Abstract
This chapter starts with a historical outline that describes the birth of supersymmetry both in String Theory and in Field Theory, touching also on the biographies and personalities of the theorists who contributed to create this entire new field through a complicated and, as usual, far from straight, path. The chapter proceeds than with the conceptual foundations of Supergravity, in particular with the notion of Free Differential Algebras and with the principle of rheonomy. Sullivan’s structural theorems are discussed and it is emphasized how the existence of p-forms, that close the supermultiplets of fundamental fields appearing in higher dimensional supergravities, is at the end of the day a consequence of the superPoincaré Lie algebras through their cohomologies. The structure of M-theory, the constructive principles to build supergravity Lagrangians and the fundamental role of Bianchi identities is emphasized. The last two sections of the chapter contain a thorough account of type IIA and type IIB supergravities in D=10, the structure of their FDAs, the rheonomic parameterization of their curvatures and the full-fledged form of their field equations.
Tiger, tiger, burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful (super) symmetry?
William Blake
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Notes
- 1.
This episode is known to the author by private conversations with Prof. Schwarz who told him this story while visiting him at his place Torino in 1981.
- 2.
The name of the algebra refers to his discoverer, the brilliant Italo-Argentinian physicist Miguel Virasoro born in Buenos Aires in 1940, who is presently full professor of Theoretical Physics at La Sapienza of Rome and from 1995 to 2002 was Director of the International Centre of Theoretical Physics of Trieste, founded by Abdus Salam.
- 3.
Born in Kharkov in 1922, Yuri Abramovich Golfand got his mathematical-physical education in that Ukrainian city. Later, since 1951, he joined the Tamm group at the Lebedev Physical Institute of the Soviet Academy of Sciences in Moscow (FIAN), an institution that collected seven Nobel Prizes in Physics in the course of sixty years. Golfand and his student Likhtman conducted there, at the end of the 1960s, the studies that led them to discover the super Poincaré Lie algebra and to construct its first field theoretical realizations, published in 1971 after a long procedure of checks and inspections by the Soviet censorship authorities (see [10] for a detailed account of these facts). The next year, in the course of a routine campaign of personnel cuts, Yuri Golfand was fired from FIAN and decided to apply for an exit visa to Israel. This put him in a very bad light in front of Soviet authorities who refused the visa and treated him as a renegate. For 7 years he lived unemployed and was readmitted to FIAN only in 1980. Golfand obtained permission to emigrate to Israel only in 1990 and there he lived his last four years in Haifa, where he died in 1994. Because of his association with the renegate Golfand, also Likhtman had very difficult times with Soviet authorities and could never get a proper academic position.
- 4.
- 5.
In this discussion the index α incorporates both the spinor index running on the dimension of the relevant spinor representation of SO(1,D−1) and the replica index related with extended supersymmetry.
- 6.
By graded symmetric we mean \(\widehat {E}^{A}_{\phantom{A}M}(x,\theta) = (-)^{f_{A}f_{B}} \widehat{E}^{M}_{\phantom{M}A}(x,\theta)\).
- 7.
For simplicity in this section we adopt a pure Lie algebra notation. Yet every definition presented here has a straightforward extension to superalgebras and indeed when we recall the discussion of how the FDA of M-theory or type II supergravity emerges from the application of Sullivan structural theorems it is within the scope of super Lie algebra cohomology.
- 8.
- 9.
According to standard nomenclature irrep means irreducible representation.
- 10.
It must also be noted that the algebras defined by (6.4.15)–(6.4.20) and by some authors named D’Auria-Frè algebras have been discussed as a possible basis for a Chern-Simons formulation of fundamental M-theory [20]. They have also been retrieved as part of a wider set of gauge algebras by Castellani [21], using his method of extended Lie derivatives.
- 11.
- 12.
- 13.
Note that our \( \mathcal{R} \) is equal to \( -\frac{1}{2}{{\mathcal{R}}^{old}} \), \( {{\mathcal{R}}^{old}} \) being the normalization of the scalar curvature usually adopted in General Relativity textbooks. The difference arises because in the traditional literature the Riemann tensor is not defined as the components of the curvature 2-form ℜab rather as −2 times such components.
- 14.
In the next Chap. 7 we will emphasize the role of the dilaton factors exp[−aφ] in front of the p-form kinetic terms.
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Frè, P.G. (2013). Supergravity: The Principles. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5443-0_6
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