Abstract
The quest for a fundamental theory of all elementary particles and their interactions has been one of the most fascinating scientific endeavors during the past century. One of the main guiding principles in the construction of the ever more refined theories of high energy physics has been the systematic use of symmetry principles.
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Notes
- 1.
Gauged supergravity refers to supergravity theories that contain also non-trivial conventional gauge interactions, as we will see in Chap. 9.
- 2.
Throughout the book, it is understood that c = 1 and ħ = 1.
- 3.
The full isometry group O(1, 3) of 4D Minkowski spacetime decomposes into four disconnected components. In this book, we mean by Lorentz group only the component, SO(1, 3)0, of Lorentz transformations that are continuously connected to the identity element. This subgroup is often called the “proper orthochronous Lorentz group”.
- 4.
As we will discuss later, an exception to this gauge invariance of the gravitino arises in the presence of Fayet–Iliopoulos terms in \(\mathcal {N}=1\) supergravity. These terms require a gauging of the R-symmetry group, under which also the gravitino is charged, but these theories are often anomalous.
- 5.
Note that the Weyl representation is irreducible as a representation of the Clifford algebra (1.12). It is only the induced representation Σ ab of the Lorentz algebra that is reducible.
- 6.
The Dirac representation corresponds to \(\gamma _{0}=i\sigma _{3}\otimes {\mathbb 1}_{2}\equiv \left ( \begin {array}{cc} i{\mathbb 1}_{2} & 0 \\ 0 & -i{\mathbb 1}_{2} \end {array} \right ) , \gamma _{i}=\sigma _{2}\otimes \sigma _{i}\), and the Majorana representation is given by γ 0 = iσ 2 ⊗ σ 3, \(\gamma _{1}= -\sigma _{1}\otimes {\mathbb 1}_{2}\), γ 2 = σ 2 ⊗ σ 2, \(\gamma _{3}=\sigma _{3}\otimes {\mathbb 1}_{2}\), but they are not really needed for this book.
- 7.
Note that imposing a Weyl or Majorana condition in 4D does not necessitate the use of the Weyl or the Majorana representation of the gamma matrices. The Weyl condition just takes on a particularly simple form in the Weyl representation, and the Majorana condition leads to a particularly simple result in the Majorana representation. We usually do not make use of these simplified forms and write down the conditions in a covariant way.
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Dall’Agata, G., Zagermann, M. (2021). Introduction. In: Supergravity. Lecture Notes in Physics, vol 991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63980-1_1
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