(L ˜ F µν + iLF µν + 2ψ i σ µν λi) + i ¯ Z λ i σ µ δz ¯ ψi + 1 µν (IG − R ˜µν G )DνL + (I 2L ˜ G µν + RG µν )Wν (4.17) − |Z| 2 UW µ + 2i ψ i σ µν ψi ∂νZ − 2iZ ψ i D µ ψi − 2i λ i σ µν ψi(Wν − iDνL) + 2 3 iZ ψiσ µ ¯ DjM ij + c.c. . When inserted into eq. (BI.2), only a few terms survive, I Dν ˜ G µν + R DνG µν = − 1 2 U ¯ Z∂ µ Z − 1 2 ¯ Z λ i σ µ δz ¯ ψi − i 12 Z Diσ µ ¯ DjM ij + c.c. = − ˜ G µν ∂νI − G µν ∂νR − Dν(LF µν + iλiσ µν ψ i − i ¯ ψi¯σµν ¯ λ i ) − ˜ F µν Wν , and the second Bianchi identity is found to read Dν ˜ G µν = − 1 2 εµνρσ FνρWσ , (4.18)
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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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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: 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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