Abstract
Relying for a complete historical account on the tale told in the twin book (Fré, A conceptual history of symmetry from Plato to the Superworld Springer, Berlin, 2018, [1]), let us summarize the steps that led, in the 1990’s to Special Geometries.
La géométrie...est une science née à propos de l’expérience...nous avons créé l’espace qu’elle etudie, mais en l’adaptant au monde où nous vivons. Nous avons choisie l’espace le plus commode...
Henri Poincaré.
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Notes
- 1.
Clarification for readers with a mostly mathematical background: in the physical literature instantons play a very important role. They are field configurations that in the Wick-rotated space-time with Euclidean signature satisfy first-order equations more restrictive than the second order Euler Lagrangian equations (the latter are implied by the former). In the path integral formulation of quantum field theory, instanton correspond to the absolute minimal of the action functional and provide the dominant contribution to quantum correlators. Depending on the type of considered fields instantons have different definitions. For gauge fields, instantons are the connections on the underlying principal fibre-bundle whose field strengths are self dual, namely satisfy Eq. (4.1.3).
- 2.
Clarification for mathematicians: the wording supermultiplets is universally used in the context of supersymmetric field theories to denote a finite set of standard fields of various spins that form a unitary irreducible representation of the supersymmetry algebra extending the Poincaré Lie algebra.
- 3.
An early review of Special Kähler Geometry was written by this author in 1996 in [21].
- 4.
- 5.
Not all non-compact, homogeneous Quaternionic Kähler manifolds which are relevant to supergravity (which are normal, i.e. exhibiting a solvable group of isometries having a free and transitive action on it) are in the image of the c-map, the only exception being the quaternionic projective spaces [14, 15].
- 6.
The chosen \(\gamma \)-matrices are a permutation of the standard pauli matrices divided by \(\sqrt{2}\) and multiplied by \(\frac{\mathrm{i}}{2}\) can be used as a basis of anti-hermitian generators for the \(\mathfrak {su}(2)\) algebra in the fundamental defining representation.
- 7.
See Sect. 3.6 for notations.
- 8.
Clarification for mathematicians: in the jargon ubiquitously utilized in the physical literature one says that a set of operators closes a Lie algebra when any of the commutators thereof belongs to the linear span of the same operators.
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Fré, P.G. (2018). Special Geometries. In: Advances in Geometry and Lie Algebras from Supergravity. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74491-9_4
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