Abstract
In the previous chapter, we discussed \(\mathcal {N}=1\) pure supergravity, which only involves the \(\mathcal {N}=1\) supergravity multiplet. In order to include also ordinary matter fields and Yang–Mills-type gauge interactions, one has to couple the pure supergravity sector to \(\mathcal {N}=1\) chiral and vector multiplets in a way consistent with local supersymmetry.
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Notes
- 1.
In terms of superfields, Φ n, this corresponds to \(\int d^4\theta \,\, \delta _{mn}\varPhi ^m(\varPhi ^{n})^\dagger +\left ( \int d^2 \theta \,\, W(\varPhi ) +\mbox{h.c.}\right )\) after integrating out auxiliary fields.
- 2.
In the following we will often denote the Kronecker symbol as \(\delta _{m \bar {n}}\) to make manifest covariance of the equations, even though one should interpret it as δ mn.
- 3.
More precisely, one defines the holonomy group, Hol(p), at a point \(p\in \mathcal {M}_{\mathrm {scalar}}\) to be the group generated by all M j i(γ) of curves that begin and end at p. This pointwise holonomy group is the same for all points on \(\mathcal {M}_{\mathrm {scalar}}\) if (as we will always assume) \(\mathcal {M}_{\mathrm {scalar}}\) is connected, so that one can elevate it to the holonomy group, \(\mbox{Hol}(\mathcal {M}_{\mathrm {scalar}})\), of the entire manifold.
- 4.
More properly speaking, they are sections in the tangent bundle of \(\mathcal {M}_{\mathrm {scalar}}\) and sections in the spacetime spinor bundle.
- 5.
The intersection of \(GL(n_{C},\mathbb {C})\) and SO(n C) inside \(GL(2n_C,\mathbb {R})\) depends on how SO(2n C) is embedded inside \(GL(2n_{C},\mathbb {R})\) relative to the embedding (5.29) of \(GL(n_{C},\mathbb {C})\). The intersection is maximal if the metric to be preserved by SO(2n C) in the basis (Re(δϕ n), Im(δϕ m)) is the unit matrix \(\mathbb {1}_{2n_{C}}\). In that case, the orthogonality condition reads \( M'{ }^{T}M'=\mathbb {1}_{2n_{C}}\), which is equivalent to \(A^\dagger A=\mathbb {1}_{n_{C}}\), which means that in that case the intersection of \(GL(n_C,\mathbb {C})\) and SO(2n C) sweeps out the full U(n C).
- 6.
Kähler manifolds can in fact equivalently be defined as manifolds that are complex and real symplectic at the same time.
- 7.
We remind the reader that this is still a toy model where we are neglecting any chirality properties of the fermions. The complete model will be discussed below.
- 8.
In classical mechanics, if the moment maps Poisson commute with the Hamiltonian, they are conserved along the time evolution generated by the Hamiltonian.
- 9.
Similar to the F-terms, the above D-terms are on-shell expressions for a certain auxiliary field, often called D(x), of an off-shell vector supermultiplet.
- 10.
In certain theories with non-gauge invariant gauge kinetic functions, generalized Chern–Simons terms may also occur, which are terms of the form A I ∧ A J ∧ dA K and A I ∧ A J ∧ A K ∧ A L [5].
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Dall’Agata, G., Zagermann, M. (2021). Matter Couplings in Global Supersymmetry. In: Supergravity. Lecture Notes in Physics, vol 991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63980-1_5
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