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Basic Ingredients

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N = 2 Supergravity in D = 4, 5, 6 Dimensions

Part of the book series: Lecture Notes in Physics ((LNP,volume 966))

Abstract

We give an introduction to the book, discussing the role of the \(\mathcal {N}=2\) theories, its geometric structure, and the superconformal tensor calculus. We also refer to other treatments. We then set out the plan of the book.

In the second part of the chapter we introduce tools that are useful for the construction of superconformal gauge theory and multiplets. We first discuss the catalogue of supersymmetric theories with 8 supercharges (Sect. 1.2) and their multiplets (Sect. 1.2.1). After a short Sect. 1.2.2 with the strategy, we discuss the conformal (Sect. 1.2.3) and then superconformal (Sect. 1.2.4) groups. The transformations of the fields under the conformal symmetry are also given in Sect. 1.2.3, while for the fermionic symmetries, this is discussed in a short Sect. 1.2.5.

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Notes

  1. 1.

    The restriction is due to interacting field theory descriptions, which e.g. in 4 dimensions does not allow fields with spin larger than 2.

  2. 2.

    We distinguish the multiplet that contains the graviton and gravitini, and is determined by specifying the dimension and the number of supersymmetries, and other multiplets, which we call ‘matter multiplets’.

  3. 3.

    In this review, we only consider Minkowski signature of spacetime. In the literature, also other signatures are discussed, e.g. Euclidean signature in a series of papers [5,6,7,8], and special geometry has also been defined for other signatures [9,10,11,12].

  4. 4.

    Strictly speaking: to quaternionic-Kähler geometry, which means that there is a metric. For the difference between these manifolds and the used terminology we refer to the review [16].

  5. 5.

    We thank G. Tartaglino-Mazzucchelli for his assistance in this overview.

  6. 6.

    Of course one can introduce 8 supercharges in 1 or 2 dimensions, where the elementary spinors have just one component. In 3 dimensions, gamma matrices are 2 × 2 matrices, and the theories with 8 supercharges are \(\mathcal {N}=4\) theories where the spinors satisfy a reality condition (a Majorana condition).

  7. 7.

    Different proofs of this statement can be given. A general argument has been given in [62]. This has been further discussed in [4, App. 6B]. For on-shell states in D = 4, see also [4, Sect. 6.4.1]. For solutions of field equations that preserve supersymmetry, a detailed explanation is in Appendix B of [63].

  8. 8.

    General coordinate transformations are also local gauge transformations that are more general than fixed translations. So the general coordinate-equivalent states should also be subtracted.

  9. 9.

    Coleman and Mandula prove that the largest spacetime group that is allowed without implying triviality of all scattering amplitudes is the conformal group. This theorem is valid under some assumptions, like the analyticity of the elastic two-body scattering amplitudes. Haag, Łopuszański and Sohnius base their analysis on the previous result and use group-theoretical arguments (essentially Jacobi identities). For an extension of the theorem to strongly coupled quantum field theories, see e.g. [73, 74].

  10. 10.

    Note that the scalar here has negative kinetic energy, and the final gravity action has positive kinetic energy.

  11. 11.

    A gauge fixing can be interpreted as choosing better coordinates such that only one field still transforms under the corresponding transformations. Then, the invariance is expressed as the absence of this field from the action. In this case we would use \(g^{\prime }_{\mu \nu } =\frac {\kappa ^2}{6}g_{\mu \nu }\phi ^2\) as D-invariant metric. One can check that this redefinition also leads to (1.12) in terms of the new field.

  12. 12.

    In Minkowski space \(z=\frac {1}{\sqrt {2}}\left ( x^1+x^0\right ) \) and \(\bar z=\frac {1}{\sqrt {2}}\left (x^1-x^0\right ) \) are not each other complex conjugates. In Euclidean signature they are complex conjugates.

  13. 13.

    In the 2-dimensional case \( \operatorname {\mathrm {SO}}(2,2)= \operatorname {\mathrm {SU}}(1,1)\times \operatorname {\mathrm {SU}}(1,1)\) is realized by the finite subgroup of the infinite dimensional conformal group, and is well known in terms of \(L_{-1}= {\textstyle \frac {1}{2}} (P_0 - P_1)\), \(L_0={\textstyle \frac {1}{2}}(D+M_{10})\), \(L_1={\textstyle \frac {1}{2}} (K_0 + K_1)\), \(\bar L_{-1}= {\textstyle \frac {1}{2}} (P_0 + P_1)\), \(\bar L_0={\textstyle \frac {1}{2}}(D-M_{10})\), \(\bar L_1={\textstyle \frac {1}{2}} (K_0 - K_1)\). Higher order L n, |n|≥ 2 have no analogs in D > 2.

  14. 14.

    The expression ξ always determines the parameters \(\{a^\mu ,\lambda ^{\mu \nu },\lambda _{\mathrm {D}},\lambda _{\mathrm {K}}^\mu \}\) as in (1.19).

  15. 15.

    Remember that, for the intrinsic part, \(\delta _K \delta _D \phi ^i =\delta _K (k_D^i(\phi ))=\partial _j k_D^i(\phi )\delta _K \phi ^j\).

  16. 16.

    The terminology reflects that the right-hand side is a constant times g ij. For a function times g ij it is just a ‘conformal Killing vector’.

  17. 17.

    However, with branes the assumptions of these papers may be too constrained. Other examples have been considered first in 10 and 11 dimensions in [77].

  18. 18.

    See Appendix B for the notations of groups and supergroups.

  19. 19.

    Note that the superalgebras that are relevant for quantum field theories, according to Coleman and Mandula [71] and Haag–Łopuszański–Sohnius [72], only exist for D ≤ 6. The maximal number of supercharges of the corresponding superconformal QFTs is bounded by the requirement that they admit a suitable stress-tensor multiplet, see e.g. [80] for a recent discussion.

  20. 20.

    To prepare these formulae, we made use of [81] for D = 4, apart from a sign change in the choice of charge conjugation, such that the anticommutators of fermionic generators have all opposite sign. For D = 5, we made use of [82], and for D = 6 of [83], but replacing there K by − K and U ij by \(-{\textstyle \frac {1}{2}} U_{ij}\).

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Authors and Affiliations

  1. CPHT, Ecole Polytechnique, Palaiseau, France

    Edoardo Lauria

  2. Institute for Theoretical Physics, KU Leuven, Leuven, Belgium

    Antoine Van Proeyen

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Lauria, E., Van Proeyen, A. (2020). Basic Ingredients. In: N = 2 Supergravity in D = 4, 5, 6 Dimensions. Lecture Notes in Physics, vol 966. Springer, Cham. https://doi.org/10.1007/978-3-030-33757-5_1

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