Abstract
In our conceptual journey from the algebraic notions to the geometrical ones, journey that may be deemed to be the very heart of the present history essay, we have vastly emphasized the idea that groups exist abstractly as algebraic structures, yet are concretely realized as symmetries of geometrical structures, in particular of smooth manifolds. Conversely, possible geometries can be characterized in terms of the symmetries they admit, i.e. of the groups of transformations that preserve some of their fundamental properties.
Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof
Girolamo Cardano
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Notes
- 1.
Some of the historical informations contained in the present section are to be credited to an unpublished note of Orlando Merino available at the following site: http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf.
- 2.
Supermultiplet is the name given to a collection of fields that form an irreducible representation of the supersymmetry algebra. The structure of supermultiplets that always involve both bosons and fermions depends on the space-time dimensions D in which we construct our field theory and on the number of supercharges (fermionic Lie algebra generators) that we include in our supersymetry algebra. This latter is usually named \(\mathscr {N}\) and in \(D=4\) can range from 1 to 8.
- 3.
Scalar multiplets and hypermultiplets are multiplets that involve only spin \({\textstyle \frac{1}{2}}\) and spin 0 fields. The connection with geometry occurs at level of the scalar fields, that are interpreted, sigma–model like as coordinates of a target manifold with an appropriate geometry (see Sect. 7.6.2).
- 4.
Albert Nijenhuis (November 21, 1926–February 13, 2015) was a Dutch-American mathematician. He wrote his Ph.D. thesis at the University of Amsterdam under the supervision of Jan Arnoldus Schouten.
- 5.
We defined complex structures as operators acting on sections of the tangent bundle, namely on vector fields. By means of the duality between the tangent bundle and the cotangent bundle, complex structures (or almost complex structures) act equally well on sections of the contangent bundle, namely on differential 1-forms: \(\omega (\mathrm {Jv})\equiv \mathbf {J}\omega (\mathbf {v})\). This is what we use here.
- 6.
We will be not too particular about the algebraic nature of the spaces \(\varOmega ^{[i]}\). What is important is that their elements can be summed and subtracted and that they form an abelian group under addition. In many instances one can take linear combinations of the cochains so that they actually form a vector space over some field, or a module over some ring, but we do not discuss the many subtleties concerning the utilized coefficients.
- 7.
By definition the kernel of a map \(\mu \, : \,\, V\, \rightarrow \, W\) from a group V to a group W is the subspace of V that is mapped by \(\mu \) mapped into the neutral element of W that for abelian groups we denote \(\mathbf {0}\).
- 8.
The non triviality of the Kähler 2-form manifests itself in the Kähler transformations that are required to connect the Kähler potential as given in one-chart with the Kähler potential as given in another one.
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Fré, P.G. (2018). Geometry Becomes Complex. In: A Conceptual History of Space and Symmetry . Springer, Cham. https://doi.org/10.1007/978-3-319-98023-2_8
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DOI: https://doi.org/10.1007/978-3-319-98023-2_8
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