N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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4 Chapter 1. Gauging the Central Charge In the N = 2 case the presence of central charges reduces this symmetry from U(2) to SU(2), under which ε ij is an invariant tensor. The algebra (1.1) with N = 2 can be represented on a so-called central charge superspace [4] with coordinates x µ , θ α i , ¯ θ ˙αi and a further bosonic complex variable z. On superfields infinitesimal supersymmetry transformations are generated by differential operators Q i α = ∂ ∂θ α i − i 2 (σµ¯ θ i )α∂µ + i with commutation relations {Q i α , Q j β } = iεαβε ij ∂z 2θi α∂z , Q¯ ˙αi = − ∂ ∂ ¯ θ {Q i α , ¯ Q ˙αj} = iδ i jσ µ α ˙α ∂µ ˙αi + i 2 (θiσ µ )α∂µ + i 2 ¯ θ ˙αi∂¯z { ¯ Q ˙αi , ¯ Q ˙ βj } = −iε ˙α ˙ β ε ij ∂¯z . The commutator of two rigid supersymmetry transformations (1.4) (1.5) ∆(ξ) = ξ α i Q i α + ¯ ξ i ˙α ¯ Q ˙α i , (ξ α i ) ∗ = ¯ ξ ˙αi , (1.6) yields global translations in the bosonic directions, [ ∆(ξ) , ∆(ζ) ] = i(ξiσ µ ¯ ζ i − ζiσ µ ¯ ξ i )∂µ + iξ i ζi∂z + i ¯ ξi ¯ ζ i ∂¯z . (1.7) Supercovariant spinor derivatives D i α = ∂ ∂θ α i + i 2 (σµ¯ θ i )α∂µ − i 2θi α∂z , D¯ ˙αi = − ∂ ∂ ¯ i − θ ˙αi 2 (θiσ µ )α∂µ − i 2 ¯ θ ˙αi∂¯z (1.8) anticommute with Q i α and ¯ Q ˙αi and therefore map superfields into superfields. Their algebra involves a minus sign relative to the algebra of the Q’s, {D i α , D j β } = −iεαβε ij ∂z {D i α , ¯ D ˙αj} = −iδ i jσ µ α ˙α ∂µ { ¯ D ˙αi , ¯ D ˙ βj } = iε ˙α ˙ β ε ij ∂¯z . (1.9) The coefficient functions in the θ-expansion of a superfield constitute a supersymmetry multiplet. Their supersymmetry transformations are generated by differential operators D i α and ¯ D ˙αi, whose action can be read off the relation D i α Φ(x, θ, ¯ θ, z) = Q i α Φ(x, θ, ¯ θ, z) , (1.10) where Di α acts only on the components and anticommutes with the θ-variables. The algebra of Di α and ¯ D ˙αi is the same as for the supercovariant derivatives. The components of a superfield Φ may be regarded as the lowest components of superfields obtained from applying an appropriate polynomial P (D, ¯ D) of supercovariant derivatives to Φ. Eq. (1.10) implies that the generators Di α, ¯ D ˙αi act on components P (D, ¯ D) Φ | θ= θ=0 ¯ according to D i α P (D, D) ¯ Φ θ= ¯ θ=0 = D i αP (D, ¯ D) Φ θ= ¯ . (1.11) θ=0
1.1. The N=2 Supersymmetry Algebra 5 If ϕ(x, z) = Φ(x, 0, 0, z) denotes the lowest component of Φ, it follows that P (D, ¯ D) Φ θ= ¯ θ=0 = P (D, ¯ D) ϕ(x, z) . (1.12) While the θ-expansion of a superfield terminates after a finite number of steps, the zdependence in general is nonpolynomial, giving rise to an infinite tower of component fields. The supercovariant derivatives may be employed to impose constraints on superfields that eliminate all but a finite number of components without restricting their x-dependence. It is convenient, and we shall make use of it from now on throughout this thesis, to consider superfields living only on the subspace parametrized by x, θ α i , ¯ θ ˙αi and to regard central charge transformations not as translations in some additional bosonic directions, but simply as transformations that map one superfield to a new superfield. Instead of ∂z we denote the generator by δz, which then maps from one coefficient in the z-expansion of a general superfield to the next. Also, we confine ourselves to only a single real central charge, as the presence of a second one usually inhibits the formulation of finite multiplets. In symbols, one has the equivalence Φ(x, θ, ¯ θ, z) = Φ(x, θ, ¯ θ, 0) + z ∂zΦ(x, θ, ¯ θ, z) |z=0 + . . . ⇐⇒ δz : Φ(x, θ, ¯ θ) ↦→ Φ (z) (x, θ, ¯ θ) ↦→ . . . , and similar for the components. The purpose of superfield constraints then is to express all but at most a finite number of the images Φ (z) , Φ (zz) , etc. in terms of the primary superfield Φ and its spacetime derivatives. The main reason why we altered our point of view concerning the central charge transformations is the similarity to (abelian) super Yang-Mills theories that arises when gauging the central charge. Let us briefly review the basics of supersymmetric gauge theories, mainly to introduce our conventions and notations. We shall denote an infinitesimal gauge transformation with spacetime dependent parameters C I (x) by ∆ g (C). Tensor fields T are characterized by their homogeneous transformation law ∆ g (C) T = C I δIT , (1.13) whereas the transformation of the gauge fields A I µ involves the derivative of the parameters C I , ∆ g (C) A I µ = −∂µC I − C J A K µ fJK I . (1.14) Here the generators δI form a basis of a Lie algebra G with corresponding structure constants fIJ K , [ δI , δJ ] = fIJ K δK , f[IJ L fK]L M = 0 . (1.15) The transformation of the gauge fields is such that the covariant derivative Dµ = ∂µ + A I µδI (1.16) of a tensor transforms again as a tensor. This implies that the generators δI commute with the Dµ, which in turn requires the gauge fields to transform in the adjoint representation of the Lie algebra G under the δI, i.e. δIA J µ = fKI J A K µ . Note that the δI do not generate the full transformations of the gauge fields.
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: Chapter 1 Gauging the Central Charg
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.
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3.5. Henneaux-Knaepen Models 55 whe
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Chapter 4 The Nonlinear Case In sec
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4.2. Transformations and Bianchi Id
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4.3. The Lagrangian 65 This of cour
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4.3. The Lagrangian 67 which introd
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70 Conclusions and Outlook likely t
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72 Appendix A. Conventions and the
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74 Appendix A. Conventions σ µν
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76 References [20] N. Dragon and S.
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N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

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