Superspace, or One thousand and one lessons in supersymmetry by S. James Gates

By S. James Gates

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By S. James Gates

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This implies that only one Lorentz component of W α is independent. The field strength describes the physical degrees of freedom: one helicity 1 2 and one helicity 1 mode. Thus W α is a suitable object for constructing an action. 21) we can compute the component action S = 1 g2 d 3 x D 2W 2 = 1 g2 d 3 x [W α D 2 W α − = 1 g2 d 3x 1 2 (D αW β ) (D αW β ) ]| λα i ∂ α β λβ − 1 2 f αβ f αβ . 22) Here (cf. 7) λα ≡ W α | while f αβ = D αW β | = D βW α | is the spinor form of the usual field strength F αβ γδ | = (∂ αβ Γ γδ − ∂ γδ Γαβ )| = = −i 1 2 1 δ (γ 2 (α f β) δ) [∂ αβ D (γ Γδ) − ∂ γδ D (α Γβ) ]| .

4. 5. 6. 7. 8. 4. Covariant derivatives 83 a. Construction 83 b. Algebraic relations 84 c. Geometry of flat superspace 86 d. 5. 6. Component expansions 92 a. θ-expansions 92 b. Projection 94 c. 7. Superintegration 97 a. Berezin integral 97 b. Dimensions 99 c. 8. Superfunctional differentiation and integration 101 a. Differentiation 101 b. 9. 10. Compensators 112 a. Stueckelberg formalism 112 b. CP(1) model 113 c. 11. Projection operators a. 1. 2. Super-Poincar´ e projectors 122 b. 1. 2. 3. 4. 12. On-shell representations and superfields 138 a.

We show this in Fig. 2: D2 Dα D2 D2 D2 D2 D2 Dα D2 Fig. 2 Only the last diagram gives a contribution. One further rule is useful in this procedure: In general, after integration by parts, various D-factors end up in different places in the final expression and one has to worry about minus signs introduced in moving them past each other. The overall sign can be fixed at the end by realizing that we start with a particular ordering of the D’s and we can examine what happened to this ordering at the end of the calculation.

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