Abstract
In this chapter, we study the coupling of 4D, \(\mathcal {N}=1\) chiral and vector multiplets to supergravity, using the geometric language developed in the previous chapter. Our main emphasis will be on the difference between the matter couplings in global and local supersymmetry.
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Notes
- 1.
This is indeed confirmed by computing the gravitino variations of \(\mathcal {L}_{RS}\) with the new Kähler covariant transformation law, \(\delta \psi _{\mu }\sim M_{P}\mathcal {D}_{\mu }\epsilon \), which leads to a new term that precisely cancels the remaining uncancelled part of Z 1 (i.e., the part of Z 1 with a derivative acting on \(\overline {\psi }_{\mu }\)).
- 2.
For the sake of simplicity, we do not introduce a new symbol for the Kähler covariantized derivative and still call it \(\mathcal {D}_{\mu }\).
- 3.
As mentioned earlier, when the gauge kinetic function is not gauge invariant, generalized Chern–Simons terms of the form A I ∧ A J ∧ dA K and A I ∧ A J ∧ A K ∧ A L may be possible. Their form, however, is the same as in global supersymmetry [5].
- 4.
Recall the definition (6.27) for the covariant derivatives, which here are taken in fields space. In the current context \(p(S) = - p( S^\ast ) = \frac 12\).
- 5.
A line bundle is simply a vector bundle where the fiber is a one-dimensional vector space, \(\mathbb {R}\) or \(\mathbb {C}\). A holomorphic line bundle is a line bundle with fiber \(\mathbb {C}\) over a complex base manifold (here our Kähler manifold \(\mathcal {M}_{\mathrm {scalar}}\)) where the transition function between two local trivializations can be chosen to be holomorphic. According to (6.93), this is the case for the superpotential W, which is hence a section of a holomorphic line bundle, \(\mathcal {L}\), over \(\mathcal {M}_{\mathrm {scalar}}\).
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Dall’Agata, G., Zagermann, M. (2021). Matter Couplings in Supergravity. In: Supergravity. Lecture Notes in Physics, vol 991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63980-1_6
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