N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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48 Chapter 3. The Linear Case Now let us consider M ij as in eq. (3.20). Using eqs. (3.21) and (3.23), we find LM = −W µ Λµ − 1 2 ˜ G µν Σµν − LR ∂µW µ − R I LAµ∂νG µν − 1 2I LUAµ ∂µ|Z| 2 + 1 2 U( ¯ Zλiψ i − Z¯ λ i ψi) ¯ − i 2 AµU(ψ i σ µ¯ λi − λ i σ µ ψi) ¯ + 2 ψ i σ µ ψi ¯ ∂µR − 1 2 L2I − iL(ψ i σ µ ∂µ ¯ λi − ∂µλ i σ µ ψi) ¯ − iYij(ψ i ψ j − ¯ ψ i ψ¯ j ) (3.60) − iFµν(ψiσ µν ψ i + ¯ ψ i ¯σ µν ψi) ¯ − 1 2I LAµ δz(Zψ i σ µ¯ λi + ¯ Zλ i σ µ ψi) ¯ + 1 4 IMijM ij − 1 2 ∂µ µ i L∂ I + iL(ψ σ µ¯ λi − λ i σ µ ψi) ¯ . When put into eq. (3.58), several terms cancel or combine into total derivatives, and we arrive at LlinVT = 1 2 I∂ µ L ∂µL − W µ Wµ − 2i ψ i ↔ µ σ ∂µ ¯ ψi + EU 2 − 1 2 L2I − 1 4 GµνIGµν − R ˜ i Gµν + 4AµWν − 2 Yij (ψ i ψ j − ¯ ψ i ψ¯ j ) − i 2 Fµν (ψiσ µν ψ i + ¯ ψ i ¯σ µν ψi) ¯ − i 2 L (ψi ↔ µ σ ∂µ ¯ λi + λ i ↔ µ σ ∂µ ¯ ψi) (3.61) + ψ i σ µ ¯ ψi ∂µR + 1 − ∂µ 4 IM ij Mij µ LRW + LG ˜µν Aν + iAν(ψiσ µν ψ i + ¯ ψ i ¯σ µν ψi) ¯ + 1 2 L2 ∂ µ I − ψ i σ µ ¯ ψiR + iL (ψ i σ µ¯ λi − λ i σ µ ¯ ψi) . At last, we have to replace W µ and Gµν by the solutions to the Bianchi identities found in the previous section. We first insert Gµν from eq. (3.34), which allows to combine the terms containing W µ in a nice way, − 1 4 GµνIGµν − R ˜ 1 Gµν + 4AµWν − 2 I W µ Wµ = = − 1 4 VµνIVµν − R ˜ I Vµν − 2|Z| 2 W µ KµνW ν , with Kµν as in eq. (3.37). Next we replace W µ using eq. (3.36). As each W µ contributes an inverse of Kµν, the result − I 2|Z| 2 W µ KµνW ν = − |Z|2 µ µρ −1 ν νσ H + V Aρ (K )µν H + V Aσ 2I (3.62) also involves the inverse matrix, giving rise to nonpolynomial but local couplings to the gauge field Aµ. When K −1 is inserted, we obtain − 1 4 GµνIGµν − R ˜ 1 Gµν + 4AµWν − 2 I W µ Wµ = = − 1 4 Vµν (IVµν − R ˜ Vµν) − |Z|2 2IE (Hµ + V µν Aν) 2 + 1 2IE (AµH µ ) 2 . (3.63)
3.4. Chern-Simons Couplings 49 To conclude this section, let us concentrate on the gauge field part of the model by freezing the scalars to constants (in particular Z = i and L = 0) and neglecting the fermions. Dropping the total derivative, the complete Lagrangian (3.61) reduces to L = − 1 4 V µν Vµν − 1 2E (Hµ + V µν Aν) 2 + 1 2E (AµH µ ) 2 − 1 4g2 F z µν Fµν , (3.64) where a kinetic term for Aµ originating from Lcc, eq. (1.48), has been added. Supersymmetry has now been broken explicitly of course, but the gauge invariances remain intact. After a rescaling Aµ → gzAµ, such that all fields have canonical dimension one 1 , we can expand the Lagrangian in powers of the coupling constant gz, which gives up to first order L = − 1 4 V µν Vµν − 1 2 Hµ Hµ − 1 4 F µν Fµν + gzAµV µν Hν + O(g 2 z) , (3.65) and we recognize the coupling of Aµ to the current J µ z from eq. (2.16). The Lagrangian (3.64) had previously been found outside the framework of supersymmetry in [24], and is actually but one example of a whole class of gauge theories known as Henneaux-Knaepen models, which we review in the last section of this chapter. 3.4 Chern-Simons Couplings Let us now consider the more general solution (3.16) to the consistency conditions (C.1–4), containing an arbitrary holomorphic function g(Z). We note that in every coefficient it is accompanied by a factor ¯ Z. This may be removed by a field redefinition L = ˆ L + f(Z) + ¯ f( ¯ Z) with ∂f = g, for the transformation rules (2.54) give (dropping the hats) C = 1 ¯Z L + h + ¯ ∂h h , D = − Z ¯Z − Z , E = ¯ ∂¯ h ¯Z − Z , (3.66) while the other coefficients are unchanged. Here h = Zg + f. The functions u and v in eqs. 25) and 26) do not vanish anymore, but one has L + ZC + ¯ Z ¯ C = −(h + ¯ h) , ZD + ¯ ZĒ = −∂h , which however is compatible with condition (C.4). We now write the constraints as D (i α ¯ D j) L = 0 ˙α D (i D j) L = 2 ¯Z − Z D (i Z D j) L + ¯ D (i ¯ Z ¯ D j) L + 1 2 L Di D j Z − i D i D j F + i ¯ D i ¯ D j ¯ F , 1 Recall that the coupling constant gz has mass dimension −1. (3.67)
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N=2 Supersymmetric Gauge Theories w
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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: 3.3. The Lagrangian 47 compute the
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.
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N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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